Research on Risk Evaluation of Hydropower Engineering EPC Project Based on Improved Fuzzy Evidence Reasoning Model

Abstract

As clean renewable energy with strong advantages, hydropower plays an extremely important role in promoting green development and energy allocation patterns. Hydropower project construction is characterized by long duration, large scale, high cost, many participants, and complex construction conditions, and is closely related to the economy, society, and ecological environment, and its construction management mode and construction risk management have become the focus of extensive attention from all walks of life. In this paper, the risk evaluation index system of hydropower engineering EPC project is constructed, and the linear weighted combination method is introduced to determine the comprehensive weights based on the calculation of weights by sequential relationship method and entropy weight method, and the improved fuzzy normal distribution is introduced as the subordinate function distribution of fuzzy evaluation level based on DS evidence theory and fuzzy theory. The risk evaluation model of a hydropower engineering EPC project is also established. Meanwhile, the model was analyzed with hydropower project examples to verify the accuracy and practicality of the model, which can guide hydropower project stakeholders to manage hydropower project risks comprehensively, collaboratively, and efficiently, and provide decision support for hydropower project construction risk management.

1. Introduction

Hydropower is a high-quality, green, and clean renewable resource. Compared to other renewable energy sources, hydropower has the most mature development technology and the most stable supply, plays an important role in national economic development and energy security, and occupies an important position in the world’s energy supply. The report of the 20th National Congress of the Communist Party of China emphasizes the coordination of hydropower development and ecological protection. The construction of hydropower projects is of great significance in promoting green development and the harmonious coexistence of man and nature.

With the accelerating transformation of the economic structure, the contract awarding model cannot meet the needs of construction projects, and the EPC model is more and more widely used in domestic and foreign engineering projects [1]. At the same time, the EPC general contracting mode has been gradually promoted and applied to the construction of water conservancy projects in China [2]. The EPC model is gradually becoming a more widely used contracting model in global engineering [3] and is often used in hydropower construction. The scale of the EPC project is relatively large, the construction period is relatively long, and the investment cost is relatively high, which makes the project more uncertain in the construction completion process. In EPC general contracting mode, when the arrangement plan conditions are not determined, the general pricing contract needs to be signed with the owner, which greatly increases the risk of the general contracting enterprise in the bidding process [4]. To strengthen the cooperation of all parties, achieve the contract objectives, and prevent the occurrence of various risk events, it is highly necessary to conduct a comprehensive evaluation of EPC project risks [5].

The project general contracting mode has achieved great development in China and is the most important contracting mode currently implemented in China [6]. However, in the process of EPC project implementation, the use of lump-sum contracts coupled with many uncertainties in hydropower projects [7], and most of the risks of the EPC model are borne by the contractor. Once potential risk factors occur and costs are overrun, the contractor faces huge risks [8], so it is necessary to identify and control risks [9]. Therefore, many scholars have begun to evaluate and study the EPC model and its risks. Ding et al. [10] developed a novel risk assessment system for key risk points during the investment stage. They conducted their research using a case study in Gansu and concluded that the most significant risk factors under the EPC model are the rationality of decision-making models and the management of risks. During the “14th Five-Year Plan” and “15th Five-Year Plan” periods, the EPC general contracting mode was confirmed to be introduced into the construction of large-scale pumped storage power stations in China [11]. In addition, Liu et al. [12] research on the cost risk evaluation of prefabricated construction projects under the EPC mode provides a new strategy for the cost risk of prefabricated construction projects. Li [13] established a fuzzy comprehensive evaluation model for bidding risks of water conservancy and hydropower EPC projects, which provides a scientific basis for project investment risk response.

Under the background of the above EPC model research, the work related to risk assessment includes: the determination of index weights and the establishment of models. The most important problem in the fuzzy comprehensive evaluation model is the determination of index weight, and previous studies generally chose single empowerment methods, such as the entropy weight method and AHP method [14]. Chen et al. [15] employed the AHP method to calculate the risk weight and determine the comprehensive level. And they introduced the cumulative risk contribution ratio to identify the main influencing factors under the risk level. Sun et al. [16] used the G1-grey evaluation method to study and reveal the relationship between the importance of indicators, and effectively quantified and evaluated the comprehensive emergency response level of water conservancy projects. Based on the construction and empirical research of the evaluation system of disruptive technologies by AHP and entropy weight method, Lv et al. [17] provide a reference for the value discovery of subsequent disruptive technologies. The relationship between each element and the final risk is ambiguous and random. Therefore, the fuzzy comprehensive evaluation model that considers this ambiguous relationship is widely used in risk evaluation [18,19]. Luo et al. [20] method based on combinatorial weighting and cloud model better combines ambiguity and randomness, which provides a basis for comprehensive evaluation of dike safety and risk elimination reinforcement. In addition, a new method based on fuzzy evidence reasoning is proposed to assess the overall risk level of construction projects [21,22,23]. Some scholars calculate the weights of each influencing factor based on the fuzzy set method of judgment matrix [24,25], and use the fuzzy comprehensive evaluation method for quantitative calculation [26,27]. Under the condition of many uncertain factors, Wei et al. [28] proposed a risk assessment method combining evidence reasoning and fuzzy set theory, and obtained the overall risk assessment value of the foundation pit reflecting different credibility levels. Jiang et al. [29] use fuzzy evidence reasoning technology to propose a specific algorithm and processing operation process of systematic risk analysis and evaluation method based on fuzzy evidence reasoning.

Existing research has provided a significant theoretical basis. While different methods should be chosen for different practical problems since each evaluation method has its unique characteristics. The risk assessment of hydropower engineering EPC projects involves many influencing factors and complex indicator systems. The fuzzy evidence reasoning method is an evidence fusion method based on the D-S evidence theory, which can effectively deal with the uncertainty and fuzziness of data information in the risk evaluation of hydropower EPC projects. Therefore, in this study, the order relation method and entropy weight method [30] are used to calculate the subjective and objective weights of indexes, and the comprehensive weights are obtained by the linear weighting group method. Based on constructing the fuzzy reliability structure model of the fuzzy normal distribution function whose evaluation level is pairwise intersecting, the improved fuzzy evidential reasoning evaluation model is obtained by combining the evidential reasoning method.

