*5.1. Proposed Method*

As stated above, the properties of boundedness and monotonicity are valid for proposed operators. Therefore, a comparison can be made among two or more GBGWA(GBGWG) operators. Let *ς* be the number of committees established comprising specialists, which are intended to classify each alternative *<sup>κ</sup>j*(*j* = 1, 2, ..., *<sup>n</sup>*), while making in account with the imperative attributes *<sup>υ</sup>i*(*<sup>i</sup>* = 1, 2, ..., *<sup>m</sup>*), and by provision of their respective grades in terms of IFSSs. Consider *d*1, *d*2, ..., *dp* be the members/experts(directors or officers), who are in-charge of constituted committees. Thereafter, the subjective information (in the form of IFSSs) from committees is collected. The senior experts will examine it and give their judgements as a group of IFSs. Then, the information of each committee comprised the GGIFSS, and there will be *ς* number of GGIFSSs. The extended union on GGIFSSs is computed, denoted as G *g* and expressed in a table. Here, two types of criteria occur in the G *g* , namely, benefit and cost criteria. To consolidate the criteria, the G *g* must be normalized through the following equation:

$$\tau\_{ji} = \begin{cases} \langle \check{t}\_{\mathcal{A}}(\upsilon\_{i}), \check{f}\_{\mathcal{A}}(\upsilon\_{i}) \rangle, & \text{if } \upsilon\_{i} \text{ is a benefit criterion,} \\ \langle \check{f}\_{\mathcal{A}}(\upsilon\_{i}), \check{t}\_{\mathcal{A}}(\upsilon\_{i}) \rangle, & \text{if } \upsilon\_{i} \text{ is a cost criterion,} \end{cases} \tag{10}$$

such that the normalized GGIFSS is denoted by G Q *g* = (T , *E*, *g* ), where (T , *E*) is the normalization of (T , *E*) and *g* is the normalization of *g*. Finally, GBGWA or GBGWG can be used to aggregate the data from G Q *g* and each *j* or *j* can be correlated through score function. Therefore, we propose our methodology as an algorithm as follows.

#### **Algorithm 1** Multi-attribute decision making on GGIFSSs

#### Input: A set of alternatives

Output: The felicitous alternative for a problem



This algorithm is depicted as a flowchart in Figure 1. The Algorithm 1 can be formulated to select the best product or alternative for *p* number of customers. In this way, the extra inputs incorporate as the demands of customers in GGIFSS, and the Algorithm 1 will conduct on a GGIFSS, F Q *g* = (S , *E*, *g*), from Step 4. To operate above methodology, we establish two case studies as below.

**Figure 1.** A flowchart for our algorithm.

#### *5.2. Case Study: Candidate Selection Problem*

In this case study, an example for evaluation of candidates is used to illustrate the applicability of the proposed method. An insurance company HG in Guangzhou, China is engaged for insurance of products, charging insurance premium, consultation on insurance, financial and other services for individuals and enterprises. Every year, this company recruits new staff for the post of insurance sales agents and consultants. To maintain the excellence and high admire reputation, the company consults with experts for their assessments and opinions to recruit the candidates. Furthermore, the insurance business department and human resources department are actively engaged in recruitment process.

Let *X* = {*<sup>κ</sup>*1, *κ*2, *κ*3, *κ*4, *<sup>κ</sup>*5} be the set of five candidates whom can be placed for the position of insurance sales consultant. A group of three senior members (directors, officers, etc.) *d*1, *d*2 and *d*3 setup a committee of specialists and experts to appoint a felicitous candidate for this position. The set of criteria for committee to select the candidate is *E* = {*<sup>υ</sup>*1, *υ*2, *υ*3, *υ*4, *υ*5, *υ*6, *<sup>υ</sup>*7}, where

*υ*1 : english level;

*υ*2 : relevant problem solving skills;


