ABSTRACT. KAZEMI TUTUNCHI, GOLBARG. A Bi-criteria Analysis in Location Problems: Similarity and Diversity. (Under the direction of Dr.

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1 ABSTRACT KAZEMI TUTUNCHI, GOLBARG. A Bi-criteria Analysis in Location Problems: Similarity and Diversity. (Under the direction of Dr.Yahya Fathi) In this dissertation we consider two bi-criteria discrete optimization problems, namely the Bi-criteria p-median p-dispersion problem (BpMD) and the Bi-criteria p-center p-dispersion problem (BpCD). Both problems arise in various applications in location theory. For the first problem, an iterative algorithm is already available in the open literature to obtain its non-dominated frontier. This algorithm is referred to as the Incremental Algorithm, and at each iteration it constructs and solves an integer programming (IP) model to obtain one non-dominated (or weakly non-dominated) point for this bi-criteria problem. Solving an IP model is usually a computation-intensive task if the size of the problem is relatively large. Since we need to solve one instance of this IP model at each iteration of the algorithm, it follows that the overall computational requirements of the Incremental Algorithm could be excessive for larger instances of the problem, especially if the number of non-dominated points in a given instance is also large. In this context we propose two strategies to reduce these computational requirements. First, we propose a Lagrangian Heuristic procedure to obtain near optimal solutions for the IP model at each iteration. We demonstrate the effectiveness of this heuristic procedure through a comprehensive computational experiment. Second, we propose to obtain only a representative subset of the non-dominated points for any given instance of the problem, hence reducing the number of iterations and consequently the number of IP models that we need to solve. To this end we propose a new measure (t-value) to evaluate the effectiveness of a representative subset of non-dominated points for a bi-criteria optimization problem (in particular for the problem BpMD). Subsequently we propose an iterative algorithm to obtain an effective representative subset, and carry out a computational experiment to assess the performance of this algorithm. To the best our knowledge, the second problem that we consider in this dissertation, namely the Bi-criteria p-center p-dispersion problem (BpCD), is not previously discussed in the open literature. In this context we first introduce and define the problem and review the related literature. Subsequently we propose an effective algorithm for obtaining the non-dominated frontier for this bi-criteria optimization problem. Similar to the Incremental Algorithm for BpMD that we mentioned above, the algorithm that we propose for this problem is also iterative, and at each iteration we construct and solve a related IP model in order to obtain one point of the corresponding non-dominated frontier. We carry out a comprehensive computational experiment to assess the effectiveness of our proposed algorithm, and present our findings. We also study the structure of the IP model that we construct and solve at each iteration of this

2 algorithm, and propose a new family of valid inequalities for this IP model. Through a limited computational experiment we demonstrate that using these valid inequalities in the context of a branch-and-cut algorithm could be quite effective for solving larger instances of this IP model.

3 A Bi-criteria Analysis in Location Problems: Similarity and Diversity by Golbarg Kazemi Tutunchi A dissertation submitted to the Graduate Faculty of North Carolina State University in partial fulfillment of the requirements for the Degree of Doctor of Philosophy Industrial Engineering Raleigh, North Carolina 201 APPROVED BY: Dr. Reha Uzsoy Dr. Maria Mayorga Dr. Osman Ozaltin Dr.Yahya Fathi Chair of Advisory Committee

4 DEDICATION To the best friend of 12 years Nima To my supportive parents Mahin and Mortaza To my lovely sister Nazak ii

5 BIOGRAPHY Golbarg Kazemi Tutunchi is a Ph.D. student in the Industrial and Systems Engineering Graduate Program at North Carolina State University. She was born in Oroumieh, Iran. She has received her Bachelor and Masters of Science degrees in Industrial Engineering from Tehran Polytechnic (Amirkabir University) in May 200 and May 2010, respectively. To further her education, she came to the United States in August She received a Master of Science degree in Engineering Management from Missouri University of Science and Technology in August Golbarg joined the Industrial and Systems Engineering program at North Carolina State University in August Currently, she is working as an operations research specialist at SAS Institute in Cary, NC. iii

