Optimization of Multiple Traveling Salesmen Problem by a Novel Representation based Genetic Algorithm

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1 Magyar Kutatók 0. Nemzetközi Szimpóziuma 0 th Iteratioal Symposium of Hugaria Researchers o Computatioal Itelligece ad Iformatics Optimizatio of Multiple Travelig Salesme Problem by a Novel Represetatio based Geetic Algorithm Adrás Király, Jáos Aboyi Departmet of Process Egieerig, Uiversity of Paoia, P.O. Box 58. H Veszprém, Hugary, kadras85@gmail.com Abstract: The Vehicle Routig Problem (VRP) 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 costumer demads; fid the set of routes with overall miimum route cost which service all the demads. The multiple travelig salesma problem (mtsp) is a geeralizatio of the well-kow travelig salesma problem (TSP), where more tha oe salesma is allowed to be used i the solutio. It is well-kow that mtsp-based algorithms ca also be utilized i several VRPs by icorporatig some additioal site costraits. The aim of this chapter is to review how geetic algorithms ca be applied to solve these problems ad to propose a ovel, iterpretable represetatio based algorithm. The elaborated heuristic algorithm is demostrated by examples cosiderig differet roud tour types determiatio of further tasks for optimal operatio of the distributio system for istace the modificatio of the vehicle capacity, ad the effects of chage of cost elemets ad data structure. Keywords: mtsp, VRP, geetic algorithm, multi-chromosome, optimizatio Itroductio The aim of logistics is to get the right materials to the right place at the right time, while optimizig a give performace measure (e.g. miimizig total operatig costs) ad satisfyig a give set of costraits (e.g. time ad capacity costraits). I most distributio systems goods are trasported from various origis to various destiatios. For example, may retail chais maage distributio systems i which goods are trasported from a umber of suppliers to a umber of retail stores. It is ofte ecoomical to cosolidate the shipmets of various origidestiatio pairs ad trasport such cosolidated shipmets i the same truck at the same time. There are may ways i which such cosolidatio ca be accomplished. I this paper tools developed for Vehicle Routig Problem (VRP) 35

2 A. Király et al. Optimizatio of Multiple Travelig Salesme Problem by a Novel Represetatio based Geetic Algorithm related to the optimizatio of oe to may distributio systems will be studied, ad a ovel geetic algorithm based solutio will be proposed. The multiple travelig salesma problem (mtsp) [4] is a geeralizatio of the well-kow travelig salesma problem (TSP) [3], 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 the case of mtsp, ambiguous umber of cities exist ad all of the cities must be visited exactly oce by the salesme who all start ad ed at the depot. The umber of cities is deoted by ad the umber of salesma by m. I this paper the sigle depot case is studied, where all salesma start from ad ed their tours at a sigle poit. The umber of salesme is assumed to be kow a priori, but it is bouded based o the maximal size of the fleet of vehicles. Sice the umber of salesme is ot fixed, each salesma has a associated fixed cost icurrig wheever is used i the solutio. This results i the miimizatio of the umber of salesma to be activated i the solutio. Time widows are ofte icorporated to the mtsp (referred as mtsptsw), where it is also defied that certai odes eed to be visited i specific time periods. I the studied case oly the time-legth of the tours are costraied, which is almost idetical costrait defied o the maximum distace a salesma ca travel. The mai goal is to miimize the total travelig cost of the above problem that is ofte formulated as assigmet based iteger liear programmig [4]: cij xij + i mi mc () so that j= 2 j = 2 i= j= j m x = m (2) j x j = m (3) x ij =,, (4) x ij =,, (5) +subtour elimiatio costraits (6) 36

3 Magyar Kutatók 0. Nemzetközi Szimpóziuma 0 th Iteratioal Symposium of Hugaria Researchers o Computatioal Itelligece ad Iformatics where x {0, } is a biary variable used to represet that a arch is used o the tour, ij c ij is the cost associated to the distaces betwee the i th ad j th odes, ad c m represets the cost of the ivolvemet of oe salesme. 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. brachad-boud [9], cuttig plaes [8], eural etwork [5] or tabu search [2]. 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. [4] 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 [0], ad the solutio i [2] 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 [22] ad a other procedure is give by Potvi et al. [5]. The algorithm of Hsu et al. [6] preseted a Neural Networkbased solutio. More recetly, geetic algorithms (GAs) are successfully implemeted to solve TSP []. Potvi presets a survey of GA approaches for the geeral TSP [2]. 2. Geetic Algorithms for 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) ([7], [9]). 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 [6], 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. You et al. [26] use GAs to solve the mtsp i path plaig. A ew approach of chromosome represetatio, the so-called two-part chromosome techique ca be foud i [7] which reduces the size of the search space by the elimiatio of redudat solutios. I the literature there ca be foud several examples that a good problem-specific represetatio ca 37

4 A. Király et al. Optimizatio of Multiple Travelig Salesme Problem by a Novel Represetatio based Geetic Algorithm dramatically improve the efficiecy of geetic algorithms. A problem-specific idividual desig ca reduce the search-space, ad with special represetatio it is eeded to implemet special operators which ca simulate the ature of the problem. These properties ca make the problem-specific geetic algorithm more effective for the give task ad it becomes more easily iterpretable. The represetatio discussed here is very similar to the characteristic of mtsp. 3 The Proposed GA-based Algorithm to Solve the mtsp GAs are relatively ew global stochastic search algorithms which based o evolutioary biology- ad computer sciece priciples [4]. Due to the effective optimizatio capabilities of GAs [3], it makes these techique suitable solvig TSP ad mtsp problems. 3. Problem Represetatio There are several represetatios of mtsp, like oe chromosome techique [25], the two chromosome techique [7, 9] ad the latest two-part chromosome techique [7]. The ew approach preseted here is a so-called multi-chromosome techique which will be discussed below. This approach is used i otoriously difficult problems to decompose complex solutio ito simpler compoets. It was used i mixed iteger problem [20] or i order problems []. A usage of routig problem optimizatio ca be see i [23] ad a lately solutio of a symbolic regressio problem i [8]. This paper discusses the usage of multichromosomal geetic programmig i the optimizatio of mtsp. Figure illustrates the ew chromosome represetatio for mtsp with 5 locatios ( = 5 ) ad with 4 salesme ( m = 4 ). chromosome Salesperso chromosome 2 8 Salesperso 2 chromosome Salesperso 3 chromosome Salesperso 4 Figure Example of multi-chromosome represetatio for = 5 ad m = 4 38

