Use of Factorial Designs and the Response Surface Methodology to Optimize a Heat Staking Process Antonio Faria Neto1,2 & Antonio Fernando Branco Costa1 & Michel Floriano de Lima2 Received: 14 February 2017 /Accepted: 20 December 2017 /Published online: 3 January 2018 # The Society for Experimental Mechanics, Inc 2018 Abstract The demand from the automotive industry for lighter and more resistant structures produced at lower costs has shifted the development focus of production processes toward hybrid components. A problem that arises from hybrid components is the necessity to join dissimilar materials, e.g., polymers and metals. A method to achieve this joining involves a process known as heat staking, in which a metal insert is heated and pushed against a thermoplastic surface. At the end of this process, the metal component may not be level with the thermoplastic surface; rather, it may be over flushed, and this discrepancy is known as the Insertion Height. This paper aims to apply the design of experiments and the response surface methodology to develop a model for the Insertion Height, considering the Heating Temperature and the Insertion Time as independent variables. The experiments revealed that the Insertion Height is most affected by the Heating Temperature. There are several combinations of the factors that can keep the Insertion Height within the specifications; therefore, it is possible to increase productivity by decreasing the Insertion Time and to save energy by reducing the Heating Temperature while considering the process constraints and specifications. Keywords Design of experiments . Factorial designs . Response surface . Heat staking . Plastic welding Introduction A problem that arises from the production of hybrid compo- nents is the necessity to join dissimilar materials, e.g., poly- mers and metals [1, 2]. In certain cases, the solution involves the use of threads. However, most thermo-plastics are too soft to be properly held by threads [3, 4]. To address this issue, brass or steel threaded inserts can be added by means of a heat induction process. The metallic threaded insert is heated by induction to a temperature higher than the melting temperature of the thermoplastic, and it is subsequently pushed for a short period by means of a mechanical device against a properly prepared cavity in the thermoplastic material. Such a process, known as heat staking, has beenwidely used in the automotive industry. At the end of the insertion process, the metal insert may not be level with the thermoplastic surface; in other words, the insert may be over flushed. The difference between the top of the insert and the thermoplastic surface is known as the Insertion (or Installation) Height. Figure 1 illus- trates a heat staking process and the Insertion (Installation) Height (h). The Insertion Height is an important design parameter from the perspective of the overall product quality. The over flushed condition decreases the tensile strength, creating unacceptable effects on the final product [5, 6]. According to [7, 8], the Insertion Height is related to the Heating Temperature of the insert, the Insertion Time, and the Insertion Load. These factors are directly linked to the mate- rial of the insert, the amount of polymer pushed during the insertion process, and the melting point of the polymer [9–11]. However, the equipment used in this research does not have an adjustment valve; as a result, the operator cannot change the value of the Insertion Load, which is kept constant at 5 bar by a lever system driven by a pneumatic cylinder. The lever pulls the insert as far as the end of the cavity. When the insert reaches the bottom of the cavity, a mechanical device stops the insertion process. * Antonio Faria Neto antfarianeto@gmail.com 1 São Paulo State University (UNESP), School of Engineering, Guaratinguetá, Av. Dr. Ariberto Pereira da Cunha, 333, Pedregulho, Guaratinguetá, SP CEP:12516-410, Brazil 2 Professional Master’s Degree Program, University of Taubaté (Unitau), R. Daniel Danelli, s/n, Vila da Juta, Taubaté, SP CEP:12060-440, Brazil Experimental Techniques (2018) 42:319–331 https://doi.org/10.1007/s40799-017-0230-1 http://crossmark.crossref.org/dialog/?doi=10.1007/s40799-017-0230-1&domain=pdf http://orcid.org/0000-0002-8773-2655 mailto:antfarianeto@gmail.com Although a great deal is known regarding the inser- tion processes, little is known concerning how much the Heating Temperature and Insertion Time affect the Insertion Height and if there is any interaction between these factors. This paper presents a model for predicting the Insertion Height for a brass insert in a thermoplastic joined by a heat staking process; the model is obtained using the design of experiments (DOE) and the response surface methodology (RSM). The Insertion Height must remain within the speci- fied tolerances to ensure the perfect assembly of the parts and to reduce the issues of