# New Multiple Attribute Decision Making Method Based on DEMATEL and TOPSIS for Multi-Valued Interval Neutrosophic Sets

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

**Definition**

**4.**

**Definition**

**5.**

**Definition**

**6.**

**Definition**

**7.**

- (1)
- $d(\tilde{a},\tilde{b})\ge 0$,
- (2)
- $d(\tilde{a},\tilde{b})=0$, if and only if $\tilde{a}=\tilde{b}$,
- (3)
- $d(\tilde{a},\tilde{b})=d(\tilde{b},\tilde{a})$.

## 3. The Multi-Valued Interval Neutrosophic Fuzzy Decision Making Method Based on DEMATEL and TOPSIS

**Step 1.**Generate the direct-relation matrix. Decision makers are asked to assess the effect among factor pairs. The direct-relation matrix is established as $A={({a}_{ij})}_{n\times n}$, where A is an $n\times n$ non-negative matrix, ${a}_{ij}$ indicates the direct effect of factor i on factor j and the diagonal elements ${a}_{ii}=0$ for $i=1,2,\dots ,n$.

**Step 2.**Normalize the direct-relation matrix by $D=A\times S$, where $S=\frac{1}{{max}_{1\le i\le n}{\sum}_{j=1}^{n}{a}_{ij}}$ and obtain the normalized initial direct-relation matrix as $D={({d}_{ij})}_{n\times n}$.

**Step 3.**Set up the total-relation matrix by the following equation:

**Step 4.**Sum each row and each column of matrix T by Equations (10) and (11). The sum of row i is denoted by ${r}_{i}$ and:

#### Algorithm

**Step 1.**Decision makers provide interval neutrosophic fuzzy evaluation values in the pairwise comparison of attributes. Then, the multi-valued interval neutrosophic fuzzy decision matrix is formed as $\tilde{M}={({\tilde{m}}_{ij})}_{n\times n}$, ${\tilde{m}}_{ij}=<{\tilde{T}}_{{\tilde{m}}_{ij}},{\tilde{I}}_{{\tilde{m}}_{ij}},{\tilde{F}}_{{\tilde{m}}_{ij}}>$. ${\tilde{T}}_{{\tilde{m}}_{ij}},{\tilde{I}}_{{\tilde{m}}_{ij}},{\tilde{F}}_{{\tilde{m}}_{ij}}$ are the sets of the truth-membership function, the indeterminacy-membership function and the falsity-membership function, respectively. $[{\mu}_{ijk}^{L},{\mu}_{ijk}^{U}]\in {\tilde{T}}_{{\tilde{m}}_{ij}}$ $[{\nu}_{ijl}^{L},{\nu}_{ijl}^{U}]\in {\tilde{I}}_{{\tilde{m}}_{ij}}$, $[{\kappa}_{ijs}^{L},{\kappa}_{ijs}^{U}]\in {\tilde{F}}_{{\tilde{m}}_{ij}}$.

**Step 2.**Normalize the initial direct-relation decision matrix. Calculate the average values of the truth-membership, the indeterminacy-membership and the real falsity-membership in each multi-valued interval neutrosophic fuzzy value and transform the multi-valued interval neutrosophic fuzzy decision matrix into single-valued interval neutrosophic fuzzy decision matrix ${\tilde{M}}^{\prime}={({\tilde{m}}_{ij}^{\prime})}_{n\times n}$ by calculating the average values, where ${\tilde{m}}_{ij}^{\prime}=<[{{\mu}^{\prime}}_{ij}^{L},{{\mu}^{\prime}}_{ij}^{U}],$ $[{{\nu}^{\prime}}_{ij}^{L},{{\nu}^{\prime}}_{ij}^{U}],[{{\kappa}^{\prime}}_{ij}^{L},{{\kappa}^{\prime}}_{ij}^{U}]>$, ${{\mu}^{\prime}}_{ij}^{L}=\frac{1}{{K}_{ij}}{\sum}_{k=1}^{{K}_{ij}}{\mu}_{ijk}^{L}$, ${{\mu}^{\prime}}_{ij}^{U}=\frac{1}{{K}_{ij}}{\sum}_{k=1}^{{K}_{ij}}{\mu}_{ijk}^{U}$, ${{\nu}^{\prime}}_{ij}^{L}=\frac{1}{{L}_{ij}}{\sum}_{l=1}^{{L}_{ij}}{\nu}_{ijl}^{L}$, ${{\nu}^{\prime}}_{ij}^{U}=\frac{1}{{L}_{ij}}{\sum}_{l=1}^{{L}_{ij}}{\nu}_{ijl}^{U}$, ${{\kappa}^{\prime}}_{ij}^{L}=\frac{1}{{S}_{ij}}{\sum}_{s=1}^{{S}_{ij}}{\kappa}_{ijs}^{L}$, ${{\kappa}^{\prime}}_{ij}^{U}=\frac{1}{{S}_{ij}}{\sum}_{s=1}^{{S}_{ij}}{\kappa}_{ijs}^{U}$.

