A Novel Approach to Solve Multiple Traveling Salesmen Problem by Genetic Algorithm

Transcription

1 A Novel Approach to Solve Multiple Travelig Salesme Problem by Geetic Algorithm Adrás Király, Jáos Aboyi Uiversity of Paoia, Departmet of Process Egieerig, P.O. Box 58. Veszprém H-8200, HUNGARY, Abstract The multiple Travelig Salesma Problem (mtsp) is a complex combiatorial optimizatio problem, which is a geeralizatio of the well-kow Travelig Salesma Problem (TSP), where oe or more salesme ca be used i the solutio. The optimizatio task ca be described as follows: give a fleet of vehicles, a commo depot ad several requests by the customers, fid the set of routes with overall miimum route cost which service all the demads. Because of the fact that TSP is already a complex, amely a NP-complete problem, heuristic optimizatio algorithms, like geetic algorithms (GAs) eed to take ito accout. The extesio of classical GA tools for mtsp is ot a trivial problem, it requires special, iterpretable ecodig to esure efficiecy. The aim of this paper is to review how geetic algorithms ca be applied to solve these problems ad propose a ovel, easily iterpretable represetatio based GA. Keywords: mtsp, VRP, geetic algorithm, multi-chromosome, optimizatio Itroductio I logistics, the mai goal is to get the right materials to the right place at the right time, while optimizig some performace measure, like the miimizatio of total operatig cost, ad satisfyig a give set of costraits (e.g. time ad capacity costraits). I logistics, several types of problems could come up; oe of the most remarkable is the set of route plaig problems. Oe of the most studied route plaig problem is the Vehicle Routig Problem (VRP), which is a complex combiatorial optimizatio problem that ca be described as follows: give a fleet of vehicles with uiform capacity, a commo depot, ad several requests by the customers, fid the set of routes with overall miimum route cost which service all the demads. The complexity of the search space ad the umber of decisio variables makes this problem otoriously difficult. The relaxatio of VRP is the multiple travelig salesma problem (mtsp) [3], which is a geeralizatio of the well-kow travelig salesma problem (TSP)

2 2 [0], where oe or more salesma ca be used i the solutio. Because of the fact that TSP belogs to the class of NP-complete problems, it is obvious that mtsp is a NP-hard problem thus it's solutio require heuristic approach. I this paper tools developed for a modified mtsp related to the optimizatio of oe to may distributio systems will be studied ad a ovel geetic algorithm based solutio will be proposed. I the case of mtsp, a set of odes (locatios or cities) are give, ad all of the cities must be visited exactly oce by the salesme who all start ad ed at the sigle depot ode. The umber of cities is deoted by ad the umber of salesma by m. The goal is to fid tours for all salesme, such that the total travellig cost (the cost of visitig all odes) is miimized. The cost metric ca be defied i terms of distace, time, etc. Some possible variatios of the problem are as follows: Multiple depots: If there exist multiple depots with a umber of salesme located at each, a salesma ca retur to ay depot with the restrictio that the iitial umber of salesme at each depot remais the same after all the travel. Number of salesme: The umber of salesme i the problem ca be a fixed umber or a bouded variable. Fixed charges: If the umber of salesme is a bouded variable, usually the usage of each salesma i the solutio has a associated fixed cost. I this case the miimizatio of this bouded variable may be ivolved i the optimizatio. Time widows: Certai cities must be visited i specific time periods, amed as time widows. This extesio of mtsp is referred to as multiple Travelig Salesma Problem with Time Widows (mtsptw). Other restrictios: These additioal restrictios ca cosist of the maximum or miimum distace or travellig duratio a salesma travels, or other special costraits. mtsp is more capable to model real life applicatios tha TSP, sice it hadles more tha oe salesme. A overview of applicatio areas ca be foud i [3] ad i [0]. I the paper, a mtsptw problem will be optimized with a ovel approach, where the umber of salesme is a upper bouded variable, ad there exist additioal costraits, like the maximum travellig distace of each salesma. Usually, mtsp is formulated by iteger programmig formulatios. Oe variatio is preseted i equatios (.)-(.7). The mtsp problem is defied o a graph G = (V,A), where V is the set of odes (vertices) ad A is the of arcs (edges). Let C = ( c ij ) be a cost (distace) matrix associated with A. The matrix C is symmetric if { 0,} c ij = c ( i, j) A ji, ad asymmetric otherwise. Here x is a biary variable used to represet that a arch is used o the tour ij ad c m represets the cost of the ivolvemet of oe salesma i the solutio. Further mathematical represetatios ca be foud i [3].

