Stochastic Production Planning Problem under Unobserved Inventory System

In this paper, an aggregate inventory, production, and workforce planning problem is formulated as a constrained stochastic linear quadratic problem under hypothesis of imperfect information of the inventory system. This stochastic problem generalizes the classical unconstrained production planning model developed by Holt, Modigliani, Muth and Simon, and known as HMMS model. Using the Kalman filter device, the conditional mean and covariance of the inventory variable can be estimated, and, as an immediate result, the certainty equivalence principle can be applied to transform the stochastic problem in an equivalent deterministic problem, which is easier to be solved then the original one. It proceeds then that, such an equivalent problem can be solved through a sub-optimal heuristics, known as open-loop updating (OLU) procedure. At last, from a simple example, it is shown that the optimal OLU policy allows the manager to get insights about the use of the aggregate resources of the company.