**Step 3.**Calculate the total-relation matrix, which can be calculated by using the above six crisp matrices and can be represented as follows.

**Step 4.**Determine the prominence and influence of each attribute. Calculate the sum ${\widehat{r}}_{i}$ of rows and sum ${\widehat{c}}_{i}$ of columns by the following equations.

**Step 5.**Calculate the weights of attributes by the following equation:

**Step 6.**Decision makers evaluate alternatives with respect to attributes with interval neutrosophic fuzzy values, and then multi-valued interval neutrosophic fuzzy decision matrix can be obtained as $\tilde{D}={({\tilde{a}}_{ij})}_{m\times n}$, where ${\tilde{a}}_{ij}=<{\tilde{T}}_{{\tilde{a}}_{ij}},{\tilde{I}}_{{\tilde{a}}_{ij}},{\tilde{F}}_{{\tilde{a}}_{ij}}>$. Extend the decision matrix according to the risk attitude of decision makers until all the multi-valued interval neutrosophic fuzzy values have the same number of truth-memberships, indeterminacy-memberships and real falsity-memberships. Then, the extended decision matrix can be obtained as ${\tilde{D}}^{\prime}={({\tilde{a}}_{ij}^{\prime})}_{m\times n}$, ${\tilde{a}}_{ij}^{\prime}=<{\tilde{T}}_{{{\tilde{a}}^{\prime}}_{ij}}^{\prime},$ ${\tilde{I}}_{{{\tilde{a}}^{\prime}}_{ij}}^{\prime},{\tilde{F}}_{{{\tilde{a}}^{\prime}}_{ij}}^{\prime}>$.

**Step 7.**Calculate the weighted decision matrix ${\tilde{D}}^{\u2033}={({\tilde{a}}_{ij}^{\u2033})}_{m\times n}$ by using the attribute weights obtained from Equation (11), where ${\tilde{a}}_{ij}^{\u2033}={w}_{j}{\tilde{a}}_{ij}^{\prime}$.

**Step 8.**Determine the multi-valued interval neutrosophic fuzzy positive ideal solution (MVINFPIS), denoted as ${\tilde{I}}^{+}$ and the multi-valued interval neutrosophic fuzzy negative ideal solution (MVINFNIS), denoted as ${\tilde{I}}^{-}$, as follows:

**Step 9.**Calculate the distances of each alternative’s weighted evaluation values to the MVINFPIS and MVINFNIS:

**Step 10.**Determine the closeness coefficients $C{C}_{i}$ of alternatives and rank alternatives according to the closeness coefficients.

## 4. Numerical Example

**Step 1.**Decision makers compare attributes pairwise by using multi-valued interval neutrosophic fuzzy values, and decision matrix $\tilde{M}={({\tilde{m}}_{ij})}_{4\times 4}$ is formed as in Table 1.

**Step 2.**Transform decision matrix $\tilde{M}$ into single-valued interval neutrosophic fuzzy decision matrix ${\tilde{M}}^{\prime}$ as in Table 2 by calculating the average values of the truth-membership, the indeterminacy-membership and the falsity-membership. Normalize decision matrix ${\tilde{M}}^{\prime}$ to obtain decision matrix ${\tilde{M}}^{\u2033}$, which can be represented as matrices ${\tilde{M}}_{{{\mu}^{\u2033}}_{ij}^{L}}^{\u2033},{\tilde{M}}_{{{\mu}^{\u2033}}_{ij}^{U}}^{\u2033},{\tilde{M}}_{{{\nu}^{\u2033}}_{ij}^{L}}^{\u2033},{\tilde{M}}_{{{\nu}^{\u2033}}_{ij}^{U}}^{\u2033},{\tilde{M}}_{{{\kappa}^{\u2033}}_{ij}^{L}}^{\u2033},{\tilde{M}}_{{{\kappa}^{\u2033}}_{ij}^{U}}^{\u2033}$. For example, since $\mu =max\{\underset{1\le i\le 4}{max}{\sum}_{j=1}^{4}{{\mu}^{\prime}}_{ij}^{U},\underset{1\le j\le 4}{max}{\sum}_{i=1}^{4}{{\mu}^{\prime}}_{ij}^{U}\}=2$ and ${{\mu}^{\u2033}}_{ij}^{L}=\frac{{{\mu}^{\prime}}_{ij}^{L}}{\mu}$, then decision matrix ${\tilde{M}}_{{{\mu}^{\u2033}}_{ij}^{L}}^{\u2033}$ can be obtained.

