A mathematical technique called data envelope analysis is used to determine the relative efficiency of decision-making units (DMU) with numerous inputs and outputs. Compared to other DMUs, it determines how efficient the DMU is at delivering a specific level of output based on the amount of input it uses. The transportation problem is a linear programming problem for reducing the net transportation cost or maximizing the net transportation profit of moving goods from a number of sources to a number of destinations. In this manuscript, a multi-objective transportation problem is examined in which all the parameters, such as transportation cost, supply, and demand are uncertain with hesitancy and triangular Fermatean fuzzy numbers are used to represent these uncertain parameters. Using Fermatean fuzzy data envelope analysis, a new technique for determining the common set of weights is presented. The fully Fermatean fuzzy multi-objective transportation problem is then solved using a novel data envelopment analysis-based approach. To this end, two different Fermatean fuzzy efficiency scores are derived, first by considering the sources as targets and changing the destinations, and second by considering the destinations as targets and changing the sources. Next, a unique Fermatean fuzzy relative efficiency is determined for each arc by combining these two different Fermatean fuzzy efficiency scores. As a result, a single-objective Fermatean fuzzy transportation problem is constructed, which can be solved using existing techniques. A numerical illustration is provided to support the suggested methodology, and the performance of the proposed method is compared with an existing technique