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Optimal multi-facility location on tree networks

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Optimal multi-facility location on tree networks
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Thesis--University of Florida.
Bibliography:
Bibliography: leaves 161-169.
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Typescript.
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Vita.
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by Barbaros C. Tansel.

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OPTIMAL MULTI-FACILITY LOCATION ON
TREE NETWORKS








By

BARBAROS C. TANSEL


A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY














UNIVERSITY OF FLORIDA


1979















ACKNOWLEDGMENTS


I am deeply indebted and grateful to Dr. Richard L. Francis, the

chairman of my supervisory committee, for his excellent guidance,

numerous suggestions, and the generosity with which he invested his time

in listening to my ideas. Dr. Francis not only initiated my interest

in location problems but also inspired many of the ideas in this dis-

sertation by asking the right questions at the right time.

I owe very special thanks to Dr. Timothy J. Lowe, the cochairman/

chairman of my committee during 1976-1978, presently of Purdue Uni-

versity, for his active interest, overall guidance, and his inspiring

suggestions.

Dr. Francis and Dr. Lowe have shown sincere care about my progress

and their encouragement has been of utmost value in bringing this

dissertation to a completion.

I would also like to express my sincere thanks and appreciation

to the other members of my committee, Dr. Ralph W. Swain, Dr. Donald W.

Hearn, Dr. Antal Majthay, and Dr. Luc G. Chalmet for their interest in

my work and their suggestions during my proposal.

I am grateful to the Department of ISE for providing me with

assistantship during my graduate studies.

Mrs. Adele Koehler has done an excellent job in typing the manu-

script. She is fast, accurate, and very observant. I sincerely

recommend her.









This research.was supported in part by NSF Grant #ENG 76-17810,

the Army Research Office, Triangle Park, N.C., under contract

DAHC04-75-G-0150, and by the Operations Research Division, National

Bureau of Standards, Washington, D.C.















TABLE OF CONTENTS


Page

ACKNOWLEDGMENTS . . . .... . ii

ABSTRACT . . . .... . . vi

CHAPTER

1 INTRODUCTION AND LITERATURE SURVEY . . ... 1

1.1 Introduction and Overview . . . 1
1.2 Terminology . . . ... .. 4
1.3 Survey of the Network Location Literature . 6

2 DUALITY AND THE NONLINEAR p-CENTER PROBLEM AND
COVERING PROBLEM ON A TREE NETWORK ..... ..... 53

2.1 Introduction and Related Work . . ... .53
2.2 Problem Statements and Duality. . . ... 56
2.3 Dual Problem Interpretation . . ... .61
2.4 Covering Algorithm. . . . .. 67
2.5 Dual Problem Solution and the Strong Duality Theorem. 73
2.6 Results for the Covering Problem. . . ... 78

3 A VECTOR-MINIMIZATION PROBLEM ON A TREE NETWORK. . 84

3.1 Introduction. . . . .... .84
3.2 Problem Statement ................. 85
3.3 Distance Constraints and Characterization of
Efficient Points. . . . . ... 87
3.4 Examples. . . . . ... .94
3.5 Further Results on the Convex Hull Property .... .96
3.6 Algorithm to Construct Efficient Location Vectors 108
3.7 Efficiency for the Case of Rectilinear or
Tchebychev Distances. . . . ... 116

4 THE BI-OBJECTIVE m-CENTER PROBLEM ON A TREE NETWORK. 122

4.1 Introduction. . . . .. 122
4.2 Problem Statement, Notation, and Definitions. ... 123
4.3 Necessary and Sufficient Conditions for Efficiency. 126
4.4 Construction of the Efficient Frontier. . .. .134









Page

5 SUMMARY AND FUTURE RESEARCH . . .... 149

5.1 Summary. . . . . ... .. .149
5.2 Generalized Multi-Center Problem . ... 150
5.3 The t-Objective m-Center Problem: Steps
Towards a Unified Theory ............ 153
5.4 Tree Networks and General Networks . ... 158

REFERENCES .. .. .. . .. . . 161

BIOGRAPHICAL SKETCH .. . . . . ... 170















Abstract of Dissertation Presented to the Graduate Council
of the University of Florida in Partial Fulfillment of the Requirements
for the Degree of Doctor of Philosophy


OPTIMAL MULTI-FACILITY LOCATION ON
TREE NETWORKS

By

Barbaros C. Tansel

December 1979

Chairman: Richard L. Francis
Major Department: Industrial and Systems Engineering

In this dissertation we develop a theory for location problems which

involve locating multiple new facilities on a tree network with respect

to existing facilities at known locations.

The first problem we consider is the nonlinear version of the

p-center location problem on a tree network for which the cost of each

served vertex is a strictly increasing continuous function of the dis-

tance between the vertex and the nearest center,and the objective is to

minimize the maximum such cost over all possible locations of the

centers. We present a dual "dispersion" problem which may be inter-

preted as the problem of choosing p + 1 (or more) vertices such that

the minimum cost to serve any two of the chosen vertices by a single

common center is as large as possible. We give a weak duality theorem

which applies to all general networks and a strong duality theorem

which applies to all tree networks. The strong duality theorem also

specifies the necessary and sufficient conditions for an optimal solu-

tion to either problem. We provide algorithms of polynomial complexity









for solving either problem provided that certain needed inverse functions

can be evaluated in a polynomial order of effort. The p-center problem

is typically solved with the aid of a nonlinear covering problem for

which we also give a dual with a physical interpretation. We provide

a covering algorithm which solves both the covering problem and its dual

simultaneously.

The second problem we consider is a vector-minimization problem

which involves as objectives the distances between specified pairs of

new and existing facilities and specified pairs of new facilities. We

relate the vector-minimization problem of interest to a distance con-

straints problem which imposes upper bounds on the distances between

specified pairs of facilities. We develop the necessary and sufficient

conditions for efficiency by making use of the theory developed for the

related distance constraints problem. Efficient solutions to the

vector-minimization problem of interest are such that in order for any

new facility to be closer to some facility than it already is, it must

in turn be placed farther from some other facility. Based on the

necessary and sufficient conditions,we provide an algorithm which

constructs an efficient location vector from a given non-efficient

solution.

The third problem we consider is a bi-objective minimax problem

which involves as objectives the maximum of the weighted distances

between specified pairs of new and existing facilities, and the maximum

of the weighted distances between specified pairs of new facilities.

We again relate the problem to the distance constraints problem and

derive the necessary and sufficient conditions for efficiency by making

use of the distance constrains. Further, we provide an (m(m +
use of the distance constraints. Further, we provide an O(m (m + n









algorithm to construct the efficient frontier, where m and n are,

respectively, the number of new and existing facilities.


v iii














CHAPTER 1

INTRODUCTION AND LITERATURE SURVEY



1.1 Introduction and Overview


Although some mathematical models of location can be traced back

to the early seventeenth century, almost all the work on operational

models for the location of facilities has taken place within the past

22 years, between 1957 and the present. An extensive annotated bibli-

ography on location-allocation problems is provided by Lea [78]. A

more recent selective bibliography is given by Francis and Goldstein

[30].

Location problems commonly involve locating a number of new

facilities (sources) in a given location space so as to provide goods

or services to a specified set of existing facilities (demands) under

one or more criteria, and, possibly, subject to a set of constraints.

The quality of the service is typically measured in terms of the dis-

tances among the facilities. The use of distances is, perhaps, the

major feature which distinguishes location problems as a special class

of optimization problems. Hence, associated with any location problem

is an underlying location space on which a "distance" is defined.

Several variations of the general location problem are possible,

depending upon the type of location space, the distance function, the

number and areal extent of the facilities, the type of interactions









between the facilities, the objective criteria used, the constraints,

the presence or lack of random elements, and possibly other factors

as well.

Among the several variants, planar location problems received

special attention in the past, starting with the earliest contribu-

tions, for example [106]. In such planar problems, one is interested

in locating new facilities in the Euclidean plane with respect to

existing facilities. For continuous planar problems, where any point

in the plane is a feasible location, the typical distance used is the

Z distance, special cases of which are the rectilinear, Euclidean,
p
and Tchebychev norms. For discrete planar problems, where there are

a finite number of candidate locations for new facilities, the distance

between any potential new facility location and any existing facility

is a specified positive number. Such discrete problems, due to the

finite nature of feasible locations, readily lend themselves to integer

programming formulations. The reader is referred to the book by

Francis and White [31] for a discussion of planar problems and a wealth

of references.

A number of real life applications suggest that, in some in-

stances, a network space can be a more faithful representation of the

reality than the Euclidean plane. For example, in a road network, a

communication network, or a pipeline system, travel occurs along the

arcs of the underlying network rather than in straight lines or recti-

linear paths. Hence, for such problems, the use of shortest path

distances along the arcs of the network can approximate the travel

distance more closely than the k distance. As opposed to planar

problems, network location problems have received much less attention









in the past. As reported by Lea [79], there are some 1500 published

papers on location-allocation problems. Among these, about 80 are on

network location problems, a ratio of a little less than 6%. Hence,

network location problems deserve well-justified attention in future

research.

In this dissertation, we develop a theory for a number of location

problems which involve locating multiple new facilities on a tree net-

work with respect to existing facilities at known locations. At this

point we give an overview of the dissertation.

In the remainder of Chapter 1, we specify our terminology and

give a survey of the network location literature. We discuss minimax and

minisum problems/and multi-objective problems involving minimax and

minisum objectives as well as other objectives. Discussed also are

problems with distance constraints. We highlight some of the convexity

properties of trees (see [22]) in relation to the problems discussed.

The chapter ends with a brief discussion of path-location problems.

In Chapter 2, we develop a theory for the nonlinear p-center

problem on a tree network. The problem is a generalization of the

linear p-center problem which involves locating p new facilities on

a network so as to minimize the maximum weighted distance from any

existing facility to its nearest new facility. Nonlinearity is ob-

tained by replacing each weight by a strictly increasing function of

the distance. We formulate a dual "dispersion" problem and prove a

weak duality and a strong duality theorem. The strong duality theorem

also specifies the necessary and sufficient conditions for an optimal

solution to either problem. We provide algorithms of polynomial com-

plexity for solving either problem. Discussed also are a covering









problem and a dual "divergence" problem. We provide a covering

algorithm which solves both the covering problem and its dual simul-

taneously.

In Chapter 3, we study a vector-minimization problem in relation

to a distance constraints problem. The problem involves as objectives

the distances between specified pairs of new and existing facilities

and specified pairs of new facilities. We extend the results of [32]

to develop a theory for identifying unique solutions to distance con-

straints, and use this theory to develop necessary and sufficient

conditions for efficient solutions to the vector-minimization problem

of interest. Further, we provide an algorithm which constructs an

efficient location vector from a given non-efficient solution.

In Chapter 4, we study a bi-objective location problem which in-

volves as objectives the maximum of the weighted distances between

specified pairs of new and existing facilities, and maximum of the

weighted distances between specified pairs of new facilities. We

characterize efficient solutions and provide an algorithm for construct-

ing the efficient frontier.

In Chapter 5, we pose a number of unresolved questions in relation

to the problems discussed and point out directions for future research.



1.2 Terminology


Before discussing the literature we specify our terminology.

An undirected network N = {V,E} is a collection of two sets V

and E, called the set of vertices and the set of edges of N, respec-

tively. Each edge in E is described by an unordered pair of vertices.





-5-


Network N is said to be edge weighted if, associated with each of its

edges, is a specified real number. Given an undirected network

N = {V,E) with positive edge weights, an imbedding of N, written as

N = {V,E}, is a geometric realization of N is some space S such that

there is a one-to-one correspondence between the members of V and V,

and E and E, respectively; each edge ecE is a rectifiable arc, and no

two edges in E intersect at more than one point, a vertex. The length

of edge e in E is defined to be the edge weight of the corresponding

member in E. A point of an imbedded network N = {V,E} is any point

along any edge in E, including the vertices. We write xeN to mean x

is a point in N. The distance d(x,y) between any two points x,ycN is

the length of a shortest path P(x,y) joining the two points. The

function d(.,.) satisfies the axioms of a metric on N so that the set

N together with d(.,.) determines a metric space.

The axioms of a metric are as follows: For any two points x,ysN,

1. d(x,y) > 0 if x # y; d(x,x) = 0,

2. d(x,y) = d(y,x),

3. d(x,y) < d(x,u) + d(u,y) for any ucN.

For a more detailed discussion of how to construct a metric space

(N,d) from a given edge weighted network N, the reader is referred to

Dearing and Francis [19], or Dearing, Francis, and Lowe [22]-

We restrict ourselves to finite undirected connected networks

that contain no loops and no multiple edges. We omit the term "im-

bedded," and simply take a network to mean an imbedded network on

which the distance d(.,.) is defined. For all other networks, we use

the terms "graph," "arcs," and "nodes" instead of network, edges, and

vertices.









Finally, for tree networks, we write T instead of N. In passing,

we note that the shortest path P(x,y) between any two points x,ysT is

unique, as otherwise T would contain a cycle.



1.3 Survey of the Network Location Literature


Historically, the earliest precise mathematical formulation of a

location problem on a network appears to be due to Hakimi [47] in 1964.

Prior to Hakimi's paper, the problem of finding the best threshing

site for harvested wheat was attacked by using a network location model

in 1962 by Hua Lo-Keng and Others [60]. This model was presented only

at an intuitive level and no mathematical formulation or properties

were given. A (correct) solution procedure was suggested (in the form

of a poem), which was to be discovered independently by Goldman [42] in

1971. Since 1964, a literature of approximately 80 papers has grown

till the present. Several new problems, as well as certain extensions

and generalizations of old problems, have been introduced.

A recent text by Handler and Mirchandani [58 ] discusses ex-

tensively a portion of the literature involving minimax and minisum

problems as well as single-facility bi-objective problems involving

the combination of these two objectives.

A "family tree" for network location problems is shown in

Figure 1.1. Although not exhaustive, the family tree covers most of

the problems formulated since 1964. With reference to the family tree

shown in Figure 1.1, network location problems can be broadly classi-

fied into two groups: point-location problems and path-location

problems. Path-location problems have been recently introduced by





































































Figure ].1. Family Tree for Network Location Problems









Slater [102]. A large portion of the literature deals with point-

location problems. Point-location problems may be classified into

three categories: single objective problems, multi-objective problems,

and a body of results of a general and unifying nature.

In the remainder of this section we give a detailed discussion

of the problems outlined in the family tree.



Point-Location Problems


Here, we consider a number of problems that involve locating new

facilities at points on a network. The general format of the dis-

cussion is as follows: For each problem type, we first define a

kernel problem. Then, we discuss the related literature on the kernel

problem, as well as several special cases and extensions of it. We

point out relations between different problem types, whenever such

relations exist.



The p-center problem

Let N be a network with a vertex set V = {vl,...,v } and an edge
1 n
set E. Denote by X a finite set of points, each of which is in N.

Let I be the set of integers 1 through n. For each vertex v., ieI,
1
define the distance D(vi,X) between vertex v. and the point set X by

D(v.,X) = min[d(vi,x): xeX]. With this definition, D(v.,X) is speci-

fied by a nearest point in X to v Let w. and a. be two given numbers
1 1
associated with vertex vi, icI. We call wi a weight and ai an addend.

We assume that each wi is nonnegative and at least one wi is positive.

For any finite point set X CN, define the function f(X) by









f(X) = max[w.D(v.,X) + a.: icl] *


The problem of interest is the following: Given a positive integer p,

find a point set X* = {x*,...,x*}, and a real number r
I p p
such that


r = f(X*) = min[f(X): |XI = p, X c N] (1.3.1)


where the symbol j*| means the cardinality of a set.

The problem defined by (1.3.1) is called the p-center problem.

Any set X* of p-points that solves (1.3.1) is called an absolute p-

center of N, and the minimum objective value r is called the p-radius.

For p = 1, an absolute 1-center is simply called an absolute center

of N.

If in (1.3.1), each xcX is restricted to a vertex location, the

resulting problem is called the vertex restricted p-center problem and

any set X* C V of p points that solves it is called a vertex restricted

p-center of N. A vertex restricted 1-center is simply called a vertex

center.

We note that the p-center problem is usually formulated in the

absence of addends. In what follows, we will assume all addends are

zero, unless we explicitly mention them. The case with all w. equal
1
to unity will be referred to as the unweighted case.

With this terminology, the p-center problem is the problem of

finding p points on a network so that the maximum (weighted) distance

between any demand point and its nearest center is as small as possible.

The problem is perhaps most applicable to the location of emergency

facilities such as fire stations, ambulance centers, and the like, as





-10-


in such problems a common objective is to provide "good" service to

each demand point by at least one facility within a least possible

distance.

In what follows, we first discuss the 1-center problem on general

networks and on tree networks. Then, we discuss the vertex restricted

1-center problem. Finally, we will discuss the p-center problem in

relation to a "covering" problem to be defined later.

1-Center problem on a general network. The absolute 1-center

problem was defined and solved by Hakimi [47] in 1964. For finding the

absolute center, Hakimi examines the function f on each edge, finds a

best local minimum on that edge, and selects the best among IEJ such

local minima. This method takes advantage of one important property

of f, namely, that it is piecewise linear and continuous on each edge

with at most n(n 1)/2 break points. A local minimum always occurs

either at a break point of f or at an end point of the edge. Hakimi,

Schmeichel, and Pierce [50] showed that Hakimi's method can be imple-

mented in 0(JIEn2logn) computational effort and gave a computational

refinement which reduces the effort to O(JElnlogn) for the unweighted

case. Further refinements of the procedure were obtained by Kariv

and Hakimi [65], resulting in an O(JIEnlogn) algorithm for the

weighted case and 0(JEJn) algorithm for the unweighted case. All

these refinements focus on finding the break points and the local

minimum of.f in the most efficient manner.

A somewhat more general version of the 1-center problem was con-

sidered by Frank [36], and (apparently) independently by Minieka [881,

as Minieka makes no reference to Frank's paper. In this modified

version, called here the continuous 1-center problem, each point on





-11-


the network is a demand point (as opposed only to vertices). The

weight of each point is unity. The objective to be minimized over all

xeN is defined by f(x) = max [d(y,x): yeN]. Both authors showed that

the problem can be reduced to a computationally finite one and pro-

posed a solution procedure which is very similar to Hakimi's.

A probabilistic version of the 1-center problem was considered

by Frank [34, 35] and a number of bounds were obtained on the expected

value of the 1-radius.

For the unweighted case, Singer [101] proved that there exists a

"critical" path, not necessarily a shortest path, connecting two cri-

tical vertices such that an absolute center of the network is at the

midpoint of this path.

1-Center problem on a tree network. We now concentrate on ab-

solute centers of tree networks. Goldman [44] solved the unweighted

case in the presence of addends. Goldman's algorithm is based on the

repeated application of a "trichotomy theorem" that either determines

the edge on which the absolute center lies, or reduces the search to

one of the subtrees obtained by removing all interior points of that

edge. Halfin [51] refined Goldman's algorithm to make it simpler and

computationally more efficient. Halfin's algorithm finds a vertex

center first, and determines the absolute center by examining all

vertices adjacent to the vertex center.

For the unweighted case with no addends, Handler [55] presents

an especially elegant algorithm. Handler's method finds a longest

path of the tree and locates the absolute center at the midpoint of

the path. To find a longest path, Handler chooses an arbitrary vertex

vi, finds a farthest vertex v from v., and then finds a farthest
S 1





-12-


vertex vt from v The path P(v ,v ) is a longest path and its mid-

point is the unique absolute center of the tree. This procedure

requires a computational effort of O(n). Handler's algorithm is

extended by Lin [81] to the unweighted case with addends. Lin showed

that the absolute center of a general network N with vertex addends

can be found by determining the absolute center of an expanded net-

work N' whose vertex addends are all zero. Network N' is obtained from

N by adding a new vertex adjacent to each old vertex, with the length

of the edge connecting the two equal to the addend associated with

the old vertex. For a tree network T, the resulting network is a

tree T' and Goldman's 0(n) algorithm can be applied to T'.

The more general case with both weights and addends was considered

by Dearing and Francis [19], and for the case of a tree network an

0(n2) algorithm was given. The Dearing-Francis paper appears to be

the first to construct a well defined metric space N with distance

d(.,.) from an arc weighted graph N. This mathematical formality per-

mits the use of such concepts as compactness, continuity, and the

extreme and intermediate value theorems. They showed that the distance

d(x,.) is continuous for each fixed x, in turn implying that f(x) is

continuous for every x. From compactness and continuity considera-

tions, they proved the existence of an absolute center for all compact

networks, and its uniqueness for all compact tree networks. They

obtained a lower bound on rl which is applicable to all networks, and

proved that it is always attainable for tree networks. Once the lower

bound is determined, it identifies two "critical" vertices, and the

absolute center can be readily located on the path joining the two.

The bound is the maximum of n(n 1)/2 terms, resulting in a





-13-


computational complexity of 0(n2), and is given by


a : max[a..: 1 < i j L n]
13

where (1.3.2)

w w. d(v.,v.) + w.a. + w.a
.. = 1 1j ji I< < ij wi + w


Hakimi, Schmeichel, and Pierce [50] proved a theorem that reduces the

computational effort for computing this lower bound. Their theorem

states that if for some a it is true that max[a .: 1 I i 5 n] = a =

max[a ti: 1 i < n] then a is the maximum of all a... A different

solution procedure is also given by Kariv and Hakimi [65] for the

same problem. Rather than computing the lower bound, their procedure

confines the search to successively smaller subtrees until an edge is

obtained. The absolute center is located at the local center (also

the global center for a tree) on this edge using Hakimi's procedure

for finding a local minimum. This algorithm is of 0(nlogn).

A nonlinear version of the 1-center problem was considered and

solved by Dearing [18], and by Francis [29]. In this version, each

weight wi is replaced by a monotone increasing function f. of the

distance d(vi,x). Both authors obtained a lower bound similar to the

one defined by (1.3.2). The bound is applicable to all networks and

is always attainable for tree networks.

A "roundtrip" version of the problem was solved by Chan and

Francis [11]. In this version each "demand point" is a pair of ver-

tices (v.,u.) and f(x) is the maximum of the roundtrip distances

defined by p.(x) W w.[d(v.,x) + d(x,u.) + a.]. A lower bound, similar
1 1 1 1 1









to the one defined by (1.3.2) is obtained. The bound is again

applicable to all networks and always attainable for tree networks.

Vertex constrained 1-center problem. The vertex constrained

1-center problem was considered as early as 1869, and perhaps earlier,

by Jordan [63] as a graph theoretic problem. This problem can be

solved by examining the distance matrix of the network, as demonstrated

by Hakimi [47]. Rosenthal, Pino, and Coulter [98] introduced a gener-

alized algorithm that solves a number of "eccentricity" problems on

tree networks, one of which is the vertex restricted 1-center problem.

In this case, the eccentricity of a vertex is defined to be the

distance from that vertex to a farthest vertex. This generalized

algorithm determines the eccentricity of each vertex by making only

two traversals of the vertices. The vertex center is that vertex

with the minimum eccentricity. Slater [103] considered the problem

of finding the vertex center of a network with respect to subnetworks.

In this version of the problem, each demand is a known collection of

vertices (or a subnetwork induced by the collection). The distance

between a vertex and any such collection is defined by a nearest

element of the collection to that vertex. For a given vertex, the

value of the objective function at that vertex is the maximum of the

distances between that vertex and any such collection. Slater showed

that a matrix D' can be constructed from the distance matrix D of the

network, so that each entry of D' is a distance from a vertex to a

nearest element of a collection. Slater demonstrated that the vertex

center with respect to collections of vertices can be found by

examining the matrix D'.





-15-


This completes the discussion of the 1-center problem. We now

concentrate on the p-center problem for p > 2.

p-Center problem on a general network. The p-center problem was

defined by Hakimi [48]. Subsequently, a number of solution procedures

have been suggested. A common characteristic of all these procedures

is that they all rely on solving a sequence of covering problems.

For completeness, we first define a set covering problem and an

r-cover problem.

Let A be a matrix of zeros and ones, y a vector of zero-one

variables yi. The problem of minimizing yi so that each row of Ay
i
is greater than or equal to one is called the (minimal) set covering

problem. Given the function f(X) = max{w.D(v.,X): 1 5 i n}, the

problem of minimizing IXI so that f(X) r for some given value of r

is called the r-cover problem.

Denoting by q(r) the minimum value of the r-cover problem, it

can be readily shown that, if q(r) = p for some r, and q(r') > p for

any r' < r, then r is the p-radius and any X which solves the r-cover

problem is an absolute p-center.

In what follows, we concentrate on the absolute p-center problem

on a general network.

Minieka [87] considered the unweighted case on a general network

and showed that the problem can be reduced to a computationally finite

one. Minieka identifies a finite point set P' such that there exists

an absolute p-center contained in P = P' U V. A point x on some edge

is a member of P' if and only if x is the unique point on its edge

such that d(v.,x) = d(x,v.) for some two distinct vertices vi and v..
Based on this result, Miniek suggested a rudimentary algorithm that
Based on this result, Minieka suggested a rudimentary algorithm that





-16-


relies on solving a finite sequence of set covering problems. Using

the framework provided by Minieka, an exact algorithm was developed

by Garfinkel, Neebe, and Rao [38] for the unweighted case. The

algorithm uses the property that the p-radius is determined by one of

a finite number of elements, namely, one of the distances between any

vertex and any point in P. Call the points in P edge bottleneck

points and let d.. be the distance between vertex v. and the jth

edge bottleneck point. Let Z and Z be a lower and upper bound on the

value of r Initially Z = 0, and Z is obtained by a trial solution.
P
Among all the distances d.. that fall within the interval [Z,Z], one

of them will determine the value of r Pick one such distance, say
p
dst, with Z < dst < Z, and let r = dst be a specified radius. Now,

we want to know if we can cover all vertices of N within this critical

distance r by using only p points. If we cannot, then clearly r is

too small a radius for p points to cover all vertices. Hence we con-

clude the p-radius r must be within the interval [r,Z]. In this

case, the lower bound is shifted to r, and the procedure is repeated.

In the other case, we find a set X of p points that cover all vertices

within r, but it is doubtful if this point set is an absolute p-center.

Clearly, then, the value of r will be within the interval [Z,f(X)].

Hence, the upper bound is shifted to f(X) for this case and the whole

procedure is repeated. Termination occurs whenever the lower and

upper bounds become equal. The r-cover part of this procedure is

solved by obtaining a feasible solution, if it exists, to a set cover-

ing problem. Let A be a IVI by JPI matrix with entries aij equal one

if vertex v. is within a distance r of the jth edge bottleneck point

and zero otherwise. Then, solving the system y i p, Ay y 1,
i






-17-


y.i{0,1} will determine whether or not at most p points (in P) can

cover all vertices of N within a radius r. Computational experience

is reported and it is found that the procedure works better for larger

values of p, as in this case the initial upper bound Z is small, and

significant computational savings result in identifying those edge

bottleneck points whose distances fall within the interval [0,Z].

The weighted case on general networks was considered by Christofides

and Viola [15], and an approximate solution procedure was given. The

procedure finds a set X of p-points whose objective value f(X) is

within some e-neighborhood of the actual p-radius r The procedure
p
obtains X by solving a sequence of r-cover problems with successively

increasing values of r. Termination occurs when the solution of an

r-cover problem generates p (or less) points the first time. In the

process, one also obtains approximate solutions for n-1, n-2,..., p+l

center problems. The solution of each r-cover problem is obtained in

two stages: First, all feasible solutions to the r-cover problem are

obtained by finding all regions on the network that can be reached by

a vertex within a radius of r. Then, among all these feasible solu-

tions, those with minimum cardinality are found by solving a set

covering problem. To find all regions on N reachable by a vertex v.,

one "penetrates" a distance of r/wi along all possible paths originating

at v.. The procedure is repeated for each vertex and the intersections
1
of these penetrations are found. Each maximal intersection defines a

connected region all of whose points are reachable by a subset of

vertices within a radius r. The subset of the vertices is that which

defines the intersection. These regions jointly cover all vertices

of N, and it is possible that a subcollection of the collection of all






-18-


these regions may also jointly cover all vertices. Hence, to find a

minimum cardinality feasible solution, one needs to choose the minimum

number of regions that jointly cover V. This choice can be made by

defining a zero-one matrix A, so that an entry aij of A is one if

vertex vi is covered by region j, and zero otherwise. Solving the

set covering problem with matrix A will provide a solution to the

r-cover problem. Computational experience is reported and it is found

that the procedure works better for small values of p, as the set

covering part of the procedure takes a significant portion of the

total computational time.

An important result is due to Kariv and Hakimi [65]. They showed

that the p-center problem on a general network is NP-complete. Kariv

and Hakimi also showed that the weighted case (as well as the un-

weighted case) can be reduced to a computationally finite one. Based

on this finiteness property, they gave an algorithm whose order of

complexity is polynomial in IEJ, but exponential in p. To show com-

putational finiteness one argues as follows: For any absolute p-center

X = {x1,...,x }, there will be a subset V. of vertices covered by the

ith center x.. If N. is the (sub)network induced by V., then it can

be shown that the absolute center x* of N. can replace x. without in-

creasing the value of the objective function, so that X* = {x*,...,x*}
1 p
is also an absolute p-center. Hence, one can restrict one's attention

to absolute p-centers every element of which is the absolute 1-center

of some subnetwork. The absolute 1-center of any subnetwork of N

will occur either at a vertex or at one of at most IEJn(n 1)/2

"suspected" points. A suspected point on an edge is a point x such

that, for some two distinct vertices vi and v., x is a break point on
J





-19-


its edge of the function f..(.) = max[w.d(vi .), w.d(v.,.)], and

that the two linear pieces defining that breakpoint have slopes of

opposite signs. There can be at most n(n 1)/2 suspected points on

each edge, resulting in a total of O(|En 2) suspected points on all

edges. If S is the set of all suspected points together with the set

of all vertices, then there is an absolute p-center contained in S.

The Kariv-Hakimi procedure selects p-i points from S and determines

all the vertices covered jointly by these p-i points. All uncovered

vertices are assigned to the pth center. Corresponding to each center,

the 1-radius is determined (with respect to the subset of vertices

covered by that point) and the maximum of these 1-radii determines

the p-radius for that trial solution. The algorithm tries every

possible combination of p-i points selected from S and chooses that

combination which minimizes the p-radius. The Kariv-Hakimi procedure

is the only exact algorithm available so far for finding an absolute

p-center of a vertex weighted general network.

A further result on the computational difficulty of the p-center

problem on a general network is given by Nemhauser and Sheu [92].

They showed that finding an approximate solution to the vertex restricted

or absolute p-center problem whose value is within 100% or 50%, respec-

tively, of the optimal value is NP-hard (i.e., at least as hard as

any NP-complete problem).

Vertex restricted p-center problem. The vertex restricted p-

center problem is considered by Toregas, Swain, ReVelle, and Bergman

[109]. A solution procedure is given which relies on solving a sequence

of minimal set covering problems, each corresponding to a specified

radius r. Given a radius r, a 0-1 matrix A can be formed with n rows






-20-


and n columns, so that an entry a.. is 1 if vertex v. is within a

distance r of v., and 0 otherwise. If one solves a set covering
1
problem using the matrix A, the variables whose values are 1 in an

optimal solution determine a feasible solution to the vertex restricted

r-cover problem. The set covering problem is solved by relaxing the

integrality constraints. In the case of non-integer termination, a

single cut produced an integer solution in a large proportion of the

cases. Their computational experience indicates that non-integer

termination seldom occurs.

p-Center problem on tree networks and duality. In what follows,

we concentrate on the p-center problem on tree networks. First, we

define the "continuous" p-center problem. In the continuous p-center

problem, each point in T is a demand point as opposed only to vertices.

Weights are absent (or unity). For any XC T, f is defined by

f(X) = max{D(y,X): yeT} and the continuous p-center problem is to

find an X*C T such that


r = f(X*) = min[f(X): IXJ = p, X C T]
P

Minieka [88] considered the continuous p-center problem on a

general network and showed that it can be reduced to a computationally

finite one.

Shier [100] considered the continuous p-center problem on a tree

network and defined a dual "dispersion" problem. The dispersion

problem is to find p+l points on T the nearest two of which are as

far apart as possible. More explicitly, let U be any finite point

set with IUI = p+l and define h(U) by


h(U) = min{d(ui,u.): 1 < i < j < p+1} .





-21-


The dispersion problem is to find a U* C T such that


h(U*) = max{h(U): UC T, Jul = p+1}


At optimality, Shier's duality result states that

1
r = h(U*)
p 2

for a tree network. The equality may not hold for general networks.

However, Shier showed that the objective value of the continuous p-

center problem is always bounded below by one-half the objective value

of the dispersion problem for any network.

Chandrasekaran and Tamir [14] observed that Shier's duality result

holds when one replaces T by any subset S of T. Chandrasekaran and

Daughety [12] described a procedure for solving the dispersion problem.

They first solve the related problem of locating the maximum number

of points on T such that any two of them are at least X distance

apart for a fixed (positive) X. This problem is solved by working

from "tips" of T to the "center" of T. The general scheme is to use

the algorithm for different values of X, until the number of points

found is p+l and a slightly larger X generates p or less points.

A number of solution procedures have been given for the p-center

problem on tree networks. We now discuss these procedures.

Handler [57] considered the continuous p-center problem on a

tree network for the special case of p = 2 and obtained an 0(n)

algorithm. Handler first finds the absolute 1-center of T, say x*,

and splits the tree at x* obtaining two disjoint subtrees T1 and T2.

Finding the absolute 1-center of each Ti, say x* and x*, determines

an absolute 2-center of T.





-22-


An algorithm of complexity 0(n2 logn) is described by Kariv and

Hakimi [65] for finding the absolute p-center of a vertex weighted

tree network. They show that there are n(n 1)/2 possible values

for r namely, the numbers a.. = w wjd(v.,v.)/(w + w.) for each

combination of vertices vi, vj. The algorithm computes all these

numbers, arranges them in increasing order, and performs a binary

search on this list of numbers. The search relies on solving an r-

cover problem for each value of r chosen from the ordered list {a..}.

The search terminates when the smallest r in the list is found for

which the r-cover problem generates at most p points. The covering

part of the algorithm requires a computational effort of 0(n) for each

r, and a total effort of O(nlogn) for all values of r tried during the

binary search. Hence, the computational effort is determined by the

initial computation and ordering of the numbers a ij and is of

0(n2logn).

A similar approach is used by Chandrasekaran and Daughety [12]

to solve the continuous p-center problem on a tree network. First,

they provided an 0(n) procedure for finding the minimum number of

points needed to cover every point of T within a given radius r.

Then, they provided a method to compute r A further refinement of

the method is given by Chandrasekaran and Tamir in [14]. They proved

that r is determined by one of the numbers d(t,t')/2k, where t and
P
t' are any two tip vertices and k is any integer between 1 and p. The

total computational effort for finding r and applying the covering
P
algorithm is of 0((nlogp) ).

A somewhat different approach, which relies on finding a clique

on a related graph, is given by Chandrasekaran and Tamir [13]. They





-23-


define an intersection graph G for a fixed value of r as follows: G
r r
has nodes corresponding to demand points v ,... ,v Two nodes of G
1 n r
are connected by an arc if the corresponding demand points can be

jointly covered by a (single) common center within a radius of r.

Once G is formed, finding a "clique cover" of G solves the r-cover

problem. A clique cover of G is a minimum number of cliques in G

such that every node is in at least one clique. The solution to the

clique cover problem in G determines a solution to the r-cover problem.

The procedure is repeated for different values of r until a smallest

value of r is found for which the clique cover solution generates at

most p cliques. The computational complexity of the procedure is

polynomial. In particular, the computational effort for finding the

minimal clique cover of G is polynomial because G satisfies the
r r
property that any circuit in G with at least four arcs contains a

chord (i.e., an arc which connects two nodes of the circuit and is

not an element of the circuit). For chordal graphs, algorithms of

linear order have been developed (see [39], [97]) for finding a

minimal clique cover.

This completes the discussion of the p-center problem.



The p-median problem

The difference between the p-center and the p-median problem is

that the objective criterion is changed from minimax to minisum. More

specifically, define the function f(X) for any finite point set X C N

by

f(X) = wiD(vi,X)
ice





-24-


The p-median problem is the following: Given a positive integer p,

find a set X* of p-points such that


f(X*) = min[f(X): IXI = p, X C N] .


Any set X* of p points that minimizes f is called an absolute p-

median of N. If each member of X is restricted to a vertex location,

the resulting problem is called a vertex restricted p-median problem.

Due to a result by Hakimi [47, 48] there exists an absolute p-median

entirely on the vertices of N. For this reason, the distinction be-

tween the vertex restricted and unrestricted versions is insignificant.

Hence, we will take the term "p-median" to mean a solution to either

version of the problem. A 1-median is simply called a median.

The p-median problem arises naturally in locating plants/ware-

houses to serve other plants/warehouses or market areas. The problem

is also motivated by ReVelle, Marks, and Liebman [96] as an example of

a public sector location model where vertices represent population

centers and facilities represent post offices, schools, public build-

ings, and the like.

The 1-median problem. Hakimi [47] appears to be the first to

define an absolute median. Hakimi proved the important result that

there exists an absolute median at a vertex of the network. This

result reduced the search to a finite number of points. The median

can be found by summing each row of the weighted-distance matrix and

choosing the vertex whose row sum is the minimum. This procedure takes

O(n3) operations to compute the distance matrix followed by 0(n2)

operations to find the median.






-25-


For tree networks, more efficient algorithms can be devised to

find a median. An 0(n) algorithm was given by Hua Lo-Keng and Others

[60] and independently by Goldman [42]. The algorithm reduces the

search to successively smaller subtrees until a median is found. At

each stage, one chooses an arbitrary tip vertex (a vertex of degree

one) of the current tree. If the (modified) weight of the selected

vertex is at least as large as half the sum of all weights, a median

is found. Otherwise, that tip vertex is eliminated from further con-

sideration together with the edge incident to it and its weight is

added to the weight of the adjacent vertex. The procedure is repeated

with the new (reduced) tree. The algorithm does not require the com-

putation of the distance matrix and uses only the incidence relation-

ships and the weights.

Goldman's algorithm is based on a "localization theorem" proved

by Goldman and Witzgall [46]. The theorem provides sufficient condi-

tions for a subset of N to contain a median. Given a compact subset

S of N, if S satisfies the two conditions (i), (ii), then it contains

at least one median. The conditions are (i) the set S must be a

"majority" set, meaning that the sum of the weights corresponding to

vertices in S must be at least as large as half the sum of all weights;

(ii) the set S must be "gated" in the sense that there must exist a

unique point g in S such that for every s c S and t c N-S, it is true

that d(t,x) = d(t,g) + d(g,s). Goldman's algorithm in essence is a

repeated application of this theorem to a tree network. Goldman [43]

also proposed an "approximate" localization theorem which somewhat

relaxes the second condition and guarantees the existence of a point

in S that approximates an actual median.





-26-


A median of a tree is shown to be the same as a centroidd" of

the tree by Zelinka [120] for the unweighted case and by Kariv and

Hakimi [65] for the weighted case. To define a centroid, consider

the subtrees T,..,T k obtained by removing vertex vi from T. Let

w(T.) be the sum of the weights of the vertices in T., and define

W(vi) to be the maximum of w(T ) for 1 : j ki. A vertex vt which

minimizes W(v.) over all v. in V is said to be a centroid of T. The

location of a centroid is independent of the distances and can be

found by using only the incidence relations. Goldman's earlier

algorithm in essence finds a centroid of T. The generalized algorithm

of Rosenthal, Pino, and Coulter [98] also finds a centroid of T by

making only two traversals of the vertices. All these algorithms are

of O(n) and solve the 1-median problem without having to compute the

distance matrix.

We now consider some generalizations of the 1-median. Minieka

[88] defined the general absolute median of a network to be any point

on the network that minimizes the sum of unweightedd) distances from it

to the point on each edge that is most distant from it. Minieka showed

that the general absolute median can be strictly interior to an edge;

hence, the search cannot be confined solely to vertices of N.

Slater [103] gave another generalization of the 1-median problem.

In this generalization, each demand is a collection of vertices. The

problem is to find a vertex such that the sum of the distances from

that vertex to a nearest element of each collection is minimum.

Slater showed that the set of vertices that solve this problem forms

a connected path in T. For a general network, the problem can be

solved by constructing a matrix that specifies the distances from each vertex





-27-


to a nearest element of each collection. Simply sum each row of this

matrix and choose the vertex whose row sum is minimum.

Frank considered a probabilistic version of the 1-median problem

in [34] where each weight is a random variable with a known distribu-

tion. A number of bounds are obtained on the expected value of the

objective function as well as its variance. Some of these results

are generalized by Frank [35] to the case where the weights are jointly

distributed random variables.

We now concentrate on the p-median problem with p > 2.

p-Median of a network and vertex optimality. A significant

theoretical contribution is due to Hakimi [48]. Hakimi proved that

there exists an absolute p-median contained in V. Certain generaliza-

tions of this result have been given in subsequent work.

Levy [80] proved that the (vertex-optimal) result holds when the

weights w. are replaced by concave cost functions c (.) of the distance

between vi and its nearest median.

Goldman [41] generalized the result to the case of a "two-stage"

commodity. More specifically, one distinguishes a vertex as being a

source or a destination. Let (Vs,Vd) be a source-destination pair,

and let x. and x. be the nearest medians to v and vd, respectively.
1 3 s d
Then the cost of transferring the commodity from source v to destina-
s
tion vd is the sum of three transport costs, namely, w dd(v ,xi) +

w d(x.,x.) + w* d(x.,v ). In general, if X = {x ,...,x } is a median
sd 1 3 sd (jd p
set, one does not know which median is the nearest to v or vd; hence,

the cost associated with a source-destination pair (s,V d) is

given by


fsd(X) = min [sdd(vsx + wsdd(Xij) + w*dd(x,vd)
xi x CX





-28-


and the objective to be minimized is f(X) = Y [fsd(X): (vsv d)cVxV].

Goldman showed that there exists an optimal X* contained in V, and

conjectured that the result holds for any multi-stage problem.

Hakimi and Maheshwari [49] proved a stronger version of Goldman's

conjecture. In this version, there are multiple commodities for each

source-destination pair, and each commodity goes through multiple

stages. Furthermore the cost of transport from one stage to the next

is a concave nondecreasing function of the distance. More specifically,

let Msd be the set of commodities to be transferred from source v to
sd s
destination vd, and let g(m) be the number of stages commodity meMsd

is to go through. For a given location set X = {x ,...,x }, denote
1 p
by yr xi(r) the location where the rth stage processing takes place.

The cost of transferring commodity m from source vs to destination vd

is given by Csdm[d(vs,y1)] + Csdm[d(y1Y2)] + ... + Csdm[d(yg(m)' d)]

where C sdm(.) is a concave nondecreasing function of the distance.

Denoting this quantity by f (Y), with YC X, IYJ = g(m), the minimum

cost of transfer for commodity m is given by f sdm(X) = min[fsd(Y):
sdm sdm
Y C X, IYi = g(m)]. The cost of transferring all commodities from v

to vd is obtained by summing over all commodities, that is,

fsd(X) = [fsdm(X): meMsd]. The total cost of the system is obtained

by summing the cost fsd(*) over all source-destination pairs, that is,

f(X) = [fsd(X): (vs,vd)CVxV]. Hakimi and Maheshevari proved that

there exists a minimum X* of f(X) contained in V.

Wendell and Hurter [111] considered a more general form of the

problem where the transportation cost functions are permitted to

differ from edge to edge. The transport cost on any edge is a non-

decreasing concave function of the distance. They proved that it is





-29-


sufficient to consider the vertices of the network under such a cost

structure. Furthermore, they obtained the conditions under which it

is necessary for the solution to occur at the vertices. In particular,

they showed that nonvertex optimal locations can occur in any given

edge, only when transportation costs are linear with distance over

that edge and in that case, when and only when the slopes of these

linear cost functions are in a special relation. Hence, if at least

one cost function over some edge is nonlinear, then no interior point

of that edge can be in an optimal solution. If the same situation

holds for every edge, then a solution must necessarily occur at the

vertices of the network.

Solution approaches. Kariv and Hakimi [66] showed that the p-

median problem on a general network is at least as hard as NP-complete

problems. For the case of tree networks, however, algorithms of

polynomial complexity have been developed. Matula and Kolde [85]
3 2
suggested an O(n p ) algorithm for finding the median of a tree net-
2 2
work. Kariv and Hakimi [66] proposed an O(n p ) algorithm for the

same problem.

For general networks, a number of solution procedures have been

developed subsequently, all based on the vertex-optimality result.

Their common characteristic is that they all confine the search to

vertex locations. The solution procedures can be grouped in three

categories: mixed-integer programming approaches, branch-and-bound

techniques, and heuristics.

ReVelle and Swain [95] formulated the problem as a linear integer

program with 0,1 variables. The solution is obtained by applying the

primal simplex algorithm to the associated linear program. In case





-30-


of non-integer termination, a branch-and-bound scheme is recommended

to resolve the problem with integers. Their computational experience

indicates that non-integer termination seldom occurs. Toregas, Swain,

ReVelle, and Bergman [109] formulated a modified version of the problem

as a mixed integer program. The modification is the presence of upper

bounds on the distance between any vertex and its nearest facility.

This formulation makes use of a related but simpler problem. This

simpler problem is to minimize the number of facilities needed to cover

all vertices of N within a specified critical distance. This problem

is formulated as a set covering problem, and solved by ignoring the

integer requirements. In case of non-integer termination, a single cut

produced an integer solution in a large proportion of the cases. A

somewhat different approach to solve the relaxed linear program is

to use a decomposition scheme rather than applying the primal simplex

algorithm. Swain [105] used a Dantzig-Wolfe decomposition approach

to solve the associated linear program. Garfinkel, Neebe, and Rao

[37] independently developed a decomposition approach similar to

Swain's. In case of non-integer termination, they used group theoretics

and a dynamic programming recursion to obtain an integer solution.

A second approach taken is to solve the problem using a branch-

and-bound technique. Khumawala [68] applied a branch-and-bound method

of Land and Doig [77] type, to solve both the set covering problem and

the modified p-median problem formulated by Toregas et al. He showed

that the branch-and-bound approach is computationally efficient for

the former but not for the latter. Narula, Ogbu, and Samuelson [91]

presented a branch-and-bound scheme which relies on obtaining the

bounds by solving the Lagrangian relaxation of the integer programming





-31-


formulation using a subgradient optimization method. Another branch-

and-bound method was developed by Jarvinen, Rajala, and Sinervo [62].

Their procedure looks for n-p vertices that do not belong to a p-

median. This method works better for larger values of p, since n-p

is smaller in this case reducing the number of possibilities. A

similar branch-and-bound procedure was given by El-Shaieb [24]. The

procedure is based on construction of a source set (i.e., p-median)

and a demand set. Starting with both sets empty, a location is added

to either set at each iteration. Whenever the number of elements in

a source set reaches p, or the number of elements in a demand set

reaches n-p, a feasible solution is obtained. An optimal solution is

eventually identified using the lower bounds.

A third approach taken is to use heuristics. A number of

heuristics have been developed by Maranzana [84], Teitz and Bart [107],

and Khumawala [69, 70].

For a discussion of a number of the solution approaches from a

computational standpoint, the reader is referred to Hillsman and Rush-

ton [59], and Khumawala, Neebe, and Dannenbring [71].

Stochastic networks and vertex-optimality. A number of pro-

babilistic versions of the p-median problem have been considered in

the literature. Mirchandani and Odoni [89, 90] extended Hakimi's

vertex optimality result to the case of a stochastic network whose

edge lengths are random variables. Berman and Larson [2] considered

a stochastic network where the availability of servers (centers) is a

random variable. They showed that under suitable conditions there

exists at least one optimal set of locations on the vertices of such

a network.


This completes the discussion of the p-median problem.





-32-


The distance constraints problem

The distance constraints problem involves locating new facilities

on a network so that they are within specified distances of existing

facilities as well as within specified distances of one another. The

distance constraints arise naturally in a locational context if one

wishes to require that a service facility be within a specified time

(distance) of any point in the region it serves. Alternatively, in a

military context, one may want to locate a number of units in such a

way that units are neither too far from their supply bases, nor too

far from one another, in order that one unit may reinforce another if

necessary.

To state the problem, let N be a network with the vertex set

V = (vl,...,v n. Denote by X = (xi,...xm) any location vector in Nm,

the m-fold Cartesian product of N by itself. Define the sets I and

IC as follows: IB = {(j,k): 1 < j < k < m), IC {(i,j): 1 r i m,

1 5 j n). Here, the pairs (j,k) and (i,j) are assumed to be un-

ordered. Let I and IC be two non-empty subsets of IB and IC,

respectively, and suppose we are given nonnegative finite numbers bjk

for each (j,k)el and c.. for each (i,j)cI .
B 13 C
The problem of interest is to find a location vector XeNm, if it

exists, such that the constraints (1.3.3) are satisfied.


d(xi,v.) 5 cij (i,j)eIC

(1.3.3)

d(xj,xk) < bjk (jk)clB


Any vector XENm satisfying (1.3.3) is called a feasible location

vector. The distance constraints are said to be consistent if there exists

at least one feasible location vector XeNm





-33-


Goldman and Dearing [45] provide a conceptual discussion of, and

a motivation for, considering such problems. The distance constraints

are formally defined by Dearing, Francis, and Lowe [22] on a network.

It was established in [22] that, in a well defined sense, the distance

constraints define convex sets under the assumption that the under-

lying network is a tree. Furthermore, the distance constraints always

define convex sets if and only if the network is a tree.

Based on the results obtained in [22], Francis, Lowe, and Ratliff

[32] considered the distance constraints on tree networks in more

detail. They established the necessary and sufficient conditions for

distance constraints to be consistent, and also devised algorithms

that find a feasible location vector whenever one exists. In what

follows we briefly discuss the results obtained in [32].

Distance constraints for a single new facility. For the case of

a single facility, Francis et al. showed that there exists a feasible

point xeT satisfying d(x,vi) < c. for ieI if and only if the in-

equalities d(v.,vk) cj + ck are all satisfied for 1 S j < k n.
j k j k
An equivalent statement of the single facility distance constraints

can be given in terms of "neighborhoods" around vi of radii ci. De-

fine the neighborhood N(u,r) around a point usT of radius r to be the

set of all points xeT for which d(u,x) S r. Then, a point x satisfies

the constraints d(vi,x) < ci, ieI,if and only if x is in each neigh-

borhood N(vi,ci), isI,if and only if x is in the intersection
n
n N(v.,c.). It follows then that the single facility distance con-
i=l
strains d(x,v.) < c., iCI,are consistent if and only if d(v.,vk) k

c. + ck for 1 I j < k < n if and only if each pairwise intersection

N(v ,cj) $ f N(vk,ck) is nonempty for 1 j < k n. Based on this





-34-


property, a "sequential intersection procedure" was developed that
n
determines the composite neighborhood N(a,r) n- N(vi,ci), with
i=l
unique center a and radius r, by intersecting the neighborhoods

N(vi,ci) one at a time in an arbitrary order. The procedure can be

implemented in 0(n) operations. The composite neighborhood N(a,r)

contains all alternate feasible points when the constraints are con-

sistent, and N(a,r) is always a convex compact subset of the tree

network. A result was also given by Francis et al. that provides a

sensitivity analysis on the constraints with no additional computa-

tional effort. Supposing that the distance constraints are consistent

with the original upper bounds c., consider an s-perturbation of the

upper bounds, i.e., for some c > 0 define the new upper bounds to be

c.-c, iel. If N(a,r) is the composite neighborhood corresponding to
1
the original upper bounds, then it can be shown that for any e with

0 5 e < r, the e-perturbed constraints remain consistent and the set

of feasible points to the s-perturbed system is given directly by

N(a,r-e).

Distance constraints for the multi-facility case. For the multi-

facility case, the necessary and sufficient conditions for the con-

sistency of distance constraints are given in terms of n(n 1)/2

inequalities called "separation conditions." The separation condi-

tions are defined by means of an auxiliary graph constructed by using

the sets I and IC. Let G be the graph with nodes N., 1 5 i < m,

corresponding to new facilities,and nodes E., 1 j 5 n,corresponding

to existing facilities. The arc set A of G contains (N.,E.) if

(i,j)CIC and (N.,Nk) if (j,k)slB. The arc length of (Ni,Ej) is ci.

and of (N.,Nk) is bjk. Under the (reasonable) assumption that G is
Jk K jk





-35-


connected, denote by L(E.,Ek) the length of a shortest path connecting

nodes E. and Ek for 1 : j < k n. It was proven in [32] that the

distance constraints are consistent on a tree network if and only if

the inequalities L(E ,Ek) 2 d(v ,vk) are satisfied for 1 S j < k 5 n.

These inequalities are called the separation conditions. The proof

of the consistency of the distance constraints implying the satisfac-

tion of the separation conditions uses only the triangle inequality

and hence is applicable to all networks. The reverse implication

always holds for tree networks, but may fail to hold for general net-

works. The proof of the reverse implication is constructive and

actually finds a feasible location vector under the assumption that

the separation conditions are satisfied. The method that constructs

such a feasible location vector is termed the "Sequential Location

Procedure" in [32]. The method can best be described with the aid of

a physical model. One may imagine that the tree is represented by

appropriately inscribing straight line segments on a board such that

each segment represents an edge. At vertex v., strings of length c..

are fastened for each new facility j such that (i,j)elC. A tip vertex

is chosen arbitrarily and all strings fastened at that vertex are

pulled tight towards the adjacent vertex. If all strings reach the

adjacent vertex, they are simply engaged there with their loose ends

free to be pulled tight in some future iteration. Also the tip vertex

together with the edge incident to it is removed from the model. The

procedure is repeated with the resulting tree. In the other case,

not all the strings reach the adjacent vertex when pulled tight. Among

those which do not reach the adjacent vertex one which is shortest is

selected, and the end point of this string determines the location of






-36-


the new facility it is associated with. All the strings pulled tight

from the chosen tip are engaged at this new facility location. The

feasibility of this location is checked with respect to all existing

facilities and all other new facilities already placed on T. If the

feasibility check is passed, new strings are fastened at this location

associated with that new facility and other unplaced new facilities for

which the distances are of concern. The procedure continues, treating

each placed new facility like an existing facility, until, either all

facilities are placed, or the current tree reduces to a point, in

which case, all remaining new facilities are placed at that point.

If the separation conditions hold, the procedure always finds a

feasible location vector. The algorithm is of O(m(m+n)) and is conjectured

to be a best order algorithm in [33], for determining the con-

sistency of the distance constraints.

Extensions of the results obtained in [32] are given by Francis,

Lowe, and Tansel [33]. These extensions focus on the analysis of

binding separation conditions which in turn determine the "uniquely"

located new facilities. A separation condition that holds at equality

is said to be a binding separation condition. If L(E.,Ek) = d(v.,vk)

is a binding separation condition, then any shortest path P(E.,Ek) in

the auxiliary graph G is said to be a tight path. New facility i is

said to be uniquely located at point Xi if in every feasible solution X to

the distance constraints the location x. is the same. It was shown

in [33] that a new facility i is uniquely located if and only if node

N. lies on at least one tight path. As an immediate consequence of
1
this property the distance constraints has a unique feasible solu-

tion if and only if each N., 1 & i m, lies on at least one tight path
1





-37-


in the graph G. Furthermore, if some path P(E.,Ek) is a tight path,

then the nodes representing facilities in the path occur with the same

ordering and spacing in the path as do the locations representing the

facilities in the path P(v.,vk) on T. This result enables one to

locate the new facilities that appear in a tight path immediately,

without having to use the Sequential Location Procedure.

A multifacility minimax application of the distance constraints

is given in [32, 33] and a multiobjective application is given in [33].

These two applications will be discussed subsequently.



m-Center problem with mutual communication

Let N be a network with vertex set V = {vl,...,vn} and edge set

E. Suppose the sets IB and IC are given with IB C {(j,k): 1 j < k < m}

and I C {(i,j): 1 < i m, 1 < j n}. We assume that we are given

positive weights Vjk for each (j,k)IB and wij for each (i,j)eIC. For

each location vector XeNm, define the functions f (X), fc(X), and

f(X) as follows:


fB(X) = max[vjkd(xj,xk): (j,k)eIB]


fC(X) = max[w ijd(xi,v ): (i,j)eIC] ,


f(X) = max[fB(X), f(X)] .


The m-center problem with mutual communication is the following:

Find a location vector X*cNm such that


Z* E f(X*) = min[f(X): XeNm] .





-38-


The problem differs from the p-center problem in two respects:

(i) the distance between any vertex v. and any new facility xi may be

of concern as opposed only to the distance between v. and the nearest

new facility to v.; (ii) certain distances between new facilities are
J
of concern, as opposed to the absence of interactions between new

facilities. For the case of a single new facility the two problems

coincide.

In this problem, the new facilities may be thought to fulfill a

supporting task to other new facilities as well as servicing those

existing facilities that are a priori assigned to them.

Certain planar cases of the multifacility minimax problem have

been studied by Dearing and Francis [20], Elzinga, Hearn, and Randolph

[25], Wendell and Peterson [113],.and Francis [28].

The problem on a network is defined by Dearing, Francis, and Lowe

[22] in the presence of distance constraints. It is established in

[22] that the function f is a convex function on a tree network. The

existence of a solution is guaranteed due to compactness and con-

tinuity considerations. Furthermore, it is shown that it suffices to

consider only new facility locations in the convex hull of the existing

facility locations in order to solve the problem.

The problem on a general network was shown to be NP-hard by Kolen

[72 ]. For the case of a tree network, the problem is solved by

Francis, Lowe, and Ratliff [32 ] by using an equivalent formulation in

terms of distance constraints (with variable right hand sides). The

solution procedure finds Z* first, by using the separation conditions.

Then an optimal feasible location vector X* is constructed by using the

Sequential Location Procedure described in [32]. To find Z* an





-39-


auxiliary graph G is formed with nodes N1,...,N ,E ,...,E Graph G

contains arcs (N.,E.) with lengths 1/w.. corresponding to pairs

(i,j)cIC, and arcs (Nj,Nk) with length 1/vjk corresponding to pairs

(j,k)eIB. It is assumed that G is connected, for otherwise the problem

decomposes into subproblems. For each pair of existing facility nodes

E Ek, define L(E ,Ek) to be the length of a shortest path in G

connecting Ej and Ek. Francis et al. showed that Z* is given by

max{d(vj,vk)/L(Ej,Ek): 1 S j < k 5 n). The distances d(vj,vk) can be

computed in 0(n2) operations for a tree network (see [23]), and the

shortest path lengths L(E.,Ek) are readily computable in 0(n3) opera-

tions. When Z* is computed, the Sequential Location Procedure de-

scribed in [32] can be applied in O(m(n+m)) operations to find a loca-

tion vector X* that solves the problem.



m-Median problem with mutual communication

Define the functions gB, gC, and g by the following expressions:

For each XENm


B(X) E [vjkd(xj,xk): (j,k)I] ,


gc(X) [wijd(xi,vj): (i,j)eIC] ,


g(X) = gB(X) + gC(X)


The m-median problem with mutual communication is the following:

Find a location vector X* in Nm such that


Z* E g(X*) = min[g(X): XNm] .






-40-


The problem differs from the p-median problem in two respects:

(i) the distance between any vertex and any new facility may be of

concern as opposed only to the distance between a vertex and the near-

est new facility to it; (ii) certain distances between new facilities

are of concern as opposed to the absence of interactions between new

facilities in the p-median problem. For the case of a single new

facility, the two problems are identical.

Planar cases of the problem using rectilinear or Euclidean dis-

tances have received considerable attention and efficient solution

procedures have been developed. A thorough discussion of these prob-

lems is given in the book by Francis and White [31]. Other references

on planar problems are Cabot, Francis, and Stary [6], Bindschedler and

Moore [3], Francis [27], Eyster, White, and Wierville [26], Pritsker

and Ghare [94], Wesolowsky and Love [115, 116], and Picard and Ratliff

[93].

The problem on a network is defined by Dearing, Francis, and Lowe

[22] in the presence of distance constraints. It was established in

[22] that the problem is a convex optimization problem for all data

choices if and only if the network is a tree. For the case of a general

network, it is known that there exists an optimal solution on the

vertices of N. This result and certain generalizations of it have

been given by Goldman [41], Levy [80], Hakimiand Maheshwari [49], and

Wendell and Hurter [111]. These references are already discussed

under the p-median problem. The problem was shown to be NP-hard by

Kolen [72 ] on a general network, and no solution procedures have

been developed yet.





-41-


For the case of a tree network, the m-median problem with mutual communi-

cation is solved by Dearing and Langford [21], and by Picard and

Ratliff [93].

The approach used by Dearing and Langford is to embed the tree T

into the Euclidean space Rp, for some p, so that the distance between

any two points on the tree is equal to the rectilinear distance between

the corresponding points in Rp. The problem in RP with rectilinear

distances decomposes into p subproblems, each of which can be solved

by using known techniques given in Francis and White [31], or, perhaps

more efficiently, by applying the network flow procedure discussed in

Cabot, Francis, and Stary [6]. For reducing the computational effort,

the embedding procedure is carried out with respect to a minimal path

decomposition of T into p edge disjoint paths (each edge is in one and

only one path). Each path in a minimal path decomposition corresponds

to a dimension in R .

The approach taken by Picard and Ratliff in [93] takes advantage

of the vertex-optimality condition and determines an optimal solution

(on the vertices of T) by solving a sequence of at most n-i minimum

cut problems, each on a graph containing at most m+2 nodes. The

method is based on a result that an optimal location vector can be

found independently of the edge lengths, by using only the incidence

relations between vertices and the weights. In this respect, the pro-

cedure is in the same spirit as Goldman's algorithm for finding a

median of a tree. Each cut problem corresponds to an edge of the

tree. To be more explicit, the removal of all interior points of an

edge e leaves two disconnected components, T1 = T (e) and T2 T(e).

Corresponding to edge e, a graph G = G(e) is constructed having nodes





-42-


1 through m corresponding to new facilities, a source s and a sink t.

Graph G contains arcs (s,i) and (i,t) for 1 5 i m and arcs (j,k) for

each pair (j,k)clB. The capacity of arc (j,k) is the weight vjk. The

capacity of arc (s,i) is given by [Wir: Vr ET, (i,r)cI], and the

capacity of arc (i,t) is given by [ [wiq: VqeT2, (i,q)Ic ]. If

(Q,Q) is a minimum capacity s-t cut of G, with scQ, tcQ, then all new

facility locations x. for which the corresponding node i is in Q are
1
in T1 in an optimal solution. Similarly, all x. for which the node j

is in Q are in T2 in an optimal solution. The procedure is a repeated

application of this minimum cut problem with respect to each edge,

until an optimal vertex location is determined for each x.. During

the process, each x. whose location is determined is treated like an
J
existing facility. The method is described originally for the

analogous rectilinear distance problem on the plane, which, in turn,

decomposes into two subproblems, each on a line.



Multi-objective location problems on networks

Multi-objective optimization problems, sometimes known as vector

optimization problems, involve decision making under two or more

criteria. More explicitly, a set (finite or infinite) S of alterna-

tives is specified and n (possibly non-commensurable) objective func-

tions are to be minimized over S. Let f ,...,f be n numerical func-
1 n
tions defined on S, and define f(x) = (fl(x),...,fn(x)) for all xeS.

The multi-objective optimization problem (VMP) is the following:

V-min f(x)
xcS

In general, the minima of the functions fl,...,f do not coincide.
In order for the minimization to be meaningful, one needs tointrodu
In order for the minimization to be meaningful, one needs to introduce





-43-


the concept of "efficient solutions." A point x in S is said to be

efficient if there does not exist a point y in S such that f.(y) 5 f (x)

for 1 i 5 n and fk () < fk(x) for at least one index k. One is

interested in finding and characterizing the set of efficient solu-

tions to (VMP).- An efficient point is sometimes known as an undominated

point. A point which is not efficient is said to be dominated.

Kuhn and Tucker [76] and Koopmans [74] are among the first to

introduce the concept of efficiency. Geoffrion [40] extended the con-

cept to "properly efficient" points and provided a comprehensive

theoretical framework for subsequent research. Necessary and suf-

ficient conditions for efficient points to be properly efficient are

given by Wendell and Lee [112]. Some of the later contributions are

due to Yu [117], Yu and Zeleny [118, 119], Bitran and Magnanti [4],

Wendell [110], and Bergstresser, Charnes, and Yu [1]. We note that

there are other approaches to multicriteria decision making, such as

goal programming, multi-attribute utility theory, construction of

outranking relations, and interactive programming techniques. For

general information on multicriteria decision making, the reader is

referred to Roy [99], Starr and Zeleny [104], Cochrane and Zeleny

[16], Keeney and Raiffa [67], and Thiriez and Zionts [108]. A survey

of multicriteria decision making is given by Chalmet [7].

Multi-objective location problems (on the plane or on networks)

have begun receiving attention only recently. Kuhn [75] appears to

be the first to consider a multi-objective location problem on the

plane. Kuhn considered the problem of minimizing the vector of

Euclidean distances from a variable point to a set of fixed points on

the plane, and showed that the set of efficient solutions is the convex





-44-


hull of the fixed points. Wendell, Hurter, and Lowe [114] considered

the same problem with rectilinear distances and provided algorithms of

0(n2 )and 0(n3 ) for generating efficient points. A most efficient

algorithm of O(nlogn) was developed by Chalmet and Francis [8] for

the same problem. McGinnis and White [83] considered the problem of

minimizing the sum of and the maximum of weighted rectilinear distances

from a variable point to a set of fixed points on the plane and formu-

lated the problem as a parametric linear program for which known solu-

tion techniques exist. Juel [64 ] considered the same problem for

the case of multiple new facilities and gave an equivalent parametric

linear program. Chalmet, Francis, and Lawrence [ 9 10 ] considered

two variants of an efficient design problem, where the location

variable (a design) is a planar region of specified positive area

but of unknown shape.

A few papers have been produced on multi-objective location

problems on networks. In what follows we discuss these problems.

The cent-dian problem. The single facility "cent-dian" problem

involves the sum of and maximum of weighted distances from a new

facility to a set of existing facilities at vertices of N. To define

the problem, let w. and h. be two positive weights associated with
1 1
vertex v., ieI = {1,...,n}. For each point xeN define:
1

m(x) {wid(vi,x): iEl} ,


c(x) E max[h.d(v.,x): iel]
1 1


f(x) E (m(x), c(x)) .





-45-


The problem of interest is to find all efficient points with

respect to f(x).

Halpern [52] is-the first to consider this problem. Halpern

formulated the problem in a slightly different manner by considering

a convex combination of m(x) and c(x). For any fixed X, 0 _< A 1,

define f(A,x) and f*(A) by


f(X,x) Xm(x) + (1 X) c(x) for xsN ,


f*(X) min[f(X,x): xcN] (1.3.4)


In Halpern's terminology, the function f(X,x) is called a cent-dian

function and any point x* x*(A) that solves (1.3.4) is called a

cent-dian point.

In [52] Halpern considered this problem on a tree network with

weights h. all equal to unity. Defining x and x to be the (vertex)
1 m c
median and the absolute center of T, respectively, Halpern proved that

for any given X, the cent-dian x*(X) is located at either x or on
c
one of the vertices located on the path P(x ,x ). This theorem pro-
m c
vides the basis for a simple and efficient algorithm to locate the

cent-dian by inspecting the vertices on P(x ,xc). Further, Halpern

showed that, if the absolute center xe is known, then the cent-dian

can be found by determining the median of a tree T' that is identical

to T except that T' contains an additional vertex v x with the
n+1 c
-1
associated weight wn1 = 1 1.

Handler [56] formulated the same problem on a tree network in a

slightly different manner by using the median function as a constraint.

In Handler's formulation one is interested in solving the problem





-46-


P for each given a, where P is defined as follows:


e(a) = min[c(x): m(x) 5 a, xcT]


Efficient solutions are obtained by parameterizing on a. Handler's

results closely parallel Halpern's.

The problem on a general network is studied by Halpern [54],

using the convex combination approach. Halpern showed that the problem

is a computationally finite one. Computational finiteness follows

from the result that f(X,x) is a continuous, piecewise linear function

of x on each edge and attains its minimum at one of a finite number of

points. Defining Q(e) to be the union of the end points of edge e

with the set of local minima of c(x) on e, the minimum of f(X,x) over

all x on edge e is a member of Q(e) for any given X, 0 5 X 5 1. De-

fining Q E U {Q(e): esE}, it follows that the cent-dian x*(X) is con-

tained in Q for any X. Further, Halpern showed that the function

f*(X) = min[f(X,x): xcN] is a continuous, piecewise linear, concave

function of X for 0 < X < 1. Based on these results, Halpern provided

an algorithm which constructs f*(X) and identifies x*(X) for

0 X < 1. To construct f*(X), the algorithm inspects each edge one

at a time and computes the set Q(e), unless a simple test indicates

that edge e cannot contain any cent-dian for any X. An upper bound

on f*(X) is carried through and improved, whenever possible, by

examining the members of Q(e).

Cent-dian problem and duality. In [53], Halpern studied the cent-

dian problem on a general network from a different angle and obtained

a duality relationship. Using an approach similar to Handler's median

constrained problem, Halpern defined two problems, a median constrained






-47-


and a center constrained one. More specifically, for real X and p

define the functions m*(A) and c*(p) as follows:


m*(X) = min[m(x): c(x) 5 A] (1.3.5)


c*(P) = min[c(x): m(x) < -] (1.3.6)


In general for some values of X (p), the constraint c(x) s X

(m(x) < p) may not admit any feasible solution. However, real inter-

vals C and M can be defined so that for any XeC and for any pcM, the

constraints in (1.3.5) and (1.3.6) admit a feasible point. To define

C, let 0 be the set of all minima to min[c(x): xsN], and let S -be
c m
the set of all minima to min[m(x): xeN]. Let x be a point in Q0 that
c
minimizes the value of m(x) over all x in Q Similarly, let y be a
c
point in 0 that minimizes the value of c(y) over all y in 0 Then
m m
C and M are defined as follows:


C = [c(x), c()]


M = [m(y), m(x)] .


With these definitions Halpern's duality theorem can be stated

as follows:

a) Given any peM, with A = c*(p), we have c*(m*(X)) = A.

b) Given any XeC, with p = m*(X), we have m*(c*(p)) = p.

For a tree network, the functions m* and c* are 1-1 and onto.

It follows from the duality theorem that the function m* and c* are

inverses of each other for a tree network. For a general network,

the functions m*, c* need not be onto, i.e., the image of the domain





-48-


may only be a proper subset of the range. Hence, the inverse property

holds only for some members of C and M for a general network.

Now, we consider a more general multi-objective problem due to

Lowe [82]. The problem involves a single facility to be located on a

tree network with respect to m convex objective functions.

Multi-objective convex location problem (on a tree). Let T be

a tree network and let fl,...,f be m convex continuous bounded func-

tions each of which is defined on T. In general, not all points in T

may be feasible with respect to f.. Let Q. be a convex compact subset

of T which contains all feasible points x with respect to the ith

optimizer. The set Q.may be defined by specifying its extreme points,
1
or by means of distance constraints, or by other means. We assume
m
that Qi is known or computable. Define Q = Qi and assume that Q
i=l
is nonempty. The problem of interest is to find all efficient points

in Q with respect to the vector minimization problem defined below:

V-min[f(x): xeQ C T]

where,

f(x) = (fl(x),...,fm(x)) for all xT .


We note that Q is a convex compact subset of T as it is the

intersection of m convex compact subsets Q of T. For a formal dis-

cussion of convexity on a network, the reader is referred to Dearing,

Francis, and Lowe [22]. Loosely speaking, Q a convex subset of T,

means Q is connected or that the (shortest, unique) path connecting

any two points in Q is contained in Q.

Lowe makes no assumptions on the specific forms of the objective

functions. Under the convexity assumptions, Lowe proves that a convex





-49-


compact subset T* of T can be identified that contains all efficient

points. To identify T*, define Rt to be the set of all minima to the
1
unconstrained problem min[f.(x): xET]. If R* intersects the feasible
1 1
set Q, define St to be this intersection. Otherwise, St is the unique
1 1
closest point in Q to R*. Having defined each S*, 1 S i m, if their
i i
intersection is non-empty, then the set of all efficient points is

given by T* = n {S: 1 5 i 5 m). If this intersection is empty, then
1
T* is the smallest compact convex subtree that intersects each St. It
1
can be shown that each RY, S* is convex, compact, and that T* is a
1 1
convex compact subset of T. Lowe's theorem assumes a knowledge of

set of minima to each f. as well as a knowledge of Qi and hence Q.

We note that the functions c(x) and m(x) in the cent-dian problem are

both convex on T. Hence, Halpern's results can be obtained by apply-

ing Lowe's theorem.

Now, we consider a multi-objective problem which involves multiple

new facilities to be located on a tree network so that the distance

between each specified pair of new and existing facilities, and each

specified pair of new facilities is, roughly speaking, "as small as

possible." The problem is defined by Francis, Lowe, and Tansel [33]

as a sequel to the distance constraints problem, and solved by making

use of the separation conditions. Here, we call the problem, the

"multifacility vector minimization problem."

The multifacility vector minimization problem (on a tree network).

Let T be a tree network and let IC, IB be given nonempty sets with

IC C {(i,j): 1 i m, 1 < j : n} and IB C {(j,k): 1 j < k : m}.

The problem of interest is to locate m new facilities on T at points

x1,...,xm so that each distance d(xi,v ) (i,j)clC and d(xj,xk) (j,k)el





-50-


is "as small as possible." More specifically, we wish to find all

efficient location vectors X = (x,...,x ) in T with respect to the
1 m
vector minimization problem


V-min[D(X): XcTm]


where D(X) is the vector of distances d(xi,vj) (i,j)clC and d(x ,xk)

(j,k)eIB. The vector is formed by assuming any convenient ordering

of the members of the sets IC and IB-

Francis, Lowe, and Tansel [33] characterized efficient points by

making use of distance constraints. By definition, a location vector

Z in Tm is efficient if an only if there does not exist a location

vector X in Tm such that D(X) S D(Z) and D(X) # D(Z). Given a location

vector Z, let b = d(z ,zk) for (j,k)IB and c.. = d(z.,v.) for
jk J k 1 1 J
(i,j)CIC, and define the distance constraints (DC) of interest by


d(xi,v.) 5 cij (i,j)EIC
1 J 1 C

d(xj,xk) bjk (jk)cIB


We note that DC is always consistent, as Z is always feasible

to DC, and hence the separation conditions are always satisfied. The

separation conditions for DC are defined by constructing a graph G

with nodes N., 1 & j 5 m, corresponding to new facilities and nodes

Ei, 1 5 i 5 n, corresponding to existing facilities. For each

(i,j)eIC, the arc (N.,E.) is in G with length c.i, and for each

(j,k)elB, the arc (Nj,Nk) is in G with length bjk. We recall that a

point xi is uniquely located in every feasible solution to DC if and

only if the corresponding node N. is in at least one tight path in G,
1





-51-


where a path of G joining any two existing facility nodes Es and Et

is said to be tight if the length of the path is equal to the distance

between the vertices v and v in T corresponding to nodes E and E ,
s t s t
respectively. For any given location vector Z, denote by A.(Z) the

collection of locations of uniquely located facilities whose nodes are

adjacent to N in G. Let H[A (Z)] be the convex hull of A (Z), i.e.,

the smallest connected subtree containing all points in A.(Z).
1
With these definitions, it was proven in [33] that the following

conditions are equivalent:

(i) Z is efficient.

(ii) Z is the unique solution to DC.

(iii) Each N. is in at least one tight path in G.
1
(iv) Each Z. is contained in H[A.(Z)], 1 5 i m.

This completes the discussion of multi-objective location problems

on networks.