**Step 3.**Calculate the total-relation matrix ${\tilde{M}}^{\u2034}=({\tilde{m}}_{ij}^{\u2034})$ by using decision matrices ${\tilde{M}}_{{{\mu}^{\u2033}}_{ij}^{L}}^{\u2033},\dots ,{\tilde{M}}_{{{\kappa}^{\u2033}}_{ij}^{U}}^{\u2033}$, where ${\tilde{m}}_{ij}^{\u2034}=<[{{\mu}^{\u2034}}_{ij}^{L},{{\mu}^{\u2034}}_{ij}^{U}],[{{\nu}^{\u2034}}_{ij}^{L},{{\nu}^{\u2034}}_{ij}^{U}],[{{\kappa}^{\u2034}}_{ij}^{L},{{\kappa}^{\u2034}}_{ij}^{U}]>$ and $[{{\mu}^{\u2034}}_{ij}^{L}]={\tilde{M}}_{{{\mu}^{\u2033}}_{ij}^{L}}^{\u2033}\times {(I-{\tilde{M}}_{{{\mu}^{\u2033}}_{ij}^{L}}^{\u2033})}^{-1}$,…, $[{{\kappa}^{\u2034}}_{ij}^{U}]={\tilde{M}}_{{{\kappa}^{\u2033}}_{ij}^{U}}^{\u2033}\times {(I-{\tilde{M}}_{{{\kappa}^{\u2033}}_{ij}^{U}}^{\u2033})}^{-1}$. The results are shown in Table 3.

**Step 4.**Calculate the average value of each row to obtain ${\widehat{r}}_{i}$ and the average of each column to obtain ${\widehat{c}}_{i}$ by using Equations (13) and (14). The results are as follows:

**Step 5.**Calculate the weights of attributes by using Equation (15) to obtain

**Step 6.**Decision makers evaluate alternatives with respect to attributes by using the multi-valued interval neutrosophic fuzzy values. Then, decision matrix $\tilde{D}={({\tilde{a}}_{ij})}_{5\times 4}$ is formed as in Table 4.

**Step 7.**Extend the decision matrix according to the risk attitude of decision makers. Assume that decision makers are risk-averse, then add the smallest interval truth-membership, the largest interval indeterminacy-membership and the largest real interval falsity-membership. The extended decision matrix ${\tilde{D}}^{\prime}={({\tilde{a}}_{ij}^{\prime})}_{5\times 4}$ is omitted due to space limitations.

**Step 8.**Calculate the extended weighted decision matrix ${\tilde{D}}^{\u2033}={({\tilde{a}}_{ij}^{\u2033})}_{5\times 4}$, ${\tilde{a}}_{ij}^{\u2033}={w}_{j}{\tilde{a}}_{ij}^{\prime}$. The results are shown in Table 5.

**Step 10.**The closeness coefficients can be calculated by using Equation (20) as

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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${\mathit{C}}_{1}$ | ${\mathit{C}}_{2}$ | |

${C}_{1}$ | 0 | $\langle \{[0.5,0.6],[0.4,0.5]\},\{[0.2,0.3]\},\{[0.3,0.4],[0.4,0.5]\}\rangle $ |

${C}_{2}$ | $\langle \{[0.6,0.7]\},\{[0.4,0.5]\},\{[0.4,0.5]\}\rangle $ | 0 |

${C}_{3}$ | $\langle \{[0.5,0.6],[0.6,0.7]\},\{[0.5,0.6]\},\{[0.3,0.4]\}\rangle $ | $\langle \{[0.5,0.6]\},\{[0.2,0.3]\},\{[0.3,0.4]\}\rangle $ |

${C}_{4}$ | $\langle \{[0.3,0.4],[0.4,0.5]\},\{[0.6,0.7]\},\{[0.5,0.6]\}\rangle $ | $\langle \{[0.3,0.4]\},\{[0.3,0.4]\},\{[0.3,0.4],[0.4,0.5]\}\rangle $ |

${\mathit{C}}_{\mathbf{3}}$ | ${\mathit{C}}_{\mathbf{4}}$ | |

${C}_{1}$ | $\langle \{[0.2,0.3],[0.3,0.4]\},\{[0.1,0.2]\},\{[0.4,0.5],[0.5,0.6]\}\rangle $ | $\langle \{[0.6,0.7]\},\{[0.3,0.4]\},\{[0.4,0.5]\}\rangle $ |

