JEFFREY ALLAN ROBBINS. Bachelor of Science. Blacksburg, Virginia

Transcription

1 A PROGRAM FOR SOLUtiON OF LARGE SCALE VEHICLE ROUTING PROBLEMS By JEFFREY ALLAN ROBBINS Bahelor of Siene Virginia Polytehni Institute and State University Blaksburg, Virginia 1974 II Submitted to the Faulty of the Graduate College of the Oklahoma State University in partial fulfillment of the requirements for the Degree of MASTER OF SCIENCE May, 1976

2 OKlAHONtA STATE LN-.JIYERS!TY LIBRl\RY AUG :.: A PROGRAM FOR SOLUTION OF LARGE SCALE VEHICLE ROUTING PROBLEMS Thesis Approved: na~ Dean of the Graduate College ii

3 PREFACE This researh is onerned with the development and implementation ' I of a man-mahine proedure to solve large sale routing problems. A user of the proedure needs no omputer programming experiene but should be familiar with the routing system under study. The author wishes to express his appreiation to his major advisor, Dr. Wayne C. Turner, for his guidane and assistane throughout this researh and studies leading to the area of routing. Appreiation is also expressed to the other ommittee members, Dr. Kenneth E. Case and Dr. Syed z. Shariq for their assistane during the researh and in the preparation of the final manusript. A note of thanks is given to Dr. David Byrd tor his assistane in the omputer programming. Thanks are also extended to Dr. Carl B. Estes for allowing the author an opportunity to develop his writing skills, to Margaret Estes for the exellene of the final draft, and to the Industrial Engineering and Management Department for the omputer funds to perform this researh. In addition, appreiation is given to my parents, Sam and Carol Robbins, for their enouragement and ontinued interest. Finally, speial gratitude is expressed to my wife, Alie, for the understanding, patiene, and enouragement given to her part-time husband. iii

4 TABLE OF CONTENTS Chapter I j I. INTRODUC'JI'ION Problem Definition Objetives of the Researh Assumptions Outline of Thesis II. LITERATURE SEARCH Introdution Traveling Salesman Problem Routing Problem Summary Page III. IV. THE CLARKE AND WRIGHT ALGORITHM Introdution The Clarke and Wright Algorithm Programming Strategies THE TWO-OPT ALGORITHM Introdution Two-Opt Requirements Programming Strategies V. COMPUTATIONAL RESULTS Introdution Bak'ground of Problems Examination of Results Statistial Analysis Additional Computational Experiene VI. MAN-MACHINE INTERACTION Desription Reasons For Use Usage Real World Implementation iv

5 Chapter VII. CONCLUSIONS AND FUTURE RESEARCH Conlusions Future Researh and Extensions BIBLIOGRAPHY Page APPENDIX A~-USER INSTRUCTIONS 58 APPENDIX B--COMP,UTER PROGRAM APPENDIX C--COORDINATES AND DEMANDS FOR SIX RANDOMLY GENERATED PROBLEMS APPENDIX D--USER INSTRUCTIONS FOR INTERACTION PHASE v

6 LIST OF TABLES Table I. II. III. IV. v. VI. VII. VIII. IX. x. XI. XII. XIII. XIV. xv. XVI. XVII. XVIII. Distane Matrix Savings Matrix Ordering of Savings List of Problems Comparison of Algorithms Solution Results and Differenes for Sign Test Exeution Times and Differenes for Sign Test Solutions to Randomly Generated Problems Node Demands. Node Distanes Data Cards Problem l Problem 2 Problem 3 Problem 4 Problem 5 Problem Interation Data Cards Page vi

7 LIST OF FIGURES Figure 1. Branh andl Bound Tree 2. Eight Ways of Conneting Three Ars 3. One Vehile Serviing Eah Node.. 4. One Vehile Serviing Two Nodes 5. Node Loations 6. Clarke and Wright Route 7. Flowhart of Clarke and Wright Algorithm 8. Two Independent Two-Opt Routes 9. Swapping Ars Between Routes Page System of Two Routes Replaement of Two Ars on One of Two Routes Initial Route and Two Swaps Sequene of Two-Optimal Changes Within One Tour Flowhart of Lih Two-Opt Algorithm Computation Time Versus Problem Size Storage Requirements Versus Problem Size 17. Yale Shool Distrit Initial Clarke-Wright/Lin Solution Interation Solution vii

8 CHAPTER I INTRODUCTION Problem Definition A well known problem to ertain segments of soiety is the "Transportation Routing Problem" or simply "Routing Problem." This problem an be stated as follows: Goods are to be distributed from a soure to a known set of destinations. These goods are arried by a fleet of arriers of known apaity. An analyst must assign eah arrier one or more destinations so that the arriers, starting at the soure, deliver the goods to eah of the assigned destinations and returns to the soure. Eah destination is traveled to only one. The objetive is to minimize the total distane traveled during delivery. [24, p. 288 J Manifestations of this problem appear in many diverse setors of the eonomy. In the publi setor, analysts are onstantly routing shool buses, street sweepers, snow plows, refuse olletion vehiles, and other servie vehiles. In the private setor, industries route vehiles to serve warehouses or branh stores. In the quasi private setor, the u.s. Post Offie Department is faed with a multitude of different routing problems. Finally, many prodution sheduling problems an be given a vehile routing formulation. The problem, therefore, is one that is presently reeiving a great deal of interest. The majority of real life situations deal with symmetrial distanes. That is, if d.. denotes the distane between destinations lj 1

