Travelling Salesman Problem

Transcription

1 Travellig Salesma Problem MIGUEL A. S. CASQUILHO Techical Uiversity of Lisbo, Ave. Rovisco Pais, Lisboa, Portugal The Travellig Salesma Problem is briefly preseted, with referece to problems that ca be assimilated to it ad solved by the same techique. Examples are show ad solved. Key words: travellig salesma problem; supply; demad; optimizatio. 1. Fudametals ad scope With optimizatio i mid, the travellig salesma problem, frequetly deoted by the iitials TSP 1, is a fudametal subject related to travellig ad trasportatio, with several geeralizatios ad with isertio i more complex situatios, ad also aki to others apparetly urelated, resoluble by the techiques used for the typical case. The TSP is kow for the strikig cotrast betwee the simplicity of its formulatio ad the difficulty of its resolutio, some eve sayig that it still does ot have a solutio. It is a so-called NP-hard 2 problem (its difficulty icreasig more tha polyomially with its size). Ayway, somethig substatial ca be preseted about the problem. The problem arises from the typical situatio of a salesma who wats to visit his cliets i a give set of cities ad retur to his ow city, thus performig a cycle. The problem ca be evisaged i this large scale, but also exists i ay other scales, such as withi a factory or o a microchip. A asymmetrical TSP ca also be the search for the optimum orderig of pait maufacturig or the preparatio of fruit juices i a commo plat, because, i these cases, setup costs (washig, etc.) depeds sigificatly o the viciity of the colours or of the flavours. The mathematical formulatio of the problem ca be as i Eq. {1}. subject to [ mi ] z = j= 1 m i= 1 i= 1 j= 1 c ij x ij xij = 1 i = 1.. m xij = 1 j = 1.. x ij = 0or1 i, j {1} Solutio must be a cycle. As i the Assigmet Problem (AP), c ij, with i, j= 1.., is the cost (or distace, etc.) to go from city i to city j, ad x ij will be 1 if the arc from i to j is used ad 0 otherwise. The problem would be a AP if the last coditio (oe ad oly oe 1 Also travelig salesma problem (America Eglish) or sometimes salesperso. 2 See, e.g., Miguel Casquilho is Assistat Professor (retired) i the Departamet of Chemical Egieerig, Istituto Superior Técico, Uiversidade Técica de Lisboa (Techical Uiversity of Lisbo), Lisbo, Portugal. address: mcasquilho@ist.utl.pt MC IST Op. Res. File={TSP_Prototype.doc}

2 2 [:6] MIGUEL CASQUILHO cycle) a fudametal differece did ot exist. I order to use the above formulatio the costs c ii, to go from oe city to itself must be made prohibitive, otherwise the (useless) solutio would be x ii = 1, others zero (ot a cycle). A brach-ad-boud (B&B) exact algorithm based o AP relaxatio will be preseted with some examples. A laguage such as Mathematica has fuctios TraveligSalesma ad FidShortestTour that solve the problem, eve i 3D, but the objective here is to clarify the method itself. A advaced algorithm by Carpaeto et al. [1995CAR] is used ad made available at the author s website [2012CAS]. Several other exact algorithms, all detailed by Lawler et al. [1995LAW], have bee costructed, as well as heuristic oes (i.e., approximate), amely for large size problems or more complex problems. 2. Examples EXAMPLE: 5 CITIES (WINSTON) A travellig salesma has to cover a set of 5 cities (his ow icluded) periodically (say, oce per week) ad retur home. The distaces betwee the cities are give i Table 1, as could have bee read o a map. Determie the most ecoomical cycle, i.e., with miimum legth (example from Wisto [2003WIN], p 530 ff). Table 1 Cost matrix (distaces i km) from oe city to aother I this problem the cost matrix is symmetric, which appears obvious. The cost matrix ca, however, be asymmetric, as i the case of air travel because of predomiat wid or i oe-way urba streets. No advatage is take from symmetry i the preset text, but symmetry may be importat i may variats of the solutio methods. RESOLUTION The strategy of AP relaxatio is to solve the problem as a AP, leavig aside the coditio of oe cycle. The solutio to the AP ca be obtaied, e.g., with Excel or Excel/Cplex, ad is the oe i Table 2, with total cost z = 495 km. Table 2 Solutio to the AP relaxatio

3 Travellig Salesma Problem [:6] 3 This is, however, ot a solutio to the TSP, because there are subtours: x 15 = x 21 = x 34 = x 43 = x 52 = 1, i.e., two subtours, ad The B&B techique will ow be used, as follows. The former problem, say, Problem 1, is replaced by others, cosiderig the shortest subtour (the oe with least arcs) to try to save computatio effort. So, will be chose. Now, two problems replace the previous oe: Problem 2, from Problem 1 but prohibitig 3 4; ad Problem 2, from Problem 1 but prohibitig 4 3. The two ew problems have the cost matrices i Table 3 ad the solutios i Table 4 Table 3 Cost matrices for Problem 2 (l.-h.) ad Problem 3 (r.-h.) Table 4 Solutios to the AP relaxatios Agai, these are ot solutios to the TSP, because there are subtours: for Problem 2, , 2 5 2; ad for Problem 3, , (Symmetry causes some redudacy.) I file TSP_Wis530.xls, the complete procedure is show util the optimum is foud. The solutio procedure is usually show as a tree, costructed progressively as the problem is solved. The oe for this problem is (from TSP_Raffesperger.ppt) give i Figure 1, with solutio ad z* = 668 km. Figure 1 Tree for the Wisto TSP problem.

