Heterogeneous Vehicle Routing Problem with profits Dynamic solving by Clustering Genetic Algorithm

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1 IJCSI Iteratioal Joural of Computer Sciece Issues, Vol. 10, Issue 4, No 1, July 2013 ISSN (Prit): ISSN (Olie): Heterogeeous Vehicle Routig Problem with profits Dyamic solvig by Clusterig Geetic Algorithm Sawsa Amous Kallel 1 ad Youes Bouelbee 2 1 UREA : Research Uit i Applied Ecoomics, Faculty of Ecoomics ad Maagemet, Airport Road Km 4, Sfax 3000, Tuisia 2 Director of the Research Uit UREA Faculty of Ecoomics ad Maagemet, Airport Road Km 4, Sfax 3000, Tuisia Abstract The trasport problem is kow as oe of the most importat combiatorial optimizatio problems that have draw the iterest of may researchers. May variats of these problems have bee studied i this decade especially the Vehicle Routig Problem. The trasport problem has bee associated with may variats such as the Heterogeeous Vehicle Routig Problem ad dyamic problem. We propose i this study dyamic performace measures added to HVRP that we call Heterogeeous Vehicle Routig Problem with Dyamic profits (HVRPD), ad we solve this problem by proposig a ew scheme based o a clusterig geetic algorithm heuristics that we will specify later. Computatioal experimets with the bechmark test istaces cofirm that our approach produces acceptable quality solutios compared with previous results i similar problems i terms of geerated solutios ad processig time. Experimetal results prove that the method of clusterig geetic algorithm heuristics is effective i solvig the HVRPD problem ad hece has a great potetial. Keywords:The Heterogeeous Vehicle Routig Problem, dyamic problems, geetic algorithm heuristics, k-meas clusterig. 1. Itroductio Withi the wide scope of logistics maagemet, trasportatio plays a cetral role ad is a crucial activity i the delivery of goods ad services. The trasport problem is oe of the maily essetial combiatorial optimizatio problems that have take the iterest of several researchers. Huge research efforts have bee devoted to the study of logistic problems ad thousads of papers have bee writte o may variats of this problem such as Travelig Salesma Problem (TSP), Vehicle Routig Problem (VRP), supply chai maagemet (SCM) ad so o [6]. The VRP is oe of the most studied combiatorial optimizatio problems ad it cosists of the optimal desig of routes to be used by a fleet of vehicles to satisfy the demads of customers. I geeral, the umber of vehicles used i VRP is a variable decisio. A variat of VRP has a dyamic ature ad ca be modelled as Dyamic Combiatorial Optimizatio Problem (DCOP). This variat has bee used i may researches o trasportatio problems. The results of these studies show that the complex real trasportatio problems are dyamic ad chage over time i terms of their obective fuctios, decisio variables ad costraits. This implies that the optimal solutio might chage at ay time due to the chages i the trasport eviromet. May other related problems are associated with VRP such as the Heterogeeous Fleet Vehicle Routig Problem (HVRP). The HVRP differs from the classical VRP i that it deals with a heterogeeous fleet of vehicles havig various capacities ad both fixed ad variable costs. Therefore, the HVRP ivolves desigig a set of vehicle routes, each startig ad edig at the depot, for a heterogeeous fleet of vehicles which services a set of customers with specific demads. Each customer is visited oly oce, ad the total demad of a route should ot exceed the loadig capacity of the vehicle assiged to it. The routig costs of a vehicle is the sum of its fixed costs ad a variable costs icurred proportioately to the travel distace. The obective is to miimize the total of such routig costs. The umber of available vehicles of each type is assumed to be ulimited [5]. I the literature, three HVRP versios have bee studied. The first oe was itroduced by Golde ad al. [10], i which variable costs are uiformly give over all vehicle types with the umber of available vehicles assumed to be ulimited for each type. This versio is also called the Vehicle Fleet Mix (VFM) [19], the Fleet Size ad Mix VRP (1)

