A method and system for factor combination experiment design under inequality constraints

Abstract

The application relates to a factor combination test design method and system limited by inequality constraints, and comprises the following steps: S1. determining response variables, influence factors, levels of the influence factors and inequality constraints among the factors in a test; S2. performing coordinate transformation of the factor levels according to the inequality constraint conditions to obtain a new coordinate system; S3. introducing a disturbance term to compensate for the missing part of the test data based on the new coordinate system; S4. converting the optimal factor combination into factor code value in the original coordinate; and S5. performing the test, calculating regression coefficients and fitting a regression equation according to the test results. The application can effectively solve the factor combination test problem with inequality constraints under the condition that test resources are limited, can consider the inequality constraints and fuse the coding space transformation method, can significantly reduce the test times, reduce resource consumption, and enhance the engineering interpretability and practicability of the obtained regression model while ensuring the model precision.

Description

Technical Field

[0001] This invention relates to the technical field of two-factor combination design methods, and in particular to a factor combination experimental design method and system constrained by inequality constraints. Background Technology

[0002] In daily production practices and scientific research, experimental design often encounters situations where factors are mutually constrained, specifically where certain factors must satisfy a particular inequality relationship. This type of problem differs fundamentally from traditional unconstrained experimental design: the experimental domain is limited by the inequality constraints between factors, making it impossible to simplify using conventional dimensionality reduction methods. Consequently, the experimental domain often exhibits irregular, asymmetrical, and complex geometric shapes, significantly different from the cubic or simplex structures commonly found in classical experimental designs.

[0003] Similar to classic regression experimental designs, these constrained experimental designs often employ coding transformation methods to map the actual values ​​of the original factors to a standard interval (such as [-1,1] or [0,1]) for processing, facilitating model construction and computational analysis. However, due to the constraints of inequalities between factors, the coded values ​​of each factor cannot independently take values ​​within the interval; instead, they must satisfy certain joint constraints. Therefore, the coding region no longer maintains the regular shape (such as a square or hypercube) of traditional orthogonal designs, but may manifest as polygons, polyhedra, or even more complex convex or non-convex regions.

[0004] Due to the constraints, the experimental region not only changes in shape but also disrupts the orthogonality foundation upon which classical experimental designs rely. Factors can no longer change independently, rendering traditional tools such as orthogonal arrays and uniform design tables unusable directly, and making model estimation and inference more complex. To restore the design's regularity and the model's predictability, appropriate coordinate transformations (such as affine transformations, principal component transformations, or projection methods under constrained optimization) are often necessary to map the original constrained region to a new space, ensuring that it meets requirements such as factor independence and region regularity in the new coordinate system, thereby reusing some of the theoretical framework of classical experimental design and modeling.

[0005] In practical engineering development and scientific research, especially in fields such as chemical formulation, material synthesis, bio-fermentation, and mixing processes, constraints often exist between factors, such as "the content of component A must not be lower than that of component B," "temperature and pressure must meet safe ranges," and "the proportions of multiple raw materials have upper and lower limits." These constraints make the formulation of experimental schemes, the establishment of predictive models, and the search for optimal formulations particularly complex. Therefore, developing two-factor or multi-factor experimental design methods that can effectively handle inequality constraints, constructing robust and reliable regression models, and achieving efficient and accurate optimization searches have become key issues for improving product quality, process efficiency, and resource utilization. Summary of the Invention

[0006] The purpose of this invention is to propose a method and system for factor combination experimental design constrained by inequality constraints, in order to solve the problems existing in the prior art. This invention focuses on the problem of factor combination experimental design constrained by inequality constraints, systematically studying how to construct scientifically reasonable experimental schemes, establish accurate prediction models, and develop efficient optimal formulation positioning strategies under realistic constraints. Through theoretical analysis, algorithm construction, and case verification, it aims to provide a practical and robust methodological support for engineering applications and scientific research in related fields, promoting the widespread application and in-depth development of constrained experimental design in actual production.