2. Risk Evaluation Index System for Hydropower Engineering EPC Project

In the EPC project general contracting mode, the risks faced by the general contractor are relatively large, and the workload requiring organization and coordination is large, the manufacturing cost risk is high, and the management cost is relatively high. The general contract project involves a large number of units and needs to deal with the relationship and contradictions of all parties, resulting in a large number of factors affecting the risk of the EPC project. Based on data collection, expert investigation, and field investigation, and in accordance with the scientific, comprehensive, independent, and important principles of indicator system construction, the factor analysis method [31] is used to optimize the EPC project risk evaluation indexes and determine the influencing factors.

2.1. Design Stage Risk

The design stage is an important stage of the EPC project, and there are many uncertain factors in the initial stage of the project. The engineering project contains value increment points in the design stage, which can provide opportunities for project value-added in the management process.

2.2. Procurement Stage Risk

The procurement management work is inseparable from the cost, schedule, and quality of the project. It undertakes the connection work between the design and construction, and the implementation of risk management in the procurement stage is an important guarantee for the smooth implementation of the EPC project.

2.3. Construction Stage Risk

The construction stage risk is objective and cannot be eliminated. In the actual construction process, the risk of an EPC general contract project is mainly affected by project factors and contractors. In the whole EPC process, risk management in the construction stage is the most basic link.

2.4. Contract Management Risk

As the contract is legally binding on the parties entering into it, the contract setting content should be described in detail, so that the construction site can take corresponding measures according to the actual situation, good contract management can improve the quality of the whole project.

2.5. Industrial Environmental Risk

Risks arising from changes in the political situation of the country and region where the EPC project is located, and the risks of the economic environment, social environment, political environment and natural environment, and other risks will increase the construction cost of the project and bring different degrees of losses to all parties of the project.

The risk evaluation index system for the hydropower engineering EPC project is shown in Figure 1.

3. Risk Evaluation Model for Hydropower Engineering EPC Project

3.1. Calculation of Weight by Combination Weighting Approach

3.1.1. Subjective Weight Determination Based on Order Relation Method

The order relation method is the G1 method [32], which combines qualitative and quantitative evaluation with the index weight decision method. Multiple experts or decision makers determine the initial qualitative order vector according to the information and knowledge they have mastered and establish a mathematical model according to the comparison of the importance of evaluation indexes. The traditional eigenvalue method calculates the weight of the index, and the judgment matrix must be the consistency matrix. The accuracy of the judgment index weight depends on the accuracy of the consistency of the judgment matrix, so it needs to be checked and calculated repeatedly. However, the order relation method can effectively solve the problem that the eigenvalue method needs the random consistency matrix, and reduce the calculation amount of importance comparison effectively. Next, combing the risk evaluation index system shown in Figure 1 for the hydropower engineering EPC project, the specific steps of subjective weight determination based on the order relation method are as follows:

Determinate order relation for every risk evaluation index.

When the general contractor’s risk evaluation index $x_i$ and $x_j$ are relative to the evaluation criteria: $x_i>x_j(i,j=1,2,\cdots ,n)$, the importance of $x_i$ is not less than $x_j$. The risk evaluation index set $\left\{x_1,x_2,\cdots x_n\right\}$ is established according to the general contractor risk index system shown in Figure 1. According to the importance evaluation principle, experts select the most important risk evaluation index in n indexes from the risk evaluation index set as $x_1^{\ast }$, and then sift out $x_2^{\ast }$, $x_3^{\ast }$, …, $x_n^{\ast }$ from the remaining n − 1 risk evaluation indexes. After n − 1 screening, the order relation of the risk evaluation indexes is recorded as: $x_1^{\ast }>x_2^{\ast }>x_3^{\ast }>\cdots >x_n^{\ast }$.

Importance grade judgment based on the order relation of risk evaluation index.

After the order relation set of risk evaluation index is determined, the ratio of importance degree $r_k$ of adjacent risk evaluation index is defined, that is, the importance degree proportional relationship between risk evaluation indexes $x_{k-1}^{\ast }$ and $x_k^{\ast }$:

(1)$r_k=\frac{{\omega }_{k-1}^{\ast }}{{\omega }_k^{\ast }}\hspace{1em}k=2,3,\cdots ,n$

From Equation (1), ${\omega }_{k-1}^{\ast }$ and ${\omega }_k^{\ast }$ are the weight coefficients of risk evaluation indexes $x_{k-1}^{\ast }$ and $x_k^{\ast }$ respectively, and $r_k$ is their proportional relation, which should be satisfied:

(2)$r_{k-1}\ge \frac{1}{r_k}\hspace{1em}k=2,3,\cdots ,n$

The $r_k$ assignment is shown in Table 1.

According to the definition of $r_k$ in Equation (1), the product is obtained:

(3)${\displaystyle \prod _{i=k}^n{r}_i=\frac{{\omega }_{k-1}^{\ast }}{{\omega }_n^{\ast }}}$

For Equation (3), the sum of the above Equation from 2 to n is obtained:

(4)${\displaystyle \sum _{k=2}^n({\displaystyle \prod _{i=k}^n{r}_i})}=\frac{1}{{\omega }_n^{\ast }}-1$

then the weight coefficient of the nth risk evaluation index is calculated using the following equation:

(5)${\omega }_n^{\ast }={\left[1+{\displaystyle \sum _{k=2}^n({\displaystyle \prod _{i=k}^n{r}_i})}\right]}^{-1}$

Therefore, the weight coefficients of the n − 1th, n − 2th, …, indexes can be obtained by combining Equation (1):

(6)${\omega }_{k-1}^{\ast }=r_k{\omega }_k^{\ast }\hspace{1em}k=n,n-1,\cdots ,3,2$

Expert group judgment weight fusion.

When J experts screened the risk evaluation index set $\left\{x_1,x_2,\cdots x_n\right\}$, due to the influence of the subjective judgment from experts, the sensitivity to the risk evaluation index was different, and then the order relation was inconsistent and the $r_k$ assignment was different. Considering the interference of subjective factors, the judgment weight of expert groups was fused. The fused weight ${\omega }_k^{″}(k=1,2,\cdots n)$ was divided into two cases:

Case 1: The judgment of initial order relation for experts’ subjective evaluation information is consistent.

According to the j-th expert, the importance ratio relation between adjacent evaluation indexes $x_{k-1,j}^{\ast }$ and $x_{k,j}^{\ast }$, which called $r_{k,j}={\omega }_{k-1,j}^{\ast }/{\omega }_{k,j}^{\ast }$, then:

(7)${\omega }_n^{″}={\left[1+{\displaystyle \sum _{k=2}^n({\displaystyle \prod _{i=k}^n{r}_i^{\ast }})}\right]}^{-1}$

(8)${\omega }_{k-1}^{″}=r_k^{\ast }{\omega }_k^{″}\hspace{1em}k=n,n-1,\cdots ,3,2$

(9)$r_k^{\ast }=\frac{1}{J}{\displaystyle \sum _{j=1}^J{r}_{k,j}\hspace{1em}k=2,3,\cdots ,n}$

Case 2: The judgment of initial order relation for experts’ subjective evaluation information is inconsistent.

