A Tabu Search Heuristic for the Vehicle Routing Problem with Time Windows and Split Deliveries

Transcription

1 A Tabu Search Heurstc for the Vehcle Routng Problem wth Tme Wndows and Splt Delveres Sn C. Ho and Dag Haugland Department of Informatcs Unversty of Bergen N-5020 Bergen, Norway 4th September 2002 Abstract In ths paper, we consder the Vehcle Routng Problem wth Tme Wndows and Splt Delveres (VRPTWSD) and present a soluton method based on Tabu Search and four dfferent neghborhood structures. Ths problem s an extenson of the Vehcle Routng Problem, where the start of servce at each customer must be wthn a tme wndow, and a customer may be servced by more than one vehcle. The heurstc gves promsng results when appled to realstc nstances of the problem. It s also adapted to the problem where splttng s not an opton, and expermental results have shown that the heurstc has mproved 5 of the 56 best publshed solutons to the Solomon benchmarks, whle matchng or mprovng the best solutons n 10 problems. Keywords: Vehcle Routng, Tme Wndows, Splt Delveres, Tabu Search 1 Introducton Vehcle Routng Problems (VRP) are concerned wth the dstrbuton of goods, people or nformaton between depots and customers. Vehcle routng problems arse n many real-lfe applcatons wthn transportaton and logstcs, such as school bus routng, postal delveres, transportaton of handcapped persons and food dstrbuton. Ths paper consders the Vehcle Routng Problem wth Tme Wndows and Splt Delveres (VRPTWSD). Gven a fleet of homogeneous vehcles statoned at a central depot and a set of customers requrng ther demands to be fulflled, the problem conssts of fndng vehcle routes startng and endng at the depot such that every customer s vsted. The routes must also meet the tme wndows defned by the customers, whch specfy when the start of servce can occur. Sometmes t s not realstc that a customer s demand must be delvered by a sngle vehcle. By allowng delveres to be splt, a customer may be servced by more than one vehcle. The obectve of the VRPTWSD conssts of mnmzng the total dstance traveled (equvalent to cost mnmzaton). Snce the VRPTWSD s NP-hard (see Lenstra and Rnnooy Kan 1981, Dror and Trudeau 1990), nstances of realstc sze are dffcult to solve to optmalty. As a way out, metaheurstcs are often used to fnd good solutons to varous routng problems n reasonable tme. Emal: sn@.ub.no Emal: dag.haugland@.ub.no 1

2 The past years, qute good results have been acheved for the Vehcle Routng Problem wth Tme Wndows (VRPTW), n both the classes of exact methods and metaheurstcs. Surveys can be found n Toth and Vgo (2002, Chapter 7) and Golden and Assad (1988). Bräysy and Gendreau (2001) gve an excellent overvew over metaheurstcs for the VRPTW. All of these consdered tradtonally vehcle routng for whch each customer s vsted exactly once. The Vehcle Routng Problem wth Splt Delveres (VRPSD) s ntroduced by Dror and Trudeau n They showed how splt delveres could result n savngs, both n the total dstance traveled and the number of vehcles utlzed. The VRPSD s a relaxaton of the classcal Vehcle Routng Problem, but t remans NP-hard (see Dror and Trudeau 1990). Dror et al. (1994) descrbed an nteger programmng formulaton of the problem and developed several classes of vald constrants. They also developed an exact constrant relaxaton branch and bound algorthm for the VRPSD. Frzzell and Gffn (1992, 1995) studed the problem wth grd network dstances, whereas they also consdered tme wndows constrants n ther second publcaton. Ther proposed heurstcs are especally talored for the problems and ths knd of network structure. Mullaserl et al. (1997) presented a heurstc for the Splt-Delvery Capactated Rural Postman Problem wth Tme Wndows on arcs. The heurstc s smlar to the one proposed by Dror and Trudeau (1989, 1990), and s appled to a real-lfe problem of managng the trucks for dstrbutng feed n a cattle ranch n Arzona. Belenguer et al. (2000) studed the polyhedron of the VRPSD and together wth a new class of vald nequaltes, a lower bound for the problem s developed. In ths paper, we develop a Tabu Search heurstc for the VRPTWSD where our strategy s dfferent from Dror and Trudeau s (1990). Ther work s a two-stage algorthm, where the frst stage constructs a VRP soluton usng node nterchanges, and the second stage mproves the VRP soluton by ntroducng splts and elmnatng splts. Our algorthm does not separate ths from one another, t looks at all ths n parallel. We have a pool of solutons that are defned by dfferent move operators, the heurstc does not specfcally choose to do node nterchanges or splt the delveres. That s left to be decded by the heurstc accordng to the pool of solutons. The best soluton n the current pool s always chosen. It could be a splt delvery, splt-elmnaton or ust a smple node nterchange. The rest of ths paper s organzed as follows: In Secton 2 we gve the formulaton of the problem, some defntons and notaton, and a theoretcal result on whch our soluton method reles. The Tabu Search heurstc based on the four dfferent neghborhood structures s descrbed n Secton 3, whereas expermental results are gven n Secton 4. Fnally, a concluson s drawn n Secton 5. 2 Problem Formulaton and Defntons In ths secton we defne the problem under study, and the notaton used throughout the paper. Customers: The problem s gven by a set of customers C = {1, 2,..., n}, resdng at n dfferent locatons. Lettng 0 denote the locaton of the depot, N = C {0} becomes the set of all locatons consdered n the problem. Every par of locatons (, ), where, N and, s assocated wth a cost of travelng d and a travel tme t. Every customer C has a demand w > 0. Vehcles: A set, V, of vehcles wth dentcal capactes, m, s gven. The vehcle set s later referred to as the fleet. Tme wndows: Each customer has a tme wndow,.e. an nterval [a, b ] R, where a and b are the earlest and latest tme to start to servce customer. A vehcle may arrve at customer locaton before a, but cannot start servcng untl the tme wndow opens at a. A vehcle cannot arrve at customer locaton after the tme wndow closes at b. The depot also has a tme wndow [a 0, b 0 ], where a 0 represents the earlest tme when the vehcles can leave the depot, and b 0 represents the latest tme when the vehcles must return to the depot. 2