reworking of the parts, setup changes in the production line, and client com- plaints. Determining the lowest Heating Temperature and Insertion Time for which the Insertion Height is within the specified tolerance enables energy saving and in- creases productivity because more parts can be assembled for a given period of time. The experiments were conducted on intake manifolds used in combustion engines, as illustrated in Fig. 2. Materials and Methods Materials The inserts used in this research were metallic bushes made of CuZn36Pb3, which can be heated from 250 °C to 370 °C. Figure 3 shows an insert employed in this research. The metal insert is housed into a polyamide base PA6-GF30. Methods The goal of this study is to determine the effects of the Heating Temperature and the Insertion Time on the Insertion Height (h). Factorial design methods are the most efficient approach to study the joint effect of two or more factors on a response. The most important class of design is the 2k factorial design, i.e., k factors, with each being at only two levels. This class of design is particularly useful in the early stages of experimental work [12]. In running a two-level factorial experiment, one assumes that the response is approximately linear over the range of the chosen factors levels. This assumption is a reasonable as- sumption at the beginning of the study; however, it is neces- sary to be alert to the possibility that a linear model for the response is not adequate because it is possible that the re- sponse variable does not behave linearly [12]. Before proceeding, it is necessary to check for curvature in the response variable by adding n runs at the center point of the 2k factorial design. Next, if the linear model is adequate, then it will be necessary to use more replicates at the factorial points to improve the estimation of the regression coefficients; otherwise a new design, with more points over the factor range, is required to fit a second-order model [12]. Factors range and response variable The range of the Heating Temperature was chosen according to the polyamide characteristics, such as the melting tempera- ture (260 °C) and the plastic decomposition temperature (340 °C). The range of the Insertion Time was chosen accord- ing to the equipment specifications, ranging from 1.00 s to 3.70 s. The center point (300 °C, 2.35 s) is equidistant from Fig. 2 Engine intake manifold collector with several brass threaded inserts Fig. 3 Metal insert (units in mm) Fig. 1 Insertion (Installation) Height [5] 320 Exp Tech (2018) 42:319–331 these extreme points. The current setup before this research was 320 °C and 3.70 s. Table 1 summarizes the level of the factors adopted for this experiment. The response variable, the Insertion Height, is the distance between the plane of the bushing and the top surface of the cavity, as illustrated in Fig. 1; this variable was measured by means of a digital dial gauge with a resolution of 0.001 mm. The specification for the Insertion Height ranges from 0.000 to 0.300 mm. Procedures and assumptions Before every new run, twenty bushes were disposed to ensure that the Heating Temperature was stabilized at the new level. Hypothesis testing is a classical statistical procedure used to determine the probability that a given hypothesis is true. This approach was used several times in this study. Hypothesis testing consists of two hypothesis: the null hy- pothesis (H0), the alternative hypothesis (H1), and a test sta- tistics, which is used to assess the truth of the null hypothesis. The hypothesis testing is associated with two types of sta- tistical errors: type I error and type II error. The type I error is the probability of rejecting the null hypothesis, given it is true, it is formally referred to as the test significance level (α). Whenever the calculated probability of type I error occurrence (P-value) is greater than α, there is no statistical evidence to reject the null hypothesis. Otherwise, if the P-value is lower than α, then there is statistical evidence to reject the null hy- pothesis because the risk of rejecting a valid hypothesis is lower than the one considered acceptable [12–14]. The type II error, also known as β risk, is the probability of accepting the null hypothesis, given it is not true. Theβ risk is inversely related to the sample size [12–14]. In all hypotheses tests performed in this research, the adopted significance level (α) was 1%; a β risk less than 10% is acceptable for the main effects of the factors, and a β risk approximately 20% is acceptable for the interaction be- tween factors, as suggested by [12–14]. Results and Discussion The experimental procedure was conducted in two stages. The first stage was a 22 factorial design with five replicates at the center point. The number of replicates at the center points should range from three to five [12]. Five replicates were chosen to be conservative. At the end of this phase, evidence of a curvature in the response function was detected over the region of exploration; thus, a new design with more points within the domain of the factors was required to fit a second-order regression model. 