A Multi-restart Deterministic Annealing Metaheuristic for the Fleet Size and Mix Vehicle Routing Problem with Time Windows

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1 A Mult-restart Determnstc Annealng Metaheurstc for the Fleet Sze and M Vehcle Routng Problem wth Tme Wndows Oll Bräysy Agora Innoroad Laboratory, P.O.Bo 35, FI Unversty of Jyväsylä, Fnland Wout Dullaert Insttute of Transport and Martme Management Antwerp, Unversty of Antwerp, Kezerstraat 64, B-2000 Antwerp, Belgum Ger Hasle SINTEF ICT, P.O. Bo 124 Blndern, NO-0314 Oslo Norway Davd Mester Insttute of Evoluton, Mathematcal and Populaton Genetcs Laboratory, Unversty of Hafa, Hafa, Israel Mchel Gendreau Centre de recherche sur les transports, Unversté de Montréal, Case postale 6128, Succursale Centre-vlle, Montréal, Canada H3C 3J7 Abstract We present a new determnstc annealng metaheurstc for the fleet sze and m vehcle routng problem wth tme wndows. The objectve s to servce a set of customers at mnmal cost wthn ther tme wndows by a heterogeneous capactated vehcle fleet. The suggested metaheurstc comprses three phases. In the frst phase, hgh qualty ntal solutons are generated by means of a savngs-based heurstc combnng dversfcaton strateges wth learnng mechansms. An attempt s then made to reduce the number of routes n the ntal soluton wth a new local search procedure n phase two. The soluton s further mproved n phase three by a set of four local search operators that are embedded n a determnstc annealng framewor. Computatonal eperments show that the suggested method outperforms the prevously publshed results. 1 Introducton The Vehcle Routng Problem (VRP) s a ey to effcent transportaton management and supply-chan coordnaton. Although often assumed n VRP theory, a vehcle fleet s rarely homogeneous n real lfe. A fleet manager typcally controls vehcles that dffer n ther equpment, carryng capacty, speed, and cost structure. There are many types of motvaton for eepng a dversfed fleet. Some customers may requre vehcles wth costly equpment. Large vehcles may be more cost effectve, but the road networ and the nature of a customer s premses often mpose physcal restrctons on vehcle sze and weght. Some types of vehcle may not be allowed n urban areas due to envronmental concerns. Vehcles 1

2 of dfferent carryng capacty gve fleblty to allocate capacty accordng to varyng demand n a more cost effectve way. The Fleet Sze and M Vehcle Routng Problem (FSMVRP) s a VRP where the homogeneous fleet assumpton of the tradtonal VRP has been lfted. In ths paper, we study the FSMVRP wth tme wndows (FSMVRPTW), where each customer has a tme wndow wthn whch servce must start. It s a natural etenson of the recently much studed VRPTW (see Bräysy and Gendreau, 2005a and 2005b). Tme wndows are crtcal n many applcatons, as more and more mportance s gven to customer servce and tmelness. We focus on the FSMVRPTW varant that was frst studed by Lu and Shen (1999). Here, the fleet s heterogeneous because each vehcle mght have ts own fed cost and ts own capacty. Drvng speed and customer servce tmes are dentcal for all vehcles. The routng cost for a soluton s defned as the sum of en route tme over all vehcles used. The man contrbuton of ths paper s to present a new effcent mult-start determnstc annealng (MSDA) heurstc for the problem. In Secton 1.1, a mathematcal formulaton of the FSMVRPTW s gven and Secton 1.2 presents a lterature revew. Secton 2 contans a descrpton of the algorthmc approach. Secton 3 descrbes our epermental study and results. Conclusons are gven n Secton Problem Formulaton We base our formulaton of the FSMVRPTW on the well-nown vehcle flow arcbased formulaton for the VRPTW. Let G= ( N,A ) be a graph, where { 0} { 1,, n} { n+ 1} N= K. 0 and 1 customers. n + represent the depot. = { 1,, n} C K s the set of A N N represents the travel possbltes between nodes. V = { 1, K, K} s the set of alternatve vehcles. There are vehcle specfc acquston / deprecaton costs capactes q. t are the vehcle ndependent travel tme between nodes. For each customer there s a tme wndow a, b, and s s the vehcle ndependent customer servce tme. For each j (, j, ) wth, j A, V, j s a bnary decson varable that epresses whether e and vehcle travels drectly from customer to customer j. For each (, ), A, V, y determnes the eact start tme of servce at ths customer f t s served by that vehcle. 2