${C}_{2}$ | $\langle \{[0.4,0.5]\},\{[0.5,0.6]\},\{[0.3,0.4],[0.4,0.5]\}\rangle $ | $\langle \{[0.7,0.8]\},\{[0.5,0.6]\},\{[0.6,0.7]\}\rangle $ |

${C}_{3}$ | 0 | $\langle \{[0.3,0.4],[0.4,0.5]\},\{[0.5,0.6]\},\{[0.4,0.5]\}\rangle $ |

${C}_{4}$ | $\langle \{[0.6,0.7]\},\{[0.4,0.5],[0.5,0.6]\},\{[0.3,0.4]\}\rangle $ | 0 |

${\mathit{C}}_{1}$ | ${\mathit{C}}_{2}$ | |

${C}_{1}$ | 0 | $\langle \{[0.4523,0.5528]\},\{[0.2000,0.3000]\},\{[0.3519,0.4523]\}\rangle $ |

${C}_{2}$ | $\langle \{[0.6000,0.7000]\},\{[0.4000,0.5000]\},\{[0.4000,0.5000]\}\rangle $ | 0 |

${C}_{3}$ | $\langle \{[0.5528,0.6536]\},\{[0.5000,0.6000]\},\{[0.3000,0.4000]\}\rangle $ | $\langle \{[0.5000,0.6000]\},\{[0.2000,0.3000]\},\{[0.3000,0.4000]\}\rangle $ |

${C}_{4}$ | $\langle \{[0.3519,0.4523]\},\{[0.6000,0.7000]\},\{[0.5000,0.6000]\}\rangle $ | $\langle \{[0.3000,0.4000]\},\{[0.3000,0.4000]\},\{[0.3519,0.4523]\}\rangle $ |

${\mathit{C}}_{\mathbf{3}}$ | ${\mathit{C}}_{\mathbf{4}}$ | |

${C}_{1}$ | $\langle \{[0.2517,0.3519]\},\{[0.1000,0.2000]\},\{[0.4523,0.5528]\}\rangle $ | $\langle \{[0.6000,0.7000]\},\{[0.3000,0.4000]\},\{[0.4000,0.5000]\}\rangle $ |

${C}_{2}$ | $\langle \{[0.4000,0.5000]\},\{[0.5000,0.6000]\},\{[0.3519,0.4523]\}\rangle $ | $\langle \{[0.7000,0.8000]\},\{[0.5000,0.6000]\},\{[0.6000,0.7000]\}\rangle $ |

${C}_{3}$ | 0 | $\langle \{[0.3519,0.4523]\},\{[0.5000,0.6000]\},\{[0.4000,0.5000]\}\rangle $ |

${C}_{4}$ | $\langle \{[0.6000,0.7000]\},\{[0.4523,0.5528]\},\{[0.3000,0.4000]\}\rangle $ | 0 |

