A Solution of Multi-Level Multi-Objective Optimization Models for Solid Transportation Problems under Fuzzy Environments

Abstract

This study presents a comprehensive multi-level, multi-objective mathematical model to address the Solid Transportation Problem (STP) under fuzzy environments. Real-world transportation systems often face significant complexity due to uncertain parameters such as fluctuating demand, varying traffic conditions, and limited conveyance capacity. To effectively manage these uncertainties, we propose a mathematical framework that incorporates hierarchical objectives and uncertain parameters using fuzzy set theory. The model is structured across three hierarchical levels, each representing a specific goal within the transportation system: 1) minimization of transportation costs, 2) minimization of total transportation time, and 3) minimization of environmental impact costs. To manage uncertainty within model parameters, we apply two robust techniques—expected constraint programming and chance constraint programming—which transform the fuzzy model into its deterministic equivalent, allowing for more practical implementation and decision-making. The proposed model is validated through a hierarchical multi-level, multi-objective numerical illustration. This example demonstrates the effectiveness and applicability of our approach in real-world scenarios. It highlights how the model supports decision-makers in navigating trade-offs among conflicting objectives while maintaining feasibility under uncertain conditions. The results underscore the model’s potential to enhance efficiency, sustainability, and resilience in solid transportation operations.