There are $J_a$ experts who have the same judgment on the order relation of the risk evaluation index set $\left\{x_1,x_2,\cdots x_n\right\}$, and the weight coefficients are ${\omega }_1^{\ast },{\omega }_2^{\ast },\cdots ,{\omega }_n^{\ast }$ in order according to Equation (7). The judgment of the initial order relation of $J-J_a$ experts is inconsistent, which are: $x_{1,j}^{\ast }>x_{2,j}^{\ast }>\cdots >x_{k,j}^{\ast }\begin{array}{cc}& j=1,2,\cdots ,J-J_a\end{array}$, respectively. Where, $x_{k,j}^{\ast }$ is the ordering of the kth element belonging to the ordered set $\left\{x_k\right\}$ by the jth expert. For the initial order relation determined by each expert j with inconsistent initial order relation judgment, the weight coefficient of index $x_{k,j}^{\ast }$ can be obtained by Equation (5) and denoted as ${\omega }_{k,j}^{\prime }(k=1,2,\cdots n)$, and its arithmetic mean value is taken as the fusion weight value of $J-J_a$ experts and denoted as ${\omega }_k^{\prime }$:

(10)${\omega }_k^{\prime }=\frac{1}{J-J_a}{\displaystyle \sum _{j=1}^{J-J_a}{\omega }_{k,j}^{\prime }}\hspace{1em}k=1,2,\cdots ,n$

Then, when the experts’ initial order relation is inconsistent, the fusion weight ${\omega }_k^{″}$ obtained is:

(11)${\omega }_k^{″}=\frac{J_a}{J}{\omega }_k^{\ast }+\frac{J-J_a}{J}{\omega }_k^{\prime }$

3.1.2. Objective Weight Determination Based on Entropy Weight Method [33]

The evaluation of an information system integration project scheme is a multi-objective decision-making problem, which requires a quantitative comprehensive analysis and comparison on whether the scheme of all bidders is reasonable, whether there is integration innovation, whether the qualification is responsive, etc. Entropy [34] is a measure of system state uncertainty [35]. When the system is in n different states and the probability of occurrence of each state is $P_i (i=1,2,\cdots ,n)$, the entropy of the evaluation system is:

(12)$E=-{\displaystyle \sum _{i=1}^n{P}_i\ln P_i}$

where $P_i$ satisfies 0 ≤ $P_i$ ≤ 1, ${\displaystyle \sum _{i=1}^n{P}_i=1}$.

Entropy has extremum properties. When the coefficient state is an equal probability, that is, $P_i=1/n (i=1,2,\cdots ,n)$, its entropy value is maximum.

(13)$E(P_1,P_2,\cdots ,P_n)\le E(\frac{1}{n},\frac{1}{n},\cdots \frac{1}{n})=\ln n$

The concept of entropy is used to measure the influence degree of an evaluation index. The index values of N experts corresponding to m indexes form the evaluation index decision matrix $Y={(y_{ij})}_{n\times m}$, that is

(14)$Y=\left[\begin{array}{cccc}{y}_{11}\hfill & y_{12}\hfill & \cdots \hfill & y_{1m}\hfill \\ y_{21}\hfill & y_{22}\hfill & \cdots \hfill & y_{2m}\hfill \\ & & \ddots \hfill & \\ y_{n1}\hfill & y_{n2}\hfill & \cdots \hfill & y_{nm}\hfill \end{array}\right]$

where, element $y_{ij}$ represents the jth index of scheme $i$. For a price index, the smaller the $Y$, the better. For performance and other indexes, the larger the $Y$, the better. Let’s say the optimal value for each column in $Y$ is $y_j^{\ast }$, that is

(15)$y_j^{\ast }=\left\{\begin{array}{l}\max \left\{y_{ij}\right\}\\ \min \left\{y_{ij}\right\}\end{array}\right.$

The proximity between $y_{ij}$ and $y_j^{\ast }$ is

(16)$D_{ij}=\left\{\begin{array}{l}{y}_{ij}/y_j^{\ast },y_j^{\ast }=\max \left\{y_{ij}\right\}\\ y_j^{\ast }/y_{ij},y_j^{\ast }=\min \left\{y_{ij}\right\}\end{array}\right.$

Normalize $D_{ij}$ and record it as

(17)$d_{ij}=D_{ij}/{\displaystyle \sum _{i=1}^n{\displaystyle \sum _{j=1}^m{D}_{ij}}}$

The entropy value of the j-th evaluation index is defined on the condition that evaluation indexes evaluate $n$ bidders.

(18)$E_j=-{\displaystyle \sum _{i=1}^n\frac{d_{ij}}{d_j}}\ln \frac{d_{ij}}{d_j}$

where:

(19)$d_j={\displaystyle \sum _{i=1}^n{d}_{ij}},j=1,2,\cdots ,m$

According to the extremum property of entropy, the closer the value of $d_{ij}/d_j$ is to equal, the greater the value of entropy is. When $d_{ij}/d_j$ is exactly equal and entropy $E_j$ reaches its maximum, $E_{\max }=\ln m$.

The larger the entropy $E_j$ of index j is, the smaller the difference between the expert’s value on this index and the optimal value. If the index with the smaller difference degree is more important, the entropy value after normalization is used as the objective weight of the index; otherwise, the complementary value of entropy can be used as the objective weight of the index after normalization. $e_j$ is used to normalize Equation (18) to obtain the entropy value representing the importance of the evaluation decision of evaluation index j.

(20)$e_j=-\frac{1}{\ln m}{E}_j$

Normalize 1 − $e_j$ to obtain the objective weight of index j [36].

(21)${\theta }_j=\frac{1-e_j}{m-{\displaystyle \sum _{j=1}^m{e}_j}}$

where:

(22)$0\le {\theta }_j\le 1,{\displaystyle \sum _{j=1}^m{\theta }_j=1}$

3.1.3. Combination Weight Determination Based on Linear Weighted Combination Method

The methods of calculating comprehensive weight include maximizing deviation method, factor multiplication method and linear weighted combination method, etc. [37]. The maximizing deviation method is suitable for the situation where the index evaluation difference is too large, factor multiplication method tends to produce a “multiplication effect”. This paper adopts the linear weighted combination method [38] to combine the subjective and objective weights.

