Fluid Inventory Models under Markovian Environment

Today's products are subject to fast changes due to market conditions, short life cycles, and technological advances. Thus, an important problem in inventory planning is how to effectively manage the inventory control in a dynamic and stochastic environment. The traditional Economic Order Quantity (EOQ) and Economic Production Quantity (EPQ) both are widely and successfully used models of inventory management. However, both models assume constant and fixed parameters over time. Unfortunately, most of these assumptions are unrealistic. In this study, we generalize the EOQ and EPQ models and study production-inventory fluid models operating in a stochastic environment. The inventory level increases or decreases according to a fluid-flow rate modulated by an n-state continuous time Markov chain (CTMC). Our main objective is to minimize the expected discounted total cost which includes ordering, purchasing, production, set up, holding, and shortage costs. Applying regenerative theory, optional sampling theorem (OST) to the multi-dimensional martingale and fluid flow techniques, we develop methods to obtain explicit formulas for these cost functionals. As such, we provide managers with a useful framework and an efficient and easy-to-implement tool to coop with different demand-supply patterns.