\A CENTRAL FACILITIES LOCATION PROBLEM INVOLVING TRAVELING SALESMAN TOURS AND EXPECTED DISTANCES/ MASTER OF SCIENCE

Transcription

1 \A CENTRAL FACILITIES LOCATION PROBLEM INVOLVING TRAVELING SALESMAN TOURS AND EXPECTED DISTANCES/ by Thesis submitted to the Graduate Faulty of the Virginia Polytehni Institute and State University in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE in Industrial Engineering and Operations Researh Approved: May 1974 Blaksburg, Virginia

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3 ACKNOWLEDGEMENTS The author wishes to express his appreiation to all those who have ontributed to the ompletion of this thesis. Dr. John A. White, as major advisor, provided timely suggestions throughout the researh. His helpful advie and ritiism are greatly appreiated. Gratitude is also expressed to the other members of the graduate ommittee, Dr. J. William Shmidt and Dr. Wayne C. Turner. The author extends his appreiation to the IEOR Department whose finanial assistane made this researh possible. Thanks are also extended to Ms. Margie Strikler for her exellent typing of the manusript under severe time limitations. ii

4 TABLE OF CONTENTS ACKNOWLEDGEMENTS LIST OF TABLES. LIST OF FIGURES. Chapter INTRODUCTION Subjet of the Researh.. Importane of the Topi. 3 Method of Approah... 3 Objetives and Purpose of the Researh. 4 Order of Disussion.. 4 PROBLEM FORMULATION 6 Introdution... 6 Notation and Mathematial Formulation.. 6 The Traveling Salesman Problem. 12 Some Simple Examples. 14 Summary SOLUTION PROCEDURES 41 Introdution.. 41 Proedure 1: Funtional Minimization Using the Hyperboli Approximation Proedure Relationship Between the Steiner-Weber Problem and the Traveling Salesman Loation Problem Presentation of the Algorithm Proedure 2: Funtional Minimization Using a Suessive Quadrati Approximation Proedure. 60 ii v vi iii

5 iv TABLE OF CONTENTS (ontinued) Chapter Presentation of the Algorithm.. Summary ,.. 4 COMPUTATIONAL EXPERIENCE. Introdution... Problem Generation. Computational Results.. Disussion. Conlusions. 5 SUMMARY AND RECOMMENDATIONS FOR FURTHER RESEARCH.. BIBLIOGRAPHY. APPENDIX A APPENDIX B VITA... Summa ry Reommendations for Further Researh. Derivation of the Least Squares Estimates for a Seond Order Model with Two Independent Variables Doumentation of Computer Programs. Page

6 Table LIST OF TABLES 2.1 Maximum and Minimum Values of Objetive Funtion of Examp 1 e Maximum and Minimum Values of Objetive Funtion of Exalllple Itineraries for the Traveling Salesman Zones... Subsets of Failities to be Visited and Trivial Itineraries for the Case Where m = Effet of Using Different Starting Points for Example Effet of Using Different Starting Points for Example Overall Performane - Proedure 1 vs. Proedure Cumulative Frequeny Distributions for Problems Involving Four Existing Failities Cumulative Frequeny Distributions for Problems Involving Six Existing Failities Cumulative Frequeny Distributions for Problems Involving Eight Existing Failities Cumulative Frequeny Distributions for Problems Involving Ten Existing Failities Exeution Time Data for Eulidean Distane Measure - Proedure 1 and Proedure Exeution Time Data for Eulidean and Retilinear Distane Measures - Proedure v

7 Figure LIST OF FIGURES A Cutaway View of f(x,y) in Example An Overall View of f(x,y) in Example 2.1. A Cutaway View of f(x,y) in Example An Overall View of f(x,y) in Example 2.2. Traveling Salesman Zones. An Overall View of f(x,y) in Example 2.3. A Cutaway View of f(x,y) when P15 =.6. An Overall View of f(x,y) when P15 =.6. Optimal Paths as Pk ~ 0 in Example Optimal Paths as Pk + 0, for m = 6. Page Optimal Paths as Pk + 0, for m = Opti~al Paths as Pk + 0, for an Asymmetri Configuration Wl th m = Opti~al Paths as Pk + 0, for an Asymmetri Configuration Wl th m = Equi va 1 ent Cheks for Optima 1 i ty in Proedure Maro Flowhart of Proedure 1. Paths to the Optimum Using Different Starting Points in Proedure Paths to the Optimum Using Different Starting Points in Proedure An Illustration of Cyling Between Traveling Salesman Zone s Formation of "Pseudo" Existing Failities from Existing Failities vi

8 Chapter 1 INTRODUCTION Subjet of the Researh The need for ost redution in the areas of distribution management, transportation, and urban planning has been ontinuously inreasing over the past several deades due among other things to the growing shortages of natural resoures. Failities loation tehniques have been employed in these areas to help determine more effiient alloations of sare resoures so as to redue osts. While most appliations have been made in industrial situations, a growing awareness in governmental areas of the advantages of loation analysis is evidened by an inreased demand for personnel familiar with failities loation tehniques. The problem to be studied in this thesis involves the determination of a minimum ost loation of a entral faility when trips taken from the entral faility to some or all of m existing failities form traveling salesman tours. The problem is probabilisti in nature in that many possible ombinations of trips involving different existing failities may our. While the problem formulated is original, muh of the disussion draws upon $1milar,well studied problems in the area of failities loation. The reader unfamiliar with the subjet of failities loation and design is referred to the text of Franis and White [10]. In addition to presenting the basi different types of loation problems they provide an in depth disussion of the appropriate, overall approah that one 1

