Abstract
The Vehicle Routing Problem with Time Windows consists of computing a minimum cost set of routes for a fleet of vehicles of limited capacity visiting a given set of customers with known demand, with the additional constraint that each customer must be visited in a specified time window. We consider the case in which time window constraints are relaxed into “soft” constraints, that is penalty terms are added to the solution cost whenever a vehicle serves a customer outside of his time window. We present a branch-and-price algorithm which is the first exact optimization algorithm for this problem.
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References
Balakrishnan N (1993) Simple heuristics for the vehicle routeing problem with soft time windows. J Oper Res Soc 44: 279–287
Baldacci R, Mingozzi A, Roberti R (2009) An exact method for the vehicle routing problem with time windows. In: 20 th International symposium of mathematical programming, Chicago
Boland N, Dethridge J, Dumitrescu I (2006) Accelerated label setting algorithms for the elementary resource constrained shortest path problem. Oper Res Lett 34: 58–68
Bramel J, Simchi-Levi D (2001) The vehicle routing problem, volume 9 of SIAM monographs on discrete mathematics and applications, chapter set-covering-based algorithms for the capacitated VRP. SIAM: Society for Industrial and Applied Mathematics, Philadelphia
Chiang WC, Russell RA (2004) A metaheuristic for the vehicle-routeing problem with soft time windows. J Oper Res Soc 55: 1298–1310
Cordeau JF, Desaulniers G, Desrosiers J, Solomon MM, Soumis F (2001) The vehicle routing problem, volume 9 of SIAM monographs on discrete mathematics and applications, chapter the VRP with time windows. SIAM: Society for Industrial and Applied Mathematics, Philadelphia
Danna E, Le Pape C (2005) Column generation, chapter branch-and-price heuristics: a case study on the vehicle routing problem with time windows. Springer, New York
Dell’Amico M, Righini G, Salani M (2006) A branch-and-price approach to the vehicle routing problem with simultaneous distribution and collection. Transport Sci 40: 235–247
Desaulniers G, Lessard F, Hadjar A (2008) Tabu search, partial elementarity, and generalized k-path inequalities for the vehicle routing problem with time windows. Transport Sci 42: 387–404
Desrochers M, Desrosiers J, Solomon MM (1992) A new optimization algorithm for the vehicle routing problem with time windows. Oper Res 40: 342–354
Dror M (1994) Note on the complexity of the shortest path models for column generation in vrptw. Oper Res 42: 977–978
Feillet D, Dejax P, Gendreau M, Gueguen C (2004) An exact algorithm for the elementary shortest path problem with resource constraints: application to some vehicle routing problems. Networks 44: 216–229
Ibaraki T, Imahori S, Kubo M, Masuda T, Uno T, Yagiura M (2005) Effective local search algorithms for routing and scheduling problems with general time-window constraints. Transport Sci 39: 206–232
Ioachim I, Gélinas S, Soumis F, Desrosiers J (1998) A dynamic programming algorithm for the shortest path problem with time windows and linear node costs. Networks 31: 193–204
Jepsen M, Petersen B, Spoorendonk S, Pisinger D (2008) Introducing subset-row inequalities in a branch-and-cut-and-price algorithm for the vehicle routing problem with time windows. Oper Res 56: 497–511
Kallehauge B, Larsen J, Madsen OBG, Solomon MM (2005) Column generation, chapter Vehicle routing problem with time windows. Springer, New York
Kohl N, Desrosiers J, Madsen OBG, Solomon MM, Soumis F (1999) 2-path cuts for the vehicle routing problem with time windows. Transport Sci 33: 101–116
Koskosidis YA, Powell WB, Solomon MM (1992) An optimization based heuristic for vehicle routing and scheduling with soft time window constraints. Transport Sci 26: 69–85
Righini G, Salani M (2006) Symmetry helps: bounded bi-directional dynamic programming for the elementary shortest path problem with resource constraints. Discrete Optim 3: 255–273
Righini G, Salani M (2008a) Decremental state space relaxation strategies and initialization heuristics for solving the orienteering problem with time windows with dynamic programming. Comput Oper Res 36: 1191–1203
Righini G, Salani M (2008b) New dynamic programming algorithms for the resource-constrained elementary shortest path problem. Networks 51: 155–170
Salani M (2006) Branch-and-price algorithms for vehicle routing problems. PhD thesis, University of Milan
Sexton T, Choi Y (1986) Pickup and delivery of partial loads with soft time windows. Am J Math Manag Sci 6: 369–398
Solomon MM (1983) Vehicle routing and scheduling with time windows constraints: models and algorithms. PhD thesis, University of Pennsylvania
Taillard E, Badeau P, Gendreau M, Guertin F, Potvin JY (1997) A tabu search heuristic for the vehicle routing problem with soft time windows. Transport Sci 31: 170–186
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Liberatore, F., Righini, G. & Salani, M. A column generation algorithm for the vehicle routing problem with soft time windows. 4OR-Q J Oper Res 9, 49–82 (2011). https://doi.org/10.1007/s10288-010-0136-6
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DOI: https://doi.org/10.1007/s10288-010-0136-6