Cattaruzza, Diego

Gianessi, Paolo

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HAL Id: hal-01651581https://hal.archives-ouvertes.fr/hal-01651581Submitted on 29 Nov 2017HAL 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.A Branch&Cut algorithm for the Multi-Trip VehicleRouting Problem with Time WindowsDiego Cattaruzza, Paolo GianessiTo cite this version:Diego Cattaruzza, Paolo Gianessi. A Branch&Cut algorithm for the Multi-Trip Vehicle RoutingProblem with Time Windows. ROADEF2017 - 18ème Conférence de la Société Française de RechercheOpérationnelle et d’Aide à la Décision, Feb 2017, Metz, France. ￿hal-01651581￿A Branch&Cut algorithm for the Multi-Trip Vehicle RoutingProblem with Time WindowsDiego Cattaruzza1, Paolo Gianessi21 Université Lille, CNRS, Centrale Lille, Inria, UMR 9189 - CRIStALCentre de Recherche en Informatique Signal et Automatique de LilleF-59000 Lille, Francediego.cattaruzza@ec-lille.fr2 Mines Saint-Étienne, FAYOL-EMSE, LIMOS, UMR CNRS 6158, Francepaolo.gianessi@emse.frKeywords : Multi-Trip VRP, Time Windows, Branch&Cut, Exact Methods1 IntroductionThe Vehicle Routing Problem with Time Windows (VRPTW) deals with the determinationof the trips of a homogeneous fleet of capacitated vehicles based at a central depot to deliver aset of customers, while complying delivery time intervals imposed by these latter. The Multi-Trip VRP with Time Windows (MTVRPTW) goes one step further by allowing vehicles toperform not one but a sequence of trips, called a journey, under a maximum overall shift lengthconstraint. Moreover, the time to load a vehicle is service-dependent, i.e. proportional to thetotal service time of the subsequent trip. Hence, the MTVRPTW consists in finding a set ofback-to-depot delivery trips and assigning them to vehicles so that the total travelled distanceis minimized and :Ieach customer is visited exactly once and service starts within the associated time window ;Ivehicle capacity is respected ;I trips assigned to one vehicle do not overlap in time and do not exceed a maximum duration.2 Short literature reviewRecently, MTVRPTW has got the attention of scholars mostly due to its application to citylogistics (see [4]). However, as far as we are aware of, the literature of exact methods is stillscarce. In [6], an algorithm based on the Branch&Price paradigm is proposed which can solvemost of the adapted Solomon instances (derived from the benchmark VRPTW instances firstproposed in [8]) with 25 customers and 2 vehicles. In [5], the same authors study a variantwhere the length of a trip is limited. Another Branch&Price algorithm is proposed in [1], whereanother variant of the problem is tackled where the visit of customers is not mandatory butrewarded with a profit. Finally, [7] proposes an exact algorithm for a variant with both profitsand limited trip duration.3 Solution methodWe propose a two-index MILP formulation for the MTVRPTW that makes use of base andreplenishment arcs. The former model the direct connection between two nodes, whereas thelatter imply a reload operation between two client nodes. Replenishment arcs, which have beenused e.g. in [3], allow to represent a journey as an elementary path with both endpoints in theinstance %r Tr TtC201_2_25 1.41 1.8 7.5C202_2_25 2.30 6.5 616.8C203_2_25 0.91 7.3 930.1C204_2_25 0.22 9.3 100.6C205_2_25 4.35 2.7 71.3C206_2_25 2.22 4.1 221.3C207_2_25 0.08 4.5 3.2C208_2_25 0.73 3.9 379.9average 1.53 5.0 291.3TAB. 1 – Results on clustered instances with 25 customers and 2 vehicles.depot. This ensures the connectivity of journeys by separating Subtour Elimination Constraintson a transformation of the graph. Further sets of two-indexed variables impose time windows,shift length, and service-dependent loading time constraints. The use of classical capacityconstraints to enforce the load limit on vehicles leads to a Branch&Cut algorithm. In order totighten the lower bound obtained from the linear relaxation of the proposed MILP model, weconsider connectivity and path valid inequalities. The latter are an adaptation to our case ofthose found in [2] : an exact procedure based on LP is proposed to efficiently separate them.4 Computational resultsTests have been conducted on a set of 54 adapted Solomon instances with 25 to 50 customersand a fleet of 2 to 4 vehicles. The algorithm is coded in C++ and makes use of CPLEX 12.6.1.Experiments are run on a Intel Core i7-5500U 2.4 Ghz machine with 15.56Gb RAM.We allocate30s to solve the root node and extra 1200s after branching on it.Table 1 shows the results on the class C (clustered) instances of of [4] with 25 customers. Forthese instances, the optimal value (which is known from [6]) is always found. Tr and %r reportthe root computation time and the gap of the root lower bound with respect to the optimalvalue, while Tt is the additional running time to explore the Branch&Bound tree.References[1] N.Azi, M.Gendreau, J.-Y.Potvin, An exact algorithm for a vehicle routing problem with time windowsand multiple use of vehicles. European Journal of Operational Research, 202(3): 756–763, 2010.[2] J.F.Bard, G.Kontoravdis, G.Yu, A Branch-and-Cut Procedure for the Vehicle Routing Problem with TimeWindows. Transportation Science, 36(2): 250–269, 2002.[3] O.J.Smith, N.Boland, H.Waterer, Solving shortest path problems with a weight constraint and replenish-ment arcs. Computers&Operations Research, 39(5): 964–984, 2012.[4] D.Cattaruzza, N.Absi, D.Feillet, Vehicle routing problems with multiple trips. 4OR A quarterly Journalof Operations Research, 14(3): 223–259, 2016.[5] F.Hernandez, D.Feillet, R.Giroudeau, O.Naud, A new exact algorithm to solve the multi-trip vehicle rou-ting problem with time windows and limited duration.. 4OR A quarterly Journal of Operations Research,12(3): 235–259, 2014.[6] F.Hernandez, D.Feillet, R.Giroudeau, O.Naud, Branch-and-price algorithms for the solution of the multi-trip vehicle routing problem with time windows.. European Journal of Operational Research, 249(2): 551–559, 2016.[7] R.Macedo, C.Alves, J.M.Valério de Carvalho, F.Clautiaux, S.Hanafi, Solving the vehicle routing problemwith time windows and multiple routes exactly using a pseudo-polynomial model.. European Journal ofOperational Research, 214(3): 536–545, 2011.[8] M.Solomon, Algorithms for the vehicle routing and scheduling problem with time window constraints..Operations Research, 35(2): 254–265, 1987.

A Branch&Cut algorithm for the Multi-Trip Vehicle Routing Problem with Time Windows

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A Branch&Cut algorithm for the Multi-Trip Vehicle Routing Problem with Time Windows

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