The commonly utilized rock mass classification systems, such as the Q system and the RMR method, consider the physical properties, structural planes occurrence, in situ stress level, and groundwater condition as the evaluation indices based on the geological conditions and evaluate the rock mass quality using the grading formulas established in the system. The calculation process and the weight of each index in these systems are intensely subjective. In order to eliminate the influence of subjective experience-based analogy on rock mass grading and objectively reflect the rock mass's actual condition, a rock mass classification method is proposed in this paper using game theory and cloud modeling (GA-CM). This method introduces the volumetric joint number to replace the RQD index for representing rock mass intactness. With the advantages of precise joint description in describing the rock mass geological information in three-dimensional space, a classification system that takes the volumetric joint number, joint spacing, joint condition, rock strength, and groundwater as the evaluation indices is established, as shown in Fig. 7.
Moreover, based on the cloud model theory, the subjective and objective weights of the rock mass quality evaluation indices are determined using the AHP and CRITIC methods. The subjective and objective weights are then coupled by applying the game theory to obtain a combined weight. As a result, the influence of subjective factors on rock mass grading is eliminated to the greatest extent. The calculation process is shown in Fig. 8.
4.1 Calculation of weights for evaluation indices
The key to rock mass quality evaluation is to properly determine the evaluation indices and the weight of each index, among which weighting is the most important part. The application of a single subjective or objective weight cannot entirely reflect the actual characteristics and information contained in the index. Currently, the AHP method is the most used method for calculating subjective weights based on users' experiences. Hence, the CRITIC algorithm is introduced to calculate the objective weight with consideration of data correlation in this study to overcome the shortcomings of the subjective method. However, data objectivity is emphasized while the data correlation is neglected in calculating the objective weight. In this study, the optimized subjective and objective weights based on the game theory combine the advantages of subjective and objective weights to obtain appropriate index weights.
4.1.1 Analytic Hierarchy Process (AHP)
AHP is a multi-objective decision analysis method proposed by Saaty et al. (1994) to combine qualitative and quantitative research. The basic idea of this method is to hierarchize and quantify complex decision problems using the human thinking process and then make multi-criteria decision-making. When applying the AHP method for weight calculation, the decision-making problem is divided into several components according to the influencing factors and their membership in the actual rock mass site conditions. The AHP model is then established for the evaluation indices (as shown in Figure. 7), with A representing the target layer and Bi representing the index one. According to the importance of each index in the traditional grading methods and the experts' opinion, the evaluation indices are ranked in pairs, and the pairwise comparison matrix for the decision-making factors is constructed (Eq. (3)). This study adopts the 1–9 scale method and its reciprocal method for comparison. Since the factors in the target layer A are correlated to the next layer B, aij represents the result for the ith factor compared to the jth factor, where aij=1/aji.
![](data:image/png;base64,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)
In order to determine whether the matrix is rational, its consistency shall be tested using Eq. (4).
\(CI{\text{=}}\frac{{\lambda \hbox{max} {\text{-n}}}}{{{\text{n-1}}}}\) \(CR{\text{=}}\frac{{CI}}{{RI}}\) (4)
where CI is the consistency index, n is the order of the matrix, CR is the random consistency ratio, and RI is the average random consistency index. When CR < 0.1, the consistency result is considered satisfactory, and the weights are reasonable.
The importance of each factor is substituted into the comparison matrix with a maximum eigenvalue λmax=5, and the consistency index CI=0 is obtained using Matlab, as shown in Table 1. The consistency is then randomly tested, and the consistency ratio CR = 0 is obtained. Therefore, the matrix consistency is satisfactory. The weights for the evaluation indices are then obtained using the AHP method, i.e., \(\gamma\)=(a1,a2,a3,a4,a5)=(0.17,0.24,0.1,0.37,0.12), as listed in Table 1.
Table 1
Weight calculation by the AHP method
\(\gamma\)
|
\(a\)1
|
\(a\)2
|
\(a\)3
|
\(a\)4
|
\(a\)5
|
Weight
|
\(a\)1
|
1
|
1/2
|
2
|
1/3
|
2
|
0.17
|
\(a\)2
|
2
|
1
|
2
|
1/2
|
2
|
0.24
|
\(a\)3
|
1/2
|
1/2
|
1
|
1/3
|
1/2
|
0.10
|
\(a\)4
|
2
|
2
|
3
|
1
|
3
|
0.37
|
\(a\)5
|
1/2
|
1/2
|
2
|
1/3
|
1
|
0.12
|
4.1.2 The CRITIC method
The CRITIC method was proposed by Diakoulaki et al. (1995) as an objective weighting method using data correlation. The weight is assigned according to the indices' information volume and correlation represented by dispersion and conflict between indices. The standard deviation indicates that the dispersion is positively correlated with the information volume; hence, the information volume is also positively correlated with weight. The conflict intensity is expressed using the correlation coefficient that is negatively correlated with the information volume. The calculation process can be mainly divided into four steps as follows:
Step 1: Since the dimensions and orders of the data for the evaluation indices are challenging; the data are standardized using Eq. (5).
$${a_{ij}}=\frac{{{x_{ij}} - {x_{j\hbox{min} }}}}{{{x_{j\hbox{max} }} - {x_{j\hbox{min} }}}}$$
5
where \({a_{ij}}\) is the standardized value of the initial data, \({x_{j\hbox{max} }}\) is the maximum value of index j in the initial data, \({x_{j\hbox{min} }}\) is the minimum value of index j in the initial data, and \({x_{ij}}\)is the value of index j in sample i.
