Improved NSGA-II-based rubber mixing process parameter optimization method

Abstract

The invention provides a rubber internal mixing process parameter optimization method based on improved NSGA-II. The method comprises the following steps: establishing a nonlinear mapping model between process parameters and quality indexes by adopting an XGBoost algorithm; the process parameters are optimized through an improved NSGA-II algorithm, the diversity of solutions is improved by introducing a normal distribution crossover (NDX) operator, an advantage set generation method is combined to ensure that initial population distribution is uniform, and finally an optimal process parameter combination is obtained. According to the method, modeling and optimization can be efficiently completed in a multi-dimensional complex parameter space, the stability of rubber material quality is effectively improved, the efficiency is improved, the problems that an internal mixing process is complex in parameter, high in nonlinearity and difficult in multi-objective optimization are solved, the method can be widely applied to rubber product manufacturing enterprises, and technical support is provided for intelligent production and process optimization.

Description

Technical Field

[0001] The present invention relates to the technical field of rubber internal mixing, and specifically but not limited to an optimization method for rubber internal mixing process parameters based on improved NSGA-II. Background Art

[0002] In the process of tire production, internal mixing is the first process and a crucial step. The internal mixing process usually involves mixing rubber raw materials with various additives to form tire rubber compounds. Its production process is complex and has significant non-linear characteristics. During this process, multiple key variables such as the Mooney viscosity, specific gravity, and hardness of the rubber interact with each other, directly determining the performance and quality of the final product.

[0003] In the internal mixing process, process parameters such as mixing time, temperature, power, rotor speed, and starting temperature all have important effects on the quality of the rubber compound. However, traditional optimization methods often rely on experience and trial-and-error, and are unable to effectively handle the high-dimensional, non-linear, and complex constraint conditions in the internal mixing process, making it difficult to achieve the production of high-efficiency and high-quality tires.

[0004] XGBoost improves the prediction accuracy of the model by integrating the prediction results of multiple decision trees, has strong fitting ability, and is particularly suitable for processing a large amount of high-dimensional and non-linear data in the internal mixing process. At the same time, the improved NSGA-II algorithm is a multi-objective optimization method based on genetic algorithms, which is widely used in complex engineering problems. By introducing a normal distribution crossover operator and good point set initialization, the search efficiency and global exploration ability can be improved, thereby further improving the convergence of the algorithm and the quality of the solution.

[0005] Combining XGBoost with the improved NSGA-II algorithm can effectively perform modeling and optimization in the high-dimensional and complex rubber internal mixing process parameter space. Optimizing the internal mixing process on the basis of fully considering various constraints and objectives in actual production helps to precisely control the quality of the rubber compound, effectively improve production efficiency, and achieve the optimal balance of multiple optimization objectives.

[0006] In view of this, a new method is needed to solve at least some of the above problems. Summary of the Invention

[0007] Aiming at one or more problems in the prior art, the present invention proposes an optimization method for rubber internal mixing process parameters based on improved NSGA-II. Aiming at the quality problems in the rubber internal mixing process, the improved NSGA-II algorithm is used to realize the optimization adjustment of process parameters and improve the product quality in the internal mixing process of tire production.

[0008] The technical solution to achieve the object of the present invention is as follows:

[0009] An optimization method for rubber mixing process parameters based on improved NSGA-II, comprising:

[0010] S1. Determine the quality indicators of the rubber mixing process, and use the Pearson correlation coefficient to screen out the rubber mixing process parameters that are strongly correlated with the quality indicators;

[0011] S2. Based on the process parameters as input and the quality indicators as output, construct a non-linear mathematical model of the process parameters and quality indicators of the rubber mixing process based on XGBoost, and obtain the relationship between the quality indicators and process parameters: Mooney viscosity model function F(x), specific gravity model function Q(x), hardness model function U(x);

[0012] S3. Based on the model functions F(x), Q(x), and U(x), with the process parameters as variables, optimize and solve the non-linear function to construct an optimization model;

[0013] S4. Use the improved NSGA-II to solve the above optimization model to obtain the optimal process parameter values.

