We consider a special case of the symmetric capacitated vehicle routing problem, in which a fleet of K identical vehicles must serve n customers, each with a given demand consisting in a set of rectangular two-dimensional weighted items. The vehicles have a two-dimensional loading surface and a maximum weight capacity. The aim is to find a partition of the customers into routes of minimum total cost and such that, for each vehicle, the weight capacity is taken into account and a feasible two-dimensional allocation of the items into the loading surface exists. The problem has several practical applications in freight transportation and it is N P-Hard in the strong sense. We propose an exact approach, based on a branchand-cut algorithm, for the minimization of the routing cost, that iteratively calls a branch-and-bound algorithm for checking the feasibility of the loadings. Heuristics

In the Orienteering Problem (OP), we are given an undirected graph with edge weights and node prizes. The problem calls for a simple cycle whose total edge weight does not exceed a given threshold, while visiting a subset of nodes with maximum total prize. This NP-hard problem arises in routing and scheduling applications. We describe a branch-and-cut algorithm for finding an optimal OP solution. The algorithm is based on several families of valid inequalities. We also introduce a family of cuts, called conditional cuts, which can cut off the optimal OP solution, and propose an effective way to use them within the overall branch-and-cut framework. Exact and heuristic separation algorithms are described, as well as heuristic procedures to produce near-optimal OP solutions. An extensive computational analysis on several classes of both real-world and random instances is reported. The algorithm proved to be able to solve to optimality large-scale instances involving up to 500 nodes, within acceptable computing time. This compares favorably with previous published methods.

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Abstract.A variety of integer programming formulations have been proposed for Vehicle Routing Problems (VRPs), including the so-called two-and three-index formulations, the set partitioning formulation, and various formulations based on extra variables representing the flow of one or more commodities. Until now, there has not been a systematic study of how these formulations relate to each other. An exception is a paper of Luis Gouveia, which shows that a one-commodity flow formulation of Gavish and Graves yields, by projection, certain 'multistar' inequalities in the two-index space.We give a survey of formulations for the capacitated VRP, and then present various results of a similar flavour to those of Gouveia. In particular, we show that:-the three-index formulation, augmented by certain families of valid inequalities, gives the same lower bound as the two-index formulation, augmented by certain simpler families of valid inequalities, -the two-commodity flow formulation of Baldacci et al. gives the same lower bound and the same multistar inequalities as the one-commodity Gavish and Graves formulation, -a certain non-standard multi-commodity flow formulation, with one commodity per customer, implies by projection certain 'hypotour-like' inequalities in the two-index space, -the set partitioning formulation implies by projection both multistar and hypotour-like inequalities in the two-index space.We also briefly look at some other variants of the VRP, such as the VRP with time windows, and derive multistar-like inequalities for them. We also present polynomial-time separation algorithms for some of the new inequalities.

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