# Rock Mass Classification Method Based on Entropy Weight–TOPSIS–Grey Correlation Analysis

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Rock Mass Classification Evaluation Index System

#### 2.1.1. Selection of Evaluation Indicators

#### 2.1.2. Evaluation System Construction

#### 2.1.3. Comparison of Classification Results

#### 2.2. Comprehensive Assessment of Rock Mass Classification

#### 2.2.1. Entropy Weight Method

- 1.
- Dimensionless processing

- 2.
- Specific gravity calculation

- 3.
- Entropy calculation

- 4.
- Entropy weighting calculation

#### 2.2.2. TOPSIS Method

- 1.
- Construction of a weighted normalization matrix

- 2.
- Determine the positive and negative ideal solution of D

- 3.
- Calculate the Euclidean distance

#### 2.2.3. Grey Correlation Analysis

- 1.
- Calculation of grey correlation coefficient

- 2.
- Calculation of grey correlation degree

- 3.
- Calculation of relative closeness

- 4.
- Calculating the relative closeness of the scheme I.

## 3. Results and Discussion

#### 3.1. Entropy Weight Analysis of Evaluation Indicators

#### 3.2. Relative Closeness Analysis of Each Section

#### 3.3. Entropy-Weighted-TOPSIS-Grey Correlation Rock Mass Classification Method

#### 3.3.1. Entropy-TOPSIS Analysis

_{7}has the largest entropy weight and R

_{6}the smallest. The information carried by the geo stress damage index is the most effective.

#### 3.3.2. Grey Correlation Analysis

_{1}-R

_{7}, were used as the sub-series, and the series were dimensionless, and the grey correlation degree of each evaluation indicator was calculated using Equations (10)–(13), and the results are shown in Table 15.

#### 3.3.3. Rock Mass Classification Method Construction

^{+}correlation are integrated, 3 groups of parameters of the same indicator are added, R

_{1}-R

_{5}indicators are normalized, R

_{6}and R

_{7}are changed in equal proportion, and the final parameter obtained is defined as the evaluation indicator importance J, as shown in Table 16, where R

_{1}-R

_{5}are moderately important and close, R6 is the least important, and in the deep region, high ground stress is one of the main causes of channel destabilization damage, so R

_{7}is the largest, at 0.3284, which is in line with the actual situation. Based on the above study, the entropy weight-TOPSIS-grey correlation rock mass grading method (referred to as “ETG” rock mass grading method) was established, with R

_{1}-R

_{5}as a positive number and a full score of 100, and R

_{6}and R

_{7}as negative numbers, which are correction indicators. To avoid “jumps” in the grading process, a linear grading method is adopted, i.e., the upper limit of the evaluation index is used as the upper limit of the score, and the detailed grading criteria are shown in Figure 6.

#### 3.3.4. Project Examples

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

- (1)
- The basic quality index of rock mass BQ is measured from the hardness of rock and the integrity of rock mass. BQ value is calculated by the following formula: $BQ=90+3{\sigma}_{CW}+250{K}_{V}$, where ${\sigma}_{CW}$ is the uniaxial compressive strength of rock; ${K}_{V}$ is the integrity coefficient of rock mass, ${K}_{V}={\left({V}_{pm}/{V}_{p}\right)}^{2}$, ${V}_{pm}$ is the p-wave velocity of rock mass, ${V}_{p}$ is the p-wave velocity of rock mass.

- (2)
- Revise according to the characteristics of specific projects. The correction formula is: $\left[BQ\right]=BQ-100\left({K}_{1}+{K}_{2}+{K}_{3}\right)$, where ${K}_{1}$ is the correction coefficient of the groundwater, ${K}_{2}$ is the correction coefficient of the main structural plane, and ${K}_{3}$ is the correction coefficient of the initial stress state.

