# Generalized Picture Fuzzy Soft Sets and Their Application in Decision Support Systems

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

**Definition**

**4.**

**Definition**

**5.**

- 1.
- $\stackrel{\u02c7}{A}\subset \stackrel{\u02c7}{B}$, if ${\xi}_{\stackrel{\u02c7}{A}}\le {\xi}_{\stackrel{\u02c7}{B}}$, ${\eta}_{\stackrel{\u02c7}{A}}\le {\eta}_{\stackrel{\u02c7}{B}}$ and ${\vartheta}_{\stackrel{\u02c7}{A}}\ge {\vartheta}_{\stackrel{\u02c7}{B}}$, $\forall f\in \stackrel{\u02c7}{X}$,
- 2.
- $\stackrel{\u02c7}{A}\cup \stackrel{\u02c7}{B}=\{(f,\mathit{max}({\xi}_{\stackrel{\u02c7}{A}},{\xi}_{\stackrel{\u02c7}{B}}),\mathit{min}({\eta}_{\stackrel{\u02c7}{A}},{\eta}_{\stackrel{\u02c7}{B}}),\mathit{min}({\vartheta}_{\stackrel{\u02c7}{A}},{\vartheta}_{\stackrel{\u02c7}{B}}))|\forall f\in \stackrel{\u02c7}{X}\}$,
- 3.
- $\stackrel{\u02c7}{A}\cap \stackrel{\u02c7}{B}=\{(f,\mathit{min}({\xi}_{\stackrel{\u02c7}{A}},{\xi}_{\stackrel{\u02c7}{B}}),\mathit{min}({\eta}_{\stackrel{\u02c7}{A}},{\eta}_{\stackrel{\u02c7}{B}}),\mathit{max}({\vartheta}_{\stackrel{\u02c7}{A}},{\vartheta}_{\stackrel{\u02c7}{B}}))|\forall f\in \stackrel{\u02c7}{X}\}$,
- 4.
- ${\stackrel{\u02c7}{A}}^{c}=\{(f,{\vartheta}_{\stackrel{\u02c7}{A}},{\eta}_{\stackrel{\u02c7}{A}},{\xi}_{\stackrel{\u02c7}{A}})|f\in \stackrel{\u02c7}{X}\}$.

**Definition**

**6.**

**Definition**

**7.**

- 1.
- $\mathit{If}\phantom{\rule{0.277778em}{0ex}}\stackrel{\u02c7}{\Theta}(q)<\stackrel{\u02c7}{\Theta}(p),\phantom{\rule{0.277778em}{0ex}}\mathit{then}\phantom{\rule{0.277778em}{0ex}}q\prec p$,
- 2.
- $\mathit{If}\phantom{\rule{0.277778em}{0ex}}\stackrel{\u02c7}{\Theta}(q)>\stackrel{\u02c7}{\Theta}(p),\phantom{\rule{0.277778em}{0ex}}\mathit{then}\phantom{\rule{0.277778em}{0ex}}q\succ p$,
- 3.
- $\mathit{If}\phantom{\rule{0.277778em}{0ex}}\stackrel{\u02c7}{\Theta}(q)=\stackrel{\u02c7}{\Theta}(p)$ and $\stackrel{\u02c7}{\varpi}(q)<\stackrel{\u02c7}{\varpi}(p),\phantom{\rule{0.277778em}{0ex}}\mathit{then}\phantom{\rule{0.277778em}{0ex}}q\prec p$,
- 4.
- $\mathit{If}\phantom{\rule{0.277778em}{0ex}}\stackrel{\u02c7}{\Theta}(q)=\stackrel{\u02c7}{\Theta}(p)$ and $\stackrel{\u02c7}{\varpi}(q)>\stackrel{\u02c7}{\varpi}(p),\phantom{\rule{0.277778em}{0ex}}\mathit{then}\phantom{\rule{0.277778em}{0ex}}q\succ p$,
- 5.
- $\mathit{If}\phantom{\rule{0.277778em}{0ex}}\stackrel{\u02c7}{\Theta}(q)=\stackrel{\u02c7}{\Theta}(p)$ and $\stackrel{\u02c7}{\varpi}(q)=\stackrel{\u02c7}{\varpi}(p),\phantom{\rule{0.277778em}{0ex}}\mathit{then}\phantom{\rule{0.277778em}{0ex}}q\sim p$.

**Definition**

**8.**

**Definition**

**9.**

## 3. Picture Fuzzy Soft Sets

**Definition**

**10.**

**Definition**

**11.**

**Definition**

**12.**

**Example**

**1.**

**Definition**

**13.**

**Definition**

**14.**

**Definition**

**15.**

**Definition**

**16.**

**Definition**

**17.**

**Definition**

**18.**

**Remark**

**1.**

**Theorem**

**1.**

- 1.
- $L{\cup}_{\u03f5}L=L{\cup}_{R}L=L$,
- 2.
- $L{\cap}_{\u03f5}L=L{\cap}_{R}L=L$.

**Proof.**

**Theorem**

**2.**

- 1.
- ${L}_{1}{\cup}_{\u03f5}{L}_{2}={L}_{2}{\cup}_{\u03f5}{L}_{1}$,
- 2.
- ${L}_{1}{\cap}_{\u03f5}{L}_{2}={L}_{2}{\cap}_{\u03f5}{L}_{1}$,
- 3.
- ${L}_{1}{\cup}_{R}{L}_{2}={L}_{2}{\cup}_{R}{L}_{1}$;
- 4.
- ${L}_{1}{\cap}_{R}{L}_{2}={L}_{2}{\cap}_{R}{L}_{1}$.

**Proof.**

**Theorem**

**3.**

- 1.
- ${[(\stackrel{\u02c7}{F},\stackrel{\u02c7}{A}){\cup}_{\u03f5}(\stackrel{\u02c7}{G},\stackrel{\u02c7}{B})]}^{c}={(\stackrel{\u02c7}{F},\stackrel{\u02c7}{A})}^{c}{\cap}_{\u03f5}{(\stackrel{\u02c7}{G},\stackrel{\u02c7}{B})}^{c}$, for all $\stackrel{\u02c7}{A},\stackrel{\u02c7}{B}\subseteq \stackrel{\u02c7}{E}$,
- 2.
- ${[(\stackrel{\u02c7}{F},\stackrel{\u02c7}{A}){\cap}_{\u03f5}(\stackrel{\u02c7}{G},\stackrel{\u02c7}{B})]}^{c}={(\stackrel{\u02c7}{F},\stackrel{\u02c7}{A})}^{c}{\cup}_{\u03f5}{(\stackrel{\u02c7}{G},\stackrel{\u02c7}{B})}^{c}$, for all $\stackrel{\u02c7}{A},\stackrel{\u02c7}{B}\subseteq \stackrel{\u02c7}{E}$.

**Proof.**

**Theorem**

**4.**

- 1.
- ${[(\stackrel{\u02c7}{F},\stackrel{\u02c7}{A}){\cup}_{R}(\stackrel{\u02c7}{G},\stackrel{\u02c7}{B})]}^{c}={(\stackrel{\u02c7}{F},\stackrel{\u02c7}{A})}^{c}{\cap}_{R}{(\stackrel{\u02c7}{G},\stackrel{\u02c7}{B})}^{c}$, for all $\stackrel{\u02c7}{A},\stackrel{\u02c7}{B}\subseteq \stackrel{\u02c7}{E}$,
- 2.
- ${[(\stackrel{\u02c7}{F},\stackrel{\u02c7}{A}){\cap}_{R}(\stackrel{\u02c7}{G},\stackrel{\u02c7}{B})]}^{c}={(\stackrel{\u02c7}{F},\stackrel{\u02c7}{A})}^{c}{\cup}_{R}{(\stackrel{\u02c7}{G},\stackrel{\u02c7}{B})}^{c}$, for all $\stackrel{\u02c7}{A},\stackrel{\u02c7}{B}\subseteq \stackrel{\u02c7}{E}$.

**Proof.**

## 4. Generalized Picture Fuzzy Soft Sets

**Definition**

**19.**

**Example**

**2.**

**Definition**

**20.**

- 1.
- $(\stackrel{\u02c7}{F},\stackrel{\u02c7}{A}){\subseteq}_{F}(\stackrel{\u02c7}{G},\stackrel{\u02c7}{B})$;
- 2.
- ${\xi}_{\rho}(h)\le {\xi}_{\sigma}(h)$, ${\eta}_{\rho}(h)\le {\eta}_{\sigma}(h)$ and ${\vartheta}_{\rho}(h)\ge {\vartheta}_{\sigma}(h)$, for all $h\in \stackrel{\u02c7}{A}$.

**Definition**

**21.**

- 1.
- $(\stackrel{\u02c7}{F},\stackrel{\u02c7}{A}){\subseteq}_{M}(\stackrel{\u02c7}{G},\stackrel{\u02c7}{B})$;
- 2.
- ${\xi}_{\rho}(h)\le {\xi}_{\sigma}(h)$, ${\eta}_{\rho}(h)\le {\eta}_{\sigma}(h)$ and ${\vartheta}_{\rho}(h)\ge {\vartheta}_{\sigma}(h)$, for all $h\in \stackrel{\u02c7}{A}$.