9 2 should take to both failities loation and plant layout problems. A omprehensive report on the more signifiant ontributions to failities loation is given by Bozok [4]. Franis and Goldstein [9] and Lea [12J reently ompiled extensive bibliographies, of failities loation literature that should enable the researher to keep abreast of the works of other researhers with similar interests, but who are in different fields. Using the taxonomy of failities loation problems presented by Franis and White [10] the loation problem onsidered in this thesis involves the loation of a single new faility relative to a known number of existing failities having known, fixed loations. A ontinuous solution spae is assumed, retilinear and Eulidean distanes are employed, and an objetive of minimizing expeted total ost is speified for the problem under study. In fat, the loation problem treated possesses the harateristis of the lassi Steiner-Weber problem, whih has been studied extensively over the past twenty years. The problem addressed herein an be referred to as the traveling salesman loation problem. It differs from the Steiner-Weber problem on the basis of the travel between the new faility and the existing failities. Speifially, the Steiner-Weber problem assumes that trips are always made between the new faility and an existing faility eah time a trip ours. The traveling salesman loation problem allows more than one existing faility to be visited during a single trip from the new faility. Additionally, the set of existing failities to be visited during a trip is onsidered to be a random variable.

10 3 Importane of the Topi Based on an extensive review of the failities loation literature it appears that the present effort is the first to formulate and study the traveling salesman loation problem. Even though the problem has not been studied previously, there exist a number of instanes for whih the traveling salesman loation problem formulation is appropriate. An obvious appliation is the problem of determining the loation of a distrit sales offie when the salesman servies a number of ustomers; the loation of a blood bank whih supplies blood to a number of hospitals is another instane where the traveling salesman loation problem ours. Method of Approah The traveling salesman loation problem involves the determination of both the loation of the new faility and the routing to be taken when visiting the existing failities. Given the loation of the new faility and the set of existing failities to be visited on a partiular trip~ the routing to be employed in minimizing the distane traveled per trip is determined by solving the assoiated traveling salesman problem. Sine the set of existing failities to be visited on a given trip is a random variable, the loation of the new faility is to be determined suh that the expeted distane traveled per trip is minimized. In onsidering methods for solving the problem, heuristi solution proedures will be used. Speifially, a step-wise proedure is employed in whih searh proedures are used to speify a loation for the new faility and, given the loation, a traveling salesman algorithm is

11 4 employed to determine the routes whih yield a minimum expeted ost. Two existing searh proedures, the Hyperboli Approximation Proedure [8 ] and the Suessive Quadrati Approximation Searh Proedure [18], are employed in determining the loation for the new faility. The assoiated traveling salesman problems are solved using the algorithm developed by Little, Murty, Sweeney, and Karel [15]. Objetives and Purpose of the Researh The objetives of the researh inlude the presentation of an original formulation of a signifiant failities loation problem whih has not been studied previously, the development of several proedures for determining minimum ost loations, and the omparison of the effetiveness of eah solution proedure. An additional objetive is to gain some insight regarding the behavior of the funtion under a variety of different onditions. The purpose of this researh effort is to expand the existing body of failities loation literature to inlude an investigation of a problem that ours quite frequently in pratie. Order of the Disussion To failitate the presentation of the results of the researh effort, Chapter 2 presents the problem formulation and disusses the speial features of the problem, e.g., those that prohibit minimization of the funtion through lassial optimization tehniques. Several examples are given to depit the psyhoti nature of the funtion under investigation. Chapter 3 presents a solution proedure whih inorporates the Hyperboli

12 5 Approximation Proedure. An alternative solution proedure partiularly well suited for problems in whih there are many existing failities is also advaned in Chapter 3. The tehnique employed here is a searh by regression through suessive quadrati approximation. Chapter 4 p'(':::sents omputational results in whih the merits of eah proedure are ompared, based on omputer run times and the effetiveness of the proedures in minimizing the funtion. Chapter 5 summarizes the researh; onlusions resulting from the study are given as well as some reommendations for further researh. Attention is devoted to the most obvious extension: the multiple new faility traveling salesman loation problem.

13 Chapter 2 PROBLEM FORMULATION Introdution This hapter onsists of a mathematial formulation of the probabilisti loation problem and a presentation of some relatively simple examples. Additionally, due to the importane of the traveling salesman problem in the researh, some of the better known solution proedures to the traveling salesman problem are onsidered. Further insight as to the nature of the funtion under investigation will be gained in the study of the example problems. Notation and Mathematial Formulation Where onvenient to do so, the notation to be used in this thesis will be onsistent with the notation found in most of the reent failities loation literature. To begin, the following notation is adopted: (x,y) m Ph = the oordinate loation of the new faility. = the number of existing failities. = a unique subset of failities, inluding the new faility, that are to be visited on trip h, h = 1, 2, 3,..., k. The order of visitation is not implied by the ordering of elements in She = the probability of visiting the existing failities ontained in subset Sh' h = 1,2,3,..., k. d.. (x,y) = the distane from faility i to faility j, 1 < i ~ m+l, lj 1.::.j < m+l 6

