A Hybrid Meta-heuristic Approach for Customer Service Level in the Vehicle Routing Problem

Transcription

1 A Hybrd Meta-heurstc Approach for Customer Servce Level n the Vehcle Routng Problem Pasquale Carotenuto, Grazano Galano, Stefano Gordan, Guseppe Stecca Isttuto d Tecnologe Industral e Automazone - Sezone d Roma Consglo azonale delle Rcerche Va del Fosso del Cavalere 00, 0033 Rome, Italy {p.carotenuto, g.galano, g.stecca}@ta.cnr.t Dpartmento d Ingegnera dell'impresa - Unverstà d Roma Tor Vergata Va del Poltecnco, 0033 Rome, Italy gordan@dsp.unroma2.t Abstract. The dstrbuton represents a crtcal factor for e-commerce. Global compettveness and ncreasng changng rate of the marketplace condtons force enterprses performng e-busness actvtes to pay more attenton to ratonalzaton of shppng processes and reducton of logstc costs. Wth modern nternet applcatons t s possble to offer dstrbuton servces wth dfferent qualty levels based on dfferent customer preferences; but one of the lacks of these tools s that they don t have a drect connecton between transportaton costs and customer choces. In ths paper we study the dstrbuton as a capactated vehcle routng problem wth tme wndow constrants. Tme wndows are used as an ndcator of customer servce level and they can be nserted n the model as hard constrants or evaluated n the objectve functon as penaltes. We propose a hybrd meta-heurstc algorthm based on the hybrdsaton of Genetc Algorthm and local search heurstc by ncorporatng an nserton heurstc able to fnd an effectve trade off between total transportaton cost and customer servce level. Keywords: vehcle routng problem, hybrd meta-heurstcs, customer servce level.

2 2 Carotenuto et Al. Introducton In recent years, because of changes n demand and offer of delvery servces, optmzaton n dstrbuton s begnnng more and more crtcal for enterprse survval. For nstance, because the Internet has provded a lower cost way of placng an order, warehouses are experencng more frequent, smaller quantty orders (Szgenda, 999; Lattmann, 2000). In these settngs optmzaton tools for dstrbuton costs reducton are very mportant as term to mprove the customer servce leve. At the hghest level, the overall performance of a dstrbuton network should be evaluated along two dmensons (S. Chopra, 2003): the frst are the customer needs that are met; the second s the cost of meetng customer needs. In ths paper we address the dstrbuton problem to customers when requests are done n terms of quanttes and delvery tmes. Delvery tme requests are tme ntervals n whch customers prefer to receve delveres, and are related to customer servce level. The objectve s to fnd the best trade off between transportaton costs and customer servce level. The modellng approach s based on the capactated vehcle routng problem wth tme wndow constrants; tme wndows are used as an ndcator of customer servce level and they can be volated f ths volaton allows a satsfyng reducton n transportaton costs. The solvng methodology used s a hybrd meta-heurstc approach based on the hybrdsaton of a genetc algorthm and local search heurstcs by ncorporatng an nserton heurstc able to fnd an effectve trade off between total transportaton cost and customer servce level. The procedure conssts of dfferent steps through whch delvery routes are generated wth dfferent tme wndow constrants volatons. These solutons are then evaluated n terms of cost and customer servce satsfacton and the best one s chosen. The paper s organzed as follows. In secton 2 we recall the more nterestng publcatons about vehcle routng problem and how, n the scentfc lterature, tme wndow constrants are consdered; n secton 3 we explan the modellng approach to fnd the soluton wth the best trade off between costs and customer servce level; n secton 4 we descrbe the solvng methodology based on a hybrd meta-heurstc approach; n secton 5 we made some prelmnary test results and dscuss about future works. 2 Vehcle routng problems wth tme wndow constrants The Capactated Vehcle Routng Problem wth Tme Wndows (VRPTW) s a well known P -hard n strong sense problem, and t s a generalzaton of the Capactated Vehcle Routng Problem (CVRP). The CVRP conssts of fndng a collecton of smple routes of mnmum cost n a connected dgraph startng from and