[0014] Further, for the optimization method of rubber mixing process parameters based on improved NSGA-II of the present invention, the Pearson correlation coefficient in S1 is:

[0015] In the mixing process, since there are numerous parameters involved and there may be complex relationships among them, it is often difficult to draw effective conclusions by directly analyzing the influence of all parameters on the results. The Pearson correlation coefficient can quantify the linear correlation between each pair of process parameters and quality indicators, and identify the parameters with stronger correlation by calculating its value. This method helps to reduce the dimensionality of data and remove redundant information, thus providing a more concise and efficient input for subsequent modeling and optimization. The Pearson correlation coefficient is usually denoted by the symbol r, and its value range is from -1 to 1. The calculation formula is as follows:

[0016]

[0017] In the formula, X and Y respectively represent the data of two variables, and respectively represent the means of X and Y.

[0018] Further, for the optimization method of rubber mixing process parameters based on improved NSGA-II of the present invention, the rubber mixing process parameters that are strongly correlated with the quality indicators include: mixing time, mixing temperature, mixing power, rotor speed, and starting temperature.

[0019] Further, for the optimization method of rubber mixing process parameters based on improved NSGA-II of the present invention, the mathematical model in S2 is:

[0020] S2-1. XGBoost is an efficient machine learning algorithm based on the gradient boosting tree framework. Its goal is to minimize an objective function composed of a loss function and a regularization term. It fits the training data through the weighted sum of trees, gradually reducing the residuals. Its objective function is:

[0021]

[0022] In the formula, is the loss function, and Ω(f k ) is the regularization term.

[0023] S2-2. XGBoost makes the final prediction through the output of the weighted decision tree, and constructs the model functions F(x), Q(x), and U(x) of the process parameters and quality indicators. For the tree in the t-th round, the prediction formula is:

[0024]

[0025] In the formula, is the predicted value of the model, and η is the learning rate. f t (x) is the prediction result of the current t-th round of the tree.

[0026] Furthermore, for the method for optimizing rubber internal mixing process parameters based on the improved NSGA-II of the present invention, the regularization term Ω(f k ) described in S2-1 is used to control the complexity of the tree and prevent overfitting, and its definition is:

[0027]

[0028] In the formula, T is the number of leaf nodes of the tree, ω j is the weight of the leaf node, and γ and λ are the hyperparameters of regularization.

[0029] Furthermore, for the method for optimizing rubber internal mixing process parameters based on the improved NSGA-II of the present invention, the optimization model described in S3 is:

[0030] Find x=[x1x2x3x4x5] T

[0031] Y 1min =min(F(x))

[0032] Y 2max =max(Q(x))

[0033] Y 3max =max(U(x))

[0034] s.t x kl ≤x k ≤xkm k = 1, 2, 3, 4, 5

[0035] Wherein, F(x), Q(x), and U(x) are respectively the relationship models between Mooney viscosity, specific gravity, hardness and process parameters. x1, x2, x3, x4, and x5 are respectively five process parameters to be optimized: mixing time, mixing temperature, mixing power, rotor speed, and starting temperature. x kl , x km are respectively the upper constraint limit and the lower constraint limit of the k-th process parameter.

[0036] Furthermore, for the rubber internal mixer process parameter optimization method based on the improved NSGA-II of the present invention, the improved NSGA-II described in S4 is as follows:

[0037] S6-1. When the NSGA-II algorithm initializes the population, it uses random generation. The disadvantage of this method is that its randomness leads to uneven distribution of the population in space. In order to eliminate this uncertainty and solve the problem of uneven spatial distribution of the population, a new way of initializing the population is introduced: generating the population with good point sets.

[0038] Assume that the spatial dimension where the population is located is n, and the population size is m. Construct a good point set with size m:

[0039] P n (i) = (r1i1, r2i2, r3i3,..., r n i n ), i = 1, 2, 3,... n

[0040] Wherein, P n (i) is the sample set, r is the good point, and n is the number of samples.

[0041] S6-2. Introduce the normal distribution crossover (NDX) operator to replace the simulated binary crossover (SBX) operator. In the crossover operation, select two individuals x1 and x2 (as the parents), and then use the NDX operator to generate two new individuals y1 and y2 (as the offspring). For the i-th variable (x 1,i and x 2,i ), its crossover process is as follows:

[0042] 1) Generate a random number u ∈ [0, 1];

[0043] 2) If u ≤ 0.5, then

[0044]

[0045] If u > 0.5, then

[0046]

[0047] Wherein, |N(0,1)| is a normally distributed random variable.