[0007] To achieve the above objectives, the present invention provides the following solution: A factor combination experimental design method constrained by inequality constraints includes: S1. Determine the response variables, influencing factors, levels of influencing factors, and inequality constraints between factors in the experiment; S2. Perform coordinate transformation of the factor levels according to the inequality constraints to obtain a new coordinate system; S3. Based on the new coordinate system, introduce a disturbance term to fill in the missing experimental data for the test points and obtain the optimal factor combination; S4. Convert the optimal factor combination into factor encoding values ​​under the original coordinates; S5. Conduct experiments based on the factor coding values, calculate regression coefficients based on the experimental results, and fit a regression equation.

[0008] Optionally, S2 performs coordinate transformation of the factor levels according to the inequality constraints, including: The original level value range is transformed based on the value range of each influencing factor; Based on the transformed interval, a variable is introduced. By taking the mutual constraints between the two factors into account in the coordinate transformation, a new coordinate system is obtained.

[0009] Optionally, the transformed interval is: in, The transformed interval, , and For components The lower and upper bounds of the original value range.

[0010] Optionally, the new coordinate system is: = ; Among them, In this case, For a non-zero point that varies freely from -1 to 1, The free variation factor is from -1 to 1. and To transform the actual values ​​of the original factors into New values ​​for the interval, for The free variation factor is a new variable constructed to take into account the mutual constraint relationship between two factors in coordinate transformation in order to handle inequality constraints.

[0011] Optionally, S3 introduces a perturbation term to compensate for missing test data, including: Experiments are conducted at test points within a preset range where data cannot be obtained, in order to supplement the experimental data.

[0012] Optionally, S4 converts the optimal factor combination into factor encoding values ​​in the original coordinates, including: Obtain the coordinates in the original coordinate system by restoring the original coordinate system. and The optimal value.

[0013] Optionally, the factor encoding value is: .

[0014] Optionally, the S5 fitted regression equation includes: Experiments were conducted based on the factor coding values ​​to collect data and establish a solution model for the regression equation. The least squares estimation method in regression analysis is used to estimate the regression coefficients in the solution model, and the significance of the solution model and regression coefficients is tested. Finally, the regression equation is given.

[0015] Optionally, the solution model is: in, For the predicted value of the response variable, b1-b3 is the intercept term, and b1-b3 are the regression coefficients.

[0016] This embodiment also proposes a factor combination experimental design system constrained by inequality constraints. The system includes: a module for determining the experimental objective and influencing factors, a coordinate transformation module, an experimental data compensation module, an optimal factor combination transformation module, and a curve fitting module. The module for determining the experimental objective and influencing factors is used to determine the experimental response variables, influencing factors, and the equality constraints and lower bound constraints between influencing factors. The coordinate transformation module is used to perform coordinate transformation of factor levels according to inequality constraints. The test data compensation module is used to introduce a disturbance term to compensate for the missing parts of the test data at the test points; The optimal factor combination transformation module is used to transform the obtained optimal factor combination into factor code values ​​under the original coordinates. The curve fitting module is used to calculate regression coefficients and fit regression equations based on experimental results.

[0017] The beneficial effects of this invention are as follows: The experimental area was expanded so that it would not shrink due to pruning, thus avoiding the artificial loss of the optimal combination of factors during the optimization experiment.

[0018] The test points in the new coordinate system fully satisfy the orthogonality requirement, making it possible to solve the linear equations using simple least squares estimation. This simplifies the solution process of this test design and facilitates its application in engineering practice.

[0019] After coordinate transformation and corresponding processing, the calculations of this experimental design method are similar to those of classical experimental design. Therefore, it is convenient for designers familiar with classical experiments to apply this design. In addition, for this reason, existing experimental design software can support its calculations with only slight modifications. Attached Figure Description

[0020] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the embodiments will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0021] Figure 1 This is a schematic diagram of a factor combination experimental design method constrained by inequality constraints according to an embodiment of the present invention. Detailed Implementation

[0022] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0023] 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.

[0024] like Figure 1 As shown, this embodiment proposes a factor combination experimental design method constrained by inequality constraints, including: S1. Determine the response variables, influencing factors, levels of influencing factors, and inequality constraints between factors in the experiment; S2. Perform coordinate transformation of the factor levels according to the inequality constraints to obtain a new coordinate system; S3. Based on the new coordinate system, introduce a disturbance term to fill in the missing experimental data for the test points and obtain the optimal factor combination; S4. Convert the optimal factor combination into factor encoding values ​​under the original coordinates; S5. Conduct experiments based on the factor coding values, calculate regression coefficients based on the experimental results, and fit a regression equation.

