Pickup and Delivery with Time Windows : Algorithms and Test Case Generation

Transcription

1 Pckup and Delvery wth Tme Wndows : Algorthms and Test ase Generaton oong hun LAU Zhe LIANG School of omputng, Natonal Unversty of Sngapore lauhc@comp.nus.edu.sg langzhe@comp.nus.edu.sg Abstract In the pckup and delvery problem wth tme wndows (PDPTW), vehcles have to transport loads from orgns to destnatons respectng capacty and tme constrants. In ths paper, we present a two-phase method to solve the PDPTW. In the frst phase, we apply a novel constructon heurs tcs to generate an ntal soluton. In the second phase, a tabu search method s proposed to mprove the soluton. Another contrbuton of ths paper s a strategy to generate good problem nstances and benchmarkng solutons for PDPTW, based on Solomon s benchmark test cases for VRPTW. Expermental results show that our approach yelds very good solutons when compared wth the benchmarkng solutons. 1. Introducton The Pckup and Delvery Problem wth Tme Wndows (PDPTW) models the stuaton n whch a fleet of vehcles must servce a collecton of transportaton requests. Each request specfes a pckup and delvery locaton. Vehcles must be routed to servce all requests, satsfyng tme wndows and vehcle capacty constrants whle optmzng a certan objectve functon such as total dstance traveled. PDPTW can be used to model many core problems arsng n logstcs and publc transt. Fndng good solutons to these problems s mportant because t enables planners to utlze the exstng fleet n the most cost-effectve fashon to meet customer demands. PDPTW s a generalzaton of well-known Vehcle Routng Problem wth Tme Wndow (VRPTW). PDPTW s an NPhard problem, snce VRP s a well-known NP-hard problem ([S1995]). Whle VRPTW s well-studed (for a comprehensve survey, see [D1995]), there s relatvely less lterature on PDPTW. Moreover, no one has developed comprehensve benchmark PDPTW nstances that facltate expermentaton of new approaches. In ths paper, we propose a new approach for PDPTW, extendng and mprovng the results of [NB000]. We also propose a method to generate good problem nstances and solutons for PDPTW, based on Solomon s benchmark data sets for VRPTW. Ths turns out to be an nterestng exercse, because good solutons for VRPTW do not mply good solutons for PDPTW, manly due to the fact that whle the vehcle load cumulatvely ncreases along each route for VRPTW, t s not the case for PDPTW as pckups and delveres occur n juxtaposton.. Problem Formulaton In our model, we assume there s an unlmted number of vehcles and all vehcles have the same capacty. Let N be the set of transportaton requests. For each request N, a load of sze q s to be transported from an orgn N to a destnaton N (postve q for pckup and negatve q for delvery). Each pckup or delvery s also referred to as a job. Defne N U N N and N U N N as the sets of all orgns and destnatons respectvely. For smplcty, assume N and N to be dsjont. Let V N U N and n = V. Let M and m denote the set and number of vehcles. Each vehcle has a capacty Q, startng from and endng wth the depot O wth no cargo. For all, j V U O, let d j denote the travel dstance and t j the travel tme. Let [ e, l ] denote the tme wndow,.e. tme nterval n whch servce at locaton must take place. Note that the servce duratons at the orgns and destnatons can be easly ncorporated n the travel tmes and hence wll not be consdered explctly n ths paper. Defnton 1 A pckup and delvery route R for vehcle k k s a drected route through a subset V k V such that: 1. R k starts and ends n O.. Both or nether N and N belongs to V for all k N. 3. If both N and N belong to V, N k s vsted before N. 4. Vehcle k vsts each locaton n V exactly once. k 5. The vehcle load at any one tme never exceeds Q. 6. The arrval tme A and departure tme D of any locaton satsfy D [ e, l ], where D = max{ A, e } (.e. f A < e, the vehcle has to wat at locaton ). Defnton A pckup and delvery plan s a set of routes R { R k M} such that k 1. R k s a pckup and delvery route for vehcle k, for all k M.