22 Factorial Design with Five Replicates at the Center Point Because the main goal of this stage of the experimental pro- cedure was to evaluate a quadratic curvature in the response function, a 22 factorial design with a single replicate of each factorial point augmented with five replicates at the center point was employed. According to Montgomery [12], the number of replicates at the center point ranges from 3 to 5; thus, to be conservative, 5 replicates were adopted. The cur- vature in the response function is evaluated by comparing the average of the runs at the factorial points,yF , to the average of the runs at the center point,yC . If the averages are different, then there is statistical evidence of a quadratic curvature in the response function over the domain of the factors. Such a check is performed by means of the hypothesis test (1) [12]. H0 : yF−yC ¼ 0 H1 : yF−yC≠0 ( ð1Þ If the averages are different, then the null hypothesis must be rejected. According to [12, 13], the appropriate test statis- tics for hypothesis test (1) is the Student’s t-score (t0), which is given by equation (2). t0 ¼ yF−yCffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi S2 1 nF þ 1 nC � �s ð2Þ Where nF and nC are the number of factorial points and the replicates at the center point, respectively; andS2is the estimate of variance obtained using the center points. The hypothesis H0is to be rejected if t0j j > tα=2;nC−1, where tα/2; nC − 1 is the critical value of the Student’s t-distribution with nC − 1degrees of freedom for a significance levelα. According to [13], for a significance level of 0.01, as stated in BProcedures and assumptions^ section, and nC = 5the criti- cal value of the Student’s t-distribution is t0:01=2;4 ¼ 4:604. The runs of this experiment are summarized in Table 2. Substituting the numerical values into equation (2), results t0 ≅ 140.77, which is greater thant0:01=2;4 ¼ 4:604; thus, the null hypothesis H0can be rejected, suggesting the existence of a quadratic curvature. Therefore, a new design with more points within the domain of the factors will be required. Table 1 Level of the factors Level Heating Temperature (°C) (Factor A) Time Insertion (s) (Factor B) -1 260 1.00 0 300 2.35 +1 340 3.70 Exp Tech (2018) 42:319–331 321 Several designs exist that are quite good for fitting second- order response models. In particular, the Central Composite Design (CCD) and the Faced-centered Cube Design (FCD) are worthy alternatives [14]. The 32 Factorial Design is also a possible choice; although it is not the most efficient method to model the quadratic relationship [12], it was employed be- cause it is a natural extension of the 2k factorial design, and this issue of the choice of design is not relevant when there are only two factors [14]. 32 Factorial Design with Five Replicates This is a factorial arrangement with 2 factors, each at three levels. The low, intermediate, and high levels are coded as −1, 0, and +1, respectively. A very important decision to make at this point was the choice of the number of replicates at each treatment combination. The number of replicates depends on the size of effects that are intended to be detected, i.e., the smaller the effect to be detected, the greater the number of replicates [12]. In general, the number of replicates ranges from 2 to 5; thus, to be conservative, 5 replicates were chosen. The results from running all combinations of the chosen factors, with each being at three levels with five replicates, are shown in Table 3. Sample size adequacy checking Before proceeding, it is worth verifying whether the number of replicates chosen is adequate to keep the β risk within the acceptable range, as previously specified in BProcedures and assumptions^ section. The β risk and the sample size are related by the operating characteristic curves [12–14]. In other words, the operating characteristic curves are employed to verify whether the selected number of replicates is adequate to make the design sensitive to a given difference between treatments. The link between the β risk and the operating characteristic curves is determined by means of a dimensionless parameter φ given by equation (3) in the case of the main effect of the factors, and by equation (4) for the interaction between factors [12]. ϕ2 ¼ nbD2 2aS2 ð3Þ ϕ2 ¼ nD2 2S2 a−1ð Þ b−1ð Þ þ 1½ � ð4Þ where n is the number of replicates, b and a are the numbers of levels of each factor, D is the minimum difference between two treatment means, and S2is the estimate variance of the response variable. From Table 3, it is possible to observe that the difference between two treatment means range from 0.054 to 0.131 mm; therefore, it is reasonable to check whether