Material mixing test method and system with limited test space
By employing a simplex centroid design method in the mixing experiment, combined with equality and lower bound constraints, the problem of limited experimental space was solved, enabling efficient regression model construction and mixing formulation optimization.
Patent Information
- Application Number
- CN202511925253.4
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2025-12-19
- Publication Date
- 2026-03-17
AI Technical Summary
In mixture experiments, due to limited experimental space, traditional methods are difficult to effectively construct regression models, leading to resource waste and information matrix degradation.
By employing a simplex centroid design method, combined with equality constraints and lower bound constraints in the experimental space, the distribution of experimental points and experimental schemes are obtained through transformation relationships, and a regression model is constructed.
This approach improves the utilization rate of experimental resources and the value of information output within a limited experimental space, while reducing the number of experiments and costs, thus achieving efficient optimization of mixture formulations.
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Figure CN121683273A_ABST
Abstract
Description
TECHNICAL FIELD
[0001] The application belongs to the technical field of mixture experiment design, and particularly relates to a mixture experiment method and system with limited experiment space. BACKGROUND
[0002] In daily life and industrial production, the problem of formula and proportion is widespread, which is usually referred to as a mixture problem. The mixture refers to the process of forming a new product by mixing several different components according to a certain proportion. For example, the proportion of components such as steel, aluminum alloy and other metal materials, the combination of active components and auxiliary materials in pharmaceutical preparations, the configuration of nutritional elements in feed, and even the optimization of fuel mixing, all belong to typical applications of mixture problems. Moreover, from a more general perspective, the allocation of resources in enterprises, such as material scheduling, capital investment, equipment layout and personnel allocation, can also be abstracted as a kind of mixture problem, and the core of which is how to determine the optimal combination proportion of each component under limited conditions.
[0003] The mixture experiment is to systematically investigate the internal relationship between each mixture component and the concerned index (such as product quality, performance, cost, etc.) through physical experiment or computer simulation experiment. Taking cake making as an example, different proportions of flour, water, oil, sugar, and leavening agent and spices will directly affect the softness, taste, appearance and other indicators of the cake. Through the design of a scientific mixture experiment, the mapping mechanism between these components and the final quality can be effectively revealed. It should be noted that the mixture experiment should at least include three variable components, and those components that remain unchanged in the experiment should not be included as mixture factors in modeling.
[0004] Unlike traditional experiment design methods, mixture experiment design is subject to a fundamental constraint: the experiment index is only related to the proportion of each component, and is not related to the total amount of mixture. The proportion of each component must be non-negative and between 0 and 1, and the sum of all component proportions must be strictly equal to 1. This equality constraint significantly limits the degree of freedom of the factor space, and also makes the direct application of traditional regression methods prone to numerical problems such as information matrix degeneration.
[0005] In addition to the equality constraint of the sum being 1, lower bound constraints are another very common restriction in mixture experiments. In practical applications, due to product performance, cost control or process feasibility considerations, the proportion of some components cannot be infinitely reduced, but must be greater than a given minimum value. The lower bound constraint further restricts the design space of the mixture experiment. In practice, due to cost, process or performance requirements, each component often needs to meet a minimum lower bound. This constraint cuts the feasible simplex space into a smaller irregular sub-region, so that many standard design points may fall into the infeasible region. Therefore, the experiment design must be re-optimized in this limited space to effectively model.
[0006] To address these challenges, mixture design has evolved into a class of regression design methods specifically for handling component ratios. It constructs specialized test point sets (such as simplex lattice designs and simplex centroid designs) to fit high-order regression models with as few trials as possible, thus systematically describing the relationship between components and responses and effectively avoiding problems such as model unidentification. This method is not only mathematically rigorous but also highly practical, significantly improving the efficiency and interpretability of formulation development.
[0007] For this reason, mixture design has been widely used in many fields such as materials science, chemical engineering, food processing, pharmaceutical industry, and agricultural research. For example, it is used to optimize the composition in alloy development to improve mechanical properties, to balance strength and cost in concrete mix design, to adjust components in coating formulations to improve leveling and weather resistance, and to improve the bioavailability of formulations through excipient ratios in the pharmaceutical field. Summary of the Invention
[0008] To address the problems existing in the prior art, this invention provides a mixing test method and system with limited experimental space, aiming to obtain experimental schemes and construct regression models under realistic constraints, thereby optimizing the mixing formulation and providing methodological support for scientific research and production in related fields.
