**2. Preliminaries**

In this section, we present the basic definitions of fuzzy sets, IFSs, soft sets and GIFSS which would be useful for subsequent discussions. Throughout the paper, *X* is the universe.

A fuzzy set *t* in *X* is usually identified as its membership function *t* : *X* −→ [0, 1] [2], each *x* ∈ *X*, where the membership grade *<sup>t</sup>*(*x*) indicates the degree to which the element *x* belongs to the fuzzy set *t*. Here, we denote by F(*X*) the collection of all fuzzy sets in *X*. The subsets intersection, union and complement of fuzzy sets follow from Zadeh [2].

In 1999, Molodtsov [3] introduced the parameterization concept soft set theory, which is different from many traditional tools for dealing with uncertainties, such as fuzzy set theory [2], rough set theory [46], IFSs [25], and hesitant fuzzy sets. The main advantage of soft set theory is that it can be freely applied to characterized parameters, sentence, words and numbers. The natural manner of parameterization of this theory was augmented by the works of Maji et al. [4] and Ali et al. [5], among others.

**Definition 1.** *[3] Let E be the set of parameters,* A ⊆ *E. A pair* (S, A) *is called a soft set over X, where* S *is a mapping given by* S : A −→ *<sup>P</sup>*(*X*)*. P*(*X*) *is the set of all power sets of X.*

The set of all soft sets over *X*, with respect to subsets of *E*, is denoted by SASS*<sup>E</sup>*(*X*).

#### *2.1. Intuitionistic Fuzzy Sets*

In the fuzzy set, only one compatibility degree exists, whereas intellectual insight in many cases suggests that non-compatibility degrees should be paired with compatibility degree. Atanassov [25] introduced the concept of IFS, which is an intellectual intuition to judge the uncertainty over the objects. Atanassov gave the definition of the IFS as follows:

**Definition 2.** *[25] An intuitionistic fuzzy set* (*IFS*) *in a universe X is defined as*

$$\mathcal{A} = \{ \langle \mathbf{x}, \widetilde{t}\_{\mathcal{A}}(\mathbf{x}), \widetilde{f}\_{\mathcal{A}}(\mathbf{x}) \rangle \mid \mathbf{x} \in X \},$$

*where the functions t*, *f* : *X* −→ [0, 1] *define, respectively, a membership function and a non-membership function of the element x* ∈ *X to the set* A*. Moreover, it is required that*

$$0 \le \tilde{t}\_{\mathcal{A}}(\mathbf{x}) + \tilde{f}\_{\mathcal{A}}(\mathbf{x}) \le 1.$$

*The function <sup>π</sup>*A = 1 − ( *<sup>t</sup>*A(*x*) + *f* A(*x*)) *is called the degree of hesitancy of x to* A*. The collection of all IFSs in X is denoted by IFS*(*X*)*.*

Let A, B ∈ *IFS*(*X*). Then,

$$\begin{array}{l} \mathcal{A} \sqcup \mathcal{B} = \{ \langle \mathbf{x}, \max \{ \tilde{t}\_{\mathcal{A}}(\mathbf{x}), \tilde{t}\_{\mathcal{B}}(\mathbf{x}) \}, \min \{ \tilde{f}\_{\mathcal{A}}(\mathbf{x}), \tilde{f}\_{\mathcal{B}}(\mathbf{x}) \} \rangle \mid \mathbf{x} \in \mathcal{X} \}, \\ \mathcal{A} \sqcap \mathcal{B} = \{ \langle \mathbf{x}, \min \{ \tilde{t}\_{\mathcal{A}}(\mathbf{x}), \tilde{t}\_{\mathcal{B}}(\mathbf{x}) \}, \max \{ \tilde{f}\_{\mathcal{A}}(\mathbf{x}), \tilde{f}\_{\mathcal{B}}(\mathbf{x}) \} \rangle \mid \mathbf{x} \in \mathcal{X} \}, \\ \mathcal{A} \sqsubseteq \mathcal{B} \iff \tilde{t}\_{\mathcal{A}}(\mathbf{x}) \le \tilde{t}\_{\mathcal{B}}(\mathbf{x}) \text{ and } \tilde{f}\_{\mathcal{A}}(\mathbf{x}) \ge \tilde{f}\_{\mathcal{B}}(\mathbf{x}) \forall \mathbf{x} \in \mathcal{X}. \end{array}$$

Deschrijver and Kerre [47] defined that IFSs can be considered as *L*-fuzzy sets with respect to the complete lattice (*V*<sup>∗</sup>, *V*∗ ), where *V*∗ = {*μ*1, *μ*2 ∈ [0, 1]2 | *μ*1 + *μ*2 ≤ <sup>1</sup>}, and the corresponding partial order *V*∗ is defined as *μ*1, *μ*2 *V*∗ *<sup>ν</sup>*1, *<sup>ν</sup>*2 ⇐⇒ (*μ*1 ≤ *<sup>ν</sup>*1) ∧ (*μ*2 ≤ *<sup>ν</sup>*2) for all *μ*1, *μ*2,*<sup>ν</sup>*1, *<sup>ν</sup>*2 ∈ *V*<sup>∗</sup>. Any ordered pair *μ*1, *μ*2 ∈ *V*∗ is called an intuitionistic fuzzy value (IFV) or intuitionistic fuzzy number (IFN).

Let *V*∗ be the set of IFVs of IFS A, such that *t*A, *f* A ∈ *V*<sup>∗</sup>. Chen and Tan [48] presented score function, which was updated by Feng et al. [42] as follows:

**Definition 3.** *[42] Let t*A, *f* A ∈ *V*∗ *be an IFV in a universe X. Then, expectation score function is a mapping δ* : *V*∗ → [0, 1]*, defined as follows:*

$$
\delta\_{\mathcal{A}} = \frac{\widetilde{t}\_{\mathcal{A}} - \widetilde{f}\_{\mathcal{A}} + 1}{2} \tag{1}
$$

*where δ*A *is called the decision value of t*A, *f* A *in* A*. In addition, fuzzy set δ*A *is called the utility fuzzy set derived from the IFS* A*.*

**Definition 4.** *[28] Let V*1 = *t*A, *f* A, *V*2 = *t* A, *f* A ∈ *V*∗ *be two IFVs in a universe X. Then we have,*


More operations and properties of IFVs (or IF numbers (IFNs)) can be seen in [27,28,42]. Let *c*1, *c*2, ..., *cm* be the IFVs and *φ* = (*φ*1, *φ*2, ..., *φm*) be the correlated weighted normalized vector, then, from Yager [28] and Xu [27], we denote and symbolize the following operators:

$$\text{IFWA}(\mathbf{c}\_1, \mathbf{c}\_2, \dots, \mathbf{c}\_m) = \phi\_1 \mathbf{c}\_1 \otimes \phi\_2 \mathbf{c}\_3 \otimes \dots \otimes \phi\_m \mathbf{c}\_m = \langle 1 - \prod\_{i}^m (1 - \tilde{t}\_{\mathbf{i}\_i})^{\Phi\_i} \prod\_{i}^m \hat{f}\_{\mathbf{i}\_i}^{\Phi\_i} \rangle \,, \tag{2}$$

$$\text{IFWG}(c\_1, c\_2, \dots, c\_m) = c\_1^{\Phi\_1} \otimes c\_3^{\Phi\_3} \otimes \dots \otimes c\_m^{\Phi\_m} = \langle \prod\_{i}^{m} \hat{f}\_{c\_i}^{\Phi\_i}, 1 - \prod\_{i}^{m} (1 - \tilde{f}\_{c\_i})^{\Phi\_i} \rangle. \tag{3}$$

IFWA and IFWG are the IF weighted averaging and geometric operators, respectively.

*2.2. Intuitionistic Fuzzy Soft Sets and Generalized Intuitionistic Fuzzy Soft Sets*

In this section, we present some basic notions in the theory of IFSS and GIFSS. The notion of IFSS is given as follows:

**Definition 5.** *[33] Let* (*<sup>X</sup>*, *E*) *be a soft universe and* A ⊆ *E. A pair* F = (S , A) *is called intuitionistic fuzzy soft set (IFSS) over X, where* S *is a mapping defined by* S : A −→ *IFS*(*X*)*.*

Formally, S : A −→ *IFS*(*X*) is referred to as the approximate function of the IFSS (S , A). It is easy to see that IFSSs extend both Atanassov's IFSs and Molodtsov's soft sets. The set of all IFSSs over *X*, with respect to subsets of *E*, is denoted by IFSS*<sup>E</sup>*(*X*). Next, the two new subsets of a IFSS are presented as follows:

**Definition 6.** *[42] Let* F = (S , A) *and* G = (T , B) *be IFSSs over X and* A, B ⊂ *E. Then,* G *is anintuitionistic fuzzy soft F-subsetof* F*, denoted by* G ⊆ *<sup>F</sup>*F*, if*

(i) B⊆A*.* (ii) T (*a*) ⊆ S (*a*) ∀*a* ∈ B*.*

**Definition 7.** *[42] Let* F = (S , A) *and* G = (T , B) *be IFSSs over X and* A, B ⊂ *E. Then,* G *is an intuitionistic fuzzy soft M-subsetof* F*, denoted by* G ⊆ *<sup>M</sup>*F*, if*

