A Solution to the Optimal Lot-Sizing Problem as a Stochastic Resource Contention Game

Abstract

We present a new way to solve the "lot-sizing" problem viewed as a stochastic noncooperative resource contention game. We develop a Stochastic Flow Model (SFM) for polling systems with non-negligible changeover times enabling us to formulate lot sizing as an optimization problem without imposing constraints on the distributional characteristics of the random processes in the system. Using Infinitesimal Perturbation Analysis (IPA) methods, we derive gradient estimators of the performance metrics of interests with respect to the lot-size parameters and prove they are unbiased. We then derive an online gradient-based algorithm for obtaining optimal lot sizes from both a system-centric and user-centric perspective. Uncharacteristically for such cases, there is no gap between the two solutions in the two-class case for which we have obtained explicit numerical results. We derive a proof of this phenomenon for a deterministic version of the problem, suggesting that lot-sizing-like scheduling policies in resource contention problems have a natural property of balancing certain user-centric and system-centric performance metrics.Note to Practitioners-This paper is motivated by the "lot-sizing" problem in manufacturing which involves the determination of the optimal number of parts combined to form a "lot" for each of several part types differing in their processing times, raw material supplying rates, etc. Using better selected lot sizes decreases the average lead time, and ultimately leads to larger throughput. In addition, lot sizing provides an inexpensive way to improve performance by controlling a simple parameter, especially when compared to complicated manufacturing reengineering processes. This paper proposes an algorithm to calculate optimal lot sizes for multiple types of manufacturing parts. This algorithm only relies on data that are either readily observable or easily calculated from operating production systems. It can be easily programmed and used for online estimation of optimal lot sizes. Further, this paper examines the problem from the point-of-view of a central coordinator aiming to optimize a system-wide objective and can select all lot sizes or as a "game" where each part type controller can individually select its own lot size to optimize its own objective. It is shown that both approaches lead to the same solution. This property is attractive since it indicates an inherent fairness in treating different part types and suggests that similar mechanisms may be used in other applications with similar resource contention features.