**1. Introduction**

In the current process of coal mining, coal mine disasters occur frequently, including landslides, gas explosion, the collapse of the roof, and rock burst, etc., whilst landslides are the dominant disaster type for the open-pit coalmine [1–3]. Slope stability directly affects coal mining safety. Especially for the high slope in the open-pit coalmine, where the height is generally over 200 m, a landslide disaster is more likely to occur because of complex geological conditions and multiple risk factors [4]. Accurate assessment and analysis of high slope stability in an open-pit mine is very important for landslide disaster prevention and control [5,6].

Slope stability analysis usually involves various uncertain factors, such as the statistical uncertainty of rock and soil parameters. However, conventional deterministic analyses, which use a factor of safety (FS) as the slope stability criterion, cannot quantitatively and comprehensively consider the influence of these uncertain factors [7–9]. Many scholars have tried to use different monitoring technologies to realize slope dynamic early warning, such as displacement information integration [10], synthetic aperture radar interference [11], ground-based synthetic aperture radar technology [12], and multi-technology

**Citation:** Wang, Z.; Hu, M.; Zhang, P.; Li, X.; Yin, S. Dynamic Risk Assessment of High Slope in Open-Pit Coalmines Based on Interval Trapezoidal Fuzzy Soft Set Method: A Case Study. *Processes* **2022**, *10*, 2168. https://doi.org/10.3390/ pr10112168

Academic Editors: Feng Du, Aitao Zhou and Bo Li

Received: 22 September 2022 Accepted: 20 October 2022 Published: 23 October 2022

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combination [13], etc. However, these monitoring techniques lack multivariate risk factors analysis. By contrast, risk assessment based on statistics and probability theory can reasonably and comprehensively quantify these uncertainties in slope stability analysis. The most frequently used analytical methods are probability theory [14–17], interval number theory [18], fuzzy theory [19–22], rough set theory, etc. [23,24], and direct MonteCarlo simulation (MCS) [25–27]. However, these methods cannot accurately solve the complex problem of slope risk assessment in which multiple factors and a large number of field data are involved. In recent years, analytic hierarchy process (AHP) [28–30] has been applied to systematically assess the slope stability. Santos et al. [31] proposed a hazard graph and generated the quantitative hazard assessment system to carry out risk analysis for open-pit mine slopes and saw the method as having a high discrimination capacity. Cheng et al. [32] used the random finite difference method (RFDM) to assess the risk of slope failure and concluded that the structure had a dual effect on the stability of the slope. Pinheiro et al. [33] developed slope quality Index and applied it to many real road slopes. Soft set theory also was used to study slope stability affected by multiple risk factors. Yang et al. [34] applied soft set theory to optimize the measures of highway slope treatment, and in their work the weights of affecting factors used in the slope safety evaluation were given by different experts. Fuzzy analytic hierarchy process (FAHP) determines the relative importance of each criterion through pairwise comparison given by domain experts or decision makers and is widely used to calculate standard weights [35–38]. In addition, Zhang and his co-researchers generalized parametric soft sets to dynamic interval trapezoidal fuzzy soft sets and made comprehensive evaluation decisions. The simultaneous feasibility of the decision-making method is proved by a case study [39]. The dynamic fuzzy soft set is also expanded into dynamic interval-valued fuzzy soft sets by He and his co-authors [40]. The operations and properties of dynamic interval-valued fuzzy soft set are studied, and the dynamic interval-valued fuzzy soft set decision is proposed. Zhu et al. enriched and improved the theory of interval trapezoidal fuzzy soft sets and promoted its practical application [41]. All of these achievements for statistics and probability theory have illustrated its capability and popularity in slope reliability analysis.