[0009] To achieve the above objectives, the present invention provides the following solution: A method for testing mixtures in a space-constrained environment, the method comprising: The experimental space is obtained based on the response variables, experimental factors, equality constraints and lower bound constraints between experimental factors in the mixture experiment; Obtain the transformation relationship between the coordinates of any point in the experimental space and the natural space and the coding space; Based on the transformation relationship, the simple centroid design method is used to obtain the distribution of test points and the test scheme; Based on the distribution of test sites and the test plan, a mixture test was conducted, test data were collected, and a regression model was constructed. The optimal combination ratio of each component of the mixture is obtained based on a regression model.
[0010] Preferably, methods for obtaining the experimental space based on response variables, experimental factors, and equality constraints and lower bound constraints between experimental factors in mixture experiments include: The purpose of the mixing test is to obtain the response variables and experimental factors of the mixing test; The experimental space is obtained based on the equality constraints and lower bound constraints of the combined values of each experimental factor.
[0011] Preferably, the method for obtaining the transformation relationship between the coordinates of any point in the experimental space and the coding space includes: The transformation formula is: ; Among them, natural spatial coordinates Encoding spatial coordinates Column vector The transformation formulas for each component: ; ; in, Represents the first in natural space i One portion, It is a component The lower bound, p The number of components. Represents the first in the coding space i Each component.
[0012] Preferably, the constructed regression models include: ; in, Represents a regression model. i , j , k Both represent index variables of the components. , , All represent regression coefficients, and regression coefficients are uniformly represented by... If it means: ; in, for p One of the components r A set of individuals for r Take from each component t All The combined values of the test indexes of each combination.
[0013] The present invention also provides a mixing test system with limited test space, the system being used to implement the aforementioned method, the system comprising: a space acquisition module, a transformation module, a scheme acquisition module, a model construction module, and an optimal ratio acquisition module; The space acquisition module is used to obtain the experimental space based on the response variables, experimental factors, and equality constraints and lower bound constraints between experimental factors in the mixture test. The transformation module is used to obtain the transformation relationship between the coordinates of any point in the test space and the natural space and the coding space. The scheme acquisition module is used to obtain the distribution of test points and test schemes based on transformation relationships and using the simple centroid design method; The model building module is used to conduct mixture experiments based on the distribution of test points and test plans, collect test data, and build regression models. The optimal ratio acquisition module is used to obtain the optimal combination ratio of each mixture component based on a regression model.
[0014] Preferably, the spatial acquisition module includes a variable factor acquisition unit and a spatial acquisition unit; The variable factor acquisition unit is used to acquire response variables and experimental factors in the mixture test according to the purpose of the mixture test; The space acquisition unit is used to obtain the experimental space based on the equality constraints and lower bound constraints of the combined values of various experimental factors.
[0015] Preferably, the method for the transformation module to obtain the transformation relationship between the coordinates of any point in the test space and the natural space and the coding space includes: The transformation formula is: ; Among them, natural spatial coordinates Encoding spatial coordinates Column vector The transformation formulas for each component: ; ; in, Represents the first in natural space i One portion, It is a component The lower bound, p The number of components. Represents the first in the coding space i Each component.
[0016] Preferably, the constructed regression models include: ; in, Represents a regression model. i , j , k Both represent index variables of the components. , , All represent regression coefficients, and regression coefficients are uniformly represented by... If it means: ; in, for p One of the components r A set of individuals for rTake from each component t All The combined values of the test indexes of each combination.
[0017] Compared with the prior art, the beneficial effects of the present invention are as follows: (1) By incorporating the constraints of the test space into the definition and transformation of the test space, the hard requirements of the actual engineering are met in advance during the test scheme design stage, thereby avoiding the waste of valuable test resources on test points that are invalid due to non-compliance with constraints, and significantly improving the utilization rate and information output value of each test under the constraints of the test space.
