Multifactor Uncertainty Analysis of Construction Risk for Deep Foundation Pits

Abstract

As it is affected by many uncertain factors, the construction risk of deep foundation subway station pits involves fuzzy and random uncertainties. Considering the fuzzy and random uncertainties involved in risk evaluation, an improved fuzzy comprehensive evaluation method combining a triangular cloud model and the probability density function (PDF) is proposed in this study. First, with reference to the actual situation of deep foundation pit construction, the sources of construction risk are identified, and a construction risk evaluation index system is established. Second, the Delphi method is used to analyse the importance of each index of the evaluation object in order to obtain the evaluation data. The fuzzy best worst method (FBWM) is used to calculate the weight of the evaluation indices. Then, the triangular cloud model is used to represent the risk grade membership function. In addition, the fuzzy comprehensive evaluation method is used to comprehensively evaluate the construction risk of deep foundation pits. The fuzzy comprehensive evaluation vector is obtained for the indices possibility (P) and loss (C), and the weighted average value of the vector’s risk grade is calculated. Finally, probability analysis is carried out using PDF to determine the risk grade of P and C, and thus, to determine the risk grade of deep foundation pit construction. This method optimises the risk evaluation process of deep foundation pit construction and realises the visualisation of the comprehensive evaluation results, making the risk evaluation process transparent and convenient for use by risk analysts. This method is applied to predict the construction risk grade of a deep foundation pit project in Nanning, China, and the prediction results are consistent with the actual situation.

1. Introduction

Deep foundation pit engineering is engineering of a complex system that is prone to accidents during construction [1]. Risk evaluation in deep foundation pit engineering is a necessary step in order to predict the risk level, and it is also an important means for reducing the probability of accidents occurring. However, risk is inevitably associated with uncertainty, which includes the fuzzy and random uncertainty involved in risk evaluation. The uncertainty increases the difficulty of accurately predicting risk in deep foundation pit construction [2]. Therefore, it is both important and difficult to carry out research on the uncertainty in deep foundation pit construction risk evaluation.

The quantitative characterisation and treatment of risk uncertainty is still a research focus in the field of risk analysis [3]. In recent years, to analyse the impact of uncertainty on risk evaluation results in deep foundation pit construction, many scholars have proposed a series of valuable evaluation methods based on fuzzy analysis, probability, Bayesian networks, and other theories. These methods include a fuzzy comprehensive evaluation method [2], a risk analysis method based on evidence theory [4], a risk coupling evaluation method [5], a fuzzy entropy and cloud theory evaluation method [6], an improved risk matrix analysis method [7], a stepwise weight evaluation ratio analysis and comprehensive risk evaluation method [8], a fuzzy comprehensive evaluation method based on a C-OWA operator [9], a dynamic risk evaluation method based on risk decomposition structure matrix and particle swarm optimisation [10], etc. Most of these evaluation models represent improvements with respect to traditional risk evaluation methods, and solve the problems of risk identification, risk analysis [11], index weight determination, fuzzy uncertainty, and risk membership encountered in risk evaluation. However, there are still some problems that need to be resolved with respect to the following aspects: (1) Generally, the indicators used to measure the nature of the evaluation object are fuzzy, but the existing weight determination methods lack consideration of the indicators’ fuzzy uncertainty. (2) The fuzzy and random uncertainties of risks are not fully considered. For example, the membership function in fuzzy mathematics can only reflect the fuzziness of the evaluation system but cannot consider the randomness of the system, which reduces the accuracy of the risk evaluation results.

As an extension of classical set theory, fuzzy set theory can solve practical problems in uncertain environments [12]. Triangular fuzzy numbers can transform uncertain evaluation language into definite values. Therefore, using a triangular fuzzy number to determine the index weight can solve the fuzzy uncertainty problem when the evaluation index is described by a natural evaluation language. The cloud model [13] is an uncertainty model for realising qualitative and quantitative transformation. The cloud model can not only solve fuzzy uncertainty problems caused by the fuzzy boundary of risk grade using natural language, it can also solve the random uncertainty problem caused by the transformation of risk grade natural language and its definite values [14]. At the same time, the cloud model can be applied to realise the visualisation of risk evaluation results and improve the intuitiveness of evaluation. The triangular membership function is the simplification of the standard normal membership function, which is simple to calculate and is often used in fuzzy control. This can be extended to triangular membership clouds by using fuzzy set theory, and provides a solution for fuzzy control problems.

