An (s, k, S) fluid inventory model with exponential leadtimes and order cancellations

Abstract

We consider a stochastic fluid inventory model based on a (s, k, S) policy. The content level W = {W(t): t ≥ 0} increases or decreases according to a fluid-flow rate modulated by an n-state continuous time Markov chain (CTMC). W starts at W(0) = S; whenever W(t) drops to level s, an order is placed to take the inventory back to level S, which the supplier will carry out after an exponential leadtime. However, if during the leadtime the content level reaches k, the order is suppressed. We obtain explicit formulas for the expected discounted costs. The derivations are based on the optional sampling theorem (OST) to the multidimensional martingale and on fluid flow techniques.

KEYWORDS:

• Markov-modulated fluid models
• leadtime
• martingale
• order cancellation
• production-inventory model
• (s, S) policy

Mathematics Subject Classification:

• Primary 60K10
• Secondary 90B22; 60J25

Acknowledgments

The author is grateful to Professor David Perry, to the editor, and to the referees for their helpful comments and suggestions that led to substantial improvements of the article.