Managing a closed-loop supply system with random returns and a cyclic delivery schedule

Motivated by an industry example, we develop a mathematical framework to address the inventory replenishment and capacity planning problem for a closed-loop supply system with random returns. The provider needs to deliver new or refurbished products to a group of clients under a fixed cyclic schedule, and also collects back a random portion of the used products in the subsequent delivery cycle for refurbishment. We first address the product replenishment strategy, in which only a random portion of the delivered products will be returned for refurbishment and the supplier must regularly purchase new products to replace the lost units. We then analyze the capacity decision problem where the provider uses his facility to refurbish the returned products for reuse, and the provider could incur extra refurbishing cost to handle the returned product at the end of each cycle due to insufficient capacity. Our models provide a simple decision support tool for making effective replenishment and capacity decisions in managing such a closed-loop supply system.