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## Material Information- Title:
- Optimal multi-facility location on tree networks
- Creator:
- Tansel, Barbaros Cetin, 1952-
- Publication Date:
- 1979
- Language:
- English
- Physical Description:
- viii, 170 leaves : ill. ; 28 cm.
## Subjects- Subjects / Keywords:
- Arc length ( jstor )
Efficient point ( jstor ) Linear programming ( jstor ) Linear transformations ( jstor ) Mathematical duality ( jstor ) Mathematical vectors ( jstor ) Mathematics ( jstor ) Minimax ( jstor ) Sufficient conditions ( jstor ) Vertices ( jstor ) Dissertations, Academic -- Industrial and Systems Engineering -- UF Electric networks ( lcsh ) Industrial and Systems Engineering thesis Ph. D System analysis ( lcsh ) - Genre:
- bibliography ( marcgt )
non-fiction ( marcgt )
## Notes- Thesis:
- Thesis--University of Florida.
- Bibliography:
- Bibliography: leaves 161-169.
- General Note:
- Typescript.
- General Note:
- Vita.
- Statement of Responsibility:
- by Barbaros C. Tansel.
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- University of Florida
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- University of Florida
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- Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
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06997979 ( OCLC ) AAL4928 ( NOTIS )
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OPTIMAL MULTI-FACILITY LOCATION ON TREE NETWORKS By BARBAROS C. TANSEL A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1979 ACKNOWLEDGMENTS I am deeply indebted and grateful to Dr. Richard L. Francis, the chairman of my supervisory committee, for his excellent guidance, numerous suggestions, and the generosity with which he invested his time in listening to my ideas. Dr. Francis not only initiated my interest in location problems but also inspired many of the ideas in this dis- sertation by asking the right questions at the right time. I owe very special thanks to Dr. Timothy J. Lowe, the cochairman/ chairman of my committee during 1976-1978, presently of Purdue Uni- versity, for his active interest, overall guidance, and his inspiring suggestions. Dr. Francis and Dr. Lowe have shown sincere care about my progress and their encouragement has been of utmost value in bringing this dissertation to a completion. I would also like to express my sincere thanks and appreciation to the other members of my committee, Dr. Ralph W. Swain, Dr. Donald W. Hearn, Dr. Antal Majthay, and Dr. Luc G. Chalmet for their interest in my work and their suggestions during my proposal. I am grateful to the Department of ISE for providing me with assistantship during my graduate studies. Mrs. Adele Koehler has done an excellent job in typing the manu- script. She is fast, accurate, and very observant. I sincerely recommend her. This research.was supported in part by NSF Grant #ENG 76-17810, the Army Research Office, Triangle Park, N.C., under contract DAHC04-75-G-0150, and by the Operations Research Division, National Bureau of Standards, Washington, D.C. TABLE OF CONTENTS Page ACKNOWLEDGMENTS . . . .... . ii ABSTRACT . . . .... . . vi CHAPTER 1 INTRODUCTION AND LITERATURE SURVEY . . ... 1 1.1 Introduction and Overview . . . 1 1.2 Terminology . . . ... .. 4 1.3 Survey of the Network Location Literature . 6 2 DUALITY AND THE NONLINEAR p-CENTER PROBLEM AND COVERING PROBLEM ON A TREE NETWORK ..... ..... 53 2.1 Introduction and Related Work . . ... .53 2.2 Problem Statements and Duality. . . ... 56 2.3 Dual Problem Interpretation . . ... .61 2.4 Covering Algorithm. . . . .. 67 2.5 Dual Problem Solution and the Strong Duality Theorem. 73 2.6 Results for the Covering Problem. . . ... 78 3 A VECTOR-MINIMIZATION PROBLEM ON A TREE NETWORK. . 84 3.1 Introduction. . . . .... .84 3.2 Problem Statement ................. 85 3.3 Distance Constraints and Characterization of Efficient Points. . . . . ... 87 3.4 Examples. . . . . ... .94 3.5 Further Results on the Convex Hull Property .... .96 3.6 Algorithm to Construct Efficient Location Vectors 108 3.7 Efficiency for the Case of Rectilinear or Tchebychev Distances. . . . ... 116 4 THE BI-OBJECTIVE m-CENTER PROBLEM ON A TREE NETWORK. 122 4.1 Introduction. . . . .. 122 4.2 Problem Statement, Notation, and Definitions. ... 123 4.3 Necessary and Sufficient Conditions for Efficiency. 126 4.4 Construction of the Efficient Frontier. . .. .134 Page 5 SUMMARY AND FUTURE RESEARCH . . .... 149 5.1 Summary. . . . . ... .. .149 5.2 Generalized Multi-Center Problem . ... 150 5.3 The t-Objective m-Center Problem: Steps Towards a Unified Theory ............ 153 5.4 Tree Networks and General Networks . ... 158 REFERENCES .. .. .. . .. . . 161 BIOGRAPHICAL SKETCH .. . . . . ... 170 Abstract of Dissertation Presented to the Graduate Council of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy OPTIMAL MULTI-FACILITY LOCATION ON TREE NETWORKS By Barbaros C. Tansel December 1979 Chairman: Richard L. Francis Major Department: Industrial and Systems Engineering In this dissertation we develop a theory for location problems which involve locating multiple new facilities on a tree network with respect to existing facilities at known locations. The first problem we consider is the nonlinear version of the p-center location problem on a tree network for which the cost of each served vertex is a strictly increasing continuous function of the dis- tance between the vertex and the nearest center,and the objective is to minimize the maximum such cost over all possible locations of the centers. We present a dual "dispersion" problem which may be inter- preted as the problem of choosing p + 1 (or more) vertices such that the minimum cost to serve any two of the chosen vertices by a single common center is as large as possible. We give a weak duality theorem which applies to all general networks and a strong duality theorem which applies to all tree networks. The strong duality theorem also specifies the necessary and sufficient conditions for an optimal solu- tion to either problem. We provide algorithms of polynomial complexity for solving either problem provided that certain needed inverse functions can be evaluated in a polynomial order of effort. The p-center problem is typically solved with the aid of a nonlinear covering problem for which we also give a dual with a physical interpretation. We provide a covering algorithm which solves both the covering problem and its dual simultaneously. The second problem we consider is a vector-minimization problem which involves as objectives the distances between specified pairs of new and existing facilities and specified pairs of new facilities. We relate the vector-minimization problem of interest to a distance con- straints problem which imposes upper bounds on the distances between specified pairs of facilities. We develop the necessary and sufficient conditions for efficiency by making use of the theory developed for the related distance constraints problem. Efficient solutions to the vector-minimization problem of interest are such that in order for any new facility to be closer to some facility than it already is, it must in turn be placed farther from some other facility. Based on the necessary and sufficient conditions,we provide an algorithm which constructs an efficient location vector from a given non-efficient solution. The third problem we consider is a bi-objective minimax problem which involves as objectives the maximum of the weighted distances between specified pairs of new and existing facilities, and the maximum of the weighted distances between specified pairs of new facilities. We again relate the problem to the distance constraints problem and derive the necessary and sufficient conditions for efficiency by making use of the distance constrains. Further, we provide an (m(m + use of the distance constraints. Further, we provide an O(m (m + n algorithm to construct the efficient frontier, where m and n are, respectively, the number of new and existing facilities. v iii CHAPTER 1 INTRODUCTION AND LITERATURE SURVEY 1.1 Introduction and Overview Although some mathematical models of location can be traced back to the early seventeenth century, almost all the work on operational models for the location of facilities has taken place within the past 22 years, between 1957 and the present. An extensive annotated bibli- ography on location-allocation problems is provided by Lea [78]. A more recent selective bibliography is given by Francis and Goldstein [30]. Location problems commonly involve locating a number of new facilities (sources) in a given location space so as to provide goods or services to a specified set of existing facilities (demands) under one or more criteria, and, possibly, subject to a set of constraints. The quality of the service is typically measured in terms of the dis- tances among the facilities. The use of distances is, perhaps, the major feature which distinguishes location problems as a special class of optimization problems. Hence, associated with any location problem is an underlying location space on which a "distance" is defined. Several variations of the general location problem are possible, depending upon the type of location space, the distance function, the number and areal extent of the facilities, the type of interactions between the facilities, the objective criteria used, the constraints, the presence or lack of random elements, and possibly other factors as well. Among the several variants, planar location problems received special attention in the past, starting with the earliest contribu- tions, for example [106]. In such planar problems, one is interested in locating new facilities in the Euclidean plane with respect to existing facilities. For continuous planar problems, where any point in the plane is a feasible location, the typical distance used is the Z distance, special cases of which are the rectilinear, Euclidean, p and Tchebychev norms. For discrete planar problems, where there are a finite number of candidate locations for new facilities, the distance between any potential new facility location and any existing facility is a specified positive number. Such discrete problems, due to the finite nature of feasible locations, readily lend themselves to integer programming formulations. The reader is referred to the book by Francis and White [31] for a discussion of planar problems and a wealth of references. A number of real life applications suggest that, in some in- stances, a network space can be a more faithful representation of the reality than the Euclidean plane. For example, in a road network, a communication network, or a pipeline system, travel occurs along the arcs of the underlying network rather than in straight lines or recti- linear paths. Hence, for such problems, the use of shortest path distances along the arcs of the network can approximate the travel distance more closely than the k distance. As opposed to planar problems, network location problems have received much less attention in the past. As reported by Lea [79], there are some 1500 published papers on location-allocation problems. Among these, about 80 are on network location problems, a ratio of a little less than 6%. Hence, network location problems deserve well-justified attention in future research. In this dissertation, we develop a theory for a number of location problems which involve locating multiple new facilities on a tree net- work with respect to existing facilities at known locations. At this point we give an overview of the dissertation. In the remainder of Chapter 1, we specify our terminology and give a survey of the network location literature. We discuss minimax and minisum problems/and multi-objective problems involving minimax and minisum objectives as well as other objectives. Discussed also are problems with distance constraints. We highlight some of the convexity properties of trees (see [22]) in relation to the problems discussed. The chapter ends with a brief discussion of path-location problems. In Chapter 2, we develop a theory for the nonlinear p-center problem on a tree network. The problem is a generalization of the linear p-center problem which involves locating p new facilities on a network so as to minimize the maximum weighted distance from any existing facility to its nearest new facility. Nonlinearity is ob- tained by replacing each weight by a strictly increasing function of the distance. We formulate a dual "dispersion" problem and prove a weak duality and a strong duality theorem. The strong duality theorem also specifies the necessary and sufficient conditions for an optimal solution to either problem. We provide algorithms of polynomial com- plexity for solving either problem. Discussed also are a covering problem and a dual "divergence" problem. We provide a covering algorithm which solves both the covering problem and its dual simul- taneously. In Chapter 3, we study a vector-minimization problem in relation to a distance constraints problem. The problem involves as objectives the distances between specified pairs of new and existing facilities and specified pairs of new facilities. We extend the results of [32] to develop a theory for identifying unique solutions to distance con- straints, and use this theory to develop necessary and sufficient conditions for efficient solutions to the vector-minimization problem of interest. Further, we provide an algorithm which constructs an efficient location vector from a given non-efficient solution. In Chapter 4, we study a bi-objective location problem which in- volves as objectives the maximum of the weighted distances between specified pairs of new and existing facilities, and maximum of the weighted distances between specified pairs of new facilities. We characterize efficient solutions and provide an algorithm for construct- ing the efficient frontier. In Chapter 5, we pose a number of unresolved questions in relation to the problems discussed and point out directions for future research. 1.2 Terminology Before discussing the literature we specify our terminology. An undirected network N = {V,E} is a collection of two sets V and E, called the set of vertices and the set of edges of N, respec- tively. Each edge in E is described by an unordered pair of vertices. -5- Network N is said to be edge weighted if, associated with each of its edges, is a specified real number. Given an undirected network N = {V,E) with positive edge weights, an imbedding of N, written as N = {V,E}, is a geometric realization of N is some space S such that there is a one-to-one correspondence between the members of V and V, and E and E, respectively; each edge ecE is a rectifiable arc, and no two edges in E intersect at more than one point, a vertex. The length of edge e in E is defined to be the edge weight of the corresponding member in E. A point of an imbedded network N = {V,E} is any point along any edge in E, including the vertices. We write xeN to mean x is a point in N. The distance d(x,y) between any two points x,ycN is the length of a shortest path P(x,y) joining the two points. The function d(.,.) satisfies the axioms of a metric on N so that the set N together with d(.,.) determines a metric space. The axioms of a metric are as follows: For any two points x,ysN, 1. d(x,y) > 0 if x # y; d(x,x) = 0, 2. d(x,y) = d(y,x), 3. d(x,y) < d(x,u) + d(u,y) for any ucN. For a more detailed discussion of how to construct a metric space (N,d) from a given edge weighted network N, the reader is referred to Dearing and Francis [19], or Dearing, Francis, and Lowe [22]- We restrict ourselves to finite undirected connected networks that contain no loops and no multiple edges. We omit the term "im- bedded," and simply take a network to mean an imbedded network on which the distance d(.,.) is defined. For all other networks, we use the terms "graph," "arcs," and "nodes" instead of network, edges, and vertices. Finally, for tree networks, we write T instead of N. In passing, we note that the shortest path P(x,y) between any two points x,ysT is unique, as otherwise T would contain a cycle. 1.3 Survey of the Network Location Literature Historically, the earliest precise mathematical formulation of a location problem on a network appears to be due to Hakimi [47] in 1964. Prior to Hakimi's paper, the problem of finding the best threshing site for harvested wheat was attacked by using a network location model in 1962 by Hua Lo-Keng and Others [60]. This model was presented only at an intuitive level and no mathematical formulation or properties were given. A (correct) solution procedure was suggested (in the form of a poem), which was to be discovered independently by Goldman [42] in 1971. Since 1964, a literature of approximately 80 papers has grown till the present. Several new problems, as well as certain extensions and generalizations of old problems, have been introduced. A recent text by Handler and Mirchandani [58 ] discusses ex- tensively a portion of the literature involving minimax and minisum problems as well as single-facility bi-objective problems involving the combination of these two objectives. A "family tree" for network location problems is shown in Figure 1.1. Although not exhaustive, the family tree covers most of the problems formulated since 1964. With reference to the family tree shown in Figure 1.1, network location problems can be broadly classi- fied into two groups: point-location problems and path-location problems. Path-location problems have been recently introduced by Figure ].1. Family Tree for Network Location Problems Slater [102]. A large portion of the literature deals with point- location problems. Point-location problems may be classified into three categories: single objective problems, multi-objective problems, and a body of results of a general and unifying nature. In the remainder of this section we give a detailed discussion of the problems outlined in the family tree. Point-Location Problems Here, we consider a number of problems that involve locating new facilities at points on a network. The general format of the dis- cussion is as follows: For each problem type, we first define a kernel problem. Then, we discuss the related literature on the kernel problem, as well as several special cases and extensions of it. We point out relations between different problem types, whenever such relations exist. The p-center problem Let N be a network with a vertex set V = {vl,...,v } and an edge 1 n set E. Denote by X a finite set of points, each of which is in N. Let I be the set of integers 1 through n. For each vertex v., ieI, 1 define the distance D(vi,X) between vertex v. and the point set X by D(v.,X) = min[d(vi,x): xeX]. With this definition, D(v.,X) is speci- fied by a nearest point in X to v Let w. and a. be two given numbers 1 1 associated with vertex vi, icI. We call wi a weight and ai an addend. We assume that each wi is nonnegative and at least one wi is positive. For any finite point set X CN, define the function f(X) by f(X) = max[w.D(v.,X) + a.: icl] * The problem of interest is the following: Given a positive integer p, find a point set X* = {x*,...,x*}, and a real number r I p p such that r = f(X*) = min[f(X): |XI = p, X c N] (1.3.1) where the symbol j*| means the cardinality of a set. The problem defined by (1.3.1) is called the p-center problem. Any set X* of p-points that solves (1.3.1) is called an absolute p- center of N, and the minimum objective value r is called the p-radius. For p = 1, an absolute 1-center is simply called an absolute center of N. If in (1.3.1), each xcX is restricted to a vertex location, the resulting problem is called the vertex restricted p-center problem and any set X* C V of p points that solves it is called a vertex restricted p-center of N. A vertex restricted 1-center is simply called a vertex center. We note that the p-center problem is usually formulated in the absence of addends. In what follows, we will assume all addends are zero, unless we explicitly mention them. The case with all w. equal 1 to unity will be referred to as the unweighted case. With this terminology, the p-center problem is the problem of finding p points on a network so that the maximum (weighted) distance between any demand point and its nearest center is as small as possible. The problem is perhaps most applicable to the location of emergency facilities such as fire stations, ambulance centers, and the like, as -10- in such problems a common objective is to provide "good" service to each demand point by at least one facility within a least possible distance. In what follows, we first discuss the 1-center problem on general networks and on tree networks. Then, we discuss the vertex restricted 1-center problem. Finally, we will discuss the p-center problem in relation to a "covering" problem to be defined later. 1-Center problem on a general network. The absolute 1-center problem was defined and solved by Hakimi [47] in 1964. For finding the absolute center, Hakimi examines the function f on each edge, finds a best local minimum on that edge, and selects the best among IEJ such local minima. This method takes advantage of one important property of f, namely, that it is piecewise linear and continuous on each edge with at most n(n 1)/2 break points. A local minimum always occurs either at a break point of f or at an end point of the edge. Hakimi, Schmeichel, and Pierce [50] showed that Hakimi's method can be imple- mented in 0(JIEn2logn) computational effort and gave a computational refinement which reduces the effort to O(JElnlogn) for the unweighted case. Further refinements of the procedure were obtained by Kariv and Hakimi [65], resulting in an O(JIEnlogn) algorithm for the weighted case and 0(JEJn) algorithm for the unweighted case. All these refinements focus on finding the break points and the local minimum of.f in the most efficient manner. A somewhat more general version of the 1-center problem was con- sidered by Frank [36], and (apparently) independently by Minieka [881, as Minieka makes no reference to Frank's paper. In this modified version, called here the continuous 1-center problem, each point on -11- the network is a demand point (as opposed only to vertices). The weight of each point is unity. The objective to be minimized over all xeN is defined by f(x) = max [d(y,x): yeN]. Both authors showed that the problem can be reduced to a computationally finite one and pro- posed a solution procedure which is very similar to Hakimi's. A probabilistic version of the 1-center problem was considered by Frank [34, 35] and a number of bounds were obtained on the expected value of the 1-radius. For the unweighted case, Singer [101] proved that there exists a "critical" path, not necessarily a shortest path, connecting two cri- tical vertices such that an absolute center of the network is at the midpoint of this path. 1-Center problem on a tree network. We now concentrate on ab- solute centers of tree networks. Goldman [44] solved the unweighted case in the presence of addends. Goldman's algorithm is based on the repeated application of a "trichotomy theorem" that either determines the edge on which the absolute center lies, or reduces the search to one of the subtrees obtained by removing all interior points of that edge. Halfin [51] refined Goldman's algorithm to make it simpler and computationally more efficient. Halfin's algorithm finds a vertex center first, and determines the absolute center by examining all vertices adjacent to the vertex center. For the unweighted case with no addends, Handler [55] presents an especially elegant algorithm. Handler's method finds a longest path of the tree and locates the absolute center at the midpoint of the path. To find a longest path, Handler chooses an arbitrary vertex vi, finds a farthest vertex v from v., and then finds a farthest S 1 -12- vertex vt from v The path P(v ,v ) is a longest path and its mid- point is the unique absolute center of the tree. This procedure requires a computational effort of O(n). Handler's algorithm is extended by Lin [81] to the unweighted case with addends. Lin showed that the absolute center of a general network N with vertex addends can be found by determining the absolute center of an expanded net- work N' whose vertex addends are all zero. Network N' is obtained from N by adding a new vertex adjacent to each old vertex, with the length of the edge connecting the two equal to the addend associated with the old vertex. For a tree network T, the resulting network is a tree T' and Goldman's 0(n) algorithm can be applied to T'. The more general case with both weights and addends was considered by Dearing and Francis [19], and for the case of a tree network an 0(n2) algorithm was given. The Dearing-Francis paper appears to be the first to construct a well defined metric space N with distance d(.,.) from an arc weighted graph N. This mathematical formality per- mits the use of such concepts as compactness, continuity, and the extreme and intermediate value theorems. They showed that the distance d(x,.) is continuous for each fixed x, in turn implying that f(x) is continuous for every x. From compactness and continuity considera- tions, they proved the existence of an absolute center for all compact networks, and its uniqueness for all compact tree networks. They obtained a lower bound on rl which is applicable to all networks, and proved that it is always attainable for tree networks. Once the lower bound is determined, it identifies two "critical" vertices, and the absolute center can be readily located on the path joining the two. The bound is the maximum of n(n 1)/2 terms, resulting in a -13- computational complexity of 0(n2), and is given by a : max[a..: 1 < i j L n] 13 where (1.3.2) w w. d(v.,v.) + w.a. + w.a .. = 1 1j ji I< < Hakimi, Schmeichel, and Pierce [50] proved a theorem that reduces the computational effort for computing this lower bound. Their theorem states that if for some a it is true that max[a .: 1 I i 5 n] = a = max[a ti: 1 i < n] then a is the maximum of all a... A different solution procedure is also given by Kariv and Hakimi [65] for the same problem. Rather than computing the lower bound, their procedure confines the search to successively smaller subtrees until an edge is obtained. The absolute center is located at the local center (also the global center for a tree) on this edge using Hakimi's procedure for finding a local minimum. This algorithm is of 0(nlogn). A nonlinear version of the 1-center problem was considered and solved by Dearing [18], and by Francis [29]. In this version, each weight wi is replaced by a monotone increasing function f. of the distance d(vi,x). Both authors obtained a lower bound similar to the one defined by (1.3.2). The bound is applicable to all networks and is always attainable for tree networks. A "roundtrip" version of the problem was solved by Chan and Francis [11]. In this version each "demand point" is a pair of ver- tices (v.,u.) and f(x) is the maximum of the roundtrip distances defined by p.(x) W w.[d(v.,x) + d(x,u.) + a.]. A lower bound, similar 1 1 1 1 1 to the one defined by (1.3.2) is obtained. The bound is again applicable to all networks and always attainable for tree networks. Vertex constrained 1-center problem. The vertex constrained 1-center problem was considered as early as 1869, and perhaps earlier, by Jordan [63] as a graph theoretic problem. This problem can be solved by examining the distance matrix of the network, as demonstrated by Hakimi [47]. Rosenthal, Pino, and Coulter [98] introduced a gener- alized algorithm that solves a number of "eccentricity" problems on tree networks, one of which is the vertex restricted 1-center problem. In this case, the eccentricity of a vertex is defined to be the distance from that vertex to a farthest vertex. This generalized algorithm determines the eccentricity of each vertex by making only two traversals of the vertices. The vertex center is that vertex with the minimum eccentricity. Slater [103] considered the problem of finding the vertex center of a network with respect to subnetworks. In this version of the problem, each demand is a known collection of vertices (or a subnetwork induced by the collection). The distance between a vertex and any such collection is defined by a nearest element of the collection to that vertex. For a given vertex, the value of the objective function at that vertex is the maximum of the distances between that vertex and any such collection. Slater showed that a matrix D' can be constructed from the distance matrix D of the network, so that each entry of D' is a distance from a vertex to a nearest element of a collection. Slater demonstrated that the vertex center with respect to collections of vertices can be found by examining the matrix D'. -15- This completes the discussion of the 1-center problem. We now concentrate on the p-center problem for p > 2. p-Center problem on a general network. The p-center problem was defined by Hakimi [48]. Subsequently, a number of solution procedures have been suggested. A common characteristic of all these procedures is that they all rely on solving a sequence of covering problems. For completeness, we first define a set covering problem and an r-cover problem. Let A be a matrix of zeros and ones, y a vector of zero-one variables yi. The problem of minimizing yi so that each row of Ay i is greater than or equal to one is called the (minimal) set covering problem. Given the function f(X) = max{w.D(v.,X): 1 5 i n}, the problem of minimizing IXI so that f(X) r for some given value of r is called the r-cover problem. Denoting by q(r) the minimum value of the r-cover problem, it can be readily shown that, if q(r) = p for some r, and q(r') > p for any r' < r, then r is the p-radius and any X which solves the r-cover problem is an absolute p-center. In what follows, we concentrate on the absolute p-center problem on a general network. Minieka [87] considered the unweighted case on a general network and showed that the problem can be reduced to a computationally finite one. Minieka identifies a finite point set P' such that there exists an absolute p-center contained in P = P' U V. A point x on some edge is a member of P' if and only if x is the unique point on its edge such that d(v.,x) = d(x,v.) for some two distinct vertices vi and v.. Based on this result, Miniek suggested a rudimentary algorithm that Based on this result, Minieka suggested a rudimentary algorithm that -16- relies on solving a finite sequence of set covering problems. Using the framework provided by Minieka, an exact algorithm was developed by Garfinkel, Neebe, and Rao [38] for the unweighted case. The algorithm uses the property that the p-radius is determined by one of a finite number of elements, namely, one of the distances between any vertex and any point in P. Call the points in P edge bottleneck points and let d.. be the distance between vertex v. and the jth edge bottleneck point. Let Z and Z be a lower and upper bound on the value of r Initially Z = 0, and Z is obtained by a trial solution. P Among all the distances d.. that fall within the interval [Z,Z], one of them will determine the value of r Pick one such distance, say p dst, with Z < dst < Z, and let r = dst be a specified radius. Now, we want to know if we can cover all vertices of N within this critical distance r by using only p points. If we cannot, then clearly r is too small a radius for p points to cover all vertices. Hence we con- clude the p-radius r must be within the interval [r,Z]. In this case, the lower bound is shifted to r, and the procedure is repeated. In the other case, we find a set X of p points that cover all vertices within r, but it is doubtful if this point set is an absolute p-center. Clearly, then, the value of r will be within the interval [Z,f(X)]. Hence, the upper bound is shifted to f(X) for this case and the whole procedure is repeated. Termination occurs whenever the lower and upper bounds become equal. The r-cover part of this procedure is solved by obtaining a feasible solution, if it exists, to a set cover- ing problem. Let A be a IVI by JPI matrix with entries aij equal one if vertex v. is within a distance r of the jth edge bottleneck point and zero otherwise. Then, solving the system y i p, Ay y 1, i -17- y.i{0,1} will determine whether or not at most p points (in P) can cover all vertices of N within a radius r. Computational experience is reported and it is found that the procedure works better for larger values of p, as in this case the initial upper bound Z is small, and significant computational savings result in identifying those edge bottleneck points whose distances fall within the interval [0,Z]. The weighted case on general networks was considered by Christofides and Viola [15], and an approximate solution procedure was given. The procedure finds a set X of p-points whose objective value f(X) is within some e-neighborhood of the actual p-radius r The procedure p obtains X by solving a sequence of r-cover problems with successively increasing values of r. Termination occurs when the solution of an r-cover problem generates p (or less) points the first time. In the process, one also obtains approximate solutions for n-1, n-2,..., p+l center problems. The solution of each r-cover problem is obtained in two stages: First, all feasible solutions to the r-cover problem are obtained by finding all regions on the network that can be reached by a vertex within a radius of r. Then, among all these feasible solu- tions, those with minimum cardinality are found by solving a set covering problem. To find all regions on N reachable by a vertex v., one "penetrates" a distance of r/wi along all possible paths originating at v.. The procedure is repeated for each vertex and the intersections 1 of these penetrations are found. Each maximal intersection defines a connected region all of whose points are reachable by a subset of vertices within a radius r. The subset of the vertices is that which defines the intersection. These regions jointly cover all vertices of N, and it is possible that a subcollection of the collection of all -18- these regions may also jointly cover all vertices. Hence, to find a minimum cardinality feasible solution, one needs to choose the minimum number of regions that jointly cover V. This choice can be made by defining a zero-one matrix A, so that an entry aij of A is one if vertex vi is covered by region j, and zero otherwise. Solving the set covering problem with matrix A will provide a solution to the r-cover problem. Computational experience is reported and it is found that the procedure works better for small values of p, as the set covering part of the procedure takes a significant portion of the total computational time. An important result is due to Kariv and Hakimi [65]. They showed that the p-center problem on a general network is NP-complete. Kariv and Hakimi also showed that the weighted case (as well as the un- weighted case) can be reduced to a computationally finite one. Based on this finiteness property, they gave an algorithm whose order of complexity is polynomial in IEJ, but exponential in p. To show com- putational finiteness one argues as follows: For any absolute p-center X = {x1,...,x }, there will be a subset V. of vertices covered by the ith center x.. If N. is the (sub)network induced by V., then it can be shown that the absolute center x* of N. can replace x. without in- creasing the value of the objective function, so that X* = {x*,...,x*} 1 p is also an absolute p-center. Hence, one can restrict one's attention to absolute p-centers every element of which is the absolute 1-center of some subnetwork. The absolute 1-center of any subnetwork of N will occur either at a vertex or at one of at most IEJn(n 1)/2 "suspected" points. A suspected point on an edge is a point x such that, for some two distinct vertices vi and v., x is a break point on J -19- its edge of the function f..(.) = max[w.d(vi .), w.d(v.,.)], and that the two linear pieces defining that breakpoint have slopes of opposite signs. There can be at most n(n 1)/2 suspected points on each edge, resulting in a total of O(|En 2) suspected points on all edges. If S is the set of all suspected points together with the set of all vertices, then there is an absolute p-center contained in S. The Kariv-Hakimi procedure selects p-i points from S and determines all the vertices covered jointly by these p-i points. All uncovered vertices are assigned to the pth center. Corresponding to each center, the 1-radius is determined (with respect to the subset of vertices covered by that point) and the maximum of these 1-radii determines the p-radius for that trial solution. The algorithm tries every possible combination of p-i points selected from S and chooses that combination which minimizes the p-radius. The Kariv-Hakimi procedure is the only exact algorithm available so far for finding an absolute p-center of a vertex weighted general network. A further result on the computational difficulty of the p-center problem on a general network is given by Nemhauser and Sheu [92]. They showed that finding an approximate solution to the vertex restricted or absolute p-center problem whose value is within 100% or 50%, respec- tively, of the optimal value is NP-hard (i.e., at least as hard as any NP-complete problem). Vertex restricted p-center problem. The vertex restricted p- center problem is considered by Toregas, Swain, ReVelle, and Bergman [109]. A solution procedure is given which relies on solving a sequence of minimal set covering problems, each corresponding to a specified radius r. Given a radius r, a 0-1 matrix A can be formed with n rows -20- and n columns, so that an entry a.. is 1 if vertex v. is within a distance r of v., and 0 otherwise. If one solves a set covering 1 problem using the matrix A, the variables whose values are 1 in an optimal solution determine a feasible solution to the vertex restricted r-cover problem. The set covering problem is solved by relaxing the integrality constraints. In the case of non-integer termination, a single cut produced an integer solution in a large proportion of the cases. Their computational experience indicates that non-integer termination seldom occurs. p-Center problem on tree networks and duality. In what follows, we concentrate on the p-center problem on tree networks. First, we define the "continuous" p-center problem. In the continuous p-center problem, each point in T is a demand point as opposed only to vertices. Weights are absent (or unity). For any XC T, f is defined by f(X) = max{D(y,X): yeT} and the continuous p-center problem is to find an X*C T such that r = f(X*) = min[f(X): IXJ = p, X C T] P Minieka [88] considered the continuous p-center problem on a general network and showed that it can be reduced to a computationally finite one. Shier [100] considered the continuous p-center problem on a tree network and defined a dual "dispersion" problem. The dispersion problem is to find p+l points on T the nearest two of which are as far apart as possible. More explicitly, let U be any finite point set with IUI = p+l and define h(U) by h(U) = min{d(ui,u.): 1 < i < j < p+1} . -21- The dispersion problem is to find a U* C T such that h(U*) = max{h(U): UC T, Jul = p+1} At optimality, Shier's duality result states that 1 r = h(U*) p 2 for a tree network. The equality may not hold for general networks. However, Shier showed that the objective value of the continuous p- center problem is always bounded below by one-half the objective value of the dispersion problem for any network. Chandrasekaran and Tamir [14] observed that Shier's duality result holds when one replaces T by any subset S of T. Chandrasekaran and Daughety [12] described a procedure for solving the dispersion problem. They first solve the related problem of locating the maximum number of points on T such that any two of them are at least X distance apart for a fixed (positive) X. This problem is solved by working from "tips" of T to the "center" of T. The general scheme is to use the algorithm for different values of X, until the number of points found is p+l and a slightly larger X generates p or less points. A number of solution procedures have been given for the p-center problem on tree networks. We now discuss these procedures. Handler [57] considered the continuous p-center problem on a tree network for the special case of p = 2 and obtained an 0(n) algorithm. Handler first finds the absolute 1-center of T, say x*, and splits the tree at x* obtaining two disjoint subtrees T1 and T2. Finding the absolute 1-center of each Ti, say x* and x*, determines an absolute 2-center of T. -22- An algorithm of complexity 0(n2 logn) is described by Kariv and Hakimi [65] for finding the absolute p-center of a vertex weighted tree network. They show that there are n(n 1)/2 possible values for r namely, the numbers a.. = w wjd(v.,v.)/(w + w.) for each combination of vertices vi, vj. The algorithm computes all these numbers, arranges them in increasing order, and performs a binary search on this list of numbers. The search relies on solving an r- cover problem for each value of r chosen from the ordered list {a..}. The search terminates when the smallest r in the list is found for which the r-cover problem generates at most p points. The covering part of the algorithm requires a computational effort of 0(n) for each r, and a total effort of O(nlogn) for all values of r tried during the binary search. Hence, the computational effort is determined by the initial computation and ordering of the numbers a ij and is of 0(n2logn). A similar approach is used by Chandrasekaran and Daughety [12] to solve the continuous p-center problem on a tree network. First, they provided an 0(n) procedure for finding the minimum number of points needed to cover every point of T within a given radius r. Then, they provided a method to compute r A further refinement of the method is given by Chandrasekaran and Tamir in [14]. They proved that r is determined by one of the numbers d(t,t')/2k, where t and P t' are any two tip vertices and k is any integer between 1 and p. The total computational effort for finding r and applying the covering P algorithm is of 0((nlogp) ). A somewhat different approach, which relies on finding a clique on a related graph, is given by Chandrasekaran and Tamir [13]. They -23- define an intersection graph G for a fixed value of r as follows: G r r has nodes corresponding to demand points v ,... ,v Two nodes of G 1 n r are connected by an arc if the corresponding demand points can be jointly covered by a (single) common center within a radius of r. Once G is formed, finding a "clique cover" of G solves the r-cover problem. A clique cover of G is a minimum number of cliques in G such that every node is in at least one clique. The solution to the clique cover problem in G determines a solution to the r-cover problem. The procedure is repeated for different values of r until a smallest value of r is found for which the clique cover solution generates at most p cliques. The computational complexity of the procedure is polynomial. In particular, the computational effort for finding the minimal clique cover of G is polynomial because G satisfies the r r property that any circuit in G with at least four arcs contains a chord (i.e., an arc which connects two nodes of the circuit and is not an element of the circuit). For chordal graphs, algorithms of linear order have been developed (see [39], [97]) for finding a minimal clique cover. This completes the discussion of the p-center problem. The p-median problem The difference between the p-center and the p-median problem is that the objective criterion is changed from minimax to minisum. More specifically, define the function f(X) for any finite point set X C N by f(X) = wiD(vi,X) ice -24- The p-median problem is the following: Given a positive integer p, find a set X* of p-points such that f(X*) = min[f(X): IXI = p, X C N] . Any set X* of p points that minimizes f is called an absolute p- median of N. If each member of X is restricted to a vertex location, the resulting problem is called a vertex restricted p-median problem. Due to a result by Hakimi [47, 48] there exists an absolute p-median entirely on the vertices of N. For this reason, the distinction be- tween the vertex restricted and unrestricted versions is insignificant. Hence, we will take the term "p-median" to mean a solution to either version of the problem. A 1-median is simply called a median. The p-median problem arises naturally in locating plants/ware- houses to serve other plants/warehouses or market areas. The problem is also motivated by ReVelle, Marks, and Liebman [96] as an example of a public sector location model where vertices represent population centers and facilities represent post offices, schools, public build- ings, and the like. The 1-median problem. Hakimi [47] appears to be the first to define an absolute median. Hakimi proved the important result that there exists an absolute median at a vertex of the network. This result reduced the search to a finite number of points. The median can be found by summing each row of the weighted-distance matrix and choosing the vertex whose row sum is the minimum. This procedure takes O(n3) operations to compute the distance matrix followed by 0(n2) operations to find the median. -25- For tree networks, more efficient algorithms can be devised to find a median. An 0(n) algorithm was given by Hua Lo-Keng and Others [60] and independently by Goldman [42]. The algorithm reduces the search to successively smaller subtrees until a median is found. At each stage, one chooses an arbitrary tip vertex (a vertex of degree one) of the current tree. If the (modified) weight of the selected vertex is at least as large as half the sum of all weights, a median is found. Otherwise, that tip vertex is eliminated from further con- sideration together with the edge incident to it and its weight is added to the weight of the adjacent vertex. The procedure is repeated with the new (reduced) tree. The algorithm does not require the com- putation of the distance matrix and uses only the incidence relation- ships and the weights. Goldman's algorithm is based on a "localization theorem" proved by Goldman and Witzgall [46]. The theorem provides sufficient condi- tions for a subset of N to contain a median. Given a compact subset S of N, if S satisfies the two conditions (i), (ii), then it contains at least one median. The conditions are (i) the set S must be a "majority" set, meaning that the sum of the weights corresponding to vertices in S must be at least as large as half the sum of all weights; (ii) the set S must be "gated" in the sense that there must exist a unique point g in S such that for every s c S and t c N-S, it is true that d(t,x) = d(t,g) + d(g,s). Goldman's algorithm in essence is a repeated application of this theorem to a tree network. Goldman [43] also proposed an "approximate" localization theorem which somewhat relaxes the second condition and guarantees the existence of a point in S that approximates an actual median. -26- A median of a tree is shown to be the same as a centroidd" of the tree by Zelinka [120] for the unweighted case and by Kariv and Hakimi [65] for the weighted case. To define a centroid, consider the subtrees T,..,T k obtained by removing vertex vi from T. Let w(T.) be the sum of the weights of the vertices in T., and define W(vi) to be the maximum of w(T ) for 1 : j ki. A vertex vt which minimizes W(v.) over all v. in V is said to be a centroid of T. The location of a centroid is independent of the distances and can be found by using only the incidence relations. Goldman's earlier algorithm in essence finds a centroid of T. The generalized algorithm of Rosenthal, Pino, and Coulter [98] also finds a centroid of T by making only two traversals of the vertices. All these algorithms are of O(n) and solve the 1-median problem without having to compute the distance matrix. We now consider some generalizations of the 1-median. Minieka [88] defined the general absolute median of a network to be any point on the network that minimizes the sum of unweightedd) distances from it to the point on each edge that is most distant from it. Minieka showed that the general absolute median can be strictly interior to an edge; hence, the search cannot be confined solely to vertices of N. Slater [103] gave another generalization of the 1-median problem. In this generalization, each demand is a collection of vertices. The problem is to find a vertex such that the sum of the distances from that vertex to a nearest element of each collection is minimum. Slater showed that the set of vertices that solve this problem forms a connected path in T. For a general network, the problem can be solved by constructing a matrix that specifies the distances from each vertex -27- to a nearest element of each collection. Simply sum each row of this matrix and choose the vertex whose row sum is minimum. Frank considered a probabilistic version of the 1-median problem in [34] where each weight is a random variable with a known distribu- tion. A number of bounds are obtained on the expected value of the objective function as well as its variance. Some of these results are generalized by Frank [35] to the case where the weights are jointly distributed random variables. We now concentrate on the p-median problem with p > 2. p-Median of a network and vertex optimality. A significant theoretical contribution is due to Hakimi [48]. Hakimi proved that there exists an absolute p-median contained in V. Certain generaliza- tions of this result have been given in subsequent work. Levy [80] proved that the (vertex-optimal) result holds when the weights w. are replaced by concave cost functions c (.) of the distance between vi and its nearest median. Goldman [41] generalized the result to the case of a "two-stage" commodity. More specifically, one distinguishes a vertex as being a source or a destination. Let (Vs,Vd) be a source-destination pair, and let x. and x. be the nearest medians to v and vd, respectively. 1 3 s d Then the cost of transferring the commodity from source v to destina- s tion vd is the sum of three transport costs, namely, w dd(v ,xi) + w d(x.,x.) + w* d(x.,v ). In general, if X = {x ,...,x } is a median sd 1 3 sd (jd p set, one does not know which median is the nearest to v or vd; hence, the cost associated with a source-destination pair (s,V d) is given by fsd(X) = min [sdd(vsx + wsdd(Xij) + w*dd(x,vd) xi x CX -28- and the objective to be minimized is f(X) = Y [fsd(X): (vsv d)cVxV]. Goldman showed that there exists an optimal X* contained in V, and conjectured that the result holds for any multi-stage problem. Hakimi and Maheshwari [49] proved a stronger version of Goldman's conjecture. In this version, there are multiple commodities for each source-destination pair, and each commodity goes through multiple stages. Furthermore the cost of transport from one stage to the next is a concave nondecreasing function of the distance. More specifically, let Msd be the set of commodities to be transferred from source v to sd s destination vd, and let g(m) be the number of stages commodity meMsd is to go through. For a given location set X = {x ,...,x }, denote 1 p by yr xi(r) the location where the rth stage processing takes place. The cost of transferring commodity m from source vs to destination vd is given by Csdm[d(vs,y1)] + Csdm[d(y1Y2)] + ... + Csdm[d(yg(m)' d)] where C sdm(.) is a concave nondecreasing function of the distance. Denoting this quantity by f (Y), with YC X, IYJ = g(m), the minimum cost of transfer for commodity m is given by f sdm(X) = min[fsd(Y): sdm sdm Y C X, IYi = g(m)]. The cost of transferring all commodities from v to vd is obtained by summing over all commodities, that is, fsd(X) = [fsdm(X): meMsd]. The total cost of the system is obtained by summing the cost fsd(*) over all source-destination pairs, that is, f(X) = [fsd(X): (vs,vd)CVxV]. Hakimi and Maheshevari proved that there exists a minimum X* of f(X) contained in V. Wendell and Hurter [111] considered a more general form of the problem where the transportation cost functions are permitted to differ from edge to edge. The transport cost on any edge is a non- decreasing concave function of the distance. They proved that it is -29- sufficient to consider the vertices of the network under such a cost structure. Furthermore, they obtained the conditions under which it is necessary for the solution to occur at the vertices. In particular, they showed that nonvertex optimal locations can occur in any given edge, only when transportation costs are linear with distance over that edge and in that case, when and only when the slopes of these linear cost functions are in a special relation. Hence, if at least one cost function over some edge is nonlinear, then no interior point of that edge can be in an optimal solution. If the same situation holds for every edge, then a solution must necessarily occur at the vertices of the network. Solution approaches. Kariv and Hakimi [66] showed that the p- median problem on a general network is at least as hard as NP-complete problems. For the case of tree networks, however, algorithms of polynomial complexity have been developed. Matula and Kolde [85] 3 2 suggested an O(n p ) algorithm for finding the median of a tree net- 2 2 work. Kariv and Hakimi [66] proposed an O(n p ) algorithm for the same problem. For general networks, a number of solution procedures have been developed subsequently, all based on the vertex-optimality result. Their common characteristic is that they all confine the search to vertex locations. The solution procedures can be grouped in three categories: mixed-integer programming approaches, branch-and-bound techniques, and heuristics. ReVelle and Swain [95] formulated the problem as a linear integer program with 0,1 variables. The solution is obtained by applying the primal simplex algorithm to the associated linear program. In case -30- of non-integer termination, a branch-and-bound scheme is recommended to resolve the problem with integers. Their computational experience indicates that non-integer termination seldom occurs. Toregas, Swain, ReVelle, and Bergman [109] formulated a modified version of the problem as a mixed integer program. The modification is the presence of upper bounds on the distance between any vertex and its nearest facility. This formulation makes use of a related but simpler problem. This simpler problem is to minimize the number of facilities needed to cover all vertices of N within a specified critical distance. This problem is formulated as a set covering problem, and solved by ignoring the integer requirements. In case of non-integer termination, a single cut produced an integer solution in a large proportion of the cases. A somewhat different approach to solve the relaxed linear program is to use a decomposition scheme rather than applying the primal simplex algorithm. Swain [105] used a Dantzig-Wolfe decomposition approach to solve the associated linear program. Garfinkel, Neebe, and Rao [37] independently developed a decomposition approach similar to Swain's. In case of non-integer termination, they used group theoretics and a dynamic programming recursion to obtain an integer solution. A second approach taken is to solve the problem using a branch- and-bound technique. Khumawala [68] applied a branch-and-bound method of Land and Doig [77] type, to solve both the set covering problem and the modified p-median problem formulated by Toregas et al. He showed that the branch-and-bound approach is computationally efficient for the former but not for the latter. Narula, Ogbu, and Samuelson [91] presented a branch-and-bound scheme which relies on obtaining the bounds by solving the Lagrangian relaxation of the integer programming -31- formulation using a subgradient optimization method. Another branch- and-bound method was developed by Jarvinen, Rajala, and Sinervo [62]. Their procedure looks for n-p vertices that do not belong to a p- median. This method works better for larger values of p, since n-p is smaller in this case reducing the number of possibilities. A similar branch-and-bound procedure was given by El-Shaieb [24]. The procedure is based on construction of a source set (i.e., p-median) and a demand set. Starting with both sets empty, a location is added to either set at each iteration. Whenever the number of elements in a source set reaches p, or the number of elements in a demand set reaches n-p, a feasible solution is obtained. An optimal solution is eventually identified using the lower bounds. A third approach taken is to use heuristics. A number of heuristics have been developed by Maranzana [84], Teitz and Bart [107], and Khumawala [69, 70]. For a discussion of a number of the solution approaches from a computational standpoint, the reader is referred to Hillsman and Rush- ton [59], and Khumawala, Neebe, and Dannenbring [71]. Stochastic networks and vertex-optimality. A number of pro- babilistic versions of the p-median problem have been considered in the literature. Mirchandani and Odoni [89, 90] extended Hakimi's vertex optimality result to the case of a stochastic network whose edge lengths are random variables. Berman and Larson [2] considered a stochastic network where the availability of servers (centers) is a random variable. They showed that under suitable conditions there exists at least one optimal set of locations on the vertices of such a network. This completes the discussion of the p-median problem. -32- The distance constraints problem The distance constraints problem involves locating new facilities on a network so that they are within specified distances of existing facilities as well as within specified distances of one another. The distance constraints arise naturally in a locational context if one wishes to require that a service facility be within a specified time (distance) of any point in the region it serves. Alternatively, in a military context, one may want to locate a number of units in such a way that units are neither too far from their supply bases, nor too far from one another, in order that one unit may reinforce another if necessary. To state the problem, let N be a network with the vertex set V = (vl,...,v n. Denote by X = (xi,...xm) any location vector in Nm, the m-fold Cartesian product of N by itself. Define the sets I and IC as follows: IB = {(j,k): 1 < j < k < m), IC {(i,j): 1 r i m, 1 5 j n). Here, the pairs (j,k) and (i,j) are assumed to be un- ordered. Let I and IC be two non-empty subsets of IB and IC, respectively, and suppose we are given nonnegative finite numbers bjk for each (j,k)el and c.. for each (i,j)cI . B 13 C The problem of interest is to find a location vector XeNm, if it exists, such that the constraints (1.3.3) are satisfied. d(xi,v.) 5 cij (i,j)eIC (1.3.3) d(xj,xk) < bjk (jk)clB Any vector XENm satisfying (1.3.3) is called a feasible location vector. The distance constraints are said to be consistent if there exists at least one feasible location vector XeNm -33- Goldman and Dearing [45] provide a conceptual discussion of, and a motivation for, considering such problems. The distance constraints are formally defined by Dearing, Francis, and Lowe [22] on a network. It was established in [22] that, in a well defined sense, the distance constraints define convex sets under the assumption that the under- lying network is a tree. Furthermore, the distance constraints always define convex sets if and only if the network is a tree. Based on the results obtained in [22], Francis, Lowe, and Ratliff [32] considered the distance constraints on tree networks in more detail. They established the necessary and sufficient conditions for distance constraints to be consistent, and also devised algorithms that find a feasible location vector whenever one exists. In what follows we briefly discuss the results obtained in [32]. Distance constraints for a single new facility. For the case of a single facility, Francis et al. showed that there exists a feasible point xeT satisfying d(x,vi) < c. for ieI if and only if the in- equalities d(v.,vk) cj + ck are all satisfied for 1 S j < k n. j k j k An equivalent statement of the single facility distance constraints can be given in terms of "neighborhoods" around vi of radii ci. De- fine the neighborhood N(u,r) around a point usT of radius r to be the set of all points xeT for which d(u,x) S r. Then, a point x satisfies the constraints d(vi,x) < ci, ieI,if and only if x is in each neigh- borhood N(vi,ci), isI,if and only if x is in the intersection n n N(v.,c.). It follows then that the single facility distance con- i=l strains d(x,v.) < c., iCI,are consistent if and only if d(v.,vk) k c. + ck for 1 I j < k < n if and only if each pairwise intersection N(v ,cj) $ f N(vk,ck) is nonempty for 1 j < k n. Based on this -34- property, a "sequential intersection procedure" was developed that n determines the composite neighborhood N(a,r) n- N(vi,ci), with i=l unique center a and radius r, by intersecting the neighborhoods N(vi,ci) one at a time in an arbitrary order. The procedure can be implemented in 0(n) operations. The composite neighborhood N(a,r) contains all alternate feasible points when the constraints are con- sistent, and N(a,r) is always a convex compact subset of the tree network. A result was also given by Francis et al. that provides a sensitivity analysis on the constraints with no additional computa- tional effort. Supposing that the distance constraints are consistent with the original upper bounds c., consider an s-perturbation of the upper bounds, i.e., for some c > 0 define the new upper bounds to be c.-c, iel. If N(a,r) is the composite neighborhood corresponding to 1 the original upper bounds, then it can be shown that for any e with 0 5 e < r, the e-perturbed constraints remain consistent and the set of feasible points to the s-perturbed system is given directly by N(a,r-e). Distance constraints for the multi-facility case. For the multi- facility case, the necessary and sufficient conditions for the con- sistency of distance constraints are given in terms of n(n 1)/2 inequalities called "separation conditions." The separation condi- tions are defined by means of an auxiliary graph constructed by using the sets I and IC. Let G be the graph with nodes N., 1 5 i < m, corresponding to new facilities,and nodes E., 1 j 5 n,corresponding to existing facilities. The arc set A of G contains (N.,E.) if (i,j)CIC and (N.,Nk) if (j,k)slB. The arc length of (Ni,Ej) is ci. and of (N.,Nk) is bjk. Under the (reasonable) assumption that G is Jk K jk -35- connected, denote by L(E.,Ek) the length of a shortest path connecting nodes E. and Ek for 1 : j < k n. It was proven in [32] that the distance constraints are consistent on a tree network if and only if the inequalities L(E ,Ek) 2 d(v ,vk) are satisfied for 1 S j < k 5 n. These inequalities are called the separation conditions. The proof of the consistency of the distance constraints implying the satisfac- tion of the separation conditions uses only the triangle inequality and hence is applicable to all networks. The reverse implication always holds for tree networks, but may fail to hold for general net- works. The proof of the reverse implication is constructive and actually finds a feasible location vector under the assumption that the separation conditions are satisfied. The method that constructs such a feasible location vector is termed the "Sequential Location Procedure" in [32]. The method can best be described with the aid of a physical model. One may imagine that the tree is represented by appropriately inscribing straight line segments on a board such that each segment represents an edge. At vertex v., strings of length c.. are fastened for each new facility j such that (i,j)elC. A tip vertex is chosen arbitrarily and all strings fastened at that vertex are pulled tight towards the adjacent vertex. If all strings reach the adjacent vertex, they are simply engaged there with their loose ends free to be pulled tight in some future iteration. Also the tip vertex together with the edge incident to it is removed from the model. The procedure is repeated with the resulting tree. In the other case, not all the strings reach the adjacent vertex when pulled tight. Among those which do not reach the adjacent vertex one which is shortest is selected, and the end point of this string determines the location of -36- the new facility it is associated with. All the strings pulled tight from the chosen tip are engaged at this new facility location. The feasibility of this location is checked with respect to all existing facilities and all other new facilities already placed on T. If the feasibility check is passed, new strings are fastened at this location associated with that new facility and other unplaced new facilities for which the distances are of concern. The procedure continues, treating each placed new facility like an existing facility, until, either all facilities are placed, or the current tree reduces to a point, in which case, all remaining new facilities are placed at that point. If the separation conditions hold, the procedure always finds a feasible location vector. The algorithm is of O(m(m+n)) and is conjectured to be a best order algorithm in [33], for determining the con- sistency of the distance constraints. Extensions of the results obtained in [32] are given by Francis, Lowe, and Tansel [33]. These extensions focus on the analysis of binding separation conditions which in turn determine the "uniquely" located new facilities. A separation condition that holds at equality is said to be a binding separation condition. If L(E.,Ek) = d(v.,vk) is a binding separation condition, then any shortest path P(E.,Ek) in the auxiliary graph G is said to be a tight path. New facility i is said to be uniquely located at point Xi if in every feasible solution X to the distance constraints the location x. is the same. It was shown in [33] that a new facility i is uniquely located if and only if node N. lies on at least one tight path. As an immediate consequence of 1 this property the distance constraints has a unique feasible solu- tion if and only if each N., 1 & i m, lies on at least one tight path 1 -37- in the graph G. Furthermore, if some path P(E.,Ek) is a tight path, then the nodes representing facilities in the path occur with the same ordering and spacing in the path as do the locations representing the facilities in the path P(v.,vk) on T. This result enables one to locate the new facilities that appear in a tight path immediately, without having to use the Sequential Location Procedure. A multifacility minimax application of the distance constraints is given in [32, 33] and a multiobjective application is given in [33]. These two applications will be discussed subsequently. m-Center problem with mutual communication Let N be a network with vertex set V = {vl,...,vn} and edge set E. Suppose the sets IB and IC are given with IB C {(j,k): 1 j < k < m} and I C {(i,j): 1 < i m, 1 < j n}. We assume that we are given positive weights Vjk for each (j,k)IB and wij for each (i,j)eIC. For each location vector XeNm, define the functions f (X), fc(X), and f(X) as follows: fB(X) = max[vjkd(xj,xk): (j,k)eIB] fC(X) = max[w ijd(xi,v ): (i,j)eIC] , f(X) = max[fB(X), f(X)] . The m-center problem with mutual communication is the following: Find a location vector X*cNm such that Z* E f(X*) = min[f(X): XeNm] . -38- The problem differs from the p-center problem in two respects: (i) the distance between any vertex v. and any new facility xi may be of concern as opposed only to the distance between v. and the nearest new facility to v.; (ii) certain distances between new facilities are J of concern, as opposed to the absence of interactions between new facilities. For the case of a single new facility the two problems coincide. In this problem, the new facilities may be thought to fulfill a supporting task to other new facilities as well as servicing those existing facilities that are a priori assigned to them. Certain planar cases of the multifacility minimax problem have been studied by Dearing and Francis [20], Elzinga, Hearn, and Randolph [25], Wendell and Peterson [113],.and Francis [28]. The problem on a network is defined by Dearing, Francis, and Lowe [22] in the presence of distance constraints. It is established in [22] that the function f is a convex function on a tree network. The existence of a solution is guaranteed due to compactness and con- tinuity considerations. Furthermore, it is shown that it suffices to consider only new facility locations in the convex hull of the existing facility locations in order to solve the problem. The problem on a general network was shown to be NP-hard by Kolen [72 ]. For the case of a tree network, the problem is solved by Francis, Lowe, and Ratliff [32 ] by using an equivalent formulation in terms of distance constraints (with variable right hand sides). The solution procedure finds Z* first, by using the separation conditions. Then an optimal feasible location vector X* is constructed by using the Sequential Location Procedure described in [32]. To find Z* an -39- auxiliary graph G is formed with nodes N1,...,N ,E ,...,E Graph G contains arcs (N.,E.) with lengths 1/w.. corresponding to pairs (i,j)cIC, and arcs (Nj,Nk) with length 1/vjk corresponding to pairs (j,k)eIB. It is assumed that G is connected, for otherwise the problem decomposes into subproblems. For each pair of existing facility nodes E Ek, define L(E ,Ek) to be the length of a shortest path in G connecting Ej and Ek. Francis et al. showed that Z* is given by max{d(vj,vk)/L(Ej,Ek): 1 S j < k 5 n). The distances d(vj,vk) can be computed in 0(n2) operations for a tree network (see [23]), and the shortest path lengths L(E.,Ek) are readily computable in 0(n3) opera- tions. When Z* is computed, the Sequential Location Procedure de- scribed in [32] can be applied in O(m(n+m)) operations to find a loca- tion vector X* that solves the problem. m-Median problem with mutual communication Define the functions gB, gC, and g by the following expressions: For each XENm B(X) E [vjkd(xj,xk): (j,k)I] , gc(X) [wijd(xi,vj): (i,j)eIC] , g(X) = gB(X) + gC(X) The m-median problem with mutual communication is the following: Find a location vector X* in Nm such that Z* E g(X*) = min[g(X): XNm] . -40- The problem differs from the p-median problem in two respects: (i) the distance between any vertex and any new facility may be of concern as opposed only to the distance between a vertex and the near- est new facility to it; (ii) certain distances between new facilities are of concern as opposed to the absence of interactions between new facilities in the p-median problem. For the case of a single new facility, the two problems are identical. Planar cases of the problem using rectilinear or Euclidean dis- tances have received considerable attention and efficient solution procedures have been developed. A thorough discussion of these prob- lems is given in the book by Francis and White [31]. Other references on planar problems are Cabot, Francis, and Stary [6], Bindschedler and Moore [3], Francis [27], Eyster, White, and Wierville [26], Pritsker and Ghare [94], Wesolowsky and Love [115, 116], and Picard and Ratliff [93]. The problem on a network is defined by Dearing, Francis, and Lowe [22] in the presence of distance constraints. It was established in [22] that the problem is a convex optimization problem for all data choices if and only if the network is a tree. For the case of a general network, it is known that there exists an optimal solution on the vertices of N. This result and certain generalizations of it have been given by Goldman [41], Levy [80], Hakimiand Maheshwari [49], and Wendell and Hurter [111]. These references are already discussed under the p-median problem. The problem was shown to be NP-hard by Kolen [72 ] on a general network, and no solution procedures have been developed yet. -41- For the case of a tree network, the m-median problem with mutual communi- cation is solved by Dearing and Langford [21], and by Picard and Ratliff [93]. The approach used by Dearing and Langford is to embed the tree T into the Euclidean space Rp, for some p, so that the distance between any two points on the tree is equal to the rectilinear distance between the corresponding points in Rp. The problem in RP with rectilinear distances decomposes into p subproblems, each of which can be solved by using known techniques given in Francis and White [31], or, perhaps more efficiently, by applying the network flow procedure discussed in Cabot, Francis, and Stary [6]. For reducing the computational effort, the embedding procedure is carried out with respect to a minimal path decomposition of T into p edge disjoint paths (each edge is in one and only one path). Each path in a minimal path decomposition corresponds to a dimension in R . The approach taken by Picard and Ratliff in [93] takes advantage of the vertex-optimality condition and determines an optimal solution (on the vertices of T) by solving a sequence of at most n-i minimum cut problems, each on a graph containing at most m+2 nodes. The method is based on a result that an optimal location vector can be found independently of the edge lengths, by using only the incidence relations between vertices and the weights. In this respect, the pro- cedure is in the same spirit as Goldman's algorithm for finding a median of a tree. Each cut problem corresponds to an edge of the tree. To be more explicit, the removal of all interior points of an edge e leaves two disconnected components, T1 = T (e) and T2 T(e). Corresponding to edge e, a graph G = G(e) is constructed having nodes -42- 1 through m corresponding to new facilities, a source s and a sink t. Graph G contains arcs (s,i) and (i,t) for 1 5 i m and arcs (j,k) for each pair (j,k)clB. The capacity of arc (j,k) is the weight vjk. The capacity of arc (s,i) is given by [Wir: Vr ET, (i,r)cI], and the capacity of arc (i,t) is given by [ [wiq: VqeT2, (i,q)Ic ]. If (Q,Q) is a minimum capacity s-t cut of G, with scQ, tcQ, then all new facility locations x. for which the corresponding node i is in Q are 1 in T1 in an optimal solution. Similarly, all x. for which the node j is in Q are in T2 in an optimal solution. The procedure is a repeated application of this minimum cut problem with respect to each edge, until an optimal vertex location is determined for each x.. During the process, each x. whose location is determined is treated like an J existing facility. The method is described originally for the analogous rectilinear distance problem on the plane, which, in turn, decomposes into two subproblems, each on a line. Multi-objective location problems on networks Multi-objective optimization problems, sometimes known as vector optimization problems, involve decision making under two or more criteria. More explicitly, a set (finite or infinite) S of alterna- tives is specified and n (possibly non-commensurable) objective func- tions are to be minimized over S. Let f ,...,f be n numerical func- 1 n tions defined on S, and define f(x) = (fl(x),...,fn(x)) for all xeS. The multi-objective optimization problem (VMP) is the following: V-min f(x) xcS In general, the minima of the functions fl,...,f do not coincide. In order for the minimization to be meaningful, one needs tointrodu In order for the minimization to be meaningful, one needs to introduce -43- the concept of "efficient solutions." A point x in S is said to be efficient if there does not exist a point y in S such that f.