6 ACKNOWLEDGEMENTS I would like to express the deepest appreciation to my advisor Dr. Yahya Fathi, who has the attitude and the substance of a genius: he continually and convincingly conveyed a spirit of adventure in regard to research and scholarship, and an excitement in regard to teaching. Without his guidance and persistent help this dissertation would not have been possible. I am deeply grateful to you for giving the opportunity to working with you, sharing your vast knowledge with me and helping me at each step of my life. I would like to thank my committee members, Dr. Reha Uzsoy, Dr. Maria Mayorga, Dr. Osman Ozaltin and Dr. Ranji Ranjithan for their willingness to serve on my committee, for evaluating my dissertation, and for offering constructive criticism to help me improve this work. I also would like to extend my thanks to all the faculties and staffs of the ISE department at North Carolina State University who contributed to my education. Also a big thanks to all my friends for making the Ph.D. life more fun and interesting. I would like to acknowledge the SAS Institute for supporting my Ph.D. education and giving me the opportunity to learn their software used in this thesis, and also providing the very valuable industrial experience. Also I would like to thank my manger Dr. Ivan Oliveira for the chance he gave me to be part of his great team. I thank him for being a great mentor during this whole process. I feel very fortunate to working for him. I would like to recognize and thank to all my colleagues more specifically Dr. Baris Kacar, Dr. Leo Lopes, Dr. Natalia Summerville and Dr. Bahadir Aral for all their supports. I would like to extend my gratitude to my former boss, Dr. Jeff Day for always having my back and helping me at each step of my life. He taught me to work with rigor but with enjoyment and pleasure. I also very much appreciate all the help and great advices I ve got from my former colleague Dr. Ming Zhao, and all his brilliant points has impacted my career and knowledge a lot. Concluding, I thank my mother Mahin and my father Mortaza for their unconditional love and the infinite support they gave me in every endeavor on which I have embarked. I thank my sister Nazak who extended her love and heartfelt support to me from miles away. The last but not the least, I would like to thank my love and best friend of past 12 years Nima for all his love, encouragement and support he always gives to me, and always believing in me. iv

7 TABLE OF CONTENTS List of Tables viii List of Figures x Chapter 1 Introduction Statement of the problem Applications Contributions Chapter 2 Background The p-median problem Brief overview of solution methods for the p-median problem The integer programming formulation The p-dispersion problem An integer programming model for the p-dispersion problem Solving the p-dispersion problem The p-center problem An integer programming model for the p-center problem Solving p-center problem Multi-objective optimization Multi-objective optimization terminology Strategies for obtaining all the non-dominated points Chapter Bi-criteria p-median p-dispersion problem The bi-criteria p-median p-dispersion problem The ε-constraint approach for computing the non-dominated points of the BpMD problem The Incremental Algorithm A Computational Experiment Data set A comparative study Solving larger instances Approximation strategies Chapter Solving the integer programming model ILP -l A Lagrangian heuristic approach for BpMD problem Lagrangian relaxation Subgradient search Primal heuristic algorithm The computational experiments v

8 Chapter Evaluating a representative subset of non-dominated points A measure of effectiveness Numeric examples Other measures of effectiveness Chapter Methods to obtain a representative subset of non-dominated points 2.1 Background Algorithms The basic procedure The Interval Search Procedure Parameter values A Computational Experiment Chapter Bi-criteria p-center p-dispersion problem Defining the bi-criteria p-center p-dispersion problem ε-constraint algorithm for solving the problem BpCD Incremental Algorithm for solving the problem BpCD A Computational Experiment Chapter An Alternative model for the problem BpCD Set Covering based reformulation Daskin s binary search for the vertex p-center problem Binary Search for solving the IP model ILP -l Computational Results The first experiment The second experiment The third experiment Chapter 9 Covering-Packing Valid Inequalities The Covering-Packing Valid Inequalities Definitions and Properties The Covering-Packing Valid Inequality is a cut A strategy for adding CP-VI constraints Adding the cuts in step Chapter 10 Computational Impact of Covering-Packing Inequalities The first experiment Effect of parameter p Effect of parameters c and l The second experiment Chapter 11 Conclusions and Future Research Future research References vi

9 Appendix Appendix vii

10 LIST OF TABLES Table.1 Uniform distributions used to generate the characeristic vectors Table.2 Execution time (seconds) of the two algorithms Table. Execution time of the Incremental Algorithm for larger instances Table.1 Execution time for the exact method Table.2 Execution time for the heuristic method Table. Value of the gap at termination for the heuristic method Table.1 All non-domminated points for the numeric example obtained by Incremental Algorithm Table.2 t-values for the three subsets in Example Table. t-values for the three subsets in Example Table. t-values for the three subsets in Example Table. Values of measures for subsets Ψ 1 and Ψ 2 in Example Table. Values of measures for subsets Ψ and Ψ in Example Table.1 Finding appropriate values for the parameter α in one-α-strategy Table.2 Finding appropriate values for the parameters (α 1,α 2 ) in two-α-strategy 1 Table. The basic procedure results for ν = Table. The basic procedure results for ν = Table. The results of the Interval Search Procedure using one-α-strategy with termination criterion of pre-specified t-value from Table Table. The results of the Interval Search Procedure using two-α-strategy with termination criterion of pre-specified t-value from Table Table. The results of the Interval Search Procedure using one-α-strategy with termination criterion of pre-specified t-value from Table Table. The results of the Interval Search Procedure using two-α-strategy with termination criterion of pre-specified t-value from Table Table.9 The results of the Interval Search Procedure using one-α-strategy with termination criterion of maximum number of IPs solved from Table.. 0 Table.10 The results of the Interval Search Procedure using two-α-strategy with termination criterion of maximum number of IPs solved from Table.. 1 Table.11 The results of the Interval Search Procedure using one-α-strategy with termination criterion of pre-specified number of IP models solved from Table Table.12 The results of the Interval Search Procedure using two-α-strategy with termination criterion of pre-specified number of IP models solved from Table Table.1 Execution time (seconds) of the two algorithms Table.2 Comparison of the two algorithms for the instances from GEO library with n= viii

