A complex product function structure configuration equilibrium solving method considering parameter misalignment

Abstract

The application discloses a kind of complex product function structure configuration equilibrium solving method considering parameter misalignment, comprising: the direct influence interval matrix between the construction product function structure attribute;Design consensus reaching process measures the judgment of expert group and establishes updating mechanism;Affine arithmetic is used instead of interval analysis to avoid interval dependency problem;The reliable weight of product function structure attribute is calculated by improving DEMATEL method;The product function structure configuration equilibrium multi-objective optimization model considering parameter misalignment disturbance is constructed;Three kinds of intelligent optimization algorithms, improved gravity search algorithm based on adaptive mechanism, improved hybrid gravity search and differential evolution algorithm and projection gradient descent algorithm, are designed to solve the model and obtain the optimal configuration solution of product function structure.The application can effectively handle the parameter misalignment problem caused by cognitive uncertainty, and improve the reliability and balance of product function structure configuration.

Description

Technical Field

[0001] This invention relates to the field of digital design and performance optimization technology for complex products, and in particular to a method for balancing the functional structure configuration of complex products considering parameter inaccuracies. Background Technology

[0002] The functional structure configuration design of complex products is a typical comprehensive problem involving multiple disciplines, objectives, and constraints. In early designs, the determination of functional structure solutions often relied on the designer's prior knowledge and intuition, employing trial-and-error or rule-based methods, which struggled to effectively handle conflicts and trade-offs among multiple objectives. With the development of computer-aided design and multidisciplinary collaborative optimization techniques, model-based functional structure mapping and optimization methods have gradually become mainstream.

[0003] However, most existing functional structure configuration methods assume precise knowledge of design parameters, neglecting the parameter inaccuracies caused by cognitive uncertainties prevalent in practical engineering. Parameter inaccuracies refer to the inability to accurately obtain the influence relationships and weight parameters between functional structure attributes during the product functional structure configuration solution process due to factors such as expert cognitive limitations, incomplete information, or subjective judgment. These relationships are often expressed in intervals or fuzzy forms, and differing or even conflicting opinions exist among different experts. If this uncertainty is not properly addressed, it will directly affect the reliability of the functional structure attribute weights, leading to performance imbalances and decreased reliability in the final configuration scheme during practical applications.

[0004] Furthermore, when quantifying such interval uncertainties, traditional interval analysis methods suffer from interval dependency issues during the calculation process, i.e., ignoring the correlation between variables, which leads to overly conservative calculation results and error accumulation, further reducing the accuracy and practicality of configuration solutions.

[0005] Therefore, there is an urgent need for a solution method that can effectively handle cognitive uncertainty, overcome the defects of interval operations, and reliably obtain the optimal balanced configuration of functional structures. Summary of the Invention

[0006] To address the aforementioned shortcomings of existing technologies, this invention provides a method for balancing the functional structure configuration of complex products while considering parameter inaccuracies. This method incorporates a consensus-building process to gather expert opinions, employs affine arithmetic to improve the traditional DEMATEL method for reliable calculation of functional structure attribute weights, constructs a multi-objective optimization model considering parameter inaccuracy disturbances, and combines various intelligent optimization algorithms for solution, thereby effectively improving the balance and reliability of complex product functional structure configuration schemes.

[0007] The specific technical solution of the present invention is as follows:

[0008] A method for solving the functional structure configuration equilibrium of complex products considering parameter misalignment includes the following steps:

[0009] (1) Based on the expert group’s judgment on the influence relationship between product functional structure attributes, a direct influence interval matrix was constructed, a consensus-reaching process was designed to measure the expert group’s judgment, and an update mechanism was established.

[0010] (2) Convert the interval variables into affine form, perform interval operations using affine arithmetic, and calculate the weights of product functional structure attributes using the CRP-based improved DEMATEL method.

[0011] (3) Construct a multi-objective optimization model for product functional structure configuration that includes parameter misalignment perturbation, and design three intelligent optimization algorithms: an improved adaptive gravity search algorithm, an improved hybrid gravity search and differential evolution algorithm, and a projection gradient descent algorithm. Solve the model to obtain the optimal configuration solution for the product functional structure.

[0012] Complex product functional structural attributes are interconnected and involve multidisciplinary knowledge. Therefore, it is necessary to invite a group of experts with different professional backgrounds, work experience and perspectives to collaborate in judging the influence relationship between product functional structural attributes.

[0013] Because the expert group comes from different professional fields, in the process of collaboratively judging the influence relationship between the functional and structural attributes of a product, there is not only an interdisciplinary knowledge gap, but also a dynamic process of constantly adjusting their own judgments due to the influence of other experts' opinions, which greatly increases the uncertainty of group decision-making.

[0014] Therefore, in order to facilitate consensus among experts, the Consensus Reaching Process (CRP) was proposed, which dynamically simulates the process by which experts provide judgments and adjust those judgments until a consensus is reached.

[0015] In step (1), based on the expert group's judgment on the influence relationship between product functional structural attributes, a direct influence interval matrix is ​​constructed. A consensus-reaching process is designed to measure the expert group's judgment and an update mechanism is established, including:

[0016] Suppose A = {a1, a2,…, a p} represents the set of product functional structure schemes, C = {c1, c2, ..., c n Let} be the set of product functional structure attributes, E = {e1, e2,…, e m For the expert group participating in the judgment of the influence relationship between product functional structural attributes, the expert group uses interval numbers to better express the judgment of the influence range between functional structural attributes, and constructs the direct influence interval matrix D as follows:

[0017] (4-1)

[0018] Where D k The product functional structure attributes provided by the k-th expert directly affect the interval matrix, k = 1, …,m. Expert e k Provided product functional structure attributes c i For c j The degree of influence.

[0019] Expert judgments are influenced by their professional knowledge, experience, and personal subjectivity. Because experts come from diverse professional fields, possess limited knowledge, and have varying levels of understanding, biases can exist between their judgments. To improve the reliability of expert judgments and reduce biases, the degree of consensus among experts is used as the basis for CRP (Consensus Reliability Assessment). The degree of consensus represents the level of reliability of expert opinions; the higher the degree of consensus, the higher the reliability of the expert opinions. k The degree of consensus can be expressed as:

[0020] (4-2)

[0021] in Represents D k With D l The Euclidean distance between them.

[0022] Assume D k With D l Given two interval matrices, the Euclidean distance between them is:

[0023] (4-3)

[0024] In the CRP process, an expert's judgment is constantly adjusted based on the opinions of other experts. To achieve a high degree of consensus within the expert group, when an expert's consensus falls below a threshold, they must dynamically update their judgment based on the opinion of the expert with the highest consensus. This method eliminates subjective uncertainty within the expert group and improves the credibility of group decision-making. k The update model for the judgment is as follows:

[0025] (4-4)

[0026] Where t is the number of iterations. and The expert with the highest consensus in generation t, e l The judgment, This is the consensus control parameter. It has been proven that, through continuous updates and iterations, the consensus of the expert group will converge to the consensus threshold.

