Method for optimizing complex decision problem of high and large aqueduct group based on AHP-FCE-TOPSIS comprehensive model

Abstract

The invention discloses an optimization method for a high and large aqueduct group complex decision problem based on an AHP-FCE-TOPSIS comprehensive model, and relates to the field of hydraulic engineering, and the method comprises the steps: S1, building an evaluation index system; s2, performing pairwise comparison on the evaluation indexes in the index layer by using an analytic hierarchy process, and calculating a weight vector W of each evaluation index in the criterion layer or the index layer by constructing a judgment matrix; s3, performing fuzzy evaluation on the performance of each scheme in the scheme layer under the evaluation index by using a fuzzy comprehensive evaluation method, and constructing a membership matrix R; s4, multiplying the membership matrix R by the weight vector W to obtain a weighted fuzzy evaluation matrix F; s5, determining an ideal solution and a negative ideal solution of the weighted fuzzy evaluation matrix F by using an approximate ideal solution sorting method; calculating the relative closeness of each scheme to the ideal solution and the negative ideal solution; and sorting the schemes according to the relative proximity, and outputting a sorting result to complete decision optimization. And the scientificity, accuracy and reliability of complex decision making are comprehensively improved.

Description

Technical Field

[0001] This invention relates to the field of water conservancy engineering technology, and in particular to an optimization method for complex decision-making problems of tall aqueduct groups based on the AHP-FCE-TOPSIS integrated model. Background Technology

[0002] In many fields such as water conservancy and civil engineering, complex decision-making problems frequently arise, such as selecting the overall construction plan for a large aqueduct complex. These problems are typically characterized by multiple objectives, multiple criteria, fuzziness, and uncertainty. Decision-makers need to comprehensively consider multiple interrelated and even conflicting factors, including cost, schedule, safety, quality reliability, and environmental impact. How to scientifically weigh these factors and ultimately select the optimal solution from multiple alternatives is an extremely challenging task.

[0003] Currently, commonly used methods for solving such complex decision-making problems include the Analytic Hierarchy Process (AHP), Fuzzy Comprehensive Evaluation (FCE), and the Top-Order Solution Approximation Method (TOPSIS).

[0004] The Analytic Hierarchy Process (AHP) can systematically determine the relative weights of each evaluation indicator by constructing a hierarchical model and pairwise comparison judgment matrices. Its process is clear and logical. However, the AHP method has limited ability to handle the large amount of fuzzy information and subjective judgments in the decision-making process, and it is difficult to effectively characterize and quantify fuzzy concepts such as "high security" and "reliable quality".

[0005] Fuzzy Comprehensive Evaluation (FCE) excels at handling this fuzziness and uncertainty. By establishing membership functions and fuzzy relation matrices, it can transform qualitative evaluation into quantitative analysis, thereby providing an overall evaluation of things constrained by multiple factors. However, the FCE method itself typically lacks the ability to intuitively and linearly rank multiple alternatives, and its evaluation results are often a fuzzy set, making it difficult to directly use for comparing the merits of different options.

[0006] The Top-Approximation-to-Ideal-Solution Ranking (TOPSIS) method ranks solutions by calculating their distances from the ideal optimal and ideal worst solutions. The concept is intuitive, the calculations are relatively simple, and the results are easy to understand. However, the TOPSIS method itself has shortcomings in determining the weights of evaluation indicators and handling the fuzziness of evaluation information. Its effectiveness largely depends on the scientific nature of the weights and the accuracy of the initial data.

[0007] In existing technologies, although the combination of AHP and TOPSIS can determine weights and handle fuzzy information, the final ranking may still not be intuitive enough; the combination of AHP and TOPSIS fails to handle the fuzziness in the original data well.

[0008] Therefore, existing single methods or pairwise combined models still have problems such as insufficient decision-making process and need to improve the scientific nature of results when dealing with highly complex decision-making scenarios with incomplete information and full of uncertainty. Summary of the Invention

[0009] The technical problem to be solved by this invention is to provide a comprehensive decision-making method that can take into account the scientific nature of weighting, the ambiguity of information, and the intuitiveness of ranking.

