Analysis of landslide risk based on fuzzy extension analytic hierarchy process

Abstract

Landslide hazard assessment generally involves multiple factors and indexes. Based on the theory of extenics with matter-element concept, a method applying the fuzzy extension analytic hierarchy process (EAHP) method is proposed for the comprehensive evaluation of landslide hazard. It can quantitatively show the relative important degree between indexes with the extension interval number. Taking the geological hazards of actual landslide as an example, based upon geological survey and characteristic analysis of landslide field, the author selected 10 influencing factors to analyze the weight of the evaluation index of each level. Through the extension interval judging matrix of evaluation index constructed by the experts, the weight of the indexes was identified and the landslide risk was evaluated step by step with membership function of fuzzy set. The correlation degree of the maximum value was calculated as 0.472, the landslide risk was valued as grade 4, in an unstable state, and at the same time, some feasible measures were put forward to control landslide. The research as described in the paper shows that the fuzzy EAHP method is valid for landslide hazard evaluation.

Keywords

Landslide fuzzy extension theory analytic hierarchy process (AHP)weight

1 Introduction

Landslide hazard assessment generally involves a comprehensive evaluation of multi-factors and multi-indexes, and the evaluation result will directly affect project safety, economic benefit and social benefits. There are many methods of research on landslide hazards, including quantitative analysis, qualitative analysis, numerical analysis and uncertainty analysis, etc. In the 1970s, the methods of landslide stability and dangerous zoning were widely used [1, 2]. Disaster assessment developed gradually from the qualitative way to semi-quantitative and quantitative way due to development of computer technology and spatial technology on carried sensing. Based on the decision support system, a variety of geological disasters on the sensitivity of the disaster factors and its own vulnerability are summed up [3]. As to landslide hazard risk assessment, there are many influencing factors, which will have a certain impact the rating system. It indicates that the index system has the hierarchy, fuzziness and difference of contribution degree [4 –7]. Many new methods have been put forward to evaluate and analyze uncertainty of disaster risks of urban and mining areas in recent years, including fuzzy comprehensive evaluation method [8], fuzzy multicriteria prioritization [9], fuzzy logic and GIS tools [10], grey clustering evaluation method [11], neural network evaluation method [12] and so on. These methods are different in focus, but all combine qualitative with quantitative risk evaluation. Despite certain effects, there are still some shortcomings and limitations for the methods.

The complexity of the landslide cannot be analyzed in a simple way, and no method can be applied to solve all the problems in landslide risk assessment. Extenics is a cross-sectional discipline founded by Prof. W. Cai in 1983, which includes the characteristics of formalization, logic and mathematics [13]. Application of extension theory in slope is relatively late; L. Wang applied the theory of extension to the stability analysis of rock slope [14]. AHP mechanism proposed by Satty [15, 16] was known as an effective tool to support the multi-attribute decision-making [15, 16]. H. Shahabi drew the zoning maps of slide hazard using the two methods analytical hierarchy process (AHP) and frequency ratio (FR) incorporating and evaluated their performance [17]. P. M. Ozfirat presented an application on roadheader selection through integration of fuzzy analytic hierarchy process and multi-objective fuzzy goal programming [18]. H. K. Dong applied fuzzy analytic hierarchy process to technology credit scorecard [19]. K.G. Li studied the relative importance of the slope stability evaluation index based on the simple correlation function in extenics, and thus put forward an evaluation method of slope stability [20].

In this paper, the author applied fuzzy extension analytic hierarchy process (EAHP) to the landslide hazard evaluation and select 10 indexes to evaluate the landslide risk, according to the characteristics of right slope of the Suzhou expressway.

2 Establishment of evaluation model based on extension analytic hierarchy process

Extension analytic hierarchy process (EAHP) theory introduces the concept of matter element R = (N, c, v), in which things to be evaluated N, their characteristics c and the specific value of the features v can be connected organically, to truly reflect the dialectical relationship between quality and quantity of things.

2.1 Determination of the classical field, the segment field and the evaluating matter element

Determination of the classical field

The classical field can be expressed in the form of matter element [13].