[0027] In step (2), the interval variable is converted into affine form, and affine arithmetic is used to perform interval operations, including:

[0028] Affine arithmetic (AA) is a model for numerical computation of uncertainty. It represents the relationships between different uncertain variables as linear combinations of noise elements and tracks the correlations between these uncertain variables through basic operations. The range of variables calculated using AA is much smaller than that calculated using interval analysis (IA), and the accuracy is correspondingly improved, achieving deconservatism. The affine form of variable x is:

[0029] (4-21)

[0030] Where x0 is the midpoint of the fluctuation range of variable x; ε i For noise elements, represent independent sources of uncertainty that have an uncertain impact on x. , where n is the number of sources of uncertainty; x i The coefficients are real, representing the corresponding noise element ε. i The magnitude and direction of the impact.

[0031] When different variables use the same noise element, it means that these variables are all affected by a certain source of uncertainty. There is a partial dependency between these variables, and the magnitude of the dependency is determined by the magnitude of their corresponding real coefficients.

[0032] When using Affine Analysis (AA) for interval range analysis, all input interval variables must first be converted to affine form, then the AA operation is performed, and finally the range analysis result is converted back to interval form. Interval variables and affine variables can be converted to each other.

[0033] When converting an interval variable to its affine form, it is necessary to consider the midpoint of the variable's range of fluctuation and the number of sources of uncertainty. Let the interval variable be denoted as... The midpoint x0 of the variable's fluctuation range and the maximum fluctuation range r of the variable about that midpoint. max They can be represented as:

[0034] (4-33) (4-34)

[0035] Assuming it has only one source of uncertainty, the transformed affine variable can be expressed as:

[0036] (4-35)

[0037] When converting an affine variable to interval form, it is necessary to consider the midpoint and maximum range of the variable's fluctuation. Let the affine variable be denoted as... Then the midpoint of its fluctuation range is x0, and its maximum fluctuation range is r. max That is, the value at which the impact of each source of uncertainty reaches its maximum, i.e.:

[0038] (4-36)

[0039] The transformed interval variable can then be represented as:

[0040] (4-37)

[0041] In step (2), the weights of the product functional structure attributes are calculated using the CRP-based improved DEMATEL method, including:

[0042] (2-i) The functional structure attributes of the product directly affect the interval matrix.

[0043] (2-ii) for D k The functional structural attributes obtained through CRP directly affect the interval matrix. .

[0044] (2-iii) Transformation The functional and structural attributes of a product directly affect the affine matrix. as follows:

[0045] (4-38)

[0046] in .

[0047] (2-iv) The average value of the product's functional structure attributes directly affects the affine matrix.

[0048] The average affine matrix directly influenced by product functional structure attributes can be obtained as follows:

[0049] (4-39)

[0050] in .

[0051] (2-v) The average of product functional structure attributes directly affects the normalization of the affine matrix, which yields:

[0052] (4-40)

[0053] in ,and .

[0054] (2-vi) Calculate the affine influence and affine affected degree of the product's functional structure attributes.

[0055] Calculate the sum of rows and columns of the affine matrix representing the average direct impact of product functional structure attributes to obtain the affine matrix of the influence degree of product functional structure attributes. And the affine matrix of influence ,and:

[0056] (4-41) (4-42)

[0057] in c represents the product's functional structure attribute i The degree of affine influence on the functional and structural attributes of other products, i.e., the degree of affine influence. c represents the product's functional structure attribute j The degree to which affines are affected by the functional and structural attributes of other products, i.e., the degree of affine influence.

[0058] (2-vii) Calculate the weights of the product's functional structure attributes.

[0059] Generally, the performance value of a product's functional structure under a certain attribute should be equal to the product of the input value of the product's functional structure attribute and the probability of fluctuation of the input value of the product's functional structure attribute (i.e., the probability that the performance of the product's functional structure attribute is affected). The affine weights of the product's functional structure attributes are obtained as follows:

[0060] (4-43)

[0061] The corresponding real weights are:

[0062] (4-44)

[0063] In step (3), a multi-objective optimization model for product functional structure configuration, including parameter misalignment perturbations, is constructed, including:

[0064] Assuming an environment of uncertainty, the multi-objective function for finding the optimal configuration equilibrium of product functional structure is:

[0065] (4-45)

[0066] Where f i (For example, product cost, the smaller the value, the better) and f j (For example, in product performance, the higher the value, the better) To optimize the objective function, w i and w j To optimize the weights corresponding to the objective function, To represent the degree to which the objective function is disturbed under uncertain conditions, f represents the independent variable and its range (e.g., the critical dimensions of the product's functional structure). lThese are constraint functions (such as the product's cost range, performance requirements, etc.).

[0067] In step (3), an improved gravity search algorithm based on an adaptive mechanism is designed, including:

[0068] Assuming the population size is popsize, the position of the i-th particle in the t-th iteration is as follows:

[0069] (4-46)

[0070] in ; n is the dimension of the search space, i.e., the number of independent variables.

[0071] The mass of the i-th particle is M i (t) is defined as:

[0072] (4-47) (4-48)

[0073] fitness i (t) represents the fitness value of the particle, worst i (t) represents the worst fitness value among all particles in the t-th iteration, best i (t) represents the optimal fitness value among all particles.

[0074] According to the law of universal gravitation, the gravitational force exerted on a particle by other particles in k-dimensional space is defined as:

[0075] (4-49)

[0076] Where τ is the compensation factor to prevent the denominator from being 0; R ij G(t) is the Euclidean distance between the two particles; G(t) is the gravitational constant, and:

[0077] (4-50)

[0078] Where G0 is the initial value of the gravitational constant, typically 100; α is the gravitational decay constant, typically 20; and H is the maximum number of iterations.

[0079] Therefore, the net gravitational force acting on the particle in k-dimensional space is:

[0080] (4-51)

[0081] The particle's acceleration at this moment is defined as:

[0082] (4-52)

[0083] The formulas for updating the particle's velocity and position can then be defined as:

[0084] (4-53) (4-54)

[0085] In step (3), an improved hybrid gravity search and differential evolution algorithm is designed, including:

[0086] (3-i) Initialize the population

[0087] Assume the population size is popsize, n is the dimension of the variable, and the i-th individual is... ,and For an initial population at t=0, the initial individual can be defined as:

[0088] (4-55)

[0089] (3-ii) Difference Variation

[0090] For the target individual X in generation t i,t Three different individuals X are randomly selected from the population. a,t X b,t and X c,t Applying a differential mutation strategy generates the next generation of mutated individuals V. i,t+1 for:

[0091] (4-56)

[0092] in and λ is the differential weight and It determines the difference variable. Size.