[0010] To address the aforementioned technical problems, this invention provides an optimization method for complex decision-making problems involving large aqueduct groups based on the AHP-FCE-TOPSIS integrated model, comprising the following steps: S1. Establish an evaluation index system, which includes an objective layer, a criterion layer, and a scheme layer; S2. Using the analytic hierarchy process (AHP), compare each evaluation index in the criterion layer or indicator layer that belongs to the same upper-level factor in pairs. By constructing a judgment matrix, calculate the weight vector W of each evaluation index in the criterion layer or indicator layer. S3. Use the fuzzy comprehensive evaluation method to perform a fuzzy evaluation on the performance of each scheme in the scheme layer under its evaluation index, and construct the membership matrix R. S4. Multiply the membership matrix R with the weight vector W to obtain the weighted fuzzy evaluation matrix F; S5. Use the approximation ideal solution ranking method to determine the ideal solution and negative ideal solution of the weighted fuzzy evaluation matrix F; calculate the relative proximity of each scheme to the ideal solution and the negative ideal solution; and rank each scheme according to the relative proximity, outputting the ranking result to complete the decision optimization.

[0011] Furthermore, S2 specifically includes: S21. Using the 1-9 scale method, pairwise importance comparisons are performed on multiple evaluation indicators belonging to the same upper-level factor to construct judgment matrix A:

[0012] In the formula, Indicates the first Factors relative to the first The importance scale of factors , In this context, 1 indicates equal importance; 9 indicates absolute importance. S22. Calculate the eigenvector corresponding to the largest eigenvalue of the judgment matrix A, or calculate the geometric mean of the elements in each row of the judgment matrix A and perform normalization processing to obtain the weight vector W representing the relative weight of each evaluation factor. S23. Calculate the consistency index of the judgment matrix A. Ratio of random consistency :

[0013]

[0014] In the formula, This represents the largest eigenvalue of the judgment matrix A. Indicates the order of the judgment matrix; This represents the random consistency index, obtained through table lookup. S24, if the random consistency ratio If the value is less than a preset threshold, the weight vector W is determined to be valid and output. If the random consistency ratio If the value is greater than or equal to the preset threshold, then return to step S21, readjust and reconstruct the judgment matrix A.

[0015] Furthermore, in step S22, the geometric mean of the elements in each row of the judgment matrix A is calculated, which satisfies the formula:

[0016] In the formula, Indicates the first Factors relative to the first The importance scale of factors; This indicates the order of the judgment matrix.

[0017] Furthermore, S3 specifically includes: Define an evaluation set, which contains multiple evaluation levels; For each evaluation indicator, the membership degree of each scheme to each evaluation level in the evaluation set is determined according to expert scores or membership functions. The membership matrix R is formed by the membership vectors of all schemes under all evaluation indicators.

[0018] Furthermore, S4 specifically includes: The weighted average operator is used to perform fuzzy synthesis calculation on the weight vector W and the membership matrix R; the weighted fuzzy evaluation matrix F satisfies the calculation formula:

[0019] The matrix elements of the weighted fuzzy evaluation matrix F Each row represents a solution, and each column represents an evaluation level. Matrix elements This indicates that the scheme, after being weighted across all indicators, belongs to the [number]th [indicator]. The overall degree of evaluation across all rating levels.

[0020] Furthermore, determining the ideal solution and the negative ideal solution in S5 specifically includes: For benefit-type indicators, the maximum value is taken from each column of the weighted fuzzy evaluation matrix F as the ideal solution S. + The corresponding component is taken as the minimum value as the negative ideal solution S. - The corresponding components; For cost-related indicators, the minimum value is taken from each column of the weighted fuzzy evaluation matrix F as the ideal solution S. + The corresponding component is taken as the negative ideal solution S. - The corresponding component.

[0021] Furthermore, in S5, the relative proximity of each solution to the ideal solution and the negative ideal solution is calculated, which satisfies the expression:

[0022] In the formula, Indicates relative proximity, and ; Indicate that each scheme is related to the negative ideal solution S. - The Euclidean distance; Indicate that each scheme is related to the ideal solution S. + The Euclidean distance.

[0023] Furthermore, the method can be applied to any scenario in construction scheme optimization, comprehensive project evaluation, risk assessment, or resource allocation optimization.