Step 2: When calculating the correlation coefficient represented by the information volume of an evaluation index, the integral difference method is often adopted, as shown in Eq. (6).
\({r_{ij}}=\frac{{\sum {\left( {{x_{ij}} - \overline {{{x_{ij}}}} } \right)\left( {{y_{ij}} - \overline {{{y_{ij}}}} } \right)} }}{{\sqrt {\sum {{{\left( {{x_{ij}} - \overline {{{x_{ij}}}} } \right)}^2}\sum {{{\left( {{y_{ij}} - \overline {{{y_{ij}}}} } \right)}^2}} } } }}{a_{ij}}{x_{ij}}\) (6)
where \(\overline {{{y_{ij}}}}\) is the average value of \({y_{ij}}\).
Step 3: The quantized coefficient that indicates the comprehensive information volume and degree of independence (i.e., the information volume Cj contained in the jth index is determined for each index) is calculated using Eq. (7).
$$cj=\sigma j\sum\limits_{{b=1}}^{m} {(1 - r{\text{i}}j)} ({\text{i}}=1,2, \cdots ,m;j=1,2, \cdots m)$$
7
where \(\sigma j\) is the standard deviation of index j.
Step 4: Finally, the weight αj of each index is calculated using Eq. (8).
$$\alpha {\text{j=cj/}}\sum\limits_{{j=1}}^{m} {{\text{cj}}}$$
8
By applying the CRITIC method to the 10 datasets obtained from field measurements in a mine, the objective weight of each evaluation index is obtained, as listed in Table 2.
Table 2
Index weight obtained by the CRITIC method
Evaluation index
|
Rock strength B1
|
Volumetric joint number B2
|
Joint spacing B3
|
Joint condition B4
|
Groundwater B5
|
Weight
|
0.24
|
0.16
|
0.20
|
0.21
|
0.19
|
4.1.3 The game theory
Game theory is a mathematical technique that deals with competitive situations. Its core idea is the Nash equilibrium, which aims to maximize the income of each participant in the combination, i.e., to find the optimal balance between subjective and objective weights. Assuming that L types of methods are used to calculate the index weight and the weight vector \(Wk=(\omega k1,\omega k2, \cdots ,\omega kn)\) (k = 1,2,\(\cdots\), L). After that, the L weight vectors are linearly combined to form the combined weight vector as follows:
$$We=\sum\limits_{{k=1}}^{L} {\delta k\cdot W{k^T}}$$
9
where We is the combined weight, and \(\delta k\) is the combination coefficient greater than 0.
In addition, the optimal combination coefficient set can be obtained by minimizing the difference between the divorce minimization We and Wk based on the solution for optimal strategy (Eq. (10)):
$$min{\left\| {\sum\limits_{{k=1}}^{L} {\delta k\cdot W{k^T}-W{l^T}} } \right\|_2}(k=1,2, \cdots ,L)$$
10
Finally, the optimal combination coefficients' set
is obtained by solving Eq. (10) with the matrix's differential property. The combined weight can then be determined by substituting the optimal combination coefficient into Eq. (11).
(11)
To sum up, the linear combination coefficient \(\delta 1\)for the AHP weighting method is 0.56, and the coefficient \(\delta 2\) for the CRITIC weighting method is 0.44. By substituting the two coefficients into Eq. (12), the combined weight for each index can be obtained, as shown in Figure. 9.
\(w=\sum\limits_{{i=1}}^{n} {\delta L\cdot {\omega ^T}}\) (12)
Figure 9. Combined weight of each evaluation index
4.2 The cloud model
The cloud model is a fuzzy evaluation model proposed by Zhang et al. (2008) to conduct qualitative and quantitative information transformation through the forward and backward cloud generators, which can fully reflect the randomness and fuzziness of language concepts. Let U{x1,x2, …, xn} be a quantitative domain expressed by exact values, and C be a qualitative concept associated with U. If the quantitative value x is a random number with stable tendency and x∈U, x is taken as a random implementation to the qualitative concept C with the degree of certainty of x to C being µ∈[0,1]. The distribution of x on the domain U is called a cloud, and each point (x, µ(x)) is a cloud drop.
