# An Extended ORESTE Approach for Evaluating Rockburst Risk under Uncertain Environments

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## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Trapezoidal Fuzzy Numbers

#### 2.2. Extended ORESTE Method

**(1)**

**Phase 1: express the evaluation information using TrFNs**

**Step 1:**Obtain the initial indicator values.

**Step 2:**Normalize the initial decision-making matrix.

**Step 3:**Convert the indicator values into TrFNs.

**(2)**

**Phase 2: determine the indicator weights**

**Step 1:**Calculate the subjective weights.

**Step 2:**Compute the objective weights.

**Step 3:**Determine the comprehensive weights.

**(3)**

**Phase 3: obtain the evaluation results with the extended ORESTE method**

**Step 1:**Compute the significance degree.

**Step 2:**Calculate the global preference score.

**Step 3:**Determine the weak rank of alternatives.

**Step 4:**Calculate the average preference intensity and the net preference intensity.

**Step 5:**Build the preference/indifference/incomparability (PIR) structure.

- (1)
- When $|\Delta {G}_{ik}|\le \epsilon $, then $\left\{\begin{array}{llll}{B}_{i}I{B}_{k},& if\hspace{1em}\left|A{G}_{ik}\right|\le \theta & and& \left|A{G}_{ki}\right|\le \theta \\ {B}_{i}R{B}_{k},& if\hspace{1em}\left|A{G}_{ik}\right|\theta & or& \left|A{G}_{ki}\right|\theta \end{array}\right.$;
- (2)
- When $|\Delta {G}_{ik}|>\epsilon $, then $\left\{\begin{array}{cc}{B}_{i}P{B}_{k},& if\hspace{1em}\Delta {G}_{ik}0\\ {B}_{k}P{B}_{i},& if\hspace{1em}\Delta {G}_{ik}0\end{array}\right.$, where $\epsilon \in [0,1]$ and $\theta \in [0,1]$ are two parameters.

**Step 6:**Obtain the strong rank of alternatives.

## 3. Case Study

#### 3.1. Project Profile

#### 3.2. Determination of Evaluation Indicators

#### 3.3. Risk Evaluation of Rockburst

## 4. Discussions

#### 4.1. Comparison Analysis

#### 4.2. Sensitivity Analysis

#### 4.3. Managerial Implication

- (1)
- Due to the influence of uncertainty on the evaluation results, some measures should be taken to avoid uncertainty in reality, such as ensuring the high quality of data.
- (2)
- Based on the evaluation results, the areas of high risk should receive more attention. For example, a monitoring system can be installed for the early warning of rockburst.
- (3)
- The technical parameters of rockburst prevention measures can be optimized according to different risk levels. For different risk levels, the prevention measures and their parameters should be different.

- (1)
- The indicator values were expressed by TrFNs after the uncertainty parameters were introduced, which can indicate the uncertain information more reasonably.
- (2)
- Game theory was used to calculate the indicator weights by combining subjective and objective weights, so that the comprehensive weights can be determined more credibly.
- (3)
- The ORESTE approach was extended with TrFNs, which can be used to solve MCDM problems under trapezoidal fuzzy environments.
- (4)
- The proposed methodology was applied to evaluate rockburst risk, and can obtain evaluation results reliably.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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Indicators | Risk Levels | |||
---|---|---|---|---|

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

${C}_{1}$ | >40 | 26.7–40 | 14.5–26.7 | <14.5 |

${C}_{2}$ | <2.0 | 2.0–3.5 | 3.5–5.0 | >5.0 |

${C}_{3}$ | <40 | 40–100 | 100–200 | >200 |

${C}_{4}$ | >14.5 | 5.5–14.5 | 2.5–5.5 | ≤2.5 |

${C}_{5}$ | <0.50 | 0.50–0.60 | 0.60–0.75 | >0.75 |

${\mathit{C}}_{1}$ | ${\mathit{C}}_{2}$ | ${\mathit{C}}_{3}$ | ${\mathit{C}}_{4}$ | ${\mathit{C}}_{5}$ | |
---|---|---|---|---|---|

