Joint assortment and inventory optimization for vertically differentiated products under consumer-driven substitution

This paper presents a solution approach to the joint assortment and inventory planning problem for vertically differentiated products considering dynamic consumer-driven substitution. The demand for each product and the stockout-based substitution rates are derived from a consumer's utility function and a random market size. We propose a two-step integral solution approach, which uses the substitution characteristic of vertically differentiated products. In a first step, the approach determines the initial purchasing probabilities of all potential products and the substitution rate matrices for all possible product-availability combinations. In a second step, the inventory levels are determined by iteratively solving a sequence of two-product problems. We prove that an optimal assortment only contains products whose critical ratios are decreasing with increasing quality levels. We further compare the proposed approach with the exact results obtained by complete enumeration and find that in almost all instances, the optimality gap is below 0.5%. Numerical experiments reveal that an increasing demand uncertainty leads to first deeper assortments and second a shift of inventory towards lower-quality products. We also compare the proposed integral approach with a sequential assortment and inventory planning approach that first optimizes the assortment only considering assortment-based substitution and subsequently, determines the optimal inventory levels constrained on this assortment. We show that a joint assortment and inventory planning considering stockout-based substitution is particularly essential when both the profit margin and demand uncertainty is high. Moreover, we provide an extension considering shelf-space constraints.