3 ( + n+ ) mnmze e y y (1) V j C j subject to: V C N N 0 j j n, + 1 = 1, C (2) d q, V ( 3) j = 1, V (4) = 0, h C, V (5) h hj = 1, V (6) ( y + s + t y ) 0, (, j) A, j j j ( ) ( ) y a 0, (, j) A, V (8 a) j b y 0, (, j) A, V (8 b) j j V (7) {0,1}, (, j) A, V (9) Constrants (2) state that all customers must be vsted by a vehcle. (3) are the vehcle capacty constrants. Constrants (4) enforce that each vehcle must leave the depot eactly once, and (6) that each vehcle must arrve at the depot eactly once. (5) ensure that f a vehcle arrves at a customer, t also departs from that customer. (7) guarantee that the arrval tmes at two consecutve customers allow for servce and travel tme. (8a) and (8b) are the tme wndow constrants. The non-lnear travel tme and tme wndow constrants can be easly lnearzed by use of the bg M trc. The FSMVRPTW s strongly NP-hard as t s a generalzaton of the CVRP. In ths paper, we focus on heurstcs Lterature revew The frst FSMVRPTW wor was reported by Lu and Shen (1999). They developed a savngs-based constructon heurstc and an mprovement heurstc nspred by the wor of Golden et al. (1984) for the FSMVRP. The approach was tested on a new benchmar developed by the authors based on a well-nown benchmar for the VRPTW (Solomon, 1987). Three sets of nstances were created, each representng a dfferent cost structure for vehcle types. Dullaert et al. (2002) proposed a sequental nserton heurstc based on Solomon s I1 heurstc for the VRPTW and tested ther approach on the Lu and Shen benchmar. More recently, the FSMVRPTW has been studed by Prvé et al. (2006), Dell Amco et al. (2006), Calvete et al. (2006), Tavaol-Moghaddam et al. (2006), and Dondo and Cerdá (2006). 3

4 2 Soluton approach Our proposed soluton approach conssts of three phases. In the Phase 1 hgh qualty ntal solutons are generated by means of a savngs-based heurstc combnng dversfcaton strateges wth learnng mechansms. In Phase 2 the focus s on reducng the number of vehcles. A set of four local search operators are embedded n a determnstc annealng framewor to gude further mprovement n Phase 3. All phases are repeated a user-defned number of tmes,.e., several ntal solutons are created and mproved separately Phase 1: Constructng ntal solutons Intal solutons are created by a probablstc modfcaton of the savngs heurstc (Clare and Wrght, 1964). The search s started by servng each customer by way of a separate route. After the frst constructon and mprovement phase, some of the arcs n the fnal soluton of the prevous mprovement phase are ept nstead of startng wth sngle customer routes n Phase 1. The route constructon heurstc features three man modfcatons compared to the orgnal savngs heurstc. Frst, the heurstc s mplemented from an nserton pont of vew to combne the strengths of savngs and nserton-based heurstcs. When mergng two routes and, all postons between consecutve customers wthn route R are consdered. R1 R2 2 Second, the algorthm accepts not only the best savngs, but also the second and thrd best savngs (f postve) wth gven probabltes, to allow for dversfcaton. Thrd, each route s ntalzed wth the smallest possble vehcle type. Savngs are calculated by consderng both fed vehcle costs and the total schedule tme. Vehcle szes are updated whenever needed, and are always set to the smallest vehcle avalable capable of servng the customers on the route. Merges are attempted untl no further mprovement can be found Phase 2: Route Elmnaton Route elmnaton s based on a depleton procedure called ELIM, whch agan uses smple nsertons. All routes of the current soluton are consdered for depleton, n random order. For a gven route, ELIM removes all customers and tres to nsert them n the remanng routes. The crtcalty measure ς defned below determnes the sequence n whch customers are renserted. The bars mean normalzaton to [ 0,1 ]. ς = + ( b a ) d c o 4

5 For a gven customer, all postons n the remanng routes are tred. The best feasble nserton accordng to the total cost objectve s selected. The algorthm starts by calculatng total savngs obtaned by elmnatng the currently selected route. It then eeps trac of the total ncrease n total costs resultng from rensertons. If the total ncrease eceeds the total savng or f any of the customers cannot be nserted n another route, the algorthm reverts and proceeds to the net route n sequence. If all the removed customers have been nserted n other routes at a lower cost, the route s elmnated. In the Route Elmnaton Phase, ELIM s run untl quescence Phase 3: Local Search Improvement The soluton from Phase 2 s further mproved by local search and metaheurstcs. Four local search operators are utlzed: ELIM descrbed above, a route splttng operator called SPLIT, along wth varants of the ICROSS and IOPT operators suggested n Bräysy (2003). The SPLIT neghborhood conssts of all solutons that result from splttng a sngle route n the current soluton nto two parts at any pont. We employ t n a greedy, frst-accept fashon, smply by loopng through all routes and all customers n them, splttng the selected route nto two parts at the poston of the current customer. The move s made f the splt reduces cost. ICROSS has been etended wth the adjustment of vehcle types. The four local search operators are employed as follows: ICROSS, IOPT, ELIM, and SPLIT (n that order) are repeated for a gven number of teratons n mprove. Before each teraton, the routes are randomly ordered. The search s embedded n a Threshold Acceptng (TA) metaheurstc (Duec and Scheurer, 1990). Our procedure starts wth threshold t = 0 (no worsenng) and s repeated wth that value untl a local mnmum s reached. If no mprovement has been found for a gven number of teratons n, t s set to a mamum value and the search s restarted from the current best soluton. The mamum value s r t ma where r s a random number n the range [0,1] and t ma s a user-defned parameter. At each nonmprovng teraton, t s reduced by untl zero s reached. The search s repeated wth zero threshold untl no more mprovements can be found. To escape the local optmum, t s set to a new mamum threshold value t r t ma. To lmt search effort for ICROSS and IOPT, we focus only on moves that nvolve customers that are close. When creatng subsequent ntal solutons, the algorthm uses nformaton gathered durng the search n a frequency-based long-term memory of arcs that appear n hgh qualty solutons. Ths learnng mechansm s based on dentfyng the common arcs n the sequence of ncumbent solutons. 5

