1. Introduction

2. Construction of a Risk Assessment Index System for High-Fill Subgrade Constructions

Figure 1. Risk assessment index of a high-fill subgrade construction.

Figure 1. Risk assessment index of a high-fill subgrade construction.

Table 1. Risk level evaluation standards for high-fill subgrade constructions.

Table 2. Index ranges of each risk level for evaluation indicators.

3. Comprehensive Risk Evaluation Model Based on the Combination weighting method based on game theory

3.1. Combination weighting method based on game theory

1. Weighting with the AHP method

The AHP method required multiple experts to compare and assess the importance degree of the evaluation indexes by virtue of their academic research and engineering experience. The subjective weight can be determined after concluding consultations and after the result has passed the consistency check. For the AHP method, the calculation steps were as follows:

Step 1: The reciprocal 1-9 scale method [33] was used to determine the quantitative scale for each pair of indexes (as shown in Table 3). The scales 2, 4, 6, and 8 indicated the corresponding importance of the relative scale. The reciprocal of these values indicated the opposite importance (i.e., j was more important than i), and it was possible to construct the judgment matrix A=(a_ij)_n×n, where a_ij is the importance of the index pairs and n is the total number of indexes.

Table 3. Relationship between scales and meanings.

Table 3. Relationship between scales and meanings.

Scales	Meanings
1	i is as important as j
3	i is slightly more important than j
5	i is more important than j
7	i is even more important than j
9	i is really more important than j

Step 2: The product normalization was adopted to obtain the weight vector w_j, where ${\overline{w}}_j$ is the proportion of weights for the j-th indexes and n is the number of indexes:

$${\overline{w}}_j=\sqrt[n]{\underset{j=1}{\overset{n}{\Pi }}{a}_{ij}}(i,j=1,2,\cdots ,n)$$

(1)

$$w_j=\frac{{\overline{w}}_j}{{\displaystyle \sum _{j=1}^n{\overline{w}}_j}}$$

(2)

Step 3: The λ_max (judgment matrix eigenvalue), $CI=\frac{{\mathsf{\lambda }}_{max}-n}{n-1}$ (consistency index), and CR (random consistency ratio) were calculated and determined, and the consistency of the judgment matrix was acceptable for CR < 0.1.

2. The entropy method

According to the influence of the degree of relative change in each index on the system as a whole, the entropy method was a way to determine the index weight coefficient by calculating the information entropy of the observed value of each index. Furthermore, according to the characteristics of the monitoring index data, the high-fill embankments of the Zhijiang-Tongren expressway were divided into several sections, which were used as different sample data. Based on the entropy method, the process of calculations was as follows:

Step 1: The dimensionless treatment of the data matrix of each monitoring index of high-fill subgrade was carried out. It was assumed that there were m objects to be evaluated with n evaluation indexes, and the data matrix was transformed into a standardized matrix B=(b_ij) _m×n.

Step 2: The entropy, e_j, of index j at the i-th section was, thereafter, calculated:

$$e_j=-\frac{1}{\ln m}{\displaystyle \sum _{i=1}^m{p}_{ij}\ln }{p}_{ij}$$

(3)

where the characteristic specific gravity $p_{ij}=b_{ij}/\sum _{i=1}^m{b}_{ij},i=\mathrm{1,2},\cdots ,m,andj=\mathrm{1,2},\cdots ,n$.

Step 3: The weight, w , of each index was then calculated:

$$w_j=\frac{1-e_j}{{\displaystyle {\sum }_{j=1}^n(1-e_j)}}$$

(4)

3. Optimized combination weighting method based on game theory

The optimized weighting game theory is a method in which the Nash equilibrium is used as the synergistic target in looking for similarities and achieving an average of each weight. That is, one has to look for the vectors that differ the least from the vectors of each basic index weight [34,35]. The combination weighting steps that were based on game theory are as follows:

Step 1: Under the assumption that the k groups of the weight vectors can be obtained by using k subjective and objective weighting methods, the basic weight vectors can be expressed as w_k={w_k1, w_k2, $\cdots$_, w_kn}, where n denotes the number of indicators and k is the number of different assignment methods. In this paper, k=2. The random linear combination of z groups of weight vectors is as follows:

$$w^{*}={\displaystyle \sum _{k=1}^z{a}_k^{*}{w}_k^T},a_k>0$$

(5)

where a_k^* is the linear combination coefficient of the k-th weight, and w* is the comprehensive weight vector of the k weight vectors w_k.

