Solving the vehicle routing problem with time windows using modified football game algorithm

The vehicle routing problem with time windows (VRPTW) is a famous problem in logistics and transportation that has many applications in real life. The objective of the problem is to find the minimum distance covered by a set of vehicles that set to start together from the warehouse and visit some prefixed customers within certain time frames. In this paper, an efficient optimization algorithm, called modified football game algorithm (MFGA), is proposed to solve the VRPTW as an NP-hard problem. MFGA imitates the behavior of football players during a game to find the best positions to score a goal under the supervision of a team coach. In the presented algorithm, a new method is presented to create an initial solution for the VRPTW problem. In addition, a more efficient method has been designed to create a new location for the player, which makes the algorithm more efficient. The performance of the proposed MFGA is validated against Solomon's VRPTW benchmark instances. Experimental results confirm that MFGA produces competitive results compared to several state-of-the-art algorithms in terms of various solution quality indicators. The proposed algorithm obtains 17 best-known solutions (BKSs) and improves 12 BKSs in the literature.