A Memetc Algorthm for the Vehcle Routng Problem wth Tme Wndows Jean Berger and Mohamed Barkaou Defence Research and Development Canada - Valcarter, Decson Support System Secton 2459 Pe-XI Blvd. North, Val-Bélar, PQ, Canada, G3J 1X5 emal: ean.berger@drdc-rddc.gc.ca, barkaou@orcom.ca Abstract Seral and parallel versons of a new memetc algorthm to address the Vehcle Routng Problem wth Tme Wndows are presented. The underlyng approach nvolves parallel co-evoluton of two populatons. The frst populaton evolves ndvduals to mnmze total traveled dstance whle the second focuses on mnmzng temporal constrant volaton to generate a feasble soluton. New genetc operators have been desgned to ncorporate key concepts emergng from recent promsng technques such as nserton heurstcs, large neghborhood search and ant colony systems to further dversfy and ntensfy the search. The parallel verson of the method s based on a master-slave message-passng paradgm. The master controls the executon of the algorthm, synchronzes atomc genetc operatons and handles parent selecton whle the slaves concurrently execute genetc operatons. Results from a computatonal experment show that the seral verson of the proposed technque matches or outperforms the best-known heurstc routng procedures, provdng sx new best-known solutons. In comparson, the method proved to be fast, cost-effectve and hghly compettve. Alternatvely, smulaton results obtaned for the parallel verson show a sgnfcant mprovement over the seral algorthm, matchng or even mprovng soluton qualty. The parallel algorthm shows a speed-up of fve n computng soluton havng near smlar qualty. 1. Introducton Vehcle routng problems are well known combnatoral optmzaton problems wth consderable economc sgnfcance. The Vehcle Routng Problem wth Tme Wndows (VRPTW) has receved a lot of attenton n the lterature recently. Ths s mostly due to the wde applcablty of tme wndow constrants n real-world cases. In VRPTW, customers wth known demands are servced by a homogeneous fleet of vehcles of lmted capacty. Routes are assumed to start and end at the central depot. Each customer provdes a tme nterval durng whch a partcular task must be completed such as loadng/unloadng the vehcle. It s worth notng that the tme wndow requrement does not prevent any vehcle from arrvng before the allowed start of servce at a customer locaton. The obectve s to mnmze the number of tours or routes, and then for the same number of tours, to mnmze the total traveled dstance, such that each customer s servced wthn ts tme wndow and the total load on any vehcle assocated wth a gven route does not exceed the vehcle capacty. A varety of algorthms ncludng exact methods and effcent heurstcs have already been proposed for VRPTW. For excellent surveys on exact, heurstc and metaheurstc methods, see [Desrosers et al., 1995], [Cordeau et al., 2001] and [Bräysy and Gendreau, 2001a and 2001b]
respectvely. In partcular, evolutonary and genetc algorthms have been among the most sutable approaches to tackle the VRPTW, and are of partcular nterest to us. Genetc algorthms [Holland, 1975]; [De Jong, 1975] and [Goldberg, 1989] are adaptve heurstc search methods that mmc evoluton through natural selecton. They work by combnng selecton, recombnaton and mutaton operatons. The selecton pressure drves the populaton toward better solutons whle recombnaton uses genes of selected parents to produce offsprng that wll form the next generaton. Mutaton s used to escape from local mnma. Routng technques based on genetc algorthms to solve VRPTW emerge from the work of [Blanton and Wanwrght,1993], [Thangah, 1995a and 1995b], [Thangah et al., 1995], [Potvn and Bengo, 1996], [Berger et al., 1998, 1999, 2000] and [Tan et al., 2001]. Alternate methods usng evolutonary metaheurstcs have been proposed by [Homberger and Gehrng, 1999], [Gehrng and Homberger, 1999 and 2001], and [Bräysy et al., 2000]. Other recent studes on varous metaheurstcs for VRPTW can be found n [Rochat and Tallard, 1995], [Tallard et al., 1997], [Chang and Russell, 1997], [Cordeau et al., 2001] (tabu searches), [Gambardella et al., 1999] (ant colony optmzaton), and [Lu and Shen, 1999]. Proposed metaheurstcs so far show sgnfcant varablty n performance. They often requre consderable computatonal effort and therefore fal to convncngly provde a sngle robust and