# A Single-Valued Neutrosophic Linguistic Combined Weighted Distance Measure and Its Application in Multiple-Attribute Group Decision-Making

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

#### 2.1. Linguistic Set

- (1)
- $Neg({s}_{i})={s}_{-i}$;
- (2)
- ${s}_{i}\le {s}_{j}\iff i\le j$;
- (3)
- $\mathrm{max}({s}_{i},{s}_{j})={s}_{j}$, if $i\le j$;
- (4)
- $\mathrm{min}({s}_{i},{s}_{j})={s}_{i}$, if $i\le j$.

- (1)
- ${s}_{\alpha}\oplus {s}_{\beta}={s}_{\alpha +\beta}$;
- (2)
- $\mu {s}_{\alpha}={s}_{\mu \alpha}$, $\mu \ge 0$;
- (3)
- ${s}_{\alpha}/{s}_{\beta}={s}_{\alpha /\beta}$.

#### 2.2. Single-Valued Neutrosophic Set (SVNS)

**Definition**

**1.**

- (1)
- $y\oplus z=({T}_{y}+{T}_{z}-{T}_{y}\ast {T}_{z},{I}_{y}\ast {T}_{z},{F}_{y}\ast {F}_{z});$
- (2)
- $\lambda y=(1-{(1-{T}_{y})}^{\lambda},{({I}_{y})}^{\lambda},{({F}_{y})}^{\lambda}),\lambda 0;$
- (3)
- ${y}^{\lambda}=({({T}_{y})}^{\lambda},1-{(1-{I}_{y})}^{\lambda},1-{(1-{F}_{y})}^{\lambda})$, $\lambda >0$.

#### 2.3. Single-Valued Neutrosophic Linguistic Set (SVNLS)

**Definition**

**2.**

**Definition**

**3.**

- (1)
- ${y}_{1}\oplus {y}_{2}=\langle {s}_{\theta ({y}_{1})+\theta ({y}_{2})},({T}_{{y}_{1}}+{T}_{{y}_{2}}-{T}_{{y}_{1}}\ast {T}_{{y}_{2}},{I}_{{y}_{1}}\ast {T}_{{y}_{2}},{F}_{{y}_{1}}\ast {F}_{{y}_{2}})\rangle ;$
- (2)
- $\lambda {y}_{1}=\langle {s}_{\lambda \theta ({y}_{1})},(1-{(1-{T}_{{y}_{1}})}^{\lambda},{({I}_{{y}_{1}})}^{\lambda},{({F}_{{y}_{1}})}^{\lambda})\rangle ,$$\lambda >0;$
- (3)
- ${y}_{{}^{1}}^{\lambda}=\langle {s}_{{\theta}^{\lambda}({y}_{1})},({({T}_{{y}_{1}})}^{\lambda},1-{(1-{I}_{{y}_{1}})}^{\lambda},1-{(1-{F}_{{y}_{1}})}^{\lambda})\rangle $, $\lambda >0$.

