*6.3. Comparative Study*

In this section, we perform some comparative analysis of the proposed method result with some of the existing approaches results in [36,46–48] under the uncertain environment. The results computed from them on to the considered problem are summarized as below:

1. In [36], authors proposed the weighted geometric Bonferroni mean operator under the type-2 fuzzy environment, denoted by IT2FWGBM, which is defined as

$$\begin{array}{rcl}d\_k &=& \text{IT2FWGBM}\_w^{p,q}(A\_1, A\_2, \dots, A\_m) \\ &=& \frac{1}{p+q} \left(\bigotimes\_{\substack{i,j=1 \\ i\neq j}}^m \left(p(A\_i)^{w\_i} \oplus q(A\_j)^{w\_j}\right)\right)^{1/m(m-1)} \end{array} \tag{28}$$

By applying Equation (28) on to the considered data, we ge<sup>t</sup> the aggregated value corresponding to each alternative as

$$\begin{array}{rcl}d\_{1}&=&\text{I}\,\text{T2FWGBM}\_{w}^{1}(A\_{11},A\_{12},A\_{13},A\_{14},A\_{15},A\_{16},A\_{17})\\&=&\text{(0.8321,0.9050,0.9050,0.9534,0.6065)}\\d\_{2}&=&\text{I}\,\text{T2FWGBM}\_{w}^{1}(A\_{21},A\_{22},A\_{23},A\_{24},A\_{25},A\_{26},A\_{27})\\&=&\text{(0.8671,0.9486,0.9486,1.0000,0.7500)}\\d\_{3}&=&\text{I}\,\text{T2FWGBM}\_{w}^{1}(A\_{31},A\_{32},A\_{33},A\_{34},A\_{35},A\_{36},A\_{37})\\&=&\text{(0.7980,0.8676,0.8676,0.9137,0.6015)}\\d\_{4}&=&\text{I}\,\text{T2FWGBM}\_{w}^{1}(A\_{41},A\_{42},A\_{43},A\_{44},A\_{45},A\_{46},A\_{47})\\&=&\text{(0.8317,0.9131,0.9131,0.9656,0.6080)}\\d\_{5}&=&\text{I}\,\text{T2FWGBM}\_{w}^{1}(A\_{51},A\_{52},A\_{53},A\_{54},A\_{55},A\_{56},A\_{57})\\&=&\text{(0.7802,0.8456,0.8456,0.8895,0.6085)}\\d\_{6}&=&\text{I}\,\text{T2FWGBM}\_{w}^{1}(A\_{61},A\_{62},A\_{63},A\_{64},A\_{65},A\_{66},A\_{66},A\_{67})\\&=&\text{(0.8318,0.9073,0.9073,0.956$$

Therefore, the score values of these aggregated numbers are *<sup>s</sup>*(*d*1) = 0.5405, *<sup>s</sup>*(*d*2) = 0.7079, *<sup>s</sup>*(*d*3) = 0.5182, *<sup>s</sup>*(*l*4) = 0.5450, *<sup>s</sup>*(*l*5) = 0.5052, and *<sup>s</sup>*(*l*6) = 0.5418 and hence the final ranking of all alternatives *Ak*(*k* = 1, 2, . . . , 6) is found as

