# Using Product Network Analysis to Optimize Product-to-Shelf Assignment Problems
## 1 Introduction  
The development of online shopping has created intense competition for traditional retail businesses. These firms have sought to respond by improving their business strategies in terms of ordering, pricing, advertising, shelving, and so on. Effective shelf-space management is a crucial factor in maximizing store profit, minimizing inventories, and building strong relationships with vendors [1]. Space elasticity has been widely used to estimate the relationship between sales and allocated space [2, 3, 4], and previous studies have used space elasticity to establish a relationship between shelf space and product demand. However, using space elasticity for shelf-space allocation requires estimating a great number of parameters, resulting in high costs and high error rates in the mathematical models [5]. Recently, advances in information technology have made it easier for retailers to collect various types of customer data [6]. Mining this data for insight into customer behavior can help retailers solidify ephemeral relationships with customers into long-term loyalty. As part of this effort, retailers havemodified their shelf-space management practices using product association analysis [7; 8; 9; 10], a powerful data mining tool that can detect significant co-purchase rules underlying a large amount of purchase transaction data. Although previous studies have demonstrated that product association analysis can improve the efficiency of shelf space usage and increase cross-selling possibility, three problems remain to be resolved. First, most previous studies simply assume that all products in the store can be re-organized based on the outcome of product association analysis. However, moving critical products might disrupt existing customer shopping patterns/rules. In reality, some products are critical to attracting customers and moving them could break existing linkages. Therefore, it might be desirable to analyze the relationships among products before deciding which products are to be re-organized. Second, previous studies have seldom considered the physical proximity of shelf locations when evaluating assignment task. For example, product A and product B are candidates to be assigned to Shelf 1. The original shelf for product A (denoted as Shelf 2) is one foot away from Shelf 1, while the original shelf for product B (denoted as Shelf 3) is ten feet away from Shelf 1. Previous studies might randomly pick a product and assign it to Shelf 1. However, Shelf 1 is much closer to Shelf 2 than to Shelf 3, and failure to consider physical proximity results in some products being reassigned to shelves far from their current locations, which not only incurs a reallocation cost but also increases inconvenience for customers trying to find specific products. Third, previous studies simply assume that products can be reassigned to any locations in the store. In reality, products belong to certain categories and should be grouped with other items within the same category to prevent customer confusion. To solve these problems, this study proposes a three-stage product-to-shelf assignment method that considers customer purchase behaviors as well as physical and category constraints. The first stage constructs a Product Relationship Network (PRN) that represents the purchase association among product items. Two types of PRNs are studied: Transaction-based networks (TBN) and customer-based networks (CBN), which will be described later in Section 3.2. The second stage uses network analysis to derive the centrality value of each product item, which reveals the importance of a product item in the network. Based on its centrality value, an item is classified as an attraction item, an opportunity item, or a trivial item (see Section 3.3.2 for definitions). Opportunity items are those that should be reorganized/re-shelved. The third stage integrates purchase association, geographic relationship, and category constraints to evaluate the location preference of each product.