Mathematical Frameworks for Uncertain Transportation Networks: Reliability, Robustness, and Stability

Transportation networks are subject to multiple sources of uncertainty, ranging from stochastic fluctuations in demand and travel times to epistemic indeterminacy in infrastructure condition, disruption risk, and user behavior. A diverse body of mathematical frameworks has emerged in response, including stochastic programming and probabilistic reliability analysis, fuzzy and possibilistic approaches, Liu’s uncertainty theory and uncertain programming, and robust or distributionally robust optimization. This article delivers a comprehensive, mathematically oriented synthesis of these paradigms for transportation networks, with emphasis on network-level structures—paths, flows, spanning trees, and network design problems—and on reliability notions including connectivity, travel-time, capacity, and max-type reliability. A central theme is that modelling choices about uncertainty representation and reliability indices are inseparable from questions of stability and sensitivity: how robust are optimal or near-optimal configurations when parameters vary within plausible ranges? Building on deterministic post-optimal analysis, this paper reviews tolerance-based stability concepts for uncertain most reliable paths, maximum reliable transmission paths, and uncertain minimum spanning trees under Liu-type uncertainty and demonstrates how inverse-distribution mappings yield exact deterministic equivalents and belief-based robustness margins. A dedicated comparative framework is developed, summarizing the data requirements, core advantages, typical limitations, and suitable engineering scenarios of each uncertainty paradigm to guide model selection in practice. The discussion extends to practical applications in post-disaster planning, infrastructure investment prioritization, and supply chain network design and identifies open research directions including network-wide travel-time reliability under belief-based uncertainty, unified stability frameworks across paradigms, and the integration of machine learning for uncertainty distribution elicitation. The emphasis throughout is on conceptual structure, modelling assumptions, and interpretability of reliability and stability indices, thereby positioning uncertain transportation networks as a rich interface between applied mathematics, operations research, and infrastructure planning.