Joint replenishment problem for multi-echelon system with planned shortage

An inventory system with one warehouse and one retailer is proposed, planned shortage inventory amounts are allowed at both the retailer and the warehouse. During the ordering process, a major order cost for the lot is incurred at each location and a minor order cost is added for each item in the lot. Demand occurs only at the retailer with a constant deterministic rate. Two policies are considered for this problem; first, the centralised non‐cooperative policy where a centralised decision making process is done on warehouse and retailer together to optimise their costs; second, the centralised cooperative policy wherein addition to the centralised decision process, the items at the retailer and the warehouse are replenished jointly to gain the advantage of reducing the major order cost. In the first policy, the objectives here are to evaluate the optimum order interval for each item (i), while in the second policy the objectives are to determine the common order interval, the associated multiple integers and that determines the time of order for each item, and the optimum inventory planned shortage amounts, for each item at each location, which minimise the total inventory cost for the echelon system.