11 Table.1 Table.2 Table. Incremental Algorithm for BpCD: comparing the two methods for solving ILP -l Comparison of the two methods for solving the IP model ILP -l for the instances from GEO library with n= The impact of parameters c and l on model SC-l for the instance with the parameters of n = 0 and p = and two extreme non-dominated points ((c cent,l cent ) and (c max, l max )) as (12.,12.) and (19.,1.) respectively Table 9.1 Comparison results for different approaches of applying cuts Table 10.1 The parameters of instances used for studying the impact of parameter p on the added CP -VIs for a data set with n = Table 10.2 The impact of parameter p on the added CP -VIs for a data set with n = Table 10. The impact of parameters c and l on the added CP -VIs for the instance with n = 00, p = 20, and the two extreme non-dominated points (c cent, l cent ) and (c max, l max ) as (1.1, 1.9) and (200., 21.1), respectively Table 10. The parameters of instances used for studying the impact of added CP - VIs on large instances Table 10. The impact of added CP -VIs on solving model SC-l for medium and large instances ix

12 LIST OF FIGURES Figure.1 Non-dominated frontier formed by 9 non-dominated points Figure.2 Non-dominated points and envelopes of subset Ψ 1 from Example 1... Figure. Non-dominated points and envelopes of subset Ψ 2 from Example 1... Figure. Non-dominated points and envelopes of subset Ψ from Example 1... Figure. Non-dominated points and envelopes of subset Ψ 1 from Example 2... Figure. Non-dominated points and envelopes of subset Ψ 2 from Example 2... Figure. Non-dominated points and envelopes of subset Ψ from Example 2... Figure. Non-dominated points and envelopes of subset Ψ 1 from Example... Figure.9 Non-dominated points and envelopes of subset Ψ 2 from Example... Figure.10 Non-dominated points and envelopes of subset Ψ from Example... Figure.11 Non-dominated points of subset Ψ 1 from Example Figure.12 Non-dominated points of subset Ψ 2 from Example Figure.1 Non-dominated points of subset Ψ from Example Figure.1 Non-dominated points of subset Ψ from Example Figure.1 The upper envelope and lower envelope formed by the efficient or weakly efficient solutions y 1, y 2 and y Figure.2 Outcome Figure. Outcome Figure. Outcome - case Figure. Outcome - case Figure 9.1 Nodes 1 and Covering-Packing constraint x

13 Chapter 1 Introduction In this research, we consider two bi-criteria location problems, namely the Bi-criteria p-median p-dispersion problem (BpMD) and the Bi-criteria p-center p-dispersion problem (BpCD). Each of the related single objective problems, namely the p-median problem, the p-dispersion problem, and the p-center problem is thoroughly investigated in the open literature and effective algorithms for solving these problems are proposed, but the study of these problems in the bi-criteria setting is relatively new. To the best of our knowledge the only article that addresses this subject is Sayyady [] where the BpMD is discussed. In this research, we extend the research of Sayyady [] and develop new heuristic algorithms for solving relatively larger instances of the bi-criteria p-median p-dispersion problem. We also carry out a computational experiment to evaluate the effectiveness of these algorithms. Later in this research, we propose a new integer programming model and algorithms for solving the bi-criteria p-center p-dispersion problem. 1.1 Statement of the problem There are two sets of problems in location theory that have been studied by many researchers in recent years. We refer to these problems as similarity problems and diversity problems, respectively. In both problems we have a collection N of n locations with a given symmetric measure of distance d ij between each pair of locations i and j, and a given integer p < n. In the similarity problems the idea is to partition the collection N into p groups so as to minimize a measure of distance among the members of each group. Examples of such problems are the p-means problem, the p-median problem, the p-mode problem, and the p-center problem. In the diversity problems, on the other hand, the idea is to select p locations from the collection N so as to maximize an appropriate measure of distance among the p selected locations. The maximin diversity problem (or the p-dispersion problem) and the maxisum diversity problem 1