Step 2: The linear combination weight coefficient was optimized according to the synergistic targets of the Nash equilibrium in achieving the minimum deviation among each group of weights, so as to obtain the optimal combination weight. The model could be deduced as follows:

$$\min ‖{\displaystyle \sum _{k=1}^z{a}_k{w}_k^T-w_k^T}‖,(k=1,2,\cdots ,z)$$

(6)

where a_k is the weight coefficient; w_k^T is the transpose matrix of the basic weight vector set.

It could be obtained from the differential properties of the matrix, and the optimized first derivative condition of the above equation was as follows:

$${\displaystyle \sum _{k=1}^z{a}_k{w}_i{w}_k^T=}{w}_i{w}_i^T,(i=1,2,\cdots ,z)$$

(7)

Its corresponding linear system of equations could be expressed as follows:

$$\left[\begin{array}{ccc}{w}_1{w}_1^T& \cdots & w_1{w}_z^T\\ ⋮& \ddots & ⋮\\ w_z{w}_1^T& \cdots & w_z{w}_z^T\end{array}\right]\left[\begin{array}{c}{a}_1\\ ⋮\\ a_z\end{array}\right]=\left[\begin{array}{c}{w}_1{w}_1^T\\ ⋮\\ w_z{w}_z^T\end{array}\right]$$

(8)

Step 3: The linear combination coefficient a_k^* could be obtained according to the above equations, and the linear coefficient was obtained as follows through normalization of the results:

$$a_i^{*}=\frac{a_i}{{\displaystyle {\sum }_{i=1}^z{a}_i}},(i=1,2,\cdots ,z)$$

(9)

The optimal combination weight, w*, was then finally obtained.

3.2. Establishment of a comprehensive risk evaluation model for the construction of a high-fill subgrade

The fuzzy comprehensive analysis method is often used to evaluate the engineering risks during the risk evaluation practice of engineering projects [36,37]. The reason is that it comprehensively considers the influence degree of all risk factors on risk events, describing the importance degree of each risk factor by adopting optimized weight game theory. It calculates various risk probabilities based on the constructed mathematical model. In the present study, the steps for the method are as follows:

Step 1: The fuzzy comment set V was determined. The fuzzy comment set was a collection of various evaluation results of risk factors. It was determined as V={v₁, v₂, v₃, v₄}= (level I, level II, level III, level IV). There were, thus, four risk levels.

Step 2: The fuzzy membership matrix was, thereafter, determined. According to the approximate evaluation index set U and comment set V (that were well established), n was set as the number of U elements in the index set, and m was set as the number of V elements in the comment set. Since the approximate evaluation elements were interval values, a trapezoidal membership function could be adopted (as shown in Table 4). According to the data characteristics of each monitoring item of a high-fill subgrade, the comprehensive membership matrix of a certain road section was calculated as R= (r_ij)_n×m.

Table 4. Expression of ladder membership functions.

Table 4. Expression of ladder membership functions.

Type of function	Partial small membership function	Intermediate type membership function	Partial large membership function
Trapezoidal membership function	$A(x)=\left\{\begin{array}{cc}1,& x\le a\\ \frac{b-x}{b-a}& a<x\le b\\ 0,& x>b\end{array}\right.$	$A(x)=\left\{\begin{array}{cc}\frac{x-a}{b-a},& a\le x<b\\ 1,& b\le x<c\\ \frac{d-x}{d-c},& c\le x\le d\\ 0,& x<a\mathrm{or}\mathrm{x}\text{>}\mathrm{d}\end{array}\right.$	$A(x)=\left\{\begin{array}{cc}0,& x<c\\ \frac{x-c}{d-c}& c\le x\le d\\ 1,& x>d\end{array}\right.$