As a result, aiming at the construction risk of deep foundation pit subway stations, this study establishes a general engineering risk analysis method in the field of engineering risk analysis. First, we determine the potential risks in the construction of deep foundation pits, identify the sources of risk during deep foundation pit construction, and establish a construction risk evaluation index system. Second, with reference to the actual construction situation of deep foundation pit engineering, we combine the FBWM and the Delphi method to determine the importance of the evaluation indices, obtain evaluation data, and determine the index weight. Then, the triangular cloud model is used to express the grade membership function, and the fuzzy comprehensive evaluation method is used to comprehensively evaluate the construction risk of deep foundation pits to obtain the fuzzy comprehensive evaluation vector of P and C. Finally, PDF is used to analyse the risk grade of P and C and to predict the level of construction risk of the project. This evaluation method uses FBWM to deal with the fuzziness of the weight of the qualitative index, which reduces the uncertainty of subjective evaluation. The triangular cloud model is used to solve the fuzzy uncertainty of the grade boundary in the fuzzy comprehensive evaluation method. Combined with PDF, the random volatility of the risk grade probability is considered, and the evaluation results are visualised.

2. Method

The construction environment of deep foundation pits is complex, and is accompanied by great uncertainty. This uncertainty makes it difficult to determine the degree of risk during engineering construction, which is not conducive to controlling the risk during deep foundation pit construction. Therefore, the quantification and treatment of risk uncertainty plays an important role in construction risk evaluation. Aiming at the problem of risk uncertainty in the construction of the foundation pits of deep subway stations, an improved fuzzy comprehensive evaluation method is proposed in this study. The advantages of this method are as follows: (1) The FBWM is used to determine the weight of the possibility and loss indices in this allocation method. Compared with the traditional analytic hierarchy process (AHP), the number of comparisons of importance is decreased, and the subjective uncertainty resulting from expert evaluation is reduced. The method can be employed in uncertain complex multifactor environments. (2) A triangular cloud model that obeys the triangular distribution is used to represent the risk grade index, which reflects the fuzzy uncertainty of the level boundary and can be used to realise the visualisation of the grade membership function. Combined with the triangular cloud model and PDF, the probability analysis of the risk grade of deep foundation pit construction is carried out, and the evaluation results reflect the random volatility of the probability of occurrence of each risk level.

The risk evaluation method for deep foundation pits proposed in this study mainly consists of the following five steps: (1) based on the results of risk identification, a clear hierarchy of the risk evaluation index system is established; (2) each risk evaluation grade is quantified according to the risk evaluation criteria; (3) the FBWM and Delphi methods are used to analyse the relative importance of P (C) between the elements in each layer of the index system and determine the index weight; (4) combined with the weight analysis results, the P and C of the primary and secondary indices are evaluated, the weighted average of the risk grade is calculated based on the fuzzy comprehensive evaluation vector, and the triangular cloud model is used to quantify the risk grade; (5) the occurrence probability of the fuzzy comprehensive evaluation vector in step 4 at each risk grade is calculated using the PDF. This process is demonstrated in Figure 1.

3. Risk Evaluation System for Deep Foundation Pit Construction

3.1. Evaluation Index

Risk identification is a prerequisite for realising comprehensive risk evaluation. Accurate and comprehensive risk identification can improve the accuracy of risk evaluation. Establishing a hierarchical evaluation index system based on the results of risk identification can make the evaluation process clearer and the evaluation scheme more organised. In this study, by comparing and verifying the risk identification results in the literature [4,5,8,9,10,15], collecting the case data of deep foundation pit risk accidents, analysing the standard named ‘Code for risk management of underground works in urban rail transit’ [16], and interviewing geotechnical engineering experts, the risk identification is carried out for deep foundation pit engineering, and reasonable risk factor identification results for deep foundation pit construction are obtained. The 29 factors influencing risk obtained on the basis of the risk identification of deep foundation pit construction can be divided into seven categories. Additionally, a three-layer evaluation index system for deep foundation pit construction risk is established, including a target layer, a criterion layer, and a sub criterion layer, as shown in Figure 2. Then, on the basis of this system, the impact of the risk factors on the risk grade of deep foundation pit construction is analysed layer by layer, and the results of the risk evaluation of deep foundation pit construction obtained in the target layer are calculated using the index weight as the hub.

3.2. Risk Evaluation Criteria and Grade Quantification

3.2.1. Risk Evaluation Criteria

A reasonable risk evaluation criterion can increase the credibility of the evaluation results, and a simple and scientific grade quantification method can be used to improve the evaluation efficiency. Therefore, the selection of risk evaluation criteria and grade quantification schemes is an important basis on which to obtain accurate risk evaluation results. In accordance with the Code for Risk Management of Underground Works in Urban Rail Transit [16], this study determines the risk evaluation criteria for deep foundation pit construction (

Table 1. Risk evaluation criteria for deep foundation pit construction.

Grade (P) Grade (C)		1	2	3	4	5
		Negligible	Need Consider	Serious	Very Serious	Catastrophic
1	Impossible	Grade I	Grade I	Grade I	Grade II	Grade II
2	Rare	Grade I	Grade I	Grade II	Grade II	Grade III
3	Occasionally	Grade I	Grade II	Grade II	Grade III	Grade IV
4	Possible	Grade II	Grade II	Grade III	Grade IV	Grade IV
5	Frequent	Grade II	Grade III	Grade IV	Grade IV	Grade IV

) and comprehensively considers the impact of risk P and C on the deep foundation pit construction risk.

