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A novel multi-dimensional cloud model coupled with connection numbers theory for evaluation of slope stability
Applied Mathematical Modelling 77 (2020) 426–438
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Applied Mathematical Modelling journal homepage: www.elsevier.com/locate/apm
A novel multi-dimensional cloud model coupled with connection numbers theory for evaluation of slope stability Mingwu Wang∗, Xiao Wang, Qiuyan Liu, Fengqiang Shen, Juliang Jin School of Civil and Hydraulic Engineering, Hefei University of Technology, 193 Tunxi Road, Hefei 230009 China
a r t i c l e
i n f o
Article history: Received 3 January 2019 Revised 17 July 2019 Accepted 23 July 2019 Available online 29 July 2019 Keywords: Slope stability Set pair analysis Cloud model Evaluation Multi-dimensional Connection number
a b s t r a c t Stability evaluation of a slope involves various fuzzy and correlation indicators randomly distributed in finite intervals. A novel multi-dimensional connection cloud model was presented here to address multiple uncertainties and distribution characteristics of indicators, and to depict the randomness and fuzziness of the measured index value belonging to the classification standard in the slope stability analysis. In the model, when simulating fuzzy and random characteristics of evaluation indicators in finite intervals, the numerical characteristics of connection cloud model were assigned on the basis of the analysis of identical-discrepancy-contrary (IDC) relationships between measured indicators and the classification standard to overcome the subjectivity. Considering the effect of indicator correlation in a unified way, the integrated connection degree of a grade was further specified for the evaluation sample. Moreover, case studies and comparisons of the proposed model with one-dimensional normal cloud model, extension model, and support vector machine (SVM) were performed to confirm the validity and reliability. The results indicate that this model employed to evaluate slope stability can clearly depict the random and fuzzy distribution features of measured data in finite intervals, and its calculation process is quicker and simpler than that of one-dimensional normal cloud model. © 2019 Elsevier Inc. All rights reserved.
1. Introduction The stability evaluation is an essential task in the design and construction of slope stabilization engineering, so to prevent damages to public and private properties, a precise estimation of slope stability is much important for decision makers and has received widely attention in geotechnical engineering field [1]. However, the work is a quite complex problem because slope stability is characterized by various uncertain factors, such as insufficient and uncertain geological information of site conditions, heterogeneous geotechnical properties of rock mass or soil mass and geomorphic characteristics [2]. Consequently, although scholars have presented numerous evaluation methods considering uncertain factors [3], such as fuzzy sets method [4], probabilistic method [5,6], artificial intelligence method [7–9], extension method [10], numerical modeling method [11,12] and so on, there still currently exist considerable challenges in accurately evaluating slope stability by these analytical methods. The evaluation method of slope stability is divided into two categories: qualitative and quantitative analysis methods. Regardless of the fact that the conventional methods have demonstrated successful performances in the slope stability evaluation, they were of their own disadvantages. For instance, the engineering geology analogy method, founded on geological ∗
Corresponding author. E-mail address: [email protected] (M. Wang).
https://doi.org/10.1016/j.apm.2019.07.043 0307-904X/© 2019 Elsevier Inc. All rights reserved.
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information and engineering experience [3], is a representative and prevalent method in the qualitative method for evaluation of slope stability, but may vary from person to person with its subjectivity because lacking uniform standards; the limit equilibrium method and numerical modeling, two of the most widely used quantitative analysis methods with the safety factor, have the advantages of clear physical meaning, but their applications for field engineers have been limited by some defects [13] because limit equilibrium methods need certain hypothetical conditions to solve, while evaluation results of numerical simulation methods rely too much on the reliability of the parameters, the simplification of the boundary conditions and model construction. In addition, they cannot describe the uncertainty of information acquisition and cognitive uncertainty in multi-index problems. Furthermore, the evaluation of slope stability will encounter subjectively ambiguities as the result of applying linguistic terms, same score prescribed in a classification rating, existence of sharp transitions between two adjacent grades [14], which sometimes may lead to a decrease in the reliability of the evaluation results. For example, the slope may indeed pose different risk levels in spite of the same value of safety factor obtained by the limit equilibrium method [15]. So recently numerous probabilistic approaches [16], random field simulation technique [17] and reliability analysis methods [18] were presented to consider limited internal and external uncertainties involved in the slope stability analysis, and the slope stability evaluation is starting from certainty to uncertainty analysis. These uncertainty approaches and reliability analysis methods provided effective ways to make use of fuzzy or random uncertainty and to estimate their effects on the safety factor in the slope stability evaluation, but the calculation methods of reliability are very different for different destructive forms [19], and the determination of membership degree function in fuzzy comprehensive method [4] is subjective and ignores the randomness of the system. In order to take multiple uncertainties into account the slope stability analysis and avoid complex failure mechanism, the artificial neural network (ANN) method and the SVM method were implemented to depict the highly non-linear relationship between physical and mechanical parameters and the safety factor of the slope [20,21]. But the ANN method is unable to present an explicit relationship between evaluation parameters and rank potential, and has some inherent drawbacks, such as a low convergence speed and over-fitting problems [2]. As for the SVM method, the determination of parameters in advance requires huge additional computational complexity [22,23]. At the same time, because of the complexity of the slope engineering, various categories of uncertainty such as fuzziness and randomness may be inevitably encountered in evaluations of slope stability, so the slope stability analysis cannot rely only on methods on the basis of a single type of uncertainty. The combination of two or more uncertainties has become a trend for the slope stability evaluation. However, the above methods cannot classify finely the features of various types of uncertainties including randomness and fuzziness in a unified way during the evaluation process. The normal cloud model proposed by Li et al. [24], which can automatically generate the certainty of the qualitative concept, is a tool for depicting fuzziness and randomness of evaluation indicators in a unified way [25], but it requires indicators obeying a normal distribution in an infinite interval that is not consistent with the actual distribution pattern of indicators because indicators are generally fuzzy and distributed randomly in finite intervals. Moreover, for the multi-index problem, the computation procedure by the one-dimensional normal cloud model might be more complicated by the increase in numbers of the evaluation indexes and samples, so some researchers proposed the multi-dimensional normal cloud model to comprehensively deal with the influence of indicators of multiple uncertainties [26], but unluckily there are few reports on the slope stability evaluation using the multi-dimensional cloud model. Therefore, the normal cloud model is useful to reflect the fuzziness and randomness of evaluation indicators, but sometimes it might fail to describe the actual distribution characteristic of evaluation indicator in a finite interval, and cannot depict the classification transformation among adjacent grades, this may lead to the deviation of the evaluation results from reality and limitation of its applications in engineering. On the other hand, due to randomness, uncertainty, and incompleteness of indicator information acquisition, the complicated geologic process of the slope deformation or failure, and the distinct inhomogeneous property of rock and soil, the design and the decision-making for the slope engineering cannot be singly depended on theoretical analyses and numerical simulations [27]. Furthermore, expert experiences are more reliable than the calculated results in most of the cases, so the engineering geological analogy based on the existing slope evaluation results is a widely used one in the earlier period for designing an appropriate slope, but it cannot give the reliability of classification rating and the possibility to other grades. In general, although the conventional engineering geological analogy method and uncertainty analysis methods are commonly applicable, they often only address a single type of uncertainty such as fuzziness or randomness, and are difficult to describe multiple uncertainties of actual slope information and information acquisition in essence. Moreover, the intervalvalued classification rating cannot be completely characterized by the normal cloud model. Consequently, a novel analytical method based on the cloud model and set pair analysis of connection numbers theory is needed to be further discussed in order to overcome the shortcomings of the conventional normal cloud model and engineering geological analogy method. The complexity of the slope system requires the employment of a new method that is efficient in depicting multiple types of uncertainties in the evaluation of slope stability. This work is aimed at introducing a novel multi-dimensional connection cloud model coupled with set pair analysis (SPA) of connection numbers theory to enhance the reliability and accuracy of evaluation of slope stability, and to dialectically express actual distribution characteristics of indicators in finite intervals. The authors intend to dialectically describe uncertainty, complexity, and correlation under multiple factors of the slope stability evaluation, the conversion tendency of classification at the boundary and the certainty and uncertainty relationships between the measured evaluation indicators and classification standards from identity, discrepancy and contrary aspects. Finally, the feasibility and validity of the proposed method were further discussed and confirmed by the case study as well as the comparisons with other methods.
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2. Methodology As discussed above, the slope stability evaluation is a nonlinear and uncertain dynamic system problem influenced by various factors, such as rock mass structure, orientation of advantage discontinuities, geometric properties of slope, construction methods and environmental conditions [3]. Here to comprehensively consider the effects of multiple uncertainties of indicators and their distribution characteristics, a multi-dimensional cloud model coupled with set pair analysis of connection numbers theory is proposed to directly and quickly analyze the slope stability, and to overcome the defects of the normal cloud model. 2.1. Multi-dimensional connection cloud model The definition of the multi-dimensional connection cloud model is given as following: Assume that there are m evaluation indicators (j = 1, 2, …, m) and n stability grades (i = 1, 2, …, n) for the slope stability problem, and cloud mapping of grade i for index j is composed of left and right half branches of connection cloud with the expected value Exji . Let C be a qualitative concept in the m-dimensional quantitative domain U with precise values. If xU, a random realization of the qualitative concept C, follows a normal distribution N(Ex, y2 ), and y satisfies a normal distribution N(En, He2 ), overall quantitative property of concept C is characterized as
k ji m 9 x ji − E x ji μ ji (x ) = exp − 3y ji , 2
(1)
j=1
where μji (x) is the connection degree, ∈ [0, 1]; Ex, En, He and y are the vector of the expected value, entropy, hyper entropy and random number satisfying normal distribution, respectively; Exji is the expected value for the grade i of the evaluation indicator j that denotes the most representative parameter of the qualitative concept; kji the order of distribution function of connection cloud for the grade i of the evaluation indicator j; yji is a random number satisfying normal distribution N(Enji , Heji 2 ); the entropy Enji denotes the uncertainty measurement and granularity of a qualitative concept, the higher Enji is, the larger scope of the universe can be accepted by the concept; Heji are the hyper entropy that reveals the correlation between randomness and fuzziness. Then the m-dimensional quantitative domain and one connection degree consists a drop of the connection cloud, and the distribution of x in the domain U is called an m-dimensional connection cloud. For the actual multi-index evaluation problem, the interval-valued classification rating of the given indicator may be different in sizes, so the width of the left and right half branches of the cloud model is different, and the connection cloud may be asymmetric, that is, the left and right half branches of the cloud model are of various numerical characteristics (see Fig. 1). Namely, cloud mapping consists of the left and right half of asymmetric connection clouds at the dividing point of the expected value, Exji . According to Eq. (1), the connection cloud model becomes a normal cloud model when kji =2. But for the unequal interval-valued classification ratings, the connection degree on the classification interface point simulated by the connection cloud model and normal cloud model is obviously different as showed in Fig. 1. The unequal interval sizes of classification grades for the evaluation indicator may be characterized by asymmetric normal clouds constructed by left and right half branches in normal clouds, but it cannot reflect the certainty and uncertainty relationships between the index and the discussed grade, and the fuzziness of classification grade when the measured index is at the threshold values as showed in Fig. 1, the distribution of the indicator simulated by a normal cloud model may be varied from the actual situation. In addition, measured indices are usually distributed in finite intervals, while the indices reflected by normal clouds are distributed in infinite intervals. This kind of deviation is mainly caused by the traditional determination method of the numerical characteristics and infinite distribution feature of indicator in the normal cloud model. Numerical characteristics of normal cloud were mainly determined on the basis of the ‘3En rule’ that cloud drops mainly locate in the basic range [Ex−3En, Ex+3En] while a small number of drops are outside of the basic range. This may neglect the fuzziness of classification grades and the finite interval distribution characteristics of measured indicators.
