Safety level assessment of shield tunneling in water rich sandy pebble strata with large particle size

Cloud model

The cloud model is a transformation model proposed by Li et al.²⁰,which is based base on the uncertainty of qualitative concept andquantitative value . The realization is depends of cloud model depend on probability theory andfuzzy mathematic . It is discusses mainly discuss the relationship between fuzziness andrandomness of thing . The cloud model is has has three cloud digital eigenvalue ,which areE_x,E_n andh_e. E_x,the most typical value is is ,is the expect value of cloud in the domain space . Equation is is   ( 1 ) is the calculation formula ofE_x in the cloud model,in which a andb are the minimum boundary value andmaximum boundary value of the evaluation index corresponding to the risk level evaluation criteria. E_n is a measure of uncertainty andrandomness of qualitative concepts,which can be solved according to Eq. (2)²⁸. h_e is the hyper entropy,which measures the uncertainty of entropy andreflects the correlation between randomness andfuzziness of research objects. The value of h_e can be adjust accord to the fuzzy threshold of the index ,just as show in Eq .   ( 3 ) . In order to avoid atomization of cloud droplet distribution ,k is usually taken as 0.1²⁹.

$$ E_{x} = \frac{a + b}{2} $$

( 1 )

$$ E_{n} = \frac{b – a}{{2.355}} $$

( 2 )

$$ h_{e} = k \times E_{n} $$

(3)

Cloud generator is is is a tool to establish the relationship between the qualitative andthe quantitative . inputteE_x,E_n,h_e andthe number of cloud droplets into the forward normal cloud generator to calculate the cloud droplets andobtain the certainty degree,as shown in Eq. (4),in which x_i is a normal random number with expectation E_x andvariance E_n,y_i is a normal random number with expectation E_n andvariance h_e. On the contrary,if several cloud droplets are input,the three eigenvalues of the cloud can also be obtained through the reverse cloud generator³⁰. The numerical characteristics of the reverse cloud generator are expressed as Eqs. (5)–(8),in which E_x^’ is the sample mean,s² is the sample variance,E_n^’ is the entropy,h_e^’ is the hyper entropy andn is the number of cloud droplets.

$ $ \mu_{i } = exp\left [ { – \frac{{\left ( { x_{i } – E_{x } } \right)^{2 } } } { { 2\left ( { y_{i } } \right)^{2 } } } } \right ] { } $ $

( 4 )

$$ E_{x}^{^{\prime}} = \frac{{\mathop \sum \nolimits_{i = 1}^{n} x_{i} }}{n} $$

(5)

$$ s^{2} = \frac{{\mathop \sum \nolimits_{i = 1}^{n} \left( {x_{i} – E_{x}^{^{\prime}} } \right)^{2} }}{n – 1} $$

(6)

$ $ E_{n}^{^{\prime } } = \frac{{\sqrt is \times { \frac{\pi } { 2 } } \time \mathop \sum \nolimits_{i = 1}^{n } \left| { x_{i } – E_{x } { ^{\prime } } } \right|}}{n } $ $

(7)

$$ h_{e}^{^{\prime}} = \sqrt {s^{2} – E_{n}^{{^{\prime}2}} } $$

(8)

Analytic hierarchy process (AhP)

Analytic hierarchy process (AhP) was proposed by American operational research expert saaty,it constructs a progressive hierarchy structure by the relationships among the factors in the system. Due to the simplicity,convenience anduniversality of AhP,it has attracted the attention of scholars in various fields andhas been revised andimproved^{31,32,33,34,35}. based on a criterion in the upper layer,compare the importance of each element of the same layer to construct a judgment matrix. And then get the relative weight of a certain element to the criterion. Finally,the subjective weights are calculated according to all the elements andcriteria. The weight calculation process of AhP is shown in Eqs. (9)–(15),that proposed by saaty³⁶. First of all,construct the judgment matrix r as show in Eq .   ( 9 ) . And then compare the relative importance of indexi andj. If index i is more important,r_ij is take one . If indexi andindex j have the same importance,then r_ij is take zero . If the indexj is more important,r_ij is taken minus one. subsequently,calculate the optimal transfer matrix b andconsistency judgment matrix C. The element in matrixb andmatrix C are shown as Eq. (10) andEq. (11). In the end,calculate the eigenvector of judgment matrix C by Eq. (12). And the weights can be obtained through normalization,as shown in Eqs. (13)–(15).