(i) B⊆A*.* (ii) T (*a*) = S (*a*) ∀*a* ∈ B*.*

The related whole IFSS is denoted as X A(1,0), where all IFVs are (1, <sup>0</sup>), and related null IFSS is denoted as I A(0,1), where all IFVs are (0, <sup>1</sup>). The other definitions of union, intersection and complements of IFSSs follow [33,42]. It is required in many cases that an extra input of moderator with IFSS could be useful. The definition of GIFSS was given by Agarwal et al. [39] as follows:

**Definition 8.** *[39] Let* (*<sup>X</sup>*, *E*) *be a soft universe and* A ⊆ *E. A generalized intuitionistic fuzzy soft set (GIFSS),* F*α over the soft universe* (*<sup>X</sup>*, *E*) *is defined as a mapping* F*α* : A −→ *IFSS*(*X*) × *IF, IFSS*(*X*) *the collection of all intuitionistic fuzzy subsets of X and the generalization parameter, α* : A −→ *IF* = (*<sup>t</sup><sup>α</sup>*, *fα*), *where IF is an IFS. The GIFSS is of the form* F*α*(*ei*)=(S (*ei*), *<sup>α</sup>*(*ei*))*.*

The model of GIFSS is very fruitful in decision making, especially the input of an extra opinion of an expert works incentively. However, in Definition 8, *IFSS*(*X*) × *IF* is not a meaningful Cartesian product and generalized parameter *α* is not well-defined. A more well-defined and flexible form of GIFSS was defined by Feng et al. [42]. They pointed out several assertions in [39], certified several notions and discussed GIFFSs theoretically. The definition of GIFSS is given as follows:

**Definition 9.** *[42] Let* (*<sup>X</sup>*, *E*) *be a soft universe and* A ⊆ *E. A triple* F<sup>Q</sup>= ( S, A, *α*) *is called generalized intuitionistic fuzzy soft set (GIFSS) over X if* (S, A) *is an IFSS over X and α is an IFS in* A.

This representation of GIFSS can be more significant to handle problems in which uncertain and unclear information are prevalent, and it enhances the accuracy and flexibility of results with opinions of experts as an IFS on the set of parameters. The two different types of subsets of GIFSS and several operations on GIFSSs are specified and categorized in [42]. Hayat et al. [43] presentes another form of GIFSSs and related notions.

#### **3. Group-Based Generalized Intuitionistic Fuzzy Soft Sets**

In this section, we clarifiy and reformulate the definition of GGIFSS presented in [44]. First, we recall the definition of GGIFSS that is given in [44];

**Definition 10.** *[44] A group-based generalized intuitionistic fuzzy soft sets (GGIFSS),* F*G, over the soft universe* (*<sup>X</sup>*, *E*) *is defined as* F*G* : *E* → *IFS*(*X*) × *IF for all υ* ∈ *E; we have* F*G*(*υ*)=( F(*υ*), *<sup>G</sup>*(*υ*))*, where* F(*υ*) ∈ *IFS*(*X*) *and <sup>G</sup>*(*υ*) ∈ *IF. Here, G* = (1, 2, ..., *p*) *are intuitionistic fuzzy subset of set of the parameter E and <sup>G</sup>*(*υ*) *denotes the opinion of experts on the elements of X in* F(*υ*)*.*

**Remark 1.** *The above definition of GGIFSS is very effective in many cases, due to its constructive scenario for decision making. However, this definition has some difficulties and dissensions on group of extra input of moderators, as well as on the mapping. Specifically, we identify the following:*


To clarify the problems mentioned in Remark 1, we reformulate the notion of GGIFSS as follows:

**Definition 11.** *Let* (*<sup>X</sup>*, *E*) *be a soft universe and* A ⊆ *E. A triple* FQ*g* = ( S, A, *g*) *is called group-based generalized intuitionistic fuzzy soft set (GGIFSS) over X, if* (S, A) *is an elementary IFSS (EIFSS) over X and g* = {*α*1, *α*2, ..., *<sup>α</sup>p*} *where α*1, *α*2, ..., *<sup>α</sup>p are the parameterized IFSs (PIFSs) of* A. *In other words, g is a group of PIFSs considered by "p" number of experts/moderators.* 

Keeping the prospects of decision making in the mind, (S, A) is basic IFS and *g* is a group of parameterized intuitionistic fuzzy sets (GPIFSs). The set of all GGIFSS over *X* obtained on *E* is denoted by GGIFSS*E*(*X*). Further, the set of all GGIFSS over *X* obtained on subset A ⊂ *E* is denoted by GGIFSS <sup>A</sup>(*X*).

**Example 1.** *Let X* = {*<sup>κ</sup>*1, *κ*2, ..., *<sup>κ</sup>*6} *be the universe set, consisting six cellphones, under consideration and E* = {*<sup>υ</sup>*1, *υ*2, *υ*3, *<sup>υ</sup>*4} *where <sup>υ</sup>ı*(*<sup>ı</sup>* = 1, 2, 3, <sup>4</sup>)*, respectively, stand for "high battery timing", "low operating cost", "high quality of voice call" and "stylish look". Consider a set of attributes* B = {*<sup>υ</sup>*1, *υ*3, *<sup>υ</sup>*4} ⊂ *E chosen by an observer* M*, which are anticipated to be most fruitful for judgment of cellphones. For* M*, the evaluation of alternatives with rating values corresponding each parameters can be defined as EIFSS,*

$$
\begin{array}{c}
\widetilde{\mathcal{S}}(\upsilon\_{1}) = \{\frac{\kappa\_{1}}{\langle 0.6, 0.4 \rangle}, \frac{\kappa\_{2}}{\langle 0.0, 3 \rangle}, \frac{\kappa\_{3}}{\langle 0.2, 0.2 \rangle}, \frac{\kappa\_{4}}{\langle 0.1 \rangle}, \frac{\kappa\_{5}}{\langle 0.1, 0.6 \rangle}, \frac{\kappa\_{6}}{\langle 0.6, 0.2 \rangle}\} \\
\widetilde{\mathcal{S}}(\upsilon\_{3}) = \{\frac{\kappa\_{1}}{\langle 0.6, 0.2 \rangle}, \frac{\kappa\_{2}}{\langle 0.7, 0.3 \rangle}, \frac{\kappa\_{3}}{\langle 0.3, 0.6 \rangle}, \frac{\kappa\_{4}}{\langle 0.4, 0.1 \rangle}, \frac{\kappa\_{5}}{\langle 0.4, 0.2 \rangle}, \frac{\kappa\_{6}}{\langle 0.3, 0.3 \rangle}\} \\
\widetilde{\mathcal{S}}(\upsilon\_{4}) = \{\frac{\kappa\_{1}}{\langle 0.9, 0 \rangle}, \frac{\kappa\_{2}}{\langle 0.5, 0.1 \rangle}, \frac{\kappa\_{3}}{\langle 0.5, 0.1 \rangle}, \frac{\kappa\_{4}}{\langle 0.2, 0.5 \rangle}, \frac{\kappa\_{5}}{\langle 0.5, 0.2 \rangle}, \frac{\kappa\_{6}}{\langle 0.6, 0.1 \rangle}\}
\end{array}
$$

*Consider three moderators d*1, *d*2, *d*3 *for assessment of rating value, such that the opinion of each moderator on each parameter of* M *is analyzed and based on opinions, PIFSs <sup>α</sup>d*1 , *<sup>α</sup>d*2 *and <sup>α</sup>d*3 *are defined on* A *as*

$$
\widetilde{\mathfrak{g}} = \begin{cases}
\widetilde{\mathfrak{a}}\_{d\_1} = \{ (\upsilon\_{1\prime} \langle 0.6, 0.2 \rangle), (\upsilon\_{3\prime} \langle 0.3, 0.4 \rangle), (\upsilon\_{4\prime} \langle 0.2, 0.2 \rangle) \}, \\
\widetilde{\mathfrak{a}}\_{d\_2} = \{ (\upsilon\_{1\prime} \langle 0.3, 0.4 \rangle), (\upsilon\_{3\prime} \langle 0.2, 0.4 \rangle), (\upsilon\_{4\prime} \langle 0.3, 0.5 \rangle) \}, \\
\widetilde{\mathfrak{a}}\_{d\_3} = \{ (\upsilon\_{1\prime} \langle 0.3, 0.4 \rangle), (\upsilon\_{3\prime} \langle 0.5, 0.4 \rangle), (\upsilon\_{4\prime} \langle 0.4, 0.1 \rangle) \}.
\end{cases}
$$

*Then, the GGIFSS is represented in Table 1.*


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

If *p* = 1, then F Q *g* = (S , A, *g*) operates as GIFSS. In general, F Q *g* can be sighted as a common form of generalized parameters with the information on IFSSs.

