The Solutions to Traveling Salesman Problem

This paper presents solutions to symmetric, asymmetric, and multiple traveling salesman problems. In the symmetric traveling salesman problem, one salesperson must go to five cities, making precisely one stop at each location. A matrix with the distances between each city is provided. Using the branch and bound algorithm and expressing it in Python, the final result is obtained. The formulations of the asymmetric traveling salesman problem and the multiple traveling salesman problem are demonstrated in the paper. The asymmetric problem in the paper is solved by transforming the asymmetric traveling salesman problem into a symmetric traveling salesman problem; then, the branch and bound algorithm has been applied to solve the problem. In the multiple traveling salesman problem, three salesmen are required to visit seven cities in total, each of which is only visited by one salesperson once. Each city is a point in an x-y plane, and the coordinates of all points are given in a graph. By formulating the MTSP problem with an assignment-based double-index integer programming and applying the constraints, the outcome is derived from Python code within a few seconds.