Solving the Capacitated Vehicle Routing Problem Based on Improved Ant-clustering Algorithm

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1 MATEC Web of Cofereces, 0 ( 05) DOI: 5/ mateccof/ 050 C Owed by the authors, published by EDP Scieces, 05 Solvig the Capacitated Vehicle Routig Problem Based o Improved At-clusterig Algorithm Jiasha Zhag*, Xiaoqu Li & Yi Ju Chogqig Vocatioal Istitute of Egieerig, Jiagji, Chogqig, Chia Qiag Li College of Busiess Admiistratio, Chogqig Sciece ad Techology Uiversity, Chogqig, Chia ABSTRACT: The capacitated vehicle routig problems (CVRP) are NP-hard Most approaches ca solve small-scale case studies to optimality Furthermore, they are time-cosumig To overcome the limitatio, this paper presets a ovel three-phase heuristic approach for the capacitated vehicle routig problem The first phase aims to idetify sets of cost-effective feasible clusters through a improved at-clusterig algorithm, i which the adaptive strategy is adopted The secod phase assigs clusters to vehicles ad sequeces them o each tour The third phase orders withi clusters for every tour ad geetic algorithm is used to order withi clusters The simulatio idicates the algorithm attais high-quality results i a short time Keywords: CVRP; three-phase heuristic; at-clusterig algorithm; geetic algorithm INTRODUCTION Vehicle routig problems (VRPs) have bee the subject of itesive research for more tha years, due to their great scietific iterest as difficult combiatorial optimizatio problems ad their importace i may applicatio fields, icludig trasportatio, logistics, commuicatios, maufacturig, military ad relief systems, ad so o Hudreds of models ad algorithms have bee developed to obtai either optimal or heuristic solutios for differet versios of VRP, i which the capacitated Vehicle Routig Problem (CVRP) is oe of the most famous ad widely studied problems The CVRP was itroduced i the semial article by Datzig ad Ramser (959) uder the ame Truc Dispatchig Problem [] The curret ame (CVRP) [] of the problem became widespread i the article by Christofides (976) The majority of curret researches o CVRP focus o the problems withi a limited size of 0 customers [3] However, it is commo for real-life vehicle routig applicatios, such as waste collectio, courier service, beverage distributio ad mil collectio ad delivery, to ivolve the daily service of hudred or eve thousad customers Lestra ad Rioy Ka (98) have aalyzed the complexity of the vehicle routig problem ad have cocluded that practically all vehicle routig problems are NP-hard because they are ot solved i polyomial time [4] The exact algorithms ad traditioal heuristic algorithms are difficult, eve impossible, to solve CVRP First, the distace i a straight lie is t able to meet problem ay loger Secod, calculatig the distace matrix is time-cosumig Actually, besides the distace betwee customers ad the distributio *Correspodig author: zh_jiasha@63com ceter, the distaces amog adjacet customers are required, whereas customers away from each other usually do t belog to the same distributio route ad there is little probability of usig them That s to say some (ot all) of the distace matrixes are used i the process of calculatig So calculatig all the distaces betwee customers are uecessary I this paper, the solutio is attaied through three-phase heuristic, which first ivolves the coversio of CVRP to TSP, usig at-clusterig algorithm ad the geerate tour ad improve it The rest of this paper is orgaized as follows Sectio itroduces the relevat literature A mathematical programmig formulatio is developed i Sectio 3 Sectio 4 proposes the heuristic algorithm for solvig the CVRP Computatioal results o Solomo istaces are reported i Sectio 5 Fially, coclusios ad future wor are preseted i Sectio 6 LITERATURE REVIEW Early, costructive heuristics, such as savig method (Clare ad Wright, 964), sweepig method (Gillet ad Miller,974), ad Mole ad Jameso heuristic (Mole ad Jameso,976) were popular for CVRP I geeral, they provide solutios at -0% above the optimum, i egligible ruig times Tabu search that costituted the most competig algorithms i the 990s is still preset via variats that iclude sophisticated memory mechaisms I 996, Glover [5] preseted the advaces, applicatios, ad challeges i tabu search ad adaptive memory programmig The mai idea is to extract a sequece of poits (called boes) from a set of solutios ad ge- This is a Ope Access article distributed uder the terms of the Creative Commos Attributio Licese 40, which permits urestricted use, distributio, ad reproductio i ay medium, provided the origial wor is properly cited Article available at or