(23)$\overline{\omega }=\alpha {\omega }_i^1+\beta {\omega }_i^2$

In the equation, ${\omega }_i^1$ and ${\omega }_i^2$ are the weights of the ith index obtained by the principal objective weighting method, respectively, and $\alpha$ and $\beta$ are the coefficients of the two weighting methods, respectively.

In order to allocate coefficients reasonably and accurately, the weighted summation optimization model is introduced:

(24)$\begin{array}{c}max\hspace{0.17em}Z={\displaystyle \sum _{i=1}^n}{\displaystyle \sum _{j=1}^m}{r}_{ij}\left(\alpha {\omega }_i^1+\beta {\omega }_i^2\right)\hfill \\ s.t.\hspace{1em}{\alpha }^2+{\beta }^2=1\hspace{1em}\hspace{1em}\alpha ,\beta \ge 0\hfill \end{array}$

The optimal solution of this model is:

(25)${\alpha }^{\ast }=\frac{{{\displaystyle \sum }}_{i=1}^n{{\displaystyle \sum }}_{j=1}^m{r}_{ij}{\omega }_i^1}{{\left[{({{\displaystyle \sum }}_{i=1}^n{{\displaystyle \sum }}_{j=1}^m{r}_{ij}{\omega }_i^1)}^2+{({{\displaystyle \sum }}_{i=1}^n{{\displaystyle \sum }}_{j=1}^m{r}_{ij}{\omega }_i^2)}^2\right]}^{\frac{1}{2}}}$

(26)${\beta }^{\ast }=\frac{{{\displaystyle \sum }}_{i=1}^n{{\displaystyle \sum }}_{j=1}^m{r}_{ij}{\omega }_i^2}{{\left[{({{\displaystyle \sum }}_{i=1}^n{{\displaystyle \sum }}_{j=1}^m{r}_{ij}{\omega }_i^1)}^2+{({{\displaystyle \sum }}_{i=1}^n{{\displaystyle \sum }}_{j=1}^m{r}_{ij}{\omega }_i^2)}^2\right]}^{\frac{1}{2}}}$

The coefficients ${\alpha }^{\ast }$ and ${\beta }^{\ast }$ can be obtained by normalizing $\alpha$ and $\beta$, and the comprehensive weight is:

3.2. Risk Evaluation Model Based on Fuzzy Evidential Reasoning [39]

3.2.1. Improved Fuzzy Reliability Structure Model

For the multi-index evaluation problem, it is assumed that there are L evaluation indexes ${\mathrm{e}}_i(i=1,2,3,\cdots ,L)$, and the weight of the indexes is ${\omega }_i(i=1,2,3,\cdots ,L)$ and meets $0\le {\omega }_i\le 1$ ${\displaystyle \sum _{i=1}^L{\omega }_i=1}$. The fuzzy evaluation level set is $H=\left\{H_n,n=1,2,\cdots ,N\right\}$, and $H_n$ is the qualitative evaluation level described by language. Generally, the affiliation function of the qualitative evaluation level H adopts triangular or trapezoidal distribution, while the fuzzy normal distribution can collect more evaluation information with a high affiliation degree and shield more evaluation information with a low affiliation degree [40], and the comparison of fuzzy normal distribution and triangular distribution affiliation function degree is shown in Figure 2, the fuzzy evaluation level is shown in Figure 3.

In this paper, the improved fuzzy normal distribution is adopted to correct the affiliation function by collecting evidence. When the evidence supports the occurrence of a high probability of the evaluation object, the peak value of the affiliation function shifts to the direction of low risk, and the corrected fuzzy evaluation level is shown in Figure 4. When the evidence supports the occurrence of a low probability of the evaluation object, the peak value of the affiliation function shifts to the high-risk direction, and the revised fuzzy evaluation level is shown in Figure 5.

The corresponding affiliation function of the fuzzy evaluation level is:

(27)$r(u)=\left\{\begin{array}{cc}\hfill 0\hfill & u<\mu -3\sigma \mathrm{or}u>\mu +3\sigma \hfill \\ e^{-\frac{{(u-\mu )}^2}{2{\sigma }^2}}\hfill & \mu -3\sigma \le u\le \mu +3\sigma \hfill \end{array}\right.$

The corrected corresponding affiliation function of the fuzzy evaluation level is:

(28)$r(u)=\left\{\begin{array}{ll}0& u<\mu -3{\sigma }_1\mathrm{or}u>\mu +3{\sigma }_2\\ & e^{-\frac{{(u-\mu )}^2}{2{\sigma }_1{}^2}}\hspace{1em}u\ge \mu -3{\sigma }_1\\ & e^{-\frac{{(u-\mu )}^2}{2{\sigma }_2{}^2}}\hspace{1em}u\le \mu +3{\sigma }_2\end{array}\right.$

Fuzzy reliability structure FBS can be expressed as

(29)$FBS(e_i)=\left\{(H_n,{\beta }_n),n=1,2,\cdots ,N\right\}$

where, N is the number of grades, ${\beta }_n$ is the confidence degree of evaluation index $e_i$ on fuzzy evaluation level $H_n$, ${\beta }_n\ge 0,{\displaystyle \sum _{n=1}^N{\beta }_n\le 1}$. ${\beta }_n$ is the description of uncertainty, if ${\displaystyle \sum _{n=1}^N{\beta }_n=1}$, shows that the information is complete. If ${\displaystyle \sum _{n=1}^N{\beta }_n<1}$, shows that information is insufficient. If ${\displaystyle \sum _{n=1}^N{\beta }_n=0}$, shows that information is completely unknown.

