Fuzzy Comprehensive Assessment Stability of Vegetated Slope with 3D Geomat Protection Using Cloud Model

2.1. Comprehensive Evaluation Index System

To evaluate the erosion stability of the vegetated eco-slope protected with 3D geomat, the influencing factors of the erosion stability should be determined at first. Many scholars have made theoretical analysis and experimental exploration to determine the influencing factors of the erosion stability. As mentioned above, the protection system is made up of 3 parts: the slope, the 3D geomat and the vegetation. Therefore, the parameters of the three parts have been used to assess the erosion stability of the protection system.

For the slope, according to Rough Set theory [12] and finite element analysis of ABAQUS [13],the main factors that affect the erosion stability are bulk density of soil, cohesion, friction angle, slope angle and ratio of pore pressure. Besides, slope length and compactness are also critical factors affecting erosion stability of artificial slope.

To the 3D geomat, the mechanism of slope protection of the 3D geomat is encapsulating the surface soil and preventing the soil particles flowing. Once there emerges tensile failure on the 3D geomat, the encapsulating function of protection system will be ineffective. Therefore, the main influencing factors are thickness and tensile strength of the 3D geomat [14]. Besides, based on Safety Factor Method [15], anchor spacing is another important factor affecting the design of similar engineering.

In the protection system, the main function of the vegetation are slope erosion prevention, reinforcement effect of roots and ecological restoration. Therefore, according to these functions, the main influencing factors are rate of vegetation coverage, roots development and planting suitability.

Precipitation condition is another important factor the erosion stability of the slope through the functions [3, 14] such as raindrop impact, runoff erosion, rainfall infiltration, especially at the initial establishment stage of the protection system. Therefore, the main influencing factors are rainfall intensity, rainfall duration and rainfall capacity.

Besides, some objective condition such as construction technology, maintenance method and environmental condition also affect the erosion stability of the slope. Especially, the factors such as temperature, humidity and PH value cannot be ignored for the vegetation restoration.

In a word, whether the evaluation indexes are too many or too few may lead to error of the evaluation. Based on previous research, seven categories of the factors are considered in this study: (a) characteristics of 3D geomat; (b) characteristics of soil; (c) characteristics of slope; (d) characteristics of vegetation; (e) characteristics of rainfall; (f) construction and maintenance; (g) environment. The factors and subfactors are as follows.

• (a).  The characteristics of 3D geomat: (1) layer number of the geomat B₁₁; (2) thickness of the geomat B₁₂; (3)  material of geomat B₁₃.
• (b).  The characteristics of soil: (1) cohesion of soil B₂₁; (2) friction angle of soil B₂₂; (3) bulk density of soil B₂₃;  (4) Permeability coefficient B₂₄; (5) Degree of compaction B₂₅; (6) pore pressure B₂₆.
• (c).  The characteristics of slope: (1) slope angel B₃₁; (2) slope length B₃₂; (3) lithology B₃₃
• (d). The characteristics of vegetation: (1) vegetation coverage B₄₁; (2) root development B₄₂; (3) planting  suitability B₄₃.
• (e).  The characteristics of rainfall: (1) rainfall intensity B₅₁; (2) rainfall duration B₅₂; (3) rainfall capacity B₅₃.
• (f).  Construction and maintenance: (1) construction technique B₆₁; (2) maintenance way B₆₂; (3) maintenance  condition B₆₃;
• (g).  Environment: (1) temperature B₇₁; (2) humidity B₇₂; (3) PH value B₇₃.

2.2. Comprehensive Evaluation Grade

The stability of slope is usually divided into five grades. Similarly, according to the stability of the slope, the erosion stability of the slope can also be divided into the following 5 grades: very stable (level I), stable (level II), basic stable (level III), unstable (level IV) and very unstable (level V). Besides, to maintain consistency with the evaluation grade, the evaluation index is divided into five grades, which is shown in the following Table 1.