$\begin{matrix} R_{j} & = & (N_{j}, c_{i}, v_{ji}) = [\begin{matrix} N_{j} & c_{1} & v_{j 1} \\ c_{2} & v_{j 2} \\ ⋮ & ⋮ \\ c_{n} & v_{jn} \end{matrix}] \\ = & [\begin{matrix} N_{j} & c_{1} & < a_{j 1}, b_{j 1} > \\ c_{2} & < a_{j 2}, b_{j 2} > \\ ⋮ & ⋮ \\ c_{n} & < a_{jn}, b_{jn} > \end{matrix}] \end{matrix}$ (1)

Where N_j means divided j grades, c_i is the ith evaluation index, and v_ji is the range of c_i values on the N_j level, that is, the classical field, and v_ji =< a_ji, b_ji >.

Determination of segment field

The segment field can be expressed as a primitive element form [13].

$\begin{matrix} R_{p} & = & (N, c_{i}, v_{pi}) = [\begin{matrix} N & c_{1} & v_{p 1} \\ c_{2} & v_{p 2} \\ ⋮ & ⋮ \\ c_{n} & v_{pn} \end{matrix}] \\ = & [\begin{matrix} N & c_{1} & < a_{p 1}, b_{p 1} > \\ c_{2} & < a_{p 2}, b_{p 2} > \\ ⋮ & ⋮ \\ c_{n} & < a_{pn}, b_{pn} > \end{matrix}] \end{matrix}$ (2)

In the equation, N means all evaluation grades, v_pi is the range of c_i values under N conditions, where i is the number of evaluation indicators (i = 1, 2, …, n).

Determination of the evaluating matter element

The matter element to be evaluated can be expressed as [13], $R_{0} = [\begin{matrix} N_{0} & c_{1} & v_{1} \\ c_{2} & v_{2} \\ ⋮ & ⋮ \\ c_{n} & v_{n} \end{matrix}]$ (3)

In the equation, N₀ is the matter to be evaluated and v_i is the value of N₀ on indicator c_i.

2.2 Definition of elementary correlation function

Setting satisfactory interval X₀ =< a₀, b₀ > and acceptable interval X =< a, b >; so X ⊇ X₀. Let x ∈ X, the optimal correlation function [13] at the midpoint of the interval is defined as

$k (x) = {\begin{matrix} \frac{ρ (x, X_{0})}{D (x, X_{0}, X)} & D (x, X_{0}, X) \neq 0 & x \in X \\ - ρ (x, X_{0}) + 1 & D (x, X_{0}, X) = 0 & x \in X_{0} \\ 0 & D (x, X_{0}, X) = 0 & x \notin X_{0}, x \in X \end{matrix}$ (4)

Where, D (x, X₀, X) = ρ (x, X) - ρ (x, X₀) ; ρ (x, X₀) is the extension distance of point x and interval X₀, that is $ρ (x, X_{0}) = | x - \frac{a + b}{2} | - \frac{b - a}{2}$ (5)

2.3 Improved Analytic Hierarchy Process (AHP) to solve weights

AHP is a decision-making method for qualitative and quantitative analysis by disintegrating indicators related to the research object into such multiple levels as programs, guidelines, goals and so on.

The result accuracy of this method depends on the construction of judgment matrix. Once the judgment matrix is constructed incorrectly, a series of subsequent calculations based on the judgment matrix will be meaningless. Therefore, the importance of each factor can be sorted by means of the simple correlation degree of the extension theory, which provides the basis for rational construction of judgment matrix, thus making the calculation of the weight more accurate.

(1) Judging the relative importance of each index based on simple correlation degree.

The calculation of the simple correlation degree can determine the maximum correlation degree K_ijmax of each index relative to each level interval. K_ijmax means that the relationship between the index i and the evaluation grade j_max is the largest. The higher the grade of the index is, the worse the stability of the system is, and the more the weight should be adopted.

(2) Constructing judgment matrix to calculate weight.

Through the calculation of the correlation degree of each evaluation index, the importance of the index is sorted, and then the judgment matrix is constructed, and the weight of the index is solved by the AHP.

3 Case analysis

In line with the above principles and methods, the right side landslide on the H_A - 4 mark K₁₉ + 700 ∼ K₁₉ + 820 section of Suzhou highway was selected as the research object. Its risk assessment was carried out based on the field investigation data and the previous studies.