3 Splt delveres: The demand of a customer may be fulflled by more than one vehcle. Ths can occur for nstance when the demand requred by a customer s larger than the capacty of a vehcle, or when t s cheaper to servce a customer more than once. The model contans three sets of decson varables x, f and s. For each arc (, ), where, N, (wth the excepton of = = 0 meanng a vehcle s drvng an empty route), and for each vehcle k, we defne x k as { 1 f vehcle k travels drectly from customer to customer x k = 0 otherwse The decson varable f k s defned for each customer and each vehcle k and denotes the fracton of demand of customer delvered by vehcle k. The last decson varable s k denotes the tme vehcle k starts to servce customer. We assume s 0k = a 0, for all k. The VRPTWSD can be stated mathematcally 1 as: mn z(x) = k V d x k (1) N N subect to x 0k = 1 k V (2) N x hk x hk = 0 h C, k V (3) N N f k = 1 C (4) k V w f k m k V (5) C x k f k C, k V (6) N s k + t K (1 x k ) s k C, N, k V (7) a s k b N, k V (8) s k + t 0 K 0 (1 x 0k ) b 0 C, k V (9) f k 0 C, k V (10) s 0k = a 0 k V (11) x k = 0 C, k V (12) x k {0, 1}, N, k V (13) The obectve (1) s to mnmze total dstance traveled. The next two sets of equatons (2) and (3) state that each vehcle leaves the depot, after arrvng at a customer the vehcle leaves agan, and t wll fnally return to the depot. The equaton set (4) ensures that every customer receves ther full demand, and (5) states that no vehcle s assgned more demand than ts capacty. Constrant set (6) ensures that a customer can only be servced by a vehcle whch vsts that customer. The nequaltes (7) state that a vehcle k cannot arrve at customer before s k + t f t s travelng drectly from customer to customer. To ths end, the constant K s defned suffcently large, e.g. K = b + t a. Constrant set (8) makes sure that every customer s servced wthn ther tme wndow. Fnally, constrant set (9) forces every vehcle to arrve at the depot before the depot s tme wndow closes. Note that summng up constrant set (6) over all vehcles yelds n combnaton wth (4) the 1 The formulaton s based on Frzzell and Gffn (1995) and Larsen (1999). 3