Path Location Problems


Here, we consider three versions of a path location problem posed

by Slater [102]. To define the problems, let P denote any path con-

necting any two vertices in a network N. For any vertex veV and any

path P, define the distance D(v,P) to be the distance from v to a

nearest vertex in P. Also define the branch weight bw(P) of a path

P to be the maximum number of vertices in any component of N-P. The

three versions of the problem are the following:


min C D(v,P) (1.3.7)
P C N veV





-52-


min max D(v,P) (1.3.8)
P CN veV



min bw(P) (1.3.9)
P C N


In Slater's terminology, any path P* that solves (1.3.7) is called

a core of N. Among all paths that solve (1.3.8), one with the fewest

vertices is called a path center of N. Similarly, among all the paths

that solve (1.3.9), one with the fewest vertices is called a spine

of N.

Slater obtained a number of properties of these problems for

tree networks. In particular, Slater showed that the path center of

T is unique and contains the vertex center of T, and that the spine of

T is unique and contains the centroid (equivalently, the vertex

median) of T. We recall that a centroid of T is any vertex v that

minimizes the maximum number of vertices in any component of T-v.

Also, Slater proposed two algorithms of linear order for determining

the path center and the spine of T.















CHAPTER 2

DUALITY AND THE NONLINEAR p-CENTER PROBLEM AND COVERING
PROBLEM ON A TREE NETWORK



2.1 Introduction and Related Work


We consider the problem of locating p new facilities on a tree

network with respect to n existing facilities at known locations so as

to minimize the maximum "loss." The problem is an extension of the

linear p-center problem to the nonlinear case. We assume a strictly

increasing, continuous "loss" function is associated with each of a

finite number of demand points (existing facilities) whose argument

is the distance between the corresponding existing facility and its

nearest new facility. Our formulation permits the use of quite general

loss functions provided that they are continuous and strictly increas-

ing with the travel distance. The term "loss" is used generically

and may refer to any form of inconvenience such as cost, disutility

of service, travel time, etc.

In locating emergency service facilities, the disutility due to

"late" service may be too great beyond a certain "threshold" response

time. Such sharp changes in the disutility of service can be re-

flected into the model by using nonlinear functions. Hurter and

Schaefer [61 ] justify and use such functions in a fire setting. As

pointed out by Dearing [18], a study by Kolesar et al. [73 ] revealed

that the travel time for fire trucks can be approximated by a particular

continuous, nonlinear, increasing function of the distance.


-53-





-54-


The literature on the p-center problem is discussed in detail

in Chapter 1. Here, we give a brief review of the more closely re-

lated work. Except for p = 1, we know of no literature on the non-

linear p-center problem. For p = 1, the only references we are aware

of which deal with the nonlinear case are Dearing [18] and Francis

[29]. Both authors showed that the minimax loss with respect to any

two existing facilities is a lower bound on the maximum loss with

respect to all existing facilities, and that the largest of the lower

bounds determines the minimax loss to all existing facilities on a

tree network. This result is an instance of the duality result we

will present in this chapter.

The linear (weighted or unweighted) p-center problem is shown to

be NP-complete on a general network by Kariv and Hakimi [65], and by

Nemhauser and Sheu [92].

The linear 1-center problem on a tree network is well solved (see

Goldman [44], Halfin [51], Lin [81], and Dearing and Francis [19]).

For p > 1, the linear p-center problem on tree networks is considered

by various authors. Handler [57] provided an 0(n) algorithm for

finding the 2-center of a tree for the unweighted case. Kariv and

Hakimi [65] gave an 0(n2logn) algorithm for tree networks which relies

on solving a sequence of covering problems for the weighted case with

p > 1. A similar procedure for the unweighted continuous p-center

problem on a tree network is given by Chandrasekaran and Daughety

[12]. A vertex-restricted version of the problem is solved by

Chandrasekaran and Tamir [13], and relies on solving a sequence of

clique covering problems on a related intersection graph.





-55-


The first duality relationship involving tree network location

problems can be found in Meir and Moon [ 86 ]. Cockayne, Hedetniemi,

and Slater [17 ] obtained a more general version of the result given

in [86 ]. The results in [ 86 ] and [17 ] closely parallel our duality

result for the covering problem and its dual. Shier [100] discovered

a "dispersion" problem which is dual to the continuous unweighted

p-center problem. The dispersion problem of Shier is to choose p+l

points in the tree network the nearest two of which are as far apart

as possible. Chandrasekaran and Tamir [14] observed that Shier's

duality holds when the problems are defined with respect to a subset

of the tree. For the case where this subset is a finite collection

of demand points, their result is an instance of the duality relation-

ship we will present in this chapter, as applied to the unweighted

linear case.

At this point we give a brief overview of the chapter. In Sec-

tion 2, we define the (nonlinear) p-center problem and a dual "dis-

persion" problem. We state and prove a weak duality theorem applicable

to all networks, and state a strong duality theorem applicable to

tree networks. In Section 3 we give a physical interpretation

of the dual dispersion problem. In Section 4 we study a covering

problem and present an algorithm, COVER, for solving it. The covering

algorithm provides the basis of our solution procedure to the p-center

problem as well as the dual dispersion problem and yields a construc-

tive approach for proving the strong duality theorem. In Section 5 we

present an algorithm, OPTKLIQUE, which provides a constructive proof

of the strong duality theorem, while solving the dual problem. Addi-

tional results for the covering problem, including a "divergence"

problem dual to the cover problem, are given in Section 6.





-56-


2.2 Problem Statements and Duality


We suppose given a finite undirected tree network with positive

arc lengths and denote by T an imbedding of the given network having

as edges rectifiable arcs. For any two points x,ycT, let d(x,y)

denote the shortest path distance between x and y.

Let J {l,...,n} and denote by V {vl,...,v n (VC T) a collec-

tion of distinct vertex locations of "demand points" or "existing

facilities." Let X = {x1,...,x } (X C T) denote a finite collection
1 P
of "centers" or "new facilities." For jeJ, define the distance of v.

to its nearest center by D(X,v.) = min{d(x.,v.): 1 5 i 5 p}, and.let
J
Sj E max{d(x,v.): xsT}. Also, for jeJ, we assume given a real valued

function f., continuous and strictly increasing, with domain [0,6.]

and (clearly) range [fj(0),f.(6.)]. For X C T, IXI < m, we define

the function f by


f(X) = max{f.(D(X,v.)): jJ} .


The Primal p-Center Problem is as follows: Find a p-center X*

for which


r = f(X*) = min{f(X): X C T, IXI = p} (2.2.1)


As discussed in Dearing and Francis [19], due to compactness of

T and continuity of d(x,.) on T for each fixed xET, an optimal solu-

tion X* to (2.2.1) exists and is contained in the convex hull of V.

With a and n defined by a = max{f.(0): jeJ} and n = min{f.(0.):

jeJ}, we shall assume a < n, for if a = fs(0) > ft (6t ) = n, say, then

the function ft would always be dominated by (strictly smaller than)





-57-


f and hence f could be deleted from the definition of f without
s t
changing f. Further, we assume p n-1, as otherwise the p-center

problem is trivial.

So as to state the dual problem, we define Bjk = kj for j,keJ by


Bjk = min max{f (d(x,v )), fk(d(x,vk))}
Jk xcT J k k


For j,keJ with j < k we define ajk max{f.(0), fk(0)} and

bjk min{f (6 ),fk(6k)}. We note that a n implies [ajk,bjk] # 0.

The following lemma, the results of which are proven in [29], provides

a closed form expression for Bjk'

Lemma 2.2.1. For any j,keJ with j _< k we have:
-1 -1
(i) The function f + fk exists, is stricly increasing, continuous,
3 k
has domain [a.j,bjk] 3 0, and range [L jk,U ], where Lj =
jk jk jk Jk jk
-1 -1 -1 -1
(f + f )o(a ) and U = (f1 + f )o(bjk).
j k jk )k j k)kk
(ii) d(v ,vk) < Ujk'

(iii) The function (f. + f ) exists, is strictly increasing and
3 k
continuous, has domain [Ljk,U.k] and range [ajk,bjk].
-1 -1 -1
(iv) Bjk = (f I + fk) o-(max{d(v. ,vk), Ljk)

We remark that either jk = ajk or .jk = (f1 + fk) o(d(v.,vk));

Bjk E [ajk,bjk], and jj. = f.(0). The closed form expression for 0jk

given in Lemma 2.2.1 facilitates construction of the dual problem.

We define the dual objective function g on subsets of V as follows:

For any K C V with IKI > 2


g(K) max{gl(K), g2(K)}

gl(K) E min{ ij: vi,vj E K, i # j}

g2(K) max{f.(0): v. c K} .
J J





-58-


The Dual Dispersion Problem is as follows: Find a subset K* of

V such that


g(K*) = max{g(K): K C V, IKI = p+1} (2.2.2)


We remark that the dispersion problem is meaningfully defined for

2 < p+l < n. The primal p-center problem is trivial for p > n. Hence,

we shall restrict p to 1 S p < n-i.

In what follows in this section, we prove a Weak Duality Theorem

(W.D.T.) and state a Strong Duality Theorem (S.D.T.) (proven in Sec-

tion 5). At the end of this section, we give an example problem

illustrating definitions and results.

In the W.D.T. we shall use the fact (readily proven as in [18]

or [29]) that a < f(X) for any XC T, |XI < m.

Theorem 2.2.1. (Weak Duality Theorem). Assume 1 < p < n-1. For any

X C T with IXI = p, and any K C V with JIK = p+1, we have f(X) > g(K).

Proof. There are two cases: g(K) < a or g(K) > a. In the former

case we have g(K) < a 5 f(X). In the latter case, we note that

g(K) = gl(K) > a > g2(K). Since jXI = p < p+l = JIK, at least two

demand points in K must be served by a single center. In other words,

for some 'v t v K with s # t, and some center xsX, we have


fs[D(X,v )] = fs[d(x,vs)] 5 f(X)
(2.2.3)
ft[D(X,v )] = ft[d(x,vt)] < f(X)


Using the definitions and the inequalities in (2.2.3), we have

g(K) = gl(K) Bst < max{fs[d(x,v )],ft[d(x,vt)]} f(X).

Remark 2.2.1. We note that the conditions JIX = p and JKJ = p+l can

be replaced by jXI < p and/or JKI > p+l, respectively, and the proof






-59-


will still apply. Furthermore, the proof applies to any network, as

no special properties of tree networks are used.

We now state the S.D.T. We remark that the S.D.T. requires the

assumption of a tree network. In effect, network cycles may create a

"duality gap."

Theorem 2.2.2. (Strong Duality Theorem). For any p, 1 5 p s n-1,

there exists an X* C T with IX*l = p and K* C V with IK*l = p+1 such

that f(X*) = g(K*).

It is evident from the W.D.T. that X* solves the primal p-center

problem and K* solves the dual dispersion problem.

Before presenting an example problem, we find it convenient to

view the dual problem as defined on "cliques" of a complete graph G.

We define G to be the undirected complete graph with node set J,

where node j of G represents vertex v. of T. To any arc (i,j) of
J
G, i # j, we assign the length i.., and, to any node j of G, we assign

the node weight j.. = f.(0). We call any complete subgraph K of G a

clique. We note that any nonempty subset of V induces a clique in G

and vice versa. For this reason, an equivalent definition of g(.) on

cliques of G can be given by defining gl(K) to be the length of a

smallest arc in a clique K of G, g2(K) to be the maximum of the

weights of nodes in K, and letting g(K) = max{gl(K), g2(K)}. If the

number of nodes of a clique K is known to be q, we call K a q-clique

and (sometimes) write K Defining C (G) to be the collection of all
q q
q-cliques of G, an equivalent statement of (2.2.2) is as follows:

Find a clique K* for which
p+l


g(K* ) = max{g(K): K c C (G)}
p+1 p+1





-60-


Whether K refers to a subset of V or a clique of G, we prefer to

call K a clique as long as it is clear from the context what K

refers to.

As an example of the nonlinear p-center problem, suppose that the

function associated with node v. is fj(y) = w (y + h ) for y c [0,6 ],

where wj h, and 0 are given parameters. Appropriate restrictions

are placed on the parameters to ensure that the f. are strictly in-

creasing on [0,6.]. We note that the linear weighted p-center problem

is a special case of this problem generated by choosing 6 = 1, h. = 0,

and w. > 0 for all j.
J
For the given form of f., the following are readily verified:
J

-1 1/1
f. (r) = (r/w.) h., r [f.(0), f.(6.)]



f (r) + f (r) = r /[/w) + (1/w.) ] (h. + h.) ,
-1i -1j 3


r e [aij, bj] ,


-1 -1 -1 w 0 6
(f. + f ) o(y) = j (y + h + h.)
i j 1/0 1/0 6
[w. + w. ]
1 J


y c [Lij, Uij]


Then, using the characterization of B.. as given in Lemma 2.2.1,

we have


ij dij if Lij d(vi',vj)
ij.. = (2.2.4)
Pmax[fi(0), f.(0)] if Lij > d(vi,vj) ,
1 j


where





-61-


w.w. 0
y = L 1/ and d..= [d(v,v.) + h + h.
(wi + w.



Consider the tree network shown in Figure 2.1, where the numbers

on the arcs represent arc lengths. The data given with Figure 2.1

corresponds to the parameters for j=1,...,6 where clearly, each f is

strictly increasing. Using (2.2.4), the .ij values for this problem

are shown in Table 2.1 along with the node weights f.(0). Figure 2.2

shows the dual graph G associated with the problem, where the number

next to each node j is the node weight and the number on the arc between

nodes i and j is 8... Using Figure 2.2 it can be verified that the

optimal cliques (specified here by their nodes) and associated g

values are K* = {3,4}, g(K*) = 13829.76; K* = {1,3,6}, g(K*) = 3600;
2 2 3 3
K* = {1,3,5,6}, g(K*) = 1664.64; K* = {1,3,4,5,6}, g(K*) = 784; and

K* = {1,2,3,4,5,6}, g(K*) = 225. Due to the duality theory, it then
6 6
follows that the r for p=1,...,5 are, respectively, 13829.76, 3600,
P
1664.64, 784, and 225.



2.3 Dual Problem Interpretation


We imagine two conservative adversaries, an aggressor A and a

defender D. Defender D has defense forces placed at vertex locations

V1,...,v Aggressor A will attack a single vertex in V. Although D

knows A will attack a vertex, he will not know the vertex attacked

until the attack occurs.

Defender D has p response forces which he must position at loca-

tions defined by a p-center X. Interpret tree distances to be travel

times, so that D(X,v.) is the minimum time to respond to v. from a
J *J





-62-


V6










Data

6 2




9
25
16
36
4
9


f(y) = w(y + h ) 0


Figure 2.1. Example Nonlinear p-Center Problem






-63-


Bi. Values and Node Weights for Example


i


1

2

(ci) = 3
ij.
4

5










j


.4


225


3600

3600


3600

3600

13829.76


3600

3600

8464

900


4356

4556.25

11664

784

1664.64


1 2 3 4 5 6


0 0 64 0 0 144


fj(0)


Table 2.1





-64-


Dual Graph for Example


3600


144/. )


Figure 2.2.






-65-


center in X. Assume A and D know functions f ...,f so that

f.(D(X,v.)) is D's loss if A attacks v. and D responds to the attack

in a time of D(X,v.). For convenience, we refer to the loss A in-

flicts on D as A's gain.

Aggressor A knows D has p response forces, but does not know how

D will position his response forces. Thus A acts conservatively and

bases his decision on a worst case analysis. If A decides to attack

v. without threatening any other vertices, A reasons that D will cor-

rectly guess v. is to be attacked and will position a response force
J
at v.. Hence A assumes his gain will be f.(0), if he decides to
J J
immediately attack v. without a prior threatening strategy. In order

to gain more, A concludes that he must threaten, i.e., pretend to

attack, q vertices, q > 1, so that even if D knows which q vertices

are threatened, D does not know which vertex A will attack until the

attack occurs. Thus D is forced to respond to the threat by position-

ing his response forces optimally with respect to these q vertices.

Hence if A threatens K C V, he assumes D will choose a p-center X

which minimizes f(X:K ) 2 max{f.(D(X,v.)): v. c K }. Thus, with
q J J J q
q p, A assumes D knows K and will position a response force at
q
every vertex in K so that A can gain at most g2(K ). The best A
q 2 q
can do in this case is to choose a K which contains some vertex v
q s
for which f (0) = a. Hence, if q 5 p, A's maximum possible gain is

at most f (0). (Parenthetically, we remark that if f (0) = r ,
s s p
p < n, then it can be shown that not all f.(0) have the same value.
J
If all f.(0) do have the same value, then r > a.) On the other hand,
J p
if A chooses a subset K with q > p, D is unable to position a response
q
force at every vertex in K even if he knows K so A will gain at
q q





-66-


least g2(K ). Hence A observes if he chooses some K with q > p which

contains a vertex v for which a = f (0), then his gain is at least
s S
a = g2(K ). However, A recognizes that there may be some other K

with q > p, which may or may not contain vs, but which yields him a

gain strictly greater than a. For this reason A restricts himself to

those subsets of V with cardinality greater than p and realizes that

if he chooses some K with q > p, then there is at least one pair of
q
vertices in K which D can cover by only a single response force. If
q
v. and v. are one such pair in K which are covered only by a single

response force, say at x, then clearly A obtains a gain of at least

.., as .. = min{max(f (d(x,v.)), f (d(x,v.))): x e T} <. max{f (d(x,vi)),
A
f.(d(x,v.))}. Since A does not know which pairs of vertices D will

cover by single response forces, once he chooses K A acts conserva-

tively, and assumes that D will cover a pair va,vb e K for which

ab = min{i..: v.,v. K i # j}. That is, by choosing a K with
ab ij ij q q
q > p, A guarantees himself a gain of at least ab = gl(K ). Hence

A's minimum gain due to threatening K is g(K ) = max{gl(K ), g2(K )},
q q 1 q 2 q
so A chooses a K* with q > p which maximizes g(K ) over all K C V
q q q
with q > p.

The question arises as to why A should choose p+l vertices to

threaten, and no more. By virtue of the W.D.T. and the remark follow-

ing it, if X* is an optimum p-center then f(X*) > g(K ) for all K
q q
with q > p+l. Thus r = f(X*) is an upper bound on A's gain due to

threatening K But the S.D.T. implies there is a p+1-clique, say
q
K* which attains this upper bound. Hence A need threaten no more

than p+1 vertices to maximize his gain, as A cannot obtain any addi-

tional gain by threatening more than p+1 vertices.





-67-


There is also the possibility that A will make a false threat,

that is, attack a vertex not among the ones he threatens. If D be-

lieves the threat is false and continues to act conservatively, he

will simply choose a p-center X* to minimize f. But since there exists

a p+1-clique K* such that g(K* ) = f(X*), the greatest loss D can
p+1 p+l
incur, given X*, is the same as if he believes A's optimal threat to

be real, and acts accordingly. Hence A cannot gain more by making a

false threat.



2.4 Covering Algorithm


In this section we study a covering problem, and present an

algorithm for solving it. Our primary interest in the algorithm is

the fact that it provides a constructive approach for proving results

about the primal and dual problem. For this reason we purposely keep

the algorithm simple, and use an analog string model to provide insight

into the algorithm. The development of both the string model and the

algorithm is motivated by an earlier string algorithm given in [32].

As in [32], an equivalent algebraic version of the algorithm is

readily obtainable. We remark that two other quite efficient algo-

rithms [14], [15], exist for solving the covering problem, but they

do not lend themselves readily to our needs.

At this point we state the Covering Problem: Given r and the

runction f, compute


q(r) = min{IXI: f(X) 5 r, X CT} (2.4.1)


It is readily seen that the covering problem has a feasible solution

if and only if a 5 r. Further, with J(r) E {j: r < f.(6.)}, we shall
*J J





-68-


assume J(r) # 0, for if J(r) = 0 then the condition f(X) 5 r holds

for all X C T and we (trivially) have q(r) = 1.

The above assumptions permit the following equivalent statement

of the covering problem:


minimize IXI

subject to
-1
D(X,v.) : fI (r), j e J(r) (2.4.2)
J J

We refer to the covering algorithm as COVER. In order to state

COVER a few definitions are convenient. We may imagine that the tree

is represented appropriately by inscribing straight line segments on a

planar surface such that each segment represents an arc. We fasten
-1
strings of length f. (r) to each node v.,j e J(r), of the inscribed
j J
tree, where, by convention, we allow strings of zero length. Every

fastened string has one end permanently affixed to the planar surface.

In addition, during the use of the algorithm we engage previously

fastened strings at various points on the tree. When a string is

engaged, some point of the string is permanently affixed to the tree

such that there is no slack in the portion of the string so far en-

gaged. When strings are removed, we imagine that they are physically

deleted from the string model.

During each iteration of the procedure, we partition the original

tree into two subsets: one green, the other brown. The green subset

is always a tree, denoted as GT (for green tree), while the brown sub-

set consists of one or more subtrees of the original tree T, each of

which is "rooted" at a node of the green tree. By convention, a root





-69-


node t will be in both GT and the associated brown subtree, denoted

as BT(t).

COVER

0) Initialize to GT = T, k = 0. For every tip vertex v. of T define
J
-1
BT(v.) = {v.}. For every j e J(r) fasten a string of length f. (r)
J J J
at v.. Define U = 0.
J o
1) Choose a tip t of GT. If GT = {t} go to 6). Else find a(t), the

vertex in GT adjacent to t.

2) If no strings are engaged or fastened at t, remove from GT the

subarc [t,a(t)] joining t and a(t), attach [t,a(t)] to BT(t), and go

to 1). Else go to 3).

3) Pull all strings at t tight towards a(t). If all tight strings

reach a(t) then engage them at a(t), remove [t,a(t)] from GT, attach

[t,a(t)] to BT(t), and go to 1). Else go to 4).

4) Add 1 to k. Choose a shortest string engaged or fastened at t.

Find the (unique) vertex, say v(k), at which the shortest string is

fastened. Construct Uk = U k- U {V(k)}. Find the farthest point, say

y, from t on [t,a(t)] to which the shortest string can reach. Locate

xk at y. Assign all strings at t to xk and remove these strings.

Attach [t,y] to BT(t) to obtain BT(xk), and remove [t,y] from GT.

Go to 5).

5) Assign to xk all other strings in GT which can reach xk, and re-

move all such strings. If no strings remain then let U = Uk and stop.

Else return to 1).

6) Add 1 to k. Locate xk at t. Assign all strings at t to xk. Of

the strings at t choose any one, and find the vertex v(k) to which

the chosen string is fastened. Let U = Uk- U {v(k) }, and stop.





-70-


Note that each time COVER places a center at xk in step 4) it

identifies an associated vertex v(k) which we call the distinguished

vertex associated with xk. When centers xl,...,xk have been placed

in step 4), we call Uk = {v(1),...v(k) } the distinguished set

associated with {xl,...,k}. If the algorithm places q centers in

total, then the set U defined by the algorithm consists of vertices

v(1)",.,V(q), the first q-1 of which are distinguished vertices

(when q > 2). The last vertex is distinguished only if x is placed

in step 4). Letting X = {x ,...,x }, we call U the primary set
q
associated with X, and call v(i) the primary vertex associated with

x., i = 1,...,q. We note that the primary vertices v(1),...,v(q) are

distinct, for as soon as a primary vertex is identified, its string

is removed, and thus the vertex is not available for any subsequent

identification. Likewise the centers xl,...,x are distinct, for if

x. = x. with i < j, then all strings assigned to x. would have been

assigned earlier to x., and so x. would not have been located. Hence

it follows that IUI = IXl = q, and U # 0, since IXj ? 1. The primary

vertices will be of theoretical significance in proving our results.

We now establish some properties of COVER.

Property 2.4.1. COVER finds a feasible solution X to the covering

problem with IXI : n.

Proof. We first note that termination is clearly finite, since at

each iteration either at least one string is removed, or some entire

arc of T becomes colored brown. Since there are at most n strings

initially, it follows that the X constructed satisfies IXI 5 n.

Choose any v.,j e J(r), and denote by x(j) the center to which

v. is assigned. Since the string fastened at v. reaches x ,
JJ J)





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-1
d(x(j,v.) < f -(r). As D(X,v.) 5 d(x j,v.) it follows that X is
(j)' J J (j) J j
a feasible solution.

Property 2.4.2. For any nonempty distinguished set Uk, with vertices

numbered so that Uk = {v,...,v k}, we have


v. E BT(x.), 1 j 5 k (2.4.3)
J J

-1
d(x.,v.) = fl (r), 1 j 5 k (2.4.4)


Proof. Expression (2.4.3) is obvious. To show (2.4.4), choose any v.
3

in Uk. Let t be the tip vertex chosen at the first of the iteration

in which x. is placed. The algorithm causes the string at v. to-be
J J
pulled tight along every edge connecting v. to t, and to be pulled

tight along [t,x.], with the string end point coinciding with x..
-1
Thus d(v.,t) + d(t,x.) = f (r). But v. e BT(t) and x. e T-BT(t) or

x. = t so that d(vj,t) + d(t,x.) = d(v.,x.). Thus, (2.4.4) follows.
JJ J J
Property 2.4.3. Let X = {x1,...,x } be the feasible solution con-

structed by COVER, with vertices numbered so that U = {v,...,v q} is

the primary set associated with X. Assume q > 1. Then

-1 -1
d(v.,v.) > f (r) + f (r) for 1 i < j q (2.4.5)
i 3 i j

Proof. We know the first q-1 members of U are distinguished vertices.

Hence Property 2.4.2 implies


v. e BT(x.), 1 < i < q-1 (2.4.6)
1 i

-1
d(v.,x.) = f (r), 1 r i < q-1 (2.4.7)
1 1 i

For i < j, x. is placed prior to x.. Since v. is assigned to x. and
1 JJ kj





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not to xi, for 1 i < j 5 q, v. was not in BT(x.), and the string at

v. did not reach x.. Hence
J I

v. e T-BT(x.), 1 < i < j < q (2.4.8)

-1
d(xi,v.) > f (r), 1 < i < j 5 q (2.4.9)


But (2.4.6) and (2.4.8) give d(vi,v.) = d(vi,x.) + d(xi,v.) for

1 5 i < j 5 q, from which, on using (2.4.7) and (2.4.9), (2.4.5)

follows.

We shall need the following remark, proven in [32]:

Remark 2.4.1. Given any a.,a. s T and s.,s. > 0, there exists a.point

x in T for which d(x,a.) 5 s. and d(x,a.) < s. if and only if d(a.,a.)
1 1 J J 1 J
5 s. + s..
1 j

We are now ready to establish the optimality of COVER.

Theorem 2.4.1. Given any r for which a < r and J(r) # 0, COVER solves

the covering problem.

Proof. Let X = {x1,...,x q be the point set found by COVER. Property

2.4.1 implies X is feasible to the problem. If q = 1, X is clearly

optimal. If q > 1, let the vertices be numbered so that U = {vl,...,v }

is a primary set associated with X. By Property 2.4.3, d(v.,v.) >
-1 -1
f. (r) + f. (r), for 1 : i < j < q. Remark 2.4.1 implies there exists
1 j
-1 -1
no x in T for which d(x,v.) < fi (r) and d(x,v.) < f (r) for any

i, j in {1,...,q} : J(r) with i < j. Hence it is impossible to cover

any two members of U with a common center. Thus, since JUI = q, any

feasible solution X to the covering problem satisfies lxi > q. Since

q = IXI and X is feasible to the problem, X is thus an optimum feasible

solution.





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We remark that the covering problem may be of as much interest,

from both a theoretical and applications point of view, as the p-center

problem. In Section 6, we will present a problem which is dual to the

covering problem and show that the primary set identified by COVER

solves the dual of the covering problem. Furthermore we will charac-

terize q(r) as a step function, and provide a formula for q(r)

assuming that r is known for 1 < p 5 n-1.



2.5 Dual Problem Solution and the Strong Duality Theorem


Based on the W.D.T. and properties of COVER we now present a

proof of the S.D.T. The proof is constructive in that we use an

algorithm called OPTKLIQUE which, given the optimal objective value

of the primal problem, constructs an optimal solution to the dual

problem. We then show that the objective values of the pair of prob-

lems are equal. As a by-product the proof also establishes that

r e R, where, for convenience, we define R E { ..: 1 i 5 j < n}.
p 1J
We find it useful to summarize Theorem 2.4.1 and Property 2.4.3

as follows:

Lemma 2.5.1. Given any r for which a 5 r and J(r) # 0, the following

assertions are true:

(a) COVER finds an optimum solution X to the covering problem with

q(r) = IXI.

(b) Whenever q = q(r) > 1, any primary set U = {v(1)"...,(q)

associated with X satisfies


g(U) = gl(U) > r


(2.5.1)





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Proof. (a) is just Theorem 2.4.1.

(b) From Property 2.4.3, for any vi,v. E U, i # j, we have d(v.,v.) >
-1 -1 -1 -1
f (r) + f- (r) > f (a) + f (a) where r a a = a.. Thus,
i j i j
-1 -1 -1
d(v.,v.) is in the domain of (f + f ) from which, upon using

Lemma 2.2.1 and the definitions of g, gl, and g2, (2.5.1) follows.

In the algorithm OPTKLIQUE we assume that r is given for some

value of p, 1 p < n-1. OPTKLIQUE constructs an optimal solution to

the associated dual problem.

OPTKLIQUE

1) If r = a, take K* to be any p+1-clique in V containing a vertex
p p+l
v for which f (0) = a, and go to 3). Else, given r > a, compute
s s p
r' = max{f.. e R: V.. < r } and choose any r for which r' < r < r .
p J1 1J P P p
Go to 2).

2) Apply COVER with the chosen value of r to find an optimum solution

X and its associated primary set U, with IXI = q = IU|. Note r < r
P
implies IXI > p, so q k p+l. Take K*+ to be any subset of U con-
p+1
sisting of p+l members of U. Go to 3). (If q > p+l, there will be

alternative optimal cliques.)

3) If K*+I is any clique found in either step 1) or 2), then g(K* ) =

r and the W.D.T. guarantees K* is an optimum solution to the dual
p p+l
problem.

Before proving the correctness of the algorithm, we note, since

a = hh for some h, that a < r implies a 5 r', and thus the r chosen

in step 2) is one for which a feasible solution exists to the covering

problem.

Theorem 2.5.1. Given r for any p, 1 p < n-1, the clique K* con-
p p+1
structed by OPTKLIQUE satisfies





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g(K* ) = r (2.5.2)


Furthermore, K* solves the dual dispersion problem.
p+1
Proof. Let X* be an optimum p-center solution to the primal problem

so that IX*I = p and f(X*) = r Since r 5 a we consider the cases
P P
r = a and r > a. Let us apply OPTKLIQUE for each case.
P P
For r = a, K* is chosen in step 1) so that IK* | = p+1 and
p p+1 p+1
a = f (0) = g(K*). The W.D.T. gives g(K* ) < f(X*). But then,
s 2 p+1- p+
a = g2(Kp+) = g(K*+) = f(X*) = r = a, establishing (2.5.2) for

this case.

For r > a, define R {(.. e R: r 5 P..} C R. Since r > r > r'
P 13 p ij P P
there exists no (.. in R for which r < B.. < r Thus 8.. > r implies
1J 1J P i3
B.. > r and so it follows that
13 P

R = {..: r < ..} (2.5.3)


Let U be the primary set identified by COVER for the chosen r,

r' < r < r By Lemma 2.5.1, U satisfies g (U) > r from which it
P P
follows that 3.. > r for v.,v. e U, i # j. Hence, (2.5.3) implies


3.i R v.,v. E U, i # j (2.5.4)
3J 1 j

Since IU| > p+l, let K* be that subset of U identified in step 2).
p+1
We have the following string of inequalities:


rp = f(X*) > g(K*) (2.5.5)

2 l(K*+1) (2.5.6)

= min{ij: vi,vj K*I+, i # j} (2.5.7)

> min{ij : vi,vj e U, i # j} (2.5.8)

> min{i.j e R} (2.5.9)

> rp (2.5.10)





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where (2.5.5) follows from the W.D.T., (2.5.6) and (2.5.7) follow

from the definitions of g and gl, (2.5.8) follows from K*+ C U,

(2.5.9) follows from (2.5.4), and (2.5.10) follows from the definition

of R. Hence, every inequality holds as an equality, establishing

(2.5.2) for this case.

The assertion that K* solves the dual problem is immediate from
p+1
f(X*) = g(K* ) and the W.D.T.
p+1l
We note that Theorem 2.5.1 provides a proof of the S.D.T. since in

the statement of the S.D.T. we take X* to be an optimum p-center solu-

tion to the primal problem and K* as constructed by OPTKLIQUE. We
p+l
also note that the duality theory provides necessary and sufficient

conditions for a p-center to be optimal, which, as far as we know, are

the first such conditions for this problem.

We remark, just as with the linear p-center problem, that if we

define Bs = min{j..: Bij R, q(.ij) : p}, then st = r Clearly

q(r ) 5 p and q( st) S p. The S.D.T. implies r e R, and thus the

definition of Bst gives st < r Let p' = q(st) and let X solve
S tSts p St
the cover problem for r = 0 so that f(X ,) st Since p p',
"st p st
append to X (if necessary) any p-p' center locations to obtain the

p-center X Clearly D(X ,v.) D(X ,,v.) for v. 6 V, and thus

f(Xp) : f(Xp,). Hence r f(X) (X (X ,) 5 8st r so s = r
P p p p p st p st p
and X is an optimum solution to the p-center problem. This remark
p
permits the use of the same procedures as discussed in [65] to compute

r efficiently, by performing a binary search over the (ordered) list
p
R, applying COVER for every r chosen from R until a smallest st in R
st
is found for which COVER finds p or less points. Once r is computed

in this manner, OPTKLIQUE requires an additional application of COVER





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for any r, r' < r < r and solves the dual dispersion problem. This
P P
approach is essentially a primal approach for solving both problems.

An alternative approach which directly works with the dual graph is

given by Chandrasekaran and Tamir [13] for the unweighted linear p-

center problem, which works directly with what would be a subgraph of

our dual graph G. Due to absence of weights and addends, their

approach does not require the use of node weights (and for that matter

the function g2) in the dual graph. For a given value of r, Chandra-

sekaran and Tamir define an intersection graph IG with node set J and

arcs (i,j) for those indices i,j e J for which 5.. 5 r. Their pro-
1j
cedure is based on a graph theoretic procedure given by Gavril [39]

and solves the covering problem by finding a minimum clique cover of

IG (minimum number of cliques such that every node is in at least one

clique). As a side result, their approach identifies a maximal anti-

clique in IG (a maximal set of nodes in IG no two of which are con-
r r
nected with an arc). Due to "chordal" properties of IG as discussed
r
in [39], the cardinality of a minimum clique cover of IG is equal to

the cardinality of a maximal anti-clique in IG This result is a
r
special instance of the duality result we will present in Section 6

for the cover problem, as applied to the linear unweighted case.

Furthermore, for r = r Chandrasekaran and Tamir [39] proved a duality

relationship for the unweighted p-center problem using the above

properties of IG We remark that their duality results can be
r
directly proven by using the algorithm OPTKLIQUE, and by appropriately

specializing our S.D.T. for the linear unweighted case.

We now demonstrate the use of OPTKLIQUE by determining K* for

the example problem. From our previous analysis, r3 = 1664.64. Since





-78-


r3 > a = 144, we compute (from Table 2.1) r-=max{Bij E R: B < r 3 = 900.
3Ij 3
We next must apply COVER using a value of r where 900 < r < 1664.64.

Figure 2.3 shows the results of using COVER with r = 1296. In the

figure, the loose ends of the strings are shown as wavy lines. Brown

subtrees are shown as crosshatched arcs of the original tree. Each

separate drawing of the tree (a)-g)) is for a subsequent iteration of

COVER. Figure 2.3a) demonstrates the initialization step, where for
-1
r = 1296, the f (r), j = 1,...,6 are 12, 7.2, 7, 6, 18, and 8, re-

spectively. The numbers next to the strings are the lengths of the

loose ends. In the figure, we indicate which tip of the green tree

is chosen at each return to step 1) of COVER. In addition, the suc-

cessive distinguished vertex sets Uk are indicated.

After the final iteration, we note that the primary vertex set

U is {v3,v1,v6,v5} which, from our previous analysis, we know to be

K*
4'



2.6 Results for the Covering Problem


In this section we present a "divergence" problem which is dual

to the covering problem. We give a weak duality and a strong duality

result and prove that the primary set identified by COVER solves the

dual problem. The term "divergence" is chosen to represent the

physical interpretation, discussed later, in which the attacker A

chooses a "divergent" set of vertices to threaten. Further, the term

permits a distinction to be made between the two different dual prob-

lems. Also, in this section, we demonstrate how having optimum solu-

tions to the p-center problem for all p, 1 5 p 5 n, enables us to

completely characterize the function q(r).
















U2 = {v 3v I
U2 3' 1
c)


Choose; e v
1 b6


U3 = v 3,vl, 6


Choose v5


c )= (V3
c)


U4 = {v3',V1,6'v5}


U = {v, vl, v6, v 5

( ( 3, I, (b, S


Figure 2.3. OPTKLIQUE for p = 3 for Example


I nIt il 1 1zat ol n


Choose vl


Chooli v3


C(lw'iii v 2





-80-


The Divergence Problem is as follows: Given r and the function

g, compute


q(r) max{lUl: g(U) > r, UC V} (2.6.1)


That is, the problem is to find the maximum number of existing facili-

ties no two of which can be jointly covered by a single center within

a radius of r. Equivalently, among all cliques of G whose gain is

larger than r, the problem is to find one with the maximum number of

nodes. The dual problem is feasible for r < rl, as, if r > rl there

does not exist a subset U of V for which g(U) > r. On the other hand,

the primal cover problem is feasible for r > a. Hence, we shall re-

strict r to a < r < rl in order to ensure feasibility to both

problems.

Theorem 2.6.1. (Weak Duality Theorem). Assume a < r < r1. For any

feasible solution X to the primal cover problem, and any feasible

solution U to the dual divergence problem, we have jIX > Iul.

Proof. By feasibility of U and the assumption of the theorem we have

g(U) = gl(U) > r > a > g2(U) from which it follows that


ij > r v,v. E U, i # j (2.6.2)


Suppose IXI < jIU. Then, the same approach as in the proof of Theorem

2.2.1 implies there exist vsvt U, s # t, such that Bst 5 f(X) < r,

contradicting at least one inequality in (2.6.2). Thus, IX| >? ul.

Theorem 2.6.2. (Strong Duality Theorem). Assume a < r < rl. Let X

be a feasible solution to the covering problem constructed by COVER.

Then, the primary set U associated with X solves the dual divergence

problem with


Ixl = q(r) = q(r) = JUI .


(2.6.3)





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Proof. By definition of a primary set we have jIX = IUI. By assump-

tion r < rl so that IXI = IUl k 2. Lemma 2.5.1 implies g(U) = gl(U) > r.

Hence U is a feasible solution to the dual problem. Theorem 2.6.1 im-

plies q(r) 1 q(r). By feasibility of X and U, and the fact that

IXJ = Iul, we have IXI : q(r) ? q(r) 2 Iju = JIX. It follows that

X solves the cover problem, U solves the dual problem, and (2.6.3)

holds.

We remark that the above proof is an alternative to the proof of

Theorem 2.4.1 for establishing the optimality of X to the covering

problem. Hence, an application of COVER solves both problems simul-

taneously.

At this point we give an interpretation of the pair of problems.

The defender D specifies an upper bound r on his loss against an attack

to any vertex and will position response forces as necessary so that

his loss will not exceed r. Each response force is an "expense" for

D. Hence, D's problem is to choose the fewest possible response

forces. The attacker A knows that D will not tolerate a loss exceeding

r. Hence, A recognizes that, no matter how many vertices he threatens,

D will have a sufficiently large number of response forces to respond

and that the loss A inflicts on D will always be less than or equal

to r. For this reason, A decides that he should not (hopelessly) try

to inflict a loss to D exceeding r, and that, instead, he should force

D into using as many of his response forces as possible. Hence,

should A choose a subset U of V with g(U) > r, he knows that no two

vertices in U can be jointly covered by a single response force by D

within the specified upper-bound r. Thus, D, not tolerating a loss

exceeding r, will have to allocate one response force for every vertex





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in U. In total, any feasible X which D chooses will satisfy IXI > IUI,

which is what the W.D.T. asserts. By virtue of the S.D.T., if U is

A's optimal choice, D can choose exactly jul response forces positioned

at, say X, with |X| = 1U| and still respond to an attack to any vertex

in U (as well as in V-U) without incurring a loss exceeding r. If A

threatens more than q(r) = J11 vertices, say, a subset U of V, then

IUI > q(r) implies g(U) < r (infeasibility). Thus, D would not be

forced into allocating a single response for every member of U. In

fact, even if A threatens every vertex in V, then D still needs ex-

actly q(r) = q(r) = IUI response forces to respond to the threat

feasibly. Thus, if each threat is an "expense" for A, he need threaten

no more than q(r) vertices. On the other hand, D adopts an optimal

strategy against A's best threat by minimizing the number of response

forces with respect to V.

Continuing our consideration of the covering problem, we now re-

verse the usual procedure, and view the p-center problem as a device

for solving the covering problem for all values of r for which the

covering problem is feasible, that is, for a : r.

The following lemma is the key to using the p-center problem to

solve the covering problem. Define r = for convenience.
o
Lemma 2.6.1. Let p e J. If r < r then
S p p-1'

q(r) = p for r < r < rp-1


Proof. We first note rn < r n- < ... rl < rO. Also, clearly,

q(r ) 5 p for p e J. Now for rl 5 r since q is non-increasing we

have 1 q(rl) 2 q(r) 1, establishing the claim if p = 1. Consider

the case p e {2,...,n}. From rp r < rp we have p > q(r ) > q(r) _
Suppose q(r) =s, with s < p. Let X,
q(rp-1). Suppose q(r) = s, with s < p, implying s < p-1. Let X,





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with IXI = s, solve the cover problem for r. We then have f(X) S r <

r 1 < r contradicting the definition of r Thus q(r) = p for

r r < r
p p-l
It now follows, if we define the set


P = {(p-l,p): p E {2,...,n}, r < r } ,
p p-i

that


Sp for r r < r-1, (p-l,p) e P
q(r) = (2.6.4)
1 for r 1 r


The formula (2.6.4) completely defines the function q(r), since r = a,
n
and the cover problem is feasible if and only if a 5 r. Hence if we

solve the p-center problem for all p and compute r2,...,r then we

have an explicit formula for q(r), and we see that the r completely

define the function q. For example, if r6 = r5 < r4 = r3 < r2 = rl,

then q(r) = 5 for r5 < r < r4, q(r) = 3 for r3 = r < r2, and q(r) = 1

for r1 < r. Also, the proof of the lemma does not require the assump-

tion that the location network is a tree. Thus the formula for q(r)

is still valid if the location network has cycles.















CHAPTER 3

A VECTOR-MINIMIZATION PROBLEM ON A TREE NETWORK



3.1 Introduction


We consider a vector-minimization problem on a tree network which

involves as objectives the distances between specified pairs of new

facilities and specified pairs of new and existing facilities. In many

location problems, especially in the public sector, it may be necessary

to build a number of public facilities which are to be shared by a number

of communities. If the optimizers cannot agree on a single objective

function, the analyst is faced with the problem of locating the facili-

ties in such a manner that all parties are satisfied with the end

result. In such a case, the optimizers can agree to rule out "dominated"

solutions and consider only "efficient" solutions.

The related literature on multi-objective location problems is

discussed in Chapter 1 under Multi-objective location problems on

networks. Here, we concentrate on characterizing efficient solutions

to the vector-minimization problem of interest. We relate efficient

solutions to a distance constraints problem studied by Francis, Lowe,

and Ratliff [32]. Extensions of results in [32] are given by Francis,

Lowe, and Tansel [33]. We use the theory developed in [32] and [33]

to establish the necessary and sufficient conditions for efficient

location vectors (parenthetically, we remark that the results we proved

in [33] are also given in our Dissertation Proposal defended on June 8,

1979).


-84-





-85-


At this point, we give an overview of the chapter. In Section 2,

necessary definitions and notation are given and the vector-minimiza-

tion problem of interest is defined. In Section 3, we relate the

problem to distance constraints, give a number of related properties

of distance constraints, and establish the necessary and sufficient

conditions for a location vector to be efficient. In Section 4, we

provide examples of efficient and non-efficient location vectors.

Section 5 is devoted to a further refinement and simplification of one

of the necessary and sufficient conditions, namely, "the convex hull

property." In Section 6, we provide an algorithm, SEVCA, which con-

structs an efficient solution from a given location vector. In Sec-

tion 7, we characterize efficient solutions for the analogous problem

in the p-dimensional Euclidean space with rectilinear (p = 2) or

Tchebychev (p 2) distances.



3.2 Problem Statement


We suppose given a finite, undirected tree network, and denote

by T an imbedding of the given network. Let V : {v ,...,v } be a set

of n distinct vertices of T. We assume existing facility i is located

at vertex vi, i E {l,...,n}. For j e {1,...,m}, denote by x. a point

to be determined in T as the location of new facility j. We define Tm

to be the m-fold Cartesian product of T by itself and define a location

vector X in Tm to be the ordered m-tuple (x,,...,x ) with each x e T,

j {1,...,m}. Sometimes, we refer to a location vector X in Tm as a

point in Tm

As in [22], given points x,y e T, we define the line L(x,y) to be

the union of all points in the shortest path connecting x and y. In





-86-


addition, given a finite point set P C T, we define the convex hull

H(P) to be the smallest (embedded) subtree of T containing all points

in P. We note that for any two points p,p' e P, the line L(p,p') is

contained in H(P).

We denote by IC the set of pairs (i,j) for which the distance

d(xiv ) is of concern. Similarly, IB is the set of pairs (j,k) for

which the distance d(x.,xk) is of concern. We remark that it need not

be the case that IC includes all possible pairs of new and existing

facility indices, nor IB includes all possible pairs of new facility

indices. With these definitions, the problem of interest is to "mini-

mize" each of the distances specified by (3.2.1);


d(x.,v.) (i,j) C IC
1 J C
(3.2.1)
d(xj,xk) (j,k) I .


For X e Tm, we denote by D(X) the vector each of whose components

is a distance specified by (3.2.1). The vector is formed by assuming

any convenient ordering of the members of IC and IB. The vector-

minimization (V-min) problem of interest is


V-min{D(X): X e Tm} (3.2.2)


With respect to (3.2.2), a location vector Z e Tm is said to

dominate a location vector X in Tm if D(Z) < D(X) and D(Z) # D(X).

A location vector Z which is not dominated by any other location vector

is said to be efficient. An equivalent definition of efficiency is as

follows: Z e Tm is efficient if and only if X e Tm and D(X) 5 D(Z)

imply D(X) = D(Z).





-87-


Our main interest is to characterize efficient location vectors

and devise an algorithm for constructing efficient location vectors

from a given (dominated) location vector.



3.3 Distance Constraints and Characterization
of Efficient Points


We make extensive use of the results obtained in [32, 33] for

distance constraints to establish the necessary and sufficient condi-

tions for efficient points. The Distance Constraints (DC) are defined

in [32] (independent of the efficiency problem) as follows: Given the

sets IC and IB and nonnegative upper bounds cij and bjk, find a point

X = (x1,...,x ) in Tm, if it exists, such that


d(xi,v.) c.. (i,j) C IC
(3.3.1)
d(xj,xk) b bjk (j,k) e IB


Corresponding to DC, we define Graph BC (GBC) as the undirected

graph having nodes E1,...,En, N1,...,N ; for every (j,k) e I there

is an arc (Nj,Nk) of length bjk between nodes Nj and Nk; for every

(i,j) C IC, there is an arc (N.,E.) of length cij between nodes N.
C 1 J ij i
and E.. We further assume that the sets IB and IC are such that GBC
J B C
is connected, as otherwise DC decomposes into independent sets of con-

straints which may be analyzed separately.

Given a node-path between any two nodes f and f in GBC, we de-
P q
note the path by P(f ,f ) and denote the length of the path by LP(f ,f ).

We define L(f ,f ) to be the length of any shortest path in GBC between
P q
nodes f and f Subsequently, unless we specify otherwise, it should
P q





-88-


be understood that any path we refer to is a simple path between some

two existing facility nodes E and E .
P q


Results on Distance Constraints


The distance constraints are said to be consistent if there exists

at least one feasible solution to (3.3.1).

The following result is established in [32].

Theorem 3.3.1. The distance constraints are consistent if and only if


d(v ,v ) < L(E ,E), 1 p < q n (3.3.2)
pq p q

The inequalities (3.3.2) are termed the Separation Conditions

[32], since each term on the right specifies an upper bound on how

separate two existing facility locations can be. Except when stated

otherwise, we assume throughout the chapter that the separation condi-

tions hold, and thus (equivalently) DC is consistent.

We call a path P(E ,E ) between E and E in GBC a tight path if
p q p q
LP(E ,E ) = d(v ,v ). We note that since we assume DC is consistent,
p q p q
it necessarily follows if P(E ,E ) is a tight path, that LP(E ,E ) =
P p P q
L(E ,E ). Any path P(E ,E ) for which LP(E ,E ) > d(v ,v ) is called
P q p q pq
a slack path.

We say that new facility i is in a tight path if there exists at

least one tight path containing N.. Every path containing N. is slack
1 1
if there is no tight path which contains N..

The motivation for the above terminology is due to a string graph

representation of GBC. This string graph is also useful for obtaining

problem insights. When knots representing nodes E and E are pulled as
P q
P pq





-89-


If then the string graph is placed upon the tree T, i.e., the strings

only lie on arcs of T, a path is tight when it is necessary to pull the

string graph tight in order to place the knots representing E and E

on v and v respectively, while a path is slack if the string path
P q
must literally be slack when the two knots are placed to coincide with

v and v
P q
A priori, one might think that the occurrence of a tight path

would be rare. However, we shall see that tight paths occur in a

quite natural way when the separation conditions are used in the analy-

sis of efficient location vectors. Further, the notion of tight paths

permits the specification of necessary and sufficient conditions for

DC to have a unique solution.

We now relate unique locations to tight paths. By definition,

new facility i is uniquely located if it has the same location in every

feasible solution to DC. Since we later refer to a collection of

facilities, which contains possibly both existing and new facilities,

being uniquely located, we note that existing facilities are uniquely

located by definition.

Theorem 3.3.2, which we proved in [33], specifies the necessary

and sufficient conditions for a new facility to be uniquely located.

Theorem 3.3.2. New facility k is uniquely located if an only if node

Nk lies in at least one tight path P(E ,E ).

Corollary 3.3.2. Distance constraints have a unique solution if and

only if node Nk lies on at least one tight path in GBC for k = l,...,m.

We now give an additional property of a tight path we proved in

[33]. The property will be used in proving our main result on efficient

points.





-90-


Property 3.3.1. If P(E ,E ) is a tight path in GBC, then
p q
(i) every facility represented by a node in P(E ,E ) is uniquely

located,

(ii) the locations of facilities corresponding to nodes in P(E ,E )

occur with the same ordering and spacing on the line L(v ,v ) in
p q
T as do the corresponding nodes in P(E ,E ).

As an illustration of Property 3.3.1, suppose P(E1,E5) is a tight

path with nodes E1, N2, N3, E5. Then, the locations v1, x2, x3, v

are unique. Furthermore, they occur in the given order on the line

L(v1,v5) with d(v1,x2) = c21, d(x2,x3) = b23, d(x3,v5) = c35, where

c21, b23, c35 are the lengths of the arcs in the path. This example

is illustrated in Figure 3.1.




C b C Tight Path
SC21 23 35 P(EGE5)









v xin T
v1 x2 x3 v5






Figure 3.1.. Illustration of Property 3.3.1.



We now consider the problem of determining when an arc lies on a

tight path. As an arc lies on a tight path if and only if it is not

the case that all paths containing the arc are slack, we consider the





-91-


equivalent problem of determining when an arc lies only on slack paths.

The following property, which we proved in [33], characterizes the con-

ditions under which an arc in GBC is not contained in any tight path.

Property 3.3.2. Let DC be consistent. Let (f.,f.) be any arc in GBC,

of positive length e.., whose length is reduced by some positive amount

C. Let DC (GBC ) be the distance constraints (graph) obtained from

DC(GBC) by replacing e.. by eij C.

(a) Evey path containing (f.,f.) in GBC is slack if and only if e can

be chosen (with s > 0) so that DC is consistent.

(b) Whenever every path containing (f.,f.) is slack, E can be chosen

(with e > 0) so that DC is consistent and at least one of the follow-

ing is true:

(i) at least one path in GBC containing (f.,f.) is tight;

(ii) the length of (f.,f.) in GBC can be reduced to zero.
1 C
Finally, we will use the following lemma proven in [33].

Lemma 3.3.1. Given points a,b e T, suppose that d(a,b) = a + 3.

Then, the inequalities d(x,a) a, d(x,b) B are consistent if and

only if they have a unique solution and the inequalities hold as

qualities.



Necessary and Sufficient Conditions for Efficiency


Given a location vector Z, we let U = D(Z) and define the distance

constraints of interest by D(X) < U, where the entries in U define the

bjk and cij by bjk = d(zj,zk) for (j,k) C I'B and cij = d(zivj) for

(i,j) c IC. We use the bjk and cij to define GBC in the customary

manner. As before, we may assume GBC is connected, for otherwise the

problem of finding efficient location vectors decomposes into





-92-


independent subproblems. Further, we note that DC is always consistent,

as Z is certainly feasible to DC, and hence, by Theorem 3.3.1, the

separation conditions are always satisfied. For convenience, for any

location vector Z, we denote by A*(Z) the collection of locations of

uniquely located facilities whose nodes are adjacent to N. in GBC. We
1
denote by 11[A*(Z)] the convex hull of A*(Z), the imbedding of the
i i
smallest subtree of T spanning all the elements of A*(Z).
1
With the above definitions we can present a family of equivalent

conditions for a location vector Z to be efficient.

Theorem 3.3.3. Given a location vector Z used to define DC and GBC,

the following are equivalent:

(a) Z is efficient;

(b) Each N. is in at least one tight path in GBC;
I
(c) Z is the unique solution to DC;

(d) z. E H[A*(Z)] for i = l,...,m.

Proof. The equivalence of (b) and (c) is a direct consequence of

Theorem 3.3.2 and the fact that Z is always a feasible solution to

DC, while (c) clearly implies (a). To show (a) implies (c), suppose

Z is not the unique solution to DC. Color every new facility node

in GBC which is not contained in any tight path blue. Color all the

other (new or existing facility) nodes red. Equivalence of (b) and

(c) implies every blue node represents a new facility which is not

uniquely located, while every red node represents a (new or existing)

facility which is uniquely located. By assumption there is at

least one blue node. By connectedness of GBC, there is at least

one arc which connects some blue colored node, say, N to some red

colored node, say, F Furthermore, arc (N ,F ) has positive
q p q




Full Text
-90-
Property 3.3.1. If P(E^,E^) is a tight path in GBC, then
(i) every facility represented by a node in P(E^,E^) is uniquely
located,
(ii) the locations of facilities corresponding to nodes in P(E^,E^)
occur with the same ordering and spacing on the line L(v ,v ) in
P 9
T as do the corresponding nodes in P(E ,E ).
P q
As an illustration of Property 3.3.1, suppose P(E^,E^) is a tight
path with nodes E^, N2, N^, E^. Then, the locations v^, x^, v^
are unique. Furthermore, they occur in the given order on the line
l(v1,v5) with d(v1,x2) = c21, d(x2,x3) = b23, d(x3,v5) = c35, where
C21 ^23 C35 are t^ie -*-enSt^ls tbe arcs in the path. This example
is illustrated in Figure 3.1.
b
23
35
Tight Path
p(e15e5)
in GBC
'21
23
C
35
x
3
Uvpvs)
in T
Figure 3.1. Illustration of Property 3.3.1.
We now consider the problem of determining when an arc lies on a
tight path. As an arc lies on a tight path if and only if it is not
the case that all paths containing the arc are slack, we consider the


-168-
102. P.J. Slater, "Central Paths in a Graph," Research Report SAND
78-0809J, Sandia Laboratories, Albuquerque, New Mexico (1978).
103. P.J. Slater, "One-Point Location of an Area Response Group,"
Research Report SAND 78-1788, Sandia Laboratories, Albuquerque,
New Mexico (1978).
104. M.K. Starr and M. Zeleny, "MCDM-State and Future of the Arts,"
TIMS Studies in Management Sciences 6, 5-29 (1977).
105. R.W. Swain, "A Parametric Decomposition Approach for the Solu
tion of Uncapacitated Location Problems," Dept, of Industrial
Engineering, The Ohio State University (1971).
106. J.J. Sylvester, "A Question in the Geometry of Situation,"
Quarterly Journal of Pure and Applied Mathematics, Vol. 1, 79
(1857).
107. M.R. Teitz and P. Bart, "Heuristic Methods for Estimating the
Generalized Vertex Median of a Weighted Graph," Opns. Res.,16,
955-961 (1968).
108. H. Thiriez and S. Zionts, Eds., Multiple Criteria Decision
Making, Proc., Jouy-en-Josas, France, 1975.
109. C. Toregas, R. Swain, C. ReVelle, and L. Bergman, "The Location
of Emergency Service Facilities," Opns. Res. 19, 1363-1373
(1971).
110. R.E. Wendell, "Efficiency and Solution Approaches to Bi-Objective
Mathematical Programs," Working Paper, Rensselaer Polytechnic
Institute, Troy, New York.
111. R.E. Wendell and A.P. Hurter, "Optimal Locations on a Network,"
Trans. Sci. 7, 18-23 (1973).
112. R.E. Wendell and D.N. Lee, "Efficiency in Multiple Objective
Optimization Problems," Math. Prog. 12, 406-414 (1977).
113. R.E. Wendell and E.L. Peterson, "Duality in Generalized Location
Problems," ORSA Bull. 20, Supp. 2, B-317 (1972).
114. R.E. Wendel, A.P. Hurter, and T.J. Lowe, "Efficient Points in
Location Problems," AIIE Transactions 9, 238-246 (1977).
115. G.O. Wesolowsky and R.F. Love, "The Optimal Location of New
Facilities Using Rectangular Distances," Opns. Res. 19, 124-
130 (1971).
116. G.O. Wesolowsky and R.F. Love, "A Nonlinear Approximation Method
for Solving a Generalized Rectangular Distance Weber Problem,"
Manag. Sci. 18, 656-663 (1972).


-15-
This completes the discussion of the 1-center problem. We now
concentrate on the p-center problem for p > 2.
p-Center problem on a general network. The p-center problem was
defined by Hakimi [48]. Subsequently, a number of solution procedures
have been suggested. A common characteristic of all these procedures
is that they all rely on solving a sequence of covering problems.
For completeness, we first define a set covering problem and an
r-cover problem.
Let A be a matrix of zeros and ones, y a vector of zero-one
variables y^. The problem of minimizing J y_^ so that each row of Ay
i
is greater than or equal to one is called the (minimal) set covering
problem. Given the function f(X) = max{w Jl(v^,X): 1 < i < n}, the
problem of minimizing |x| so that f(X) < r for some given value of r
is called the r-cover problem.
Denoting by q(r) the minimum value of the r-cover problem, it
can be readily shown that, if q(r) = p for some r, and q(r') > p for
any r' < r, then r is the p-radius and any X which solves the r-cover
problem is an absolute p-center.
In what follows, we concentrate on the absolute p-center problem
on a general network.
Minieka [87] considered the unweighted case on a general network
and showed that the problem can be reduced to a computationally finite
one. Minieka identifies a finite point set P* such that there exists
an absolute p-center contained in P = P' U V. A point x on some edge
is a member of P' if and only if x is the unique point on its edge
such that d(v.,x) = d(x,v.) for some two distinct vertices v, and v..
i J i J
Based on this result, Minieka suggested a rudimentary algorithm that


-60-
Whether K refers to a subset of V or a clique of G, we prefer to
call K a clique as long as it is clear from the context what K
refers to.
As an example of the nonlinear p-center problem, suppose that the
function associated with node v is f^ (y) = (y + h^)8 for y e [0j ]
where w^ h^, and 0 are given parameters. Appropriate restrictions
are placed on the parameters to ensure that the f^ are strictly in
creasing on [0,6j]. We note that the linear weighted p-center problem
is a special case of this problem generated by choosing 0 = 1, h^ =0,
and Wj > 0 for all j.
For the given form of f the following are readily verified:
f ^(r) = (r/w.)1^6 h., r e [f.(0), f.(6.)] ,
J 1 3 J 3 3
fT1(r) + fT1(r) = r1/8[(l/w )1/6 + (1/w )1/8] (h + h ) ,
1 J 1 J 1 J
r e [a.., b..] ,
13 13
w .w.
i 3
(fi + fj ) (y) r ^ ]./ j_ l/e^
0
(y + h + h ) ,
[w.x'" + w.x/v]" 3
y e [L.., U..]
Then, using the characterization of 3 as given in Lemma 2.2.1,
we have
3.. =
13
.. d.. if L £d(v. ,v.)
13 13 3 i 3
(2.2.4)
max[fi(0), f.(0)] if L^. > div^v.)
where


-166-
73. P. Kolesar, W. Walker, and J. Hausner, "Determining the Relation
Between Fire Engine Travel Times and Travel Distances in New York
City," Opns. Res. 23, 614-627 (1975).
74. T.C. Koopmans, Activity Analysis of Production and Allocation,
Cowles Commission for Research in Economics, Monograph No. 13,
John Wiley and Sons, New York, 1951.
75. H.W. Kuhn, "On a Pair of Dual Nonlinear Problems," Nonlinear
Programming, Chapter 3, J. Abadie, editor, John Wiley and Sons,
New York, 1967.
76. H.W. Kuhn and A.W. Tucker, "Nonlinear Programming," Proc. of the
2nd Berkeley Symposium on Math., Stat. and Probability, Univ. of
California Press, Berkeley, California (1951).
77. A.H. Land and A.G. Doig, "An Automatic Method for Solving Dis
crete Programming Problems," Econometrica 28, 497-520 (1960).
78. A.C. Lea, "Location-Allocation Systems: An Annotated Bibliography,"
Discussion Paper No. 13, Dept, of Geography, University of Toronto,
Toronto, Canada (1973).
79. A.C. Lea, "A Model Taxonomy and a View of Research Frontiers in
Normative Locational Modelling," Paper presented to International
Symposium on Locational Decisions, Banff, Alberta (1978).
80. J. Levy, "An Extended Theorem for Location on a Network," Opnl.
Res. Q. 18, 433-442 (1967).
81.C.C. Lin, "On Vertex Addends in Minimax Location Problems,"
Trans. Sci. 9, 165-168 (1975).
82.T.J. Lowe, "Efficient Solutions in Multiobjective Tree Network
Location Problems," Trans. Sci. 12, 298-316 (1979).
83. L.F. McGinnis and J.A. White, "A Single Facility Rectilinear
Location Problem with Multiple Criteria," Trans. Sci. 12,
217-231 (1978).
84. F.E. Maranzana, "On the Location of Supply Points to Minimize
Transport Costs," Opnl. Res. Q. 15, 261-270 (1964).
85. D.W. Matula and R. Kolde, "Efficient Multi-Median Location in
Acyclic Networks," ORSA/TIMS Bulletin, No. 2 (1976).
86. A. Meir and J.W. Moon, "Relations Between Packing and Covering
Numbers of a Tree," Pac. J. Math. 61, 225-233 (1975).
87.E. Minieka, "The m-Center Problem," SIAM Review 12, 138-139
(1970).


-106-
We can readily use induction to obtain the following generaliza
tion of Lemma 3.5.3.
Lemma 3.5.4. Given r points p ,... ,p £ T with r > 4, if
p e L(P.ji_iP1+i) for 2 ^ i < r-1, and if p 4 p+1 for 2 < i < r-2,
r-1
then d(p1,pr) = £
i=l
We are now ready to prove the sufficient conditions for irredu
cible location vectors. We remark that the arc lengths of GBC are
defined by the entries of D(Z), so that if N,N, *,E is a sub-
Vi) OO p
path P(N^,Ep) connecting to Ep, then the length of the subpath
is given by LP(N^,Ep) = d(z^^.z^) + ... + d(z^,v ).
Lemma 3.5.5 (Sufficiency). Suppose Z is irreducible. If, for every
j e {1,... ,m}, Zj e H[A^.(Z)], then every z^ is uniquely located.
Furthermore Z is efficient.
Proof. For notational brevity, let S = A (Z).
J 3
Choose any j in
{l,...,m}. Either N^. is adjacent to exactly one node or more than one
node. In the former case, S^ is a singleton, say, {y}. Since
Zj e H[Sj], Zj = y. By irreducibility of Z, y is an existing facility
location. Hence z_. is in the convex hull of uniquely located facili
ties so that Theorem 3.3.3 implies z^ is uniquely located in this
case.
For the other case, N. is adjacent to at least two nodes in GBC.
J
The hypothesis z_. e H[S^] implies there exist p,q e S_. with
£ L(pq)
(3.5.1)
If p and q are both existing facility locations, Theorem 3.3.3 implies
Zj is uniquely located. Hence, suppose, without loss of generality,
that q is an existing facility location, but p is a new facility


-8-
Slater [102]. A large portion of the literature deals with point-
location problems. Point-location problems may be classified into
three categories: single objective problems, multi-objective problems,
and a body of results of a general and unifying nature.
In the remainder of this section we give a detailed discussion
of the problems outlined in the family tree.
Point-Location Problems
Here, we consider a number of problems that involve locating new
facilities at points on a network. The general format of the dis
cussion is as follows: For each problem type, we first define a
kernel problem. Then, we discuss the related literature on the kernel
problem, as well as several special cases and extensions of it. We
point out relations between different problem types, whenever such
relations exist.
The p-center problem
Let N be a network with a vertex set V = {v,,...,v } and an edge
1 n
set E. Denote by X a finite set of points, each of which is in N.
Let I be the set of integers 1 through n. For each vertex v., iel,
define the distance D(v^,X) between vertex v and the point set X by
D(v^,X) = min[d(v^,x) : xeX]. With this definition, D(v_^,X) is speci
fied by a nearest point in X to vJ. Let w. and a. be two given numbers
i 11
associated with vertex v^, iel. We call wi a weight and aan addend.
We assume that each w^ is nonnegative and at least one w_^ is positive.
For any finite point set X CD, define the function f (X) by


-142-
To show e(z ) > tCz.), let a.,a. be arcs in A for which
1 i i J G
t(z,) = x..(z,). Suppose a. = (N ,E ) and a. = (N ,E ). By step 1)
1 lj 1 l s p 3 t q J
of E-FRONT we have
VZP
d(vv)
p q
(l/w + l/w )
sp tq
m
st
m
(4.4.7)
st
But m is the length of a shortest path in Gg connecting Ng to N .
Let (N ,N, ...,N,.,N ) be such a shortest path in G. Let P(E ,E ) be
s k f t r B p q
the path (E ,N ,N, ,...,N£,N,E ). Define z = (z,,z0) with z = e(z.).
p s k f t q 12 2 1
Since z is feasible to P DC is consistent so that for the path
2 z^ z v
P(E ,E ) identified above we have
P q
LP(E >E ) > d(v ,v ) .
z p q p q
(4.4.8)
But LzP(E^,E^) = z^(l/wSp) + z2mst + Zl^^Wtq^ aS Pat^ consists of
the arcs (E ,N ) (N ,N. ),...,(N,,N),(N ,E ). It follows then from
ps sk. it tq
(4.4.8) that z.(l/w + 1A* ) + z0m ^ > d(v ,v ), or, equivalently,
1 sp tq 2 st p q
Z2
d(v ,v )
JB SL
(1/w + l/w )
sp tq
m
st
m
(4.4.9)
st
But the right side of (4.4.9) is = x(z^) while z^ is e(z^) by
definition, hence, e(z^) > x(z^).
The inequalities e(z^) < t(z^) and e(z^) > t(z^) imply e(z^) = i(z^)
for every e [a,b]. Hence, Z* = {(z^tCz^)): a < z^ < b), com
pleting the proof.
2 2
We now show that the computational order of E-FRONT is 0(m (m + n ))
The algorithm constructs Z* by identifying no more than r(r l)/2
linear functions. To identify the linear functions one must first


-167-
88. E. Mlnieka, "The Centers and Medians of a Graph," Opns. Res. 25,
641-650 (1977).
89. P.B. Mirchandani and A.R. Odoni, "Locating New Passenger Facili
ties on a Transportation Network," Working Paper, Electrical and
Systems Engineering Dept., Rensselaer Polytechnic Inst., Troy,
New York (1977).
90. P.B. Mirchandani and A.R. Odoni, "Locations of Medians on Sto
chastic Networks," Working Paper OR-065-77, Operations Research
Center, M.I.T., Cambridge, Massachusetts (1977).
91. S.C. Narula, V.I. Ogbu, and H.M. Samuelson, "An Algorithm for the
p-Median Problem," Opns. Res. 25, 709-712 (1977).
92. G.L. Nemhauser and W.L. Sheu, "Easy and Hard Bottleneck Location
Problems," Technical Report 386, School of Operations Research
and Industrial Engineering, Cornell University, Ithaca, New
York (1979).
93. J.C. Picard and H.D. Ratliff, "A Cut Approach to the Rectilinear
Distance Location Problem," Opns. Res. 26, 422-434 (1978).
94. A.B. Pritsker and P.M. Ghare, "Locating New Facilities with
Respect to Existing Facilities," AIIE Transactions 12, 290-297
(1970).
95. C. ReVelle and R. Swain, "Central Facilities Location," Geog.
Anal. 2, 30-42 (1970).
96. C. ReVelle, D. Marks, and J.C. Liebman, "An Analysis of Private
and Public Sector Location Models," Manag. Sci. 16, 692-707
(1970).
97. D.J. Rose, R.E. Tarjan, and G.S. Lueker, "Algorithmic Aspects of
Vertex Elimination in Graphs," SIAM J. Comput. 5, 266-283 (1976).
98. A. Rosenthal, J. Pino, and M. Coulter, "A Generalized Algorithm
for Centrality Problems on Trees," Working Paper, Dept, of
Computer and Communication Sciences, Univ. of Michigan, Ann
Arbor, Michigan (1978).
99. B. Roy,"Problems and Methods with Multiple Objective Functions,"
Math. Prog. 1, 239-266 (1971).
100.D.R. Shier, "A Min-Max Theorem for p-Center Problems on a Tree,"
Trans. Sci. 11, 243-252 (1977).
101.S. Singer, "Multi-Centers and Multi-Medians of a Graph with
Application to Optimal Warehouse Location," Unpublished Paper,
Dunlap and Associates, Inc., Darien, Conn. (1968).


-45-
The problem of interest is to find all efficient points with
respect to f(x).
Halpem [52] is the first to consider this problem. Halpern
formulated the problem in a slightly different manner by considering
a convex combination of m(x) and c(x). For any fixed X, 0 < X 1,
define f(X,x) and f*(X) by
f(X,x) = Xm(x) + (1 X) c(x) for xeN }
f*(X) = min[f(X,x): xeN] (1.3.4)
In Halpem's terminology, the function f(X,x) is called a cent-dian
function and any point x* = x*(X) that solves (1.3.4) is called a
cent-dian point.
In [52] Halpem considered this problem on a tree network with
weights h all equal to unity. Defining x^ and x^ to be the (vertex)
median and the absolute center of T, respectively, Halpem proved that
for any given X, the cent-dian x*(X) is located at either x^ or on
one of the vertices located on the path P(x ,x ). This theorem pro-
m c
vides the basis for a simple and efficient algorithm to locate the
cent-dian by inspecting the vertices on P(x ,x ). Further, Halpern
m c
showed that, if the absolute center x is known, then the cent-dian
c
can be found by determining the median of a tree T' that is identical
to T except that T' contains an additional vertex v = x with the
n+1 c
associated weight w = X 1.
n+1
Handler [56] formulated the same problem on a tree network in a
slightly different manner by using the median function as a constraint.
In Handler's formulation one is interested in solving the problem


-23-
define an intersection graph for a fixed value of r as follows:
has nodes corresponding to demand points v^,...,v^. Two nodes of G^
are connected by an arc if the corresponding demand points can be
jointly covered by a (single) common center within a radius of r.
Once Gr is formed, finding a "clique cover" of G^ solves the r-cover
problem. A clique cover of G^ is a minimum number of cliques in G^
such that every node is in at least one clique. The solution to the
clique cover problem in G^_ determines a solution to the r-cover problem.
The procedure is repeated for different values of r until a smallest
value of r is found for which the clique cover solution generates at
most p cliques. The computational complexity of the procedure is
polynomial. In particular, the computational effort for finding the
minimal clique cover of G^ is polynomial because G^ satisfies the
property that any circuit in G^_ with at least four arcs contains a
chord (i.e., an arc which connects two nodes of the circuit and is
not an element of the circuit). For chordal graphs, algorithms of
linear order have been developed (see [39], [97]) for finding a
minimal clique cover.
This completes the discussion of the p-center problem.
The p-median problem
The difference between the p-center and the p-median problem is
that the objective criterion is changed from minimax to minisum. More
specifically, define the function f (X) for any finite point set XCN
by
f (X) = l w D (v ,X) .
iel 1


-119-
of the separation conditions for the distance constraints to be con
sistent. Furthermore, this equivalence is the only property that is
used. Hence, the theorem holds for any distance for which it is true
that the distance constraints are consistent if and only if the separa
tion conditions hold.
In Theorem 3.7.1 one does not have to worry about arcs with zero
lengths. To see this, partition D(Z) into subvectors D^(Z) and (Z)
so that D (Z) contains the zero entries, while (Z) contains the
positive entries. Clearly, in every feasible solution X to the con
straints D(X) < D(Z), the constraints corresponding to entries of
Dq(Z) will hold at equality. Hence the only way Z can be dominated is
by having (X) < (Z) and (X) f D^(Z). Thus, one needs to test only
those arcs with positive lengths in GBC to determine whether or not Z
is efficient. For this reason, Theorem 3.7.1 is applicable to both
irreducible location vectors and reducible location vectors.
2
We remark that for rectilinear distances in R and Tchebychev
1c
distances in R k > 2, the equivalences stated in Theorem 3.3.3
need not hold. An example of such a case is given in Figure 3.8 for
2
Tchebychev distances in R With reference to Figure 3.8, it is direct
to verify that both and are contained in the tight path
0^2 N^, E^) Clearly (z^,z^) is not efficient, as, z^ can be
moved to (2,2) and z^ can be moved to (3,1) thereby reducing the dis
tance between new facility 1 and existing facility 1 without increasing
any of the other distances. The resulting location vector is shown
in Figure 3.9. It Is direct to verify that every positive arc in GBC
of Figure 3.9 is contained in a tight path and hence the location
vector is efficient.