14 7 (a.,b.) = the oordinate loation of the ith existing faility, 1 1 i = 1,2,3,..., m k qh = 2m-l, the total number of possible distint subsets = the number of elements ontained in She Note that Sh an ontain at most (m+l) elements sine the subset that inludes all existing failities an be written as Sh = {l, 2, 3,..., m, (m+l)~ where (111+1) denotes the new faility. Also, only one subset an ontain (m+l) elements sine (~) = 1. Note further that faility (m+l) must be a member of all subsets sine all trips begin and end with the new faility. Sh must ontain at least two elements, an existing faility and the new faility. It is easy to see that there are (~) = m unique subsets ontaining two elements. They are {l,(m+l )}, {2,(m+l )}, {3,(m+l)},..., {m,(m+l)}. In general, for subsets ontaining j+l elements, j ~ m, there are (j> unique subsets that an be formed. The total number of subsets that must be onsidered, k, equals 2 m - 1. The probabilisti loation problem an now be expressed as (PO) min f(x,y) = X,Y k=2m-l I h=l (2. 1 ) s.t. I z.. lj ie:sh l z.. lj je:sh = 1 V je:sh h = 1, 2,.., = 1 V ie:sh h = 1, 2,.., (2.la) k (2.1b) k u. 1 - u. J + qhzij < qh - 1, V ;,jesr irj, i;, jrl, h = 1, 2,.., k (2.l)

15 8 Z.. = 0,1 lj, 11 i,jesh h = 1, 2,., k where u i and Uj are arbitrary real numbers that satisfy (2.l). For eah h the term in brakets along with the four onstraints represents the partiular traveling salesman problem orresponding to She interpretation of the first onstraint, (2.1a), is that eah faility ontained in Sh must be entered only one. The seond onstraint indiates that eah faility ontained in Sh must be left only one. The first two onstraints together impose the restrition that eah faility ontained in Sh an be visited one and only one.. The third onstraint fores the solution to be a Hamiltonian iruit, i.e., there an be no subtours in the solution. The last onstraint defines the problem to be a zero-one, integer programming problem. The If the order of visitation is from faility i to faility j, then Z. = 1. If the lj minimum distane solution does not ditate a routing from i to j, then Z.. = O. lj Problem (PO) is embedded with the minimum distane solutions to k different traveling salesman problems, one for eah trip, h: (2.2) s.t. I Z. = 1. S lj Je: h

16 9 u - u. + qh Z" < qh J lj -, 'tj i,jesh' ilj, ill, jll Z.. = 0,1 lj where u i and u j are arbitrary real numbers that satisfy the third onstraint. (2.2) is a slight variation of the integer programming formulation of the traveling salesman problem presented by Miller, Tuker, and Zemlin. [17J. Observe that the solution (2.2) is simply the solution to the traveling salesman problem orresponding to the failities ontained in subset Sh with the new faility loated at (x,y). Notie that the deision variables ;n (2.2) are the Z.. IS rather than x and y. lj (PO) an now be rewritten as k (Pl) min f(x,y) =.~ (x,y) h=l Ph[th*(x,y)] (2.3) Before disussing further the objetive funtion, f(x,y), attention must be given to the distane measure that is to be used. measure may be Eulidean or retilinear. Eulidean, then The distane If the distane measure is, i = m+ 1, j = 1,2,3,.. ~,m 2 2 1/2 [(a.-a.) +(b.-b.)], i = 1,2,3,...,m 1 J 1 J JO = 1 23m If the distane measure is retilinear, then,,,..., (2.4)

17 10 d.. (x,y) = lj lx-a 1+ly-b 1 J J, i = m+l, j = l t 2,3,... t m (2.5) I either ase it is assumed that dij(x,y) = dji(x,y). Notie that the distane from faility i to faility j depends on the loation of the new faility, (x,y), only when either faility i or faility j is the new faility. Letting D be the (m+l)x(m+l) distane matrix, a hange in the loation of the new faility will only hange the (m+l)th row and olumn of the distane matrix. In Equation (2.3) the objetive funtion, f(x,y), is a onvex ombination of solutions to k = 2 m - 1 traveling salesman problems. The objetive is to find a value of x and y for whih the onvex ombination is at a minimum. There is no apparent relationship between Ph' the probability of visiting the failities ontained in Sh' and the term in brakets, namely, the minimum distane solution to the traveling salesman problem for Sh" The relative frequeny of ourene of subset Sh is given initially and does not hange if the oordinate loation of the new faility is hanged. To minimize the objetive funtion it would seem that primary attention must be foused on the solution of the 2 m - 1 traveling salesman problems in suh a manner that their weighted sum is minimized. Before examining some of the existing solution proedures to the traveling salesman problem, several other features of (Pl) must be presented. First, note that one funtional evaluation of f(x,y) requires the solution of 2 m - 1 traveling salesman problems so that as m beomes

18 large the solution time required for even one evaluation of the funtion may beome quite long. 11 Hene, for problems involving many existing failities, proedures that minimize f(x,y), but require many funtional evaluations, may not be feasible due to the enormous omputation time involved. For only ten existing failities, for example, one funtional evaluation requires the solution of = 1,023 distint traveling salesman problems. However, some of the traveling salesman problems are solved trivially. When only one or two existing failities are to be visited, there is only one route that an be seleted. Thus, (~) + (~) solutions are trivial. This redues the number of nontrivial traveling salesman problems to 2 m (~) - (~) = 2 m - ~m2+m+2). The seond proedure for minimizing f(x,y) that is developed in Chapter 3 has a strong point in that it is designed to minimize the number of funtional evaluations required and onsequently is well suited for problems involving many existing failities. There;s a trade-off in _. --. this ase, though, sine the proedure will only generate near optimal solutions. A seond diffiulty presented by (Pl) is that there is no reason to believe apriori that f(x,y) is a onvex funtion. The suessful use of quasi-enumerative tehniques is heavily dependent on the onvexity of a funtion. As will be seen in the examples presented at the end of this hapter, f{x,y) ;s not always onvex. The first tehnique developed in Chapter 3, however, takes advantage of a speial struture of the problem suh that it ;s apable of reognizing a loal minimum.