However, the studies rarely considered the factor of time in their sensitivity evaluation. Thus, the risk assessment of high slope cannot fully reflect the dynamic change of high slope in an open-pit coal mine during a certain period of time. For a slope in an open-pit coal mine, the change of slope stability at different time points is of great importance in risk management. In this work, an interval trapezoidal fuzzy soft set method is developed to achieve the high slope dynamic risk evaluation. The proposed dynamic interval trapezoidal fuzzy soft set method for risk assessment of high slope is developed by integrating time points and the weights of slope risk factors. More specifically, the time period is firstly clarified when the risk factors affecting the slope stability of open-pit mine are strongly superimposed. Then, the risk evaluation grade of the high slope is determined by the FAHP method, which determines slope risk factors by referring to the relevant specifications of the open-pit mine. The risk evaluation grade of the high slope is determined by the FAHP method. Again, the slope risk factors are quantified by expert scoring, and the weight of each risk factor is calculated by FAHP. The weight is independent of time. Then, the dynamic interval trapezoidal fuzzy soft set method is used to integrate the time and weights of different risk factors, and the entropy matrix is used to calculate the evaluation value of risk factors at different moments. The following, the multi-attribute decision integration and evaluation of all risk factor parameters in all time periods are carried out by the dynamic interval trapezoidal fuzzy soft set method. Finally, the high slope in the Shengli open- pit coal mine was taken as a case study, three time points were selected, and the parameter values of 17 secondary indicators were obtained. The risk factors of the high slope at different time points were evaluated and then verified with the field data monitored.

#### **2. Expansion of Interval Trapezoidal Fuzzy Soft Sets**

*2.1. Interval Trapezoidal Fuzzy Soft Sets and Their Properties*

In this section, we introduced some basic definitions and their properties in interval trapezoidal fuzzy soft set theory [42,43].

**Definition 1.** *Let U be an initial universe set and E be a set of parameters. A denotes an interval trapezoidal fuzzy soft set over a common universe U. A is a subset of E, and A* ⊂ *E. A pair* (*F*, *A*) *is called an interval trapezoidal fuzzy soft set over a common universe U, where F is a mapping given by F: A* → Γ(*U*).

*That is to say, for* ∀*e* ∈ *A, we can obtain*

$$F(\mathfrak{e}) = \left\{ \left< \mathbf{x}, S\_{F(\mathfrak{e})}(\mathfrak{x}) | \mathfrak{x} \in \mathcal{U} \right> \right\} \tag{1}$$

*where SF*(*e*)(*x*) *is the interval trapezoidal fuzzy number corresponding to x in F*(*e*)*.*

**Definition 2.** *Let* (*F*, *A*) *and* (*G*, *B*) *be interval trapezoidal fuzzy soft sets over a common universe U.* (*H*, *C*) *is the intersection of* (*F*, *A*) *and* (*G*, *B*)*, where C* = *A* ∩ *B and* (*F*, *A*) ∩ (*G*, *B*)=(*H*, *C*)*. For* ∀*ε* ∈ *C, we can obtain,*

$$F(\xi) = F(\xi) \cap G(\xi) = \left\{ \left< \mathbf{x}, \mathbf{S}\_{F(\xi)}(\mathbf{x}) \cap \mathbf{S}\_{G(\xi)}(\mathbf{x}) \right> \mathbf{x} \in \mathcal{U} \right\} \tag{2}$$

*where SF*(*ξ*)(*x*) *and SG*(*ξ*)(*x*) *are the interval trapezoidal fuzzy Numbers corresponding to x in F*(*ξ*) *and G*(*ξ*)*.*

**Definition 3.** *Let* (*F*, *A*) *and* (*G*, *B*) *be interval trapezoidal fuzzy soft sets over a common universe U. The union of* (*F*, *A*) *and* (*G*, *B*) *is defined as the soft set* (*M*, *D*)*, where D* = *A* ∪ *B and* (*F*, *A*) ∪ (*G*, *B*)=(*M*, *D*)*. For* ∀*ξ* ∈ *D, we can have*

$$M(\varepsilon) = F(\varepsilon) \cup G(\varepsilon) = \left\{ \left< \mathbf{x}, \mathbf{S}\_{M(\varepsilon)}(\mathbf{x}) \right> \mathbf{x} \in \mathcal{U} \right\} \tag{3}$$

*Among them*

$$S\_{M(\varepsilon)}(\mathbf{x}) = \begin{cases} S\_{F(\varepsilon)}(\mathbf{x}), \varepsilon = A - B \\ S\_{G(\varepsilon)}(\mathbf{x}), \varepsilon = B - A \\ S\_{F(\varepsilon)}(\mathbf{x}) \cup S\_{G(\varepsilon)}(\mathbf{x}), \varepsilon = A \cap B \end{cases} \tag{4}$$

*where SF*(*ε*)(*x*) *and SG*(*ε*)(*x*) *are the interval trapezoidal fuzzy numbers corresponding to x in F*(*ε*) *and G*(*ε*)*, respectively.*