2 IJCSI Iteratioal Joural of Computer Sciece Issues, Vol. 10, Issue 4, No 1, July 2013 ISSN (Prit): ISSN (Olie): [11] or the Fleet Size ad Compositio VRP [8]. This versio that we are cosider i this paper. The secod versio cosiders the variable costs, depedet o the vehicle type, which are igored i the first versio. This versio is referred to as the HVRP [7], the VFM with variable uit ruig costs [18]. The third oe, called the VRP with a heterogeeous fleet of vehicles [20] or the heterogeeous fixed fleet VRP [21], geeralizes the secod versio by limitig the umber of available vehicles of each type. The remaiig part of the paper is orgaized as follows: the ext sectio itroduces the otatio used throughout the paper ad describes the variats of heterogeeous VRPs studied i the literature the we itroduce our problem: heterogeeous vehicle routig problem with dyamic profits associated to the customer. I sectio 3 we propose a hybrid algorithm that we call Clusterig Geetic Algorithm. Fially, Sectio 4 reviews the experimetal results. The remaiig part of the paper is orgaized as follows: the ext sectio itroduces the otatio used throughout the paper ad describes the variats of heterogeeous VRPs studied i the literature the we itroduce our problem: heterogeeous vehicle routig problem with dyamic profits associated to the customer. I sectio 3 we propose a hybrid algorithm that we call Clusterig Geetic Algorithm. Fially, Sectio 4 reviews the experimetal results. 2. Heterogeeous Vehicle Routig Problem with Dyamic profits associated to the customer priority I this part, we describe the formulatio of HVRP the we explai our problem ad we propose a ew mathematic formulatio for HVRPD. 2.1 The Heterogeeous Vehicle Routig Problem: formulatio of the problem The Heterogeeous Vehicle Routig Problem is G V, A defied as follows [2]. Let s V v v,..., v graph where 0, 1 A vi, v : viv V, i ad is a directed is the set of 1 odes is the arc set. Node v0 represets a depot o which a fleet of vehicles is based, while the remaiig ode set correspods to cities or customers. Each customer v i V ' V ' V \ 0 vertices has a o egative demad q i. The vehicle fleet is composed of m differet vehicle M 1,...,m types, with. For each type k M, mk vehicles are available at the depot, which have a Q capacity equal to k. Each vehicle type is also associated F with a fixed cost, equal to k ad a variable cost Gk per distace uit. The umber of vehicles of each type is assumed to be v i, v ulimited. With each arc is associated a distace or C i cost. The HVRP aims at desigig a set of vehicle routes, each startig ad edig at the depot, visitig each customer oly oce, limitig the total demad of a route to the loadig capacity of the vehicle assiged to it, ad miimizig the total cost of the route [7]. All the HVRP variats are NP-hard as they iclude the classical VRP as a special case. All the preseted studies have thus focused o developig heuristic algorithms as a substitute for exact algorithms. They ca be geerally grouped ito two kids: classical heuristics [11], [18], [16], frequetly derived from the classical VRP heuristics, ad meta-heuristics like the tabu search method [7], [21]. For more iformatio, see [18] ad [21] for a review of HVRP variats. I our approach, we propose a ew problem that we called HVRPD it is to itroduce the priority of the customers with dyamic profits i the formulatio of HVRP. 2.2 Proposed problem DHVRP Practically, dyamism ca be attributed to several factors, such as ew customer order, cacellatio of old demads Demads of customers that have bee studied by Taillard [20], Gedreau ad al. [7], Ismail ad Irhamah [12] have already used dyamism i their study. Other factors are stochastic like travel times betwee customers, customers to be visited, locatios of customers, capacities of vehicles, ad umber of vehicles available have bee dealt with by Pris [15], Reaud [16] I the static vehicle routig problem, iformatio is assumed to be kow, icludig all attributes of the customers such as the geographical locatio, the service time ad all the details about the customer s demad. However, i the dyamic applicatios, iformatio o the problem is ot completely kow i advace. (2)