-136
We shall first state a theorem due to Wendell [110] which gives
a global characterization of the efficient frontier. Then we will
exploit the result of the theorem to construct Z*.
m /s
Let a be the minimum value of f^ on T b be the minimum value of
f^ on Tm, and b be the minimum value of f^ over all minima to f^. The
A
values a, b, and b are displayed in Figure 4.3 for an arbitrary bi
objective problem with convex objectives. For each z^ e [a,b] define
the function e^) to be the minimum value of the problem P defined below
Z1
e(Zl) = minf2(X): f (X) < z X e Tm) .
Wendell [110] showed that whenever f^,f2 are lower semicontinuous
convex functions defined over a nonempty convex compact set S, the
efficient frontier is the set {(z^,e(z^)): a < z^ < b}. Wendells
theorem is applicable to the bi-objective m-center problem, as Tm is
convex, compact, and nonempty, and f^ and f2 are continuous convex
functions (see [22]) over Tm. For an arbitrary choice of z^, the
value of e(z^) is marked in Figure 4.3.
The computation of a, b, and b presents no difficulties and will
be given subsequently.
Using the definitions of f^ and an equivalent definition of
e(Zj) is as follows:
e(zp = min Z2
s. t.
d(x,Vj) < z^w (i,j) e Ic
d(Xj,xk) z2/vjk e IB
where z^ is understood to be fixed to any value in [a,b]
(4.4.1)
The constraints


-117-
1c
points x,y e R with x = (x^,...,x^) and y = (y^,...,y^), the rectilinear
distance between x and y is given by |x^ y^| + ... + |x^ + y^J, while
the Tchebychev distance between x and y is given by max{|x^ y^|,...,
|x y |}, where the symbol |*| denotes the absolute value sign.
K. K.
It is known that (proven in [32]) the distance constraints with
2 k
rectilinear distances in R or with Tchebychev distances in R k 2,
are consistent if and only if the separation conditions hold.
Based on this result, we characterize efficient location vectors
for the analogous vector-minimization problem which uses the recti
linear or Tchebychev distances.
Theorem 3.7.1. Let D(Z) be the vector of objectives with all entries
2
of D(Z) either the rectilinear distances in R or the Tchebychev
distances in R k > 2, as specified in (3.2.1). Let GBC be the graph
with arc lengths defined by D(Z). The following are equivalent:
(a) Z is efficient;
(b) Every arc in GBC of positive length is in a tight path.
Proof. To show (a) implies (b) suppose that Z is efficient. Let DC
be the distance constraints D(X) < U = D (Z). Since Z is a feasible
solution to DC, DC is consistent and hence the separation conditions
hold. Assume that there exists at least one arc in GBC with positive
length which is not in any tight path. Let (f ,f ) be such an arc with
P q
length e Since (f ,f ) is not in any tight path and since the
pq p q
separation conditions hold, every path which contains (f ,f ) is slack.
P q
Hence, for any path P(E.,E.) containing (f ,f ) we have
1 J p q
LP(E ,E ) d (v ,v ) > 0 (3.7.1)
J J
Define e' to be the minimum of the left side of (3.7.1) over all paths


I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
Richard L. Francis, Chairman
Professor of Industrial and Systems
Engineering
I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
X/
Donald U. Hearn
Associate Professor of Industrial and
Systems Engineering
I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
Ra llph W'. Swa in
Associate Professor of Industrial and
Systems Engineering


-li
the network is a demand point (as opposed only to vertices). The
weight of each point is unity. The objective to be minimized over all
xeN is defined by f(x) = max [d(y,x): yeN]. Both authors showed that
the problem can be reduced to a computationally finite one and pro
posed a solution procedure which is very similar to Hakimi's.
A probabilistic version of the 1-center problem was considered
by Frank [34, 35] and a number of bounds were obtained on the expected
value of the 1-radius.
For the unweighted case, Singer [101] proved that there exists a
"critical" path, not necessarily a shortest path, connecting two cri
tical vertices such that an absolute center of the network is at the
midpoint of this path.
1-Center problem on a tree network. We now concentrate on ab
solute centers of tree networks. Goldman [44] solved the unweighted
case in the presence of addends. Goldman's algorithm is based on the
repeated application of a "trichotomy theorem" that either determines
the edge on which the absolute center lies, or reduces the search to
one of the subtrees obtained by removing all interior points of that
edge. Halfin [51] refined Goldman's algorithm to make it simpler and
computationally more efficient. Halfin's algorithm finds a vertex
center first, and determines the absolute center by examining all
vertices adjacent to the vertex center.
For the unweighted case with no addends, Handler [55] presents
an especially elegant algorithm. Handler's method finds a longest
path of the tree and locates the absolute center at the midpoint of
the path. To find a longest path, Handler chooses an arbitrary vertex
v finds a farthest vertex v from v., and then finds a farthest
i si


-131-
corresponding to E and E will be violated, as, A _. (E ,E ) <
P Q z A p q
L ,P(E,E) Z A p (J Z p CJ p CJ
DC is inconsistent. Lemma 4.3.1 then implies Y is efficient.
z-A
We remark that Theorem 4.3.1 considers only those tight paths
which pass through Ag. The reason is as follows: Any path in GBC^
passes through A so that if there exists a tight path which contains
an arc in Ag, then the same path necessarily contains an arc in A^,.
However, an arbitrarily chosen path need not pass through Ag. For this
reason, the assumption that there exists at least one arc in A^, which
is contained in a tight path does not imply that at least one arc in
Ag is contained in a tight path. Hence, if a location vector Y is
efficient then there is at least one arc in A^ which is contained in
a tight path while the reverse implication does not hold.
Further, we remark that the proof of Lemma 4.3.2 is based on the
necessity and sufficiency of the separation conditions. Hence, Lemma
4.3.2 is applicable to tree networks as well as the analogous problems
with rectilinear distances on the plane, or, the Tchebychev distances
in the k-dimensional Euclidean space with k > 2. Further, Theorem
4.3.1 uses Lemmas 4.3.1 and 4.3.2 for its proof. Hence, the theorem
is applicable to tree networks as well as rectilinear distances on the
lc
plane and the Tchebychev distances in R k > 2.
At this point we give an example of a non-efficient and an
efficient location vector. In Figure 4.1 the tree network is shown
along with the distance matrix and the weights w.. and vfor
13 Jk
(i,j) e Ic = {(1,1),(1,2),(2,3),(2,4),(3,4),(3,5)} and (j,k) e Ifi -
{(1,2),(1,3),(2,3)} for the example bi-objective m-center problem.
In Figure 4.2a) we give an example of a dominated location vector X.


-141-
facility node Ng. Let X be any location vector for which f^(X) = a.
Hence, w d(x ,v ) < a and w d(x ,v ) < a. But a < z. and the length
sp s p sq s q 1
of P(E ,E ) is z.(1/w + 1/w ) so that we have, upon using the tri-
p q 1 sp sq
angle inequality, L P(E ,E ) = z (1/w + 1/w ) > a(l/w + 1/w ) >
& n 3 z p q 1 sp sq sp sq
d(x ,v ) + d(x ,v ) > d(v ,v ). Thus, for any nd-path P(E ,E ) which
s p s q p q 3 1 p q
does not pass through A^, we have
L P (E ,E ) > d(v v )
z p q p q
(4.4.3)
For the other case, P(E ,E ) passes through A,, so that its length is
P q B
given by L P(E ,E ) = z,WP(E ,E ) + zVP(E ,E ). Since the path is an
zpq 1 p q 2 p q
nd-path it passes through exactly two arcs in A^, say, (E^,Ns) and
(N ,E ) with s t. Thus, WP(E ,E ) = 1/w + 1/w while
t q p q sp tq
VP(E ,E ) ra by the definitions of m and VP(E ,E ). It follows
p q st st p q
that
L P(E ,E ) > z.(l/w + 1/w ) + z m
zpq 1 sp tq 2 st
Due to steps 1) and 2) of E-FRONT, we have
(4.4.4)
z2 = T(zl) -
L_£_ z
m ^ Z1
st
(1/w + 1/w )
Sp tq
ra
st
(4.4.5)
Using (4.4.4) and (4.4.5), for any nd-path which passes through A^ we
have
IP(E E ) > d(v ,v ) (4.4.6)
z p q p q
From (4.4.3) and (4.4.6), L P(E ,E ) > d(v ,v ) for every nd-path in
zpq-pq
GBC¡z so that Lemma 4.4.1 implies the separation conditions on GBCz
hold. Hence, z is feasible to P and thus e(z.) x(z.) = z.
z. 112


-40-
The problem differs from the p-median problem in two respects:
(i) the distance between any vertex and any new facility may be of
concern as opposed only to the distance between a vertex and the near
est new facility to it; (ii) certain distances between new facilities
are of concern as opposed to the absence of interactions between new
facilities in the p-median problem. For the case of a single new
facility, the two problems are identical.
Planar cases of the problem using rectilinear or Euclidean dis
tances have received considerable attention and efficient solution
procedures have been developed. A thorough discussion of these prob
lems is given in the book by Francis and White [31]. Other references
on planar problems are Cabot, Francis, and Stary [6], Bindschedler and
Moore [3], Francis [27], Eyster, White, and Wierville [26], Pritsker
and Ghare [94], Wesolowsky and Love [115, 116], and Picard and Ratliff
[93].
The problem on a network is defined by Dearing, Francis, and Lowe
[22] in the presence of distance constraints. It was established in
[22] that the problem is a convex optimization problem for all data
choices if and only if the network is a tree. For the case of a general
network, it is known that there exists an optimal solution on the
vertices of N. This result and certain generalizations of it have
been given by Goldman [41 ], Levy [80], Hakimiand Maheshwari [49], and
Wendell and Hurter [ill]. These references are already discussed
under the p-median problem. The problem was shown to be NP-hard by
Kolen [72 ] on a general network, and no solution procedures have
been developed yet.


-151-
Let be m-collections of centers with |x_^| = p_^ for
1 < i m, where each p^ is a given positive integer. Let V^,...,Vn
be n collections of existing facility locations. The elements of X^
are x^,...^1 with each x^ e T. The elements of V. are
1 p 2 i 1 n
i X X
with each v. a vertex of T. Let X = {X,,...,X }. For any two finite
J 1 m 7
subsets P and Q of T let D(P,Q) = min[d(p,q): p e P, q e Q]. Define
the function f by
f(X) = max{max{w D(X ,V ): (i,j) e I~} ,
1J 1 J V-
max{Vj^D(Xj >X^): (j,k) e Ig}} .
The Generalized Multi-Center Problem (GMCP) is as follows:
min[f(X): |x ¡ = p^, X_^ C T for 1 < i < m]
An equivalent statement of GMCP in terms of distance constraints is
as follows:
mm
s. t.
D(X.,V.) z/w..
i J 1J
j'V s z/vjk
X. p.
1 X 1 1
(ij) e Ic
(j,k) e IB
1 i ^ m
For the case with m = 1 and each = {v..}, GMCP specializes to
the p-center problem. For the case with each p^ = 1 and each = {v^},
GMCP specializes to the m-center problem with mutual communication.
We pose the following questions for future research.
Ql. What special cases of GMCP are tractable? Some of the special
cases are obtained by taking each weight unity, or,


-120-
4
3
2
1
0
v,
v.
Z2
a) Facility Locations
b) Graph GBC
L(E1SE2) = 4
L(El5E3) = 5
L(ErE4) = 6
L(E2,E3) = 3
L(E2>E4) = 4
L(E3,E4) = 3
> 2 = d(v1}v2)
> 4 = d(v1#v3)
> 5 = d(v1,v4)
3 = d(v2>V3)
> 3 = d(v2,v4>
= 3 = d(v3,v4)
c) Separation Conditions
Figure 3.8. Example of a Dominated Vector with Tchebychev Distances
in


-116-
X £ Tm whose irreducible representation is X* is efficient and satisfies
D (X) < D(Z) (3.6. 1)
where Z is the given vector in Tm to which SEVCA is applied.
Proof. Due to the Reduction routine it is evident that X*, the loca
tion vector at the termination of SEVCA, is irreducible. Let K* be
the list of composite indices at the termination. The Termination
Test implies every member of K* is scanned. But a composite index
can be labeled scanned only in either step 6) or step 7). In either
case, we have x* £ H[Ap(X*)] for every P e K*. Property 3.5.2 then
implies X* is efficient since every component of X* is in the convex
hull associated with it, and X* is irreducible.
To show (3.6.1), let x^ = x* for every i e P with P e K*. Thus,
X = (x^,...,xm) is the vector in Tm whose irreducible representation
is X*. Clearly, the entries of D(X) are either zeros or the arc
lengths of the reduced graph, say, GBC* at termination. Since the
arc lengths of GBC either remain the same or decrease by a positive
amount (in step 7)) from one iteration to the next, it follows that
D(X) < D(Z). The assertion that X is efficient follows immediately
from Property 3.5.3 and the fact that x* e H[Ap(X*)] for every
P z K*.
3.7 Efficiency for the Case of Rectilinear or
Tchebychev Distances
In this section we consider the analogous vector-minimization
k
problem in the k-dimensional Euclidean space, denoted by R for the
case of rectilinear (k = 2) or Tchebychev (k > 2) distances. Given


-20-
and n columns, so that an entry a., is 1 if vertex v. is within a
ij J
distance r of v and 0 otherwise. If one solves a set covering
i
problem using the matrix A, the variables whose values are 1 in an
optimal solution determine a feasible solution to the vertex restricted
r-cover problem. The set covering problem is solved by relaxing the
integrality constraints. In the case of non-integer termination, a
single cut produced an integer solution in a large proportion of the
cases. Their computational experience indicates that non-integer
termination seldom occurs.
p-Center problem on tree networks and duality. In what follows,
we concentrate on the p-center problem on tree networks. First, we
define the "continuous" p-center problem. In the continuous p-center
problem, each point in T is a demand point as opposed only to vertices.
Weights are absent (or unity). For any XC T, f is defined by
f(X) = max{D(y,X): yeT} and the continuous p-center problem is to
find an X*d T such that
rp = f(X*) = min[f(X): |x| = p, X C T] .
Minieka [88] considered the continuous p-center problem on a
general network and showed that it can be reduced to a computationally
finite one.
Shier [100'] considered the continuous p-center problem on a tree
network and defined a dual "dispersion" problem. The dispersion
problem is to find p+1 points on T the nearest two of which are as
far apart as possible. More explicitly, let U be any finite point
set with |U| = p+1 and define h(U) by
h(U) = mindu^jUj): 1 < i < j < p+l} .


-146-
Figure 4.4. Data for Example


-123-
[32]. At the end of the section we provide an example problem.
Section 4 is devoted to the development of a procedure, E-FRONT,
for constructing the "efficient frontier." We prove the correctness
2 2
of the procedure and show that it is of 0(m (m + n )), where m and n
are, respectively, the number of new and existing facilities. The
section ends with an example application of the procedure.
4.2 Problem Statement, Notation, and Definitions
Let T be an imbedding of a finite undirected tree network with
existing facilities located at distinct vertex locations v^,...,vn. It
is of interest to locate m new facilities at points x^,...,x in T
under two objective criteria to be defined below. We suppose given
positive weights w and v k and denote by 1^ and Ig the nonempty sets of pairs
(i,j) and (j,k), respectively, for which the weighted distances
w..d(x.,v.) and v...d(x.,x, ) are of concern. We remark that it need
13 i 3 jk 3 k
not be the case that 1^, includes all possible pairs of new and existing
facility indices nor Ig includes all possible pairs of new facility
indices. For each location vector X = (x^,...,xm) in Tm we define two
objective functions f^ and f^ by
f ^X) = max(w_d(xi,Vj): (i,j) e Ic> ,
f2(X) = max{v^kd(x^. ,xk): (j,k) e Ig} ,
and we let f(X) = (f (X), f2(X)).
(4.2.1)
The Bi-objective m-Center Problem (with Mutual Communication)
is as follows:
V min{f(X): X e Tm} .
(4.2.2)


-114-
An example application of SEVCA is given in Figure 3.7. For every
iteration, GBC and the current location vector is given. For iterations
6 and 7, the location vectors at the end of these iterations are shown
separately. For the other iterations, the location vector does not
change. Iterations 1 and 5 perform the Reduction routine. For the
other iterations, the node chosen in that iteration is the one inci
dent to every thickly-drawn arc. The associated convex hull is shown
by cross-hatched lines in the tree network. For any iteration the
circular-shaped new facility nodes of GBC are the unscanned nodes,
while the rectangular-shaped new facility nodes are the ones which
have been scanned prior to the given iteration or during that itera
tion. For any iteration, the numbers on the arcs of GBC show the arc
lengths at the beginning of that iteration. If the arc lengths change
during that iteration, the new arc lengths are indicated by the numbers
in parentheses.
By one iteration of SEVCA, we shall mean the execution of step 1)
through the last step. The last step of any given iteration is either
step 3), step 4), step 6), or step 7). Define, for i = 3, 4, 6, 7, t^
to be the total number of iterations which used step i) as the last
step. Clearly, t^ = 1. Since any given iteration uses only one of
these steps as the last step, the total number of iterations, denoted
by t, will be given by t = t^ + tg + t^ + 1. We want to show that
(k)
t 3m. For convenience, denote by K the list of composite indices
at the first of iteration k.
Property 3.6.1. The algorithm SEVCA terminates in at most 3m-l
iterations.


-139-
Lemma 4.4.1. Let DC be the distance constraints specified in (3.3.1)
and let GBC be the associated graph. The separation conditions on GBC
hold if and only if for every nd-path P(E ,E ), LP(E ,E ) > d(v ,v ).
pq p q p q
Proof. Suppose the separation conditions hold. Choose any nd-path
P(E ,E ). We have LP(E ,E ) > L(E ,E ) > d(v ,v ) as the length of
p q p q p q p q
P(Ep,E^) is at least as large as the shortest path length between E^
and E .
q
Suppose for any nd-path P(E^,E^) we have LP(E^,E^) > d(Vp,v^).
Choose any two existing facility nodes, say, Eg and E^, with
1 s < t < n. Let P(Eg,Et) be any shortest path connecting Eg and E^.
If P(E ,E ) is an nd-path then clearly L(E ,E ) = LP(E ,E ) > d(v ,v )
so that the separation condition for Eg and Efc is satisfied in this
case. Consider the case when P(Eg,Et) is a d-path. Decompose P(Eg,Et)
into its nd-paths, say, P(E ,E,P(E. .,E ). Hence,
S^IJ (V) t
LP(Eg,E^) > d(vg,v^^) ,... ,LP(E^rj ,Efc) > d(v(r)vt)> as the Paths
are nd-paths. Further, the length of P(Eg,Et) is the sum of the
lengths of P(Eg,E^j) ,... ,P(E^ jE^). Hence, upon using the triangle
inequality, we have d(v ,v ) d(v ,vnv) + ... +d(v. .,v ) <
St s v. X) ) t
LP (E ,E...) + ... + LP(E. ,E ) = LP(E ,E ) = L (E ,E ) so that the
s (1) (r) t s t s t
separation condition for E and E is satisfied for this case. Since
s t
the choice of Eg and Efc is arbitrary, the proof is complete.
We are now ready to present the procedure for constructing the
efficient frontier. We define G^ to be the graph with nodes
N^,...,Nm and the arc set A^. To every arc (N^,N^) of G^ we assign
the length 1/v For 1 < s < t m we denote by mgt the length of a
shortest path connecting the nodes Ng and Nfc in GThe computation of
3
mgt_, 1 < s < t < m, can be achieved in 0(m ) operations by using known
algorithms (see Dreyfus [23]).


-152-
taking p = for each i, or taking Vj = {v^} for
each j.
Q2. Can we use (or modify) COVER and/or the Sequential Location
Procedure of [32] to solve either of the related covering problems de
fined by
min p, + ... + p
rl rm
D(X,V )
< z/w..
ij
(i,j) e Ic
D(Xj,Xk)
- 2/vjk
. (j >k) e IB
or,
min max{p ^ ,... p }
s. t.
D(X,V ) < z/w (i,j) e I
z/vjk (j,k) e I
C
B
where z is fixed?
Q3. Are the separation conditions of direct use for determining the
consistency of the distance constraints of GMCP?
Q4. Can we extend the duality results of Chapter 2 to GMCP?
Q5. Is there a dual to either of the related covering problems?
Q6. What kind of applications may GMCP find?
Q7. Can the search for the minimum objective value be confined to
a finite set of numbers?
We remark that the minimum objective value of the m-center problem
with mutual communication is the maximum of a finite number of ratios
with the numerators distances between existing facility locations while
the denominators are sums of reciprocal weights which correspond to


-153-
shortest path lengths in a related graph. Hence, there appears to be
hope for extending the duality results of Chapter 2 to GMCP.
5.3 The t-Objective m-Center Problem: Steps Towards
a Unified Theory
Here we define a location problem which involves t minimax type
objectives. Special cases of the problem are the m-center problem with
mutual communication, the vector-minimization problem of Chapter 3, and
the bi-objective m-center problem of Chapter 4. We give a theorem
which unifies the independent results for each of these problems. An
outline of the proof of the theorem is also provided.
Given sets I and I let kn = {(N.,E.): (ij) e I0} and
C d G 1 j G
Ag = {(N^jN^): (j,k) e 1^}. On defining A = A^ U A^, we suppose given
t nonempty, mutually disjoint, exhaustive subsets of A, enumerated as
A, ,...,A Associated with A 1 < r < t, the rth objective f is
I t r J r
defined by
f (X) = max[max{w. ,d(x. ,v.): (N.,E.) eA ) A.} ,
rv iJ i j i j r C
max{vjkd(Xj,xk) : £ Ar /I A^}]
where, by convention, the value of either of the inner maximizations
is understood to be zero if the maximum is taken over an empty set.
Letting f(X) = (f^(X),...,f (X)), the t-Objective m-Center Problem
is as follows:
V min[f(X): X e Tm] .
For the case t = 1 and A^ = A^ U A^, the problem specializes to the
m-center problem with mutual communication. For the case with each A^


-105-
Proof. Since and are adjacent, is in A^(Z). By Lemma 3.5.2,
we have H[A2(Z)] = U{L(z^,y): y e A^CZ)}. Since z2 e H[A^(Z)], for
at least one facility location y in A^(Z), z^ e L(z^,y). Also
z^ f y, for otherwise, z^ = y and z^ e L(z^,y) imply z^ = z2, contra~
dieting the irreducibility of Z. Hence, a) is established. Part
b) follows immediately from a) and the irreducibility of Z.
We remark that the irreducibility assumption cannot be relaxed
in Lemma 3.5.2, for otherwise we may have z^ = = y. Figure 3.5
illustrates such a case.
We will subsequently use Lemma 3.5.2 repeatedly to identify a
sequence of locations z z z. v such that they all lie
(1) (2.) (r)5 p y
in the line L(z^j,v ) in the given order. The corresponding sequence
of nodes N N. ,E in GBC will form a subpath connecting N.,.
(1) (r) p v B (1)
to with the length of that subpath equal to d(z^^,v^). By the
same token, we will find another node E in GBC with the subpath
q
connecting E^ to having length d(v^,z^). Then, we will show
that the two subpaths when connected at form a tight path which
contains N
(1)'
First we give the following result given in [82].
Lemma 3.5,3. Given four points p^ ,p^ jP^jP^ e T, if p2 e LPj^)
P3 e L(p2,p4) and p2 f p3> then d(Pl>p4) = d(p ,p ).
i=l


-39-
auxiliary graph G is formed with nodes N ,...,Nm,E^,...,E Graph G
contains arcs (N^,E^) with lengths l/w corresponding to pairs
(i,j)el and arcs (N.,N ) with length 1/v corresponding to pairs
G J K J K
(j,k)eIB. It is assumed that G is connected, for otherwise the problem
decomposes into subproblems. For each pair of existing facility nodes
Ej, E^, define L(Ej,E^) to be the length of a shortest path in G
connecting E. and E,. Francis et al. showed that Z* is given by
J k
max{d(v. ,v, )/£(E. ,E, ) : 1 j < k n}. The distances d(v.,v.) can be
J K. J K J k
2
computed in 0(n ) operations for a tree network (see [23]), and the
shortest path lengths Zr(E.,E,) are readily computable in 0(n ) opera-
1 k
tions. When Z* is computed, the Sequential Location Procedure de
scribed in [32] can be applied in 0(m(n+m)) operations to find a loca
tion vector X* that solves the problem.
m-Median problem with mutual communication
Define the functions g g and g by the following expressions:
D L
For each XeNm
gB(X) = l [v^dCx^Xjp: (j,k)eIB] ,
grQO = l [w d(x ,v ): (i,j)el ] ,
XJ 1 J o
g(X) = gB(X) + gc(X) .
The m-median problem with mutual communication is the following:
Find a location vector X* in Nm such that
Z* = g(X*) = min[g(X): XeNm] .


-112-
e) Iteration 4
f) Iteration 5
Figure 3.7. Continued


-154-
corresponding to exactly one arc in U A^, the problem specializes to
the vector-minimization problem considered in Chapter 3. For the case
t = 2, = A^, and = A^, the problem specializes to the bi-objective
m-center problem of Chapter 4.
Consider the related distance constraints DC where z = (z. z ),
z It
defined below:
d(x.,v.) < z /w. (N.,E.) e A PI A, 1 < r < t ,
i 3 r xj x j r C-
d(x. ,x, ) z /vM (N. ,N, ) e A' H A,., 1 £ r < t
j k r jk j k r a
It is direct to verify the following assertion: Let X be given and
define z = f(X). The location vector X is efficient if and only if for
every X ) > 0, X 4- 0, DC is inconsistent. The proof of
1 L ZA
the assertion is very similar to the proof of Lemma 4.3.1 in Chapter 4.
Based on the above property we give the following theorem for
characterizing efficient solutions.
Theorem 5.3.1. Given X used to define DC and GBC with z = f(X),
z z
the following are equivalent:
(a) X is efficient.
(b) For every r with r e { 1,... ,t} .and z > 0, at least one arc in
A^ is in some tight path in GBCz.
(Equivalently, there exists a collection of tight paths in GBCz
such that at least one tight path passes through Afor every r
for which r e {l,...,t} and z^ > 0.)
Outline of the proof. To show (a) implies (b) suppose X is efficient.
Assume that for some r for which z^_ > 0, no arc in A is in a tight
path. Clearly DC^ is consistent so that every path which passes through
Af is slack. Let PCE^jE^) be any path which passes through A^. Define


-5-
Network N is said to be edge weighted if, associated with each of its
edges, is a specified real number. Given an undirected network
N = {V,E} with positive edge weights, an imbedding of N, written as
N = {V,E}, is a geometric realization of N is some space S such that
there is a one-to-one correspondence between the members of V and k,
and E and E, respectively; each edge eeE is a rectifiable arc, and no
two edges in E intersect at more than one point, a vertex. The length
of edge e in E is defined to be the edge weight of the corresponding
member in E. A point of an imbedded network N = {V,E} is any point
along any edge in E, including the vertices. We write xeN to mean x
is a point in N. The distance d(x,y) between any two points x,yeN is
the length of a shortest path P(x,y) joining the two points. The
function d(.,.) satisfies the axioms of a metric on N so that the set
N together with d(.,.) determines a metric space.
The axioms of a metric are as follows: For any two points x,y£N,
1. d(x,y) > 0 if x j y; d(x,x) = 0,
2. d(x,y) = d(y,x),
3. d(x,y) 1 d(x,u) + d(u,y) for any ueN.
For a more detailed discussion of how to construct a metric space
(N,d) from a given edge weighted network N, the reader is referred to
Dearing and Francis [19], or Dearing, Francis, and Lowe [22]*
We restrict ourselves to finite undirected connected networks
that contain no loops and no multiple edges. We omit the term "im
bedded," and simply take a network to mean an imbedded network on
which the distance d(.,.) is defined. For all other networks, we use
the terms "graph," "arcs," and "nodes" instead of network, edges, and
vertices.


-lu
c) Iteration 3
Figure 3.7. Continued


-16-
relies on solving a finite sequence of set covering problems. Using
the framework provided by Minieka, an exact algorithm was developed
by Garfinkel, Neebe, and Rao [38] for the unweighted case. The
algorithm uses the property that the p-radius is determined by one of
a finite number of elements, namely, one of the distances between any
vertex and any point in P. Call the points in P edge bottleneck
points and let d be the distance between vertex v^, and the jth
edge bottleneck point. Let and Z be a lower and upper bound on the
value of r^. Initially Z^ = 0, and Z is obtained by a trial solution.
Among all the distances d that fall within the interval [_Z,Z], one
of them will determine the value of r Pick one such distance, say
P
d with Z < d < Z, and let r = d be a specified radius. Now,
st st st
we want to know if we can cover all vertices of N within this critical
distance r by using only p points. If we cannot, then clearly r is
too small a radius for p points to cover all vertices. Hence we con
clude the p-radius r^ must be within the interval [r,Z], In this
case, the lower bound is shifted to r, and the procedure is repeated.
In the other case, we find a set X of p points that cover all vertices
within r, but it is doubtful if this point set is an absolute p-center.
Clearly, then, the value of r^ will be within the interval [_Z,f(X)].
Hence, the upper bound is shifted to f(X) for this case and the whole
procedure is repeated. Termination occurs whenever the lower and
upper bounds become equal. The r-cover part of this procedure is
solved by obtaining a feasible solution, if it exists, to a set cover
ing problem. Let A be a |v| by |p| matrix with entries a_^. equal one
if vertex v is within a distance r of the jth edge bottleneck point
and zero otherwise. Then, solving the system £ y^ £ p, Ay > 1,


CHAPTER 2
DUALITY AND THE NONLINEAR p-CENTER PROBLEM AND COVERING
PROBLEM ON A TREE NETWORK
2.1 Introduction and Related Work
We consider the problem of locating p new facilities on a tree
network with respect to n existing facilities at known locations so as
to minimize the maximum "loss." The problem is an extension of the
linear p-center problem to the nonlinear case. We assume a strictly
increasing, continuous "loss" function is associated with each of a
finite number of demand points (existing facilities) whose argument
is the distance between the corresponding existing facility and its
nearest new facility. Our formulation permits the use of quite general
loss functions provided that they are continuous and strictly increas
ing with the travel distance. The term "loss" is used generically
and may refer to any form of inconvenience such as cost, disutility
of service, travel time, etc.
In locating emergency service facilities, the disutility due to
"late" service may be too great beyond a certain "threshold" response
time. Such sharp changes in the disutility of service can be re
flected into the model by using nonlinear functions. Hurter and
Schaefer [61 ] justify and use such functions in a fire setting. As
pointed out by Dearing [183 a study by Kolesar et al. [73 ] revealed
that the travel time for fire trucks can be approximated by a particular
continuous, nonlinear, increasing function of the distance.
-5V


-69-
node t will be in both GT and the associated brown subtree, denoted
as BT(t).
COVER
0) Initialize to GT = T, k =
BT(v.) = iv.}. For every j e
3 J
at v.. Define U = 0.
0. For every tip vertex v of T define
J(r) fasten a string of length f ^(r)
3 o
1) Choose a tip t of GT. If GT = {t} go to 6). Else find a(t), the
vertex in GT adjacent to t.
2) If no strings are engaged or fastened at t, remove from GT the
subarc [t,a(t)] joining t and a(t), attach [t,a(t)] to BT(t), and go
to 1) Else go to 3).
3) Pull all strings at t tight towards a(t). If all tight strings
reach a(t) then engage them at a(t), remove [t,a(t)] from GT, attach
[t,a(t)] to BT(t), and go to 1). Else go to 4).
4) Add 1 to k. Choose a shortest string engaged or fastened at t.
Find the (unique) vertex, say v., at which the shortest string is
(k)
fastened. Construct ^ U {v^^}. Find the farthest point, say
y, from t on [t,a(t)] to which the shortest string can reach. Locate
x^ at y. Assign all strings at t to x^ and remove these strings.
Attach [t,y] to BT(t) to obtain BT(x ), and remove [t,y] from GT.
Go to 5).
5) Assign to x^ all other strings in GT which can reach x^, and re
move all such strings. If no strings remain then let U = U and stop.
K
Else return to 1).
6) Add 1 to k. Locate x^ at t. Assign all strings at t to x^. Of
the strings at t choose any one, and find the vertex v^ to which
the chosen string is fastened. Let U = U, 1 1/ {v,,,.}, and stop.
k-1 (k)


-81-
Proof. By definition of a primary set we have |x| = |u|. By assump
tion r < r^ so that |x| = |u| k 2. Lemma 2.5.1 implies g(U) = g^(U) > r.
Hence U is a feasible solution to the dual problem. Theorem 2.6.1 im
plies q(r) > q(r). By feasibility of X and U, and the fact that
|X| = |U|, we have |x| > q(r) > q(r) £ |u| = |x|. It follows that
X solves the cover problem, U solves the dual problem, and (2.6.3)
holds.
We remark that the above proof is an alternative to the proof of
Theorem 2.4.1 for establishing the optimality of X to the covering
problem. Hence, an application of COVER solves both problems simul
taneously.
At this point we give an interpretation of the pair of problems.
The defender D specifies an upper bound r on his loss against an attack
to any vertex and will position response forces as necessary so that
his loss will not exceed r. Each response force is an "expense" for
D. Hence, D's problem is to choose the fewest possible response
forces. The attacker A knows that D will not tolerate a loss exceeding
r. Hence, A recognizes that, no matter how many vertices he threatens,
D will have a sufficiently large number of response forces to respond
and that the loss A inflicts on D will always be less than or equal
to r. For this reason, A decides that he should not (hopelessly) try
to inflict a loss to D exceeding r, and that, instead, he should force
D into using as many of his response forces as possible. Hence,
should A choose a subset U of V with g(U) > r, he knows that no two
vertices in U can be jointly covered by a single response force by D
within the specified upper-bound r. Thus, D, not tolerating a loss
exceeding r, will have to allocate one response force for every vertex


-157-
once in DC. Clearly, the effective upper bound for the distance
between the locations corresponding to F and F is the minimum of the
P q
upper bounds which involve these two facilities. Thus, the effective
arc length to be assigned to (F ,F^) is the reciprocal weight associ
ated with Fp and F^ multiplied by where z^ is the minimum z^ over
all indices r for which (F ,F ) e A Let GBC be the graph with arc
lengths appropriately assigned as described above. Partition A into
A A
mutually disjoint subsets A^,...,A (s £ t) such that every arc in any
A A A .
A^ has the same multiplier, say, z^ (where z £ iz^,...,z }). De-
A A A
fining z = ) it is direct to verify that DC is equivalent
1. s z
to DC* defined below:
z
d(x.,v.) < z /w..
i y r i j
(N.,E.) eA 0 A., 1 r £ s
l j r C
d(x.,x. ) < z /w., (N.,N, ) e A A 1 < r < s
J k rjk jk r TB
In other words, DC^ is obtained from DCz by choosing the minimum effec
tive upper bound for any constraint which appears more than once in DCz<
As a result of the equivalence of DC and DC^ and the fact that
z z
Aj,...,A are mutually disjoint and exhaustive subsets of A, we make
the following proposition:
Proposition 5.3.1. Given X and z with z = f(X), let DCg be the equi
valent representation of DCz as described in the previous paragraph.
The following are equivalent:
(a) X is efficient.
(b) For every r e {l,...,s} with z^ > 0, at least one arc in Ar is
in a tight path in GBC~.


-108-
facilty j. Since j is arbitrary, Z is the unique solution to DC, and,
thus, upon using Theorem 3.3.3, Z is efficient.
3.6 Algorithm to Construct Efficient
Location Vectors
To this point we have presented a family of conditions for char
acterizing efficient points. Theorem 3.3.3 provides the necessary and
sufficient conditions in terms of uniquely located facilities, tight
paths in GBC, and the convex hulls of uniquely located facilities.
Property 3.5.2 provides the sufficient conditions for irreducible
location vectors without requiring the identification of uniquely
located facilities. Property 3.5.3 extends the results of Property
3.5.2 to the case of reducible vectors.
Based on Properties 3.5.2 and 3.5.3, we now present the Sequential
Efficient Vector Construction Algorithm (SEVCA). Given a location
vector Z, the algorithm first finds the irreducible representation Z*
of Z by using RP. Then each component of Z* is checked to see if it
satisfies the convex hull containment property. If some component is
found which is not within the convex hull associated with it, its loca
tion is moved to the closest point in the convex hull. The procedure
is repeated with the resulting location vector. Termination occurs
when every component of the current irreducible vector is within the
convex hull associated with it. In order to prove finite termination
(in 0(m) iterations), we use a labeling scheme for the current com
posite indices. The list K is the list of composite indices during
K
any given iteration, while Z denotes the location vector whose com
ponents are indexed by the members of K.


-33-
Goldman and Dearing [45] provide a conceptual discussion of, and
a motivation for, considering such problems. The distance constraints
are formally defined by Dearing, Francis, and Lowe [22] on a network.
It was established in [22] that, in a well defined sense, the distance
constraints define convex sets under the assumption that the under
lying network is a tree. Furthermore, the distance constraints always
define convex sets if and only if the network is a tree.
Based on the results obtained in [22], Francis, Lowe, and Ratliff
[32] considered the distance constraints on tree networks in more
detail. They established the necessary and sufficient conditions for
distance constraints to be consistent, and also devised algorithms
that find a feasible location vector whenever one exists. In what
follows we briefly discuss the results obtained in [32].
Distance constraints for a single new facility. For the case of
a single facility, Francis et al. showed that there exists a feasible
point xeT satisfying d(x,v.) < c. for iel if and only if the in-
i 1
equalities d(v.,v, ) c. + c, are all satisfied for 1 1 1 < k < n.
j k j k j -
An equivalent statement of the single facility distance constraints
can be given in terms of "neighborhoods" around v_^ of radii c^,. De
fine the neighborhood N(u,r) around a point ueT of radius r to be the
set of all points xeT for which d(u,x) < r. Then, a point x satisfies
the constraints d(v^,x) < c^> iel, if and only if x is in each neigh
borhood N(v.,c ), iel,if and only if x is in the intersection
n 1
n N(v.,c.). It follows then that the single facility distance con-
i=l 1 1
straints d(x,v.) Sc., iel,are consistent if and only if d(v.,v.) S
ii j k
Cj + ck for 1 j < k 5 n if and only if each pairwise intersection
N

-65-
center in X. Assume A and D know functions f,,...,f so that
1 n
f.(D(X,v.)) is D's loss if A attacks v. and D responds to the attack
3 3 J
in a time of D(X,v^). For convenience, we refer to the loss A in
flicts on D as A's gain.
Aggressor A knows D has p response forces, but does not know how
D will position his response forces. Thus A acts conservatively and
bases his decision on a worst case analysis. If A decides to attack
Vj without threatening any other vertices, A reasons that D will cor
rectly guess v is to be attacked and will position a response force
at v.. Hence A assumes his gain will be f (0), if he decides to
J J
immediately attack v^ without a prior threatening strategy. In order
to gain more, A concludes that he must threaten, i.e., pretend to
attack, q vertices, q > 1, so that even if D knows which q vertices
are threatened, D does not know which vertex A will attack until the
attack occurs. Thus D is forced to respond to the threat by position
ing his response forces optimally with respect to these q vertices.
Hence if A threatens K C V, he assumes D will choose a p-center X
q
which minimizes f(X:K ) = max{f.(D(X,v.)): v e K }. Thus, with
q j j j q
q p, A assumes D knows and will position a response force at
every vertex in K so that A can gain at most g (K ) The best A
q Z q
can do in this case is to choose a K which contains some vertex v
q s
for which fg(0) = a. Hence, if q < p, A's maximum possible gain is
at most f (0). (Parenthetically, we remark that if f (0) = r ,
s s p
p < n, then it can be shown that not all f.(0) have the same value.
3
If all f.(0) do have the same value, then r > a.) On the other hand,
3 p
if A chooses a subset with q > p, D is unable to position a response
force at every vertex in K even if he knows K so A will gain at
q q


-50-
is "as small as possible." More specifically, we wish to find all
efficient location vectors X = (x,,...,x ) in Tm with respect to the
i m
vector minimization problem
V-min[D(X): XeT]
where D(X) is the vector of distances d(x^,v^) (i,j)el^, and d(Xj,x^)
(i,k)el The vector is formed by assuming any convenient ordering
D
of the members of the sets I_ and I.
L> D
Francis, Lowe, and Tansel [33] characterized efficient points by
making use of distance constraints. By definition, a location vector
Z in T is efficient if an only if there does not exist a location
vector X in Tm such that D(X) < D(Z) and D(X) D(Z). Given a location
vector Z, let b., = d(z.,z,) for (jjk^I,, and c.. = d(z.,v.) for
jk 3 k J B xj 1 3
(i,j)el^,, and define the distance constraints (DC) of interest by
d(xi*vj) cij (i,j)elc
d(xj"xk)iV
We note that DC is always consistent, as Z is always feasible
to DC, and hence the separation conditions are always satisfied. The
separation conditions for DC are defined by constructing a graph G
with nodes 1 S j £ m, corresponding to new facilities and nodes
E^, 1 i < n, corresponding to existing facilities. For each
(i,j)elr,, the arc (N.,E.) is in G with length c.., and for each
c 1 1 ij
(j,k)el the arc (N.,N, ) is in G with length b.. We recall that a
B j k jk
point x_^ is uniquely located in every feasible solution to DC if and
only if the corresponding node N is in at least one tight path in G,


-148-
Figure 4.5. Efficient Frontier for Example


-66-
least (K ). Hence A observes if he chooses some K with q > p which
2 q q
contains a vertex v for which a = f (0), then his gain is at least
s s
a = g (K ). However, A recognizes that there may be some other K
2 q q
with q > p, which may or may not contain v but which yields him a
gain strictly greater than a. For this reason A restricts himself to
those subsets of V with cardinality greater than p and realizes that
if he chooses some K with q > p, then there is at least one pair of
q
vertices in K which D can cover by only a single response force. If
q
v^ and v_. are one such pair in which are covered only by a single
response force, say at x, then clearly A obtains a gain of at least
3.., as 3.. = min{max(f (d(x,v.)), f.(d(x,v.))): x e T} < max{f (d(x,v )),
ij 13 i 1 3 1 11
A
fj(d(x,Vj))}. Since A does not know which pairs of vertices D will
cover by single response forces, once he chooses K^, A acts conserva
tively, and assumes that D will cover a pair v ,v, e K for which
a b q
3 = min{3..: v.,v. e K i ^ 3}. That is, by choosing a K with
ab 13 1 3 q q
q > p, A guarantees himself a gain of at least 3 = g. (K ). Hence
clD J- CJ
A's minimum gain due to threatening is g(K^) = max{g^(K^), g^CK^)},
so A chooses a K* with q > p which maximizes g(K ) over all K C V
q q q
with q > p.
The question arises as to why A should choose p+1 vertices to
threaten, and no more. By virtue of the W.D.T. and the remark follow
ing it, if X* is an optimum p-center then f(X*) > g(K ) for all K
q q
with q > p+1. Thus r^ = f(X*) is an upper bound on A's gain due to
threatening But the S.D.T. implies there is a p+l-clique, say
K*+^, which attains this upper bound. Hence A need threaten no more
than p+1 vertices to maximize his gain, as A cannot obtain any addi
tional gain by threatening more than p+1 vertices.


-127-
Lemma 4.3.1. Given a location vector Y used to define DC and GBC
z z
with z = (zpZ^ = (f ^ (Y) f 2 (Y) ) the following are equivalent:
(a) Y is efficient.
(b) Por any X = (X,,X ) > 0 and X 0, DC is inconsistent.
Proof. Using the definition of efficiency, f^, f2> and the fact
z = f(Y) we have the following equivalences. The location vector Y
is efficient if and only if f(X) < z implies f(X) = z if and only if
there does not exist X such that f(X) z and f (X) 5s z if and only if
for any X £ 0 and X ^ 0 there does not exist X for which f^(X) < z^ X^
for i = 1,2 if and only if for every X > 0 and X ^ 0 there does not
exist X such that max{wd (x^,v^) : (i,j) e 1^,} z^ X^ and
max{vM d(x. ,x, ) : (j,k) e 1^.} < z X if and only if for every X £ 0
jk j K a 2 2
and X £ 0 there does not exist X such that d(x.,v.) < (z, X,)/w..
i J 1 1 ij
for all (i,j) e Ic and dCx^.x^) < (z2 *2^Vjk for G *b
and only if for every X 0 and X ^ 0, DC is inconsistent, com-
Z" A
pleting the proof.
Corollary 4.3.1. Given Y with z = f(Y), Y is dominated if and only if
there exists X z 0, X 4- 0 such that DC is consistent.
z-A
We remark that the proof of Lemma 4.3.1 does not use any special
properties of tree networks. Hence the lemma is applicable to any
metric provided that f^ and 2 are the maximum of the pairwise weighted
distances.
The following lemma provides the sufficient conditions for DCr
to be consistent in terms of the slack paths in GBCz.
Lemma 4.3.2. Suppose DC is consistent. If every path in GBC which
2 Z
passes through Ag is slack then X = (Xj^) can be chosen with X £ 0
and X ^ 0 such that DC is consistent.
z-X
z-X


OPTIMAL MULTI-FACILITY LOCATION ON
TREE NETWORKS
By
BARBAROS C. TANSEL
A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1979

ACKNOWLEDGMENTS
I am deeply indebted and grateful to Dr. Richard L. Francis, the
chairman of my supervisory committee, for his excellent guidance,
numerous suggestions, and the generosity with which he invested his time
in listening to my ideas. Dr. Francis not only initiated my interest
in location problems but also inspired many of the ideas in this dis
sertation by asking the right questions at the right time.
I owe very special thanks to Dr. Timothy J. Lowe, the cochairman/
chairman of my committee during 1976-1978, presently of Purdue Uni
versity, for his active interest, overall guidance, and his inspiring
suggestions.
Dr. Francis and Dr. Lowe have shown sincere care about my progress
and their encouragement has been of utmost value in bringing this
dissertation to a completion.
I would also like to express my sincere thanks and appreciation
to the other members of my committee, Dr. Ralph W. Swain, Dr. Donald W.
Hearn, Dr. Antal Majthay, and Dr. Luc G. Chalmet for their interest in
my work and their suggestions during my proposal.
I am grateful to the Department of ISE for providing me with
assistantship during my graduate studies.
Mrs. Adele Koehler has done an excellent job in typing the manu
script. She is fast, accurate, and very observant. I sincerely
recommend her.
ii

This research was supported in part by NSF Grant //ENG 76-17810,
the Army Research Office, Triangle Park, N.C., under contract
DAHC04-75-G-0150, and by the Operations Research Division, National
Bureau of Standards, Washington, D.C.

TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS ii
ABSTRACT vi
CHAPTER
1 INTRODUCTION AND LITERATURE SURVEY 1
1.1 Introduction and Overview . 1
1.2 Terminology 4
1.3 Survey of the Network Location Literature 6
2 DUALITY AND THE NONLINEAR p-CENTER PROBLEM AND
COVERING PROBLEM ON A TREE NETWORK ...... 53
2.1 Introduction and Related Work 53
2.2 Problem Statements and Duality 56
2.3 Dual Problem Interpretation 61
2.4 Covering Algorithm 67
2.5 Dual Problem Solution and the Strong Duality Theorem. 73
2.6 Results for the Covering Problem 78
3 A VECTOR-MINIMIZATION PROBLEM ON A TREE NETWORK 84
3.1 Introduction 84
3.2 Problem Statement 85
3.3 Distance Constraints and Characterization of
Efficient Points 87
3.4 Examples 94
3.5 Further Results on the Convex Hull Property 96
3.6 Algorithm to Construct Efficient Location Vectors . 108
3.7 Efficiency for the Case of Rectilinear or
Tchebychev Distances. 116
4 THE BI-OBJECTIVE m-CENTER PROBLEM ON A TREE NETWORK. ... 122
4.1 Introduction 122
4.2 Problem Statement, Notation, and Definitions 123
4.3 Necessary and Sufficient Conditions for Efficiency. 126
4.4 Construction of the Efficient Frontier 134
Iv

Page
5 SUMMARY AND FUTURE RESEARCH 149
5.1 Summary 149
5.2 Generalized Multi-Center Problem 150
5.3 The t-Objective m-Center Problem: Steps
Towards a Unified Theory 153
5.4 Tree Networks and General Networks 158
REFERENCES 161
BIOGRAPHICAL SKETCH . 170
v

Abstract of Dissertation Presented to the Graduate Council
of the University of Florida in Partial Fulfillment of the Requirements
for the Degree of Doctor of Philosophy
OPTIMAL MULTI-FACILITY LOCATION ON
TREE NETWORKS
By
Barbaros C. Tansel
December 1979
Chairman: Richard L. Francis
Major Department: Industrial and Systems Engineering
In this dissertation we develop a theory for location problems which
involve locating multiple new facilities on a tree network with respect
to existing facilities at known locations.
The first problem we consider is the nonlinear version of the
p-center location problem on a tree network for which the cost of each
served vertex is a strictly increasing continuous function of the dis
tance between the vertex and the nearest center,and the objective is to
minimize the maximum such cost over all possible locations of the
centers. We present a dual "dispersion" problem which may be inter
preted as the problem of choosing p + 1 (or more) vertices such that
the minimum cost to serve any two of the chosen vertices by a single
common center is as large as possible. We give a weak duality theorem
which applies to all general networks and a strong duality theorem
which applies to all tree networks. The strong duality theorem also
specifies the necessary and sufficient conditions for an optimal solu
tion to either problem. We provide algorithms of polynomial complexity
v

for solving either problem provided that certain needed inverse functions
can be evaluated in a polynomial order of effort. The p-center problem
is typically solved with the aid of a nonlinear covering problem for
which we also give a dual with a physical interpretation. We provide
a covering algorithm which solves both the covering problem and its dual
simultaneously.
The second problem we consider is a vector-minimization problem
which involves as objectives the distances between specified pairs of
new and existing facilities andspecified pairs of new facilities. We
relate the vector-minimization problem of interest to a distance con
straints problem which imposes upper bounds on the distances between
specified pairs of facilities. We develop the necessary and sufficient
conditions for efficiency by making use of the theory developed for the
related distance constraints problem. Efficient solutions to the
vector-minimization problem of interest are such that in order for any
new facility to be closer to some facility than it already is, it must
in turn be placed farther from some other facility. Based on the
necessary and sufficient conditions, we provide an algorithm which
constructs an efficient location vector from a given non-efficient
solution.
The third problem we consider is a bi-objective minimax problem
which involves as objectives the maximum of the weighted distances
between specified pairs of new and existing facilities, and the maximum
of the weighted distances between specified pairs of new facilities.
We again relate the problem to the distance constraints problem and
derive the necessary and sufficient conditions for efficiency by making
2 2
use of the distance constraints. Further, we provide an 0(m (m + n ))

algorithm to construct the efficient frontier, where m and n are,
respectively, the number of new and existing facilities.
v i i i

CHAPTER 1
INTRODUCTION AND LITERATURE SURVEY
1.1 Introduction and Overview
Although some mathematical models of location can be traced back
to the early seventeenth century, almost all the work on operational
models for the location of facilities has taken place within the past
22 years, between 1957 and the present. An extensive annotated bibli
ography on location-allocation problems is provided by Lea [78]. A
more recent selective bibliography is given by Francis and Goldstein
[30],
Location problems commonly involve locating a number of new
facilities (sources) in a given location space so as to provide goods
or services to a specified set of existing facilities (demands) under
one or more criteria, and, possibly, subject to a set of constraints.
The quality of the service is typically measured in terms of the dis
tances among the facilities. The use of distances is, perhaps, the
major feature which distinguishes location problems as a special class
of optimization problems. Hence, associated with any location problem
is an underlying location space on which a "distance" is defined.
Several variations of the general location problem are possible,
depending upon the type of location space, the distance function, the
number and areal extent of the facilities, the type of interactions

-2-
between the facilities, the objective criteria used, the constraints,
the presence or lack of random elements, and possibly other factors
as well.
Among the several variants, planar location problems received
special attention in the past, starting with the earliest contribu
tions, for example [106]. In such planar problems, one is interested
in locating new facilities in the Euclidean plane with respect to
existing facilities. For continuous planar problems, where any point
in the plane is a feasible location, the typical distance used is the
distance, special cases of which are the rectilinear, Euclidean,
and Tchebychev norms. For discrete planar problems, where there are
a finite number of candidate locations for new facilities, the distance
between any potential new facility location and any existing facility
is a specified positive number. Such discrete problems, due to the
finite nature of feasible locations, readily lend themselves to integer
programming formulations. The reader is referred to the book by
Francis and White [31] for a discussion of planar problems and a wealth
of references.
A number of real life applications suggest that, in some in
stances, a network space can be a more faithful representation of the
reality than the Euclidean plane. For example, in a road network, a
communication network, or a pipeline system, travel occurs along the
arcs of the underlying network rather than in straight lines or recti
linear paths. Hence, for such problems, the use of shortest path
distances along the arcs of the network can approximate the travel
distance more closely than the X. distance. As opposed to planar
problems, network location problems have received much less attention

-3-
in the past. As reported by Lea [79], there are some 1500 published
papers on location-allocation problems. Among these, about 80 are on
network location problems, a ratio of a little less than 6%. Hence,
network location problems deserve well-justified attention in future
research.
In this dissertation, we develop a theory for a number of location
problems which involve locating multiple new facilities on a tree net
work with respect to existing facilities at known locations. At this
point we give an overview of the dissertation.
In the remainder of Chapter 1, we specify our terminology and
give a survey of the network location literature. We discuss minimax and
minisum problems.and multi-objective problems involving minimax and
minisum objectives as well as other objectives. Discussed also are
problems with distance constraints. We highlight some of the convexity
properties of trees (see [22]) in relation to the problems discussed.
The chapter ends with a brief discussion of path-location problems.
In Chapter 2, we develop a theory for the nonlinear p-center
problem on a tree network. The problem is a generalization of the
linear p-center problem which involves locating p new facilities on
a network so as to minimize the maximum weighted distance from any
existing facility to its nearest new facility. Nonlinearity is ob
tained by replacing each weight by a strictly increasing function of
the distance. We formulate a dual "dispersion" problem and prove a
weak duality and a strong duality theorem. The strong duality theorem
also specifies the necessary and sufficient conditions for an optimal
solution to either problem. We provide algorithms of polynomial com
plexity for solving either problem. Discussed also are a covering

-4
problem and a dual "divergence" problem. We provide a covering
algorithm which solves both the covering problem and its dual simul
taneously.
In Chapter 3, we study a vector-minimization problem in relation
to a distance constraints problem. The problem involves as objectives
the distances between specified pairs of new and existing facilities
and specified pairs of new facilities. We extend the results of [32]
to develop a theory for identifying unique solutions to distance con
straints, and use this theory to develop necessary and sufficient
conditions for efficient solutions to the vector-minimization problem
of interest. Further, we provide an algorithm which constructs an
efficient location vector from a given non-efficient solution.
In Chapter 4, we study a bi-objective location problem which in
volves as objectives the maximum of the weighted distances between
specified pairs of new and existing facilities, and maximum of the
weighted distances between specified pairs of new facilities. We
characterize efficient solutions and provide an algorithm for construct
ing the efficient frontier.
In Chapter 5, we pose a number of unresolved questions in relation
to the problems discussed and point out directions for future research.
1.2 Terminology
Before discussing the literature we specify our terminology.
An undirected network N = {V,E} is a collection of two sets V
and E, called the set of vertices and the set of edges of N, respec
tively. Each edge in E is described by an unordered pair of vertices.

-5-
Network N is said to be edge weighted if, associated with each of its
edges, is a specified real number. Given an undirected network
N = {V,E} with positive edge weights, an imbedding of N, written as
N = {V,E}, is a geometric realization of N is some space S such that
there is a one-to-one correspondence between the members of V and k,
and E and E, respectively; each edge eeE is a rectifiable arc, and no
two edges in E intersect at more than one point, a vertex. The length
of edge e in E is defined to be the edge weight of the corresponding
member in E. A point of an imbedded network N = {V,E} is any point
along any edge in E, including the vertices. We write xeN to mean x
is a point in N. The distance d(x,y) between any two points x,yeN is
the length of a shortest path P(x,y) joining the two points. The
function d(.,.) satisfies the axioms of a metric on N so that the set
N together with d(.,.) determines a metric space.
The axioms of a metric are as follows: For any two points x,y£N,
1. d(x,y) > 0 if x j y; d(x,x) = 0,
2. d(x,y) = d(y,x),
3. d(x,y) 1 d(x,u) + d(u,y) for any ueN.
For a more detailed discussion of how to construct a metric space
(N,d) from a given edge weighted network N, the reader is referred to
Dearing and Francis [19], or Dearing, Francis, and Lowe [22]*
We restrict ourselves to finite undirected connected networks
that contain no loops and no multiple edges. We omit the term "im
bedded," and simply take a network to mean an imbedded network on
which the distance d(.,.) is defined. For all other networks, we use
the terms "graph," "arcs," and "nodes" instead of network, edges, and
vertices.

-6-
Finally, for tree networks, we write T instead of N. In passing,
we note that the shortest path P(x,y) between any two points x,yeT is
unique, as otherwise T would contain a cycle.
1.3 Survey of the Network Location Literature
Historically, the earliest precise mathematical formulation of a
location problem on a network appears to be due to Hakimi [47] in 1964.
Prior to Hakimi's paper, the problem of finding the best threshing
site for harvested wheat was attacked by using a network location model
in 1962 by Hua Lo-Keng and Others [60]. This model was presented only
at an intuitive level and no mathematical formulation or properties
were given. A (correct) solution procedure was suggested (in the form
of a poem), which was to be discovered independently by Goldman [42] in
1971. Since 1964, a literature of approximately 80 papers has grown
till the present. Several new problems, as well as certain extensions
and generalizations of old problems, have been introduced.
A recent text by Handler and Mirchandani [ 58 ] discusses ex
tensively a portion of the literature involving minimax and minisum
problems as well as single-facility bi-objective problems involving
the combination of these two objectives.
A "family tree" for network location problems is shown in
Figure 1.1. Although not exhaustive, the family tree covers most of
the problems formulated since 1964. With reference to the family tree
shown in Figure 1.1, network location problems can be broadly classi
fied into two groups: point-location problems and path-location
problems. Path-location problems have been recently introduced by

-7-
Figure 1.1.
Family Tree for Network Location Problems

-8-
Slater [102]. A large portion of the literature deals with point-
location problems. Point-location problems may be classified into
three categories: single objective problems, multi-objective problems,
and a body of results of a general and unifying nature.
In the remainder of this section we give a detailed discussion
of the problems outlined in the family tree.
Point-Location Problems
Here, we consider a number of problems that involve locating new
facilities at points on a network. The general format of the dis
cussion is as follows: For each problem type, we first define a
kernel problem. Then, we discuss the related literature on the kernel
problem, as well as several special cases and extensions of it. We
point out relations between different problem types, whenever such
relations exist.
The p-center problem
Let N be a network with a vertex set V = {v,,...,v } and an edge
1 n
set E. Denote by X a finite set of points, each of which is in N.
Let I be the set of integers 1 through n. For each vertex v., iel,
define the distance D(v^,X) between vertex v and the point set X by
D(v^,X) = min[d(v^,x) : xeX]. With this definition, D(v_^,X) is speci
fied by a nearest point in X to vJ. Let w. and a. be two given numbers
i 11
associated with vertex v^, iel. We call wi a weight and aan addend.
We assume that each w^ is nonnegative and at least one w_^ is positive.
For any finite point set X CD, define the function f (X) by

-9-
f(X) = max[wJ)(v^,X) + a^: iel]
The problem of Interest is the following: Given a positive integer p,
find a point set X* = {x*,...,x*}, and a real number r
1 p p
such that
rp = f(X*) = min [ f (X) : |x| =p,XcN] (1.3.1)
where the symbol j*| means the cardinality of a set.
The problem defined by (1.3.1) is called the p-center problem.
Any set X* of p-points that solves (1.3.1) is called an absolute p-
center of N, and the minimum objective value r^ is called the p-radius.
For p = 1, an absolute 1-center is simply called an absolute center
of N.
If in (1.3.1), each xeX is restricted to a vertex location, the
resulting problem is called the vertex restricted p-center problem and
any set X* C V of p points that solves it is called a vertex restricted
p-center of N. A vertex restricted 1-center is simply called a vertex
center.
We note that the p-center problem is usually formulated in the
absence of addends. In what follows, we will assume all addends are
zero, unless we explicitly mention them. The case with all w^ equal
to unity will be referred to as the unweighted case.
With this terminology, the p-center problem is the problem of
finding p points on a network so that the maximum (weighted) distance
between any demand point and its nearest center is as small as possible.
The problem is perhaps most applicable to the location of emergency
facilities such as fire stations, ambulance centers, and the like, as

-10-
ln such problems a common objective is to provide "good" service to
each demand point by at least one facility within a least possible
distance.
In what follows, we first discuss the 1-center problem on general
networks and on tree networks. Then, we discuss the vertex restricted
1-center problem. Finally, we will discuss the p-center problem in
relation to a "covering" problem to be defined later.
1-Center problem on a general network. The absolute 1-center
problem was defined and solved by Hakimi [47] in 1964. For finding the
absolute center, Hakimi examines the function f on each edge, finds a
best local minimum on that edge, and selects the best among |e| such
local minima. This method takes advantage of one important property
of f, namely, that it is piecewise linear and continuous on each edge
with at most n(n l)/2 break points. A local minimum always occurs
either at a break point of f or at an end point of the edge. Hakimi,
Schmeichel, and Pierce [50] showed that Hakimi's method can be imple-
mented in 0(|E|n logn) computational effort and gave a computational
refinement which reduces the effort to 0(|E|nlogn) for the unweighted
case. Further refinements of the procedure were obtained by Kariv
and Hakimi [65], resulting in an 0(|E|nlogn) algorithm for the
weighted case and 0(|E|n) algorithm for the unweighted case. All
these refinements focus on finding the break points and the local
minimum of.f in the most efficient manner.
A somewhat more general version of the 1-center problem was con
sidered by Frank [36], and (apparently) independently by Minieka [88],
as Minieka makes no reference to Frank's paper. In this modified
version, called here the continuous 1-center problem, each point on

-li
the network is a demand point (as opposed only to vertices). The
weight of each point is unity. The objective to be minimized over all
xeN is defined by f(x) = max [d(y,x): yeN]. Both authors showed that
the problem can be reduced to a computationally finite one and pro
posed a solution procedure which is very similar to Hakimi's.
A probabilistic version of the 1-center problem was considered
by Frank [34, 35] and a number of bounds were obtained on the expected
value of the 1-radius.
For the unweighted case, Singer [101] proved that there exists a
"critical" path, not necessarily a shortest path, connecting two cri
tical vertices such that an absolute center of the network is at the
midpoint of this path.
1-Center problem on a tree network. We now concentrate on ab
solute centers of tree networks. Goldman [44] solved the unweighted
case in the presence of addends. Goldman's algorithm is based on the
repeated application of a "trichotomy theorem" that either determines
the edge on which the absolute center lies, or reduces the search to
one of the subtrees obtained by removing all interior points of that
edge. Halfin [51] refined Goldman's algorithm to make it simpler and
computationally more efficient. Halfin's algorithm finds a vertex
center first, and determines the absolute center by examining all
vertices adjacent to the vertex center.
For the unweighted case with no addends, Handler [55] presents
an especially elegant algorithm. Handler's method finds a longest
path of the tree and locates the absolute center at the midpoint of
the path. To find a longest path, Handler chooses an arbitrary vertex
v finds a farthest vertex v from v., and then finds a farthest
i si

-12-
vertex v from v The path P(v .v^) is a longest path and its mid-
t s s t
point is the unique absolute center of the tree. This procedure
requires a computational effort of 0(n). Handler's algorithm is
extended by Lin [81] to the unweighted case with addends. Lin showed
that the absolute center of a general network N with vertex addends
can be found by determining the absolute center of an expanded net
work N' whose vertex addends are all zero. Network N' is obtained from
N by adding a new vertex adjacent to each old vertex, with the length
of the edge connecting the two equal to the addend associated with
the old vertex. For a tree network T, the resulting network is a
tree T' and Goldman's 0(n) algorithm can be applied to T'.
The more general case with both weights and addends was considered
by Dearing and Francis [19], and for the case of a tree network an
2
0(n ) algorithm was given. The Dearing-Francis paper appears to be
the first to construct a well defined metric space N with distance
d(.,.) from an arc weighted graph N. This mathematical formality per
mits the use of such concepts as compactness, continuity, and the
extreme and intermediate value theorems. They showed that the distance
d(x,.) is continuous for each fixed x, in turn implying that f(x) is
continuous for every x. From compactness and continuity considera
tions, they proved the existence of an absolute center for all compact
networks, and its uniqueness for all compact tree networks. They
obtained a lower bound on r^ which is applicable to all networks, and
proved that it is always attainable for tree networks. Once the lower
bound is determined, it identifies two "critical" vertices, and the
absolute center can be readily located on the path joining the two.
The bound is the maximum of n(n l)/2 terms, resulting in a

-13-
2
computational complexity of 0(n ), and is given by
a = max[a..: 1 i j n]
where
(1.3.2)
W-W.VAIV.JV,/ W C* I W t-4. .
= i J 1 J 1 J J 1
w w.d(v.,v.) +w.a. + w.a
a
ij
w. + w,
1 jj
Hakimi, Schmeichel, and Pierce [50] proved a theorem that reduces the
computational effort for computing this lower bound. Their theorem
states that if for some a it is true that max[a l5i st si st
max[a 1 K i < n] then a is the maximum of all a... A different
ti J st 13
solution procedure is also given by Kariv and Hakimi [65] for the
same problem. Rather than computing the lower bound, their procedure
confines the search to successively smaller subtrees until an edge is
obtained. The absolute center is located at the local center (also
the global center for a tree) on this edge using Hakimi's procedure
for finding a local minimum. This algorithm is of O(nlogn).
A nonlinear version of the 1-center problem was considered and
solved by Dearing [18], and by Francis [29]. In this version, each
weight w_^ is replaced by a monotone increasing function f of the
distance d(v_^,x). Both authors obtained a lower bound similar to the
one defined by (1.3.2). The bound is applicable to all networks and
is always attainable for tree networks.
A "roundtrip" version of the problem was solved by Chan and
Francis [ 11 ]. In this version each "demand point" is a pair of ver
tices (v^,^) and f(x) is the maximum of the roundtrip distances
defined by p^(x) = w^dCv^x) + d(x,u^) + a^]. A lower bound, similar

-14-
to the one defined by (1.3.2) is obtained. The bound is again
applicable to all networks and always attainable for tree networks.
Vertex constrained 1-center problem. The vertex constrained
1-center problem was considered as early as 1869, and perhaps earlier,
by Jordan [63] as a graph theoretic problem. This problem can be
solved by examining the distance matrix of the network, as demonstrated
by Hakimi [47], Rosenthal, Pino, and Coulter [98] introduced a gener
alized algorithm that solves a number of "eccentricity" problems on
tree networks, one of which is the vertex restricted 1-center problem.
In this case, the eccentricity of a vertex is defined to be the
distance from that vertex to a farthest vertex. This generalized
algorithm determines the eccentricity of each vertex by making only
two traversals of the vertices. The vertex center is that vertex
with the minimum eccentricity. Slater [103] considered the problem
of finding the vertex center of a network with respect to subnetworks.
In this version of the problem, each demand is a known collection of
vertices (or a subnetwork induced by the collection). The distance
between a vertex and any such collection is defined by a nearest
element of the collection to that vertex. For a given vertex, the
value of the objective function at that vertex is the maximum of the
distances between that vertex and any such collection. Slater showed
that a matrix D' can be constructed from the distance matrix D of the
network, so that each entry of D' is a distance from a vertex to a
nearest element of a collection. Slater demonstrated that the vertex
center with respect to collections of vertices can be found by
examining the matrix D'.

-15-
This completes the discussion of the 1-center problem. We now
concentrate on the p-center problem for p > 2.
p-Center problem on a general network. The p-center problem was
defined by Hakimi [48]. Subsequently, a number of solution procedures
have been suggested. A common characteristic of all these procedures
is that they all rely on solving a sequence of covering problems.
For completeness, we first define a set covering problem and an
r-cover problem.
Let A be a matrix of zeros and ones, y a vector of zero-one
variables y^. The problem of minimizing J y_^ so that each row of Ay
i
is greater than or equal to one is called the (minimal) set covering
problem. Given the function f(X) = max{w Jl(v^,X): 1 < i < n}, the
problem of minimizing |x| so that f(X) < r for some given value of r
is called the r-cover problem.
Denoting by q(r) the minimum value of the r-cover problem, it
can be readily shown that, if q(r) = p for some r, and q(r') > p for
any r' < r, then r is the p-radius and any X which solves the r-cover
problem is an absolute p-center.
In what follows, we concentrate on the absolute p-center problem
on a general network.
Minieka [87] considered the unweighted case on a general network
and showed that the problem can be reduced to a computationally finite
one. Minieka identifies a finite point set P* such that there exists
an absolute p-center contained in P = P' U V. A point x on some edge
is a member of P' if and only if x is the unique point on its edge
such that d(v.,x) = d(x,v.) for some two distinct vertices v, and v..
i J i J
Based on this result, Minieka suggested a rudimentary algorithm that

-16-
relies on solving a finite sequence of set covering problems. Using
the framework provided by Minieka, an exact algorithm was developed
by Garfinkel, Neebe, and Rao [38] for the unweighted case. The
algorithm uses the property that the p-radius is determined by one of
a finite number of elements, namely, one of the distances between any
vertex and any point in P. Call the points in P edge bottleneck
points and let d be the distance between vertex v^, and the jth
edge bottleneck point. Let and Z be a lower and upper bound on the
value of r^. Initially Z^ = 0, and Z is obtained by a trial solution.
Among all the distances d that fall within the interval [_Z,Z], one
of them will determine the value of r Pick one such distance, say
P
d with Z < d < Z, and let r = d be a specified radius. Now,
st st st
we want to know if we can cover all vertices of N within this critical
distance r by using only p points. If we cannot, then clearly r is
too small a radius for p points to cover all vertices. Hence we con
clude the p-radius r^ must be within the interval [r,Z], In this
case, the lower bound is shifted to r, and the procedure is repeated.
In the other case, we find a set X of p points that cover all vertices
within r, but it is doubtful if this point set is an absolute p-center.
Clearly, then, the value of r^ will be within the interval [_Z,f(X)].
Hence, the upper bound is shifted to f(X) for this case and the whole
procedure is repeated. Termination occurs whenever the lower and
upper bounds become equal. The r-cover part of this procedure is
solved by obtaining a feasible solution, if it exists, to a set cover
ing problem. Let A be a |v| by |p| matrix with entries a_^. equal one
if vertex v is within a distance r of the jth edge bottleneck point
and zero otherwise. Then, solving the system £ y^ £ p, Ay > 1,

-17-
y.e{0,l} will determine whether or not at most p points (in P) can
cover all vertices of N within a radius r. Computational experience
is reported and it is found that the procedure works better for larger
values of p, as in this case the initial upper bound Z is small, and
significant computational savings result in identifying those edge
bottleneck points whose distances fall within the interval [0,Z].
The weighted case on general networks was considered by Christofides
and Viola [15], and an approximate solution procedure was given. The
procedure finds a set X of p-points whose objective value f(X) is
within some e-neighborhood of the actual p-radius r The procedure
P
oi
obtains X by solving a sequence of r-cover problems with successively
increasing values of r. Termination occurs when the solution of an
r-cover problem generates p (or less) points the first time. In the
process, one also obtains approximate solutions for n-1, n-2,..., p+1
center problems. The solution of each r-cover problem is obtained in
two stages: First, all feasible solutions to the r-cover problem are
obtained by finding all regions on the network that can be reached by
a vertex within a radius of r. Then, among all these feasible solu
tions, those with minimum cardinality are found by solving a set
covering problem. To find all regions on N reachable by a vertex v_^,
one "penetrates" a distance of r/w_^ along all possible paths originating
at v_^. The procedure is repeated for each vertex and the intersections
of these penetrations are found. Each maximal intersection defines a
connected region all of whose points are reachable by a subset of
vertices within a radius r. The subset of the vertices is that which
defines the intersection. These regions jointly cover all vertices
of N, and it is possible that a subcollection of the collection of all

-18-
these regions may also jointly cover all vertices. Hence, to find a
minimum cardinality feasible solution, one needs to choose the minimum
number of regions that jointly cover V. This choice can be made by
defining a zero-one matrix A, so that an entry a^ of A is one if
vertex v^ is covered by region j, and zero otherwise. Solving the
set covering problem with matrix A will provide a solution to the
r-cover problem. Computational experience is reported and it is found
that the procedure works better for small values of p, as the set
covering part of the procedure takes a significant portion of the
total computational time.
An important result is due to Kariv and Hakimi [6.5] They showed
that the p-center problem on a general network is NP-complete. Kariv
and Hakimi also showed that the weighted case (as well as the un
weighted case) can be reduced to a computationally finite one. Based
on this finiteness property, they gave an algorithm whose order of
complexity is polynomial in |e|, but exponential in p. To show com
putational finiteness one argues as follows: For any absolute p-center
X = {x^,...,Xp}, there will be a subset of vertices covered by the
ith center x.. If N. is the (sub)network induced by V., then it can
xi 1
be shown that the absolute center x* of N. can replace x without in-
1 i i
creasing the value of the objective function, so that X* = {x*,...,x*}
1 p
is also an absolute p-center. Hence, one can restrict one's attention
to absolute p-centers every element of which is the absolute 1-center
of some subnetwork. The absolute 1-center of any subnetwork of N
will occur either at a vertex or at one of at most | E |n (n l)/2
"suspected" points. A suspected point on an edge is a point x such
that, for some two distinct vertices v_^ and v., x is a break point on

-19-
its edge of the function f..(-) = max[w.d(v.,.), w.d(v.,.)], and
ij i i J J
that the two linear pieces defining that breakpoint have slopes of
opposite signs. There can be at most n(n l)/2 suspected points on
each edge, resulting in a total of 0(|E|n^) suspected points on all
edges. If S is the set of all suspected points together with the set
of all vertices, then there is an absolute p-center contained in S.
The Kariv-Hakimi procedure selects p-1 points from S and determines
all the vertices covered jointly by these p-1 points. All uncovered
vertices are assigned to the pth center. Corresponding to each center,
the I-radius is determined (with respect to the subset of vertices
covered by that point) and the maximum of these 1-radii determines
the p-radius for that trial solution. The algorithm tries every
possible combination of p-1 points selected from S and chooses that
combination which minimizes the p-radius. The Kariv-Hakimi procedure
is the only exact algorithm available so far for finding an absolute
p-center of a vertex weighted general network.
A further result on the computational difficulty of the p-center
problem on a general network is given by Nemhauser and Sheu [92].
They showed that finding an approximate solution to the vertex restricted
or absolute p-center problem whose value is within 100% or 50%, respec
tively, of the optimal value is NP-hard (i.e., at least as hard as
any NP-complete problem).
Vertex restricted p-center problem. The vertex restricted p-
center problem is considered by Toregas, Swain, ReVelle, and Bergman
[109]. A solution procedure is given which relies on solving a sequence
of minimal set covering problems, each corresponding to a specified
radius r. Given a radius r, a 0-1 matrix A can be formed with n rows

-20-
and n columns, so that an entry a., is 1 if vertex v. is within a
ij J
distance r of v and 0 otherwise. If one solves a set covering
i
problem using the matrix A, the variables whose values are 1 in an
optimal solution determine a feasible solution to the vertex restricted
r-cover problem. The set covering problem is solved by relaxing the
integrality constraints. In the case of non-integer termination, a
single cut produced an integer solution in a large proportion of the
cases. Their computational experience indicates that non-integer
termination seldom occurs.
p-Center problem on tree networks and duality. In what follows,
we concentrate on the p-center problem on tree networks. First, we
define the "continuous" p-center problem. In the continuous p-center
problem, each point in T is a demand point as opposed only to vertices.
Weights are absent (or unity). For any XC T, f is defined by
f(X) = max{D(y,X): yeT} and the continuous p-center problem is to
find an X*d T such that
rp = f(X*) = min[f(X): |x| = p, X C T] .
Minieka [88] considered the continuous p-center problem on a
general network and showed that it can be reduced to a computationally
finite one.
Shier [100'] considered the continuous p-center problem on a tree
network and defined a dual "dispersion" problem. The dispersion
problem is to find p+1 points on T the nearest two of which are as
far apart as possible. More explicitly, let U be any finite point
set with |U| = p+1 and define h(U) by
h(U) = mindu^jUj): 1 < i < j < p+l} .

