Comprehensive support hanger section multi-objective optimization method based on NSGA-II and optimal determination method

Abstract

The invention discloses a comprehensive support hanger section multi-objective optimization method based on NSGA-II and an optimal determination method. The comprehensive support hanger section multi-objective optimization method comprises the following four steps: establishing a multi-objective optimization mathematical model of a comprehensive support hanger steel section; establishing a non-dominated sorting genetic algorithm; the NSGA-II is adopted to solve the multi-objective optimization mathematical model; establishing implementation of a multi-objective decision-making method; evaluating, sorting and selecting solutions in the approximate Pareto non-inferior solution set by adopting a multi-objective decision-making method; according to the optimized design scheme, final design parameters of the comprehensive support hanger steel section are determined. The method is directed to comprehensive supports and hangers used in construction machinery, electrical and piping systems, aiming at simultaneously achieving material cost minimization and bending performance maximization. According to the method, the defects in the aspects of multi-objective optimization and comprehensive decision making in the prior art are overcome, and an efficient and practical solution with multi-objective comprehensive performance is provided for the design of the steel section of the comprehensive support hanger.

Description

Technical Field

[0001] This invention belongs to the field of building engineering technology, and in particular relates to a multi-objective optimization design method for integrated support and hanger cross sections based on non-dominated sorting genetic algorithm II and optimal solution determination method. Background Technology

[0002] Building Information Modeling (BIM) technology, as a digital information model, is widely used throughout the planning, design, construction, and operation of building projects. BIM technology enables 3D visualization of building components, information integration, and collaborative work, significantly improving design collaboration efficiency and information transmission accuracy, and aiding in the discovery of design conflicts and preliminary quantity surveying. Non-dominated sorting genetic algorithm II (NSGA-II) is a widely used and high-performance multi-objective genetic algorithm. By introducing concepts such as non-dominated sorting and crowding distance, NSGA-II can generate a relatively uniformly distributed Pareto optimal solution set in a single run, effectively revealing the trade-offs between different design objectives, making it a powerful tool for handling complex multi-objective optimization problems. In addition, various multi-criteria decision-making methods exist, such as expert scoring, analytic hierarchy process (AHP), and TOPSIS, to select the optimal solution from multiple alternatives.

[0003] However, existing technologies in the design of steel sections for integrated supports and hangers in MEP systems still face the following challenges and shortcomings: 1) There is a lack of a systematic design method that combines multi-objective optimization algorithms with the characteristics of this type of component, making it difficult to efficiently generate Pareto optimal solution sets that satisfy multiple design objectives (such as cost and performance). 2) Although various multi-criteria decision-making methods exist, how to effectively and seamlessly integrate the Pareto optimal solution sets obtained by multi-objective optimization algorithms with these decision-making methods, and construct a scientific and objective decision-making process to select the unique optimal design scheme from a large number of Pareto optimal solutions, remains a challenge in this field. Existing decision-making processes often rely on subjective judgment or fail to fully utilize the rich information in the optimization results. 3) In existing technologies, few studies or applications specifically address the integrated supports and hangers of MEP systems, comprehensively considering their unique constraints and design objectives, and providing a solution that integrates optimization and decision-making.

[0004] Therefore, there is an urgent need for a systematic approach that can overcome the above-mentioned shortcomings and effectively combine advanced multi-objective optimization technology with multi-criteria decision-making methods to achieve multi-objective optimization design of steel sections for integrated supports and hangers, so as to improve design quality, reduce costs and increase design efficiency. Summary of the Invention

[0005] To address the aforementioned problems, this invention aims to provide a multi-objective optimization design method for integrated support and hanger cross-sections based on a non-dominated sorting genetic algorithm II and an optimal solution determination method. This method includes the following steps:

[0006] Step 1: Establish a multi-objective optimization mathematical model for the steel section of the integrated support and hanger. By identifying the key geometric parameters of the steel section of the integrated support and hanger as design variables, two conflicting optimization objectives, namely cost and performance, are defined, and the structural and construction constraints that the design scheme must meet are determined. These variables, objectives, and constraints are expressed mathematically, thus constructing a multi-objective optimization mathematical model for subsequent algorithmic solutions.

[0007] Step 2: Establish the non-dominated sorting genetic algorithm. Based on the characteristics of the multi-objective optimization problem for the steel cross-section of the integrated support frame, the non-dominated sorting genetic algorithm-II (NSGA-II) was configured and implemented. This includes determining the encoding method of the design variables, setting various operating parameters of the algorithm such as population size and number of iterations, constructing a fitness function for evaluating the design scheme, and implementing the core genetic operations of the algorithm: non-dominated sorting, crowding calculation, selection, crossover, and mutation.

[0008] Step 3: Solve the multi-objective optimization mathematical model using NSGA-II. Execute the NSGA-II algorithm implemented in Step 2 and apply it to the multi-objective optimization mathematical model established in Step 1. Through the iterative optimization process of the algorithm, search and evolve the population in the solution space defined by the design variables. The output of this step is a set of approximate Pareto nondominated solutions, which represent the optimal set of trade-offs among different design schemes under the given optimization objectives and constraints.

[0009] Step 4: Implement the Multi-Objective Decision-Making (MDOS) method. Construct or configure a multi-objective decision-making method for final solution selection from the Pareto non-dominated solution set obtained in Step 3, preferably a solution-ranking-based multi-objective decision-making (MDOS) method. This step defines the decision-making principles or criteria for evaluating and comparing different solutions on the Pareto front, and establishes the corresponding evaluation, ranking, and selection mechanisms.