${\mathit{C}}_{1}$ | |

${C}_{1}$ | $<\{[0.4330,1.2162]\},\{[0.2093,0.6268]\},\{[0.4389,1.5669]\}>$ |

${C}_{2}$ | $<\{[0.7688,1.7019]\},\{[0.6125,1.2709]\},\{[0.6885,1.9523]\}>$ |

${C}_{3}$ | $<\{[0.6894,1.5117]\},\{[0.5852,1.1798]\},\{[0.5306,1.5928]\}>$ |

${C}_{4}$ | $<\{[0.5743,1.3708]\},\{[0.6522,1.2774]\},\{[0.6594,1.8002]\}>$ |

${\mathit{C}}_{\mathbf{2}}$ | |

${C}_{1}$ | $<\{[0.5578,1.3127]\},\{[0.2100,0.5311]\},\{[0.5502,0.6134]\}>$ |

${C}_{2}$ | $<\{[0.4641,1.3004]\},\{[0.2238,0.6509]\},\{[0.4290,1.5442]\}>$ |

${C}_{3}$ | $<\{[0.6083,1.3799]\},\{[0.2907,0.7167]\},\{[0.4768,1.4449]\}>$ |

${C}_{4}$ | $<\{[0.5003,1.2400]\},\{[0.3471,0.7935]\},\{[0.5418,1.5864]\}>$ |

${\mathit{C}}_{\mathbf{3}}$ | |

${C}_{1}$ | $<\{[0.5063,1.2744]\},\{[0.2052,0.5828]\},\{[0.6604,1.8433]\}>$ |

${C}_{2}$ | $<\{[0.6467,1.5260]\},\{[0.5215,1.0551]\},\{[0.7369,2.0188]\}>$ |

${C}_{3}$ | $<\{[0.4087,1.5260]\},\{[0.2450,0.6976]\},\{[0.3858,1.4259]\}>$ |

${C}_{4}$ | $<\{[0.6087,1.3496]\},\{[0.4682,0.9824]\},\{[0.5914,1.7585]\}>$ |

${\mathit{C}}_{\mathbf{4}}$ | |

${C}_{1}$ | $<\{[0.7142,1.5870],\{[0.3169,0.7327]\},\{[0.6558,1.8664]\}>$ |

${C}_{2}$ | $<\{[0.8569,1.8610],\{[0.5870,1.1843]\},\{[0.7557,2.0658]\}>$ |

${C}_{3}$ | $<\{[0.6676,1.5697],\{[0.5242,1.0668]\},\{[0.5912,1.6918]\}>$ |

${C}_{4}$ | $<\{[0.4545,1.2810],\{[0.3352,0.8757]\},\{[0.4536,1.6064]\}>$ |

Alternative | ${\mathit{C}}_{1}$ | ${\mathit{C}}_{2}$ |

${A}_{1}$ | $<\{[0.4,0.5],[0.5,0.6]\},\{[0.3,0.4]\},\{[0.3,0.4],$ | $<\{[0.2,0.3],[0.3,0.4]\},\{[0.1,0.2]\},\{[0.4,0.5],$ |

$[0.4,0.5]\}>$ | $[0.5,0.6]\}>$ | |

${A}_{2}$ | $<\{[0.6,0.7]\},\{[0.1,0.2]\},\{[0.2,0.3]\}>$ | $<\{[0.6,0.7]\},\{[0.1,0.2]\},\{[0.1,0.2]\}>$ |

${A}_{3}$ | $<\{[0.5,0.6]\},\{[0.2,0.3]\},\{[0.3,0.4]\}>$ | $<\{[0.5,0.6]\},\{[0.2,0.3]\},\{[0.2,0.3]\}>$ |

${A}_{4}$ | $<\{[0.6,0.7],[0.7,0.8]\},\{[0.1,0.2]\},\{[0.2,0.3]\}>$ | $<\{[0.3,0.4]\},\{[0.3,0.4]\},\{[0.1,0.2],[0.3,0.4]\}>$ |

${A}_{5}$ | $<\{[0.8,0.9]\},\{[0.3,0.4]\},\{[0.1,0.2]\}>$ | $<\{[0.4,0.5]\},\{[0.4,0.5]\},\{[0.5,0.6]\}>$ |

${\mathit{C}}_{\mathbf{3}}$ | ${\mathit{C}}_{\mathbf{4}}$ | |

${A}_{1}$ | $<\{[0.4,0.5],[0.5,0.6]\},\{[0.1,0.2]\},\{[0.3,0.4]\}>$ | $<\{[0.7,0.8],[0.8,0.9]\},\{[0.1,0.2]\},\{[0.2,0.3]\}>$ |

${A}_{2}$ | $<\{[0.3,0.4]\},\{[0.1,0.2]\},\{[0.1,0.2],[0.2,0.3]\}>$ | $<\{[0.3,0.4]\},\{[0.3,0.4],[0.4,0.5]\},\{[0.4,0.5],$ |

$[0.5,0.6]\}>$ | ||

${A}_{3}$ | $<\{[0.3,0.4],[0.5,0.7]\},\{[0.2,0.3]\},\{[0.2,0.3]\}>$ | $<\{[0.8,0.9]\},\{[0.5,0.6]\},\{[0.3,0.4]\}>$ |

${A}_{4}$ | $<\{[0.7,0.8]\},\{[0.1,0.2]\},\{[0.1,0.2]\}>$ | $<\{[0.3,0.5]\},\{[0.2,0.3],[0.3,0.5]\},\{[0.5,0.6]\}>$ |

${A}_{5}$ | $<\{[0.2,0.3]\},\{[0.4,0.5]\},\{[0.3,0.4]\}>$ | $<\{[0.4,0.5]\},\{[0.1,0.2],[0.2,0.3]\},\{[0.2,0.3]\}>$ |

Alternative | ${\mathit{C}}_{1}$ |

${A}_{1}$ | $<\{[0.1045,0.1391],[0.1391,0.1796]\},\{[0.6081,0.7064],[0.8609,0.8955]\},\{[0.7064,0.7710],[0.7710,0.8204]\}>$ |

${A}_{2}$ | $<\{[0.0471,0.0741],[0.1796,0.2290]\},\{[0.6081,0.7064],[0.8609,0.8955]\},\{[0.7064,0.7710],[0.8609,0.8955]\}>$ |

