Pessoa, Artur,

Sadykov, Ruslan

Uchoa, Eduardo

Pessoa, Artur, Alves

INRIA a CCSD electronic archive server

HAL Id: hal-01676020https://hal.inria.fr/hal-01676020Submitted on 5 Jan 2018HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestinée au dépôt et à la diffusion de documentsscientifiques de niveau recherche, publiés ou non,émanant des établissements d’enseignement et derecherche français ou étrangers, des laboratoirespublics ou privés.Improved Branch-Cut-and-Price algorithm for theHeterogeneous Vehicle Routing ProblemArtur Pessoa, Ruslan Sadykov, Eduardo UchoaTo cite this version:Artur Pessoa, Ruslan Sadykov, Eduardo Uchoa. Improved Branch-Cut-and-Price algorithm for theHeterogeneous Vehicle Routing Problem. 18ème congrès annuel de la Société Française de RechercheOpérationnelle et d’Aide à la Décision, Feb 2017, Metz, France. ￿hal-01676020￿Improved Branch-Cut-and-Price algorithm for theHeterogeneous Vehicle Routing ProblemArtur Pessoa1, Ruslan Sadykov2, Eduardo Uchoa11 Universidade Federal Fluminense, Departamento de Engenharia de ProduçãoRua Passo da Pátria, 156, 24210-2140, Niterói, Brazil{pessoa,uchoa}@producao.uff.br2 INRIA Bordeaux — Sud-Ouest,200 avenue de la Veille Tour, 33405 Talence, France ruslan.Sadykov@inria.frMots-clÃ c©s : vehicle routing, branch-and-price, branch-and-cut, column generation1 IntroductionIn this problem, a fleet of heterogeneous vehicles of different capacity is available to servea set of customers with different demands. Each vehicle has to start and end its tour at thesame depot. The cost of a tour of a vehicle is the sum of the fixed cost of the vehicle and thetotal distance of this tour multiplied by the “weight” of the vehicle. The problem consists infinding a set of vehicle routes that serve all the customers such that the sum of the quantitiesdelivered in each tour does not exceed the capacity of the vehicle used in the tour and the totalcost is minimized. The problem has several different variants depending of whether 1) thereare fixed vehicle costs or not, 2) there are different vehicle “weights” or they are the same, 3)there are upper bounds of the number of available vehicles of the same type of not.To our knowledge, the best exact approaches for the problem are proposed in [5] and in [2].The later considers five variants of the problem and presents computational results on the ins-tances of Taillard [7], with up to 100 customers. All but five instances were solved to optimality.The best published heuristic for the problem was proposed in [6] and provides the current bestknown solutions for all Taillard instances and also for a set of Brandão instances [1] with upto 199 customers.2 Branch-Cut-and-Price AlgorithmIn the work, we present an extension of the branch-cut-and-price algorithm of [4] to thecase of heterogeneous vehicles. It uses a state-of-the-art bi-directional labeling algorithm tosolve the pricing problem which decomposes to a set of non-elementary resource-constrainedshortest path problems, one for each vehicle type. The labeling algorithm supports ng-routesand limited memory rank-1 cuts, which are used to strengthen the relaxation of the problem.Additionally we use reduced cost fixing and enumeration of elementary routes techniques tospeed-up pricing problem resolution. We branch first on the number of vehicles of a same typeand then on the arcs of the graph of customers. Multi-phase strong-branching with pseudo-costsis used to improve the selection of branching candidates.The original contributions of this work in comparison with [4] are : 1) the use of cuts overthe extended variables introduced in [5], even with the rank-1 cuts they are shown to be stillimportant for solving some instances ; 2) the concept of subproblem dependent memory forrank-1 cuts, that sharper memory definitions allows more cuts to be added without makingthe pricing too expensive ; 3) We propose a way of lifting the coefficients of columns-routesin rank-1 cuts that takes advantage of the fact that the enumeration of routes is subproblemdependent.3 Computational ResultsOur branch-cut-and-price algorithm was able to solve all Taillard instances considered in [2]to optimality. The maximum instance solution time is 1 hour and 35 minutes (all instancesexcept one were solved in less than 30 minutes). Improved feasible solutions were found for twoinstances.Our algorithm was able to solve seven out of ten Brandão instances to optimality. The largestsolved instance has 199 customers and took 12 hours. The smallest unsolved instance has 120customers. Improved feasible solutions were found for four instances.We have also tested our algorithm with a time limit of 24 hours on 96 DLP instances [3]which have from 20 to 256 customers. Each instance is based on the map of one of departmentsof France. Our algorithm solved to optimality 14 (out of 15) instances up to 100 customers, 25(out of 38) instances with 101-150 customers, 9 (out of 30) instances with 151-200 customers.Improved feasible solutions were found for 23 instances.Références[1] José Brand ao. A tabu search algorithm for the heterogeneous fixed fleet vehicle routingproblem. Computers and Operations Research, 38(1) :140 – 151, 2011.[2] Roberto Baldacci and Aristide Mingozzi. A unified exact method for solving differentclasses of vehicle routing problems. Mathematical Programming, 120(2) :347–380, 2009.[3] Christophe Duhamel, Philippe Lacomme, and Caroline Prodhon. Efficient frameworks forgreedy split and new depth first search split procedures for routing problems. Computersand Operations Research, 38(4) :723 – 739, 2011.[4] Diego Pecin, Artur Pessoa, Marcus Poggi, and Eduardo Uchoa. Improved branch-cut-and-price for capacitated vehicle routing. Mathematical Programming Computation, pages 1–40,2016.[5] Artur Pessoa, Eduardo Uchoa, and Marcus Poggi de Aragão. A robust branch-cut-and-pricealgorithm for the heterogeneous fleet vehicle routing problem. Networks, 54(4) :167–177,2009.[6] Anand Subramanian, Puca Huachi Vaz Penna, Eduardo Uchoa, and Luiz Satoru Ochi. Ahybrid algorithm for the heterogeneous fleet vehicle routing problem. European Journal ofOperational Research, 221(2) :285 – 295, 2012.[7] Taillard, E. D. A heuristic column generation method for the heterogeneous fleet vrp.RAIRO-Oper. Res., 33(1) :1–14, 1999.

