An Efficient Algorithm for the Joint Replenishment Problem with Quantity Discounts, Minimum Order Quantity and Transport Capacity Constraints

The joint replenishment problem has been extensively studied and the joint replenishment strategy has been adopted by a large variety of retailers in recent years. However, the joint replenishment problem under minimum order quantity and other constraints does not receive sufficient attention. This paper analyzes a retailing supply chain involving a supplier that provides quantity discount schedules and limits the order quantity. The order quantity constraints include minimum order requirements for each item and as to the total quantity; additionally, the latter cannot exceed the transport capacity constraint. These are common constraints in the retail industry today and create greater complexity and difficulty in the retailer's decision-making. To analyze the problem, an integer nonlinear programming model is set up to maximize retailers' profit with all practical constraints. A two-layer efficient algorithm named the Marginal and Cumulative Profit-Based Algorithm (MCPB) is then proposed to find whether to order and the optimal order quantity for each item. The results of computational experiments show that the proposed algorithm can find near-optimal solutions to the problem efficiently and is a reference for retailers to solve practical joint replenishment problems.