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A New Roof Strata Cavability Index (RSCi) for Longwall Mining Incorporating New Rating System

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This paper presents a new index to assess caving potential of roof strata above longwall panels. Nine inherent parameters were chosen as significant affecting factors in three categories, including roof strata characteristics, roof discontinuities properties and local features. Fuzzy hybrid multi criteria decision making was used by combining fuzzy analytic network process technique and fuzzy decision making trial and evaluation laboratory method to develop a new rating system. Subsequently, roof strata cavability index (RSCi) was defined as a simple summation of ratings for all parameters, indicating the cavability level qualitatively. The RSCi was applied to determine the cavability of various actual cases of worldwide longwall panels. In addition, the correlation between RSCi and span of main caving was examined where reliable performance of the new index was noted. It was concluded that RSCi is a simple and efficient tool to assess the cavability of immediate roof and evaluation of caving intervals in longwall mining.

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Appendix 1

Fuzzy DEMATEL steps is described as following (Mohammadi et al. 2018):

Step 1: Acquiring the initial and average direct relationship matrices. Experts indicated the direct influence that factor i exerts on the factor j, using a fuzzy triangular number (TFN). The TFN corresponding to each of the linguistic variables are given in Table 12.

Table 12 Correspondences of linguistic terms and linguistic values

Elements of \(\tilde{z}_{ij}\) (called initial direct relationship matrix) are (lij, mij, uij) in which the first, second, and third terms represent the lower, middle, and upper bounds of TFN. The average direct relationship matrix \(\tilde{A}\) is calculated by taking the average of h expert’s value matrices as follows:

$$\tilde{A} = \frac{{(\tilde{Z}^{1} \oplus \tilde{Z}^{2} \oplus \cdots \oplus \tilde{Z}^{h} )}}{h}$$

Step 2: Calculating the normalized direct relationship matrix. The normalized direct relationship matrix \(\tilde{X}\) is obtained through normalizing the matrix \(\tilde{A}\) as follow:

$$\tilde{X} = \frac{{\tilde{A}}}{r}$$
$$r = \hbox{max} [\max_{1 \le i \le n} \sum\nolimits_{j = 1}^{n} {u_{ij} } ,\max_{1 \le j \le n} \sum\nolimits_{i = 1}^{n} {u_{ij} } ], \, i,j = 1,2, \ldots ,n.$$

\(\tilde{x}_{ij} = (l^{\prime}_{ij} ,m^{\prime}_{ij} ,u^{\prime}_{ij} )\) are elements of \(\tilde{X}\) and define three crisp matrices, whose elements are extracted from \(\tilde{X}\) as follows:

$$X_{l} = \left[ {\begin{array}{*{20}c} 0 & {l^{\prime}_{12} } & \cdots & {l^{\prime}_{1n} } \\ {l^{\prime}_{21} } & 0 & \cdots & {l^{\prime}_{2n} } \\ \vdots & \vdots & \ddots & \vdots \\ {l^{\prime}_{n1} } & {l^{\prime}_{n2} } & \cdots & 0 \\ \end{array} } \right];\;X_{m} = \left[ {\begin{array}{*{20}c} 0 & {m^{\prime}_{12} } & \cdots & {m^{\prime}_{1n} } \\ {m^{\prime}_{21} } & 0 & \cdots & {m^{\prime}_{2n} } \\ \vdots & \vdots & \ddots & \vdots \\ {m^{\prime}_{n1} } & {m^{\prime}_{n2} } & \cdots & 0 \\ \end{array} } \right];\;X_{u} = \left[ {\begin{array}{*{20}c} 0 & {u^{\prime}_{12} } & \cdots & {u^{\prime}_{1n} } \\ {u^{\prime}_{21} } & 0 & \cdots & {u^{\prime}_{2n} } \\ \vdots & \vdots & \ddots & \vdots \\ {u^{\prime}_{n1} } & {u^{\prime}_{n2} } & \cdots & 0 \\ \end{array} } \right]$$

Step 3: Drive total relationship matrix. The total relationship matrix \(\tilde{T}\) is obtained using following formulas in which I is denoted as identity matrix:

$$\tilde{T} = \tilde{X}(I - \tilde{X})^{ - 1}$$

Elements of \(\tilde{T}\) are \(\tilde{t}_{ij} = (l^{\prime\prime}_{ij} ,m^{\prime\prime}_{ij} ,u^{\prime\prime}_{ij} )\). According to the crisp case, crisp elements of total relationship matrices are calculated as:

\(T_{l} = [l^{\prime\prime}_{ij} ] = X_{l} (I - X_{l} )^{ - 1} ;\;T_{m} = [m^{\prime\prime}_{ij} ] = X_{m} (I - X_{m} )^{ - 1} ;\;T_{u} = [u^{\prime\prime}_{ij} ] = X_{u} (I - X_{u} )^{ - 1}\)

This matrix is the interdependencies matrix that is substituted in the super matrix.

Appendix 2

In general, ANP has two main steps as follows (Saaty 2005):

Step 1: Problem network establishment: Firstly, it is necessary to state the problem clearly and to construct its corresponding network accordingly. For this purpose, the decision maker’s opinion through brainstorming or other appropriate methods such as DEMATEL is incorporated.

Step 2: Forming the supermatrix: To form the supermatrix, system criteria are compared by determining the importance of each criterion in comparing to another criterion with respect to its controlled criteria. The relative importance is determined using a scale of 1–9, representing equal importance to extreme importance. The general form of supermatrix is shown in Fig. 15.

Fig. 15
figure 15

General form of supermatrix (Saaty 2005)

where Cm denotes mth cluster, emn denotes nth element in mth cluster, and matrix Wij is the principal eigenvector compared to jth and ith cluster.

Subsequently, the weighted supermatrix is derived by equating the normalized summation of each column to 1. The weighted supermatrix is raised to limiting powers as per Eq. (15) to calculate weights and overall priorities.

$$\mathop {\lim }\limits_{k \to \infty } \, W^{k}$$

where W is the supermatrix.

Appendix 3

The procedure to calculate \(S_{{c_{i} }}\) is illustrated in Fig. 16

Fig. 16
figure 16

Calculation procedure of \(S_{{c_{i} }}\) for two different cases

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Mohammadi, S., Ataei, M., Kakaie, R. et al. A New Roof Strata Cavability Index (RSCi) for Longwall Mining Incorporating New Rating System. Geotech Geol Eng 37, 3619–3636 (2019).

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