# Simplified Neutrosophic Sets Based on Interval Dependent Degree for Multi-Criteria Group Decision-Making Problems

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## Abstract

**:**

## 1. Introduction

## 2. Interval Number and Interval Dependent Function

#### 2.1. Interval Number

**Definition**

**1.**

#### 2.2. Simplified Dependent Function and Interval Dependent Function

**Definition**

**2.**

**Definition**

**3.**

**Definition**

**4.**

**Property**

**1.**

**Property**

**2.**

**Property**

**3.**

**Property**

**4.**

**Property**

**5.**

## 3. Interval Transformation Operator and Interval Dependent Function of SNS

#### 3.1. NSs and SNSs

**Definition**

**5**

**.**Let X be a space of points (objects), with a generic element in X, denoted by x. An NS A in X is characterized by a truth-membership function ${T}_{A}(x)$, an indeterminacy membership function ${I}_{A}(x)$ and a falsity-membership function ${F}_{A}(x)$. ${T}_{A}(x)$, ${I}_{A}(x)$, and ${F}_{A}(x)$ are standard or non-standard subsets of $]{0}^{-},{1}^{+}[$, that is, ${T}_{A}(x)$: $X\to ]{0}^{-},{1}^{+}[$, ${I}_{A}(x)$: $X\to ]{0}^{-},{1}^{+}[$ and ${F}_{A}(x)$: $X\to ]{0}^{-},{1}^{+}[$. There is no restriction on the sum of ${T}_{A}(x),{I}_{A}(x),and{F}_{A}(x)$, therefore ${0}^{-}\le sup{T}_{A}(x)+sup{I}_{A}(x)+sup{F}_{A}(x)\le {3}^{+}$.

**Definition**

**6**

**.**An NS A is contained in another NS B, denoted by $A\subseteq B$, if and only if $inf{T}_{A}(x)\le inf{T}_{B}(x)$, $sup{T}_{A}(x)\le sup{T}_{B}(x)$, $inf{I}_{A}(x)\ge inf{I}_{B}(x)$, $sup{I}_{A}(x)\ge sup{I}_{B}(x)$, $inf{F}_{A}(x)\ge inf{F}_{B}(x)$, and $sup{F}_{A}(x)\ge sup{F}_{B}(x)$ for $x\in X$. Since it is difficult to apply NSs to practical problems, Ye (2014a) reduced NSs of non-standard intervals into SNSs of standard intervals that would preserve the operations of NSs.

**Definition**

**7**

**.**Let X be a space of points (objects), with a generic element in X, denoted by x. An NS A in X is characterised by ${T}_{A}(x)$, ${I}_{A}(x)$, and ${F}_{A}(x)$, which are subintervals/subsets in the standard interval $[0,1]$, that is, ${T}_{A}(x)$: $X\to [0,1]$, ${I}_{A}(x):X\to [0,1]$ and ${F}_{A}(x):X\to [0,1]$. Then, a simplification of A is denoted by $A\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}\{<x,{T}_{A}(x),{I}_{A}(x),{F}_{A}(x)>|x\in X\}$, which is called an SNS. In particular, if X has only one element, $A\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}<{T}_{A}(x),{I}_{A}(x),{F}_{A}(x)>$ is called an SNN. For convenience, a SNN is denoted by $A\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}<{T}_{A},{I}_{A},{F}_{A}>$. Clearly, SNSs are a subclass of NSs.

**Definition**

**8**

**.**An SNS A is contained in another SNS B, denoted by $A\subseteq B$, if and only if ${T}_{A}(x)\le {T}_{B}(x)$, ${I}_{A}(x)\ge {I}_{B}(x)$, and ${F}_{A}(x)\ge {F}_{B}(x)$ for any $x\in X$.

**Definition**

**9**

**.**The complement of an SNS A is denoted by ${A}^{C}$ and is defined as

#### 3.2. Interval Transformation Operator of SNNs

**Definition**

**10.**

**Example**

**1.**

#### 3.3. Interval Dependent Function of SNNs

## 4. The MCGDM Method Based on the Interval Dependent Degrees of SNNs

**Step 1.**Normalize the decision matrix. Generally, there are two kinds criteria including maximizing criteria and minimizing criteria in MCDM problems. For the maximizing criteria, it remains unchanged. For the minimizing criteria, it can be transformed into maximizing criteria by taking its complement as ${r}_{ij}^{k}={({r}_{ij}^{k})}^{C}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}<{F}_{{r}_{ij}^{k}},1-{I}_{{r}_{ij}^{k}},{T}_{{r}_{ij}^{k}}>$ in Definition 9.

