A decomposition technique for solving the shortest path problem under bipolar fuzzy environment

The shortest path problem is a fundamental optimization problem with numerous applications in transportation, communication, and logistics networks. In many real-world situations, arc lengths are subject to uncertainty and may contain both positive and negative aspects reflecting satisfaction and dissatisfaction. Bipolar fuzzy sets provide an effective framework for modeling such dual uncertainty. In this paper, we investigate the n-sided bipolar fuzzy shortest path problem (NBFSPP) in which arc lengths is represented by n-sided bipolar fuzzy numbers. A decomposition-based framework is proposed to transform the NBFSPP into two satisfaction models corresponding to positive and negative membership components. These models are further converted into deterministic multi-objective shortest path problems. To obtain a single optimal solution, three solution strategies are employed: a bipolar fuzzy approach, the weighted-sum method, and an n-sided bipolar Euclidean distance technique. Two numerical examples involving triangular and trapezoidal bipolar fuzzy numbers are presented to demonstrate the applicability of the proposed framework. The obtained results are compared with existing ranking-based methods, showing that the proposed approaches can effectively determine optimal paths under bipolar fuzzy uncertainty. The proposed framework provides a flexible decision-making tool for shortest path problems in uncertain environments where both positive and negative preferences must be considered simultaneously.