Optimization of Single-valued Triangular Neutrosophic Fuzzy Travelling Salesman Problem

The travelling salesman problem(TSP) is a classic optimization puzzle, widely studied and celebrated for its significance in operations research, mathematics and computer science. It can also be described as an evolution from a mathematical curiosity to a problem that challenges the computation boundaries, sparks algorithmic innovation, and finds practical applications in various industries. The neutrosophic TSP(NTSP) extends the problem by introducing neutrosophy, handling indeterminacy and inconsistency with distances represented by neutrosophic numbers (NNs). The single-valued triangular fuzzy neutrosophic TSP(SVTFNTSP) goes a step further by incorporating both single-valued triangular fuzzy numbers(SVTFNs) and neutrosophy, representing distances with SVTFNNs. The single-valued triangular fuzzy neutrosophic numbers(SVTFNNs) provide a way to model uncertainty via triangular membership functions, offering a more nuanced representation of uncertain and vague distances. This arises the need to use them and enhances realism in solving complex real-world optimization problems. These extensions adapt the TSP to varying uncertain and vague data, ideal for intricate real-world optimization scenarios. This research article delves into the SVTFNTSP, expressed as a single-valued triangular fuzzy neutrosophic distance matrix(SVTFNDM) with SVTFNNs as its core elements, accounting for both uncertainty and imprecision. The investigation encompasses the formulation and examination of this specialized problem by incorporating a score function to assess defuzzification and optimality, alongside the utilization of a proposed systematic stepwise approach to efficiently ascertain optimal solutions. This approach is practically demonstrated through its application to real-world scenarios, effectively showcasing its feasibility and real-world relevance. Subsequently, through a rigorous comparative analysis with the established methodologies, the superior effectiveness and value of the proposed approach are highlighted, specifically in terms of minimizing total travelling costs. This reaffirms its potential as a robust solution for tackling the SVTFNTSP by underlining its practical utility and enhanced performance. Keywords: Neutrosophic set, Neutrosophic number, Single-valued triangular fuzzy neutrosophic number, Single-valued triangular fuzzy neutrosophic distance matrix, Travelling salesman problem, Single-valued triangular fuzzy neutrosophic travelling salesman problem, Sco