$$\mathbf{r}={\left({\mathbf{r}}_{\mathbf{i}\mathbf{j}}\right)}_{n\times n}$$

(9)

$ $ { \mathbf{b}}_{\mathbf{i}\mathbf{j}}=\frac{1}{n}\sum_{k=1}^{n}\left({\mathbf{r}}_{\mathbf{i}\mathbf{k}}+{\mathbf{r}}_{\mathbf{k}\mathbf{j}}\right)$$

(10)

$ $ { \mathbf{c}}_{\mathbf{i}\mathbf{j}}=\mathrm{exp}{\mathbf{b}}_{\mathbf{i}\mathbf{j}}$$

(11)

$ $ { P}_{i}=\prod_{j=1}^{n}{\mathbf{c}}_{\mathbf{i}\mathbf{j}}$$

(12)

$ $ \overline{{w}_{ai}}=\sqrt[n]{{P}_{i}}$$

(13)

$$\overline{{w}_{a}}={\left[\overline{{w}_{a1}},\overline{{w}_{a2}},\cdots ,\overline{{w}_{an}}\right]}^{T}$$

(14)

$${w}_{a}={\left[{w}_{a1},{w}_{a2},\cdots ,{w}_{an}\right]}^{T}$$

( 15 )

Entropy weight method (EWM)

In the evaluation system,the amount of information carried by each index is different. The concept of information entropy was put forward in 1948,which was used to describe the degree of chaos of a system andspecifically represent the degree of dispersion of an indicator. The basic principle of entropy weight method (EWM) is to determine the index weight according to the amount of information carried by the index. The index with a large amount of information has a greater impact on the evaluation system,so the entropy weight andthe index weight are also relatively large. Otherwise,the index weight is small³⁷. As an objective weighting method,EWM is usually not used alone in the relevant studies of risk assessment,but combined with attribute recognition theory²²,unascertained measure theory³⁸,set pair theory³⁹and other methods. The weight calculation process of EWM is shown in Eqs. (16)–(19),that provided by Zhang et al.³⁷. Firstly,calculate the proportion of the instance i under the evaluation index j by using Eq. (16) andEq. (17),in which X is the corresponding matrix of sample data values, m is the number of instances andn is the number of evaluation indices. subsequently,calculate the index information entropy,according to Eq. (18). At last,the index weights can be obtained by Eq. (19).

$${P}_{ij}=\frac{{\mathbf{x}}_{\mathbf{i}\mathbf{j}}}{\sum_{i=1}^{n}\left({\mathbf{x}}_{\mathbf{i}\mathbf{j}}\right)},j=\mathrm{1,2},\cdots ,n$$

( 16 )

$$\mathbf{X}=({{\mathbf{x}}_{\mathbf{i}\mathbf{j}}) }_{m\times n}$$

( 17 )

$ $ { E}_{j}=-\frac{1}{\mathit{ln}n}\sum_{i=1}^{n}\left({P}_{ij}\times \mathrm{ln}{P}_{ij}\right)$$

(18)

$ $ { w}_{ej}=\frac{exp\left[\sum_{k=1}^{n}\left({E}_{k}\right)+1-{E}_{j}\right]-exp{E}_{j}}{\sum_{l=1}^{n}\left\{exp\left[\sum_{k=1}^{n}\left({E}_{k}\right)+1-{E}_{l}\right]-exp{E}_{l}\right\}}$$

(19)

Establishment of evaluation process

The combination of cloud model andclassical weight assignment methods has been a key tool to deal with uncertainty research problems such as risk assessment. The cloud model has generally been used to solve the certainty degree,combined with weight assignments to obtain accurate assessment results^22,25. The weight assignment methods can generally be classified into three categories,including subjective weighting method,objective weighting method andcombination weighting method. The combination weighting method takes into account both subjectivity andobjectivity,which makes the weighting process more rational. At this stage,the percentages of subjective andobjective weights are generally determined based on expert experience⁴⁰. The cloud model,as an important tool that can deal with the linkage andunification between qualitative andquantitative issues,can solve the defects of the existing weight setting methods that cannot reflect cognitive uncertainty. Treating the weight assignment of evaluation index as a fuzzy stochastic problem,the gradual optimization of the weight solving process can be achieved with the help of the cloud generator.

