Shortest Path Problem with Subnormal Gaussian Fuzzy Costs

Abstract: This paper addresses the fuzzy shortest path problem in directed graphs, where edge costs are modeled as generalized fuzzy numbers with Gaussian membership functions. We interpret height as an indicator of information reliability. Based on this view, we introduce a weighted geometric mean to aggregate heights during the addition of generalized Gaussian fuzzy numbers. We employ a reliability-aware ranking that jointly considers the core, height, and standard deviation of fuzzy edge costs to determine the shortest path, thereby capturing their central tendency, reliability, and variability while keeping Dijkstra-level complexity per relaxation. The method yields routes that are not only cost-efficient but also supported by highly reliable information. To assess robustness, we construct a crisp baseline from the ranking and conduct Monte Carlo alpha-cut sampling--drawing membership levels uniformly and then sampling within the induced intervals--to recompute path costs and quantify sensitivity via the mean percentage deviation and its standard deviation. Finally, a large-scale case study on the FAA air traffic network demonstrates that the proposed GGFN--SPP framework scales efficiently to real-world networks, balances cost and reliability through $α$--cut aggregation and risk-aware ranking, and exhibits stable performance under Monte Carlo simulations with subnormal fuzzy costs.