y ¼ b0 þ b1x1 þ b2x2 þ b11x2 þ b

x2 þ b

x1x2  þ e ðEq 6Þ

1 22  2 12

x3 ¼ x12; x4 ¼ x22; x5 ¼ x1x2;   b3 ¼ b11; b4 ¼ b11; b5 ¼ b12

Y2 ¼ b0 þ b1x1 þ b2x2 þ b3x3 þ b4x4 þ b5x5 þ e (Eq 7)

The method of the least squares is typically used to estimate the regression coefficients (b0, b1, b2) in the multiple linear regression models. Y1  is  the  response  of  the first-order model and Y2 is the second-order model, respectively. The method of the least squares can be defined by the following equations:

not, R2 (R-sq) is the correlation coefficient, R2 (adj) is the ad- justed statistic, MSE is the mean square error, RMSE is the

root mean square error, and APE (%) is the average absolute percentage error.

4. Results and Discussion

The results obtained from the statistical and Moldflow analyses in this study have been investigated in two categories:

the first is the statistical results consisting of multiple regression and ANOVA analysis, and the second is the simulation results involving Moldflow analysis during the plastic injection molding process.

4.1 Statistical Results

A multiple regression analysis using Minitab (Ref 20) have performed to determine the mathematical relationship between the process parameters and the shrinkage obtained from the values of the FE analyses based on the plastic injection molding. For this purpose, the first-order and the second-order regression models have developed to find the values of the shrinkage when compared to the values from the FE analyses taken from Moldflow. After that, the ANOVA analyses have conducted to check the adequacy of the developed regression models and to evaluate the effects of the process parameters on the shrinkage.

The first-order regression model in terms of coded values of process parameters is given below:

Shrinkage ¼ 1:01 þ 0:00127Tmold þ 0:0375Tmelt

— 0:0003Pinj — 0:389Itime — 0:0133Ctime

ðEq 17Þ

Table 5 presents the results of the first-order regression model for  the  shrinkage.  In  Table 5,  R2,  which  is  expressed  in Eq 13, is the correlation coefficient and is called R-sq. When R2 approaches the value of 1, the multiple regression models match very well with the experimental results. R2 is calcu- lated to be 0.959 (95.9%) by utilizing the first-order regres- sion model given in Eq 17. The T value of the process parameters  can  be  utilized  to  control  whether  the predictor

Shrinkage ¼ 1:14 þ 0:0375 Tmelt — 0:489 Itime

— 0:0133 Ctime þ 0:0251 Itime (Eq 18)

Table 7 demonstrates the results of the second-order regres- sion model for the shrinkage. In Table 7, R2 is calculated to be 0.958 (95.8%) using the second-order regression model, as given in Eq 18. In Table 7, Tmelt, Itime, Ctime, and I2 are sta- tistically the most significant process parameters for the shrinkage. The other process parameters (Pinj and Tmold) and their interactions are not significant because their P value is bigger than 0.05. Table 8 illustrates the results of the ANOVA analysis verifying that the process parameters for the shrink- age are statistically significant. The F value of  124.45  in Table 8 indicates that the second-order regression model is significant.

In the comparison of the results in Table 5 and 7, it can be seen that R2 (R-sq) values are almost the same, both for the first-order and the second-order regression models. However, five process parameters, Tmold, Tmelt, Pinj, Itime, and Ctime, are used to develop the first-order regression model, whereas three process parameters, including Tmelt, Itime, and Ctime, are employed to develop the second-order regression model. The comparison of the ANOVA results in Table 6 and 8 shows that the F value (124.45) of the second-order regression model is greater than the F value (98.72) of the first-order regression model. This result indicates that the second-order regression model is more significant than the first-order regression model in terms of the statistical approach.