${B}_{{L}_{1}}$ | 40.0 | 0 | 0 | 14.5 | 0 |

${B}_{{L}_{2}}$ | 26.7 | 2.0 | 40 | 5.5 | 0.50 |

${B}_{{L}_{3}}$ | 14.5 | 3.5 | 100 | 2.5 | 0.60 |

${B}_{{L}_{4}}$ | 0 | 5.0 | 200 | 0 | 0.75 |

${B}_{1}$ | 13.09 | 1.39 | 10.09 | 1.78 | 0.45 |

${B}_{2}$ | 23.24 | 5.1 | 165.51 | 5.41 | 0.62 |

${B}_{3}$ | 15.07 | 2.03 | 103.99 | 1.50 | 0.59 |

${B}_{4}$ | 29.7 | 6.31 | 290.97 | 5.61 | 0.69 |

${\mathit{C}}_{1}$ | ${\mathit{C}}_{2}$ | ${\mathit{C}}_{3}$ | ${\mathit{C}}_{4}$ | ${\mathit{C}}_{5}$ | |
---|---|---|---|---|---|

${B}_{{L}_{1}}$ | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |

${B}_{{L}_{2}}$ | 0.333 | 0.317 | 0.137 | 0.621 | 0.667 |

${B}_{{L}_{3}}$ | 0.638 | 0.555 | 0.344 | 0.828 | 0.800 |

${B}_{{L}_{4}}$ | 1.000 | 0.792 | 0.687 | 1.000 | 1.000 |

${B}_{1}$ | 0.673 | 0.220 | 0.035 | 0.877 | 0.600 |

${B}_{2}$ | 0.419 | 0.808 | 0.569 | 0.627 | 0.827 |

${B}_{3}$ | 0.623 | 0.322 | 0.357 | 0.897 | 0.787 |

${B}_{4}$ | 0.258 | 1.000 | 1.000 | 0.613 | 0.920 |

${\mathit{C}}_{1}$ | ${\mathit{C}}_{2}$ | ${\mathit{C}}_{3}$ | ${\mathit{C}}_{4}$ | ${\mathit{C}}_{5}$ | |
---|---|---|---|---|---|

${B}_{{L}_{1}}$ | (0.00, 0.00, 0.00, 0.00) | (0.00, 0.00, 0.00, 0.00) | (0.00, 0.00, 0.00, 0.00) | (0.00, 0.00, 0.00, 0.00) | (0.00, 0.00, 0.00, 0.00) |

${B}_{{L}_{2}}$ | (0.27, 0.30, 0.37, 0.40) | (0.25, 0.29, 0.35, 0.38) | (0.11, 0.12, 0.15, 0.17) | (0.50, 0.56, 0.68, 0.74) | (0.53, 0.60, 0.73, 0.80) |

${B}_{{L}_{3}}$ | (0.51, 0.57, 0.70, 0.77) | (0.44, 0.50, 0.61, 0.67) | (0.27, 0.31, 0.38, 0.41) | (0.66, 0.74, 0.91, 0.99) | (0.64, 0.72, 0.88, 0.96) |

${B}_{{L}_{4}}$ | (0.80, 0.90, 1.10, 1.200) | (0.63, 0.71, 0.87, 0.95) | (0.55, 0.62, 0.76, 0.82) | (0.80, 0.90, 1.10, 1.20) | (0.80, 0.90, 1.10, 1.20) |

${B}_{1}$ | (0.54, 0.61, 0.74, 0.81) | (0.18, 0.20, 0.24, 0.26) | (0.028, 0.031, 0.038, 0.042) | (0.70, 0.79, 0.97, 1.05) | (0.48, 0.54, 0.66, 0.72) |

${B}_{2}$ | (0.34, 0.38, 0.46, 0.50) | (0.65, 0.73, 0.89, 0.97) | (0.46, 0.51, 0.63, 0.68) | (0.50, 0.56, 0.69, 0.75) | (0.66, 0.74, 0.91, 0.99) |

