Assortment planning and inventory decisions under a locational choice model

We consider a single-period assortment planning and inventory management problem for a retailer, using a locational choice model to represent consumer demand. We first determine the optimal variety, product location, and inventory decisions under static substitution, and show that the optimal assortment consists of products equally spaced out such that there is no substitution among them regardless of the distribution of consumer preferences. The optimal solution can be such that some customers prefer not to buy any product in the assortment, and such that the most popular product is not offered. We then obtain bounds on profit when customers dynamically substitute, using the static substitution for the lower bound, and a retailer-controlled substitution for the upper bound. We thus define two heuristics to solve the problem under dynamic substitution and numerically evaluate their performance. This analysis shows the value of modeling dynamic substitution and identifies conditions in which the static substitution solution serves as a good approximation.