9 2 (nodes) i and j, then d.. = d.. vi and j l.j Jl. ( 1.1) The asymetri ase allows for the possibility of d.. I d.. l.j Jl. (1.2) This is not frequently enountered in real world situations exept when the problem involves one way streets or bridges. This researh is appliable only to the symmetri ase although it would be simple to alter for asymmetri problems. Regardless of symmetry, all routing problems are ombinatori in nature. This means an exeedingly large number of ombinations is possible. Assuming eah pair of nodes is linked and distanes are symmetri, the total number of different possible routes through N points is ~1. For example, a group of 12 nodes an be servied by any one of 239,500,800 routes. As an be seen, exhaustive enumeration is infeasible for all but the smallest problems. Objetives of the Researh Many people have developed proedures, manual and omputer, that will optimally or heuristially solve routing problems. Optimal seeking proedures are interesting intelletually but, so far, unrewarding realistially. A problem involving 20 to JO nodes an be solved optimally but with large storage and omputational time requirements. Problems of more than about 50 nodes simply annot be solved optimally.with today's tehnology. Heuristi proedures, however, seem to offer hope for good solutions to large sale problems. Thus, the basi objetives of this

10 3 researh are to explore existing heuristi proedures and attempt to disern those that offer the most promise. Then, the next objetive is to effiiently program the hosen tehnique or ombination of tehniques in order to produe a proedure that: (a) (b) () handles "large" symmetri problems, produes good solutions, is easy and eonomial to use. Another objetive is to allow the analyst to interat with the solution proedure so that he does not have to blindly aept an answer. This inreases the validity of the solution and the faith of the analyst in the final result. A omputer program is developed in this thesis that ombines two of the better heuristi proedures into one effetive and effiient program. Interation is allowed and enouraged. Computation results are presented. Assumptions The following assumptions are made: 1) Capaity of eah arrier is known. (Usually all arriers have the same apaity but this is not a requirement.) 2) Distane between any two nodes, inluding the soure (depot), is known. The distanes are usually put into the form of a \ matrix and must be symmetri. 3) Demand at eah node is known. ~) All arriers or vehiles start and end at the depot. There an be only one depot. 5) Eah node is servied one and only one by some vehile. l

11 The above assumptions are onsistent with the problem definition previously given and the literature searh to be overed in Chapter II. Outline of Thesis The results are presented in seven hapters. Chapter I, this hapter, defines the problem and states the objetives of the researh. Chapter II reviews the existing literature on Routing Problems. Chapter III disusses the Clarke and Wright route building algorithm and programming proedures. Chapter IV presents the Lin route improvement proedure. Chapter V details the omputational experiene. Chapter VI disusses the "interation phase" and its use. Chapter VII presents the summary, onlusions, and ideas on extensions and future researh.

12 CHAPTER II LITERATURE SEARCH Introdution.The vehile routing problem was probably first formulated by Dantzig and Ramser [9]. In this early paper, they showed the vehile routing problem to be a generalization of the lassial traveling salesman problem (TSP). Sine muh of the work on vehile routing problems draws heavily from traveling salesman literature, a brief review is given below and is followed by the review of vehile routing literature. Traveling Salesman Problem The TSP is one where a salesman, starting in his home ity, wishes to establish an itinerary suh that he visits eah of N other ities one and only one and travels a minimum distane. Many solution proedures have been developed but all an be lassified as either an "optimal seeking" or "heuristi" proedure. Qptimal Seeking Proedures Optimal seeking proedures are those methods that guarantee an optimal solution. Sine the problem is ombinatori in nature, however, optimal seeking proedures require exessive omputational 5

13 6 time and storage requirements exept for small textbook type problems. Eastman [lo] was the first to solve the TSP exatly. His method is based on a branh and bound strategy where the assignment algorithm is used for omputing bounds. Little, et al. [19], use an approah similar to Eastman's in that bounds are used on the assignment problem. The proedure begins by reduing the assoiated ost or distane matrix until a zero exists in every row and olumn. The total redution is a lower bound on the solution. Penalties are then reated for eah zero in the matrix by alulating a ost, or penalty, assoiated with~ hoosing the orresponding (i,j) of a zero entry. The zero entry with the highest penalty is then plaed in a "tree". The tree (Figure 1) ontains two sets of branhes. One branh ontains all tours inluding (i,j); the other ontains all tours exluding (i,j), i.e., (i,j). As branhes are explored, the penalty assoiated with not traveling (i,j) is added to the previous lower bound. In addition, as (i,j) is added to the branhing tree, the orresponding row and olumn are deleted from the ost matrix. A ontinuation of this "branh and bound" tehnique results in the optimal tour. Bellman [2] disusses the formulation of the TSP as a dynami programming problem. The proedure begins by onsidering the problem as a 'multistage" deision problem. Then, starting at any node and using dynami programming, Bellman shows the resulting tour to be optimal.

14 Figure 1. Branh and Bound Tree 7

15 8 Heuristi Traveling Salesman Algorithms Bellman, Eastman, and Little, et al., all were able to solve the TSP optimally. However, the omputation time and storage requirements inreased exponentially in eah ase. For this reason, only problems with a small number of nodes ould be solved. Large problems, approximately 50 nodes or more, must be handled by heuristi tehniques. Ashour and Parker [1] suggest using a heuristi where, after starting with some initial node, the.~~~-!_ unvisited node is traveled to next. This proedure ontinues until a yle exists produing the route of the traveling salesman. The proedure is repeated with different initial nodes until all have been used as a starting point. The best of the tours is hosen as the solution. The most importan t heuristi, with respet to this thesis, was. proposed by Croes [8] and extended by Lin [18]. Croes' algorithm revolves around the idea of removi~g two ars from a route and replaing with two different ars suh that the distane of the route is redued. The Lin algorithm is disussed thoroughly in Chapter IV. Other proedures for solving the TSP, exat and heuristi, have been found and are disussed by Bellmore and Nemhauser [4]. It is reommended as a referene. Routing Problem Optimal Seeking Proedures As previously stated, omputation time for solving the TSP inreases quite rapidly as the number of nodes inreases. This same problem ours in routing problem algorithms. Still, optimal seeking

