Robust Parameter Design and Optimization for Quality Engineering

Abstract

This paper proposes a methodology to determine the optimal settings of key decision variables that affect the resilience of an engineering design against uncertainty. Uncertainty in quality engineering is often caused by environmental factors, and scarcity of data due to limitations in the experimentation phase amplifies the level of ambiguity. The proposed robust parameter design and optimization approach utilizes the Taguchi method to find critical variables to be used in the optimization, and it utilizes robust optimization to immunize the obtained solution against uncertainty. To demonstrate our approach, we focus on design optimization of an injection molding product, a refrigerator door cap, made from thermoplastic raw material and its key quality characteristic, warpage. The near-optimal designs found by the robust parameter design and optimization approach are implemented in a real-life manufacturing environment. The numerical experiments show that the new designs significantly improve the warpage quality characteristic and the total production cycle time compared to the current design used in the manufacturing company.

Appendices

Appendix 1

1.1 Technical Product/Process and Company Descriptions

Engineering Design

A refrigerator door consists of both plastic and metal parts that are assembled together during mass production using a polyurethane-based insulation material. As polyurethane is a polymer of high viscosity, assembled parts should not have a free space in between them in order to prevent leakage. The refrigerator doors that have polyurethane leakage are separated and have to be reworked or cleaned in order to be reused in mass production. Leakage has a negative effect on the production throughput and labor cost. One of the main causes of polyurethane leakage is that the outer metallic doors and the inner plastic door caps are not properly assembled due to warpage of the door caps. 3D drawings of the refrigerator door and the door cap as well as the assembly relationship of the parts are given in Fig. 3.

Fig. 3

3D drawings of the refrigerator door (left), the door cap (middle), and the assembly relation (right)

Injection Molding and Process Variables

The core task in injection molding is forcing the melted plastic raw material into the mold. The injection process starts with preparing the raw plastic material in the form of grains. A coloring agent called ‘masterbatch’ is added, and the raw material composition is fed into the injection barrel. As the granules are slowly pushed forward by a screw, the plastic is forced into a chamber and heated to the melting point. During the process, the injection molding machine is exposed to the clamping force applied by the clamping unit of the injection molding machine to keep the mold closed while the liquefied plastic material forced into the mold by the injection unit applies a separating force. The mold remains at the prescribed temperature so that the plastic can solidify almost as soon as the mold is filled. Once the cooling phase is over, the two halves of the mold are separated from each other, and the plastic can be taken out of the mold by an operator.

Complex parts can be produced in short cycle times using injection molding. On the other hand, the attained quality characteristic of the product constitutes a trade-off with the production cycle time, namely, improving one leads to deterioration of the other. Therefore, it is of critical importance to find the optimal production design settings that ensure that the final product satisfies the given quality characteristic(s) while the associated production cycle time is kept at the minimum level. Because the manufacturer uses injection molding in mass production, the wrong operating conditions may result in extensive time and money loss due to the scale of the production (see Fig. 4).

Fig. 4

3D drawing of the injection mold

Injection Molding Process Variables

In this paper, we focus on thirteen variables, and we conduct extensive experiments to assess their effect on the warpage. The selected variables are as follows:

• injection pressure: the pressure required to fill the molten plastic raw material (mprm) in the mold cavity.

• injection speed: defines the speed of mprm while it is being injected into the mold.

• injection time: the time it takes to fill the mold completely with mprm.

• holding pressure: denotes the outside pressure applied to the mold until mprm is solidified.

• holding speed: the speed of the raw material when it is injected inside the mold during the holding phase. Due to shrinkage, the mold accommodates higher volumes of mprm than its actual volume.

• holding time: denotes the time that the raw material flows inside the mold during the holding phase.

• back suction offset: a parameter necessary to ensure uniformity of mprm for maximum part performance.

• cavity coolant temperature: the temperature of the coolant material that runs through the cooling channels of the cavity.

• cooling time: the period over which the molded part is left inside the mold to solidify. The cooling time starts when the holding time ends.

• refill speed: the speed at which plastic raw material is poured inside the barrel.

• refill pressure: the pressure used to pour plastic raw material from the hopper to the barrel.

• holding transaction: a parameter used to adjust the position of the screw inside the barrel and regulate the raw material inside the mold during injection and holding phases.