14 are examples of such problems. Each of these problems arise from one or more applications, and various algorithmic strategies are discussed in the open literature to solve each problem. There are some applications however, where both criteria need to be considered simultaneously. Such applications naturally lend themselves to a bi-criteria analysis. In this dissertation, we propose to undertake the task of developing and testing appropriate algorithms for solving such bi-criteria problems. In particular, we focus on two such bi-criteria problems. In the first problem, the objective functions correspond to those of the p-median problem and the p- dispersion problem, respectively, and in the second problem, we consider the objective functions of the p-center problem and the p-dispersion problem, respectively, in a similar setting. In the remaining sections of this chapter, we discuss various applications and settings where such a bi-criteria analysis would be appropriate. In chapter 2 we briefly introduce the p-median problem, the p-dispersion problem and the p-center problem, and give a brief review of the relevant literature of each problem. In this chapter, we also introduce the appropriate terminologies for multi-objective analysis. In chapter we focus on the bi-criteria p-median p-dispersion problem and discuss appropriate strategies for solving this problem. First, we briefly review the work of Sayyady [] in this problem, i.e., we present an integer programming model for this bi-criteria problem and discuss an algorithm for finding its non-dominated frontier. This model and algorithm can be used to solve small to moderate size instances of the problem. Subsequently, we introduce two strategies to solve larger instances of this problem, namely a heuristic approach to solve the IP model, and an appropriate algorithm to obtain a representative subset of the non-dominated points. In chapter we discuss a Lagrangian heuristic for solving the IP model and present the results of the computational experiment. In chapter we present appropriate measures to evaluate the effectiveness of a representative subset of non-dominated points. In chapter we propose an effective methodology to obtain a representative subset of the non-dominated points. In chapter, we introduce the bi-criteria p-center p-dispersion problem, construct a bi-criteria mathematical model for this problem, and discuss conventional strategies for solving this problem. Later, in chapter we propose an alternative IP model for the bi-criteria p-center p-dispersion problem, and present an appropriate strategy to solve this model. In this chapter we also present the results of a computational experiment that we carried out to evaluate the effectiveness of our proposed model and algorithmic strategy. In chapter 9, we study the structure of the IP model proposed in chapter, and propose a new family of valid inequalities for this model. We then employ these valid inequalities to devise a branch-and-cut algorithm for solving this IP model. Finally, in chapter 10, we present the results of a computational experiment where we demonstrate the effectiveness of the proposed branch-and-cut algorithm. 2

15 1.2 Applications There are various applications in which minimizing total dissimilarity and maximizing minimum dispersion should be considered simultaneously. One application discussed in [] is finding optimal locations for traffic sensors in a highway network. As discussed in [], in this application we wish to pick p locations out of N possible road segments to install the sensors to measure various traffic characteristics. The ideal case would be to install one sensor at each road segment, but the corresponding cost would be prohibitively large. Therefore, the number of sensors are limited to p. We wish to pick these p locations such that the given traffic vector for each of the remaining locations is similar to (close to) the traffic vector for at least one of the selected locations. This problem can be formulated as a p-median problem where we are interested in partitioning all the collection of road segments to p groups (group medians) to minimize the total dissimilarity between each location and its closest selected group medians. Alternatively, we can minimize the maximum dissimilarity between each location and its closest selected location. In this case we can model this problem as a p-center problem. On the other hand we wish to select the p locations for the sensors so as to achieve maximum diversity in the corresponding traffic vectors. Now, the problem can be formulated as the p-dispersion problem. Considering each objective separately would lead to the selection of two distinct set of locations. Therefore, this problem could be analyzed as a bi-criteria problem. One other similar application is finding optimal locations for radio transceivers which are serving cellular phones. Assume that we have N candidate locations for locating the radio transceivers. Each of these locations has a traffic characteristic and we have to pick p of these locations. The placement of these transceivers in close proximity to each other produces cosite interference between them. When the interference between the transceivers is high, the provided radio communications are totally disrupted. Therefore, we want to locate them as dispersed as possible in order to minimize interference between them. We are also interested in minimizing the total cost of this communication network. Each of the remaining N p locations should be connected to the closest selected p locations. The p locations should be selected such that the total distance between each location and its closest selected locations is minimized. This problem also lends itself to a bi-criteria analysis. One of the other interesting applications of bi-criteria location problems is finding the location of the business franchise in an urban area. Here the set N would be all possible locations (customer centers) that we can locate a franchise in. We would like to select p locations out of all possible N locations to open the franchises. Potential customers from each of the remaining N p locations (customer centers) would then use the nearest open franchise to that location. Now, the question is that which p locations could serve our objectives the best. On one hand, if the franchises are far from each other the total sales would be higher, since they are not