**Definition**

**22.**

**Definition**

**23.**

## 5. Basic Operations of Generalized Picture Fuzzy Soft Sets

**Definition**

**24.**

- $(\stackrel{\u02c7}{H},\stackrel{\u02c7}{C})=(\stackrel{\u02c7}{F},\stackrel{\u02c7}{A}){\cup}_{\u03f5}(\stackrel{\u02c7}{G},\stackrel{\u02c7}{B})$, where $\stackrel{\u02c7}{C}=\stackrel{\u02c7}{A}\cup \stackrel{\u02c7}{B}$.
- For all $h\in \stackrel{\u02c7}{C}=\stackrel{\u02c7}{A}\cup \stackrel{\u02c7}{B}$,$${\xi}_{\tau}(h)=\left\{\begin{array}{c}{\xi}_{\rho}(h),\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}if\phantom{\rule{0.277778em}{0ex}}h\in \stackrel{\u02c7}{A}\setminus \stackrel{\u02c7}{B},\hfill \\ {\xi}_{\sigma}(h),\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}if\phantom{\rule{0.277778em}{0ex}}h\in \stackrel{\u02c7}{B}\setminus \stackrel{\u02c7}{A},\hfill \\ max\{{\xi}_{\rho}(h),{\xi}_{\sigma}(h)\},\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}if\phantom{\rule{0.277778em}{0ex}}h\in \stackrel{\u02c7}{A}\cap \stackrel{\u02c7}{B},\hfill \end{array}\right.$$
- for all $h\in \stackrel{\u02c7}{C}=\stackrel{\u02c7}{A}\cup \stackrel{\u02c7}{B}$,$${\eta}_{\tau}(h)=\left\{\begin{array}{c}{\eta}_{\rho}(h),\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}if\phantom{\rule{0.277778em}{0ex}}h\in \stackrel{\u02c7}{A}\setminus \stackrel{\u02c7}{B},\hfill \\ {\eta}_{\sigma}(h),\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}if\phantom{\rule{0.277778em}{0ex}}h\in \stackrel{\u02c7}{B}\setminus \stackrel{\u02c7}{A},\hfill \\ min\{{\eta}_{\rho}(h),{\eta}_{\sigma}(h)\},\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}if\phantom{\rule{0.277778em}{0ex}}h\in \stackrel{\u02c7}{A}\cap \stackrel{\u02c7}{B},\hfill \end{array}\right.$$
- for all $h\in \stackrel{\u02c7}{C}=\stackrel{\u02c7}{A}\cup \stackrel{\u02c7}{B}$,$${\vartheta}_{\tau}(h)=\left\{\begin{array}{c}{\vartheta}_{\rho}(h),\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}if\phantom{\rule{0.277778em}{0ex}}h\in \stackrel{\u02c7}{A}\setminus \stackrel{\u02c7}{B},\hfill \\ {\vartheta}_{\sigma}(h),\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}if\phantom{\rule{0.277778em}{0ex}}h\in \stackrel{\u02c7}{B}\setminus \stackrel{\u02c7}{A},\hfill \\ min\{{\vartheta}_{\rho}(h),{\eta}_{\sigma}(h)\},\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}if\phantom{\rule{0.277778em}{0ex}}h\in \stackrel{\u02c7}{A}\cap \stackrel{\u02c7}{B}.\hfill \end{array}\right.$$

**Definition**

**25.**

- $(\stackrel{\u02c7}{H},\stackrel{\u02c7}{C})=(\stackrel{\u02c7}{F},\stackrel{\u02c7}{A}){\cap}_{\u03f5}(\stackrel{\u02c7}{G},\stackrel{\u02c7}{B})$, where $\stackrel{\u02c7}{C}=\stackrel{\u02c7}{A}\cup \stackrel{\u02c7}{B}$.
- For all $h\in \stackrel{\u02c7}{C}=\stackrel{\u02c7}{A}\cup \stackrel{\u02c7}{B}$,$${\xi}_{\tau}(h)=\left\{\begin{array}{c}{\xi}_{\rho}(h),\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}if\phantom{\rule{0.277778em}{0ex}}h\in \stackrel{\u02c7}{A}\setminus \stackrel{\u02c7}{B},\hfill \\ {\xi}_{\sigma}(h),\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}if\phantom{\rule{0.277778em}{0ex}}h\in \stackrel{\u02c7}{B}\setminus \stackrel{\u02c7}{A},\hfill \\ min\{{\xi}_{\rho}(h),{\xi}_{\sigma}(h)\},\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}if\phantom{\rule{0.277778em}{0ex}}h\in \stackrel{\u02c7}{A}\cap \stackrel{\u02c7}{B},\hfill \end{array}\right.$$
- for all $h\in \stackrel{\u02c7}{C}=\stackrel{\u02c7}{A}\cup \stackrel{\u02c7}{B}$,$${\eta}_{\tau}(h)=\left\{\begin{array}{c}{\eta}_{\rho}(h),\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}if\phantom{\rule{0.277778em}{0ex}}h\in \stackrel{\u02c7}{A}\setminus \stackrel{\u02c7}{B},\hfill \\ {\eta}_{\sigma}(h),\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}if\phantom{\rule{0.277778em}{0ex}}h\in \stackrel{\u02c7}{B}\setminus \stackrel{\u02c7}{A},\hfill \\ min\{{\eta}_{\rho}(h),{\eta}_{\sigma}(h)\},\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}if\phantom{\rule{0.277778em}{0ex}}h\in \stackrel{\u02c7}{A}\cap \stackrel{\u02c7}{B},\hfill \end{array}\right.$$
- for all $h\in \stackrel{\u02c7}{C}=\stackrel{\u02c7}{A}\cup \stackrel{\u02c7}{B}$,$${\vartheta}_{\tau}(h)=\left\{\begin{array}{c}{\vartheta}_{\rho}(h),\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}if\phantom{\rule{0.277778em}{0ex}}h\in \stackrel{\u02c7}{A}\setminus \stackrel{\u02c7}{B},\hfill \\ {\vartheta}_{\sigma}(h),\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}if\phantom{\rule{0.277778em}{0ex}}h\in \stackrel{\u02c7}{B}\setminus \stackrel{\u02c7}{A},\hfill \\ max\{{\vartheta}_{\rho}(h),{\vartheta}_{\sigma}(h)\},\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}if\phantom{\rule{0.277778em}{0ex}}h\in \stackrel{\u02c7}{A}\cap \stackrel{\u02c7}{B}.\hfill \end{array}\right.$$

**Definition**

**26.**

- $(\stackrel{\u02c7}{H},\stackrel{\u02c7}{C})=(\stackrel{\u02c7}{F},\stackrel{\u02c7}{A}){\cup}_{R}(\stackrel{\u02c7}{G},\stackrel{\u02c7}{B})$;
- for all $h\in \stackrel{\u02c7}{H}$, ${\xi}_{\tau}(h)=$ max$\{{\xi}_{\rho}(h),{\xi}_{\sigma}(h)\}$, ${\eta}_{\tau}(h)=$ min$\{{\eta}_{\rho}(h),{\eta}_{\sigma}(h)\}$ and$${\vartheta}_{\tau}=min\{{\vartheta}_{\rho}(h),{\vartheta}_{\sigma}(h)\}.$$

**Definition**

**27.**

- $(\stackrel{\u02c7}{H},\stackrel{\u02c7}{C})=(\stackrel{\u02c7}{F},\stackrel{\u02c7}{A}){\cap}_{R}(\stackrel{\u02c7}{G},\stackrel{\u02c7}{B})$;
- for all $h\in \stackrel{\u02c7}{H}$, ${\xi}_{\tau}(h)=$ min$\{{\xi}_{\rho}(h),{\xi}_{\sigma}(h)\}$, ${\eta}_{\tau}(h)=$ min$\{{\eta}_{\rho}(h),{\eta}_{\sigma}(h)\}$ and$${\vartheta}_{\tau}=max\{{\vartheta}_{\rho}(h),{\vartheta}_{\sigma}(h)\}.$$

**Example**

**3.**

**Remark**

**2.**

**Theorem**

**5.**

- 1.
- $\Gamma {\bigsqcup}_{\u03f5}\Gamma =\Gamma {\bigsqcup}_{R}\Gamma =\Gamma $;
- 2.
- $\Gamma {\sqcap}_{\u03f5}\Gamma =\Gamma {\sqcap}_{R}\Gamma =\Gamma $.

