A production–transportation problem with stochastic demand and concave production costs

Well known extensions of the classical transportation problem are obtained by including fixed costs for the production of goods at the supply points (facility location) and/or by introducing stochastic demand, modeled by convex nonlinear costs, at the demand points (the stochastic transportation problem, [STP]). However, the simultaneous use of concave and convex costs is not very well treated in the literature. Economies of scale often yield concave cost functions other than fixed charges, so in this paper we consider a problem with general concave costs at the supply points, as well as convex costs at the demand points. The objective function can then be represented as the difference of two convex functions, and is therefore called a d.c. function. We propose a solution method which reduces the problem to a d.c. optimization problem in a much smaller space, then solves the latter by a branch and bound procedure in which bounding is based on solving subproblems of the form of [STP]. We prove convergence of the method and report computational tests that indicate that quite large problems can be solved efficiently. Problems up to the size of 100 supply points and 500 demand points are solved.