4 followng constrants: x k 1 C (14) k V N mplyng that every customer receves at least one delvery. The followng equaton set statng that all vehcles have to return to the depot s omtted from the model x 0k = 1 k V (15) N because t s mpled by equaton sets (2) and (3). 2.1 Property of VRPTWSD solutons For a gven feasble soluton (x, f, s) B N N v R N v + R N v + to VRPTWSD, each k V defnes a unque ordered set R k = (0, c k 1, ck 2,...,ck g, 0) such that x 0c k 1 k = x c k 1 ck 2 k =... = x c k g 0k = 1. We shall refer to R k as the route of vehcle k (nduced by x). For notatonal convenence, we shall n the remander of the paper assume that c k =, although ths generally s not the case. Hence refers to the poston of the customer n R k, and 1 and + 1 are respectvely the predecessor and the successor of n ths partcular route. The followng property s proved by Dror and Trudeau (1989, 1990) n the case of VRPSD, and we wll prove that t also apples to VRPTWSD. Proposton 1 If VRPTWSD s feasble and {t }, N satsfy the trangle nequalty, then the problem has an optmal soluton where no two of the nduced routes have more than one common customer. R k R k Fgure 1: The two-route two-splt example: two-splt (left) and one-splt (rght). Squares represent the depot (whch s duplcated at each end) and crcles represent customers n the route. Proof Assume that (ˆx, ˆf, ŝ) s an optmal soluton to VRPTWSD, and let ˆR 1,..., ˆR v be the routes nduced by ˆx. Also assume that, ˆR k ˆ where, C and 1 k < l v,.e. the delveres at both and are splt between vehcles k and l (see Fgure 1, left). We shall construct a feasble soluton (x, f, s) wth nduced routes R 1,...,R v such that z(x) z(ˆx), R k, and such that R k ˆR k ˆ (e.g. Fgure 1, rght). If such a soluton can be found, the proposton s proven. 4

5 We let ˆq k = ˆf k w denote the quantty delvered to customer by ˆR k. Wthout loss of generalty, t can be assumed that ˆq k = mn{ˆq k, ˆq l, ˆq k, ˆq l }. Defne q equal to ˆq and x equal to ˆx n all components wth the followng exceptons: q k = ˆq k + ˆq k, q l = ˆq l ˆq k, q l = ˆq l + ˆq k, q k = 0, x 1,+1,k = 1, x 1,,k = 0 and x,+1,k = 0. Lettng f k = q k /w for all (, k) C V, t follows from Dror and Trudeau (1989, 1990) that (x, f) satsfes (2)-(6), (10) and (12)-(13), and that z(x) z(ˆx). Snce ŝ satsfes (8) and (11), puttng s = ŝ ensures that s satsfes these constrants. It remans to show that ths choce of s also satsfes (7) and (9). But the only affected constrants are those ndexed by ( 1,, k), (, +1, k) and ( 1, +1, k). The frst two are trvally satsfed snce x 1,,k = x,+1,k = 0. If + 1 C, we have s 1,k + t 1, s k and s k + t,+1 s +1,k, snce (ŝ, ˆx) satsfes (7) and ˆx 1,,k = ˆx,+1,k = 1. Snce t 1,+1 t 1, + t,+1, we thus get s 1,k + t 1,+1 s 1,k + t 1, + t,+1 s k + t,+1 s +1,k. Snce x 1,+1,k = 1, we have s 1,k + t 1,+1 K 1,+1 (1 x 1,+1,k ) s +1,k, and (7) s satsfed. If + 1 s the depot, a smlar argument s used n order to prove that (9) s satsfed. 3 Tabu Search Tabu Search s a memory-based search strategy, orgnally proposed by Glover (see Glover and Laguna 1997), to gude the local search method to contnue ts search beyond a local optmum. One way of achevng ths s to keep track of recent moves or solutons made n the past. Tabu lst records recently made moves or vsted solutons. Whenever the algorthm attempts to make a move lsted n the tabu lst, the move s banned. By ths, the algorthm forces other solutons to be explored. However, ths feature s not strct, t can be overrdden when some aspraton crtera are satsfed. A popular aspraton crteron s that the target functon value be the best ever seen. If ths s the case, t s obvous that ths soluton has never been encountered before. Ths s the reason for acceptng the soluton, although t s forbdden by the tabu lst. A soluton to the problem expressed by (1)-(13) s gven by a set of routes, an assgnment of customers to routes, and a servce start tme and delvery fracton for each customer on each route. We let σ = {R 1,...,R v } denote the set of routes. Let S be the set of route sets for whch feasble servce start tmes and delvery fractons exst. Each soluton σ S has an assocated set of neghbors, N(σ) S, called the neghborhood of σ. Each soluton σ N(σ) can be reached drectly from σ by a move. A move s a transton from σ to σ by means of a move operator descrbed n Secton 3.2. Our soluton method s a three-step process. Frst we compute an ntal feasble soluton by smple consderaton of the combnaton of travel tme and watng tme. Next we try to mprove t by Tabu Search, and fnally a post-optmzaton phase s appled to t. 3.1 Intal soluton The ntal soluton s computed by appendng the nearest unrouted neghbor ĵ to customer (the latest routed customer) wth respect to mnmum sum of travel tme and watng tme from to ĵ. Ths process repeats untl all customers are routed, as descrbed n Algorthm 1, where θ s the tme the vehcle starts to servce customer, and ĵ arg mn{t + max{a θ t, 0} C} (16) s the crteron for choosng ĵ. Notce that f neghbor ĵ s pcked and ts demand exceeds the vehcle s capacty, ĵ s stll appended and the excess demand s left to some other vehcle(s) to handle. In ths way an ntal soluton wth splt delveres s made. Algorthm 1 does not determne whether the splt s proftable or not. Note that Algorthm 1 may end up wth more routes than the actual number of vehcles avalable. 5

