Inventory System with Scheduled Demand and Distributed Supply

Abstract

We consider a simple supply chain model constituted of a retailer facing end- consumers’ demand while being supplied by a large number of suppliers. We model the retailer as a single server queue while the independent suppliers are assumed to form an infinite server system. Many applications can fit this setting. We focus in this work on the case where the market demand is predictable (i.e., orders arriv- ing following a deterministic sequence) while concentrating the uncertainty on the supply side through the processing time of each “server”. In this setting, suppliers decide first on their capacity level followed by the retailer who decides on the ade- quate base-stock level. From a queueing perspective, suppliers can be represented by a D/G/∞ queue. The retailer’s queue turns out to be an S/D/1 queue (the S denotes a scheduled traffic as defined in Araman et al. (2021)), where the positive perturbation is the supplier’s processing time. To analyze this system, we consider first the centralized system as a benchmark where the retailer sets both the capacity level as well as the base-stock level. For the decentralized setting, we consider two cases. In the first one, all suppliers are under one supply function and decide their capacity levels as one entity. In the more interesting case, we assume that suppliers decide individually on their capacity. The objective function is the inventory cost rate that suppliers and retailer are each minimizing. Even under exponential per- turbations, the problem is intractable. We therefore suggest, through an asymptotic analysis, a full characterization of the optimal centralized and Nash solutions under a heavy traffic regime. Moreover, we perform a numerical analysis to validate these approximations through a Monte Carlo simulation.