4 4 [:6] MIGUEL CASQUILHO About the B&B, otice that: (a) (disadvatage) Every replacemet of a problem gives rise to two or more childre ad ca ever improve the value of the objective fuctio (495, 652, 668, icreasig, while a miimum was sought); but (b) (advatage) It is, hopefully, ot ecessary to ivestigate all the combiatios, this beig the real merit of the B&B. The systematic procedure of B&B permits to avoid the ivestigatio of all the combiatios. For a mere = 18 cities the umber of Portugal s mai cities ( district capitals 3 ), these combiatios would be 17!, which is 3, (11 years of computig at oe combiatio per microsecod). The motive it is cosidered that the TSP still has o solutio is that the size of the tree is ot predictable. So, the amout of memory or disk storage occupied ca icrease beyod the availability, makig this ot oly a matter of time, but ofte more so oe of space. EXAMPLE: GRID 2 I a maufacture of grids havig 4 poits i a square arragemet of 2 2, as i Figure 2, all the poits have to be treated (e.g., welded, coected, paited) i a certai order. Determie the most ecoomic order. Figure 2 Grid with 2 by 2 poits i a square arragemet. RESOLUTION It is obvious that the aswer is , with z* = 4, if, as is implicit, the distaces betwee the adjacet poits (horizotally or vertically) is oe. If, however, the problem is to be solved by a coveiet algorithm 4, the cost matrix has to be provided ad is give i Table 5. I order to use the web site, supply 1 as ifiity (diagoal etries). Table 5 Cost matrix for the Grid The district ( distrito ), of which there are 18 (average km 2 ) i cotietal Portugal, is roughly equivalet i area to oe Frech départemet (totallig 96) or two Italia provice (110). 4 The oe o the web, based o Carpaeto et al..

5 Travellig Salesma Problem [:6] 5 EXAMPLE: GRID 3 Now suppose that (previous example) it is the maufacture of grids havig 9 poits i a square arragemet of 3 3, as i Figure 3. Determie the most ecoomic order. Figure 3 Grid with 3 by 3 poits i a square arragemet. RESOLUTION Now the solutio is by o meas obvious. See TSP_grids.xls. The problem is solved assumig both a taxicab 5 geometry 6 ad a Euclidea geometry. The taxicab geometry assumes a grid layout, such as the arragemet of streets i certai zoes of the cities, hece the metio to the taxicab, so (as i the previous example) oly vertical ad horizotal movemets are possible. The cost matrix is give i Table 6. Table 6 Taxicab cost matrix for the Grid The solutio is , with z* = 10. Lookig at these results, oe may woder if allowig other movemets may result i a better value. This leads to the commo, Euclidea geometry. Now the cost matrix is give i Table 7. The solutio to the Euclidea geometry Grid 3 problem, as show i Table 7, is , with z* = 9,41. As expected, allowig a less costraied geometry has led to a better result. 5 From taximeter cabriolet. 6 See, e.g.,

6 6 [:6] MIGUEL CASQUILHO Table 7 Solutio for the Euclidea cost matrix for the Grid , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , Coclusios The TSP is a remarkable problem both for the cotrast betwee the simplicity of its formulatio ad the complexity of its resolutio, ad the variety of its applicatios. The exact resolutio was preseted usig the brach-ad-boud techique applied to Assigmet Problem relaxatios. A more advaced algorithm by Carpaeto et al. is made available for the solutio of this type of problems. Examples of typical situatios were preseted, amely, a problem with taxicab ad Euclidea distaces as costs. Ackowledgemets This work was doe at Cetro de Processos Químicos do IST (Chemical Process Research Cetre of IST), Departmet of Chemical Egieerig, Techical Uiversity of Lisbo. Computatios were doe o the cetral system of CIIST (Computig Cetre of IST). Bibliography [1995CAR] CARPANETO, G., M. DELL AMICO ad P. TOTH, 1995, Exact solutio of large-scale asymmetric travelig salesma problems, ACM Trasactios o Mathematical Software, 21: [2012CAS] CASQUILHO, M., accessed May [1985LAW] LAWLER, E. L., J. K. LENSTRA, A. H. G. RINNOOY KAN, D. B. Shmoys, The travellig salesma problem, editors, Chichester (UK). ISBN [2003WIN] WINSTON, Waye L., ad M. VENKATARAMANAN, 2003, Itroductio to Mathematical Programmig, Thomso, New York, NY (USA). ISBN