2.1 Bridge pile foundation engineering in karst area

Currently, due to the complexity of karst geological formations, bridge pile foundation engineering has been extensively studied. On one hand, existing research is concentrated on innovative construction processes, investigations into structural mechanical properties, analysis of construction risks, and comprehensive evaluations of technical indicators. For instance, Huang et al. [7] introduced a novel theoretical model for a large-diameter stepped-tapered hollow pile, addressing the issue of instability in the bearing layer due to karst geological conditions. He et al. [8] established models to calculate the side resistance, frictional resistance, and pile-end resistance of rock-socketed piles in the karst area. Xing et al. [9] assessed the impact of various pile driving techniques and grouting methods on the mechanical properties of rock-socketed piles. Ji et al. [10] based on the Yinzhou Lake Bridge project, identified sources of risk within divisional work and conducted risk prioritization. Li et al. [11] constructed evaluation models to assess the construction safety risks of highway bridges. Jin et al. [12] conducted an evaluation of grouting effectiveness in karst areas, considering the construction technology, apparent parameters, observational data from inspection holes and geophysical exploration techniques. On the other hand, prior to construction, risk reduction is achieved through the detection of karst features with the aim of mitigating construction risks. Detection methods encompass ground-penetrating radar (GPR) [13], electrical resistivity tomography (ERT) [14], transient electromagnetic method (TEM) [15], crosshole seismic Computed Tomography (CT) [16] and so on. However, GPR, ERT, and TEM are suitable for karst exploration closer to the surface, and traditional surveying techniques have limitations when it comes to deeper or larger-scale karst features [17]. Hence, research into the prediction of karst features has been developed [18]. For instance, Zhang et al. [19] proposed the construction of an assessment system to predict the state of karst development. This approach involves using fuzzy evaluation methods to provide initial karst predictions before construction, and during construction, combining various geological survey techniques to refine the exploration of karst features.

2.2 Karst treatment techniques and scheme decision-making

Due to the complexities associated with karst geological exploration, the karst treatment methods during the pile foundation drilling process have become a technological challenge. LI et al.[20] based on the foundation construction examples of the Zhonglao Railway bridge in karst area, summarized the applicable treatment techniques, selection principles, and control points for different types of karst development. Dong et al. [21] analyzed the bearing capacity of pile foundation characteristics of the pile foundation for the double-layer retaining wall by using numerical simulation verified the feasibility of the double-layer steel casing retaining wall in super-large karst caves. LI et al. [22] putted forward a set of management procedures for karst gushing water treatment, and introduced further-more the precise exploration method, grouting material, construction steps and effectiveness check for the karst gushing water treatment in detail [23].

The literature mentioned above is focused on the exploration of construction techniques. However, in engineering construction, it is essential to consider not only meeting technical requirements but also addressing multiple constraints, including economic factors. For instance, Chen [24] conducted a quantitative comparison of karst area pile foundation construction schemes, considering both project duration and cost aspects. Pan [25] posits that selecting an appropriate bridge construction method is a process of balancing multiple factors, including cost, quality, project duration, safety, and bridge design. For the lean management of construction schemes, Li et al. [26] utilizes a management framework comprising five dimensions: quality, cost, time, safety, and organization during the construction process. The Analytic Network Process (ANP) method is considered highly suitable for evaluating systems that involve complexity, nonlinearity, as well as a mix of qualitative and quantitative indicators. Similarly, considering the complex geological characteristics and construction challenges in karst areas, pile foundation construction schemes can be defined as a Multiple Attribute Decision Making (MADM) problem involving multiple factors, objectives, and criteria.

