# Another View of Aggregation Operators on Group-Based Generalized Intuitionistic Fuzzy Soft Sets: Multi-Attribute Decision Making Methods

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Definition**

**1.**

#### 2.1. Intuitionistic Fuzzy Sets

**Definition**

**2.**

**Definition**

**3.**

**Definition**

**4.**

- (i)
- ${V}_{1}\oplus {V}_{2}=\langle {\tilde{t}}_{\mathcal{A}}+{\tilde{t}}_{\mathcal{A}}^{\prime}-{\tilde{t}}_{\mathcal{A}}{\tilde{t}}_{\mathcal{A}}^{\prime},{\tilde{f}}_{\mathcal{A}}{\tilde{f}}_{\mathcal{A}}^{\prime}\rangle $.
- (ii)
- ${V}_{1}\otimes {V}_{2}=\langle {\tilde{t}}_{\mathcal{A}}{\tilde{t}}_{\mathcal{A}}^{\prime},{\tilde{f}}_{\mathcal{A}}+{\tilde{f}}_{\mathcal{A}}^{\prime}-{\tilde{f}}_{\mathcal{A}}{\tilde{f}}_{\mathcal{A}}^{\prime}\rangle $.
- (iii)
- $\u03f5{V}_{1}=\langle 1-{(1-{\tilde{t}}_{\mathcal{A}})}^{\u03f5},{({\tilde{f}}_{\mathcal{A}})}^{\u03f5}\rangle $, where ϵ is a positive real number.

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

**Definition**

**5.**

**Definition**

**6.**

- (i)
- $\mathcal{B}\subseteq \mathcal{A}$.
- (ii)
- $\tilde{\mathcal{T}}(a)\subseteq \tilde{\mathcal{S}}(a)$$\forall a\in \mathcal{B}$.

**Definition**

**7.**

- (i)
- $\mathcal{B}\subseteq \mathcal{A}$.
- (ii)
- $\tilde{\mathcal{T}}(a)=\tilde{\mathcal{S}}(a)$$\forall a\in \mathcal{B}$.

**Definition**

**8.**

**Definition**

**9.**

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

**Definition**

**10.**

**Remark**

**1.**

- (i)
- On the point that ${G}_{\varrho}(\upsilon )$ is an IFS, stated in Definition 10, but ${G}_{\varrho}(\upsilon )$ is a group of IFSs. $IFS(X)\times IF$ is not a meaningful product. In this way, mapping ${\mathcal{F}}_{G}:E\to IFS(X)\times IF$ is not well-defined.
- (ii)
- As the Definition 10 is stated on group of extra opinions, which is an intuitionistic fuzzy subset of set of the parameter E, therefore ${G}_{\varrho}(\upsilon )$ is not defined in a precise way.
- (iii)
- The extra inputs can be seen as another IFVs based data of alternatives.

**Definition**

**11.**

**Example**

**1.**

## 4. Operations on GGIFSSs and Aggregation Operators

**Definition**

**12.**

**Definition**

**13.**

- (i)
- $(\tilde{\mathcal{S}},\mathcal{A}){\tilde{\subseteq}}_{F}(\tilde{\mathcal{T}},\mathcal{B})$.
- (ii)
- ${\tilde{g}}_{1}\u22d8{\tilde{g}}_{2}$.

**Example**

**2.**

**Definition**

**14.**

- (i)
- $(\tilde{\mathcal{S}},\mathcal{A}){\tilde{\subseteq}}_{M}(\tilde{\mathcal{T}},\mathcal{B})$.
- (ii)
- ${\tilde{g}}_{1}\u22d8{\tilde{g}}_{2}$.

**Definition**

**15.**

**Example**

**3.**

**Definition**

**16.**

- (i)
- $(\tilde{\mathcal{H}},\mathcal{C})=(\tilde{\mathcal{S}},\mathcal{A}){\cup}_{\mathcal{E}}(\tilde{\mathcal{T}},\mathcal{B})$.
- (ii)
- For each moderator ${d}_{k}$, ${\tilde{\gamma}}_{{d}_{k}}$ $(k=1,2,\dots ,p)$ can be defined $\forall \upsilon \in \mathcal{C}$,$${\tilde{t}}_{{\tilde{\gamma}}_{{d}_{k}}}(\upsilon )=\left\{\begin{array}{cc}{\tilde{t}}_{{\tilde{\alpha}}_{{d}_{k}}}(\upsilon ),\hfill & \mathit{if}\phantom{\rule{4.pt}{0ex}}\upsilon \in \mathcal{A}\backslash \mathcal{B},\hfill \\ {\tilde{t}}_{{\tilde{\beta}}_{{d}_{k}}}(\upsilon ),\hfill & \mathit{if}\phantom{\rule{4.pt}{0ex}}\upsilon \in \mathcal{B}\backslash \mathcal{A},\hfill \\ max\{{\tilde{t}}_{{\tilde{\alpha}}_{{d}_{k}}}(\upsilon ),{\tilde{t}}_{{\tilde{\beta}}_{{d}_{k}}}(\upsilon )\},\hfill & \mathit{if}\phantom{\rule{4.pt}{0ex}}\upsilon \in \mathcal{A}\cap \mathcal{B};\hfill \end{array}\right.$$$${\tilde{f}}_{{\tilde{\gamma}}_{{d}_{k}}}(\upsilon )=\left\{\begin{array}{cc}{\tilde{f}}_{{\tilde{\alpha}}_{{d}_{k}}}(\upsilon ),\hfill & \mathit{if}\phantom{\rule{4.pt}{0ex}}\upsilon \in \mathcal{A}\backslash \mathcal{B},\hfill \\ {\tilde{f}}_{{\tilde{\beta}}_{{d}_{k}}}(\upsilon ),\hfill & \mathit{if}\phantom{\rule{4.pt}{0ex}}\upsilon \in \mathcal{B}\backslash \mathcal{A},\hfill \\ min\{{\tilde{f}}_{{\tilde{\alpha}}_{{d}_{k}}}(\upsilon ),{\tilde{f}}_{{\tilde{\beta}}_{{d}_{k}}}(\upsilon )\},\hfill & \mathit{if}\phantom{\rule{4.pt}{0ex}}\upsilon \in \mathcal{A}\cap \mathcal{B}.\hfill \end{array}\right.$$

**Definition**

**17.**

- (i)
- $(\tilde{\mathcal{R}},\mathcal{C})=(\tilde{\mathcal{S}},\mathcal{A}){\cap}_{\mathcal{E}}(\tilde{\mathcal{T}},\mathcal{B})$.
- (ii)
- For each moderator ${d}_{k}$, ${\tilde{\gamma}}_{{d}_{k}}$ $(k=1,2,\dots ,p)$ can be defined $\forall \upsilon \in \mathcal{A}\cup \mathcal{B}$,$${\tilde{t}}_{{\tilde{\gamma}}_{{d}_{k}}}(\upsilon )=\left\{\begin{array}{cc}{\tilde{t}}_{{\tilde{\alpha}}_{{d}_{k}}}(\upsilon ),\hfill & \mathit{if}\phantom{\rule{4.pt}{0ex}}\upsilon \in \mathcal{A}\backslash \mathcal{B},\hfill \\ {\tilde{t}}_{{\tilde{\beta}}_{{d}_{k}}}(\upsilon ),\hfill & \mathit{if}\phantom{\rule{4.pt}{0ex}}\upsilon \in \mathcal{B}\backslash \mathcal{A},\hfill \\ min\{{\tilde{t}}_{{\tilde{\alpha}}_{{d}_{k}}}(\upsilon ),{\tilde{t}}_{{\tilde{\beta}}_{{d}_{k}}}(\upsilon )\},\hfill & \mathit{if}\phantom{\rule{4.pt}{0ex}}\upsilon \in \mathcal{A}\cap \mathcal{B};\hfill \end{array}\right.$$$${\tilde{f}}_{{\tilde{\gamma}}_{{d}_{k}}}(\upsilon )=\left\{\begin{array}{cc}{\tilde{f}}_{{\tilde{\alpha}}_{{d}_{k}}}(\upsilon ),\hfill & \mathit{if}\phantom{\rule{4.pt}{0ex}}\upsilon \in \mathcal{A}\backslash \mathcal{B},\hfill \\ {\tilde{f}}_{{\tilde{\beta}}_{{d}_{k}}}(\upsilon ),\hfill & \mathit{if}\phantom{\rule{4.pt}{0ex}}\upsilon \in \mathcal{B}\backslash \mathcal{A},\hfill \\ max\{{\tilde{f}}_{{\tilde{\alpha}}_{{d}_{k}}}(\upsilon ),{\tilde{f}}_{{\tilde{\beta}}_{{d}_{k}}}(\upsilon )\},\hfill & \mathit{if}\phantom{\rule{4.pt}{0ex}}\upsilon \in \mathcal{A}\cap \mathcal{B}.\hfill \end{array}\right.$$