5 Magyar Kutatók 0. Nemzetközi Szimpóziuma 0 th Iteratioal Symposium of Hugaria Researchers o Computatioal Itelligece ad Iformatics The figure above illustrates a idividual of the populatio. Each idividual represets a 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. 3.2 Operators Because of our ew represetatio, implemetatio of ew geetic operators became ecessary, like mutatio operators. Oly a overview of the operators are give i this subchapter. There are two sets of mutatio operators, the so-called I-route mutatios ad the Cross-route mutatios. I-route mutatio operators work iside oe chromosome. The first operator chooses a radom subsectio of a chromosome ad iverts the order of the gees iside it (Figure 2). The secod operator reverses two radomly chose gees i the give chromosome (Figure 3) ad the third put a radomly chose gee ito a give place as it ca be see i Figure Figure 2 I-route mutatio gee sequece iversio Figure 3 I-route mutatio gee traspositio Figure 4 I-route mutatio gee isertio 39

6 A. Király et al. Optimizatio of Multiple Travelig Salesme Problem by a Novel Represetatio based Geetic Algorithm 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. Figure 5 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 Figure 5 Cross-route mutatio gee sequece traspositio O Figure 6 it ca be see a cotractio of two chromosomes. I this situatio the umber of routes i the ewly created idividual is decreased by oe Figure 6 Cross-route mutatio chromosome cotractio Figure 7 illustrates the iverse operatio of chromosome cotractio. I this case a sigle chromosome is partitioed ito two ew chromosomes i the ewly created idividual, thus the umber of salesme is icremeted by oe. 320

7 Magyar Kutatók 0. Nemzetközi Szimpóziuma 0 th Iteratioal Symposium of Hugaria Researchers o Computatioal Itelligece ad Iformatics Figure 7 Cross-route mutatio chromosome partitio 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 (see also i previous chapter) ad selectio method 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. 32

8 A. Király et al. Optimizatio of Multiple Travelig Salesme Problem by a Novel Represetatio based Geetic Algorithm 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, m let the legth of the secod be k 2 ad so o. Of course k i = i =. 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! ( k )! ( k)! ( k k )! distict assigmet. For the secod chromosome this umber is 2 formulated as equatio (7).! ( k)! * * ( k )! ( k k )! ad so o. So the total search space of the problem ca be 2 ( k k * ( k k m )!! = )! ( )! m =! (7) It is ecessary to determie the legth of each chromosome too. It ca be represeted as a positive vector of the legths,,, ) k k k that must sum to ( 2 m. There are distict positive iteger-valued vectors that satisfy this m requiremet [24]. Thus, the solutio space of the ew represetatio is!. It is equal with the solutio space i [7], but this approach is more m similar to the characteristic of the mtsp, so it ca be more problem-specific therefore more effective. 322

9 Magyar Kutatók 0. Nemzetközi Szimpóziuma 0 th Iteratioal Symposium of Hugaria Researchers o Computatioal Itelligece ad Iformatics 4 Illustrative Example Although the algorithm was tested with a big umber of problems, oly two illustrative results is give i this article. The algorithm has implemeted i MATLAB, tiy refiemets i costraits are i progress. Figure 8 below illustrates the result of the optimizatio with 9 (right) ad 25 (left) cities. It results that 2 salesme is the optimal for this problem i both cases. I this test issue, the maximal route legth was 500 kilometers ad the maximal travelig duratio was 9 hours. The resulted 2 tour with 9 cities were 425 km ad 298 km log, the duratios were 50 miutes ad 366 miutes, respectively. These values were 346 km, 456 km ad 430 miutes, 539 miutes with 25 cities respectively. Figure 8 Illustrative result of the optimizatio 25 ad 9 cities The secod example o Figure 9 illustrates the result of the optimizatio with 25 cities ad with 450 km maximal route legth costrait. It results that 3 salesme is the optimal for this problem. The resulted 3 tour were 360, 357 ad 390 kilometers log, respectively. It ca be see that the approach is more sesitive for the legth of the tours tha the umber of cities. I every case, the ruig time was betwee ad 2 miutes. The geetic algorithm has made 00 iteratios, because experieces have show that this umber is sufficiet for the optimizatio. 323

10 A. Király et al. Optimizatio of Multiple Travelig Salesme Problem by a Novel Represetatio based Geetic Algorithm Figure 8 Illustrative result of the optimizatio 450 km maximal route legth 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. 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 mtsp. The mai beefit is the trasparecy of the represetatio that allows the effective icorporatio of heuristics ad costrais ad allows easy implemetatio. After some fial touches, the supportig MATLAB code will be also available at the website of the authors. Ackowledgemet The fiacial support from the TAMOP // (Élhetőbb köryezet, egészségesebb ember - Bioiováció és zöldtechológiák kutatása a Pao Egyeteme, MK/2) project is gratefully ackowledged. 324

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