On the set parameters, the weight vector is given and denoted by

*w* = (*w*1/0.12, *w*2/0.13, *w*3/0.15, *w*4/0.15, *w*5/0.17, *w*6/0.11, *w*7/0.17)*<sup>T</sup>* such that ∑<sup>7</sup>*i*=<sup>1</sup> *wi* = 1. The three senior members arrange specialists into two groups; the first group consists of the specialists of insurance business managemen<sup>t</sup> and the second group consists of the specialists of human resource management. The set of parameters A = {*<sup>υ</sup>*2, *υ*3, *<sup>υ</sup>*5} is assigned for first group and the set of parameters B = {*<sup>υ</sup>*1, *υ*4, *υ*6, *<sup>υ</sup>*7} is assigned for second group. These two groups give their judgments as IFSSs (S 1, A) and (S 2, B), respectively. Then, the group of senior members examine the data of IFSSs and then provide the two groups of IFSs, *g*1 = {*αd*1 , *<sup>α</sup>d*2 , *<sup>α</sup>d*3 } and *g*2 = {*β d*1 , *β d*2 , *β d*3 } to complete the GGIFSSs, F Q *g* 1 = (S 1, A, *g*1) and F Q *g* 2 = (S 2, B, *g*2), as shown in Tables 5 and 6, respectively.


**Table 5.** Tabular representation of the GGIFSS F *g* 1 = (S 1, A, *g*1)

Q

**Table 6.** Tabular representation of the GGIFSS F Q *g* 2 = (S 2, B, *g*2)


To evaluate most felicitous candidate on provided information in Tables 5 and 6, the extended intersection of F Q *g* 1 and F Q *g* 2 is contemplated as follows:

$$
\hat{\mathcal{H}}\_{\overline{\mathcal{S}}} = \overline{\mathcal{P}}\_{\overline{\mathcal{S}}1} \amalg\_{\mathcal{E}} \overline{\mathcal{P}}\_{\overline{\mathcal{S}}2} = (\widetilde{\mathcal{T}}, E, \widetilde{\mathcal{g}}) = (\widetilde{\mathcal{S}}\_1, \mathcal{A}, \widetilde{\mathcal{g}}\_1) \amalg\_{\mathcal{E}} (\widetilde{\mathcal{S}}\_2, \mathcal{B}, \widetilde{\mathcal{g}}\_2),
$$

and shown in Table 7.


**Table 7.** Tabular representation of the GGIFSS F Q *g* 1+ <sup>E</sup>.F Q *g* 2.

As the all criterion are benefit type, the normalized GGIFSS, G *g* , is not needed. The weight vector for three senior members is given by = (1/0.33, 2/0.34, 3/0.33)*<sup>T</sup>* such that ∑<sup>3</sup>*k*=<sup>1</sup> *k* = 1.The GBGWA operator is used on integrated data in Table 7, and given as follows: 1 = GGWA(*<sup>c</sup>*11, *c*12, ..., *<sup>c</sup>*17) = IFWA*k*(0.390932, 0.353894,0.260704, 0.518795,0.325209, 0.489552) = 0.327079, 0.448601 2 = GGWA(*<sup>c</sup>*21, *c*22, ..., *<sup>c</sup>*27) = IFWA*k*(0.355421, 0.494395,0.237022, 0.623437,0.295668, 0.600553) = 0.297119, 0.570421 3 = GGWA(*<sup>c</sup>*31, *c*32, ..., *<sup>c</sup>*37) = IFWA*k*(0.390064, 0.455207,0.260125, 0.594251,0.324487, 0.569593) = 0.326346, 0.536655 4 = GGWA(*<sup>c</sup>*41, *c*42, ..., *<sup>c</sup>*47) =

IFWA*k*(0.374141, 0.435074,0.249507, 0.579256,0.311241, 0.553687) = 0.312904, 0.519253 5 = GGWA(*<sup>c</sup>*51, *c*52, ..., *<sup>c</sup>*57) = IFWA*k*(0.422383, 0.383619,0.281678, 0.540934,0.351372, 0.513036) = 0.353679, 0.474575

Now, the score functions are calculated on above five operators and given as in the following: *δ*( 1) = 0.439239, *δ*( 2) = 0.363349, *δ*( 3) = 0.394845, *δ*( 4) = 0.396825, and *δ*( 5) = 0.439552. The descending order is acquired as *κ*5 > *κ*1 > *κ*4 > *κ*3 > *κ*2; thus, *κ*5 is the felicitous candidate for the position because *δ*( 5) = 0.439552 is the maximum score.