3.2.2. Quantification of Risk Grade

Table 2. Classification standard for engineering risk possibility.

Grade	1	2	3	4	5
Description	Impossible	Rare	Occasionally	Possible	Frequent
Score interval	(0.8, 1]	(0.6, 0.8]	(0.4, 0.6]	(0.2, 0.4]	(0, 0.2]

and

Table 3. Classification standard for engineering risk loss.

Grade	1	2	3	4	5
Description	Negligible	Needs consideration	Serious	Very serious	Catastrophic
Score interval	(0.8, 1]	(0.6, 0.8]	(0.4, 0.6]	(0.2, 0.4]	(0, 0.2]

show the classification standards of project risk probability P and risk loss C, respectively, providing a unified measure for evaluating the data required for the study. In accordance with Table 2 and Table 3, and with reference to construction data obtained from actual engineering cases, the Delphi method is used to invite experts in relevant fields to determine the importance of the evaluation indices for possibility and loss in an actual project, thus providing effective evaluation data for the risk evaluation of deep foundation pit construction. In combination with the classification standards in Table 2 and Table 3, the triangular membership function [17] (Equations (1)–(5)) is used to represent the degree of membership of the construction risk of deep foundation pits to each of the risk grades. The midpoint of the risk grade score interval is taken as the dividing point. At the midpoint of the score interval, the degree of membership of the index to the grade is 1, and it is 0 in the adjacent interval. By using fuzzy set theory, the triangular membership function is extended to the membership cloud triangular cloud model, which is used to realise the visualisation of the risk grades and reflects the random and fuzzy uncertainty of each risk grade.

$$\mathrm{Grade} \mathrm{V}: \{\begin{array}{l}1,x\le 0.1\\ \frac{0.3-x}{0.2},0.1<x<0.3\\ 0,0.3\le x\end{array}$$

(1)

$$\mathrm{Grade} \mathrm{IV}: \{\begin{array}{l}0,x\le 0.1\\ \frac{x-0.1}{0.2},0.1<x<0.3\\ \frac{0.5-x}{0.2},0.3\le x<0.5\\ 0,0.5\le x\end{array}$$

(2)

$$\mathrm{Grade} \mathrm{III}: \{\begin{array}{l}0,x\le 0.3\\ \frac{x-0.3}{0.2},0.3<x<0.5\\ \frac{0.7-x}{0.2},0.5\le x<0.7\\ 0,0.7\le x\end{array}$$

(3)

$$\mathrm{Grade} \mathrm{II}: \{\begin{array}{l}0,x\le 0.5\\ \frac{x-0.5}{0.2},0.5<x<0.7\\ \frac{0.9-x}{0.2},0.7<x<0.9\\ 0,0.9\le x\end{array}$$

(4)

$$\mathrm{Grade} \mathrm{\text{I}}: \{\begin{array}{l}0,x\le 0.7\\ \frac{x-0.7}{0.2},0.7<x<0.9\\ 1,0.9\le x\end{array}$$

(5)

3.3. Weight Determination Method of the Index

FBWM is a new method that was improved by Guo et al. [12] on the basis of the best-worst method proposed by Rezaei [18], and can be used to analyse the index weight in uncertain environments. In this study, the FBWM is used to analyse the weight of risk probability P and loss C. Compared with the AHP, the application of FBWM makes it possible to take the divergence degree of experts on each evaluation index into consideration, giving each evaluation index a certain range of change, and solving the problem of uncertainty in subjective evaluation. In addition, when the engineering construction risk is affected by n indicators, the AHP method needs to carry out n × (n − 1) importance comparison processes, while the FBWM only needs to perform 2n − 3 comparisons, thus representing a reduction in the number of importance comparisons. Therefore, for deep foundation pit projects with many construction risk factors, it is more feasible to use FBWM to determine the index weight.

3.3.1. Fuzzy Sets and Triangular Fuzzy Numbers

In 1965, Prof. L. A. Zadeh proposed fuzzy set theory [19]. Given a universe U, a mapping from U to the unit interval [0, 1] is called a fuzzy set on U (${\mu }_{\tilde{a}}:U\to \left[0, 1\right]$). ${\mu }_{\tilde{a}}\left(x\right)\in \left[0, 1\right]$ is the membership function of fuzzy set $\tilde{a}$. For $\forall x\in U$, ${\mu }_{\tilde{a}}\left(x\right)$ represents the membership of element x to fuzzy set $\tilde{a}$.