Fig. 1. Comparison of connection cloud and normal cloud.
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Fig. 2. IDC relationships between cloud drops and classification grades.
2.2. Set pair analysis of connection numbers theory To express the actual distribution characteristic of the indicator, and the certainty and uncertainty relationships between the indicator and classification standard, the SPA of connection numbers theory was introduced here to improve the generation algorithm of cloud model, and presented a novel asymmetric connection cloud model. The connection numbers theory is an uncertainty analysis theory proposed by Zhao [28] for dealing with systematic uncertainty problem based on the essence of Chinese philosophy ‘One divided into three’ [29] and the principle of universal connection [30,31], its mathematical model is
μ = a + bI + cJ,
(2)
where μ is the connection degree denoting the certainty and uncertainty relationships of a set pair H(A, B), which consists of two related sets A and B; a, b, and c are measurements of identity, discrepancy, and contrary relationships, respectively; I and J are the discrepancy and contrary coefficients, respectively. Obviously, the connection numbers theory is of advantage in depicting the development from ‘all are the same in the same group’ to quantitative differences in the same group. The complexity and diversity of uncertainty and certainty relationships and transformation can be obtained by the microscopic analysis of SPA and quantitatively depicted as Eq. (2) from a unified view of the connection degree. So the connection numbers theory provides a new idea for the micro-analysis of cloud drop distribution and extra thoughts to overcome the defects in the traditional normal cloud model. As we know, during the generation of a cloud, the analysis of the relationship between measured indicators and classification standard plays a significant role in the determination of the numerical characteristic parameter. Here, to determine the numerical characteristic parameters of the connection cloud model more reasonably, the fuzzy random uncertainty that the threshold values belong to the given classification grade is deeply analyzed from three relationships including the identity, discrepancy and contrary by the set pair analysis. According to the identical-discrepancy-contrary rules of SPA, the farther away from Exji the cloud drops are, the more discrete and sparse they are, reflecting the more unstable and inconsistent perception of concepts, so their transformation can be also characterized by the connection degree. So the connection degree was utilized here to depict the possibility that the sample belongs to the classification grade and to deal with the certainty and uncertainty relationships between classification standard and measured indicators distributed in finite intervals. In fact, Eq. (1) skillfully reflects the maximum random uncertainty of threshold values belonging to adjacent levels through kji parameters. The identical-discrepancy-contrary relationships between the measured evaluation indicators and classification standard in the connection cloud model are defined as showed in Fig. 2. 3. The evaluation of slope stability with multi-dimensional connection cloud The basic evaluation principle of the connection cloud model is depicted in the following: A rational evaluation indicator system is first set up; then, on the basis of IDC relationships analysis, the numerical characteristics (Ex, En, He, k) are determined for each classification rating of evaluation indicator, respectively; finally, the integrated connection degree gained with indicator weights is adopted to specify the grade of the evaluation sample according to the maximum membership rule. The procedure for the evaluation of slope stability based on the multi-dimensional connection cloud model is illustrated as showed in Fig. 3. Detailed evaluation procedure is consists of 5 steps as following. Step 1: Set up an appropriate evaluation system for the slope stability with selected indicators and classification standard. Previous studies have shown that there are various factors affecting the slope stability, which are either human factors or natural factors; and for an actual slope each factor often plays a different role from others. Therefore, the selection of
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Fig. 3. The process of slope stability evaluation by the multi-dimensional connection cloud model.
evaluation index is very important, and the selected evaluation index should be simple in quantitative form and easy to obtain. The corresponding interval-valued classification is rated for each grade. Step 2: Assign the m-dimensional connection cloud descriptors, (Ex, En, He, k) and the cloud drop number λ. As noted above, the asymmetric connection cloud can dialectically express the actual distribution characteristic of the indicator and the certainty and uncertainty relationships between the indicator and classification standard, and is dependent upon the reasonable determination of numerical characteristic parameters. According to the connection numbers theory, corresponding IDC relationships for the measured indicator j and classification grade i of a sample are specified as follows: an identical relationship between measured indicator and classification rating is specified when the measured value of indicator j locates in the interval-valued classification grade i; a discrepancy relationship when the measure value locates in the adjacent grades (grade i + 1 or grade i−1); and a contrary relationship when it locates in other grades. The numerical characteristic parameters are given by
E x ji =
C min ji + C max ji , 2
(3)
H e ji = β , E n ji =
k ji =
(4)
a ji , 3 ln
(5)
ln 4
9 , C −E x ln ji3E n ji ji
(6)
where Exji , Enji , Heji and kji are connection cloud descriptors for the ith grade of the evaluation indicator j, respectively; Cmaxji and Cminji are thresholds of the interval-valued stability grade i of the indicator j; β denotes the atomized feature of the connection cloud, here is a constant taken; Cji represents the lower limitation of discussed classification rating Cminji or the upper limitation Cmaxji ; α ji indicates the width of the connection cloud. For the performance index, the corresponding parameters α ji of intermediate classification grades (i = 2, 3, …, n−1) for the left and right branch of the connection cloud are
a_left ji = E x ji − C min j (i−1)
a_right ji = C max j (i+1) − E x ji
,
(7)
where α _leftji and α _rightji are the width of the left and right branch of connection cloud, respectively. While for the cost index, the corresponding values of intermediate classification grades are
a_left ji = E x ji − C min j (i+1)
a_right ji = C max j (i−1) − E x ji
.
(8)
The distribution form of the evaluation index of grades (i = 1, n) at both ends is often different from that of the intermediate grades, and has asymmetric characteristics. For example, the distribution form of the indicator for the grade at both ends is often uniform distribution with a certainty degree of 1.0, so it is difficult to describe the transition characteristics between the both ends grades and the adjacent grades by using the traditional normal cloud model, and the corresponding parameters of numerical characteristics for the normal cloud model are also difficult to determine; while the corresponding
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Table 1 Classification standard in case A [10]. Evaluation indicator
Stable (I)
Relatively stable (II)
Basically stable (III)
Unstable (IV)
Extremely unstable (V)
Weights
C1 C2 C3 C4 C5 C6 C7
0–75 0–10 0–20 55–100 35–50 80–100 0–2
75–175 10–20 20–30 40–55 30–35 60–80 2–4
175–300 20–30 30–35 25–40 20–30 40–60 4–6
300–500 30–40 35–45 20–25 15–20 20–40 6–8
500–1000 40–90 45–90 0–20 0–15 0–20 8–12
0.083 0.053 0.053 0.221 0.148 0.338 0.104
(m) (°) (°) (kPa) (°)
parameters of the connection cloud model can be determined according to the measured lower limit value or upper limit value of the index. Step 3: Generate the m-dimensional connection cloud for the classification rating i. The generation of cloud drops is the fundamental content to embody uncertainty problem with characteristics of fuzziness and the stochasticity problem. The generation algorithm for the connection cloud is: firstly, generate random numbers y following normal distribution based on the entropy En and hyper entropy He, and get x from the normal distribution with expected values Ex and random standard deviations y; then the connection degree is specified by Eq. (1) to set up a cloud drop of the m-dimensional connection cloud as (x, μ(x)); λ cloud drops can be attained by the above process. Step 4: Combining together weights of indicators with each other, wj , the assignment of the connection degree of the grade i for the sample of identical relationship or discrepancy relationship with the relevant grades assumes the following formula.