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**Figure 1.**RMR classification and Q classification corresponding to the results of the boxplot. Where: (

**a**) is the boxplot when RMR = 81, at the same time, a partial magnification of the Y-axis is performed for ease of observation; (

**b**) is the boxplot when RMR = 60; (

**c**) is the boxplot when RMR = 41; (

**d**) is the boxplot when RMR = 20.

Classification Method | Influencing Factors | |||||
---|---|---|---|---|---|---|

Q | Rock quality designation (RQD) | Joint set number (J_{n}) | Joint roughness number (J_{r}) | Joint alteration number (J_{a}) | Joint water reduction factor (J_{w}) | Stress Reduction factor (SRF) |

RMR | Rock compressive strength (R_{1}) | Rock quality designation (R_{2}) | Spacing of discontinuities (R_{3}) | Condition of discontinuities (R_{4}) | Ground water conditions (R_{5}) | Orientation of discontinuities (R_{6}) |

[BQ] * | Rock compressive strength ($\mathsf{\sigma}$_{cw}) | Rock mass integrity (K_{V}) | Correction coefficient of the groundwater (K_{1}) | Correction coefficient of the main structural plane (K_{2}) | Correction coefficient of the initial stress state (K_{3}) |

_{1}K

_{2}K

_{3}are the correction coefficient, both of which come with BQ method.

Middle Section | Evaluation Indicators | |||||
---|---|---|---|---|---|---|

RQD | J_{n} | J_{r} | J_{a} | J_{w} | SRF | |

$-$780-1 * | 87.4 | 7 | 2 | 4 | 0.65 | 1 |

$-$795-1 | 91.7 | 7 | 1.5 | 4 | 0.6 | 1.5 |

$-$825-1 | 75.9 | 9 | 2 | 4 | 0.6 | 2 |

$-$870-1 | 22.5 | 7 | 2 | 4 | 0.65 | 2 |

$-$885-1 | 82.1 | 8 | 2 | 4 | 0.58 | 2 |

$-$900-1 | 82.6 | 10 | 2 | 4 | 0.55 | 2 |

$-$915-1 | 64.6 | 9 | 2 | 4 | 0.58 | 2 |

Middle Section | Evaluation Indicators | |||||
---|---|---|---|---|---|---|

R_{1} * | R_{2} * | R_{3} * | R_{4} | R_{5} | R_{6} | |

$-$780-1 | 9.8 | 16.4 | 11.9 | 17 | 0 | −5 |

$-$795-1 | 9.7 | 17.6 | 12.2 | 20 | 7 | −5 |

$-$825-1 | 10.0 | 13.4 | 12.8 | 22 | 7 | −5 |

$-$870-1 | 9.4 | 3.4 | 12.2 | 18 | 12 | −5 |

$-$885-1 | 10.1 | 15.0 | 12.5 | 20 | 4 | 0 |

$-$900-1 | 10.4 | 15.1 | 12.7 | 19 | 7 | −10 |

$-$915-1 | 9.3 | 10.7 | 11.5 | 16 | 0 | −5 |

_{1}, R

_{2}and R

_{3}are obtained according to the equation provided by the paper [18], which about Sanshandao mine, thereinto, ${R}_{1}=-0.0003{\mathsf{\sigma}}_{cw}^{2}+0.135{\sigma}_{cw}+0.9023$, ${R}_{2}=0.0012RQ{D}^{2}+0.0692\mathrm{RQD}+1.23$, ${R}_{3}=3.5411\mathrm{ln}{D}_{V}+0.9023$, ${D}_{V}$ is the distance between joints, and its unit is m.