19 12 The Traveling Salesman Problem The well studied traveling salesman problem, while simply stated, has yet to have been solved by any omputationally effiient solution tehnique. Given n ities and known distanes between all ities, the minimum distane route must be determined suh that the salesman starts at the first ity, visits all other ities only one, and finally returns to the first ity. In other words, there an be no subtours. One omplete tour is known in graph theory as a Hamiltonian iruit. Starting from the first ity there are (n-l)! possible routes that may be taken. Note that n = m+l so that there are m!possible round trips for the subset in whih all existing failities are visited. The integer programming formulation of the traveling salesman problem was used in the mathematial formulation of the traveling salesman loation problem for the sole purpose of making the formulation preise. The solution proedures to be used will not use integer programming tehniques. Unfortunately, the omputational results reported on the use of integer programming to solve large traveling salesman problems have been unfavorable. There have been numerous optimal seeking proedures developed for the solution of the traveling salesman problem, the most famous of whih is the branh and bound algorithm developed by Little, Murty, Sweeney, and Karel [15J. The solution proedures for the probabilisti loation problem will utilize this algorithm. Another algorithm developed by Eastman [ 5,6] is a branh and bound proedure using subtour eliminations. This method is not known to have

20 13 been programmed, however. Bellman [2] has developed a dynami programming formulation of the traveling salesman problem. The shortoming of this approah is that for large problems storage requirements are exessive. A more omprehensive disussion of branh and bound tehniques and strategies is given by Lawler and Wood [11], A more in depth treatment of optimal seeking proedures an be found in Bellmore and Malone [3] and Turner, Ghare, and Foulds [19]. Though they will not be used in this thesis, some heuristi methods that generate near optimal solutions to the traveling salesman problem will be mentioned. Suh methods may have to be used for large sale probabilisti loation problems due to the ombinatoris involved with the loation problem itself. With regard to traveling salesman problems it has been found that the problem is ompounded by the fat that for optimal seeking methods the omputation time required for solution of traveling salesman problems inreases exponentially with n. One heuristi method is that of Ashour and Parker [1] where a look-ahead proedure is ombined with the proedure of seleting the nearest unvisited ity. The 1t2-optU proedure of Lin [13] and the Ilk-opt" proedure of Lin and Kernighan [14], based on their reported omputational results for 200 ity problems, seems to be another effetive heuristi. Other heuristi proedures are di sussed by Ei 1 on, Wa tson-gandy, and Chri s tofi des [7]. The solution tehnique to be used in this thesis, Little1s method, is a branh and bound method based on penalty tour building. A omplete disussion of the algorithm, inluding an example, is given by Wagner [20]. Little, et al., report mean omputation times on an IBM 7090 of 58.5 seonds for 100 thirty-ity problems and 8.37 minutes for 5

21 14 forty-ity problems. Though they argue in favor of branhing from the newest bounding problem t their omputational results are based on the poliy of branhing from the node having the least lower bound. The solution proedure to be used in this thesis will adopt the reommended branhing strategy. Another refinement espeially effetive on larger problems is suggested by Eilon t Watson-GandYt and Christofides [7 J. They found a redution in omputation times when the penalties were determined by weighting the seond and third smallest row and olumn elements. Their suggested refinement was not used in this researh. Some Simple Examples In order to illustrate the omplexity of the problem that has been formulated, four relatively simple examples will be presented. These examples will also be used in later hapters to illustrate the various solution proedures. Consider a situation where four existing failities are loated at (al,b l ) = (0,0), (a 2,b 2 ) = (loo,o), (a 3,b 3 ) = (100,100), and (a 4t b 4 ) = (0,100). The new faility, faility number 5, must be loated so as to minimize the expeted distane traveled. The distane measure an be Eulidean or retilinear. Two of the examples use Eulidean distane measures and two use retilinear distane. The relative frequeny of ourrene of subsets of failities to "be visited will be either the same over all subsets or the same among subsets involving the same number of existing failities. Example 2.1 subset probabilities as follows: Let the distane measure be Eulidean and define the

22 h Ph 15 Sh 1 1/15 1,5 2 1/15 2,5 3 1/15 3,5 4 1/15 4,5 5 1/15 1,2,5 6 1/15 1,3,5 7 1/15 1,4,5 8 1/15 2,3,5 9 1/15 2,4,5 10 1/15 3,4,5 11 1/15 1,2,3,5 12 1/15 1,2,4,5 13 1/15 1,3,4,5 14 1/15 2,3,4,5 15 1/15 1,2,3,4,5 Sine the existing failities are loated at the verties of a square and sine the subset probabilities give equal weight to all the existing failities, intuition might lead one to believe that the point (50,50) is the minimum distane loation for the new faility. Figure 2.1 provides a utaway view of the three dimensional graph of the funtion, z = f(x,y), for 0 ~ x ~ 50, 50 ~ y ~ 100. Close examination of the figure reveals that the funtion is at a minimum somewhere between the points (42,50) and (46,50),with another minimum ourring between the points (50,42) and (50,46). a relative maximum. AtuallY,at the point (SO,SO) f{x,y) ;s at Figure 2.2 is a plot of f(x,y) over the entire square. The maximum and minimum values of f(x,y) are given in Table 2.1. The last four entri'es in Table 2.1 are not extreme points, but f(x,y) appears to be stritly onvex in a neighborhood of eah of the four points so that it is safe to onlude that a relative minimum will our at x* = 50, and within either the interval 42 < y < 46 or the interval 54 < y < 58, and by symmetry at y* = 50, and within either the interval

