# Single-Valued Neutrosophic Linguistic-Induced Aggregation Distance Measures and Their Application in Investment Multiple Attribute Group Decision Making

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

#### 2.1. The Single-Valued Neutrosophic Set (SVNS)

**Definition**

**1**

**.**Let$u$be an element in a finite set$U$. A single-valued neutrosophic set (SVNS)$A$in$U$can be defined as in (1):

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

#### 2.2. The Linguistic Set

- (1)
- ${s}_{i}\le {s}_{j}\iff i\le j$;
- (2)
- $Neg({s}_{i})={s}_{-i}$;
- (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)
- $\mu {s}_{\alpha}={s}_{\mu \alpha}$, $\mu \ge 0$.

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

**Definition**

**2**

**.**Let$U$be a finite universe set and$\overline{S}$be a continuous linguistic set, a SVNLS$B$in$U$is defined as in (3):

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

**Definition**

**3**

**.**Given two SVNLNs${u}_{i}=\langle {s}_{\theta ({u}_{i})},({T}_{{u}_{i}},{I}_{{u}_{i}},{F}_{{u}_{i}})\rangle (i=1,2)$, their distance measure is defined using the following formula:

**Definition**

**4.**

#### 2.4. The Single-Valued Neutrosophic Linguistic Set (SVNLS)

**Definition**

**5.**

## 3. Single-Valued Neutrosophic Linguistic-Induced Aggregation Distance Measures

#### 3.1. SVNLIOWAD Measure

**Definition**

**6.**

**Theorem**

**1**

**.**If${d}_{i}=d({u}_{i},{u}_{i}^{\prime})=\left|{u}_{i}-{u}_{i}^{\prime}\right|=d$for all$i$, then

**Theorem**

**2**

**.**Let$\underset{i}{\mathrm{min}}(\left|{u}_{i}-{u}_{i}^{\prime}\right|)=x$and$\underset{i}{\mathrm{max}}(\left|{u}_{i}-{u}_{i}^{\prime}\right|)=y$, then

**Theorem**

**3**

**.**If$\left|{u}_{i}-{u}_{i}^{\prime}\right|\ge \left|{v}_{i}-{v}_{i}^{\prime}\right|$for all$i$, then

**Theorem**

**4**

**.**Let$\left(\langle {t}_{1},{u}_{1},{u}_{1}^{\prime}\rangle ,\dots ,\langle {t}_{n},{u}_{n},{u}_{n}^{\prime}\rangle \right)$$(i=1,2,\dots ,n)$be any possible permutation of the argument vector$\left(\langle {t}_{1},{v}_{1},{v}_{1}^{\prime}\rangle ,\dots ,\langle {t}_{n},{v}_{n},{v}_{n}^{\prime}\rangle \right)$, then

- If ${w}_{1}=\cdots ={w}_{n}=\frac{1}{n}$, we obtain the SVNLWD;
- If the ordering of weight ${w}_{j}$ is same as the order-inducing ${t}_{j}$ for all $j$, then the SVNLIOWAD reduces to the SVNLOWAD measure [15];
- If $T=(t,0,\cdots ,0)$, then$$SVNLIOWAD\left(\langle {t}_{1},{u}_{1},{u}_{1}^{\prime}\rangle ,\dots ,\langle {t}_{n},{u}_{n},{u}_{n}^{\prime}\rangle \right)={D}_{1}.$$

