# Decision-Making Model under Risk Assessment Based on Entropy

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Classification of Risk Problem

**Case**

**1.**

**Case**

**2.**

## 3. Discussion of the Risk Assessment Model

#### 3.1. Applicability of the R = P × C Classical Model

- The line of extra-long tunnel is designed to be of a gable slope for drainage.
- Comprehensive advanced geological forecasting, which includes conventional geological methods, comprehensive advanced geophysical exploration, and horizontal borehole advanced forecasting (five holes of every main tunnel section, and three holes of every parallel guide-pit tunnel), is applied in high-risk active fault regions and fault fracture zones.
- Comprehensive advanced geological forecasting, which includes conventional geological methods, comprehensive advanced geophysical exploration, and horizontal borehole advanced forecasting (three holes for every main tunnel section, and one hole for every parallel guide-pit tunnel), is applied in the syncline and anticline of the soluble rock region.
- According to the surrounding rock and water enrichment situation, a parallel driftway between the main tunnel and water enrichment area is excavated to reduce the water inrush hazard of the groundwater for the main tunnel.

#### 3.2. Applicability Discussion of the Expected Utility-Entropy Model

- a: ${p}_{1},{p}_{2},{p}_{3},{p}_{4}=(0.25,0.25,0.25,0.25)$
- b: ${p}_{1},{p}_{2},{p}_{3},{p}_{4}=(0,0.5,0.5,0)$

- The expected utility and entropy model is different from the typical risk assessment model in application. The former is for the choice between different schemes. The objective of risk assessment is to provide a basis for risk disposition.
- Entropy represents the uncertainty of different outcomes in the expected utility and entropy model. However, the first step in risk assessment is hazard classification. Risk control measures correspond with risk levels. Thus, entropy in risk assessment should be used to represent the uncertainty of risk levels. In addition, the calculation of entropy should be different from that of the expected utility and entropy model.
- The value of entropy is not related to the expected value in the expected utility and entropy model. However, in this way, risk assessment does not make sense for the following reasons:
- (a)
- When the hazard is relatively small, people would not take measures to reduce the uncertainty no matter how large the uncertainty is. For example, people must choose between two ways ahead. No useful information for decision-making currently exists. People who make a wrong choice will suffer great losses. Under this circumstance, people would pay the price to obtain useful information. However, if the price of making a wrong choice is acceptable, people would not take measures to reduce the uncertainty of this state. In the expected utility and entropy model, high entropy means high risk. It does not make sense under this circumstance.
- (b)
- When the hazard is large, people would pay the price to obtain further information on uncertainty reduction. The more serious the consequences are when people make a wrong choice, the more people would spend on obtaining information.

## 4. Our Recommend Approach

#### 4.1. Entropy-Hazard Risk Assessment Model

_{1}. Entropy represents the uncertainty of risk, which is denoted by R

_{2}. Risk assessment of the incident is conducted by analyzing the above three elements.

- When the hazard is relatively large, people would pay the price to reduce the uncertainty for risk control.
- No matter how large the uncertainty is, people would not take measures to reduce uncertainty when the hazard is small.
- If the hazard and uncertainty can be reduced to the same degree at the same cost, people would prefer choosing to reduce the hazard. For example, we assume that the hazard value of one incident is R
_{1}and the uncertainty value of that is R_{2}. When taking measures respectively, but at the same cost T, the hazard and uncertainty are reduced by 100%. People choose measures that reduce the hazard because risk is absent when the hazard value decreases to 0 regardless of the size of the uncertainty. By contrast, a risk still exists even when the entropy decreases to 0.

- When ${T}_{H}>{T}_{H}^{\prime}$, measures should not be taken;
- When ${T}_{H}<{T}_{H}^{\prime}$, measures can be taken.

#### 4.2. Calculation Method of Entropy

#### 4.2.1. Calculation of the Entropy Value

_{1}belonging to grade i.

_{1}is [2, 10]. However, under this circumstance, decision-makers can definitely know that the incident will occur. Thus, reducing $H(X)$ is not needed, and $H(A)$ is taken as the value of R

_{2}.