**Step 2.**Interval transformation. Performing interval transformation operator $Z({r}_{ij}^{k})$ to SNN ${r}_{ij}^{k}$ according Definition 10. Then, the corresponding subinterval is obtained as

**Step 3.**Select the simplified dependent function and distribution function. According to the preference of decision makers and the actual requirements, the forms of dependent function Definition 2 and distribution function as in Definition 4 should be decided.

**Step 4.**Calculate the interval dependent degree of each SNN of the decision matrix. According to Equation (16), the dependent degree of SNN ${r}_{ij}^{k}$ as

**Step 5.**Calculate the comprehensive dependent degree of each alternative. The comprehensive dependent degree of each alternative ${a}_{i}$ is obtained as

**Step 6.**Stability analysis. By selecting different distribution functions in Definition 4, the stability analysis is performed to the decision results. It can be seen whether the sorting result will change under different distributions of truth-membership values.

## 5. An Illustrative Example

#### 5.1. The Decision Making Procedure

**Step 1.**Normalize the decision matrix. Because all the criteria are maximizing criteria, all the SNNs should remain unchanged.

**Step 2.**Interval transformation. By interval transformation operator Z, the corresponding subinterval is obtained as ${X}_{0}^{{r}_{ij}^{1}}$, ${X}_{0}^{{r}_{ij}^{2}}$, and ${X}_{0}^{{r}_{ij}^{3}}$ in Table 4, Table 5 and Table 6.

**Step 3.**Select the simplified dependent function and distribution function. To facilitate a clearer comparison, a linear simplified dependent function as in Equation (5) and a nonlinear simplified dependent function as in Equation (6) $(\alpha =2.0)$ are selected. In addition, triangular distribution as in Figure 3b is used as the distribution function.

**Step 4.**Calculate interval dependent degree of each SNN of the decision matrix. According to Equation (18), the dependent degrees $k({r}_{ij}^{k})$ of SNNs are obtained as in Table 7 and Table 8, in which Table 7 shows the dependent degrees with linear simplified dependent function as in Equation (5), and Table 8 shows those with nonlinear simplified dependent function as in Equation (6).

**Step 5.**Calculate the comprehensive dependent degree of each alternative. According to Equation (19), experts weight vector $\lambda $ and criteria weight vector w, there are comprehensive dependent degree $K({a}_{i})$ of each alternative ${a}_{i}$ as in Table 9. Table 9 shows the values of $K({a}_{i})$ under different simplified dependent functions which include linear function and nonlinear functions with $\alpha =3.0$, $\alpha =2.0$, $\alpha =1.5$, and $\alpha =1.2$. As we can see, although the dependent degrees under different dependent functions are different from each other, the sorting result remains unchanged. In fact, the simplified dependent function can reflect the psychology status of decision makers. For example, Equation (6) describes risk aversion psychology, which means the curve slope will change with different evaluation values. The smaller the parameter $\alpha $, the greater the extent of regret evasion from decision makers. Nevertheless, the result shows that it is not influenced by the risk aversion psychology changing in the decision makers. So it exhibits high stability.

**Step 6.**Uncertainty analysis. Table 10 and Table 11 show the sorting result under different distribution functions as in Figure 3. Although the dependent degree values are slightly different as the distribution function changes, the result remain unchanged, which illustrates the lower uncertainty and sensibility of the ranking result. Therefore, for decision makers in this case, the sorting result is sufficiently certain and stable.