6 Algorthm 1 Intal Soluton k = 1 repeat Begn wth an empty route k startng from the depot, R k = (0, 0). Set ĵ = 0 and θ 0 = a 0. repeat Set = ĵ. k V f ĵ k < 1) neghbor ĵ to customer w.r.t. (16) and Fnd a nearest unrouted (.e. feasblty n terms of tme constrants. Insert ĵ n the current route k, and set θĵ = θ + t + max{a ĵ ĵ θ t, 0}. ĵ Compute the spare capacty u k. untl u k = 0 or no more nsertons are vald k = k + 1 untl all customers are routed If the succeedng soluton method s unable to reduce the number of routes to v or less, t reports that no feasble soluton s found. We wll now revew some features used n the Tabu Search algorthm. 3.2 Neghborhoods The neghborhood of our tabu search algorthm s based on some common move operators; relocate, exchange and 2-opt* (these are modfed a lttle for handlng splt delveres) and one move operator called relocate splt. Followng s a short descrpton of each one of them. 1. Relocate operator (see Fgure 2): For locatons R k C and α α, place after α n α, where α = 1,...,β, and β s the number of necessary vehcles servcng wth β 1. The new vehcle routes are R k = (0,..., 1, + 1,...,0) and R l α = (0,..., α,, α + 1,...,0). In cases where R k α C, a splt delvery may be elmnated, and the poston of n α remans unchanged. R k R k Fgure 2: A relocate operaton 2. Relocate splt operator (see Fgure 3): For customers R k C and C: Remove customer from R k, and let ˆR k be (0,..., 1, + 1,...,0). Whle s servced by, let also be servced by ˆR k. The new vehcle routes are R l = (but wth a hgher quantty delvered to and less delvered to ) and R k = ˆR k {}. 6

7 R k R k Fgure 3: A relocate splt operaton 3. Exchange operator (see Fgure 4): The exchange s between a par of routes R k and, and s an exchange of customers R k C and C. Customer s nserted n \{}, but not necessarly n the poston held by. Smlarly, s nserted n some poston n R k \ {}. In cases where R k C,, a splt delvery may be elmnated. R k R k Fgure 4: An exchange operaton 4. 2-opt* operator (see Fgure 5): For locatons R k and, let the new vehcle routes be R k = (0,...,, + 1, + 2,...,0) and R l = (0,...,, + 1, + 2,...,0). In cases where R k C and concdes wth + 1, a splt delvery between R k and may be elmnated. The neghborhoods defned by the above operators are denoted N 1, N 2, N 3 and N 4, respectvely. Some other phases are also used n the heurstc: Route savng phase: In ths phase we only consder vehcle routes wth three or less customers. The heurstc tres to nsert each of the route s customer nto the other routes usng the relocate operator. If the routes can successfully be empted, and stll obtan dstance savngs, then these routes are removed from the soluton, otherwse the heurstc keeps the orgnal soluton. US: Ths s an mprovement phase that s a part of the GENIUS algorthm developed by Gendreau et al. (1998). The US algorthm removes each customer n turn (startng wth the frst customer of the route), and renserts t whle performng a local reoptmzaton of the route. Both the renserton and the reoptmzaton part have two dfferent types of unstrngng/strngng processes, but we only consder type 1 unstrngng/strngng. Type 1 nsertons (.e. strngng) are equvalent 7