Improved Branch-Cut-and-Price algorithm for the Heterogeneous Vehicle Routing Problem

PESSOA, Artur

SADYKOV, Ruslan

UCHOA, Eduardo

Oskar Bordeaux

An improved Branch-Cut-and-Price algorithm for heterogeneous vehicle routing problems

HAL Id: hal-01676022https://hal.inria.fr/hal-01676022Submitted on 5 Jan 2018HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestinée au dépôt et à la diffusion de documentsscientifiques de niveau recherche, publiés ou non,émanant des établissements d’enseignement et derecherche français ou étrangers, des laboratoirespublics ou privés.An improved Branch-Cut-and-Price algorithm forheterogeneous vehicle routing problemsArtur Pessoa, Ruslan Sadykov, Eduardo UchoaTo cite this version:Artur Pessoa, Ruslan Sadykov, Eduardo Uchoa. An improved Branch-Cut-and-Price algorithm forheterogeneous vehicle routing problems. VeRoLog 2017 - Annual Workshop of the EURO WorkingGroup on Vehicle Routing and Logistics optimization, Jul 2017, Amsterdam, Netherlands. ￿hal-01676022￿An improved Branch-Cut-and-Price Algorithmfor Heterogeneous Vehicle Routing ProblemsArtur Pessoa3 Ruslan Sadykov1,2 Eduardo Uchoa31Inria Bordeaux,France2Université Bordeaux,France3Universidade FederalFluminense, BrazilVerolog 2017Amsterdam, July 111 / 23Heterogeneous Vehicle RoutingSet I of n customers, eachi ∈ I with demand di .Set U of vehicle types,each u ∈ U has a depotwith Ku vehicles ofcapacity Qu, with fixedcost fu and travel costs cuafor each edge a.Objective: minimize thetotal fixed and travel cost.VariantsI Multi-depot VRPI Site-dependent VRPInstance HVRPFV-20-1006 , 4 , 3Optimum 4760.68 (BKS 4761.26)2 / 23Set partitioning (master) formulationI Ru — set of q−routes feasible for a vehicle of type uI ari — number of times that customer i appears in route r .I cr — cost of route r .I Binary variable λru = 1 if and only if a vehicle of type uuses route rmin∑u∈U∑r∈Rucrλr∑u∈U∑r∈Ruari λr = 1, ∀i ∈ I,∑r∈Ruλr ≤ Ku, ∀u ∈ U,λr ∈ {0,1}, ∀r ∈ Ru,∀u ∈ U.3 / 23Pricing subproblem for a vehicle typex030x020x010x152x342x203x404x505q = 0            1            2            3            4           554321i = 0Figure: |I| = Q = 5, d1 = d3 = d4 = 2,d2 = d5 = 3; routes 0-1-5-0,0-2-0, 0-3-4-0 are shown4 / 23The labeling algorithm5 / 23The labeling algorithm5 / 23The labeling algorithm5 / 23The labeling algorithm5 / 23The labeling algorithm5 / 23Subset Row Cuts (SRCs)Given C ⊆ V+ and a multiplier p, the (C,p)-Subset Row Cut is:∑u∈U∑r∈Ru⌊p∑i∈Cari⌋λr ≤ bp|C|cSpecial case of Chvátal-Gomory rank-1 cuts obtained byrounding of |C| constraints in the masterEach cut adds an additional resource in the shortest pathpricing problemMads Jepsen and Bjorn Petersen and Simon Spoorendonk and David Pisinger(2008).Subset-Row Inequalities Applied to the Vehicle-Routing Problem with TimeWindows.Operations Research, 56(2):497–511.6 / 23Example of 3-Subset Row Cut, |C| = 3, p = 12123If λ1 = 0.5, λ2 = 0.5, and λ3 = 0.5, cut C = {1,2,3} is violated.7 / 23Example of 3-Subset Row Cut, |C| = 3, p = 12123Less possibilities for domination after adding the cut.7 / 23Example of 3-Subset Row Cut, |C| = 3, p = 12123add to memoryConcept of limited memory cuts [Pecin et al., 2017b].Pecin, D., Pessoa, A., Poggi, M., and Uchoa, E. (2017a).Improved branch-cut-and-price for capacitated vehicle routing.Mathematical Programming Computation, 9(1):61–100.7 / 23Example of 3-Subset Row Cut, |C| = 3, p = 12123add to memoryConcept of limited memory cuts [Pecin et al., 2017b].Pecin, D., Pessoa, A., Poggi, M., and Uchoa, E. (2017a).Improved branch-cut-and-price for capacitated vehicle routing.Mathematical Programming Computation, 9(1):61–100.7 / 23Example of 3-Subset Row Cut, |C| = 3, p = 12123add to memoryConcept of limited memory cuts [Pecin et al., 2017b].Pecin, D., Pessoa, A., Poggi, M., and Uchoa, E. (2017a).Improved branch-cut-and-price for capacitated vehicle routing.Mathematical Programming Computation, 9(1):61–100.7 / 23Example of 3-Subset Row Cut, |C| = 3, p = 12123memory setDashed partial path “forgot” the cut (cut state in the label is 0)⇒ larger domination probability.7 / 23Example of 3-Subset Row Cut, |C| = 3, p = 12123memory setDifferent memory setsI Node memoryI Arc memory [Pecin et al., 2017a]I Subproblem dependent memory (this work)7 / 23Example of 3-Subset Row Cut, |C| = 3, p = 12123Different memory setsI Node memoryI Arc memory [Pecin et al., 2017a]I Subproblem dependent memory (this work)Pecin, D., Contardo, C., Desaulniers, G., and Uchoa, E. (2017a).New enhancements for the exact solution of the vehicle routing problem with timewindows.INFORMS Journal on Computing, 29(3):489–502.7 / 23Example of 3-Subset Row Cut, |C| = 3, p = 12123Different memory setsI Node memoryI Arc memory [Pecin et al., 2017a]I Subproblem dependent memory (this work)Pecin, D., Contardo, C., Desaulniers, G., and Uchoa, E. (2017a).New enhancements for the exact solution of the vehicle routing problem with timewindows.INFORMS Journal on Computing, 29(3):489–502.7 / 23Arbitrary cuts of Rank 1Rounding using a vector p instead of single value:∑u∈U∑r∈Ru⌊∑i∈Cpiari⌋λr ≤⌊∑i∈Cpi⌋All facet-defining vectors p for cuts up to 5 rows[Pecin et al., 2017c]:I |C| = 1, p =( 12)I |C| = 3, p =( 12 ,12 ,12)I |C| = 4, p =( 23 ,13 ,13 ,13)I |C| = 5, p =( 13 ,13 ,13 ,13 ,13)I |C| = 5, p =( 24 ,24 ,14 ,14 ,14)I |C| = 5, p =( 34 ,14 ,14 ,14 ,14)I |C| = 5, p =( 35 ,25 ,25 ,15 ,15)I |C| = 5, p =( 12 ,12 ,12 ,12 ,12)I |C| = 5, p =( 23 ,23 ,13 ,13 ,13)I |C| = 5, p =( 34 ,34 ,24 ,24 ,14)Pecin, D., Pessoa, A., Poggi, M., Uchoa, E., and Santos, H. (2017).Limited memory rank-1 cuts for vehicle routing problems.Operations Research Letters, 45(3):206 – 209.8 / 23Extended Capacity CutsDefinitionAn Extended Capacity Cut (ECC) [Pessoa et al., 2009] oversubset C of customers is any inequality valid for P(C), thepolyhedron given by the convex hull of the 0− 1 solutions of∑u∈U ∑a∈δ−u (C)Q∑q=1dxqa −∑a∈δ+u (C)Q−1∑q=0dxqa = d(C)Pessoa, Artur and Uchoa, Eduardo and Poggi, Marcus (2009).A robust branch-cut-and-price algorithm for the heterogeneous fleet vehiclerouting problem.Networks, 54(4):167–177.9 / 23Homogeneous Extended Capacity Cutsym,q =∑aq∈δ−u (C)xqa , zq =∑aq∈δ+u (C)xm,qa , (q = 0, . . . ,Q).DefinitionA Homogeneous Extended Capacity Cut (HECC) over set C ofcustomers is any inequality valid for the polyhedron given bythe convex hull of the integral solutions of∑u∈U Q∑q=1dym,q −Q−1∑q=0dzm,q = d(C). (1)SeparationI Cuts obtained by applying integer rounding of (1).I Heuristic separation of [Pessoa et al., 2009] is used.10 / 23Labeling algorithm enhancementsI ng-routes to impose partial elementarity[Baldacci et al., 2011].I Bi-directional labelling [Righini and Salani, 2006]I Reduced cost fixing