#### 5.2. A Comparison Analysis

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Libo, X.; Xingsen, L.; Chaoyi, P. Uncertain Multiattribute Decision-Making Based on Interval Number with Extension-Dependent Degree and Regret Aversion. Math. Probl. Eng.
**2018**. [Google Scholar] [CrossRef] - Zadeh, L.A. Fuzzy sets. Inf. Control
**1965**, 8, 338–356. [Google Scholar] [CrossRef] - Zadeh, L.A. Probability measures of fuzzy events. J. Math. Anal. Appl.
**1968**, 23, 421–427. [Google Scholar] [CrossRef] - Zadeh, L.A. Fuzzy logic and approximate reasoning. Synthese
**1975**, 30, 407–428. [Google Scholar] [CrossRef] - Bellman, R.; Zadeh, L.A. Decision making in a fuzzy environment. Manag. Sci.
**1970**, 17, 141–164. [Google Scholar] [CrossRef] - Atanassov, K. Intuitionistic fuzzy sets. Fuzzy Sets Syst.
**1986**, 20, 87–96. [Google Scholar] [CrossRef] - Atanassov, K. New operations defined over the intuitionistic fuzzy sets. Fuzzy Sets Syst.
**1994**, 61, 137–142. [Google Scholar] [CrossRef] - Atanassov, K. Two theorems for intuitionistic fuzzy sets. Fuzzy Sets Syst.
**2000**, 110, 267–269. [Google Scholar] [CrossRef] - Gau, W.L.; Buehrer, D.J. Vague sets. IEEE Trans. Syst. Man Cybern.
**1993**, 23, 610–614. [Google Scholar] [CrossRef] - Bustince, H.; Burillo, P. Vague sets are intuitionistic fuzzy sets. Fuzzy Sets Syst.
**1996**, 79, 403–405. [Google Scholar] [CrossRef] - Chen, Y.T. A outcome-oriented approach to multicriteria decision analysis with intuitionistic fuzzy optimistic pessimistic operators. Expert Syst. Appl.
**2010**, 37, 7762–7774. [Google Scholar] [CrossRef] - Xu, Z.S. Intuitionistic fuzzy multiattribute decision making: An interactive method. IEEE Trans. Fuzzy Syst.
**2012**, 20, 514–525. [Google Scholar] - Chen, S.M.; Cheng, S.H.; Chiou, C.H. Fuzzymulti-attribute group decision making based on intuitionistic fuzzy sets and evidential reasoning methodology. Inf. Fusion
**2016**, 27, 215–227. [Google Scholar] [CrossRef] - Cao, Q.W.; Wu, J.; Liang, C.Y. An intuitionistic fuzzy judgement matrix and TOPSIS integrated multi-criteria decision-making method for green supplier selection. J. Intell. Fuzzy Syst.
**2015**, 28, 117–126. [Google Scholar] - 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, 905–919. [Google Scholar] - Yue, Z.L. TOPSIS-based group decision-making methodology in intuitionistic fuzzy setting. Inf. Sci.
**2014**, 277, 141–153. [Google Scholar] [CrossRef] - Chen, T.Y. Interval-valued intuitionistic fuzzy QUALIFLEX method with a likelihood-based comparison approach for multiple criteria decision analysis. Inf. Sci.
**2014**, 261, 149–169. [Google Scholar] [CrossRef] - Liu, B.S.; Shen, Y.H.; Zhang, W.; Chen, X.H.; Wang, X.Q. An interval-valued intuitionistic fuzzy principal component analysis model-based method for complex multi-attribute large-group decision-making. Eur. J. Oper. Res.
**2015**, 245, 209–225. [Google Scholar] [CrossRef] - Atanassov, K.T.; Gargov, G. Interval valued intuitionistic fuzzy sets. Fuzzy Sets Syst.
**1989**, 31, 343–349. [Google Scholar] [CrossRef] - Joshi, D.; Kumar, S. Interval-valued intuitionistic hesitant fuzzy Choquet integral based TOPSIS method for multi-criteria group decision making. Eur. J. Oper. Res.