2 . { V k k M } s a partton of V. Defne f (R) as the cost of plan R correspondng to a certan objectve functon f. PDPTW s defned as: mn{ f ( R) R s a pckup and delvery plan. } There are a wde varety of objectve functons for PDPTW. In ths paper, we consder the followng: 1. Mnmze the number of vehcles, whch s almost always the most domnant part of the cost.. Mnmze travel dstance. That s, the sum of lengths of all the routes n the plan. To model PDPTW as an nteger lnear program (ILP), two types of bnary varables are ntroduced. Let z k ( V, k M ) become true ff request s assgned to vehcle k, x jk ( V, j V, k M ) s true ff vehcle k s travelng from node to node j. Let y j denote an ntermedate varable that stores the total load of the vehcle vstng job j. PDPTW s to mnmze f (x) subject to the followng constrants: N, z = 1 (1) k M V, x = 1 () k M V j V k j V k M, x = 1 (3) k M, x = 1 (4) ( h V )( k M), x = 0 (5) Ok Ojk V hk x j V hjk j V )( k M ), x V jk Q y j ( (6) (, j V O)( k M ), x = 1 y q = y (7) jk y = 0 (8) O V, y 0 (9) (, j V)( k M ), xjk = 1 D tj Dj (10) N p = N, q = N, D p D q j, (11) D = 0 (1) O onstrant (1) ensures that each request s assgned to exactly one vehcle. onstrant () ensures that each job s vsted exactly once. onstrants (3) and (4) ensure that each vehcle departs from and arrves at the depot. (5) ensures that f a vehcle arrves at a node then t must also depart from that node. (6)-(9) together form the capacty constrants. The tme wndows and precedence constrants are ensured by (10)-(1). To model the duo-objectve of mnmzng (a) the total number of vehcles and (b) total travel dstance as a lnear functon, we multply a coeffcent for each objectve and jk then add them together. Snce the number of vehcles s more mportant than the total dstance of a plan, the cost of each vehcle (route) s penalzed wth a coeffcent P, whch s set to be greater than the maxmum possble total travel dstance. ence, the objectve functon of the problem s: P m d x j mnmze k M V j V Ths formulaton of the problem has O ( n m) constrants and O ( n m) varables. For large-scale problems, an ILP solver almost always experences combnatoral exploson. 3. Lterature Revew Most prevous work focused on the sngle vehcle dal-arde problem wth tme wndows (1-PDPTW). For the objectve to mnmze the total customer nconvenence, Psarafs ([P1980], [P1983]) developed a dynamc programmng algorthm wth a O(n 3 n ) tme complexty, whch could only solve small-szed problems wth 10 or fewer requests. In Sexton and Bodn ([SB1985a], [SB1985b]), the problem was de-coupled nto a coordnatng routng master problem formulated as an nteger program, and a schedulng subproblem for a fxed route, whch was formulated as lnear program. By usng a heurstc verson of Benders' decomposton, the routng master problem and the schedulng subproblem were solved ndvdually. Real problems wth szes from 7 to 0 could be solved n an average of 18 seconds of UNIVA 1100/81A PU tme. Sexton and ho [S1986] used a smlar approach to mnmze a lnear combnaton of total vehcle operatng tme and total customer penalty due to mssng any of the tme wndows. For mnmzng the schedule duraton, Van der Bruggen et al. [VLB1993] developed a two-phase heurstc algorthm based on arcexchange procedures and an alternatve algorthm based on smulated annealng. Ther approaches produced hgh qualty solutons on real-lfe problems n reasonable computatonal tme. Fnally, for mnmzng the total travel cost, a forward dynamc programmng approach was developed by Dumas et al. [D1986]. The effcency of the algorthm s mproved by elmnatng states that are ncompatble wth vehcle capacty, precedence and tme wndow constrants. The multple vehcle pckup and delvery problem wth tme wndows has receved few attenton untl recently. The only optmal algorthm to our knowledge developed by Dumas et al. [D1991] who employed a column generaton scheme wth a shortest path subproblem wth capacty, tme wndow, precedence and couplng constrants. Ther algorthm can solve 1-PDPTW problems up to 55 pared requests and multple- vehcle PDPTW wth a small sze of pared requests per vehcle. In [S95], Savelsbergh dvded the General Pckup and Delvery Problem nto four categores, whch are Statc Sngle-Vehcle PDP, Statc Mult-Vehcle PDP, Dynamc Sngle-Vehcle PDP and Dynamc Mult-Vehcle PDP. e presented a general model that can handle the practcal jk

