# Fuzzy Parameterized Complex Neutrosophic Soft Expert Set for Decision under Uncertainty

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Definition**

**1.**

**Definition**

**2.**

- 1.
- The complement of A, denoted as $\tilde{c}(A)$, is specified by functions:${T}_{\tilde{c}(A)}(u)={p}_{\tilde{c}(A)}(u).{e}^{j{\mu}_{\tilde{c}(A)}(u)}={r}_{A}(u).{e}^{j(2\pi -{\mu}_{A}(u))}$,${I}_{\tilde{c}(A)}(u)={q}_{\tilde{c}(A)}(u).{e}^{j{\nu}_{\tilde{c}(A)}(u)}=(1-{q}_{A}(u)).{e}^{j(2\pi -{\nu}_{A}(u))}$, and${F}_{\tilde{c}(A)}(u)={r}_{\tilde{c}(A)}(u).{e}^{j{\omega}_{\tilde{c}(A)}(u)}={p}_{A}(u).{e}^{j(2\pi -{\omega}_{A}(u))}$.
- 2.
- A is said to be complex neutrosophic subset of B ($A\subseteq B$) if and only if the following conditions are satisfied:(a) ${T}_{A}(u)\le {T}_{B}(u)$ such that ${p}_{A}(u)\le {p}_{B}(u)$ and ${\mu}_{A}(u)\le {\mu}_{B}(u)$.(b) ${I}_{A}(u)\ge {I}_{B}(u)$ such that ${q}_{A}(u)\ge {q}_{B}(u)$ and ${\nu}_{A}(u)\ge {\nu}_{B}(u)$.(c) ${F}_{A}(u)\ge {F}_{B}(u)$ such that ${r}_{A}(u)\ge {r}_{B}(u)$ and ${\omega}_{A}(u)\ge {\omega}_{B}(u)$.
- 3.
- The union(intersection) of A and B, denoted as $A\cup (\cap )B$ and the truth membership function ${T}_{A\cup (\cap )B}(u)$, the indeterminacy membership function ${I}_{A\cup (\cap )B}(u)$, and the falsity membership function ${F}_{A\cup (\cap )B}(u)$ are defined as:${T}_{A\cup (\cap )B}(u)=\left[({p}_{A}(u)\vee (\wedge ){p}_{B}(u))\right].{e}^{j({\mu}_{A}(u)\vee (\wedge ){\mu}_{B}(u))},$${I}_{A\cup (\cap )B}(u)=\left[({q}_{A}(u)\wedge (\vee ){q}_{B}(u))\right].{e}^{j({\nu}_{A}(u)\wedge (\vee ){\nu}_{B}(u))}$ and${F}_{A\cup (\cap )B}(u)=\left[({r}_{A}(u)\wedge (\vee ){r}_{B}(u))\right].{e}^{j({\omega}_{A}(u)\wedge (\vee ){\omega}_{B}(u))}$,where ∨ = max and ∧ = min.

**Definition**

**3.**

**Definition**

**4.**

## 3. Fuzzy Parameterized Complex Neutrosophic Soft Expert Set

**Definition**

**5.**

**Example**

**1.**

**Definition**

**6.**

**Definition**

**7.**

**Definition**

**8.**

**Definition**

**9.**

**Definition**

**10.**

**Example**

**2.**

**Proposition**

**1.**

**Proof.**

**Definition**

**11.**

**Definition**

**12.**

**Example**

**3.**

**Proposition**

**2.**

- 1.
- $({(H,A)}_{\mathrm{\Gamma}}\phantom{\rule{4pt}{0ex}}\tilde{\cup}\phantom{\rule{4pt}{0ex}}{(G,B)}_{\Delta})\tilde{\cap}{(S,W)}_{\mathrm{\Theta}}=({(H,A)}_{\mathrm{\Gamma}}\phantom{\rule{4pt}{0ex}}\tilde{\cap}\phantom{\rule{4pt}{0ex}}{(S,W)}_{\mathrm{\Theta}})\tilde{\cup}({(G,B)}_{\Delta}\tilde{\cap}{(S,W)}_{\mathrm{\Theta}});$
- 2.
- $({(H,A)}_{\mathrm{\Gamma}}\phantom{\rule{4pt}{0ex}}\tilde{\cap}\phantom{\rule{4pt}{0ex}}{(G,B)}_{\Delta})\tilde{\cup}{(S,W)}_{\mathrm{\Theta}}=({(H,A)}_{\mathrm{\Gamma}}\phantom{\rule{4pt}{0ex}}\tilde{\cup}\phantom{\rule{4pt}{0ex}}{(S,W)}_{\mathrm{\Theta}})\tilde{\cap}({(G,B)}_{\Delta}\tilde{\cup}{(S,W)}_{\mathrm{\Theta}})$.

