Assortment optimization under the multinomial logit choice model with product-specific capacities

We consider assortment optimization with product-specific capacities under the multinomial logit (MNL) choice model, wherein each product has a finite capacity, i.e., a limited number of units for sale. The number of customers served by each product is the smaller of its demand and capacity. We assume the demand of each product is deterministic as a function of the assortment offered. We show that this assortment optimization problem is NP- hard. We devise a 1/2-approximation algorithm and a fully polynomial-time approximation scheme (FPTAS) by exploiting its connection to a series of knapsack problems.