[0025] Specifically, in this embodiment S1, the response variable, influencing factors, levels of influencing factors, and inequality constraints between factors are determined in the experiment: First, the response variable needs to be determined according to the purpose of the experiment. Then, appropriate influencing factors are selected for study based on the actual situation and the purpose of the experiment, and the value levels of the influencing factors are determined. In addition, the inequality constraints between the influencing factors need to be determined.

[0026] In this embodiment S2, the coordinate transformation of the factor levels is performed according to the inequality constraints: firstly, based on each influence... The factor value range transforms the original level value range into As shown in equation (1): (1); in and It is a component The lower and upper bounds of the original value range are then defined. Then a variable is introduced. Taking the mutual constraint relationship between the two factors into account in the coordinate transformation, as shown in equation (2): = (2); It is now known that, In this case, It is a non-zero point that varies freely from -1 to 1, and at the same time It can also be a free variation factor that can range from -1 to 1. This completes a problem concerning... and The establishment of a new coordinate system.

[0027] In this embodiment S3, a perturbation term is introduced to compensate for the missing test data at the test points: As can be seen from the new coordinate system obtained in S2, and The enclosed figure is a standard square that is symmetrical about the origin in the new coordinate system. However, due to... point Since it cannot take any value, the shape is not yet a complete square. Since it's impossible to conduct experiments at designated test sites, we can introduce a perturbation term to allow us to obtain data at nearby test sites where data can be obtained. Specifically, in... Since it is impossible to take test points at certain locations, tests can be conducted at those points. A perturbation term is introduced at the point, namely: Generally speaking, take It is a normal random distribution with an expected value of 0 and a variance of 0.1. Its values ​​can be used to approximate the... point.

[0028] In this embodiment S4, the obtained optimal factor combination is transformed into factor encoding values ​​under the original coordinate system: specifically, the theoretical values ​​under the new coordinate system are obtained through regression design. and After finding the optimal value, the coordinates can be restored to their original state to obtain the value in the original coordinate system. and The optimal value is shown in equation (3): (3); In this embodiment S5, experiments are arranged, data is collected, and regression equations are fitted. Specifically, the experimental points selected under the new coordinate system fully conform to the orthogonality principle of the experiments. Therefore, experiments need to be conducted first, and then data needs to be collected. The task is to solve the model as shown in equation (3): (4); The model to be finally solved is shown in equation (4): (5); At this point, the least squares estimation in regression analysis can be used to estimate each regression coefficient, and the significance of the regression equation and regression coefficients can be tested. Finally, the regression equation is given.

[0029] The following embodiment uses the identification of a mixture that produces an elastic modulus greater than 3000 as its experimental objective to illustrate the detailed implementation process: S1. Determine the response variables, influencing factors, levels of influencing factors, and inequality constraints between factors in the experiment; In S1, the objective of the experiment is to identify the mixture that produces an elastic modulus greater than 3000, with the amount of binder being as low as possible. The level value of factor B is... Not higher than the level of factor C The values ​​differ between [0, 90] and [50, 70], respectively; Factor B and Factor C can be represented as the amount of filler added and the amount of reinforcing fiber added. The constraint is then given by equation (6): (6); S2. Perform coordinate transformation of factor levels based on inequality constraints; In S2, the original level values ​​of the factors are first transformed so that the transformed values ​​are in the range [-1, 1]. That is, as shown in equation (7).

[0030] , (7); Secondly, the constraints on the level values ​​of factors B and C can be expressed as shown in equation (8): (8); Therefore, = (9); It is now known that, In this case, It is a non-zero point that varies freely from -1 to 1, and at the same time It can also be a free variation factor that can range from -1 to 1. This completes a problem concerning... and The establishment of a new coordinate system. S3. Introduce a disturbance term to fill in the missing test data for the test points; In S3, since X1′ cannot take a value at X2=-1, the graph is not a complete square, meaning that test points cannot be selected at X2=-1. Therefore, for cases where data cannot be obtained at certain test points, a perturbation term can be introduced to allow experiments to be conducted at nearby test points. Specifically, in Since it is impossible to take test points at certain locations, tests can be conducted at those points. A perturbation term is introduced at the point, namely: Generally speaking, take It is a normal random distribution with an expected value of 0 and a variance of 0.1. Its values ​​can be used to approximate the... point.

