## 3 Methods  
To boost the genetic algorithm's effectiveness, the IAM-TSP+ [41] method was used as an initial population seeding method for the genetic algorithm. The method has proven effective in finding high-quality solutions to the TSP. The IAM-TSP+ is a greedy algorithm for the TSP, and it is an improvement of the IAM-TSP [41] method. The IAM-TSP beginsby selecting four cities within the input set to act as locations in the east, west, north, and south. The order in which these cities are added to the route list is as follows: east, north, west, and south, with an eastward return. Then, the algorithm recursively adds each of the remainder cities to the route one at a time. Every iteration of the technique determines the cost of adding a city at different places along the route. It thoroughly looks into every possible position for every location, selecting the best position and city for the least cost. Lastly, this method calculates the overall cost of the route and outputs the result. The Euclidean distance between each city along the route is used to compute the cost. The IAM-TSP pseudocode is shown in Algorithm 1. ```
0: Cities \(xy\) (list of cities with \(x\)- and \(y\)-coordinates) of size \(n\) (number of cities)
0: Cost (total cost of the route using Euclidean distance) and Route (the best possible sequence of cities)
1: Create an empty list Route to store the order of visited cities. 2: Obtain the East, North, West, and South cities and add them to Route. 3: Initialize a Boolean array Visited of size \(n\) and set all elements to \(false\). 4: Set the Visited status of the cities in Route to true (East, North, West, and South). 5: Create an empty list Open. 6: Add all cities in Cities\(xy\) to Open if their Visited status is false. 7:while Open is not empty do
8: Create an empty list PossibleSolutions to store the possible solutions (routes). 9:for each city \(i\) in Open
10: find the best location to insert \(i\) into Route
11: add the new Route to PossibleSolutions
12:endfor
13: Find the best solution Best in PossibleSolutions and keep track of the city \(ID\). 14: Update Route by Best. 15: Remove the city \(ID\) from Open that satisfies Best. 16:endwhile
17: Calculate the cost of the Route using the Euclidean distance and store it in Cost. 18:return Cost, Route. ```  
**Algorithm 1** The proposed IAM-TSP algorithm  
With a predetermined number of local cities (\(k\)), an improved version of the IAM-TSP called IAM-TSP+ computes every permutation sequence from the first city in the output route to the \(k\)th city, identifying the best solution and updating the route in the process. Following that, the method goes into a loop where it iterates through each city until it finds the next \(k\) permutations until \(n-k\). The procedure of the suggested IAM-TSP+ is displayed in Algorithm 2. ```
0: Cities \(xy\) of size \(n\) and \(k=5\) (local permutations)
0: Cost, Route
1:\(Route\) = Algorithm1(\(Citiesxy\)). 2:for\(i=1\) to \(n-k\)do
3: create an empty list of lists PossibleSolutions to store the possible solutions (routes). 4:for each\(j\) from \(1\) to \(k\), generate all \(k\) local permutations of Route from \(j\) to \(j+k\). 5: Update the cost of each solution in PossibleSolutions. 6:\(Route\) = the minimum cost solution in PossibleSolutions. 7:endfor
8:return Cost(Route), RouteThe IAM-TSP+ route results for a280 and tsp225 from the TSPLIB [48] are shown in Figures 2 and 3. This method provides a single solution; because the population requires many solutions to initiate the GA, we utilized the seed solution from the previous algorithm to generate n solutions using the mutation operator, which mutates the seed solution \(n-1\) times, in which n is the size of the population. To boost the diversity of the starting population, we employed the mutation operator rather than the crossover operator in this case. The GA will then follow up to optimize the solutions after the initial population is finished. In this work, we used several types of mutations to achieve a kind of diversity.