Some scholars have engaged in discussions regarding decision-making for bridge pile foundation construction schemes. Zheng [27] focused on the decision-making for bridge construction schemes across rivers. They established an evaluation system based on four aspects: technical performance, economic benefits, implementation effectiveness, and environmental effects. The weightings were determined through expert rating, and the selection of the scheme was carried out using fuzzy evaluation. Yu and Pan [28] developed an evaluation model using the Analytic Hierarchy Process (AHP), facilitating the comparative analysis of pile foundation construction schemes for mountainous area renovation and expansion projects. Among these methods, AHP is better suited for linear evaluation systems. It exhibits a high degree of subjectivity, which can lead to potential biases in decision-making due to individual differences. Wu [29] formulated a fuzzy comprehensive evaluation model for comparing bridge pile foundation construction schemes near the operating railway lines in the western mountainous regions. They applied the Maximum Deviation Method for objective weighting, intending to mitigate the influence of subjective factors on weight allocation. However, the decision-making methods in these studies are relatively simplistic, and there is room for further enhancement in modeling approaches. The limited research volume in the field of pile foundation engineering decision-making in karst areas has led to a knowledge gap. Particularly, in managing the challenges of multi-attribute decision-making, issues persist regarding incomplete evaluation systems, one-sided and inefficient decision-making methods. Therefore, it is necessary to establish a scientific, efficient, and practical decision-making model.

2.3 Multiple attribute decision making

Compared to Multi-Objective Decision Making (MODM) methods, Multiple Attribute Decision Making (MADM) is better suited for decisions involving a limited set of alternative schemes [30]. In order to address intricate decision-making challenges, Multiple Attribute Decision Making (MADM) methods have found extensive applications across various domains, including engineering decision support systems [31], finance [32], industrial design [33], and other fields. For instance, Więckowski [34] posited that supplier evaluation constitutes a Multi-Criteria Decision Analysis (MCDA) problem. Using Triangular Fuzzy Numbers in combination with five MCDA methods, suppliers are ranked and selected, with the robustness of the results examined through sensitivity analysis. Jagtap [35] innovatively improved the Simos’ method for weight calculation in the issue of selecting non-traditional machining processes. The results demonstrated greater stability in comparison to the AHP method. Specially, a dynamic directed graph was employed to visualize the preferences among alternative options. The decision-making process was conducted using the M-Polar Fuzzy Set ELECTRE-I method. Ors [36] addressed the issue of selecting bridge pier construction schemes and designed an AHP-TOPSIS model for choosing the optimal construction technique.

Additionally, a combination of weighted integrated evaluation was proposed to overcome the limitations of a single weight. Zheng et al. [37] considered the comparative intensity and conflicts among evaluation values, using the CRITIC method for objective weight calculation and minimum discrimination information for weight aggregation. Xu [38] considered the information content contained in each indicator and employed the entropy weight method to calculate objective weights, which is particularly suitable for quantitative metrics. However, objective weighting is calculated solely based on subjective score values, which has certain limitations.

Then, the process of selecting a solution requires both qualitative and quantitative analyses, and many pieces of information exhibit fuzziness and incompleteness. Fuzzy theory [39] represents one of the widely applied methodologies in such cases. Mahmood Shafiee [40] considered wind energy site selection as a complex decision problem with a high degree of uncertainty, and proposed the FANP-TOSIS model to evaluate potential sites for wind power development. In response to the challenges in selecting an optimal blasting plan for a certain bauxite deposit with significant dimensions, Wu [41] et al. integrated Analytic Network Process (ANP), Fuzzy Theory, and TOPSIS theory to make the blasting plan selection.

Xin et al. [42] based on the grey theory, calculated the evaluation grey category for each rating value using the whitenization weight function. This effectively illustrates the degree of membership of each indicator to different levels, addressing the issue of quantitative assessment in the bridge inspection system. In the multi-objective decision-making problem for the selection of railway route schemes, Li et al. [43] replaced the evaluation of qualitative indicators with interval fuzzy numbers and transformed them into cloud model features. This approach effectively addresses the limitations related to the fuzziness and randomness in the evaluation method. Hence, the paramount consideration in addressing multi-attribute decision-making challenges is to ascertain the optimal methodologies for enhancing the efficiency and applicability of decision processes.

2.4 Comparison of MADM methods

Weight calculation methods can be primarily categorized into two major classes: subjective weighting, which determines weights through comprehensive consultation-based scoring, and objective weighting, which calculates weights based on interrelationships or variations among indicator values. Prominent integrated decision-making methodologies encompass TOPSIS, cloud model, GRA, ELECTRE, VIKOR, among others. Drawing from the existing research, these approaches will be synthetically elucidated and compared, as shown in Table 1.

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Table 1. Comparison of methodologies in scheme selection.