[0093] (3-iii) Cross

[0094] To increase population diversity, experimental individuals were generated by crossover operations between target individuals and mutant individuals. ,in:

[0095] (4-57)

[0096] Where CR is the crossover probability and .

[0097] (3-iv) Choice

[0098] Based on the differential mutation and crossover operations described above, the winners from the target individuals and experimental individuals are selected as the next generation of target individuals, i.e.:

[0099] (4-58)

[0100] In step (3), the projected gradient descent algorithm is designed, including:

[0101] (a) Initialize the point;

[0102] (b) Calculate the gradient of the objective function with the current parameters;

[0103] (c) Update the temporary parameters based on the gradient values, and project the temporary parameters into the feasible region to obtain new parameter points;

[0104] (d) Repeat steps (b) and (c) until the iteration termination condition is met. Attached Figure Description

[0105] Figure 1 The attributes that need to be considered when performing functional structure configuration balance solution for a certain type of tunneling machine cutting head;

[0106] Figure 2 A three-dimensional visualization view of the objective function for optimal configuration and balanced solution of the cutting head function structure of a tunneling machine;

[0107] Figures 3a-3c The graph shows the iterative curves for solving the objective function, considering the perturbation of the objective function. Figure 3a For IGSA, Figure 3b GSA+DE Figure 3c It is PGD. Detailed Implementation

[0108] The present invention will now be described in further detail with reference to specific embodiments. It should be noted that the following embodiments are intended to facilitate understanding of the present invention and are not intended to limit it in any way.

[0109] The specific implementation process of the present invention for the balanced solution of complex product functional structure configuration considering parametric misalignment is as follows:

[0110] 1. Construction and updating of expert judgment matrix based on consensus

[0111] When seeking the optimal product functional structure solution, it is necessary to quantify the contribution values ​​of different functional structure solutions to functional structure attributes and the weights of these attributes, thereby deriving the performance values ​​of different functional structure solutions. Accurately obtaining the reasonable weights of each complex product functional structure attribute is crucial for finding the optimal solution. Since complex product functional structure attributes are interdependent and involve multidisciplinary knowledge, it is necessary to invite a group of experts with different professional backgrounds, work experience, and perspectives to collaboratively determine the influence relationships between these attributes. However, because the expert group comes from different professional fields, the collaborative determination of these relationships not only involves interdisciplinary knowledge gaps but also a dynamic process of constantly adjusting their judgments due to the influence of other experts' opinions, significantly increasing the uncertainty of group decision-making. Therefore, to facilitate consensus among experts, a Consensus Reaching Process (CRP) is proposed to dynamically simulate the process by which the expert group provides and adjusts its judgments until a consensus is reached.

[0112] Suppose A = {a1, a2,…, a p} represents the set of product functional structure schemes, C = {c1, c2, ..., c n Let} be the set of product functional structure attributes, E = {e1, e2,…, e m For the expert group participating in the judgment of the influence relationship between product functional structural attributes, the expert group uses interval numbers to better express the judgment of the influence range between functional structural attributes, and constructs the direct influence interval matrix D as follows:

[0113] （4-1）

[0114] Where D k The product functional structure attributes provided by the k-th expert directly affect the interval matrix, k = 1, …,m. Expert e k Provided product functional structure attributes c i For c j The degree of influence.

[0115] Expert judgments are influenced by their professional knowledge, experience, and personal subjectivity. Because experts come from diverse professional fields, possess limited knowledge, and have varying levels of understanding, biases can exist between their judgments. To improve the reliability of expert judgments and reduce biases, the degree of consensus among experts is used as the basis for CRP (Consensus Reliability Assessment). The degree of consensus represents the level of reliability of expert opinions; the higher the degree of consensus, the higher the reliability of the expert opinions. k The degree of consensus can be expressed as:

[0116] （4-2）

[0117] in Represents D k With D l The Euclidean distance between them.

[0118] Assume D k With D l Given two interval matrices, the Euclidean distance between them is:

[0119] （4-3）

[0120] In the CRP process, an expert's judgment is constantly adjusted based on the opinions of other experts. To achieve a high degree of consensus within the expert group, when an expert's consensus falls below a threshold, they must dynamically update their judgment based on the opinion of the expert with the highest consensus. This method eliminates subjective uncertainty within the expert group and improves the credibility of group decision-making. k The update model for the judgment is as follows:

[0121] (4-4)

[0122] Where t is the number of iterations. and The expert with the highest consensus in generation t, e l The judgment, This is the consensus control parameter. It has been proven that, through continuous updates and iterations, the consensus of the expert group will converge to the consensus threshold.

[0123] By adjusting the expert group's judgment on the influence relationship between product functional and structural attributes through CRP, experts who deviate from the group consensus and have low reliability are brought closer to experts with high consensus through continuous iteration. This improves the overall consensus of the expert group and provides an indispensable key prerequisite for accurately obtaining the weight of product functional and structural attributes in the future.

[0124] 2. Attribute weight calculation based on affine arithmetic and improved Dematel

[0125] 2.1 Analysis of the working principle and limitations of the traditional DEMATEL method

[0126] The DEMATEL method, proposed by Japanese scholar Hiroshi Tanaka, is a decision analysis tool used to analyze the relationships between factors in complex systems. The working principle of this method is as follows:

[0127] (1) The functional structure attributes of the product directly affect the interval matrix.

[0128] We invited m experts to assess the influence relationships between the product's functional and structural attributes, and constructed the direct influence interval matrix D as follows:

[0129] （4-5）

[0130] Where D k The product functional structure attributes provided by the k-th expert directly affect the interval matrix. ; Expert e k Provided product functional structure attributes c i For c j The degree of influence.

[0131] (2) Constructing the interval matrix of the average direct influence of product functional structure attributes

[0132] The average direct influence interval matrix of product functional structure attributes can be obtained from the interval matrix:

[0133] （4-6）

[0134] in .

[0135] (3) The average value of product functional structure attributes directly affects the normalization of the interval matrix.

[0136] Normalizing the interval matrix of the average direct influence of product functional and structural attributes yields:

[0137] （4-7）

[0138] in ,and .

[0139] (4) Construct a comprehensive influence interval matrix of product functional structure attributes

[0140] Based on the normalized average direct influence interval matrix of product functional structure attributes, the comprehensive influence interval matrix of product functional structure attributes can be obtained by adding the direct influence and all indirect influences, as shown below:

[0141] （4-8）

[0142] Where I is the identity matrix.