**Proof.**

**Theorem**

**6.**

- 1.
- ${\Gamma}_{1}{\bigsqcup}_{\u03f5}{\Gamma}_{2}={\Gamma}_{2}{\bigsqcup}_{\u03f5}{\Gamma}_{1}$;
- 2.
- ${\Gamma}_{1}{\sqcap}_{\u03f5}{\Gamma}_{2}={\Gamma}_{2}{\sqcap}_{\u03f5}{\Gamma}_{1}$;
- 3.
- ${\Gamma}_{1}{\bigsqcup}_{R}{\Gamma}_{2}={\Gamma}_{2}{\bigsqcup}_{R}{\Gamma}_{1}$;
- 4.
- ${\Gamma}_{1}{\sqcap}_{R}{\Gamma}_{2}={\Gamma}_{2}{\sqcap}_{R}{\Gamma}_{1}$.

**Proof.**

**Theorem**

**7.**

- 1.
- ${[(\stackrel{\u02c7}{F},\stackrel{\u02c7}{A},\rho ){\bigsqcup}_{\u03f5}(\stackrel{\u02c7}{G},\stackrel{\u02c7}{B},\sigma )]}^{c}={(\stackrel{\u02c7}{F},\stackrel{\u02c7}{A},\rho )}^{c}{\sqcap}_{\u03f5}{(\stackrel{\u02c7}{G},\stackrel{\u02c7}{B},\sigma )}^{c}$, for all $\stackrel{\u02c7}{A},\stackrel{\u02c7}{B}\subseteq \stackrel{\u02c7}{E}$;
- 2.
- ${[(\stackrel{\u02c7}{F},\stackrel{\u02c7}{A},\rho ){\sqcap}_{\u03f5}(\stackrel{\u02c7}{G},\stackrel{\u02c7}{B},\sigma )]}^{c}={(\stackrel{\u02c7}{F},\stackrel{\u02c7}{A},\rho )}^{c}{\bigsqcup}_{\u03f5}{(\stackrel{\u02c7}{G},\stackrel{\u02c7}{B},\sigma )}^{c}$, for all $\stackrel{\u02c7}{A},\stackrel{\u02c7}{B}\subseteq \stackrel{\u02c7}{E}$.

**Proof.**

**Theorem**

**8.**

- 1.
- ${[(\stackrel{\u02c7}{F},\stackrel{\u02c7}{A},\rho ){\bigsqcup}_{R}(\stackrel{\u02c7}{G},\stackrel{\u02c7}{B},\sigma )]}^{c}={(\stackrel{\u02c7}{F},\stackrel{\u02c7}{A},\rho )}^{c}{\sqcap}_{R}{(\stackrel{\u02c7}{G},\stackrel{\u02c7}{B},\sigma )}^{c}$, for all $\stackrel{\u02c7}{A},\stackrel{\u02c7}{B}\subseteq \stackrel{\u02c7}{E}$;
- 2.
- ${[(\stackrel{\u02c7}{F},\stackrel{\u02c7}{A},\rho ){\sqcap}_{R}(\stackrel{\u02c7}{G},\stackrel{\u02c7}{B},\sigma )]}^{c}={(\stackrel{\u02c7}{F},\stackrel{\u02c7}{A},\rho )}^{c}{\bigsqcup}_{R}{(\stackrel{\u02c7}{G},\stackrel{\u02c7}{B},\sigma )}^{c}$, for all $\stackrel{\u02c7}{A},\stackrel{\u02c7}{B}\subseteq \stackrel{\u02c7}{E}$.

**Proof.**

## 6. Substitution Operations of Generalized Picture Fuzzy Soft Sets

**Definition**

**28.**

**Definition**

**29.**

**Theorem**

**9.**

**Proof.**

**Theorem**

**10.**

- 1.
- ${[U(\rho )]}^{c}=L(\rho )$,
- 2.
- ${[L(\rho )]}^{c}=U(\rho )$,
- 3.
- $U({\rho}^{c})=U(\rho )$,
- 4.
- $L({\rho}^{c})=L(\rho )$.

**Proof.**

**Theorem**

**11.**

- 1.
- $U(\rho )=\rho \cup {\rho}^{c}$,
- 2.
- $L(\rho )=\rho \cap {\rho}^{c}$.

**Proof.**

**Theorem**

**12.**

- 1.
- $U(\rho \cup \sigma )=(U(\rho )\cup \sigma )\cap (\rho \cup U(\sigma ))$,
- 2.
- $U(\rho \cap \sigma )=(U(\rho )\cup {\sigma}^{c})\cap ({\rho}^{c}\cup U(\sigma ))$.

**Proof.**

**Theorem**

**13.**

- 1.
- $L(\rho \cup \sigma )=(L(\rho )\cap {\sigma}^{c})\cup ({\rho}^{c}\cap L(\sigma ))$,
- 2.
- $L(\rho \cap \sigma )=(L(\rho )\cap \sigma )\cup (\rho \cap L(\sigma ))$.

**Proof.**

**Definition**

**30.**

- 1.
- $\sigma =U(\rho )$,
- 2.
- $\stackrel{\u02c7}{G}(h)=U(\stackrel{\u02c7}{F}(h))$, for all $h\in \stackrel{\u02c7}{A}$.

**Definition**

**31.**

- 1.
- $\sigma =L(\rho )$,
- 2.
- $\stackrel{\u02c7}{G}(h)=L(\stackrel{\u02c7}{F}(h))$, for all $h\in \stackrel{\u02c7}{A}$.

**Theorem**

**14.**

**Proof.**

**Theorem**

**15.**

- 1.
- $\Gamma {\bigsqcup}_{\u03f5}{\Gamma}^{c}=\Gamma {\bigsqcup}_{R}{\Gamma}^{c}=\stackrel{\u02c7}{U}(\Gamma )$;
- 2.
- $\Gamma {\sqcap}_{\u03f5}{\Gamma}^{c}=\Gamma {\sqcap}_{R}{\Gamma}^{c}=\stackrel{\u02c7}{L}(\Gamma )$.

**Proof.**

## 7. A Generalized Picture Fuzzy Soft Sets Based MADM Process

**Definition**

**32.**

**Definition**

**33.**

**Definition**

**34.**

#### 7.1. Algorithm

- Step 1.
- Let $\stackrel{\u02c7}{X}=\{{f}_{1},{f}_{2},\dots ,{f}_{n}\}$, and $\stackrel{\u02c7}{E}=\stackrel{\u02c7}{A}\cup \stackrel{\u02c7}{B}=\{{h}_{1},{h}_{2},\dots ,{h}_{n}\}$. Two expert groups construct two $BPFSSs$ $(\stackrel{\u02c7}{F},\stackrel{\u02c7}{A})$ and $(\stackrel{\u02c7}{G},\stackrel{\u02c7}{B})$ over $\stackrel{\u02c7}{X}$ separately. Two $PPFSs$ $\rho $ and $\sigma $ are given by the head or director, which completes the construction of two $GPFSSs$ ${\Gamma}_{1}=(\stackrel{\u02c7}{F},\stackrel{\u02c7}{A},\rho )$ and ${\Gamma}_{2}=(\stackrel{\u02c7}{G},\stackrel{\u02c7}{B},\sigma )$.
- Step 2.
- By using Definition 25, calculate extended intersection $\Gamma ={\Gamma}_{1}{\sqcap}_{\u03f5}{\Gamma}_{2}=(\stackrel{\u02c7}{H},\stackrel{\u02c7}{C},\tau )$, of ${\Gamma}_{1}$ and ${\Gamma}_{2}$.
- Step 3.
- Calculate the Dombi aggregated picture fuzzy decision values $(DAPFDVs)$ by using picture fuzzy Dombi weighted average operator $(PFDWA)$ as follows,$${W}_{\Gamma}({f}_{i})={\oplus}_{j=1}^{m}\frac{{\stackrel{\u02c7}{\delta}}_{\tau ({h}_{j})}}{b}\stackrel{\u02c7}{H}({h}_{j})({f}_{i}).$$
- Step 4.
- Ascendingly rank ${W}_{\Gamma}({f}_{i})$ according to Definition 7.
- Step 5.
- Rank ${f}_{i}$ $(i=1,2,3,\dots ,n)$ ascendingly according to the rank of ${W}_{\Gamma}({f}_{i})$ and output ${f}_{i}$ as the optimal decision if it is the largest $PFV$ according to Definition 7.