The triangular fuzzy number [11] (TFN) (s, m, u) is used to represent the membership function of fuzzy language variables (Equation (6)). Let $\tilde{a}$ = (s, m, u), where s < m < u, s and u represent the lower and upper bounds of the fuzzy number, respectively, and m represents the value with the greatest probability of occurrence. In this research method, TFNs are used to express the index importance comments, and when calculating the index weight, let $\tilde{\omega }$ = (s, m, u). However, the weight obtained by this method is a fuzzy number, and the subsequent evaluation process still needs to defuzzify the result. Generally, the result of defuzzification of the jth index weight is expressed by the graded mean integration (GMI) (Equation (7)), recorded as $R\left({\tilde{\omega }}_j\right)$.

$$\{\begin{array}{l}{\mu }_{\tilde{a}}\left(x\right)=0,x\in [0,s]\\ {\mu }_{\tilde{a}}\left(x\right)=\frac{x-s}{m-s},x\in [s,m]\\ {\mu }_{\tilde{a}}\left(x\right)=\frac{u-x}{u-m},x\in [m,u]\\ {\mu }_{\tilde{a}}\left(x\right)=0,x\in [u,+\infty ]\end{array}$$

(6)

$$R\left({\tilde{\omega }}_j\right)=\frac{s_j+4m_j+u_j}{6}$$

(7)

3.3.2. Determination of Index Weight on the Basis of FBWM

The steps for determining the weight of the evaluation indices using FBWM are as follows:

Step1: For n evaluation indices (c₁, c₂, …, c_n), with reference to the index importance evaluation results obtained by the Delphi method, the optimal index c_B (in this study, this refers to the index with the lowest probability of occurrence or loss) and the worst index c_W (in this study, this refers to the index with the highest probability of occurrence or loss) are chosen.

Step 2: First, the optimal index c_B is compared with other indices c_J, and then the other indices c_J are compared with the worst index c_B.

Step 3: In line with the importance comparison results obtained in Step 2, in consideration of the properties of the TFN, a nonlinear constrained optimisation problem is constructed as shown in Equation (8), and the standard fuzzy weight of each index is calculated as follows:

$$\begin{array}{l}\min \max \{\left|\frac{{\omega }_B}{{\omega }_j}-{\tilde{a}}_{Bj}\right|,\left|\frac{{\omega }_j}{{\omega }_W}-{\tilde{a}}_{jW}\right|\},p^{\ast }\le s_p\le m_p\le u_p\\ s.t.\{\begin{array}{l}\left|\frac{\left(s_B^{\omega },m_B^{\omega },u_B^{\omega }\right)}{\left(s_j^{\omega },m_j^{\omega },u_j^{\omega }\right)}-\left(s_{Bj},m_{Bj},u_{Bj}\right)\right|\le \left(p^{\ast },p^{\ast },p^{\ast }\right)\\ \left|\frac{\left(s_j^{\omega },m_j^{\omega },u_j^{\omega }\right)}{\left(h_W^{\omega },k_W^{\omega },l_W^{\omega }\right)}-\left(s_{jW},m_{jW},u_{jW}\right)\right|\le \left(p^{\ast },p^{\ast },p^{\ast }\right)\\ {\displaystyle \sum _{j=1}^{n}R\left({\tilde{\omega }}_j\right)=1}\\ s_j^{\omega }\le m_j^{\omega }\le u_j^{\omega }\\ s_j^{\omega }\ge 0\\ j=1,2,\dots ,n\end{array}\end{array}$$

(8)

where $R\left({\tilde{\omega }}_j\right)$ represents the defuzzification value of the TFN of the jth index weight. $\left(p^{\ast },p^{\ast },p^{\ast }\right)$ represents the deviation value between the actual weight ratio and the ratio obtained by the decision maker’s subjective determination, which represents the objective function value in Equation (8).

Step 4: The defuzzification value of the fuzzy weight of each index (Equation (7)) is calculated, and the final weight $\omega =\left({\omega }_1,{\omega }_2\dots {\omega }_n\right)$ of the index is obtained.

3.3.3. Consistency Test

Because this weight determination method relies on the experienced determinations of decision makers, consistent reasoning among decision makers can improve the validity of the results of weight determination. Therefore, consistency testing the weight calculation results is an important step in determining whether the calculated weight is of practical importance.

