Optimizing Sustainable Two-Stage Bi-Objective Fixed-Charge 4-Dimensional Transportation Problem under Fermatean Fuzzy Environment

The transportation problem is an important field in supply chain management and international trade. In transportation problem, the goods are carried from origin to final destination by using a specific mode of shipping. In reality, goods are often delivered using a variety of modes of shipping and various routes. So, by integrating supply, demand, conveyance, and route together, it becomes a Four-Dimensional Transportation Problem. The entrepreneurs often prefer two-stage transportation which will always reduce multiple payments of fixed charges and also transportation of goods by using appropriate conveyance, route, and fuel type while optimizing multiple objectives together will always preferably maximize the company’s profit and simultaneously minimize the environmental pollution. Due to market fluctuations, variations in supply and so on, the problem parameters are considered as Fermatean fuzzy numbers. By consideration of all economic and environmental aspects together we focus on developing a Two-Stage Bi-Objective Fixed-Charge Four-Dimensional Transportation Problem under a Fermatean fuzzy environment. Uncertainty associated with the parameters of the developed model has been resolved by employing the (α, β)-cut technique and accuracy function. Now, the deterministic multi-objective problem is transformed into single objective problem by using the proposed Fermatean hesitant fuzzy programming technique. Then LINGO software is employed to obtain optimal compromise solution for the reduced problem. The competency of our model is clarified with a numerical illustration. The results offer managerial insights for selecting optimal routes and fuel types based on available resources, leading to sustainable development. A comparison is made with the existing method to demonstrate the efficacy of our proposed method. Lastly, sensitivity analysis and conclusions with future scopes are presented.