[0024] Compared with the prior art, the embodiments of the present invention have the following beneficial effects: This invention organically links and deeply integrates the scientific weights determined by the analytic hierarchy process, the fuzzy information processed by the fuzzy comprehensive evaluation method, and the intuitive ranking ability of the TOPSIS method, effectively overcoming the limitations of single decision-making models or pairwise combination models, and comprehensively improving the scientificity, accuracy, and reliability of complex decisions. Attached Figure Description

[0025] To more clearly illustrate the technical solutions in the embodiments of the present invention, the accompanying drawings used in the description of the embodiments will be briefly introduced below. Obviously, the accompanying drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0026] Figure 1 This is a flowchart disclosed in this invention. Detailed Implementation

[0027] To enable those skilled in the art to better understand the present invention, the technical solutions of the present invention will be clearly and completely described below with reference to the accompanying drawings of the embodiments of the present invention. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0028] This invention aims to provide an optimization method for complex decision-making problems of large aqueduct groups based on the AHP-FCE-TOPSIS integrated model. By scientifically determining weights, handling fuzziness, and intuitively ranking the data based on the integrated model of AHP, FCE, and TOPSIS, the method achieves optimization and selection of complex decision-making problems.

[0029] Please see Figure 1 It mainly includes the following steps: S1. Establish an evaluation index system, which includes the target layer, criterion layer, and scheme layer.

[0030] Specifically, the decision-making objectives are clearly defined, and a hierarchical model is constructed, including an objective layer, a criterion layer, and a solution layer. The objective layer represents the ultimate goal of the decision; the criterion layer outlines the intermediate steps and core dimensions involved in achieving the objective; and the solution layer comprises the various alternative solutions to be evaluated.

[0031] S2. Using the Analytic Hierarchy Process (AHP), compare each evaluation index in the criterion layer or indicator layer that belongs to the same upper-level factor in pairs. By constructing a judgment matrix, calculate the weight vector W of each evaluation index in the criterion layer or indicator layer.

[0032] This step provides a scientifically quantifiable basis for the importance of indicators in subsequent evaluations, specifically: S21. Using the 1-9 scale method, pairwise importance comparisons are performed on multiple evaluation indicators belonging to the same upper-level factor to construct judgment matrix A:

[0033] In the formula, Indicates the first Factors relative to the first The importance scale of factors , In this system, 1 indicates equal importance and 9 indicates absolute importance.

[0034] S22. Calculate the eigenvector corresponding to the largest eigenvalue of the judgment matrix A, or calculate the geometric mean of the elements in each row of the judgment matrix A and perform normalization processing to obtain the weight vector W representing the relative weight of each evaluation factor.

[0035] Taking the geometric mean method as an example, we first calculate the geometric mean of each row of elements, and then normalize the resulting vector. The geometric mean method satisfies the following formula:

[0036] In the formula, Indicates the first Factors relative to the first The importance scale of factors; This indicates the order of the matrix.

[0037] Finally, the weight vector is obtained. .

[0038] S23. Calculate the consistency index of judgment matrix A. Ratio of random consistency :

[0039]

[0040] In the formula, This represents the largest eigenvalue of the judgment matrix A. The order of the judgment matrix represents the number of rows or columns, which is equal to the number of factors compared in the criterion or indicator layer. For example, when comparing three factors (cost, quality, and environment), then... ; The random consistency index is obtained by looking up a table, such as... When =3, =0.58.

[0041] S24, If the random consistency ratio If the value is less than the preset threshold, the weight vector W is deemed valid and output. If the random consistency ratio If the value is greater than or equal to the preset threshold, return to S21, readjust and reconstruct the judgment matrix A.

[0042] In a specific example, if If the value is less than 0.1, the consistency of the judgment matrix A is considered acceptable and the weight vector W is valid; otherwise, the judgment matrix A needs to be readjusted until the consistency test is passed.

[0043] S3. Use the fuzzy comprehensive evaluation method to perform fuzzy evaluation on the performance of each scheme in the scheme layer under its evaluation index, and construct the membership matrix R.

[0044] This step is used to quantify the ambiguity and subjective judgment in decision-making, specifically: S31, Setting the Evaluation Set Evaluation set It includes multiple rating levels, such as: {Excellent, Good, Average, Poor}.

[0045] S32. For each evaluation indicator, based on expert scores or membership functions, determine the impact of each scheme on the evaluation set under that indicator. The membership degrees of each evaluation level are used to form the membership vector of the indicator. .

[0046] S33. The membership matrix R is constructed from the membership vectors of all schemes under all evaluation indicators:

[0047] in, The plan indicates the indicators The following were rated as level Membership degree ( ).

[0048] S4. Multiply the membership matrix R with the weight vector W to obtain the weighted fuzzy evaluation matrix F.

