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Volume 31 Issue 6
Dec.  2020
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Peng Xia, Xinli Hu, Shuangshuang Wu, Chunye Ying, Chang Liu. Slope Stability Analysis Based on Group Decision Theory and Fuzzy Comprehensive Evaluation. Journal of Earth Science, 2020, 31(6): 1121-1132. doi: 10.1007/s12583-020-1101-8
Citation: Peng Xia, Xinli Hu, Shuangshuang Wu, Chunye Ying, Chang Liu. Slope Stability Analysis Based on Group Decision Theory and Fuzzy Comprehensive Evaluation. Journal of Earth Science, 2020, 31(6): 1121-1132. doi: 10.1007/s12583-020-1101-8

Slope Stability Analysis Based on Group Decision Theory and Fuzzy Comprehensive Evaluation

doi: 10.1007/s12583-020-1101-8
More Information
  • A slope stability evaluation method is proposed combining group decision theory, the analytic hierarchy process and fuzzy comprehensive evaluation. The index weight assignment of each evaluation element is determined by group decision theory and the analytic hierarchy process, and the membership degree of each indicator is determined based on fuzzy set theory. According to the weights and memberships, the membership degrees of the criterion layer are obtained by fuzzy operations to evaluate the slope stability. The results show that (1) the evaluation method comprehensively combines the effects of multiple factors on the slope stability, and the evaluation results are accurate; (2) the evaluation method can fully leverage the experience of the expert group and effectively avoid evaluation errors caused by the subjective bias of a single expert; (3) based on a group decision theory entropy model, this evaluation method can quantitatively evaluate the reliability of expert decisions and effectively improve the efficiency of expert group discussion; and (4) the evaluation method can transform the originally fuzzy and subjective slope stability evaluation into a quantitative evaluation.
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Slope Stability Analysis Based on Group Decision Theory and Fuzzy Comprehensive Evaluation

doi: 10.1007/s12583-020-1101-8

Abstract: A slope stability evaluation method is proposed combining group decision theory, the analytic hierarchy process and fuzzy comprehensive evaluation. The index weight assignment of each evaluation element is determined by group decision theory and the analytic hierarchy process, and the membership degree of each indicator is determined based on fuzzy set theory. According to the weights and memberships, the membership degrees of the criterion layer are obtained by fuzzy operations to evaluate the slope stability. The results show that (1) the evaluation method comprehensively combines the effects of multiple factors on the slope stability, and the evaluation results are accurate; (2) the evaluation method can fully leverage the experience of the expert group and effectively avoid evaluation errors caused by the subjective bias of a single expert; (3) based on a group decision theory entropy model, this evaluation method can quantitatively evaluate the reliability of expert decisions and effectively improve the efficiency of expert group discussion; and (4) the evaluation method can transform the originally fuzzy and subjective slope stability evaluation into a quantitative evaluation.

Peng Xia, Xinli Hu, Shuangshuang Wu, Chunye Ying, Chang Liu. Slope Stability Analysis Based on Group Decision Theory and Fuzzy Comprehensive Evaluation. Journal of Earth Science, 2020, 31(6): 1121-1132. doi: 10.1007/s12583-020-1101-8
Citation: Peng Xia, Xinli Hu, Shuangshuang Wu, Chunye Ying, Chang Liu. Slope Stability Analysis Based on Group Decision Theory and Fuzzy Comprehensive Evaluation. Journal of Earth Science, 2020, 31(6): 1121-1132. doi: 10.1007/s12583-020-1101-8
  • With the rapid development of China's economy, the development of large-scale infrastructure projects across a large number of rock slopes has brought about great economic benefits. The stability of these slopes is directly related to both the safety of the lives and property of local residents and the safe operation of infrastructure such as railways, highways, waterways and hydropower stations. It is necessary to carry out research on the stability evaluation methods of such slopes.

    At present, commonly used methods for slope stability evaluation are the limit equilibrium method, the strength reduction method (SRM) and numerical simulation (Deng, 2020; Yuan et al., 2020; Iqbal et al., 2018; Liu et al., 2015; Lu et al., 2015; Ni et al., 2014; Sun et al., 2013; Hu et al., 2012). In addition to these three commonly used analytical methods, some scholars use neural networks, extreme learning machines, fuzzy comprehensive evaluation, gray relational analysis, reliability theory and the analytic hierarchy process (AHP) to analyze the stability of slopes (Deng et al., 2020; Ray et al., 2020; Tan et al., 2018; Jiang et al., 2015; Liu et al., 2014; Bi et al., 2012; Li et al., 2012; Daftaribesheli et al., 2011; Rozos et al., 2011; Suh et al., 2011).

    However, there are still some shortcomings of these methods. For example, the most commonly used limit equilibrium method is the slicing method. The slicing method is based on the assumption that soil strips are separable; however, slope stability is a statically indeterminate problem. It is necessary to assume the force between the soil pieces to solve the problem. These assumptions are completely artificial. If the assumptions used are different, the results are different (Liu et al., 2015; Griffiths and Fenton, 2004). The disadvantage of the SRM method is that it cannot locate other "slip" surfaces (i.e., local minima) (Cheng et al., 2007). Slope stability analysis and design can usually use data with only a limited density, and more routine use of advanced numerical simulation programs such as FLAC and UDEC is generally impractical and not always necessary (Bar et al., 2020). The penalty parameters of the support vector machine (SVM) method are robust and difficult to identify (Liu et al., 2013). For an artificial neural network, the problems include: it is difficult to determine the network structure, the training speed is slow, and it is possible to fall into a local minimum solution (Sinha et al., 2010).