16 sharing the same customers. This problem could be formulated as p-dispersion problem where we want to select p locations such that the minimum distance between them is maximized. But on the other hand, we would like to minimize the distance between the customer centers and their respective nearest open franchise store. To this end, we can model this problem as a p-center problem where we seek to partition all the possible locations for franchise to p groups, with a center location for each group so as to minimize the maximum distance between the customer locations and their respective group centers. Other similar applications could be locating schools, banks and emergency medical services in the cities, etc. In any of these applications, simultaneous achievement of both objectives is not possible. Therefore, it would be desirable to study the problem of selecting the p locations in a bi-criteria optimization context. If the application calls for minimizing total dissimilarity between the locations and their respective selected locations (group centers) while maximizing the diversity among the selected locations themselves, we refer to the problem as the bi-criteria p-median p- dispersion problem. Alternatively if the application calls for minimizing maximum dissimilarity between the locations and their respective group centers while maximizing the diversity among the selected locations, we refer to this problem as the bi-criteria p-center p-dispersion problem. 1. Contributions In this dissertation we propose effective models and algorithms for solving the bi-criteria location problems that we discussed above, more specifically the bi-criteria p-median p-dispersion problem and also the bi-criteria p-center p-dispersion problem. As mentioned earlier the bicriteria p-median p-dispersion problem was studied by Sayyady in []. We propose several solution procedures to solve the large size instances of this problem. To the best of our knowledge the bi-criteria p-center p-dispersion problem has not been previously discussed in the open literature. Therefore, in this dissertation we define the problem, propose a mathematical model for the problem and offer various algorithmic strategies for solving the problem. Our specific contributions are as follows: 1. We propose a Lagrangian heuristic approach for solving large instances of the bi-criteria p-median p-dispersion problem, and demonstrate the efficiency of this algorithm through a numerical experiment. 2. For a given instance of the bi-criteria p-median p-dispersion problem, we propose an appropriate measure to evaluate the effectiveness of a given representative subset of nondominated points as an approximation (representation) of the entire set of non-dominated points.

17 . We propose an iterative algorithm to obtain a subset of non-dominated points as a representation of the entire non-dominated frontier for the bi-criteria p-median p-dispersion problem.. We study the bi-criteria p-center p-dispersion problem, and propose an integer programming model to formulate this problem. We solve this problem with different strategies and present the effectiveness of these strategies through a numerical experiment.. We propose an alternative integer programming model and a solution strategy to solve the bi-criteria p-center p-dispersion problem, using its relationship with the set covering problem.. We further study the structure of the IP model proposed for the bi-criteria p-center p- dispersion problem, and propose appropriate valid inequalities for this model. Using these valid inequalities we propose a branch-and-cut algorithm for solving this problem.. We carry out a computational experiment to demonstrate the potential impact of the proposed valid inequalities on solving the IP model of the bi-criteria p-center p-dispersion problem.

18 Chapter 2 Background In the first three sections of this chapter, we discuss the p-median problem, the p-dispersion problem, and the p-center problem, respectively, and give a brief review of the corresponding literature. In the last section, we present a brief review of appropriate terminologies and solution methods of multi-objective optimization. 2.1 The p-median problem The p-median problem is also known as minisum location-allocation problem in the literature. Given a collection N of n locations with symmetric measure of distance d ij between each pair of locations the p-median problem seeks to identify p of these locations so as to minimizes the distance between the locations in N and their respective nearest selected locations as measured by d ij (ties are broken arbitrarily). Any instance of the p-median problem is denoted by a triplet (N, D, p), and any subset γ of size p from the collection N defines a feasible solution of the p-median problem. Associated with each feasible solution γ of the p-median problem, we define the objective function of the p-median problem as f 1 (γ) = iɛn min j γ d ij which represents the total dissimilarity between the locations in collection N and their respective nearest location in set γ. Therefore, for any given instance (N, D, p) of the p-median problem, we seek a subset γ of size p from the collection N such that f 1 (γ) is minimized. In the context of the p-median problem, the feasible solution γ 1 is said to be better than feasible solution γ 2 if f 1 (γ 1 ) < f 1 (γ 2 ). The optimal solution of the p-median problem is denoted as γmed. The p-median problem is extensively studied in the literature. Many applications of the p-median problem are discussed in [], [2], [], [], [21], [] and [0], and various methods are proposed for solving the problem.

19 2.1.1 Brief overview of solution methods for the p-median problem Over the past few decades various solution methods are proposed for solving the p-median problem, and several comprehensive survey articles on the formulation and solution methods of the p-median problem are written. In Handler and Mirchandani (199) categorized the solution methods for solving the p- median problem into categories: (1) enumeration, (2) graph theoretic, () heuristic, () primal-based LP-methods, () dual-based LP methods. They also provided a survey of these solution methods [22]. Since that time several other survey papers on these solution methods have been written, some of these papers are Tansel et. al (19) [], Mirchandani (1990) [2], Labbe et. al (199) [2], Daskin (199) [12]. In (200), Reese provided a more detailed categorization of these solution methods compared to [22]. These categories are IP formulation and reductions, LP relaxation, approximation, heuristic, metaheuristic, graph theoretic, enumeration and surrogate methods [1]. Metaheuristic approaches like variable neighborhood search, heuristic concentration, genetic algorithms, GRASP (greedy randomized adaptive search procedure) metaheuristic, scatter search, tabu search, simulated annealing, neural networks and combination of these methods are used for solving the p-median problem recently. Meladenovic et al. provided an extensive survey on the metaheuristic methods applied to the p-median problem [1] The integer programming formulation In this subsection, we present the mathematical model from [] for the p-median problem. Later we employ a similar constraint structure to construct a mathematical programming model for the bi-criteria problem that we study in this dissertation. In this model for each j N we define a decision variable y j = 1 if location j is selected (i.e., if location j is in the set γ) and y j = 0 otherwise; and for each pair i, j N we define x ij = 1 if location i is assigned to location j, and x ij = 0 otherwise. We can now write the corresponding integer linear programming model that we call ILP 1 as follows.

