# A Novel Single-Valued Neutrosophic Set Similarity Measure and Its Application in Multicriteria Decision-Making

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

#### 2.1. Neutrosophic Sets

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

**Definition**

**4.**

**Definition**

**5.**

#### 2.2. Single-Valued Neutrosophic Set

**Definition**

**6.**

**Definition**

**7.**

**Definition**

**8.**

**Definition**

**9.**

**Definition**

**10.**

**Definition**

**11.**

#### 2.3. Dempster–Shafer Evidence Theory

**Definition**

**12.**

**Definition**

**13.**

**Definition**

**14.**

#### 2.4. A Correlation Coefficient

**Definition**

**15.**

## 3. A New Similarity Measures for SNVS

**Definition**

**16.**

**Definition**

**17.**

**Step 1**: According to A and B, two groups of BPAs ${m}_{A}$ and ${m}_{B}$ can be obtained by Equation (22);**Step 3**: The similarity measure ${S}_{r}(A,B)$ can be obtained as:$${S}_{r}(A,B)={r}_{BPA}({m}_{A},{m}_{B})$$

**Example**

**1.**

**Step 1**: According to A and B of SVNS, we can know$${T}_{A}\left(x\right)=0.7,{I}_{A}\left(x\right)=0.8,{F}_{A}\left(x\right)=0.2$$$${T}_{B}\left(x\right)=0.6,{I}_{B}\left(x\right)=0.8,{F}_{B}\left(x\right)=0.1$$So, two groups of BPAs ${m}_{A}$ and ${m}_{B}$ can be obtained by Equation (22).$${m}_{A}(\varnothing )=0,\phantom{\rule{0ex}{0ex}}{m}_{A}\left({T}_{x}\right)={T}_{A}\left(x\right)/({T}_{A}\left(x\right)+{F}_{A}\left(x\right)+(1-{I}_{A}\left(x\right)))=0.6364,\phantom{\rule{0ex}{0ex}}{m}_{A}\left({F}_{x}\right)={T}_{A}\left(x\right)/({T}_{A}\left(x\right)+{F}_{A}\left(x\right)+(1-{I}_{A}\left(x\right)))=0.1818,\phantom{\rule{0ex}{0ex}}{m}_{A}({T}_{x},{F}_{x})={T}_{A}\left(x\right)/({T}_{A}\left(x\right)+{F}_{A}\left(x\right)+(1-{I}_{A}\left(x\right)))=0.1818$$$${m}_{B}(\varnothing )=0,\phantom{\rule{0ex}{0ex}}{m}_{B}\left({T}_{x}\right)={T}_{B}\left(x\right)/({T}_{B}\left(x\right)+{F}_{B}\left(x\right)+(1-{I}_{B}\left(x\right)))=0.6667,\phantom{\rule{0ex}{0ex}}{m}_{B}\left({F}_{x}\right)={T}_{B}\left(x\right)/({T}_{B}\left(x\right)+{F}_{B}\left(x\right)+(1-{I}_{B}\left(x\right)))=0.1111,\phantom{\rule{0ex}{0ex}}{m}_{B}({T}_{x},{F}_{x})={T}_{B}\left(x\right)/({T}_{B}\left(x\right)+{F}_{B}\left(x\right)+(1-{I}_{B}\left(x\right)))=0.2222$$**Step 3**: The similarity measure of SNVSs can be obtained as$${S}_{r}(A,B)=0.9965.$$

## 4. Test and Analysis

**Example**

**2.**

**Example**

**3.**

## 5. Multicriteria Decision-Making

**Example**

**4.**

**Step 1**: ${\alpha}_{i}(i=1,2,3,4)$ can be obtained by Equation (12). The computing results are:$$\begin{array}{c}{\alpha}_{1}=\u23290.3268,0.2000,0.3881\u232a\hfill \\ {\alpha}_{2}=\u23290.5627,0.1414,0.2000\u232a\hfill \\ {\alpha}_{3}=\u23290.4375,0.2416,0.2616\u232a\hfill \\ {\alpha}_{4}=\u23290.5746,0.1555,0.1663\u232a\hfill \end{array}$$**Step 2**: Correlation measure between each alternative ${\alpha}_{i}$ and the ideal alternative ${\alpha}^{*}$ were calculated by the method proposed in Section 3. The results are as follows:$$\begin{array}{c}{S}_{r}({\alpha}^{*},{\alpha}_{1})=0.8975\hfill \\ {S}_{r}({\alpha}^{*},{\alpha}_{2})=0.9745\hfill \\ {S}_{r}({\alpha}^{*},{\alpha}_{3})=0.9510\hfill \\ {S}_{r}({\alpha}^{*},{\alpha}_{4})=0.9804\hfill \end{array}$$**Step 3**: According to the results in the second step, the ranking order of four alternatives is:$${A}_{4}>{A}_{2}>{A}_{3}>{A}_{1}$$

