Optimal composition of number and size of machines in a multi-stage make-to-order system with due dates

We develop an optimisation model to minimise costs for work-in-process (WIP), finished goods inventory (FGI), backorders, and capacity for a multi-stage production system applying a work-ahead window work release policy. Customers arrive to a make-to-order production system and have stochastic due dates. The parameters to optimise are the capacity invested at each stage, consisting of the number of machines and the processing rate (defined by a set of possible processing rates), as well as the work-ahead window. An optimality condition is provided showing when it is optimal to invest into a single machine at each stage if processing rates are continuous decision variables. The optimality of increasing capacity towards the customer end of the production line is proven under certain conditions. For a special case consisting of two stages with M/M/s queues with exponentially distributed customer required lead time, explicit expressions for WIP, FGI and backorders are developed. A set of numerical examples illustrates the influence of predefined processing rates on the optimal number and selection of machines. A simple solution heuristic for the problem with a predefined set of processing rates is proposed and its performance is shown in a numerical example. Additionally, the influence of uncertain input rates into the production system is discussed.