[0143] (5) Calculate the degree of influence and the degree of being influenced by the product's functional structure attributes.

[0144] Calculate the sum of rows and columns of the comprehensive influence interval matrix of product functional structure attributes to obtain the influence matrix R and the affected matrix Q of product functional structure attributes, and:

[0145] （4-9） （4-10）

[0146] Where r i c represents the product's functional structure attribute i The degree of influence on the functional and structural attributes of other products, i.e., the degree of influence; q j c represents the product's functional structure attribute j The degree to which a product is affected by the functional and structural attributes of other products is called the degree of influence.

[0147] (6) Calculate and obtain the weights of the product's functional structure attributes.

[0148] Based on the influence matrix R and the influenced matrix Q of the product functional structure attributes, the weights of the product functional structure attributes can be obtained as follows:

[0149] （4-11）

[0150] The DEMATEL method can quickly and comprehensively obtain the weights of each attribute of the product's functional structure; however, this method has several shortcomings:

[0151] (1) Step 4 in this method is valid only if the matrix is ​​invertible: if and only if Formula (4-8) only holds true when the condition is met; otherwise, the formula is invalid.

[0152] (2) This method has obvious limitations in handling two-dimensional problems. For a two-dimensional matrix, that is, when there are only two product functional structure attributes, regardless of the numerical value of the relationship between the two product functional structure attributes as input, the weight of the two attributes calculated by this method is always 0.5. Obviously, this method cannot accurately reflect the possible differences between different functional structure attributes when dealing with two-dimensional problems;

[0153] (3) This method has conceptual ambiguity and vagueness. When solving for the optimal product functional structure scheme, the performance value of the product functional structure under a certain attribute should be equal to the input value of the product functional structure attribute and the fluctuation probability of the input value of the product functional structure attribute. The fluctuation probability of the input value of the product functional structure attribute should be the degree of influence caused by external factors, i.e., the degree of influence. The traditional DEMATEL method uses both the degree of influence and the degree of influence to measure the weight of the product functional structure attribute, which on the one hand contributes to the limitation in dealing with two-dimensional problems, and on the other hand wastes solution resources;

[0154] (4) Interval numbers are a kind of uncertainty language tool that can well characterize the range and fluctuation of the relationship between product functional structural attributes. However, when the expert judgment is presented in the form of interval numbers, the DEMATEL method will be overly conservative in the process of interval operation, resulting in the calculated interval being much larger than the actual range of the interval, that is, the interval arithmetic dependency problem.

[0155] Therefore, based on the above analysis, affine arithmetic (AA) is used for interval operations, and the steps of the traditional DEMATEL method are improved to overcome the above problems. Finally, reliable weights of product functional structure attributes are calculated based on the consensus reached by experts, laying the foundation for the final optimal configuration equilibrium solution of product functional structure.

[0156] 2.2 Working Principles and Comparative Analysis of Interval Analysis and Affine Arithmetic

[0157] Interval analysis (IA) is an important method for describing uncertain product data. It uses a floating-point interval to represent the range of product characteristics, and then calculates the (unknown) interval value that guarantees the inclusion of the product characteristic it represents by adding, subtracting, and multiplying intervals using basic interval arithmetic. This provides a powerful tool for analyzing the uncertainty range in product optimization solutions. The working principle of interval arithmetic is as follows:

[0158] Let the interval number be . ,in This is the lower bound of the interval. Let be the upper bound of the interval, and .remember and Given two interval numbers, the basic operational rules between them are as follows:

[0159] (1) Rules of addition:

[0160] （4-12）

[0161] (2) Subtraction rules:

[0162] （4-13）

[0163] (3) Multiplication rules:

[0164] 1) When and hour:

[0165] （4-14）

[0166] 2) When and hour:

[0167] （4-15）

[0168] 3) When neither 1) nor 2) is satisfied:

[0169] （4-16）

[0170] in:

[0171] , ;

[0172] 4) When multiplied by a constant k:

[0173] （4-17）

[0174] (4) Division rule: if and only if hour:

[0175] 1) When and hour:

[0176] （4-18）

[0177] 2) When and hour:

[0178] （4-19）

[0179] 3) When neither 1) nor 2) is satisfied:

[0180] （4-20）

[0181] in:

[0182] , .

[0183] IA can quantify product characteristic parameters into possible fluctuation ranges, thus effectively handling the impact of uncertain characteristic variables in the product system.

[0184] However, in product design, especially when dealing with complex systems, uncertainty analysis (IA) is frequently used to handle uncertainties. As product design calculations progress, overly conservative IA can lead to a gradual explosion of errors and an exponential decrease in the precision of product characteristic parameter ranges. This reduces the feasibility of the design and may even prevent the product from meeting performance requirements. Therefore, uncertainty analysis (AA) is introduced to consider the correlation between variables and calculate the uncertainty range, avoiding the problem of over-conservatism.

[0185] Affine Algebra (AA) is a model used for numerical computation of uncertainties. It represents the correlation between different uncertain variables as a linear combination of noise elements and tracks the correlation between these uncertain variables through basic operations. The range of variables calculated by AA is much smaller than that of Algebra (IA), and the accuracy is correspondingly improved, achieving deconservatism. The affine form of variable x is:

[0186] （4-21）

[0187] Where x0 is the midpoint of the fluctuation range of variable x; ε i For noise elements, represent independent sources of uncertainty that have an uncertain impact on x. , where n is the number of sources of uncertainty; x i The coefficients are real, representing the corresponding noise element ε. i The magnitude and direction of the impact.

[0188] When different variables use the same noise element, it means that these variables are all affected by a certain source of uncertainty. There is a partial dependency between these variables, and the magnitude of the dependency is determined by the magnitude of their corresponding real coefficients.

[0189] To evaluate the deconservatism effect of AA, it is necessary to understand the operational rules between affine variables. and Given two affine variables, the basic operational rules between them are as follows:

[0190] (1) Rules of addition:

[0191] （4-22） （4-23）

[0192] (2) Subtraction rules:

[0193] （4-24） （4-25）

[0194] (3) Multiplication rules:

[0195] （4-26） （4-27）

[0196] (4) Division rules:

[0197] （4-28）

[0198] In the formula , , ε n+1 This is the generated new noise element.

[0199] As can be seen from the basic operational rules of affine variables, since these noise elements are treated as independent sources of uncertainty, the influence of each uncertainty source can be considered independently during the calculation process, effectively handling the correlation between variables and thus reducing the conservatism of interval analysis calculations. Therefore, using AA can better consider the correlation between variables and obtain more accurate range analysis results.

