*3.3. Integration and Decision Method of Interval Trapezoidal Fuzzy Soft Sets*

Let *U* = {*x*1, *x*2, ··· *xm* } be an initial universe set and *A* = {*e*1,*e*2, ···*en* } be a set of parameters. The matrix of *F*0 *<sup>k</sup>* = (*SFk* (*ej*)(*xi*))*m*×*<sup>n</sup>* is corresponding to the interval trapezoidal fuzzy soft set of (*Fk*, *A*). Where, *i* = 1, 2, ... *m*; *j* = 1, 2, ... *n*; *k =* 1, 2, ... *K*. *SFk* (*ej*)(*xi*) is the corresponding interval trapezoidal fuzzy number of *Fk*(*ej*) and *xi*, the interval trapezoidal fuzzy soft matrix between the algorithm is given as follows.

If *<sup>F</sup>*0<sup>1</sup> = (*SF*1(*ej*)(*xi*))*m*×*<sup>n</sup>* and *<sup>F</sup>*0<sup>2</sup> = (*SF*2(*ej*)(*xi*))*m*×*<sup>n</sup>* are interval trapezoidal fuzzy soft matrix, and *c* > 0, we can obtain

$$
\tilde{F}\_1 + \tilde{F}\_2 = \left( S\_{F\_1(\mathfrak{e}\_j)}(\mathfrak{x}\_i) \right)\_{m \times n} + \left( S\_{F\_2(\mathfrak{e}\_j)}(\mathfrak{x}\_i) \right)\_{m \times n} \tag{27}
$$

$$c\tilde{F}\_1 = \left(cS\_{F\_1\left(\mathfrak{e}\_{\tilde{f}}\right)}\left(\mathfrak{x}\_{\tilde{i}}\right)\right)\_{m\times n} \tag{28}$$

If *F*0 *<sup>k</sup>*(*k* = 1, 2 ··· *K*) is the interval trapezoidal fuzzy soft matrix and the weight satisfies the conditions of *<sup>ω</sup>* <sup>∈</sup> [0, 1] and *<sup>K</sup>* ∑ *k*=1 *ω<sup>k</sup>* = 1.

The weighted average operator of the interval trapezoidal fuzzy matrix can be denoted by

$$f\_{\omega}(\vec{F}\_1, \vec{F}\_{2\prime}, \cdots, \vec{F}\_K) = \sum\_{k=1}^{K} \omega\_k \vec{F}\_k \tag{29}$$

We can obtain

$$f\_{\omega}(\vec{F}\_1, \vec{F}\_2, \dots, \vec{F}\_K) = \sum\_{k=1}^K \left(\omega\_k S\_{\vec{F}\_k(c\_j)}(\mathbf{x}\_i)\right)\_{m \times n} \tag{30}$$

The integrated synthetic interval trapezoidal fuzzy soft matrix of *F*0*K*(*k* = 1, 2, ··· , *K*) is denoted as *R*0 = (*rij*) *<sup>m</sup>*×*n*.

The decision values are used to deal with the decision problems of the interval trapezoidal fuzzy soft sets, and the decision values for *xi* in the universe set based on interval trapezoidal fuzzy soft sets in the domain are denoted by

$$\eta\_{i} = \sum\_{j=1}^{m} \left( (\pi\_{i}^{-} - \pi\_{j}^{-}) + (\pi\_{i}^{+} - \pi\_{j}^{+}) + (\varphi\_{i} - \varphi\_{j}) + (\varepsilon\_{i} - \varepsilon\_{j}) + (\gamma\_{i}^{-} - \gamma\_{j}^{-}) + (\gamma\_{i}^{+} - \gamma\_{j}^{+}) \right) \tag{31}$$

where the corresponding interval trapezoidal fuzzy number of *xi* is

$$S(\mathfrak{x}\_i) = [(\pi\_i^- \wr \pi\_i^+); \mathfrak{q}\_i; \mathfrak{e}\_{i\prime}(\gamma\_i^- \wr \gamma\_i^+)] \tag{32}$$

*3.4. Case Study*

3.4.1. Dynamic Risk Assessment Based on Trapezoidal Fuzzy Soft Set in Shengli Open-Pit Mine

(1) Decision-making steps and methods

Based on above interval trapezoidal fuzzy soft set model of high slope in an open-pit coal mine, the dynamic evaluation risk steps of high slope in an open-pit coal mine are divided into four steps:

(1) Determine the times when the risk factors affecting the slope stability of open-pit mine are strong superimposed;

(2) Determine the weight of each parameter using Equations (22)–(26);

(3) Integrate interval trapezoidal fuzzy soft matrix at different times using Equation (30);

(4) Calculate the decision values corresponding to different risks using Equation (31). 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.