-21-
The dispersion problem is to find a U*C T such that
h(U*) =max{h(U): U C T, |u| = p+1} .
At optimality, Shier's duality result states that
rp-{h(U*)
for a tree network. The equality may not hold for general networks.
However, Shier showed that the objective value of the continuous p-
center problem is always bounded below by one-half the objective value
of the dispersion problem for any network.
Chandrasekaran and Tamir [14] observed that Shier's duality result
holds when one replaces T by any subset S of T. Chandrasekaran and
Daughety [12] described a procedure for solving the dispersion problem.
They first solve the related problem of locating the maximum number
of points on T such that any two of them are at least A distance
apart for a fixed (positive) A. This problem is solved by working
from "tips" of T to the "center" of T. The general scheme is to use
the algorithm for different values of A, until the number of points
found is p+1 and a slightly larger A generates p or less points.
A number of solution procedures have been given for the p-center
problem on tree networks. We now discuss these procedures.
Handler [57] considered the continuous p-center problem on a
tree network for the special case of p = 2 and obtained an 0(n)
algorithm. Handler first finds the absolute 1-center of T, say x*,
and splits the tree at x* obtaining two disjoint subtrees T^ and T^.
Finding the absolute 1-center of each T say x* and x*, determines
an absolute 2-center of T.

-22-
2
An algorithm of complexity 0(n logn) is described by Kariv and
Hakimi [65] for finding the absolute p-center-of a vertex weighted
tree network. They show that there are n(n l)/2 possible values
for r namely, the numbers a.. = w.w.d(v.,v.)/(w. + w.) for each
P 1JJ1JJ
combination of vertices v^, v The algorithm computes all these
numbers, arranges them in increasing order, and performs a binary
search on this list of numbers. The search relies on solving an r-
cover problem for each value of r chosen from the ordered list {a..},
ij
The search terminates when the smallest r in the list is found for
which the r-cover problem generates at most p points. The covering
part of the algorithm requires a computational effort of 0(n) for each
r, and a total effort of O(nlogn) for all values of r tried during the
binary search. Hence, the computational effort is determined by the
initial computation and ordering of the numbers ay> and is of
2
0(n logn).
A similar approach is used by Chandrasekaran and Daughety [12]
to solve the continuous p-center problem on a tree network. First,
they provided an 0(n) procedure for finding the minimum number of
points needed to cover every point of T within a given radius r.
Then, they provided a method to compute r A further refinement of
the method is given by Chandrasekaran and Tamir in [14]. They proved
that r^ is determined by one of the numbers d(t,t')/2k, where t and
t* are any two tip vertices and k is any integer between 1 and p. The
total computational effort for finding r and applying the covering
P
algorithm is of O((nlogp)^).
A somewhat different approach, which relies on finding a clique
on a related graph, is given by Chandrasekaran and Tamir [13]. They

-23-
define an intersection graph for a fixed value of r as follows:
has nodes corresponding to demand points v^,...,v^. Two nodes of G^
are connected by an arc if the corresponding demand points can be
jointly covered by a (single) common center within a radius of r.
Once Gr is formed, finding a "clique cover" of G^ solves the r-cover
problem. A clique cover of G^ is a minimum number of cliques in G^
such that every node is in at least one clique. The solution to the
clique cover problem in G^_ determines a solution to the r-cover problem.
The procedure is repeated for different values of r until a smallest
value of r is found for which the clique cover solution generates at
most p cliques. The computational complexity of the procedure is
polynomial. In particular, the computational effort for finding the
minimal clique cover of G^ is polynomial because G^ satisfies the
property that any circuit in G^_ with at least four arcs contains a
chord (i.e., an arc which connects two nodes of the circuit and is
not an element of the circuit). For chordal graphs, algorithms of
linear order have been developed (see [39], [97]) for finding a
minimal clique cover.
This completes the discussion of the p-center problem.
The p-median problem
The difference between the p-center and the p-median problem is
that the objective criterion is changed from minimax to minisum. More
specifically, define the function f (X) for any finite point set XCN
by
f (X) = l w D (v ,X) .
iel 1

-24-
The p-median problem is the following: Given a positive integer p,
find a set X* of p-points such that
f(X*) = min[f(X): |x| = p, X C N] .
Any set X* of p points that minimizes f is called an absolute p-
median of N. If each member of X is restricted to a vertex location,
the resulting problem is called a vertex restricted p-median problem.
Due to a result by Hakimi [47, 48] there exists an absolute p-median
entirely on the vertices of N. For this reason, the distinction be
tween the vertex restricted and unrestricted versions is insignificant.
Hence, we will take the term "p-median" to mean a solution to either
version of the problem. A 1-median is simply called a median.
The p-median problem arises naturally in locating plants/ware
houses to serve other plants/warehouses or market areas. The problem
is also motivated by ReVelle, Marks, and Liebman [96] as an example of
a public sector location model where vertices represent population
centers and facilities represent post offices, schools, public build
ings, and the like.
The 1-median problem. Hakimi [47] appears to be the first to
define an absolute median. Hakimi proved the important result that
there exists an absolute median at a vertex of the network. This
result reduced the search to a finite number of points. The median
can be found by summing each row of the weighted-distance matrix and
choosing the vertex whose row sum is the minimum. This procedure takes
3 2
0(n ) operations to compute the distance matrix followed by 0(n )
operations to find the median.

-25-
For tree networks, more efficient algorithms can be devised to
find a median. An 0(n) algorithm was given by Hua Lo-Keng and Others
[60] and independently by Goldman [42]. The algorithm reduces the
search to successively smaller subtrees until a median is found. At
each stage, one chooses an arbitrary tip vertex (a vertex of degree
one) of the current tree. If the (modified) weight of the selected
vertex is at least as large as half the sum of all weights, a median
is found. Otherwise, that tip vertex is eliminated from further con
sideration together with the edge incident to it and its weight is
added to the weight of the adjacent vertex. The procedure is repeated
with the new (reduced) tree. The algorithm does not require the com
putation of the distance matrix and uses only the incidence relation
ships and the weights.
Goldman's algorithm is based on a "localization theorem" proved
by Goldman and Witzgall [46]. The theorem provides sufficient condi
tions for a subset of N to contain a median. Given a compact subset
S of N, if S satisfies the two conditions (i), (ii) then it contains
at least one median. The conditions are (i) the set S must be a
"majority" set, meaning that the sum of the weights corresponding to
vertices in S must be at least as large as half the sum of all weights
(ii) the set S must be "gated" in the sense that there must exist a
unique point g in S such that for every s e S and t e N-S, it is true
that d(t,x) = d(t,g) + d(g,s). Goldman's algorithm in essence is a
repeated application of this theorem to a tree network. Goldman [43]
also proposed an "approximate" localization theorem which somewhat
relaxes the second condition and guarantees the existence of a point
in S that approximates an actual median.

-26-
A median of a tree is shown to be the same as a "centroid" of
the tree by Zelinka [120] for the unweighted case and by Kariv and
Hakimi [65] for the weighted case. To define a centroid, consider
the subtrees T-,...,T. obtained by removing vertex v from T. Let
1 X
w(T ) be the sum of the weights of the vertices in T^., and define
W(v
.) to be the maximum of w(T.) for 1 i i k., A vertex v which
i j J i t
minimizes W(v.) over all v. in V is said to be a centroid of T. The
i i
location of a centroid is independent of the distances and can be
found by using only the incidence relations. Goldman's earlier
algorithm in essence finds a centroid of T. The generalized algorithm
of Rosenthal, Pino, and Coulter [98] also finds a centroid of T by
making only two traversals of the vertices. All these algorithms are
of 0(n) and solve the 1-median problem without having to compute the
distance matrix.
We now consider some generalizations of the 1-median. Minieka
[88] defined the general absolute median of a network to be any point
on the network that minimizes the sum of (unweighted) distances from it
to the point on each edge that is most distant from it. Minieka showed
that the general absolute median can be strictly interior to an edge;
hence, the search cannot be confined solely to vertices of N.
Slater [103] gave another generalization of the 1-median problem.
In this generalization, each demand is a collection of vertices. The
problem is to find a vertex such that the sum of the distances from
that vertex to a nearest element of each collection is minimum.
Slater showed that the set of vertices that solve this problem forms
a connected path in T. For a general network, the problem can be
solved by constructing a matrix that specifies the distances from each vertex

-li
to a nearest element of each collection. Simply sum each row of this
matrix and choose the vertex whose row sum is minimum.
Frank considered a probabilistic version of the 1-median problem
in [34] where each weight is a random variable with a known distribu
tion. A number of bounds are obtained on the expected value of the
objective function as well as its variance. Some of these results
are generalized by Frank [35] to the case where the weights are jointly
distributed random variables.
We now concentrate on the p-median problem with p > 2.
p-Median of a network and vertex optimality. A significant
theoretical contribution is due to Hakimi [48]. Hakimi proved that
there exists an absolute p-median contained in V. Certain generaliza
tions of this result have been given in subsequent work.
Levy [80] proved that the (vertex-optimal) result holds when the
weights w^ are replaced by concave cost functions c^(*) of the distance
between v_^ and its nearest median.
Goldman [41] generalized the result to the case of a "two-stage"
commodity. More specifically, one distinguishes a vertex as being a
source or a destination. Let (v ,v.) be a source-destination pair,
S Cl
and let x^ and x_. be the nearest medians to v and v^, respectively.
Then the cost of transferring the commodity from source v to destina-
s
tion V, is the sum of three transport costs, namely, w .d(v ,x ) +
a sd s 1
r\j
wsdd(xi,x^.) + w*dd(x_. ,v) In general, if X = {x^...^} is a median
set, one does not know which median is the nearest to v or v,; hence,
s d
the cost associated with a source-destination pair (v ,v,) is
s d
given by
fsd(x) = min Kd^VV +"sdd(xi*xj) + Wsdd(xjVd)]

-28-
and the objective to be minimized is f(X) = J [f (X) : (v ,v,)eVxV],
u sd s d
Goldman showed that there exists an optimal X* contained in V, and
conjectured that the result holds for any multi-stage problem.
Hakimi and Maheshwari [49] proved a stronger version of Goldman's
conjecture. In this version, there are multiple commodities for each
source-destination pair, and each commodity goes through multiple
stages. Furthermore the cost of transport from one stage to the next
is a concave nondecreasing function of the distance. More specifically,
let M be the set of commodities to be transferred from source v to
sd s
destination v^, and let g(m) be the number of stages commodity meM^
is to go through. For a given location set X = {x.,...,x }, denote
1 P
by y^ = the location where the rth stage processing takes place.
The cost of transferring commodity m from source v to destination v,
s d
is given by C^Jd (v^yp ] + ] + ... + C^d (yg(n),vd) ],
where () is a concave nondecreasing function of the distance.
Denoting this quantity by f^^(Y), with Y C X, |y| = g(m), the minimum
cost of transfer for commodity m is given by f (X) = min[f (Y):
sdm sdm
Y C. X, |Y| = g(m)]. The cost of transferring all commodities from vg
to Vj is obtained by summing over all commodities, that is,
fgd(X) = J [fsdm(X): meM d]. The total cost of the system is obtained
by summing the cost f ^(*) over all source-destination pairs, that is,
f(X) = £ [f^W: (v ,Vj)eVxV]. Hakimi and Maheshevari proved that
there exists a minimum X* of f(X) contained in V.
Wendell and Hurter [111] considered a more general form of the
problem where the transportation cost functions are permitted to
differ from edge to edge. The transport cost on any edge is a non
decreasing concave function of the distance. They proved that it is

-29-
sufficient to consider the vertices of the network under such a cost
structure. Furthermore, they obtained the conditions under which it
is necessary for the solution to occur at the vertices. In particular,
they showed that nonvertex optimal locations can occur in any given
edge, only when transportation costs are linear with distance over
that edge and in that case, when and only when the slopes of these
linear cost functions are in a special relation. Hence, if at least
one cost function over some edge is nonlinear, then no interior point
of that edge can be in an optimal solution. If the same situation
holds for every edge, then a solution must necessarily occur at the
vertices of the network.
Solution approaches. Kariv and Hakimi [66] showed that the p-
median problem on a general network is at least as hard as NP-complete
problems. For the case of tree networks, however, algorithms of
polynomial complexity have been developed. Matula and Kolde [85]
3 2
suggested an 0(n p ) algorithm for finding the median of a tree net-
2 2
work. Kariv and Hakimi [66] proposed an 0(n p ) algorithm for the
same problem.
For general networks, a number of solution procedures have been
developed subsequently, all based on the vertex-optimality result.
Their common characteristic is that they all confine the search to
vertex locations. The solution procedures can be grouped in three
categories: mixed-integer programming approaches, branch-and-bound
techniques, and heuristics.
ReVelle and Swain [95] formulated the problem as a linear integer
program with 0,1 variables. The solution is obtained by applying the
primal simplex algorithm to the associated linear program. In case

-30-
of non-integer termination, a branch-and-bound scheme is recommended
to resolve the problem with integers. Their computational experience
indicates that non-integer termination seldom occurs. Toregas, Swain,
ReVelle, and Bergman [109] formulated a modified version of the problem
as a mixed integer program. The modification is the presence of upper
bounds on the distance between any vertex and its nearest facility.
This formulation makes use of a related but simpler problem. This
simpler problem is to minimize the number of facilities needed to cover
all vertices of N within a specified critical distance. This problem
is formulated as a set covering problem, and solved by ignoring the
integer requirements. In case of non-integer termination, a single cut
produced an integer solution in a large proportion of the cases. A
somewhat different approach to solve the relaxed linear program is
to use a decomposition scheme rather than applying the primal simplex
algorithm. Swain [105] used a Dantzig-Wolfe decomposition approach
to solve the associated linear program. Garfinkel, Neebe, and Rao
[37] independently developed a decomposition approach similar to
Swain's. In case of non-integer termination, they used group theoretics
and a dynamic programming recursion to obtain an integer solution.
A second approach taken is to solve the problem using a branch-
and-bound technique. Khumawala [68] applied a branch-and-bound method
of Land and Doig [77] type, to solve both the set covering problem and
the modified p-median problem formulated by Toregas et al. He showed
that the branch-and-bound approach is computationally efficient for
the former but not for the latter. Narula, ,0gbu, and Samuelson [91]
presented a branch-and-bound scheme which relies on obtaining the
bounds by solving the Lagrangian relaxation of the integer programming

-31-
formulation using a subgradient optimization method. Another branch-
and-bound method was developed by Jarvinen, Rajala, and Sinervo [62].
Their procedure looks for n-p vertices that do not belong to a p-
median. This method works better for larger values of p, since n-p
is smaller in this case reducing the number of possibilities. A
similar branch-and-bound procedure was given by El-Shaieb [24]. The
procedure is based on construction of a source set (i.e., p-median)
and a demand set. Starting with both sets empty, a location is added
to either set at each iteration. Whenever the number of elements in
a source set reaches p, or the number of elements in a demand set
reaches n-p, a feasible solution is obtained. An optimal solution is
eventually identified using the lower bounds.
A third approach taken is to use heuristics. A number of
heuristics have been developed by Maranzana [84], Teitz and Bart [107],
and Khumawala [69, 70].
For a discussion of a number of the solution approaches from a
computational standpoint, the reader is referred to Hillsman and Rush-
ton [59], and Khumawala, Neebe, and Dannenbring [71].
Stochastic networks and vertex-optimality. A number of pro
babilistic versions of the p-median problem have been considered in
the literature. Mirchandani and Odoni [89, 90] extended Hakimis
vertex optimality result to the case of a stochastic network whose
edge lengths are random variables. Berman and Larson [2] considered
a stochastic network where the availability of servers (centers) is a
random variable. They showed that under suitable conditions there
exists at least one optimal set of locations on the vertices of such
a network.
This completes the discussion of the p-median problem.

-32-
The distance constraints problem
The distance constraints problem involves locating new facilities
on a network so that they are within specified distances of existing
facilities as well as within specified distances of one another. The
distance constraints arise naturally in a locational context if one
wishes to require that a service facility be within a specified time
(distance) of any point in the region it serves. Alternatively, in a
military context, one may want to locate a number of units in such a
way that units are neither too far from their supply bases, nor too
far from one another, in order that one unit may reinforce another if
necessary.
To state the problem, let N be a network with the vertex set
V = iv.,...,v ). Denote by X = (x,,...x ) any location vector in Nm,
the m-fold Cartesian product of N by itself. Define the sets and
I as follows: I = (j,k): 1 < j < k < m>, I = {(i,j): 1 i S m,
1 j n}. Here, the pairs (j,k) and (i,j) are assumed to be un
ordered. Let I and I be two non-empty subsets of I' and I,
d L B G
respectively, and suppose we are given nonnegative finite numbers b
jk
for each (j,k)el and c.. for each (i,j)el_.
a ij C
The problem of interest is to find a location vector XeNm, if it
exists, such that the constraints (1.3.3) are satisfied.
d(x ,v ) < c (i,j)el
1 J 1J ^
(1.3.3)
d(Xj,xk)-bjk (j,k)cIB
Any vector XeNm satisfying (1.3.3) is called a feasible location
vector. The distance constraints are said to be consistent if there exists
at least one feasible location vector XeNm.

-33-
Goldman and Dearing [45] provide a conceptual discussion of, and
a motivation for, considering such problems. The distance constraints
are formally defined by Dearing, Francis, and Lowe [22] on a network.
It was established in [22] that, in a well defined sense, the distance
constraints define convex sets under the assumption that the under
lying network is a tree. Furthermore, the distance constraints always
define convex sets if and only if the network is a tree.
Based on the results obtained in [22], Francis, Lowe, and Ratliff
[32] considered the distance constraints on tree networks in more
detail. They established the necessary and sufficient conditions for
distance constraints to be consistent, and also devised algorithms
that find a feasible location vector whenever one exists. In what
follows we briefly discuss the results obtained in [32].
Distance constraints for a single new facility. For the case of
a single facility, Francis et al. showed that there exists a feasible
point xeT satisfying d(x,v.) < c. for iel if and only if the in-
i 1
equalities d(v.,v, ) c. + c, are all satisfied for 1 1 1 < k < n.
j k j k j -
An equivalent statement of the single facility distance constraints
can be given in terms of "neighborhoods" around v_^ of radii c^,. De
fine the neighborhood N(u,r) around a point ueT of radius r to be the
set of all points xeT for which d(u,x) < r. Then, a point x satisfies
the constraints d(v^,x) < c^> iel, if and only if x is in each neigh
borhood N(v.,c ), iel,if and only if x is in the intersection
n 1
n N(v.,c.). It follows then that the single facility distance con-
i=l 1 1
straints d(x,v.) Sc., iel,are consistent if and only if d(v.,v.) S
ii j k
Cj + ck for 1 j < k 5 n if and only if each pairwise intersection
N
-34-
property, a "sequential intersection procedure" was developed that
n
determines the composite neighborhood N(a,r) = O N(v.,c.), with
i=l 1
unique center a and radius r, by intersecting the neighborhoods
N(v ,c ) one at a time in an arbitrary order. The procedure can be
implemented in 0(n) operations. The composite neighborhood N(a,r)
contains all alternate feasible points when the constraints are con
sistent, and N(a,r) is always a convex compact subset of the tree
network. A result was also given by Francis et al. that provides a
sensitivity analysis on the constraints with no additional computa
tional effort. Supposing that the distance constraints are consistent
with the original upper bounds c^, consider an e-perturbation of the
upper bounds, i.e., for some e > 0 define the new upper bounds to be
c^-e, iel. If N(a,r) is the composite neighborhood corresponding to
the original upper bounds, then it can be shown that for any e with
0 e ~ r, the e-perturbed constraints remain consistent and the set
of feasible points to the e-perturbed system is given directly by
N(a,r-e).
Distance constraints for the multi-facility case. For the multi
facility case, the necessary and sufficient conditions for the con
sistency of distance constraints are given in terms of n(n l)/2
inequalities called "separation conditions." The separation condi
tions are defined by means of an auxiliary graph constructed by using
the sets I and I Let G be the graph with nodes N., 1 5 i < m,
ij v 1
corresponding to new facilities,and nodes E ^, 1 < j < n,corresponding
to existing facilities. The arc set A of G contains (N ,E ) if
i j
(i,j)cl-, and (N.,N ) if (j,k)el. The arc length of (N,,E.) is c_,.
G j k B i j ij
and of (N.,N^) is b.^. Under the (reasonable) assumption that G is

-35-
connected, denote by L(E.,E ) the length of a shortest path connecting
J k
nodes E. and E. for 1< j 3 k
distance constraints are consistent on a tree network if and only if
the inequalities (E^.E^) £ d(vj,v]P are satisfied for 1 < j < k < n.
These inequalities are called the separation conditions. The proof
of the consistency of the distance constraints implying the satisfac
tion of the separation conditions uses only the triangle inequality
and hence is applicable to all networks. The reverse implication
always holds for tree networks, but may fail to hold for general net
works. The proof of the reverse implication is constructive and
actually finds a feasible location vector under the assumption that
the separation conditions are satisfied. The method that constructs
such a feasible location vector is termed the "Sequential Location
Procedure" in [32]. The method can best be described with the aid of
a physical model. One may imagine that the tree is represented by
appropriately inscribing straight line segments on a board such that
each segment represents an edge. At vertex v_^, strings of length c
are fastened for each new facility j such that (i,j)el A tip vertex
Li
is chosen arbitrarily and all strings fastened at that vertex are
pulled tight towards the adjacent vertex. If all strings reach the
adjacent vertex, they are simply engaged there with their loose ends
free to be pulled tight in some future iteration. Also the tip vertex
together with the edge incident to it is removed from the model. The
procedure is repeated with the resulting tree. In the other case,
not all the strings reach the adjacent vertex when pulled tight. Among
those which do not reach the adjacent vertex one which is shortest is
selected, and the end point of this string determines the location of

-36-
the new facility it is associated with. All the strings pulled tight
from the chosen tip are engaged at this new facility location. The
feasibility of this location is checked with respect to all existing
facilities and all other new facilities already placed on T. If the
feasibility check is passed, new strings are fastened at this location
associated with that new facility and other unplaced new facilities for
which the distances are of concern. The procedure continues, treating
each placed new facility like an existing facility, until, either all
facilities are placed, or the current tree reduces to a point, in
which case, all remaining new facilities are placed at that point.
If the separation conditions hold, the procedure always finds a
feasible location vector. The algorithm is of 0(m(m+n)) and is conjectured
to be a best order algorithm in [33], for determining the con
sistency of the distance constraints.
Extensions of the results obtained in [32] are given by Francis,
Lowe, and Tansel [33]. These extensions focus on the analysis of
binding separation conditions which in turn determine the "uniquely"
located new facilities. A separation condition that holds at equality
is said to be a binding separation condition. If Z,(E.,E.) = d(v.,v,)
3 k J k
is a binding separation condition, then any shortest path P(E.,E,) in
J k
the auxiliary graph G is said to be a tight path. New facility i is
said to be uniquely located at point if in every feasible solution X to
the distance constraints the location x. is the same. It was shown
i
in [33] that a new facility i is uniquely located if and only if node
N_^ lies on at least one tight path. As an immediate consequence of
this property the distance constraints has a unique feasible solu
tion if and only if each N_^, 1 i < m, lies on at least one tight path

-37-
in the graph G. Furthermore, if some path P(E.,E. ) is a tight path,
J k
then the nodes representing facilities in the path occur with the same
ordering and spacing in the path as do the locations representing the
facilities in the path P(v.,v.) on T. This result enables one to
J k
locate the new facilities that appear in a tight path immediately,
without having to use the Sequential Location Procedure.
A multifacility minimax application of the distance constraints
is given in [32, 33] and a multiobjective application is given in [33].
These two applications will be discussed subsequently.
m-Center problem with mutual communication
Let N be a network with vertex set V = {v,,...,v } and edge set
1 n
E. Suppose the sets I and In are given with I,, C {(j,k): 1 j < k m}
BO B
and I C {(i,j): 1 < i S m, 1 < j < n}. We assume that we are given
positive weights v ^ for each (j,k)elg and w for each (i,j)el^. For
each location vector XeN, define the functions f (X), f_(X), and
B 0
f(X) as follows:
fB(X) = max[Vjkd(x^.,xk) : (j,k)eIB] ,
fc(X) = maxtw^dCx^v ) : (i,j)elc] ,
f(X) =max[fB(X), fc(X)] .
The m-center problem with mutual communication is the following:
Find a location vector X*eNm such that
Z* = f(X*) = min[f(X): XeN] .

-38-
The problem differs from the p-center problem in two respects:
(i) the distance between any vertex v and any new facility x_^ may be
of concern as opposed only to the distance between v and the nearest
new facility to v.; (ii) certain distances between new facilities are
of concern, as opposed to the absence of interactions between new
facilities. For the case of a single new facility the two problems
coincide.
In this problem, the new facilities may be thought to fulfill a
supporting task to other new facilities as well as servicing those
existing facilities that are a priori assigned to them.
Certain planar cases of the multifacility minimax problem have
been studied by Dearing and Francis [20]> Elzinga, Hearn, and Randolph
[25], Wendell and Peterson [113],. and Francis [28l*
The problem on a network is defined by Dearing, Francis, and Lowe
[22] in the presence of distance constraints. It is established in
[22] that the function f is a convex function on a tree network. The
existence of a solution is guaranteed due to compactness and con
tinuity considerations. Furthermore, it is shown that it suffices to
consider only new facility locations in the convex hull of the existing
facility locations in order to solve the problem.
The problem on a general network was shown to be NP-hard by Kolen
[72 ]. For the case of a tree network, the problem is solved by
Francis, Lowe, and Ratliff [32] by using an equivalent formulation in
terms of distance constraints (with variable right hand sides). The
solution procedure finds Z* first, by using the separation conditions.
Then an optimal feasible location vector X* is constructed by using the
Sequential Location Procedure described in [32]* To find Z* an

-39-
auxiliary graph G is formed with nodes N ,...,Nm,E^,...,E Graph G
contains arcs (N^,E^) with lengths l/w corresponding to pairs
(i,j)el and arcs (N.,N ) with length 1/v corresponding to pairs
G J K J K
(j,k)eIB. It is assumed that G is connected, for otherwise the problem
decomposes into subproblems. For each pair of existing facility nodes
Ej, E^, define L(Ej,E^) to be the length of a shortest path in G
connecting E. and E,. Francis et al. showed that Z* is given by
J k
max{d(v. ,v, )/£(E. ,E, ) : 1 j < k n}. The distances d(v.,v.) can be
J K. J K J k
2
computed in 0(n ) operations for a tree network (see [23]), and the
shortest path lengths Zr(E.,E,) are readily computable in 0(n ) opera-
1 k
tions. When Z* is computed, the Sequential Location Procedure de
scribed in [32] can be applied in 0(m(n+m)) operations to find a loca
tion vector X* that solves the problem.
m-Median problem with mutual communication
Define the functions g g and g by the following expressions:
D L
For each XeNm
gB(X) = l [v^dCx^Xjp: (j,k)eIB] ,
grQO = l [w d(x ,v ): (i,j)el ] ,
XJ 1 J o
g(X) = gB(X) + gc(X) .
The m-median problem with mutual communication is the following:
Find a location vector X* in Nm such that
Z* = g(X*) = min[g(X): XeNm] .

-40-
The problem differs from the p-median problem in two respects:
(i) the distance between any vertex and any new facility may be of
concern as opposed only to the distance between a vertex and the near
est new facility to it; (ii) certain distances between new facilities
are of concern as opposed to the absence of interactions between new
facilities in the p-median problem. For the case of a single new
facility, the two problems are identical.
Planar cases of the problem using rectilinear or Euclidean dis
tances have received considerable attention and efficient solution
procedures have been developed. A thorough discussion of these prob
lems is given in the book by Francis and White [31]. Other references
on planar problems are Cabot, Francis, and Stary [6], Bindschedler and
Moore [3], Francis [27], Eyster, White, and Wierville [26], Pritsker
and Ghare [94], Wesolowsky and Love [115, 116], and Picard and Ratliff
[93].
The problem on a network is defined by Dearing, Francis, and Lowe
[22] in the presence of distance constraints. It was established in
[22] that the problem is a convex optimization problem for all data
choices if and only if the network is a tree. For the case of a general
network, it is known that there exists an optimal solution on the
vertices of N. This result and certain generalizations of it have
been given by Goldman [41 ], Levy [80], Hakimiand Maheshwari [49], and
Wendell and Hurter [ill]. These references are already discussed
under the p-median problem. The problem was shown to be NP-hard by
Kolen [72 ] on a general network, and no solution procedures have
been developed yet.

-41-
For the case of a tree network, the m-median problem with mutual communi
cation is solved by Dearing and Langford [21] and by Picard and
Ratliff [93].
The approach used by Dearing and Langford is to embed the tree T
into the Euclidean space R^, for some p, so that the distance between
any two points on the tree is equal to the rectilinear distance between
the corresponding points in R^. The problem in R^ with rectilinear
distances decomposes into p subproblems, each of which can be solved
by using known techniques given in Francis and White [31 ], or, perhaps
more efficiently, by applying the network flow procedure discussed in
Cabot, Francis, and Stary [6]. For reducing the computational effort,
the embedding procedure is carried out with respect to a minimal path
decomposition of T into p edge disjoint paths (each edge is in one and
only one path). Each path in a minimal path decomposition corresponds
to a dimension in R*5.
The approach taken by Picard and Ratliff in [93] takes advantage
of the vertex-optimality condition and determines an optimal solution
(on the vertices of T) by solving a sequence of at most n-1 minimum
cut problems, each on a graph containing at most m+2 nodes. The
method is based on a result that an optimal location vector can be
found independently of the edge lengths, by using only the incidence
relations between vertices and the weights. In this respect, the pro
cedure is in the same spirit as Goldman's algorithm for finding a
median of a tree. Each cut problem corresponds to an edge of the
tree. To be more explicit, the removal of all interior points of an
edge e leaves two disconnected components, T^ = T^(e) and T^ = 12(e).
Corresponding to edge e, a graph G = G(e) is constructed having nodes

-42-
1 through m corresponding to new facilities, a source s and a sink t.
Graph G contains arcs (s,i) and (i,t) for 1 < i < m and arcs (j,k) for
each pair (j,k)eIB. The capacity of arc (j ,k) is the weight v^. The
capacity of arc (s,i) is given by J [w : v eT., (i,r)el ], and the
ir v i o
capacity of arc (i,t) is given by J [w. : v eT, (ijqjel.,]. If
xq q u
(Q,Q) is a minimum capacity s-t cut of G, with seQ, teQ, then all new
facility locations x^ for which the corresponding node i is in Q are
in T^ in an optimal solution. Similarly, all x_. for which the node j
is in Q are in T^ in an optimal solution. The procedure is a repeated
application of this minimum cut problem with respect to each edge,
until an optimal vertex location is determined for each x^. During
the process, each x^ whose location is determined is treated like an
existing facility. The method is described originally for the
analogous rectilinear distance problem on the plane, which, in turn,
decomposes into two subproblems, each on a line.
Multi-objective location problems on networks
Multi-objective optimization problems, sometimes known as vector
optimization problems, involve decision making under two or more
criteria. More explicitly, a set (finite or infinite) S of alterna
tives is specified and n (possibly non-commensurable) objective func
tions are to be minimized over S. Let f,,...,f be n numerical func-
1 n
tions defined on S, and define f(x) = (f,(x),...,f (x)) for all xeS.
1 n
The multi-objective optimization problem (VMP) is the following:
V-min f(x)
xeS
In general, the minima of the functions f_,...,f do not coincide.
1 n
In order for the minimization to be meaningful, one needs to introduce

-43-
the concept of "efficient solutions." A point x in S is said to be
efficient if there does not exist a point y in S such that f^(y) < f_^(x)
for 1 i < n and f^(y) < f^Cx) for at least one index k. One is
interested in finding and characterizing the set of efficient solu
tions to (VMP)An efficient point is sometimes known as an undominated
point. A point which is not efficient is said to be dominated.
Kuhn and Tucker [76] and Koopmans [74] are among the first to
introduce the concept of efficiency. Geoffrion [40] extendd the con
cept to "properly efficient" points and provided a comprehensive
theoretical framework for subsequent research. Necessary and suf
ficient conditions for efficient points to be properly efficient are
given by Wendell and Lee [112]. Some of the later contributions are
due to Yu [117], Yu and Zeleny [118, 119], Bitran and Magnanti [4],
Wendell [110], and Bergstresser, Chames, and Yu [l ] We note that
there are other approaches to multicriteria decision making, such as
goal programming, multi-attribute utility theory, construction of
outranking relations, and interactive programming techniques. For
general information on multicriteria decision making, the reader is
referred to Roy [99], Starr and Zeleny [104], Cochrane and Zeleny
[16], Keeney and Raiffa [67], and Thiriez and Zionts [l08]- A survey
of multicriteria decision making is given by Chalmet [7].
Multi-objective location problems (on the plane or on networks)
have begun receiving attention only recently. Kuhn [75] appears to
be the first to consider a multi-objective location problem on the
plane. Kuhn considered the problem of minimizing the vector of
Euclidean distances from a variable point to a set of fixed points on
the plane, and showed that the set of efficient solutions is the convex

-44-
hull of the fixed points. Wendell, Hurter, and Lowe [114] considered
the same problem with rectilinear distances and provided algorithms of
2 3
0(n ) and 0(n ) for generating efficient points. A most efficient
algorithm of O(nlogn) was developed by Chalmet and Francis [8] for
the same problem. McGinnis and White [83] considered the problem of
minimizing the sum of and the maximum of weighted rectilinear distances
from a variable point to a set of fixed points on the plane and formu
lated the problem as a parametric linear program for which known solu
tion techniques exist. Juel [64 ] considered the same problem for
the case of multiple new facilities and gave an equivalent parametric
linear program. Chalmet, Francis, and Lawrence [ 9 10 ] considered
two variants of an efficient design problem, where the location
variable (a design) is a planar region of specified positive area
but of unknown shape.
A few papers have been produced on multi-objective location
problems on networks. In what follows we discuss these problems.
The cent-dian problem. The single facility "cent-dian" problem
involves the sum of and maximum of weighted distances from a new
facility to a set of existing facilities at vertices of N. To define
the problem, let w^ and h_^ be two positive weights associated with
vertex v iel = {1,...,n}. For each point xeN define:
m(x) = J {w_jd(v^,x): iel} ,
c(x) = max[h^d(v^jx): iel]
f(x) = (m(x), c(x))

-45-
The problem of interest is to find all efficient points with
respect to f(x).
Halpem [52] is the first to consider this problem. Halpern
formulated the problem in a slightly different manner by considering
a convex combination of m(x) and c(x). For any fixed X, 0 < X 1,
define f(X,x) and f*(X) by
f(X,x) = Xm(x) + (1 X) c(x) for xeN }
f*(X) = min[f(X,x): xeN] (1.3.4)
In Halpem's terminology, the function f(X,x) is called a cent-dian
function and any point x* = x*(X) that solves (1.3.4) is called a
cent-dian point.
In [52] Halpem considered this problem on a tree network with
weights h all equal to unity. Defining x^ and x^ to be the (vertex)
median and the absolute center of T, respectively, Halpem proved that
for any given X, the cent-dian x*(X) is located at either x^ or on
one of the vertices located on the path P(x ,x ). This theorem pro-
m c
vides the basis for a simple and efficient algorithm to locate the
cent-dian by inspecting the vertices on P(x ,x ). Further, Halpern
m c
showed that, if the absolute center x is known, then the cent-dian
c
can be found by determining the median of a tree T' that is identical
to T except that T' contains an additional vertex v = x with the
n+1 c
associated weight w = X 1.
n+1
Handler [56] formulated the same problem on a tree network in a
slightly different manner by using the median function as a constraint.
In Handler's formulation one is interested in solving the problem

-46-
P for each given a, where P is defined as follows:
a b a
e(a) = min[c(x): m(x) £ a, xeT]
Efficient solutions are obtained by parameterizing on a. Handler's
results closely parallel Halpem's.
The problem on a general network is studied by Halpern [54].
using the convex combination approach. Halpern showed that the problem
is a computationally finite one. Computational finiteness follows
from the result that f(X,x) is a continuous, piecewise linear function
of x on each edge and attains its minimum at one of a finite number of
points. Defining Q(e) to be the union of the end points of edge e
with the set of local minima of c(x) on e, the minimum of f(X,x) over
all x on edge e is a member of Q(e) for any given X, 0 < X < 1. De
fining Q = U {Q(e) : eeE}, it follows that the cent-dian x*(X) is con
tained in Q for any X. Further, Halpern showed that the function
f*(X) = min[f(X,x): xeN] is a continuous, piecewise linear, concave
function of X for 0 < X < 1. Based on these results, Halpern provided
an algorithm which constructs f*(X) and identifies x*(X) for
0 X < 1. To construct f*(X), the algorithm inspects each edge one
at a time and computes the set Q(e), unless a simple test indicates
that edge e cannot contain any cent-dian for any X. An upper bound
on f*(X) is carried through and improved, whenever possible, by
examining the members of Q(e).
Cent-dian problem and duality. In [53], Halpern studied the cent-
dian problem on a general network from a different angle and obtained
a duality relationship. Using an approach similar to Handler's median
constrained problem, Halpern defined two problems, a median constrained

-47-
and a center constrained one. More specifically, for real A and y
define the functions m*(A) and c*(y) as follows:
m*(A) = min[m(x): c(x) < A]
(1.3.5)
c*(y) = min[c(x): m(x) y]
(1.3.6)
In general for some values of A (y), the constraint c(x) £ A
(m(x) £ y) may not admit any feasible solution. However, real inter
vals C and M can be defined so that for any AeC and for any yeM, the
constraints in (1.3.5) and (1.3.6) admit a feasible point. To define
C, let Q be the set of all minima to min[c(x): xeN], and let 2 -be
c m
the set of all minima to min[m(x): xeN]. Let x be a point in that
c
minimizes the value of m(x) over all x in Similarly, let y be a
point in £! that minimizes the value of c (y) over all y in ft Then
m J J m
C and M are defined as follows:
C = [c(x), c(y)]
M = [m(y) m(x) ]
With these definitions Halpem's duality theorem can be stated
as follows:
a) Given any yeM, with A = c*(y), we have c*(m*(A)) = A.
b) Given any AeC, with y = m*(A), we have m*(c*(y)) = y.
For a tree network, the functions m* and c* are 1-1 and onto.
It follows from the duality theorem that the function m* and c* are
inverses of each other for a tree network. For a general network,
the functions m*, c* need not be onto, i.e., the image of the domain

-48-
may only be a proper subset of the range. Hence, the inverse property
holds only for some members of C and M for a general network.
Now, we consider a more general multi-objective problem due to
Lowe [82]. The problem involves a single facility to be located on a
tree network with respect to m convex objective functions.
Multi-objective convex location problem (on a tree). Let T be
a tree network and let f,,...,f be m convex continuous bounded func-
1 m
tions each of which is defined on T. In general, not all points in T
may be feasible with respect to f_^. Let be a convex compact subset
of T which contains all feasible points x with respect to the ith
optimizer. The set Q, may be defined by specifying its extreme points,
or by means of distance constraints, or by other means. We assume
m
that Q. is known or computable. Define Q = D Q. and assume that Q
1 i=l 1
is nonempty. The problem of interest is to find all efficient points
in Q with respect to the vector minimization problem defined below:
V-min[f(x): xeQ C T]
where,
f(x) = (f1(x),...,f (x)) for all xeT
i m
We note that Q is a convex compact subset of T as it is the
intersection of m convex compact subsets of T. For a formal dis
cussion of convexity on a network, the reader is referred to Dearing,
Francis, and Lowe [22]. Loosely speaking, Q a convex subset of T,
means Q is connected or that the (shortest, unique) path connecting
any two points in Q is contained in Q.
Lowe makes no assumptions on the specific forms of the objective
functions. Under the convexity assumptions, Lowe proves that a convex

-49-
compact subset T* of T can be identified that contains all efficient
points. To identify T*, define R* to be the set of all minima to the
unconstrained problem min[f,(x): x£Tl. If R* intersects the feasible
1 1
set Q, define S* to be this intersection. Otherwise, S* is the unique
i i
closest point in Q to R*. Having defined each S*, 1 < i m, if their
intersection is non-empty, then the set of all efficient points is
given by T* = H{S*: 1 i < m}. If this intersection is empty, then
T* is the smallest compact convex subtree that intersects each S*. It
can be shown that each R*, S* is convex, compact, and that T* is a
li
convex compact subset of T. Lowe's theorem assumes a knowledge of
set of minima to each f as well as a knowledge of and hence Q.
We note that the functions c(x) and m(x) in the cent-dian problem are
both convex on T. Hence, Halpem's results can be obtained by apply
ing Lowe's theorem.
Now, we consider a multi-objective problem which involves multiple
new facilities to be located on a tree network so that the distance
between each specified pair of new and existing facilities, and each
specified pair of new facilities is, roughly speaking, "as small as
possible." The problem is defined by Francis, Lowe, and Tansel [33]
as a sequel to the distance constraints problem, and solved by making
use of the separation conditions. Here, we call the problem, the
"multifacility vector minimization problem."
The multifacility vector minimization problem (on a tree network).
Let T be a tree network and let I I be given nonempty sets with
Iq c (ij): 1 £ i S m, 1 < j < n} and IB C {(j,k): 1 5 j < k < m}.
The problem of interest is to locate m new facilities on T at points
x ,...,x so that each distance d(x ,v.) (i,j)el and d(x.,x.) (j.k^I^
kjlcB

-50-
is "as small as possible." More specifically, we wish to find all
efficient location vectors X = (x,,...,x ) in Tm with respect to the
i m
vector minimization problem
V-min[D(X): XeT]
where D(X) is the vector of distances d(x^,v^) (i,j)el^, and d(Xj,x^)
(i,k)el The vector is formed by assuming any convenient ordering
D
of the members of the sets I_ and I.
L> D
Francis, Lowe, and Tansel [33] characterized efficient points by
making use of distance constraints. By definition, a location vector
Z in T is efficient if an only if there does not exist a location
vector X in Tm such that D(X) < D(Z) and D(X) D(Z). Given a location
vector Z, let b., = d(z.,z,) for (jjk^I,, and c.. = d(z.,v.) for
jk 3 k J B xj 1 3
(i,j)el^,, and define the distance constraints (DC) of interest by
d(xi*vj) cij (i,j)elc
d(xj"xk)iV
We note that DC is always consistent, as Z is always feasible
to DC, and hence the separation conditions are always satisfied. The
separation conditions for DC are defined by constructing a graph G
with nodes 1 S j £ m, corresponding to new facilities and nodes
E^, 1 i < n, corresponding to existing facilities. For each
(i,j)elr,, the arc (N.,E.) is in G with length c.., and for each
c 1 1 ij
(j,k)el the arc (N.,N, ) is in G with length b.. We recall that a
B j k jk
point x_^ is uniquely located in every feasible solution to DC if and
only if the corresponding node N is in at least one tight path in G,

-51-
where a path of G joining any two existing facility nodes E and E
s t
is said to be tight if the length of the path is equal to the distance
between the vertices v and v in T corresponding to nodes E and E ,
S t 1 w St
respectively. For any given location vector Z, denote by A^(Z) the
collection of locations of uniquely located facilities whose nodes are
adjacent to N_^ in G. Let H[A^(Z)] be the convex hull of A^(Z), i.e.,
the smallest connected subtree containing all points in A^(Z).
With these definitions, it was proven in [33] that the following
conditions are equivalent:
(i)Z is efficient.
(ii)Z is the unique solution to DC.
(iii)Each is in at least one tight path in G.
(iv)Each Z. is contained in H[A.(Z)], 1 < i £ m.
1 1
This completes the discussion of multi-objective location problems
on networks.
Path Location Problems
Here, we consider three versions of a path location problem posed
by Slater [102]. To define the problems, let P denote any path con
necting any two vertices in a network N. For any vertex veV and any
path P, define the distance D(v,P) to be the distance from v to a
nearest vertex in P. Also define the branch weight bw(P) of a path
P to be the maximum number of vertices in any component of N-P. The
three versions of the problem are the following:
min l D(v,P) (1.3.7)
P C N veV

-52-
min max D(v,P)
(1.3.8)
P C N veV
min bw(P)
(1.3.9)
P C N
In Slater's terminology, any path P* that solves (1.3.7) is called
a core of N. Among all paths that solve (1.3.8), one with the fewest
vertices is called a path center of N. Similarly, among all the paths
that solve (1.3.9), one with the fewest vertices is called a spine
of N.
Slater obtained a number of properties of these problems for
tree networks. In particular, Slater showed that the path center of
T is unique and contains the vertex center of T, and that the spine of
T is unique and contains the centroid (equivalently, the vertex
median) of T. We recall that a centroid of T is any vertex v that
minimizes the maximum number of vertices in any component of T-v.
Also, Slater proposed two algorithms of linear order for determining
the path center and the spine of T.

CHAPTER 2
DUALITY AND THE NONLINEAR p-CENTER PROBLEM AND COVERING
PROBLEM ON A TREE NETWORK
2.1 Introduction and Related Work
We consider the problem of locating p new facilities on a tree
network with respect to n existing facilities at known locations so as
to minimize the maximum "loss." The problem is an extension of the
linear p-center problem to the nonlinear case. We assume a strictly
increasing, continuous "loss" function is associated with each of a
finite number of demand points (existing facilities) whose argument
is the distance between the corresponding existing facility and its
nearest new facility. Our formulation permits the use of quite general
loss functions provided that they are continuous and strictly increas
ing with the travel distance. The term "loss" is used generically
and may refer to any form of inconvenience such as cost, disutility
of service, travel time, etc.
In locating emergency service facilities, the disutility due to
"late" service may be too great beyond a certain "threshold" response
time. Such sharp changes in the disutility of service can be re
flected into the model by using nonlinear functions. Hurter and
Schaefer [61 ] justify and use such functions in a fire setting. As
pointed out by Dearing [183 a study by Kolesar et al. [73 ] revealed
that the travel time for fire trucks can be approximated by a particular
continuous, nonlinear, increasing function of the distance.
-5V

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The literature on the p-center problem is discussed in detail
in Chapter 1. Here, we give a brief review of the more closely re
lated work. Except for p = 1, we know of no literature on the non
linear p-center problem. For p = 1, the only references we are aware
of which deal with the nonlinear case are Dearing [18] and Francis
[29]. Both authors showed that the minimax loss with respect to any
two existing facilities is a lower bound on the maximum loss with
respect to all existing facilities, and that the largest of the lower
bounds determines the minimax loss to all existing facilities on a
tree network. This result is an instance of the duality result we
will present in this chapter.
The linear (weighted or unweighted) p-center problem is shown to
be NP-complete on a general network by Kariv and Hakimi [65], and by
Nemhauser and Sheu [92].
The linear 1-center problem on a tree network is well solved (see
Goldman [44], Halfin [51], Lin [81], and Dearing and Francis [19]).
For p > 1, the linear p-center problem on tree networks is considered
by various authors. Handler [57] provided an 0(n) algorithm for
finding the 2-center of a tree for the unweighted case. Kariv and
2
Hakimi [65] gave an 0(n logn) algorithm for tree networks which relies
on solving a sequence of covering problems for the weighted case with
p > 1. A similar procedure for the unweighted continuous p-center
problem on a tree network is given by Chandrasekaran and Daughety
[12]. A vertex-restricted version of the problem is solved by
Chandrasekaran and Tamir [13], and relies on solving a sequence of
clique covering problems on a related intersection graph.

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The first duality relationship involving tree network location
problems can be found in Meir and Moon [ 86 ]. Cockayne, Hedetniemi,
and Slater [ 17 ] obtained a more general version of the result given
in [ 86 ]. The results in [ 86 ] and [ 17 ] closely parallel our duality
result for the covering problem and its dual. Shier [100] discovered
a "dispersion" problem which is dual to the continuous unweighted
p-center problem. The dispersion problem of Shier is to choose p+1
points in the tree network the nearest two of which are as far apart
as possible. Chandrasekaran and Tamir [14] observed that Shier's
duality holds when the problems are defined with respect to a subset
of the tree. For the case where this subset is a finite collection
of demand points, their result is an instance of the duality relation
ship we will present in this chapter, as applied to the unweighted
linear case.
At this point we give a brief overview of the chapter. In Sec
tion 2, we define the (nonlinear) p-center problem and a dual "dis
persion" problem. We state and prove a weak duality theorem applicable
to all networks, and state a strong duality theorem applicable to
tree networks. In Section 3 we give a physical interpretation
of the dual dispersion problem. In Section 4 we study a covering
problem and present an algorithm, COVER, for solving it. The covering
algorithm provides the basis of our solution procedure to the p-center
problem as well as the dual dispersion problem and yields a construc
tive approach for proving the strong duality theorem. In Section 5 we
present an algorithm, OPTKLIQUE, which provides a constructive proof
of the strong duality theorem, while solving the dual problem. Addi
tional results for the covering problem, including a "divergence"
problem dual to the cover problem, are given in Section 6.

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2.2 Problem Statements and Duality
We suppose given a finite undirected tree network with positive
arc lengths and denote by T an imbedding of the given network having
as edges rectifiable arcs. For any two points x,yeT, let d(x,y)
denote the shortest path distance between x and y.
Let J = {1,...,n} and denote by V = {v^,...,vn) (V C T) a collec
tion of distinct vertex locations of "demand points" or "existing
facilities." Let X = {x.,...,x } (X C T) denote a finite collection
1 P
of "centers" or "new facilities." For ieJ, define the distance of v.
1
to its nearest center by D(X,v^) = min{d(x^,v_.) : 1 < i < p}, and. let
Sj = maxid(x,v.): xeT}. Also, for jeJ, we assume given a real valued
function f continuous and strictly increasing, with domain [0,6^]
and (clearly) range [f (0) ,f^ (6^)] For X C T, |x| < o, we define
the function f by
f(X) = max{f.(D(X,v^)): jeJ}
The Primal p-Center Problem is as follows: Find a p-center X*
for which
rp = f(X*) = mini f (X) : XCT, ¡X | = p} .
(2.2.1)
As discussed in Dearing and Francis [19], due to compactness of
T and continuity of d(x,.) on T for each fixed xeT, an optimal solu
tion X* to (2.2.1) exists and is contained in the convex hull of V.
With a and p defined by a = max{f.(0): jeJ} and n = min{f (5 ):
J J J
jeJ}, we shall assume a < n, for if a = f (0) > f (6 ) = n. say, then
s t t
the function f would always be dominated by (strictly smaller than)

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f and hence f could be deleted from the definition of f without
s t
changing f. Further, we assume p < n-1, as otherwise the p-center
problem is trivial.
So as to state the dual problem, we define 8., =8,. for j,keJ by
j K Kj
8., = min maxif (d (x ,v.) ) f (d(x,v,))}
3k 3 3 K K
For j,kej with j < k we define a., = maxif .(0), f, (0)} and
3k 3 k
b#, = min{f. (5.) ,f (6 ) }. We note that a 5 n implies [a., ,b ., ] ^ 0.
jK 3 3 k k JK jK
The following lemma, the results of which are proven in [29] provides
a closed form expression for 8jk
Lemma 2.2.1. For any j,keJ with j 5 k we have:
(i) The function f.^ + f exists, is stricly increasing, continuous
3 k
has domain [a., ,b., ] ^ 0, and range [L., ,U., ], where L =
JK 3K jK JK jk
ifT1 + f^)o(ajk) and Ujk (f'1 +
(ii) d(v.,vk) < U.k.
(iii) The function (f.^ + f ^ exists, is strictly increasing and
3 k
continuous, has domain [L ,U., ] and range [a., ,bM ].
JK. jk Jk jk
(iv) 8_.k = (f"1 + fj^1) 1o(max{d(v_. ,vk> L^}) .
We remark that either 8., = a., or 8., = (f. ^ + f, *) ^o(d(v ,v, ));
jk jk jk j k j k"
8.. e [a., ,b.. ], and 8.. = f.(0). The closed form expression for 8.,
JK 3k jk jj j r jk
given in Lemma 2.2.1 facilitates construction of the dual problem.
We define the dual objective function g on subsets of V as follows
For any K C V with K1 t 2
g(K) = max{g1(K), g2(K)}
(K) = min(8ij: v^v^ e K, i j}
g2 (K) .= max{fj(0): v^ e K}

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The Dual Dispersion Problem is as follows: Find a subset K* of
V such that
g(K*) = max{g(K) : K^V, |k| = p+1} (2.2.2)
We remark that the dispersion problem is meaningfully defined for
2 p+1 < n. The primal p-center problem is trivial for p > n. Hence,
we shall restrict p to 1 p < n-1.
In what follows in this section, we prove a Weak Duality Theorem
(W.D.T.) and state a Strong Duality Theorem (S.D.T.) (proven in Sec
tion 5). At the end of this section, we give an example problem
illustrating definitions and results.
In the W.D.T. we shall use the fact (readily proven as in [18]
or [29]) that a f (X) for any XC T, |x| < .
Theorem 2.2.1. (Weak Duality Theorem). Assume 1 p n-1. For any
X C T with |X| = p, and any K C V with |k| = p+1, we have f(X) > g(K).
Proof. There are two cases: g(K) < a or g(K) > a. In the former
case we have g(K) 1 a < f(X). In the latter case, we note that
g(K) = g^(K) > a > g^(K). Since |x| = p < p+1 = |k|, at least two
demand points in K must be served by a single center. In other words,
for some v ,v £ K with s ^ t, and some center xeX, we have
s t
f [D(X,v )] = f [d(x,v )] 5 f(X)
s s s s
(2.2.3)
ft[D(X,vt)] = ft[d(x,vt)] < f(X) .
Using the definitions and the inequalities in (2.2.3), we have
g(K) = g^(K) < 3gt 2 max{fg[d(x,vs)],ft[d(x,vt)]} < f(X).
Remark 2.2.1. We note that the conditions |x| = p and jK¡ = p+1 can
be replaced by |x| 2 p and/or |k| > p+1, respectively, and the proof

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will still apply. Furthermore, the proof applies to any network, as
no special properties of tree networks are used.
We now state the S.D.T. We remark that the S.D.T. requires the
assumption of a tree network. In effect, network cycles may create a
"duality gap."
Theorem 2.2.2. (Strong Duality Theorem). For any p, 1 £ p n-1,
there exists an X* C T with |x*| = p and K* C V with |K*| = p+1 such
that f(X*) = g(K*).
It is evident from the W.D.T. that X* solves the primal p-center
problem and K* solves the dual dispersion problem.
Before presenting an example problem, we find it convenient to
view the dual problem as defined on "cliques" of a complete graph G.
We define G to be the undirected complete graph with node set J,
where node j of G represents vertex v of T. To any arc (i,j) of
G, i ^ j, we assign the length and, to any node j of G, we assign
the node weight g = f^(0). We call any complete subgraph K of G a
clique. We note that any nonempty subset of V induces a clique in G
and vice versa. For this reason, an equivalent definition of g(.) on
cliques of G can be given by defining g^(K) to be the length of a
smallest arc in a clique K of G, g2(K) to be the maximum of the
weights of nodes in K, and letting g(K) = maxig^K), g2(K)}. If the
number of nodes of a clique K is known to be q, we call K a q-clique
and (sometimes) write K Defining C (G) to be the collection of all
q q
q-cliques of G, an equivalent statement of (2.2.2) is as follows:
Find a clique K* for which
p+1
g(K*+l) = max{g(K): K e Cp+1(G)} .

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Whether K refers to a subset of V or a clique of G, we prefer to
call K a clique as long as it is clear from the context what K
refers to.
As an example of the nonlinear p-center problem, suppose that the
function associated with node v is f^ (y) = (y + h^)8 for y e [0j ]
where w^ h^, and 0 are given parameters. Appropriate restrictions
are placed on the parameters to ensure that the f^ are strictly in
creasing on [0,6j]. We note that the linear weighted p-center problem
is a special case of this problem generated by choosing 0 = 1, h^ =0,
and Wj > 0 for all j.
For the given form of f the following are readily verified:
f ^(r) = (r/w.)1^6 h., r e [f.(0), f.(6.)] ,
J 1 3 J 3 3
fT1(r) + fT1(r) = r1/8[(l/w )1/6 + (1/w )1/8] (h + h ) ,
1 J 1 J 1 J
r e [a.., b..] ,
13 13
w .w.
i 3
(fi + fj ) (y) r ^ ]./ j_ l/e^
0
(y + h + h ) ,
[w.x'" + w.x/v]" 3
y e [L.., U..]
Then, using the characterization of 3 as given in Lemma 2.2.1,
we have
3.. =
13
.. d.. if L £d(v. ,v.)
13 13 3 i 3
(2.2.4)
max[fi(0), f.(0)] if L^. > div^v.)
where

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Y. .
ij
w .w.
1 3
+ w.
3
1/0)0
and
d..
13
[d(v^,Vj) + h^ + hj]^
Consider the tree network shown in Figure 2.1, where the numbers
on the arcs represent arc lengths. The data given with Figure 2.1
corresponds to the parameters for j=l,...,6 where clearly, each f is
strictly increasing. Using (2.2.4), the 3 values for this problem
are shown in Table 2.1 along with the node weights f^(0). Figure 2.2
shows the dual graph G associated with the problem, where the number
next to each node j is the node weight and the number on the arc between
nodes i and j is 3^ Using Figure 2.2 it can be verified that the
optimal cliques (specified here by their nodes) and associated g
values are K* = (3,4), g(K*) = 13829.76; K* = {1,3,6}, g(K*) = 3600;
K* = {1,3,5,6}, g(K*) = 1664.64; K* = {1,3,4,5,6}, g(K*) = 784; and
K* = {1,2,3,4,5,6}, g(K*) = 225. Due to the duality theory, it then
o o
follows that the r^ for p=l,...,5 are, respectively, 13829.76, 3600,
1664.64, 784, and 225.
2.3 Dual Problem Interpretation
We imagine two conservative adversaries, an aggressor A and a
defender D. Defender D has defense forces placed at vertex locations
Vl,',,Vn' Aggressor A will attack a single vertex in V. Although D
knows A will attack a vertex, he will not know the vertex attacked
until the attack occurs.
Defender D has p response forces which he must position at loca
tions defined by a p-center X. Interpret tree distances to be travel
times, so that D(X,v.) is the minimum time to respond to v. from a

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n V
T f
20
22
-Cr
10
t)V
6
fj(y) = wj(y + Hj)0
Data
0 n 2
19 0
2 25 0
3 16 2
6 36 0
5 4 0
6 9 4
Figure 2.1. Example Nonlinear p-Center Problem

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Table 2.1
Values and Node Weights for Example
i
1
2
3
4
5
2
3
4
5
6
225
3600
3600
3600
4356
3600
3600
3600
4556.25
13829.76
8464
11664
900
784
1664.64
j
1
V0)
0
2 3 4
0 64 0
5 6
0 144

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Figure 2.2. Dual Graph for Example

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center in X. Assume A and D know functions f,,...,f so that
1 n
f.(D(X,v.)) is D's loss if A attacks v. and D responds to the attack
3 3 J
in a time of D(X,v^). For convenience, we refer to the loss A in
flicts on D as A's gain.
Aggressor A knows D has p response forces, but does not know how
D will position his response forces. Thus A acts conservatively and
bases his decision on a worst case analysis. If A decides to attack
Vj without threatening any other vertices, A reasons that D will cor
rectly guess v is to be attacked and will position a response force
at v.. Hence A assumes his gain will be f (0), if he decides to
J J
immediately attack v^ without a prior threatening strategy. In order
to gain more, A concludes that he must threaten, i.e., pretend to
attack, q vertices, q > 1, so that even if D knows which q vertices
are threatened, D does not know which vertex A will attack until the
attack occurs. Thus D is forced to respond to the threat by position
ing his response forces optimally with respect to these q vertices.
Hence if A threatens K C V, he assumes D will choose a p-center X
q
which minimizes f(X:K ) = max{f.(D(X,v.)): v e K }. Thus, with
q j j j q
q p, A assumes D knows and will position a response force at
every vertex in K so that A can gain at most g (K ) The best A
q Z q
can do in this case is to choose a K which contains some vertex v
q s
for which fg(0) = a. Hence, if q < p, A's maximum possible gain is
at most f (0). (Parenthetically, we remark that if f (0) = r ,
s s p
p < n, then it can be shown that not all f.(0) have the same value.
3
If all f.(0) do have the same value, then r > a.) On the other hand,
3 p
if A chooses a subset with q > p, D is unable to position a response
force at every vertex in K even if he knows K so A will gain at
q q

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least (K ). Hence A observes if he chooses some K with q > p which
2 q q
contains a vertex v for which a = f (0), then his gain is at least
s s
a = g (K ). However, A recognizes that there may be some other K
2 q q
with q > p, which may or may not contain v but which yields him a
gain strictly greater than a. For this reason A restricts himself to
those subsets of V with cardinality greater than p and realizes that
if he chooses some K with q > p, then there is at least one pair of
q
vertices in K which D can cover by only a single response force. If
q
v^ and v_. are one such pair in which are covered only by a single
response force, say at x, then clearly A obtains a gain of at least
3.., as 3.. = min{max(f (d(x,v.)), f.(d(x,v.))): x e T} < max{f (d(x,v )),
ij 13 i 1 3 1 11
A
fj(d(x,Vj))}. Since A does not know which pairs of vertices D will
cover by single response forces, once he chooses K^, A acts conserva
tively, and assumes that D will cover a pair v ,v, e K for which
a b q
3 = min{3..: v.,v. e K i ^ 3}. That is, by choosing a K with
ab 13 1 3 q q
q > p, A guarantees himself a gain of at least 3 = g. (K ). Hence
clD J- CJ
A's minimum gain due to threatening is g(K^) = max{g^(K^), g^CK^)},
so A chooses a K* with q > p which maximizes g(K ) over all K C V
q q q
with q > p.
The question arises as to why A should choose p+1 vertices to
threaten, and no more. By virtue of the W.D.T. and the remark follow
ing it, if X* is an optimum p-center then f(X*) > g(K ) for all K
q q
with q > p+1. Thus r^ = f(X*) is an upper bound on A's gain due to
threatening But the S.D.T. implies there is a p+l-clique, say
K*+^, which attains this upper bound. Hence A need threaten no more
than p+1 vertices to maximize his gain, as A cannot obtain any addi
tional gain by threatening more than p+1 vertices.

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There is also the possibility that A will make a false threat,
that is, attack a vertex not among the ones he threatens. If D be
lieves the threat is false and continues to act conservatively, he
will simply choose a p-center X* to minimize f. But since there exists
a p+l-clique KA+^ such that g(K*+p = f(X*), the greatest loss D can
incur, given Xa, is the same as if he believes A's optimal threat to
be real, and acts accordingly. Hence A cannot gain more by making a
false threat.
2.4 Covering Algorithm
In this section we study a covering problem, and present an
algorithm for solving it. Our primary interest in the algorithm is
the fact that it provides a constructive approach for proving results
about the primal and dual problem. For this reason we purposely keep
the algorithm simple, and use an analog string model to provide insight
into the algorithm. The development of both the string model and the
algorithm is motivated by an earlier string algorithm given in [32].
As in [32], an equivalent algebraic version of the algorithm is
readily obtainable. We remark that two other quite efficient algo
rithms [14], [15], exist for solving the covering problem, but they
do not lend themselves readily to our needs.
At this point we state the Covering Problem: Given r and the
runction f, compute
q(r) = mini|x|: f(X) S r, X C T} (2.4.1)
It is readily seen that the covering problem has a feasible solution
if and only if a < r. Further, with J(r) = {j: r < f^(6 )}, we shall

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assume J(r) ^ 0, for if J(r) = 0 then the condition f(X) < r holds
for all X C T and we (trivially) have q(r) = 1.
The above assumptions permit the following equivalent statement
of the covering problem:
minimize |x|
subject to
D(X,v.) < f.^Cr), j e J(r) (2.4.2)
2 2
We refer to the covering algorithm as COVER. In order to state
COVER a few definitions are convenient. We may imagine that the tree
is represented appropriately by inscribing straight line segments on a
planar surface such that each segment represents an arc. We fasten
strings of length f ^(r) to each node vjj e J(r), of the inscribed
tree, where, by convention, we allow strings of zero length. Every
fastened string has one end permanently affixed to the planar surface.
In addition, during the use of the algorithm we engage previously
fastened strings at various points on the tree. When a string is
engaged, some point of the string is permanently affixed to the tree
such that there is no slack in the portion of the string so far en
gaged. When strings are removed, we imagine that they are physically
deleted from the string model.
During each iteration of the procedure, we partition the original
tree into two subsets: one green, the other brown. The green subset
is always a tree, denoted as GT (for green tree), while the brown sub
set consists of one or more subtrees of the original tree T, each of
which is "rooted" at a node of the green tree. By convention, a root

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node t will be in both GT and the associated brown subtree, denoted
as BT(t).
COVER
0) Initialize to GT = T, k =
BT(v.) = iv.}. For every j e
3 J
at v.. Define U = 0.
0. For every tip vertex v of T define
J(r) fasten a string of length f ^(r)
3 o
1) Choose a tip t of GT. If GT = {t} go to 6). Else find a(t), the
vertex in GT adjacent to t.
2) If no strings are engaged or fastened at t, remove from GT the
subarc [t,a(t)] joining t and a(t), attach [t,a(t)] to BT(t), and go
to 1) Else go to 3).
3) Pull all strings at t tight towards a(t). If all tight strings
reach a(t) then engage them at a(t), remove [t,a(t)] from GT, attach
[t,a(t)] to BT(t), and go to 1). Else go to 4).
4) Add 1 to k. Choose a shortest string engaged or fastened at t.
Find the (unique) vertex, say v., at which the shortest string is
(k)
fastened. Construct ^ U {v^^}. Find the farthest point, say
y, from t on [t,a(t)] to which the shortest string can reach. Locate
x^ at y. Assign all strings at t to x^ and remove these strings.
Attach [t,y] to BT(t) to obtain BT(x ), and remove [t,y] from GT.
Go to 5).
5) Assign to x^ all other strings in GT which can reach x^, and re
move all such strings. If no strings remain then let U = U and stop.
K
Else return to 1).
6) Add 1 to k. Locate x^ at t. Assign all strings at t to x^. Of
the strings at t choose any one, and find the vertex v^ to which
the chosen string is fastened. Let U = U, 1 1/ {v,,,.}, and stop.
k-1 (k)

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Note that each time COVER places a center at x^ in step 4) it
identifies an associated vertex v^ which we call the distinguished
vertex associated with x^. When centers x^,...,x^. have been placed
in step 4), we call = iv^ ,... ,v^ } the distinguished set
associated with {x^,...,^}. If the algorithm places q centers in
total, then the set U defined by the algorithm consists of vertices
v,,.,...,v, N, the first q-1 of which are distinguished vertices
(i) (q)
(when q >2). The last vertex is distinguished only if x^ is placed
in step 4). Letting X = {x^,...,x }, we call U the primary set
associated with X, and call the primary vertex associated with
x^, i = l,...,q. We note that the primary vertices v(i)*,v(q) are
distinct, for as soon as a primary vertex is identified, its string
is removed, and thus the vertex is not available for any subsequent
identification. Likewise the centers x ,. .. ,x are distinct, for if
1 q
x. = x. with i < i, then all strings assigned to x. would have been
i J J
assigned earlier to x^, and so x^. would not have been located. Hence
it follows that JU| = JXJ = q, and U 0, since JXJ k 1. The primary
vertices will be of theoretical significance in proving our results.
We now establish some properties of COVER.
Property 2.4.1. COVER finds a feasible solution X to the covering
problem with |x| < n.
Proof. We first note that termination is clearly finite, since at
each iteration either at least one string is removed, or some entire
arc of T becomes colored brown. Since there are at most n strings
initially, it follows that the X constructed satisfies |x| < n.
Choose any v^,j £ J(r), and denote by x^ the center to which
v. is assigned. Since the string fastened at v. reaches x,..,
J (j /

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d(x,..,v.) < f/(r). As D(X,v.) d (x ,.. ,v.) it follows that X is
(j) J 3 3 U) 3
a feasible solution.
Property 2.A.2. For any nonempty distinguished set U^, with vertices
numbered so that U, = {v. ,... ,v, }, we have
k 1 k
v. e BT(x.) ,
3 3
1 < j < k
.-I.
d(Xj,Vj) = fj (r), 1 5 j < k .
(2.4.3)
(2.4.4)
Proof. Expression (2.4.3) is obvious. To show (2.4.4), choose any v
in U. Let t be the tip vertex chosen at the first of the iteration
tC
in which x^. is placed. The algorithm causes the string at v^. to-be
pulled tight along every edge connecting v^ to t, and to be pulled
tight along [t,x.], with the string end point coinciding with x..
J 1
Thus d(v.,t) + d(t,x.) = f.^(r). But v. e BT(t) and x. e T-BT(t) or
J J 3 3 3
x. = t so that d(v.,t) + d(t,x.) = d(v.,x.). Thus, (2.4.4) follows.
3 1 3 3 3
Property 2.4.3. Let X = ix^,...,x ) be the feasible solution con
structed by COVER, with vertices numbered so that U = {v^,...,v } is
the primary set associated with X. Assume q > 1. Then
d(vivj) > + for 1 i < j £ q (2.4.5)
Proof. We know the first q-1 members of U are distinguished vertices.
Hence Property 2.4.2 implies
v. e BT(x.),
i i
1 < i < q-1
.-1
d(v.,x.) = f (r), 1 i S q-1 .
(2.4.6)
(2.4.7)
For i < j, x^ is placed prior to x^.. Since v^ is assigned to x^
and

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no t
to x^ for 1 < i < j < q, v. was not in BTix^, and the string at
v. did not reach x.. Hence
J i
v. e T-BT(x.),
J i
1 < i < j ^ q
d(xi,Vj) > f (r), 1 < i < j < q .
(2.4.8)
(2.4.9)
But (2.4.6) and (2.4.8) give d(v ,v.) = d(v.,x.) + d(x.,v.) for
i j 11 1 J
1 < i < j < q, from which, on using (2.4.7) and (2.4.9), (2.4.5)
follows.
We shall need the following remark, proven in [32]:
Remark 2.4.1. Given any a.,a. e T and s.,s. > 0, there exists a-point
i 3 i J
x in T for which d(x,a^) < s^ and d(x,a^.) < s^ if and only if d(a^,aj)
< s + s..
i 3
We are now ready to establish the optimality of COVER.
Theorem 2.4.1. Given any r for which a < r and J(r) 4 0, COVER solves
the covering problem.
Proof. Let X = {x^,...,x^} be the point set found by COVER. Property
2.4.1 implies X is feasible to the problem. If q = 1, X is clearly
optimal. If q > 1, let the vertices be numbered so that U = {v,,...,v }
1 q
is a primary set associated with X. By Property 2.4.3, d(v_^,Vj) >
f^(r) +* f ^(r), for 1 <, i < j < q. Remark 2.4.1 implies there exists
no x in T for which d(x,v.) < f.^(r) and d(x,v.) < f .^(r) for any
i i 13
i, j in {l,...,q} e J(r) with i < j. Hence it is impossible to cover
any two members of U with a common center. Thus, since |u| = q, any
feasible solution X to the covering problem satisfies |x| > q. Since
q = |X| and X is feasible to the problem, X is thus an optimum feasible
solution.

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We remark that the covering problem may be of as much interest,
from both a theoretical and applications point of view, as the p-center
problem. In Section 6, we will present a problem which is dual to the
covering problem and show that the primary set identified by COVER
solves the dual of the covering problem. Furthermore we will charac
terize q(r) as a step function, and provide a formula for q(r)
assuming that r^ is known for 1 <, p n-1.
2.5 Dual Problem Solution and the Strong Duality Theorem
Based on the W.D.T. and properties of COVER we now present a
proof of the S.D.T. The proof is constructive in that we use an
algorithm called OPTKLIQUE which, given the optimal objective value
of the primal problem, constructs an optimal solution to the dual
problem. We then show that the objective values of the pair of prob
lems are equal. As a by-product the proof also establishes that
r e R, where, for convenience, we define R ={£..: l p ij J
We find it useful to summarize Theorem 2.4.1 and Property 2.4.3
as follows:
Lemma 2.5.1. Given any r for which a < r and J(r) ^ 0, the following
assertions are true:
(a) COVER finds an optimum solution X to the covering problem with
q(r) = |X|.
(b) Whenever q = q(r) > 1, any primary set U = ^v(l)
associated with X satisfies
g(U) = g^U) > r
(2.5.1)

-74
Proof. (a) is just Theorem 2.4.1.
(b) From Property 2.4.3, for any v_^,v^ e U, i ^ j, we have d(v^,v^) >
f.^(r) + f.^ir) > f.^Ca) + f.^a) where r £ a > a = a. .. Thus,
i J i J. iJ
d(v^,Vj) is in the domain of (f_^ + f ^) from which, upon using
Lemma 2.2.1 and the definitions of g, g^, and g^, (2.5.1) follows.
In the algorithm OPTKLIQUE we assume that r^ is given for some
value of p, 1 g p n-1. OPTKLIQUE constructs an optimal solution to
the associated dual problem.
OPTKLIQUE
1) If r = a, take K*+^ to be any p+l-clique in V containing a vertex
v for which f (0) = a, and go to 3). Else, given r > a, compute
s s p
r' = maxifS. e R: < r } and choose any r for which r' < r < r .
P iJ P P P
Go to 2) .
2) Apply COVER with the chosen value of r to find an optimum solution
X and its associated primary set U, with |x| = q = |u|. Note r < r
implies |X| > p, so q £ p+1. Take K*+^ to be any subset of U con
sisting of p+1 members of U. Go to 3). (If q > p+1, there will be
alternative optimal cliques.)
3) If K*+j, is any clique found in either step 1) or 2), then g(K*+p =
r and the W.D.T. guarantees K* is an optimum solution to the dual
P P+1
problem.
Before proving the correctness of the algorithm, we note, since
a = for some h, that a < r implies a < r', and thus the r chosen
hh P P
in step 2) is one for which a feasible solution exists to the covering
problem.
Theorem 2.5.1. Given r for any p, 1 g p < n-1, the clique K* con-
P P+1
structed by OPTKLIQUE satisfies

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S(Kp*+l> = rp
(2.5.2)
Furthermore, K* solves the dual dispersion problem.
p+1
Proof. Let X* be an optimum p-center solution to the primal problem
so that X* = p and f(X*) = r Since r S a we consider the cases
P P
r^ = a and r^ > a. Let us apply OPTKLIQUE for each case.
For r^ = a, K*+^ is chosen in step 1) so that |K*_y| = p+1 and
a = fg(0) = g2^Kp+l^ T^e W*D,Tt Sives S(K*+1) f(X*). But then,
a = 82^K*+1^ = = f(X*) = r = a, establishing (2.5.2) for
p+1"1
this case.
c'
P
For r > a, define R = {3. e R: r < g..} C- R. Since r > r >
P iJ P ij P
there exists no g. in R for which r < g ,. < r Thus g.. > r implies
ij iJ P 1J
3. > r and so it follows that
1J P
R={g..:r ij ij
(2.5.3)
Let U be the primary set identified by COVER for the chosen r,
r' < r < r By Lemma 2.5.1, U satisfies g,(U) > r from which it
p p J bl
follows that gy > r for v^*vj e U, i j. Hence, (2.5.3) implies
By e R v.,v. e U, i j (2.5.A)
J 3
Since |U| > p+1, let be that subset of U identified in step 2).
We have the following string of inequalities:
r
P
f(X*) > g(K*+1)
2 Mkh>
= mlniByi Vj.Tj E K*+1, i j* j)
> minig..: v.,v. e U, 1 i}
ij i J
> minig.. e R}
> r
(2.5.5)
(2.5.6)
(2.5.7)
(2.5.8)
(2.5.9)
(2.5.10)

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where (2.5.5) follows from the W.D.T., (2.5.6) and (2.5.7) follow
from the definitions of g and g^, (2.5.8) follows from K*+^ c U,
(2.5.9) follows from (2.5.4), and (2.5.10) follows from the definition
of R. Hence, every inequality holds as an equality, establishing
(2.5.2) for this case.
The assertion that K*., solves the dual problem is immediate from
p+i
f(X*) g(K*+1) and the W.D.T.
We note that Theorem 2.5.1 provides a proof of the S.D.T. since in
the statement of the S.D.T. we take X* to be an optimum p-center solu
tion to the primal problem and K*+^ as constructed by OPTKLIQUE. We
also note that the duality theory provides necessary and sufficient
conditions for a p-center to be optimal, which, as far as we know, are
the first such conditions for this problem.
We remark, just as with the linear p-center problem, that if we
define 6 = minig..: g e R, q(B..) < p}, then 8 ^ = r Clearly
st ij ij ij st p
q(r ) < p and q(8 ) < p. The S.D.T. implies r e R, and thus the
P st p
definition of g ^ gives g ^ < r Let p' = q(g ) and let X solve
st st p st p
the cover problem for r = g so that f(X .) < g Since p > p',
st p st
append to X^, (if necessary) any p-p' center locations to obtain the
p-center X^. Clearly D(X^,v^) D(X^,,Vj) for v e V, and thus
f(X ) < f(X ,). Hence r f(X ) < f(X ,) < g < r so g = r
P p p p p st p st p
and X^ is an optimum solution to the p-center problem. This remark
permits the use of the same procedures as discussed in [65] to compute
r^ efficiently, by performing a binary search over the (ordered) list
R, applying COVER for every r chosen from R until a smallest g in R
st
is found for which COVER finds p or less points. Once r^ is computed
in this manner, OPTKLIQUE requires an additional application of COVER

-li
ter any r, r' < r < r and solves the dual dispersion problem. This
P P
approach is essentially a primal approach for solving both problems.
An alternative approach which directly works with the dual graph is
given by Chandrasekaran and Tamir [13] for the unweighted linear p-
center problem, which works directly with what would be a subgraph of
our dual graph G. Due to absence of weights and addends, their
approach does not require the use of node weights (and for that matter
the function g^) in the dual graph. For a given value of r, Chandra-
sekaran and Tamir define an intersection graph IG with node set J and
r
arcs (i,j) for those indices i,j e J for which 8 < r. Their pro
cedure is based on a graph theoretic procedure given by Gavril [39]
and solves the covering problem by finding a minimum clique cover of
IG^ (minimum number of cliques such that every node is in at least one
clique). As a side result, their approach identifies a maximal anti
clique in IG (a maximal set of nodes in IG no two of which are con-
r r
nected with an arc). Due to "chordal" properties of IG^ as discussed
in [39], the cardinality of a minimum clique cover of IG^ is equal to
the cardinality of a maximal anti-clique in IG^. This result is a
special instance of the duality result we will present in Section 6
for the cover problem, as applied to the linear unweighted case.
Furthermore, for r = r Chandrasekaran and Tamir [39] proved a duality
P
relationship for the unweighted p-center problem using the above
properties of IG^. We remark that their duality results can be
directly proven by using the algorithm OPTKLIQUE, and by appropriately
specializing our S.D.T. for the linear unweighted case.
We now demonstrate the use of OPTKLIQUE by determining K* for
4
the example problem. From our previous analysis, r^ = 1664.64. Since

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r > a = 144, we compute (from Table 2.1) rl=max{8.. e R: 3.. < r} = 900.
3 3 ij ij 3
We next must apply COVER using a value of r where 900 < r < 1664.64.
Figure 2.3 shows the results of using COVER with r = 1296. In the
figure, the loose ends of the strings are shown as wavy lines. Brown
subtrees are shown as crosshatched arcs of the original tree. Each
separate drawing of the tree (a)-g)) is for a subsequent iteration of
COVER. Figure 2.3a) demonstrates the initialization step, where for
r = 1296, the f.^(r), j = 1,...,6 are 12, 7.2, 7, 6, 18, and 8, re
spectively. The numbers next to the strings are the lengths of the
loose ends. In the figure, we indicate which tip of the green tree
is chosen at each return to step 1) of COVER. In addition, the suc
cessive distinguished vertex sets are indicated.
After the final iteration, we note that the primary vertex set
U is {v,v, ,v- ,v,-} which, from our previous analysis, we know to be
J 1 D 5
2.6 Results for the Covering Problem
In this section we present a "divergence" problem which is dual
to the covering problem. We give a weak duality and a strong duality
result and prove that the primary set identified by COVER solves the
dual problem. The term "divergence" is chosen to represent the
physical interpretation, discussed later, in which the attacker A
chooses a "divergent" set of vertices to threaten. Further, the term
permits a distinction to be made between the two different dual prob
lems. Also, in this section, we demonstrate how having optimum solu
tions to the p-center problem for all p, 1 < p < n, enables us to
completely characterize the function q(r).

