# Entropy-Based Risk Control of Geological Disasters in Mountain Tunnels under Uncertain Environments

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

_{1}, and the dynamic relationship between them makes risk-control more complicated. By considering these two factors, we constructed a reasonable risk control process (Figure 1).

## 2. Risk Control Considering Uncertainty

_{1}). Only those variables with a larger influence will be considered. This preliminary judgment can improve the efficiency of decision-making and reduce blindness.

**Step 1:**Use entropy to calculate the sensitivity of variables, and narrow the scope of alternatives according to sensitivity, so as to improve decision-making efficiency.

**Step 2:**The necessity of reducing uncertainty.

**Step 3:**Decision-making with different attributes.

## 3. Scheme Selection under Multi-Source Attributes

_{9}) < Var(a) < Var(b

_{1}~b

_{8}), $a$ reasonable decision can be made based on variance.

**The first situation**: $R(a)=R(b)$, the decision result cannot be obtained based on EU-E model. If scheme $a$ is (2, 1/4; 4, 1/4; 6, 1/4; 8, 1/4), then scheme $b$ can be ($0,1/2$; $x,1/2$), ($x=10(1-\frac{0.7\lambda}{1-\lambda})$, $0<\lambda <0.5882$ for $x>0$, $E(a)>E(b)$ and $R(a)=R(b)$). $0<\lambda <0.5882$ indicates that the decision-maker is not absolutely risk-seeking, and is more likely to be conservative or neutral. Assuming that the decision-maker is risk neutral, $\lambda =1/2$, then scheme $b$ is (0, 1/2; 8.6, 1/2). According to the variance: $Var(a)=5<Var(b)=18.5$, we should choose $a$, which is consistent with the attitude to risk of the decision-maker. For the risk-averse, scheme $a$ is more reasonable.

**The second situation**: $R(a)>R(b)$. If scheme $a$ is (2, 1/4; 4, 1/4; 6, 1/4; 8, 1/4), then scheme $b$ can be (0, 1/2; $x$, 1/2), ($0<x<10$, $x>10(1-\frac{0.7\lambda}{1-\lambda})$, $0<\lambda <1$ for $E(a)>E(b)$ and $R(a)>R(b)$). When $0<\lambda <0.5882$, $10(1-\frac{0.7\lambda}{1-\lambda})>0$, then we get $10(1-\frac{0.7\lambda}{1-\lambda})<x<10$; when $0.5882<\lambda <1$, $10(1-\frac{0.7\lambda}{1-\lambda})<0$, then we have $0<x<10$.

**The third situation**: $R(a)<R(b)$. We still let scheme $a$ have (2, 1/4; 4, 1/4; 6, 1/4; 8, 1/4), then scheme $b$ can be (0, 1/2; $x$, 1/2), ($0<x<10(1-\frac{0.7\lambda}{1-\lambda})$, $0<\lambda <0.5882$ for $E(a)>E(b)$ and $R(a)<R(b)$). In such a situation, it is more reasonable for decision-makers who are risk-averse or risk-neutral.

**Problem 1**: Select ${a}_{1}$ or ${b}_{1}$?

**Problem 2**: select ${a}_{2}$ or ${b}_{2}$?

## 4. Case Analysis and Discussion

#### 4.1. Engineering Background

#### 4.2. Engineering Application and Discussion

## 5. Conclusions

- (1)
- Through case analysis, possible defects in the existing entropy-hazard decision model in engineering application are discussed, and an improved model of uncertainty evaluation and control is proposed.
- (2)
- Tolerance cost ${T}_{H}^{\prime}$ is an important factor used to screen plans to improve the efficiency of decision-making. Through analysis, it is found that there are deficiencies in the existing calculation method, and a new calculation model is proposed.
- (3)
- The existing expected utility-entropy model is only valid under certain conditions, and there are limitations to its application. Multiple-attribute decision making problems are classified based on factors such as attitude to risk, uncertainty measures, expected utility, variance, etc., and corresponding decision-making methods are proposed according to different decision types.
- (4)
- By analysing different decision-making issues, it is found that the attitude to risk of the decision-makers exerts an important influence on the decision-making results; attitudes to risk will be influenced by the decision-maker’s own experience and the problem itself. Based on this idea, the well-known Allais paradox is reasonably explained by use of the proposed methods.
- (5)
- Taking the Zhiziyuan tunnel as a research object, the method is applied to a real engineering task. The application results show that the proposed method is effective with regard to decision-making about geological disaster risk control schemes, and that it can remedy the deficiencies in the existing method.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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Serial Number | $\mathit{U}$ | ${\mathit{R}}_{1}$ | Serial Number | $\mathit{U}$ | ${\mathit{R}}_{1}$ |
---|---|---|---|---|---|

1 | Small | Small | 3 | Large | Large |

2 | Small | Large | 4 | Large | Small |

Condition 1 | Condition 2 | Condition 3 | Risk Attitude | Suggested Methods |
---|---|---|---|---|

$H(a)=H(b)$ | $E(a)=E(b)$ | $R(a)=R(b)$ | Risk-averse | Variance |

Risk-neutral | Variance | |||

Risk-seeking | If $p{a}_{\mathrm{max}}>q{b}_{\mathrm{max}}$, then choose a, otherwise choose b | |||

$E(a)<E(b)$ ($E(a)>E(b)$) | $R(a)>R(b)$ ($R(a)<R(b)$) | Risk-averse | If $l{a}_{\mathrm{min}}>m{b}_{\mathrm{min}}$, choose a, otherwise choose b | |