#### **4. Operations on GGIFSSs and Aggregation Operators**

In this section, several new operations on GGIFSSs and their examples are presented. As in Remark 1, it is pointed out that the group of extra assessments of experts in [36] is not defined in a precise way. In this scenario, we define two different subsets of a GGIFSS; for this purpose, a notion on group of generalized parameters of two GGIFSSs is defined as follows:

**Definition 12.** *Let* (*<sup>X</sup>*, *E*) *be a soft universe and* A, B ⊆ *E. Suppose that* F Q *g* = (S , A, *g*1) *and* G *g* = (T , B, *g*2) *are two GGIFSSs over X, where* A⊆B*, g*1 = {*αd*1 , *<sup>α</sup>d*2 , ..., *<sup>α</sup>dp* } *and g*2 = {*β d*1 , *β d*2 , ..., *β dp* } *and d*1, *d*2, ..., *dp are "p" number of senior experts/members. If g*1 *is the group intuitionistic fuzzy subset of g*2*, then it is denoted and defined by g*1 ≪ *g*2 *if and only if <sup>t</sup>αd*1 (*<sup>υ</sup>i*) ≤ *tβd*1 (*<sup>υ</sup>i*), *f <sup>α</sup>d*1 (*<sup>υ</sup>i*) ≥ *f β d*1 (*<sup>υ</sup>i*), *<sup>t</sup>αd*2 (*<sup>υ</sup>i*) ≤ *tβd*2 (*<sup>υ</sup>i*), *f <sup>α</sup>d*2 (*<sup>υ</sup>i*) ≥ *f β d*2 (*<sup>υ</sup>i*)*,... , <sup>t</sup><sup>α</sup>dp* (*<sup>υ</sup>i*) ≤ *tβdp* (*<sup>υ</sup>i*), *f <sup>α</sup>dp* (*<sup>υ</sup>i*) ≥ *f β dp* (*<sup>υ</sup>i*)*,* ∀ *i* = 1, 2, ..., *m and υi* ∈ A*.*

Based on Definition 12, the following two different kinds of group-based generalized intuitionistic fuzzy soft subsets can be presented.

**Definition 13.** *Let* F Q *g* = (S , A, *g*1) *and* G *g* = (T , B, *g*2) *be two GGIFSSs over X and* A, B ⊆ *E. Then,* F Q *g is a group-based generalized intuitionistic fuzzy soft F-subsetof* G *g , denoted by* F Q *g* - *F*G *g , if*


Now, an example is given to clarify group-based generalized intuitionistic fuzzy soft F-subset of a GGIFSS.

**Example 2.** *Let X* = {*<sup>κ</sup>*1, *κ*2, ..., *<sup>κ</sup>*6} *be the universe set, consisting six robots under consideration and E* = {*<sup>υ</sup>*1, *υ*2, *υ*3, *<sup>υ</sup>*4} *where υı, respectively, stand for "high capacity", "low degree of freedom", "high memory capacity" and "high repeatability". Consider two sets of parameters* A = {*<sup>υ</sup>*1, *<sup>υ</sup>*4} ⊂ *E, and* B = {*<sup>υ</sup>*1, *υ*3, *<sup>υ</sup>*4} ⊂ *E chosen by observers* M1 *and* M<sup>2</sup>*, respectively, which are anticipated to be most fruitful for evaluation of robots. For* M1 *and* M<sup>2</sup>*, evaluation of alternatives with their ratting values corresponding each parameters can be defined, respectively, as EIFSSs,*

 $\widetilde{S}(\upsilon\_{1}) = \{\frac{\kappa\_{1}}{\langle 0.6, 0.3 \rangle}, \frac{\kappa\_{2}}{\langle 0.0, 0.)}, \frac{\kappa\_{3}}{\langle 0.2, 0.2 \rangle}, \frac{\kappa\_{4}}{\langle 0.1 \rangle}, \frac{\kappa\_{5}}{\langle 0.1, 0.6 \rangle}, \frac{\kappa\_{6}}{\langle 0.6, 0.2 \rangle}\}$  $\widetilde{S}(\upsilon\_{4}) = \{\frac{\kappa\_{1}}{\langle 0.9, 0 \rangle}, \frac{\kappa\_{2}}{\langle 0.5, 0.1 \rangle}, \frac{\kappa\_{3}}{\langle 0.5, 0.1 \rangle}, \frac{\kappa\_{4}}{\langle 0.3, 0.5 \rangle}, \frac{\kappa\_{5}}{\langle 0.5, 0.2 \rangle}, \frac{\kappa\_{6}}{\langle 0.6, 0.1 \rangle}\}$ 

*and*

$$\begin{array}{l} \widetilde{\mathcal{T}}(\upsilon\_{1}) = \{ \frac{\underline{\kappa\_{1}}}{\langle 0.6, 0.4 \rangle}, \frac{\underline{\kappa\_{2}}}{\langle 0.1, 0.5 \rangle}, \frac{\underline{\kappa\_{3}}}{\langle 0.3, 0.2 \rangle}, \frac{\underline{\kappa\_{4}}}{\langle 0.1 \rangle}, \frac{\underline{\kappa\_{5}}}{\langle 0.2, 0.7 \rangle}, \frac{\underline{\kappa\_{6}}}{\langle 0.7, 0.3 \rangle} \} \\ \widetilde{\mathcal{T}}(\upsilon\_{3}) = \{ \frac{\underline{\kappa\_{1}}}{\langle 0.6, 0.2 \rangle}, \frac{\underline{\kappa\_{2}}}{\langle 0.7, 0.3 \rangle}, \frac{\underline{\kappa\_{3}}}{\langle 0.3, 0.6 \rangle}, \frac{\underline{\kappa\_{4}}}{\langle 0.4, 0.1 \rangle}, \frac{\underline{\kappa\_{5}}}{\langle 0.4, 0.2 \rangle}, \frac{\underline{\kappa\_{6}}}{\langle 0.3, 0.3 \rangle} \} \\ \widetilde{\mathcal{T}}(\upsilon\_{4}) = \{ \frac{\underline{\kappa\_{1}}}{\langle 0.9, 0.1 \rangle}, \frac{\underline{\kappa\_{2}}}{\langle 0.6, 0.2 \rangle}, \frac{\underline{\kappa\_{3}}}{\langle 0.5, 0.2 \rangle}, \frac{\underline{\kappa\_{4}}}{\langle 0.3, 0.5 \rangle}, \frac{\underline{\kappa\_{5}}}{\langle 0.6, 0.3 \rangle}, \frac{\underline{\kappa\_{6}}}{\langle 0.6, 0.3 \rangle} \} \end{array}$$

*Consider three moderator, d*1 *from engineering department, d*2 *from production department and d*3 *from quality inspection department; their additional opinions for assessments of each observer are analyzed and, based on their opinions, PIFSs on* M1*: <sup>α</sup>d*1 , *<sup>α</sup>d*2 *, <sup>α</sup>d*3 *and IFSs of* M2*: β d*1 , *β d*2 *, β d*3 *are defined.*

$$
\widetilde{\mathcal{g}}\_1 = \begin{cases}
\widetilde{\boldsymbol{\alpha}}\_{d\_1} = \{ (\boldsymbol{\upsilon}\_{1}, \langle 0.3, 0.2 \rangle), (\boldsymbol{\upsilon}\_{4}, \langle 0.3, 0.4 \rangle) \}, \\
\widetilde{\boldsymbol{\alpha}}\_{d\_2} = \{ (\boldsymbol{\upsilon}\_{1}, \langle 0.3, 0.4 \rangle), (\boldsymbol{\upsilon}\_{4}, \langle 0.2, 0.4 \rangle) \}, \\
\widetilde{\boldsymbol{\alpha}}\_{d\_3} = \{ (\boldsymbol{\upsilon}\_{1}, \langle 0.3, 0.4 \rangle), (\boldsymbol{\upsilon}\_{4}, \langle 0.5, 0.4 \rangle) \}, \\
\widetilde{\boldsymbol{\beta}}\_2 = \begin{cases}
\widetilde{\boldsymbol{\beta}}\_{d\_1} = \{ (\boldsymbol{\upsilon}\_{1}, \langle 0.6, 0.2 \rangle), (\boldsymbol{\upsilon}\_{3}, \langle 0.4, 0.5 \rangle), (\boldsymbol{\upsilon}\_{4}, \langle 0.4, 0.4 \rangle) \}, \\
\widetilde{\boldsymbol{\beta}}\_{d\_2} = \{ (\boldsymbol{\upsilon}\_{1}, \langle 0.3, 0.2 \rangle), (\boldsymbol{\upsilon}\_{3}, \langle 0.4, 0.2 \rangle), (\boldsymbol{\upsilon}\_{4}, \langle 0.4, 0.3 \rangle) \}, \\
\widetilde{\boldsymbol{\beta}}\_{d\_3} = \{ (\boldsymbol{\upsilon}\_{1}, \langle 0.4, 0.2 \rangle), (\boldsymbol{\upsilon}\_{3}, \langle 0.4, 0.4 \rangle), (\boldsymbol{\upsilon}\_{4}, \langle 0.4, 0.2 \rangle) \}.
\end{cases}
$$

*Then, the GGIFSSs* F Q *g and* G *g are tabulated in Tables 2, and 3, respectively.*

**Table 2.** Tabular representation of the GGIFSS F Q *g* = (S , A, *g*1).



**Table 3.** Tabular representation of the GGIFSS, G *g* = (T , B, *g*2).

*One can easily check that* (S , A)⊆ *F*(T , B) *and g*1 ≪ *g*2*. Thus,* F Q *g* = (S , A, *g*1) *is group-based generalized intuitionistic fuzzy soft F-subset of* G *g* = (T , B, *g*2)*.*

**Definition 14.** *Let* F Q *g* = (S , A, *g*1) *and* G *g* = (T , B, *g*2) *be two GGIFSSs over X and* A, B ⊂ *E. Then,* F Q *g is a group-based generalized intuitionistic fuzzy soft M-subsetof* G *g , denoted by* F Q *g* - *M*G *g , if*

(i) (S , A)⊆ *M*(T , B)*.* (ii) *g*1 ≪ *g*2*.*

> The complement of a GGIFSS is given as follows:

**Definition 15.** *Let* G *g* = (T , A, *g*) *be GGIFSS over X. The complement of* G *g is defined as the GGIFSS* G *c g* = (T *c* , A, *gc*) *where* (G *c* , A) *is the complement of the EIFSS* (G , A) *and gc* = {*αcd*1 , *αcd*2 , ..., *αcdp* } *is the complement of g* = {*αd*1 , *<sup>α</sup>d*2 , ..., *<sup>α</sup>dp* }*.*

Now, an example is given to clarify complement of a GGIFSS.