μi
m x ji − E x ji k ji 9 . = exp − w j · 2 3y ji
(9)
j=1
As discussed above, generated cloud drops mainly focus in the range [Ex-3En, Ex+3En] in the cloud model (see Fig. 2), and those drops outside of the basic range might not be considered for their contributions to the qualitative concepts. At the same time, to reduce the computational error caused by the multiplication of near zero numbers, the connection degree is approximately e − 9/2 when there is a contrary relationship between classification standard and measured value xji . Step 5: Specify the grade of slope stability for the sample according to the maximum membership rule. 4. Case study 4.1. Data and evaluation system 4.1.1. Case A To verify the feasibility and reliability of the model, the data of literature [10] were used here to analyze and conduct comparisons with extension model and one-dimensional normal cloud model. In case A, slope height (m) C1 , slope angle (°) C2 , dip angle (°) C3 , cohesive strength of weak intercalation (kPa) C4 , internal friction angle of weak intercalation (°) C5 , rock mass rating (RMR) C6 , and seismic intensity C7 were chosen as evaluation indicators. The slope stability was classified into five grades: stable (I), relatively stable (II), basically stable (III), unstable (IV) and extremely unstable (V). The classification standard and measured indicator values of samples were shown in Tables 1 and 2, respectively. The assignment of indicator weights plays a critical role in the evaluation result, so in order to better compare the results obtained by the other models, the same indicator weights assigned by an analytic hierarchy process method were adopted here as the reference [10]. Namely, according to the above evaluation procedure, the sample values and weights of wj ={0.083, 0.053, 0.053, 0.221, 0.148, 0.338, 0.104} were substituted for Eq. (9) to obtain their connection degrees at grades as showed in Table 3. 4.1.2. Case B As we know, there is a lack of clear and unified classification standard for the slope stability analysis so far. Different classification standards of slope stability may lead to changes in results obtained by the same method. Data from the literature [23] named as case B were therefore carried out to further verify the validity of the proposed model. In case B, the slope gradient D1 , slope height (m) D2 , slope type D3 , weathering degree D4 , vegetation cover D5 , and hardness of rock and soil mass D6 were chosen as evaluation indicators. The classification grade of slope stability was from stable (I), basically stable (II), unstable (III) to extremely unstable (IV). The classification standard and measured values of indicators were listed in Tables 4 and 5. 4.2. Model validation and algorithm efficiency analysis To clearly highlight the simulation process of the multi-dimensional connection cloud model, an example of twodimensional connection cloud for the evaluation indicators of the slope angle C2 and seismic intensity C7 , was illustrated
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Table 2 Measured indicator values of case A [10]. Sample
C1
C2
C3
C4
C5
C6
C7
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
140 145 100 225 250 120 105 120 40 180 150 217 133 210 205 500 130 400
20 30 45 16 14 40 46 50 45 31 35 40 42 45 25 17 35 20
5 60 25 5 7 50 25 6 30 9 45 10 4 45 30 20 25 25
3 6 85 63 58 3 11 20 20 40 33 60 80 21 45 67 23 70
19 27 30 31 11 22 27 32 17 10 32 40 21 26 19 41 11 20
15 34 87 83 32 32 40 71 38 14 23 89 35 31 29 82 28 68
6 6 7 7 7 7 7 6 6 4 6 7 7 7 6 7 7 6
Table 3 Connection degrees of samples for case A. Sample
μI
μII
μIII
μIV
μV
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
0.0159 0.0124 0.2003 0.1948 0.0398 0.0134 0.0160 0.0531 0.0160 0.0140 0.0166 0.3160 0.0442 0.0111 0.0166 0.3053 0.0148 0.0633
0.0206 0.0238 0.1191 0.2119 0.0383 0.0181 0.0304 0.1496 0.0166 0.0573 0.0509 0.1076 0.0257 0.0202 0.0521 0.1457 0.0204 0.1348
0.0421 0.1307 0.0304 0.0383 0.0538 0.0766 0.1231 0.0593 0.0907 0.0639 0.1263 0.0202 0.0917 0.1155 0.2259 0.0164 0.0757 0.1135
0.0975 0.1566 0.0211 0.0209 0.1511 0.2073 0.1409 0.0445 0.3385 0.0549 0.1518 0.0250 0.1529 0.4140 0.1729 0.0243 0.3998 0.0421
0.1518 0.0614 0.0179 0.0143 0.0515 0.1107 0.0486 0.0325 0.0582 0.1327 0.0493 0.0175 0.0266 0.0934 0.0300 0.0196 0.1239 0.0136
Table 4 Classification standard and weights in case B [23]. Evaluation indicator Slope gradient D1 Standard value Normalized value Slope height (m) D2 Standard value Normalized value Slope type D3 Standard value Normalized value Weathering degree D4 Standard value Normalized value Vegetation cover D5 Standard value Normalized value Hardness of rock and soil mass D6 Standard value Normalized value
Stable (I)
Basically stable (II)
Unstable (III)
Extremely unstable (IV)
0–1 0–0.25
1–2 0.25–0.50
2–3 0.50–0.75
>3 0.75–1.00
0–8 0–0.25
8–20 0.25–0.50
20–30 0.50–0.75
>30 0.75–1.00
Concave 0–0.25
Straight 0.25–0.50
Stepped 0.50–0.75
Convex 0.75–1.00
No 0–0.25
Slight 0.25–0.50
Moderate 0.50–0.75
Severe 0.75–1.00
Good 0–0.25
Better 0.25–0.50
Average 0.50–0.75
Bad 0.75–1.00
Hard
Hard rock; Sandwiched soft rock 0.25–0.50
Soft rock; Sandwiched hard soil 0.50–0.75
Soft
Weights 0.1742
0.1659
0.1648
0.1684
0.1571
0.1696
0–0.25
0.75–1.00
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Table 5 Measured values of indicators for case B [23]. Sample
D1
D2
D3
D4
D5
D6
19 20 21 22 23 24 25 26 27 28 29 30
2.3 2.7 1.6 1.6 3.1 0.8 1.4 1.6 3.2 1.4 3.1 1.7
16 7 18 14 18 7 12 7 28 14 31 26
0.63 0.89 0.64 0.39 0.83 0.20 0.75 0.43 0.38 0.34 0.92 0.81
0.63 0.89 0.41 0.32 0.64 0.15 0.50 0.66 0.90 0.29 0.92 0.62
0.80 0.91 0.83 0.91 0.59 0.82 0.88 0.77 0.94 0.78 0.83 0.85
0.48 0.59 0.85 0.52 0.61 0.71 0.83 0.61 0.23 0.30 0.92 0.86
Fig. 4. The two-dimensional connection cloud for indicators of the slope angle and seismic intensity.
for case A. On the basis of the discussed simulation steps in Section 3, the numerical characteristics (Ex2 , En2 , He2 , k2 ; Ex7 , En7 , He7 , k7 ) of the slope angle and seismic intensity at different grades were first gained by Eqs. from (3) to (8). The corresponding process of determining the parameters of numerical characteristics for the given grade III of the evaluation index C2 by means of the IDC principles of SPA was described as following. First, the mean value 25 of the classification standard interval of the given grade III was taken as the expected value, Ex23 , of the connection cloud. Then corresponding widths of the left and right half branches of cloud determined through Eq. (7) were both 15. Next, according to the IDC principle illustrated as showed in Fig. 2, it was an identical relationship between the measured indicator C2 value and discussed classification standard III when the value of indicator C2 was within the interval [20, 30], and a discrepancy relationship when the indicator C2 value was [10, 20] or [30, 40]. Thus, the orders k23 of the left and the right half branches of the connection cloud were both obtained as 1.703 according to Eq. (6). Orders of indicator C7 were also obtained depending on the similar procedure. The constant value of He was 0.01. Finally, according to the numerical characteristics, a two-dimensional connection cloud responding to the two indicators was obtained as showed in Fig. 4. It is also found that the simulation process of the multi-dimensional connection cloud model is much simpler for the multi-index evaluation problem than that of the one-dimensional normal cloud. The efficiency issue for the slope stability evaluation of m indicators and n classification grades was discussed here in order to illustrate the advanced nature of the multi-dimensional connection clouds relative to the one-dimensional cloud model. According to the production principle of the one-dimensional cloud model, the number of simulation cloud composed of λ drops for all the classification grades is I1 =nm, and the corresponding cloud drops are required to randomly generate I1 =nmλ times by a one-dimensional cloud model. But by the multi-dimensional cloud model, only n multi-dimensional clouds are needed, and the corresponding generation number of cloud drops is only nλ. Obviously, for the multi-index problem, the calculation amount of the
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M. Wang, X. Wang and Q. Liu et al. / Applied Mathematical Modelling 77 (2020) 426–438 Table 6 Evaluation results and comparisons for case A. Sample
Proposed model
One-dimensional normal cloud model
Extension model [10]
Actual situation on site [10]
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
Ⅴ IV I II IV IV IV II IV Ⅴ IV I IV IV IV I IV II
Ⅴ IV II II IV IV IV II IV IV IV I IV IV IV II IV II
Ⅴ Ⅴ I I IV IV IV I IV Ⅴ IV I IV IV IV I Ⅴ II
Unstable Unstable Stable Stable Unstable Unstable Creep Stable Unstable Unstable Creep Stable Unstable Unstable Creep Stable Unstable Stable
one-dimensional cloud model is m−1 times more than that of the multi-dimensional cloud model. Moreover, the interaction among indicators is neglected to some extent in the one-dimensional cloud model, which may lead to the evaluation result being more dependent on the index weight. 5. Comparisons and discussions Comparisons of evaluation results by the proposed model, the one-dimensional normal cloud model, the extension model, actual situation on site and support vector machine method [10] for cases A and B were performed to confirm the reliability and validity. The evaluation results obtained from the proposed model and comparisons with other methods were listed in Tables 6 and 7. As listed in Table 6, the evaluation results from the model proposed were basically in agreement with those from the extension evaluation method except for samples 8 and 17, and from the one-dimensional normal cloud model except for samples 3 and 16. The results of the proposed model were almost same as those of the actual situation on site. For sample 8, there was one indicator at grade I, two indicators at grade II, and other indicators at grades III, IV and IV, respectively, so it was more reasonable to rate sample 8 as II than I. It was seen in Table 2 that there was no indicator at grade V for sample 17, so it was obviously more at grade IV than at grade V. These results indicate that the proposed model is more feasible and reliable than the extension method. Sample 2 was taken as an example to analyze the reasons for the difference between the evaluation result obtained by the multi-dimensional connection cloud and one-dimensional normal cloud model in the following. For sample 2, measured values of indicators C2 , C5 and C7 were at grade III; the grade of indicators C1 and C6 was at grade II and grade IV, respectively; indicators C3 and C4 were at grade V, so if the importance of each indicator was equal, that is to say, the influence of each index on the evaluation result was same, the overall stability grade of sample 2 was reasonable to be specified as grade IV. Based on weights used in Ref. [10], the classification rating obtained by the proposed model was also grade IV, which was consistent with the actual state of the slope on site, while the classification rating obtained by the extension method was grade V and obviously in favor of unsafety.