16 9 routines have been developed. Eilon, et al. [11], use a proedure similar to Little, et al. [19], to optimally solve small routing problems. Svestka and Hudkfeldt [22] extended the work of Bellmore and Malone [J] to solve a multiple traveling salesman problem (MTSP), whih is very similar to the vehile routing problem. There are other optimal seeking proedures, but this researh onentrates on heuristi proedur-es for solving large sale vehile. ' routing problems. Therefore, the majority of the literature searh is in the next setion where heuristi algorithms are overed. Heuristi Proedures The solution of a problem with a large number of nodes requires a tehnique that: {a) (b) () is fast, produes good results, and. uses reasonable omputer time and storage spae. There are many heuristi solution proedures to the Routing Problem that meet at least one and possibly all of the above requirements. The algorithms relevant to this thesis are disussed here. For a more exhaustive survey of the Routing Problem see Turner, et al. [24], or Bodin [5 ]. As previously mentioned, Dantzig and Ramser [9] were the first to formulate the Routing Problem. In addition to their formulation, Dantzig and Ramser also proposed a heuristi solution proedure. Their heuristi was based on building routes that filled truks to apaity rather than minimizing the total distane. Clarke and Wright [7] extended this work to onsider the minimization of distane as the sole

17 10 objetive. Clarke and Wright first assume the existene of one arrier for eah node. Then, if one truk servies any two nodes, the following savings an be alulated for that pair of nodes: S.. 1J d. + d. - d.., V i and j, i -J j 01 JO 1J (2.1) In the savings equation, d.. is the distane or ost from node i to 1J node j and o denotes the depot. All savings are then arranged in desending order. The proedure, starting with the pair of nodes having the largest saving, builds routes by ombining feasible pairs of nodes in the above order. At every ombination, apaity and distane onstraints are heked and the proedu're ontinues until all nodes are on a route. A more detailed examination of the Clarke and Wright heuristi follows in Chapter III. Gaskell [12] proposed a heuristi that is a slight modifiation of the Clarke and Wright algorithm. He proeeds in the same manner as Clarke and Wright exept the savings are alulated as either: A.. = S.. ( D + I d. - d. \ - d.. ), or 1J 1J 01 OJ 1J TT. s.. - d.. 1J 1J 1J (2.2) (2.3) where N \ d. L o1 i=l N and S.. is the Clarke and Wright savings. 1J Gaskell alulations give more weight to nodes with high d. 's. 10 Robbins, et al. [21 ], have shown, using randomly generated problems,.th.e Clarke and Wright method to be at least as good as Gaskell 1 s first savings alulations on the problems examined.

18 11 Tillman and Cohran [23] also extended the work of Clarke and Wright. Their method hooses the pair of nodes with the best savings suh that the seon.d best feasible pair may be hosen. This manner of hoosing the best two feasible pairs of nodes maximizes the savings over four nodes, not two. Golden, et al. [14], proposed a heuristi that modifies the Clarke and Wright algorithm in three ways: 1) by using a 'route shape' parameter Y to alulate a new savings s.. = d. + d. - Yd.. 1J 01 JO 1J The value of Y is varied over some range and the best set of routes are seleted; 2) by only alulating a savings between nodes lose to eah other; and J) by storing the savings alulations in a 11heap struture" to redue omparisons and improve the speed of the algorithm. Robbins C2oJ has shown it possible to generate better Clarke and Wright solutions using the seond modifiation above. It should be stated,-though, that this happens rarely and the quality of the solution will in general be worse than when all savings are alulated. Christofides and Eilon [6] developed an algorithm that solves routing problems by a three-optimal tour method. The three-opt method, as it is alled in the literature, begins with a set of random routes. The proedure ontinues by removing three ars from a route and replaing With three d1"fferent ars. F1"gure 2 1 f th t d, an examp e o ree unonne e ars, demonstrates eight ways they an be reonneted to form a route. When this is repeated for all ombinations of three ars, a three-optimal tour is obtained.

19 12 ~\ I I ; I. ~/ /,l, < /"., ',., '., ' ' I ~~. I ~---~~.~ I Figure 2. Eight Ways of Conneting Three. Ars

20 13 Gillett and Miller [13] reently proposed a proedure alled the ~~~~~P C:llg()!:i_t]:~m"~, The method involves.l.al>.e:!!ing every node with its polar oordinates. The nodes are then numbered aording to polar oordinate with the depot onsidered as loation 1. A sweep is made suh that nodes 2, 3, 4,, K are plaed on the first route. Node K is.the last node on the route that an be servied without exeeding the arrier's apaity. This proess ontinues until all nodes are on a route. A swapping routine then follows where stops are transferred between wedges if a redution in distane is realized. A useful aspet of the algorithm is the time it takes to solve a problem. The authors report a linear inrease in time with respet to the number of nodes if the number of nodes per route remains approximately onstant. Krolak, Felts, and Nelson [16] have developed a man-mahine two phased proedure for routing problems. The proedure is very similar to that proposed by Krolak, Felts, and Marble [15] for the TSP. The first phase onsists of a heuristi, alled the "Truk-Route Generator", to aggregate n odes aording to their loation. Grouping sizes are kept small to reate many lusters. Aggregates are ombined where the total demand does not exeed the apaity of the largest vehile. Results are probably not feasible but a TSP is solved for eah group. Swaps between routes are then analyzed and made where possible. Finally, feasible one and two ar swaps are made within eah route. The seond phase of solution takes plae at a athode ray tube. The CRT displays the solution so an analyst an alter the routes in any way he desires as long as they remain feasible. This sort of interation allows experiene and intuitive knowledge to ahieve better solutions. This,

21 in turn, makes the results more aeptable to an analyst as he plays a part in deriving the routes. International Business Mahines (IBM) [17] has a omputer pakage alled Vehile Sheduling Program (VSP) that has been used to route vehiles. The progra!ll is primarily based on the Clarke and Wright proedure disussed earlier. Different fators an be minimized suh as time, distane, or number of vehiles used. Where the designated fators annot be minimized, the program tries to ahieve a balane among them. In addition, VSP is well programmed and an handle over 1000 nodes. Summary Routing problems an be solved by many algorithms. Some proedures are exat while others are heuristi. Exat solution pro~ edures generate optimal answers but are only pratial for problems up to about 50 nodes. Large sale problems must be solved by heuristi tehniques. Of the heuristis, the Clarke and Wright [7] method has been given the most attention. Gaskell C12J, Tillman and Cohran [23], Golden [14], and IBM [17] have extended Clarke and Wright to produe proedures of their own. Other methods inlude those proposed by Gillett and Miller bjj, Krolak, et al. b6], and Christofides and Eilon [6]. The next hapter presents the Clarke and Wright algorithm in detail. Also, an effiient program of the proedure is outlined.