[0049] This step is crucial for connecting AHP and FCE to achieve model fusion. Specifically: This scheme preferably uses a weighted average operator to perform fuzzy synthesis calculation on the weight vector W and the membership matrix R; the weighted fuzzy evaluation matrix F satisfies the calculation formula:

[0050] Matrix elements of the weighted fuzzy evaluation matrix F Each row represents a solution, and each column represents an evaluation level. Matrix elements This indicates that the scheme, after being weighted across all indicators, belongs to the [number]th [indicator]. The overall degree of evaluation of each level, of which:

[0051] It can be seen that the weighted fuzzy evaluation matrix F integrates the weight information of the indicators and the fuzzy evaluation information of the scheme. Each row of the matrix represents the comprehensive degree to which a scheme belongs to each evaluation level after being weighted on all indicators.

[0052] S5. Use the approximation ideal solution ranking method to determine the ideal solution and negative ideal solution of the weighted fuzzy evaluation matrix F; calculate the relative closeness of each scheme to the ideal solution and negative ideal solution; rank each scheme according to the relative closeness, and output the ranking results to complete the decision optimization.

[0053] This step utilizes the TOPSIS method to clearly sort the fused results, specifically: S51. Use the weighted fuzzy evaluation matrix F as the input decision matrix of TOPSIS.

[0054] S52. Determine the ideal solution S + and negative ideal solution S - : For benefit-type indicators, the maximum value in each column of the weighted fuzzy evaluation matrix F is taken as the ideal solution S. + The corresponding component is taken as the minimum value as the negative ideal solution S. - The corresponding components; For cost-related indicators, the minimum value in each column of the weighted fuzzy evaluation matrix F is taken as the ideal solution S. + The corresponding component is taken as the negative ideal solution S. - The corresponding component.

[0055] S53. Calculate the sum of the ideal solution and the solution S for each scheme. + Euclidean distance d + , and the negative ideal solution S - Euclidean distance d - :

[0056]

[0057] In the formula, Representation scheme In indicators The weighted evaluation value; Indicates the ideal solution / negative ideal solution at the th... The value of the indicator.

[0058] S54. Calculate the relative similarity of each scheme. :

[0059] in, The larger the value, the closer the solution is to the ideal state. Based on this, all solutions are ranked according to their merits.

[0060] This solution can be applied to any scenario in construction plan optimization, comprehensive project evaluation, risk assessment, or resource allocation optimization.

[0061] Example The following is a calculation example of the optimal construction scheme for a large aqueduct group based on the AHP-FCE-TOPSIS integrated model, using a common pier as an example. The calculation process is shown in detail, incorporating actual engineering parameters and average social data.

[0062] A certain aqueduct project includes two types of structures: 1. Simply supported aqueduct: span 42m, pier height is divided into two categories, namely ordinary pier (15m) and high pier (40m), both of which include prefabrication hoisting, scaffolding cast-in-place, and mobile formwork.

[0063] 2. Continuous rigid frame aqueduct: main span 120m, cantilever construction only using hanging baskets (fixed scheme, not included in the selection process).

[0064] The AHP-FCE-TOPSIS calculation process is as follows: Step 1: Establish an evaluation index system, as shown in Table 1 below: Table 1

[0065] Step 2: Determine the weights using the AHP method The judgment matrix A (using the 1-9 scale) is shown in Table 2 below: Table 2

[0066] 2.1 Based on the geometric mean method, the weight vector W is calculated as follows: Cost: (1×0.5×3×3×2)^(1 / 5)=1.552 Construction period: (2×1×3×3×2)^(1 / 5)=2.048 Safety: (0.333 × 0.333 × 1 × 0.5 × 0.333)^(1 / 5) = 0.457 Mass: (0.333 × 0.333 × 2 × 1 × 0.5)^(1 / 5) = 0.644 Environment: (0.5×0.5×3×2×1)^(1 / 5)=1.084 2.2 Normalized weight vector W Σ=1.552+2.048+0.457+0.644+1.084=5.785 The weight vector W = [0.268, 0.354, 0.079, 0.111, 0.188] corresponds to cost, construction period, safety, quality, and environment. 2.3 Consistency Check: =5.213, =(5.213-5) / (5-1)=0.053, =1.12 (n=5), =0.053 / 1.12=0.047<0.1, indicating that the consistency test is passed.

