Solving utility-maximization selection problems with Multinomial Logit demand: Is the First-Choice model a good approximation?

Abstract

We investigate First-Choice (FC) assignment models, a simple type of choice model where customers are allocated to their highest utility option, as a heuristic or starting point for the Multinomial Logit (MNL) model in the context of selection problems with a utility maximization objective. This type of problem occurs in a variety of applications, from location problems to assortment planning or transportation planning. FC assignment models are less refined but computationally more tractable than the more commonly used MNL. MNL suffers from tractability issues due to its nonlinear structure when used within a large size optimization problem with binary decision variables. We design the first comparison of the two modeling frameworks in a context of customer utility maximization for selection problems with binary variables. We provide a probabilistic analysis of the expected customer choice probabilities, document the computational challenges faced by the MNL model in our setting and show in numerical experiments that the FC model exhibits excellent performance as an approximation of the MNL model with an average gap for instance of at most 2.2% for uniformly distributed utilities and of at most 1.4% for normally distributed utilities (and below 1% in a majority of test cases). The key contribution of this paper is to build the case for the FC model as a tractable, high-quality approximation of the MNL model for binary selection problems with utility maximization.