3 IJCSI Iteratioal Joural of Computer Sciece Issues, Vol. 10, Issue 4, No 1, July 2013 ISSN (Prit): ISSN (Olie): The problem is said dyamic whe ot all iformatio related to the plaig of the routes is kow by the plaer whe the routig process begis. Besides, the iformatio is progressively reveled to the decisio maker ad it is likely to chage accordig to the dyamic ature of the trasport eviromet. For the problem uder study, we will combie HVRP ad dyamic VRP icludig the priority of customers. We will preset i this paper, a mathematical model for HVRPD. Hece, the mathematical formulatio is similar to that of geeral HVRP [9] with the itroductio of the dyamism ad priorities by cliets. Note z ir = 1 if the cliet i is at the rak r ad 0 otherwise. The modelig of the problem is similar to stadard HVRP [8] ad is as follows: Subect to (1) (2) (3) (4) (5) (6) (7) With the additio of other costraits to itroduce the priciple of dyamic gais are: r1 rz ir 1 k 1 xi r1 rz r (8) k rz r rzir 1 1 xi r1 r1 (9) Ad z ir r1 1 (10) q Give the radom character of the eeds i, chagig cotiually, ad availability of vehicles, we assume a probability distributio accordig to the history of the customer's request. I the above formulatio, costrait (1) esures that the customer is visited oly oce ad the costrait (2) esures the cotiuity of a tour that meas: if a vehicle visits a customer i, it marks the ed for this customer ad the start of the ext oe. The maximum umber available for each type of vehicles is imposed by costrait (3). Costrait (4) states that the differece betwee the quatities of goods that a vehicle carries raised before ad after visitig a customer is equal to the demad of this customer. Costrait (5) esures that the vehicle capacity does ever exceed the demad of customers. The two costraits (6) ad (7) show that the quatity trasported must be positive ad the value of ca take oly 0 or 1. For the two ew costraits (8) ad (9), we itroduce the profits: the lower the rak of the cliet is the higher his gais are. The rakig of cliet ca replace the priciple of time widow ad it makes it easier to implemet ad reduce the computatio time with the same efficiecy. It will be beeficial to have a small place. I this paper, ad to solve this problem, we will develop hybrid approaches that icorporate the best features of the metaheuristic "geetic algorithms" ad the method of classificatio "k-meas" that we called "Clusterig Geetic Algorithm". So, we develop five steps, that we will metio later, that begi by clusterig method ad fiish with fidig the optimal tours at the same time, they satisfy the demad of customers. 3. Clusterig Geetic Algorithm The growig importace of combiatorial optimizatio problems requires the search for efficiet algorithms to fid optimal ad ear optimal solutios. Clusterig methods are importat tool to aalyze ad maage data. They are used to classify data ito homogeeous classes. Ispired by the high performace of Geetic Algorithms (GA); we propose to itroduce a clusterig process ito a geetic algorithm i order to desig a efficiet tool to deal with complex optimizatio data. I this domai, some proposed works have appeared recetly [4]. To solve the HVRPD, takig ito accout the ew capacity costraits ad priorities for customers, a hybrid algorithm will be iitialized amed Clusterig Geetic Algorithm (CGA), which must make appropriate chages o the structural represetatio of the elemets of the problem ad reset, ad that crossig operatios with regard to the specifics of the problem i five steps. First, cliet groups withi the capacity of the vehicles are formed. They are based o clusterig techiques of k- meas. The, a travelig salesma problem is solved i each group by the GA. I the third phase, the vehicles are assiged to tours obtaied withi the costraits of capacity. They should also aim at miimizig the costs. The fourth phase is to itroduce two variables that show the (3)

4 IJCSI Iteratioal Joural of Computer Sciece Issues, Vol. 10, Issue 4, No 1, July 2013 ISSN (Prit): ISSN (Olie): degree of certaity with whish all the requiremets must be satisfied by a give vehicle k. Fially, the fifth phase aims to idetify the customers who are satisfied, takig ito accout the levels of priority, by which groups of customers are aggregated ito superpriority odes. 3.1 Clusterig Method Clusterig ca be cosidered the most importat usupervised learig problem; so, as every other problem of this kid, it deals with fidig a structure i a collectio of ulabeled data. A loose defiitio of clusterig could be the process of orgaizig obects ito groups whose members are similar i some way. A cluster is therefore