Next, a case study in a different scenario is given as follows.

#### *5.3. Case Study: Alternative Evaluation on Customer Demands*

Nowadays, the markets possess immense competition for the quality of service, besides the demands of customers are increased and widened in the different prospects. The service industries are booming and upgrading by entertainment, catering, tourism, and auction. Indeed, there is a fierce competition among the service industries, but currently film industry is in the most competitive position as customers always classify and compare cinemas on different parameters, such as convenience, environment, quality of service, upcoming movies, and expenses.

Let *X* = {*<sup>κ</sup>*1, *κ*2, *κ*3, *<sup>κ</sup>*4} be the set of four cinemas. The set of attributes *E* = {*<sup>υ</sup>*1, *υ*2, *υ*3, *υ*4, *<sup>υ</sup>*5}, where

*υ*1 : quality of service;

*υ*2 : quality of expected films;

*υ*3 : environment in cinema;

*υ*4 : price reasonability; and

*υ*5 : convenience and luxuriousness.

A committee of experts and specialists from a cinema managemen<sup>t</sup> organization give the judgment for cinemas on provided attributes as an IFSSs (S, *E*) (Table 8).


**Table 8.** Tabular representation of the IFSS, (S, *<sup>E</sup>*).

Now, the two customers *d*1 and *d*2 desire to choose a most suitable cinema to watch movies; their demands comprise IFSs,

$$\begin{aligned} \widetilde{\boldsymbol{\pi}}\_{d\_1} &= \{ \boldsymbol{\upsilon}\_1 / \langle 0.4, 0.6 \rangle, \boldsymbol{\upsilon}\_2 / \langle 0.5, 0.3 \rangle, \boldsymbol{\upsilon}\_3 / \langle 0.5, 0.4 \rangle, \boldsymbol{\upsilon}\_4 / \langle 0.4, 0.6 \rangle, \boldsymbol{\upsilon}\_5 / \langle 0.4, 0.4 \rangle \}, \\ \widetilde{\boldsymbol{\pi}}\_{d\_2} &= \{ \boldsymbol{\upsilon}\_1 / \langle 0.5, 0.4 \rangle, \boldsymbol{\upsilon}\_2 / \langle 0.4, 0.5 \rangle, \boldsymbol{\upsilon}\_3 / \langle 0.5, 0.3 \rangle, \boldsymbol{\upsilon}\_4 / \langle 0.5, 0.4 \rangle, \boldsymbol{\upsilon}\_5 / \langle 0.4, 0.2 \rangle \}. \end{aligned}$$

The attribute *υ*4 belongs to cost criteria, therefore corresponding IFVs can be normalized using Equation (10). Let *g* = {*αd*1 , *<sup>α</sup>d*2 } and the normalization of *g* is expressed as *g* = {*α d*1 , *α d*2 }. Thereafter, Table 8 can be normalized. Then, the information can be extended into GGIFSS F Q *g* = ( S , *E*, *g* ), and specified in Table 9.


**Table 9.** Tabular representation of the GGIFSS, (S , *E*, *g* ).

Q

Let *w* = (*w*1/0.18, *w*2/0.19, *w*3/0.21, *w*4/0.22, *w*5/0.2)*<sup>T</sup>* be the weight vector for attributes and = (1/0.52, 2/0.48)*<sup>T</sup>* be the weight vector for customers. The GBGWA operator is used on integrated data in Table 9, and given as follows:

 1 = GGWA(*<sup>c</sup>*11, *c*21, ..., *<sup>c</sup>*15) = IFWA*k*(0.220934, 0.635540,0.199004, 0.605904) = 0.210483, 0.621138 2 = GGWA(*<sup>c</sup>*21, *c*22, ..., *<sup>c</sup>*52) = IFWA*k*(0.265324, 0.619281,0.238988, 0.588324) = 0.252798, 0.604223 3 = GGWA(*<sup>c</sup>*31, *c*32, ..., *<sup>c</sup>*35) = IFWA*k*(0.209414, 0.678382,0.188627, 0.652231) = 0.199503, 0.665701 4 = GGWA(*<sup>c</sup>*41, *c*42, ..., *<sup>c</sup>*45) = IFWA*k*(0.223041, 0.608588,0.200901, 0.576761) = 0.212491, 0.593098 5 = GGWA(*<sup>c</sup>*51, *c*52, ..., *<sup>c</sup>*55) = IFWA*k*(0.227867, 0.654431,0.205248, 0.626332) = 0.217091, 0.640789