This slope is located on the southeast slope of Yuping Mountain. The average slope gradient is about 12°. Its surface is sparsely planted, mainly with masson pines. On its east side, there are some bamboos and the surface is messily covered with little turf. While on the west side, the terrain is complex. There is a spring in the middle, with water in it all the year. The surface is covered with vegetations, and a few sabre trees and drunkard woods. The stratum is divided into two layers, the upper layer is composed of soil containing rock fragments, and the lower one is clay within gravel soil. The upper formation contains many small gravels, While the depth increased, the number of gravel decreases but the block diameter increases. Local gravel content is up to 40%. The layer is not sorted and the structure is poor, the maximum thickness is 16.10 m. The soil of clay with gravel is more compact, good structured, mainly with Piedmont slope deposits.

The slope soil is loose, the surface fractures are developed, and the permeability of the slope soil is good. Slope soil moisture is small in sunny days, while in rainy days, landslide soil is saturated, with groundwater overflow in the middle and bottom, and the groundwater infiltration line is at the same level as the hillside. The vegetation coverage is low.

3.1 Selection and quantification of evaluation factors

3.1.1 Selection of evaluation factors

The evaluation index should be quantified when the evaluation model is used to evaluate the index. In general, the evaluation indexes are divided into two categories, one is continuous index (quantitative index) and the other is discrete index (qualitative index). Firstly, the qualitative and semi quantitative indexes are assigned values to quantitative indexes, so as to realize the unification of the indexes. According to the field investigation and the occurrence and evolution mechanism of landslide disasters, 10 main factors are selected as the research objects. Domestic and foreign scholars have different standards in dealing with the classification of landslide hazards. In this paper, the slope stability level is classified into such five categories, based on previous researches, as stable, basically stable (the vegetation coverage is good and no large cracks in the slope), potentially unstable (many cracks in the slope may develop into landslide surface), unstable (there is a landslide surface but the slip surface is not obvious) and extremely unstable (landslide has a clear slip surface), respectively represented by the symbol for I, II, III, IV and V.

The evaluation criteria are shown in Table 1.

Table 1
Evaluation indexes and quantitative criteria of landslide hazards

Slope grade I II III IV V

B₁ Rock C₁ Slope angle (°) [0,15) [15,25) [25,45) [45,65) [65,90)

characteristics C₂ Ground water No water Less water Medium water Abundant Extremely abundant

C₃ Slope height (m) <30 30–60 60–100 100–200 >200

A Factors affecting C₄ Lithology Reverse slope, Reverse slope, Skew slope Consequent slope, Consequent slope,

slope stability rock slope semi-rocky slope rock slope semi-rocky slope

B₂ Internal C₅ Angle of internal friction (°) >35 35–28 28–21 21–14 <14

factors C₆ Cohesion (MPa) >0.25 0.25–0.15 0.15–0.1 0.1–0.05 <0.05

C₇ Bulk density φ (kN/m³) >32 32–26 26–22 22–18 <18

B₃ Environmental C₈ Annual rainfall (mm) <400 400–700 700–1000 1000–1500 >1500

factor C₉ Vegetation cover (% ) [80,100) [60,80) [40,60) [20,40) [10,20)

C₁₀ Artificial destruction No tiny weak Strong Very strong

	Slope grade	I	II	III	IV	V
	B₁ Rock	C₁ Slope angle (°)	[0,15)	[15,25)	[25,45)	[45,65)	[65,90)
	characteristics	C₂ Ground water	No water	Less water	Medium water	Abundant	Extremely abundant
		C₃ Slope height (m)	<30	30–60	60–100	100–200	>200
A Factors affecting		C₄ Lithology	Reverse slope,	Reverse slope,	Skew slope	Consequent slope,	Consequent slope,
slope stability			rock slope	semi-rocky slope		rock slope	semi-rocky slope
	B₂ Internal	C₅ Angle of internal friction (°)	>35	35–28	28–21	21–14	<14
	factors	C₆ Cohesion (MPa)	>0.25	0.25–0.15	0.15–0.1	0.1–0.05	<0.05
		C₇ Bulk density φ (kN/m³)	>32	32–26	26–22	22–18	<18
	B₃ Environmental	C₈ Annual rainfall (mm)	<400	400–700	700–1000	1000–1500	>1500
	factor	C₉ Vegetation cover (% )	[80,100)	[60,80)	[40,60)	[20,40)	[10,20)
		C₁₀ Artificial destruction	No	tiny	weak	Strong	Very strong

3.1.2 Standardization of risk evaluation index

The binary relationship between the evaluation index and the quantitative is shown in Table 2 rand it provides classification standards. The units of each index are not consistent rwhich makes it necessary to eliminate the dimensional effect of the index variables.