If (*G*1, *A*1) and (*G*2, *A*2) are two interval trapezoidal fuzzy soft sets over a common universe *U*, then (*G*1, *A*1) and (*G*2, *A*2) is defined as

$$(G\_{1\prime}A\_{1\prime}) \wedge (G\_{2\prime}A\_{2\prime}) = (H\_{\prime}A\_1 \times A\_2) \tag{5}$$

We can obtain

$$H(\mathfrak{a}, \mathfrak{k}) = \mathcal{G}\_1(\mathfrak{a}) \cap \mathcal{G}\_2(\mathfrak{k}) = \left\{ \left< \mathbf{x}, \mathcal{S}\_{\mathcal{G}\_1(\mathfrak{a})}(\mathbf{x}) \cap \mathcal{S}\_{\mathcal{G}\_2(\mathfrak{k})}(\mathbf{x}) \right> | \mathbf{x} \in \mathcal{U} \right\} \tag{6}$$

for ∀(*α*, *β*) ∈ *A*<sup>1</sup> × *A*2, where *SG*1(*α*)(*x*) and *SG*2(*α*)(*x*) are the interval trapezoidal fuzzy numbers of *x* in the *G*1(*α*) and *G*2(*β*), respectively.

If (*G*1, *A*1) and (*G*2, *A*2) are two interval trapezoidal fuzzy soft sets over a common universe *U*, (*G*1, *A*1) or (*G*2, *A*2) is defined as

$$(\mathcal{G}\_1, A\_1) \vee (\mathcal{G}\_2, A\_2) = (M, A\_1 \times A\_2) \tag{7}$$

We can obtain

$$M(\mathbf{a}, \boldsymbol{\beta}) = \mathbb{G}\_1(\mathbf{a}) \cup \mathbb{G}\_2(\boldsymbol{\beta}) = \left\{ \left< \mathbf{x}, \mathbb{S}\_{\mathbb{G}\_1(\mathbf{a})}(\mathbf{x}) \cup \mathbb{S}\_{\mathbb{G}\_2(\mathbf{a})}(\mathbf{x}) \right> | \mathbf{x} \in \mathcal{U} \right\} \\ \text{for } \forall (\mathbf{a}, \boldsymbol{\beta}) \in A\_1 \times A\_2 \tag{8}$$

where *SG*1(*α*)(*x*) and *SG*2(*α*)(*x*) are the interval trapezoidal fuzzy numbers of *x* in the *G*1(*α*) and *G*2(*β*), respectively.

*2.2. Correlation Theorem of Interval Trapezoidal Fuzzy Soft Sets*

**Theorem 1.** *If* (*G*1, *A*1)*,* (*G*2, *A*2) *and* (*G*3, *A*3) *are three interval trapezoidal fuzzy soft sets over a common universe U, we can obtain*

$$(\left(\left(\mathbf{G}\_1, A\_1\right) \lor \left(\mathbf{G}\_2, A\_2\right)\right) \lor \left(\mathbf{G}\_3, A\_3\right) = \left(\mathbf{G}\_1, A\_1\right) \lor \left(\left(\mathbf{G}\_2, A\_2\right) \lor \left(\mathbf{G}\_3, A\_3\right)\right) \tag{9}$$

$$(\left(\left(\mathbf{G}\_1, A\_1\right) \land \left(\mathbf{G}\_2, A\_2\right)\right) \land \left(\mathbf{G}\_3, A\_3\right) = \left(\mathbf{G}\_1, A\_1\right) \land \left(\left(\mathbf{G}\_2, A\_2\right) \land \left(\mathbf{G}\_3, A\_3\right)\right) \tag{10}$$

**Theorem 2.** *If* (*G*1, *A*1) *and* (*G*2, *A*2) *are two interval trapezoidal fuzzy soft sets over a common universe U, we can obtain*

$$((G\_1, A\_1) \vee (G\_2, A\_2))^\varepsilon = (G\_1, A\_1)^\varepsilon \wedge (G\_2, A\_2)^\varepsilon \tag{11}$$

$$\left( \left( \mathbf{G}\_1 \, A\_1 \right) \wedge \left( \mathbf{G}\_2 \, A\_2 \right) \right)^c = \left( \mathbf{G}\_1 \, A\_1 \right)^c \vee \left( \mathbf{G}\_2 \, A\_2 \right)^c \tag{12}$$