**Proof.**

## 4. MAPPING ON FP-CNSESs

**Definition**

**13.**

**Definition**

**14.**

**Definition**

**15.**

**Definition**

**16.**

**Definition**

**17.**

**Proposition**

**3.**

- 1.
- $F(\tilde{\varphi})=\tilde{\varphi}$.
- 2.
- $F(\tilde{\psi})=\tilde{\psi}$.
- 3.
- $F\left({\left(H,A\right)}_{\mathrm{\Gamma}}\tilde{\cup}{\left({H}^{\prime},{A}^{\prime}\right)}_{\Delta}\right)=F{\left(H,A\right)}_{\mathrm{\Gamma}}\tilde{\cup}F{\left({H}^{\prime},{A}^{\prime}\right)}_{\Delta}$.
- 4.
- $F\left({\left(H,A\right)}_{\mathrm{\Gamma}}\tilde{\cap}{\left({H}^{\prime},{A}^{\prime}\right)}_{\Delta}\right)\subseteq F{\left(H,A\right)}_{\mathrm{\Gamma}}\tilde{\cap}F{\left({H}^{\prime},{A}^{\prime}\right)}_{\Delta}$.
- 5.
- If ${\left(H,A\right)}_{\mathrm{\Gamma}}\subseteq {\left({H}^{\prime},{A}^{\prime}\right)}_{\Delta}$, then $F{\left(H,A\right)}_{\mathrm{\Gamma}}\subseteq F{\left({H}^{\prime},{A}^{\prime}\right)}_{\Delta}$.

**Proof.**

**Proposition**

**4.**

- 1.
- ${F}^{-1}(\tilde{\varphi})=\tilde{\varphi}$.
- 2.
- ${F}^{-1}(\tilde{\psi})=\tilde{\psi}$.
- 3.
- ${F}^{-1}\left({\left(G,B\right)}_{\Sigma}\tilde{\cup}{\left({G}^{\prime},{B}^{\prime}\right)}_{\mathsf{\Omega}}\right)={F}^{-1}{\left(G,B\right)}_{\Sigma}\tilde{\cup}{F}^{-1}{\left({G}^{\prime},{B}^{\prime}\right)}_{\mathsf{\Omega}}$.
- 4.
- ${F}^{-1}\left({\left(G,B\right)}_{\Sigma}\tilde{\cap}{\left({G}^{\prime},{B}^{\prime}\right)}_{\mathsf{\Omega}}\right)={F}^{-1}{\left(G,B\right)}_{\Sigma}\tilde{\cap}{F}^{-1}{\left({G}^{\prime},{B}^{\prime}\right)}_{\mathsf{\Omega}}$.
- 5.
- If ${\left(G,B\right)}_{\Sigma}\subseteq {\left({G}^{\prime},{B}^{\prime}\right)}_{\mathsf{\Omega}}$, then ${F}^{-1}{\left(G,B\right)}_{\Sigma}\subseteq {F}^{-1}{\left({G}^{\prime},{B}^{\prime}\right)}_{\mathsf{\Omega}}$.

**Proof.**

## 5. An Application of Fuzzy Parameterized Complex Neutrosophic Soft Expert Set

**Example**

**4.**

Algorithm 1: Fuzzy parameterized complex neutrosophic soft expert method (FP-CNSEM). |