successful technque. Recently, a new memetc or parallel hybrd genetc algorthm (PHGA) for the VRPTW has been successfully developed [Berger, Barkaou and Bräysy, 2002]. Memetc Algorthms s a populaton-based approach for heurstc search n optmzaton problems [Moscato, 1989]. They have shown that they are orders of magntude faster than tradtonal genetc algorthms for some problem domans. Bascally, they combne local search heurstcs wth mutaton and crossover operators. For ths reason, some researchers have vewed them as hybrd genetc algorthms or genetc local search. Our approach s based on a new concept that combnes constraned parallel co-evoluton of two populatons and partal temporal constrant relaxaton to mprove soluton qualty. The frst populaton evolves ndvduals to mnmze the total traveled dstance whle the second focuses on mnmzng temporal constrant volaton n tryng to generate a feasble soluton. Imposng a constant number of tours for each soluton of a gven populaton, temporal constrant relaxaton allows escapng local mnma whle progressvely movng toward a better soluton. Populatons nteract wth one another whenever a new feasble soluton emerges, reducng by one the number of tours mposed on future solutons. New genetc operators have been desgned to maxmze the number of customers served wthn ther tme ntervals frst, and then temporal constrant relaxaton s used to nsert remanng unvsted customers. Key prncples and varants emergng from recent promsng technques are also captured to further dversfy and ntensfy the search. As a result, even though the algorthm s more robust, effcent, stable and hghly compettve, prohbtve computatonal cost of key genetc operators and overall run-tme stll reman a senstve ssue to be satsfactorly addressed. The man contrbuton of ths paper s to further mprove the PHGA technque by developng an effcent parallel mplementaton based upon a master-slave message-passng paradgm (networked parallel computng) n order to sgnfcantly reduce run-tme. The master processng element controls the executon of the algorthm, synchronzes atomc genetc operatons and handles the parent selecton process whle the slave processng elements concurrently execute
reproducton and mutaton operators. Current and new genetc operators have been desgned and revsted to reduce processor starvaton over each generaton. The paper s outlned as follows. Secton 2 ntroduces the basc concepts of the proposed parallel hybrd genetc algorthm. The basc prncples and features of the algorthm are frst descrbed. Detals on the parallel mplementaton of the algorthm are then gven. Secton 3 presents the results of a computatonal experment to assess the value of the proposed approach and reports a comparatve performance analyss to alternate methods. Fnally, a summary s presented n Secton 4. 2. Parallel Hybrd Genetc Algorthm 2.1 General Descrpton The proposed algorthm reles upon constraned parallel co-evoluton and partal constrant relaxaton. Two populatons Pop 1 and Pop 2, prmarly formed of non-feasble soluton ndvduals, are evolvng concurrently, each wth ther own obectve functons. Pop 1 contans at least one feasble soluton and s used to mnmze total traveled dstance whle Pop 2 focuses on mnmzng constrant volaton. Constraned to a fxed number of tours over the same populaton, soluton ndvduals dffer by exactly one route across both populatons. Parallel evoluton s nterrupted whenever a new best feasble soluton s obtaned. Populatons are then rentalzed and co-evoluton resumed, whle decreasng the number of routes assocated wth soluton ndvduals by one. The number of tours mposed on soluton ndvduals n Pop 1 and Pop 2 are R mn and R mn 1, respectvely. R mn refers to the number of routes found n the best feasble soluton obtaned so far. As a new feasble soluton emerges from Pop 2, populaton Pop 1 s replaced by Pop 2, R mn s updated and, Pop 2 s rentalzed wth the revsed number of tours (R mn -1), usng the RSS_M mutaton operator. In addton, a post-processng procedure (RC_M) amed at reorderng customers, s appled to further mprove the new best soluton. The evolutonary process s repeated untl a predefned stoppng condton s met. The proposed approach uses a steady-state genetc algorthm that nvolves overlappng populatons. At frst, new ndvduals are generated and added to the current populaton Pop p. The process contnues untl the overlappng populaton outnumbers the ntal populaton by n p. Then, the n p worst ndvduals are elmnated to mantan populaton sze usng the followng ndvdual evaluaton: where Eval = E + CV, (1) d E = r rm +, (2) max{ d, d } CV n =1 m = α max{ 0, b l } + β Vol (3) r = number of routes n ndvdual, r m = lower bound for number of routes (rato of total demand over vehcle capacty), d = total traveled dstance related to ndvdual, d m = average traveled dstance over the ndvduals formng the ntal populaton,