**Definition**

**4.**

**Definition**

**5.**

**Definition**

**6.**

**Definition**

**7.**

## 3. SVNL Combined Weighted Distance (SVNLCWD) Operator

**Definition**

**8.**

**Definition**

**9.**

**Example**

**3.1.**

- (1)
- Compute the individual distances $d({y}_{i},{{y}^{\prime}}_{i})(i=1,2,\dots ,5)$ (let $\lambda =1$) according to Equation (5):$$d({y}_{1},{{y}^{\prime}}_{1})=\left|2\times 0.5-5\times 0.2\right|+\left|2\times 0.3-5\times 0.9\right|+\left|2\times 0.4-5\times 0\right|=4.7.$$$$\begin{array}{c}d({y}_{2},{{y}^{\prime}}_{2})=2.4,\text{}d({y}_{3},{{y}^{\prime}}_{3})=1.5,\\ d({y}_{4},{{y}^{\prime}}_{4})=3.2,\text{}d({y}_{5},{{y}^{\prime}}_{5})=7.7.\end{array}$$
- (2)
- Sort the $d({y}_{i},{{y}^{\prime}}_{i})(i=1,2,\dots ,5)$ in decreasing order:$$\begin{array}{c}{D}_{1}=d({y}_{5},{{y}^{\prime}}_{5})=7.7,\text{}{D}_{2}=d({y}_{1},{{y}^{\prime}}_{1})=4.7,\text{}{D}_{3}=d({y}_{4},{{y}^{\prime}}_{4})=3.2,\\ {D}_{4}=d({y}_{2},{{y}^{\prime}}_{2})=2.4,\text{}{D}_{5}=d({y}_{3},{{y}^{\prime}}_{3})=1.5.\end{array}$$
- (3)
- Let the weighting vector $\omega ={(0.1,0.15,0.2,0.35,0.2)}^{T}$ and $\delta =0.4,$ calculate the integrated weights ${\overline{w}}_{j}$ according to Equation (10):$$\begin{array}{c}{\overline{w}}_{1}=0.4\times 0.2+(1-0.4)\times 0.15=0.17,{\overline{w}}_{2}=0.4\times 0.1+(1-0.4)\times 0.3=0.22,\\ {\overline{w}}_{3}=0.4\times 0.35+(1-0.4)\times 0.2=0.26,\text{}{\widehat{w}}_{4}=0.4\times 0.15+(1-0.4)\times 0.25=0.21,\\ {\overline{w}}_{5}=0.4\times 0.2+(1-0.4)\times 0.1=0.14.\end{array}$$
- (4)
- Use the SVNLCWD measure defined in Equation (9) to perform the following aggregation:$$\begin{array}{c}SVNLCWD(Y,{Y}^{\prime})\\ =0.17\times 7.7+0.22\times 4.7+0.26\times 3.2+0.21\times 2.4+0.14\times 1.5\\ =3.889\end{array}$$

- If ${w}_{1}=1$, ${w}_{2}=\cdots ={w}_{n}=0$, then max-SVNLWD (SVNLMaxD) is formed.
- If ${w}_{1}=\cdots ={w}_{n-1}=0$, ${w}_{n}=1$, then the min-SVNLWD (SVNLMinD) is obtained.
- The step-SVNLCWD operator is rendered by imposing ${w}_{1}=\cdots ={w}_{k-1}=0$, ${w}_{k}=1$ and ${w}_{k+1}=\cdots ={w}_{n}=0$.

**Theorem**

**1.**

**Theorem**

**2.**

**Theorem**

**3.**

**Theorem**

**4.**

**Theorem**

**5.**

**Theorem**

**6.**

## 4. New MAGDM Method Using the SVNLCWD Operator

**Step 1:**Let each decision-maker (DM) ${e}_{k}(k=1,2,\dots ,t)$ (whose weight is ${\epsilon}_{k}$, meeting ${\epsilon}_{k}\ge 0$ and $\sum _{k=1}^{t}{\epsilon}_{k}=1$) provide his/her evaluation on the attributes expressed by the SVNLVs, and then form the individual matrix ${Y}^{k}={\left({y}_{{}_{ij}}^{(k)}\right)}_{m\times n}$.

**Step 2:**Aggregate all evaluations of the individual DMs into a collective one, and then construct the group matrix:

**Step 3:**Construct the ideal levels for each attribute to establish the ideal scheme (see Table 1).

**Step 4:**Utilize the SVNLCWD to compute the distances between the ideal scheme $I$ and the different alternatives ${C}_{i}(i=1,2,\dots ,m)$.

**Step 5:**Sort all alternatives and identify the best alternative(s) according to the results derived from Step 4.

## 5. An Illustrative Example: Low-Carbon Supplier Selection

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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${\mathit{A}}_{1}$ | ${\mathit{A}}_{2}$ | $\cdots $ | ${\mathit{A}}_{\mathit{n}}$ | |
---|---|---|---|---|

$I$ | ${\tilde{y}}_{1}$ | ${\tilde{y}}_{2}$ | $\dots $ | ${\tilde{y}}_{n}$ |

${\mathit{A}}_{1}$ | ${\mathit{A}}_{2}$ | ${\mathit{A}}_{3}$ | ${\mathit{A}}_{4}$ | |
---|---|---|---|---|

${C}_{1}$ | $\langle {s}_{{}_{5}}^{(1)},(0.7,0.0,0.1)\rangle $ | $\langle {s}_{4}^{(1)},(0.6,0.1,0.2)\rangle $ | $\langle {s}_{{}_{3}}^{(1)},(0.3,0.1,0.2)\rangle $ | $\langle {s}_{6}^{(1)},(0.6,0.1,0.2)\rangle $ |