Table 7 Evaluation results of the multi-dimensional connection cloud and comparison with the support vector machine method for case B. Sample
μI
μII
μIII
μIV
Proposed model
Support vector machine method [23]
19 20 21 22 23 24 25 26 27 28 29 30
0.0151 0.0217 0.0230 0.0504 0.0126 0.1913 0.0276 0.0373 0.0312 0.1207 0.0111 0.0142
0.1565 0.0408 0.1416 0.4074 0.0730 0.1072 0.0957 0.1784 0.0508 0.4327 0.0111 0.0480
0.5204 0.1276 0.2490 0.1212 0.5267 0.0360 0.1607 0.2287 0.0943 0.0678 0.1258 0.3190
0.0597 0.2246 0.0707 0.0241 0.1279 0.0390 0.0886 0.0464 0.1730 0.0210 0.8328 0.2167
III IV III II III I III III IV II IV III
III IV III II III I III III IV II IV III
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Fig. 5. Comparison of two-dimensional connection cloud and normal cloud for classification standard of indicators of the internal friction angle of weak intercalation and RMR in case A: (a) connection clouds; (b) normal clouds.
It can be seen from case A study that cloud model, as a cognitive model to realize the conversion of qualitative concepts and quantitative data, can convert the fuzziness and randomness of slope stability into the quantitative data of the connection degree, but it may be difficult to evaluate slope stability with various types of uncertainty factors only by onedimensional normal cloud model. As shown in sample 10, the indicators C5 and C6 belonged to extremely unstable level, the indicators C1 , C2 , C3 , C4 , and C7 belonged to basically stable level, unstable level, stable level, basically stable and relatively stable level, respectively. Obviously, measured values of indicators scattered in five levels, so the integrated connection degree at each level was not so high. As noted above, the normal cloud model cannot depict the characteristics of interval distribution and transformation of classification grade of evaluation indicator. So there were differences in results obtained by the proposed multi-dimensional cloud model and the conventional normal cloud model for sample 10. Figs. 5 and 6 illustrate the comparison of normal cloud and connection cloud for indicators of the internal friction angle of weak intercalation and RMR. It was seen that the normal cloud cannot reflect the actual distribution characteristics of evaluation indicators in finite intervals and at the highest and lowest classification grades, as well as the fuzziness and convertibility of classification grades. It is also found that the differences in the evaluation results, obtained by the multi-dimensional connection cloud
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Fig. 6. Clouds for the single indicator of the internal friction angle of weak intercalation and RMR: (a) and (c) connection clouds; (b) and (d) normal clouds.
model and normal cloud model, may result from the fact that the traditional normal cloud model cannot take into account the actual distribution characteristics of the indicators, but the connection cloud model can overcome this defect, and depict the transformation tendency of classification rating among adjacent grades based on the identical-discrepancy-contrary principles of SPA. In addition, the one-dimensional cloud model needed more computation time and more complicated process of generating clouds of classification standard for the multi-index problems. Interaction among indicators was omitted to some extent in one-dimensional cloud model. So its evaluation result was more dependent on the indicator weight. Namely, different determination methods of indicator weight may lead to different stable grades, and partly reduce the reliability of evaluation by one-dimensional cloud model. However, the multi-dimensional connection cloud model not only reflects the interaction of multiple indicators in the calculation process, but also promotes calculation efficiency. As expected, the evaluation results of case B obtained by the proposed model were also consistent with those of the SVM method as listed in Table 7. These results and comparisons indicate that the evaluation model proposed here has good stability and reliability in spite of different classification standards and evaluation indicators. The multi-dimensional connection cloud model with numerical characteristics specified by the IDC analysis of connection numbers theory can describe the effect of multiple attributes on evaluation results and has some advantages relative to the normal cloud model as showed in the following: (1) Numerical characteristic parameters determined by the SPA in the multi-dimensional connection cloud model can describe well the actual distribution of indicators relative to conventional normal cloud model. Moreover, many clouds have to be built separately for each evaluation indicator in the one-dimensional cloud model when considering the effects of all indicators, while the multi-dimensional connection cloud model just needs to establish only one cloud. In case A, five connection clouds were simulated for the corresponding stable grades in a proposed multi-dimensional connection cloud model, however, 35 clouds were needed to be generated for each classification rating of every indicator respectively in the one-dimensional cloud model. So for multi-index problems, the evaluation process of the multi-dimensional connection cloud is precise and convenient than that of the one-dimensional normal cloud and the calculation time of the multi-dimensional cloud model might decrease more sharply than that of the one-dimensional normal cloud model. (2) The proposed multi-dimensional connection cloud model provides a novel cloud model considering the characteristics of the interval-valued classification grades, and the actual distribution of fuzzy and random indicators. It enables a quantitative description of the certainty and uncertainty relationships from a whole perspective view, and the conversion trend of adjacent boundaries, because the proposed model suggests that indicators outside the given grade should have less impact on evaluation results of the sample than indicators located in the given grade. The connection degree in the multi-dimensional connection cloud model is adopted to express the conversion possibility to other
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grades, namely, an identity relationship is defined when the obtained connection degree is in [0.5, 1] according to the SPA of connection numbers theory, a discrepancy relationship when the acquired connection degree is in [e-9/2 , 0.5], and a contrary relationship when the connection degree is less than e-9/2 . It means that the given sample behaves higher certainty degree to the certain grade and less possibility transferring into the adjacent grades when the connection degrees to each classification rating are of large differences; while it behaves a lower certainty and greater conversion tendency to adjacent grades when the differences among connection degrees are small. For instance, the connection degree of the stability grade is μI =0.1948, μII =0.2119, μIII =0.0383, μIV =0.0209, and μV =0.0143 in sample 4, respectively, so the maximum membership principle implies that the grade of this sample belongs to the grade II, but the connection degrees to μI and μII are almost same, so it had a tendency to be grade I. However, for sample 5, the obtained connection degree to each grade is μI = 0.0398, μII =0.0383, μIII =0.0538, μIV =0.1511, and μV = 0.0515, respectively. The connection degree of μIV is the maximum value and much larger than other values, hence, it had a less conversion tendency to other grades. Obviously, the integrated connection degree can effectively describe the certainty and uncertainty relationships between measured value of the sample and the discussed classification grade, and the transformation tendency of classification grade among adjacent grades. Thus, the model proposed here provides a useful basis for the depiction of the measured indicators to classification standard from three aspects embracing identity, discrepancy, and contrary. Slope stability involving a large number of uncertain factors is a complex system problem, so up to present there is still no unified evaluation method, and now uncertainty evaluation methods get more and more attention from engineers and researchers. For the above reasons, the multi-dimensional connection cloud model, which is easy to be understood and operated for engineers, was proposed here to transform qualitative result of deterministic method into quantitative evaluation, the randomness and fuzziness of multiple indicators in finite intervals were also considered into the evaluation process to improve the accuracy and reliability of slope stability evaluation. Therefore, the proposed evaluation method of slope stability based on the multi-dimensional connection cloud model is a concise method and can take into account the uncertainty of the actual nature of the slope, the uncertainties of evaluation information acquisition and personnel knowledge. On the other hand, the discussed evaluation method here is difficult to give the safety factor of the corresponding slope as those methods based on the limit equilibrium theory or numerical analysis method. But the evaluation results related to the safety factor can be also achieved when the classification standard including the safety factor is constructed based on the existing evaluation results. 6. Conclusions Engineers and scholars have paid thorough attention to the evaluation of slope stability since the slope instability often leads to human injuries and huge economic losses. But up to the present, the problem has not yet been solved very well; this may be due to that previous studies performed are mainly focused on a single type of uncertainty of evaluation indicator, while the classification of slope stability involves fuzzy and stochastic factors distributed in finite intervals. To depict random and fuzzy uncertainty features of evaluation indicator distributed in finite intervals and the transformation tendency of classification rating, this work presents a multi-dimensional cloud model coupled with the SPA of connection numbers theory for the evaluation of slope stability. Some conclusions are drawn as follows. (1) Aiming at the uncertainty and the complex correlation of multiple factors, the novel multi-dimensional connection cloud model is presented to analyze slope stability. The reliability and validity of application of the proposed model in the slope stability evaluation are further investigated by 30 slope examples and the comparisons of results with other models. In addition, considering the intrinsic relationships of all the evaluation indicators, the method of determining numerical characteristics based on IDC analysis of connection numbers theory is submitted to the generation of cloud drops in the more precise interval. Case study shows that the novel multi-dimensional cloud model is reliable and acceptable for evaluating slope stability considering multiple uncertainties and interval distribution characteristics of indicators. And the multi-dimensional connection cloud model allows us to express the certainty and uncertainty relationships between the evaluation indicators and each classification standard in a unified way. (2) Compared with one-dimensional and multi-dimensional normal cloud models, the presented multi-dimensional connection cloud model can overcome the defects of the normal cloud model that the indicator must obey the normal distribution, and its processing procedure is more accurate and reliable to characterize the distribution of measured indicator values and more convenient for practical applications. The obtained results of the proposed model are more intuitive and accurate than those by the extension method and normal cloud model since it can depict the actual distribution of indicators and the conversion tendency of obtained grade among the classification grades. So the multi-dimensional connection cloud model can be viewed as a concise and effective alternative method for depicting multiple uncertainties of evaluation indicators in a unified way and transformation tendency among different grades. (3) It is just an attempt at the application of the multi-dimensional connection cloud model in the slope stability evaluation, but difficult to give the evaluation results related to safety factors as the limit equilibrium methods and numerical simulation. In addition, how to better couple indicator weight into the multi-dimensional connection cloud model and how to consider comprehensively the effects of the weight on the evaluation still need to clarify in future for a further accurate evaluation of slope stability.