Table 2
Classification of landslide hazard evaluation indexes

Evaluation index Stable Basically stable Potentially instable Instable Extremely unstable

Slope angle [0, 0.17) [0.17, 0.28) [0.28, 0.50) [0.50, 0.72) [0.72, 1)

Groundwater [0, 0.20) [0.20, 0.40) [0.40, 0.60) [0.60, 0.80) [0.80, 1)

Slope height [0, 0.1) [0.1, 0.2) [0.2, 0.33) [0.33, 0.667) [0.667, 1)

Lithology [0, 0.20) [0.20, 0.40) [0.40, 0.60) [0.60, 0.80) [0.80, 1)

Angle of internal friction [0, 0.22) [0.22, 0.378) [0.378, 0.533) [0.533, 0.689) [0.689, 1)

Cohesion [0, 0.167) [0.167, 0.5) [0.5, 0.667) [0.667, 0.833) [0.833, 1)

Bulk density [0, 0.2) [0.2, 0.35) [0.35, 0.45) [0.45, 0.55) [0.55, 1)

Annual maximum rainfall [0, 0.16) [0.16, 0.28) [0.28, 0.4) [0.4, 0.6) [0.6, 1)

Vegetation cover [0, 0.20) [0.20, 0.40) [0.40, 0.60) [0.60, 0.80) [0.80, 1)

Human influence [0, 0.20) [0.20, 0.40) [0.40, 0.60) [0.60, 0.80) [0.80, 1)

Evaluation index	Stable	Basically stable	Potentially instable	Instable	Extremely unstable
Slope angle	[0, 0.17)	[0.17, 0.28)	[0.28, 0.50)	[0.50, 0.72)	[0.72, 1)
Groundwater	[0, 0.20)	[0.20, 0.40)	[0.40, 0.60)	[0.60, 0.80)	[0.80, 1)
Slope height	[0, 0.1)	[0.1, 0.2)	[0.2, 0.33)	[0.33, 0.667)	[0.667, 1)
Lithology	[0, 0.20)	[0.20, 0.40)	[0.40, 0.60)	[0.60, 0.80)	[0.80, 1)
Angle of internal friction	[0, 0.22)	[0.22, 0.378)	[0.378, 0.533)	[0.533, 0.689)	[0.689, 1)
Cohesion	[0, 0.167)	[0.167, 0.5)	[0.5, 0.667)	[0.667, 0.833)	[0.833, 1)
Bulk density	[0, 0.2)	[0.2, 0.35)	[0.35, 0.45)	[0.45, 0.55)	[0.55, 1)
Annual maximum rainfall	[0, 0.16)	[0.16, 0.28)	[0.28, 0.4)	[0.4, 0.6)	[0.6, 1)
Vegetation cover	[0, 0.20)	[0.20, 0.40)	[0.40, 0.60)	[0.60, 0.80)	[0.80, 1)
Human influence	[0, 0.20)	[0.20, 0.40)	[0.40, 0.60)	[0.60, 0.80)	[0.80, 1)

In regard to the index application where the smaller the risk assessment, the more favorable the index, the equation used is as follows [21], $y_{ij}^{'} = \frac{y_{ij} - y_{i min}}{y_{i max} - y_{i min}}$ (6)

In regard to the index application where the greater the risk assessment, the more favorable the index, the equation used is as follows [21], $y_{ij}^{'} = \frac{y_{i max} - y_{ij}}{y_{i max} - y_{i min}}$ (7)

In the equation, $y_{ij}^{'}$ is range data, y_ij is samples data, y_imax is the maximum of the ith line data and y_imin is the minimum of the ith line data.

Since there is no maximum value of the rainfall, cohesion and bulk density in the above evaluation index, in order to facilitate data processing, interval will be divided according to the actual situation. Taking rainfall as an example, a very high risk is normalized to the [1500, 2500] mm interval. As to the slope height, a very high risk is normalized to the [200, 300] m interval. Other indexes, such as the friction angle, cohesion and bulk density, are normalized to low risk intervals [35, 45], [0.25, 0.35] and [32, 40]. In addition, such discrete variables as the stratum lithology and the artificial destruction are assigned according to the level of risk as follows: [0, 0.2], (0.2, 0.4], (0.4, 0.6], (0.6, 0.8], [0.8, 1.0).

