Initial order quantities in a multistage production system with backlogging

The problem considered is that of determining optimal initial order quantities and backlogged amounts in a multistage serial production system with lost sales allowed. Items are processed into end products at finite rates within each production stage and thereafter delivered into an inventory of end products. Before each entire batch of end products is completed, any demand appearing from the start of the production period cannot be met when there is a lack of initial inventory. In order to reduce lost sales, smaller batches and/or larger backlogs than the stationary ones may be preferred, but this would incur additional costs. The net present value principle is applied to evaluate the performance of the system. This procedure automatically takes capital costs and the effects of lost sales and backlogging into account. The optimal decision policy developed by dynamic programming determines the optimal sequences of order quantities and backlogs over time, and provides conditions for whether or not the process starts in a transient phase to be followed by a stationary phase. Under certain circumstances batch sizes will increase and backlogged amounts decrease during the former phase, after which both remain constant during the subsequent stationary phase. The model incorporates backlogging costs both per unit backlogged and per unit and time unit. Properties of the stationary solution are derived. An interior solution will normally be obtained also, with positive backlogging costs per unit in contrast to corresponding results when applying the traditional average cost approach. Numerical examples illustrating different basic cases are provided.