3 3 mi c x + mc (.) so that j=2 j=2 i= j= x x i= j= j = m ij ij m (.2) j = m (.3) xij =, j = 2, K, xij =, i = 2, K, (.4) (.5) + subtour elimiatio costratis (.6) 0,, i, j (.7) x ij { } ( ) A 2 Literature review I the last two decades the travelig salesma problem received quite big attetio, ad various approaches have proposed to solve the problem, e.g. brach-adboud [7], cuttig plaes [7], eural etwork [4] or tabu search [9]. Some of these methods are exact algorithms, while others are ear-optimal or approximate algorithms. The exact algorithms use iteger liear programmig approaches with additioal costraits. The mtsp is much less studied like TSP. [3] gives a comprehesive review of the kow approaches. There are several exact algorithms of the mtsp with relaxatio of some costraits of the problem, like [5], ad the solutio i [] is based o Brach-ad-Boud algorithm. Due to the combiatorial complexity of mtsp, it is ecessary to apply some heuristic i the solutio, especially i real-sized applicatios. Oe of the first heuristic approach were published by Russell [23] ad aother procedure is give by Potvi et al. [20]. The algorithm of Hsu et al. [2] preseted a Neural Networkbased solutio. More recetly, geetic algorithms (GAs) are successfully implemeted to solve TSP [8]. Potvi presets a survey of GA approaches for the geeral TSP [2].

4 4 2. Applicatio of geetic algorithms to solve mtsp Lately GAs are used for the solutio of mtsp too. The first result ca be boud to Zhag et al. [25]. Most of the work o solvig mtsps usig GAs has focused o the vehicle schedulig problem (VSP) ([6, 8]). VSP typically icludes additioal costraits like the capacity of a vehicle (it also determies the umber of cities each vehicle ca visit), or time widows for the duratio of loadigs. Recet applicatio ca be foud i [4], where GAs were developed for hot rollig schedulig. It coverts the mtsp ito a sigle TSP ad apply a modified GA to solve the problem. A ew approach of chromosome represetatio, the so-called two-part chromosome techique ca be foud i [5] which reduces the size of the search space by the elimiatio of redudat solutios. Accordig to the referred paper, this represetatio is the most effective oe so far. There are several represetatios of mtsp, like oe chromosome techique [20], the two chromosome techique [6, 8] ad the latest two-part chromosome techique. Each of the previous approaches has used oly a sigle chromosome to represet the whole problem, although salesme are physically separated from each other. The ovel approach preseted i the ext chapter use multiple chromosomes to model the tours. 3 The proposed GA-based approach to solve the mtsp GAs are relatively ew global stochastic search algorithms which based o evolutioary biology- ad computer sciece priciples []. Due to the effective optimizatio capabilities of GAs [2], it makes these techique suitable solvig TSP ad mtsp problems. 3. The ovel geetic represetatio for mtsp As metioed i the previous chapter, every GA-based approach for solvig the mtsp has used sigle chromosome for represetatio so far. The ew approach preseted here is a so-called multi-chromosome techique, which separates the salesme from each other thus may preset a more effective approach. This approach is used i otoriously difficult problems to decompose complex solutio ito simpler compoets. It was used i mixed iteger problem [9], a usage of routig problem optimizatio ca be see i [8] ad a lately solutio of a symbolic regressio problem i [6]. This sectio discusses the usage of multichromosomal geetic programmig i the optimizatio of mtsp.

