Knowledge that Transforms

We consider the asymmetric capacitated vehicle routing problem (CVRP), a particular case of the standard asymmetric vehicle routing problem in which only the vehicle capacity constraints are imposed. CVRP is known to be NP-hard and finds practical applications in distribution and scheduling. We describe two new bounding procedures for CVRP, based on the so-called additive approach. Each procedure computes a sequence of nondecreasing lower bounds, obtained by solving different relaxations of CVRP. Effective implementations of the procedures are also outlined which considerably reduce the computational effort. The two procedures are combined into an overall bounding algorithm. A branch-and-bound exact algorithm is then proposed, whose performance is enhanced by means of reduction procedures, dominance criteria, and feasibility checks. Extensive computational results on both real-world and random test problems are presented, showing that the proposed approach favorably compares with previous algorithms from the literature.

Several recent papers have suggested using a product estimator in Monte Carlo Markov chain sampling for estimating the volume of a convex body, the permanent of a matrix and the distribution of first-passage time for a positive recurrent Markov chain. The present paper analyzes the properties of this estimator when each replication starts in an arbitrarily selected state. In particular, it describes a procedure for determining optimal warm-up intervals and optimal sample sizes to achieve a specified level of statistical accuracy at minimal cost. Also, it examines the variation in the optimal solution in response to changes in the parameters of the problem.