(y) 5 f (x) for 1 i 5 n and fk () < fk(x) for at least one index k. One is interested in finding and characterizing the set of efficient solu- tions to (VMP).- An efficient point is sometimes known as an undominated point. A point which is not efficient is said to be dominated. Kuhn and Tucker [76] and Koopmans [74] are among the first to introduce the concept of efficiency. Geoffrion [40] extended the con- cept to "properly efficient" points and provided a comprehensive theoretical framework for subsequent research. Necessary and suf- ficient conditions for efficient points to be properly efficient are given by Wendell and Lee [112]. Some of the later contributions are due to Yu [117], Yu and Zeleny [118, 119], Bitran and Magnanti [4], Wendell [110], and Bergstresser, Charnes, and Yu [1]. We note that there are other approaches to multicriteria decision making, such as goal programming, multi-attribute utility theory, construction of outranking relations, and interactive programming techniques. For general information on multicriteria decision making, the reader is referred to Roy [99], Starr and Zeleny [104], Cochrane and Zeleny [16], Keeney and Raiffa [67], and Thiriez and Zionts [108]. A survey of multicriteria decision making is given by Chalmet [7]. Multi-objective location problems (on the plane or on networks) have begun receiving attention only recently. Kuhn [75] appears to be the first to consider a multi-objective location problem on the plane. Kuhn considered the problem of minimizing the vector of Euclidean distances from a variable point to a set of fixed points on the plane, and showed that the set of efficient solutions is the convex -44- hull of the fixed points. Wendell, Hurter, and Lowe [114] considered the same problem with rectilinear distances and provided algorithms of 0(n2 )and 0(n3 ) for generating efficient points. A most efficient algorithm of O(nlogn) was developed by Chalmet and Francis [8] for the same problem. McGinnis and White [83] considered the problem of minimizing the sum of and the maximum of weighted rectilinear distances from a variable point to a set of fixed points on the plane and formu- lated the problem as a parametric linear program for which known solu- tion techniques exist. Juel [64 ] considered the same problem for the case of multiple new facilities and gave an equivalent parametric linear program. Chalmet, Francis, and Lawrence [ 9 10 ] considered two variants of an efficient design problem, where the location variable (a design) is a planar region of specified positive area but of unknown shape. A few papers have been produced on multi-objective location problems on networks. In what follows we discuss these problems. The cent-dian problem. The single facility "cent-dian" problem involves the sum of and maximum of weighted distances from a new facility to a set of existing facilities at vertices of N. To define the problem, let w. and h. be two positive weights associated with 1 1 vertex v., ieI = {1,...,n}. For each point xeN define: 1 m(x) {wid(vi,x): iEl} , c(x) E max[h.d(v.,x): iel] 1 1 f(x) E (m(x), c(x)) . -45- The problem of interest is to find all efficient points with respect to f(x). Halpern [52] is-the first to consider this problem. Halpern formulated the problem in a slightly different manner by considering a convex combination of m(x) and c(x). For any fixed X, 0 _< A 1, define f(A,x) and f*(A) by f(X,x) Xm(x) + (1 X) c(x) for xsN , f*(X) min[f(X,x): xcN] (1.3.4) In Halpern's terminology, the function f(X,x) is called a cent-dian function and any point x* x*(A) that solves (1.3.4) is called a cent-dian point. In [52] Halpern considered this problem on a tree network with weights h. all equal to unity. Defining x and x to be the (vertex) 1 m c median and the absolute center of T, respectively, Halpern proved that for any given X, the cent-dian x*(X) is located at either x or on c one of the vertices located on the path P(x ,x ). This theorem pro- m c vides the basis for a simple and efficient algorithm to locate the cent-dian by inspecting the vertices on P(x ,xc). Further, Halpern showed that, if the absolute center xe is known, then the cent-dian can be found by determining the median of a tree T' that is identical to T except that T' contains an additional vertex v x with the n+1 c -1 associated weight wn1 = 1 1. Handler [56] formulated the same problem on a tree network in a slightly different manner by using the median function as a constraint. In Handler's formulation one is interested in solving the problem -46- P for each given a, where P is defined as follows: e(a) = min[c(x): m(x) 5 a, xcT] Efficient solutions are obtained by parameterizing on a. Handler's results closely parallel Halpern's. The problem on a general network is studied by Halpern [54], using the convex combination approach. Halpern showed that the problem is a computationally finite one. Computational finiteness follows from the result that f(X,x) is a continuous, piecewise linear function of x on each edge and attains its minimum at one of a finite number of points. Defining Q(e) to be the union of the end points of edge e with the set of local minima of c(x) on e, the minimum of f(X,x) over all x on edge e is a member of Q(e) for any given X, 0 5 X 5 1. De- fining Q E U {Q(e): esE}, it follows that the cent-dian x*(X) is con- tained in Q for any X. Further, Halpern showed that the function f*(X) = min[f(X,x): xcN] is a continuous, piecewise linear, concave function of X for 0 < X < 1. Based on these results, Halpern provided an algorithm which constructs f*(X) and identifies x*(X) for 0 X < 1. To construct f*(X), the algorithm inspects each edge one at a time and computes the set Q(e), unless a simple test indicates that edge e cannot contain any cent-dian for any X. An upper bound on f*(X) is carried through and improved, whenever possible, by examining the members of Q(e). Cent-dian problem and duality. In [53], Halpern studied the cent- dian problem on a general network from a different angle and obtained a duality relationship. Using an approach similar to Handler's median constrained problem, Halpern defined two problems, a median constrained -47- and a center constrained one. More specifically, for real X and p define the functions m*(A) and c*(p) as follows: m*(X) = min[m(x): c(x) 5 A] (1.3.5) c*(P) = min[c(x): m(x) < -] (1.3.6) In general for some values of X (p), the constraint c(x) s X (m(x) < p) may not admit any feasible solution. However, real inter- vals C and M can be defined so that for any XeC and for any pcM, the constraints in (1.3.5) and (1.3.6) admit a feasible point. To define C, let 0 be the set of all minima to min[c(x): xsN], and let S -be c m the set of all minima to min[m(x): xeN]. Let x be a point in Q0 that c minimizes the value of m(x) over all x in Q Similarly, let y be a c point in 0 that minimizes the value of c(y) over all y in 0 Then m m C and M are defined as follows: C = [c(x), c()] M = [m(y), m(x)] . With these definitions Halpern's duality theorem can be stated as follows: a) Given any peM, with A = c*(p), we have c*(m*(X)) = A. b) Given any XeC, with p = m*(X), we have m*(c*(p)) = p. For a tree network, the functions m* and c* are 1-1 and onto. It follows from the duality theorem that the function m* and c* are inverses of each other for a tree network. For a general network, the functions m*, c* need not be onto, i.e., the image of the domain -48- may only be a proper subset of the range. Hence, the inverse property holds only for some members of C and M for a general network. Now, we consider a more general multi-objective problem due to Lowe [82]. The problem involves a single facility to be located on a tree network with respect to m convex objective functions. Multi-objective convex location problem (on a tree). Let T be a tree network and let fl,...,f be m convex continuous bounded func- tions each of which is defined on T. In general, not all points in T may be feasible with respect to f.. Let Q. be a convex compact subset of T which contains all feasible points x with respect to the ith optimizer. The set Q.may be defined by specifying its extreme points, 1 or by means of distance constraints, or by other means. We assume m that Qi is known or computable. Define Q = Qi and assume that Q i=l is nonempty. The problem of interest is to find all efficient points in Q with respect to the vector minimization problem defined below: V-min[f(x): xeQ C T] where, f(x) = (fl(x),...,fm(x)) for all xT . We note that Q is a convex compact subset of T as it is the intersection of m convex compact subsets Q of T. For a formal dis- cussion of convexity on a network, the reader is referred to Dearing, Francis, and Lowe [22]. Loosely speaking, Q a convex subset of T, means Q is connected or that the (shortest, unique) path connecting any two points in Q is contained in Q. Lowe makes no assumptions on the specific forms of the objective functions. Under the convexity assumptions, Lowe proves that a convex -49- compact subset T* of T can be identified that contains all efficient points. To identify T*, define Rt to be the set of all minima to the 1 unconstrained problem min[f.(x): xET]. If R* intersects the feasible 1 1 set Q, define St to be this intersection. Otherwise, St is the unique 1 1 closest point in Q to R*. Having defined each S*, 1 S i m, if their i i intersection is non-empty, then the set of all efficient points is given by T* = n {S: 1 5 i 5 m). If this intersection is empty, then 1 T* is the smallest compact convex subtree that intersects each St. It 1 can be shown that each RY, S* is convex, compact, and that T* is a 1 1 convex compact subset of T. Lowe's theorem assumes a knowledge of set of minima to each f. as well as a knowledge of Qi and hence Q. We note that the functions c(x) and m(x) in the cent-dian problem are both convex on T. Hence, Halpern's results can be obtained by apply- ing Lowe's theorem. Now, we consider a multi-objective problem which involves multiple new facilities to be located on a tree network so that the distance between each specified pair of new and existing facilities, and each specified pair of new facilities is, roughly speaking, "as small as possible." The problem is defined by Francis, Lowe, and Tansel [33] as a sequel to the distance constraints problem, and solved by making use of the separation conditions. Here, we call the problem, the "multifacility vector minimization problem." The multifacility vector minimization problem (on a tree network). Let T be a tree network and let IC, IB be given nonempty sets with IC C {(i,j): 1 i m, 1 < j : n} and IB C {(j,k): 1 j < k : m}. The problem of interest is to locate m new facilities on T at points x1,...,xm so that each distance d(xi,v ) (i,j)clC and d(xj,xk) (j,k)el -50- is "as small as possible." More specifically, we wish to find all efficient location vectors X = (x,...,x ) in T with respect to the 1 m vector minimization problem V-min[D(X): XcTm] where D(X) is the vector of distances d(xi,vj) (i,j)clC and d(x ,xk) (j,k)eIB. The vector is formed by assuming any convenient ordering of the members of the sets IC and IB- Francis, Lowe, and Tansel [33] characterized efficient points by making use of distance constraints. By definition, a location vector Z in Tm is efficient if an only if there does not exist a location vector X in Tm such that D(X) S D(Z) and D(X) # D(Z). Given a location vector Z, let b = d(z ,zk) for (j,k)IB and c.. = d(z.,v.) for jk J k 1 1 J (i,j)CIC, and define the distance constraints (DC) of interest by d(xi,v.) 5 cij (i,j)EIC 1 J 1 C d(xj,xk) bjk (jk)cIB We note that DC is always consistent, as Z is always feasible to DC, and hence the separation conditions are always satisfied. The separation conditions for DC are defined by constructing a graph G with nodes N., 1 & j 5 m, corresponding to new facilities and nodes Ei, 1 5 i 5 n, corresponding to existing facilities. For each (i,j)eIC, the arc (N.,E.) is in G with length c.i, and for each (j,k)elB, the arc (Nj,Nk) is in G with length bjk. We recall that a point xi is uniquely located in every feasible solution to DC if and only if the corresponding node N. is in at least one tight path in G, 1 -51- where a path of G joining any two existing facility nodes Es and Et is said to be tight if the length of the path is equal to the distance between the vertices v and v in T corresponding to nodes E and E , s t s t respectively. For any given location vector Z, denote by A.(Z) the collection of locations of uniquely located facilities whose nodes are adjacent to N in G. Let H[A (Z)] be the convex hull of A (Z), i.e., the smallest connected subtree containing all points in A.(Z). 1 With these definitions, it was proven in [33] that the following conditions are equivalent: (i) Z is efficient. (ii) Z is the unique solution to DC. (iii) Each N. is in at least one tight path in G. 1 (iv) Each Z. is contained in H[A.(Z)], 1 5 i m. This completes the discussion of multi-objective location problems on networks. Path Location Problems Here, we consider three versions of a path location problem posed by Slater [102]. To define the problems, let P denote any path con- necting any two vertices in a network N. For any vertex veV and any path P, define the distance D(v,P) to be the distance from v to a nearest vertex in P. Also define the branch weight bw(P) of a path P to be the maximum number of vertices in any component of N-P. The three versions of the problem are the following: min C D(v,P) (1.3.7) P C N veV -52- min max D(v,P) (1.3.8) P CN veV min bw(P) (1.3.9) P C N In Slater's terminology, any path P* that solves (1.3.7) is called a core of N. Among all paths that solve (1.3.8), one with the fewest vertices is called a path center of N. Similarly, among all the paths that solve (1.3.9), one with the fewest vertices is called a spine of N. Slater obtained a number of properties of these problems for tree networks. In particular, Slater showed that the path center of T is unique and contains the vertex center of T, and that the spine of T is unique and contains the centroid (equivalently, the vertex median) of T. We recall that a centroid of T is any vertex v that minimizes the maximum number of vertices in any component of T-v. Also, Slater proposed two algorithms of linear order for determining the path center and the spine of T. CHAPTER 2 DUALITY AND THE NONLINEAR p-CENTER PROBLEM AND COVERING PROBLEM ON A TREE NETWORK 2.1 Introduction and Related Work We consider the problem of locating p new facilities on a tree network with respect to n existing facilities at known locations so as to minimize the maximum "loss." The problem is an extension of the linear p-center problem to the nonlinear case. We assume a strictly increasing, continuous "loss" function is associated with each of a finite number of demand points (existing facilities) whose argument is the distance between the corresponding existing facility and its nearest new facility. Our formulation permits the use of quite general loss functions provided that they are continuous and strictly increas- ing with the travel distance. The term "loss" is used generically and may refer to any form of inconvenience such as cost, disutility of service, travel time, etc. In locating emergency service facilities, the disutility due to "late" service may be too great beyond a certain "threshold" response time. Such sharp changes in the disutility of service can be re- flected into the model by using nonlinear functions. Hurter and Schaefer [61 ] justify and use such functions in a fire setting. As pointed out by Dearing [18], a study by Kolesar et al. [73 ] revealed that the travel time for fire trucks can be approximated by a particular continuous, nonlinear, increasing function of the distance. -53- -54- The literature on the p-center problem is discussed in detail in Chapter 1. Here, we give a brief review of the more closely re- lated work. Except for p = 1, we know of no literature on the non- linear p-center problem. For p = 1, the only references we are aware of which deal with the nonlinear case are Dearing [18] and Francis [29]. Both authors showed that the minimax loss with respect to any two existing facilities is a lower bound on the maximum loss with respect to all existing facilities, and that the largest of the lower bounds determines the minimax loss to all existing facilities on a tree network. This result is an instance of the duality result we will present in this chapter. The linear (weighted or unweighted) p-center problem is shown to be NP-complete on a general network by Kariv and Hakimi [65], and by Nemhauser and Sheu [92]. The linear 1-center problem on a tree network is well solved (see Goldman [44], Halfin [51], Lin [81], and Dearing and Francis [19]). For p > 1, the linear p-center problem on tree networks is considered by various authors. Handler [57] provided an 0(n) algorithm for finding the 2-center of a tree for the unweighted case. Kariv and Hakimi [65] gave an 0(n2logn) algorithm for tree networks which relies on solving a sequence of covering problems for the weighted case with p > 1. A similar procedure for the unweighted continuous p-center problem on a tree network is given by Chandrasekaran and Daughety [12]. A vertex-restricted version of the problem is solved by Chandrasekaran and Tamir [13], and relies on solving a sequence of clique covering problems on a related intersection graph. -55- The first duality relationship involving tree network location problems can be found in Meir and Moon [ 86 ]. Cockayne, Hedetniemi, and Slater [17 ] obtained a more general version of the result given in [86 ]. The results in [ 86 ] and [17 ] closely parallel our duality result for the covering problem and its dual. Shier [100] discovered a "dispersion" problem which is dual to the continuous unweighted p-center problem. The dispersion problem of Shier is to choose p+l points in the tree network the nearest two of which are as far apart as possible. Chandrasekaran and Tamir [14] observed that Shier's duality holds when the problems are defined with respect to a subset of the tree. For the case where this subset is a finite collection of demand points, their result is an instance of the duality relation- ship we will present in this chapter, as applied to the unweighted linear case. At this point we give a brief overview of the chapter. In Sec- tion 2, we define the (nonlinear) p-center problem and a dual "dis- persion" problem. We state and prove a weak duality theorem applicable to all networks, and state a strong duality theorem applicable to tree networks. In Section 3 we give a physical interpretation of the dual dispersion problem. In Section 4 we study a covering problem and present an algorithm, COVER, for solving it. The covering algorithm provides the basis of our solution procedure to the p-center problem as well as the dual dispersion problem and yields a construc- tive approach for proving the strong duality theorem. In Section 5 we present an algorithm, OPTKLIQUE, which provides a constructive proof of the strong duality theorem, while solving the dual problem. Addi- tional results for the covering problem, including a "divergence" problem dual to the cover problem, are given in Section 6. -56- 2.2 Problem Statements and Duality We suppose given a finite undirected tree network with positive arc lengths and denote by T an imbedding of the given network having as edges rectifiable arcs. For any two points x,ycT, let d(x,y) denote the shortest path distance between x and y. Let J {l,...,n} and denote by V {vl,...,v n (VC T) a collec- tion of distinct vertex locations of "demand points" or "existing facilities." Let X = {x1,...,x } (X C T) denote a finite collection 1 P of "centers" or "new facilities." For jeJ, define the distance of v. to its nearest center by D(X,v.) = min{d(x.,v.): 1 5 i 5 p}, and.let J Sj E max{d(x,v.): xsT}. Also, for jeJ, we assume given a real valued function f., continuous and strictly increasing, with domain [0,6.] and (clearly) range [fj(0),f.(6.)]. For X C T, IXI < m, we define the function f by f(X) = max{f.(D(X,v.)): jJ} . The Primal p-Center Problem is as follows: Find a p-center X* for which r = f(X*) = min{f(X): X C T, IXI = p} (2.2.1) As discussed in Dearing and Francis [19], due to compactness of T and continuity of d(x,.) on T for each fixed xET, an optimal solu- tion X* to (2.2.1) exists and is contained in the convex hull of V. With a and n defined by a = max{f.(0): jeJ} and n = min{f.(0.): jeJ}, we shall assume a < n, for if a = fs(0) > ft (6t ) = n, say, then the function ft would always be dominated by (strictly smaller than) -57- f and hence f could be deleted from the definition of f without s t changing f. Further, we assume p n-1, as otherwise the p-center problem is trivial. So as to state the dual problem, we define Bjk = kj for j,keJ by Bjk = min max{f (d(x,v )), fk(d(x,vk))} Jk xcT J k k For j,keJ with j < k we define ajk max{f.(0), fk(0)} and bjk min{f (6 ),fk(6k)}. We note that a n implies [ajk,bjk] # 0. The following lemma, the results of which are proven in [29], provides a closed form expression for Bjk' Lemma 2.2.1. For any j,keJ with j _< k we have: -1 -1 (i) The function f + fk exists, is stricly increasing, continuous, 3 k has domain [a.j,bjk] 3 0, and range [L jk,U ], where Lj = jk jk jk Jk jk -1 -1 -1 -1 (f + f )o(a ) and U = (f1 + f )o(bjk). j k jk )k j k)kk (ii) d(v ,vk) < Ujk' (iii) The function (f. + f ) exists, is strictly increasing and 3 k continuous, has domain [Ljk,U.k] and range [ajk,bjk]. -1 -1 -1 (iv) Bjk = (f I + fk) o-(max{d(v. ,vk), Ljk) We remark that either jk = ajk or .jk = (f1 + fk) o(d(v.,vk)); Bjk E [ajk,bjk], and jj. = f.(0). The closed form expression for 0jk given in Lemma 2.2.1 facilitates construction of the dual problem. We define the dual objective function g on subsets of V as follows: For any K C V with IKI > 2 g(K) max{gl(K), g2(K)} gl(K) E min{ ij: vi,vj E K, i # j} g2(K) max{f.(0): v. c K} . J J -58- The Dual Dispersion Problem is as follows: Find a subset K* of V such that g(K*) = max{g(K): K C V, IKI = p+1} (2.2.2) We remark that the dispersion problem is meaningfully defined for 2 < p+l < n. The primal p-center problem is trivial for p > n. Hence, we shall restrict p to 1 S p < n-i. In what follows in this section, we prove a Weak Duality Theorem (W.D.T.) and state a Strong Duality Theorem (S.D.T.) (proven in Sec- tion 5). At the end of this section, we give an example problem illustrating definitions and results. In the W.D.T. we shall use the fact (readily proven as in [18] or [29]) that a < f(X) for any XC T, |XI < m. Theorem 2.2.1. (Weak Duality Theorem). Assume 1 < p < n-1. For any X C T with IXI = p, and any K C V with JIK = p+1, we have f(X) > g(K). Proof. There are two cases: g(K) < a or g(K) > a. In the former case we have g(K) < a 5 f(X). In the latter case, we note that g(K) = gl(K) > a > g2(K). Since jXI = p < p+l = JIK, at least two demand points in K must be served by a single center. In other words, for some 'v t v K with s # t, and some center xsX, we have fs[D(X,v )] = fs[d(x,vs)] 5 f(X) (2.2.3) ft[D(X,v )] = ft[d(x,vt)] < f(X) Using the definitions and the inequalities in (2.2.3), we have g(K) = gl(K) Bst < max{fs[d(x,v )],ft[d(x,vt)]} f(X). Remark 2.2.1. We note that the conditions JIX = p and JKJ = p+l can be replaced by jXI < p and/or JKI > p+l, respectively, and the proof -59- will still apply. Furthermore, the proof applies to any network, as no special properties of tree networks are used. We now state the S.D.T. We remark that the S.D.T. requires the assumption of a tree network. In effect, network cycles may create a "duality gap." Theorem 2.2.2. (Strong Duality Theorem). For any p, 1 5 p s n-1, there exists an X* C T with IX*l = p and K* C V with IK*l = p+1 such that f(X*) = g(K*). It is evident from the W.D.T. that X* solves the primal p-center problem and K* solves the dual dispersion problem. Before presenting an example problem, we find it convenient to view the dual problem as defined on "cliques" of a complete graph G. We define G to be the undirected complete graph with node set J, where node j of G represents vertex v. of T. To any arc (i,j) of J G, i # j, we assign the length i.., and, to any node j of G, we assign the node weight j.. = f.(0). We call any complete subgraph K of G a clique. We note that any nonempty subset of V induces a clique in G and vice versa. For this reason, an equivalent definition of g(.) on cliques of G can be given by defining gl(K) to be the length of a smallest arc in a clique K of G, g2(K) to be the maximum of the weights of nodes in K, and letting g(K) = max{gl(K), g2(K)}. If the number of nodes of a clique K is known to be q, we call K a q-clique and (sometimes) write K Defining C (G) to be the collection of all q q q-cliques of G, an equivalent statement of (2.2.2) is as follows: Find a clique K* for which p+l g(K* ) = max{g(K): K c C (G)} p+1 p+1 -60- Whether K refers to a subset of V or a clique of G, we prefer to call K a clique as long as it is clear from the context what K refers to. As an example of the nonlinear p-center problem, suppose that the function associated with node v. is fj(y) = w (y + h ) for y c [0,6 ], where wj h, and 0 are given parameters. Appropriate restrictions are placed on the parameters to ensure that the f. are strictly in- creasing on [0,6.]. We note that the linear weighted p-center problem is a special case of this problem generated by choosing 6 = 1, h. = 0, and w. > 0 for all j. J For the given form of f., the following are readily verified: J -1 1/1 f. (r) = (r/w.) h., r [f.(0), f.(6.)] f (r) + f (r) = r /[/w) + (1/w.) ] (h. + h.) , -1i -1j 3 r e [aij, bj] , -1 -1 -1 w 0 6 (f. + f ) o(y) = j (y + h + h.) i j 1/0 1/0 6 [w. + w. ] 1 J y c [Lij, Uij] Then, using the characterization of B.. as given in Lemma 2.2.1, we have ij dij if Lij d(vi',vj) ij.. = (2.2.4) Pmax[fi(0), f.(0)] if Lij > d(vi,vj) , 1 j where -61- w.w. 0 y = L 1/ and d..= [d(v,v.) + h + h. (wi + w. Consider the tree network shown in Figure 2.1, where the numbers on the arcs represent arc lengths. The data given with Figure 2.1 corresponds to the parameters for j=1,...,6 where clearly, each f is strictly increasing. Using (2.2.4), the .ij values for this problem are shown in Table 2.1 along with the node weights f.(0). Figure 2.2 shows the dual graph G associated with the problem, where the number next to each node j is the node weight and the number on the arc between nodes i and j is 8... Using Figure 2.2 it can be verified that the optimal cliques (specified here by their nodes) and associated g values are K* = {3,4}, g(K*) = 13829.76; K* = {1,3,6}, g(K*) = 3600; 2 2 3 3 K* = {1,3,5,6}, g(K*) = 1664.64; K* = {1,3,4,5,6}, g(K*) = 784; and K* = {1,2,3,4,5,6}, g(K*) = 225. Due to the duality theory, it then 6 6 follows that the r for p=1,...,5 are, respectively, 13829.76, 3600, P 1664.64, 784, and 225. 2.3 Dual Problem Interpretation We imagine two conservative adversaries, an aggressor A and a defender D. Defender D has defense forces placed at vertex locations V1,...,v Aggressor A will attack a single vertex in V. Although D knows A will attack a vertex, he will not know the vertex attacked until the attack occurs. Defender D has p response forces which he must position at loca- tions defined by a p-center X. Interpret tree distances to be travel times, so that D(X,v.) is the minimum time to respond to v. from a J *J -62- V6 Data 6 2 9 25 16 36 4 9 f(y) = w(y + h ) 0 Figure 2.1. Example Nonlinear p-Center Problem -63- Bi. Values and Node Weights for Example i 1 2 (ci) = 3 ij. 4 5 j .4 225 3600 3600 3600 3600 13829.76 3600 3600 8464 900 4356 4556.25 11664 784 1664.64 1 2 3 4 5 6 0 0 64 0 0 144 fj(0) Table 2.1 -64- Dual Graph for Example 3600 144/. ) Figure 2.2. -65- center in X. Assume A and D know functions f ...,f so that f.(D(X,v.)) is D's loss if A attacks v. and D responds to the attack in a time of D(X,v.). For convenience, we refer to the loss A in- flicts on D as A's gain. Aggressor A knows D has p response forces, but does not know how D will position his response forces. Thus A acts conservatively and bases his decision on a worst case analysis. If A decides to attack v. without threatening any other vertices, A reasons that D will cor- rectly guess v. is to be attacked and will position a response force J at v.. Hence A assumes his gain will be f.(0), if he decides to J J immediately attack v. without a prior threatening strategy. In order to gain more, A concludes that he must threaten, i.e., pretend to attack, q vertices, q > 1, so that even if D knows which q vertices are threatened, D does not know which vertex A will attack until the attack occurs. Thus D is forced to respond to the threat by position- ing his response forces optimally with respect to these q vertices. Hence if A threatens K C V, he assumes D will choose a p-center X which minimizes f(X:K ) 2 max{f.(D(X,v.)): v. c K }. Thus, with q J J J q q p, A assumes D knows K and will position a response force at q every vertex in K so that A can gain at most g2(K ). The best A q 2 q can do in this case is to choose a K which contains some vertex v q s for which f (0) = a. Hence, if q 5 p, A's maximum possible gain is at most f (0). (Parenthetically, we remark that if f (0) = r , s s p p < n, then it can be shown that not all f.(0) have the same value. J If all f.(0) do have the same value, then r > a.) On the other hand, J p if A chooses a subset K with q > p, D is unable to position a response q force at every vertex in K even if he knows K so A will gain at q q -66- least g2(K ). Hence A observes if he chooses some K with q > p which contains a vertex v for which a = f (0), then his gain is at least s S a = g2(K ). However, A recognizes that there may be some other K with q > p, which may or may not contain vs, but which yields him a gain strictly greater than a. For this reason A restricts himself to those subsets of V with cardinality greater than p and realizes that if he chooses some K with q > p, then there is at least one pair of q vertices in K which D can cover by only a single response force. If q v. and v. are one such pair in K which are covered only by a single response force, say at x, then clearly A obtains a gain of at least .., as .. = min{max(f (d(x,v.)), f (d(x,v.))): x e T} <. max{f (d(x,vi)), A f.(d(x,v.))}. Since A does not know which pairs of vertices D will cover by single response forces, once he chooses K A acts conserva- tively, and assumes that D will cover a pair va,vb e K for which ab = min{i..: v.,v. K i # j}. That is, by choosing a K with ab ij ij q q q > p, A guarantees himself a gain of at least ab = gl(K ). Hence A's minimum gain due to threatening K is g(K ) = max{gl(K ), g2(K )}, q q 1 q 2 q so A chooses a K* with q > p which maximizes g(K ) over all K C V q q q with q > p. The question arises as to why A should choose p+l vertices to threaten, and no more. By virtue of the W.D.T. and the remark follow- ing it, if X* is an optimum p-center then f(X*) > g(K ) for all K q q with q > p+l. Thus r = f(X*) is an upper bound on A's gain due to threatening K But the S.D.T. implies there is a p+1-clique, say q K* which attains this upper bound. Hence A need threaten no more than p+1 vertices to maximize his gain, as A cannot obtain any addi- tional gain by threatening more than p+1 vertices. -67- There is also the possibility that A will make a false threat, that is, attack a vertex not among the ones he threatens. If D be- lieves the threat is false and continues to act conservatively, he will simply choose a p-center X* to minimize f. But since there exists a p+1-clique K* such that g(K* ) = f(X*), the greatest loss D can p+1 p+l incur, given X*, is the same as if he believes A's optimal threat to be real, and acts accordingly. Hence A cannot gain more by making a false threat. 2.4 Covering Algorithm In this section we study a covering problem, and present an algorithm for solving it. Our primary interest in the algorithm is the fact that it provides a constructive approach for proving results about the primal and dual problem. For this reason we purposely keep the algorithm simple, and use an analog string model to provide insight into the algorithm. The development of both the string model and the algorithm is motivated by an earlier string algorithm given in [32]. As in [32], an equivalent algebraic version of the algorithm is readily obtainable. We remark that two other quite efficient algo- rithms [14], [15], exist for solving the covering problem, but they do not lend themselves readily to our needs. At this point we state the Covering Problem: Given r and the runction f, compute q(r) = min{IXI: f(X) 5 r, X CT} (2.4.1) It is readily seen that the covering problem has a feasible solution if and only if a 5 r. Further, with J(r) E {j: r < f.(6.)}, we shall *J J -68- assume J(r) # 0, for if J(r) = 0 then the condition f(X) 5 r holds for all X C T and we (trivially) have q(r) = 1. The above assumptions permit the following equivalent statement of the covering problem: minimize IXI subject to -1 D(X,v.) : fI (r), j e J(r) (2.4.2) J J We refer to the covering algorithm as COVER. In order to state COVER a few definitions are convenient. We may imagine that the tree is represented appropriately by inscribing straight line segments on a planar surface such that each segment represents an arc. We fasten -1 strings of length f. (r) to each node v.,j e J(r), of the inscribed j J tree, where, by convention, we allow strings of zero length. Every fastened string has one end permanently affixed to the planar surface. In addition, during the use of the algorithm we engage previously fastened strings at various points on the tree. When a string is engaged, some point of the string is permanently affixed to the tree such that there is no slack in the portion of the string so far en- gaged. When strings are removed, we imagine that they are physically deleted from the string model. During each iteration of the procedure, we partition the original tree into two subsets: one green, the other brown. The green subset is always a tree, denoted as GT (for green tree), while the brown sub- set consists of one or more subtrees of the original tree T, each of which is "rooted" at a node of the green tree. By convention, a root -69- node t will be in both GT and the associated brown subtree, denoted as BT(t). COVER 0) Initialize to GT = T, k = 0. For every tip vertex v. of T define J -1 BT(v.) = {v.}. For every j e J(r) fasten a string of length f. (r) J J J at v.. Define U = 0. J o 1) Choose a tip t of GT. If GT = {t} go to 6). Else find a(t), the vertex in GT adjacent to t. 2) If no strings are engaged or fastened at t, remove from GT the subarc [t,a(t)] joining t and a(t), attach [t,a(t)] to BT(t), and go to 1). Else go to 3). 3) Pull all strings at t tight towards a(t). If all tight strings reach a(t) then engage them at a(t), remove [t,a(t)] from GT, attach [t,a(t)] to BT(t), and go to 1). Else go to 4). 4) Add 1 to k. Choose a shortest string engaged or fastened at t. Find the (unique) vertex, say v(k), at which the shortest string is fastened. Construct Uk = U k- U {V(k)}. Find the farthest point, say y, from t on [t,a(t)] to which the shortest string can reach. Locate xk at y. Assign all strings at t to xk and remove these strings. Attach [t,y] to BT(t) to obtain BT(xk), and remove [t,y] from GT. Go to 5). 5) Assign to xk all other strings in GT which can reach xk, and re- move all such strings. If no strings remain then let U = Uk and stop. Else return to 1). 6) Add 1 to k. Locate xk at t. Assign all strings at t to xk. Of the strings at t choose any one, and find the vertex v(k) to which the chosen string is fastened. Let U = Uk- U {v(k) }, and stop. -70- Note that each time COVER places a center at xk in step 4) it identifies an associated vertex v(k) which we call the distinguished vertex associated with xk. When centers xl,...,xk have been placed in step 4), we call Uk = {v(1),...v(k) } the distinguished set associated with {xl,...,k}. If the algorithm places q centers in total, then the set U defined by the algorithm consists of vertices v(1)",.,V(q), the first q-1 of which are distinguished vertices (when q > 2). The last vertex is distinguished only if x is placed in step 4). Letting X = {x ,...,x }, we call U the primary set q associated with X, and call v(i) the primary vertex associated with x., i = 1,...,q. We note that the primary vertices v(1),...,v(q) are distinct, for as soon as a primary vertex is identified, its string is removed, and thus the vertex is not available for any subsequent identification. Likewise the centers xl,...,x are distinct, for if x. = x. with i < j, then all strings assigned to x. would have been assigned earlier to x., and so x. would not have been located. Hence it follows that IUI = IXl = q, and U # 0, since IXj ? 1. The primary vertices will be of theoretical significance in proving our results. We now establish some properties of COVER. Property 2.4.1. COVER finds a feasible solution X to the covering problem with IXI : n. Proof. We first note that termination is clearly finite, since at each iteration either at least one string is removed, or some entire arc of T becomes colored brown. Since there are at most n strings initially, it follows that the X constructed satisfies IXI 5 n. Choose any v.,j e J(r), and denote by x(j) the center to which v. is assigned. Since the string fastened at v. reaches x , JJ J) -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 -72- 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) -76- where (2.5.5) follows from the W.D.T., (2.5.6) and (2.5.7) follow from the definitions of g and 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 -82- 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, -83- with IXI = s, solve the cover problem for r. We then have f(X) S r < r 1 < r contradicting the definition of r Thus q(r) = p for r r < r p p-l It now follows, if we define the set P = {(p-l,p): p E {2,...,n}, r < r } , p p-i that Sp for r r < r-1, (p-l,p) e P q(r) = (2.6.4) 1 for r 1 r The formula (2.6.4) completely defines the function q(r), since r = a, n and the cover problem is feasible if and only if a 5 r. Hence if we solve the p-center problem for all p and compute r2,...,r then we have an explicit formula for q(r), and we see that the r completely define the function q. For example, if r6 = r5 < r4 = r3 < r2 = rl, then q(r) = 5 for r5 < r < r4, q(r) = 3 for r3 = r < r2, and q(r) = 1 for r1 < r. Also, the proof of the lemma does not require the assump- tion that the location network is a tree. Thus the formula for q(r) is still valid if the location network has cycles. CHAPTER 3 A VECTOR-MINIMIZATION PROBLEM ON A TREE NETWORK 3.1 Introduction We consider a vector-minimization problem on a tree network which involves as objectives the distances between specified pairs of new facilities and specified pairs of new and existing facilities. In many location problems, especially in the public sector, it may be necessary to build a number of public facilities which are to be shared by a number of communities. If the optimizers cannot agree on a single objective function, the analyst is faced with the problem of locating the facili- ties in such a manner that all parties are satisfied with the end result. In such a case, the optimizers can agree to rule out "dominated" solutions and consider only "efficient" solutions. The related literature on multi-objective location problems is discussed in Chapter 1 under Multi-objective location problems on networks. Here, we concentrate on characterizing efficient solutions to the vector-minimization problem of interest. We relate efficient solutions to a distance constraints problem studied by Francis, Lowe, and Ratliff [32]. Extensions of results in [32] are given by Francis, Lowe, and Tansel [33]. We use the theory developed in [32] and [33] to establish the necessary and sufficient conditions for efficient location vectors (parenthetically, we remark that the results we proved in [33] are also given in our Dissertation Proposal defended on June 8, 1979). -84- -85- At this point, we give an overview of the chapter. In Section 2, necessary definitions and notation are given and the vector-minimiza- tion problem of interest is defined. In Section 3, we relate the problem to distance constraints, give a number of related properties of distance constraints, and establish the necessary and sufficient conditions for a location vector to be efficient. In Section 4, we provide examples of efficient and non-efficient location vectors. Section 5 is devoted to a further refinement and simplification of one of the necessary and sufficient conditions, namely, "the convex hull property." In Section 6, we provide an algorithm, SEVCA, which con- structs an efficient solution from a given location vector. In Sec- tion 7, we characterize efficient solutions for the analogous problem in the p-dimensional Euclidean space with rectilinear (p = 2) or Tchebychev (p 2) distances. 3.2 Problem Statement We suppose given a finite, undirected tree network, and denote by T an imbedding of the given network. Let V : {v ,...,v } be a set of n distinct vertices of T. We assume existing facility i is located at vertex vi, i E {l,...,n}. For j e {1,...,m}, denote by x. a point to be determined in T as the location of new facility j. We define Tm to be the m-fold Cartesian product of T by itself and define a location vector X in Tm to be the ordered m-tuple (x,,...,x ) with each x e T, j {1,...,m}. Sometimes, we refer to a location vector X in Tm as a point in Tm As in [22], given points x,y e T, we define the line L(x,y) to be the union of all points in the shortest path connecting x and y. In -86- addition, given a finite point set P C T, we define the convex hull H(P) to be the smallest (embedded) subtree of T containing all points in P. We note that for any two points p,p' e P, the line L(p,p') is contained in H(P). We denote by IC the set of pairs (i,j) for which the distance d(xiv ) is of concern. Similarly, IB is the set of pairs (j,k) for which the distance d(x.,xk) is of concern. We remark that it need not be the case that IC includes all possible pairs of new and existing facility indices, nor IB includes all possible pairs of new facility indices. With these definitions, the problem of interest is to "mini- mize" each of the distances specified by (3.2.1); d(x.,v.) (i,j) C IC 1 J C (3.2.1) d(xj,xk) (j,k) I . For X e Tm, we denote by D(X) the vector each of whose components is a distance specified by (3.2.1). The vector is formed by assuming any convenient ordering of the members of IC and IB. The vector- minimization (V-min) problem of interest is V-min{D(X): X e Tm} (3.2.2) With respect to (3.2.2), a location vector Z e Tm is said to dominate a location vector X in Tm if D(Z) < D(X) and D(Z) # D(X). A location vector Z which is not dominated by any other location vector is said to be efficient. An equivalent definition of efficiency is as follows: Z e Tm is efficient if and only if X e Tm and D(X) 5 D(Z) imply D(X) = D(Z). -87- Our main interest is to characterize efficient location vectors and devise an algorithm for constructing efficient location vectors from a given (dominated) location vector. 3.3 Distance Constraints and Characterization of Efficient Points We make extensive use of the results obtained in [32, 33] for distance constraints to establish the necessary and sufficient condi- tions for efficient points. The Distance Constraints (DC) are defined in [32] (independent of the efficiency problem) as follows: Given the sets IC and IB and nonnegative upper bounds cij and bjk, find a point X = (x1,...,x ) in Tm, if it exists, such that d(xi,v.) c.. (i,j) C IC (3.3.1) d(xj,xk) b bjk (j,k) e IB Corresponding to DC, we define Graph BC (GBC) as the undirected graph having nodes E1,...,En, N1,...,N ; for every (j,k) e I there is an arc (Nj,Nk) of length bjk between nodes Nj and Nk; for every (i,j) C IC, there is an arc (N.,E.) of length cij between nodes N. C 1 J ij i and E.. We further assume that the sets IB and IC are such that GBC J B C is connected, as otherwise DC decomposes into independent sets of con- straints which may be analyzed separately. Given a node-path between any two nodes f and f in GBC, we de- P q note the path by P(f ,f ) and denote the length of the path by LP(f ,f ). We define L(f ,f ) to be the length of any shortest path in GBC between P q nodes f and f Subsequently, unless we specify otherwise, it should P q -88- be understood that any path we refer to is a simple path between some two existing facility nodes E and E . P q Results on Distance Constraints The distance constraints are said to be consistent if there exists at least one feasible solution to (3.3.1). The following result is established in [32]. Theorem 3.3.1. The distance constraints are consistent if and only if d(v ,v ) < L(E ,E), 1 p < q n (3.3.2) pq p q The inequalities (3.3.2) are termed the Separation Conditions [32], since each term on the right specifies an upper bound on how separate two existing facility locations can be. Except when stated otherwise, we assume throughout the chapter that the separation condi- tions hold, and thus (equivalently) DC is consistent. We call a path P(E ,E ) between E and E in GBC a tight path if p q p q LP(E ,E ) = d(v ,v ). We note that since we assume DC is consistent, p q p q it necessarily follows if P(E ,E ) is a tight path, that LP(E ,E ) = P p P q L(E ,E ). Any path P(E ,E ) for which LP(E ,E ) > d(v ,v ) is called P q p q pq a slack path. We say that new facility i is in a tight path if there exists at least one tight path containing N.. Every path containing N. is slack 1 1 if there is no tight path which contains N.. The motivation for the above terminology is due to a string graph representation of GBC. This string graph is also useful for obtaining problem insights. When knots representing nodes E and E are pulled as P q P pq -89- If then the string graph is placed upon the tree T, i.e., the strings only lie on arcs of T, a path is tight when it is necessary to pull the string graph tight in order to place the knots representing E and E on v and v respectively, while a path is slack if the string path P q must literally be slack when the two knots are placed to coincide with v and v P q A priori, one might think that the occurrence of a tight path would be rare. However, we shall see that tight paths occur in a quite natural way when the separation conditions are used in the analy- sis of efficient location vectors. Further, the notion of tight paths permits the specification of necessary and sufficient conditions for DC to have a unique solution. We now relate unique locations to tight paths. By definition, new facility i is uniquely located if it has the same location in every feasible solution to DC. Since we later refer to a collection of facilities, which contains possibly both existing and new facilities, being uniquely located, we note that existing facilities are uniquely located by definition. Theorem 3.3.2, which we proved in [33], specifies the necessary and sufficient conditions for a new facility to be uniquely located. Theorem 3.3.2. New facility k is uniquely located if an only if node Nk lies in at least one tight path P(E ,E ). Corollary 3.3.2. Distance constraints have a unique solution if and only if node Nk lies on at least one tight path in GBC for k = l,...,m. We now give an additional property of a tight path we proved in [33]. The property will be used in proving our main result on efficient points. -90- Property 3.3.1. If P(E ,E ) is a tight path in GBC, then p q (i) every facility represented by a node in P(E ,E ) is uniquely located, (ii) the locations of facilities corresponding to nodes in P(E ,E ) occur with the same ordering and spacing on the line L(v ,v ) in p q T as do the corresponding nodes in P(E ,E ). As an illustration of Property 3.3.1, suppose P(E1,E5) is a tight path with nodes E1, N2, N3, E5. Then, the locations v1, x2, x3, v are unique. Furthermore, they occur in the given order on the line L(v1,v5) with d(v1,x2) = c21, d(x2,x3) = b23, d(x3,v5) = c35, where c21, b23, c35 are the lengths of the arcs in the path. This example is illustrated in Figure 3.1. C b C Tight Path SC21 23 35 P(EGE5) v xin T v1 x2 x3 v5 Figure 3.1.. Illustration of Property 3.3.1. We now consider the problem of determining when an arc lies on a tight path. As an arc lies on a tight path if and only if it is not the case that all paths containing the arc are slack, we consider the -91- equivalent problem of determining when an arc lies only on slack paths. The following property, which we proved in [33], characterizes the con- ditions under which an arc in GBC is not contained in any tight path. Property 3.3.2. Let DC be consistent. Let (f.,f.) be any arc in GBC, of positive length e.., whose length is reduced by some positive amount C. Let DC (GBC ) be the distance constraints (graph) obtained from DC(GBC) by replacing e.. by eij C. (a) Evey path containing (f.,f.) in GBC is slack if and only if e can be chosen (with s > 0) so that DC is consistent. (b) Whenever every path containing (f.,f.) is slack, E can be chosen (with e > 0) so that DC is consistent and at least one of the follow- ing is true: (i) at least one path in GBC containing (f.,f.) is tight; (ii) the length of (f.,f.) in GBC can be reduced to zero. 1 C Finally, we will use the following lemma proven in [33]. Lemma 3.3.1. Given points a,b e T, suppose that d(a,b) = a + 3. Then, the inequalities d(x,a) a, d(x,b) B are consistent if and only if they have a unique solution and the inequalities hold as qualities. Necessary and Sufficient Conditions for Efficiency Given a location vector Z, we let U = D(Z) and define the distance constraints of interest by D(X) < U, where the entries in U define the bjk and cij by bjk = d(zj,zk) for (j,k) C I'B and cij = d(zivj) for (i,j) c IC. We use the bjk and cij to define GBC in the customary manner. As before, we may assume GBC is connected, for otherwise the problem of finding efficient location vectors decomposes into -92- independent subproblems. Further, we note that DC is always consistent, as Z is certainly feasible to DC, and hence, by Theorem 3.3.1, the separation conditions are always satisfied. For convenience, for any location vector Z, we denote by A*(Z) the collection of locations of uniquely located facilities whose nodes are adjacent to N. in GBC. We 1 denote by 11[A*(Z)] the convex hull of A*(Z), the imbedding of the i i smallest subtree of T spanning all the elements of A*(Z). 1 With the above definitions we can present a family of equivalent conditions for a location vector Z to be efficient. Theorem 3.3.3. Given a location vector Z used to define DC and GBC, the following are equivalent: (a) Z is efficient; (b) Each N. is in at least one tight path in GBC; I (c) Z is the unique solution to DC; (d) z. E H[A*(Z)] for i = l,...,m. Proof. The equivalence of (b) and (c) is a direct consequence of Theorem 3.3.2 and the fact that Z is always a feasible solution to DC, while (c) clearly implies (a). To show (a) implies (c), suppose Z is not the unique solution to DC. Color every new facility node in GBC which is not contained in any tight path blue. Color all the other (new or existing facility) nodes red. Equivalence of (b) and (c) implies every blue node represents a new facility which is not uniquely located, while every red node represents a (new or existing) facility which is uniquely located. By assumption there is at least one blue node. By connectedness of GBC, there is at least one arc which connects some blue colored node, say, N to some red colored node, say, F Furthermore, arc (N ,F ) has positive q p q |

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-90-
Property 3.3.1. If P(E^,E^) is a tight path in GBC, then (i) every facility represented by a node in P(E^,E^) is uniquely located, (ii) the locations of facilities corresponding to nodes in P(E^,E^) occur with the same ordering and spacing on the line L(v ,v ) in P 9 T as do the corresponding nodes in P(E ,E ). P q As an illustration of Property 3.3.1, suppose P(E^,E^) is a tight path with nodes E^, N2, N^, E^. Then, the locations v^, x^, v^ are unique. Furthermore, they occur in the given order on the line l(v1,v5) with d(v1,x2) = c21, d(x2,x3) = b23, d(x3,v5) = c35, where C21 ^23 C35 are t^ie -*-enSt^ls tbe arcs in the path. This example is illustrated in Figure 3.1. b 23 35 Tight Path p(e15e5) in GBC '21 23 C 35 x 3 Uvpvs) in T Figure 3.1. Illustration of Property 3.3.1. We now consider the problem of determining when an arc lies on a tight path. As an arc lies on a tight path if and only if it is not the case that all paths containing the arc are slack, we consider the -168- 102. P.J. Slater, "Central Paths in a Graph," Research Report SAND 78-0809J, Sandia Laboratories, Albuquerque, New Mexico (1978). 103. P.J. Slater, "One-Point Location of an Area Response Group," Research Report SAND 78-1788, Sandia Laboratories, Albuquerque, New Mexico (1978). 104. M.K. Starr and M. Zeleny, "MCDM-State and Future of the Arts," TIMS Studies in Management Sciences 6, 5-29 (1977). 105. R.W. Swain, "A Parametric Decomposition Approach for the Solu tion of Uncapacitated Location Problems," Dept, of Industrial Engineering, The Ohio State University (1971). 106. J.J. Sylvester, "A Question in the Geometry of Situation," Quarterly Journal of Pure and Applied Mathematics, Vol. 1, 79 (1857). 107. M.R. Teitz and P. Bart, "Heuristic Methods for Estimating the Generalized Vertex Median of a Weighted Graph," Opns. Res.,16, 955-961 (1968). 108. H. Thiriez and S. Zionts, Eds., Multiple Criteria Decision Making, Proc., Jouy-en-Josas, France, 1975. 109. C. Toregas, R. Swain, C. ReVelle, and L. Bergman, "The Location of Emergency Service Facilities," Opns. Res. 19, 1363-1373 (1971). 110. R.E. Wendell, "Efficiency and Solution Approaches to Bi-Objective Mathematical Programs," Working Paper, Rensselaer Polytechnic Institute, Troy, New York. 111. R.E. Wendell and A.P. Hurter, "Optimal Locations on a Network," Trans. Sci. 7, 18-23 (1973). 112. R.E. Wendell and D.N. Lee, "Efficiency in Multiple Objective Optimization Problems," Math. Prog. 12, 406-414 (1977). 113. R.E. Wendell and E.L. Peterson, "Duality in Generalized Location Problems," ORSA Bull. 20, Supp. 2, B-317 (1972). 114. R.E. Wendel, A.P. Hurter, and T.J. Lowe, "Efficient Points in Location Problems," AIIE Transactions 9, 238-246 (1977). 115. G.O. Wesolowsky and R.F. Love, "The Optimal Location of New Facilities Using Rectangular Distances," Opns. Res. 19, 124- 130 (1971). 116. G.O. Wesolowsky and R.F. Love, "A Nonlinear Approximation Method for Solving a Generalized Rectangular Distance Weber Problem," Manag. Sci. 18, 656-663 (1972). -15- This completes the discussion of the 1-center problem. We now concentrate on the p-center problem for p > 2. p-Center problem on a general network. The p-center problem was defined by Hakimi [48]. Subsequently, a number of solution procedures have been suggested. A common characteristic of all these procedures is that they all rely on solving a sequence of covering problems. For completeness, we first define a set covering problem and an r-cover problem. Let A be a matrix of zeros and ones, y a vector of zero-one variables y^. The problem of minimizing J y_^ so that each row of Ay i is greater than or equal to one is called the (minimal) set covering problem. Given the function f(X) = max{w Jl(v^,X): 1 < i < n}, the problem of minimizing |x| so that f(X) < r for some given value of r is called the r-cover problem. Denoting by q(r) the minimum value of the r-cover problem, it can be readily shown that, if q(r) = p for some r, and q(r') > p for any r' < r, then r is the p-radius and any X which solves the r-cover problem is an absolute p-center. In what follows, we concentrate on the absolute p-center problem on a general network. Minieka [87] considered the unweighted case on a general network and showed that the problem can be reduced to a computationally finite one. Minieka identifies a finite point set P* such that there exists an absolute p-center contained in P = P' U V. A point x on some edge is a member of P' if and only if x is the unique point on its edge such that d(v.,x) = d(x,v.) for some two distinct vertices v, and v.. i J i J Based on this result, Minieka suggested a rudimentary algorithm that -60- Whether K refers to a subset of V or a clique of G, we prefer to call K a clique as long as it is clear from the context what K refers to. As an example of the nonlinear p-center problem, suppose that the function associated with node v is f^ (y) = (y + h^)8 for y e [0j ] where w^ h^, and 0 are given parameters. Appropriate restrictions are placed on the parameters to ensure that the f^ are strictly in creasing on [0,6j]. We note that the linear weighted p-center problem is a special case of this problem generated by choosing 0 = 1, h^ =0, and Wj > 0 for all j. For the given form of f the following are readily verified: f ^(r) = (r/w.)1^6 h., r e [f.(0), f.(6.)] , J 1 3 J 3 3 fT1(r) + fT1(r) = r1/8[(l/w )1/6 + (1/w )1/8] (h + h ) , 1 J 1 J 1 J r e [a.., b..] , 13 13 w .w. i 3 (fi + fj ) (y) r ^ ]./ j_ l/e^ 0 (y + h + h ) , [w.x'" + w.x/v]" 3 y e [L.., U..] Then, using the characterization of 3 as given in Lemma 2.2.1, we have 3.. = 13 .. d.. if L Â£d(v. ,v.) 13 13 3 i 3 (2.2.4) max[fi(0), f.(0)] if L^. > div^v.) where -166- 73. P. Kolesar, W. Walker, and J. Hausner, "Determining the Relation Between Fire Engine Travel Times and Travel Distances in New York City," Opns. Res. 23, 614-627 (1975). 74. T.C. Koopmans, Activity Analysis of Production and Allocation, Cowles Commission for Research in Economics, Monograph No. 13, John Wiley and Sons, New York, 1951. 75. H.W. Kuhn, "On a Pair of Dual Nonlinear Problems," Nonlinear Programming, Chapter 3, J. Abadie, editor, John Wiley and Sons, New York, 1967. 76. H.W. Kuhn and A.W. Tucker, "Nonlinear Programming," Proc. of the 2nd Berkeley Symposium on Math., Stat. and Probability, Univ. of California Press, Berkeley, California (1951). 77. A.H. Land and A.G. Doig, "An Automatic Method for Solving Dis crete Programming Problems," Econometrica 28, 497-520 (1960). 78. A.C. Lea, "Location-Allocation Systems: An Annotated Bibliography," Discussion Paper No. 13, Dept, of Geography, University of Toronto, Toronto, Canada (1973). 79. A.C. Lea, "A Model Taxonomy and a View of Research Frontiers in Normative Locational Modelling," Paper presented to International Symposium on Locational Decisions, Banff, Alberta (1978). 80. J. Levy, "An Extended Theorem for Location on a Network," Opnl. Res. Q. 18, 433-442 (1967). 81.C.C. Lin, "On Vertex Addends in Minimax Location Problems," Trans. Sci. 9, 165-168 (1975). 82.T.J. Lowe, "Efficient Solutions in Multiobjective Tree Network Location Problems," Trans. Sci. 12, 298-316 (1979). 83. L.F. McGinnis and J.A. White, "A Single Facility Rectilinear Location Problem with Multiple Criteria," Trans. Sci. 12, 217-231 (1978). 84. F.E. Maranzana, "On the Location of Supply Points to Minimize Transport Costs," Opnl. Res. Q. 15, 261-270 (1964). 85. D.W. Matula and R. Kolde, "Efficient Multi-Median Location in Acyclic Networks," ORSA/TIMS Bulletin, No. 2 (1976). 86. A. Meir and J.W. Moon, "Relations Between Packing and Covering Numbers of a Tree," Pac. J. Math. 61, 225-233 (1975). 87.E. Minieka, "The m-Center Problem," SIAM Review 12, 138-139 (1970). -106- We can readily use induction to obtain the following generaliza tion of Lemma 3.5.3. Lemma 3.5.4. Given r points p ,... ,p Â£ T with r > 4, if p e L(P.ji_iP1+i) for 2 ^ i < r-1, and if p 4 p+1 for 2 < i < r-2, r-1 then d(p1,pr) = Â£ i=l We are now ready to prove the sufficient conditions for irredu cible location vectors. We remark that the arc lengths of GBC are defined by the entries of D(Z), so that if N,N, *,E is a sub- Vi) OO p path P(N^,Ep) connecting to Ep, then the length of the subpath is given by LP(N^,Ep) = d(z^^.z^) + ... + d(z^,v ). Lemma 3.5.5 (Sufficiency). Suppose Z is irreducible. If, for every j e {1,... ,m}, Zj e H[A^.(Z)], then every z^ is uniquely located. Furthermore Z is efficient. Proof. For notational brevity, let S = A (Z). J 3 Choose any j in {l,...,m}. Either N^. is adjacent to exactly one node or more than one node. In the former case, S^ is a singleton, say, {y}. Since Zj e H[Sj], Zj = y. By irreducibility of Z, y is an existing facility location. Hence z_. is in the convex hull of uniquely located facili ties so that Theorem 3.3.3 implies z^ is uniquely located in this case. For the other case, N. is adjacent to at least two nodes in GBC. J The hypothesis z_. e H[S^] implies there exist p,q e S_. with Â£ L(pq) (3.5.1) If p and q are both existing facility locations, Theorem 3.3.3 implies Zj is uniquely located. Hence, suppose, without loss of generality, that q is an existing facility location, but p is a new facility -8- Slater [102]. A large portion of the literature deals with point- location problems. Point-location problems may be classified into three categories: single objective problems, multi-objective problems, and a body of results of a general and unifying nature. In the remainder of this section we give a detailed discussion of the problems outlined in the family tree. Point-Location Problems Here, we consider a number of problems that involve locating new facilities at points on a network. The general format of the dis cussion is as follows: For each problem type, we first define a kernel problem. Then, we discuss the related literature on the kernel problem, as well as several special cases and extensions of it. We point out relations between different problem types, whenever such relations exist. The p-center problem Let N be a network with a vertex set V = {v,,...,v } and an edge 1 n set E. Denote by X a finite set of points, each of which is in N. Let I be the set of integers 1 through n. For each vertex v., iel, define the distance D(v^,X) between vertex v and the point set X by D(v^,X) = min[d(v^,x) : xeX]. With this definition, D(v_^,X) is speci fied by a nearest point in X to vJ. Let w. and a. be two given numbers i 11 associated with vertex v^, iel. We call wi a weight and aan addend. We assume that each w^ is nonnegative and at least one w_^ is positive. For any finite point set X CD, define the function f (X) by -142- To show e(z ) > tCz.), let a.,a. be arcs in A for which 1 i i J G t(z,) = x..(z,). Suppose a. = (N ,E ) and a. = (N ,E ). By step 1) 1 lj 1 l s p 3 t q J of E-FRONT we have VZP d(vv) p q (l/w + l/w ) sp tq m st m (4.4.7) st But m is the length of a shortest path in Gg connecting Ng to N . Let (N ,N, ...,N,.,N ) be such a shortest path in G. Let P(E ,E ) be s k f t r B p q the path (E ,N ,N, ,...,NÂ£,N,E ). Define z = (z,,z0) with z = e(z.). p s k f t q 12 2 1 Since z is feasible to P DC is consistent so that for the path 2 z^ z v P(E ,E ) identified above we have P q LP(E >E ) > d(v ,v ) . z p q p q (4.4.8) But LzP(E^,E^) = z^(l/wSp) + z2mst + Zl^^Wtq^ aS Pat^ consists of the arcs (E ,N ) (N ,N. ),...,(N,,N),(N ,E ). It follows then from ps sk. it tq (4.4.8) that z.(l/w + 1A* ) + z0m ^ > d(v ,v ), or, equivalently, 1 sp tq 2 st p q Z2 d(v ,v ) JB SL (1/w + l/w ) sp tq m st m (4.4.9) st But the right side of (4.4.9) is = x(z^) while z^ is e(z^) by definition, hence, e(z^) > x(z^). The inequalities e(z^) < t(z^) and e(z^) > t(z^) imply e(z^) = i(z^) for every e [a,b]. Hence, Z* = {(z^tCz^)): a < z^ < b), com pleting the proof. 2 2 We now show that the computational order of E-FRONT is 0(m (m + n )) The algorithm constructs Z* by identifying no more than r(r l)/2 linear functions. To identify the linear functions one must first -167- 88. E. Mlnieka, "The Centers and Medians of a Graph," Opns. Res. 25, 641-650 (1977). 89. P.B. Mirchandani and A.R. Odoni, "Locating New Passenger Facili ties on a Transportation Network," Working Paper, Electrical and Systems Engineering Dept., Rensselaer Polytechnic Inst., Troy, New York (1977). 90. P.B. Mirchandani and A.R. Odoni, "Locations of Medians on Sto chastic Networks," Working Paper OR-065-77, Operations Research Center, M.I.T., Cambridge, Massachusetts (1977). 91. S.C. Narula, V.I. Ogbu, and H.M. Samuelson, "An Algorithm for the p-Median Problem," Opns. Res. 25, 709-712 (1977). 92. G.L. Nemhauser and W.L. Sheu, "Easy and Hard Bottleneck Location Problems," Technical Report 386, School of Operations Research and Industrial Engineering, Cornell University, Ithaca, New York (1979). 93. J.C. Picard and H.D. Ratliff, "A Cut Approach to the Rectilinear Distance Location Problem," Opns. Res. 26, 422-434 (1978). 94. A.B. Pritsker and P.M. Ghare, "Locating New Facilities with Respect to Existing Facilities," AIIE Transactions 12, 290-297 (1970). 95. C. ReVelle and R. Swain, "Central Facilities Location," Geog. Anal. 2, 30-42 (1970). 96. C. ReVelle, D. Marks, and J.C. Liebman, "An Analysis of Private and Public Sector Location Models," Manag. Sci. 16, 692-707 (1970). 97. D.J. Rose, R.E. Tarjan, and G.S. Lueker, "Algorithmic Aspects of Vertex Elimination in Graphs," SIAM J. Comput. 5, 266-283 (1976). 98. A. Rosenthal, J. Pino, and M. Coulter, "A Generalized Algorithm for Centrality Problems on Trees," Working Paper, Dept, of Computer and Communication Sciences, Univ. of Michigan, Ann Arbor, Michigan (1978). 99. B. Roy,"Problems and Methods with Multiple Objective Functions," Math. Prog. 1, 239-266 (1971). 100.D.R. Shier, "A Min-Max Theorem for p-Center Problems on a Tree," Trans. Sci. 11, 243-252 (1977). 101.S. Singer, "Multi-Centers and Multi-Medians of a Graph with Application to Optimal Warehouse Location," Unpublished Paper, Dunlap and Associates, Inc., Darien, Conn. (1968). -45- The problem of interest is to find all efficient points with respect to f(x). Halpem [52] is the first to consider this problem. Halpern formulated the problem in a slightly different manner by considering a convex combination of m(x) and c(x). For any fixed X, 0 < X 1, define f(X,x) and f*(X) by f(X,x) = Xm(x) + (1 X) c(x) for xeN } f*(X) = min[f(X,x): xeN] (1.3.4) In Halpem's terminology, the function f(X,x) is called a cent-dian function and any point x* = x*(X) that solves (1.3.4) is called a cent-dian point. In [52] Halpem considered this problem on a tree network with weights h all equal to unity. Defining x^ and x^ to be the (vertex) median and the absolute center of T, respectively, Halpem proved that for any given X, the cent-dian x*(X) is located at either x^ or on one of the vertices located on the path P(x ,x ). This theorem pro- m c vides the basis for a simple and efficient algorithm to locate the cent-dian by inspecting the vertices on P(x ,x ). Further, Halpern m c showed that, if the absolute center x is known, then the cent-dian c can be found by determining the median of a tree T' that is identical to T except that T' contains an additional vertex v = x with the n+1 c associated weight w = X 1. n+1 Handler [56] formulated the same problem on a tree network in a slightly different manner by using the median function as a constraint. In Handler's formulation one is interested in solving the problem -23- define an intersection graph for a fixed value of r as follows: has nodes corresponding to demand points v^,...,v^. Two nodes of G^ are connected by an arc if the corresponding demand points can be jointly covered by a (single) common center within a radius of r. Once Gr is formed, finding a "clique cover" of G^ solves the r-cover problem. A clique cover of G^ is a minimum number of cliques in G^ such that every node is in at least one clique. The solution to the clique cover problem in G^_ determines a solution to the r-cover problem. The procedure is repeated for different values of r until a smallest value of r is found for which the clique cover solution generates at most p cliques. The computational complexity of the procedure is polynomial. In particular, the computational effort for finding the minimal clique cover of G^ is polynomial because G^ satisfies the property that any circuit in G^_ with at least four arcs contains a chord (i.e., an arc which connects two nodes of the circuit and is not an element of the circuit). For chordal graphs, algorithms of linear order have been developed (see [39], [97]) for finding a minimal clique cover. This completes the discussion of the p-center problem. The p-median problem The difference between the p-center and the p-median problem is that the objective criterion is changed from minimax to minisum. More specifically, define the function f (X) for any finite point set XCN by f (X) = l w D (v ,X) . iel 1 -119- of the separation conditions for the distance constraints to be con sistent. Furthermore, this equivalence is the only property that is used. Hence, the theorem holds for any distance for which it is true that the distance constraints are consistent if and only if the separa tion conditions hold. In Theorem 3.7.1 one does not have to worry about arcs with zero lengths. To see this, partition D(Z) into subvectors D^(Z) and (Z) so that D (Z) contains the zero entries, while (Z) contains the positive entries. Clearly, in every feasible solution X to the con straints D(X) < D(Z), the constraints corresponding to entries of Dq(Z) will hold at equality. Hence the only way Z can be dominated is by having (X) < (Z) and (X) f D^(Z). Thus, one needs to test only those arcs with positive lengths in GBC to determine whether or not Z is efficient. For this reason, Theorem 3.7.1 is applicable to both irreducible location vectors and reducible location vectors. 