A HYBRID DIFFERENTIAL EVOLUTION -ITERATIVE GREEDY SEARCH ALGORITHM FOR CAPACITATED VEHICLE ROUTING PROBLEM

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1 IJCMA: Vol. 6, No. 1, January-June 2012, pp Global Research Publcatons A HYBRID DIFFERENTIAL EVOLUTION -ITERATIVE GREEDY SEARCH ALGORITHM FOR CAPACITATED VEHICLE ROUTING PROBLEM S. Kavtha and Nrmala P. Ratchagar Abstract: Ths paper presents a novel and effcent approach to solve the capactated vehcle routng problem (CVRP) ncludng dstance constrants. CVRP s an optmzaton problem wth an objectve to determne the optmal set of routes for a set of vehcles to delver goods to the customers n order to mnmze the total cost whle satsfyng varous operatonal constrants. The decson varables of ths problem are frst encoded as a strng called decson vector. The decson vector of the proposed algorthm conssts of a sequence of nteger numbers representng the customers for each vehcle n CVRP. The proposed method s developed n such a way that dfferental evoluton s actng as a man optmzer to dentfy the optmal set of customers for a fleet of homogeneous vehcles. The destructon and constructon phases of teratve greedy algorthm are ncorporated to dentfy the optmal sequence of vst for each vehcle n the proposed algorthm whch mproves the ftness of each vector n the populaton of dfferental evoluton algorthm. A computatonal study s carred out on Chrstofde and Elon benchmark problems to llustrate the performance of the proposed method. The feasblty of the proposed method s demonstrated and compared wth reported algorthm n the lterature. The results are promsng and show the effectveness of the proposed method. Keywords: optmzaton capactated vehcle routng problem dfferental evoluton greedy search 1. INTRODUCTION The dstrbuton of goods plays an mportant role n logstcs because the relevant costs consttute a large porton of the overall feld expense of the dstrbutor. The vehcle routng problem s an mportant problem for dstrbuton management and has motvated ntense theoretcal work for development of effectve models and sophstcated algorthms. A typcal vehcle routng problem can be stated as follows: The group of vehcles leaves from the depot and serve customers n the network, after completon of ther routes, vehcles return to the depot. Each customer s descrbed by a certan demand. Other nformaton ncludes the co-ordnates of the depot and customers, the dstance between them and the capacty of vehcles provdng the servce. All these nformaton are known n advance for the purpose of plannng a set of routes whch mnmzes the transportaton cost whle satsfyng other practcal constrants such as capacty constrants, dstance constrants etc. The vehcle routng problem was frst formulated by Dantzg and Ramser [1]. The

2 2 S. Kavtha and Nrmala P. Ratchagar standard verson of the vehcle routng problem called as capactated vehcle routng problem (CVRP). It s a problem to desgn a set of vehcle routes n whch a fxed fleet of delvery vehcles of unform capacty must serve known customer demands from a common depot at mnmum cost. The CVRP s an mportant operatonal problem of vehcle routng problems that must be solved n the daly operaton of physcal dstrbuton and logstc. The CVRP research started wth exact methodologes lke lnear programmng, dynamc programmng and branch & bound algorthms. The lmtatons of these methods are computaton tmes are extreme, addng new real constrants s a challenge and cannot be mplemented for realstc large scale problems [2, 3]. The detaled revew of exact methodologes for the soluton of VRP problems are gven n references [4, 5]. As the CVRP s a very complex NP-hard problem, solvng the real-lfe CVRP to optmalty s often not possble wthn the lmted computng tme avalable n practcal stuaton. Therefore the most of the research has focused on heurstc soluton methods desgned to produce solutons of reasonably good qualty n a reasonable amount of tme. Heurstc approaches can be dvded nto the classcal heurstc approaches and the metaheurstc approaches. The classcal heurstc approaches can fnd one feasble soluton quckly, but ths feasble soluton may have a large dsparty compared wth the best soluton [6]. In meta-heurstcs, the emphass s on performng a deep exploraton of the most promsng regons of the soluton space. Meta-heurstc approaches make no assumptons about the problem beng optmzed and can search very large spaces of canddate solutons. These methods typcally combne sophstcated neghborhood search rules, memory structures, and recombnaton of solutons. The qualty of solutons produced by these methods s much hgher than that obtaned by classcal heurstcs, but the prce to pay s ncreased computaton tme [7]. The meta-heurstc approaches can be used for combnatoral optmzaton problem n whch an optmal soluton s sought over a dscrete search-space. The advantage of these approaches s ther flexblty for handlng numerous constrants that are common n the VRP. Several meta-heurstc algorthms have been dentfed as havng potental to solve practcal dscrete optmzaton problems nclude Genetc algorthm (GA) [8], Tabu Search (TS) [9] and Partcle Swarm Optmzaton (PSO) [10]. Recently, t has become evdent that the use of a sole meta-heurstc s rather restrctve n solvng combnatoral optmzaton problem. Therefore, a new class of methods called hybrd meta-heurstcs has attracted more and more concern, whch s more effcent and flexble [11]. Many hybrd meta-heurstc approaches have been recently publshed for the soluton of CVRP n the lterature. Juan et al. [12] presented a hybrd algorthm that combnes a CVRP classcal heurstc wth Monte Carlo smulaton. Ln et al. [13] appled a hybrd algorthm of smulated annealng and tabu search to solve CVRP. Ths algorthm takes the advantages of smulated annealng and tabu search for solvng CVRP. A hybrd algorthm whch combnes the partcle swarm optmzaton and smulated annealng algorthm for CVRP was proposed by Chen A-Lng et al.[14]. Marnaks et al. [15] adopted the

