# Genetic Algorithm for the Retailers' Shelf Space Allocation Profit Maximization Problem
## 2 Theoretical Background  
Many types of research have focused on SSAP in the context of retailers' profit maximization as efficient shelf-space management influence on product sales [18; 23; 24]. According to [PERSON] et al. [25] and [PERSON] et al. [26], attention has been given to shelf space allocation models [27]. [PERSON] and [PERSON] [28; 29] provided a model which combines assortment and shelf space planning decisions and considered stochastic and space-elastic demand as well as out-of-assortment and out-of-stock substitution effects. Practical guidelines of the impact of cross-space elasticities and product substitutions on shelf space decisions are given to retailers by [PERSON] and [PERSON] [30]. [PERSON] [16] models the demand function, including the inventory level dependent decisions, and divided the backroom and the showroom inventory. [PERSON] and [PERSON] [28] incorporated the optimization model with in-store replenishment processes. The demand increased if there was more shelf space designated for a product because of the higher visibility while replenishment was decreasing and inventory costs were also increasing. Generating planograms is a complicated and time-consuming task due to the vast number of retail and marketing constraints, the majority of which are non-standard. The problem is already reduced to a multi-knapsack, one of which is NP-hard [4]. Commercial software uses heuristics rules to solve the SSAP to help retailers [10; 13; 14; 16; 28; 29; 31]. Researchers apply specialized heuristic and metaheuristic search algorithms to the SSAP to reduce the time to solve the problem [21]. Unfortunately, such algorithms do not guarantee an optimal solution globally. Heuristic solutions are also used for planogram evaluation, which involves comparing the product positioning on the planogram and the template planogram. [PERSON] et al. [9] proposed a heuristic approach that finds and counts multiple instances of the same product in planogram maintenance. [PERSON] et al. [32] introduced a model that combines shelf space allocation, assortment, and inventory processes using a heuristic solution approach based on simulated annealing. [PERSON] and [PERSON] [33] introduced a simulation-optimization approach for replenishment planning and shelf space allocation, taking into account customers' shopping paths with the demand for various products. They developed a genetic algorithm and hybrid genetic algorithm with an imperialist competitive algorithm. [PERSON] [16] generalized inventory management and SSAPs in retail management and proposed a greedy heuristic and genetic algorithm for the consolidated problem. [PERSON] et al. [5] developed a model to evaluate the effects of reallocating the product on the shelf and its influence on sales of individual brands within the product category. [PERSON] [18] used a knapsack formulation problem model and demonstrated a heuristic approach for solving some examples of the SSAP. [PERSON] and [PERSON] [4] investigated automatic planogram generation using a simulated annealing-based hyper-heuristic framework and a set of low-level heuristics. They formulated a model as an extension of the multi-knapsack problem. [PERSON] and [PERSON] [17] proposed a non-linear integer programming model for shelf space allocation that could be used in retail. They categorized shelf space management strategies (dominance, adaptation, passiveness) based on the questionnaire approach. [PERSON] and [PERSON] [34] formulated a non-linear integer SSAP problem taking into account the dependency between profitability and customer service factors. [PERSON] and [PERSON] [22] provided retail SSAP with a linear profit function and developed heuristics to solve the model. [PERSON], [PERSON], and [PERSON] [2] extended the basic, linear shelf space allocation model with product grouping and non-linear profit functions, developed a network-flow model and used local search with a metaheuristic to find near-optimal solutions.