1. Input the FP-CNSES ${(H,A)}_{\mathrm{\Gamma}}$ 2. Convert the FP-CNSES ${(H,A)}_{\mathrm{\Gamma}}$ to the FPSVNSES ${(\widehat{H},A)}_{\mathrm{\Gamma}}$ by obtaining the weighted aggregation values of ${T}_{{\widehat{H}}_{\mathrm{\Gamma}}({a}_{i})}^{}({u}_{j}),{I}_{{\widehat{H}}_{\mathrm{\Gamma}}({a}_{i})}^{}({u}_{j})$ and ${F}_{{\widehat{H}}_{\mathrm{\Gamma}}({a}_{i})}^{}({u}_{j})$, $\forall {a}_{i}\in A$ and $\forall {u}_{j}\in U$ as the following formulas:
$${T}_{{\widehat{H}}_{\mathrm{\Gamma}}({a}_{i})}^{}({u}_{j})={w}_{1}{p}_{{H}_{\mathrm{\Gamma}}({a}_{i})}^{}({u}_{j})+{w}_{2}(1/2\pi ){\mu}_{{H}_{\mathrm{\Gamma}}({a}_{i})}({u}_{j}),$$
$${I}_{{\widehat{H}}_{\mathrm{\Gamma}}({a}_{i})}^{}({u}_{j})={w}_{1}{q}_{{H}_{\mathrm{\Gamma}}({a}_{i})}^{}({u}_{j})+{w}_{2}(1/2\pi ){\nu}_{{H}_{\mathrm{\Gamma}}({a}_{i})}({u}_{j}),$$
$${F}_{{\widehat{H}}_{\mathrm{\Gamma}}({a}_{i})}^{}({u}_{j})={w}_{1}{r}_{{H}_{\mathrm{\Gamma}}({a}_{i})}^{}({u}_{j})+{w}_{2}(1/2\pi ){\omega}_{{H}_{\mathrm{\Gamma}}({a}_{i})}({u}_{j}),$$
3. Find the values of ${c}_{ij}={T}_{{\widehat{H}}_{\mathrm{\Gamma}}({a}_{i})}^{}({u}_{j})-{I}_{{\widehat{H}}_{\mathrm{\Gamma}}({a}_{i})}^{}({u}_{j})-{F}_{{\widehat{H}}_{\mathrm{\Gamma}}({a}_{i})}^{}({u}_{j}),\forall {u}_{j}\in U$ and $\forall {a}_{i}\in A$. 4. Compute the score of each element ${u}_{j}\phantom{\rule{4pt}{0ex}}\in U$ by the following formulas:
$${K}_{j}=\sum _{x\in X}\sum _{i=1}^{n}{c}_{ij}({\mu}_{\mathrm{\Gamma}}({e}_{i})),\phantom{\rule{4pt}{0ex}}{S}_{j}=\sum _{x\in X}\sum _{i=1}^{n}{c}_{ij}({\mu}_{\mathrm{\Gamma}}({e}_{i}))$$
5. Find the values of the score ${r}_{j}={K}_{j}-{S}_{j}$ for each element ${u}_{j}\phantom{\rule{4pt}{0ex}}\in U$. 6. Determine the value of the highest score $m=ma{x}_{{u}_{j\phantom{\rule{4pt}{0ex}}\in U}}\left\{{r}_{j}\right\}$. Then, the decision is to choose element ${u}_{j}$ as the optimal solution to the problem. If there are more than one elements with the highest ${r}_{j}$ score, then any one of those elements can be chosen as the optimal solution. |

## 6. Comparison between FP-CNSES and Other Existing Methods

## 7. Weighted Fuzzy Parameterized Complex Neutrosophic Soft Expert Set

**Definition**

**18.**

**Definition**

**19.**

**Example**

**5.**

Algorithm 2: Weighted fuzzy parameterized complex neutrosophic soft expert method (WFP-CNSEM). |

