Optimal Production Planning in a Multi-Product Stochastic Manufacturing System with Long-Run Average Cost

Abstract

This paper is concerned with the problem of production planning in a flexible manufacturing system consisting of a single or parallel failure-prone machines producing a number of different products. The objective is to choose the rates of production of the various products over time in order to meet their demands at the minimum long-run average cost of production and surplus. The analysis proceeds with a study of the corresponding problem with a discounted cost. It is shown using the vanishing discount approach for the average cost problem that the Hamilton-Jacobi-Bellman equation in terms of directional derivatives has a solution consisting of the minimal average cost and the so-called potential function. The result helps in establishing a verification theorem, and in specifying an optimal control policy in terms of the potential function. The results settle a hitherto open problem as well as generalize known results.

Keywords: Production planning, stochastic dynamic programming, vanishing discount approach, optimal control, long-run average cost

JEL Classification: C61, M11, M20