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The literature review presented above illustrates that the construction technology for bridge pile foundations in karst areas and the exploration techniques have been consistently developing and making breakthroughs. However, there has been limited exploration regarding the multi-attribute decision-making for pile foundation construction schemes. While the methodological framework for multi-attribute decision-making problems is relatively comprehensive, each approach still exhibits its own inherent limitations. This study contributes to the fusion of theoretical and practical aspects concerning project management issues in the field of bridge pile foundation construction. It is crucial to emphasize that, with the aim of promoting sustainability, in research representing the technical disciplines, the importance lies not only in innovative technical methods but also in the paramount practicality of research outcomes [49]. Theoretical research can provide a foundation for the implementation of new technologies, and it is also essential for construction preparations to include assessments of costs, quality, and safety. Therefore, research on scheme decision-making is meaningful. In practical engineering construction, traditional karst treatment methods are determined by a single influencing factor. The decision-making process is characterized by strong subjectivity, expert reliance, and difficulties in harmonizing diverse opinions. The constructed evaluation criteria system still faces issues related to comprehensiveness and limited applicability. To address these challenges, the adoption of appropriate decision models becomes essential.

Then, a novel approach is proposed to study the decision-making process for pile foundation construction schemes in karst areas. Firstly, an evaluation criteria system for construction schemes is constructed, taking into account technical, economic, quality, and environmental aspects. An important point to consider is the utilization of directed graphs within the system structure model and the Bellman-Ford algorithm in the expert rating phase of the Analytic Network Process (ANP). In the form of distributing survey questionnaires, aim to seek subjective ratings from domain experts for the indicators, and then input the obtained ratings into the Super Decision software to derive the indicator weights. Drawing on the concept of dynamic weighting, the introduction of universal weights serves as objective weights. Subsequently, The data from relevant research studies is analyzed. Multiple linear regression is applied to these data, and the correlation coefficients among four indicators within similar projects are calculated. This enables the establishment of linear relationships between the decision-making and control layer indicators, which are then weighted to determine their subjective importance at the control level. Based on the concept of dynamic weights, it is employed as a universal weight and integrated with the control layer indicators’ weights to constitute composite weights. Finally, by integrating grey fuzzy evaluation, the scores for each alternative scheme are calculated, which facilitates the decision-making process for the construction plan. A case study of a railway bridge project in southern China is conducted to validate the feasibility of this research methodology.

3.1 Construction of the evaluation indicator system

Based on the characteristics of pile foundation construction in karst regions and engineering practices, and with reference to similar literature [50–63], four primary indicators and sixteen secondary indicators were constructed.

Taking into account the four major management elements at the construction site, the technical criteria, economic criteria, working environment, and safety and quality were used as primary evaluation indicators. Following comprehensive, objective, and scientific principles, the assessment of the construction schemes’ merits was conducted.

Focusing on the 16 secondary indicators, decision analysis was conducted. Considering the characteristics of these indicators, U₁₃, U₃₁, U₃₂, and U₃₅ are quantitative indicators, while the remaining indicators are qualitative.

The evaluation index system is presented in Table 2.

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Table 2. The evaluation index system.

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3.2 Improved Analytic Network Process (ANP) method

The Analytic Network Process (ANP) [64], proposed by Professor Saaty as an extension of the Analytic Hierarchy Process (AHP), is specifically designed to accommodate non-independent hierarchical structures. This approach takes into consideration the interrelationships among various factors, employing nonlinear structures instead of linear structures and accounting for the dominance relationships among different hierarchical elements.

During the construction process, there exist interdependencies and inseparable relationships among the technical, economic, environmental, and quality criteria. Therefore, the adoption of the ANP method for analysis is more appropriate. The ANP comprises a control layer and a network layer. It involves establishing network relationships among different criteria to assess the pairwise comparisons between indicators and construct an evaluation matrix. By calculating the weighted hypermatrix and the limit hypermatrix, which consider both relative preferences and importance degrees, the weight values for each second-level indicator are determined.

The comparative judgment matrix used in the ANP is a positive reciprocal matrix that possesses transitivity. Therefore, in the evaluation process, it is necessary to examine the consistency of the judgment matrix through the Consistency Ratio (CR). If the CR value is greater than 0.1, it is considered that the judgment matrix satisfies consistency, and the test is passed.

