Note---On the Optimality of Integer Lot Size Ratios in "Economic Lot Size Determination in Multi-Stage Assembly Systems"

The proof of a well known theorem by Crowston and Wagner in Crowston, Wagner, and Williams (Crowston, W. B., M. H. Wagner, J. F., Williams. 1973. Economic lot size determination in multi-stage assembly systems. Management Sci. 19 (5, January) 517--527.) is shown here to be defective. According to this theorem, an optimal solution to the batch size determination problem for multi-echelon production/inventory assembly structures is characterized by a set of lot sizes, such that the lot size at each stage must be an integer multiple of the lot size at its successor stage. The theorem has been said to apply to uncapacitated assembly systems having constant final product demand over an infinite horizon, with no backlogs or lost sales permitted, where lot sizes must be rational numbers and time invariant. While the theorem supposedly applied to the case of both finite production rates and infinite production rates, the proof dealt only with the case of instantaneous production. Also, the proof implicitly assumed that lots were processed at any given stage only after unchanging spans of time. In this paper we show that with or without this implicit assumption, the theorem does hold true for the special case of instantaneous processing in two-level assembly systems; that is, configurations where there is only one successor stage in the entire system. However, we also show by counterexample that without this implicit assumption, the theorem is invalid for more general assembly structures. Furthermore, in the appendix of this paper we show that the proof is defective at the point where Crowston and Wagner extend their results for two-level systems to more general assembly systems. Thus it is an open question whether or not the theorem is valid for all assembly structures when processing at any given stage must occur only after unchanging spans of time.