Optimal aggregate production planning with minimum inventory limit affected by demand uncertainties

This paper deals with the determination of an optimal decision policy for a production planning problem with inventory and production constraints. The planning time horizon is finite, from 1 to 2 years approximately at monthly periods, which means that all data involved with the problem are totally aggregated and the fluctuating demand for each one period is stochastic, with probability distribution function assumed as gaussian. Thus the problem studied here is a stochastic planning one with probabilistic constraint at the inventory level variable. It is shown that it is possible to obtain, by means of appropriate transformations, a deterministic equivalent formulation, for which an open-loop solution to stochastic problems can be generated, using any applicable mathematical programming algorithm. It is also shown that the uncertainties concerning demand fluctuation are explicitly presented in the deterministic formulation through a constraint function that represents the minimum inventory level limit or safety stock. This function is essentially concave and increases during time and depends on the variance of inventory level and probabilistic degree supply by the user to the inventory constraint. To illustrate the theoretical developments, a simple example of a single product system is proposed and solved by dynamic programming. The open-loop solution (i.e. approximate solution from an equivalent problem) is compared with the true solution obtained directly from the stochastic problem via stochastic dynamic programming.