Underground and open-pit mines are complex undertakings, mainly underground mines, for stability and economy. In addition to mining, many engineering projects are closely related to rock stability, such as tunneling, hydropower engineering, traffic engineering, and others. For all these endeavors, the stability assessment of surrounding rock is vital in designing and constructing such engineering objects.

The significance of the assessment of rock mass quality is supported by the fact that there are different approaches to solving this problem. Many scholars implemented various methods and approaches for rock mass classification and rock mass analysis, such as fuzzy sets theory [1–4], fuzzy RMR method [5,6], RMR method [7], Gaussian cloud model [8], modified RMR method [9], fuzzy inference system [10,11], multi-criteria decision–making (MCDM) methods [12–15], Least squares support vector machine [16], Support vector machine [17], Deep learning approach [18,19], Artificial neural networks [20], Deep neural network [21], Support vector regression [22], Stacking ensemble learning [23], Convolutional neural networks [24,25], Machine learning [26], Cloud model [27], Discrete element method [28] and others. Also, several authors analyzed and investigated rock fracture and crack conditions that highly influenced rock mass classification [29–32].

A comprehensive literature review indicated that no studies have investigated the Hausdorff distance for rock mass classification. In addition, the integrated Gromov-Hausdorff distance was applied as a discriminant function (classifier) of rock mass quality. Accordingly, the GHDQR method seeks to fill a gap in the literature review by representing an authentic, innovative, and efficient mechanism for rock mass quality optimization.

The innovative aspects and significance of this study can be summarized as follows:

This study proposes a methodology based on the discriminant function with uncertainty. Firstly, the algorithm transforms interval values of selected evaluation indices into crisp values by calculating the Hausdorff distance between rock samples and indices. This indicates the capability of treating the data uncertainties, where the interval numbers quantify uncertainties. The normalization of data creates a nondimensional environment. The developed methodology includes the influence (weight) of each evaluation index. Weights are defined by the Standard deviation method. Weighted-normalized data are further processed. The algorithm creates an artificial class of rock mass and partitions each space of real and artificial class into min and max subspaces. For min and max subspaces separately, the Gromov-Hausdorff function on metric spaces defines the distance between each real and artificial class. Finally, the algorithm calculates the aggregated intensities of distances of min and max subspaces for each real class and sorts them in descending order. The maximum value of aggregated intensity defines the class of the rock mass sample. The method’s efficiency is validated by comparing it to seven existing methods. The results showed that the model enables objective, efficient, and reliable assessment of rock mass quality.

The structure of this study is as follows. Section 1 presents the background of the conducted research. Section 2 indicates the developed methodology in detail. Section 3 contains comparisons of the GHDQR method to seven existing methods, sensitivity analysis, and a discussion of the results for efficiency estimation of the developed model. Section 4 provides conclusions and further direction of research.

This study uses the Gromov-Hausdorff distance to match rock samples and a defined set of classes for a finite set of evaluation indexes. Samples and classes present the metric spaces, and the algorithm calculates the matching intensity between them. The calculated intensity between the rock sample and each class presents a discriminant value employed to determine the class to which the sample belongs. The developed classifier selects the maximum intensity value and assigns the class to the rock sample. Experts can define a finite set of evaluation indices based on available information about the rock sampled and form the evaluation matrix. In the following part, the mathematical formulation of the developed methodology is presented in twelve successive steps.

Step 1: Define the vector of characteristics of the rock mass sample.

The rock sample is obtained by core-drilling operations, and it is characterized by the following vector Y: (1)Y=(y1y2…yn)where:

n is the total number of characteristics of rock mass.

Each element of the vector Y presents one characteristic of the sampled rock, which is defined by the adequate test.

Step 2: Creation of interval rock mass evaluation matrix.

The interval rock mass evaluation matrix is created according to the information obtained from different tests that were conducted on the rock mass sample, and the characteristics are presented by the vector Y. Several matrix columns equal n.