Taking the slope angle as an example rthe smaller the risk assessment rthe more favorable the index rso the Equation (6) is used to calculate. The maximum value is 90 and the minimum value is 0. $\begin{matrix} y_{11}^{'} & = & \frac{y_{ij} - y_{i min}}{y_{i max} - y_{i min}} = \frac{0 - 0}{90 - 0} = 0 \\ y_{12}^{'} & = & \frac{y_{ij} - y_{i min}}{y_{i max} - y_{i min}} = \frac{15 - 0}{90 - 0} = 0.17 \end{matrix}$

For the angle of internal friction rthe greater the risk assessment rthe more favorable the index rso the Equation (7) is used to calculate. The maximum value is 45 and the minimum value is 0. $\begin{matrix} y_{51}^{'} & = & \frac{y_{i max} - y_{ij}}{y_{i max} - y_{i min}} = \frac{45 - 45}{45 - 0} = 0 \\ y_{52}^{'} & = & \frac{y_{i max} - y_{ij}}{y_{i max} - y_{i min}} = \frac{45 - 35}{45 - 0} = 0.22 \\ y_{53}^{'} & = & \frac{y_{i max} - y_{ij}}{y_{i max} - y_{i min}} = \frac{45 - 28}{45 - 0} = 0.378 \end{matrix}$

Other calculations are the same, and so on.

The index values, after standardized, are shown in Table 2.

3.2 Determination of membership

Based on the membership function, the fuzzy set is established, and the concept of fuzzy logic is expressed quantitatively by using the fuzzy set. The experience assignment methods are applied to deal with the membership degree of discrete index. As to the solution of membership degree of continuous index, the gradient membership function is established by referring to the research results of domestic and foreign scholars and considering the local landslide disaster conditions. The membership function of the index with different risk levels is as follows [22]: $\begin{matrix} u_{1} & = & {\begin{matrix} 0 & x > a_{2} \\ \frac{a_{2} - x}{a_{2} - a_{1}} & a_{1} < x < a_{2} \\ 1 & x \leq a_{1} \end{matrix} \\ u_{2} & = & {\begin{matrix} 0 & x \geq a_{3} \\ \frac{x - a_{0}}{a_{1} - a_{0}} & a_{0} \leq x < a_{1} \\ 1 & a_{1} \leq x < a_{2} \\ \frac{a_{3} - x}{a_{3} - a_{2}} & a_{2} \leq x < a_{3} \end{matrix} \\ u_{3} & = & {\begin{matrix} 0 & x < a_{1}, x > a_{4} \\ \frac{x - a_{1}}{a_{2} - a_{1}} & a_{1} \leq x < a_{2} \\ 1 & a_{2} \leq x < a_{3} \\ \frac{a_{4} - x}{a_{4} - a_{3}} & a_{3} \leq x \leq a_{4} \end{matrix} \\ u_{4} & = & {\begin{matrix} 0 & x < a_{2}, x > a_{5} \\ \frac{x - a_{2}}{a_{3} - a_{2}} & a_{2} \leq x < a_{3} \\ 1 & a_{3} \leq x < a_{4} \\ \frac{a_{5} - x}{a_{5} - a_{4}} & a_{4} \leq x \leq a_{5} \end{matrix} \\ u_{5} & = & {\begin{matrix} 0 & x < a_{3} \\ \frac{a_{4} - x}{a_{4} - a_{3}} & a_{3} \leq x < a_{4} \\ 1 & x \geq a_{1} \end{matrix} \end{matrix}$ (8)

Where, a₁, a₂, a₃, a₄ and a₅ are the limit value of each grade standard interval.

3.3 Model establishment

The physical and mechanical parameters of the slope obtained through field investigation and experiments are shown in Table 3.

