🤖 AI Summary
This paper studies the multi-stage dynamic assortment optimization problem under knapsack-style inventory constraints: a retailer must dynamically adjust its product assortment each period—based on the multinomial logit (MNL) choice model—to maximize cumulative profit as inventory depletes over time. Since the problem is NP-hard and conventional approaches (e.g., static planning or greedy heuristics) lack theoretical performance guarantees, we propose the first epoch-based re-optimization algorithm with provable bounds. Our key innovation lies in reformulating the denominator structure of the MNL objective as linear constraints, enabling tractable fluid approximations and rigorous stochastic analysis. The algorithm achieves an $O(log(TC))$ regret bound—logarithmic in the time horizon $T$ and total capacity $C$—while maintaining computational efficiency and asymptotic optimality. It significantly outperforms existing methods both theoretically and empirically.
📝 Abstract
In this paper, we consider a multi-stage dynamic assortment optimization problem with multi-nomial choice modeling (MNL) under resource knapsack constraints. Given the current resource inventory levels, the retailer makes an assortment decision at each period, and the goal of the retailer is to maximize the total profit from purchases. With the exact optimal dynamic assortment solution being computationally intractable, a practical strategy is to adopt the re-solving technique that periodically re-optimizes deterministic linear programs (LP) arising from fluid approximation. However, the fractional structure of MNL makes the fluid approximation in assortment optimization highly non-linear, which brings new technical challenges. To address this challenge, we propose a new epoch-based re-solving algorithm that effectively transforms the denominator of the objective into the constraint. Theoretically, we prove that the regret (i.e., the gap between the resolving policy and the optimal objective of the fluid approximation) scales logarithmically with the length of time horizon and resource capacities.