2 We remark that for rectilinear distances in R and Tchebychev 1c distances in R k > 2, the equivalences stated in Theorem 3.3.3 need not hold. An example of such a case is given in Figure 3.8 for 2 Tchebychev distances in R With reference to Figure 3.8, it is direct to verify that both and are contained in the tight path 0^2 N^, E^) Clearly (z^,z^) is not efficient, as, z^ can be moved to (2,2) and z^ can be moved to (3,1) thereby reducing the dis tance between new facility 1 and existing facility 1 without increasing any of the other distances. The resulting location vector is shown in Figure 3.9. It Is direct to verify that every positive arc in GBC of Figure 3.9 is contained in a tight path and hence the location vector is efficient. -136 We shall first state a theorem due to Wendell [110] which gives a global characterization of the efficient frontier. Then we will exploit the result of the theorem to construct Z*. m /s Let a be the minimum value of f^ on T b be the minimum value of f^ on Tm, and b be the minimum value of f^ over all minima to f^. The A values a, b, and b are displayed in Figure 4.3 for an arbitrary bi objective problem with convex objectives. For each z^ e [a,b] define the function e^) to be the minimum value of the problem P defined below Z1 e(Zl) = minf2(X): f (X) < z X e Tm) . Wendell [110] showed that whenever f^,f2 are lower semicontinuous convex functions defined over a nonempty convex compact set S, the efficient frontier is the set {(z^,e(z^)): a < z^ < b}. Wendells theorem is applicable to the bi-objective m-center problem, as Tm is convex, compact, and nonempty, and f^ and f2 are continuous convex functions (see [22]) over Tm. For an arbitrary choice of z^, the value of e(z^) is marked in Figure 4.3. The computation of a, b, and b presents no difficulties and will be given subsequently. Using the definitions of f^ and an equivalent definition of e(Zj) is as follows: e(zp = min Z2 s. t. d(x,Vj) < z^w (i,j) e Ic d(Xj,xk) z2/vjk e IB where z^ is understood to be fixed to any value in [a,b] (4.4.1) The constraints -117- 1c points x,y e R with x = (x^,...,x^) and y = (y^,...,y^), the rectilinear distance between x and y is given by |x^ y^| + ... + |x^ + y^J, while the Tchebychev distance between x and y is given by max{|x^ y^|,..., |x y |}, where the symbol |*| denotes the absolute value sign. K. K. It is known that (proven in [32]) the distance constraints with 2 k rectilinear distances in R or with Tchebychev distances in R k 2, are consistent if and only if the separation conditions hold. Based on this result, we characterize efficient location vectors for the analogous vector-minimization problem which uses the recti linear or Tchebychev distances. Theorem 3.7.1. Let D(Z) be the vector of objectives with all entries 2 of D(Z) either the rectilinear distances in R or the Tchebychev distances in R k > 2, as specified in (3.2.1). Let GBC be the graph with arc lengths defined by D(Z). The following are equivalent: (a) Z is efficient; (b) Every arc in GBC of positive length is in a tight path. Proof. To show (a) implies (b) suppose that Z is efficient. Let DC be the distance constraints D(X) < U = D (Z). Since Z is a feasible solution to DC, DC is consistent and hence the separation conditions hold. Assume that there exists at least one arc in GBC with positive length which is not in any tight path. Let (f ,f ) be such an arc with P q length e Since (f ,f ) is not in any tight path and since the pq p q separation conditions hold, every path which contains (f ,f ) is slack. P q Hence, for any path P(E.,E.) containing (f ,f ) we have 1 J p q LP(E ,E ) d (v ,v ) > 0 (3.7.1) J J Define e' to be the minimum of the left side of (3.7.1) over all paths I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Richard L. Francis, Chairman Professor of Industrial and Systems Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. X/ Donald U. Hearn Associate Professor of Industrial and Systems Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Ra llph W'. Swa in Associate Professor of Industrial and Systems Engineering -li the network is a demand point (as opposed only to vertices). The weight of each point is unity. The objective to be minimized over all xeN is defined by f(x) = max [d(y,x): yeN]. Both authors showed that the problem can be reduced to a computationally finite one and pro posed a solution procedure which is very similar to Hakimi's. A probabilistic version of the 1-center problem was considered by Frank [34, 35] and a number of bounds were obtained on the expected value of the 1-radius. For the unweighted case, Singer [101] proved that there exists a "critical" path, not necessarily a shortest path, connecting two cri tical vertices such that an absolute center of the network is at the midpoint of this path. 1-Center problem on a tree network. We now concentrate on ab solute centers of tree networks. Goldman [44] solved the unweighted case in the presence of addends. Goldman's algorithm is based on the repeated application of a "trichotomy theorem" that either determines the edge on which the absolute center lies, or reduces the search to one of the subtrees obtained by removing all interior points of that edge. Halfin [51] refined Goldman's algorithm to make it simpler and computationally more efficient. Halfin's algorithm finds a vertex center first, and determines the absolute center by examining all vertices adjacent to the vertex center. For the unweighted case with no addends, Handler [55] presents an especially elegant algorithm. Handler's method finds a longest path of the tree and locates the absolute center at the midpoint of the path. To find a longest path, Handler chooses an arbitrary vertex v finds a farthest vertex v from v., and then finds a farthest i si -131- corresponding to E and E will be violated, as, A _. (E ,E ) < P Q z A p q L ,P(E,E) DC is inconsistent. Lemma 4.3.1 then implies Y is efficient. z-A We remark that Theorem 4.3.1 considers only those tight paths which pass through Ag. The reason is as follows: Any path in GBC^ passes through A so that if there exists a tight path which contains an arc in Ag, then the same path necessarily contains an arc in A^,. However, an arbitrarily chosen path need not pass through Ag. For this reason, the assumption that there exists at least one arc in A^, which is contained in a tight path does not imply that at least one arc in Ag is contained in a tight path. Hence, if a location vector Y is efficient then there is at least one arc in A^ which is contained in a tight path while the reverse implication does not hold. Further, we remark that the proof of Lemma 4.3.2 is based on the necessity and sufficiency of the separation conditions. Hence, Lemma 4.3.2 is applicable to tree networks as well as the analogous problems with rectilinear distances on the plane, or, the Tchebychev distances in the k-dimensional Euclidean space with k > 2. Further, Theorem 4.3.1 uses Lemmas 4.3.1 and 4.3.2 for its proof. Hence, the theorem is applicable to tree networks as well as rectilinear distances on the lc plane and the Tchebychev distances in R k > 2. At this point we give an example of a non-efficient and an efficient location vector. In Figure 4.1 the tree network is shown along with the distance matrix and the weights w.. and vfor 13 Jk (i,j) e Ic = {(1,1),(1,2),(2,3),(2,4),(3,4),(3,5)} and (j,k) e Ifi - {(1,2),(1,3),(2,3)} for the example bi-objective m-center problem. In Figure 4.2a) we give an example of a dominated location vector X. -141- facility node Ng. Let X be any location vector for which f^(X) = a. Hence, w d(x ,v ) < a and w d(x ,v ) < a. But a < z. and the length sp s p sq s q 1 of P(E ,E ) is z.(1/w + 1/w ) so that we have, upon using the tri- p q 1 sp sq angle inequality, L P(E ,E ) = z (1/w + 1/w ) > a(l/w + 1/w ) > & n 3 z p q 1 sp sq sp sq d(x ,v ) + d(x ,v ) > d(v ,v ). Thus, for any nd-path P(E ,E ) which s p s q p q 3 1 p q does not pass through A^, we have L P (E ,E ) > d(v v ) z p q p q (4.4.3) For the other case, P(E ,E ) passes through A,, so that its length is P q B given by L P(E ,E ) = z,WP(E ,E ) + zVP(E ,E ). Since the path is an zpq 1 p q 2 p q nd-path it passes through exactly two arcs in A^, say, (E^,Ns) and (N ,E ) with s t. Thus, WP(E ,E ) = 1/w + 1/w while t q p q sp tq VP(E ,E ) ra by the definitions of m and VP(E ,E ). It follows p q st st p q that L P(E ,E ) > z.(l/w + 1/w ) + z m zpq 1 sp tq 2 st Due to steps 1) and 2) of E-FRONT, we have (4.4.4) z2 = T(zl) - L_Â£_ z m ^ Z1 st (1/w + 1/w ) Sp tq ra st (4.4.5) Using (4.4.4) and (4.4.5), for any nd-path which passes through A^ we have IP(E E ) > d(v ,v ) (4.4.6) z p q p q From (4.4.3) and (4.4.6), L P(E ,E ) > d(v ,v ) for every nd-path in zpq-pq GBCÂ¡z so that Lemma 4.4.1 implies the separation conditions on GBCz hold. Hence, z is feasible to P and thus e(z.) x(z.) = z. z. 112 -40- The problem differs from the p-median problem in two respects: (i) the distance between any vertex and any new facility may be of concern as opposed only to the distance between a vertex and the near est new facility to it; (ii) certain distances between new facilities are of concern as opposed to the absence of interactions between new facilities in the p-median problem. For the case of a single new facility, the two problems are identical. Planar cases of the problem using rectilinear or Euclidean dis tances have received considerable attention and efficient solution procedures have been developed. A thorough discussion of these prob lems is given in the book by Francis and White [31]. Other references on planar problems are Cabot, Francis, and Stary [6], Bindschedler and Moore [3], Francis [27], Eyster, White, and Wierville [26], Pritsker and Ghare [94], Wesolowsky and Love [115, 116], and Picard and Ratliff [93]. The problem on a network is defined by Dearing, Francis, and Lowe [22] in the presence of distance constraints. It was established in [22] that the problem is a convex optimization problem for all data choices if and only if the network is a tree. For the case of a general network, it is known that there exists an optimal solution on the vertices of N. This result and certain generalizations of it have been given by Goldman [41 ], Levy [80], Hakimiand Maheshwari [49], and Wendell and Hurter [ill]. These references are already discussed under the p-median problem. The problem was shown to be NP-hard by Kolen [72 ] on a general network, and no solution procedures have been developed yet. -151- Let be m-collections of centers with |x_^| = p_^ for 1 < i m, where each p^ is a given positive integer. Let V^,...,Vn be n collections of existing facility locations. The elements of X^ are x^,...^1 with each x^ e T. The elements of V. are 1 p 2 i 1 n i X X with each v. a vertex of T. Let X = {X,,...,X }. For any two finite J 1 m 7 subsets P and Q of T let D(P,Q) = min[d(p,q): p e P, q e Q]. Define the function f by f(X) = max{max{w D(X ,V ): (i,j) e I~} , 1J 1 J V- max{Vj^D(Xj >X^): (j,k) e Ig}} . The Generalized Multi-Center Problem (GMCP) is as follows: min[f(X): |x Â¡ = p^, X_^ C T for 1 < i < m] An equivalent statement of GMCP in terms of distance constraints is as follows: mm s. t. D(X.,V.) z/w.. i J 1J j'V s z/vjk X. p. 1 X 1 1 (ij) e Ic (j,k) e IB 1 i ^ m For the case with m = 1 and each = {v..}, GMCP specializes to the p-center problem. For the case with each p^ = 1 and each = {v^}, GMCP specializes to the m-center problem with mutual communication. We pose the following questions for future research. Ql. What special cases of GMCP are tractable? Some of the special cases are obtained by taking each weight unity, or, -120- 4 3 2 1 0 v, v. Z2 a) Facility Locations b) Graph GBC L(E1SE2) = 4 L(El5E3) = 5 L(ErE4) = 6 L(E2,E3) = 3 L(E2>E4) = 4 L(E3,E4) = 3 > 2 = d(v1}v2) > 4 = d(v1#v3) > 5 = d(v1,v4) 3 = d(v2>V3) > 3 = d(v2,v4> = 3 = d(v3,v4) c) Separation Conditions Figure 3.8. Example of a Dominated Vector with Tchebychev Distances in -116- X Â£ Tm whose irreducible representation is X* is efficient and satisfies D (X) < D(Z) (3.6. 1) where Z is the given vector in Tm to which SEVCA is applied. Proof. Due to the Reduction routine it is evident that X*, the loca tion vector at the termination of SEVCA, is irreducible. Let K* be the list of composite indices at the termination. The Termination Test implies every member of K* is scanned. But a composite index can be labeled scanned only in either step 6) or step 7). In either case, we have x* Â£ H[Ap(X*)] for every P e K*. Property 3.5.2 then implies X* is efficient since every component of X* is in the convex hull associated with it, and X* is irreducible. To show (3.6.1), let x^ = x* for every i e P with P e K*. Thus, X = (x^,...,xm) is the vector in Tm whose irreducible representation is X*. Clearly, the entries of D(X) are either zeros or the arc lengths of the reduced graph, say, GBC* at termination. Since the arc lengths of GBC either remain the same or decrease by a positive amount (in step 7)) from one iteration to the next, it follows that D(X) < D(Z). The assertion that X is efficient follows immediately from Property 3.5.3 and the fact that x* e H[Ap(X*)] for every P z K*. 3.7 Efficiency for the Case of Rectilinear or Tchebychev Distances In this section we consider the analogous vector-minimization k problem in the k-dimensional Euclidean space, denoted by R for the case of rectilinear (k = 2) or Tchebychev (k > 2) distances. Given -20- and n columns, so that an entry a., is 1 if vertex v. is within a ij J distance r of v and 0 otherwise. If one solves a set covering i problem using the matrix A, the variables whose values are 1 in an optimal solution determine a feasible solution to the vertex restricted r-cover problem. The set covering problem is solved by relaxing the integrality constraints. In the case of non-integer termination, a single cut produced an integer solution in a large proportion of the cases. Their computational experience indicates that non-integer termination seldom occurs. p-Center problem on tree networks and duality. In what follows, we concentrate on the p-center problem on tree networks. First, we define the "continuous" p-center problem. In the continuous p-center problem, each point in T is a demand point as opposed only to vertices. Weights are absent (or unity). For any XC T, f is defined by f(X) = max{D(y,X): yeT} and the continuous p-center problem is to find an X*d T such that rp = f(X*) = min[f(X): |x| = p, X C T] . Minieka [88] considered the continuous p-center problem on a general network and showed that it can be reduced to a computationally finite one. Shier [100'] considered the continuous p-center problem on a tree network and defined a dual "dispersion" problem. The dispersion problem is to find p+1 points on T the nearest two of which are as far apart as possible. More explicitly, let U be any finite point set with |U| = p+1 and define h(U) by h(U) = mindu^jUj): 1 < i < j < p+l} . -146- Figure 4.4. Data for Example -123- [32]. At the end of the section we provide an example problem. Section 4 is devoted to the development of a procedure, E-FRONT, for constructing the "efficient frontier." We prove the correctness 2 2 of the procedure and show that it is of 0(m (m + n )), where m and n are, respectively, the number of new and existing facilities. The section ends with an example application of the procedure. 4.2 Problem Statement, Notation, and Definitions Let T be an imbedding of a finite undirected tree network with existing facilities located at distinct vertex locations v^,...,vn. It is of interest to locate m new facilities at points x^,...,x in T under two objective criteria to be defined below. We suppose given positive weights w and v k and denote by 1^ and Ig the nonempty sets of pairs (i,j) and (j,k), respectively, for which the weighted distances w..d(x.,v.) and v...d(x.,x, ) are of concern. We remark that it need 13 i 3 jk 3 k not be the case that 1^, includes all possible pairs of new and existing facility indices nor Ig includes all possible pairs of new facility indices. For each location vector X = (x^,...,xm) in Tm we define two objective functions f^ and f^ by f ^X) = max(w_d(xi,Vj): (i,j) e Ic> , f2(X) = max{v^kd(x^. ,xk): (j,k) e Ig} , and we let f(X) = (f (X), f2(X)). (4.2.1) The Bi-objective m-Center Problem (with Mutual Communication) is as follows: V min{f(X): X e Tm} . (4.2.2) -114- An example application of SEVCA is given in Figure 3.7. For every iteration, GBC and the current location vector is given. For iterations 6 and 7, the location vectors at the end of these iterations are shown separately. For the other iterations, the location vector does not change. Iterations 1 and 5 perform the Reduction routine. For the other iterations, the node chosen in that iteration is the one inci dent to every thickly-drawn arc. The associated convex hull is shown by cross-hatched lines in the tree network. For any iteration the circular-shaped new facility nodes of GBC are the unscanned nodes, while the rectangular-shaped new facility nodes are the ones which have been scanned prior to the given iteration or during that itera tion. For any iteration, the numbers on the arcs of GBC show the arc lengths at the beginning of that iteration. If the arc lengths change during that iteration, the new arc lengths are indicated by the numbers in parentheses. By one iteration of SEVCA, we shall mean the execution of step 1) through the last step. The last step of any given iteration is either step 3), step 4), step 6), or step 7). Define, for i = 3, 4, 6, 7, t^ to be the total number of iterations which used step i) as the last step. Clearly, t^ = 1. Since any given iteration uses only one of these steps as the last step, the total number of iterations, denoted by t, will be given by t = t^ + tg + t^ + 1. We want to show that (k) t 3m. For convenience, denote by K the list of composite indices at the first of iteration k. Property 3.6.1. The algorithm SEVCA terminates in at most 3m-l iterations. -139- Lemma 4.4.1. Let DC be the distance constraints specified in (3.3.1) and let GBC be the associated graph. The separation conditions on GBC hold if and only if for every nd-path P(E ,E ), LP(E ,E ) > d(v ,v ). pq p q p q Proof. Suppose the separation conditions hold. Choose any nd-path P(E ,E ). We have LP(E ,E ) > L(E ,E ) > d(v ,v ) as the length of p q p q p q p q P(Ep,E^) is at least as large as the shortest path length between E^ and E . q Suppose for any nd-path P(E^,E^) we have LP(E^,E^) > d(Vp,v^). Choose any two existing facility nodes, say, Eg and E^, with 1 s < t < n. Let P(Eg,Et) be any shortest path connecting Eg and E^. If P(E ,E ) is an nd-path then clearly L(E ,E ) = LP(E ,E ) > d(v ,v ) so that the separation condition for Eg and Efc is satisfied in this case. Consider the case when P(Eg,Et) is a d-path. Decompose P(Eg,Et) into its nd-paths, say, P(E ,E,P(E. .,E ). Hence, S^IJ (V) t LP(Eg,E^) > d(vg,v^^) ,... ,LP(E^rj ,Efc) > d(v(r)vt)> as the Paths are nd-paths. Further, the length of P(Eg,Et) is the sum of the lengths of P(Eg,E^j) ,... ,P(E^ jE^). Hence, upon using the triangle inequality, we have d(v ,v ) d(v ,vnv) + ... +d(v. .,v ) < St s v. X) ) t LP (E ,E...) + ... + LP(E. ,E ) = LP(E ,E ) = L (E ,E ) so that the s (1) (r) t s t s t separation condition for E and E is satisfied for this case. Since s t the choice of Eg and Efc is arbitrary, the proof is complete. We are now ready to present the procedure for constructing the efficient frontier. We define G^ to be the graph with nodes N^,...,Nm and the arc set A^. To every arc (N^,N^) of G^ we assign the length 1/v For 1 < s < t m we denote by mgt the length of a shortest path connecting the nodes Ng and Nfc in GThe computation of 3 mgt_, 1 < s < t < m, can be achieved in 0(m ) operations by using known algorithms (see Dreyfus [23]). -152- taking p = for each i, or taking Vj = {v^} for each j. Q2. Can we use (or modify) COVER and/or the Sequential Location Procedure of [32] to solve either of the related covering problems de fined by min p, + ... + p rl rm D(X,V ) < z/w.. ij (i,j) e Ic D(Xj,Xk) - 2/vjk . (j >k) e IB or, min max{p ^ ,... p } s. t. D(X,V ) < z/w (i,j) e I z/vjk (j,k) e I C B where z is fixed? Q3. Are the separation conditions of direct use for determining the consistency of the distance constraints of GMCP? Q4. Can we extend the duality results of Chapter 2 to GMCP? Q5. Is there a dual to either of the related covering problems? Q6. What kind of applications may GMCP find? Q7. Can the search for the minimum objective value be confined to a finite set of numbers? We remark that the minimum objective value of the m-center problem with mutual communication is the maximum of a finite number of ratios with the numerators distances between existing facility locations while the denominators are sums of reciprocal weights which correspond to -153- shortest path lengths in a related graph. Hence, there appears to be hope for extending the duality results of Chapter 2 to GMCP. 5.3 The t-Objective m-Center Problem: Steps Towards a Unified Theory Here we define a location problem which involves t minimax type objectives. Special cases of the problem are the m-center problem with mutual communication, the vector-minimization problem of Chapter 3, and the bi-objective m-center problem of Chapter 4. We give a theorem which unifies the independent results for each of these problems. An outline of the proof of the theorem is also provided. Given sets I and I let kn = {(N.,E.): (ij) e I0} and C d G 1 j G Ag = {(N^jN^): (j,k) e 1^}. On defining A = A^ U A^, we suppose given t nonempty, mutually disjoint, exhaustive subsets of A, enumerated as A, ,...,A Associated with A 1 < r < t, the rth objective f is I t r J r defined by f (X) = max[max{w. ,d(x. ,v.): (N.,E.) eA ) A.} , rv iJ i j i j r C max{vjkd(Xj,xk) : Â£ Ar /I A^}] where, by convention, the value of either of the inner maximizations is understood to be zero if the maximum is taken over an empty set. Letting f(X) = (f^(X),...,f (X)), the t-Objective m-Center Problem is as follows: V min[f(X): X e Tm] . For the case t = 1 and A^ = A^ U A^, the problem specializes to the m-center problem with mutual communication. For the case with each A^ -105- Proof. Since and are adjacent, is in A^(Z). By Lemma 3.5.2, we have H[A2(Z)] = U{L(z^,y): y e A^CZ)}. Since z2 e H[A^(Z)], for at least one facility location y in A^(Z), z^ e L(z^,y). Also z^ f y, for otherwise, z^ = y and z^ e L(z^,y) imply z^ = z2, contra~ dieting the irreducibility of Z. Hence, a) is established. Part b) follows immediately from a) and the irreducibility of Z. We remark that the irreducibility assumption cannot be relaxed in Lemma 3.5.2, for otherwise we may have z^ = = y. Figure 3.5 illustrates such a case. We will subsequently use Lemma 3.5.2 repeatedly to identify a sequence of locations z z z. v such that they all lie (1) (2.) (r)5 p y in the line L(z^j,v ) in the given order. The corresponding sequence of nodes N N. ,E in GBC will form a subpath connecting N.,. (1) (r) p v B (1) to with the length of that subpath equal to d(z^^,v^). By the same token, we will find another node E in GBC with the subpath q connecting E^ to having length d(v^,z^). Then, we will show that the two subpaths when connected at form a tight path which contains N (1)' First we give the following result given in [82]. Lemma 3.5,3. Given four points p^ ,p^ jP^jP^ e T, if p2 e LPj^) P3 e L(p2,p4) and p2 f p3> then d(Pl>p4) = d(p ,p ). i=l -39- auxiliary graph G is formed with nodes N ,...,Nm,E^,...,E Graph G contains arcs (N^,E^) with lengths l/w corresponding to pairs (i,j)el and arcs (N.,N ) with length 1/v corresponding to pairs G J K J K (j,k)eIB. It is assumed that G is connected, for otherwise the problem decomposes into subproblems. For each pair of existing facility nodes Ej, E^, define L(Ej,E^) to be the length of a shortest path in G connecting E. and E,. Francis et al. showed that Z* is given by J k max{d(v. ,v, )/Â£(E. ,E, ) : 1 j < k n}. The distances d(v.,v.) can be J K. J K J k 2 computed in 0(n ) operations for a tree network (see [23]), and the shortest path lengths Zr(E.,E,) are readily computable in 0(n ) opera- 1 k tions. When Z* is computed, the Sequential Location Procedure de scribed in [32] can be applied in 0(m(n+m)) operations to find a loca tion vector X* that solves the problem. m-Median problem with mutual communication Define the functions g g and g by the following expressions: D L For each XeNm gB(X) = l [v^dCx^Xjp: (j,k)eIB] , grQO = l [w d(x ,v ): (i,j)el ] , XJ 1 J o g(X) = gB(X) + gc(X) . The m-median problem with mutual communication is the following: Find a location vector X* in Nm such that Z* = g(X*) = min[g(X): XeNm] . -112- e) Iteration 4 f) Iteration 5 Figure 3.7. Continued -154- corresponding to exactly one arc in U A^, the problem specializes to the vector-minimization problem considered in Chapter 3. For the case t = 2, = A^, and = A^, the problem specializes to the bi-objective m-center problem of Chapter 4. Consider the related distance constraints DC where z = (z. z ), z It defined below: d(x.,v.) < z /w. (N.,E.) e A PI A, 1 < r < t , i 3 r xj x j r C- d(x. ,x, ) z /vM (N. ,N, ) e A' H A,., 1 Â£ r < t j k r jk j k r a It is direct to verify the following assertion: Let X be given and define z = f(X). The location vector X is efficient if and only if for every X ) > 0, X 4- 0, DC is inconsistent. The proof of 1 L ZA the assertion is very similar to the proof of Lemma 4.3.1 in Chapter 4. Based on the above property we give the following theorem for characterizing efficient solutions. Theorem 5.3.1. Given X used to define DC and GBC with z = f(X), z z the following are equivalent: (a) X is efficient. (b) For every r with r e { 1,... ,t} .and z > 0, at least one arc in A^ is in some tight path in GBCz. (Equivalently, there exists a collection of tight paths in GBCz such that at least one tight path passes through Afor every r for which r e {l,...,t} and z^ > 0.) Outline of the proof. To show (a) implies (b) suppose X is efficient. Assume that for some r for which z^_ > 0, no arc in A is in a tight path. Clearly DC^ is consistent so that every path which passes through Af is slack. Let PCE^jE^) be any path which passes through A^. Define -5- Network N is said to be edge weighted if, associated with each of its edges, is a specified real number. Given an undirected network N = {V,E} with positive edge weights, an imbedding of N, written as N = {V,E}, is a geometric realization of N is some space S such that there is a one-to-one correspondence between the members of V and k, and E and E, respectively; each edge eeE is a rectifiable arc, and no two edges in E intersect at more than one point, a vertex. The length of edge e in E is defined to be the edge weight of the corresponding member in E. A point of an imbedded network N = {V,E} is any point along any edge in E, including the vertices. We write xeN to mean x is a point in N. The distance d(x,y) between any two points x,yeN is the length of a shortest path P(x,y) joining the two points. The function d(.,.) satisfies the axioms of a metric on N so that the set N together with d(.,.) determines a metric space. The axioms of a metric are as follows: For any two points x,yÂ£N, 1. d(x,y) > 0 if x j y; d(x,x) = 0, 2. d(x,y) = d(y,x), 3. d(x,y) 1 d(x,u) + d(u,y) for any ueN. For a more detailed discussion of how to construct a metric space (N,d) from a given edge weighted network N, the reader is referred to Dearing and Francis [19], or Dearing, Francis, and Lowe [22]* We restrict ourselves to finite undirected connected networks that contain no loops and no multiple edges. We omit the term "im bedded," and simply take a network to mean an imbedded network on which the distance d(.,.) is defined. For all other networks, we use the terms "graph," "arcs," and "nodes" instead of network, edges, and vertices. -lu c) Iteration 3 Figure 3.7. Continued -16- relies on solving a finite sequence of set covering problems. Using the framework provided by Minieka, an exact algorithm was developed by Garfinkel, Neebe, and Rao [38] for the unweighted case. The algorithm uses the property that the p-radius is determined by one of a finite number of elements, namely, one of the distances between any vertex and any point in P. Call the points in P edge bottleneck points and let d be the distance between vertex v^, and the jth edge bottleneck point. Let and Z be a lower and upper bound on the value of r^. Initially Z^ = 0, and Z is obtained by a trial solution. Among all the distances d that fall within the interval [_Z,Z], one of them will determine the value of r Pick one such distance, say P d with Z < d < Z, and let r = d be a specified radius. Now, st st st we want to know if we can cover all vertices of N within this critical distance r by using only p points. If we cannot, then clearly r is too small a radius for p points to cover all vertices. Hence we con clude the p-radius r^ must be within the interval [r,Z], In this case, the lower bound is shifted to r, and the procedure is repeated. In the other case, we find a set X of p points that cover all vertices within r, but it is doubtful if this point set is an absolute p-center. Clearly, then, the value of r^ will be within the interval [_Z,f(X)]. Hence, the upper bound is shifted to f(X) for this case and the whole procedure is repeated. Termination occurs whenever the lower and upper bounds become equal. The r-cover part of this procedure is solved by obtaining a feasible solution, if it exists, to a set cover ing problem. Let A be a |v| by |p| matrix with entries a_^. equal one if vertex v is within a distance r of the jth edge bottleneck point and zero otherwise. Then, solving the system Â£ y^ Â£ p, Ay > 1, CHAPTER 2 DUALITY AND THE NONLINEAR p-CENTER PROBLEM AND COVERING PROBLEM ON A TREE NETWORK 2.1 Introduction and Related Work We consider the problem of locating p new facilities on a tree network with respect to n existing facilities at known locations so as to minimize the maximum "loss." The problem is an extension of the linear p-center problem to the nonlinear case. We assume a strictly increasing, continuous "loss" function is associated with each of a finite number of demand points (existing facilities) whose argument is the distance between the corresponding existing facility and its nearest new facility. Our formulation permits the use of quite general loss functions provided that they are continuous and strictly increas ing with the travel distance. The term "loss" is used generically and may refer to any form of inconvenience such as cost, disutility of service, travel time, etc. In locating emergency service facilities, the disutility due to "late" service may be too great beyond a certain "threshold" response time. Such sharp changes in the disutility of service can be re flected into the model by using nonlinear functions. Hurter and Schaefer [61 ] justify and use such functions in a fire setting. As pointed out by Dearing [183 a study by Kolesar et al. [73 ] revealed that the travel time for fire trucks can be approximated by a particular continuous, nonlinear, increasing function of the distance. -5V -69- node t will be in both GT and the associated brown subtree, denoted as BT(t). COVER 0) Initialize to GT = T, k = BT(v.) = iv.}. For every j e 3 J at v.. Define U = 0. 0. For every tip vertex v of T define J(r) fasten a string of length f ^(r) 3 o 1) Choose a tip t of GT. If GT = {t} go to 6). Else find a(t), the vertex in GT adjacent to t. 2) If no strings are engaged or fastened at t, remove from GT the subarc [t,a(t)] joining t and a(t), attach [t,a(t)] to BT(t), and go to 1) Else go to 3). 3) Pull all strings at t tight towards a(t). If all tight strings reach a(t) then engage them at a(t), remove [t,a(t)] from GT, attach [t,a(t)] to BT(t), and go to 1). Else go to 4). 4) Add 1 to k. Choose a shortest string engaged or fastened at t. Find the (unique) vertex, say v., at which the shortest string is (k) fastened. Construct ^ U {v^^}. Find the farthest point, say y, from t on [t,a(t)] to which the shortest string can reach. Locate x^ at y. Assign all strings at t to x^ and remove these strings. Attach [t,y] to BT(t) to obtain BT(x ), and remove [t,y] from GT. Go to 5). 5) Assign to x^ all other strings in GT which can reach x^, and re move all such strings. If no strings remain then let U = U and stop. K Else return to 1). 6) Add 1 to k. Locate x^ at t. Assign all strings at t to x^. Of the strings at t choose any one, and find the vertex v^ to which the chosen string is fastened. Let U = U, 1 1/ {v,,,.}, and stop. k-1 (k) -81- Proof. By definition of a primary set we have |x| = |u|. By assump tion r < r^ so that |x| = |u| k 2. Lemma 2.5.1 implies g(U) = g^(U) > r. Hence U is a feasible solution to the dual problem. Theorem 2.6.1 im plies q(r) > q(r). By feasibility of X and U, and the fact that |X| = |U|, we have |x| > q(r) > q(r) Â£ |u| = |x|. It follows that X solves the cover problem, U solves the dual problem, and (2.6.3) holds. We remark that the above proof is an alternative to the proof of Theorem 2.4.1 for establishing the optimality of X to the covering problem. Hence, an application of COVER solves both problems simul taneously. At this point we give an interpretation of the pair of problems. The defender D specifies an upper bound r on his loss against an attack to any vertex and will position response forces as necessary so that his loss will not exceed r. Each response force is an "expense" for D. Hence, D's problem is to choose the fewest possible response forces. The attacker A knows that D will not tolerate a loss exceeding r. Hence, A recognizes that, no matter how many vertices he threatens, D will have a sufficiently large number of response forces to respond and that the loss A inflicts on D will always be less than or equal to r. For this reason, A decides that he should not (hopelessly) try to inflict a loss to D exceeding r, and that, instead, he should force D into using as many of his response forces as possible. Hence, should A choose a subset U of V with g(U) > r, he knows that no two vertices in U can be jointly covered by a single response force by D within the specified upper-bound r. Thus, D, not tolerating a loss exceeding r, will have to allocate one response force for every vertex -157- once in DC. Clearly, the effective upper bound for the distance between the locations corresponding to F and F is the minimum of the P q upper bounds which involve these two facilities. Thus, the effective arc length to be assigned to (F ,F^) is the reciprocal weight associ ated with Fp and F^ multiplied by where z^ is the minimum z^ over all indices r for which (F ,F ) e A Let GBC be the graph with arc lengths appropriately assigned as described above. Partition A into A A mutually disjoint subsets A^,...,A (s Â£ t) such that every arc in any A A A . A^ has the same multiplier, say, z^ (where z Â£ iz^,...,z }). De- A A A fining z = ) it is direct to verify that DC is equivalent 1. s z to DC* defined below: z d(x.,v.) < z /w.. i y r i j (N.,E.) eA 0 A., 1 r Â£ s l j r C d(x.,x. ) < z /w., (N.,N, ) e A A 1 < r < s J k rjk jk r TB In other words, DC^ is obtained from DCz by choosing the minimum effec tive upper bound for any constraint which appears more than once in DCz< As a result of the equivalence of DC and DC^ and the fact that z z Aj,...,A are mutually disjoint and exhaustive subsets of A, we make the following proposition: Proposition 5.3.1. Given X and z with z = f(X), let DCg be the equi valent representation of DCz as described in the previous paragraph. The following are equivalent: (a) X is efficient. (b) For every r e {l,...,s} with z^ > 0, at least one arc in Ar is in a tight path in GBC~. -108- facilty j. Since j is arbitrary, Z is the unique solution to DC, and, thus, upon using Theorem 3.3.3, Z is efficient. 3.6 Algorithm to Construct Efficient Location Vectors To this point we have presented a family of conditions for char acterizing efficient points. Theorem 3.3.3 provides the necessary and sufficient conditions in terms of uniquely located facilities, tight paths in GBC, and the convex hulls of uniquely located facilities. Property 3.5.2 provides the sufficient conditions for irreducible location vectors without requiring the identification of uniquely located facilities. Property 3.5.3 extends the results of Property 3.5.2 to the case of reducible vectors. Based on Properties 3.5.2 and 3.5.3, we now present the Sequential Efficient Vector Construction Algorithm (SEVCA). Given a location vector Z, the algorithm first finds the irreducible representation Z* of Z by using RP. Then each component of Z* is checked to see if it satisfies the convex hull containment property. If some component is found which is not within the convex hull associated with it, its loca tion is moved to the closest point in the convex hull. The procedure is repeated with the resulting location vector. Termination occurs when every component of the current irreducible vector is within the convex hull associated with it. In order to prove finite termination (in 0(m) iterations), we use a labeling scheme for the current com posite indices. The list K is the list of composite indices during K any given iteration, while Z denotes the location vector whose com ponents are indexed by the members of K. -33- Goldman and Dearing [45] provide a conceptual discussion of, and a motivation for, considering such problems. The distance constraints are formally defined by Dearing, Francis, and Lowe [22] on a network. It was established in [22] that, in a well defined sense, the distance constraints define convex sets under the assumption that the under lying network is a tree. Furthermore, the distance constraints always define convex sets if and only if the network is a tree. Based on the results obtained in [22], Francis, Lowe, and Ratliff [32] considered the distance constraints on tree networks in more detail. They established the necessary and sufficient conditions for distance constraints to be consistent, and also devised algorithms that find a feasible location vector whenever one exists. In what follows we briefly discuss the results obtained in [32]. Distance constraints for a single new facility. For the case of a single facility, Francis et al. showed that there exists a feasible point xeT satisfying d(x,v.) < c. for iel if and only if the in- i 1 equalities d(v.,v, ) c. + c, are all satisfied for 1 1 1 < k < n. j k j k j - An equivalent statement of the single facility distance constraints can be given in terms of "neighborhoods" around v_^ of radii c^,. De fine the neighborhood N(u,r) around a point ueT of radius r to be the set of all points xeT for which d(u,x) < r. Then, a point x satisfies the constraints d(v^,x) < c^> iel, if and only if x is in each neigh borhood N(v.,c ), iel,if and only if x is in the intersection n 1 n N(v.,c.). It follows then that the single facility distance con- i=l 1 1 straints d(x,v.) Sc., iel,are consistent if and only if d(v.,v.) S ii j k Cj + ck for 1 j < k 5 n if and only if each pairwise intersection N -65- center in X. Assume A and D know functions f,,...,f so that 1 n f.(D(X,v.)) is D's loss if A attacks v. and D responds to the attack 3 3 J in a time of D(X,v^). For convenience, we refer to the loss A in flicts on D as A's gain. Aggressor A knows D has p response forces, but does not know how D will position his response forces. Thus A acts conservatively and bases his decision on a worst case analysis. If A decides to attack Vj without threatening any other vertices, A reasons that D will cor rectly guess v is to be attacked and will position a response force at v.. Hence A assumes his gain will be f (0), if he decides to J J immediately attack v^ without a prior threatening strategy. In order to gain more, A concludes that he must threaten, i.e., pretend to attack, q vertices, q > 1, so that even if D knows which q vertices are threatened, D does not know which vertex A will attack until the attack occurs. Thus D is forced to respond to the threat by position ing his response forces optimally with respect to these q vertices. Hence if A threatens K C V, he assumes D will choose a p-center X q which minimizes f(X:K ) = max{f.(D(X,v.)): v e K }. Thus, with q j j j q q p, A assumes D knows and will position a response force at every vertex in K so that A can gain at most g (K ) The best A q Z q can do in this case is to choose a K which contains some vertex v q s for which fg(0) = a. Hence, if q < p, A's maximum possible gain is at most f (0). (Parenthetically, we remark that if f (0) = r , s s p p < n, then it can be shown that not all f.(0) have the same value. 3 If all f.(0) do have the same value, then r > a.) On the other hand, 3 p if A chooses a subset with q > p, D is unable to position a response force at every vertex in K even if he knows K so A will gain at q q -50- is "as small as possible." More specifically, we wish to find all efficient location vectors X = (x,,...,x ) in Tm with respect to the i m vector minimization problem V-min[D(X): XeT] where D(X) is the vector of distances d(x^,v^) (i,j)el^, and d(Xj,x^) (i,k)el The vector is formed by assuming any convenient ordering D of the members of the sets I_ and I. L> D Francis, Lowe, and Tansel [33] characterized efficient points by making use of distance constraints. By definition, a location vector Z in T is efficient if an only if there does not exist a location vector X in Tm such that D(X) < D(Z) and D(X) D(Z). Given a location vector Z, let b., = d(z.,z,) for (jjk^I,, and c.. = d(z.,v.) for jk 3 k J B xj 1 3 (i,j)el^,, and define the distance constraints (DC) of interest by d(xi*vj) cij (i,j)elc d(xj"xk)iV We note that DC is always consistent, as Z is always feasible to DC, and hence the separation conditions are always satisfied. The separation conditions for DC are defined by constructing a graph G with nodes 1 S j Â£ m, corresponding to new facilities and nodes E^, 1 i < n, corresponding to existing facilities. For each (i,j)elr,, the arc (N.,E.) is in G with length c.., and for each c 1 1 ij (j,k)el the arc (N.,N, ) is in G with length b.. We recall that a B j k jk point x_^ is uniquely located in every feasible solution to DC if and only if the corresponding node N is in at least one tight path in G, -148- Figure 4.5. Efficient Frontier for Example -66- least (K ). Hence A observes if he chooses some K with q > p which 2 q q contains a vertex v for which a = f (0), then his gain is at least s s a = g (K ). However, A recognizes that there may be some other K 2 q q with q > p, which may or may not contain v but which yields him a gain strictly greater than a. For this reason A restricts himself to those subsets of V with cardinality greater than p and realizes that if he chooses some K with q > p, then there is at least one pair of q vertices in K which D can cover by only a single response force. If q v^ and v_. are one such pair in which are covered only by a single response force, say at x, then clearly A obtains a gain of at least 3.., as 3.. = min{max(f (d(x,v.)), f.(d(x,v.))): x e T} < max{f (d(x,v )), ij 13 i 1 3 1 11 A fj(d(x,Vj))}. Since A does not know which pairs of vertices D will cover by single response forces, once he chooses K^, A acts conserva tively, and assumes that D will cover a pair v ,v, e K for which a b q 3 = min{3..: v.,v. e K i ^ 3}. That is, by choosing a K with ab 13 1 3 q q q > p, A guarantees himself a gain of at least 3 = g. (K ). Hence clD J- CJ A's minimum gain due to threatening is g(K^) = max{g^(K^), g^CK^)}, so A chooses a K* with q > p which maximizes g(K ) over all K C V q q q with q > p. The question arises as to why A should choose p+1 vertices to threaten, and no more. By virtue of the W.D.T. and the remark follow ing it, if X* is an optimum p-center then f(X*) > g(K ) for all K q q with q > p+1. Thus r^ = f(X*) is an upper bound on A's gain due to threatening But the S.D.T. implies there is a p+l-clique, say K*+^, which attains this upper bound. Hence A need threaten no more than p+1 vertices to maximize his gain, as A cannot obtain any addi tional gain by threatening more than p+1 vertices. -127- Lemma 4.3.1. Given a location vector Y used to define DC and GBC z z with z = (zpZ^ = (f ^ (Y) f 2 (Y) ) the following are equivalent: (a) Y is efficient. (b) Por any X = (X,,X ) > 0 and X 0, DC is inconsistent. Proof. Using the definition of efficiency, f^, f2> and the fact z = f(Y) we have the following equivalences. The location vector Y is efficient if and only if f(X) < z implies f(X) = z if and only if there does not exist X such that f(X) z and f (X) 5s z if and only if for any X Â£ 0 and X ^ 0 there does not exist X for which f^(X) < z^ X^ for i = 1,2 if and only if for every X > 0 and X ^ 0 there does not exist X such that max{wd (x^,v^) : (i,j) e 1^,} z^ X^ and max{vM d(x. ,x, ) : (j,k) e 1^.} < z X if and only if for every X Â£ 0 jk j K a 2 2 and X Â£ 0 there does not exist X such that d(x.,v.) < (z, X,)/w.. i J 1 1 ij for all (i,j) e Ic and dCx^.x^) < (z2 *2^Vjk for G *b and only if for every X 0 and X ^ 0, DC is inconsistent, com- Z" A pleting the proof. Corollary 4.3.1. Given Y with z = f(Y), Y is dominated if and only if there exists X z 0, X 4- 0 such that DC is consistent. z-A We remark that the proof of Lemma 4.3.1 does not use any special properties of tree networks. Hence the lemma is applicable to any metric provided that f^ and 2 are the maximum of the pairwise weighted distances. The following lemma provides the sufficient conditions for DCr to be consistent in terms of the slack paths in GBCz. Lemma 4.3.2. Suppose DC is consistent. If every path in GBC which 2 Z passes through Ag is slack then X = (Xj^) can be chosen with X Â£ 0 and X ^ 0 such that DC is consistent. z-X z-X OPTIMAL MULTI-FACILITY LOCATION ON TREE NETWORKS By BARBAROS C. TANSEL A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1979 ACKNOWLEDGMENTS I am deeply indebted and grateful to Dr. Richard L. Francis, the chairman of my supervisory committee, for his excellent guidance, numerous suggestions, and the generosity with which he invested his time in listening to my ideas. Dr. Francis not only initiated my interest in location problems but also inspired many of the ideas in this dis sertation by asking the right questions at the right time. I owe very special thanks to Dr. Timothy J. Lowe, the cochairman/ chairman of my committee during 1976-1978, presently of Purdue Uni versity, for his active interest, overall guidance, and his inspiring suggestions. Dr. Francis and Dr. Lowe have shown sincere care about my progress and their encouragement has been of utmost value in bringing this dissertation to a completion. I would also like to express my sincere thanks and appreciation to the other members of my committee, Dr. Ralph W. Swain, Dr. Donald W. Hearn, Dr. Antal Majthay, and Dr. Luc G. Chalmet for their interest in my work and their suggestions during my proposal. I am grateful to the Department of ISE for providing me with assistantship during my graduate studies. Mrs. Adele Koehler has done an excellent job in typing the manu script. She is fast, accurate, and very observant. I sincerely recommend her. ii This research was supported in part by NSF Grant //ENG 76-17810, the Army Research Office, Triangle Park, N.C., under contract DAHC04-75-G-0150, and by the Operations Research Division, National Bureau of Standards, Washington, D.C. TABLE OF CONTENTS Page ACKNOWLEDGMENTS ii ABSTRACT vi CHAPTER 1 INTRODUCTION AND LITERATURE SURVEY 1 1.1 Introduction and Overview . 1 1.2 Terminology 4 1.3 Survey of the Network Location Literature 6 2 DUALITY AND THE NONLINEAR p-CENTER PROBLEM AND COVERING PROBLEM ON A TREE NETWORK ...... 53 2.1 Introduction and Related Work 53 2.2 Problem Statements and Duality 56 2.3 Dual Problem Interpretation 61 2.4 Covering Algorithm 67 2.5 Dual Problem Solution and the Strong Duality Theorem. 73 2.6 Results for the Covering Problem 78 3 A VECTOR-MINIMIZATION PROBLEM ON A TREE NETWORK 84 3.1 Introduction 84 3.2 Problem Statement 85 3.3 Distance Constraints and Characterization of Efficient Points 87 3.4 Examples 94 3.5 Further Results on the Convex Hull Property 96 3.6 Algorithm to Construct Efficient Location Vectors . 108 3.7 Efficiency for the Case of Rectilinear or Tchebychev Distances. 116 4 THE BI-OBJECTIVE m-CENTER PROBLEM ON A TREE NETWORK. ... 122 4.1 Introduction 122 4.2 Problem Statement, Notation, and Definitions 123 4.3 Necessary and Sufficient Conditions for Efficiency. 126 4.4 Construction of the Efficient Frontier 134 Iv Page 5 SUMMARY AND FUTURE RESEARCH 149 5.1 Summary 149 5.2 Generalized Multi-Center Problem 150 5.3 The t-Objective m-Center Problem: Steps Towards a Unified Theory 153 5.4 Tree Networks and General Networks 158 REFERENCES 161 BIOGRAPHICAL SKETCH . 170 v Abstract of Dissertation Presented to the Graduate Council of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy OPTIMAL MULTI-FACILITY LOCATION ON TREE NETWORKS By Barbaros C. Tansel December 1979 Chairman: Richard L. Francis Major Department: Industrial and Systems Engineering In this dissertation we develop a theory for location problems which involve locating multiple new facilities on a tree network with respect to existing facilities at known locations. The first problem we consider is the nonlinear version of the p-center location problem on a tree network for which the cost of each served vertex is a strictly increasing continuous function of the dis tance between the vertex and the nearest center,and the objective is to minimize the maximum such cost over all possible locations of the centers. We present a dual "dispersion" problem which may be inter preted as the problem of choosing p + 1 (or more) vertices such that the minimum cost to serve any two of the chosen vertices by a single common center is as large as possible. We give a weak duality theorem which applies to all general networks and a strong duality theorem which applies to all tree networks. The strong duality theorem also specifies the necessary and sufficient conditions for an optimal solu tion to either problem. We provide algorithms of polynomial complexity v for solving either problem provided that certain needed inverse functions can be evaluated in a polynomial order of effort. The p-center problem is typically solved with the aid of a nonlinear covering problem for which we also give a dual with a physical interpretation. We provide a covering algorithm which solves both the covering problem and its dual simultaneously. The second problem we consider is a vector-minimization problem which involves as objectives the distances between specified pairs of new and existing facilities andspecified pairs of new facilities. We relate the vector-minimization problem of interest to a distance con straints problem which imposes upper bounds on the distances between specified pairs of facilities. We develop the necessary and sufficient conditions for efficiency by making use of the theory developed for the related distance constraints problem. Efficient solutions to the vector-minimization problem of interest are such that in order for any new facility to be closer to some facility than it already is, it must in turn be placed farther from some other facility. Based on the necessary and sufficient conditions, we provide an algorithm which constructs an efficient location vector from a given non-efficient solution. The third problem we consider is a bi-objective minimax problem which involves as objectives the maximum of the weighted distances between specified pairs of new and existing facilities, and the maximum of the weighted distances between specified pairs of new facilities. We again relate the problem to the distance constraints problem and derive the necessary and sufficient conditions for efficiency by making 2 2 use of the distance constraints. Further, we provide an 0(m (m + n )) algorithm to construct the efficient frontier, where m and n are, respectively, the number of new and existing facilities. v i i i CHAPTER 1 INTRODUCTION AND LITERATURE SURVEY 1.1 Introduction and Overview Although some mathematical models of location can be traced back to the early seventeenth century, almost all the work on operational models for the location of facilities has taken place within the past 22 years, between 1957 and the present. An extensive annotated bibli ography on location-allocation problems is provided by Lea [78]. A more recent selective bibliography is given by Francis and Goldstein [30], Location problems commonly involve locating a number of new facilities (sources) in a given location space so as to provide goods or services to a specified set of existing facilities (demands) under one or more criteria, and, possibly, subject to a set of constraints. The quality of the service is typically measured in terms of the dis tances among the facilities. The use of distances is, perhaps, the major feature which distinguishes location problems as a special class of optimization problems. Hence, associated with any location problem is an underlying location space on which a "distance" is defined. Several variations of the general location problem are possible, depending upon the type of location space, the distance function, the number and areal extent of the facilities, the type of interactions -2- between the facilities, the objective criteria used, the constraints, the presence or lack of random elements, and possibly other factors as well. Among the several variants, planar location problems received special attention in the past, starting with the earliest contribu tions, for example [106]. In such planar problems, one is interested in locating new facilities in the Euclidean plane with respect to existing facilities. For continuous planar problems, where any point in the plane is a feasible location, the typical distance used is the distance, special cases of which are the rectilinear, Euclidean, and Tchebychev norms. For discrete planar problems, where there are a finite number of candidate locations for new facilities, the distance between any potential new facility location and any existing facility is a specified positive number. Such discrete problems, due to the finite nature of feasible locations, readily lend themselves to integer programming formulations. The reader is referred to the book by Francis and White [31] for a discussion of planar problems and a wealth of references. A number of real life applications suggest that, in some in stances, a network space can be a more faithful representation of the reality than the Euclidean plane. For example, in a road network, a communication network, or a pipeline system, travel occurs along the arcs of the underlying network rather than in straight lines or recti linear paths. Hence, for such problems, the use of shortest path distances along the arcs of the network can approximate the travel distance more closely than the X. distance. As opposed to planar problems, network location problems have received much less attention -3- in the past. As reported by Lea [79], there are some 1500 published papers on location-allocation problems. Among these, about 80 are on network location problems, a ratio of a little less than 6%. Hence, network location problems deserve well-justified attention in future research. In this dissertation, we develop a theory for a number of location problems which involve locating multiple new facilities on a tree net work with respect to existing facilities at known locations. At this point we give an overview of the dissertation. In the remainder of Chapter 1, we specify our terminology and give a survey of the network location literature. We discuss minimax and minisum problems.and multi-objective problems involving minimax and minisum objectives as well as other objectives. Discussed also are problems with distance constraints. We highlight some of the convexity properties of trees (see [22]) in relation to the problems discussed. The chapter ends with a brief discussion of path-location problems. In Chapter 2, we develop a theory for the nonlinear p-center problem on a tree network. The problem is a generalization of the linear p-center problem which involves locating p new facilities on a network so as to minimize the maximum weighted distance from any existing facility to its nearest new facility. Nonlinearity is ob tained by replacing each weight by a strictly increasing function of the distance. We formulate a dual "dispersion" problem and prove a weak duality and a strong duality theorem. The strong duality theorem also specifies the necessary and sufficient conditions for an optimal solution to either problem. We provide algorithms of polynomial com plexity for solving either problem. Discussed also are a covering -4 problem and a dual "divergence" problem. We provide a covering algorithm which solves both the covering problem and its dual simul taneously. In Chapter 3, we study a vector-minimization problem in relation to a distance constraints problem. The problem involves as objectives the distances between specified pairs of new and existing facilities and specified pairs of new facilities. We extend the results of [32] to develop a theory for identifying unique solutions to distance con straints, and use this theory to develop necessary and sufficient conditions for efficient solutions to the vector-minimization problem of interest. Further, we provide an algorithm which constructs an efficient location vector from a given non-efficient solution. In Chapter 4, we study a bi-objective location problem which in volves as objectives the maximum of the weighted distances between specified pairs of new and existing facilities, and maximum of the weighted distances between specified pairs of new facilities. We characterize efficient solutions and provide an algorithm for construct ing the efficient frontier. In Chapter 5, we pose a number of unresolved questions in relation to the problems discussed and point out directions for future research. 1.2 Terminology Before discussing the literature we specify our terminology. An undirected network N = {V,E} is a collection of two sets V and E, called the set of vertices and the set of edges of N, respec tively. Each edge in E is described by an unordered pair of vertices. -5- Network N is said to be edge weighted if, associated with each of its edges, is a specified real number. Given an undirected network N = {V,E} with positive edge weights, an imbedding of N, written as N = {V,E}, is a geometric realization of N is some space S such that there is a one-to-one correspondence between the members of V and k, and E and E, respectively; each edge eeE is a rectifiable arc, and no two edges in E intersect at more than one point, a vertex. The length of edge e in E is defined to be the edge weight of the corresponding member in E. A point of an imbedded network N = {V,E} is any point along any edge in E, including the vertices. We write xeN to mean x is a point in N. The distance d(x,y) between any two points x,yeN is the length of a shortest path P(x,y) joining the two points. The function d(.,.) satisfies the axioms of a metric on N so that the set N together with d(.,.) determines a metric space. The axioms of a metric are as follows: For any two points x,yÂ£N, 1. d(x,y) > 0 if x j y; d(x,x) = 0, 2. d(x,y) = d(y,x), 3. d(x,y) 1 d(x,u) + d(u,y) for any ueN. For a more detailed discussion of how to construct a metric space (N,d) from a given edge weighted network N, the reader is referred to Dearing and Francis [19], or Dearing, Francis, and Lowe [22]* We restrict ourselves to finite undirected connected networks that contain no loops and no multiple edges. We omit the term "im bedded," and simply take a network to mean an imbedded network on which the distance d(.,.) is defined. For all other networks, we use the terms "graph," "arcs," and "nodes" instead of network, edges, and vertices. -6- Finally, for tree networks, we write T instead of N. In passing, we note that the shortest path P(x,y) between any two points x,yeT is unique, as otherwise T would contain a cycle. 1.3 Survey of the Network Location Literature Historically, the earliest precise mathematical formulation of a location problem on a network appears to be due to Hakimi [47] in 1964. Prior to Hakimi's paper, the problem of finding the best threshing site for harvested wheat was attacked by using a network location model in 1962 by Hua Lo-Keng and Others [60]. This model was presented only at an intuitive level and no mathematical formulation or properties were given. A (correct) solution procedure was suggested (in the form of a poem), which was to be discovered independently by Goldman [42] in 1971. Since 1964, a literature of approximately 80 papers has grown till the present. Several new problems, as well as certain extensions and generalizations of old problems, have been introduced. A recent text by Handler and Mirchandani [ 58 ] discusses ex tensively a portion of the literature involving minimax and minisum problems as well as single-facility bi-objective problems involving the combination of these two objectives. A "family tree" for network location problems is shown in Figure 1.1. Although not exhaustive, the family tree covers most of the problems formulated since 1964. With reference to the family tree shown in Figure 1.1, network location problems can be broadly classi fied into two groups: point-location problems and path-location problems. Path-location problems have been recently introduced by -7- Figure 1.1. Family Tree for Network Location Problems -8- Slater [102]. A large portion of the literature deals with point- location problems. Point-location problems may be classified into three categories: single objective problems, multi-objective problems, and a body of results of a general and unifying nature. In the remainder of this section we give a detailed discussion of the problems outlined in the family tree. Point-Location Problems Here, we consider a number of problems that involve locating new facilities at points on a network. The general format of the dis cussion is as follows: For each problem type, we first define a kernel problem. Then, we discuss the related literature on the kernel problem, as well as several special cases and extensions of it. We point out relations between different problem types, whenever such relations exist. The p-center problem Let N be a network with a vertex set V = {v,,...,v } and an edge 1 n set E. Denote by X a finite set of points, each of which is in N. Let I be the set of integers 1 through n. For each vertex v., iel, define the distance D(v^,X) between vertex v and the point set X by D(v^,X) = min[d(v^,x) : xeX]. With this definition, D(v_^,X) is speci fied by a nearest point in X to vJ. Let w. and a. be two given numbers i 11 associated with vertex v^, iel. We call wi a weight and aan addend. We assume that each w^ is nonnegative and at least one w_^ is positive. For any finite point set X CD, define the function f (X) by -9- f(X) = max[wJ)(v^,X) + a^: iel] The problem of Interest is the following: Given a positive integer p, find a point set X* = {x*,...,x*}, and a real number r 1 p p such that rp = f(X*) = min [ f (X) : |x| =p,XcN] (1.3.1) where the symbol j*| means the cardinality of a set. The problem defined by (1.3.1) is called the p-center problem. Any set X* of p-points that solves (1.3.1) is called an absolute p- center of N, and the minimum objective value r^ is called the p-radius. For p = 1, an absolute 1-center is simply called an absolute center of N. If in (1.3.1), each xeX is restricted to a vertex location, the resulting problem is called the vertex restricted p-center problem and any set X* C V of p points that solves it is called a vertex restricted p-center of N. A vertex restricted 1-center is simply called a vertex center. We note that the p-center problem is usually formulated in the absence of addends. In what follows, we will assume all addends are zero, unless we explicitly mention them. The case with all w^ equal to unity will be referred to as the unweighted case. With this terminology, the p-center problem is the problem of finding p points on a network so that the maximum (weighted) distance between any demand point and its nearest center is as small as possible. The problem is perhaps most applicable to the location of emergency facilities such as fire stations, ambulance centers, and the like, as -10- ln such problems a common objective is to provide "good" service to each demand point by at least one facility within a least possible distance. In what follows, we first discuss the 1-center problem on general networks and on tree networks. Then, we discuss the vertex restricted 1-center problem. Finally, we will discuss the p-center problem in relation to a "covering" problem to be defined later. 1-Center problem on a general network. The absolute 1-center problem was defined and solved by Hakimi [47] in 1964. For finding the absolute center, Hakimi examines the function f on each edge, finds a best local minimum on that edge, and selects the best among |e| such local minima. This method takes advantage of one important property of f, namely, that it is piecewise linear and continuous on each edge with at most n(n l)/2 break points. A local minimum always occurs either at a break point of f or at an end point of the edge. Hakimi, Schmeichel, and Pierce [50] showed that Hakimi's method can be imple- mented in 0(|E|n logn) computational effort and gave a computational refinement which reduces the effort to 0(|E|nlogn) for the unweighted case. Further refinements of the procedure were obtained by Kariv and Hakimi [65], resulting in an 0(|E|nlogn) algorithm for the weighted case and 0(|E|n) algorithm for the unweighted case. All these refinements focus on finding the break points and the local minimum of.f in the most efficient manner. A somewhat more general version of the 1-center problem was con sidered by Frank [36], and (apparently) independently by Minieka [88], as Minieka makes no reference to Frank's paper. In this modified version, called here the continuous 1-center problem, each point on -li the network is a demand point (as opposed only to vertices). The weight of each point is unity. The objective to be minimized over all xeN is defined by f(x) = max [d(y,x): yeN]. Both authors showed that the problem can be reduced to a computationally finite one and pro posed a solution procedure which is very similar to Hakimi's. A probabilistic version of the 1-center problem was considered by Frank [34, 35] and a number of bounds were obtained on the expected value of the 1-radius. For the unweighted case, Singer [101] proved that there exists a "critical" path, not necessarily a shortest path, connecting two cri tical vertices such that an absolute center of the network is at the midpoint of this path. 1-Center problem on a tree network. We now concentrate on ab solute centers of tree networks. Goldman [44] solved the unweighted case in the presence of addends. Goldman's algorithm is based on the repeated application of a "trichotomy theorem" that either determines the edge on which the absolute center lies, or reduces the search to one of the subtrees obtained by removing all interior points of that edge. Halfin [51] refined Goldman's algorithm to make it simpler and computationally more efficient. Halfin's algorithm finds a vertex center first, and determines the absolute center by examining all vertices adjacent to the vertex center. For the unweighted case with no addends, Handler [55] presents an especially elegant algorithm. Handler's method finds a longest path of the tree and locates the absolute center at the midpoint of the path. To find a longest path, Handler chooses an arbitrary vertex v finds a farthest vertex v from v., and then finds a farthest i si -12- vertex v from v The path P(v .v^) is a longest path and its mid- t s s t point is the unique absolute center of the tree. This procedure requires a computational effort of 0(n). Handler's algorithm is extended by Lin [81] to the unweighted case with addends. Lin showed that the absolute center of a general network N with vertex addends can be found by determining the absolute center of an expanded net work N' whose vertex addends are all zero. Network N' is obtained from N by adding a new vertex adjacent to each old vertex, with the length of the edge connecting the two equal to the addend associated with the old vertex. For a tree network T, the resulting network is a tree T' and Goldman's 0(n) algorithm can be applied to T'. The more general case with both weights and addends was considered by Dearing and Francis [19], and for the case of a tree network an 2 0(n ) algorithm was given. The Dearing-Francis paper appears to be the first to construct a well defined metric space N with distance d(.,.) from an arc weighted graph N. This mathematical formality per mits the use of such concepts as compactness, continuity, and the extreme and intermediate value theorems. They showed that the distance d(x,.) is continuous for each fixed x, in turn implying that f(x) is continuous for every x. From compactness and continuity considera tions, they proved the existence of an absolute center for all compact networks, and its uniqueness for all compact tree networks. They obtained a lower bound on r^ which is applicable to all networks, and proved that it is always attainable for tree networks. Once the lower bound is determined, it identifies two "critical" vertices, and the absolute center can be readily located on the path joining the two. The bound is the maximum of n(n l)/2 terms, resulting in a -13- 2 computational complexity of 0(n ), and is given by a = max[a..: 1 i j n] where (1.3.2) W-W.VAIV.JV,/ W C* I W t-4. . = i J 1 J 1 J J 1 w w.d(v.,v.) +w.a. + w.a a ij w. + w, 1 jj Hakimi, Schmeichel, and Pierce [50] proved a theorem that reduces the computational effort for computing this lower bound. Their theorem states that if for some a it is true that max[a l5i max[a 1 K i < n] then a is the maximum of all a... A different ti J st 13 solution procedure is also given by Kariv and Hakimi [65] for the same problem. Rather than computing the lower bound, their procedure confines the search to successively smaller subtrees until an edge is obtained. The absolute center is located at the local center (also the global center for a tree) on this edge using Hakimi's procedure for finding a local minimum. This algorithm is of O(nlogn). A nonlinear version of the 1-center problem was considered and solved by Dearing [18], and by Francis [29]. In this version, each weight w_^ is replaced by a monotone increasing function f of the distance d(v_^,x). Both authors obtained a lower bound similar to the one defined by (1.3.2). The bound is applicable to all networks and is always attainable for tree networks. A "roundtrip" version of the problem was solved by Chan and Francis [ 11 ]. In this version each "demand point" is a pair of ver tices (v^,^) and f(x) is the maximum of the roundtrip distances defined by p^(x) = w^dCv^x) + d(x,u^) + a^]. A lower bound, similar -14- to the one defined by (1.3.2) is obtained. The bound is again applicable to all networks and always attainable for tree networks. Vertex constrained 1-center problem. The vertex constrained 1-center problem was considered as early as 1869, and perhaps earlier, by Jordan [63] as a graph theoretic problem. This problem can be solved by examining the distance matrix of the network, as demonstrated by Hakimi [47], Rosenthal, Pino, and Coulter [98] introduced a gener alized algorithm that solves a number of "eccentricity" problems on tree networks, one of which is the vertex restricted 1-center problem. In this case, the eccentricity of a vertex is defined to be the distance from that vertex to a farthest vertex. This generalized algorithm determines the eccentricity of each vertex by making only two traversals of the vertices. The vertex center is that vertex with the minimum eccentricity. Slater [103] considered the problem of finding the vertex center of a network with respect to subnetworks. In this version of the problem, each demand is a known collection of vertices (or a subnetwork induced by the collection). The distance between a vertex and any such collection is defined by a nearest element of the collection to that vertex. For a given vertex, the value of the objective function at that vertex is the maximum of the distances between that vertex and any such collection. Slater showed that a matrix D' can be constructed from the distance matrix D of the network, so that each entry of D' is a distance from a vertex to a nearest element of a collection. Slater demonstrated that the vertex center with respect to collections of vertices can be found by examining the matrix D'. -15- This completes the discussion of the 1-center problem. We now concentrate on the p-center problem for p > 2. p-Center problem on a general network. The p-center problem was defined by Hakimi [48]. Subsequently, a number of solution procedures have been suggested. A common characteristic of all these procedures is that they all rely on solving a sequence of covering problems. For completeness, we first define a set covering problem and an r-cover problem. Let A be a matrix of zeros and ones, y a vector of zero-one variables y^. The problem of minimizing J y_^ so that each row of Ay i is greater than or equal to one is called the (minimal) set covering problem. Given the function f(X) = max{w Jl(v^,X): 1 < i < n}, the problem of minimizing |x| so that f(X) < r for some given value of r is called the r-cover problem. Denoting by q(r) the minimum value of the r-cover problem, it can be readily shown that, if q(r) = p for some r, and q(r') > p for any r' < r, then r is the p-radius and any X which solves the r-cover problem is an absolute p-center. In what follows, we concentrate on the absolute p-center problem on a general network. Minieka [87] considered the unweighted case on a general network and showed that the problem can be reduced to a computationally finite one. Minieka identifies a finite point set P* such that there exists an absolute p-center contained in P = P' U V. A point x on some edge is a member of P' if and only if x is the unique point on its edge such that d(v.,x) = d(x,v.) for some two distinct vertices v, and v.. i J i J Based on this result, Minieka suggested a rudimentary algorithm that -16- relies on solving a finite sequence of set covering problems. Using the framework provided by Minieka, an exact algorithm was developed by Garfinkel, Neebe, and Rao [38] for the unweighted case. The algorithm uses the property that the p-radius is determined by one of a finite number of elements, namely, one of the distances between any vertex and any point in P. Call the points in P edge bottleneck points and let d be the distance between vertex v^, and the jth edge bottleneck point. Let and Z be a lower and upper bound on the value of r^. Initially Z^ = 0, and Z is obtained by a trial solution. Among all the distances d that fall within the interval [_Z,Z], one of them will determine the value of r Pick one such distance, say P d with Z < d < Z, and let r = d be a specified radius. Now, st st st we want to know if we can cover all vertices of N within this critical distance r by using only p points. If we cannot, then clearly r is too small a radius for p points to cover all vertices. Hence we con clude the p-radius r^ must be within the interval [r,Z], In this case, the lower bound is shifted to r, and the procedure is repeated. In the other case, we find a set X of p points that cover all vertices within r, but it is doubtful if this point set is an absolute p-center. Clearly, then, the value of r^ will be within the interval [_Z,f(X)]. Hence, the upper bound is shifted to f(X) for this case and the whole procedure is repeated. Termination occurs whenever the lower and upper bounds become equal. The r-cover part of this procedure is solved by obtaining a feasible solution, if it exists, to a set cover ing problem. Let A be a |v| by |p| matrix with entries a_^. equal one if vertex v is within a distance r of the jth edge bottleneck point and zero otherwise. Then, solving the system Â£ y^ Â£ p, Ay > 1, -17- y.e{0,l} will determine whether or not at most p points (in P) can cover all vertices of N within a radius r. Computational experience is reported and it is found that the procedure works better for larger values of p, as in this case the initial upper bound Z is small, and significant computational savings result in identifying those edge bottleneck points whose distances fall within the interval [0,Z]. The weighted case on general networks was considered by Christofides and Viola [15], and an approximate solution procedure was given. The procedure finds a set X of p-points whose objective value f(X) is within some e-neighborhood of the actual p-radius r The procedure P oi obtains X by solving a sequence of r-cover problems with successively increasing values of r. Termination occurs when the solution of an r-cover problem generates p (or less) points the first time. In the process, one also obtains approximate solutions for n-1, n-2,..., p+1 center problems. The solution of each r-cover problem is obtained in two stages: First, all feasible solutions to the r-cover problem are obtained by finding all regions on the network that can be reached by a vertex within a radius of r. Then, among all these feasible solu tions, those with minimum cardinality are found by solving a set covering problem. To find all regions on N reachable by a vertex v_^, one "penetrates" a distance of r/w_^ along all possible paths originating at v_^. The procedure is repeated for each vertex and the intersections of these penetrations are found. Each maximal intersection defines a connected region all of whose points are reachable by a subset of vertices within a radius r. The subset of the vertices is that which defines the intersection. These regions jointly cover all vertices of N, and it is possible that a subcollection of the collection of all -18- these regions may also jointly cover all vertices. Hence, to find a minimum cardinality feasible solution, one needs to choose the minimum number of regions that jointly cover V. This choice can be made by defining a zero-one matrix A, so that an entry a^ of A is one if vertex v^ is covered by region j, and zero otherwise. Solving the set covering problem with matrix A will provide a solution to the r-cover problem. Computational experience is reported and it is found that the procedure works better for small values of p, as the set covering part of the procedure takes a significant portion of the total computational time. An important result is due to Kariv and Hakimi [6.5] They showed that the p-center problem on a general network is NP-complete. Kariv and Hakimi also showed that the weighted case (as well as the un weighted case) can be reduced to a computationally finite one. Based on this finiteness property, they gave an algorithm whose order of complexity is polynomial in |e|, but exponential in p. To show com putational finiteness one argues as follows: For any absolute p-center X = {x^,...,Xp}, there will be a subset of vertices covered by the ith center x.. If N. is the (sub)network induced by V., then it can xi 1 be shown that the absolute center x* of N. can replace x without in- 1 i i creasing the value of the objective function, so that X* = {x*,...,x*} 1 p is also an absolute p-center. Hence, one can restrict one's attention to absolute p-centers every element of which is the absolute 1-center of some subnetwork. The absolute 1-center of any subnetwork of N will occur either at a vertex or at one of at most | E |n (n l)/2 "suspected" points. A suspected point on an edge is a point x such that, for some two distinct vertices v_^ and v., x is a break point on -19- its edge of the function f..