3 A Hybrd Dfferental Evoluton - Iteratve Greedy Search Algorthm for Capactated... 3 combnaton of nature nspred methodology namely hybrd genetc - partcle swarm optmzaton algorthm for the soluton of vehcle routng problem. A hybrd heurstc method terated varable neghborhood descent algorthm wth varable neghborhood descent based on mult-operator optmzaton for CVRP was proposed by Chen et al. [16]. Genetc algorthm s the frst evolutonary algorthm appled to the combnatoral optmzaton problems. Evolutonary algorthms are stochastc search methods that operate on a populaton of solutons by smulatng, at a hgh level of abstracton, the evoluton of speces observed n nature. In a Genetc algorthm, populaton of solutons evolves over a number of generatons through the applcaton of operators, lke selecton, crossover and mutaton that mmc the correspondng genetc processes observed n nature. Dfferental evoluton s a populaton based meta-heurstc algorthm for global optmzaton [17]. It s one of the most promnent new generaton evolutonary algorthms for global optmzaton. Despte the smplcty and the hgh effcency of dfferental evoluton, ts applcaton on the soluton of dscrete optmzaton problem s lmted. CVRP s a dscrete optmzaton problem, therefore the conventonal dfferental evoluton algorthm cannot be drectly appled for the soluton wthout modfcaton. A novel hybrd method s proposed n ths paper by combnng dfferental evoluton wth teratve greedy algorthm to solve the CVRP. In the proposed method, dfferental evoluton dentfes the set of customers for each vehcle and then the optmal sequence of vst for each vehcle s determned by applyng teratve greedy algorthm. In an teratve greedy approach, some customers are elmnated from the ncumbent soluton and elmnated customers are renserted nto the sequence usng the constructon heurstc. Constructon heurstc buld a feasble soluton by addng one component to the current partal soluton untl a complete soluton s obtaned. The advantage of teratve greedy algorthm s ts smple concept, whch makes t easy tunable [18]. A combnaton of a global search optmzaton method wth a local search optmzaton method mproves the performance of the algorthm. The advantages of the proposed method for solvng CVRP are ts smple concept, great robustness and easy mplementaton. 2. FORMULATION OF CAPACITATED VEHICLE ROUTING PROBLEM The am of the CVRP s to determne the delvery routes whch mnmze the total cost and should satsfy the requrements of the dstrbuton ponts. In practce, mnmzng the total cost s equvalent to total dstance traveled for the number of gven vehcles. The problem defntons of the capactated vehcle routng problem are as descrbed follows. Each vehcle has the same loadng capacty. All customers have known demands. Each customer s vsted exactly by one vehcle. Each vehcle route starts and ends at the same depot. Each vehcle's route can only pass through one depot exactly once. The loadng and travelng dstance of each vehcle should not exceed the loadng capacty and the maxmum travelng dstance of vehcle, that s the sum of demands and total length on any vehcle route should not exceed the vehcle capacty and the specfed vehcle range [14]. Assume that the delvery depot s node 0.

