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.

No abstract

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

Abstract-We develop a genetic algorithm for the topological design of survivable optical transport networks with minimum capital expenditure. Using the developed genetic algorithm we can obtain near-optimal topologies in a short time. The quality of the obtained solutions is assessed using an integer linear programming model. Two initial population generators, two selection methods, two crossover operators, and two population sizes are analyzed. Computational results obtained using real telecommunications networks show that by using an initial population that resembles real optical transport networks a good convergence is achieved.

In a previous article, using underlying graph theoretical properties, Gouveia and Magnanti (2003) described several network flow-based formulations for diameter-constrained tree problems. Their computational results showed that, even with several enhancements, models for situations when the tree diameter D is odd proved to be more difficult to solve than those when D is even. In this article we provide an alternative modeling approach for the situation when D is odd. The approach views the diameter-constrained minimum spanning tree as being composed of a variant of a directed spanning tree (from an artificial root node) together with two constrained paths, a shortest and a longest path, from the root node to any node in the tree. We also show how to view the feasible set of the linear programming relaxation of the new formulation as the intersection of two integer polyhedra, a so-called triangle-tree polyhedron and a constrained path polyhedron. This characterization improves upon a model of Gouveia and Magnanti (2003) whose linear programming relaxation feasible set is the intersection of three rather than two integer polyhedra. The linear programming gaps for the tightened model are very small, typically less than 0.5%, and are usually one third to one tenth of the gaps of the best previous model described in Gouveia and Magnanti (2003). Moreover, using the new model, we have been able to solve large Euclidean problem instances that are not solvable by the previous approaches.

No abstract

The p-median problem seeks for the location of p facilities on the vertices (customers) of a graph to minimize the sum of transportation costs for satisfying the demands of the customers from the facilities. In many real applications of the p-median problem the underlying graph is disconnected. That is the case of p-median problem defined over split administrative regions or regions geographically apart (e.g. archipelagos), and the case of problems coming from industry such as the optimal diversity management problem. In such cases the problem can be decomposed into smaller p-median problems which are solved in each component k for different feasible values of p k , and the global solution is obtained by finding the best combination of p k medians. This approach has the advantage that it permits to solve larger instances since only the sizes of the connected components are important and not the size of the whole graph. However, since the optimal number of facilities to select from each component is not known, it is necessary to solve p-median problems for every feasible number of facilities on each component. In this paper we give a decomposition algorithm that uses a procedure to reduce the number of subproblems to solve. Computational tests on real instances of the optimal diversity management problem and on simulated instances are reported showing that the reduction of subproblems is significant, and that optimal solutions were found within reasonable time.