[0200] Meanwhile, during the use of AA, in some cases the calculation results are no longer in a linear form and do not conform to the definition of affine variables, thus preventing the affine calculation from continuing. In such cases, we employ the optimal affine approximation method, increasing ε... k The term represents the approximation error, and a new affine approximation variable is formed with the objective of minimizing the maximum error. That is, assuming The result is the result of the affine calculation, and:

[0201] （4-29）

[0202] Where ε k This is a new noise element, including additional sources of uncertainty introduced during the calculation process, such as approximation errors and discrete errors; z k This represents the maximum value of the error.

[0203] This optimal affine approximation method allows for the derivation and computation of all fundamental operations and functions. For example, suppose... ,but:

[0204] （4-30） （4-31） （4-32）

[0205] in , .

[0206] When using Affine Analysis (AA) for interval range analysis, all input interval variables must first be converted to affine form, then the AA operation is performed, and finally the range analysis result is converted back to interval form. Interval variables and affine variables can be converted to each other.

[0207] When converting an interval variable to its affine form, it is necessary to consider the midpoint of the variable's range of fluctuation and the number of sources of uncertainty. Let the interval variable be denoted as... The midpoint x0 of the variable's fluctuation range and the maximum fluctuation range r of the variable about that midpoint. max They can be represented as:

[0208] （4-33） （4-34）

[0209] Assuming it has only one source of uncertainty, the transformed affine variable can be expressed as:

[0210] （4-35）

[0211] When converting an affine variable to interval form, it is necessary to consider the midpoint and maximum range of the variable's fluctuation. Let the affine variable be denoted as... Then the midpoint of its fluctuation range is x0, and its maximum fluctuation range is r. max That is, the value at which the impact of each source of uncertainty reaches its maximum, i.e.:

[0212] （4-36）

[0213] The transformed interval variable can then be represented as:

[0214] （4-37）

[0215] Combining AA with other design optimization techniques can better calculate the uncertainty range in the product design optimization process and improve calculation accuracy.

[0216] 2.3 Improving the Acquisition of Functional Structure Attribute Weights in DEMATEL Based on CRP

[0217] (1) The functional structure attributes of the product directly affect the interval matrix.

[0218] (2) Regarding D k The functional structural attributes obtained through CRP directly affect the interval matrix. .

[0219] (3) Conversion The functional and structural attributes of a product directly affect the affine matrix. as follows:

[0220] （4-38）

[0221] in .

[0222] (4) The average value of the product's functional structure attributes directly affects the affine matrix.

[0223] The average affine matrix directly influenced by product functional structure attributes can be obtained as follows:

[0224] （4-39）

[0225] in .

[0226] (5) The average of product functional and structural attributes directly affects the normalization of the affine matrix.

[0227] Normalizing the affine matrix that directly influences the average functional and structural attributes of the product, we obtain:

[0228] （4-40）

[0229] in ,and .

[0230] (6) Calculate the affine influence and affine affected degree of the product's functional structural attributes.

[0231] Calculate the sum of rows and columns of the affine matrix representing the average direct impact of product functional structure attributes to obtain the affine matrix of the influence degree of product functional structure attributes. And the affine matrix of influence ,and:

[0232] （4-41） （4-42）

[0233] in c represents the product's functional structure attribute i The degree of affine influence on the functional and structural attributes of other products, i.e., the degree of affine influence. c represents the product's functional structure attribute j The degree to which affines are affected by the functional and structural attributes of other products, i.e., the degree of affine influence.

[0234] (7) Calculate and obtain the weights of the product's functional structure attributes.

[0235] Generally, the performance value of a product's functional structure under a certain attribute should be equal to the product of the input value of the product's functional structure attribute and the probability of fluctuation of the input value of the product's functional structure attribute (i.e., the probability that the performance of the product's functional structure attribute is affected). Based on this mathematical meaning, the affine weight of the product's functional structure attribute can be obtained as:

[0236] （4-43）

[0237] The corresponding real weights are:

[0238] （4-44）

[0239] 3. Construction and solution of a multi-objective optimization model considering parameter misalignment perturbations

[0240] Assuming an environment of uncertainty, the multi-objective function for finding the optimal configuration equilibrium of product functional structure is:

[0241] （4-45）

[0242] Where f i (For example, product cost, the smaller the value, the better) and f j (For example, in product performance, the higher the value, the better) To optimize the objective function, w i and w jTo optimize the weights corresponding to the objective function, To represent the degree to which the objective function is disturbed under uncertain conditions, f represents the independent variable and its range (e.g., the critical dimensions of the product's functional structure). l These are constraint functions (such as the product's cost range, performance requirements, etc.).

[0243] To address this multi-objective optimization problem, this study designed three algorithms for solving it and compared their performance and robustness in practical applications. The solution principles of the three algorithms are as follows:

[0244] 3.1 Improved Gravity Search Algorithm Based on Adaptive Mechanism

[0245] The algorithm's theoretical foundation stems from the combination of the law of universal gravitation and Newton's second law. It represents feasible solutions using the position of each particle, treating each feasible solution as an object with mass, the mass of which is determined by the solution's fitness value. These feasible solutions attract each other, with the attraction proportional to their masses and inversely proportional to the distance between them. Each feasible solution updates its position and velocity based on the total gravitational force exerted on it by other feasible solutions, thus effectively exploring the solution space. The final solution tends towards the object with the largest mass, i.e., the optimal solution. The algorithm's solution principle is as follows:

[0246] Assuming the population size is popsize, the position of the i-th particle in the t-th iteration is as follows:

[0247] （4-46）

[0248] in ; n is the dimension of the search space, i.e., the number of independent variables.

[0249] The mass of the i-th particle is M i (t) is defined as:

[0250] （4-48） （4-49）

[0251] fitness i (t) represents the fitness value of the particle, worst i (t) represents the worst fitness value among all particles in the t-th iteration, best i (t) represents the optimal fitness value among all particles.

[0252] According to the law of universal gravitation, the gravitational force exerted on a particle by other particles in k-dimensional space is defined as:

[0253] （4-50）

[0254] Where τ is the compensation factor to prevent the denominator from being 0; R ij G(t) is the Euclidean distance between the two particles; G(t) is the gravitational constant, and:

[0255] （4-51）

[0256] Where G0 is the initial value of the gravitational constant, typically 100; α is the gravitational decay constant, typically 20; and H is the maximum number of iterations.

