Generating subtour elimination constraints for the Traveling Salesman Problem

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1 IOSR Joural of Egieerig (IOSRJEN) ISSN (e): , ISSN (p): Vol. 08, Issue 7 (July. 2018), V (I) PP Geeratig subtour elimiatio costraits for the Travelig Salesma Problem Soufia Behida 1, Ahmed Mir 2 1, 2 (Laboratory of Idustrial Egieerig ad Computer Sciece, Natioal School of Applied Scieces ENSA, Aveue Tamesoult Agadir 80000, AGADIR, MOROCCO) Correspodig Author: Soufia Behida 1 Abstract: -.The travelig salesma problem (TSP) has commaded much attetio from mathematicias ad computer scietists specifically because it is so easy to describe ad so difficult to solve. I this work we solved the Travelig Salesma Problem, with three differet formulatios, the formulatio DFJ (Dazig-Fulkerso- Johso), MTZ formulatio (Miller-Tucker-Zemli) ad DL formulatio (Desrochers-Laporte). The goal of this work is to solve the Travelig Salesma Problem with a big size of etwork, i the first we eplai the resolutio method ad we will preset some umerical result Keywords: - geeratio of costrait, Liear iteger programmig, Travelig Salesma Problem Date of Submissio: Date of acceptace: I. INTRODUCTION Whe we talk about operatioal research, we ecessarily talk about the Travelig Salesma problem (TSP). I this work we are iterested to solve with eact method usig costrait geeratio approach, ad for that we will use three basic formulatios for the travelig Salesma Problem. The travelig salesma problem was formulated i two ways; the first is from Dazig et al. (1954) [1], who proposed the DFJ model usig 2 biary variables, for the ATSP, the secod is of Miller, Tucker & Zemli (1960) [2], who rewrote the problem with differet way, MTZ was proposed for a Vehicle routig problem, the other formulatios appeared reiforcig the MTZ as the DL formulatio [3]. I this work we will solve the travelig salesma Problem with large istaces, usig the three formulatios (DFJ, MTZ ad DL), ad see which of the formulatios is stroger by applyig a strategy of costrait geeratio. I the literature Padberg ad Sug [4] showed that the MTZ formulatio is weaker tha the DFJ formulatio. The weakess of the MTZ formulatio has also bee demostrated by Lagevi et al. [5]. Desrochers ad Laporte [6] reiforced the MTZ formulatio, while Sherali ad Driscoll [7] proposed a reformulatio of the MTZ costraits, the aim of this work is ot oly to compare the formulatios, but also to apply a Costrait geeratio strategy to solve the DFJ model that geerates a epoetial umber of costraits, which makes the resolutio a bit difficult. Figure 1: Solutio of the travelig salesma problem: the red lie is the shortest path that coects all the black poits (cities). Iteratioal orgaizatio of Scietific Research 17 Page

