A hybrid metaheuristic for the vehicle routing problem with stochastic demand and duration constraints

Abstract

The vehicle routing problem with stochastic demands (VRPSD) consists in designing optimal routes to serve a set of customers with random demands following known probability distributions. Because of demand uncertainty, a vehicle may arrive at a customer without enough capacity to satisfy its demand and may need to apply a recourse to recover the route’s feasibility. Although travel times are assumed to be deterministic, because of eventual recourses the total duration of a route is a random variable. We present two strategies to deal with route-duration constraints in the VRPSD. In the first, the duration constraints are handled as chance constraints, meaning that for each route, the probability of exceeding the maximum duration must be lower than a given threshold. In the second, violations to the duration constraint are penalized in the objective function. To solve the resulting problem, we propose a greedy randomized adaptive search procedure (GRASP) enhanced with heuristic concentration (HC). The GRASP component uses a set of randomized route-first, cluster-second heuristics to generate starting solutions and a variable-neighborhood descent procedure for the local search phase. The HC component assembles the final solution from the set of all routes found in the local optima reached by the GRASP. For each strategy, we discuss extensive computational experiments that analyze the impact of route-duration constraints on the VRPSD. In addition, we report state-of-the-art solutions for a established set of benchmarks for the classical VRPSD.

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Notes

  1. 1.

    The mechanism has been given different names, but we believe the term heuristic concentration best encapsulates the spirit of the idea.

  2. 2.

    SA was tested on Intel Xeon X5660 2.8 GHz processors with 12Gb of RAM (running CentOS 5.3), MSH was tested on a PC with an Intel Xeon 2.4 GHz and 12 Gb of RAM (running Windows Server 2008 64 bit).

  3. 3.

    In fact, customer 2 violates one of the basic assumptions of the problem since \(Pr(\tilde{\xi }_{2}>Q)=0.1573\). Because of the high failure probability and the travel time to the depot, it is impossible to include customer 2 in a route, even the trivial route \((0,2,0)\), without violating the DC for \(\beta <0.1573\).

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Acknowledgments

This research was partially funded by the Region Pays de la Loire (France) through project LigéRO, Universidad de Antioquia (Colombia) through project CODI MDC11-01-09, and École Polytechnique de Montréal (Canada). The authors would like to thank Charles Gauvin at CIRRELT (Montreal) for providing the optimal solutions for the VRSPD instances used in Sect. 4.1.

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Correspondence to Jorge E. Mendoza.

Appendices

Appendix 1: Detailed results for VRPSD instances

See Table 8

Table 8 Results for the Christiansen and Lysgaard (2007) instances

Appendix 2: Detailed CPU times

See Table 9

Table 9 Average running times (in seconds) over ten runs of GRASP + HC for the different VRPSD-DC formulations

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Mendoza, J.E., Rousseau, LM. & Villegas, J.G. A hybrid metaheuristic for the vehicle routing problem with stochastic demand and duration constraints. J Heuristics 22, 539–566 (2016). https://doi.org/10.1007/s10732-015-9281-6

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Keywords

  • Distance-constrained vehicle routing problem
  • Vehicle routing problem with stochastic demands
  • Two-stage stochastic programming
  • GRASP
  • Heuristic concentration