3.2.2. Fuzzy Evidential Reasoning Algorithm

The basic credibility mass value of the evaluation index $e_i$ is

(30)${\mathrm{m}}_{\mathrm{i}}\left\{H_n\right\}=\omega {\beta }_{\mathrm{n}}$

(31)${\mathrm{m}}_{\mathrm{i}}\{H\}=1-{\displaystyle \sum _{n=1}^N{\mathrm{m}}_{\mathrm{i}}\left\{H_n\right\}}$

Then ${\mathrm{m}}_{\mathrm{i}}\left\{H_n\right\}$ is the basic reliability allocation function of the evaluation index $e_i$ at the evaluation level of $H_n$; ${\mathrm{m}}_{\mathrm{i}}\left\{H\right\}$ is the unassigned reliability, indicating the degree to which all evidence has not been allocated after being synthesized. Let

(32)${\overline{m}}_{\mathrm{i}}\left\{H\right\}=1-{\omega }_i$

The L evaluation indexes included in the evaluation object are fused with evidence. The specific algorithm is as follows

(33)$m_{1-L}\{H_n\}=k\{{\displaystyle \prod _{i=1}^L[m_i\{H_n\}+m_i\{H\}]-{\displaystyle \prod _{i=1}^L{m}_i\{H\}}}\},n=1,2,\dots ,N$

(34)$m_{1-L}\{H\}=k\{{\displaystyle \prod _{i=1}^L[m_i\{H\}}\}$

(35)${\overline{m}}_{1-L}\{H\}=k\{{\displaystyle \prod _{i=1}^L[{\overline{m}}_i\{H\}}\}$

(36)$\begin{array}{cc}\hfill m_{1-L}\{{\overline{H}}_{n(n+t)}\}& =k{\mu }_{H_{n(n+t)}}^{\max }\{{\displaystyle \prod _{i=1}^L[m_i\{H_n\}+m_i\{H_{n+t}\}+m_i\{H\}]}-{\displaystyle \prod _{i=1}^L[m_i\{H_n\}+m_i\{H\}}]\hfill \\ & -{\displaystyle \prod _{i=1}^L{m}_i\{H_{n+t}\}+m_i\{H\}}]+{\displaystyle \prod _{i=1}^L{m}_i\{H\}}\}\hfill \\ & n=1,2,\cdots ,N-1\hfill \\ & t=1,2,\cdots ,N-1,\mathrm{n}+t\le N\hfill \end{array}$

where

(37)$\begin{array}{cc}\hfill k=& \{{\displaystyle \sum _{n=1}^N\{{\displaystyle \prod _{i=1}^L[m_i\{H_n\}+m_i\{H\}]-{\displaystyle \prod _{i=1}^L{m}_i\{H\}}}\}}\hfill \\ \hfill & +{\displaystyle \sum _{t=1}^{N-1}{\displaystyle \sum _{n=1}^{N-t}{\mu }_{H_{n(n+t)}}^{\max }\{{\displaystyle \prod _{i=1}^L[m_i\{H_n\}+m_i\{H_{n+t}\}+m_i\{H\}]-{\displaystyle \prod _{i=1}^L[m_i\{H_n\}+m_i\{H\}]}}}}\hfill \\ & -{\displaystyle \prod _{i=1}^L[m_i\{H_{n+t}\}+m_i\{H\}]+{\displaystyle \prod _{i=1}^L{m}_i\{H\}}}\}+{\displaystyle \prod _{i=1}^L{m}_i\{H\}}{\}}^{-1}\hfill \end{array}$

The total risk evaluation mass value $m_{1-L}\left\{H\right\}$ of the evaluation object was obtained according to fusion Operations (32)–(36), $m_{1-L}\left\{{\overline{H}}_{n(n+t)}\right\}$ is the mass value at the intersection $H_{n,n+t}$ of the fuzzy evaluation level, ${\mu }_{H_{n(n+t)}}^{\max }$ is the maximum ordinate of the intersection of evaluation level $H_n$ and $H_{n+t}$, and k is the normalization coefficient.

3.2.3. Fuzzy Intersection Reliability Allocation [41]

After L evaluation indexes are synthesized, the combination reliability ${\beta }_n$ and ${\beta }_{n(n+t)}$ of the evaluation object can be obtained as

(38)${\beta }_n=\frac{m_{1-L}\left\{H_n\right\}}{1-{\overline{m}}_{1-L}\left\{H\right\}},n=1,2,\cdots ,N$

(39)$\begin{array}{cc}\hfill {\beta }_{n(\mathrm{n}+\mathrm{t})}& =\frac{m_{1-L}\left\{{\overline{H}}_{n(n+t)}\right\}}{1-{\overline{m}}_{1-L}\left\{H\right\}}\hfill \\ \hfill n& =1,2,\cdots ,N-1\hfill \\ \hfill t& =1,2,\cdots ,N-1,\mathrm{and}n+t\le N\hfill \end{array}$

The reliability ${\beta }_{n(n+t)}$ on $H_{n,n+t}$ is assigned to the reliability ${\beta }_n^{n(n+t)}$ and ${\beta }_{(n+t)}^{n(n+t)}$ on $H_n$ and $H_{n+t}$, respectively, as follows

(40)${\beta }_n^{n(n+t)}=\frac{S_n+A{F}_n^{n(n+t)}{S}_{n(n+t)}}{S_n+S_{n(n+t)}+S_{(n+t)}}{\beta }_{n(n+t)}$

(41)${\beta }_{n+t}^{n(n+t)}=\frac{S_{n+t}+A{F}_{n+t}^{n(n+t)}{S}_{n(n+t)}}{S_n+S_{n(n+t)}+S_{(n+t)}}{\beta }_{n(n+t)}$

where

(42)$A{F}_n^{n(n+t)}=\frac{1}{2}\left[(1-\frac{d_n}{d_n+d_{n+t}})+\frac{S_n}{S_n+S_{n+t}}\right]$

(43)$A{F}_{n+t}^{n(n+t)}=\frac{1}{2}\left[(1-\frac{d_{n+t}}{d_n+d_{n+t}})+\frac{S_{n+t}}{S_n+S_{n+t}}\right]$

where, $d_n$ and $d_{n+t}$ are the minimum distance between $H_{n(n+t)}$ and the abscissa corresponding to the maximum affiliation degree of evaluation levels $H_n$ and $H_{n+t}$, respectively. $S_n+S_{n(n+t)}$ and $S_{n+t}+S_{n(n+t)}$ represent the intersection area of $H_{n(n+t)}$ and evaluation level: $H_n$ and $H_{n+t}$, respectively. ${\beta }_n^{n(n+t)}$, ${\beta }_{(n+t)}^{n(n+t)}$ is called redistribution reliability; $A{F}_n^{n(n+t)}$, $A{F}_{n+t}^{n(n+t)}$ is the reliability sub factor. Fuzzy intersection reliability allocation is shown in Figure 6.

4. Case Analysis

4.1. Project Overview

A city proposed to build an XX water conservancy and hydropower project. The planned construction period is 72 months, the maximum dam height is 102 m, the top of the dam is 303.8 m long, the normal total reservoir capacity is 437 million m³, and the effective capacity is 255 million m³ for the multi-year regulation reservoir. The power plant is of the behind-the-dam type, with an installed capacity of 3 × 20,000 kW and a maximum quoted flow of 111.6 m³/s. This hydropower project was built using the EPC turnkey model, with a total investment of RMB 3.276 billion, and when the project is put into use, it will be of great significance to the rational allocation of water resources in the region, helping the people of the region to increase their income and improve their living standards.