3 A Hybrd Meta-heurstc Method for Customer Servce Level n the VRP 3 endng to a common depot, such that each customer s vsted exactly by a route, and vehcle capacty constrants are satsfed. In most papers the objectve s to fnd the mnmum number of tours K * ; a secondary objectve s often ether to mnmze the total dstance travelled or the duraton of the routes. In the VRPTW, servce at each customer must start wthn an assocated tme wndow and the vehcle must reman at the customer locaton durng servce. Soft tme wndows can be volated at a cost, whle hard tme wndows do not allow for a vehcle to arrve at a customer after the latest tme to begn servce. In the latter case, f t arrves before the customer s ready to begn servce, t wats (Toth & Vgo, 2000). The VRPTW s one of the most studed varatons of the VRP and recent surveys can be found n (Bräysy and Gendreau, 2002a, Bräysy and Gendreau, 2002b, Laporte et Al., 2000, Tan et Al., 2000); useful applcaton of the VRPTW are bank delveres, postal delveres, ndustral refuse collecton, Just In Tme dstrbuton, and recently studed (Az et Al., 2004) are pershable goods dstrbuton problems. When tme wndows can be thought as no hard constrants, ther volaton s penalzed wth penalty functons proportonal to the entty of volaton. In the lterature, Vehcle Routng Problem wth Soft Tme Wndows s referred wth the notaton VRPSTW and s used to closely model stuatons found n practce, when t s hard to fnd feasble solutons, or when t s necessary to fnd a good trade-off between fleet sze and servce qualty to customers. Properly settng penalty values, VRPSTW can also be used to solve VRPTW. The problem s addressed wth dfferent technques usng both heurstc (Ioannou et Al., 2003) and meta-heurstc approaches (Tallard et Al., 997). 3 Problem formulaton and best-trade-off model In ths work, we address a dstrbuton problem n whch, there are a set of customers, wth customers requestng for servce of a good of quantty q n a tme wndow [e, l ]. The startng tme t of the servce outsde the tme wndow [e, l ] leads to an extra cost s, or penalty for the dstrbuton. The objectve s to fnd a collecton of routes and a set of start tmes t of customers servces, for, such that all customers are served and the sum of travelled dstance and extra costs s s mnmzed. The startng tme t of the servce not always equals the arrval tme a. In fact, f a e and no tme wndow volaton s allowed, a (watng) tme wt = e a has to be defned at the node before startng the servce at tme t = e. The objectve s the mnmzaton of the sum of three terms: K 0 j= 0, j k= 0 c j x jk + K k= w k y k + α s () where a dgraph G = (V, A) wth V = + nodes s consdered (node 0 s the depot center node) together wth a tme wndow [e, l ] for each customer, and: K s an upper bound on the vehcles number to be used;