(-) = max[w.d(v.,.), w.d(v.,.)], and ij i i J J that the two linear pieces defining that breakpoint have slopes of opposite signs. There can be at most n(n l)/2 suspected points on each edge, resulting in a total of 0(|E|n^) suspected points on all edges. If S is the set of all suspected points together with the set of all vertices, then there is an absolute p-center contained in S. The Kariv-Hakimi procedure selects p-1 points from S and determines all the vertices covered jointly by these p-1 points. All uncovered vertices are assigned to the pth center. Corresponding to each center, the I-radius is determined (with respect to the subset of vertices covered by that point) and the maximum of these 1-radii determines the p-radius for that trial solution. The algorithm tries every possible combination of p-1 points selected from S and chooses that combination which minimizes the p-radius. The Kariv-Hakimi procedure is the only exact algorithm available so far for finding an absolute p-center of a vertex weighted general network. A further result on the computational difficulty of the p-center problem on a general network is given by Nemhauser and Sheu [92]. They showed that finding an approximate solution to the vertex restricted or absolute p-center problem whose value is within 100% or 50%, respec tively, of the optimal value is NP-hard (i.e., at least as hard as any NP-complete problem). Vertex restricted p-center problem. The vertex restricted p- center problem is considered by Toregas, Swain, ReVelle, and Bergman [109]. A solution procedure is given which relies on solving a sequence of minimal set covering problems, each corresponding to a specified radius r. Given a radius r, a 0-1 matrix A can be formed with n rows -20- and n columns, so that an entry a., is 1 if vertex v. is within a ij J distance r of v and 0 otherwise. If one solves a set covering i problem using the matrix A, the variables whose values are 1 in an optimal solution determine a feasible solution to the vertex restricted r-cover problem. The set covering problem is solved by relaxing the integrality constraints. In the case of non-integer termination, a single cut produced an integer solution in a large proportion of the cases. Their computational experience indicates that non-integer termination seldom occurs. p-Center problem on tree networks and duality. In what follows, we concentrate on the p-center problem on tree networks. First, we define the "continuous" p-center problem. In the continuous p-center problem, each point in T is a demand point as opposed only to vertices. Weights are absent (or unity). For any XC T, f is defined by f(X) = max{D(y,X): yeT} and the continuous p-center problem is to find an X*d T such that rp = f(X*) = min[f(X): |x| = p, X C T] . Minieka [88] considered the continuous p-center problem on a general network and showed that it can be reduced to a computationally finite one. Shier [100'] considered the continuous p-center problem on a tree network and defined a dual "dispersion" problem. The dispersion problem is to find p+1 points on T the nearest two of which are as far apart as possible. More explicitly, let U be any finite point set with |U| = p+1 and define h(U) by h(U) = mindu^jUj): 1 < i < j < p+l} . -21- The dispersion problem is to find a U*C T such that h(U*) =max{h(U): U C T, |u| = p+1} . At optimality, Shier's duality result states that rp-{h(U*) for a tree network. The equality may not hold for general networks. However, Shier showed that the objective value of the continuous p- center problem is always bounded below by one-half the objective value of the dispersion problem for any network. Chandrasekaran and Tamir [14] observed that Shier's duality result holds when one replaces T by any subset S of T. Chandrasekaran and Daughety [12] described a procedure for solving the dispersion problem. They first solve the related problem of locating the maximum number of points on T such that any two of them are at least A distance apart for a fixed (positive) A. This problem is solved by working from "tips" of T to the "center" of T. The general scheme is to use the algorithm for different values of A, until the number of points found is p+1 and a slightly larger A generates p or less points. A number of solution procedures have been given for the p-center problem on tree networks. We now discuss these procedures. Handler [57] considered the continuous p-center problem on a tree network for the special case of p = 2 and obtained an 0(n) algorithm. Handler first finds the absolute 1-center of T, say x*, and splits the tree at x* obtaining two disjoint subtrees T^ and T^. Finding the absolute 1-center of each T say x* and x*, determines an absolute 2-center of T. -22- 2 An algorithm of complexity 0(n logn) is described by Kariv and Hakimi [65] for finding the absolute p-center-of a vertex weighted tree network. They show that there are n(n l)/2 possible values for r namely, the numbers a.. = w.w.d(v.,v.)/(w. + w.) for each P 1JJ1JJ combination of vertices v^, v The algorithm computes all these numbers, arranges them in increasing order, and performs a binary search on this list of numbers. The search relies on solving an r- cover problem for each value of r chosen from the ordered list {a..}, ij The search terminates when the smallest r in the list is found for which the r-cover problem generates at most p points. The covering part of the algorithm requires a computational effort of 0(n) for each r, and a total effort of O(nlogn) for all values of r tried during the binary search. Hence, the computational effort is determined by the initial computation and ordering of the numbers ay> and is of 2 0(n logn). A similar approach is used by Chandrasekaran and Daughety [12] to solve the continuous p-center problem on a tree network. First, they provided an 0(n) procedure for finding the minimum number of points needed to cover every point of T within a given radius r. Then, they provided a method to compute r A further refinement of the method is given by Chandrasekaran and Tamir in [14]. They proved that r^ is determined by one of the numbers d(t,t')/2k, where t and t* are any two tip vertices and k is any integer between 1 and p. The total computational effort for finding r and applying the covering P algorithm is of O((nlogp)^). A somewhat different approach, which relies on finding a clique on a related graph, is given by Chandrasekaran and Tamir [13]. They -23- define an intersection graph for a fixed value of r as follows: has nodes corresponding to demand points v^,...,v^. Two nodes of G^ are connected by an arc if the corresponding demand points can be jointly covered by a (single) common center within a radius of r. Once Gr is formed, finding a "clique cover" of G^ solves the r-cover problem. A clique cover of G^ is a minimum number of cliques in G^ such that every node is in at least one clique. The solution to the clique cover problem in G^_ determines a solution to the r-cover problem. The procedure is repeated for different values of r until a smallest value of r is found for which the clique cover solution generates at most p cliques. The computational complexity of the procedure is polynomial. In particular, the computational effort for finding the minimal clique cover of G^ is polynomial because G^ satisfies the property that any circuit in G^_ with at least four arcs contains a chord (i.e., an arc which connects two nodes of the circuit and is not an element of the circuit). For chordal graphs, algorithms of linear order have been developed (see [39], [97]) for finding a minimal clique cover. This completes the discussion of the p-center problem. The p-median problem The difference between the p-center and the p-median problem is that the objective criterion is changed from minimax to minisum. More specifically, define the function f (X) for any finite point set XCN by f (X) = l w D (v ,X) . iel 1 -24- The p-median problem is the following: Given a positive integer p, find a set X* of p-points such that f(X*) = min[f(X): |x| = p, X C N] . Any set X* of p points that minimizes f is called an absolute p- median of N. If each member of X is restricted to a vertex location, the resulting problem is called a vertex restricted p-median problem. Due to a result by Hakimi [47, 48] there exists an absolute p-median entirely on the vertices of N. For this reason, the distinction be tween the vertex restricted and unrestricted versions is insignificant. Hence, we will take the term "p-median" to mean a solution to either version of the problem. A 1-median is simply called a median. The p-median problem arises naturally in locating plants/ware houses to serve other plants/warehouses or market areas. The problem is also motivated by ReVelle, Marks, and Liebman [96] as an example of a public sector location model where vertices represent population centers and facilities represent post offices, schools, public build ings, and the like. The 1-median problem. Hakimi [47] appears to be the first to define an absolute median. Hakimi proved the important result that there exists an absolute median at a vertex of the network. This result reduced the search to a finite number of points. The median can be found by summing each row of the weighted-distance matrix and choosing the vertex whose row sum is the minimum. This procedure takes 3 2 0(n ) operations to compute the distance matrix followed by 0(n ) operations to find the median. -25- For tree networks, more efficient algorithms can be devised to find a median. An 0(n) algorithm was given by Hua Lo-Keng and Others [60] and independently by Goldman [42]. The algorithm reduces the search to successively smaller subtrees until a median is found. At each stage, one chooses an arbitrary tip vertex (a vertex of degree one) of the current tree. If the (modified) weight of the selected vertex is at least as large as half the sum of all weights, a median is found. Otherwise, that tip vertex is eliminated from further con sideration together with the edge incident to it and its weight is added to the weight of the adjacent vertex. The procedure is repeated with the new (reduced) tree. The algorithm does not require the com putation of the distance matrix and uses only the incidence relation ships and the weights. Goldman's algorithm is based on a "localization theorem" proved by Goldman and Witzgall [46]. The theorem provides sufficient condi tions for a subset of N to contain a median. Given a compact subset S of N, if S satisfies the two conditions (i), (ii) then it contains at least one median. The conditions are (i) the set S must be a "majority" set, meaning that the sum of the weights corresponding to vertices in S must be at least as large as half the sum of all weights (ii) the set S must be "gated" in the sense that there must exist a unique point g in S such that for every s e S and t e N-S, it is true that d(t,x) = d(t,g) + d(g,s). Goldman's algorithm in essence is a repeated application of this theorem to a tree network. Goldman [43] also proposed an "approximate" localization theorem which somewhat relaxes the second condition and guarantees the existence of a point in S that approximates an actual median. -26- A median of a tree is shown to be the same as a "centroid" of the tree by Zelinka [120] for the unweighted case and by Kariv and Hakimi [65] for the weighted case. To define a centroid, consider the subtrees T-,...,T. obtained by removing vertex v from T. Let 1 X w(T ) be the sum of the weights of the vertices in T^., and define W(v .) to be the maximum of w(T.) for 1 i i k., A vertex v which i j J i t minimizes W(v.) over all v. in V is said to be a centroid of T. The i i location of a centroid is independent of the distances and can be found by using only the incidence relations. Goldman's earlier algorithm in essence finds a centroid of T. The generalized algorithm of Rosenthal, Pino, and Coulter [98] also finds a centroid of T by making only two traversals of the vertices. All these algorithms are of 0(n) and solve the 1-median problem without having to compute the distance matrix. We now consider some generalizations of the 1-median. Minieka [88] defined the general absolute median of a network to be any point on the network that minimizes the sum of (unweighted) distances from it to the point on each edge that is most distant from it. Minieka showed that the general absolute median can be strictly interior to an edge; hence, the search cannot be confined solely to vertices of N. Slater [103] gave another generalization of the 1-median problem. In this generalization, each demand is a collection of vertices. The problem is to find a vertex such that the sum of the distances from that vertex to a nearest element of each collection is minimum. Slater showed that the set of vertices that solve this problem forms a connected path in T. For a general network, the problem can be solved by constructing a matrix that specifies the distances from each vertex -li to a nearest element of each collection. Simply sum each row of this matrix and choose the vertex whose row sum is minimum. Frank considered a probabilistic version of the 1-median problem in [34] where each weight is a random variable with a known distribu tion. A number of bounds are obtained on the expected value of the objective function as well as its variance. Some of these results are generalized by Frank [35] to the case where the weights are jointly distributed random variables. We now concentrate on the p-median problem with p > 2. p-Median of a network and vertex optimality. A significant theoretical contribution is due to Hakimi [48]. Hakimi proved that there exists an absolute p-median contained in V. Certain generaliza tions of this result have been given in subsequent work. Levy [80] proved that the (vertex-optimal) result holds when the weights w^ are replaced by concave cost functions c^(*) of the distance between v_^ and its nearest median. Goldman [41] generalized the result to the case of a "two-stage" commodity. More specifically, one distinguishes a vertex as being a source or a destination. Let (v ,v.) be a source-destination pair, S Cl and let x^ and x_. be the nearest medians to v and v^, respectively. Then the cost of transferring the commodity from source v to destina- s tion V, is the sum of three transport costs, namely, w .d(v ,x ) + a sd s 1 r\j wsdd(xi,x^.) + w*dd(x_. ,v) In general, if X = {x^...^} is a median set, one does not know which median is the nearest to v or v,; hence, s d the cost associated with a source-destination pair (v ,v,) is s d given by fsd(x) = min Kd^VV +"sdd(xi*xj) + Wsdd(xjVd)] -28- and the objective to be minimized is f(X) = J [f (X) : (v ,v,)eVxV], u sd s d Goldman showed that there exists an optimal X* contained in V, and conjectured that the result holds for any multi-stage problem. Hakimi and Maheshwari [49] proved a stronger version of Goldman's conjecture. In this version, there are multiple commodities for each source-destination pair, and each commodity goes through multiple stages. Furthermore the cost of transport from one stage to the next is a concave nondecreasing function of the distance. More specifically, let M be the set of commodities to be transferred from source v to sd s destination v^, and let g(m) be the number of stages commodity meM^ is to go through. For a given location set X = {x.,...,x }, denote 1 P by y^ = the location where the rth stage processing takes place. The cost of transferring commodity m from source v to destination v, s d is given by C^Jd (v^yp ] + ] + ... + C^d (yg(n),vd) ], where () is a concave nondecreasing function of the distance. Denoting this quantity by f^^(Y), with Y C X, |y| = g(m), the minimum cost of transfer for commodity m is given by f (X) = min[f (Y): sdm sdm Y C. X, |Y| = g(m)]. The cost of transferring all commodities from vg to Vj is obtained by summing over all commodities, that is, fgd(X) = J [fsdm(X): meM d]. The total cost of the system is obtained by summing the cost f ^(*) over all source-destination pairs, that is, f(X) = Â£ [f^W: (v ,Vj)eVxV]. Hakimi and Maheshevari proved that there exists a minimum X* of f(X) contained in V. Wendell and Hurter [111] considered a more general form of the problem where the transportation cost functions are permitted to differ from edge to edge. The transport cost on any edge is a non decreasing concave function of the distance. They proved that it is -29- sufficient to consider the vertices of the network under such a cost structure. Furthermore, they obtained the conditions under which it is necessary for the solution to occur at the vertices. In particular, they showed that nonvertex optimal locations can occur in any given edge, only when transportation costs are linear with distance over that edge and in that case, when and only when the slopes of these linear cost functions are in a special relation. Hence, if at least one cost function over some edge is nonlinear, then no interior point of that edge can be in an optimal solution. If the same situation holds for every edge, then a solution must necessarily occur at the vertices of the network. Solution approaches. Kariv and Hakimi [66] showed that the p- median problem on a general network is at least as hard as NP-complete problems. For the case of tree networks, however, algorithms of polynomial complexity have been developed. Matula and Kolde [85] 3 2 suggested an 0(n p ) algorithm for finding the median of a tree net- 2 2 work. Kariv and Hakimi [66] proposed an 0(n p ) algorithm for the same problem. For general networks, a number of solution procedures have been developed subsequently, all based on the vertex-optimality result. Their common characteristic is that they all confine the search to vertex locations. The solution procedures can be grouped in three categories: mixed-integer programming approaches, branch-and-bound techniques, and heuristics. ReVelle and Swain [95] formulated the problem as a linear integer program with 0,1 variables. The solution is obtained by applying the primal simplex algorithm to the associated linear program. In case -30- of non-integer termination, a branch-and-bound scheme is recommended to resolve the problem with integers. Their computational experience indicates that non-integer termination seldom occurs. Toregas, Swain, ReVelle, and Bergman [109] formulated a modified version of the problem as a mixed integer program. The modification is the presence of upper bounds on the distance between any vertex and its nearest facility. This formulation makes use of a related but simpler problem. This simpler problem is to minimize the number of facilities needed to cover all vertices of N within a specified critical distance. This problem is formulated as a set covering problem, and solved by ignoring the integer requirements. In case of non-integer termination, a single cut produced an integer solution in a large proportion of the cases. A somewhat different approach to solve the relaxed linear program is to use a decomposition scheme rather than applying the primal simplex algorithm. Swain [105] used a Dantzig-Wolfe decomposition approach to solve the associated linear program. Garfinkel, Neebe, and Rao [37] independently developed a decomposition approach similar to Swain's. In case of non-integer termination, they used group theoretics and a dynamic programming recursion to obtain an integer solution. A second approach taken is to solve the problem using a branch- and-bound technique. Khumawala [68] applied a branch-and-bound method of Land and Doig [77] type, to solve both the set covering problem and the modified p-median problem formulated by Toregas et al. He showed that the branch-and-bound approach is computationally efficient for the former but not for the latter. Narula, ,0gbu, and Samuelson [91] presented a branch-and-bound scheme which relies on obtaining the bounds by solving the Lagrangian relaxation of the integer programming -31- formulation using a subgradient optimization method. Another branch- and-bound method was developed by Jarvinen, Rajala, and Sinervo [62]. Their procedure looks for n-p vertices that do not belong to a p- median. This method works better for larger values of p, since n-p is smaller in this case reducing the number of possibilities. A similar branch-and-bound procedure was given by El-Shaieb [24]. The procedure is based on construction of a source set (i.e., p-median) and a demand set. Starting with both sets empty, a location is added to either set at each iteration. Whenever the number of elements in a source set reaches p, or the number of elements in a demand set reaches n-p, a feasible solution is obtained. An optimal solution is eventually identified using the lower bounds. A third approach taken is to use heuristics. A number of heuristics have been developed by Maranzana [84], Teitz and Bart [107], and Khumawala [69, 70]. For a discussion of a number of the solution approaches from a computational standpoint, the reader is referred to Hillsman and Rush- ton [59], and Khumawala, Neebe, and Dannenbring [71]. Stochastic networks and vertex-optimality. A number of pro babilistic versions of the p-median problem have been considered in the literature. Mirchandani and Odoni [89, 90] extended Hakimis vertex optimality result to the case of a stochastic network whose edge lengths are random variables. Berman and Larson [2] considered a stochastic network where the availability of servers (centers) is a random variable. They showed that under suitable conditions there exists at least one optimal set of locations on the vertices of such a network. This completes the discussion of the p-median problem. -32- The distance constraints problem The distance constraints problem involves locating new facilities on a network so that they are within specified distances of existing facilities as well as within specified distances of one another. The distance constraints arise naturally in a locational context if one wishes to require that a service facility be within a specified time (distance) of any point in the region it serves. Alternatively, in a military context, one may want to locate a number of units in such a way that units are neither too far from their supply bases, nor too far from one another, in order that one unit may reinforce another if necessary. To state the problem, let N be a network with the vertex set V = iv.,...,v ). Denote by X = (x,,...x ) any location vector in Nm, the m-fold Cartesian product of N by itself. Define the sets and I as follows: I = (j,k): 1 < j < k < m>, I = {(i,j): 1 i S m, 1 j n}. Here, the pairs (j,k) and (i,j) are assumed to be un ordered. Let I and I be two non-empty subsets of I' and I, d L B G respectively, and suppose we are given nonnegative finite numbers b jk for each (j,k)el and c.. for each (i,j)el_. a ij C The problem of interest is to find a location vector XeNm, if it exists, such that the constraints (1.3.3) are satisfied. d(x ,v ) < c (i,j)el 1 J 1J ^ (1.3.3) d(Xj,xk)-bjk (j,k)cIB Any vector XeNm satisfying (1.3.3) is called a feasible location vector. The distance constraints are said to be consistent if there exists at least one feasible location vector XeNm. -33- Goldman and Dearing [45] provide a conceptual discussion of, and a motivation for, considering such problems. The distance constraints are formally defined by Dearing, Francis, and Lowe [22] on a network. It was established in [22] that, in a well defined sense, the distance constraints define convex sets under the assumption that the under lying network is a tree. Furthermore, the distance constraints always define convex sets if and only if the network is a tree. Based on the results obtained in [22], Francis, Lowe, and Ratliff [32] considered the distance constraints on tree networks in more detail. They established the necessary and sufficient conditions for distance constraints to be consistent, and also devised algorithms that find a feasible location vector whenever one exists. In what follows we briefly discuss the results obtained in [32]. Distance constraints for a single new facility. For the case of a single facility, Francis et al. showed that there exists a feasible point xeT satisfying d(x,v.) < c. for iel if and only if the in- i 1 equalities d(v.,v, ) c. + c, are all satisfied for 1 1 1 < k < n. j k j k j - An equivalent statement of the single facility distance constraints can be given in terms of "neighborhoods" around v_^ of radii c^,. De fine the neighborhood N(u,r) around a point ueT of radius r to be the set of all points xeT for which d(u,x) < r. Then, a point x satisfies the constraints d(v^,x) < c^> iel, if and only if x is in each neigh borhood N(v.,c ), iel,if and only if x is in the intersection n 1 n N(v.,c.). It follows then that the single facility distance con- i=l 1 1 straints d(x,v.) Sc., iel,are consistent if and only if d(v.,v.) S ii j k Cj + ck for 1 j < k 5 n if and only if each pairwise intersection N -34- property, a "sequential intersection procedure" was developed that n determines the composite neighborhood N(a,r) = O N(v.,c.), with i=l 1 unique center a and radius r, by intersecting the neighborhoods N(v ,c ) one at a time in an arbitrary order. The procedure can be implemented in 0(n) operations. The composite neighborhood N(a,r) contains all alternate feasible points when the constraints are con sistent, and N(a,r) is always a convex compact subset of the tree network. A result was also given by Francis et al. that provides a sensitivity analysis on the constraints with no additional computa tional effort. Supposing that the distance constraints are consistent with the original upper bounds c^, consider an e-perturbation of the upper bounds, i.e., for some e > 0 define the new upper bounds to be c^-e, iel. If N(a,r) is the composite neighborhood corresponding to the original upper bounds, then it can be shown that for any e with 0 e ~ r, the e-perturbed constraints remain consistent and the set of feasible points to the e-perturbed system is given directly by N(a,r-e). Distance constraints for the multi-facility case. For the multi facility case, the necessary and sufficient conditions for the con sistency of distance constraints are given in terms of n(n l)/2 inequalities called "separation conditions." The separation condi tions are defined by means of an auxiliary graph constructed by using the sets I and I Let G be the graph with nodes N., 1 5 i < m, ij v 1 corresponding to new facilities,and nodes E ^, 1 < j < n,corresponding to existing facilities. The arc set A of G contains (N ,E ) if i j (i,j)cl-, and (N.,N ) if (j,k)el. The arc length of (N,,E.) is c_,. G j k B i j ij and of (N.,N^) is b.^. Under the (reasonable) assumption that G is -35- connected, denote by L(E.,E ) the length of a shortest path connecting J k nodes E. and E. for 1< j distance constraints are consistent on a tree network if and only if the inequalities (E^.E^) Â£ d(vj,v]P are satisfied for 1 < j < k < n. These inequalities are called the separation conditions. The proof of the consistency of the distance constraints implying the satisfac tion of the separation conditions uses only the triangle inequality and hence is applicable to all networks. The reverse implication always holds for tree networks, but may fail to hold for general net works. The proof of the reverse implication is constructive and actually finds a feasible location vector under the assumption that the separation conditions are satisfied. The method that constructs such a feasible location vector is termed the "Sequential Location Procedure" in [32]. The method can best be described with the aid of a physical model. One may imagine that the tree is represented by appropriately inscribing straight line segments on a board such that each segment represents an edge. At vertex v_^, strings of length c are fastened for each new facility j such that (i,j)el A tip vertex Li is chosen arbitrarily and all strings fastened at that vertex are pulled tight towards the adjacent vertex. If all strings reach the adjacent vertex, they are simply engaged there with their loose ends free to be pulled tight in some future iteration. Also the tip vertex together with the edge incident to it is removed from the model. The procedure is repeated with the resulting tree. In the other case, not all the strings reach the adjacent vertex when pulled tight. Among those which do not reach the adjacent vertex one which is shortest is selected, and the end point of this string determines the location of -36- the new facility it is associated with. All the strings pulled tight from the chosen tip are engaged at this new facility location. The feasibility of this location is checked with respect to all existing facilities and all other new facilities already placed on T. If the feasibility check is passed, new strings are fastened at this location associated with that new facility and other unplaced new facilities for which the distances are of concern. The procedure continues, treating each placed new facility like an existing facility, until, either all facilities are placed, or the current tree reduces to a point, in which case, all remaining new facilities are placed at that point. If the separation conditions hold, the procedure always finds a feasible location vector. The algorithm is of 0(m(m+n)) and is conjectured to be a best order algorithm in [33], for determining the con sistency of the distance constraints. Extensions of the results obtained in [32] are given by Francis, Lowe, and Tansel [33]. These extensions focus on the analysis of binding separation conditions which in turn determine the "uniquely" located new facilities. A separation condition that holds at equality is said to be a binding separation condition. If Z,(E.,E.) = d(v.,v,) 3 k J k is a binding separation condition, then any shortest path P(E.,E,) in J k the auxiliary graph G is said to be a tight path. New facility i is said to be uniquely located at point if in every feasible solution X to the distance constraints the location x. is the same. It was shown i in [33] that a new facility i is uniquely located if and only if node N_^ lies on at least one tight path. As an immediate consequence of this property the distance constraints has a unique feasible solu tion if and only if each N_^, 1 i < m, lies on at least one tight path -37- in the graph G. Furthermore, if some path P(E.,E. ) is a tight path, J k then the nodes representing facilities in the path occur with the same ordering and spacing in the path as do the locations representing the facilities in the path P(v.,v.) on T. This result enables one to J k locate the new facilities that appear in a tight path immediately, without having to use the Sequential Location Procedure. A multifacility minimax application of the distance constraints is given in [32, 33] and a multiobjective application is given in [33]. These two applications will be discussed subsequently. m-Center problem with mutual communication Let N be a network with vertex set V = {v,,...,v } and edge set 1 n E. Suppose the sets I and In are given with I,, C {(j,k): 1 j < k m} BO B and I C {(i,j): 1 < i S m, 1 < j < n}. We assume that we are given positive weights v ^ for each (j,k)elg and w for each (i,j)el^. For each location vector XeN, define the functions f (X), f_(X), and B 0 f(X) as follows: fB(X) = max[Vjkd(x^.,xk) : (j,k)eIB] , fc(X) = maxtw^dCx^v ) : (i,j)elc] , f(X) =max[fB(X), fc(X)] . The m-center problem with mutual communication is the following: Find a location vector X*eNm such that Z* = f(X*) = min[f(X): XeN] . -38- The problem differs from the p-center problem in two respects: (i) the distance between any vertex v and any new facility x_^ may be of concern as opposed only to the distance between v and the nearest new facility to v.; (ii) certain distances between new facilities are of concern, as opposed to the absence of interactions between new facilities. For the case of a single new facility the two problems coincide. In this problem, the new facilities may be thought to fulfill a supporting task to other new facilities as well as servicing those existing facilities that are a priori assigned to them. Certain planar cases of the multifacility minimax problem have been studied by Dearing and Francis [20]> Elzinga, Hearn, and Randolph [25], Wendell and Peterson [113],. and Francis [28l* The problem on a network is defined by Dearing, Francis, and Lowe [22] in the presence of distance constraints. It is established in [22] that the function f is a convex function on a tree network. The existence of a solution is guaranteed due to compactness and con tinuity considerations. Furthermore, it is shown that it suffices to consider only new facility locations in the convex hull of the existing facility locations in order to solve the problem. The problem on a general network was shown to be NP-hard by Kolen [72 ]. For the case of a tree network, the problem is solved by Francis, Lowe, and Ratliff [32] by using an equivalent formulation in terms of distance constraints (with variable right hand sides). The solution procedure finds Z* first, by using the separation conditions. Then an optimal feasible location vector X* is constructed by using the Sequential Location Procedure described in [32]* To find Z* an -39- auxiliary graph G is formed with nodes N ,...,Nm,E^,...,E Graph G contains arcs (N^,E^) with lengths l/w corresponding to pairs (i,j)el and arcs (N.,N ) with length 1/v corresponding to pairs G J K J K (j,k)eIB. It is assumed that G is connected, for otherwise the problem decomposes into subproblems. For each pair of existing facility nodes Ej, E^, define L(Ej,E^) to be the length of a shortest path in G connecting E. and E,. Francis et al. showed that Z* is given by J k max{d(v. ,v, )/Â£(E. ,E, ) : 1 j < k n}. The distances d(v.,v.) can be J K. J K J k 2 computed in 0(n ) operations for a tree network (see [23]), and the shortest path lengths Zr(E.,E,) are readily computable in 0(n ) opera- 1 k tions. When Z* is computed, the Sequential Location Procedure de scribed in [32] can be applied in 0(m(n+m)) operations to find a loca tion vector X* that solves the problem. m-Median problem with mutual communication Define the functions g g and g by the following expressions: D L For each XeNm gB(X) = l [v^dCx^Xjp: (j,k)eIB] , grQO = l [w d(x ,v ): (i,j)el ] , XJ 1 J o g(X) = gB(X) + gc(X) . The m-median problem with mutual communication is the following: Find a location vector X* in Nm such that Z* = g(X*) = min[g(X): XeNm] . -40- The problem differs from the p-median problem in two respects: (i) the distance between any vertex and any new facility may be of concern as opposed only to the distance between a vertex and the near est new facility to it; (ii) certain distances between new facilities are of concern as opposed to the absence of interactions between new facilities in the p-median problem. For the case of a single new facility, the two problems are identical. Planar cases of the problem using rectilinear or Euclidean dis tances have received considerable attention and efficient solution procedures have been developed. A thorough discussion of these prob lems is given in the book by Francis and White [31]. Other references on planar problems are Cabot, Francis, and Stary [6], Bindschedler and Moore [3], Francis [27], Eyster, White, and Wierville [26], Pritsker and Ghare [94], Wesolowsky and Love [115, 116], and Picard and Ratliff [93]. The problem on a network is defined by Dearing, Francis, and Lowe [22] in the presence of distance constraints. It was established in [22] that the problem is a convex optimization problem for all data choices if and only if the network is a tree. For the case of a general network, it is known that there exists an optimal solution on the vertices of N. This result and certain generalizations of it have been given by Goldman [41 ], Levy [80], Hakimiand Maheshwari [49], and Wendell and Hurter [ill]. These references are already discussed under the p-median problem. The problem was shown to be NP-hard by Kolen [72 ] on a general network, and no solution procedures have been developed yet. -41- For the case of a tree network, the m-median problem with mutual communi cation is solved by Dearing and Langford [21] and by Picard and Ratliff [93]. The approach used by Dearing and Langford is to embed the tree T into the Euclidean space R^, for some p, so that the distance between any two points on the tree is equal to the rectilinear distance between the corresponding points in R^. The problem in R^ with rectilinear distances decomposes into p subproblems, each of which can be solved by using known techniques given in Francis and White [31 ], or, perhaps more efficiently, by applying the network flow procedure discussed in Cabot, Francis, and Stary [6]. For reducing the computational effort, the embedding procedure is carried out with respect to a minimal path decomposition of T into p edge disjoint paths (each edge is in one and only one path). Each path in a minimal path decomposition corresponds to a dimension in R*5. The approach taken by Picard and Ratliff in [93] takes advantage of the vertex-optimality condition and determines an optimal solution (on the vertices of T) by solving a sequence of at most n-1 minimum cut problems, each on a graph containing at most m+2 nodes. The method is based on a result that an optimal location vector can be found independently of the edge lengths, by using only the incidence relations between vertices and the weights. In this respect, the pro cedure is in the same spirit as Goldman's algorithm for finding a median of a tree. Each cut problem corresponds to an edge of the tree. To be more explicit, the removal of all interior points of an edge e leaves two disconnected components, T^ = T^(e) and T^ = 12(e). Corresponding to edge e, a graph G = G(e) is constructed having nodes -42- 1 through m corresponding to new facilities, a source s and a sink t. Graph G contains arcs (s,i) and (i,t) for 1 < i < m and arcs (j,k) for each pair (j,k)eIB. The capacity of arc (j ,k) is the weight v^. The capacity of arc (s,i) is given by J [w : v eT., (i,r)el ], and the ir v i o capacity of arc (i,t) is given by J [w. : v eT, (ijqjel.,]. If xq q u (Q,Q) is a minimum capacity s-t cut of G, with seQ, teQ, then all new facility locations x^ for which the corresponding node i is in Q are in T^ in an optimal solution. Similarly, all x_. for which the node j is in Q are in T^ in an optimal solution. The procedure is a repeated application of this minimum cut problem with respect to each edge, until an optimal vertex location is determined for each x^. During the process, each x^ whose location is determined is treated like an existing facility. The method is described originally for the analogous rectilinear distance problem on the plane, which, in turn, decomposes into two subproblems, each on a line. Multi-objective location problems on networks Multi-objective optimization problems, sometimes known as vector optimization problems, involve decision making under two or more criteria. More explicitly, a set (finite or infinite) S of alterna tives is specified and n (possibly non-commensurable) objective func tions are to be minimized over S. Let f,,...,f be n numerical func- 1 n tions defined on S, and define f(x) = (f,(x),...,f (x)) for all xeS. 1 n The multi-objective optimization problem (VMP) is the following: V-min f(x) xeS In general, the minima of the functions f_,...,f do not coincide. 1 n In order for the minimization to be meaningful, one needs to introduce -43- the concept of "efficient solutions." A point x in S is said to be efficient if there does not exist a point y in S such that f^(y) < f_^(x) for 1 i < n and f^(y) < f^Cx) for at least one index k. One is interested in finding and characterizing the set of efficient solu tions to (VMP)An efficient point is sometimes known as an undominated point. A point which is not efficient is said to be dominated. Kuhn and Tucker [76] and Koopmans [74] are among the first to introduce the concept of efficiency. Geoffrion [40] extendd the con cept to "properly efficient" points and provided a comprehensive theoretical framework for subsequent research. Necessary and suf ficient conditions for efficient points to be properly efficient are given by Wendell and Lee [112]. Some of the later contributions are due to Yu [117], Yu and Zeleny [118, 119], Bitran and Magnanti [4], Wendell [110], and Bergstresser, Chames, and Yu [l ] We note that there are other approaches to multicriteria decision making, such as goal programming, multi-attribute utility theory, construction of outranking relations, and interactive programming techniques. For general information on multicriteria decision making, the reader is referred to Roy [99], Starr and Zeleny [104], Cochrane and Zeleny [16], Keeney and Raiffa [67], and Thiriez and Zionts [l08]- A survey of multicriteria decision making is given by Chalmet [7]. Multi-objective location problems (on the plane or on networks) have begun receiving attention only recently. Kuhn [75] appears to be the first to consider a multi-objective location problem on the plane. Kuhn considered the problem of minimizing the vector of Euclidean distances from a variable point to a set of fixed points on the plane, and showed that the set of efficient solutions is the convex -44- hull of the fixed points. Wendell, Hurter, and Lowe [114] considered the same problem with rectilinear distances and provided algorithms of 2 3 0(n ) and 0(n ) for generating efficient points. A most efficient algorithm of O(nlogn) was developed by Chalmet and Francis [8] for the same problem. McGinnis and White [83] considered the problem of minimizing the sum of and the maximum of weighted rectilinear distances from a variable point to a set of fixed points on the plane and formu lated the problem as a parametric linear program for which known solu tion techniques exist. Juel [64 ] considered the same problem for the case of multiple new facilities and gave an equivalent parametric linear program. Chalmet, Francis, and Lawrence [ 9 10 ] considered two variants of an efficient design problem, where the location variable (a design) is a planar region of specified positive area but of unknown shape. A few papers have been produced on multi-objective location problems on networks. In what follows we discuss these problems. The cent-dian problem. The single facility "cent-dian" problem involves the sum of and maximum of weighted distances from a new facility to a set of existing facilities at vertices of N. To define the problem, let w^ and h_^ be two positive weights associated with vertex v iel = {1,...,n}. For each point xeN define: m(x) = J {w_jd(v^,x): iel} , c(x) = max[h^d(v^jx): iel] f(x) = (m(x), c(x)) -45- The problem of interest is to find all efficient points with respect to f(x). Halpem [52] is the first to consider this problem. Halpern formulated the problem in a slightly different manner by considering a convex combination of m(x) and c(x). For any fixed X, 0 < X 1, define f(X,x) and f*(X) by f(X,x) = Xm(x) + (1 X) c(x) for xeN } f*(X) = min[f(X,x): xeN] (1.3.4) In Halpem's terminology, the function f(X,x) is called a cent-dian function and any point x* = x*(X) that solves (1.3.4) is called a cent-dian point. In [52] Halpem considered this problem on a tree network with weights h all equal to unity. Defining x^ and x^ to be the (vertex) median and the absolute center of T, respectively, Halpem proved that for any given X, the cent-dian x*(X) is located at either x^ or on one of the vertices located on the path P(x ,x ). This theorem pro- m c vides the basis for a simple and efficient algorithm to locate the cent-dian by inspecting the vertices on P(x ,x ). Further, Halpern m c showed that, if the absolute center x is known, then the cent-dian c can be found by determining the median of a tree T' that is identical to T except that T' contains an additional vertex v = x with the n+1 c associated weight w = X 1. n+1 Handler [56] formulated the same problem on a tree network in a slightly different manner by using the median function as a constraint. In Handler's formulation one is interested in solving the problem -46- P for each given a, where P is defined as follows: a b a e(a) = min[c(x): m(x) Â£ a, xeT] Efficient solutions are obtained by parameterizing on a. Handler's results closely parallel Halpem's. The problem on a general network is studied by Halpern [54]. using the convex combination approach. Halpern showed that the problem is a computationally finite one. Computational finiteness follows from the result that f(X,x) is a continuous, piecewise linear function of x on each edge and attains its minimum at one of a finite number of points. Defining Q(e) to be the union of the end points of edge e with the set of local minima of c(x) on e, the minimum of f(X,x) over all x on edge e is a member of Q(e) for any given X, 0 < X < 1. De fining Q = U {Q(e) : eeE}, it follows that the cent-dian x*(X) is con tained in Q for any X. Further, Halpern showed that the function f*(X) = min[f(X,x): xeN] is a continuous, piecewise linear, concave function of X for 0 < X < 1. Based on these results, Halpern provided an algorithm which constructs f*(X) and identifies x*(X) for 0 X < 1. To construct f*(X), the algorithm inspects each edge one at a time and computes the set Q(e), unless a simple test indicates that edge e cannot contain any cent-dian for any X. An upper bound on f*(X) is carried through and improved, whenever possible, by examining the members of Q(e). Cent-dian problem and duality. In [53], Halpern studied the cent- dian problem on a general network from a different angle and obtained a duality relationship. Using an approach similar to Handler's median constrained problem, Halpern defined two problems, a median constrained -47- and a center constrained one. More specifically, for real A and y define the functions m*(A) and c*(y) as follows: m*(A) = min[m(x): c(x) < A] (1.3.5) c*(y) = min[c(x): m(x) y] (1.3.6) In general for some values of A (y), the constraint c(x) Â£ A (m(x) Â£ y) may not admit any feasible solution. However, real inter vals C and M can be defined so that for any AeC and for any yeM, the constraints in (1.3.5) and (1.3.6) admit a feasible point. To define C, let Q be the set of all minima to min[c(x): xeN], and let 2 -be c m the set of all minima to min[m(x): xeN]. Let x be a point in that c minimizes the value of m(x) over all x in Similarly, let y be a point in Â£! that minimizes the value of c (y) over all y in ft Then m J J m C and M are defined as follows: C = [c(x), c(y)] M = [m(y) m(x) ] With these definitions Halpem's duality theorem can be stated as follows: a) Given any yeM, with A = c*(y), we have c*(m*(A)) = A. b) Given any AeC, with y = m*(A), we have m*(c*(y)) = y. For a tree network, the functions m* and c* are 1-1 and onto. It follows from the duality theorem that the function m* and c* are inverses of each other for a tree network. For a general network, the functions m*, c* need not be onto, i.e., the image of the domain -48- may only be a proper subset of the range. Hence, the inverse property holds only for some members of C and M for a general network. Now, we consider a more general multi-objective problem due to Lowe [82]. The problem involves a single facility to be located on a tree network with respect to m convex objective functions. Multi-objective convex location problem (on a tree). Let T be a tree network and let f,,...,f be m convex continuous bounded func- 1 m tions each of which is defined on T. In general, not all points in T may be feasible with respect to f_^. Let be a convex compact subset of T which contains all feasible points x with respect to the ith optimizer. The set Q, may be defined by specifying its extreme points, or by means of distance constraints, or by other means. We assume m that Q. is known or computable. Define Q = D Q. and assume that Q 1 i=l 1 is nonempty. The problem of interest is to find all efficient points in Q with respect to the vector minimization problem defined below: V-min[f(x): xeQ C T] where, f(x) = (f1(x),...,f (x)) for all xeT i m We note that Q is a convex compact subset of T as it is the intersection of m convex compact subsets of T. For a formal dis cussion of convexity on a network, the reader is referred to Dearing, Francis, and Lowe [22]. Loosely speaking, Q a convex subset of T, means Q is connected or that the (shortest, unique) path connecting any two points in Q is contained in Q. Lowe makes no assumptions on the specific forms of the objective functions. Under the convexity assumptions, Lowe proves that a convex -49- compact subset T* of T can be identified that contains all efficient points. To identify T*, define R* to be the set of all minima to the unconstrained problem min[f,(x): xÂ£Tl. If R* intersects the feasible 1 1 set Q, define S* to be this intersection. Otherwise, S* is the unique i i closest point in Q to R*. Having defined each S*, 1 < i m, if their intersection is non-empty, then the set of all efficient points is given by T* = H{S*: 1 i < m}. If this intersection is empty, then T* is the smallest compact convex subtree that intersects each S*. It can be shown that each R*, S* is convex, compact, and that T* is a li convex compact subset of T. Lowe's theorem assumes a knowledge of set of minima to each f as well as a knowledge of and hence Q. We note that the functions c(x) and m(x) in the cent-dian problem are both convex on T. Hence, Halpem's results can be obtained by apply ing Lowe's theorem. Now, we consider a multi-objective problem which involves multiple new facilities to be located on a tree network so that the distance between each specified pair of new and existing facilities, and each specified pair of new facilities is, roughly speaking, "as small as possible." The problem is defined by Francis, Lowe, and Tansel [33] as a sequel to the distance constraints problem, and solved by making use of the separation conditions. Here, we call the problem, the "multifacility vector minimization problem." The multifacility vector minimization problem (on a tree network). Let T be a tree network and let I I be given nonempty sets with Iq c (ij): 1 Â£ i S m, 1 < j < n} and IB C {(j,k): 1 5 j < k < m}. The problem of interest is to locate m new facilities on T at points x ,...,x so that each distance d(x ,v.) (i,j)el and d(x.,x.) (j.k^I^ kjlcB -50- is "as small as possible." More specifically, we wish to find all efficient location vectors X = (x,,...,x ) in Tm with respect to the i m vector minimization problem V-min[D(X): XeT] where D(X) is the vector of distances d(x^,v^) (i,j)el^, and d(Xj,x^) (i,k)el The vector is formed by assuming any convenient ordering D of the members of the sets I_ and I. L> D Francis, Lowe, and Tansel [33] characterized efficient points by making use of distance constraints. By definition, a location vector Z in T is efficient if an only if there does not exist a location vector X in Tm such that D(X) < D(Z) and D(X) D(Z). Given a location vector Z, let b., = d(z.,z,) for (jjk^I,, and c.. = d(z.,v.) for jk 3 k J B xj 1 3 (i,j)el^,, and define the distance constraints (DC) of interest by d(xi*vj) cij (i,j)elc d(xj"xk)iV We note that DC is always consistent, as Z is always feasible to DC, and hence the separation conditions are always satisfied. The separation conditions for DC are defined by constructing a graph G with nodes 1 S j Â£ m, corresponding to new facilities and nodes E^, 1 i < n, corresponding to existing facilities. For each (i,j)elr,, the arc (N.,E.) is in G with length c.., and for each c 1 1 ij (j,k)el the arc (N.,N, ) is in G with length b.. We recall that a B j k jk point x_^ is uniquely located in every feasible solution to DC if and only if the corresponding node N is in at least one tight path in G, -51- where a path of G joining any two existing facility nodes E and E s t is said to be tight if the length of the path is equal to the distance between the vertices v and v in T corresponding to nodes E and E , S t 1 w St respectively. For any given location vector Z, denote by A^(Z) the collection of locations of uniquely located facilities whose nodes are adjacent to N_^ in G. Let H[A^(Z)] be the convex hull of A^(Z), i.e., the smallest connected subtree containing all points in A^(Z). With these definitions, it was proven in [33] that the following conditions are equivalent: (i)Z is efficient. (ii)Z is the unique solution to DC. (iii)Each is in at least one tight path in G. (iv)Each Z. is contained in H[A.(Z)], 1 < i Â£ m. 1 1 This completes the discussion of multi-objective location problems on networks. Path Location Problems Here, we consider three versions of a path location problem posed by Slater [102]. To define the problems, let P denote any path con necting any two vertices in a network N. For any vertex veV and any path P, define the distance D(v,P) to be the distance from v to a nearest vertex in P. Also define the branch weight bw(P) of a path P to be the maximum number of vertices in any component of N-P. The three versions of the problem are the following: min l D(v,P) (1.3.7) P C N veV -52- min max D(v,P) (1.3.8) P C N veV min bw(P) (1.3.9) P C N In Slater's terminology, any path P* that solves (1.3.7) is called a core of N. Among all paths that solve (1.3.8), one with the fewest vertices is called a path center of N. Similarly, among all the paths that solve (1.3.9), one with the fewest vertices is called a spine of N. Slater obtained a number of properties of these problems for tree networks. In particular, Slater showed that the path center of T is unique and contains the vertex center of T, and that the spine of T is unique and contains the centroid (equivalently, the vertex median) of T. We recall that a centroid of T is any vertex v that minimizes the maximum number of vertices in any component of T-v. Also, Slater proposed two algorithms of linear order for determining the path center and the spine of T. CHAPTER 2 DUALITY AND THE NONLINEAR p-CENTER PROBLEM AND COVERING PROBLEM ON A TREE NETWORK 2.1 Introduction and Related Work We consider the problem of locating p new facilities on a tree network with respect to n existing facilities at known locations so as to minimize the maximum "loss." The problem is an extension of the linear p-center problem to the nonlinear case. We assume a strictly increasing, continuous "loss" function is associated with each of a finite number of demand points (existing facilities) whose argument is the distance between the corresponding existing facility and its nearest new facility. Our formulation permits the use of quite general loss functions provided that they are continuous and strictly increas ing with the travel distance. The term "loss" is used generically and may refer to any form of inconvenience such as cost, disutility of service, travel time, etc. In locating emergency service facilities, the disutility due to "late" service may be too great beyond a certain "threshold" response time. Such sharp changes in the disutility of service can be re flected into the model by using nonlinear functions. Hurter and Schaefer [61 ] justify and use such functions in a fire setting. As pointed out by Dearing [183 a study by Kolesar et al. [73 ] revealed that the travel time for fire trucks can be approximated by a particular continuous, nonlinear, increasing function of the distance. -5V -54- The literature on the p-center problem is discussed in detail in Chapter 1. Here, we give a brief review of the more closely re lated work. Except for p = 1, we know of no literature on the non linear p-center problem. For p = 1, the only references we are aware of which deal with the nonlinear case are Dearing [18] and Francis [29]. Both authors showed that the minimax loss with respect to any two existing facilities is a lower bound on the maximum loss with respect to all existing facilities, and that the largest of the lower bounds determines the minimax loss to all existing facilities on a tree network. This result is an instance of the duality result we will present in this chapter. The linear (weighted or unweighted) p-center problem is shown to be NP-complete on a general network by Kariv and Hakimi [65], and by Nemhauser and Sheu [92]. The linear 1-center problem on a tree network is well solved (see Goldman [44], Halfin [51], Lin [81], and Dearing and Francis [19]). For p > 1, the linear p-center problem on tree networks is considered by various authors. Handler [57] provided an 0(n) algorithm for finding the 2-center of a tree for the unweighted case. Kariv and 2 Hakimi [65] gave an 0(n logn) algorithm for tree networks which relies on solving a sequence of covering problems for the weighted case with p > 1. A similar procedure for the unweighted continuous p-center problem on a tree network is given by Chandrasekaran and Daughety [12]. A vertex-restricted version of the problem is solved by Chandrasekaran and Tamir [13], and relies on solving a sequence of clique covering problems on a related intersection graph. -55- The first duality relationship involving tree network location problems can be found in Meir and Moon [ 86 ]. Cockayne, Hedetniemi, and Slater [ 17 ] obtained a more general version of the result given in [ 86 ]. The results in [ 86 ] and [ 17 ] closely parallel our duality result for the covering problem and its dual. Shier [100] discovered a "dispersion" problem which is dual to the continuous unweighted p-center problem. The dispersion problem of Shier is to choose p+1 points in the tree network the nearest two of which are as far apart as possible. Chandrasekaran and Tamir [14] observed that Shier's duality holds when the problems are defined with respect to a subset of the tree. For the case where this subset is a finite collection of demand points, their result is an instance of the duality relation ship we will present in this chapter, as applied to the unweighted linear case. At this point we give a brief overview of the chapter. In Sec tion 2, we define the (nonlinear) p-center problem and a dual "dis persion" problem. We state and prove a weak duality theorem applicable to all networks, and state a strong duality theorem applicable to tree networks. In Section 3 we give a physical interpretation of the dual dispersion problem. In Section 4 we study a covering problem and present an algorithm, COVER, for solving it. The covering algorithm provides the basis of our solution procedure to the p-center problem as well as the dual dispersion problem and yields a construc tive approach for proving the strong duality theorem. In Section 5 we present an algorithm, OPTKLIQUE, which provides a constructive proof of the strong duality theorem, while solving the dual problem. Addi tional results for the covering problem, including a "divergence" problem dual to the cover problem, are given in Section 6. -56- 2.2 Problem Statements and Duality We suppose given a finite undirected tree network with positive arc lengths and denote by T an imbedding of the given network having as edges rectifiable arcs. For any two points x,yeT, let d(x,y) denote the shortest path distance between x and y. Let J = {1,...,n} and denote by V = {v^,...,vn) (V C T) a collec tion of distinct vertex locations of "demand points" or "existing facilities." Let X = {x.,...,x } (X C T) denote a finite collection 1 P of "centers" or "new facilities." For ieJ, define the distance of v. 1 to its nearest center by D(X,v^) = min{d(x^,v_.) : 1 < i < p}, and. let Sj = maxid(x,v.): xeT}. Also, for jeJ, we assume given a real valued function f continuous and strictly increasing, with domain [0,6^] and (clearly) range [f (0) ,f^ (6^)] For X C T, |x| < o, we define the function f by f(X) = max{f.(D(X,v^)): jeJ} The Primal p-Center Problem is as follows: Find a p-center X* for which rp = f(X*) = mini f (X) : XCT, Â¡X | = p} . (2.2.1) As discussed in Dearing and Francis [19], due to compactness of T and continuity of d(x,.) on T for each fixed xeT, an optimal solu tion X* to (2.2.1) exists and is contained in the convex hull of V. With a and p defined by a = max{f.(0): jeJ} and n = min{f (5 ): J J J jeJ}, we shall assume a < n, for if a = f (0) > f (6 ) = n. say, then s t t the function f would always be dominated by (strictly smaller than) -57- f and hence f could be deleted from the definition of f without s t changing f. Further, we assume p < n-1, as otherwise the p-center problem is trivial. So as to state the dual problem, we define 8., =8,. for j,keJ by j K Kj 8., = min maxif (d (x ,v.) ) f (d(x,v,))} 3k 3 3 K K For j,kej with j < k we define a., = maxif .(0), f, (0)} and 3k 3 k b#, = min{f. (5.) ,f (6 ) }. We note that a 5 n implies [a., ,b ., ] ^ 0. jK 3 3 k k JK jK The following lemma, the results of which are proven in [29] provides a closed form expression for 8jk Lemma 2.2.1. For any j,keJ with j 5 k we have: (i) The function f.^ + f exists, is stricly increasing, continuous 3 k has domain [a., ,b., ] ^ 0, and range [L., ,U., ], where L = JK 3K jK JK jk ifT1 + f^)o(ajk) and Ujk (f'1 + (ii) d(v.,vk) < U.k. (iii) The function (f.^ + f ^ exists, is strictly increasing and 3 k continuous, has domain [L ,U., ] and range [a., ,bM ]. JK. jk Jk jk (iv) 8_.k = (f"1 + fj^1) 1o(max{d(v_. ,vk> L^}) . We remark that either 8., = a., or 8., = (f. ^ + f, *) ^o(d(v ,v, )); jk jk jk j k j k" 8.. e [a., ,b.. ], and 8.. = f.(0). The closed form expression for 8., JK 3k jk jj j r jk given in Lemma 2.2.1 facilitates construction of the dual problem. We define the dual objective function g on subsets of V as follows For any K C V with K1 t 2 g(K) = max{g1(K), g2(K)} (K) = min(8ij: v^v^ e K, i j} g2 (K) .= max{fj(0): v^ e K} -58- The Dual Dispersion Problem is as follows: Find a subset K* of V such that g(K*) = max{g(K) : K^V, |k| = p+1} (2.2.2) We remark that the dispersion problem is meaningfully defined for 2 p+1 < n. The primal p-center problem is trivial for p > n. Hence, we shall restrict p to 1 p < n-1. In what follows in this section, we prove a Weak Duality Theorem (W.D.T.) and state a Strong Duality Theorem (S.D.T.) (proven in Sec tion 5). At the end of this section, we give an example problem illustrating definitions and results. In the W.D.T. we shall use the fact (readily proven as in [18] or [29]) that a f (X) for any XC T, |x| < . Theorem 2.2.1. (Weak Duality Theorem). Assume 1 p n-1. For any X C T with |X| = p, and any K C V with |k| = p+1, we have f(X) > g(K). Proof. There are two cases: g(K) < a or g(K) > a. In the former case we have g(K) 1 a < f(X). In the latter case, we note that g(K) = g^(K) > a > g^(K). Since |x| = p < p+1 = |k|, at least two demand points in K must be served by a single center. In other words, for some v ,v Â£ K with s ^ t, and some center xeX, we have s t f [D(X,v )] = f [d(x,v )] 5 f(X) s s s s (2.2.3) ft[D(X,vt)] = ft[d(x,vt)] < f(X) . Using the definitions and the inequalities in (2.2.3), we have g(K) = g^(K) < 3gt 2 max{fg[d(x,vs)],ft[d(x,vt)]} < f(X). Remark 2.2.1. We note that the conditions |x| = p and jKÂ¡ = p+1 can be replaced by |x| 2 p and/or |k| > p+1, 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 Â£ p n-1, there exists an X* C T with |x*| = p and K* C V with |K*| = p+1 such that f(X*) = g(K*). It is evident from the W.D.T. that X* solves the primal p-center problem and K* solves the dual dispersion problem. Before presenting an example problem, we find it convenient to view the dual problem as defined on "cliques" of a complete graph G. We define G to be the undirected complete graph with node set J, where node j of G represents vertex v of T. To any arc (i,j) of G, i ^ j, we assign the length and, to any node j of G, we assign the node weight g = f^(0). We call any complete subgraph K of G a clique. We note that any nonempty subset of V induces a clique in G and vice versa. For this reason, an equivalent definition of g(.) on cliques of G can be given by defining g^(K) to be the length of a smallest arc in a clique K of G, g2(K) to be the maximum of the weights of nodes in K, and letting g(K) = maxig^K), g2(K)}. If the number of nodes of a clique K is known to be q, we call K a q-clique and (sometimes) write K Defining C (G) to be the collection of all q q q-cliques of G, an equivalent statement of (2.2.2) is as follows: Find a clique K* for which p+1 g(K*+l) = max{g(K): K e Cp+1(G)} . -60- Whether K refers to a subset of V or a clique of G, we prefer to call K a clique as long as it is clear from the context what K refers to. As an example of the nonlinear p-center problem, suppose that the function associated with node v is f^ (y) = (y + h^)8 for y e [0j ] where w^ h^, and 0 are given parameters. Appropriate restrictions are placed on the parameters to ensure that the f^ are strictly in creasing on [0,6j]. We note that the linear weighted p-center problem is a special case of this problem generated by choosing 0 = 1, h^ =0, and Wj > 0 for all j. For the given form of f the following are readily verified: f ^(r) = (r/w.)1^6 h., r e [f.(0), f.(6.)] , J 1 3 J 3 3 fT1(r) + fT1(r) = r1/8[(l/w )1/6 + (1/w )1/8] (h + h ) , 1 J 1 J 1 J r e [a.., b..] , 13 13 w .w. i 3 (fi + fj ) (y) r ^ ]./ j_ l/e^ 0 (y + h + h ) , [w.x'" + w.x/v]" 3 y e [L.., U..] Then, using the characterization of 3 as given in Lemma 2.2.1, we have 3.. = 13 .. d.. if L Â£d(v. ,v.) 13 13 3 i 3 (2.2.4) max[fi(0), f.(0)] if L^. > div^v.) where -61- Y. . ij w .w. 1 3 + w. 3 1/0)0 and d.. 13 [d(v^,Vj) + h^ + hj]^ Consider the tree network shown in Figure 2.1, where the numbers on the arcs represent arc lengths. The data given with Figure 2.1 corresponds to the parameters for j=l,...,6 where clearly, each f is strictly increasing. Using (2.2.4), the 3 values for this problem are shown in Table 2.1 along with the node weights f^(0). Figure 2.2 shows the dual graph G associated with the problem, where the number next to each node j is the node weight and the number on the arc between nodes i and j is 3^ Using Figure 2.2 it can be verified that the optimal cliques (specified here by their nodes) and associated g values are K* = (3,4), g(K*) = 13829.76; K* = {1,3,6}, g(K*) = 3600; K* = {1,3,5,6}, g(K*) = 1664.64; K* = {1,3,4,5,6}, g(K*) = 784; and K* = {1,2,3,4,5,6}, g(K*) = 225. Due to the duality theory, it then o o follows that the r^ for p=l,...,5 are, respectively, 13829.76, 3600, 1664.64, 784, and 225. 2.3 Dual Problem Interpretation We imagine two conservative adversaries, an aggressor A and a defender D. Defender D has defense forces placed at vertex locations Vl,',,Vn' Aggressor A will attack a single vertex in V. Although D knows A will attack a vertex, he will not know the vertex attacked until the attack occurs. Defender D has p response forces which he must position at loca tions defined by a p-center X. Interpret tree distances to be travel times, so that D(X,v.) is the minimum time to respond to v. from a -62- n V T f 20 22 -Cr 10 t)V 6 fj(y) = wj(y + Hj)0 Data 0 n 2 19 0 2 25 0 3 16 2 6 36 0 5 4 0 6 9 4 Figure 2.1. Example Nonlinear p-Center Problem -63- Table 2.1 Values and Node Weights for Example i 1 2 3 4 5 2 3 4 5 6 225 3600 3600 3600 4356 3600 3600 3600 4556.25 13829.76 8464 11664 900 784 1664.64 j 1 V0) 0 2 3 4 0 64 0 5 6 0 144 -64- Figure 2.2. Dual Graph for Example -65- center in X. Assume A and D know functions f,,...,f so that 1 n f.(D(X,v.)) is D's loss if A attacks v. and D responds to the attack 3 3 J in a time of D(X,v^). For convenience, we refer to the loss A in flicts on D as A's gain. Aggressor A knows D has p response forces, but does not know how D will position his response forces. Thus A acts conservatively and bases his decision on a worst case analysis. If A decides to attack Vj without threatening any other vertices, A reasons that D will cor rectly guess v is to be attacked and will position a response force at v.. Hence A assumes his gain will be f (0), if he decides to J J immediately attack v^ without a prior threatening strategy. In order to gain more, A concludes that he must threaten, i.e., pretend to attack, q vertices, q > 1, so that even if D knows which q vertices are threatened, D does not know which vertex A will attack until the attack occurs. Thus D is forced to respond to the threat by position ing his response forces optimally with respect to these q vertices. Hence if A threatens K C V, he assumes D will choose a p-center X q which minimizes f(X:K ) = max{f.(D(X,v.)): v e K }. Thus, with q j j j q q p, A assumes D knows and will position a response force at every vertex in K so that A can gain at most g (K ) The best A q Z q can do in this case is to choose a K which contains some vertex v q s for which fg(0) = a. Hence, if q < p, A's maximum possible gain is at most f (0). (Parenthetically, we remark that if f (0) = r , s s p p < n, then it can be shown that not all f.(0) have the same value. 3 If all f.(0) do have the same value, then r > a.) On the other hand, 3 p if A chooses a subset with q > p, D is unable to position a response force at every vertex in K even if he knows K so A will gain at q q -66- least (K ). Hence A observes if he chooses some K with q > p which 2 q q contains a vertex v for which a = f (0), then his gain is at least s s a = g (K ). However, A recognizes that there may be some other K 2 q q with q > p, which may or may not contain v but which yields him a gain strictly greater than a. For this reason A restricts himself to those subsets of V with cardinality greater than p and realizes that if he chooses some K with q > p, then there is at least one pair of q vertices in K which D can cover by only a single response force. If q v^ and v_. are one such pair in which are covered only by a single response force, say at x, then clearly A obtains a gain of at least 3.., as 3.. = min{max(f (d(x,v.)), f.(d(x,v.))): x e T} < max{f (d(x,v )), ij 13 i 1 3 1 11 A fj(d(x,Vj))}. Since A does not know which pairs of vertices D will cover by single response forces, once he chooses K^, A acts conserva tively, and assumes that D will cover a pair v ,v, e K for which a b q 3 = min{3..: v.,v. e K i ^ 3}. That is, by choosing a K with ab 13 1 3 q q q > p, A guarantees himself a gain of at least 3 = g. (K ). Hence clD J- CJ A's minimum gain due to threatening is g(K^) = max{g^(K^), g^CK^)}, so A chooses a K* with q > p which maximizes g(K ) over all K C V q q q with q > p. The question arises as to why A should choose p+1 vertices to threaten, and no more. By virtue of the W.D.T. and the remark follow ing it, if X* is an optimum p-center then f(X*) > g(K ) for all K q q with q > p+1. Thus r^ = f(X*) is an upper bound on A's gain due to threatening But the S.D.T. implies there is a p+l-clique, say K*+^, which attains this upper bound. Hence A need threaten no more than p+1 vertices to maximize his gain, as A cannot obtain any addi tional gain by threatening more than p+1 vertices. -67- There is also the possibility that A will make a false threat, that is, attack a vertex not among the ones he threatens. If D be lieves the threat is false and continues to act conservatively, he will simply choose a p-center X* to minimize f. But since there exists a p+l-clique KA+^ such that g(K*+p = f(X*), the greatest loss D can incur, given Xa, is the same as if he believes A's optimal threat to be real, and acts accordingly. Hence A cannot gain more by making a false threat. 2.4 Covering Algorithm In this section we study a covering problem, and present an algorithm for solving it. Our primary interest in the algorithm is the fact that it provides a constructive approach for proving results about the primal and dual problem. For this reason we purposely keep the algorithm simple, and use an analog string model to provide insight into the algorithm. The development of both the string model and the algorithm is motivated by an earlier string algorithm given in [32]. As in [32], an equivalent algebraic version of the algorithm is readily obtainable. We remark that two other quite efficient algo rithms [14], [15], exist for solving the covering problem, but they do not lend themselves readily to our needs. At this point we state the Covering Problem: Given r and the runction f, compute q(r) = mini|x|: f(X) S r, X C T} (2.4.1) It is readily seen that the covering problem has a feasible solution if and only if a < r. Further, with J(r) = {j: r < f^(6 )}, we shall -68- assume J(r) ^ 0, for if J(r) = 0 then the condition f(X) < r holds for all X C T and we (trivially) have q(r) = 1. The above assumptions permit the following equivalent statement of the covering problem: minimize |x| subject to D(X,v.) < f.^Cr), j e J(r) (2.4.2) 2 2 We refer to the covering algorithm as COVER. In order to state COVER a few definitions are convenient. We may imagine that the tree is represented appropriately by inscribing straight line segments on a planar surface such that each segment represents an arc. We fasten strings of length f ^(r) to each node vjj e J(r), of the inscribed tree, where, by convention, we allow strings of zero length. Every fastened string has one end permanently affixed to the planar surface. In addition, during the use of the algorithm we engage previously fastened strings at various points on the tree. When a string is engaged, some point of the string is permanently affixed to the tree such that there is no slack in the portion of the string so far en gaged. When strings are removed, we imagine that they are physically deleted from the string model. During each iteration of the procedure, we partition the original tree into two subsets: one green, the other brown. The green subset is always a tree, denoted as GT (for green tree), while the brown sub set consists of one or more subtrees of the original tree T, each of which is "rooted" at a node of the green tree. By convention, a root -69- node t will be in both GT and the associated brown subtree, denoted as BT(t). COVER 0) Initialize to GT = T, k = BT(v.) = iv.}. For every j e 3 J at v.. Define U = 0. 0. For every tip vertex v of T define J(r) fasten a string of length f ^(r) 3 o 1) Choose a tip t of GT. If GT = {t} go to 6). Else find a(t), the vertex in GT adjacent to t. 2) If no strings are engaged or fastened at t, remove from GT the subarc [t,a(t)] joining t and a(t), attach [t,a(t)] to BT(t), and go to 1) Else go to 3). 3) Pull all strings at t tight towards a(t). If all tight strings reach a(t) then engage them at a(t), remove [t,a(t)] from GT, attach [t,a(t)] to BT(t), and go to 1). Else go to 4). 4) Add 1 to k. Choose a shortest string engaged or fastened at t. Find the (unique) vertex, say v., at which the shortest string is (k) fastened. Construct ^ U {v^^}. Find the farthest point, say y, from t on [t,a(t)] to which the shortest string can reach. Locate x^ at y. Assign all strings at t to x^ and remove these strings. Attach [t,y] to BT(t) to obtain BT(x ), and remove [t,y] from GT. Go to 5). 5) Assign to x^ all other strings in GT which can reach x^, and re move all such strings. If no strings remain then let U = U and stop. K Else return to 1). 6) Add 1 to k. Locate x^ at t. Assign all strings at t to x^. Of the strings at t choose any one, and find the vertex v^ to which the chosen string is fastened. Let U = U, 1 1/ {v,,,.}, and stop. k-1 (k) -70- Note that each time COVER places a center at x^ in step 4) it identifies an associated vertex v^ which we call the distinguished vertex associated with x^. When centers x^,...,x^. have been placed in step 4), we call = iv^ ,... ,v^ } the distinguished set associated with {x^,...,^}. If the algorithm places q centers in total, then the set U defined by the algorithm consists of vertices v,,.,...,v, N, the first q-1 of which are distinguished vertices (i) (q) (when q >2). The last vertex is distinguished only if x^ is placed in step 4). Letting X = {x^,...,x }, we call U the primary set associated with X, and call the primary vertex associated with x^, i = l,...,q. We note that the primary vertices v(i)*,v(q) are distinct, for as soon as a primary vertex is identified, its string is removed, and thus the vertex is not available for any subsequent identification. Likewise the centers x ,. .. ,x are distinct, for if 1 q x. = x. with i < i, then all strings assigned to x. would have been i J J assigned earlier to x^, and so x^. would not have been located. Hence it follows that JU| = JXJ = q, and U 0, since JXJ k 1. The primary vertices will be of theoretical significance in proving our results. We now establish some properties of COVER. Property 2.4.1. COVER finds a feasible solution X to the covering problem with |x| < n. Proof. We first note that termination is clearly finite, since at each iteration either at least one string is removed, or some entire arc of T becomes colored brown. Since there are at most n strings initially, it follows that the X constructed satisfies |x| < n. Choose any v^,j Â£ J(r), and denote by x^ the center to which v. is assigned. Since the string fastened at v. reaches x,.., J (j / -71- d(x,..,v.) < f/(r). As D(X,v.) d (x ,.. ,v.) it follows that X is (j) J 3 3 U) 3 a feasible solution. Property 2.A.2. For any nonempty distinguished set U^, with vertices numbered so that U, = {v. ,... ,v, }, we have k 1 k v. e BT(x.) , 3 3 1 < j < k .-I. d(Xj,Vj) = fj (r), 1 5 j < k . (2.4.3) (2.4.4) Proof. Expression (2.4.3) is obvious. To show (2.4.4), choose any v in U. Let t be the tip vertex chosen at the first of the iteration tC in which x^. is placed. The algorithm causes the string at v^. to-be pulled tight along every edge connecting v^ to t, and to be pulled tight along [t,x.], with the string end point coinciding with x.. J 1 Thus d(v.,t) + d(t,x.) = f.^(r). But v. e BT(t) and x. e T-BT(t) or J J 3 3 3 x. = t so that d(v.,t) + d(t,x.) = d(v.,x.). Thus, (2.4.4) follows. 3 1 3 3 3 Property 2.4.3. Let X = ix^,...,x ) be the feasible solution con structed by COVER, with vertices numbered so that U = {v^,...,v } is the primary set associated with X. Assume q > 1. Then d(vivj) > + for 1 i < j Â£ q (2.4.5) Proof. We know the first q-1 members of U are distinguished vertices. Hence Property 2.4.2 implies v. e BT(x.), i i 1 < i < q-1 .-1 d(v.,x.) = f (r), 1 i S q-1 . (2.4.6) (2.4.7) For i < j, x^ is placed prior to x^.. Since v^ is assigned to x^ and -72- no t to x^ for 1 < i < j < q, v. was not in BTix^, and the string at v. did not reach x.. Hence J i v. e T-BT(x.), J i 1 < i < j ^ q d(xi,Vj) > f (r), 1 < i < j < q . (2.4.8) (2.4.9) But (2.4.6) and (2.4.8) give d(v ,v.) = d(v.,x.) + d(x.,v.) for i j 11 1 J 1 < i < j < q, from which, on using (2.4.7) and (2.4.9), (2.4.5) follows. We shall need the following remark, proven in [32]: Remark 2.4.1. Given any a.,a. e T and s.,s. > 0, there exists a-point i 3 i J x in T for which d(x,a^) < s^ and d(x,a^.) < s^ if and only if d(a^,aj) < s + s.. i 3 We are now ready to establish the optimality of COVER. Theorem 2.4.1. Given any r for which a < r and J(r) 4 0, COVER solves the covering problem. Proof. Let X = {x^,...,x^} be the point set found by COVER. Property 2.4.1 implies X is feasible to the problem. If q = 1, X is clearly optimal. If q > 1, let the vertices be numbered so that U = {v,,...,v } 1 q is a primary set associated with X. By Property 2.4.3, d(v_^,Vj) > f^(r) +* f ^(r), for 1 <, i < j < q. Remark 2.4.1 implies there exists no x in T for which d(x,v.) < f.^(r) and d(x,v.) < f .^(r) for any i i 13 i, j in {l,...,q} e J(r) with i < j. Hence it is impossible to cover any two members of U with a common center. Thus, since |u| = q, any feasible solution X to the covering problem satisfies |x| > q. Since q = |X| and X is feasible to the problem, X is thus an optimum feasible solution. -73- We remark that the covering problem may be of as much interest, from both a theoretical and applications point of view, as the p-center problem. In Section 6, we will present a problem which is dual to the covering problem and show that the primary set identified by COVER solves the dual of the covering problem. Furthermore we will charac terize q(r) as a step function, and provide a formula for q(r) assuming that r^ is known for 1 <, p n-1. 2.5 Dual Problem Solution and the Strong Duality Theorem Based on the W.D.T. and properties of COVER we now present a proof of the S.D.T. The proof is constructive in that we use an algorithm called OPTKLIQUE which, given the optimal objective value of the primal problem, constructs an optimal solution to the dual problem. We then show that the objective values of the pair of prob lems are equal. As a by-product the proof also establishes that r e R, where, for convenience, we define R ={Â£..: l