[0010] Step 5: Employ a multi-objective decision-making method to evaluate, rank, and select solutions in the approximate Pareto non-dominated solution set. Using the decision-making method implemented in Step 4, process the approximate Pareto non-dominated solution set obtained in Step 3 via the NSGA-II algorithm. By applying preset evaluation criteria and ranking rules, analyze and compare each candidate solution in the solution set. Finally, based on the established selection logic, determine one or more optimal design solutions that best meet the actual requirements from the trade-off solution set.

[0011] Step 6: Based on the preferred design scheme, determine the final design parameters of the integrated support and hanger steel section. The optimization results corresponding to one or more preferred design schemes selected through the multi-objective decision-making process in Step 5 are transformed into specific geometric parameters of the integrated support and hanger steel section. These parameters include, but are not limited to, web thickness, web height, flange thickness, and flange width. These finalized design parameters can be directly used for engineering design, manufacturing, and installation.

[0012] The method for optimizing the cross-section design of building electromechanical pipe supports and hangers based on NSGA-II and MDOS as described in claim 1 is characterized in that, in step 1, the following steps can be performed sequentially:

[0013] Step 1.1: Determine Design Variables. Identify and define the key geometric parameters that affect the cost and performance of the steel section as design variables for the optimization process. The range of values ​​for these variables is limited by standard specifications, manufacturing processes, and construction requirements.

[0014] Step 1.2: Define the objective function. Clearly define the conflicting objectives that the optimization aims to achieve. This typically includes at least one objective aimed at reducing costs, and at least one objective aimed at improving structural performance.

[0015] Step 1.3: Establish Constraints: Determine the various restrictions that the design scheme must meet to ensure the safety, feasibility, and compliance with relevant specifications of the structure. Constraints typically include, but are not limited to, the strength verification of steel, the stability requirements of the structure, the stiffness (deformation) limits of components, and the construction requirements for actual manufacturing and installation.

[0016] Step 1.4: Construct a mathematical model. Express the design variables, objective function, and constraints using mathematical formulas or functional relationships to form a standard form of multi-objective optimization mathematical model.

[0017] Furthermore, in step 2, the following steps can be performed sequentially:

[0018] Step 2.1: Determine the chromosome encoding method: Define how to map or encode the design variables in Step 1 into individuals or chromosomes in the NSGA-II algorithm for genetic operations.

[0019] Step 2.2: Configure algorithm parameters: Set the key operating parameters of the NSGA-II algorithm, such as population size, maximum number of iterations (or convergence criterion), crossover probability, mutation probability, etc.

[0020] Step 2.3: Construct a fitness evaluation mechanism: Establish an evaluation function or module to calculate the fitness value of each individual (design scheme) based on the objective function and constraints in Step 1, and determine whether it meets the constraints.

[0021] Step 2.4: Implement core genetic operations: Program the genetic operations of the NSGA-II algorithm, such as non-dominated sorting, crowding calculation, selection (e.g., tournament selection), crossover, and mutation.

[0022] Furthermore, in step 3, the following steps can be performed sequentially:

[0023] Step 3.1: Initialize the population: Based on the set population size and the range of design variables, generate an initial set of individuals randomly or through other means.

[0024] Step 3.2: Iterative Optimization: The core iterative process of the NSGA-II algorithm is executed iteratively, including: performing non-dominated sorting and crowding calculation on the current population; performing selection operations based on the sorting and crowding information; applying crossover and mutation operations to generate the offspring population; and merging the parent and offspring generations for the next round of non-dominated sorting and selection.

[0025] Step 3.3: Termination condition judgment: Check whether the preset termination conditions are met, such as reaching the maximum number of iterations, slowing down the change in population diversity, etc.

[0026] Step 3.4: Output Pareto nondominated solution set: When the termination condition is met, the algorithm outputs a set of solutions found in the target space that have good nondominated performance, forming an approximate Pareto front.

[0027] Furthermore, in step 4, the following steps can be performed sequentially:

[0028] Step 4.1: Determine decision principles or criteria: Define decision principles or criteria used to evaluate and compare different solutions on the Pareto front. This may be based on engineer preferences, project priorities, cost sensitivity, or performance focus, etc.

[0029] Step 4.2: Construct an evaluation and ranking mechanism: Establish a mechanism or algorithm to quantitatively evaluate and rank each solution in the Pareto non-dominated solution set according to the principles determined in Step 4.1. The MDOS method typically involves some form of scoring or distance calculation.

[0030] Step 4.3: Implement the selection logic: Define how to select one or more final preferred design solutions based on the evaluation and ranking results. This could be through threshold filtering, selecting the top-ranked solutions, or allowing users to interactively select from the ranking results.

[0031] Furthermore, in step 5, the following steps can be performed sequentially:

[0032] Step 5.1: Input Pareto nondominated solution set: Obtain the approximate Pareto nondominated solution set output by the NSGA-II algorithm in Step 3.

[0033] Step 5.2: Perform evaluation and ranking: Using the evaluation and ranking mechanism established in Step 4, process each solution in the input Pareto solution set to obtain the evaluation result and ranking of each solution in the solution set.

[0034] Step 5.3: Select the preferred solution: Based on the preset selection logic or user interaction input, select one or more preferred design solutions that best meet the requirements from the evaluated and sorted solution set.