4 4 S. Kavtha and Nrmala P. Ratchagar Notatons used n capactated vehcle routng problem N - Number of customers V - Number of vehcles k d j - Travelng dstance from the th customer to the jth customer by kth vehcle Q k - Maxmum loadng capacty of kth vehcle q - Demand of the th customer D k - Maxmum allowed travelng dstance of the kth vehcle. x j k - Assgnment condton of dstance from the th customer to the jth customer k k by kth vehcle; x j = 1 s the assgnment, and x j s not an assgnment. The mathematcal model of the CVRP can be stated as follows: Mnmze Subject to V N N k k dj x j (1) k 1 0 j 0 V N k 1 0 V N k 1 j 0 k xj 1 for j 1, 2,..., N (2) k xj 1 for 1, 2,..., N (3) N N 0 j 0 k k dj xj Dk for k 1, 2,..., V (4) N N 0 j 0 k k xjq Q for k 1, 2,..., V (5) N N k j j 1 j 1 k x xj 1 for 0 and k 1,2,,..., V (6) V N k 1 j 1 k xj V for 0 (7) Equaton (1) s the objectve functon of the problem whch seeks to mnmze total dstance traveled. Equatons (2) and (3) ensure that each customer s vsted by exactly one

5 A Hybrd Dfferental Evoluton - Iteratve Greedy Search Algorthm for Capactated... 5 vehcle. Equaton (4) represents the total length of each route cannot exceed the maxmum travelng dstance of the vehcle. Equaton (5) shows that the total demand of any route should not exceed the capacty of the vehcle. Equaton (6) makes sure every route starts and ends at the delvery depot. Equaton (7) specfes that there are maxmum V routes gong out of the delvery depot. 3. OVERVIEW OF DIFFERENTIAL EVOLUTION Dfferental Evoluton developed by Storn and Prce s one of the excellent evolutonary algorthms for optmzaton problem. Dfferental evoluton was developed n 1995 as a populaton-based stochastc evolutonary optmzaton algorthm. In the ntalzaton, a populaton of NP vectors X G ; = 1, 2,, NP (8) s randomly generated wthn user-defned bounds. The sze of the populaton s specfed by the parameter NP that has to be set by the user. Usually t s kept fxed durng an optmzaton run. The populaton members are real-valued vectors wth dmenson D that equals the number of decson varable n the optmzaton problem. For convenence, G the decson vector, X s represented as (X G 1, X G 2,..., X G D ). The ftness of each ndvdual n the populaton s evaluated. The evolutonary operators mutaton, recombnaton and selecton are appled to every populaton member to generate a new generaton. Frst, a mutant vector s bult by addng a vector dfferental to a populaton vector of ndvdual accordng to the followng equaton: Z X F. ( X X ) (9) G 1 G G G r1 r2 where = 1, 2,, NP s the ndvdual's ndex of populaton; G s the generaton; The mutaton factor F s a control parameter of DE that has to be set by the user. The ndexes r 1, r 2 represents the random and mutually dfferent ntegers generated wthn the range [1, NP] and also dfferent from the runnng ndex. The specalty n DE les n the mutaton step whereby two vectors are randomly selected from the populaton and the vector dfference between them s establshed. The dfference s multpled by a mutaton factor, F and added to a thrd randomly chosen vector from the populaton. Ths step s known as dfferental varaton and the result s known as mutant vector. The mutaton factor controls the amplfcaton of the dfference between two ndvduals so as to avod search stagnaton and s usually taken from the range [0.1, 1]. DE s senstve to the choce of mutaton factor. Followng the mutaton operaton, recombnaton s appled to the populaton. Recombnaton s employed to generate a tral vector by replacng certan parameters of the target wth the correspondng parameters of a randomly selected donor vector. In the recombnaton operaton, each gene of the th ndvdual s reproduced from the mutant vectors Z G ( G 1, G 2,..., G Z Z Z D ) and the current ndvdual X G ( G 1, G 2,..., G X X X D ) as follows:

6 6 S. Kavtha and Nrmala P. Ratchagar U G j 1 X G j G 1 j f a random number CR Z otherwse; j 1, 2,... D, 1,..., NP (10) where CR s a crossover or recombnaton rate n the range [0, 1] and has to be set by the user. The selecton operaton selects accordng to the ftness value of the populaton vector and ts correspondng tral vector, whch vector wll survve to be a member of the generaton. If f denotes the objectve functon under mnmzaton, then X G 1 G 1 T G 1 U f f ( U ) f ( X ) G X otherwse In ths case, the cost of each tral vector U G+1 s compared wth that of ts parent target vector X G. If the cost f of the target vector X G s lower than that of the tral vector, the target s allowed to advance to the next generaton. Otherwse, the target vector s replaced by the tral vector n the next generaton. The mutaton, recombnaton and selecton are repeated for NP tmes to complete one teraton. The above teratve process of mutaton, recombnaton and selecton on the populaton wll contnue untl a user-specfed stoppng crteron s met. (11) 4. HYBRID DIFFERENTIAL EVOLUTION ALGORITHM FOR CVRP PROBLEM The detaled mplementaton of hybrd dfferental evoluton algorthm to fnd soluton for capactated vehcle routng problem s gven below: Intalzaton: DE uses NP D-dmensonal parameter vectors P h, G; h = 1, 2,..., NP, = 1, 2,..., D (12) n a generaton G, wth NP beng constant over the entre optmzaton process. At the start of the procedure,.e., generaton G = 1, the populaton vectors have to be generated randomly wthn the lmts. The mnmum lmt s one and the maxmum lmt s total number of vehcles employed for servng some commodty to a gven number of customers. Assumng that the dstrbuton centre has two vehcles and eght customers to serve, the populaton vector s represented by the followng strng: Customer node : Populaton vector : In the above example, the populaton vector means that nodes 1, 4, 6 and 7 wll be vsted by vehcle 1 and node 2, 3, 5 and 8 by vehcle 2. The ntal populaton vectors are randomly generated such that all the vehcles should be ncluded n each vector and the vehcle capacty of each vehcle should not volate ther lmts.

7 A Hybrd Dfferental Evoluton - Iteratve Greedy Search Algorthm for Capactated... 7 Mutaton: Mutaton s an operaton that adds vector dfferentals to a populaton vector of ndvduals. For the followng generaton G + 1, new vectors V h, G + 1 are generated accordng to the followng mutaton scheme V h, = nt(p, + F. (P, P )) for h = 1, 2,..., NP (13) G+1 h G r1 G r2, G The ntegers r1 and r2 are chosen randomly over [1, NP] and should be mutually dfferent from the runnng ndex h. F s a scalng factor, whch controls the amplfcaton of the dfference between two ndvduals so as to avod search stagnaton. Crossover operaton: In order to ncrease the dversty of the perturbed parameter vectors, crossover operaton s ntroduced after the mutaton operaton. Crossover operaton s appled to generate a tral vector by replacng certan parameters of the mutant vectors V h, G+1 by the correspondng parameters of a present ndvdual vector P h,g. That s U h, G 1 V h, G 1 P h, G f (rand (0,1) CR) otherwse In the above equaton, rand (0, 1) s the th evaluaton of a unform random number generator wth outcome [0, 1]. CR s the crossover rate [0, 1] whch has to be fxed by the user. Evaluaton of each agent: Each ndvdual n the populaton s evaluated usng the ftness functon of the problem to mnmze the objectve functon. The maxmum travelng dstance constrant s augmented wth the objectve to form a generalzed ftness functon f h as gven below V N N V k k h j j m k k 1 1 j 1 m 1 2 (14) f d x ( D D ) (15) where µ s penalty parameter, D k s the travelng dstance of kth vehcle and D m s the maxmum allowed travel dstance of kth vehcle. If the travelng dstance of mth vehcle s less than ts allowed travelng dstance then D k s fxed to D m n the ftness functon equaton. The penalty term reflects the volaton of the nequalty constrant and assgns a hgh cost of penalty functon to canddate pont far from feasble regon. Estmaton and Selecton: The parent s replaced by ts chld f the ftness of the chld s better than that of ts parent. Explctly, the parent s retaned n the next generaton f the ftness of the chld s worse than that of ts parent. DE selecton scheme s based on local competton only..e., a chld U h, G+1 wll compete aganst one populaton member P h, G and survvor wll enter the new populaton. Chld or tral vector dentfes the dstrbuton ponts for each vehcle. The number NC of tral vector that may be produced for parent vector to satsfy the vehcle capacty constrants should be chosen suffcently hgh so that a suffcent number of chldren wll enter the new populaton. If any one vehcle capacty of tral vector s volated, a new tral vector s generated usng the vector generaton process