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Declaration of Competing Interest The authors declare that there is no conflict of interests regarding the publication of this paper. Acknowledgments This research has been partially supported by the National Key Research and Development Program of China under Grant (No. 2017YFC1502405 and 2016YFC0401303) and the National Natural Sciences Foundation of China (41172274) is gratefully acknowledged. The authors also thank the reviewers for their thorough reviews and suggestions that helped to improve this paper. References [1] C.S. Juang, W.P. Gong, J.R. Martin, Q.S. Chen, Model selection in geological and geotechnical engineering in the face of uncertainty – does a complex model always outperform a simple model? Eng. Geol. 242 (2018) 184–196. [2] H.B. Wang, W.Y. Xu, R.C. Xu, Slope stability evaluation using back propagation neural networks, Eng. Geol. 80 (2005) 302–315. [3] H. Su, J. Li, J. Cao, Z. 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Applied Mathematical Modelling journal homepage: www.elsevier.com/locate/apm
A novel multi-dimensional cloud model coupled with connection numbers theory for evaluation of slope stability Mingwu Wang∗, Xiao Wang, Qiuyan Liu, Fengqiang Shen, Juliang Jin School of Civil and Hydraulic Engineering, Hefei University of Technology, 193 Tunxi Road, Hefei 230009 China
a r t i c l e
i n f o
Article history: Received 3 January 2019 Revised 17 July 2019 Accepted 23 July 2019 Available online 29 July 2019 Keywords: Slope stability Set pair analysis Cloud model Evaluation Multi-dimensional Connection number
a b s t r a c t Stability evaluation of a slope involves various fuzzy and correlation indicators randomly distributed in finite intervals. A novel multi-dimensional connection cloud model was presented here to address multiple uncertainties and distribution characteristics of indicators, and to depict the randomness and fuzziness of the measured index value belonging to the classification standard in the slope stability analysis. In the model, when simulating fuzzy and random characteristics of evaluation indicators in finite intervals, the numerical characteristics of connection cloud model were assigned on the basis of the analysis of identical-discrepancy-contrary (IDC) relationships between measured indicators and the classification standard to overcome the subjectivity. Considering the effect of indicator correlation in a unified way, the integrated connection degree of a grade was further specified for the evaluation sample. Moreover, case studies and comparisons of the proposed model with one-dimensional normal cloud model, extension model, and support vector machine (SVM) were performed to confirm the validity and reliability. The results indicate that this model employed to evaluate slope stability can clearly depict the random and fuzzy distribution features of measured data in finite intervals, and its calculation process is quicker and simpler than that of one-dimensional normal cloud model. © 2019 Elsevier Inc. All rights reserved.
1. Introduction The stability evaluation is an essential task in the design and construction of slope stabilization engineering, so to prevent damages to public and private properties, a precise estimation of slope stability is much important for decision makers and has received widely attention in geotechnical engineering field [1]. However, the work is a quite complex problem because slope stability is characterized by various uncertain factors, such as insufficient and uncertain geological information of site conditions, heterogeneous geotechnical properties of rock mass or soil mass and geomorphic characteristics [2]. Consequently, although scholars have presented numerous evaluation methods considering uncertain factors [3], such as fuzzy sets method [4], probabilistic method [5,6], artificial intelligence method [7–9], extension method [10], numerical modeling method [11,12] and so on, there still currently exist considerable challenges in accurately evaluating slope stability by these analytical methods. The evaluation method of slope stability is divided into two categories: qualitative and quantitative analysis methods. Regardless of the fact that the conventional methods have demonstrated successful performances in the slope stability evaluation, they were of their own disadvantages. For instance, the engineering geology analogy method, founded on geological ∗
Corresponding author. E-mail address: [email protected] (M. Wang).
https://doi.org/10.1016/j.apm.2019.07.043 0307-904X/© 2019 Elsevier Inc. All rights reserved.
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information and engineering experience [3], is a representative and prevalent method in the qualitative method for evaluation of slope stability, but may vary from person to person with its subjectivity because lacking uniform standards; the limit equilibrium method and numerical modeling, two of the most widely used quantitative analysis methods with the safety factor, have the advantages of clear physical meaning, but their applications for field engineers have been limited by some defects [13] because limit equilibrium methods need certain hypothetical conditions to solve, while evaluation results of numerical simulation methods rely too much on the reliability of the parameters, the simplification of the boundary conditions and model construction. In addition, they cannot describe the uncertainty of information acquisition and cognitive uncertainty in multi-index problems. Furthermore, the evaluation of slope stability will encounter subjectively ambiguities as the result of applying linguistic terms, same score prescribed in a classification rating, existence of sharp transitions between two adjacent grades [14], which sometimes may lead to a decrease in the reliability of the evaluation results. For example, the slope may indeed pose different risk levels in spite of the same value of safety factor obtained by the limit equilibrium method [15]. So recently numerous probabilistic approaches [16], random field simulation technique [17] and reliability analysis methods [18] were presented to consider limited internal and external uncertainties involved in the slope stability analysis, and the slope stability evaluation is starting from certainty to uncertainty analysis. These uncertainty approaches and reliability analysis methods provided effective ways to make use of fuzzy or random uncertainty and to estimate their effects on the safety factor in the slope stability evaluation, but the calculation methods of reliability are very different for different destructive forms [19], and the determination of membership degree function in fuzzy comprehensive method [4] is subjective and ignores the randomness of the system. In order to take multiple uncertainties into account the slope stability analysis and avoid complex failure mechanism, the artificial neural network (ANN) method and the SVM method were implemented to depict the highly non-linear relationship between physical and mechanical parameters and the safety factor of the slope [20,21]. But the ANN method is unable to present an explicit relationship between evaluation parameters and rank potential, and has some inherent drawbacks, such as a low convergence speed and over-fitting problems [2]. As for the SVM method, the determination of parameters in advance requires huge additional computational complexity [22,23]. At the same time, because of the complexity of the slope engineering, various categories of uncertainty such as fuzziness and randomness may be inevitably encountered in evaluations of slope stability, so the slope stability analysis cannot rely only on methods on the basis of a single type of uncertainty. The combination of two or more uncertainties has become a trend for the slope stability evaluation. However, the above methods cannot classify finely the features of various types of uncertainties including randomness and fuzziness in a unified way during the evaluation process. The normal cloud model proposed by Li et al. [24], which can automatically generate the certainty of the qualitative concept, is a tool for depicting fuzziness and randomness of evaluation indicators in a unified way [25], but it requires indicators obeying a normal distribution in an infinite interval that is not consistent with the actual distribution pattern of indicators because indicators are generally fuzzy and distributed randomly in finite intervals. Moreover, for the multi-index problem, the computation procedure by the one-dimensional normal cloud model might be more complicated by the increase in numbers of the evaluation indexes and samples, so some researchers proposed the multi-dimensional normal cloud model to comprehensively deal with the influence of indicators of multiple uncertainties [26], but unluckily there are few reports on the slope stability evaluation using the multi-dimensional cloud model. Therefore, the normal cloud model is useful to reflect the fuzziness and randomness of evaluation indicators, but sometimes it might fail to describe the actual distribution characteristic of evaluation indicator in a finite interval, and cannot depict the classification transformation among adjacent grades, this may lead to the deviation of the evaluation results from reality and limitation of its applications in engineering. On the other hand, due to randomness, uncertainty, and incompleteness of indicator information acquisition, the complicated geologic process of the slope deformation or failure, and the distinct inhomogeneous property of rock and soil, the design and the decision-making for the slope engineering cannot be singly depended on theoretical analyses and numerical simulations [27]. Furthermore, expert experiences are more reliable than the calculated results in most of the cases, so the engineering geological analogy based on the existing slope evaluation results is a widely used one in the earlier period for designing an appropriate slope, but it cannot give the reliability of classification rating and the possibility to other grades. In general, although the conventional engineering geological analogy method and uncertainty analysis methods are commonly applicable, they often only address a single type of uncertainty such as fuzziness or randomness, and are difficult to describe multiple uncertainties of actual slope information and information acquisition in essence. Moreover, the intervalvalued classification rating cannot be completely characterized by the normal cloud model. Consequently, a novel analytical method based on the cloud model and set pair analysis of connection numbers theory is needed to be further discussed in order to overcome the shortcomings of the conventional normal cloud model and engineering geological analogy method. The complexity of the slope system requires the employment of a new method that is efficient in depicting multiple types of uncertainties in the evaluation of slope stability. This work is aimed at introducing a novel multi-dimensional connection cloud model coupled with set pair analysis (SPA) of connection numbers theory to enhance the reliability and accuracy of evaluation of slope stability, and to dialectically express actual distribution characteristics of indicators in finite intervals. The authors intend to dialectically describe uncertainty, complexity, and correlation under multiple factors of the slope stability evaluation, the conversion tendency of classification at the boundary and the certainty and uncertainty relationships between the measured evaluation indicators and classification standards from identity, discrepancy and contrary aspects. Finally, the feasibility and validity of the proposed method were further discussed and confirmed by the case study as well as the comparisons with other methods.
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2. Methodology As discussed above, the slope stability evaluation is a nonlinear and uncertain dynamic system problem influenced by various factors, such as rock mass structure, orientation of advantage discontinuities, geometric properties of slope, construction methods and environmental conditions [3]. Here to comprehensively consider the effects of multiple uncertainties of indicators and their distribution characteristics, a multi-dimensional cloud model coupled with set pair analysis of connection numbers theory is proposed to directly and quickly analyze the slope stability, and to overcome the defects of the normal cloud model. 2.1. Multi-dimensional connection cloud model The definition of the multi-dimensional connection cloud model is given as following: Assume that there are m evaluation indicators (j = 1, 2, …, m) and n stability grades (i = 1, 2, …, n) for the slope stability problem, and cloud mapping of grade i for index j is composed of left and right half branches of connection cloud with the expected value Exji . Let C be a qualitative concept in the m-dimensional quantitative domain U with precise values. If xU, a random realization of the qualitative concept C, follows a normal distribution N(Ex, y2 ), and y satisfies a normal distribution N(En, He2 ), overall quantitative property of concept C is characterized as
k ji m 9 x ji − E x ji μ ji (x ) = exp − 3y ji , 2
(1)
j=1
where μji (x) is the connection degree, ∈ [0, 1]; Ex, En, He and y are the vector of the expected value, entropy, hyper entropy and random number satisfying normal distribution, respectively; Exji is the expected value for the grade i of the evaluation indicator j that denotes the most representative parameter of the qualitative concept; kji the order of distribution function of connection cloud for the grade i of the evaluation indicator j; yji is a random number satisfying normal distribution N(Enji , Heji 2 ); the entropy Enji denotes the uncertainty measurement and granularity of a qualitative concept, the higher Enji is, the larger scope of the universe can be accepted by the concept; Heji are the hyper entropy that reveals the correlation between randomness and fuzziness. Then the m-dimensional quantitative domain and one connection degree consists a drop of the connection cloud, and the distribution of x in the domain U is called an m-dimensional connection cloud. For the actual multi-index evaluation problem, the interval-valued classification rating of the given indicator may be different in sizes, so the width of the left and right half branches of the cloud model is different, and the connection cloud may be asymmetric, that is, the left and right half branches of the cloud model are of various numerical characteristics (see Fig. 1). Namely, cloud mapping consists of the left and right half of asymmetric connection clouds at the dividing point of the expected value, Exji . According to Eq. (1), the connection cloud model becomes a normal cloud model when kji =2. But for the unequal interval-valued classification ratings, the connection degree on the classification interface point simulated by the connection cloud model and normal cloud model is obviously different as showed in Fig. 1. The unequal interval sizes of classification grades for the evaluation indicator may be characterized by asymmetric normal clouds constructed by left and right half branches in normal clouds, but it cannot reflect the certainty and uncertainty relationships between the index and the discussed grade, and the fuzziness of classification grade when the measured index is at the threshold values as showed in Fig. 1, the distribution of the indicator simulated by a normal cloud model may be varied from the actual situation. In addition, measured indices are usually distributed in finite intervals, while the indices reflected by normal clouds are distributed in infinite intervals. This kind of deviation is mainly caused by the traditional determination method of the numerical characteristics and infinite distribution feature of indicator in the normal cloud model. Numerical characteristics of normal cloud were mainly determined on the basis of the ‘3En rule’ that cloud drops mainly locate in the basic range [Ex−3En, Ex+3En] while a small number of drops are outside of the basic range. This may neglect the fuzziness of classification grades and the finite interval distribution characteristics of measured indicators.