8 R k R k Fgure 5: A 2-opt* operaton to selectng the best of several moves, each consstng of a smple nserton followed by only one 3-opt exchange. Type 1 unstrngng refers to removng a customer from a route, and rearrange the rest of the customers of the same route n a dfferent way (usng somethng smlar to 3-opt). Detals can be found n Gendreau et al. (1992, 1998). 3.3 Tabu Search for VRPTWSD At each teraton the four dfferent move operators are appled to the current soluton. The best feasble neghbor soluton found s chosen. The soluton s ether tabu or non-tabu. It s tabu f the move that lead to the soluton s tabu, but the tabu status may be overrdden by an aspraton crteron (whch s the best total cost seen so far). The nverse move s then set tabu for the next p teratons, and wll not be classfed as a vald move untl the tabu status s expred or s overrdden by the aspraton crteron. For any feasble route set σ, let z(σ) be the travel cost. The move operators relocate, relocate splt, exchange and 2-opt* are assocated wth 1,...,4 respectvely. We let T (σ) be the set of solutons to whch the move defned by operator s tabu, where = 1,...,4. We also defne A (σ) T (σ) to be the set of solutons wth tabu moves overrdden by the aspraton crteron. What knd of moves are consdered tabu? Assume we have routes R k and where R k and, by applyng the dfferent move operators we obtan the followng tabu moves: 1. Relocate operator: We place R k C after α n α. Then the tabu lst records the tabu status of customer n R k, after beng moved to α. That s, we set TABU(, R k ) = p, meanng movng α to R k s tabu for the next p teratons. We let T 1 (σ) consst of all soluton n N 1 (σ) where R k and TABU(, R k ) > 0 for some and k. 2. Relocate splt operator: We assume that R k C, and after applyng the move operator, customer R k C and / R k. The tabu lst records the tabu status of customer n R k, after beng moved to. That s, we set TABU(, R k ) = p, meanng movng to R k s tabu for the next p teratons. We let T 2 (σ) be defned n a way analogous to T 1 (σ). 3. Exchange operator: We exchange customer wth customer. The tabu lst records the tabu status of customer n R k, after beng moved to, and t also records the tabu status of customer n, after beng moved to R k. That s, we set TABU(, R k ) = p and TABU(, ) = p, meanng movng to R k and movng to are tabu for the next p teratons. We let T 3 (σ) be defned n a way analogous to T 1 (σ). 8

9 4. 2-opt* operator: We exchange s descendants wth s descendants. The tabu lst records the tabu status of and + 1, and of and + 1. That s, we set TABU(, + 1) = p and TABU(, + 1) = p, meanng the arcs (, + 1) and (, + 1) are forbdden to remove for the next p teratons. We let T 4 (σ) consst of all solutons n N 4 (σ) where h s the successor of n some route, and TABU(, h) > 0. Note that T 4 (σ) dffers n notaton from the frst three tabu lsts. The second entry s a customer and no longer a route. In other words, the tabu lst for the frst three operators deals wth the vertces (.e. customers), whle the second tabu lst deals wth the arcs. The route savng phase s performed once n every q teratons. The heurstc ends when y consecutve teratons are performed wthout any mprovement to the best known soluton. Fnally, a post-optmzaton phase s appled to the best soluton found n the search. Thus, the heurstc can be summarzed as n Algorthm 2. Algorthm 2 Tabu Search heurstc Obtan an ntal soluton by Algorthm 1, followed by US and a route savng phase, and denotes ths soluton σ. Set σ = σ. Intate tabu lsts. repeat for = 1,...,4 do σ = arg mn{z(γ) γ (N (σ) T (σ)) A (σ)} =argmn{z(σ ) = 1,...,4} σ = σ Set the nverse move of σ tabu for the next p teratons, and decrement prevous entres n the tabu lsts by one. f z(σ) < z(σ ) then σ = σ Perform route savng phase once n q teratons. untl y consecutve teratons are performed wthout any mprovement to the best known soluton Apply post-optmzaton phase (US) on the best soluton. 3.4 VRPTW verson of the heurstc We have also developed a VRPTW verson of the heurstc proposed n the prevous subsecton. That s, we make changes to Algorthms 1 and 2 necessary to produce a feasble soluton to VRPTW, where no splt delveres are allowed. Concernng the ntal soluton, Algorthm 1 shows the VRPTWSD verson where a customer s appended to the route wthout determnng f the vehcle s capable to handle the whole demand, only whether t s full-loaded or not. In the VRPTW verson, the vehcle s spare capacty must be at least as large as the customer s demand n order to let the customer be servced by ths vehcle. There are also a few mnor adaptons concernng the neghborhood structures. In the case of the relocate operator, customer can only be moved to one vehcle, contrary to Secton 3.2 where customer s allowed to be transferred to more vehcles. Snce we do not allow any splt delveres to be made, the relocate splt operator s never used (because ts precondton s a splt delvery between two routes). The other two move operators reman unchanged. The fnal adapton s made n the route savng phase, whch s due to the relocate operator mentoned above. These are all the changes needed to be ncorporated n the Tabu Search heurstc n order to have a VRPTW verson of the heurstc. 9

