T-spherical fuzzy sets, the direct extension of fuzzy sets, intuitionistic fuzzy sets and picture fuzzy sets are examined in this composition, and a mathematical examination among them is set up. A T-spherical fuzzy set can demonstrate phenomenon like choice utilizing four trademark capacities indicating the level of choice of inclusion, restraint, resistance, and exclusion, another example of such situation is that human opinion cannot be restricted to yes or no but it can be yes, abstain, no and refusal. T-spherical fuzzy set can deal the said situation with a boundless space. With the assistance of some mathematical outcomes, it is talked about that current similarity measures have a few drawbacks and could not be implemented where the data is in T-spherical fuzzy mode. Thus, some new similarity measures in T-spherical fuzzy environment are proposed, with the assistance of certain outcomes, it is demonstrated that the suggested similarity measures are generalization of current ones. Further the proposed similarity measures are applied in pattern recognition with numerical supportive examples. The maximum spanning tree clustering algorithm has been extended into T-spherical fuzzy context and supports our theory with numerical examples. A parallel investigation of fresh and existing similarity measures have been made and some of the benefits of designated work have been discussed.

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In this work, ambiguity and unclarity are coped with the effective tools of picture fuzzy sets (PFSs), especially where the conditions demand simulation of various dimensions for evaluation, for example decision making. PFS requires operators to measure the coordination of two PFSs. As far as this paper is concerned, we bring new operators to PFSs with an application, validating this as the generalization of the concept of Fuzzy Sets (FSs) and Intuitionistic Fuzzy Sets (IFSs). The hybrid structure of PFSs has been incorporated with other operators to develop picture fuzzy Dombi Hamy mean (PFDHM) operator, picture fuzzy weighted Dombi Hamy mean (PFWDHM) operator, picture fuzzy Dombi dual Hamy mean (PFDDHM) operator, and PF weighted Dombi dual Hamy mean (PFWDDHM) operator. Further, the properties such as Idempotency, Monotonicity, Boundedness, and Commutativity related to each proposed operator have been discussed. By using these operators the multiple attribute group decision-making (MAGDM) methods are proposed. Moreover, we have explained the application by providing an example of a car supplier. The results are concluded by selecting the best car on the basis of attributes such as quality, production, service efficiency, and risk factors using operators defined on PFSs. A comparative study is also conducted to study the significance of the developed work.

T-spherical fuzzy set is an effective tool to deal with vagueness and uncertainty in real-life problems, especially in a situation when there are more than two circumstances, like in casting a ballot. The correlation coefficient of T-spherical fuzzy sets is a tool to calculate the association of two T-spherical fuzzy sets. It has several applications in various disciplines like science, management, and engineering. The noticeable applications incorporate pattern analysis, decision-making, medical diagnosis, and clustering. The aim of this article is to introduce some correlation coefficients for T-spherical fuzzy sets with their applications in pattern recognition and decision-making. The under discussion correlation coefficients are far more advantageous than the existing and many other tools used for T-spherical fuzzy sets, because they are used completely and demonstrate the nature as well as the limit of association between two T-spherical fuzzy sets. Further, an application of proposed correlation coefficients in pattern analysis is discussed here. In addition to it, the proposed correlation coefficients are applied to a multi-attribute decision-making problem, in which the selection of a suitable COVID-19 mask is presented. A comparative analysis has also been made to check the effectiveness of the proposed work with the existing correlation coefficients.

The aim of this study is to introduce an innovative concept of T-spherical fuzzy matrix, which is a hybrid structure of fuzzy matrix and T-spherical fuzzy set. This article introduces the square T-spherical fuzzy matrix and constant T-spherical fuzzy matrix and discusses related properties. Determinant and the adjoint of a square T-spherical fuzzy matrix are also established, and some related properties are investigated. An application of the T-spherical fuzzy matrix in decision-making problem with an illustrative example is discussed here. Then, in the end, to check capability and viability, a practical demonstration of the planned approach has also been explained.