According to the definition of consistency, if the decision results meet the conditions $\left(s_{Bj},m_{Bj},u_{Bj}\right)\times \left(s_{jW},m_{jW},u_{jW}\right)=\left(s_{BW},m_{BW},u_{BW}\right)$, then they are completely consistent. If there is a deviation (△=$\left(p^{*},p^{*},p^{*}\right)$) between the actual weight ratio ($\frac{{\omega }_B}{{\omega }_j}=\frac{\left(s_B^{\omega },m_B^{\omega },u_B^{\omega }\right)}{\left(s_j^{\omega },m_j^{\omega },u_j^{\omega }\right)}$ or $\frac{{\omega }_j}{{\omega }_W}=\frac{\left(s_j^{\omega },m_j^{\omega },u_j^{\omega }\right)}{\left(s_W^{\omega },m_W^{\omega },u_W^{\omega }\right)}$) and the weight ratio obtained on the basis of expert experience ($\left(s_{Bj},m_{Bj},u_{Bj}\right)$ or $\left(s_{jW},m_{jW},u_{jW}\right)$), then the evaluation results are inconsistent. At this time

$$\{\begin{array}{l}\left(s_{Bj},m_{Bj},u_{Bj}\right)-\Delta =\frac{\left(s_B^{\omega },m_B^{\omega },u_B^{\omega }\right)}{\left(s_j^{\omega },m_j^{\omega },u_j^{\omega }\right)}\\ \left(s_{jW},m_{jW},u_{jW}\right)-\Delta =\frac{\left(s_j^{\omega },m_j^{\omega },u_j^{\omega }\right)}{\left(s_W^{\omega },m_W^{\omega },u_W^{\omega }\right)}\\ \frac{\left(s_B^{\omega },m_B^{\omega },u_B^{\omega }\right)}{\left(s_W^{\omega },m_W^{\omega },u_W^{\omega }\right)}-\Delta =\left(s_{BW},m_{BW},u_{BW}\right)\end{array}$$

(9)

because

$$\frac{{\omega }_B}{{\omega }_j}\times \frac{{\omega }_j}{{\omega }_W}=\frac{{\omega }_B}{{\omega }_W}$$

(10)

By combining the conditions under which maximum inconsistency can be achieved with Equations (9) and (10), the equation of C.I. value with respect to the deviation △ under different levels of comparative importance can be obtained (Equation (11)). By substituting the C.I. value calculated using Equation (11) into Equation (12), the consistency test can be carried out on the results of the weight calculation. If C.R. < 0.1, the results meet the consistency test. In contrast, if this condition is not satisfied, then relevant experts need to be consulted in order to adjust the comparison results of the indicators’ importance until the results pass the consistency test.

Table 4. Importance grade and consistency index C.I. value.

Fuzzy Linguistic Variables	Membership Function (Triangular Fuzzy Number)	Consistency Index C.I.
Equally important	(1, 1, 1)	3
More important	(2/3, 1, 3/2)	3.802
Important	(3/2, 2, 5/2)	5.291
Very important	(5/2, 3, 7/2)	6.692
Particularly important	(7/2, 4, 9/2)	8.041

shows the TFN and consistency index C.I. values corresponding to importance grades.

$${\Delta }^2-\left(1+2R\left({\tilde{\omega }}_{BW}\right)\right)\Delta +{\left(R\left({\tilde{\omega }}_{BW}\right)\right)}^2-R\left({\tilde{\omega }}_{BW}\right)=0$$

(11)

$$C.R.=\frac{{\Delta }^{*}}{C.I.}$$

(12)

where $\left(s_B^{\omega },m_B^{\omega },u_B^{\omega }\right),\left(s_j^{\omega },m_j^{\omega },u_j^{\omega }\right),\left(s_W^{\omega },m_W^{\omega },u_W^{\omega }\right)$ represents the triangular blur result of ${\omega }_B,{\omega }_j,{\omega }_W$.$\left(s_{Bj},m_{Bj},u_{Bj}\right),\left(s_{BW},m_{BW},u_{BW}\right),\left(s_{jW},m_{jW},u_{jW}\right)$ represent the results of the importance comparison of the best index and the other indices, the best index and the worst index, and the other indices and the worst index, respectively. $R\left({\tilde{\omega }}_{Bj}\right)$, $R\left({\tilde{\omega }}_{BW}\right)$, $R\left({\tilde{\omega }}_{Wj}\right)$ represent the defuzzification value of the results of the importance comparison.

3.4. Improved Fuzzy Comprehensive Evaluation Method

3.4.1. Triangle Cloud Model

Three numerical characteristics, expectation (Ex), entropy (En), and hyperentropy (He), are introduced into cloud theory to represent the expected cloud droplet distribution in the universe space, the uncertainty of qualitative concepts, and the uncertainty of entropy (i.e., the thickness of the clouds). These numerical characteristics organically combine the fuzziness and the randomness of their concepts, thereby reflecting the quantitative characteristics of qualitative concepts [14]. The cloud model is composed of a forward cloud generator and a backward cloud generator, with the forward cloud generator converting the numerical characterisation of qualitative concepts into quantitative data in order to generate cloud droplets. The triangular membership function is a simplification of the standard normal membership function that is simple to calculate and which is commonly used in fuzzy control [12]. In this study, the triangular cloud model generated on the basis of the triangular membership function is used to represent the risk grade; the steps to achieve this are as follows:

Step 1: The parameters a, b, and c of the triangular membership function are determined according to the boundary quantisation results of the P and C risk grades.

