Temporally repeated fuzzy maximum dynamic flow with intermediate storage

The classical network flow models have been extensively studied in deterministic ways without addressing uncertainty, which is not able to capture the real-world systems. Unlike classical models, the fuzzy maximum network flow problem with intermediate storage incorporates uncertainty in arc capacities (due to congestion or other factors) and accounts for storage constraints at intermediate nodes. Recent trends in research are focusing on solving maximum dynamic network flow problems with intermediate storage in fuzzy environments. The solution strategies presented in the literature are not efficient, as the algorithms rely on a time-expanded network rather than the flow decomposition idea to use temporally repeated formulae. In this paper, an efficient algorithm is proposed to solve this problem by taking the time-invariant arc capacities over the given time horizon in fuzzy environment. For this, we model arc capacities, demands, and storage capacities using triangular fuzzy numbers to address real-world variability. We formally define the fuzzy maximum flow problem and the earliest arrival flow problem with intermediate storage, assuming crisp transit times at all time steps. Our solutions employ a temporally repeated flow algorithm, where flow-balancing paths are extracted from an excess flow network derived from static intermediate storage solutions. Additionally, we apply this approach to solve the earliest arrival flow problem with intermediate storage in a series-parallel network.