2.3. Comprehensive Evaluation Model

Multiple-level comprehensive evaluation is a decision making technology that has been widely used for evaluation system with multiple factors and objectives. In this method, the evaluation index system is divided into a multiple-level structure, and the index weight of each layer is determined by AHP. The single index evaluation should be taken first and then the comprehensive evaluation is carried out to get the overall evaluation results by Calculating the single index evaluation result and its weight coefficient [16]. With the help of multilevel comprehensive evaluation, we can judge the erosion stability of the slope protected by the 3D geomat according to the following formula:

In the formula, W means weight coefficient of all the evaluation indexes. R means the single index evaluation result matrix. B means the final evaluation result matrix of comprehensive evaluation.

In fact, the comprehensive evaluation index system of erosion stability of vegetated slope protected with 3D geomat is complex and uncertain. The system not only contains the fuzziness, but also contains the uncertainty such as the randomness and the discreteness, which the multiple-level comprehensive evaluation does not take into consideration. Therefore, to evaluate the erosion stability more comprehensively and objectively, the Qualitative Rule Generator based on Cloud Model is introduced into the comprehensive evaluation.

3.1. The Theory of Cloud Model [17]

Cloud model, proposed by Academician Li Deyi in 1990s, is uncertainty conversion model between the qualitative concept and the quantitative data, which mainly reflects uncertainty in the objective world and knowledge: fuzziness and randomness [12]. The cloud model integrates the two characteristics and constitutes the mutual mapping between the qualitative concept and quantitative data.

Based on the probability theory and fuzzy mathematics, Cloud Model expresses a concept with three numeral characteristic [18]: Ex (expected value), En(entropy) and He(hyper entropy), which combine the fuzziness, randomness and discreteness organically and realizes the uncertainty conversion between qualitative knowledge and quantitative data. Ex (expected value)is the most representative value of the qualitative concept and reflects the central position the cloud model; En (entropy), the uncertainty measure of the qualitative concept, reflects the degree of discreteness of cloud droplets which represent the randomness and fuzziness of qualitative concept; He(hyper entropy), determined by randomness and uncertainty of entropy, is an uncertainty measure of entropy, which reveals the relationship between the fuzziness and randomness.

Definition 3.1: Suppose U is a quantitative theory which is expressed by a certain value and C is a qualitative concept on U. If the quantitative value x is a random realization of the qualitative concept C and the certainty degree, μ (x) ,of x is a random number with stable tendency for C,which is to say:

In this case, the distribution of X in the domain U is cloud. Each value of x is called a cloud droplet. Especially, if the number x satisfies x = R_N (E_x, y ) (R_N means Gaussian distribution) while y = R_N (E_n, H_e ) and the certainty degree of x on U is , the cloud is called Normal Cloud Model. Normal Cloud Model is the most widely used cloud model and it has been proved to be generic [12]. The following picture show a cloud model that represents a qualitative concept (Fig. 1).

3.2. Qualitative Rule Generator of Cloud Model [18]

Definition 3.2: Forward Normal Cloud Generator (FCG) is a mapping from the qualitative concept to the quantitative data, which generates the cloud droplets according to the numerical characteristics (Ex, En, He).

Definition 3.3: Front Member Cloud Generator (FMCG) refers to the Forward Cloud Generator that generates membership degree distribution of a given point according to the point and numerical characteristics of cloud model.

Definition 3.4: Behind Member Cloud Generator (BMCG) refers to the Forward Cloud Generator that generates cloud droplet distribution according to the given membership degree and numerical characteristics of cloud model.

Definition 3.5: Backward Normal Cloud Generator (BCG) is a mapping from the quantitative to the qualitative, which converts a certain number of accurate data into a qualitative concept represented by numerical characteristics (Ex, En, He).

