Employment Smoothing (Capital Accumulation) with Production for Stochastic Demand

We consider the problem of periodically choosing production quantity and manufacturing capacity with the latter constraining the former. It is assumed that capacity can be altered at a cost that is proportional to the amount of change and that other capacity costs are proportional to the capacity. Other manufacturing costs are assumed to be proportional to the quantity manufactured. We summarise inventory costs with a convex expected holding and penalty cost function; excess demand is backlogged. The model includes a (possibly) random change in capacity (labor force attrition or depreciation of capital) each period. We characterize optimal policies when the criterion is the minimization of expected discounted costs. The characterizations also apply when the unit costs of capacity change are stochastic with a known distribution that may be correlated in any period with the demand in the preceding period.