Fig. 1. Comparison of connection cloud and normal cloud.
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Fig. 2. IDC relationships between cloud drops and classification grades.
2.2. Set pair analysis of connection numbers theory To express the actual distribution characteristic of the indicator, and the certainty and uncertainty relationships between the indicator and classification standard, the SPA of connection numbers theory was introduced here to improve the generation algorithm of cloud model, and presented a novel asymmetric connection cloud model. The connection numbers theory is an uncertainty analysis theory proposed by Zhao [28] for dealing with systematic uncertainty problem based on the essence of Chinese philosophy ‘One divided into three’ [29] and the principle of universal connection [30,31], its mathematical model is
μ = a + bI + cJ,
(2)
where μ is the connection degree denoting the certainty and uncertainty relationships of a set pair H(A, B), which consists of two related sets A and B; a, b, and c are measurements of identity, discrepancy, and contrary relationships, respectively; I and J are the discrepancy and contrary coefficients, respectively. Obviously, the connection numbers theory is of advantage in depicting the development from ‘all are the same in the same group’ to quantitative differences in the same group. The complexity and diversity of uncertainty and certainty relationships and transformation can be obtained by the microscopic analysis of SPA and quantitatively depicted as Eq. (2) from a unified view of the connection degree. So the connection numbers theory provides a new idea for the micro-analysis of cloud drop distribution and extra thoughts to overcome the defects in the traditional normal cloud model. As we know, during the generation of a cloud, the analysis of the relationship between measured indicators and classification standard plays a significant role in the determination of the numerical characteristic parameter. Here, to determine the numerical characteristic parameters of the connection cloud model more reasonably, the fuzzy random uncertainty that the threshold values belong to the given classification grade is deeply analyzed from three relationships including the identity, discrepancy and contrary by the set pair analysis. According to the identical-discrepancy-contrary rules of SPA, the farther away from Exji the cloud drops are, the more discrete and sparse they are, reflecting the more unstable and inconsistent perception of concepts, so their transformation can be also characterized by the connection degree. So the connection degree was utilized here to depict the possibility that the sample belongs to the classification grade and to deal with the certainty and uncertainty relationships between classification standard and measured indicators distributed in finite intervals. In fact, Eq. (1) skillfully reflects the maximum random uncertainty of threshold values belonging to adjacent levels through kji parameters. The identical-discrepancy-contrary relationships between the measured evaluation indicators and classification standard in the connection cloud model are defined as showed in Fig. 2. 3. The evaluation of slope stability with multi-dimensional connection cloud The basic evaluation principle of the connection cloud model is depicted in the following: A rational evaluation indicator system is first set up; then, on the basis of IDC relationships analysis, the numerical characteristics (Ex, En, He, k) are determined for each classification rating of evaluation indicator, respectively; finally, the integrated connection degree gained with indicator weights is adopted to specify the grade of the evaluation sample according to the maximum membership rule. The procedure for the evaluation of slope stability based on the multi-dimensional connection cloud model is illustrated as showed in Fig. 3. Detailed evaluation procedure is consists of 5 steps as following. Step 1: Set up an appropriate evaluation system for the slope stability with selected indicators and classification standard. Previous studies have shown that there are various factors affecting the slope stability, which are either human factors or natural factors; and for an actual slope each factor often plays a different role from others. Therefore, the selection of
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Fig. 3. The process of slope stability evaluation by the multi-dimensional connection cloud model.
evaluation index is very important, and the selected evaluation index should be simple in quantitative form and easy to obtain. The corresponding interval-valued classification is rated for each grade. Step 2: Assign the m-dimensional connection cloud descriptors, (Ex, En, He, k) and the cloud drop number λ. As noted above, the asymmetric connection cloud can dialectically express the actual distribution characteristic of the indicator and the certainty and uncertainty relationships between the indicator and classification standard, and is dependent upon the reasonable determination of numerical characteristic parameters. According to the connection numbers theory, corresponding IDC relationships for the measured indicator j and classification grade i of a sample are specified as follows: an identical relationship between measured indicator and classification rating is specified when the measured value of indicator j locates in the interval-valued classification grade i; a discrepancy relationship when the measure value locates in the adjacent grades (grade i + 1 or grade i−1); and a contrary relationship when it locates in other grades. The numerical characteristic parameters are given by
E x ji =
C min ji + C max ji , 2
(3)
H e ji = β , E n ji =
k ji =
(4)
a ji , 3 ln
(5)
ln 4
9 , C −E x ln ji3E n ji ji
(6)
where Exji , Enji , Heji and kji are connection cloud descriptors for the ith grade of the evaluation indicator j, respectively; Cmaxji and Cminji are thresholds of the interval-valued stability grade i of the indicator j; β denotes the atomized feature of the connection cloud, here is a constant taken; Cji represents the lower limitation of discussed classification rating Cminji or the upper limitation Cmaxji ; α ji indicates the width of the connection cloud. For the performance index, the corresponding parameters α ji of intermediate classification grades (i = 2, 3, …, n−1) for the left and right branch of the connection cloud are
a_left ji = E x ji − C min j (i−1)
a_right ji = C max j (i+1) − E x ji
,
(7)
where α _leftji and α _rightji are the width of the left and right branch of connection cloud, respectively. While for the cost index, the corresponding values of intermediate classification grades are
a_left ji = E x ji − C min j (i+1)
a_right ji = C max j (i−1) − E x ji
.
(8)
The distribution form of the evaluation index of grades (i = 1, n) at both ends is often different from that of the intermediate grades, and has asymmetric characteristics. For example, the distribution form of the indicator for the grade at both ends is often uniform distribution with a certainty degree of 1.0, so it is difficult to describe the transition characteristics between the both ends grades and the adjacent grades by using the traditional normal cloud model, and the corresponding parameters of numerical characteristics for the normal cloud model are also difficult to determine; while the corresponding
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Table 1 Classification standard in case A [10]. Evaluation indicator
Stable (I)
Relatively stable (II)
Basically stable (III)
Unstable (IV)
Extremely unstable (V)
Weights
C1 C2 C3 C4 C5 C6 C7
0–75 0–10 0–20 55–100 35–50 80–100 0–2
75–175 10–20 20–30 40–55 30–35 60–80 2–4
175–300 20–30 30–35 25–40 20–30 40–60 4–6
300–500 30–40 35–45 20–25 15–20 20–40 6–8
500–1000 40–90 45–90 0–20 0–15 0–20 8–12
0.083 0.053 0.053 0.221 0.148 0.338 0.104
(m) (°) (°) (kPa) (°)
parameters of the connection cloud model can be determined according to the measured lower limit value or upper limit value of the index. Step 3: Generate the m-dimensional connection cloud for the classification rating i. The generation of cloud drops is the fundamental content to embody uncertainty problem with characteristics of fuzziness and the stochasticity problem. The generation algorithm for the connection cloud is: firstly, generate random numbers y following normal distribution based on the entropy En and hyper entropy He, and get x from the normal distribution with expected values Ex and random standard deviations y; then the connection degree is specified by Eq. (1) to set up a cloud drop of the m-dimensional connection cloud as (x, μ(x)); λ cloud drops can be attained by the above process. Step 4: Combining together weights of indicators with each other, wj , the assignment of the connection degree of the grade i for the sample of identical relationship or discrepancy relationship with the relevant grades assumes the following formula.