**Example 3.** *Consider GGIFSS* G *g* = (T , A, *g*) *defined in Example 1. Then, complement of* G *g is defined in Table 4.*


**Table 4.** Tabular representation of the GGIFSS G *c g* = (T *c* , A, *g<sup>c</sup>*).

Next, the definitions of *extended union*, *extended intersection*, *restricted union* and*restricted intersection* are provided below.

**Definition 16.** *Let* F Q *g* = (S , A, *g*1) *and* G *g* = (T , B, *g*2) *be two GGIFSSs over X,* A, B ⊆ *E,* C = A∪B *and g*1 = {*αd*1 , *<sup>α</sup>d*2 , ..., *<sup>α</sup>dp* }*, g*2 = {*β d*1 , *β d*2 , ..., *β dp* }*. The extended union of* F Q *g and* G *g is defined as the GGIFSS* 

$$(\mathcal{H}\_{\prime}\mathcal{C}\_{\prime}\widetilde{\mathcal{S}}) = (\mathcal{S}\_{\prime}\mathcal{A}\_{\prime}\widetilde{\mathcal{S}}\_{1})\widetilde{\sqcup}\_{\mathcal{E}}\left(\mathcal{T}\_{\prime}\mathcal{B}\_{\prime}\widetilde{\mathcal{S}}\_{2}\right)$$

*such that*

$$\begin{array}{ll} \text{(i)} & (\overleftarrow{\mathcal{H}}, \mathcal{C}) = (\overleftarrow{\mathcal{S}}, \mathcal{A}) \cup\_{\mathcal{E}} (\overleftarrow{\mathcal{T}}, \mathcal{B}). \\\text{(ii)} & \text{For each moderate } d\_{k\prime} \ \overleftarrow{\gamma}\_{d\_{k}} \ (k = 1, 2, \dots, p) \text{ can be defined } \forall v \in \mathcal{C}, \end{array}$$

$$\widetilde{\mathfrak{f}}\_{\widetilde{\mathbb{J}}\_{k}}(\upsilon) = \begin{cases} \widetilde{\mathfrak{f}}\_{\overline{\mathfrak{a}}\_{d\_{k}}}(\upsilon), & \text{if } \upsilon \in \mathcal{A} \backslash \mathcal{B}, \\\widetilde{\mathfrak{f}}\_{\overline{\mathfrak{f}}\_{d\_{k}}}(\upsilon), & \text{if } \upsilon \in \mathcal{B} \backslash \mathcal{A}, \\\max\{\widetilde{\mathfrak{f}}\_{\overline{\mathfrak{a}}\_{d\_{k}}}(\upsilon), \widetilde{\mathfrak{f}}\_{\overline{\mathfrak{f}}\_{d\_{k}}}(\upsilon)\}, & \text{if } \upsilon \in \mathcal{A} \cap \mathcal{B}; \end{cases}$$

*and*

$$
\widetilde{f}\_{\overline{\gamma}\_{d\_k}}(\upsilon) = \begin{cases}
\widetilde{f}\_{\overline{a}\_{d\_k}}(\upsilon), & \text{if } \upsilon \in \mathcal{A} \backslash \mathcal{B}, \\
\widetilde{f}\_{\overline{\beta}\_{d\_k}}(\upsilon), & \text{if } \upsilon \in \mathcal{B} \backslash \mathcal{A}, \\
\min\{\widetilde{f}\_{\overline{a}\_{d\_k}}(\upsilon), \widetilde{f}\_{\overline{\beta}\_{d\_k}}(\upsilon)\}, & \text{if } \upsilon \in \mathcal{A} \cap \mathcal{B}.
\end{cases}
$$

**Definition 17.** *Let* F Q *g* = (S , A, *g*1) *and* G *g* = (T , B, *g*2) *be two GGIFSSs over X, where* A, B ⊆ *E,* C = A∩B *and g*1 = {*αd*1 , *<sup>α</sup>d*2 , ..., *<sup>α</sup>dp* }*, g*2 = {*β d*1 , *β d*2 , ..., *β dp* }*. The extended intersection of* F Q *g and* G *g is defined as the GGIFSS*

$$(\widetilde{\mathcal{R}}, \mathcal{C}, \widetilde{\mathfrak{F}}) = (\widetilde{\mathcal{S}}, \mathcal{A}, \widetilde{\mathfrak{F}}\_1) \upharpoonright\_{\mathcal{E}} (\widetilde{\mathcal{T}}, \mathcal{B}, \widetilde{\mathfrak{F}}\_2) \upharpoonright\_{\mathcal{E}}$$

*such that*

$$\begin{array}{ll} \text{(i)} & (\vec{\mathcal{R}}, \mathcal{C}) = (\vec{\mathcal{S}}, \mathcal{A}) \cap\_{\mathcal{E}} (\vec{\mathcal{T}}, \mathcal{B}). \\ \text{(ii)} & \text{For each moderate } d\_{k'} \,\, \widetilde{\gamma}\_{d\_k} \, (k = 1, 2, \dots, p) \text{ can be defined } \forall \upsilon \in \mathcal{A} \cup \mathcal{B}, \end{array}$$

$$
\check{t}\_{\widetilde{\boldsymbol{\beta}}\_{k}}(\boldsymbol{\upsilon}) = \begin{cases}
\check{t}\_{\overline{\boldsymbol{a}}\_{\boldsymbol{d}\_{k}}}(\boldsymbol{\upsilon}), & \text{if } \boldsymbol{\upsilon} \in \mathcal{A} \backslash \mathcal{B}, \\
\check{t}\_{\overline{\boldsymbol{\beta}}\_{\boldsymbol{d}\_{k}}}(\boldsymbol{\upsilon}), & \text{if } \boldsymbol{\upsilon} \in \mathcal{B} \backslash \mathcal{A}, \\
\min\{\check{t}\_{\overline{\boldsymbol{a}}\_{\boldsymbol{d}\_{k}}}(\boldsymbol{\upsilon}), \check{t}\_{\overline{\boldsymbol{\beta}}\_{\boldsymbol{d}\_{k}}}(\boldsymbol{\upsilon})\}, & \text{if } \boldsymbol{\upsilon} \in \mathcal{A} \cap \mathcal{B}; \\
\end{cases}
$$

*and*

$$
\check{f}\_{\overline{\gamma}\_{d\_k}}(\upsilon) = \begin{cases}
\check{f}\_{\overline{\alpha}\_{d\_k}}(\upsilon), & \text{if } \upsilon \in \mathcal{A} \backslash \mathcal{B}, \\
\check{f}\_{\overline{\beta}\_{d\_k}}(\upsilon), & \text{if } \upsilon \in \mathcal{B} \backslash \mathcal{A}, \\
\max\{\check{f}\_{\overline{\alpha}\_{d\_k}}(\upsilon), \check{f}\_{\overline{\beta}\_{d\_k}}(\upsilon)\}, & \text{if } \upsilon \in \mathcal{A} \cap \mathcal{B}.
\end{cases}
$$

**Definition 18.** *Let* F Q *g* = (S , A, *g*1) *and* G *g* = (T , B, *g*2) *be two GGIFSSs over X, where* A, B ⊆ *E,* C = A∩B *and g*1 = {*αd*1 , *<sup>α</sup>d*2 , ..., *<sup>α</sup>dp* }*, g*2 = {*β d*1 , *β d*2 , ..., *β dp* }*. The restricted union of* F Q *g and* G *g is defined as the GGIFSS*

$$(\tilde{\mathcal{R}}, \mathcal{C}, \mathfrak{g}) = (\tilde{\mathcal{S}}, \mathcal{A}, \tilde{\mathfrak{g}}\_1) \sqcup\_r (\tilde{\mathcal{T}}, \mathcal{B}, \tilde{\mathfrak{g}}\_2)$$