[0048] Compared with the prior art by adopting the above technical solution, the present invention has the following technical effects:

[0049] 1. The method for optimizing the rubber mixing process parameters based on the improved NSGA-II of the present invention uses XGBoost modeling to accurately capture the non-linear characteristics and complex multi-variable relationships in the rubber mixing process, provides a more accurate prediction of the properties of rubber compounds, and thus provides a scientific basis for the optimization of process parameters.

[0050] 2. The method for optimizing the rubber mixing process parameters based on the improved NSGA-II of the present invention uses the improved NSGA-II algorithm to find the optimal solution in a large parameter space, accurately adjusts the key process parameters in the rubber mixing process, and significantly improves the accuracy and efficiency of process optimization. Description of the Drawings

[0051] The drawings are used to provide a further understanding of the present invention, and together with the description are used to explain the embodiments of the present invention, and do not constitute a limitation to the present invention. In the drawings:

[0052] Figure 1 Shows the flow chart of the method for optimizing the rubber mixing process parameters based on the improved NSGA-II of the present invention.

[0053] Figure 2 Shows the flow chart of the improved NSGA-II algorithm of the present invention.

[0054] Figure 3 Shows the correlation analysis diagram of the present invention.

[0055] Figure 4 Shows the XGBoost model training result diagram of the present invention.

[0056] Figure 5 Shows the multi-objective optimization result diagram of the present invention.

[0057] Figure 6 Shows the iterative convergence diagram of the present invention. Detailed Embodiments

[0058] To further understand the present invention, the preferred embodiments of the present invention will be described below in conjunction with embodiments. However, it should be understood that these descriptions are only for further explaining the features and advantages of the present invention, rather than limiting the claims of the present invention.

[0059] The description of this part is only for typical embodiments, and the present invention is not limited to the scope described in the embodiments. Combinations of different embodiments, mutual replacement of some technical features in different embodiments, and mutual replacement of the same or similar prior art means and some technical features in the embodiments are also within the scope of description and protection of the present invention.

[0060] The present invention proposes an optimization method for rubber mixing process parameters based on improved NSGA-II, as Figure 1 shown, including:

[0061] S1. Determine the quality indicators of the rubber mixing process, and use the Pearson correlation coefficient to screen out the process parameters strongly correlated with the quality indicators.

[0062] Determine the quality indicators of the rubber mixing process as: Mooney viscosity, specific gravity, and hardness. There are many process parameters in the mixing process, including temperature, rotation speed, time, power, etc. Due to the highly complex interaction and influence among the parameters, it is often difficult to draw effective conclusions by directly analyzing the influence of all parameters on the results. Therefore, it is necessary to screen out the most critical parameters for the quality indicators among these variables and optimize them effectively.

[0063] The Pearson correlation coefficient can quantify the linear correlation between each pair of process parameters and quality indicators, and identify the parameters with strong correlation by calculating its value. This method helps to reduce the dimension of the data and remove redundant information, thus providing a more concise and efficient input for subsequent modeling and optimization.

[0064] The Pearson correlation coefficient is usually represented by the symbol r, and its value range is from -1 to 1. The calculation formula is as follows:

[0065]

[0066] In the formula, X and Y respectively represent the data of two variables, and respectively represent the means of X and Y.

[0067] The results of the correlation analysis are as Figure 3 shown. Finally, five strongly correlated process parameters, namely mixing time, mixing temperature, mixing power, rotor speed, and starting temperature, are determined.

[0068] S2. Based on XGBoost, construct a non-linear mathematical model of the mixing process quality indicators and process parameters to obtain the model functions F(x), Q(x), and U(x): Taking the five process parameters of mixing time, mixing temperature, mixing power, rotor speed, and starting temperature as inputs, based on XGBoost, construct a non-linear mathematical model of the mixing process quality indicators and process parameters to obtain the model functions F(x), Q(x), and U(x).

[0069] S2-1. Set initial model parameters, including the depth of the tree, learning rate, number of iterations, etc. Select 0.1 of the dataset as the test set, and the rest as the training set. Train to obtain the initial model, and evaluate the initial model using the test set.