20 Minimize d ij x ij (ILP 1) (i,j) N subject to x ij = 1 i N (2.1) j N x ij y j (i, j) N (2.2) y j = p (2.) j N x ij {0, 1} (i, j) N (2.) y j {0, 1} j N (2.) The objective in this model is to minimize the total dissimilarity between all locations in the set N and their corresponding group medians. The first constraint implies that each location should be assigned to one group median. The second constraint allows assignment only to locations that are already selected as group medians. The third constraint implies that there are only p locations selected as group medians. 2.2 The p-dispersion problem The p-dispersion problem is also known as the maximum diversity problem in the literature. Given a collection N of n locations with a symmetric measure of distance d ij between each pair of locations, the p-dispersion problem seeks to identify p of these locations such that the minimum distance between all pairs of selected locations is maximized. Any subset γ of size p from the collection N defines the feasible solution of the p-dispersion problem. For each feasible solution (γ) associated with the p-dispersion problem, the objective function is defined as f 2 (γ) = min i,j γ,i j d ij. Therefore, the p-dispersion problem seeks to identify a subset γ of size p from the collection N such that f 2 (γ) is maximized. In the context of the p-dispersion problem, a feasible solution γ 1 is said to be better than an other feasible solution γ 2 if f 2 (γ 1 ) > f 2 (γ 2 ). We denote the optimal solution of the p-dispersion problem as γdisp, and its corresponding value of the objective function is denoted by lmax. Various applications of the p-dispersion problem are discussed in the open literature. The interested reader is referred to [], [10], [1] and [2] for the detailed information on the applications of the p-dispersion problem.

21 2.2.1 An integer programming model for the p-dispersion problem In this subsection, we present an integer programming model proposed in [] for the p- dispersion problem. We use the same notation employed in subsection y j = model. 1 if location j is selected 0 otherwise The p-dispersion problem can now be written as the following nonlinear integer programming subject to Maximize {minimum i,j N d ij y i y j } (disp) y j = p (2.) j N y j {0, 1} j N (2.) The objective in this model is to maximize the minimum pairwise dissimilarity among the selected locations. The constraint 2. restricts the number of selected locations to p. The last constraint implies that the decision variables should be binary Solving the p-dispersion problem Various attempts have been made to linearize the disp model, and different solution procedures are proposed for solving the p-dispersion problem. In the next two subsubsections we briefly present these models and algorithms. We first present the exact algorithms, and then briefly discuss the heuristic methods. Exact algorithms In an attempt to solve the model disp, in (19) Kuby [2] proposed a linear mixed integer formulation for the p-dispersion problem. For this purpose, the nonlinear objective is replaced with a new variable r, and a new group of constraints is added to the model. In addition to, the notations used in model disp, a new parameter M is also used in this model. This parameter represents a large number, and it could be set equal to the largest distance d ij between all pairs of locations in the set N, i.e. M = max i,j N {d ij }. We refer this linear model as disp IP throughout this document, and it is presented below: 9

22 Maximize r (disp IP ) subject to r M(2 y i y j ) + d ij (i, j) N, i < j (2.) y j = p (2.9) j N y j {0, 1} j N (2.10) The objective function of this model is to maximize r which is the minimum pairwise dissimilarity among the selected collections. Constraint 2. places an upper bound on r equal to d ij only if locations i and j are selected. If either or both locations i and j are not selected the corresponding upper bound would be larger than the largest d ij, and this constraint becomes redundant. The rest of the constraints are the same as the disp model. The integer programming model disp IP can be solved to optimality for small to medium-size instances of the p-dispersion problem in a reasonable amount of time using branch and bound approach. For larger instances, however either the execution time is prohibitively large or the memory requirements of this approach is excessive (or both). Various attempts have been made afterwards to solve larger instances of the p-dispersion problem. In (1990) Erkut proposed two branch and bound procedures for solving the p- dispersion problem. The proposed branch and bound procedures can only solve small size instances of the problem [1]. Later in (199) Kuo et al. proposed a new linearization of the disp model which was flexible enough to accept other side constraints [2]. In (200) Pisinger solved the p-dispersion problem by transforming it to a number of cliques [9]. In (2012) Sayyady and Fathi [] proposed an efficient binary search procedure for solving the p-dispersion problem. This procedure can solve larger instances of the problem as compared with the previous methods. In this algorithm they use an integer programming model that depends on a parameter l, hence we denote it as ILP 2-l. Here is a brief explanation of the model; subject to ν(l) = maximize y j j N (ILP 2-l) y j + y i 1 (i, j) N such that d ij < l (2.11) y j {0, 1} j N (2.12) 10

