Decision-making under uncertainty is consistently an essential fear and the most challenging circle of exploration. To manage the uncertainty, the intuitionistic fuzzy set (IFS) assumes a critical part in taking care of the conditions wherein decision-makers furnish an alternative with a grade of membership and a nonmembership. Distance measures of IFSs are apparatuses used in different decision-making problems, such as medical investigation, pattern recognition, multicriteria decision-making, clustering problems, and other real-world problems. As such, various distance measures were developed by different researchers and applied to decision-making problems with situation-based deficiencies. Motivated by this, in this paper, a symmetric distance formula is being proposed for effectively determining the distance between the information held by IFSs. The distance formula involves membership degree, nonmembership degree, the difference of the minimum of the cross-evaluation factor, and the difference of the maximum of the cross-evaluation factor. Furthermore, it is being proved that the proposed distance formula follows all the axiomatic definitions of a distance measure. Numerical examples depict the efficiency of the proposed distance measure. Hence, this measure is being applied to practical problems of decision-making, pattern recognition, and clustering problems. This measure is not restricted to a particular domain of study; it can be effectively applied in diverse decision-making problems.

Pythagorean fuzzy sets are an extension of intuitionistic fuzzy sets and are more efficient from an application perspective. Though the Pythagorean fuzzy sets are more informative, not much work on similarity measures is available in the literature. Furthermore, existing similarity measures are not efficient. Also, the containment property in Pythagorean fuzzy units is not correctly defined or ineffective. As a result, the existing similarity measures do not reflect appropriate information between the Pythagorean fuzzy sets. The scalar function of the Yager is mainly used for verifying the validity of similarity measures. Most of the existing similarity measures do not conform to the Yager scalar function. Hence, the existing similarity measures exhibit some discrepancies. Furthermore, the existing similarity measures are inconsistent in determining the similarity in intuitionsitic and Pythagorean fuzzy sets. In some real-world modeling issues, past, present, and cross-time information are essential. However, such information is missing in the existing similarity measures. Therefore, in this paper, two new measures of similarity are being developed based on the deviation of the parameters: membership degree, nonmembership degree, strength of commitment, direction of commitment, and cross-time evaluation factors. Under this construction, the proposed similarity measures effectively measure the similarity between the Pythagorean fuzzy sets. Furthermore, the newly defined containment property is also reflected in the proposed similarity measures, which were a limitation in most cases. Moreover, Yager's scalar function is also reflected by the proposed similarity measures. The complement of given information is also essential in some real-world problems. However, such information is incomplete in the theory of Pythagorean fuzzy sets. Hence, the complement of the Pythagorean fuzzy set is being redefined, and a few related results on similarity measures are proposed. Finally, the proposed similarity measures are tested for applicability to medical diagnosis and clustering problems through some hypothetical case studies.

In this paper, two new tools of decision-making problems, namely similarity measures between intuitionistic fuzzy sets, are being forwarded. In general, parameters like the difference of membership degrees and the difference of nonmembership degrees are very prominent factors while constructing these tools. Various studies are evident which tried to incorporate concepts like the cross-evaluation factor and the difference of hesitancy factors. In some studies, it is claimed that the use of the hesitancy parameter is not proper. In this study, the hesitancy factor is being incorporated effectively. Furthermore, another prominent parameter is the difference in the minimum of cross-evaluation factor along with the difference in the maximum of cross-evaluation factor, which is not being used earlier. The incorporation of these parameters produces outperforming results, and the limitations of the existing methods are overcome. The numerical examples discussed to show the performance of the proposed distance measure. Furthermore, the applicability of the proposed similarity measures is exhibited by various applications in pattern recognition, face-mask selection, and clustering problems.