p ij J We find it useful to summarize Theorem 2.4.1 and Property 2.4.3 as follows: Lemma 2.5.1. Given any r for which a < r and J(r) ^ 0, the following assertions are true: (a) COVER finds an optimum solution X to the covering problem with q(r) = |X|. (b) Whenever q = q(r) > 1, any primary set U = ^v(l) associated with X satisfies g(U) = g^U) > r (2.5.1) -74 Proof. (a) is just Theorem 2.4.1. (b) From Property 2.4.3, for any v_^,v^ e U, i ^ j, we have d(v^,v^) > f.^(r) + f.^ir) > f.^Ca) + f.^a) where r Â£ a > a = a. .. Thus, i J i J. iJ d(v^,Vj) is in the domain of (f_^ + f ^) from which, upon using Lemma 2.2.1 and the definitions of g, g^, and g^, (2.5.1) follows. In the algorithm OPTKLIQUE we assume that r^ is given for some value of p, 1 g p n-1. OPTKLIQUE constructs an optimal solution to the associated dual problem. OPTKLIQUE 1) If r = a, take K*+^ to be any p+l-clique in V containing a vertex v for which f (0) = a, and go to 3). Else, given r > a, compute s s p r' = maxifS. e R: < r } and choose any r for which r' < r < r . P iJ P P P Go to 2) . 2) Apply COVER with the chosen value of r to find an optimum solution X and its associated primary set U, with |x| = q = |u|. Note r < r implies |X| > p, so q Â£ p+1. Take K*+^ to be any subset of U con sisting of p+1 members of U. Go to 3). (If q > p+1, there will be alternative optimal cliques.) 3) If K*+j, is any clique found in either step 1) or 2), then g(K*+p = r and the W.D.T. guarantees K* is an optimum solution to the dual P P+1 problem. Before proving the correctness of the algorithm, we note, since a = for some h, that a < r implies a < r', and thus the r chosen hh P P in step 2) is one for which a feasible solution exists to the covering problem. Theorem 2.5.1. Given r for any p, 1 g p < n-1, the clique K* con- P P+1 structed by OPTKLIQUE satisfies -75- S(Kp*+l> = rp (2.5.2) Furthermore, K* solves the dual dispersion problem. p+1 Proof. Let X* be an optimum p-center solution to the primal problem so that X* = p and f(X*) = r Since r S a we consider the cases P P r^ = a and r^ > a. Let us apply OPTKLIQUE for each case. For r^ = a, K*+^ is chosen in step 1) so that |K*_y| = p+1 and a = fg(0) = g2^Kp+l^ T^e W*D,Tt Sives S(K*+1) f(X*). But then, a = 82^K*+1^ = = f(X*) = r = a, establishing (2.5.2) for p+1"1 this case. c' P For r > a, define R = {3. e R: r < g..} C- R. Since r > r > P iJ P ij P there exists no g. in R for which r < g ,. < r Thus g.. > r implies ij iJ P 1J 3. > r and so it follows that 1J P R={g..:r (2.5.3) Let U be the primary set identified by COVER for the chosen r, r' < r < r By Lemma 2.5.1, U satisfies g,(U) > r from which it p p J bl follows that gy > r for v^*vj e U, i j. Hence, (2.5.3) implies By e R v.,v. e U, i j (2.5.A) J 3 Since |U| > p+1, let be that subset of U identified in step 2). We have the following string of inequalities: r P f(X*) > g(K*+1) 2 Mkh> = mlniByi Vj.Tj E K*+1, i j* j) > minig..: v.,v. e U, 1 i} ij i J > minig.. e R} > r (2.5.5) (2.5.6) (2.5.7) (2.5.8) (2.5.9) (2.5.10) -76- where (2.5.5) follows from the W.D.T., (2.5.6) and (2.5.7) follow from the definitions of g and g^, (2.5.8) follows from K*+^ c U, (2.5.9) follows from (2.5.4), and (2.5.10) follows from the definition of R. Hence, every inequality holds as an equality, establishing (2.5.2) for this case. The assertion that K*., solves the dual problem is immediate from p+i f(X*) g(K*+1) and the W.D.T. We note that Theorem 2.5.1 provides a proof of the S.D.T. since in the statement of the S.D.T. we take X* to be an optimum p-center solu tion to the primal problem and K*+^ as constructed by OPTKLIQUE. We also note that the duality theory provides necessary and sufficient conditions for a p-center to be optimal, which, as far as we know, are the first such conditions for this problem. We remark, just as with the linear p-center problem, that if we define 6 = minig..: g e R, q(B..) < p}, then 8 ^ = r Clearly st ij ij ij st p q(r ) < p and q(8 ) < p. The S.D.T. implies r e R, and thus the P st p definition of g ^ gives g ^ < r Let p' = q(g ) and let X solve st st p st p the cover problem for r = g so that f(X .) < g Since p > p', st p st append to X^, (if necessary) any p-p' center locations to obtain the p-center X^. Clearly D(X^,v^) D(X^,,Vj) for v e V, and thus f(X ) < f(X ,). Hence r f(X ) < f(X ,) < g < r so g = r P p p p p st p st p and X^ is an optimum solution to the p-center problem. This remark permits the use of the same procedures as discussed in [65] to compute r^ efficiently, by performing a binary search over the (ordered) list R, applying COVER for every r chosen from R until a smallest g in R st is found for which COVER finds p or less points. Once r^ is computed in this manner, OPTKLIQUE requires an additional application of COVER -li ter any r, r' < r < r and solves the dual dispersion problem. This P P approach is essentially a primal approach for solving both problems. An alternative approach which directly works with the dual graph is given by Chandrasekaran and Tamir [13] for the unweighted linear p- center problem, which works directly with what would be a subgraph of our dual graph G. Due to absence of weights and addends, their approach does not require the use of node weights (and for that matter the function g^) in the dual graph. For a given value of r, Chandra- sekaran and Tamir define an intersection graph IG with node set J and r arcs (i,j) for those indices i,j e J for which 8 < r. Their pro cedure is based on a graph theoretic procedure given by Gavril [39] and solves the covering problem by finding a minimum clique cover of IG^ (minimum number of cliques such that every node is in at least one clique). As a side result, their approach identifies a maximal anti clique in IG (a maximal set of nodes in IG no two of which are con- r r nected with an arc). Due to "chordal" properties of IG^ as discussed in [39], the cardinality of a minimum clique cover of IG^ is equal to the cardinality of a maximal anti-clique in IG^. This result is a special instance of the duality result we will present in Section 6 for the cover problem, as applied to the linear unweighted case. Furthermore, for r = r Chandrasekaran and Tamir [39] proved a duality P relationship for the unweighted p-center problem using the above properties of IG^. We remark that their duality results can be directly proven by using the algorithm OPTKLIQUE, and by appropriately specializing our S.D.T. for the linear unweighted case. We now demonstrate the use of OPTKLIQUE by determining K* for 4 the example problem. From our previous analysis, r^ = 1664.64. Since -78- r > a = 144, we compute (from Table 2.1) rl=max{8.. e R: 3.. < r} = 900. 3 3 ij ij 3 We next must apply COVER using a value of r where 900 < r < 1664.64. Figure 2.3 shows the results of using COVER with r = 1296. In the figure, the loose ends of the strings are shown as wavy lines. Brown subtrees are shown as crosshatched arcs of the original tree. Each separate drawing of the tree (a)-g)) is for a subsequent iteration of COVER. Figure 2.3a) demonstrates the initialization step, where for r = 1296, the f.^(r), j = 1,...,6 are 12, 7.2, 7, 6, 18, and 8, re spectively. The numbers next to the strings are the lengths of the loose ends. In the figure, we indicate which tip of the green tree is chosen at each return to step 1) of COVER. In addition, the suc cessive distinguished vertex sets are indicated. After the final iteration, we note that the primary vertex set U is {v,v, ,v- ,v,-} which, from our previous analysis, we know to be J 1 D 5 2.6 Results for the Covering Problem In this section we present a "divergence" problem which is dual to the covering problem. We give a weak duality and a strong duality result and prove that the primary set identified by COVER solves the dual problem. The term "divergence" is chosen to represent the physical interpretation, discussed later, in which the attacker A chooses a "divergent" set of vertices to threaten. Further, the term permits a distinction to be made between the two different dual prob lems. Also, in this section, we demonstrate how having optimum solu tions to the p-center problem for all p, 1 < p < n, enables us to completely characterize the function q(r). 1 n U 1 ji J I /..i t Ion (I M M > I (* V ? Choose v 1 Choose v 18 9 x. U4 = {v3WV5} K) u = {v3, v6, v5! k* o, i, <>, r> OPTKLIQUE for p = 3 for Example -80- The Divergence Problem is as follows: Given r and the function g, compute q(r) = max{ |u| : g(U) >r,UCV}. (2.6.1) That is, the problem is to find the maximum number of existing facili ties no two of which can be jointly covered by a single center within a radius of r. Equivalently, among all cliques of G whose gain is larger than r, the problem is to find one with the maximum number of nodes. The dual problem is feasible for r < r^, as, if r > r^ there does not exist a subset U of V for which g(U) > r. On the other hand, the primal cover problem is feasible for r > a. Hence, we shall re strict r to a < r < r^ in order to ensure feasibility to both problems. Theorem 2.6.1. (Weak Duality Theorem). Assume a < r < r^. For any feasible solution X to the primal cover problem, and any feasible solution U to the dual divergence problem, we have |x| > |u|. Proof. By feasibility of U and the assumption of the theorem we have g(U) = gl(U) > r > a > g2(U) from which it follows that 8.. > r v.,v. e U, i ^ j (2.6.2) ij i J Suppose |X| < Ju|. Then, the same approach as in the proof of Theorem 2.2.1 implies there exist v ,v e U, s ^ t, such that 8 < f(X) < r, st st contradicting at least one inequality in (2.6.2). Thus, |x| > |u|. Theorem 2.6.2. (Strong Duality Theorem). Assume a < r < r^. Let X be a feasible solution to the covering problem constructed by COVER. Then, the primary set U associated with X solves the dual divergence problem with X| = q(r) = q(r) = |U (2.6.3) -81- Proof. By definition of a primary set we have |x| = |u|. By assump tion r < r^ so that |x| = |u| k 2. Lemma 2.5.1 implies g(U) = g^(U) > r. Hence U is a feasible solution to the dual problem. Theorem 2.6.1 im plies q(r) > q(r). By feasibility of X and U, and the fact that |X| = |U|, we have |x| > q(r) > q(r) Â£ |u| = |x|. It follows that X solves the cover problem, U solves the dual problem, and (2.6.3) holds. We remark that the above proof is an alternative to the proof of Theorem 2.4.1 for establishing the optimality of X to the covering problem. Hence, an application of COVER solves both problems simul taneously. At this point we give an interpretation of the pair of problems. The defender D specifies an upper bound r on his loss against an attack to any vertex and will position response forces as necessary so that his loss will not exceed r. Each response force is an "expense" for D. Hence, D's problem is to choose the fewest possible response forces. The attacker A knows that D will not tolerate a loss exceeding r. Hence, A recognizes that, no matter how many vertices he threatens, D will have a sufficiently large number of response forces to respond and that the loss A inflicts on D will always be less than or equal to r. For this reason, A decides that he should not (hopelessly) try to inflict a loss to D exceeding r, and that, instead, he should force D into using as many of his response forces as possible. Hence, should A choose a subset U of V with g(U) > r, he knows that no two vertices in U can be jointly covered by a single response force by D within the specified upper-bound r. Thus, D, not tolerating a loss exceeding r, will have to allocate one response force for every vertex -82- in U. In total, any feasible X which D chooses will satisfy |x| > |u|, which is what the W.D.T. asserts. By virtue of the S.D.T., if U is A's optimal choice, D can choose exactly || response forces positioned at, say X, with |x| = |u| and still respond to an attack to any vertex in U (as well as in V-U) without incurring a loss exceeding r. If A threatens more than q(r) = |u| vertices, say, a subset U of V, then |UÂ¡ > q(r) implies g(U) < r (infeasibility). Thus, D would not be forced into allocating a single response for every member of U. In fact, even if A threatens every vertex in V, then D still needs ex actly q(r) = q(r) = || response forces to respond to the threat feasibly. Thus, if each threat is an "expense" for A, he need threaten no more than q(r) vertices. On the other hand, D adopts an optimal strategy against A's best threat by minimizing the number of response forces with respect to V. Continuing our consideration of the covering problem, we now re verse the usual procedure, and view the p-center problem as a device for solving the covering problem for all values of r for which the covering problem is feasible, that is, for a < r. The following lemma is the key to using the p-center problem to solve the covering problem. Define r = < for convenience. o Lemma 2.6.1. Let p e J. If r < r ,, then : r p p-1 q(r) = p for r < r < r . P P-1 Proof. We first note r < r < ... S r. < r_. Also, clearlv. n n-1 10 J q(r ) < p for p e J. Now for r^ Â£ r since q is non-increasing we have 1 > q(r^) > q(r) > 1, establishing the claim if p = 1. Consider the case p e {2,...,n}. From r < r < r we have p > q(r ) > q(r) > p p i p q(r ]_). Suppose q(r) = s, with s < p, implying s < p-1. Let X, -83- with |X| = s, solve the cover problem for r. r < r contradicting the definition of r pi s s r < r < r . P P"1 It now follows, if we define the set We then have f(X) < r < Thus q(r) = p for P = {(p-1,p): p e {2,...,n}, r^ < r^} > that q(r) r P for rp < r < r j, (p-l,p) e P 1 for r^ < r . (2.6.4) The formula (2.6.4) completely defines the function q(r), since r = a, n and the cover problem is feasible if and only if a < r. Hence if we solve the p-center problem for all p and compute r ...,r then we 2 n have an explicit formula for q(r), and we see that the r^ completely define the function q. For example, if r, = rc < r. = r < r_ = r,, then q(r) = 5 for r^ ^ r < r^, q(r) = 3 for r^ r < r^, and q(r) = 1 for r^ ~ r. Also, the proof of the lemma does not require the assump tion that the location network is a tree. Thus the formula for q(r) is still valid if the location network has cycles. CHAPTER 3 A VECTOR-MINIMIZATION PROBLEM ON A TREE NETWORK 3.1 Introduction We consider a vector-minimization problem on a tree network which involves as objectives the distances between specified pairs of new facilities and specified pairs of new and existing facilities. In many location problems, especially in the public sector, it may be necessary to build a number of public facilities which are to be shared by a number of communities. If the optimizers cannot agree on a single objective function, the analyst is faced with the problem of locating the facili ties in such a manner that all parties are satisfied with the end result. In such a case, the optimizers can agree to rule out "dominated" solutions and consider only "efficient" solutions. The related literature on multi-objective location problems is discussed in Chapter 1 under Multi-objective location problems on networks. Here, we concentrate on characterizing efficient solutions to the vector-minimization problem of interest. We relate efficient solutions to a distance constraints problem studied by Francis, Lowe, and Ratliff [32]. Extensions of results in [32] are given by Francis, Lowe, and Tansel [33]. We use the theory developed in [32] and [33] to establish the necessary and sufficient conditions for efficient location vectors (parenthetically, we remark that the results we proved in [33] are also given in our Dissertation Proposal defended on June 8, 1979). -84- -85- At this point, we give an overview of the chapter. In Section 2, necessary definitions and notation are given and the vector-minimiza tion problem of interest is defined. In Section 3, we relate the problem to distance constraints, give a number of related properties of distance constraints, and establish the necessary and sufficient conditions for a location vector to be efficient. In Section 4, we provide examples of efficient and non-efficient location vectors. Section 5 is devoted to a further refinement and simplification of one of the necessary and sufficient conditions, namely, "the convex hull property." In Section 6, we provide an algorithm, SEVCA, which con structs an efficient solution from a given location vector. In Sec tion 7, we characterize efficient solutions for the analogous problem in the p-dimensional Euclidean space with rectilinear (p 2). or Tchebychev (p > 2) distances. 3.2 Problem Statement We suppose given a finite, undirected tree network, and denote by T an imbedding of the given network. Let V = {v,,...,v } be a set of n distinct vertices of T. We assume existing facility i is located at vertex v^, i e {l,...,n}. For j e {l,...,m}, denote by x^ a point to be determined in T as the location of new facility j. We define Tm to be the m-fold Cartesian product of T by itself and define a location vector X in Tm to be the ordered m-tuple (x,,...,x ) with each x e T, 1 m j j e {l,...,m}. Sometimes, we refer to a location vector X in Tm as a point m T As in [22], given points x,y e T, we define the line L(x,y) to be the union of all points in the shortest path connecting x and y. In -86- addition, given a finite point set P C T, we define the convex hull H(P) to be the smallest (embedded) subtree of T containing all points in P. We note that for any two points p,p' e P, the line L(p,p') is contained in H(P). We denote by I the set of pairs (i,j) for which the distance u d(x^,Vj) is of concern. Similarly, 1^ is the set of pairs (j,k) for which the distance d(x.,x, ) is of concern. We remark that it need not 3 k be the case that 1^ includes all possible pairs of new and existing facility indices, nor I includes all possible pairs of new facility indices. With these definitions, the problem of interest is to "mini mize" each of the distances specified by (3.2.1); d(x >v ) (i,j) e I , 1 J O (3.2.1) d(x.,x ) (j ,k) e I . J K. d For X e Tm, we denote by D(X) the vector each of whose components is a distance specified by (3.2.1). The vector is formed by assuming any convenient ordering of the members of 1 and I. The vector- C B minimization (V-min) problem of interest is V-min{D(X): X e Tm} (3.2.2) With respect to (3.2.2), a location vector Z e Tm is said to dominate a location vector X in Tm if D(Z) < D(X) and D(Z) D(X). A location vector Z which is not dominated by any other location vector is said to be efficient. An equivalent definition of efficiency is as follows: Z e Tm is efficient if and only if X e Tm and D(X) < D(Z) imply D(X) = D(Z). -87- Our main interest is to characterize efficient location vectors and devise an algorithm for constructing efficient location vectors from a given (dominated) location vector. 3.3 Distance Constraints and Characterization of Efficient Points We make extensive use of the results obtained in [32, 33] for distance constraints to establish the necessary and sufficient condi tions for efficient points. The Distance Constraints (DC) are defined in [32] (independent of the efficiency problem) as follows: Given the sets 1^ and 1^ and nonnegative upper bounds c^. and b^ find a point X = (xx ) in Tm, if it exists, such that 1 m d(x.,v.) < c . i 3 ij d(x. ,x. ) Â£ b., 3 k J)k (i,j) e I (j,k) e I, (3.3.1) Corresponding to DC, we define Graph BC (GBC) as the undirected graph having nodes E.,...,E N,,...,N ; for every (i,k) e I,,, there i n i m is is an arc (N. ,N, ) of length b., between nodes N. and N, : for every J k' jk j k J (i,j) e I, there is an arc (N.,E.) of length c.. between nodes N. C i J ij i and E.. We further assume that the sets I,, and I are such that GBC J B C is connected, as otherwise DC decomposes into independent sets of con straints which may be analyzed separately. Given a node-path between any two nodes f and f in GBC, we de- P q note the path by P(f ,f ) and denote the length of the path by LP(f ,f ) p q p q We define (f^jf^) to be the length of any shortest path in GBC between nodes f and f Subsequently, unless we specify otherwise, it should p q -88- be understood that any path we refer to is a simple path between some two existing facility nodes and E^. Results on Distance Constraints The distance constraints are said to be consistent if there exists at least one feasible solution to (3.3.1). The following result is established in [32]. Theorem 3.3.1. The distance constraints are consistent if and only if d(v ,v ) < Â£(E ,E ), 1 Â£ p < q n (3.3.2) p q p q The inequalities (3.3.2) are termed the Separation Conditions [32], since each term on the right specifies an upper bound on how separate two existing facility locations can be. Except when stated otherwise, we assume throughout the chapter that the separation condi tions hold, and thus (equivalently) DC is consistent. We call a path P(E^,E^) between E^ and E^ in GBC a tight path if LP(E ,E ) = d (v ,v ). We note that since we assume DC is consistent, P q p q it necessarily follows if P(E ,E ) is a tight path, that LP(E ,E ) = p q p q L(E ,E ). Any path P(E ,E ) for which LP(E ,E ) > d(v ,v ) is called pq p q p q p q a slack path. We say that new facility i is in a tight path if there exists at least one tight path containing Ik. Every path containing Ik is slack if there is no tight path which contains Ik . The motivation for the above terminology is due to a string graph representation of GBC. This string graph is also useful for obtaining problem insights. When knots representing nodes E^ and E^ are pulled as far apart as possible, the distance between the two knots is L(E ,E ). P q -89- If then the string graph is placed upon the tree T, i.e., the strings only lie on arcs of T, a path is tight when it is necessary to pull the string graph tight in order to place the knots representing and on v and v respectively, while a path is slack if the string path P 9 must literally be slack when the two knots are placed to coincide with v and v . P q A priori, one might think that the occurrence of a tight path would be rare. However, we shall see that tight paths occur in a quite natural way when the separation conditions are used in the analy sis of efficient location vectors. Further, the notion of tight paths permits the specification of necessary and sufficient conditions for DC to have a unique solution. We now relate unique locations to tight paths. By definition, new facility i is uniquely located if it has the same location in every feasible solution to DC. Since we later refer to a collection of facilities, which contains possibly both existing and new facilities, being uniquely located, we note that existing facilities are uniquely located by definition. Theorem 3.3.2, which we proved in [33], specifies the necessary and sufficient conditions for a new facility to be uniquely located. Theorem 3.3.2. New facility k is uniquely located if an only if node lies in at least one tight path P(E^,E^). Corollary 3.3.2. Distance constraints have a unique solution if and only if node lies on at least one tight path in GBC for k = l,...,m. We now give an additional property of a tight path we proved in [33]. The property will be used in proving our main result on efficient points. -90- Property 3.3.1. If P(E^,E^) is a tight path in GBC, then (i) every facility represented by a node in P(E^,E^) is uniquely located, (ii) the locations of facilities corresponding to nodes in P(E^,E^) occur with the same ordering and spacing on the line L(v ,v ) in P 9 T as do the corresponding nodes in P(E ,E ). P q As an illustration of Property 3.3.1, suppose P(E^,E^) is a tight path with nodes E^, N2, N^, E^. Then, the locations v^, x^, v^ are unique. Furthermore, they occur in the given order on the line l(v1,v5) with d(v1,x2) = c21, d(x2,x3) = b23, d(x3,v5) = c35, where C21 ^23 C35 are t^ie -*-enSt^ls tbe arcs in the path. This example is illustrated in Figure 3.1. b 23 35 Tight Path p(e15e5) in GBC '21 23 C 35 x 3 Uvpvs) in T Figure 3.1. Illustration of Property 3.3.1. We now consider the problem of determining when an arc lies on a tight path. As an arc lies on a tight path if and only if it is not the case that all paths containing the arc are slack, we consider the -91- equivalent problem of determining when an arc lies only on slack paths. The following property, which we proved in [33], characterizes the con ditions under which an arc in GBC is not contained in any tight path. Property 3.3.2. Let DC be consistent. Let (f^,fj) be any arc in GBC, of positive length e^., whose length is reduced by some positive amount e. Let DC^CGBC^) be the distance constraints (graph) obtained from DC (GBC) by replacing e by e e. (a) Evey path containing (f_^,f_.) in GBC is slack if and only if e can be chosen (with e > 0) so that DC is consistent. e (b) Whenever every path containing (ff) is slack, e can be chosen (with e > 0) so that DCÂ£ is consistent and at least one of the follow ing is true: (i) at least one path in GBC containing (f.,f.) is tight; Â£ 1 J (ii) the length of (f.,f.) in GBC can be reduced to zero. i j e Finally, we will use the following lemma proven in [33]. Lemma 3.3.1. Given points a,b e T, suppose that d(a,b) = a +3. Then, the inequalities d(x,a) a, d(x,b) Â£ 3 are consistent if and only if they have a unique solution and the inequalities hold as equalities. Necessary and Sufficient Conditions for Efficiency Given a location vector Z, we let U = D(Z) and define the distance constraints of interest by D(X) < U, where the entries in U define the bjk and Cij by bjk = d('ZyZk) fr Â£ IB and Cij = d(zivj) for (ij) Â£ 1^. We use the b^ and c to define GBC in the customary manner. As before, we may assume GBC is connected, for otherwise the problem of finding efficient location vectors decomposes into -92- independent subproblems. Further, we note that DC is always consistent, as Z is certainly feasible to DC, and hence, by Theorem 3.3.1, the separation conditions are always satisfied. For convenience, for any location vector Z, we denote by A*(Z) the collection of locations of uniquely located facilities whose nodes are adjacent to N in GBC. We denote by 11[A*(Z)] the convex hull of A*(Z), the imbedding of the smallest subtree of T spanning all the elements of A*(Z). With the above definitions we can present a family of equivalent conditions for a location vector Z to be efficient. Theorem 3.3.3. Given a location vector Z used to define DC and GBC, the following are equivalent: (a) Z is efficient; (b) Each FL is in at least one tight path in GBC; (c) Z is the unique solution to DC; (d) z^ e H[A*(Z)] for i = l,...,m. Proof. The equivalence of (b) and (c) is a direct conseqeuence of Theorem 3.3.2 and the fact that Z is always a feasible solution to DC, while (c) clearly implies (a). To show (a) implies (c) suppose Z is not the unique solution to DC. Color every new facility node in GBC which is not contained in any tight path blue. Color all the other (new or existing facility) nodes red. Equivalence of (b) and (c) implies every blue node represents a new facility which is not uniquely located, while every red node represents a (new or existing) facility which is uniquely located. By assumption there is at least one blue node. By connectedness of GBC, there is at least one arc which connects some blue colored node, say, N to some red P colored node, say, F Furthermore, arc (N ,F ) has positive 9 P q * -93- length; for otherwise, in every feasible solution to DC, the location of new facility p would be the same as the location of the uniquely located facility represented by node F contradicting the fact that N is colored blue.. Property 3.3.2 then implies that the entry in P U = D(Z) corresponding to arc (N ,F ) can be reduced by a positive P 9 amount and the resultant distance constraints will still have a feasible solution, say Y. But then clearly D(Y) < D(Z) and D(Y) 4- D(Z), contra dicting the fact that Z is efficient. Hence (a), (b), and (c) are equivalent. It can be seen that the proof will be complete if we show (b) implies (d) and (d) implies (c). To show (b) implies (d), suppose N_^ is in some tight path P. Let f^ and f^ be the nodes adjacent to N_^ in P, so that ((f^,N^), (N^jf^)) is a subpath of P. Since f^ and f2 are in the tight path P, by Theorem 3.3.2 the facilities represented by f^ and f are uniquely located. We may let y^ and y2 denote the unique locations of f^ and f^, respectively. Thus it is clear that y^ and y^ are elements of A*(Z). By Property 3.3.1, z^ e L(y^yy^), and by definition of the convex hull, L(y^,y^) C H[A*(Z)]. Thus it follows that z^ e H[A*(Z)J. To show (d) implies (c), suppose e H[A*(Z)] and let f^ and f^ be nodes adjacent to N_^ in GBC, where f^ and f^ represent facilities with unique locations y^ and y respectively, such that z. e L(y ,y ) C T. Thus d(y ,y ) = ^ 1 i. m 1 Z d(y ,z ) + d^z^y^). Now for any feasible solution X to DC we know dCy^Xi) d(y^,z^) and d(y2>x^) < d(y^,z^). But then because f^ and f2 are uniquely located, Lemma 3.3.1 implies x. = z^, for i = l,...,m. Hence X = Z, so Z is the unique solution to DC, completing the proof. -94- 3.4 Examples Here, we give examples of efficient and non-efficient points. Ex. 1. For a single new facility, D(z) is the vector (d(z,v ),..., d(z,v )). Any point z in T is efficient since T is the convex hull of iv . ,V }. 1 n Ex. 2. Consider the tree T shown in Figure 3.2. Each arc length in the corresponding graph GBC corresponds to an entry of D(Z). In this case Z is efficient. Notice that and are both contained in the tight path P = (E^, ^, E^). Also, both z^ and z2 satisfy the con vex hull property, i.e. z Htiv^ v2, z2}] and z2 e H[{v3> v^-Zj}]. Figure 3.2. Example of an Efficient Location Vector (a) Graph BC, (b) Tree T. Ex. 3. Consider the same tree as in Example 2 except that the location of z2 is changed to the midpoint of edge (v^v^. In this case -95- (z^.z^) is not efficient as is not contained in any tight path. Also, t Hv^, v^, z^}. This example is shown in Figure 3.3. 1.5 1.5 / 4 7 c ? / Figure 3.3. Example of a Non-Efficient Location Vector Ex. 4. Again consider the same tree with z^ = z^ located at the mid point of edge (v^,v^). In this case, both z^ and z^ are uniquely located and both satisfy the convex hull property. Thus Z = is efficient in this case. This example is shown in Figure 3.4. -96- Figure 3.4. Example of an Efficient Location Vector 3.5 Further Results on the Convex Hull Property In this section we concentrate on the last statement of Theorem 3.3.3, namely, that Z is efficient if and only if each is contained in the convex hull H[A*(Z)], where A*(Z) contains the locations of those uniquely located facilities whose nodes are adjacent to N^. Our main interest is to delete the phrase "uniquely located" from the definition of A*(Z) and still have the equivalence hold under the new (relaxed) definition. From a computational standpoint, this deletion would make it unnecessary to identify the uniquely located new facilities, which, in turn, requires the identification of tight paths in GBC. -97- With this motivation in mind, define, for 1 < j < m, A(N_.) to be the collection of nodes in GBC which are adjacent to N and denote by A (Z) the collection of the locations of the new and existing facili- j ties whose nodes are in A(N). We remark that N, is not a member of J J A(N.) and hence z^ f/ A^ (Z) . The following property states the necessary conditions for Z to be efficient. Property 3.5.1. Suppose Z is efficient. Then ze H[A^(Z)] for every j Â£ {1,... ,mj. Proof. From Theorem 3.3.3, whenever Z is efficient, z^ e H[A*(Z).] for each j e {l,...,m}. But A*(Z) is clearly a subset of A(Z) implying that z e H[*(Z)] C H[A (Z)], completing the proof. In general, the reverse implication in Property 3.5.1 need not hold for certain (pathological) cases. Such occurrences correspond to the case where Z is such that for some two adjacent nodes N. and N, , J k the locations z and z^ coincide. We provide an example of such a case in Figure 3.5. With reference to Figure 3.5, observe that every z. is contained in the associated convex hull. In particular, 7. and J ^ z^ are contained in their respective convex hulls because their loca tions are the same. The location vector is clearly a non-efficient one, since and z^ can both be moved to v^> thereby reducing the distances associated with them. Sufficiency for Irreducible Location Vectors At this point we distinguish two classes of location vectors and show that the reverse implication (sufficiency) in Property 3.5.1 holds for one class ("irreducible" location vectors) while it need not hold -98- z2=z3 # v 1.5 1.5 a) Graph GBC b) Tree T A^(Z) = (z2,vi,v2,v4) Zj e H[ A^(Z)] = T A2(Z) = {z^ZyV^} z2 e H[A2(Z)] = L(zl5z3) A3(Z) = {z2>v^} z3 e H[A3(Z)] = L(z2,v^) c) Sets A^. (Z) d) Convex Hulls Figure 3.5. Example of a Non-Efficient Location Vector -99- for the other class ("reducible" location vectors). We say a pair of facilities interact if their nodes are adjacent in GBC. We define a location vector Z = (z^,...,z^) to be irreducible if for every pair of interacting new facilities i and j, their locations z^ and zare dis tinct; Z is said to be reducible if there exists at least one pair of interacting new facilities i and j for which z^ = zj ^he lcatin vector of Figure 3.5 is an example of a reducible location vector. The following property gives the sufficient conditions for an ir reducible location vector to be efficient. Property 3.5.2. Suppose Z e T is an irreducible location vector. If for every j, 1 < j < m, z^ e H[A^(Z)], then Z is efficient. The proof of Property 3.5.2 requires a number of preliminary results. To preserve the continuity of the discussion, we leave the proof until the end of section 5. From a computational standpoint, Property 3.5.2 provides an ap proach for determining whether or not an irreducible location vector is efficient, and constructing one if it is not. To check if Z is efficient, we only need to determine the nodes adjacent to N in GBC and form the convex hull (the smallest subtree) which spans the loca tions of these adjacent nodes. If it is the case that every z. is within its convex hull, then Z is efficient. Otherwise, we can choose a z. which is not in the convex hull associated with it, and move its location to the closest point in the convex hull. The procedure can be employed repeatedly until every new facility satisfies the convex hull containment property. However, during such a procedure, the current location vector may change its status from an irreducible one to a reducible one, as the locations of new facilities change. For -100- this reason, it becomes necessary to develop the sufficient conditions for reducible location vectors. Sufficiency for Reducible Location Vectors The basis of our approach for establishing sufficiency for redu cible location vectors is to represent a reducible location vector by an irreducible one and apply Property 3.5.2. Suppose Z is reducible. Then at least one arc in GBC connecting two new facility nodes has length zero. In general, there may be several arcs of length zero connecting new facility nodes. Let GB be the subgraph of GBC with nodes N,,...,N and arcs (N. ,N ) for (j ,k) 1 m J k e ID. If arc (N.,N, ) in GBC has length zero, then combining these B j k two nodes into a single (super) node will not affect the length of any path containing this arc. If the resulting graph (with one less node) has an arc in GB of length zero, again the two nodes connected by that arc can be combined into a single node without affecting the path lengths. In general, this graph transformation can be applied as many times as necessary (clearly, at most m 1 times) to obtain a new graph GBC* from GBC so that no arc in GBC* connecting two new facility nodes has length zero. With this transformation, a node N of GBC* P will actually be representing a number of the original nodes in GBC. We may define the index p as a composite index for the indices of new facility nodes represented by in GBC*. Hence, if p is the composite index, say, {j,k,l}, we can define z* to be the common location P z^ = z^ = z^ of new facilities j, k, and 1. Thus, if GBC* has, say, t new facility nodes, then the location vector Z* with components corresponding to new facility nodes of GBC* will be an irreducible -101- location vector, as GBC*, by definition, has no arc of length zero connecting two new facility nodes. Hence, sufficient conditions for Z can be expressed in terms of the sufficient conditions (given in Property 3.5.2) for Z*. The following procedure, RP (Reduction Procedure), transforms GBC into GBC* by applying successive elementary transformations as de scribed in the above paragraph. During the procedure, we also keep a list K which contains as members the composite indices. RP. 0) Given Z, set up GBC with arc lengths defined by entries of D(Z). Define K = {{1},. .. ,{m}}. Label new facility node as N^j, 1 < i < m. 1) If, for some P,Q e K, P f Q, there is an arc (Np,N^) of length zero in GBC go to 2). Else go to 4). 2) Superimpose node on together with all arcs incident to Np. Remove arc (Np,N^) from GBC. (If parallel arcs occur due to this transformation they will clearly have equal lengths. Parallel arcs may optionally be represented by a single arc.) 3) Remove P and Q from K, insert P U Q in if and go to 1). 4) Stop with K* = K and GBC* = GBC. The algorithm RP terminates in at most m 1 iterations as each iteration reduces the number of elements of K by one. An example application of RP is given in Figure 3.6. For each composite index P in K* we define z* to be the common location of every new facility i for which i e P. For the example of Figure 3.6, let K* = {P^P^ with P = {1}, ?2 = {2,3,4}. Then zPt Z1 and ZP2 z2 z3 z4 We let Z* be the location vector -102- c) Iteration 1 b) Graph GBC d) Iteration 2 and Termination Figure 3.6. Example Application of RP -103- wit h components z*, P e K*, and call Z* the irreducible representation of Z. Corresponding to GBC*, define DC* to be the distance con straints with every constraint corresponding to exactly one arc in GBC*. It will be convenient to refer to the triplet (Z*, DC*, GBC*) as the reduction of (Z, DC, GBC). We remark that for an irreducible location vector Z, the reduction of (Z, DC, GBC) is Identical to (Z, DC, GBC), as RP terminates immediately in this case. For P e K*, define A(N^) to be the set of adjacent nodes to . in GBC*, and let Ap(Z*) be the collection of locations of facili ties whose nodes are members of A(Np). The following property gives the sufficient conditions for reducible location vectors (as well as irreducible ones). Property 3.5.3. Let (Z*, DC*, GBC*) be the reduction of (Z, DC, GBC) and let K* be the list of composite indices for new facility nodes of GBC*. If Zp e H[Ap(Z*)] for every P e K*, then Z is efficient. Proof. By definition Z* is irreducible. Hence, the hypotheses of the property imply, upon using Property 3.5.2, that Z* is efficient with respect to the reduced constraints DC*. From Theorem 3.3.3, for every P e K*, node Np is in a tight path in GBC*. Now, we want to show that the original nodes i e P, are all in tight paths in GBC. Recover GBC from GBC* by decomposing every node Np of GBC* into its original nodes N^, i e P, and connect these nodes to one another by arcs of zero length by adding those arcs which were removed by RP. Since the added arcs have lengths of zero, the shortest path lengths cannot change. Hence, the shortest path length between any two existing facility nodes of GBC* is the same as the -104- shortest path length between the same existing facility nodes in GBC. Since Np is in a tight path in GBC*, then every original node Ih for which i e P will be in a tight path in GBC, as the shortest path lengths in GBC* and GBC are the same. But then every N^, 1 < i < m, is in a tight path in GBC, as is in a tight path in GBC for every P c K*, and U{P: P e '*} = {l,...,m}. Thus, upon using Theorem 3.3.3, Z is the unique solution to DC and Z is efficient. Proof of Sufficiency for Irreducible Location Vectors We now return to the proof of Property 3.5.2. After presenting a number of preliminary results, we will show that if Z is irreducible and z_. e H[A^(Z)] for j e {l,...,m}, then every new facility node is in a tight path in GBC. The following lemma is proven in [22]. Lemma 3.5.1. Let P be a finite set of points each of which is in T. For any p e P, we have H[P] = UL(p,p): p e P). That is, the convex hull of P can be constructed by finding the line segments joining an arbitrary element of P to every point in P. Next, we have the following lemma. Lemma 3.5.2. Suppose Z is irreducible. Let and be two adjacent new facility nodes in GBC. If z2 e H[A2(Z)] then there exists a facility location y in A2(Z) such that a) z2 e L(z^,y) and z^ ^ y, b) whenever y is a new facility location, z2 f y. -105- Proof. Since and are adjacent, is in A^(Z). By Lemma 3.5.2, we have H[A2(Z)] = U{L(z^,y): y e A^CZ)}. Since z2 e H[A^(Z)], for at least one facility location y in A^(Z), z^ e L(z^,y). Also z^ f y, for otherwise, z^ = y and z^ e L(z^,y) imply z^ = z2, contra~ dieting the irreducibility of Z. Hence, a) is established. Part b) follows immediately from a) and the irreducibility of Z. We remark that the irreducibility assumption cannot be relaxed in Lemma 3.5.2, for otherwise we may have z^ = = y. Figure 3.5 illustrates such a case. We will subsequently use Lemma 3.5.2 repeatedly to identify a sequence of locations z z z. v such that they all lie (1) (2.) (r)5 p y in the line L(z^j,v ) in the given order. The corresponding sequence of nodes N N. ,E in GBC will form a subpath connecting N.,. (1) (r) p v B (1) to with the length of that subpath equal to d(z^^,v^). By the same token, we will find another node E in GBC with the subpath q connecting E^ to having length d(v^,z^). Then, we will show that the two subpaths when connected at form a tight path which contains N (1)' First we give the following result given in [82]. Lemma 3.5,3. Given four points p^ ,p^ jP^jP^ e T, if p2 e LPj^) P3 e L(p2,p4) and p2 f p3> then d(Pl>p4) = d(p ,p ). i=l -106- We can readily use induction to obtain the following generaliza tion of Lemma 3.5.3. Lemma 3.5.4. Given r points p ,... ,p Â£ T with r > 4, if p e L(P.ji_iP1+i) for 2 ^ i < r-1, and if p 4 p+1 for 2 < i < r-2, r-1 then d(p1,pr) = Â£ i=l We are now ready to prove the sufficient conditions for irredu cible location vectors. We remark that the arc lengths of GBC are defined by the entries of D(Z), so that if N,N, *,E is a sub- Vi) OO p path P(N^,Ep) connecting to Ep, then the length of the subpath is given by LP(N^,Ep) = d(z^^.z^) + ... + d(z^,v ). Lemma 3.5.5 (Sufficiency). Suppose Z is irreducible. If, for every j e {1,... ,m}, Zj e H[A^.(Z)], then every z^ is uniquely located. Furthermore Z is efficient. Proof. For notational brevity, let S = A (Z). J 3 Choose any j in {l,...,m}. Either N^. is adjacent to exactly one node or more than one node. In the former case, S^ is a singleton, say, {y}. Since Zj e H[Sj], Zj = y. By irreducibility of Z, y is an existing facility location. Hence z_. is in the convex hull of uniquely located facili ties so that Theorem 3.3.3 implies z^ is uniquely located in this case. For the other case, N. is adjacent to at least two nodes in GBC. J The hypothesis z_. e H[S^] implies there exist p,q e S_. with Â£ L(pq) (3.5.1) If p and q are both existing facility locations, Theorem 3.3.3 implies Zj is uniquely located. Hence, suppose, without loss of generality, that q is an existing facility location, but p is a new facility -107- location (the case with both p and q new facility locations is very similar to the proof we will give below and hence will not be con sidered) Define z. = p. Find a sequence of locations z.,z. ,... z. ,v for some r, 1 < r < m-1, by applying Lemma 3.5.2 to the pairs Jr (z.,z: ), (z, ),...,(z. ,v ) one at a time in the given order so '1 j t Jr that the family of conditions in (3.5.2) is satisfied: z. e L(z.,z. ) and N. N. are adjacent, Jx J 32 J1 J2 z. e L(z. z. ) and N. N. are adjacent, J2 Ji J3 ^2 J3 (3.5.2) z. e L(z. Jr 3 r-1 ,vfc) and N E are adjacent. We remark that the irreducibility of Z and the conclusion of Lemma 3.5.2 guarantees that such a sequence can be found and will end with an existing facility location v as, r can be at most m-1 and for the last z. we must have some y e S. such that z. e L(z ,y) with lr Jr Jr Jr-1 y necessarily an existing facility location. Let q in (3.5.1) be the location v of existing facility s, s s ^ t. Then, the sequence v z.,z. ,...,z. ,v satisfies the assump- s J Jr t tions of Lemma 3.5.4 as a result of (3.5.1), (3.5.2), and the irredu cibility of Z. Hence, we have d(v ,v ) = d(v ,z.) + d(z.,z. ) + ... + d(z. ,v ) s c s J J Ji Jr (3.5.3) where the right hand side of (3.5.3) is clearly the length of the path Eg,N.,N. ,...,N. ,Et> Hence, the path is a tight path due to J J ^ J r (3.5.3) and contains N Thus, z^ is the unique location of new -108- facilty j. Since j is arbitrary, Z is the unique solution to DC, and, thus, upon using Theorem 3.3.3, Z is efficient. 3.6 Algorithm to Construct Efficient Location Vectors To this point we have presented a family of conditions for char acterizing efficient points. Theorem 3.3.3 provides the necessary and sufficient conditions in terms of uniquely located facilities, tight paths in GBC, and the convex hulls of uniquely located facilities. Property 3.5.2 provides the sufficient conditions for irreducible location vectors without requiring the identification of uniquely located facilities. Property 3.5.3 extends the results of Property 3.5.2 to the case of reducible vectors. Based on Properties 3.5.2 and 3.5.3, we now present the Sequential Efficient Vector Construction Algorithm (SEVCA). Given a location vector Z, the algorithm first finds the irreducible representation Z* of Z by using RP. Then each component of Z* is checked to see if it satisfies the convex hull containment property. If some component is found which is not within the convex hull associated with it, its loca tion is moved to the closest point in the convex hull. The procedure is repeated with the resulting location vector. Termination occurs when every component of the current irreducible vector is within the convex hull associated with it. In order to prove finite termination (in 0(m) iterations), we use a labeling scheme for the current com posite indices. The list K is the list of composite indices during K any given iteration, while Z denotes the location vector whose com ponents are indexed by the members of K. -109- SEVCA Initial 0) Given Z e Tm, set up GBC with arc lengths defined by entries of D (Z). Define K = {{!},...,{m}}. Label each member of K unscanned Reduction 1) If for some P,Q e K, P f Q, there exists an arc (N^.N^) in GBC of length zero go to 2). Else go to 4). 2) Superimpose node Np on together with all arcs incident to N^. Remove arc (Np,N ) from GBC. (If parallel arcs occur due to this transformation, they will have equal lengths. Parallel arcs may optionally be represented by a single arc.) 3) Remove P and Q from K, insert P U Q in K and label P U Q unscanned Define zpyg to be the common location of Zp and z^ and go to 1). Termination Test 4) If every member of K is scanned, stop. Else, choose an unscanned composite index P in K and go to 5). Check for Convex Hull Containment 5) Find A(Np), the set of nodes adjacent to in (current) GBC, and ,K. define Ap (Z ) to be the set of current locations of new existing facilities whose nodes are members of A(Np). % 6)If Zp e H[Ap(Z )], label P scanned and go to 1). Else go and to 7). Movement 7)Find the closest point, say, y to z^ in H[Ap(Z )]. Define e(P) = d(Zp,y). Move zp to y. Update the arc lengths of GBC by subtracting the amount e(P) from every arc incident to Np. Label P scanned and go to 1). -110- b) Iteration 1 Figure 3.7. Example Application of SEVCA -lu c) Iteration 3 Figure 3.7. Continued -112- e) Iteration 4 f) Iteration 5 Figure 3.7. Continued -113- g) Iteration 6 and Termination (in Iteration 7) Figure 3.7. Continued -114- An example application of SEVCA is given in Figure 3.7. For every iteration, GBC and the current location vector is given. For iterations 6 and 7, the location vectors at the end of these iterations are shown separately. For the other iterations, the location vector does not change. Iterations 1 and 5 perform the Reduction routine. For the other iterations, the node chosen in that iteration is the one inci dent to every thickly-drawn arc. The associated convex hull is shown by cross-hatched lines in the tree network. For any iteration the circular-shaped new facility nodes of GBC are the unscanned nodes, while the rectangular-shaped new facility nodes are the ones which have been scanned prior to the given iteration or during that itera tion. For any iteration, the numbers on the arcs of GBC show the arc lengths at the beginning of that iteration. If the arc lengths change during that iteration, the new arc lengths are indicated by the numbers in parentheses. By one iteration of SEVCA, we shall mean the execution of step 1) through the last step. The last step of any given iteration is either step 3), step 4), step 6), or step 7). Define, for i = 3, 4, 6, 7, t^ to be the total number of iterations which used step i) as the last step. Clearly, t^ = 1. Since any given iteration uses only one of these steps as the last step, the total number of iterations, denoted by t, will be given by t = t^ + tg + t^ + 1. We want to show that (k) t 3m. For convenience, denote by K the list of composite indices at the first of iteration k. Property 3.6.1. The algorithm SEVCA terminates in at most 3m-l iterations. -115- Proof. We must show that t, the total number of Iterations, is at most 3m-l. Define P =\J{K^^: 1 < k < t}. With this definition P contains as members distinct (but not necessarily disjoint) subsets of { 1,. .. ,m). Since = {{ 1{m} } every set {j}, 1 < j m, is an element of P. If step 3) is never executed, then P = as A'^ = ... = K^ in this case. Otherwise, whenever step 3) is executed, (k) say, in some iteration k, some two distinct members of K are removed from A^), and their union is inserted in to obtain Hence, if iteration k performs step 3), we know that the cardinality of (k+1) (k) a y is one less than that of Av and that exactly one member of (k+1) (k) K (the one inserted) differs from every element of K Clearly then, step 3) can be executed at most m 1 times, and therefore P contains at most m+(m-l)=2m-l distinct members. Hence t^, the total number of iterations which used step 3), satisfies t^ < m- 1. Now, imagine that we apply SEVCA a second time in exactly the same order as the first application. In the second application, each time step 6) or step 7) is used as the last step of an iteration, one member of P will be labeled scanned (P is available as a result of the first application). Since P contains at most 2m 1 distinct members, clearly, tg + t^, the total number of iterations which used either one of step 6) or step 7), will satisfy tg + t^ < 2m 1. It follows then that t = t_ + t. + t, + t_ < 3m 1 as t_ Â£ m 1, t, =1, 3 4 6 7 3 4 and tg + t^ < 2m 1, completing the proof. Next we have the following property. Let X* be the location vector at the termination of SEVCA. Property 3.6.2. The algorithm SEVCA terminates with an irreducible location vector X* which is efficient. Furthermore the location vector -116- X Â£ Tm whose irreducible representation is X* is efficient and satisfies D (X) < D(Z) (3.6. 1) where Z is the given vector in Tm to which SEVCA is applied. Proof. Due to the Reduction routine it is evident that X*, the loca tion vector at the termination of SEVCA, is irreducible. Let K* be the list of composite indices at the termination. The Termination Test implies every member of K* is scanned. But a composite index can be labeled scanned only in either step 6) or step 7). In either case, we have x* Â£ H[Ap(X*)] for every P e K*. Property 3.5.2 then implies X* is efficient since every component of X* is in the convex hull associated with it, and X* is irreducible. To show (3.6.1), let x^ = x* for every i e P with P e K*. Thus, X = (x^,...,xm) is the vector in Tm whose irreducible representation is X*. Clearly, the entries of D(X) are either zeros or the arc lengths of the reduced graph, say, GBC* at termination. Since the arc lengths of GBC either remain the same or decrease by a positive amount (in step 7)) from one iteration to the next, it follows that D(X) < D(Z). The assertion that X is efficient follows immediately from Property 3.5.3 and the fact that x* e H[Ap(X*)] for every P z K*. 3.7 Efficiency for the Case of Rectilinear or Tchebychev Distances In this section we consider the analogous vector-minimization k problem in the k-dimensional Euclidean space, denoted by R for the case of rectilinear (k = 2) or Tchebychev (k > 2) distances. Given -117- 1c points x,y e R with x = (x^,...,x^) and y = (y^,...,y^), the rectilinear distance between x and y is given by |x^ y^| + ... + |x^ + y^J, while the Tchebychev distance between x and y is given by max{|x^ y^|,..., |x y |}, where the symbol |*| denotes the absolute value sign. K. K. It is known that (proven in [32]) the distance constraints with 2 k rectilinear distances in R or with Tchebychev distances in R k 2, are consistent if and only if the separation conditions hold. Based on this result, we characterize efficient location vectors for the analogous vector-minimization problem which uses the recti linear or Tchebychev distances. Theorem 3.7.1. Let D(Z) be the vector of objectives with all entries 2 of D(Z) either the rectilinear distances in R or the Tchebychev distances in R k > 2, as specified in (3.2.1). Let GBC be the graph with arc lengths defined by D(Z). The following are equivalent: (a) Z is efficient; (b) Every arc in GBC of positive length is in a tight path. Proof. To show (a) implies (b) suppose that Z is efficient. Let DC be the distance constraints D(X) < U = D (Z). Since Z is a feasible solution to DC, DC is consistent and hence the separation conditions hold. Assume that there exists at least one arc in GBC with positive length which is not in any tight path. Let (f ,f ) be such an arc with P q length e Since (f ,f ) is not in any tight path and since the pq p q separation conditions hold, every path which contains (f ,f ) is slack. P q Hence, for any path P(E.,E.) containing (f ,f ) we have 1 J p q LP(E ,E ) d (v ,v ) > 0 (3.7.1) J J Define e' to be the minimum of the left side of (3.7.1) over all paths -118- whlch contain (f ,f ), giving e' > 0. Choose any e such that p q 0 < e < min(e',e ). Reduce the length of arc (f ,f ) from e to Pq p q pq e e and define the resulting graph to be GBC^. Due to our choice of e clearly every separation condition on GBC^ holds, as, the length of every path which contains (f ,f ) is reduced by an amount smaller than the P q difference between the path length and the distance between the existing facility locations corresponding to the terminal nodes of the path, while the length of any path which does not contain (f ,f^) remains the same. Let DC^ be the distance constraints corresponding to GBC^.. Since the separation conditions on GBC^ hold, DC^ is consistent. Letting Y be any feasible solution to DC ', it follows that D(Y) < D(Z) and the entry of D(Y) corresponding to (f ,f ) is strictly smaller than the corresponding entry of D(Z). Hence, Y dominates Z, contra dicting that Z is efficient. Thus, (a) implies (b). To show (b) implies (a) suppose every arc in GBC of positive length is in a tight path. Let (f ,f^) be any arc with positive length e For e > 0, let GBC^CDC^) be the graph (distance constraints) ob tained from GBC(DC) by replacing e by e e. Since (f ,f ) is in pq pq p q a tight path, for any choice of e > 0, at least one separation con dition is violated. Since the violation of a separation condition implies the inconsistency of the distance constraints, there does not exist e, e > 0, for which DC^ is consistent. Clearly, then, there does not exist Y such that D(Y) < D(Z) and D(Y) ^ D (Z) which is the definition of efficiency. We remark that Theorem 3.7.1 holds for tree networks as well as If rectilinear distances on the plane, or, Tchebychev distances in R , k Â£ 2. The proof of the theorem relies on the necessity and sufficiency -119- of the separation conditions for the distance constraints to be con sistent. Furthermore, this equivalence is the only property that is used. Hence, the theorem holds for any distance for which it is true that the distance constraints are consistent if and only if the separa tion conditions hold. In Theorem 3.7.1 one does not have to worry about arcs with zero lengths. To see this, partition D(Z) into subvectors D^(Z) and (Z) so that D (Z) contains the zero entries, while (Z) contains the positive entries. Clearly, in every feasible solution X to the con straints D(X) < D(Z), the constraints corresponding to entries of Dq(Z) will hold at equality. Hence the only way Z can be dominated is by having (X) < (Z) and (X) f D^(Z). Thus, one needs to test only those arcs with positive lengths in GBC to determine whether or not Z is efficient. For this reason, Theorem 3.7.1 is applicable to both irreducible location vectors and reducible location vectors. 2 We remark that for rectilinear distances in R and Tchebychev 1c distances in R k > 2, the equivalences stated in Theorem 3.3.3 need not hold. An example of such a case is given in Figure 3.8 for 2 Tchebychev distances in R With reference to Figure 3.8, it is direct to verify that both and are contained in the tight path 0^2 N^, E^) Clearly (z^,z^) is not efficient, as, z^ can be moved to (2,2) and z^ can be moved to (3,1) thereby reducing the dis tance between new facility 1 and existing facility 1 without increasing any of the other distances. The resulting location vector is shown in Figure 3.9. It Is direct to verify that every positive arc in GBC of Figure 3.9 is contained in a tight path and hence the location vector is efficient. -120- 4 3 2 1 0 v, v. Z2 a) Facility Locations b) Graph GBC L(E1SE2) = 4 L(El5E3) = 5 L(ErE4) = 6 L(E2,E3) = 3 L(E2>E4) = 4 L(E3,E4) = 3 > 2 = d(v1}v2) > 4 = d(v1#v3) > 5 = d(v1,v4) 3 = d(v2>V3) > 3 = d(v2,v4> = 3 = d(v3,v4) c) Separation Conditions Figure 3.8. Example of a Dominated Vector with Tchebychev Distances in -121- Figure 3. 4 2 o 1 1 O v z, z. V, a) Facility Locations b) Graph GBC L(E1SE2) = 3 > MEltE3) = 4 = L(E15E4) = 5 = L(E2,E3) = 3 = L(E2,E4) = 4 > L(E3,E4) = 3 = 2 = d(vlSv2) 4 = d(vl9v3) 5 = d(v^ >v4> 3 = d(v2,v3) 3 = d(v2,v4) 3 = d(v3,v4) c) Separation Conditions 9. Example of an Efficient Vector with Tchebychev Distances in CHAPTER 4 THE BI-OBJECTIVE m-CENTER PROBLEM ON A TREE NETWORK 4.1 Introduction In this chapter we consider a bi-objective problem on a tree network which involves as objectives the maximum of the weighted dis tances between specified pairs of new and existing facilities and' the maximum of the weighted distances between specified pairs of new facilities. Such a vector-minimization problem may find applications in locating emergency service units for the case when service units are required to support one another in addition to providing service to potential hazard zones (existing facilities). The related literature on multi-objective location problems is discussed in Chapter 1. Here, we concentrate on characterizing efficient location vectors to the bi-objective minimax problem and constructing the "efficient frontier," the set of objective values corresponding to efficient points. At this point we give an overview of the chapter. In Section 2 we give the necessary definitions and notation, and define the bi objective problem of interest. In Section 3 we relate the bi-objec tive problem to the distance constraints studied by Francis et al. [32] and develop the necessary and sufficient conditions for efficiency by making use of the results given for the distance constraints in -123- [32]. At the end of the section we provide an example problem. Section 4 is devoted to the development of a procedure, E-FRONT, for constructing the "efficient frontier." We prove the correctness 2 2 of the procedure and show that it is of 0(m (m + n )), where m and n are, respectively, the number of new and existing facilities. The section ends with an example application of the procedure. 4.2 Problem Statement, Notation, and Definitions Let T be an imbedding of a finite undirected tree network with existing facilities located at distinct vertex locations v^,...,vn. It is of interest to locate m new facilities at points x^,...,x in T under two objective criteria to be defined below. We suppose given positive weights w and v k and denote by 1^ and Ig the nonempty sets of pairs (i,j) and (j,k), respectively, for which the weighted distances w..d(x.,v.) and v...d(x.,x, ) are of concern. We remark that it need 13 i 3 jk 3 k not be the case that 1^, includes all possible pairs of new and existing facility indices nor Ig includes all possible pairs of new facility indices. For each location vector X = (x^,...,xm) in Tm we define two objective functions f^ and f^ by f ^X) = max(w_d(xi,Vj): (i,j) e Ic> , f2(X) = max{v^kd(x^. ,xk): (j,k) e Ig} , and we let f(X) = (f (X), f2(X)). (4.2.1) The Bi-objective m-Center Problem (with Mutual Communication) is as follows: V min{f(X): X e Tm} . (4.2.2) -124- As in Chapter 3, a location vector Y in Tm is said to be efficient with respect to (4.2.2) if and only if X e Tm and f(X) < f(Y) imply f(X) = f(Y). A location vector which is not efficient is said to be dominated. Our main interest is to relate the bi-objective m-center problem to the distance constraints problem studied by Francis, Lowe, and Ratliff [32], We shall characterize efficient points by making use of the separation conditions (defined in Chapter 3) which are known to be necessary and sufficient for the distance constraints to be consistent. To define the distance constraints of interest, let z = (z^,z^) be any two-tuple (with z > (0,0)) and consider the constraints given in (4.2.3): d(x^,Vj ) < Zj/w^ (i, j ) e Ic (4.2.3) d(xjxk) Z2/Vjk (j>k) e Ig . We shall refer to the family of constraints in (4.2.3) as DC^. The constraints DC are said to be consistent if there exists at least z one feasible solution X = (x, x ) to (4.2.3). 1 m Corresponding to DC^ we define GBCz to be the undirected graph with nodes ,...,Nm,E^,...,En- For every (i,j) e 1^ there is an arc (N.,E.) of length z./w.. and for every (j,k) s ID there is an arc 1 3 i ij a (N.,N,) of length z/v., We partition the arc set of GBC into two j K z j k. z sets Ab and Ac with Ag = {(N.,Nk): (j,k) e Ig} and Ac = {(N,Ej): (i,j) e Iq}. We shall assume that the sets 1^ and Ig are such that GBC is connected, as otherwise DC decomposes into independent sets z z of constraints which may be analyzed separately. -125- As in Chapter 3, denote by P(Fp,F^) any path in GBC^ connecting nodes Fp and F A path is specified either by the sequence of nodes in the path, or, by the sequence of arcs in the path. We denote by L P(F ,F ) the length of the path P(F ,F ) and define L (F ,F ) to be z p q w * p q z p q the length of any shortest path connecting nodes Fp and F^. We say P(Fp,F^) passes through an arc (F^,F^) if (F^,F^) is an arc in the path. We say P(F ,F ) passes through AT1 (A) if the path passes p ^ iJ u through at least one arc in A^ (A^). Associated with any path P(Fp,F^) we define two more terms, namely, WP(F ,F ) and VP(F ,F ). The first term WP(F ,F ) is the sum of the p q p q p q reciprocal weights 1/w^ where the summation is taken over all arcs (N.,E.) which are contained in the path P(F ,F ). If the summation 13 P q is taken over an empty set, then WP(Fp,F^) = 0. Similarly, VP(Fp,F^) is the sum of the reciprocal weights l/v^ over all pairs (N^.,N^) which are contained in'P(F ,Fq). Again, VP(Fp,Fq) = 0 if Ag fl P(Fp,Fq) = 0. The motivation for these two quantities can be given by observing the relation L P(F ,F ) = z -WP(F ,F ) + z_VP (F ,F ) . z p q 1 p q 2 p q (4.2.4) The relation in (4.2.4) can be readily verified by observing that the arc lengths of GBC^ are defined by the quantities and z2^vj^ so that the length of any given path is the sum of the reciprocal weights multiplied by or z^, whichever is applicable. In what follows any path (in GBCz) we refer to is a path connecting some two existing facility nodes Ep and E^. All other paths (for which one or both of the terminal nodes are new facility nodes) will be referred to as subpaths. -126- It was proven in [32] and stated in Theorem 3.3.1 of Chapter 3 that DC is consistent if and only if L (E E ) > d(v ,v ) for z z p q p q 1 Â£ p < q n. The inequalities L^CE^E^) dCv^jV^) are called the separation conditions and the separation conditions are said to hold if every separation condition is satisfied. It is direct to verify that whenever the separation conditions hold (equivalently, whenever DCz is consistent) it necessarily follows that L P(E ,E ) > d(v ,v ) for any path P(E ,E ). Conversely, whenever zpq p q p q L P(E ,E ) > d(v ,v ) for all paths P(E ,E ), it necessarily follows zpq p q p q that L (E ,E ) > d(v ,v ). z p q p q The definitions for tight and slack paths are given in Chapter 3 and will not be repeated here. 4.3 Necessary and Sufficient Conditions for Efficiency In this section we develop the necessary and sufficient conditions for efficiency by making use of the distance constraints. Our main theorem states that a location vector Y is efficient if and only if at least one arc in is contained in a tight path in GBC^, where GBC^ is the graph corresponding to DC^ obtained by letting z = f(Y). Notationally, for any X = (X.,X0), DC is the distance con- X Z ZA straints with right hand sides (z1 X )/w.., (i,j) c I and i I xj C (z^ ^2^vjk e Ig* The graph GBCz ^ is the graph associated with DC z-X' Before proving our main theorem, we first prove two lemmas relating DC to DC and GBC to GBC ... We remark that "0" denotes Z ZA Z ZA either the scalar zero or the two-tuple (0,0). It will be clear from the context what "0" refers to. -127- Lemma 4.3.1. Given a location vector Y used to define DC and GBC z z with z = (zpZ^ = (f ^ (Y) f 2 (Y) ) the following are equivalent: (a) Y is efficient. (b) Por any X = (X,,X ) > 0 and X 0, DC is inconsistent. Proof. Using the definition of efficiency, f^, f2> and the fact z = f(Y) we have the following equivalences. The location vector Y is efficient if and only if f(X) < z implies f(X) = z if and only if there does not exist X such that f(X) z and f (X) 5s z if and only if for any X Â£ 0 and X ^ 0 there does not exist X for which f^(X) < z^ X^ for i = 1,2 if and only if for every X > 0 and X ^ 0 there does not exist X such that max{wd (x^,v^) : (i,j) e 1^,} z^ X^ and max{vM d(x. ,x, ) : (j,k) e 1^.