The comparison and selection of pile foundation construction schemes are complex decision problems involving multiple attributes and criteria. During the decision-making process, when experts are invited to participate in group decision-making, the following issues typically arise:

1. Different experts provide similar or conflicting evaluation opinions, and after collecting the rating data, the decision matrix may not even pass the consistency test, leading to inefficiencies in the indicator evaluation process.
2. When there is a large number of experts, there are certain difficulties in achieving consensus and integration of the final scoring values.
3. The conventional ANP method requires experts to pairwise compare indicators, and it becomes challenging for experts to intuitively assess the relative importance of factors from an overall perspective.

To address the above-mentioned issues, directed graphs and the Bellman-Ford algorithm are introduced into the system structure to improve the ANP method. A directed graph, originating from graph theory, consists of nodes and directed edges, where nodes represent elements, and directed edges represent the influence relationships among these elements. HUANG [65] based on the Interpretative Structural Modeling Method to elucidate the rules for describing the relationships among elements using a directed graph. They also illustrated the process of transforming a directed graph into an adjacency matrix, wherein, if element A exerts an influence on element B, the corresponding element in the adjacency matrix is denoted as 1. Hence, the directed graph has become an effective tool for visualizing connections among elements. GAO [66] used directed graphs to represent the cause-and-effect relationships between explicit and implicit factors in the enterprise environmental assessment information system, with the direction of the edges indicating positive or negative influences and reachability relationships. Bellman-Ford algorithm [67] originates from the computational method for the shortest path in graph theory. In comparison to the Dijkstra algorithm, it has the advantage of being able to handle negative weights.

The introduction of directed graphs primarily aims to visualize the evaluation objects, enabling experts to have a holistic perspective and make intuitive judgments of the evaluation objects, aligning with the central philosophy of the ANP method. Following the principles of G1, experts are required to convey the importance of evaluation criteria in an ordered relationship from left to right, ensuring the consistency of the judgment matrix. The numerical values along the edges directly reflect the significant relationships between adjacent criteria. In order to maintain consistency with the 1–9 scale scoring method, the improvement is made by adopting a five-segment scoring scale, thus providing a clear quantification of the importance values for pairwise comparisons of factors. Subsequently, BF algorithm performs pairwise weight calculations between the criteria, achieving the transformation from a directed graph to a judgment matrix, serving as a bridge between graphical and matrix representations.

Step1: Inviting experts to rank the importance of factors in the network layer and criterion layer, denoted as . Quantifying the importance values between pairwise comparisons of metrics using the set {0.5, 1, 1.5, 2, 2.5} corresponding to progressively increasing levels of importance.

(1)

Where is the rating value of the j_k indicator in the i-th layer; is the importance value between the two indicators.

Step2: Converting different evaluation orders into ascending sequences, where the assignment of values between elements is determined based on the cumulative summation operation using the Bellman-Ford algorithm.

Step3: For a single expert, the judgment matrix can be directly obtained from the ascending evaluations. In the case of multiple experts, multiple sets of ascending scores are collected, and their average values are rounded to obtain the judgment matrix. The evaluation process for a single expert is illustrated in Fig 1.

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Fig 1. Expert scoring flow chart.

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Step4: Calculate the Consistency Ratio (CR) for each judgment matrix. If CR < 0.1, it passes the consistency test.

Step5: Utilizing the Super Decision software, the hypermatrix matrix and limit matrix were calculated to obtain the indicator weights.

3.3 Prediction of universal weights based on MLR

Weights are typically computed by experts around a specific instance or evaluation system, reflecting a static perspective. Jiang [68] recognized often disregarded dynamic changes within indicators and introduced a dynamically weighted grey optimization model, suggesting that the weight assigned to indicators is dependent on their amplitude of change across distinct categories. Wang [69] employed historical data pertaining to aqueduct engineering as a universal reference weight, thereby facilitating the dynamization of evaluation indicator weights. Li [70] He conducted a frequency analysis of the occurrences of risk impact factors to compute universal weights, emphasizing the reference value of historical data from similar construction projects. Therefore, the introduction of universal weights is conducive to reflecting the objectivity and dynamic changes of indicator weights.