    The AHP and the group eigenvalue method (GEM) are practical decision-making methods. These two methods combine qualitative and quantitative methods to address various decision-making problems and have the advantages of being systematic, flexible and concise. They are widely used in different fields (Bahrami et al., 2020; Qiao and Chen, 2015; Shi et al., 2015; Zang et al., 2015; Kayastha et al., 2013; Yu and Wen, 2012; Gao et al., 2011; Lin et al., 2010). However, each has its own shortcomings. For example, regarding the AHP, due to the diversity of the objective world and human judgment errors, it is very difficult to construct a consistent judgment matrix. The GEM is too general and fuzzy when considering problems (Hong and Qiu, 2000).

    The fuzzy set theory was founded in the 1960s using the concept of a membership function to describe the epitaxial ambiguity of a set (Zadeh, 1965). The characteristics of slopes and the boundary of slope stability are not very clear, and they are quite fuzzy. It is difficult to use a classic mathematical model to measure them uniformly. The fuzzy comprehensive evaluation method can comprehensively consider the relevant factors of the object or attribute being evaluated and then perform a grade or category evaluation. Fuzzy comprehensive evaluation provides an effective method for slope stability analysis with multiple variables and multiple factors (Zhang et al., 2010). At present, fuzzy set theory is widely used in slope stability analysis (Azarafza et al., 2020; Cho, 2013; Park et al., 2012; Daftaribesheli et al., 2011).

    Despite the above-mentioned advantages of slope stability evaluation methods, group decision theory, and fuzzy comprehensive evaluation methods, few existing studies combine group decision theory and fuzzy comprehensive evaluation methods for slope stability evaluation.

    A new method has been developed in this work to evaluate slope stability. This method introduces the group decision theory and makes full use of the experience of the expert group. It can effectively avoid evaluation errors caused by differences in individual knowledge structures and subjective judgment deviations. Meanwhile, a fuzzy comprehensive evaluation method is used to comprehensively consider the influence of various factors on slope stability, solving the problem of the fuzzification of the characteristics of a landslide and the boundary of the stability state. AHP and GEM are combined to determine the weight of each indicator. It retains the scientific analysis process of AHP to establish the hierarchical analysis structure and avoids the inconsistency of the AHP judgment matrix. The entropy model (Qiu, 1995) is introduced into the group discussion, realizing the quantitative evaluation of the reliability of expert group decision- making and improving the efficiency of group discussion.

  • The AHP proposed by Saaty(1990, 1979) collects decision- making problems into a hierarchy in which the highest elements represent the overall decision-making goal. The elements are decomposed into subgoals at the next highest level and are further decomposed into their respective subgoals until the final hierarchy is sufficient to represent the relevant targets; then, the complex problems are solved according to the hierarchy.

    Group decision-making involves grouping different members' preferences about each program in the program set into a consistent or compromised group preference of the decision- making group according to certain rules. This definition characterizes some of the characteristics of normative group decision-making; that is, it aims to find a rule that is fair to decision-making groups to aggregate the preferences of individual decision-makers (Hwang and Lin, 2012). An eigenvalue method for group decisions (Qiu, 1997) was introduced into the stability evaluation of slopes. This method is superior to the traditional AHP in that it does not need to construct a judgment matrix of pairwise comparisons, and it avoids the inconsistency of the target sequence that the judgment matrix can easily lead to. Experts provide scores according to their own habits to obtain the optimal ranking results of the group. Therefore, it is more convenient to use the eigenvalue method than the traditional AHP. An entropy model for a group decision system (Qiu, 1995; Gu and Qiu, 1992) was used to measure the reliability of individual and group decision-making.

    The implementation steps are as follows.

    ① Determine the stability evaluation indicator and grading standard of the slope.

    ② Calculate the weight and expert reliability of each evaluation indicator based on group decision theory.

    ③ Calculate the membership degree based on the fuzzy comprehensive evaluation method.

    ④ Evaluate the stability of the slope.

    The flow chart of the evaluation method is shown in Fig. 1.

    Figure 1.  Flow chart of the slope stability evaluation method based on group decision theory and fuzzy comprehensive evaluation.