the design is sen- sitive to a difference ofD= 0.050 mm between any two treat- ment means. The estimate variance of the Insertion Height is the average variance of the treatments, which is approximately 0.000288 mm2. Both factors are studied in three levels. The results for equations (3) and (4) are summarized in Table 4, considering that the number of replicates (n) is equal to five. Referring to the operating characteristic curves [12, 13] for the values of φ presented in Table 4, it can be seen that five replicates are sufficient to obtain a β risk less than 1%, i.e., 99% chance of rejecting the null hypothesis if the difference in mean Insertion Height at two Heating Temperatures or two Insertion Times is as high as 0.050 mm. In the same manner, five replicates provide a β risk of approximately 20% for the same difference between any two interactions effects, as seen in Table 4. Graphical interpretation of the results To assist in interpreting the results in Table 3, it is helpful to construct a graph of the average Insertion Height at each treat- ment combination, as shown in Fig. 4. The Insertion Height decreases from low to intermediate temperature (260 to 300 °C), because, at low temperatures, the polyamide is not adequately melted to enable a perfect inser- tion [15]. The minimum value for the Insertion Height occurs at approximately 300 °C. From the intermediate to high tem- perature (300 to 340 °C), the Insertion Height increases, be- cause as the temperature increases, the amount of gas pro- duced in the polyamide also increases, pushing out the insert [5]. Thus, the insertion height follows the order of 1.00 s > 2.35 s > 3.70 s for all temperature. A shorter Table 2 22 factorial design with five replicates at the center point Heating Temp. (°C) Insertion Time (seconds) Insertion Height (mm) Factorial Points nF = 4 260 1.00 0.862 260 3.70 0.638 340 1.00 0.574 340 3.70 0.493 y̅F = 0.64175 Replicates at the Center Point nC = 5 300 2.35 0.192 300 2.35 0.206 300 2.35 0.216 300 2.35 0.219 300 2.35 0.188 y̅C = 0.2042 S2 = 0.000193 322 Exp Tech (2018) 42:319–331 Insertion Height is attained at a longer Insertion Time, regard- less of the Heating Temperatures. To confirm the conclusions from the graphical analysis above, it is necessary to compare all the treatments means; the analysis of variance is the appropriate procedure for testing the equality of several means. Analysis of variance The analysis of variance (ANOVA) for data in Table 4 is summarized in Table 5. It can be seen in Table 5 that the P-value for the main effects of Heating Temperature and for Insertion Time, as well as for the interaction between these two factors are less than 0.01; thus, they are statistically significant. The analysis of variance is the decomposition of the vari- ability in the observations through a purely algebraic relation- ship. However, such a procedure requires that experimental data can be represented by an empirical model, known as the Fixed Effect Model, which expresses the relationship between the observations (response variable) and the factors [12]. Therefore, before adopting the conclusions from the analysis of variance, it is necessary to check the adequacy of such a model by means of the analysis of the residuals. Model adequacy checking The primary diagnostic tool for the model adequacy checking is the residual analysis. The residuals (or errors) are defined as the difference between the actual and predicted values for the response variable. Table 6 presents the standardized residuals for the Insertion Height data in Table 3. The residual analysis is performed by means of a graphical analysis of the residuals to check whether they are normally and independently distributed with a mean zero and constant variance [12]. A check of the normality assumption can be made by con- structing a normal probability plot of the standardized resid- uals, as shown in Fig. 5. If the underlying error distribution is normal, then this plot will resemble a straight line. The normal probability plot shows a reasonably linear pat- tern in the center of the data. However, the tails show devia- tion from the fitted line. In other words, the middle of the Table 3 Insertion Height (mm) for the 32 design Heating Insertion Time (s) Temperature (°C) 1.00 2.35 3.70 (−1) 0 (+1) 0.851 Average 0.734 Average 0.646 Average 0.862 0.869 0.739 0.738 0.668 0.650 260 (−1) 0.866 0.754 0.673 0.876 Variance 0.745 Variance 0.621 Variance 0.891 0.000229 0.716 0.000201 0.641 0.000448 0.281 Average 0.217 Average 0.134 Average 0.307 0.301 0.209 0.204 0.139 0.139 300 (0) 0.328 0.193 0.161 0.304 Variance 0.181 Variance 0.147 Variance 0.285 0.000358 0.218 0.000260 0.116 0.000275 0.591 Average 0.554 Average 0.504 Average 0.586 0.586 0.551 0.531 0.479 0.477 340 (+1) 0.588 0.516 0.465 0.584 Variance 0.525 Variance 0.484 Variance 0.582 0.000012. 