*5.3. Comparative Analysis*

The significance of the proposed new geometric operators lies in the fact that the result obtained by using these operations were more justifiable than those developed earlier (i.e., [34,37,38]). Such operators could not deal with situations where if membership and abstinence value of any number becomes zero then the membership and abstinence value of their aggregated value is also zero. Hence the existing operators of PFSs and T-SFSs did possess the capability of dealing with any kinds of information. But, on the other hand, the new geometric operators of T-SFSs can deal with any type of data justifiably. This point is demonstrated in the case study described in Section 5.1.

The second main advantage of our proposed work is that it has the ability to aggregate the data available in the form of IFSs, PyFSs, PFSs, and SFSs. But, conversely, the existing operators could not handle the data provided in the T-spherical fuzzy environment. For example, if we look at Example 2, its data is purely in the form of T-SFNs based on four grades, being membership, abstinence, non-membership, and refusal degree with *t* = 3, which shows that the aggregation operators of IFSs, PyFSs, PFSs, and SFSs could not aggregate this data. But if we look at Example 3, its data is in the form of IFNs, and our proposed operators easily aggregated this type of data with *t* = 1 and *i* = 0.

Hence, by all means, the proposed work had superiority over the existing work.

**Example 3.** *Let P*1 = (0, 0.5), *P*2 = (0.5, 0.4), *P*3 = (0.4, 0.2), *P*4 = (0.3, 0.3) *and P*5 = (0.7, 0.1) ∈ *IFN. The weight vector for Pi*(*i* = 1, 2, . . . , 5) *is ω* = (0.18, 0.22, 0.16, 0.21, 0.23) *T.*

$$\begin{aligned} P\_1 &= \left( \left( 1 - 0.5 \right)^{5 \times 0.18} - \left( 1 - \left( 0 + 0.5 \right) \right)^{5 \times 0.18}, 1 - \left( 1 - 0.5 \right)^{5 \times 0.18} \right) \\ &= \left( 0, 0.5796 \right) \\ P\_2 &= \left( \left( 1 - 0.4 \right)^{5 \times 0.22} - \left( 1 - \left( 0.5 + 0.4 \right) \right)^{5 \times 0.22}, 1 - \left( 1 - 0.4 \right)^{5 \times 0.22} \right) \\ &= \left( 0.5039, 0.3183 \right) \\ P\_3 &= \left( \left( 1 - 0.2 \right)^{5 \times 0.16} - \left( 1 - \left( 0.4 + 0.2 \right) \right)^{5 \times 0.16}, 1 - \left( 1 - 0.2 \right)^{5 \times 0.16} \right) \right) \\ P\_4 &= \left( \left( 1 - 0.3 \right)^{5 \times 0.21} - \left( 1 - \left( 0.3 + 0.3 \right) \right)^{5 \times 0.21}, 1 - \left( 1 - 0.3 \right)^{5 \times 0.21} \right) \right) \\ &= \left( 0.2870, 0.2746 \right) \\ P\_5 &= \left( \left( 1 - 0.1 \right)^{5 \times 0.23} - \left( 1 - \left( 0.7 + 0.1 \right) \right)^{5 \times 0.23}, 1 - \left( 1 - 0.1 \right)^{5 \times 0.23} \right) \right) \\ &= \left( 0.7203, 0.1094 \right) \end{aligned}$$

Scores values were

*SC*(*<sup>P</sup>*1) = −0.5796, *SC*(*<sup>P</sup>*2) = 0.1856, *SC*(*<sup>P</sup>*3) = 0.2000, *SC*(*<sup>P</sup>*4) = 0.0125, *SC*(*<sup>P</sup>*5) = 0.6109.

Thus, *SC*(*<sup>P</sup>*5) > *SC*(*<sup>P</sup>*3) > *SC*(*<sup>P</sup>*2) > *SC*(*<sup>P</sup>*4) > *SC*(*<sup>P</sup>*1) and we had

> *<sup>P</sup>σ*(1) = (0.7203, 0.1094) *<sup>P</sup>σ*(2) = (0.4000, 0.2000) *<sup>P</sup>σ*(3) = (0.5039, 0.3183) *<sup>P</sup>σ*(4) = (0.2870, 0.2746) *<sup>P</sup>σ*(5)= (0, 0.5796)

By using the normal distribution-based method, we found *w* = (0.1117, 0.2365, 0.3036, 0.2365, 0.1117) *T*.

Now, by using the definition of the T-SFHGIA operator, we found

$$(T - SFHGIA\_{\omega, w}(P\_1, P\_2, P\_3, P\_4, P\_5) = (0.4093, 0.2919)^{\circ}$$

Here we go<sup>t</sup> the same result as in [9,10,39]. Thus, the proposed new operators had the capability to solve the problems that lie in the existing structures.
