4.2. The Process of Establishing a Bayesian Risk Source Analysis Model
The risk assessment process for the entire cantilever casting arch bridge construction process is shown in
Figure 3.
The Bayesian network structure diagram for Bridge A relies on the project’s structural system and the corresponding characteristics of the cantilever casting concrete arch bridge. Using the risk source identification method proposed in this paper, the corresponding Bayesian network was established, as shown in
Figure 4.
After a Bayesian network was established, it was necessary to solve for the prior probability and conditional probability values of the risks.
Firstly, the prior probability was calculated.
The hanging basket construction risk is taken as an example from the list of risk sources for cantilever casting concrete arch bridge construction to illustrate the method for determining the prior probability of risk events.
The concept of prior probability is of great significance in the process of building Bayesian networks, and it also directly affects the accuracy of probabilistic risk results. There are various styles of risk events, each with its own unique characteristics; therefore, the probability distribution form also has various types. Therefore, the prerequisite for an accurate risk probability assessment is to select the appropriate prior probability of risk events. In most cases, accident data are lacking, but data can generally be obtained based on expert experience combined with relevant engineering data analysis.
According to
Table 1, the values given by experts are triangular fuzzy numbers (
a,
b, and
c), which are difficult to directly use as prior probability values for calculating the risk probability of hanging basket construction in the application process. Therefore, it is necessary to adopt appropriate methods to convert these opinions into exact values, corresponding specifically to solving fuzzy numbers. In this study, the triangular membership function was selected to defuzzify the expert opinions; specifically, the mean area method was adopted. The defuzzification formula in this method can be specifically expressed as
.
During the research process, the evaluation opinions of five experts were collected, taking the prior probability calculation of “inadequate anchoring after hanging the basket” as an example. Experts A–E rated the risk sources of inadequate anchoring after the construction of the hanging basket as “slightly high”, “slightly high”, “high”, “high”, and “high”. The mean area method was then used to defuzzify these ratings and obtain clear scores. According to the relationship between risk assessments and triangular fuzzy numbers in
Table 1, the corresponding scores
Sk for experts A–E were 0.7, 0.7, 0.875, 0.875, and 0.875, respectively.
Similarly, the arithmetic mean values of the fuzzy numbers (
av,
bv, and
cv) provided by the experts were calculated according to
Table 1; these were 0.62, 0.82, and 0.96, respectively. The similarity
SL of the average fuzzy values was applied to calculate the weights for the experts’ responses, using the calculation formula shown in Equation (8). The resulting similarity
SL values were 0.75, 0.75, 0.833, 0.833, and 0.833, respectively. The expert weight coefficients were then calculated using these
SL values, as shown in Equations (9) and (10).
where
av,
bv, and
cv are the mean values of
ak,
bk, and
ck, respectively.
Sk is the expert evaluation value, and
Sv is the arithmetic mean of fuzzy numbers evaluated by experts.
where
is the expert weight coefficient, and
k = A, B, C, D, and E.
The prior probability for the risk source in which the rear anchoring of the hanging basket
is insufficient is shown in Equation (11).
The risk sources of the construction hanging basket were then taken as the research object to determine the conditional probabilities of risk events for each sub-node risk source.
Firstly, a relative attribute judgment matrix for the sub-node risk sources was constructed. Based on the relative magnitudes of the seven levels in
Table 1, the sub-node risk sources B51~B56 were evaluated. Based on the obtained results, the corresponding attribute judgment matrix was constructed, as shown in
Figure 5.
According to
Table 1, the seven evaluation levels were converted into the corresponding triangular fuzzy numbers to obtain the corresponding fuzzy measure matrix. Then, the scores for each evaluation object were calculated according to Equations (2)–(4), as shown in Equation (12). The relative fuzzy weight calculation result is shown in Equation (13).
The final risk weights (the conditional probabilities of the Bayesian network nodes)
were then obtained via defuzzification using the mean area method, as shown in Equation (14).
The risk probability for the construction hanging basket risk source
was obtained by combining the prior probability
and the conditional probability
with the Bayesian total probability formula.
For the other six primary risk sources, the same method was used to analyze their impact on the overall construction risk of Bridge A. The relative attribute judgment matrix for the primary risk sources was determined through expert scoring, as shown in
Table 6.
After the score calculation and defuzzification, the weights of the primary risk sources were obtained, as shown in Equation (16). According to Equation (16), the importance of the seven primary risk sources during the construction process of Bridge A is in the order B5 > B3 > B6 > B2 > B4 > B1 > B7. The weight coefficients are the values of conditional probability
in the Bayesian network.
The calculation process for the conditional probabilities of the other secondary risk sources is similar to that for the seven primary risk source conditional probabilities mentioned above. The conditional probabilities of secondary risk sources during the construction process of this bridge were thereby obtained, as shown in
Figure 6.
Combining the conditional probabilities and prior probabilities for the primary risk sources, the construction risk for Bridge A
was ultimately obtained.
A construction risk assessment of Bridge A was carried out using the risk loss assessment model established in
Section 3.3. The risk level assessment results for the primary risk sources for Bridge A are shown in
Table 7.
From
Table 7, it can be seen that the risk frequency level for Bridge A is largely “occasional”, the level of loss is “serious”, and the corresponding risk level is largely level III. The evaluation results indicate that there are significant risks during the construction process. Strict control of risks in the project should be implemented, and reasonable and effective risk reduction measures should be formulated.