} < z X if and only if for every X Â£ 0 jk j K a 2 2 and X Â£ 0 there does not exist X such that d(x.,v.) < (z, X,)/w.. i J 1 1 ij for all (i,j) e Ic and dCx^.x^) < (z2 *2^Vjk for G *b and only if for every X 0 and X ^ 0, DC is inconsistent, com- Z" A pleting the proof. Corollary 4.3.1. Given Y with z = f(Y), Y is dominated if and only if there exists X z 0, X 4- 0 such that DC is consistent. z-A We remark that the proof of Lemma 4.3.1 does not use any special properties of tree networks. Hence the lemma is applicable to any metric provided that f^ and 2 are the maximum of the pairwise weighted distances. The following lemma provides the sufficient conditions for DCr to be consistent in terms of the slack paths in GBCz. Lemma 4.3.2. Suppose DC is consistent. If every path in GBC which 2 Z passes through Ag is slack then X = (Xj^) can be chosen with X Â£ 0 and X ^ 0 such that DC is consistent. z-X z-X -128- Proof. Since DC is consistent clearly z. 0, i = 1,2. We consider z 1 the cases z^ > 0 and z^ = 0 separately. Case with ^2 > 0. Let P(E^,E^) be any path which passes through Ag. By hypothesis the path is slack so that L^PE^E^) d(vp>Vq) > 0. Further VP(E^,E^) is positive since the path passes through A^. Hence we have [L P(E ,E ) d(v v )]/VP(E ,E ) > 0 (4.3.1) z p q p q p q Let e be the minimum of the left side of (4.3.1) over all paths which min(e,Z2). Let GBCz_^ be the graph with arc lengths (ij) e and (Z2 ^2^vjk* (j,k) el. We want to show that the separation conditions defined on D GBC are satisfied. z-A Choose any two nodes E^ and E^. Let P(E^,E^) be a shortest path in GBC connecting E and E Hence, we have z-A p q L (E ,E) = L ,P(E ,E) (4.3.2) ZA p C[ ZA p CJ Either P(E ,E ) passes through A or it does not. In the latter case p q U clearly the length of P(E ,E ) in GBC and GBC is the same, as every p q z z-A arc in A^ has the same length in both graphs. Since DC^ is consistent, we have L ^P(E ,E) =LP(E ,E) > L (E ,E) >d(v ,v) so that the z-A pq zpq zpq pq separation condition for E and E is satisfied in this case. For the P q other case, P(E ,E ) passes through A^ so that its length on GBC is given by pass through A_. Choose A = (A^^2) with A^ = 0 and 0 < A2 L ,P(E ,E ) Z-A p C| = z WP(E ,E ) + (z A)VP(E ,E ) 1 P q 2 2 p q (4.3.3) -129- But L P(E ,E ) = z. WP(E ,E ) + zVP(E ,E ) so that from (4.3.3) we zpq lpq 2pq have L ,P(E ,E ) = Li(E ,E ) X -VP(E ,E ) . z-x pq zpq z pq (4.3.4) By our choice of X, we have 0 < X e [L P(E ,E ) d(v ,v )]/ J 2 2 zpq p q VP(EpjE^). It follows then, upon using (4.3.4), that L P(E ,E ) > L P(E ,E ) [ z-X p q ~ z p q' 1 L P(E ,E ) d(v ,v ) zpq P 9 i up/F F \ VP (E ,E ) * ^ ^Ep5 Eq^ p q = d(v v ) p q (4.3.5) From (4.3.2) and (4.3.5) it follows that L ,(E ,E ) > d(v ,v ) for z-X p q p q this case. Since the choice of E^ and E^ is arbitrary, every separation condition holds on GBC so that DC is consistent with X = (0,X), ZA Z~A X2 > 0. Case with z2 = 0 By hypothesis every path which passes through Ag is slack. Choose any path P(E^,E^) which does not pass through Ag. Consistency of DC^ implies either LzP(Ep,E^) = d(v^,v^) or L P(E ,E ) > d(v ,v ). The former case is not possible since a z p q p q subpath of length zero can be chosen from the arcs in Ag and this subpath can be appended to P(E^,E^) to obtain a new path, say, P'(E^,E^) without increasing the length of the path. Hence L^PiE^jE^) = L^P'(Ep,Eq) = d(Vp,v^) contradicting that every path which passes through Ag is slack. Thus, every path which passes through A^, is also slack. Define e to be the minimum of [L P(E ,E ) d(v ,v )1/WP(E ,E ) zpq p q p q over all paths in GBC2. Clearly e > 0, since every path is slack and every path necessarily passes through A so that WP(E ,E ) > 0. Choose > C P q (X, ,0) with 0 1 i i I p q on -130- GBC The length of P(E ,E ) on GBC is L ,P(E ,E ) = (z, A.)* z-X & p q z-X z-X p q 1 1 p q WP(E ,E ) + zVP(E ,E ). But z. = 0 so that L ,P(E ,E ) = L P(E ,E ) - p q p q / zA p q z p q X,WP(E ,E ). By our choice of X. we have X, < [L P(E ,E ) - 1 p q J 1 1 z p q d(v ,v )]/WP(E ,E ). Hence, L ,P(E ,E ) > L P(E ,E ) {[L P(E ,E ) - p q p q z-X p q z p q z p q d(v,v )]/WP(E,E )}-WP(E ,E ) = d(v ,v ). Thus, L ,P(E ,E ) > d(v ,v ) pq pq pq pq z-x p q p q for any path P(E^,E^) so that the separation conditions on GBCz_^ hold and is consistent with X ^ 0, X ^ 0, completing the proof. Next, we have the necessary and sufficient conditions for efficiency. Theorem 4.3.1. Given a location vector Y used to define DC and GBC z z with z = (z^,Z2) = f(Y), the following are equivalent: (a) The location vector Y is efficient. (b) At least one arc in is contained in a tight path. (Equivalently, there exists at least one tight path which passes through Ad.) 15 Proof. To show (a) implies (b), suppose Y is efficient. Assume that no arc in A^ is contained in a tight path. Hence every path which passes through A^g is slack as DCz is certainly consistent. Lemma 4.3.2 implies X = (X,,X0) can be chosen with X > 0 and X ^ 0 so that DC is con- 1 l z-X sistent. Corollary 4.3.1 then implies Y is dominated, contradicting that Y is efficient. To show (b) implies (a) suppose at least one arc in A^ is in a tight path. Let P(E ,E ) be such a path which passes through A^ and P q B which is tight. Clearly, P(E ,E ) also passes through A For any p q c X = (X ,X) >0, X 0, the length of P(E ,E ) in GBC .. will be i z p q za strictly smaller than its length in GBC as at least one of z. and z Z 1 z is reduced by a positive amount due to X being different from (0,0). Hence, for any X ^ 0, X ^ 0, the separation condition on GBC -131- corresponding to E and E will be violated, as, A _. (E ,E ) < P Q z A p q L ,P(E,E) DC is inconsistent. Lemma 4.3.1 then implies Y is efficient. z-A We remark that Theorem 4.3.1 considers only those tight paths which pass through Ag. The reason is as follows: Any path in GBC^ passes through A so that if there exists a tight path which contains an arc in Ag, then the same path necessarily contains an arc in A^,. However, an arbitrarily chosen path need not pass through Ag. For this reason, the assumption that there exists at least one arc in A^, which is contained in a tight path does not imply that at least one arc in Ag is contained in a tight path. Hence, if a location vector Y is efficient then there is at least one arc in A^ which is contained in a tight path while the reverse implication does not hold. Further, we remark that the proof of Lemma 4.3.2 is based on the necessity and sufficiency of the separation conditions. Hence, Lemma 4.3.2 is applicable to tree networks as well as the analogous problems with rectilinear distances on the plane, or, the Tchebychev distances in the k-dimensional Euclidean space with k > 2. Further, Theorem 4.3.1 uses Lemmas 4.3.1 and 4.3.2 for its proof. Hence, the theorem is applicable to tree networks as well as rectilinear distances on the lc plane and the Tchebychev distances in R k > 2. At this point we give an example of a non-efficient and an efficient location vector. In Figure 4.1 the tree network is shown along with the distance matrix and the weights w.. and vfor 13 Jk (i,j) e Ic = {(1,1),(1,2),(2,3),(2,4),(3,4),(3,5)} and (j,k) e Ifi - {(1,2),(1,3),(2,3)} for the example bi-objective m-center problem. In Figure 4.2a) we give an example of a dominated location vector X. -132- y<~ 7- 2 y +- 9 vr y 4 a) Tree T v2 v3 V4 V5 W11 = 1/5 V12 1/3 V1 6 4 8 10 W12 = 1 V13 = 1/6 V2 2 6 8 W23= = 1/3 V23 = 1/2 V3 4 6 W24 = 1/2 V4 2 W34 = 1/4 W35 = 1/3 b) Distance Matrix c) Weights Figure 4.1. Data for Example Bi-Objective m-Center Problem -133- a) A Non-Efficient Location Vector with f(X) = (1.5,1) b) An Efficient Location Vector with f(Y) = (1.5,0.5) Figure 4.2. Example of Non-Efficient and Efficient Location Vectors -134- Using the weights given, the value of f(X) is (1.5,1). Graph GBC^ with z = (1.5,1) is also shown with arc lengths as indicated. It is direct to verify that GBC^ in Figure 4.2a) does not have any tight paths. Hence, X is dominated. The location vector Y in Figure 4.2b) is an efficient one and dominates X. The value of f(Y) is (1.5,0.5). The thickly drawn arcs in GBC^ of Figure 4.2b) form a tight path. We remark that in every feasible solution to DC^ for z = (1.5,0.5), the locations y^ and y^ ate the same, as and ^ are contained in a tight path. 4.4 Construction of the Efficient Frontier Let S be the set of all efficient location vectors in Tm. Define Z and Z* by Z = {(z^,Z2): 3X e T such that f(X) = (z^,Z2)} , Z* = {(z^,z^): 2X c S such that f(X) = (z^,z^)} That is, Z = f(Tm), the image of Tm under f, and Z* = f(S), the image of the efficient set S under f. We call Z the objective space and Z* the efficient frontier. Our main interest in this section is to develop a method to construct the efficient frontier. One can display Z* graphically on the (z^^) plane and obtain much of the insight about efficient points. In general, for any convex bi-objective problem, the efficient frontier and the objective space may look like the illustration given in Figure 4.3. The objective space is the shaded region and the efficient frontier is the thickly drawn part of the boundary of Z. -135- Figure 4.3. Illustration of Z and Z* for a Convex Bi-Objective Problem -136 We shall first state a theorem due to Wendell [110] which gives a global characterization of the efficient frontier. Then we will exploit the result of the theorem to construct Z*. m /s Let a be the minimum value of f^ on T b be the minimum value of f^ on Tm, and b be the minimum value of f^ over all minima to f^. The A values a, b, and b are displayed in Figure 4.3 for an arbitrary bi objective problem with convex objectives. For each z^ e [a,b] define the function e^) to be the minimum value of the problem P defined below Z1 e(Zl) = minf2(X): f (X) < z X e Tm) . Wendell [110] showed that whenever f^,f2 are lower semicontinuous convex functions defined over a nonempty convex compact set S, the efficient frontier is the set {(z^,e(z^)): a < z^ < b}. Wendells theorem is applicable to the bi-objective m-center problem, as Tm is convex, compact, and nonempty, and f^ and f2 are continuous convex functions (see [22]) over Tm. For an arbitrary choice of z^, the value of e(z^) is marked in Figure 4.3. The computation of a, b, and b presents no difficulties and will be given subsequently. Using the definitions of f^ and an equivalent definition of e(Zj) is as follows: e(zp = min Z2 s. t. d(x,Vj) < z^w (i,j) e Ic d(Xj,xk) z2/vjk e IB where z^ is understood to be fixed to any value in [a,b] (4.4.1) The constraints -137- pf (4.4.1) are just DCÂ¡z. We remark that for any z^ e [a,b], there exists at least one feasible solution to (4.4.1), as there exists an X such that f^(X) = a z^ and can be chosen large enough so that f200 z2. Let z^ be fixed with z^ e [a,b] and let (Y,z2) be any feasible solution to (4.4.1). Define z = (z ,z) and let DC be the distance J. z z constraints in (4.4.1). Since (Y,z2) is feasible to (4.4.1), the separation conditions on GBC hold so that every path in GBC has z z length at least as large as the distance between the locations of existing facilities corresponding to the terminal nodes of the path. Hence, for any path P(E^,E^) we have L P(E ,E ) = z WP(En,E ) + zVP(E ,E_) > d(v,v ) , zpq lpq 2pq pq or, equivalently, d(v ,v ) WP(E ,E ) 2 lVP(E ,E )J llVP(E ,E )J p q p q (4.4.2) Defining x(z^) to be the maximum of right side of (4.4.2) over all paths in GBC^, it follows that z^ > x(z^) whenever z2 is feasible to (4.4.1). Hence, the minimum value of z2 which solves (4.4.1) is x(z^). We observe that the right side of (4.4.2) is the value of a linear func tion (of z^) evaluated at z^. There are as many such linear functions as there are paths in GBCz. Further x(z^) is the maximum of these functions at z^. Thus, defining x(*) to be the pointwise maximum of these linear functions over the interval [a,b] we have e(zp = x(zp for every z^ Â£ [a,b]. Hence, a brute-force method to construct the efficient frontier is to enumerate all paths on GBC, compute the parameters (slope and intercept) for each linear function corresponding -138- to each distinct path, and equate e() to the pointwise maximum of these linear functions over the interval [a,b]. Such a method is not computationally efficient as there may be a very large number of paths For achieving computational efficiency we shall restrict our attention to a certain subset of the set of all possible paths and then evalute e(*) by taking the pointwise maximum of the linear functions cor responding to these paths. In general, an arbitrarily chosen path P(E^,E^) may pass through several existing facility nodes distinct from E^ and E^. First, we want to show that paths of this type need not be considered. Define any path to be a decomposable path (d-path) if the path passes through at least three (distinct) existing facility nodes. An example of a d-path which passes through four existing facility nodes is (E^, Nj-, Eg, N^, E^, N^, E^) Define any path to be a non- decomposable path (nd-path) if the only existing facility nodes the path passes through are its terminal nodes. Every d-path can be de composed into a (unique) collection of nd-paths which, when appended end to end, gives the original d-path. The decomposition of the aforementioned example d-path into its nd-paths is {(E,, Nc, N1ft, E,), 1 j 10 o (Eg, N^, E^), (E^, N^, E^)}. Clearly, any nd-path uses exactly two arcs in A^,, while any d-path uses at least four arcs of A^,. An nd- path may or may not use arcs of A^. Next, we have the following lemma, which permits us to check the separation conditions by only evaluating nd-paths. The lemma is applicable to any distance constraints problem defined in Chapter 3 by (3.3.1). We use the notation of Chapter 3 for the lemma. -139- Lemma 4.4.1. Let DC be the distance constraints specified in (3.3.1) and let GBC be the associated graph. The separation conditions on GBC hold if and only if for every nd-path P(E ,E ), LP(E ,E ) > d(v ,v ). pq p q p q Proof. Suppose the separation conditions hold. Choose any nd-path P(E ,E ). We have LP(E ,E ) > L(E ,E ) > d(v ,v ) as the length of p q p q p q p q P(Ep,E^) is at least as large as the shortest path length between E^ and E . q Suppose for any nd-path P(E^,E^) we have LP(E^,E^) > d(Vp,v^). Choose any two existing facility nodes, say, Eg and E^, with 1 s < t < n. Let P(Eg,Et) be any shortest path connecting Eg and E^. If P(E ,E ) is an nd-path then clearly L(E ,E ) = LP(E ,E ) > d(v ,v ) so that the separation condition for Eg and Efc is satisfied in this case. Consider the case when P(Eg,Et) is a d-path. Decompose P(Eg,Et) into its nd-paths, say, P(E ,E,P(E. .,E ). Hence, S^IJ (V) t LP(Eg,E^) > d(vg,v^^) ,... ,LP(E^rj ,Efc) > d(v(r)vt)> as the Paths are nd-paths. Further, the length of P(Eg,Et) is the sum of the lengths of P(Eg,E^j) ,... ,P(E^ jE^). Hence, upon using the triangle inequality, we have d(v ,v ) d(v ,vnv) + ... +d(v. .,v ) < St s v. X) ) t LP (E ,E...) + ... + LP(E. ,E ) = LP(E ,E ) = L (E ,E ) so that the s (1) (r) t s t s t separation condition for E and E is satisfied for this case. Since s t the choice of Eg and Efc is arbitrary, the proof is complete. We are now ready to present the procedure for constructing the efficient frontier. We define G^ to be the graph with nodes N^,...,Nm and the arc set A^. To every arc (N^,N^) of G^ we assign the length 1/v For 1 < s < t m we denote by mgt the length of a shortest path connecting the nodes Ng and Nfc in GThe computation of 3 mgt_, 1 < s < t < m, can be achieved in 0(m ) operations by using known algorithms (see Dreyfus [23]). -140- The following algorithm, E-FRONT, constructs the efficient frontier. We assume m has been computed for 1 A s < t < m. E-FRONT 0) Label the arcs in as a^,...,ar where r is the cardinality of A Define A' = {(a.,a.): 1 < i < j < r}. Delete from A' every 0 x j pair (a^,a^) for which a^, and a^ are incident to the same new facility node. Let A be the resulting subset of A' after the deletions. 1) For every (a.,a.) e A define the linear function t.,(z.) as i 3 iJ 1 follows: Suppose a. = (N ,E ) and a. = (N ,E ). Due to step i s p/ j t q clearly s / t and thus m > 0. For z^ e [a,b] 0) d(Vr,Vr,) (1 + 1/..) X. .(Z.) = E_3_ Â§Â£ tq_ y 1 st 1 st 2) Define x(z^) = max{x (z^) : (a^a^) e A} for z^ e [a,b]. The efficient frontier is given by Z* = {(z^,x(z^)): a Â£ b}. The next theorem establishes the correctness of the algorithm. 2 2 Then we will show that the algorithm is 0(m (m + n )). Theorem 4.4.1. The algorithm E-FRONT constructs the efficient frontier for the bi-objective m-center problem. Proof. By Wendell's theorem Z* = {(z^,e(z^)): a < z^ < b}. Hence, it suffices to show that e(z^) < x(z^) and e(z^) > x(z^) for a < z^ < b. To show e(zp x(z^), choose any z^ e [a,b] and define z = (z1,z) with z = xCz,). Let DC and GBC be the constraints of 12 2 1 z z the problem P and the associated graph, respectively. Choose any Z1 nd-path P(E ,E ). Either the path passes through A^ or it does not. p q a In the latter case P(E ,E ) is the path (E ,N ),(N ,E ) for some new -141- facility node Ng. Let X be any location vector for which f^(X) = a. Hence, w d(x ,v ) < a and w d(x ,v ) < a. But a < z. and the length sp s p sq s q 1 of P(E ,E ) is z.(1/w + 1/w ) so that we have, upon using the tri- p q 1 sp sq angle inequality, L P(E ,E ) = z (1/w + 1/w ) > a(l/w + 1/w ) > & n 3 z p q 1 sp sq sp sq d(x ,v ) + d(x ,v ) > d(v ,v ). Thus, for any nd-path P(E ,E ) which s p s q p q 3 1 p q does not pass through A^, we have L P (E ,E ) > d(v v ) z p q p q (4.4.3) For the other case, P(E ,E ) passes through A,, so that its length is P q B given by L P(E ,E ) = z,WP(E ,E ) + zVP(E ,E ). Since the path is an zpq 1 p q 2 p q nd-path it passes through exactly two arcs in A^, say, (E^,Ns) and (N ,E ) with s t. Thus, WP(E ,E ) = 1/w + 1/w while t q p q sp tq VP(E ,E ) ra by the definitions of m and VP(E ,E ). It follows p q st st p q that L P(E ,E ) > z.(l/w + 1/w ) + z m zpq 1 sp tq 2 st Due to steps 1) and 2) of E-FRONT, we have (4.4.4) z2 = T(zl) - L_Â£_ z m ^ Z1 st (1/w + 1/w ) Sp tq ra st (4.4.5) Using (4.4.4) and (4.4.5), for any nd-path which passes through A^ we have IP(E E ) > d(v ,v ) (4.4.6) z p q p q From (4.4.3) and (4.4.6), L P(E ,E ) > d(v ,v ) for every nd-path in zpq-pq GBCÂ¡z so that Lemma 4.4.1 implies the separation conditions on GBCz hold. Hence, z is feasible to P and thus e(z.) x(z.) = z. z. 112 -142- To show e(z ) > tCz.), let a.,a. be arcs in A for which 1 i i J G t(z,) = x..(z,). Suppose a. = (N ,E ) and a. = (N ,E ). By step 1) 1 lj 1 l s p 3 t q J of E-FRONT we have VZP d(vv) p q (l/w + l/w ) sp tq m st m (4.4.7) st But m is the length of a shortest path in Gg connecting Ng to N . Let (N ,N, ...,N,.,N ) be such a shortest path in G. Let P(E ,E ) be s k f t r B p q the path (E ,N ,N, ,...,NÂ£,N,E ). Define z = (z,,z0) with z = e(z.). p s k f t q 12 2 1 Since z is feasible to P DC is consistent so that for the path 2 z^ z v P(E ,E ) identified above we have P q LP(E >E ) > d(v ,v ) . z p q p q (4.4.8) But LzP(E^,E^) = z^(l/wSp) + z2mst + Zl^^Wtq^ aS Pat^ consists of the arcs (E ,N ) (N ,N. ),...,(N,,N),(N ,E ). It follows then from ps sk. it tq (4.4.8) that z.(l/w + 1A* ) + z0m ^ > d(v ,v ), or, equivalently, 1 sp tq 2 st p q Z2 d(v ,v ) JB SL (1/w + l/w ) sp tq m st m (4.4.9) st But the right side of (4.4.9) is = x(z^) while z^ is e(z^) by definition, hence, e(z^) > x(z^). The inequalities e(z^) < t(z^) and e(z^) > t(z^) imply e(z^) = i(z^) for every e [a,b]. Hence, Z* = {(z^tCz^)): a < z^ < b), com pleting the proof. 2 2 We now show that the computational order of E-FRONT is 0(m (m + n )) The algorithm constructs Z* by identifying no more than r(r l)/2 linear functions. To identify the linear functions one must first -143- calculate m for 1 < s < t < m, which requires 0(m ) operations. Every linear function is determined by computing its slope and inter- 3 2 cept so that steps 0), 1), and 2) require 0(m + r ) operations. But r can be at most mn so that excluding step 3) and the computation of a 3 2 2 2 and b, the algorithm is 0(m + (mn) ) =0(m (m+n )). Each linear function has positive intercepts and negative slope. Clearly, their pointwise maximum is a piecewise linear decreasing function over the interval [a,b]. Hence, t(*) can be constructed by finding its break points. Each break point is determined by the intersection of some two linear functions. Since each linear function is strictly decreasing, any linear function can determine at most two (consecutive) break points. Thus, there are at most 2*(r)(r l)/2 = r(r 1) break points. Hence, excluding the computation of a and b, the algorithm requires 2 2 0(m (m+n )) operations, as the computational effort for constructing the linear functions dominates the computational effort for finding the break points of t(0- To compute a, define, for every new facility index i, the set 1^ by ^ = ij : (i,j) e ICL Letting g^x^ = max{w^d (x^v..) : j c 1^, it is direct to verify that f^(X) = max{g^(x_^): 1 i < m). Hence, fj is separable and its minimum value is given by a = max{g*: 1 < i < m) where g* = min{g^(x): x e T}. The Kariv-Hakimi procedure in [65] com putes g* in 0(iI|log|I|). Since |l | < n, the computation of a requires no more than mnlogn operations. Hence, the computational effort for identifying the linear functions again dominates the com putational effort for computing a. To compute b, we must first compute b. Clearly, b = 0 as it is the minimum value of f 2 (X) = maxi'v fcd (x^ ,xfc) : (j,k) e IB>. It is direct to verify the following equalities: -144- b = minf^(X): ^2^^ b) = minify (X) : v^dCx^.x^) < 0 (j ,k) e Ig} = minif^(x,..,x): xeT) = min max g (x) xeT l
= min max max{w..d(x,v.): j e I. }xeT l
= minxeT max{w_.d (x,Vj) 1 < j < n} where w. = max(w,,: over all i for which i e I.} J ij i (4.4.10) (4.4.11) Thus, the value of b is obtained by solving the absolute 1-center problem defined by (4.4.10) and (4.4.11), and will require O(nlogn) opera 2 2 tions. -Therefore, the computational effort for E-FRONT is 0(m (m + n )) and is determined by the computational effort for steps 1), 2), and 3) of E-FRONT. Once the efficient frontier is constructed, efficient location vectors can be identified as follows: Choose z = (z^,zÂ£) in Z* with z2 = x(z^) = TÂ£j(z^)> say, and identify the arcs a^ and a^. Supposing a. = (N ,E ) and a. = (N. ,E ), let P(N ,N 1 be a path in whose length i s p j t q s t r B & is mgt_. Then, clearly, (Ep,P(Ng,Nt) ,E^) will be a tight path in GBCz with length L P(E ,E ) = z,WP(E ,E ) + z0m Every facility whose zpq l pq zst node is in the path P(E^,E^) is uniquely located on the line L(v^.v^) in T with the same ordering and spacing as the nodes which lie in P(Ep,E^). Hence, the new facility locations corresponding to new facility nodes in P(E^,E^) can be readily identified (see Property 3.3.1 of Chapter 3). The locations of other new facilities can be -145- found by applying the Sequential Location Procedure of [32] to the constraints DCz after fixing the locations of the uniquely located facilities. Any such feasible solution Y to DC^ is clearly an effi cient location vector. At this point we give an example application of E-FRONT. We apply the algorithm to the tree network for which the necessary data are given (earlier) in Figure 4.1. The arcs in A^ are labeled a^ through a^ as shown in Figure 4.4a). Figure 4.4b) shows the graph GBC^j ^ with arc lengths l/w and l/v^. The thickly drawn subgraph of GBC,, is the graph GD. The values of m 1 Â£ s < t Â£ 3, are \I j I / D St also given in the same figure. Corresponding to each pair of arcs (a^,a^) e A the linear function T^j(zp is specified in Table 4.1. These functions are plotted in Figure 4.5, and t(z^) is indicated by the cross-hatched line in the same figure. The values of a and b are computed by using the techniques given earlier. We remark that t(z^) = T24^zl^ fr zl e for this example problem. Since a2 = anc* a4 = ^2^4^ anc* m12 = t^ie -^-en8t^ f arc ^N1N2^ t^ie Path (E2^i^2^4^ as a path for any choice of (z^,Z2) in Z*. For example, for z^ = 1.5 and Z2 = 0.5, (z^,Z2) is on the efficient frontier. The tight path corresponding to this choice of (z^,Z2) is shown earlier in Figure 4.2b), together with the corresponding location vector Y = an t^ie same figure. Note that y^ and are uniquely located as N^ and N2 are on the tight path. -146- Figure 4.4. Data for Example -147- Table 4.1. Example Linear Functions T13(zl> = 1.33 2.67z1 t14(z1^ = 2.67 2.33zl t15(z1) = 1.6 1.8z1 t16(z1) = 2 1.6z1 t23(z1> = 0.67 1.33z1 t24(z1* = 2 21 t25(zP = 1.2 Z]L T26 W = 2 3.5Zl T36(z1) = 3 3z^ t46(zI> = 1 2.5zj -148- Figure 4.5. Efficient Frontier for Example CHAPTER 5 SUMMARY AND FUTURE RESEARCH 5.1 Summary In this dissertation we considered problems which involve locating multiple new facilities on a tree network with respect to n existing facilities at known locations. In Chapter 2, we solved the nonlinear p-center problem where the objective is to minimize the maximum cost associated with any existing facility. The cost (disutility) of service associated with any existing facility is a nonlinear (strictly increasing and continuous) function of the distance between that existing facility and its nearest new facility. We gave a weak and a strong duality theorem and pro vided a physical interpretation of the dual problem. Our approach for solving the nonlinear p-center problem and its dual was to solve a sequence of covering problems which involve minimizing the number of new facilities subject to a family of n distance constraints which impose upper bounds on the distances between any existing facility and its nearest new facility. In Chapter 3, we considered a vector-minimization problem which involves as objectives the distances between specified pairs of new and existing facilities and specified pairs of new facilities. We developed the necessary and sufficient conditions for efficiency and provided an algorithm for constructing efficient solutions. Our L 49 -150- approach to the vector-minimization problem was to reformulate the problem in terms of a family of distance constraints which impose upper bounds on the distances between specified pairs of new and existing facilities and specified pairs of new facilities. In Chapter 4, we considered the bi-objective m-center problem (with mutual communication) which involves as objectives the maximum of the weighted distances between specified pairs of new and existing facilities and maximum of the weighted distances between specified pairs of new facilities. We developed the necessary and sufficient condi tions for efficient solutions and provided a procedure for construct ing the efficient frontier. Our approach was to reformulate the problem in terms of a. family of distance constraints which impose upper bounds on the distances between specified pairs of new and existing facilities and specified pairs of new facilities. In what follows we give certain generalizations of the problems considered in this dissertation as well as other location problems considered in the literature. We point out some possible directions for future research. 5.2 Generalized Multi-Center Problem Here, we define a problem which generalizes the p-center problem and the m-center problem with mutual communication. For convenience, we consider the weighted case. Nonlinearity can be obtained by re placing each weight by a strictly increasing continuous function of the associated distance. -151- Let be m-collections of centers with |x_^| = p_^ for 1 < i m, where each p^ is a given positive integer. Let V^,...,Vn be n collections of existing facility locations. The elements of X^ are x^,...^1 with each x^ e T. The elements of V. are 1 p 2 i 1 n i X X with each v. a vertex of T. Let X = {X,,...,X }. For any two finite J 1 m 7 subsets P and Q of T let D(P,Q) = min[d(p,q): p e P, q e Q]. Define the function f by f(X) = max{max{w D(X ,V ): (i,j) e I~} , 1J 1 J V- max{Vj^D(Xj >X^): (j,k) e Ig}} . The Generalized Multi-Center Problem (GMCP) is as follows: min[f(X): |x Â¡ = p^, X_^ C T for 1 < i < m] An equivalent statement of GMCP in terms of distance constraints is as follows: mm s. t. D(X.,V.) z/w.. i J 1J j'V s z/vjk X. p. 1 X 1 1 (ij) e Ic (j,k) e IB 1 i ^ m For the case with m = 1 and each = {v..}, GMCP specializes to the p-center problem. For the case with each p^ = 1 and each = {v^}, GMCP specializes to the m-center problem with mutual communication. We pose the following questions for future research. Ql. What special cases of GMCP are tractable? Some of the special cases are obtained by taking each weight unity, or, -152- taking p = for each i, or taking Vj = {v^} for each j. Q2. Can we use (or modify) COVER and/or the Sequential Location Procedure of [32] to solve either of the related covering problems de fined by min p, + ... + p rl rm D(X,V ) < z/w.. ij (i,j) e Ic D(Xj,Xk) - 2/vjk . (j >k) e IB or, min max{p ^ ,... p } s. t. D(X,V ) < z/w (i,j) e I z/vjk (j,k) e I C B where z is fixed? Q3. Are the separation conditions of direct use for determining the consistency of the distance constraints of GMCP? Q4. Can we extend the duality results of Chapter 2 to GMCP? Q5. Is there a dual to either of the related covering problems? Q6. What kind of applications may GMCP find? Q7. Can the search for the minimum objective value be confined to a finite set of numbers? We remark that the minimum objective value of the m-center problem with mutual communication is the maximum of a finite number of ratios with the numerators distances between existing facility locations while the denominators are sums of reciprocal weights which correspond to -153- shortest path lengths in a related graph. Hence, there appears to be hope for extending the duality results of Chapter 2 to GMCP. 5.3 The t-Objective m-Center Problem: Steps Towards a Unified Theory Here we define a location problem which involves t minimax type objectives. Special cases of the problem are the m-center problem with mutual communication, the vector-minimization problem of Chapter 3, and the bi-objective m-center problem of Chapter 4. We give a theorem which unifies the independent results for each of these problems. An outline of the proof of the theorem is also provided. Given sets I and I let kn = {(N.,E.): (ij) e I0} and C d G 1 j G Ag = {(N^jN^): (j,k) e 1^}. On defining A = A^ U A^, we suppose given t nonempty, mutually disjoint, exhaustive subsets of A, enumerated as A, ,...,A Associated with A 1 < r < t, the rth objective f is I t r J r defined by f (X) = max[max{w. ,d(x. ,v.): (N.,E.) eA ) A.} , rv iJ i j i j r C max{vjkd(Xj,xk) : Â£ Ar /I A^}] where, by convention, the value of either of the inner maximizations is understood to be zero if the maximum is taken over an empty set. Letting f(X) = (f^(X),...,f (X)), the t-Objective m-Center Problem is as follows: V min[f(X): X e Tm] . For the case t = 1 and A^ = A^ U A^, the problem specializes to the m-center problem with mutual communication. For the case with each A^ -154- corresponding to exactly one arc in U A^, the problem specializes to the vector-minimization problem considered in Chapter 3. For the case t = 2, = A^, and = A^, the problem specializes to the bi-objective m-center problem of Chapter 4. Consider the related distance constraints DC where z = (z. z ), z It defined below: d(x.,v.) < z /w. (N.,E.) e A PI A, 1 < r < t , i 3 r xj x j r C- d(x. ,x, ) z /vM (N. ,N, ) e A' H A,., 1 Â£ r < t j k r jk j k r a It is direct to verify the following assertion: Let X be given and define z = f(X). The location vector X is efficient if and only if for every X ) > 0, X 4- 0, DC is inconsistent. The proof of 1 L ZA the assertion is very similar to the proof of Lemma 4.3.1 in Chapter 4. Based on the above property we give the following theorem for characterizing efficient solutions. Theorem 5.3.1. Given X used to define DC and GBC with z = f(X), z z the following are equivalent: (a) X is efficient. (b) For every r with r e { 1,... ,t} .and z > 0, at least one arc in A^ is in some tight path in GBCz. (Equivalently, there exists a collection of tight paths in GBCz such that at least one tight path passes through Afor every r for which r e {l,...,t} and z^ > 0.) Outline of the proof. To show (a) implies (b) suppose X is efficient. Assume that for some r for which z^_ > 0, no arc in A is in a tight path. Clearly DC^ is consistent so that every path which passes through Af is slack. Let PCE^jE^) be any path which passes through A^. Define -155- S(P(Ep,E^)jA^) to be the sum of the reciprocal weights where the sum mation is taken over all arcs which are contained both in P(E ,E ) P q and in A Define e to be the minimum of r L P(E ,E ) d(v ,v ) z P q P q S(P(E ,E ),A ) p q r over all paths P(E^,E^) which pass through A^. Clearly e > 0. Choose X = (A,,...,!) with A. = 0 for i I r, and 0 < A < e. It is direct I t i r to verify, by using arguments similar to the ones given in the proof of Lemma 4.3.2, that such a choice of A is a valid choice for DC to z-A be consistent. But consistency of DCz_^ and the fact that A > 0, A 0, imply X is dominated, contradicting that X is efficient. To show (b) implies (a) suppose for any r for which > 0, at least one arc in A^ is in a tight path. Hence, for any A = (A^,...,A ) > 0, A 0, the length of at least one tight path in GBC .. will be strictly smaller than the distance between the locations of the exist ing facilities corresponding to the terminal nodes of the path. Thus, at least one separation condition on GBC is violated so that DC .. z-A z-A is inconsistent for any A > 0, A ^ 0. It follows that there does not exist Y for which f(Y) < z = f(X) and f(Y) ^ f(X), which is the definition of efficiency. The theorem holds for the problems considered in Chapters 3 and 4 as well as the m-center problem with mutual communication considered in [32]. We remark that the condition > 0 may appear to be somewhat superfluous. Its omission will not affect the equivalence of (a) and (b). The reason we included this condition is that it is unnecessary -156- to check those arcs for which the lengths are zero, as the lengths of these arcs cannot be reduced, and in any feasible solution to DC2, the constraints corresponding to arcs of zero lengths will definitely hold at equality. The assumption that the sets A^,...,A are disjoint is needed for the following reason: Given a path P(E^,E^) which passes through A^_, clearly z^ is the common multiplier for every arc which is contained in the intersection of P(E^,E^) and A^. That is, the length of that part of the path P(E^,E^) consisting of the arcs chosen from A^ is the quantity zr*S(P(E^,E^) ,A^). If it were the case that A^fl A^ ^ 0 for some j r, then the above assertion would not necessarily be true, as an arc in the intersection A^_ 0 A^ will have at least two multipliers in this case, namely, z^ and Zy We will consider this case in the conclusion of this section. The following questions seem worth investigating for future research. Ql. Is there a computationally efficient way of checking whether or not arcs of GBC are contained in tight paths? Q2. How can we construct efficient solutions efficiently? Q3. Can the results of Theorem 5.3.1 be extended to the case when some of the A_^ are not disjoint? Q4. How tractable is the t-objective m-center problem if we generalize it by replacing each x^ by a collection of centers? With respect to Q3, suppose that A^/l A^. ^ 0 for at least two indices i and j for which 1 i < j < t. Let (F ,F ) be any arc in A. P If (F ,F ) is contained in at least two members of {A,,...,A }, then p q I t the distance constraint corresponding to (F ,F ) will appear more than -157- once in DC. Clearly, the effective upper bound for the distance between the locations corresponding to F and F is the minimum of the P q upper bounds which involve these two facilities. Thus, the effective arc length to be assigned to (F ,F^) is the reciprocal weight associ ated with Fp and F^ multiplied by where z^ is the minimum z^ over all indices r for which (F ,F ) e A Let GBC be the graph with arc lengths appropriately assigned as described above. Partition A into A A mutually disjoint subsets A^,...,A (s Â£ t) such that every arc in any A A A . A^ has the same multiplier, say, z^ (where z Â£ iz^,...,z }). De- A A A fining z = ) it is direct to verify that DC is equivalent 1. s z to DC* defined below: z d(x.,v.) < z /w.. i y r i j (N.,E.) eA 0 A., 1 r Â£ s l j r C d(x.,x. ) < z /w., (N.,N, ) e A A 1 < r < s J k rjk jk r TB In other words, DC^ is obtained from DCz by choosing the minimum effec tive upper bound for any constraint which appears more than once in DCz< As a result of the equivalence of DC and DC^ and the fact that z z Aj,...,A are mutually disjoint and exhaustive subsets of A, we make the following proposition: Proposition 5.3.1. Given X and z with z = f(X), let DCg be the equi valent representation of DCz as described in the previous paragraph. The following are equivalent: (a) X is efficient. (b) For every r e {l,...,s} with z^ > 0, at least one arc in Ar is in a tight path in GBC~. -158- 5.4 Tree Networks and General Networks For the nonlinear p-center problem we pose the following questions. Ql. For the nonlinear p-center problem defined on a general network N, does there exist a spanning tree T of N such thnt the solution of the p-center problem on T determines the solution to the p-center problem on N? Q2. If such a spanning tree exists, is there a way to find it without having to enumerate on all spanning trees of N? Dearing and Francis [19] showed that the 1-center problem on N can be solved by solving a sequence of 1-center problems on spanning trees of N. The basic theme of the approach is as follows: Suppose that we know an optimal 1-center, say, x which solves the problem on N. Using the procedure given in Busacker and Saaty [5], find a shortest path tree rooted at x, by identifying the shortest paths (on N) con necting x to any vertex v^. If T(x) is such a shortest path tree, then clearly x is also the optimal 1-center of T(x). Thus, for the 1-center problem the answer to Ql is in the affirmative while Q2 remains un answered as the proof of the existence of such a tree is based on the knowledge of an optimal 1-center of N. For the case with p > 1 we propose a similar approach. Suppose we know an optimal p-center X = ix. x } which solves the problem ^ P on N. Partition the vertex set V = (v^,...,vn} of N into p disjoint, exhaustive subsets V,,...,V such that a nearest center to any vertex 1 p v in V, is the ith center x.. If ties occur we break the ties i i i appropriately so as to satisfy the condition that V^,...,V are mutually disjoint. For 1 < i < p, define T^(x^) to be a shortest path (sub)tree which is constructed by finding the shortest paths -159- Con N) which connect x. to any vertex v. in V.. Thus, T.(x.) is a i J 1111 subtree which spans the subset of V. We make the following con jecture: Conjecture 5.4.1. Given an optimal p-center X of N, there exists a collection of p shortest path subtrees (T^(X^),...,1^(x^)} such that (a) for every i, 1 <, i p, x^ is a closest center to any vertex v^ which is in T.(x.), and i i (b) T^(x^) Tp(Xp) are mutually disjoint. We remark that in order for the conjecture to be true, it is necessary to show that any shortest path tree T^(x_^) contains as ver tices only the members of V^, where Vj,...,V are the mutually disjoint subsets of V which are identified as described in the paragraph pre ceding the conjecture. In other words, if v^ is an arbitrarily chosen vertex from V^, it is necessary to show that, among all the shortest paths connecting x^ to v^, there exists at least one shortest path, say, P(XÂ£>V^) such that P(x^,v^) passes through only those vertices which are members of V^. Otherwise, if every shortest path P(x^,v^) passes through some v^. for which v^ i V^, then- clearly any shortest path tree T.(x.) contains at least one vertex v. for which v. V., 11 j J1 so that T,(x.) H T.(x.) ^ 0 for at least one index j with j ^ i. But i i J J if the intersection of Tk(x^) and Tj(x^) is nonempty then the union of T^(x^) and T^Cx^) may have cycles. If the conjecture holds, then {T,(x,),...,T (x )} is a collection 11 p p of disjoint subtrees, i.e., a forest of N. If we had a knowledge of such a collection of subtrees without having had to know X, then clearly the 1-center of each subtree could be determined by known techniques, and the collection of those one-centers would be an optimal p-center -160- of N. Hence, one possible approach to solve the p-center problem on N is to enumerate on all possible forests of N consisting of p disjoint subtrees and determine the one center of each subtree in any given forest. 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Sci. 18, 656-663 (1972). -169- 117. P.L. Yu, "Cone Convexity, Cone Extreme Points, and Nondominated Solutions in Decision Problems with Multi-Objectives," J. of Opt. Theory and App. 14, 319-377 (1974). 118. P.L. Yu and M. Zeleny, "The Set of All Nondominated Solutions in Linear Cases and a Multicriteria Simplex Method," J. Math. Anal, and AÂ£Â£. 49, 430-468 (1975). 119. P.L. Yu and M. Zeleny, "Linear Multiparametric Programming by Multicriteria Simplex Method," Manag. Sci. 23, 159-170 (1977). 120. B. Zelinka, "Medians and Peripherians of Trees," Archivum Mathematicum (Brno), 87-95 (1968). BIOGRAPHICAL SKETCH Barbaros Tansel was born on January 10, 1952, in Ankara, Turkey, where he received his early education. For his high school education he attended the Robert Academy in Istanbul and graduated in June 1970. In September 1970, he began his undergraduate study in the Middle East Technical University in Ankara and was awarded the Kennedy Scholarship in 1971. He graduated from the Middle East Technical University in June 1974 with a B.S. degree in industrial engineering. In 1975, he was awarded the Fullbright Scholarship and began his graduate study in the University of Florida. He received his M.Sc. degree in December 1976 and Ph.D. in December 1979. During his graduate study he worked as a teaching and research assistant in the Department of Industrial and Systems Engineering. Barbaros's hobbies include classical music, chess, philosophy, and folk dancing. -170- I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Richard L. Francis, Chairman Professor of Industrial and Systems Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. X/ Donald U. Hearn Associate Professor of Industrial and Systems Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Ra llph W'. Swa in Associate Professor of Industrial and Systems Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Antal Maj thay Associate Professor of Managemen This dissertation was submitted to the Graduate Faculty of the College of Engineering and to the Graduate Council, and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy. December 1979 Dean, Graduate School Page 2 of 2 Internet Distribution Consent Agreement In reference to the following dissertation: AUTHOR: Tansel, Barbaros TITLE: Optimal multi-facility location on tree networks / (record number: 99473) PUBLICATION DATE: 1979 i. 'R nms, Tinsel , as copyright holder for the aforementioned dissertation, hereby grant specific and limited archive and distribution rights to the Board of Trustees of the University of Florida and its agents. I authorize the University of Florida to digitize and distribute the dissertation described above for nonprofit, educational purposes via the Internet or successive technologies. This is a non-exclusive grant of permissions for specific off-line and on-line uses for an indefinite term. Off-line uses shall be limited to those specifically allowed by-.'^ir Use" as prescribed by the terms of United States copyright legislation (cf, Title 17, U.S. Code) as well as to the maintenance and preservation of a digital archive copy. Digitization allows the' fiversity of Florida to generate image- and text-based versions as appropriate and to provide and^enhance access using search software. This grant of permissions prohibits use of the digitized versionsjbr cmmercial use or profit. Signature of Copyright Holder 13 L r 1 "HULj tii. Personal information blurred Date of Signature Please print, sign and return to: Cathleen Martyniak UF Dissertation Project Preservation Department University of Florida Libraries P.O. Box 117007 Gainesville, FL 32611-7007 11.06.2008 -64- Figure 2.2. Dual Graph for Example -3- in the past. As reported by Lea [79], there are some 1500 published papers on location-allocation problems. Among these, about 80 are on network location problems, a ratio of a little less than 6%. Hence, network location problems deserve well-justified attention in future research. In this dissertation, we develop a theory for a number of location problems which involve locating multiple new facilities on a tree net work with respect to existing facilities at known locations. At this point we give an overview of the dissertation. In the remainder of Chapter 1, we specify our terminology and give a survey of the network location literature. We discuss minimax and minisum problems.and multi-objective problems involving minimax and minisum objectives as well as other objectives. Discussed also are problems with distance constraints. We highlight some of the convexity properties of trees (see [22]) in relation to the problems discussed. The chapter ends with a brief discussion of path-location problems. In Chapter 2, we develop a theory for the nonlinear p-center problem on a tree network. The problem is a generalization of the linear p-center problem which involves locating p new facilities on a network so as to minimize the maximum weighted distance from any existing facility to its nearest new facility. Nonlinearity is ob tained by replacing each weight by a strictly increasing function of the distance. We formulate a dual "dispersion" problem and prove a weak duality and a strong duality theorem. The strong duality theorem also specifies the necessary and sufficient conditions for an optimal solution to either problem. We provide algorithms of polynomial com plexity for solving either problem. Discussed also are a covering Page 5 SUMMARY AND FUTURE RESEARCH 149 5.1 Summary 149 5.2 Generalized Multi-Center Problem 150 5.3 The t-Objective m-Center Problem: Steps Towards a Unified Theory 153 5.4 Tree Networks and General Networks 158 REFERENCES 161 BIOGRAPHICAL SKETCH . 170 v -38- The problem differs from the p-center problem in two respects: (i) the distance between any vertex v and any new facility x_^ may be of concern as opposed only to the distance between v and the nearest new facility to v.; (ii) certain distances between new facilities are of concern, as opposed to the absence of interactions between new facilities. For the case of a single new facility the two problems coincide. In this problem, the new facilities may be thought to fulfill a supporting task to other new facilities as well as servicing those existing facilities that are a priori assigned to them. Certain planar cases of the multifacility minimax problem have been studied by Dearing and Francis [20]> Elzinga, Hearn, and Randolph [25], Wendell and Peterson [113],. and Francis [28l* The problem on a network is defined by Dearing, Francis, and Lowe [22] in the presence of distance constraints. It is established in [22] that the function f is a convex function on a tree network. The existence of a solution is guaranteed due to compactness and con tinuity considerations. Furthermore, it is shown that it suffices to consider only new facility locations in the convex hull of the existing facility locations in order to solve the problem. The problem on a general network was shown to be NP-hard by Kolen [72 ]. For the case of a tree network, the problem is solved by Francis, Lowe, and Ratliff [32] by using an equivalent formulation in terms of distance constraints (with variable right hand sides). The solution procedure finds Z* first, by using the separation conditions. Then an optimal feasible location vector X* is constructed by using the Sequential Location Procedure described in [32]* To find Z* an -41- For the case of a tree network, the m-median problem with mutual communi cation is solved by Dearing and Langford [21] and by Picard and Ratliff [93]. The approach used by Dearing and Langford is to embed the tree T into the Euclidean space R^, for some p, so that the distance between any two points on the tree is equal to the rectilinear distance between the corresponding points in R^. The problem in R^ with rectilinear distances decomposes into p subproblems, each of which can be solved by using known techniques given in Francis and White [31 ], or, perhaps more efficiently, by applying the network flow procedure discussed in Cabot, Francis, and Stary [6]. For reducing the computational effort, the embedding procedure is carried out with respect to a minimal path decomposition of T into p edge disjoint paths (each edge is in one and only one path). Each path in a minimal path decomposition corresponds to a dimension in R*5. The approach taken by Picard and Ratliff in [93] takes advantage of the vertex-optimality condition and determines an optimal solution (on the vertices of T) by solving a sequence of at most n-1 minimum cut problems, each on a graph containing at most m+2 nodes. The method is based on a result that an optimal location vector can be found independently of the edge lengths, by using only the incidence relations between vertices and the weights. In this respect, the pro cedure is in the same spirit as Goldman's algorithm for finding a median of a tree. Each cut problem corresponds to an edge of the tree. To be more explicit, the removal of all interior points of an edge e leaves two disconnected components, T^ = T^(e) and T^ = 12(e). Corresponding to edge e, a graph G = G(e) is constructed having nodes -100- this reason, it becomes necessary to develop the sufficient conditions for reducible location vectors. Sufficiency for Reducible Location Vectors The basis of our approach for establishing sufficiency for redu cible location vectors is to represent a reducible location vector by an irreducible one and apply Property 3.5.2. Suppose Z is reducible. Then at least one arc in GBC connecting two new facility nodes has length zero. In general, there may be several arcs of length zero connecting new facility nodes. Let GB be the subgraph of GBC with nodes N,,...,N and arcs (N. ,N ) for (j ,k) 1 m J k e ID. If arc (N.,N, ) in GBC has length zero, then combining these B j k two nodes into a single (super) node will not affect the length of any path containing this arc. If the resulting graph (with one less node) has an arc in GB of length zero, again the two nodes connected by that arc can be combined into a single node without affecting the path lengths. In general, this graph transformation can be applied as many times as necessary (clearly, at most m 1 times) to obtain a new graph GBC* from GBC so that no arc in GBC* connecting two new facility nodes has length zero. With this transformation, a node N of GBC* P will actually be representing a number of the original nodes in GBC. We may define the index p as a composite index for the indices of new facility nodes represented by in GBC*. Hence, if p is the composite index, say, {j,k,l}, we can define z* to be the common location P z^ = z^ = z^ of new facilities j, k, and 1. Thus, if GBC* has, say, t new facility nodes, then the location vector Z* with components corresponding to new facility nodes of GBC* will be an irreducible ACKNOWLEDGMENTS I am deeply indebted and grateful to Dr. Richard L. Francis, the chairman of my supervisory committee, for his excellent guidance, numerous suggestions, and the generosity with which he invested his time in listening to my ideas. Dr. Francis not only initiated my interest in location problems but also inspired many of the ideas in this dis sertation by asking the right questions at the right time. I owe very special thanks to Dr. Timothy J. Lowe, the cochairman/ chairman of my committee during 1976-1978, presently of Purdue Uni versity, for his active interest, overall guidance, and his inspiring suggestions. Dr. Francis and Dr. Lowe have shown sincere care about my progress and their encouragement has been of utmost value in bringing this dissertation to a completion. I would also like to express my sincere thanks and appreciation to the other members of my committee, Dr. Ralph W. Swain, Dr. Donald W. Hearn, Dr. Antal Majthay, and Dr. Luc G. Chalmet for their interest in my work and their suggestions during my proposal. I am grateful to the Department of ISE for providing me with assistantship during my graduate studies. Mrs. Adele Koehler has done an excellent job in typing the manu script. She is fast, accurate, and very observant. I sincerely recommend her. ii Page 2 of 2 Internet Distribution Consent Agreement In reference to the following dissertation: AUTHOR: Tansel, Barbaros TITLE: Optimal multi-facility location on tree networks / (record number: 99473) PUBLICATION DATE: 1979 i. 'R nms, Tinsel , as copyright holder for the aforementioned dissertation, hereby grant specific and limited archive and distribution rights to the Board of Trustees of the University of Florida and its agents. I authorize the University of Florida to digitize and distribute the dissertation described above for nonprofit, educational purposes via the Internet or successive technologies. This is a non-exclusive grant of permissions for specific off-line and on-line uses for an indefinite term. Off-line uses shall be limited to those specifically allowed by-.'^ir Use" as prescribed by the terms of United States copyright legislation (cf, Title 17, U.S. Code) as well as to the maintenance and preservation of a digital archive copy. Digitization allows the' fiversity of Florida to generate image- and text-based versions as appropriate and to provide and^enhance access using search software. This grant of permissions prohibits use of the digitized versionsjbr cmmercial use or profit. Signature of Copyright Holder 13 L r 1 "HULj tii. Personal information blurred Date of Signature Please print, sign and return to: Cathleen Martyniak UF Dissertation Project Preservation Department University of Florida Libraries P.O. Box 117007 Gainesville, FL 32611-7007 11.06.2008 -18- these regions may also jointly cover all vertices. Hence, to find a minimum cardinality feasible solution, one needs to choose the minimum number of regions that jointly cover V. This choice can be made by defining a zero-one matrix A, so that an entry a^ of A is one if vertex v^ is covered by region j, and zero otherwise. Solving the set covering problem with matrix A will provide a solution to the r-cover problem. Computational experience is reported and it is found that the procedure works better for small values of p, as the set covering part of the procedure takes a significant portion of the total computational time. An important result is due to Kariv and Hakimi [6.5] They showed that the p-center problem on a general network is NP-complete. Kariv and Hakimi also showed that the weighted case (as well as the un weighted case) can be reduced to a computationally finite one. Based on this finiteness property, they gave an algorithm whose order of complexity is polynomial in |e|, but exponential in p. To show com putational finiteness one argues as follows: For any absolute p-center X = {x^,...,Xp}, there will be a subset of vertices covered by the ith center x.. If N. is the (sub)network induced by V., then it can xi 1 be shown that the absolute center x* of N. can replace x without in- 1 i i creasing the value of the objective function, so that X* = {x*,...,x*} 1 p is also an absolute p-center. Hence, one can restrict one's attention to absolute p-centers every element of which is the absolute 1-center of some subnetwork. The absolute 1-center of any subnetwork of N will occur either at a vertex or at one of at most | E |n (n l)/2 "suspected" points. A suspected point on an edge is a point x such that, for some two distinct vertices v_^ and v., x is a break point on -91- equivalent problem of determining when an arc lies only on slack paths. The following property, which we proved in [33], characterizes the con ditions under which an arc in GBC is not contained in any tight path. Property 3.3.2. Let DC be consistent. Let (f^,fj) be any arc in GBC, of positive length e^., whose length is reduced by some positive amount e. Let DC^CGBC^) be the distance constraints (graph) obtained from DC (GBC) by replacing e by e e. (a) Evey path containing (f_^,f_.) in GBC is slack if and only if e can be chosen (with e > 0) so that DC is consistent. e (b) Whenever every path containing (ff) is slack, e can be chosen (with e > 0) so that DCÂ£ is consistent and at least one of the follow ing is true: (i) at least one path in GBC containing (f.,f.) is tight; Â£ 1 J (ii) the length of (f.,f.) in GBC can be reduced to zero. i j e Finally, we will use the following lemma proven in [33]. Lemma 3.3.1. Given points a,b e T, suppose that d(a,b) = a +3. Then, the inequalities d(x,a) a, d(x,b) Â£ 3 are consistent if and only if they have a unique solution and the inequalities hold as equalities. Necessary and Sufficient Conditions for Efficiency Given a location vector Z, we let U = D(Z) and define the distance constraints of interest by D(X) < U, where the entries in U define the bjk and Cij by bjk = d('ZyZk) fr Â£ IB and Cij = d(zivj) for (ij) Â£ 1^. We use the b^ and c to define GBC in the customary manner. As before, we may assume GBC is connected, for otherwise the problem of finding efficient location vectors decomposes into -121- Figure 3. 4 2 o 1 1 O v z, z. V, a) Facility Locations b) Graph GBC L(E1SE2) = 3 > MEltE3) = 4 = L(E15E4) = 5 = L(E2,E3) = 3 = L(E2,E4) = 4 > L(E3,E4) = 3 = 2 = d(vlSv2) 4 = d(vl9v3) 5 = d(v^ >v4> 3 = d(v2,v3) 3 = d(v2,v4) 3 = d(v3,v4) c) Separation Conditions 9. Example of an Efficient Vector with Tchebychev Distances in TABLE OF CONTENTS Page ACKNOWLEDGMENTS ii ABSTRACT vi CHAPTER 1 INTRODUCTION AND LITERATURE SURVEY 1 1.1 Introduction and Overview . 1 1.2 Terminology 4 1.3 Survey of the Network Location Literature 6 2 DUALITY AND THE NONLINEAR p-CENTER PROBLEM AND COVERING PROBLEM ON A TREE NETWORK ...... 53 2.1 Introduction and Related Work 53 2.2 Problem Statements and Duality 56 2.3 Dual Problem Interpretation 61 2.4 Covering Algorithm 67 2.5 Dual Problem Solution and the Strong Duality Theorem. 73 2.6 Results for the Covering Problem 78 3 A VECTOR-MINIMIZATION PROBLEM ON A TREE NETWORK 84 3.1 Introduction 84 3.2 Problem Statement 85 3.3 Distance Constraints and Characterization of Efficient Points 87 3.4 Examples 94 3.5 Further Results on the Convex Hull Property 96 3.6 Algorithm to Construct Efficient Location Vectors . 108 3.7 Efficiency for the Case of Rectilinear or Tchebychev Distances. 116 4 THE BI-OBJECTIVE m-CENTER PROBLEM ON A TREE NETWORK. ... 122 4.1 Introduction 122 4.2 Problem Statement, Notation, and Definitions 123 4.3 Necessary and Sufficient Conditions for Efficiency. 126 4.4 Construction of the Efficient Frontier 134 Iv -56- 2.2 Problem Statements and Duality We suppose given a finite undirected tree network with positive arc lengths and denote by T an imbedding of the given network having as edges rectifiable arcs. For any two points x,yeT, let d(x,y) denote the shortest path distance between x and y. Let J = {1,...,n} and denote by V = {v^,...,vn) (V C T) a collec tion of distinct vertex locations of "demand points" or "existing facilities." Let X = {x.,...,x } (X C T) denote a finite collection 1 P of "centers" or "new facilities." For ieJ, define the distance of v. 1 to its nearest center by D(X,v^) = min{d(x^,v_.) : 1 < i < p}, and. let Sj = maxid(x,v.): xeT}. Also, for jeJ, we assume given a real valued function f continuous and strictly increasing, with domain [0,6^] and (clearly) range [f (0) ,f^ (6^)] For X C T, |x| < o, we define the function f by f(X) = max{f.(D(X,v^)): jeJ} The Primal p-Center Problem is as follows: Find a p-center X* for which rp = f(X*) = mini f (X) : XCT, Â¡X | = p} . (2.2.1) As discussed in Dearing and Francis [19], due to compactness of T and continuity of d(x,.) on T for each fixed xeT, an optimal solu tion X* to (2.2.1) exists and is contained in the convex hull of V. With a and p defined by a = max{f.(0): jeJ} and n = min{f (5 ): J J J jeJ}, we shall assume a < n, for if a = f (0) > f (6 ) = n. say, then s t t the function f would always be dominated by (strictly smaller than) -147- Table 4.1. Example Linear Functions T13(zl> = 1.33 2.67z1 t14(z1^ = 2.67 2.33zl t15(z1) = 1.6 1.8z1 t16(z1) = 2 1.6z1 t23(z1> = 0.67 1.33z1 t24(z1* = 2 21 t25(zP = 1.2 Z]L T26 W = 2 3.5Zl T36(z1) = 3 3z^ t46(zI> = 1 2.5zj -58- The Dual Dispersion Problem is as follows: Find a subset K* of V such that g(K*) = max{g(K) : K^V, |k| = p+1} (2.2.2) We remark that the dispersion problem is meaningfully defined for 2 p+1 < n. The primal p-center problem is trivial for p > n. Hence, we shall restrict p to 1 p < n-1. In what follows in this section, we prove a Weak Duality Theorem (W.D.T.) and state a Strong Duality Theorem (S.D.T.) (proven in Sec tion 5). At the end of this section, we give an example problem illustrating definitions and results. In the W.D.T. we shall use the fact (readily proven as in [18] or [29]) that a f (X) for any XC T, |x| < . Theorem 2.2.1. (Weak Duality Theorem). Assume 1 p n-1. For any X C T with |X| = p, and any K C V with |k| = p+1, we have f(X) > g(K). Proof. There are two cases: g(K) < a or g(K) > a. In the former case we have g(K) 1 a < f(X). In the latter case, we note that g(K) = g^(K) > a > g^(K). Since |x| = p < p+1 = |k|, at least two demand points in K must be served by a single center. In other words, for some v ,v Â£ K with s ^ t, and some center xeX, we have s t f [D(X,v )] = f [d(x,v )] 5 f(X) s s s s (2.2.3) ft[D(X,vt)] = ft[d(x,vt)] < f(X) . Using the definitions and the inequalities in (2.2.3), we have g(K) = g^(K) < 3gt 2 max{fg[d(x,vs)],ft[d(x,vt)]} < f(X). Remark 2.2.1. We note that the conditions |x| = p and jKÂ¡ = p+1 can be replaced by |x| 2 p and/or |k| > p+1, respectively, and the proof -145- found by applying the Sequential Location Procedure of [32] to the constraints DCz after fixing the locations of the uniquely located facilities. Any such feasible solution Y to DC^ is clearly an effi cient location vector. At this point we give an example application of E-FRONT. We apply the algorithm to the tree network for which the necessary data are given (earlier) in Figure 4.1. The arcs in A^ are labeled a^ through a^ as shown in Figure 4.4a). Figure 4.4b) shows the graph GBC^j ^ with arc lengths l/w and l/v^. The thickly drawn subgraph of GBC,, is the graph GD. The values of m 1 Â£ s < t Â£ 3, are \I j I / D St also given in the same figure. Corresponding to each pair of arcs (a^,a^) e A the linear function T^j(zp is specified in Table 4.1. These functions are plotted in Figure 4.5, and t(z^) is indicated by the cross-hatched line in the same figure. The values of a and b are computed by using the techniques given earlier. We remark that t(z^) = T24^zl^ fr zl e for this example problem. Since a2 = anc* a4 = ^2^4^ anc* m12 = t^ie -^-en8t^ f arc ^N1N2^ t^ie Path (E2^i^2^4^ as a path for any choice of (z^,Z2) in Z*. For example, for z^ = 1.5 and Z2 = 0.5, (z^,Z2) is on the efficient frontier. The tight path corresponding to this choice of (z^,Z2) is shown earlier in Figure 4.2b), together with the corresponding location vector Y = an t^ie same figure. Note that y^ and are uniquely located as N^ and N2 are on the tight path. -li to a nearest element of each collection. Simply sum each row of this matrix and choose the vertex whose row sum is minimum. Frank considered a probabilistic version of the 1-median problem in [34] where each weight is a random variable with a known distribu tion. A number of bounds are obtained on the expected value of the objective function as well as its variance. Some of these results are generalized by Frank [35] to the case where the weights are jointly distributed random variables. We now concentrate on the p-median problem with p > 2. p-Median of a network and vertex optimality. A significant theoretical contribution is due to Hakimi [48]. Hakimi proved that there exists an absolute p-median contained in V. Certain generaliza tions of this result have been given in subsequent work. Levy [80] proved that the (vertex-optimal) result holds when the weights w^ are replaced by concave cost functions c^(*) of the distance between v_^ and its nearest median. Goldman [41] generalized the result to the case of a "two-stage" commodity. More specifically, one distinguishes a vertex as being a source or a destination. Let (v ,v.) be a source-destination pair, S Cl and let x^ and x_. be the nearest medians to v and v^, respectively. Then the cost of transferring the commodity from source v to destina- s tion V, is the sum of three transport costs, namely, w .d(v ,x ) + a sd s 1 r\j wsdd(xi,x^.) + w*dd(x_. ,v) In general, if X = {x^...^} is a median set, one does not know which median is the nearest to v or v,; hence, s d the cost associated with a source-destination pair (v ,v,) is s d given by fsd(x) = min Kd^VV +"sdd(xi*xj) + Wsdd(xjVd)] -159- Con N) which connect x. to any vertex v. in V.. Thus, T.(x.) is a i J 1111 subtree which spans the subset of V. We make the following con jecture: Conjecture 5.4.1. Given an optimal p-center X of N, there exists a collection of p shortest path subtrees (T^(X^),...,1^(x^)} such that (a) for every i, 1 <, i p, x^ is a closest center to any vertex v^ which is in T.(x.), and i i (b) T^(x^) Tp(Xp) are mutually disjoint. We remark that in order for the conjecture to be true, it is necessary to show that any shortest path tree T^(x_^) contains as ver tices only the members of V^, where Vj,...,V are the mutually disjoint subsets of V which are identified as described in the paragraph pre ceding the conjecture. In other words, if v^ is an arbitrarily chosen vertex from V^, it is necessary to show that, among all the shortest paths connecting x^ to v^, there exists at least one shortest path, say, P(XÂ£>V^) such that P(x^,v^) passes through only those vertices which are members of V^. Otherwise, if every shortest path P(x^,v^) passes through some v^. for which v^ i V^, then- clearly any shortest path tree T.(x.) contains at least one vertex v. for which v. V., 11 j J1 so that T,(x.) H T.(x.) ^ 0 for at least one index j with j ^ i. But i i J J if the intersection of Tk(x^) and Tj(x^) is nonempty then the union of T^(x^) and T^Cx^) may have cycles. If the conjecture holds, then {T,(x,),...,T (x )} is a collection 11 p p of disjoint subtrees, i.e., a forest of N. If we had a knowledge of such a collection of subtrees without having had to know X, then clearly the 1-center of each subtree could be determined by known techniques, and the collection of those one-centers would be an optimal p-center -163 26. J.W. Eyster, J.A. White, and W.W. Wierville, "On Solving Multi facility Location Problems Using a Hyperboloid Approximation Procedure," AIIE Transactions 5, 1-6 (1973). 27. R.L. Francis, "A Note on the Optimum Location of New Machines in Existing Plant Layouts," .J. Ind. Engr. 14, 57-58 (1963). 28. R.L. Francis, "Some Aspects of a Minimax Location Problem," Opns. Res. 15, 1163-1169 (1967). 29. R.L. Francis, "A Note on a Nonlinear Minimax Location Problem in a Tree Network," J. Res. Nat. Bur, of Stds. 82, 73-80 (1977). 30. R.L. Francis and J.M. Goldstein, "Location Theory: A Selective Bibliography," Opns. Res. 22, 400-410 (1974). 31. R.L. Francis and J.A. White, Facility Layout and Location: An Analytical Approach, Prentice Hall, Inc., Englewood Cliffs, New Jersey, 1974. 32. R.L. Francis, T.J. Lowe, and H.D. Ratliff, "Distance Constraints for Tree Network Multifacility Location Problems," Opns. Res. 26, 570-596 (1978). 33. R.L. Francis, T.J. Lowe, and B.C. Tansel, "Binding Inequalities for Tree Network Location Problems with Distance Constraints," Research Report No. 78-10, Dept, of Industrial and Systems Engineering, Univ. of Florida, Gainesville, Florida (1978). 34. H. Frank, "Optimum Locations on a Graph with Probabilistic Demands," Opns. Res. 14, 409-421 (1966). 35. H. Frank, "Optimum Locations on a Graph with Correlated Normal Demands," Opns. Res. 15, 552-556 (1967). 36. H. Frank, "A Note on a Graph Theoretic Game of Hakimi's," Opns. Res. 15, 567-570 (1967). 37. R. Garfinkel, A. Neebe, and M. Rao, "An Algorithm for the m- Median Plant Location Problem," Trans. Sci. 8, 217-236 (1974). 38. R. Garfinkel, A. Neebe, and M. Rao, "The m-Center Problem: Minimax Facility Location," Manag. Sci.23, 1133-1142 (1977). 39. F. Gavril, "Algorithms for Minimum Coloring, Maximum Clique, Minimum Covering by Cliques, and Maximum Independent Set of a Chordal Graph," SIAM. J. Comp., Vol. 1, 180-187 (1972). 40. A.M. Geoffrion, "Proper Efficiency and the Theory of Vector Maximization," Â£. of Math. Anal, and Appl. 22, 618-630 (1968). 41. A.J. Goldman, "Optimum Locations for Centers in a Network," Trans. Sci. 3, 352-360 (1969). -42- 1 through m corresponding to new facilities, a source s and a sink t. Graph G contains arcs (s,i) and (i,t) for 1 < i < m and arcs (j,k) for each pair (j,k)eIB. The capacity of arc (j ,k) is the weight v^. The capacity of arc (s,i) is given by J [w : v eT., (i,r)el ], and the ir v i o capacity of arc (i,t) is given by J [w. : v eT, (ijqjel.,]. If xq q u (Q,Q) is a minimum capacity s-t cut of G, with seQ, teQ, then all new facility locations x^ for which the corresponding node i is in Q are in T^ in an optimal solution. Similarly, all x_. for which the node j is in Q are in T^ in an optimal solution. The procedure is a repeated application of this minimum cut problem with respect to each edge, until an optimal vertex location is determined for each x^. During the process, each x^ whose location is determined is treated like an existing facility. The method is described originally for the analogous rectilinear distance problem on the plane, which, in turn, decomposes into two subproblems, each on a line. Multi-objective location problems on networks Multi-objective optimization problems, sometimes known as vector optimization problems, involve decision making under two or more criteria. More explicitly, a set (finite or infinite) S of alterna tives is specified and n (possibly non-commensurable) objective func tions are to be minimized over S. Let f,,...,f be n numerical func- 1 n tions defined on S, and define f(x) = (f,(x),...,f (x)) for all xeS. 1 n The multi-objective optimization problem (VMP) is the following: V-min f(x) xeS In general, the minima of the functions f_,...,f do not coincide. 