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 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.

Risk plays an important role in the decision making process. To assess the severity and likelihood of impairment to human health from exposure to a substance or activity that under plausible circumstances can cause harm to human health is the main purpose of risk assessment. It is important to know the nature and characteristic of all available information, data or model parameters which are more generally tainted with aleatory and epistemic uncertainty or both type of uncertainty. In some situation, model parameters are affected by aleatory uncertainty and simultaneously other some parameters are affected by epistemic uncertainty, how far computation of the risk is concern, two ways to deal with the situation either transform all the uncertainties to one type of format or need for joint propagation of uncertainties. In this paper, an effort has been made to combine probability distributions, generalized fuzzy numbers, normal fuzzy numbers and generalized interval valued fuzzy numbers within the same framework.

Abstract Uncertainty comes in many forms in the real world and is an unavoidable component of human life. Generally, two types of uncertainties arise, namely, aleatory and epistemic uncertainty. Probability is a well established mathematical tool to handle aleatory uncertainty and fuzzy set theory is a tool to handle epistemic uncertainty. However, in certain situations, parameters of probability distributions may be tainted with epistemic uncertainty; and so, representation of parameters of probability distributions may be treated as fuzzy numbers (may be of different shapes). A probability box (P‐box) can be constructed when parameters are not precisely known. In this paper, an attempt has been made to construct families of P‐boxes when parameters of probability distributions are bell shaped or normal fuzzy numbers; and from these families of P‐boxes, membership functions are generated at different fractiles for different alpha levels.

This study attempts to perform arithmetic operations of generalized triangular fuzzy numbers apart from the existing approaches. Also, average width of generalized fuzzy numbers is defined. Numerical examples are illustrated and uncertainty is measured by using defined average width, and finally, the results are compared with the existing approaches. A case study for health risk assessment is carried out in this comparison.