**Remark**

**3.**

**Example**

**4.**

**Step 1.**- First, we find the extended intersection by using Definition 25.Let$$\Gamma =(\stackrel{\u02c7}{H},\stackrel{\u02c7}{C},\tau )=(\stackrel{\u02c7}{F},\stackrel{\u02c7}{A},\rho ){\sqcap}_{\u03f5}(\stackrel{\u02c7}{G},\stackrel{\u02c7}{B},\sigma ).$$For all $h\in \stackrel{\u02c7}{C}=\stackrel{\u02c7}{A}\cup \stackrel{\u02c7}{B}$, we have$$\begin{array}{c}\tau =\{(0.5,0.1,0.3)/{h}_{1},(0.3,0.4,0.2)/{h}_{2},(0.2,0.3,0.4)/{h}_{3},(0.3,0.2,0.4)/{h}_{4},\hfill \\ \hfill (0.1,0.4,0.4)/{h}_{5}\}.\end{array}$$Moreover, we have$$\stackrel{\u02c7}{H}({h}_{1})=\{(0.3,0.2,0.1)/{f}_{1},(0.4,0.2,0.2)/{f}_{2},(0.2,0.3,0.4)/{f}_{3},(0.1,0.2,0.5)/{f}_{4}\},$$$$\stackrel{\u02c7}{H}({h}_{2})=\{(0.4,0.5,0.1)/{f}_{1},(0.2,0.1,0.5)/{f}_{2},(0.3,0.4,0.2)/{f}_{3},(0.1,0.6,0.1)/{f}_{4}\},$$$$\stackrel{\u02c7}{H}({h}_{3})=\{(0.6,0.1,0.1)/{f}_{1},(0.2,0.1,0.5)/{f}_{2},(0.2,0.1,0.4)/{f}_{3},(0.1,0.5,0.1)/{f}_{4}\},$$$$\stackrel{\u02c7}{H}({h}_{4})=\{(0.6,0.2,0.1)/{f}_{1},(0.5,0.2,0.2)/{f}_{2},(0.4,0.4,0.2)/{f}_{3},(0.2,0.4,0.2)/{f}_{4}\},$$$$\stackrel{\u02c7}{H}({h}_{5})=\{(0.3,0.2,0.4)/{f}_{1},(0.6,0.1,0.1)/{f}_{2},(0.5,0.1,0.3)/{f}_{3},(0.2,0.2,0.5)/{f}_{4},\}.$$The tabular representation of extended union is shown in Table 7.
**Step 2.**- Now, we calculate Dombi aggregated picture fuzzy decision values $(DAPFDVs)$ by Definition 33, using $PFDWA$ for $k=1$. First, we calculate weight vectors from the picture fuzzy set by using expectation score function ${\stackrel{\u02c7}{\delta}}_{\tau ({h}_{j})}$ $(j=1,2,\dots ,5)$ using Definition 32, where the expectation score functions are ${\stackrel{\u02c7}{\delta}}_{1}=0.65$, ${\stackrel{\u02c7}{\delta}}_{2}=0.75$, ${\stackrel{\u02c7}{\delta}}_{3}=0.55$, ${\stackrel{\u02c7}{\delta}}_{4}=0.55$, ${\stackrel{\u02c7}{\delta}}_{5}=0.55$ and their sum is $b\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}{\sum}_{h\in \stackrel{\u02c7}{A}}{\stackrel{\u02c7}{\delta}}_{\tau (h)}=3.05$. Following is the weight vector$$\stackrel{\u02c7}{\omega}={(0.2131,0.2459,0.1803,0.1803,0.1803)}^{T},$$Now using these weight vector, the $DAPFDVs$ can be calculated as:$${W}_{\Gamma}({f}_{i})=PFDW{A}_{\stackrel{\u02c7}{\omega}}(\stackrel{\u02c7}{H}({h}_{1})({f}_{i}),\stackrel{\u02c7}{H}({h}_{2})({f}_{i}),\stackrel{\u02c7}{H}({h}_{3})({f}_{i}),\stackrel{\u02c7}{H}({h}_{4})({f}_{i}),\stackrel{\u02c7}{H}({h}_{5})({f}_{i}))$$$$={\oplus}_{j=1}^{5}{\stackrel{\u02c7}{\omega}}_{j}\stackrel{\u02c7}{H}({h}_{j})({f}_{i}).$$So, the $DAPFDVs$ are$${W}_{\Gamma}({f}_{1})=(0.46622,0.19367,0.11565),$$$${W}_{\Gamma}({f}_{2})=(0.41155,0.12450,0.21633),$$$${W}_{\Gamma}({f}_{3})=(0.33521,0.18581,0.26914),$$$${W}_{\Gamma}({f}_{4})=(0.13881,0.31365,0.16806).$$
**Step 3.**- Find score function of ${W}_{\Gamma}({f}_{i})$ $(i=1,2,3,4)$ as$$\stackrel{\u02c7}{\Theta}({W}_{\Gamma}({f}_{1}))=0.466221-0.115649=0.350572.$$Similarly, we get $\stackrel{\u02c7}{\Theta}({W}_{\Gamma}({f}_{2}))=0.195212$, $\stackrel{\u02c7}{\Theta}({W}_{\Gamma}({f}_{3}))=0.066075$ and $\stackrel{\u02c7}{\Theta}({W}_{\Gamma}({f}_{4}))=-0.029246$. More detail founds in Table 9.
**Step 4.**- Ranking the DAPFDVs according to Definition 7, we have$$\stackrel{\u02c7}{\Theta}({W}_{\Gamma}({f}_{4}))\prec \stackrel{\u02c7}{\Theta}({W}_{\Gamma}({f}_{3}))\prec \stackrel{\u02c7}{\Theta}({W}_{\Gamma}({f}_{2}))\prec \stackrel{\u02c7}{\Theta}({W}_{\Gamma}({f}_{1})).$$
**Step 5.**- From above calculations, alternatives have the order$${f}_{4}\prec {f}_{3}\prec {f}_{2}\prec {f}_{1}.$$

**Remark**

**4.**

**Remark**

**5.**

**Example**

**5.**

**Remark**

**6.**

## 8. Case Study: A Tower Construction Problem

**Remark**

**7.**

**Example**

**6.**

## 9. Comparison

- First, we compare our method with the method proposed in [40]. In his paper he did not give any information about how he calculated the weight vector, but in our proposed method we give a proper way to find the weight vector by using the expectation score function $\stackrel{\u02c7}{\omega}={\stackrel{\u02c7}{\delta}}_{\tau (h)}/{\sum}_{h\in \stackrel{\u02c7}{A}}{\stackrel{\u02c7}{\delta}}_{\tau (h)}$. For this, we actually use the parametric picture fuzzy soft sets $(PPFSs)$, $\rho $ and $\sigma $ which are given by the head or director who is responsible for firm or department in the form of $PFSs$, which is actually an additional judgment about the general quality of work done by the specialists groups.
- Secondly, if we compare our method with the method proposed in [30], we also find that they did not give any information about the weight vector. Also, in case Section 8, when we use the operator defined in [30], we get the same optimal decision and in addition, we are working in a more general situation.
- In [32], De Morgan’s laws hold with restricted conditions, while in this paper we relaxed the conditions for De Morgan’s laws by defining the new operations, like the extended union, extended intersection, restricted union, and restricted intersection.