1 n U 1 ji J I /..i t Ion
(I M M > I (*
V
?
Choose v
1
Choose v
18
9 x.
U4 = {v3WV5}
K)
u = {v3, v6, v5!
k* o, i, <>, r>
OPTKLIQUE for p = 3 for Example

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The Divergence Problem is as follows: Given r and the function
g, compute
q(r) = max{ |u| : g(U) >r,UCV}. (2.6.1)
That is, the problem is to find the maximum number of existing facili
ties no two of which can be jointly covered by a single center within
a radius of r. Equivalently, among all cliques of G whose gain is
larger than r, the problem is to find one with the maximum number of
nodes. The dual problem is feasible for r < r^, as, if r > r^ there
does not exist a subset U of V for which g(U) > r. On the other hand,
the primal cover problem is feasible for r > a. Hence, we shall re
strict r to a < r < r^ in order to ensure feasibility to both
problems.
Theorem 2.6.1. (Weak Duality Theorem). Assume a < r < r^. For any
feasible solution X to the primal cover problem, and any feasible
solution U to the dual divergence problem, we have |x| > |u|.
Proof. By feasibility of U and the assumption of the theorem we have
g(U) = gl(U) > r > a > g2(U) from which it follows that
8.. > r v.,v. e U, i ^ j (2.6.2)
ij i J
Suppose |X| < Ju|. Then, the same approach as in the proof of Theorem
2.2.1 implies there exist v ,v e U, s ^ t, such that 8 < f(X) < r,
st st
contradicting at least one inequality in (2.6.2). Thus, |x| > |u|.
Theorem 2.6.2. (Strong Duality Theorem). Assume a < r < r^. Let X
be a feasible solution to the covering problem constructed by COVER.
Then, the primary set U associated with X solves the dual divergence
problem with
X| = q(r) = q(r) = |U
(2.6.3)

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Proof. By definition of a primary set we have |x| = |u|. By assump
tion r < r^ so that |x| = |u| k 2. Lemma 2.5.1 implies g(U) = g^(U) > r.
Hence U is a feasible solution to the dual problem. Theorem 2.6.1 im
plies q(r) > q(r). By feasibility of X and U, and the fact that
|X| = |U|, we have |x| > q(r) > q(r) £ |u| = |x|. It follows that
X solves the cover problem, U solves the dual problem, and (2.6.3)
holds.
We remark that the above proof is an alternative to the proof of
Theorem 2.4.1 for establishing the optimality of X to the covering
problem. Hence, an application of COVER solves both problems simul
taneously.
At this point we give an interpretation of the pair of problems.
The defender D specifies an upper bound r on his loss against an attack
to any vertex and will position response forces as necessary so that
his loss will not exceed r. Each response force is an "expense" for
D. Hence, D's problem is to choose the fewest possible response
forces. The attacker A knows that D will not tolerate a loss exceeding
r. Hence, A recognizes that, no matter how many vertices he threatens,
D will have a sufficiently large number of response forces to respond
and that the loss A inflicts on D will always be less than or equal
to r. For this reason, A decides that he should not (hopelessly) try
to inflict a loss to D exceeding r, and that, instead, he should force
D into using as many of his response forces as possible. Hence,
should A choose a subset U of V with g(U) > r, he knows that no two
vertices in U can be jointly covered by a single response force by D
within the specified upper-bound r. Thus, D, not tolerating a loss
exceeding r, will have to allocate one response force for every vertex

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in U. In total, any feasible X which D chooses will satisfy |x| > |u|,
which is what the W.D.T. asserts. By virtue of the S.D.T., if U is
A's optimal choice, D can choose exactly || response forces positioned
at, say X, with |x| = |u| and still respond to an attack to any vertex
in U (as well as in V-U) without incurring a loss exceeding r. If A
threatens more than q(r) = |u| vertices, say, a subset U of V, then
|U¡ > q(r) implies g(U) < r (infeasibility). Thus, D would not be
forced into allocating a single response for every member of U. In
fact, even if A threatens every vertex in V, then D still needs ex
actly q(r) = q(r) = || response forces to respond to the threat
feasibly. Thus, if each threat is an "expense" for A, he need threaten
no more than q(r) vertices. On the other hand, D adopts an optimal
strategy against A's best threat by minimizing the number of response
forces with respect to V.
Continuing our consideration of the covering problem, we now re
verse the usual procedure, and view the p-center problem as a device
for solving the covering problem for all values of r for which the
covering problem is feasible, that is, for a < r.
The following lemma is the key to using the p-center problem to
solve the covering problem. Define r = < for convenience.
o
Lemma 2.6.1. Let p e J. If r < r ,, then
: r p p-1
q(r) = p for r < r < r .
P P-1
Proof. We first note r < r < ... S r. < r_. Also, clearlv.
n n-1 10 J
q(r ) < p for p e J. Now for r^ £ r since q is non-increasing we
have 1 > q(r^) > q(r) > 1, establishing the claim if p = 1. Consider
the case p e {2,...,n}. From r < r < r we have p > q(r ) > q(r) >
p p i p
q(r ]_). Suppose q(r) = s, with s < p, implying s < p-1. Let X,

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with |X| = s, solve the cover problem for r.
r < r contradicting the definition of r
pi s s
r < r < r .
P P"1
It now follows, if we define the set
We then have f(X) < r <
Thus q(r) = p for
P = {(p-1,p): p e {2,...,n}, r^ < r^} >
that
q(r)
r P for rp < r < r j, (p-l,p) e P

1 for r^ < r .
(2.6.4)
The formula (2.6.4) completely defines the function q(r), since r = a,
n
and the cover problem is feasible if and only if a < r. Hence if we
solve the p-center problem for all p and compute r ...,r then we
2 n
have an explicit formula for q(r), and we see that the r^ completely
define the function q. For example, if r, = rc < r. = r < r_ = r,,
then q(r) = 5 for r^ ^ r < r^, q(r) = 3 for r^ r < r^, and q(r) = 1
for r^ ~ r. Also, the proof of the lemma does not require the assump
tion that the location network is a tree. Thus the formula for q(r)
is still valid if the location network has cycles.

CHAPTER 3
A VECTOR-MINIMIZATION PROBLEM ON A TREE NETWORK
3.1 Introduction
We consider a vector-minimization problem on a tree network which
involves as objectives the distances between specified pairs of new
facilities and specified pairs of new and existing facilities. In many
location problems, especially in the public sector, it may be necessary
to build a number of public facilities which are to be shared by a number
of communities. If the optimizers cannot agree on a single objective
function, the analyst is faced with the problem of locating the facili
ties in such a manner that all parties are satisfied with the end
result. In such a case, the optimizers can agree to rule out "dominated"
solutions and consider only "efficient" solutions.
The related literature on multi-objective location problems is
discussed in Chapter 1 under Multi-objective location problems on
networks. Here, we concentrate on characterizing efficient solutions
to the vector-minimization problem of interest. We relate efficient
solutions to a distance constraints problem studied by Francis, Lowe,
and Ratliff [32]. Extensions of results in [32] are given by Francis,
Lowe, and Tansel [33]. We use the theory developed in [32] and [33]
to establish the necessary and sufficient conditions for efficient
location vectors (parenthetically, we remark that the results we proved
in [33] are also given in our Dissertation Proposal defended on June 8,
1979).
-84-

-85-
At this point, we give an overview of the chapter. In Section 2,
necessary definitions and notation are given and the vector-minimiza
tion problem of interest is defined. In Section 3, we relate the
problem to distance constraints, give a number of related properties
of distance constraints, and establish the necessary and sufficient
conditions for a location vector to be efficient. In Section 4, we
provide examples of efficient and non-efficient location vectors.
Section 5 is devoted to a further refinement and simplification of one
of the necessary and sufficient conditions, namely, "the convex hull
property." In Section 6, we provide an algorithm, SEVCA, which con
structs an efficient solution from a given location vector. In Sec
tion 7, we characterize efficient solutions for the analogous problem
in the p-dimensional Euclidean space with rectilinear (p 2). or
Tchebychev (p > 2) distances.
3.2 Problem Statement
We suppose given a finite, undirected tree network, and denote
by T an imbedding of the given network. Let V = {v,,...,v } be a set
of n distinct vertices of T. We assume existing facility i is located
at vertex v^, i e {l,...,n}. For j e {l,...,m}, denote by x^ a point
to be determined in T as the location of new facility j. We define Tm
to be the m-fold Cartesian product of T by itself and define a location
vector X in Tm to be the ordered m-tuple (x,,...,x ) with each x e T,
1 m j
j e {l,...,m}. Sometimes, we refer to a location vector X in Tm as a
point m T
As in [22], given points x,y e T, we define the line L(x,y) to be
the union of all points in the shortest path connecting x and y. In

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addition, given a finite point set P C T, we define the convex hull
H(P) to be the smallest (embedded) subtree of T containing all points
in P. We note that for any two points p,p' e P, the line L(p,p') is
contained in H(P).
We denote by I the set of pairs (i,j) for which the distance
u
d(x^,Vj) is of concern. Similarly, 1^ is the set of pairs (j,k) for
which the distance d(x.,x, ) is of concern. We remark that it need not
3 k
be the case that 1^ includes all possible pairs of new and existing
facility indices, nor I includes all possible pairs of new facility
indices. With these definitions, the problem of interest is to "mini
mize" each of the distances specified by (3.2.1);
d(x >v ) (i,j) e I ,
1 J O
(3.2.1)
d(x.,x ) (j ,k) e I .
J K. d
For X e Tm, we denote by D(X) the vector each of whose components
is a distance specified by (3.2.1). The vector is formed by assuming
any convenient ordering of the members of 1 and I. The vector-
C B
minimization (V-min) problem of interest is
V-min{D(X): X e Tm} (3.2.2)
With respect to (3.2.2), a location vector Z e Tm is said to
dominate a location vector X in Tm if D(Z) < D(X) and D(Z) D(X).
A location vector Z which is not dominated by any other location vector
is said to be efficient. An equivalent definition of efficiency is as
follows: Z e Tm is efficient if and only if X e Tm and D(X) < D(Z)
imply D(X) = D(Z).

-87-
Our main interest is to characterize efficient location vectors
and devise an algorithm for constructing efficient location vectors
from a given (dominated) location vector.
3.3 Distance Constraints and Characterization
of Efficient Points
We make extensive use of the results obtained in [32, 33] for
distance constraints to establish the necessary and sufficient condi
tions for efficient points. The Distance Constraints (DC) are defined
in [32] (independent of the efficiency problem) as follows: Given the
sets 1^ and 1^ and nonnegative upper bounds c^. and b^ find a point
X = (xx ) in Tm, if it exists, such that
1 m
d(x.,v.) < c .
i 3 ij
d(x. ,x. ) £ b.,
3 k J)k
(i,j) e I
(j,k) e I,
(3.3.1)
Corresponding to DC, we define Graph BC (GBC) as the undirected
graph having nodes E.,...,E N,,...,N ; for every (i,k) e I,,, there
i n i m is
is an arc (N. ,N, ) of length b., between nodes N. and N, : for every
J k' jk j k J
(i,j) e I, there is an arc (N.,E.) of length c.. between nodes N.
C i J ij i
and E.. We further assume that the sets I,, and I are such that GBC
J B C
is connected, as otherwise DC decomposes into independent sets of con
straints which may be analyzed separately.
Given a node-path between any two nodes f and f in GBC, we de-
P q
note the path by P(f ,f ) and denote the length of the path by LP(f ,f )
p q p q
We define (f^jf^) to be the length of any shortest path in GBC between
nodes f and f Subsequently, unless we specify otherwise, it should
p q

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be understood that any path we refer to is a simple path between some
two existing facility nodes and E^.
Results on Distance Constraints
The distance constraints are said to be consistent if there exists
at least one feasible solution to (3.3.1).
The following result is established in [32].
Theorem 3.3.1. The distance constraints are consistent if and only if
d(v ,v ) < £(E ,E ), 1 £ p < q n (3.3.2)
p q p q
The inequalities (3.3.2) are termed the Separation Conditions
[32], since each term on the right specifies an upper bound on how
separate two existing facility locations can be. Except when stated
otherwise, we assume throughout the chapter that the separation condi
tions hold, and thus (equivalently) DC is consistent.
We call a path P(E^,E^) between E^ and E^ in GBC a tight path if
LP(E ,E ) = d (v ,v ). We note that since we assume DC is consistent,
P q p q
it necessarily follows if P(E ,E ) is a tight path, that LP(E ,E ) =
p q p q
L(E ,E ). Any path P(E ,E ) for which LP(E ,E ) > d(v ,v ) is called
pq p q p q p q
a slack path.
We say that new facility i is in a tight path if there exists at
least one tight path containing Ik. Every path containing Ik is slack
if there is no tight path which contains Ik .
The motivation for the above terminology is due to a string graph
representation of GBC. This string graph is also useful for obtaining
problem insights. When knots representing nodes E^ and E^ are pulled as
far apart as possible, the distance between the two knots is L(E ,E ).
P q

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If then the string graph is placed upon the tree T, i.e., the strings
only lie on arcs of T, a path is tight when it is necessary to pull the
string graph tight in order to place the knots representing and
on v and v respectively, while a path is slack if the string path
P 9
must literally be slack when the two knots are placed to coincide with
v and v .
P q
A priori, one might think that the occurrence of a tight path
would be rare. However, we shall see that tight paths occur in a
quite natural way when the separation conditions are used in the analy
sis of efficient location vectors. Further, the notion of tight paths
permits the specification of necessary and sufficient conditions for
DC to have a unique solution.
We now relate unique locations to tight paths. By definition,
new facility i is uniquely located if it has the same location in every
feasible solution to DC. Since we later refer to a collection of
facilities, which contains possibly both existing and new facilities,
being uniquely located, we note that existing facilities are uniquely
located by definition.
Theorem 3.3.2, which we proved in [33], specifies the necessary
and sufficient conditions for a new facility to be uniquely located.
Theorem 3.3.2. New facility k is uniquely located if an only if node
lies in at least one tight path P(E^,E^).
Corollary 3.3.2. Distance constraints have a unique solution if and
only if node lies on at least one tight path in GBC for k = l,...,m.
We now give an additional property of a tight path we proved in
[33]. The property will be used in proving our main result on efficient
points.

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Property 3.3.1. If P(E^,E^) is a tight path in GBC, then
(i) every facility represented by a node in P(E^,E^) is uniquely
located,
(ii) the locations of facilities corresponding to nodes in P(E^,E^)
occur with the same ordering and spacing on the line L(v ,v ) in
P 9
T as do the corresponding nodes in P(E ,E ).
P q
As an illustration of Property 3.3.1, suppose P(E^,E^) is a tight
path with nodes E^, N2, N^, E^. Then, the locations v^, x^, v^
are unique. Furthermore, they occur in the given order on the line
l(v1,v5) with d(v1,x2) = c21, d(x2,x3) = b23, d(x3,v5) = c35, where
C21 ^23 C35 are t^ie -*-enSt^ls tbe arcs in the path. This example
is illustrated in Figure 3.1.
b
23
35
Tight Path
p(e15e5)
in GBC
'21
23
C
35
x
3
Uvpvs)
in T
Figure 3.1. Illustration of Property 3.3.1.
We now consider the problem of determining when an arc lies on a
tight path. As an arc lies on a tight path if and only if it is not
the case that all paths containing the arc are slack, we consider the

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equivalent problem of determining when an arc lies only on slack paths.
The following property, which we proved in [33], characterizes the con
ditions under which an arc in GBC is not contained in any tight path.
Property 3.3.2. Let DC be consistent. Let (f^,fj) be any arc in GBC,
of positive length e^., whose length is reduced by some positive amount
e. Let DC^CGBC^) be the distance constraints (graph) obtained from
DC (GBC) by replacing e by e e.
(a) Evey path containing (f_^,f_.) in GBC is slack if and only if e can
be chosen (with e > 0) so that DC is consistent.
e
(b) Whenever every path containing (ff) is slack, e can be chosen
(with e > 0) so that DC£ is consistent and at least one of the follow
ing is true:
(i) at least one path in GBC containing (f.,f.) is tight;
£ 1 J
(ii) the length of (f.,f.) in GBC can be reduced to zero.
i j e
Finally, we will use the following lemma proven in [33].
Lemma 3.3.1. Given points a,b e T, suppose that d(a,b) = a +3.
Then, the inequalities d(x,a) a, d(x,b) £ 3 are consistent if and
only if they have a unique solution and the inequalities hold as
equalities.
Necessary and Sufficient Conditions for Efficiency
Given a location vector Z, we let U = D(Z) and define the distance
constraints of interest by D(X) < U, where the entries in U define the
bjk and Cij by bjk = d('ZyZk) fr £ IB and Cij = d(zivj) for
(ij) £ 1^. We use the b^ and c to define GBC in the customary
manner. As before, we may assume GBC is connected, for otherwise the
problem of finding efficient location vectors decomposes into

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independent subproblems. Further, we note that DC is always consistent,
as Z is certainly feasible to DC, and hence, by Theorem 3.3.1, the
separation conditions are always satisfied. For convenience, for any
location vector Z, we denote by A*(Z) the collection of locations of
uniquely located facilities whose nodes are adjacent to N in GBC. We
denote by 11[A*(Z)] the convex hull of A*(Z), the imbedding of the
smallest subtree of T spanning all the elements of A*(Z).
With the above definitions we can present a family of equivalent
conditions for a location vector Z to be efficient.
Theorem 3.3.3. Given a location vector Z used to define DC and GBC,
the following are equivalent:
(a) Z is efficient;
(b) Each FL is in at least one tight path in GBC;
(c) Z is the unique solution to DC;
(d) z^ e H[A*(Z)] for i = l,...,m.
Proof. The equivalence of (b) and (c) is a direct conseqeuence of
Theorem 3.3.2 and the fact that Z is always a feasible solution to
DC, while (c) clearly implies (a). To show (a) implies (c) suppose
Z is not the unique solution to DC. Color every new facility node
in GBC which is not contained in any tight path blue. Color all the
other (new or existing facility) nodes red. Equivalence of (b) and
(c) implies every blue node represents a new facility which is not
uniquely located, while every red node represents a (new or existing)
facility which is uniquely located. By assumption there is at
least one blue node. By connectedness of GBC, there is at least
one arc which connects some blue colored node, say, N to some red
P
colored node, say, F Furthermore, arc (N ,F ) has positive
9 P q
*

-93-
length; for otherwise, in every feasible solution to DC, the location
of new facility p would be the same as the location of the uniquely
located facility represented by node F contradicting the fact that
N is colored blue.. Property 3.3.2 then implies that the entry in
P
U = D(Z) corresponding to arc (N ,F ) can be reduced by a positive
P 9
amount and the resultant distance constraints will still have a feasible
solution, say Y. But then clearly D(Y) < D(Z) and D(Y) 4- D(Z), contra
dicting the fact that Z is efficient. Hence (a), (b), and (c) are
equivalent. It can be seen that the proof will be complete if we show
(b) implies (d) and (d) implies (c).
To show (b) implies (d), suppose N_^ is in some tight path P. Let
f^ and f^ be the nodes adjacent to N_^ in P, so that ((f^,N^), (N^jf^))
is a subpath of P. Since f^ and f2 are in the tight path P, by Theorem
3.3.2 the facilities represented by f^ and f are uniquely located. We
may let y^ and y2 denote the unique locations of f^ and f^, respectively.
Thus it is clear that y^ and y^ are elements of A*(Z). By Property
3.3.1, z^ e L(y^yy^), and by definition of the convex hull, L(y^,y^) C
H[A*(Z)]. Thus it follows that z^ e H[A*(Z)J. To show (d) implies
(c), suppose e H[A*(Z)] and let f^ and f^ be nodes adjacent to N_^
in GBC, where f^ and f^ represent facilities with unique locations y^
and y respectively, such that z. e L(y ,y ) C T. Thus d(y ,y ) =
^ 1 i. m 1 Z
d(y ,z ) + d^z^y^). Now for any feasible solution X to DC we know
dCy^Xi) d(y^,z^) and d(y2>x^) < d(y^,z^). But then because f^ and
f2 are uniquely located, Lemma 3.3.1 implies x. = z^, for i = l,...,m.
Hence X = Z, so Z is the unique solution to DC, completing the proof.

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3.4 Examples
Here, we give examples of efficient and non-efficient points.
Ex. 1. For a single new facility, D(z) is the vector (d(z,v ),...,
d(z,v )). Any point z in T is efficient since T is the convex hull
of iv . ,V }.
1 n
Ex. 2. Consider the tree T shown in Figure 3.2. Each arc length in
the corresponding graph GBC corresponds to an entry of D(Z). In this
case Z is efficient. Notice that and are both contained in the
tight path P = (E^, ^, E^). Also, both z^ and z2 satisfy the con
vex hull property, i.e. z Htiv^ v2, z2}] and z2 e H[{v3> v^-Zj}].
Figure 3.2. Example of an Efficient Location Vector
(a) Graph BC, (b) Tree T.
Ex. 3. Consider the same tree as in Example 2 except that the location
of z2 is changed to the midpoint of edge (v^v^. In this case

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(z^.z^) is not efficient as is not contained in any tight path.
Also, t Hv^, v^, z^}. This example is shown in Figure 3.3.
1.5
1.5
/
4
7
c ?
/
Figure 3.3. Example of a Non-Efficient Location Vector
Ex. 4. Again consider the same tree with z^ = z^ located at the mid
point of edge (v^,v^). In this case, both z^ and z^ are uniquely
located and both satisfy the convex hull property. Thus Z =
is efficient in this case. This example is shown in Figure 3.4.

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Figure 3.4. Example of an Efficient Location Vector
3.5 Further Results on the Convex Hull Property
In this section we concentrate on the last statement of Theorem
3.3.3, namely, that Z is efficient if and only if each is contained
in the convex hull H[A*(Z)], where A*(Z) contains the locations of
those uniquely located facilities whose nodes are adjacent to N^.
Our main interest is to delete the phrase "uniquely located" from
the definition of A*(Z) and still have the equivalence hold under the
new (relaxed) definition. From a computational standpoint, this deletion
would make it unnecessary to identify the uniquely located new facilities,
which, in turn, requires the identification of tight paths in GBC.

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With this motivation in mind, define, for 1 < j < m, A(N_.) to be
the collection of nodes in GBC which are adjacent to N and denote by
A (Z) the collection of the locations of the new and existing facili-
j
ties whose nodes are in A(N). We remark that N, is not a member of
J J
A(N.) and hence z^ f/ A^ (Z) .
The following property states the necessary conditions for Z to
be efficient.
Property 3.5.1. Suppose Z is efficient. Then ze H[A^(Z)] for every
j £ {1,... ,mj.
Proof. From Theorem 3.3.3, whenever Z is efficient, z^ e H[A*(Z).] for
each j e {l,...,m}. But A*(Z) is clearly a subset of A(Z) implying
that z e H[*(Z)] C H[A (Z)], completing the proof.
In general, the reverse implication in Property 3.5.1 need not
hold for certain (pathological) cases. Such occurrences correspond to
the case where Z is such that for some two adjacent nodes N. and N, ,
J k
the locations z and z^ coincide. We provide an example of such a
case in Figure 3.5. With reference to Figure 3.5, observe that every
z. is contained in the associated convex hull. In particular, 7. and
J ^
z^ are contained in their respective convex hulls because their loca
tions are the same. The location vector is clearly a non-efficient
one, since and z^ can both be moved to v^> thereby reducing the
distances associated with them.
Sufficiency for Irreducible Location Vectors
At this point we distinguish two classes of location vectors and
show that the reverse implication (sufficiency) in Property 3.5.1 holds
for one class ("irreducible" location vectors) while it need not hold

-98-
z2=z3
#
v
1.5
1.5
a) Graph GBC
b) Tree T
A^(Z) = (z2,vi,v2,v4)
Zj e H[ A^(Z)] = T
A2(Z) = {z^ZyV^}
z2 e H[A2(Z)] = L(zl5z3)
A3(Z) = {z2>v^}
z3 e H[A3(Z)] = L(z2,v^)
c) Sets A^. (Z)
d) Convex Hulls
Figure 3.5. Example of a Non-Efficient Location Vector

-99-
for the other class ("reducible" location vectors). We say a pair of
facilities interact if their nodes are adjacent in GBC. We define a
location vector Z = (z^,...,z^) to be irreducible if for every pair of
interacting new facilities i and j, their locations z^ and zare dis
tinct; Z is said to be reducible if there exists at least one pair of
interacting new facilities i and j for which z^ = zj ^he lcatin
vector of Figure 3.5 is an example of a reducible location vector.
The following property gives the sufficient conditions for an ir
reducible location vector to be efficient.
Property 3.5.2. Suppose Z e T is an irreducible location vector. If
for every j, 1 < j < m, z^ e H[A^(Z)], then Z is efficient.
The proof of Property 3.5.2 requires a number of preliminary
results. To preserve the continuity of the discussion, we leave the
proof until the end of section 5.
From a computational standpoint, Property 3.5.2 provides an ap
proach for determining whether or not an irreducible location vector
is efficient, and constructing one if it is not. To check if Z is
efficient, we only need to determine the nodes adjacent to N in GBC
and form the convex hull (the smallest subtree) which spans the loca
tions of these adjacent nodes. If it is the case that every z. is
within its convex hull, then Z is efficient. Otherwise, we can choose
a z. which is not in the convex hull associated with it, and move its
location to the closest point in the convex hull. The procedure can
be employed repeatedly until every new facility satisfies the convex
hull containment property. However, during such a procedure, the
current location vector may change its status from an irreducible one
to a reducible one, as the locations of new facilities change. For

-100-
this reason, it becomes necessary to develop the sufficient conditions
for reducible location vectors.
Sufficiency for Reducible Location Vectors
The basis of our approach for establishing sufficiency for redu
cible location vectors is to represent a reducible location vector by
an irreducible one and apply Property 3.5.2.
Suppose Z is reducible. Then at least one arc in GBC connecting
two new facility nodes has length zero. In general, there may be
several arcs of length zero connecting new facility nodes. Let GB be
the subgraph of GBC with nodes N,,...,N and arcs (N. ,N ) for (j ,k)
1 m J k
e ID. If arc (N.,N, ) in GBC has length zero, then combining these
B j k
two nodes into a single (super) node will not affect the length of any
path containing this arc. If the resulting graph (with one less node)
has an arc in GB of length zero, again the two nodes connected by that
arc can be combined into a single node without affecting the path
lengths. In general, this graph transformation can be applied as many
times as necessary (clearly, at most m 1 times) to obtain a new
graph GBC* from GBC so that no arc in GBC* connecting two new facility
nodes has length zero. With this transformation, a node N of GBC*
P
will actually be representing a number of the original nodes in GBC.
We may define the index p as a composite index for the indices of new
facility nodes represented by in GBC*. Hence, if p is the composite
index, say, {j,k,l}, we can define z* to be the common location
P
z^ = z^ = z^ of new facilities j, k, and 1. Thus, if GBC* has, say,
t new facility nodes, then the location vector Z* with components
corresponding to new facility nodes of GBC* will be an irreducible

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location vector, as GBC*, by definition, has no arc of length zero
connecting two new facility nodes. Hence, sufficient conditions for
Z can be expressed in terms of the sufficient conditions (given in
Property 3.5.2) for Z*.
The following procedure, RP (Reduction Procedure), transforms GBC
into GBC* by applying successive elementary transformations as de
scribed in the above paragraph. During the procedure, we also keep a
list K which contains as members the composite indices.
RP.
0) Given Z, set up GBC with arc lengths defined by entries of D(Z).
Define K = {{1},. .. ,{m}}. Label new facility node as N^j,
1 < i < m.
1) If, for some P,Q e K, P f Q, there is an arc (Np,N^) of length zero
in GBC go to 2). Else go to 4).
2) Superimpose node on together with all arcs incident to Np.
Remove arc (Np,N^) from GBC. (If parallel arcs occur due to this
transformation they will clearly have equal lengths. Parallel arcs
may optionally be represented by a single arc.)
3) Remove P and Q from K, insert P U Q in if and go to 1).
4) Stop with K* = K and GBC* = GBC.
The algorithm RP terminates in at most m 1 iterations as each
iteration reduces the number of elements of K by one.
An example application of RP is given in Figure 3.6.
For each composite index P in K* we define z* to be the common
location of every new facility i for which i e P. For the example of
Figure 3.6, let K* = {P^P^ with P = {1}, ?2 = {2,3,4}. Then
zPt Z1 and ZP2 z2 z3 z4
We let Z* be the location vector

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c) Iteration 1
b) Graph GBC
d) Iteration 2 and Termination
Figure 3.6. Example Application of RP

-103-
wit h components z*, P e K*, and call Z* the irreducible representation
of Z. Corresponding to GBC*, define DC* to be the distance con
straints with every constraint corresponding to exactly one arc
in GBC*. It will be convenient to refer to the triplet (Z*, DC*, GBC*)
as the reduction of (Z, DC, GBC). We remark that for an irreducible
location vector Z, the reduction of (Z, DC, GBC) is Identical to
(Z, DC, GBC), as RP terminates immediately in this case.
For P e K*, define A(N^) to be the set of adjacent nodes to .
in GBC*, and let Ap(Z*) be the collection of locations of facili
ties whose nodes are members of A(Np). The following property gives
the sufficient conditions for reducible location vectors (as well as
irreducible ones).
Property 3.5.3. Let (Z*, DC*, GBC*) be the reduction of (Z, DC, GBC)
and let K* be the list of composite indices for new facility nodes
of GBC*. If Zp e H[Ap(Z*)] for every P e K*, then Z is efficient.
Proof. By definition Z* is irreducible. Hence, the hypotheses of
the property imply, upon using Property 3.5.2, that Z* is efficient
with respect to the reduced constraints DC*. From Theorem 3.3.3,
for every P e K*, node Np is in a tight path in GBC*. Now, we want
to show that the original nodes i e P, are all in tight paths
in GBC. Recover GBC from GBC* by decomposing every node Np of GBC*
into its original nodes N^, i e P, and connect these nodes to one
another by arcs of zero length by adding those arcs which were
removed by RP. Since the added arcs have lengths of zero, the
shortest path lengths cannot change. Hence, the shortest path length
between any two existing facility nodes of GBC* is the same as the

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shortest path length between the same existing facility nodes in GBC.
Since Np is in a tight path in GBC*, then every original node Ih for
which i e P will be in a tight path in GBC, as the shortest path
lengths in GBC* and GBC are the same. But then every N^, 1 < i < m,
is in a tight path in GBC, as is in a tight path in GBC for every
P c K*, and U{P: P e '*} = {l,...,m}. Thus, upon using Theorem 3.3.3,
Z is the unique solution to DC and Z is efficient.
Proof of Sufficiency for Irreducible Location Vectors
We now return to the proof of Property 3.5.2. After presenting
a number of preliminary results, we will show that if Z is irreducible
and z_. e H[A^(Z)] for j e {l,...,m}, then every new facility node
is in a tight path in GBC.
The following lemma is proven in [22].
Lemma 3.5.1. Let P be a finite set of points each of which is in T.
For any p e P, we have H[P] = UL(p,p): p e P).
That is, the convex hull of P can be constructed by finding the
line segments joining an arbitrary element of P to every point
in P.
Next, we have the following lemma.
Lemma 3.5.2. Suppose Z is irreducible. Let and be two adjacent
new facility nodes in GBC. If z2 e H[A2(Z)] then there exists a
facility location y in A2(Z) such that
a) z2 e L(z^,y) and z^ ^ y,
b) whenever y is a new facility location, z2 f y.

-105-
Proof. Since and are adjacent, is in A^(Z). By Lemma 3.5.2,
we have H[A2(Z)] = U{L(z^,y): y e A^CZ)}. Since z2 e H[A^(Z)], for
at least one facility location y in A^(Z), z^ e L(z^,y). Also
z^ f y, for otherwise, z^ = y and z^ e L(z^,y) imply z^ = z2, contra~
dieting the irreducibility of Z. Hence, a) is established. Part
b) follows immediately from a) and the irreducibility of Z.
We remark that the irreducibility assumption cannot be relaxed
in Lemma 3.5.2, for otherwise we may have z^ = = y. Figure 3.5
illustrates such a case.
We will subsequently use Lemma 3.5.2 repeatedly to identify a
sequence of locations z z z. v such that they all lie
(1) (2.) (r)5 p y
in the line L(z^j,v ) in the given order. The corresponding sequence
of nodes N N. ,E in GBC will form a subpath connecting N.,.
(1) (r) p v B (1)
to with the length of that subpath equal to d(z^^,v^). By the
same token, we will find another node E in GBC with the subpath
q
connecting E^ to having length d(v^,z^). Then, we will show
that the two subpaths when connected at form a tight path which
contains N
(1)'
First we give the following result given in [82].
Lemma 3.5,3. Given four points p^ ,p^ jP^jP^ e T, if p2 e LPj^)
P3 e L(p2,p4) and p2 f p3> then d(Pl>p4) = d(p ,p ).
i=l

-106-
We can readily use induction to obtain the following generaliza
tion of Lemma 3.5.3.
Lemma 3.5.4. Given r points p ,... ,p £ T with r > 4, if
p e L(P.ji_iP1+i) for 2 ^ i < r-1, and if p 4 p+1 for 2 < i < r-2,
r-1
then d(p1,pr) = £
i=l
We are now ready to prove the sufficient conditions for irredu
cible location vectors. We remark that the arc lengths of GBC are
defined by the entries of D(Z), so that if N,N, *,E is a sub-
Vi) OO p
path P(N^,Ep) connecting to Ep, then the length of the subpath
is given by LP(N^,Ep) = d(z^^.z^) + ... + d(z^,v ).
Lemma 3.5.5 (Sufficiency). Suppose Z is irreducible. If, for every
j e {1,... ,m}, Zj e H[A^.(Z)], then every z^ is uniquely located.
Furthermore Z is efficient.
Proof. For notational brevity, let S = A (Z).
J 3
Choose any j in
{l,...,m}. Either N^. is adjacent to exactly one node or more than one
node. In the former case, S^ is a singleton, say, {y}. Since
Zj e H[Sj], Zj = y. By irreducibility of Z, y is an existing facility
location. Hence z_. is in the convex hull of uniquely located facili
ties so that Theorem 3.3.3 implies z^ is uniquely located in this
case.
For the other case, N. is adjacent to at least two nodes in GBC.
J
The hypothesis z_. e H[S^] implies there exist p,q e S_. with
£ L(pq)
(3.5.1)
If p and q are both existing facility locations, Theorem 3.3.3 implies
Zj is uniquely located. Hence, suppose, without loss of generality,
that q is an existing facility location, but p is a new facility

-107-
location (the case with both p and q new facility locations is very
similar to the proof we will give below and hence will not be con
sidered) Define z. = p. Find a sequence of locations z.,z. ,...
z. ,v for some r, 1 < r < m-1, by applying Lemma 3.5.2 to the pairs
Jr
(z.,z: ), (z,
),...,(z. ,v ) one at a time in the given order so
'1
j t
Jr
that the family of conditions in (3.5.2) is satisfied:
z. e L(z.,z. ) and N. N. are adjacent,
Jx J 32 J1 J2
z. e L(z. z. ) and N. N. are adjacent,
J2 Ji J3 ^2 J3
(3.5.2)
z. e L(z.
Jr 3
r-1
,vfc) and N
E are adjacent.
We remark that the irreducibility of Z and the conclusion of Lemma
3.5.2 guarantees that such a sequence can be found and will end with
an existing facility location v as, r can be at most m-1 and for
the last z. we must have some y e S. such that z. e L(z ,y) with
lr Jr Jr Jr-1
y necessarily an existing facility location.
Let q in (3.5.1) be the location v of existing facility s,
s
s ^ t. Then, the sequence v z.,z. ,...,z. ,v satisfies the assump-
s J Jr t
tions of Lemma 3.5.4 as a result of (3.5.1), (3.5.2), and the irredu
cibility of Z. Hence, we have
d(v ,v ) = d(v ,z.) + d(z.,z. ) + ... + d(z. ,v )
s c s J J Ji Jr
(3.5.3)
where the right hand side of (3.5.3) is clearly the length of the
path Eg,N.,N. ,...,N. ,Et> Hence, the path is a tight path due to
J J ^ J r
(3.5.3) and contains N Thus, z^ is the unique location of new

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facilty j. Since j is arbitrary, Z is the unique solution to DC, and,
thus, upon using Theorem 3.3.3, Z is efficient.
3.6 Algorithm to Construct Efficient
Location Vectors
To this point we have presented a family of conditions for char
acterizing efficient points. Theorem 3.3.3 provides the necessary and
sufficient conditions in terms of uniquely located facilities, tight
paths in GBC, and the convex hulls of uniquely located facilities.
Property 3.5.2 provides the sufficient conditions for irreducible
location vectors without requiring the identification of uniquely
located facilities. Property 3.5.3 extends the results of Property
3.5.2 to the case of reducible vectors.
Based on Properties 3.5.2 and 3.5.3, we now present the Sequential
Efficient Vector Construction Algorithm (SEVCA). Given a location
vector Z, the algorithm first finds the irreducible representation Z*
of Z by using RP. Then each component of Z* is checked to see if it
satisfies the convex hull containment property. If some component is
found which is not within the convex hull associated with it, its loca
tion is moved to the closest point in the convex hull. The procedure
is repeated with the resulting location vector. Termination occurs
when every component of the current irreducible vector is within the
convex hull associated with it. In order to prove finite termination
(in 0(m) iterations), we use a labeling scheme for the current com
posite indices. The list K is the list of composite indices during
K
any given iteration, while Z denotes the location vector whose com
ponents are indexed by the members of K.

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SEVCA
Initial
0) Given Z e Tm, set up GBC with arc lengths defined by entries of
D (Z). Define K = {{!},...,{m}}. Label each member of K unscanned
Reduction
1) If for some P,Q e K, P f Q, there exists an arc (N^.N^) in GBC of
length zero go to 2). Else go to 4).
2) Superimpose node Np on together with all arcs incident to N^.
Remove arc (Np,N ) from GBC. (If parallel arcs occur due to this
transformation, they will have equal lengths. Parallel arcs may
optionally be represented by a single arc.)
3) Remove P and Q from K, insert P U Q in K and label P U Q unscanned
Define zpyg to be the common location of Zp and z^ and go to 1).
Termination Test
4) If every member of K is scanned, stop. Else, choose an unscanned
composite index P in K and go to 5).
Check for Convex Hull Containment
5) Find A(Np), the set of nodes adjacent to in (current) GBC, and
,K.
define Ap (Z ) to be the set of current locations of new
existing facilities whose nodes are members of A(Np).
%
6)If Zp e H[Ap(Z )], label P scanned and go to 1). Else go
and
to 7).
Movement
7)Find the closest point, say, y to z^ in H[Ap(Z )]. Define
e(P) = d(Zp,y). Move zp to y. Update the arc lengths of GBC by
subtracting the amount e(P) from every arc incident to Np. Label
P scanned and go to 1).

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b) Iteration 1
Figure 3.7. Example Application of SEVCA

-lu
c) Iteration 3
Figure 3.7. Continued

-112-
e) Iteration 4
f) Iteration 5
Figure 3.7. Continued

-113-
g) Iteration 6 and Termination (in
Iteration 7)
Figure 3.7. Continued

-114-
An example application of SEVCA is given in Figure 3.7. For every
iteration, GBC and the current location vector is given. For iterations
6 and 7, the location vectors at the end of these iterations are shown
separately. For the other iterations, the location vector does not
change. Iterations 1 and 5 perform the Reduction routine. For the
other iterations, the node chosen in that iteration is the one inci
dent to every thickly-drawn arc. The associated convex hull is shown
by cross-hatched lines in the tree network. For any iteration the
circular-shaped new facility nodes of GBC are the unscanned nodes,
while the rectangular-shaped new facility nodes are the ones which
have been scanned prior to the given iteration or during that itera
tion. For any iteration, the numbers on the arcs of GBC show the arc
lengths at the beginning of that iteration. If the arc lengths change
during that iteration, the new arc lengths are indicated by the numbers
in parentheses.
By one iteration of SEVCA, we shall mean the execution of step 1)
through the last step. The last step of any given iteration is either
step 3), step 4), step 6), or step 7). Define, for i = 3, 4, 6, 7, t^
to be the total number of iterations which used step i) as the last
step. Clearly, t^ = 1. Since any given iteration uses only one of
these steps as the last step, the total number of iterations, denoted
by t, will be given by t = t^ + tg + t^ + 1. We want to show that
(k)
t 3m. For convenience, denote by K the list of composite indices
at the first of iteration k.
Property 3.6.1. The algorithm SEVCA terminates in at most 3m-l
iterations.

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Proof. We must show that t, the total number of Iterations, is at
most 3m-l. Define P =\J{K^^: 1 < k < t}. With this definition P
contains as members distinct (but not necessarily disjoint) subsets
of { 1,. .. ,m). Since = {{ 1{m} } every set {j}, 1 < j m,
is an element of P. If step 3) is never executed, then P = as
A'^ = ... = K^ in this case. Otherwise, whenever step 3) is executed,
(k)
say, in some iteration k, some two distinct members of K are removed
from A^), and their union is inserted in to obtain Hence,
if iteration k performs step 3), we know that the cardinality of
(k+1) (k)
a y is one less than that of Av and that exactly one member of
(k+1) (k)
K (the one inserted) differs from every element of K Clearly
then, step 3) can be executed at most m 1 times, and therefore P
contains at most m+(m-l)=2m-l distinct members. Hence t^,
the total number of iterations which used step 3), satisfies t^ < m- 1.
Now, imagine that we apply SEVCA a second time in exactly the same
order as the first application. In the second application, each time
step 6) or step 7) is used as the last step of an iteration, one
member of P will be labeled scanned (P is available as a result of
the first application). Since P contains at most 2m 1 distinct
members, clearly, tg + t^, the total number of iterations which used
either one of step 6) or step 7), will satisfy tg + t^ < 2m 1. It
follows then that t = t_ + t. + t, + t_ < 3m 1 as t_ £ m 1, t, =1,
3 4 6 7 3 4
and tg + t^ < 2m 1, completing the proof.
Next we have the following property. Let X* be the location
vector at the termination of SEVCA.
Property 3.6.2. The algorithm SEVCA terminates with an irreducible
location vector X* which is efficient. Furthermore the location vector

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X £ Tm whose irreducible representation is X* is efficient and satisfies
D (X) < D(Z) (3.6. 1)
where Z is the given vector in Tm to which SEVCA is applied.
Proof. Due to the Reduction routine it is evident that X*, the loca
tion vector at the termination of SEVCA, is irreducible. Let K* be
the list of composite indices at the termination. The Termination
Test implies every member of K* is scanned. But a composite index
can be labeled scanned only in either step 6) or step 7). In either
case, we have x* £ H[Ap(X*)] for every P e K*. Property 3.5.2 then
implies X* is efficient since every component of X* is in the convex
hull associated with it, and X* is irreducible.
To show (3.6.1), let x^ = x* for every i e P with P e K*. Thus,
X = (x^,...,xm) is the vector in Tm whose irreducible representation
is X*. Clearly, the entries of D(X) are either zeros or the arc
lengths of the reduced graph, say, GBC* at termination. Since the
arc lengths of GBC either remain the same or decrease by a positive
amount (in step 7)) from one iteration to the next, it follows that
D(X) < D(Z). The assertion that X is efficient follows immediately
from Property 3.5.3 and the fact that x* e H[Ap(X*)] for every
P z K*.
3.7 Efficiency for the Case of Rectilinear or
Tchebychev Distances
In this section we consider the analogous vector-minimization
k
problem in the k-dimensional Euclidean space, denoted by R for the
case of rectilinear (k = 2) or Tchebychev (k > 2) distances. Given

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1c
points x,y e R with x = (x^,...,x^) and y = (y^,...,y^), the rectilinear
distance between x and y is given by |x^ y^| + ... + |x^ + y^J, while
the Tchebychev distance between x and y is given by max{|x^ y^|,...,
|x y |}, where the symbol |*| denotes the absolute value sign.
K. K.
It is known that (proven in [32]) the distance constraints with
2 k
rectilinear distances in R or with Tchebychev distances in R k 2,
are consistent if and only if the separation conditions hold.
Based on this result, we characterize efficient location vectors
for the analogous vector-minimization problem which uses the recti
linear or Tchebychev distances.
Theorem 3.7.1. Let D(Z) be the vector of objectives with all entries
2
of D(Z) either the rectilinear distances in R or the Tchebychev
distances in R k > 2, as specified in (3.2.1). Let GBC be the graph
with arc lengths defined by D(Z). The following are equivalent:
(a) Z is efficient;
(b) Every arc in GBC of positive length is in a tight path.
Proof. To show (a) implies (b) suppose that Z is efficient. Let DC
be the distance constraints D(X) < U = D (Z). Since Z is a feasible
solution to DC, DC is consistent and hence the separation conditions
hold. Assume that there exists at least one arc in GBC with positive
length which is not in any tight path. Let (f ,f ) be such an arc with
P q
length e Since (f ,f ) is not in any tight path and since the
pq p q
separation conditions hold, every path which contains (f ,f ) is slack.
P q
Hence, for any path P(E.,E.) containing (f ,f ) we have
1 J p q
LP(E ,E ) d (v ,v ) > 0 (3.7.1)
J J
Define e' to be the minimum of the left side of (3.7.1) over all paths

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whlch contain (f ,f ), giving e' > 0. Choose any e such that
p q
0 < e < min(e',e ). Reduce the length of arc (f ,f ) from e to
Pq p q pq
e e and define the resulting graph to be GBC^. Due to our choice
of e clearly every separation condition on GBC^ holds, as, the length of every
path which contains (f ,f ) is reduced by an amount smaller than the
P q
difference between the path length and the distance between the existing
facility locations corresponding to the terminal nodes of the path,
while the length of any path which does not contain (f ,f^) remains
the same. Let DC^ be the distance constraints corresponding to GBC^..
Since the separation conditions on GBC^ hold, DC^ is consistent.
Letting Y be any feasible solution to DC ', it follows that D(Y) < D(Z)
and the entry of D(Y) corresponding to (f ,f ) is strictly smaller
than the corresponding entry of D(Z). Hence, Y dominates Z, contra
dicting that Z is efficient. Thus, (a) implies (b).
To show (b) implies (a) suppose every arc in GBC of positive
length is in a tight path. Let (f ,f^) be any arc with positive length
e For e > 0, let GBC^CDC^) be the graph (distance constraints) ob
tained from GBC(DC) by replacing e by e e. Since (f ,f ) is in
pq pq p q
a tight path, for any choice of e > 0, at least one separation con
dition is violated. Since the violation of a separation condition
implies the inconsistency of the distance constraints, there does not
exist e, e > 0, for which DC^ is consistent. Clearly, then, there
does not exist Y such that D(Y) < D(Z) and D(Y) ^ D (Z) which is the
definition of efficiency.
We remark that Theorem 3.7.1 holds for tree networks as well as
If
rectilinear distances on the plane, or, Tchebychev distances in R ,
k £ 2. The proof of the theorem relies on the necessity and sufficiency

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of the separation conditions for the distance constraints to be con
sistent. Furthermore, this equivalence is the only property that is
used. Hence, the theorem holds for any distance for which it is true
that the distance constraints are consistent if and only if the separa
tion conditions hold.
In Theorem 3.7.1 one does not have to worry about arcs with zero
lengths. To see this, partition D(Z) into subvectors D^(Z) and (Z)
so that D (Z) contains the zero entries, while (Z) contains the
positive entries. Clearly, in every feasible solution X to the con
straints D(X) < D(Z), the constraints corresponding to entries of
Dq(Z) will hold at equality. Hence the only way Z can be dominated is
by having (X) < (Z) and (X) f D^(Z). Thus, one needs to test only
those arcs with positive lengths in GBC to determine whether or not Z
is efficient. For this reason, Theorem 3.7.1 is applicable to both
irreducible location vectors and reducible location vectors.
2
We remark that for rectilinear distances in R and Tchebychev
1c
distances in R k > 2, the equivalences stated in Theorem 3.3.3
need not hold. An example of such a case is given in Figure 3.8 for
2
Tchebychev distances in R With reference to Figure 3.8, it is direct
to verify that both and are contained in the tight path
0^2 N^, E^) Clearly (z^,z^) is not efficient, as, z^ can be
moved to (2,2) and z^ can be moved to (3,1) thereby reducing the dis
tance between new facility 1 and existing facility 1 without increasing
any of the other distances. The resulting location vector is shown
in Figure 3.9. It Is direct to verify that every positive arc in GBC
of Figure 3.9 is contained in a tight path and hence the location
vector is efficient.

-120-
4
3
2
1
0
v,
v.
Z2
a) Facility Locations
b) Graph GBC
L(E1SE2) = 4
L(El5E3) = 5
L(ErE4) = 6
L(E2,E3) = 3
L(E2>E4) = 4
L(E3,E4) = 3
> 2 = d(v1}v2)
> 4 = d(v1#v3)
> 5 = d(v1,v4)
3 = d(v2>V3)
> 3 = d(v2,v4>
= 3 = d(v3,v4)
c) Separation Conditions
Figure 3.8. Example of a Dominated Vector with Tchebychev Distances
in

-121-
Figure 3.
4
2 o 1
1
O
v
z,
z.
V,
a) Facility Locations
b) Graph GBC
L(E1SE2) = 3 >
MEltE3) = 4 =
L(E15E4) = 5 =
L(E2,E3) = 3 =
L(E2,E4) = 4 >
L(E3,E4) = 3 =
2 = d(vlSv2)
4 = d(vl9v3)
5 = d(v^ >v4>
3 = d(v2,v3)
3 = d(v2,v4)
3 = d(v3,v4)
c) Separation Conditions
9. Example of an Efficient Vector with Tchebychev Distances
in

CHAPTER 4
THE BI-OBJECTIVE m-CENTER PROBLEM ON A
TREE NETWORK
4.1 Introduction
In this chapter we consider a bi-objective problem on a tree
network which involves as objectives the maximum of the weighted dis
tances between specified pairs of new and existing facilities and' the
maximum of the weighted distances between specified pairs of new
facilities. Such a vector-minimization problem may find applications
in locating emergency service units for the case when service units are
required to support one another in addition to providing service to
potential hazard zones (existing facilities).
The related literature on multi-objective location problems is
discussed in Chapter 1. Here, we concentrate on characterizing efficient
location vectors to the bi-objective minimax problem and constructing
the "efficient frontier," the set of objective values corresponding to
efficient points.
At this point we give an overview of the chapter. In Section
2 we give the necessary definitions and notation, and define the bi
objective problem of interest. In Section 3 we relate the bi-objec
tive problem to the distance constraints studied by Francis et al.
[32] and develop the necessary and sufficient conditions for efficiency
by making use of the results given for the distance constraints in

-123-
[32]. At the end of the section we provide an example problem.
Section 4 is devoted to the development of a procedure, E-FRONT,
for constructing the "efficient frontier." We prove the correctness
2 2
of the procedure and show that it is of 0(m (m + n )), where m and n
are, respectively, the number of new and existing facilities. The
section ends with an example application of the procedure.
4.2 Problem Statement, Notation, and Definitions
Let T be an imbedding of a finite undirected tree network with
existing facilities located at distinct vertex locations v^,...,vn. It
is of interest to locate m new facilities at points x^,...,x in T
under two objective criteria to be defined below. We suppose given
positive weights w and v k and denote by 1^ and Ig the nonempty sets of pairs
(i,j) and (j,k), respectively, for which the weighted distances
w..d(x.,v.) and v...d(x.,x, ) are of concern. We remark that it need
13 i 3 jk 3 k
not be the case that 1^, includes all possible pairs of new and existing
facility indices nor Ig includes all possible pairs of new facility
indices. For each location vector X = (x^,...,xm) in Tm we define two
objective functions f^ and f^ by
f ^X) = max(w_d(xi,Vj): (i,j) e Ic> ,
f2(X) = max{v^kd(x^. ,xk): (j,k) e Ig} ,
and we let f(X) = (f (X), f2(X)).
(4.2.1)
The Bi-objective m-Center Problem (with Mutual Communication)
is as follows:
V min{f(X): X e Tm} .
(4.2.2)

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As in Chapter 3, a location vector Y in Tm is said to be
efficient with respect to (4.2.2) if and only if X e Tm and f(X) < f(Y)
imply f(X) = f(Y). A location vector which is not efficient is said
to be dominated.
Our main interest is to relate the bi-objective m-center problem
to the distance constraints problem studied by Francis, Lowe, and
Ratliff [32], We shall characterize efficient points by making use
of the separation conditions (defined in Chapter 3) which are known
to be necessary and sufficient for the distance constraints to be
consistent.
To define the distance constraints of interest, let z = (z^,z^)
be any two-tuple (with z > (0,0)) and consider the constraints given
in (4.2.3):
d(x^,Vj ) < Zj/w^ (i, j ) e Ic
(4.2.3)
d(xjxk) Z2/Vjk (j>k) e Ig .
We shall refer to the family of constraints in (4.2.3) as DC^.
The constraints DC are said to be consistent if there exists at least
z
one feasible solution X = (x, x ) to (4.2.3).
1 m
Corresponding to DC^ we define GBCz to be the undirected graph
with nodes ,...,Nm,E^,...,En- For every (i,j) e 1^ there is an arc
(N.,E.) of length z./w.. and for every (j,k) s ID there is an arc
1 3 i ij a
(N.,N,) of length z/v., We partition the arc set of GBC into two
j K z j k. z
sets Ab and Ac with Ag = {(N.,Nk): (j,k) e Ig} and Ac = {(N,Ej):
(i,j) e Iq}. We shall assume that the sets 1^ and Ig are such that
GBC is connected, as otherwise DC decomposes into independent sets
z z
of constraints which may be analyzed separately.

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As in Chapter 3, denote by P(Fp,F^) any path in GBC^ connecting
nodes Fp and F A path is specified either by the sequence of nodes
in the path, or, by the sequence of arcs in the path. We denote by
L P(F ,F ) the length of the path P(F ,F ) and define L (F ,F ) to be
z p q w * p q z p q
the length of any shortest path connecting nodes Fp and F^. We say
P(Fp,F^) passes through an arc (F^,F^) if (F^,F^) is an arc in the
path. We say P(F ,F ) passes through AT1 (A) if the path passes
p ^ iJ u
through at least one arc in A^ (A^).
Associated with any path P(Fp,F^) we define two more terms, namely,
WP(F ,F ) and VP(F ,F ). The first term WP(F ,F ) is the sum of the
p q p q p q
reciprocal weights 1/w^ where the summation is taken over all arcs
(N.,E.) which are contained in the path P(F ,F ). If the summation
13 P q
is taken over an empty set, then WP(Fp,F^) = 0. Similarly, VP(Fp,F^)
is the sum of the reciprocal weights l/v^ over all pairs (N^.,N^) which
are contained in'P(F ,Fq). Again, VP(Fp,Fq) = 0 if Ag fl P(Fp,Fq) = 0.
The motivation for these two quantities can be given by observing the
relation
L P(F ,F ) = z -WP(F ,F ) + z_VP (F ,F ) .
z p q 1 p q 2 p q
(4.2.4)
The relation in (4.2.4) can be readily verified by observing that the
arc lengths of GBC^ are defined by the quantities and z2^vj^ so
that the length of any given path is the sum of the reciprocal weights
multiplied by or z^, whichever is applicable.
In what follows any path (in GBCz) we refer to is a path connecting
some two existing facility nodes Ep and E^. All other paths (for which
one or both of the terminal nodes are new facility nodes) will be
referred to as subpaths.

-126-
It was proven in [32] and stated in Theorem 3.3.1 of Chapter 3
that DC is consistent if and only if L (E E ) > d(v ,v ) for
z z p q p q
1 £ p < q n. The inequalities L^CE^E^) dCv^jV^) are called the
separation conditions and the separation conditions are said to hold
if every separation condition is satisfied.
It is direct to verify that whenever the separation conditions
hold (equivalently, whenever DCz is consistent) it necessarily follows
that L P(E ,E ) > d(v ,v ) for any path P(E ,E ). Conversely, whenever
zpq p q p q
L P(E ,E ) > d(v ,v ) for all paths P(E ,E ), it necessarily follows
zpq p q p q
that L (E ,E ) > d(v ,v ).
z p q p q
The definitions for tight and slack paths are given in Chapter 3
and will not be repeated here.
4.3 Necessary and Sufficient Conditions for Efficiency
In this section we develop the necessary and sufficient conditions
for efficiency by making use of the distance constraints. Our main
theorem states that a location vector Y is efficient if and only if at
least one arc in is contained in a tight path in GBC^, where GBC^
is the graph corresponding to DC^ obtained by letting z = f(Y).
Notationally, for any X = (X.,X0), DC is the distance con-
X Z ZA
straints with right hand sides (z1 X )/w.., (i,j) c I and
i I xj C
(z^ ^2^vjk e Ig* The graph GBCz ^ is the graph associated
with DC
z-X'
Before proving our main theorem, we first prove two lemmas
relating DC to DC and GBC to GBC ... We remark that "0" denotes
Z ZA Z ZA
either the scalar zero or the two-tuple (0,0). It will be clear from
the context what "0" refers to.

-127-
Lemma 4.3.1. Given a location vector Y used to define DC and GBC
z z
with z = (zpZ^ = (f ^ (Y) f 2 (Y) ) the following are equivalent:
(a) Y is efficient.
(b) Por any X = (X,,X ) > 0 and X 0, DC is inconsistent.
Proof. Using the definition of efficiency, f^, f2> and the fact
z = f(Y) we have the following equivalences. The location vector Y
is efficient if and only if f(X) < z implies f(X) = z if and only if
there does not exist X such that f(X) z and f (X) 5s z if and only if
for any X £ 0 and X ^ 0 there does not exist X for which f^(X) < z^ X^
for i = 1,2 if and only if for every X > 0 and X ^ 0 there does not
exist X such that max{wd (x^,v^) : (i,j) e 1^,} z^ X^ and
max{vM d(x. ,x, ) : (j,k) e 1^.} < z X if and only if for every X £ 0
jk j K a 2 2
and X £ 0 there does not exist X such that d(x.,v.) < (z, X,)/w..
i J 1 1 ij
for all (i,j) e Ic and dCx^.x^) < (z2 *2^Vjk for G *b
and only if for every X 0 and X ^ 0, DC is inconsistent, com-
Z" A
pleting the proof.
Corollary 4.3.1. Given Y with z = f(Y), Y is dominated if and only if
there exists X z 0, X 4- 0 such that DC is consistent.
z-A
We remark that the proof of Lemma 4.3.1 does not use any special
properties of tree networks. Hence the lemma is applicable to any
metric provided that f^ and 2 are the maximum of the pairwise weighted
distances.
The following lemma provides the sufficient conditions for DCr
to be consistent in terms of the slack paths in GBCz.
Lemma 4.3.2. Suppose DC is consistent. If every path in GBC which
2 Z
passes through Ag is slack then X = (Xj^) can be chosen with X £ 0
and X ^ 0 such that DC is consistent.
z-X
z-X

-128-
Proof. Since DC is consistent clearly z. 0, i = 1,2. We consider
z 1
the cases z^ > 0 and z^ = 0 separately.
Case with ^2 > 0. Let P(E^,E^) be any path which passes through
Ag. By hypothesis the path is slack so that L^PE^E^) d(vp>Vq) > 0.
Further VP(E^,E^) is positive since the path passes through A^. Hence
we have
[L P(E ,E ) d(v v )]/VP(E ,E ) > 0 (4.3.1)
z p q p q p q
Let e be the minimum of the left side of (4.3.1) over all paths which
min(e,Z2). Let
GBCz_^ be the graph with arc lengths (ij) e and (Z2 ^2^vjk*
(j,k) el. We want to show that the separation conditions defined on
D
GBC are satisfied.
z-A
Choose any two nodes E^ and E^. Let P(E^,E^) be a shortest path
in GBC connecting E and E Hence, we have
z-A p q
L (E ,E) = L ,P(E ,E) (4.3.2)
ZA p C[ ZA p CJ
Either P(E ,E ) passes through A or it does not. In the latter case
p q U
clearly the length of P(E ,E ) in GBC and GBC is the same, as every
p q z z-A
arc in A^ has the same length in both graphs. Since DC^ is consistent,
we have L ^P(E ,E) =LP(E ,E) > L (E ,E) >d(v ,v) so that the
z-A pq zpq zpq pq
separation condition for E and E is satisfied in this case. For the
P q
other case, P(E ,E ) passes through A^ so that its length on GBC is
given by
pass through A_.
Choose A = (A^^2) with A^ = 0 and 0 < A2
L ,P(E ,E )
Z-A p C|
= z WP(E ,E ) + (z A)VP(E ,E )
1 P q 2 2 p q
(4.3.3)

-129-
But L P(E ,E ) = z. WP(E ,E ) + zVP(E ,E ) so that from (4.3.3) we
zpq lpq 2pq
have
L ,P(E ,E ) = Li(E ,E ) X -VP(E ,E ) .
z-x pq zpq z pq
(4.3.4)
By our choice of X, we have 0 < X e [L P(E ,E ) d(v ,v )]/
J 2 2 zpq p q
VP(EpjE^). It follows then, upon using (4.3.4), that
L P(E ,E ) > L P(E ,E ) [
z-X p q ~ z p q' 1
L P(E ,E ) d(v ,v )
zpq P 9 i up/F F \
VP (E ,E ) * ^ ^Ep5 Eq^
p q
= d(v v )
p q
(4.3.5)
From (4.3.2) and (4.3.5) it follows that L ,(E ,E ) > d(v ,v ) for
z-X p q p q
this case. Since the choice of E^ and E^ is arbitrary, every separation
condition holds on GBC so that DC is consistent with X = (0,X),
ZA Z~A
X2 > 0.
Case with z2 = 0 By hypothesis every path which passes through
Ag is slack. Choose any path P(E^,E^) which does not pass through Ag.
Consistency of DC^ implies either LzP(Ep,E^) = d(v^,v^) or
L P(E ,E ) > d(v ,v ). The former case is not possible since a
z p q p q
subpath of length zero can be chosen from the arcs in Ag and this
subpath can be appended to P(E^,E^) to obtain a new path, say, P'(E^,E^)
without increasing the length of the path. Hence L^PiE^jE^) =
L^P'(Ep,Eq) = d(Vp,v^) contradicting that every path which passes
through Ag is slack. Thus, every path which passes through A^, is also
slack. Define e to be the minimum of [L P(E ,E ) d(v ,v )1/WP(E ,E )
zpq p q p q
over all paths in GBC2. Clearly e > 0, since every path is slack and
every path necessarily passes through A so that WP(E ,E ) > 0. Choose >
C P q
(X, ,0) with 00. Consider any path P(E ,E )
1 i i I p q
on

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GBC The length of P(E ,E ) on GBC is L ,P(E ,E ) = (z, A.)*
z-X & p q z-X z-X p q 1 1
p q
WP(E ,E ) + zVP(E ,E ). But z. = 0 so that L ,P(E ,E ) = L P(E ,E ) -
p q p q / zA p q z p q
X,WP(E ,E ). By our choice of X. we have X, < [L P(E ,E ) -
1 p q J 1 1 z p q
d(v ,v )]/WP(E ,E ). Hence, L ,P(E ,E ) > L P(E ,E ) {[L P(E ,E ) -
p q p q z-X p q z p q z p q
d(v,v )]/WP(E,E )}-WP(E ,E ) = d(v ,v ). Thus, L ,P(E ,E ) > d(v ,v )
pq pq pq pq z-x p q p q
for any path P(E^,E^) so that the separation conditions on GBCz_^ hold
and is consistent with X ^ 0, X ^ 0, completing the proof.
Next, we have the necessary and sufficient conditions for efficiency.
Theorem 4.3.1. Given a location vector Y used to define DC and GBC
z z
with z = (z^,Z2) = f(Y), the following are equivalent:
(a) The location vector Y is efficient.
(b) At least one arc in is contained in a tight path.
(Equivalently, there exists at least one tight path which passes
through Ad.)
15
Proof. To show (a) implies (b), suppose Y is efficient. Assume that
no arc in A^ is contained in a tight path. Hence every path which passes
through A^g is slack as DCz is certainly consistent. Lemma 4.3.2 implies
X = (X,,X0) can be chosen with X > 0 and X ^ 0 so that DC is con-
1 l z-X
sistent. Corollary 4.3.1 then implies Y is dominated, contradicting
that Y is efficient.
To show (b) implies (a) suppose at least one arc in A^ is in a
tight path. Let P(E ,E ) be such a path which passes through A^ and
P q B
which is tight. Clearly, P(E ,E ) also passes through A For any
p q c
X = (X ,X) >0, X 0, the length of P(E ,E ) in GBC .. will be
i z p q za
strictly smaller than its length in GBC as at least one of z. and z
Z 1 z
is reduced by a positive amount due to X being different from (0,0).
Hence, for any X ^ 0, X ^ 0, the separation condition on GBC

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corresponding to E and E will be violated, as, A _. (E ,E ) <
P Q z A p q
L ,P(E,E) Z A p (J Z p CJ p CJ
DC is inconsistent. Lemma 4.3.1 then implies Y is efficient.
z-A
We remark that Theorem 4.3.1 considers only those tight paths
which pass through Ag. The reason is as follows: Any path in GBC^
passes through A so that if there exists a tight path which contains
an arc in Ag, then the same path necessarily contains an arc in A^,.
However, an arbitrarily chosen path need not pass through Ag. For this
reason, the assumption that there exists at least one arc in A^, which
is contained in a tight path does not imply that at least one arc in
Ag is contained in a tight path. Hence, if a location vector Y is
efficient then there is at least one arc in A^ which is contained in
a tight path while the reverse implication does not hold.
Further, we remark that the proof of Lemma 4.3.2 is based on the
necessity and sufficiency of the separation conditions. Hence, Lemma
4.3.2 is applicable to tree networks as well as the analogous problems
with rectilinear distances on the plane, or, the Tchebychev distances
in the k-dimensional Euclidean space with k > 2. Further, Theorem
4.3.1 uses Lemmas 4.3.1 and 4.3.2 for its proof. Hence, the theorem
is applicable to tree networks as well as rectilinear distances on the
lc
plane and the Tchebychev distances in R k > 2.
At this point we give an example of a non-efficient and an
efficient location vector. In Figure 4.1 the tree network is shown
along with the distance matrix and the weights w.. and vfor
13 Jk
(i,j) e Ic = {(1,1),(1,2),(2,3),(2,4),(3,4),(3,5)} and (j,k) e Ifi -
{(1,2),(1,3),(2,3)} for the example bi-objective m-center problem.
In Figure 4.2a) we give an example of a dominated location vector X.

-132-
y<~
7-
2
y
+-
9
vr
y
4
a) Tree T
v2 v3
V4
V5
W11
= 1/5
V12
1/3
V1
6 4
8
10
W12
= 1
V13 =
1/6
V2
2
6
8
W23=
= 1/3
V23 =
1/2
V3
4
6
W24
= 1/2
V4
2
W34
= 1/4
W35
= 1/3
b)
Distance
Matrix
c)
Weights
Figure 4.1. Data for Example Bi-Objective m-Center Problem

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a) A Non-Efficient Location Vector
with f(X) = (1.5,1)
b) An Efficient Location Vector
with f(Y) = (1.5,0.5)
Figure 4.2. Example of Non-Efficient and Efficient Location
Vectors

-134-
Using the weights given, the value of f(X) is (1.5,1). Graph GBC^
with z = (1.5,1) is also shown with arc lengths as indicated. It is
direct to verify that GBC^ in Figure 4.2a) does not have any tight
paths. Hence, X is dominated. The location vector Y in Figure 4.2b)
is an efficient one and dominates X. The value of f(Y) is (1.5,0.5).
The thickly drawn arcs in GBC^ of Figure 4.2b) form a tight path. We
remark that in every feasible solution to DC^ for z = (1.5,0.5), the
locations y^ and y^ ate the same, as and ^ are contained in a
tight path.
4.4 Construction of the Efficient Frontier
Let S be the set of all efficient location vectors in Tm. Define
Z and Z* by
Z = {(z^,Z2): 3X e T such that f(X) = (z^,Z2)} ,
Z* = {(z^,z^): 2X c S such that f(X) = (z^,z^)}
That is, Z = f(Tm), the image of Tm under f, and Z* = f(S), the image
of the efficient set S under f. We call Z the objective space and Z*
the efficient frontier. Our main interest in this section is to develop
a method to construct the efficient frontier. One can display Z*
graphically on the (z^^) plane and obtain much of the insight about
efficient points. In general, for any convex bi-objective problem,
the efficient frontier and the objective space may look like the
illustration given in Figure 4.3. The objective space is the shaded
region and the efficient frontier is the thickly drawn part of the
boundary of Z.

-135-
Figure 4.3. Illustration of Z and Z* for a Convex
Bi-Objective Problem

-136
We shall first state a theorem due to Wendell [110] which gives
a global characterization of the efficient frontier. Then we will
exploit the result of the theorem to construct Z*.
m /s
Let a be the minimum value of f^ on T b be the minimum value of
f^ on Tm, and b be the minimum value of f^ over all minima to f^. The
A
values a, b, and b are displayed in Figure 4.3 for an arbitrary bi
objective problem with convex objectives. For each z^ e [a,b] define
the function e^) to be the minimum value of the problem P defined below
Z1
e(Zl) = minf2(X): f (X) < z X e Tm) .
Wendell [110] showed that whenever f^,f2 are lower semicontinuous
convex functions defined over a nonempty convex compact set S, the
efficient frontier is the set {(z^,e(z^)): a < z^ < b}. Wendells
theorem is applicable to the bi-objective m-center problem, as Tm is
convex, compact, and nonempty, and f^ and f2 are continuous convex
functions (see [22]) over Tm. For an arbitrary choice of z^, the
value of e(z^) is marked in Figure 4.3.
The computation of a, b, and b presents no difficulties and will
be given subsequently.
Using the definitions of f^ and an equivalent definition of
e(Zj) is as follows:
e(zp = min Z2
s. t.
d(x,Vj) < z^w (i,j) e Ic
d(Xj,xk) z2/vjk e IB
where z^ is understood to be fixed to any value in [a,b]
(4.4.1)
The constraints

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pf (4.4.1) are just DC¡z. We remark that for any z^ e [a,b], there
exists at least one feasible solution to (4.4.1), as there exists an
X such that f^(X) = a z^ and can be chosen large enough so that
f200 z2.
Let z^ be fixed with z^ e [a,b] and let (Y,z2) be any feasible
solution to (4.4.1). Define z = (z ,z) and let DC be the distance
J. z z
constraints in (4.4.1). Since (Y,z2) is feasible to (4.4.1), the
separation conditions on GBC hold so that every path in GBC has
z z
length at least as large as the distance between the locations of
existing facilities corresponding to the terminal nodes of the path.
Hence, for any path P(E^,E^) we have
L P(E ,E ) = z WP(En,E ) + zVP(E ,E_) > d(v,v ) ,
zpq lpq 2pq pq
or, equivalently,
d(v ,v ) WP(E ,E )
2 lVP(E ,E )J llVP(E ,E )J
p q p q
(4.4.2)
Defining x(z^) to be the maximum of right side of (4.4.2) over all paths
in GBC^, it follows that z^ > x(z^) whenever z2 is feasible to (4.4.1).
Hence, the minimum value of z2 which solves (4.4.1) is x(z^). We
observe that the right side of (4.4.2) is the value of a linear func
tion (of z^) evaluated at z^. There are as many such linear functions
as there are paths in GBCz. Further x(z^) is the maximum of these
functions at z^. Thus, defining x(*) to be the pointwise maximum of
these linear functions over the interval [a,b] we have e(zp = x(zp
for every z^ £ [a,b]. Hence, a brute-force method to construct the
efficient frontier is to enumerate all paths on GBC, compute the
parameters (slope and intercept) for each linear function corresponding

-138-
to each distinct path, and equate e() to the pointwise maximum of
these linear functions over the interval [a,b]. Such a method is not
computationally efficient as there may be a very large number of paths
For achieving computational efficiency we shall restrict our attention
to a certain subset of the set of all possible paths and then evalute
e(*) by taking the pointwise maximum of the linear functions cor
responding to these paths.
In general, an arbitrarily chosen path P(E^,E^) may pass through
several existing facility nodes distinct from E^ and E^. First, we
want to show that paths of this type need not be considered. Define
any path to be a decomposable path (d-path) if the path passes through
at least three (distinct) existing facility nodes. An example of a
d-path which passes through four existing facility nodes is
(E^, Nj-, Eg, N^, E^, N^, E^) Define any path to be a non-
decomposable path (nd-path) if the only existing facility nodes the
path passes through are its terminal nodes. Every d-path can be de
composed into a (unique) collection of nd-paths which, when appended
end to end, gives the original d-path. The decomposition of the
aforementioned example d-path into its nd-paths is {(E,, Nc, N1ft, E,),
1 j 10 o
(Eg, N^, E^), (E^, N^, E^)}. Clearly, any nd-path uses exactly two
arcs in A^,, while any d-path uses at least four arcs of A^,. An nd-
path may or may not use arcs of A^.
Next, we have the following lemma, which permits us to check the
separation conditions by only evaluating nd-paths. The lemma is
applicable to any distance constraints problem defined in Chapter 3
by (3.3.1). We use the notation of Chapter 3 for the lemma.

