When Location Shapes Choice: Placement Optimization of Substitutable Products

TL;DR

This work tackles optimal placement of substitutable products across multiple display locations under a two-stage browse-and-choose customer process. It develops a general, oracle-driven algorithm that, given an α-approximate cardinality-constrained assortment solver, achieves a revenue-approximation of Θ(α)/log m for Placement, with ρ capturing variation in location-visiting behavior; a tighter, deterministic algorithm is obtained for Markov choice models, and a (1−1/e) benchmark is proven when prices are identical. The paper also proves hardness results indicating limits of approximation in the general heterogeneous-price setting and validates the approach through extensive simulations on line and grid layouts, highlighting the value of product repetition and the practical scalability of the method. Overall, the results offer a versatile framework for revenue optimization via product placement under broad browsing and choice models, with strong theoretical guarantees and practical performance. Key insights show how leveraging existing assortment-optimization oracles can yield near-optimal placements in complex, distribution-driven environments. The work informs both theoretical and applied avenues in retail analytics, online marketplaces, and display optimization.

Abstract

Strategic product placement can have a strong influence on customer purchase behavior in physical stores as well as online platforms. Motivated by this, we consider the problem of optimizing the placement of substitutable products in designated display locations to maximize the expected revenue of the seller. We model the customer behavior as a two-stage process: first, the customer visits a subset of display locations according to a browsing distribution; second, the customer chooses at most one product from the displayed products at those locations according to a choice model. Our goal is to design a general algorithm that can select and place the products optimally for any browsing distribution and choice model, and we call this the Placement problem. We give a randomized algorithm that utilizes an $α$-approximate algorithm for cardinality constrained assortment optimization and outputs a $\frac{Θ(α)}{\log m}$-approximate solution (in expectation) for Placement with $m$ display locations, i.e., our algorithm outputs a solution with value at least $\frac{Ω(α)}{\log m}$ factor of the optimal and this is tight in the worst case. We also give algorithms with stronger guarantees in some special cases. In particular, we give a deterministic $\frac{Ω(1)}{\log m}$-approximation algorithm for the Markov choice model, and a tight $(1-1/e)$-approximation algorithm for the problem when products have identical prices.

Figures (7)

• Figure 1: Journeys of two different customers through a store. Figure reproduced from flamand2023store with permission.
• Figure 2: Screenshot from Amazon.com. The product on the extreme left (Amazon Essentials) is the same as the product in middle (also Amazon Essentials) and links to the same product page.
• Figure 3: Examples of different customer browsing behavior. There are 6 display locations. In $(a)$ the customer looks at products displayed on locations $\{2,3,4,5\}$ and in $(b)$ the customer looks at locations $\{1,2,3,6\}$. The sequence in which customer visits the locations is not important in our model.
• Figure 4: Example of Placement on a Line
• Figure 5: Example of a random walk on the two-dimensional grid.

Theorems & Definitions (55)

• Remark 1: Product Repetition
• Remark 2: Filling empty locations with highest price product
• Remark 3: Multiple products in a location
• Remark 4: Placement on a Line.
• Lemma 1
• Corollary 1
• Lemma 2
• Remark 5: Usage of Exact Revenue Oracle
• Remark 6: Deterministic vs. Randomized Algorithms
• Remark 7: Constructing the Sampling Oracle from Data