The stability of the surrounding rock is defined by classes, where each class signifies a closed region that contains a description of the rock. Each class property is influenced by the finite set of rock mass evaluation indices, which form a closed region. Accordingly, it is more appropriate to consider indices as interval numbers when determining the class of surrounding rock. Hence, the index is treated as an interval number.

Definition 1: The number a^ is an interval or grey number which is defined in real line ℝ and expressed as a^=[al,au], where al and au are the lower and upper bound of an interval number. If al=au, then a^ is a real number [33].

The relationship between classes and rock mass evaluation indexes can be presented in the following interval matrix form: (2)X^=[[xijl,xiju]]i=1,2,…,mj=1,2,…,n=[C,II1I2…Ijtargetmax∨minmax∨min…max∨minC1[x11l,x11u][x12l,x12u]…[x1jl,x1ju]C2[x21l,x21u][x22l,x22u]…[x2jl,x2ju]⋮⋮⋮⋱…Ci[xi1l,xi1u][xi2l,xi2u]…[xmnl,xmnu]]where elements of the matrix are as follows:

X is the interval matrix of classes,

C=[C1,C2,…,Cm] is a finite set of classes,

I=[I1,I2,…,In] is a finite set of evaluation indexes,

[xijl,xiju] is the interval value of the i-th class for the j-th evaluation index,

i is the total number of classes,

j is the total number of evaluation indices,

target is the desired extreme value of the evaluation index.

Analyzing the matrix X^ indicates that each matrix row presents a class vector.

Step 3: Transformation of an interval matrix to crisp one.

The crisp rock mass evaluation matrix is obtained by calculating the Hausdorff distance between each element of the i-th class vector and each vector Y. The Hausdorff distance is the maximum distance of a set to the nearest point in the other set [34].

Definition 2: Let A=[a1,a2,…,ak] and B=[b1,b2,…,bq] be two finite sets, the Hausdorff distance H(A,B) is defined as follows: (3)H(A,B)=max{h(A,B),h(B,A)}where: (4)h(A,B)=max⏟a∈Amin⏟b∈B d(a,b)where:

a,b are elements of sets A and B, respectively,

d(a,b) is any metric between these elements,

h(A,B),h(B,A)–the directed Hausdorff distances.

The function h(A,B) is ranking each element of set A using its distance to the nearest element of set B, and then the highest ranked element presents the distance value between A and B.

Let a^=[al,au] and b^=[bl,bu] be two interval-grey numbers, then Hausdorff distance between them is defined by the following equation: (5)H(a^,b^)=max{|al−bl|,|au−bu|}

The Hausdorff distance between each element of the i-th class vector and each element of the vector Y is calculated using Eq. (5) as follows: (6)Hi(C^i,Y)=Zi=(zi1zi2…zij),∀i∈[1,m],j=1,2,…,nwhere (7)(zi1=max{|xi1l−y1|,|xi1u−y1|}zi2=max{|xi2l−y2|,|xi2u−y2|}⋮zij=max{|xijl−yj|,|xiju−yj|})T,∀i∈[1,m],j=1,2,…,n

The calculation’s outcome of the Hausdorf distance is the following matrix with crisp elements: (8)Z=[zij]i=1,2,…,mj=1,2,…,n=[C,II1I2…Ijtargetminmin…minC1z11z12…z1jC2z21z22…z2j⋮⋮⋮⋱…Cizi1zi2…zmn]

Step 4: Creation of min and max targets.

Since all desired values are uniformly distributed, i.e., all indices tend to get min target, one of the indices must be transformed into the opposite target value. Hence, there must exist min and max targets. It is a precondition to the GHDQR method working correctly. The calculation of reciprocal values of a selected index does it. Index I1 is selected to be transformed, and its reciprocal values are 1zi1,∀i=1,2,…,m. The rest of the elements of the matrix Z remain the same. The new transformed matrix is: (9)Z=[zij]i=1,2,…,mj=1,2,…,n=[C,II1I2…Ijtargetmaxmin…minC1z11=1z11z12…z1jC2z21=1z21z22…z2j⋮⋮⋮⋱…Cizi1=1zi1zi2…zmn]

Step 5: Normalization of index data.