3.3.2 Prediction of universal weights.

In research concerning decision-making, it is common to establish an indicator system to assess the strengths and weaknesses of various schemes. The calculated weights for different indicators reflect the relative importance of each criterion in the project scheme evaluation. Moreover, in the context of research related to the evaluation of similar schemes, the weight values of criteria for different projects hold certain reference significance.

The control layer indicators also serve as common objectives for all engineering constructions, rendering the evaluation system comparable. Utilizing the assessments of these indicators’ significance by other scholars as references, the objective nature of weight determination is maximally pursued.

Data acquisition entailed a systematic literature search, utilizing key terms such as "construction scheme selection," "construction scheme optimization," and "scheme comparison" to identify relevant papers in the same domain. Twenty-three papers [27–29,76–95] were gathered using the commonality of control layer indicators as the filtering criterion. Weight data was organized, analyzed, and then merged with criteria of the same type, resulting in the formation of the sample dataset, as shown in Fig 2. Subsequently, by utilizing multiple linear regression, the regression equations between different criteria are calculated to establish the linear relationship between decision-making schemes and the four evaluation criteria, obtained the universal weights for similar projects, thus enabling the prediction of weights for this project. The calculation steps are as follows:

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Fig 2. Weights of sample data.

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Step1: Establishing the overall regression linear equation.

(2)

Where are respectively the dependent variable and the independent variable., β₀ is the parameter. is the error term, ε~N(0,σ²).

(3)

Where X_t,X_e,X_c,X_q are respectively technical criterion, economic criterion, construction environment and quality and safety criterion.

(4)

Then t cross product matrix of the variable augmented matrix be shown as: (5)

Step2: Calculation of the regression coefficients.

(6)

Step3: Joint significance test of the regression equation.

(7)

Where SST is Sum of total difference squares; SSE is residual sum of squares; is the average value of y; is the fitting value of the i-th sample; y_i is the regression value of the i-th sample.

By individually setting the technical criterion (Xi), economic criterion (Xc), construction environment (Xe), and quality and safety criterion (Xq) as dependent variables, and the remaining three criteria as independent variables. Based on multiple linear regression, identify the correlation between the dependent variable and independent variables, and calculate the four regression equations. The regression coefficients are then used to assign weights to different criteria.

The results of the joint significance test are presented in Table 3.

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Table 3. Parameters of joint significance test.

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Clearly, p < 0.05, indicating that at a 95% confidence level, the regression coefficients of the four equations are significantly different from zero. Therefore, the null hypothesis can be rejected, indicating the existence of a correlation between the dependent variable and independent variables.

By combining the four regression coefficient equations, the correlation coefficient equations for four independent variables can be solved based on the sample data from the multi-attribute decision-making research of similar schemes. These coefficients can be considered as the objective weight of the four criteria based on the sample data. The equation is as follows: (8)

3.4 Grey-fuzzy comprehensive decision model

Fuzzy comprehensive evaluation can comprehensively and integratively assess all factors of the evaluated object based on the principle of maximum membership degree, thereby determining the membership degree level of the evaluated object [96].Based on expert ratings, the introduction of whitenization weight functions is utilized to determine the membership degrees of evaluation values with respect to the grades. This approach leverages the advantages of membership degree to describe the boundaries of scheme evaluation grades, achieving the integration of qualitative and quantitative research and resolving the qualitative and quantitative issues of multi-attribute evaluation. It overcomes the subjective bias of individual decision-making and enhances the effectiveness and practicality of group decision-making.

3.4.1 Determination of the indicators sets and the evaluation sets.

Investigating engineering instances of pile foundation construction projects in karst areas, establishing four criteria as the control layer, and delineating 16 evaluation factor indicators as the evaluation indicator set U = {u1, u2, …, u16}. The evaluation set V = {Ⅰ, Ⅱ, Ⅲ, Ⅳ, Ⅴ} is divided into five levels, corresponding to excellent, good, moderate, pass, and fail. This reflects the quality of the pending construction schemes and establishes the score value ranges for each level [97], as shown in Table 4.

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Table 4. Score of evaluation level.

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3.4.3 Construction of the triangular whitenization weight function.

