Optimizing Expected Utility in a Multinomial Logit Model with Position Bias and Social Influence

Motivated by applications in retail, online advertising, and cultural markets, this paper studies how to find the optimal assortment and positioning of products subject to a capacity constraint. We prove that the optimal assortment and positioning can be found in polynomial time for a multinomial logit model capturing utilities, position bias, and social influence. Moreover, in a dynamic market, we show that the policy that applies the optimal assortment and positioning and leverages social influence outperforms in expectation any policy not using social influence

An Exact Method for Assortment Optimization under the Nested Logit Model

We study the problem of finding an optimal assortment of products maximizing the expected revenue, in which customer preferences are modeled using a Nested Logit choice model. This problem is known to be polynomially solvable in a specific case and NP-hard otherwise, with only approximation algorithms existing in the literature. For the NP-hard cases, we provide a general exact method that embeds a tailored Branch-and-Bound algorithm into a fractional programming framework. Contrary to the existing literature, in which assumptions are imposed on either the structure of nests or the combination and characteristics of products, no assumptions on the input data are imposed, and hence our approach can solve the most general problem setting. We show that the parameterized subproblem of the fractional programming scheme, which is a binary highly non-linear optimization problem, is decomposable by nests, which is a main advantage of the approach. To solve the subproblem for each nest, we propose a two-stage approach. In the first stage, we identify those products that are undoubtedly beneficial to offer, or not, which can significantly reduce the problem size. In the second stage, we design a tailored Branch-and-Bound algorithm with problem-specific upper bounds. Numerical results show that the approach is able to solve assortment instances with up to 5,000 products per nest. The most challenging instances for our approach are those in which the dissimilarity parameters of nests can be either less or greater than one

Combinatorial Assortment Optimization

Assortment optimization refers to the problem of designing a slate of products to offer potential customers, such as stocking the shelves in a convenience store. The price of each product is fixed in advance, and a probabilistic choice function describes which product a customer will choose from any given subset. We introduce the combinatorial assortment problem, where each customer may select a bundle of products. We consider a model of consumer choice where the relative value of different bundles is described by a valuation function, while individual customers may differ in their absolute willingness to pay, and study the complexity of the resulting optimization problem. We show that any sub-polynomial approximation to the problem requires exponentially many demand queries when the valuation function is XOS, and that no FPTAS exists even for succinctly-representable submodular valuations. On the positive side, we show how to obtain constant approximations under a "well-priced" condition, where each product's price is sufficiently high. We also provide an exact algorithm for $k$-additive valuations, and show how to extend our results to a learning setting where the seller must infer the customers' preferences from their purchasing behavior