23 16 Table 2.1 Maximum and Minimum Values of Objetive Funtion of Example 2.1 (x,y) f(x,y) Nature of Extreme Point (0,0) Global maximum (0,100) Global maximum (100,100 ) Global maximum (100,0 ) Global maximum (50,50) Loal maximum (50,44) Minimum value obtained in grid searh (50,56) Minimum value obtained in grid searh (44,50) Minimum value obtained in grid searh (56,50) Minimum value obtained in grid searh

24 17 Fi gure 2.1 A Cutaway View of f(x,y) in Example 2.1

25 18 x Figure 2.2 An Overall View of f(x,y) in Example 2.1

26 19 42 < x < 46 or the interval 54 < x < 58. Notie in Figure 2.2 the ridges that run from the existing failities to the point (50,50). The existene of these ridges by themselves are enough to make f(x,y) non-onvex over the entire region. Now onsider a triangular region in the xy plane formed by any two of the four existing failities and the point (50,50). It an be seen that f(x,y} is nononvex in this region also. Using the triangular region formed by the points (0,0), (0,100), and (50,50) and then examining f(x,y = 50), a ~ x < 50, in Figure 2.1, an infletion point an be seen at f(l8,50). f(x,y) appears to be onvex beyond the point (18,50), however. ExamEle 2.2 As a seond example we will maintain the Eulidean distane measure and redefine the subset probabilities as follows: h Ph Sh , , , , ,2, ,3, ,4, ,3, ,4, ,4, ,2,3, ,2,4, ,3,4, ,3,4, ,2,3,4,5 Thi s example an be onsidered to be another speial ase sine the subset probabilities are equal among subsets involving the same number of failities. As a result, no partiular existing faility is given an extra weight due to the assignment of subset probabilities. As in the

27 20 previous example, a good intuitive guess at the minimum distane loation would be the point (50,50). Figures 2.3 and 2.4 provide a utaway view of the funtion and a view over the entire region, respetively, analagous to the figures used in Example 2.1. Notie that the four minimum points have shifted slightly outward in relation to the minimum points of Example 2.1, and that the relative maximum ourring at the point (50,50) is now more pronouned. The explanation for these shifts an be made by examining Pk' the probability of visiting all four existing failities, in eah example. Notie first of all, however, that if only all four existing failities are to be visited, i.e., P15 = 1, the minimum distane solution is immediately determined by loating the new faility anywhere on the boundary of the square region formed by the existing failities. Now Pk in Example 2.2 is slightly greater than Pk in Example 2.1 so that the distane resulting from visiting all four existing failities is given a higher" weight in the objetive funtion. Consequently, it is expeted that the minimum distane loation will shift toward the minimum distane solution that results when Pk = 1. The shifts our along the lines x = 50 and y = 50 beause no sing~ existing faility is weighted higher than another. Only the subsets ontaining a different number of existing failities are weighted unequally. The shifting of the minimum points of f(x,y) is entirely due to the hanges made in the weighting of groups of existing failities. As an example in whih unequal weighting of existing failities ours suppose that for 5", 5'2' 5 13, and 5'4' the 'probabilities of visiting these subsets involving three existing failities are hanged to Pll = P12 = P13 =.08 and P14 =.16. $14 involves all existing failities exept faility number one. Hene, faility one is now weighted lower than

28 21 x Fig~re 2.3 A Cutaway View of f(x,y) in Example 2.2

29 22 x Figure 2.4 An Overall View of f(x~y) in Example 2.2

30 23 failities two, three, and four. The minimum points an be expeted to shift away from faility one somewhat, sine the expeted distane traveled has dereased for faility one, and has inreased for the other existing failities. Table 2.2 lists the maximum and minimum values of f(x,y) that were obtained in plotting Figures 2.3 and 2.4. The relative minima our at x* = 50, and within either the interval 38 < Y < 42 or the interval 58 < y <62, and by symmetry at y* = 50, and within either the interval 38 < x < 42 or the interval 58 < x < 62. In Example 2.2 the shifts in the minimum points and the values of f(x,y) have hanged due to the new probability assignments. It should be reognized, however, that the loations of the minimum points are dependent, not only on the probabilities of visiting various subsets of existing failities, but also on the traveling salesman zones resulting from the onfiguration of the existing failities. A traveling salesman zone an be defined as a region in whih a new faility an be loated suh that when all other existing faility loations are held fixed, the set of optimal routes resulting from the solution of the traveling salesman problems remains the same. The symmetry of f(x,y), as given in Examples 2.1 and 2.2, an be destroyed either by assigning subset probabilities so as to weight the existing failities unequally or by altering the onfiguration of existing failities, thus hanging the traveling salesman zones. Figure 2.5 shows, approximately, the different traveling salesman zones that result when the distane measure is Eulidean and the four existing failities have the same loations as in the previous examples.