of subproblem arc variables x[Irnich et al., 2010]Baldacci, R., Mingozzi, A., and Roberti, R. (2011).New route relaxation and pricing strategies for the vehicle routing problem.Operations Research, 59(5):1269–1283.Righini, G. and Salani, M. (2006).Symmetry helps: Bounded bi-directional dynamic programming for theelementary shortest path problem with resource constraints.Discrete Optimization, 3(3):255 – 273.Irnich, S., Desaulniers, G., Desrosiers, J., and Hadjar, A. (2010).Path-reduced costs for eliminating arcs in routing and scheduling.INFORMS Journal on Computing, 22(2):297–313.11 / 23Elementary routes enumerationWe try to enumerate all elementary routes whose reduced costis smaller than the current gap [Baldacci et al., 2008], possibly toa pool [Contardo and Martinelli, 2014].This work contributionsI Subproblem dependent enumerationI If succeeded, a subproblem passes to the enumeratedstate:I Pricing is performed by inspectionI Cut coefficients of columns in the master are liftedBaldacci, R., Christofides, N., and Mingozzi, A. (2008).An exact algorithm for the vehicle routing problem based on the set partitioningformulation with additional cuts.Mathematical Programming, 115:351–385.12 / 23Lifting of cuts in enumerated stateRank-1 cutsLimited memory is extended to full memoryHomogeneous Extended Capacity CutsInteger rounding of∑u∈EU∑r∈Rudr (C)λr +∑u∈U\EU Q∑q=1dym,q −Q−1∑q=0dzm,q = d(C).For a particular rounding multiplier and EU = U (allsubproblems are in the enumerated state), equivalent to StrongCapacity Cuts [Baldacci et al., 2008].13 / 23Other enhancementsI Heuristic pricing (keeping one label per bucket)I Automatic dual price smoothing stabilization[Pessoa et al., 2017].I Rollback mechanism [Pecin et al., 2017b]Pessoa, A., Sadykov, R., Uchoa, E., and Vanderbeck, F. (2017).Automation and combination of linear-programming based stabilizationtechniques in column generation.INFORMS Journal on Computing, (Forthcoming).14 / 23BranchingStrong branching [Pecin et al., 2017b]I Multi-strategyI Branching history (pseudo-costs)I Multi-phaseBranching strategiesI Number of vehiclesI Assignment of customers to vehicle typesI Participation of arcs in routes15 / 23Results for classic Heterogeneous VRP instances40 instances with 50-100 customers by [Taillard, E. D., 1999]10s 1m 10m 1h 10h 36h17/2435/4040/40solved to optimality in at most X time (log scale)NumberofinstancesOur algorithm [Baldacci and Mingozzi, 2009] [Pessoa et al., 2009]Baldacci, R. and Mingozzi, A. (2009).A unified exact method for solving different classes of vehicle routing problems.Mathematical Programming, 120(2):347–380.16 / 23Results for larger Heterogeneous VRP instances10 instances with 100-200 customers [Brandao, 2011]1m 10m 1h 10h 36h8/10solved to optimality in at most X time (log scale)NumberofinstancesOur algorithmBrandao, J. (2011).A tabu search algorithm for the heterogeneous fixed fleet vehicle routing problem.Computers and Operations Research, 38(1):140 – 151.17 / 23Results for standard Site-Dependent VRP instancesInstances with 27-324 customers by [Cordeau and Laporte, 2001]10s 1m 10m 1h 10h 36h9/1321/23solved to optimality in at most X time (log scale)NumberofinstancesOur algorithm [Baldacci and Mingozzi, 2009]Largest solved instance has 216 customers[Baldacci and Mingozzi, 2009] solved only 1 of 5 instances with100 