Now, the score functions are calculated on above five operators and given as in the following: *δ*( 1) = 0.294672, *δ*( 2) = 0.324287, *δ*( 3) = 0.266901, *δ*( 4) = 0.309696, and *δ*( 5) = 0.288151. One can check that *κ*2 is the suitable cinema for both customers as *δ*( 2) = 0.324287 is the maximum score.

Now, based on our results, comparisons with other methods are given in next section.

#### **6. Comparisons and Discussions**

In this section, we compare our framework and results with existing methodologies. At first, we make a comparison of our method with the framework presented in [44]. Then, we discuss the advantages of proposed technique.

#### *6.1. Comparisons with the Method of Garg*

Garg et al. [44] defined geometric and averaging operators on the GGIFSSs and then provided an algorithm for decision making methodology. Let *X* = {*<sup>κ</sup>*1, *κ*2, ..., *<sup>κ</sup>n* } be the set of alternatives and *E* = {*<sup>υ</sup>*1, *υ*2, ..., *<sup>υ</sup>m* } be the set of criteria. To evaluate *<sup>κ</sup>j*(*j* = 1, ..., *n* ) as a optimal choice, IFSS on *E* are given and assessments of moderators are given as an IFSs *<sup>G</sup>*(*e*), where *G* = (1, 2, ..., *p*) and *<sup>G</sup>*(*e*) denotes the opinion of experts on the elements of *X* by virtue of IFSS on *E*. We recall the algorithm contemplated in [44] and given as follows:

#### **Algorithm 2** Grag's Algorithm for faculty appointment



Under the approach established in [44], we provide some key points and compare Algorithm 1, with Algorithm 2 :


#### *6.2. Comparisons with the Results of GIFSSs*

The obtained results on the case studies in Sections 5.2 and 5.3 are compared with the outcomes that are achieved on GIFSSs as given below.

(i) As discussed earlier, GGIFSS with only single generalized parameter is known as GIFSS. Then, Algorithm 1 can be separated for each senior moderator/customer in the case studies in Sections 5.2 and 5.3.

1. Using the Lemma 1 and Algorithm 1, we obtained the results separately for each senior experts/members in the case study in Section 5.2. If only Senior Member 1 is taken into account during selection process, then *δ*( 1) = 0.5185, *δ*( 2) = 0.4305, *δ*( 3) = 0.4674, *δ*( 4) = 0.4695, and *δ*( 5) = 0.5194. The descending order is acquired as *κ*5 > *κ*1 > *κ*4 > *κ*3 > *κ*2; thus, *κ*5 is the felicitous candidate for the position.

If only Senior Member 2 is taken into account during selection process in the case study in Section 5.2, then *δ*( 1) = 0.3709, *δ*( 2) = 0.3068, *δ*( 3) = 0.3329, *δ*( 4) = 0.3351, and *δ*( 5) = 0.3704. The descending order is acquired as *κ*1 > *κ*5 > *κ*3 > *κ*4 > *κ*2; thus, *κ*1 is the felicitous candidate for the position.

If only Senior Member 3 is taken into account during selection process in the case study in Section 5.2, then *δ*( 1) = 0.4178, *δ*( 2) = 0.3475, *δ*( 3) = 0.3774, *δ*( 4) = 0.3788, and *δ*( 5) = 0.4192. The descending order is acquired as *κ*5 > *κ*1 > *κ*4 > *κ*2 > *κ*3; thus, *κ*1 is the felicitous candidate for the position.

It can be observed from above discussion that *κ*5 is the most suitable candidate as per individual opinions of Senior Members 1 and 3. Similarly, *κ*1 is the most felicitous candidate on individual

opinion of Senior Member 2, while *κ*5 is on second place in descending order. Thus, in general, *κ*5 is the most suitable candidate.