Due to the subjectivity of the ratings provided by management personnel, determining the rating levels is an urgent issue that needs to be addressed. Typically, in fuzzy comprehensive evaluation, it is difficult to determine the membership functions. Therefore, the concept of gray numbers is adopted to assess the membership of rating values to different gray classes.

The utilization of the central point whitenization weight function helps to avoid the issue of multiple gray classes covered by the same rating value, thereby preventing ambiguity in membership to different levels [98]. The calculation steps are as follows: (1) Assuming there are n candidate schemes, the value range of the 16 evaluation criteria is divided into five gray classes. (2) Let the rating value be defined as the centroid λk of the k-th gray class. The most probable range of rating values for the k-th gray class is [λ_k-1, λ_k+1], k = 1,2,…,5. The left endpoint of the first gray class and the right endpoint of the fifth gray class are obtained by extending the rating value range to the left and right, respectively. Subsequently, by connecting the starting point of class k-1 with the center point of class k, the triangular whitenization weight function fj can be constructed. The expression is shown in Table 5, and the function graph is depicted in Fig 3. (3) Calculate the membership degree Rij of each rating value dij belonging to class k.

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Fig 3. Diagram of triangular whitenization weight function.

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Table 5. Expression of triangular whitenization weight function.

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3.4.4 The comprehensive evaluation result.

The second-level indicator comprehensive evaluation set bj is calculated using the fuzzy operator M(·,+) according to the following formula: (9)

The comprehensive evaluation set Bj of the control layer indicators is calculated using the following formula: (10)

The comprehensive scores Z for each candidate scheme are calculated according to the following formula, based on which comprehensive ranking can be conducted.

(11)

4 Empirical analysis

4.1 Engineering situations.

The bridge project is located in the southern part of China, with a total length of approximately 3600 meters. The pile foundation lengths range from 13.5 to 74.5 meters, with a total quantity of about 1500 piles. The selection of this project for study is motivated by its unique characteristics. (1) The magnitude of the engineering quantities and the corresponding substantial financial investment accentuate the necessity for the most appropriate construction approach. Furthermore, the methodology should be amenable to iterative enhancements and validation through recurrent applications. (2) Intricate geological conditions are prevalent. The engineering site is covered by a geological layer of sand with a thickness ranging from 4 to 40 meters. The majority of the pile foundations are concentrated in karst regions, constituting 54.9% of the total. (3) Karst characteristics display a diverse range of features, characterized by the existence of caves with varying vertical extents, spanning from 1 to 10 meters in height.

Representative foundation piles within this project have been selected. Taking Pile 94–4# of the project as an example, the geotechnical investigation indicates the presence of karst caves with depths of 8.7 meters and 7.7 meters, and a sand layer thickness of 37.7 meters. Based on this information, three alternative schemes [20,21,24] are proposed as shown in Table 6.

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Table 6. Pile foundation construction options.

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4.2 Calculation of subjective weights based on the improved ANP

Experts were invited to engage in the research questions through interviews and questionnaire completion, demographic characteristics of experts as shown in Table 7. Particularly, with due consideration for the domain-specific expertise of experts across different fields, the foremost five experts listed in the table were solicited to undertake the assessment of criteria, while the last five experts from the table were engaged for the selection of scheme decision, aiming to enhance the professionalism and rigor of the evaluation outcomes.

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Table 7. Demographic characteristics of experts in this study.

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Subsequently, based on the correlation analysis of the indicators, the establishment of a ANP network hierarchical structure was carried out, as shown in Fig 4. Using the secondary indicators in the technical criteria as an example, questionnaires were collected to obtain ratings from five experts for different indicators, as shown in Table 8. The judgment matrix was computed according to the method described in Section 2.2, and its corresponding consistency ratio (CR) was calculated using the MATLAB software, as shown in Tables 9–13. Furthermore, subjective weights were calculated using AHP and G1 methods and compared with those derived from the improved ANP method, as shown in Fig 5.

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Fig 4. Diagram of ANP network structure.

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Fig 5. Comparison of three methods to calculate weight values.

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Table 8. Expert scoring values of criterion layer U₁.

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Table 9. Judgment matrix of criterion layer U₁.