Middle Section | Evaluation Indicators | ||||
---|---|---|---|---|---|

$\mathit{\sigma}$_{cw} | K_{V} | K_{1} | K_{2} | K_{3} | |

$-$780-1 | 80.23 | 0.461 | 0.3 | 0.3 | 1.0 |

$-$795-1 | 79.38 | 0.555 | 0.3 | 0.2 | 1.0 |

$-$825-1 | 82.74 | 0.741 | 0.3 | 0.2 | 1.0 |

$-$870-1 | 76.04 | 0.693 | 0.3 | 0.05 | 1.0 |

$-$885-1 | 88.90 | 0.401 | 0.1 | 0.25 | 1.0 |

$-$900-1 | 87.30 | 0.412 | 0.5 | 0.2 | 1.0 |

$-$915-1 | 75.76 | 0.453 | 0.3 | 0.3 | 1.0 |

**Table 5.**The correlation equation of RMR classification and Q classification [27].

Number * | Author(s) and Time | Equation | |
---|---|---|---|

1 | Bieniawski (1976) | RMR = 9lnQ + 44 | (1) |

2 | Rutledge and Preston (1978) | RMR = 5.9lnQ + 43 | (2) |

3 | Moreno (1980) | RMR = 5.4lnQ + 55.2 | (3) |

4 | Cameron-Clarke and Budavari (1981) | RMR = 5lnQ + 60.8 | (4) |

5 | Abad et al. (1983) | RMR = 10.5lnQ + 41.8 | (5) |

6 | Kaiser and Gale (1985) | RMR = 8.7lnQ + 38 | (6) |

7 | Al-Harthi (1993) | RMR = 9lnQ + 49 | (7) |

8 | Choquet and Hadjigogiu (1993,2016) | RMR = 10lnQ + 39 | (8) |

9 | El-Naqa (1994) | RMR = 7lnQ + 44 | (9) |

10 | Barton (1995) | RMR = 15lnQ + 50 | (10) |

11 | Tugrul (1998) | RMR = 7lnQ + 36 | (11) |

12 | Sari and Pasamehmetoglu (2004) | RMR = 3.7lnQ + 53.1 | (12) |

13 | Kumar et al. (2004) | RMR = 6.4lnQ + 49.6 | (13) |

14 | Cosar (2004) | RMR = 2.8lnQ + 45.19 | (14) |

15 | Hashemi et al. (2010) | RMR = 5.37lnQ + 40.48 | (15) |

16-1 | Laderian and Abaspoor (2012) | RMR = 8.15lnQ + 44.88 | (16) |

16-2 | Laderian and Abaspoor (2012) | RMR = 42.87Q^{0.162} | (17) |

17 | Ranasooriya and Nikraz (2012) | RMR = 6.3lnQ + 43 | (18) |

18 | Rafiee (2013) | RMR = 8.09lnQ + 43.08 | (19) |

19 | Castro Caicedo and Pérez Pérez (2013) | RMR = 5.7lnQ + 43.65 | (20) |

20 | Ali et al. (2014) | RMR = 2.87lnQ + 48.71 | (21) |

21 | Senra (2016) | RMR = 6.55lnQ + 59.53 | (22) |

22 | Sayeed and Khanna (2015) | RMR = 4.52lnQ + 43.635 | (23) |

Number | I | II | III | IV | V |
---|---|---|---|---|---|

RMR | 81~100 | 60~81 | 41~60 | 20~41 | <20 |

Q1 | >61.01 | 5.92~61.01 | 1.40~5.92 | 0.07~1.40 | <0.07 |

Q2 | >626.83 | 17.84~626.83 | 0.71~17.84 | 0.02~0.71 | <0.02 |

Q3 | >118.84 | 2.43~118.84 | 0.07~2.43 | 0.0001~0.07 | <0.0001 |

Q4 | >56.83 | 0.85~56.83 | 0.02~0.85 | 0.0003~0.02 | <0.0003 |

Q5 | >41.82 | 5.66~41.82 | 0.93~5.66 | 0.13~0.93 | <0.13 |

Q6 | >140.12 | 12.54~140.12 | 1.41~12.54 | 0.13~1.41 | <0.13 |

Q7 | >35.01 | 3.39~35.01 | 0.41~3.39 | 0.04~0.41 | <0.04 |

Q8 | >66.69 | 8.17~66.69 | 1.22~8.17 | 0.15~1.22 | <0.15 |

Q9 | >197.50 | 9.83~197.50 | 0.65~9.83 | 0.03~0.65 | <0.03 |

Q10 | >7.90 | 1.95~7.90 | 0.55~1.95 | 0.14~0.55 | <0.14 |