[0070] XGBoost is an efficient machine learning algorithm based on the gradient boosting tree framework. Its goal is to minimize an objective function composed of a loss function and a regularization term. It fits the training data through the weighted sum of trees, gradually reducing the residuals. The objective function is:

[0071]

[0072] In the formula, is the loss function, and Ω(f k ) is the regularization term. Among them, the regularization term Ω(f k ) serves to control the complexity of the tree and prevent overfitting. Its definition is:

[0073]

[0074] In the formula, T is the number of leaf nodes of the tree, ω j is the weight of the leaf node, and γ and λ are the hyperparameters of regularization.

[0075] S2-2. XGBoost makes the final prediction through the output of the weighted decision tree. For the tree in the t-th round, the prediction formula is:

[0076]

[0077] In the formula, is the predicted value of the model, η is the learning rate. f t (x) is the prediction result of the current tree in the t-th round.

[0078] Through the above XGBoost, model functions F(x), Q(x), and U(x) of quality indicators (Mooney viscosity, specific gravity, hardness) and process parameters (mixing time, mixing temperature, mixing power, rotor speed, starting temperature) are constructed. The training results of the model are as Figure 4 shown.

[0079] S3. Based on the model functions F(x), Q(x), and U(x), taking the process parameters of the internal mixer as variables, optimize and solve with the non-linear function of process parameters and quality indicators.

[0080] The Mooney viscosity directly affects the fluidity and processing performance of rubber. A lower Mooney viscosity generally means better fluidity, enabling more uniform dispersion of fillers during mixing. Considering the ease of processing of subsequent semi-finished products and the mutual adhesion and penetration of rubber compounds, the smaller the Mooney viscosity within the standard, the better. Specific gravity refers to the mass of rubber material per unit volume, i.e., the density of rubber. The lower the specific gravity, the lower its weight and cost. A higher specific gravity can increase the quality, stability, and wear resistance of tires. Considering the quality and stability of tire products, the larger the specific gravity within the index standard, the better. Hardness is a rigidity index of rubber products and is directly related to its service life and wear resistance. Therefore, the larger the hardness within the index standard, the better.

[0081] The optimization model constructed from the above analysis is as follows:

[0082] Find x=[x1 x2 x3 x4 x5] T

[0083] Y 1min =min(F(x))

[0084] Y 2max =max(Q(x))

[0085] Y 3max =max(U(x))

[0086] s.t x kl ≤x k ≤x km k=1,2,3,4,5

[0087] In the formula, F(x), Q(x), and U(x) are the relationship models between Mooney viscosity, specific gravity, hardness, and process parameters respectively. x1, x2, x3, x4, and x5 are the five process parameters to be optimized, namely mixing time, mixing temperature, mixing power, rotor speed, and starting temperature. x kl ,x km are the upper and lower bounds of the k-th process parameter constraint respectively.

[0088] S4. The improved NSGA-II is used to solve the above optimization model to obtain the optimal process parameter values for the rubber internal mixing process, realizing the multi-objective optimization of the process parameters in the internal mixing process.

[0089] The flow chart of the improved NSGA-II algorithm is as Figure 2 shown. By introducing the good point set initialization and normal distribution crossover operator, the optimization efficiency and effect of the NSGA-II algorithm are improved. Specifically:

[0090] S4-1. When initializing the population, the NSGA-II algorithm uses random generation. The disadvantage of this method is that its randomness leads to uneven distribution of the population in space. To eliminate this uncertainty and solve the problem of uneven spatial distribution of the population, a new way of initializing the population is introduced: generating the population with good point sets.

[0091] Suppose the spatial dimension where the population is located is n, and the population size is m. Construct a good point set with a quantity of m:

[0092] P n (i) = (r1i1, r2i2, r3i3,..., r n i n ), i = 1, 2, 3,... n

[0093] In the formula, P n (i) is the sample set, r is the good point, and n is the number of samples.

[0094] S4-2. Introduce the normal distribution crossover (NDX) operator to replace the simulated binary crossover (SBX) operator. In the crossover operation, select two individuals x1 and x2 (as parents), and then use the NDX operator to generate two new individuals y1 and y2 (as offspring).

[0095] For the i-th variable (x 1,i and x 2,i ), its crossover process is as follows:

[0096] 1) Generate a random number u ∈ [0, 1];

[0097] 2) If u ≤ 0.5, then

[0098]

[0099] If u > 0.5, then

[0100]

[0101] In the formula, |N(0, 1)| is a normal distribution random variable.