**2016**, 248, 183–191. [Google Scholar] [CrossRef] - Liu, P.D.; Shi, L.L. The generalized hybrid weighted average operator based on interval neutrosophic hesitant set and its application to multiple attribute decision making. Neural Comput. Appl.
**2015**, 26, 451–471. [Google Scholar] [CrossRef] - Zhang, H.Y.; Wang, J.Q.; Chen, X.H. An outranking approach for multi-criteria decision-making problems with interval-valued neutrosophic sets. Neural Comput. Appl.
**2016**, 27, 615–627. [Google Scholar] [CrossRef] - Wang, H.; Smarandache, F.; Zhang, Y.Q.; Sunderraman, R. Single valued neutrosophic sets. Multispace Multistruct.
**2010**, 4, 410–413. [Google Scholar] - Smarandache, F. Neutrosophy: Neutrosophic Probability, Set, and Logic; American Research Press: Rehoboth, DE, USA, 1998; pp. 1–105. [Google Scholar]
- Smarandache, F. A Unifying Field in Logics. Neutrosophy: Neutrosophic Probability, Set and Logic; American Research Press: Rehoboth, DE, USA, 1999; pp. 1–141. [Google Scholar]
- Smarandache, F. A Unifying Field in Logics: Neutrosophic Logic. Neutrosophy, Neutrosophic Set, Neutrosophic Probability: Neutrsophic Logic. Neutrosophy, Neutrosophic Set, Neutrosophic Probability and Statistics; American Research Press: Rehoboth, DE, USA, 2005; pp. 1–156. [Google Scholar]
- Khan, M.; Son, L.; Ali, M.; Chau, H.; Na, N.; Smarandache, F. Systematic Review of Decision Making Algorithms in Extended Neutrosophic Sets. Symmetry
**2018**, 10, 314. [Google Scholar] [CrossRef] - Rivieccio, U. Neutrosophic logics: Prospects and problems. Fuzzy Sets Syst.
**2008**, 159, 1860–1868. [Google Scholar] [CrossRef] - Deli, I.; Broumi, S.; Smarandache, F. On neutrosophic refined sets and their applications in medical diagnosis. J. New Theory
**2015**, 6, 88–98. [Google Scholar] - Ali, M.; Son, L.H.; Khan, M.; Tung, N.T. Segmentation of dental X-ray images in medical imaging using neutrosophic orthogonal matrices. Expert Syst. Appl.
**2018**, 91, 434–441. [Google Scholar] [CrossRef] - Mondal, K.; Surapati, P.; Giri, B.C. Role of Neutrosophic Logic in Data Mining. New Trends in Neutrosophic Theory and Applications. 2016, pp. 15–23. Available online: https://books.google.com.hk/books?hl=zh-CN&lr=&id=s7OdDQAAQBAJ&oi=fnd&pg=PA15&dq=Role+of+Neutrosophic+Logic+in+Data+Mining&ots=GHfYqos9XC&sig=qA_uHSJ6geF1qVI7UT3g0Kj5MKU&redir_esc=y#v=onepage&q=Role%20of%20Neutrosophic%20Logic%20in%20Data%20Mining&f=false (accessed on 1 October 2018).
- Radwan, N.M. Neutrosophic Applications in E-learning: Outcomes, Challenges and Trends. New Trends in Neutrosophic Theory and Applications. 2016, pp. 177–184. Available online: https://books.google.com.hk/books?hl=zh-CN&lr=&id=s7OdDQAAQBAJ&oi=fnd&pg=PA177&dq=Neutrosophic+Applications+in+E-learning:+Outcomes,+Challenges+and+Trends&ots=GHfYqosaTC&sig=h0MqRrBaY2Aq2FHOMU7PN40LPxQ&redir_esc=y#v=onepage&q=Neutrosophic%20Applications%20in%20E-learning%3A%20Outcomes%2C%20Challenges%20and%20Trends&f=false (accessed on 1 October 2018).
- Ali, M.; Deli, I.; Smarandache, F. The theory of neutrosophic cubic sets and their applications in pattern recognition. J. Intell. Fuzzy Syst.