Definition 3.6: Qualitative Rule Generator (QRG), important application of Cloud Model theory, is consisted of Front Member Cloud Generator and Behind Member Cloud Generator to realize the uncertain conversion between the qualitative knowledge and quantitative data.

Qualitative Rule Generator based on Cloud Model may have multiple rules and multiple conditions. For the evaluation system of erosion stability, the article chooses the Qualitative Rule Generator with single condition and single rule. Its conceptual diagram and algorithm is shown in the following part (Fig. 2).

Algorithm 3.1

Input: Three numeral characteristics (Ex_A, En_A He_A ) of front member A and the quantitative value x_A, three numeral characteristics (Ex_B, En_B He_B ) of behind member B.

Output: cloud droplet x_B of behind member and its membership grade μ.

(1) Generate a random number En'_A that meets Rn( En _A, He _A );

(2) Calculate the membership grade μ:

(1)

(3) Generate a random number En'_B that meets Rn( En_B, He_B );

(4) If the inputted value activates the rising phase of the front member, the behind member choose the rising phase: If x_A ≤ Ex_A, then ;

(5) If the inputted value activates the decline phase of the front member, the behind member choose the decline phase: If x_A > Ex_A, then ;

After the Cloud Model theory is applied into the erosion stability evaluation system, this study will take the uncertainty such as fuzziness and randomness of the evaluation system into consideration, making the result of the evaluation more objective and accurate.

3.3. Improved AHP with Cloud Model [19]

Analytic Hierarchy Process(AHP), put forward by American operation research scientist Satty.T.L [20], is a multiple criteria decision analysis method. It is flexible and can combine the qualitative and quantitative analysis. The traditional AHP use classical Satty scale (1-9) to describe the relative importance of different factors, and the weight of each index is calculated by constructing the comparison judgment matrix. However, using accurate number to express the relative importance may ignore the uncertainty of the relative importance. Besides, there may has randomness and fuzziness in the process of determining the relative importance value by the experts. Considering these factors, this study introduces the Cloud Model into the AHP by using the Backward Normal Cloud Generator to calculate the relative importance and optimize the the result of weight calculation .

(2)

(3)

(4)

(1) Invite N experts to judge the relative importance of the index. With the help of the Backward Normal Cloud Generator, we can generate a decision cloud about the judgment of experts.

(2) Build the judgment matrix of relative importance based on cloud model. First, the judgment matrix of relative importance is obtained according to group decision-making:

On the diagonal, the hyper entropy and the entropy of the cloud model is 0 and the expected value is 1. A_ij is the reciprocal of A_ji, and according to the reciprocal algorithm of cloud model they satisfy the following formula:

(5)

(3) Calculate the relative weight coefficient of the evaluation elements according to root method: . According to the multiplication of the cloud model, the calculation results of are listed as the following:

(6)

(7)

(8)

(4) Take consistency test. The consistency test method for the three numerical characteristics of cloud model is consistent. Usually, we only take consistency test for the numerical characteristic Ex. The index of conformance test C.I. is

The index of consistency text is consistent with the classical AHP. C.R.,the proportion of conformance test, satisfies the following formula: C.R. < 0.1.

3.4. The Procedure of Comprehensive Evaluation

(1) Classify the indexes of the comprehensive evaluation system according to the classification of comment set. Normalize [21] the variables with different dimensionless and predetermine the range of variation for the different grades.

For the influencing factor that the bigger the better, data normalization can be implemented as:

(9)

For the influencing factor that the smaller the better, data normalization can be implemented as:

(10)

(2) Format the cloud model of the comments set. According to the range of the comments set and the definition of the score of comments set, the three numerical characteristics of the cloud model are determined [22].

a) As mentioned above, E_X is the expected value of sample points that meet normal distribution, therefore,

(11)

In the formula, Ex_{i, j} means the expected value of factor ‘i’ of grade ‘j’. x¹_{i, j} means the maximum value and x²_{i, j} means minimum value in the section of different grades.

b) In order to show the boundary fuzziness of adjacent grades, the membership of the two adjacent grades is equal. Therefore, the En (entropy)is determined by the following equation:

(12)

(13)

c) The He(hyper entropy)is usually determined by experience. The value of He(hyper entropy) is usually less than 0.05. The paper make the value as He = 0.01.