[0102] Adopting the normal distribution crossover operator to replace the simulated binary crossover operator further enhances the spatial search ability of the algorithm and speeds up the convergence rate of the algorithm.

[0103] Finally, a non-linear mathematical model of process parameters and quality indicators is established through XGBoost, and the improved NSGA-II optimization algorithm is used to realize the actual industrial optimization problem of rubber internal mixer production. During the algorithm solving process, the population size and the number of iterations are set to 100 and 200 respectively, and the distribution of the population generated by the algorithm is as Figure 5 shown. The convergence situation of the three optimization objectives with the number of iterations is asFigure 6 As shown. The optimal combination of process parameters is: mixing time 108.73 s, mixing temperature 100.21 °C, mixing power 290.58 KW / h, rotor speed 23.09 r / s, starting temperature 80.09 °C. The optimization results are Mooney viscosity of 50.214, specific gravity of 1.169, and hardness of 59.981, which are better than the historical values of the data samples at the corresponding time, indicating that the improved algorithm can improve the product quality of rubber tires, effectively optimize the internal mixing process, and improve production efficiency and effect.

[0104] The description and application of the present invention here are illustrative and not intended to limit the scope of the present invention to the above embodiments. The related descriptions of effects or advantages in the specification may not be reflected in actual experimental examples due to uncertainties in specific condition parameters or other factors, and the related descriptions of effects or advantages are not used to limit the scope of the invention. Deformations and changes of the disclosed embodiments here are possible, and various replacements and equivalent components of the embodiments are known to those of ordinary skill in the art. It should be clear to those skilled in the art that the present invention can be implemented in other forms, structures, arrangements, proportions, and with other components, materials, and parts without departing from the spirit or essential characteristics of the present invention. Other deformations and changes can be made to the disclosed embodiments here without departing from the scope and spirit of the present invention.

Claims

1. A method for optimizing rubber mixing process parameters based on improved NSGA-II, characterized in that: include: S1. Determine the quality index of the rubber mixing process, and use the Pearson correlation coefficient to screen out the rubber mixing process parameters that are strongly correlated with the quality index, wherein the quality index includes: Mooney viscosity, specific gravity and hardness; S2. Taking the process parameters as input and the quality index as output, a nonlinear mathematical model of the process parameters and quality indexes of the rubber mixing process is constructed based on XGBoost to obtain a relationship model between the quality index and the process parameters, wherein the relationship model includes a Mooney viscosity model function F(x), a specific gravity model function Q(x) and a hardness model function U(x); S3, based on the Mooney viscosity model function F(x), the specific gravity model function Q(x) and the hardness model function U(x), taking the process parameters as variables, optimizing and solving the nonlinear function to construct an optimization model; S4. Use improved NSGA-II to solve the above optimization model and obtain the optimal process parameter values.

2. The method for optimizing rubber mixing process parameters based on improved NSGA-II according to claim 1, characterized in that: The Pearson correlation coefficient r described in S1 is: In the formula, n represents the total number of samples, X i is the process parameter data of the i-th sample, Y i is the quality index data of the i-th sample, and Respectively represent X i and Y i The mean of .

3. The method for optimizing rubber mixing process parameters based on improved NSGA-II according to claim 1, characterized in that: The rubber mixing process parameters that are strongly related to the quality index include: mixing time, mixing temperature, mixing power, rotor speed and starting temperature.

4. The method for optimizing rubber mixing process parameters based on improved NSGA-II according to claim 1, characterized in that: The nonlinear mathematical model of process parameters and quality indicators based on XGBoost in S2 specifically includes: S2-1. Set the initial model parameters, train the initial model, and build the objective function O based on XGBoost. bj for: In the formula, y i is the true value of the i-th sample, is the predicted value of the i-th sample, n represents the total number of samples, f k is the kth decision tree model, K is the total number of decision trees, is the loss function, Ω(f k ) is the regularization term; S2-2. The final prediction is made through the output of the weighted decision tree to obtain the Mooney viscosity model function F(x), specific gravity model function Q(x), and hardness model function U(x) between the quality index and the process parameters. The prediction formula of the decision tree of the tth iteration is: In the formula, is the t-th round prediction value of the model, η is the learning rate, f t (x) is the prediction result of the current t-th round decision tree, and x represents the input feature of the sample.