[0035] Furthermore, in step 6, the following steps can be performed sequentially:

[0036] Step 6.1: Extract design variable values: Extract the specific values ​​of the corresponding design variables from the preferred schemes selected in Step 5.

[0037] Step 6.2: Output final design parameters: Use the extracted design variable values ​​as the final geometric parameters of the steel section of the integrated support and hanger, namely web thickness, web height, flange thickness, and flange width.

[0038] Step 6.3: Verification: Perform a verification step to confirm whether the final determined design parameters meet all the constraints defined in Step 1, so as to ensure the effectiveness of the design scheme.

[0039] The advantages and positive effects of this invention are as follows:

[0040] (1) Multi-objective collaborative optimization through NSGA-II algorithm: Effective trade-offs and collaborative optimization among multiple (often conflicting) design objectives (such as cost, strength, stiffness, and stability) are achieved.

[0041] (2) Integration and application of multi-objective decision-making method (MDOS): It realizes the transformation of numerous optimization solutions generated by NSGA-II into practically feasible engineering decisions.

[0042] (3) Simultaneous and refined optimization of steel section usage cost and overall structural performance: achieving better resource allocation and significant cost control.

[0043] (4) An integrated design process of “NSGA-II optimization iteration + MDOS scientific decision-making” was proposed: a standardized and systematic methodology for the optimization design of steel sections of integrated supports and hangers was constructed.

[0044] Based on this, this invention proposes a multi-objective optimization design method for integrated support and hanger sections based on NSGA-II and the optimal solution determination method. It revolutionarily combines the advanced NSGA-II multi-objective optimization algorithm with the scientific multi-objective decision-making method (MDOS), providing a complete, efficient, and systematic solution for the optimization design of integrated support and hanger steel sections in complex systems (such as building electromechanical engineering). Specifically, its core value lies in: solving the pain point of traditional design methods in balancing multiple conflicting objectives (such as cost, strength, stiffness, and stability); by generating a Pareto optimal solution set, it makes it possible to quantitatively weigh and collaboratively optimize among numerous objectives, thereby obtaining a globally superior design solution. It significantly improves design efficiency and quality; through automated optimization iteration and scientific decision support, it greatly shortens the design cycle, reduces excessive reliance on experience, and frees designers from tedious calculations, allowing them to focus on innovation and decision-making, while ensuring the scientific rigor, reliability, and engineering practicality of the design solution. It promotes resource conservation and cost control; through refined synchronous optimization, it enables more efficient material utilization and effective reduction of engineering costs, providing strong support for the economic benefits of the project. This enhances the safety and reliability of complex system designs. Through rigorous control of various performance indicators and system optimization, it ensures the long-term stability and safe operation of the support and hanger system in practical applications. It introduces a standardized and intelligent new paradigm to the field of support and hanger design. The proposed integrated design process not only improves current engineering practices but also lays a solid methodological foundation for the future introduction of advanced technologies such as AI-assisted design and automated design, playing a significant role in promoting technological progress in the industry.

[0045] This invention is not limited to the above-described embodiments. Anyone can derive other products in various forms under the guidance of this invention. However, regardless of any changes in shape or structure, any technical solution that is the same as or similar to this application falls within the protection scope of this invention. Attached Figure Description

[0046] The above and / or other aspects and advantages of the present invention will become clearer and more readily understood from the following detailed description taken in conjunction with the accompanying drawings, which are merely illustrative and do not limit the invention, wherein:

[0047] Figure 1 This is a flowchart of a multi-objective optimization design method for integrated support and hanger sections based on NSGA-II and the optimal solution determination method.

[0048] Figure 2 This is the algorithm implementation framework for NSGA-II.

[0049] Figure 3 This is a schematic diagram of the Pareto front.

[0050] Figure 4 This is a non-dominated ordination graph.

[0051] Figure 5 This is a cross diagram of NSGA-II.

[0052] Figure 6 This is a diagram of NSGA-II variants.

[0053] Figure 7 The distribution of all feasible solutions and Pareto fronts.

[0054] Figure 8 This is the final drawing for the integrated support and hanger scheme. Detailed Implementation

[0055] To make the above-mentioned objects, features, and advantages of the present invention more apparent and understandable, the present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments. The multi-objective optimization design method for integrated support and hanger sections based on NSGA-II and the optimal solution determination method is as follows: Figure 1 As shown.

[0056] Step 1: Establish a multi-objective optimization mathematical model for the steel section of the integrated support and hanger.

[0057] Step 1.1: Determine design variables

[0058] Design variables are variable parameters that affect the properties and cost of steel sections. For the steel sections of integrated supports and hangers, the core design variables typically include the geometric dimensions of the section. For a typical I-beam or H-shaped steel section, the design variables are defined as follows:

[0059] in:

[0060] x={h w ,t w ,b f ,t f}(mm)

[0061] h w Indicates the cross-sectional height;

[0062] b f Indicates the wing width;

[0063] t w Indicates the thickness of the web;

[0064] t f This indicates the flange thickness.

[0065] Step 1.2: Define the objective function

[0066] The objective function is the performance metric to be optimized. This invention focuses on at least two conflicting objectives. Typical objective functions include cost minimization and structural performance optimization. The objective function is defined as follows:

[0067] F(x) = [f1(x), f2(x)] where: f1(x) represents the cost target, and f2(x) represents the structural performance target.

[0068] Step 1.3: Establish constraints

[0069] Constraints are the technical requirements that the design scheme must meet, including:

[0070] Strength constraint: Ensure that the stress of the section under various load combinations does not exceed the allowable value of the material strength.

