### Abstract

We consider a discrete state, discrete decision epoch inventory replenishment control problem under supply uncertainty. We assume that there is no backlogging, the single period demand d is deterministic, and once an item is placed in inventory, it will not perish. If a units of the product are ordered, then α units are placed into inventory with probability P(α{pipe}a), where ∑_{α=0}^{a}P(α{pipe}a) = 1. Let z=d-x, where x is the current inventory level. For the infinite horizon, total discounted cost criterion, we present conditions that guarantee that an optimal replenishment policy δ ^{*} is such that δ ^{*}(z)=0 for z≤0, δ ^{*}(z)≥z≥0, and δ ^{*}(z)-z is monotonically non-decreasing for z≥0. Such a "staircase" structure has a simple parametric description, which can help to accelerate value iteration and policy iteration.

Original language | English |
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Pages (from-to) | 581-592 |

Number of pages | 12 |

Journal | Annals of Operations Research |

Volume | 208 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2013 Sep 1 |

Externally published | Yes |

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### Keywords

- Inventory systems
- Markov decision processes
- Periodic review
- Random yields
- Supply uncertainty

### ASJC Scopus subject areas

- Decision Sciences(all)
- Management Science and Operations Research

### Cite this

*Annals of Operations Research*,

*208*(1), 581-592. https://doi.org/10.1007/s10479-011-0929-9