2 MATEC Web of Cofereces erate a route usig adaptive memory Further, the adaptive large eighborhood search (ALNLS) [6] is preseted by Pisiger ad Rope (007), i which a cotrol layer adaptively chooses amog a umber of removal ad isertio heuristics to itesify ad diversify the search However, the quality of tabu search depeds o the quality of iitial solutio Evolutioary algorithms are proved efficiet for the CVRP [7] presets the first hybrid GA for the VRP, which becomes a effective algorithm available for the large-scale istaces geerated by Golde et al [8] presets a improved geetic algorithm for solvig the largest existig bechmar istaces of CVRP [9] presets a Parallel Simulated Aealig for large-scale istaces However, the EA is slower tha may TS algorithms [] presets a efficiet variable eighborhood search heuristic for the capacitated vehicle routig problem A strategy of the guided local search metaheuristic is used to help escape local miima Proposed by Fisher ad Jaiumar [], Cluster First-route Secod method, is a effective way to deal with CVRP, especially large scale CVRP The approach, first creates customer clusters, ad the optimizes the order of visits for each cluster as a TSP sub-problem I [], the customers were firstly divided ito districts accordig to the mai road grid system The the customer districts were assiged to vehicles usig the vehicle flow formulatio model Experimets show that the method ca foud high-quality solutios I the method, clusterig efficietly is the ey of solvig problems 3 PROBLEM DESCRIPTION AND FORMULA- TION Let G ( V, E) be a complete udirected graph with V =+ The ode v 0 V represets a depot, where a fleet of idetical vehicles is based, ad where the product to be distributed is stored The other v i V\{ v 0 }, for i{,, }, represet the customers, characterized by demads for o-egative amouts of product q i Edges {i,j}e represet the possibility of travelig directly from a ode (customer or depot) v i V to a differet ode v j V for a trasportatio cost of c ij The CVRP aims to fid m or less vehicle routes, that is, sequeces of deliveries to customers, to visit each customer oe time exactly while miimizig the total travel distace The sum of demads should ot exceed o ay route of value Q assimilated to the vehicle capacity The decisio variables of the model are: x ij, if customer j is supplied after customer i by a vehicle of type 0, otherwise y j, if vehicle visits cliet j 0, else The cost of a vehicle of type traversig the pair (i, j) is deoted by c ij The objective fuctio ca be writte as follows: mi m m fxoj cijxij j i0 j Subjected to oj j ip i j0 i i i m j0 ij x,,, m () x x 0, p0, () pj qy Q,,, m (3) y i, i,, m, i 0 (4) x y, i,, (5) ij i x {0,}, i, jv;,, m Costraits () ad () state that each vehicle leaves the depot, after arrivig at a customer, the vehicle leaves agai, ad fially returs to the depot Costrait (3) guaratees that the vehicle capacity will ot be exceeded Costrait (4) ad (5) esure that each cliet s demad is fulfilled by exactly oe vehicle 3 Improved At-Clusterig Algorithm The improved At-Clusterig Algorithm is based o [3] Iitially, the are scattered radomly o a discrete D board The board ca be cosidered as a matrix of m m cells, where m = 4, ad is the total umber of to be clustered At first, K-meas is used to cluster the objects () to form heaps, which iclude or more objects i a sigle cell Give heap H with H objects, the parameters are defied as follows: The maximum distace betwee two objects i the heap: d ( H) max d(o, o ) max oi, ojh Where do (, o) i j is euclidia distace The ceter of heap: oce( H) oi H oi H i j 0-p