## 10. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Zadeh, L.A. Fuzzy sets. Inf. Contr.
**1965**, 8, 338–353. [Google Scholar] [CrossRef] [Green Version] - Pawlak, Z. Rough sets. Int. J. Comput. Inf. Sci.
**1982**, 11, 341–356. [Google Scholar] [CrossRef] - Gau, W.L.; Buehrer, D.J. Vague sets. IEEE Trans. Syst. Man Cybernet.
**1993**, 23, 610–614. [Google Scholar] [CrossRef] - Atanassov, K.T. Intuitionistic fuzzy sets. Fuzzy Sets Syst.
**1986**, 20, 87–96. [Google Scholar] [CrossRef] - Molodtsov, Soft set theory-first results. Comput. Math. Appl.
**1999**, 37, 19–31. [CrossRef] - Maji, P.K.; Biswas, R.; Roy, A.R. Fuzzy soft sets. J. Fuzzy Math.
**2001**, 9, 589–602. [Google Scholar] - Maji, P.K.; Biswas, R.; Roy, A.R. Intuitionistic fuzzy soft sets. J. Fuzzy Math.
**2001**, 9, 677–692. [Google Scholar] - Yang, X.B.; Lin, T.Y.; Yang, J.Y.; Li, Y.; Yu, D.Y. Combination of interval-valued fuzzy set and soft set. Comput. Math. Appl.
**2009**, 58, 521–527. [Google Scholar] [CrossRef] [Green Version] - Majumdar, P.; Samanta, S.K. Generalised fuzzy soft sets. Comput. Math. Appl.
**2010**, 59, 1425–1432. [Google Scholar] [CrossRef] [Green Version] - Xu, W.; Ma, J.; Wang, S.; Hao, G. Vague soft sets and their properties. Comput. Math. Appl.
**2010**, 59, 787–794. [Google Scholar] [CrossRef] [Green Version] - Ali, M.I. A note on soft sets, rough soft sets and fuzzy soft sets. Appl. Soft Comput.
**2011**, 11, 3329–3332. [Google Scholar] - Xiao, Z.; Xia, S.; Gong, K.; Li, D. The trapezoidal fuzzy soft set and its application in MCDM. Appl. Math. Model.
**2012**, 36, 5844–5855. [Google Scholar] [CrossRef] - Maji, P.K. Neutrosophic soft set. Ann. Fuzzy Math. Inform.
**2013**, 5, 57–168. [Google Scholar] - Broumi, S.; Smarandache, F. Intuitionistic neutrosophic soft set. J. Inf. Comput. Sci.
**2013**, 8, 130–140. [Google Scholar] - Yang, Y.; Tan, X.; Meng, C.C. The multi-fuzzy soft set and its application in decision making. Appl. Math. Model.
**2013**, 37, 4915–4923. [Google Scholar] [CrossRef] - Wang, F.; Li, X.; Chen, X. Hesitant fuzzy soft set and its applications in multicriteria decision making. J. Appl. Math.
**2014**, 2014, 643785. [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] - Feng, F.; Fujita, H.; Ali, M.I.; Yager, R.R. Another view on generalized intuitionistic fuzzy soft sets and related multi attribute decision making methods. IEEE Trans. Fuzzy Syst.
**2018**, 27, 474–488. [Google Scholar] [CrossRef] - Cagman, N.; Enginoglu, S. Soft matrix theory and its decision making. Comput. Math. Appl.
**2010**, 59, 3308–3314. [Google Scholar] [CrossRef] [Green Version] - Feng, Q.; Zhou, Y. Soft discernibility matrix and its applications in decision making. Appl. Soft Comput.
**2014**, 24, 749–756. [Google Scholar] [CrossRef] - Cuong, B.C. Picture fuzzy sets. J. Comput. Sci. Cybern.
**2014**, 30, 409–420. [Google Scholar] - Singh, P. Correlation coefficients for picture fuzzy sets. J. Intell. Fuzzy Syst.
**2014**, 27, 2857–2868. [Google Scholar] - Son, L.H. DPFCM: A novel distributed picture fuzzy clustering method on picture fuzzy sets. Expert Syst. Appl.
**2015**, 2, 51–66. [Google Scholar] [CrossRef] - Thong, P.H.; Son, L.H. A new approach to multi-variables fuzzy forecasting using picture fuzzy clustering and picture fuzzy rules interpolation method. In Proceedings of the 6th International Conference on Knowledge and Systems Engineering, Hanoi, Vietnam, 9–11 January 2015; pp. 679–690. [Google Scholar]
- Son, L.H. Generalized picture distance measure and applications to picture fuzzy clustering. Appl. Soft Comput.
**2016**, 46, 284–295. [Google Scholar] [CrossRef] - Son, L.H. Measuring analogousness in picture fuzzy sets: From picture distance measures to picture association measures. Fuzzy Optim. Decis. Mak.
**2017**, 16, 1–20. [Google Scholar] [CrossRef] - Son, L.H.; Viet, P.; Hai, P. Picture inference system: A new fuzzy inference system on picture fuzzy set. Appl. Intell.
**2016**, 46, 652–669. [Google Scholar] [CrossRef] - Thong, P.H.; Son, L.H. Picture fuzzy clustering for complex data. Eng. Appl. Artif. Intell.
**2016**, 56, 121–130. [Google Scholar] [CrossRef] - Thong, P.H.; Son, L.H. A novel automatic picture fuzzy clustering method based on particle swarm optimization and picture composite cardinality. Knowl. Based Syst.
**2016**, 109, 48–60. [Google Scholar] [CrossRef] - Wei, G. Picture fuzzy aggregation operator and their application to multiple attribute decision making. J. Int. Fuzzy Syst.
**2017**, 33, 713–724. [Google Scholar] [CrossRef] - Wei, G.W. Picture fuzzy cross-entropy for multiple attribute decision making problems. J. Bus. Econ. Manag.
**2016**, 17, 491–502. [Google Scholar] [CrossRef] - Yang, Y.; Liang, C.; Ji, S.; Liu, T. Adjustable soft discernibility matrix based on picture fuzzy soft sets and its application in decision making. J. Int. Fuzzy Syst.
**2015**, 29, 1711–1722. [Google Scholar] [CrossRef] - Garg, H. Some picture fuzzy aggregation operators and their applications to multi criteria decision-making. Arab. J. Sci. Eng.
**2017**, 42, 5275–5290. [Google Scholar] [CrossRef] - Peng, X.; Dai, J. Algorithm for picture fuzzy multiple attribute decision making based on new distance measure. Int. J. Uncertain. Quant.
**2017**, 7, 177–187. [Google Scholar] [CrossRef] - Liu, Z.; Qin, K.; Pei, Z. A Method for Fuzzy Soft Sets in Decision-Making Based on an Ideal Solution. Symmetry
**2017**, 9, 246. [Google Scholar] [CrossRef] - Ashraf, S.; Mahmood, T.; Abdullah, S.; Khan, Q. Different approaches to multi-criteria group decision making problems for picture fuzzy environment. Bull. Braz. Math. Soc. New Ser.
**2018**, 1–25. [Google Scholar] [CrossRef] - Ashraf, S.; Abdullah, S.; Qadir, A. Novel concept of cubic picture fuzzy sets. J. New Theory
**2018**, 24, 59–72. [Google Scholar] - Zeng, S.; Asharf, S.; Arif, M.; Abdullah, S. Application of Exponential Jensen Picture Fuzzy Divergence Measure in Multi-Criteria Group Decision Making. Mathematics
**2019**, 7, 191. [Google Scholar] [CrossRef] - Muhammad, Q.; Abdullah, S.; Asharf, S. Solution of multi-criteria group decision making problem based on picture linguistic informations. Int. J. Algebra Stat.
**2019**, 8, 1–11. [Google Scholar] - Jana, C.; Senapati, T.; Pal, M.; Yager, R.R. Picture fuzzy Dombi aggregation operator: Application to MADM process. Appl. Soft Comput. J.
**2019**, 74, 99–109. [Google Scholar] [CrossRef] - Chen, J.; Ye, J. Some single-valued neutrosophic Dombi weighted aggregation operators for multiple attribute decision-making. Symmetry
**2017**, 9, 82. [Google Scholar] [CrossRef] - Liu, P.; Liu, J.; Chen, S.M. Some intuitionistic fuzzy Dombi bonferroni mean operators and their application to multi-attribute group decision making. J. Oper. Res. Soc.
**2018**, 69, 1–24. [Google Scholar] [CrossRef]

$\stackrel{\u02c7}{\mathit{X}}$ | ${\mathit{h}}_{1}$ | ${\mathit{h}}_{2}$ | ${\mathit{h}}_{3}$ | ${\mathit{h}}_{5}$ |
---|---|---|---|---|

${f}_{1}$ | (0.7, 0.1, 0.1) | (0.5, 0.1, 0.3) | (0.4, 0.1, 0.5) | (0.5, 0.2, 0.2) |

${f}_{2}$ | (0.3, 0.2, 0.4) | (0.3, 0.2, 0.4) | (0.1, 0.2, 0.5) | (0.3, 0.1, 0.4) |

${f}_{3}$ | (0.1, 0.5, 0.3) | (0.2, 0.3, 0.4) | (0.5, 0.1, 0.3) | (0.6, 0.1, 0.2) |

${f}_{4}$ | (0.4, 0.1, 0.3) | (0.6, 0.2, 0.2) | (0.4, 0.1, 0.5) | (0.3, 0.2, 0.3) |

${f}_{5}$ | (0.2, 0.5, 0.2) | (0.5, 0.2, 0.3) | (0.7, 0.1, 0.2) | (0.4, 0.1, 0.3) |

${f}_{6}$ | (0.6, 0.1, 0.2) | (0.6, 0.1, 0.2) | (0.3, 0.2, 0.4) | (0.2, 0.1, 0.5) |

**Table 2.**The generalized picture fuzzy soft set, $GPFSS=(\stackrel{\u02c7}{F},\stackrel{\u02c7}{A},\rho )$.

$\stackrel{\u02c7}{\mathit{X}}$ | ${\mathit{h}}_{1}$ | ${\mathit{h}}_{2}$ | ${\mathit{h}}_{3}$ | ${\mathit{h}}_{5}$ |
---|---|---|---|---|

${f}_{1}$ | (0.7, 0.1, 0.1) | (0.5, 0.1, 0.3) | (0.4, 0.1, 0.5) | (0.5, 0.2, 0.2) |

${f}_{2}$ | (0.3, 0.2, 0.4) | (0.3, 0.2, 0.4) | (0.1, 0.2, 0.5) | (0.3, 0.1, 0.4) |

${f}_{3}$ | (0.1, 0.5, 0.3) | (0.2, 0.3, 0.4) | (0.5, 0.1, 0.3) | (0.6, 0.1, 0.2) |

${f}_{4}$ | (0.4, 0.1, 0.3) | (0.6, 0.2, 0.2) | (0.4, 0.1, 0.5) | (0.3, 0.2, 0.3) |

${f}_{5}$ | (0.2, 0.5, 0.2) | (0.5, 0.2, 0.3) | (0.7, 0.1, 0.2) | (0.4, 0.1, 0.3) |

${f}_{6}$ | (0.6, 0.1, 0.2) | (0.6, 0.1, 0.2) | (0.3, 0.2, 0.4) | (0.2, 0.1, 0.5) |

$\rho $ | (0.3, 0.3, 0.2) | (0.5, 0.2, 0.3) | (0.2, 0.2, 0.5) | (0.7, 0.1, 0.2) |