Step 3: A comprehensive evaluation was then conducted. The weight of the evaluation factors was set to w*, which was a comprehensive weight value based on game theory. The approximate comprehensive evaluation result B was obtained as follows:

$$B_i=w^{*}\circ R_i=\left[\begin{array}{ccc}{w}_1^{*}& \cdots & w_n^{*}\end{array}\right]\left[\begin{array}{ccc}{r}_{i11}& \cdots & r_{i1m}\\ ⋮& \ddots & ⋮\\ r_{in1}& \cdots & r_{inm}\end{array}\right]=\left[\begin{array}{ccc}{b}_{i1}& \cdots & b_{im}\end{array}\right]$$

(10)

where ∘ is the approximate synthesis operator, which is the weighted average from which (M(∙, ⊕)) was selected; R_i is the fuzzy evaluation matrix of the i-th road section; and r_inm is denoted as the affiliation of the n-th indicator of the i-th road section at the m-th risk level.

The analysis was mainly conducted by adopting the maximum membership principle. This means that the risk level that corresponds to the maximum value in the integrated affiliation matrix B was used as the risk assessment result of the evaluated object. The comprehensive evaluation result of the construction risk at the i-th section of high-fill subgrade was:

$$O_i=\max \left\{B_i\right\}=\max \{b_{i1},b_{i2},\cdots ,b_{im}\}$$

(11)

4. Case Study

4.1. Data selection and processing

The monitoring equipment (including the settlement plate, inclinometer tube, side pile, and pore water pressure gauge) was used to monitor the data of each index (Figure 1), and the equipment was buried, as shown in Figure 2. To better evaluate the stability of the high-fill subgrade, the worst index value in the past monitoring data was adopted according to the most unfavorable principle for the single index monitoring of a certain road section. The corresponding index value was obtained by calculating and processing the original monitoring data of the equipment (as shown in Table 6).

Figure 2. Schematic diagram of the buried position of the monitoring equipment.

Figure 2. Schematic diagram of the buried position of the monitoring equipment.

Table 5. Breakdown of effective measurement points for high fill roadbed monitoring.

Table 5. Breakdown of effective measurement points for high fill roadbed monitoring.

Road section	Construction stake number	Shoulder settlement plate	Central settlement plate	Inclino-meter	Border Pile	Vibrating wire piezometer	Earth pressure cell
L1	K27+060~ K27+707	4	2	3	3	3	3
L2	ZK31+730~ ZK31+810	4	2	3	3	3	3
L3	ZK31+830~ ZK32+100	4	2	2	3	3	3

Table 6. Sampled data of evaluation indexes for the Zhitong Expressway.

Table 6. Sampled data of evaluation indexes for the Zhitong Expressway.

Road section	u1 (mm/d)	u2 (cm)	u3	u4 (mm/d)	u5 (%)
L1	4.333	6.47	1.9	0.500	0.541
L2	0.717	2.21	1.5	0.428	0.560
L3	2.667	3.37	1.6	0.571	0.209

It could be analyzed and concluded from Table 6 that the sedimentation rate, lateral differential settlement, and deep horizontal displacement of L₁ were the highest, while all indexes of L₂ were the lowest, with the exception of the excess pore water pressure. It could also be seen from the risk level classification of the evaluation indexes that the original values of each index in the above table belonged to the minimization indexes and did not have to be made consistent. Since each index unit and evaluation scale differed, the proportional transformation method was adopted to carry out a dimensionless treatment of the original index data (as shown in eq. 12).

$$m_{ij}^{*}=\frac{\underset{1\le i\le n}{\min }{m}_{ij}}{m_{ij}},1\le i\le n,1\le j\le m$$

(12)

where $m_{ij}$ is the original data of the j-th index at the i-th road section, and $m_{ij}^{*}$ is the data after a dimensionless treatment of the original data. After that, the data in Table 6 were substituted into $m_{ij}$ and $m_{ij}^{*}$, and the standardized matrix M was obtained as follows:

Table 7. Value of the standardized matrix M.

Table 7. Value of the standardized matrix M.

Road section	u1	u2	u3	u4	u5
L1	0.165	0.342	0.789	0.856	0.386
L2	1	1	1	1	0.373
L3	0.269	0.656	0.938	0.750	1

4.2. Establishment of optimized combination weights

To evaluate the importance of the monitoring indexes of high-fill subgrades, 14 experts and engineers in the field of road engineering were invited to grade the importance of each indicator. Thus, 10 of the 14 experts and engineers had intermediate and higher-level titles, which ensured the authority of the survey results. In the research study, the questionnaire format was used to collect data, which used a Likert five-scale method to rate each monitoring indicator to assess its importance to the safety of high-fill subgrade. The experts rated each indicator on a scale of 1-5 depending on their academic experience. Higher scores represent a higher level of importance of the indicator to the stability of high-fill roadbeds.