${A}_{3}$ | $<\{[0.0471,0.0741],[0.1391,0.1796]\},\{[0.7064,0.7710],[0.8609,0.8955]\},\{[0.7710,0.8204],[0.8609,0.8955]\}>$ |

${A}_{4}$ | $<\{[0.1796,0.2290],[0.2290,0.2936]\},\{[0.7064,0.7710],[0.8609,0.8955]\},\{[0.7064,0.7710],[0.8609,0.8955]\}>$ |

${A}_{5}$ | $<\{[0.0471,0.0741],[0.2936,0.3919]\},\{[0.7710,0.8204],[0.8609,0.8955]\},\{[0.6081,0.7064],[0.8609,0.8955]\}>$ |

${\mathit{C}}_{\mathbf{2}}$ | |

${A}_{1}$ | $<\{[0.0607,0.0953],[0.0953,0.1336]\},\{[0.7131,0.7731],[0.8231,0.8664]\},\{[0.7731,0.8231],[0.8231,0.8664]\}>$ |

${A}_{2}$ | $<\{[0.0607,0.0953],[0.2269,0.2869]\},\{[0.5236,0.6364],[0.8231,0.8664]\},\{[0.5238,0.6364],[0.8231,0.8664]\}>$ |

${A}_{3}$ | $<\{[0.0607,0.0953],[0.1769,0.2269]\},\{[0.6364,0.7131],[0.8231,0.8664]\},\{[0.6364,0.7131],[0.8231,0.8664]\}>$ |

${A}_{4}$ | $<\{[0.0607,0.0953],[0.0953,0.1336]\},\{[0.7131,0.7731],[0.8231,0.8664]\},\{[0.5238,0.6364],[0.7131,0.7731]\}>$ |

${A}_{5}$ | $<\{[0.0607,0.0953],[0.1336,0.1769]\},\{[0.7731,0.8231],[0.8231,0.8664]\},\{[0.8231,0.8664],[0.8231,0.8664]\}>$ |

${\mathit{C}}_{\mathbf{3}}$ | |

${A}_{1}$ | $<\{[0.1280,0.1696],[0.1696,0.2179]\},\{[0.5393,0.6494],[0.8304,0.8720]\},\{[0.7240,0.7821],[0.8304,0.8720]\}>$ |

${A}_{2}$ | $<\{[0.0581,0.0912],[0.0912,0.1280]\},\{[0.5393,0.6494],[0.8304,0.8720]\},\{[0.5393,0.6494],[0.8304,0.8720]\}>$ |

${A}_{3}$ | $<\{[0.0912,0.1280],[0.1696,0.2760]\},\{[0.6494,0.7240],[0.8304,0.8720]\},\{[0.6494,0.7240],[0.8304,0.8720]\}>$ |

${A}_{4}$ | $<\{[0.0581,0.0912],[0.2760,0.3506]\},\{[0.5393,0.6494],[0.8304,0.8720]\},\{[0.5393,0.6494],[0.8304,0.8720]\}>$ |

${A}_{5}$ | $<\{[0.0581,0.0912],[0.0581,0.0912]\},\{[0.7821,0.8304],[0.8304,0.8720]\},\{[0.7240,0.7821],[0.8304,0.8720]\}>$ |

${\mathit{C}}_{\mathbf{4}}$ | |

${A}_{1}$ | $<\{[0.1937,0.2464],[0.2464,0.3149]\},\{[0.5821,0.6851],[0.8497,0.8869]\},\{[0.6851,0.7536],[0.8497,0.8869]\}>$ |

${A}_{2}$ | $<\{[0.0511,0.0804],[0.0804,0.1131]\},\{[0.7536,0.8063],[0.8063,0.8497]\},\{[0.8063,0.8497],[0.8497,0.8869]\}>$ |

${A}_{3}$ | $<\{[0.0511,0.0804],[0.3149,0.4179]\},\{[0.8497,0.8869],[0.8497,0.8869]\},\{[0.7536,0.8063],[0.8497,0.8869]\}>$ |

${A}_{4}$ | $<\{[0.0511,0.0804],[0.0804,0.1503]\},\{[0.6851,0.7536],[0.7536,0.8063]\},\{[0.8497,0.8869],[0.8497,0.8869]\}>$ |

${A}_{5}$ | $<\{[0.0511,0.0804],[0.1131,0.1503]\},\{[0.5821,0.6851],[0.6851,0.7536]\},\{[0.6851,0.7536],[0.8497,0.8869]\}>$ |

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

${C}_{1}$ | 0 | $\langle \{0.5,0.4\},\{0.2\},\{0.3,0.4\}\rangle $ | $\langle \{0.2,0.3\},\{0.1\},\{0.4,0.5\}\rangle $ | $\langle \{0.6\},\{0.3\},\{0.4\}\rangle $ |