[0071] Stiffness constraint: Restricts the deformation (deflection, rotation) of a component to within the allowable range.

[0072] Stability constraints: prevent overall or local instability of components (such as buckling, bending-torsional buckling, local instability of web or flange).

[0073] Construction constraints: To meet the construction requirements for manufacturing, installation and use.

[0074] Step 2: Establish a non-dominated sorting genetic algorithm

[0075] Based on the characteristics of the multi-objective optimization model, this step configures and implements the NSGA-II algorithm, enabling it to effectively search for the Pareto front. Figure 2 Framework for implementing algorithms for NSGA-II

[0076] Step 2.1: Determine the chromosome encoding method: Use real number encoding to encode the design variable x in Step 1.1 into a chromosome in the genetic algorithm.

[0077] Step 2.2: Configure algorithm parameters: Set the key operating parameters of the NSGA-II algorithm, which affect the performance and convergence of the algorithm.

[0078] Population size (N) pop ): The number of individuals in each generation.

[0079] Maximum number of iterations (Gmax): The total number of iterations the algorithm runs.

[0080] Crossover probability (Pc): Determines the probability that individuals will cross over.

[0081] Probability of mutation (Pm): Determines the probability of an individual undergoing mutation.

[0082] Step 2.3 Constructing a Fitness Evaluation Mechanism: Establish a mechanism to evaluate the performance of each individual. In NSGA-II, the evaluation primarily relies on the objective function value and the satisfaction of constraints. For constrained optimization problems, the penalty function method is typically used to transform constraint violations into impacts on the objective function, employing specific methods for handling constraints. For each individual x:

[0083] Calculate its objective function value F(x).

[0084] Check if all constraints are met. If constraints are violated, the degree of violation can be calculated.

[0085] Step 2.4 Implement core genetic operations: Program the core operations of NSGA-II, which drive the evolution of the population.

[0086] Step 2.5 Non-dominated ranking: Individuals in the population are stratified according to their objective function values ​​to form different non-dominated fronts (ranks). Individuals at lower ranks are considered superior to those at higher ranks.

[0087] Step 2.6 Crowding Calculation: For individuals with the same non-dominated front, calculate the "crowding" of the individuals around them to distinguish individuals with similar objective function values ​​during selection, thus maintaining population diversity.

[0088] Step 2.7 Selection: Based on non-dominance level and crowding information, select individuals with better performance to enter the next generation or to be used as parents for breeding. A commonly used selection method is tournament selection.

[0089] Step 2.8 Crossover: Simulating the biological reproduction process, partial gene exchange is performed between two parent individuals to generate new offspring. For real-number encoding, methods such as simulated binary crossover can be used.

[0090] Step 2.9 Mutation: Randomly alter individual genes with a certain probability to increase population diversity and help escape local optima. For real-number encoding, methods such as polynomial mutation can be used.

[0091] Step 3. Solve the multi-objective optimization mathematical model using a non-dominated sorting genetic algorithm.

[0092] This step is the execution process of the NSGA-II algorithm, which finds an approximate Pareto non-dominated solution set through iterative optimization. Figure 3 This is a schematic diagram of the Pareto front.

[0093] Step 3.1 Initialize the population: Based on the population size N set in Step 2.2 pop And the range of design variables, randomly generate N pop The initial individuals constitute the first generation population P0.

[0094] Step 3.2 Iterative optimization: For generations 0 to G... max-1 Each generation of G:

[0095] Generate offspring population Q G For the current population P G Perform selection, crossover, and mutation operations to generate N. pop Individual offspring. Figure 4 and Figure 5 These are the NSGA-II cross plot and the variant plot, respectively.

[0096] Step 3.3 Merge populations: Merge the parent population P G and offspring population Q G Merge to form a size of 2N pop A mixed population.

[0097] Step 3.4 Non-dominated ranking and crowding calculation: For the mixed population R G Individuals within the group are non-dominated and ranked, and assigned to different frontiers F1, F2, ... For individuals within the same frontier, their crowding distance is calculated. Figure 6 This is a non-dominated ordination graph.

[0098] Step 3.5 Select the next generation population P G+1 Individuals are selected to enter the next generation population P according to their non-dominant hierarchy from low to high (i.e., starting from F1). G+1 until the population size reaches N pop If the addition of all individuals at a certain frontier causes the population size to exceed N, then... pop Within this frontier, individuals are selected from highest to lowest crowding level until N is reached. pop .

[0099] Step 3.6 Determine the termination condition: Check if the maximum number of iterations G has been reached. max If the condition is met, the loop terminates.

[0100] Step 3.7 Output Approximate Pareto Nondominated Solution Set: After the algorithm terminates, the last nondominated front (or the nondominated solutions among all nondominated fronts) is output as the approximate Pareto nondominated solution set found by the algorithm. This solution set contains a series of mutually nondominated design schemes in the target space. Figure 7 The distribution of all feasible solutions and Pareto fronts.

[0101] Step 4: Establish the implementation of the multi-objective decision-making (MDOS) method.

[0102] This step aims to build or configure the MDOS method for selecting the final design from the Pareto non-dominated solution set.

[0103] Step 4.1 Determine decision factors or criteria: Based on actual engineering needs, expert experience, or decision-makers' preferences, determine the decision factors used to evaluate and rank Pareto solutions.

