Multi-level, multi-stage capacity-constrained production – inventory systems in discrete time with non-zero lead times using materials requirements planning theory

A substantial amount of research has been carried out on capacity-constrained production–inventory systems. This has mostly dealt with models assuming deterministic demand and single-item systems. In previous work, one of the authors together with a co-researcher designed a basic theoretical model for systems with multiple items and stochastic external demand. These developments were presented within a discrete time framework. Lead times were assumed to be given constants and the net present value principle was applied. Although the theory developed concerned situations in which the lead times could be any non-zero constants, in order to design an analytical solution procedure, the assumption was made that lead times were zero, in order to be able to apply dynamic programming. Cumulative production and cumulative demand were taken as state variables. In this paper, we remain in the discrete time framework and develop a methodology for the case that lead times are non-zero, whereas demand is deterministic. Our emphasis is on the design of the state space, the properties of which depend on the product structures (the input matrix), the distribution of lead times among the production processes (the lead time matrix), and on the historical sequences of the production vectors. Once an efficient state space is designed, dynamic programming may be applied as a solution method. The net present value principle is again applied.