Joint Assortment and Quality Consistent Discrete Pricing Under the Nested Logit Model

Discrete choice models have been used for several decades to capture central features of customer preferences. In particular, they provide a compact expression of how product demand changes as the offered assortment changes in composition or price. Some models, such as the basic attraction model, have been justified with axioms; see [14]. Alternatively, models like the multinomial logit (MNL) model have been justified with random utility theory; see [15]. This paper considers an important extension to the MNL model: the nested logit (NL) model introduced by [23]. Justifications and extensions for the NL model are provided in [5, 16]. The NL model was developed primarily to avoid the independence of irrelevant alternatives property suffered by the MNL model; see [4]. Under the NL model customers first select a nest and then a product within the selected nest. The nesting structure is designed to capture correlations among product utilities, where closely correlated products are in the same nest. A nest can be a particular product category, products with closely related features, or a distinct sales channel. In this paper we study the joint assortment and quality consistent pricing problem where customer choice is governed by the NL model. A key piece of input to the problem is a quality ranking among the products where higher ranked items are higher quality. In addition to the quality ordering we are given a cost for each product, a discrete collection of prices at which products can be priced, and the parameters that govern the NL model. Our objective is to find an assortment of products to offer, and quality consistent prices to set, so that expected revenue is maximized. Prices are quality consistent when higher ranked items have higher prices. We assume that each item has some fixed cost per sale and that higher quality items have higher costs. We make only a mild assumption on how the parameters of the NL model depend on price; we require that the probability of a customer purchasing a product does not increase when the price of the product is increased. The joint assortment and quality consistent pricing problem using the MNL model has been studied in [6]. In [6] they present a polynomial time algorithm that gives an optimal assortment and prices. Their technique uses a general result: any constrained assortment optimization problem under the MNL model where the constraints are totally unimodular can be modelled with a linear program. For assortment optimization problems under the NL model there is no such linear program. This greatly complicates assortment optimization problems under the NL model and, in particular, prevents us from applying the techniques of [6] to our problem. Additionally, the quality consistent constraint in the NL model does not appear to be captured by any totally unimodular matrix.