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Table 10. Judgment matrix of criterion layer U₂.

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Table 11. Judgment matrix of criterion layer U₃.

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Table 12. Judgment matrix of criterion layer U₄.

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Table 13. Judgment matrix of control layer U.

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After calculation, it was found that the average CR of the judgment matrix was less than 0.1, indicating that the consistency test has passed. The subjective weights of the control layer indicators can be computed using the Super Decision software.

According to the analysis of expert subjective evaluations, it can be observed that in the technical criteria, time limit of construction and the difficulty of construction are relatively important. In the economic criteria, the construction cost of karst cave and the construction cost are considered significant. Regarding the operational and environmental criteria, attention should be given to the filling degree of karst caves and the maximum height of karst cavity. In terms of safety and quality criteria, the focus should be on the construction technology control measures and the emergency response plans. Among the four control criteria, experts believe that the technical criteria occupy a crucial position.

4.3 Computation of combined weights

Universal weights, predicted through the multiple linear regression method based on sample data, are used as objective weights. By combining the subjective weights determined through the improved Analytic Network Process (ANP) to obtain the combined weights ω_Combined. Considering the limited sample size and the non-replicability of each engineering project, a weighted average approach is employed to determine the combined weights of the control layer indicators. This approach takes into account the inadequacy of data samples and the unique characteristics of each project, with the mean value serving as the composite weight for the control layer indicators.

When the subjective evaluation concludes that the technical criteria hold a significant position, in this case, further statistical analysis and data mining can be conducted. It becomes evident that in the decision-making process regarding pile foundation construction plans in karst areas, the technical, quality and safety, and economic criteria all possess undeniable importance that cannot be overlooked.

4.4 Grey-fuzzy evaluation method

4.4.1 Categorizations of indicator evaluation levels.

Drawing upon the existing literature and the practical engineering context, the classification of indicator evaluation levels was established. Regarding quantitative indicators, taking thickness of sand layer(U₃₁) as an example, the interval partition sets for Ruan [50] are {5, 10, 15, 20}, for Huang [63] {20, 30; 40}, and Hao [57] divides 5 levels into with 3 intervals {10, 30}. Clearly, there is a lack of uniformity in the delineation ranges for the same indicator across different studies, which is in accordance with the variations in practical engineering scenarios. There are notable disparities in karst geological conditions across different regions. For instance, indicators like thickness of the first rock layer(U₃₂) allow for a graded classification by calculating mechanical parameters. Conversely, metrics such as "thickness of sand layer(U₃₁), the maximum height of karst cavity(U₃₅), and time limit of construction(U₁₃) prove challenging to generalize, given their strong contextual relevance to specific engineering conditions. Therefore, the K-Means algorithm is suggested for data clustering based on the project’s unique geological survey information. Subsequently, the results are compared with established standards in existing literature to effectively determine the classification of indicator evaluation levels.

The provided pile foundation geological survey maps were taken as the sample set, with the Euclidean distance used as a metric to measure the similarity between data objects. Based on the evaluation requirements, a total of 760 pile foundation data were selected as samples and input into the SPSS software for analysis and calculations. Firstly, the number of clusters is set to 5. Subsequently, an iterative algorithm continuously updates the positions of cluster centers by minimizing the square sum of errors within each cluster, until the objective function converges, indicating the completion of the clustering process. Finally, the clustering result is used to assign levels to the quantitative indicators. The sample data is shown in Fig 6, and the classification range of the secondary index is shown in Table 14.

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Fig 6. Sample data of K-means clustering.

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Table 14. Category range of secondary indicators.

https://doi.org/10.1371/journal.pone.0295296.t014

4.4.2 Construction of evaluation matrix.

The data from the survey questionnaire were analyzed, and the average rating scores for each alternative solution are presented in Table 15. The data were incorporated into a whitenization weighting function to calculate the degree of membership of the rating values for different levels. Furthermore, employing MADM analytical approaches including cloud models, ELECTRE-II, and VIKOR methodologies. The computational processes are presented in supporting information.

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Table 15. Evaluation matrix.

https://doi.org/10.1371/journal.pone.0295296.t015

The comprehensive evaluation set was calculated based on Formula (10). The specific data are presented as follows: