Perturbation heuristics for the pickup and delivery traveling salesman problem

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1 Computers & Operatios Research 29 (2002) 1129}1141 Perturbatio heuristics for the pickup ad delivery travelig salesma problem Jacques Reaud, Fayez F. Boctor, Gilbert Laporte * Te&le& -Uiversite&, 2600 boulevard Laurier, Tour de la Cite&,7ee& tage, C.P , Saite-Foy, Caada G1V 4V9 Cetre de recherche sur les techologies de l'orgaisatio re&seau, Uiversite& Laval, Que., Caada G1K 7P4 Faculte& des scieces de l'admiistratio, Uiversite& Laval, Que., Caada G1K 7P4 E! cole des Hautes E! tudes Commerciales, 3000 chemi de la CoLte-Saite-Catherie, Motre& al, Que., Caada H3T 2A7 Cetre de recherche sur les trasports, Uiversite& de Motre& al, C.P. 6128, Succursale Cetre-ville, Motre& al, Que., Caada H3C 3J7 Abstract Received 1 December 1999; received i revised form 1 May 2000; accepted 1 October 2000 This article describes ad compares seve perturbatio heuristics for the Pickup ad DeliveryTravelig Salesma Problem (PDTSP). I this problem, a shortest Hamiltoia cycle is sought through a depot ad several pickup ad delivery pairs. Perturbatio heuristics are diversi"catio schemes which help a local search process move away from a local optimum. Three such schemes have bee implemeted ad compared: Istace Perturbatio, Algorithmic Perturbatio, ad Solutio Perturbatio. Computatioal results o PDTSP istaces idicate that the latter scheme yields the best results. O istaces for which the optimum is kow, it cosistetly produces optimal or ear-optimal solutios. Scope ad purpose I several distributio maagemet cotexts, it is ecessary to costruct a shortest tour startig at a depot ad makig several pickup ad deliveries. I the Travelig Salesma Problem with Pickup ad Delivery, to each pickup poit is associated a delivery poit later i the tour. Like several routig problems, the PDTSP is very hard to solve to optimality ad local search heuristics ofte get trapped i local optima. Perturbatio heuristics provide a meas of escapig from local optima. This paper describes ad compares three types of perturbatio heuristic. It shows that the best scheme cosistetly yields high-quality solutios Elsevier Sciece Ltd. All rights reserved. Keywords: Pickup ad delivery travelig salesma problem; Perturbatio heuristics * Correspodece address. Cetre de recherche sur les trasports, UiversiteH de MotreH al, C.P. 6128, Succursale Cetre-ville, MotreH al, Que., Caada H3C 3J7. Tel.: # ; fax: # address: gilbert@crt.umotreal.ca (G. Laporte) /02/$ - see frot matter 2002 Elsevier Sciece Ltd. All rights reserved. PII: S (00)00109-X

2 1130 J. Reaud et al. / Computers & Operatios Research 29 (2002) 1129} Itroductio The purpose of this article is to describe ad compare several perturbatio heuristics for the Pickup ad DeliveryTravelig Salesma Problem (PDTSP) de"ed as follows. Let G"(<, E) bea udirected graph where <" v,2, v is the vertex set, is odd, v represets a depot, ad E" (v, v ): i(j, v, v 3< is the edge set. The set < v is partitioed ito P, D where P is a set of pickup customers, D is a set of deliverycustomers, ad P " D "(!1)/2. These two sets are twied i the sese that to each pickup customer, v 3P correspods exactly oe delivery customer d(v )3D, ad to each delivery customer correspods exactly oe pickup customer. A distace matrix C"(c )isde"ed o E. I what follows, c must be iterpreted as c wheever i'j. The PDTSP cosists of determiig a shortest Hamiltoia cycle o G startig ad edig at the depot, ad such that each pickup customer is visited before its associated delivery customer. The PDTSP is NP-hard sice it reduces to the travelig salesma problem (TSP) wheever each pickup customer v coicides with its associated delivery customer d(v ). The PDTSP is related to, but di!eret from, the dial-a-ride problem (DARP) i the followig way: i the DARP there may be several capacitated vehicles istead of oly oe, time widows are ofte preset, ad the objective fuctio may be di!eret. For example, oe may wish to miimize the total distace traveled betwee each pickup poit ad its associated delivery poit. For refereces o the DARP, see, e.g., Psaraftis [1}3], Sexto ad Bodi [4,5], Sexto ad Choi [6], Desrosiers et al. [7], Jaw et al. [8], Dumas et al. [9], Va der Brugge et al. [10], Savelsbergh ad Sol [11]. Two other problems related to the PDTSP are the vehicle routig problem with backhauls [12}14], ad the TSP with Backhauls [15,16]. I both problems, all pickup customers must be visited before delivery customers but there is o relatioship betwee these two sets. Relatively few algorithms have bee proposed for the PDTSP. Kalatari et al. [17] have described a exact brach-ad-boud procedure for this problem ad have applied to istaces ivolvig o more tha 40 vertices. Savelsbergh [18] ad Healy ad Moll [19] have proposed several heuristic improvemet schemes based o edge iterchages. More recetly, Reaud Boctor ad Oueiche [20] have implemeted a edge-iterchage scheme as well as a vertex deletio ad reisertio method. As far as we are aware, o heuristic method has bee assessed by makig comparisos with optimal solutios. Our aim is to develop several types of perturbatio heuristics for the PDTSP. These ca be viewed as diversixcatio schemes that ca help a improvemet algorithm to escape from a local optimum, similar to what is sometimes doe i tabu search [21]. A "rst type of perturbatio heuristic, called Istace Perturbatio (IP) was itroduced by Storer et al. [22], Charo ad Hudry [23] ad later implemeted by Codeetti et al. [24] i the cotext of the TSP. I IP, whe a local optimum is reached, the istace data are margially perturbed, a improvemet algorithm is the applied to the modi"ed data, ad the local optimum of the perturbed istace is traslated back ito the origial data. The same process ca be applied iteratively. A secod type of perturbatio is called Algorithmic Perturbatio (AP). This cocept ca be applied to a costructio or to a improvemet heuristic. I the "rst case, the criterio used to geerate a feasible solutio ca be modi"ed from oe iteratio to the ext. I the secod case, it is useful to view a improvemet algorithm as a process that iteratively moves from a solutio to aother solutio i its eighbourhood. For example, the eighbour of a TSP solutio ca be ay solutio reachable by removig ad reitroducig k edges. To avoid local optima, oe ca occasioally

3 J. Reaud et al. / Computers & Operatios Research 29 (2002) 1129} modify the rule that govers the de"itio of the eighbourhood, e.g., by goig from 3-opt to 4-opt. Examples of AP are multistart methods [25] ad variable eighbourhood search [26,27]. A third type of perturbatio is Solutio Perturbatio (SP). Here, a local optimum is modi"ed ad the improvemet procedure is reapplied to the perturbed solutio. A commo example of SP is the mutatio process i geetic algorithms [28]. For a iterestig applicatio of SP to the Quadratic Assigmet Problem, see Fleuret ad Ferlad [29]. The remaider of this article is orgaized as follows. I Sectio 2, we describe several implemetatios of IP, AP ad SP to the PDTSP. Extesive computatioal results are preseted i Sectio 3, followed by the coclusio i Sectio Perturbatio heuristics We have developed a total of seve perturbatio heuristics for the PDTSP. All proceed accordig to the followig geeral framework: Step 1(Iitializatio): Costruct a feasible PDTSP solutio ad improve it by meas of 4-optHH, a adaptatio of the 4-optH heuristic developed by Reaud et al. [30]. Basically, 4-optH uses eight of the forty-eight potetial 4-opt [31] moves while esurig that at least oe added edge will be shorter tha a removed edge. Our adaptatio of 4-optH is such that v 3P always appears before d(v ) whe the tour is iitialized at v. Step 2(Perturbatio): Apply a perturbatio scheme (either IP, AP, or SP). Step 3(Postoptimizatio): Apply the 4-optHH heuristic to oe or several solutios obtaied at the ed of Step 2. If a stoppig criterio is satis"ed, stop. Otherwise go to Step 2. If IP or SP are used, the applicatio of Step 2 at the ed of Step 3 is iitiated from either the icumbet or from the curret solutio, at the choice of the user. We have tested these two possibilities. I all cases, the best-kow solutio is kept i memory ad the algorithm termiates whe the icumbet has ot improved for λ successive applicatios of Steps 2 ad 3, where λ is a usercotrolled parameter. We ow describe each step of the geeral framework Iitializatio The costructio heuristic starts by determiig the pair (v H, v H"d(v H)) yieldig the logest tour, ad the sequetially iserts the remaiig pairs (v, d(v )) yieldig at each step the miimum value of a score fuctio. Step 1 (Iitial subtour): Determie the vertex pair (v H, v H"d(v H)) yieldig max c #c #c, where v "d(v ). This is the MAX TRIANGLE rule used i the TSP heuristic I for o-euclidea istaces, fully described i Reaud et al. [30]. Set P :"P v H. Step 2 (Vertex isertio): If P", go to Step 3. Otherwise determie the vertex pair (v H, v H"d(v H)) yieldig the best score value. Two cases are possible. Case 1: v H ad v H are iserted cosecutively betwee vertices v ad v l. Let F be the set of all edges of the curret subtour. The SCORE 1" mi αc #c #(2!α)c l!c l, l where v "d(v ) ad α is a user-cotrolled parameter (0)α)2).