-139-
Lemma 4.4.1. Let DC be the distance constraints specified in (3.3.1)
and let GBC be the associated graph. The separation conditions on GBC
hold if and only if for every nd-path P(E ,E ), LP(E ,E ) > d(v ,v ).
pq p q p q
Proof. Suppose the separation conditions hold. Choose any nd-path
P(E ,E ). We have LP(E ,E ) > L(E ,E ) > d(v ,v ) as the length of
p q p q p q p q
P(Ep,E^) is at least as large as the shortest path length between E^
and E .
q
Suppose for any nd-path P(E^,E^) we have LP(E^,E^) > d(Vp,v^).
Choose any two existing facility nodes, say, Eg and E^, with
1 s < t < n. Let P(Eg,Et) be any shortest path connecting Eg and E^.
If P(E ,E ) is an nd-path then clearly L(E ,E ) = LP(E ,E ) > d(v ,v )
so that the separation condition for Eg and Efc is satisfied in this
case. Consider the case when P(Eg,Et) is a d-path. Decompose P(Eg,Et)
into its nd-paths, say, P(E ,E,P(E. .,E ). Hence,
S^IJ (V) t
LP(Eg,E^) > d(vg,v^^) ,... ,LP(E^rj ,Efc) > d(v(r)vt)> as the Paths
are nd-paths. Further, the length of P(Eg,Et) is the sum of the
lengths of P(Eg,E^j) ,... ,P(E^ jE^). Hence, upon using the triangle
inequality, we have d(v ,v ) d(v ,vnv) + ... +d(v. .,v ) <
St s v. X) ) t
LP (E ,E...) + ... + LP(E. ,E ) = LP(E ,E ) = L (E ,E ) so that the
s (1) (r) t s t s t
separation condition for E and E is satisfied for this case. Since
s t
the choice of Eg and Efc is arbitrary, the proof is complete.
We are now ready to present the procedure for constructing the
efficient frontier. We define G^ to be the graph with nodes
N^,...,Nm and the arc set A^. To every arc (N^,N^) of G^ we assign
the length 1/v For 1 < s < t m we denote by mgt the length of a
shortest path connecting the nodes Ng and Nfc in GThe computation of
3
mgt_, 1 < s < t < m, can be achieved in 0(m ) operations by using known
algorithms (see Dreyfus [23]).

-140-
The following algorithm, E-FRONT, constructs the efficient frontier.
We assume m has been computed for 1 A s < t < m.
E-FRONT
0) Label the arcs in as a^,...,ar where r is the cardinality of
A Define A' = {(a.,a.): 1 < i < j < r}. Delete from A' every
0 x j
pair (a^,a^) for which a^, and a^ are incident to the same new
facility node. Let A be the resulting subset of A' after the
deletions.
1) For every (a.,a.) e A define the linear function t.,(z.) as
i 3 iJ 1
follows: Suppose a. = (N ,E ) and a. = (N ,E ). Due to step
i s p/ j t q
clearly s / t and thus m > 0. For z^ e [a,b]
0)
d(Vr,Vr,) (1 + 1/..)
X. .(Z.) = E_3_ §£ tq_
y 1 st 1 st
2) Define x(z^) = max{x (z^) : (a^a^) e A} for z^ e [a,b]. The
efficient frontier is given by Z* = {(z^,x(z^)): a £ b}.
The next theorem establishes the correctness of the algorithm.
2 2
Then we will show that the algorithm is 0(m (m + n )).
Theorem 4.4.1. The algorithm E-FRONT constructs the efficient frontier
for the bi-objective m-center problem.
Proof. By Wendell's theorem Z* = {(z^,e(z^)): a < z^ < b}. Hence, it
suffices to show that e(z^) < x(z^) and e(z^) > x(z^) for a < z^ < b.
To show e(zp x(z^), choose any z^ e [a,b] and define
z = (z1,z) with z = xCz,). Let DC and GBC be the constraints of
12 2 1 z z
the problem P and the associated graph, respectively. Choose any
Z1
nd-path P(E ,E ). Either the path passes through A^ or it does not.
p q a
In the latter case P(E ,E ) is the path (E ,N ),(N ,E ) for some new

-141-
facility node Ng. Let X be any location vector for which f^(X) = a.
Hence, w d(x ,v ) < a and w d(x ,v ) < a. But a < z. and the length
sp s p sq s q 1
of P(E ,E ) is z.(1/w + 1/w ) so that we have, upon using the tri-
p q 1 sp sq
angle inequality, L P(E ,E ) = z (1/w + 1/w ) > a(l/w + 1/w ) >
& n 3 z p q 1 sp sq sp sq
d(x ,v ) + d(x ,v ) > d(v ,v ). Thus, for any nd-path P(E ,E ) which
s p s q p q 3 1 p q
does not pass through A^, we have
L P (E ,E ) > d(v v )
z p q p q
(4.4.3)
For the other case, P(E ,E ) passes through A,, so that its length is
P q B
given by L P(E ,E ) = z,WP(E ,E ) + zVP(E ,E ). Since the path is an
zpq 1 p q 2 p q
nd-path it passes through exactly two arcs in A^, say, (E^,Ns) and
(N ,E ) with s t. Thus, WP(E ,E ) = 1/w + 1/w while
t q p q sp tq
VP(E ,E ) ra by the definitions of m and VP(E ,E ). It follows
p q st st p q
that
L P(E ,E ) > z.(l/w + 1/w ) + z m
zpq 1 sp tq 2 st
Due to steps 1) and 2) of E-FRONT, we have
(4.4.4)
z2 = T(zl) -
L_£_ z
m ^ Z1
st
(1/w + 1/w )
Sp tq
ra
st
(4.4.5)
Using (4.4.4) and (4.4.5), for any nd-path which passes through A^ we
have
IP(E E ) > d(v ,v ) (4.4.6)
z p q p q
From (4.4.3) and (4.4.6), L P(E ,E ) > d(v ,v ) for every nd-path in
zpq-pq
GBC¡z so that Lemma 4.4.1 implies the separation conditions on GBCz
hold. Hence, z is feasible to P and thus e(z.) x(z.) = z.
z. 112

-142-
To show e(z ) > tCz.), let a.,a. be arcs in A for which
1 i i J G
t(z,) = x..(z,). Suppose a. = (N ,E ) and a. = (N ,E ). By step 1)
1 lj 1 l s p 3 t q J
of E-FRONT we have
VZP
d(vv)
p q
(l/w + l/w )
sp tq
m
st
m
(4.4.7)
st
But m is the length of a shortest path in Gg connecting Ng to N .
Let (N ,N, ...,N,.,N ) be such a shortest path in G. Let P(E ,E ) be
s k f t r B p q
the path (E ,N ,N, ,...,N£,N,E ). Define z = (z,,z0) with z = e(z.).
p s k f t q 12 2 1
Since z is feasible to P DC is consistent so that for the path
2 z^ z v
P(E ,E ) identified above we have
P q
LP(E >E ) > d(v ,v ) .
z p q p q
(4.4.8)
But LzP(E^,E^) = z^(l/wSp) + z2mst + Zl^^Wtq^ aS Pat^ consists of
the arcs (E ,N ) (N ,N. ),...,(N,,N),(N ,E ). It follows then from
ps sk. it tq
(4.4.8) that z.(l/w + 1A* ) + z0m ^ > d(v ,v ), or, equivalently,
1 sp tq 2 st p q
Z2
d(v ,v )
JB SL
(1/w + l/w )
sp tq
m
st
m
(4.4.9)
st
But the right side of (4.4.9) is = x(z^) while z^ is e(z^) by
definition, hence, e(z^) > x(z^).
The inequalities e(z^) < t(z^) and e(z^) > t(z^) imply e(z^) = i(z^)
for every e [a,b]. Hence, Z* = {(z^tCz^)): a < z^ < b), com
pleting the proof.
2 2
We now show that the computational order of E-FRONT is 0(m (m + n ))
The algorithm constructs Z* by identifying no more than r(r l)/2
linear functions. To identify the linear functions one must first

-143-
calculate m for 1 < s < t < m, which requires 0(m ) operations.
Every linear function is determined by computing its slope and inter-
3 2
cept so that steps 0), 1), and 2) require 0(m + r ) operations. But
r can be at most mn so that excluding step 3) and the computation of a
3 2 2 2
and b, the algorithm is 0(m + (mn) ) =0(m (m+n )). Each linear
function has positive intercepts and negative slope. Clearly, their
pointwise maximum is a piecewise linear decreasing function over the
interval [a,b]. Hence, t(*) can be constructed by finding its break
points. Each break point is determined by the intersection of some two
linear functions. Since each linear function is strictly decreasing,
any linear function can determine at most two (consecutive) break
points. Thus, there are at most 2*(r)(r l)/2 = r(r 1) break points.
Hence, excluding the computation of a and b, the algorithm requires
2 2
0(m (m+n )) operations, as the computational effort for constructing
the linear functions dominates the computational effort for finding
the break points of t(0-
To compute a, define, for every new facility index i, the set 1^
by ^ = ij : (i,j) e ICL Letting g^x^ = max{w^d (x^v..) : j c 1^,
it is direct to verify that f^(X) = max{g^(x_^): 1 i < m). Hence,
fj is separable and its minimum value is given by a = max{g*: 1 < i < m)
where g* = min{g^(x): x e T}. The Kariv-Hakimi procedure in [65] com
putes g* in 0(iI|log|I|). Since |l | < n, the computation of a
requires no more than mnlogn operations. Hence, the computational
effort for identifying the linear functions again dominates the com
putational effort for computing a.
To compute b, we must first compute b. Clearly, b = 0 as it is
the minimum value of f 2 (X) = maxi'v fcd (x^ ,xfc) : (j,k) e IB>. It is direct
to verify the following equalities:

-144-
b = minf^(X): ^2^^ b)
= minify (X) : v^dCx^.x^) < 0 (j ,k) e Ig}
= minif^(x,..,x): xeT)
= min max g (x)
xeT l = min max max{w..d(x,v.): j e I. }
xeT l = min
xeT
max{w_.d (x,Vj)
1 < j < n}
where
w. = max(w,,: over all i for which i e I.}
J ij i
(4.4.10)
(4.4.11)
Thus, the value of b is obtained by solving the absolute 1-center
problem defined by (4.4.10) and (4.4.11), and will require O(nlogn) opera
2 2
tions. -Therefore, the computational effort for E-FRONT is 0(m (m + n ))
and is determined by the computational effort for steps 1), 2), and 3)
of E-FRONT.
Once the efficient frontier is constructed, efficient location
vectors can be identified as follows: Choose z = (z^,z£) in Z* with
z2 = x(z^) = T£j(z^)> say, and identify the arcs a^ and a^. Supposing
a. = (N ,E ) and a. = (N. ,E ), let P(N ,N 1 be a path in whose length
i s p j t q s t r B &
is mgt_. Then, clearly, (Ep,P(Ng,Nt) ,E^) will be a tight path in GBCz
with length L P(E ,E ) = z,WP(E ,E ) + z0m Every facility whose
zpq l pq zst
node is in the path P(E^,E^) is uniquely located on the line L(v^.v^)
in T with the same ordering and spacing as the nodes which lie in
P(Ep,E^). Hence, the new facility locations corresponding to new
facility nodes in P(E^,E^) can be readily identified (see Property
3.3.1 of Chapter 3). The locations of other new facilities can be

-145-
found by applying the Sequential Location Procedure of [32] to the
constraints DCz after fixing the locations of the uniquely located
facilities. Any such feasible solution Y to DC^ is clearly an effi
cient location vector.
At this point we give an example application of E-FRONT. We
apply the algorithm to the tree network for which the necessary data
are given (earlier) in Figure 4.1. The arcs in A^ are labeled a^
through a^ as shown in Figure 4.4a). Figure 4.4b) shows the graph
GBC^j ^ with arc lengths l/w and l/v^. The thickly drawn subgraph
of GBC,, is the graph GD. The values of m 1 £ s < t £ 3, are
\I j I / D St
also given in the same figure. Corresponding to each pair of arcs
(a^,a^) e A the linear function T^j(zp is specified in Table 4.1.
These functions are plotted in Figure 4.5, and t(z^) is indicated by
the cross-hatched line in the same figure. The values of a and b are
computed by using the techniques given earlier. We remark that
t(z^) = T24^zl^ fr zl e for this example problem. Since
a2 = anc* a4 = ^2^4^ anc* m12 = t^ie -^-en8t^ f arc
^N1N2^ t^ie Path (E2^i^2^4^ as a path for any choice of
(z^,Z2) in Z*. For example, for z^ = 1.5 and Z2 = 0.5, (z^,Z2) is on
the efficient frontier. The tight path corresponding to this choice
of (z^,Z2) is shown earlier in Figure 4.2b), together with the
corresponding location vector Y = an t^ie same figure.
Note that y^ and are uniquely located as N^ and N2 are on the
tight path.

-146-
Figure 4.4. Data for Example

-147-
Table 4.1.
Example Linear Functions
T13(zl>
= 1.33 2.67z1
t14(z1^
= 2.67 2.33zl
t15(z1)
= 1.6 1.8z1
t16(z1)
= 2 1.6z1
t23(z1>
= 0.67 1.33z1
t24(z1*
= 2 21
t25(zP
= 1.2 Z]L
T26 =1.6- 0.8z^
W
= 2 3.5Zl
T36(z1)
= 3 3z^
t46(zI>
= 1 2.5zj

-148-
Figure 4.5. Efficient Frontier for Example

CHAPTER 5
SUMMARY AND FUTURE RESEARCH
5.1 Summary
In this dissertation we considered problems which involve locating
multiple new facilities on a tree network with respect to n existing
facilities at known locations.
In Chapter 2, we solved the nonlinear p-center problem where the
objective is to minimize the maximum cost associated with any existing
facility. The cost (disutility) of service associated with any
existing facility is a nonlinear (strictly increasing and continuous)
function of the distance between that existing facility and its nearest
new facility. We gave a weak and a strong duality theorem and pro
vided a physical interpretation of the dual problem. Our approach for
solving the nonlinear p-center problem and its dual was to solve a
sequence of covering problems which involve minimizing the number of
new facilities subject to a family of n distance constraints which impose
upper bounds on the distances between any existing facility and its
nearest new facility.
In Chapter 3, we considered a vector-minimization problem which
involves as objectives the distances between specified pairs of new
and existing facilities and specified pairs of new facilities. We
developed the necessary and sufficient conditions for efficiency and
provided an algorithm for constructing efficient solutions. Our
L 49

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approach to the vector-minimization problem was to reformulate the
problem in terms of a family of distance constraints which impose upper
bounds on the distances between specified pairs of new and existing
facilities and specified pairs of new facilities.
In Chapter 4, we considered the bi-objective m-center problem
(with mutual communication) which involves as objectives the maximum
of the weighted distances between specified pairs of new and existing
facilities and maximum of the weighted distances between specified pairs
of new facilities. We developed the necessary and sufficient condi
tions for efficient solutions and provided a procedure for construct
ing the efficient frontier. Our approach was to reformulate the problem
in terms of a. family of distance constraints which impose upper bounds
on the distances between specified pairs of new and existing facilities
and specified pairs of new facilities.
In what follows we give certain generalizations of the problems
considered in this dissertation as well as other location problems
considered in the literature. We point out some possible directions
for future research.
5.2 Generalized Multi-Center Problem
Here, we define a problem which generalizes the p-center problem
and the m-center problem with mutual communication. For convenience,
we consider the weighted case. Nonlinearity can be obtained by re
placing each weight by a strictly increasing continuous function of
the associated distance.

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Let be m-collections of centers with |x_^| = p_^ for
1 < i m, where each p^ is a given positive integer. Let V^,...,Vn
be n collections of existing facility locations. The elements of X^
are x^,...^1 with each x^ e T. The elements of V. are
1 p 2 i 1 n
i X X
with each v. a vertex of T. Let X = {X,,...,X }. For any two finite
J 1 m 7
subsets P and Q of T let D(P,Q) = min[d(p,q): p e P, q e Q]. Define
the function f by
f(X) = max{max{w D(X ,V ): (i,j) e I~} ,
1J 1 J V-
max{Vj^D(Xj >X^): (j,k) e Ig}} .
The Generalized Multi-Center Problem (GMCP) is as follows:
min[f(X): |x ¡ = p^, X_^ C T for 1 < i < m]
An equivalent statement of GMCP in terms of distance constraints is
as follows:
mm
s. t.
D(X.,V.) z/w..
i J 1J
j'V s z/vjk
X. p.
1 X 1 1
(ij) e Ic
(j,k) e IB
1 i ^ m
For the case with m = 1 and each = {v..}, GMCP specializes to
the p-center problem. For the case with each p^ = 1 and each = {v^},
GMCP specializes to the m-center problem with mutual communication.
We pose the following questions for future research.
Ql. What special cases of GMCP are tractable? Some of the special
cases are obtained by taking each weight unity, or,

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taking p = for each i, or taking Vj = {v^} for
each j.
Q2. Can we use (or modify) COVER and/or the Sequential Location
Procedure of [32] to solve either of the related covering problems de
fined by
min p, + ... + p
rl rm
D(X,V )
< z/w..
ij
(i,j) e Ic
D(Xj,Xk)
- 2/vjk
. (j >k) e IB
or,
min max{p ^ ,... p }
s. t.
D(X,V ) < z/w (i,j) e I
z/vjk (j,k) e I
C
B
where z is fixed?
Q3. Are the separation conditions of direct use for determining the
consistency of the distance constraints of GMCP?
Q4. Can we extend the duality results of Chapter 2 to GMCP?
Q5. Is there a dual to either of the related covering problems?
Q6. What kind of applications may GMCP find?
Q7. Can the search for the minimum objective value be confined to
a finite set of numbers?
We remark that the minimum objective value of the m-center problem
with mutual communication is the maximum of a finite number of ratios
with the numerators distances between existing facility locations while
the denominators are sums of reciprocal weights which correspond to

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shortest path lengths in a related graph. Hence, there appears to be
hope for extending the duality results of Chapter 2 to GMCP.
5.3 The t-Objective m-Center Problem: Steps Towards
a Unified Theory
Here we define a location problem which involves t minimax type
objectives. Special cases of the problem are the m-center problem with
mutual communication, the vector-minimization problem of Chapter 3, and
the bi-objective m-center problem of Chapter 4. We give a theorem
which unifies the independent results for each of these problems. An
outline of the proof of the theorem is also provided.
Given sets I and I let kn = {(N.,E.): (ij) e I0} and
C d G 1 j G
Ag = {(N^jN^): (j,k) e 1^}. On defining A = A^ U A^, we suppose given
t nonempty, mutually disjoint, exhaustive subsets of A, enumerated as
A, ,...,A Associated with A 1 < r < t, the rth objective f is
I t r J r
defined by
f (X) = max[max{w. ,d(x. ,v.): (N.,E.) eA ) A.} ,
rv iJ i j i j r C
max{vjkd(Xj,xk) : £ Ar /I A^}]
where, by convention, the value of either of the inner maximizations
is understood to be zero if the maximum is taken over an empty set.
Letting f(X) = (f^(X),...,f (X)), the t-Objective m-Center Problem
is as follows:
V min[f(X): X e Tm] .
For the case t = 1 and A^ = A^ U A^, the problem specializes to the
m-center problem with mutual communication. For the case with each A^

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corresponding to exactly one arc in U A^, the problem specializes to
the vector-minimization problem considered in Chapter 3. For the case
t = 2, = A^, and = A^, the problem specializes to the bi-objective
m-center problem of Chapter 4.
Consider the related distance constraints DC where z = (z. z ),
z It
defined below:
d(x.,v.) < z /w. (N.,E.) e A PI A, 1 < r < t ,
i 3 r xj x j r C-
d(x. ,x, ) z /vM (N. ,N, ) e A' H A,., 1 £ r < t
j k r jk j k r a
It is direct to verify the following assertion: Let X be given and
define z = f(X). The location vector X is efficient if and only if for
every X ) > 0, X 4- 0, DC is inconsistent. The proof of
1 L ZA
the assertion is very similar to the proof of Lemma 4.3.1 in Chapter 4.
Based on the above property we give the following theorem for
characterizing efficient solutions.
Theorem 5.3.1. Given X used to define DC and GBC with z = f(X),
z z
the following are equivalent:
(a) X is efficient.
(b) For every r with r e { 1,... ,t} .and z > 0, at least one arc in
A^ is in some tight path in GBCz.
(Equivalently, there exists a collection of tight paths in GBCz
such that at least one tight path passes through Afor every r
for which r e {l,...,t} and z^ > 0.)
Outline of the proof. To show (a) implies (b) suppose X is efficient.
Assume that for some r for which z^_ > 0, no arc in A is in a tight
path. Clearly DC^ is consistent so that every path which passes through
Af is slack. Let PCE^jE^) be any path which passes through A^. Define

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S(P(Ep,E^)jA^) to be the sum of the reciprocal weights where the sum
mation is taken over all arcs which are contained both in P(E ,E )
P q
and in A Define e to be the minimum of
r
L P(E ,E ) d(v ,v )
z P q P q
S(P(E ,E ),A )
p q r
over all paths P(E^,E^) which pass through A^. Clearly e > 0. Choose
X = (A,,...,!) with A. = 0 for i I r, and 0 < A < e. It is direct
I t i r
to verify, by using arguments similar to the ones given in the proof
of Lemma 4.3.2, that such a choice of A is a valid choice for DC to
z-A
be consistent. But consistency of DCz_^ and the fact that A > 0,
A 0, imply X is dominated, contradicting that X is efficient.
To show (b) implies (a) suppose for any r for which > 0, at
least one arc in A^ is in a tight path. Hence, for any A = (A^,...,A )
> 0, A 0, the length of at least one tight path in GBC .. will be
strictly smaller than the distance between the locations of the exist
ing facilities corresponding to the terminal nodes of the path. Thus,
at least one separation condition on GBC is violated so that DC ..
z-A z-A
is inconsistent for any A > 0, A ^ 0. It follows that there does not
exist Y for which f(Y) < z = f(X) and f(Y) ^ f(X), which is the
definition of efficiency.
The theorem holds for the problems considered in Chapters 3 and 4
as well as the m-center problem with mutual communication considered
in [32].
We remark that the condition > 0 may appear to be somewhat
superfluous. Its omission will not affect the equivalence of (a) and
(b). The reason we included this condition is that it is unnecessary

-156-
to check those arcs for which the lengths are zero, as the lengths of
these arcs cannot be reduced, and in any feasible solution to DC2, the
constraints corresponding to arcs of zero lengths will definitely hold
at equality.
The assumption that the sets A^,...,A are disjoint is needed for
the following reason: Given a path P(E^,E^) which passes through A^_,
clearly z^ is the common multiplier for every arc which is contained
in the intersection of P(E^,E^) and A^. That is, the length of that
part of the path P(E^,E^) consisting of the arcs chosen from A^ is the
quantity zr*S(P(E^,E^) ,A^). If it were the case that A^fl A^ ^ 0 for
some j r, then the above assertion would not necessarily be true, as
an arc in the intersection A^_ 0 A^ will have at least two multipliers
in this case, namely, z^ and Zy We will consider this case in the
conclusion of this section.
The following questions seem worth investigating for future
research.
Ql. Is there a computationally efficient way of checking whether or
not arcs of GBC are contained in tight paths?
Q2. How can we construct efficient solutions efficiently?
Q3. Can the results of Theorem 5.3.1 be extended to the case when
some of the A_^ are not disjoint?
Q4. How tractable is the t-objective m-center problem if we generalize
it by replacing each x^ by a collection of centers?
With
respect to Q3, suppose that A^/l A^. ^ 0 for at least two
indices i and j for which 1 i < j < t. Let (F ,F ) be any arc in A.
P
If (F ,F ) is contained in at least two members of {A,,...,A }, then
p q I t
the distance constraint corresponding to (F ,F ) will appear more than

-157-
once in DC. Clearly, the effective upper bound for the distance
between the locations corresponding to F and F is the minimum of the
P q
upper bounds which involve these two facilities. Thus, the effective
arc length to be assigned to (F ,F^) is the reciprocal weight associ
ated with Fp and F^ multiplied by where z^ is the minimum z^ over
all indices r for which (F ,F ) e A Let GBC be the graph with arc
lengths appropriately assigned as described above. Partition A into
A A
mutually disjoint subsets A^,...,A (s £ t) such that every arc in any
A A A .
A^ has the same multiplier, say, z^ (where z £ iz^,...,z }). De-
A A A
fining z = ) it is direct to verify that DC is equivalent
1. s z
to DC* defined below:
z
d(x.,v.) < z /w..
i y r i j
(N.,E.) eA 0 A., 1 r £ s
l j r C
d(x.,x. ) < z /w., (N.,N, ) e A A 1 < r < s
J k rjk jk r TB
In other words, DC^ is obtained from DCz by choosing the minimum effec
tive upper bound for any constraint which appears more than once in DCz<
As a result of the equivalence of DC and DC^ and the fact that
z z
Aj,...,A are mutually disjoint and exhaustive subsets of A, we make
the following proposition:
Proposition 5.3.1. Given X and z with z = f(X), let DCg be the equi
valent representation of DCz as described in the previous paragraph.
The following are equivalent:
(a) X is efficient.
(b) For every r e {l,...,s} with z^ > 0, at least one arc in Ar is
in a tight path in GBC~.

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5.4 Tree Networks and General Networks
For the nonlinear p-center problem we pose the following questions.
Ql. For the nonlinear p-center problem defined on a general network N,
does there exist a spanning tree T of N such thnt the solution of
the p-center problem on T determines the solution to the p-center
problem on N?
Q2. If such a spanning tree exists, is there a way to find it without
having to enumerate on all spanning trees of N?
Dearing and Francis [19] showed that the 1-center problem on N
can be solved by solving a sequence of 1-center problems on spanning
trees of N. The basic theme of the approach is as follows: Suppose
that we know an optimal 1-center, say, x which solves the problem on N.
Using the procedure given in Busacker and Saaty [5], find a shortest
path tree rooted at x, by identifying the shortest paths (on N) con
necting x to any vertex v^. If T(x) is such a shortest path tree, then
clearly x is also the optimal 1-center of T(x). Thus, for the 1-center
problem the answer to Ql is in the affirmative while Q2 remains un
answered as the proof of the existence of such a tree is based on the
knowledge of an optimal 1-center of N.
For the case with p > 1 we propose a similar approach. Suppose
we know an optimal p-center X = ix. x } which solves the problem
^ P
on N. Partition the vertex set V = (v^,...,vn} of N into p disjoint,
exhaustive subsets V,,...,V such that a nearest center to any vertex
1 p
v in V, is the ith center x.. If ties occur we break the ties
i i i
appropriately so as to satisfy the condition that V^,...,V are
mutually disjoint. For 1 < i < p, define T^(x^) to be a shortest
path (sub)tree which is constructed by finding the shortest paths

-159-
Con N) which connect x. to any vertex v. in V.. Thus, T.(x.) is a
i J 1111
subtree which spans the subset of V. We make the following con
jecture:
Conjecture 5.4.1. Given an optimal p-center X of N, there exists a
collection of p shortest path subtrees (T^(X^),...,1^(x^)} such that
(a) for every i, 1 <, i p, x^ is a closest center to any vertex v^
which is in T.(x.), and
i i
(b) T^(x^) Tp(Xp) are mutually disjoint.
We remark that in order for the conjecture to be true, it is
necessary to show that any shortest path tree T^(x_^) contains as ver
tices only the members of V^, where Vj,...,V are the mutually disjoint
subsets of V which are identified as described in the paragraph pre
ceding the conjecture. In other words, if v^ is an arbitrarily chosen
vertex from V^, it is necessary to show that, among all the shortest
paths connecting x^ to v^, there exists at least one shortest path,
say, P(X£>V^) such that P(x^,v^) passes through only those vertices
which are members of V^. Otherwise, if every shortest path P(x^,v^)
passes through some v^. for which v^ i V^, then- clearly any shortest
path tree T.(x.) contains at least one vertex v. for which v. V.,
11 j J1
so that T,(x.) H T.(x.) ^ 0 for at least one index j with j ^ i. But
i i J J
if the intersection of Tk(x^) and Tj(x^) is nonempty then the union of
T^(x^) and T^Cx^) may have cycles.
If the conjecture holds, then {T,(x,),...,T (x )} is a collection
11 p p
of disjoint subtrees, i.e., a forest of N. If we had a knowledge of
such a collection of subtrees without having had to know X, then clearly
the 1-center of each subtree could be determined by known techniques,
and the collection of those one-centers would be an optimal p-center

-160-
of N. Hence, one possible approach to solve the p-center problem on
N is to enumerate on all possible forests of N consisting of p disjoint
subtrees and determine the one center of each subtree in any given
forest. Such an approach is in the same spirit as Kariv and Hakimi's
in [65] with the only difference being that they enumerate on all
possible collections of p subnetworks of N while we propose to enumerate
on all possible collections of p disjoint subtrees.
It is evident that if the conjecture is true, then the disjoint
subtrees (x^),...,T (x^) can be connected to one another by adding
edges appropriately without creating cycles. With the addition of the
new edges the forest becomes a spanning tree of N, so that the answer
to Q1 would be in the affirmative (provided that the conjecture
holds).
The second question still remains unresolved as we assumed a
knowledge of a p-center X to construct such a forest (assuming that
we can).

REFERENCES
1. K. Bergstresser, A. Chames, and P.L. Yu, "Generalization of
Domination Structures and Nondominated Structures in Multicriteria
Decision Making," Research Report No. JS185, Center for Cyber
netic Studies, Univ. of Texas, Austin, Texas (1974).
2. 0. Berman and R. Larson, "The Congested Median Problem," Working
Paper OR-076-78, MIT, OR Center (1978).
3. A.E. Bindschedler and J.M. Moore, "Optimal Locations of New
Machines in Existing Plant Layouts," J_. Ind. Engr. 12, 41-48
(1961).
4. G.R. Bitran and T.L. Magnanti, "The Structure of Admissable Points
with Respect to Cone Dominance," C.O.R.E. DP 7716, Louvain-la-
Neuve (1977).
5. R.G. Busacker and T.L. Saaty, Finite Graphs and Networks, McGraw-
Hill, New York, N.Y., 1965.
6. A.V. Cabot, R.L. Francis, and M.A. Stary, "A Network Flow Solu
tion to a Rectilinear Distance Facility Location Problem," AIIE
Transactions 2, 132-141 (1970).
7. L.G. Chalmet, Efficiency in Multi-Objective Location, Design, and
Layout Problems, Ph.D. Dissertation, Katholieke Universiteit te
Leuven (1978).
8. L.G. Chalmet and R.L. Francis, "Finding Efficient Solutions for
Rectilinear Distance Location Problems Efficiently," Research
Report No. 77-3, Dept, of Industrial and Systems Engineering,
Univ. of Florida, Gainesville, Florida (1977).
9. L.G. Chalmet, R.L. Francis, and J.F. Lawrence, "On Characterizing
Supremum-Efficient Facility Designs," Research Report No. 78-9,
Dept, of Industrial and Systems Engineering, Univ. of Florida,
Gainesville, Florida (1978).
10. L.G. Chalmet, R.L. Francis, and J.F. Lawrence, "Efficiency in
Integral Facility Design Problems," Research Report No. 78-11,
Dept, of Industrial and Systems Engineering, Univ. of Florida,
Gainesville, Florida (1978).
11. A. Chan and R.L. Francis, "A Round-Trip Location Problem on a
Tree Graph," Trans. Sci. 10, 35-51 (1976).
-161-

-162-
12. R. Chandrasekaran and A. Daughety, "Problems of Location on Trees,"
Discussion Paper No. 357, The Center for Mathematical Studies in
Economics and Management Science, Northwestern University,
Evanston, Illinois (1978).
13. R. Chandrasekaran and A. Tamir, "Polynomially Bounded Algorithms
for Locating p-Centers on a Tree," Discussion Paper No. 358, The
Center for Mathematical Studies in Economics and Management
Science, Northwestern University, Evanston, Illinois (1978).
2
14. R. Chandrasekaran and A. Tamir, "An 0((nlogp) ) Algorithm for the
Continuous p-Center Problem on a Tree," Discussion Paper No. 367,
The Center for Mathematical Studies in Economics and Management
Science, Northwestern University, Evanston, Illinois (1978).
15. N. Christofides and P. Viola, "The Optimum Location of Multi-
Centers on a Graph," Opnl. Res. Q;. 22, 145-154 (1971).
16. J.L. Cochrane and M. Zeleny, Eds., Multiple Criteria Decision
Making, Univ. of S. Carolina Press, Columbia, South Carolina,
1973.
17. E.J. Cockayne, S.T. Hedetniemi, and P.J. Slater, "Matchings and
Transversals in Hypergraphs, Domination and Independence in Trees,"
^J. Combinatorial Theory, Series B26, 78-80 (1979).
18. P.M. Dearing, "Minimax Location Problems with Nonlinear Costs,"
J. Res. Nat. Bur. of Stds. 82, 65-72 (1977).
19.P.M. Dearing and R.L. Francis, "A Minimax Location Problem on a
Network," Trans. Sci. 8, 333-343 (1974).
20.P.M. Dearing and R.L. Francis, "A Network Flow Solution to a
Multifacility Minimax Location Problem Involving Rectilinear
Distances," Trans. Sci. 8, 126-141 (1974).
21. P.M. Dearing and G.J. Langford, "The Multifacility Total Cost
Location Problem on a Tree Network," Technical Report No. 209,
Dept, of Mathematical Sciences, Clemson University, Clemson,
South Carolina (1975).
22. P.M. Dearing, R.L. Francis, and T.J. Lowe, "Convex Location
Problems on Tree Networks," Opns.Res. 24, 628-642 (1976).
23. S.E. Dreyfus, "An Appraisal of Some Shortest Path Algorithms,"
Opns. Res. 17, 395-412 (1969).
24. A.M. El-Shaieb, "A New Algorithm for Locating Sources Among
Destinations," Manag. Sci. 20, 221-231 (1973).
25. D.J. Elzinga, D.W. Hearn, and W.D. Randolph, "Minimax Multifacility
Location with Euclidean Distances," Trans. Sci. 10, 321-336
(1976).

-163
26. J.W. Eyster, J.A. White, and W.W. Wierville, "On Solving Multi
facility Location Problems Using a Hyperboloid Approximation
Procedure," AIIE Transactions 5, 1-6 (1973).
27. R.L. Francis, "A Note on the Optimum Location of New Machines in
Existing Plant Layouts," .J. Ind. Engr. 14, 57-58 (1963).
28. R.L. Francis, "Some Aspects of a Minimax Location Problem,"
Opns. Res. 15, 1163-1169 (1967).
29. R.L. Francis, "A Note on a Nonlinear Minimax Location Problem in
a Tree Network," J. Res. Nat. Bur, of Stds. 82, 73-80 (1977).
30. R.L. Francis and J.M. Goldstein, "Location Theory: A Selective
Bibliography," Opns. Res. 22, 400-410 (1974).
31. R.L. Francis and J.A. White, Facility Layout and Location: An
Analytical Approach, Prentice Hall, Inc., Englewood Cliffs, New
Jersey, 1974.
32. R.L. Francis, T.J. Lowe, and H.D. Ratliff, "Distance Constraints
for Tree Network Multifacility Location Problems," Opns. Res. 26,
570-596 (1978).
33. R.L. Francis, T.J. Lowe, and B.C. Tansel, "Binding Inequalities
for Tree Network Location Problems with Distance Constraints,"
Research Report No. 78-10, Dept, of Industrial and Systems
Engineering, Univ. of Florida, Gainesville, Florida (1978).
34. H. Frank, "Optimum Locations on a Graph with Probabilistic
Demands," Opns. Res. 14, 409-421 (1966).
35. H. Frank, "Optimum Locations on a Graph with Correlated Normal
Demands," Opns. Res. 15, 552-556 (1967).
36. H. Frank, "A Note on a Graph Theoretic Game of Hakimi's," Opns.
Res. 15, 567-570 (1967).
37. R. Garfinkel, A. Neebe, and M. Rao, "An Algorithm for the m-
Median Plant Location Problem," Trans. Sci. 8, 217-236 (1974).
38. R. Garfinkel, A. Neebe, and M. Rao, "The m-Center Problem:
Minimax Facility Location," Manag. Sci.23, 1133-1142 (1977).
39. F. Gavril, "Algorithms for Minimum Coloring, Maximum Clique,
Minimum Covering by Cliques, and Maximum Independent Set of a
Chordal Graph," SIAM. J. Comp., Vol. 1, 180-187 (1972).
40. A.M. Geoffrion, "Proper Efficiency and the Theory of Vector
Maximization," £. of Math. Anal, and Appl. 22, 618-630 (1968).
41. A.J. Goldman, "Optimum Locations for Centers in a Network,"
Trans. Sci. 3, 352-360 (1969).

-164-
42.A.J. Goldman, "Optimal Center Location in Simple Networks,"
Trans. Sci. 5, 212-221 (1971).
43. A.J. Goldman, "Approximate Localization Theorems for Optimal
Facility Placement," Trans. Sci. 6, 195-201 (1972).
44. A.J. Goldman, "Minimax Location of a Facility in a Network,"
Trans. Sci. 6, 407-418 (1972).
45. A.J. Goldman and P.M. Dearing, "Concepts of Optimal Location for
Partially Noxious Facilities," Bull. Opns. Res. Soc. Am. 23,
Suppl. 1, B-31 (1975).
46. A.J. Goldman and C.J. Witzgall, "A Localization Theorem for
Optimal Facility Placement," Trans. Sci. 4, 406-409 (1970).
47. S.L. Hakimi, "Optimal Locations of Switching Centers and the
Absolute Centers and Medians of a Graph," Opns. Res. 12, 450-459
(1964).
48. S.L. Hakimi, "Optimum Distribution of Switching Centers in a
Communication Network and Some Related Graph Theoretic Problems,"
Opns. Res. 13, 462-475 (1965).
49.S.L. Hakimi and S.N. Maheshwari, "Optimum Locations of Centers
in Networks," Opns. Res. 20, 967-973 (1972).
50.S.L. Hakimi, E.F. Schmeichel, and J.G. Pierce, "On p-Centers in
Networks," Trans. Sci. 12, 1-15 (1978).
51.S. Halfin, "On Finding the Absolute and Vertex Centers of a Tree
with Distances," Trans. Sci. 8, 75-77 (1974).
52. J. Halpem, "The Location of a Center-Median Convex Combination
on an Undirected Tree," J. Reg. Sci. 16, 237-245 (1976).
53. J. Halpem, "Duality in the Cent-Dian of a Graph," Working
Paper No. WP-08-77, Univ. of Calgary, Calgary, Canada (1977).
54. J. Halpem, "Finding Minimal Center-Median Convex Combinations
(Cent-Dian) of a Graph," Manag. Sci. 24, 535-544 (1978).
55.G.Y. Handler, "Minimax Location of a Facility in an Undirected
Tree Graph," Trans. Sci. 7, 287-293 (1973).
56. G.Y. Handler, "Medi-Centers of a Tree," Working Paper 278/76,
Recanati Graduate School of Business Administration, Tel-Aviv
University, Israel (1976).
57. G.Y. Handler, "Finding Two Centers of a Tree; The Continuous
Case," Working Paper, Recanati Graduate School of Business Ad
ministration, Tel-Aviv University, Israel (1977).

-165-
58. G.Y. Handler and P.B. Mirchandani, Location on Networks, The MIT
Press, Cambridge, Massachusetts, 1979.
59. E.L. Hillsman and G. Rushton, "The p-Median Problem with Maximum
Distance Constraints: A Comment," Geog. Anal. 1, 85-89 (1975).
60. L.-K. Hua and Others, "Applications of Mathematical Methods to
Wheat Harvesting," Chinese Math. 2, 77-91 (1962).
61. A.P. Hurter and M.K. Schaefer, "The Regional Allocation of Fire
Resources: A Damage Minimizing Approach," Working Paper, North
western Univ., Evanston, Illinois (1979).
62. P. Jarvinen, J. Rajala, and H. Sinervo, "A Branch-and-Bound
Algorithm for Seeking the p-Median," Opns. Res. 20, 173-182
(1972).
63. C. Jordan, "Sur les Assemblages des Lignes,"^!. Reine Angew. Math.
70, 185-190 (1969).
64. H. Juel, "Bi-Objective Location Problems with Rectangular Dis
tances," Working Paper, Michigan Technological University (1979).
65. 0. Kariv and S.L. Hakimi, "An Algorithmic Approach to Network
Location Problems. Part 1: The p-Centers," Working Paper,
Northwestern Univ., Evanston, Illinois (1976).
66. 0. Kariv and S.L. Hakimi, "An Algorithmic Approach to Network
Location Problems. Part 2: The p-Medians," Working Paper,
Northwestern Univ., Evanston, Illinois (1976).
67. R.L. Keeney and H. Raiffa, Decisions with Multiple Objectives:
Preference and Value Trade-Offs, John Wiley and Sons, New York,
1976.
68. B.M. Khumawala, "Branch-and-Bound Algorithms for Locating
Emergency Service Facilities," Krannert Institute Paper, No.
355, Purdue University, West Lafayette, Indiana (1972).
69. B.M. Khumawala, "An Efficient Algorithm for the p-Median Problem
with Maximum Distance Constraints," Geog. Anal. 5, 309-321
(1973).
70. B.M. Khumawala, "Algorithm for the p-Median Problem with Maximum
Distance Constraints: Extension and Reply," Geog. Anal. 7,
91-95 (1975).
71. B.M. Khumawala, A.W. Neebe, and D.G. Dannenbring, "A Note on
El-Shaieb's New Algorithm for Locating Sources Among Destina
tions, Manag. Sci. 21, 230-233 (1974).
72. A. Kolen, "Complexity of Location Problems on Networks," Working
Paper, Stitching Mathematisch Centrum, Tweede Boerhaavestraat 49,
1091, A1 Amsterdam, The Netherlands (1979).

-166-
73. P. Kolesar, W. Walker, and J. Hausner, "Determining the Relation
Between Fire Engine Travel Times and Travel Distances in New York
City," Opns. Res. 23, 614-627 (1975).
74. T.C. Koopmans, Activity Analysis of Production and Allocation,
Cowles Commission for Research in Economics, Monograph No. 13,
John Wiley and Sons, New York, 1951.
75. H.W. Kuhn, "On a Pair of Dual Nonlinear Problems," Nonlinear
Programming, Chapter 3, J. Abadie, editor, John Wiley and Sons,
New York, 1967.
76. H.W. Kuhn and A.W. Tucker, "Nonlinear Programming," Proc. of the
2nd Berkeley Symposium on Math., Stat. and Probability, Univ. of
California Press, Berkeley, California (1951).
77. A.H. Land and A.G. Doig, "An Automatic Method for Solving Dis
crete Programming Problems," Econometrica 28, 497-520 (1960).
78. A.C. Lea, "Location-Allocation Systems: An Annotated Bibliography,"
Discussion Paper No. 13, Dept, of Geography, University of Toronto,
Toronto, Canada (1973).
79. A.C. Lea, "A Model Taxonomy and a View of Research Frontiers in
Normative Locational Modelling," Paper presented to International
Symposium on Locational Decisions, Banff, Alberta (1978).
80. J. Levy, "An Extended Theorem for Location on a Network," Opnl.
Res. Q. 18, 433-442 (1967).
81.C.C. Lin, "On Vertex Addends in Minimax Location Problems,"
Trans. Sci. 9, 165-168 (1975).
82.T.J. Lowe, "Efficient Solutions in Multiobjective Tree Network
Location Problems," Trans. Sci. 12, 298-316 (1979).
83. L.F. McGinnis and J.A. White, "A Single Facility Rectilinear
Location Problem with Multiple Criteria," Trans. Sci. 12,
217-231 (1978).
84. F.E. Maranzana, "On the Location of Supply Points to Minimize
Transport Costs," Opnl. Res. Q. 15, 261-270 (1964).
85. D.W. Matula and R. Kolde, "Efficient Multi-Median Location in
Acyclic Networks," ORSA/TIMS Bulletin, No. 2 (1976).
86. A. Meir and J.W. Moon, "Relations Between Packing and Covering
Numbers of a Tree," Pac. J. Math. 61, 225-233 (1975).
87.E. Minieka, "The m-Center Problem," SIAM Review 12, 138-139
(1970).

-167-
88. E. Mlnieka, "The Centers and Medians of a Graph," Opns. Res. 25,
641-650 (1977).
89. P.B. Mirchandani and A.R. Odoni, "Locating New Passenger Facili
ties on a Transportation Network," Working Paper, Electrical and
Systems Engineering Dept., Rensselaer Polytechnic Inst., Troy,
New York (1977).
90. P.B. Mirchandani and A.R. Odoni, "Locations of Medians on Sto
chastic Networks," Working Paper OR-065-77, Operations Research
Center, M.I.T., Cambridge, Massachusetts (1977).
91. S.C. Narula, V.I. Ogbu, and H.M. Samuelson, "An Algorithm for the
p-Median Problem," Opns. Res. 25, 709-712 (1977).
92. G.L. Nemhauser and W.L. Sheu, "Easy and Hard Bottleneck Location
Problems," Technical Report 386, School of Operations Research
and Industrial Engineering, Cornell University, Ithaca, New
York (1979).
93. J.C. Picard and H.D. Ratliff, "A Cut Approach to the Rectilinear
Distance Location Problem," Opns. Res. 26, 422-434 (1978).
94. A.B. Pritsker and P.M. Ghare, "Locating New Facilities with
Respect to Existing Facilities," AIIE Transactions 12, 290-297
(1970).
95. C. ReVelle and R. Swain, "Central Facilities Location," Geog.
Anal. 2, 30-42 (1970).
96. C. ReVelle, D. Marks, and J.C. Liebman, "An Analysis of Private
and Public Sector Location Models," Manag. Sci. 16, 692-707
(1970).
97. D.J. Rose, R.E. Tarjan, and G.S. Lueker, "Algorithmic Aspects of
Vertex Elimination in Graphs," SIAM J. Comput. 5, 266-283 (1976).
98. A. Rosenthal, J. Pino, and M. Coulter, "A Generalized Algorithm
for Centrality Problems on Trees," Working Paper, Dept, of
Computer and Communication Sciences, Univ. of Michigan, Ann
Arbor, Michigan (1978).
99. B. Roy,"Problems and Methods with Multiple Objective Functions,"
Math. Prog. 1, 239-266 (1971).
100.D.R. Shier, "A Min-Max Theorem for p-Center Problems on a Tree,"
Trans. Sci. 11, 243-252 (1977).
101.S. Singer, "Multi-Centers and Multi-Medians of a Graph with
Application to Optimal Warehouse Location," Unpublished Paper,
Dunlap and Associates, Inc., Darien, Conn. (1968).

-168-
102. P.J. Slater, "Central Paths in a Graph," Research Report SAND
78-0809J, Sandia Laboratories, Albuquerque, New Mexico (1978).
103. P.J. Slater, "One-Point Location of an Area Response Group,"
Research Report SAND 78-1788, Sandia Laboratories, Albuquerque,
New Mexico (1978).
104. M.K. Starr and M. Zeleny, "MCDM-State and Future of the Arts,"
TIMS Studies in Management Sciences 6, 5-29 (1977).
105. R.W. Swain, "A Parametric Decomposition Approach for the Solu
tion of Uncapacitated Location Problems," Dept, of Industrial
Engineering, The Ohio State University (1971).
106. J.J. Sylvester, "A Question in the Geometry of Situation,"
Quarterly Journal of Pure and Applied Mathematics, Vol. 1, 79
(1857).
107. M.R. Teitz and P. Bart, "Heuristic Methods for Estimating the
Generalized Vertex Median of a Weighted Graph," Opns. Res.,16,
955-961 (1968).
108. H. Thiriez and S. Zionts, Eds., Multiple Criteria Decision
Making, Proc., Jouy-en-Josas, France, 1975.
109. C. Toregas, R. Swain, C. ReVelle, and L. Bergman, "The Location
of Emergency Service Facilities," Opns. Res. 19, 1363-1373
(1971).
110. R.E. Wendell, "Efficiency and Solution Approaches to Bi-Objective
Mathematical Programs," Working Paper, Rensselaer Polytechnic
Institute, Troy, New York.
111. R.E. Wendell and A.P. Hurter, "Optimal Locations on a Network,"
Trans. Sci. 7, 18-23 (1973).
112. R.E. Wendell and D.N. Lee, "Efficiency in Multiple Objective
Optimization Problems," Math. Prog. 12, 406-414 (1977).
113. R.E. Wendell and E.L. Peterson, "Duality in Generalized Location
Problems," ORSA Bull. 20, Supp. 2, B-317 (1972).
114. R.E. Wendel, A.P. Hurter, and T.J. Lowe, "Efficient Points in
Location Problems," AIIE Transactions 9, 238-246 (1977).
115. G.O. Wesolowsky and R.F. Love, "The Optimal Location of New
Facilities Using Rectangular Distances," Opns. Res. 19, 124-
130 (1971).
116. G.O. Wesolowsky and R.F. Love, "A Nonlinear Approximation Method
for Solving a Generalized Rectangular Distance Weber Problem,"
Manag. Sci. 18, 656-663 (1972).

-169-
117. P.L. Yu, "Cone Convexity, Cone Extreme Points, and Nondominated
Solutions in Decision Problems with Multi-Objectives," J. of
Opt. Theory and App. 14, 319-377 (1974).
118. P.L. Yu and M. Zeleny, "The Set of All Nondominated Solutions
in Linear Cases and a Multicriteria Simplex Method," J. Math.
Anal, and A££. 49, 430-468 (1975).
119. P.L. Yu and M. Zeleny, "Linear Multiparametric Programming by
Multicriteria Simplex Method," Manag. Sci. 23, 159-170 (1977).
120. B. Zelinka, "Medians and Peripherians of Trees," Archivum
Mathematicum (Brno), 87-95 (1968).

BIOGRAPHICAL SKETCH
Barbaros Tansel was born on January 10, 1952, in Ankara, Turkey,
where he received his early education. For his high school education
he attended the Robert Academy in Istanbul and graduated in June 1970.
In September 1970, he began his undergraduate study in the Middle East
Technical University in Ankara and was awarded the Kennedy Scholarship
in 1971. He graduated from the Middle East Technical University in
June 1974 with a B.S. degree in industrial engineering. In 1975, he
was awarded the Fullbright Scholarship and began his graduate study in
the University of Florida. He received his M.Sc. degree in December
1976 and Ph.D. in December 1979. During his graduate study he worked
as a teaching and research assistant in the Department of Industrial
and Systems Engineering.
Barbaros's hobbies include classical music, chess, philosophy,
and folk dancing.
-170-

I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
Richard L. Francis, Chairman
Professor of Industrial and Systems
Engineering
I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
X/
Donald U. Hearn
Associate Professor of Industrial and
Systems Engineering
I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
Ra llph W'. Swa in
Associate Professor of Industrial and
Systems Engineering

I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
Antal Maj thay
Associate Professor of Managemen
This dissertation was submitted to the Graduate Faculty of the College
of Engineering and to the Graduate Council, and was accepted as partial
fulfillment of the requirements for the degree of Doctor of Philosophy.
December 1979
Dean, Graduate School

Page 2 of 2
Internet Distribution Consent Agreement
In reference to the following dissertation:
AUTHOR: Tansel, Barbaros
TITLE: Optimal multi-facility location on tree networks / (record number:
99473)
PUBLICATION DATE: 1979
i. 'R nms, Tinsel
, as copyright holder for the aforementioned
dissertation, hereby grant specific and limited archive and distribution rights to the Board of Trustees
of the University of Florida and its agents. I authorize the University of Florida to digitize and
distribute the dissertation described above for nonprofit, educational purposes via the Internet or
successive technologies.
This is a non-exclusive grant of permissions for specific off-line and on-line uses for an indefinite
term. Off-line uses shall be limited to those specifically allowed by-.'^ir Use" as prescribed by the
terms of United States copyright legislation (cf, Title 17, U.S. Code) as well as to the maintenance
and preservation of a digital archive copy. Digitization allows the' fiversity of Florida to generate
image- and text-based versions as appropriate and to provide and^enhance access using search
software.
This grant of permissions prohibits use of the digitized versionsjbr cmmercial use or profit.
Signature of Copyright Holder
13 L r 1
"HULj
tii.
Personal information blurred
Date of Signature
Please print, sign and return to:
Cathleen Martyniak
UF Dissertation Project
Preservation Department
University of Florida Libraries
P.O. Box 117007
Gainesville, FL 32611-7007
11.06.2008



-64-
Figure 2.2. Dual Graph for Example


-3-
in the past. As reported by Lea [79], there are some 1500 published
papers on location-allocation problems. Among these, about 80 are on
network location problems, a ratio of a little less than 6%. Hence,
network location problems deserve well-justified attention in future
research.
In this dissertation, we develop a theory for a number of location
problems which involve locating multiple new facilities on a tree net
work with respect to existing facilities at known locations. At this
point we give an overview of the dissertation.
In the remainder of Chapter 1, we specify our terminology and
give a survey of the network location literature. We discuss minimax and
minisum problems.and multi-objective problems involving minimax and
minisum objectives as well as other objectives. Discussed also are
problems with distance constraints. We highlight some of the convexity
properties of trees (see [22]) in relation to the problems discussed.
The chapter ends with a brief discussion of path-location problems.
In Chapter 2, we develop a theory for the nonlinear p-center
problem on a tree network. The problem is a generalization of the
linear p-center problem which involves locating p new facilities on
a network so as to minimize the maximum weighted distance from any
existing facility to its nearest new facility. Nonlinearity is ob
tained by replacing each weight by a strictly increasing function of
the distance. We formulate a dual "dispersion" problem and prove a
weak duality and a strong duality theorem. The strong duality theorem
also specifies the necessary and sufficient conditions for an optimal
solution to either problem. We provide algorithms of polynomial com
plexity for solving either problem. Discussed also are a covering


Page
5 SUMMARY AND FUTURE RESEARCH 149
5.1 Summary 149
5.2 Generalized Multi-Center Problem 150
5.3 The t-Objective m-Center Problem: Steps
Towards a Unified Theory 153
5.4 Tree Networks and General Networks 158
REFERENCES 161
BIOGRAPHICAL SKETCH . 170
v


-38-
The problem differs from the p-center problem in two respects:
(i) the distance between any vertex v and any new facility x_^ may be
of concern as opposed only to the distance between v and the nearest
new facility to v.; (ii) certain distances between new facilities are
of concern, as opposed to the absence of interactions between new
facilities. For the case of a single new facility the two problems
coincide.
In this problem, the new facilities may be thought to fulfill a
supporting task to other new facilities as well as servicing those
existing facilities that are a priori assigned to them.
Certain planar cases of the multifacility minimax problem have
been studied by Dearing and Francis [20]> Elzinga, Hearn, and Randolph
[25], Wendell and Peterson [113],. and Francis [28l*
The problem on a network is defined by Dearing, Francis, and Lowe
[22] in the presence of distance constraints. It is established in
[22] that the function f is a convex function on a tree network. The
existence of a solution is guaranteed due to compactness and con
tinuity considerations. Furthermore, it is shown that it suffices to
consider only new facility locations in the convex hull of the existing
facility locations in order to solve the problem.
The problem on a general network was shown to be NP-hard by Kolen
[72 ]. For the case of a tree network, the problem is solved by
Francis, Lowe, and Ratliff [32] by using an equivalent formulation in
terms of distance constraints (with variable right hand sides). The
solution procedure finds Z* first, by using the separation conditions.
Then an optimal feasible location vector X* is constructed by using the
Sequential Location Procedure described in [32]* To find Z* an


-41-
For the case of a tree network, the m-median problem with mutual communi
cation is solved by Dearing and Langford [21] and by Picard and
Ratliff [93].
The approach used by Dearing and Langford is to embed the tree T
into the Euclidean space R^, for some p, so that the distance between
any two points on the tree is equal to the rectilinear distance between
the corresponding points in R^. The problem in R^ with rectilinear
distances decomposes into p subproblems, each of which can be solved
by using known techniques given in Francis and White [31 ], or, perhaps
more efficiently, by applying the network flow procedure discussed in
Cabot, Francis, and Stary [6]. For reducing the computational effort,
the embedding procedure is carried out with respect to a minimal path
decomposition of T into p edge disjoint paths (each edge is in one and
only one path). Each path in a minimal path decomposition corresponds
to a dimension in R*5.
The approach taken by Picard and Ratliff in [93] takes advantage
of the vertex-optimality condition and determines an optimal solution
(on the vertices of T) by solving a sequence of at most n-1 minimum
cut problems, each on a graph containing at most m+2 nodes. The
method is based on a result that an optimal location vector can be
found independently of the edge lengths, by using only the incidence
relations between vertices and the weights. In this respect, the pro
cedure is in the same spirit as Goldman's algorithm for finding a
median of a tree. Each cut problem corresponds to an edge of the
tree. To be more explicit, the removal of all interior points of an
edge e leaves two disconnected components, T^ = T^(e) and T^ = 12(e).
Corresponding to edge e, a graph G = G(e) is constructed having nodes


-100-
this reason, it becomes necessary to develop the sufficient conditions
for reducible location vectors.
Sufficiency for Reducible Location Vectors
The basis of our approach for establishing sufficiency for redu
cible location vectors is to represent a reducible location vector by
an irreducible one and apply Property 3.5.2.
Suppose Z is reducible. Then at least one arc in GBC connecting
two new facility nodes has length zero. In general, there may be
several arcs of length zero connecting new facility nodes. Let GB be
the subgraph of GBC with nodes N,,...,N and arcs (N. ,N ) for (j ,k)
1 m J k
e ID. If arc (N.,N, ) in GBC has length zero, then combining these
B j k
two nodes into a single (super) node will not affect the length of any
path containing this arc. If the resulting graph (with one less node)
has an arc in GB of length zero, again the two nodes connected by that
arc can be combined into a single node without affecting the path
lengths. In general, this graph transformation can be applied as many
times as necessary (clearly, at most m 1 times) to obtain a new
graph GBC* from GBC so that no arc in GBC* connecting two new facility
nodes has length zero. With this transformation, a node N of GBC*
P
will actually be representing a number of the original nodes in GBC.
We may define the index p as a composite index for the indices of new
facility nodes represented by in GBC*. Hence, if p is the composite
index, say, {j,k,l}, we can define z* to be the common location
P
z^ = z^ = z^ of new facilities j, k, and 1. Thus, if GBC* has, say,
t new facility nodes, then the location vector Z* with components
corresponding to new facility nodes of GBC* will be an irreducible


ACKNOWLEDGMENTS
I am deeply indebted and grateful to Dr. Richard L. Francis, the
chairman of my supervisory committee, for his excellent guidance,
numerous suggestions, and the generosity with which he invested his time
in listening to my ideas. Dr. Francis not only initiated my interest
in location problems but also inspired many of the ideas in this dis
sertation by asking the right questions at the right time.
I owe very special thanks to Dr. Timothy J. Lowe, the cochairman/
chairman of my committee during 1976-1978, presently of Purdue Uni
versity, for his active interest, overall guidance, and his inspiring
suggestions.
Dr. Francis and Dr. Lowe have shown sincere care about my progress
and their encouragement has been of utmost value in bringing this
dissertation to a completion.
I would also like to express my sincere thanks and appreciation
to the other members of my committee, Dr. Ralph W. Swain, Dr. Donald W.
Hearn, Dr. Antal Majthay, and Dr. Luc G. Chalmet for their interest in
my work and their suggestions during my proposal.
I am grateful to the Department of ISE for providing me with
assistantship during my graduate studies.
Mrs. Adele Koehler has done an excellent job in typing the manu
script. She is fast, accurate, and very observant. I sincerely
recommend her.
ii


Page 2 of 2
Internet Distribution Consent Agreement
In reference to the following dissertation:
AUTHOR: Tansel, Barbaros
TITLE: Optimal multi-facility location on tree networks / (record number:
99473)
PUBLICATION DATE: 1979
i. 'R nms, Tinsel
, as copyright holder for the aforementioned
dissertation, hereby grant specific and limited archive and distribution rights to the Board of Trustees
of the University of Florida and its agents. I authorize the University of Florida to digitize and
distribute the dissertation described above for nonprofit, educational purposes via the Internet or
successive technologies.
This is a non-exclusive grant of permissions for specific off-line and on-line uses for an indefinite
term. Off-line uses shall be limited to those specifically allowed by-.'^ir Use" as prescribed by the
terms of United States copyright legislation (cf, Title 17, U.S. Code) as well as to the maintenance
and preservation of a digital archive copy. Digitization allows the' fiversity of Florida to generate
image- and text-based versions as appropriate and to provide and^enhance access using search
software.
This grant of permissions prohibits use of the digitized versionsjbr cmmercial use or profit.
Signature of Copyright Holder
13 L r 1
"HULj
tii.
Personal information blurred
Date of Signature
Please print, sign and return to:
Cathleen Martyniak
UF Dissertation Project
Preservation Department
University of Florida Libraries
P.O. Box 117007
Gainesville, FL 32611-7007
11.06.2008


-18-
these regions may also jointly cover all vertices. Hence, to find a
minimum cardinality feasible solution, one needs to choose the minimum
number of regions that jointly cover V. This choice can be made by
defining a zero-one matrix A, so that an entry a^ of A is one if
vertex v^ is covered by region j, and zero otherwise. Solving the
set covering problem with matrix A will provide a solution to the
r-cover problem. Computational experience is reported and it is found
that the procedure works better for small values of p, as the set
covering part of the procedure takes a significant portion of the
total computational time.
An important result is due to Kariv and Hakimi [6.5] They showed
that the p-center problem on a general network is NP-complete. Kariv
and Hakimi also showed that the weighted case (as well as the un
weighted case) can be reduced to a computationally finite one. Based
on this finiteness property, they gave an algorithm whose order of
complexity is polynomial in |e|, but exponential in p. To show com
putational finiteness one argues as follows: For any absolute p-center
X = {x^,...,Xp}, there will be a subset of vertices covered by the
ith center x.. If N. is the (sub)network induced by V., then it can
xi 1
be shown that the absolute center x* of N. can replace x without in-
1 i i
creasing the value of the objective function, so that X* = {x*,...,x*}
1 p
is also an absolute p-center. Hence, one can restrict one's attention
to absolute p-centers every element of which is the absolute 1-center
of some subnetwork. The absolute 1-center of any subnetwork of N
will occur either at a vertex or at one of at most | E |n (n l)/2
"suspected" points. A suspected point on an edge is a point x such
that, for some two distinct vertices v_^ and v., x is a break point on


-91-
equivalent problem of determining when an arc lies only on slack paths.
The following property, which we proved in [33], characterizes the con
ditions under which an arc in GBC is not contained in any tight path.
Property 3.3.2. Let DC be consistent. Let (f^,fj) be any arc in GBC,
of positive length e^., whose length is reduced by some positive amount
e. Let DC^CGBC^) be the distance constraints (graph) obtained from
DC (GBC) by replacing e by e e.
(a) Evey path containing (f_^,f_.) in GBC is slack if and only if e can
be chosen (with e > 0) so that DC is consistent.
e
(b) Whenever every path containing (ff) is slack, e can be chosen
(with e > 0) so that DC£ is consistent and at least one of the follow
ing is true:
(i) at least one path in GBC containing (f.,f.) is tight;
£ 1 J
(ii) the length of (f.,f.) in GBC can be reduced to zero.
i j e
Finally, we will use the following lemma proven in [33].
Lemma 3.3.1. Given points a,b e T, suppose that d(a,b) = a +3.
Then, the inequalities d(x,a) a, d(x,b) £ 3 are consistent if and
only if they have a unique solution and the inequalities hold as
equalities.
Necessary and Sufficient Conditions for Efficiency
Given a location vector Z, we let U = D(Z) and define the distance
constraints of interest by D(X) < U, where the entries in U define the
bjk and Cij by bjk = d('ZyZk) fr £ IB and Cij = d(zivj) for
(ij) £ 1^. We use the b^ and c to define GBC in the customary
manner. As before, we may assume GBC is connected, for otherwise the
problem of finding efficient location vectors decomposes into


-121-
Figure 3.
4
2 o 1
1
O
v
z,
z.
V,
a) Facility Locations
b) Graph GBC
L(E1SE2) = 3 >
MEltE3) = 4 =
L(E15E4) = 5 =
L(E2,E3) = 3 =
L(E2,E4) = 4 >
L(E3,E4) = 3 =
2 = d(vlSv2)
4 = d(vl9v3)
5 = d(v^ >v4>
3 = d(v2,v3)
3 = d(v2,v4)
3 = d(v3,v4)
c) Separation Conditions
9. Example of an Efficient Vector with Tchebychev Distances
in


TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS ii
ABSTRACT vi
CHAPTER
1 INTRODUCTION AND LITERATURE SURVEY 1
1.1 Introduction and Overview . 1
1.2 Terminology 4
1.3 Survey of the Network Location Literature 6
2 DUALITY AND THE NONLINEAR p-CENTER PROBLEM AND
COVERING PROBLEM ON A TREE NETWORK ...... 53
2.1 Introduction and Related Work 53
2.2 Problem Statements and Duality 56
2.3 Dual Problem Interpretation 61
2.4 Covering Algorithm 67
2.5 Dual Problem Solution and the Strong Duality Theorem. 73
2.6 Results for the Covering Problem 78
3 A VECTOR-MINIMIZATION PROBLEM ON A TREE NETWORK 84
3.1 Introduction 84
3.2 Problem Statement 85
3.3 Distance Constraints and Characterization of
Efficient Points 87
3.4 Examples 94
3.5 Further Results on the Convex Hull Property 96
3.6 Algorithm to Construct Efficient Location Vectors . 108
3.7 Efficiency for the Case of Rectilinear or
Tchebychev Distances. 116
4 THE BI-OBJECTIVE m-CENTER PROBLEM ON A TREE NETWORK. ... 122
4.1 Introduction 122
4.2 Problem Statement, Notation, and Definitions 123
4.3 Necessary and Sufficient Conditions for Efficiency. 126
4.4 Construction of the Efficient Frontier 134
Iv


-56-
2.2 Problem Statements and Duality
We suppose given a finite undirected tree network with positive
arc lengths and denote by T an imbedding of the given network having
as edges rectifiable arcs. For any two points x,yeT, let d(x,y)
denote the shortest path distance between x and y.
Let J = {1,...,n} and denote by V = {v^,...,vn) (V C T) a collec
tion of distinct vertex locations of "demand points" or "existing
facilities." Let X = {x.,...,x } (X C T) denote a finite collection
1 P
of "centers" or "new facilities." For ieJ, define the distance of v.
1
to its nearest center by D(X,v^) = min{d(x^,v_.) : 1 < i < p}, and. let
Sj = maxid(x,v.): xeT}. Also, for jeJ, we assume given a real valued
function f continuous and strictly increasing, with domain [0,6^]
and (clearly) range [f (0) ,f^ (6^)] For X C T, |x| < o, we define
the function f by
f(X) = max{f.(D(X,v^)): jeJ}
The Primal p-Center Problem is as follows: Find a p-center X*
for which
rp = f(X*) = mini f (X) : XCT, ¡X | = p} .
(2.2.1)
As discussed in Dearing and Francis [19], due to compactness of
T and continuity of d(x,.) on T for each fixed xeT, an optimal solu
tion X* to (2.2.1) exists and is contained in the convex hull of V.
With a and p defined by a = max{f.(0): jeJ} and n = min{f (5 ):
J J J
jeJ}, we shall assume a < n, for if a = f (0) > f (6 ) = n. say, then
s t t
the function f would always be dominated by (strictly smaller than)


-147-
Table 4.1.
Example Linear Functions
T13(zl>
= 1.33 2.67z1
t14(z1^
= 2.67 2.33zl
t15(z1)
= 1.6 1.8z1
t16(z1)
= 2 1.6z1
t23(z1>
= 0.67 1.33z1
t24(z1*
= 2 21
t25(zP
= 1.2 Z]L
T26 =1.6- 0.8z^
W
= 2 3.5Zl
T36(z1)
= 3 3z^
t46(zI>
= 1 2.5zj


-58-
The Dual Dispersion Problem is as follows: Find a subset K* of
V such that
g(K*) = max{g(K) : K^V, |k| = p+1} (2.2.2)
We remark that the dispersion problem is meaningfully defined for
2 p+1 < n. The primal p-center problem is trivial for p > n. Hence,
we shall restrict p to 1 p < n-1.
In what follows in this section, we prove a Weak Duality Theorem
(W.D.T.) and state a Strong Duality Theorem (S.D.T.) (proven in Sec
tion 5). At the end of this section, we give an example problem
illustrating definitions and results.
In the W.D.T. we shall use the fact (readily proven as in [18]
or [29]) that a f (X) for any XC T, |x| < .
Theorem 2.2.1. (Weak Duality Theorem). Assume 1 p n-1. For any
X C T with |X| = p, and any K C V with |k| = p+1, we have f(X) > g(K).
Proof. There are two cases: g(K) < a or g(K) > a. In the former
case we have g(K) 1 a < f(X). In the latter case, we note that
g(K) = g^(K) > a > g^(K). Since |x| = p < p+1 = |k|, at least two
demand points in K must be served by a single center. In other words,
for some v ,v £ K with s ^ t, and some center xeX, we have
s t
f [D(X,v )] = f [d(x,v )] 5 f(X)
s s s s
(2.2.3)
ft[D(X,vt)] = ft[d(x,vt)] < f(X) .
Using the definitions and the inequalities in (2.2.3), we have
g(K) = g^(K) < 3gt 2 max{fg[d(x,vs)],ft[d(x,vt)]} < f(X).
Remark 2.2.1. We note that the conditions |x| = p and jK¡ = p+1 can
be replaced by |x| 2 p and/or |k| > p+1, respectively, and the proof


-145-
found by applying the Sequential Location Procedure of [32] to the
constraints DCz after fixing the locations of the uniquely located
facilities. Any such feasible solution Y to DC^ is clearly an effi
cient location vector.
At this point we give an example application of E-FRONT. We
apply the algorithm to the tree network for which the necessary data
are given (earlier) in Figure 4.1. The arcs in A^ are labeled a^
through a^ as shown in Figure 4.4a). Figure 4.4b) shows the graph
GBC^j ^ with arc lengths l/w and l/v^. The thickly drawn subgraph
of GBC,, is the graph GD. The values of m 1 £ s < t £ 3, are
\I j I / D St
also given in the same figure. Corresponding to each pair of arcs
(a^,a^) e A the linear function T^j(zp is specified in Table 4.1.
These functions are plotted in Figure 4.5, and t(z^) is indicated by
the cross-hatched line in the same figure. The values of a and b are
computed by using the techniques given earlier. We remark that
t(z^) = T24^zl^ fr zl e for this example problem. Since
a2 = anc* a4 = ^2^4^ anc* m12 = t^ie -^-en8t^ f arc
^N1N2^ t^ie Path (E2^i^2^4^ as a path for any choice of
(z^,Z2) in Z*. For example, for z^ = 1.5 and Z2 = 0.5, (z^,Z2) is on
the efficient frontier. The tight path corresponding to this choice
of (z^,Z2) is shown earlier in Figure 4.2b), together with the
corresponding location vector Y = an t^ie same figure.
Note that y^ and are uniquely located as N^ and N2 are on the
tight path.