0.509 0.000419 0.455 0.000352 Table 4 Choice of sample size (S2 = 0.000288 mm2) φ2 n φ Numerator Denominator β risk degrees of degrees of freedom freedom Main effects 5n 5 4.658 2 36 < 0.01 Interaction effects 1n 5 2.083 4 36 ≈ 0.20 Exp Tech (2018) 42:319–331 323 residuals shows a mild S-like pattern. The first few points show increasing deviation from the fitted line below the line, and the last few points show increasing deviation from the fitted line above the line; thus, this pattern suggests the possi- bility of a distribution with long tails relative to the normal distribution. However, according to [12], moderate deviation from normality is of little concern in the fixed effects analysis of variance because the analysis of variance is robust to the normality assumption. Thus, there were no considerable vio- lations of normality. To be more objective, three classical goodness-of-fit tests were used to check whether the residuals come from a nor- mally distributed population, the Kolmogorov-Smirnov test [16], the Anderson-Darling test [16], and the Ryan-Joiner test [17]. All of these tests are hypothesis testing, as stated in (5). H0 : the data are normally distributed H1 : the data are not normally distributed � ð5Þ In this paper, the statistics for the Kolmogorov-Smirnov test, the Anderson-Darling test, and the Ryan-Joiner test are denominated KS, AD, and RJ, respectively. Although these statistics can be compared to the respective critical values, most of the data analysis software give the P-values associated with them. Therefore, if the P-value is greater than the signif- icance level (α) adopted for this test (which is 0.01 in this paper), there is no statistical evidence to reject the null hypoth- esis, and the residuals should be considered normally distrib- uted [12, 14]. Using the software Minitab, the Komolgorov-Smirnov test produced the statistics KS = 0.065 with a P-value >0.150, the Anderson-Darling test produced the statistics AD = 0.361 with a P-value = 0.430, and the Ryan-Joiner produced RJ = 0.998 with a P-value >0.100. Thus, according to the previous paragraph, the normality assumption of the residuals has not been violated. The independence assumption can be evaluated by means of a plot of residuals in the time order of the data collection, as shown in Fig. 6. The plot of the residuals versus the observation order is helpful in detecting a correlation between the residuals. Positive serial correlation exists when residuals tend to be followed by residuals of the same sign and approximately the same magni- tude. Negative serial correlation exists when residuals of one signal tends to be followed by residuals of the opposite sign. If such a correlation exists, then the independence assumption on Fig. 4 Average Insertion Height at each treatment combination Table 5 Analysis of variance for the insertion height data Source of variation Degrees of freedom Sum of Squares Mean square F0 ¼ Mean Square Mean Square Error P- value Heating Temperature 2 2.19020 1.09510 3861.27 0.000 Insertion Time 2 0.20164 0.10082 355.49 0.000 Heating Temp. x Time Insert. 4 0.01610 0.00403 14.19 0.000 Error 36 0.01021 0.00028 Total 44 2.41815 324 Exp Tech (2018) 42:319–331 the errors has been violated [12]. Figure 6 shows that the resid- uals bounce randomly around the zero line, suggesting that there is no correlation. The homoscedasticity assumption can be verified bymeans of the plot of the residuals versus the fitted value [12], as shown in Fig. 7. A constant variance implies the variation of observation is approximately constant as the magnitude of the observation increases. Figure 7 reveals that the residuals are structureless, i.e., they are unrelated to the predicted response. As a result, the assumption of homogeneity of variances has not been violated. Because none of the assumptions was violated, the conclu- sions based on ANOVA (Table 5) remain valid, including the fact that the interaction between the factors is significant. In such cases, comparisons between the means of one factor may be obscured by the interaction. Therefore, it is recommended to apply a multiple comparisons procedure, such as Tukey’s test, to evaluate the effect of each factor [12]. Tukey’s test Tukey’s test is used in conjunction with Analysis of Variance to find the means that are significantly different from each other. The test compares all possible pairs of means by fixing one factor at a specific level and testing the means of the other factors at that level. This test is based on a studentized range distribution (q). Therefore, first, it is necessary to determine the differences among the means of all factors combinations, which are shown in Table 7, and afterwards, it is necessary to test these means. For the data in Table 4, Tukey’s test declares that two means are significantly different if the absolute value of their sample mean differences exceeds T0.01 ≈ 0.033 mm, which can be calculated according to Montgomery [12] for q0.01(3;36) ≈ 4.39 (by interpolation) and