Nowadays, extensive research is being done in the area of revenue management, with applications across industries. In the center of this area lays the assortment problem, which amounts to find a subset of products to offer in order to maximise revenue, provided that customers follow a certain model of choice. Most studied models satisfy the following property: whenever the offered set is enlarged, then the probability of selecting a specific product decreases. This property is called regularity in the literature. However, customer behaviour often shows violations of this condition such as the decoy effect, where adding extra options sometimes leads to a positive effect for some products, whose probabilities of being selected increase relative to other products (e.g., including a medium size popcorn slightly cheaper than the large one, with the purpose of making the latter more attractive by comparison). We study two models of customer choice where regularity violations can be accommodated (hence the non-conventionality), and show that the assortment optimisation problem can still be solved in polynomial time.
First we analyse the Sequential Multinomial Logit (SML). Under the SML model, products are partitioned into two levels, to capture differences in attractiveness, brand awareness and, or visibility of the products in the market. When a consumer is presented with an assortment of products, she first considers products on the first level and, if none of them is purchased, products in the second level are considered. This model is a special case of the Perception-Adjusted Luce Model (PALM) recently proposed by Echenique et al.(2018). It can explain many behavioural phenomena such as the attraction, compromise, similarity effects and choice overload which cannot be explained by the Multinomial Logit (MNL) model or any discrete choice model based on random utility. We show that the concept of revenue-ordered assortment sets, which contain an optimal assortment under the MNL model, can be generalized to the SML model. More precisely, we show that all optimal assortments under the SML are revenue-ordered by level, a natural generalization of revenue-ordered assortments that contains, at most, a quadratic number of assortments. As a corollary, assortment optimization under the SML is polynomial-time solvable
Secondly, the Two-Stage Luce model (2SLM), is a discrete choice model introduced by Echenique and Saito (2018) that generalizes the standard multinomial logit model (MNL). The 2SLM does not satisfy the Independence of Irrelevant Alternatives (IIA) property nor regularity, and to model customer behaviour, each product has an intrinsic utility, and uses a dominance relation between products.
Given a proposed assortment S, consumers first discard all dominated products in S before using an MNL model on the remaining products. As a result, the model can capture behaviour that cannot be replicated by any discrete choice model based on random utilities. We show that the assortment problem under the 2SLM is polynomially-solvable. Moreover, we prove that the capacitated assortment optimization problem is NP-hard and present polynomial-time algorithms for the cases where (1) the dominance relation is attractiveness correlated and (2) its transitive reduction is a forest. The proofs exploit a strong connection between assortments under the 2SLM and independent sets in comparability graphs.
The third and final contribution is an in-depth study of the pricing problem under the 2SLM. We first note that changes in prices should be reflected in the dominance relation if the differences between the resulting attractiveness are large enough. This is formalised by solving the joint assortment and pricing problem under the Threshold Luce model, where one product dominates another if the ratio between their attractiveness is greater than a fixed threshold. In this setting, we show that this problem can be solved in polynomial time