${B}_{3}$ | (0.50, 0.56, 0.69, 0.75) | (0.26, 0.29, 0.35, 0.39) | (0.29, 0.32, 0.39, 0.43) | (0.72, 0.81, 0.99, 1.08) | (0.63, 0.71, 0.87, 0.94) |

${B}_{4}$ | (0.21, 0.23, 0.28, 0.31) | (0.80, 0.90, 1.10, 1.20) | (0.80, 0.90, 1.10, 1.20) | (0.49, 0.55, 0.67, 0.74) | (0.74, 0.83, 1.01, 1.10) |

Indicators | Linguistic Ratings of Indicators | ${\tilde{\mathit{\omega}}}_{\mathit{j}}$ | ${\mathit{\omega}}_{\mathit{j}}^{\mathit{s}}$ | ||||
---|---|---|---|---|---|---|---|

${\mathit{E}}_{1}$ | ${\mathit{E}}_{2}$ | ${\mathit{{\rm E}}}_{3}$ | ${\mathit{{\rm E}}}_{4}$ | ${\mathit{E}}_{5}$ | |||

${C}_{1}$ | VH | H | M | FH | H | (3.1, 3.6, 3.8, 4.2) | 0.1883 |

${C}_{2}$ | H | VH | VH | H | FH | (3.5, 4.0, 4.3, 4.6) | 0.2099 |

${C}_{3}$ | H | VH | H | FH | H | (3.4, 3.9, 4.1, 4.5) | 0.2037 |

${C}_{4}$ | FH | FH | VH | VH | H | (3.3, 3.8, 4.2, 4.5) | 0.2023 |

${C}_{5}$ | H | M | H | H | VH | (3.3, 3.8, 3.9, 4.3) | 0.1959 |

${\mathit{C}}_{1}$ | ${\mathit{C}}_{2}$ | ${\mathit{C}}_{3}$ | ${\mathit{C}}_{4}$ | ${\mathit{C}}_{5}$ | |
---|---|---|---|---|---|

${B}_{{L}_{1}}$ | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |

${B}_{{L}_{2}}$ | 0.333 | 0.317 | 0.137 | 0.621 | 0.667 |

${B}_{{L}_{3}}$ | 0.638 | 0.555 | 0.344 | 0.828 | 0.800 |

${B}_{{L}_{4}}$ | 1.000 | 0.792 | 0.687 | 1.000 | 1.000 |

${B}_{1}$ | 0.673 | 0.220 | 0.035 | 0.877 | 0.600 |

${B}_{2}$ | 0.419 | 0.808 | 0.569 | 0.627 | 0.827 |

${B}_{3}$ | 0.623 | 0.322 | 0.357 | 0.897 | 0.787 |

${B}_{4}$ | 0.258 | 1.000 | 1.000 | 0.613 | 0.920 |

${\mathit{C}}_{1}$ | ${\mathit{C}}_{2}$ | ${\mathit{C}}_{3}$ | ${\mathit{C}}_{4}$ | ${\mathit{C}}_{5}$ | |
---|---|---|---|---|---|

${B}_{{L}_{1}}$ | 0.130 | 0.163 | 0.188 | 0.114 | 0.112 |

${B}_{{L}_{2}}$ | 0.269 | 0.277 | 0.212 | 0.454 | 0.484 |

${B}_{{L}_{3}}$ | 0.469 | 0.425 | 0.307 | 0.596 | 0.577 |

${B}_{{L}_{4}}$ | 0.719 | 0.583 | 0.521 | 0.716 | 0.716 |

${B}_{1}$ | 0.493 | 0.225 | 0.190 | 0.631 | 0.439 |

${B}_{2}$ | 0.324 | 0.594 | 0.444 | 0.458 | 0.595 |

${B}_{3}$ | 0.460 | 0.280 | 0.315 | 0.644 | 0.567 |

${B}_{4}$ | 0.224 | 0.726 | 0.732 | 0.448 | 0.660 |

${\mathit{B}}_{{\mathit{L}}_{1}}$ | ${\mathit{B}}_{{\mathit{L}}_{2}}$ | ${\mathit{B}}_{{\mathit{L}}_{3}}$ | ${\mathit{B}}_{{\mathit{L}}_{4}}$ | ${\mathit{B}}_{1}$ | ${\mathit{B}}_{2}$ | ${\mathit{B}}_{3}$ | ${\mathit{B}}_{4}$ | |
---|---|---|---|---|---|---|---|---|

