Fixed vs. random proportions demand models for the assortment planning problem under stockout-based substitution

We consider the problem of determining the optimal assortment of products to offer in a given product category when each customer is characterized by a type, which is a list of products he is willing to buy in decreasing order of preference. We assume consumer-driven, dynamic, stockout-based substitution and random proportions of each type. No efficient method to obtain the optimal solution for this problem is known to our knowledge. However, if the number of customers of each type is a fixed proportion of demand, there exists an efficient algorithm for solving for the optimal assortment. We show that the fixed proportions model gives an upper bound to the optimal expected profit for the random proportions model. This bound allows us to obtain a measure of the absolute performance of heuristic solutions. We also provide a bound for the component-wise absolute difference in expected sales between the two models, which is asymptotically tight as the inventory vector is made large, while keeping the number of products fixed. This result provides us with a lower bound to the optimal expected profit and a performance guarantee for the fixed proportions solution in the random proportions model.