Imprecise Maximum-Flow Analysis in Real-World Systems: Methodology and Decision Insights

In real-world systems such as the Internet of Things (IoT), intelligent transportation, and energy distribution, addressing fuzziness is crucial for making reliable decisions in maximum flow analysis. This paper investigates the generalized maximum-flow problem in which the lower and upper bounds on arc flows are represented as fuzzy numbers, including single- and double-directed arcs, and the flows can be either real- or integer-valued. Rather than a singleton solution, this paper presents an analysis method based on Zadeh's extension principle and the duality theorem for determining a likely interval of the fuzzy maximum-flow value with a possibility level (degree of certainty). By formulating and transforming two-level parametric programs, the lower and upper bounds of fuzzy maximum-flow values at each possibility level are calculated, and the maximum flow membership function is constructed. This methodology can be extended to integer-valued fuzzy maximum-flow problems. Expressing fuzzy maximum flow with a membership function rather than a crisp value yields reasonable results suitable for various scenarios and managerial implications, providing richer decision-making insights from the most likely to the least likely cases. A benchmark transportation case study and additional sensitivity analyses further demonstrate that the proposed α-cut formulation is consistent with crisp maximum-flow results while preserving uncertainty-dependent decision information.