[0067] Step 3: Fuzzy evaluation using the FCE method Evaluation set: V = {Excellent (4), Good (3), Average (2), Poor (1)} Quantitative indicator conversion rules: Cost (ten thousand yuan): Excellent <100, Good 100-130, Average 130-160, Poor >160; Construction period (days): Excellent <30, Good 30-40, Average 40-50, Poor >50.

[0068] The original data for the ordinary pier scheme are shown in Table 3 below: Table 3

[0069] The fuzzy evaluation matrix (membership vector) is shown below: Prefabrication and hoisting: Cost → [0,1,0,0] (Good) Construction period → [1,0,0,0] (Excellent) Safety → [0,1,0,0] (Good) Quality → [1,0,0,0] (Excellent) Environment → [0,0,1,0] (Medium) Cast-in-place support: Cost → [0,0,1,0] (Medium) Construction period → [0,0,1,0] (Medium) Security → [0,0,1,0] (Medium) Quality → [0,1,0,0] (Good) Environment → [0,0,0,1] (Poor) Mobile mold frame: Cost → [0,0,0,1] (difference) Construction period → [0,0,0,1] (difference) Security → [1,0,0,0] (Excellent) Quality → [0,1,0,0] (Good) Environment → [0,1,0,0] (Good) Step 4: Calculate the weighted fuzzy evaluation matrix Defuzzification (fraction vector V=[4,3,2,1]) is shown in Table 4 below: Table 4

[0070] Weighted fuzzy evaluation matrix F: Prefabrication and hoisting: [3×0.268,4×0.354,3×0.079,4×0.111,2×0.188]=[0.804,1.416,0.237,0.444,0.376] Cast-in-place support: [2×0.268,2×0.354,2×0.079,3×0.111,1×0.188]=[0.536,0.708,0.158,0.333, 0.188] Mobile mold frame: [1×0.268,1×0.354,4×0.079,3×0.111,3×0.188]=[0.268,0.354,0.316,0.333, 0.564] Step 5: Sort using TOPSIS method Ideal solution S + =[max(0.804,0.536,0.268),max(1.416,0.708,0.354),...]=[0.804,1.416,0.316,0.444,0.564] Negative ideal solution S - =[min(0.804,0.536,0.268),min(1.416,0.708,0.354),...]=[0.268, 0.354,0.158,0.333,0.188] Calculation of Euclidean distance for prefabricated hoisting: d + =√[(0.804-0.804)²+(1.416-1.416)²+...]=0.203 d - =√[(0.804-0.268)²+(1.416-0.354)²+...]=1.212 Calculation of Euclidean distance for cast-in-place supports: d + =√[(0.536-0.804)²+(0.708-1.416)²+...]=0.866 d - =√[(0.536-0.268)²+(0.708-0.354)²+...]=0.444 Calculation of Euclidean distance for the moving formwork: d + =√[(0.268-0.804)²+( 0.354-1.416)²+...]= 1.195 d - =√[(0.268-0.268)²+( 0.354-0.354)²+...]= 0.406 Relative proximity of prefabricated hoisting: C1 = 1.212 / (0.203 + 1.212) = 0.856 The relative proximity of the cast-in-place support structure: C2 = 0.444 / (0.866 + 0.444) = 0.339 The relative proximity of the moving mold base: C3 = 0.406 / (1.195 + 0.406) = 0.254 The sorted output schemes are shown in Table 5 below: Table 5

[0071] The order of computation is: precast hoisting (C1=0.856) > cast-in-place with support (C2=0.339) > movable formwork (C3=0.254). Therefore, "precast hoisting" is the optimal solution for ordinary piers.

[0072] Although embodiments of the invention have been shown and described, it will be understood by those skilled in the art that various changes, modifications, substitutions and alterations can be made to these embodiments without departing from the principles and spirit of the invention, the scope of which is defined by the appended claims and their equivalents.

Claims

1. An optimization method for complex decision-making problems of tall aqueduct groups based on the AHP-FCE-TOPSIS integrated model, characterized in that, Includes the following steps: S1. Establish an evaluation index system, which includes an objective layer, a criterion layer, and a scheme layer; S2. Using the analytic hierarchy process (AHP), compare each evaluation index in the criterion layer or indicator layer that belongs to the same upper-level factor in pairs. By constructing a judgment matrix, calculate the weight vector W of each evaluation index in the criterion layer or indicator layer. S3. Use the fuzzy comprehensive evaluation method to perform a fuzzy evaluation on the performance of each scheme in the scheme layer under its evaluation index, and construct the membership matrix R. S4. Multiply the membership matrix R with the weight vector W to obtain the weighted fuzzy evaluation matrix F; S5. Use the approximation ideal solution ranking method to determine the ideal solution and negative ideal solution of the weighted fuzzy evaluation matrix F; calculate the relative proximity of each scheme to the ideal solution and the negative ideal solution; and rank each scheme according to the relative proximity, outputting the ranking result to complete the decision optimization.