22 CHAPTER III THE CLARKE AND WRIGHT ALGORITHM Introdution This hapter disusses the Clarke and Wright algorithm [?] mentioned in Chapter II. The theory behind the algorithm.and an example showing its use are presented along with the advantages and disadvantages. In addition, a omputer program of the Clarke and Wright method is outlined. The Clarke and Wright Algorithm The Clarke and Wright routing proedure is based upon the alulation of a "saving" between every pair of nodes. The method initially assumes the existene of one vehile to servie eah node. This is unrealisti as a vehile usually an serve a great number of nodes. Clarke and Wright realized this and asked the question, 11How muh distane will be saved ifone truk servies two nodes instead of one?" As shown in Figures 3 and 4:, the savings assoiated with pairs of two nodes an be alulated as: s.. l.j d. + d.. - d.. Ol. JO l.j Vi and j, i /: j (J.1) where S.. is the savings assoiated with pairing i and j on one route, l.j d.. is the distane from ito j, and o denotes the depot. "Distane" l.j may be replaed by 'money, time," or any other sare resoure. 15

23 16 7 I J O(DEPOT) Figure 3. One Vehile Serviing Eah Node I J O(DEPOT) Figure ~. One Vehile Serviing Two Nodes

24 17 Equation 3.1 is valid for symmetri or asymmetri distanes. Sine this researh deals with symmetri distanes, the asymmetri ase will not be disussed further. For symmetri distanes, the savings equation may be rewritten as: s. = d oi + d.. - d.., or 1j OJ 1J s. = d. + d. - d. 1j 10 JO 1j (3.2) (3.3) One the S.. are alulated for all pairs of nodes, the savings 1J are ranked in non-inreasing order. Going down the savings list, eah pair of nodes is examined.* For any given pair, an attempt is made to: 1) reate a new route, or 2) add a node to the front or bak of a route, or 3) join two routes to form one. If none of the above an be aomplished, the nodes are disarded. This proedure ontinues until all pairs of nodes have been onsidered. An example is given below. Example Table I is a matrix of symmetri distanes. Sine the distanes are symmetri, only half the matrix is needed. From the distane matrix the savings in Table II an be generated. For example, the savings for nodes 3-4 is: * = = 16 s 4,3 Ties are broken by randomly seleting a pair.

25 18 TABLE I DISTANCE MATRIX j i llj, lj, 10 TABlE II SAVINGS MATRIX llj, llj, lj, 8 The savings of Table II are arranged in non-inreasing order in Table III. The pair of nodes with the largest saving is 1-2. These stops beome the first ombination and form a route of

26 19 TABLE III ORDERING OF SAVINGS Stops Savings Stops Savings Next on the list is the ombination 2-1. The addition of 2-1 to J. the first route reates a "subtour" whih is not permitted. The next pair is 3-4. Sine route 1 has no stop in ommon with this pair, a new route must be formed. Thus forms route 2. Thus far, the routes appear as follows: Route # Stops o The next pair, 4-3, is dropped due to subtours. 1-3 is dropped sine stop 1 has previously been traveled from. Likewise, 3-1 is dropped. If 1-2 were onsidered as 2-1, sine the distane is symmetrial, the link 1-3 ould be used to reate one route from two: o. This flexibility is diffiult to inorporate into a program. Thus, a rule for the program is established: One an order of ars is estabiished, the sequene will not be reversed.

27 20 The next pair, 2-3 allows two routes to beome one: o. Continuing down the list, all pairs are dropped exept 5~1. Stop 5 is appended to the front of the route. Thus, after adding the depot, the path.beomes o with a distane of 54 units. The loation of nodes is shown in Figure 5 with the tour in Figure 6. Table I ontains retalinear distanes while Figure 6 shows the route with direted ars. Advantages and Disadvantages The Clarke and Wright algorithm has been very popular as indiated in Chapter II. Beause of this exposure, many strengths and weaknesses have ome to light. The following is a summary of the advantages and disadvantages of the proedure: Advantages: 1) The proedure is simple to use. 2) A realisti onstraint an be added easily. 3) The proedure provides a "good" starting solution whih an be used as input to an improvement algorithm. Disadvantages! 1) One an ar or link is reated it annot be broken. 2) Results an be, but generally are not, optimal. In some ases where the onstraints are tight, results are far from optimal. 3) A omputer is required for most problems due to size (but this is true for most proedures). The first advantage needs little elaboration. The proedure is straightforward and based on simple alulations. Realisti..