10 4 Expermental results 4.1 Problem sets Standard test cases for the VRPTW are used for expermentaton. The Solomon test problems consst of 100 customers wth Eucldean dstance. In these problems, the travel tmes are equal to the correspondng Eucldean dstances. We consder the travel tmes as a composton of the correspondng Eucldean dstances and servce tme. There are sx sets of problems where the geographcal data are ether randomly generated accordng to a unform dstrbuton (problem sets R1 and R2), clustered n groups (problem sets C1 and C2), or semclustered (problem sets RC1 and RC2). By a semclustered problem, we mean one that contans a mx of randomly generated data and clusters. Further, problem sets R1, C1 and RC1 have a short schedulng horzon, and combned wth a constraned vehcle capacty, t allows only a few (3-8) customers to be servced by the same vehcle. The sets R2, C2 and RC2 have a long schedulng horzon, and larger vehcle capacty yeldng routes wth more (.e. 10+) customers. A thorough descrpton of the test sets s gven n Solomon (1987). The computatonal experments also nclude modfed Solomon test problems whch are descrbed n Subsecton Comparson wth best publshed VRPTW results In ths subsecton we compare the VRPTW verson of our heurstc wth other heurstcs found n the lterature. Several benchmarks for ths problem exst, and several authors compete to publsh the best solutons. A sample of them are (abbrevatons have been used n Tables 2 and 3): BV=Bent and Van Hentenryck (2001), CLM=Cordeau et al. (2001), DFSKP=De Backer et al. (2000), GTA=Gambardella et al. (1999), HG=Homberger and Gehrng (1999), RGP=Rousseau et al. (2002), RT=Rochat and Tallard (1995), S=Shaw (1998) and TBGGP=Tallard et al. (1997). The results n ths and the followng subsectons were obtaned wth the parameter settngs presented n Table 1. Note that the parameter settngs were chosen n an arbtrary fashon and we have not tuned ther values. The heurstc descrbed n the prevous secton s coded n C++, compled usng Sun C++ compler and run on a Sun Ultra 10 (UltraSPARC-II 440 MHz) workstaton. Tables 2 and 3 show the comparson between the best publshed results and the results from our heurstc. The column labeled Best shows the total dstance traveled and the number of vehcles utlzed. Ref. gves references to where the best publshed solutons can be found. The columns labeled VRPTW and VRPTWSD report the results obtaned by the VRPTW and VRPTWSD versons of our heurstc. Although VRPTWSD s an nterestng problem n ts own rght, t can also be consdered as a relaxaton of VRPTW. Hence, our VRPTWSD heurstc produces solutons to a relaxaton of VRPTW, and n cases where the splt delvery opton s not taken advantage of, the relaxed soluton s actually a feasble VRPTW-soluton. Ths turned out to be true for all experments reported n Tables 2 and 3. The boldfaced results ndcate matchngs of the best results, whle talzed results represent mprovements over the best known solutons. Table 1: Parameter settngs for the Heurstc Tabu tenure: p = 15 Stoppng crteron: y = 100 Performng route savng phase crteron: q = 5 Parameters used n Gendreau et al. (1998): p 1 = 4, p 2 = 5 10

11 Table 2: Comparson between the best publshed results of Solomon s test cases (class 1) and ours Problem Best Ref. VRPTW VRPTWSD R RT R RT R S R S R RT R RT R S R BV R HG R BV R RGP R GTA C RT C RT C RT C RT C RT C RT C RT C RT C RT RC TBGGP RC TBGGP RC S RC S RC BV RC BV RC S RC TBGGP

12 Table 3: Comparson between the best publshed results of Solomon s test cases (class 2) and ours Problem Best Ref. VRPTW VRPTWSD R HG R RGP R BV R BV R RGP R RT R BV R GTA R BV R DFSKP R BV C RT C RT C RT C RT C RT C RT C RT C RT RC CLM RC GTA RC HG RC GTA RC BV RC BV RC BV RC BV