8 8 S. Kavtha and Nrmala P. Ratchagar defned by equatons (13) and (14). If the capacty constrants of vehcles are satsfed then terated greedy algorthm s mplemented to mprove the ndvdual routes of capactated vehcle routng problem. Iterated greedy algorthm generates a sequence of solutons by teratng over greedy constructve heurstcs usng two man phases: destructon and constructon. Durng the destructon phase some soluton components are removed from a prevously constructed complete canddate soluton. The constructon procedure then apples a greedy constructve heurstc to reconstruct a complete canddate soluton. Once a canddate soluton has been completed, an acceptance crteron decdes whether the newly constructed soluton wll replace the ncumbent soluton. The process of destructon and constructon contnues untl the predefned teratons NT are reached. The functonng of terated greedy algorthm for one teraton s llustrated n the followng example: Customer node : Tral vector : In the above example, customer nodes 1, 2, 3, 5 and 7 are vsted by vehcle 1, and nodes 4, 6 and 8 by vehcle 2. The depot s node 0. The example teraton for frst vehcle s llustrated below wth number of nodes removed n destructon phase d = 2. The ntal route for vehcle 1 s Intal route of vehcle 1, Dstance = 48 Km DESTRUCTION PHASE Choose 2 nodes at random Partal sequence to reconstruct 5 1 Customer nodes to rensert CONSTRUCTION PHASE After rensertng customer node 5 nto all the possble postons of the partal sequence of the nodes that are already scheduled, the best locaton s selected based on the mnmum dstance After rensertng node 5, Dstance = 43.5 Km After rensertng node 1, Dstance = 42.5 Km

9 A Hybrd Dfferental Evoluton - Iteratve Greedy Search Algorthm for Capactated... 9 The example shows that startng from the soluton for the frst vehcle gven by the tral vector (dstance = 48 km), a new soluton (dstance = 42.5 Km) s reached by removng customer nodes 5 and 1 and rensertng them n the way descrbed above. The ftness of newly constructed soluton s less than the startng soluton; therefore ncumbent soluton s replaced by the newly constructed soluton. Ths new sequence s known to be optmal soluton at the end frst teraton of terated greedy algorthm. Ths new sequence wll be taken as a startng soluton for the next teraton of terated greedy algorthm. The teratve procedure wll be contnued untl a predefned teraton s reached. The advantage of the terated greedy algorthm s that t has only two control parameters, NT maxmum number of teratons, and d number of nodes removed and renserted durng the destructon and constructon phases. These control parameters are selected based on the number of customer nodes assgned for each vehcle. The optmal sequence of customer nodes for each vehcle s dentfed by the terated greedy algorthm, whch are used to fnd the ftness of tral vector U h,g+1 usng equaton (15). f U h, G+1 s better than that of ts parent, P h, G+1 wll be set to U h, G+1. Stoppng Crteron: The above teratve process of mutaton, crossover, and selecton on the populaton wll contnue untl there s no apprecable mprovement n the mnmum ftness value or predefned maxmum number of teratons reached. The flow chart of the proposed algorthm s gven n Fgure NUMERICAL EXPERIMENT AND DISCUSSION Hybrd dfferental evoluton-teratve greedy search algorthm for the CVRP descrbed above has been appled to 8-customers CVRP to llustrate the mplementaton of the proposed algorthm. A set of computatonal experment s conducted on the Chrstofde and Elon benchmark problems, that had been used by Wang et al [19], to assess the performance of the proposed algorthm for a large sze nstances and to compare the results of proposed method wth hybrd genetc algorthm (HGA) reported n the lterature. The smulatons are carred out on a PC wth Pentum IV 2.8-GHZ processor. The software s developed usng MATLAB 6.5. The proposed DE algorthm wth terated greedy search algorthm uses the followng control varables.e. populaton sze NP, maxmum number of generatons NG, crossover rate CR, scalng factor F, maxmum number of chld vectors NC produced for each parent vector to satsfy capacty constrants of vehcles, number of nodes selected for destructon and constructon phase n terated greedy search, and number of teratons NT n terated greedy search to fnd optmal sequence of vst by the vehcles. The number of trals has been conducted wth changes n the sze of populaton, number of generatons, crossover rate and scalng factor n order to obtan the best values to acheve the overall mnmum travelng dstance. In 8-customer vehcle routng problem wth dstance constrants, 2-vehcles from a sngle depot are allotted to serve 8-customers. The capacty of each vehcle s 8-unts and