  • Indicators and grading standards were studied by He et al. (2012). The indicators and grading standards for the stability evaluation of slopes include 19 indicators of 3 levels and 5 projects that are important for research. The three levels are the criteria layer, the project layer and the indicator layer. The project layer consists of five projects: the slope geometry, the state of the rock mass, the factors affecting seepage, the landslide history and the reinforcement measures, which are represented by AE. There are 19 indicators in the indicator layer, which are the rock slope height and rock slope angle, represented by A1 and A2. The types of discontinuities, spacing of the discontinuities, degree of closure of a discontinuity, degree of discontinuity, camber angle of the main discontinuity, angle between the slope strike and structural plane strike, angle between the two sets of discontinuities, lithology, weathering degree of the rock mass, and boulder or suspended rock block distribution are represented by B1B10, respectively. The drainage system, maximum daily rainfall, groundwater level, and seepage are represented by C1C4. The landslide that has occurred is represented by D1. The protective structure state and reinforcement structure state are represented by E1 and E2. After numbering 19 indicators of 3 levels and 5 projects, the AHP structure chart is established, as shown in Fig. 2.

    Figure 2.  AHP structure for the slope stability evaluation.

    There are four different failure modes of high rock slopes (He et al., 2012): the collapse and shedding failure mode, toppling failure model, plane failure mode and wedge failure mode. These four failure modes consider the impacts of different indicators. The indicators considered for the collapse and shedding failure mode are A1, A2, B8B10, C1C4, D, E1, and E2. The indicators considered for the toppling failure mode, plane failure mode and wedge failure mode are A1, A2, B1B8, C1C4, D, E1, and E2.

  • The influences of the 19 indicators included in the indicator layer on the stability of the slope are different, that is, the weight of each indicator is different. The determination of the weight value directly affects the accuracy of the final evaluation. The slope is a complex dynamic system whose stability is affected by many factors. The expert scoring method is used to construct the scoring matrix and calculate the weight value. To prevent the subjective differences of the experts from leading to errors in the evaluation results, group decision theory is introduced to determine the weights. In contrast to the traditional AHP, the group decision eigenvalue method (Qiu, 1997) does not need to construct a judgment matrix of pairwise comparisons, and the problem of the inconsistency of the judgment matrix is avoided (Hong and Qiu, 2000). Entropy is used to describe the uncertainty of information or choices (Shannon, 1948). A group decision entropy model is used to calculate the reliability of expert group decisions and individual decisions (Qiu, 1995).

    Experts use their own engineering experience to score the importance of each indicator to construct a score matrix, and based on the score matrix, the optimal weight value of each indicator is obtained through the GEM; the reliability of expert decision-making is obtained through the decision entropy model, and whether the weight value of each indicator is accepted is based on the judgment of expert decision reliability value.

  • S1, S2, ..., Sm are group decision systems G composed of m experts, and the evaluation objects are B1, B2, ..., Bn. The score of expert Si on the evaluated indicator Bj is denoted as xij (i=1, 2, ..., m; j=1, 2, ..., n). The larger the value of xij is, the more important the target Bj. The score order matrix X composed of the scores of group G is as follows

    where xmn represents the score of expert Sm for evaluation subject Bn. The higher the score of the indicator is, the more important the subject is.

    Assuming that the score is as accurate as possible (100% reliability), the highest decision-making-level expert S*, whose score vector is x*=(x*1, x*2, ..., x*n)T, is defined as the ideal expert, and S* is the expert who has the highest consistency with group G. The decision of S* is completely consistent with that of G, and the difference between S* and the other experts is the smallest among the experts.

    Considering square matrix F, x* formed by the score matrix of the group expert is the feature vector corresponding to the largest eigenvalue of the square matrix F (Qiu, 1997).

    where X is the scoring matrix scored by the group expert.

    We use the power method to solve for the eigenvector corresponding to the largest eigenvalue (Qiu, 1997). The solution steps are as follows

    ① Let k=0, ${\mathit{\boldsymbol{y}}_0} = {\left({\frac{1}{n}, \frac{1}{n}, ..., \frac{1}{n}} \right)^{\rm{T}}}$, y1=F·y0, and ${\mathit{\boldsymbol{z}}_1} = \frac{{{\mathit{\boldsymbol{y}}_1}}}{{{{\left\| {{\mathit{\boldsymbol{y}}_1}} \right\|}_2}}}$

    where n is the number of schemes evaluated and F is the square matrix obtained by multiplying the score matrix X and its inverse.

    ② Let yk+1=F·zk and ${\mathit{\boldsymbol{z}}_{k + 1}} = \frac{{{\mathit{\boldsymbol{y}}_{k + 1}}}}{{{{\left\| {{\mathit{\boldsymbol{y}}_{k + 1}}} \right\|}_2}}}$, where k=1, 2, ...

    ③ Let |zkk+1| represent the largest absolute value of the difference between the components corresponding to zk and zk+1. When the accuracy requirement is ε, it is judged whether |zkk+1| is smaller than ε. If |zkk+1| is less than ε, then zk+1 is the requested x*; otherwise, move to step ② to calculate zk+1 again until the accuracy requirement is reached.

    According to the optimal scoring vector x*=(x*1, x*2, ..., x*n)T discussed in Section 1.3, the indicator weight qi is solved for as follows: qi= x*i/(x*1+x*2+...+ x*n), where qi is the weight value of scheme i for the group decision and n is the total number of schemes.

  • We solve the expert decision level vector as shown in the following steps.

    ① Construct a normalized scoring matrix

    where i represents expert i, * represents the ideal expert assumed in Section 1.3.2 of this paper, j represents scheme j, and xij represents the scoring value of expert i for scheme j. dij represents an element of the normalized processed score matrix corresponding to xij. Di represents the score of scheme n by expert i and the normalized score matrix. D represents a normalized score matrix composed of m experts and n schemes.

