A Stochastic Transportation Problem

Abstract

When the market demands for a commodity are not known with certainty, the problem of scheduling shipments to a number of demand points from several supply points is a stochastic transportation problem. If the dynamic aspects of the problem may be neglected, i.e., if the effects of “stock outs” and “inventory carry-overs” can be reflected in terms of linear penalty costs (in the case of undersupply) and linear salvage values (in case of oversupply), one obtains a convex nonlinear programming problem. We show that the author's algorithm for the case of known demands can be generalized to obtain a solution algorithm for this stochastic problem. In case the joint cumulative distribution function for the demand is continuous, however, the algorithm is better described as a special (constructive) case of Dantzig's general method for convex programming. Each of these methods is, in turn, based on the decomposition algorithm. No assumptions are made as to the independence or dependence of the probability distributions of the various demands. If F(ξ1, …, ξn) is the joint cumulative distribution function for the demands, we assume only that each of the first moments of F exist (are finite), and that F is regular enough for the integration by parts formula to be valid. The analysis here is applicable (subject to a qualification) to the case of indivisible commodities, as well as the case of infinitely divisible ones.