We are concerned with obtaining the optimal schedule of N products over a single facility in the finite and infinite horizons. This is a generalization of the classical cyclical "Economic Manufacturing Quantity" formulations. Through number-theoretic arguments we discuss questions of feasibility, the existence of communicating classes, and sensitivity to "saturation". In the finite horizon case, we present an algorithm that generates the production plan, and another algorithm that generates the production schedule through a shortest-path interpretation. In the infinite horizon case, we do likewise: the optimal production plan is a Turnpike cycle. Due to the immensity of the state space, approximate procedures are also discussed.
Historical PerspectiveThis paper is concerned with obtaining the optimal schedule of N products over a single facility in the finite and infinite horizons. In a real sense, it is a generalization of the classical cyclical EMQ, for which we substitute a cycle of a different nature, but a cycle nevertheless.The scheduling of a single multi-product facility for the purpose of controlling the production and inventory of N products is a problem that boasts a long history, dating back to the famous "square root" formula of Harris (1915) and Raymond (1931) [23). The "EMQ" formula, as it has come to be known, received wide dissemination, and was elaborated upon in a variety of ways to incorporate pricequantity discounts, backlogging of demand, finite horizon, etc. For almost forty years it reigned supreme as the model for economic production and stock management. The problem of concern to us here is that of scheduling N products on a single facility when the demand for item j is a fixed rate, rj ~ 0, integer, per period j=1,2, ••• ,N. The limited capacity of the facility is a major restriction since in any period t, the facility is devoted to the production of one, and only one, product. But while the production of that product is in progress for one or several periods, demand for the other products is continuing, hence their inventory is depleted at their respective demand rates.Let Pj be the production rate per period, assumed integer, when the facility is devoted to the production of item j. To have a meaningful problem we assume that P j > rj for all j. Such is not true in the case of two or more products. Furthermore, the case of several products raises the spectre of infeasibility, which was completely absent in the single product case (as long as p > r).These, and other questions, indicate that there exist some novel considerations peculiar to the problem under investigation. In particular, we must consider the following: Pj: rate of production of j when facility is devoted to it; Pj > r j , all j, integer.number of periods (in a given planning horizon) devoted to the production of item j; Xj ~ 0, all j.the set a production plan.We shall call It: vector of inventories at time t, the ~ vector, The necessity of the condition is obvious for otherwise the available capacity in an...