    ② Calculate the expert decision level vector.

    Determine the expert level decision vector Ei=(ei1, ei2, ..., ein) according to the normalized scoring matrix D

    where eij represents the component of the decision vector of level Ei of expert Si with respect to scheme j. Nij represents the ranking of the evaluated program B1, B2, ..., Bi determined by the score of expert Si. The scheme with the highest score is taken as 1, and the scheme with the lowest score is taken as j.

    The group decision-entropy model measures an expert's decision level by the inaccuracy or uncertainty of the expert conclusions (Qiu, 1995; Gu and Qiu, 1992). The calculation method is as follows

    where Hi represents the decision entropy of expert i. eij represents the component of the decision vector of level Ei of expert i with respect to scheme j. HG represents the decision entropy of the expert group. m represents the number of experts.

    Combining the decision entropy value and the "Decision Reliability and Decision Entropy Value Table", the reliability of expert decisions and group decisions is obtained (Qiu, 1995).

  • A group discussion is a process of brainstorming, from shallow to deep, and is consistent. In the discussion, a group of highly reliable opinions is gradually collected.

    The process of implementing variable weights is to calculate the reliability of the overall group decision and the reliability of each expert decision and judge whether the reliability meets the preset reliability requirements. If the requirements are not met, then further discussion and analysis of the reasons and rescoring are performed. Then, the reliability is calculated again and judged. This cycle continues until the required reliability score matrix is obtained and a relatively consistent high- reliability weight is finally obtained, which is the weight value used for the final stability evaluation. This method can more reasonably determine weight values than other methods can. The flow chart of the group discussion is shown in Fig. 3.

    Figure 3.  Flowchart of the group discussion of changing weights.

  • Fuzzy comprehensive evaluation is a comprehensive decision-making method to solve multivariate problems in complex decision situations.

    An important mathematical means of performing fuzzy comprehensive evaluation is fuzzy transformation. The fuzzy comprehensive evaluation method proceeds by using fuzzy transformations to solve a practical comprehensive evaluation problem. The evaluation process is as follows.

    There are two universes: the factor set U={u1, u2, ..., un}, where ui is the judgment factor, and the evaluation set V={v1, v2, ..., vm}, where vi is the grade judgment.

    If f(ui) is evaluated for each element ui in U, it can be regarded as a fuzzy mapping of U~V. From the fuzzy mapping f, the fuzzy matrix R can be derived: R=(rij)n×m, 0≤rij≤1, where R is a single-factor judgment matrix from U to V. If there is a subset A={a1, a2, ..., an} of the set U, then A is represented in the form of a vector, and the following equation is guaranteed

    If ai is the weight of factor i, then the fuzzy transformation B of U~V can be uniquely determined, and B is the fuzzy synthesis result: B=A·R, where B={b1, b2, ..., bm} and bi (j=1, 2, ..., m) reflect the membership degree of vi and the fuzzy set B.° is a fuzzy operator. The weighted average fuzzy operator M(°, ⊕) is used.

    According to the principle of the maximum membership degree, the largest degree is selected in B, and the corresponding level is the final result of the fuzzy comprehensive evaluation.

    Fuzzy comprehensive evaluation model

    According to fuzzy set theory and the AHP structure of slope stability evaluation established in Section 1.1 of this paper, a two-level fuzzy comprehensive evaluation calculation model is adopted, as shown in Eq. (8)

    where Ui is the first fuzzy evaluation matrix, composed of the stability grade membership degree vector the indicator layer elements contained in each element of the project layer. W1~Wi are the weight vectors of the indicator layer elements contained in each element of the project layer; W0 is the weight vector of each element of the project layer; Ni=Wi·Ui is the first comprehensive evaluation result vector of the indicator layer elements contained in each element of the project layer; Ni is also a row vector in the secondary evaluation matrix; and N0 is the two- level fuzzy comprehensive evaluation result vector, that is, the membership degree vector of the slope stability.

    Fuzzy comprehensive evaluation process

    ① Determination of the set of factors and the set of evaluations.

    The first-order factor set and the secondary factor set for the slope stability evaluation are determined by the AHP structure described in Section 1.2.

    After numbering the elements of the indicator layer, the 19 indicators that affect the stability of the slope constitute five sets of first-order factors.

    The five project layer elements constitute a secondary factor set: U={F1, F2, F3, F4, F5}.

    The stability of the slope is divided into three levels—stable (Level 1), basically stable (Level 2), and unstable (Level 3) (He et al., 2012)—and the evaluation set V is V={V1, V2, V3}.

    ② Determination of the weight of each element in the indicator layer and the project layer.

    These weights are determined by the method described in Section 1.3 of this paper.

    ③ Determination of the first fuzzy evaluation matrix.

    The slope stability analysis proceeds as follows: The first evaluation matrix is composed of the indicator elements of the elements included in the project layer with respect to the membership degree of the three evaluation levels. Therefore, the determination of the first evaluation matrix can be converted into the determination of the membership degree of each grade of the slope stability of each element of the indicator layer.

    The scores are based on the indicators and grading standards (He et al., 2012). Substituting the score into the membership function yields the membership degree of each element of the indicator layer corresponding to each slope stability evaluation level.

    The functional relationship between the membership degree and indicator score is established by using a "lower semitrapezoid" distribution (Mao et al., 2020; Peng et al., 2011; Zhao et al., 2004). The membership degree calculation function of the three evaluation grades is obtained by combining the indicators and grading standards (He et al., 2012).

    where r ~r are the membership degrees of the elements of the indicator layer corresponding to the three slope stability evaluation levels and Q is the actual score of each element of the indicator layer.

    The first fuzzy evaluation matrix can be obtained from the membership degree of each element in the indicator layer as follows

    where Ui is the first fuzzy evaluation matrix composed of the element i of the project layer and includes the membership degree of the indicator layer element and rn1 to rn3 indicates that the element n in the indicator layer included in the element i in the project layer corresponds to the membership degree 1–3 of the level.

    ④ Determination of the secondary fuzzy comprehensive evaluation matrix and evaluation result vector.

    The formula for the secondary fuzzy comprehensive evaluation matrix is as follows

    where S is the secondary fuzzy evaluation matrix; Ni is a first- order comprehensive evaluation result vector of the indicator layer elements included in the project layer element i; Ui is the element i in the project layer, and the included indicator element membership degree is the first fuzzy evaluation matrix; and Wi is the weight vector of each element in the indicator layer contained by the element i in the project layer.

    The two-level fuzzy comprehensive evaluation result vector is as follows

    where N0 is the two-level fuzzy comprehensive evaluation result vector, W0 is the weight of each element in the project layer, and S is the secondary fuzzy evaluation matrix.

    According to the two-level fuzzy evaluation result vector described in this section, the stability grade of the slope is determined according to the principle of the maximum membership degree. That is, in the two-level fuzzy comprehensive evaluation result vector, the level corresponding to the element with the largest membership degree is the stability level of the slope.

  • To fully verify the rationality of this method, the Leiyi (Leiyang, Hunan-Yizhang) highway excavation slope project was selected as a case study. The project is approximately 230 km away from Changsha. This case considers a typical high rock slope with many influencing factors (19 index layer elements), and multiple failure modes may be expressed there, so this case can show whether the evaluation method can determine the correct failure mode and its corresponding slope stability level in this complex slope project.

    This engineering case is referred to in He et al. (2012). The rationality of the proposed slope stability evaluation method is verified. The scores of each factor are shown in Table 1. The length of the high slope of the excavation is 160 m, the average slope height of the slope is 36.0 m (A1=50), and the rock slope angle is 45° (A2=50). A 1.5 m high cutting wall is installed at the foot of the first grade slope and is partially damaged (E2=80). The first grade slope is protected by mortar flagstone, and there are small cracks in the area (E1=80); above the first grade slope, the area is bare, the upper part of the rock is loose (B10=80), and the rock is scattered and broken (D=60). According to the on-site investigation, the geotechnical properties of the slope are as follows: the strongly weathered shale is grayish black with a carbonaceous structure and medium-thin layers of highly variable thickness and exhibits joint fissure development (B1=80); the thick moderately weathered shale is grayish black with medium-thin layers of a carbonate mud, and the joints are relatively developed. Some of the inclusions are brownish yellow, brown, slightly dense, and slightly wet, and the particle size is generally 20–60 mm (B8=80, B9=80); the spacing of the discontinuities is 0.5 m (B2=50), the discontinuities are rough but well developed (B4=80), with a crack opening of 8 mm (B3=60). The camber angle is greater than the slope angle (B5=50). The angle between the strikes of the slope and the structural planes is 15° (B6=50). The angle between the two sets of discontinuities is 10° (B7=50), and slight-moderate seepage occurs along the two sets of discontinuities (C4=80). Slope scouring is ongoing. The slope scouring is mainly due to direct slope scour by atmospheric precipitation and slope runoff, forming a gully along the flow direction of the slope (C1=80). The maximum daily rainfall is 80 mm (C2=70), and the groundwater level is above the slope (C3=90). After continuous rainfall in the first half of 2010, the phenomenon of rolling stones on the slope arose, which seriously affected driving safety.

    Indicator layer element A1 A2 B1 B2 B3 B4 B5 B6 B7 B8 B9 B10 C1 C2 C3 C4 D E1 E2
    Actual score 50 50 80 50 60 80 50 50 50 80 80 80 80 70 90 80 60 80 80

    Table 1.  Values of each element of the indicator layer

  • According to the engineering situation, the calculation parameters of each indicator of the slope are shown in Table 1.

  • In this case, the reliability of the weight determination is required to be no less than 90%. Six experts were invited to initially score the importance of the elements of the indicator layer contained in the five elements of the project layers of the four failure modes. Since the landslide history element in the project layer only contains one indicator layer element, there is no need for experts to score the importance of the indicator layer element. When determining the weight value of each element of the project layer, the landslide history element needs to be considered.

    According to the group decision eigenvalue method and group decision entropy model described in Section 1.3, the optimal weight vector and the reliability of each expert and group are obtained.