Risk-neutral | EU-E model | |||

Risk-seeking | EU-E model | |||

$H(a)>H(b)$ or ($H(a)<H(b)$) | $E(a)=E(b)$ | $R(a)>R(b)$ ($R(a)<R(b)$) | Risk-averse | Variance |

Risk-neutral | Variance | |||

Risk-seeking | If $p{a}_{\mathrm{max}}<q{b}_{\mathrm{max}}$, then choose b, otherwise choose a | |||

$E(a)>E(b)$ ($E(a)<E(b)$) | $R(a)=R(b)$ | Risk-averse | Variance | |

Risk-neutral | Variance | |||

$R(a)>R(b)$ | Risk-averse | Variance | ||

Risk-neutral | There is no uniform criterion, the recommendation is similar to that when risk-averse | |||

Risk-seeking | If $p{a}_{\mathrm{max}}<q{b}_{\mathrm{max}}$, then choose b, otherwise choose a | |||

$R(a)<R(b)$ | Risk-averse | EU-E model | ||

Risk-neutral | EU-E model | |||

$E(a)<E(b)$ ($E(a)>E(b)$) | $R(a)>R(b)$ ($R(a)<R(b)$) | Risk-averse | If $l{a}_{\mathrm{min}}>m{b}_{\mathrm{min}}$, choose a, otherwise choose b | |

Risk-neutral | There is no uniform criterion, the recommendation is similar to that when risk-averse | |||

Risk pursuit | EU or EU-E model |

Schemes | Probability | Consequence | Probability | Consequence | Probability | Consequence |
---|---|---|---|---|---|---|

${a}_{1}$ | 1 | 1 (million) | - | - | - | - |

${b}_{1}$ | 0.01 | 0 | 0.89 | 1 (million) | 0.1 | 5 (million) |

${a}_{2}$ | 0.89 | 0 | 0.11 | 1 (million) | - | - |

${b}_{2}$ | 0.9 | 0 | 0.1 | 5 (million) | - | - |

Schemes | Expense | Schemes | Expense |
---|---|---|---|

${a}_{1}({x}_{1})$ | 103,000 | ${a}_{4}({x}_{1},{x}_{5})$ | 120,000 |

${a}_{2}({x}_{1})$ | 115,000 | ${a}_{5}({x}_{2},{x}_{5})$ | 98,000 |

${a}_{3}({x}_{1},{x}_{2})$ | 110,000 | ${a}_{6}({x}_{1},{x}_{2},{x}_{5})$ | 135,000 |

Attributes | C_{1} (Economic Cost) | C_{2} (Time Cost) | C_{3} (Environment Impact) | C_{4} (Execution Effect) | |
---|---|---|---|---|---|

Schemes | |||||

${a}_{1}({x}_{1})$ | 0.5829 | (0.5, 30%; 0.75, 60%; 1, 10%) | (0.5385, 20%; 0.3846, 50%; 0.2308, 30%) | (0, 20%; 0.3429, 40%; 0.7143, 40%) | |

${a}_{3}({x}_{1},{x}_{2})$ | 0 | (0, 15%; 0.25, 62%; 0.5, 23%) | (0.3846, 40%; 0.1538, 50%; 0, 10%) | (0.5714, 30%; 0.7143, 50%; 1, 20%) | |

${a}_{5}({x}_{2},{x}_{5})$ | 1 | (0.75, 25%; 0.875, 70%; 1, 5%) | (1, 10%, 0.8462, 50%; 0.6923, 40%) | (0.2857, 28%; 0.5714, 52%; 0.8571, 20%) |

Attributes | C_{1} (Economic Cost) | C_{2} (Time Cost) | C_{3} (Environment Impact) | C_{4} (Execution Effect) | Aggregation | ||
---|---|---|---|---|---|---|---|

Schemes | |||||||

${a}_{1}({x}_{1})$ | Entropy | 0 | 0.8979 | 1.0297 | 1.0549 | 0.6995 | |

EU | 0.5829 | 0.7 | 0.3692 | 0.4229 | 0.5102 | ||

${a}_{3}({x}_{1},{x}_{2})$ | Entropy | 0 | 0.9190 | 0.9433 | 1.0297 | 0.6727 | |

EU | 0 | 0.27 | 0.2307 | 0.7286 | 0.2689 | ||

${a}_{5}({x}_{2},{x}_{5})$ | Entropy | 0 | 0.7460 | 0.9433 | 1.0184 | 0.6359 | |

EU | 1 | 0.85 | 0.8 | 0.5485 | 0.8197 |

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## Share and Cite

**MDPI and ACS Style**

Xia, Y.; Xiong, Z.; Wen, Z.; Lu, H.; Dong, X.
Entropy-Based Risk Control of Geological Disasters in Mountain Tunnels under Uncertain Environments. *Entropy* **2018**, *20*, 503.
https://doi.org/10.3390/e20070503

**AMA Style**

Xia Y, Xiong Z, Wen Z, Lu H, Dong X.
Entropy-Based Risk Control of Geological Disasters in Mountain Tunnels under Uncertain Environments. *Entropy*. 2018; 20(7):503.
https://doi.org/10.3390/e20070503

**Chicago/Turabian Style**

Xia, Yuanpu, Ziming Xiong, Zhu Wen, Hao Lu, and Xin Dong.
2018. "Entropy-Based Risk Control of Geological Disasters in Mountain Tunnels under Uncertain Environments" *Entropy* 20, no. 7: 503.
https://doi.org/10.3390/e20070503