${C}_{2}$ | $\langle {s}_{{}_{6}}^{(1)},(0.6,0.1,0.2)\rangle $ | $\langle {s}_{{}_{5}}^{(1)},(0.6,0.1,0.2)\rangle $ | $\langle {s}_{{}_{4}}^{(1)},(0.5,0.2,0.2)\rangle $ | $\langle {s}_{3}^{(1)},(0.6,0.2,0.4)\rangle $ |

${C}_{3}$ | $\langle {s}_{{}_{4}}^{(1)},(0.3,0.2,0.3)\rangle $ | $\langle {s}_{{}_{4}}^{(1)},(0.5,0.2,0.3)\rangle $ | $\langle {s}_{{}_{3}}^{(1)},(0.5,0.3,0.1)\rangle $ | $\langle {s}_{5}^{(1)},(0.3,0.5,0.2)\rangle $ |

${C}_{4}$ | $\langle {s}_{{}_{5}}^{(1)},(0.4,0.2,0.3)\rangle $ | $\langle {s}_{{}_{5}}^{(1)},(0.4,0.2,0.3)\rangle $ | $\langle {s}_{3}^{(1)},(0.3,0.2,0.5)\rangle $ | $\langle {s}_{4}^{(1)},(0.5,0.3,0.3)\rangle $ |

${\mathit{A}}_{1}$ | ${\mathit{A}}_{2}$ | ${\mathit{A}}_{3}$ | ${\mathit{A}}_{4}$ | |
---|---|---|---|---|

${C}_{1}$ | $\langle {s}_{{}_{4}}^{(3)},(0.6,0.1,0.2)\rangle $ | $\langle {s}_{4}^{(3)},(0.5,0.2,0.2)\rangle $ | $\langle {s}_{3}^{(3)},(0.4,0.1,0.1)\rangle $ | $\langle {s}_{5}^{(3)},(0.7,0.2,0.1)\rangle $ |

${C}_{2}$ | $\langle {s}_{{}_{5}}^{(3)},(0.5,0.2,0.3)\rangle $ | $\langle {s}_{4}^{(3)},(0.7,0.2,0.2)\rangle $ | $\langle {s}_{5}^{(3)},(0.7,0.2,0.1)\rangle $ | $\langle {s}_{6}^{(3)},(0.4,0.6,0.2)\rangle $ |

${C}_{3}$ | $\langle {s}_{{}_{6}}^{(3)},(0.5,0.1,0.3)\rangle $ | $\langle {s}_{5}^{(3)},(0.6,0.1,0.3)\rangle $ | $\langle {s}_{4}^{(3)},(0.6,0.2,0.1)\rangle $ | $\langle {s}_{4}^{(3)},(0.3,0.6,0.2)\rangle $ |

${C}_{4}$ | $\langle {s}_{{}_{6}}^{(3)},(0.5,0.2,0.3)\rangle $ | $\langle {s}_{6}^{(3)},(0.6,0.2,0.4)\rangle $ | $\langle {s}_{5}^{(3)},(0.2,0.1,0.6)\rangle $ | $\langle {s}_{4}^{(3)},(0.5,0.2,0.3)\rangle $ |

${\mathit{A}}_{1}$ | ${\mathit{A}}_{2}$ | ${\mathit{A}}_{3}$ | ${\mathit{A}}_{4}$ | |
---|---|---|---|---|

${C}_{1}$ | $\langle {s}_{{}_{4}}^{(2)},(0.8,0.1,0.2)\rangle $ | $\langle {s}_{{}_{5}}^{(2)},(0.7,0.2,0.3)\rangle $ | $\langle {s}_{4}^{(2)},(0.4,0.2,0.2)\rangle $ | $\langle {s}_{{}_{6}}^{(2)},(0.6,0.3,0.3)\rangle $ |

${C}_{2}$ | $\langle {s}_{{}_{6}}^{(2)},(0.7,0.2,0.3)\rangle $ | $\langle {s}_{{}_{6}}^{(2)},(0.7,0.2,0.3)\rangle $ | $\langle {s}_{5}^{(2)},(0.6,0.2,0.2)\rangle $ | $\langle {s}_{4}^{(2)},(0.5,0.4,0.2)\rangle $ |

