Proof of the variability propagation principle in a pull serial line: Existence and measurement

In this study, we consider infinite supply of raw materials and backlogged demands as given two boundary conditions. And we need not make any specific assumptions about the inter-arrival of external demand and service time distributions. Under these situations, the ultimate objective of this study is to prove the variability propagation principle in a pull serial line and is to measure it in terms of the first two moments of the inter-departure process subject to number of cards in each cell. Two preparations are required to achieve this objective: The one is to derive a true lower bound of variance of the inter-departure process. The other is to establish a constrained discrete min–max problem for the no backorder (backlogging) probabilities in each cell. We may get some fundamental results necessary to a completion for the proof through the necessary and sufficient conditions for existence of optimal solution of a constrained discrete minimax problem and the implicit function theorem. Finally, we propose a numeric model to measure the variability propagation principle. Numeric examples show the validity and applicability of our study.