**Definition**

**18.**

- (i)
- $(\tilde{\mathcal{R}},\mathcal{C})=(\tilde{\mathcal{S}},\mathcal{A}){\cup}_{r}(\tilde{\mathcal{T}},\mathcal{B})$;
- (ii)
- For each moderator ${d}_{k}$, ${\tilde{\gamma}}_{{d}_{k}}$ $(k=1,2,\dots ,p)$ can be defined $\forall \upsilon \in \mathcal{C}$,$${\tilde{t}}_{{\tilde{\gamma}}_{{d}_{k}}}(\upsilon )=max\{{\tilde{t}}_{{\tilde{\alpha}}_{{d}_{k}}}(\upsilon ),{\tilde{t}}_{{\tilde{\beta}}_{{d}_{k}}}(\upsilon )\},\phantom{\rule{4.pt}{0ex}}\mathit{for}\phantom{\rule{4.pt}{0ex}}\mathit{all}\phantom{\rule{4.pt}{0ex}}\upsilon \in \mathcal{A}\cap \mathcal{B};$$$${\tilde{f}}_{{\tilde{\gamma}}_{{d}_{k}}}(\upsilon )=min\{{\tilde{f}}_{{\tilde{\alpha}}_{{d}_{k}}}(\upsilon ),{\tilde{f}}_{{\tilde{\beta}}_{{d}_{k}}}(\upsilon )\},\phantom{\rule{4.pt}{0ex}}\mathit{for}\phantom{\rule{4.pt}{0ex}}\mathit{all}\phantom{\rule{4.pt}{0ex}}\upsilon \in \mathcal{A}\cap \mathcal{B}.$$

**Definition**

**19.**

- (i)
- $(\tilde{\mathcal{R}},\mathcal{C})=(\tilde{\mathcal{S}},\mathcal{A}){\cap}_{r}(\tilde{\mathcal{T}},\mathcal{B})$.
- (ii)
- For each moderator ${d}_{k}$, ${\tilde{\gamma}}_{{d}_{k}}$ $(k=1,2,\dots ,p)$ can be defined $\forall \upsilon \in \mathcal{C}$,$${\tilde{t}}_{{\tilde{\gamma}}_{{d}_{k}}}(\upsilon )=min\{{\tilde{t}}_{{\tilde{\alpha}}_{{d}_{k}}}(\upsilon ),{\tilde{t}}_{{\tilde{\beta}}_{{d}_{k}}}(\upsilon )\},\phantom{\rule{4.pt}{0ex}}\mathit{for}\phantom{\rule{4.pt}{0ex}}\mathit{all}\phantom{\rule{4.pt}{0ex}}\upsilon \in \mathcal{A}\cap \mathcal{B};$$$${\tilde{f}}_{{\tilde{\gamma}}_{{d}_{k}}}(\upsilon )=max\{{\tilde{f}}_{{\tilde{\alpha}}_{{d}_{k}}}(\upsilon ),{\tilde{f}}_{{\tilde{\beta}}_{{d}_{k}}}(\upsilon )\},\phantom{\rule{4.pt}{0ex}}\mathit{for}\phantom{\rule{4.pt}{0ex}}\mathit{all}\phantom{\rule{4.pt}{0ex}}\upsilon \in \mathcal{A}\cap \mathcal{B}.$$

**Definition**

**20.**

- (1)
- $(\tilde{\mathcal{S}},\mathcal{A})={\tilde{\mathcal{I}}}^{{\mathcal{A}}^{(0,1)}}$.
- (2)
- For each moderator ${d}_{k}$, ${\tilde{t}}_{{\tilde{\alpha}}_{{d}_{k}}}(\upsilon )=0$ and ${\tilde{f}}_{{\tilde{\alpha}}_{{d}_{k}}}(\upsilon )=1$ for all $\upsilon \in \mathcal{A}$.

**Definition**

**21.**

- (1)
- $(\tilde{\mathcal{S}},\mathcal{A})={\tilde{\mathcal{X}}}^{{\mathcal{A}}^{(1,0)}}$.
- (2)
- For each moderator ${d}_{k}$, ${\tilde{t}}_{{\tilde{\alpha}}_{{d}_{k}}}(\upsilon )=1$ and ${\tilde{f}}_{{\tilde{\alpha}}_{{d}_{k}}}(\upsilon )=0$ for all $\upsilon \in \mathcal{A}$.

**Proposition**

**1.**

- (i)
- ${\tilde{\mathcal{G}}}_{\tilde{g}}{\tilde{\bigsqcup}}_{\mathcal{E}}{\tilde{\mathcal{G}}}_{\tilde{g}}={\tilde{\mathcal{G}}}_{\tilde{g}}{\tilde{\bigsqcup}}_{r}{\tilde{\mathcal{G}}}_{\tilde{g}}={\tilde{\mathcal{G}}}_{\tilde{g}}$.
- (ii)
- ${\tilde{\mathcal{G}}}_{\tilde{g}}{\tilde{\sqcap}}_{\mathcal{E}}{\tilde{\mathcal{G}}}_{\tilde{g}}={\tilde{\mathcal{G}}}_{\tilde{g}}{\tilde{\sqcap}}_{r}{\tilde{\mathcal{G}}}_{\tilde{g}}={\tilde{\mathcal{G}}}_{\tilde{g}}$.
- (iii)
- ${\tilde{\mathcal{G}}}_{\tilde{g}}{\tilde{\bigsqcup}}_{\mathcal{E}}{\tilde{\mathcal{N}}}_{\tilde{g}}^{\mathcal{A}}={\tilde{\mathcal{G}}}_{\tilde{g}}{\tilde{\bigsqcup}}_{r}{\tilde{\mathcal{N}}}_{\tilde{g}}^{\mathcal{A}}={\tilde{\mathcal{G}}}_{\tilde{g}}$.
- (iv)
- ${\tilde{\mathcal{G}}}_{\tilde{g}}{\tilde{\sqcap}}_{\mathcal{E}}{\tilde{\mathcal{N}}}_{\tilde{g}}^{\mathcal{A}}={\tilde{\mathcal{G}}}_{\tilde{g}}{\tilde{\sqcap}}_{r}{\tilde{\mathcal{N}}}_{\tilde{g}}^{\mathcal{A}}={\tilde{\mathcal{N}}}_{\tilde{g}}^{\mathcal{A}}$.
- (v)
- ${\tilde{\mathcal{G}}}_{\tilde{g}}{\tilde{\bigsqcup}}_{\mathcal{E}}{\tilde{\mathcal{W}}}_{\tilde{g}}^{\mathcal{A}}=\tilde{\mathcal{G}}{\tilde{\bigsqcup}}_{r}{\tilde{\mathcal{W}}}_{\mathcal{A}}={\tilde{\mathcal{W}}}_{\mathcal{A}}$.
- (vi)
- ${\tilde{\mathcal{G}}}_{\tilde{g}}{\tilde{\sqcap}}_{\mathcal{E}}{\tilde{\mathcal{W}}}_{\tilde{g}}^{\mathcal{A}}={\tilde{\mathcal{G}}}_{\tilde{g}}{\tilde{\sqcap}}_{r}{\tilde{\mathcal{W}}}_{\tilde{g}}^{\mathcal{A}}={\tilde{\mathcal{G}}}_{\tilde{g}}$.