${C}_{2}$ | $\langle \{0.6\},\{0.4\},\{0.4\}\rangle $ | 0 | $\langle \{0.4\},\{0.5\},\{0.3,0.4\}\rangle $ | $\langle \{0.7\},\{0.5\},\{0.6\}\rangle $ |

${C}_{3}$ | $\langle \{0.5,0.6\},\{0.5\},\{0.3\}\rangle $ | $\langle \{0.5\},\{0.2\},\{0.3\}\rangle $ | 0 | $\langle \{0.3,0.4\},\{0.5\},\{0.4\}\rangle $ |

${C}_{4}$ | $\langle \{0.3,0.4\},\{0.6\},\{0.5\}\rangle $ | $\langle \{0.3\},\{0.3\},\{0.3,0.4\}\rangle $ | $\langle \{0.6\},\{0.4,0.5\},\{0.3\}\rangle $ | 0 |

${\mathit{C}}_{1}$ | ${\mathit{C}}_{2}$ | |

${C}_{1}$ | $<\{0.9927\},\{0.4213\},\{1.2141\}>$ | $<\{1.0715\},\{0.3422\},\{1.0696\}>$ |

${C}_{2}$ | $<\{1.4835\},\{1.0615\},\{1.5491\}>$ | $<\{1.0610\},\{0.4505\},\{0.9694\}>$ |

${C}_{3}$ | $<\{1.3203\},\{0.9732\},\{1.4587\}>$ | $<\{1.1606\},\{0.5000\},\{1.0434\}>$ |

${C}_{4}$ | $<\{1.1403\},\{1.0742\},\{1.5408\}>$ | $<\{0.9952\},\{0.5777\},\{1.0985\}>$ |

${\mathit{C}}_{\mathbf{3}}$ | ${\mathit{C}}_{\mathbf{4}}$ | |

${C}_{1}$ | $<\{1.0261\},\{0.3657\},\{1.1854\}>$ | $<\{1.3569\},\{0.5202\},\{1.3345\}>$ |

${C}_{2}$ | $<\{1.2826\},\{0.8494\},\{1.2351\}>$ | $<\{1.6377\},\{0.96788\},\{1.5303\}>$ |

${C}_{3}$ | $<\{0.9437\},\{0.4902\},\{0.9466\}>$ | $<\{1.3462\},\{0.8580\},\{1.3264\}>$ |

${C}_{4}$ | $<\{1.1247\},\{0.7655\},\{1.1539\}>$ | $<\{1.0450\},\{0.6625\},\{1.1581\}>$ |

Alternative | ${\mathit{C}}_{1}$ | ${\mathit{C}}_{2}$ |

${A}_{1}$ | $<\{0.4,0.5\},\{0.3\},\{0.3,0.4\}>$ | $<\{0.2,0.3\},\{0.1\},\{0.4,0.5\}>$ |

${A}_{2}$ | $<\{0.6\},\{0.1\},\{0.2\}>$ | $<\{0.6\},\{0.1\},\{0.1\}>$ |

${A}_{3}$ | $<\{0.5\},\{0.2\},\{0.3\}>$ | $<\{0.5\},\{0.2\},\{0.2\}>$ |

${A}_{4}$ | $<\{0.6,0.7\},\{0.1\},\{0.2\}>$ | $<\{0.3\},\{0.3\},\{0.1,0.3\}>$ |

${A}_{5}$ | $<\{0.8\},\{0.3\},\{0.1\}>$ | $<\{0.4\},\{0.4\},\{0.5\}>$ |

${\mathit{C}}_{\mathbf{3}}$ | ${\mathit{C}}_{\mathbf{4}}$ | |

${A}_{1}$ | $<\{0.4,0.5\},\{0.1\},\{0.3\}>$ | $<\{0.7,0.8\},\{0.1\},\{0.2\}>$ |

${A}_{2}$ | $<\{0.3\},\{0.1\},\{0.1,0.2\}>$ | $<\{0.3\},\{0.3,0.4\},\{0.4,0.5\}>$ |

${A}_{3}$ | $<\{0.3,0.5\},\{0.2\},\{0.2\}>$ | $<\{0.8\},\{0.5\},\{0.3\}>$ |

${A}_{4}$ | $<\{0.7\},\{0.1\},\{0.1\}>$ | $<\{0.3\},\{0.2,0.3\},\{0.5\}>$ |

${A}_{5}$ | $<\{0.2\},\{0.4\},\{0.3\}>$ | $<\{0.4\},\{0.1,0.2\},\{0.2\}>$ |

Alternative | ${\mathit{C}}_{1}$ | ${\mathit{C}}_{2}$ |

${A}_{1}$ | $<\{0.1339,0.1773\},\{0.7125,0.8227\},\{0.7125,0.7726\}>$ | $<\{0.0463,0.0730\},\{0.6132,0.8631\},\{0.8231,0.8631\}>$ |