The cloud model has three critical characteristic parameters, namely, the expectation Ex, the entropy En, and the super entropy He, among which Ex is the distribution function expectation that reflects the cloud model's qualitative characteristics. On the other hand, En represents the range of cloud drop values acceptable for qualitative concepts in the domain, i.e., the uncertainty of the qualitative concept. Besides, He is an objective representation of the cloud model stability, which reflects the degree of the predicted index's dispersion, as shown in Fig. 10. The equations are as follows:
$$\left\{ \begin{gathered} Ex=\frac{{C\hbox{min} +C\hbox{max} }}{{2.355}} \hfill \\ E{\text{n}}=\frac{{C\hbox{max} - C\hbox{min} }}{{2.355}} \hfill \\ He={\text{cons}}\tan {\text{t}} \hfill \\ \end{gathered} \right.$$
13
where Cmax and Cmin are the upper and lower bounds of a rock mass grade, respectively.
In short, for a given data sample x, the certainty of the cloud model can be obtained using Eq. (14):
$$\mu =exp\left[ {-\frac{{{{(x-Ex)}^2}}}{{2En{'^2}}}} \right]$$
14
where x is the data sample, µ∈[0,1] is the cloud model certainty, and Enis the Gaussian random number generated from (En, He).
The cloud model performs a two-way cognitive transformation between qualitative concepts and quantitative data through forward and backward cloud generators. The forward cloud generator generates quantitative values from the numerical characteristics of the cloud. In contrast, the backward cloud generator transforms qualitative concepts into quantitative values, as illustrated in Fig. 11. This paper employs the forward cloud generator to develop a model for each evaluation index, as shown in Fig. 11. The X-coordinate represents the value of the evaluation index, while the Y-coordinate stands for the corresponding certainty.
Based on the forward cloud generator algorithm, for a given data sample x, the certainty is obtained using Eq. (14). Together with the combined weight of each evaluation index obtained from the game theory, the rock mass quality can be determined using the principle of maximum membership degree given in Eq. (15).
$$\beta =\sum\limits_{{i=1}}^{n} {\mu i \cdot w{\text{j}}}$$
15
where µi is the certainty of the ith evaluation index, and ωj is the weight of the ith evaluation index.
4.3 Principle for determining the index values and evaluation criteria for rock mass classification
Rock masses are classified using the evaluation indices that are already determined. The influence of each evaluation index on rock mass quality is rated using a hundred-mark system. The rating principle is determined according to the traditional methods in the geotechnical field, such as the RMR method, the Q-system, and the BQ method. Some hardly quantified indices, such as joint conditions and groundwater, can be quantitatively classified after the experts' qualitative evaluation, as shown in Table 3.
Table 3
Rating of evaluation indexes
Rock strength B1/Mpa
|
Volumetric joint number B2/(No./m3)
|
Joint spacing
B3/cm
|
Joint condition
B4
|
B4 rating
|
Groundwater B5
|
B5 rating
|
Grade
|
> 250
|
< 3
|
> 200
|
Rough joint surface, discontinuous and closed joints, hard rock.
(no fillings in borehole and no obvious joint distribution)
|
100
|
Dry
|
100
|
Ⅰ
|
100ཞ250
|
3 ~ 10
|
60 ~ 200
|
Moderately rough joint surface, discontinuous joints with width < 1mm, hard rock.
(no fillings in borehole, few closed joints)
|
80 ~ 100
|
Wet
|
70 ~ 100
|
Ⅱ
|
50ཞ100
|
10 ~ 20
|
20 ~ 60
|
Moderately rough joint surface, joints with low continuity and width < 1mm, moderately weak rock.
(no fillings in borehole, number of joints < 3/m, width < 1mm)
|
60 ~ 80
|
Dripping water
|
40 ~ 70
|
Ⅲ
|
25ཞ50
|
20 ~ 30
|
6 ~ 20
|
Smooth joint surface or containing weak interbed (thickness < 5mm), continuous joints with width 1 ~ 5mm.
(fillings in borehole, many open joints)
|
30 ~ 60
|
Fissure water
|
10 ~ 40
|
Ⅳ
|
< 25
|
> 35
|
< 6
|
Containing weak interbed (thickness > 5mm), continuous joint with width > 1mm.
(weak interbed in borehole, unordered joint number and open joints)
|
0
|
Flowing water
|
0
|
V
|
By referring to the traditional classification systems criteria and integrating the research results with rock mass quality classification in the international geotechnical evaluation field, the rock mass classification model proposed in this study classifies the rock mass quality characteristics description into five grades. The grading criteria are shown in Table 4.
Table 4
Grading criteria for rock mass classification
Grade
|
Rock mass evaluation
|
Description of rock mass characteristics
|
Ⅰ
|
Excellent
|
Hard rock, few joints on the structural plane, high intactness. Stable in the long term with occasional block falling
|
Ⅱ
|
Good
|
Moderately hard rock, several joints on the structural plane, moderately high intactness, or fair rock strength but high intactness. Stable with local block falling.
|
Ⅲ
|
Moderate
|
Moderately hard rock, many joints on the structural plane, moderately fractured rock mass, or low rock strength but high intactness. Poor stability with local block dislocation.
|
Ⅳ
|
Poor
|
Weak rock, many joints on the structural plane, low intactness. Generally no self-stability capability, the occurrence of local collapse.
|
Ⅴ
|
Very poor
|
Many joints on the structural plane, fractured rock with extremely low intactness and no self-stability capacity.
|