**Example**

**5.**

**Example**

**6.**

**Step 1**: According to the data in Table 3, the D matrix can be obtained, which is not listed. Then, ${\alpha}_{i}(i=1,2,\dots ,10)$ can be obtained by Equation (12). The computing results are:$$\begin{array}{c}{\alpha}_{1}=\u23290.1974,0.0234,1\u232a,{\alpha}_{2}=\u23290.1349,0.0301,1\u232a\hfill \\ {\alpha}_{3}=\u23290.1003,0.0674,1\u232a,{\alpha}_{4}=\u23290.1717,0.0101,1\u232a\hfill \\ {\alpha}_{5}=\u23290.0950,0.0367,0.8713\u232a,{\alpha}_{6}=\u23290.1088,0.0260,1\u232a\hfill \\ {\alpha}_{7}=\u23290.2037,0.0125,1\u232a,{\alpha}_{8}=\u23290.1224,0.0193,1\u232a\hfill \\ {\alpha}_{9}=\u23290.1975,0.0137,1\u232a,{\alpha}_{10}=\u23290.1703,0.0113,1\u232a\hfill \end{array}$$**Step 2**: According to SVNS of the real testing samples B, $\beta =\u23290.2347,0,1\u232a$ can be obtained by Equation (12). Additionally, correlation measure between each fault diagnosis problem ${\alpha}_{i}$ and the sample $\beta $ were calculated by the method we proposed in Section 3 and that of Ye [5] individually. Therefore, the two method ranking order of all faults is as follows:$$Ours:{A}_{7}>{A}_{9}>{A}_{1}>{A}_{4}>{A}_{10}>{A}_{2}>{A}_{3}>{A}_{5}>{A}_{8}>{A}_{6}$$$$Y{e}^{\prime}s:{A}_{7}>{A}_{9}>{A}_{1}>{A}_{4}>{A}_{10}>{A}_{2}>{A}_{8}>{A}_{6}>{A}_{5}>{A}_{3}$$