considering that the error variance is estimated as the ErrorMean Square in Table 5 (MSE = 0.000280). Table 6 Standardized residuals for the Insertion Height (the numbers in parentheses indicate the order of data collection) Heating Insertion Time (s) Temp. (°C) 1.00 2.35 3.70 −1.208(1) 0.451(29) −0.239(44) 0.491(15) −0.252(8) −1.912(13) 260 −0.478(22) 1.447(23) 0.093(19) −1.434(40) 1.208(32) −0.584(28) −0.212(24) 1.089(37) 1.540(43) −1.328(6) 0.199(17) 0.889(4) −1.500(5) −0.358(12) 0.504(45) 300 0.398(25) −1.062(34) 0.358(9) 0.956(21) −0.027(27) −1.553(38) 1.792(39) −0.704(26) 1.434(42) 0.319(11) 0.119(14) 1.527(2) −0.996(7) 1.766(3) −0.823(10) 340 −0.013(30) −0.146(36) 1.328(16) −0.398(20) 0.106(18) 0.438(31) −0.279(41) −1.460(35) −1.487(33) Fig. 5 Normal probability plot of the residuals from the fixed effects model Exp Tech (2018) 42:319–331 325 Therefore, Tukey’s test indicates that the Insertion Height is different for all treatment combinations in Table 4; thus, the previous conclusions are confirmed. Recall that the tolerance for the Insertion Height for the process under analysis ranges from 0.000 to 0.300 mm; as a result, a trade-off exists between the Heating Temperature and the Insertion Time to increase productivity and save energy while ensuring that the Insertion Height remains within the specifications (see Fig. 4). To perform a better analysis of this trade-off between Insertion Time and Heating Temperature, a regression model for the Insertion Height as a function of these factors is constructed. Response Surface Model A response surface or regression model is useful for prediction, process optimization, or control processes. Equation (6) is the second-order regression model for two variables [14]. y ¼ β0 þ β1x1 þ β2x2 þ β12x1x2 þ β11x 2 1 þ β22x 2 2 þ ε ð6Þ where y represents the response function, and x1 and x2 are the factors of the experiment, with the independent variables often called the predictor variables or the re- gressors. The parameters βi, (i = 0, 1,⋯) are called the regression coefficients, and ε is the model error. Equation (6) is called the multiple linear regression model; the most popular method to estimate the regression coeffi- cients β’s is the least squares method. The fitted regression model can be written as equation (7) [14]: ŷ̂¼ B0 þ B1x1 þ B2x2 þ B12x1x2 þ B11x21 þ B22x22 ð7Þ where ŷ estimates y, Bi (i = 0, 1,⋯) estimatesβi (i = 0, 1,⋯), x1 is the Heating Temperature, and x2 is the Insertion Time. Fig. 6 Residuals versus observation order Fig. 7 Residuals versus fitted value 326 Exp Tech (2018) 42:319–331 Table 8 presents a summary of the information concerning the regression model for the data in Table 3, listed in coded variables. The regression coefficients are presented followed by the respective P-value. In this case, the P-value is associated with a hypothesis test for which the null hypothesis state that the regression coefficient is zero [14]. Therefore, if P-value > α, then the null hypothesis cannot be rejected, i.e., the corresponding population regression coefficient is zero. The adjusted R2 statistics, R2(adj), the maximum value of which is 1, is used to assess the model performance. R2(adj) is a measure of the amount of reduction in the variability of the estimated variable obtained by using the regressor variables in the model. In opposition to the classical statistics R2, its value does not always increases as variables are added to the model. In fact, if unnecessary terms are added, its value decreases [12, 14]. Thus, the model exhibits a good performance because R2(adj) is very close to 1 and the residual error is notably low, approximately 0.0168. The performance analysis of the model can be complemented by the ANOVA presented in Table 9. The ANOVA indicates that the model fits the data well; in other words, the model is adequate to describe the Insertion Height. Before relying on ANOVA conclusions, it is neces- sary to investigate the model adequacy. Model adequacy checking As performed previously, this verification can be accom- plished by performing a residual analysis based upon the stan- dardized residuals presented in Table 10. The full graphical analysis is presented in Fig. 8. In Fig. 8(a), the normal probability plot shows a moderate deviation from the fitted line. However, according to the normality tests of Kolmogorov-Smirnov (KS = 0.106; P-value >0.1500), Anderson-Darling (AD =0.457; P-val- ue =0.254), and Ryan-Joiner (RJ = 0.986; P-value >0.100), the residuals above can be considered normally distributed, i.e., the normality assumption of the resid- uals has not been violated. Figure 8(b), (c) and (d) indicate that the variance of the observed Insertion Height is stable with respect to the predicted Insertion Height, Heating Temperature and Insertion Time, re- spectively. Thus, the homoscedasticity of the residuals has not been violated. Once the model presented in Table 8 is confirmed, the