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Product weight is one of the most important properties for an injection-molded part. The determination of process parameters for obtaining an accurate weight is therefore essential. This study proposed a new optimization strategy for the injection-molding process in which the parameter optimization problem is converted to a weight classification problem. Injection-molded parts are produced under varying parameters and labeled as positive or negative compared with the standard weight, and the weight error of each sample is calculated. A support vector classifier (SVC) method is applied to construct a classification hyperplane in which the weight error is supposed to be zero. A particle swarm optimization (PSO) algorithm contributes to the tuning of the hyperparameters of the SVC model in order to minimize the error between the SVC prediction results and the experimental results. The proposed method is verified to be highly accurate, and its average weight error is 0.0212%. This method only requires a small amount of experiment samples and thus can reduce cost and time. This method has the potential to be widely promoted in the optimization of injection-molding process parameters.

Injection molding is regarded as the most important method for mass-producing plastic parts [

For a specific material and product, process parameters are the most important factors affecting product weight, and many studies have been conducted to explore the relationship between process parameters and product weights. Hassan [

Because of these challenges, researchers have conducted many studies to explore the approaches to take in order to optimize injection-molding parameters. The design of experiments (DOE), such as the Taguchi method, is applied to improve the quality of manufactured goods by analyzing the signal-to-noise (S/N) ratio and the analysis of variance (ANOVA) [

In this study, the optimization objective is to generate process parameters for the molding of optimal products of a standard weight. Because the weight of the molded product has an either positive or negative error, the optimization problem can be converted into a classification problem. The support vector machine (SVM) method is now widely applied in classification problems, such as separating defective and nondefective products. Yu [

The implementation procedure for the proposed optimization method is described in Section

The standard weight of a plastic product is determined by technologists prior to molding. The product weight can be regarded as a function of the process parameters throughout the injection-molding process. Each experiment sample under different process parameters could be heavier or lighter than the standard weight and can therefore be labeled as a positive sample (heavier) or a negative sample (lighter). The molding parameters for the desired product with a standard weight should be located between the molding parameters for the positive and negative samples. Hence, this study intends to search for the parameter classification boundary between the positive and negative classes, as shown in Figure

Schematic diagram of converting the parameter optimization problem to a classification problem.

In this study, an SVC model is employed to classify products by their weights. Products under different process parameters are injection-molded and weighed. These products are labeled as either positive or negative samples, and they are employed as training data for the SVC. As illustrated in Figure

Step 1: initialize parameter range. Choose molding parameters, and then set their initial range. The initial parameter range should cover all the feasible parameters.

Step 2: perform experiments. Carry out experiments under each process parameter. Then, evaluate experimental results by comparing the product weight with the standard weight.

Step 3: train the SVC model. Label these samples, and prepare the data sets for the SVC. Select proper kernel function, and train the SVC with sample data.

Step 4: tune SVC hyperparameters. Apply PSO to determine hyperparameters. A hyperplane that separates underweight and overweight samples can be obtained. The optimal process parameters are located on that hyperplane.

Step 5: perform verification experiments.

The support vector classifier is a machine learning algorithm based on statistical learning theory. It minimizes the structural risk and improves the generalization ability of the learning machine. The target of a classification is to estimate a function

Considering a linear case as shown in Figure

The margin between two classes is

Moreover, the target is to find the hyperplane that makes the margin the largest. For mathematic convenience, maximize

Because of the linear inseparability of the data set, a kernel function must be adopted to map the data (input space) to a high-dimensional space (feature space) where a linear separating hyperplane can be constructed. One kernel commonly used in training nonlinear SVCs is the Gaussian kernel which maps data to an infinite-dimensional space. The Gaussian kernel on two samples

Although the probability of linear separation is increased by introducing the kernel function, it is still difficult to deal with the noise in a data set. Noisy data, also known as “outliers,” have a great influence on the SVC model, especially when outliers become support vectors. The slack variable

The slack variable means that the accurate classification of the outliers is abandoned, which is an additional loss to the classifier and should be added to the objective function. Thus, the optimization problem becomes