[0257] Therefore, the net gravitational force acting on the particle in k-dimensional space is:

[0258] （4-52）

[0259] The particle's acceleration at this moment is defined as:

[0260] （4-53）

[0261] The formulas for updating the particle's velocity and position can then be defined as:

[0262] （4-54） （4-55）

[0263] 3.2 Improved Hybrid Gravity Search and Differential Evolution Algorithm

[0264] Differential evolutionary algorithm generates new individuals by performing differential mutation and crossover operations on the population, and then updates the population using a selection strategy. The solution principle of this algorithm is as follows:

[0265] Step 1: Initialize the population

[0266] Assume the population size is popsize, n is the dimension of the variable, and the i-th individual is... ,and For an initial population at t=0, the initial individual can be defined as:

[0267] （4-56）

[0268] Step 2: Differential Mutation

[0269] For the target individual X in generation t i,t Three different individuals X are randomly selected from the population. a,t X b,t and X c,t Applying a differential mutation strategy generates the next generation of mutated individuals V. i,t+1 for:

[0270] （4-57）

[0271] in and λ is the differential weight and It determines the difference variable. Size.

[0272] Step 3: Cross

[0273] To increase population diversity, experimental individuals were generated by crossover operations between target individuals and mutant individuals. ,in:

[0274] （4-58）

[0275] Where CR is the crossover probability and .

[0276] Step 4: Select

[0277] Based on the differential mutation and crossover operations described above, the winners from the target individuals and experimental individuals are selected as the next generation of target individuals, i.e.:

[0278] （4-59）

[0279] Combining GSA with DE methods, first using GSA to update the velocity and position of the search particles, and then using differential mutation and crossover operations of DE to update the population, can improve the algorithm's local search capability while ensuring global search capability.

[0280] 3.3 Projected Gradient Descent Algorithm

[0281] Projective gradient descent is an optimization algorithm for constrained optimization problems. It iteratively updates temporary points at the current point by projecting gradients into the feasible region until the optimal parameters are found. The algorithm's solution principle is as follows:

[0282] Step 1: Initialize the point;

[0283] Step 2: Calculate the gradient of the objective function with the current parameters;

[0284] Step 3: Update the temporary parameters based on the gradient value, and project the temporary parameters into the feasible region to obtain new parameter points;

[0285] Step 4: Repeat steps 2 and 3 until the iteration termination condition is met.

[0286] The projective gradient descent algorithm is simple and easy to implement. However, when the dataset is too large, the gradient calculation will be very slow. Therefore, the projective gradient descent algorithm is suitable for situations where the dataset is relatively small.

[0287] Example

[0288] This invention proposes a solution method for balancing the functional structure configuration of complex products that considers parameter inaccuracies. By integrating fuzzy uncertainty and random uncertainty factors, a multi-level, multi-attribute functional structure configuration balance model is constructed to optimize the performance of complex equipment throughout its entire life cycle.

[0289] A specific example is as follows:

[0290] This chapter takes the functional structure configuration of a certain type of tunneling machine's cutting head as the research object to verify the rationality and superiority of the technology presented in this chapter. Through preliminary design, 40 sets of data covering the cutting line spacing φ have been obtained. ) and helix angle β ( The design parameter combination of the tunneling machine cutter head is as follows. Given that these parameters have a critical impact on the performance of the cutter head, it is still necessary to use a reliable and accurate functional structure optimal configuration equilibrium solution process to solve for the parameter combination with the best working performance from these 40 parameter combinations. Figure 1 When performing a functional-structural configuration equilibrium solution for a certain type of tunneling machine cutting head, the attributes that need to be considered include cutting cost T and cutting efficiency V. Cutting cost T includes manufacturing cost T0. m Usage cost T u and scrap cost T d Manufacturing cost T m Including raw material procurement costs Material transportation and storage costs Machining costs Heat treatment cost and assembly costs Cost of use T u Including cutting power consumption costs Auxiliary system energy consumption cost Cost of replacing easily damaged parts and regular maintenance costs ; Scrapping cost T d Including manual dismantling costs Cost of tool and equipment depreciation The value of recycled metal materials and the value of reusing parts The relationship between cutting cost and cutting efficiency is denoted as Layer A; the relationship between manufacturing cost, usage cost, and scrap cost is denoted as Layer B; the relationship between raw material procurement cost, material transportation and storage cost, machining cost, heat treatment cost, and assembly cost is denoted as Layer C1; the relationship between cutting power consumption cost, auxiliary system energy consumption cost, vulnerable parts replacement cost, and periodic maintenance cost is denoted as Layer C2; and the relationship between manual dismantling cost, tool and equipment wear cost, metal material recycling value, and component reuse value is denoted as Layer C3.

[0291] The cutting cost of a tunneling machine's cutting head can be defined as:

[0292]

[0293] in

[0294] （4-60） （4-61） （4-62）

[0295] Assuming that the values ​​of each cost are uniformly abstracted and normalized to the range of [0.1, 1], based on the approximate linear relationship between the slit spacing and the spiral angle and each cost, the specific values ​​of the slit spacing and the spiral angle are mapped to the corresponding approximate cost values, and the specific approximate values ​​of each cost can be obtained, as shown in Table 1.

[0296]

[0297] The cutting efficiency (V) of a tunneling machine's cutting head is significantly affected by the cutting tooth spacing and helix angle. Different cutting tooth spacings result in different tooth distributions, which in turn affect the depth and angle at which the teeth cut into the coal and rock each time. If the cutting tooth spacing is too large, it can lead to insufficient cutting and incomplete crushing of some coal and rock, thus reducing cutting efficiency. If the cutting tooth spacing is too small, it can cause interference between the cutting teeth, further affecting cutting efficiency. As for the helix angle, different helix angles affect the extent to which the crushed coal and rock are discharged. A suitable helix angle allows the crushed coal and rock to be discharged smoothly, ensuring cutting efficiency. If the helix angle is too small, it can lead to the accumulation of a large amount of crushed coal and rock, increasing cutting resistance and reducing cutting efficiency.

[0298] Based on the above analysis, assuming the radius of the tunneling machine's cutting head is R, the cutting line spacing is φ, the helix angle is β, the rotational speed during cutting is n (r / min), and the effective cutting width of the cutting teeth is b, the approximate volume of coal and rock cut by a single cutting tooth per revolution is:

[0299] （4-63）

[0300] in The cutting depth is related to the sectional spacing and the helix angle. Assume... The format is as follows:

[0301] （4-64）

[0302] in This is a coefficient related to the hardness of the coal and the geometry of the cutting teeth. The harder the coal and the blunter the cutting teeth, the less favorable it is for cutting, and the smaller k1 will be; conversely, the harder the coal and the sharper the cutting teeth, the larger k1 will be. For medium-hardness coal and conventionally shaped cutting teeth, the value of k1 ranges from 0.1 to 0.5. It is a coefficient related to the ratio of the cutting line spacing to the cutting head radius. When φ increases relative to R, the cutting depth of the cutting teeth is suppressed, and the cutting efficiency decreases.