28 ' "O(DEPOT) 2 3 i ~ I - I - ' - -4 Figure 5. Node Loations - -~ v ~ _... ~ ~ 5 / ~-"" ~ ~ 1 / i' " '" ' ~...,... ~ r'\.0( DEPOT) ~ ~ - ~ ~ ~ ~ I "~ 3 '\ 1\.. I ~ I - ' 2 ' I " 4 Figure 6. Clarke and Wright Route

29 22 onstraints, suh as maximum route length or required times of pikup, are easy to add. The ability to provide a good initial solution to a following improvement algorithm is ruial to this thesis. The Clarke and Wright algorithm generates a starting solution for use by the Lin two-opt proedure whih is outlined in the next hapter. The two-opt proedure attempts to improve an initial solution. If the starting solution is poor, the final solution may be poor. Likewise, a good starting solution will result in at least a good final solution. Primarily, it is this reason why the Clarke and Wright algorithm was hosen to build routes. The disadvantages, while valid for the Clarke and Wright proedure, are partially' remedied by the Lin two-opt algorithm. With respet to the first disadvantage, the two-opt proedure, whih is a within-route swapping or perturbation routine, is used to break up links. Also, the third phase, man-mahine interation, allows further alteration of routes. The seond disadvantage must be kept in proper ontext. As previously disussed, optimal seeking algorithms use exessive omputer time and normally are not pratial. Thus, any attempt to find the optimal solution to a large sale routing problem is usually not feasible. As the third advantage states, this proedure provides a 11good 11 starting solution for an improvement algorithm. The-Lin two-opt proedure will, in general, improve the results enough to make the solution reasonable. The man-mahine interative phase may improve it even more.

30 23 Programming Strategies In this setion the programming strategies inorporated in oding the Clarke-Wright algorithm are disussed. The program is written in FORTRAN with partiular eff'ort direted at reduing storage requirements and improving exeution speed and effiieny. A flowhart of the program appears in Figure 7. Usage instrutions are given in Appendix A while the program appears in Appendix B. In order to effiiently arry out the Clarke-Wright proedure, it is neessary to retrieve speifi information about nodes, distanes, and routes without extensive searhing. In order to aomplish thi~ the program inorporates a onept known as "linked list proessing". Simply stated, linked list proessing employs a set of pointers whih provide rapid aess to stored data. Use of these pointers generally results in less data manipulation and therefore faster exeution times. The program requires as input the number of nodes in the system, demand at eah node, and a distane matrix. This distane matrix onsists of the distanes between all nodes in the system. Sine these systems may be quite large, all distanes and related route information \. are stored in 'half-word integer variables, i.e., INTEGER*2. This,J_J redues storage requirements by roughly one-half. Also by limiting omputations to stritly integer variable~ exeution speed is greatly enhaned. After the data has been read in, the savings alulations are performed. Nex~ these savings must be ordered. During this sorting proess, an initial set of pointers is established. Through the remainder of the program, these and other pointers will be revised to

31 ENTER NUMBER OF NODES AND DISTANCE MATRIX CALCULATE SAVINGS FOR ALL PAIRS OF NODES ORDER SAVINGS IN DESCENDING ORDER PICK BEST LINK AND CREATE ROUTE 5 J--.., ~ SCRATCH PREVIOUS BEST LINK AND CONSIDER NEXT BEST LINK CREATE NEW NO ROUTE Figure 7. Flowhart of Clarke and Wright Algorithm

32 25 yield suh information as: (a) Is a node on a route? (b) Whih route is a node on? () Where on a route is the node loated? This information is then used to generate or improve potential routes. The updating of routes and pointers ontinues until the proedure iterates to ompletion.

33 CHAPTER IV THE TWO-OPT ALGORITHM Introdution As originally oneived, the Lin two-opt algorithm [Is] attempts to improve a traveling salesman tour. Given a route through N nodes, two ars are removed from the iruit and replaed with two different links. If a redution in total distane is realized by this swapping of ars, the new links are retained and the tour is reexamined. Otherwise, the proposed links are disarded and examination of ars ontinues. The proedure stops when a tour has been totally examined and no swaps have been made. The two-opt proedure an be easily extended to routing problems. A feasible solution to a routing problem ontains M routes, where M >1. The ase where M = 1 is a traveling salesman problem. By onsidering eah route independently of the others, a two-opt proedure an be used for possible improvement. Figure 8 shows two routes whih, when onsidered independently, annot be improved further by the two-opt method. If a swap of ars between routes is onsidered though (7-0 and 9-q to replae 7-q and 0-9), the total distane might be redued. Figure 9 shows the results of making the proposed swap. Unfortunately, between-route swapping is diffiult to program. Thus, it is not inluded in this thesis. Instead, the man-mahine interation phase is inorporated whih allows for analysis of between-route swapping. ' 26

34 (DEPOT). _.,.,.,.,.,., Figure 8. Two Independent Two-Opt Routes (DEPOT) 6 Figure 9. Swapping Ars Between Routes

35 28 Figure 10 shows two routes, one of whih an be improved with the two-opt proedure. Route A is two-optimal (when onsidered independently) while route B an be improved sine it has two links that interset, 6-4 and 3-0. Lin [18] states that any route ontaining an intersetion of two or more links annot be two-optimal. By swapping iinks 6-4 and 3-0 (the dashed lines, Figure 10) for 6-3 and 4-0 (Figure 11), route B beomes independently two-optimal. Two-Opt Requirements The two-opt proedure, like most other improvement algorithms, starts with an initial feasible solution, whih in this ase is the Clarke and Wright algorithm. In addition to the starting solution, a distane matrix must also be provided so the feasibility of swaps an be determined. A swap of links an be made two ways within a route, but only one of the swaps is realistially possible as shown in Figure 12. The feasible swap in the middle of Figure 12 may result in a better route while the infeasible swap at the bottom of Figure 12 results in subtours, whih are not allowed. Assuming I-J and K-L are the ars onsidered for replaement, the initial route is L-N-M-I-J-T-S-K-L while the possible replaement route is L-N-M-I-K-S-T-J-L. These two routes are idential exept for the links between I and L whih inludes those being onsidered. (The links I-J and K-L are being onsidered for replaement by I-K and J-L.) The remaining links in the replaement route are K-S-T-J. Upon examination of the initial route, the links J-T-S-K are found, whih are the reverse of those in the replaement route. Sine distanes are symmetri, the links from K to J are the same