Q11 | >619.29 | 30.83~619.29 | 2.04~30.83 | 0.10~2.04 | <0.10 |

Q12 | >1882.85 | 6.46~1882.85 | 0.04~6.46 | 0.0001~0.04 | <0.0001 |

Q13 | >135.13 | 5.08~135.13 | 0.26~5.08 | 0.01~0.26 | <0.01 |

Q14 | >358,357.26 | 198.20~358,357.26 | 0.22~198.20 | 0.0001~0.22 | <0.0001 |

Q15 | >1892.44 | 37.90~1892.44 | 1.10~37.90 | 0.02~1.10 | <0.02 |

Q16-1 | >84.09 | 6.39~84.09 | 0.62~6.39 | 0.05~0.62 | <0.05 |

Q16-2 | >50.79 | 7.97~50.79 | 0.76~7.97 | 0.01~0.76 | <0.01 |

Q17 | >416.44 | 14.86~416.44 | 0.73~14.86 | 0.03~0.73 | <0.03 |

Q18 | >108.56 | 8.10~108.56 | 0.77~8.10 | 0.06~0.77 | <0.06 |

Q19 | >701.09 | 17.61~701.09 | 0.63~17.61 | 0.02~0.63 | <0.02 |

Q20 | >76,946.92 | 51.10~76,946.92 | 0.07~51.10 | 0.00005~0.07 | <0.00005 |

Q21 | >26.52 | 1.07~26.52 | 0.06~1.07 | 0.002~0.06 | <0.002 |

Q22 | >3891.67 | 37.36~3891.67 | 0.56~37.36 | 0.01~0.56 | <0.01 |

I | II | III | IV | V | |
---|---|---|---|---|---|

Q | >135.13 | 8.1~135.13 | 0.63~8.1 | 0.02~0.63 | <0.02 |

RMR | 81~100 | 60~81 | 41~60 | 20~41 | <20 |

[BQ] | >550 | 451~550 | 351~450 | 251~350 | <250 |

Middle Section | Classification Results | |||||
---|---|---|---|---|---|---|

Q Value | Grade | RMR Value | Grade | [BQ] Value | Grade | |

$-$780-1 | 2.86 | III | 50.1 | III | 419.72 | III |

$-$795-1 | 28.21 | II | 61.5 | II | 466.89 | II |

$-$825-1 | 5.55 | III | 60.2 | II | 523.47 | II |

$-$870-1 | 0.41 | IV | 50.0 | III | 491.37 | II |

$-$885-1 | 7.94 | III | 61.6 | II | 388.52 | III |

$-$900-1 | 10.00 | II | 54.2 | III | 394.24 | III |

$-$915-1 | 0.02 | V | 42.5 | III | 415.56 | III |

Q | RMR | BQ | |||
---|---|---|---|---|---|

Indicators | Type | Indicators | Type | Indicators | Type |

RQD | Positive | R_{1} | Positive | $\mathsf{\sigma}$_{cw} | Positive |

J_{n} | Negative | R_{2} | Positive | K_{V} | Positive |

J_{r} | Positive | R_{3} | Positive | K_{1} | Negative |

J_{a} | Negative | R_{4} | Positive | K_{2} | Negative |

J_{w} | Positive | R_{5} | Positive | K_{3} | Negative |

SRF | Negative | R_{6} | Positive |

Q | RMR | BQ | ||||||
---|---|---|---|---|---|---|---|---|

Indicators | B | C | Indicators | B | C | Indicators | B | C |

RQD | 0.9148 | 0.0771 | R_{1} | 0.8391 | 0.1985 | $\mathsf{\sigma}$_{cw} | 0.7840 | 0.2607 |

J_{n} | 0.8725 | 0.1155 | R_{2} | 0.9105 | 0.1105 | K_{v} | 0.7349 | 0.3200 |

J_{r} | 0.9208 | 0.0717 | R_{3} | 0.8867 | 0.1398 | K_{1} | 0.8982 | 0.1228 |

J_{a} | 1.0000 | 0.0000 | R_{4} | 0.8580 | 0.1752 | K_{2} | 0.7543 | 0.2965 |

J_{w} | 0.8604 | 0.1265 | R_{5} | 0.7969 | 0.2505 | K_{3} | 1.0000 | 0.0000 |

SRF | 0.3273 | 0.6092 | R_{6} | 0.8982 | 0.1255 |

Classification Method | Parameters | Middle Section | ||||||
---|---|---|---|---|---|---|---|---|