3 constrants. The paper amed to solate and dscuss some of the characterstcs that dfferentate pckup and delvery problems from tradtonal vehcle routng problem, Recently, Wllam and Barnes [WB000] proposed a reactve tabu search approach to mnmzng the travel cost by usng a penalty objectve functon n terms of travel tme, penalty for volaton of overload and tme wndow constrants. The approach was tested on 5-customer nstances, 50-customer nstances and customer nstances constructed from Solomon's 1 VRPTW benchmark nstances. Fgure 1 shows a PDPTW nstance. Usng the above nserton heurstc, the soluton generated s shown n Fgure. Observe that, n ths nstance, all the jobs close to depot (.e. A, A, B, B ) are also close to one another; hence, they wll be served by the same vehcle. onsequently, all the jobs further away are far from one another, to the extent that each vehcle can only serve one par of jobs at a tme. In order to avod such knd of mbalance (some very good routes and some very bad routes), another constructon heurstc, the Sweep eurstc, s often used. In [NB000], the authors proposed a reactve tabu search to solve the PDPTW. They frst used a greedy nserton method to construct a feasble PDPTW plan. Then, a reactve tabu search method was used to mprove the plan. They proposed three neghborhoods, namely, Sngle pared nserton (SPI), Swappng pars between routes (SBR) and Wthn route nserton (WRI). In ther work, the data sets were bult based on Solomon test cases for VRPTW. The vehcle capacty, the spatal nformaton and tme wndow of each locaton s the same as the orgnal Solomon test cases. Jobs are pared randomly based on the optmal soluton provded by [K1995], whle assurng that feasblty was mantaned. owever, t s not clear from ther work how the load of each pckup-delvery par s set. Ths s an mportant consderaton because t would determne whether the gven solutons (from [K1995] n ths case) are stll the optmal solutons for the correspondng PDPTW test cases. In ths paper, we wll address ths ssue by proposng a more rgorous test case generaton strategy. D D A A X X Depot Pckup Job Delvery Job B B Fgure 1. Example PDPTW nstance D E E E 4. Two Phase Method D B E In ths secton, we propose our two-phase method for solvng PDPTW. Ths two-phase method comprses the onstructon heurstc and the Tabu Search. A B 4.1 onstructon heurstcs A Our constructon heurstc, whch we name as parttoned nserton heurstc, s a hybrd heurstc combnng the advantages of the standard nserton heurstc and sweep heurstc. Inserton eurstc Inserton eurstc s one of the most commonly used constructon heurstcs for VRP (see [SS94]). To solve PDPTW, the Inserton heurstc can be adapted as shown below: 1. Let all vehcles have empty routes.. Let L be the lst of unassgned requests. 3. Take a job par v n L. 4. Insert v n a route at a feasble poston where there s the least ncrease n cost. 5. Remove v from L. 6. If L s not empty, go to 3. Fgure. Soluton Usng Inserton eurstc Sweep eurstc Sweep heurstc s the other well-known constructon heurstc method for VRP. It bulds routes by a sweep technque around the depot. The Sweep heurstc for VRP s shown below: 1. Let O be a ste from whch vehcles leave (usually the depot), and let A (dfferent from O) be another locaton, whch serves as a reference.. Sort jobs by ncreasng angle AOS where S s the job locaton. Put the result n a lst L. 3. The jobs n L wll be allocated to the vehcles n that order as long as constrants are respected.