${B}_{{L}_{1}}$ | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |

${B}_{{L}_{2}}$ | 0.198 | 0.000 | 0.000 | 0.000 | 0.024 | 0.000 | 0.000 | 0.010 |

${B}_{{L}_{3}}$ | 0.333 | 0.136 | 0.000 | 0.000 | 0.091 | 0.057 | 0.033 | 0.079 |

${B}_{{L}_{4}}$ | 0.510 | 0.312 | 0.176 | 0.000 | 0.256 | 0.170 | 0.198 | 0.164 |

${B}_{1}$ | 0.254 | 0.080 | 0.012 | 0.000 | 0.000 | 0.069 | 0.007 | 0.090 |

${B}_{2}$ | 0.342 | 0.144 | 0.065 | 0.002 | 0.156 | 0.000 | 0.094 | 0.022 |

${B}_{3}$ | 0.312 | 0.114 | 0.011 | 0.000 | 0.064 | 0.064 | 0.000 | 0.086 |

${B}_{4}$ | 0.417 | 0.229 | 0.162 | 0.071 | 0.253 | 0.097 | 0.191 | 0.000 |

${\mathit{B}}_{{\mathit{L}}_{1}}$ | ${\mathit{B}}_{{\mathit{L}}_{2}}$ | ${\mathit{B}}_{{\mathit{L}}_{3}}$ | ${\mathit{B}}_{{\mathit{L}}_{4}}$ | ${\mathit{B}}_{1}$ | ${\mathit{B}}_{2}$ | ${\mathit{B}}_{3}$ | ${\mathit{B}}_{4}$ | |
---|---|---|---|---|---|---|---|---|

${B}_{{L}_{1}}$ | 0.000 | −0.198 | −0.333 | −0.510 | −0.254 | −0.342 | −0.312 | −0.417 |

${B}_{{L}_{2}}$ | 0.198 | 0.000 | −0.136 | −0.312 | −0.056 | −0.144 | −0.114 | −0.219 |

${B}_{{L}_{3}}$ | 0.333 | 0.136 | 0.000 | −0.176 | 0.079 | −0.008 | 0.022 | −0.083 |

${B}_{{L}_{4}}$ | 0.510 | 0.312 | 0.176 | 0.000 | 0.256 | 0.168 | 0.198 | 0.093 |

${B}_{1}$ | 0.254 | 0.056 | −0.079 | −0.256 | 0.000 | −0.087 | −0.058 | −0.162 |

${B}_{2}$ | 0.342 | 0.144 | 0.008 | −0.168 | 0.087 | 0.000 | 0.030 | −0.075 |

${B}_{3}$ | 0.312 | 0.114 | −0.022 | −0.198 | 0.058 | −0.030 | 0.000 | −0.105 |

${B}_{4}$ | 0.417 | 0.219 | 0.083 | −0.093 | 0.162 | 0.075 | 0.105 | 0.000 |

${\mathit{B}}_{{\mathit{L}}_{1}}$ | ${\mathit{B}}_{{\mathit{L}}_{2}}$ | ${\mathit{B}}_{{\mathit{L}}_{3}}$ | ${\mathit{B}}_{{\mathit{L}}_{4}}$ | ${\mathit{B}}_{1}$ | ${\mathit{B}}_{2}$ | ${\mathit{B}}_{3}$ | ${\mathit{B}}_{4}$ | |
---|---|---|---|---|---|---|---|---|