-li
to a nearest element of each collection. Simply sum each row of this
matrix and choose the vertex whose row sum is minimum.
Frank considered a probabilistic version of the 1-median problem
in [34] where each weight is a random variable with a known distribu
tion. A number of bounds are obtained on the expected value of the
objective function as well as its variance. Some of these results
are generalized by Frank [35] to the case where the weights are jointly
distributed random variables.
We now concentrate on the p-median problem with p > 2.
p-Median of a network and vertex optimality. A significant
theoretical contribution is due to Hakimi [48]. Hakimi proved that
there exists an absolute p-median contained in V. Certain generaliza
tions of this result have been given in subsequent work.
Levy [80] proved that the (vertex-optimal) result holds when the
weights w^ are replaced by concave cost functions c^(*) of the distance
between v_^ and its nearest median.
Goldman [41] generalized the result to the case of a "two-stage"
commodity. More specifically, one distinguishes a vertex as being a
source or a destination. Let (v ,v.) be a source-destination pair,
S Cl
and let x^ and x_. be the nearest medians to v and v^, respectively.
Then the cost of transferring the commodity from source v to destina-
s
tion V, is the sum of three transport costs, namely, w .d(v ,x ) +
a sd s 1
r\j
wsdd(xi,x^.) + w*dd(x_. ,v) In general, if X = {x^...^} is a median
set, one does not know which median is the nearest to v or v,; hence,
s d
the cost associated with a source-destination pair (v ,v,) is
s d
given by
fsd(x) = min Kd^VV +"sdd(xi*xj) + Wsdd(xjVd)]


-159-
Con N) which connect x. to any vertex v. in V.. Thus, T.(x.) is a
i J 1111
subtree which spans the subset of V. We make the following con
jecture:
Conjecture 5.4.1. Given an optimal p-center X of N, there exists a
collection of p shortest path subtrees (T^(X^),...,1^(x^)} such that
(a) for every i, 1 <, i p, x^ is a closest center to any vertex v^
which is in T.(x.), and
i i
(b) T^(x^) Tp(Xp) are mutually disjoint.
We remark that in order for the conjecture to be true, it is
necessary to show that any shortest path tree T^(x_^) contains as ver
tices only the members of V^, where Vj,...,V are the mutually disjoint
subsets of V which are identified as described in the paragraph pre
ceding the conjecture. In other words, if v^ is an arbitrarily chosen
vertex from V^, it is necessary to show that, among all the shortest
paths connecting x^ to v^, there exists at least one shortest path,
say, P(X£>V^) such that P(x^,v^) passes through only those vertices
which are members of V^. Otherwise, if every shortest path P(x^,v^)
passes through some v^. for which v^ i V^, then- clearly any shortest
path tree T.(x.) contains at least one vertex v. for which v. V.,
11 j J1
so that T,(x.) H T.(x.) ^ 0 for at least one index j with j ^ i. But
i i J J
if the intersection of Tk(x^) and Tj(x^) is nonempty then the union of
T^(x^) and T^Cx^) may have cycles.
If the conjecture holds, then {T,(x,),...,T (x )} is a collection
11 p p
of disjoint subtrees, i.e., a forest of N. If we had a knowledge of
such a collection of subtrees without having had to know X, then clearly
the 1-center of each subtree could be determined by known techniques,
and the collection of those one-centers would be an optimal p-center


-163
26. J.W. Eyster, J.A. White, and W.W. Wierville, "On Solving Multi
facility Location Problems Using a Hyperboloid Approximation
Procedure," AIIE Transactions 5, 1-6 (1973).
27. R.L. Francis, "A Note on the Optimum Location of New Machines in
Existing Plant Layouts," .J. Ind. Engr. 14, 57-58 (1963).
28. R.L. Francis, "Some Aspects of a Minimax Location Problem,"
Opns. Res. 15, 1163-1169 (1967).
29. R.L. Francis, "A Note on a Nonlinear Minimax Location Problem in
a Tree Network," J. Res. Nat. Bur, of Stds. 82, 73-80 (1977).
30. R.L. Francis and J.M. Goldstein, "Location Theory: A Selective
Bibliography," Opns. Res. 22, 400-410 (1974).
31. R.L. Francis and J.A. White, Facility Layout and Location: An
Analytical Approach, Prentice Hall, Inc., Englewood Cliffs, New
Jersey, 1974.
32. R.L. Francis, T.J. Lowe, and H.D. Ratliff, "Distance Constraints
for Tree Network Multifacility Location Problems," Opns. Res. 26,
570-596 (1978).
33. R.L. Francis, T.J. Lowe, and B.C. Tansel, "Binding Inequalities
for Tree Network Location Problems with Distance Constraints,"
Research Report No. 78-10, Dept, of Industrial and Systems
Engineering, Univ. of Florida, Gainesville, Florida (1978).
34. H. Frank, "Optimum Locations on a Graph with Probabilistic
Demands," Opns. Res. 14, 409-421 (1966).
35. H. Frank, "Optimum Locations on a Graph with Correlated Normal
Demands," Opns. Res. 15, 552-556 (1967).
36. H. Frank, "A Note on a Graph Theoretic Game of Hakimi's," Opns.
Res. 15, 567-570 (1967).
37. R. Garfinkel, A. Neebe, and M. Rao, "An Algorithm for the m-
Median Plant Location Problem," Trans. Sci. 8, 217-236 (1974).
38. R. Garfinkel, A. Neebe, and M. Rao, "The m-Center Problem:
Minimax Facility Location," Manag. Sci.23, 1133-1142 (1977).
39. F. Gavril, "Algorithms for Minimum Coloring, Maximum Clique,
Minimum Covering by Cliques, and Maximum Independent Set of a
Chordal Graph," SIAM. J. Comp., Vol. 1, 180-187 (1972).
40. A.M. Geoffrion, "Proper Efficiency and the Theory of Vector
Maximization," £. of Math. Anal, and Appl. 22, 618-630 (1968).
41. A.J. Goldman, "Optimum Locations for Centers in a Network,"
Trans. Sci. 3, 352-360 (1969).


-42-
1 through m corresponding to new facilities, a source s and a sink t.
Graph G contains arcs (s,i) and (i,t) for 1 < i < m and arcs (j,k) for
each pair (j,k)eIB. The capacity of arc (j ,k) is the weight v^. The
capacity of arc (s,i) is given by J [w : v eT., (i,r)el ], and the
ir v i o
capacity of arc (i,t) is given by J [w. : v eT, (ijqjel.,]. If
xq q u
(Q,Q) is a minimum capacity s-t cut of G, with seQ, teQ, then all new
facility locations x^ for which the corresponding node i is in Q are
in T^ in an optimal solution. Similarly, all x_. for which the node j
is in Q are in T^ in an optimal solution. The procedure is a repeated
application of this minimum cut problem with respect to each edge,
until an optimal vertex location is determined for each x^. During
the process, each x^ whose location is determined is treated like an
existing facility. The method is described originally for the
analogous rectilinear distance problem on the plane, which, in turn,
decomposes into two subproblems, each on a line.
Multi-objective location problems on networks
Multi-objective optimization problems, sometimes known as vector
optimization problems, involve decision making under two or more
criteria. More explicitly, a set (finite or infinite) S of alterna
tives is specified and n (possibly non-commensurable) objective func
tions are to be minimized over S. Let f,,...,f be n numerical func-
1 n
tions defined on S, and define f(x) = (f,(x),...,f (x)) for all xeS.
1 n
The multi-objective optimization problem (VMP) is the following:
V-min f(x)
xeS
In general, the minima of the functions f_,...,f do not coincide.
1 n
In order for the minimization to be meaningful, one needs to introduce


-103-
wit h components z*, P e K*, and call Z* the irreducible representation
of Z. Corresponding to GBC*, define DC* to be the distance con
straints with every constraint corresponding to exactly one arc
in GBC*. It will be convenient to refer to the triplet (Z*, DC*, GBC*)
as the reduction of (Z, DC, GBC). We remark that for an irreducible
location vector Z, the reduction of (Z, DC, GBC) is Identical to
(Z, DC, GBC), as RP terminates immediately in this case.
For P e K*, define A(N^) to be the set of adjacent nodes to .
in GBC*, and let Ap(Z*) be the collection of locations of facili
ties whose nodes are members of A(Np). The following property gives
the sufficient conditions for reducible location vectors (as well as
irreducible ones).
Property 3.5.3. Let (Z*, DC*, GBC*) be the reduction of (Z, DC, GBC)
and let K* be the list of composite indices for new facility nodes
of GBC*. If Zp e H[Ap(Z*)] for every P e K*, then Z is efficient.
Proof. By definition Z* is irreducible. Hence, the hypotheses of
the property imply, upon using Property 3.5.2, that Z* is efficient
with respect to the reduced constraints DC*. From Theorem 3.3.3,
for every P e K*, node Np is in a tight path in GBC*. Now, we want
to show that the original nodes i e P, are all in tight paths
in GBC. Recover GBC from GBC* by decomposing every node Np of GBC*
into its original nodes N^, i e P, and connect these nodes to one
another by arcs of zero length by adding those arcs which were
removed by RP. Since the added arcs have lengths of zero, the
shortest path lengths cannot change. Hence, the shortest path length
between any two existing facility nodes of GBC* is the same as the


-2-
between the facilities, the objective criteria used, the constraints,
the presence or lack of random elements, and possibly other factors
as well.
Among the several variants, planar location problems received
special attention in the past, starting with the earliest contribu
tions, for example [106]. In such planar problems, one is interested
in locating new facilities in the Euclidean plane with respect to
existing facilities. For continuous planar problems, where any point
in the plane is a feasible location, the typical distance used is the
distance, special cases of which are the rectilinear, Euclidean,
and Tchebychev norms. For discrete planar problems, where there are
a finite number of candidate locations for new facilities, the distance
between any potential new facility location and any existing facility
is a specified positive number. Such discrete problems, due to the
finite nature of feasible locations, readily lend themselves to integer
programming formulations. The reader is referred to the book by
Francis and White [31] for a discussion of planar problems and a wealth
of references.
A number of real life applications suggest that, in some in
stances, a network space can be a more faithful representation of the
reality than the Euclidean plane. For example, in a road network, a
communication network, or a pipeline system, travel occurs along the
arcs of the underlying network rather than in straight lines or recti
linear paths. Hence, for such problems, the use of shortest path
distances along the arcs of the network can approximate the travel
distance more closely than the X. distance. As opposed to planar
problems, network location problems have received much less attention


-47-
and a center constrained one. More specifically, for real A and y
define the functions m*(A) and c*(y) as follows:
m*(A) = min[m(x): c(x) < A]
(1.3.5)
c*(y) = min[c(x): m(x) y]
(1.3.6)
In general for some values of A (y), the constraint c(x) £ A
(m(x) £ y) may not admit any feasible solution. However, real inter
vals C and M can be defined so that for any AeC and for any yeM, the
constraints in (1.3.5) and (1.3.6) admit a feasible point. To define
C, let Q be the set of all minima to min[c(x): xeN], and let 2 -be
c m
the set of all minima to min[m(x): xeN]. Let x be a point in that
c
minimizes the value of m(x) over all x in Similarly, let y be a
point in £! that minimizes the value of c (y) over all y in ft Then
m J J m
C and M are defined as follows:
C = [c(x), c(y)]
M = [m(y) m(x) ]
With these definitions Halpem's duality theorem can be stated
as follows:
a) Given any yeM, with A = c*(y), we have c*(m*(A)) = A.
b) Given any AeC, with y = m*(A), we have m*(c*(y)) = y.
For a tree network, the functions m* and c* are 1-1 and onto.
It follows from the duality theorem that the function m* and c* are
inverses of each other for a tree network. For a general network,
the functions m*, c* need not be onto, i.e., the image of the domain


-76-
where (2.5.5) follows from the W.D.T., (2.5.6) and (2.5.7) follow
from the definitions of g and g^, (2.5.8) follows from K*+^ c U,
(2.5.9) follows from (2.5.4), and (2.5.10) follows from the definition
of R. Hence, every inequality holds as an equality, establishing
(2.5.2) for this case.
The assertion that K*., solves the dual problem is immediate from
p+i
f(X*) g(K*+1) and the W.D.T.
We note that Theorem 2.5.1 provides a proof of the S.D.T. since in
the statement of the S.D.T. we take X* to be an optimum p-center solu
tion to the primal problem and K*+^ as constructed by OPTKLIQUE. We
also note that the duality theory provides necessary and sufficient
conditions for a p-center to be optimal, which, as far as we know, are
the first such conditions for this problem.
We remark, just as with the linear p-center problem, that if we
define 6 = minig..: g e R, q(B..) < p}, then 8 ^ = r Clearly
st ij ij ij st p
q(r ) < p and q(8 ) < p. The S.D.T. implies r e R, and thus the
P st p
definition of g ^ gives g ^ < r Let p' = q(g ) and let X solve
st st p st p
the cover problem for r = g so that f(X .) < g Since p > p',
st p st
append to X^, (if necessary) any p-p' center locations to obtain the
p-center X^. Clearly D(X^,v^) D(X^,,Vj) for v e V, and thus
f(X ) < f(X ,). Hence r f(X ) < f(X ,) < g < r so g = r
P p p p p st p st p
and X^ is an optimum solution to the p-center problem. This remark
permits the use of the same procedures as discussed in [65] to compute
r^ efficiently, by performing a binary search over the (ordered) list
R, applying COVER for every r chosen from R until a smallest g in R
st
is found for which COVER finds p or less points. Once r^ is computed
in this manner, OPTKLIQUE requires an additional application of COVER


-43-
the concept of "efficient solutions." A point x in S is said to be
efficient if there does not exist a point y in S such that f^(y) < f_^(x)
for 1 i < n and f^(y) < f^Cx) for at least one index k. One is
interested in finding and characterizing the set of efficient solu
tions to (VMP)An efficient point is sometimes known as an undominated
point. A point which is not efficient is said to be dominated.
Kuhn and Tucker [76] and Koopmans [74] are among the first to
introduce the concept of efficiency. Geoffrion [40] extendd the con
cept to "properly efficient" points and provided a comprehensive
theoretical framework for subsequent research. Necessary and suf
ficient conditions for efficient points to be properly efficient are
given by Wendell and Lee [112]. Some of the later contributions are
due to Yu [117], Yu and Zeleny [118, 119], Bitran and Magnanti [4],
Wendell [110], and Bergstresser, Chames, and Yu [l ] We note that
there are other approaches to multicriteria decision making, such as
goal programming, multi-attribute utility theory, construction of
outranking relations, and interactive programming techniques. For
general information on multicriteria decision making, the reader is
referred to Roy [99], Starr and Zeleny [104], Cochrane and Zeleny
[16], Keeney and Raiffa [67], and Thiriez and Zionts [l08]- A survey
of multicriteria decision making is given by Chalmet [7].
Multi-objective location problems (on the plane or on networks)
have begun receiving attention only recently. Kuhn [75] appears to
be the first to consider a multi-objective location problem on the
plane. Kuhn considered the problem of minimizing the vector of
Euclidean distances from a variable point to a set of fixed points on
the plane, and showed that the set of efficient solutions is the convex


-17-
y.e{0,l} will determine whether or not at most p points (in P) can
cover all vertices of N within a radius r. Computational experience
is reported and it is found that the procedure works better for larger
values of p, as in this case the initial upper bound Z is small, and
significant computational savings result in identifying those edge
bottleneck points whose distances fall within the interval [0,Z].
The weighted case on general networks was considered by Christofides
and Viola [15], and an approximate solution procedure was given. The
procedure finds a set X of p-points whose objective value f(X) is
within some e-neighborhood of the actual p-radius r The procedure
P
oi
obtains X by solving a sequence of r-cover problems with successively
increasing values of r. Termination occurs when the solution of an
r-cover problem generates p (or less) points the first time. In the
process, one also obtains approximate solutions for n-1, n-2,..., p+1
center problems. The solution of each r-cover problem is obtained in
two stages: First, all feasible solutions to the r-cover problem are
obtained by finding all regions on the network that can be reached by
a vertex within a radius of r. Then, among all these feasible solu
tions, those with minimum cardinality are found by solving a set
covering problem. To find all regions on N reachable by a vertex v_^,
one "penetrates" a distance of r/w_^ along all possible paths originating
at v_^. The procedure is repeated for each vertex and the intersections
of these penetrations are found. Each maximal intersection defines a
connected region all of whose points are reachable by a subset of
vertices within a radius r. The subset of the vertices is that which
defines the intersection. These regions jointly cover all vertices
of N, and it is possible that a subcollection of the collection of all


CHAPTER 5
SUMMARY AND FUTURE RESEARCH
5.1 Summary
In this dissertation we considered problems which involve locating
multiple new facilities on a tree network with respect to n existing
facilities at known locations.
In Chapter 2, we solved the nonlinear p-center problem where the
objective is to minimize the maximum cost associated with any existing
facility. The cost (disutility) of service associated with any
existing facility is a nonlinear (strictly increasing and continuous)
function of the distance between that existing facility and its nearest
new facility. We gave a weak and a strong duality theorem and pro
vided a physical interpretation of the dual problem. Our approach for
solving the nonlinear p-center problem and its dual was to solve a
sequence of covering problems which involve minimizing the number of
new facilities subject to a family of n distance constraints which impose
upper bounds on the distances between any existing facility and its
nearest new facility.
In Chapter 3, we considered a vector-minimization problem which
involves as objectives the distances between specified pairs of new
and existing facilities and specified pairs of new facilities. We
developed the necessary and sufficient conditions for efficiency and
provided an algorithm for constructing efficient solutions. Our
L 49


-28-
and the objective to be minimized is f(X) = J [f (X) : (v ,v,)eVxV],
u sd s d
Goldman showed that there exists an optimal X* contained in V, and
conjectured that the result holds for any multi-stage problem.
Hakimi and Maheshwari [49] proved a stronger version of Goldman's
conjecture. In this version, there are multiple commodities for each
source-destination pair, and each commodity goes through multiple
stages. Furthermore the cost of transport from one stage to the next
is a concave nondecreasing function of the distance. More specifically,
let M be the set of commodities to be transferred from source v to
sd s
destination v^, and let g(m) be the number of stages commodity meM^
is to go through. For a given location set X = {x.,...,x }, denote
1 P
by y^ = the location where the rth stage processing takes place.
The cost of transferring commodity m from source v to destination v,
s d
is given by C^Jd (v^yp ] + ] + ... + C^d (yg(n),vd) ],
where () is a concave nondecreasing function of the distance.
Denoting this quantity by f^^(Y), with Y C X, |y| = g(m), the minimum
cost of transfer for commodity m is given by f (X) = min[f (Y):
sdm sdm
Y C. X, |Y| = g(m)]. The cost of transferring all commodities from vg
to Vj is obtained by summing over all commodities, that is,
fgd(X) = J [fsdm(X): meM d]. The total cost of the system is obtained
by summing the cost f ^(*) over all source-destination pairs, that is,
f(X) = £ [f^W: (v ,Vj)eVxV]. Hakimi and Maheshevari proved that
there exists a minimum X* of f(X) contained in V.
Wendell and Hurter [111] considered a more general form of the
problem where the transportation cost functions are permitted to
differ from edge to edge. The transport cost on any edge is a non
decreasing concave function of the distance. They proved that it is


-6-
Finally, for tree networks, we write T instead of N. In passing,
we note that the shortest path P(x,y) between any two points x,yeT is
unique, as otherwise T would contain a cycle.
1.3 Survey of the Network Location Literature
Historically, the earliest precise mathematical formulation of a
location problem on a network appears to be due to Hakimi [47] in 1964.
Prior to Hakimi's paper, the problem of finding the best threshing
site for harvested wheat was attacked by using a network location model
in 1962 by Hua Lo-Keng and Others [60]. This model was presented only
at an intuitive level and no mathematical formulation or properties
were given. A (correct) solution procedure was suggested (in the form
of a poem), which was to be discovered independently by Goldman [42] in
1971. Since 1964, a literature of approximately 80 papers has grown
till the present. Several new problems, as well as certain extensions
and generalizations of old problems, have been introduced.
A recent text by Handler and Mirchandani [ 58 ] discusses ex
tensively a portion of the literature involving minimax and minisum
problems as well as single-facility bi-objective problems involving
the combination of these two objectives.
A "family tree" for network location problems is shown in
Figure 1.1. Although not exhaustive, the family tree covers most of
the problems formulated since 1964. With reference to the family tree
shown in Figure 1.1, network location problems can be broadly classi
fied into two groups: point-location problems and path-location
problems. Path-location problems have been recently introduced by


-63-
Table 2.1
Values and Node Weights for Example
i
1
2
3
4
5
2
3
4
5
6
225
3600
3600
3600
4356
3600
3600
3600
4556.25
13829.76
8464
11664
900
784
1664.64
j
1
V0)
0
2 3 4
0 64 0
5 6
0 144


-162-
12. R. Chandrasekaran and A. Daughety, "Problems of Location on Trees,"
Discussion Paper No. 357, The Center for Mathematical Studies in
Economics and Management Science, Northwestern University,
Evanston, Illinois (1978).
13. R. Chandrasekaran and A. Tamir, "Polynomially Bounded Algorithms
for Locating p-Centers on a Tree," Discussion Paper No. 358, The
Center for Mathematical Studies in Economics and Management
Science, Northwestern University, Evanston, Illinois (1978).
2
14. R. Chandrasekaran and A. Tamir, "An 0((nlogp) ) Algorithm for the
Continuous p-Center Problem on a Tree," Discussion Paper No. 367,
The Center for Mathematical Studies in Economics and Management
Science, Northwestern University, Evanston, Illinois (1978).
15. N. Christofides and P. Viola, "The Optimum Location of Multi-
Centers on a Graph," Opnl. Res. Q;. 22, 145-154 (1971).
16. J.L. Cochrane and M. Zeleny, Eds., Multiple Criteria Decision
Making, Univ. of S. Carolina Press, Columbia, South Carolina,
1973.
17. E.J. Cockayne, S.T. Hedetniemi, and P.J. Slater, "Matchings and
Transversals in Hypergraphs, Domination and Independence in Trees,"
^J. Combinatorial Theory, Series B26, 78-80 (1979).
18. P.M. Dearing, "Minimax Location Problems with Nonlinear Costs,"
J. Res. Nat. Bur. of Stds. 82, 65-72 (1977).
19.P.M. Dearing and R.L. Francis, "A Minimax Location Problem on a
Network," Trans. Sci. 8, 333-343 (1974).
20.P.M. Dearing and R.L. Francis, "A Network Flow Solution to a
Multifacility Minimax Location Problem Involving Rectilinear
Distances," Trans. Sci. 8, 126-141 (1974).
21. P.M. Dearing and G.J. Langford, "The Multifacility Total Cost
Location Problem on a Tree Network," Technical Report No. 209,
Dept, of Mathematical Sciences, Clemson University, Clemson,
South Carolina (1975).
22. P.M. Dearing, R.L. Francis, and T.J. Lowe, "Convex Location
Problems on Tree Networks," Opns.Res. 24, 628-642 (1976).
23. S.E. Dreyfus, "An Appraisal of Some Shortest Path Algorithms,"
Opns. Res. 17, 395-412 (1969).
24. A.M. El-Shaieb, "A New Algorithm for Locating Sources Among
Destinations," Manag. Sci. 20, 221-231 (1973).
25. D.J. Elzinga, D.W. Hearn, and W.D. Randolph, "Minimax Multifacility
Location with Euclidean Distances," Trans. Sci. 10, 321-336
(1976).


-29-
sufficient to consider the vertices of the network under such a cost
structure. Furthermore, they obtained the conditions under which it
is necessary for the solution to occur at the vertices. In particular,
they showed that nonvertex optimal locations can occur in any given
edge, only when transportation costs are linear with distance over
that edge and in that case, when and only when the slopes of these
linear cost functions are in a special relation. Hence, if at least
one cost function over some edge is nonlinear, then no interior point
of that edge can be in an optimal solution. If the same situation
holds for every edge, then a solution must necessarily occur at the
vertices of the network.
Solution approaches. Kariv and Hakimi [66] showed that the p-
median problem on a general network is at least as hard as NP-complete
problems. For the case of tree networks, however, algorithms of
polynomial complexity have been developed. Matula and Kolde [85]
3 2
suggested an 0(n p ) algorithm for finding the median of a tree net-
2 2
work. Kariv and Hakimi [66] proposed an 0(n p ) algorithm for the
same problem.
For general networks, a number of solution procedures have been
developed subsequently, all based on the vertex-optimality result.
Their common characteristic is that they all confine the search to
vertex locations. The solution procedures can be grouped in three
categories: mixed-integer programming approaches, branch-and-bound
techniques, and heuristics.
ReVelle and Swain [95] formulated the problem as a linear integer
program with 0,1 variables. The solution is obtained by applying the
primal simplex algorithm to the associated linear program. In case


Abstract of Dissertation Presented to the Graduate Council
of the University of Florida in Partial Fulfillment of the Requirements
for the Degree of Doctor of Philosophy
OPTIMAL MULTI-FACILITY LOCATION ON
TREE NETWORKS
By
Barbaros C. Tansel
December 1979
Chairman: Richard L. Francis
Major Department: Industrial and Systems Engineering
In this dissertation we develop a theory for location problems which
involve locating multiple new facilities on a tree network with respect
to existing facilities at known locations.
The first problem we consider is the nonlinear version of the
p-center location problem on a tree network for which the cost of each
served vertex is a strictly increasing continuous function of the dis
tance between the vertex and the nearest center,and the objective is to
minimize the maximum such cost over all possible locations of the
centers. We present a dual "dispersion" problem which may be inter
preted as the problem of choosing p + 1 (or more) vertices such that
the minimum cost to serve any two of the chosen vertices by a single
common center is as large as possible. We give a weak duality theorem
which applies to all general networks and a strong duality theorem
which applies to all tree networks. The strong duality theorem also
specifies the necessary and sufficient conditions for an optimal solu
tion to either problem. We provide algorithms of polynomial complexity
v


-55-
The first duality relationship involving tree network location
problems can be found in Meir and Moon [ 86 ]. Cockayne, Hedetniemi,
and Slater [ 17 ] obtained a more general version of the result given
in [ 86 ]. The results in [ 86 ] and [ 17 ] closely parallel our duality
result for the covering problem and its dual. Shier [100] discovered
a "dispersion" problem which is dual to the continuous unweighted
p-center problem. The dispersion problem of Shier is to choose p+1
points in the tree network the nearest two of which are as far apart
as possible. Chandrasekaran and Tamir [14] observed that Shier's
duality holds when the problems are defined with respect to a subset
of the tree. For the case where this subset is a finite collection
of demand points, their result is an instance of the duality relation
ship we will present in this chapter, as applied to the unweighted
linear case.
At this point we give a brief overview of the chapter. In Sec
tion 2, we define the (nonlinear) p-center problem and a dual "dis
persion" problem. We state and prove a weak duality theorem applicable
to all networks, and state a strong duality theorem applicable to
tree networks. In Section 3 we give a physical interpretation
of the dual dispersion problem. In Section 4 we study a covering
problem and present an algorithm, COVER, for solving it. The covering
algorithm provides the basis of our solution procedure to the p-center
problem as well as the dual dispersion problem and yields a construc
tive approach for proving the strong duality theorem. In Section 5 we
present an algorithm, OPTKLIQUE, which provides a constructive proof
of the strong duality theorem, while solving the dual problem. Addi
tional results for the covering problem, including a "divergence"
problem dual to the cover problem, are given in Section 6.


-124-
As in Chapter 3, a location vector Y in Tm is said to be
efficient with respect to (4.2.2) if and only if X e Tm and f(X) < f(Y)
imply f(X) = f(Y). A location vector which is not efficient is said
to be dominated.
Our main interest is to relate the bi-objective m-center problem
to the distance constraints problem studied by Francis, Lowe, and
Ratliff [32], We shall characterize efficient points by making use
of the separation conditions (defined in Chapter 3) which are known
to be necessary and sufficient for the distance constraints to be
consistent.
To define the distance constraints of interest, let z = (z^,z^)
be any two-tuple (with z > (0,0)) and consider the constraints given
in (4.2.3):
d(x^,Vj ) < Zj/w^ (i, j ) e Ic
(4.2.3)
d(xjxk) Z2/Vjk (j>k) e Ig .
We shall refer to the family of constraints in (4.2.3) as DC^.
The constraints DC are said to be consistent if there exists at least
z
one feasible solution X = (x, x ) to (4.2.3).
1 m
Corresponding to DC^ we define GBCz to be the undirected graph
with nodes ,...,Nm,E^,...,En- For every (i,j) e 1^ there is an arc
(N.,E.) of length z./w.. and for every (j,k) s ID there is an arc
1 3 i ij a
(N.,N,) of length z/v., We partition the arc set of GBC into two
j K z j k. z
sets Ab and Ac with Ag = {(N.,Nk): (j,k) e Ig} and Ac = {(N,Ej):
(i,j) e Iq}. We shall assume that the sets 1^ and Ig are such that
GBC is connected, as otherwise DC decomposes into independent sets
z z
of constraints which may be analyzed separately.


-35-
connected, denote by L(E.,E ) the length of a shortest path connecting
J k
nodes E. and E. for 1< j 3 k
distance constraints are consistent on a tree network if and only if
the inequalities (E^.E^) £ d(vj,v]P are satisfied for 1 < j < k < n.
These inequalities are called the separation conditions. The proof
of the consistency of the distance constraints implying the satisfac
tion of the separation conditions uses only the triangle inequality
and hence is applicable to all networks. The reverse implication
always holds for tree networks, but may fail to hold for general net
works. The proof of the reverse implication is constructive and
actually finds a feasible location vector under the assumption that
the separation conditions are satisfied. The method that constructs
such a feasible location vector is termed the "Sequential Location
Procedure" in [32]. The method can best be described with the aid of
a physical model. One may imagine that the tree is represented by
appropriately inscribing straight line segments on a board such that
each segment represents an edge. At vertex v_^, strings of length c
are fastened for each new facility j such that (i,j)el A tip vertex
Li
is chosen arbitrarily and all strings fastened at that vertex are
pulled tight towards the adjacent vertex. If all strings reach the
adjacent vertex, they are simply engaged there with their loose ends
free to be pulled tight in some future iteration. Also the tip vertex
together with the edge incident to it is removed from the model. The
procedure is repeated with the resulting tree. In the other case,
not all the strings reach the adjacent vertex when pulled tight. Among
those which do not reach the adjacent vertex one which is shortest is
selected, and the end point of this string determines the location of


-158-
5.4 Tree Networks and General Networks
For the nonlinear p-center problem we pose the following questions.
Ql. For the nonlinear p-center problem defined on a general network N,
does there exist a spanning tree T of N such thnt the solution of
the p-center problem on T determines the solution to the p-center
problem on N?
Q2. If such a spanning tree exists, is there a way to find it without
having to enumerate on all spanning trees of N?
Dearing and Francis [19] showed that the 1-center problem on N
can be solved by solving a sequence of 1-center problems on spanning
trees of N. The basic theme of the approach is as follows: Suppose
that we know an optimal 1-center, say, x which solves the problem on N.
Using the procedure given in Busacker and Saaty [5], find a shortest
path tree rooted at x, by identifying the shortest paths (on N) con
necting x to any vertex v^. If T(x) is such a shortest path tree, then
clearly x is also the optimal 1-center of T(x). Thus, for the 1-center
problem the answer to Ql is in the affirmative while Q2 remains un
answered as the proof of the existence of such a tree is based on the
knowledge of an optimal 1-center of N.
For the case with p > 1 we propose a similar approach. Suppose
we know an optimal p-center X = ix. x } which solves the problem
^ P
on N. Partition the vertex set V = (v^,...,vn} of N into p disjoint,
exhaustive subsets V,,...,V such that a nearest center to any vertex
1 p
v in V, is the ith center x.. If ties occur we break the ties
i i i
appropriately so as to satisfy the condition that V^,...,V are
mutually disjoint. For 1 < i < p, define T^(x^) to be a shortest
path (sub)tree which is constructed by finding the shortest paths


-67-
There is also the possibility that A will make a false threat,
that is, attack a vertex not among the ones he threatens. If D be
lieves the threat is false and continues to act conservatively, he
will simply choose a p-center X* to minimize f. But since there exists
a p+l-clique KA+^ such that g(K*+p = f(X*), the greatest loss D can
incur, given Xa, is the same as if he believes A's optimal threat to
be real, and acts accordingly. Hence A cannot gain more by making a
false threat.
2.4 Covering Algorithm
In this section we study a covering problem, and present an
algorithm for solving it. Our primary interest in the algorithm is
the fact that it provides a constructive approach for proving results
about the primal and dual problem. For this reason we purposely keep
the algorithm simple, and use an analog string model to provide insight
into the algorithm. The development of both the string model and the
algorithm is motivated by an earlier string algorithm given in [32].
As in [32], an equivalent algebraic version of the algorithm is
readily obtainable. We remark that two other quite efficient algo
rithms [14], [15], exist for solving the covering problem, but they
do not lend themselves readily to our needs.
At this point we state the Covering Problem: Given r and the
runction f, compute
q(r) = mini|x|: f(X) S r, X C T} (2.4.1)
It is readily seen that the covering problem has a feasible solution
if and only if a < r. Further, with J(r) = {j: r < f^(6 )}, we shall


-96-
Figure 3.4. Example of an Efficient Location Vector
3.5 Further Results on the Convex Hull Property
In this section we concentrate on the last statement of Theorem
3.3.3, namely, that Z is efficient if and only if each is contained
in the convex hull H[A*(Z)], where A*(Z) contains the locations of
those uniquely located facilities whose nodes are adjacent to N^.
Our main interest is to delete the phrase "uniquely located" from
the definition of A*(Z) and still have the equivalence hold under the
new (relaxed) definition. From a computational standpoint, this deletion
would make it unnecessary to identify the uniquely located new facilities,
which, in turn, requires the identification of tight paths in GBC.


-155-
S(P(Ep,E^)jA^) to be the sum of the reciprocal weights where the sum
mation is taken over all arcs which are contained both in P(E ,E )
P q
and in A Define e to be the minimum of
r
L P(E ,E ) d(v ,v )
z P q P q
S(P(E ,E ),A )
p q r
over all paths P(E^,E^) which pass through A^. Clearly e > 0. Choose
X = (A,,...,!) with A. = 0 for i I r, and 0 < A < e. It is direct
I t i r
to verify, by using arguments similar to the ones given in the proof
of Lemma 4.3.2, that such a choice of A is a valid choice for DC to
z-A
be consistent. But consistency of DCz_^ and the fact that A > 0,
A 0, imply X is dominated, contradicting that X is efficient.
To show (b) implies (a) suppose for any r for which > 0, at
least one arc in A^ is in a tight path. Hence, for any A = (A^,...,A )
> 0, A 0, the length of at least one tight path in GBC .. will be
strictly smaller than the distance between the locations of the exist
ing facilities corresponding to the terminal nodes of the path. Thus,
at least one separation condition on GBC is violated so that DC ..
z-A z-A
is inconsistent for any A > 0, A ^ 0. It follows that there does not
exist Y for which f(Y) < z = f(X) and f(Y) ^ f(X), which is the
definition of efficiency.
The theorem holds for the problems considered in Chapters 3 and 4
as well as the m-center problem with mutual communication considered
in [32].
We remark that the condition > 0 may appear to be somewhat
superfluous. Its omission will not affect the equivalence of (a) and
(b). The reason we included this condition is that it is unnecessary


-97-
With this motivation in mind, define, for 1 < j < m, A(N_.) to be
the collection of nodes in GBC which are adjacent to N and denote by
A (Z) the collection of the locations of the new and existing facili-
j
ties whose nodes are in A(N). We remark that N, is not a member of
J J
A(N.) and hence z^ f/ A^ (Z) .
The following property states the necessary conditions for Z to
be efficient.
Property 3.5.1. Suppose Z is efficient. Then ze H[A^(Z)] for every
j £ {1,... ,mj.
Proof. From Theorem 3.3.3, whenever Z is efficient, z^ e H[A*(Z).] for
each j e {l,...,m}. But A*(Z) is clearly a subset of A(Z) implying
that z e H[*(Z)] C H[A (Z)], completing the proof.
In general, the reverse implication in Property 3.5.1 need not
hold for certain (pathological) cases. Such occurrences correspond to
the case where Z is such that for some two adjacent nodes N. and N, ,
J k
the locations z and z^ coincide. We provide an example of such a
case in Figure 3.5. With reference to Figure 3.5, observe that every
z. is contained in the associated convex hull. In particular, 7. and
J ^
z^ are contained in their respective convex hulls because their loca
tions are the same. The location vector is clearly a non-efficient
one, since and z^ can both be moved to v^> thereby reducing the
distances associated with them.
Sufficiency for Irreducible Location Vectors
At this point we distinguish two classes of location vectors and
show that the reverse implication (sufficiency) in Property 3.5.1 holds
for one class ("irreducible" location vectors) while it need not hold


-34-
property, a "sequential intersection procedure" was developed that
n
determines the composite neighborhood N(a,r) = O N(v.,c.), with
i=l 1
unique center a and radius r, by intersecting the neighborhoods
N(v ,c ) one at a time in an arbitrary order. The procedure can be
implemented in 0(n) operations. The composite neighborhood N(a,r)
contains all alternate feasible points when the constraints are con
sistent, and N(a,r) is always a convex compact subset of the tree
network. A result was also given by Francis et al. that provides a
sensitivity analysis on the constraints with no additional computa
tional effort. Supposing that the distance constraints are consistent
with the original upper bounds c^, consider an e-perturbation of the
upper bounds, i.e., for some e > 0 define the new upper bounds to be
c^-e, iel. If N(a,r) is the composite neighborhood corresponding to
the original upper bounds, then it can be shown that for any e with
0 e ~ r, the e-perturbed constraints remain consistent and the set
of feasible points to the e-perturbed system is given directly by
N(a,r-e).
Distance constraints for the multi-facility case. For the multi
facility case, the necessary and sufficient conditions for the con
sistency of distance constraints are given in terms of n(n l)/2
inequalities called "separation conditions." The separation condi
tions are defined by means of an auxiliary graph constructed by using
the sets I and I Let G be the graph with nodes N., 1 5 i < m,
ij v 1
corresponding to new facilities,and nodes E ^, 1 < j < n,corresponding
to existing facilities. The arc set A of G contains (N ,E ) if
i j
(i,j)cl-, and (N.,N ) if (j,k)el. The arc length of (N,,E.) is c_,.
G j k B i j ij
and of (N.,N^) is b.^. Under the (reasonable) assumption that G is


-13-
2
computational complexity of 0(n ), and is given by
a = max[a..: 1 i j n]
where
(1.3.2)
W-W.VAIV.JV,/ W C* I W t-4. .
= i J 1 J 1 J J 1
w w.d(v.,v.) +w.a. + w.a
a
ij
w. + w,
1 jj
Hakimi, Schmeichel, and Pierce [50] proved a theorem that reduces the
computational effort for computing this lower bound. Their theorem
states that if for some a it is true that max[a l5i st si st
max[a 1 K i < n] then a is the maximum of all a... A different
ti J st 13
solution procedure is also given by Kariv and Hakimi [65] for the
same problem. Rather than computing the lower bound, their procedure
confines the search to successively smaller subtrees until an edge is
obtained. The absolute center is located at the local center (also
the global center for a tree) on this edge using Hakimi's procedure
for finding a local minimum. This algorithm is of O(nlogn).
A nonlinear version of the 1-center problem was considered and
solved by Dearing [18], and by Francis [29]. In this version, each
weight w_^ is replaced by a monotone increasing function f of the
distance d(v_^,x). Both authors obtained a lower bound similar to the
one defined by (1.3.2). The bound is applicable to all networks and
is always attainable for tree networks.
A "roundtrip" version of the problem was solved by Chan and
Francis [ 11 ]. In this version each "demand point" is a pair of ver
tices (v^,^) and f(x) is the maximum of the roundtrip distances
defined by p^(x) = w^dCv^x) + d(x,u^) + a^]. A lower bound, similar


I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
Antal Maj thay
Associate Professor of Managemen
This dissertation was submitted to the Graduate Faculty of the College
of Engineering and to the Graduate Council, and was accepted as partial
fulfillment of the requirements for the degree of Doctor of Philosophy.
December 1979
Dean, Graduate School


-70-
Note that each time COVER places a center at x^ in step 4) it
identifies an associated vertex v^ which we call the distinguished
vertex associated with x^. When centers x^,...,x^. have been placed
in step 4), we call = iv^ ,... ,v^ } the distinguished set
associated with {x^,...,^}. If the algorithm places q centers in
total, then the set U defined by the algorithm consists of vertices
v,,.,...,v, N, the first q-1 of which are distinguished vertices
(i) (q)
(when q >2). The last vertex is distinguished only if x^ is placed
in step 4). Letting X = {x^,...,x }, we call U the primary set
associated with X, and call the primary vertex associated with
x^, i = l,...,q. We note that the primary vertices v(i)*,v(q) are
distinct, for as soon as a primary vertex is identified, its string
is removed, and thus the vertex is not available for any subsequent
identification. Likewise the centers x ,. .. ,x are distinct, for if
1 q
x. = x. with i < i, then all strings assigned to x. would have been
i J J
assigned earlier to x^, and so x^. would not have been located. Hence
it follows that JU| = JXJ = q, and U 0, since JXJ k 1. The primary
vertices will be of theoretical significance in proving our results.
We now establish some properties of COVER.
Property 2.4.1. COVER finds a feasible solution X to the covering
problem with |x| < n.
Proof. We first note that termination is clearly finite, since at
each iteration either at least one string is removed, or some entire
arc of T becomes colored brown. Since there are at most n strings
initially, it follows that the X constructed satisfies |x| < n.
Choose any v^,j £ J(r), and denote by x^ the center to which
v. is assigned. Since the string fastened at v. reaches x,..,
J (j /


-10-
ln such problems a common objective is to provide "good" service to
each demand point by at least one facility within a least possible
distance.
In what follows, we first discuss the 1-center problem on general
networks and on tree networks. Then, we discuss the vertex restricted
1-center problem. Finally, we will discuss the p-center problem in
relation to a "covering" problem to be defined later.
1-Center problem on a general network. The absolute 1-center
problem was defined and solved by Hakimi [47] in 1964. For finding the
absolute center, Hakimi examines the function f on each edge, finds a
best local minimum on that edge, and selects the best among |e| such
local minima. This method takes advantage of one important property
of f, namely, that it is piecewise linear and continuous on each edge
with at most n(n l)/2 break points. A local minimum always occurs
either at a break point of f or at an end point of the edge. Hakimi,
Schmeichel, and Pierce [50] showed that Hakimi's method can be imple-
mented in 0(|E|n logn) computational effort and gave a computational
refinement which reduces the effort to 0(|E|nlogn) for the unweighted
case. Further refinements of the procedure were obtained by Kariv
and Hakimi [65], resulting in an 0(|E|nlogn) algorithm for the
weighted case and 0(|E|n) algorithm for the unweighted case. All
these refinements focus on finding the break points and the local
minimum of.f in the most efficient manner.
A somewhat more general version of the 1-center problem was con
sidered by Frank [36], and (apparently) independently by Minieka [88],
as Minieka makes no reference to Frank's paper. In this modified
version, called here the continuous 1-center problem, each point on


-86-
addition, given a finite point set P C T, we define the convex hull
H(P) to be the smallest (embedded) subtree of T containing all points
in P. We note that for any two points p,p' e P, the line L(p,p') is
contained in H(P).
We denote by I the set of pairs (i,j) for which the distance
u
d(x^,Vj) is of concern. Similarly, 1^ is the set of pairs (j,k) for
which the distance d(x.,x, ) is of concern. We remark that it need not
3 k
be the case that 1^ includes all possible pairs of new and existing
facility indices, nor I includes all possible pairs of new facility
indices. With these definitions, the problem of interest is to "mini
mize" each of the distances specified by (3.2.1);
d(x >v ) (i,j) e I ,
1 J O
(3.2.1)
d(x.,x ) (j ,k) e I .
J K. d
For X e Tm, we denote by D(X) the vector each of whose components
is a distance specified by (3.2.1). The vector is formed by assuming
any convenient ordering of the members of 1 and I. The vector-
C B
minimization (V-min) problem of interest is
V-min{D(X): X e Tm} (3.2.2)
With respect to (3.2.2), a location vector Z e Tm is said to
dominate a location vector X in Tm if D(Z) < D(X) and D(Z) D(X).
A location vector Z which is not dominated by any other location vector
is said to be efficient. An equivalent definition of efficiency is as
follows: Z e Tm is efficient if and only if X e Tm and D(X) < D(Z)
imply D(X) = D(Z).


-48-
may only be a proper subset of the range. Hence, the inverse property
holds only for some members of C and M for a general network.
Now, we consider a more general multi-objective problem due to
Lowe [82]. The problem involves a single facility to be located on a
tree network with respect to m convex objective functions.
Multi-objective convex location problem (on a tree). Let T be
a tree network and let f,,...,f be m convex continuous bounded func-
1 m
tions each of which is defined on T. In general, not all points in T
may be feasible with respect to f_^. Let be a convex compact subset
of T which contains all feasible points x with respect to the ith
optimizer. The set Q, may be defined by specifying its extreme points,
or by means of distance constraints, or by other means. We assume
m
that Q. is known or computable. Define Q = D Q. and assume that Q
1 i=l 1
is nonempty. The problem of interest is to find all efficient points
in Q with respect to the vector minimization problem defined below:
V-min[f(x): xeQ C T]
where,
f(x) = (f1(x),...,f (x)) for all xeT
i m
We note that Q is a convex compact subset of T as it is the
intersection of m convex compact subsets of T. For a formal dis
cussion of convexity on a network, the reader is referred to Dearing,
Francis, and Lowe [22]. Loosely speaking, Q a convex subset of T,
means Q is connected or that the (shortest, unique) path connecting
any two points in Q is contained in Q.
Lowe makes no assumptions on the specific forms of the objective
functions. Under the convexity assumptions, Lowe proves that a convex


-118-
whlch contain (f ,f ), giving e' > 0. Choose any e such that
p q
0 < e < min(e',e ). Reduce the length of arc (f ,f ) from e to
Pq p q pq
e e and define the resulting graph to be GBC^. Due to our choice
of e clearly every separation condition on GBC^ holds, as, the length of every
path which contains (f ,f ) is reduced by an amount smaller than the
P q
difference between the path length and the distance between the existing
facility locations corresponding to the terminal nodes of the path,
while the length of any path which does not contain (f ,f^) remains
the same. Let DC^ be the distance constraints corresponding to GBC^..
Since the separation conditions on GBC^ hold, DC^ is consistent.
Letting Y be any feasible solution to DC ', it follows that D(Y) < D(Z)
and the entry of D(Y) corresponding to (f ,f ) is strictly smaller
than the corresponding entry of D(Z). Hence, Y dominates Z, contra
dicting that Z is efficient. Thus, (a) implies (b).
To show (b) implies (a) suppose every arc in GBC of positive
length is in a tight path. Let (f ,f^) be any arc with positive length
e For e > 0, let GBC^CDC^) be the graph (distance constraints) ob
tained from GBC(DC) by replacing e by e e. Since (f ,f ) is in
pq pq p q
a tight path, for any choice of e > 0, at least one separation con
dition is violated. Since the violation of a separation condition
implies the inconsistency of the distance constraints, there does not
exist e, e > 0, for which DC^ is consistent. Clearly, then, there
does not exist Y such that D(Y) < D(Z) and D(Y) ^ D (Z) which is the
definition of efficiency.
We remark that Theorem 3.7.1 holds for tree networks as well as
If
rectilinear distances on the plane, or, Tchebychev distances in R ,
k £ 2. The proof of the theorem relies on the necessity and sufficiency


-li
ter any r, r' < r < r and solves the dual dispersion problem. This
P P
approach is essentially a primal approach for solving both problems.
An alternative approach which directly works with the dual graph is
given by Chandrasekaran and Tamir [13] for the unweighted linear p-
center problem, which works directly with what would be a subgraph of
our dual graph G. Due to absence of weights and addends, their
approach does not require the use of node weights (and for that matter
the function g^) in the dual graph. For a given value of r, Chandra-
sekaran and Tamir define an intersection graph IG with node set J and
r
arcs (i,j) for those indices i,j e J for which 8 < r. Their pro
cedure is based on a graph theoretic procedure given by Gavril [39]
and solves the covering problem by finding a minimum clique cover of
IG^ (minimum number of cliques such that every node is in at least one
clique). As a side result, their approach identifies a maximal anti
clique in IG (a maximal set of nodes in IG no two of which are con-
r r
nected with an arc). Due to "chordal" properties of IG^ as discussed
in [39], the cardinality of a minimum clique cover of IG^ is equal to
the cardinality of a maximal anti-clique in IG^. This result is a
special instance of the duality result we will present in Section 6
for the cover problem, as applied to the linear unweighted case.
Furthermore, for r = r Chandrasekaran and Tamir [39] proved a duality
P
relationship for the unweighted p-center problem using the above
properties of IG^. We remark that their duality results can be
directly proven by using the algorithm OPTKLIQUE, and by appropriately
specializing our S.D.T. for the linear unweighted case.
We now demonstrate the use of OPTKLIQUE by determining K* for
4
the example problem. From our previous analysis, r^ = 1664.64. Since


CHAPTER 1
INTRODUCTION AND LITERATURE SURVEY
1.1 Introduction and Overview
Although some mathematical models of location can be traced back
to the early seventeenth century, almost all the work on operational
models for the location of facilities has taken place within the past
22 years, between 1957 and the present. An extensive annotated bibli
ography on location-allocation problems is provided by Lea [78]. A
more recent selective bibliography is given by Francis and Goldstein
[30],
Location problems commonly involve locating a number of new
facilities (sources) in a given location space so as to provide goods
or services to a specified set of existing facilities (demands) under
one or more criteria, and, possibly, subject to a set of constraints.
The quality of the service is typically measured in terms of the dis
tances among the facilities. The use of distances is, perhaps, the
major feature which distinguishes location problems as a special class
of optimization problems. Hence, associated with any location problem
is an underlying location space on which a "distance" is defined.
Several variations of the general location problem are possible,
depending upon the type of location space, the distance function, the
number and areal extent of the facilities, the type of interactions


CHAPTER 3
A VECTOR-MINIMIZATION PROBLEM ON A TREE NETWORK
3.1 Introduction
We consider a vector-minimization problem on a tree network which
involves as objectives the distances between specified pairs of new
facilities and specified pairs of new and existing facilities. In many
location problems, especially in the public sector, it may be necessary
to build a number of public facilities which are to be shared by a number
of communities. If the optimizers cannot agree on a single objective
function, the analyst is faced with the problem of locating the facili
ties in such a manner that all parties are satisfied with the end
result. In such a case, the optimizers can agree to rule out "dominated"
solutions and consider only "efficient" solutions.
The related literature on multi-objective location problems is
discussed in Chapter 1 under Multi-objective location problems on
networks. Here, we concentrate on characterizing efficient solutions
to the vector-minimization problem of interest. We relate efficient
solutions to a distance constraints problem studied by Francis, Lowe,
and Ratliff [32]. Extensions of results in [32] are given by Francis,
Lowe, and Tansel [33]. We use the theory developed in [32] and [33]
to establish the necessary and sufficient conditions for efficient
location vectors (parenthetically, we remark that the results we proved
in [33] are also given in our Dissertation Proposal defended on June 8,
1979).
-84-


-30-
of non-integer termination, a branch-and-bound scheme is recommended
to resolve the problem with integers. Their computational experience
indicates that non-integer termination seldom occurs. Toregas, Swain,
ReVelle, and Bergman [109] formulated a modified version of the problem
as a mixed integer program. The modification is the presence of upper
bounds on the distance between any vertex and its nearest facility.
This formulation makes use of a related but simpler problem. This
simpler problem is to minimize the number of facilities needed to cover
all vertices of N within a specified critical distance. This problem
is formulated as a set covering problem, and solved by ignoring the
integer requirements. In case of non-integer termination, a single cut
produced an integer solution in a large proportion of the cases. A
somewhat different approach to solve the relaxed linear program is
to use a decomposition scheme rather than applying the primal simplex
algorithm. Swain [105] used a Dantzig-Wolfe decomposition approach
to solve the associated linear program. Garfinkel, Neebe, and Rao
[37] independently developed a decomposition approach similar to
Swain's. In case of non-integer termination, they used group theoretics
and a dynamic programming recursion to obtain an integer solution.
A second approach taken is to solve the problem using a branch-
and-bound technique. Khumawala [68] applied a branch-and-bound method
of Land and Doig [77] type, to solve both the set covering problem and
the modified p-median problem formulated by Toregas et al. He showed
that the branch-and-bound approach is computationally efficient for
the former but not for the latter. Narula, ,0gbu, and Samuelson [91]
presented a branch-and-bound scheme which relies on obtaining the
bounds by solving the Lagrangian relaxation of the integer programming


-24-
The p-median problem is the following: Given a positive integer p,
find a set X* of p-points such that
f(X*) = min[f(X): |x| = p, X C N] .
Any set X* of p points that minimizes f is called an absolute p-
median of N. If each member of X is restricted to a vertex location,
the resulting problem is called a vertex restricted p-median problem.
Due to a result by Hakimi [47, 48] there exists an absolute p-median
entirely on the vertices of N. For this reason, the distinction be
tween the vertex restricted and unrestricted versions is insignificant.
Hence, we will take the term "p-median" to mean a solution to either
version of the problem. A 1-median is simply called a median.
The p-median problem arises naturally in locating plants/ware
houses to serve other plants/warehouses or market areas. The problem
is also motivated by ReVelle, Marks, and Liebman [96] as an example of
a public sector location model where vertices represent population
centers and facilities represent post offices, schools, public build
ings, and the like.
The 1-median problem. Hakimi [47] appears to be the first to
define an absolute median. Hakimi proved the important result that
there exists an absolute median at a vertex of the network. This
result reduced the search to a finite number of points. The median
can be found by summing each row of the weighted-distance matrix and
choosing the vertex whose row sum is the minimum. This procedure takes
3 2
0(n ) operations to compute the distance matrix followed by 0(n )
operations to find the median.


-80-
The Divergence Problem is as follows: Given r and the function
g, compute
q(r) = max{ |u| : g(U) >r,UCV}. (2.6.1)
That is, the problem is to find the maximum number of existing facili
ties no two of which can be jointly covered by a single center within
a radius of r. Equivalently, among all cliques of G whose gain is
larger than r, the problem is to find one with the maximum number of
nodes. The dual problem is feasible for r < r^, as, if r > r^ there
does not exist a subset U of V for which g(U) > r. On the other hand,
the primal cover problem is feasible for r > a. Hence, we shall re
strict r to a < r < r^ in order to ensure feasibility to both
problems.
Theorem 2.6.1. (Weak Duality Theorem). Assume a < r < r^. For any
feasible solution X to the primal cover problem, and any feasible
solution U to the dual divergence problem, we have |x| > |u|.
Proof. By feasibility of U and the assumption of the theorem we have
g(U) = gl(U) > r > a > g2(U) from which it follows that
8.. > r v.,v. e U, i ^ j (2.6.2)
ij i J
Suppose |X| < Ju|. Then, the same approach as in the proof of Theorem
2.2.1 implies there exist v ,v e U, s ^ t, such that 8 < f(X) < r,
st st
contradicting at least one inequality in (2.6.2). Thus, |x| > |u|.
Theorem 2.6.2. (Strong Duality Theorem). Assume a < r < r^. Let X
be a feasible solution to the covering problem constructed by COVER.
Then, the primary set U associated with X solves the dual divergence
problem with
X| = q(r) = q(r) = |U
(2.6.3)


-133-
a) A Non-Efficient Location Vector
with f(X) = (1.5,1)
b) An Efficient Location Vector
with f(Y) = (1.5,0.5)
Figure 4.2. Example of Non-Efficient and Efficient Location
Vectors


-134-
Using the weights given, the value of f(X) is (1.5,1). Graph GBC^
with z = (1.5,1) is also shown with arc lengths as indicated. It is
direct to verify that GBC^ in Figure 4.2a) does not have any tight
paths. Hence, X is dominated. The location vector Y in Figure 4.2b)
is an efficient one and dominates X. The value of f(Y) is (1.5,0.5).
The thickly drawn arcs in GBC^ of Figure 4.2b) form a tight path. We
remark that in every feasible solution to DC^ for z = (1.5,0.5), the
locations y^ and y^ ate the same, as and ^ are contained in a
tight path.
4.4 Construction of the Efficient Frontier
Let S be the set of all efficient location vectors in Tm. Define
Z and Z* by
Z = {(z^,Z2): 3X e T such that f(X) = (z^,Z2)} ,
Z* = {(z^,z^): 2X c S such that f(X) = (z^,z^)}
That is, Z = f(Tm), the image of Tm under f, and Z* = f(S), the image
of the efficient set S under f. We call Z the objective space and Z*
the efficient frontier. Our main interest in this section is to develop
a method to construct the efficient frontier. One can display Z*
graphically on the (z^^) plane and obtain much of the insight about
efficient points. In general, for any convex bi-objective problem,
the efficient frontier and the objective space may look like the
illustration given in Figure 4.3. The objective space is the shaded
region and the efficient frontier is the thickly drawn part of the
boundary of Z.


-61-
Y. .
ij
w .w.
1 3
+ w.
3
1/0)0
and
d..
13
[d(v^,Vj) + h^ + hj]^
Consider the tree network shown in Figure 2.1, where the numbers
on the arcs represent arc lengths. The data given with Figure 2.1
corresponds to the parameters for j=l,...,6 where clearly, each f is
strictly increasing. Using (2.2.4), the 3 values for this problem
are shown in Table 2.1 along with the node weights f^(0). Figure 2.2
shows the dual graph G associated with the problem, where the number
next to each node j is the node weight and the number on the arc between
nodes i and j is 3^ Using Figure 2.2 it can be verified that the
optimal cliques (specified here by their nodes) and associated g
values are K* = (3,4), g(K*) = 13829.76; K* = {1,3,6}, g(K*) = 3600;
K* = {1,3,5,6}, g(K*) = 1664.64; K* = {1,3,4,5,6}, g(K*) = 784; and
K* = {1,2,3,4,5,6}, g(K*) = 225. Due to the duality theory, it then
o o
follows that the r^ for p=l,...,5 are, respectively, 13829.76, 3600,
1664.64, 784, and 225.
2.3 Dual Problem Interpretation
We imagine two conservative adversaries, an aggressor A and a
defender D. Defender D has defense forces placed at vertex locations
Vl,',,Vn' Aggressor A will attack a single vertex in V. Although D
knows A will attack a vertex, he will not know the vertex attacked
until the attack occurs.
Defender D has p response forces which he must position at loca
tions defined by a p-center X. Interpret tree distances to be travel
times, so that D(X,v.) is the minimum time to respond to v. from a


-138-
to each distinct path, and equate e() to the pointwise maximum of
these linear functions over the interval [a,b]. Such a method is not
computationally efficient as there may be a very large number of paths
For achieving computational efficiency we shall restrict our attention
to a certain subset of the set of all possible paths and then evalute
e(*) by taking the pointwise maximum of the linear functions cor
responding to these paths.
In general, an arbitrarily chosen path P(E^,E^) may pass through
several existing facility nodes distinct from E^ and E^. First, we
want to show that paths of this type need not be considered. Define
any path to be a decomposable path (d-path) if the path passes through
at least three (distinct) existing facility nodes. An example of a
d-path which passes through four existing facility nodes is
(E^, Nj-, Eg, N^, E^, N^, E^) Define any path to be a non-
decomposable path (nd-path) if the only existing facility nodes the
path passes through are its terminal nodes. Every d-path can be de
composed into a (unique) collection of nd-paths which, when appended
end to end, gives the original d-path. The decomposition of the
aforementioned example d-path into its nd-paths is {(E,, Nc, N1ft, E,),
1 j 10 o
(Eg, N^, E^), (E^, N^, E^)}. Clearly, any nd-path uses exactly two
arcs in A^,, while any d-path uses at least four arcs of A^,. An nd-
path may or may not use arcs of A^.
Next, we have the following lemma, which permits us to check the
separation conditions by only evaluating nd-paths. The lemma is
applicable to any distance constraints problem defined in Chapter 3
by (3.3.1). We use the notation of Chapter 3 for the lemma.


-19-
its edge of the function f..(-) = max[w.d(v.,.), w.d(v.,.)], and
ij i i J J
that the two linear pieces defining that breakpoint have slopes of
opposite signs. There can be at most n(n l)/2 suspected points on
each edge, resulting in a total of 0(|E|n^) suspected points on all
edges. If S is the set of all suspected points together with the set
of all vertices, then there is an absolute p-center contained in S.
The Kariv-Hakimi procedure selects p-1 points from S and determines
all the vertices covered jointly by these p-1 points. All uncovered
vertices are assigned to the pth center. Corresponding to each center,
the I-radius is determined (with respect to the subset of vertices
covered by that point) and the maximum of these 1-radii determines
the p-radius for that trial solution. The algorithm tries every
possible combination of p-1 points selected from S and chooses that
combination which minimizes the p-radius. The Kariv-Hakimi procedure
is the only exact algorithm available so far for finding an absolute
p-center of a vertex weighted general network.
A further result on the computational difficulty of the p-center
problem on a general network is given by Nemhauser and Sheu [92].
They showed that finding an approximate solution to the vertex restricted
or absolute p-center problem whose value is within 100% or 50%, respec
tively, of the optimal value is NP-hard (i.e., at least as hard as
any NP-complete problem).
Vertex restricted p-center problem. The vertex restricted p-
center problem is considered by Toregas, Swain, ReVelle, and Bergman
[109]. A solution procedure is given which relies on solving a sequence
of minimal set covering problems, each corresponding to a specified
radius r. Given a radius r, a 0-1 matrix A can be formed with n rows


-110-
b) Iteration 1
Figure 3.7. Example Application of SEVCA


1 n U 1 ji J I /..i t Ion
(I M M > I (*
V
?
Choose v
1
Choose v
18
9 x.
U4 = {v3WV5}
K)
u = {v3, v6, v5!
k* o, i, <>, r>
OPTKLIQUE for p = 3 for Example


-54-
The literature on the p-center problem is discussed in detail
in Chapter 1. Here, we give a brief review of the more closely re
lated work. Except for p = 1, we know of no literature on the non
linear p-center problem. For p = 1, the only references we are aware
of which deal with the nonlinear case are Dearing [18] and Francis
[29]. Both authors showed that the minimax loss with respect to any
two existing facilities is a lower bound on the maximum loss with
respect to all existing facilities, and that the largest of the lower
bounds determines the minimax loss to all existing facilities on a
tree network. This result is an instance of the duality result we
will present in this chapter.
The linear (weighted or unweighted) p-center problem is shown to
be NP-complete on a general network by Kariv and Hakimi [65], and by
Nemhauser and Sheu [92].
The linear 1-center problem on a tree network is well solved (see
Goldman [44], Halfin [51], Lin [81], and Dearing and Francis [19]).
For p > 1, the linear p-center problem on tree networks is considered
by various authors. Handler [57] provided an 0(n) algorithm for
finding the 2-center of a tree for the unweighted case. Kariv and
2
Hakimi [65] gave an 0(n logn) algorithm for tree networks which relies
on solving a sequence of covering problems for the weighted case with
p > 1. A similar procedure for the unweighted continuous p-center
problem on a tree network is given by Chandrasekaran and Daughety
[12]. A vertex-restricted version of the problem is solved by
Chandrasekaran and Tamir [13], and relies on solving a sequence of
clique covering problems on a related intersection graph.


-113-
g) Iteration 6 and Termination (in
Iteration 7)
Figure 3.7. Continued


-144-
b = minf^(X): ^2^^ b)
= minify (X) : v^dCx^.x^) < 0 (j ,k) e Ig}
= minif^(x,..,x): xeT)
= min max g (x)
xeT l = min max max{w..d(x,v.): j e I. }
xeT l = min
xeT
max{w_.d (x,Vj)
1 < j < n}
where
w. = max(w,,: over all i for which i e I.}
J ij i
(4.4.10)
(4.4.11)
Thus, the value of b is obtained by solving the absolute 1-center
problem defined by (4.4.10) and (4.4.11), and will require O(nlogn) opera
2 2
tions. -Therefore, the computational effort for E-FRONT is 0(m (m + n ))
and is determined by the computational effort for steps 1), 2), and 3)
of E-FRONT.
Once the efficient frontier is constructed, efficient location
vectors can be identified as follows: Choose z = (z^,z£) in Z* with
z2 = x(z^) = T£j(z^)> say, and identify the arcs a^ and a^. Supposing
a. = (N ,E ) and a. = (N. ,E ), let P(N ,N 1 be a path in whose length
i s p j t q s t r B &
is mgt_. Then, clearly, (Ep,P(Ng,Nt) ,E^) will be a tight path in GBCz
with length L P(E ,E ) = z,WP(E ,E ) + z0m Every facility whose
zpq l pq zst
node is in the path P(E^,E^) is uniquely located on the line L(v^.v^)
in T with the same ordering and spacing as the nodes which lie in
P(Ep,E^). Hence, the new facility locations corresponding to new
facility nodes in P(E^,E^) can be readily identified (see Property
3.3.1 of Chapter 3). The locations of other new facilities can be


-51-
where a path of G joining any two existing facility nodes E and E
s t
is said to be tight if the length of the path is equal to the distance
between the vertices v and v in T corresponding to nodes E and E ,
S t 1 w St
respectively. For any given location vector Z, denote by A^(Z) the
collection of locations of uniquely located facilities whose nodes are
adjacent to N_^ in G. Let H[A^(Z)] be the convex hull of A^(Z), i.e.,
the smallest connected subtree containing all points in A^(Z).
With these definitions, it was proven in [33] that the following
conditions are equivalent:
(i)Z is efficient.
(ii)Z is the unique solution to DC.
(iii)Each is in at least one tight path in G.
(iv)Each Z. is contained in H[A.(Z)], 1 < i £ m.
1 1
This completes the discussion of multi-objective location problems
on networks.
Path Location Problems
Here, we consider three versions of a path location problem posed
by Slater [102]. To define the problems, let P denote any path con
necting any two vertices in a network N. For any vertex veV and any
path P, define the distance D(v,P) to be the distance from v to a
nearest vertex in P. Also define the branch weight bw(P) of a path
P to be the maximum number of vertices in any component of N-P. The
three versions of the problem are the following:
min l D(v,P) (1.3.7)
P C N veV


REFERENCES
1. K. Bergstresser, A. Chames, and P.L. Yu, "Generalization of
Domination Structures and Nondominated Structures in Multicriteria
Decision Making," Research Report No. JS185, Center for Cyber
netic Studies, Univ. of Texas, Austin, Texas (1974).
2. 0. Berman and R. Larson, "The Congested Median Problem," Working
Paper OR-076-78, MIT, OR Center (1978).
3. A.E. Bindschedler and J.M. Moore, "Optimal Locations of New
Machines in Existing Plant Layouts," J_. Ind. Engr. 12, 41-48
(1961).
4. G.R. Bitran and T.L. Magnanti, "The Structure of Admissable Points
with Respect to Cone Dominance," C.O.R.E. DP 7716, Louvain-la-
Neuve (1977).
5. R.G. Busacker and T.L. Saaty, Finite Graphs and Networks, McGraw-
Hill, New York, N.Y., 1965.
6. A.V. Cabot, R.L. Francis, and M.A. Stary, "A Network Flow Solu
tion to a Rectilinear Distance Facility Location Problem," AIIE
Transactions 2, 132-141 (1970).
7. L.G. Chalmet, Efficiency in Multi-Objective Location, Design, and
Layout Problems, Ph.D. Dissertation, Katholieke Universiteit te
Leuven (1978).
8. L.G. Chalmet and R.L. Francis, "Finding Efficient Solutions for
Rectilinear Distance Location Problems Efficiently," Research
Report No. 77-3, Dept, of Industrial and Systems Engineering,
Univ. of Florida, Gainesville, Florida (1977).
9. L.G. Chalmet, R.L. Francis, and J.F. Lawrence, "On Characterizing
Supremum-Efficient Facility Designs," Research Report No. 78-9,
Dept, of Industrial and Systems Engineering, Univ. of Florida,
Gainesville, Florida (1978).
10. L.G. Chalmet, R.L. Francis, and J.F. Lawrence, "Efficiency in
Integral Facility Design Problems," Research Report No. 78-11,
Dept, of Industrial and Systems Engineering, Univ. of Florida,
Gainesville, Florida (1978).
11. A. Chan and R.L. Francis, "A Round-Trip Location Problem on a
Tree Graph," Trans. Sci. 10, 35-51 (1976).
-161-


CHAPTER 4
THE BI-OBJECTIVE m-CENTER PROBLEM ON A
TREE NETWORK
4.1 Introduction
In this chapter we consider a bi-objective problem on a tree
network which involves as objectives the maximum of the weighted dis
tances between specified pairs of new and existing facilities and' the
maximum of the weighted distances between specified pairs of new
facilities. Such a vector-minimization problem may find applications
in locating emergency service units for the case when service units are
required to support one another in addition to providing service to
potential hazard zones (existing facilities).
The related literature on multi-objective location problems is
discussed in Chapter 1. Here, we concentrate on characterizing efficient
location vectors to the bi-objective minimax problem and constructing
the "efficient frontier," the set of objective values corresponding to
efficient points.
At this point we give an overview of the chapter. In Section
2 we give the necessary definitions and notation, and define the bi
objective problem of interest. In Section 3 we relate the bi-objec
tive problem to the distance constraints studied by Francis et al.
[32] and develop the necessary and sufficient conditions for efficiency
by making use of the results given for the distance constraints in


-26-
A median of a tree is shown to be the same as a "centroid" of
the tree by Zelinka [120] for the unweighted case and by Kariv and
Hakimi [65] for the weighted case. To define a centroid, consider
the subtrees T-,...,T. obtained by removing vertex v from T. Let
1 X
w(T ) be the sum of the weights of the vertices in T^., and define
W(v
.) to be the maximum of w(T.) for 1 i i k., A vertex v which
i j J i t
minimizes W(v.) over all v. in V is said to be a centroid of T. The
i i
location of a centroid is independent of the distances and can be
found by using only the incidence relations. Goldman's earlier
algorithm in essence finds a centroid of T. The generalized algorithm
of Rosenthal, Pino, and Coulter [98] also finds a centroid of T by
making only two traversals of the vertices. All these algorithms are
of 0(n) and solve the 1-median problem without having to compute the
distance matrix.
We now consider some generalizations of the 1-median. Minieka
[88] defined the general absolute median of a network to be any point
on the network that minimizes the sum of (unweighted) distances from it
to the point on each edge that is most distant from it. Minieka showed
that the general absolute median can be strictly interior to an edge;
hence, the search cannot be confined solely to vertices of N.
Slater [103] gave another generalization of the 1-median problem.
In this generalization, each demand is a collection of vertices. The
problem is to find a vertex such that the sum of the distances from
that vertex to a nearest element of each collection is minimum.
Slater showed that the set of vertices that solve this problem forms
a connected path in T. For a general network, the problem can be
solved by constructing a matrix that specifies the distances from each vertex


-44-
hull of the fixed points. Wendell, Hurter, and Lowe [114] considered
the same problem with rectilinear distances and provided algorithms of
2 3
0(n ) and 0(n ) for generating efficient points. A most efficient
algorithm of O(nlogn) was developed by Chalmet and Francis [8] for
the same problem. McGinnis and White [83] considered the problem of
minimizing the sum of and the maximum of weighted rectilinear distances
from a variable point to a set of fixed points on the plane and formu
lated the problem as a parametric linear program for which known solu
tion techniques exist. Juel [64 ] considered the same problem for
the case of multiple new facilities and gave an equivalent parametric
linear program. Chalmet, Francis, and Lawrence [ 9 10 ] considered
two variants of an efficient design problem, where the location
variable (a design) is a planar region of specified positive area
but of unknown shape.
A few papers have been produced on multi-objective location
problems on networks. In what follows we discuss these problems.
The cent-dian problem. The single facility "cent-dian" problem
involves the sum of and maximum of weighted distances from a new
facility to a set of existing facilities at vertices of N. To define
the problem, let w^ and h_^ be two positive weights associated with
vertex v iel = {1,...,n}. For each point xeN define:
m(x) = J {w_jd(v^,x): iel} ,
c(x) = max[h^d(v^jx): iel]
f(x) = (m(x), c(x))


-98-
z2=z3
#
v
1.5
1.5
a) Graph GBC
b) Tree T
A^(Z) = (z2,vi,v2,v4)
Zj e H[ A^(Z)] = T
A2(Z) = {z^ZyV^}
z2 e H[A2(Z)] = L(zl5z3)
A3(Z) = {z2>v^}
z3 e H[A3(Z)] = L(z2,v^)
c) Sets A^. (Z)
d) Convex Hulls
Figure 3.5. Example of a Non-Efficient Location Vector


-132-
y<~
7-
2
y
+-
9
vr
y
4
a) Tree T
v2 v3
V4
V5
W11
= 1/5
V12
1/3
V1
6 4
8
10
W12
= 1
V13 =
1/6
V2
2
6
8
W23=
= 1/3
V23 =
1/2
V3
4
6
W24
= 1/2
V4
2
W34
= 1/4
W35
= 1/3
b)
Distance
Matrix
c)
Weights
Figure 4.1. Data for Example Bi-Objective m-Center Problem


-89-
If then the string graph is placed upon the tree T, i.e., the strings
only lie on arcs of T, a path is tight when it is necessary to pull the
string graph tight in order to place the knots representing and
on v and v respectively, while a path is slack if the string path
P 9
must literally be slack when the two knots are placed to coincide with
v and v .
P q
A priori, one might think that the occurrence of a tight path
would be rare. However, we shall see that tight paths occur in a
quite natural way when the separation conditions are used in the analy
sis of efficient location vectors. Further, the notion of tight paths
permits the specification of necessary and sufficient conditions for
DC to have a unique solution.
We now relate unique locations to tight paths. By definition,
new facility i is uniquely located if it has the same location in every
feasible solution to DC. Since we later refer to a collection of
facilities, which contains possibly both existing and new facilities,
being uniquely located, we note that existing facilities are uniquely
located by definition.
Theorem 3.3.2, which we proved in [33], specifies the necessary
and sufficient conditions for a new facility to be uniquely located.
Theorem 3.3.2. New facility k is uniquely located if an only if node
lies in at least one tight path P(E^,E^).
Corollary 3.3.2. Distance constraints have a unique solution if and
only if node lies on at least one tight path in GBC for k = l,...,m.
We now give an additional property of a tight path we proved in
[33]. The property will be used in proving our main result on efficient
points.