4 The advantage of sweep heurstc s that near and far jobs are mxed n the same route. Ths makes the soluton more balanced,.e. there are no extremely good routes and extremely bad routes. Notce that n PDPTW, geographcally close destnatons may have orgns that are far away; consequently, a pckup and delvery par may not be served by the same vehcle usng the above algorthm! The followng modfcatons are done to adapt the sweep heurstc for PDPTW: 1. Let O be a ste from whch vehcles leave, and let A (dfferent from O) be another locaton, whch serves as a reference.. Sort pckup jobs by ncreasng angle AOS where S s the job locaton. Put result n a lst L. 3. Pck a pckup job n L wth locaton I and ts delvery job wth locaton J and create a new route wth ths job par. 4. Untl no more jobs can be added the route do: a. If there are unnserted pckup jobs located n the sector IOJ, nsert the par that s best feasble. Otherwse, nsert an unnserted pckup-delvery job par, n whch the pckup job s at locaton K where JOK s smallest and all the constrants are respected. b. Remove ths pckup job from L. 5. If L s not empty, goto 3. Fgure 3 shows the soluton generated usng the modfed Sweep eurstc. D D A A B B E Fgure 3. Soluton Usng Sweep eurstc Parttoned Inserton eurstc We now present our constructon heurstc, the Parttoned Inserton eurstc: 1. Set all vehcles to empty routes.. Let L be the lst of unassgned vsts. 3. Sort jobs by ncreasng angle AOS where S s the job locaton. Put the result n a lst L. 4. Dvde L nto K sub-lsts such that [ 1, K ], all the jobs n the th sub-lst satsfes AOS [ / π, ( 1) / π ). 5. Randomly fnd a partton and nsert the farthest job v n L. 6. Insert v n a route at a feasble poston where there wll be the least ncrease n cost. E 7. Use the Inserton eurstc to form a route. 8. If L s not empty, go to 5. In our algorthm, both advantages of the nserton heurstc and modfed sweep heurstc are merged. The furthest job wthn a sub-lst s always selected as the frst job to be nserted n a new route. Ths wll ensure that the bad jobs (snce they are far) are taken care of at the onset, thus avodng the formaton of mbalance routes. The number of the partton s set to the number of the establshed routes needed. In Secton 6, we wll llustrate the proposed algorthm n terms of both speed and qualty of solutons. 4. Tabu Search We ntroduce three dfferent neghborhood moves, namely, Sngle Par Inserton (SPI), Swap Pars between Routes (SBR) and Wthn Routes Inserton (WRI). These moves are adapted from [NB000]. The Noton of luster In [T1996], multple consecutve jobs exchange was presented as a local move. Ths s because often a segment wth consecutve jobs s a good component to formng a good route. Ths move s extended to PDPTW as follows. onsder a stuaton that pckup jobs A, B and delvery jobs A, B are n the same route (Fg. 4). If we move both pars together as done n [T1996], they must be consecutve. We defne such a consecutve segment as a cluster. In other words, the vehcle can enter and leave the cluster wth no other jobs nvolved. Another property of cluster s that vehcle wll enter and leave the cluster wth the same load. A B A Fgure 4. Example of luster Sngle Par Inserton (SPI) The frst move neghborhood attempts to move a pckupdelvery par or a cluster from ts current vehcle route to another vehcle route n the soluton. SPI performs the followng process for all n/ pckup-delvery job pars n the current soluton. Once a pckup-delvery job par or a cluster s dentfed, the method attempts to place t on another route. An admssble placement s one where both jobs (pckup and delvery) satsfy both tme wndow and capacty constrants. There are n/ ways of choosng pckup-delvery job par. There are O (n) postons to place the pckup and delvery jobs respectvely. ence, SPI has an O ( n 3 ) search neghborhood. To reduce the number of routes, the search process should be based such that t tres to remove the job pars from the shorter routes and nsert them nto longer routes. B

5 Assume that a pckup-delvery job par s selected from route r, and are nserted nto route 1 r. The routes after ' ' ths move are denoted as r 1, r. Orgnally, there are n 1 jobs n r 1 and n jobs n r. The cost of a route r s denoted as f (r) and the pure savng cost (PS) s ' ' defned as PS( r1, r ) = f ( r1 ) f ( r ) f ( r1 ) f ( r ). To bas the search, we ntroduce a new savng cost known as the bas route savng cost (BRS), defned as P P BRS ( r1, r ) = PS( r1, r ). learly, f n1 / ( n ) / n < n 1, BRS ( r, r 1 ) wll lkely become postve; f n 1 = n, BRS ( r, r 1 ) and BRS ( r, r 1) wll only depend on PS ( r, r 1 ) and PS ( r, r 1) ; f n > n, 1 BRS r, r ) wll lkely become postve. ( 1 Swappng Par Between Routes (SBR) The second move neghborhood nvolves exchanges of pckup-delvery pars and/or clusters between two dfferent routes. Assume that n jobs are evenly dstrbuted n m routes, the computatonal