Assortment Optimization under the Paired Combinatorial Logit Model

We consider uncapacitated and capacitated assortment problems under the paired combinatorial logit model, where the goal is to find a set of products to maximize the expected revenue obtained from each customer. In the uncapacitated setting, we can offer any set of products, whereas in the capacitated setting, there is a limit on the number of products that we can offer. We establish that even the uncapacitated assortment problem is strongly NP-hard. To develop an approximation framework for our assortment problems, we transform the assortment problem into an equivalent problem of finding the fixed point of a function, but computing the value of this function at any point requires solving a nonlinear integer program. Using a suitable linear programming relaxation of the nonlinear integer program and randomized rounding, we obtain a 0.6-approximation algorithm for the uncapacitated assortment problem. Using randomized rounding on a semidefinite programming relaxation, we obtain an improved, but a more complicated, 0.79-approximation algorithm. Finally, using iterative variable fixing and coupled randomized rounding, we obtain a 0.25-approximation algorithm for the capacitated assortment problem. Our computational experiments demonstrate that our approximation algorithms, on average, yield expected revenues that are within 3.6% of a tractable upper bound on the optimal expected revenues.