Step 2: The numerical characteristics Ex, En, and He are calculated according to Equation (13) (He is the constant reflecting the fuzzy domain degree of the comment, and the thicknesses of different value clouds are inconsistent). To consider a variety of random situations and more intuitively display the cloud thickness, the triangular cloud in this study is taken as He = 0.2 [20]).

$$a=Ex-En,b=Ex+En,c=Ex$$

(13)

Step 3: x_k~Tri(a, b, c) generates random numbers with a triangular distribution x_k, where k is the number of random numbers, that is, the number of triangular cloud droplets (k = 3000).

Step 4: a_k~Tri(a − He, a + He, a), b_k~Tri(b − He, b + He, b), which generates random numbers ai, bi of triangular distribution.

Step 5: We calculate ${\mu }_k=\{\begin{array}{l}\frac{x_k-a_k}{c-a_k},a_k<x_k\le c\\ \frac{b_k-x_k}{b_k-c},c<x_k\le b_k\\ 0,else\end{array}$; let (x_k, μ_k) represent the cloud droplets (k = 3000).

3.4.2. Fuzzy Comprehensive Evaluation

The fuzzy comprehensive evaluation method is a commonly used risk evaluation method in engineering. In this study, the triangular cloud model and PDF are used to improve it, and the uncertainty of the boundary when the triangular membership function is used to represent the risk grade is considered. First, the importance of the evaluation results and the results of the weight analysis of the indices for accident loss and accident probability are sorted. Then, the fuzzy relationship matrix R is used to express the fuzzy relationship between the factor set (P, H) and the evaluation set T (Grade I, Grade II, Grade III, Grade IV, Grade V). R_ji in matrix R refers to the probability that the value of the jth index is rated as Grade I, that is, its degree of membership of Grade I, which is calculated using the triangular membership function (Equations (1)–(5)). Then, according to the fuzzy comprehensive evaluation model, B = wR (w is the weight of the index loss degree (index occurrence possibility)), the fuzzy comprehensive evaluation vector B can be calculated, and through the weighted average principle, B* can be calculated (Equation (15)). Finally, a PDF is built based on random numbers using the triangular cloud model (to more intuitively express the risk grade and to better incorporate the triangular membership function, it is necessary to unify the PDF and the measurement of B* according to

Table 5. Quantification unification of B* and the risk grade of the indices P and C.

Grade	1	2	3	4	5
P	Impossible	Rare	Occasionally	Possible	Frequent
C	Negligible	Need consider	Serious	Very serious	Catastrophic
Grade quantification	(−0.5, 1.5]	[0.5, 2.5]	[1.5, 3.5]	[2.5, 4.5]	[3.5, 5.5)

). The B* values of P and C are substituted into the PDF, the probability of the corresponding B* value is calculated for each risk grade, and the risk evaluation grade of deep foundation pit construction is determined according to Table 1.

$$B=wR=\left(w_1,w_2,\dots ,w_m\right)\left[\begin{array}{cccccc}{r}_{11}& r_{12}& \cdots & r_{1i}& \cdots & r_{1n}\\ r_{21}& r_{22}& \cdots & r_{2i}& \cdots & r_{2n}\\ ⋮& ⋮& \ddots & ⋮& ⋮& ⋮\\ r_{j1}& r_{j2}& \cdots & r_{ji}& \cdots & r_{jn}\\ ⋮& ⋮& ⋮& ⋮& \ddots & ⋮\\ r_{m1}& r_{m2}& \cdots & r_{mi}& \cdots & r_{mn}\end{array}\right]=\left(b_1,b_2,\cdots ,b_m\right)$$

(14)

$$B^{*}=\frac{{\displaystyle \sum _{i=1}^n{b}_i.i}}{{\displaystyle \sum _{i=1}^n{b}_i}}$$

(15)

where i represents the risk grade (i = Grade I, Grade II, Grade III, Grade IV, Grade V), b_i represents the degree of membership of each evaluation index to the different risk grades, and B* represents the comprehensive evaluation value of the construction risk of the research object, i.e., the evaluation grade.

4. Project Case Analysis

4.1. Project Overview

Employing the risk evaluation method for deep foundation pit construction proposed in this study, taking the deep foundation pit of a city subway as the evaluation object and considering the complex factors influencing the related risk, a risk evaluation is carried out for deep foundation pit construction. Figure 3a presents the distribution of the deep foundation pit, which is irregular in shape (i.e., L-shaped). The excavation depth of the foundation pit is 9.25 m~15.3 m (i.e., >5 m; it is a deep foundation pit), and the area is about 1500 m².