4 1132 J. Reaud et al. / Computers & Operatios Research 29 (2002) 1129}1141 Case 2: v H ad v H are iserted betwee vertices v, v l ad v, v, respectively, where (v, v ) appears after (v, v l ) o the subtour. The SCORE 2" mi α(c #c l!c l)#(2!α)(c #c!c ), l where v "d(v ) ad α is a user-cotrolled parameter (0)α)2). These two score fuctios were adopted after some computatioal experimetatio. The vertex pair (v H, d(v H)) yieldig mi SCORE 1, SCORE 2 is the iserted i its appropriate positio i the subtour ad P :"P v H. Repeat this step. Step 3 (4-optHH): Attempt to improve the curret solutio by meas of 4-optHH. Several values of α were tested (α"0.5, 0.75, 1.00, 1.25 ad 1.50) ad the value α"1.25 was retaied Perturbatio We have developed two IP schemes (IP1 ad IP2), two AP schemes (AP1 ad AP2), ad three SP schemes (SP1, SP2 ad SP3). Their descriptios ow follow. Perturbatio scheme IP1 : This perturbatio scheme applies to plaar istaces oly. It operates with two iput user-cotrolled parameters β ad γ, where 0(β)1 ad γ'0. Step 1(Vertex moves): Move each vertex v 3< v with probability β. Whe vertex v is selected for a move, it is radomly relocated withi a circle of radius γc cetered at its curret positio. Compute the modi"ed distace matrix C, ad update the tour legth. Step 2(4-optHH): Apply 4-optHH to the perturbed istace. Step 3(Mappig): Map each vertex of the tour obtaied at the ed of Step 2 oto its iitial positio, ad update the tour legth. Perturbatio scheme IP2 : Oly Step 1 di!ers from the previous scheme. Step 1(Vertex moves): Cosider the curret solutio (v,v, v,2, v, v ). For t"2,2,, radomly relocate vertex v with probability β i the crow cetered at v, determied by the two radii c ad (1#γ)c. Perturbatio scheme AP1 : I this scheme, Step 2 of the geeral framework is used to geerate iitial solutios di!eretly (ad later postoptimize them i Step 3) i the hope of geeratig a better local optimum tha the iitial solutio obtaied at the ed of Step 1. The costructio algorithm described i Sectio 2.1 is perturbed as follows. Steps 1 ad 3 are idetical. I Step 2, istead of seekig the vertex pair (v H, d(v H)) miimizig mi SCORE 1, SCORE 2, this choice is ow made radomly amog all o-iserted vertices v 3P. The selected pair is the iserted at miimum cost by computig mi SCORE 1, SCORE 2 with α"1. Perturbatio scheme AP2 : Here the costructio algorithm of Sectio 2.1 is perturbed by usig a radomly selected value of α i the iterval [0, 2].