${B}_{{L}_{1}}$ | - | O | O | O | O | O | O | O |

${B}_{{L}_{2}}$ | P | - | O | O | I | O | O | O |

${B}_{{L}_{3}}$ | P | P | - | O | I | I | I | O |

${B}_{{L}_{4}}$ | P | P | P | - | P | P | P | P |

${B}_{1}$ | P | I | I | O | - | O | I | O |

${B}_{2}$ | P | P | I | O | P | - | I | I |

${B}_{3}$ | P | P | I | O | I | I | - | O |

${B}_{4}$ | P | P | P | O | P | I | P | - |

Authors | Indicators | Evaluation Criteria | Evaluation Results | |||
---|---|---|---|---|---|---|

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

Peng et al. [49] | ${C}_{1}$ | The intervals of ${L}_{1}$, ${L}_{2}$, ${L}_{3}$ and ${L}_{4}$ are $\left[40,+\infty \right]$, $\left[26.7,40.0\right]$, $\left[14.5,26.7\right]$ and $\left[0,14.5\right]$, respectively. | ${L}_{4}$ | ${L}_{3}$ | ${L}_{3}$ | ${L}_{2}$ |

Kidybiński [13] | ${C}_{2}$ | The intervals of ${L}_{1}$, ${L}_{2}$, ${L}_{3}$ and ${L}_{4}$ are $\left[0,2.0\right]$, $\left[2.0,3.5\right]$, $\left[3.5,5.0\right]$ and $\left[5.0,+\infty \right]$, respectively. | ${L}_{1}$ | ${L}_{4}$ | ${L}_{2}$ | ${L}_{4}$ |

Kwasniewski et al. [50] | ${C}_{3}$ | The intervals of ${L}_{1}$, ${L}_{2}$, ${L}_{3}$ and ${L}_{4}$ are $\left[0,40\right]$, $\left[40,100\right]$, $\left[100,200\right]$ and $\left[200,+\infty \right]$, respectively. | ${L}_{1}$ | ${L}_{3}$ | ${L}_{3}$ | ${L}_{4}$ |

Tao [15] | ${C}_{4}$ | The intervals of ${L}_{1}$, ${L}_{2}$, ${L}_{3}$ and ${L}_{4}$ are $\left[14.5,+\infty \right]$, $\left[5.5,14.5\right]$, $\left[2.5,5.5\right]$ and $\left[0,2.5\right]$, respectively. | ${L}_{4}$ | ${L}_{3}$ | ${L}_{4}$ | ${L}_{2}$ |

Wang et al. [28] | ${C}_{5}$ | The intervals of ${L}_{1}$, ${L}_{2}$, ${L}_{3}$ and ${L}_{4}$ are $\left[0,0.50\right]$, $\left[0.50,0.60\right]$, $\left[0.60,0.75\right]$ and $\left[0.75,+\infty \right]$, respectively. | ${L}_{1}$ | ${L}_{3}$ | ${L}_{2}$ | ${L}_{3}$ |

Authors | Evaluation Methods | Evaluation Results |
---|---|---|

Mahdavi et al. [44] | Trapezoidal fuzzy TOPSIS method | ${B}_{{L}_{4}}\succ {B}_{4}\succ {B}_{2}\succ {B}_{{L}_{3}}\succ {B}_{3}\succ {B}_{1}\succ {B}_{{L}_{2}}\succ {B}_{{L}_{1}}$ |

Wang and Li [51] | Trapezoidal fuzzy TODIM method | ${B}_{{L}_{4}}\succ {B}_{4}\succ {B}_{{L}_{3}}\succ {B}_{2}\succ {B}_{3}\succ {B}_{1}\succ {B}_{{L}_{2}}\succ {B}_{{L}_{1}}$ |

The proposed method | Trapezoidal fuzzy ORESTE method | Weak rank: ${B}_{{L}_{4}}\succ {B}_{4}\succ {B}_{2}\succ {B}_{{L}_{3}}\succ {B}_{3}\succ {B}_{1}\succ {B}_{{L}_{2}}\succ {B}_{{L}_{1}}$ |

