Supply management with concave cost

This paper is divided in two parts. In the first part, we consider the case where several providers feed a single manufacturing unit periodically for certain specified 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 do not deliver any thing. We are introducing some properties of 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. A heuristic algorithm is proposed. The results provided by this algorithm are compared with the optimal solution for the costs that are linear for a strictly positive quantity and equal to zero otherwise.