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Smarandache, F. A Unifying Field in Logics: Neutrosophic Logic. Philosophy
**1999**, 8, 1–141. [Google Scholar] - Wang, H.; Smarandache, F.; Sunderraman, R.; Zhang, Y.Q. Interval Neutrosophic Sets and Logic: Theory and Applications in Computing; Infinite Study: Hexis, AZ, USA, 2005. [Google Scholar]
- Wang, H.; Smarandache, F.; Zhang, Y.; Sunderraman, R. Single valued neutrosophic sets. Rev. Air Force Acad.
**2010**, 17, 10–14. [Google Scholar] - Zhang, H.Y.; Wang, J.Q.; Chen, X.H. Interval neutrosophic sets and their application in multicriteria decision making problems. Sci. World J.
**2014**, 2014, 645953. [Google Scholar] [CrossRef] [PubMed] - Ye, J. A multicriteria decision-making method using aggregation operators for simplified neutrosophic sets. J. Intell. Fuzzy Syst.
**2014**, 26, 2459–2466. [Google Scholar] - 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] - Peng, J.J.; Wang, J.Q.; Wang, J.; Zhang, H.Y.; Chen, X.H. Simplified neutrosophic sets and their applications in multi-criteria group decision-making problems. Int. J. Syst. Sci.
**2016**, 47, 2342–2358. [Google Scholar] [CrossRef] - Zhang, H.; Wang, J.; Chen, X. An outranking approach for multi-criteria decision-making problems with interval-valued neutrosophic sets. Neural Comput. Appl.
**2016**, 27, 615–627. [Google Scholar] [CrossRef] - Liu, P.; Wang, Y. Multiple attribute decision-making method based on single-valued neutrosophic normalized weighted Bonferroni mean. Neural Comput. Appl.
**2014**, 25, 2001–2010. [Google Scholar] [CrossRef] - Guo, Y.; Cheng, H.D. New neutrosophic approach to image segmentation. Pattern Recognit.
**2009**, 42, 587–595. [Google Scholar] [CrossRef] - Broumi, S.; Smarandache, F. Correlation coefficient of interval neutrosophic set. Appl. Mech. Mater.
**2013**, 436, 511–517. [Google Scholar] [CrossRef] - Guo, Y.; Sengur, A. A novel color image segmentation approach based on neutrosophic set and modified fuzzy c-means. Circuits Syst. Signal Process.
**2013**, 32, 1699–1723. [Google Scholar] [CrossRef] - Ma, Y.X.; Wang, J.Q.; Wang, J.; Wu, X.H. An interval neutrosophic linguistic multi-criteria group decision-making method and its application in selecting medical treatment options. Neural Comput. Appl.
**2016**, 1–21. [Google Scholar] [CrossRef] - Ye, J. Single-valued neutrosophic similarity measures based on cotangent function and their application in the fault diagnosis of steam turbine. Soft Comput.
**2017**, 21, 1–9. [Google Scholar] [CrossRef] - Ji, P.; Zhang, H.Y.; Wang, J.Q. A projection-based TODIM method under multi-valued neutrosophic environments and its application in personnel selection. Neural Comput. Appl.
**2016**, 1–14. [Google Scholar] [CrossRef] - Fatimah, F.; Rosadi, D.; Hakim, R.F.; Alcantud, J.C.R. Probabilistic soft sets and dual probabilistic soft sets in decision-making. Neural Comput. Appl.
**2017**, 1–11. [Google Scholar] [CrossRef] - Alcantud, J.C.R.; Santos-García, G. A New Criterion for Soft Set Based Decision Making Problems under Incomplete Information; Technical Report; Mimeo: New York, NY, USA, 2015. [Google Scholar]
- Majumdar, P.; Samanta, S.K. On similarity and entropy of neutrosophic sets. J. Intell. Fuzzy Syst. Appl. Eng. Technol.
**2014**, 26, 1245–1252. [Google Scholar] - Zadeh, L.A. A simple view of the Dempster-Shafer theory of evidence and its implication for the rule of combination. AI Mag.
**1986**, 7, 85–90. [Google Scholar] - Jiang, W.; Xie, C.; Luo, Y.; Tang, Y. Ranking Z-numbers with an improved ranking method for generalized fuzzy numbers. J. Intell. Fuzzy Syst.
**2017**, 32, 1931–1943. [Google Scholar] [CrossRef] - Jiang, W.; Wei, B.; Tang, Y.; Zhou, D. Ordered visibility graph average aggregation operator: An application in produced water management. Chaos
**2017**, 27, 023117. [Google Scholar] [CrossRef] [PubMed] - Yang, J.B.; Xu, D.L. Evidential reasoning rule for evidence combination. Artif. Intell.
**2013**, 205, 1–29. [Google Scholar] [CrossRef] - Chin, K.S.; Fu, C. Weighted cautious conjunctive rule for belief functions combination. Inf. Sci.
