Polynomial algorithm of inventory model with complete backordering and correlated demand caused by cross-selling

In a paper published in the International Journal of Production Economics (IJPE) [Zhang, R., Kaku, I., Xiao, Y., 2012. Model and heuristic algorithm of the joint replenishment problem with complete backordering and correlated demand. International Journal of Production Economics 139 (1), 33–41], the authors proposed a joint replenishment problem (JRP) model with complete backordering and correlated demand caused by cross-selling. The model was transformed into minimizing a function with respect to multiples of a major item's order cycle, and a heuristic algorithm was developed for near-optimal solutions. In this paper, we reinvestigate the problem and analyze the mathematical property of the model to develop an exact algorithm. The algorithm can obtain global optima and exhibits polynomial complexity.