next step is to study the behavior of the response function. Study of the response surface The regression model presented in Table 8 can be rewritten as follows: ŷ̂¼ 0:2059−0:1103x1−0:08163x2 þ 0:02765x1x2 þ 0:42720x21 þ 0:01310x22 ð8Þ and, for the uncoded variable, as ŷ̂¼ 25:6062−0:1642x1−0:2479x2 þ 0:0005120x1x2 þ 0:0002670x21 þ 0:007188 x22 ð9Þ as illustrated in Fig. 9. The response surface is a parabolic cylinder, which has one critical point solely, and it is at 301.17 °C and 6.51 s. Although this point is a minimum point, it is outside the domain of the Insertion Time (1.00–3.70 s); therefore, it is necessary to identify the extreme value ofŷ at the response function boundary. The minimum point is the vertex of the parabola resulting from the intersection of the surface with a plane parallel to the Insertion-Height - Heating-Temperature x1Oŷð Þ plane at the Insertion Time equal to 3.70 s. Thus, substituting Table 7 Differences among the means of all factor combinations Insertion Time (s) Heating Temperature (°C) 260 300 340 1.00→ 2.35 0.131 0.097 0.055 1.00→ 3.70 0.219 0.162 0.109 2.35→ 3.70 0.088 0.065 0.054 Heating Temperature (°C) Insertion Time (s) 1.00 2.35 3.70 260→ 300 0.568 0.534 0.511 260→ 340 0.283 0.207 0.173 300→ 340 0.285 0.327 0.338 Table 8 Regression model in coded variables for data in Table 4 Regressor Bi P-value Intercept 0.20593 < 0.001 x1 −0.11033 < 0.001 x2 −0.08163 < 0.001 x1 · x2 0.02765 < 0.001 x1 2 0.42720 < 0.001 x2 2 0.01310 = 0.018 R2(adj) = 0.9949 Res. Error = 0.0168 Reg. P-value= < 0.001 Exp Tech (2018) 42:319–331 327 x2by 3.70 s in equation (9), it will become a single variate func- tion, with a local minimum at x1 = 303.87°C which is ŷmin ¼ 0:133mm. The Insertion Height contour, Fig. 10, is useful to deter- mine the region over which the Insertion Height is within the specified tolerance (0 to 0.300 mm). Several combinations of the Heating Temperature and the Insertion Time correspond to an Insertion Height less than or equal to 0.300 mm. The combination that maximizes produc- tivity and minimizes energy consumption is the Heating Temperature of approximately 300 °C and the Insertion Time equal to 1.00 s. Although the optimum point has been achieved, it is nec- essary to check the capability of the process. Process Capability Among the several statistics that can be used to measure the capability of a process, the Process Capability Index, Cpk, will be used in this work. According to Montgomery [18], the index Cpk can be estimated as follows: Cpk ¼ min USL−ŷ̂ 3S ; ŷ̂−LSL 3S � � ð10Þ Table 9 ANOVA for the regression model presented in Table 8 Source of Variation Degrees of Freedom Sum of Squares Mean Square F0 ¼ Mean Square Mean Square Error P- value Regression 5 2.40713 0.48143 761.15 0.000 Error 39 0.01103 0.00028 Lack-of-Fit 3 0.00081 0.00027 0.95 0.423 Pure Error 36 0.01021 0.00028 Total 44 2.41816 Table 10 Residuals for the second-order model in Table 8 Order yi ŷi Residual Stand. Order yi ŷi Residual Stand. Res. Res. 1 0.851 0.866 −0.015 −0.964 24 0.866 0.866 0.000 0.010 2 0.554 0.523 0.031 1.968 25 0.307 0.301 0.006 0.399 3 0.504 0.482 0.022 1.434 26 0.193 0.206 −0.013 −0.816 4 0.217 0.206 0.011 0.698 27 0.139 0.137 0.002 0.101 5 0.181 0.206 −0.025 −1.573 28 0.641 0.647 −0.006 −0.408 6 0.281 0.301 −0.020 −1.241 29 0.876 0.866 0.010 0.659 7 0.516 0.523 −0.007 −0.429 30 0.586 0.590 −0.004 −0.252 8 0.646 0.647 −0.001 −0.083 31 0.484 0.482 0.002 0.135 9 0.209 0.206 0.003 0.193 32 0.668 0.647 0.021 1.345 10 0.465 0.482 −0.017 −1.098 33 0.455 0.482 −0.027 −1.748 11 0.591 0.590 0.001 0.073 34 0.285 0.301 −0.016 −0.988 12 0.134 0.137 −0.003 −0.214 35 0.509 0.523 −0.014 −0.870 13 0.621 0.647 −0.026 −1.707 36 0.584 0.590 −0.006 −0.382 14 0.588 0.590 −0.002 −0.122 37 0.754 0.743 0.011 0.664 15 0.745 0.743 0.002 0.097 38 0.116 0.137 −0.021 −1.350 16 0.551 0.523 0.028 1.779 39 0.328 0.301 0.027 1.724 17 0.304 0.301 0.003 0.210 40 0.716 0.743 −0.027 −1.733 18 0.479 0.482 −0.003 −0.189 41 0.582 0.590 −0.008 −0.512 19 0.739 0.743 −0.004 −0.282 42 0.161 0.137 0.024 1.489 20 0.525 0.523 0.002 0.139 43 0.673 0.647 0.026 1.670 21 0.218 0.206 0.012 0.761 44 0.734 0.743 −0.009 −0.597 22 0.862 0.866 −0.004 −0.250 45 0.147 0.137 0.010 0.606 23 0.891 0.866 0.025 1.633 328 Exp Tech (2018) 42:319–331 Considering the upper limit specification (USL) is equal to 0.300 mm, the lower specification limit is equal to 0.000, and the standard deviation estimate (S), as the residual error of the model from Table 8, is 0.0168 mm, (10) can be rewritten as follows: Cpk ¼ min 0:300−ŷ̂ 0:0504 ; ŷ̂ 0:0504 � � ð11Þ Unfortunately, the process is not capable of achieving the optimal setup (300 °C, 1.00 s), once the Cpk is very low, i.e., the probability that the Insertion Height for an insert bush is out of specification is very high. A standard acceptance crite- rion for a process in the automotive industry [19] is Cpk is equal to 1.33, which corresponds to an Insertion Height of 0.233 mm. By replacing ŷ with 0.233 in equation (9), one obtains equation (12), which represents the ellipse illustrated in Fig. 11. 