36 Figure 10. System of Two Routes 5 2 Figure 11. Replaement of Two Ars on One of Two Routes

37 JO 5 T K J I L M N INITIAL ROUTE: L-N M-I-J-T-5-K-L s T K J I L M N FEASIBLE SWAP: L-N-M-1-K-S-T-J-L s T '-J IN FEA S I BL E SWAP : L- N - M- I - L AND J - K- S - T- J Figure 12. Initial Route and Two Swaps

38 31 distane as from J to K. Thus, the links from K to J need not be onsidered when examining a swap. This leaves the ars I-J and K-L as the ~s for a swap. Sine I-K and J-L are being onsidered to take the plae of I-J and K-L, the following equation must be satisfied: If equation 4.1 is satisfied, a swapping of ars takes plae. Otherwise, the route remains in its present form. Swapping ars results in just two new ars. All other links retain their initial sequene or are the reverse of their original order. It should be pointed out that the entire examination proess ould be reversed sine the distanes are symmetri; but the end result wohld be idential. Example Figure 13 is a diagram showing how a route may be improved by the two-opt method. The initial solution (iteration 1) is a route of 68 miles. In iteration 1, the ars 0-1 and 3-2 are onsidered for removal. The replaement links are 0-3 and 1-2. Sine a redution in distane an be made (from 68 to 63 miles) the swap is made in iteration 2. This swapping of ars ontinues, where swapped ars are dashed lines, in iterations 3 and 4 until the minimum distane of 5~ miles is reahed. Programming Strategies Sine the first phase of the program is a Clarke and Wright proedure, a good initial solution is obtained. This is then used to feed into the two-opt within-route proedure, a flowhart of whih is shown in Figure 14. The two-opt algorithm makes use of the pointers established in the Clarke and Wright proedure. Whenever a swap is deemed feasible, the pointers are rearranged and the ro~te is reexamined starting from the beginning. This reexamination is neessary beause

39 ITERATION 1 68 MILES 5,, I I I I 3 4 I.. O(DEPOT) ITERATION 2 63 MILES I I I I -.. I..._..._, I I......_..., 3 ITERATION 3 55 MILES ITERATION 4 54 MILES NO FURTHER IMPROVEMENT Figure 13. Sequene of Two-Optimal Changes Within One Tour

40 33 INPUT INITIAL SOLUTION AND DISTANCE MATRIX NO NO NO Figure 14. Flowhart of Lin Two-Opt Algorithm

41 new links are introdued into the route whih may make more swaps feasible. The algorithm ends when a route is examined and no swaps are made.

42 C1IAPTER V COMPUTATIONAL RESULTS Introdution This hapter presents some omputational experiene of the program developed in this thesis. Problems solved by Christofides and Eilon [6], Gaskell [12], and Gillett and Miller [13] are examined. Bakground of Problems Christofides and Eilon [6] solve ten vehile routing problems with their three-opt proedure (and other proedures) and ompare the results to those obtained by one of Gaskell's savings algorithms [12]. In most ases, the three-opt proedure produes better results but takes onsiderably more time for solution. Of the ten problems, seven have less than fifty nodes and are not examined here. The three remaining problems (50, 75, and 100 nodes) are solved by Gillett and Miller [13] along with five others. These eight problems are presented in Table IV. Problems 3, ~' and 5 are the same as 2 exept the maximum load is altered. Likewise, problem 7 is the same as 6 exept for the load onstraint. Solution results are presented in Table V. 35

43 36 TABLE IV LIST OF PROBLEMS Problem Number a Maximum Author Number of Nodes Load 1 Christofides and Eilon [6] Gillett and Miller [13] Christofides and Eilon [6] Gillett and Miller [13 J Gillett and Miller [13 J Gillett and Miller [13 J 100 ll2 7 Christofides and Eilon [6] Gillett and Miller [13] ~xludes depot.

44 TABlE V COMPARISON OF ALGORITHMS Three-Opt Gaskell Sayings Swee;12 Algorithm (Proposed A1 gori thm) Clarke-Wri~htLLin Problem Number Sol. Rts. Min. Sol. Rts. Min. Sol. Rts. Min. Avg. No. nodes/route Sol. Rts. Min. Avg. No. nodes/route ~ ~.o ~ 75~ ~ ~ ~ ~ ~ ~.12 5-~ ~ ~ ~ 9.6 \,;.) --J

45 38 Examination of Results Table V shows the results of four different sol uti on proedures on eight problems. The four proedures are: (a) Three-Opt Proedure (results available on only three problems), (b) Gaskell Savings Proedure (results available on only three problems), () Gillett and Miller Sweep Algorithm, (d) Clarke and Wright-Lin Algorithm (proedure developed in this thesis). The sweep algorithm outperforms the three-opt and Gaskell savings routine for problems where omparisons an be made (1, 3, and 7). Computer times are diffiult to ontrast sine eah algorithm was programmed on a different omputer. Thus, one might onjeture (unsientifially) that the sweep algorithm is at least as good on the eight problems as the other two proedures. For that reason, the Clarke and Wright-Lin program will be ompared to the sweep algorithm. The Clarke and Wright-Lin program was run on an IBM at Oklahoma State University,. Gillett and Miller ran the sweep program on an IBM The only signifiant advantage the!em has over the IBM is the ability to handle more programs. Storage apability and exeution speed are virtually the same. Thus, for pratial purposes, the two programs were exeuted on the same omputer. As shown in Table V, the sweep algorithm produes better solutions for all problems exept 2, 6, and 8. The maximum differene in solutions is 5.8% (problem 8) while the sweep algorithm produes