customers and more18 / 23Results for standard Multi-Depot VRP instancesInstances with 50–360 customers [Cordeau et al., 1997]10s 1m 10m 1h 10h7/910/1011/11solved to optimality in at most X time (log scale)NumberofinstancesOur algorithm [Baldacci and Mingozzi, 2009] [Contardo and Martinelli, 2014]Contardo, C. and Martinelli, R. (2014).A new exact algorithm for the multi-depot vehicle routing problem under capacityand route length constraints.Discrete Optimization, 12:129 – 146.19 / 23Impact of Extended Capacity CutsRank-1 cuts Root Root Nodes TotalCutsmemory gap time num.SolvedtimeR1C only arc, sp.dep 0.323% 93 67.8 87/90 181R1C+ECC node, sp.dep 0.133% 106 38.7 86/90 178R1C+ECC arc, sp.dep 0.105% 113 29.6 88/90 1701 2 5 10 20102030405060708090for which variant is at most X times slower than the bestNumberofinstancesRank-1 cuts only with arc memoryECC and rank-1 cuts with node memoryECC and rank-1 cuts with arc memory20 / 23Improved Best Known SolutionsPrevious ImprovedProblem Instance SizeBKSReferencevalueHVRP BrandaoN1fsmd 150 2212.77 [SPUS12] 2211.63BrandaoN1hd 150 2234.13 [S16] 2233.90BrandaoN2fsmd 199 2823.75 [SPUS12] 2810.20BrandaoN2hd 199 2859.82 [S16] 2851.94c100_20fsmf 100 4032.81 [SPUS12] 4029.61c100_20hvrp 100 4761.26 [SPUS12] 4760.68MDVRP n200-k16-3-80 200 1757.86 [BM09] 1756.48SDVRP p16 216 3393.55 [CM12] 3393.31p18 324 4751.27 [CM12] 4747.751p21 209 1263.71 [CM12] 1260.011 optimality is not proved, other values are optimal[SPUS12] [Subramanian et al., 2012][S16] [Subramanian, 2016][BM09] [Baldacci and Mingozzi, 2009][CM12] [Cordeau and Maischberger, 2012]21 / 23ContributionsI Large computational improvement over the state-of-the-artalgorithms for the problemI Showed importance of Extended Capacity CutsI New concept of subproblem dependent memory for rank-1cutsI New concept of “enumerated state” for pricingsubproblemsI New family of lifted Extended Capacity Cuts22 / 23Case of multiple and “non-discretizable” resourcesPresentation at IFORS in Quebec next weekA Bucket Graph Based Labelling Algorithm for the RCSPPwith Applications to Vehicle Routing23 / 23Case of multiple and “non-discretizable” resourcesPresentation at IFORS in Quebec next weekA Bucket Graph Based Labelling Algorithm for the RCSPPwith Applications to Vehicle RoutingA glimpse of the resultsI Solved 5/9 open VRPTW instances of[Gehring and Homberger, 2002] with 200 customersI Solved 6/7 distance-constrained CVRP instances of[Christofides et al., 1979] (CMT) with up to 200 customersI Solved all 22 distance-constrained MDVRP instances of[Cordeau et al., 1997] with up to 288 customersI Solved 7/10 distance-constrained SDVRP instances of[Cordeau and Laporte, 2001] with up to 216 customersI Solved 56/96 “nightmare” HFVRP instances of[Duhamel et al., 2011] with up to 186 customers.23 / 23References IBaldacci, R., Christofides, N., and Mingozzi, A. (2008).An exact algorithm for the vehicle routing problem based on the setpartitioning formulation with additional cuts.Mathematical Programming, 115:351–385.Baldacci, R. and Mingozzi, A. (2009).A unified exact method for solving different classes of vehicle routingproblems.Mathematical Programming, 120(2):347–380.Baldacci, R., Mingozzi, A., and Roberti, R. (2011).New route relaxation and pricing strategies for the vehicle routingproblem.Operations Research, 59(5):1269–1283.Brandao, J. (2011).A