13 Tables 2 and 3 show that our heurstc has mproved (n terms of total dstance traveled) 5 (8.93%) of the 56 best publshed solutons to the Solomon benchmarks, whle matchng or mprovng the best solutons n 10 problems (17.86%). However, our solutons utlze more vehcles than the best publshed n almost two thrds of the nstances. 4.3 Dscusson The demand by vehcle capacty ratos, or demand ratos for short, for the orgnal Solomon benchmarks are the followng: R1: C1: RC1: R2: C2: RC2: The ratos are very small and together wth a short schedulng horzon t s most unlkely that any splt delveres wll be made. Ths s confrmed by the experments reported n Tables 2 and 3. From Table 2 we can see that most of the VRPTWSD solutons concde wth the VRPTW solutons. There are two nstances where the VRPTW verson of the heurstc does gve better solutons than VRPTWSD, and there are eght nstances where VRPTWSD s better than VRPTW. Why s t that VRPTW s better than VRPTWSD for two of the nstances? Ths can be explaned by dfferent ntal solutons for the two dfferent versons of the problem. One ntal soluton contans splt delveres, whle the other does not. Other factors are the relocate splt operator, splt delveres and spare capactes, all these combned wll gve dfferent search traectores. We have also conducted experments that used the VRPTW verson of the heurstc to generate the ntal soluton, and the VRPTWSD verson dong the search. All results concde wth the VRPTW results. The only exceptons are two nstances that yelded better results than the VRPTW verson. Table 4: Mean values of results R1 R2 C1 C2 RC1 RC2 VRPTW VRPTWSD Table 4 shows the mean values of the number of vehcles utlzed and total dstance traveled. As reported n ths table, VRPTWSD s dong slghtly better than VRPTW, even wthout takng advantage of ts splttng opton. All splt delveres that occurred n the ntal solutons are elmnated durng the search. The reported fndngs n the prevous subsecton have shown that our heurstc does gve better solutons for the VRPTWSD than for VRPTW, when the demand ratos are low. As demonstrated n the next subsecton, ths effect s amplfed as the demand rato grows. 4.4 Experments on Solomon s problems wth augmented demands We have slghtly modfed Solomon s test nstances n order to study how the value of the splttng opton vares wth the demand rato. All demands are transformed affnely to the nterval [lm, um], where l < u are gven numbers n the nterval (0, 1]. That s, for all C we defne the new demand w u l = lm+m w w (w w) where w = mn{w : C} and w = max{w : C}. Results from experments wth Solomon s problems for varous values of l and u are reported n Table 5. 13

14 Table 5 shows the mean values of the number of vehcles utlzed and total dstance traveled of the VRPTW and VRPTWSD verson of the heurstc. To conduct ths experment we let the ntal solutons be the same (.e. we do not allow splt delveres n the ntal soluton) for both of the versons of the heurstc. We choose to do ths because of the larger demands and the larger neghborhood for the VRPTWSD (as a result from hgh demand ratos). Snce the demand ratos are hgher, also the varatons between VRPTW and VRPTWSD are bgger when choosng the most proftable move at each teraton. As reported n Table 5, the overall results favor VRPTWSD, both n terms of total dstance traveled and number of vehcles utlzed. Table 5: Comparson of results for the modfed problem nstances from Solomon l, u R1 R2 C1 0.01,0.50 VRPTW 18.67/ / / VRPTWSD 18.25/ / / ,1.00 VRPTW 36.92/ / / VRPTWSD 35.17/ / / ,1.00 VRPTW 99.08/ / / VRPTWSD 67.83/ / / ,1.00 VRPTW / / / VRPTWSD 80.50/ / / l, u C2 RC1 RC2 0.01,0.50 VRPTW 12.25/ / / VRPTWSD 12.00/ / / ,1.00 VRPTW 27.00/ / / VRPTWSD 23.00/ / / ,1.00 VRPTW 74.00/ / / VRPTWSD 61.00/ / / ,1.00 VRPTW / / / VRPTWSD 77.13/ / / Only for problem class R2 and RC2 for values of 0.01 and 0.5 of l and u, the VRPTW verson s better than the VRPTWSD verson. The bggest savngs, both n terms of dstance and the number of vehcles are yelded n the last two categores. When the demand ratos are at least 0.5, there are not many combnatons to arrange a sequence of the customers of the VRPTW verson. But for the VRPTWSD verson of the heurstc, there are consderable more choces. Ths s due to the numerous splts generated by the route savng phase, relocate and relocate splt operators. The number of splts decreased as the demand by vehcle capacty rato decreased. Total customer vsts are n = 100 for all problem nstances of the VRPTW snce each customer can be servced only once. For VRPTWSD, ths fgure s the sum of the degrees of all delveres (an unsplt delvery has degree 1, a delvery splt between β vehcles has degree β). The total number of vsts tends to grow as the demand rato ncreases. Table 6 shows average customer vsts for the sx problem classes (wth 1=R1, 2=R2, 3=C1, 4=C2, 5=RC1 and 6=RC2) and varous values of l and u. Ths growth s due to the reducton of vehcles obtaned by applyng the relocate and relocate splt operators. When the respectve demands occupy much of the vehcles, applyng the relocate operator wll result n numerous splts for the customers. As demand ratos grow, the heurstc tends to operate wth the relocate splt operator more often than when the ratos are small. Ths operator wll also generate splts n certan cases. Table 7 shows the mean values of the dstrbuton of moves of varous values of l and u for 14