(2) Determination of values for dynamic model parameter set

The north end slope of Shengli #1 open-pit mine is taken in this case study (see Figure 2). The slope height and maximum slope angle are, respectively, 200 m and 57◦. Three moments were selected, i.e., *T* = {*t*1, *t*2, *t*3} , *t*<sup>1</sup> is 1 July 2016; *t*<sup>2</sup> is 1 August 2016 and *t*<sup>3</sup> is 1 September 2016. The three time points (July, August and September) selected are the times with strong superposition of risk factors for open-pit mine slope. In this time period, there are many unstable factors inducing landslides. On the one hand, these three months are the rainy season for the open-pit mines in northern China, with a relatively concentrated and large rainfall, and the slope stability is greatly affected by hydrogeological factors. On the other hand, the open-pit mines in northern China can only be mined but not stripped in winter due to weather reasons. In order to leave enough earthwork and coal for winter mining, the blasting frequency during the three months (July, August and September) is relatively large, and the slope stability is greatly affected by mining factors. The risk decision of high slope in open-pit mine is evaluated from four primary risk factors (S), the hydrological and climatic conditions (*B*1), the slope internal geological structure (*B*2), the slope geometric conditions (*B*3), and landslide risk factors (*B*4). The risk level may be labeled as low, general and high, which are denoted by (*x*1), (*x*2), and (*x*3), respectively. In addition, 17 secondary indicators are selected to carry out risk assessment in four primary indicators. This will be described in the third part later on. The risk grade is divided by considering the influence of comprehensive factors. Low and medium are classified as low decision risk, high as general decision risk, and dangerous and extremely dangerous as high decision risk. The expert evaluation method was used to evaluate the slope risk factors of open-pit mine. Data were collected mainly through questionnaires. A sample of the questionnaire is attached. The data were mainly collected from the production technicians, stripping workers and management personnel of Shengli Open-pit coal mine. It covers all positions of frontline production, management and technology in open-pit coal mines. Among them, 80 questionnaires were sent out by Shengli open-pit coal mine, and 72 questionnaires were effectively recovered. The parameters of slope risk factor *B*1, *B*2, *B*<sup>3</sup> and *B*<sup>4</sup> at time *t*1, *t*<sup>2</sup> and *t*<sup>3</sup> are listed in Tables A2–A12 in Appendix A.

(3) Calculation of weights for dynamic model

**Figure 2.** High slopes of Shengli #1 open-pit mine.

The values of the four primary risk indicators of *B*1, *B*2, *B*<sup>3</sup> and *B*<sup>4</sup> are calculated by Equation (22), subsequently the entropy matrix of *E<sup>k</sup> Bq* = (*E*(*e* (*k*) *ij* ))*m*×*<sup>n</sup>* corresponding to *tk*

=

= = =

= <sup>×</sup>

at different moments can be obtained. For example, the calculation results of the entropy matrix of *B*<sup>1</sup> at different times are shown in Tables 1–3.