Appendix 2

1.1 Taguchi Method

Taguchi adopts a judgmental sampling, where the experiments are selected using the idea of an orthogonal array instead of being randomly selected. The method decreases the number of experiments to a set of decision parameter combinations of the process instead of a full factorial design. Taguchi uses coded, also called scaled or standardized, values for the decision factors. For further details, we refer the reader to [45] that explains the core steps of coding in the design of experiments. After the number of levels has been decided and the variables have been coded, the design experiment consists of, e.g., N combinations of the coded decision factors, where these combinations are decided according to the OA. As a statistical heuristic, the total number of experiments (N) must be greater than the total degrees of freedom of the experiment; we delve into details of these measures in the next subsection.

The aim of the Taguchi methodology is to find the settings of the decision factors that minimize the response (i.e., warpage in our case) variance while keeping the mean response at the desired level. The desired level could be a minimum/maximum or a given target value. More precisely, Taguchi uses signal-to-noise ratios (SNRs) to measure the following goals of the experiment: 1) the larger the better (e.g., maximizing the total throughput), 2) the smaller the better (e.g., minimizing the warpage), and 3) the target is the best (e.g., keeping the surface area of the mating parts in an assembly at a given target). SNRs are used to evaluate the condition of the quality characteristic in a controlled environment. In the Taguchi approach, the ‘signal’ represents the desirable effect (the mean) for the output characteristic, and the term ‘noise’ represents the undesirable effect(s) (signal disturbance) on the output due to environmental or noise factors.

Table 10 Orthogonal array table for $\text {L}_{27}(3^{13})$

The SNR for the ‘the lower the better’ goal of the experiment is defined as

$$\begin{aligned} \text {S/N}_i \ \text {ratio} = \eta _i = -10 \ \log (\text {MSD}_i), \end{aligned}$$

(23)

where “MSD” stands for the mean squared deviation of the output, given by the following formula:

$$\begin{aligned} \text {MSD}_i = 1/N \sum _{j=1}^N y_{ij}^2, \end{aligned}$$

(24)

where $y_{ij}$ is the output or response, i.e., the measured warpage in millimeters (mm), of the jth trial of the ith experiment, and N is the total number of trials for the given experiment. Importantly, a higher SNR means a better quality property regardless of the goals mentioned above. Ultimately, [46] summarizes the core steps of the Taguchi methodology as follows: 1) choosing the injection molding variables, 2) coding the variables and selecting the OA, 3) conducting the physical experiments and calculating SNRs, and 4) analyzing and interpreting the outcomes. In Sect. 4, we implement the related steps of the Taguchi methodology in our injection molding problem.

Calculating the SNRs

The SNR for each experiment was calculated using formulas (23) and (24). For example, the MSD of the first experiment in Table 2 was calculated as

$$\begin{aligned} \text {MSD}_{1}\ = \left( {0.75}^{2} + {0.9}^{2} \right) /\ 2 = \ 1.372, \end{aligned}$$

where the corresponding SNR is equivalent to

$$\begin{aligned} \text {S/N}_{1}\ = \ - 10\ \log \ (1.372)\ \ = \ 1.635, \end{aligned}$$

Note that the goal of our experiment is to minimize the warpage, which is why we set the goal of our experiment to ‘the lower the better’ and use the corresponding SNR. The SNRs of the experiments are reported in the last column of Table 2.

Robust Parameter Design

To find the ‘best’ level of each parameter, the average SNR for each parameter level must be calculated. For example, the average SNR of parameter A at level-1 (A₁) is given by

$$\begin{aligned} \text {S/N}_{\text {A}_{1}} = \frac{1}{9}\sum _{i = 1}^{9}{S/N_{i}} = \ 2.67, \end{aligned}$$

where the terms inside the summation denote the SNRs of the first nine experiments conducted at the A₁ level (see Tables 2 and 10).

Each row of the Table 10 represents an experiment with a different combination of the variables, e.g., in the first experiment of $\text {L}_{27}(3^{13})$, the first variable (A) is at level-1, the second variable (B) is at level-2 and so on. However, the order in which these experiments are carried out is randomized.

Appendix 3

1.1 Variable Selection

In this section we present several heuristic approaches to assess the percentage contribution of production settings on the variability of the attained warpage values. We first present analysis of variance (ANOVA) along with calculation details. Then results for two widely preferred machine learning based feature selection techniques, namely random forest based SHAP values and Lasso regression based feature selection, are shared.

Table 11 presents the ANOVA results. The results reveal that the top three variables are injection time, holding pressure, and cavity coolant temperature according to the percentage contributions.

Table 11 Analysis of variance

The degrees of freedom for each factor are calculated as shown below (also see [47]). The total degrees of freedom are equivalent to

$$\begin{aligned} f_T = N - 1, \end{aligned}$$

where N is the total number of experiments and is equal to 27. The degrees of freedom for the ‘injection pressure’ parameter are

$$\begin{aligned} f_A = k_A -1, \end{aligned}$$

where k is the number of levels for factor A and is equal to 3. Therefore,

$$\begin{aligned} f_A = f_B = f_C = \ldots = f_M = 2. \end{aligned}$$

The sums of squares for all factors and the total sum of squares are given below.

