Computational procedures for stochastic multi-echelon production systems

This paper is concerned with the numerical evaluation of multi-echelon production systems. Each stage requires a fixed predetermined leadtime; furthermore, we assume a stochastic, stauonary eLd.tlmc demand process. In a previous paper, we have developed an analytical framework for determining optimal control pohclcs for such systems under an average COSt criterion The current paper Is based on this analytical theory but discusses computational aspects, m particular for serial and assembly systems A hierarchical (exact) decomposition of these systems can be obtained by consldcnng echelon stocks and by transforming penalty and holding costs accordingly. The one-dimensional problems arising after this decomposition however involve Incomplete convolutions of distribution functions, which are only recurslvcly defined. We develop numerical procedures for analyzing these incomplete consolations; these procedures are based on approximations of distnbuuon functions by mixtures of Erlang dlstnbutmns Combining the analytic.ally obtained (exact) decomposition re. suits with these numerical procedures enables us to quickly determine optmJal order-up-to levels for all stages. Moreover, expressions for the customer ser~ lee level of such a multi-stage system are obtained, yielding the possibility to determine policies which minimize average mve.-~ory holding costs, given a service level constraint