10 10 S. Kavtha and Nrmala P. Ratchagar Fgure 1: Flow Chart of Hybrd Dfferental Evoluton - Iteratve Greedy Search Algorthm for CVRP

11 A Hybrd Dfferental Evoluton - Iteratve Greedy Search Algorthm for Capactated the total length of each route should not exceed 40 km. The dstance between customers (n Km) and demand of each customer are gven n Table-1. The objectve s to mnmze the total travelng dstance of commtted vehcles subject to satsfyng vehcle capacty and dstance constrants. Table 1 Dstance Between Customers n km and Demand of Each Customer Customer Demand Customer The parameters of the dfferental evoluton chosen for the sample problem are NP = 50, NG = 25, NC = 2, NT = 10, F = 2, CR = 0.6 and number of nodes selected n terated greedy search approach s two. In order to show the consstency n gettng optmal solutons of the proposed approach, 10 ndependent trals have been conducted. The depot s node 0. The optmal routes obtaned from smulaton are route 1: , the total number unts delvered by the frst vehcle are 7 and the total travel dstance s 34 km; route 2: , the total number unts delvered by the second vehcle are 8 and the total travel Fgure 2: Graphcal Representaton of Capactated Vehcle Routng Vehcle Routng Problem Soluton for 8 Customers of 2 Vehcles [0 Denotes the Depot]

12 12 S. Kavtha and Nrmala P. Ratchagar dstance s 33.5 km. The total travelng dstance by the two vehcles s 67.5 km. The average smulaton tme taken by the proposed approach s 21 seconds. The smulaton results are compared wth the results reported usng double populaton genetc algorthm (DGA) [20]. The DGA provdes global optmal soluton of 67.5 km for ths problem for two trals out of ten trals. But the proposed approach provdes the same optmal soluton of 67.5 km for all ndependent trals, whch shows the consstency of the algorthm for solvng dscrete optmzaton problem. The graphcal representaton of optmal routes for 8-custmers CVRP s shown n Fgure 2. In Chrstofdes and Elon benchmark problems, the total number of customers s varyng from 22 to 100 customers, and the total number of vehcles s varyng from 3 to 14 vehcles. In these benchmark problems, X-Y coordnates of the depot and customers, and the demand of each customer are specfed n the dataset. The node 1 represents X-Y coordnates of the depot and the remanng nodes represents coordnates of the customers. The dstance from depot to dstrbuton ponts and dstance between dstrbuton ponts are calculated from the X-Y coordnates. The populaton vector dentfes the group of dstrbuton ponts for each vehcle n the capactated vehcle routng problem. The X and Y coordnates of all dstrbuton ponts relatve to the depot are calculated as follows X x x Y y y Where (X,Y ) s the x and y coordnates of the th dstrbuton pont relatve to the depot, (x 0, y 0 ) s the coordnate of the depot and (x, y ) s the orgnal coordnate for the th dstrbuton pont. The X and Y coordnates of all dstrbuton ponts are converted to polar angles wth respect to depot usng the followng formula Y 1 tan, X 0, Y 0 X Where represents the polar angle of th dstrbuton pont. For a large sze problem, the dstrbuton ponts of each vehcle are ntally arranged n ascendng order of the polar angles so as to generate the structured sequence of vst by the vehcle. The constructon and destructon phases are then employed to mprove the ftness of the populaton vectors. The followng control parameters have been chosen for the benchmark problems after a number of trals: NP = 100, NG = 200, NC = 10, NT = 20, F = 0.5, CR = 0.6 and number of nodes d selected n terated greedy search approach depends upon the length of the strng for each vehcle. By tral and error method, the value of d s assgned to three for a vehcle havng more than fve customer nodes and two for vehcle havng three to fve nodes whch provdes optmal soluton for the test problems. The program s termnated when the best found soluton has not been updated to 20 consecutve generatons. To evaluate the performance of the proposed algorthm for large nstances, 10 ndependent trals are carred out for all the benchmark problems and the best operaton plan soluton obtaned are gven n Appendx (16) (17)