[0104] Step 4.2 Constructing the Evaluation Function or Rules: Establish a quantitative evaluation function or a set of rules to calculate the comprehensive evaluation value or ranking index of each solution in the Pareto optimal solution set. The specific steps of the MDOS method are as follows: First, use expert scoring to determine the importance that project participants and owners attach to different objectives, thereby obtaining subjective weights. Then, introduce the Continuous Ordered Weighted Average (COWA) operator to determine the objective weights of the two objectives. The COWA operator is a fuzzy mathematics method used to evaluate the merits of a criterion and its contribution to the overall objective, quantifying the importance of the criterion as a weight. This step effectively reduces the impact of unreliable solutions at both ends of the Pareto optimal solution set on the objective weighting of each objective. Then, based on the idea of ​​game theory, coordinate and integrate the objective weights and subjective weights to obtain the comprehensive weight. Finally, use the TOPSIS method to calculate the proximity index R of the optimal level. m This allows for the selection of the design scheme that best meets the project requirements from the Pareto solution set. The specific calculation steps are as follows:

[0105] Step 4.2.1 Calculate the subjective weights β of the two indicators using the expert rating method. n (A total of g experts):

[0106]

[0107] In the formula: A an It is the score given by the a-th expert for the n-th indicator.

[0108] Step 4.2.1 The Pareto optimal solution set has t solutions. Using two objective functions as evaluation metrics, the metric matrix t can be obtained:

[0109] S=(s mn ) t×2 (m = 1, 2, ..., t; n = 1, 2)

[0110] Step 4.2.1 Standardize each element of the processed index matrix:

[0111]

[0112] Wherein: S max It is the largest element in each column.

[0113] Step 4.2.1 Since the two indicators have different dimensions and units of measurement, directly applying weights to the target will affect the representation of the data itself. Therefore, in order to preserve the essence of the data, Formula 14 is used to standardize the indicators in the matrix.

[0114]

[0115] Step 4.2.1 Use the COWA operator to calculate the objective weights ω of the two indicators. n :

[0116]

[0117] in: variable This represents the number of possible combinations when m-1 elements are randomly selected from a set of t-1 elements.

[0118] Step 4.2.1 Based on game theory, two linear weighting systems are used to combine objective and subjective weights. The aim is to minimize the deviation between the obtained comprehensive weight and the objective and subjective weights. Then, the comprehensive weight ψ is obtained. n .

[0119] ψ n =φ1·ω n +φ2·β n

[0120] In the formula: φ1 and φ2 are linear combination coefficients.

[0121] Step 4.2.1 Weight each element of the index matrix to obtain the weighting matrix K:

[0122] K = (k mn ) t×2 (m = 1, 2, ..., t; n = 1, 2)

[0123] In the formula:

[0124] Step 4.2.1 Take the smallest element in the weighted matrix as the optimal solution. Take the largest element as the worst solution

[0125]

[0126] Step 4.2.1 Calculate each element k in the weighting matrix mn With the optimal solution Y + and the negative ideal solution Y - The distance Z between them + and Z - They are respectively:

[0127]

[0128] Step 4.2.1 Calculate the approximate index Rm of the m-th solution in the Pareto optimal solution set, and sort them in descending order (the larger Rm is, the closer it is to the optimal level):

[0129]

[0130] Through the above process, a solution corresponding to the maximum Rm can be found, and this solution is the only optimal solution applicable to the project. Figure 8 Final drawing for integrated support and hanger scheme

[0131] Step 5. Using a multi-objective decision-making method, evaluate, rank, and select the solutions in the approximate Pareto non-dominated solution set.

[0132] This step applies the established MDOS method to the optimization results of NSGA-II to make a final decision on the solution.

[0133] Step 5.1 Input Pareto nondominated solution set: Obtain the approximate Pareto nondominated solution set output by the NSGA-II algorithm in Step 3. This solution set contains multiple nondominated design schemes and their corresponding objective function values ​​and design variable values.

[0134] Step 5.2 Perform evaluation and ranking: For each solution in the Pareto solution set, use the evaluation function or rule established in Step 4 to calculate its comprehensive evaluation value or determine its ranking.

[0135] Step 5.3 Selecting the Preferred Solution: Based on the selection logic set in Step 4.3, one or more preferred design solutions that meet the requirements are determined from the evaluated and ranked Pareto solution set. These solutions represent the design options considered to have the greatest engineering application value after considering all optimization objectives and constraints.

[0136] Step 6. Based on the preferred design scheme, determine the final design parameters of the steel section of the integrated support and hanger.

[0137] This step transforms the selected optimal solution into concrete engineering design results.

[0138] Step 6.1 Extract design variable values: Extract the specific values ​​of the corresponding design variables from one or more preferred design schemes selected in Step 5.

[0139] Step 6.2 Output final design parameters: Output the extracted design variable values ​​as the final design parameters of the integrated support and hanger steel section. These parameters can be directly used to draw engineering drawings and guide material procurement and component fabrication.

[0140] Step 6.3 Final Verification and Validation: After generating the final design parameters, a final verification can be performed to confirm whether the scheme strictly meets all strength, stiffness, stability and structural constraints under specific load and boundary conditions, ensuring its safety and reliability.