SPSS (Statistical Product Service Solutions) were used to test the reliability of the survey data. After completing the questionnaire data collection, the validity and reliability tests of the questionnaire results from 14 experts were considered to ensure the reliability and validity of the data. According to the results of the validity test, the KMO (Kaiser–Meyer–Olkin) is 0.702, which is greater than 0.7, indicating that the validity of the data in this study is better and, therefore, suitable information to extract. Bartlett’s spherical test p-value is 0.001, which is less than 0.05, indicating that the data form the questionnaire passed the validity test. According to the results of the reliability test, the corresponding Cronbach Alpha coefficient was 0.809, and the CITC (corrected item-total correlation) values were 0.546, 0.819, 0.494, 0.579, and 0.618; all the CITC values were greater than 0.3, which indicated that the data in this study were reliable, with a high degree of credibility and validity.

Furthermore, it was shown that the high-fill subgrade construction risk index system, which was established in the present study, is reasonable. The average score of each indicator was calculated based on the score of each indicator of the questionnaire, and then the judgment matrix was obtained based on the average score of each indicator, as shown in Table 8 and Table 9.

Table 8. Evaluation of assessment indexes by experts.

Table 8. Evaluation of assessment indexes by experts.

Risk Index	u1	u2	u3	u4	u5
Mean	3.571	2.571	3.071	3.714	2.429

The index judgment matrix A that was constructed using the 1-9 scale method is as follows:

Table 9. Value of the index judgment matrix A.

Table 9. Value of the index judgment matrix A.

A	u1	u2	u3	u4	u5
u1	1	3	2	1/2	4
u2	1/3	1	1/2	1/4	2
u3	1/2	2	1	1/3	3
u4	2	4	3	1	5
u5	1/4	1/2	1/3	1/5	1

As a result of the calculation steps in the AHP method, the CR value of the evaluation index set was 0.027, which was less than 0.1. It, thus, indicated that the judgment matrix passed the consistency test and that the weighting was reasonable. Furthermore, the calculations of eqs. 1 and 2 showed that the subjective weight was w₁ = 0.259, 0.113, 0.171, 0.380, and 0.077 for risk indicators u₁, u₂, u₃, u₄, and u₅, respectively. In addition, the calculations of the standardized matrix, M, showed that the objective weight was w₂ = 0.573, 0.170, 0.010, 0.014, and 0.233 for risk indicators u₁, u₂, u₃, u₄, and u₅, respectively. According to the entropy method, the standardized matrix was obtained from the monitoring of the data processing of road sections L₁, L₂, and L₃.

It was also found that the linear optimal coefficients a₁ and a₂ were 0.326 and 0.674, respectively. According to game theory, these values were based on the weight vector of the subjective and objective weighting method. And, finally, the optimized combination weight value was w* = 0.470, 0.152, 0.062, 0.133, and 0.182 for risk indicators u₁, u₂, u₃, u₄, and u₅, respectively. As can be seen in Figure 3, the comprehensive weight factor w* was positioned between the subjective weight factor and the objective weight factor for each risk indicator, which indicated that the designed comprehensive weight method undermined the deviation degree of some indexes according to the entropy weight method. In addition, it also reduced the subjective influence of the AHP method, making the calculation results more accurate.

Figure 3. Comparison of the three weights factors for the risk indicator system.

Figure 3. Comparison of the three weights factors for the risk indicator system.

4.3. Comprehensive assessment

By substituting the index data in Table 2 into a trapezoidal membership function, the approximate matrix for L₁, L₂, and L₃ was calculated as follows:

Table 10. Results of each index membership matrix of road sections.

Table 10. Results of each index membership matrix of road sections.

