Fuzzy multi objective solid transportation model for optimal distribution

This study demonstrates a novel fuzzy multi-objective solid transportation problem (FMOSTP) which optimally manages logistics chain under uncertainty by simultaneously addressing multiple conflicting criteria cost, transit time, and security represented as fuzzy numbers. The proposed framework uses trapezoidal fuzzy numbers to model the parameters including costs, capacities, demands, and transit times, which allows the model to better represent the actual situation. A centroid-based defuzzification method is used to transform fuzzy data into their crisp equivalents so that classical linear programming procedures can be applied, and fuzzy ranking procedures are applied to make the solutions express some degree of tolerance to the uncertainties that may remain. The weighted Tchebycheff scalarization is used to generate a well-distributed Pareto front of optimal solutions, enabling decision-makers to investigate the trade-offs and to assign weights to the objectives according to their strategic preferences. Its effectiveness, computational efficiency, and flexibility under data uncertainty are demonstrated through two hypothetical numerical examples of multi-modal transportation in manufacturing supply chains and disaster-relief logistics. The finding of this study demonstrates how multi-objective optimization combined with fuzzy uncertainty modeling can support supply-chain and transportation management by producing dependable transportation plans. The principal novelty of this work lies in the simultaneous integration of three features that prior studies treat only in isolation: (i) a solid (three-dimensional) transportation structure with explicit conveyance modes; (ii) three conflicting objectives, namely cost, transit time, and route security; and (iii) a trapezoidal-fuzzy representation of all imprecise parameters. Centroid defuzzification is adopted for its linearity-preserving property, which keeps the scalarized model a linear program and yields a convex, computationally tractable feasible region. Beyond the numerical study, the feasibility, convexity, and stability of the formulation are established analytically, and the model is benchmarked against classical deterministic and single-objective fuzzy models to demonstrate measurable gains in robustness.