5 5 Fig. Example of the multi-chromosome represetatio for a 5 city mtsp with 4 salesme. Fig. illustrates the ew chromosome represetatio for mtsp with 5 locatios (=5) ad with 4 salesperso (m=4). The figure above illustrates a sigle idividual of the populatio. Each idividual represets a sigle solutio of the problem. The first chromosome represets the first salesma itself so each gee deotes a city (depot is ot preseted here, it is the first ad the last statio of each salesma). This ecodig is so-called permutatio ecodig. It ca be see i the example that salesperso visits 4 cities: city 2,5,4 ad 6, respectively. I the same way, chromosome 2 represets salesperso 2 ad so o. This represetatio is much similar to the characteristic of the problem, because salesme are separated from each other "physically". 3.2 Special geetic operators Because of our ew represetatio, implemetatio of ew geetic operators became ecessary, like mutatio operators. There are two sets of mutatio operators, the so-called I-route mutatios ad the Cross-route mutatios. Oly some example of the ewly created operators are give i this sectio. Further iformatio with several examples about the ovel operators ca be foud i [3]. I-route mutatio operators work iside oe chromosome. A example is illustrated o Fig. 2. The operator chooses a radom subsectio of a chromosome ad iverts the order of the gees iside it.

6 6 Fig. 2 I-route mutatio gee sequece iversio. Cross-route mutatio operates o multiple chromosomes. If we thik about the distict chromosomes as idividuals, this method could be similar to the regular crossover operator. Fig. 3 illustrates the method whe radomly chose subparts of two chromosomes are trasposed. If the legth of oe of the chose subsectios is equal to zero, the operator could trasform ito a iterpolatio. Fig. 3 Cross-route mutatio gee sequece traspositio. 3.3 Geetic algorithm Every geetic algorithm starts with a iitial solutio set cosists of radomly created chromosomes. This is called populatio. The idividuals i the ew populatio are geerated from the previous populatio s idividuals by the predetermied geetic operators. The algorithm fiishes if the stop criteria is satisfied. Obviously for a specific problem it is a much more complex task, we eed to defie the ecodig, the specific operators ad selectio method. The ecodig is the so-called permutatio ecodig (see previous sectio). Detailed descriptio of the related operators ca be foud i [4] ad a example ca be see i the previous sectio.

7 Fitess fuctio The fitess fuctio assigs a umeric value to each idividual i the populatio. This value defie some kid of goodess, thus it determies the rakig of the idividuals. The fitess fuctio is always problem depedet. I this case the fitess value is the total cost of the trasportatio, i.e. the total legth of each roud trip. The fitess fuctio calculates the total legth for each chromosome, ad summarizes these values for each idividual. This sum is the fitess value of a solutio. Obviously it is a miimizatio problem, thus the smallest value is the best Selectio Idividuals are selected accordig to their fitess. The better the chromosomes are, the more chaces to be selected they have. The selected idividuals ca be preseted i the ew populatio without ay chages (usually with the best fitess), or ca be selected to be a paret for a crossover. We use the so-called touramet selectio because of its efficiecy. I the course of touramet selectio, a few (touramet size, mi. 2) idividuals are selected from the populatio radomly. The wier of the touramet is the idividual with the best fitess value. Some of the first participats i the rakig are selected ito the ew populatio (directly or as a paret). 3.4 Complexity aalysis Usig the multi-chromosome techique for the mtsp reduces the size of the overall search space of the problem. Let the legth of the first chromosome be k, let m i i the legth of the secod be k 2 ad so o. Of course = k =. Determiig the gees of the first chromosome is equal to the problem of obtaiig a ordered subset of k elemet from a set of elemets. There are distict assigmet. This umber is ( k)! ( k k )! 2! ( k )! for the secod chromosome, ad so o. Thus, the total search space of the problem ca be formulated as equatio (3.).! ( k)! * * ( k ) ( k k )! 2 ( k K k * ( k K k m )!! = =! )! ( )! m K (3.)