4.2. Determination of Comprehensive Weight of Risk Indexes

4.2.1. Subjective Weights Based on the Order Relation Method

The risk index system of the hydropower engineering EPC project constructed in this paper is used to calculate the weight of the index. Five experts in related fields rank and score each risk index, and use them as the original data to calculate the index weight. Five experts evaluated the project risk from five aspects: design risk $(C_1)$, procurement stage risk $(C_2)$, construction stage risk $(C_3)$, contract management risk $(C_4)$, and industry environmental risk $(C_5)$, with different evaluation indexes for each aspect. Five experts ranked the priority of the risk indexes and judged the importance of the indexes by referring to Table 1. The priority and importance of the indexes obtained are shown in Table 2.

According to Equations (3)–(6), the weights of the five experts’ indexes are calculated, as shown in Table 3.

If the order relation given by different experts is inconsistent, the expert group judgment is fused according to Equations (7)–(11), and the subjective weight of C₁ is calculated as shown in Table 4 and Table 5.

4.2.2. Objective Weights Based on the Entropy Weight Method

The entropy of the j-th evaluation index is obtained according to Equations (15)–(18). The entropy of each evaluation index of the first level index is shown in Table 6.

The entropy value representing the importance of the evaluation decision of evaluation index j is obtained through normalization according to Equation (20). The objective weight of the first-level indexes can be obtained according to Equation (21), as shown in Table 7. The objective weight of the second-level indexes can be obtained in the same way.

4.2.3. Determination of Comprehensive Weight

According to Equations (23)–(26), the comprehensive weight is obtained by combining subjective weight and objective weight, as shown in Table 8.

4.3. Comprehensive Risk Evaluation

The risks are divided into five levels by analyzing the emergencies that the EPC project general contractor may encounter and the impact of the risks. That is $H=\left\{H_1,H_2,H_3,H_4,H_5\right\}$ = {‘low risk’, ‘lower risk’, ‘medium risk’, ‘higher risk’, ‘high risk’}. The initial risk classification is shown in Table 9. The initial general contractor risk fuzzy evaluation level is shown in Figure 7.

By collecting data, the affiliation function is modified. When the evidence supports the occurrence of the high probability of the evaluation object, the peak value of the affiliation function shifts to the direction of low risk. The risk classification after modification is shown in Table 10. The revised fuzzy risk evaluation level of the general contractor is shown in Figure 8.

When the evidence supports the occurrence of a low probability of the evaluation object, the peak value of the affiliation function shifts to the high-risk direction, and the initial risk classification is shown in Table 11. The fuzzy evaluation level of the initial general contractor risk is shown in Figure 9.

According to Table 9 and Figure 7, the maximum affiliation at the intersection of the initial evaluation levels is calculated, as shown in Table 12.

According to Table 10 and Figure 8, when the evidence supports the occurrence of the high probability of the evaluation object, the maximum affiliation degree ${\mu }_{n(n+l)}^{\max }$ of the intersection of all evaluation levels after modification is calculated, as shown in Table 13.

According to Table 11 and Figure 9, when the evidence supports the occurrence of the low probability of the evaluation object, the maximum affiliation degree of the intersection of all evaluation levels after modification is calculated, as shown in Table 14.

Through data collection, the design stage risk, the procurement stage risk, and the industrial environmental risk are subject to the revised risk level classification standard as shown in Figure 9, the construction stage risk and the contract management risk are subject to the revised risk grade classification standard as shown in Figure 8, and the comprehensive risk is subject to the initial risk grade classification mark as shown in Figure 7.

The basic reliability and undistributed reliability on all risk re-identification frameworks were calculated according to Equations (30) and (31), and the combined distribution functions of all risk evaluations were calculated from Equations (32)–(37), as shown in Table 15.

The combined reliability ${\beta }_n$ and ${\beta }_{n(n+t)}$ of risks is calculated from Equations (38) and (39), and ${\beta }_{n(n+t)}$ is redistributed from Equations (40) to (43). Finally, the reliability structure of each risk level is obtained, as shown in Table 16.

According to the results in the table, the general contractor risk for this water resources and hydropower EPC project is at medium risk, with contract management risk at a high risk level, so the general contractor needs to strengthen its preventive measures for this risk.

5. Discussion

The fuzzy evidence reasoning evaluation model demonstrates significant advantages in the risk evaluation of hydropower EPC projects selected in this study. From the results in Section 4, some discussions are obtained as follows.

• (I).. Design phase risk is a type of risk that needs to be highly valued in hydropower EPC projects. Due to the early involvement of the design unit and the advantage of providing feasibility study and design consulting services for the owner, the design unit can deeply participate in the owner’s early project planning and strengthen the review and supervision of the design stage. During the owner’s bidding process, the design unit will also undertake project management work, so it is necessary for the general contracting unit to have more strength and bear greater risks. When bidding, it is necessary to objectively analyze and accurately evaluate the bidding conditions of the owner, and sign as comprehensive an agreement as possible in case of disputes. In the process of project implementation, a perfect EPC contract should be signed as far as possible, a perfect design Change control process should be formulated, and a corresponding EPC design management system and project management organization should be established.

• (II).. The procurement process plays a connecting role in EPC general contracting projects, so the risks in the procurement stage must also be taken seriously. There are various risk factors in the procurement process of planning, implementation, and execution. The procurement process, procurement sources, procurement coordination, etc. will all be risk points, especially with the high and low prices of equipment and materials procurement directly affecting the comprehensive benefits of hydropower projects. Therefore, in the EPC project of hydropower engineering, it is necessary to strengthen procurement management, do a good job in procurement layout and planning, strictly control and manage the procurement process, fully investigate and evaluate the supply channels, and timely follow up on market changes.

• (III).. Due to the long construction period of hydropower projects, they are affected by many factors during the construction process, so construction risks are also relatively high. Preventing construction risks is crucial. In the preparation stage, it is necessary to strictly review the construction drawings, repeatedly test the corresponding technologies, and select the most suitable technology for hydropower engineering construction, in order to more accurately determine the construction technology. Before the construction of the project, the contractor should invite professional technical personnel and consulting experts to comprehensively investigate the surrounding environment of the project, and then write a detailed investigation and analysis report. During the construction process, a sound quality management system should be established, strict management methods and norms should be established, process monitoring and stable control should be strengthened, and problems discovered during construction should be corrected in a timely manner. At the same time, it is necessary to enhance the safety awareness of all personnel, establish a systematic safety management system, and take necessary measures in a timely manner to eliminate or reduce safety hazards.