a collectio of obects which are similar betwee them ad are dissimilar to the obects belogig to other clusters [22]. There are may clusterig methods available, ad each of them may give a differet groupig of a dataset. The choice of a particular method will deped o the type of output desired, The kow performace of method with particular types of data, the hardware ad software facilities available ad the size of the dataset. I geeral, clusterig methods may be divided ito two categories based o the cluster structure which they produce. The o-hierarchical or usupervised methods divide a dataset of N obects ito M clusters, with or without overlap ad the hierarchical clusterig algorithm is based o the uio betwee the two earest clusters. The begiig coditio is realized by settig every datum as a cluster. After a few iteratios it reaches the fial clusters wated. The specific method used i our approach is the k meas. It s oe of the simplest usupervised learig algorithms that solve the will-kow clusterig problem. k-meas cosists i partitioig the data ito groups, or "clusters". These clusters are obtaied by positioig "cetroids" i regios of space that have the largest umber of populatio. Each observatio is the assiged to the closest prototype. Each class therefore cotais observatios that are earer to a prototype tha ay other. The, the cetroids are positioed by a iterative procedure (the cetroid of the classes is calculated: the weight of a ceter of gravity is equal to the sum of the weights of the class of idividual forms) which leads progressively to their fial stable positio. The weight of a ceter of gravity aims at miimizig a obective fuctio, i this case a square error fuctio. The obective fuctio [22]: x ( ) i c 2 where is a chose distace measure ( ) x betwee a data poit i c ad the cetroid, is a idicator of the distace of the data poits from their respective cluster ceter. At this state, the fial umber of classes will be defiig. k-meas is a algorithm that has bee adapted to may problem domais. As we are goig to see, it is a good cadidate for extesio to work with fuzzy feature vectors. Oce the classes are idetified i the fial partitio of the data space, it is possible, to ru the Geetic Algorithm cluster by cluster. 3.2 Geetic Algorithm The Geetic Algorithm (GA) starts from a iitial populatio of cadidate solutios or idividuals ad proceeds for a certai umber of iteratios util oe or more stoppig criteria is /are satisfied. This evolutio is directed by a fitess measure fuctio that assigs to each solutio (represeted by a chromosome) a quality value. Oce the populatio is evaluated, the selectio operator chooses which chromosomes i the populatio will be allowed to reproduce. The stroger a idividual is, the greater chace of cotributig to the productio of ew idividuals it has. The ew idividuals iherit the properties of their parets ad they may be created by crossover (the probabilistic exchage of values betwee chromosomes) or mutatio (the radom replacemet of values i a chromosome). Cotiuatio of this process through a umber of geeratios will result i a group of solutios with better fitess i which optimal or ear-optimal solutios ca be foud [1]. Whe the size of the problem is becomig wider, geetic algorithms fid a difficulty i idetifyig the optimal solutio. Nevertheless, the questio is why sacrifyig a approach that ca have a good opportuity to develop a ew area of expertise ad keep the problem i touch with reality? The purpose of the implemetatio of geetic algorithms is to get a hamiltoia cycle for the clusters idetified. This meas the geetic algorithms are applied cluster by cluster ad the outcome of this step is the tours the clusters K,..., 1, K2 K. T,..., 1, T2 T for (4)