**2016**, 30, 1957–1963. [Google Scholar] [CrossRef] - Broumi, S.; Talea, M.; Bakali, A.; Smarandache, F. Single valued neutrosophic graphs. J. New Theory
**2016**, 10, 86–101. [Google Scholar] - Broumi, S.; Talea, M.; Bakali, A.; Smarandache, F. On bipolar single valued neutrosophic graphs. J. New Theory
**2016**, 11, 84–102. [Google Scholar] - Majumdar, P.; Samant, S.K. On similarity and entropy of neutrosophic sets. J. Intell. Fuzzy Syst.
**2014**, 26, 1245–1252. [Google Scholar] - Ye, J. Multicriteria decision-making method using the correlation coefficient under single-value neutrosophic environment. Int. J. Gen. Syst.
**2013**, 42, 386–394. [Google Scholar] [CrossRef] - Huang, H. New distance measure of single-valued neutrosophic sets and its application. Int. J. Intell. Syst.
**2016**, 31, 1021–1032. [Google Scholar] [CrossRef] - Thanh, N.D.; Ali, M.; Son, L.H. A novel clustering algorithm in a neutrosophic recommender system for medical diagnosis. Cogn. Comput.
**2017**, 9, 526–544. [Google Scholar] [CrossRef] - Karaaslan, F. Correlation coefficients of single-valued neutrosophic refined soft sets and their applications in clustering analysis. Neural Comput. Appl.
**2017**, 28, 2781–2793. [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, 817–825. [Google Scholar] [CrossRef] - 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. Similarity measures between interval neutrosophic sets and their applications in multicriteria decision-making. J. Intell. Fuzzy Syst.
**2014**, 26, 165–172. [Google Scholar] - Peng, J.J.; Wang, J.Q.; Zhang, H.Y.; Chen, X.H. An outranking approach for multi-criteria decision-making problems with simplified neutrosophic sets. Appl. Soft Comput.
**2014**, 25, 336–346. [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] - Ye, J.; Cui, W. Exponential Entropy for Simplified Neutrosophic Sets and Its Application in Decision Making. Entropy
**2018**, 20, 357. [Google Scholar] [CrossRef] - Han, L.; Wei, C. Group decision making method based on single valued neutrosophic choquet inttegral operator. Oper. Res. Trans.
**2017**, 21, 110–118. [Google Scholar] - Ye, J.; Du, S.G. Some distances, similarity and entropy measures for interval-valued neutrosophic sets and their relationship. Int. J. Mach. Learn. Cybern.
**2017**, 1–9. [Google Scholar] [CrossRef] - Wang, H.B.; Smarandache, F.; Zhang, Y.Q.; Sunderraman, R. Interval Neutrosophic Sets and Logic: Theory and Applications in Computing; Hexis: Phoenix, AZ, USA, 2005. [Google Scholar]
- Wang, J.Q.; Li, X.E. The TODIM method with multi-valued neutrosophic sets. Control Decis.
**2015**, 30, 1139–1142. [Google Scholar] - Peng, J.J.; Wang, J.Q.; Wu, X.H.; Wang, J.; Chen, X.H. Multi-valued neutrosophic sets and power aggregation operators with their applications in multi-criteria group decision-making problems. Int. J. Comput. Intell. Syst.
**2015**, 8, 345–363. [Google Scholar] [CrossRef] - Ali, M.; Smarandache, F. Complex neutrosophic set. Neural Comput. Appl.
**2017**, 28, 1817–1834. [Google Scholar] [CrossRef] - Arena, P.; Fazzino, S.; Fortuna, L.; Maniscalco, P. Game theory and non-linear dynamics: The Parrondo Paradox case study. Chaos Solitons Fractals
**2003**, 17, 545–555. [Google Scholar] [CrossRef] - Chen, Y.W.; Larbani, M. Two-person zero-sum game approach for fuzzy multiple attribute decision making problems. Fuzzy Sets Syst.
**2006**, 157, 34–51. [Google Scholar] [CrossRef]