Because the dimensions of the index values are not consistent, the data in the matrix Z must be transformed into dimensionless values. The following way of normalization is used for transformation.

For max target value of the index, the normalization is: (10)rij(max)=max(zij)−zijmax(zij)−min(zij),∀i∈[1,m],j=1,2,…,n

For min target value of the index, the normalization is: (11)rij(min)=zij−min(zij)max(zij)−min(zij),∀i∈[1,m],j=1,2,…,n

The output of normalization is a normalized matrix of the following form: (12)R=[rij]i=1,2,…,mj=1,2,…,n=[C,II1I2…Ijtargetmaxmin…minC1r11r12…r1jC2r21r22…r2j⋮⋮⋮⋱…Ciri1ri2…rmn]

Step 6: Influence of rock mass evaluation index.

The standard deviation method is employed to determine the influence of the rock mass evaluation index on the classification process. This method belongs to the group of objective methods, and it excludes the subjectivity of the experts. The following equation is utilized to determine the influence of the index: (13)W=[w1w2…wj]=[σ1∑j=1nσjσ2∑j=1nσj…σj∑j=1nσj]where

σj is the standard deviation of the normalized Hausdorff distance of the j-th index.

The influence of the index can be called the index’s weight by applying decision-making terminology.

Step 7: Weighing the normalized matrix.

Each element of a matrix R is weighted by the corresponding weight. The outcome of the weighting process is a weighted normalized matrix as follows: (14)Q=[qij]i=1,2,…,mj=1,2,…,n=[C,II1I2…Ijtargetmaxmin…minC1q11q12…q1jC2q21q22…q2j⋮⋮⋮⋱…Ciqi1qi2…qmn]

Element qij of a matrix Q is calculated by: (15)qij=wjrij,∀i∈[1,2,…,m],∀j∈[1,2,…,n]

Step 8: Artificial class creation.

In this step, an artificial class is created, composed of the max values of each class index. Elements of an artificial class are as follows: (16)K=(k1k2…kj),∀j∈[1,2,…,n]where: (17)kj=max(qij|1≤j≤n),∀i∈[1,m]

The artificial class creation process presents the creation of an optimal class space of Hausdorff distances based on extracting the best value from the existing distance values for each index’s desired target.

Step 9: Creation of min and max subspaces of an artificial class space.

This step implies dividing an artificial class space into two subspaces. An artificial class space is the union of min and max subspaces: (18)K=K(min)∪K(max)

Let γ and β=1 be the total number of indices targeted to min and max desired values, respectively. Accordingly, an artificial class space can be presented as a union of min subspace and max subspace: (19)K=[k1,k2,…,kγ]∪[kβ],∀i∈[1,m],γ+β=j

Step 10: Creation of min and max subspaces of each class space.

Each class in the original matrix X^, and therefore, each class in the matrix Q, is treated as a space. Let γ and β=1 be the total number of indices targeted to min and max desired values, respectively. Accordingly, each class space can be presented as a union of min subspace and max subspace: (20)Ci=Ci(min)∪Ci(max),∀i∈[1,m],γ+β=j (21)Ci=[qi1,qi2,…,qiγ]∪[qiβ],∀i∈[1,m],γ+β=j

Step 11: Calculation the “intensity” of subspaces.

Intensities of min and max subspaces of an artificial class are calculated as follows: (22)K(min)=k12+k22+…+kγ2 (23)K(max)=kβ

The same approach is applied for the min and max subspaces of each class: (24)Ci(min)=qi12+qi22+…+qiγ2,∀i∈[1,m] (25)Ci(max)=qiβ,∀i∈[1,m]

Step 12: Rock mass quality classifier.

A classifier was developed based on the Gromov-Hausdorff distance between two metric spaces to assess to which class a rock mass sample belongs. For basic properties of the Gromov-Hausdorff distance, see [35–37].