Table 3
Physical and mechanical properties of soil

Parameters Moisture Unit weight Saturated unit weight Specific gravity G_s Void ratio e

content ω γ (kN/m³) γ_sat (kN/m³)

1 23 19.6 19.8 2.75 0.726

2 21 20.0 20.2 2.76 0.664

Parameters Degree of Coefficient of Compression Peak strength Residual strength

saturation S_r compressibility a_1 - 2 MPa^–1 modulus E_s MPa C (kPa), φ (°) C (kPa), φ (°)

1 99 0.34 5.2 26.6, 8.8 16.9, 4.7

2 87 0.31 6.5 33.8, 13.8 25.0, 9.8

Parameters	Moisture	Unit weight	Saturated unit weight	Specific gravity G_s	Void ratio e
1	23	19.6	19.8	2.75	0.726
2	21	20.0	20.2	2.76	0.664
Parameters	Degree of	Coefficient of	Compression	Peak strength	Residual strength
	saturation S_r	compressibility a_1 - 2 MPa^–1	modulus E_s MPa	C (kPa), φ (°)	C (kPa), φ (°)
1	99	0.34	5.2	26.6, 8.8	16.9, 4.7
2	87	0.31	6.5	33.8, 13.8	25.0, 9.8

The soil parameters of the slope are obtained through the geotechnical tests. The first line is the soil parameters of upper layer (landslide body); the second line is the soil parameters of slop bedsoil. In first line, the natural unit weight γ= 19.6 kN/m³, the saturated unit weight γ_sat= 19.8 kN/m³, the cohesion C = 16.9 kPa, the angle of internal friction φ= 4.7°. In this example, the slope angle β= 21°, the slope height H = 20 m. It takes 0.65 for obvious stratification in soil, which makes it easy to form landslides, 0.9 for rich groundwater of 1300 mm annual rainfall, 0.8 for sparse vegetation coverage, and 0.7 for unreasonable artificial activities. The original data is standardized as shown in Table 4.

Table 4

Standardization of original data

Serial number	1	2	3	4	5	6	7	8	9	10
Evaluation index	Slope	Ground	Slope	Lithology	Angle of internal	Cohesion	Bulk	Annual	Vegetation	Artificial
	angle	water	Height		friction		density	rainfall	cover	destruction
Initial data	21	V	20	IV	4.7	16.9	19.8	1300	IV	V
Standardized data	0.233	0.9	0.07	0.65	0.9	0.94	0.505	0.52	0.8	0.7

Put the standardized data into Equation (8) of the membership degree calculation to calculate membership of each index. Taking the bulk density as an example, the calculation process is as follows. $\begin{matrix} u_{1 (x = 0, 505)} & = & 0 \\ u_{2 (x = 0, 505)} & = & 0 \\ u_{3 (x = 0, 505)} & = & \frac{0.55 - 0.505}{0.55 - 0.45} = 0.45 \\ u_{4 (x = 0, 505)} & = & 1 \\ u_{5 (x = 0, 505)} & = & \frac{0.55 - 0.505}{0.55 - 0.45} = 0.45 \end{matrix}$ (9)

Similarly, we can calculate the membership degree of the other index corresponding risk grade as shown in Table 5.

Table 5

Calculation of membership degree of landslide evaluation index

Serial number	Evaluating indicator	Standard value	Risk grade
			I	II	III	IV	V
1	Slope angle	0.23	0.43	1	0.57	0	0
2	Groundwater	0.9	0	0	0	0.5	1
3	Slope height	0.07	1	0.67	0	0	0
4	Lithology	0.65	0	0	0.7	1	0.7
5	Angle of internal friction	0.9	0	0	0	0.32	1
6	Cohesion	0.94	0	0	0	0.34	1
7	Bulk density	0.51	0	0	0.45	1	0.45
8	Annual rainfall	0.52	0	0	0.4	1	0.4
9	Vegetation cover	0.7	0	0	0.5	1	0.5
10	Artificial destruction	0.7	0	0	0.5	1	0.5

3.4 Establishment of judgment matrix based on extenics

3.4.1 Calculation of correlation degree

According to the above description, 10 indexes are calculated. Taking the slope angle as an example, the calculation process is as follows: $\begin{matrix} k_{1} & = & \frac{b_{1} - x}{b_{1} - a_{1}} = \frac{0.17 - 0.233}{0.17} = - 0.37 \\ k_{2} & = & \frac{x - a_{2}}{b_{2} - a_{2}} = \frac{0.233 - 0.17}{0.28 - 0.17} = 0.57 \\ k_{3} & = & \frac{x - a_{3}}{b_{3} - a_{3}} = \frac{0.233 - 0.28}{0.50 - 0.28} = - 0.21 \\ k_{4} & = & \frac{x - a_{4}}{b_{4} - a_{4}} = \frac{0.233 - 0.50}{0.72 - 0.50} = - 1.21 \\ k_{5} & = & \frac{x - a_{5}}{b_{5} - a_{5}} = \frac{0.233 - 0.72}{1 - 0.72} = - 1.74 \end{matrix}$

The calculation process of the other index is the same. Results are shown in Table 6.