$\stackrel{\u02c7}{\mathit{X}}$ | ${\mathit{h}}_{1}$ | ${\mathit{h}}_{3}$ | ${\mathit{h}}_{4}$ |
---|---|---|---|

${f}_{1}$ | (0.4, 0.2, 0.1) | (0.6, 0.1, 0.1) | (0.6, 0.2, 0.1) |

${f}_{2}$ | (0.5, 0.2, 0.2) | (0.6, 0.1, 0.2) | (0.5, 0.2, 0.2) |

${f}_{3}$ | (0.3, 0.3, 0.3) | (0.2, 0.3, 0.4) | (0.4, 0.4, 0.2) |

${f}_{4}$ | (0.2, 0.2, 0.5) | (0.3, 0.5, 0.1) | (0.2, 0.4, 0.2) |

$\rho $ | (0.7, 0.1, 0.1) | (0.5, 0.3, 0.1) | (0.3, 0.2, 0.4) |

$\stackrel{\u02c7}{\mathit{X}}$ | ${\mathit{h}}_{1}$ | ${\mathit{h}}_{2}$ | ${\mathit{h}}_{3}$ | ${\mathit{h}}_{5}$ |
---|---|---|---|---|

${f}_{1}$ | (0.3, 0.5, 0.1) | (0.4, 0.5, 0.1) | (0.7, 0.1, 0.1) | (0.3, 0.2, 0.4) |

${f}_{2}$ | (0.4, 0.3, 0.2) | (0.2, 0.1, 0.5) | (0.2, 0.2, 0.5) | (0.6, 0.1, 0.1) |

${f}_{3}$ | (0.2, 0.3, 0.4) | (0.3, 0.4, 0.2) | (0.4, 0.1, 0.3) | (0.5, 0.1, 0.3) |

${f}_{4}$ | (0.1, 0.5, 0.3) | (0.1, 0.6, 0.1) | (0.1, 0.7, 0.1) | (0.2, 0.2, 0.5) |

$\rho $ | (0.5, 0.2, 0.3) | (0.3, 0.4, 0.2) | (0.2, 0.3, 0.4) | (0.1, 0.4, 0.4) |

**Table 5.**The $GPFSS$ $({\stackrel{\u02c7}{H}}_{3},\stackrel{\u02c7}{A}\cap \stackrel{\u02c7}{B},{\tau}_{3})=(\stackrel{\u02c7}{F},\stackrel{\u02c7}{A},\rho ){\bigsqcup}_{R}(\stackrel{\u02c7}{G},\stackrel{\u02c7}{B},\sigma )$.

$\stackrel{\u02c7}{\mathit{X}}$ | ${\mathit{h}}_{1}$ | ${\mathit{h}}_{3}$ |
---|---|---|

${f}_{1}$ | (0.4, 0.2, 0.1) | (0.7, 0.1, 0.1) |

${f}_{2}$ | (0.5, 0.2, 0.2) | (0.6, 0.2, 0.2) |

${f}_{3}$ | (0.3, 0.3, 0.3) | (0.4, 0.1, 0.3) |

${f}_{4}$ | (0.2, 0.2, 0.3) | (0.3, 0.5, 0.1) |

${\tau}_{3}$ | (0.7, 0.1, 0.1) | (0.5, 0.3, 0.1) |

**Table 6.**The $GPFSS$ $({\stackrel{\u02c7}{H}}_{4},\stackrel{\u02c7}{A}\cap \stackrel{\u02c7}{B},{\tau}_{4})=(\stackrel{\u02c7}{F},\stackrel{\u02c7}{A},\rho ){\sqcap}_{R}(\stackrel{\u02c7}{G},\stackrel{\u02c7}{B},\sigma )$.

$\stackrel{\u02c7}{\mathit{X}}$ | ${\mathit{h}}_{1}$ | ${\mathit{h}}_{3}$ |
---|---|---|

${f}_{1}$ | (0.3, 0.2, 0.1) | (0.6, 0.1, 0.1) |

${f}_{2}$ | (0.4, 0.2, 0.2) | (0.2, 0.1, 0.5) |

${f}_{3}$ | (0.2, 0.3, 0.4) | (0.2, 0.1, 0.4) |

${f}_{4}$ | (0.1, 0.2, 0.5) | (0.1, 0.5, 0.1) |

${\tau}_{4}$ | (0.5, 0.1, 0.3) | (0.2, 0.3, 0.4) |

**Table 7.**The $GPFSS$ $\Gamma =(\stackrel{\u02c7}{H},\stackrel{\u02c7}{A}\cup \stackrel{\u02c7}{B},\tau )=(\stackrel{\u02c7}{F},\stackrel{\u02c7}{A},\rho ){\sqcap}_{\u03f5}(\stackrel{\u02c7}{G},\stackrel{\u02c7}{B},\sigma )$.

$\stackrel{\u02c7}{\mathit{X}}$ | ${\mathit{h}}_{1}$ | ${\mathit{h}}_{2}$ | ${\mathit{h}}_{3}$ | ${\mathit{h}}_{4}$ | ${\mathit{h}}_{5}$ |
---|---|---|---|---|---|

${f}_{1}$ | (0.3, 0.2, 0.1) | (0.4, 0.5, 0.1) | (0.6, 0.1, 0.1) | (0.6, 0.2, 0.1) | (0.3, 0.2, 0.4) |

${f}_{2}$ | (0.4, 0.2, 0.2) | (0.2, 0.1, 0.5) | (0.2, 0.1, 0.5) | (0.5, 0.2, 0.2) | (0.6, 0.1, 0.1) |

${f}_{3}$ | (0.2, 0.3, 0.4) | (0.3, 0.4, 0.2) | (0.2, 0.1, 0.4) | (0.4, 0.4, 0.2) | (0.5, 0.1, 0.3) |

${f}_{4}$ | (0.1, 0.2, 0.5) | (0.1, 0.6, 0.1) | (0.1, 0.5, 0.1) | (0.2, 0.4, 0.2) | (0.2, 0.2, 0.5) |

$\rho $ | (0.5, 0.1, 0.3) | (0.3, 0.4, 0.2) | (0.2, 0.3, 0.4) | (0.3, 0.2, 0.4) | (0.1, 0.4, 0.4) |

$\stackrel{\u02c7}{\mathit{X}}$ | ${\mathit{h}}_{1}$ | ${\mathit{h}}_{2}$ | ${\mathit{h}}_{3}$ | ${\mathit{h}}_{4}$ | ${\mathit{h}}_{5}$ |
---|---|---|---|---|---|

$\tau $ | $(0.5,0.1,0.3)$ | (0.3, 0.4, 0.2) | (0.2, 0.3, 0.4) | (0.3, 0.2, 0.4) | (0.1, 0.4, 0.4) |

${\stackrel{\u02c7}{\delta}}_{\tau ({h}_{j})}$ | 0.65 | 0.75 | 0.55 | 0.55 | 0.55 |

${\stackrel{\u02c7}{\omega}}_{j}$ | 0.2131 | 0.2459 | 0.1803 | 0.1803 | 0.1803 |

$\stackrel{\u02c7}{\mathit{X}}$ | $\mathit{DAPFDVs}$ | $\stackrel{\u02c7}{\mathsf{\Theta}}({\mathit{W}}_{\mathbf{\Gamma}}({\mathit{f}}_{\mathit{i}}))$ |
---|---|---|

${f}_{1}$ | (0.46622, 0.19367, 0.11564) | 0.35057 |

${f}_{2}$ | (0.41155, 0.12450, 0.21633) | 0.19521 |

${f}_{3}$ | (0.33521, 0.18581, 0.26914) | 0.06607 |

${f}_{4}$ | (0.13881, 0.31365, 0.16806) | −0.02925 |

$\stackrel{\u02c7}{\mathit{X}}$ | $\mathit{DAPFDVs}$ | $\stackrel{\u02c7}{\mathsf{\Theta}}({\mathit{W}}_{\mathbf{\Gamma}}({\mathit{f}}_{\mathit{i}}))$ |
---|---|---|

${f}_{1}$ | (0.49911, 0.16944, 0.10902) | 0.39009 |

${f}_{2}$ | (0.45682, 0.11842, 0.17800) | 0.27882 |

${f}_{3}$ | (0.36493, 0.15158, 0.25334) | 0.11158 |

${f}_{4}$ | (0.14853, 0.27444, 0.13990) | 0.00864 |

$\stackrel{\u02c7}{\mathit{X}}$ | $\mathit{DAPFDVs}$ | $\stackrel{\u02c7}{\mathsf{\Theta}}({\mathit{W}}_{\mathbf{\Gamma}}({\mathit{f}}_{\mathit{i}}))$ |
---|---|---|