**Figure 1.**Dependent function of Equation (6).

**Figure 2.**dependent function of Equation (7).

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

${s}_{1}$ | $<0.65,0.10,0.25>$ | $<0.50,0.18,0.32>$ | $<0.68,0.12,0.20>$ | $<0.50,0.10,0.25>$ |

${s}_{2}$ | $<0.83,0.12,0.05>$ | $<0.65,0.15,0.20>$ | $<0.50,0.10,0.40>$ | $<0.67,0.18,0.15>$ |

${s}_{3}$ | $<0.67,0.13,0.20>$ | $<0.50,0.15,0.35>$ | $<0.68,0.12,0.20>$ | $<0.50,0.20,0.30>$ |

${s}_{4}$ | $<0.66,0.14,0.20>$ | $<0.50,0.16,0.34>$ | $<0.70,0.10,0.20>$ | $<0.50,0.15,0.35>$ |

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

${s}_{1}$ | $<0.90,0.02,0.08>$ | $<0.10,0.10,0.80>$ | $<0.15,0.15,0.70>$ | $<0.10,0.05,0.85>$ |

${s}_{2}$ | $<0.75,0.15,0.10>$ | $<0.85,0.05,0.10>$ | $<0.50,0.10,0.40>$ | $<0.68,0.10,0.22>$ |

${s}_{3}$ | $<0.50,0.05,0.45>$ | $<0.40,0.15,0.45>$ | $<0.68,0.12,0.20>$ | $<0.15,0.05,0.80>$ |

${s}_{4}$ | $<0.50,0.10,0.40>$ | $<0.50,0.10,0.40>$ | $<0.60,0.10,0.30>$ | $<0.50,0.05,0.45>$ |

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

${s}_{1}$ | $<0.65,0.15,0.20>$ | $<0.30,0.10,0.60>$ | $<0.65,0.20,0.15>$ | $<0.50,0.10,0.40>$ |

${s}_{2}$ | $<0.85,0.05,0.10>$ | $<0.85,0.05,0.10>$ | $<0.34,0.16,0.50>$ | $<0.60,0.10,0.30>$ |

${s}_{3}$ | $<0.61,0.18,0.21>$ | $<0.67,0.13,0.20>$ | $<0.68,0.22,0.10>$ | $<0.30,0.10,0.60>$ |

${s}_{4}$ | $<0.62,0.28,0.10>$ | $<0.68,0.22,0.10>$ | $<0.68,0.12,0.20>$ | $<0.50,0.10,0.40>$ |

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

${X}_{0}^{{r}_{1j}^{1}}$ | $[0.650,0.750]$ | $[0.500,0.680]$ | $[0.680,0.800]$ | $[0.588,0.706]$ |

${X}_{0}^{{r}_{2j}^{1}}$ | $[0.830,0.950]$ | $[0.650,0.800]$ | $[0.500,0.600]$ | $[0.670,0.850]$ |

${X}_{0}^{{r}_{3j}^{1}}$ | $[0.670,0.800]$ | $[0.500,0.650]$ | $[0.680,0.800]$ | $[0.500,0.700]$ |

${X}_{0}^{{r}_{4j}^{1}}$ | $[0.660,0.800]$ | $[0.500,0.660]$ | $[0.700,0.800]$ | $[0.500,0.650]$ |

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

${X}_{0}^{{r}_{1j}^{2}}$ | $[0.900,0.920]$ | $[0.100,0.200]$ | $[0.150,0.300]$ | $[0.100,0.150]$ |

${X}_{0}^{{r}_{2j}^{2}}$ | $[0.750,0.900]$ | $[0.850,0.900]$ | $[0.500,0.600]$ | $[0.680,0.780]$ |

${X}_{0}^{{r}_{3j}^{2}}$ | $[0.500,0.550]$ | $[0.400,0.550]$ | $[0.680,0.800]$ | $[0.150,0.200]$ |

${X}_{0}^{{r}_{4j}^{2}}$ | $[0.500,0.600]$ | $[0.500,0.600]$ | $[0.600,0.700]$ | $[0.500,0.550]$ |

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

${X}_{0}^{{r}_{1j}^{3}}$ | $[0.650,0.800]$ | $[0.300,0.400]$ | $[0.650,0.850]$ | $[0.500,0.600]$ |

${X}_{0}^{{r}_{2j}^{3}}$ | $[0.850,0.900]$ | $[0.850,0.900]$ | $[0.340,0.500]$ | $[0.600,0.700]$ |

${X}_{0}^{{r}_{3j}^{3}}$ | $[0.610,0.790]$ | $[0.670,0.800]$ | $[0.680,0.900]$ | $[0.300,0.400]$ |

${X}_{0}^{{r}_{4j}^{3}}$ | $[0.620,0.900]$ | $[0.680,0.900]$ | $[0.680,0.800]$ | $[0.500,0.600]$ |

Expert | |||
---|---|---|---|

${\mathit{d}}_{\mathbf{1}}$ | ${\mathit{d}}_{\mathbf{2}}$ | ${\mathit{d}}_{\mathbf{3}}$ | |

$k({r}_{1j}^{k})$ | $0.7000,0.5900,0.7400,0.6471$ | $0.9100,0.1500,0.2250,0.1250$ | $0.7250,0.3500,0.7500,0.5500$ |

$k({r}_{2j}^{k})$ | $0.8900,0.7250,0.5500,0.7600$ | $0.8250,0.8750,0.5500,0.7300$ | $0.8750,0.8750,0.4200,0.6500$ |