-115-
Proof. We must show that t, the total number of Iterations, is at
most 3m-l. Define P =\J{K^^: 1 < k < t}. With this definition P
contains as members distinct (but not necessarily disjoint) subsets
of { 1,. .. ,m). Since = {{ 1{m} } every set {j}, 1 < j m,
is an element of P. If step 3) is never executed, then P = as
A'^ = ... = K^ in this case. Otherwise, whenever step 3) is executed,
(k)
say, in some iteration k, some two distinct members of K are removed
from A^), and their union is inserted in to obtain Hence,
if iteration k performs step 3), we know that the cardinality of
(k+1) (k)
a y is one less than that of Av and that exactly one member of
(k+1) (k)
K (the one inserted) differs from every element of K Clearly
then, step 3) can be executed at most m 1 times, and therefore P
contains at most m+(m-l)=2m-l distinct members. Hence t^,
the total number of iterations which used step 3), satisfies t^ < m- 1.
Now, imagine that we apply SEVCA a second time in exactly the same
order as the first application. In the second application, each time
step 6) or step 7) is used as the last step of an iteration, one
member of P will be labeled scanned (P is available as a result of
the first application). Since P contains at most 2m 1 distinct
members, clearly, tg + t^, the total number of iterations which used
either one of step 6) or step 7), will satisfy tg + t^ < 2m 1. It
follows then that t = t_ + t. + t, + t_ < 3m 1 as t_ £ m 1, t, =1,
3 4 6 7 3 4
and tg + t^ < 2m 1, completing the proof.
Next we have the following property. Let X* be the location
vector at the termination of SEVCA.
Property 3.6.2. The algorithm SEVCA terminates with an irreducible
location vector X* which is efficient. Furthermore the location vector


-75-
S(Kp*+l> = rp
(2.5.2)
Furthermore, K* solves the dual dispersion problem.
p+1
Proof. Let X* be an optimum p-center solution to the primal problem
so that X* = p and f(X*) = r Since r S a we consider the cases
P P
r^ = a and r^ > a. Let us apply OPTKLIQUE for each case.
For r^ = a, K*+^ is chosen in step 1) so that |K*_y| = p+1 and
a = fg(0) = g2^Kp+l^ T^e W*D,Tt Sives S(K*+1) f(X*). But then,
a = 82^K*+1^ = = f(X*) = r = a, establishing (2.5.2) for
p+1"1
this case.
c'
P
For r > a, define R = {3. e R: r < g..} C- R. Since r > r >
P iJ P ij P
there exists no g. in R for which r < g ,. < r Thus g.. > r implies
ij iJ P 1J
3. > r and so it follows that
1J P
R={g..:r ij ij
(2.5.3)
Let U be the primary set identified by COVER for the chosen r,
r' < r < r By Lemma 2.5.1, U satisfies g,(U) > r from which it
p p J bl
follows that gy > r for v^*vj e U, i j. Hence, (2.5.3) implies
By e R v.,v. e U, i j (2.5.A)
J 3
Since |U| > p+1, let be that subset of U identified in step 2).
We have the following string of inequalities:
r
P
f(X*) > g(K*+1)
2 Mkh>
= mlniByi Vj.Tj E K*+1, i j* j)
> minig..: v.,v. e U, 1 i}
ij i J
> minig.. e R}
> r
(2.5.5)
(2.5.6)
(2.5.7)
(2.5.8)
(2.5.9)
(2.5.10)


-99-
for the other class ("reducible" location vectors). We say a pair of
facilities interact if their nodes are adjacent in GBC. We define a
location vector Z = (z^,...,z^) to be irreducible if for every pair of
interacting new facilities i and j, their locations z^ and zare dis
tinct; Z is said to be reducible if there exists at least one pair of
interacting new facilities i and j for which z^ = zj ^he lcatin
vector of Figure 3.5 is an example of a reducible location vector.
The following property gives the sufficient conditions for an ir
reducible location vector to be efficient.
Property 3.5.2. Suppose Z e T is an irreducible location vector. If
for every j, 1 < j < m, z^ e H[A^(Z)], then Z is efficient.
The proof of Property 3.5.2 requires a number of preliminary
results. To preserve the continuity of the discussion, we leave the
proof until the end of section 5.
From a computational standpoint, Property 3.5.2 provides an ap
proach for determining whether or not an irreducible location vector
is efficient, and constructing one if it is not. To check if Z is
efficient, we only need to determine the nodes adjacent to N in GBC
and form the convex hull (the smallest subtree) which spans the loca
tions of these adjacent nodes. If it is the case that every z. is
within its convex hull, then Z is efficient. Otherwise, we can choose
a z. which is not in the convex hull associated with it, and move its
location to the closest point in the convex hull. The procedure can
be employed repeatedly until every new facility satisfies the convex
hull containment property. However, during such a procedure, the
current location vector may change its status from an irreducible one
to a reducible one, as the locations of new facilities change. For


-128-
Proof. Since DC is consistent clearly z. 0, i = 1,2. We consider
z 1
the cases z^ > 0 and z^ = 0 separately.
Case with ^2 > 0. Let P(E^,E^) be any path which passes through
Ag. By hypothesis the path is slack so that L^PE^E^) d(vp>Vq) > 0.
Further VP(E^,E^) is positive since the path passes through A^. Hence
we have
[L P(E ,E ) d(v v )]/VP(E ,E ) > 0 (4.3.1)
z p q p q p q
Let e be the minimum of the left side of (4.3.1) over all paths which
min(e,Z2). Let
GBCz_^ be the graph with arc lengths (ij) e and (Z2 ^2^vjk*
(j,k) el. We want to show that the separation conditions defined on
D
GBC are satisfied.
z-A
Choose any two nodes E^ and E^. Let P(E^,E^) be a shortest path
in GBC connecting E and E Hence, we have
z-A p q
L (E ,E) = L ,P(E ,E) (4.3.2)
ZA p C[ ZA p CJ
Either P(E ,E ) passes through A or it does not. In the latter case
p q U
clearly the length of P(E ,E ) in GBC and GBC is the same, as every
p q z z-A
arc in A^ has the same length in both graphs. Since DC^ is consistent,
we have L ^P(E ,E) =LP(E ,E) > L (E ,E) >d(v ,v) so that the
z-A pq zpq zpq pq
separation condition for E and E is satisfied in this case. For the
P q
other case, P(E ,E ) passes through A^ so that its length on GBC is
given by
pass through A_.
Choose A = (A^^2) with A^ = 0 and 0 < A2
L ,P(E ,E )
Z-A p C|
= z WP(E ,E ) + (z A)VP(E ,E )
1 P q 2 2 p q
(4.3.3)


-143-
calculate m for 1 < s < t < m, which requires 0(m ) operations.
Every linear function is determined by computing its slope and inter-
3 2
cept so that steps 0), 1), and 2) require 0(m + r ) operations. But
r can be at most mn so that excluding step 3) and the computation of a
3 2 2 2
and b, the algorithm is 0(m + (mn) ) =0(m (m+n )). Each linear
function has positive intercepts and negative slope. Clearly, their
pointwise maximum is a piecewise linear decreasing function over the
interval [a,b]. Hence, t(*) can be constructed by finding its break
points. Each break point is determined by the intersection of some two
linear functions. Since each linear function is strictly decreasing,
any linear function can determine at most two (consecutive) break
points. Thus, there are at most 2*(r)(r l)/2 = r(r 1) break points.
Hence, excluding the computation of a and b, the algorithm requires
2 2
0(m (m+n )) operations, as the computational effort for constructing
the linear functions dominates the computational effort for finding
the break points of t(0-
To compute a, define, for every new facility index i, the set 1^
by ^ = ij : (i,j) e ICL Letting g^x^ = max{w^d (x^v..) : j c 1^,
it is direct to verify that f^(X) = max{g^(x_^): 1 i < m). Hence,
fj is separable and its minimum value is given by a = max{g*: 1 < i < m)
where g* = min{g^(x): x e T}. The Kariv-Hakimi procedure in [65] com
putes g* in 0(iI|log|I|). Since |l | < n, the computation of a
requires no more than mnlogn operations. Hence, the computational
effort for identifying the linear functions again dominates the com
putational effort for computing a.
To compute b, we must first compute b. Clearly, b = 0 as it is
the minimum value of f 2 (X) = maxi'v fcd (x^ ,xfc) : (j,k) e IB>. It is direct
to verify the following equalities:


-126-
It was proven in [32] and stated in Theorem 3.3.1 of Chapter 3
that DC is consistent if and only if L (E E ) > d(v ,v ) for
z z p q p q
1 £ p < q n. The inequalities L^CE^E^) dCv^jV^) are called the
separation conditions and the separation conditions are said to hold
if every separation condition is satisfied.
It is direct to verify that whenever the separation conditions
hold (equivalently, whenever DCz is consistent) it necessarily follows
that L P(E ,E ) > d(v ,v ) for any path P(E ,E ). Conversely, whenever
zpq p q p q
L P(E ,E ) > d(v ,v ) for all paths P(E ,E ), it necessarily follows
zpq p q p q
that L (E ,E ) > d(v ,v ).
z p q p q
The definitions for tight and slack paths are given in Chapter 3
and will not be repeated here.
4.3 Necessary and Sufficient Conditions for Efficiency
In this section we develop the necessary and sufficient conditions
for efficiency by making use of the distance constraints. Our main
theorem states that a location vector Y is efficient if and only if at
least one arc in is contained in a tight path in GBC^, where GBC^
is the graph corresponding to DC^ obtained by letting z = f(Y).
Notationally, for any X = (X.,X0), DC is the distance con-
X Z ZA
straints with right hand sides (z1 X )/w.., (i,j) c I and
i I xj C
(z^ ^2^vjk e Ig* The graph GBCz ^ is the graph associated
with DC
z-X'
Before proving our main theorem, we first prove two lemmas
relating DC to DC and GBC to GBC ... We remark that "0" denotes
Z ZA Z ZA
either the scalar zero or the two-tuple (0,0). It will be clear from
the context what "0" refers to.


OPTIMAL MULTI-FACILITY LOCATION ON
TREE NETWORKS
By
BARBAROS C. TANSEL
A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1979


-169-
117. P.L. Yu, "Cone Convexity, Cone Extreme Points, and Nondominated
Solutions in Decision Problems with Multi-Objectives," J. of
Opt. Theory and App. 14, 319-377 (1974).
118. P.L. Yu and M. Zeleny, "The Set of All Nondominated Solutions
in Linear Cases and a Multicriteria Simplex Method," J. Math.
Anal, and A££. 49, 430-468 (1975).
119. P.L. Yu and M. Zeleny, "Linear Multiparametric Programming by
Multicriteria Simplex Method," Manag. Sci. 23, 159-170 (1977).
120. B. Zelinka, "Medians and Peripherians of Trees," Archivum
Mathematicum (Brno), 87-95 (1968).


-12-
vertex v from v The path P(v .v^) is a longest path and its mid-
t s s t
point is the unique absolute center of the tree. This procedure
requires a computational effort of 0(n). Handler's algorithm is
extended by Lin [81] to the unweighted case with addends. Lin showed
that the absolute center of a general network N with vertex addends
can be found by determining the absolute center of an expanded net
work N' whose vertex addends are all zero. Network N' is obtained from
N by adding a new vertex adjacent to each old vertex, with the length
of the edge connecting the two equal to the addend associated with
the old vertex. For a tree network T, the resulting network is a
tree T' and Goldman's 0(n) algorithm can be applied to T'.
The more general case with both weights and addends was considered
by Dearing and Francis [19], and for the case of a tree network an
2
0(n ) algorithm was given. The Dearing-Francis paper appears to be
the first to construct a well defined metric space N with distance
d(.,.) from an arc weighted graph N. This mathematical formality per
mits the use of such concepts as compactness, continuity, and the
extreme and intermediate value theorems. They showed that the distance
d(x,.) is continuous for each fixed x, in turn implying that f(x) is
continuous for every x. From compactness and continuity considera
tions, they proved the existence of an absolute center for all compact
networks, and its uniqueness for all compact tree networks. They
obtained a lower bound on r^ which is applicable to all networks, and
proved that it is always attainable for tree networks. Once the lower
bound is determined, it identifies two "critical" vertices, and the
absolute center can be readily located on the path joining the two.
The bound is the maximum of n(n l)/2 terms, resulting in a


-93-
length; for otherwise, in every feasible solution to DC, the location
of new facility p would be the same as the location of the uniquely
located facility represented by node F contradicting the fact that
N is colored blue.. Property 3.3.2 then implies that the entry in
P
U = D(Z) corresponding to arc (N ,F ) can be reduced by a positive
P 9
amount and the resultant distance constraints will still have a feasible
solution, say Y. But then clearly D(Y) < D(Z) and D(Y) 4- D(Z), contra
dicting the fact that Z is efficient. Hence (a), (b), and (c) are
equivalent. It can be seen that the proof will be complete if we show
(b) implies (d) and (d) implies (c).
To show (b) implies (d), suppose N_^ is in some tight path P. Let
f^ and f^ be the nodes adjacent to N_^ in P, so that ((f^,N^), (N^jf^))
is a subpath of P. Since f^ and f2 are in the tight path P, by Theorem
3.3.2 the facilities represented by f^ and f are uniquely located. We
may let y^ and y2 denote the unique locations of f^ and f^, respectively.
Thus it is clear that y^ and y^ are elements of A*(Z). By Property
3.3.1, z^ e L(y^yy^), and by definition of the convex hull, L(y^,y^) C
H[A*(Z)]. Thus it follows that z^ e H[A*(Z)J. To show (d) implies
(c), suppose e H[A*(Z)] and let f^ and f^ be nodes adjacent to N_^
in GBC, where f^ and f^ represent facilities with unique locations y^
and y respectively, such that z. e L(y ,y ) C T. Thus d(y ,y ) =
^ 1 i. m 1 Z
d(y ,z ) + d^z^y^). Now for any feasible solution X to DC we know
dCy^Xi) d(y^,z^) and d(y2>x^) < d(y^,z^). But then because f^ and
f2 are uniquely located, Lemma 3.3.1 implies x. = z^, for i = l,...,m.
Hence X = Z, so Z is the unique solution to DC, completing the proof.


-4
problem and a dual "divergence" problem. We provide a covering
algorithm which solves both the covering problem and its dual simul
taneously.
In Chapter 3, we study a vector-minimization problem in relation
to a distance constraints problem. The problem involves as objectives
the distances between specified pairs of new and existing facilities
and specified pairs of new facilities. We extend the results of [32]
to develop a theory for identifying unique solutions to distance con
straints, and use this theory to develop necessary and sufficient
conditions for efficient solutions to the vector-minimization problem
of interest. Further, we provide an algorithm which constructs an
efficient location vector from a given non-efficient solution.
In Chapter 4, we study a bi-objective location problem which in
volves as objectives the maximum of the weighted distances between
specified pairs of new and existing facilities, and maximum of the
weighted distances between specified pairs of new facilities. We
characterize efficient solutions and provide an algorithm for construct
ing the efficient frontier.
In Chapter 5, we pose a number of unresolved questions in relation
to the problems discussed and point out directions for future research.
1.2 Terminology
Before discussing the literature we specify our terminology.
An undirected network N = {V,E} is a collection of two sets V
and E, called the set of vertices and the set of edges of N, respec
tively. Each edge in E is described by an unordered pair of vertices.


-21-
The dispersion problem is to find a U*C T such that
h(U*) =max{h(U): U C T, |u| = p+1} .
At optimality, Shier's duality result states that
rp-{h(U*)
for a tree network. The equality may not hold for general networks.
However, Shier showed that the objective value of the continuous p-
center problem is always bounded below by one-half the objective value
of the dispersion problem for any network.
Chandrasekaran and Tamir [14] observed that Shier's duality result
holds when one replaces T by any subset S of T. Chandrasekaran and
Daughety [12] described a procedure for solving the dispersion problem.
They first solve the related problem of locating the maximum number
of points on T such that any two of them are at least A distance
apart for a fixed (positive) A. This problem is solved by working
from "tips" of T to the "center" of T. The general scheme is to use
the algorithm for different values of A, until the number of points
found is p+1 and a slightly larger A generates p or less points.
A number of solution procedures have been given for the p-center
problem on tree networks. We now discuss these procedures.
Handler [57] considered the continuous p-center problem on a
tree network for the special case of p = 2 and obtained an 0(n)
algorithm. Handler first finds the absolute 1-center of T, say x*,
and splits the tree at x* obtaining two disjoint subtrees T^ and T^.
Finding the absolute 1-center of each T say x* and x*, determines
an absolute 2-center of T.


-46-
P for each given a, where P is defined as follows:
a b a
e(a) = min[c(x): m(x) £ a, xeT]
Efficient solutions are obtained by parameterizing on a. Handler's
results closely parallel Halpem's.
The problem on a general network is studied by Halpern [54].
using the convex combination approach. Halpern showed that the problem
is a computationally finite one. Computational finiteness follows
from the result that f(X,x) is a continuous, piecewise linear function
of x on each edge and attains its minimum at one of a finite number of
points. Defining Q(e) to be the union of the end points of edge e
with the set of local minima of c(x) on e, the minimum of f(X,x) over
all x on edge e is a member of Q(e) for any given X, 0 < X < 1. De
fining Q = U {Q(e) : eeE}, it follows that the cent-dian x*(X) is con
tained in Q for any X. Further, Halpern showed that the function
f*(X) = min[f(X,x): xeN] is a continuous, piecewise linear, concave
function of X for 0 < X < 1. Based on these results, Halpern provided
an algorithm which constructs f*(X) and identifies x*(X) for
0 X < 1. To construct f*(X), the algorithm inspects each edge one
at a time and computes the set Q(e), unless a simple test indicates
that edge e cannot contain any cent-dian for any X. An upper bound
on f*(X) is carried through and improved, whenever possible, by
examining the members of Q(e).
Cent-dian problem and duality. In [53], Halpern studied the cent-
dian problem on a general network from a different angle and obtained
a duality relationship. Using an approach similar to Handler's median
constrained problem, Halpern defined two problems, a median constrained


-102-
c) Iteration 1
b) Graph GBC
d) Iteration 2 and Termination
Figure 3.6. Example Application of RP


-32-
The distance constraints problem
The distance constraints problem involves locating new facilities
on a network so that they are within specified distances of existing
facilities as well as within specified distances of one another. The
distance constraints arise naturally in a locational context if one
wishes to require that a service facility be within a specified time
(distance) of any point in the region it serves. Alternatively, in a
military context, one may want to locate a number of units in such a
way that units are neither too far from their supply bases, nor too
far from one another, in order that one unit may reinforce another if
necessary.
To state the problem, let N be a network with the vertex set
V = iv.,...,v ). Denote by X = (x,,...x ) any location vector in Nm,
the m-fold Cartesian product of N by itself. Define the sets and
I as follows: I = (j,k): 1 < j < k < m>, I = {(i,j): 1 i S m,
1 j n}. Here, the pairs (j,k) and (i,j) are assumed to be un
ordered. Let I and I be two non-empty subsets of I' and I,
d L B G
respectively, and suppose we are given nonnegative finite numbers b
jk
for each (j,k)el and c.. for each (i,j)el_.
a ij C
The problem of interest is to find a location vector XeNm, if it
exists, such that the constraints (1.3.3) are satisfied.
d(x ,v ) < c (i,j)el
1 J 1J ^
(1.3.3)
d(Xj,xk)-bjk (j,k)cIB
Any vector XeNm satisfying (1.3.3) is called a feasible location
vector. The distance constraints are said to be consistent if there exists
at least one feasible location vector XeNm.


-68-
assume J(r) ^ 0, for if J(r) = 0 then the condition f(X) < r holds
for all X C T and we (trivially) have q(r) = 1.
The above assumptions permit the following equivalent statement
of the covering problem:
minimize |x|
subject to
D(X,v.) < f.^Cr), j e J(r) (2.4.2)
2 2
We refer to the covering algorithm as COVER. In order to state
COVER a few definitions are convenient. We may imagine that the tree
is represented appropriately by inscribing straight line segments on a
planar surface such that each segment represents an arc. We fasten
strings of length f ^(r) to each node vjj e J(r), of the inscribed
tree, where, by convention, we allow strings of zero length. Every
fastened string has one end permanently affixed to the planar surface.
In addition, during the use of the algorithm we engage previously
fastened strings at various points on the tree. When a string is
engaged, some point of the string is permanently affixed to the tree
such that there is no slack in the portion of the string so far en
gaged. When strings are removed, we imagine that they are physically
deleted from the string model.
During each iteration of the procedure, we partition the original
tree into two subsets: one green, the other brown. The green subset
is always a tree, denoted as GT (for green tree), while the brown sub
set consists of one or more subtrees of the original tree T, each of
which is "rooted" at a node of the green tree. By convention, a root


-78-
r > a = 144, we compute (from Table 2.1) rl=max{8.. e R: 3.. < r} = 900.
3 3 ij ij 3
We next must apply COVER using a value of r where 900 < r < 1664.64.
Figure 2.3 shows the results of using COVER with r = 1296. In the
figure, the loose ends of the strings are shown as wavy lines. Brown
subtrees are shown as crosshatched arcs of the original tree. Each
separate drawing of the tree (a)-g)) is for a subsequent iteration of
COVER. Figure 2.3a) demonstrates the initialization step, where for
r = 1296, the f.^(r), j = 1,...,6 are 12, 7.2, 7, 6, 18, and 8, re
spectively. The numbers next to the strings are the lengths of the
loose ends. In the figure, we indicate which tip of the green tree
is chosen at each return to step 1) of COVER. In addition, the suc
cessive distinguished vertex sets are indicated.
After the final iteration, we note that the primary vertex set
U is {v,v, ,v- ,v,-} which, from our previous analysis, we know to be
J 1 D 5
2.6 Results for the Covering Problem
In this section we present a "divergence" problem which is dual
to the covering problem. We give a weak duality and a strong duality
result and prove that the primary set identified by COVER solves the
dual problem. The term "divergence" is chosen to represent the
physical interpretation, discussed later, in which the attacker A
chooses a "divergent" set of vertices to threaten. Further, the term
permits a distinction to be made between the two different dual prob
lems. Also, in this section, we demonstrate how having optimum solu
tions to the p-center problem for all p, 1 < p < n, enables us to
completely characterize the function q(r).


-22-
2
An algorithm of complexity 0(n logn) is described by Kariv and
Hakimi [65] for finding the absolute p-center-of a vertex weighted
tree network. They show that there are n(n l)/2 possible values
for r namely, the numbers a.. = w.w.d(v.,v.)/(w. + w.) for each
P 1JJ1JJ
combination of vertices v^, v The algorithm computes all these
numbers, arranges them in increasing order, and performs a binary
search on this list of numbers. The search relies on solving an r-
cover problem for each value of r chosen from the ordered list {a..},
ij
The search terminates when the smallest r in the list is found for
which the r-cover problem generates at most p points. The covering
part of the algorithm requires a computational effort of 0(n) for each
r, and a total effort of O(nlogn) for all values of r tried during the
binary search. Hence, the computational effort is determined by the
initial computation and ordering of the numbers ay> and is of
2
0(n logn).
A similar approach is used by Chandrasekaran and Daughety [12]
to solve the continuous p-center problem on a tree network. First,
they provided an 0(n) procedure for finding the minimum number of
points needed to cover every point of T within a given radius r.
Then, they provided a method to compute r A further refinement of
the method is given by Chandrasekaran and Tamir in [14]. They proved
that r^ is determined by one of the numbers d(t,t')/2k, where t and
t* are any two tip vertices and k is any integer between 1 and p. The
total computational effort for finding r and applying the covering
P
algorithm is of O((nlogp)^).
A somewhat different approach, which relies on finding a clique
on a related graph, is given by Chandrasekaran and Tamir [13]. They


-25-
For tree networks, more efficient algorithms can be devised to
find a median. An 0(n) algorithm was given by Hua Lo-Keng and Others
[60] and independently by Goldman [42]. The algorithm reduces the
search to successively smaller subtrees until a median is found. At
each stage, one chooses an arbitrary tip vertex (a vertex of degree
one) of the current tree. If the (modified) weight of the selected
vertex is at least as large as half the sum of all weights, a median
is found. Otherwise, that tip vertex is eliminated from further con
sideration together with the edge incident to it and its weight is
added to the weight of the adjacent vertex. The procedure is repeated
with the new (reduced) tree. The algorithm does not require the com
putation of the distance matrix and uses only the incidence relation
ships and the weights.
Goldman's algorithm is based on a "localization theorem" proved
by Goldman and Witzgall [46]. The theorem provides sufficient condi
tions for a subset of N to contain a median. Given a compact subset
S of N, if S satisfies the two conditions (i), (ii) then it contains
at least one median. The conditions are (i) the set S must be a
"majority" set, meaning that the sum of the weights corresponding to
vertices in S must be at least as large as half the sum of all weights
(ii) the set S must be "gated" in the sense that there must exist a
unique point g in S such that for every s e S and t e N-S, it is true
that d(t,x) = d(t,g) + d(g,s). Goldman's algorithm in essence is a
repeated application of this theorem to a tree network. Goldman [43]
also proposed an "approximate" localization theorem which somewhat
relaxes the second condition and guarantees the existence of a point
in S that approximates an actual median.


-104-
shortest path length between the same existing facility nodes in GBC.
Since Np is in a tight path in GBC*, then every original node Ih for
which i e P will be in a tight path in GBC, as the shortest path
lengths in GBC* and GBC are the same. But then every N^, 1 < i < m,
is in a tight path in GBC, as is in a tight path in GBC for every
P c K*, and U{P: P e '*} = {l,...,m}. Thus, upon using Theorem 3.3.3,
Z is the unique solution to DC and Z is efficient.
Proof of Sufficiency for Irreducible Location Vectors
We now return to the proof of Property 3.5.2. After presenting
a number of preliminary results, we will show that if Z is irreducible
and z_. e H[A^(Z)] for j e {l,...,m}, then every new facility node
is in a tight path in GBC.
The following lemma is proven in [22].
Lemma 3.5.1. Let P be a finite set of points each of which is in T.
For any p e P, we have H[P] = UL(p,p): p e P).
That is, the convex hull of P can be constructed by finding the
line segments joining an arbitrary element of P to every point
in P.
Next, we have the following lemma.
Lemma 3.5.2. Suppose Z is irreducible. Let and be two adjacent
new facility nodes in GBC. If z2 e H[A2(Z)] then there exists a
facility location y in A2(Z) such that
a) z2 e L(z^,y) and z^ ^ y,
b) whenever y is a new facility location, z2 f y.


-101-
location vector, as GBC*, by definition, has no arc of length zero
connecting two new facility nodes. Hence, sufficient conditions for
Z can be expressed in terms of the sufficient conditions (given in
Property 3.5.2) for Z*.
The following procedure, RP (Reduction Procedure), transforms GBC
into GBC* by applying successive elementary transformations as de
scribed in the above paragraph. During the procedure, we also keep a
list K which contains as members the composite indices.
RP.
0) Given Z, set up GBC with arc lengths defined by entries of D(Z).
Define K = {{1},. .. ,{m}}. Label new facility node as N^j,
1 < i < m.
1) If, for some P,Q e K, P f Q, there is an arc (Np,N^) of length zero
in GBC go to 2). Else go to 4).
2) Superimpose node on together with all arcs incident to Np.
Remove arc (Np,N^) from GBC. (If parallel arcs occur due to this
transformation they will clearly have equal lengths. Parallel arcs
may optionally be represented by a single arc.)
3) Remove P and Q from K, insert P U Q in if and go to 1).
4) Stop with K* = K and GBC* = GBC.
The algorithm RP terminates in at most m 1 iterations as each
iteration reduces the number of elements of K by one.
An example application of RP is given in Figure 3.6.
For each composite index P in K* we define z* to be the common
location of every new facility i for which i e P. For the example of
Figure 3.6, let K* = {P^P^ with P = {1}, ?2 = {2,3,4}. Then
zPt Z1 and ZP2 z2 z3 z4
We let Z* be the location vector


-7-
Figure 1.1.
Family Tree for Network Location Problems


-36-
the new facility it is associated with. All the strings pulled tight
from the chosen tip are engaged at this new facility location. The
feasibility of this location is checked with respect to all existing
facilities and all other new facilities already placed on T. If the
feasibility check is passed, new strings are fastened at this location
associated with that new facility and other unplaced new facilities for
which the distances are of concern. The procedure continues, treating
each placed new facility like an existing facility, until, either all
facilities are placed, or the current tree reduces to a point, in
which case, all remaining new facilities are placed at that point.
If the separation conditions hold, the procedure always finds a
feasible location vector. The algorithm is of 0(m(m+n)) and is conjectured
to be a best order algorithm in [33], for determining the con
sistency of the distance constraints.
Extensions of the results obtained in [32] are given by Francis,
Lowe, and Tansel [33]. These extensions focus on the analysis of
binding separation conditions which in turn determine the "uniquely"
located new facilities. A separation condition that holds at equality
is said to be a binding separation condition. If Z,(E.,E.) = d(v.,v,)
3 k J k
is a binding separation condition, then any shortest path P(E.,E,) in
J k
the auxiliary graph G is said to be a tight path. New facility i is
said to be uniquely located at point if in every feasible solution X to
the distance constraints the location x. is the same. It was shown
i
in [33] that a new facility i is uniquely located if and only if node
N_^ lies on at least one tight path. As an immediate consequence of
this property the distance constraints has a unique feasible solu
tion if and only if each N_^, 1 i < m, lies on at least one tight path


-49-
compact subset T* of T can be identified that contains all efficient
points. To identify T*, define R* to be the set of all minima to the
unconstrained problem min[f,(x): x£Tl. If R* intersects the feasible
1 1
set Q, define S* to be this intersection. Otherwise, S* is the unique
i i
closest point in Q to R*. Having defined each S*, 1 < i m, if their
intersection is non-empty, then the set of all efficient points is
given by T* = H{S*: 1 i < m}. If this intersection is empty, then
T* is the smallest compact convex subtree that intersects each S*. It
can be shown that each R*, S* is convex, compact, and that T* is a
li
convex compact subset of T. Lowe's theorem assumes a knowledge of
set of minima to each f as well as a knowledge of and hence Q.
We note that the functions c(x) and m(x) in the cent-dian problem are
both convex on T. Hence, Halpem's results can be obtained by apply
ing Lowe's theorem.
Now, we consider a multi-objective problem which involves multiple
new facilities to be located on a tree network so that the distance
between each specified pair of new and existing facilities, and each
specified pair of new facilities is, roughly speaking, "as small as
possible." The problem is defined by Francis, Lowe, and Tansel [33]
as a sequel to the distance constraints problem, and solved by making
use of the separation conditions. Here, we call the problem, the
"multifacility vector minimization problem."
The multifacility vector minimization problem (on a tree network).
Let T be a tree network and let I I be given nonempty sets with
Iq c (ij): 1 £ i S m, 1 < j < n} and IB C {(j,k): 1 5 j < k < m}.
The problem of interest is to locate m new facilities on T at points
x ,...,x so that each distance d(x ,v.) (i,j)el and d(x.,x.) (j.k^I^
kjlcB


BIOGRAPHICAL SKETCH
Barbaros Tansel was born on January 10, 1952, in Ankara, Turkey,
where he received his early education. For his high school education
he attended the Robert Academy in Istanbul and graduated in June 1970.
In September 1970, he began his undergraduate study in the Middle East
Technical University in Ankara and was awarded the Kennedy Scholarship
in 1971. He graduated from the Middle East Technical University in
June 1974 with a B.S. degree in industrial engineering. In 1975, he
was awarded the Fullbright Scholarship and began his graduate study in
the University of Florida. He received his M.Sc. degree in December
1976 and Ph.D. in December 1979. During his graduate study he worked
as a teaching and research assistant in the Department of Industrial
and Systems Engineering.
Barbaros's hobbies include classical music, chess, philosophy,
and folk dancing.
-170-


-125-
As in Chapter 3, denote by P(Fp,F^) any path in GBC^ connecting
nodes Fp and F A path is specified either by the sequence of nodes
in the path, or, by the sequence of arcs in the path. We denote by
L P(F ,F ) the length of the path P(F ,F ) and define L (F ,F ) to be
z p q w * p q z p q
the length of any shortest path connecting nodes Fp and F^. We say
P(Fp,F^) passes through an arc (F^,F^) if (F^,F^) is an arc in the
path. We say P(F ,F ) passes through AT1 (A) if the path passes
p ^ iJ u
through at least one arc in A^ (A^).
Associated with any path P(Fp,F^) we define two more terms, namely,
WP(F ,F ) and VP(F ,F ). The first term WP(F ,F ) is the sum of the
p q p q p q
reciprocal weights 1/w^ where the summation is taken over all arcs
(N.,E.) which are contained in the path P(F ,F ). If the summation
13 P q
is taken over an empty set, then WP(Fp,F^) = 0. Similarly, VP(Fp,F^)
is the sum of the reciprocal weights l/v^ over all pairs (N^.,N^) which
are contained in'P(F ,Fq). Again, VP(Fp,Fq) = 0 if Ag fl P(Fp,Fq) = 0.
The motivation for these two quantities can be given by observing the
relation
L P(F ,F ) = z -WP(F ,F ) + z_VP (F ,F ) .
z p q 1 p q 2 p q
(4.2.4)
The relation in (4.2.4) can be readily verified by observing that the
arc lengths of GBC^ are defined by the quantities and z2^vj^ so
that the length of any given path is the sum of the reciprocal weights
multiplied by or z^, whichever is applicable.
In what follows any path (in GBCz) we refer to is a path connecting
some two existing facility nodes Ep and E^. All other paths (for which
one or both of the terminal nodes are new facility nodes) will be
referred to as subpaths.


-130-
GBC The length of P(E ,E ) on GBC is L ,P(E ,E ) = (z, A.)*
z-X & p q z-X z-X p q 1 1
p q
WP(E ,E ) + zVP(E ,E ). But z. = 0 so that L ,P(E ,E ) = L P(E ,E ) -
p q p q / zA p q z p q
X,WP(E ,E ). By our choice of X. we have X, < [L P(E ,E ) -
1 p q J 1 1 z p q
d(v ,v )]/WP(E ,E ). Hence, L ,P(E ,E ) > L P(E ,E ) {[L P(E ,E ) -
p q p q z-X p q z p q z p q
d(v,v )]/WP(E,E )}-WP(E ,E ) = d(v ,v ). Thus, L ,P(E ,E ) > d(v ,v )
pq pq pq pq z-x p q p q
for any path P(E^,E^) so that the separation conditions on GBCz_^ hold
and is consistent with X ^ 0, X ^ 0, completing the proof.
Next, we have the necessary and sufficient conditions for efficiency.
Theorem 4.3.1. Given a location vector Y used to define DC and GBC
z z
with z = (z^,Z2) = f(Y), the following are equivalent:
(a) The location vector Y is efficient.
(b) At least one arc in is contained in a tight path.
(Equivalently, there exists at least one tight path which passes
through Ad.)
15
Proof. To show (a) implies (b), suppose Y is efficient. Assume that
no arc in A^ is contained in a tight path. Hence every path which passes
through A^g is slack as DCz is certainly consistent. Lemma 4.3.2 implies
X = (X,,X0) can be chosen with X > 0 and X ^ 0 so that DC is con-
1 l z-X
sistent. Corollary 4.3.1 then implies Y is dominated, contradicting
that Y is efficient.
To show (b) implies (a) suppose at least one arc in A^ is in a
tight path. Let P(E ,E ) be such a path which passes through A^ and
P q B
which is tight. Clearly, P(E ,E ) also passes through A For any
p q c
X = (X ,X) >0, X 0, the length of P(E ,E ) in GBC .. will be
i z p q za
strictly smaller than its length in GBC as at least one of z. and z
Z 1 z
is reduced by a positive amount due to X being different from (0,0).
Hence, for any X ^ 0, X ^ 0, the separation condition on GBC


-14-
to the one defined by (1.3.2) is obtained. The bound is again
applicable to all networks and always attainable for tree networks.
Vertex constrained 1-center problem. The vertex constrained
1-center problem was considered as early as 1869, and perhaps earlier,
by Jordan [63] as a graph theoretic problem. This problem can be
solved by examining the distance matrix of the network, as demonstrated
by Hakimi [47], Rosenthal, Pino, and Coulter [98] introduced a gener
alized algorithm that solves a number of "eccentricity" problems on
tree networks, one of which is the vertex restricted 1-center problem.
In this case, the eccentricity of a vertex is defined to be the
distance from that vertex to a farthest vertex. This generalized
algorithm determines the eccentricity of each vertex by making only
two traversals of the vertices. The vertex center is that vertex
with the minimum eccentricity. Slater [103] considered the problem
of finding the vertex center of a network with respect to subnetworks.
In this version of the problem, each demand is a known collection of
vertices (or a subnetwork induced by the collection). The distance
between a vertex and any such collection is defined by a nearest
element of the collection to that vertex. For a given vertex, the
value of the objective function at that vertex is the maximum of the
distances between that vertex and any such collection. Slater showed
that a matrix D' can be constructed from the distance matrix D of the
network, so that each entry of D' is a distance from a vertex to a
nearest element of a collection. Slater demonstrated that the vertex
center with respect to collections of vertices can be found by
examining the matrix D'.


-140-
The following algorithm, E-FRONT, constructs the efficient frontier.
We assume m has been computed for 1 A s < t < m.
E-FRONT
0) Label the arcs in as a^,...,ar where r is the cardinality of
A Define A' = {(a.,a.): 1 < i < j < r}. Delete from A' every
0 x j
pair (a^,a^) for which a^, and a^ are incident to the same new
facility node. Let A be the resulting subset of A' after the
deletions.
1) For every (a.,a.) e A define the linear function t.,(z.) as
i 3 iJ 1
follows: Suppose a. = (N ,E ) and a. = (N ,E ). Due to step
i s p/ j t q
clearly s / t and thus m > 0. For z^ e [a,b]
0)
d(Vr,Vr,) (1 + 1/..)
X. .(Z.) = E_3_ §£ tq_
y 1 st 1 st
2) Define x(z^) = max{x (z^) : (a^a^) e A} for z^ e [a,b]. The
efficient frontier is given by Z* = {(z^,x(z^)): a £ b}.
The next theorem establishes the correctness of the algorithm.
2 2
Then we will show that the algorithm is 0(m (m + n )).
Theorem 4.4.1. The algorithm E-FRONT constructs the efficient frontier
for the bi-objective m-center problem.
Proof. By Wendell's theorem Z* = {(z^,e(z^)): a < z^ < b}. Hence, it
suffices to show that e(z^) < x(z^) and e(z^) > x(z^) for a < z^ < b.
To show e(zp x(z^), choose any z^ e [a,b] and define
z = (z1,z) with z = xCz,). Let DC and GBC be the constraints of
12 2 1 z z
the problem P and the associated graph, respectively. Choose any
Z1
nd-path P(E ,E ). Either the path passes through A^ or it does not.
p q a
In the latter case P(E ,E ) is the path (E ,N ),(N ,E ) for some new


-129-
But L P(E ,E ) = z. WP(E ,E ) + zVP(E ,E ) so that from (4.3.3) we
zpq lpq 2pq
have
L ,P(E ,E ) = Li(E ,E ) X -VP(E ,E ) .
z-x pq zpq z pq
(4.3.4)
By our choice of X, we have 0 < X e [L P(E ,E ) d(v ,v )]/
J 2 2 zpq p q
VP(EpjE^). It follows then, upon using (4.3.4), that
L P(E ,E ) > L P(E ,E ) [
z-X p q ~ z p q' 1
L P(E ,E ) d(v ,v )
zpq P 9 i up/F F \
VP (E ,E ) * ^ ^Ep5 Eq^
p q
= d(v v )
p q
(4.3.5)
From (4.3.2) and (4.3.5) it follows that L ,(E ,E ) > d(v ,v ) for
z-X p q p q
this case. Since the choice of E^ and E^ is arbitrary, every separation
condition holds on GBC so that DC is consistent with X = (0,X),
ZA Z~A
X2 > 0.
Case with z2 = 0 By hypothesis every path which passes through
Ag is slack. Choose any path P(E^,E^) which does not pass through Ag.
Consistency of DC^ implies either LzP(Ep,E^) = d(v^,v^) or
L P(E ,E ) > d(v ,v ). The former case is not possible since a
z p q p q
subpath of length zero can be chosen from the arcs in Ag and this
subpath can be appended to P(E^,E^) to obtain a new path, say, P'(E^,E^)
without increasing the length of the path. Hence L^PiE^jE^) =
L^P'(Ep,Eq) = d(Vp,v^) contradicting that every path which passes
through Ag is slack. Thus, every path which passes through A^, is also
slack. Define e to be the minimum of [L P(E ,E ) d(v ,v )1/WP(E ,E )
zpq p q p q
over all paths in GBC2. Clearly e > 0, since every path is slack and
every path necessarily passes through A so that WP(E ,E ) > 0. Choose >
C P q
(X, ,0) with 00. Consider any path P(E ,E )
1 i i I p q
on


-71-
d(x,..,v.) < f/(r). As D(X,v.) d (x ,.. ,v.) it follows that X is
(j) J 3 3 U) 3
a feasible solution.
Property 2.A.2. For any nonempty distinguished set U^, with vertices
numbered so that U, = {v. ,... ,v, }, we have
k 1 k
v. e BT(x.) ,
3 3
1 < j < k
.-I.
d(Xj,Vj) = fj (r), 1 5 j < k .
(2.4.3)
(2.4.4)
Proof. Expression (2.4.3) is obvious. To show (2.4.4), choose any v
in U. Let t be the tip vertex chosen at the first of the iteration
tC
in which x^. is placed. The algorithm causes the string at v^. to-be
pulled tight along every edge connecting v^ to t, and to be pulled
tight along [t,x.], with the string end point coinciding with x..
J 1
Thus d(v.,t) + d(t,x.) = f.^(r). But v. e BT(t) and x. e T-BT(t) or
J J 3 3 3
x. = t so that d(v.,t) + d(t,x.) = d(v.,x.). Thus, (2.4.4) follows.
3 1 3 3 3
Property 2.4.3. Let X = ix^,...,x ) be the feasible solution con
structed by COVER, with vertices numbered so that U = {v^,...,v } is
the primary set associated with X. Assume q > 1. Then
d(vivj) > + for 1 i < j £ q (2.4.5)
Proof. We know the first q-1 members of U are distinguished vertices.
Hence Property 2.4.2 implies
v. e BT(x.),
i i
1 < i < q-1
.-1
d(v.,x.) = f (r), 1 i S q-1 .
(2.4.6)
(2.4.7)
For i < j, x^ is placed prior to x^.. Since v^ is assigned to x^
and


-92-
independent subproblems. Further, we note that DC is always consistent,
as Z is certainly feasible to DC, and hence, by Theorem 3.3.1, the
separation conditions are always satisfied. For convenience, for any
location vector Z, we denote by A*(Z) the collection of locations of
uniquely located facilities whose nodes are adjacent to N in GBC. We
denote by 11[A*(Z)] the convex hull of A*(Z), the imbedding of the
smallest subtree of T spanning all the elements of A*(Z).
With the above definitions we can present a family of equivalent
conditions for a location vector Z to be efficient.
Theorem 3.3.3. Given a location vector Z used to define DC and GBC,
the following are equivalent:
(a) Z is efficient;
(b) Each FL is in at least one tight path in GBC;
(c) Z is the unique solution to DC;
(d) z^ e H[A*(Z)] for i = l,...,m.
Proof. The equivalence of (b) and (c) is a direct conseqeuence of
Theorem 3.3.2 and the fact that Z is always a feasible solution to
DC, while (c) clearly implies (a). To show (a) implies (c) suppose
Z is not the unique solution to DC. Color every new facility node
in GBC which is not contained in any tight path blue. Color all the
other (new or existing facility) nodes red. Equivalence of (b) and
(c) implies every blue node represents a new facility which is not
uniquely located, while every red node represents a (new or existing)
facility which is uniquely located. By assumption there is at
least one blue node. By connectedness of GBC, there is at least
one arc which connects some blue colored node, say, N to some red
P
colored node, say, F Furthermore, arc (N ,F ) has positive
9 P q
*


-83-
with |X| = s, solve the cover problem for r.
r < r contradicting the definition of r
pi s s
r < r < r .
P P"1
It now follows, if we define the set
We then have f(X) < r <
Thus q(r) = p for
P = {(p-1,p): p e {2,...,n}, r^ < r^} >
that
q(r)
r P for rp < r < r j, (p-l,p) e P

1 for r^ < r .
(2.6.4)
The formula (2.6.4) completely defines the function q(r), since r = a,
n
and the cover problem is feasible if and only if a < r. Hence if we
solve the p-center problem for all p and compute r ...,r then we
2 n
have an explicit formula for q(r), and we see that the r^ completely
define the function q. For example, if r, = rc < r. = r < r_ = r,,
then q(r) = 5 for r^ ^ r < r^, q(r) = 3 for r^ r < r^, and q(r) = 1
for r^ ~ r. Also, the proof of the lemma does not require the assump
tion that the location network is a tree. Thus the formula for q(r)
is still valid if the location network has cycles.


-94-
3.4 Examples
Here, we give examples of efficient and non-efficient points.
Ex. 1. For a single new facility, D(z) is the vector (d(z,v ),...,
d(z,v )). Any point z in T is efficient since T is the convex hull
of iv . ,V }.
1 n
Ex. 2. Consider the tree T shown in Figure 3.2. Each arc length in
the corresponding graph GBC corresponds to an entry of D(Z). In this
case Z is efficient. Notice that and are both contained in the
tight path P = (E^, ^, E^). Also, both z^ and z2 satisfy the con
vex hull property, i.e. z Htiv^ v2, z2}] and z2 e H[{v3> v^-Zj}].
Figure 3.2. Example of an Efficient Location Vector
(a) Graph BC, (b) Tree T.
Ex. 3. Consider the same tree as in Example 2 except that the location
of z2 is changed to the midpoint of edge (v^v^. In this case


-87-
Our main interest is to characterize efficient location vectors
and devise an algorithm for constructing efficient location vectors
from a given (dominated) location vector.
3.3 Distance Constraints and Characterization
of Efficient Points
We make extensive use of the results obtained in [32, 33] for
distance constraints to establish the necessary and sufficient condi
tions for efficient points. The Distance Constraints (DC) are defined
in [32] (independent of the efficiency problem) as follows: Given the
sets 1^ and 1^ and nonnegative upper bounds c^. and b^ find a point
X = (xx ) in Tm, if it exists, such that
1 m
d(x.,v.) < c .
i 3 ij
d(x. ,x. ) £ b.,
3 k J)k
(i,j) e I
(j,k) e I,
(3.3.1)
Corresponding to DC, we define Graph BC (GBC) as the undirected
graph having nodes E.,...,E N,,...,N ; for every (i,k) e I,,, there
i n i m is
is an arc (N. ,N, ) of length b., between nodes N. and N, : for every
J k' jk j k J
(i,j) e I, there is an arc (N.,E.) of length c.. between nodes N.
C i J ij i
and E.. We further assume that the sets I,, and I are such that GBC
J B C
is connected, as otherwise DC decomposes into independent sets of con
straints which may be analyzed separately.
Given a node-path between any two nodes f and f in GBC, we de-
P q
note the path by P(f ,f ) and denote the length of the path by LP(f ,f )
p q p q
We define (f^jf^) to be the length of any shortest path in GBC between
nodes f and f Subsequently, unless we specify otherwise, it should
p q


-164-
42.A.J. Goldman, "Optimal Center Location in Simple Networks,"
Trans. Sci. 5, 212-221 (1971).
43. A.J. Goldman, "Approximate Localization Theorems for Optimal
Facility Placement," Trans. Sci. 6, 195-201 (1972).
44. A.J. Goldman, "Minimax Location of a Facility in a Network,"
Trans. Sci. 6, 407-418 (1972).
45. A.J. Goldman and P.M. Dearing, "Concepts of Optimal Location for
Partially Noxious Facilities," Bull. Opns. Res. Soc. Am. 23,
Suppl. 1, B-31 (1975).
46. A.J. Goldman and C.J. Witzgall, "A Localization Theorem for
Optimal Facility Placement," Trans. Sci. 4, 406-409 (1970).
47. S.L. Hakimi, "Optimal Locations of Switching Centers and the
Absolute Centers and Medians of a Graph," Opns. Res. 12, 450-459
(1964).
48. S.L. Hakimi, "Optimum Distribution of Switching Centers in a
Communication Network and Some Related Graph Theoretic Problems,"
Opns. Res. 13, 462-475 (1965).
49.S.L. Hakimi and S.N. Maheshwari, "Optimum Locations of Centers
in Networks," Opns. Res. 20, 967-973 (1972).
50.S.L. Hakimi, E.F. Schmeichel, and J.G. Pierce, "On p-Centers in
Networks," Trans. Sci. 12, 1-15 (1978).
51.S. Halfin, "On Finding the Absolute and Vertex Centers of a Tree
with Distances," Trans. Sci. 8, 75-77 (1974).
52. J. Halpem, "The Location of a Center-Median Convex Combination
on an Undirected Tree," J. Reg. Sci. 16, 237-245 (1976).
53. J. Halpem, "Duality in the Cent-Dian of a Graph," Working
Paper No. WP-08-77, Univ. of Calgary, Calgary, Canada (1977).
54. J. Halpem, "Finding Minimal Center-Median Convex Combinations
(Cent-Dian) of a Graph," Manag. Sci. 24, 535-544 (1978).
55.G.Y. Handler, "Minimax Location of a Facility in an Undirected
Tree Graph," Trans. Sci. 7, 287-293 (1973).
56. G.Y. Handler, "Medi-Centers of a Tree," Working Paper 278/76,
Recanati Graduate School of Business Administration, Tel-Aviv
University, Israel (1976).
57. G.Y. Handler, "Finding Two Centers of a Tree; The Continuous
Case," Working Paper, Recanati Graduate School of Business Ad
ministration, Tel-Aviv University, Israel (1977).


-57-
f and hence f could be deleted from the definition of f without
s t
changing f. Further, we assume p < n-1, as otherwise the p-center
problem is trivial.
So as to state the dual problem, we define 8., =8,. for j,keJ by
j K Kj
8., = min maxif (d (x ,v.) ) f (d(x,v,))}
3k 3 3 K K
For j,kej with j < k we define a., = maxif .(0), f, (0)} and
3k 3 k
b#, = min{f. (5.) ,f (6 ) }. We note that a 5 n implies [a., ,b ., ] ^ 0.
jK 3 3 k k JK jK
The following lemma, the results of which are proven in [29] provides
a closed form expression for 8jk
Lemma 2.2.1. For any j,keJ with j 5 k we have:
(i) The function f.^ + f exists, is stricly increasing, continuous
3 k
has domain [a., ,b., ] ^ 0, and range [L., ,U., ], where L =
JK 3K jK JK jk
ifT1 + f^)o(ajk) and Ujk (f'1 +
(ii) d(v.,vk) < U.k.
(iii) The function (f.^ + f ^ exists, is strictly increasing and
3 k
continuous, has domain [L ,U., ] and range [a., ,bM ].
JK. jk Jk jk
(iv) 8_.k = (f"1 + fj^1) 1o(max{d(v_. ,vk> L^}) .
We remark that either 8., = a., or 8., = (f. ^ + f, *) ^o(d(v ,v, ));
jk jk jk j k j k"
8.. e [a., ,b.. ], and 8.. = f.(0). The closed form expression for 8.,
JK 3k jk jj j r jk
given in Lemma 2.2.1 facilitates construction of the dual problem.
We define the dual objective function g on subsets of V as follows
For any K C V with K1 t 2
g(K) = max{g1(K), g2(K)}
(K) = min(8ij: v^v^ e K, i j}
g2 (K) .= max{fj(0): v^ e K}


-72-
no t
to x^ for 1 < i < j < q, v. was not in BTix^, and the string at
v. did not reach x.. Hence
J i
v. e T-BT(x.),
J i
1 < i < j ^ q
d(xi,Vj) > f (r), 1 < i < j < q .
(2.4.8)
(2.4.9)
But (2.4.6) and (2.4.8) give d(v ,v.) = d(v.,x.) + d(x.,v.) for
i j 11 1 J
1 < i < j < q, from which, on using (2.4.7) and (2.4.9), (2.4.5)
follows.
We shall need the following remark, proven in [32]:
Remark 2.4.1. Given any a.,a. e T and s.,s. > 0, there exists a-point
i 3 i J
x in T for which d(x,a^) < s^ and d(x,a^.) < s^ if and only if d(a^,aj)
< s + s..
i 3
We are now ready to establish the optimality of COVER.
Theorem 2.4.1. Given any r for which a < r and J(r) 4 0, COVER solves
the covering problem.
Proof. Let X = {x^,...,x^} be the point set found by COVER. Property
2.4.1 implies X is feasible to the problem. If q = 1, X is clearly
optimal. If q > 1, let the vertices be numbered so that U = {v,,...,v }
1 q
is a primary set associated with X. By Property 2.4.3, d(v_^,Vj) >
f^(r) +* f ^(r), for 1 <, i < j < q. Remark 2.4.1 implies there exists
no x in T for which d(x,v.) < f.^(r) and d(x,v.) < f .^(r) for any
i i 13
i, j in {l,...,q} e J(r) with i < j. Hence it is impossible to cover
any two members of U with a common center. Thus, since |u| = q, any
feasible solution X to the covering problem satisfies |x| > q. Since
q = |X| and X is feasible to the problem, X is thus an optimum feasible
solution.


-73-
We remark that the covering problem may be of as much interest,
from both a theoretical and applications point of view, as the p-center
problem. In Section 6, we will present a problem which is dual to the
covering problem and show that the primary set identified by COVER
solves the dual of the covering problem. Furthermore we will charac
terize q(r) as a step function, and provide a formula for q(r)
assuming that r^ is known for 1 <, p n-1.
2.5 Dual Problem Solution and the Strong Duality Theorem
Based on the W.D.T. and properties of COVER we now present a
proof of the S.D.T. The proof is constructive in that we use an
algorithm called OPTKLIQUE which, given the optimal objective value
of the primal problem, constructs an optimal solution to the dual
problem. We then show that the objective values of the pair of prob
lems are equal. As a by-product the proof also establishes that
r e R, where, for convenience, we define R ={£..: l p ij J
We find it useful to summarize Theorem 2.4.1 and Property 2.4.3
as follows:
Lemma 2.5.1. Given any r for which a < r and J(r) ^ 0, the following
assertions are true:
(a) COVER finds an optimum solution X to the covering problem with
q(r) = |X|.
(b) Whenever q = q(r) > 1, any primary set U = ^v(l)
associated with X satisfies
g(U) = g^U) > r
(2.5.1)


-74
Proof. (a) is just Theorem 2.4.1.
(b) From Property 2.4.3, for any v_^,v^ e U, i ^ j, we have d(v^,v^) >
f.^(r) + f.^ir) > f.^Ca) + f.^a) where r £ a > a = a. .. Thus,
i J i J. iJ
d(v^,Vj) is in the domain of (f_^ + f ^) from which, upon using
Lemma 2.2.1 and the definitions of g, g^, and g^, (2.5.1) follows.
In the algorithm OPTKLIQUE we assume that r^ is given for some
value of p, 1 g p n-1. OPTKLIQUE constructs an optimal solution to
the associated dual problem.
OPTKLIQUE
1) If r = a, take K*+^ to be any p+l-clique in V containing a vertex
v for which f (0) = a, and go to 3). Else, given r > a, compute
s s p
r' = maxifS. e R: < r } and choose any r for which r' < r < r .
P iJ P P P
Go to 2) .
2) Apply COVER with the chosen value of r to find an optimum solution
X and its associated primary set U, with |x| = q = |u|. Note r < r
implies |X| > p, so q £ p+1. Take K*+^ to be any subset of U con
sisting of p+1 members of U. Go to 3). (If q > p+1, there will be
alternative optimal cliques.)
3) If K*+j, is any clique found in either step 1) or 2), then g(K*+p =
r and the W.D.T. guarantees K* is an optimum solution to the dual
P P+1
problem.
Before proving the correctness of the algorithm, we note, since
a = for some h, that a < r implies a < r', and thus the r chosen
hh P P
in step 2) is one for which a feasible solution exists to the covering
problem.
Theorem 2.5.1. Given r for any p, 1 g p < n-1, the clique K* con-
P P+1
structed by OPTKLIQUE satisfies


This research was supported in part by NSF Grant //ENG 76-17810,
the Army Research Office, Triangle Park, N.C., under contract
DAHC04-75-G-0150, and by the Operations Research Division, National
Bureau of Standards, Washington, D.C.


-31-
formulation using a subgradient optimization method. Another branch-
and-bound method was developed by Jarvinen, Rajala, and Sinervo [62].
Their procedure looks for n-p vertices that do not belong to a p-
median. This method works better for larger values of p, since n-p
is smaller in this case reducing the number of possibilities. A
similar branch-and-bound procedure was given by El-Shaieb [24]. The
procedure is based on construction of a source set (i.e., p-median)
and a demand set. Starting with both sets empty, a location is added
to either set at each iteration. Whenever the number of elements in
a source set reaches p, or the number of elements in a demand set
reaches n-p, a feasible solution is obtained. An optimal solution is
eventually identified using the lower bounds.
A third approach taken is to use heuristics. A number of
heuristics have been developed by Maranzana [84], Teitz and Bart [107],
and Khumawala [69, 70].
For a discussion of a number of the solution approaches from a
computational standpoint, the reader is referred to Hillsman and Rush-
ton [59], and Khumawala, Neebe, and Dannenbring [71].
Stochastic networks and vertex-optimality. A number of pro
babilistic versions of the p-median problem have been considered in
the literature. Mirchandani and Odoni [89, 90] extended Hakimis
vertex optimality result to the case of a stochastic network whose
edge lengths are random variables. Berman and Larson [2] considered
a stochastic network where the availability of servers (centers) is a
random variable. They showed that under suitable conditions there
exists at least one optimal set of locations on the vertices of such
a network.
This completes the discussion of the p-median problem.


algorithm to construct the efficient frontier, where m and n are,
respectively, the number of new and existing facilities.
v i i i


-88-
be understood that any path we refer to is a simple path between some
two existing facility nodes and E^.
Results on Distance Constraints
The distance constraints are said to be consistent if there exists
at least one feasible solution to (3.3.1).
The following result is established in [32].
Theorem 3.3.1. The distance constraints are consistent if and only if
d(v ,v ) < £(E ,E ), 1 £ p < q n (3.3.2)
p q p q
The inequalities (3.3.2) are termed the Separation Conditions
[32], since each term on the right specifies an upper bound on how
separate two existing facility locations can be. Except when stated
otherwise, we assume throughout the chapter that the separation condi
tions hold, and thus (equivalently) DC is consistent.
We call a path P(E^,E^) between E^ and E^ in GBC a tight path if
LP(E ,E ) = d (v ,v ). We note that since we assume DC is consistent,
P q p q
it necessarily follows if P(E ,E ) is a tight path, that LP(E ,E ) =
p q p q
L(E ,E ). Any path P(E ,E ) for which LP(E ,E ) > d(v ,v ) is called
pq p q p q p q
a slack path.
We say that new facility i is in a tight path if there exists at
least one tight path containing Ik. Every path containing Ik is slack
if there is no tight path which contains Ik .
The motivation for the above terminology is due to a string graph
representation of GBC. This string graph is also useful for obtaining
problem insights. When knots representing nodes E^ and E^ are pulled as
far apart as possible, the distance between the two knots is L(E ,E ).
P q


-95-
(z^.z^) is not efficient as is not contained in any tight path.
Also, t Hv^, v^, z^}. This example is shown in Figure 3.3.
1.5
1.5
/
4
7
c ?
/
Figure 3.3. Example of a Non-Efficient Location Vector
Ex. 4. Again consider the same tree with z^ = z^ located at the mid
point of edge (v^,v^). In this case, both z^ and z^ are uniquely
located and both satisfy the convex hull property. Thus Z =
is efficient in this case. This example is shown in Figure 3.4.


-107-
location (the case with both p and q new facility locations is very
similar to the proof we will give below and hence will not be con
sidered) Define z. = p. Find a sequence of locations z.,z. ,...
z. ,v for some r, 1 < r < m-1, by applying Lemma 3.5.2 to the pairs
Jr
(z.,z: ), (z,
),...,(z. ,v ) one at a time in the given order so
'1
j t
Jr
that the family of conditions in (3.5.2) is satisfied:
z. e L(z.,z. ) and N. N. are adjacent,
Jx J 32 J1 J2
z. e L(z. z. ) and N. N. are adjacent,
J2 Ji J3 ^2 J3
(3.5.2)
z. e L(z.
Jr 3
r-1
,vfc) and N
E are adjacent.
We remark that the irreducibility of Z and the conclusion of Lemma
3.5.2 guarantees that such a sequence can be found and will end with
an existing facility location v as, r can be at most m-1 and for
the last z. we must have some y e S. such that z. e L(z ,y) with
lr Jr Jr Jr-1
y necessarily an existing facility location.
Let q in (3.5.1) be the location v of existing facility s,
s
s ^ t. Then, the sequence v z.,z. ,...,z. ,v satisfies the assump-
s J Jr t
tions of Lemma 3.5.4 as a result of (3.5.1), (3.5.2), and the irredu
cibility of Z. Hence, we have
d(v ,v ) = d(v ,z.) + d(z.,z. ) + ... + d(z. ,v )
s c s J J Ji Jr
(3.5.3)
where the right hand side of (3.5.3) is clearly the length of the
path Eg,N.,N. ,...,N. ,Et> Hence, the path is a tight path due to
J J ^ J r
(3.5.3) and contains N Thus, z^ is the unique location of new


-9-
f(X) = max[wJ)(v^,X) + a^: iel]
The problem of Interest is the following: Given a positive integer p,
find a point set X* = {x*,...,x*}, and a real number r
1 p p
such that
rp = f(X*) = min [ f (X) : |x| =p,XcN] (1.3.1)
where the symbol j*| means the cardinality of a set.
The problem defined by (1.3.1) is called the p-center problem.
Any set X* of p-points that solves (1.3.1) is called an absolute p-
center of N, and the minimum objective value r^ is called the p-radius.
For p = 1, an absolute 1-center is simply called an absolute center
of N.
If in (1.3.1), each xeX is restricted to a vertex location, the
resulting problem is called the vertex restricted p-center problem and
any set X* C V of p points that solves it is called a vertex restricted
p-center of N. A vertex restricted 1-center is simply called a vertex
center.
We note that the p-center problem is usually formulated in the
absence of addends. In what follows, we will assume all addends are
zero, unless we explicitly mention them. The case with all w^ equal
to unity will be referred to as the unweighted case.
With this terminology, the p-center problem is the problem of
finding p points on a network so that the maximum (weighted) distance
between any demand point and its nearest center is as small as possible.
The problem is perhaps most applicable to the location of emergency
facilities such as fire stations, ambulance centers, and the like, as


-150-
approach to the vector-minimization problem was to reformulate the
problem in terms of a family of distance constraints which impose upper
bounds on the distances between specified pairs of new and existing
facilities and specified pairs of new facilities.
In Chapter 4, we considered the bi-objective m-center problem
(with mutual communication) which involves as objectives the maximum
of the weighted distances between specified pairs of new and existing
facilities and maximum of the weighted distances between specified pairs
of new facilities. We developed the necessary and sufficient condi
tions for efficient solutions and provided a procedure for construct
ing the efficient frontier. Our approach was to reformulate the problem
in terms of a. family of distance constraints which impose upper bounds
on the distances between specified pairs of new and existing facilities
and specified pairs of new facilities.
In what follows we give certain generalizations of the problems
considered in this dissertation as well as other location problems
considered in the literature. We point out some possible directions
for future research.
5.2 Generalized Multi-Center Problem
Here, we define a problem which generalizes the p-center problem
and the m-center problem with mutual communication. For convenience,
we consider the weighted case. Nonlinearity can be obtained by re
placing each weight by a strictly increasing continuous function of
the associated distance.


-52-
min max D(v,P)
(1.3.8)
P C N veV
min bw(P)
(1.3.9)
P C N
In Slater's terminology, any path P* that solves (1.3.7) is called
a core of N. Among all paths that solve (1.3.8), one with the fewest
vertices is called a path center of N. Similarly, among all the paths
that solve (1.3.9), one with the fewest vertices is called a spine
of N.
Slater obtained a number of properties of these problems for
tree networks. In particular, Slater showed that the path center of
T is unique and contains the vertex center of T, and that the spine of
T is unique and contains the centroid (equivalently, the vertex
median) of T. We recall that a centroid of T is any vertex v that
minimizes the maximum number of vertices in any component of T-v.
Also, Slater proposed two algorithms of linear order for determining
the path center and the spine of T.


-135-
Figure 4.3. Illustration of Z and Z* for a Convex
Bi-Objective Problem


for solving either problem provided that certain needed inverse functions
can be evaluated in a polynomial order of effort. The p-center problem
is typically solved with the aid of a nonlinear covering problem for
which we also give a dual with a physical interpretation. We provide
a covering algorithm which solves both the covering problem and its dual
simultaneously.
The second problem we consider is a vector-minimization problem
which involves as objectives the distances between specified pairs of
new and existing facilities andspecified pairs of new facilities. We
relate the vector-minimization problem of interest to a distance con
straints problem which imposes upper bounds on the distances between
specified pairs of facilities. We develop the necessary and sufficient
conditions for efficiency by making use of the theory developed for the
related distance constraints problem. Efficient solutions to the
vector-minimization problem of interest are such that in order for any
new facility to be closer to some facility than it already is, it must
in turn be placed farther from some other facility. Based on the
necessary and sufficient conditions, we provide an algorithm which
constructs an efficient location vector from a given non-efficient
solution.
The third problem we consider is a bi-objective minimax problem
which involves as objectives the maximum of the weighted distances
between specified pairs of new and existing facilities, and the maximum
of the weighted distances between specified pairs of new facilities.
We again relate the problem to the distance constraints problem and
derive the necessary and sufficient conditions for efficiency by making
2 2
use of the distance constraints. Further, we provide an 0(m (m + n ))


-59-
will still apply. Furthermore, the proof applies to any network, as
no special properties of tree networks are used.
We now state the S.D.T. We remark that the S.D.T. requires the
assumption of a tree network. In effect, network cycles may create a
"duality gap."
Theorem 2.2.2. (Strong Duality Theorem). For any p, 1 £ p n-1,
there exists an X* C T with |x*| = p and K* C V with |K*| = p+1 such
that f(X*) = g(K*).
It is evident from the W.D.T. that X* solves the primal p-center
problem and K* solves the dual dispersion problem.
Before presenting an example problem, we find it convenient to
view the dual problem as defined on "cliques" of a complete graph G.
We define G to be the undirected complete graph with node set J,
where node j of G represents vertex v of T. To any arc (i,j) of
G, i ^ j, we assign the length and, to any node j of G, we assign
the node weight g = f^(0). We call any complete subgraph K of G a
clique. We note that any nonempty subset of V induces a clique in G
and vice versa. For this reason, an equivalent definition of g(.) on
cliques of G can be given by defining g^(K) to be the length of a
smallest arc in a clique K of G, g2(K) to be the maximum of the
weights of nodes in K, and letting g(K) = maxig^K), g2(K)}. If the
number of nodes of a clique K is known to be q, we call K a q-clique
and (sometimes) write K Defining C (G) to be the collection of all
q q
q-cliques of G, an equivalent statement of (2.2.2) is as follows:
Find a clique K* for which
p+1
g(K*+l) = max{g(K): K e Cp+1(G)} .


-156-
to check those arcs for which the lengths are zero, as the lengths of
these arcs cannot be reduced, and in any feasible solution to DC2, the
constraints corresponding to arcs of zero lengths will definitely hold
at equality.
The assumption that the sets A^,...,A are disjoint is needed for
the following reason: Given a path P(E^,E^) which passes through A^_,
clearly z^ is the common multiplier for every arc which is contained
in the intersection of P(E^,E^) and A^. That is, the length of that
part of the path P(E^,E^) consisting of the arcs chosen from A^ is the
quantity zr*S(P(E^,E^) ,A^). If it were the case that A^fl A^ ^ 0 for
some j r, then the above assertion would not necessarily be true, as
an arc in the intersection A^_ 0 A^ will have at least two multipliers
in this case, namely, z^ and Zy We will consider this case in the
conclusion of this section.
The following questions seem worth investigating for future
research.
Ql. Is there a computationally efficient way of checking whether or
not arcs of GBC are contained in tight paths?
Q2. How can we construct efficient solutions efficiently?
Q3. Can the results of Theorem 5.3.1 be extended to the case when
some of the A_^ are not disjoint?
Q4. How tractable is the t-objective m-center problem if we generalize
it by replacing each x^ by a collection of centers?
With
respect to Q3, suppose that A^/l A^. ^ 0 for at least two
indices i and j for which 1 i < j < t. Let (F ,F ) be any arc in A.
P
If (F ,F ) is contained in at least two members of {A,,...,A }, then
p q I t
the distance constraint corresponding to (F ,F ) will appear more than


-165-
58. G.Y. Handler and P.B. Mirchandani, Location on Networks, The MIT
Press, Cambridge, Massachusetts, 1979.
59. E.L. Hillsman and G. Rushton, "The p-Median Problem with Maximum
Distance Constraints: A Comment," Geog. Anal. 1, 85-89 (1975).
60. L.-K. Hua and Others, "Applications of Mathematical Methods to
Wheat Harvesting," Chinese Math. 2, 77-91 (1962).
61. A.P. Hurter and M.K. Schaefer, "The Regional Allocation of Fire
Resources: A Damage Minimizing Approach," Working Paper, North
western Univ., Evanston, Illinois (1979).
62. P. Jarvinen, J. Rajala, and H. Sinervo, "A Branch-and-Bound
Algorithm for Seeking the p-Median," Opns. Res. 20, 173-182
(1972).
63. C. Jordan, "Sur les Assemblages des Lignes,"^!. Reine Angew. Math.
70, 185-190 (1969).
64. H. Juel, "Bi-Objective Location Problems with Rectangular Dis
tances," Working Paper, Michigan Technological University (1979).
65. 0. Kariv and S.L. Hakimi, "An Algorithmic Approach to Network
Location Problems. Part 1: The p-Centers," Working Paper,
Northwestern Univ., Evanston, Illinois (1976).
66. 0. Kariv and S.L. Hakimi, "An Algorithmic Approach to Network
Location Problems. Part 2: The p-Medians," Working Paper,
Northwestern Univ., Evanston, Illinois (1976).
67. R.L. Keeney and H. Raiffa, Decisions with Multiple Objectives:
Preference and Value Trade-Offs, John Wiley and Sons, New York,
1976.
68. B.M. Khumawala, "Branch-and-Bound Algorithms for Locating
Emergency Service Facilities," Krannert Institute Paper, No.
355, Purdue University, West Lafayette, Indiana (1972).
69. B.M. Khumawala, "An Efficient Algorithm for the p-Median Problem
with Maximum Distance Constraints," Geog. Anal. 5, 309-321
(1973).
70. B.M. Khumawala, "Algorithm for the p-Median Problem with Maximum
Distance Constraints: Extension and Reply," Geog. Anal. 7,
91-95 (1975).
71. B.M. Khumawala, A.W. Neebe, and D.G. Dannenbring, "A Note on
El-Shaieb's New Algorithm for Locating Sources Among Destina
tions, Manag. Sci. 21, 230-233 (1974).
72. A. Kolen, "Complexity of Location Problems on Networks," Working
Paper, Stitching Mathematisch Centrum, Tweede Boerhaavestraat 49,
1091, A1 Amsterdam, The Netherlands (1979).


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SEVCA
Initial
0) Given Z e Tm, set up GBC with arc lengths defined by entries of
D (Z). Define K = {{!},...,{m}}. Label each member of K unscanned
Reduction
1) If for some P,Q e K, P f Q, there exists an arc (N^.N^) in GBC of
length zero go to 2). Else go to 4).
2) Superimpose node Np on together with all arcs incident to N^.
Remove arc (Np,N ) from GBC. (If parallel arcs occur due to this
transformation, they will have equal lengths. Parallel arcs may
optionally be represented by a single arc.)
3) Remove P and Q from K, insert P U Q in K and label P U Q unscanned
Define zpyg to be the common location of Zp and z^ and go to 1).
Termination Test
4) If every member of K is scanned, stop. Else, choose an unscanned
composite index P in K and go to 5).
Check for Convex Hull Containment
5) Find A(Np), the set of nodes adjacent to in (current) GBC, and
,K.
define Ap (Z ) to be the set of current locations of new
existing facilities whose nodes are members of A(Np).
%
6)If Zp e H[Ap(Z )], label P scanned and go to 1). Else go
and
to 7).
Movement
7)Find the closest point, say, y to z^ in H[Ap(Z )]. Define
e(P) = d(Zp,y). Move zp to y. Update the arc lengths of GBC by
subtracting the amount e(P) from every arc incident to Np. Label
P scanned and go to 1).


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At this point, we give an overview of the chapter. In Section 2,
necessary definitions and notation are given and the vector-minimiza
tion problem of interest is defined. In Section 3, we relate the
problem to distance constraints, give a number of related properties
of distance constraints, and establish the necessary and sufficient
conditions for a location vector to be efficient. In Section 4, we
provide examples of efficient and non-efficient location vectors.
Section 5 is devoted to a further refinement and simplification of one
of the necessary and sufficient conditions, namely, "the convex hull
property." In Section 6, we provide an algorithm, SEVCA, which con
structs an efficient solution from a given location vector. In Sec
tion 7, we characterize efficient solutions for the analogous problem
in the p-dimensional Euclidean space with rectilinear (p 2). or
Tchebychev (p > 2) distances.
3.2 Problem Statement
We suppose given a finite, undirected tree network, and denote
by T an imbedding of the given network. Let V = {v,,...,v } be a set
of n distinct vertices of T. We assume existing facility i is located
at vertex v^, i e {l,...,n}. For j e {l,...,m}, denote by x^ a point
to be determined in T as the location of new facility j. We define Tm
to be the m-fold Cartesian product of T by itself and define a location
vector X in Tm to be the ordered m-tuple (x,,...,x ) with each x e T,
1 m j
j e {l,...,m}. Sometimes, we refer to a location vector X in Tm as a
point m T
As in [22], given points x,y e T, we define the line L(x,y) to be
the union of all points in the shortest path connecting x and y. In


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n V
T f
20
22
-Cr
10
t)V
6
fj(y) = wj(y + Hj)0
Data
0 n 2
19 0
2 25 0
3 16 2
6 36 0
5 4 0
6 9 4
Figure 2.1. Example Nonlinear p-Center Problem


-37-
in the graph G. Furthermore, if some path P(E.,E. ) is a tight path,
J k
then the nodes representing facilities in the path occur with the same
ordering and spacing in the path as do the locations representing the
facilities in the path P(v.,v.) on T. This result enables one to
J k
locate the new facilities that appear in a tight path immediately,
without having to use the Sequential Location Procedure.
A multifacility minimax application of the distance constraints
is given in [32, 33] and a multiobjective application is given in [33].
These two applications will be discussed subsequently.
m-Center problem with mutual communication
Let N be a network with vertex set V = {v,,...,v } and edge set
1 n
E. Suppose the sets I and In are given with I,, C {(j,k): 1 j < k m}
BO B
and I C {(i,j): 1 < i S m, 1 < j < n}. We assume that we are given
positive weights v ^ for each (j,k)elg and w for each (i,j)el^. For
each location vector XeN, define the functions f (X), f_(X), and
B 0
f(X) as follows:
fB(X) = max[Vjkd(x^.,xk) : (j,k)eIB] ,
fc(X) = maxtw^dCx^v ) : (i,j)elc] ,
f(X) =max[fB(X), fc(X)] .
The m-center problem with mutual communication is the following:
Find a location vector X*eNm such that
Z* = f(X*) = min[f(X): XeN] .


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pf (4.4.1) are just DC¡z. We remark that for any z^ e [a,b], there
exists at least one feasible solution to (4.4.1), as there exists an
X such that f^(X) = a z^ and can be chosen large enough so that
f200 z2.
Let z^ be fixed with z^ e [a,b] and let (Y,z2) be any feasible
solution to (4.4.1). Define z = (z ,z) and let DC be the distance
J. z z
constraints in (4.4.1). Since (Y,z2) is feasible to (4.4.1), the
separation conditions on GBC hold so that every path in GBC has
z z
length at least as large as the distance between the locations of
existing facilities corresponding to the terminal nodes of the path.
Hence, for any path P(E^,E^) we have
L P(E ,E ) = z WP(En,E ) + zVP(E ,E_) > d(v,v ) ,
zpq lpq 2pq pq
or, equivalently,
d(v ,v ) WP(E ,E )
2 lVP(E ,E )J llVP(E ,E )J
p q p q
(4.4.2)
Defining x(z^) to be the maximum of right side of (4.4.2) over all paths
in GBC^, it follows that z^ > x(z^) whenever z2 is feasible to (4.4.1).
Hence, the minimum value of z2 which solves (4.4.1) is x(z^). We
observe that the right side of (4.4.2) is the value of a linear func
tion (of z^) evaluated at z^. There are as many such linear functions
as there are paths in GBCz. Further x(z^) is the maximum of these
functions at z^. Thus, defining x(*) to be the pointwise maximum of
these linear functions over the interval [a,b] we have e(zp = x(zp
for every z^ £ [a,b]. Hence, a brute-force method to construct the
efficient frontier is to enumerate all paths on GBC, compute the
parameters (slope and intercept) for each linear function corresponding


-160-
of N. Hence, one possible approach to solve the p-center problem on
N is to enumerate on all possible forests of N consisting of p disjoint
subtrees and determine the one center of each subtree in any given
forest. Such an approach is in the same spirit as Kariv and Hakimi's
in [65] with the only difference being that they enumerate on all
possible collections of p subnetworks of N while we propose to enumerate
on all possible collections of p disjoint subtrees.
It is evident that if the conjecture is true, then the disjoint
subtrees (x^),...,T (x^) can be connected to one another by adding
edges appropriately without creating cycles. With the addition of the
new edges the forest becomes a spanning tree of N, so that the answer
to Q1 would be in the affirmative (provided that the conjecture
holds).
The second question still remains unresolved as we assumed a
knowledge of a p-center X to construct such a forest (assuming that
we can).


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in U. In total, any feasible X which D chooses will satisfy |x| > |u|,
which is what the W.D.T. asserts. By virtue of the S.D.T., if U is
A's optimal choice, D can choose exactly || response forces positioned
at, say X, with |x| = |u| and still respond to an attack to any vertex
in U (as well as in V-U) without incurring a loss exceeding r. If A
threatens more than q(r) = |u| vertices, say, a subset U of V, then
|U¡ > q(r) implies g(U) < r (infeasibility). Thus, D would not be
forced into allocating a single response for every member of U. In
fact, even if A threatens every vertex in V, then D still needs ex
actly q(r) = q(r) = || response forces to respond to the threat
feasibly. Thus, if each threat is an "expense" for A, he need threaten
no more than q(r) vertices. On the other hand, D adopts an optimal
strategy against A's best threat by minimizing the number of response
forces with respect to V.
Continuing our consideration of the covering problem, we now re
verse the usual procedure, and view the p-center problem as a device
for solving the covering problem for all values of r for which the
covering problem is feasible, that is, for a < r.
The following lemma is the key to using the p-center problem to
solve the covering problem. Define r = < for convenience.
o
Lemma 2.6.1. Let p e J. If r < r ,, then
: r p p-1
q(r) = p for r < r < r .
P P-1
Proof. We first note r < r < ... S r. < r_. Also, clearlv.
n n-1 10 J
q(r ) < p for p e J. Now for r^ £ r since q is non-increasing we
have 1 > q(r^) > q(r) > 1, establishing the claim if p = 1. Consider
the case p e {2,...,n}. From r < r < r we have p > q(r ) > q(r) >
p p i p
q(r ]_). Suppose q(r) = s, with s < p, implying s < p-1. Let X,