5 IJCSI Iteratioal Joural of Computer Sciece Issues, Vol. 10, Issue 4, No 1, July 2013 ISSN (Prit): ISSN (Olie): Clusterig Geetic Algorithm I our approach CGA, we have first formed the clusters usig k-meas the we have built the route by solvig TSP problems for each tour. So, we have divided our vehicle routig problem (search space) ito areas to locate the task of GA. Therefore, we will use a two-phase method. It has show i may applicatios [3], [16] of the vehicle routig problem ad its geeralizatios that the set of routes is the most importat phase of the algorithm. The two-phase method uses the priciple of "cluster first - route secod" [5]. I the case of TSP, such a method comprises, first, splittig the set of vertices (customers) ito sub-classes (sets) of customers, ad the issuig a routig process o each class to have a solutio. The geeral outlie of the two-phase method is as follows: Geeral diagram of the two-phase method step 1 : i = 0, iitialize imax. step 2 : Repeat steps 3 through 4 util i = imax. step 3 : Geerate a Phase1 partitio ito classes of all customers usig a specific method of classificatio. step 4 : Phase2 Apply a routig procedure o each class of peaks geerated by step 3, i++. The diversificatio of class divisios allows the two phase method to provide several solutios to the problem by applyig alteratively the classificatio procedures ad routig. After clusterig ad geetic algorithm, the ext step is to fid the optimal tour. Therefore, we must fid the complete tour by elimiatig some edges ad oiig clusters. We will take the adacet cetroids ad choose the shorts edges that we have to elimiate ad repeat the same operatio util we have oly oe tour ad ultimately fid the miimal cost of the whole tour. Oce the tours have bee completed, we will assig the vehicles i a way that makes the fixed costs associated with the trucks the lowest possible. The ext step of our algorithm is to itroduce two variables that allow to idetify the degree of certaity with which a give vehicle k will meet all requiremets must be satisfied. Fially, the fifth phase aims to idetify the satisfied customers, takig ito accout the levels of priorities i which groups of customers are aggregated ito superpriority odes. 4. Experimet results I our research, we iclude the k-meas method to classify data ito idetical classes i order to facilitate the procedure of the geetic algorithm ad it tackles the travellig salesma problem with may cities. The performaces of the heuristics are tested, with a few exceptios, o two sets of bechmark istaces: the first oe cosists of the 20 VFM istaces [10], ad the secod cotais 8 istaces oly with fixed costs [19]. The followig table illustrates the results foud by comparig the clusterig geetic algorithm (Clu.GA) with a classical geetic algorithm (Cla.GA) (two-fuctio poit mutatio by iversio), icludig the miimum legth of the tour ad fially the ratio = average / optimal for differet istaces of bechmarks from TSP library LIB see table 1. Istaces Table 1: Examples of istaces per cluster Best Miimal tour kow Cla. GA Clu. GA Cla. solutios GA Ratio Clu. GA A Brg Ch Ch Eil Eil Eil Gil Fl Fl Results from the experimet i table 1 show the advatages of the proposed algorithm cocerig the quality of the solutio (the ratio is: Average/optimale). The proposed algorithm, Clu. GA, is compared with stadard geetic algorithms cosiderig a set of istaces from the TSP LIB. Our experimetal results prove the efficiecy of the proposed algorithm i may istaces whe compared to the best-kow solutios of Taillard [19]. We have also tested these istaces with Lido ad CPLEX algorithm that gives optimal solutios but oly for small istaces. Both tests have take much time to fid these solutios. (5)

6 IJCSI Iteratioal Joural of Computer Sciece Issues, Vol. 10, Issue 4, No 1, July 2013 ISSN (Prit): ISSN (Olie): Eve though, our algorithm gives solutios that are ot optimal but it has proved to be more efficiet the Lido ad CPLEX oes i terms of processig time. The resolutio of the problem shows that our algorithm eeds less time to solve big istaces. Let be the followig istaces with a umber of differet cities. We compare our algorithm with [12], [19] ad [10]. The results are summarized i the followig table: Ist. Table 2. Comparable istaces Osma ad Salhi Taillard Golde Our Avreage Solutio Time The results preseted i table 2 show that our algorithm produces solutios that are comparable quality to those of Osma ad Salhi[12] ad Taillard [19] ad it improves the computatioal time (i secodes). Comparisos with other authors are the oly way to evaluate our solutios because it is a ew problem i the literature. Our average results foud by Clu.GA are close to those of Osma ad Salhi [12], but they are bit lower tha those of Taillard [19]. Is Veh. Vehicles ature Table 3. Our average results Vehicles capacity Our average value Fixed costs Time Our algorithm has better performace o istaces with fixed costs. Here, our average solutio values are lower tha those of Taillard i most cases, but they allow a good classificatio of vehicles ad their fixed costs ad also our algorithm allocate customers i a better way tha that preseted by Golde (See table 3). 