A global optimum search algorithm for the joint replenishment problem under power-of-two policy

Abstract

In this study, we perform theoretical analysis and derive a global optimum search algorithm for the joint replenishment problem (JRP) under power-of-two (PoT) policy. The JRP models concern how to determine lot sizes and to schedule replenishment times for products so as to minimize the total costs per unit time. PoT policy requires replenishment frequency of each product, denoted by k i , to be a PoT integer, i.e., k i =2 p where p=0,1,2,… . By utilizing a 10-product example, we graphically present the curve of the optimal total cost with respect to the values of basic period. Under PoT policy, we prove that the optimality structure of the JRP is piece-wise convex. By making use of the junction points in the optimality structure, we derive an effective search algorithm to secure a global optimal solution for the JRP under PoT policy. Evidently, we provide a numerical example to demonstrate the efficiency of the proposed algorithm. Scope and purpose The joint replenishment problem (JRP) is concerned with the determination of lot sizes and schedule of n products in single-facility production/inventory systems. In the JRP, a major setup cost is incurred whenever the production facility sets up to jointly replenish a subset of products. Major setups often incur significant setup times and costs in certain industries, for instance, pharmaceutical, chemical processes, and textile companies. If the executive managers may effectively apply the concept of the JRP in their production/inventory systems, they could join the replenishment schedule of the products to satisfactorily meet customers’ demand and importantly, to reduce significant cost in the meanwhile. In this study, we perform theoretical analysis and derive a global optimum search algorithm for the JRP under power-of-two (PoT) policy where PoT policy requires replenishment frequency of each product to be a power-of-two integer. By making use of the optimality structure, we derive an effective search algorithm to secure a global optimal solution for the JRP under PoT policy. We provide a numerical example to demonstrate the efficiency of the proposed algorithm.