[0031] S4. Convert the obtained optimal factor combination into factor encoding values ​​under the original coordinates; In S4, the specific experimental design scheme is shown in Table 1: Table 1 Then, according to formula (3), the values ​​of the factors in Table 1 are restored to their original values ​​to obtain the experimental design scheme in Table 2.

[0032] Table 2 S5. Arrange the experiment, collect data, and fit the regression equation; In S5, the regression coefficients are fitted using the least squares method according to equations (4) and (5), and the regression equation is obtained.

[0033] The following uses a certain composite material preparation process as an example to illustrate the specific implementation process of this embodiment: The experimental objective is to minimize the dosage of additive A to save costs while ensuring the tensile strength of the material is not less than 400 MPa. Let the dosage of additive A be XA and the dosage of additive B be XB, with values ​​ranging from XA ∈ [10, 30] (g) to XB ∈ [20, 40] (g). Additives A and B can be represented as high-performance reinforcing fibers and calcium carbonate as the base filler. According to process requirements, the dosage of additive A must not exceed the dosage of additive B, thus satisfying the constraint: (10); S2. Perform coordinate transformation of factor levels based on inequality constraints; First, transform the original factor levels to the standard interval [-1, 1]: , (11); At this point, constraint X A ≤X B It can be converted into: (12); To establish a new coordinate system, a transformation variable X1′ is introduced: = (13); When X2≠-1, both X2 and X1′ can vary freely in the interval [-1,1], thus forming a regular square test area in the new coordinate system (X2,X1′).

[0034] S3. Introduce a disturbance term to fill in the missing test data for the test points; Since X1′ is undefined at X2 = -1, this point cannot be tested. Therefore, a perturbation term is introduced near X2 = -1: (14); ξ follows a normal distribution with an expected value of 0 and a variance of 0.1, and is used to approximate the test point X2=-1, thereby ensuring the integrity of the test area.

[0035] S4. Convert the obtained optimal factor combination into factor encoding values ​​under the original coordinates; After completing the experimental design and obtaining the optimal factor combination in the new coordinate system, the original coordinates are restored through inverse transformation: (15); Substitute into equation (11) to find X. A With X B : (16); S5. Arrange the experiment, collect data, and fit the regression equation; Eight sets of orthogonal test points were designed under the new coordinate system. The test scheme is shown in Table 3. Table 3 Based on equations (15) and (16), the original factor levels are restored, and the actual experimental scheme is shown in Table 4: Table 4 After completing the experiment and collecting response data (tensile strength), a regression model was established, the coefficients were estimated using the least squares method, the fitted equation was obtained, and the significance of the model and coefficients was tested. Finally, a regression equation that can be used for prediction and optimization was given.

[0036] This embodiment incorporates inequality constraints into the experimental design process, enabling precise definition of feasible experimental regions and the construction of corresponding coded spatial transformation relationships. It also estimates methods for adding perturbation terms to untestable points. Based on this, a distribution scheme for a limited number of experimental points is scientifically designed. Even under extremely limited experimental conditions, this method can still efficiently establish accurate mixing models, achieving optimization of key performance indicators and coordinated control of component costs. While improving the efficiency of experimental information acquisition and reducing resource consumption, it significantly improves the overall cost-effectiveness of the experiment, providing a systematic and practical technical means to solve multi-constraint mixing optimization problems under strictly limited resources in fields such as formulation development and materials development.

[0037] The following embodiment also proposes a factor combination experimental design system constrained by inequality, including: an experimental objective and influencing factor determination module, a coordinate transformation module, an experimental data compensation module, an optimal factor combination transformation module, and a curve fitting module; The module for determining the experimental objective and influencing factors is used to determine the experimental response variables, influencing factors, and the equality constraints and lower bound constraints between influencing factors. The coordinate transformation module is used to perform coordinate transformation of factor levels according to inequality constraints. The test data compensation module is used to introduce a disturbance term to compensate for the missing parts of the test data at the test points; The optimal factor combination transformation module is used to transform the obtained optimal factor combination into factor code values ​​under the original coordinates. The curve fitting module is used to calculate regression coefficients and fit regression equations based on experimental results.