It should be noted that the performance of an SVC model depends largely on the parameters

Particle swarm optimization is an algorithm that simulates the foraging behavior of a flock of birds. The basic idea is to find the optimal solution via collaboration and information sharing among individuals in the group. A group of particles (candidate solutions) is randomly initialized in the solution space. Each particle is evaluated by calculating its fitness value. The optimal solution is then found by iteration. In each iteration, the particle updates its own velocity and position by tracking “

In this study, PSO is applied to improve the accuracy of the SVC model by determining its hyperparameters. The key to applying PSO in this optimization problem is to construct a proper fitness function. In order to acquire the most precise result, the prediction value of the SVC should be the same as the actual value from experiments. By tuning the hyperparameters of the SVC model, the position of the hyperplane can be modified. Thus, the distance between the hyperplane and the samples is as close as possible to the actual weight error. The calculation of fitness of particles is

Through PSO, all particles converge to the position where the fitness value is reduced to the minimum. The optimal position corresponds to the modified combination of hyperparameters of the SVC. With this method, the parameters

To demonstrate the accuracy of the proposed method, an experiment using a plastic lenses is presented in this section. A high-precision electrical injection-molding machine, Zhafir VE400 (Zhafir Plastics Machinery GmbH, China), is employed in this study and is depicted in Figure

Experimental setup: (a) injection-molding machine; (b) injection mold; (c) molded plastic lenses.

Many process parameters have large effect on product weight, including packing pressure, packing time, injection pressure, injection time, and injection temperature. The effect of these parameters depends on the type of the resin [

The product weight under the condition of each process parameter is presented in Table

Weight under each parameter set.

No. | Process parameters | Product weight (g) | Weight error (g) | |
---|---|---|---|---|

Packing pressure (MPa) | Injection temperature (°C) | |||

1 | 70 | 235 | 8.0968 | −0.0632 |

2 | 80 | 235 | 8.1324 | −0.0276 |

3 | 90 | 235 | 8.1040 | −0.0560 |

4 | 100 | 235 | 8.0920 | −0.0680 |

5 | 110 | 235 | 8.1456 | −0.0144 |

6 | 120 | 235 | 8.1664 | 0.0064 |

7 | 70 | 245 | 8.0944 | −0.0656 |

8 | 80 | 245 | 8.0896 | −0.0704 |

9 | 90 | 245 | 8.1100 | −0.0500 |

10 | 100 | 245 | 8.1248 | −0.0352 |

11 | 110 | 245 | 8.1716 | 0.0116 |

12 | 120 | 245 | 8.2104 | 0.0504 |

13 | 70 | 255 | 8.1316 | −0.0284 |

14 | 80 | 255 | 8.1624 | 0.0024 |

15 | 90 | 255 | 8.2028 | 0.0428 |

16 | 100 | 255 | 8.2404 | 0.0804 |

17 | 110 | 255 | 8.2836 | 0.1236 |

18 | 120 | 255 | 8.3288 | 0.1688 |

19 | 70 | 265 | 8.4116 | 0.2516 |

20 | 80 | 265 | 8.5212 | 0.3612 |

21 | 90 | 265 | 8.7520 | 0.5920 |

22 | 100 | 265 | 8.7608 | 0.6008 |

23 | 110 | 265 | 8.6756 | 0.5156 |

24 | 120 | 265 | 8.8624 | 0.7024 |

Product weight under varying process parameters.

From the experimental results, we can observe that the product weight has a positive correlation with both the packing pressure and injection temperature. This phenomenon agrees well with the research of Hamdy [

As the SVC is a supervised machine learning model, the data set must be labeled. In this study, we regard all the negative values with a -1 label and the positive values with a +1 label. The two key parameters (

PSO convergence procedure: (a) random initialization; (b) 8^{th} iteration step; (c) 54^{th} iteration step; (d) 261^{st} iteration step, converge at

Classification hyperplane (green curve); red dots are overweight samples; blue dots are underweight samples.