2 Paper preparatio guidelies for IOSR Joural of Egieerig II. DESCRIPTION OF THE PROBLEM The Travelig Salesma Problem is defied o a complete ad directed graph G= (V, A), because we treat Asymmetric TSP (ATSP) (See sectio 3), the set V = {1... } is the verte set (cities) with is the umber of cities, ad is a arc set; a cost matri d is defied o A. The cost matri satisfies the triagle iequality wheever, for all i,, k. III. MATHEMATICAL FORMULATIONS OF TRAVELING SALESMAN PROBLEM: A. The Dazig-Fulkerso-Johso formulatio: The DFJ formulatio ad may ATSP formulatios cosist of a assigmet problem with itegrality costrait ad sub-tour elimiatio costraits (SECs) [1], they use a biary variable equal to 1 if ad oly if arc i, belogs to the optimal solutio ad otherwise it would be equal to 0, S is the sub set of vertices (cities) i the sub tour. The basic model is as follows: i, S Mi 1 i1 d i 1 ( i V, i ) 1 ( V, i ) S -1 (S V, 2 S 2) 0;1 i, A (9) (5) (6) (7) (8) (5) represets the obective fuctio, (6) ad (7) represet the assigmet costraits, (8) is the sub-tour elimiatio costraits (9) is a itegrality costraits, This formulatio becomes difficult i practice, as soo as the size of the model icreases, ad it is due to the epoetial umber of subtour elimiatio costraits 1 geerated, this formulatio of the TSP cotais (-1) variables ad 2 2 costraits. B. The Miller -Tucker-Zemli formulatio: This formulatio was origially proposed by Miller ad all, for a travelig salesma problem, where the umber of vertices of each route is limited [2], they use oe biary variable ad for each arc i, A, which takes U 1 if the city is visited immediately after i i the tour, otherwise it takes 0, they uses i variables to defie the order i which each verte i is visited o a tour, so the MTZ formulatio of the TSP is : mi i 1 i1 d 1 ( i V, i 1 0;1, ) ( V, i ) i A (13) U U 1,, 2.. i i (14) 1 Ui 1 i 2.. (10) Represets the obective fuctio, (11) ad (12) are the assigmet costraits, (13) is the itegrality costrait, (14) are the sub-tour elimiatio costraits, (15) is the itegrality costrait. This TSP formulatio cotais 11 2 variables, ad 2 costraits. (10) (11) (12) (15) Iteratioal orgaizatio of Scietific Research 18 Page

3 Paper preparatio guidelies for IOSR Joural of Egieerig C. The Desrochers-Laporte formulatio: Amog the advatages of the MTZ formulatio is that the sub-tour elimiatio costraits (14) ad (15) ca be icorporated ito other types of problem formulatios together with stroger costraits [3]. Motivated by these facts, Desrochers ad Laporte have lifted sub-tour elimiatio costraits (14) ad (15) to obtai the stroger forms: Mi d i 1 i 1 V i 0;1 i, A ( i V, i ) 1 (, ) U U + 2, i, = 2.. (20) i i 1 U 1, i = 2.. (21) i (16) (17) (18) (19) (16) is the obective fuctio, the assigmet costraits (17) ad (18) esure that each city is visited oe ad oly oce, (19)ad (21) are the itegrality costraits, (20) is the sub-tour elimiatio costraits. IV. RESOLUTION METHODS FOR THE TRAVELING SALESMAN PROBLEM : I our case, for the resolutio we used eact methods [8], usig a liear programmig solver CPLEX versio which by default uses the method Brach ad Cut, it is a eact method for solve iteger liear optimizatio problems, usig the Brach ad Boud method, ad the secat pla method. A. Resolutio of MTZ ad DL formulatio : For the resolutio with this formulatio we used a iteger liear programmig solver that uses the default Brach ad Cut method. B. Resolutio of DFJ Formulatio For the DFJ (Dazig-Fulkerso-Johso) formulatio, takig ito accout the epoetial umber of subtour elimiatio of costraits (SEC) of the order of 2, we proceed by costrait geeratio. C. The geeratio of costrait method : The costrait geeratio approach works as follows: First of all, we solve the problem without ay costraits of Elimiatio of sub-turs. ad we solve the remaiig ILP model. The we check if the resultig whole solutio cotais sub-turs. Otherwise, the solutio is a optimal TSP visit. Ad if we have sub-towers, we fid all the sub-towers i the itegral solutio ad add the correspodig SEC to the model. The resultig epaded ILP model is agai resolved to optimality. Iteratig this process clearly leads to a optimal TSP tour. V. RESULTS AND DISCUSSTION Eperimets were performed o a processor Itel Celero CPU 847 of 1.10 GHz, 1.10 GHz with 4 GB Memory. The solutio has bee provided by IBM ILOG CPLEX versio ; we have developed programs i OPL laguage (Laguage Modeler of the CPLEX Solver). To solve the TSP with the tree formulatios MTZ, DL ad DFJ we have geerated small radom istaces similar to some of the TSPLIB istaces to obtai 10 complete graphs from 10 to 950 odes. These odes have take betwee 0 ad 99 coordiates radomly; the distaces betwee odes are the Euclidea distaces (d). We have calculated the Cotiuous Relaatio Value (R) (obtaied by relaig the itegrality costraits), ad the Optimal Value (Vopt), with T(s) is the calculatio time i secod ad Relaatio quality (RQ), (Table 1 shows the results.) with QR=(Vopt-R)*100/Vopt. Iteratioal orgaizatio of Scietific Research 19 Page

