Assortment optimization given basket shopping behavior using the Ising model

TL;DR

Addressing basket-based assortment optimization, the paper develops an Ising-model-based Markov random field to capture item-level attractiveness $\theta_{ii}$ and pairwise dependencies $\theta_{ij}$, yielding choice probabilities $p_{\theta}(x|S)$ for baskets. It proves the problem is APX-hard and introduces a graph-based preprocessing to reduce problem size, plus parameter estimation methods (approximate sparse maximum likelihood and density consistency) and a Gibbs-sampling based evaluation. It then proposes several heuristics, notably Simulated Annealing, achieving about $15\%$ profit gain over offering all products and around $5\%$ over revenue-ordered baselines, highlighting practical impact for basket-aware retailers. The work further connects Ising models to MVL, suggesting extensions via network analytics and more general MRFs for richer basket phenomena.

Abstract

In markets where customers tend to purchase baskets of products rather than single products, assortment optimization is a major challenge for retailers. Removing a product from a retailer's assortment can result in a severe drop in aggregate demand if this product is a complement to other products. Therefore, accounting for the complementarity effect is essential when making assortment decisions. In this paper, we develop a modeling framework designed to address this problem. We model customers' choices using a Markov random field -- in particular, the Ising model -- which captures pairwise demand dependencies as well as the individual attractiveness of each product. Using the Ising model allows us to leverage existing methodologies for various purposes including parameter estimation and efficient simulation of customer choices. We formulate the assortment optimization problem under this model and show that it is APX-hard. We also provide multiple theoretical insights into the structure of the optimal assortments based on the graphical representation of the Ising model, and propose several heuristic algorithms that can be used to obtain high-quality solutions to the assortment optimization problem. Our numerical analysis demonstrates that the developed simulated annealing procedure leads to an expected profit gain of 15% compared to offering an unoptimized assortment (where all products are included) and around 5% compared to using a revenue-ordered heuristic algorithm.

Figures (10)

• Figure 1: Sparse graphical representation of the Ising model for the Bakery dataset.
• Figure 2: Ising model parameters estimated using (a) approximate maximum likelihood estimation and (b) Density Consistency estimation for the Bakery dataset.
• Figure 3: Example of when it is profitable to exclude a product from the assortment even though there are only positive edges connected to it.
• Figure 4: Distribution of edge weights (Ising model couplings) for real-world datasets.
• Figure 5: Number of isolated nodes as a function of the proportion of edges removed.

Theorems & Definitions (14)

• Definition 1
• Definition 2
• Theorem : Hammersley-Clifford
• Proposition 1
• Corollary 1
• Theorem 1
• Remark 1
• Proposition 2
• Example 1
• Proposition 3