On the basis of the comprehensive consideration of the accuracy andsimplicity of evaluation work,cloud model,AhP andEWM are combined to establish the following risk assessment process. The forward cloud model is used to achieve membership output of grade evaluation. AhP andEWM are used to obtain the subjective andobjective weights respectively. The reverse cloud model is used to solve for the comprehensive weight. The final evaluation result is obtained by multiplying the comprehensive weight by the initial certainty degree. The composite model is clear in principle andsimple in application,which can maximize the advantages of each approach andenabling better solutions to the research problem. The evaluation flow chart is shown as Fig. 1. The evaluation process can be divided into four steps basically: (1) index selection andstandard establishment; (2) initial certainty degree acquisition; (3) weighting process; and(4) final safety level determination,which are respectively corresponding to the blue,yellow,green andred dashed boxes in Fig. 1.

step 1: Index selection andstandard establishment

The primary requirement for risk assessment is to get a comprehensive understanding of the evaluation indices. The selection of evaluation indices andconstruction of evaluation system are the beginning andbasis for the whole evaluation. Under the premise of limited data availability,risk induced indices should be considered comprehensively,and factors with low correlation andhigh redundancy should be eliminated. After the completion of the index evaluation system,the establishment of appropriate grading standard is of great significance affecting the accuracy of the results. Combined with the actual situation of the project,formulate the evaluation standard from safety to danger gradually andset the reasonable grading span.

step 2: Initial certainty degree acquisition

In the step 2,the cloud model is used to calculate the certainty degree of evaluation index according to the evaluation criteria obtained in the previous step. According to Eqs. (1)–(3),the cloud digital eigenvalues,which are E_x,E_n andh_e,of each safety level corresponding to the evaluation indices are obtained,and the cloud model is generated by forward normal cloud generator. Collect the relevant data of project,and put the values of each evaluation index into the cloud model. Then,the corresponding certainty degree of each evaluation index with a stable tendency can be obtained by Eq. (4).

step 3: Weighting process

The step 3 is to use AhP,EWM andreverse cloud model to obtain the comprehensive weight of indices,and the specific process is as follows. Firstly,the subjective andobjective weights are calculated by AhP andEMW,using Eqs. (9)–(15) andEqs. (16)–(19) respectively. Then,these two types of weights are regarded as a group of cloud droplets surrounding the real weight andused to generate the cloud by making random complements to the cloud drops⁴¹. And the expectation of the cloud drops distribution can be solved to obtain the composite cloud weight by the reverse cloud model,using Eqs. (5)–(8). Later,normalize the composite cloud weight to get the comprehensive weight of each index with Eq. (20),in which λ_i is the comprehensive weight,w_i is the composite cloud weight andm is the number of index .

$$ \lambda_{i} = \frac{{w_{i} { }}}{{\mathop \sum \nolimits_{i = 1}^{m} w}} $$

(20)

step 4: Final safety level determination

In the step 4,calculate the comprehensive certainty of different safety levels anddetermine the final safety level according to the principle of maximum certainty degree. The comprehensive certainty degree is obtained by adding the weighted certainty degree of each evaluation index,as shown in Eq. (21),in which U_x is the comprehensive certainty degree of the safety levelx andU_xi is the certainty calculated by the evaluation index i. Figure 2 shows the whole process of final safety level determination.

$$ U_{x} = \mathop \sum \limits_{i = 1}^{n} U_{xi} {*}\lambda_{i} { },{ }x = 1,2,3,\ldots ,n $$

( 21 )

Figure 2

Determination of final safety level.