**Theorem 3.** *If* (*G*1, *A*1)*,* (*G*2, *A*2) *and* (*G*3, *A*3) *are three interval trapezoidal fuzzy soft sets over a common universe U, we can obtain*

$$(\left( (\mathcal{G}\_1, A\_1) \lor (\mathcal{G}\_2, A\_2) \right) \land (\mathcal{G}\_3, A\_3) = (\left( \mathcal{G}\_1, A\_1 \right) \land \left( \mathcal{G}\_3, A\_3 \right)) \lor (\left( \mathcal{G}\_2, A\_2 \right) \land \left( \mathcal{G}\_3, A\_3 \right)) \tag{13}$$

$$(\left( (\mathcal{G}\_1, A\_1) \land (\mathcal{G}\_2, A\_2) \right) \land (\mathcal{G}\_3, A\_3) = (\left( \mathcal{G}\_1, A\_1 \right) \lor \left( \mathcal{G}\_3, A\_3 \right)) \land \left( \left( \mathcal{G}\_2, A\_2 \right) \lor \left( \mathcal{G}\_3, A\_3 \right) \right) \tag{14}$$

Based on the above expansion of interval trapezoidal fuzzy soft sets, the dynamic risk evaluation model of high slope in an open-pit mine are established in the next section.

### **3. Establishing the Dynamic Risk Evaluation Model of High Slope in Open-Pit Mine**

In this section, the following steps are used to integrate the time points and weights of slope risk factors (see Figure 1).

Step 1: Make clear the time period when the risk factors affecting the slope stability of open-pit mine are strong superimposed.

Step 2: Determine slope risk factors by referring to the relevant specifications of openpit mine. They are technical specification for annual evaluation of the stability of open-pit coal mine slopes (GB/T 37573-2019) and code for the design of open-pit mines in the coal industry (GB 50197-2015). Then, the risk evaluation grade of high slope is determined by the FAHP method.

Step 3: The slope risk factors are quantified by expert scoring, and the weight of each risk factor is determined by FAHP method.

Step 4: The time and weight of risk factors are integrated by dynamic interval trapezoidal fuzzy soft sets, and the evaluation value of risk factors at different moments is calculated by entropy matrix. Finally, the multi-attribute decision integration and evaluation of all risk factor parameters in all time periods are carried out by dynamic interval trapezoidal fuzzy soft sets.

**Figure 1.** Intuitionistic trapezoidal fuzzy soft set method for risk assessment of high slopes.

For the dynamic evaluation of high slope risk in open-pit mine, the time point sets of monitoring data are given, which is denoted as

$$T = \{t\_1, t\_2, \dots, t\_k\} \tag{15}$$

where *t* is the time point of different risk factors; *k* is the number of time points.

To determine four primary indexes, we mainly refer to the relevant specifications of open-pit mine, which are technical specification for annual evaluation of the stability of open-pit coal mine slopes (GB/T 37573-2019) and specifications for design of open pit mine of coal industry (GB 50197-2015). To determine 17 primary indexes (see Table A1), we not only refer to the relevant specifications of open pit mine but also add to concern factors for high slope stability by technical and managerial personnel of the open pit mine.

The parameter sets of risk assessment are

$$A = \{B\_1, B\_2, B\_3, B\_4\} \tag{16}$$

= { where *B*<sup>1</sup> is hydrological-climatic conditions, and *B*<sup>2</sup> is internal geological structure of slope. *B*<sup>3</sup> and *B*<sup>4</sup> are slope geometry and inducing factors of landslide, respectively.