    First, element B of the project layer in the toppling failure mode, elements A and B of the project layer in the plane failure mode, and elements B and C of the project layer in the wedge failure mode are discussed. The decision of the weights of the index layer elements corresponding to these elements is less than 90% reliable. The scoring scheme for the importance of these elements in the indicator layer can be discussed in a targeted way and regraded after the discussion, which can effectively improve the discussion efficiency. The indicators that were discussed for a second time are marked in red in Tables 2–4. On the basis of the second discussion, the optimal weight of each index, the decision entropy and the reliability of the experts and expert groups are obtained. These are shown in Tables 2–4.

    Element Collapse and shedding failure mode Toppling failure mode Plane failure mode Wedge failure mode
    A1 0.250 5 0.247 0.415 2 0.435 7
    A2 0.749 5 0.753 0.584 5 0.564 3
    B1 0 0.065 1 0.056 5 0.074 0
    B2 0 0.114 6 0.062 5 0.092 2
    B3 0 0.096 0.086 0.139 2
    B4 0 0.144 9 0.082 5 0.168 6
    B5 0 0.321 9 0 0
    B6 0 0 0.463 1 0
    B7 0 0 0 0.264 9
    B8 0.179 3 0.257 4 0.249 4 0.261 0
    B9 0.732 7 0 0 0
    B10 0.088 0 0 0 0
    C1 0.338 6 0.366 0 0.346 1 0.359 7
    C2 0.164 8 0.184 8 0.197 5 0.193 6
    C3 0.051 4 0.117 6 0.129 7 0.131 4
    C4 0.441 5 0.331 7 0.326 7 0.315 3
    D 1 1 1 1
    E1 0.195 6 0.212 1 0.228 2 0.198 9
    E2 0.804 4 0.781 9 0.771 8 0.801 1
    A 0.078 2 0.077 2 0.077 9 0.078 1
    B 0.178 0 0.176 1 0.178 5 0.171 5
    C 0.057 7 0.061 2 0.059 7 0.058 0
    D 0.227 4 0.227 7 0.224 8 0.232 1
    E 0.458 7 0.457 7 0.459 1 0.460 2
    The elements marked in red are those that did not meet the reliability requirements after the experts' first discussion.

    Table 2.  Summary table of the optimal weight vector based on the second discussion

    Failure mode Expert A B C E Five elements of the project layer
    Collapse and shedding failure mode S1 0.020 8 0.059 2 0.041 9 0.007 7 0.013 3
    S2 0.043 8 0.140 5 0.118 1 0.041 4 0.068 4
    S3 0.006 4 0.124 8 0.077 9 0.007 7 0.067 0
    S4 0.014 7 0.069 3 0.049 8 0.039 7 0.086 3
    S5 0.020 8 0.205 7 0.012 4 0.032 6 0.022 5
    S6 0.011 6 0.208 0 0.039 6 0.011 3 0.029 5
    Expert group 0.019 7 0.134 6 0.056 6 0.023 4 0.047 8
    Toppling failure mode S1 0.007 7 0.055 5 0.062 1 0.035 2 0.019 7
    S2 0.041 4 0.152 4 0.058 2 0.038 3 0.065 5
    S3 0.007 7 0.053 4 0.067 4 0.013 6 0.052 4
    S4 0.039 7 0.073 5 0.120 1 0.107 0 0.123 0
    S5 0.032 6 0.092 3 0.104 4 0.004 0 0.033 4
    S6 0.011 3 0.103 0 0.039 5 0.040 4 0.036 6
    Expert group 0.023 4 0.088 3 0.075 3 0.039 7 0.055 1
    Plane failure mode S1 0.094 9 0.086 6 0.041 6 0.008 2 0.021 2
    S2 0.045 7 0.108 7 0.085 6 0.036 7 0.046 2
    S3 0.101 4 0.099 5 0.114 5 0.109 3 0.026 0
    S4 0.034 2 0.082 9 0.042 7 0.067 2 0.069 2
    S5 0.098 0 0.039 9 0.172 4 0.165 9 0.023 0
    S6 0.061 7 0.095 2 0.020 5 0.026 1 0.048 1
    Expert group 0.072 7 0.085 5 0.079 6 0.068 9 0.039 0
    Wedge failure mode S1 0.009 9 0.045 4 0.033 3 0.014 7 0.025 3
    S2 0.033 3 0.064 9 0.061 7 0.057 9 0.095 2
    S3 0.053 5 0.078 6 0.175 6 0.040 5 0.093 3
    S4 0.037 9 0.125 0 0.097 1 0.002 0 0.091 9
    S5 0.049 6 0.021 5 0.178 2 0.010 1 0.010 9
    S6 0.014 2 0.070 1 0.054 7 0.006 6 0.037 7
    Expert group 0.033 1 0.067 6 0.100 1 0.022 0 0.059 1
    Elements A~C and E represent the decision schemes of the indicator layer element weights contained in each of them. The elements marked in red are those that did not meet the reliability requirements after the experts' first discussion.