**Example**

**1.**

- (1)
- Calculate the individual distances$d({u}_{i},{v}_{i})$$(i=1,2,\dots ,5)$(let$\lambda =1$) according to Equation (5):$$d({u}_{1},{v}_{1})=\left|2\times 0.5-3\times 0.7\right|+\left|2\times 0.3-3\times 0.8\right|+\left|2\times 0.4-3\times 0\right|=3.7.$$Similarly, we get$$d({u}_{2},{v}_{2})=1.5,\text{}d({u}_{3},{v}_{3})=2.4,\text{}d({u}_{4},{v}_{4})=7.7,\text{}d({u}_{5},{v}_{5})=3.2;$$
- (2)
- Sort the$d({u}_{i},{v}_{i})$$(i=1,2,\dots ,5)$according to the decreasing order of the order-inducing variable:$$\begin{array}{c}{D}_{1}=d({u}_{2},{v}_{2})=1.5,\text{}{D}_{2}=d({u}_{5},{v}_{5})=3.2,\text{}{D}_{3}=d({u}_{1},{v}_{1})=3.7,\\ {D}_{4}=d({u}_{3},{v}_{3})=2.4,\text{}d({u}_{4},{v}_{4})=7.7;\end{array}$$
- (3)
- Utilize the SVNLIOWAD operator defined in Equation (8) to perform the following aggregation:$$\begin{array}{c}SVNLIOWAD(U,V)\\ =0.20\times 1.5+0.30\times 3.2+0.15\times 3.7+0.10\times 2.4+0.25\times 7.7=3.71.\end{array}$$

#### 3.2. SVNLWIOWAD Measure

**Definition**

**7.**

**Example**

**2**

**.**To utilize the SVNLWIOWAD operator, we calculated the moderated weight${\varpi}_{j}$defined in Equation (16):

**Theorem**

**5**

**.**Let$Q$be the SVNLWIOWAD operator, if all${d}_{i}=\left|{u}_{i}-{u}_{i}^{\prime}\right|=d$for all$i$, then:

**Proof.**

**Theorem**

**6**

**.**Let$\underset{i}{\mathrm{min}}(\left|{u}_{i}-{u}_{i}^{\prime}\right|)=x$and$\underset{i}{\mathrm{max}}(\left|{u}_{i}-{u}_{i}^{\prime}\right|)=y$, then:

**Proof.**

**Theorem**

**7**

**.**If$\left|{u}_{i}-{u}_{i}^{\prime}\right|\ge \left|{v}_{i}-{v}_{i}^{\prime}\right|$for all$i$, then:

**Proof.**

**Theorem**

**8**

**.**Let$\left(\langle {t}_{1},{u}_{1},{u}_{1}^{\prime}\rangle ,\dots ,\langle {t}_{n},{u}_{n},{u}_{n}^{\prime}\rangle \right)$$(i=1,2,\dots ,n)$be any possible permutation of the argument vector$\left(\langle {t}_{1},{v}_{1},{v}_{1}^{\prime}\rangle ,\dots ,\langle {t}_{n},{v}_{n},{v}_{n}^{\prime}\rangle \right)$, then:

**Proof.**

## 4. A New MAGDM Approach Based on the SVNLWIOWAD Operator

#### 4.1. Steps of the MAGDM Method Based on the SVNWIOWAD Operator

**Step 1:**Each expert ${d}_{k}(k=1,2,\dots ,l)$ (whose weight is ${\epsilon}_{k}$, meeting ${\epsilon}_{k}\ge 0$ and $\sum _{k=1}^{l}{\epsilon}_{k}=1$) provides his or her performance of attributes by the SVNLNs. Afterwards, the individual decision matrix ${U}^{k}={\left({u}_{ij}^{(k)}\right)}_{m\times n}$ is obtained, where ${u}_{ij}^{(k)}$ is the k-th expert’s evaluation of the alternative ${A}_{j}$ with respect to the attribute ${C}_{i}$;

**Step 2:**Aggregate all performances of the individual experts into a collective one and then form the group decision matrix:

**Step 3:**Find the ideal levels for each attribute to construct the ideal scheme, listed in the Table 1;

**Step 4:**Utilize Equation (15) to calculate the distance $SVNLWIOWAD({A}_{i},I)$ between different alternatives ${A}_{i}(i=1,2,\dots ,m)$ and the ideal scheme $I$;

**Step 5:**Rank the alternatives and identify the best one(s) according to $SVNLWIOWAD({A}_{i},I)$, where the smaller the value of $SVNLWIOWAD({A}_{i},I)$, the better the alternative ${A}_{i}(i=1,2,\dots ,m)$.