${f}_{1}$ | (0.52317, 0.15332, 0.10610) | 0.41708 |

${f}_{2}$ | (0.48730, 0.11415, 0.15637) | 0.33093 |

${f}_{3}$ | (0.38935, 0.13442, 0.24167) | 0.14768 |

${f}_{4}$ | (0.15735, 0.25227, 0.12725) | 0.03010 |

$\stackrel{\u02c7}{\mathit{X}}$ | $\mathit{APFDVs}$ | $\stackrel{\u02c7}{\mathsf{\Theta}}({\mathit{Y}}_{\mathbf{\Gamma}}({\mathit{f}}_{\mathit{i}}))$ |
---|---|---|

${f}_{1}$ | (0.44918, 0.22115, 0.12843) | 0.32075 |

${f}_{2}$ | (0.38991, 0.13138, 0.26087) | 0.12904 |

${f}_{3}$ | (0.32468, 0.22823, 0.28267) | 0.04201 |

${f}_{4}$ | (0.13742, 0.35031, 0.21348) | −0.07606 |

$\stackrel{\u02c7}{\mathit{X}}$ | ${\mathit{h}}_{2}$ | ${\mathit{h}}_{3}$ | ${\mathit{h}}_{4}$ | ${\mathit{h}}_{6}$ |
---|---|---|---|---|

${f}_{1}$ | (0.4, 0.3, 0.2) | (0.2, 0.3, 0.4) | (0.5, 0.1, 0.3) | (0.3, 0.2, 0.5) |

${f}_{2}$ | (0.3, 0.4, 0.3) | (0.3, 0.2, 0.4) | (0.4, 0.2, 0.3) | (0.4, 0.1, 0.4) |

${f}_{3}$ | (0.2, 0.2, 0.5) | (0.5, 0.3, 0.1) | (0.6, 0.1, 0.2) | (0.4, 0.2, 0.3) |

${f}_{4}$ | (0.5, 0.1, 0.3) | (0.6, 0.1, 0.2) | (0.1, 0.1, 0.7) | (0.2, 0.4, 0.3) |

${f}_{5}$ | (0.6, 0.1, 0.2) | (0.2, 0.2, 0.5) | (0.2, 0.2, 0.5) | (0.2, 0.2, 0.4) |

${f}_{6}$ | (0.2, 0.2, 0.5) | (0.1, 0.3, 0.4) | (0.3, 0.1, 0.4) | (0.5, 0.1, 0.3) |

${f}_{7}$ | (0.3, 0.1, 0.5) | (0.3, 0.3, 0.3) | (0.3, 0.2, 0.4) | (0.3, 0.1, 0.5) |

${f}_{8}$ | (0.4, 0.2, 0.3) | (0.4, 0.2, 0.3) | (0.5, 0.3, 0.1) | (0.4, 0.1, 0.5) |

$\rho $ | (0.4, 0.2, 0.3) | (0.5, 0.1, 0.3) | (0.3, 0.2, 0.5) | (0.2, 0.2, 0.5) |

$\stackrel{\u02c7}{\mathit{X}}$ | ${\mathit{h}}_{1}$ | ${\mathit{h}}_{2}$ | ${\mathit{h}}_{3}$ | ${\mathit{h}}_{5}$ |
---|---|---|---|---|

${f}_{1}$ | (0.1, 0.3, 0.5) | (0.2, 0.3, 0.4) | (0.3, 0.2, 0.5) | (0.4, 0.1, 0.4) |

${f}_{2}$ | (0.5, 0.1, 0.3) | (0.5, 0.1, 0.3) | (0.6, 0.1, 0.2) | (0.3, 0.2, 0.5) |

${f}_{3}$ | (0.2, 0.4, 0.3) | (0.3, 0.3, 0.3) | (0.5, 0.3, 0.1) | (0.5, 0.1, 0.3) |

${f}_{4}$ | (0.6, 0.1, 0.2) | (0.1, 0.3, 0.4) | (0.2, 0.2, 0.5) | (0.2, 0.2, 0.5) |

${f}_{5}$ | (0.2, 0.2, 0.5) | (0.3, 0.2, 0.4) | (0.4, 0.1, 0.3) | (0.6, 0.1, 0.2) |

${f}_{6}$ | (0.5, 0.1, 0.3) | (0.6, 0.2, 0.1) | (0.3, 0.2, 0.3) | (0.3, 0.1, 0.5) |

${f}_{7}$ | (0.4, 0.2, 0.3) | (0.4, 0.3, 0.2) | (0.7, 0.1, 0.2) | (0.4, 0.2, 0.3) |

${f}_{8}$ | (0.3, 0.4, 0.3) | (0.1, 0.5, 0.2) | (0.2, 0.3, 0.4) | (0.3, 0.1, 0.4) |

$\sigma $ | (0.4, 0.2, 0.3) | (0.3, 0.4, 0.3) | (0.4, 0.1, 0.4) | (0.5, 0.2, 0.3) |

**Table 15.**The $GPFSS$ $(\stackrel{\u02c7}{{F}^{{}^{\prime}}},\stackrel{\u02c7}{A},{\rho}^{{}^{\prime}})$.

$\stackrel{\u02c7}{\mathit{X}}$ | ${\mathit{h}}_{2}$ | ${\mathit{h}}_{3}$ | ${\mathit{h}}_{4}$ | ${\mathit{h}}_{6}$ |
---|---|---|---|---|

${f}_{1}$ | (0.4, 0.3, 0.2) | (0.2, 0.3, 0.4) | (0.3, 0.1, 0.5) | (0.3, 0.2, 0.5) |

${f}_{2}$ | (0.3, 0.4, 0.3) | (0.3, 0.2, 0.4) | (0.3, 0.2, 0.4) | (0.4, 0.1, 0.4) |

${f}_{3}$ | (0.2, 0.2, 0.5) | (0.5, 0.3, 0.1) | (0.2, 0.1, 0.6) | (0.4, 0.2, 0.3) |

${f}_{4}$ | (0.5, 0.1, 0.3) | (0.6, 0.1, 0.2) | (0.7, 0.1, 0.1) | (0.2, 0.4, 0.3) |

${f}_{5}$ | (0.6, 0.1, 0.2) | (0.2, 0.2, 0.5) | (0.5, 0.2, 0.2) | (0.2, 0.2, 0.4) |

${f}_{6}$ | (0.2, 0.2, 0.5) | (0.1, 0.3, 0.4) | (0.4, 0.1, 0.3) | (0.5, 0.1, 0.3) |

${f}_{7}$ | (0.3, 0.1, 0.5) | (0.3, 0.3, 0.3) | (0.4, 0.2, 0.3) | (0.3, 0.1, 0.5) |

${f}_{8}$ | (0.4, 0.2, 0.3) | (0.4, 0.2, 0.3) | (0.1, 0.3, 0.5) | (0.4, 0.1, 0.5) |

${\rho}^{{}^{\prime}}$ | (0.4, 0.2, 0.3) | (0.5, 0.1, 0.3) | (0.5, 0.2, 0.3) | (0.2, 0.2, 0.5) |

**Table 16.**The $GPFSS$ $\Gamma =(\stackrel{\u02c7}{{F}^{{}^{\prime}}},\stackrel{\u02c7}{A},{\rho}^{{}^{\prime}}){\sqcap}_{\u03f5}(\stackrel{\u02c7}{G},\stackrel{\u02c7}{B},\sigma )$.

$\stackrel{\u02c7}{\mathit{X}}$ | ${\mathit{h}}_{1}$ | ${\mathit{h}}_{2}$ | ${\mathit{h}}_{3}$ | ${\mathit{h}}_{4}$ | ${\mathit{h}}_{5}$ | ${\mathit{h}}_{6}$ |
---|---|---|---|---|---|---|

${f}_{1}$ | (0.1, 0.3, 0.5) | (0.2, 0.3, 0.4) | (0.2, 0.2, 0.5) | (0.3, 0.1, 0.5) | (0.4, 0.1, 0.4) | (0.3, 0.2, 0.5) |

${f}_{2}$ | (0.5, 0.1, 0.3) | (0.3, 0.1, 0.3) | (0.3, 0.1, 0.4) | (0.3, 0.2, 0.4) | (0.3, 0.2, 0.5) | (0.4, 0.1, 0.4) |

${f}_{3}$ | (0.2, 0.4, 0.3) | (0.2, 0.2, 0.5) | (0.5, 0.3, 0.1) | (0.2, 0.1, 0.6) | (0.5, 0.1, 0.3) | (0.4, 0.2, 0.3) |

${f}_{4}$ | (0.6, 0.1, 0.2) | (0.1, 0.1, 0.4) | (0.2, 0.1, 0.5) | (0.7, 0.1, 0.1) | (0.2, 0.2, 0.5) | (0.2, 0.4, 0.3) |

${f}_{5}$ | (0.2, 0.2, 0.5) | (0.3, 0.1, 0.4) | (0.2, 0.1, 0.5) | (0.5, 0.2, 0.2) | (0.6, 0.1, 0.2) | (0.2, 0.2, 0.4) |

${f}_{6}$ | (0.5, 0.1, 0.3) | (0.2, 0.2, 0.5) | (0.1, 0.2, 0.4) | (0.4, 0.1, 0.3) | (0.3, 0.1, 0.5) | (0.5, 0.1, 0.3) |

