Optimal Aggregate Production Plans via a Constrained LQG Model

Abstract

In this paper, a single product, multi-period, aggregate production planning problem is formulated as a linear-quadratic Gaussian (LQG) optimal control model with chance constraints on state and control variables. Such formulation is based on a classical production planning model developed in 1960 by Holt, Modigliani, Muth and Simon, and known, since then, as the HMMS model [1]. The proposed LQG model extends the HMMS model, taking into account both chance-constraints on the decision variables and data generating process, based on ARMA model, to represent the fluctuation of demand. Using the certainty-equivalence principle, the constrained LQG model can be transformed into an equivalent, but deterministic model, which is called here as Mean Value Problem (MVP). This problem preserves the main properties of the original model such as convexity and some statistical moments. Besides, it is easier to be implemented and solved numerically than its stochastic version. In addition, two very simple suboptimal procedures from stochastic control theory are briefly discussed. Finally, an illustrative example is introduced to show how the extended HMMS model can be used to develop plans and to generate production scenarios.