complexty of 4 swap neghborhood s n O ( ). m Wthn Route Inserton (WRI) WRI s used to mprove routes by movng ndvdual nodes forward or backward wthn ther respectve routes. Note however that snce there are ( n / m)! possble ways to sequence the jobs, local search s used agan. For each route, do the followng: 1. Move one pckup and delvery par n the route.. If the cost s reduced and all constrants are satsfed, goto Step When all such moves have been tred, move clusters consstng of two pars. 4. Eventually, move clusters consstng of three pars. omposte Neghborhood Of the three move neghborhoods, SPI has the greatest potental for mprovement n the objectve functon, and t s the only move that can reduce the number of routes. When SPI reaches a barrer where no more admssble SPI exsts, SBR s used to overcome the barrer. Fnally, WRI s appled, whch s especally helpful when large tme wndows are prevalent. The applcaton of three neghborhood moves n our tabu search s shown below: 1. Fnd SPI move wth hghest PS and mplement the move.. If no more SPI move wth postve PS exsts, fnd the best SPI move wth BRS and mplement the move. Goto If no more SPI move wth postve BRS, fnd the SBR move wth the greatest savng and mplement the move. Goto If no more SBR move wth postve savng, fnd the best WRI move and mplement the move. Goto If no WRI move found, stop. 5. Test ase Generaton We performed a careful lterature survey, and to our knowledge, no comprehensve benchmark test cases for PDPTW are avalable. Fortunately, from the VRPTW lterature, there are well-establshed benchmark test cases for VRPTW by Solomon [S1987], as well as good solutons to those nstances. In ths secton, we present how we adapt Solomon nstances and the best-publshed solutons to generate good PDPTW nstances and ther correspondng benchmark solutons. Our strategy s to reuse the best-publshed VRPTW solutons as benchmark solutons for PDPTW nstances. In essence, two ssues need to be resolved. Frst, how we ensure that a VRPTW soluton remans feasble for the PDPTW nstance, gven that the latter s more constraned than a VRPTW nstance. Second, gven that pckup and dropoff occur throughout the route, how to ensure that the VRPTW soluton remans to be good (n the sense of ts optmalty) for the PDPTW nstance. 5.1 Preservng Feasblty Unlke VRPTW n whch jobs have no precedence constrants, a PDPTW nstance does. ence, any feasble soluton y for a gven VRPTW nstance X may not be feasble on a PDPTW nstance resultng from randomly or arbtrarly desgnatng the jobs n X as pckup or delvery jobs. ence, rather than parng jobs on X, we par jobs based on y. We do so n such a way that y remans feasble under the generated PDPTW nstance, as shown below: Algorthm GENERATE: For each route r n y do a. Randomly select two jobs (j 1, j ) n r to be pared b. Randomly select ether j 1 or j s load as pckup and delvery load for both j 1 and j c. If there are stll jobs not pared. If the number of jobs s more than 1, go to step a.. If number of jobs s 1, set t as a pckup job; create a dummy delvery job, whose tme wndow s set to the largest possble tme wndow, servce tme s set to 0, and load s equal to the load of the remanng pckup job. 5. Preservng Optmalty Unlke VRPTW where the total load (sum of job loads) on each route remans statc, each route n PDPTW wll have dfferent cumulatve loads as the vehcle pcks up and drops off the loads throughout the route. ence, f we keep the vehcle capacty as t s (as done n [NB000]), the vehcle capacty constrant s no longer as tght as ntended for the gven VRPTW nstance. Ths makes t unclear whether an optmal soluton for a VRPTW nstance

6 s stll optmal, or there exs ts even better solutons for the correspondng PDPTW test case generated from the VRPTW nstance. On the other hand, however, f the vehcle capacty were changed to become too tght, then the neghborhood space wll be naturally lmted. ence, the key ssue s to adjust the vehcle capacty so that a good (.e. near optmal) soluton for a gven VRPTW nstance remans to be good for the correspondng PDPTW nstance. Gven a PDPTW nstance and ts soluton, we frst compute the maxmum load of each route on the soluton (whch s the maxmum possble load that the vehcle s carryng at any one pont throughout the load). The vehcle capacty for that nstance s then set as the hghest maxmum load over all routes. From Algorthm GENERATE presented n Secton 5.1, we see that numerous PDPTW nstances can be generated from a gven VRPTW nstance. ence, to ensure that the vehcle capacty s suffcently tght over the many possble PDPTW nstances derved from a gven VRPTW nstance, we apply the followng procedure: 1. Apply Algorthm GENERATE to generate 100 dfferent PDPTW nstances. ompute the average vehcle capacty by averagng over the vehcle capactes for all nstances Usng the above approach, we present statstcs on the average vehcle capactes of several