Optimizing traveling salesman problem using tabu search metaheuristic algorithm with Pythagorean fuzzy uncertainty

The traveling salesman problem is a well-known combinatorial optimization problem. Solving the traveling salesman problem efficiently becomes more challenging when considering uncertainties in the problem parameters, which are prevalent in real-world scenarios. Pythagorean fuzzy uncertain variables combine the strengths of fuzzy logic with the principles of uncertainty theory, allowing for a more balanced and comprehensive representation of uncertainty. This paper defines the theoretical foundations of discrete, linear, and zigzag Pythagorean fuzzy uncertainty distributions, including their mathematical formulation and operational laws. Moreover, it proposes a novel approach to tackle the traveling salesman problem with Pythagorean fuzzy uncertainty distribution using the tabu search metaheuristic. By integrating this uncertainty representation into the tabu search metaheuristic, the proposed method effectively explores the solution space while considering the potential variations in the Pythagorean fuzzy distance matrix. The detailed steps of the algorithm are demonstrated via a numerical exemplification. Subsequently, a comprehensive case study is presented, wherein the objective is to determine the optimal touring sequence among the largest cities of China. This investigation is founded upon data source from Google Maps. We conduct a sensitivity analysis to assess the sensitivity of the model’s output by varying the input parameters. The proposed algorithm’s performance is compared with traditional tabu search and other existing methods. The results highlight the potential of incorporating Pythagorean fuzzy uncertainty distribution into metaheuristic algorithms for solving combinatorial optimization problems.