# Risky Multi-Attribute Decision-Making Method Based on the Interval Number of Normal Distribution

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theoretical Basis

#### 2.1. Interval Number

**Definition**

**1.**

**Definition**

**2.**

#### 2.2. Interval Number of Normal Distribution

**Definition**

**3.**

#### 2.3. Sequencing for the Interval Number of Normal Distribution

**Theorem**

**1.**

**Definition**

**4.**

**Definition**

**5.**

**Theorem**

**2.**

## 3. Risky Multi-Attribute Decision-Making Method

#### 3.1. Problem Description

#### 3.2. Normalization of Decision-Making Matrix

#### 3.3. Decision-Making Steps

**Step 1**According to Formulas (3) and (4), conduct the normalization to risky decision-making matrix by using the range transformation method, and obtain the decision-making matrix $R$.

**Step 2**In view of the limitations of the common normalization method, according to Formula (5), transform the attribute value again based on the preference of the decision makers, and obtain the decision-making matrix ${R}_{1}$.

**Step 3**For the decision-making matrix after normalization, according to Formula (6), conduct weighting operation to attribute value through occurrence probability ${P}_{k}$ in different natural states ${\theta}_{k}$, obtain decision-making matrix ${R}_{2}=\left({r}_{xy}^{L},{r}_{xy}^{U}\right)$ [25].

**Step 4**Combine with the weight vector of attribute $w$, calculate the deviation ${V}_{x}$ of each scheme according to Formula (7).

**Step 5**According to the sequencing method for the interval value of normal distribution derived from the above steps, establish the possibility degree matrix of pairwise comparison $P$, and hereafter, conduct the sequencing of all schemes.

## 4. Calculating Example Analysis

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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Scheme | C1 | C2 | C3 | ||||||
---|---|---|---|---|---|---|---|---|---|

${\mathit{\theta}}_{1}$ | ${\mathit{\theta}}_{2}$ | ${\mathit{\theta}}_{3}$ | ${\mathit{\theta}}_{1}$ | ${\mathit{\theta}}_{2}$ | ${\mathit{\theta}}_{3}$ | ${\mathit{\theta}}_{1}$ | ${\mathit{\theta}}_{2}$ | ${\mathit{\theta}}_{3}$ | |

A1 | [80,90] | [90,100] | [90,110] | [100,120] | [80,100] | [70,80] | [12,16] | [9,12] | [6,8] |

A2 | [90,100] | [100,110] | [110,120] | [110,120] | [90,100] | [80,90] | [12,18] | [10,15] | [7,10] |

A3 | [90,110] | [100,120] | [110,130] | [120,130] | [100,110] | [80,100] | [15,22] | [13,20] | [8,12] |

A4 | [100,110] | [110,130] | [120,130] | [100,110] | [80,90] | [60,80] | [18,23] | [15,20] | [6,10] |

A5 | [110,120] | [115,130] | [120,140] | [120,150] | [100,120] | [90,100] | [20,25] | [12,18] | [8,10] |

Scheme | C1 | C2 | C3 | ||||||
---|---|---|---|---|---|---|---|---|---|

${\mathit{\theta}}_{1}$ | ${\mathit{\theta}}_{2}$ | ${\mathit{\theta}}_{3}$ | ${\mathit{\theta}}_{1}$ | ${\mathit{\theta}}_{2}$ | ${\mathit{\theta}}_{3}$ | ${\mathit{\theta}}_{1}$ | ${\mathit{\theta}}_{2}$ | ${\mathit{\theta}}_{3}$ | |

A1 | [0.75,1] | [0.75,1] | [0.6,1] | [0,0.4] | [0,0.5] | [0.25,0.5] | [0,0.308] | [0,0.273] | [0,0.333] |

A2 | [0.5,0.75] | [0.5,0.75] | [0.4,0.6] | [0.2,0.4] | [0.25,0.5] | [0.5,0.75] | [0,0.462] | [0.091,0.545] | [0.167,0.667] |

