**1. Introduction**

The term fuzzy set (FS) was developed by Zadeh [1] based on a characteristic function that described the degree of membership of an element. Atanassov [2] established the theory of intuitionistic fuzzy set (IFS) as a generalization of FS with the help of two characteristic functions, known as membership and non-membership functions, describing the positive and negative aspects of an element or object. In the framework of IFSs, there was a constraint on two characteristic functions, in that their sum must not exceed the unit interval, which restricted the selection of membership and non-membership grades. Accordingly, Atanassov and Gargov [3] extended the IFS to the interval-valued intuitionistic fuzzy sets (IVIFSs), which contain the degrees of agreeing and disagreeing as interval values instead of single digits. Keeping in mind the constraint on IFSs, Yager [4,5] introduced a new generalization of IFSs, known as Pythagorean fuzzy set (PyFS), with a condition that the sum of squares of membership and non-membership grades must not exceed the unit interval.

The frameworks of IFSs and PyFSs have importance in situations where the structure of FSs fails to be applied. But these structures have their own limitations, as in the circumstances of voting where opinion cannot be restricted to "yes" or "no" but some refusal degree and abstinence is also involved. Therefore, Cuong [6,7] developed a novel concept of picture fuzzy sets (PFSs), which is based on four characteristic functions known as membership, non-membership, abstinence, and

refusal grades. Cuong's structure of PFSs is diverse in nature but, similar to IFSs, there is also a restriction in PFSs that the sum of all three membership grades must not exceed the unit interval. In the above-stated environments, various researchers have constructed their methodologies for solving the multi-attribute decision-making (MADM) problems, focusing on information measures, aggregation operators, etc. For instance, Xu [8] presented some weighted averaging aggregation operators (AOs) for intuitionistic fuzzy numbers (IFNs). Garg [9,10] presented some improved interactive AOs for IFNs. Wang and Liu [11] gave interval-valued intuitionistic fuzzy hybrid weighted AOs based on Einstein operations. Wang and Liu [12] presented some hybrid weighted AOs using Einstein norm operations for IFNs. Wang et al. [13] presented some AOs to aggregate various interval-valued intuitionistic fuzzy (IVIF) numbers (IVIFNs). Garg [14,15] presented generalized AOs using Einstein norm operations for Pythagorean fuzzy sets. Xu and Xia [16] proposed induced generalized aggregation tools and applied them in MADM. Garg and Kumar [17] presented some new similarity measures for IVIFNs based on the connection number of the set pair analysis theory. However, apart from these, a comprehensive overview of the different approaches under the IFSs, IVIFSs, PyFSs, etc., to solve MCDM problems are summarized in [18–33].

Apart from the above theories, the concept of the spherical fuzzy sets (SFSs) has been introduced by Mahmood et al. [34], which consists of three membership degrees with a condition that the sum of squares of all degrees must not exceeds one. Further, the concepts of SFSs are extended to T-spherical fuzzy sets T-SFSs, where there are no restrictions on their constants and, hence, T-SFSs can handle all the situations where the frameworks of FSs, IFSs, PyFSs, PFSs, and SFSs failed. For this environment, Mahmood et al. [34] presented some aggregation operators for T-SFSs. Later on, Ullah et al. [35] presented the concept of the symmetry measures for handling the uncertainties under the T-SFSs environment, and applied it to solve the decision-making problems. However, from the existing work, it is noticeable that the existing AOs under the IFSs, PFSs, etc., have failed to handle the situations under some certain cases. For instance, under the IFS environment, if we consider the two IFNs, *A* = (0, *nA*) and *B* = (*mB*, *nB*) where *mB*, *nA*, *nB* represented degrees of membership grades that lies between zero and one, by applying the geometric AOs, as defined in [36], to such numbers we then ge<sup>t</sup> the aggregated numbers as (0, *nA* + *nB* − *nAnB*). Thus, the final aggregated value of membership degree is zero, irrespective of value of *mB*. Similarly, for T-SFSs, if we assume *A* = (*mA*, 0, *nA*) and *B* = (*mB*, *iB*, *nB*) then, using the geometric aggregation operators of PFSs [37,38] and T-SFSs [34], we obtain the result of type (*some value*, 0,*some value*). This shows that the abstinence value of *B* is not accounted for in aggregation. Further, by taking *A* = (0, 0, *nA*) and *B* = (*mB*, *iB*, *nB*), then using the operators defined in [34,37,38], we ge<sup>t</sup> (*some value*, 0,*some value*). This shows that the membership and abstinence value is not accounted for in aggregation. These examples clearly point out the shortcomings that exist in the aggregation operators of PFSs and T-SFSs.

In order to overcome such shortcomings, and by utilizing the advantages of the T-SFSs over the several other existing theories, in this manuscript we have presented some new, improved geometric interactive aggregation operators. For it, firstly, we define some new operational laws by adding the degree of the hesitation into the operations. To do this, the concept of probability membership, non-membership, and heterogeneous are introduced and then some of their desirable properties are studied. Then, based on these proposed operational laws, some weighted, ordered weighted, and hybrid geometric aggregation operators, namely, T-spherical fuzzy weighted geometric interaction averaging (T-SFWGIA), T-spherical fuzzy ordered weighted geometric interaction averaging (T-SFOWGIA), and T-spherical fuzzy hybrid geometric interaction averaging (T-SFHGIA) operators are introduced in the paper. The desirable properties of such operators are investigated in detail. Then, based on such operators, we developed an algorithm for solving the decision-making problem under the T-SFS environment. The practical utility of the proposed approach is demonstrated through a numerical example, and comparative studies investigate the superiority of the approach. Finally, a counter example is provided to show the supremacy of the proposed operators with respect to the existing operators. Therefore, motivated from it, the objectives of the paper are summarized as follows:


To do so, the organization of this manuscript is summarized as follows: Section 2 gives a basic overview of the basic concepts of IFSs, PFSs, SFSs, and T-SFSs; Section 3 deals with some new multiplication operations laws and their corresponding weighed geometric AOs; inn Section 4, we present a MADM approach for solving the decision-making problem by using the proposed AOs (here the preferences related to each alternative are summarized in the form of T-SFS information); Section 5 presents a numerical example to illustrate the proposed approach and the comparative analysis; and finally, Section 6 concludes the paper with some concluding remarks.