$k({r}_{3j}^{k})$ | $0.7350,0.5750,0.7400,0.6000$ | $0.5250,0.4750,0.7400,0.1750$ | $0.7000,0.7350,0.7900,0.3500$ |

$k({r}_{4j}^{k})$ | $0.7300,0.5800,0.7500,0.5750$ | $0.5500,0.5500,0.6500,0.5250$ | $0.7600,0.7900,0.7400,0.5500$ |

Expert | |||
---|---|---|---|

${\mathit{d}}_{\mathbf{1}}$ | ${\mathit{d}}_{\mathbf{2}}$ | ${\mathit{d}}_{\mathbf{3}}$ | |

$k({r}_{1j}^{k})$ | $0.8234,0.7415,0.8503,0.7855$ | $0.9529,0.2603,0.3663,0.2221$ | $0.8402,0.5182,0.8565,0.7095$ |

$k({r}_{2j}^{k})$ | $0.9416,0.8402,0.7095,0.8631$ | $0.9038,0.9333,0.7095,0.8438$ | $0.9333,0.9333,0.5908,0.7877$ |

$k({r}_{3j}^{k})$ | $0.8470,0.7297,0.8503,0.7492$ | $0.6885,0.6435,0.8503,0.2977$ | $0.8230,0.8470,0.8820,0.5182$ |

$k({r}_{4j}^{k})$ | $0.8436,0.7336,0.8570,0.7297$ | $0.7095,0.7095,0.7877,0.6885$ | $0.8624,0.8820,0.8503,0.7095$ |

${\mathit{s}}_{1}$ | ${\mathit{s}}_{2}$ | ${\mathit{s}}_{3}$ | ${\mathit{s}}_{4}$ | Sorting Result | |
---|---|---|---|---|---|

$K({a}_{i})$ (linear) | $0.5591$ | $0.7276$ | $0.6193$ | $0.6549$ | ${s}_{2}\succ {s}_{4}\succ {s}_{3}\succ {s}_{1}$ |

$K({a}_{i})$ (nonlinear $\alpha =3.0$) | $0.6325$ | $0.7928$ | $0.6995$ | $0.7367$ | ${s}_{2}\succ {s}_{4}\succ {s}_{3}\succ {s}_{1}$ |

$K({a}_{i})$ (nonlinear $\alpha =2.0$) | $0.6811$ | $0.8322$ | $0.7500$ | $0.7868$ | ${s}_{2}\succ {s}_{4}\succ {s}_{3}\succ {s}_{1}$ |

$K({a}_{i})$ (nonlinear $\alpha =1.5$) | $0.7434$ | $0.8778$ | $0.8111$ | $0.8453$ | ${s}_{2}\succ {s}_{4}\succ {s}_{3}\succ {s}_{1}$ |

$K({a}_{i})$ (nonlinear $\alpha =1.2$) | $0.8323$ | $0.9321$ | $0.8886$ | $0.9148$ | ${s}_{2}\succ {s}_{4}\succ {s}_{3}\succ {s}_{1}$ |

$\mathit{\alpha}=2.0$ | ${\mathit{s}}_{1}$ | ${\mathit{s}}_{2}$ | ${\mathit{s}}_{3}$ | ${\mathit{s}}_{4}$ | Sorting Result |
---|---|---|---|---|---|

$K({a}_{i})$ (uniform) | $0.6806$ | $0.8318$ | $0.7496$ | $0.7864$ | ${s}_{2}\succ {s}_{4}\succ {s}_{3}\succ {s}_{1}$ |

$K({a}_{i})$ (normal) | $0.6793$ | $0.8300$ | $0.7481$ | $0.7848$ | ${s}_{2}\succ {s}_{4}\succ {s}_{3}\succ {s}_{1}$ |

$K({a}_{i})$ (triangular) | $0.6811$ | $0.8322$ | $0.7500$ | $0.7868$ | ${s}_{2}\succ {s}_{4}\succ {s}_{3}\succ {s}_{1}$ |

$K({a}_{i})$ (appro-triangular) | $0.6808$ | $0.8320$ | $0.7498$ | $0.7866$ | ${s}_{2}\succ {s}_{4}\succ {s}_{3}\succ {s}_{1}$ |

