# Genetic Algorithm for the Retailers' Shelf Space Allocation Profit Maximization Problem
## 1 Introduction  
Retailing is a very competitive industry highly dependent on profit [1]. Efficient shelf space allocation is critical in gaining a competitive advantage [2]. The shelf on which products are being displayed is one of the most critical resources in the retail environment as retailers can increase their profit and simultaneously decrease cost by proper management of shelf space and product display [3]. Customers are also influenced by floor and space planning or assortment planning [4]. Retailers work with the goal of profit and sales maximization, allocating a limited amount of shelf space in a practicable way [5]. Retailers use a planogram to plan product allocation on shelves, which is a diagram or model that indicates the placement of offered products on shelves [6, 7, 8, 9]. Of course, maximization of sales is dependent not only on efficient category planning but also on marketing and merchandising of products. Parameters such as unusual product shapes, colors, promotional additions, etc., may disturb the perception of shelves and product allocation. Still, the retailer's regular work lies in consecutive decisions regarding the following problems: to determine the assortment size, allocate the assortment product items to the shelves, specify prices and discounts, and manage in-store shelf replenishment. Retail shelf management involves quickly and cost-effectively matching retail operations with customer demand [10]. Shelf space seems to be a scarce and expensive resource in the retail industry as a large number of products compete for limited display space. Therefore, shelf-space allocation is frequently implemented in shops to increase product sales and profits [11]. The problem of optimally allocating products presentation spaces to maximize revenues is of great importance to retailers [12]. Shelf space planning designates the position of products on a shelf and the number of facings of each selected product under limited shelf size constraints. Actual product sales may depend on the product's quantity and location on the shelf [13]. The shelf space allocation problem (SSAP) is a decision problem where the goal is to achieve the best possible result under various operational constraints in a retail store. It is regarded as an extension of the knapsack problem [14]. The SSAP aims to determine the optimal mix of product displays to maximize profitability. Decisions on the right product at the right location with the proper space allocation are necessary when shelf space is limited [15]. The main SSAP literature research gaps are:  
* considering only length limit as shelf capacity parameter;
* considering only basic constraints;
* omitting capping and nesting allocation parameters in all models;
* not investigating the shelf segments of flexible size. Most of the SSAP models in literature treat available shelf space one-dimensionally, estimating it in shelf length or in volume [2; 4; 5; 16; 17; 18; 19; 20; 21; 22]. Shelf capacity is usually defined as a single, real value. Another key point is that the SSAP literature omits the possibility of product capping (placing the product on the top of the product with rotation of the upper product to push in more products on the shelf space even if the product on its main orientation does not fit) and nesting (placing the product into the product as a bucket). There is no model that considers such decisions. Motivated by those findings, we focused on the SSAP to deal with how to efficiently allocate shelf space to each product to maximize the total profit volume. The aim of this paper is to develop a mathematical model and a genetic algorithm for improving the shelf space allocation with the criteria of retailers' profit maximization. The main contribution is related to the following issues. The majority of SSAP models use a basic set of constraints such as capacity, face size, lower and upper bound, allocation, and profit. However, in retail real-life, there are significantly more constraints.