The Multi-Item Joint Replenishment Problem with Transportation and Container Effects

We address the problem of scheduling the delivery of multiple items from a single supplier to a manufacturer. The items are packaged into containers, and the containers are shipped by truck. There is a fixed charge per truck shipment, and inventory holding costs are charged on end-of-period inventory. We seek to minimize the sum of transportation and inventory costs. The problem is a combination of a bin-packing problem (due to the presence of containers and finite-capacity trucks) and a multi-item joint replenishment problem. We present a heuristic solution procedure which starts with the optimal solution of the problem in which the integrality of the containers is relaxed. (A solution procedure for this relaxed problem appears in N. Ben-Kheder and C. A. Yano, “The Multi-Item Joint Replenishment Problem with Volume-Sensitive Transportation Costs,” Technical Report 89-19, Department of Industrial & Operations Engineering, The University of Michigan, Ann Arbor, MI (1989a). We develop a method to modify this solution to account for the integrality of containers. This modification scheme involves sequentially considering each item and optimally scheduling the fractional containers in the relaxed solution. To solve this single-item problem, we devise a procedure that accounts for the availability of “free” remaining capacity of trucks that have been partially filled with other items. In a computational study, our heuristic is compared with a lower bound, with variations of our heuristic, and with simple rule-of-thumb policies. The results suggest that our heuristic performs very well, especially in problems where considering tradeoffs between inventory and transportation costs is important.