$-$780-1 | $-$795-1 | $-$825-1 | $-$870-1 | $-$885-1 | $-$900-1 | $-$915-1 | ||

Q | E^{+} | 0.0048 | 0.3192 | 0.6175 | 0.6140 | 0.6169 | 0.6329 | 0.6211 |

E^{−} | 0.6409 | 0.3407 | 0.1190 | 0.1857 | 0.1301 | 0.0982 | 0.1013 | |

G^{+} | 0.6985 | 0.4762 | 0.4212 | 0.5238 | 0.4475 | 0.3988 | 0.3913 | |

G^{−} | 0.7143 | 0.2402 | 0.3810 | 0.4038 | 0.4286 | 0.3935 | 0.5284 | |

I | 0.5551 | 0.6379 | 0.4552 | 0.5044 | 0.4504 | 0.4321 | 0.3784 | |

RMR | E^{+} | 0.3304 | 0.1959 | 0.1453 | 0.2579 | 0.1889 | 0.1866 | 0.3990 |

E^{−} | 0.1582 | 0.2490 | 0.3123 | 0.2759 | 0.2773 | 0.3056 | 0.0847 | |

G^{+} | 0.4231 | 0.5151 | 0.6075 | 0.4481 | 0.5845 | 0.5693 | 0.3344 | |

G^{−} | 0.8571 | 0.5352 | 0.3977 | 0.3548 | 0.5373 | 0.3633 | 0.4320 | |

I | 0.4054 | 0.5836 | 0.6925 | 0.6248 | 0.6138 | 0.6861 | 0.4505 | |

BQ | E^{+} | 0.4367 | 0.3190 | 0.2244 | 0.2662 | 0.3983 | 0.3790 | 0.4828 |

E^{−} | 0.1217 | 0.2098 | 0.3734 | 0.4090 | 0.2942 | 0.2580 | 0.0785 | |

G^{+} | 0.2346 | 0.2629 | 0.3529 | 0.3740 | 0.3883 | 0.2761 | 0.2197 | |

G^{−} | 0.5714 | 0.4049 | 0.3179 | 0.2677 | 0.3086 | 0.3401 | 0.4082 | |

I | 0.3563 | 0.4796 | 0.6284 | 0.6668 | 0.6125 | 0.5233 | 0.3889 |

Middle Section | $-$780-1 | $-$795-1 | $-$825-1 | $-$870-1 | $-$885-1 | $-$900-1 | $-$915-1 |
---|---|---|---|---|---|---|---|

Mass grade | III | II | II | II | II | III | III |

Classification method | Q | Q | RMR | BQ | RMR | RMR | RMR |

Relative closeness | 0.5551 | 0.6379 | 0.6925 | 0.6668 | 0.6138 | 0.6861 | 0.4505 |

Serial Number | Location of Measurement Points | Evaluation Indicators | ||||||
---|---|---|---|---|---|---|---|---|