Membership matrix	Evaluation index	Risk Level
		I	II	III	IV
R1	u1	0.374	0.626	0	0
	u2	0	0.653	0.815	0
	u3	0.063	0.533	0.831	0
	u4	0.700	0.300	0	0
	u5	0.218	0.782	0	0
R2	u1	0.802	0.198	0	0
	u2	0.231	0.769	0	0
	u3	0.010	0.990	0.251	0
	u4	0.678	0.322	0	0
	u5	0.232	0.768	0	0
R3	u1	0.657	0.343	0	0
	u2	0.090	0.910	0.109	0
	u3	0.027	0.973	0.451	0
	u4	0.674	0.326	0	0
	u5	0.343	0.657	0	0

The B_i (i=1,2,3) was obtained as follows according to the comprehensive membership evaluation Equation (10):

Table 11. Results of comprehensive membership.

Table 11. Results of comprehensive membership.

Comprehensive membership	Risk Level
	I	II	III	IV
B1	0.313	0.609	0.175	0
B2	0.545	0.455	0.016	0
B3	0.477	0.523	0.045	0

Based on the traditional AHP method and the entropy weight method, the results of the comprehensive membership of the three high-fill road sections (L₁, L₂, and L₃) are shown in Table 12. In these results, the w₁ factor was calculated using the traditional AHP method, and the w₂ factor was calculated according to the entropy weight method and eq. 10.

Table 12. Evaluation results using the AHP method and the entropy method.

Table 12. Evaluation results using the AHP method and the entropy method.

Road section	AHP method	Entropy method
L1	[0.390 0.501 0.234 0]	[0.275 0.662 0.147 0]
L2	[0.511 0.489 0.043 0]	[0.562 0.438 0.002 0]
L3	[0.468 0.532 0.089 0]	[0.481 0.519 0.023 0]

The membership of the three road sections (L₁, L₂, and L₃) of the Zhijiang-Tongren expressway, which was calculated and obtained according to the evaluation model, is shown in Figure 4. According to the maximum membership principle, L₂ had a maximum membership of 0.545 and belonged to level I (safe), while L₁ and L₃ had a maximum membership of 0.609 and 0.523, respectively, and belonged to level II (relatively safe). The results of the three evaluation methods were consistent, indicating that the high-fill embankments were in a safe state as a whole. Among these, the lateral differential settlement and deep horizontal displacement of L₁ were at level III (less safe), which indicated that L₁ was the most unstable among the three road sections. Thus, it required concentrated monitoring, while the other two were in a stage of steady state.

The comprehensive membership matrix of each road section was normalized. The weighted average method was then used to score the safety condition of each road section as a percentage, as shown in Eqs. 13-14. The results from the scoring of each road section, as derived from each assignment method, are shown in Table 13.

$$B_i^{\text{'}}=\frac{b_{ij}}{{\displaystyle {\sum }_{j=1}^m{b}_{ij}}},1\le j\le m$$

(13)

$$S{\mathrm{core}}_i=B_i^{\text{'}}\cdot {\left[\begin{array}{cccc}100& 75& 25& 0\end{array}\right]}^T$$

(14)

where B_i^’ denotes the comprehensive membership matrix of the i-th road section after normalization; b_ij denotes the affiliation of the i-th road section at the j-th risk level; and m is the risk level.

Table 13. Results from the scoring of each road segment for each assignment method.

Table 13. Results from the scoring of each road segment for each assignment method.

Road section	Combined method	AHP method	Entropy method
L1	78.145	78.467	77.952
L2	88.017	86.218	88.972
L3	85.335	83.701	86.193

Through a comparison of the evaluation results of the above three methods, it could be concluded that it is difficult to ensure weight scientificity and accuracy using the traditional AHP method since the level of expertise is uneven. It is also difficult to reflect the importance of each index in the actual setting using the entropy weight method. In the present study, the Nash equilibrium of game theory was used to initially combine the subjective and objective indexes in the consideration of the importance of various risk factors as scientifically as possible. As can be concluded from the results presented in Table 13, the comprehensive safety evaluation results of the three road sections, as calculated using the three assignment methods, were L₂ > L₃ > L₁, and the evaluation results did not change much. It basically maintained a trend for each road section in the comprehensive score ranking. The evaluation results were compared with the sample data of the three road sections in Table 6. It can be understood from Table 6 that the L₂ road section performed the best and the L₁ road section performed the worst for the sample data. This fully indicates that the model evaluation results were both accurate and valid.