$$B\_1 = \{c\_{11}, c\_{12}, c\_{13}, c\_{14}\} \tag{17}$$

$$B\_2 = \{e\_{21}, e\_{22}, e\_{23}, e\_{24}, e\_{25}\} \tag{18}$$

$$B\_3 = \{ \mathfrak{e}\_{31}, \mathfrak{e}\_{32}, \mathfrak{e}\_{33}, \mathfrak{e}\_{34} \} \tag{19}$$

$$B\_4 = \{e\_{41'}e\_{42'}e\_{43'}e\_{44}\} \tag{20}$$

where *e*<sup>11</sup> is weathering and freeze-thaw, and *e*<sup>12</sup> is the state of groundwater. *e*<sup>13</sup> is permeability of rock and soil layer, and *e*<sup>14</sup> is annual rainfall. *e*<sup>21</sup> and *e*<sup>22</sup> are lithology and geological structure, respectively. *e*<sup>23</sup> and *e*<sup>24</sup> are slope structure and internal friction angle, respectively. *e*<sup>25</sup> is cohesion of slope. *e*<sup>31</sup> and *e*<sup>32</sup> are slope angle and slope height, respectively. *e*<sup>33</sup> is relationship between soft surface and slope surface, and *e*<sup>34</sup> is slope morphology. *e*<sup>41</sup> and *e*<sup>42</sup> are human factors and impact of blasting, respectively. *e*<sup>43</sup> and *e*<sup>44</sup> are slope angle of excavation and earthquake intensity, respectively. *B*1~*B*<sup>4</sup> are primary indicators and *e*11~*e*<sup>44</sup> are secondary indicators.

The slope risk evaluation result set is

$$X = \{\mathbf{x}\_1, \mathbf{x}\_2, \dots, \mathbf{x}\_n\} \tag{21}$$

where *n* is the number of evaluation grades.

#### *3.1. Determining the Weights in Different Time Range*

If *s* = [(*a*−, *a*+); *b*; *c*;(*d*−, *d*+)] is the interval trapezoidal fuzzy number, its entropy value is

$$E(\mathbf{s}) = \left| \frac{1}{4} (\frac{a^- + a^+}{2} + b + c + \frac{d^- + d^+}{2}) - 0.5 \right| \tag{22}$$

The entropy matrix of *E<sup>k</sup> Bq* = (*E<sup>k</sup> Bq* (*e* (*k*) *ij* ))*m*×*<sup>n</sup>* at a different range is obtained by calculating the values of *B*1, *B*2, *B*<sup>3</sup> using Equation (22). (*e* (*k*) *ij* ) *<sup>n</sup>*×*<sup>m</sup>* is the *<sup>k</sup>*th evaluation matrix. *e* (*k*) *ij* (*i* = 1, 2, 3 ··· *n*; *j* = 1, 2, 3 ··· *m*) is the value of attribute *cj* in the form of interval trapezoidal fuzzy number about evaluation grade *ti* at time point of *tk*.

Where *i* = 1, 2, 3; *k* = 1, 2 ... ; *j* = *q*1, *q*2 ... *ql*; *q* = 1, 2, 3, 4. *ql* represents the number of parameters of the index *Bq*. Then, all the entropy values in the corresponding entropy matrix are added.

The entropy matrix can be obtained

$$E(B\_q^k) = \frac{1}{mn} \sum\_{i=1}^m \sum\_{j=1}^n E(e\_{ij}^{(k)}) \tag{23}$$

The corresponding weight *w*(*B<sup>k</sup> <sup>q</sup>*) at the moment *tk* of *Bq*

$$w(B\_q^k) = \frac{1 - E(B\_q^k)}{\sum\_{i=1}^m (1 - E(B\_q^k))}\tag{24}$$

#### *3.2. Determining the Index Weight by FAHP Method*

In order to determine the weight of each influencing factor, FAHP [44–46] method is used and a fuzzy consistent matrix is firstly constructed, and the matrix *A* = (*aij*) *n*×*n* satisfies the following conditions:

$$\begin{cases} \ 0 \le a\_{ij} \le 1 \ (i, j = 1, 2, 3 \cdots \cdot, n) \\\ a\_{ij} + a\_{ji} = 1 \ (i, j = 1, 2, 3 \cdots \cdot, n) \\\ \forall i, j, a\_{ij} = \omega\_i - \omega\_j + 0.5 \end{cases} \tag{25}$$

If *A* is a fuzzy consistent matrix, the weight *ω* can be denoted by

$$
\omega\_i = \frac{1}{n} (\sum\_{j=1}^n a\_{ij} + 1 - \frac{n}{2}) \tag{26}
$$

It is worth noting that when *<sup>n</sup>* ∑ *j*=1 *aij* <sup>≤</sup> *<sup>n</sup>* <sup>2</sup> − 1, it needs to be modified until *ω<sup>i</sup>* ≥ 0

is satisfied.