2. Using the Lemma 1 and Algorithm 1, we obtained the results separately for each customer for the case study in Section 5.3. If it is required to select cinema only for Customer 1, then *δ*( 1) = 0.2927, *δ*( 2) = 0.3230, *δ*( 3) = 0.2655, *δ*( 4) = 0.3072, and *δ*( 5) = 0.2867. The order is acquired as *κ*2 > *κ*4 > *κ*1 > *κ*5 > *κ*3; thus, *κ*2 is the best cinema for Customer 1.

If it is required to select cinema only for the Customer 2, then *δ*( 1) = 0.2965, *δ*( 2) = 0.3253, *δ*( 3) = 0.2682, *δ*( 4) = 0.3121, and *δ*( 5) = 0.2894. The order is acquired as *κ*2 > *κ*4 > *κ*1 > *κ*5 > *κ*3; thus, *κ*2 is the best cinema for Customer 2. Thus,ingeneral,*κ*2isthemostsuitableforbothcustomers.

(ii) Feng et al. [42] introduced a framework of decision makings on GIFSSs. We correlate proposed results with their method as below. We acquired the results separately for each customer for the case study in Section 5.3. If it is required to select cinema only for Customer 1, then *<sup>δ</sup>*(*ZJ*(*<sup>κ</sup>*1)) = 0.5344, *<sup>δ</sup>*(*ZJ*(*<sup>κ</sup>*2)) = 0.5928, *<sup>δ</sup>*(*ZJ*(*<sup>κ</sup>*3)) = 0.4857, *<sup>δ</sup>*(*ZJ*(*<sup>κ</sup>*1)) = 0.5549, and *<sup>δ</sup>*(*ZJ*(*<sup>κ</sup>*1)) = 0.5251. The descending order acquired as *κ*2 > *κ*4 > *κ*1 > *κ*5 > *κ*3; thus, *κ*2 is the best cinema for Customer 1. If it is require to select cinema only for Customer 2, then *<sup>δ</sup>*(*ZJ*(*<sup>κ</sup>*1)) = 0.5311, *<sup>δ</sup>*(*ZJ*(*<sup>κ</sup>*2)) = 0.5961, *<sup>δ</sup>*(*ZJ*(*<sup>κ</sup>*3)) = 0.4894, *<sup>δ</sup>*(*ZJ*(*<sup>κ</sup>*1)) = 0.5568, and *<sup>δ</sup>*(*ZJ*(*<sup>κ</sup>*1)) = 0.5262. The descending order acquired as *κ*2 > *κ*4 > *κ*1 > *κ*5 > *κ*3; thus, *κ*2is the best cinema for Customer 2.

(iii) A framework for the best concept selection in design process has been computed in [43], where GIFSSs are utilized to acquire integrated information on customers demands and design concepts. To meet their objectives, they introduced an algorithm, which we updated as follows:

#### **Algorithm 3** Updated form of Algorithm in [43]


The case study in Section 5.3 can be contemplated through Algorithm 3. On this prospect, GGIFSS given in Table 9 can be separated into two GIFSSs. After adopting all steps of Algorithm 3, <sup>Δ</sup>(*<sup>κ</sup>*1) = 0.2252, <sup>Δ</sup>(*<sup>κ</sup>*2) = 0.2678, <sup>Δ</sup>(*<sup>κ</sup>*3) = 0.2112, <sup>Δ</sup>(*<sup>κ</sup>*4) = 0.2419, and <sup>Δ</sup>(*<sup>κ</sup>*5) = 0.2261. The descending order is acquired as *κ*2 > *κ*4 > *κ*5 > *κ*1 > *κ*3; thus, *κ*2 is the best cinema for both customers.

The superiorities and advantages of our method are given in next section.

#### **7. Superiority of Proposed Method**

In this section, we give some counter-examples to show the superiority of proposed method over recent approaches [42–44].