[0303] Equation (4-64) shows that as the spacing between the cutting teeth increases, the cutting depth initially increases. However, once the spacing reaches a certain point, interference between the cutting teeth due to the influence of radius R causes the cutting depth to decrease. A larger helix angle results in a greater cutting depth. Considering the coal and rock crushing characteristics and the impact of discharge on cutting efficiency, equation (4-64) can be further optimized as follows:

[0304] （4-65）

[0305] Where k3 is the coal and rock accumulation coefficient after crushing, and k4 is the coal and rock discharge coefficient, and k3 is related to the looseness and particle size distribution of the coal and rock. If the coal and rock are loose and the particles are small, it will not cause serious coal and rock accumulation, and k3 can be taken as a smaller value. k4 is related to the geometry of the cutting head. The more conducive the cutting head is to coal and rock discharge, the larger k4 will be. Let , where k5 is the functional structure correlation coefficient of the cutting section.

[0306] Furthermore, the number of cutting teeth on the cutting head that participate in the cutting is related to the cutting head radius, the spacing between cutting lines, and the helix angle, and can be approximately expressed as:

[0307] （4-66）

[0308] The volume of coal and rock cut by the cutting head per unit time, i.e., the cutting efficiency, can be approximated as:

[0309] （4-67）

[0310] Therefore, it can be concluded that the cutting efficiency of the cutting head is mainly related to the cutting head's radius, rotation speed, effective cutting width, cutting line spacing, and helix angle.

[0311] During the cutting process of the tunneling machine's cutting head, it is subject to the constraint of the cutting force, which cannot be too large or too small. Since the cutting force constraint has been met in the preliminary design of the functional structural parameters of the tunneling machine's cutting section, it will not be considered here.

[0312] Based on the above, the objective function for the optimal configuration equilibrium solution of the tunneling machine cutting head's functional structure can be constructed as follows:

[0313] （4-68）

[0314] Next, it is necessary to obtain the weights of each functional structural attribute of the tunnel boring machine's cutting head, and invite 20 experts to judge the influence relationships between these attributes. Regarding the influence relationship between cutting cost T and cutting efficiency V, the direct influence interval matrices provided by the 20 experts are as follows:

[0315] …

[0316] …

[0317] …

[0318] …

[0319] A consensus was reached on the opinions of 20 experts, making =0.7, the consensus threshold is 0.88, and the interval matrix directly affected by the functional structure attributes of high consensus is:

[0320]

[0321]

[0322]

[0323] …

[0324]

[0325]

[0326]

[0327] Transforming the direct influence of high-consensus functional structural properties on interval matrices into the direct influence of functional structural properties on affine matrices yields:

[0328]

[0329]

[0330]

[0331] …

[0332]

[0333]

[0334]

[0335] The average direct influence of functional structure properties on the affine matrix can be obtained as follows:

[0336]

[0337] Normalizing the affine matrix directly influenced by the average functional structural properties, we obtain:

[0338]

[0339] The sum of the rows and columns of the affine matrix directly influenced by the average functional structural attributes is calculated to obtain the influence degree affine matrix of the functional structural attributes. And the affine matrix of influence for:

[0340]

[0341]

[0342] Therefore, the affine weights for severance cost and severance efficiency are as follows:

[0343]

[0344]

[0345] The weights converted to real numbers are respectively and .

[0346] Based on the solution results, the ranges of the two attribute weights calculated using the improved DEMATEL method that considers consensus are as follows: and The traditional DEMATEL method yields two attribute weight ranges, which are respectively... and This demonstrates that, compared to traditional methods, the improved method for calculating the weights of functional structural attributes significantly narrows the range and improves the accuracy and reliability of the calculation.

[0347] Similarly, the weights of other functional structural attributes were calculated, and the weights between the functional structural attributes of each layer are shown in Table 2. Table 2 shows that the range of weights for each layer of functional structural attributes is significantly reduced compared to traditional methods, demonstrating the effectiveness and reliability of the improved DEMATEL method that considers consensus.

[0348]

[0349] After obtaining the weight values ​​of the functional structure attributes at each layer, three designed algorithms are used to solve the multi-objective optimization problem of the optimal configuration of the cutting head of the tunnel boring machine. Let the objective function be f=TV, popsize=100, H=50, λ=0.5, CR=0.9, R=347.5mm, b=12.5mm, n=90r / min, P=260kW, k1=0.5, k2=1, k3=0.1, k5=1, k6=0.6, k7=0.6, α=54°. Random numbers are generated to follow a standard normal distribution with a standard deviation of 120,000. A three-dimensional visualization method is used to characterize the objective function of the optimal configuration equilibrium solution for the functional structure of the tunnel boring machine's cutting head, such as... Figure 2 As shown.

[0350] Considering the perturbation of the objective function, the results obtained by solving the objective function using the three designed intelligent algorithms are shown in Table 3, and the iteration curves are as follows. Figures 3a-3c As shown.

[0351]

[0352] From Table 3 and Figures 3a-3c It can be seen that when considering the objective function perturbation, IGSA's solution capability is superior to GSA+DE, which is superior to PGD. By using IGSA, the optimal configuration of the functional structure under reasonable perturbation was obtained, where φ=36mm and β=24°. This proves that IGSA is the best method for solving the optimal configuration equilibrium problem of the current tunneling machine cutter head functional structure, and the optimal configuration of the tunneling machine cutter head functional structure is a cut-off spacing φ of 36 mm and a helix angle β of 24°.

[0353] The embodiments described above provide a detailed explanation of the technical solutions and beneficial effects of the present invention. It should be understood that the above descriptions are merely specific embodiments of the present invention and are not intended to limit the present invention. Any modifications, additions, and equivalent substitutions made within the scope of the principles of the present invention should be included within the protection scope of the present invention.

Claims

1. A method for balancing the functional structure configuration of complex products considering parameter misalignment, characterized in that, Includes the following steps: (1) Based on the expert group’s judgment on the influence relationship between the product’s functional structure attributes, construct a direct influence interval matrix, design a consensus-reaching process to measure the expert group’s judgment, and establish an update mechanism; (2) Convert the interval variables into affine form, perform interval operations using affine arithmetic, and calculate the weights of product functional structure attributes using the CRP-based improved DEMATEL method. (3) Construct a multi-objective optimization model for product functional structure configuration that includes parameter misalignment perturbation, and design an improved gravity search algorithm based on adaptive mechanism, an improved hybrid gravity search and differential evolution algorithm and a projection gradient descent algorithm to solve the model and obtain the optimal configuration solution of product functional structure.