**2015**, 325, 70–86. [Google Scholar] [CrossRef] - Wang, J.; Xiao, F.; Deng, X.; Fei, L.; Deng, Y. Weighted evidence combination based on distance of evidence and entropy function. Int. J. Distrib. Sens. Netw.
**2016**, 12, 3218784. [Google Scholar] [CrossRef] - Yang, Y.; Liu, Y. Iterative Approximation of Basic Belief Assignment Based on Distance of Evidence. PLoS ONE
**2016**, 11, e0147799. [Google Scholar] [CrossRef] [PubMed] - Yang, Y.; Han, D.; Han, C.; Cao, F. A novel approximation of basic probability assignment based on rank-level fusion. Chin. J. Aeronaut.
**2013**, 26, 993–999. [Google Scholar] [CrossRef] - Yang, Y.; Han, D. A new distance-based total uncertainty measure in the theory of belief functions. Knowl. Based Syst.
**2016**, 94, 114–123. [Google Scholar] [CrossRef] - Deng, X.; Xiao, F.; Deng, Y. An improved distance-based total uncertainty measure in belief function theory. Appl. Intell.
**2017**, 46, 898–915. [Google Scholar] [CrossRef] - Jiang, W.; Zhuang, M.; Xie, C.; Wu, J. Sensing Attribute Weights: A Novel Basic Belief Assignment Method. Sensors
**2017**, 17, 721. [Google Scholar] [CrossRef] [PubMed] - He, Y.; Hu, L.; Guan, X.; Han, D.; Deng, Y. New conflict representation model in generalized power space. J. Syst. Eng. Electron.
**2012**, 23, 1–9. [Google Scholar] [CrossRef] - Jiang, W.; Zhan, J. A modified combination rule in generalized evidence theory. Appl. Intell.
**2017**, 46, 630–640. [Google Scholar] [CrossRef] - Mo, H.; Lu, X.; Deng, Y. A generalized evidence distance. J. Syst. Eng. Electron.
**2016**, 27, 470–476. [Google Scholar] [CrossRef] - Dong, J.; Zhuang, D.; Huang, Y.; Fu, J. Advances in multi-sensor data fusion: Algorithms and applications. Sensors
**2009**, 9, 7771–7784. [Google Scholar] [CrossRef] [PubMed] - Yang, F.; Wei, H. Fusion of infrared polarization and intensity images using support value transform and fuzzy combination rules. Infrared Phys. Technol.
**2013**, 60, 235–243. [Google Scholar] [CrossRef] - Jiang, W.; Xie, C.; Zhuang, M.; Shou, Y.; Tang, Y. Sensor Data Fusion with Z-Numbers and Its Application in Fault Diagnosis. Sensors
**2016**, 16, 1509. [Google Scholar] [CrossRef] [PubMed] - Ma, J.; Liu, W.; Miller, P.; Zhou, H. An evidential fusion approach for gender profiling. Inf. Sci.
**2016**, 333, 10–20. [Google Scholar] [CrossRef] [Green Version] - Islam, M.S.; Sadiq, R.; Rodriguez, M.J.; Najjaran, H.; Hoorfar, M. Integrated Decision Support System for Prognostic and Diagnostic Analyses of Water Distribution System Failures. Water Resour. Manag.
**2016**, 30, 2831–2850. [Google Scholar] [CrossRef] - Jiang, W.; Xie, C.; Zhuang, M.; Tang, Y. Failure Mode and Effects Analysis based on a novel fuzzy evidential method. Appl. Soft Comput.
**2017**, 57, 672–683. [Google Scholar] [CrossRef] - Zhang, X.; Deng, Y.; Chan, F.T.S.; Adamatzky, A.; Mahadevan, S. Supplier selection based on evidence theory and analytic network process. J. Eng. Manuf.
**2016**, 230, 562–573. [Google Scholar] [CrossRef] - Deng, Y. Deng entropy. Chaos Solitons Fractals
**2016**, 91, 549–553. [Google Scholar] [CrossRef] - Deng, X.; Jiang, W.; Zhang, J. Zero-sum matrix game with payoffs of Dempster-Shafer belief structures and its applications on sensors. Sensors
**2017**, 17, 922. [Google Scholar] [CrossRef] [PubMed] - Zhang, X.; Mahadevan, S.; Deng, X. Reliability analysis with linguistic data: An evidential network approach. Reliab. Eng. Syst. Saf.
**2017**, 162, 111–121. [Google Scholar] [CrossRef] - Jiang, W.; Wei, B.; Zhan, J.; Xie, C.; Zhou, D. A visibility graph power averaging aggregation operator: A methodology based on network analysis. Comput. Ind. Eng.
**2016**, 101, 260–268. [Google Scholar] [CrossRef] - Jousselme, A.L.; Grenier, D.; Bossé, É. A new distance between two bodies of evidence. Inf. Fusion
**2001**, 2, 91–101. [Google Scholar] [CrossRef] - Jiang, W. A Correlation Coefficient of Belief Functions. Available online: http://arxiv.org/abs/1612.05497 (accessed on 2 February 2017).
- Ye, J. Fault diagnosis of turbine based on fuzzy cross entropy of vague sets. Expert Syst. Appl.
**2009**, 36, 8103–8106. [Google Scholar] [CrossRef]