Strong rank: ${B}_{{L}_{4}}\succ {B}_{4}\succ \{{B}_{{L}_{3}},{B}_{2},{B}_{3}\}\succ \{{B}_{{L}_{2}},{B}_{1}\}\succ {B}_{{L}_{1}}$ |

Value of $\alpha $ | Value of $\beta $ | Evaluation Results |
---|---|---|

0.1 | 0.2 | Weak rank: ${B}_{{L}_{4}}\succ {B}_{4}\succ {B}_{2}\succ {B}_{{L}_{3}}\succ {B}_{3}\succ {B}_{1}\succ {B}_{{L}_{2}}\succ {B}_{{L}_{1}}$ |

Strong rank: ${B}_{{L}_{4}}\succ {B}_{4}\succ \{{B}_{{L}_{3}},{B}_{2},{B}_{3}\}\succ \{{B}_{{L}_{2}},{B}_{1}\}\succ {B}_{{L}_{1}}$ | ||

0.2 | 0.3 | Weak rank: ${B}_{{L}_{4}}\succ {B}_{4}\succ {B}_{2}\succ {B}_{{L}_{3}}\succ {B}_{3}\succ {B}_{1}\succ {B}_{{L}_{2}}\succ {B}_{{L}_{1}}$ |

Strong rank: ${B}_{{L}_{4}}\succ \{{B}_{4},{B}_{2}\}\succ \{{B}_{{L}_{3}},{B}_{3}\}\succ {B}_{1}\succ {B}_{{L}_{2}}\succ {B}_{{L}_{1}}$ | ||

0.3 | 0.4 | |

Strong rank: ${B}_{{L}_{4}}\succ {B}_{4}\succ {B}_{2}\succ \{{B}_{{L}_{3}},{B}_{3}\}\succ {B}_{1}\succ {B}_{{L}_{2}}\succ {B}_{{L}_{1}}$ | ||

0.4 | 0.5 | |

Strong rank: ${B}_{{L}_{4}}\succ {B}_{4}\succ {B}_{2}\succ \{{B}_{{L}_{3}},{B}_{3}\}\succ {B}_{1}\succ {B}_{{L}_{2}}\succ {B}_{{L}_{1}}$ | ||

0.1 | 0.3 | |

Strong rank: ${B}_{{L}_{4}}\succ {B}_{4}\succ {B}_{2}\succ {B}_{{L}_{3}}\succ \{{B}_{3},{B}_{1}\}\succ {B}_{{L}_{2}}\succ {B}_{{L}_{1}}$ | ||

0.1 | 0.4 | |

Strong rank: ${B}_{{L}_{4}}\succ \{{B}_{4},{B}_{2}\}\succ \{{B}_{{L}_{3}},{B}_{3}\}\succ {B}_{1}\succ {B}_{{L}_{2}}\succ {B}_{{L}_{1}}$ | ||

0.1 | 0.5 | |

Strong rank: ${B}_{{L}_{4}}\succ {B}_{4}\succ {B}_{2}\succ \{{B}_{{L}_{3}},{B}_{3}\}\succ {B}_{1}\succ {B}_{{L}_{2}}\succ {B}_{{L}_{1}}$ | ||

0.2 | 0.4 | |

0.2 | 0.5 | |

0.3 | 0.5 | |

Number | Value of $\mathit{\alpha}$ | Value of $\mathit{\beta}$ | Evaluation Results | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

${\mathit{C}}_{1}$ | ${\mathit{C}}_{2}$ | ${\mathit{C}}_{3}$ | ${\mathit{C}}_{4}$ | ${\mathit{C}}_{5}$ | ${\mathit{C}}_{1}$ | ${\mathit{C}}_{2}$ | ${\mathit{C}}_{3}$ | ${\mathit{C}}_{4}$ | ${\mathit{C}}_{5}$ | ||

${N}_{1}$ | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | |

Strong rank:${B}_{{L}_{4}}\succ \{{B}_{4},{B}_{2}\}\succ \{{B}_{{L}_{3}},{B}_{3}\}\succ {B}_{1}\succ {B}_{{L}_{2}}\succ {B}_{{L}_{1}}$ | |||||||||||

