### Abstract

This article analyzes a continuous time back-ordered inventory system with stochastic demand and stochastic delivery lags for placed orders. This problem in general has an infinite dimensional state space and is hence intractable. We first obtain the set of minimal conditions for reducing such a system's state space to one dimension and show how this reduction is done. Next, by modeling demand as a diffusion process, we reformulate the inventory control problem as an impulse control problem. We simplify the impulse control problem to a Quasi-Variation Inequality (QVI). Based on the QVI formulation, we obtain the optimality of the (s1 S) policy and the limiting distribution of the inventory level. We also obtain the long run average cost of such an inventory system. Finally, we provide a method to solve the QVI formulation. Using a set of computational experiments, we show that significant losses are incurred in approximating a stochastic lead-time system with a fixed lead-time system, thereby highlighting the need for such stochastic lead-time models. We also provide insights into the dependence of this value loss on various problem parameters.

Original language | English (US) |
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Pages (from-to) | 302-327 |

Number of pages | 26 |

Journal | Mathematics of Operations Research |

Volume | 40 |

Issue number | 2 |

DOIs | |

State | Published - May 1 2015 |

Externally published | Yes |

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### Keywords

- Free boundary problem
- Impulse control
- Inventory control
- Stochastic lead times

### ASJC Scopus subject areas

- Mathematics(all)
- Computer Science Applications
- Management Science and Operations Research

### Cite this

*Mathematics of Operations Research*,

*40*(2), 302-327. https://doi.org/10.1287/moor.2014.0671