Bixby & Lee/truck Dispatching Scheduling Using Branch-and-cut

1 In this paper we present a branch-and-cut algorithm for a structured 0/1 integer programming problem referred to as the \truck dispatching scheduling (TDS) problem." The TDS problem is a vehicle routing and scheduling problem that arose from an oil company's desire to better utilize regional truck eets and drivers. Many studies on diierent variations of vehicle routing and scheduling have appeared in the literature (Bodin 1990, Golden and Assad 1988). In most of these, the emphasis has been on developing fast heuristic algorithms to nd good feasible solutions (Bodin 1990, p. 575). In contrast, the branch-and-cut algorithm developed herein nds optimal solutions. A classical approach for solving integer programming problems is branch-and-bound. However, for large scale problems, such an approach often fails due to excessive memory usage. In 1983 Crowder, Johnson and Padberg incorporated preprocessing and cut generation prior to initiating the branch-and-bound procedure, resulting in remarkable solution times for some diicult instances (some which were once thought to be intractable). By the end of the 1980's, the term branch-and-cut was being used to describe methods that utilize cutting-plane techniques not only at the root node, but at nodes within the search tree itself. In these methods, cuts generated at a given node are valid globally across the entire search tree. The branch-and-cut approach has been responsible for solving important hard combinatorial optimization problems. In particular, large-scale instances of the traveling salesman problem and the airline crew scheduling problem have been solved to proven optimality in reasonably short amounts of time Certainly, an important aspect of a successful application of branch-and-cut is the ability to generate deep cuts in a computationally eecient way. This ability is often enhanced by knowledge of the facet structure of the convex hull of 0/1 solutions, or of certain approximating polytopes to this convex hull. Thus, an important component of this research involved a study of the polyhedral structure of the TDS problem. Polyhedral results pertaining to the TDS problem are summarized in Section 2. First, we consider approximating knapsack equality polytopes associated with individual constraints of the TDS model. These knapsack equality polytopes are of course intimately related to their inequality counterparts. We consider various families of facet deening inequalities for the associated knapsack inequality polytope and state conditions when these inequalities also deene facets of the equality polytope. We also consider an approximating polytope derived from the entire system of constraints. In …