Hybridizing simulated annealing and genetic algorithms with Pythagorean fuzzy uncertainty for traveling salesman problem optimization

The traveling salesman problem is a classic combinatorial optimization challenge with profound implications for various industries. While significant progress has been made in solving traveling salesman problem instances, real-world applications often involve uncertainties that challenge the accuracy and robustness of traditional approaches. 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 normal, lognormal, and empirical Pythagorean fuzzy uncertainty distributions, including their mathematical formulation and operational laws. Moreover, it presents a novel hybrid optimization approach that leverages the strengths of simulated annealing and genetic algorithms while incorporating Pythagorean fuzzy uncertain variables to address the traveling salesman problem under uncertain conditions. The synergy of these two techniques enables effective exploration and exploitation of solution candidates, leading to improved traveling salesman problem solutions. The detailed steps of the algorithm are demonstrated through a numerical example. A case study of a decision support system for optimizing a beverage logistics vehicle routing problem is discussed to find out the best possible route in the distribution zones. The incorporation of Pythagorean fuzzy uncertain variables enhances the algorithm's robustness in uncertain environments, resulting in higher-quality solutions and improved adaptability to different levels of uncertainty.