$K({a}_{i})$ (trapezoid 1) | $0.6864$ | $0.8362$ | $0.7553$ | $0.7919$ | ${s}_{2}\succ {s}_{4}\succ {s}_{3}\succ {s}_{1}$ |

$K({a}_{i})$ (trapezoid 2) | $0.6748$ | $0.8276$ | $0.7439$ | $0.7809$ | ${s}_{2}\succ {s}_{4}\succ {s}_{3}\succ {s}_{1}$ |

$\mathit{\alpha}=1.2$ | ${\mathit{s}}_{1}$ | ${\mathit{s}}_{2}$ | ${\mathit{s}}_{3}$ | ${\mathit{s}}_{4}$ | Sorting Result |
---|---|---|---|---|---|

$K({a}_{i})$ (uniform) | $0.8316$ | $0.9319$ | $0.8882$ | $0.9144$ | ${s}_{2}\succ {s}_{4}\succ {s}_{3}\succ {s}_{1}$ |

$K({a}_{i})$ (normal) | $0.8303$ | $0.9297$ | $0.8864$ | $0.9124$ | ${s}_{2}\succ {s}_{4}\succ {s}_{3}\succ {s}_{1}$ |

$K({a}_{i})$ (triangular) | $0.8323$ | $0.9321$ | $0.8886$ | $0.9148$ | ${s}_{2}\succ {s}_{4}\succ {s}_{3}\succ {s}_{1}$ |

$K({a}_{i})$ (appro-triangular) | $0.8318$ | $0.9319$ | $0.8884$ | $0.9145$ | ${s}_{2}\succ {s}_{4}\succ {s}_{3}\succ {s}_{1}$ |

$K({a}_{i})$ (trapezoid 1) | $0.8356$ | $0.9339$ | $0.8911$ | $0.9169$ | ${s}_{2}\succ {s}_{4}\succ {s}_{3}\succ {s}_{1}$ |

$K({a}_{i})$ (trapezoid 2) | $0.8234$ | $0.9283$ | $0.8844$ | $0.9136$ | ${s}_{2}\succ {s}_{4}\succ {s}_{3}\succ {s}_{1}$ |

Sorting Result | Best One | Worst One | |
---|---|---|---|

SNNWA [42] | ${s}_{2}\succ {s}_{4}\succ {s}_{3}\succ {s}_{1}$ | ${s}_{2}$ | ${s}_{1}$ |

Entropy of Euclidean [48] | ${s}_{2}\succ {s}_{4}\succ {s}_{3}\succ {s}_{1}$ | ${s}_{2}$ | ${s}_{1}$ |

SNEE [46] | ${s}_{2}\succ {s}_{4}\succ {s}_{3}\succ {s}_{1}$ | ${s}_{2}$ | ${s}_{1}$ |

SNNCI [47] | ${s}_{2}\succ {s}_{4}\succ {s}_{3}\succ {s}_{1}$ | ${s}_{2}$ | ${s}_{1}$ |

GSNNWA [45] | ${s}_{2}\succ {s}_{4}\succ {s}_{3}\succ {s}_{1}$ | ${s}_{2}$ | ${s}_{1}$ |

the proposed method | ${s}_{2}\succ {s}_{4}\succ {s}_{3}\succ {s}_{1}$ | ${s}_{2}$ | ${s}_{1}$ |

© 2018 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**

Xu, L.; Li, X.; Pang, C.; Guo, Y.
Simplified Neutrosophic Sets Based on Interval Dependent Degree for Multi-Criteria Group Decision-Making Problems. *Symmetry* **2018**, *10*, 640.
https://doi.org/10.3390/sym10110640

**AMA Style**

Xu L, Li X, Pang C, Guo Y.
Simplified Neutrosophic Sets Based on Interval Dependent Degree for Multi-Criteria Group Decision-Making Problems. *Symmetry*. 2018; 10(11):640.
https://doi.org/10.3390/sym10110640

**Chicago/Turabian Style**

Xu, Libo, Xingsen Li, Chaoyi Pang, and Yan Guo.
2018. "Simplified Neutrosophic Sets Based on Interval Dependent Degree for Multi-Criteria Group Decision-Making Problems" *Symmetry* 10, no. 11: 640.
https://doi.org/10.3390/sym10110640