Table 6
Correlation degree value of each risk grade

Evaluation Stable Basically Potentially Instable Extremely

index stable instable unstable

Slope angle –0.37 0.57 –0.21 –1.21 –1.74

Groundwater effect –4.5 –2.5 –1.5 –0.5 0.5

Slope height 0.7 –0.3 –1 –0.76 –1.82

Lithology –2.25 –1.25 –0.25 0.25 –0.75

Angle of internal friction –3.09 –3.25 –2.47 –1.31 0.68

Cohesion –4.53 –1.33 –1.69 –0.69 0.65

Bulk density –1.55 –0.9 –0.6 0.6 –0.09

Annual rainfall –3.09 –2 –1 0.6 –0.98

Vegetation cover –2.5 –1.5 –0.5 0.5 –0.5

Human influence –2.5 –1.5 –0.5 0.5 –0.5

Evaluation	Stable	Basically	Potentially	Instable	Extremely
Slope angle	–0.37	0.57	–0.21	–1.21	–1.74
Groundwater effect	–4.5	–2.5	–1.5	–0.5	0.5
Slope height	0.7	–0.3	–1	–0.76	–1.82
Lithology	–2.25	–1.25	–0.25	0.25	–0.75
Angle of internal friction	–3.09	–3.25	–2.47	–1.31	0.68
Cohesion	–4.53	–1.33	–1.69	–0.69	0.65
Bulk density	–1.55	–0.9	–0.6	0.6	–0.09
Annual rainfall	–3.09	–2	–1	0.6	–0.98
Vegetation cover	–2.5	–1.5	–0.5	0.5	–0.5
Human influence	–2.5	–1.5	–0.5	0.5	–0.5

Based on correlation grade (or risk grade) and correlation degree, the importance of the above indicators is sorted, shown in Table 7.

Table 7

Rank of the corresponding indexes

Index	Slope angle	Groundwater effect	Slope height	Lithology	Angle of internal friction
Grade	II	V	I	IV	V
Correlation degree	0.57	0.5	0.7	0.25	0.68
Index	Cohesion	Bulk density	Annual rainfall	Vegetation cover	Human factor
Grade	V	IV	IV	IV	IV
Correlation degree	0.65	0.6	0.6	0.5	0.5

3.4.2 Construction and evaluation of judgment matrix

(1) Construction of judgment matrix of index layer

The judgment matrix is constructed according to the correlation degree. The total order and consistency test of judgment matrix: $\begin{matrix} R = [\begin{matrix} 1 & 1 / 6 & 2 & 1 / 4 & 1 / 6 & 1 / 7 & 1 / 3 & 1 / 3 & 1 / 5 & 1 / 5 \\ 6 & 1 & 9 & 3 & 1 & 1 & 4 & 4 & 2 & 2 \\ 1 / 2 & 1 / 9 & 1 & 1 / 6 & 1 / 8 & 1 / 9 & 1 / 5 & 1 / 5 & 1 / 7 & 1 / 7 \\ 4 & 1 / 3 & 6 & 1 & 1 / 2 & 1 / 3 & 2 & 2 & 1 / 2 & 1 / 2 \\ 6 & 1 & 8 & 2 & 1 & 1 & 4 & 4 & 2 & 2 \\ 7 & 1 & 9 & 3 & 1 & 1 & 4 & 4 & 2 & 2 \\ 3 & 1 / 4 & 5 & 1 / 2 & 1 / 4 & 1 / 4 & 1 & 1 & 1 / 3 & 1 / 3 \\ 3 & 1 / 4 & 5 & 1 / 2 & 1 / 4 & 1 / 4 & 1 & 1 & 1 / 2 & 1 / 2 \\ 5 & 1 / 2 & 7 & 2 & 1 / 2 & 1 / 2 & 3 & 2 & 1 & 1 \\ 5 & 1 / 2 & 7 & 2 & 1 / 2 & 1 / 2 & 3 & 2 & 1 & 1 \end{matrix}] \end{matrix}$

Each column is normalized, and then each line is summed as follows, ${[0.322 1.778 0.244 0.785 1.692 1.803 0.549 0.584 1.109 1.109]}^{T}$

The normalization of the matrix is as follows, $[0.032 0.178 0.024 0.079 0.170 0.181 0.055 0.059 0.111 0.111]$

These are the weights of the corresponding factors to the slope stability.

