Production Smoothing with Stochastic Demand I: Finite Horizon Case

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Abstract
Beckmann [1] and Mills [7] consider production smoothing problems in which demands are random variables. This paper generalizes and extends Beckmann's results which predicate backlogging of excess demand. Convex expected holding and penalty cost functions pertain to inventory and the cost of changing the production rate is proportional to the change. The model has a finite horizon and permits nonstationary costs and demand distributions. Beckmann shows that two curves in the plane determine an optimal policy (minimum expected discounted cost) each period. It is shown here that the curves have slopes between minus one and zero, are differentiable, and are bounded by two straight lines with a slope of minus one. The results are not changed by an upper bound on the quantity produced or by a lag between the time a product is manufactured and the time it is available to satisfy demand.

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