**Table 1.** Entropy of each parameter of hydrological and climatic conditions *B*<sup>1</sup> at time *t*1.

**Table 2.** Entropy of each parameter of hydrological and climatic conditions *B*<sup>1</sup> at time *t*2.


**Table 3.** Entropy of each parameter of hydrological and climatic conditions *B*<sup>1</sup> at time *t*3.


The weight of *B*<sup>1</sup> is calculated by Equations (23) and (24), and the result is

$$w\_{B\_1} = (0.219, 0.395, 0.386)\tag{33}$$

Similarly, the weights of *B*2, *B*<sup>3</sup> and *B*<sup>4</sup> can be obtained

$$w\_{B\_2} = (0.214, 0.396, 0.390)\tag{34}$$

$$w\_{B\_3} = (0.229, 0.394, 0.377) \tag{35}$$

$$w\_{B\_4} = (0.235, 0.385, 0.380)\tag{36}$$

#### 3.4.2. Risk Evaluation

Based on the above interval valued fuzzy set model, the dynamic risk of the north end slope in Shengli #1 open-pit coalmine is evaluated comprehensively. The above three time points are selected to evaluate the risk from four aspects, i.e., hydro-climatic conditions, slope internal geological structure, slope geometric conditions and induced factors. The risk evaluation of high slope is divided into three levels: low (*x*1), general (*x*2) and high (*x*3). Combined with the above theory, the weights at different times are obtained and then the comprehensive interval trapezoidal fuzzy soft set evaluation information for different parameter sets in all time range can be obtained by Equation (29). That is, the comprehensive evaluation values of influencing factors *B*1, *B*2, *B*<sup>3</sup> and *B*<sup>4</sup> in the whole time range subjected to different risks can be obtained as listed in Tables A14–A17 in Appendix A.

Meanwhile, the fuzzy symmetry matrix of each parameter for the risk factors (i.e., hydro-climatic conditions, internal geological structure, geometric conditions and induced factors) are obtained by FAHP theory in Tables A18–A21, as shown in Appendix A. The corresponding weight vectors are calculated as follows

$$w\_{B\_1} = (0.225, 0.250, 0.250, 0.275) \tag{37}$$

$$w\_{B\_2} = (0.160, 0.260, 0.220, 0.180, 0.180)\tag{38}$$

$$w\_{B3} = (0.150, 0.250, 0.250, 0.350)\tag{39}$$

$$w\_{B4} = (0.225, 0.375, 0.300, 0.100)\tag{40}$$

The fuzzy symmetric matrix for primary risk indicators is obtained in Table A22 in Appendix A. The weight value is obtained by Equation (41)

$$
\omega\_A = (0.150, 0.150, 0.325, 0.375)\tag{41}
$$

The weights of risk parameters for the risk evaluation system of high slope in open-pit mine are given in Table 4.


**Table 4.** Weights of risk parameters for the risk evaluation system.

The interval trapezoidal fuzzy soft set of different parameters are integrated by Equation (30). The integrated evaluation value of different parameters belonging to openpit slope risk and the integrated value of surface mine slope risk relative to different risk grades are obtained in Tables A23 and A24 in Appendix A.

The comprehensive evaluation value of the relative risk level, which is either low, general or high, can be obtained. The evaluation value which represents low risk for high slope in Shengli open-pit mine is

$$\mu\_1 = [(0.225, 0.401); 0.427; 0.491; (0.548, 0.704)] \tag{42}$$

The evaluation value which represents general risk for high slope is

$$\mu\_2 = \left[ (0.243, 0.408); 0.414; 0.481; (0.546, 0.699) \right] \tag{43}$$

The evaluation value which represents high risk for high slope is

$$\mu\_{\mathbb{B}} = [(0.295, 0.434); 0.450; 0.512; (0.567, 0.681)] \tag{44}$$

Further, the decision value of the high slope risk in the open-pit which is subject to low risk, general risk, and higher risk can be obtained. The integrated evaluation value is calculated according to Equation (28). The results are shown in Table 5.


**Table 5.** Different decision factors relative to open-pit slope risk.

As shown in Table 5, for the decision-making results of the high slope in Shengli #1 open-pit mine, the values of *B*<sup>1</sup> and *B*<sup>3</sup> indicate a high risk level, while the values of *B*<sup>2</sup> and *B*<sup>4</sup> indicate a general risk level. Thus, among the many factors affecting the instability of the slope of Shengli #1 open-pit coal mine, hydro-climatic conditions and slope geometric conditions belong to high risk factors. Special attention should be paid to hydrology and slope geometry during slope stability maintenance. For example, slope reinforcement and radar displacement monitoring should be planned and performed during the rainy season. The stability evaluation should be performed when designing the slope angle.