[0141] This invention innovatively proposes a new integrated design paradigm that deeply integrates the NSGA-II multi-objective optimization algorithm with the MDOS multi-objective decision-making method, and successfully applies it to the optimization design of the steel section of integrated supports and hangers. This effectively supplements and improves the shortcomings of traditional design methods in multi-objective trade-offs and scientific decision-making, and solves key practical problems faced in engineering practice such as cost control, performance conflicts, low design efficiency, and difficulty in comprehensively evaluating the reliability of the scheme. Through this innovation, the synergistic optimization of integrated supports and hangers among multiple objectives such as cost, strength, stiffness, and stability is finally achieved, significantly improving design efficiency and quality, reducing engineering costs and resource consumption, and laying a solid foundation for the standardization and intelligent development of this field.

Claims

1. A multi-objective optimization method for integrated support and hanger cross-sections based on NSGA-II and the optimal determination method, characterized in that, The method includes the following steps: Step 1: Establish a multi-objective optimization mathematical model for the steel section of the integrated support and hanger; by identifying the key geometric parameters of the steel section of the integrated support and hanger as design variables, define two conflicting optimization objectives, namely cost and performance, and determine the structural and construction constraints that the design scheme must meet. Through the mathematical expression of variables, objectives and constraints, a multi-objective optimization mathematical model for subsequent algorithm solution is constructed. Step 2: Establish a non-dominated sorting genetic algorithm; based on the characteristics of the multi-objective optimization problem of the steel section of the integrated support frame, the non-dominated sorting genetic algorithm-II, namely NSGA-II, was configured and implemented; including determining the encoding method of the design variables, setting the various running parameters of the algorithm, namely the population size and the number of iterations, constructing the fitness function for evaluating the design scheme, and implementing the core genetic operations of the algorithm, namely non-dominated sorting, crowding calculation, selection, crossover, and mutation; Step 3: Solve the multi-objective optimization mathematical model using NSGA-II; execute the NSGA-II algorithm implemented in Step 2 and apply it to the multi-objective optimization mathematical model established in Step 1; search and evolve the population in the solution space defined by the design variables through the iterative optimization process of the algorithm; the output of Step 3 is a set of approximate Pareto nondominated solutions, which represent the optimal set of trade-offs among different design schemes under the given optimization objectives and constraints; Step 4: Implement the multi-objective decision-making MDOS method; construct or configure a multi-objective decision-making method for final solution selection from the Pareto non-dominated solution set obtained in Step 3; implement a solution ranking-based multi-objective decision-making MDOS method; this step defines the decision-making principles or criteria for evaluating and comparing different solutions on the Pareto front, and establishes the corresponding evaluation, ranking and selection mechanisms. Step 5: Using a multi-objective decision-making method, evaluate, rank, and select the solutions in the approximate Pareto non-dominated solution set; use the decision-making method implemented in step 4 to process the approximate Pareto non-dominated solution set obtained in step 3 through the NSGA-II algorithm; analyze and compare each candidate scheme in the solution set by applying preset evaluation criteria and ranking rules, and finally determine one or more preferred design schemes that best meet the actual needs from the trade-off solution set according to the set selection logic; Step 6: Based on the preferred design scheme, determine the final design parameters of the integrated support steel section; transform the optimization results corresponding to one or more preferred design schemes selected through the multi-objective decision-making process in Step 5 into specific geometric parameters of the integrated support steel section, including web thickness, web height, flange thickness, and flange width; the final determined design parameters are used for engineering design, manufacturing, and installation.

2. The multi-objective optimization method for integrated support and hanger cross-sections based on NSGA-II and the optimal determination method according to claim 1, characterized in that, In step 1, the following steps are performed in sequence: Step 1.1: Determine design variables; identify and define key geometric parameters that affect the cost and performance of steel sections as design variables for the optimization process; The range of values ​​for these design variables is limited by standard specifications, manufacturing processes, and construction requirements; Step 1.2: Define the objective function; Clearly define the conflicting objectives that need to be achieved in the optimization; including at least one objective aimed at reducing costs and at least one objective aimed at improving structural performance; Step 1.3: Establish constraints: Determine the various restrictions that the design scheme must meet to ensure the safety, feasibility and compliance with relevant specifications of the structure; constraints include the strength check of steel, the stability requirements of the structure, the stiffness limits of components, and the construction requirements for actual manufacturing and installation; Step 1.4: Construct a mathematical model; express the design variables, objective function, and constraints using mathematical formulas or functional relationships to form a standard form of multi-objective optimization mathematical model.

3. The multi-objective optimization method for integrated support and hanger cross-sections based on NSGA-II and the optimal determination method according to claim 1, characterized in that, In step 2, the following steps can be performed sequentially: Step 2.1: Determine the chromosome encoding method: Define how to map or encode the design variables in Step 1 into individuals or chromosomes in the NSGA-II algorithm for genetic operations; Step 2.2: Configure algorithm parameters: Set the key operating parameters of the NSGA-II algorithm, including population size, maximum number of iterations, crossover probability, and mutation probability; Step 2.3: Construct a fitness evaluation mechanism: Establish an evaluation function or module to calculate the fitness value of each individual based on the objective function and constraints in Step 1, and determine whether it meets the constraints. Step 2.4: Implement core genetic operations: Program the non-dominated sorting, crowding calculation, selection, crossover and mutation genetic operations of the NSGA-II algorithm.