1. Input the WFP-CNSES ${(H,A)}_{\mathrm{\Gamma},W}.$ 2. Convert the WFP-CNSES ${(H,A)}_{\mathrm{\Gamma},W}$ to the WFP-SVNSES ${(\widehat{H},A)}_{\mathrm{\Gamma},W}$ as it was illustrated in step 2 of Algorithm 1. Note that the WFP-CNSES ${(H,A)}_{\mathrm{\Gamma},W}$ and the FP-CNSES ${(H,A)}_{\mathrm{\Gamma}}$ has the same evaluation information and the difference between them lies in the structure of the expert set which does not affect the conversion process. 3. Find the values of ${c}_{ij}$ for agree WFP-SVNSES and disagree WFP-SVNSES respectively, where ${c}_{ij}={T}_{{\widehat{H}}_{\mathrm{\Gamma},W}({a}_{i})}^{}({u}_{j})-{I}_{{\widehat{H}}_{\mathrm{\Gamma},W}({a}_{i})}^{}({u}_{j})-{F}_{{\widehat{H}}_{\mathrm{\Gamma},W}({a}_{i})}^{}({u}_{j}),\forall {u}_{j}\in U$ and $\forall {a}_{i}\in A$. 4. Compute the score of each element ${u}_{j}\phantom{\rule{4pt}{0ex}}\in U$ by the following formulas: ${K}_{j}=\sum _{x\in X}\sum _{i}{c}_{ij}({\mu}_{\mathrm{\Gamma}}(e))({\mu}_{W}(x)),\phantom{\rule{4pt}{0ex}}{S}_{j}=\sum _{x\in X}\sum _{i}{c}_{ij}({\mu}_{\mathrm{\Gamma}}(e))({\mu}_{W}(x)),$ for the agree WFP-SVNSES and disagree WFP-SVNSES, where ${\mu}_{\mathrm{\Gamma}}(e)$ and ${\mu}_{W}(x)$ are the corresponding membership functions of the fuzzy sets $\mathrm{\Gamma}$ and W, respectively. 5. Find the values of the score ${r}_{j}={K}_{j}-{S}_{j}$ for each element ${u}_{j}\phantom{\rule{4pt}{0ex}}\in U$. 6. Determine the value of the highest score $m={\mathrm{max}}_{{u}_{j\phantom{\rule{4pt}{0ex}}\in U}}\left\{{r}_{j}\right\}$. Then, the decision is to choose element ${u}_{j}$ as the optimal solution to the problem. |

## 8. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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${\mathit{u}}_{\mathbf{1}}$ | ${\mathit{u}}_{\mathbf{2}}$ | |
---|---|---|

$\left(\frac{{e}_{1}}{0.3},{x}_{1},1\right)$ | $\langle 0.43,0.34,0.17\rangle $ | $\langle 0.67,0.43,0.73\rangle $ |

$\left(\frac{{e}_{1}}{0.3},{x}_{2},1\right)$ | $\langle 0.53,0.46,0.24\rangle $ | $\langle 0.83,0.57,0.53\rangle $ |

$\left(\frac{{e}_{2}}{0.6},{x}_{1},1\right)$ | $\langle 0.33,0.77,0.73\rangle $ | $\langle 0.75,0.41,0.71\rangle $ |

$\left(\frac{{e}_{2}}{0.6},{x}_{2},1\right)$ | $\langle 0.87,0.33,0.49\rangle $ | $\langle 0.43,0.66,0.32\rangle $ |

$\left(\frac{{e}_{3}}{0.8},{x}_{1},1\right)$ | $\langle 0.20,0.30,0.63\rangle $ | $\langle 0.29,0.35,0.38\rangle $ |

$\left(\frac{{e}_{3}}{0.8},{x}_{2},1\right)$ | $\langle 0.31,0.23,0.35\rangle $ | $\langle 0.45,0.34,0.74\rangle $ |

$\left(\frac{{e}_{1}}{0.3},{x}_{1},0\right)$ | $\langle 0.50,0.43,0.48\rangle $ | $\langle 0.83,0.77,0.64\rangle $ |

$\left(\frac{{e}_{1}}{0.3},{x}_{2},0\right)$ | $\langle 0.54,0.49,0.77\rangle $ | $\langle 0.43,0.43,0.44\rangle $ |

$\left(\frac{{e}_{2}}{0.6},{x}_{1},0\right)$ | $\langle 0.23,0.32,0.41\rangle $ | $\langle 0.49,0.71,0.74\rangle $ |

$\left(\frac{{e}_{2}}{0.6},{x}_{2},0\right)$ | $\langle 0.44,0.16,0.48\rangle $ | $\langle 0.58,0.73,0.63\rangle $ |

$\left(\frac{{e}_{3}}{0.8},{x}_{1},0\right)$ | $\langle 0.43,0.47,0.33\rangle $ | $\langle 0.38,0.23,0.29\rangle $ |

