Capacitated Vehicle Routing Problem (CVRP) is considered as one of the most famous specialized forms of VRP that has attracted considerable attention from researchers. This problem belongs to complex combinatorial optimization problems included in the NP-Hard Problem category, which is a problem that needs difficult computation. This paper presents an improvement of Ant Colony System (ACS) to solve this problem. In this study, the problem deals with a few vehicles which are used for transporting products to specific places. Each vehicle starts from a main location at different times every day. The capacitated vehicle routing problem (CVRP) is defined to serve a group of delivery customers with known demands. The proposed study seeks to find the best solution of CVRP by using improvement ACS with the accompanying targets: (1) To decrease the distance as long distances negatively affect the course of the process since it consumes a great time to visit all customers. (2) To implement the improvement of ACS algorithm on new data from the database of CVRP. Through the implementation of the proposed algorithm better results were obtained from the results of other methods and the results were compared.

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.