n = number of customers, α = penalty assocated wth temporal constrant volaton, b = scheduled tme to vst customer n ndvdual, l = latest tme to vst customer, β = penalty assocated wth number of volated temporal constrants, Vol = number of temporal constrants volated n ndvdual. The proposed evaluaton expresson ndcates that better ndvduals generally (but not necessarly) nclude fewer routes, and smaller total traveled dstance, whle satsfyng temporal constrants. The general algorthm s as follows: Intalzaton Repeat p=1 Repeat {evolve populaton Pop p - new generaton} For =1..n p do Select two parents from Pop p Generate a new soluton S usng recombnaton and mutaton operators assocated wth Pop p Add S to Pop p end for Remove the n p worst ndvduals from Pop p usng the evaluaton functon (1). p=p+1 Untl (all populatons Pop p have been consdered) f (Pop 2 ncludes a new best feasble soluton) then {elmnate all Pop 1 ndvduals} Set Pop 1 = Pop 2 Modfy Pop 2 solutons by applyng RSS_M {reduces number of routes by one}. endf Apply RC_M on the best soluton {customer reorderng} Untl(convergence crtera or max number of generatons) Feasble solutons for ntal populatons are frst generated usng a sequental nserton heurstc n whch customers are nserted n random order at randomly chosen nserton postons wthn routes. The ntalzaton procedure then proceeds as follows: For p = 1..2 do {revst Pop 1 and Pop 2 } For = 1..n p do Generate a new soluton S usng the EE_M mutator (defned n Secton 2.3.2) Add S n Pop p end for Remove the n p worst ndvduals from Pop p usng Eval end for Determne R mn, the mnmum number of tours assocated wth a feasble soluton n Pop 1 or Pop 2. Replcate (f needed) best feasble soluton (R mn routes) n Pop 1. Replace Pop 1 ndvduals wth R mn -route solutons usng the procedure RI(R mn ). Replace Pop 2 members wth R mn -1 route solutons usng the procedure RI(R mn -1). RI(r) s a re-ntalzaton procedure creatng an r-route soluton. It frst generates r one-customer routes formed from randomly selected customers. Then, t uses the nserton procedure proposed by Lu and Shen [Lu and Shen, 1999] to nsert as many customers as possble wthout volatng tme wndow constrants. Accordngly, customer route-neghborhoods are repeatedly examned
for nserton. The next customer for nserton s selected by maxmzng a so-called regret cost functon that accounts for multple route nserton opportuntes: regret cost = r RN ( ) { c ( r) c ( r*)}, (4) where RN () = route-neghborhood of customer, c (r) = mnmum nserton cost of customer wthn route r, c (r*) = mnmum nserton cost of customer over ts route-neghborhood. Remanng unvsted customers (f any) are then nserted n the r-tour soluton maxmzng an extended nserton regret cost functon, n whch c (r) ncludes an addtonal contrbuton reflectng temporal constrant volatons: nr =1 α max{ 0, b l } + β Vol (5) n whch n r = current number of customers n route r, α = penalty assocated wth temporal constrant volaton, β = penalty assocated wth the number of volated temporal constrants, b = scheduled tme to vst customer n route r, l = latest tme to vst customer, Vol r = current number of temporal constrants volated n route r. 2.2 Selecton r The selecton process conssts of choosng two ndvduals (parent solutons) wthn the populaton for matng purposes. The selecton procedure s stochastc and based toward the best solutons usng a roulette-wheel scheme [Goldberg, 1989]. In ths scheme, the probablty of selectng an ndvdual s proportonal to ts ftness value. An ndvdual ftness value s computed as follows: Populaton Pop 1 : ftness n = d + =1 α max{ 0, b l } + β Vol (6) Populaton Pop 2 : ftness n = α max{ 0, b l } + β Vol (7) =1 The notatons are the same as n equatons 1 3. Better ndvduals generally (but not necessarly) tend to nclude short total traveled dstance n Pop 1 and satsfy as many temporal constrants as possble n Pop 2.