2. The optimization method for complex decision-making problems of tall aqueduct groups based on the AHP-FCE-TOPSIS integrated model according to claim 1, characterized in that, S2 specifically includes: S21. Using the 1-9 scale method, pairwise importance comparisons are performed on multiple evaluation indicators belonging to the same upper-level factor to construct judgment matrix A: In the formula, Indicates the first Factors relative to the first The importance scale of factors , In this context, 1 indicates equal importance; 9 indicates absolute importance. S22. Calculate the eigenvector corresponding to the largest eigenvalue of the judgment matrix A, or calculate the geometric mean of the elements in each row of the judgment matrix A and perform normalization processing to obtain the weight vector W representing the relative weight of each evaluation factor. S23. Calculate the consistency index of the judgment matrix A. Ratio of random consistency : In the formula, This represents the largest eigenvalue of the judgment matrix A. Indicates the order of the judgment matrix; This represents the random consistency index, obtained through table lookup. S24, if the random consistency ratio If the value is less than a preset threshold, the weight vector W is determined to be valid and output. If the random consistency ratio If the value is greater than or equal to the preset threshold, then return to step S21, readjust and reconstruct the judgment matrix A.

3. The optimization method for complex decision-making problems of tall aqueduct groups based on the AHP-FCE-TOPSIS integrated model according to claim 2, characterized in that, In step S22, the geometric mean of the elements in each row of the judgment matrix A is calculated, which satisfies the formula: In the formula, Indicates the first Factors relative to the first The importance scale of factors; This indicates the order of the judgment matrix.

4. The optimization method for complex decision-making problems of tall aqueduct groups based on the AHP-FCE-TOPSIS integrated model according to claim 1, characterized in that, S3 specifically includes: Define an evaluation set, which contains multiple evaluation levels; For each evaluation indicator, the membership degree of each scheme to each evaluation level in the evaluation set is determined according to expert scores or membership functions. The membership matrix R is formed by the membership vectors of all schemes under all evaluation indicators.

5. The optimization method for complex decision-making problems of tall aqueduct groups based on the AHP-FCE-TOPSIS integrated model according to claim 1, characterized in that, S4 specifically includes: The weighted average operator is used to perform fuzzy synthesis calculation on the weight vector W and the membership matrix R; the weighted fuzzy evaluation matrix F satisfies the calculation formula: The matrix elements of the weighted fuzzy evaluation matrix F Each row represents a solution, and each column represents an evaluation level. Matrix elements This indicates that the scheme, after being weighted across all indicators, belongs to the [number]th [indicator]. The overall degree of evaluation across all rating levels.

6. The optimization method for complex decision-making problems of tall aqueduct groups based on the AHP-FCE-TOPSIS integrated model according to claim 5, characterized in that, Determining the ideal solution and the negative ideal solution in S5 specifically includes: For benefit-type indicators, the maximum value is taken from each column of the weighted fuzzy evaluation matrix F as the ideal solution S. + The corresponding component is taken as the minimum value as the negative ideal solution S. - The corresponding components; For cost-related indicators, the minimum value is taken from each column of the weighted fuzzy evaluation matrix F as the ideal solution S. + The corresponding component is taken as the negative ideal solution S. - The corresponding components.

7. The optimization method for complex decision-making problems of tall aqueduct groups based on the AHP-FCE-TOPSIS integrated model according to claim 6, characterized in that, In step S5, the relative proximity of each solution to the ideal solution and the negative ideal solution is calculated, which satisfies the expression: In the formula, Indicates relative proximity, and ; Indicate that each scheme is related to the negative ideal solution S. - The Euclidean distance; Indicate that each scheme is related to the ideal solution S. + The Euclidean distance.

8. The optimization method for complex decision-making problems of tall aqueduct groups based on the AHP-FCE-TOPSIS integrated model according to claim 1, characterized in that, The method can be applied to any scenario in construction scheme optimization, comprehensive project evaluation, risk assessment, or resource allocation optimization.

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