*such that*

$$\begin{array}{ll} \text{(i)} & (\widetilde{\mathcal{R}}, \mathcal{C}) = (\widetilde{\mathcal{S}}', \mathcal{A}) \cup\_r (\widetilde{\mathcal{T}}, \mathcal{B}); \\ \text{(ii)} & \text{For each moderate } d\_{k\prime} \,\widetilde{\gamma}\_{d\_k} \,(k=1,2,...,p) \text{ can be defined } \forall \upsilon \in \mathcal{C}, \end{array}$$

$$\widetilde{t}\_{\overline{\gamma}\_{d\_k}}(\upsilon) = \max \{ \widetilde{t}\_{\overline{a}\_{d\_k}}(\upsilon), \widetilde{t}\_{\overline{\beta}\_{d\_k}}(\upsilon) \}, \text{ for all } \upsilon \in \mathcal{A} \cap \mathcal{B} \}$$

*and*

$$\widetilde{f}\_{\widetilde{\gamma}\_{d\_k}}(\upsilon) = \min \{ \widetilde{f}\_{\overline{a}\_{d\_k}}(\upsilon), \widetilde{f}\_{\overline{\beta}\_{d\_k}}(\upsilon) \}, \text{ for all } \upsilon \in \mathcal{A} \cap \mathcal{B}.$$

**Definition 19.** *Let* F Q *g* = (S , A, *g*1) *and* G *g* = (T , B, *g*2) *be two GGIFSSs over X, where* A, B ⊆ *E,* C = A∩B *and g*1 = {*αd*1 , *<sup>α</sup>d*2 , ..., *<sup>α</sup>dp* }*, g*2 = {*β d*1 , *β d*2 , ..., *β dp* }*. The restricted intersection of* F Q *g and* G *g is defined as the GGIFSS*

$$(\tilde{\mathcal{R}}, \mathcal{C}, \underline{\mathfrak{g}}) = (\tilde{\mathcal{S}}, \mathcal{A}, \tilde{\mathfrak{g}}\_1) \uplus\_r (\tilde{\mathcal{T}}, \mathcal{B}, \tilde{\mathfrak{g}}\_2)$$

*such that*

(i) (R , C)=(S , A) ∩*r* (T , B)*.* (ii) *For each moderator dk, <sup>γ</sup>dk* (*k* = 1, 2, ..., *p*) *can be defined* ∀*υ* ∈ C*,*

$$\widetilde{t}\_{\overline{\gamma}\_{d\_k}}(\upsilon) = \min \{ \widetilde{t}\_{\overline{\alpha}\_{d\_k}}(\upsilon), \widetilde{t}\_{\overline{\beta}\_{d\_k}}(\upsilon) \}, \text{ for all } \upsilon \in \mathcal{A} \cap \mathcal{B};$$

*and*

$$\widetilde{f}\_{\overline{\gamma}\_{\mathcal{d}\_k}}(\upsilon) = \max \{ \widetilde{f}\_{\overline{a}\_{\mathcal{d}\_k}}(\upsilon), \widetilde{f}\_{\overline{\beta}\_{\mathcal{d}\_k}}(\upsilon) \}, \text{ for all } \upsilon \in \mathcal{A} \cap \mathcal{B}.$$

The definition of null GGIFSS and whole GGIFSS are specified below.

**Definition 20.** *Let* G *g* = (S , A, *g*) *be a GGIFSS over X, where* A ⊆ *E, and g* = {*αd*1 , *<sup>α</sup>d*2 , ..., *<sup>α</sup>dp* }*. Then,* G *g is called the group-based generalized relative null intuitionistic fuzzy soft set, denoted by* N QA *g , if*

(1) (S , A) = I A(0,1)

*.*

*.*

(2) *For each moderator dk, <sup>t</sup><sup>α</sup>dk*(*υ*) = 0 *and f <sup>α</sup>dk*(*υ*) = 1 *for all υ* ∈ A*.*

*g*

**Definition 21.** *Let* G *g* = (S , A, *g*) *be a GGIFSS over X, where* A ⊆ *E and g* = {*αd*1 , *<sup>α</sup>d*2 , ..., *<sup>α</sup>dp* }*. Then,* G *g is called the group-based generalized relative whole intuitionistic fuzzy soft set, denoted by* W QA *g, if*

(1) (S , A) = X A(1,0)

$$\text{(2)}\quad \text{For each moderate } d\_{k\prime} \widetilde{\mathfrak{t}}\_{\overline{a}\_{d\_k}}(\upsilon) = 1 \text{ and } \widetilde{f}\_{\overline{a}\_{d\_k}}(\upsilon) = 0 \text{ for all } \upsilon \in \mathcal{A}.$$

*g*

*g*

*.*

*.*

*.*

**Proposition 1.** *Let* G *g* = (S , A, *g*) *be a GGIFSS over X, where* A ⊆ *E and g* = {*αd*1 , *<sup>α</sup>d*2 , ..., *<sup>α</sup>dp* }*. Then,*


*g*

Now, we introduce group-based generalized weighted averaging (GBGWA) and group-based generalized weighted geometric (GBGWG) operators on GGIFSSs. On these operators, we contemplate and discussed some properties as well. The definition of GBGWA operator is specified below.

**Definition 22. GBGWA;** *Let* F Q *g* = (S , A, *g*) *be a GGIFSS over X, where g* = {*αd*1 , *<sup>α</sup>d*2 , ..., *<sup>α</sup>dp* } *be the group of PIFSs. Assume that w* = (*<sup>w</sup>*1, *w*2, ..., *wm*)*<sup>T</sup> is the normalized weight vector for* A*, such that wi* > 0 *and* ∑*mi*=<sup>1</sup> *wi* = 1*. Let IFV* R (*<sup>κ</sup>j*) = {*cj*1, *cj*2, ..., *cjm*} (*j* = 1 *to n*) *be the set of IFVs in EIFSS* (S , A) *for all κj* ∈ *X. For each senior moderator/ prospector, <sup>α</sup>dk* (*υ*) = { *<sup>t</sup><sup>α</sup>dk* (*υ*), *f <sup>α</sup>dk* (*υ*) | *υ* ∈ A} (*k* = 1 *to p*) *be the PIFS, it can be represented as* IF*k* = {*ak*1, *ak*2, ..., *akm*} (*k* = 1 *to p*) *and* = (1, 2, ..., *p*)*<sup>T</sup> is the set of* *weights for moderators, such that k* > 0 *and* <sup>∑</sup>*pk*=<sup>1</sup> *k* = 1*. Define* GBGWA*:* Γ*ms* −→ Γ*s,* IFWA*:* Γ*m* −→ Γ*, where*

$$\text{IGGVA}(c\_{\text{j1}}, c\_{\text{j2}}, \dots, c\_{\text{jm}}) = \text{IFWA}\_{k} \left( \begin{array}{l} (\text{IFWA}\_{i}(a\_{11}, a\_{12}, \dots, a\_{1m}) \otimes \text{IFWA}\_{i}(c\_{\text{j1}}, c\_{\text{j2}}, \dots, c\_{\text{jm}})), \\ (\text{IFWA}\_{i}(a\_{21}, a\_{22}, \dots, a\_{2m}) \otimes \text{IFWA}\_{i}(c\_{\text{j1}}, c\_{\text{j2}}, \dots, c\_{\text{jm}})), \dots, \\ (\text{IFWA}\_{i}(a\_{p1}, a\_{p2}, \dots, a\_{pm}) \otimes \text{IFWA}\_{i}(c\_{\text{j1}}, c\_{\text{j2}}, \dots, c\_{\text{jm}})) \end{array} \right) \tag{4}$$

*where* GBGWA *is known as GGIFSS weighted averaging operator, then the set of all* GBGWAs *is denoted L* = { 1, 2, ..., *n*}*. In addition,* IFWA*k and* IFWA*i are* IFWA *operators on set of moderators/prospectors and set of parameters, respectively. Note that* Γ*ms and* Γ *are families of GGIFSS and IFSs, respectively.*

**Lemma 1.** *Let* F Q *g* = (S , A, *g*) *be a GGIFSS over X, where g* = {*αd*1 , *<sup>α</sup>d*2 , ..., *<sup>α</sup>dp* } *be the group of IFSs. If p* = 1*, then* F Q *g is a GIFSS and* GBGWA *is given as follows:*

$$\text{GBGVA}(c\_{j1}, c\_{j2}, \dots, c\_{jm}) = \text{IFVA}\_i(a\_{11}, a\_{12}, \dots, a\_{1m}) \otimes \text{IFVA}\_i(c\_{j1}, c\_{j2}, \dots, c\_{jm}).\tag{5}$$

**Theorem 1.** *If cji* = *tji*, *f ji and aki* = *tki*, *f ki* (*i* = 1, 2, ..., *m*, *j* = 1, 2, ..., *n*, *k* = 1, 2, ..., *p*)*, be the IFVs, then the accumulated value by* GBGWA *operator is given by*

GBGWA(*cj*1, *cj*2, ..., *cjm*) = 1 − <sup>∏</sup>*pk*=<sup>1</sup>(<sup>1</sup> − (1 − ∏*mi*=<sup>1</sup>(<sup>1</sup> − *taki*)) · (1 − ∏*mi*=<sup>1</sup>(<sup>1</sup> − *tcji*))), <sup>∏</sup>*pk*=<sup>1</sup>(<sup>1</sup> − (1 − ∏*mi*=<sup>1</sup> *f aki*)(<sup>1</sup> − ∏*mi*=<sup>1</sup> *f cji*))*.*