23 As shown above we denote the optimal value of this model by ν(l). If we have l 1 and l 2 > l 1 such that ν(l 1 ) p and ν(l 2 ) < p, it follows that l 1 l max < l 2. Using this property, Sayyady and Fathi [] propose to divide the interval between the smallest d ij and the largest d ij into 2 q smaller intervals (q is sufficiently large so that the length of each resulting interval is smaller than the smallest difference between any two distinct d ij values). They would then carry out a binary search among these intervals to identify the interval [l 1, l 2 ) that satisfies the above property, i.e., contains the optimal value l max. The value of l max would then be equal to the only d ij value that is in this interval. Throughout this dissertation in all occasions that we need to solve numeric instances of the p-dispersion problem, we employ this integer programming model and the corresponding binary search procedure as proposed in []. Inexact algorithms for solving the p-dispersion problem Various heuristic approaches are proposed to solve larger instances of the p-dispersion problem. In [1] Erkut proposed a two-stage heuristic method for solving the larger instances of the p- dispersion problem. Unlike the proposed branch and bound algorithms by Erkut, the heuristic procedure could find optimal or near optimal solutions very fast. In [19] Ghosh provided a greedy randomized heuristic method for solving the p-dispersion problem. Later in [1] Agca et al. proposed a new linearization of the p-dispersion problem, and solved the problem with a heuristic approach. In this heuristic, the lower bound is obtained by an effective Lagrangian relaxation formulation, and a good heuristic is applied to find the upper bound. Later in [] Corce et al. reduced the p-dispersion problem to the maximum clique problem. They applied an efficient heuristic approach from the literature for solving the maximum clique problem. One of the very successful heuristics was proposed by Resende et al. []. The authors proposed a hybrid heuristic of GRASP and path relinking to solve the maximin diversity problem. Their computational results indicate that this hybrid method outperforms previously proposed metaheuristics for solving this problem such as simulated annealing and tabu search. 2. The p-center problem In this section we specifically focus on a particular type of the p-center problem that is known as the vertex p-center problem. We briefly explain this problem in this section. The p-center problem is also known as minimax problem in the literature. Given a collection N of n locations with symmetric measure of distance d ij between each pair of locations the vertex p-center problem seeks to identify p of these locations so as to minimize the maximum distance between the locations in N and their respective nearest selected locations as measured 11

24 by d ij (ties are broken arbitrarily). Any instance of the vertex p-center problem is denoted by a triplet (N, D, p), and any subset γ of size p from the collection N defines a feasible solution of the vertex p-center problem. Associated with each feasible solution γ of the vertex p-center problem, we define the objective function of the vertex p-center problem as f 1 (γ) = max i N minimum j γ d ij which represents the maximum dissimilarity between the locations in collection N and their respective nearest location in set γ. For ease of reference, we refer to f 1 (γ) as max-dissimilarity for the set γ. For any given instance (N, D, p) of the vertex p-center problem, we seek a subset γ of size p from the collection N with minimum max-dissimilarity. In the context of the vertex p-center problem, the feasible solution γ 1 is said to be better than feasible solution γ 2 if f 1 (γ 1) < f 1 (γ 2). The optimal solution of the vertex p-center problem is denoted as γ cent. Various integer programming models and solution methods have been proposed for modeling the vertex p-center problem in the open literature. In the following subsections, we present some of these IP models and solution methods An integer programming model for the p-center problem The problem can now be stated as the following integer programming model; subject to Minimize maximum i,j N d ij x ij (V P IP ) x ij = 1 i N (2.1) j N x ij y j (i, j) N (2.1) y j = p (2.1) j N x ij {0, 1} (i, j) N (2.1) y j {0, 1} j N (2.1) All of the decision variables used in this model are similar to those defined earlier in disp IP and ILP 1. x ij = 1 if location i is assigned to the selected location j i, j N x ij = 0 otherwise 12

25 y j = 1 if location j is selected j N y j = 0 otherwise Daskin [12] proposed a linear integer programming model for the model V P IP. Throughout this dissertation, we refer to this model as IP. Let z=maximum distance between any location in collection N and its nearest selected location The model V P IP can be written as the following integer linear programming (ILP) model. subject to Minimize z (IP ) d ij x ij z i N (2.1) j N x ij = 1 i N (2.19) j N x ij y j (i, j) N (2.20) y j = p (2.21) j N x ij {0, 1} (i, j) N (2.22) y j {0, 1} j N (2.2) Constraints group 2.1 indicate that z must be greater than the distance between any location i in collection N and its nearest selected location j. Constraints group 2.19 state that each location in collection N should be assigned to one of the selected locations. Third constraint allows only assignment to the selected locations in collection N. Constraint 2.21 limits the number of selected locations to p Solving p-center problem For the fixed values of p in the problem BpCD, we can enumerate each possible candidate set of selected locations in O(n p ) time. Even for small values of n and p, the enumeration method is not realistic [12]. Based on the computational results reported by Ilhan and Pinar in [2], solving even small instances of the p-center problem using the IP model IP, is very time consuming. They show that for n = 0 and p = 10, it takes more than hours to solve model IP by CPLEX.0 on a SUN Sparc 10. So various other solution methods are used to solve the p-center problem. In [12] Daskin proposes a binary search (bisection) algorithm over an interval between an upper bound and a lower bound on the optimal value of the vertex p-center problem, and solves a sequence of set covering problems for the selected value of the coverage 1