A comprehensive survey method was adopted, combining geological surveys with surveying and mapping, drilling, geophysical prospecting, in situ testing, and hydrogeological tests and indoor testing of soil, rock, and water samples. Investigations prior to excavation are relatively comprehensive, and monitoring points are arranged within the influence range of the excavation site of the foundation pit excavation (i.e., 1~4 times the excavation depth of the deep foundation pit) for real-time monitoring. Technical risk in engineering construction is low. There is a round gravel layer in the deep groove of the foundation pit, which has strong water permeability, and the project faces adverse geological conditions. The surrounding environment is complex, and environmental risks are worth considering. The normal groundwater level of the foundation pit is approximately 10 m underground. Before excavation, a safety appraisal of the surrounding buildings (A and B in Figure 3a) was carried out, and the results show that the design risk of the engineering construction is low.

Figure 3b depicts the profile of the supporting structure of the deep foundation pit. The main body of the foundation pit adopts an 800-mm-thick diaphragm wall, and three internal supports are used for foundation pit support (reinforced concrete supports, and the foundation pit is locally equipped with plate supports). To reduce the impact of foundation pit construction on the surrounding buildings, buildings A and B, which are close to the foundation pit, are equipped with isolation piles (the distance between the diaphragm wall and the isolation piles is 1.6 m, and they are connected by pull beams). The design of the supporting structure meets the safety requirements. Risk factors such as pit bottom heave and surface subsidence are factors to which the construction is prone, and construction safety risks urgently need to be considered. The overall construction scheme and the technology employed in the project are relatively mature, and the construction risk is low. No harmful gas that could jeopardise the project was found, and there are few extreme bad weather events in Nanning; therefore, accidental risk is not considered in this scheme. Therefore, on the basis of the survey results obtained for this project, a total of 26 risk factors from R₁ to R₆ are selected as the evaluation indices, and the Delphi method was used to invite relevant experts (i.e., five valid questionnaires were selected) to evaluate the importance of the indices for occurrence possibility and loss in light of the actual situation.

4.2. Index Weight Results and Analysis

The risk evaluation index system was constructed on the basis of the factors influencing risk in deep foundation pit construction, and the importance of each of the indices was evaluated. Using FBWM, on the basis of the nonlinear objective optimisation model (Equation (8)), the objective optimisation analysis of index weight was carried out in MATLAB. The weight determination results for each secondary index are shown in Figure 4a–l. Figure 4b,d,f,h,j,l demonstrate that, among the various risks affecting the construction safety of deep foundation pits, the factors influencing risk that cause high loss severity are R₁₅ (structural deformation), R₂₄ (lack of timely monitoring), R₃₃ (poor construction quality and nonstandard construction (uneven pouring, inadequate grouting, etc.), R₄₄ (groundwater and the surrounding buildings not being considered in the design), R₅₂ (force majeure), and R₆₃ (insufficient investigation before excavation). To reduce construction risk for deep foundation pits and to save money, it is recommended that protective measures mainly be taken against the above risk factors. At the same time, preventive measures should be taken in a timely fashion for indices that have a high probability of occurrence. Figure 5 presents the weight of the primary indices for P and C (i.e., the risk of deep foundation pit construction). Safety risk is the most likely to occur, and once it occurs, the severity of the loss will be the highest.

4.3. Comprehensive Evaluation and Probability Analysis

The risk grade of deep foundation pit construction was systematically and comprehensively evaluated. First, the results of the importance scoring of the evaluation indices were substituted into Equations (1)–(5) in order to calculate the degree of membership of each index to Gades I–V, and membership degree matrices R(P) and R(C) were constructed that are able to reflect the fuzzy relationship between the factor set and the evaluation set. Then, in combiantion with the results of the weight analysis obtained by FBWM, the fuzzy comprehensive evaluation vector B of the P and C of each index was calculated using Equation (14). Moreover, Equation (14) was used to calculate the weighted average of the risk grade of B, from which the value of B* was obtained, which represents the construction risk grade.

Figure 6a and Figure 7a show the triangular cloud model of risk grade, which obeys the triangular distribution and reflects the comprehensive evaluation results of the occurrence possibility (P) and loss (C) of all of the factors influencing risk in the project case (B*(P) = 2.500, B*(C) = 2.6583), respectively. The triangular cloud model is well able to reflect the fuzzy uncertainty of the grade boundary and to represent the random uncertainty of the degree of membership of evaluation data to different risk grades under different circumstances. Since risk grade is a qualitative concept, the value of its quantification is uncertain, and its PDF is not unique. therefore, the occurrence probability of the risk grade corresponding to B* also exists in many cases. Furthermore, the probability of risk at each grade changes in accordance with different data combinations, reflecting the random volatility of probability. As a result, this method generates the PDF of the evaluation data many times on the basis of randomly generated triangular cloud droplets (quantitative risk grade data). We substitute the values of B*(P) and B*(C) into the PDF to calculate the probability distribution results, and determine the risk grades of P and C of the project case. The comprehensive risk grade can be determined on the basis of a comparison with Table 1.