5 J. Reaud et al. / Computers & Operatios Research 29 (2002) 1129} Perturbatio scheme SP1 : This scheme works with two parameters δ ad θ. The curret solutio is perturbed by radomly removig betwee δ ad θ vertices from the curret tour ad reisertig each of them i a radom but feasible positio i the tour. Perturbatio scheme SP2 : The curret solutio is agai perturbed by radomly removig betwee δ ad θ vertices from the curret tour. However, reisertios are performed by usig a least isertio legth criterio while maitaiig feasibility. This is doe very simply i the followig Table 1 Combiatios ad variats tested for the perturbatio heuristics Heuristics List Number of combiatios Recommeded selectio IP IP1, IP2, β"0.5, 1; γ"0.1, 0.2; 64 IP1, β"0.5; γ"0.2; λ"50; icumbet. λ"5, 10, 20, 50; Step 2: icumbet, curret. IP2, β"1.0; γ"0.2; λ"50; curret. AP AP1, AP2; λ"5, 10, 20, AP1: λ"50. AP2: λ"50. SP SP1, SP2, SP3; (δ, θ)"(5, 10),(10, 15), 96 SP1: (δ, θ)"(10, 15); λ"50; icumbet. (0.05, 0.10), (0.10, 0.15); λ"5, 10, 20, 50; SP2: (δ, θ)"(0.10, 0.15); λ"50; icumbet. Step 2: icumbet, curret. SP3: (δ, θ)"(0.05, 0.10); λ"50; icumbet. Table 2 Istace perturbatio, Class 1 istaces Proximity rule Number of istaces IP1 IP2 RBO Value/Best Secods Value/Best Secods Value/Best A }99 B C A }199 B C A }299 B C A }499 B C A All B C Global average

6 1134 J. Reaud et al. / Computers & Operatios Research 29 (2002) 1129}1141 maer. The removed vertices v are sequetially reiserted betwee two cosecutive vertices v ad v i order to miimize c #c!c, while maitaiig feasibility. The e!ect of this perturbatio rule is likely to be less importat tha that of SP1 sice the isertio criterio may relocate vertices i their origial positio. Perturbatio scheme SP3 : This scheme is ispired from the cross-over operatio i geetic algorithms. Two ew solutios are created by combiig the solutio S produced by SP1 with the best kow solutio S di!eret from S. A cross-over positio u is radomly selected betwee π ad σ, where π ad σ are two user-cotrolled parameters take as π"0.3 ad σ"0.6 i our tests. To geerate a ew solutio S, the "rst u vertices v,v, v,2, v of S are kept i this order, ad S is completed by cosiderig i tur the vertices of S. Wheever a vertex of S ot already preset i S is ideti"ed, it is itroduced after the vertices of S, i the same sequece i which they appear i S. A secod solutio S is also created by keepig this time the "rst u vertices of S ad completig the solutio by usig i tur the vertices of S. Either S or S may the yield a ew icumbet. Table 3 Istace perturbatio, Class 2 istaces Istace umber Opt IP1 IP2 Value Value/Opt Secods Value Value/Opt Secods Average Average Global average

7 2.3. Postoptimizatio J. Reaud et al. / Computers & Operatios Research 29 (2002) 1129} The postoptimizatio heuristic 4-optHH is applied to the solutio obtaied at the ed of the perturbatio step (i the case of IP1, IP2, AP1, AP2, SP1 ad SP2), or to the two solutios S ad S geerated by SP3. 3. Computatioal results The algorithms just described were coded i Borlad Delphi 3.0 ad ru o a PC Petium II, 200 MHz uder Widows 95. We have tested the perturbatio heuristics o two classes of istaces. The "rst class was geerated from 36 of the TSPLIB [32] Euclidea istaces, ragig from 51 to 441 vertices, with distaces rouded up or dow to the earest iteger. If the umber of vertices is eve, the last oe is dropped. The depot is always the "rst vertex. The each pickup-delivery pair was de"ed as follows: radomly select a pickup vertex ad associate to it a delivery vertex accordig to oe of three proximity rules. Rule A: Select the delivery vertex from amog the "ve closest eighbours (ot yet selected) of the pickup vertex. Rule B: Select the delivery vertex from amog the 10 closest eighbours (ot yet selected) of the pickup vertex. Table 4 Algorithm perturbatio, Class 1 istaces Proximity rule Number of istaces AP1 AP2 RBO Value/Best Secods Value/Best Secods Value/Best A }99 B C A }199 B C A }299 B C A }499 B C A All B C Global average ,