3 ICETA 05 The average distace betwee the objects ad the ceter of heap: davg ( H) d( oi, oce( H)) H oi H odissimiar ( H ) is the object farthest from the ceter of heap At secod, At-Clusterig Algorithm is used as follows: The ats are radomly scattered throughout the board do ( i, o j) measures the similarity betwee the pair of elemets ( oi, o j) 0, oi is similar with o j do ( i, oj) (6), otherwise The ormalizig term s equals the total umber of sites i the local area, ad itroduces similarity desity fuctio f(o i ), which is a local estimatio of the desity of ad their similarity to o i do ( i, oj) [ ], if f 0 f( o) o j i (7) s 0, otherwise Where the costat α scales the similarities If α is too large, it becomes difficult that ats pic up the object, but it is easy to put dow Thus, objects, which are dissimilar, are clustered i the same cluster Otherwise, α is too small, objects, which are similar, are ot clustered i the same cluster I this paper, the adaptive strategy is adopted to determie the value of α to improve the clusterig Quality It is show as follows The iitial value of α is set to 0 Durig the cotiuous times iteratio, times of at s failure to lay dow the object is N f, ad the ratio of N f ad N is α is give by = (8) I the process of clusterig, parameters are maitaied adaptive chages, which mae clusterig process more robust Whe the at is ot carryig ay objects, it loos for possible objects to pic up by looig at the eight eighborig cells aroud its curret positio The algorithm for picig up is show as follows () Label the 8 eighborig cells aroud the at as uexplored () Repeat Cosider the ext uexplored cell aroud the at If the cell is empty, the the object carried by the at, o i is dropped with a probability If the cell is ot empty, ad the cell cotais a sigle object o i, the the object o i is piced up with probability, p( i) ( ) f( oi ) P o, Where ad are threshold costats If the cell cotais a heap of two objects, the the heap is destroyed by picig up a radom object with a probability P destr If the cell cotais a heap H of more tha objects, the the most dissimilar object of H is removed oly if Do ( dissimiar ( H), oce ( H)) p remove D ( H) avg Label the cell as explored (3) Util all the eighborig cells have bee explored Whe the at is carryig a object, the it examies the 8 cells surroudig its curret locatio The algorithm for droppig the object is show as follows () Label the 8 eighborig cells aroud the at as uexplored () Repeat Cosider the ext uexplored cell aroud the at If the cell is empty the the object carried by the at, o i is dropped with a probability, f( oi), if f( oi) Pd( oi), if f ( oi ) If the cell cotais a sigle object o j, the a heap of two objects is created by droppig o i o o j oly if Do ( i, oj) P create Dmax If the cell cotais a heap H the o i is dropped o H oly if Do (, O ( H)) Do ( ( H), o ( H)) i ce dissimiar ce Label the cell as explored (3) Util all the eighborig cells have bee explored Meawhile, the total load of cluster caot exceed the vehicle capacity Otherwise, the will merge i aother cluster 4 ALGORITHM FLOW Phase I is iteded to reduce the computatioal burde of the subsequet solutio phase By establishig the mathematical model i terms of a few clusters rather tha a huge umber of customers, the CVRP size ca be decreased evidetly Phase II ad III are ot differet from other two phase method, which is show i [4] That is, Phase II aims to assig clusters to vehicles ad the sequece them o the same tour by solvig a compact 0-p3

