Model and heuristic algorithm of the joint replenishment problem with complete backordering and correlated demand

In practice, demands for items are often correlated with each other because of cross-selling effects. Traditional inventory models hold the assumption that demands are independent, which may lead to the risk of misestimating the inventory cost of associated items. This paper addresses a multi-item inventory system where demands for some minor items are correlated with that for a major item. Because of the cross-selling, demand for minor items may either be raised by the sales of the major item or be pulled down by the unsatisfied demands for the major item. Thus, a joint inventory policy considering correlations across items should be pursued. Considering general integer policy, we propose a joint replenishment problem (JRP) model with complete backordering and correlated demand. Through the use of derivatives the model is transformed to minimize a cost function with respect to multiples of the major items order cycle. Given that the optimal solution should offer a trade-off between the ordering cost and the inventory holding cost of each item, we develop a heuristic algorithm by adjusting the replenishment frequencies of minor items to solve the model. The heuristic is tested by simulated numerical examples and shows satisfactory results.