Optimal production mix

This paper deals with a production plant in which two different products can be produced. The plant consists of three subsystemsS_i. Before or after a phase of separate processing in subsystemsS₁ andS₂, the two products have to be processed in subsystemS₃. Each of these subsystems has a limited capacity.In the first part, we assume empty stocks at the beginning; at a fixed timeT in the future, certain quantitiesX_i of the two products have to be delivered to the customers. Facing linear holding costs, convex production costs, and stringent capacity constraints, the problem is to decide when to produce which product at what rate.It is shown that the optimal solution consists of up to six different regimes and that the time paths of the production rates need not be monotonic. These results, which can be obtained analytically, are also illustrated in several numerical examples.Finally, the case is considered where the terminal demand at timeT is replaced by a continuous and seasonally fluctuating demand rate. It is demonstrated that the optimal production rates show an interesting and nontrivial behavior. In particular, it may happen that, on intervals where the demand for the one product increases, the optimal production rate decreases. This is also demonstrated by computer plots in some numerical examples.The first author gratefully acknowledges support from the Austrian Science Foundation under Grant S3204 and the second author from Stiftung Volkswagenwerk. An earlier version of this paper was presented at the DGOR-NSOR Joint Conference, Eindhoven, Holland, September 23–25, 1987.