8 8 It is ecessary to determie the legth of each chromosome too. It ca be represeted as a positive vector of the legths (k, k 2,, k m ) that must sum to. There are m distict positive iteger-valued vectors that satisfy this requiremet [22]. Thus, the solutio space of the ew represetatio is! m. It is equal with the solutio space i [5], but this approach is more similar to the characteristic of the mtsp, so it ca be more problem-specific therefore more effective. 4 Implemetatio issues To aalyze the ew represetatio, a ovel geetic algorithm usig this approach was developed i MATLAB. This ovel approach was compared with the most effective oe so far (the two-part chromosome) which is available o MATLAB Cetral. The ovel algorithm ca optimize the traditioal mtsp problems, furthermore, it is capable to hadle the additioal costraits ad time widows (see Sect. ). It requires two iput sets, like the coordiates of the cities ad the distace table which cotais the travellig distaces betwee ay pair of cities. Naturally, the determiatio of the costraits, time widows ad the parameters of the geetic algorithms are also ecessary. The fitess fuctio simply summarizes the overall route legths for each salesma iside a idividual. The selectio is touramet selectio, where touramet size i.e. the umber of idividuals who compete for survival is 8. Therefore populatio size must be divisible by 8. The wier of the touramet is the member with the smallest fitess, this idividual is selected for ew idividual creatio, ad this member will get ito the ew populatio without ay modificatio. The pealty of the too log routes (over the defied costrait) istead of a proportioally large fitess value assigmet is implemeted by a split operator, which separates the route ito smaller routes, which do ot exceed the costraits (but the umber of salesme is icremeted). Because there exists a costrait for the umber of the salesme, the algorithm ivolves the miimizatio of this amout, hece this pealty has a remarkable effect i the optimizatio process. Further iformatio about the implemeted algorithm ca be foud i [4].

9 9 5 Illustrative example Although the algorithm was tested with a big umber of problems, oly a illustrative result is preseted here. As it was metioed earlier, the algorithm has implemeted i MATLAB, tiy refiemets i costraits are i progress. The exmaple represets a whole process of a real problem s solutio. The iitial iput is give i a Google Maps map, ad the fial output is a route system defied by a Google Maps map also. The first step is the determiatio of the distace matrix. The iput data is give by a map as it ca see o Fig. 4 ad a portio of the resulted distace table is show o Table. It cotais 25 locatios (with the depot). The task is to determie the optimal routes for these locatios with the followig costraits: the maximum umber of salesme is 5 ad the maximum travellig distace of each salesma is 450 km. Fig. 4 The map of the example applicatio (iitial iput). Kilometers Adoy Celldömölk Kapuvár Adoy Celldömölk Kapuvár Table Example distace table - kilometers. After distace table determiatio, the optimizer algorithm ca be executed to determie the optimal routes usig the ovel represetatio. The GA ra with a populatio size 320 ad it did 200 iteratios. The result of the optimizatio is

10 0 show o Fig. 5. It resulted that 4 salesma is eough to satisfy the costraits. After the optimizatio, we ca visualize the results o a Google Maps map, as it is show o Fig. 6. The legth of the routes are 364 km, 424 km, 398 km ad 49 km respectively, i.e. they satisfy the costraits, thus the algorithm provided a feasible solutio of the problem. I every case, the ruig time was betwee ad 2 miutes. The geetic algorithm has made 200 iteratios, because experieces have show that this umber is sufficiet for the optimizatio. Fig. 5 The result of the optimizatio by MATLAB. Fig. 6 Result of the optimizatio o a Google Maps map for 25 locatios with at most 5 salesme ad at most 450 km tour legth per salesma.

11 Obviously the algorithm is highly sesitive for the umber of iteratios. The ruig time is directly proportioal to the iteratio umber, but the resulted best solutio ca t get better after a specific time. If the costraits become tighter, the duratio time will icrease slightly. With 500 maximal tour legths, it is about 90 secods, ad with 450 it is about 0 secods. The maximal tour legth (or equivaletly the maximal duratio per tour) has a big effect of the umber of salesma eeded. The tighter the costraits are, the bigger the umber of salesma we eed. However arrower restrictios forth more square roud trips. Furthermore, the resulted optima ca deped o the iitial populatio. O Fig. 7 it ca be see that the algorithm ca fid a ear optimal solutio i more tha 80% of the cases. The effectiveess of the calculatio ca be ehaced by applyig additioal heuristics. Obviously these results ca be further improved by executig more iteratio also. Fig. 7 Results of the optimizatio from differet iitial values. 6. Coclusios I this paper a detailed overview was give about the applicatio of geetic algorithms i vehicle routig problems. It has bee show that the problem is closely related to the multiple Travelig Salesma Problem. A ovel represetatio based geetic algorithm has bee developed to the specific oe depot versio of mtsptw. The mai beefit is the trasparecy of the represetatio that allows the effective icorporatio of heuristics ad costrais ad allows easy implemetatio. Some heuristics ca be applied to improve the effectiveess of the algorithm, like the appropriate choice of the iitial populatio. After some fial touches, the supportig MATLAB code will be also available at the website of the authors.

