A solution of mathematical multi-objective transportation problems using the fermatean fuzzy programming approach

In this paper, we present a mathematical model of a traditional transportation problem (TTP) using the fermatean fuzzy parameters (FFPs) and convert it into a crisp form using a new fermatean fuzzy score function (NFFSF). Additionally, we proposed a mathematical model of the multi-objective transportation problem (MOTP) incorporating FFPs, which is similarly transformed into a crisp form using NFFSF under fermatean fuzzy environments (FFE). We extend this approach to develop a mathematical model of a multi-level, multi-objective solid transportation problem (MLMOSTP) using FFPs under FFE. It is also converted into crisp form using NFFSF. The mathematical model of MOTP and MLMOSTP aims to simultaneously minimize three objective functions: total transportation cost, total transportation time, and total carbon emissions. The parameters of mathematical models, including objectives, costs, supply, and demands, are considered as FFPs. The mathematical model of MOTP and MLMOSTP is solved using the fermatean fuzzy programming approach (FFPA) under FFE. The FFPA is particularly suitable for addressing the MOTP and MLMOSTP due to its ability to handle higher uncertainty and vagueness inherent in multi-objective optimization problems. We provided numerical examples to demonstrate the proposed problems' efficiency and practicality. The SciPy optimization library in Python was used to solve these numerical examples and obtain the best compromise solutions. Managerial and practical implications are discussed.