#### 4.2. An Illustrative Example: Investment Selection

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Zadeh, L.A. Fuzzy sets. Inf. Control.
**1965**, 18, 338–353. [Google Scholar] [CrossRef] [Green Version] - Atanassov, K.T. Intuitionistic fuzzy sets. Fuzzy Sets Syst.
**1986**, 20, 87–96. [Google Scholar] [CrossRef] - Cuong, B.C. Picture fuzzy sets. J. Comput. Sci. Cybern.
**2014**, 30, 409–420. [Google Scholar] - Wei, G.W. Picture fuzzy aggregation operators and their application to multiple attribute decision making. J. Intell. Fuzzy Syst.
**2017**, 33, 713–724. [Google Scholar] [CrossRef] - Herrera, F.; Herrera-Viedma, E. Linguistic decision analysis: Steps for solving decision problems under linguistic information. Fuzzy Sets Syst.
**2000**, 115, 67–82. [Google Scholar] [CrossRef] - Smarandache, F. A Unifying Field in Logics. Neutrosophy: Neutrosophic Probability, Set and Logic; American Research Press: Rehoboth, DE, USA, 1999. [Google Scholar]
- Ye, J. An extended TOPSIS method for multiple attribute group decision making based on single valued neutrosophic linguistic numbers. J. Intell. Fuzzy Syst.
**2015**, 28, 247–255. [Google Scholar] [CrossRef] - Wang, H.; Smarandache, F.; Zhang, Y.Q.; Sunderraman, R. Single valued neutrosophic sets. Multispace Multistruct.
**2010**, 4, 410–413. [Google Scholar] - Hwang, C.L.; Yoon, K. Multiple Attribute Decision Making: Methods and Applications. In A State-of-the-Art Survey; Springer: Berlin, Germany, 1981. [Google Scholar]
- Wang, J.Q.; Yang, Y.; Li, L. Multi-criteria decision-making method based on single-valued neutrosophic linguistic Maclaurin symmetric mean operators. Neural Comput. Appl.
**2018**, 30, 1529–1547. [Google Scholar] [CrossRef] - Chen, J.; Zeng, S.Z.; Zhang, C.H. An OWA Distance-Based, Single-Valued Neutrosophic Linguistic TOPSIS Approach for Green Supplier Evaluation and Selection in Low-Carbon Supply Chains. Int. J. Environ. Res. Public Health
**2018**, 15, 1439. [Google Scholar] [CrossRef] [Green Version] - Wu, Q.; Wu, P.; Zhou, L. Some new Hamacher aggregation operators under single-valued neutrosophic 2-tuple linguistic environment and their applications to multi-attribute group decision making. Comput. Ind. Eng.
**2018**, 116, 144–162. [Google Scholar] [CrossRef] [Green Version] - Kazimieras, Z.E.; Bausys, R.; Lazauskas, M. Sustainable Assessment of Alternative Sites for the Construction of a Waste Incineration Plant by Applying WASPAS Method with Single-Valued Neutrosophic Set. Sustainability
**2015**, 7, 15923–15936. [Google Scholar] [CrossRef] [Green Version] - Garg, H.; Nancy. Linguistic single-valued neutrosophic prioritized aggregation operators and their applications to multiple-attribute group decision-making. J. Ambient. Intell. Humaniz. Comput.
**2018**, 9, 1975–1997. [Google Scholar] [CrossRef] - Cao, C.D.; Zeng, S.Z.; Luo, D.D. A Single-Valued Neutrosophic Linguistic Combined Weighted Distance Measure and Its Application in Multiple-Attribute Group Decision-Making. Symmetry
**2019**, 11, 275. [Google Scholar] [CrossRef] [Green Version] - Xu, Z.S.; Chen, J. Ordered weighted distance measure. J. Syst. Sci. Syst. Eng.
**2008**, 16, 529–555. [Google Scholar] [CrossRef] - Merigó, J.M.; Gil-Lafuente, A.M. New decision-making techniques and their application in the selection of financial products. Inf. Sci.
**2010**, 180, 2085–2094. [Google Scholar] [CrossRef] - Zeng, S.Z.; Xiao, Y. A method based on TOPSIS and distance measures for hesitant fuzzy multiple attribute decision making. Technol. Econ. Dev. Econ.