**Definition**

**22.**

**GBGWA**; Let ${\tilde{\mathcal{F}}}_{\tilde{g}}=(\tilde{\mathcal{S}},\mathcal{A},\tilde{g})$ be a GGIFSS over X, where $\tilde{g}=\{{\tilde{\alpha}}_{{d}_{1}},{\tilde{\alpha}}_{{d}_{2}},\dots ,{\tilde{\alpha}}_{{d}_{p}}\}$ be the group of PIFSs. Assume that $w={({w}_{1},{w}_{2},\dots ,{w}_{m})}^{T}$ is the normalized weight vector for $\mathcal{A}$, such that ${w}_{i}>0$ and ${\sum}_{i=1}^{m}{w}_{i}=1$. Let $\tilde{IFV}({\kappa}_{j})=\{{c}_{j1},{c}_{j2},\dots ,{c}_{jm}\}$ $(j=1$ to $n)$ be the set of IFVs in EIFSS $(\tilde{\mathcal{S}},\mathcal{A})$ for all ${\kappa}_{j}\in X$. For each senior moderator/prospector, ${\tilde{\alpha}}_{{d}_{k}}(\upsilon )=\{\langle {\tilde{t}}_{{\tilde{\alpha}}_{{d}_{k}}}(\upsilon ),{\tilde{f}}_{{\tilde{\alpha}}_{{d}_{k}}}(\upsilon )\rangle \mid \upsilon \in \mathcal{A}\}$ $(k=1$ to $p)$ be the PIFS, it can be represented as ${\mathcal{IF}}_{k}=\{{a}_{k1},{a}_{k2},\dots ,{a}_{km}\}$ $(k=1$ to $p)$ and $\varpi ={({\varpi}_{1},{\varpi}_{2},\dots ,{\varpi}_{p})}^{T}$ is the set of weights for moderators, such that ${\varpi}_{k}>0$ and ${\sum}_{k=1}^{p}{\varpi}_{k}=1$. Define GBGWA: ${\mathsf{\Gamma}}_{s}^{m}\u27f6{\mathsf{\Gamma}}_{s}$, IFWA: ${\mathsf{\Gamma}}^{m}\u27f6\mathsf{\Gamma}$, where

**Lemma**

**1.**

**Theorem**

**1.**

**Proof.**

**Example**

**4.**

**Property**

**23.**

**Idempotency;**If ${c}_{ji}={c}_{j}$ and ${a}_{ki}={a}_{k}=a$ for all i, then GBGWA$({c}_{j1},{c}_{j2},\dots ,{c}_{jm})=(a\otimes {c}_{j}).$

**Proof.**

**Property**

**24.**

**Boundedness;**If ${c}_{j}^{+}=\langle {\tilde{t}}_{({a}_{ki}\otimes {c}_{ji})}^{max},{\tilde{f}}_{({a}_{ki}\otimes {c}_{ji})}^{min}\rangle $ and ${c}_{j}^{-}=\langle {\tilde{t}}_{({a}_{ki}\otimes {c}_{ji})}^{min},{\tilde{f}}_{({a}_{ki}\otimes {c}_{ji})}^{max}\rangle $, then ${c}_{j}^{-}\le \mathrm{GBGWA}({c}_{ji},{c}_{j2},\dots ,{c}_{jm})\le {c}_{j}^{+}$.

**Proof.**

**Property**

**25.**

**Monotonicity;**If ${c}_{ji}^{\prime}$ and ${c}_{ji}$ are two IFVs such that ${c}_{ji}^{\prime}\le {c}_{ji}$, then $\mathrm{GBGWA}({c}_{j1}^{\prime},{c}_{j2}^{\prime},\dots ,{c}_{jm}^{\prime})\le \mathrm{GBGWA}({c}_{j1},{c}_{j2},\dots ,{c}_{jm})$.

**Proof.**

**Proposition**

**2.**

- (i)
- If the assessments of each moderator/prospector on $\mathcal{A}$, are ${\mathcal{IF}}_{k}=\{\langle 1,0\rangle ,\langle 1,0\rangle ,\dots ,\langle 1,0\rangle \}$, $k=1,2,\dots ,p$, then $\mathrm{GBGWA}({c}_{j1},{c}_{j2},\dots ,{c}_{jm})={\mathrm{IFWA}}_{i}({c}_{j1},{c}_{j2},\dots ,{c}_{jm}).$
- (ii)
- If the assessments of each moderator/prospector on $\mathcal{A}$, are ${\mathcal{IF}}_{k}=\{\langle 0,1\rangle ,\langle 0,1\rangle ,\dots ,\langle 0,1\rangle \}$, $k=1,2,\dots ,p$, then $\mathrm{GBGWA}({c}_{j1},{c}_{j2},\dots ,{c}_{jm})=\langle 0,1\rangle $,
- (iii)
- If $(\tilde{\mathcal{S}},\mathcal{A})={\tilde{\mathcal{X}}}^{{\mathcal{A}}^{(1,0)}}$, then$$\mathrm{GBGWA}({c}_{j1},{c}_{j2},\dots ,{c}_{jm})={\mathrm{IFWA}}_{k}\left(\begin{array}{c}{\mathrm{IFWA}}_{i}({a}_{11},{a}_{12},\dots ,{a}_{1m}),\\ {\mathrm{IFWA}}_{i}({a}_{21},{a}_{22},\dots ,{a}_{2m}),\dots ,{\mathrm{IFWA}}_{i}({a}_{p1},{a}_{p2},\dots ,{a}_{pm})\end{array}\right).$$
- (iv)
- If $(\tilde{\mathcal{S}},\mathcal{A})={\tilde{\mathcal{I}}}^{{\mathcal{A}}^{(0,1)}}$, then $\mathrm{GBGWA}({c}_{j1},{c}_{j2},\dots ,{c}_{jm})=\langle 0,1\rangle $.

**Proof.**

**Definition**

**26.**

**GBGWG;**Let ${\tilde{\mathcal{F}}}_{\tilde{g}}=(\tilde{\mathcal{S}},\mathcal{A},\tilde{g})$ be a GGIFSS over X, where $\tilde{g}=\{{\tilde{\alpha}}_{{d}_{1}},{\tilde{\alpha}}_{{d}_{2}},\dots ,{\tilde{\alpha}}_{{d}_{p}}\}$ be the group of PIFSs. Assume that $w={({w}_{1},{w}_{2},\dots ,{w}_{m})}^{T}$ be the normalized weight vector for $\mathcal{A}$, such that ${w}_{i}>0$ and ${\sum}_{i=1}^{m}{w}_{i}=1$. Let $\tilde{IFV}({\kappa}_{j})=\{{c}_{j1},{c}_{j2},\dots ,{c}_{jm}\}$ $(j=1$ to $n)$ be the set of IFVs in EIFSS $(\tilde{\mathcal{S}},\mathcal{A})$ for all ${\kappa}_{j}\in X$. For each senior moderator/prospector, ${\tilde{\alpha}}_{{d}_{k}}(\upsilon )=\{\langle {\tilde{t}}_{{\tilde{\alpha}}_{{d}_{k}}}(\upsilon ),{\tilde{f}}_{{\tilde{\alpha}}_{{d}_{k}}}(\upsilon )\rangle \mid \upsilon \in \mathcal{A}\}$ $(k=1$ to $p)$ be the PIFS, it can be represented as ${\mathcal{IF}}_{k}=\{{a}_{k1},{a}_{k2},\dots ,{a}_{km}\}$ $(k=1$ to $p)$ and $\varpi ={({\varpi}_{1},{\varpi}_{2},\dots ,{\varpi}_{p})}^{T}$ is the set of weights for moderators, such that ${\varpi}_{k}>0$ and ${\sum}_{k=1}^{p}{\varpi}_{k}=1$. Define GBGWG: ${\mathsf{\Gamma}}_{s}^{m}\u27f6{\mathsf{\Gamma}}_{s}$, IFWG: ${\mathsf{\Gamma}}^{m}\u27f6\mathsf{\Gamma}$, where

**Lemma**

**2.**

**Theorem**

**2.**

**Proof.**

**Proposition**

**3.**

- (i)
- If the assessments of each moderator on $\mathcal{A}$, are ${\mathcal{IF}}_{k}=\{\langle 1,0\rangle ,\langle 1,0\rangle ,\dots ,\langle 1,0\rangle \}$, $k=1,2,\dots ,p$, then $\mathrm{GBGWG}({c}_{j1},{c}_{j2},\dots ,{c}_{jm})={\mathrm{IFWG}}_{i}({c}_{j1},{c}_{j2},\dots ,{c}_{jm}).$
- (ii)
- If the assessments of each moderator on $\mathcal{A}$, are ${\mathcal{IF}}_{k}=\{\langle 0,1\rangle ,\langle 0,1\rangle ,\dots ,\langle 0,1\rangle \}$, $k=1,2,\dots ,p$, then $\mathrm{GBGWG}({c}_{j1},{c}_{j2},\dots ,{c}_{jm})=\langle 0,1\rangle $.
- (iii)
- If $(\tilde{\mathcal{S}},\mathcal{A})={\tilde{\mathcal{X}}}^{{\mathcal{A}}^{(1,0)}}$, then$$\mathrm{GBGWG}({c}_{j1},{c}_{j2},\dots ,{c}_{jm})={\mathrm{IFWG}}_{k}\left(\begin{array}{c}{\mathrm{IFWG}}_{i}({a}_{11},{a}_{12},\dots ,{a}_{1m}),\\ {\mathrm{IFWG}}_{i}({a}_{21},{a}_{22},\dots ,{a}_{2m}),\dots ,{\mathrm{IFWG}}_{i}({a}_{p1},{a}_{p2},\dots ,{a}_{pm})\end{array}\right).$$
- (iv)
- If $(\tilde{\mathcal{S}},\mathcal{A})={\tilde{\mathcal{I}}}^{{\mathcal{A}}^{(0,1)}}$, then $\mathrm{GBGWG}({c}_{j1},{c}_{j2},\dots ,{c}_{jm})=\langle 0,1\rangle $.