In order to evaluate the hyperplane obtained from the proposed method, a set of verification experiments are carried out. The verification experiment scheme is illustrated in Figure

Verification experiments scheme: a0, b1, and b2 are process parameters on the hyperplane, while a1, a2, ……, a8 are process parameters not on the hyperplane.

To verify that a single point located on the hyperplane is the best parameter combination among the other points around it, the center point (a0) with a packing pressure of 95 MPa and an injection temperature of 248°C is selected as the verification point. A series of experiments (a1, a2, a3, and a4) is first conducted with a fixed packing pressure of 95 MPa and varying injection temperatures from 242°C to 254°C in increments of 3°C. Then, several experiments (a5, a6, a7, and a8) were conducted with a fixed injection temperature of 248°C and varying packing pressures from 89 MPa to 101 MPa in increments of 3 MPa. In total, 9 sets of experiments are conducted. For each parameter combination, 30 products are molded and weighed. The results are provided in Figure

Verification results: product weight under (a) varying injection temperature; (b) varying packing pressure.

Product weight under varying injection temperature.

Injection temperature (°C) | Average weight (g) | Error (%) | Standard deviation (g) |
---|---|---|---|

242 | 8.1050 | −0.6746 | 0.00401 |

245 | 8.1194 | −0.4973 | 0.00445 |

248 | 8.1617 | 0.0212 | 0.00453 |

251 | 8.1823 | 0.2736 | 0.00564 |

254 | 8.2147 | 0.6705 | 0.00705 |

Product weight under varying packing pressure.

Packing pressure (MPa) | Average weight (g) | Error (%) | Standard deviation (g) |
---|---|---|---|

242 | 8.1286 | −0.3852 | 0.00562 |

245 | 8.1437 | −0.1998 | 0.00310 |

248 | 8.1617 | 0.0212 | 0.00453 |

251 | 8.1787 | 0.2288 | 0.00464 |

254 | 8.1979 | 0.4647 | 0.00623 |

In the case in which a single point on the hyperplane is known as an optimal parameter set, the second step is to verify that other points on the hyperplane are also feasible. Experiments under two other parameter combinations on the hyperplane (b1: 85 MPa/252.5°C and b2: 105 MPa/244.8°C) are carried out. Their weights are 8.1645 g and 8.1634 g, respectively, and the results are presented in Figure

Product weight under three parameters on the hyperplane.

This study proposed a new method for optimizing process parameters to mold products with a standard weight. The parameter optimization problem is converted into a weight classification problem according to whether the sample is heavier or lighter than the standard weight. The support vector classifier and particle swarm optimization algorithm are adopted to construct the classification hyperplane, which separates samples. A new criterion for the classification model is introduced in order to improve accuracy. The product weight under parameter sets located on the classification hyperplane should be the same as the standard weight. Based on the results obtained in this study, the following conclusions can be drawn: (1) The product weight under optimized parameters is rather close to the standard weight. Experimental results indicate that the weight error can reach 0.0212%. The hyperplane obtained from the SVC has a high level of correspondence with the verification result. (2) The idea of converting an optimization problem to a classification problem is proven useful for process parameter optimization. (3) The accuracy of the classification model can be improved by applying the PSO method and the criterion proposed in this study. (4) In contrast to the shortcomings in the traditional Taguchi method, this approach for the optimization of process parameters can deal with the situation in which the process parameters are continuous and nonlinear. In general, the proposed method has the advantages of a small data set requirement, high accuracy, and the ability to deal with nonlinear problems.

The ongoing studies will serve to solve the following problems: (1) apply this method to multiple parameters optimization problems; (2) drive the mathematical model to a tool for design algorithms to determine the process window; (3) promote the proposed method to other parameter optimization problems for other polymer processing techniques, such as extrusion molding and blow molding.

The data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that they have no conflicts of interest.

The authors would like to acknowledge the financial support of the Zhejiang Provincial Natural Science Foundation of China (No. LZ18E050002), the Key Research and Development Plan of Zhejiang Province (No. 2020C01113), and the National Natural Science Foundation Council of China (No. 51875519 and No. 51635006).