4 MATEC Web of Cofereces versio of CVRP Replacig the customer, the clusters geerated i Phase I mae problem size shri sharply Phase III orders withi clusters for every tour, ad the will be visited i sequece To reach the goal, a low-size CVRP will be solved as may times as the umber of tours foud i Phase II Actually, each sigle-tour schedulig problem is tacled as TSP, which icludes the same i the clusters The mai at-based clusterig algorithm is preseted as follows ) Iitially, radomly place the ats o the board Radomly place o the board at most oe per cell ) K-meas is used to cluster the objects () to form heaps 3) Repeat For each at Do (a) Move the at; (b) If the at does ot carry ay object the: If there is a object i the 8 eighborig cells of the at, the at possibly pics up the object; Else, The at possibly drops a carried object, by looig at the 8 eighborig cells aroud it Ed if, updated α accordig to (8);Util stoppig criteria 4) Assig clusters to vehicles ad the sequece them o the same tour 5) Geetic algorithm (GA) [5] is implemeted withi each cluster to order 5 EXPERIMENTS I this sectio, we report our computatioal results ad compare them with those from the existig literature The proposed algorithm has bee executed o a Itel Petium 4 machie with GB of memory, ruig widows Our computatioal experimet is based o the Solomo istaces developed by Solomo et al (987) Table (a) Clusters geerated for example C-5( ) i the first phase Assiged C C C C C Cluster load Travel distace Loadig rate % % % % % Example C-5(), R-() ad RC-4 () have bee derived from the origial Solomo problem C-5, R-() ad RC-4 by just cosiderig the first customers I the first phase, the clusterig procedure is applied to problem C-5(), R-() ad RC-4() The origial have bee merged ito customer clusters i a very short time The i each cluster are show i Table Table (b) Clusters geerated for example R-( ) i the first phase Assiged Cluster load Travel distace Loadig rate C % C % C % C % 3 4 C % C % 3464 Discrete ,3,6, Table (c) Clusters geerated for example RC-4( ) i the first phase Assiged C C C C C Discrete 5 (merged i C5) Cluster load Travel distace Loadig rate % % % % % --- 5% 0-p4

5 ICETA Cluster C Cluster C Cluster C cluster C (a) (b) The improved at-clusterig algorithm has a good performace i clusterig, as deoted i Table (a-c) It maes full preparatios for Phase II ad Phase III Figure (a-c) shows the truly problem optimum is already foud through the exact approach The method is very successful for clustered istaces, such as C-class problems, ad the optimum for may of them is retaied See Figure (a) It is also effective for RC-class problems, which is show i Figure (c), but ot succeed i R-class problems See Figure (b) 6 CONCLUSION AND FUTURE WORK Cluster C Cluster C Cluster C This paper itroduces a ew at-clusterig algorithm for the capacitated Vehicle Routig Problem The method aims to itegrate a heuristic clusterig algorithm ito a optimizatio framewor The itroductio of preprocessig phase to gather ito a few clusters maes the CVRP size decreased sharply The Cluster C Cluster C Cluster C Cluster C proposed method ca retai optimum i a reasoable time, especially doig well i solvig large-scale CVRP with more tha 0 It is robust as the optimizatio method Experimets show that at-clusterig algorithm ca succeed i solvig a variety of Solomo problems Cluster C Cluster C (c) Figure Best solutio foud for problem Real-life vehicle routig applicatio is more complicated For example, the requiremet of customers is ofte ucertai The extesio of the method to these more difficult problems is worth further research ACKNOWLEDGEMENT This paper is Supported by the Natioal Natural Sciece Foudatio of Chia (No 90403) ad the Educatio Departmet of Liaoig provice sciece ad techology research project (NoL077) REFERENCES [] Datzig, G & Ramser, J, 959 The truc dispatchig problem Maagemet Sciece, 6 (), 80-9 [] Christofides, N 976 The vehicle routig problem RAIRO Operatios Research, (): [3] PaoloT & Daiele V 00 Models, relaxatios ad exact approaches for the capacitated vehicle routig problem Discrete Applied Mathematics, 3: [4] Lestra, J K, & Rioy Ka, A H G 98 Complexity of Vehicle ad Schedulig Problems Networs, : -7 [5] F Glover 996 Tabu search ad adaptive memory programmig-advaces, applicatios, ad challeges Cluster C4 Cluster C3 5 8 Cluster C5 0-p5

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