8 1136 J. Reaud et al. / Computers & Operatios Research 29 (2002) 1129}1141 Rule C: Select the delivery vertex radomly from the set of all uselected vertices. No optimal solutios are kow for these problems. The oly available compariso is with the heuristic of Reaud et al. [20]. We also produced a secod class of istaces for which the optimum could be computed. Te istaces of size "101, ad 10 istaces of size "201 we geerated as follows. First, vertices were radomly geerated i the [0, 100] square, accordig to a cotiuous uiform distributio ad Euclidea distaces, rouded up or dow to the earest iteger, were computed betwee these vertices. The TSP associated with each istace was the solved optimally usig the Padberg ad Rialdi [33] brach-ad-cut algorithm. The "rst geerated vertex was desigated as the depot. The, pickupdelivery pairs were obtaied by followig the tour. The pickup vertex was take as the ext uselected vertex, ad its associated delivery vertex was radomly selected from amog the uselected vertices. This procedure esures that the optimal TSP solutio always yields a optimal PDTSP solutio. For the "rst class of istaces, the best kow solutio was obtaied from 136 rus i the Reaud et al. [20] experimets, ad 168 applicatios of our perturbatio heuristics, ru uder various rules ad with various sets of parameters. Table 5 Algorithm perturbatio, Class 2 istaces Istace umber Opt AP1 AP2 Value Value/Opt Secods Value Value/Opt Secods Average Average Global average

9 J. Reaud et al. / Computers & Operatios Research 29 (2002) 1129} Table 6 Solutio perturbatio, Class 1 istaces Proximity rule Number of istaces SP1 SP2 SP3 RBO Value/Best Secods Value/Best Secods Value/Best Secods Value/Best A }99 B C A }199 B C A }299 B C A }499 B C A All B C Global average Several idepedet rus of the IP, AP ad SP heuristics were performed o the two classes of istaces usig several combiatios of parameters ad variats (applyig Step 2 to the icumbet or to the curret solutio). These combiatios are listed i Table 1, together with our recommeded selectio. Computatioal results are preseted i Tables 2}7usig the recommeded selectio of parameters ad variats. The parameters i the table are de"ed as follows: umber of vertices Proximity rule rule employed to geerate the delivery customers i class 1 istaces Value solutio value produced by the heuristic Best best-kow solutio value Opt Secods IP1, IP2, AP1, AP2, SP1, SP2, SP3 RBO optimal solutio value for class 2 istaces computig time i secods for the perturbatio ad postoptimizatio steps, excludig the iitializatio phase (Step 1 described at the begiig of Sectio 2) perturbatio heuristic Reaud, Boctor ad Oueiche [20] heuristic For Class 1 istaces, statistics are averages over the umber of istaces. For class two istaces, statistics relate to a sigle istace.

10 1138 J. Reaud et al. / Computers & Operatios Research 29 (2002) 1129}1141 Table 7 Solutio perturbatio, Class 2 istaces Istace umber Opt SP1 SP2 SP3 Value Value/Opt Secods Value Value/Opt Secods Value Value/Opt Secods Average Average Global average Results preseted i Tables 2 ad 3 idicate that IP1 is better but slower tha IP2. O all classes ad sizes of istaces, IP1 yields a lower deviatio from the best kow or optimal solutio value tha IP2 does. O the twety class 2 istaces, the solutio values produced by IP1 are withi 0.5% of the optimum, ad the optimum is reached 12 times of 20. Moreover, our prelimiary tests have show that IP1 eve with λ"5 typically performs better tha IP2 with λ"50. The two AP heuristics are compared i Tables 4 ad 5. For most istaces, heuristic AP2 is better tha AP1, but there are several cases where this is ot true. However, AP2 is cosiderably more time cosumig tha AP1. Overall, either AP1 or AP2 fares as well as IP1. The solutio perturbatio scheme is really the best of all three, except perhaps for SP2 which did ot always perform very well. Heuristics SP3, which is obtaied from SP1 by itroducig a cross-over operatio, teds to be better tha the latter approach. O class 1 istaces, it takes about twice as log as SP1. O class 2 istaces, SP1 ad SP3 have about the same computig time sice SP3 coverges quickly towards the optimum ad thus performs fewer iteratios. O these