$\left(\frac{{e}_{3}}{0.8},{x}_{2},0\right)$ | $\langle 0.54,0.27,0.45\rangle $ | $\langle 0.09,0.46,0.16\rangle $ |

U | ${\mathit{u}}_{\mathbf{1}}$ | ${\mathit{u}}_{\mathbf{2}}$ |
---|---|---|

$\left(\frac{{e}_{1}}{0.3},{x}_{1}\right)$ | −0.08 | −0.49 |

$\left(\frac{{e}_{1}}{0.3},{x}_{2}\right)$ | −0.17 | −0.27 |

$\left(\frac{{e}_{2}}{0.6},{x}_{1}\right)$ | −1.17 | −0.37 |

$\left(\frac{{e}_{2}}{0.6},{x}_{2}\right)$ | 0.05 | −0.55 |

$\left(\frac{{e}_{3}}{0.8},{x}_{1}\right)$ | −0.73 | −0.44 |

$\left(\frac{{e}_{3}}{0.8},{x}_{2}\right)$ | −0.27 | −0.63 |

${K}_{j}=\sum _{x\in X}\sum _{i=1}^{3}{c}_{ij}({\mu}_{\mathrm{\Gamma}}({e}_{i}))$ | ${K}_{1}=-1.547$ | ${K}_{2}=-1.636$ |

U | ${\mathit{u}}_{\mathbf{1}}$ | ${\mathit{u}}_{\mathbf{2}}$ |
---|---|---|

$\left(\frac{{e}_{1}}{0.3},{x}_{1}\right)$ | −0.41 | −0.58 |

$\left(\frac{{e}_{1}}{0.3},{x}_{2}\right)$ | −0.72 | −0.44 |

$\left(\frac{{e}_{2}}{0.6},{x}_{1}\right)$ | −0.50 | −0.96 |

$\left(\frac{{e}_{2}}{0.6},{x}_{2}\right)$ | −0.20 | −0.78 |

$\left(\frac{{e}_{3}}{0.8},{x}_{1}\right)$ | −0.37 | −0.14 |

$\left(\frac{{e}_{3}}{0.8},{x}_{2}\right)$ | −0.18 | −0.53 |

${S}_{j}=\sum _{x\in X}\sum _{i=1}^{3}{c}_{ij}({\mu}_{\mathrm{\Gamma}}({e}_{i}))$ | ${S}_{1}=-1.199$ | ${S}_{2}=-1.886$ |

$\mathit{K}}_{\mathit{j}}\mathbf{=}\mathbf{\sum}_{\mathit{x}\mathbf{\in}\mathit{X}}\mathbf{\sum}_{\mathit{i}\mathbf{=}\mathbf{1}}^{\mathbf{3}}{\mathit{c}}_{\mathit{ij}}\mathbf{(}{\mathit{\mu}}_{\mathbf{\Gamma}}\mathbf{(}{\mathit{e}}_{\mathit{i}}\mathbf{)}\mathbf{)$ | $\mathit{S}}_{\mathit{j}}\mathbf{=}\mathbf{\sum}_{\mathit{x}\mathbf{\in}\mathit{X}}\mathbf{\sum}_{\mathit{i}\mathbf{=}\mathbf{1}}^{\mathbf{3}}{\mathit{c}}_{\mathit{ij}}\mathbf{(}{\mathit{\mu}}_{\mathbf{\Gamma}}\mathbf{(}{\mathit{e}}_{\mathit{i}}\mathbf{)}\mathbf{)$ | ${\mathit{r}}_{\mathit{j}}\mathbf{=}{\mathit{K}}_{\mathit{j}}\mathbf{-}{\mathit{S}}_{\mathit{j}}$ |
---|---|---|