${A}_{2}$ | $<\{0.0609,0.2274\},\{0.5230,0.8227\},\{0.6357,0.8227\}>$ | $<\{0.0463,0.1769\},\{0.6132,0.8631\},\{0.6132,0.8631\}>$ |

${A}_{3}$ | $<\{0.0609,0.1773\},\{0.6357,0.8227\},\{0.7125,0.8227\}>$ | $<\{0.0463,0.1369\},\{0.7105,0.8631\},\{0.7105,0.7744\}>$ |

${A}_{4}$ | $<\{0.2274,0.2875\},\{0.5230,0.8227\},\{0.6357,0.8227\}>$ | $<\{0.0463,0.0730\},\{0.7744,0.8631\},\{0.6132,0.8632\}>$ |

${A}_{5}$ | $<\{0.0609,0.3643\},\{0.7125,0.8227\},\{0.5230,0.8227\}>$ | $<\{0.0463,0.1028\},\{0.8231,0.8631\},\{0.8631,0.8631\}>$ |

${\mathit{C}}_{\mathbf{3}}$ | ${\mathit{C}}_{\mathbf{4}}$ | |

${A}_{1}$ | $<\{0.1090,0.1450\},\{0.5943,0.8550\},\{0.7618,0.8550\}>$ | $<\{0.2863,0.3630\},\{0.5246,0.8235\},\{0.6370,0.8235\}>$ |

${A}_{2}$ | $<\{0.0492,0.0774\},\{0.5943,0.8550\},\{0.5943,0.6951\}>$ | $<\{0.0606,0.0951\},\{0.7131,0.7736\},\{0.7736,0.8235\}>$ |

${A}_{3}$ | $<\{0.0774,0.1450\},\{0.6951,0.8550\},\{0.6951,0.8550\}>$ | $<\{0.0606,0.3630\},\{0.8235,0.8235\},\{0.7137,0.8235\}>$ |

${A}_{4}$ | $<\{0.0492,0.2382\},\{0.5943,0.8550\},\{0.5943,0.8550\}>$ | $<\{0.0606,0.0951\},\{0.6370,0.8235\},\{0.8235,0.8235\}>$ |

${A}_{5}$ | $<\{0.0492,0.0492\},\{0.8130,0.8550\},\{0.7618,0.8550\}>$ | $<\{0.0606,0.1334\},\{0.5246,0.8235\},\{0.6370,0.8235\}>$ |

Method | Information by Multi-Valued Interval Neutrosophic Fuzzy Value | Whether the Weights Are Determined by DEMATEL | Whether Alternatives Are Ranked by TOPSIS |
---|---|---|---|

Method 1 | No | Yes | No |

Method 2 | Yes | No | No |

Our proposed method | Yes | Yes | Yes |

Method | Alternatives Ranking | Optimal Alternative |
---|---|---|

Zhao et al. [45] | ${A}_{4}\succ {A}_{2}\succ {A}_{3}\succ {A}_{1}\succ {A}_{5}$ | ${A}_{4}$ |

Chi and Liu [29] | ${A}_{4}\succ {A}_{2}\succ {A}_{3}\succ {A}_{1}\succ {A}_{5}$ | ${A}_{4}$ |

Our proposed method | ${A}_{4}\succ {A}_{1}\succ {A}_{2}\succ {A}_{3}\succ {A}_{5}$ | ${A}_{4}$ |

© 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**

Yang, W.; Pang, Y.
New Multiple Attribute Decision Making Method Based on DEMATEL and TOPSIS for Multi-Valued Interval Neutrosophic Sets. *Symmetry* **2018**, *10*, 115.
https://doi.org/10.3390/sym10040115

**AMA Style**

Yang W, Pang Y.
New Multiple Attribute Decision Making Method Based on DEMATEL and TOPSIS for Multi-Valued Interval Neutrosophic Sets. *Symmetry*. 2018; 10(4):115.
https://doi.org/10.3390/sym10040115

**Chicago/Turabian Style**

Yang, Wei, and Yongfeng Pang.
2018. "New Multiple Attribute Decision Making Method Based on DEMATEL and TOPSIS for Multi-Valued Interval Neutrosophic Sets" *Symmetry* 10, no. 4: 115.
https://doi.org/10.3390/sym10040115