6 When the teraton lmt n mprove for the current ntal soluton s reached, a startng pont s created for the savngs heurstc to create a new ntal soluton based on arc frequency nformaton. If an arc s not present n a user-defned share of the best solutons, t s removed. Remanng arcs are randomly deleted or ept accordng to a gven user-defned probablty. The resultng arc set forms a new startng pont for the savngs algorthm. The creaton of ntal solutons and subsequent mprovement s repeated for a gven number of tmes, n specfed by the user. 3 Computatonal eperments The proposed algorthm was mplemented n Java (JDK 5.0) and tested on an AMD Athlon (512 MB RAM) computer. Eperments were performed on the nstances proposed by Lu and Shen (1999), who ntroduced several vehcle types wth dfferent capactes and costs to the 56 Solomon 100 cases. In addton, three dfferent vehcle cost structures A, B and C were suggested, resultng n 168 problem nstances. After senstvty analyss the followng parameter values were used: = 1000 ; n mprove nnt = 2 ; t = 0. ma 04 ; t = ; n = 40. It appeared to be mportant to create several ntal nt, solutons, but the gan from addtonal solutons becomes small after 5-6 solutons. Three runs were used for each nstance. Results show that the MSDA procedure s consstently better than the competton for all data sets. It found 157 best-nown solutons, where 6 are tes and 151 are new solutons. Consderng computer speed and the number of runs, computng tmes are slghtly hgher than the competton. A quc varant wth a sngle run and 200 teratons s sgnfcantly faster and gves better average qualty than the competton for all data sets. 4 Summary and Conclusons The Fleet Sze and M Vehcle Routng Problem wth Tme Wndows (FSMVRPTW) s an ndustrally mportant problem that has not been studed much n the lterature. We have developed a novel 3-phase algorthm for solvng the FSMVRPTW. Wth a moderate ncrease of computatonal effort relatve to the competton, our procedure sgnfcantly outperforms the competton wth a small ncrease of effort and has found 151 new best nown solutons. References Bräysy, O A reactve varable neghborhood search for the vehcle routng problem wth tme wndows. INFORMS J. Comput

7 Bräysy O., M. Gendreau. 2005a. Vehcle routng problem wth tme wndows, Part I: route constructon and local search algorthms. Transportaton Sc Bräysy O., M. Gendreau. 2005b. Vehcle routng problem wth tme wndows, Part II: metaheurstcs. Transportaton Sc Calvete, H.I, C. Galé, M.-J. Olveros, B. Sánchez-Valverde A goal programmng approach to vehcle routng problems wth soft tme wndows. To appear Eur. J. of Oper. Res. Clare G., J.W. Wrght Schedulng of vehcles from a central depot to a number of delvery ponts. Oper. Res Dell Amco M., M. Monac, C. Pagan, D. Vgo Heurstc approaches for the Fleet Sze and M Vehcle Routng Problem wth Tme Wndows. Worng paper, Unversty of Modena, Italy. Dondo R, J. Cerdá A cluster-based optmzaton approach for the mult-depot heterogeneous fleet vehcle routng problem wth tme wndows. To appear Eur. J. Oper. Res. Duec G, T. Scheurer Threshold acceptng: a general purpose optmzaton algorthm. J. Comput. Phys Dullaert W, G.K. Janssens, K. Sörensen, B. Vernmmen New heurstcs for the fleet sze and m vehcle routng problem wth tme wndows. J. Oper. Res. Soc Golden B, A. Assad, L. Levy, F. Gheysens The fleet sze and m vehcle routng problem. Comput. Oper. Res Lu F-H, S.-Y. Shen The fleet sze and m vehcle routng problem wth tme wndows. J. Oper. Res. Soc Prvé J, J. Renaud, F.F. Boctor, G. Laporte Solvng a vehcle routng problem arsng n soft drn dstrbuton. J. Oper. Res. Soc Solomon MM Algorthms for the vehcle routng and schedulng problems wth tme wndow constrants. Oper. Res Tavaol-Moghaddam R, N. Safae, Y. Gholpour A hybrd smulated annealng for capactated vehcle routng problems wth the ndependent route length. Appl. Math. Comput