1 n In order for the minimization to be meaningful, one needs to introduce -103- wit h components z*, P e K*, and call Z* the irreducible representation of Z. Corresponding to GBC*, define DC* to be the distance con straints with every constraint corresponding to exactly one arc in GBC*. It will be convenient to refer to the triplet (Z*, DC*, GBC*) as the reduction of (Z, DC, GBC). We remark that for an irreducible location vector Z, the reduction of (Z, DC, GBC) is Identical to (Z, DC, GBC), as RP terminates immediately in this case. For P e K*, define A(N^) to be the set of adjacent nodes to . in GBC*, and let Ap(Z*) be the collection of locations of facili ties whose nodes are members of A(Np). The following property gives the sufficient conditions for reducible location vectors (as well as irreducible ones). Property 3.5.3. Let (Z*, DC*, GBC*) be the reduction of (Z, DC, GBC) and let K* be the list of composite indices for new facility nodes of GBC*. If Zp e H[Ap(Z*)] for every P e K*, then Z is efficient. Proof. By definition Z* is irreducible. Hence, the hypotheses of the property imply, upon using Property 3.5.2, that Z* is efficient with respect to the reduced constraints DC*. From Theorem 3.3.3, for every P e K*, node Np is in a tight path in GBC*. Now, we want to show that the original nodes i e P, are all in tight paths in GBC. Recover GBC from GBC* by decomposing every node Np of GBC* into its original nodes N^, i e P, and connect these nodes to one another by arcs of zero length by adding those arcs which were removed by RP. Since the added arcs have lengths of zero, the shortest path lengths cannot change. Hence, the shortest path length between any two existing facility nodes of GBC* is the same as the -2- between the facilities, the objective criteria used, the constraints, the presence or lack of random elements, and possibly other factors as well. Among the several variants, planar location problems received special attention in the past, starting with the earliest contribu tions, for example [106]. In such planar problems, one is interested in locating new facilities in the Euclidean plane with respect to existing facilities. For continuous planar problems, where any point in the plane is a feasible location, the typical distance used is the distance, special cases of which are the rectilinear, Euclidean, and Tchebychev norms. For discrete planar problems, where there are a finite number of candidate locations for new facilities, the distance between any potential new facility location and any existing facility is a specified positive number. Such discrete problems, due to the finite nature of feasible locations, readily lend themselves to integer programming formulations. The reader is referred to the book by Francis and White [31] for a discussion of planar problems and a wealth of references. A number of real life applications suggest that, in some in stances, a network space can be a more faithful representation of the reality than the Euclidean plane. For example, in a road network, a communication network, or a pipeline system, travel occurs along the arcs of the underlying network rather than in straight lines or recti linear paths. Hence, for such problems, the use of shortest path distances along the arcs of the network can approximate the travel distance more closely than the X. distance. As opposed to planar problems, network location problems have received much less attention -47- and a center constrained one. More specifically, for real A and y define the functions m*(A) and c*(y) as follows: m*(A) = min[m(x): c(x) < A] (1.3.5) c*(y) = min[c(x): m(x) y] (1.3.6) In general for some values of A (y), the constraint c(x) Â£ A (m(x) Â£ y) may not admit any feasible solution. However, real inter vals C and M can be defined so that for any AeC and for any yeM, the constraints in (1.3.5) and (1.3.6) admit a feasible point. To define C, let Q be the set of all minima to min[c(x): xeN], and let 2 -be c m the set of all minima to min[m(x): xeN]. Let x be a point in that c minimizes the value of m(x) over all x in Similarly, let y be a point in Â£! that minimizes the value of c (y) over all y in ft Then m J J m C and M are defined as follows: C = [c(x), c(y)] M = [m(y) m(x) ] With these definitions Halpem's duality theorem can be stated as follows: a) Given any yeM, with A = c*(y), we have c*(m*(A)) = A. b) Given any AeC, with y = m*(A), we have m*(c*(y)) = y. For a tree network, the functions m* and c* are 1-1 and onto. It follows from the duality theorem that the function m* and c* are inverses of each other for a tree network. For a general network, the functions m*, c* need not be onto, i.e., the image of the domain -76- where (2.5.5) follows from the W.D.T., (2.5.6) and (2.5.7) follow from the definitions of g and g^, (2.5.8) follows from K*+^ c U, (2.5.9) follows from (2.5.4), and (2.5.10) follows from the definition of R. Hence, every inequality holds as an equality, establishing (2.5.2) for this case. The assertion that K*., solves the dual problem is immediate from p+i f(X*) g(K*+1) and the W.D.T. We note that Theorem 2.5.1 provides a proof of the S.D.T. since in the statement of the S.D.T. we take X* to be an optimum p-center solu tion to the primal problem and K*+^ as constructed by OPTKLIQUE. We also note that the duality theory provides necessary and sufficient conditions for a p-center to be optimal, which, as far as we know, are the first such conditions for this problem. We remark, just as with the linear p-center problem, that if we define 6 = minig..: g e R, q(B..) < p}, then 8 ^ = r Clearly st ij ij ij st p q(r ) < p and q(8 ) < p. The S.D.T. implies r e R, and thus the P st p definition of g ^ gives g ^ < r Let p' = q(g ) and let X solve st st p st p the cover problem for r = g so that f(X .) < g Since p > p', st p st append to X^, (if necessary) any p-p' center locations to obtain the p-center X^. Clearly D(X^,v^) D(X^,,Vj) for v e V, and thus f(X ) < f(X ,). Hence r f(X ) < f(X ,) < g < r so g = r P p p p p st p st p and X^ is an optimum solution to the p-center problem. This remark permits the use of the same procedures as discussed in [65] to compute r^ efficiently, by performing a binary search over the (ordered) list R, applying COVER for every r chosen from R until a smallest g in R st is found for which COVER finds p or less points. Once r^ is computed in this manner, OPTKLIQUE requires an additional application of COVER -43- the concept of "efficient solutions." A point x in S is said to be efficient if there does not exist a point y in S such that f^(y) < f_^(x) for 1 i < n and f^(y) < f^Cx) for at least one index k. One is interested in finding and characterizing the set of efficient solu tions to (VMP)An efficient point is sometimes known as an undominated point. A point which is not efficient is said to be dominated. Kuhn and Tucker [76] and Koopmans [74] are among the first to introduce the concept of efficiency. Geoffrion [40] extendd the con cept to "properly efficient" points and provided a comprehensive theoretical framework for subsequent research. Necessary and suf ficient conditions for efficient points to be properly efficient are given by Wendell and Lee [112]. Some of the later contributions are due to Yu [117], Yu and Zeleny [118, 119], Bitran and Magnanti [4], Wendell [110], and Bergstresser, Chames, and Yu [l ] We note that there are other approaches to multicriteria decision making, such as goal programming, multi-attribute utility theory, construction of outranking relations, and interactive programming techniques. For general information on multicriteria decision making, the reader is referred to Roy [99], Starr and Zeleny [104], Cochrane and Zeleny [16], Keeney and Raiffa [67], and Thiriez and Zionts [l08]- A survey of multicriteria decision making is given by Chalmet [7]. Multi-objective location problems (on the plane or on networks) have begun receiving attention only recently. Kuhn [75] appears to be the first to consider a multi-objective location problem on the plane. Kuhn considered the problem of minimizing the vector of Euclidean distances from a variable point to a set of fixed points on the plane, and showed that the set of efficient solutions is the convex -17- y.e{0,l} will determine whether or not at most p points (in P) can cover all vertices of N within a radius r. Computational experience is reported and it is found that the procedure works better for larger values of p, as in this case the initial upper bound Z is small, and significant computational savings result in identifying those edge bottleneck points whose distances fall within the interval [0,Z]. The weighted case on general networks was considered by Christofides and Viola [15], and an approximate solution procedure was given. The procedure finds a set X of p-points whose objective value f(X) is within some e-neighborhood of the actual p-radius r The procedure P oi obtains X by solving a sequence of r-cover problems with successively increasing values of r. Termination occurs when the solution of an r-cover problem generates p (or less) points the first time. In the process, one also obtains approximate solutions for n-1, n-2,..., p+1 center problems. The solution of each r-cover problem is obtained in two stages: First, all feasible solutions to the r-cover problem are obtained by finding all regions on the network that can be reached by a vertex within a radius of r. Then, among all these feasible solu tions, those with minimum cardinality are found by solving a set covering problem. To find all regions on N reachable by a vertex v_^, one "penetrates" a distance of r/w_^ along all possible paths originating at v_^. The procedure is repeated for each vertex and the intersections of these penetrations are found. Each maximal intersection defines a connected region all of whose points are reachable by a subset of vertices within a radius r. The subset of the vertices is that which defines the intersection. These regions jointly cover all vertices of N, and it is possible that a subcollection of the collection of all CHAPTER 5 SUMMARY AND FUTURE RESEARCH 5.1 Summary In this dissertation we considered problems which involve locating multiple new facilities on a tree network with respect to n existing facilities at known locations. In Chapter 2, we solved the nonlinear p-center problem where the objective is to minimize the maximum cost associated with any existing facility. The cost (disutility) of service associated with any existing facility is a nonlinear (strictly increasing and continuous) function of the distance between that existing facility and its nearest new facility. We gave a weak and a strong duality theorem and pro vided a physical interpretation of the dual problem. Our approach for solving the nonlinear p-center problem and its dual was to solve a sequence of covering problems which involve minimizing the number of new facilities subject to a family of n distance constraints which impose upper bounds on the distances between any existing facility and its nearest new facility. In Chapter 3, we considered a vector-minimization problem which involves as objectives the distances between specified pairs of new and existing facilities and specified pairs of new facilities. We developed the necessary and sufficient conditions for efficiency and provided an algorithm for constructing efficient solutions. Our L 49 -28- and the objective to be minimized is f(X) = J [f (X) : (v ,v,)eVxV], u sd s d Goldman showed that there exists an optimal X* contained in V, and conjectured that the result holds for any multi-stage problem. Hakimi and Maheshwari [49] proved a stronger version of Goldman's conjecture. In this version, there are multiple commodities for each source-destination pair, and each commodity goes through multiple stages. Furthermore the cost of transport from one stage to the next is a concave nondecreasing function of the distance. More specifically, let M be the set of commodities to be transferred from source v to sd s destination v^, and let g(m) be the number of stages commodity meM^ is to go through. For a given location set X = {x.,...,x }, denote 1 P by y^ = the location where the rth stage processing takes place. The cost of transferring commodity m from source v to destination v, s d is given by C^Jd (v^yp ] + ] + ... + C^d (yg(n),vd) ], where () is a concave nondecreasing function of the distance. Denoting this quantity by f^^(Y), with Y C X, |y| = g(m), the minimum cost of transfer for commodity m is given by f (X) = min[f (Y): sdm sdm Y C. X, |Y| = g(m)]. The cost of transferring all commodities from vg to Vj is obtained by summing over all commodities, that is, fgd(X) = J [fsdm(X): meM d]. The total cost of the system is obtained by summing the cost f ^(*) over all source-destination pairs, that is, f(X) = Â£ [f^W: (v ,Vj)eVxV]. Hakimi and Maheshevari proved that there exists a minimum X* of f(X) contained in V. Wendell and Hurter [111] considered a more general form of the problem where the transportation cost functions are permitted to differ from edge to edge. The transport cost on any edge is a non decreasing concave function of the distance. They proved that it is -6- Finally, for tree networks, we write T instead of N. In passing, we note that the shortest path P(x,y) between any two points x,yeT is unique, as otherwise T would contain a cycle. 1.3 Survey of the Network Location Literature Historically, the earliest precise mathematical formulation of a location problem on a network appears to be due to Hakimi [47] in 1964. Prior to Hakimi's paper, the problem of finding the best threshing site for harvested wheat was attacked by using a network location model in 1962 by Hua Lo-Keng and Others [60]. This model was presented only at an intuitive level and no mathematical formulation or properties were given. A (correct) solution procedure was suggested (in the form of a poem), which was to be discovered independently by Goldman [42] in 1971. Since 1964, a literature of approximately 80 papers has grown till the present. Several new problems, as well as certain extensions and generalizations of old problems, have been introduced. A recent text by Handler and Mirchandani [ 58 ] discusses ex tensively a portion of the literature involving minimax and minisum problems as well as single-facility bi-objective problems involving the combination of these two objectives. A "family tree" for network location problems is shown in Figure 1.1. Although not exhaustive, the family tree covers most of the problems formulated since 1964. With reference to the family tree shown in Figure 1.1, network location problems can be broadly classi fied into two groups: point-location problems and path-location problems. Path-location problems have been recently introduced by -63- Table 2.1 Values and Node Weights for Example i 1 2 3 4 5 2 3 4 5 6 225 3600 3600 3600 4356 3600 3600 3600 4556.25 13829.76 8464 11664 900 784 1664.64 j 1 V0) 0 2 3 4 0 64 0 5 6 0 144 -162- 12. R. Chandrasekaran and A. Daughety, "Problems of Location on Trees," Discussion Paper No. 357, The Center for Mathematical Studies in Economics and Management Science, Northwestern University, Evanston, Illinois (1978). 13. R. Chandrasekaran and A. Tamir, "Polynomially Bounded Algorithms for Locating p-Centers on a Tree," Discussion Paper No. 358, The Center for Mathematical Studies in Economics and Management Science, Northwestern University, Evanston, Illinois (1978). 2 14. R. Chandrasekaran and A. Tamir, "An 0((nlogp) ) Algorithm for the Continuous p-Center Problem on a Tree," Discussion Paper No. 367, The Center for Mathematical Studies in Economics and Management Science, Northwestern University, Evanston, Illinois (1978). 15. N. Christofides and P. Viola, "The Optimum Location of Multi- Centers on a Graph," Opnl. Res. Q;. 22, 145-154 (1971). 16. J.L. Cochrane and M. Zeleny, Eds., Multiple Criteria Decision Making, Univ. of S. Carolina Press, Columbia, South Carolina, 1973. 17. E.J. Cockayne, S.T. Hedetniemi, and P.J. Slater, "Matchings and Transversals in Hypergraphs, Domination and Independence in Trees," ^J. Combinatorial Theory, Series B26, 78-80 (1979). 18. P.M. Dearing, "Minimax Location Problems with Nonlinear Costs," J. Res. Nat. Bur. of Stds. 82, 65-72 (1977). 19.P.M. Dearing and R.L. Francis, "A Minimax Location Problem on a Network," Trans. Sci. 8, 333-343 (1974). 20.P.M. Dearing and R.L. Francis, "A Network Flow Solution to a Multifacility Minimax Location Problem Involving Rectilinear Distances," Trans. Sci. 8, 126-141 (1974). 21. P.M. Dearing and G.J. Langford, "The Multifacility Total Cost Location Problem on a Tree Network," Technical Report No. 209, Dept, of Mathematical Sciences, Clemson University, Clemson, South Carolina (1975). 22. P.M. Dearing, R.L. Francis, and T.J. Lowe, "Convex Location Problems on Tree Networks," Opns.Res. 24, 628-642 (1976). 23. S.E. Dreyfus, "An Appraisal of Some Shortest Path Algorithms," Opns. Res. 17, 395-412 (1969). 24. A.M. El-Shaieb, "A New Algorithm for Locating Sources Among Destinations," Manag. Sci. 20, 221-231 (1973). 25. D.J. Elzinga, D.W. Hearn, and W.D. Randolph, "Minimax Multifacility Location with Euclidean Distances," Trans. Sci. 10, 321-336 (1976). -29- sufficient to consider the vertices of the network under such a cost structure. Furthermore, they obtained the conditions under which it is necessary for the solution to occur at the vertices. In particular, they showed that nonvertex optimal locations can occur in any given edge, only when transportation costs are linear with distance over that edge and in that case, when and only when the slopes of these linear cost functions are in a special relation. Hence, if at least one cost function over some edge is nonlinear, then no interior point of that edge can be in an optimal solution. If the same situation holds for every edge, then a solution must necessarily occur at the vertices of the network. Solution approaches. Kariv and Hakimi [66] showed that the p- median problem on a general network is at least as hard as NP-complete problems. For the case of tree networks, however, algorithms of polynomial complexity have been developed. Matula and Kolde [85] 3 2 suggested an 0(n p ) algorithm for finding the median of a tree net- 2 2 work. Kariv and Hakimi [66] proposed an 0(n p ) algorithm for the same problem. For general networks, a number of solution procedures have been developed subsequently, all based on the vertex-optimality result. Their common characteristic is that they all confine the search to vertex locations. The solution procedures can be grouped in three categories: mixed-integer programming approaches, branch-and-bound techniques, and heuristics. ReVelle and Swain [95] formulated the problem as a linear integer program with 0,1 variables. The solution is obtained by applying the primal simplex algorithm to the associated linear program. In case Abstract of Dissertation Presented to the Graduate Council of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy OPTIMAL MULTI-FACILITY LOCATION ON TREE NETWORKS By Barbaros C. Tansel December 1979 Chairman: Richard L. Francis Major Department: Industrial and Systems Engineering In this dissertation we develop a theory for location problems which involve locating multiple new facilities on a tree network with respect to existing facilities at known locations. The first problem we consider is the nonlinear version of the p-center location problem on a tree network for which the cost of each served vertex is a strictly increasing continuous function of the dis tance between the vertex and the nearest center,and the objective is to minimize the maximum such cost over all possible locations of the centers. We present a dual "dispersion" problem which may be inter preted as the problem of choosing p + 1 (or more) vertices such that the minimum cost to serve any two of the chosen vertices by a single common center is as large as possible. We give a weak duality theorem which applies to all general networks and a strong duality theorem which applies to all tree networks. The strong duality theorem also specifies the necessary and sufficient conditions for an optimal solu tion to either problem. We provide algorithms of polynomial complexity v -55- The first duality relationship involving tree network location problems can be found in Meir and Moon [ 86 ]. Cockayne, Hedetniemi, and Slater [ 17 ] obtained a more general version of the result given in [ 86 ]. The results in [ 86 ] and [ 17 ] closely parallel our duality result for the covering problem and its dual. Shier [100] discovered a "dispersion" problem which is dual to the continuous unweighted p-center problem. The dispersion problem of Shier is to choose p+1 points in the tree network the nearest two of which are as far apart as possible. Chandrasekaran and Tamir [14] observed that Shier's duality holds when the problems are defined with respect to a subset of the tree. For the case where this subset is a finite collection of demand points, their result is an instance of the duality relation ship we will present in this chapter, as applied to the unweighted linear case. At this point we give a brief overview of the chapter. In Sec tion 2, we define the (nonlinear) p-center problem and a dual "dis persion" problem. We state and prove a weak duality theorem applicable to all networks, and state a strong duality theorem applicable to tree networks. In Section 3 we give a physical interpretation of the dual dispersion problem. In Section 4 we study a covering problem and present an algorithm, COVER, for solving it. The covering algorithm provides the basis of our solution procedure to the p-center problem as well as the dual dispersion problem and yields a construc tive approach for proving the strong duality theorem. In Section 5 we present an algorithm, OPTKLIQUE, which provides a constructive proof of the strong duality theorem, while solving the dual problem. Addi tional results for the covering problem, including a "divergence" problem dual to the cover problem, are given in Section 6. -124- As in Chapter 3, a location vector Y in Tm is said to be efficient with respect to (4.2.2) if and only if X e Tm and f(X) < f(Y) imply f(X) = f(Y). A location vector which is not efficient is said to be dominated. Our main interest is to relate the bi-objective m-center problem to the distance constraints problem studied by Francis, Lowe, and Ratliff [32], We shall characterize efficient points by making use of the separation conditions (defined in Chapter 3) which are known to be necessary and sufficient for the distance constraints to be consistent. To define the distance constraints of interest, let z = (z^,z^) be any two-tuple (with z > (0,0)) and consider the constraints given in (4.2.3): d(x^,Vj ) < Zj/w^ (i, j ) e Ic (4.2.3) d(xjxk) Z2/Vjk (j>k) e Ig . We shall refer to the family of constraints in (4.2.3) as DC^. The constraints DC are said to be consistent if there exists at least z one feasible solution X = (x, x ) to (4.2.3). 1 m Corresponding to DC^ we define GBCz to be the undirected graph with nodes ,...,Nm,E^,...,En- For every (i,j) e 1^ there is an arc (N.,E.) of length z./w.. and for every (j,k) s ID there is an arc 1 3 i ij a (N.,N,) of length z/v., We partition the arc set of GBC into two j K z j k. z sets Ab and Ac with Ag = {(N.,Nk): (j,k) e Ig} and Ac = {(N,Ej): (i,j) e Iq}. We shall assume that the sets 1^ and Ig are such that GBC is connected, as otherwise DC decomposes into independent sets z z of constraints which may be analyzed separately. -35- connected, denote by L(E.,E ) the length of a shortest path connecting J k nodes E. and E. for 1< j distance constraints are consistent on a tree network if and only if the inequalities (E^.E^) Â£ d(vj,v]P are satisfied for 1 < j < k < n. These inequalities are called the separation conditions. The proof of the consistency of the distance constraints implying the satisfac tion of the separation conditions uses only the triangle inequality and hence is applicable to all networks. The reverse implication always holds for tree networks, but may fail to hold for general net works. The proof of the reverse implication is constructive and actually finds a feasible location vector under the assumption that the separation conditions are satisfied. The method that constructs such a feasible location vector is termed the "Sequential Location Procedure" in [32]. The method can best be described with the aid of a physical model. One may imagine that the tree is represented by appropriately inscribing straight line segments on a board such that each segment represents an edge. At vertex v_^, strings of length c are fastened for each new facility j such that (i,j)el A tip vertex Li is chosen arbitrarily and all strings fastened at that vertex are pulled tight towards the adjacent vertex. If all strings reach the adjacent vertex, they are simply engaged there with their loose ends free to be pulled tight in some future iteration. Also the tip vertex together with the edge incident to it is removed from the model. The procedure is repeated with the resulting tree. In the other case, not all the strings reach the adjacent vertex when pulled tight. Among those which do not reach the adjacent vertex one which is shortest is selected, and the end point of this string determines the location of -158- 5.4 Tree Networks and General Networks For the nonlinear p-center problem we pose the following questions. Ql. For the nonlinear p-center problem defined on a general network N, does there exist a spanning tree T of N such thnt the solution of the p-center problem on T determines the solution to the p-center problem on N? Q2. If such a spanning tree exists, is there a way to find it without having to enumerate on all spanning trees of N? Dearing and Francis [19] showed that the 1-center problem on N can be solved by solving a sequence of 1-center problems on spanning trees of N. The basic theme of the approach is as follows: Suppose that we know an optimal 1-center, say, x which solves the problem on N. Using the procedure given in Busacker and Saaty [5], find a shortest path tree rooted at x, by identifying the shortest paths (on N) con necting x to any vertex v^. If T(x) is such a shortest path tree, then clearly x is also the optimal 1-center of T(x). Thus, for the 1-center problem the answer to Ql is in the affirmative while Q2 remains un answered as the proof of the existence of such a tree is based on the knowledge of an optimal 1-center of N. For the case with p > 1 we propose a similar approach. Suppose we know an optimal p-center X = ix. x } which solves the problem ^ P on N. Partition the vertex set V = (v^,...,vn} of N into p disjoint, exhaustive subsets V,,...,V such that a nearest center to any vertex 1 p v in V, is the ith center x.. If ties occur we break the ties i i i appropriately so as to satisfy the condition that V^,...,V are mutually disjoint. For 1 < i < p, define T^(x^) to be a shortest path (sub)tree which is constructed by finding the shortest paths -67- There is also the possibility that A will make a false threat, that is, attack a vertex not among the ones he threatens. If D be lieves the threat is false and continues to act conservatively, he will simply choose a p-center X* to minimize f. But since there exists a p+l-clique KA+^ such that g(K*+p = f(X*), the greatest loss D can incur, given Xa, is the same as if he believes A's optimal threat to be real, and acts accordingly. Hence A cannot gain more by making a false threat. 2.4 Covering Algorithm In this section we study a covering problem, and present an algorithm for solving it. Our primary interest in the algorithm is the fact that it provides a constructive approach for proving results about the primal and dual problem. For this reason we purposely keep the algorithm simple, and use an analog string model to provide insight into the algorithm. The development of both the string model and the algorithm is motivated by an earlier string algorithm given in [32]. As in [32], an equivalent algebraic version of the algorithm is readily obtainable. We remark that two other quite efficient algo rithms [14], [15], exist for solving the covering problem, but they do not lend themselves readily to our needs. At this point we state the Covering Problem: Given r and the runction f, compute q(r) = mini|x|: f(X) S r, X C T} (2.4.1) It is readily seen that the covering problem has a feasible solution if and only if a < r. Further, with J(r) = {j: r < f^(6 )}, we shall -96- Figure 3.4. Example of an Efficient Location Vector 3.5 Further Results on the Convex Hull Property In this section we concentrate on the last statement of Theorem 3.3.3, namely, that Z is efficient if and only if each is contained in the convex hull H[A*(Z)], where A*(Z) contains the locations of those uniquely located facilities whose nodes are adjacent to N^. Our main interest is to delete the phrase "uniquely located" from the definition of A*(Z) and still have the equivalence hold under the new (relaxed) definition. From a computational standpoint, this deletion would make it unnecessary to identify the uniquely located new facilities, which, in turn, requires the identification of tight paths in GBC. -155- S(P(Ep,E^)jA^) to be the sum of the reciprocal weights where the sum mation is taken over all arcs which are contained both in P(E ,E ) P q and in A Define e to be the minimum of r L P(E ,E ) d(v ,v ) z P q P q S(P(E ,E ),A ) p q r over all paths P(E^,E^) which pass through A^. Clearly e > 0. Choose X = (A,,...,!) with A. = 0 for i I r, and 0 < A < e. It is direct I t i r to verify, by using arguments similar to the ones given in the proof of Lemma 4.3.2, that such a choice of A is a valid choice for DC to z-A be consistent. But consistency of DCz_^ and the fact that A > 0, A 0, imply X is dominated, contradicting that X is efficient. To show (b) implies (a) suppose for any r for which > 0, at least one arc in A^ is in a tight path. Hence, for any A = (A^,...,A ) > 0, A 0, the length of at least one tight path in GBC .. will be strictly smaller than the distance between the locations of the exist ing facilities corresponding to the terminal nodes of the path. Thus, at least one separation condition on GBC is violated so that DC .. z-A z-A is inconsistent for any A > 0, A ^ 0. It follows that there does not exist Y for which f(Y) < z = f(X) and f(Y) ^ f(X), which is the definition of efficiency. The theorem holds for the problems considered in Chapters 3 and 4 as well as the m-center problem with mutual communication considered in [32]. We remark that the condition > 0 may appear to be somewhat superfluous. Its omission will not affect the equivalence of (a) and (b). The reason we included this condition is that it is unnecessary -97- With this motivation in mind, define, for 1 < j < m, A(N_.) to be the collection of nodes in GBC which are adjacent to N and denote by A (Z) the collection of the locations of the new and existing facili- j ties whose nodes are in A(N). We remark that N, is not a member of J J A(N.) and hence z^ f/ A^ (Z) . The following property states the necessary conditions for Z to be efficient. Property 3.5.1. Suppose Z is efficient. Then ze H[A^(Z)] for every j Â£ {1,... ,mj. Proof. From Theorem 3.3.3, whenever Z is efficient, z^ e H[A*(Z).] for each j e {l,...,m}. But A*(Z) is clearly a subset of A(Z) implying that z e H[*(Z)] C H[A (Z)], completing the proof. In general, the reverse implication in Property 3.5.1 need not hold for certain (pathological) cases. Such occurrences correspond to the case where Z is such that for some two adjacent nodes N. and N, , J k the locations z and z^ coincide. We provide an example of such a case in Figure 3.5. With reference to Figure 3.5, observe that every z. is contained in the associated convex hull. In particular, 7. and J ^ z^ are contained in their respective convex hulls because their loca tions are the same. The location vector is clearly a non-efficient one, since and z^ can both be moved to v^> thereby reducing the distances associated with them. Sufficiency for Irreducible Location Vectors At this point we distinguish two classes of location vectors and show that the reverse implication (sufficiency) in Property 3.5.1 holds for one class ("irreducible" location vectors) while it need not hold -34- property, a "sequential intersection procedure" was developed that n determines the composite neighborhood N(a,r) = O N(v.,c.), with i=l 1 unique center a and radius r, by intersecting the neighborhoods N(v ,c ) one at a time in an arbitrary order. The procedure can be implemented in 0(n) operations. The composite neighborhood N(a,r) contains all alternate feasible points when the constraints are con sistent, and N(a,r) is always a convex compact subset of the tree network. A result was also given by Francis et al. that provides a sensitivity analysis on the constraints with no additional computa tional effort. Supposing that the distance constraints are consistent with the original upper bounds c^, consider an e-perturbation of the upper bounds, i.e., for some e > 0 define the new upper bounds to be c^-e, iel. If N(a,r) is the composite neighborhood corresponding to the original upper bounds, then it can be shown that for any e with 0 e ~ r, the e-perturbed constraints remain consistent and the set of feasible points to the e-perturbed system is given directly by N(a,r-e). Distance constraints for the multi-facility case. For the multi facility case, the necessary and sufficient conditions for the con sistency of distance constraints are given in terms of n(n l)/2 inequalities called "separation conditions." The separation condi tions are defined by means of an auxiliary graph constructed by using the sets I and I Let G be the graph with nodes N., 1 5 i < m, ij v 1 corresponding to new facilities,and nodes E ^, 1 < j < n,corresponding to existing facilities. The arc set A of G contains (N ,E ) if i j (i,j)cl-, and (N.,N ) if (j,k)el. The arc length of (N,,E.) is c_,. G j k B i j ij and of (N.,N^) is b.^. Under the (reasonable) assumption that G is -13- 2 computational complexity of 0(n ), and is given by a = max[a..: 1 i j n] where (1.3.2) W-W.VAIV.JV,/ W C* I W t-4. . = i J 1 J 1 J J 1 w w.d(v.,v.) +w.a. + w.a a ij w. + w, 1 jj Hakimi, Schmeichel, and Pierce [50] proved a theorem that reduces the computational effort for computing this lower bound. Their theorem states that if for some a it is true that max[a l5i max[a 1 K i < n] then a is the maximum of all a... A different ti J st 13 solution procedure is also given by Kariv and Hakimi [65] for the same problem. Rather than computing the lower bound, their procedure confines the search to successively smaller subtrees until an edge is obtained. The absolute center is located at the local center (also the global center for a tree) on this edge using Hakimi's procedure for finding a local minimum. This algorithm is of O(nlogn). A nonlinear version of the 1-center problem was considered and solved by Dearing [18], and by Francis [29]. In this version, each weight w_^ is replaced by a monotone increasing function f of the distance d(v_^,x). Both authors obtained a lower bound similar to the one defined by (1.3.2). The bound is applicable to all networks and is always attainable for tree networks. A "roundtrip" version of the problem was solved by Chan and Francis [ 11 ]. In this version each "demand point" is a pair of ver tices (v^,^) and f(x) is the maximum of the roundtrip distances defined by p^(x) = w^dCv^x) + d(x,u^) + a^]. A lower bound, similar I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Antal Maj thay Associate Professor of Managemen This dissertation was submitted to the Graduate Faculty of the College of Engineering and to the Graduate Council, and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy. December 1979 Dean, Graduate School -70- Note that each time COVER places a center at x^ in step 4) it identifies an associated vertex v^ which we call the distinguished vertex associated with x^. When centers x^,...,x^. have been placed in step 4), we call = iv^ ,... ,v^ } the distinguished set associated with {x^,...,^}. If the algorithm places q centers in total, then the set U defined by the algorithm consists of vertices v,,.,...,v, N, the first q-1 of which are distinguished vertices (i) (q) (when q >2). The last vertex is distinguished only if x^ is placed in step 4). Letting X = {x^,...,x }, we call U the primary set associated with X, and call the primary vertex associated with x^, i = l,...,q. We note that the primary vertices v(i)*,v(q) are distinct, for as soon as a primary vertex is identified, its string is removed, and thus the vertex is not available for any subsequent identification. Likewise the centers x ,. .. ,x are distinct, for if 1 q x. = x. with i < i, then all strings assigned to x. would have been i J J assigned earlier to x^, and so x^. would not have been located. Hence it follows that JU| = JXJ = q, and U 0, since JXJ k 1. The primary vertices will be of theoretical significance in proving our results. We now establish some properties of COVER. Property 2.4.1. COVER finds a feasible solution X to the covering problem with |x| < n. Proof. We first note that termination is clearly finite, since at each iteration either at least one string is removed, or some entire arc of T becomes colored brown. Since there are at most n strings initially, it follows that the X constructed satisfies |x| < n. Choose any v^,j Â£ J(r), and denote by x^ the center to which v. is assigned. Since the string fastened at v. reaches x,.., J (j / -10- ln such problems a common objective is to provide "good" service to each demand point by at least one facility within a least possible distance. In what follows, we first discuss the 1-center problem on general networks and on tree networks. Then, we discuss the vertex restricted 1-center problem. Finally, we will discuss the p-center problem in relation to a "covering" problem to be defined later. 1-Center problem on a general network. The absolute 1-center problem was defined and solved by Hakimi [47] in 1964. For finding the absolute center, Hakimi examines the function f on each edge, finds a best local minimum on that edge, and selects the best among |e| such local minima. This method takes advantage of one important property of f, namely, that it is piecewise linear and continuous on each edge with at most n(n l)/2 break points. A local minimum always occurs either at a break point of f or at an end point of the edge. Hakimi, Schmeichel, and Pierce [50] showed that Hakimi's method can be imple- mented in 0(|E|n logn) computational effort and gave a computational refinement which reduces the effort to 0(|E|nlogn) for the unweighted case. Further refinements of the procedure were obtained by Kariv and Hakimi [65], resulting in an 0(|E|nlogn) algorithm for the weighted case and 0(|E|n) algorithm for the unweighted case. All these refinements focus on finding the break points and the local minimum of.f in the most efficient manner. A somewhat more general version of the 1-center problem was con sidered by Frank [36], and (apparently) independently by Minieka [88], as Minieka makes no reference to Frank's paper. In this modified version, called here the continuous 1-center problem, each point on -86- addition, given a finite point set P C T, we define the convex hull H(P) to be the smallest (embedded) subtree of T containing all points in P. We note that for any two points p,p' e P, the line L(p,p') is contained in H(P). We denote by I the set of pairs (i,j) for which the distance u d(x^,Vj) is of concern. Similarly, 1^ is the set of pairs (j,k) for which the distance d(x.,x, ) is of concern. We remark that it need not 3 k be the case that 1^ includes all possible pairs of new and existing facility indices, nor I includes all possible pairs of new facility indices. With these definitions, the problem of interest is to "mini mize" each of the distances specified by (3.2.1); d(x >v ) (i,j) e I , 1 J O (3.2.1) d(x.,x ) (j ,k) e I . J K. d For X e Tm, we denote by D(X) the vector each of whose components is a distance specified by (3.2.1). The vector is formed by assuming any convenient ordering of the members of 1 and I. The vector- C B minimization (V-min) problem of interest is V-min{D(X): X e Tm} (3.2.2) With respect to (3.2.2), a location vector Z e Tm is said to dominate a location vector X in Tm if D(Z) < D(X) and D(Z) D(X). A location vector Z which is not dominated by any other location vector is said to be efficient. An equivalent definition of efficiency is as follows: Z e Tm is efficient if and only if X e Tm and D(X) < D(Z) imply D(X) = D(Z). -48- may only be a proper subset of the range. Hence, the inverse property holds only for some members of C and M for a general network. Now, we consider a more general multi-objective problem due to Lowe [82]. The problem involves a single facility to be located on a tree network with respect to m convex objective functions. Multi-objective convex location problem (on a tree). Let T be a tree network and let f,,...,f be m convex continuous bounded func- 1 m tions each of which is defined on T. In general, not all points in T may be feasible with respect to f_^. Let be a convex compact subset of T which contains all feasible points x with respect to the ith optimizer. The set Q, may be defined by specifying its extreme points, or by means of distance constraints, or by other means. We assume m that Q. is known or computable. Define Q = D Q. and assume that Q 1 i=l 1 is nonempty. The problem of interest is to find all efficient points in Q with respect to the vector minimization problem defined below: V-min[f(x): xeQ C T] where, f(x) = (f1(x),...,f (x)) for all xeT i m We note that Q is a convex compact subset of T as it is the intersection of m convex compact subsets of T. For a formal dis cussion of convexity on a network, the reader is referred to Dearing, Francis, and Lowe [22]. Loosely speaking, Q a convex subset of T, means Q is connected or that the (shortest, unique) path connecting any two points in Q is contained in Q. Lowe makes no assumptions on the specific forms of the objective functions. Under the convexity assumptions, Lowe proves that a convex -118- whlch contain (f ,f ), giving e' > 0. Choose any e such that p q 0 < e < min(e',e ). Reduce the length of arc (f ,f ) from e to Pq p q pq e e and define the resulting graph to be GBC^. Due to our choice of e clearly every separation condition on GBC^ holds, as, the length of every path which contains (f ,f ) is reduced by an amount smaller than the P q difference between the path length and the distance between the existing facility locations corresponding to the terminal nodes of the path, while the length of any path which does not contain (f ,f^) remains the same. Let DC^ be the distance constraints corresponding to GBC^.. Since the separation conditions on GBC^ hold, DC^ is consistent. Letting Y be any feasible solution to DC ', it follows that D(Y) < D(Z) and the entry of D(Y) corresponding to (f ,f ) is strictly smaller than the corresponding entry of D(Z). Hence, Y dominates Z, contra dicting that Z is efficient. Thus, (a) implies (b). To show (b) implies (a) suppose every arc in GBC of positive length is in a tight path. Let (f ,f^) be any arc with positive length e For e > 0, let GBC^CDC^) be the graph (distance constraints) ob tained from GBC(DC) by replacing e by e e. Since (f ,f ) is in pq pq p q a tight path, for any choice of e > 0, at least one separation con dition is violated. Since the violation of a separation condition implies the inconsistency of the distance constraints, there does not exist e, e > 0, for which DC^ is consistent. Clearly, then, there does not exist Y such that D(Y) < D(Z) and D(Y) ^ D (Z) which is the definition of efficiency. We remark that Theorem 3.7.1 holds for tree networks as well as If rectilinear distances on the plane, or, Tchebychev distances in R , k Â£ 2. The proof of the theorem relies on the necessity and sufficiency -li ter any r, r' < r < r and solves the dual dispersion problem. This P P approach is essentially a primal approach for solving both problems. An alternative approach which directly works with the dual graph is given by Chandrasekaran and Tamir [13] for the unweighted linear p- center problem, which works directly with what would be a subgraph of our dual graph G. Due to absence of weights and addends, their approach does not require the use of node weights (and for that matter the function g^) in the dual graph. For a given value of r, Chandra- sekaran and Tamir define an intersection graph IG with node set J and r arcs (i,j) for those indices i,j e J for which 8 < r. Their pro cedure is based on a graph theoretic procedure given by Gavril [39] and solves the covering problem by finding a minimum clique cover of IG^ (minimum number of cliques such that every node is in at least one clique). As a side result, their approach identifies a maximal anti clique in IG (a maximal set of nodes in IG no two of which are con- r r nected with an arc). Due to "chordal" properties of IG^ as discussed in [39], the cardinality of a minimum clique cover of IG^ is equal to the cardinality of a maximal anti-clique in IG^. This result is a special instance of the duality result we will present in Section 6 for the cover problem, as applied to the linear unweighted case. Furthermore, for r = r Chandrasekaran and Tamir [39] proved a duality P relationship for the unweighted p-center problem using the above properties of IG^. We remark that their duality results can be directly proven by using the algorithm OPTKLIQUE, and by appropriately specializing our S.D.T. for the linear unweighted case. We now demonstrate the use of OPTKLIQUE by determining K* for 4 the example problem. From our previous analysis, r^ = 1664.64. Since CHAPTER 1 INTRODUCTION AND LITERATURE SURVEY 1.1 Introduction and Overview Although some mathematical models of location can be traced back to the early seventeenth century, almost all the work on operational models for the location of facilities has taken place within the past 22 years, between 1957 and the present. An extensive annotated bibli ography on location-allocation problems is provided by Lea [78]. A more recent selective bibliography is given by Francis and Goldstein [30], Location problems commonly involve locating a number of new facilities (sources) in a given location space so as to provide goods or services to a specified set of existing facilities (demands) under one or more criteria, and, possibly, subject to a set of constraints. The quality of the service is typically measured in terms of the dis tances among the facilities. The use of distances is, perhaps, the major feature which distinguishes location problems as a special class of optimization problems. Hence, associated with any location problem is an underlying location space on which a "distance" is defined. Several variations of the general location problem are possible, depending upon the type of location space, the distance function, the number and areal extent of the facilities, the type of interactions CHAPTER 3 A VECTOR-MINIMIZATION PROBLEM ON A TREE NETWORK 3.1 Introduction We consider a vector-minimization problem on a tree network which involves as objectives the distances between specified pairs of new facilities and specified pairs of new and existing facilities. In many location problems, especially in the public sector, it may be necessary to build a number of public facilities which are to be shared by a number of communities. If the optimizers cannot agree on a single objective function, the analyst is faced with the problem of locating the facili ties in such a manner that all parties are satisfied with the end result. In such a case, the optimizers can agree to rule out "dominated" solutions and consider only "efficient" solutions. The related literature on multi-objective location problems is discussed in Chapter 1 under Multi-objective location problems on networks. Here, we concentrate on characterizing efficient solutions to the vector-minimization problem of interest. We relate efficient solutions to a distance constraints problem studied by Francis, Lowe, and Ratliff [32]. Extensions of results in [32] are given by Francis, Lowe, and Tansel [33]. We use the theory developed in [32] and [33] to establish the necessary and sufficient conditions for efficient location vectors (parenthetically, we remark that the results we proved in [33] are also given in our Dissertation Proposal defended on June 8, 1979). -84- -30- of non-integer termination, a branch-and-bound scheme is recommended to resolve the problem with integers. Their computational experience indicates that non-integer termination seldom occurs. Toregas, Swain, ReVelle, and Bergman [109] formulated a modified version of the problem as a mixed integer program. The modification is the presence of upper bounds on the distance between any vertex and its nearest facility. This formulation makes use of a related but simpler problem. This simpler problem is to minimize the number of facilities needed to cover all vertices of N within a specified critical distance. This problem is formulated as a set covering problem, and solved by ignoring the integer requirements. In case of non-integer termination, a single cut produced an integer solution in a large proportion of the cases. A somewhat different approach to solve the relaxed linear program is to use a decomposition scheme rather than applying the primal simplex algorithm. Swain [105] used a Dantzig-Wolfe decomposition approach to solve the associated linear program. Garfinkel, Neebe, and Rao [37] independently developed a decomposition approach similar to Swain's. In case of non-integer termination, they used group theoretics and a dynamic programming recursion to obtain an integer solution. A second approach taken is to solve the problem using a branch- and-bound technique. Khumawala [68] applied a branch-and-bound method of Land and Doig [77] type, to solve both the set covering problem and the modified p-median problem formulated by Toregas et al. He showed that the branch-and-bound approach is computationally efficient for the former but not for the latter. Narula, ,0gbu, and Samuelson [91] presented a branch-and-bound scheme which relies on obtaining the bounds by solving the Lagrangian relaxation of the integer programming -24- The p-median problem is the following: Given a positive integer p, find a set X* of p-points such that f(X*) = min[f(X): |x| = p, X C N] . Any set X* of p points that minimizes f is called an absolute p- median of N. If each member of X is restricted to a vertex location, the resulting problem is called a vertex restricted p-median problem. Due to a result by Hakimi [47, 48] there exists an absolute p-median entirely on the vertices of N. For this reason, the distinction be tween the vertex restricted and unrestricted versions is insignificant. Hence, we will take the term "p-median" to mean a solution to either version of the problem. A 1-median is simply called a median. The p-median problem arises naturally in locating plants/ware houses to serve other plants/warehouses or market areas. The problem is also motivated by ReVelle, Marks, and Liebman [96] as an example of a public sector location model where vertices represent population centers and facilities represent post offices, schools, public build ings, and the like. The 1-median problem. Hakimi [47] appears to be the first to define an absolute median. Hakimi proved the important result that there exists an absolute median at a vertex of the network. This result reduced the search to a finite number of points. The median can be found by summing each row of the weighted-distance matrix and choosing the vertex whose row sum is the minimum. This procedure takes 3 2 0(n ) operations to compute the distance matrix followed by 0(n ) operations to find the median. -80- The Divergence Problem is as follows: Given r and the function g, compute q(r) = max{ |u| : g(U) >r,UCV}. (2.6.1) That is, the problem is to find the maximum number of existing facili ties no two of which can be jointly covered by a single center within a radius of r. Equivalently, among all cliques of G whose gain is larger than r, the problem is to find one with the maximum number of nodes. The dual problem is feasible for r < r^, as, if r > r^ there does not exist a subset U of V for which g(U) > r. On the other hand, the primal cover problem is feasible for r > a. Hence, we shall re strict r to a < r < r^ in order to ensure feasibility to both problems. Theorem 2.6.1. (Weak Duality Theorem). Assume a < r < r^. For any feasible solution X to the primal cover problem, and any feasible solution U to the dual divergence problem, we have |x| > |u|. Proof. By feasibility of U and the assumption of the theorem we have g(U) = gl(U) > r > a > g2(U) from which it follows that 8.. > r v.,v. e U, i ^ j (2.6.2) ij i J Suppose |X| < Ju|. Then, the same approach as in the proof of Theorem 2.2.1 implies there exist v ,v e U, s ^ t, such that 8 < f(X) < r, st st contradicting at least one inequality in (2.6.2). Thus, |x| > |u|. Theorem 2.6.2. (Strong Duality Theorem). Assume a < r < r^. Let X be a feasible solution to the covering problem constructed by COVER. Then, the primary set U associated with X solves the dual divergence problem with X| = q(r) = q(r) = |U (2.6.3) -133- a) A Non-Efficient Location Vector with f(X) = (1.5,1) b) An Efficient Location Vector with f(Y) = (1.5,0.5) Figure 4.2. Example of Non-Efficient and Efficient Location Vectors -134- Using the weights given, the value of f(X) is (1.5,1). Graph GBC^ with z = (1.5,1) is also shown with arc lengths as indicated. It is direct to verify that GBC^ in Figure 4.2a) does not have any tight paths. Hence, X is dominated. The location vector Y in Figure 4.2b) is an efficient one and dominates X. The value of f(Y) is (1.5,0.5). The thickly drawn arcs in GBC^ of Figure 4.2b) form a tight path. We remark that in every feasible solution to DC^ for z = (1.5,0.5), the locations y^ and y^ ate the same, as and ^ are contained in a tight path. 4.4 Construction of the Efficient Frontier Let S be the set of all efficient location vectors in Tm. Define Z and Z* by Z = {(z^,Z2): 3X e T such that f(X) = (z^,Z2)} , Z* = {(z^,z^): 2X c S such that f(X) = (z^,z^)} That is, Z = f(Tm), the image of Tm under f, and Z* = f(S), the image of the efficient set S under f. We call Z the objective space and Z* the efficient frontier. Our main interest in this section is to develop a method to construct the efficient frontier. One can display Z* graphically on the (z^^) plane and obtain much of the insight about efficient points. In general, for any convex bi-objective problem, the efficient frontier and the objective space may look like the illustration given in Figure 4.3. The objective space is the shaded region and the efficient frontier is the thickly drawn part of the boundary of Z. -61- Y. . ij w .w. 1 3 + w. 3 1/0)0 and d.. 13 [d(v^,Vj) + h^ + hj]^ Consider the tree network shown in Figure 2.1, where the numbers on the arcs represent arc lengths. The data given with Figure 2.1 corresponds to the parameters for j=l,...,6 where clearly, each f is strictly increasing. Using (2.2.4), the 3 values for this problem are shown in Table 2.1 along with the node weights f^(0). Figure 2.2 shows the dual graph G associated with the problem, where the number next to each node j is the node weight and the number on the arc between nodes i and j is 3^ Using Figure 2.2 it can be verified that the optimal cliques (specified here by their nodes) and associated g values are K* = (3,4), g(K*) = 13829.76; K* = {1,3,6}, g(K*) = 3600; K* = {1,3,5,6}, g(K*) = 1664.64; K* = {1,3,4,5,6}, g(K*) = 784; and K* = {1,2,3,4,5,6}, g(K*) = 225. Due to the duality theory, it then o o follows that the r^ for p=l,...,5 are, respectively, 13829.76, 3600, 1664.64, 784, and 225. 2.3 Dual Problem Interpretation We imagine two conservative adversaries, an aggressor A and a defender D. Defender D has defense forces placed at vertex locations Vl,',,Vn' Aggressor A will attack a single vertex in V. Although D knows A will attack a vertex, he will not know the vertex attacked until the attack occurs. Defender D has p response forces which he must position at loca tions defined by a p-center X. Interpret tree distances to be travel times, so that D(X,v.) is the minimum time to respond to v. from a -138- to each distinct path, and equate e() to the pointwise maximum of these linear functions over the interval [a,b]. Such a method is not computationally efficient as there may be a very large number of paths For achieving computational efficiency we shall restrict our attention to a certain subset of the set of all possible paths and then evalute e(*) by taking the pointwise maximum of the linear functions cor responding to these paths. In general, an arbitrarily chosen path P(E^,E^) may pass through several existing facility nodes distinct from E^ and E^. First, we want to show that paths of this type need not be considered. Define any path to be a decomposable path (d-path) if the path passes through at least three (distinct) existing facility nodes. An example of a d-path which passes through four existing facility nodes is (E^, Nj-, Eg, N^, E^, N^, E^) Define any path to be a non- decomposable path (nd-path) if the only existing facility nodes the path passes through are its terminal nodes. Every d-path can be de composed into a (unique) collection of nd-paths which, when appended end to end, gives the original d-path. The decomposition of the aforementioned example d-path into its nd-paths is {(E,, Nc, N1ft, E,), 1 j 10 o (Eg, N^, E^), (E^, N^, E^)}. Clearly, any nd-path uses exactly two arcs in A^,, while any d-path uses at least four arcs of A^,. An nd- path may or may not use arcs of A^. Next, we have the following lemma, which permits us to check the separation conditions by only evaluating nd-paths. The lemma is applicable to any distance constraints problem defined in Chapter 3 by (3.3.1). We use the notation of Chapter 3 for the lemma. -19- its edge of the function f..(-) = max[w.d(v.,.), w.d(v.,.)], and ij i i J J that the two linear pieces defining that breakpoint have slopes of opposite signs. There can be at most n(n l)/2 suspected points on each edge, resulting in a total of 0(|E|n^) suspected points on all edges. If S is the set of all suspected points together with the set of all vertices, then there is an absolute p-center contained in S. The Kariv-Hakimi procedure selects p-1 points from S and determines all the vertices covered jointly by these p-1 points. All uncovered vertices are assigned to the pth center. Corresponding to each center, the I-radius is determined (with respect to the subset of vertices covered by that point) and the maximum of these 1-radii determines the p-radius for that trial solution. The algorithm tries every possible combination of p-1 points selected from S and chooses that combination which minimizes the p-radius. The Kariv-Hakimi procedure is the only exact algorithm available so far for finding an absolute p-center of a vertex weighted general network. A further result on the computational difficulty of the p-center problem on a general network is given by Nemhauser and Sheu [92]. They showed that finding an approximate solution to the vertex restricted or absolute p-center problem whose value is within 100% or 50%, respec tively, of the optimal value is NP-hard (i.e., at least as hard as any NP-complete problem). Vertex restricted p-center problem. The vertex restricted p- center problem is considered by Toregas, Swain, ReVelle, and Bergman [109]. A solution procedure is given which relies on solving a sequence of minimal set covering problems, each corresponding to a specified radius r. Given a radius r, a 0-1 matrix A can be formed with n rows -110- b) Iteration 1 Figure 3.7. Example Application of SEVCA 1 n U 1 ji J I /..i t Ion (I M M > I (* V ? Choose v 1 Choose v 18 9 x. U4 = {v3WV5} K) u = {v3, v6, v5! k* o, i, <>, r> OPTKLIQUE for p = 3 for Example -54- The literature on the p-center problem is discussed in detail in Chapter 1. Here, we give a brief review of the more closely re lated work. Except for p = 1, we know of no literature on the non linear p-center problem. For p = 1, the only references we are aware of which deal with the nonlinear case are Dearing [18] and Francis [29]. Both authors showed that the minimax loss with respect to any two existing facilities is a lower bound on the maximum loss with respect to all existing facilities, and that the largest of the lower bounds determines the minimax loss to all existing facilities on a tree network. This result is an instance of the duality result we will present in this chapter. The linear (weighted or unweighted) p-center problem is shown to be NP-complete on a general network by Kariv and Hakimi [65], and by Nemhauser and Sheu [92]. The linear 1-center problem on a tree network is well solved (see Goldman [44], Halfin [51], Lin [81], and Dearing and Francis [19]). For p > 1, the linear p-center problem on tree networks is considered by various authors. Handler [57] provided an 0(n) algorithm for finding the 2-center of a tree for the unweighted case. Kariv and 2 Hakimi [65] gave an 0(n logn) algorithm for tree networks which relies on solving a sequence of covering problems for the weighted case with p > 1. A similar procedure for the unweighted continuous p-center problem on a tree network is given by Chandrasekaran and Daughety [12]. A vertex-restricted version of the problem is solved by Chandrasekaran and Tamir [13], and relies on solving a sequence of clique covering problems on a related intersection graph. -113- g) Iteration 6 and Termination (in Iteration 7) Figure 3.7. Continued -144- b = minf^(X): ^2^^ b) = minify (X) : v^dCx^.x^) < 0 (j ,k) e Ig} = minif^(x,..,x): xeT) = min max g (x) xeT l
= min max max{w..d(x,v.): j e I. }xeT l
= minxeT max{w_.d (x,Vj) 1 < j < n} where w. = max(w,,: over all i for which i e I.} J ij i (4.4.10) (4.4.11) Thus, the value of b is obtained by solving the absolute 1-center problem defined by (4.4.10) and (4.4.11), and will require O(nlogn) opera 2 2 tions. -Therefore, the computational effort for E-FRONT is 0(m (m + n )) and is determined by the computational effort for steps 1), 2), and 3) of E-FRONT. Once the efficient frontier is constructed, efficient location vectors can be identified as follows: Choose z = (z^,zÂ£) in Z* with z2 = x(z^) = TÂ£j(z^)> say, and identify the arcs a^ and a^. Supposing a. = (N ,E ) and a. = (N. ,E ), let P(N ,N 1 be a path in whose length i s p j t q s t r B & is mgt_. Then, clearly, (Ep,P(Ng,Nt) ,E^) will be a tight path in GBCz with length L P(E ,E ) = z,WP(E ,E ) + z0m Every facility whose zpq l pq zst node is in the path P(E^,E^) is uniquely located on the line L(v^.v^) in T with the same ordering and spacing as the nodes which lie in P(Ep,E^). Hence, the new facility locations corresponding to new facility nodes in P(E^,E^) can be readily identified (see Property 3.3.1 of Chapter 3). The locations of other new facilities can be -51- where a path of G joining any two existing facility nodes E and E s t is said to be tight if the length of the path is equal to the distance between the vertices v and v in T corresponding to nodes E and E , S t 1 w St respectively. For any given location vector Z, denote by A^(Z) the collection of locations of uniquely located facilities whose nodes are adjacent to N_^ in G. Let H[A^(Z)] be the convex hull of A^(Z), i.e., the smallest connected subtree containing all points in A^(Z). With these definitions, it was proven in [33] that the following conditions are equivalent: (i)Z is efficient. (ii)Z is the unique solution to DC. (iii)Each is in at least one tight path in G. (iv)Each Z. is contained in H[A.(Z)], 1 < i Â£ m. 1 1 This completes the discussion of multi-objective location problems on networks. Path Location Problems Here, we consider three versions of a path location problem posed by Slater [102]. To define the problems, let P denote any path con necting any two vertices in a network N. For any vertex veV and any path P, define the distance D(v,P) to be the distance from v to a nearest vertex in P. Also define the branch weight bw(P) of a path P to be the maximum number of vertices in any component of N-P. The three versions of the problem are the following: min l D(v,P) (1.3.7) P C N veV REFERENCES 1. K. Bergstresser, A. Chames, and P.L. Yu, "Generalization of Domination Structures and Nondominated Structures in Multicriteria Decision Making," Research Report No. JS185, Center for Cyber netic Studies, Univ. of Texas, Austin, Texas (1974). 2. 0. Berman and R. Larson, "The Congested Median Problem," Working Paper OR-076-78, MIT, OR Center (1978). 3. A.E. Bindschedler and J.M. Moore, "Optimal Locations of New Machines in Existing Plant Layouts," J_. Ind. Engr. 12, 41-48 (1961). 4. G.R. Bitran and T.L. Magnanti, "The Structure of Admissable Points with Respect to Cone Dominance," C.O.R.E. DP 7716, Louvain-la- Neuve (1977). 5. R.G. Busacker and T.L. Saaty, Finite Graphs and Networks, McGraw- Hill, New York, N.Y., 1965. 6. A.V. Cabot, R.L. Francis, and M.A. Stary, "A Network Flow Solu tion to a Rectilinear Distance Facility Location Problem," AIIE Transactions 2, 132-141 (1970). 7. L.G. Chalmet, Efficiency in Multi-Objective Location, Design, and Layout Problems, Ph.D. Dissertation, Katholieke Universiteit te Leuven (1978). 8. L.G. Chalmet and R.L. Francis, "Finding Efficient Solutions for Rectilinear Distance Location Problems Efficiently," Research Report No. 77-3, Dept, of Industrial and Systems Engineering, Univ. of Florida, Gainesville, Florida (1977). 9. L.G. Chalmet, R.L. Francis, and J.F. Lawrence, "On Characterizing Supremum-Efficient Facility Designs," Research Report No. 78-9, Dept, of Industrial and Systems Engineering, Univ. of Florida, Gainesville, Florida (1978). 10. L.G. Chalmet, R.L. Francis, and J.F. Lawrence, "Efficiency in Integral Facility Design Problems," Research Report No. 78-11, Dept, of Industrial and Systems Engineering, Univ. of Florida, Gainesville, Florida (1978). 11. A. Chan and R.L. Francis, "A Round-Trip Location Problem on a Tree Graph," Trans. Sci. 10, 35-51 (1976). -161- CHAPTER 4 THE BI-OBJECTIVE m-CENTER PROBLEM ON A TREE NETWORK 4.1 Introduction In this chapter we consider a bi-objective problem on a tree network which involves as objectives the maximum of the weighted dis tances between specified pairs of new and existing facilities and' the maximum of the weighted distances between specified pairs of new facilities. Such a vector-minimization problem may find applications in locating emergency service units for the case when service units are required to support one another in addition to providing service to potential hazard zones (existing facilities). The related literature on multi-objective location problems is discussed in Chapter 1. Here, we concentrate on characterizing efficient location vectors to the bi-objective minimax problem and constructing the "efficient frontier," the set of objective values corresponding to efficient points. At this point we give an overview of the chapter. In Section 2 we give the necessary definitions and notation, and define the bi objective problem of interest. In Section 3 we relate the bi-objec tive problem to the distance constraints studied by Francis et al. [32] and develop the necessary and sufficient conditions for efficiency by making use of the results given for the distance constraints in -26- A median of a tree is shown to be the same as a "centroid" of the tree by Zelinka [120] for the unweighted case and by Kariv and Hakimi [65] for the weighted case. To define a centroid, consider the subtrees T-,...,T. obtained by removing vertex v from T. Let 1 X w(T ) be the sum of the weights of the vertices in T^., and define W(v .) to be the maximum of w(T.) for 1 i i k., A vertex v which i j J i t minimizes W(v.) over all v. in V is said to be a centroid of T. The i i location of a centroid is independent of the distances and can be found by using only the incidence relations. Goldman's earlier algorithm in essence finds a centroid of T. The generalized algorithm of Rosenthal, Pino, and Coulter [98] also finds a centroid of T by making only two traversals of the vertices. All these algorithms are of 0(n) and solve the 1-median problem without having to compute the distance matrix. We now consider some generalizations of the 1-median. Minieka [88] defined the general absolute median of a network to be any point on the network that minimizes the sum of (unweighted) distances from it to the point on each edge that is most distant from it. Minieka showed that the general absolute median can be strictly interior to an edge; hence, the search cannot be confined solely to vertices of N. Slater [103] gave another generalization of the 1-median problem. In this generalization, each demand is a collection of vertices. The problem is to find a vertex such that the sum of the distances from that vertex to a nearest element of each collection is minimum. Slater showed that the set of vertices that solve this problem forms a connected path in T. For a general network, the problem can be solved by constructing a matrix that specifies the distances from each vertex -44- hull of the fixed points. Wendell, Hurter, and Lowe [114] considered the same problem with rectilinear distances and provided algorithms of 2 3 0(n ) and 0(n ) for generating efficient points. A most efficient algorithm of O(nlogn) was developed by Chalmet and Francis [8] for the same problem. McGinnis and White [83] considered the problem of minimizing the sum of and the maximum of weighted rectilinear distances from a variable point to a set of fixed points on the plane and formu lated the problem as a parametric linear program for which known solu tion techniques exist. Juel [64 ] considered the same problem for the case of multiple new facilities and gave an equivalent parametric linear program. Chalmet, Francis, and Lawrence [ 9 10 ] considered two variants of an efficient design problem, where the location variable (a design) is a planar region of specified positive area but of unknown shape. A few papers have been produced on multi-objective location problems on networks. In what follows we discuss these problems. The cent-dian problem. The single facility "cent-dian" problem involves the sum of and maximum of weighted distances from a new facility to a set of existing facilities at vertices of N. To define the problem, let w^ and h_^ be two positive weights associated with vertex v iel = {1,...,n}. For each point xeN define: m(x) = J {w_jd(v^,x): iel} , c(x) = max[h^d(v^jx): iel] f(x) = (m(x), c(x)) -98- z2=z3 # v 1.5 1.5 a) Graph GBC b) Tree T A^(Z) = (z2,vi,v2,v4) Zj e H[ A^(Z)] = T A2(Z) = {z^ZyV^} z2 e H[A2(Z)] = L(zl5z3) A3(Z) = {z2>v^} z3 e H[A3(Z)] = L(z2,v^) c) Sets A^. (Z) d) Convex Hulls Figure 3.5. Example of a Non-Efficient Location Vector -132- y<~ 7- 2 y +- 9 vr y 4 a) Tree T v2 v3 V4 V5 W11 = 1/5 V12 1/3 V1 6 4 8 10 W12 = 1 V13 = 1/6 V2 2 6 8 W23= = 1/3 V23 = 1/2 V3 4 6 W24 = 1/2 V4 2 W34 = 1/4 W35 = 1/3 b) Distance Matrix c) Weights Figure 4.1. Data for Example Bi-Objective m-Center Problem -89- If then the string graph is placed upon the tree T, i.e., the strings only lie on arcs of T, a path is tight when it is necessary to pull the string graph tight in order to place the knots representing and on v and v respectively, while a path is slack if the string path P 9 must literally be slack when the two knots are placed to coincide with v and v . P q A priori, one might think that the occurrence of a tight path would be rare. However, we shall see that tight paths occur in a quite natural way when the separation conditions are used in the analy sis of efficient location vectors. Further, the notion of tight paths permits the specification of necessary and sufficient conditions for DC to have a unique solution. We now relate unique locations to tight paths. By definition, new facility i is uniquely located if it has the same location in every feasible solution to DC. Since we later refer to a collection of facilities, which contains possibly both existing and new facilities, being uniquely located, we note that existing facilities are uniquely located by definition. Theorem 3.3.2, which we proved in [33], specifies the necessary and sufficient conditions for a new facility to be uniquely located. Theorem 3.3.2. New facility k is uniquely located if an only if node lies in at least one tight path P(E^,E^). Corollary 3.3.2. Distance constraints have a unique solution if and only if node lies on at least one tight path in GBC for k = l,...,m. We now give an additional property of a tight path we proved in [33]. The property will be used in proving our main result on efficient points. -115- Proof. We must show that t, the total number of Iterations, is at most 3m-l. Define P =\J{K^^: 1 < k < t}. With this definition P contains as members distinct (but not necessarily disjoint) subsets of { 1,. .. ,m). Since = {{ 1{m} } every set {j}, 1 < j m, is an element of P. If step 3) is never executed, then P = as A'^ = ... = K^ in this case. Otherwise, whenever step 3) is executed, (k) say, in some iteration k, some two distinct members of K are removed from A^), and their union is inserted in to obtain Hence, if iteration k performs step 3), we know that the cardinality of (k+1) (k) a y is one less than that of Av and that exactly one member of (k+1) (k) K (the one inserted) differs from every element of K Clearly then, step 3) can be executed at most m 1 times, and therefore P contains at most m+(m-l)=2m-l distinct members. Hence t^, the total number of iterations which used step 3), satisfies t^ < m- 1. Now, imagine that we apply SEVCA a second time in exactly the same order as the first application. In the second application, each time step 6) or step 7) is used as the last step of an iteration, one member of P will be labeled scanned (P is available as a result of the first application). Since P contains at most 2m 1 distinct members, clearly, tg + t^, the total number of iterations which used either one of step 6) or step 7), will satisfy tg + t^ < 2m 1. It follows then that t = t_ + t. + t, + t_ < 3m 1 as t_ Â£ m 1, t, =1, 3 4 6 7 3 4 and tg + t^ < 2m 1, completing the proof. Next we have the following property. Let X* be the location vector at the termination of SEVCA. Property 3.6.2. The algorithm SEVCA terminates with an irreducible location vector X* which is efficient. Furthermore the location vector -75- S(Kp*+l> = rp (2.5.2) Furthermore, K* solves the dual dispersion problem. p+1 Proof. Let X* be an optimum p-center solution to the primal problem so that X* = p and f(X*) = r Since r S a we consider the cases P P r^ = a and r^ > a. Let us apply OPTKLIQUE for each case. For r^ = a, K*+^ is chosen in step 1) so that |K*_y| = p+1 and a = fg(0) = g2^Kp+l^ T^e W*D,Tt Sives S(K*+1) f(X*). But then, a = 82^K*+1^ = = f(X*) = r = a, establishing (2.5.2) for p+1"1 this case. c' P For r > a, define R = {3. e R: r < g..