    Table 3.  Decision entropy summary table of expert decisions based on the secondary discussion

    Failure mode Expert A B C E Five elements of the project layer
    Collapse and shedding failure mode S1 95 99 95 99 99
    S2 95 95 95 95 95
    S3 99 95 95 99 95
    S4 99 99 95 95 95
    S5 95 95 99 95 99
    S6 99 95 95 99 99
    Expert group 95 95 95 95 95
    Toppling failure mode S1 99 99 95 95 99
    S2 95 95 95 95 95
    S3 99 95 95 99 95
    S4 95 99 95 90 95
    S5 95 99 95 99 99
    S6 99 95 95 95 99
    Expert group 95 99 95 95 95
    Plane failure mode S1 95 99 95 99 99
    S2 95 95 95 95 99
    S3 90 95 95 90 99
    S4 95 99 95 95 95
    S5 90 99 95 90 99
    S6 95 95 99 95 95
    Expert group 95 99 95 95 99
    Wedge failure mode S1 99 99 99 99 99
    S2 95 99 95 95 95
    S3 95 99 95 95 95
    S4 95 95 95 99 95
    S5 95 99 95 99 99
    S6 99 99 95 99 99
    Expert group 95 99 95 95 95
    Elements A~C and E represent the decision schemes of the indicator layer element weights contained in each of them. The elements marked in red are those that did not meet the reliability requirements after the experts' first discussion.

    Table 4.  Reliability summary table of expert decisions based on the secondary discussion

    The calculation results show that the reliability of the decision is greater than 90%, which meets the predetermined reliability requirements. Therefore, the optimal weight vector based on the second discussion, as shown in Table 2, is adopted as the weight value in this case study.

    Figure 4.  Sketch map showing the location (a) and sketch diagram showing the cross-section (b) of the study area. Figure 4a after the Standard Map Service of the Ministry of Natural Resources of the People's Republic of China (http://bzdt.ch.mnr.gov.cn/), No. GS(2016)1549.

  • The fuzzy comprehensive evaluation model of this case is determined by using the fuzzy comprehensive evaluation model established in Section 2.4 of this paper. The weights of each element in the indicator layer and project layer are the values determined in Section 2.3 of this article.

    The five first-order factor sets determined according to the four different failure modes of high rock slopes are as follows:

    The set of first-order factors for the stability evaluation of the slope in the collapse and shedding failure modes is

    The set of first-order factors for the stability evaluation of the slope in the toppling failure mode, plane failure mode and wedge failure mode is

    The first fuzzy evaluation matrices can be obtained by combining each measurement parameter in Table 1 and the comprehensive evaluation model, as shown in Table 5.

    Element of the project layer Element of the indicator layer The first fuzzy evaluation matrix
    A A1 0.285 7 0.714 3 0.000 0
    A2 0.285 7 0.714 3 0.000 0
    B B1 0.000 0 0.200 0 0.800 0
    B2 0.285 7 0.714 3 0.000 0
    B3 0.000 0 1.000 0 0.000 0
    B4 0.000 0 0.200 0 0.800 0
    B5 0.285 7 0.714 3 0.000 0
    B6 0.285 7 0.714 3 0.000 0
    B7 0.285 7 0.714 3 0.000 0
    B8 0.000 0 0.200 0 0.800 0
    B9 0.000 0 0.200 0 0.800 0
    B10 0.000 0 0.200 0 0.800 0
    C C1 0.000 0 0.200 0 0.800 0
    C2 0.000 0 0.600 0 0.400 0
    C3 0.000 0 0.000 0 1.000 0
    C4 0.000 0 0.200 0 0.800 0
    D D 0.000 0 1.000 0 0.000 0
    E E1 0.000 0 0.200 0 0.800 0
    E2 0.000 0 0.200 0 0.800 0
    Elements A~C and E of the project layer represent the first fuzzy evaluation matrix composed of the membership degree of the indicator layer elements included in each.

    Table 5.  Summary table of the first fuzzy evaluation matrix

    According to the first fuzzy evaluation matrix determined in this section, the weight of each element, and the first-order factor set of the four failure modes, four secondary fuzzy evaluation matrices can be obtained, as shown in Table 6.

    Failure mode Element The secondary fuzzy evaluation matrix
    Collapse and shedding failure mode A 0.285 7 0.714 3 0.000 0
    B 0.000 0 0.200 0 0.800 0
    C 0.000 0 0.254 9 0.741 4
    D 0.000 0 1.000 0 0.000 0
    E 0.000 0 0.200 0 0.800 0
    Toppling failure mode A 0.285 7 0.714 3 0.000 0
    B 0.124 7 0.501 3 0.373 9
    C 0.000 0 0.250 4 0.749 7
    D 0.000 0 1.000 0 0.000 0
    E 0.000 0 0.198 8 0.795 2
    Plane failure mode A 0.285 6 0.714 1 0.000 0
    B 0.150 2 0.539 1 0.310 7
    C 0.000 0 0.253 1 0.746 9
    D 0.000 0 1.000 0 0.000 0
    E 0.000 0 0.200 0 0.800 0
    Wedge failure mode A 0.285 7 0.714 3 0.000 0
    B 0.102 0 0.495 0 0.402 9
    C 0.000 0 0.251 2 0.748 8
    D 0.000 0 1.000 0 0.000 0
    E 0.000 0 0.200 0 0.800 0

    Table 6.  Summary table of the secondary fuzzy evaluation matrix

    The resulting vectors of the two-level fuzzy comprehensive evaluation of the four failure modes can be summarized as shown in Table 7.