31 24 Table 2.2 Maximum and Minimum Values of Objetive Funtion of Example 2.2 (x,y) f(x,y) Nature of Extreme Point (0,0) Global maximum (0,100) Global maximum (100,100) Global maximum (100,0) Global maximum (50,50) Loal maximum (50,40) Minimum value obtained in grid searh (50,60) Minimum value obtained in grid searh (40,50) Minimum value obtained in grid searh (60,50) Minimum value obtained in grid searh

32 25 (a,,b,) = (0,0) Figure 2.5 Traveling Salesman Zones

33 26 Table 2.3 lists the non-trivial itineraries for eah zone. I(Sh) is the optimal route or itinerary for the hth subset of fail ities to be visited. Faility number S, the new faility, is not given in the routes, I(Sh)' h = 11, 12, 13, 14, 15, sine it is assumed that all tours begin and end with the new faility. It is interesting to note that when only all four existing failities are visited, the traveling salesman zones turn out to be the four triangles formed by the diagonals of the square. This an be seen by omparing zone numbers in Figure 2.5 with the zones in Table 2.3 that have idential routes for 1(5 1S ). When the four ombinations of unique subsets ontaining ~hree existing failities are onsidered, the zones beome distorted. Notie in Table 2.3 that traveling salesman zones hange when at least one route, I(5 h ), hanges. It is expeted that as the total number of existing failities is inreased, resulting in many more subsets of existing failities to be visited, the zones will beome even more distorted. Furthermore, any hange in the onfiguration of the existing failities will tend to distort the traveling salesman zones. The importane of traveling salesman zones in relation to the traveling salesman loation problem will beome more apparent in Chapter 3. Example 2.3 Consider as a third example using the equal subset probabilities of Example 2.1 and employing a retilinear distane measure. The objetive funtion now beomes pyramidal.in shape. Figure 2.6 is a plot of f(x,y) over the entire square region. The point (50,50) has now beome a global maximum with f(so,so) = Any point lying on the boundary of the square region is a global minimum, i.e.,

34 27 Table 2.3 Itineraries for the Traveling Salesman Zones Zone 1(5 11 ).1(5 12 ) 1(5 13 ) 1(5,4) 1(5 15 )

35 2R x Figure 2.6 An Overall View of f(x,y) in Example 2.3

36 29 (x*,y*) = {(x,y) I 0 ~ x* ~ 100, y* = 0 or y* = 100, and x* = 0 or x* = 100, 0 ~y* < loo} with f(x*,y*) = 320. Example 2.4 As the fourth example onsider using the subset probabilities of Example 2.2 and employing retilinear distanes. The objetive funtion forms a pyramid similar to the one in the previous example. The global maximum is still at the point (50,50) with f(50,50) = 350. The minimum points again lie on the boundary of the square region with f(x*,y*) = 340. It ;s interesting to note that there are only four different traveling salesman zones in Examples 2.3 and 2.4 whereas 16 different zones were formed in Examples 2.1 and 2.2. When the retilinear distane measure is used, the four traveling salesman zones are defined by the four right isoseles triangles formed by the diagonals of the square, e.g., one omplete zone for the retilinear problem is given by zones 1,2, 3, and 4 in Figure 2.5. Returning now to the results obtained in Examples 2.1 and 2.2, reall that as Pk inreased from to 0.1 that the four minimum points moved outward in diretions perpendiular to the sides of the ~quare region. Figures 2.7 and 2.8 provide a utaway view and an overall view of f(x,y), respetively, with P15 = 0.6, and Ph~ h = 1,2,...,14. The minimum points have now moved even loser to the sides of the square. When Pk = 1.0, the optimal loation for the new faility will lie anywhere on the boundary of the square. The reason behind this is simple. The boundary of the square represents the path of the optimal routes for travel between all four existing failities given that all four existing are visited in one trip. Hene, any loation of the new faility off

37 30 x Figure 2.7 A Cutaway View of f(x,y) when P15 =.6

38 31 x Figure 2.8 An Overall View of f(x,y) when P15 =.6

39 32 the boundary of the square an only inrease the travel distane sine it is known with ertainty that all four existing failities will be visited eah trip. Notie in Figures 2.7 and 2.8 that the point (50,50) now represents a global maximum for f(x,y), and in addition, the ridges have beome muh sharper. As was pointed out earlier, the symmetry of the objetive funtion an be distorted either by assigning subset probabilities so as to unequally weight existing failities or by hanging the onfiguration of the existing failities. It is instrutive to examine the paths of the relative minima of f{x,y), for a ~ Pk ~ 1, for different onfigurations of existing failities. Consider the square onfiguration given by the four existing faility loations in Examples 2.1 and 2.2. When all Ph' h < k, are held equal, and Pk + 0, the resulting minimum points of f(x,y) trae out the paths given in Figure 2.9. Eah point on an arrow represents a global minimum point for f(x,y) for a partiular Pk. When Pk = 0, the single global minimum is at point (50,50). As previously mentioned, when Pk = 1, any point on the boundary of the square represents a global minimum. For a partiular Pk' 0 < Pk < 1, there are four global minimum points, one on eah arrow, with eah global minimum point loated the same distane away from the point (50,50). It should be emphasized here that all four paths trae out global minimum points as Pk + O. In general, if the onfiguration of m existing failities forms a regular polygon, and all m failities are weighted eq~ally, then as Pk + 0, there will be m paths traed out by the minimum points of f(x,y), all m of whih represent global minima. A regular polygon is defined to