**Proof.** Let *p* = 1 and *m* = 2. Firstly, we apply mathematical induction on *m*, we have GBGWA(*cj*1, *cj*2) = IFWA*k*(IFWA(*<sup>a</sup>*11, *<sup>a</sup>*12) ⊗ IFWA(*cj*1, *cj*2)) = (IFWA(*<sup>a</sup>*11, *<sup>a</sup>*12) ⊗ IFWA(*cj*1, *cj*2)) = 1 − (1 − *ta*11 )*<sup>w</sup>*1 · (1 − *ta*12 )*<sup>w</sup>*2 , *f w*1 *a*11 · *f w*2 *a*12 ⊗1 − (1 − *tcj*1 )*<sup>w</sup>*1 · (1 − *tcj*2 )*<sup>w</sup>*2 , *f w*1 *cj*1 · *f w*2 *cj*2 = (1 − (1 − *ta*11 )*<sup>w</sup>*1 · (1 − *ta*12 )*<sup>w</sup>*2 ) · (1 − (1 − *tcj*1 )*<sup>w</sup>*1 · (1 − *tcj*2 )*<sup>w</sup>*2 ), *f w*1 *a*11 · *f w*2 *a*12 + *f w*1 *cj*1 . *f w*2 *cj*2 − *f w*1 *a*11 . *f w*2 *a*12 . *f w*1 *cj*1 · *f w*2 *cj*2 = (1 − ∏<sup>2</sup>*i*=<sup>1</sup>(<sup>1</sup> <sup>−</sup>*ta*1*i*)*wi*) · (1 − ∏<sup>2</sup>*i*=<sup>1</sup>(<sup>1</sup> <sup>−</sup>*tcji*)*wi*), ∏<sup>2</sup>*i*=<sup>1</sup> *fwi <sup>a</sup>*1*i* + ∏<sup>2</sup>*i*=<sup>1</sup> *fwi cji* − ∏<sup>2</sup>*i*=<sup>1</sup> *fwi <sup>a</sup>*1*i* . ∏<sup>2</sup>*i*=<sup>1</sup> *fwi cji* .

Thus, theorem is true for *m* = 2; assuming that the result is true for *m* = *s* , that is, GBGWA(*cj*1, *cj*2, ..., *cjs* ) = (1 − ∏*s <sup>i</sup>*=<sup>1</sup>(<sup>1</sup> − *ta*1*i*) · (1 − ∏*s <sup>i</sup>*=<sup>1</sup>(<sup>1</sup> − *tcji*))), 1 − ((1 − ∏*s i*=1 *fa*1*i*)(<sup>1</sup> − ∏*s i*=1 *<sup>f</sup>cji*)). then, for *m* = *s* + 1, GBGWA(*cj*1, *cj*2, ..., *cj*(*s* +<sup>1</sup>)) = (1 − ∏*s* +1 *i*=1 (1 <sup>−</sup>*ta*1*i*) · (1 − ∏*s* +1 *i*=1 (1 − *tcji*))), 1 − ((1 − ∏*s* +1 *i*=1 *fa*1*i*)(<sup>1</sup> − ∏*s* +1 *i*=1 *<sup>f</sup>cji*)). Thus, by mathematical induction, Theorem 1 holds for all positive integer *m*. Similarly, we can prove this theorem for *k* = 2, 3, ..., *p*.

#### **Example 4.** *Consider Example 1, where*

R

*IFV* (*<sup>κ</sup>*1) = {*<sup>c</sup>*11, *c*12, *<sup>c</sup>*13} = {0.6, 0.4,0.6, 0.2,0.9, 0} *is a family of IFVs in second row of Table 1. The three IFSs of moderator's assessments are*

> IF1 = {*<sup>a</sup>*11, *a*12, *<sup>a</sup>*13} = {0.6, 0.2,0.3, 0.4,0.2, 0.2} IF2 = {*<sup>a</sup>*21, *a*22, *<sup>a</sup>*23} = {0.3, 0.4,0.2, 0.4,0.3, 0.5} IF3 = {*<sup>a</sup>*31, *a*32, *<sup>a</sup>*33} = {0.3, 0.4,0.5, 0.4,0.4, 0.1}*,*

*respectively. Let w* = {*w*1/0.29, *w*2/0.35, *w*3/0.36} *be the weighted vector over E and* = {1/0.25, 2/0.40, 3/0.35} *be the weighted vector for three senior experts. Now, the GBGWA is given below.*

$$\ell\_1' = \text{GBGWA}(c\_{11}, c\_{12}, c\_{13}) = \text{IFWA}\_k \left( \begin{array}{l} (\text{IFWA}\_i(a\_{11}, a\_{12}, a\_{13}) \otimes \text{IFWA}\_i(c\_{11}, c\_{12}, c\_{13})), \\ (\text{IFWA}\_i(a\_{21}, a\_{22}, a\_{23}) \otimes \text{IFWA}\_i(c\_{11}, c\_{12}, c\_{13})), \\ (\text{IFWA}\_i(a\_{31}, a\_{32}, a\_{33}) \otimes \text{IFWA}\_i(c\_{11}, c\_{12}, c\_{13})) \end{array} \right)$$

*Next, calculate,* IFWA*i*(*<sup>c</sup>*11, *c*12, *<sup>c</sup>*13) = IFWA*i*(0.6, 0.4,0.6, 0.2,0.9, 0.0) = 0.7572, 0.0000, IFWA*i*(*<sup>a</sup>*11, *a*12, *<sup>a</sup>*13) = IFWA*i*(0.6, 0.2,0.3, 0.4,0.2, 0.2) = 0.3755, 0.2549, IFWA*i*(*<sup>a</sup>*21, *a*22, *<sup>a</sup>*23) = IFWA*i*(0.3, 0.4,0.2, 0.4,0.3, 0.5) = 0.2665, 0.4334,

IFWA*i*(*<sup>a</sup>*31, *a*32, *<sup>a</sup>*33) = IFWA*i*(0.3, 0.4,0.5, 0.4,0.4, 0.1) = 0.4113, 0.2428.

*Then,*

$$\ell\_1' = \text{GBGVA}(\langle 0.6, 0.4 \rangle, \langle 0.6, 0.2 \rangle, \langle 0.9, 0 \rangle) = \text{IFWA}\_k \left( \begin{array}{c} \langle (0.2843, 0.2549) \rangle \\ \langle 0.2018, 0.4334 \rangle, \\ \langle 0.3114, 0.2428 \rangle \end{array} \right) = \langle 0.7441, 0.3099 \rangle.$$

*Similarly, we can calculate* 2, 3, 4, 5 *and* 6*.*

**Property 23. Idempotency;** *If cji* = *cj and aki* = *ak* = *a for all i, then GBGWA*(*cj*1, *cj*2, ..., *cjm*)=(*a* ⊗ *cj*). 

**Proof.** Since *cji* = *cj* ∀*j*, that is, *tcji* = *tcj* and *f cji* = *f cj* . Therefore, for *p* = 2, *a*1*i* = *a* and *a*2*i* = *a*, this implies that *ta*1*i* = *ta*, *f <sup>a</sup>*1*i* = *f a* and *ta*2*i* = *ta*, *f <sup>a</sup>*2*i* = *f a*. Then,