15 Table 6: Average customer vsts l, u R1 R2 C1 C2 RC1 RC2 0.01, , , , the VRPTWSD. For the frst category we see that the relocate operator clearly domnates the dstrbuton. When the demand ratos are relatvely small, t s proftable to move customers from one route to another usng relocate operator. The relocate splt operator takes over as the domnator for the other values of l and u (except for problem classes C1 and C2 of 0.2 and 1). The reason for ths can be found n the constraned spare capactes and the relatvely small demand ratos. When applyng the relocate operator, t wll result n numerous splts and wll not be proftable. A better alternatve s the relocate splt operator, although n certan cases a splt wll be made. Table 7: Average dstrbuton of moves l, u operator R1 R2 C1 C2 RC1 RC2 0.01,0.50 relocate relocate splt exchange opt* ,1.00 relocate relocate splt exchange opt* ,1.00 relocate relocate splt exchange opt* ,1.00 relocate relocate splt exchange opt* Computaton tme Table 8 shows the mean values of the CPU tme the workstaton consumed n order to perform the experments descrbed n Subsecton 4.4. The VRPTW verson s faster when the demand ratos are hgh. Conversely for the VRPTWSD verson, t s computatonally more expensve to run on problems wth hgh demand ratos. The reason for ths les n the possblty of movng customers between the routes wthout volatng constrants. The set of feasble solutons to VRPTW becomes very restrcted as the rato grows, resultng n a rapd search. 15

16 Table 8: Computaton tme (mn:sec) for the modfed problem nstances from Solomon l, u R1 R2 C1 C2 RC1 RC2 0.01,0.50 VRPTW 8:50 11:12 6:29 6:25 3:16 5:07 VRPTWSD 11:31 27:08 8:25 13:48 8:04 12: ,1.00 VRPTW 5:48 6:20 4:48 5:07 4:15 5:16 VRPTWSD 19:18 32:42 12:29 16:51 13:50 25: ,1.00 VRPTW 4:34 5:00 4:46 4:45 4:54 5:03 VRPTWSD 27:08 23:10 15:48 25:09 23:02 22: ,1.00 VRPTW 4:25 4:52 4:52 4:45 4:50 4:55 VRPTWSD 18:40 18:46 20:58 27:54 18:28 16:30 Table 9 shows the average number of teratons needed to run a test problem of varous classes. As depcted n Tables 8 and 9, the teraton number decreased as the demand rato ncreased. However, the computaton tme has not decreased as a result of ths, and t rather ncreased due to more tme needed to perform one sngle teraton. Ths can be explaned by more legal moves due to the splttng opton. Table 9: Iteratons needed for the modfed problem nstances from Solomon l, u R1 R2 C1 C2 RC1 RC2 0.01,0.50 VRPTW VRPTWSD ,1.00 VRPTW VRPTWSD ,1.00 VRPTW VRPTWSD ,1.00 VRPTW VRPTWSD Conclusons We have proposed a Tabu Search heurstc for the VRPTWSD. The heurstc mnmzes the total dstance traveled usng Tabu Search wth four dfferent neghborhood structures. Results on both the VRPTW and VRPTWSD versons of the heurstc are reported. Expermental results have shown that the heurstc has mproved (n terms of total dstance) 5 (8.93%) of the 56 best publshed solutons to the Solomon benchmarks, whle matchng or mprovng the best solutons n 10 problems (17.86%). We have also conducted experments on how the two versons of the heurstc perform to dfferent demand by vehcle capacty ratos. The results have shown that the VRPTWSD verson s capable of gettng lower dstance and number of requred vehcles than the VRPTW verson. Ths s especally remarkable for test problems where the demand by vehcle capacty ratos are at least 0.5, but these test nstances requre consderable more tme to run. These problems also resulted n more customer vsts. 16

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