Computational procedures for stochastic multi-echelon production systems

This paper is concerned with the numerical analysis of multi-echelon production systems. In these systems, materials and components are ordered from outside suppliers and next proceed through a number of manufacturing and/or assembly stages. Each stage requires a fixed predetermined leadtime; furthermore, we assume a stochastic, stationary end-item demand process. In a previous paper, we have presented an exact analysis of such multiechelon systems under an average cost criterion. All three basic structures, i.e. serial, assembly and distribution systems, have been considered. In particular, it has been shown that, by transforming penalty and holding costs into appropriate echelon cost functions, an exact decomposition of these systems can be obtained, thus reducing complex multi-dimensional problems to a series of more simple one-dimensional problems. The current paper is based on this analytical theory but discusses numerical aspects, in particular for serial and assembly systems. The one-dimensional problems arising after the (exact) decomposition of a multi-echelon system involve incomplete convolutions of distribution functions, which are only recursively defined. We develop numerical procedures for analyzing these incomplete convolutions; these procedures are based on approximations of distribution functions by mixtures of Erlang distributions. The combination of the analytically obtained (exact) decomposition results with these numerical procedures enables us to analyze fairly complex systems in only a few seconds on a microcomputer.