Capacitated assortment and price optimization under the nested logit model

We study the capacitated assortment and price optimization problem, where a retailer sells categories of substitutable products subject to a capacity constraint. The goal of the retailer is to determine the subset of products as well as their selling prices so as to maximize the expected revenue. We model the customer purchase behavior using the nested logit model and formulate this problem as a non-linear binary integer program. For this NP-complete problem, we show that there exists a pseudo polynomial time approximation scheme that finds its $$\epsilon $$ ϵ -approximate solution. We first convert the original problem into an equivalent fixed point problem. We then show that finding an $$\epsilon $$ ϵ -approximate solution to the fixed point problem can be achieved by binary search, where a non-linear auxiliary problem is repeatedly approximated by a dynamic programing based algorithm involving an approximation to a series of multiple-choice parametric knapsack problems. For the special case when the capacity constraints are cardinal and nest-specific, we develop an algorithm that finds the optimal solution in strongly polynomial time. Moreover, our algorithm can be directly applied to find an $$\epsilon $$ ϵ -approximate solution to the capacitated assortment optimization problem under the nested logit model, which is the first approximate algorithm that is polynomial with respect to the number of nests in the literature.