R1 type test cases (rounded to the nearest nteger). These fgures were presented usng the solutons presented n Data R103 R104 R107 R108 R109 R110 R111 R11 Avg Vehcle apacty Table 1. Average Vehcle apacty for PDPTW R1 nstances Notce that for Solomon s VRPTW test cases, all problem nstances belongng to the same type category (R1, for example) have the same vehcle capacty. Lkewse, to be consstent wth ths standard, we compute the average of all R1 test nstances, whch turns out to be ence, we set the R1 type vehcle capacty as 85 (rounded to the nearest 5 or 10, lke Solomon test cases). Lkewse, the vehcle capactes for R1, R and R are computed, as shown n Table. In ths table, we also lst the mnmum and maxmum vehcle capacty computed over dfferent PDPTW test cases wthn each category. We observe that even wthn each type category, there s a large gap of between the mnmum and maxmum vehcle capactes over all generated PDPTW nstances. Ths means that t s good to set a Type vehcle capacty n order to ensure that the problem nstances generated under each type s consstently tght. 6 Expermental Results In ths secton, we present an analyss the effectveness of three constructon heurstcs presented above and a comparson of our approach aganst publshed PDPTW algorthms. 6.1 Analyss of onstructon eurstcs Results Usng the above test generaton algorthm, we generate 4 types of test cases (R1, R, R1, R) for PDPTW, whch conssts of 7 test cases. We dd not generate type test cases. Instead, we used those provded by [NB000] 1. Each of the test cases was run 100 tmes aganst each constructon heurstc and the best solutons returned were pcked. The detaled result s lsted n Table 3. The four columns respectvely represent the results obtaned by the Inserton eurstc, Parttoned Inserton eurstc, Sweep eurstc and the best publshed results. Data Best I Best PI Best S Best Pub R R R R R Data R1 R1 R R Type Vehcle apacty 85 (86.13) 95 (96.07) 05 (05.55) 10 (11.38) Mn Veh ap Max Veh ap Table. Average Vehcle apactes for all PDPTW Test ases 1 Durng our experments, we found some bugs for the 10 and 103 test cases reported n [BN000]. In partcular, the pckup and delvery jobs n these cases do not match. We have done some patchng to ensure the correctness of the test cases. These are the objectve values of the VRPTW solutons obtaned from based on the objectve functon (*) defned n Secton.

7 R R R R R R R R R R R R R R R Table 3. onstructon heurstcs Results In ths table, the bold fgures represent the best among the three heurstcs. The Italc fonts represent that the results use the same number of vehcles n best result and the underlne fonts represent that the soluton use less number of vehcles than other constructon heurstcs. Several observatons can be made. Frst, t shows that whle the Parttoned Inserton eurstc yeld best results n 18 out of 7 nstances, the Sweep eurstc has no good effect on the PDPTW nstances. Ths s attrbuted to the tme wndows constrants. In fact, the tghter the tme wndows are, the worse the solutons obtaned by Sweep. Second, we observe that both Inserton eurstc and Parttoned Inserton eurstc behave qute well n type of the test cases. Both of them can acheve the best number of vehcles. Notce also that the Parttoned Inserton eurstc gves even better result n terms of dstance traveled. Another nterestng observaton s that the soluton for 104 s even better than the optmal soluton. Ths shows that the soluton of test cases gven by [NB000] s no longer optmal under the correspondng PDPTW nstance. The optmalty s destroyed because they dd not pay attenton to settng the capacty constrants approprately. 6. Tabu Search Results In ths secton, we wll present the results produced by our tabu search approach. The constructon heurstc we used s the Parttoned Inserton eurstc. The tabu length was set to 50. Our expermental results are lsted below: Data Tabu NB000 Best Pub (0 Iter.) (5 Iter.) (300 Iter) (0 Iter) (0 Iter.) (75 Iter) (83 Iter) (91 Iter) R Not reported R Not reported R Not reported R Not reported R Not reported R Not reported R Not reported R Not reported R Not reported R Not reported R Not reported R Not reported R Not reported R Not reported R Not reported 76.8 R Not reported R Not reported R Not reported R Not reported R Not reported Table 4. Tabu Search Results From the table above, we can see that our proposed tabu search yelds solutons that are very close to the best soluton for most of the cases. onsderng the fact that we have pad careful attenton n settng the vehcle capactes, we beleve that the best-publshed results reman to be near optmal solutons for the PDPTW nstances. Ths mples that our results are capable of generatng near optmal solutons for PDPTW. In fact, there are 17 out of 1 non--type cases that attan the number of vehcles gven n the best soluton (as shown n bold fgures). The others requre exactly one vehcle more.

