Discrete Optimization
An exact algorithm for a single-vehicle routing problem with time windows and multiple routes

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Abstract

This paper describes an exact algorithm for solving a problem where the same vehicle performs several routes to serve a set of customers with time windows. The motivation comes from the home delivery of perishable goods, where vehicle routes are short and must be combined to form a working day. A method based on an elementary shortest path algorithm with resource constraints is proposed to solve this problem. The method is divided into two phases: in the first phase, all non-dominated feasible routes are generated; in the second phase, some routes are selected and sequenced to form the vehicle workday. Computational results are reported on Euclidean problems derived from benchmark instances of the classical vehicle routing problem with time windows.

Introduction

In this work, we consider a variant of the vehicle routing problem with time windows (VRPTW) where the same vehicle can perform several routes during its workday. Surprisingly, this problem has received little attention in the literature in spite of its importance in practice. For example, in the home delivery of perishable goods, like food, routes are of short duration and must be combined to form a complete workday. We believe that this type of problem will become increasingly important in the future with the advent of electronic services, like e-groceries, where customers can order goods through the internet and have them delivered at home.

The vehicle routing problem with multiple uses of vehicles, but no time windows, has been addressed through heuristic means in [7], [14]. In [14], different solutions to the classical vehicle routing problem are generated using a tabu search heuristic. The routes obtained are then combined to produce workdays for the vehicles by solving a bin packing problem, an idea previously introduced in [7]. A recent work in [3] reports about insertion heuristics that can efficiently handle different types of constraints including time windows and multiple uses of vehicles. In [2], the authors introduce the home delivery problem, which is more closely related to real-world applications. Here, a probability of occurrence and a revenue are associated with each potential customer. When a new request occurs, a decision to accept or reject it must be taken in real-time, and a time window for service is determined. Although vehicle routes are generated and used to decide about the acceptance or rejection of a particular request, the “real” routes are executed later. Logistics and socio-economic considerations about different types of home delivery problems, with a particular emphasis on electronic groceries, can also be found in [8], [9], [10], [11], [15].

In this paper, an exact algorithm for solving a single-vehicle routing problem with time windows and multiple routes is reported. To the best of our knowledge, this is the first time that an exact algorithm is devised for this kind of problem. The outline of the paper is as follows. In Section 2, a mathematical programming formulation is proposed. The problem-solving approach, based on an elementary shortest path algorithm with resource constraints, is then presented in Section 3. Computational results on problem instances derived from Solomon’s VRPTW testbed [13] are reported in Section 4. Finally, concluding remarks follow in Section 5.

Section snippets

Problem formulation

The problem considered can be stated as follows. We have a single vehicle of capacity Q delivering perishable goods from a depot to a set of customer nodes N = {1, 2,  , n} in a complete directed graph with arc set A. A distance dij and a travel time tij are associated with every arc (i, j)  A. Each customer i  N is characterized by a demand qi, a service or dwell time si and a time window [ai, bi], where ai is the earliest time to begin service and bi the latest time. Accordingly, the vehicle must wait

Problem-solving approach

The problem is addressed via an approach that exploits an elementary shortest path algorithm with resource constraints, noted FDGG in the following [6]. The latter extends Desrochers’ algorithm [5] for the shortest path problem with resource constraints to generate elementary paths only. After briefly introducing FDGG, our problem-solving methodology is described in the next subsections.

Computational results

We have used the classical VRPTW instances of Solomon [13] for this study. Those are 100-customer Euclidean problems where distances and travel times are the same. There are six different classes of problems depending on the geographic location of customers (R: random; C: clustered; RC: mixed) and length of scheduling horizon (1: short horizon; 2: long horizon). In this study, problems of type 1 have been discarded because the short horizon does not allow the vehicle to serve many routes during

Conclusion

This paper has described an exact algorithm for solving a routing problem where a vehicle performs several routes over the scheduling horizon. The results indicate that this algorithm is very sensitive to the deadline constraint. When this constraint is not tight enough, the number of feasible routes “explodes” and becomes too large to allow the algorithm to produce a solution. Future developments will now be aimed at considering problems that are closer to real-world applications. First, a

Acknowledgements

Financial support for this work was provided by the Canadian Natural Sciences and Engineering Research Council (NSERC) and by the Quebec Fonds pour la Formation de Chercheurs et l’Aide à la Recherche (FCAR). This support is gratefully acknowledged.

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