Let a and b be two non-negative numbers, and a,b∈ℜ≥0, then absolute p–difference between them is defined by the following expression: (26)Λp:=|ap−bp|1p,p∈[1,∞)

The main idea of the rock mass quality classifier is to use the Gromov-Hausdorff distance between an artificial class space and i-th class space to find out the class of the rock sample. Based on the decomposition process, there are two spaces, K=[K(min),K(max)] and Ci=[Ci(min),Ci(max)], where each space contains min and max subspace. All relations between K and Ci can be presented as a graph G(V,E), where V is a set of vertices, and E is a set of edges. Within this study, a set of vertices equals a set of subspaces, while a set of edges equals the relations between subspaces (Fig. 1).

No crossing edges are allowed in the graph. It implies the existence of only one-to-one relationships between K and Ci spaces. In order to meet this condition, the original graph is partitioned into two subgraphs, G(min) and G(max), as depicted in Fig. 2.

Let K(min) and Ci(min) be a minimum class subspace of an artificial class and minimum subspace of the i-th class space, respectively, then the absolute p–difference between these subspaces equals: (27)Λi,p(min,min):=|Kp(min)−Cip(min)|1p,∀i∈[1,m]

Analogically, this study defines absolute p–difference for max subspaces: (28)Λi,p(max,max):=|Kp(max)−Cip(max)|1p,∀i∈[1,m]

For rock mass quality assessment, the value of p is p≥2. This study adopts p=2. The intensity of the integrated absolute p–difference is defined as follows: (29)Λi,2=Λi,22(min,min)+Λi,22(max,max)=|K2(min)−Ci2(min)|+|K2(max)−Ci2(max)|,∀i∈[1,m]

Finally, the discriminant function (classifier) is defined as follows: (30)C=max{Λi,2}=max{|K2(min)−Ci2(min)|+|K2(max)−Ci2(max)|},i=1,2,…,m

A class with the highest value of {Λi,2},i=1,2,…,m is a class of a sample Y.

The methodology for determining the rock mass class of surrounding rock is shown in the following graphical way (Fig. 3).

The developed algorithm for the same purpose is depicted in Fig. 4.

This section uses three numerical examples from different studies to validate the developed method. Classes obtained by the GHDQR method are compared to classes obtained by other methods.

This study develops an innovative approach for rock mass quality rating in a multi-criteria grey environment named GHDQR. It proposes a new discriminant function (classifier) based on the Gromov-Hausdorff metric space distance for assessing rock mass quality. These metric spaces are the rock mass evaluation matrix and rock sample vector. The algorithm can deal with uncertainties of input data, as interval-grey numbers express the elements of the matrix. The performance of the proposed methodology is verified by comprehensive comparative analysis through the three numerical examples, showing its strength and stability. Several advantages can be observed from the obtained results: o)

A developed classifier objectively defines the class of rock samples, with no experts interfering in the classification process.

The study has several limitations that can be analyzed in the future. One of the limitations of the proposed methodology can be reflected in the fact that a case study with actual data is not presented but only through a comparative analysis. Mining engineers must understand this developed model and compare the results obtained with our original values. Another limitation can be observed in the number of parameters evaluating the rock mass. Only five-rock mass evaluation indices are considered in the proposed model, which should be improved for more parameters such as discontinuity and groundwater conditions.

Future research will focus on developing a model capable of processing more rock samples and hydraulic radius simultaneously to assess the conditions for bulk underground mining methods, such as block caving, sublevel caving, and longwall mining. Another direction of future research will include actual data, i.e., a case study where the results will verify the proposed methodology’s practical implication and efficiency. Also, the specific numerical example should be extended to include the number of parameters evaluating the rock mass in future research. From the point of view of mathematical calculation, future research can include other techniques for calculating the distance metrics such as Euclidean distance, Chebyshev distance, Manhattan distance, and others. By performing a detailed comparative analysis of these methods, the most suitable distance methodology can be selected as the best one. Also, by combining these approaches, hybrid methods for rock mass classification can be created.