Intuitionistic fuzzy modal operators in an entropy-weighted TOPSIS framework

Parametric intuitionistic fuzzy modal operators redistribute the degree of uncertainty in intuitionistic fuzzy evaluations by transforming their membership and non-membership degrees. This paper examines the effect of applying the E_(α,β) and Z_(α,β)^ω operators to the decision matrix in the entropy-weighted heuristic fuzzy TOPSIS method, where the criterion weights are derived from the Ye, Wang, Wu–Zhang, and Guo entropy measures. Since both operators are affine transformations with a common contraction coefficient of αβ, we show that their direct application to the evaluations in the Euclidean-based TOPSIS method does not alter the proximity coefficients; thus, the effect of these operators manifests solely through the recalculation of entropy weights and the nonlinear weighting step. A numerical example with four alternatives and six criteria, in which all details are defined, shows that in 38 of the 44 operator-entropy combinations considered, the transformations altered the ranking and, in each case, replaced the preferred alternative with a different option; the primary source of these changes is the recalculation of the weights. Analyses examining the parameters individually reveal how the values of α, β, and ω trigger these changes. The results suggest that modal operators should be considered more as a sensitivity analysis tool for the entropy-weighted TOPSIS method rather than as tools that improve the method in a way that preserves the ranking.