13 A Hybrd Dfferental Evoluton - Iteratve Greedy Search Algorthm for Capactated A comparson of the solutons obtaned through the proposed algorthm and HGA reported n the recent lterature s gven n Table 2. The effcency of the proposed algorthm s measured by the qualty of the produced solutons. The qualty s gven n terms of the relatve devaton from the best known soluton. From the comparson wth the prevous best known soluton, t s apparent that the proposed hybrd method produces best soluton to 5 problems out of the 10 problems n ths dataset and three of them are new best soluton. The maxmum devaton from best known soluton s 1.11, whch shows the potental of the algorthm to solve large customer nstances. The performance of proposed method s superor to that of HGA n terms of the soluton qualty. The convergence result on the shortest dstance of the proposed hybrd algorthm for E-n23-k3 problem s shown n Fgure 3. It s clear from the characterstcs that the soluton by the proposed method has converged to qualty soluton at early teratons. Table 2 Comparson of Results for Benchmark Problems Problem No. of No. of Best known HGA Proposed Percentage Comp. tme number Customers vehcles soluton [19] [19] method Devaton (%) (s) E-n23-k E-n30-k * E-n33-k E-n51-k E-n76-k E-n76-k E-n76-k E-n76-k E-n101-k E-n101-k * No. of vehcles exceed the lmt. Fgure 3: Convergence Characterstcs of the Proposed Algorthm for the 22-customer Capactated Vehcle Routng Problem

14 14 S. Kavtha and Nrmala P. Ratchagar A qualty result s obtaned by the proposed method wth reasonable computaton tme because a number of trals are appled to get a feasble soluton vector whch satsfes the capacty constrants of vehcles, thus the computatonal efforts on local search for unfeasble soluton s avoded n the algorthm..e. A capacty constrant for each vehcle has already been done whle constructng the route usng teratve greedy search algorthm. Further, the soluton qualty s mproved from destructon and constructon phases of teratve search algorthm. The combnaton of these efforts s potental for obtanng good solutons for capactated vehcle routng problems. 6. CONCLUSION A novel dfferental evoluton based algorthm has been developed for the soluton of capactated vehcle routng problem. In ths approach, nteger valued soluton vectors are employed n dfferental evoluton algorthm and evolutonary process are carred out n dscrete space. The constructon and destructon phases of terated greedy algorthm have been mplemented as local search algorthm. The proposed algorthm provdes a global or near global optmal soluton for the benchmark test systems. The computatonal result shows that the proposed hybrd algorthm s an alternatve for solvng practcal capactated vehcle routng problems. The applcaton of proposed algorthm can be extended to other varants of vehcle routng problems wth sutable modfcatons. REFERENCES [1] Dantzg G.B., Ramser J.H.: The Truck Dspatchng Problem. Management Scence, 6, (1959). [2] Descrochers M., Desrosers J., Solomon M.: A New Optmzaton Algorthms for the Vehcle Routng Problem wth Tme Wndows. Operaton Research, 40(2), (1992) [3] Kallehauge B.: Formulatons and Exact Algorthms for the Vehcle Routng Problem wth Tme Wndows. Computers and Operaton Research, 35(7), (2008). [4] Laporye G., Nobert Y.: Exact Algorthms for the Vehcle Routng Problems. Annals of Dscrete Mathematcs, 31, (1987). [5] Toth P., Vgo D.: Exact Soluton of the Vehcle Routng Problem n Fleet Management and Logstcs. Kluwer, Boston, 1-31 (2003). [6] Shh-We L., Zne-Jung L., Kuo-ChngY., Chou-Yuan L., Applyng Hybrd Meta-heurstcs for Capactated Vehcle Routng Problem. Expert Systems wth Applcatons, 36, (2009). [7] Toth P., Vgo D., The Vehcle Routng Problem. SIAM Monographs on Dscrete Mathematcs and Applcatons, Phladelpha. [8] Baker B.M., Ayechew M.A.: A Genetc Algorthm for the Vehcle Routng Problem. Computers and Operaton Research, 30, (2003).