_{1}) to represent R

_{2}in Case 2.

#### 4.2.2. Calculation Method of ${H}^{\prime}$

_{1}is predicted to be 0.2 after taking measures, then the value of x

_{1}is in the interval of [a, a + 0.2]. If we assume that the initial range of x

_{1}is [0, 2], then the range of a is [0, 1.8]. The uncertainty of a leads to the uncertainty of y. Hence, determining the value of entropy $H(A)$ is difficult. However, the probability distribution of $H(A)$ can be obtained by the probability distribution of a and the formula between y and x

_{1}. $E[H(A)]$, the expected value of $H(A)$, can also be obtained through the probability distribution of $H(A)$. Taking the expected entropy value as ${H}^{\prime}$ for entropy is reasonable after taking measures, such as the predictive entropy. Thus, ${H}^{\prime}$ can be expressed as follows:

## 5. Assessment Process and Case Study

- Step 1: Analyze risk incidents and obtain information about risk incidents, including influencing factors ${x}_{1},{x}_{2},{x}_{3},\cdots ,{x}_{n}$ and their values.
- Step 2: Enter the initial parameters: C, ${T}_{H,i}$ (the cost of ith exploration measures) and ${x}_{1},{x}_{2},{x}_{3},\cdots ,{x}_{n}$.
- Step 3: Determine the risk category to which the incident belongs and calculate P, H.If it belongs to Case 1, the probability P is calculated using formula (2), and H is calculated using formula (9);If it belongs to Case 2, the probability P is calculated using formula (3), and H is calculated using formula (12).
- Step 4: Calculate ${H}^{\prime}$ using formula (13) and calculate ${T}_{H,i}^{\prime}$ using formula (7). If ${T}_{H,i}$ < ${T}_{H,i}^{\prime}$ (the tolerance cost of i-th exploration measures), continue to step 5; otherwise, skip to step 6.
- Step 5: After taking measures, update initial parameters: ${x}_{1},{x}_{2},{x}_{3},\cdots ,{x}_{n}$, C, ${T}_{H,i}$ and $\alpha $. Calculate the updated probability P using formula (2) or (3).
- Step 6: Determine whether to take risk control measures according to the value ${R}_{1}$ and acceptance criteria. In addition, implement the next risk management plan. A simple example is made to provide a specific explanation.

^{6}. The value ranges of ${x}_{1},{x}_{2},{x}_{3}$ are ${x}_{1}\in (0.005,0.03)$, ${x}_{2}\in (0,0.3)$, and ${x}_{3}\in (0.05,0.2)$, respectively. The occurrence probability of risk P and H(R

_{1}) can be calculated using the Mote Carlo method. As the probability is uncertain, P is replaced with the expected probability. Then, we can obtain P = 0.0236 and H(R

_{1}) = 0.6754. The probabilities of second-grade and third-grade risk levels are 0.6 and 0.4, respectively.

- Measure 1: ${H}^{\prime}$ = 0.6743, ${T}_{H}^{\prime}$ =38.44
- Measure 2: ${H}^{\prime}$ = 0.1842, ${T}_{H}^{\prime}$ = 17,164
- Measure 3: ${H}^{\prime}$ = 0.3173, ${T}_{H}^{\prime}$ = 12,502
- Measure 4: ${H}^{\prime}$ = 0.6667, ${T}_{H}^{\prime}$ = 303

_{1}) = 0.0358. The entropy is rather small, and the risk level is in the second-grade risk level, the probability of which is 0.9942.