26 distance. Later in [11] Daskin makes some improvements to his previously proposed binary search algorithm for solving vertex p-center problem. He improves the performance of his 199 binary search by solving a maximal covering problem at each iteration of the binary search algorithm. This formulation also allows us to solve the weighted p-center problem. In this article, Daskin compares the regression results for CPU time versus number of nodes (locations) and number of selected locations for the p-center problem. For the p-center problem, the execution time grows roughly with square of number of nodes, and goes down with square root of number of selected facilities. Ilhan and Pinar [2] propose a two phase algorithm for solving the vertex p-center problem. This two phase algorithm could be considered as extension of Daskin 199 binary search algorithm. In the first phase a sequence of LP relaxation of the set covering problem with the constraint 2.21 added is solved as feasibility problems to obtain a lower bound for the vertex p-center problem. In the next phase, the IP formulation of the feasibility problem is solved by systematically changing the coverage value starting from the phase 1 lower bound. Ilhan and Pinar compare their algorithm with solving model IP through a comprehensive computational analysis, and their algorithm is more efficient than model IP. Also the proposed two phase algorithm produce tighter lower bounds compared to the LP relaxation of the IP model IP. Al-khedhiari and Salhi [2] propose some improvements to the Daskin s algorithm and the Ilhan and Pinar algorithm. The improvements to Daskin 199 algorithm consists of tightening the upper and lower bounds and applying some improvements to Daskin s binary search. They also reduce the number of iterations required for Ilhan and Pinar s algorithm by setting new bounds and modifying the coverage distance at each iteration of the algorithm. Their improvements reduces the CPU time compared to the Ilhan and Pinar algorithm, and Daskin s binary search CPU time. Elloumi et al. [] propose a new formulation for the vertex p-center problem. They state that the LP relaxation of this model provides a tighter lower bound compared to the LP relaxation of IP model IP. They propose a binary search algorithm and solve a group of set covering problems at each iteration of their algorithm. Their computational results show that their algorithm outperforms Daskin [11] binary search with respect to both execution time, and capability in solving larger instances of the problem. Later Calik and Tansel [20] propose a reformulation of the IP model proposed in []. They state that the LP relaxation of their IP model provides a tighter lower bound compared to the LP relaxation of the IP model proposed in []. 1

27 2. Multi-objective optimization Due to the multiple and sometimes conflicting objective functions of most real-world decision making problems, the area of multi-criteria decision making has gained a lot of interest during the past forty years [1]. The interested reader could refer to various survey papers written in the area of multi-objective decision making such as [2] and []. Since many real world problems have combinatorial nature, multi-objective combinatorial optimization problems has gained lots of attention over the time. In 2000 Erghott [1] wrote a comprehensive survey paper and annotated bibliography on multi-objective combinatorial optimization. In this dissertation, we study the bi-criteria location problems which can be classified as multi-objective combinatorial optimization problems. Therefore, in the following subsections of this chapter, first we present the terminologies of the multi-objective problems. In the subsequent subsections we briefly explain some of the strategies proposed for solving multiobjective optimization problems Multi-objective optimization terminology Since in this dissertation we deal with bi-criteria location problems, all the notations and definitions that we state here are in the context of a bi-criteria optimization problem. Let s assume that we have two conflicting objective functions; f 1 and f 2. We would like to minimize both objective functions, but simultaneous achievement of both of them is not possible. Therefore, we model this problem as a bi-criteria optimization problem (BCOP). We denote the feasible region of the BCOP by X, and each feasible solution of this problem is denoted by Υ. The problem can now be stated as following. min (f 1(Υ), f 2 (Υ)) (BCOP ) Υ X Definition 1. (Domination) For a given bi-criteria optimization problem (BCOP), a feasible solution Υ 1 dominates a feasible solution Υ 2 if Υ 1 is at least as good as Υ 2 with respect to one objective function and it is strictly better than Υ 2 with respect to the other objective. Definition 2. (Non-dominated Point and Efficient Solution) For a given bi-criteria optimization problem (BCOP), a feasible solution Υ is said to be an efficient solution, if there is no other feasible solution Υ 1 that dominates Υ. The corresponding point in the x y plane, where x and y are the values of the two objective functions, f 1 and f 2, respectively, is said to be a non-dominated point. Definition. (Weakly Non-dominated Point and Weakly Efficient Solution) For a given bicriteria optimization problem (BCOP), a feasible solution Υ is said to be a weakly efficient 1