# Genetic Algorithm for the Retailers' Shelf Space Allocation Profit Maximization Problem
## 8 Conclusions  
In this paper, an example of using combinatorial optimization in solving a practical problem is presented. We studied the problem of SSAP to maximize the total planogram profit. It used a practical mathematical model to generate automatic planograms that may be of different lengths in each store but allocate the same product set. We found no previous articles examining so many constraints in SSAP simultaneously; only facings had been investigated, despite real-world retailers having ever more restrictions to satisfy. This study augmented the basic SSAP model with retail merchandising tactics, resulting in a set of constraints grouped into four categories: shelf (capacity); shelf type (vertical level effect); product (regional, complementary, etc.); and virtual segment (location, grouping) constraints. It further enriched the previous models with complex product representation of facings, cappings per facings group, and nestings per facings position.  
This problem is an extension of a knapsack problem classified as NP-hard, so heuristic and metaheuristic approaches were used. A GA was implemented; the most valuable part of it is the improvement procedure that provides the practical guidelines for solution enhancement. The quality of the solution was compared to the CPLEX solver.  
Based on the experimental results, the GA proposed in this paper rapidly gives high-quality results for different problem instances: on average, 2.52 min for GA\({}^{+}\) with Tournament, 2.81 min for GA\({}^{+}\) with Two Rankings, 3.29 min for GA\({}^{+}\) with Roulette Wheel. Interestingly, GA\({}^{+}\) with Two Rankings and GA\({}^{+}\) with Roulette Wheel were better than the CPLEX in 18 cases out of 22, and GA\({}^{+}\) with Tournament was better in 16 cases out of 22 the solver within the same predefined solution time limitation. Notably, GA\({}^{+}\) finds a more stable solution set than GA\({}^{-}\), which is better by approximately 60-65% due to the improvement procedure. Moreover, GA\({}^{+}\) gives 23-32% and 95-97% less result deviation, which is not highly dependent on the initial solution or random values selected.  
The proposed recursive algorithm for allocating products on a shelf may be used in solving other knapsack-like problems. GA is a promising approach for solving combinatorial SSAP optimization tasks. Regarding its limitations, it only finds the possibility that the products can be physically allocated on the shelves in the defined virtual segments. Still, it does not find the products' exact coordinates on the shelves. Another limitation is that the depth (of the shelf and product) parameter was omitted. The solution could be further expanded, also with the possibility of sorting products on the shelf according to their price, brand, category, and so on. Therefore, the directions of our further investigation include enhancing the model with the effects of uncertain demand as well as space- and cross-space elasticity effects, and including depth of products and shelves.  
**Author Contributions:** Conceptualization, K.C., K.S.-W., and M.H.; Data curation, K.C.; formal analysis, K.C., P.S., and A.R.; funding acquisition, M.H.; investigation, K.C.; methodology, K.C.; project administration, M.H.; resources, K.C. and M.H.; software, K.C.; supervision, K.S.-W. and M.H.; validation, K.C. and M.H.; visualization, K.C.; writing--original draft, K.C.; writing--review and editing, K.C., K.S.-W., P.S., and A.R. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the Ministry of Science and Higher Education in Poland under the program \"Regional Initiative of Excellence\" 2019-2022 project number 015/RID/2018/19, total funding amount 10,721,040,00 PLN.  
Not applicable.  
The authors declare no conflict of interest.