0:0002670 x21 þ 0:007188 x22−0:1642 x1−0:2479 x2 þ 0:0005120 x1x2 þ 25:3732 ¼ 0 ð12Þ The contour line in Fig. 11 shows several combinations of the Heating Temperature and the Insertion Time, for which the Insertion Height is 0.233 mm, and therefore, it is within the specifications. Thus, it is possible to choose the combination that maximizes the productivity or minimize the energy con- sumption by moving the operating point along such a curve. Productivity Improvement and Energy Saving The current operating point is (320 °C, 3.70 s), and it can move along the curve in Fig. 11, depending on the production requirements. According to the plant manager, a decrease of 1 s in the Insertion Time saves approximately 366 h/yr. Thus, if the main concern is to improve the process productivity, the Insertion Time must be minimized in equation (12), leading to Fig. 8 (a) Normal probability plot. (b) Residual versus fitted value. (c) Residual versus Heating Temperature. (d) Residual versus Insertion Time Fig. 9 Insertion height surface Exp Tech (2018) 42:319–331 329 the minimum point (305.9 °C, 1.62 s), as illustrated in Fig. 12, resulting in a time saving of approximately 760 h/yr. However, if the main concern is energy saving, a decrease of approximately 0.020 kWh/yr. in the electric energy con- sumption accompanies a reduction of 1 °C in the Heating Temperature. Thus, the operating point can be moved along the curve towards to the minimum value of the Heating Temperature, 283.5 °C, which corresponds to 3.70 s, resulting in a saving of 0.730 kWh/yr as illustrated in Fig. 13. Conclusions This paper presented a model for predicting the Insertion Height for the joining of a brass insert to a thermoplastic form using a heat staking process. First, factorial designs were ap- plied to investigate the behavior of the Insertion Height with respect to the Heating Temperature and Insertion Time, which were the two controllable factors of the process. Afterwards, the RSM was used to obtain the desired model. A 22 factorial design with a central point with five repli- cates demonstrated the existence of curvature in the response surface. As a result, it was necessary to run a new design to provide extra points to model the response function. The 32 factorial design was chosen. Although the Insertion Height was affected by both factors and their interaction, the experi- ments revealed that the Insertion Height was affected primar- ily by the heating. From the Insertion Height contour plot, it was possible to define a region where the Insertion Height is within the speci- fications, i.e., less than or equal to 0.300 mm. The combination of 300 °C and 1 s was found to correspond to an Insertion Height of 0.300 mm. Although this set point allows for the highest productivity, the process cannot be implemented be- cause it showed a very low Cpk. Cpk equal to 1.33 is a standard acceptance criterion for processes in the automotive industry. A Cpk equal to 1.33 Fig. 10 Insertion height contour plot (mm) Fig. 11 Contour plot for insertion height 0.233 mm Fig. 12 Point of maximum productivity 330 Exp Tech (2018) 42:319–331 corresponds to an Insertion Height equal to 0.233 mm; as a result, a contour line for the Insertion Height equal to 0.233 mm (Cpk equal to 1.33) was determined. Thus, the op- erating point can be set on this line, and it can be chosen based on the optimization criterion, maximizing energy saving or productivity. 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Luleå University of Technology - Department of Mathematics, ISSN 1402–1757 Fig. 13 Minimum power consumption point Exp Tech (2018) 42:319–331 331 https://doi.org/10.1002/pen.21424 https://doi.org/10.1002/pen.21424 https://doi.org/10.1080/10426914.2013.792413 https://doi.org/10.1016/j.jmatprotec.2007.06.058 https://doi.org/10.1080/10426914.2014.961479 https://doi.org/10.1080/10426914.2011.647940 https://doi.org/10.1080/10426914.2010.529587 https://doi.org/10.1080/08982112.2013.852679 https://doi.org/10.1080/10426914.2011.593232 https://doi.org/10.1080/10426914.2011.593232 https://doi.org/10.1016/j.compstruct.2017.04.003 https://doi.org/10.1016/j.compstruct.2017.04.003 Use of Factorial Designs and the Response Surface Methodology to Optimize a Heat Staking Process Abstract Introduction Materials and Methods Materials Methods Factors range and response variable Procedures and assumptions Results and Discussion 22 Factorial Design with Five Replicates at the Center Point 32 Factorial Design with Five Replicates Sample size adequacy checking Graphical interpretation of the results Analysis of variance Model adequacy checking Tukey’s test Response Surface Model Model adequacy checking Study of the response surface Process Capability Productivity Improvement and Energy Saving Conclusions References