GBGWA(*cj*1, *cj*2, ..., *cjm*) = IFWA*k* ⎛⎜⎜⎜⎝ (1 − ∏*mi*=<sup>1</sup>(<sup>1</sup> − *ta*1*i*)*wi*) · (1 − ∏*mi*=<sup>1</sup>(<sup>1</sup> − *tcji*)*wi*), ∏*mi*=<sup>1</sup> *f wi <sup>a</sup>*1*i* + ∏*mi*=<sup>1</sup> *f wi cji* − ∏*mi*=<sup>1</sup> *f wi <sup>a</sup>*1*i* .∏*mi*=<sup>1</sup> *f wi cji* ,(<sup>1</sup> − ∏*mi*=<sup>1</sup>(<sup>1</sup> − *ta*2*i*)*wi*) · (1 − ∏*mi*=<sup>1</sup>(<sup>1</sup> − *tcji*)*wi*), ∏*mi*=<sup>1</sup> *f wi <sup>a</sup>*2*i* + ∏*mi*=<sup>1</sup> *f wi cji* − ∏*mi*=<sup>1</sup> *f wi <sup>a</sup>*2*i* .∏*mi*=<sup>1</sup> *f wi cji* ⎞⎟⎟⎟⎠ = IFWA*k* ⎛⎜⎜⎜⎜⎝ (1 − (1 −*ta*)<sup>∑</sup>*mi*=<sup>1</sup> *wi*) · (1 − (1 −*tcj*)<sup>∑</sup>*mi*=<sup>1</sup> *wi*), *<sup>f</sup>*∑*mi*=<sup>1</sup> *wi a* + *<sup>f</sup>*∑*mi*=<sup>1</sup> *wi cj* − *<sup>f</sup>*∑*mi*=<sup>1</sup> *wi a* . *<sup>f</sup>*∑*mi*=<sup>1</sup> *wi cj* ,(<sup>1</sup> − (1 −*ta*)<sup>∑</sup>*mi*=<sup>1</sup> *wi*) · (1 − (1 −*tcj*)<sup>∑</sup>*mi*=<sup>1</sup> *wi*), *<sup>f</sup>*∑*mi*=<sup>1</sup> *wi a* + *<sup>f</sup>*∑*mi*=<sup>1</sup> *wi cj* − *<sup>f</sup>*∑*mi*=<sup>1</sup> *wi a* . *<sup>f</sup>*∑*mi*=<sup>1</sup> *wi cj* ⎞⎟⎟⎟⎟⎠ = IFWA*k* (1 − (1 −*ta*)) · (1 − (1 <sup>−</sup>*tcj*)), *fa* + *fcj* − *fa*. *fcj* ,(<sup>1</sup> − (1 −*ta*)) · (1 − (1 <sup>−</sup>*tcj*)), *fa* + *fcj* − *fa*. *fcj* = IFWA*k ta*.*tcj* , *fa* + *fcj* − *fa*. *fcj* ,*ta*.*tcj* , *fa* + *fcj* − *fa*. *fcj* = 1 − (1 <sup>−</sup>*ta*.*tcj*)<sup>1</sup> · (1 <sup>−</sup>*ta*.*tcj*)<sup>2</sup> , (*fa* + *fcj* − *fa*. *<sup>f</sup>cj*)<sup>1</sup> .(*f<sup>a</sup>* + *fcj* − *fa*. *<sup>f</sup>cj*)<sup>2</sup> = 1 − (1 <sup>−</sup>*ta*.*tcj*)<sup>∑</sup><sup>2</sup>*k*=<sup>1</sup> *k* , (*fa* + *fcj* − *fa*. *<sup>f</sup>cj*)<sup>∑</sup><sup>2</sup>*k*=<sup>1</sup> *k* = 1 − (1 <sup>−</sup>*ta*.*tcj*),(*f<sup>a</sup>* + *fcj* − *fa*. *<sup>f</sup>cj*) = *a* ⊗ *cj*

Now, using operation laws between IFVs, assume that results hold for *p* = *p* , that is,

$$\text{GBGWA}(c\_{j1}, c\_{j2}, \dots, c\_{jm}) = (a \otimes c\_{j})\dots$$

Then, for *p* = *p* + 1,

$$\text{GBGWA}(c\_{j1}, c\_{j2}, \dots, c\_{jm}) = (a \otimes c\_j)\text{.}$$

Thus, by mathematical induction, Theorem 23 holds for all positive integer *p*.

**Property 24. Boundedness;** *If c*+*j* = *t*max (*aki*⊗*cji*), *f*min (*aki*⊗*cji*) *and c* − *j* = *t*min (*aki*⊗*cji*), *f*max (*aki*⊗*cji*)*, then c*−*j* ≤ GBGWA(*cji*, *cj*2, ..., *cjm*) ≤ *c*+*j .*

**Proof.** Let *cji* = *tcji* , *f cji* and *aki* = *taki* , *f aki* be IFVs, for all *i*, *j*, *k*. Then, *aki* ⊗ *cji* = *taki* · *tcji* , 1 − (1 − *f aki*)(<sup>1</sup> − *f cji*) and denote *t*max (*aki*⊗*cji*) = *t*max *aki* · *t*max *cji* , *t*min (*aki*⊗*cji*) = *t*min *aki* · *t*min *cji* , *f*max (*aki*⊗*cji*) = 1 − (1 − *f*max *aki* ) · (1 − *f* max *cji* ), *f* min (*aki*⊗*cji*) = 1 − (1 − *f* min *aki* ) · (1 − *f* min *cji* ).

Now, *t*min *cji* ≤ *tcji* ≤ *t*max *cji* ⇐⇒ (1 − *t*max *cji* ) ≤ (1 − *tcji*) ≤ (1 − *t*min *cji* ) ⇐⇒ (1 − *t*max *cji* )<sup>∑</sup>*mi*=<sup>1</sup> *wi* ≤ ∏*mi*=<sup>1</sup>(<sup>1</sup> − *tcji*) ≤ (1 − *t*min *cji* )<sup>∑</sup>*mi*=<sup>1</sup> *wi* ⇐⇒ 1 − (1 − *t*min *cji* ) ≤ 1 − ∏*mi*=<sup>1</sup>(<sup>1</sup> − *tcji*) ≤ 1 − (1 − *t*max *cji* ) ⇐⇒ *t*min *cji* ≤ 1 − ∏*mi*=<sup>1</sup>(<sup>1</sup> −*tcji*) ≤ *t*max *cji* . Similarly, we obtain *t*min *aki* ≤ 1 − ∏*mi*=<sup>1</sup>(<sup>1</sup> −*taki*) ≤ *t*max *aki* . 

Therefore, *t*min *aki* · *t*min *cji* ≤ (1 − ∏*mi*=<sup>1</sup>(<sup>1</sup> − *taki*))(1 − ∏*mi*=<sup>1</sup>(<sup>1</sup> − *tcji*)) ≤ *t*max *aki* · *t*max *cji* ⇐⇒ *t*min (*aki*⊗*cji*) ≤ (1 − ∏*mi*=<sup>1</sup>(<sup>1</sup> − *taki*))(1 − ∏*mi*=<sup>1</sup>(<sup>1</sup> − *tcji*)) ≤ *t*max (*aki*⊗*cji*) ⇐⇒ 1 − *t*max (*aki*⊗*cji*) ≤ 1 − (1 − ∏*mi*=<sup>1</sup>(<sup>1</sup> − *taki*)) · (1 − ∏*mi*=<sup>1</sup>(<sup>1</sup> − *tcji*)) ≤ 1 − *t*min (*aki*⊗*cji*) ⇐⇒ (1 − *t*max (*aki*⊗*cji*))<sup>∑</sup>*pk*=<sup>1</sup> *k* ≤ <sup>∏</sup>*pk*=<sup>1</sup>(<sup>1</sup> − (1 − ∏*mi*=<sup>1</sup>(<sup>1</sup> − *taki*)) · (1 − ∏*mi*=<sup>1</sup>(<sup>1</sup> − *tcji*))) ≤ (1 − *t*min (*aki*⊗*cji*))<sup>∑</sup>*pk*=<sup>1</sup> *k* ⇐⇒ 1 − (1 −*t*min (*aki*⊗*cji*)) ≤ 1 − <sup>∏</sup>*pk*=<sup>1</sup>(<sup>1</sup> − (1 − ∏*mi*=<sup>1</sup>(<sup>1</sup> −*taki*)) · (1 − ∏*mi*=<sup>1</sup>(<sup>1</sup> − *tcji*))) ≤ 1 − (1 − *t*max (*aki*⊗*cji*)) ⇐⇒

$$\hat{I}\_{(a\_{li}\otimes c\_{ji})}^{\text{min}} \le 1 - \prod\_{k=1}^{p} (1 - \left( (1 - \prod\_{i=1}^{m} (1 - \hat{I}\_{a\_{ki}}) \right) \cdot (1 - \prod\_{i=1}^{m} (1 - \hat{I}\_{c\_{ji}})))) \le \hat{I}\_{(a\_{li}\otimes c\_{ji})}^{\text{max}} \tag{6}$$

In addition, *f* min *cji* ≤ *f cji* ≤ *f* max *cji* ⇐⇒ *f* min *cji* ≤ ∏*mi*=<sup>1</sup> *f cji* ≤ *f* max *cji* ⇐⇒ 1 − *f* max *cji* ≤ 1 − ∏*mi*=<sup>1</sup> *f cji* ≤ 1 − *f* min *cji* . Similarly, we obtain 1 − *f* max *aki* ≤ 1 − ∏*mi*=<sup>1</sup> *f aki* ≤ 1 − *f* min *aki* . 