μi
m x ji − E x ji k ji 9 . = exp − w j · 2 3y ji
(9)
j=1
As discussed above, generated cloud drops mainly focus in the range [Ex-3En, Ex+3En] in the cloud model (see Fig. 2), and those drops outside of the basic range might not be considered for their contributions to the qualitative concepts. At the same time, to reduce the computational error caused by the multiplication of near zero numbers, the connection degree is approximately e − 9/2 when there is a contrary relationship between classification standard and measured value xji . Step 5: Specify the grade of slope stability for the sample according to the maximum membership rule. 4. Case study 4.1. Data and evaluation system 4.1.1. Case A To verify the feasibility and reliability of the model, the data of literature [10] were used here to analyze and conduct comparisons with extension model and one-dimensional normal cloud model. In case A, slope height (m) C1 , slope angle (°) C2 , dip angle (°) C3 , cohesive strength of weak intercalation (kPa) C4 , internal friction angle of weak intercalation (°) C5 , rock mass rating (RMR) C6 , and seismic intensity C7 were chosen as evaluation indicators. The slope stability was classified into five grades: stable (I), relatively stable (II), basically stable (III), unstable (IV) and extremely unstable (V). The classification standard and measured indicator values of samples were shown in Tables 1 and 2, respectively. The assignment of indicator weights plays a critical role in the evaluation result, so in order to better compare the results obtained by the other models, the same indicator weights assigned by an analytic hierarchy process method were adopted here as the reference [10]. Namely, according to the above evaluation procedure, the sample values and weights of wj ={0.083, 0.053, 0.053, 0.221, 0.148, 0.338, 0.104} were substituted for Eq. (9) to obtain their connection degrees at grades as showed in Table 3. 4.1.2. Case B As we know, there is a lack of clear and unified classification standard for the slope stability analysis so far. Different classification standards of slope stability may lead to changes in results obtained by the same method. Data from the literature [23] named as case B were therefore carried out to further verify the validity of the proposed model. In case B, the slope gradient D1 , slope height (m) D2 , slope type D3 , weathering degree D4 , vegetation cover D5 , and hardness of rock and soil mass D6 were chosen as evaluation indicators. The classification grade of slope stability was from stable (I), basically stable (II), unstable (III) to extremely unstable (IV). The classification standard and measured values of indicators were listed in Tables 4 and 5. 4.2. Model validation and algorithm efficiency analysis To clearly highlight the simulation process of the multi-dimensional connection cloud model, an example of twodimensional connection cloud for the evaluation indicators of the slope angle C2 and seismic intensity C7 , was illustrated
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Table 2 Measured indicator values of case A [10]. Sample
C1
C2
C3
C4
C5
C6
C7
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
140 145 100 225 250 120 105 120 40 180 150 217 133 210 205 500 130 400
20 30 45 16 14 40 46 50 45 31 35 40 42 45 25 17 35 20
5 60 25 5 7 50 25 6 30 9 45 10 4 45 30 20 25 25
3 6 85 63 58 3 11 20 20 40 33 60 80 21 45 67 23 70
19 27 30 31 11 22 27 32 17 10 32 40 21 26 19 41 11 20
15 34 87 83 32 32 40 71 38 14 23 89 35 31 29 82 28 68
6 6 7 7 7 7 7 6 6 4 6 7 7 7 6 7 7 6
Table 3 Connection degrees of samples for case A. Sample
μI
μII
μIII
μIV
μV
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
0.0159 0.0124 0.2003 0.1948 0.0398 0.0134 0.0160 0.0531 0.0160 0.0140 0.0166 0.3160 0.0442 0.0111 0.0166 0.3053 0.0148 0.0633
0.0206 0.0238 0.1191 0.2119 0.0383 0.0181 0.0304 0.1496 0.0166 0.0573 0.0509 0.1076 0.0257 0.0202 0.0521 0.1457 0.0204 0.1348
0.0421 0.1307 0.0304 0.0383 0.0538 0.0766 0.1231 0.0593 0.0907 0.0639 0.1263 0.0202 0.0917 0.1155 0.2259 0.0164 0.0757 0.1135
0.0975 0.1566 0.0211 0.0209 0.1511 0.2073 0.1409 0.0445 0.3385 0.0549 0.1518 0.0250 0.1529 0.4140 0.1729 0.0243 0.3998 0.0421
0.1518 0.0614 0.0179 0.0143 0.0515 0.1107 0.0486 0.0325 0.0582 0.1327 0.0493 0.0175 0.0266 0.0934 0.0300 0.0196 0.1239 0.0136
Table 4 Classification standard and weights in case B [23]. Evaluation indicator Slope gradient D1 Standard value Normalized value Slope height (m) D2 Standard value Normalized value Slope type D3 Standard value Normalized value Weathering degree D4 Standard value Normalized value Vegetation cover D5 Standard value Normalized value Hardness of rock and soil mass D6 Standard value Normalized value
Stable (I)
Basically stable (II)
Unstable (III)
Extremely unstable (IV)
0–1 0–0.25
1–2 0.25–0.50
2–3 0.50–0.75
>3 0.75–1.00
0–8 0–0.25
8–20 0.25–0.50
20–30 0.50–0.75
>30 0.75–1.00
Concave 0–0.25
Straight 0.25–0.50
Stepped 0.50–0.75
Convex 0.75–1.00
No 0–0.25
Slight 0.25–0.50
Moderate 0.50–0.75
Severe 0.75–1.00
Good 0–0.25
Better 0.25–0.50
Average 0.50–0.75
Bad 0.75–1.00
Hard
Hard rock; Sandwiched soft rock 0.25–0.50
Soft rock; Sandwiched hard soil 0.50–0.75
Soft
Weights 0.1742
0.1659
0.1648
0.1684
0.1571
0.1696
0–0.25
0.75–1.00
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Table 5 Measured values of indicators for case B [23]. Sample
D1
D2
D3
D4
D5
D6
19 20 21 22 23 24 25 26 27 28 29 30
2.3 2.7 1.6 1.6 3.1 0.8 1.4 1.6 3.2 1.4 3.1 1.7
16 7 18 14 18 7 12 7 28 14 31 26
0.63 0.89 0.64 0.39 0.83 0.20 0.75 0.43 0.38 0.34 0.92 0.81
0.63 0.89 0.41 0.32 0.64 0.15 0.50 0.66 0.90 0.29 0.92 0.62
0.80 0.91 0.83 0.91 0.59 0.82 0.88 0.77 0.94 0.78 0.83 0.85
0.48 0.59 0.85 0.52 0.61 0.71 0.83 0.61 0.23 0.30 0.92 0.86
Fig. 4. The two-dimensional connection cloud for indicators of the slope angle and seismic intensity.
for case A. On the basis of the discussed simulation steps in Section 3, the numerical characteristics (Ex2 , En2 , He2 , k2 ; Ex7 , En7 , He7 , k7 ) of the slope angle and seismic intensity at different grades were first gained by Eqs. from (3) to (8). The corresponding process of determining the parameters of numerical characteristics for the given grade III of the evaluation index C2 by means of the IDC principles of SPA was described as following. First, the mean value 25 of the classification standard interval of the given grade III was taken as the expected value, Ex23 , of the connection cloud. Then corresponding widths of the left and right half branches of cloud determined through Eq. (7) were both 15. Next, according to the IDC principle illustrated as showed in Fig. 2, it was an identical relationship between the measured indicator C2 value and discussed classification standard III when the value of indicator C2 was within the interval [20, 30], and a discrepancy relationship when the indicator C2 value was [10, 20] or [30, 40]. Thus, the orders k23 of the left and the right half branches of the connection cloud were both obtained as 1.703 according to Eq. (6). Orders of indicator C7 were also obtained depending on the similar procedure. The constant value of He was 0.01. Finally, according to the numerical characteristics, a two-dimensional connection cloud responding to the two indicators was obtained as showed in Fig. 4. It is also found that the simulation process of the multi-dimensional connection cloud model is much simpler for the multi-index evaluation problem than that of the one-dimensional normal cloud. The efficiency issue for the slope stability evaluation of m indicators and n classification grades was discussed here in order to illustrate the advanced nature of the multi-dimensional connection clouds relative to the one-dimensional cloud model. According to the production principle of the one-dimensional cloud model, the number of simulation cloud composed of λ drops for all the classification grades is I1 =nm, and the corresponding cloud drops are required to randomly generate I1 =nmλ times by a one-dimensional cloud model. But by the multi-dimensional cloud model, only n multi-dimensional clouds are needed, and the corresponding generation number of cloud drops is only nλ. Obviously, for the multi-index problem, the calculation amount of the
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Proposed model
One-dimensional normal cloud model
Extension model [10]
Actual situation on site [10]
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
Ⅴ IV I II IV IV IV II IV Ⅴ IV I IV IV IV I IV II
Ⅴ IV II II IV IV IV II IV IV IV I IV IV IV II IV II
Ⅴ Ⅴ I I IV IV IV I IV Ⅴ IV I IV IV IV I Ⅴ II
Unstable Unstable Stable Stable Unstable Unstable Creep Stable Unstable Unstable Creep Stable Unstable Unstable Creep Stable Unstable Stable
one-dimensional cloud model is m−1 times more than that of the multi-dimensional cloud model. Moreover, the interaction among indicators is neglected to some extent in the one-dimensional cloud model, which may lead to the evaluation result being more dependent on the index weight. 5. Comparisons and discussions Comparisons of evaluation results by the proposed model, the one-dimensional normal cloud model, the extension model, actual situation on site and support vector machine method [10] for cases A and B were performed to confirm the reliability and validity. The evaluation results obtained from the proposed model and comparisons with other methods were listed in Tables 6 and 7. As listed in Table 6, the evaluation results from the model proposed were basically in agreement with those from the extension evaluation method except for samples 8 and 17, and from the one-dimensional normal cloud model except for samples 3 and 16. The results of the proposed model were almost same as those of the actual situation on site. For sample 8, there was one indicator at grade I, two indicators at grade II, and other indicators at grades III, IV and IV, respectively, so it was more reasonable to rate sample 8 as II than I. It was seen in Table 2 that there was no indicator at grade V for sample 17, so it was obviously more at grade IV than at grade V. These results indicate that the proposed model is more feasible and reliable than the extension method. Sample 2 was taken as an example to analyze the reasons for the difference between the evaluation result obtained by the multi-dimensional connection cloud and one-dimensional normal cloud model in the following. For sample 2, measured values of indicators C2 , C5 and C7 were at grade III; the grade of indicators C1 and C6 was at grade II and grade IV, respectively; indicators C3 and C4 were at grade V, so if the importance of each indicator was equal, that is to say, the influence of each index on the evaluation result was same, the overall stability grade of sample 2 was reasonable to be specified as grade IV. Based on weights used in Ref. [10], the classification rating obtained by the proposed model was also grade IV, which was consistent with the actual state of the slope on site, while the classification rating obtained by the extension method was grade V and obviously in favor of unsafety.
