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Lower and upper bounds for the m-peripatetic vehicle routing problem

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The m-Peripatetic Vehicle Routing Problem (m-PVRP) consists in finding a set of routes of minimum total cost over m periods so that two customers are never sequenced consecutively during two different periods. It models for example money transports or cash machines supply, and the aim is to minimize the total cost of the routes chosen. The m-PVRP can be considered as a generalization of two well-known NP-hard problems: the Vehicle Routing Problem (VRP or 1-PVRP) and the m-Peripatetic Salesman Problem (m-PSP). In this paper we discuss some complexity results of the problem before presenting upper and lower bounding procedures. Good results are obtained not only on the m-PVRP in general, but also on the VRP and the m-PSP using classical VRP instances and TSPLIB instances.

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  • Augerat P (1995) Approche polyédrale du problème de tournées de véhicules, Ph.D. thesis, Institut National Polytechnique de Grenoble, France

  • Baldacci R, Christofides N, Mingozzi A (2008) An exact algorithm for the vehicle routing problem based on the set partitioning formulation with additional cuts. Mathe Program 115(2): 351–385

    Article  Google Scholar 

  • Baum EB (1986) Towards practical “neural” computation for combinatorial optimization problems. In: Denker J (ed) AIP conferences proceedings on Neural Networks for Computing, vol 151. Utah, pp 53–58

  • Christofides N, Mingozzi A, Toth P (1979) The vehicle routing problem. In: Christofides N, Mingozzi A, Toth P, Sandi C (eds) Combinatorial optimization. Wiley, Chichester, pp 315–338

    Google Scholar 

  • Clarke G, Wright JW (1964) Scheduling of vehicles from a central depot to a number of delivery points. Oper Res 12(4): 568–581

    Article  Google Scholar 

  • Cordeau J-F, Gendreau M, Hertz A, Laporte G, Sormany J-S (2005) New heuristics for the vehicle routing problem. In: Langevin A, Riopel D (eds) Logistics systems: design and optimization. Springer, New York, pp 279–297

    Chapter  Google Scholar 

  • Cowling PI, Keuthen R (2005) Embedded local search approaches for routing optimization. Comput Oper Res 32(3): 465–490

    Article  Google Scholar 

  • De Kort JBJM (1991) Lower bounds for symmetric k-peripatetic salesman problems. Optimization 22(1): 113–122

    Article  Google Scholar 

  • De Kort JBJM (1993) A branch-and-bound algorithm for symmetric 2-peripatetic salesman problems. Eur J Oper Res 70(2): 229–243

    Article  Google Scholar 

  • Duchenne E, Laporte G, Semet F (2005) Branch-and-cut algorithms for the undirected m-peripatetic salesman problem. Eur J Oper Res 162(3): 700–712

    Article  Google Scholar 

  • Duchenne E, Laporte G, Semet F (2007) The undirected m-peripatetic salesman problem: Polyhedral results and new algorithms. Oper Res 55(5): 949–965

    Article  Google Scholar 

  • Edmonds J (1965) Paths, trees, and flowers. Can J Mathe 17: 449–467

    Article  Google Scholar 

  • Glover F (1986) Future paths for integer programming and links to artificial intelligence. Comput Oper Res 13(5): 533–549

    Article  Google Scholar 

  • Glover F, Laguna M (1993) Tabu search. In: Reeves CR (eds) Modern heuristic techniques for combinatorial problems. Blackwell, Oxford, pp 70–150

    Google Scholar 

  • Held M, Karp RM (1970) The traveling salesman problem and the minimum spanning trees. Oper Res 18(6): 1138–1162

    Article  Google Scholar 

  • Krarup J (1975) The peripatetic salesman and some related unsolved problems. In: Roy B (eds) Combinatorial programming methods and applications. Reidel, Dordrecht, pp 173–178

    Google Scholar 

  • Kruskal JB (1956) On the shortest spanning subtree of a graph and the traveling salesman problem. Proc Am Mathe Soc 7(1): 48–50

    Article  Google Scholar 

  • Miller DL, Wright J (1995) A matching based exact algorithm for capacitated vehicle routing problems. ORSA J Comput 7(1): 298–320

    Google Scholar 

  • Ngueveu SU, Prins C, Wolfler Calvo R (2008) Bornes supérieures et inférieures pour le probléme de tournées de véhicules m-péripatétiques. In: S. e. a. Lamouri (ed) Actes de la 7ème conférence internationale de MOdélisation et SIMulation vol 3. Paris, pp 1617–1625

  • Prins C (2004) A simple and effective evolutionary algorithm for the vehicle routing problem. Comput Oper Res 31(12): 1985–2002

    Article  Google Scholar 

  • Reinelt G (1995)

  • Roskind J, Tarjan RE (1985) A note on finding minimum-cost edge disjoint spanning trees. Mathe Oper Res 10(4): 701–708

    Article  Google Scholar 

  • Taillard ED (1993) Parallel iterative search methods for vehicle routing problems. Networks 23(8): 661–673

    Article  Google Scholar 

  • Toth P, Vigo D (2002) The vehicle routing problem. SIAM, Philadelphia

    Google Scholar 

  • Toth P, Vigo D (2003) The granular tabu search (and its application to the vehicle routing problem). INFORMS J Comput 15(4): 333–346

    Article  Google Scholar 

  • Wolfler Calvo R, Cordone R (2003) A heuristic approach to the overnight security service problem. Comput Oper Res 30(9): 1269–1287

    Article  Google Scholar 

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Correspondence to Sandra Ulrich Ngueveu.

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Ngueveu, S.U., Prins, C. & Wolfler Calvo, R. Lower and upper bounds for the m-peripatetic vehicle routing problem. 4OR-Q J Oper Res 8, 387–406 (2010).

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