${f}_{7}$ | (0.4, 0.2, 0.3) | (0.3, 0.1, 0.5) | (0.3, 0.1, 0.3) | (0.4, 0.2, 0.3) | (0.4, 0.2, 0.3) | (0.3, 0.1, 0.5) |

${f}_{8}$ | (0.3, 0.4, 0.3) | (0.1, 0.2, 0.3) | (0.2, 0.2, 0.4) | (0.1, 0.3, 0.5) | (0.3, 0.1, 0.4) | (0.4, 0.1, 0.5) |

$\tau $ | (0.4, 0.2, 0.3) | (0.3, 0.2, 0.3) | (0.4, 0.1, 0.4) | (0.5, 0.2, 0.3) | (0.5, 0.2, 0.3) | (0.2, 0.2, 0.5) |

$\stackrel{\u02c7}{\mathit{X}}$ | ${\mathit{h}}_{1}$ | ${\mathit{h}}_{2}$ | ${\mathit{h}}_{3}$ | ${\mathit{h}}_{4}$ | ${\mathit{h}}_{5}$ | ${\mathit{h}}_{6}$ |
---|---|---|---|---|---|---|

$\tau $ | (0.4, 0.2, 0.3) | (0.3, 0.2, 0.3) | (0.4, 0.1, 0.4) | (0.5, 0.2, 0.3) | (0.5, 0.2, 0.3) | (0.2, 0.2, 0.5) |

${\stackrel{\u02c7}{\delta}}_{\tau ({h}_{j})}$ | 0.65 | 0.6 | 0.55 | 0.7 | 0.7 | 0.45 |

${\stackrel{\u02c7}{\omega}}_{j}$ | 0.1780 | 0.1644 | 0.1507 | 0.1918 | 0.1918 | 0.1233 |

$\stackrel{\u02c7}{\mathit{X}}$ | $\mathit{DAPFDVs}$ | $\stackrel{\u02c7}{\mathsf{\Theta}}({\mathit{W}}_{\mathbf{\Gamma}}({\mathit{f}}_{\mathit{i}}))$ |
---|---|---|

${f}_{1}$ | (0.265496, 0.157546, 0.459116) | −0.193620 |

${f}_{2}$ | (0.358828, 0.123732, 0.371826) | −0.012998 |

${f}_{3}$ | (0.358254, 0.160724, 0.263218) | 0.095037 |

${f}_{4}$ | (0.459240, 0.123209, 0.231749) | 0.227491 |

${f}_{5}$ | (0.398662, 0.132723, 0.303522) | 0.095139 |

${f}_{6}$ | (0.362737, 0.118701, 0.365923) | −0.003186 |

${f}_{7}$ | (0.359912, 0.139043, 0.339014) | 0.020899 |

${f}_{8}$ | (0.241239, 0.172092, 0.380549) | −0.139310 |

$\mathit{k}\ge 1$ | Rank |
---|---|

$k=1$ | ${f}_{1}\prec {f}_{8}\prec {f}_{2}\prec {f}_{6}\prec {f}_{7}\prec {f}_{3}\prec {f}_{5}\prec {f}_{4}$ |

$k=2$ | ${f}_{1}\prec {f}_{8}\prec {f}_{2}\prec {f}_{7}\prec {f}_{6}\prec {f}_{5}\prec {f}_{3}\prec {f}_{4}$ |

$k=3$ | ${f}_{1}\prec {f}_{8}\prec {f}_{2}\prec {f}_{7}\prec {f}_{6}\prec {f}_{5}\prec {f}_{3}\prec {f}_{4}$ |

$k=4$ | ${f}_{1}\prec {f}_{8}\prec {f}_{7}\prec {f}_{2}\prec {f}_{6}\prec {f}_{5}\prec {f}_{3}\prec {f}_{4}$ |

$k=5$ | ${f}_{1}\prec {f}_{8}\prec {f}_{7}\prec {f}_{2}\prec {f}_{6}\prec {f}_{5}\prec {f}_{3}\prec {f}_{4}$ |

$k=6$ | ${f}_{1}\prec {f}_{8}\prec {f}_{7}\prec {f}_{2}\prec {f}_{6}\prec {f}_{5}\prec {f}_{3}\prec {f}_{4}$ |

$k=7$ | ${f}_{1}\prec {f}_{8}\prec {f}_{7}\prec {f}_{2}\prec {f}_{6}\prec {f}_{5}\prec {f}_{3}\prec {f}_{4}$ |

$k=8$ | ${f}_{1}\prec {f}_{8}\prec {f}_{7}\prec {f}_{2}\prec {f}_{6}\prec {f}_{5}\prec {f}_{3}\prec {f}_{4}$ |

$k=9$ | ${f}_{1}\prec {f}_{8}\prec {f}_{7}\prec {f}_{2}\prec {f}_{6}\prec {f}_{5}\prec {f}_{3}\prec {f}_{4}$ |

$k=10$ | ${f}_{1}\prec {f}_{8}\prec {f}_{7}\prec {f}_{2}\prec {f}_{6}\prec {f}_{5}\prec {f}_{3}\prec {f}_{4}$ |

$\mathit{k}\ge 1$ | ${\mathit{f}}_{1}$ | ${\mathit{f}}_{2}$ | ${\mathit{f}}_{3}$ | ${\mathit{f}}_{4}$ | ${\mathit{f}}_{5}$ | ${\mathit{f}}_{6}$ | ${\mathit{f}}_{7}$ | ${\mathit{f}}_{8}$ |
---|---|---|---|---|---|---|---|---|

$k=1$ | −0.19362 | −0.01299 | 0.09503 | 0.22749 | 0.09513 | −0.00318 | 0.02089 | −0.13931 |

$k=2$ | −0.16522 | 0.01328 | 0.19178 | 0.36488 | 0.17957 | 0.04482 | 0.03606 | −0.09816 |

$k=3$ | −0.14220 | 0.03890 | 0.25160 | 0.43441 | 0.23462 | 0.07727 | 0.04737 | −0.06694 |

$k=4$ | −0.12349 | 0.06179 | 0.28713 | 0.47314 | 0.26955 | 0.09985 | 0.05584 | −0.04302 |

$k=5$ | −0.11259 | 0.07497 | 0.29812 | 0.48843 | 0.28479 | 0.11045 | 0.05913 | −0.03117 |

$k=6$ | −0.09550 | 0.09665 | 0.32461 | 0.51417 | 0.30931 | 0.12821 | 0.06734 | −0.00906 |

$k=7$ | −0.08501 | 0.10925 | 0.33545 | 0.52632 | 0.32153 | 0.13747 | 0.07132 | 0.00328 |

$k=8$ | −0.07625 | 0.11945 | 0.34360 | 0.53555 | 0.33093 | 0.14474 | 0.07451 | 0.01346 |

$k=9$ | −0.06890 | 0.12778 | 0.34995 | 0.54278 | 0.33839 | 0.15058 | 0.07712 | 0.02194 |

$k=10$ | −0.06270 | 0.13465 | 0.35502 | 0.54859 | 0.34444 | 0.15535 | 0.07928 | 0.02906 |

$\stackrel{\u02c7}{\mathit{X}}$ | $\mathit{DAPFDVs}$ | $\stackrel{\u02c7}{\mathsf{\Theta}}({\mathit{W}}_{\mathbf{\Gamma}}({\mathit{f}}_{\mathit{i}}))$ |
---|---|---|

${f}_{1}$ | (0.258760, 0.176136, 0.461797) | −0.203036 |

${f}_{2}$ | (0.353107, 0.130459, 0.378328) | −0.025221 |

${f}_{3}$ | (0.342684, 0.184364, 0.315812) | 0.026872 |

${f}_{4}$ | (0.402669, 0.135510, 0.282350) | 0.120318 |

${f}_{5}$ | (0.373864, 0.140747, 0.329943) | 0.043921 |

${f}_{6}$ | (0.348055, 0.124409, 0.375813) | −0.027759 |

${f}_{7}$ | (0.358051, 0.147591, 0.347494) | 0.010557 |

${f}_{8}$ | (0.233592, 0.196577, 0.388882) | −0.155289 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Khan, M.J.; Kumam, P.; Ashraf, S.; Kumam, W.
Generalized Picture Fuzzy Soft Sets and Their Application in Decision Support Systems. *Symmetry* **2019**, *11*, 415.
https://doi.org/10.3390/sym11030415

**AMA Style**

Khan MJ, Kumam P, Ashraf S, Kumam W.
Generalized Picture Fuzzy Soft Sets and Their Application in Decision Support Systems. *Symmetry*. 2019; 11(3):415.
https://doi.org/10.3390/sym11030415

**Chicago/Turabian Style**

Khan, Muhammad Jabir, Poom Kumam, Shahzaib Ashraf, and Wiyada Kumam.
2019. "Generalized Picture Fuzzy Soft Sets and Their Application in Decision Support Systems" *Symmetry* 11, no. 3: 415.
https://doi.org/10.3390/sym11030415