${C}_{3}$ | $\langle {s}_{{}_{6}}^{(2)},(0.4,0.2,0.4)\rangle $ | $\langle {s}_{{}_{6}}^{(2)},(0.6,0.3,0.4)\rangle $ | $\langle {s}_{4}^{(2)},(0.6,0.1,0.3)\rangle $ | $\langle {s}_{5}^{(2)},(0.4,0.4,0.1)\rangle $ |

${C}_{4}$ | $\langle {s}_{{}_{5}}^{(2)},(0.4,0.3,0.4)\rangle $ | $\langle {s}_{{}_{6}}^{(2)},(0.5,0.1,0.2)\rangle $ | $\langle {s}_{5}^{(2)},(0.3,0.1,0.6)\rangle $ | $\langle {s}_{3}^{(2)},(0.7,0.1,0.1)\rangle $ |

${\mathit{A}}_{1}$ | ${\mathit{A}}_{2}$ | ${\mathit{A}}_{3}$ | ${\mathit{A}}_{4}$ | |
---|---|---|---|---|

${C}_{1}$ | $\langle {s}_{{}_{4.37}}^{},(0.714,0.000,0.155)\rangle $ | $\langle {s}_{4.33}^{},(0.611,0.155,0.229)\rangle $ | $\langle {s}_{3.67}^{},(0.365,0.128,0.163)\rangle $ | $\langle {s}_{5.70}^{},(0.633,0.180,0.186)\rangle $ |

${C}_{2}$ | $\langle {s}_{{}_{5.70}}^{},(0.611,0.155,0.258)\rangle $ | $\langle {s}_{4.70}^{},(0.666,0.155,0.229)\rangle $ | $\langle {s}_{2.37}^{},(0.602,0.200,0.162)\rangle $ | $\langle {s}_{4.23}^{},(0.514,0.350,0.258)\rangle $ |

${C}_{3}$ | $\langle {s}_{{}_{5.26}}^{},(0.399,0.163,0.330)\rangle $ | $\langle {s}_{4.96}^{},(0.566,0.186,0.330)\rangle $ | $\langle {s}_{3.37}^{},(0.566,0.185,0.144)\rangle $ | $\langle {s}_{4.70}^{},(0.335,0.491,0.159)\rangle $ |

${C}_{4}$ | $\langle {s}_{{}_{5.30}}^{},(0.432,0.229,0.330)\rangle $ | $\langle {s}_{5.63}^{},(0.450,0.159,0.286)\rangle $ | $\langle {s}_{2.37}^{},(0.271,0.129,0.561)\rangle $ | $\langle {s}_{3.67}^{},(0.578,0.185,0.209)\rangle $ |

${\mathit{A}}_{1}$ | ${\mathit{A}}_{2}$ | ${\mathit{A}}_{3}$ | ${\mathit{A}}_{4}$ | |
---|---|---|---|---|

$I$ | $\langle {s}_{{}_{7}}^{},(0.9,0,0)\rangle $ | $\langle {s}_{{}_{7}}^{},(1,0,0.1)\rangle $ | $\langle {s}_{7}^{},(0.9,0,0.1)\rangle $ | $\langle {s}_{6},(0.9,0.1,0)\rangle $ |

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**MDPI and ACS Style**

Cao, C.; Zeng, S.; Luo, D.
A Single-Valued Neutrosophic Linguistic Combined Weighted Distance Measure and Its Application in Multiple-Attribute Group Decision-Making. *Symmetry* **2019**, *11*, 275.
https://doi.org/10.3390/sym11020275

**AMA Style**

Cao C, Zeng S, Luo D.
A Single-Valued Neutrosophic Linguistic Combined Weighted Distance Measure and Its Application in Multiple-Attribute Group Decision-Making. *Symmetry*. 2019; 11(2):275.
https://doi.org/10.3390/sym11020275

**Chicago/Turabian Style**

Cao, Chengdong, Shouzhen Zeng, and Dandan Luo.
2019. "A Single-Valued Neutrosophic Linguistic Combined Weighted Distance Measure and Its Application in Multiple-Attribute Group Decision-Making" *Symmetry* 11, no. 2: 275.
https://doi.org/10.3390/sym11020275