Group Number | A | B | ${\mathit{S}}_{\mathit{r}}$ | S [6] |
---|---|---|---|---|

1 | <1 0 0> | <0 0 0> | 0.8660 | 1 |

2 | <1 0 0> | <0.1 0 0> | 0.9042 | 1 |

3 | <1 0 0> | <0.2 0 0> | 0.9333 | 1 |

4 | <1 0 0> | <0.3 0 0> | 0.9549 | 1 |

5 | <1 0 0> | <0.4 0 0> | 0.9707 | 1 |

6 | <1 0 0> | <0.5 0 0> | 0.9820 | 1 |

7 | <1 0 0> | <0.6 0 0> | 0.9897 | 1 |

8 | <1 0 0> | <0.7 0 0> | 0.9948 | 1 |

9 | <1 0 0> | <0.8 0 0> | 0.9979 | 1 |

10 | <1 0 0> | <0.9 0 0> | 0.9995 | 1 |

11 | <1 0 0> | <1 0 0> | 1 | 1 |

Group Number | A | B | ${\mathit{S}}_{\mathit{r}}$ | S [6] |
---|---|---|---|---|

1 | <0.6 0.2 0.8> | <0.3 0.1 0.4> | 0.9862 | 1 |

2 | <0.7 0.8 0.2> | <0.6 0.8 0.1> | 0.9965 | 0.9935 |

3 | <0.2 0.1 0.5> | <0.2 0.1 0.5> | 1 | 1 |

4 | <0.9 0.8 0.7> | <0.1 0.2 0.1> | 0.8440 | 0.9379 |

${\mathit{A}}_{\mathit{i}}$(Fault Knowledge) | ${\mathit{C}}_{1}$(0.01–0.39 f) | ${\mathit{C}}_{2}$(0.4–0.49 f) | ${\mathit{C}}_{3}\left(0.5\mathit{f}\right)$ | ${\mathit{C}}_{4}$(0.51–0.99 f) | ${\mathit{C}}_{5}\left(\mathit{f}\right)$ | ${\mathit{C}}_{6}\left(2\mathit{f}\right)$ | ${\mathit{C}}_{7}$(3–5 f) | ${\mathit{C}}_{8}$(Odd Times of f) | ${\mathit{C}}_{9}$(High Frequency > 5 f) |
---|---|---|---|---|---|---|---|---|---|

${A}_{1}$(unbalance) | <0 0 1> | <0 0 1> | <0 0 1> | <0 0 1> | <0.85 0.15 0> | <0.04 0.02 0.94> | <0.04 0.03 0.93> | <0 0 1> | <0 0 1> |

${A}_{2}$(pneumatic force couple) | <0 0 1> | <0.03 0.28 0.69> | <0.9 0.3 0.88> | <0.55 0.15 0.3> | <0 0 1> | <0 0 1> | <0 0 1> | <0 0 1> | <0.08 0.05 0.87> |

${A}_{3}$(offset center) | <0 0 1> | <0 0 1> | <0 0 1> | <0 0 1> | <0.3 0.28 0.42> | <0.40 0.22 0.38> | <0.08 0.05 0.87> | <0 0 1> | <0 0 1> |

${A}_{4}$(oil-membrane oscillation) | <0.09 0.22 0.89> | <0.78 0.04 0.18> | <0 0 1> | <0.08 0.03 0.89> | <0 0 1> | <0 0 1> | <0 0 1> | <0 0 1> | <0 0 1> |

${A}_{5}$(radial impact friction of rotor) | <0.09 0.03 0.88> | <0.09 0.02 0.89> | <0.08 0.04 0.88> | <0.09 0.03 0.88> | <0.18 0.03 0.79> | <0.08 0.05 0.87> | <0.08 0.05 0.87> | <0.08 0.04 0.88> | <0.08 0.04 0.88> |

${A}_{6}$(symbiosis looseness) | <0 0 1> | <0 0 1> | <0 0 1> | <0 0 1> | <0.18 0.04 0.78> | <0.12 0.05 0.83> | <0.37 0.08 0.55> | <0 0 1> | <0.22, 0.06, 0.72> |

${A}_{7}$(damage of antithrust bearing) | <0 0 1> | <0 0 1> | <0.08 0.04 0.88> | <0.86 0.07 0.07> | <0 0 1> | <0 0 1> | <0 0 1> | <0 0 1> | <0 0 1> |

${A}_{8}$(surge) | <0 0 1> | <0.27 0.05 0.68> | <0.08 0.04 0.88> | <0.54 0.08 0.38> | <0 0 1> | <0 0 1> | <0 0 1> | <0 0 1> | <0 0 1> |

${A}_{9}$(looseness of bearing block) | <0.85, 0.08, 0.07> | <0 0 1> | <0 0 1> | <0 0 1> | <0 0 1> | <0 0 1> | <0 0 1> | <0.08 0.04 0.88> | <0 0 1> |

${A}_{10}$(non-uniform bearing stiffness) | <0 0 1> | <0 0 1> | <0 0 1> | <0 0 1> | <0 0 1> | <0.77 0.06 0.17> | <0.19 0.04 0.7> | <0 0 1> | <0 0 1> |

© 2017 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**

Jiang, W.; Shou, Y.
A Novel Single-Valued Neutrosophic Set Similarity Measure and Its Application in Multicriteria Decision-Making. *Symmetry* **2017**, *9*, 127.
https://doi.org/10.3390/sym9080127

**AMA Style**

Jiang W, Shou Y.
A Novel Single-Valued Neutrosophic Set Similarity Measure and Its Application in Multicriteria Decision-Making. *Symmetry*. 2017; 9(8):127.
https://doi.org/10.3390/sym9080127

**Chicago/Turabian Style**

Jiang, Wen, and Yehang Shou.
2017. "A Novel Single-Valued Neutrosophic Set Similarity Measure and Its Application in Multicriteria Decision-Making" *Symmetry* 9, no. 8: 127.
https://doi.org/10.3390/sym9080127