A joint replenishment problem with the (T,ki) policy under obsolescence

Companies are frequently confronted with the need to order different types of items from a single supplier or to manufacture the items in a production line. Indeed, coordinated ordering of multiple items may lead to important savings whenever a family of items can be ordered from a common supplier, produced in a common facility, or use a common mode of transportation. The Joint Replenishment Problem (JRP) tackles the coordinated replenishment of multiple items by minimizing the total cost, composed of ordering (or setup) costs and holding costs, while satisfying the demand. On the other hand, when items are subject to obsolescence, they may face an abrupt decline in demand as they are no longer needed. This decline can be caused by reasons such as rapid advancements in technology, going out of fashion, or ceasing to be economically viable. The present article develops an extension of the JRP where the items may suddenly become obsolete during an infinite planning horizon. The point at which an item becomes obsolete is uncertain. The lifetimes of the items are assumed to follow independent negative exponential distributions. A model is proposed by using the total expected discounted cost as the minimization criterion. The time value of money is considered through an appropriate discount rate. Extensive tests were performed to assess the impact of obsolescence rates and discount rates on the ordering policies. The progressive increase of the obsolescence rates determines smaller periods between successive replenishments, while the progressive increase of the discount rate determines smaller lot sizes.