Fractional Multi-Commodity Network Flow Problem in a Fuzzy Stochastic Hybrid Environment

Abstract

This study introduces a novel approach to solving fractional multi commodity network flow (FMCNF) problems under a hybrid fuzzy-stochastic environment with multi-choice parameters. The proposed model addresses uncertainties by incorpo rating both fuzzy and stochastic elements into key components of the objective function and network constraints. Specifically, the coefficients in the numerator and denominator of the fractional objective function are modelled using multi choice fuzzy-stochastic and fuzzy param eters, respectively, while arc capacities are also treated as fuzzy-stochastic variables. The aim is to equip decision-makers with a flexible and realistic framework for optimizing flows in complex, uncertain net works. To handle the hybrid uncertainty and convert the problem into a solvable deterministic form, techniques such as the probability-possibility transformation, Lagrange interpolation, and Charnes-Cooper transformation are employed. The practical use of the proposed model is shown through numerical examples, especially in areas like maritime and air transport where costs and routes often change. For example, the model helps in deciding how to load cargo and choose ports while considering changing transport and selling costs. This research is the first to combine multi-choice fractional modelling with hybrid fuzzy-stochastic uncertainty.