Unlike its deterministic counterpart, static and stochastic vehicle routing problems (SS-VRP) aim at modeling and solving real-life operational problems by considering uncertainty on data. We consider the SS-VRPTW-CR introduced in Saint- (2017). Like the SS-VRP introduced by Bertsimas (1992), we search for optimal first stage routes for a fleet of vehicles to handle a set of stochastic customer demands, i.e., demands are uncertain and we only know their probabilities. In addition to capacity constraints, customer demands are also constrained by time windows. Unlike all SS-VRP variants, the SS-VRPTW-CR does not make any assumption on the time at which a stochastic demand is revealed, i.e., the reveal time is stochastic as well. To handle this new problem, we introduce waiting locations: Each vehicle is assigned a sequence of waiting locations from which it may serve some associated demands, and the objective is to minimize the expected number of demands that cannot be satisfied in time. In this paper, we propose two new recourse strategies for the SS-VRPTW-CR, together with their closed-form expressions for efficiently computing their expectations: The first one allows us to take vehicle capacities into account; The second one allows us to optimize routes by avoiding some useless trips. We propose two algorithms for searching for routes with optimal expected costs: The first one is an extended branch-and-cut algorithm, based on a stochastic integer formulation, and the second one is a local search based heuristic method. We also introduce a new public benchmark for the SS-VRPTW-CR, based on real-world data coming from the city of Lyon. We evaluate our two algorithms on this benchmark and empirically demonstrate the expected superiority of the SS-VRPTW-CR anticipative actions over a basic "wait-and-serve" policy.
@cite_45 introduced the Dial-a-Ride Problem (DARP) with stochastic customer delays. The DARP is a generalization of the VRPTW that distinguishes between pickup and delivery locations and involves customer ride time constraints. Each customer is present at its pickup location with a stochastic delay. A customer is then skipped if it is absent when the vehicle visits the corresponding location, involving the cost of fulfilling the request by an alternative service (, a taxi). In a sense, stochastic delays imply that each request is revealed at some uncertain time during the planning horizon. That study is thus related to our problem, although in the SS-VRPTW-CR only a subset of the requests are actually revealed. Similarly, @cite_51 studied a probabilistic DARP where a priori routes are modified by removing absent customers at the beginning of the day, and proposed local search based heuristics.
@cite_45 @cite_51
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This paper considers a single-vehicle Dial-a-Ride Problem in which customers may experience stochastic delays at their pickup locations. If a customer is absent when the vehicle serves the pickup location, the request is fulfilled by an alternative service (e.g., a taxi) whose cost is added to the total cost of the tour. In this case, the vehicle skips the corresponding delivery location, which yields a reduction in the total tour cost. The aim of the problem is to determine an a priori Hamiltonian tour minimizing the expected cost of the solution. This problem is solved by means of an integer L-shaped algorithm. Computational experiments show that the algorithm yields optimal solutions on several instances within reasonable CPU times. It is also shown that the actual cost of an optimal solution obtained with this algorithm can be significantly smaller than that of an optimal solution obtained with a deterministic formulation. This paper introduces the probabilistic dial-a-ride problem, and describes an efficient request-relocation neighborhood evaluation procedure for the problem. The running time of the procedure is @math , compared to @math for a straightforward approach. For solving the problem we embed the suggested evaluation procedure in a pure local search heuristic and in a tabu search heuristic. The quality of the solutions obtained by the two heuristics have been compared experimentally. Computational results confirm that our neighborhood evaluation technique is much faster than the straightforward one, and for cases with 144 users and 4 vehicles it is demonstrated that the computation time can be reduced by a factor larger than 27.