4 Vopt, R Paper preparatio guidelies for IOSR Joural of Egieerig INSTANCES Vopt RDFJ RMTZ RDL RQ df % RQ mtz % RQ dl % TSP10 337,67 322,5 256,78 263,76 4,49 23,96 21,89 TSP20 412,79 411,13 344,37 366,97 0,40 16,58 11,10 TSP35 518,96 495,65 390,31 460,6 4,49 24,79 11,25 TSP45 537,68 489,28 371,96 402,8 9,00 30,82 25,09 TSP60 603,42 578,33 481,69 510,5 4,16 20,17 15,40 TSP75 645,76 588,10 476,77 489,8 8,93 26,17 24,15 TSP85 678,97 678, , TSP90 691,41 691, , TSP ,09 738, , TSP ,64 888, , TSP , , , TSP , , , TSP , , , TSP , , , TSP , , , TSP , , , TSP , , , Table 1: Optimal Value (Vopt), Solutio time T(s), Cotiuous Relaatio Value(R), ad Relaatio quality (RQ) for the DFJ, MTZ ad DL formulatios Vopt Rdf Rmtz Rdl TSP20 TSP45 TSP75 TSP90 TSP150 TSP250 Istaces Figure 2.The Cotiuous Relaatio Value (R) of the DFJ, MTZ ad DL formulatios It is apparet from the results that: The DFJ Relaatio value is better tha the MTZ ad DL Relaatio value. From 35 cities, the formulatio MTZ ad DL become very slow i calculatio time, we took as limit time 30 mi. VI. CONCLUSION The study leads to the followig coclusios: The MTZ formulatio is seductive because it is easy to implemet but it gives a low cotiuous relaatio. I practice it is more iterestig to use the DFJ formulatio, eve this requires to implemet the geeratio of costraits. The DL formulatio, although sigificatly stregthes the MTZ formulatio, does ot compete with the DFJ formulatio. REFERENCES [1]. Datzig, G.B.; Fulkerso, D.R. & Johso, S.M. (1954). Solutio of a large-scale travelig salesma problem. Operatios Research, Vol. 2, pp [2]. Miller, C.E.; Tucker, A.W. & Zemli, R.A.(1960). Iteger programmig formulatio of travelig salesma problems. Joural of Associatio for Computig Machiery, Vol. 7, pp Iteratioal orgaizatio of Scietific Research 20 Page

5 Paper preparatio guidelies for IOSR Joural of Egieerig [3]. Marti Desrochers, Gilbert Laporte (1991). Improvemets ad etesios to the Miller-Tucker-Zemli subtour elimiatio costraits. Volume 10, Issue 1, pp [4]. Padberg M, Sug T. A aalytical compariso of differet formulatios of the travellig salesma problem. Mathematical Programmig 1991;52: [5]. Lagevi A, Soumis F, Desrosiers J. Classificatio of travellig salesma problem formulatios. Operatios Research Letters 1990;9: [6]. Desrochers M, Laporte G. Improvemets ad etesios to the Miller Tucker Zemli subtour elimiatio costraits. Operatios Research Letters 1991;10:27 36 [7]. Sherali HD, Driscoll PJ. O tighteig the relaatios of Miller Tucker Zemli formulatios for asymmetric travelig salesma problems. Operatios Research 2002;50: [8]. Rya J. O Neil, Karla Hoffma, Eact Methods for Solvig Travelig Salesma Problems with Pickup ad Delivery i Real Time, INFORMS Joural o Computig mauscript Soufia Behida1 Geeratig subtour elimiatio costraits for the Travelig Salesma Problem"IOSR Joural of Egieerig (IOSRJEN), vol. 08, o. 7, 2018, pp Iteratioal orgaizatio of Scietific Research 21 Page