The Combinatorial Optimization Problem (COPs) is one of the branches of applied mathematics and computer sciences, which is accompanied by many problems such as Facility Layout Problem (FLP), Vehicle Routing Problem (VRP), etc. Even though the use of several mathematical formulations is employed for FLP, Quadratic Assignment Problem (QAP) is one of the most commonly used. One of the major problems of Combinatorial NP-hard Optimization Problem is QAP mathematical model. Consequently, many approaches have been introduced to solve this problem, and these approaches are classified as Approximate and Exact methods. With QAP, each facility is allocated to just one location, thereby reducing cost in terms of aggregate distances weighted by flow values. The primary aim of this study is to propose a hybrid approach which combines Discrete Differential Evolution (DDE) algorithm and Tabu Search (TS) algorithm to enhance solutions of QAP model, to reduce the distances between the locations by finding the best distribution of N facilities to N locations, and to implement hybrid approach based on discrete differential evolution (HDDETS) on many instances of QAP from the benchmark. The performance of the proposed approach has been tested on several sets of instances from the data set of QAP and the results obtained have shown the effective performance of the proposed algorithm in improving several solutions of QAP in reasonable time. Afterwards, the proposed approach is compared with other recent methods in the literature review. Based on the computation results, the proposed hybrid approach outperforms the other methods.

The main objective of facility location problem is the utilization of the facility by maximum number of possible customers so that the profit is maximized. For instance, in some services like wireless sensor networks, Wi-Fi, repeaters, etc., where the service area is limited, some specific equipment is installed in such a way that it could be used by maximum number of users. Here, the number of users for a particular facility is optimized with the help of clustering technique. The study develops a model for facility allocation problem. For the solution algorithm, a hybrid approach which is based on clustering and mixed integer linear programming (MILP) is proposed. The proposed method consists of two parts where in the first part, the K-means clustering technique is used and in the second part, for each cluster an MILP technique is implemented so that the facility which yields the maximum profit is obtained. Numerical examples for clustering and without clustering are presented. Analysis shows that due to clustering the average distance between facility and customer is significantly reduced.