[0038] The embodiments described above are merely preferred embodiments of the present invention and are not intended to limit the scope of the present invention. Various modifications and improvements made to the technical solutions of the present invention by those skilled in the art without departing from the spirit of the present invention should fall within the protection scope defined by the claims of the present invention.

Claims

1. A factor combination experimental design method constrained by inequality constraints, characterized in that, include: S1. Determine the response variables, influencing factors, levels of influencing factors, and inequality constraints between factors in the experiment; S2. Perform coordinate transformation of the factor levels according to the inequality constraints to obtain a new coordinate system; S3. Based on the new coordinate system, introduce a disturbance term to fill in the missing experimental data for the test points and obtain the optimal factor combination; S4. Convert the optimal factor combination into factor encoding values ​​under the original coordinates; S5. Conduct experiments based on the factor coding values, calculate regression coefficients based on the experimental results, and fit a regression equation.

2. The factor combination experimental design method constrained by inequality restrictions according to claim 1, characterized in that, S2 performs coordinate transformation of the factor levels according to the inequality constraints, including: The original level value range is transformed based on the value range of each influencing factor; Based on the transformed interval, a variable is introduced. By taking the mutual constraints between the two factors into account in the coordinate transformation, a new coordinate system is obtained.

3. The factor combination experimental design method constrained by inequality restrictions according to claim 2, characterized in that, The transformed interval is: ； in, The transformed interval, , and For components The lower and upper bounds of the original value range.

4. The factor combination experimental design method constrained by inequality restrictions according to claim 2, characterized in that, The new coordinate system is: = ； Among them, In this case, For a non-zero point that varies freely from -1 to 1, The free variation factor is from -1 to 1. and To transform the actual values ​​of the original factors into New values ​​for the interval, for The free variation factor.

5. The factor combination experimental design method constrained by inequality restrictions according to claim 1, characterized in that, S3 introduces a perturbation term to fill in the missing parts of the test data, including: Experiments are conducted at test points within a preset range where data cannot be obtained, in order to supplement the experimental data.

6. The factor combination experimental design method constrained by inequality restrictions according to claim 1, characterized in that, S4 converts the optimal factor combination into factor encoding values ​​in the original coordinate system, including: Obtain the coordinates in the original coordinate system by restoring the original coordinate system. and The optimal value.

7. The factor combination experimental design method constrained by inequality restrictions according to claim 1, characterized in that, The factor encoding value is: 。 8. The factor combination experimental design method constrained by inequality restrictions according to claim 1, characterized in that, The S5 fitted regression equation includes: Experiments were conducted based on the factor coding values ​​to collect data and establish a solution model for the regression equation. The least squares estimation method in regression analysis is used to estimate the regression coefficients in the solution model, and the significance of the solution model and regression coefficients is tested. Finally, the regression equation is given.

9. The factor combination experimental design method constrained by inequality restrictions according to claim 8, characterized in that, The solution model is as follows: ； in, For the predicted value of the response variable, b1-b3 is the intercept term, and b1-b3 are the regression coefficients.

10. A factor combination experimental design system constrained by inequality constraints, characterized in that, The system for implementing the method as described in any one of claims 1-9 includes: a module for determining the experimental objective and influencing factors, a coordinate transformation module, an experimental data compensation module, an optimal factor combination transformation module, and a curve fitting module; The module for determining the experimental objective and influencing factors is used to determine the experimental response variables, influencing factors, and the equality constraints and lower bound constraints between influencing factors. The coordinate transformation module is used to perform coordinate transformation of factor levels according to inequality constraints. The test data compensation module is used to introduce a disturbance term to compensate for the missing parts of the test data at the test points; The optimal factor combination transformation module is used to transform the obtained optimal factor combination into factor code values ​​under the original coordinates. The curve fitting module is used to calculate regression coefficients and fit regression equations based on experimental results.

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