46 39 results averaging 1% better than the Clarke and Wright-Lin proedure. Thus, on the eight problems examined, these two methods perform almost equally as far as solution is onerned. Another fator of interest is the omputer time of the two programs. Computer time is defined as the exeution time of the program. Gillett and Miller report a linear inrease in omputer time with an inrease in the number of nodes if the number of nodes per route remains relatively onstant. Also, for arty given route, the omputer time inreases as the number of nodes per route inreases. This an be seen in Table V for the 75 node problem. Computer time ranges from.68 minutes to 3.68 minutes while the average number of nodes per route varies from 5 to Thus, omputer time may beome exessive for a problem with a large number of nodes per route. Computer time for the Clarke and Wright-Lin proedure is independent of the number of nodes per route. Thus, although the average number of nodes per route inreases from 5.4 -to 10.7 (Table V), the time for solution of the four 75 node problems remains onstant (.12 minutes). Additionally, omputer time for the Clarke and Wright- Lin proedure is signifiantly less than the sweep algorithm for eah of the eight problems onsidered. As in most heuristi solutions though, omputer time inreases exponentially with the number of nodes. Table V shows omputer time inreasing' from.07 minutes for 50 nodes to 2.74 minutes for a 249 node problem. For some number of nqdes, the exeution time for the two proedures should be approximately equal. These results show that this probably ours somewhere around 350 nodes. Thus, for problems ontaining less than 350 nodes, the Clarke and Wright-Lin algorithm is probably best to us~ as omputer time will be \')I

47 4:0 smaller than for the sweep algorithm. Past 350 nodes, the sweep algorithm is probably best to use. Statistial Analysis For a statistial omparison of the proposed algorithm and the sweep proedure, it is probably best to turn to non-parametri tests. The sign test is used to test two hypotheses. The first, whih is based on solutions generated by the eight problems, an be stated as: H 0 : P(X 1 > X 2 ) = P(X 1 < X 2 ) =.5 H 1 : P(X 2 < X 1 ) >.5 where x 1 and x 2 are the solutions of the Clarke and Wright-Lin program and sweep algorithm resp.:etively. This is a one tailed test and the ritial test value, for a 5% signifiane level, is 3.84:. Table VI shows three positive and five negative signs. The experiment's statisti an be alulated as: == 5 Sine.5 is less than 3.84:, the null hypothesis, H 0, annot be rejeted. Thus, there is no statistial differene between the solutions of the Clarke and Wright-Lin algorithm and the sweep algorithm. The seond hypothesis an be stated as: H 0 : P(X 1 > X 2 ) = P(X 1 < X 2 ) =.5 H 1 : P ( x 1 < x 2 ) > 5 where x 1 and x 2 are the exeution times of the Clarke and Wright-Lin program and sweep algorithm respetively. This is also a one tailed

48 TABLE VI SOLUTION RESULTS AND DIFFERENCES FOR SIGN TEST Problem Sweep Clarke-Wright/ Sign of Number Algorithm Lin Differene ~

49 4:2 test with a ritial test value of 3.84: for a 5% signifiane level. Sine Table VII shows eight positive signs, the following statisti an be alulated: = B.o Sine 8.0 is greater than 3.84:, the null hypothesis an be rejeted. Thus, it seems safe to suggest the Clarke and Wright-Lin program is at least as fast in exeution time as the sweep algorithm. Ad'di tional Computational Experiene Computational experiene using the program proposed in this thesis has also been gained by solving six randomly generated problems. Node oordinates and demands are given in Appendix C and the solution results are presented in Table VIII. Note that as the number of nodes inreases, the exeution time and required storage inrease exponentially. Figure 15 is a graph of the exeution time versus number of nodes while Figure 16 shows the graph of storage requirements versus number of nodes. In both ases, the urvature is the lassial exponential urve.

50 TABLE VII EXECUTION TIMES AND DIFFERENCES FOR SIGN TEST Problem Sweep Clarke-Wright/ Sign of' Number Algorithm Lin Dif'f'erene TABI.E VIII SOLUTIONS TO RANDOMLY GENERATED PROBLEMS Problem Number of' Exeution Time Number Nodes Solution Routes (seonds) Memory Requirements (Bytes) , , , , , ,254

51 0 I.C) 0 (1) 0 1"- 0 I.C) 0 rt') 0 V> w 0 0 z LJ :: w m ~ ::> z Cll N.-i Ul 8 Cll...-!.a 0 ~ p.. Ul ::l Ul ~ Cll > Cll 8.-i E-l s:: 0.-i +' m +' ::l ~ 0 (.). lf\...-! Cll ~ ::l Cl.-i "" 0 I.C) ~ rt') C\J (SaN 0J3 S ) 3 ~ LL NO I.L '1.L n d ~OJ 0

52 w... m... 2 w ~ w 0: :::> 0 w 0: w ~ 0:... 0 (/) I I,' H= Vl 2 50,0QQ.---r r----r------r--r , /,' NUMBER OF NODES Figure 16. Storage Requirements Versus Problem Size -(/) -(/) >- 200,00

53 CHAPTER VI MAN-MACHINE INTERACTION Desription Solution.of a large sale routing problem generally requires the use of a omputer. Information is fed in, an algorithm exeuted, and final.results are printed. In other words, an analyst gathers the data and the omputer does the work. The analyst, by examining the omputer solution, may be able to reommend hanges that result in a redution of distane sine large sale omputer solutions usually are not optimal. This implementation of hanges in a disourse between analyst and omputer is alled "man-mahine interati9n 11 In this interation, an analyst (man) examines the results of a omputer (mahine) and tries to reate a better solution. This is then given to the omputer whih tries to improve the solution further. This interation an be ontinued as long as desired. This hapter disusses the reasons for using an interative phase and how it ours. Reasons For Use This third phase of the omputer program allows an analyst to interat with the omputer to improve the first generated solution. After examination of the results provided by the Clarke and Wright-Lin phases, hanges may be desired. For example, a route may be too long 4:6