12 2 Ackowledgmets The fiacial support from the TAMOP // (Élhetőbb köryezet, egészségesebb ember - Bioiováció és zöldtechológiák kutatása a Pao Egyeteme, MK/2) project is gratefully ackowledged. Refereces [] Ali AI, Keigto JL (986) The asymmetric m-travelig salesme problem: a duality based brach-ad-boud algorithm. Discrete Applied Mathematics 3: [2] Back T (996) Evolutioary algorithms i theory ad practice: evolutio strategies, evolutioary programmig, geetic algorithms. Oxford Uiversity Press [3] Bektas T (2006) The multiple travelig salesma problem: a overview of formulatios ad solutio procedures. Omega, 34: [4] Bhide S, Joh N, Kabuka MR (993) A boolea eural etwork approach for the travelig salesma problem. IEEE Trasactios o Computers 42(0):27 [5] Carter AE, Ragsdale CT (2006) A ew approach to solvig the multiple travelig salesperso problem usig geetic algorithms. Europea Joural of Operatioal Research 75: [6] Cavill R, Smith S, Tyrrell A (2005) Multi-chromosomal geetic programmig. I: Proceedigs of the 2005 coferece o Geetic ad evolutioary computatio, ACM New York, NY, USA, pp [7] Fike G, Claus A ad Gu E (984) A two-commodity etwork flow approach to the travelig salesma problem. Cogressus Numeratium, 4:67 78 [8] Ge M, Cheg R (997) Geetic algorithms ad egieerig desig. Wiley-Itersciece [9] Glover F (990) Artificial itelligece, heuristic frameworks ad tabu search. Maagerial ad Decisio Ecoomics (5) [0] Guti G, Pue AP (2002) The Travelig Salesma Problem ad Its Variatios. Combiatorial Optimizatio, Kluwer Academic Publishers, Dordrecht, The Nederlads [] Hollad JH (975) Adaptatio i atural ad artificial systems. The Uiversity of Michiga Press [2] Hsu CY, Tsai MH, Che WM (99) A study of feature-mapped approach to the multiple travellig salesme problem. IEEE Iteratioal Symposium o Circuits ad Systems 3: [3] Király A, Aboyi J (2009) Optimizatio of multiple travelig salesme problem by a ovel represetatio based geetic algorithm. I 0th Iteratioal Symposium of Hugaria Researchers o Computatioal Itelligece ad Iformatics, Budapest, Hugary [4] Király A, Aboyi J (excepted to 200) Optimizatio of multiple travelig salesme problem by a ovel represetatio based geetic algorithm. I M. Koeppe, G. Schaefer, A. Abraham, ad L. Nolle, editors, Itelliget Computatioal Optimizatio i Egieerig: Techiques & Applicatios, Advaces i Itelliget ad Soft Computig. Spriger [5] Laporte G, Nobert Y (980) A cuttig plaes algorithm for the m-salesme problem. Joural of the Operatioal Research Society 3: [6] Malmborg CJ (996) A geetic algorithm for service level based vehicle schedulig. Europea Joural of Operatioal Research 93():2 34 [7] Miliotis P (978) Usig cuttig plaes to solve the symmetric travellig salesma problem. Mathematical Programmig 5():77 88 [8] Park YB (200) A hybrid geetic algorithm for the vehicle schedulig problem with due times ad time deadlies. Iteratioal Joural of Productios Ecoomics 73(2):75 88 [9] Pierrot HJ, Hiterdig R (997) Multi-chromosomal geetic programmig, Lecture Notes i Computer Sciece, vol 342/997, Spriger Berli / Heidelberg, chap Usig multichromosomes to solve a simple mixed iteger problem, pp 37 46

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