    Failure mode Level 1 Level 2 Level 3 Stability risk score
    Collapse and shedding failure mode 0.022 3 0.425 3 0.552 1 73.008 1
    Toppling failure mode 0.044 0.615 9 0.282 9 62.103 1
    Plane failure mode 0.049 0.639 3 0.242 7 60.213
    Wedge failure mode 0.039 8 0.615 2 0.298 63.235 4

    Table 7.  The result vector summary table of the two-level fuzzy comprehensive evaluation and stability risk score

    According to the evaluation standard of slope stability grades (He et al., 2012), the standard scores corresponding to levels 1, 2, and 3 are determined to be 25, 60, and 85 points, respectively. The membership degree of each level in the two-level fuzzy comprehensive evaluation result vector is multiplied by the standard score of the corresponding level, and then the stability risk scores of the four failure modes are obtained as shown in Eq. (16)

    where Rs is the stability risk score and N0 is the two-level fuzzy comprehensive evaluation result vector.

    The two-level fuzzy comprehensive evaluation result vectors of the four failure modes are substituted into Eq. (16), and the stability risk score summary table is shown in Table 7.

    Table 7 shows that the collapse and shedding failure mode has the greatest risk in this case, so the result vector of the two-level fuzzy comprehensive evaluation of the collapse and shedding failure mode is selected for slope stability evaluation. According to the maximum membership degree criteria and Table 7, the stability level of this case is 3.

    The evaluation results are consistent with those of He et al. (2012). Therefore, the evaluation method is reasonable and feasible and can accurately evaluate the actual state of a slope.

    Compared with the method of He et al. (2012), this evaluation method can more fully absorb the experience of the expert group and effectively avoid the error of the evaluation results caused by the subjective differences of individual experts. The reliability of each expert decision and the reliability of the group decision can be intuitively obtained, the efficiency of the expert group discussion can be effectively improved, and the weight of each element can be obtained with high reliability. This evaluation method can obtain the risk scores of the four failure modes of the slope and the membership degree of the three levels of the four failure modes. The evaluation results are more intuitive and convenient for the application of slope evaluation.

  • This work selects a high rock slope with four failure modes as a case to illustrate the rationality of the evaluation method. However, due to different types of slopes, the impact indicators on slope stability will be different. when the evaluation method is applied to other types of slopes, such as reservoir slopes, it is necessary to add relevant indicators of reservoir water level changes to the rebuilt indicator system. In addition, unlike the limit equilibrium method and SRM, this method can quickly evaluate the stability of multiple slopes. For example, the stability of dozens of slopes along a highway needs to be evaluated. Using the proposed method, a set of index system is established, the corresponding indexes are calculated and analyzed, and then the stability of these slopes can be quickly evaluated. For slopes with poor stability, the limit equilibrium method or SRM can help to further evaluate the slope stabilities.

    This method makes full use of AHP to establish a hierarchical analysis structure when constructing the index system. Figure 2 shows how the hierarchical analysis structure was established for high rock slopes. This hierarchical analysis structure is also fully utilized when the fuzzy comprehensive evaluation of the slope in Section 1.4. GEM is used to solve the optimal weight vector, which solves the inconsistency of the judgment matrix in the AHP. Therefore, during solving the optimal weight vector in Section 1.3, the solution process is simplified. In addition, the entropy model is introduced into group decision-making to quantitatively evaluate the reliability of expert decision-making, as shown in Table 4. It obtains the reliability of each expert and group decision-making. The second discussion is only for decisions with low reliability, which improves the efficiency of group decision-making. Moreover, the method of constructing the evaluation index system will be further studied, and more different types of slope projects will be introduced to verify the accuracy of the evaluation method.

  • (1) The proposed evaluation method of slope stability comprehensively considers the influence of many factors on slope stability, and the evaluation result is accurate.

    (2) The slope stability evaluation method is based on group decision theory to determine the weights of each element, which can fully utilize the experience of the expert group and effectively avoid evaluation errors caused by the subjective deviation of a single expert. At the same time, a group decision theory entropy model is introduced to perform a quantitative evaluation of the reliability of expert decisions, which can intuitively obtain the reliability of each expert and the expert group decision, improve the efficiency of expert group discussions, and provide a more consistent weight with higher reliability.

    (3) This method can not only fully utilize the experience of the expert group but also avoid the inconsistency in the judgment matrix that will arise when using the AHP. The calculation process of this evaluation method is simple. The evaluation method has a clear evaluation process and provides clear intuitive evaluation results.

  • This study was supported by the National Key Research and Development Program of China (No. 2017YFC1501302), the National Natural Science Foundation of China (No. 41630643) and the Fundamental Research Funds for the Central Universities (No. CUGCJ 1701). The final publication is available at Springer via https://doi.org/10.1007/s12583-020-1101-8.

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