[0018] (2) Optimizing the placement of points based on the transformed coding space can efficiently construct an accurate regression model with far fewer trials than the classical method, greatly reducing the economic and time costs of conducting mixed experiments under limited experimental space conditions. Attached Figure Description
[0019] To more clearly illustrate the technical solution of the present invention, the drawings used in the embodiments are 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.
[0020] Figure 1 This is a schematic diagram of the mixing test method with limited test space according to an embodiment of the present invention; Figure 2 This is a schematic diagram of the test space in an embodiment of the present invention, wherein ①-⑦ represent the coordinates corresponding to the actual proportions of the 7 tests. Detailed Implementation
[0021] 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.
[0022] 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.
[0023] Example 1 like Figure 1 As shown, the present invention provides a mixture testing method with limited testing space, comprising: The experimental space is obtained based on the response variables, experimental factors, equality constraints and lower bound constraints between experimental factors in the mixture experiment; Obtain the transformation relationship between the coordinates of any point in the experimental space and the natural space and the coding space; Based on the transformation relationship, the simple centroid design method is used to obtain the distribution of test points and the test scheme; Based on the distribution of test sites and the test plan, a mixture test was conducted, test data were collected, and a regression model was constructed. The optimal combination ratio of each component of the mixture is obtained based on a regression model.
[0024] The present invention has been specifically implemented as follows: Based on the response variables, experimental factors, equality constraints, and lower bound constraints among experimental factors in the mixture experiment, the experimental space is obtained, including: The purpose of the mixing test is to obtain the response variables and experimental factors of the mixing test; The experimental space is obtained based on the equality constraints and lower bound constraints of the combined values of each experimental factor.
[0025] Specifically, the response variables need to be determined based on the purpose of the mixture test. Then, appropriate experimental factors are selected for study based on the actual situation and the experimental objective. In addition, it is necessary to determine whether there are equality constraints and lower bound constraints among the various experimental factors.
[0026] Based on the equality constraints and lower bound constraints for the combined values of each experimental factor, the experimental space is determined as follows: ; in, Represents the first in natural space i One portion, It is a component (experimental factor). The lower bound of the value, that is, the minimum value that the component can actually achieve. p This refers to the number of components, specifically the number of different initial components in the mixture. (Regarding the lower bound...) The following conditions must be met: 。
[0027] Furthermore, the transformation relationship of the coordinates of any point in the experimental space with respect to the natural space and the coding space is obtained, including: The transformation formula between the coordinates of any point in the experimental space and the natural space and the coding space is as follows: ; Among them, natural spatial coordinates Encoding spatial coordinates Column vector , Represents the first in the coding space i Each component has a transformation formula: ; .
[0028] Furthermore, based on the transformation relationship, a simplex centroid design method is used to obtain the experimental scheme, specifically: The generation of test points is based on a core barycentric coordinate calculation rule. For any non-empty subset... Its corresponding coding space coordinates [ x 1′, x 2′,..., x p ′] T Determined by the following function: ; Among them, | S | represents a subset S Size.
[0029] Generate a first-order centroid: Take all subsets of size 1, i.e., | S |=1. Total There are such subsets. Each subset S ={ i Generate a test point, the first test point of which is... i One component is 1, and the rest are 0. These points form the simplex. p A vertex, for example: (1,0,...,0)(1,0,...,0),(0,1,0,...,0)(0,1,0,...,0), ...,(0,...,0,1)(0,...,0,1).
[0030] Generate second-order centroids: Take all subsets of size 2, i.e., | S |=2. Total There are such subsets. Each subset S ={ i , j Generate a test point, the first test point of which is... i and the j Each component is The remaining components are 0. These points constitute the simplex. p A vertex, for example: .
[0031] Generate the final center of gravity: Take the unique size as p A subset of, i.e. S ={1,2,..., p This subset generates a test point where all components are... , that is, point ( , ,..., ).
[0032] For example, when containing With one unknown parameter, the simplex lattice design at this point consists of two types of points: one type of point is... p One type of vertex has two simple vertices, and the other type of point has two coordinates. Other points with coordinates of 0, there are a total of such points. The design schemes are shown in Table 1: Table 1 Test Protocol Furthermore, based on the experimental protocol, mixture experiments were conducted, experimental data were collected, and a regression model was constructed, including: The constructed regression model includes: ; in, Represents a regression model. i , j , k Both represent index variables of the components. , , All represent regression coefficients, and regression coefficients are uniformly represented by... If it means: ; in, for One of the components r A set of individuals for r Take from each component t All The combined values of the test indexes of each combination.