15 A Hybrd Dfferental Evoluton - Iteratve Greedy Search Algorthm for Capactated [9] Barbarosoglu G., Ozgur D.: A Tabu Search Algorthm for the Vehcle Routng Problem. Computers and Operaton Research, 26, (1999). [10] A T.J., Kachtvchyanukul V., A Partcle Swarm Optmzaton for the Capactated Vehcle Routng Problem. Internatonal Journal of Logstcs and SCM Systems, 2, (2007). [11] Chen P., Huang H., Dong X., Iterated Neghborhood Descent Algorthm for the Capactated Vehcle Routng Problem. Expert Systems wth Applcatons, 37, (2010). [12] Juan A.A., Fauln J., Ruz R., Barros B., Caballe S.: The SR-GCWS Hybrd Algorthm for Solvng the Capactated Vehcle Routng Problem. Appled Soft Computng, 10, (2010). [13] Ln S., Lee Z., Yng K., Lee C.: Applyng Hybrd Meta-heurstcs for Capactated Vehcle Routng Problem. Expert Systems wth Applcatons, 36, (2009). [14] Chen A-Lng, Yang Gen-ke, WU Zh-mng,: Hybrd Dscrete Partcle Swarm Optmzaton Algorthm for Capactated Vehcle Routng Problem. Journal of Zhejang Unversty SCIENCE, 7, (2006). [15] Marnaks Y., Marnak M.: A Hybrd Genetc-Partcle Swarm Optmzaton Algorthm for the Vehcle Routng Problem. Expert Systems wth Applcatons, 37, (2010). [16] Chen P., Huang H., Dong X.: Iterated Varable Neghborhood Descent Algorthm for the Capactated Vehcle Routng Problem. Expert Systems wth Applcatons, 37, (2010). [17] Storn R., Prce K.: Dfferental Evoluton - A Smple and Effcent Heurstc for Global Optmzaton Over Contnuous Spaces. Journal of Global Optmzaton, 11, (1997). [18] Ruz R., Stutzle T.: A Smple and Effectve Iterated Greedy Algorthm for the Permutaton Flowshop Schedulng Problem. European Journal of Operatonal Research, 177, (2007) [19] Wang C.H., Lu J.Z.: A Hybrd Genetc Algorthm that Optmzes Capactated Vehcle Routng Problems. Expert Systems wth Applcatons, Elsever, 36, (2009). [20] Zhao Y.W., Wu B., Jang L., Dong H.Z., Wang W.L.: Study of the Optmzng of Physcal Dstrbuton Routng Problem on Genetc Algorthm. Computer Integrated Manufacturng Systems, 109(3), (2004). Appendx-1 Problem number E-n23-k3 Vehcle Capacty: 4500 Route No Nodes Demand Dstance Total

16 16 S. Kavtha and Nrmala P. Ratchagar Problem number E-n30-k3 Vehcle Capacty: 4500 Route No Nodes Demand Dstance Problem number E-n33-k4 Vehcle Capacty: 8000 Total Route No Nodes Demand Dstance Problem number E-n51-k5 Vehcle Capacty: 160 Total Route No Nodes Demand Dstance Problem number E-n76-k7 Vehcle Capacty: 220 Total Route No Nodes Demand Dstance Total

17 A Hybrd Dfferental Evoluton - Iteratve Greedy Search Algorthm for Capactated Problem number E-n76-k8 Vehcle Capacty: 180 Route No Nodes Demand Dstance Total Problem number E-n76-k10 Vehcle Capacty: 140 Route No Nodes Demand Dstance Total Problem number E-n76-k14 Vehcle Capacty: 100 Route No Nodes Demand Dstance Cont d

18 18 S. Kavtha and Nrmala P. Ratchagar Problem number E-n101-k8 Vehcle Capacty: 200 Total Route No Nodes Demand Dstance Total Problem number E-n101-k14 Vehcle Capacty: 112 Route No Nodes Demand Dstance Total

19 A Hybrd Dfferental Evoluton - Iteratve Greedy Search Algorthm for Capactated S. Kavtha and Nrmala P. Ratchagar Mathematcs Secton, Faculty of Engneerng & Technology, Annamala Unversty, Annamalanagar , Inda, E-mal: kavtha_aucdm@yahoo.com