**2018**, 24, 969–983. [Google Scholar] [CrossRef] [Green Version] - Merigó, J.M.; Casanovas, M. Decision making with distance measures and induced aggregation operators. Comput. Ind. Eng.
**2011**, 60, 66–76. [Google Scholar] [CrossRef] [Green Version] - Zeng, Z.S.; Li, W.; Merigó, J.M. Extended induced ordered weighted averaging distance operators and their application to group decision-making. Int. J. Inf. Technol. Decis. Mak.
**2013**, 12, 1973–6845. [Google Scholar] [CrossRef] - Xian, S.D.; Sun, W.J. Fuzzy linguistic induced Euclidean OWA distance operator and its application in group linguistic decision making. Int. J. Intell. Syst.
**2014**, 29, 478–491. [Google Scholar] [CrossRef] - Zeng, S.Z.; Merigó, J.M.; Palacios-Marques, D.; Jin, H.H.; Gu, F.J. Intuitionistic fuzzy induced ordered weighted averaging distance operator and its application to decision making. J. Intell. Fuzzy Syst.
**2017**, 32, 11–22. [Google Scholar] [CrossRef] - Li, C.G.; Zeng, S.Z.; Pan, T.J.; Zheng, L.N. A method based on induced aggregation operators and distance measures to multiple attribute decision making under 2-tuple linguistic environment. J. Comput. Syst. Sci.
**2014**, 80, 1339–1349. [Google Scholar] [CrossRef] - Ye, J. Multicriteria decision-making method using the correlation coefficient under single-valued neutrosophic environment. Int. J. Gen. Syst.
**2013**, 42, 386–394. [Google Scholar] [CrossRef] - Xu, Z.S. A note on linguistic hybrid arithmetic averaging operator in multiple attribute group decision making with linguistic information. Group Decis. Negot.
**2006**, 15, 593–604. [Google Scholar] [CrossRef] - Yager, R.R.; Filev, D.P. Induced ordered weighted averaging operators. IEEE Trans. Syst. Man Cybern. Part B
**1999**, 29, 141–150. [Google Scholar] [CrossRef] [PubMed] - Yu, L.P.; Zeng, S.Z.; Merigo, J.M.; Zhang, C.H. A new distance measure based on the weighted induced method and its application to Pythagorean fuzzy multiple attribute group decision making. Int. J. Intell. Syst.
**2019**, 34, 1440–1454. [Google Scholar] [CrossRef] - Zhou, L.; Tao, Z.; Chen, H.; Liu, J. Generalized ordered weighted logarithmic harmonic averaging operators and their applications to group decision making. Soft Comput.
**2014**, 19, 715–730. [Google Scholar] [CrossRef] - Aggarwal, M. A new family of induced OWA operators. Int. J. Intell. Syst.
**2015**, 30, 170–205. [Google Scholar] [CrossRef] - Merigó, J.M.; Palacios-Marqués, D.; Soto-Acosta, P. Distance measures, weighted averages, OWA operators and Bonferroni means. Appl. Soft Comput.
**2017**, 50, 356–366. [Google Scholar] [CrossRef] - Balezentis, T.; Streimikiene, D.; Melnikienė, R.; Zeng, S.Z. Prospects of green growth in the electricity sector in Baltic States: Pinch analysis based on ecological footprint. Resour. Conserv. Recycl.
**2019**, 142, 37–48. [Google Scholar] [CrossRef] - Zeng, S.Z.; Mu, Z.M.; Balezentis, T. A novel aggregation method for Pythagorean fuzzy multiple attribute group decision making. Int. J. Intell. Syst.
**2018**, 33, 573–585. [Google Scholar] [CrossRef] - Zeng, S.Z.; Chen, S.M.; Kuo, L.W. Multiattribute decision making based on novel score function of intuitionistic fuzzy values and modified VIKOR method. Inf. Sci.
**2019**, 488, 76–92. [Google Scholar] [CrossRef] - Zeng, S.Z.; Peng, X.M.; Baležentis, T.; Streimikiene, D. Prioritization of low-carbon suppliers based on Pythagorean fuzzy group decision making with self-confidence level. Econ. Res. Ekon. Istraž.
**2019**, 32, 1073–1087. [Google Scholar] [CrossRef]

