Joint replenishment model for multiple products with substitution

We extend the classical economic order quantity model to address the joint replenishment of multiple products under substitution. The proposed model optimizes ordering quantities for each product under substitution effects with the objective of minimizing the total cost associated with the setup, holding, and shortage of products, while partially meeting demand. First, the special case of three substitutable products is examined in detail. Then, a nonlinear mathematical programming formulation is presented as a general-purpose solution approach for any number of substitutable products. The convexity of the model is discussed. We find that the objective function to be convex in the important special case of products having equal unit holding costs, which typically holds for substitutable products in practice. Sensitivity analysis is conducted in order to determine the impact of cost parameters variations on the ordering policy. We focus on identifying conditions that favor substitution among products. We find that allowing substitution among products is an effective vehicle for cost cutting in supply chain settings involving high fixed costs, low holding costs, and low shortage costs.