**Proof.**

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

#### 5.1. Proposed Method

Algorithm 1 Multi-attribute decision making on GGIFSSs | |

Input: A set of alternatives | |

Output: The felicitous alternative for a problem | |

1: | Let $X=\{{\kappa}_{1},{\kappa}_{2},\dots ,{\kappa}_{n}\}$ be the set of alternatives and $E={\mathcal{A}}_{1}\cup {\mathcal{A}}_{2}\cup ,\dots ,\cup {\mathcal{A}}_{\varsigma}=\{{\upsilon}_{1},{\upsilon}_{2},\dots ,{\upsilon}_{m}\}$ be the set of attributes. Constitute a mechanism of the specialists’ judgements on attributes in the form of IFVs and establish IFSSs on each committee of specialists. |

2: | Obtain $\varsigma $ number of GGIFSSs, ${\tilde{\mathcal{F}}}_{\tilde{g}1}=({\tilde{\mathcal{S}}}_{1},{\mathcal{A}}_{1},{\tilde{g}}_{1})$, ${\tilde{\mathcal{F}}}_{\tilde{g}2}=({\tilde{\mathcal{S}}}_{2},{\mathcal{A}}_{2},{\tilde{g}}_{2})$, …, ${\tilde{\mathcal{F}}}_{\tilde{g}\varsigma}=({\tilde{\mathcal{S}}}_{\varsigma},{\mathcal{A}}_{\varsigma},{\tilde{g}}_{\varsigma})$ over X, which are handled by $\varsigma $ number of committees of experts and specialists. Each group ${\tilde{g}}_{1},{\tilde{g}}_{2},\dots ,{\tilde{g}}_{\varsigma}$ of IFSs is constituted by p number of senior members/moderators for available information on each ${\mathcal{A}}_{{i}^{\prime}}({i}^{\prime}=1,2,\dots ,\varsigma )$, respectively. |

3: | Compute extended union ${\tilde{\mathcal{G}}}_{\tilde{g}}={({\tilde{\bigsqcup}}_{\mathcal{E}})}_{{i}^{\prime}=1}^{\varsigma}{\tilde{\mathcal{F}}}_{\tilde{g}{i}^{\prime}}$ of GGIFSSs. Represent ${\tilde{\mathcal{G}}}_{\tilde{g}}$ in a table. |

4: | Normalized the data in ${\tilde{\mathcal{G}}}_{\tilde{g}}$ using Equation (3), and represent ${\tilde{{\mathcal{G}}^{\prime}}}_{\tilde{g}}$ in a table. |

5: | Calculate GBGWA ${\ell}_{j}^{\prime}(j=1,2,\dots ,n)$ or GBGWG ${\ell}_{j}^{\u2033}(j=1,2,\dots ,n)$ operators on GGIFFS ${\tilde{{\mathcal{G}}^{\prime}}}_{\tilde{g}}$. There will be n operators. |

6: | Obtain score function on each operator ${\ell}_{j}^{\prime}$ or ${\ell}_{j}^{\u2033}$ using Definition 3. |

7: | Rank the alternatives on score function; the best choice is obtained on a maximum score. |

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

- ${\upsilon}_{1}:$ english level;
- ${\upsilon}_{2}:$ relevant problem solving skills;
- ${\upsilon}_{3}:$ relevant working experience;
- ${\upsilon}_{4}:$ communication skills;
- ${\upsilon}_{5}:$ finance and insurance professional;
- ${\upsilon}_{6}:$ score obtained in a college degree; and
- ${\upsilon}_{7}:$ interpersonal skills.

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

- ${\upsilon}_{1}:$ quality of service;
- ${\upsilon}_{2}:$ quality of expected films;
- ${\upsilon}_{3}:$ environment in cinema;
- ${\upsilon}_{4}:$ price reasonability; and
- ${\upsilon}_{5}:$ convenience and luxuriousness.

## 6. Comparisons and Discussions

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

Algorithm 2 Grag’s Algorithm for faculty appointment | |

1: | Make a framework of the specialists’ judgment related to each possible choice (alternatives) in the form IFVs and then construct their corresponding decision matrix ${\left[h\right]}^{{n}^{\prime}\times {m}^{\prime}}.$ |

2: | Get the generalization matrix ${\left[{G}_{\varrho}\right]}^{{n}^{\prime}\times 1}$ by using the perceptions of the senior members/experts’ committee on each ${\kappa}_{j}$ $(j=1,2,\dots ,{n}^{\prime})$. |

3: | Construct the new matrix ${\left[{h}_{G}\right]}^{{n}^{\prime}\times ({m}^{\prime}+1)}$ by placing ${\left[{G}_{\varrho}\right]}^{{n}^{\prime}\times 1}$ in ${\left[h\right]}^{{n}^{\prime}\times {m}^{\prime}}$ with respect to ${\kappa}_{j}$ $(j=1,2,\dots ,{n}^{\prime})$. |

4: | Apply operators with respect to ${\kappa}_{j}$, and the results are denoted by ${r}_{j}^{\prime},j=1,2,\dots ,{n}^{\prime}.$ |

5: | Rank the ${\kappa}_{j}(j=1,2,\dots ,{n}^{\prime})$ in descending order on their score values ${r}_{j}^{\prime}$ $(j=1,2,\dots ,{n}^{\prime})$. |

- (i)
- In Algorithm 2, the generalized parameter matrix is obtained by incorporating preferences of experts on alternatives. In other words, information from moderator on alternatives is given; nevertheless, extra input can be sighted as another information over IFSS, and both information types (IFSS and extra inputs) deal with alternatives. Conversely, in Algorithm 1, clear and well-defined GGIFSSs are taken into account by incorporating IFSS and IFSs.
- (ii)
- Operation of extended union is used in Algorithm 1 on two GGIFSSs, while, in Algorithm 2, there are some difficulties in defining an operation of extended union on two or more GGIFSSs.
- (iii)
- It seems that GGWA or GGWG operators in Algorithm 2 are applied contrarily on two different information types, however, in Algorithm 1, an integrated manner is adopted to compile results through GBGWA or GBGWG.
- (iv)
- In Algorithm 1, the generalized parameters can be applied as the demands of customers, and thus an integrated framework can be employed in industries. However, Algorithm 2 lacks creating such frameworks.