The results of multiple calculations show that the probability of the P of the project case being a member of level II is the highest, and the probability of the C of the project case being a member of level III is the highest (Figure 6b and Figure 7b), on the basis of the probability distribution results (P_II(P) = 0.5537, P_III(P) = 0.1838; P_III(C) = 0.5537, P_IV(C) = 0.1838)). Therefore, it can be seen from Table 1 that the construction risk grade of this deep foundation pit project is Grade II.

According to the risk acceptance criteria in [16] the Code for Risk Management of Underground Works in Urban Rail Transit, risk management measures should be taken for the project, and risk treatment measures can be taken to strengthen daily management and monitoring. Combined with the actual situation of the project, the following treatment suggestions are presented: (1) harden the ground within the construction enclosure around the foundation pit, cut off the recharging of groundwater by surface water and atmospheric precipitation, and perform the drainage of the ground and the pit thoroughly and at the same time; (2) obey the rules of layer excavation, timely support, strictly prohibit overexcavation, and provide steel supports with anti-stripping and prestressed reinforcement measures; (3) during construction, perform monitoring of the foundation pit and the observation and treatment of the diving level thoroughly, and strengthen the waterproof design and treatment of the bottom of the foundation pit. There is no major construction risk during the construction of the deep foundation pit, and the risk prediction results obtained using this method are consistent with the actual situation.

5. Comparison with Combination Weight—Traditional Fuzzy Comprehensive Evaluation Method

AHP and the entropy weight method are used to calculate the combined weight, and the traditional fuzzy comprehensive evaluation method [2] is used to analyse the risk during the construction of a deep foundation pit. The comprehensive evaluation results consist of the calculated product of accident possibility P and loss C, as well as the grade membership calculated according to Equations (16)–(19). According to the fuzzy comprehensive evaluation model B = wR, the fuzzy comprehensive evaluation vector B of each index can be calculated, and B* can be calculated by means of the weighted average principle. After calculation, the comprehensive evaluation results can be obtained, whereby B* = B*’(C) × B*’(P) = 2.1753 × 1.584851688 = 3.447527877 and the membership degree of each grade is (0, 0.763, 0.237, 0); therefore, the risk grade is Grade II (due to space limitations, the detailed calculation process is not listed here). These results are consistent with the results obtained using the multifactor uncertainty analysis method for subway deep foundation pit construction risk proposed in this study, thereby showing the reliability of the proposed method.

$$\mathrm{Grade} \mathrm{I}:\{\begin{array}{l}1,x\le 0.5\\ \frac{2.5-x}{2},0.5<x<2.5\\ 0,2.5\le x\end{array}$$

(16)

$$\mathrm{Grade} \mathrm{II}:\{\begin{array}{l}0,x\le 0.5\\ \frac{x-0.5}{2},0.5<x<2.5\\ \frac{6.5-x}{4},2.5\le x<6.5\\ 0,6.5\le x\end{array}$$

(17)

$$\mathrm{Grade} \mathrm{III}:\{\begin{array}{l}0,x\le 2.5\\ \frac{x-2.5}{4},2.5<x<6.5\\ \frac{12.5-x}{6},6.5\le x<12.5\\ 0,12.5\le x\end{array}$$

(18)

$$\mathrm{Grade} \mathrm{IV}:\{\begin{array}{l}0,x\le 6.5\\ \frac{x-6.5}{6},6.5<x<12.5\\ 1,12.5\le x\end{array}$$

(19)

6. Conclusions

The new method of risk evaluation of deep foundation pit construction proposed in this study uses FBWM to determine the importance weight of evaluation indices, adopts a triangle cloud model and PDF-improved fuzzy comprehensive evaluation method, and comprehensively evaluates the risk occurrence possibility P and risk loss C of evaluation indices. The results of application in engineering examples show that this method can be used for risk assessment in the early stages of deep foundation pit construction. This method optimises the risk evaluation process of deep foundation pit construction and visualises the comprehensive evaluation results. It also provides a theoretical basis for the development of construction safety measures and early risk warning procedures. At the same time, this method is universal, and can be applied to the risk evaluation of other fields of engineering construction, making the risk evaluation process transparent and convenient for use by risk analysts, with certain application prospects.

Compared to the traditional subjective weight determination method, the FBWM method requires fewer comparisons of importance between indices, and is suitable for the evaluation of construction risk under the influence of multiple and complex factors. TFN well able to reflect the fuzziness of the index importance evaluation language and solve the fuzzy uncertainty of subjective evaluation.

The triangular cloud model is used to represent the membership function of the risk grades of P and C, and the fuzzy uncertainty of the risk grade boundary are considered with the random uncertainty of membership of the comprehensive results of grade evaluation. The occurrence probability of the weighted average value of vector B* in each grade was analysed using PDF, reflecting the random volatility of risk probability, and the evaluation results are reliable.