• (IV).. Contract management risk and industry environmental risk are different from the first three types of risks. They run through various stages of hydropower EPC projects and are also two very important types of risks. The EPC project of hydropower engineering involves multiple participants, and the task of contract management is heavy. Contract risks are directly related to factors such as the total contract amount, technical difficulty, construction period, and management quality. In contract management risks, a precise grasp of contract terms, dynamic supervision of contract performance, and strict control of contract changes are all important risk points. It is necessary to ensure that all contract terms are clear and operable and to develop unified change management rules and processes to ensure contract performance. For all parties involved, we should attach importance to win-win cooperation, establish a correct overall view of the project, have a strong sense of performance, ensure the quality of performance, and strive to minimize the risks caused by execution. In industry environmental risks, policy environment, economic environment, market environment, construction environment, etc. are all important risk points. It is necessary to closely monitor policy dynamics, make timely adjustments and improvements, maintain competitiveness and sustainability, and create a good construction environment.

6. Conclusions

This paper proposes a risk evaluation model for hydropower engineering EPC projects based on improved fuzzy evidence-based reasoning and empirically demonstrates it with examples of hydropower engineering construction risk management. The model helps with risk analysis and risk response of hydropower engineering EPC projects, promotes orderly development of hydropower in terms of hydropower engineering risk management, accelerates green energy allocation, and promotes green and low-carbon development.

(1) Combining the characteristics of EPC projects, this paper identifies 32 evaluation indicators for five types of risks: design stage risk, procurement stage risk, construction stage risk, contract management risk, and industry environmental risk, based on the analysis of risk influencing factors of hydropower engineering EPC projects. A total of 20 evaluation indicators were preferentially selected through factor analysis to construct a risk evaluation index system for hydropower engineering EPC projects.

(2) The subjective and objective weights were determined based on the ordinal relationship method and the binary semantics of disparity maximization, and the linear weighted combination method was introduced to determine the comprehensive weights and determine the weight calculation model of risk indicators for hydropower engineering EPC projects.

(3) Considering the uncertainty in the process of the comprehensive evaluation of risk in hydropower engineering EPC projects, in order to effectively solve this problem, DS evidence theory and fuzzy theory are adopted, and an improved fuzzy normal distribution is introduced as the distribution of the affiliation function of fuzzy evaluation grade, and a risk evaluation model for water conservancy and hydropower engineering EPC projects for general contractors is established. Moreover, through the empirical analysis of risk management of hydropower project construction, the risk evaluation grade of the hydropower project EPC project is calculated as medium risk, which verifies the accuracy and practicality of the model.

Although this study has achieved certain research results, there are still some limitations. This study determines the risk assessment indicators by conducting a literature study, expert interviews, and field investigations. While these indicators are obtained through scientific methods and comprehensive analysis of literature and consultation with experts, further research and improvements are necessary to enhance the combination of specific components and stages of hydropower engineering EPC projects. Furthermore, in future research, it is important to consider both the iterative updating of the EPC project risk evaluation index system and the application of big data technology in the evaluation process.

Author Contributions

Conceptualization, Q.L. and B.W.; methodology, Y.G.; software, J.X.; formal analysis, Q.L. and B.W.; investigation, Y.G. and Y.C.; data curation, J.X.; writing—original draft preparation, Q.L., Y.G. and J.X.; writing—review and editing, B.W. and C.W.; supervision, C.W. and B.W.; project administration, Q.L. All authors have read and agreed to the published version of the manuscript.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

The data presented in this study are available on request from the corresponding author (J.X).

Acknowledgments

The author would like to thank the journal’s reviewers for providing helpful comments, and the staff for their careful and professional work.

Conflicts of Interest

The authors declare no conflict of interest.

Footnotes

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

Figures and Tables

Figure 1. Risk evaluation index system for hydropower engineering EPC project.

Figure 2. Comparison of fuzzy normal distribution and triangular distribution affiliation function degree.

Figure 3. Risk fuzzy evaluation level.

Figure 4. Modified risk fuzzy evaluation level.

Figure 5. Modified risk fuzzy evaluation level.

Figure 6. Fuzzy intersection reliability assignment.

Figure 7. Initial general contractor risk fuzzy evaluation level.

Figure 8. Modified general contractor risk fuzzy evaluation level.

Figure 9. Modified general contractor risk fuzzy evaluation level.

Table 1

Proportion of index importance.

${\mathit{r}}_{\mathit{k}}$	Meaning
1.0	Index $x_{k-1}^{\ast }$ and index $x_k^{\ast }$ are equally important
1.2	Index $x_{k-1}^{\ast }$ and index $x_k^{\ast }$ are slightly important
1.4	Index $x_{k-1}^{\ast }$ and index $x_k^{\ast }$ are obviously important
1.6	Index $x_{k-1}^{\ast }$ and index $x_k^{\ast }$ are strongly important
1.8	Index $x_{k-1}^{\ast }$ and index $x_k^{\ast }$ are extremely important
Intermediate value	The above neighbors determine the intermediate value

Calculation of risk evaluation indexes’ weight coefficients.

Table 2

Priority and Importance of Expert Ranking Index.

Sort	${\mathit{J}}_1$	${\mathit{J}}_2$	${\mathit{J}}_3$	${\mathit{J}}_4$	${\mathit{J}}_5$
1	$C_4$, 0	$C_3$, 0	$C_4$, 0	$C_4$, 0	$C_3$, 0
2	$C_3$, 1.1	$C_4$, 1.2	$C_3$, 1.1	$C_3$, 1.1	$C_4$, 1.2
3	$C_5$, 1.2	$C_5$, 1.1	$C_5$, 1.2	$C_5$, 1.2	$C_5$, 1.2
4	$C_1$, 1.2	$C_1$, 1.2	$C_1$, 1.2	$C_1$, 1.2	$C_1$, 1.2
5	$C_2$, 1.3	$C_2$, 1.3	$C_2$, 1.2	$C_2$, 1.3	$C_2$, 1.3

Table 3

Weight of experts on secondary indexes.