**Example 5.** *Assume a decision making problem by letting the two alternatives κ*1 *and κ*2*, which have to be evaluated by the committee of specialists over set of parameters E* = {*<sup>υ</sup>*1, *υ*2, *<sup>υ</sup>*3}*. The committee of specialists provide the judgments in the form of IFSS, given in Table 10;*


**Table 10.** Tabular representation of the IFSS, (S , *<sup>E</sup>*).

*Here, we apply the approach provided by Feng et al. [42], by letting an extra input β* = {0.4, 0.2,0.5, 0.3,0.6, 0.4} *of a moderator. Then, the GIFSS is consolidated as in Table 11;*

**Table 11.** Tabular representation of the IFSS, (S , *E*, *β* ).


*The score function on IFVs in β are* 0.6, 0.6*, and* 0.6 *and the weights are* 0.33, 0.33*, and* 0.33*, respectively. It can be seen that, when we convert extra input into weights in initial stages of decision making, the importance of membership and non-membership diminish. Using the method of Feng et al. [42], we get <sup>δ</sup>*(*ZJ*(*<sup>κ</sup>*1)) = 0.503 > *<sup>δ</sup>*(*ZJ*(*<sup>κ</sup>*2)) = 0.498 *such that κ*1 > *κ*2. *Let w* = {*w*1/0.32, *w*2/0.33, *w*3/0.35} *be the weighted vector over E. Then, by proposed method, δ*( 1) = 0.310 < *δ*( 2) = 0.314 *such that κ*1 < *κ*2. *Therefore, the conversation of extra input into weighted vector in early process of decision making diminish the importance of membership and non-membership. Thus, proposed approach is better then the method of Feng et al. [42].*

**Example 6.** *Assume that κ*1*,κ*2 *and κ*3 *are three products and E* = {*<sup>υ</sup>*1, *υ*2, *<sup>υ</sup>*3} *is the set of parameters. The dependencies of products on criteria are provided in IFSS* (S , *E*) *and given in Table 12.*


**Table 12.** Tabular representation of the IFSS, (S , *<sup>E</sup>*).

*Here, we consider the methodology of Hayat et al. [43]. To select best product for two customers d*1, *d*2*, their demands are investigated as β d*1 = {0.3, 0.5,0.4, 0.4,0.6, 0.2}*, β d*2 = {0.3, 0.6,0.3, 0.4,0.5, 0.4}*, respectively. Then, the GIFSSs for d*1 *and d*2 *are given in Tables 13 and 14, respectively.*

**Table 13.** GIFSS (S , *E*, *β d*1).


**Table 14.** GIFSS (S , *E*, *β d*2).


*In [43], AND operation is computed on two GIFSSs for product for two customers. One can check that*

$$\widetilde{\mathcal{S}}(\boldsymbol{\upsilon}\_{1}) \wedge \widetilde{\mathcal{S}}(\boldsymbol{\upsilon}\_{2}) = \widetilde{\mathcal{S}}(\boldsymbol{\upsilon}\_{1}) \text{ with } \widetilde{t}\_{\widetilde{\boldsymbol{\beta}}\_{\mathcal{d}\_{1}}}(\boldsymbol{\upsilon}\_{1}) \wedge \widetilde{t}\_{\widetilde{\boldsymbol{\beta}}\_{\mathcal{d}\_{2}}}(\boldsymbol{\upsilon}\_{2}) = \widetilde{t}\_{\widetilde{\boldsymbol{\beta}}\_{\mathcal{d}\_{1}}}(\boldsymbol{\upsilon}\_{1}), \widetilde{f}\_{\widetilde{\boldsymbol{\beta}}\_{\mathcal{d}\_{1}}}(\boldsymbol{\upsilon}\_{1}) \vee \widetilde{f}\_{\widetilde{\boldsymbol{\beta}}\_{\mathcal{d}\_{2}}}(\boldsymbol{\upsilon}\_{2}) = \widetilde{f}\_{\widetilde{\boldsymbol{\beta}}\_{\mathcal{d}\_{1}}}(\boldsymbol{\upsilon}\_{1}) \tag{11}$$