} C- R. Since r > r > P iJ P ij P there exists no g. in R for which r < g ,. < r Thus g.. > r implies ij iJ P 1J 3. > r and so it follows that 1J P R={g..:r (2.5.3) Let U be the primary set identified by COVER for the chosen r, r' < r < r By Lemma 2.5.1, U satisfies g,(U) > r from which it p p J bl follows that gy > r for v^*vj e U, i j. Hence, (2.5.3) implies By e R v.,v. e U, i j (2.5.A) J 3 Since |U| > p+1, let be that subset of U identified in step 2). We have the following string of inequalities: r P f(X*) > g(K*+1) 2 Mkh> = mlniByi Vj.Tj E K*+1, i j* j) > minig..: v.,v. e U, 1 i} ij i J > minig.. e R} > r (2.5.5) (2.5.6) (2.5.7) (2.5.8) (2.5.9) (2.5.10) -99- for the other class ("reducible" location vectors). We say a pair of facilities interact if their nodes are adjacent in GBC. We define a location vector Z = (z^,...,z^) to be irreducible if for every pair of interacting new facilities i and j, their locations z^ and zare dis tinct; Z is said to be reducible if there exists at least one pair of interacting new facilities i and j for which z^ = zj ^he lcatin vector of Figure 3.5 is an example of a reducible location vector. The following property gives the sufficient conditions for an ir reducible location vector to be efficient. Property 3.5.2. Suppose Z e T is an irreducible location vector. If for every j, 1 < j < m, z^ e H[A^(Z)], then Z is efficient. The proof of Property 3.5.2 requires a number of preliminary results. To preserve the continuity of the discussion, we leave the proof until the end of section 5. From a computational standpoint, Property 3.5.2 provides an ap proach for determining whether or not an irreducible location vector is efficient, and constructing one if it is not. To check if Z is efficient, we only need to determine the nodes adjacent to N in GBC and form the convex hull (the smallest subtree) which spans the loca tions of these adjacent nodes. If it is the case that every z. is within its convex hull, then Z is efficient. Otherwise, we can choose a z. which is not in the convex hull associated with it, and move its location to the closest point in the convex hull. The procedure can be employed repeatedly until every new facility satisfies the convex hull containment property. However, during such a procedure, the current location vector may change its status from an irreducible one to a reducible one, as the locations of new facilities change. For -128- Proof. Since DC is consistent clearly z. 0, i = 1,2. We consider z 1 the cases z^ > 0 and z^ = 0 separately. Case with ^2 > 0. Let P(E^,E^) be any path which passes through Ag. By hypothesis the path is slack so that L^PE^E^) d(vp>Vq) > 0. Further VP(E^,E^) is positive since the path passes through A^. Hence we have [L P(E ,E ) d(v v )]/VP(E ,E ) > 0 (4.3.1) z p q p q p q Let e be the minimum of the left side of (4.3.1) over all paths which min(e,Z2). Let GBCz_^ be the graph with arc lengths (ij) e and (Z2 ^2^vjk* (j,k) el. We want to show that the separation conditions defined on D GBC are satisfied. z-A Choose any two nodes E^ and E^. Let P(E^,E^) be a shortest path in GBC connecting E and E Hence, we have z-A p q L (E ,E) = L ,P(E ,E) (4.3.2) ZA p C[ ZA p CJ Either P(E ,E ) passes through A or it does not. In the latter case p q U clearly the length of P(E ,E ) in GBC and GBC is the same, as every p q z z-A arc in A^ has the same length in both graphs. Since DC^ is consistent, we have L ^P(E ,E) =LP(E ,E) > L (E ,E) >d(v ,v) so that the z-A pq zpq zpq pq separation condition for E and E is satisfied in this case. For the P q other case, P(E ,E ) passes through A^ so that its length on GBC is given by pass through A_. Choose A = (A^^2) with A^ = 0 and 0 < A2 L ,P(E ,E ) Z-A p C| = z WP(E ,E ) + (z A)VP(E ,E ) 1 P q 2 2 p q (4.3.3) -143- calculate m for 1 < s < t < m, which requires 0(m ) operations. Every linear function is determined by computing its slope and inter- 3 2 cept so that steps 0), 1), and 2) require 0(m + r ) operations. But r can be at most mn so that excluding step 3) and the computation of a 3 2 2 2 and b, the algorithm is 0(m + (mn) ) =0(m (m+n )). Each linear function has positive intercepts and negative slope. Clearly, their pointwise maximum is a piecewise linear decreasing function over the interval [a,b]. Hence, t(*) can be constructed by finding its break points. Each break point is determined by the intersection of some two linear functions. Since each linear function is strictly decreasing, any linear function can determine at most two (consecutive) break points. Thus, there are at most 2*(r)(r l)/2 = r(r 1) break points. Hence, excluding the computation of a and b, the algorithm requires 2 2 0(m (m+n )) operations, as the computational effort for constructing the linear functions dominates the computational effort for finding the break points of t(0- To compute a, define, for every new facility index i, the set 1^ by ^ = ij : (i,j) e ICL Letting g^x^ = max{w^d (x^v..) : j c 1^, it is direct to verify that f^(X) = max{g^(x_^): 1 i < m). Hence, fj is separable and its minimum value is given by a = max{g*: 1 < i < m) where g* = min{g^(x): x e T}. The Kariv-Hakimi procedure in [65] com putes g* in 0(iI|log|I|). Since |l | < n, the computation of a requires no more than mnlogn operations. Hence, the computational effort for identifying the linear functions again dominates the com putational effort for computing a. To compute b, we must first compute b. Clearly, b = 0 as it is the minimum value of f 2 (X) = maxi'v fcd (x^ ,xfc) : (j,k) e IB>. It is direct to verify the following equalities: -126- It was proven in [32] and stated in Theorem 3.3.1 of Chapter 3 that DC is consistent if and only if L (E E ) > d(v ,v ) for z z p q p q 1 Â£ p < q n. The inequalities L^CE^E^) dCv^jV^) are called the separation conditions and the separation conditions are said to hold if every separation condition is satisfied. It is direct to verify that whenever the separation conditions hold (equivalently, whenever DCz is consistent) it necessarily follows that L P(E ,E ) > d(v ,v ) for any path P(E ,E ). Conversely, whenever zpq p q p q L P(E ,E ) > d(v ,v ) for all paths P(E ,E ), it necessarily follows zpq p q p q that L (E ,E ) > d(v ,v ). z p q p q The definitions for tight and slack paths are given in Chapter 3 and will not be repeated here. 4.3 Necessary and Sufficient Conditions for Efficiency In this section we develop the necessary and sufficient conditions for efficiency by making use of the distance constraints. Our main theorem states that a location vector Y is efficient if and only if at least one arc in is contained in a tight path in GBC^, where GBC^ is the graph corresponding to DC^ obtained by letting z = f(Y). Notationally, for any X = (X.,X0), DC is the distance con- X Z ZA straints with right hand sides (z1 X )/w.., (i,j) c I and i I xj C (z^ ^2^vjk e Ig* The graph GBCz ^ is the graph associated with DC z-X' Before proving our main theorem, we first prove two lemmas relating DC to DC and GBC to GBC ... We remark that "0" denotes Z ZA Z ZA either the scalar zero or the two-tuple (0,0). It will be clear from the context what "0" refers to. OPTIMAL MULTI-FACILITY LOCATION ON TREE NETWORKS By BARBAROS C. TANSEL A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1979 -169- 117. P.L. Yu, "Cone Convexity, Cone Extreme Points, and Nondominated Solutions in Decision Problems with Multi-Objectives," J. of Opt. Theory and App. 14, 319-377 (1974). 118. P.L. Yu and M. Zeleny, "The Set of All Nondominated Solutions in Linear Cases and a Multicriteria Simplex Method," J. Math. Anal, and AÂ£Â£. 49, 430-468 (1975). 119. P.L. Yu and M. Zeleny, "Linear Multiparametric Programming by Multicriteria Simplex Method," Manag. Sci. 23, 159-170 (1977). 120. B. Zelinka, "Medians and Peripherians of Trees," Archivum Mathematicum (Brno), 87-95 (1968). -12- vertex v from v The path P(v .v^) is a longest path and its mid- t s s t point is the unique absolute center of the tree. This procedure requires a computational effort of 0(n). Handler's algorithm is extended by Lin [81] to the unweighted case with addends. Lin showed that the absolute center of a general network N with vertex addends can be found by determining the absolute center of an expanded net work N' whose vertex addends are all zero. Network N' is obtained from N by adding a new vertex adjacent to each old vertex, with the length of the edge connecting the two equal to the addend associated with the old vertex. For a tree network T, the resulting network is a tree T' and Goldman's 0(n) algorithm can be applied to T'. The more general case with both weights and addends was considered by Dearing and Francis [19], and for the case of a tree network an 2 0(n ) algorithm was given. The Dearing-Francis paper appears to be the first to construct a well defined metric space N with distance d(.,.) from an arc weighted graph N. This mathematical formality per mits the use of such concepts as compactness, continuity, and the extreme and intermediate value theorems. They showed that the distance d(x,.) is continuous for each fixed x, in turn implying that f(x) is continuous for every x. From compactness and continuity considera tions, they proved the existence of an absolute center for all compact networks, and its uniqueness for all compact tree networks. They obtained a lower bound on r^ which is applicable to all networks, and proved that it is always attainable for tree networks. Once the lower bound is determined, it identifies two "critical" vertices, and the absolute center can be readily located on the path joining the two. The bound is the maximum of n(n l)/2 terms, resulting in a -93- length; for otherwise, in every feasible solution to DC, the location of new facility p would be the same as the location of the uniquely located facility represented by node F contradicting the fact that N is colored blue.. Property 3.3.2 then implies that the entry in P U = D(Z) corresponding to arc (N ,F ) can be reduced by a positive P 9 amount and the resultant distance constraints will still have a feasible solution, say Y. But then clearly D(Y) < D(Z) and D(Y) 4- D(Z), contra dicting the fact that Z is efficient. Hence (a), (b), and (c) are equivalent. It can be seen that the proof will be complete if we show (b) implies (d) and (d) implies (c). To show (b) implies (d), suppose N_^ is in some tight path P. Let f^ and f^ be the nodes adjacent to N_^ in P, so that ((f^,N^), (N^jf^)) is a subpath of P. Since f^ and f2 are in the tight path P, by Theorem 3.3.2 the facilities represented by f^ and f are uniquely located. We may let y^ and y2 denote the unique locations of f^ and f^, respectively. Thus it is clear that y^ and y^ are elements of A*(Z). By Property 3.3.1, z^ e L(y^yy^), and by definition of the convex hull, L(y^,y^) C H[A*(Z)]. Thus it follows that z^ e H[A*(Z)J. To show (d) implies (c), suppose e H[A*(Z)] and let f^ and f^ be nodes adjacent to N_^ in GBC, where f^ and f^ represent facilities with unique locations y^ and y respectively, such that z. e L(y ,y ) C T. Thus d(y ,y ) = ^ 1 i. m 1 Z d(y ,z ) + d^z^y^). Now for any feasible solution X to DC we know dCy^Xi) d(y^,z^) and d(y2>x^) < d(y^,z^). But then because f^ and f2 are uniquely located, Lemma 3.3.1 implies x. = z^, for i = l,...,m. Hence X = Z, so Z is the unique solution to DC, completing the proof. -4 problem and a dual "divergence" problem. We provide a covering algorithm which solves both the covering problem and its dual simul taneously. In Chapter 3, we study a vector-minimization problem in relation to a distance constraints problem. The problem involves as objectives the distances between specified pairs of new and existing facilities and specified pairs of new facilities. We extend the results of [32] to develop a theory for identifying unique solutions to distance con straints, and use this theory to develop necessary and sufficient conditions for efficient solutions to the vector-minimization problem of interest. Further, we provide an algorithm which constructs an efficient location vector from a given non-efficient solution. In Chapter 4, we study a bi-objective location problem which in volves as objectives the maximum of the weighted distances between specified pairs of new and existing facilities, and maximum of the weighted distances between specified pairs of new facilities. We characterize efficient solutions and provide an algorithm for construct ing the efficient frontier. In Chapter 5, we pose a number of unresolved questions in relation to the problems discussed and point out directions for future research. 1.2 Terminology Before discussing the literature we specify our terminology. An undirected network N = {V,E} is a collection of two sets V and E, called the set of vertices and the set of edges of N, respec tively. Each edge in E is described by an unordered pair of vertices. -21- The dispersion problem is to find a U*C T such that h(U*) =max{h(U): U C T, |u| = p+1} . At optimality, Shier's duality result states that rp-{h(U*) for a tree network. The equality may not hold for general networks. However, Shier showed that the objective value of the continuous p- center problem is always bounded below by one-half the objective value of the dispersion problem for any network. Chandrasekaran and Tamir [14] observed that Shier's duality result holds when one replaces T by any subset S of T. Chandrasekaran and Daughety [12] described a procedure for solving the dispersion problem. They first solve the related problem of locating the maximum number of points on T such that any two of them are at least A distance apart for a fixed (positive) A. This problem is solved by working from "tips" of T to the "center" of T. The general scheme is to use the algorithm for different values of A, until the number of points found is p+1 and a slightly larger A generates p or less points. A number of solution procedures have been given for the p-center problem on tree networks. We now discuss these procedures. Handler [57] considered the continuous p-center problem on a tree network for the special case of p = 2 and obtained an 0(n) algorithm. Handler first finds the absolute 1-center of T, say x*, and splits the tree at x* obtaining two disjoint subtrees T^ and T^. Finding the absolute 1-center of each T say x* and x*, determines an absolute 2-center of T. -46- P for each given a, where P is defined as follows: a b a e(a) = min[c(x): m(x) Â£ a, xeT] Efficient solutions are obtained by parameterizing on a. Handler's results closely parallel Halpem's. The problem on a general network is studied by Halpern [54]. using the convex combination approach. Halpern showed that the problem is a computationally finite one. Computational finiteness follows from the result that f(X,x) is a continuous, piecewise linear function of x on each edge and attains its minimum at one of a finite number of points. Defining Q(e) to be the union of the end points of edge e with the set of local minima of c(x) on e, the minimum of f(X,x) over all x on edge e is a member of Q(e) for any given X, 0 < X < 1. De fining Q = U {Q(e) : eeE}, it follows that the cent-dian x*(X) is con tained in Q for any X. Further, Halpern showed that the function f*(X) = min[f(X,x): xeN] is a continuous, piecewise linear, concave function of X for 0 < X < 1. Based on these results, Halpern provided an algorithm which constructs f*(X) and identifies x*(X) for 0 X < 1. To construct f*(X), the algorithm inspects each edge one at a time and computes the set Q(e), unless a simple test indicates that edge e cannot contain any cent-dian for any X. An upper bound on f*(X) is carried through and improved, whenever possible, by examining the members of Q(e). Cent-dian problem and duality. In [53], Halpern studied the cent- dian problem on a general network from a different angle and obtained a duality relationship. Using an approach similar to Handler's median constrained problem, Halpern defined two problems, a median constrained -102- c) Iteration 1 b) Graph GBC d) Iteration 2 and Termination Figure 3.6. Example Application of RP -32- The distance constraints problem The distance constraints problem involves locating new facilities on a network so that they are within specified distances of existing facilities as well as within specified distances of one another. The distance constraints arise naturally in a locational context if one wishes to require that a service facility be within a specified time (distance) of any point in the region it serves. Alternatively, in a military context, one may want to locate a number of units in such a way that units are neither too far from their supply bases, nor too far from one another, in order that one unit may reinforce another if necessary. To state the problem, let N be a network with the vertex set V = iv.,...,v ). Denote by X = (x,,...x ) any location vector in Nm, the m-fold Cartesian product of N by itself. Define the sets and I as follows: I = (j,k): 1 < j < k < m>, I = {(i,j): 1 i S m, 1 j n}. Here, the pairs (j,k) and (i,j) are assumed to be un ordered. Let I and I be two non-empty subsets of I' and I, d L B G respectively, and suppose we are given nonnegative finite numbers b jk for each (j,k)el and c.. for each (i,j)el_. a ij C The problem of interest is to find a location vector XeNm, if it exists, such that the constraints (1.3.3) are satisfied. d(x ,v ) < c (i,j)el 1 J 1J ^ (1.3.3) d(Xj,xk)-bjk (j,k)cIB Any vector XeNm satisfying (1.3.3) is called a feasible location vector. The distance constraints are said to be consistent if there exists at least one feasible location vector XeNm. -68- assume J(r) ^ 0, for if J(r) = 0 then the condition f(X) < r holds for all X C T and we (trivially) have q(r) = 1. The above assumptions permit the following equivalent statement of the covering problem: minimize |x| subject to D(X,v.) < f.^Cr), j e J(r) (2.4.2) 2 2 We refer to the covering algorithm as COVER. In order to state COVER a few definitions are convenient. We may imagine that the tree is represented appropriately by inscribing straight line segments on a planar surface such that each segment represents an arc. We fasten strings of length f ^(r) to each node vjj e J(r), of the inscribed tree, where, by convention, we allow strings of zero length. Every fastened string has one end permanently affixed to the planar surface. In addition, during the use of the algorithm we engage previously fastened strings at various points on the tree. When a string is engaged, some point of the string is permanently affixed to the tree such that there is no slack in the portion of the string so far en gaged. When strings are removed, we imagine that they are physically deleted from the string model. During each iteration of the procedure, we partition the original tree into two subsets: one green, the other brown. The green subset is always a tree, denoted as GT (for green tree), while the brown sub set consists of one or more subtrees of the original tree T, each of which is "rooted" at a node of the green tree. By convention, a root -78- r > a = 144, we compute (from Table 2.1) rl=max{8.. e R: 3.. < r} = 900. 3 3 ij ij 3 We next must apply COVER using a value of r where 900 < r < 1664.64. Figure 2.3 shows the results of using COVER with r = 1296. In the figure, the loose ends of the strings are shown as wavy lines. Brown subtrees are shown as crosshatched arcs of the original tree. Each separate drawing of the tree (a)-g)) is for a subsequent iteration of COVER. Figure 2.3a) demonstrates the initialization step, where for r = 1296, the f.^(r), j = 1,...,6 are 12, 7.2, 7, 6, 18, and 8, re spectively. The numbers next to the strings are the lengths of the loose ends. In the figure, we indicate which tip of the green tree is chosen at each return to step 1) of COVER. In addition, the suc cessive distinguished vertex sets are indicated. After the final iteration, we note that the primary vertex set U is {v,v, ,v- ,v,-} which, from our previous analysis, we know to be J 1 D 5 2.6 Results for the Covering Problem In this section we present a "divergence" problem which is dual to the covering problem. We give a weak duality and a strong duality result and prove that the primary set identified by COVER solves the dual problem. The term "divergence" is chosen to represent the physical interpretation, discussed later, in which the attacker A chooses a "divergent" set of vertices to threaten. Further, the term permits a distinction to be made between the two different dual prob lems. Also, in this section, we demonstrate how having optimum solu tions to the p-center problem for all p, 1 < p < n, enables us to completely characterize the function q(r). -22- 2 An algorithm of complexity 0(n logn) is described by Kariv and Hakimi [65] for finding the absolute p-center-of a vertex weighted tree network. They show that there are n(n l)/2 possible values for r namely, the numbers a.. = w.w.d(v.,v.)/(w. + w.) for each P 1JJ1JJ combination of vertices v^, v The algorithm computes all these numbers, arranges them in increasing order, and performs a binary search on this list of numbers. The search relies on solving an r- cover problem for each value of r chosen from the ordered list {a..}, ij The search terminates when the smallest r in the list is found for which the r-cover problem generates at most p points. The covering part of the algorithm requires a computational effort of 0(n) for each r, and a total effort of O(nlogn) for all values of r tried during the binary search. Hence, the computational effort is determined by the initial computation and ordering of the numbers ay> and is of 2 0(n logn). A similar approach is used by Chandrasekaran and Daughety [12] to solve the continuous p-center problem on a tree network. First, they provided an 0(n) procedure for finding the minimum number of points needed to cover every point of T within a given radius r. Then, they provided a method to compute r A further refinement of the method is given by Chandrasekaran and Tamir in [14]. They proved that r^ is determined by one of the numbers d(t,t')/2k, where t and t* are any two tip vertices and k is any integer between 1 and p. The total computational effort for finding r and applying the covering P algorithm is of O((nlogp)^). A somewhat different approach, which relies on finding a clique on a related graph, is given by Chandrasekaran and Tamir [13]. They -25- For tree networks, more efficient algorithms can be devised to find a median. An 0(n) algorithm was given by Hua Lo-Keng and Others [60] and independently by Goldman [42]. The algorithm reduces the search to successively smaller subtrees until a median is found. At each stage, one chooses an arbitrary tip vertex (a vertex of degree one) of the current tree. If the (modified) weight of the selected vertex is at least as large as half the sum of all weights, a median is found. Otherwise, that tip vertex is eliminated from further con sideration together with the edge incident to it and its weight is added to the weight of the adjacent vertex. The procedure is repeated with the new (reduced) tree. The algorithm does not require the com putation of the distance matrix and uses only the incidence relation ships and the weights. Goldman's algorithm is based on a "localization theorem" proved by Goldman and Witzgall [46]. The theorem provides sufficient condi tions for a subset of N to contain a median. Given a compact subset S of N, if S satisfies the two conditions (i), (ii) then it contains at least one median. The conditions are (i) the set S must be a "majority" set, meaning that the sum of the weights corresponding to vertices in S must be at least as large as half the sum of all weights (ii) the set S must be "gated" in the sense that there must exist a unique point g in S such that for every s e S and t e N-S, it is true that d(t,x) = d(t,g) + d(g,s). Goldman's algorithm in essence is a repeated application of this theorem to a tree network. Goldman [43] also proposed an "approximate" localization theorem which somewhat relaxes the second condition and guarantees the existence of a point in S that approximates an actual median. -104- shortest path length between the same existing facility nodes in GBC. Since Np is in a tight path in GBC*, then every original node Ih for which i e P will be in a tight path in GBC, as the shortest path lengths in GBC* and GBC are the same. But then every N^, 1 < i < m, is in a tight path in GBC, as is in a tight path in GBC for every P c K*, and U{P: P e '*} = {l,...,m}. Thus, upon using Theorem 3.3.3, Z is the unique solution to DC and Z is efficient. Proof of Sufficiency for Irreducible Location Vectors We now return to the proof of Property 3.5.2. After presenting a number of preliminary results, we will show that if Z is irreducible and z_. e H[A^(Z)] for j e {l,...,m}, then every new facility node is in a tight path in GBC. The following lemma is proven in [22]. Lemma 3.5.1. Let P be a finite set of points each of which is in T. For any p e P, we have H[P] = UL(p,p): p e P). That is, the convex hull of P can be constructed by finding the line segments joining an arbitrary element of P to every point in P. Next, we have the following lemma. Lemma 3.5.2. Suppose Z is irreducible. Let and be two adjacent new facility nodes in GBC. If z2 e H[A2(Z)] then there exists a facility location y in A2(Z) such that a) z2 e L(z^,y) and z^ ^ y, b) whenever y is a new facility location, z2 f y. -101- location vector, as GBC*, by definition, has no arc of length zero connecting two new facility nodes. Hence, sufficient conditions for Z can be expressed in terms of the sufficient conditions (given in Property 3.5.2) for Z*. The following procedure, RP (Reduction Procedure), transforms GBC into GBC* by applying successive elementary transformations as de scribed in the above paragraph. During the procedure, we also keep a list K which contains as members the composite indices. RP. 0) Given Z, set up GBC with arc lengths defined by entries of D(Z). Define K = {{1},. .. ,{m}}. Label new facility node as N^j, 1 < i < m. 1) If, for some P,Q e K, P f Q, there is an arc (Np,N^) of length zero in GBC go to 2). Else go to 4). 2) Superimpose node on together with all arcs incident to Np. Remove arc (Np,N^) from GBC. (If parallel arcs occur due to this transformation they will clearly have equal lengths. Parallel arcs may optionally be represented by a single arc.) 3) Remove P and Q from K, insert P U Q in if and go to 1). 4) Stop with K* = K and GBC* = GBC. The algorithm RP terminates in at most m 1 iterations as each iteration reduces the number of elements of K by one. An example application of RP is given in Figure 3.6. For each composite index P in K* we define z* to be the common location of every new facility i for which i e P. For the example of Figure 3.6, let K* = {P^P^ with P = {1}, ?2 = {2,3,4}. Then zPt Z1 and ZP2 z2 z3 z4 We let Z* be the location vector -7- Figure 1.1. Family Tree for Network Location Problems -36- the new facility it is associated with. All the strings pulled tight from the chosen tip are engaged at this new facility location. The feasibility of this location is checked with respect to all existing facilities and all other new facilities already placed on T. If the feasibility check is passed, new strings are fastened at this location associated with that new facility and other unplaced new facilities for which the distances are of concern. The procedure continues, treating each placed new facility like an existing facility, until, either all facilities are placed, or the current tree reduces to a point, in which case, all remaining new facilities are placed at that point. If the separation conditions hold, the procedure always finds a feasible location vector. The algorithm is of 0(m(m+n)) and is conjectured to be a best order algorithm in [33], for determining the con sistency of the distance constraints. Extensions of the results obtained in [32] are given by Francis, Lowe, and Tansel [33]. These extensions focus on the analysis of binding separation conditions which in turn determine the "uniquely" located new facilities. A separation condition that holds at equality is said to be a binding separation condition. If Z,(E.,E.) = d(v.,v,) 3 k J k is a binding separation condition, then any shortest path P(E.,E,) in J k the auxiliary graph G is said to be a tight path. New facility i is said to be uniquely located at point if in every feasible solution X to the distance constraints the location x. is the same. It was shown i in [33] that a new facility i is uniquely located if and only if node N_^ lies on at least one tight path. As an immediate consequence of this property the distance constraints has a unique feasible solu tion if and only if each N_^, 1 i < m, lies on at least one tight path -49- compact subset T* of T can be identified that contains all efficient points. To identify T*, define R* to be the set of all minima to the unconstrained problem min[f,(x): xÂ£Tl. If R* intersects the feasible 1 1 set Q, define S* to be this intersection. Otherwise, S* is the unique i i closest point in Q to R*. Having defined each S*, 1 < i m, if their intersection is non-empty, then the set of all efficient points is given by T* = H{S*: 1 i < m}. If this intersection is empty, then T* is the smallest compact convex subtree that intersects each S*. It can be shown that each R*, S* is convex, compact, and that T* is a li convex compact subset of T. Lowe's theorem assumes a knowledge of set of minima to each f as well as a knowledge of and hence Q. We note that the functions c(x) and m(x) in the cent-dian problem are both convex on T. Hence, Halpem's results can be obtained by apply ing Lowe's theorem. Now, we consider a multi-objective problem which involves multiple new facilities to be located on a tree network so that the distance between each specified pair of new and existing facilities, and each specified pair of new facilities is, roughly speaking, "as small as possible." The problem is defined by Francis, Lowe, and Tansel [33] as a sequel to the distance constraints problem, and solved by making use of the separation conditions. Here, we call the problem, the "multifacility vector minimization problem." The multifacility vector minimization problem (on a tree network). Let T be a tree network and let I I be given nonempty sets with Iq c (ij): 1 Â£ i S m, 1 < j < n} and IB C {(j,k): 1 5 j < k < m}. The problem of interest is to locate m new facilities on T at points x ,...,x so that each distance d(x ,v.) (i,j)el and d(x.,x.) (j.k^I^ kjlcB BIOGRAPHICAL SKETCH Barbaros Tansel was born on January 10, 1952, in Ankara, Turkey, where he received his early education. For his high school education he attended the Robert Academy in Istanbul and graduated in June 1970. In September 1970, he began his undergraduate study in the Middle East Technical University in Ankara and was awarded the Kennedy Scholarship in 1971. He graduated from the Middle East Technical University in June 1974 with a B.S. degree in industrial engineering. In 1975, he was awarded the Fullbright Scholarship and began his graduate study in the University of Florida. He received his M.Sc. degree in December 1976 and Ph.D. in December 1979. During his graduate study he worked as a teaching and research assistant in the Department of Industrial and Systems Engineering. Barbaros's hobbies include classical music, chess, philosophy, and folk dancing. -170- -125- As in Chapter 3, denote by P(Fp,F^) any path in GBC^ connecting nodes Fp and F A path is specified either by the sequence of nodes in the path, or, by the sequence of arcs in the path. We denote by L P(F ,F ) the length of the path P(F ,F ) and define L (F ,F ) to be z p q w * p q z p q the length of any shortest path connecting nodes Fp and F^. We say P(Fp,F^) passes through an arc (F^,F^) if (F^,F^) is an arc in the path. We say P(F ,F ) passes through AT1 (A) if the path passes p ^ iJ u through at least one arc in A^ (A^). Associated with any path P(Fp,F^) we define two more terms, namely, WP(F ,F ) and VP(F ,F ). The first term WP(F ,F ) is the sum of the p q p q p q reciprocal weights 1/w^ where the summation is taken over all arcs (N.,E.) which are contained in the path P(F ,F ). If the summation 13 P q is taken over an empty set, then WP(Fp,F^) = 0. Similarly, VP(Fp,F^) is the sum of the reciprocal weights l/v^ over all pairs (N^.,N^) which are contained in'P(F ,Fq). Again, VP(Fp,Fq) = 0 if Ag fl P(Fp,Fq) = 0. The motivation for these two quantities can be given by observing the relation L P(F ,F ) = z -WP(F ,F ) + z_VP (F ,F ) . z p q 1 p q 2 p q (4.2.4) The relation in (4.2.4) can be readily verified by observing that the arc lengths of GBC^ are defined by the quantities and z2^vj^ so that the length of any given path is the sum of the reciprocal weights multiplied by or z^, whichever is applicable. In what follows any path (in GBCz) we refer to is a path connecting some two existing facility nodes Ep and E^. All other paths (for which one or both of the terminal nodes are new facility nodes) will be referred to as subpaths. -130- GBC The length of P(E ,E ) on GBC is L ,P(E ,E ) = (z, A.)* z-X & p q z-X z-X p q 1 1 p q WP(E ,E ) + zVP(E ,E ). But z. = 0 so that L ,P(E ,E ) = L P(E ,E ) - p q p q / zA p q z p q X,WP(E ,E ). By our choice of X. we have X, < [L P(E ,E ) - 1 p q J 1 1 z p q d(v ,v )]/WP(E ,E ). Hence, L ,P(E ,E ) > L P(E ,E ) {[L P(E ,E ) - p q p q z-X p q z p q z p q d(v,v )]/WP(E,E )}-WP(E ,E ) = d(v ,v ). Thus, L ,P(E ,E ) > d(v ,v ) pq pq pq pq z-x p q p q for any path P(E^,E^) so that the separation conditions on GBCz_^ hold and is consistent with X ^ 0, X ^ 0, completing the proof. Next, we have the necessary and sufficient conditions for efficiency. Theorem 4.3.1. Given a location vector Y used to define DC and GBC z z with z = (z^,Z2) = f(Y), the following are equivalent: (a) The location vector Y is efficient. (b) At least one arc in is contained in a tight path. (Equivalently, there exists at least one tight path which passes through Ad.) 15 Proof. To show (a) implies (b), suppose Y is efficient. Assume that no arc in A^ is contained in a tight path. Hence every path which passes through A^g is slack as DCz is certainly consistent. Lemma 4.3.2 implies X = (X,,X0) can be chosen with X > 0 and X ^ 0 so that DC is con- 1 l z-X sistent. Corollary 4.3.1 then implies Y is dominated, contradicting that Y is efficient. To show (b) implies (a) suppose at least one arc in A^ is in a tight path. Let P(E ,E ) be such a path which passes through A^ and P q B which is tight. Clearly, P(E ,E ) also passes through A For any p q c X = (X ,X) >0, X 0, the length of P(E ,E ) in GBC .. will be i z p q za strictly smaller than its length in GBC as at least one of z. and z Z 1 z is reduced by a positive amount due to X being different from (0,0). Hence, for any X ^ 0, X ^ 0, the separation condition on GBC -14- to the one defined by (1.3.2) is obtained. The bound is again applicable to all networks and always attainable for tree networks. Vertex constrained 1-center problem. The vertex constrained 1-center problem was considered as early as 1869, and perhaps earlier, by Jordan [63] as a graph theoretic problem. This problem can be solved by examining the distance matrix of the network, as demonstrated by Hakimi [47], Rosenthal, Pino, and Coulter [98] introduced a gener alized algorithm that solves a number of "eccentricity" problems on tree networks, one of which is the vertex restricted 1-center problem. In this case, the eccentricity of a vertex is defined to be the distance from that vertex to a farthest vertex. This generalized algorithm determines the eccentricity of each vertex by making only two traversals of the vertices. The vertex center is that vertex with the minimum eccentricity. Slater [103] considered the problem of finding the vertex center of a network with respect to subnetworks. In this version of the problem, each demand is a known collection of vertices (or a subnetwork induced by the collection). The distance between a vertex and any such collection is defined by a nearest element of the collection to that vertex. For a given vertex, the value of the objective function at that vertex is the maximum of the distances between that vertex and any such collection. Slater showed that a matrix D' can be constructed from the distance matrix D of the network, so that each entry of D' is a distance from a vertex to a nearest element of a collection. Slater demonstrated that the vertex center with respect to collections of vertices can be found by examining the matrix D'. -140- The following algorithm, E-FRONT, constructs the efficient frontier. We assume m has been computed for 1 A s < t < m. E-FRONT 0) Label the arcs in as a^,...,ar where r is the cardinality of A Define A' = {(a.,a.): 1 < i < j < r}. Delete from A' every 0 x j pair (a^,a^) for which a^, and a^ are incident to the same new facility node. Let A be the resulting subset of A' after the deletions. 1) For every (a.,a.) e A define the linear function t.,(z.) as i 3 iJ 1 follows: Suppose a. = (N ,E ) and a. = (N ,E ). Due to step i s p/ j t q clearly s / t and thus m > 0. For z^ e [a,b] 0) d(Vr,Vr,) (1 + 1/..) X. .(Z.) = E_3_ Â§Â£ tq_ y 1 st 1 st 2) Define x(z^) = max{x (z^) : (a^a^) e A} for z^ e [a,b]. The efficient frontier is given by Z* = {(z^,x(z^)): a Â£ b}. The next theorem establishes the correctness of the algorithm. 2 2 Then we will show that the algorithm is 0(m (m + n )). Theorem 4.4.1. The algorithm E-FRONT constructs the efficient frontier for the bi-objective m-center problem. Proof. By Wendell's theorem Z* = {(z^,e(z^)): a < z^ < b}. Hence, it suffices to show that e(z^) < x(z^) and e(z^) > x(z^) for a < z^ < b. To show e(zp x(z^), choose any z^ e [a,b] and define z = (z1,z) with z = xCz,). Let DC and GBC be the constraints of 12 2 1 z z the problem P and the associated graph, respectively. Choose any Z1 nd-path P(E ,E ). Either the path passes through A^ or it does not. p q a In the latter case P(E ,E ) is the path (E ,N ),(N ,E ) for some new -129- But L P(E ,E ) = z. WP(E ,E ) + zVP(E ,E ) so that from (4.3.3) we zpq lpq 2pq have L ,P(E ,E ) = Li(E ,E ) X -VP(E ,E ) . z-x pq zpq z pq (4.3.4) By our choice of X, we have 0 < X e [L P(E ,E ) d(v ,v )]/ J 2 2 zpq p q VP(EpjE^). It follows then, upon using (4.3.4), that L P(E ,E ) > L P(E ,E ) [ z-X p q ~ z p q' 1 L P(E ,E ) d(v ,v ) zpq P 9 i up/F F \ VP (E ,E ) * ^ ^Ep5 Eq^ p q = d(v v ) p q (4.3.5) From (4.3.2) and (4.3.5) it follows that L ,(E ,E ) > d(v ,v ) for z-X p q p q this case. Since the choice of E^ and E^ is arbitrary, every separation condition holds on GBC so that DC is consistent with X = (0,X), ZA Z~A X2 > 0. Case with z2 = 0 By hypothesis every path which passes through Ag is slack. Choose any path P(E^,E^) which does not pass through Ag. Consistency of DC^ implies either LzP(Ep,E^) = d(v^,v^) or L P(E ,E ) > d(v ,v ). The former case is not possible since a z p q p q subpath of length zero can be chosen from the arcs in Ag and this subpath can be appended to P(E^,E^) to obtain a new path, say, P'(E^,E^) without increasing the length of the path. Hence L^PiE^jE^) = L^P'(Ep,Eq) = d(Vp,v^) contradicting that every path which passes through Ag is slack. Thus, every path which passes through A^, is also slack. Define e to be the minimum of [L P(E ,E ) d(v ,v )1/WP(E ,E ) zpq p q p q over all paths in GBC2. Clearly e > 0, since every path is slack and every path necessarily passes through A so that WP(E ,E ) > 0. Choose > C P q (X, ,0) with 0 1 i i I p q on -71- d(x,..,v.) < f/(r). As D(X,v.) d (x ,.. ,v.) it follows that X is (j) J 3 3 U) 3 a feasible solution. Property 2.A.2. For any nonempty distinguished set U^, with vertices numbered so that U, = {v. ,... ,v, }, we have k 1 k v. e BT(x.) , 3 3 1 < j < k .-I. d(Xj,Vj) = fj (r), 1 5 j < k . (2.4.3) (2.4.4) Proof. Expression (2.4.3) is obvious. To show (2.4.4), choose any v in U. Let t be the tip vertex chosen at the first of the iteration tC in which x^. is placed. The algorithm causes the string at v^. to-be pulled tight along every edge connecting v^ to t, and to be pulled tight along [t,x.], with the string end point coinciding with x.. J 1 Thus d(v.,t) + d(t,x.) = f.^(r). But v. e BT(t) and x. e T-BT(t) or J J 3 3 3 x. = t so that d(v.,t) + d(t,x.) = d(v.,x.). Thus, (2.4.4) follows. 3 1 3 3 3 Property 2.4.3. Let X = ix^,...,x ) be the feasible solution con structed by COVER, with vertices numbered so that U = {v^,...,v } is the primary set associated with X. Assume q > 1. Then d(vivj) > + for 1 i < j Â£ q (2.4.5) Proof. We know the first q-1 members of U are distinguished vertices. Hence Property 2.4.2 implies v. e BT(x.), i i 1 < i < q-1 .-1 d(v.,x.) = f (r), 1 i S q-1 . (2.4.6) (2.4.7) For i < j, x^ is placed prior to x^.. Since v^ is assigned to x^ and -92- independent subproblems. Further, we note that DC is always consistent, as Z is certainly feasible to DC, and hence, by Theorem 3.3.1, the separation conditions are always satisfied. For convenience, for any location vector Z, we denote by A*(Z) the collection of locations of uniquely located facilities whose nodes are adjacent to N in GBC. We denote by 11[A*(Z)] the convex hull of A*(Z), the imbedding of the smallest subtree of T spanning all the elements of A*(Z). With the above definitions we can present a family of equivalent conditions for a location vector Z to be efficient. Theorem 3.3.3. Given a location vector Z used to define DC and GBC, the following are equivalent: (a) Z is efficient; (b) Each FL is in at least one tight path in GBC; (c) Z is the unique solution to DC; (d) z^ e H[A*(Z)] for i = l,...,m. Proof. The equivalence of (b) and (c) is a direct conseqeuence of Theorem 3.3.2 and the fact that Z is always a feasible solution to DC, while (c) clearly implies (a). To show (a) implies (c) suppose Z is not the unique solution to DC. Color every new facility node in GBC which is not contained in any tight path blue. Color all the other (new or existing facility) nodes red. Equivalence of (b) and (c) implies every blue node represents a new facility which is not uniquely located, while every red node represents a (new or existing) facility which is uniquely located. By assumption there is at least one blue node. By connectedness of GBC, there is at least one arc which connects some blue colored node, say, N to some red P colored node, say, F Furthermore, arc (N ,F ) has positive 9 P q * -83- with |X| = s, solve the cover problem for r. r < r contradicting the definition of r pi s s r < r < r . P P"1 It now follows, if we define the set We then have f(X) < r < Thus q(r) = p for P = {(p-1,p): p e {2,...,n}, r^ < r^} > that q(r) r P for rp < r < r j, (p-l,p) e P 1 for r^ < r . (2.6.4) The formula (2.6.4) completely defines the function q(r), since r = a, n and the cover problem is feasible if and only if a < r. Hence if we solve the p-center problem for all p and compute r ...,r then we 2 n have an explicit formula for q(r), and we see that the r^ completely define the function q. For example, if r, = rc < r. = r < r_ = r,, then q(r) = 5 for r^ ^ r < r^, q(r) = 3 for r^ r < r^, and q(r) = 1 for r^ ~ r. Also, the proof of the lemma does not require the assump tion that the location network is a tree. Thus the formula for q(r) is still valid if the location network has cycles. -94- 3.4 Examples Here, we give examples of efficient and non-efficient points. Ex. 1. For a single new facility, D(z) is the vector (d(z,v ),..., d(z,v )). Any point z in T is efficient since T is the convex hull of iv . ,V }. 1 n Ex. 2. Consider the tree T shown in Figure 3.2. Each arc length in the corresponding graph GBC corresponds to an entry of D(Z). In this case Z is efficient. Notice that and are both contained in the tight path P = (E^, ^, E^). Also, both z^ and z2 satisfy the con vex hull property, i.e. z Htiv^ v2, z2}] and z2 e H[{v3> v^-Zj}]. Figure 3.2. Example of an Efficient Location Vector (a) Graph BC, (b) Tree T. Ex. 3. Consider the same tree as in Example 2 except that the location of z2 is changed to the midpoint of edge (v^v^. In this case -87- Our main interest is to characterize efficient location vectors and devise an algorithm for constructing efficient location vectors from a given (dominated) location vector. 3.3 Distance Constraints and Characterization of Efficient Points We make extensive use of the results obtained in [32, 33] for distance constraints to establish the necessary and sufficient condi tions for efficient points. The Distance Constraints (DC) are defined in [32] (independent of the efficiency problem) as follows: Given the sets 1^ and 1^ and nonnegative upper bounds c^. and b^ find a point X = (xx ) in Tm, if it exists, such that 1 m d(x.,v.) < c . i 3 ij d(x. ,x. ) Â£ b., 3 k J)k (i,j) e I (j,k) e I, (3.3.1) Corresponding to DC, we define Graph BC (GBC) as the undirected graph having nodes E.,...,E N,,...,N ; for every (i,k) e I,,, there i n i m is is an arc (N. ,N, ) of length b., between nodes N. and N, : for every J k' jk j k J (i,j) e I, there is an arc (N.,E.) of length c.. between nodes N. C i J ij i and E.. We further assume that the sets I,, and I are such that GBC J B C is connected, as otherwise DC decomposes into independent sets of con straints which may be analyzed separately. Given a node-path between any two nodes f and f in GBC, we de- P q note the path by P(f ,f ) and denote the length of the path by LP(f ,f ) p q p q We define (f^jf^) to be the length of any shortest path in GBC between nodes f and f Subsequently, unless we specify otherwise, it should p q -164- 42.A.J. Goldman, "Optimal Center Location in Simple Networks," Trans. Sci. 5, 212-221 (1971). 43. A.J. Goldman, "Approximate Localization Theorems for Optimal Facility Placement," Trans. Sci. 6, 195-201 (1972). 44. A.J. Goldman, "Minimax Location of a Facility in a Network," Trans. Sci. 6, 407-418 (1972). 45. A.J. Goldman and P.M. Dearing, "Concepts of Optimal Location for Partially Noxious Facilities," Bull. Opns. Res. Soc. Am. 23, Suppl. 1, B-31 (1975). 46. A.J. Goldman and C.J. Witzgall, "A Localization Theorem for Optimal Facility Placement," Trans. Sci. 4, 406-409 (1970). 47. S.L. Hakimi, "Optimal Locations of Switching Centers and the Absolute Centers and Medians of a Graph," Opns. Res. 12, 450-459 (1964). 48. S.L. Hakimi, "Optimum Distribution of Switching Centers in a Communication Network and Some Related Graph Theoretic Problems," Opns. Res. 13, 462-475 (1965). 49.S.L. Hakimi and S.N. Maheshwari, "Optimum Locations of Centers in Networks," Opns. Res. 20, 967-973 (1972). 50.S.L. Hakimi, E.F. Schmeichel, and J.G. Pierce, "On p-Centers in Networks," Trans. Sci. 12, 1-15 (1978). 51.S. Halfin, "On Finding the Absolute and Vertex Centers of a Tree with Distances," Trans. Sci. 8, 75-77 (1974). 52. J. Halpem, "The Location of a Center-Median Convex Combination on an Undirected Tree," J. Reg. Sci. 16, 237-245 (1976). 53. J. Halpem, "Duality in the Cent-Dian of a Graph," Working Paper No. WP-08-77, Univ. of Calgary, Calgary, Canada (1977). 54. J. Halpem, "Finding Minimal Center-Median Convex Combinations (Cent-Dian) of a Graph," Manag. Sci. 24, 535-544 (1978). 55.G.Y. Handler, "Minimax Location of a Facility in an Undirected Tree Graph," Trans. Sci. 7, 287-293 (1973). 56. G.Y. Handler, "Medi-Centers of a Tree," Working Paper 278/76, Recanati Graduate School of Business Administration, Tel-Aviv University, Israel (1976). 57. G.Y. Handler, "Finding Two Centers of a Tree; The Continuous Case," Working Paper, Recanati Graduate School of Business Ad ministration, Tel-Aviv University, Israel (1977). -57- f and hence f could be deleted from the definition of f without s t changing f. Further, we assume p < n-1, as otherwise the p-center problem is trivial. So as to state the dual problem, we define 8., =8,. for j,keJ by j K Kj 8., = min maxif (d (x ,v.) ) f (d(x,v,))} 3k 3 3 K K For j,kej with j < k we define a., = maxif .(0), f, (0)} and 3k 3 k b#, = min{f. (5.) ,f (6 ) }. We note that a 5 n implies [a., ,b ., ] ^ 0. jK 3 3 k k JK jK The following lemma, the results of which are proven in [29] provides a closed form expression for 8jk Lemma 2.2.1. For any j,keJ with j 5 k we have: (i) The function f.^ + f exists, is stricly increasing, continuous 3 k has domain [a., ,b., ] ^ 0, and range [L., ,U., ], where L = JK 3K jK JK jk ifT1 + f^)o(ajk) and Ujk (f'1 + (ii) d(v.,vk) < U.k. (iii) The function (f.^ + f ^ exists, is strictly increasing and 3 k continuous, has domain [L ,U., ] and range [a., ,bM ]. JK. jk Jk jk (iv) 8_.k = (f"1 + fj^1) 1o(max{d(v_. ,vk> L^}) . We remark that either 8., = a., or 8., = (f. ^ + f, *) ^o(d(v ,v, )); jk jk jk j k j k" 8.. e [a., ,b.. ], and 8.. = f.(0). The closed form expression for 8., JK 3k jk jj j r jk given in Lemma 2.2.1 facilitates construction of the dual problem. We define the dual objective function g on subsets of V as follows For any K C V with K1 t 2 g(K) = max{g1(K), g2(K)} (K) = min(8ij: v^v^ e K, i j} g2 (K) .= max{fj(0): v^ e K} -72- no t to x^ for 1 < i < j < q, v. was not in BTix^, and the string at v. did not reach x.. Hence J i v. e T-BT(x.), J i 1 < i < j ^ q d(xi,Vj) > f (r), 1 < i < j < q . (2.4.8) (2.4.9) But (2.4.6) and (2.4.8) give d(v ,v.) = d(v.,x.) + d(x.,v.) for i j 11 1 J 1 < i < j < q, from which, on using (2.4.7) and (2.4.9), (2.4.5) follows. We shall need the following remark, proven in [32]: Remark 2.4.1. Given any a.,a. e T and s.,s. > 0, there exists a-point i 3 i J x in T for which d(x,a^) < s^ and d(x,a^.) < s^ if and only if d(a^,aj) < s + s.. i 3 We are now ready to establish the optimality of COVER. Theorem 2.4.1. Given any r for which a < r and J(r) 4 0, COVER solves the covering problem. Proof. Let X = {x^,...,x^} be the point set found by COVER. Property 2.4.1 implies X is feasible to the problem. If q = 1, X is clearly optimal. If q > 1, let the vertices be numbered so that U = {v,,...,v } 1 q is a primary set associated with X. By Property 2.4.3, d(v_^,Vj) > f^(r) +* f ^(r), for 1 <, i < j < q. Remark 2.4.1 implies there exists no x in T for which d(x,v.) < f.^(r) and d(x,v.) < f .^(r) for any i i 13 i, j in {l,...,q} e J(r) with i < j. Hence it is impossible to cover any two members of U with a common center. Thus, since |u| = q, any feasible solution X to the covering problem satisfies |x| > q. Since q = |X| and X is feasible to the problem, X is thus an optimum feasible solution. -73- We remark that the covering problem may be of as much interest, from both a theoretical and applications point of view, as the p-center problem. In Section 6, we will present a problem which is dual to the covering problem and show that the primary set identified by COVER solves the dual of the covering problem. Furthermore we will charac terize q(r) as a step function, and provide a formula for q(r) assuming that r^ is known for 1 <, p n-1. 2.5 Dual Problem Solution and the Strong Duality Theorem Based on the W.D.T. and properties of COVER we now present a proof of the S.D.T. The proof is constructive in that we use an algorithm called OPTKLIQUE which, given the optimal objective value of the primal problem, constructs an optimal solution to the dual problem. We then show that the objective values of the pair of prob lems are equal. As a by-product the proof also establishes that r e R, where, for convenience, we define R ={Â£..: l
p ij J We find it useful to summarize Theorem 2.4.1 and Property 2.4.3 as follows: Lemma 2.5.1. Given any r for which a < r and J(r) ^ 0, the following assertions are true: (a) COVER finds an optimum solution X to the covering problem with q(r) = |X|. (b) Whenever q = q(r) > 1, any primary set U = ^v(l) associated with X satisfies g(U) = g^U) > r (2.5.1) -74 Proof. (a) is just Theorem 2.4.1. (b) From Property 2.4.3, for any v_^,v^ e U, i ^ j, we have d(v^,v^) > f.^(r) + f.^ir) > f.^Ca) + f.^a) where r Â£ a > a = a. .. Thus, i J i J. iJ d(v^,Vj) is in the domain of (f_^ + f ^) from which, upon using Lemma 2.2.1 and the definitions of g, g^, and g^, (2.5.1) follows. In the algorithm OPTKLIQUE we assume that r^ is given for some value of p, 1 g p n-1. OPTKLIQUE constructs an optimal solution to the associated dual problem. OPTKLIQUE 1) If r = a, take K*+^ to be any p+l-clique in V containing a vertex v for which f (0) = a, and go to 3). Else, given r > a, compute s s p r' = maxifS. e R: < r } and choose any r for which r' < r < r . P iJ P P P Go to 2) . 2) Apply COVER with the chosen value of r to find an optimum solution X and its associated primary set U, with |x| = q = |u|. Note r < r implies |X| > p, so q Â£ p+1. Take K*+^ to be any subset of U con sisting of p+1 members of U. Go to 3). (If q > p+1, there will be alternative optimal cliques.) 3) If K*+j, is any clique found in either step 1) or 2), then g(K*+p = r and the W.D.T. guarantees K* is an optimum solution to the dual P P+1 problem. Before proving the correctness of the algorithm, we note, since a = for some h, that a < r implies a < r', and thus the r chosen hh P P in step 2) is one for which a feasible solution exists to the covering problem. Theorem 2.5.1. Given r for any p, 1 g p < n-1, the clique K* con- P P+1 structed by OPTKLIQUE satisfies This research was supported in part by NSF Grant //ENG 76-17810, the Army Research Office, Triangle Park, N.C., under contract DAHC04-75-G-0150, and by the Operations Research Division, National Bureau of Standards, Washington, D.C. -31- formulation using a subgradient optimization method. Another branch- and-bound method was developed by Jarvinen, Rajala, and Sinervo [62]. Their procedure looks for n-p vertices that do not belong to a p- median. This method works better for larger values of p, since n-p is smaller in this case reducing the number of possibilities. A similar branch-and-bound procedure was given by El-Shaieb [24]. The procedure is based on construction of a source set (i.e., p-median) and a demand set. Starting with both sets empty, a location is added to either set at each iteration. Whenever the number of elements in a source set reaches p, or the number of elements in a demand set reaches n-p, a feasible solution is obtained. An optimal solution is eventually identified using the lower bounds. A third approach taken is to use heuristics. A number of heuristics have been developed by Maranzana [84], Teitz and Bart [107], and Khumawala [69, 70]. For a discussion of a number of the solution approaches from a computational standpoint, the reader is referred to Hillsman and Rush- ton [59], and Khumawala, Neebe, and Dannenbring [71]. Stochastic networks and vertex-optimality. A number of pro babilistic versions of the p-median problem have been considered in the literature. Mirchandani and Odoni [89, 90] extended Hakimis vertex optimality result to the case of a stochastic network whose edge lengths are random variables. Berman and Larson [2] considered a stochastic network where the availability of servers (centers) is a random variable. They showed that under suitable conditions there exists at least one optimal set of locations on the vertices of such a network. This completes the discussion of the p-median problem. algorithm to construct the efficient frontier, where m and n are, respectively, the number of new and existing facilities. v i i i -88- be understood that any path we refer to is a simple path between some two existing facility nodes and E^. Results on Distance Constraints The distance constraints are said to be consistent if there exists at least one feasible solution to (3.3.1). The following result is established in [32]. Theorem 3.3.1. The distance constraints are consistent if and only if d(v ,v ) < Â£(E ,E ), 1 Â£ p < q n (3.3.2) p q p q The inequalities (3.3.2) are termed the Separation Conditions [32], since each term on the right specifies an upper bound on how separate two existing facility locations can be. Except when stated otherwise, we assume throughout the chapter that the separation condi tions hold, and thus (equivalently) DC is consistent. We call a path P(E^,E^) between E^ and E^ in GBC a tight path if LP(E ,E ) = d (v ,v ). We note that since we assume DC is consistent, P q p q it necessarily follows if P(E ,E ) is a tight path, that LP(E ,E ) = p q p q L(E ,E ). Any path P(E ,E ) for which LP(E ,E ) > d(v ,v ) is called pq p q p q p q a slack path. We say that new facility i is in a tight path if there exists at least one tight path containing Ik. Every path containing Ik is slack if there is no tight path which contains Ik . The motivation for the above terminology is due to a string graph representation of GBC. This string graph is also useful for obtaining problem insights. When knots representing nodes E^ and E^ are pulled as far apart as possible, the distance between the two knots is L(E ,E ). P q -95- (z^.z^) is not efficient as is not contained in any tight path. Also, t Hv^, v^, z^}. This example is shown in Figure 3.3. 1.5 1.5 / 4 7 c ? / Figure 3.3. Example of a Non-Efficient Location Vector Ex. 4. Again consider the same tree with z^ = z^ located at the mid point of edge (v^,v^). In this case, both z^ and z^ are uniquely located and both satisfy the convex hull property. Thus Z = is efficient in this case. This example is shown in Figure 3.4. -107- location (the case with both p and q new facility locations is very similar to the proof we will give below and hence will not be con sidered) Define z. = p. Find a sequence of locations z.,z. ,... z. ,v for some r, 1 < r < m-1, by applying Lemma 3.5.2 to the pairs Jr (z.,z: ), (z, ),...,(z. ,v ) one at a time in the given order so '1 j t Jr that the family of conditions in (3.5.2) is satisfied: z. e L(z.,z. ) and N. N. are adjacent, Jx J 32 J1 J2 z. e L(z. z. ) and N. N. are adjacent, J2 Ji J3 ^2 J3 (3.5.2) z. e L(z. Jr 3 r-1 ,vfc) and N E are adjacent. We remark that the irreducibility of Z and the conclusion of Lemma 3.5.2 guarantees that such a sequence can be found and will end with an existing facility location v as, r can be at most m-1 and for the last z. we must have some y e S. such that z. e L(z ,y) with lr Jr Jr Jr-1 y necessarily an existing facility location. Let q in (3.5.1) be the location v of existing facility s, s s ^ t. Then, the sequence v z.,z. ,...,z. ,v satisfies the assump- s J Jr t tions of Lemma 3.5.4 as a result of (3.5.1), (3.5.2), and the irredu cibility of Z. Hence, we have d(v ,v ) = d(v ,z.) + d(z.,z. ) + ... + d(z. ,v ) s c s J J Ji Jr (3.5.3) where the right hand side of (3.5.3) is clearly the length of the path Eg,N.,N. ,...,N. ,Et> Hence, the path is a tight path due to J J ^ J r (3.5.3) and contains N Thus, z^ is the unique location of new -9- f(X) = max[wJ)(v^,X) + a^: iel] The problem of Interest is the following: Given a positive integer p, find a point set X* = {x*,...,x*}, and a real number r 1 p p such that rp = f(X*) = min [ f (X) : |x| =p,XcN] (1.3.1) where the symbol j*| means the cardinality of a set. The problem defined by (1.3.1) is called the p-center problem. Any set X* of p-points that solves (1.3.1) is called an absolute p- center of N, and the minimum objective value r^ is called the p-radius. For p = 1, an absolute 1-center is simply called an absolute center of N. If in (1.3.1), each xeX is restricted to a vertex location, the resulting problem is called the vertex restricted p-center problem and any set X* C V of p points that solves it is called a vertex restricted p-center of N. A vertex restricted 1-center is simply called a vertex center. We note that the p-center problem is usually formulated in the absence of addends. In what follows, we will assume all addends are zero, unless we explicitly mention them. The case with all w^ equal to unity will be referred to as the unweighted case. With this terminology, the p-center problem is the problem of finding p points on a network so that the maximum (weighted) distance between any demand point and its nearest center is as small as possible. The problem is perhaps most applicable to the location of emergency facilities such as fire stations, ambulance centers, and the like, as -150- approach to the vector-minimization problem was to reformulate the problem in terms of a family of distance constraints which impose upper bounds on the distances between specified pairs of new and existing facilities and specified pairs of new facilities. In Chapter 4, we considered the bi-objective m-center problem (with mutual communication) which involves as objectives the maximum of the weighted distances between specified pairs of new and existing facilities and maximum of the weighted distances between specified pairs of new facilities. We developed the necessary and sufficient condi tions for efficient solutions and provided a procedure for construct ing the efficient frontier. Our approach was to reformulate the problem in terms of a. family of distance constraints which impose upper bounds on the distances between specified pairs of new and existing facilities and specified pairs of new facilities. In what follows we give certain generalizations of the problems considered in this dissertation as well as other location problems considered in the literature. We point out some possible directions for future research. 5.2 Generalized Multi-Center Problem Here, we define a problem which generalizes the p-center problem and the m-center problem with mutual communication. For convenience, we consider the weighted case. Nonlinearity can be obtained by re placing each weight by a strictly increasing continuous function of the associated distance. -52- min max D(v,P) (1.3.8) P C N veV min bw(P) (1.3.9) P C N In Slater's terminology, any path P* that solves (1.3.7) is called a core of N. Among all paths that solve (1.3.8), one with the fewest vertices is called a path center of N. Similarly, among all the paths that solve (1.3.9), one with the fewest vertices is called a spine of N. Slater obtained a number of properties of these problems for tree networks. In particular, Slater showed that the path center of T is unique and contains the vertex center of T, and that the spine of T is unique and contains the centroid (equivalently, the vertex median) of T. We recall that a centroid of T is any vertex v that minimizes the maximum number of vertices in any component of T-v. Also, Slater proposed two algorithms of linear order for determining the path center and the spine of T. -135- Figure 4.3. Illustration of Z and Z* for a Convex Bi-Objective Problem for solving either problem provided that certain needed inverse functions can be evaluated in a polynomial order of effort. The p-center problem is typically solved with the aid of a nonlinear covering problem for which we also give a dual with a physical interpretation. We provide a covering algorithm which solves both the covering problem and its dual simultaneously. The second problem we consider is a vector-minimization problem which involves as objectives the distances between specified pairs of new and existing facilities andspecified pairs of new facilities. We relate the vector-minimization problem of interest to a distance con straints problem which imposes upper bounds on the distances between specified pairs of facilities. We develop the necessary and sufficient conditions for efficiency by making use of the theory developed for the related distance constraints problem. Efficient solutions to the vector-minimization problem of interest are such that in order for any new facility to be closer to some facility than it already is, it must in turn be placed farther from some other facility. Based on the necessary and sufficient conditions, we provide an algorithm which constructs an efficient location vector from a given non-efficient solution. The third problem we consider is a bi-objective minimax problem which involves as objectives the maximum of the weighted distances between specified pairs of new and existing facilities, and the maximum of the weighted distances between specified pairs of new facilities. We again relate the problem to the distance constraints problem and derive the necessary and sufficient conditions for efficiency by making 2 2 use of the distance constraints. Further, we provide an 0(m (m + n )) -59- will still apply. Furthermore, the proof applies to any network, as no special properties of tree networks are used. We now state the S.D.T. We remark that the S.D.T. requires the assumption of a tree network. In effect, network cycles may create a "duality gap." Theorem 2.2.2. (Strong Duality Theorem). For any p, 1 Â£ p n-1, there exists an X* C T with |x*| = p and K* C V with |K*| = p+1 such that f(X*) = g(K*). It is evident from the W.D.T. that X* solves the primal p-center problem and K* solves the dual dispersion problem. Before presenting an example problem, we find it convenient to view the dual problem as defined on "cliques" of a complete graph G. We define G to be the undirected complete graph with node set J, where node j of G represents vertex v of T. To any arc (i,j) of G, i ^ j, we assign the length and, to any node j of G, we assign the node weight g = f^(0). We call any complete subgraph K of G a clique. We note that any nonempty subset of V induces a clique in G and vice versa. For this reason, an equivalent definition of g(.) on cliques of G can be given by defining g^(K) to be the length of a smallest arc in a clique K of G, g2(K) to be the maximum of the weights of nodes in K, and letting g(K) = maxig^K), g2(K)}. If the number of nodes of a clique K is known to be q, we call K a q-clique and (sometimes) write K Defining C (G) to be the collection of all q q q-cliques of G, an equivalent statement of (2.2.2) is as follows: Find a clique K* for which p+1 g(K*+l) = max{g(K): K e Cp+1(G)} . -156- to check those arcs for which the lengths are zero, as the lengths of these arcs cannot be reduced, and in any feasible solution to DC2, the constraints corresponding to arcs of zero lengths will definitely hold at equality. The assumption that the sets A^,...,A are disjoint is needed for the following reason: Given a path P(E^,E^) which passes through A^_, clearly z^ is the common multiplier for every arc which is contained in the intersection of P(E^,E^) and A^. That is, the length of that part of the path P(E^,E^) consisting of the arcs chosen from A^ is the quantity zr*S(P(E^,E^) ,A^). If it were the case that A^fl A^ ^ 0 for some j r, then the above assertion would not necessarily be true, as an arc in the intersection A^_ 0 A^ will have at least two multipliers in this case, namely, z^ and Zy We will consider this case in the conclusion of this section. The following questions seem worth investigating for future research. Ql. Is there a computationally efficient way of checking whether or not arcs of GBC are contained in tight paths? Q2. How can we construct efficient solutions efficiently? Q3. Can the results of Theorem 5.3.1 be extended to the case when some of the A_^ are not disjoint? Q4. How tractable is the t-objective m-center problem if we generalize it by replacing each x^ by a collection of centers? With respect to Q3, suppose that A^/l A^. ^ 0 for at least two indices i and j for which 1 i < j < t. Let (F ,F ) be any arc in A. P If (F ,F ) is contained in at least two members of {A,,...,A }, then p q I t the distance constraint corresponding to (F ,F ) will appear more than -165- 58. G.Y. Handler and P.B. Mirchandani, Location on Networks, The MIT Press, Cambridge, Massachusetts, 1979. 59. E.L. Hillsman and G. Rushton, "The p-Median Problem with Maximum Distance Constraints: A Comment," Geog. Anal. 1, 85-89 (1975). 60. L.-K. Hua and Others, "Applications of Mathematical Methods to Wheat Harvesting," Chinese Math. 2, 77-91 (1962). 61. A.P. Hurter and M.K. Schaefer, "The Regional Allocation of Fire Resources: A Damage Minimizing Approach," Working Paper, North western Univ., Evanston, Illinois (1979). 62. P. Jarvinen, J. Rajala, and H. Sinervo, "A Branch-and-Bound Algorithm for Seeking the p-Median," Opns. Res. 20, 173-182 (1972). 63. C. Jordan, "Sur les Assemblages des Lignes,"^!. Reine Angew. Math. 70, 185-190 (1969). 64. H. Juel, "Bi-Objective Location Problems with Rectangular Dis tances," Working Paper, Michigan Technological University (1979). 65. 0. Kariv and S.L. Hakimi, "An Algorithmic Approach to Network Location Problems. Part 1: The p-Centers," Working Paper, Northwestern Univ., Evanston, Illinois (1976). 66. 0. Kariv and S.L. Hakimi, "An Algorithmic Approach to Network Location Problems. Part 2: The p-Medians," Working Paper, Northwestern Univ., Evanston, Illinois (1976). 67. R.L. Keeney and H. Raiffa, Decisions with Multiple Objectives: Preference and Value Trade-Offs, John Wiley and Sons, New York, 1976. 68. B.M. Khumawala, "Branch-and-Bound Algorithms for Locating Emergency Service Facilities," Krannert Institute Paper, No. 355, Purdue University, West Lafayette, Indiana (1972). 69. B.M. Khumawala, "An Efficient Algorithm for the p-Median Problem with Maximum Distance Constraints," Geog. Anal. 5, 309-321 (1973). 70. B.M. Khumawala, "Algorithm for the p-Median Problem with Maximum Distance Constraints: Extension and Reply," Geog. Anal. 7, 91-95 (1975). 71. B.M. Khumawala, A.W. Neebe, and D.G. Dannenbring, "A Note on El-Shaieb's New Algorithm for Locating Sources Among Destina tions, Manag. Sci. 21, 230-233 (1974). 72. A. Kolen, "Complexity of Location Problems on Networks," Working Paper, Stitching Mathematisch Centrum, Tweede Boerhaavestraat 49, 1091, A1 Amsterdam, The Netherlands (1979). -109- SEVCA Initial 0) Given Z e Tm, set up GBC with arc lengths defined by entries of D (Z). Define K = {{!},...,{m}}. Label each member of K unscanned Reduction 1) If for some P,Q e K, P f Q, there exists an arc (N^.N^) in GBC of length zero go to 2). Else go to 4). 2) Superimpose node Np on together with all arcs incident to N^. Remove arc (Np,N ) from GBC. (If parallel arcs occur due to this transformation, they will have equal lengths. Parallel arcs may optionally be represented by a single arc.) 3) Remove P and Q from K, insert P U Q in K and label P U Q unscanned Define zpyg to be the common location of Zp and z^ and go to 1). Termination Test 4) If every member of K is scanned, stop. Else, choose an unscanned composite index P in K and go to 5). Check for Convex Hull Containment 5) Find A(Np), the set of nodes adjacent to in (current) GBC, and ,K. define Ap (Z ) to be the set of current locations of new existing facilities whose nodes are members of A(Np). % 6)If Zp e H[Ap(Z )], label P scanned and go to 1). Else go and to 7). Movement 7)Find the closest point, say, y to z^ in H[Ap(Z )]. Define e(P) = d(Zp,y). Move zp to y. Update the arc lengths of GBC by subtracting the amount e(P) from every arc incident to Np. Label P scanned and go to 1). -85- At this point, we give an overview of the chapter. In Section 2, necessary definitions and notation are given and the vector-minimiza tion problem of interest is defined. In Section 3, we relate the problem to distance constraints, give a number of related properties of distance constraints, and establish the necessary and sufficient conditions for a location vector to be efficient. In Section 4, we provide examples of efficient and non-efficient location vectors. Section 5 is devoted to a further refinement and simplification of one of the necessary and sufficient conditions, namely, "the convex hull property." In Section 6, we provide an algorithm, SEVCA, which con structs an efficient solution from a given location vector. In Sec tion 7, we characterize efficient solutions for the analogous problem in the p-dimensional Euclidean space with rectilinear (p 2). or Tchebychev (p > 2) distances. 3.2 Problem Statement We suppose given a finite, undirected tree network, and denote by T an imbedding of the given network. Let V = {v,,...,v } be a set of n distinct vertices of T. We assume existing facility i is located at vertex v^, i e {l,...,n}. For j e {l,...,m}, denote by x^ a point to be determined in T as the location of new facility j. We define Tm to be the m-fold Cartesian product of T by itself and define a location vector X in Tm to be the ordered m-tuple (x,,...,x ) with each x e T, 1 m j j e {l,...,m}. Sometimes, we refer to a location vector X in Tm as a point m T As in [22], given points x,y e T, we define the line L(x,y) to be the union of all points in the shortest path connecting x and y. In -62- n V T f 20 22 -Cr 10 t)V 6 fj(y) = wj(y + Hj)0 Data 0 n 2 19 0 2 25 0 3 16 2 6 36 0 5 4 0 6 9 4 Figure 2.1. Example Nonlinear p-Center Problem -37- in the graph G. Furthermore, if some path P(E.,E. ) is a tight path, J k then the nodes representing facilities in the path occur with the same ordering and spacing in the path as do the locations representing the facilities in the path P(v.,v.) on T. This result enables one to J k locate the new facilities that appear in a tight path immediately, without having to use the Sequential Location Procedure. A multifacility minimax application of the distance constraints is given in [32, 33] and a multiobjective application is given in [33]. These two applications will be discussed subsequently. m-Center problem with mutual communication Let N be a network with vertex set V = {v,,...,v } and edge set 1 n E. Suppose the sets I and In are given with I,, C {(j,k): 1 j < k m} BO B and I C {(i,j): 1 < i S m, 1 < j < n}. We assume that we are given positive weights v ^ for each (j,k)elg and w for each (i,j)el^. For each location vector XeN, define the functions f (X), f_(X), and B 0 f(X) as follows: fB(X) = max[Vjkd(x^.,xk) : (j,k)eIB] , fc(X) = maxtw^dCx^v ) : (i,j)elc] , f(X) =max[fB(X), fc(X)] . The m-center problem with mutual communication is the following: Find a location vector X*eNm such that Z* = f(X*) = min[f(X): XeN] . -137- pf (4.4.1) are just DCÂ¡z. We remark that for any z^ e [a,b], there exists at least one feasible solution to (4.4.1), as there exists an X such that f^(X) = a z^ and can be chosen large enough so that f200 z2. Let z^ be fixed with z^ e [a,b] and let (Y,z2) be any feasible solution to (4.4.1). Define z = (z ,z) and let DC be the distance J. z z constraints in (4.4.1). Since (Y,z2) is feasible to (4.4.1), the separation conditions on GBC hold so that every path in GBC has z z length at least as large as the distance between the locations of existing facilities corresponding to the terminal nodes of the path. Hence, for any path P(E^,E^) we have L P(E ,E ) = z WP(En,E ) + zVP(E ,E_) > d(v,v ) , zpq lpq 2pq pq or, equivalently, d(v ,v ) WP(E ,E ) 2 lVP(E ,E )J llVP(E ,E )J p q p q (4.4.2) Defining x(z^) to be the maximum of right side of (4.4.2) over all paths in GBC^, it follows that z^ > x(z^) whenever z2 is feasible to (4.4.1). Hence, the minimum value of z2 which solves (4.4.1) is x(z^). We observe that the right side of (4.4.2) is the value of a linear func tion (of z^) evaluated at z^. There are as many such linear functions as there are paths in GBCz. Further x(z^) is the maximum of these functions at z^. Thus, defining x(*) to be the pointwise maximum of these linear functions over the interval [a,b] we have e(zp = x(zp for every z^ Â£ [a,b]. Hence, a brute-force method to construct the efficient frontier is to enumerate all paths on GBC, compute the parameters (slope and intercept) for each linear function corresponding -160- of N. Hence, one possible approach to solve the p-center problem on N is to enumerate on all possible forests of N consisting of p disjoint subtrees and determine the one center of each subtree in any given forest. Such an approach is in the same spirit as Kariv and Hakimi's in [65] with the only difference being that they enumerate on all possible collections of p subnetworks of N while we propose to enumerate on all possible collections of p disjoint subtrees. It is evident that if the conjecture is true, then the disjoint subtrees (x^),...,T (x^) can be connected to one another by adding edges appropriately without creating cycles. With the addition of the new edges the forest becomes a spanning tree of N, so that the answer to Q1 would be in the affirmative (provided that the conjecture holds). The second question still remains unresolved as we assumed a knowledge of a p-center X to construct such a forest (assuming that we can). -82- in U. In total, any feasible X which D chooses will satisfy |x| > |u|, which is what the W.D.T. asserts. By virtue of the S.D.T., if U is A's optimal choice, D can choose exactly || response forces positioned at, say X, with |x| = |u| and still respond to an attack to any vertex in U (as well as in V-U) without incurring a loss exceeding r. If A threatens more than q(r) = |u| vertices, say, a subset U of V, then |UÂ¡ > q(r) implies g(U) < r (infeasibility). Thus, D would not be forced into allocating a single response for every member of U. In fact, even if A threatens every vertex in V, then D still needs ex actly q(r) = q(r) = || response forces to respond to the threat feasibly. Thus, if each threat is an "expense" for A, he need threaten no more than q(r) vertices. On the other hand, D adopts an optimal strategy against A's best threat by minimizing the number of response forces with respect to V. Continuing our consideration of the covering problem, we now re verse the usual procedure, and view the p-center problem as a device for solving the covering problem for all values of r for which the covering problem is feasible, that is, for a < r. The following lemma is the key to using the p-center problem to solve the covering problem. Define r = < for convenience. o Lemma 2.6.1. Let p e J. If r < r ,, then : r p p-1 q(r) = p for r < r < r . P P-1 Proof. We first note r < r < ... S r. < r_. Also, clearlv. n n-1 10 J q(r ) < p for p e J. Now for r^ Â£ r since q is non-increasing we have 1 > q(r^) > q(r) > 1, establishing the claim if p = 1. Consider the case p e {2,...,n}. From r < r < r we have p > q(r ) > q(r) > p p i p q(r ]_). Suppose q(r) = s, with s < p, implying s < p-1. Let X, |