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

- (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 Section 5.2 and Section 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 $\delta ({\ell}_{1}^{\prime})=0.5185$, $\delta ({\ell}_{2}^{\prime})=0.4305$, $\delta ({\ell}_{3}^{\prime})=0.4674$, $\delta ({\ell}_{4}^{\prime})=0.4695$, and $\delta ({\ell}_{5}^{\prime})=0.5194$. The descending order is acquired as ${\kappa}_{5}>{\kappa}_{1}>{\kappa}_{4}>{\kappa}_{3}>{\kappa}_{2}$; thus, ${\kappa}_{5}$ 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 $\delta ({\ell}_{1}^{\prime})=0.3709$, $\delta ({\ell}_{2}^{\prime})=0.3068$, $\delta ({\ell}_{3}^{\prime})=0.3329$, $\delta ({\ell}_{4}^{\prime})=0.3351$, and $\delta ({\ell}_{5}^{\prime})=0.3704$. The descending order is acquired as ${\kappa}_{1}>{\kappa}_{5}>{\kappa}_{3}>{\kappa}_{4}>{\kappa}_{2}$; thus, ${\kappa}_{1}$ 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 $\delta ({\ell}_{1}^{\prime})=0.4178$, $\delta ({\ell}_{2}^{\prime})=0.3475$, $\delta ({\ell}_{3}^{\prime})=0.3774$, $\delta ({\ell}_{4}^{\prime})=0.3788$, and $\delta ({\ell}_{5}^{\prime})=0.4192$. The descending order is acquired as ${\kappa}_{5}>{\kappa}_{1}>{\kappa}_{4}>{\kappa}_{2}>{\kappa}_{3}$; thus, ${\kappa}_{1}$ is the felicitous candidate for the position.It can be observed from above discussion that ${\kappa}_{5}$ is the most suitable candidate as per individual opinions of Senior Members 1 and 3. Similarly, ${\kappa}_{1}$ is the most felicitous candidate on individual opinion of Senior Member 2, while ${\kappa}_{5}$ is on second place in descending order. Thus, in general, ${\kappa}_{5}$ 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 $\delta ({\ell}_{1}^{\prime})=0.2927$, $\delta ({\ell}_{2}^{\prime})=0.3230$, $\delta ({\ell}_{3}^{\prime})=0.2655$, $\delta ({\ell}_{4}^{\prime})=0.3072$, and $\delta ({\ell}_{5}^{\prime})=0.2867$. The order is acquired as ${\kappa}_{2}>{\kappa}_{4}>{\kappa}_{1}>{\kappa}_{5}>{\kappa}_{3}$; thus, ${\kappa}_{2}$ is the best cinema for Customer 1.If it is required to select cinema only for the Customer 2, then $\delta ({\ell}_{1}^{\prime})=0.2965$, $\delta ({\ell}_{2}^{\prime})=0.3253$, $\delta ({\ell}_{3}^{\prime})=0.2682$, $\delta ({\ell}_{4}^{\prime})=0.3121$, and $\delta ({\ell}_{5}^{\prime})=0.2894$. The order is acquired as ${\kappa}_{2}>{\kappa}_{4}>{\kappa}_{1}>{\kappa}_{5}>{\kappa}_{3}$; thus, ${\kappa}_{2}$ is the best cinema for Customer 2.Thus, in general, ${\kappa}_{2}$ is the most suitable for both customers.
- (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 $\delta ({Z}_{J}({\kappa}_{1}))=0.5344$, $\delta ({Z}_{J}({\kappa}_{2}))=0.5928$, $\delta ({Z}_{J}({\kappa}_{3}))=0.4857$, $\delta ({Z}_{J}({\kappa}_{1}))=0.5549$, and $\delta ({Z}_{J}({\kappa}_{1}))=0.5251$. The descending order acquired as ${\kappa}_{2}>{\kappa}_{4}>{\kappa}_{1}>{\kappa}_{5}>{\kappa}_{3}$; thus, ${\kappa}_{2}$ is the best cinema for Customer 1. If it is require to select cinema only for Customer 2, then $\delta ({Z}_{J}({\kappa}_{1}))=0.5311$, $\delta ({Z}_{J}({\kappa}_{2}))=0.5961$, $\delta ({Z}_{J}({\kappa}_{3}))=0.4894$, $\delta ({Z}_{J}({\kappa}_{1}))=0.5568$, and $\delta ({Z}_{J}({\kappa}_{1}))=0.5262$. The descending order acquired as ${\kappa}_{2}>{\kappa}_{4}>{\kappa}_{1}>{\kappa}_{5}>{\kappa}_{3}$; thus, ${\kappa}_{2}$ is 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]1: The demands of p number of customers are represented as IFSs ${\tilde{\alpha}}_{1},{\tilde{\alpha}}_{2},\dots ,{\tilde{\alpha}}_{p}$. 2: Represent IFSS $(\tilde{\mathcal{S}},E)$ over set of all possible choices X. 3: Represent GIFSSs for each customer ${\tilde{\mathcal{F}}}_{k}=(\tilde{\mathcal{S}},E,{\tilde{\alpha}}_{k})(k=1,\dots ,p)$. 4: Compute $int-AND-product$ operation on ${\tilde{\mathcal{F}}}_{1},{\tilde{\mathcal{F}}}_{2},\dots ,{\tilde{\mathcal{F}}}_{p}$, obtain GIFSS $\tilde{\mathcal{F}}$ and show it in tabular form. 5: Derive the utility fuzzy set ${\Delta}_{\tilde{\mathcal{F}}}$ from the GIFSS $\tilde{\mathcal{F}}$. 6: Output ${\kappa}_{{j}^{\prime}}$ as the optimal decision if ${\Delta}_{\tilde{\mathcal{F}}}({\kappa}_{{j}^{\prime}})=max\{{\Delta}_{\tilde{\mathcal{F}}}({\kappa}_{j})\mid {\kappa}_{j}\in X\}$. 7: If ${j}^{\prime}$ has more than one values then any one of ${\kappa}_{j}$ may be chosen. 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, $\Delta ({\kappa}_{1})=0.2252$, $\Delta ({\kappa}_{2})=0.2678$, $\Delta ({\kappa}_{3})=0.2112$, $\Delta ({\kappa}_{4})=0.2419$, and $\Delta ({\kappa}_{5})=0.2261$. The descending order is acquired as ${\kappa}_{2}>{\kappa}_{4}>{\kappa}_{5}>{\kappa}_{1}>{\kappa}_{3}$; thus, ${\kappa}_{2}$ is the best cinema for both customers.

## 7. Superiority of Proposed Method

**Example**

**5.**

**Example**

**6.**

#### Advantages of Proposed Method

- (i)
- The case study indicated in [44] is implemented on two IFVs based matrix but not for the GGIFSSs. The extra inputs are located as in IFVs type weights on alternatives and operators presumed to be collected on the two different IFVs based data. In this prospect, the proposed approach is based on well-defined GGIFSSs.
- (ii)
- In [42], an extra input turns into the weighted vector in initial stages of decision making after calculation of score functions but it is not integrated with the information of experts to achieve better results. In the proposed method, extra inputs are taken into account in an accurate way using GBGWA or GBGWG.
- (iii)
- In [43], the $AND$ operation is used on several GIFSSs. In many cases, $AND$ or $OR$ operations on IFVs provide instantaneous results but do not give comprehensively aggregated results.
- (iv)
- The judgements/demands of senior prospectors/customers in GGIFSSs as managed with proposed operators are useful to rank the alternatives. The proposed framework can be correlated with the shortening of any number of existing senior prospectors/customers.