4. The multi-objective optimization method for integrated support and hanger cross-sections based on NSGA-II and the optimal determination method according to claim 1, characterized in that, In step 3, the following steps can be performed sequentially: Step 3.1: Initialize the population: Based on the set population size and the range of design variables, generate an initial set of individuals randomly or through other means; Step 3.2: Iterative Optimization: The core iterative process of the NSGA-II algorithm is executed repeatedly, including: performing non-dominated sorting and crowding calculation on the current population; performing selection operations based on the sorting and crowding information; applying crossover and mutation operations to generate offspring populations; merging the parent and offspring generations for the next round of non-dominated sorting and selection; Step 3.3: Termination condition judgment: Check whether the preset termination conditions are met, such as reaching the maximum number of iterations or the population diversity change slowing down; Step 3.4: Output Pareto nondominated solution set: When the termination condition is met, the algorithm outputs a set of solutions found in the target space that have good nondominated performance, forming an approximate Pareto front.

5. The multi-objective optimization method for integrated support and hanger cross-sections based on NSGA-II and the optimal determination method according to claim 1, characterized in that, In step 4, the following steps are performed sequentially: Step 4.1: Determine decision principles or criteria: Define decision principles or criteria for evaluating and comparing different solutions on the Pareto front; This includes factors such as engineer preferences, project priorities, cost sensitivity, or performance focus; Step 4.2: Construct an evaluation and ranking mechanism: Establish a mechanism or algorithm to quantitatively evaluate and rank each solution in the Pareto non-dominated solution set according to the principles determined in Step 4.1; the MDOS method involves some form of scoring or distance calculation; Step 4.3: Implement the selection logic: Define how to select one or more final preferred design solutions based on the evaluation and ranking results; through threshold filtering, selecting the top-ranked solutions, or through interactive selection by the user from the ranking results.

6. The multi-objective optimization method for integrated support and hanger cross-sections based on NSGA-II and the optimal determination method according to claim 1, characterized in that, In step 5, the following steps are performed sequentially: Step 5.1: Input Pareto non-dominated solution set: Obtain the approximate Pareto non-dominated solution set output by the NSGA-II algorithm in Step 3; Step 5.2: Perform evaluation and ranking: Using the evaluation and ranking mechanism established in Step 4, process each solution in the input Pareto solution set to obtain the evaluation result and ranking of each solution in the solution set; Step 5.3: Select the preferred solution: Based on the preset selection logic or user interaction input, select one or more preferred design solutions that best meet the requirements from the evaluated and sorted solution set.

7. The multi-objective optimization method for integrated support and hanger cross-sections based on NSGA-II and the optimal determination method according to claim 1, characterized in that, In step 6, the following steps are performed sequentially: Step 6.1: Extract design variable values: Extract the specific values ​​of the corresponding design variables from the preferred schemes selected in Step 5; Step 6.2: Output final design parameters: Use the extracted design variable values ​​as the final geometric parameters of the steel section of the integrated support and hanger, namely web thickness, web height, flange thickness, and flange width; Step 6.3: Verification: Perform a verification step to confirm whether the final determined design parameters meet all the constraints defined in Step 1, in order to ensure the effectiveness of the design scheme.

8. The multi-objective optimization method for integrated support and hanger cross-sections based on NSGA-II and the optimal determination method according to claim 1, characterized in that, In step 1, a multi-objective optimization mathematical model for the steel section of the integrated support and hanger is established; Step 1.1: Determine the design variables; Design variables are variable parameters that affect the properties and cost of steel sections. For the steel sections of integrated supports and hangers, the core design variables typically include the geometric dimensions of the section. For a typical I-beam or H-beam steel section, the design variables are defined as follows: in: x={h w ,t w ,b f ,t f } h w Indicates the cross-sectional height; b f Indicates the wing width; t w Indicates the thickness of the web; t f Indicates the flange thickness; Step 1.2: Define the objective function; The objective function is the performance metric to be optimized, focusing on at least two conflicting objectives; typical objective functions include cost minimization and structural performance optimization; the objective function is defined as follows: F(x) = [f1(x), f2(x)] where: f1(x) represents the cost target, and f2(x) represents the structural performance target; Step 1.3: Establish constraints; Constraints are the technical requirements that the design scheme must meet, including: Strength constraints: Ensure that the stress on the cross section under various load combinations does not exceed the allowable value of the material strength; Stiffness constraint: Restricts the deformation of a component within allowable limits; Stability constraints: prevent overall or local instability of components; Construction constraints: Meet the construction requirements for manufacturing, installation, and use; Step 2: Establish a non-dominated sorting genetic algorithm; Based on the characteristics of multi-objective optimization models, the NSGA-II algorithm is configured and implemented, which can effectively search for Pareto fronts. Step 2.1: Determine the chromosome encoding method: Use real number encoding to encode the design variable x in Step 1.1 into a chromosome in the genetic algorithm; Step 2.2: Configure algorithm parameters: Set the key operating parameters of the NSGA-II algorithm; Population size N pop The number of individuals in each generation; Maximum number of iterations Gmax: The total number of iterations the algorithm runs; Crossover probability Pc: determines the probability that individuals will cross over; Probability of mutation Pm: determines the probability of an individual undergoing mutation; Step 2.3 Constructing a fitness evaluation mechanism: Establish a mechanism to evaluate the performance of each individual; in NSGA-II, the evaluation depends on the objective function value and the satisfaction of constraints; for constrained optimization problems, the penalty function method is used to transform constraint violations into impacts on the objective function, and specific methods for handling constraints are employed; for each individual x: Calculate its objective function value F(x); Check if all constraints are met; if constraints are violated, calculate the degree of violation. Step 2.4 Implement core genetic operations: Program the core operations of NSGA-II, which drive the evolution of the population; Step 2.5 Non-dominated ranking: Individuals in the population are stratified according to their objective function values ​​to form different non-dominated fronts; individuals at lower ranks are superior to individuals at higher ranks; Step 2.6 Crowding Calculation: For individuals with the same non-dominated front, calculate the "crowding" of the individuals around them to distinguish individuals with similar objective function values ​​during selection, thus maintaining population diversity; Step 2.7 Selection: Based on non-dominance level and crowding information, select individuals with better performance to enter the next generation or to reproduce as parents; Step 2.8 Crossover: Simulating the biological reproduction process, some genes of two parent individuals are exchanged to generate new offspring individuals; for real number encoding, a simulated binary crossover method is used; Step 2.9 Mutation: Randomly alter individual genes with a certain probability to increase population diversity and help escape local optima; use polynomial mutation for real number encoding.

