# A Method of Determining Multi-Attribute Weights Based on Single-Valued Neutrosophic Numbers and Its Application in TODIM

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

#### 2.1. Neutrosophic Sets

**Definition**

**1**

**.**Let X be a space of points(objects), where a generic element in X is denoted by x. A neutrosophic set(NS) A in X is characterized by truth-membership function ${T}_{A(x)}$, indeterminacy-membership function ${I}_{A(x)}$ and falsity-membership function ${F}_{A(x)}$. ${T}_{A(x)}$, ${I}_{A(x)}$, ${F}_{A(x)}$ are the function of finite discrete subset of $[{0}^{-},{1}^{+}]$. It means ${T}_{A(x)}$, ${I}_{A(x)}$, ${F}_{A(x)}$:$X\to [{0}^{-}$,${1}^{+}]$.

**Definition**

**2.**

**Definition**

**3.**

#### 2.2. Single-Valued Neutrosophic Sets

**Definition**

**4**

**.**Let A is a NS, if ${T}_{A(x)}$:$X\to [0,1]$, ${I}_{A(x)}:X\to [0,1]$, ${F}_{A(x)}:X\to [0,1]$, and $0\le {T}_{A}(x)+{I}_{A}(x)+{F}_{A}(x)\le 3$, then A is a single-valued neutrosophic set(SVNS), and is characterized by $A=\{\langle x,{T}_{A}(x),{I}_{A}(x),{F}_{A}(x)\rangle |x\in X\}$.

**Definition**

**5**

**.**${A}^{C}$ is the complement of A and is defined as

**Definition**

**6**

**.**The normalized Euclidean distance between A and B is

**Definition**

**8**

**.**The entropy of A is

## 3. Problem Description

- (1)
- In the initial stage of operation, enterprises are generally small and medium-sized enterprises, most of which are high-tech enterprises.
- (2)
- The investment period of a venture capital company is at least three to five years, and the investment mode is equity investment. Overall, these shares account for about 30% of the total shares of the enterprise. However, the investor has no control right, and the EN does not need to provide any guarantee or mortgage to the investor.
- (3)
- Investment must be highly specialized and procedural;
- (4)
- Generally, investors will actively participate in the operation and management of EN to provide value-added services.
- (5)
- The withdrawal of capital through listing, mergers and acquisitions or other forms of equity transfer can enable investors to achieve value-added and excess returns.

- (1)
- Team management.Choosing a good investment project mainly depends on whether the company’s team is excellent, especially the leader who leads the team.
- (2)
- Industry prospect.VC firms generally prefer industries with future development potential.
- (3)
- Competitiveness. Enterprises with core competitiveness have more advantages than other competitors in the same industry.
- (4)
- Professional business model, excellent profit model and unique marketing model.
- (5)
- Both operating income and the proportion of operating profit are high.
- (6)
- The structures of ownership, top management, enterprises, customers and suppliers are very clear.

## 4. TODIM Method for Single-Valued Neutrosophic Multiple Attribute Decision-Making with Optimal Weight

## 5. Determination of Optimal Weight

#### 5.1. Entropy Weight Method

#### 5.2. Weight Optimization Model

## 6. Practical Example

#### 6.1. Method 1

#### 6.2. Method 2

#### 6.3. Analysis

## 7. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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$\mathit{\lambda}$ | $\mathit{\mu}$ | ${\mathit{w}}_{1}$ | ${\mathit{w}}_{2}$ | ${\mathit{w}}_{3}$ | ${\mathit{A}}_{1}$ | ${\mathit{A}}_{2}$ | ${\mathit{A}}_{3}$ | Ranking Order | FC |
---|---|---|---|---|---|---|---|---|---|

0.1 | 0.9 | 0.398 | 0.299 | 0.303 | 1 | 0.604 | 0 | ${A}_{3}$ < ${A}_{2}$ < ${A}_{1}$ | ${A}_{1}$ |

0.3 | 0.7 | 0.394 | 0.297 | 0.309 | 1 | 0.550 | 0 | ${A}_{3}$ < ${A}_{2}$ < ${A}_{1}$ | ${A}_{1}$ |

0.5 | 0.5 | 0.391 | 0.294 | 0.315 | 1 | 0.512 | 0 | ${A}_{3}$ < ${A}_{2}$ < ${A}_{1}$ | ${A}_{1}$ |

0.7 | 0.3 | 0.387 | 0.292 | 0.321 | 1 | 0.484 | 0 | ${A}_{3}$ < ${A}_{2}$ < ${A}_{1}$ | ${A}_{1}$ |

0.9 | 0.1 | 0.383 | 0.290 | 0.327 | 1 | 0.464 | 0 | ${A}_{3}$ < ${A}_{2}$ < ${A}_{1}$ | ${A}_{1}$ |

$\mathit{\alpha}$ | $\mathit{\beta}$ | ${\mathit{w}}_{1}$ | ${\mathit{w}}_{2}$ | ${\mathit{w}}_{3}$ | ${\mathit{A}}_{1}$ | ${\mathit{A}}_{2}$ | ${\mathit{A}}_{3}$ | Ranking Order | FC |
---|---|---|---|---|---|---|---|---|---|

0.1 | 0.9 | 0.572 | 0.296 | 0.132 | 1 | 0.605 | 0 | ${A}_{3}$ < ${A}_{2}$ < ${A}_{1}$ | ${A}_{1}$ |

0.3 | 0.7 | 0.515 | 0.288 | 0.196 | 1 | 0.552 | 0 | ${A}_{3}$ < ${A}_{2}$ < ${A}_{1}$ | ${A}_{1}$ |

0.5 | 0.5 | 0.459 | 0.280 | 0.261 | 1 | 0.516 | 0 | ${A}_{3}$ < ${A}_{2}$ < ${A}_{1}$ | ${A}_{1}$ |

0.7 | 0.3 | 0.402 | 0.273 | 0.325 | 1 | 0.489 | 0 | ${A}_{3}$ < ${A}_{2}$ < ${A}_{1}$ | ${A}_{1}$ |

0.9 | 0.1 | 0.346 | 0.265 | 0.389 | 1 | 0.471 | 0 | ${A}_{3}$ < ${A}_{2}$ < ${A}_{1}$ | ${A}_{1}$ |

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**MDPI and ACS Style**

Xu, D.; Hong, Y.; Xiang, K.
A Method of Determining Multi-Attribute Weights Based on Single-Valued Neutrosophic Numbers and Its Application in TODIM. *Symmetry* **2019**, *11*, 506.
https://doi.org/10.3390/sym11040506

**AMA Style**

Xu D, Hong Y, Xiang K.
A Method of Determining Multi-Attribute Weights Based on Single-Valued Neutrosophic Numbers and Its Application in TODIM. *Symmetry*. 2019; 11(4):506.
https://doi.org/10.3390/sym11040506

**Chicago/Turabian Style**

Xu, Dongsheng, Yanran Hong, and Kaili Xiang.
2019. "A Method of Determining Multi-Attribute Weights Based on Single-Valued Neutrosophic Numbers and Its Application in TODIM" *Symmetry* 11, no. 4: 506.
https://doi.org/10.3390/sym11040506