## 8. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

GGIFSSs | Group-based generalized intuitionistic fuzzy soft sets |

GIFSSs | Generalized intuitionistic fuzzy soft sets |

GBGWA | Group-based generalized weighted averaging operators |

GBGWG | Group-based generalized weighted geometric operators |

IFSs | Intuitionistic fuzzy sets |

## References

- Maji, P.K.; Biswas, R.; Roy, A.R. Fuzzy soft sets. J. Fuzzy Math.
**2001**, 9, 589–602. [Google Scholar] - Zadeh, L.A. Fuzzy Sets. Inf. Control
**1965**, 8, 338–353. [Google Scholar] [CrossRef] - Molodtsov, D. Soft set theory-first results. Comput. Math. Appl.
**1999**, 37, 19–31. [Google Scholar] [CrossRef] - Maji, P.K.; Biswas, R.; Roy, A.R. Soft set theory. Comput. Math. Appl.
**2003**, 45, 555–562. [Google Scholar] [CrossRef] [Green Version] - Ali, M.I.; Feng, F.; Liu, X.; Min, W.K.; Shabir, M. On some new operations in soft set theory. Comput. Math. Appl.
**2009**, 57, 1547–1553. [Google Scholar] [CrossRef] [Green Version] - Roy, A.R.; Maji, P.K. A fuzzy soft set theoretic approach to decision making problems. J. Comput. Appl. Math.
**2007**, 203, 412–418. [Google Scholar] [CrossRef] - Alcantud, J.C.R.; Mathew, T.J. Separable fuzzy soft sets and decision making with positive and negative attributes. Appl. Soft Comput.
**2017**, 59, 586–595. [Google Scholar] [CrossRef] - Alcantud, J.C.R.; Torra, V. Decomposition theorems and extension principles for hesitant fuzzy sets. Inf. Fusion
**2018**, 41, 48–56. [Google Scholar] [CrossRef] - Khameneh, A.Z.; Kılıçman, A. Parameter reduction of fuzzy soft sets: An adjustable approach based on the three-way decision. Int. J. Fuzzy Syst.
**2018**, 20, 928–942. [Google Scholar] [CrossRef] - Hassan, N.; Sayed, O.R.; Khalil, A.M.; Ghany, M.A. Fuzzy soft expert system in prediction of coronary artery disease. Int. J. Fuzzy Syst.
**2017**, 19, 1546–1559. [Google Scholar] [CrossRef] - Hayat, K.; Ali, M.I.; Cao, B.Y.; Karaaslan, F.; Qin, Z. Characterizations of certain types of type-2 soft graphs. Discrete Dyn. Nat. Soc.
**2018**, 2018, 8535703. [Google Scholar] [CrossRef] - Hayat, K.; Ali, M.I.; Cao, B.; Karaaslan, F. New results on type-2 soft sets. J. Math. Stat.
**2018**, 484, 47. [Google Scholar] [CrossRef] - Hayat, K.; Ali, M.I.; Cao, B.-Y.; Yang, X.-P. A new type-2 soft set: Type-2 soft graphs and their applications. Adv. Fuzzy Sys.
**2017**. [Google Scholar] [CrossRef] - Kumar, A.; Kumar, D.; Jarial, S.K. A hybrid clustering method based on improved artificial bee colony and fuzzy C-means algorithm. Int. J. Artif. Intell.
**2017**, 15, 24–44. [Google Scholar] - Liu, Y.; Qin, K.; Martınez, L. Improving decision making approaches based on fuzzy soft sets and rough soft sets. Appl. Soft Comput.
**2018**, 65, 320–332. [Google Scholar] [CrossRef] - Medina, J.; Ojeda-Aciego, M. Multi-adjoint t-concept lattices. Inf. Sci.
**2010**, 180, 712–725. [Google Scholar] [CrossRef] - Pozna, C.; Minculete, N.; Precup, R.E.; Koczy, L.T.; Ballagi, A. Signatures: Definitions, operators and applications to fuzzy modelling. Fuzzy Sets Syst.
**2012**, 201, 86–104. [Google Scholar] [CrossRef] - Qinrong, F.; Fenfen, W. A discernibility matrix approach to fuzzy soft sets based decision making problems. J. Intell. Fuzzy Syst.
**2018**, 22, 59–74. [Google Scholar] - Vildan, C.; Halis, A. A topological view on L-fuzzy soft sets: Connectedness degree. J. Intell. Fuzzy Syst.
**2018**, 34, 1975–1983. [Google Scholar] - Vimala, J.; Arockia, J.R.; Anusuya, I.V.S. A study on fuzzy soft cardinality in lattice ordered fuzzy soft group and its application in decision making problems. J. Intell. Fuzzy. Syst.
**2018**, 34, 1535–1542. [Google Scholar] [CrossRef] - Xiao, F. A hybrid fuzzy soft sets decision making method in medical diagnosis. IEEE Acess
**2018**, 6, 25300–25312. [Google Scholar] [CrossRef] - Beg, I.; Rashid, T.; Jamil, R.N. Human attitude analysis based on fuzzy soft differential equations with Bonferroni mean. Comput. Appl. Math.
**2018**, 37, 2632–2647. [Google Scholar] [CrossRef] - Ma, X.; Zhan, J.; Ali, M.I. Applications of a kind of novel Z-soft fuzzy rough ideals to hemirings. J. Intell. Fuzzy Syst.
**2017**, 32, 2071–2080. [Google Scholar] [CrossRef] - Sun, B.; Ma, W. Soft fuzzy rough sets and its application in decision making. Artif. Intell. Rev.
**2014**, 41, 67–80. [Google Scholar] [CrossRef] - Atanassov, K. Intuitionistic fuzzy sets. Fuzzy Sets Syst.
**1986**, 20, 87–96. [Google Scholar] [CrossRef] - Atanassov, K. Intuitionistic Fuzzy Sets, Theory, and Applications; Series in Fuzziness and Soft Computing; Phisica-Verlag: Heidelberg, Germany, 1999. [Google Scholar]
- Xu, Z. Intuitionistic fuzzy aggregation operators. IEEE Trans. Fuzzy Syst.
**2007**, 15, 1179–1187. [Google Scholar] - Xu, Z.; Yager, R.R. Some geometric aggregation operators based on intuitionistic fuzzy sets. Int. J. Gen. Syst.
**2006**, 35, 417–433. [Google Scholar] [CrossRef] - Jemal, H.; Kechaou, Z.; Ayed, M.B. Enhanced decision support systems in intensive care unit based on intuitionistic fuzzy sets. Adv. Fuzzy Syst.
**2017**, 2017, 7371634. [Google Scholar] [CrossRef] - Mukherjee, S. Selection of alternative fuels for sustainable urban transportation under multi-criteria intuitionistic fuzzy environment. Fuzzy Inf. Eng.
**2017**, 9, 117–135. [Google Scholar] [CrossRef] - Ren, H.P.; Chen, H.H.; Fei, W.; Li, D.F. A MAGDM method considering the amount and reliability information of interval-valued intuitionistic fuzzy sets. Int. J. Fuzzy Syst.
**2017**, 19, 15–25. [Google Scholar] [CrossRef] - Yun, S.M.; Lee, S.J. Intuitionistic fuzzy topologies induced by intuitionistic fuzzy approximation spaces. Int. J. Fuzzy Syst.
**2017**, 9, 285–291. [Google Scholar] [CrossRef] - Maji, P.K.; Biswas, R.; Roy, A.R. Intuitionistic fuzzy soft sets. J. Fuzzy Math.
**2001**, 9, 677–692. [Google Scholar] - Deli, I.; Karataş, S. Interval valued intuitionistic fuzzy parameterized soft set theory and its decision making. J. Intell. Fuzzy Syst.
**2016**, 30, 2073–2082. [Google Scholar] [CrossRef] - Akram, M.; Shahzadi, S. Novel intuitionistic fuzzy soft multiple-attribute decision-making methods. Neural Comput. Appl.
**2018**, 29, 435–447. [Google Scholar] [CrossRef] - Garg, H.; Arora, R. A nonlinear-programming methodology for multi-attribute decision-making problem with interval-valued intuitionistic fuzzy soft sets information. Appl. Intell.
**2018**, 48, 2031–2046. [Google Scholar] [CrossRef] - Garg, H.; Arora, R. Bonferroni mean aggregation operators under intuitionistic fuzzy soft set environment and their applications to decision-making. J. Oper. Res. Soc.
**2018**, 1–4. [Google Scholar] [CrossRef] - Garg, H.; Arora, R. Novel scaled prioritized intuitionistic fuzzy soft interaction averaging aggregation operators and their application to multi criteria decision making. Eng. Appl. Artif. Intell.
**2018**, 71, 100–112. [Google Scholar] [CrossRef] - Agarwal, M.; Biswas, K.K.; Hanmandlu, M. Generalized intuitionistic fuzzy soft sets with applications in decision-making. Appl. Soft Comput.
**2013**, 13, 3552–3566. [Google Scholar] [CrossRef] - Khalil, A.M. Commentary on Generalized intuitionistic fuzzy soft sets with applications in decision-making. Appl. Soft Comput.
**2015**, 37, 519–520. [Google Scholar] [CrossRef] - Yang, Y.; Wang, Y.; Zhang, Y.; Zhang, D. Commentary on generalized intuitionistic fuzzy soft sets with applications in decision-making [Appl. Soft Comput. 37 (2015) 519–520]. Appl. Soft Comput.
**2016**, 40, 427–428. [Google Scholar] [CrossRef] - Feng, F.; Fujita, H.; Ali, M.I.; Yager, R.R.; Liu, X. Another View on generalized Intuitionistic Fuzzy Soft Sets and Related Multiattribute Decision Making Methods. IEEE Trans. Fuzzy Syst.
**2018**. [Google Scholar] [CrossRef] - Hayat, K.; Ali, M.I.; Alcantud, J.C.R.; Cao, B.Y.; Tariq, K.U. Best concept selection in design process: An application of generalized intuitionistic fuzzy soft sets. J. Intell. Fuzzy Syst.
**2018**. [Google Scholar] [CrossRef] - Garg, H.; Arora, R. Generalized and group-based generalized intuitionistic fuzzy soft sets with applications in decision-making. Appl. Intell.
**2018**, 48, 343–356. [Google Scholar] [CrossRef] - Selvachandran, G.; Maji, P.K.; Faisal, R.Q.; Salleh, A.R. Distance and distance induced intuitionistic entropy of generalized intuitionistic fuzzy soft sets. Appl. Intell.
**2017**, 47, 132–147. [Google Scholar] [CrossRef] - Pawlak, Z. Rough sets. Int. J. Comput. Inf. Sci.
**1982**, 11, 341–356. [Google Scholar] [CrossRef] - Deschrijver, G.; Kerre, E. On the relationship between some extensions of fuzzy set theory. Fuzzy Sets Syst.
**2003**, 133, 227–235. [Google Scholar] [CrossRef] - Chen, S.M.; Tan, J.M. Handling multicriteria fuzzy decision-making problems based on vague set theory. Fuzzy Sets Syst.
**1994**, 67, 163–172. [Google Scholar] [CrossRef]

**Table 1.**Tabular representation of the GGIFSS, ${\tilde{\mathcal{F}}}_{\tilde{g}}=(\tilde{\mathcal{S}},\mathcal{A},\tilde{g})$.

