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Top2. Introduction
The travelling salesman problem is the challenge of finding the shortest route for a traveller to a list of specific destinations. It is a well-known algorithmic problem in the fields of computer science and operations research. A salesman and a collection of cities are both components of the travelling salesman problem. Each city must be visited by the salesperson, who must begin in a particular place and end in that same city.
In fact, TSP belongs to the class of combinatorial optimization problems known as NP-complete. It is noteworthy that the most studied and well-known routing problem is the TSP. An artificial bee colony algorithm has been developed by Pandiri and Singh (2019) to solve the TSP. A quanta TSP is solved by Silva et al. (2020) using an Ant Colony Optimization (ACO) based algorithm. A 2-Opt Heuristic for the metric Travelling Salesman Problem has been developed by Hougardy, Zaiser, and Zhong (2020). A reinforcement learning approach was developed by Hu, Yao, and Lee (2020) for TSP. An exact algorithm was developed by Cavani, Iori, and Roberti (2021) for TSP with multiple drones. Zhao, Xiong, and Shu (2015) have developed a hybrid local search algorithm to solve TSP. Another hybrid algorithm based on particle swarm optimization, ant colony optimization, and 3-opt algorithm have developed by Mahi, Baykan, and Kodaz (2015). Osaba et al. (2016) have proposed a bat algorithm for both the asymmetric and symmetric TSP. A new genetic algorithm has been proposed by Nagata and Soler (2012) for asymmetric TSP. A parallelized genetic ant colony optimization based algorithm has been proposed for TSP by Chen and Chien (2011). Time is also an important objective for TSP. Different types of time window constant have been proposed by different researchers (Lopez-Ibanez et al., 2013; Silva et al., 2010; Kara et al., 2013). Recently, Kara and Deryaa (2015) have proposed a TSP with time window to minimize the tour duration of a traveller. Ardalan et al. (2015) have developed an effective meta-heuristic method hybridized with a local search procedure to solve the generalized TSP. Wang (2014) has developed a hybrid algorithm using two local optimization based methods to solve TSP. The electric travelling salesman problem with time windows is proposed by Roberti and Wen (2016).