[0033] Furthermore, the optimal combination ratio of each component is obtained based on a regression model, thereby optimizing the mixture formulation.
[0034] In summary, this invention integrates the constraints of experimental space into the experimental design process, enabling precise definition of feasible experimental space and the construction of corresponding coded spatial transformation relationships. Based on this, a distribution scheme for a limited number of experimental points can be scientifically designed. Even under extremely limited experimental conditions, this method can still efficiently establish an accurate regression model for mixture experiments, 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 mixture optimization problems under strictly limited resources in fields such as formulation development and materials development.
[0035] Example 2 Based on the same inventive concept, this embodiment illustrates the method described in the foregoing embodiment using specific experimental data.
[0036] S1. Determine the response variables, experimental factors, and equality constraints and lower bound constraints between experimental factors in the mixture test; The objective of the mixing test in this embodiment is to find a mixture with an elastic modulus greater than 1. p The mixing should be done carefully, and the amount of adhesive used should be as small as possible. x 1>= a 1. Oxidizing agent x 2>= a 2. Fuel x 3>= a 3, a 1. a 2. a 3 represents the corresponding threshold constant, so the constraint is: ; Its experimental space is a small triangle inside a large equilateral triangle, such as Figure 2 As shown, Figure 2 In the diagram, ①-⑦ represent the coordinates corresponding to the actual proportions of the seven experiments.
[0037] S2. Determine the transformation relationship between the coordinates of any point in the test space and the natural space and the coding space according to the constraints. The transformation relationship between the coordinates of any point in the experimental space and the natural space and the coding space is as follows: ; Where c represents the constant factor of the linear transformation.
[0038] S3. Determine the distribution of test points and the test plan according to the simplex centroid design method; In this embodiment, a three-component, third-order simplex centroid design {3,3} is used, and the number of trials is 7. The actual proportions for the 7 trials are calculated using the above formula, and the trials are conducted. The experimental scheme is shown in Table 2: Table 2 Test Protocol In Table 2, d 11 - d 73 , - All of these are variables used to represent the numerical values in the experiment matrix.
[0039] S4. Arrange the experiment, collect data, and calculate the regression coefficients to obtain the regression model; The elastic modulus was obtained by conducting the test according to the test plan in S3: ; in, y 1. y 2. y 3. y 12 , y 13 , y 23 , y 123 Both are variables, representing values in the elastic modulus.
[0040] The regression coefficients were then calculated and are shown in Table 3. Table 3 Regression Coefficients The regression model of the response variable y with respect to the small equilateral triangle coordinate system is obtained as follows: .
[0041] Finally, the optimal combination ratio of each component is obtained based on the regression model, thereby optimizing the mixture formulation.
[0042] Example 3 Based on the same inventive concept, the present invention also provides a mixing test system with limited test space, used to implement the method described in the foregoing embodiments. The system includes: a space acquisition module, a transformation module, a scheme acquisition module, a model construction module, and an optimal ratio acquisition module. The space acquisition module is used to obtain the experimental space based on the response variables, experimental factors, and equality constraints and lower bound constraints between experimental factors in the mixture test. The transformation module is used to obtain the transformation relationship between the coordinates of any point in the test space and the natural space and the coding space. The scheme acquisition module is used to obtain the distribution of test points and test schemes based on transformation relationships and using the simple centroid design method; The model building module is used to conduct mixture experiments based on the distribution of test points and test plans, collect test data, and build regression models. The optimal ratio acquisition module is used to obtain the optimal combination ratio of each mixture component based on a regression model.
[0043] Preferably, the spatial acquisition module includes a variable factor acquisition unit and a spatial acquisition unit; The variable factor acquisition unit is used to acquire response variables and experimental factors in the mixture test according to the purpose of the mixture test; The space acquisition unit is used to obtain the experimental space based on the equality constraints and lower bound constraints of the combined values of various experimental factors.