40 33 (a 3,b 3 ) (100,100)... Figure 2.9 Optimal Paths as Pk + 0 in Example 2.2

41 34 be a losed figure with all sides of equal length loated symmetrially about the IIenter ll of the polygon. If the polygon is not regular, then the paths may not all represent global minima. This is the ase in Figure In Figure 2.10 the onfiguration of the six existing failities shown forms a hexagon.. Sine the two vertial sides of the hexagon are shorter than the other sides, the hexagon is not regular. The broken lines starting from the shorter sides represent loal minimum paths traed out by f(x,y) as Pk -+ 0, while the solid lines represent global minimum paths. Sine the hexagon ;s not regular, the traveling salesman zones are not symmetri, and therefore f(x,y) is not symmetri. The three existing failities in Figure 2.11 are loated so as to form an equilateral triangle. The traveling salesman zones are symmetri about the enter of the triangle, and as a result, f(x,y) has three global minimum paths that onverge on the "enter U of the triangle as Pk -+ O. Moving any existing faility will distort the symmetry of the traveling salesman zones, and therefore, will ause f(x,y) to be asymmetri. This, in turn, will redue the number of global minimum paths to either two or one) depending on the diretion in whih the existing faility is moved. For example, if (a 3,b 3 ) is moved toward (a,,b l ) and (a 2,b 2 ) in any fashion, only one global minimum path ours, its loation being near (a 3,b 3 ). On the other hand, if (a 3,b 3 ) is moved away from (a l,b,) and (a 2,b 2 ) in any fashion, two global minimum paths will result, one loated near (a l,b l ), and one near (a 2,b 2 ). Not only an a non-regular onfiguration of existing failities redue the number of global minimum paths, but also it may alter the diretion of the paths as well. The global minimum paths in Figures 2.12

42 ~ (a 6,b 6 ) (0,28.9) (a 2,b 2 ) (100,28.9) Figure 2.10 Optimal Paths as Pk + 0, for m = 6

43 36 (a 3,b 3 ) (50,86.6) Fi gure Optimal Paths as Pk + 0, for m = 3

44 37 and 2.13 were approximated by letting Pk go to zero and employing the first searh proedure, presented in the next hapter, to loate relative and global minimum points. Figure 2.12 shows the resulting distortion of the optimal paths when existing faility number three (Examples 2.1 and 2.2) is moved from the point (100,100) to the point (150,150). As before, the broken lines depit relative minimum paths, and the solid lines are global minimum paths. Notie that when Pk = 0, the single optimal loation is still at the point (50,50), just as it would be if the onfiguration of existing failities had formed a square. The global minimum paths tend to favor the three existing failities that have not been moved. When the third existing faility ;s moved to the point (25,25), the optimal (global) paths remain inside the losed figure. Observe in Figure 2.13 that the loal minimum paths fall within the onvex hull formed by the four existing failities, however. At Pk = 0, the global minimum is at (25,25). It should now be lear that the loation of the minimum point, or minimum points for f(x,y) is heavily dependent not only on the subset probabilities, but also on the onfiguration of the existing failities. Determining the minimum points for f(x,y), even for small problems suh as those presented in this hapter, turns out to be an exeedingly diffiult task. Many of the results obtained in the examples thus far have been ounter-intuitive. Fortunately, however, one of the searh proedures advaned in the next hapter is able to deal with the psyhoti nature of f(x,y) through the use of information given by the traveling salesman zones.

45 38 (a 3,b 3 ) (150,150 ), \ \ +. ~, " \ Fi gure Optimal Paths as Pk ~ 0, for an Asymmetri Configuration with m = 4

46 39 Figure 2.13 Optimal Paths as Pk + 0, for an Asymmetri Configuration with m = 4

47 40 Summary Chapter 2 has presented the mathematial formulation of the traveling salesman loation problem. Additionally, several simple example problems were provided to better familiarize the reader with the nature of the problem under investigation. The onept of a traveling salesman zone, a term that will be used throughout this thesis, was introdued via the example problems. The next hapter is devoted to the implementation of two existing searh proedures as a means of solving the traveling salesman loation problem. The example problems disussed in Chapter 2 will be used again in Chapter 3 to verify the two solution proedures.

48 Chapter 3 SOLUTION PROCEDURES Introdution In the previous hapter the traveling salesman loation problem was introdued, first by way of a mathematial formulation of the problem, and then by examining several relatively simple example problems. Even for a simple problem, it was seen that the objetive funtion, f(x,y), is non-onvex. The non-onvexity of f(x,y) was shown to be aused by the different traveling salesman zones formed at various oordinate loations. Two heuristi proedures for solving the traveling salesman loation problem will be given in this hapter. The first proedure is based on a strutural relationship between the traveling salesman loation problem and the well known Steiner-Weber problem. The seond proedure proposed utilizes a searh by regression. After the presentation and disussion of the algorithms, the effetiveness of eah will be demonstrated by solving the problems given in Examples 2.1, 2.2, 2.3, and 2.4. Proedure 1: Funtional Minimization Using the Hyperboli Approximation Proedure The proedure for solving the traveling salesman loation problem developed in this setion is apable of produing solutions that represent loal minima on f(x,y). As is usually the ase when searh tehniques are employed to minimize non-onvex funtions, there an be no absolute 41

49 42 guarantee that a global minimum will be found. It will be shown by way of the example problems that a judiious seletion of several starting points for the new faility greatly inreases the hanes of obtaining a global minimum. solutions. All effetive heuristi proedures are apable of obtaining optimal The strength of Proedure 1 lies in its ability both to produe an optimal solution and to immediately reognize that solution as being an optimal one. trait. Relationship Between the Steiner-Weber Problem Many heuristi proedures do not possess the latter and the Traveling Salesman Loation Problem The development of Proedure 1, based on an interesting relationship between the Steiner-Weber problem and the traveling salesman loation problem, is an extension of an approah originally suggested by Lohmar [16]. This relationship is best exposed by first disussing the Steiner Weber problem. The single faility loation problem, a speial ase of the Steiner Weber problem, an be stated mathematially as where m m minimize g(x,y) =.r Widi(x,y) 1=1 = the number of existing failities di(x,y) = the distane from the new faility, m+l, loated at oordinate position (x,y), to existing faility i, 1 < i < m. (3.1)