54 4:7 or a vehile may not serve enough nodes. These are only two of the many reasons a dispather may wish to alter the omputer's results. Usually, for example, it is human nature for a dispather to have more faith in a solution he helps derive than in the results of a mahine. Also, a dispather is usually very familiar with his territory as he may have spent years learning his trade and gaining experiene. For example, he may know a ertain road is dangerous in high water and should be avoided or a bridge may be overloaded with a full bus but not an empty one. The interation phase allows this experiene to be put to use. Also, many times a dispather may be interested in experimenting with his routes in the hope of finding a better solution. Experimentation an be done easily through the interation phase. The effets of adding or deleting nodes, adding additional routes, or transferring nodes within the system an be realized very quikly and at low ost. With the use of a omputer's speed and memory, many solutions an be generated with the best result hosen for use. Appendix D outlines the steps to be taken when altering the solution generated by the Clarke and Wright-Lin phases. Usage The first two phases of the program provide a solution to the routing problem. The Clarke and Wright algorithm (first phase) generates a solution and the Lin proedure (seond phase) attempts to improve it. The interation phase is also used for route improvement purposes. An analyst first reeives the Clarke and Wright-Lin solution and examines it. If any hanges are desired, the alterations are made and punhed on data ards whih are fed bak into the omputer. The

55 48 omputer then skips the Clarke and Wright phase and attempts t() improve the inputted solution using the within-route two-opt proedure. Results are again printed and the analyst deides what hanges, if any, need to be made. This proess ontinues until the analyst is satisfied with the results provided by the omputer. Real World Implementation Reently, the program of this thesis was used to find new routes for the shool distrit of Yale, Oklahoma. Figure 17 is a piture of the shool distrit with the loation of nodes. Presently, six buses, eah with a fifty-five student apaity, travel 166 miles to pik up two hundred nineteen hildren. Sine the loation and demand of nodes (shool bus stops) is known, the only data needed is the distane matrix. Using a map and roughly forty man-hours, the matrix was generated and oded on data ards. After oding the other neessary data, the program was ready for use. The initial result, provided by the Clarke and Wright-Lin phases and shown in Figure 18, yielded a total distane of miles and five buses. Every route is reasonable exept route one whih is too long (52.5 miles). Thus, the solution is deemed impratial. Were an interation phase not available, the proedure would be a failure for this problem or at least. manual hanges would be neessary. Using the interation phase, an examination of routes one and two (Figure 18) reveals some unneessary overlapping near nodes 72 and 73. Keeping this in mind and the fat that route one must somehow be shortened, a new solution an be proposed. Route one an be broken into two routes. The first route onsists of nodes 67, 66, 64,, 76, 74 in addition to 73 and 72.

56 SCHOOL e NODE >I="" -.!) Figure 17. Yale Shool Distrit

57 j;:--; 65 I 0 L 62~ ,.?Z ~ liz --r 4~ I I.74 :,.;;; L_ L~8... J r ~.!J ROUTE NUMBER SCHOOL NODE Vl 0 rii"..,._....fl r- I"" e:r--- _j. -.,.. I I ~~ I l \ ;-.J 88 I I \\\\ 64 i. ~. 5~ , :r!u ~" u r~. ~ L....J,... ' o1\l L. ril.. - ~ 'i.[_ ~. 47 I ' Ill... Li- 76 n ~15... l.,i Le78 rte : 13il... _ : ~--/ I = I -,._ 1 5 ~ 79 I 83,- 6 : 93 I r- -. ~ ~ :: T'5 ooe L~ l _ ~~! : "T 81,-:-> IJ_:II ' I ~. la W ~.9Z... ~~~ r-: r ~ Figure 18. Initial Clarke-Wright/Lin Solution

58 51 Nodes 73 and 72 formally were part of old route two. route is omposed of nodes 89, 91, 93,, 79 and 75. The seond new These hanges are fed into the omputer and reappliation of the Lin proedure yields another solution. This revised solution is shown in Figure 19 with a distane of miles and six buses. Even though an extra route ie:; added to the system, the total distane inreases by only four miles and the solution is aeptable. This is a good example of an analyst reognizing a onstraint of the system and altering the solution aordingly.

59 (.i.u-- ~. 3,. 62fJ " r+- ~! I " '. ~ -- j. r--e. I,.. I, 63 r , L. -, : : I \._j - tj,.~ I 70 ~ \ I -, L ' -... ' ' ' :_-;;;;] 5I ~. ----, 'l, L \\ r-.,..,. r- ' ' 4_9- - _j'- -~2 1 I le11 ' \ 61 L '~ ~: L...!To L.7 p-- '<; I" L 7! i r.5 ~ --~~r 50,,. r:-_.tl ~- -t-... : ":i' I j ~4 79 I 86 : 93 t..re..., 1' ~J! :,,.., 78 9: I 13WI -~j,l:: :! ~. 82.J "'---~ 2 94 es 96. :... gg... lt7~ 98~ 10, 7. ae I. 17 ROUTE NUMBER SCHOOL e NODE Figure 19. Interation Solution Vl N

60 CHAPTER VII CONCLUSIONS AND FUTURE RESEARCH Conlusions The first six hapters of this thesis present a three phase proedure alled the Clarke and Wright-Lin Interation Program for solution of routing problems. Problems presented in papers by Gillett and Miller [13] and Christofides and Eilon [6] have been solved with the proposed algorithm. Results presented in Chapter V show the proposed proedure to be ompetitive with the Gillett-Miller sweep algorithm as far as quality of solution is onerned. Both perform well on the same problems ru1d no statistial differene an be determined for a signifiane level of 5%. The program developed in this researh, however, has been shown, statistially, to be faster than the sweep algorithm for the problems solved. This should be taken in proper ontext as the proposed algorithm was intended to be programmed effiiently while the sweep algorit~ may or may not have been. Chapter VI disusses the third phase of the proposed program, the interation phase. This phase opens up many alternatives for an analyst. Probably one of the most important aspets is the addition of another resoure to help solve routing problems: the analyst's experiene. This is a virtually untapped resoure that an be turned into an asset if properly used. The Clarke and Wright-Lin Interation 53