Similarity of generalized trapezoidal fuzzy numbers with different left and right heights

Within the realm of fuzzy multicriteria decision-making, the measure of similarity in generalized trapezoidal fuzzy numbers (GTFNs) is vital for electing the most suitable option. Several studies focusing on similarity measures have been addressed in scholarly works. Nonetheless, the current methods lack adequate results for similarity calculations or fail to compute the differing left and right heights. The aim of this paper is to create a novel measure for assessing the similarity of GTFNs varying in their left and right heights. Initially, our analysis revealed that if both the left and right heights of a pair of GTFNs equal zero, their similarity is discernible solely through geometric distance. Conversely, if these heights differ from zero, we incorporated elements like center point distance, edge lengths, area, and both left and right heights into the similarity computation equation. The scope of this measure extends beyond GTFNs of varying left and right heights, encompassing the handling of random fuzzy numbers as well. Subsequently, certain properties of the suggested similarity measure are examined. Concentrates on ten properties, including translation, symmetry, folding, and the proportional increase and decrease in size. Furthermore, to confirm the new method's effectiveness, fifteen unique test sets are provided to evaluate the performance of three current techniques for determining similarity across various heights using the new approach.