The concept of Z-number, is very new in the literature, proposed by Zadeh in 2011. The Z-number Z = ( A , B ) is a pair of fuzzy numbers where the first component represents the restriction and the second component represents the certainty of the first component. Further, its application is evident in decision-making problems, risk assessment, linear programming problems, and so forth. Hence, under such circumstances ranking of Z-numbers need an utmost attention, as such, a few methods of ranking Z-numbers are proposed by various researcher. However, in many situations the existing methods depict drawbacks and limitations as discussed in this paper. Hence, a new method of ranking Z-numbers is essential for an appropriate decision-making. In this paper, a new method of ranking Z-numbers based on the concept of value and ambiguity at levels of decision-making have been proposed. The method seems to deliver a reasonable decision and proper ranking of Z-numbers of various types. A few numerical examples are discussed which show the out-performance of the proposed method.

Ranking methods of intuitionistic fuzzy numbers are abundant in literature. However, an adequately accepted approach of ordering intuitionistic fuzzy numbers is not apparent. This motivates to develop a new method of ordering intuitionistic fuzzy numbers. In many cases, studies of ranking intuitionistic fuzzy numbers do not investigate the reasonable properties that a ranking method should follow. Furthermore, the majority of existing methods do not meet the reasonable properties. Furthermore, investigating the consistency in image ordering was never a concern in previous studies of ranking intuitionistic fuzzy numbers. Hence, in this paper, an innovative method of ranking intuitionistic fuzzy numbers has been developed. The method is based on the index of optimism, that is, a convex combination of value and multiple of ambiguity inclusion function with ambiguity of membership function and value and multiple of ambiguity inclusion function with ambiguity of nonmembership function. The present method exclusively complies with the reasonable properties as it satisfies all the reasonable properties of a ranking method. Further, newer properties that depicts the consistency in ranking the intuitionistic fuzzy numbers with their corresponding images are also being developed in this study. It is worth mentioning that the present method consistently ranks the intuitionistic fuzzy numbers and their corresponding images.

Undoubtedly, most of the information conveyed is full of impreciseness or vagueness. In such situation, fuzzy set theory is a proper tool to handle these types of impreciseness or vagueness. Fuzzy numbers are also of different types based on the presence of uncertainty. Neutrosphic numbers are one type of fuzzy number that consists of three types of membership functions, namely, truth, indeterminacy and falsity. In fact, neutrosophic numbers are generalization of intuitionistic fuzzy numbers. These numbers might be used in decision making problems, where it is essential to rank those numbers. So, ordering of these numbers becomes a new trends with more challenges. In this paper, an innovative ranking approach of single-valued neutrosophic number has been put forwarded. The method is based on value, ambiguity and θ function. Value and ambiguity are defined on three functions of the neutrosophic numbers. Further, the reliability of the proposed method is established by considering the reasonable properties of ranking method. One important aspect of ranking neutrosophic numbers is to consider the consistency in ordering the neutrosophic numbers and their corresponding images. It is worth to mention that the current approach consistently ranks the single-valued neutrosophic numbers as well as their corresponding images.

The distance measure between intuitionistic fuzzy sets (IFSs) is a concept of very contemporary interest among the researchers in the field of decision-makings, such as pattern recognition, medical diagnosis, and multiattribute decision-making (MADM) problems. Consequently, diverse distance measures are developed and used in determining the similarity and dissimilarity between IFSs. In the existing methods, the distance measures are calculated based on the geometry of the IFSs. However, the IFSs hold information about the elements in a set. As such, some of the existing distance measures are misleading and unreasonable. Hence, in this paper, a nonlinear distance formula is devised to follow the problem definition. Further, by explicitly proving the distance properties, it is being established that the distance formula is a distance measure. Further, theories for the construction of distance measures are developed. The convex combination of two distance measures is also a distance measure is being proved explicitly. Furthermore, based on the proposed distance measures, similarity measures have been developed. Aside from that, an intriguing idea has been introduced, namely, that an infinite number of distance measures can be constructed from a given pair of distance measures. Additionally, the proposed distance and similarity measures are applied to a variety of problems, including medical diagnosis, pattern recognition, and a MADM problem in COVID-19 face mask selection, where the legitimacy and applicability of the proposed advanced distance measure is demonstrated.