$$\tilde{\mathcal{S}}(\boldsymbol{\upsilon}\_{1}) \wedge \tilde{\mathcal{S}}(\boldsymbol{\upsilon}\_{3}) = \tilde{\mathcal{S}}(\boldsymbol{\upsilon}\_{1}) \text{ with } \tilde{t}\_{\tilde{\boldsymbol{\beta}}\_{\tilde{\boldsymbol{d}}\_{1}}}(\boldsymbol{\upsilon}\_{1}) \wedge \tilde{t}\_{\tilde{\boldsymbol{\beta}}\_{\tilde{\boldsymbol{d}}\_{2}}}(\boldsymbol{\upsilon}\_{3}) = \tilde{t}\_{\tilde{\boldsymbol{\beta}}\_{\tilde{\boldsymbol{d}}\_{1}}}(\boldsymbol{\upsilon}\_{1}), \tilde{f}\_{\tilde{\boldsymbol{\beta}}\_{\tilde{\boldsymbol{d}}\_{1}}}(\boldsymbol{\upsilon}\_{1}) \vee \tilde{f}\_{\tilde{\boldsymbol{\beta}}\_{\tilde{\boldsymbol{d}}\_{2}}}(\boldsymbol{\upsilon}\_{3}) = \tilde{f}\_{\tilde{\boldsymbol{\beta}}\_{\tilde{\boldsymbol{d}}\_{1}}}(\boldsymbol{\upsilon}\_{1}) \tag{12}$$

$$\tilde{\mathcal{S}}(\boldsymbol{\upsilon}\_{2}) \wedge \tilde{\mathcal{S}}(\boldsymbol{\upsilon}\_{3}) = \tilde{\mathcal{S}}(\boldsymbol{\upsilon}\_{2}) \text{ with } \tilde{\mathfrak{t}}\_{\tilde{\boldsymbol{\beta}}\_{\tilde{\boldsymbol{\beta}}\_{1}}}(\boldsymbol{\upsilon}\_{2}) \wedge \tilde{\mathfrak{t}}\_{\tilde{\boldsymbol{\beta}}\_{\tilde{\boldsymbol{\beta}}\_{2}}}(\boldsymbol{\upsilon}\_{3}) = \tilde{\mathfrak{t}}\_{\tilde{\boldsymbol{\beta}}\_{\tilde{\boldsymbol{\beta}}\_{1}}}(\boldsymbol{\upsilon}\_{2}) \lrcorner \tilde{f}\_{\tilde{\boldsymbol{\beta}}\_{\tilde{\boldsymbol{\beta}}\_{1}}}(\boldsymbol{\upsilon}\_{2}) \vee \tilde{f}\_{\tilde{\boldsymbol{\beta}}\_{\tilde{\boldsymbol{\beta}}\_{2}}}(\boldsymbol{\upsilon}\_{3}) = \tilde{f}\_{\tilde{\boldsymbol{\beta}}\_{\tilde{\boldsymbol{\beta}}\_{1}}}(\boldsymbol{\upsilon}\_{2}).\tag{13}$$

*It can be seen that, using AND operation, the importance of IFVs for υ*2 *and υ*3 *are diminished in Equations (11) and (12). The importance of IFVs for υ*3 *are diminished in Equation (13). Thus, such an approach is not valid in the initial stages of decision making; therefore, for this prospect, the proposed approach is better then Hayat et al. [43].*

In [44], GWA is computed on two information; one from a committee of experts (in form of IFSS) and other from group of senior persons. The extra inputs can be seen as a so-called IFSS of a group of senior persons over alternatives. Consider Example 5, where IFSS from a committee of experts is given in Table 10. The extra input is given in Table 15, and can be seen as a so-called IFSS on alternatives. In the prospect of Garg et al. [44], the extra opinions of the two senior experts *d*1, *d*2 can be merged with IFSS in Table 10.

**Table 15.** Opinions of experts on alternatives.


Clearly, the combination of two data ((i) IFSS and (ii) IFVs of experts) based matrix is analyzed as GGIFSS in [44]. In another way, if it might be recognized that the group of extra inputs of senior experts is a summarization of the data (IFSS obtained from a committee of specialists), then the results can be obtained from Table 15, thus why would we contemplate two data based matrix over alternatives? Nevertheless, there exist some serious difficulties in [44]. Noteworthily, the proposed results are superior in certain aspects and a well-defined manner is considered.

#### *Advantages of Proposed Method*

Based on correlative and comparative research, the following benefits of present framework are acquired and emphasized:


better results. In the proposed method, extra inputs are taken into account in an accurate way using GBGWA or GBGWG.