$\mathit{X}\setminus \mathcal{A}$ | ${\mathit{\upsilon}}_{1}$ | ${\mathit{\upsilon}}_{3}$ | ${\mathit{\upsilon}}_{4}$ |
---|---|---|---|

${\kappa}_{1}$ | $\langle 0.6,0.4\rangle $ | $\langle 0.6,0.2\rangle $ | $\langle 0.9,0\rangle $ |

${\kappa}_{2}$ | $\langle 0,0.3\rangle $ | $\langle 0.7,0.3\rangle $ | $\langle 0.5,0.1\rangle $ |

${\kappa}_{3}$ | $\langle 0.2,0.2\rangle $ | $\langle 0.3,0.6\rangle $ | $\langle 0.5,0.1\rangle $ |

${\kappa}_{4}$ | $\langle 0,1\rangle $ | $\langle 0.4,0.1\rangle $ | $\langle 0.2,0.5\rangle $ |

${\kappa}_{5}$ | $\langle 0.1,0.6\rangle $ | $\langle 0.4,0.2\rangle $ | $\langle 0.5,0.2\rangle $ |

${\kappa}_{6}$ | $\langle 0.6,0.2\rangle $ | $\langle 0.3,0.3\rangle $ | $\langle 0.6,0.1\rangle $ |

${\tilde{\alpha}}_{{d}_{1}}$ | $\langle 0.6,0.2\rangle $ | $\langle 0.3,0.4\rangle $ | $\langle 0.2,0.2\rangle $ |

${\tilde{\alpha}}_{{d}_{2}}$ | $\langle 0.3,0.4\rangle $ | $\langle 0.2,0.4\rangle $ | $\langle 0.3,0.5\rangle $ |

${\tilde{\alpha}}_{{d}_{3}}$ | $\langle 0.3,0.4\rangle $ | $\langle 0.5,0.4\rangle $ | $\langle 0.4,0.1\rangle $ |

**Table 2.**Tabular representation of the GGIFSS ${\tilde{\mathcal{F}}}_{\tilde{g}}=(\tilde{\mathcal{S}},\mathcal{A},{\tilde{g}}_{1})$.

$\mathit{X}\setminus \mathcal{A}$ | ${\mathit{\upsilon}}_{1}$ | ${\mathit{\upsilon}}_{4}$ |
---|---|---|

${\kappa}_{1}$ | $\langle 0.6,0.3\rangle $ | $\langle 0.9,0\rangle $ |

${\kappa}_{2}$ | $\langle 0,0.3\rangle $ | $\langle 0.5,0.1\rangle $ |

${\kappa}_{3}$ | $\langle 0.2,0.2\rangle $ | $\langle 0.5,0.1\rangle $ |

${\kappa}_{4}$ | $\langle 0,1\rangle $ | $\langle 0.3,0.5\rangle $ |

${\kappa}_{5}$ | $\langle 0.1,0.6\rangle $ | $\langle 0.5,0.2\rangle $ |

${\kappa}_{6}$ | $\langle 0.6,0.2\rangle $ | $\langle 0.6,0.1\rangle $ |

${\tilde{\alpha}}_{{d}_{1}}$ | $\langle 0.3,0.2\rangle $ | $\langle 0.3,0.4\rangle $ |

${\tilde{\alpha}}_{{d}_{2}}$ | $\langle 0.3,0.4\rangle $ | $\langle 0.2,0.4\rangle $ |

${\tilde{\alpha}}_{{d}_{3}}$ | $\langle 0.3,0.4\rangle $ | $\langle 0.5,0.4\rangle $ |

**Table 3.**Tabular representation of the GGIFSS, ${\tilde{\mathcal{G}}}_{\tilde{g}}=(\tilde{\mathcal{T}},\mathcal{B},{\tilde{g}}_{2})$.

$\mathit{X}\setminus \mathcal{B}$ | ${\mathit{\upsilon}}_{1}$ | ${\mathit{\upsilon}}_{3}$ | ${\mathit{\upsilon}}_{4}$ |
---|---|---|---|

${\kappa}_{1}$ | $\langle 0.6,0.2\rangle $ | $\langle 0.6,0.2\rangle $ | $\langle 0.9,0\rangle $ |

${\kappa}_{2}$ | $\langle 0.1,0.1\rangle $ | $\langle 0.7,0.3\rangle $ | $\langle 0.6,0.1\rangle $ |

${\kappa}_{3}$ | $\langle 0.3,0.2\rangle $ | $\langle 0.3,0.6\rangle $ | $\langle 0.5,0\rangle $ |

${\kappa}_{4}$ | $\langle 0,1\rangle $ | $\langle 0.4,0.1\rangle $ | $\langle 0.4,0.5\rangle $ |

${\kappa}_{5}$ | $\langle 0.2,0.3\rangle $ | $\langle 0.4,0.2\rangle $ | $\langle 0.6,0.1\rangle $ |

${\kappa}_{6}$ | $\langle 0.7,0.2\rangle $ | $\langle 0.3,0.3\rangle $ | $\langle 0.8,0.1\rangle $ |

${\tilde{\beta}}_{{d}_{1}}$ | $\langle 0.6,0.2\rangle $ | $\langle 0.4,0.5\rangle $ | $\langle 0.3,0.4\rangle $ |

${\tilde{\beta}}_{{d}_{2}}$ | $\langle 0.3,0.2\rangle $ | $\langle 0.4,0.2\rangle $ | $\langle 0.3,0.3\rangle $ |

${\tilde{\beta}}_{{d}_{3}}$ | $\langle 0.4,0.2\rangle $ | $\langle 0.4,0.4\rangle $ | $\langle 0.5,0.2\rangle $ |

**Table 4.**Tabular representation of the GGIFSS ${\tilde{\mathcal{G}}}_{\tilde{g}}^{c}=({\tilde{\mathcal{T}}}^{c},\mathcal{A},{\tilde{g}}^{c})$.

$\mathit{X}\setminus \mathcal{A}$ | ${\mathit{\upsilon}}_{1}$ | ${\mathit{\upsilon}}_{3}$ | ${\mathit{\upsilon}}_{4}$ |
---|---|---|---|

${\kappa}_{1}$ | $\langle 0.4,0.6\rangle $ | $\langle 0.2,0.6\rangle $ | $\langle 0,0.9\rangle $ |

${\kappa}_{2}$ | $\langle 0.3,0\rangle $ | $\langle 0.3,0.7\rangle $ | $\langle 0.1,0.5\rangle $ |

${\kappa}_{3}$ | $\langle 0.2,0.2\rangle $ | $\langle 0.6,0.3\rangle $ | $\langle 0.1,0.5\rangle $ |

${\kappa}_{4}$ | $\langle 1,0\rangle $ | $\langle 0.1,0.4\rangle $ | $\langle 0.5,0.2\rangle $ |

${\kappa}_{5}$ | $\langle 0.6,0.1\rangle $ | $\langle 0.2,0.4\rangle $ | $\langle 0.2,0.5\rangle $ |

${\kappa}_{6}$ | $\langle 0.2,0.6\rangle $ | $\langle 0.3,0.3\rangle $ | $\langle 0.1,0.6\rangle $ |

${\tilde{\alpha}}_{{d}_{1}}^{c}$ | $\langle 0.2,0.6\rangle $ | $\langle 0.4,0.3\rangle $ | $\langle 0.2,0.2\rangle $ |

${\tilde{\alpha}}_{{d}_{2}}^{c}$ | $\langle 0.4,0.3\rangle $ | $\langle 0.4,0.2\rangle $ | $\langle 0.5,0.3\rangle $ |

${\tilde{\alpha}}_{{d}_{3}}^{c}$ | $\langle 0.4,0.3\rangle $ | $\langle 0.4,0.5\rangle $ | $\langle 0.1,0.4\rangle $ |

**Table 5.**Tabular representation of the GGIFSS ${\tilde{\mathcal{F}}}_{\tilde{g}1}=({\tilde{\mathcal{S}}}_{1},\mathcal{A},{\tilde{g}}_{1})$.

$\mathit{X}\setminus \mathcal{A}$ | ${\mathit{\upsilon}}_{2}$ | ${\mathit{\upsilon}}_{3}$ | ${\mathit{\upsilon}}_{5}$ |
---|---|---|---|

${\kappa}_{1}$ | $\langle 0.7,0.1\rangle $ | $\langle 0.7,0.1\rangle $ | $\langle 0.6,0.1\rangle $ |

${\kappa}_{2}$ | $\langle 0.5,0.3\rangle $ | $\langle 0.6,0.3\rangle $ | $\langle 0.6,0.4\rangle $ |

${\kappa}_{3}$ | $\langle 0.5,0.4\rangle $ | $\langle 0.5,0.4\rangle $ | $\langle 0.9,0.1\rangle $ |

${\kappa}_{4}$ | $\langle 0.3,0.4\rangle $ | $\langle 0.7,0.2\rangle $ | $\langle 0.7,0.2\rangle $ |

${\kappa}_{5}$ | $\langle 0.6,0.4\rangle $ | $\langle 0.6,0.4\rangle $ | $\langle 0.8,0.1\rangle $ |

${\tilde{\alpha}}_{{d}_{1}}$ | $\langle 0.5,0.2\rangle $ | $\langle 0.5,0.4\rangle $ | $\langle 0.7,0.2\rangle $ |

${\tilde{\alpha}}_{{d}_{2}}$ | $\langle 0.4,0.4\rangle $ | $\langle 0.4,0.6\rangle $ | $\langle 0.3,0.5\rangle $ |

${\tilde{\alpha}}_{{d}_{3}}$ | $\langle 0.6,0.4\rangle $ | $\langle 0.3,0.5\rangle $ | $\langle 0.5,0.5\rangle $ |

**Table 6.**Tabular representation of the GGIFSS ${\tilde{\mathcal{F}}}_{\tilde{g}2}=({\tilde{\mathcal{S}}}_{2},\mathcal{B},{\tilde{g}}_{2})$.

