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 Title Page
 Acknowledgement
 Table of Contents
 Abstract
 Introduction and literature...
 Duality and the nonlinear p-center...
 A vector-minimization problem on...
 The bi-objective m-center problem...
 Summary and future research
 References
 Biographical sketch
 Copyright






Title: Optimal multi-facility location on tree networks
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 Material Information
Title: Optimal multi-facility location on tree networks
Physical Description: viii, 170 leaves : ill. ; 28 cm.
Language: English
Creator: Tansel, Barbaros Cetin, 1952-
Publication Date: 1979
 Subjects
Subject: System analysis   ( lcsh )
Electric networks   ( lcsh )
Industrial and Systems Engineering thesis Ph. D
Dissertations, Academic -- Industrial and Systems Engineering -- UF
Genre: bibliography   ( marcgt )
non-fiction   ( marcgt )
 Notes
Thesis: Thesis--University of Florida.
Bibliography: Bibliography: leaves 161-169.
Statement of Responsibility: by Barbaros C. Tansel.
General Note: Typescript.
General Note: Vita.
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Bibliographic ID: UF00089541
Volume ID: VID00001
Source Institution: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: aleph - 000099473
oclc - 06997979
notis - AAL4928

Table of Contents
    Title Page
        Page i
    Acknowledgement
        Page ii
        Page iii
    Table of Contents
        Page iv
        Page v
    Abstract
        Page vi
        Page vii
        Page viii
    Introduction and literature survey
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    Duality and the nonlinear p-center problem and covering problem on a tree network
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    A vector-minimization problem on a tree network
        Page 84
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    The bi-objective m-center problem on a tree network
        Page 122
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    Summary and future research
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    References
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    Biographical sketch
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    Copyright
        Copyright
Full Text













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.





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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)





-71-


-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.





-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 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)





-74-


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





-75-


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





-77-


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)





-81-


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).


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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 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





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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 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).





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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





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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





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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 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.





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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





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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.,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





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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
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




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