2.3 Genetc Operators The proposed genetc operators mostly rely on two basc prncples. Frst, for a gven number of tours, an attempt s made to construct feasble solutons wth as many customer vsts as possble. Second, the remanng customers are nserted nto exstng routes through temporal constrant relaxaton. Constrant volaton s used to restrct the total number of routes to a constant value. The proposed genetc operators ncorporate some key features of the best heurstc routng technques such as Solomon s nsertons heurstc I1 [Solomon, 1987] large neghborhood search [Shaw, 1998] and the route neghborhood-based two-stage metaheurstc (RNETS) [Lu and Shen, 1999]. Detals on the recombnaton and mutaton operators used are gven n the next sectons. 2.3.1 Recombnaton The nserton-based IB_X(k) recombnaton operator creates an offsprng by combnng, one at a tme, k routes of parent soluton P 1 wth a subset of customers, formed by nearest-neghbor routes {r 2 } n parent soluton P 2. The k routes ({r 1 }) are selected ether randomly, wth a probablty proportonal to the relatve number of customers or based on the average dstance separatng consecutve customers on the routes. A removal procedure s frst carred out to remove from r 1 some key customers beleved to be most sutably relocated wthn some alternate routes. More precsely, the stochastc customer removal procedure removes ether randomly some customers, customers rather dstant from ther successors, or customers wth watng tmes. Then, a modfed nserton heurstc of [Solomon, 1987] s appled to buld a feasble route, consderng the modfed partal route r 1 as the ntal soluton and unrouted customers n routes r 2 for nserton. The I1 standard nserton heurstc of [Solomon, 1987] s coupled to a random customer selecton procedure, to choose the next canddate customer to be routed. Once the constructon of the chld route s completed, and renserton s no longer possble, a new route constructon cycle s ntated. The overall process s repeated for the k routes selected from P 1. Fnally, f necessary, the chld nherts the remanng dmnshed routes of P 1. If unrouted customers stll reman, addtonal routes are bult usng a nearest-neghbor procedure of [Solomon, 1987]. The whole process s then terated once more to generate a second chld by nterchangng the roles of P 1 and P 2. Further detals of the operator may be found n [Berger and Barkaou, 2000]. In order to keep the number of routes of a chld soluton dentcal to ts parents, a post-processng procedure s appled. If the soluton has a larger number of routes than expected, the RSS_M (Secton 2.3.2) procedure s used repeatedly to reduce the number of routes. Conversely, for solutons havng a smaller number of routes, new feasble routes are constructed repeatedly by breakng the most populated route n two untl the targeted number of routes s obtaned. 2.3.2 Mutaton A sute of fve mutaton operators s proposed, namely LNSB_M, EE_M, IEE_M, RS_M, RSS_M and RC_M. The LNSB_M (Large Neghborhood Search -based) mutaton operator reles on the concepts of the Large Neghborhood Search (LNS) proposed by [Shaw, 1998]. The LNS conssts of explorng the search space by repeatedly removng related customers and rensertng them usng constrant-based tree search (constrant programmng). As n [Shaw, 1998], a set of
related customers s frst removed. In addton, LNSB_M removes customers volatng temporal constrants from ther routes. The proposed customer re-nserton method dffers from the procedure proposed by [Shaw, 1998] n two respects, namely, the nserton cost functon used, and the order n whch customers are consdered for nserton (varable orderng scheme) durng the branch-and-bound search process. Unvsted customers (f any) are then renserted usng the same customer re-nserton method whle relaxng temporal constrants. Inserton cost s defned by the sum of key contrbutons referrng respectvely to ncreased traveled dstance, and delayed servce tme, as specfed n Solomon s procedure I1 (c 11 +c 12 ), as well as to constrant volaton (equaton (5)). Concernng customer vst orderng, customers ({c}) are sorted (CustOrd) accordng to a composte rankng. The rankng s defned as an addtve combnaton of two separate rankngs, prevously acheved over best nserton costs (Rank Cost (c)) on the one hand, and number of feasble nserton postons (Rank Pos (c)) on the other hand: CustOrd Sort{ c} ( Rank ( c) Rank ( c) ) (8) Cost + The smaller the nserton cost (short total dstance, traveled tme) and the number of postons (opportuntes), the better (smaller) the rankng. The next customer to be vsted wthn the search process s selected accordng to the followng expresson D customer CustOrd[ INTEGER( L rand )] (9) n whch L = current number of customers to be nserted, rand = real number over the nterval [0,1] (unform random number generator), D = parameter controllng determnsm. If D=1 then selecton s purely random (default: D=15). Once a customer s selected, tree search s carred out over ts dfferent nserton postons as specfed n [Shaw, 1998]. However, the search tree expanson s ntated usng a non-constant dscrepancy factor, selected randomly over the set {1,2,3}. The EE_M (edge exchange) and RS_M (repar soluton) mutators focus on nter-route mprovement. EE_M uses the λ-nterchange mechansm of [Osman, 1993], performng rensertons of customer sets over two neghborng routes. Here, route neghborhood s determned by route centrod proxmty. Customer exchanges take place as soon as the soluton mproves,.e., we use the frst-accept strategy. Assumng the notaton (x,y) to descrbe the dfferent szes of customer sets (λ) ssued from the neghborng routes, the current operator explores values runnng over the range (x=1, y=0,1,2). The RS_M mutaton operator focuses on exchanges nvolvng one llegal customer. Each llegal customer n a route s exchanged wth an alternate legal one or two-customer sequence n order to generate a new set of customers wth ether volated or non-volated temporal constrants. The obectve s to further explore the soluton space (dversty) whle possbly mprovng qualty. The IEE_M mutaton operator s smlar to EE_M except for customer reorderng n whch customer permutatons are restrcted to the same route. Pos
The RSS_M (rensert shortest Solomon) mutaton operator elmnates the shortest route (smallest number of customers) of the soluton, decreasng by one the total number of routes. Customers from the shortest route are frst removed. Then, followng an teratve process, unvsted customers are re-nserted nto exstng routes usng the nserton procedure proposed by [Lu and Shen, 1999] n whch the regret cost functon (equaton (4)) has been extended to nclude a constrant volaton contrbuton (equaton (5)). The entre teratve process s repeated over I dfferent sets (e.g. I=20) of randomly generated parameter values. The RC_M (reorder customers) mutaton operator s an ntensfcaton procedure that tres to reduce the total dstance of feasble solutons by reorderng customers wthn a route. The procedure conssts of repeatedly reconstructng a new tour usng the sequental nserton procedure I1 of Solomon [Solomon, 1987] over I dfferent sets (e.g. I=2) of randomly generated parameter values. 2.4 Parallel Implementaton The parallel mplementaton conssts n usng the parallel vrtual machne PVM [Gest and al., 1994] software based on a master-slave message-passng paradgm that enables a collecton of heterogeneous computers to be used as a sngle coherent and flexble computatonal resource supportng concurrency. In the current verson, the master processng element supervses and controls the executon of the algorthm, handles the parent selecton process, populaton replacement and the emergence of a new feasble soluton, selects and synchronzes (actvaton and completon) atomc genetc operatons, fnalzes constructon of new generatons whle the slave processng elements supervsed by the master concurrently execute reproducton and mutaton operators. Current and new genetc operators have been revsted or confgured to reduce processor starvaton over each generaton. In a nutshell, the master component of the networked parallel mplementaton presents smlar characterstcs to the sequental verson of the algorthm except that genetc operators are concurrently executed as atomc operatons on multple processors. The algorthm has been mplemented n C++, usng a modfed verson of the GAlb genetc algorthm lbrary of [Wall, 1995], on a 19-computer cluster archtecture: Lnux platform envronment, 19 Athlon 1.2 GHz processors (a master and 18 slaves) wth 768M of RAM wth a 100M/sec communcaton bandwdth, and a 3 module swtch ncludng 41 ports. The overall run-tme for the algorthm has been lmted to two mnutes. 3. Computatonal Experment A computatonal experment has been conducted to compare the performance of the parallel verson of the proposed algorthm wth some of the best technques desgned recently for VRPTW. The algorthm has been tested wth 56 VRPTW benchmark problems of Solomon [Solomon, 1987]. Each problem nvolves 100 customers, randomly dstrbuted over a geographcal area. The travel tme separatng two customers corresponds to ther relatve Eucldean dstance. Customer locatons for a problem nstance are ether generated randomly usng a unform dstrbuton (problem data sets R1 and R2), clustered (problem data sets C1 and C2) or mxed, combnng randomly dstrbuted and clustered customers (problem data sets RC1 and RC2). The experment conssted n performng three smulaton runs for each problem nstance n a gven data set.
