Near-Optimal Algorithms for the Assortment Planning Problem Under Dynamic Substitution and Stochastic Demand

Abstract

Assortment planning of substitutable products is a major operational issue that arises in many industries such as retailing, airlines, and consumer electronics. We consider a single-period joint assortment and inventory planning problem under dynamic substitution with stochastic demands, and provide complexity and algorithmic results as well as insightful structural characterizations of near-optimal solutions for important variants of the problem. First, we show that the assortment planning problem is NP-hard even for a very simple consumer choice model, where each consumer is willing to buy only two products. In fact, we show that the problem is hard to approximate within a factor better than 1 − 1/e. Second, we show that for several interesting and practical customer choice models, one can devise a polynomial-time approximation scheme (PTAS), i.e., the problem can be solved efficiently to within any level of accuracy. To the best of our knowledge, this is the first efficient algorithm with provably near-optimal performance guarantees for assortment planning problems under dynamic substitution. Quite surprisingly, the algorithm we propose stocks only a constant number of different product types; this constant depends only on the desired accuracy level. This provides an important managerial insight that assortments with a relatively small number of product types can obtain almost all of the potential revenue. Furthermore, we show that our algorithm can be easily adapted for more general choice models, and we present numerical experiments to show that it performs significantly better than other known approaches.

Product Assortment and Price Competition under Multinomial Logit Demand

“…The case of customers looking for substitutes if their choice is stocked out, known as stock‐out‐based substitution, was studied in Smith and Agrawal (), Mahajan and van Ryzin () and more recently Goyal et al. (). We also refer the reader to Rooderkerk et al.…”

Ranking an Assortment of Products Via Sequential Submodular Optimization

“…Kök et al 2008a, Lancaster 1990, Ho and Tang 1998 for a comprehensive study. Later work have considered various consumer choice models, e.g., multinomial logit models (Talluri and van Ryzin 2004, Liu and van Ryzin 2008, Topaloglu 2013, the Lancaster choice model (Gaur and Honhon 2006), ranked-list preference (Honhon et al 2010, Goyal et al 2016, and non-parametric (data-driven) choice model (Farias et al 2013). The main difference of our model with the assortment optimization literature is that we optimize over permutations of products rather than subsets.…”

An Exact Method for Assortment Optimization Under the Nested Logit Model

“…In a different context, Goyal et al [16] considered a problem where the quality of different products is similar and customers differentiate products based on their price. Therefore, customers' preferences have a form of nested lists which means products are ordered by increasing selling price per unit and customers always prefer the cheapest product.…”

“…This class of problems has important applications in retail, online advertising, or revenue management, see, e.g. [3,18,16]. The research of Talluri and Van Ryzin [27] was the first of its kind to demonstrate the importance of incorporating the choice behavior of customers when deciding which products to offer in an assortment.…”