Multi-stage and Multi-customer Assortment Optimization with Inventory Constraints

TL;DR

This paper develops approximation algorithms for multi-stage assortment optimization with inventory constraints, achieving improved ratios over previous methods for special and general cases, including online stochastic matching.

Contribution

It introduces the first constant-factor approximation ratios for multi-customer assortment problems with inventory constraints, improving existing bounds and establishing tight ratios.

Findings

Achieved a 0.51 approximation ratio for the special case, improving previous 0.46.

Established a 0.09 approximation ratio for diverse customer valuations.

Developed LP rounding algorithms that improve previous randomized schemes.

Abstract

We consider an assortment optimization problem where a customer chooses a single item from a sequence of sets shown to her, while limited inventories constrain the items offered to customers over time. In the special case where all of the assortments have size one, our problem captures the online stochastic matching with timeouts problem. For this problem, we derive a polynomial-time approximation algorithm which earns at least 1-ln(2-1/e), or 0.51, of the optimum. This improves upon the previous-best approximation ratio of 0.46, and furthermore, we show that it is tight. For the general assortment problem, we establish the first constant-factor approximation ratio of 0.09 for the case that different types of customers value items differently, and an approximation ratio of 0.15 for the case that different customers value each item the same. Our algorithms are based on rounding an LP relaxation for multi-stage assortment optimization, and improve upon previous randomized rounding schemes to derive the tight ratio of 1-ln(2-1/e).