Index	J 1	J 2	J 3	J 4	J 5
C 1	0.163	0.179	0.175	0.175	0.171
C 2	0.148	0.148	0.146	0.146	0.143
C 3	0.235	0.216	0.231	0.231	0.226
C 4	0.258	0.260	0.254	0.255	0.272
C 5	0.196	0.197	0.194	0.193	0.188

Table 4

Risk index sequence relation weight.

First Level Index	Weight	Second Level Index	Weight
C 1	0.173	C 11	0.167
		C 12	0.136
		C 13	0.239
		C 14	0.248
		C 15	0.210

Table 5

The subjective weight of the first level index.

Index	C 1	C 2	C 3	C 4	C 5
Weight	0.173	0.146	0.228	0.260	0.193

Table 6

Entropy of the first order index.

Index	C 1	C 2	C 3	C 4	C 5
Entropy	1.6090	1.6088	1.6089	1.6088	1.6086

Table 7

Objective weights of first-level indexes.

Index	C 1	C 2	C 3	C 4	C 5
Weight	0.156	0.191	0.186	0.192	0.275

Table 8

Comprehensive weight of first-level indexes.

Index	C 1	C 2	C 3	C 4	C 5
Weight	0.163	0.172	0.203	0.221	0.241

Table 9

Initial risk classification.

Risk Level	Severity	Risk Value Range	Affiliation Function μ σ
Level 1	low	0 < u ≤ 45	μ = 0, σ = 15
Level 2	lower	0 < u ≤ 60	μ = 30, σ = 10
Level 3	medium	5 < u ≤ 95	μ = 50, σ = 15
Level 4	higher	30 < u ≤ 100	μ = 75, σ = 15
Level 5	high	70 < u ≤ 100	μ = 100, σ = 10

Table 10

Revised risk classification.

Risk Level	Severity	Risk Value Range	Affiliation Function μ σ
Level 1	Low	0 < u ≤ 30	μ = 0, σ = 10
Level 2	Lower	0 < u ≤ 20	μ = 20, σ = 20/3
		20 < u ≤ 60	μ = 20, σ = 40/3
Level 3	Medium	5 < u ≤ 40	μ = 40, σ = 35/3
		40 < u ≤ 95	μ = 40, σ = 55/3
Level 4	Higher	30 < u ≤ 65	μ = 65, σ = 35/3
		65 < u ≤ 100	μ = 65, σ = 55/3
Level 5	High	80 < u ≤ 100	μ = 100, σ = 20/3

Table 11

The initial risk classification.

Risk Level	Severity	Risk Value Range	Affiliation Function μ σ
Level 1	Low	0 < u ≤ 60	μ = 0, σ = 20
Level 2	Lower	0 < u ≤ 40	μ = 40, σ = 40/3
		40 < u ≤ 60	μ = 40, σ = 20/3
Level 3	Medium	5 < u ≤ 60	μ = 60, σ = 55/3
		60 < u ≤ 95	μ = 60, σ = 35/3
Level 4	Higher	30 < u ≤ 85	μ = 85, σ = 55/3
		85 < u ≤ 100	μ = 85, σ = 35/3
Level 5	High	60 < u ≤ 100	μ = 100, σ = 40/3

Table 12

List of the maximum degree of affiliation of the intersection of each evaluation level.

${\mathit{\mu }}_{\mathit{n}(\mathit{n}+\mathit{l})}^{\mathbf{max}}$	${\mathit{H}}_1$	${\mathit{H}}_2$	${\mathit{H}}_3$	${\mathit{H}}_4$	${\mathit{H}}_5$
$H_1$	1	0.4868	0.2494	0.0439	0
$H_2$		1	0.726	0.198	0
$H_3$			1	0.707	0.135
$H_4$				1	0.6065
$H_5$					1

Table 13

List of the maximum degree of affiliation of the intersection of each evaluation level.

${\mathit{\mu }}_{\mathit{n}(\mathit{n}+\mathit{l})}^{\mathbf{max}}$	${\mathit{H}}_1$	${\mathit{H}}_2$	${\mathit{H}}_3$	${\mathit{H}}_4$	${\mathit{H}}_5$
$H_1$	1	0.4868	0.1819	0	0
$H_2$		1	0.7261	0.1979	0
$H_3$			1	0.7066	0.0561
$H_4$				1	0.1617
$H_5$					1

Table 14

List of the maximum degree of affiliation of the intersection of each evaluation level.

${\mathit{\mu }}_{\mathit{n}(\mathit{n}+\mathit{l})}^{\mathbf{max}}$	${\mathit{H}}_1$	${\mathit{H}}_2$	${\mathit{H}}_3$	${\mathit{H}}_4$	${\mathit{H}}_5$
$H_1$	1	0.487	0.294	0.0439	0
$H_2$		1	0.726	0.086	0
$H_3$			1	0.707	0.278
$H_4$				1	0.835
$H_5$					1

Table 15

Combination probability distribution function of risk evaluation in the design stage.

n	$\mathbf{m}\left\{{\mathit{H}}_{\mathit{n}}\right\}$	$\mathbf{m}\left\{{\overline{\mathit{H}}}_{1\mathit{n}}\right\}$	$\mathbf{m}\left\{{\overline{\mathit{H}}}_{2\mathit{n}}\right\}$	$\mathbf{m}\left\{{\overline{\mathit{H}}}_{3\mathit{n}}\right\}$	$\mathbf{m}\left\{{\overline{\mathit{H}}}_{4\mathit{n}}\right\}$
1	0.0611
2	0.1291	0.0075
3	0.1629	0.0058	0.029
4	0.1354	0.0014	0.0065	0.0275
5	0.0575	0.0000	0.0000	0.0046	0.0114
k = 1.151; $m\left\{H\right\}$ = 0.3773; $\overline{\mathrm{m}}\left\{H\right\}$ = 0.3773

Table 16

Reliability structure of risk level.

	${\mathit{H}}_1$	${\mathit{H}}_2$	${\mathit{H}}_3$	${\mathit{H}}_4$	${\mathit{H}}_5$
${\beta }_{{\mathrm{C}}_1}$	0.107	0.24	0.298	0.242	0.097
${\beta }_{{\mathrm{C}}_2}$	0.114	0.299	0.348	0.162	0.058
${\beta }_{{\mathrm{C}}_3}$	0.117	0.302	0.365	0.149	0.048
${\beta }_{{\mathrm{C}}_4}$	0.083	0.143	0.273	0.319	0.167
${\beta }_{{\mathrm{C}}_5}$	0.116	0.237	0.315	0.223	0.095
${\beta }_{\mathrm{C}}$	0.093	0.235	0.324	0.227	0.091

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