5. The method for optimizing rubber mixing process parameters based on improved NSGA-II according to claim 4, characterized in that: The regularization term Ω(f k )for: Where T is the number of leaf nodes in the decision tree, ω j is the weight of leaf node j, and γ and λ are regularization hyperparameters.

6. The method for optimizing rubber mixing process parameters based on improved NSGA-II according to claim 1, characterized in that: The optimization model described in S3 is: Find x=[x1x2x3x4x5] T Y 1min =min(F(x)) Y 2max =max(Q(x)) Y 3max =max(U(x)) s.t x kl ≤x k ≤x km k=1,2,3,4,5 Where Y 1min represents the minimization of the Mooney viscosity model function, Y 2max Represents the maximum weight model function, Y 3max represents the maximization hardness model function, F(x), Q(x), and U(x) are the relationship models between Mooney viscosity, specific gravity, hardness, and process parameters, respectively. x1, x2, x3, x4, and x5 are the five process parameters to be optimized, namely, mixing time, mixing temperature, mixing power, rotor speed, and starting temperature, respectively. kl 、x km are the upper and lower constraints of the kth process parameter respectively.

7. The method for optimizing rubber mixing process parameters based on improved NSGA-II according to claim 1, characterized in that: The improved NSGA-II described in S4 is: S4-1. When r∈G s , there exists a point set P n (i) are as follows: P n (i)=(r1i1,r2i2,r3i3,...,r n i n ),i=1,2,3,...n In the formula, G s is an S-dimensional Euclidean geometric space, P n (i) represents the sample set, r represents the best point, i represents the i-th sample, and n represents the number of samples; If P n (i) Deviation If the following conditions are met, it is called P(k) good point set: In the formula, C(r,ε) is a constant related only to r and ε, and ε is an arbitrary positive number; or r = {e j i}, k is the smallest prime number satisfying (k-3) / 2≥s, j is the dimension index, and s is the Euclidean geometric space G s Dimensional restrictions; Use the good point set P(k) to generate the initialization population P0, the size of the initialization population is N; S4-2. Perform non-dominated sorting on the initialized population P0 to obtain multiple non-dominated frontiers, and calculate the congestion degree of the individuals in each non-dominated front. S4-3, Selection: Based on non-dominated sorting and crowding, select the parent individuals from P0. Crossover: Introduce the normal distribution crossover NDX operator, select two individuals x1 and x2 as parent individuals in the crossover operation, and use the NDX operator to generate two new individuals y1 and y2 as offspring. For the i-th variable x 1,i and x 2,i The crossover process is as follows: 1) Generate a random number u∈[0,1]; 2) If u≤0.5, then If u>0.5, then Where |N(0,1)| is a normally distributed random variable, x 1,i and x 2,i Represents the value of the i-th variable of the parent individuals x1 and x2, y 1,i ,y 2,i Respectively represent the value of the i-th variable of the offspring individuals y1 and y2; S4-4, after the crossover process described in 4-2, a mutation operation is performed to form a child population Q0, which is merged with the parent population P0 to obtain a merged population R0, the size of which is 2N; S4-5, perform non-dominated sorting on the merged population R0 to obtain multiple non-dominated frontiers, and calculate the crowding degree of each individual in the non-dominated frontier; S4-6, select the first N individuals from the merged population R0 based on non-dominated sorting and crowding to generate a new population P1; S4-7. Repeat steps S4-2 to S4-6 until the termination condition is met.

8. The method for optimizing rubber mixing process parameters based on improved NSGA-II according to claim 7, characterized in that: The generation of the initialization population P0 from the good point set P(k) in S4-1 specifically includes: 1) Calculate the r value: r = (r1, r2, ... r t ),in mod is the modulus operation, m i represents the i-th individual; 2) Construct a set of good points of number m: P n (i)=(r1i1,r2i2,r3i3,...,r n i n ),i=1,2,3,...n; 3) P n Mapped to the feasible domain where the population is located: in, represents the value of the i-th individual in the j-th dimension, a j Indicates the lower limit of the current dimension, b j Indicates the upper limit of the current dimension; 4) Generate an initialization population P0, the size of the initialization population is N.

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