50 43 w. = non-negative weight between the new faility and the 1 ith existing faility, 1.::. i.::. m. di(x,y) is given by the first equation in (2.4) for Eulidean distanes and is given by the first equation in (2.5) for retilinear distanes. The weighting fator, wi' reflets the perentage of diret movement per unit time between the ith existing faility and the new faility. The oordinate positions of the existing failities are (ai,b i ), i = 1, 2, 3,..., m, as given in the previous hapter. A suffiient ondition for g to have a minimum at a point (x*,y*) is that the first partial derivatives of g, with respet to x, and with respet to y, evaluated at (x*.,y*) both be zero. distane measure is used, the first partial derivatives are When the Eulidean ag{x,l) ax m 2 2 1/2 =.r w.(x-a }/[(x-a.) +(y-b.) ] ', =1 (3.2) ag{x,l) 'Oy m 2 2 1/2 = I w.(y-b.)/[(x-a.) +(y-b.) ] i =1 (3.3) Equations (3.2) and (3.3) ould be solved by numerial tehniques if the partial derivatives were defined at all points. Notie, however, that if the new faility loation oinides with the loation of an existing faility, the partial derivatives are undefined. Several solution tehniques have been developed to handle the problem of undefined derivatives in the Steiner-Weber problem. The Hyperboli Approximation Proedure, proposed by Eyster, Wh i te, and Wi erwi 11 e [8], has proven to be an espeially effetive algorithm for solving the failities

51 44 loation problem. Their proedure will be utilized in Proedure 1 to assist in the solution of the traveling salesman loation problem. Essentially, the Hyperboli Approximation Proedure minimizes g(x,y) by roughly approximating the right irular ones in Equation (3.1) with hyperboloids, thus foring the derivatives to be defined over the entire spae, minimizing the resulting funtion, and then sequentially making loser approximations until an optimal solution is obtained. A omplete desription of the algorithm and a presentation of omputational experi ene is gi ven in [8]. The relationship between the single faility loation problem and the traveling salesman loation problem will now be given. Consider an initial loation, (xo,yo), for the new faility in the traveling salesman loation problem. In the proess of determining the expeted total distane traveled, f(xo,yo), the optimal routing for eah distint subset of failities to be visited is also determined. Reall that eah of these routes, or itineraries, was previously defined in Chapter 2 by I(Sh)' h = 1,2,3,..., k, and that while the order of visitation was given by I(Sh)' the new faility was exluded from I(Sh)' Sine it is assumed that all tours begin and end with the new faility, there is no need to inlude it in I(Sh). For a speifi subset, Sh' the only existing failities that interat diretly with the new faility, m+l, are the first and last existing failities in the optimal route. These existing failities are given by the first and last elements of I(Sh). In order to failitate the disussion that follows, some additional notation is required. Speifially, let

52 45 1, if the first or last element of I{Sh) equals i 0, otherwise. Note that R = «r ih )) is an m x k matrix with eah of the first m olumns having (m-l) zero entries and a single entry of one, and with eah of the remaining (k-m) olumns having (m-2) zero entries and two entries of one. (3.4) The first m olumns have only one entry of one beause the first m subsets of failities to be visited ontain only one existing faility, and onsequently, the first element of I(Sh)' h = 1, 2, 3,. 0' m, is also the last element of I(Sh)' Therefore, for eah of the first m subsets of existing failities to be visited the new faility will interat diretly with a partiular existing faility twie, one on the trip to the existing faility, and one on the return trip. After the first m subsets, however, there are two or more existing failities ontained in Sh' and sine traveling salesman tours are required, the new faility must interat diretly with two, and only two, existing failities. The diret interation takes plae between the first existing faility visited and the last existing faility visited. This explains why the R matrix is omposed of all zeroes and ones. The R matrix is determined by solving the "string" of traveling salesman problems when the new faility is loated at the oordinate position (x,y). For example, onsider the ase where m = 4, and k = 2m-l or 15. The R matrix may appear as follows: (i a a 1 0 a R = a I)

53 46 The first 10 olumns of R are stritly determined sine the solutions to the traveling salesman problems involving one or two existing failities are trivial. The entri'es in the last five olumns of R are not stritly determined, but rather are dependent on the optimal routes given by the solutions to the traveling salesman problems for 5 11, 5 12, 5 13, 5'4' and 5'5' It was demonstrated in Chapter 2 that the determination of the non-trivial routes is dependent on the onfiguration of existing failities and the partiular oordinate loation of the new faility. Reall that in this ase, the non-trivial routes are given by one of 16 possible traveling salesman zones, depending on the new faility loation. Given the initial loation of the new faility, (xo,yo), and the resulting itineraries, I O (5 h ), h = 1,2,3,..., k, the new faility an be optimally loated, inasmuh as the diret interationoetween-new and existing failities is onerned. This is aomplished by minimizing g(x,y) in Equation (3.1), the single faility loation problem. Eah of the appropriate weights to be used is given by k Wi = 2Pi + L Phr h h=m+l 1,i = 1, 2,3,..., m (3.5) where Ph is the probability of visiting subset 5 h, and r ih, Equation (3.4), is the indiator variable for diret interation between the new faility and existing faility i of 5 h. To illustrate the method of forming the weights onsider existing faility number 1 in the ase where m = = {1,5}, and 51 is visited with probability Pl' The result"ing itinerary is 5-1-5, i.e., 1(5 1 ) = ill, so that for this subset, existing faility 1 interats diretly with the