5. Coclusios I this paper, we suggest a ew mathematic model for HVRPD ad we solve this problem by proposig a hybrid metaheuristic based o GA ad k-meas clusterig. The various experimets give idicate o the sesitivity of the algorithms i differet cofiguratios. They ca exhibit the advatage of usig a defiitio of compromise proposed to be specifically icorporated ito a evolutioary algorithm. I this study, we have ivestigated a class of dyamic optimizatio problems. These algorithms eed to be evaluated o other classes of problems icludig other variats ad costraits. We have used the clusterig approach as a additioal meas to cotrol itesificatio i order to obtai robust algorithms able to produce good solutios. The results are ecouragig especially i relatio to processig time if compared with similar problems ad similar algorithms. Refereces [1] Amous K, S., Loukil, T. Elaoud, S. ad Dhaees, C A New geetic algorithm applied to the travelig salesma problem. Iteratioal Joural of Pure ad Applied Mathematics, Vol. 48, 0 2, pp , [2] Baldacci, R., Christofides, N., Migozzi, A A exact algorithm for the vehicle routig problem based o the set partitioig formulatio with additioal cuts. M athematical Programmig Ser. A 2007 ( [3] Choi, E. ad Tcha, D.W. A colum geeratio approach to the heterogeeous fleet vehicle routig problem. Computers & Operatios Research; (i press). [4] Fisher, ML. ad Jaikumar, R A geeralized assigmet heuristic for vehicle routig. Networks 11: , [5] Ge, M. ad Cheg, R Geetic algorithms ad egieerig optimizatio. Joh Wiley &Sos, New York, [6] Gedreau, M. Laport, G. Musaragayi, C. ad Taillard, E.D A tabu search heuristic for the heterogeeous fleet vehicle routig problem. Computers & Operatios Research. 26, , [7] Gheyses, F. Golde, B. Assad, A A ew heuristic for determiig fleet size ad compositio. Mathematical Programmig Study.;26:233 6, [8] Gheyses, F.G. Golde, B.L. ad Assad, A A compariso of techiques for solvig the feet size ad mix vehicle routig problem. OR Spectrum, 6(4): , (6)

7 IJCSI Iteratioal Joural of Computer Sciece Issues, Vol. 10, Issue 4, No 1, July 2013 ISSN (Prit): ISSN (Olie): [9] Golde, B. ad Assad, A Vehicle Routig: Methods ad Studies. Editors, North-Hollad, Amsterdam, pp , [10] Golde, B. Assad, A. Levy, L. ad Gheyses, F The fleet size ad mix vehicle routig problem. Computers ad Operatios Research;11:49 66, [11] Ismail, Z. ad Irhamah Solvig the Vehicle Routig Problem with Stochastic Demads via Hybrid Geetic Algorithm-Tabu Search. Joural of Mathematics ad Statistics: 4(3): , 2008 ISSN: , [12] Osma, IH. ad Salhi, S Local search strategies for the vehicle fleet mix problem. I V.J. Rayward-Smith, I.H. Osma, C.R. Reeves, ad G.D. Smith, editors, Moder Heuristic Search Methods, pages Wiley: Chichester, [13] Pris C. A simple ad effective evolutioary algorithm for the vehicle routig problem, Computers ad Operatios Research, Volume 31, Issue 12, , [14] Pris, C Efficiet heuristics for the heterogeeous fleet multitrip VRP with applicatios to a large-scale real case. Joural of Mathematical Modelig ad Algorithms, 1(2): , [15] Reaud, J. Boctor, FF A sweep-based algorithm for the fleet size ad mix vehicle routig problem. Europea Joural of Operatioal Research ;140:618 28, [16] Roberto, Baldacci, Maria, Battarra ad Daiele, Vigo Routig a Heterogeeous Fleet of Vehicles. DEIS, Uiversity Bologa via Veezia 52, Cesea, Italy, [17] Salhi, S. Sari, M. Saidi, D. ad Touati, N Adaptatio of some vehicle fleet mix heuristics. Omega, 20: , [18] Taillard, ED A heuristic colum geeratio method for the heterogeeous fleet VRP. RAIRO; 33:1 14, [19] Taratilis, CD. Kiraoudis, CT. Vassiliadis, VS A threshold acceptig metaheuristic for the heterogeeous fixed fleet vehicle routig problem. Europea Joural of Operatioal Research. 152:148 58, [20] Toth, P. & Vigo, D Models, relaxatios ad exact approaches for the capacitated vehicle routig problem. Discrete Applied M athematics. 123, 1-3: , [21] Wassa, NA. Osma, IH Tabu search variats for the mix fleet vehicle routig problem. Joural of Operatioal Research Society;53:768 82, (7)