4 4 Carotenuto et Al. x jk {0,} s equal to f arc (, j) A s traversed by vehcle k, 0 otherwse;, j = 0,,, ; c j s the cost to travel the arc (, j) A;, j = 0,,, ; y k, {0,} s equal to f the vehcle k s actvated, 0 otherwse, k = 0,, K ; w k s the cost for vehcle actvaton, k = 0,, K ; w k s hgher than travellng cost, ndeed the number of used vehcles K s a prmary objectve; for the same K, total travelled dstance s a secondary objectve; s s the tme wndow volaton measure: s = max{t l, 0, e - t } assocated to each one of customers, =, 2,,. Moreover, n order to fnd a feasble soluton, the followng constrants have to be satsfed,.e.: a) every route must start and end at a central depot; b) every customer must be vsted by exactly one vehcle; c) there s a capacty constrants for each vehcle; e) there s a maxmum travel tme or dstance for each vehcle; d) s S, =, 2,,, where S s the maxmum volaton of tme wndow allowed. The frst 2 terms n formula () represent the dstrbuton cost whle the thrd term s a measure of the customer servce level. The customer servce level s the maxmum when there are no tme wndow volatons; n ths case we have the thrd term of formula () equal to 0. The maxmum volaton of the tme wndows correspond to a value of the thrd term of formula () equal to α S. Ths s a bound for the mnmum level of customer satsfacton. We can then defne the level of customer satsfacton LS and as LS α = α ( S s ) S, LS [0, ]. In correspondence of LS = 0 and LS = we have the worst and the best case performance wth respect to tme wndows. As a consequence we wll have a greater travellng cost to obtan a reduced value of LS. We can defne the maxmum value of travellng cost as Z max, correspondng to fnd a soluton of vehcle routng problem wth hard tme wndows and such that LS = ; f we relax some or all the tme wndows we could fnd solutons wth a reduced value of travellng cost (specally because there s possblty to use a reduced number of vehcles). Let Z(Ch): K 0 j= 0, j k = 0 K Z( Ch) = c x + w y be the travellng cost of soluton Ch; j jk k = we can then defne a sort of gan ndex : =, [0, ). k k Z(Ch) Z max

5 A Hybrd Meta-heurstc Method for Customer Servce Level n the VRP 5 For = 0 we obtan LS = and for = we obtan LS = 0. The objectve s to fnd the best trade-off between the values of and LS can be defned as to fnd the combnaton of values maxmzng the sum of plus LS, weghted wth α S and Z max, max(ls α S + Z max ). It s also mportant to fnd a set of no domnated combnatons of LS and, and to draw the Pareto effcent boundary by solvng the mult objectve functon max( LSα S ; Z max ). In the followng secton we descrbe a hybrd genetc algorthm that at same tme generate a near optmal soluton for the CVRP wth hard tme wndow constrants and a contaner of solutons wth reduced value of travellng cost at mnmum tme wndow volaton. 4 The hybrd meta-heurstc approach To solve the problem descrbed n prevous sectons, a meta-heurstc for vehcle routng problem wth tme wndows can be used. In ths secton we propose a hybrd genetc algorthm that at same tme generates a near optmal soluton for the CVRP wth hard tme wndow constrants and a contaner of solutons wth reduced value of travellng cost at mnmum tme wndow volaton (Fgure ). The genetc algorthm (GA) s an adaptve heurstc search method based on populaton genetcs. The basc concepts are developed by (Holland, 975) and (Goldberg, 989). Generally a smple GA s composed of three operators whch are appled to the current populaton of chromosomes generaton after generaton. The representaton of the soluton space conssts of encodng of a soluton as a chromosome, defnng an ndvdual member of a populaton. In our problem we use a double strng of length as the representaton of chromosome. A gene of a GA chromosome s formed by two alleles: the former allele represents the tme wndow volaton of a customer (s ), and the latter represents a customer node number (). The sequence of the genes s the order of vstng these customers. Snce all routes must start and end at the central depot, the number 0 can be omtted. To decode ths chromosome nto route confguraton, we nsert the genes nto new routes sequentally.