## 6. Conclusions and Discussion

- The occurrence of risk incidents is assumed to be determined by a certain formula and influencing factors. On the basis of this assumption, risk incidents are classified into two categories.
- Uncertainty is taken as a part of the risk assessment model, and it is expressed by entropy. Importantly, the calculation formula of entropy is given under different circumstances.
- Entropy may be used to represent uncertainty under risk. With regard to the interrelation between hazard and uncertainty, a new risk assessment model based on entropy is developed.
- Decision strategies for reducing uncertainty are based on the value of information. Specifically, the comparison between reducing hazard and reducing uncertainty is expressed in terms of cost. If the value of collecting information, namely reducing uncertainty, is larger than that of reducing hazard, then the measures taken to reduce uncertainty are tolerable and available.
- Risk assessment under uncertainty involves a decision to reduce uncertainty or reduce hazard. The general assessment process of our model is based on the criterion that uncertainty is first reduced, followed by the hazard.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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Risk Factors | Risk Value | ||||
---|---|---|---|---|---|

${\mathit{x}}_{\mathbf{1}}$ | ${\mathit{x}}_{\mathbf{2}}$ | Probability | Outcome | Risk Value | |

Before taking exploration measures | (0, 2) | 1 | 0.5 | C | 0.5C |

After taking exploration measures | (0.9, 1.4) | 1 | 0.8 | C | 0.8C |

Frequency | Consequence | ||||
---|---|---|---|---|---|

Disastrous | Severe | Serious | Considerable | Insignificant | |

Very likely | Unacceptable | Unacceptable | Unacceptable | Unwanted | Unwanted |

Likely | Unacceptable | Unacceptable | Unacceptable | Unwanted | Acceptable |

Occasional | Unacceptable | Unwanted | Unwanted | Acceptable | Acceptable |

Unlikely | Unwanted | Unwanted | Acceptable | Acceptable | Negligible |

Very unlikely | Unwanted | Acceptable | Acceptable | Negligible | Negligible |

Case 1 | Index Type | Influencing Factors | Risk | |||

Risk Incident | Occurrence Probability | Consequence | Risk Level | |||

Range | $({x}_{1},{x}_{2},{x}_{3},\dots ,{x}_{n})$ | $P(y>0),P(y\le 0)$ | $P(y>0)$ | C | $({R}_{1}^{1},{R}_{1}^{2},{R}_{1}^{3},{R}_{1}^{4})$ | |

Entropy Value | >0 | 0~−ln(0.5) | 0 | 0 | 0 | |

Case 2 | Index Type | Influencing Factors | Risk | |||

Risk Incident | Occurrence Probability | Consequence | Risk Level | |||

Range | $({x}_{1},{x}_{2},{x}_{3},\dots ,{x}_{n})$ | $P(y>0),P(y\le 0)$ | $P(y>0)$ | C | $({R}_{1}^{1},{R}_{1}^{2},{R}_{1}^{3},{R}_{1}^{4})$ | |

Entropy Value | >0 | >0 | 0 | 0 | 0~ln(0.25) |

Risk Level | Scope of Risk Component R_{1} | Risk-Acceptable Criterion |
---|---|---|

${R}_{1}^{1}$ | >10^{6} | Unacceptable |

${R}_{1}^{2}$ | 10^{5}~10^{6} | Unwanted |

${R}_{1}^{3}$ | 5 × 10^{3}~10^{5} | Acceptable |

${R}_{1}^{4}$ | <5 × 10^{3} | Negligible |

Number of Measure | Target Factor | Expense | Expected Precision Range |
---|---|---|---|

1 | ${x}_{1}$ | 13,000 | 0.005 |

2 | ${x}_{2}$ | 20,000 | 0.05 |

3 | ${x}_{2}$ | 10,000 | 0.1 |

4 | ${x}_{3}$ | 30,000 | 0.01 |

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**MDPI and ACS Style**

Dong, X.; Lu, H.; Xia, Y.; Xiong, Z.
Decision-Making Model under Risk Assessment Based on Entropy. *Entropy* **2016**, *18*, 404.
https://doi.org/10.3390/e18110404

**AMA Style**

Dong X, Lu H, Xia Y, Xiong Z.
Decision-Making Model under Risk Assessment Based on Entropy. *Entropy*. 2016; 18(11):404.
https://doi.org/10.3390/e18110404

**Chicago/Turabian Style**

Dong, Xin, Hao Lu, Yuanpu Xia, and Ziming Xiong.
2016. "Decision-Making Model under Risk Assessment Based on Entropy" *Entropy* 18, no. 11: 404.
https://doi.org/10.3390/e18110404