As a new extended form of intuitionistic fuzzy sets, Fermatean fuzzy sets are powerful tools for describing vagueness and uncertainty in complex problems. In the method of handling Fermatean fuzzy information, the distance measure is an essential tool to depict the difference between two Fermatean fuzzy sets. However, how to accurately measure the distance between two Fermatean fuzzy sets is still a problem to be solved. In this paper, we devise two novel distance measure methods for Fermatean fuzzy sets. One is the distance measure of Fermatean fuzzy sets based on the Hellinger distance, which is called the FFSH distance. The other is the distance measure of Fermatean fuzzy sets based on the triangular divergence, which is called the FFSTD distance. Then, we prove that the proposed distance measure methods satisfy the axiomatic requirements of the distance function. Afterward, numerical examples are given to reveal that the proposed distance measures are more effective and reasonable than the normalized Euclidean distance measure, which can overcome the counter-intuitive situation. Besides, we utilize the proposed distance measure methods to address the problems of pattern recognition and medical diagnosis under Fermatean fuzzy environment and achieved excellent results. The experimental results illustrate that the proposed distance measure methods can efficiently handle the practical application under Fermatean fuzzy environment, and are more reliable than the normalized Euclidean distance measure.

Dempster–Shafer (DS) evidence theory is the most significant and effective method for uncertainty modeling and reasoning. How to measure the uncertainty in DS evidence theory precisely remains an outstanding problem. Various types of uncertainty measures for evidence have been presented. However, they all suffer some limitations. To address this issue, we propose a novel total uncertainty measure for the DS evidence theory framework that can quantify the uncertainty in the evidence. The new total uncertainty measure uses the Hellinger distance between the belief interval of every singleton and the most uncertain interval. Compared with the existing uncertainty measurement methods, the proposed approach is more sensitive to changes in evidence. The effectiveness and rationality of the brand new total uncertainty measure are illustrated by numerical examples and practical applications.

Fermatean fuzzy set (FFS) is an effective tool to depict expert reasoning information in the decision-making process. In this study, we first propose a novel Fermatean fuzzy entropy measure to describe the fuzziness degree of FFSs. The new Fermatean fuzzy entropy takes into account the uncertainty information and the indeterminacy degree of FFSs. Subsequently, we prove that Fermatean fuzzy entropy satisfies the axiom requirement of fuzzy entropy measure. Thereafter, a novel Fermatean fuzzy multicriteria decision-making approach is developed based on Dempster–Shafer theory with the help of the Fermatean fuzzy entropy. The proposed method modeled each Fermatean fuzzy number as a piece of evidence, and the weights of criteria are determined by the entropy measure of FFSs. Then, the weighted average evidence for the alternatives under all criteria is computed from the weights of criteria. Later, Dempster's combination rule is leveraged to combine the weighted average evidence of the alternatives to obtain the final evaluation information about each alternative. The proposed approach can effectively deal with the uncertain information in decision-making problems and help reduce the information loss in the decision-making process. Ultimately, the feasibility and validity of the proposed approach are demonstrated through two practical instances.