Explainable Data-driven Share-of-choice Product Line Design Optimization

Abstract

The share-of-choice (SOC) problem is a widely studied problem for product line design (PLD) where representative customers are sampled from a target population and the percentage of the ones who choose the offered products over outside options over the sample is maximized. The utility maximization framework captures individual choices. A significant challenge is that these utilities are not directly observable and must be estimated from other primitive data. Conjoint analysis is a commonly applied technique for generating such data, where sampled customers rate, rank, or choose between different product alternatives. With the responses, various methods, such as the hierarchical Bayesian method and polyhedral estimation, are employed to estimate the utilities. However, this " estimate-then-optimize" procedure disconnects the decision-making process from the primitive data and thus cannot attribute the PLD decision to the unresolved survey data. To fill the gap, we integrate the polyhedral estimation method into the PLD optimization, which constructs a polyhedron set containing all utilities consistent with the responses. Specifically, we proposed a robust model that maximizes the share-of-choice calculated by the worst-case utility over the polyhedron. The model brings threefold explainability to the PLD problem. First, it enables us to explore the influence of both estimation error and the number of sampled customers through an out-of-sample performance guarantee of the proposed model. Secondly, the linearly tractable equivalent reformulation constructed by the duality allows us to attribute the optimal product line to the survey data through the dual variables. Third, inspired by the fact that the survey questions are associated with columns in the dual reformulation, we propose a column generation approach to identify new questions that most effectively improve the model's objective.