3.1 Confguraton Parameter settng and smulaton confguraton for the nvestgated algorthm are specfed as follows: Populatons: 2 (Pop 1 and Pop 2 ) Run-tme: 120 seconds Recombnaton and mutaton rates: 100% Recombnaton operator: IB_X Mutaton operators: LNSB_M(d), EE_M, IEE_M, RS_M, RSS_M and RC_M Wthn the LNSB_M(d) mutaton operator the number of customers consdered for elmnaton vares wthn the range [12, 17]. The dscrepancy factor d s randomly chosen over {1,2,3}. In ftness, evaluaton and nserton cost functons: α = 100, α 0 = 1000 β = 100 The probabltes and parameter values for the proposed genetc operators are defned as follows. For all data sets except C1 and C2: Populaton sze: 25 Pop 1 : Populaton overlap per generaton: n 1 =1 LNSB_M(d) (100%) EE_M (50%) + IEE_M(50%) Pop 2 : Populaton overlap per generaton n 2 =17. LNSB_M(d) (100%) RS_M + EE_M + IEE_M (33%), RS_M + EE_M (33%) and RS_M + IEE_M (33%) For data sets C1 and C2: Populaton sze: 25 Pop 1 : Populaton overlap per generaton: n 1 =15 IB_X(k=2) (100%) (for C2: k=1) RC_M(I=2) (100%) Pop 2 : Populaton overlap per generaton n 2 =3. IB_X(k=2) (100%) (for C2: k=1) RC_M(I=2) (100%) Because of lmted computatonal resources, the parameter values were determned by tryng ust a few ntutvely selected combnatons, and selectng the one that yelded the best average output. Ths s ustfed by the fact that the senstvty of the results wth respect to changes n the parameter values such as recombnaton and mutaton rates was found to be generally qute small. Populaton sze was chosen emprcally to balance ntensfcaton and dversfcaton.
Populaton overlaps (n 1 and n 2 ) were selected such that the sum (n 1 + n 2 ) matches the maxmum number of slave processors n order to generate a new populaton as quckly as possble whle mnmzng processor starvaton and, allevate the mpact of the ntrnsc seral part of the parallel program. Populaton overlaps have been determned accordng to the most promnent characterstc of a data set. As we amed at computng the mnmum number of routes frst, over a lmted tme, the dea conssts n allocatng a maxmum number of processors to Pop 2, as number of tours mnmzaton generally represents the most dffcult task to acheve. But, for cases where computng the mnmum number of routes does not present a maor challenge, more computatonal resources are allocated to reduce total traveled dstance. Consequently, more processng power were devoted to evolve Pop 1 for clustered data sets (n 1 =15, n 2 =3), emphaszng total traveled dstance mnmzaton. Alternatvely, computatonal resources were prmarly dedcated to Pop 2 evoluton for alternate problem nstances (n 1 =1, n 2 =17), stressng number of routes mnmzaton. Varatons over computed soluton qualty regardng populaton overlap parameters are neglgble as far as we explot basc data set characterstcs (clustered versus nonclustered dstrbuton). Communcaton and synchronzaton cost combned to processor starvaton, followng genetc operatons when constructng an entre new generaton, mpose nevtable performance lmts on parallel computaton. In order to counter ths adverse condton, processor starvaton was mnmzed by confgurng genetc operators to keep processng tme as unform as possble through genetc operatons sute durng chld computaton. For a matter of run-tme convenence, dfferent parameter settngs are proposed for C1 and C2, as opposed to other data sets. The parameters nstantaton was nspred from a prevous genetc algorthm by [Berger and Barkaou, 2000]. In fact, ths class of problem nstances does not present a real challenge for most VRPTW metaheurstcs as convergence generally occurs very quckly. 