A model for performance evaluation and stock optimization in a kit management problem

In this paper we consider a kit planning problem where demand occurrences are not for individual items, but for kits (a group of items). Each kit contains an arbitrary number of items. Kit demands occur according to a Poisson process. Whenever a kit demand occurs, only one item from the kit is used and the rest is returned as unused. The item that will be used from the kit is not known in advance and the whole kit has to stay at the demand site for the whole duration. The used item is replenished through a stochastic supply system, with possible capacity limitation. This model has applications in health care (planning surgical implant inventories), and repair kit management systems. As a demand for a kit triggers simultaneous demands for the items within the kit, the individual demand arrival processes for the items in that kit are correlated. Therefore, finding the joint probability distribution of the number of items that are outstanding, and hence finding the probability of kit availability, is generally difficult. We can obtain these terms in a fairly explicit form under the assumption that an item which is not in stock when a kit demand occurs can be obtained through borrowing from an emergency supply channel. As soon as a unit of such an item becomes available, it is returned back to its original supply source. We also formulate an optimization problem where the expected holding cost of items is minimized, and pre-specified kit availability constraints are satisfied. Since the optimization problem is hard to solve, we provide a heuristic procedure for obtaining the stock levels, and test the quality of the heuristic.