2. The method for balancing the functional structure configuration of complex products considering parameter misalignment according to claim 1, characterized in that, In step (1), the expert group uses interval numbers to express the judgment on the influence range between functional structural attributes, and constructs the direct influence interval matrix D as follows: （4-1） Where D k The product functional structure attributes provided by the k-th expert directly affect the interval matrix, k = 1, …, m; Expert e k Provided product functional structure attributes c i For c j The degree of influence, C = {c1, c2,…,c n Let} be the set of product functional structure attributes, E = {e1, e2,…, e m This refers to a group of experts who participate in determining the influence relationships between product functional and structural attributes. Expert e k The degree of consensus can be expressed as: （4-2） in Represents D k With D l Euclidean distance between them: （4-3） Expert e k The update model for the judgment is as follows: （4-4） Where t is the number of iterations. and The expert with the highest consensus in generation t, e l The judgment, These are consensus control parameters.

3. The method for balancing the functional structure configuration of complex products considering parameter misalignment according to claim 1, characterized in that, In step (2), when using affine arithmetic AA to perform interval range analysis calculation, first convert all input interval variables into affine form, then perform AA operation on them, and finally convert the range analysis calculation result back into interval form. Assume the interval variable is denoted as The midpoint x0 of the variable's fluctuation range and the maximum fluctuation range r of the variable about that midpoint. max They can be represented as: (4-33) (4-34) Assuming there is only one source of uncertainty, the transformed affine variable can be expressed as: (4-35) When converting an affine variable to interval form, consider the midpoint and maximum fluctuation range of the variable's range; assuming the affine variable is denoted as... Then the midpoint of its fluctuation range is x0, and the maximum fluctuation range is r. max That is, the value at which the impact of each source of uncertainty reaches its maximum, i.e.: (4-36) The transformed interval variable can then be represented as: (4-37)。 4. The method for balancing the functional structure configuration of complex products considering parameter misalignment according to claim 1, characterized in that, In step (2), the CRP-based improved DEMATEL method includes the following steps: (4-i) The structural attributes of product functions directly affect the interval matrix; (4-ii) for D k The functional structural attributes obtained through CRP directly affect the interval matrix. ; (4-iii) Transformation The functional and structural attributes of a product directly affect the affine matrix. as follows: (4-38) in ; (4-iv) The average value of the product's functional structure attributes directly affects the affine matrix. The average affine matrix directly influenced by product functional structure attributes can be obtained as follows: (4-39) in ; (4-v) The average of product functional structure attributes directly affects the affine matrix normalization, from which we can obtain: (4-40) in ,and ; (4-vi) Calculate the affine influence and affine affected degree of the product's functional structure attributes. Calculate the sum of rows and columns of the affine matrix representing the average direct impact of product functional structure attributes to obtain the affine matrix of the influence degree of product functional structure attributes. And the affine matrix of influence ,and: (4-41) (4-42) in c represents the product's functional structure attribute i The degree of affine influence on the functional and structural attributes of other products, i.e., the degree of affine influence. c represents the product's functional structure attribute j The degree to which affines are affected by the functional and structural attributes of other products, i.e., the degree of affine influence. (4-vii) Calculate the weights of the product's functional structure attributes. The performance value of a product's functional structure under a certain attribute should be equal to the product of the input value of the product's functional structure attribute and the probability of fluctuation of the input value of the product's functional structure attribute. The affine weight of the product's functional structure attribute is obtained as follows: (4-43) The corresponding real weights are: (4-44)。 5. The method for balancing the functional structure configuration of complex products considering parameter misalignment according to claim 1, characterized in that, In step (3), it is assumed that under an uncertain environment, the multi-objective function for solving the optimal configuration equilibrium of the product functional structure is: (4-45) Where f i and f j To optimize the objective function, w i and w j To optimize the weights corresponding to the objective function, To represent the degree to which the objective function is disturbed under uncertain conditions, For the independent variable and its range, f l For constraint functions.

6. The method for balancing the functional structure configuration of complex products considering parameter misalignment according to claim 1, characterized in that, In step (3), the improved gravity search algorithm based on the adaptive mechanism includes: Assuming the population size is popsize, the position of the i-th particle in the t-th iteration is as follows: (4-46) in n represents the dimension of the search space, i.e., the number of independent variables. The mass of the i-th particle is M i (t) is defined as: (4-47) (4-48) fitness i (t) represents the fitness value of the particle, worst i (t) represents the worst fitness value among all particles in the t-th iteration, best i (t) represents the optimal fitness value among all particles; According to the law of universal gravitation, the gravitational force exerted on a particle by other particles in k-dimensional space is defined as: (4-49) Where τ is the compensation factor to prevent the denominator from being 0; R ij G(t) is the Euclidean distance between the two particles; G(t) is the gravitational constant, and: (4-50) Where G0 is the initial value of the gravitational constant, typically 100; α is the gravitational decay constant, typically 20; and H is the maximum number of iterations. Therefore, the net gravitational force acting on the particle in k-dimensional space is: (4-51) The particle's acceleration at this moment is defined as: (4-52) The formulas for updating the particle's velocity and position can then be defined as: (4-53) (4-54)。 7. The method for balancing the functional structure configuration of complex products considering parameter misalignment according to claim 1, characterized in that, In step (3), the improved hybrid gravity search and differential evolution algorithm solution steps include: (7-i) Initialize the population Assume the population size is popsize, n is the dimension of the variable, and the i-th individual is... ,and For an initial population at t=0, the initial individual can be defined as: (4-55) (7-ii) Difference Variation For the target individual X in generation t i,t Three different individuals X are randomly selected from the population. a,t X b,t and X c,t Applying a differential mutation strategy generates the next generation of mutated individuals V. i,t+1 for: (4-56) in and λ is the differential weight and It determines the difference variable. Size; (7-iii) Cross To increase population diversity, experimental individuals were generated by crossover operations between target individuals and mutant individuals. ,in: (4-57) Where CR is the crossover probability and ; (7-iv) Choice Based on the differential mutation and crossover operations described above, the winners from the target individuals and experimental individuals are selected as the next generation of target individuals, i.e.: (4-58)。 8. The method for balancing the functional structure configuration of complex products considering parameter misalignment according to claim 1, characterized in that, In step (3), the projected gradient descent algorithm includes the following steps: (8-i) Initialization point; (8-ii) Calculate the gradient of the objective function under the current parameters; (8-iii) Update the temporary parameters based on the gradient values, and project the temporary parameters into the feasible region to obtain new parameter points; (8-iv) Repeat steps (8-ii) and (8-iii) until the iteration termination condition is met.

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