Figure 4. The comprehensive membership of each road section at various risk levels.

Figure 4. The comprehensive membership of each road section at various risk levels.

4.4. Model stability analysis

Deviations were inevitable in the process of field monitoring or data statistics. Although the model had a certain fault tolerance, it was impossible to know the fault tolerance range of the model for some specific data. In the present study, the data noise test was used to add a certain range of noise to the monitoring data. Furthermore, the model performance was analyzed to obtain the fault-tolerant range of the model with respect to the data. The data reliability was, thereafter, evaluated. First, the sample data were added with different amplitudes of noise, and the specific experimental scheme was as follows: the S sample feature data were randomly selected to add different degrees of noise, and ε was the upper bound of the ratio of noise value to the original number. The values of the ratio of noise were: 5%, 10%, 20%, 30%, 40%, and 50%. Data set C after adding noise was as follows: C’=C·(1+ε). When the scale of noise data exceeded 40% of the total data, the monitoring data did not reflect the actual pattern well. Therefore, noise scale S was selected to be less than or equal to 40%, and the noise amplitude ε was selected to be less than or equal to 50%.

The processed data were, thereafter, used for the model calculations in the derivation of safety scores for each road segment and for a comparison of original scores for each road segment. MAE (mean absolute error) and MAPE (mean absolute percentage error) were selected as error evaluation indicators to measure the superiority of the model results under different noise addition amplitudes. The model results without noise are denoted as $Y=\{y_1,y_2,\cdots ,y_i\}$. The model results under noise addition are denoted as $\widehat{Y}=\{{\widehat{y}}_1,{\widehat{y}}_2,\cdots ,{\widehat{y}}_i\}$. The MAE and MAPE can be represented as Eqs. 15-16, and the results are shown in Table 14.

$$MAE=\frac{1}{n}{\displaystyle \sum _{i=1}^n\left|(y_i-{\widehat{y}}_i)\right|}$$

(15)

$$MAPE=\frac{1}{n}{\displaystyle \sum _{i=1}^n\left|\frac{y_i-{\widehat{y}}_i}{y_i}\right|}$$

(16)

Table 14. Error situations under different noise amplitudes.

Table 14. Error situations under different noise amplitudes.

Random number	Error index	Noise amplitude (%)
		5	10	20	30	40	50
1	MAE	0.545	1.057	1.742	3.289	5.661	7.964
	MAPE	0.616	1.191	1.967	3.777	6.663	9.685
2	MAE	0.410	0.671	1.363	2.490	4.354	6.182
	MAPE	0.465	0.760	1.574	2.871	5.050	7.345
3	MAE	0.531	0.996	1.821	3.187	5.287	7.512
	MAPE	0.598	1.126	2.104	3.721	6.275	9.156
4	MAE	0.287	0.590	1.258	2.385	4.616	6.764
	MAPE	0.319	0.657	1.482	2.864	5.577	8.388

Experimentally, due to the addition of noise to the whole sample data, it could be seen that the model error index behaved differently for different noise amplitudes. With an increase in the noise amplitude, the values of MAE and MAPE gradually increased. This indicated that after the addition of noise, the evaluation results of the model gradually became inaccurate with an increase in noise amplitude. In addition, the error also increased. At the same time, it could be seen that the increase in random times had little effect on the error index. That is, the evaluation results of the model were relatively stable. According to the MAE curve in Figure 5, when the experimental data deviated from the true value by 20%, the growth trend of the error value clearly became faster. This indicated that the reliability of the evaluation results was reduced, the stability was weakened, and the overall performance of the model was reduced. When the noise amplitude was 20%, the MAPE values were all within 3%. This indicated that when the noise amplitude was 20%, the deviation of the evaluation results was within 3%, which had a small influence on the model evaluation results. Therefore, when the error range of the monitoring data index set was less than 20%, the evaluation model was able to provide the correct feedback for the evaluation results. When the deviation reached more than 20% of the true value, it was necessary to check the experimental data and eliminate the problem data.

Figure 5. The MAE results for different noise amplitudes.

Figure 5. The MAE results for different noise amplitudes.

5. Conclusions

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