3.2 Results The results for the sx problem data sets are summarzed n Tables 1-2 for some of the best reported methods for VRPTW, namely, GTA [Gambardella et al., 1999], RT [Rochat and Tallard, 1995], SW [Shaw, 1998], TB [Tallard et al., 1997], CR [Chang and Russell, 1997], LS [Lu and Shen, 1999], HG [Homberger and Gehrng, 1999], CLM [Cordeau et al., 2001] and, Table 1: Best performance comparson among VRPTW algorthms. Problem RT LS CR TB GTA HG (ES1) HG (ES2) BB1-2 R1 Vehcles Dstance 12.25 1208.50 12.17 1249.57 12.17 1204.19 12.17 1209.35 12.00 1217.73 11.92 1228.06 12.00 1226.38 11.92 1221.1 R2 Vehcles Dstance 2.91 961.72 2.82 1016.58 2.73 986.32 2.82 980.27 2.73 967 2.73 969.95 2.73 1033.58 2.73 975.43 C1 Vehcles Dstance 828.38 830.06 828.38 828.38 828.38 828.38 828.38 828.48 C2 Vehcles Dstance 589.86 591.03 591.42 589.86 589.86 589.86 589.86 589.93 RC1 Vehcles Dstance 11.88 1377.39 11.88 1412.87 11.88 1397.44 11.50 1389.22 11.63 1382.42 11.63 1392.57 11.50 1406.58 11.50 1389.89 RC2 Vehcles Dstance 3.38 1119.59 3.25 1204.87 3.25 1229.54 3.38 1117.44 3.25 1129.19 3.25 1144.43 3.25 1175.98 3.25 1159.37 ALL Vehcles Dstance 415 57231 412 59317 411 58502 410 57522 407 57516 406 57876 406 58921 405 57952
BB1 [Berger, Barkaou and Bräysy, 2002] and BB2 for the sequental and parallel versons of the parallel hybrd genetc algorthm respectvely. The results are usually ranked accordng to a herarchcal obectve functon, where the number of vehcles s the prmary obectve and, for the same number of vehcles, the secondary obectve s total traveled dstance. The best computed results are shown n Table 1. Each entry refers to the best performance obtaned wth a specfc technque over a partcular data set. The frst column descrbes the varous data sets and correspondng measures of performance defned by the average number of routes (or vehcles), and total traveled dstance. The followng columns refer to partcular problem-solvng methods. The performance of our PHGA s depcted n the last column (BB1-2). Results ndcate that PHGA matches or outperforms the best-known heurstc routng procedures. The last row refers to the cumulatve number of routes and traveled dstance over all problem nstances. The total number of tours computed over all problem data sets outperform by one the best-computed result so far, reported by Homberger and Gehrng [Homberger and Gehrng, 1999]. In addton, PHGA s the only method that found the mnmum number of tours consstently for all problem data sets. PHGA also succeeded n mprovng sx of the best-known solutons. Accordngly, Table 2 provdes sx new best-known solutons and compares them wth the prevous best-known solutons. Detals of the new solutons can be made avalable by the authors. Table 2: New best computed solutons for some Solomon problem nstances Problem Best-Known Solutons New Best Solutons Vehcles Dstance Reference Vehcles Dstance R108 9 963.99 SW 9 960.88 R110 10 1125.04 CLM 10 1119 RC105 13 1637.15 HG 13 1629.44 RC106 11 1427.13 CLM 11 1424.73 R210 3 955.39 HG 3 954.12 R211 2 910.09 HG 2 906.19 Overall, computatonal results for BB2 have shown a sgnfcant mprovement over the sequental algorthm (BB1), mostly matchng or even mprovng soluton qualty. The parallel mplementaton of the PHGA algorthm has generally shown a speed-up of fve over the sequental verson, n computng solutons havng near smlar qualty, that s solutons presentng the same mnmum number of tours and comparable traveled dstance. Even though soluton qualty s expected to grow wth the number of processng elements, the parallel mplementaton nvolves mportant lmtatons such as processor watng tme (starvaton), communcaton cost and the nherent rato of the seral/parallel code attached to the proposed method. Intrnsc sequental contrbutons nclude synchronzaton constrants nvolved n the computaton and completon of a new generaton, ftness computaton and populaton replacement scheme. However, the current experment shows that run-tme assocated wth computaton of solutons presentng smlar or comparable qualty, steadly decreases wth the number of processors. Saturaton on performance s eventually expected to happen as the soluton space s further explored, but ths dd not occur wthn the context of the current experment (19-processor (maxmum number) cluster envronment).