A3 | [0.25,0.75] | [0.25,0.75] | [0.2,0.6] | [0.4,0.6] | [0.5,0.75] | [0.5,1] | [0.231,0.769] | [0.364,1] | [0.333,1] |

A4 | [0.25,0.5] | [0,0.5] | [0.2,0.4] | [0,0.2] | [0,0.25] | [0,0.5] | [0.462,846] | [0.545,1] | [0,0.667] |

A5 | [0,0.25] | [0,0.375] | [0,0.4] | [0.4,1] | [0.5,1] | [0.75,1] | [0.615,1] | [0.273,0.818] | [0.333,0.667] |

Scheme | C1 | C2 | C3 | ||||||
---|---|---|---|---|---|---|---|---|---|

${\mathit{\theta}}_{1}$ | ${\mathit{\theta}}_{2}$ | ${\mathit{\theta}}_{3}$ | ${\mathit{\theta}}_{1}$ | ${\mathit{\theta}}_{2}$ | ${\mathit{\theta}}_{3}$ | ${\mathit{\theta}}_{1}$ | ${\mathit{\theta}}_{2}$ | ${\mathit{\theta}}_{3}$ | |

A1 | [0.779,1.000] | [0.779,1.000] | [0.670,1.000] | [0.368,0.549] | [0.368,0.607] | [0.472,0.607] | [0.368,0.500] | [0.368,0.0.483] | [0.368,0.513] |

A2 | [0.607,0.779] | [0.607,0.779] | [0.549,0.670] | [0.449,0.549] | [0.472,0.607] | [0.607,0.779] | [0.368,0.584] | [0.403,0.635] | [0.435,0.717] |

A3 | [0.472,0.779] | [0.472,0.779] | [0.449,0.670] | [0.549,0.670] | [0.607,0.779] | [0.607,1.000] | [0.463,0.794] | [0.529,1.000] | [0.513,1.000] |

A4 | [0.472,0.607] | [0.368,0.607] | [0.449,0.549] | [0.368,0.449] | [0.368,0.472] | [0.368,0.607] | [0.584,0.857] | [0.635,1.000] | [0.368,0.717] |

A5 | [0.368,0.472] | [0.368,0.535] | [0.368,0.549] | [0.549,1.000] | [0.607,1.000] | [0.779,1.000] | [0.681,1.000] | [0.483,0.834] | [0.513,0.717] |

${\mathit{k}}_{\mathit{i}}\mathbf{Value}$ | Sequencing Result of Each Scheme | ||
---|---|---|---|

K1=1 | K2=1 | K3=1 | A3> A5> A1> A2> A4 |

K1=1 | K2=1 | K3=3 | A3> A5> A1> A4> A2 |

K1=1 | K2=3 | K3=1 | A3> A5> A1> A4> A2 |

K1=1 | K2=3 | K3=3 | A3> A5> A1> A2> A4 |

K1=3 | K2=1 | K3=1 | A5> A3> A2> A1> A4 |

K1=3 | K2=1 | K3=3 | A5> A3> A2> A1> A4 |

K1=3 | K2=3 | K3=1 | A3> A5> A2> A4> A1 |

K1=3 | K2=3 | K3=3 | A3> A5> A1> A2> A4 |

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**MDPI and ACS Style**

Fu, S.; Qu, X.-L.; Xiao, Y.-Z.; Zhou, H.-J.; Fan, G.-B.
Risky Multi-Attribute Decision-Making Method Based on the Interval Number of Normal Distribution. *Symmetry* **2020**, *12*, 264.
https://doi.org/10.3390/sym12020264

**AMA Style**

Fu S, Qu X-L, Xiao Y-Z, Zhou H-J, Fan G-B.
Risky Multi-Attribute Decision-Making Method Based on the Interval Number of Normal Distribution. *Symmetry*. 2020; 12(2):264.
https://doi.org/10.3390/sym12020264

**Chicago/Turabian Style**

Fu, Sha, Xi-Long Qu, Ye-Zhi Xiao, Hang-Jun Zhou, and Guo-Bing Fan.
2020. "Risky Multi-Attribute Decision-Making Method Based on the Interval Number of Normal Distribution" *Symmetry* 12, no. 2: 264.
https://doi.org/10.3390/sym12020264