The total sum of squares is $SA = (( 6.882^2 + 7.152^2 + 7.682^2 ) / 9) - ((0.83 + 0.63 + 0.90+....+0.78)^2 /27)= 0.03$, where $N_{A_1}(=9)$ is the number of experiments conducted under the $A_1$ setting.

Similarly, the sums of squares for all factors are calculated, and the results are given in Table 11. The error estimation is given by

$$\begin{aligned} S_{\epsilon } = S_T - (S_A + S_B + \ldots + S_M)= 0.03. \end{aligned}$$

Next, the variance for factor A is calculated as shown below.

$$\begin{aligned} V_A = S_A / f_A= 0.03/2 = 0.015. \end{aligned}$$

The percentage contribution for factor A is

$$\begin{aligned} P_A = (S_A / S_T) \times 100 = \%3.30. \end{aligned}$$

As a second approach we use random forest based SHAP values in order to assess the importance of features. Figures 5 and 6 present the resulting ranking of features. It is seen that, similar to the findings of above ANOVA, the most important features turn out to be Injection Time, Cavity Cooling Temperature and Holding Pressure.

Fig. 5

Feature importance with respect to SHAP value

Fig. 6

SHAP values with respect to nominal values of features

Lasso regression is applied with several different penalty coefficient values. Lasso results heavily depends on the selected penalty value. Restrictive (or loose) penalty values result in poor feature selection performance. After a sensitivity analysis, it is decided to select penalty coefficient as 0.15. With this selection, lasso regression choose to drop every feature except Injection Time, Cavity Cooling Temperature, Holding Pressure, Injection Pressure. All selected features, except Injection Pressure, is coherent with the choices of ANOVA and random forest based selection.

Appendix 4

1.1 Regression Analysis

To select the best regression model, the following metrics are used:

• R ² is a statistical measure of the percentage of variation present in the data explained by the fitted model. It is also known as the coefficient of determination.

• R ²-adjusted is a modified version of R², where the coefficient of determination is penalized with increasing number of predictors used in the model. R²-adjusted increases with increasing number of predictors only if the new terms improve the model more than would be expected by chance.

• AIC (Akaike information criterion) is an estimator of the relative quality of statistical models for a given set of data. Given a collection of models for the data, AIC estimates the quality of each model, relative to each of the other models.

• f-statistic in the context of our analysis refers to the f-statistic value of the analysis of variance (ANOVA) table, that is, the f-statistics value of the null hypothesis of whether the regression model is significant or not. In other words, the test shows whether the ratio of the mean variation explained by the model to the mean variation not explained by the model is sufficiently large.

• p-values show whether the regression coefficients are statistically significant or not.

• VIFs (variance inflation factors) measure how much the variance of the estimated regression coefficients are inflated compared to when the predictor variables are not linearly related. In other words, it is a measure of multicollinearity, that is, the linear dependency among regressors (i.e., used to detect whether correlated regressors exist).

• testMAPE is used to indicate the mean absolute percentage deviation of predicted values with respect to actual values in the holdout (test) data set.

The selected regression models with respect to the given metrics are presented below (see Table 12).

Table 12 Regression models and selection metric

Appendix 5

1.1 Statistical Tests

A series of statistical tests are conducted to assess various attributes of the data sets in Tables 2 and 5. The details and the results of the tests are presented in Table 13. The conducted analysis is composed of several steps. In analysis 1, the trial-1 and trial-2 samples listed in Table 2 are tested for normality (using the Jarque-Bera test, [48]). Then, the equal mean (using the Wilcoxon rank-sum test, [49]) and equal variance (using the Bartlett test, [50]) hypotheses are tested. The results show that the normality assumption cannot be denied for trial-1 and trial-2. It is also found that trial-1 and trial-2 are likely drawn from populations with an equal mean and an equal variance. In analysis 2, similar tests (normality, equality of means and variances) are conducted for the data presented in Table 5. It is seen that the RPD, R1 and R2 data sets likely follow the normal distribution. The most valuable observation of all evaluations is that mean value of RPD is statistically smaller than those of R1 and R2. Similarly, the mean value of R1 is smaller than that of R2.

Table 13 Statistical tests

Appendix 6

2.1 Sampling Algorithm

Below, the sampling algorithm used in the Monte Carlo simulation is shown.