A Unified Framework to Impose Market Share Constraints for Selected Product Classes: Randomized and Deterministic Assortments Under the Multinomial Logit Model.

The article focuses on assortment optimization problems under the multinomial logit model with selective market share constraints imposed only on product classes represented in the offered assortment. It formulates two variants: a randomized variant allowing probabilistic offering of assortments, and a deterministic variant offering a single fixed assortment. The study establishes the NP-hardness of these problems, develops a unified approximation framework, and provides a fully polynomial-time approximation scheme (FPTAS) for the randomized variant, as well as approximation algorithms for the deterministic variant with controlled market share violations and a pseudopolynomial-time constant-factor approximation algorithm satisfying constraints exactly. Computational experiments using real-world retail data demonstrate that imposing market share constraints can significantly increase the minimum market shares of product classes with minimal loss in expected revenue, thereby balancing demand volumes across classes more effectively than hard constraints.