Nonparametric choice modeling: applications to operations management

We consider two broad setups: (1) solving decision problems using choice models, and (2) learning the choice models. In both setups, data available corresponds to marginal information about the underlying distribution over rankings. So the problems essentially boil down to designing the 'right' criterion to pick a model from one of the (several) distributions that are consistent with the available marginal information. First, we consider a central decision problem in operations management (OM): find an assortment of products that maximizes the revenues subject to a capacity constraint on the size of the assortment. Solving this problem requires two components: (a) predicting revenues for assortments and (b) searching over all subsets of a certain size for the optimal assortment. In order to predict revenues for an assortment, of all models consistent with the data, we use the choice model that results in the 'worst-case' revenue. We derive theoretical guarantees for the predictions, and show that the accuracy of predictions is good for the cases when the choice data comes from several different parametric models. Finally, by applying our approach to real-world sales transaction data from a major US automaker, we demonstrate an improvement in accuracy of around 20% over state-of-the-art parametric approaches. Once we have revenue predictions, we consider the problem of finding the optimal assortment. It has been shown that this problem is provably hard for most of the important families of parametric of choice models, except the multinomial logit (MNL) model. In addition, most of the approximation schemes proposed in the literature are tailored to a specific parametric structure. We deviate from this and propose a general algorithm to find the optimal assortment assuming access to only a subroutine that gives revenue predictions; this means that the algorithm can be applied with any choice model. We prove that when the underlying choice model is the MNL model, our algorithm can find the optimal assortment efficiently. Next, we consider the problem of learning the underlying distribution from the given marginal information. For that, of all the models consistent with the data, we propose to select the sparsest or simplest model, where we measure sparsity as the support size of the distribution. Finding the sparsest distribution is hard in general, so we restrict our search to what we call the 'signature family' to obtain an algorithm that is computationally efficient compared to the brute-force approach. We show that the price one pays for restricting the search to the signature family is minimal by establishing that for a large class of models, there exists a "sparse enough" model in the signature family that fits the given marginal information well. We demonstrate the efficacy of learning sparse models on the well-known American Psychological Association (APA) dataset by showing that our sparse approximation manages to capture useful structural properties of the underlying model. Finally, our results suggest that signature condition can be considered an alternative to the recently popularized Restricted Null Space condition for efficient recovery of sparse models. (Copies available exclusively from MIT Libraries, Rm. 14-0551, Cambridge, MA 02139-4307. Ph. 617-253-5668; Fax 617-253-1690.) (Abstract shortened by UMI.)