${K}_{1}=-1.547$ | ${S}_{1}$ = −1.199 | −0.348 |

${K}_{2}=-1.636$ | ${S}_{2}$ = −1.886 | 0.25 |

U | ${\mathit{u}}_{\mathbf{1}}$ | ${\mathit{u}}_{\mathbf{2}}$ |
---|---|---|

$\left(\frac{{e}_{1}}{0.3},\frac{{x}_{1}}{0.5}\right)$ | $-0.08$ | $-0.49$ |

$\left(\frac{{e}_{1}}{0.3},\frac{{x}_{2}}{0.8}\right)$ | $-0.17$ | $-0.27$ |

$\left(\frac{{e}_{2}}{0.6},\frac{{x}_{1}}{0.5}\right)$ | $-1.17$ | $-0.37$ |

$\left(\frac{{e}_{2}}{0.6},\frac{{x}_{2}}{0.8}\right)$ | $0.05$ | $-0.55$ |

$\left(\frac{{e}_{3}}{0.8},\frac{{x}_{1}}{0.5}\right)$ | $-0.73$ | $-0.44$ |

$\left(\frac{{e}_{3}}{0.8},\frac{{x}_{2}}{0.8}\right)$ | $-0.27$ | $-0.63$ |

${K}_{j}=\sum _{x\in X}\sum _{i}{c}_{ij}({\mu}_{\mathrm{\Gamma}}(e))({\mu}_{W}(x))$ | ${K}_{1}=-0.845$ | ${K}_{2}=-1.093$ |

U | ${\mathit{u}}_{\mathbf{1}}$ | ${\mathit{u}}_{\mathbf{2}}$ |
---|---|---|

$\left(\frac{{e}_{1}}{0.3},\frac{{x}_{1}}{0.5}\right)$ | $-0.41$ | $-0.58$ |

$\left(\frac{{e}_{1}}{0.3},\frac{{x}_{2}}{0.8}\right)$ | $-0.72$ | $-0.44$ |

$\left(\frac{{e}_{2}}{0.6},\frac{{x}_{1}}{0.5}\right)$ | $-0.50$ | $-0.96$ |

$\left(\frac{{e}_{2}}{0.6},\frac{{x}_{2}}{0.8}\right)$ | $-0.20$ | $-0.78$ |

$\left(\frac{{e}_{3}}{0.8},\frac{{x}_{1}}{0.5}\right)$ | $-0.37$ | $-0.14$ |

$\left(\frac{{e}_{3}}{0.8},\frac{{x}_{2}}{0.8}\right)$ | $-0.18$ | $-0.53$ |

${S}_{j}=\sum _{x\in X}\sum _{i}{c}_{ij}({\mu}_{\mathrm{\Gamma}}(e))({\mu}_{W}(x))$ | ${S}_{1}=-0.744$ | ${S}_{2}=-1.250$ |

${\mathit{K}}_{\mathit{j}}=\sum _{\mathit{x}\in \mathit{X}}\sum _{\mathit{i}}{\mathit{c}}_{\mathit{ij}}({\mathit{\mu}}_{\mathbf{\Gamma}}(\mathit{e}))({\mathit{\mu}}_{\mathit{W}}(\mathit{x}))$ | ${\mathit{S}}_{\mathit{j}}=\sum _{\mathit{x}\in \mathit{X}}\sum _{\mathit{i}}{\mathit{c}}_{\mathit{ij}}({\mathit{\mu}}_{\mathbf{\Gamma}}(\mathit{e}))({\mathit{\mu}}_{\mathit{W}}(\mathit{x}))$ | ${\mathit{r}}_{\mathit{j}}={\mathit{K}}_{\mathit{j}}-{\mathit{S}}_{\mathit{j}}$ |
---|---|---|

${K}_{1}=-0.845$ | ${S}_{1}$ = $-0.744$ | $-0.101$ |

${K}_{2}=-1.093$ | ${S}_{2}$ = $-1.250$ | $0.157$ |

© 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**

Al-Quran, A.; Hassan, N.; Alkhazaleh, S.
Fuzzy Parameterized Complex Neutrosophic Soft Expert Set for Decision under Uncertainty. *Symmetry* **2019**, *11*, 382.
https://doi.org/10.3390/sym11030382

**AMA Style**

Al-Quran A, Hassan N, Alkhazaleh S.
Fuzzy Parameterized Complex Neutrosophic Soft Expert Set for Decision under Uncertainty. *Symmetry*. 2019; 11(3):382.
https://doi.org/10.3390/sym11030382

**Chicago/Turabian Style**

Al-Quran, Ashraf, Nasruddin Hassan, and Shawkat Alkhazaleh.
2019. "Fuzzy Parameterized Complex Neutrosophic Soft Expert Set for Decision under Uncertainty" *Symmetry* 11, no. 3: 382.
https://doi.org/10.3390/sym11030382