9. The multi-objective optimization method for integrated support and hanger cross-sections based on NSGA-II and the optimal determination method according to claim 1, characterized in that, In step 3, the NSGA-II algorithm finds an approximate Pareto non-dominated solution set through iterative optimization; Step 3.1 Initialize the population: Based on the population size N set in Step 2.2 pop And the range of design variables, randomly generate N pop The initial individuals constitute the first generation population P0; Step 3.2 Iterative optimization: For generations 0 to G... max-1 Each generation of G: Generate offspring population Q G For the current population P G Perform selection, crossover, and mutation operations to generate N. pop Individual offspring; Step 3.3 Merge populations: Merge the parent population P G and offspring population Q G Merge to form a size of 2N pop A mixed population; Step 3.4 Non-dominated ranking and crowding calculation: For the mixed population R G Individuals in the data are non-dominated and ordinated, and assigned to different frontiers F1, F2, ...; for individuals in the same frontier, their crowding distance is calculated; Figure 6 shows the non-dominated ordination diagram; Step 3.5 Select the next generation population P G+1 Individuals are selected to enter the next generation of the population P according to the non-dominant hierarchy from low to high, starting from F1. G+1 until the population size reaches N pop If the addition of all individuals at a certain frontier causes the population size to exceed N, then... pop Within this frontier, individuals are selected from highest to lowest crowding level until N is reached. pop ; Step 3.6 Determine the termination condition: Check if the maximum number of iterations G has been reached. max If the condition is met, the loop terminates. Step 3.7 Output Approximate Pareto Nondominated Solution Set: After the algorithm terminates, the last layer of nondominated fronts is output as the approximate Pareto nondominated solution set found by the algorithm.

10. The multi-objective optimization method for integrated support and hanger cross-sections based on NSGA-II and the optimal determination method according to claim 1, characterized in that, Step 4: Establish the implementation of the multi-objective decision-making MDOS method; Step 4.1 Determine decision factors or criteria: Based on actual engineering needs, expert experience, or decision-makers' preferences, determine the decision factors used to evaluate and rank Pareto solutions; Step 4.2 Constructing the evaluation function or rules: Establish a quantitative evaluation function or a set of rules to calculate the comprehensive evaluation value or ranking index of each solution in the Pareto solution set; The specific steps of the MDOS method are as follows: First, use the expert scoring method to determine the degree of importance that project participants and owners attach to different objectives, thereby obtaining subjective weights; then, introduce the continuous ordered weighted average operator COWA to determine the objective weights of the two objectives. Based on game theory, objective and subjective weights are coordinated and integrated to obtain a comprehensive weight. Finally, the TOPSIS method is used to calculate the proximity index R of the optimal level. m This allows for the selection of the design scheme that best meets the project requirements from the Pareto solution set; the specific calculation steps are as follows: Step 4.2.1 Calculate the subjective weights β of the two indicators using the expert rating method. n : In the formula: A an It is the score given by the a-th expert for the n-th indicator; Step 4.2.1 The Pareto optimal solution set has t solutions; using two objective functions as evaluation indicators, the indicator matrix S is obtained: S=(s mn ) t×2 m=1,2,…,t;n=1,2 Step 4.2.1 Standardize each element of the processed index matrix: Wherein: S max It is the largest element in each column; Step 4.2.1 To preserve the essence of the data, standardize the indicators in the matrix using Formula 14; Step 4.2.1 Use the COWA operator to calculate the objective weights ω of the two indicators. n : in: variable This represents the number of possible combinations when m-1 elements are randomly selected from a set of t-1 elements; Step 4.2.1 Based on game theory, two linear weighting systems are used to combine objective and subjective weights; the aim is to minimize the deviation between the obtained comprehensive weight and the objective and subjective weights; then, the comprehensive weight ψ is obtained. n ; ψ n =φ1·ω n +φ2·β n In the formula: φ1 and φ2 are linear combination coefficients; Step 4.2.1 Weight each element of the index matrix to obtain the weighting matrix K: K=(k mn ) t×2 m=1,2,…,t;n=1,2 In the formula: Step 4.2.1 Take the smallest element in the weighted matrix as the optimal solution. Take the largest element as the worst solution Step 4.2.1 Calculate each element k in the weighting matrix mn With the optimal solution Y + and the negative ideal solution Y - The distance Z between them + and Z - They are respectively: Step 4.2.1 Calculate the approximate index Rm of the m-th solution in the Pareto optimal solution set, and sort them in descending order. The larger Rm is, the closer it is to the optimal level. This leads to the solution corresponding to the maximum Rm, and this solution is the only optimal solution applicable to the project.

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