Advancing green logistics through Fermatean fuzzy optimization

Abstract

Modern supply chains are increasingly required to balance multiple, often conflicting objectives such as economic efficiency, environmental responsibility, blended-product quality, and delivery performance, all under significant uncertainty. Ensuring the consistent quality of blended products is particularly critical in sectors such as energy, food processing, and chemicals, where multiple raw materials must be combined in precise proportions to satisfy regulatory and performance requirements. Achieving high-quality blending while minimizing transportation costs and environmental impact becomes considerably more complex when products flow through multiple stages and transport modes. This paper proposes a comprehensive optimization framework for a multi-stage green transportation problem with product blending, spanning the flow from suppliers through manufacturers and retailers to end consumers. The model incorporates alternative transportation routes and modes—such as trucks, rail, and maritime transport—each exhibiting distinct cost, time, and emission characteristics. By jointly integrating blending decisions with route and mode selection, the framework captures the real trade-offs among economic, environmental, and quality-oriented objectives. The formulation seeks to ensure that the final blended product satisfies required specifications while simultaneously minimizing (i) total transportation cost, (ii) carbon emissions, (iii) product degradation, and (iv) delivery time. Uncertainty associated with transportation and blending operations is represented using a Fermatean fuzzy framework, where key parameters—including transportation costs, emissions, blending quality, travel times, and the total transported or blended amount are modeled as Triangular Fermatean Fuzzy Numbers (TrFFNs). To enhance tractability, a ranking index is developed to transform the fuzzy multi-objective problem into an equivalent deterministic formulation, enabling the use of efficient optimization algorithms while preserving the inherent uncertainty in the system. To generate Pareto-optimal solutions, several multi-objective optimization techniques are employed, including fuzzy TOPSIS, the ε-constraint method, the augmented Tchebycheff approach, and weighted Tchebycheff metric programming. A TOPSIS-based ranking algorithm is then applied solely to rank and compare the four resulting solution methods. A real-worldinspired case study demonstrates the applicability of the proposed framework for identifying energy-efficient routes, optimizing blending strategies, and enhancing operational performance under uncertainty. The results confirm that the proposed methodology constitutes a robust decision-support tool for advancing sustainable and resilient supply chain management.