8 For the type of the cases, our approach yelds solutons that are almost equal to the best-publshed solutons. An observaton s the results provded by [NB000] (last column n the table). In brackets are the number of the teratons ther approach requred to obtan ther results. There, 4 out of 8 test cases whch, wthout teraton, yelded the best results. In other words, these results were obtaned smply by the constructon heurstc. In ther paper, the authors clamed that the nserton algorthm s used as a constructon heurstc. ence, we suspect that there could be somethng wrong wth ther test cases. After a careful comparson between ther test cases and the optmal solutons of VRPTW, we found that n ther test cases, most pckup and delvery jobs were adjacent wth each other! They were not well randomly pared and hence the problem nstances were very easy to solve. 6. oncluson In ths paper, we presented a two-phase approach to solve the Pckup and Delvery Problem wth Tme Wndows (PDPTW). We desgned a set of good (.e. reasonably hard) benchmark test cases and solutons for PDPTW based on the full sute of Solomon test cases, pavng the way for future PDPTW research. We conducted expermental comparsons over dfferent constructon heurstcs on these data sets. Our expermental results show that our tabu search approach yelds solutons that are very close to the benchmark solutons. Reference [D1995] J. Desrosers et al., Tme onstraned Routng and Schedulng. In andbooks n Operatons Research and Management Scence: Network Routng. Elsever Scence Publ., , (1995). [D1986] Y. Dumas, J. Desrosers, F. Soums, A dynamc programmng soluton of the large-scale sngle vehcle dal-a-rde problem wth tme wndows. Amercan Journal of Mathematcal and Management Scence 16, , (1986). [D1991] Y. Dumas, J. Desrosers, F. Soums, The pckup and delvery problem wth tme wndows. European Journal of Operatonal Research 54, 7-, (1991). [K1995] [P1980] N. Kohl, Exact Methods for Tme onstraned Routng and Related Schedulng Problems, Ph.D. Dssertaton, Insttute of Mathematcal Modelng, Techncal Unversty of Denmark.. Psarafs, A dyanmc programmng soluton to the sngle vehcle many-to-many mmedate request dal-a-rde problem. Transportaton Scence 14, , (1980). [P1983]. Psarafs, An exact algorthm for the sngle vehcle many-to-many mmedate request dala-rde problem. Transportaton Scence 17 (4), , (1983). [NB000] W. P. Nanry, J. W. Barnes, Solvng the Pckup and Delvery Problem wth Tme Wndows Usng Reactve Tabu Search, Transportaton Research (Part B), 34, , (000). [RT1995] Y. Rochat, E. D. Tallard, Probablstc Dversfcaton and Intensfcaton n Local Search for Vehcle Routng, Journal of eurstcs, 1, , (1995). [S1995] [S1987] [SS1995] M.W.P. Savelsbergh, Local Search for Routng Problems wth Tme Wndows, Annals of Operatons Research 4, , (1985). M.M. Solomon, Algorthms for the vehcle Routng and Schedulng Problem wth Tme Wndow onstrants, Operatons Research, 41, , (1987). M.W.P. Savelsbergh, M. Solomon, The General Pckup and Delvery Problem, Transportaton Scence, 9, 17-9, (1995). [SB1985a] T.R. Sexton,, L. D. Bodn, Optmzng sngle vehcle many-to-many dal-a-rde problem wth desred delvery tme: I Schedulng. Transportaton Scence 19, , (1985). [SB1985b] T.R., Sexton, L. D. Bodn, Optmzng sngle vehcle many-to-many dal-a-rde problem wth desred delvery tme: II Routng. Transportaton Scence 19, , (1985). [S1986] T.R Sexton,, Y.Y. ho,, Pckup and delvery partal loads wth soft tme wndows. Amercan Journal of Mathematcal and Management Scence 6, , (1986). [T1996] E. Tallard et al., A Tabu Search eurstc for the Vehcle Routng Problem wth Soft Tme Wndows, Transportaton Scence, 31, , (1996). [VLS1993] L.J.J. Van der Bruggen, J.K. Lenstra,, P.. Schuur, Varable-depth search for the sngle vehcle pckup and delvery problem wth tme wndows. Transportaton Scence 7, , (1993). [WB000] P.N. Wllam, J.W.Barnes, Solvng the pckup and delvery problem wth tme wndows usng tabu search. Transportaton Research Part B 34, , (000).