The concave cost supply problem

This paper is divided into two parts. In the first part, we consider the case where several providers feed a single manufacturing unit periodically in order to satisfy a given demand. The cost incurred when a provider feeds the manufacturing unit is a concave function of the quantity delivered. Furthermore, each provider either delivers a quantity that lies between a minimum and maximum value or does not deliver anything. We are introducing some properties of an optimal solution and derive a heuristic algorithm from these properties. A numerical example illustrates this approach. In the second part of the paper, we consider the case where several providers feed periodically several manufacturing units in order to satisfy given demands. A heuristic algorithm is proposed. The results provided by this algorithm are compared with the optimal solution in the case when the costs are linear for a strictly positive quantity and equal to zero otherwise.