${\mathit{C}}_{\mathbf{1}}$ | ${\mathit{C}}_{\mathbf{2}}$ | $\mathbf{\cdots}$ | ${\mathit{C}}_{\mathit{n}}$ | |

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

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

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

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

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

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

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

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

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

${A}_{3}$ | $\langle {s}_{4}^{(2)},(0.8,0.1,0.2)\rangle $ | $\langle {s}_{4}^{(2)},(0.4,0.2,0.2)\rangle $ | $\langle {s}_{5}^{(2)},(0.7,0.2,0.3)\rangle $ | $\langle {s}_{6}^{(2)},(0.6,0.3,0.3)\rangle $ |

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

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

${A}_{1}$ | $\langle {s}_{6}^{(3)},(0.5,0.1,0.3)\rangle $ | $\langle {s}_{4}^{(3)},(0.6,0.2,0.1)\rangle $ | $\langle {s}_{5}^{(3)},(0.6,0.1,0.3)\rangle $ | $\langle {s}_{4}^{(3)},(0.3,0.6,0.2)\rangle $ |

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

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

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

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

${A}_{1}$ | $\langle {s}_{5.26},(0.399,0.163,0.330)\rangle $ | $\langle {s}_{3.37},(0.566,0.185,0.144)\rangle $ | $\langle {s}_{4.96},(0.566,0.186,0.330)\rangle $ | $\langle {s}_{4.70},(0.335,0.491,0.159)\rangle $ |

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

${A}_{3}$ | $\langle {s}_{4.37},(0.714,0.000,0.155)\rangle $ | $\langle {s}_{3.67},(0.365,0.128,0.163)\rangle $ | $\langle {s}_{4.33},(0.611,0.155,0.229)\rangle $ | $\langle {s}_{5.70},(0.633,0.180,0.186)\rangle $ |

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

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

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

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

$SVNLWD({A}_{i},I)$ | 6.828 | 5.836 | 5.048 | 6.444 | ${A}_{3}\succ {A}_{2}\succ {A}_{4}\succ {A}_{1}$ |

$SVNLOWAD({A}_{i},I)$ | 6.466 | 5.652 | 4.802 | 6.460 | ${A}_{3}\succ {A}_{2}\succ {A}_{4}\succ {A}_{1}$ |

$SVNLIOWAD({A}_{i},I)$ | 6.770 | 5.788 | 4.833 | 6.460 | ${A}_{3}\succ {A}_{2}\succ {A}_{1}\succ {A}_{4}$ |

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## Share and Cite

**MDPI and ACS Style**

Yu, G.; Zeng, S.; Zhang, C.
Single-Valued Neutrosophic Linguistic-Induced Aggregation Distance Measures and Their Application in Investment Multiple Attribute Group Decision Making. *Symmetry* **2020**, *12*, 207.
https://doi.org/10.3390/sym12020207

**AMA Style**

Yu G, Zeng S, Zhang C.
Single-Valued Neutrosophic Linguistic-Induced Aggregation Distance Measures and Their Application in Investment Multiple Attribute Group Decision Making. *Symmetry*. 2020; 12(2):207.
https://doi.org/10.3390/sym12020207

**Chicago/Turabian Style**

Yu, Guansheng, Shouzhen Zeng, and Chonghui Zhang.
2020. "Single-Valued Neutrosophic Linguistic-Induced Aggregation Distance Measures and Their Application in Investment Multiple Attribute Group Decision Making" *Symmetry* 12, no. 2: 207.
https://doi.org/10.3390/sym12020207