## Abstract

Interaction effect across complementary products plays an important role in characterizing the optimal inventory policy. The inventory levels of complementary products are interrelated due to interaction between demand streams. In this paper, we consider a periodic review base-stock policy in the presence of two complementary products with interrelated demands and joint replenishment. Demands are modeled by a Poisson process and any unmet demand is lost. Demands can be in sets of one unit of each or jointly. If an arrival demand requests two products jointly and one of the products is not in stock, then the whole demand is lost. We aim to investigate how this interrelated demand phenomenon influences the optimal base-stock levels and the period length of a periodic review policy.

We utilize the renewal reward theorem to derive the explicit expression of the expected profit rate in the system. The goal is to determine the optimal period length and the base-stock levels such that the expected profit rate is maximized. Enumeration and approximation algorithms are employed to find the optimal and near-optimal solutions, respectively. The approximation algorithm is based on a scenario with independent demand processes which results in an explicit expression for the long-run profit per time unit and leads to analytical solutions for optimal policies. Our numerical results reveal that the solutions obtained by the approximation algorithm are close to optimal solutions. Numerical experiences show that the maximum profit in the system is achieved if the proportion of customers with jointly demand increases. Moreover, the interaction effect between demand processes has a significant impact on the control policy performance when the units lost sales and unit holding costs are high, and the demand rare is low.

We utilize the renewal reward theorem to derive the explicit expression of the expected profit rate in the system. The goal is to determine the optimal period length and the base-stock levels such that the expected profit rate is maximized. Enumeration and approximation algorithms are employed to find the optimal and near-optimal solutions, respectively. The approximation algorithm is based on a scenario with independent demand processes which results in an explicit expression for the long-run profit per time unit and leads to analytical solutions for optimal policies. Our numerical results reveal that the solutions obtained by the approximation algorithm are close to optimal solutions. Numerical experiences show that the maximum profit in the system is achieved if the proportion of customers with jointly demand increases. Moreover, the interaction effect between demand processes has a significant impact on the control policy performance when the units lost sales and unit holding costs are high, and the demand rare is low.

Original language | English |
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Journal | Annals of Operations Research |

Volume | XX |

Issue number | XX |

Publication status | Published - 12 Jan 2021 |

## Keywords

- Inventory
- Complementary products
- Interaction effect
- Periodic review policy
- Renewal reward theorem