This paper addresses the robust vehicle routing problem with time windows. We are motivated by a problem that arises in maritime transportation where delays are frequent and should be taken into account. Our model only allows routes that are feasible for all values of the travel times in a predetermined uncertainty polytope, which yields a robust optimization problem. We propose two new formulations for the robust problem, each based on a different robust approach. The first formulation extends the well-known resource inequalities formulation by employing adjustable robust optimization. We propose two techniques, which, using the structure of the problem, allow to reduce significantly the number of extreme points of the uncertainty polytope. The second formulation generalizes a path inequalities formulation to the uncertain context. The uncertainty appears implicitly in this formulation, so that we develop a new cutting plane technique for robust combinatorial optimization problems with complicated constraints. In particular, efficient separation procedures are discussed. We compare the two formulations on a test bed composed of maritime transportation instances. These results show that the solution times are similar for both formulations while being significantly faster than the solutions times of a layered formulation recently proposed for the problem.

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Abstract. This paper studies the vehicle routing problem with time windows where travel times are uncertain and belong to a predetermined polytope. The objective of the problem is to find a set of routes that services all nodes of the graph and that are feasible for all values of the travel times in the uncertainty polytope. The problem is motivated by maritime transportation where delays are frequent and must be taken into account. We present an extended formulation for the vehicle routing problem with time windows that allows us to apply the classical (static) robust programming approach to the problem. The formulation is based on a layered representation of the graph, which enables to track the position of each arc in its route. We test our formulation on a test bed composed of maritime transportation instances.

We present a new Lagrangean relaxation for the hop-constrained minimum spanning tree problem (HMST). The HMST is NP-hard and models the design of centralized telecommunication networks with quality of service constraints. The linear programming (LP) relaxation of a hop-indexed¯ow-based model recently presented in the literature (see Gouveia, L., 1998. Using variable rede®nition for computing lower bounds for minimum spanning and Steiner trees with hop constraints. INFORMS Journal on Computing 10, 180±188) produces very tight bounds but has the disadvantage of being very time consuming, especially for dense graphs. In this paper, we present a new Lagrangean relaxation which is derived from the hop-indexed¯ow based formulation. Our computational results indicate that the lower bounds given by the new relaxation dominate the lower bounds given by previous Lagrangean relaxations. Our results also show that for dense graphs the new Lagrangean relaxation proves to be a reasonable alternative to solving the LP relaxation of the hop-indexed model. Ó