Therefore, 1 − (1 − *f* min *aki* )(1 − *f* min *cji* ) ≤ 1 − (1 − ∏*mi*=<sup>1</sup> *f aki*)(<sup>1</sup> − ∏*mi*=<sup>1</sup> *f cji*) ≤ 1 − (1 − *f* max *aki* )(1 − *f* max *cji* ) ⇐⇒ *f* min (*aki*⊗*cji*) ≤ 1 − (1 − ∏*mi*=<sup>1</sup> *f aki*)(<sup>1</sup> − ∏*mi*=<sup>1</sup> *f cji*) ≤ *f* max (*aki*⊗*cji*) ⇐⇒ (*f* min (*aki*⊗*cji*))<sup>∑</sup>*pk*=<sup>1</sup> *k* ≤ <sup>∏</sup>*pk*=<sup>1</sup>(<sup>1</sup> − (1 − ∏*mi*=<sup>1</sup> *faki*)(<sup>1</sup> − ∏*mi*=<sup>1</sup> *<sup>f</sup>cji*)) ≤ (*f*max (*aki*⊗*cji*))<sup>∑</sup>*pk*=<sup>1</sup> *k* ⇐⇒

$$\hat{f}\_{(a\_{\hat{k}}\odot c\_{\hat{\boldsymbol{\mu}}})}^{\min} \leq \prod\_{k=1}^{p} (1 - (1 - \prod\_{i=1}^{m} \check{f}\_{a\_{\hat{k}}}) (1 - \prod\_{i=1}^{m} \check{f}\_{c\_{\hat{\boldsymbol{\mu}}}})) \leq \hat{f}\_{(a\_{\hat{k}}\odot c\_{\hat{\boldsymbol{\mu}}})}^{\max}.\tag{7}$$

If GBGWA(*cj*1, *cj*2, ..., *cjm*) = *tρ*, *f ρ*, therefore from Equations (6) and (7), we have *t*min (*aki*⊗*cji*) ≤ *tρ* ≤ *t*max (*aki*⊗*cji*) and *f* min (*aki*⊗*cji*) ≤ *f ρ* ≤ *f* max (*aki*⊗*cji*). Further using score function *<sup>δ</sup>*(GBGWA(*cj*1, *cj*2, ..., *cjm*)) = *tρ* − *f ρ* ≤ *t*max (*aki*⊗*cji*) − *t*min (*aki*⊗*cji*) = *<sup>δ</sup>*(*c*+*j* ), *<sup>δ</sup>*(GBGWA(*cj*1, *cj*2, ..., *cjm*) = *tρ* − *fρ* ≥ *t*min (*aki*⊗*cji*) − *t*max (*aki*⊗*cji*) = *<sup>δ</sup>*(*c*<sup>−</sup>*j* ). Hence, by order relation *c*−*j* ≤ GBGWA(*cj*1, *cj*2, ..., *cjm*) ≤ *c*+*j* .

**Property 25. Monotonicity;** *If c ji and cji are two IFVs such that c ji* ≤ *cji, then* GBGWA(*c j*1, *c j*2, ..., *c jm*) ≤ GBGWA(*cj*1, *cj*2, ..., *cjm*)*.*

**Proof.** It follows from Theorem 24, thus it is omitted from here.

**Proposition 2.** *Let* G *g* = (S , A, *g*) *be a GGIFSS over X. Then,*

, - (i) *If the assessments of each moderator/prospector on* A*, are* IF*k* = {1, <sup>0</sup>,1, <sup>0</sup>, ...,1, <sup>0</sup>}*, k* = 1, 2, ..., *p, then* GBGWA(*cj*1, *cj*2, ..., *cjm*) = IFWA*i*(*cj*1, *cj*2, ..., *cjm*).

 ...,

**Proof.** It is straightforward, thus it is omitted from here.

*,*  Now, the definition of GBGWG operator is specified as follows:

**Definition 26. GBGWG;** *Let* F Q *g* = (S , A, *g*) *be a GGIFSS over X, where g* = {*αd*1 , *<sup>α</sup>d*2 , ..., *<sup>α</sup>dp* } *be the group of PIFSs. Assume that w* = (*<sup>w</sup>*1, *w*2, ..., *wm*)*<sup>T</sup> be the normalized weight vector for* A*, such that wi* > 0 *and* ∑*mi*=<sup>1</sup> *wi* = 1*. Let IFV* R (*<sup>κ</sup>j*) = {*cj*1, *cj*2, ..., *cjm*} (*j* = 1 *to n*) *be the set of IFVs in EIFSS* (S , A) *for all κj* ∈ *X. For each senior moderator/prospector, <sup>α</sup>dk* (*υ*) = { *<sup>t</sup><sup>α</sup>dk* (*υ*), *f <sup>α</sup>dk* (*υ*) | *υ* ∈ A} (*k* = 1 *to p*) *be the PIFS, it can be represented as* IF*k* = {*ak*1, *ak*2, ..., *akm*} (*k* = 1 *to p*) *and* = (1, 2, ..., *p*)*<sup>T</sup> is the set of weights for moderators, such that k* > 0 *and* <sup>∑</sup>*pk*=<sup>1</sup> *k* = 1*. Define* GBGWG*:* Γ*ms* −→ Γ*s,* IFWG*:* Γ*m* −→ Γ*, where*

$$\text{GBGWG}(c\_{j1}, c\_{j2}, \dots, c\_{jm}) = \text{IFWG}\_{\mathbf{k}}\left( \begin{array}{l} (\text{IFWG}\_{\mathbf{i}}(a\_{11}, a\_{12}, \dots, a\_{1m}) \otimes \text{IFWG}\_{\mathbf{i}}(c\_{j1}, c\_{j2}, \dots, c\_{jm})), \\ (\text{IFWG}\_{\mathbf{i}}(a\_{21}, a\_{22}, \dots, a\_{2m}) \otimes \text{IFWG}\_{\mathbf{i}}(c\_{j1}, c\_{j2}, \dots, c\_{jm})), \dots, \\ (\text{IFWG}\_{\mathbf{i}}(a\_{p1}, a\_{p2}, \dots, a\_{pm}) \otimes \text{IFWG}\_{\mathbf{i}}(c\_{j1}, c\_{j2}, \dots, c\_{jm})) \end{array} \right) \tag{8}$$

*where* GBGWG *is known as GGIFSS weighted geometric operator, then the set of all* GBGWGs *is denoted L* = { 1 , 2 , ..., *n*}*. In addition,* IFWA*k and* IFWA*i are* IFWG *operators on set of senior moderators/prospectors and set of parameters, respectively. Note that* Γ*ms and* Γ *are families of GGIFSS and IFSs, respectively.*

**Lemma 2.** *Let* F Q *g* = (S , A, *g*) *be a GGIFSS over X, where g* = {*αd*1 , *<sup>α</sup>d*2 , ..., *<sup>α</sup>dp* } *be the group of IFSs. If p* = 1*, then* F Q *g is a GIFSS and* GBGWG *is specified below.*

$$\text{GBGNG}(c\_{j1}, c\_{j2}, \dots, c\_{jm}) = \text{IFWG}\_i(a\_{11}, a\_{12}, \dots, a\_{1m}) \otimes \text{IFWG}\_i(c\_{j1}, c\_{j2}, \dots, c\_{jm})\,\big) . \tag{9}$$

**Theorem 2.** *If cij* = ( *tji*, *f ji*) *and aki* = ( *tki*, *f ki*) (*i* = 1, 2, ..., *m*, *j* = 1, 2, ..., *n*, *k* = 1, 2, ..., *p*)*, be an IFV, then the aggregated value by GBGWG operator is given by* GBGWG(*cj*1, *cj*2, ..., *cjm*) = ∏*pk*=<sup>1</sup>(<sup>1</sup><sup>−</sup> (<sup>1</sup>− ∏*mi*=<sup>1</sup> *taki*)(<sup>1</sup>− ∏*mi*=<sup>1</sup> *tcji*)), 1− <sup>∏</sup>*pk*=<sup>1</sup>(<sup>1</sup><sup>−</sup> (<sup>1</sup>− ∏*mi*=<sup>1</sup>(<sup>1</sup><sup>−</sup> *faki*))· (1 − ∏*mi*=<sup>1</sup>(<sup>1</sup> − *f cji*)))*.*

**Proof.** It follows from Theorem 1, thus it is omitted from here.

In addition, the properties of idempotent, bounding and monotonicity for GBGWGs can be stated and proved in a similar manner as for GBGWAs.

**Proposition 3.** *Let* G *g* = (S , A, *g*) *be a GGIFSS over X. Then,*


$$\begin{aligned} \text{(iii)} \quad &lf \left(\tilde{\mathcal{S}}, \mathcal{A}\right) = \tilde{\mathcal{X}}^{\mathcal{A}^{(1,0)}}, \text{ then} \\ &\text{GBGWG}(c\_{j1}, c\_{j2}, \dots, c\_{jm}) = \text{IFWG}\_{k} \left( \begin{array}{c} \text{IFWG}\_{i}(a\_{11}, a\_{12}, \dots, a\_{1m}), \\ \text{IFWG}\_{i}(a\_{21}, a\_{22}, \dots, a\_{2m}), \dots \text{IFWG}\_{i}(a\_{p1}, a\_{p2}, \dots, a\_{pm}) \end{array} \right). \end{aligned}$$

$$\text{(iv)}\quad \text{If } (\tilde{\mathcal{S}}, \mathcal{A}) = \tilde{\mathcal{T}}^{\mathcal{A}^{(0,1)}}, \text{ then } \text{GBGNGG}(\mathcal{c}\_{j1}^{\cdot}, \mathcal{c}\_{j2}, \dots, \mathcal{c}\_{jm}) = \langle 0, 1 \rangle.$$

**Proof.** It is straightforward, thus is omitted here.

As aggregation operators are used to create MCDM frameworks, based on proposed GBGWA or GBGWG operators, some multi-criteria decision making methods are discussed in next section.

#### **5. Multi-Attribute Decision Making under GGIFSSs Environment**

In this section, firstly we present our approach comprising of an algorithm by virtue of GGIFSSs, and GBGWA or GBGWG operators. Then, we conduct two illustrations on proposed method as in the case studies: (1) candidates evaluation for an insurance company; and (2) cinema selection for the customers.