4. Concluson A parallel mplementaton of a promsng hybrd genetc algorthm (PHGA) targeted to the vehcle routng problem wth tme wndows has been successfully developed. The mplementaton of the technque s based upon a master-slave message-passng paradgm (networked parallel computng) usng the PVM software wthn a 19-computer cluster envronment. The master processng element controls the executon of the algorthm, synchronzes atomc genetc operatons and handles the parent selecton process whle the slave processng elements concurrently execute reproducton and mutaton operators. Genetc operators have been desgned and revsted to further reduce processor starvaton over each generaton. Results from a computatonal experment show a sgnfcant speed-up of the method over ts sequental verson mostly matchng or even mprovng soluton qualty. Accordngly, a new best result has been computed for the R1 Solomon's data set, whle mprovng best-known solutons for some nstances of the R2 data set. Future work wll focus on comparatve average performance analyss whle extendng computatonal experments to larger VRPTW problem nstances to better characterze the lmtatons of the approach. Varants of the parallel algorthm, addressng desgn lmtatons wth respect to processor starvaton, communcaton cost and current seral/parallel code rato, mght be nvestgated as well. 5. References Berger, J., M. Salos and R. Begn (1998), A Hybrd Genetc Algorthm for the Vehcle Routng Problem wth Tme Wndows, Lecture Notes n Artfcal Intellgence 1418, AI 98, Advances n Artfcal Intellgence, Vancouver, Canada, 114 127. Berger J., M. Sass and M. Salos (1999), A Hybrd Genetc Algorthm for the Vehcle Routng Problem wth Tme Wndows and Itnerary Constrants, In Proceedngs of the Genetc and Evolutonary Computaton Conference, Orlando, USA, 44 51. Berger, J. and M. Barkaou (2000), An Improved Hybrd Genetc Algorthm for the Vehcle Routng Problem wth Tme Wndows, Internatonal ICSC Symposum on Computatonal Intellgence, part of the Internatonal ICSC Congress on Intellgent Systems and Applcatons (ISA'2000), Unversty of Wollongong, Wollongong, Australa. Berger J., M. Barkaou and O. Bräysy (2002), "A Parallel Hybrd Genetc Algorthm for the Vehcle Routng Problem wth Tme Wndows", to be publshed. Blanton, J.L. and R.L. Wanwrght (1993), Multple Vehcle Routng wth Tme and Capacty Constrants usng Genetc Algorthms, In Proceedngs of the 5th Internatonal Conference on Genetc Algorthms, S. Forrest (edtor), 452 459 Morgan Kaufmann, San Francsco. Bräysy, O., J. Berger and M. Barkaou (2000), A New Hybrd Evolutonary Algorthm for the Vehcle Routng Problem wth Tme Wndows, Presented n Route 2000 Workshop, Skodsborg, Denmark. Bräysy, O. and M. Gendreau (2001a), Vehcle Routng Problem wth Tme Wndows, Part I: Route Constructon and Local Search Algorthms, Workng Paper, SINTEF Appled Mathematcs, Department of Optmsaton, Norway.
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