So the maximum eigenvalue of the judgment matrix: $λ_{\max} = \sum_{i = 1}^{n} \frac{(AW)_{i}}{n ω_{i}} = 10.520$

Consistency index: $C = \frac{λ_{max} - n}{n - 1} = \frac{10.520 - 10}{10 - 1} = 0.058$

Random consistency ratio: $C_{R} = \frac{C}{R} = \frac{0.058}{1.49} = 0.039 < 0.1$

Among them, the value of the random consistency index R is taken as 1.49 according to the order of 10 in this case [15 , 23], and shown in the Table 8 [23].

Table 8
The value of index R under different order n

n 3 4 5 6 7 8 9 10 11

R 0.58 0.90 1.12 1.24 1.32 1.41 1.45 1.49 1.51

n	3	4	5	6	7	8	9	10	11
R	0.58	0.90	1.12	1.24	1.32	1.41	1.45	1.49	1.51

Consistency check is completed and passed.

(2) Evaluation of target layer

According to the membership degree of risk grade (Table 5) and the weight of the above factors, the risk grade of landslide hazard can be evaluated. The calculation process is as follows, $\begin{matrix} A = [0.032 0.178 0.024 0.079 0.170 0.181 0.055 0.059 0.111 0.111] \cdot \\ [\begin{matrix} 0.43 & 1 & 0.57 & 0 & 0 \\ 0 & 0 & 0 & 0.5 & 1 \\ 1 & 0.67 & 0 & 0 & 0 \\ 0 & 0 & 0.7 & 1 & 0.7 \\ 0 & 0 & 0 & 0.32 & 1 \\ 0 & 0 & 0 & 0.34 & 1 \\ 0 & 0 & 0.45 & 1 & 0.45 \\ 0 & 0 & 0.4 & 1 & 0.4 \\ 0 & 0 & 0.5 & 1 & 0.5 \\ 0 & 0 & 0.5 & 1 & 0.5 \end{matrix}] = [0.038 0.048 0.233 1.147 0.966] \end{matrix}$

Through normalizing the results, the correlation degree value of each grade is [0.016 0.020 0.096 0.472 0.397].

Therefore, according to the principle of maximum membership degree: C_k = max A = max [C₁, C₂, …, C_k] = max [0.016 0.020 0.096 0.472 0.397] = 0.472.

Therefore, the risk of landslide hazard is determined as grade 4, unstable.

4 Conclusions

The factors affecting the landslide risk is complex. As a formalized tool, take the control of multiple factors into consideration, when supplemented with fuzzy EAHP, provides a method a good method for landslide hazard evaluation, for it reasonable in landslide hazard analysis result evaluable and easy to operate in calculation.

The determination of the weight is one of the key steps in the fuzzy EAHP of landslide risk, which determines the authenticity of the evaluation results. In this paper, based on the actual situation of the landslide case, 10 evaluation indexes, affecting landslide hazards, are determined, to calculate the weight of evaluation index on each level, which overcomes the relative subjectivity of the previous weighting coefficient methods and thus makes the evaluation method more objective and accurate.

Based on the extension theory of the matter-element concept, the fuzzy EAHP evaluation model of landslide hazard comprehensive evaluation is constructed. The maximum of the correlation degree is calculated as 0.472. The risk of landslide hazard is determined as grade 4, unstable.

In the research work, the choice of membership function in the above method is essential, so how to rationally construct the actual membership function in the future need to do a lot of work.

According to the characteristics and mechanism of landslides, comprehensive control measures should be taken with supporting and retaining structures (such as sheet-pile wall and anchored pile) as main treatment, aided by anchorage, slope protection and drainage. The landslide monitoring should also be taken seriously.

Footnotes

Acknowledgments

Financial support for this work is provided by the Fundamental Research Funds for the Central Universities (No. 2017CXNL03), the National Key Research and Development Program of China (2017YFC0804101), the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD), the Science and technology planning project of Jiangxi Provincial Education Department (GJJ150601) and the Doctoral Science Foundation of ECUT (DHBK2015101), all of which are gratefully acknowledged. We also would like to express our acknowledgments to editors and the anonymous reviewers who had gave us useful comments to help improved the earlier version of the manuscript.

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