On the Linear Integration of Attraction Choice Models in Business Optimization Problems

Decision problems from various fields (e.g., assortment optimization, product line selection, location planning) require to endogenously incorporate probabilistic choice behavior in dependence of the availability of given choice alternatives. A widely spread demand model in marketing and econometrics to represent such choices is the attraction choice model. Of this model, the well-known multinomial logit model and—in case of multiple latent customer segments—the finite-mixture logit model are special cases. However, integrating such models in optimization problems results in non-linear formulations. Thus, in recent years, several exact linearization approaches have been proposed. These approaches are based on different ideas, and they have appeared independently from each other in different fields of research. Thus, the question arises how these approaches differ and how they relate to each other. In this short communication, we settle this question by arguing that many of the proposed approaches—even though they might seem different at first glance—can be traced back to one of two underlying linearization ideas. Establishing a generic problem, we discuss the two ideas in a unified way by presenting two corresponding general model formulations that are shown to be equivalent. Based upon this, we are able to classify the major publications which integrate some type of attraction choice model in detail. In particular, for each formulation of the analyzed literature, we explain to which extent it is a special case of (one of) the presented generic formulations. This also makes clear under which context-specific conditions certain elements of the generic linearization can be omitted, potentially serving as helpful guideline for future applications of such linearizations.