(3) Evaluate the single factor of the evaluation system by qualitative rules generator based on cloud model according to the uncertain reasoning method.

(4) Determine the weight coefficient of the index of the evaluation system. The article selects AHP improved with Cloud Model, considering the fuzziness and randomness of the subjective factors in the evaluation process.

(5) Calculate the score of each evaluation index with its weight coefficient, then we can obtain the final score of the comprehensive evaluation, further we can obtain the evaluation grade of the erosion stability.

4.1. The Classification and the Normalization of the Evaluation Index

In this paper, the erosion stability of the slope protected by 3D geomat is divided into 5 grades: Very stable (grade I), stable (gradeII), basically stable (gradeIII), unstable (gradeIV), very unstable (grade V). In order to keep the consistency of the forward cloud model and backward cloud model, classify the evaluation index into 5 grades and normalize the date of the evaluation index. The result of the classification and normalization is given in the Table 3.

4.2. Build the Front Member Cloud Model and Behind Member Cloud Model

According to the variation range of the classification for each index and the formula (3-9) ~ (3-12), the front member cloud model is developed. For example, for the index ‘the thickness of 3D geomat’, the cloud model (E_x,E_n,H_e) are: (0.975, 0.05/2.355, 0.01), (0.8, 0.3/2.355, 0.01), (0.5, 0.3/2.355, 0.01), (0.2, 0.3/2.355, 0.01), (0.025, 0.05/2.355, 0.01) corresponding to its qualitative comment ‘very stable, stable, generally stable, unstable and very unstable’.

For the behind member cloud model, assuming that the better the index is, the higher the score is. If the full mark is 100, the cloud model are: (90, 20/2.355, 0.01), (70, 20/2.355, 0.01), (50, 20/2.355, 0.01), (30, 20/2.355, 0.01), (10, 20/2.355, 0.01) [23] corresponding to its qualitative comment ‘very high, high, medium, low and very low’.

4.3. Determine Weight Coefficient

The study uses AHP developed with Cloud Model (CM-AHP) to calculate the weight coefficient of the evaluation index to make the calculation results more objective. The calculation procedure is shown in the chapter 3.3. The article take the index B₁ “the characteristics of the 3D geomat “ as an example to show the calculation procedure, the result is given in the Table 4. And the calculation results of the weights of all indexes are given in the Table 5.

Table 4.

B	B11   B12   B13
B11 B11 B13	(1,0,0) (4, 0.836, 0.050) (0.260, 0.054, 0.011) (0.25, 0.053, 0.003) (1, 0, 0) (0.145, 0.018, 0.003) (3.846, 0.799, 0.163) (6.897, 0.856, 0.143) (1,0,0)	(1.013,0.674, 0.358) (0.331, 0.207, 0.095) (2.982, 1.858,1.078)	(0.234, 0.246, 0.234) (0.077, 0.076, 0.062) (0.689, 0.678, 0.704)

4.4. Single Factor Evaluation

According to the uncertainty reasoning based on cloud model, the qualitative language description of evaluation index is quantified as the certain score with the help of Qualitative Rule Generator based on cloud model. The calculation results are provided in the Table 5.

4.5. The Comprehensive Evaluation

Multiply the score of the evaluation index with the weight coefficient to get comprehensive evaluation score. The score of comprehensive evaluation based on cloud model is 58.921 and the grade of the erosion stability is grade III. To prove the feasibility of the cloud model in comprehensive evaluation, we use the fuzzy matter-element [24] to evaluate the same example. The result shows consistent with the result of the comprehensive evaluation using cloud model.