6 6 Carotenuto et Al. Genetc Algorthm: Defnton: populaton P of ndvduals Ch. Set the dmenson of populaton P to P_sze. Intal Populaton: Fll the set P wth P_sze ntal ndvduals obtaned by the randomzed verson of Push Forward Inserton Heurstcs (PFIH r ). Whle (generaton_number < MAX_GEERATIO ) { Selecton: select an ndvdual from P usng a based roulette wheel; ftness functon s /(Z(S) gap). Crossover: The sequence based crossover (SBX) s appled. A populaton control mechansm s appled n order to mantan a prefxed rate r of unfeasble ndvduals on populaton P. Mutaton Phase Feasble Hybrdsaton: wth a probablty pm pck an ndvdual Ch and do: a. SBX(Ch, Ch). Durng the offsprng generaton consder the best feasble or-opt local search moves. b. Route Savng phase. Apply a or-opt local search heurstc to Ch n order to remove tours wth customers C T C mn. Apply the move only f the feasblty of Ch s preserved. Unfeasble Hybrdsaton (TW manpulaton): wth a probablty pm 2 pck a ndvdual Ch and do: a. Route savng phase. Apply a or-opt local search heurstc to Ch not consderng tme wndow feasblty. Ths soluton s nserted n P and n the Best Unfeasble Soluton Contaner (BUSC) } return (the best feasble ndvdual and the contaner BUSC) Fgure : Hybrd genetc algorthm wth tme wndow manpulaton (HTW) The algorthm, startng from an ntal populaton generated by a randomzed verson of the Push Forward nserton Heurstcs (PFIH, Solomon, 987), progressvely nsert n the populaton unfeasble ndvduals n term of tme wndow constrants.

7 A Hybrd Meta-heurstc Method for Customer Servce Level n the VRP 7 Selecton operaton s used to select the potental chromosome from the current populaton n accordance wth ther ftness values. We have mplemented the reproductve operator creatng a based roulette wheel where each chromosome n the current populaton has a slot szed proportonally to ts ftness functon value and proportonally to ts reproducton probablty. The ftness functon f(ch) to evaluate the ndvduals s obtaned from the objectve functon of the optmzaton problem: f(s) = / (Z(Ch) gap, where gap s parameter of the algorthm and t s used to equally penalze unfeasble solutons. The mplemented crossover s the Sequence Based Crossover (SBX) descrbed n (Potvn and Bengo, 996). In the proposed algorthm unfeasble offsprng can be nserted n the populaton f the rate r u of unfeasble offsprng n the populaton s less than a prefxed rate r. An eltsm strategy was adopted; hence, the best soluton at a gven generaton s preserved n the next generaton. The eltsm guarantees the presence of at least one feasble soluton even f r =. Three dfferent hybrdsed mutaton operators have been mplemented. Frst two mutaton operators are desgned to work wth feasble soluton; the thrd mutaton operator s appled n order to manpulate tme wndow volatons. The hybrdsaton conssts on a local search nserton mprovement based on the or-opt move (Or, 976). In partcular, n the thrd mutaton operator, local search perform the nserton wth the best cost mprovement at mnmum tme wndow volaton. All other constrans, (n partcular the depot tme wndow constraned) are always respected. Best found unfeasble solutons are nserted n a Best Unfeasble Soluton Contaner BUSC and ranked by the travelng cost Z. 5 Prelmnary test results and conclusons The algorthm was appled on Solomon s VRPTW benchmark problems and we report the solutons of the best run of our algorthm on ths test set, usng the followng settngs: The populaton sze s set to 00. The number of generatons s set to 00. Generaton replacement wth eltsm s appled. The mutaton rates, pm and pm 2, are both equal to 0.. Crossover rate s equal to 0.5. The prelmnary results of our algorthm were compared to the best known solutons dentfed by heurstcs. We solved some problems n Solomon s test set, namely, problems R0, R02, R03, R04, and R05, (havng narrower tme wndows). The computatonal results are shown n Table.

8 8 Carotenuto et Al. The two numbers n each cell are the number of routes and total dstance, respectvely. The best known solutons are shown n the frst column; the best feasble and TW unfeasble solutons produced by our algorthm are shown n the second and thrd columns, respectvely. Problem Best F-HTW U-HTW R R R R R Table. On problem R0, our algorthm found the TW unfeasble solutons shown n Table 2. In partcular, the best TW unfeasble soluton has three less routes than the best known soluton n respect to a modest decrease of the customer servce level (the tme wndows volaton s equal to 2). Soluton umber of Total TW Routes Dstance Volaton Table 2. Beng our research at early stage, for the future we wll produce more testng results on standard test nstances.

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