Assortment Optimization and the Sample Average Approximation
Hassaan Khalid, Bradley Sturt
TL;DR
This work tackles the computational bottleneck in solving large-scale assortment optimization problems under random utility maximization by leveraging the sample average approximation (SAA) and Monte Carlo methods. It introduces the Exclusion Set MILP formulation, a stronger and more compact alternative to existing MISIC formulations, and couples it with an accelerated Benders decomposition that yields Pareto-optimal cuts via a novel reformulation tied to separable convex optimization and isotonic regression. The paper develops two fast algorithms for cut generation (Phase 1: $O(L_k\log L_k)$ and Phase 2: $O(L_k)$) and provides a complete Pareto-cut transformation procedure, significantly reducing solve times on real-world datasets with thousands of products and rankings. Empirical results on synthetic and real data show speedups up to orders of magnitude, validating the practicality of SAA for assortment optimization when distribution-specific algorithms are lacking. Overall, the contributions expand the viability and scalability of SAA-based approaches for large-scale, complex choice models in revenue management and marketing.
Abstract
We consider a simple approach to solving assortment optimization under the random utility maximization model. The approach uses Monte-Carlo simulation to construct a ranking-based choice model that serves as a proxy for the true choice model, followed by finding an assortment that is optimal with respect to that proxy. In this paper, we make that approach more viable by developing faster algorithms for finding assortments that are optimal under ranking-based choice models. Our algorithms are based on mixed-integer programming and consist of stronger formulations as well as new structural and algorithmic results related to Benders cuts. We demonstrate that our algorithms - without any heuristics or parameter tuning - can offer more than a 20x speedup in real-world settings with thousands of products and samples. Equipped with our algorithms, we showcase the value of using the sample average approximation to solve assortment optimization problems for which no practically efficient algorithms are known.