tabu search algorithm for the heterogeneous fixed fleet vehicle routingproblem.Computers and Operations Research, 38(1):140 – 151.24 / 23References IIChristofides, N., Mingozzi, A., and Toth, P. (1979).Combinatorial Optimization, chapter The vehicle routing problem, pages315–338.Wiley, Chichester.Contardo, C. and Martinelli, R. (2014).A new exact algorithm for the multi-depot vehicle routing problem undercapacity and route length constraints.Discrete Optimization, 12:129 – 146.Cordeau, J.-F., Gendreau, M., and Laporte, G. (1997).A tabu search heuristic for periodic and multi-depot vehicle routingproblems.Networks, 30(2):105–119.Cordeau, J.-F. and Laporte, G. (2001).A tabu search algorithm for the site dependent vehicle routing problemwith time windows.INFOR: Information Systems and Operational Research, 39(3):292–298.25 / 23References IIICordeau, J.-F. C. and Maischberger, M. (2012).A parallel iterated tabu search heuristic for vehicle routing problems.Computers and Operations Research, 39(9):2033 – 2050.Duhamel, C., Lacomme, P., and Prodhon, C. (2011).Efficient frameworks for greedy split and new depth first search splitprocedures for routing problems.Computers and Operations Research, 38(4):723 – 739.Gehring, H. and Homberger, J. (2002).Parallelization of a two-phase metaheuristic for routing problems withtime windows.Journal of Heuristics, 8(3):251–276.Irnich, S., Desaulniers, G., Desrosiers, J., and Hadjar, A. (2010).Path-reduced costs for eliminating arcs in routing and scheduling.INFORMS Journal on Computing, 22(2):297–313.26 / 23References IVPecin, D., Contardo, C., Desaulniers, G., and Uchoa, E. (2017a).New enhancements for the exact solution of the vehicle routing problemwith time windows.INFORMS Journal on Computing, 29(3):489–502.Pecin, D., Pessoa, A., Poggi, M., and Uchoa, E. (2017b).Improved branch-cut-and-price for capacitated vehicle routing.Mathematical Programming Computation, 9(1):61–100.Pecin, D., Pessoa, A., Poggi, M., Uchoa, E., and Santos, H. (2017c).Limited memory rank-1 cuts for vehicle routing problems.Operations Research Letters, 45(3):206 – 209.Pessoa, A., Sadykov, R., Uchoa, E., and Vanderbeck, F. (2017).Automation and combination of linear-programming based stabilizationtechniques in column generation.INFORMS Journal on Computing, (Forthcoming).27 / 23References VPessoa, A., Uchoa, E., and Poggi de Aragão, M. (2009).A robust branch-cut-and-price algorithm for the heterogeneous fleetvehicle routing problem.Networks, 54(4):167–177.Righini, G. and Salani, M. (2006).Symmetry helps: Bounded bi-directional dynamic programming for theelementary shortest path problem with resource constraints.Discrete Optimization, 3(3):255 – 273.Subramanian, A. (2016).Personal communication.Subramanian, A., Penna, P. H. V., Uchoa, E., and Ochi, L. S. (2012).A hybrid algorithm for the heterogeneous fleet vehicle routing problem.European Journal of Operational Research, 221(2):285 – 295.Taillard, E. D. (1999).A heuristic column generation method for the heterogeneous fleet vrp.RAIRO-Oper. Res., 33(1):1–14.28 / 23

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Improved Branch-Cut-and-Price algorithm for the Heterogeneous Vehicle Routing Problem

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Oskar Bordeaux

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