[0044] Preferably, the method for the transformation module to obtain the transformation relationship between the coordinates of any point in the test space and the natural space and the coding space includes: The transformation formula is: ; Among them, natural spatial coordinates Encoding spatial coordinates Column vector The transformation formulas for each component: ; ; in, Represents the first in natural space i One portion, It is a component The lower bound, p The number of components. Represents the first in the coding space i Each component.
[0045] Preferably, the constructed regression models include: ; in, Represents a regression model. i , j , k Both represent index variables of the components. , , All represent regression coefficients, and regression coefficients are uniformly represented by... If it means: ; in, for p One of the components r A set of individuals for rTake from each component t All The combined values of the test indexes of each combination.
[0046] 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 compounding test method for a test space-limited compounding test, characterized by, The method comprises: obtaining a test space based on response variables, test factors, equation constraints and lower bound constraints among test factors in a mixture experiment; obtaining a transformation relationship of coordinates of any point in the test space with respect to a natural space and a coding space; obtaining distribution of test points and a test scheme based on the transformation relationship by using a simplex centroid design method; performing the mixture experiment based on the distribution of test points and the test scheme, collecting test data, and constructing a regression model; obtaining an optimal combination ratio of each mixture component based on the regression model.
2. The method of claim 1, wherein, The method for obtaining the test space based on response variables, test factors, equation constraints and lower bound constraints among test factors in a mixture experiment comprises: obtaining response variables and test factors in the mixture experiment according to a purpose of the mixture experiment; obtaining the test space according to equation constraints and lower bound constraints of combination values of each test factor.
3. The method of claim 1, wherein, The method for obtaining the transformation relationship of coordinates of any point in the test space with respect to the natural space and the coding space comprises: the transformation relationship formula is: ； wherein the natural space coordinates , the encoding space coordinates , the column vector , the transformation of the individual components: ； ； in, Represents the first in natural space i One portion, It is a component The lower bound, p The number of components. Represents the first in the coding space i Each component.
4. The method of claim 3, wherein, the constructed regression model comprises: ； wherein represents a regression model, i , j , k each represents an index variable of a component, , , each represents a regression coefficient, the regression coefficients being uniformly denoted by , then: ； wherein, is p a set of certain r of the components, is r all of the components t all of the components a combination of the test index values.
5. A space-limited compounding test system for carrying out the method according to any one of claims 1 to 4, characterized in that The system comprises a space obtaining module, a transformation module, a scheme obtaining module, a model constructing module and an optimal ratio obtaining module; the space obtaining module is configured to obtain a test space based on response variables, test factors, equation constraints and lower bound constraints among test factors in a mixture experiment; the transformation module is configured to obtain a transformation relationship of coordinates of any point in the test space with respect to a natural space and a coding space; the scheme obtaining module is configured to obtain distribution of test points and a test scheme based on the transformation relationship by using a simplex centroid design method; the model constructing module is configured to perform the mixture experiment based on the distribution of test points and the test scheme, collect test data, and construct a regression model; the optimal ratio obtaining module is configured to obtain an optimal combination ratio of each mixture component based on the regression model.
6. The system of claim 5, wherein, The space obtaining module comprises a variable factor obtaining unit and a space obtaining unit; the variable factor obtaining unit is configured to obtain response variables and test factors in the mixture experiment according to a purpose of the mixture experiment; the space obtaining unit is configured to obtain the test space according to equation constraints and lower bound constraints of combination values of each test factor.
7. The system of claim 5, wherein, The transformation module obtains the transformation relationship of coordinates of any point in the test space with respect to the natural space and the coding space by using the following method: the transformation relationship formula is: ； wherein the natural space coordinates , the encoding space coordinates , the column vector , the transformation of the individual components: ； ； in, Represents the first in natural space i One portion, It is a component The lower bound, p The number of components. Represents the first in the coding space i Each component.
8. The system of claim 7, wherein, the constructed regression model comprises: ； wherein represents a regression model, i , j , k each represents an index variable of a component, , , each represents a regression coefficient, the regression coefficients being uniformly denoted by , then: ； wherein, is p a set of certain r of the components, is r extracted from t all of the components a combination of the test index values.