11 J. Reaud et al. / Computers & Operatios Research 29 (2002) 1129} Table 8 Compariso of the seve perturbatio schemes Heuristics Class 1 istaces Class 2 istaces Value/Best Secods Value/Opt Secods IP IP AP AP SP SP SP Table 9 Improvemets as a fuctio of λ as obtaied by SP3, Class 1 istaces SP3 Iitial solutio obtaied λ"5 λ"10 λ"20 λ"50 Value Value/Best Secods (cumulative) istaces, SP3 ideti"es the optimum 16 times out of 20. We have summarized i Table 8 the behavior of the seve perturbatio schemes over all istaces. A "al commet cocers the e!ect of applyig a perturbatio scheme to a locally optimal solutio. This is best illustrated by studyig the e!ect of λ (umber of cosecutive applicatios of the perturbatio ad postoptimizatio steps without improvemet) whe SP3 is applied with the recommeded selectio of parameters. Our aalysis is preseted i Table 9 for class 1 istaces. The "rst colum of this table correspods to the locally optimal solutio obtaied at the ed of Step 1. It has a Value/Best ratio equal to Executig SP3 with λ"5 brigs this ratio dow to ad requires approximately 45 s. If oe is willig to ivest more time, this ratio ca be brought dow eve further, but the margial bee"t evetually levels o!. 4. Coclusio We have coducted a computatioal compariso of seve perturbatio heuristics for the PDTSP. This study makes two distict cotributios. It "rst provides a classi"catio ad performace aalysis of several types of perturbatio schemes applicable to a wide rage of combiatorial optimizatio problems. These provide a e$ciet meas of escapig from local optima i iterative search algorithms. Our experimets show that applyig solutio perturbatio

12 1140 J. Reaud et al. / Computers & Operatios Research 29 (2002) 1129}1141 ca be quite powerful. A secod cotributio of this study is the developmet of a highly e$ciet heuristic for the PDTSP. O class 1 istaces, the best of our perturbatio schemes improves the previous results of Reaud et al. [20] by approximately 4% ad o class 2 istaces it cosistetly yields optimal or ear-optimal solutios. Ackowledgemets This research was partly supported by the Caadia Natural Scieces ad Egieerigs Research Coucil uder grats OGP , OGP ad OGP This support is gratefully ackowledged. Thaks are also due to two aoymous referees for their valuable commets. Refereces [1] Psaraftis H. A dyamic programmig solutio to the sigle vehicle may-to-may immediate request dial-a-ride problem. Trasportatio Sciece 1980;14:130}54. [2] Psaraftis H. Aalysis of a O(N ) heuristic for the sigle vehicle may-to-may Euclidea dial-a-ride problem. Trasportatio Research 1983;17B:133}45. [3] Psaraftis HN. Schedulig large-scale advace-request dial-a-ride systems. America Joural of Mathematical ad Maagemet Scieces 1986;6:327}67. [4] Sexto TR, Bodi LD. Optimizig sigle vehicle may-to-may operatios with desired delivery times: I. Schedulig. Trasportatio Sciece 1985;19:378}410. [5] Sexto TR, Bodi LD. Optimizig sigle vehicle may-to-may operatios with desired delivery times: II. Routig. Trasportatio Sciece 1985;19:411}35. [6] Sexto TR, Choi Y. Pickup ad delivery of partial loads with time widows. America Joural of Mathematical ad Maagemet Scieces 1986;6:369}98. [7] Desrosiers J, Dumas Y, Soumis F. A dyamic programmig solutio to the large-scale sigle-vehicle dial-a-ride problem with time widows. America Joural of Mathematics ad Maagemet Sciece 1986;6:301}25. [8] Jaw J, Odoi AR, Psaraftis HN, Wilso NHM. A heuristic algorithm for the multi-vehicle advace-request dial-a-ride problem with time widows. Trasportatio Research B 1986;20:243}57. [9] Dumas Y, Desrosiers J, Soumis F. The pick-up ad delivery problem with time widows. Europea Joural of Operatioal Research 1991;54:7}22. [10] Va der Brugge LJJ, Lestra JK, Schuur PC. Variable-depth search for the sigle-vehicle pickup ad delivery problem with time widows. Trasportatio Sciece 1993;27:298}311. [11] Savelsbergh MWP, Sol M. The geeral pickup ad delivery problem. Trasportatio Sciece 1995;29:17}29. [12] Casco DO, Golde BL, Wasil EA. Vehicle routig with backhauls: models, algorithms ad case studies, I: Golde BL, Assad AA, editors. Vehicle routig: methods ad studies. Amsterdam: North-Hollad, p. 121}47. [13] Goetschalckx M, Jacobs-Blecha C. The vehicle routig problem with backhauls. Europea Joural of Operatioal Research 1989;42:39}51. [14] Migozzi A, Giorgi S, Baldacci R. A exact method for the vehicle routig problem with backhaul. Trasportatio Sciece 1999;33:315}29. [15] Gedreau M, Hertz A, Laporte G. A approximatio algorithm for the travelig salesma problem with backhauls. Operatios Research 1997;45:639}41. [16] Gedreau M, Hertz A, Laporte G. The travelig salesma problem with backhauls. Computers & Operatios Research 1996;23:501}8. [17] Kalatari B, Hill AV, Arora SR. A algorithm for the travelig salesma problem with pickup ad delivery customers. Europea Joural of Operatioal Research 1985;22:377}86.

