Four-dimensional green transportation problem considering multiple objectives and product blending in Fermatean fuzzy environment

This paper presents a study on the multi-objective green four-dimensional transportation problem (MOG4DTP) with product blending. Due to uncontrollable circumstances and globalization, it is not always practical to exactly determine the parameters of the MO4DGTP. In such situations, decision experts sometimes have to deal with data that can be described by a membership degree (MD) and a non-membership degree (NMD), such that their total does not fall within the range \(\left[ 0,1\right] \). Such a situation cannot be addressed by fuzzy set theory or intuitionistic fuzzy set (IFS) theory. However, there are cases where the sum of the cubes of the MD and the NMD of the data lies within the range \(\left[ 0,1\right] \), even though their sum is greater than 1. Fermatean fuzzy sets (FFSs) can deal with such ambiguous data. Thus, we consider parameters such as transportation cost, time, availability, demand, conveyance capacity and carbon emission as triangular Fermatean fuzzy numbers (TrFFNs). Also, since greenhouse gas emission is the most controversial issue in present times, we have considered carbon emission as one of the objectives of our problem. Both these considerations make our problem more realistic. Additionally, we propose a ranking index for TrFFNs and, by utilizing its linearity, transform the Fermatean fuzzy model into its corresponding deterministic form. Further, we obtain the Pareto-optimal solution of this model by four methods, namely, fuzzy TOPSIS, \(\epsilon \)-constraint method, augmented Tchebycheff method (ATM) and weighted Tchebycheff metrics programming (WTMP) method. We describe a real-world industrial transportation problem (TP) and compare the solutions obtained using different techniques in order to show the value and applicability of the suggested model. The proposed algorithm’s performance is validated through comparisons with state-of-the-art multi-objective algorithms, ensuring credibility and demonstrating its effectiveness in solving complex optimization problems. Further, a comprehensive sensitivity analysis is conducted to assess the robustness of the proposed algorithm, ensuring its reliability across varying parameter settings and problem instances. Lastly, we present key conclusions along with the limitations of the proposed approach, and suggest directions for future research building upon this work.