$\mathit{X}\setminus \mathcal{B}$ | ${\mathit{\upsilon}}_{1}$ | ${\mathit{\upsilon}}_{4}$ | ${\mathit{\upsilon}}_{6}$ | ${\mathit{\upsilon}}_{7}$ |
---|---|---|---|---|

${\kappa}_{1}$ | $\langle 0.5,0.5\rangle $ | $\langle 0.6,0.1\rangle $ | $\langle 0.6,0.4\rangle $ | $\langle 0.7,0.1\rangle $ |

${\kappa}_{2}$ | $\langle 0.5,0.4\rangle $ | $\langle 0.3,0.4\rangle $ | $\langle 0.5,0.4\rangle $ | $\langle 0.8,0.2\rangle $ |

${\kappa}_{3}$ | $\langle 0.4,0.4\rangle $ | $\langle 0.7,0.2\rangle $ | $\langle 0.6,0.3\rangle $ | $\langle 0.4,0.4\rangle $ |

${\kappa}_{4}$ | $\langle 0.5,0.2\rangle $ | $\langle 0.7,0.3\rangle $ | $\langle 0.7,0.2\rangle $ | $\langle 0.5,0.3\rangle $ |

${\kappa}_{5}$ | $\langle 0.7,0.2\rangle $ | $\langle 0.8,0.1\rangle $ | $\langle 0.6,0.3\rangle $ | $\langle 0.6,0.1\rangle $ |

${\tilde{\beta}}_{{d}_{1}}$ | $\langle 0.4,0.3\rangle $ | $\langle 0.7,0.3\rangle $ | $\langle 0.6,0.2\rangle $ | $\langle 0.7,0.2\rangle $ |

${\tilde{\beta}}_{{d}_{2}}$ | $\langle 0.6,0.4\rangle $ | $\langle 0.3,0.4\rangle $ | $\langle 0.3,0.6\rangle $ | $\langle 0.5,0.3\rangle $ |

${\tilde{\beta}}_{{d}_{3}}$ | $\langle 0.5,0.5\rangle $ | $\langle 0.4,0.5\rangle $ | $\langle 0.4,0.4\rangle $ | $\langle 0.7,0.2\rangle $ |

**Table 7.**Tabular representation of the GGIFSS ${\tilde{\mathcal{F}}}_{\tilde{g}1}{\tilde{\bigsqcup}}_{\mathcal{E}.}{\tilde{\mathcal{F}}}_{\tilde{g}2}$.

$\mathit{X}\setminus \mathit{E}$ | ${\mathit{\upsilon}}_{1}$ | ${\mathit{\upsilon}}_{2}$ | ${\mathit{\upsilon}}_{3}$ | ${\mathit{\upsilon}}_{4}$ | ${\mathit{\upsilon}}_{5}$ | ${\mathit{\upsilon}}_{6}$ | ${\mathit{\upsilon}}_{7}$ |
---|---|---|---|---|---|---|---|

${\kappa}_{1}$ | $\langle 0.5,0.5\rangle $ | $\langle 0.7,0.1\rangle $ | $\langle 0.7,0.1\rangle $ | $\langle 0.6,0.1\rangle $ | $\langle 0.6,0.1\rangle $ | $\langle 0.6,0.4\rangle $ | $\langle 0.7,0.1\rangle $ |

${\kappa}_{2}$ | $\langle 0.5,0.4\rangle $ | $\langle 0.5,0.3\rangle $ | $\langle 0.6,0.3\rangle $ | $\langle 0.3,0.4\rangle $ | $\langle 0.6,0.4\rangle $ | $\langle 0.5,0.4\rangle $ | $\langle 0.8,0.2\rangle $ |

${\kappa}_{3}$ | $\langle 0.4,0.4\rangle $ | $\langle 0.5,0.4\rangle $ | $\langle 0.5,0.4\rangle $ | $\langle 0.7,0.2\rangle $ | $\langle 0.9,0.1\rangle $ | $\langle 0.6,0.3\rangle $ | $\langle 0.4,0.4\rangle $ |

${\kappa}_{4}$ | $\langle 0.5,0.2\rangle $ | $\langle 0.3,0.4\rangle $ | $\langle 0.7,0.2\rangle $ | $\langle 0.7,0.3\rangle $ | $\langle 0.7,0.2\rangle $ | $\langle 0.7,0.2\rangle $ | $\langle 0.5,0.3\rangle $ |

${\kappa}_{5}$ | $\langle 0.7,0.2\rangle $ | $\langle 0.6,0.4\rangle $ | $\langle 0.6,0.4\rangle $ | $\langle 0.8,0.1\rangle $ | $\langle 0.8,0.1\rangle $ | $\langle 0.6,0.3\rangle $ | $\langle 0.6,0.1\rangle $ |

${\tilde{\gamma}}_{{d}_{1}}$ | $\langle 0.4,0.3\rangle $ | $\langle 0.5,0.2\rangle $ | $\langle 0.5,0.4\rangle $ | $\langle 0.7,0.3\rangle $ | $\langle 0.7,0.2\rangle $ | $\langle 0.6,0.2\rangle $ | $\langle 0.7,0.2\rangle $ |

${\tilde{\gamma}}_{{d}_{2}}$ | $\langle 0.6,0.4\rangle $ | $\langle 0.4,0.4\rangle $ | $\langle 0.4,0.6\rangle $ | $\langle 0.3,0.4\rangle $ | $\langle 0.3,0.5\rangle $ | $\langle 0.3,0.6\rangle $ | $\langle 0.5,0.3\rangle $ |

${\tilde{\gamma}}_{{d}_{3}}$ | $\langle 0.5,0.5\rangle $ | $\langle 0.6,0.4\rangle $ | $\langle 0.3,0.5\rangle $ | $\langle 0.4,0.5\rangle $ | $\langle 0.5,0.5\rangle $ | $\langle 0.4,0.4\rangle $ | $\langle 0.7,0.2\rangle $ |

$\mathit{X}\setminus \mathit{E}$ | ${\mathit{\upsilon}}_{1}$ | ${\mathit{\upsilon}}_{2}$ | ${\mathit{\upsilon}}_{3}$ | ${\mathit{\upsilon}}_{4}$ | ${\mathit{\upsilon}}_{5}$ |
---|---|---|---|---|---|

${\kappa}_{1}$ | $\langle 0.4,0.5\rangle $ | $\langle 0.5,0.5\rangle $ | $\langle 0.6,0.4\rangle $ | $\langle 0.3,0.4\rangle $ | $\langle 0.3,0.3\rangle $ |

${\kappa}_{2}$ | $\langle 0.6,0.4\rangle $ | $\langle 0.6,0.3\rangle $ | $\langle 0.5,0.4\rangle $ | $\langle 0.4,0.4\rangle $ | $\langle 0.6,0.3\rangle $ |

${\kappa}_{3}$ | $\langle 0.3,0.4\rangle $ | $\langle 0.4,0.4\rangle $ | $\langle 0.5,0.4\rangle $ | $\langle 0.6,0.4\rangle $ | $\langle 0.5,0.5\rangle $ |

${\kappa}_{4}$ | $\langle 0.5,0.3\rangle $ | $\langle 0.3,0.3\rangle $ | $\langle 0.3,0.4\rangle $ | $\langle 0.4,0.6\rangle $ | $\langle 0.5,0.3\rangle $ |

${\kappa}_{5}$ | $\langle 0.6,0.4\rangle $ | $\langle 0.5,0.3\rangle $ | $\langle 0.5,0.5\rangle $ | $\langle 0.5,0.4\rangle $ | $\langle 0.3,0.4\rangle $ |

**Table 9.**Tabular representation of the GGIFSS, $(\tilde{{\mathcal{S}}^{\prime}},E,{\tilde{g}}^{\prime})$.

$$ |
---|