13 J. Reaud et al. / Computers & Operatios Research 29 (2002) 1129} [18] Savelsbergh MWP. A e$ciet implemetatio of local search algorithms for costraied routig problems. Europea Joural of Operatioal Research 1990;47:75}85. [19] Healy P, Moll R. A ew extesio of local search applied to the dial-a-ride problem. Europea Joural of Operatioal Research 1995;83:83}104. [20] Reaud J, Boctor FF, Oueiche I. A heuristic for the pickup ad delivery travelig salesma problem. Computers & Operatios Research 2000;27:905}16. [21] Glover F, Lagua M. Tabu search. Bosto: Kluwer, [22] Storer R, Wu SD, Vaccari R. New search spaces for sequecig problems with applicatio to job shop schedulig. Maagemet Sciece 1992;38:1495}509. [23] Charo I, Hudry O. The oisig method: a ew method for combiatorial optimizatio. Operatios Research Letters 1993;14:133}7. [24] Codeotti B, Mazii G, Margara L, Resta G. Perturbatio: a e$ciet techique for the solutio of very large istaces of Euclidea TSP. INFORMS Joural o Computig 1996;8:125}33. [25] Eglese RW. Simulated aealig: a tool for operatioal research. Europea Joural of Operatioal Research 1990;46:271}81. [26] Boctor FF. Discrete optimizatio ad multi-eighbourhood local improvemet heuristics. Workig paper 93-35, FaculteH des scieces de l'admiistratio, UiversiteH Laval, [27] Mladeovic N, Hase P. Variable eighbourhood search. Computers & Operatios Research 1997;24:1097}100. [28] Dowslad K. Geetic algorithms * a tool for OR? Joural of the Operatioal Research Society 1996;47:550}61. [29] Fleuret C, Ferlad J. A hybrid geetic algorithms i combiatorial optimizatio. RAIRO 1996;30:373}98. [30] Reaud J, Boctor FF, Laporte G. A fast composite heuristic for the symmetric travelig salesma problem. INFORMS Joural o Computig 1996;8:134}43. [31] Li S. Computer solutios to the travelig salesma problem. Bell Systems Techical Joural 1965;44:2245}69. [32] Reielt G. TSPLIB-A travelig salesma problem library. ORSA Joural o Computig 1991;3:376}84. [33] Padberg MW, Rialdi G. A brach ad cut algorithm for the resolutio of large-scale symmetric travelig salesma problems. SIAM Review 1991;33:60}100. Jacques Reaud is Associate Professor at TeH leh -UiversiteH, QueH bec city. His research iterests iclude routig, distributio ad supply chai maagemet. Fayez F. Boctor is Professor at the Faculty of Admiistrative Scieces, UiversiteH Laval. His research iterests iclude productio ad logistics maagemet. Gilbert Laporte is Professor at the ED cole des Hautes ED tudes Commerciales de MotreH al ad member of the Cetre for Research o Trasportatio ad of the GERAD. His research iterests iclude vehicle routig, locatio ad schedulig.