$$A\_2 \succ A\_4 \succ A\_6 \succ A\_1 \succ A\_3 \succ A\_5$$

2. If we use the existing WSTIT2FHM operator as proposed by Qin [46] under the T2FS environment

$$\begin{array}{rcl} l\_{p} &=& \text{WSTIT2FHM}^{(k)}(A\_{1}, A\_{2}, \ldots, A\_{n}) \\ &=& \begin{pmatrix} \sum\_{\begin{subarray}{c} \sum\_{i \in \zeta\_{i}} \end{subarray}} \left(1 - \sum\_{j=1}^{k} w\_{i\_{j}}\right) \left(\prod\_{j=1}^{k} \zeta\_{w\_{i\_{j}}}\right)^{\frac{1}{k}} & \sum\_{1 \leq i\_{1} < \cdots < i\_{k} \leq n} \left(1 - \sum\_{j=1}^{k} w\_{i\_{j}}\right) \left(\prod\_{j=1}^{k} \varrho\_{w\_{i\_{j}}}\right)^{\frac{1}{k}} \\ & \xrightarrow{\cdots < \underline{k} \leq \underline{n}} \left(\prod\_{i \in \underline{k} \leq \underline{n}} \left(1 - \left(\prod\_{j=1}^{k} \left(1 - \varrho\_{a\_{i\_{j}}}\right)\right)^{\frac{1}{k}}\right)^{\frac{1}{\binom{k}{m} - 1}} \\ & \left(\prod\_{1 \leq i\_{1} < \cdots < i\_{k} \leq n} \left(1 - \left(\prod\_{j=1}^{k} \varrho\_{a\_{i\_{j}}}^{\*}\right)^{\frac{1}{k}}\right)^{\frac{1}{\binom{k}{m-1}}} \right)^{\frac{1}{\binom{k}{m-1}}} \\ & \left(1 - \left(\prod\_{1 \leq i\_{1} \leq n} \left(1 - \left(\prod\_{j=1}^{k} \varrho\_{a\_{i\_{j}}}^{\*}\right)^{\frac{1}{k}}\right)^{\frac{1}{\binom{k}{m-1}}}\right)^{\frac{1}{\binom{k}{m-1}}} \\ \end{array} \end{array} \\ \begin{pmatrix} \prod\_{i \in \underline{k} \leq$$

then, the aggregated values corresponding to each alternative (by taking *k* = 2) are obtained as

$$\begin{aligned} l\_1 &= (0.5154, 0.2276, 0.7820, 0.8314); l\_2 = (0.6950, 0.3239, 0.8546, 0.9054); \\ l\_3 &= (0.3846, 0.1681, 0.7166, 0.7633); l\_4 = (0.5481, 0.2612, 0.7951, 0.8449); \\ l\_5 &= (0.3201, 0.1342, 0.6769, 0.7244); l\_6 = (0.5272, 0.2390, 0.7914, 0.8414) \end{aligned}$$

Thus, the score values are

$$\begin{array}{rcl} s(l\_1) &=& (0.2077, 0.8067); s(l\_2) = (0.3055, 0.8799); s(l\_3) = (0.1422, 0.7400) \\ s(l\_4) &=& (0.2245, 0.8200); s(l\_5) = (0.1120, 0.7006); s(l\_6) = (0.2150, 0.8164) \end{array}$$

and hence ordering is

$$A\_2 \succ A\_4 \succ A\_6 \succ A\_1 \succ A\_3 \succ A\_5$$

From the above examinations, it is revealed that the ranking order of the alternatives stays same ye<sup>t</sup> the computational procedure is altogether unique. For instance, in [36,46] authors have introduced AOs under TIT2FNs by considering just the degree of membership during an examination. But it is quite recognizable that the level of non-membership likewise assumes a predominant part during the aggregation process. Thus, the outcomes processes by these methodologies [36,46] might be unreasonable under some specific constraints where the degree of non-membership pays a more significance than the degree of agreement.

However, apart from these, we give some characteristics comparison of our proposed method and the aforementioned methods, which are listed in Table 4.


**Table 4.** The characteristic comparisons of different methods.

In [47], authors presented an analytical method for solving the problems by using the fuzzy weighted average. In [36], the authors have presented the BM by considering simultaneously the values of UMF and LMF to aggregate IT2FS information. On the other hand, the present study is based on the HM operator which is more adaptable and robustness in process of information fusion than others such as BM, GBM. The outstanding characteristic of the HM operator is to catch the inter-relationship between more than two input arguments with a parameter *k* from the finite integer set. Furthermore, in [46], the author developed HM operator by taking into account the membership degree only but in practical problems, it is sometimes not possible for DM to give their preferences in terms of acceptance degree only. Therefore, the non-membership degree is required for handling the problems in which rejection degree is not equal to one minus acceptance degree. Also by comparing with the AHP-based method [48], the proposed method does not require any software package to compute the results while the technique proposed in [48] requires it. Thus, the computation complexity of the proposed technique is comparatively easy. Furthermore, the AHP-based technique is usually dependent on various parameters and thus the final ranking may some time suffers from inconsistency, in the case of inappropriate parameter selection. On the other hand, the proposed method draws up a more authentic ranking result as it can terminate the difference, draws up for the flaws of already existing aggregation methods that do not capture experts utility or decision preference and achieves more stationary and commendable interrelationships result with less information loss. The proposed method takes into consideration the uniformity of the alternatives as well as highlights the significance and interactions in association with any solutions of alternatives. On the other hand, the AHP-based technique is good at calculating only the optimal ranking values of the alternatives beyond inter-relationships.