R_{1} | R_{2} | R_{3} | R_{4} | R_{5} | R_{6} | R_{7} | ||

1 | $-$780-1 | 9.8 | 16.4 | 11.9 | 17 | 0 | −5 | −4.3 |

2 | $-$780-2 | 9.8 | 16.4 | 12 | 17 | 0 | −5 | −4.3 |

3 | $-$795-1 | 9.7 | 17.6 | 12.2 | 20 | 7 | −5 | −4.3 |

4 | $-$795-2 | 9.7 | 17.6 | 12.2 | 19 | 7 | −5 | −4.2 |

5 | $-$825-1 | 10 | 13.4 | 12.8 | 22 | 7 | −5 | −9.5 |

6 | $-$825-2 | 10 | 13.4 | 12.4 | 21 | 7 | −5 | −9.5 |

7 | $-$825-3 | 10 | 13.4 | 12.8 | 20 | 7 | −5 | −8.2 |

8 | $-$825-4 | 10 | 13.4 | 12.4 | 20 | 7 | −5 | −9 |

9 | $-$870-1 | 9.4 | 3.4 | 12.2 | 18 | 12 | −5 | −9.5 |

10 | $-$885-1 | 10.1 | 15 | 12.5 | 20 | 4 | 0 | −9.5 |

11 | $-$900-1 | 10.4 | 15.1 | 12.7 | 19 | 7 | −10 | −9.2 |

12 | $-$915-1 | 9.3 | 10.7 | 11.5 | 16 | 0 | −5 | −9.2 |

13 | $-$915-2 | 9.3 | 10.7 | 11.5 | 17 | 1 | −6 | −9.2 |

Evaluation Indicators | R_{1} | R_{2} | R_{3} | R_{4} | R_{5} | R_{6} | R_{7} |
---|---|---|---|---|---|---|---|

B | 0.8971 | 0.9605 | 0.9124 | 0.9160 | 0.8615 | 0.9572 | 0.6800 |

C | 0.1263 | 0.0485 | 0.1074 | 0.1030 | 0.1699 | 0.0525 | 0.3925 |

Evaluation Indicators | R_{1} | R_{2} | R_{3} | R_{4} | R_{5} | R_{6} | R_{7} |
---|---|---|---|---|---|---|---|

RMR correlation | 0.1774 | 0.1763 | 0.1036 | 0.1071 | 0.1402 | 0.1503 | 0.1451 |

E^{+} relevance | 0.1457 | 0.1467 | 0.1881 | 0.1834 | 0.1048 | 0.1031 | 0.1284 |

Evaluation Indicators | R_{1} | R_{2} | R_{3} | R_{4} | R_{5} | R_{6} | R_{7} |
---|---|---|---|---|---|---|---|

J | 0.2216 | 0.1831 | 0.1967 | 0.1941 | 0.2046 | 0.1507 | 0.3284 |

Rating value | 22.16 | 18.31 | 19.67 | 19.41 | 20.46 | 15.07 | 32.84 |

Serial Number | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

IRMR value | 45.8 | 45.9 | 57.2 | 56.3 | 50.7 | 49.3 | 50 | 48.8 | 40.5 | 52.1 | 45 | 33.3 | 34.3 |

Grade | Ⅲ | Ⅲ | Ⅲ | Ⅲ | Ⅲ | Ⅲ | Ⅲ | Ⅲ | Ⅲ | Ⅲ | Ⅲ | Ⅳ | Ⅳ |

ETG value | 43.2 | 43.3 | 54.6 | 53.8 | 45.1 | 43.7 | 45.3 | 43.6 | 35.9 | 45.1 | 40.8 | 28.1 | 29.3 |

Grade | Ⅲ | Ⅲ | Ⅲ | Ⅲ | Ⅲ | Ⅲ | Ⅲ | Ⅲ | Ⅳ | Ⅲ | Ⅲ | Ⅳ | Ⅳ |

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## Share and Cite

**MDPI and ACS Style**

Dai, B.; Li, D.; Zhang, L.; Liu, Y.; Zhang, Z.; Chen, S.
Rock Mass Classification Method Based on Entropy Weight–TOPSIS–Grey Correlation Analysis. *Sustainability* **2022**, *14*, 10500.
https://doi.org/10.3390/su141710500

**AMA Style**

Dai B, Li D, Zhang L, Liu Y, Zhang Z, Chen S.
Rock Mass Classification Method Based on Entropy Weight–TOPSIS–Grey Correlation Analysis. *Sustainability*. 2022; 14(17):10500.
https://doi.org/10.3390/su141710500

**Chicago/Turabian Style**

Dai, Bing, Danli Li, Lei Zhang, Yong Liu, Zhijun Zhang, and Shirui Chen.
2022. "Rock Mass Classification Method Based on Entropy Weight–TOPSIS–Grey Correlation Analysis" *Sustainability* 14, no. 17: 10500.
https://doi.org/10.3390/su141710500