${N}_{2}$ | 0.5 | 0.4 | 0.3 | 0.2 | 0.1 | 0.6 | 0.5 | 0.4 | 0.3 | 0.2 | |

Strong rank: ${B}_{{L}_{4}}\succ \{{B}_{4},{B}_{2},{B}_{{L}_{3}}\}\succ {B}_{3}\succ {B}_{1}\succ {B}_{{L}_{2}}\succ {B}_{{L}_{1}}$ | |||||||||||

${N}_{3}$ | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | |

${N}_{4}$ | 0.6 | 0.5 | 0.4 | 0.3 | 0.2 | 0.7 | 0.6 | 0.5 | 0.4 | 0.3 | |

Strong rank: ${B}_{{L}_{4}}\succ {B}_{4}\succ \{{B}_{2},{B}_{{L}_{3}}\}\succ {B}_{3}\succ {B}_{1}\succ {B}_{{L}_{2}}\succ {B}_{{L}_{1}}$ | |||||||||||

${N}_{5}$ | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | |

Strong rank: ${B}_{{L}_{4}}\succ \{{B}_{4},{B}_{2}\}\succ {B}_{{L}_{3}}\succ {B}_{3}\succ {B}_{1}\succ {B}_{{L}_{2}}\succ {B}_{{L}_{1}}$ | |||||||||||

${N}_{6}$ | 0.7 | 0.6 | 0.5 | 0.4 | 0.3 | 0.8 | 0.7 | 0.6 | 0.5 | 0.4 | |

Strong rank:${B}_{{L}_{4}}\succ \{{B}_{4},{B}_{2},{B}_{{L}_{3}}\}\succ {B}_{3}\succ {B}_{1}\succ {B}_{{L}_{2}}\succ {B}_{{L}_{1}}$ | |||||||||||

${N}_{7}$ | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 | Weak rank: ${B}_{{L}_{4}}\succ {B}_{2}\succ {B}_{4}\succ {B}_{{L}_{3}}\succ {B}_{3}\succ {B}_{1}\succ {B}_{{L}_{2}}\succ {B}_{{L}_{1}}$ |

Strong rank: ${B}_{{L}_{4}}\succ \{{B}_{2},{B}_{4},{B}_{{L}_{3}}\}\succ \{{B}_{3},{B}_{1}\}\succ {B}_{{L}_{2}}\succ {B}_{{L}_{1}}$ | |||||||||||

${N}_{8}$ | 0.8 | 0.7 | 0.6 | 0.5 | 0.4 | 0.9 | 0.8 | 0.7 | 0.6 | 0.5 | Weak rank: ${B}_{{L}_{4}}\succ {B}_{2}\succ {B}_{4}\succ {B}_{{L}_{3}}\succ {B}_{3}\succ {B}_{1}\succ {B}_{{L}_{2}}\succ {B}_{{L}_{1}}$ |

Strong rank: ${B}_{{L}_{4}}\succ \{{B}_{2},{B}_{4},{B}_{{L}_{3}}\}\succ {B}_{3}\succ \{{B}_{1},{B}_{{L}_{2}}\}\succ {B}_{{L}_{1}}$ |

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## Share and Cite

**MDPI and ACS Style**

Shi, K.; Liu, Y.; Liang, W.
An Extended ORESTE Approach for Evaluating Rockburst Risk under Uncertain Environments. *Mathematics* **2022**, *10*, 1699.
https://doi.org/10.3390/math10101699

**AMA Style**

Shi K, Liu Y, Liang W.
An Extended ORESTE Approach for Evaluating Rockburst Risk under Uncertain Environments. *Mathematics*. 2022; 10(10):1699.
https://doi.org/10.3390/math10101699

**Chicago/Turabian Style**

Shi, Keyou, Yong Liu, and Weizhang Liang.
2022. "An Extended ORESTE Approach for Evaluating Rockburst Risk under Uncertain Environments" *Mathematics* 10, no. 10: 1699.
https://doi.org/10.3390/math10101699