Pricing Problems Under the Nested Logit Model with a Quality Consistency Constraint

We consider pricing problems when customers choose among the products according to the nested logit model and there is a quality consistency constraint on the prices charged for the products. We consider two types of quality consistency constraints. In the first type of constraint, there is an inherent ordering between the qualities of the products in a particular nest and the price for the product of a higher quality level should be larger. In the second type of constraint, different nests correspond to different quality levels and the price of any product that is in a nest corresponding to a higher quality level should be larger than the price of any product that is in a nest corresponding to a lower quality level. The prices of the products are chosen within a finite set of possible prices. We develop algorithms to find the prices to charge for the products to maximize the expected revenue obtained from a customer, while adhering to a quality consistency constraint. Our algorithms are based on solving linear programs whose sizes scale polynomially with the number of nests, number of products and number of possible prices for the products. Numerical experiments indicate that our algorithms can effectively compute the optimal prices even when there is a large number of products in consideration. Authors are affiliated with School of Operations Research and Information Engineering, Cornell University, Ithaca, New York 14853, USA. Their email addresses are [email protected], [email protected] and [email protected]. In many retail environments, there are multiple substitutable products that can serve the needs of a customer and customers make a choice among the available products by comparing them with respect to attributes such as price, quality, richness of features and durability. When such substitution possibilities are present, the demand for a product depends not only on its own attributes, but also on the attributes of the other products, creating interactions between the demands for different products. Discrete choice models become useful to capture such demand interactions, since they represent the demand for a particular product as a joint function of the attributes of all available products. Capturing the interactions between the demands for the products has recently become more important than ever, as online retailers and travel agencies bring a large variety of options to customers. Nevertheless, optimization models that try to find the right prices to charge for the products quickly become complicated when one uses sophisticated choice models to capture the interaction between the demands for the products. These optimization models become even more complicated when one tries to impose operational constraints on the prices charged for the products. In this paper, we consider pricing problems when customers choose according to the nested logit model and there is a quality consistency constraint on the prices charged for the products. Under the nested logit model, the products are grouped into nests. The choice process of the customer proceeds in two stages. In the first stage, the customer decides either to make a purchase in one of the nests or to leave the system without making a purchase. In the second stage, if the customer decides to make a purchase in one of the nests, then the customer chooses one of the products in the chosen nest. This choice process is shown in Figure 1.a. The customer starts from the root node of the tree. In the first stage, she chooses one of the nests or the no purchase option. In the second stage, if she has chosen one of the nests in the first stage, then she selects one of the products in the chosen nest. In the quality consistency constraint that we impose on the prices, there is an intrinsic ordering between the qualities of the products. The quality consistency constraint ensures that the prices charged for the products of higher quality are also larger. The goal is to find the prices to charge for the products to maximize the expected revenue obtained from a customer, while making sure that the prices satisfy the quality consistency constraint. We consider two types of quality consistency constraint. In the first type of constraint, there is an intrinsic ordering between the qualities of the products in each nest. We refer to this quality consistency constraint as price ladders inside nests. Figure 1.b illustrates this quality consistency constraint with three products in each nest and the price of product j in nest i is denoted by pij . The products in each nest are indexed such that the third product is of higher quality than the second product in the same nest, which is, in turn, of higher quality than the first product. Therefore, the price of the third product should be larger than the price of the second product, which should, in turn, be larger than the price of the first product. There is no dictated ordering between the qualities or prices of the products in different nest. In the second type of constraint, there is an intrinsic ordering between the qualities of the nests, but there is no clear ordering between the qualities of the products in the same nest. We refer to this quality consistency constraint as price