Table 7 Evaluation results of the multi-dimensional connection cloud and comparison with the support vector machine method for case B. Sample
μI
μII
μIII
μIV
Proposed model
Support vector machine method [23]
19 20 21 22 23 24 25 26 27 28 29 30
0.0151 0.0217 0.0230 0.0504 0.0126 0.1913 0.0276 0.0373 0.0312 0.1207 0.0111 0.0142
0.1565 0.0408 0.1416 0.4074 0.0730 0.1072 0.0957 0.1784 0.0508 0.4327 0.0111 0.0480
0.5204 0.1276 0.2490 0.1212 0.5267 0.0360 0.1607 0.2287 0.0943 0.0678 0.1258 0.3190
0.0597 0.2246 0.0707 0.0241 0.1279 0.0390 0.0886 0.0464 0.1730 0.0210 0.8328 0.2167
III IV III II III I III III IV II IV III
III IV III II III I III III IV II IV III
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Fig. 5. Comparison of two-dimensional connection cloud and normal cloud for classification standard of indicators of the internal friction angle of weak intercalation and RMR in case A: (a) connection clouds; (b) normal clouds.
It can be seen from case A study that cloud model, as a cognitive model to realize the conversion of qualitative concepts and quantitative data, can convert the fuzziness and randomness of slope stability into the quantitative data of the connection degree, but it may be difficult to evaluate slope stability with various types of uncertainty factors only by onedimensional normal cloud model. As shown in sample 10, the indicators C5 and C6 belonged to extremely unstable level, the indicators C1 , C2 , C3 , C4 , and C7 belonged to basically stable level, unstable level, stable level, basically stable and relatively stable level, respectively. Obviously, measured values of indicators scattered in five levels, so the integrated connection degree at each level was not so high. As noted above, the normal cloud model cannot depict the characteristics of interval distribution and transformation of classification grade of evaluation indicator. So there were differences in results obtained by the proposed multi-dimensional cloud model and the conventional normal cloud model for sample 10. Figs. 5 and 6 illustrate the comparison of normal cloud and connection cloud for indicators of the internal friction angle of weak intercalation and RMR. It was seen that the normal cloud cannot reflect the actual distribution characteristics of evaluation indicators in finite intervals and at the highest and lowest classification grades, as well as the fuzziness and convertibility of classification grades. It is also found that the differences in the evaluation results, obtained by the multi-dimensional connection cloud
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Fig. 6. Clouds for the single indicator of the internal friction angle of weak intercalation and RMR: (a) and (c) connection clouds; (b) and (d) normal clouds.
model and normal cloud model, may result from the fact that the traditional normal cloud model cannot take into account the actual distribution characteristics of the indicators, but the connection cloud model can overcome this defect, and depict the transformation tendency of classification rating among adjacent grades based on the identical-discrepancy-contrary principles of SPA. In addition, the one-dimensional cloud model needed more computation time and more complicated process of generating clouds of classification standard for the multi-index problems. Interaction among indicators was omitted to some extent in one-dimensional cloud model. So its evaluation result was more dependent on the indicator weight. Namely, different determination methods of indicator weight may lead to different stable grades, and partly reduce the reliability of evaluation by one-dimensional cloud model. However, the multi-dimensional connection cloud model not only reflects the interaction of multiple indicators in the calculation process, but also promotes calculation efficiency. As expected, the evaluation results of case B obtained by the proposed model were also consistent with those of the SVM method as listed in Table 7. These results and comparisons indicate that the evaluation model proposed here has good stability and reliability in spite of different classification standards and evaluation indicators. The multi-dimensional connection cloud model with numerical characteristics specified by the IDC analysis of connection numbers theory can describe the effect of multiple attributes on evaluation results and has some advantages relative to the normal cloud model as showed in the following: (1) Numerical characteristic parameters determined by the SPA in the multi-dimensional connection cloud model can describe well the actual distribution of indicators relative to conventional normal cloud model. Moreover, many clouds have to be built separately for each evaluation indicator in the one-dimensional cloud model when considering the effects of all indicators, while the multi-dimensional connection cloud model just needs to establish only one cloud. In case A, five connection clouds were simulated for the corresponding stable grades in a proposed multi-dimensional connection cloud model, however, 35 clouds were needed to be generated for each classification rating of every indicator respectively in the one-dimensional cloud model. So for multi-index problems, the evaluation process of the multi-dimensional connection cloud is precise and convenient than that of the one-dimensional normal cloud and the calculation time of the multi-dimensional cloud model might decrease more sharply than that of the one-dimensional normal cloud model. (2) The proposed multi-dimensional connection cloud model provides a novel cloud model considering the characteristics of the interval-valued classification grades, and the actual distribution of fuzzy and random indicators. It enables a quantitative description of the certainty and uncertainty relationships from a whole perspective view, and the conversion trend of adjacent boundaries, because the proposed model suggests that indicators outside the given grade should have less impact on evaluation results of the sample than indicators located in the given grade. The connection degree in the multi-dimensional connection cloud model is adopted to express the conversion possibility to other
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grades, namely, an identity relationship is defined when the obtained connection degree is in [0.5, 1] according to the SPA of connection numbers theory, a discrepancy relationship when the acquired connection degree is in [e-9/2 , 0.5], and a contrary relationship when the connection degree is less than e-9/2 . It means that the given sample behaves higher certainty degree to the certain grade and less possibility transferring into the adjacent grades when the connection degrees to each classification rating are of large differences; while it behaves a lower certainty and greater conversion tendency to adjacent grades when the differences among connection degrees are small. For instance, the connection degree of the stability grade is μI =0.1948, μII =0.2119, μIII =0.0383, μIV =0.0209, and μV =0.0143 in sample 4, respectively, so the maximum membership principle implies that the grade of this sample belongs to the grade II, but the connection degrees to μI and μII are almost same, so it had a tendency to be grade I. However, for sample 5, the obtained connection degree to each grade is μI = 0.0398, μII =0.0383, μIII =0.0538, μIV =0.1511, and μV = 0.0515, respectively. The connection degree of μIV is the maximum value and much larger than other values, hence, it had a less conversion tendency to other grades. Obviously, the integrated connection degree can effectively describe the certainty and uncertainty relationships between measured value of the sample and the discussed classification grade, and the transformation tendency of classification grade among adjacent grades. Thus, the model proposed here provides a useful basis for the depiction of the measured indicators to classification standard from three aspects embracing identity, discrepancy, and contrary. Slope stability involving a large number of uncertain factors is a complex system problem, so up to present there is still no unified evaluation method, and now uncertainty evaluation methods get more and more attention from engineers and researchers. For the above reasons, the multi-dimensional connection cloud model, which is easy to be understood and operated for engineers, was proposed here to transform qualitative result of deterministic method into quantitative evaluation, the randomness and fuzziness of multiple indicators in finite intervals were also considered into the evaluation process to improve the accuracy and reliability of slope stability evaluation. Therefore, the proposed evaluation method of slope stability based on the multi-dimensional connection cloud model is a concise method and can take into account the uncertainty of the actual nature of the slope, the uncertainties of evaluation information acquisition and personnel knowledge. On the other hand, the discussed evaluation method here is difficult to give the safety factor of the corresponding slope as those methods based on the limit equilibrium theory or numerical analysis method. But the evaluation results related to the safety factor can be also achieved when the classification standard including the safety factor is constructed based on the existing evaluation results. 6. Conclusions Engineers and scholars have paid thorough attention to the evaluation of slope stability since the slope instability often leads to human injuries and huge economic losses. But up to the present, the problem has not yet been solved very well; this may be due to that previous studies performed are mainly focused on a single type of uncertainty of evaluation indicator, while the classification of slope stability involves fuzzy and stochastic factors distributed in finite intervals. To depict random and fuzzy uncertainty features of evaluation indicator distributed in finite intervals and the transformation tendency of classification rating, this work presents a multi-dimensional cloud model coupled with the SPA of connection numbers theory for the evaluation of slope stability. Some conclusions are drawn as follows. (1) Aiming at the uncertainty and the complex correlation of multiple factors, the novel multi-dimensional connection cloud model is presented to analyze slope stability. The reliability and validity of application of the proposed model in the slope stability evaluation are further investigated by 30 slope examples and the comparisons of results with other models. In addition, considering the intrinsic relationships of all the evaluation indicators, the method of determining numerical characteristics based on IDC analysis of connection numbers theory is submitted to the generation of cloud drops in the more precise interval. Case study shows that the novel multi-dimensional cloud model is reliable and acceptable for evaluating slope stability considering multiple uncertainties and interval distribution characteristics of indicators. And the multi-dimensional connection cloud model allows us to express the certainty and uncertainty relationships between the evaluation indicators and each classification standard in a unified way. (2) Compared with one-dimensional and multi-dimensional normal cloud models, the presented multi-dimensional connection cloud model can overcome the defects of the normal cloud model that the indicator must obey the normal distribution, and its processing procedure is more accurate and reliable to characterize the distribution of measured indicator values and more convenient for practical applications. The obtained results of the proposed model are more intuitive and accurate than those by the extension method and normal cloud model since it can depict the actual distribution of indicators and the conversion tendency of obtained grade among the classification grades. So the multi-dimensional connection cloud model can be viewed as a concise and effective alternative method for depicting multiple uncertainties of evaluation indicators in a unified way and transformation tendency among different grades. (3) It is just an attempt at the application of the multi-dimensional connection cloud model in the slope stability evaluation, but difficult to give the evaluation results related to safety factors as the limit equilibrium methods and numerical simulation. In addition, how to better couple indicator weight into the multi-dimensional connection cloud model and how to consider comprehensively the effects of the weight on the evaluation still need to clarify in future for a further accurate evaluation of slope stability.
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Declaration of Competing Interest The authors declare that there is no conflict of interests regarding the publication of this paper. Acknowledgments This research has been partially supported by the National Key Research and Development Program of China under Grant (No. 2017YFC1502405 and 2016YFC0401303) and the National Natural Sciences Foundation of China (41172274) is gratefully acknowledged. The authors also thank the reviewers for their thorough reviews and suggestions that helped to improve this paper. References [1] C.S. Juang, W.P. Gong, J.R. Martin, Q.S. Chen, Model selection in geological and geotechnical engineering in the face of uncertainty – does a complex model always outperform a simple model? Eng. Geol. 242 (2018) 184–196. [2] H.B. Wang, W.Y. Xu, R.C. Xu, Slope stability evaluation using back propagation neural networks, Eng. Geol. 80 (2005) 302–315. [3] H. Su, J. Li, J. Cao, Z. 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