65 An algorithm for the transportation problem with given frequencies L. Bertazzi, M. G. Speranza Dept. of Quantitative Methods - University of Brescia Corso G. Mameli, 27 - 1-25122 Brescia, Italy. Tel: 39-30-2988.1. Fax: 39-30-2400925. e-mail: speranza@master. cci. unibs. it W. Ukovich D.E.E.I.- University of Trieste Via A. Valerio, 10 - I - 34127 Trieste, Italy. Tel: 39-40-6767135. Fax: 39-40-6763460. e-mail: ukovich@univ.trieste.it Abstract In this paper we consider the problem of shipping several products from an origin to a destination, when a discrete set of shipping frequencies is available only, in such a way that the sum of the transportation and inventory cost is minimized. This problem, which is known to be NP-hard, has applications in transportation planning and in location analysis. We derive some dominance rules for the problem solutions which allow to tighten the bounds on the problem variables. Moreover, we present a branch-and-bound algorithm and we evaluate its performance on randomly generated problem instances. Keywords Transportation, inventory, dominance rules, branch-and-bound. 1 INTRODUCTION We consider the situation in which each product of a given set is made available at an origin and needed at a destination at a constant rate. A discrete set of available shipping frequencies is given and the shipment is carried out by trucks with given capacity. The problem is to decide how much of each product to ship at each frequency in such a way that the sum of the transportation and inventory cost is minimized. This problem was introduced in Speranza and Ukovich (1994) together with other models for shipping products from an origin to a destination. Similar models, where the shipping frequencies are allowed to take any continuous value, are considered in Burns et al (1984), Blumenfeld et al (1985), Anily and Federgruen (1990). The problem was shown to be NP-hard in Speranza and Ukovich (1991), where an optimal branch-and- J. Doležal et al. (eds.), System Modelling and Optimization © Springer Science+Business Media Dordrecht 1996