- Email: [email protected]
Application of the Box–Behnken design to the optimization of process parameters in foam cup molding
Expert Systems with Applications 39 (2012) 8059–8065
Contents lists available at SciVerse ScienceDirect
Expert Systems with Applications journal homepage: www.elsevier.com/locate/eswa
Application of the Box–Behnken design to the optimization of process parameters in foam cup molding Long Wu a, Kit-lun Yick a,⇑, Sun-pui Ng b, Joanne Yip a a b
Institute of Textiles & Clothing, The Hong Kong Polytechnic University, Hong Kong Hong Kong Community College, The Hong Kong Polytechnic University, Hong Kong
a r t i c l e
i n f o
Keywords: Box–Behnken Response surface methodology Optimization Shape conformity Foam
a b s t r a c t Currently, foam molding technologies are widely adopted for most bra styles, which demonstrate the incomparable advantages in the contemporary intimate apparel industry. The determination of proper molding conditions, such as molding temperatures and length of time on the basis of cup sizes and styles, is crucial in achieving the required cup shape with high stability, which is regarded as the most challenging part of the molded bra making process. To determine the optimal process parameter settings, numerous process trials are generally required to evaluate the molding variables and their interactions. This study proposes a novel systematic methodology to identify the optimal molding process parameters based on design of experiment (DOE) and a parameterization-based remesh method to evaluate the 3D shape conformity of molded cups. By solving the regression equation obtained from a Box–Behnken design (BBD) and analyzing the response surface plots, the results prove that molding temperature has greater influence than the length of the dwell time on the 3D shape conformity of molded cups. The optimal molding conditions can be determined for the cup depths of different sized mold heads, which are validated by the experimental results. Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction The optimization of process parameters is routinely performed in the manufacturing industry so as to solve the challenges in product quality and/or optimize the cost effectiveness of manufacturing processes (Chang, 2008). In the intimate apparel industry, foam cup molding is a critical process in the course of making seamless molded bras. Due to the smooth surface and natural configuration under the outer garment, seamless molded bras nowadays represent almost 80–90% of the overall bra market (Haycock, 1978). Nevertheless, the determination of optimal molding conditions is a long standing problem in controlling the process of foam cup molding. The process has been carried out on a flat polyurethane (PU) foam sheet with a certain thickness by a hot male mold head press which is actuated by means of compressed air, and maintained for a certain dwell time before the male mold is automatically released back (Yu, Fan, Harlock, & Ng, 2006). The PU foam is then softened by heat and forms the required shape and thickness that are defined by a cavity between the pair of molds (Yick, Wu, Yip, Ng, & Yu, 2010). The process and the ultimate quality of the molded cups are greatly influenced by the applied molding temperature, length of time and clamping pressure. Factors such as cup size, style, and foam physical ⇑ Corresponding author. Tel.: +852 27666551; fax: +852 27731432. E-mail address: [email protected] (K.-l. Yick). 0957-4174/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.eswa.2012.01.137
and thermal mechanical properties can also increase difficulties in the determination of optimal molding conditions for the required quality and shape conformity. For over 40 years since garment molding was first introduced in the 1970s, the molding conditions are roughly determined by the experience of the molder on a trial-and-error basis. As the molded cups are soft and readily deform, the quality and shape of the molded cups are visually inspected through comparisons with a plastic cup template. The judgment depends on the experience of the inspectors. Problems have further increased in the new century due to the complex design of mold cups, the extreme size and softness requirements of bras, and a vast number of foam materials introduced into the market. As a consequence of the increasing demand of seamless molded bras and the short cycle time in manufacturing, the optimization of the foam cup molding process has become a very important subject for researchers in order to provide accurate and reliable procedures that will assure molding quality and improve production efficiency (Yu, Yeung, Harlock, & Leaf, 1998). The optimization approach provided by the Box–Behnken design (BBD), which is a response surface methodology (RSM) that uses Design-Expert software (Version 7.0.0, Stat-Ease Inc., Minneapolis, USA), is proposed in this study. As a collection of statistical and mathematical techniques for developing, improving, and optimizing processes, RSM is specifically applied in the industrial world, particularly in situations where several input variables potentially influence a performance measure or
8060
L. Wu et al. / Expert Systems with Applications 39 (2012) 8059–8065
quality characteristic of the product or process (Myers & Montgomery, 2002). On the basis of the BBD, the process parameters (molding temperatures, length of dwell time and cup depths of mold heads) in the foam cup molding process can be optimized with a minimum number of experimental runs and achieve competitive advantages in product quality and costs.
2. Response surface methodology and Box–Behnken design In recent years, design of experiment (DOE) has been frequently applied to optimize analytical methods due to its advantages, such as a reduction in the number of experiments that need to be executed, which results in lower consumption and considerably less laboratory work (Ferreira et al., 2007). Although many experimental design methods are available to optimize product quality and the manufacturing process, there are two mainstreams in the view of development. One is the Taguchi method conceived and developed by Dr. Genichi Taguchi, a Japanese quality control expert, in the 1950s. The other is presented from the classical western method which consists of factorial design and RSM. Both methods realize robust design and process optimization of products by using statistical methods, but the procedures and principles are different. The Taguchi experimental design is considered a highly effective method to determine the optimal values for various parameters involved in a given manufacturing system (Fowlkes & Creveling, 1995). It entails system, parameter and tolerance design processes to achieve a robust design and result for the best product quality. The purpose of system design is to determine the suitable working levels of design factors (Hou, Su, & Liu, 2007). Parameter design determines the factor levels while tolerance design is used to fine-tune the results of parameter design to seek the least combination of fluctuation. This method can recognize significant factors that warrant attention at the next stage to improve product quality and obtain optimal collocation in designated levels, so it is generally used for screening tests (Beauregard, Mikulak, & Olson, 2000). Box and Wilson (1951) used RSM to explore the relationships between explanatory and response variables in statistics. When many factors and interactions affect the desired response, RSM is an effective tool for optimizing the process (Sun, Li, Yan, & Liu, 2010). The RSM is a form of visualization that allows engineers to better comprehend the characteristics of a process and the law, providing support to track suboptimal points (Khuri, 2006). In most RSM problems, the relationship between the response and the independent variables is unknown. The objective of the RSM is to find a suitable approximation for the true functional relationship between them (Montgomery, 2005). Since the quality characteristics of a production process may not be linear with the input variables, the polynomial of the second order is employed in some regions of the independent variables if there is curvature in the system. A great deal of empirical evidence has shown that the quadratic model (or very occasionally, some higher-order polynomial) is usually sufficient for the optimum region (Myers & Montgomery, 2002). The response surface can be graphically used to make judgments about the relationship between explanatory and response variables (Tong, Chang, & Lin, 2011). Knowledge of this relationship is important to find the treatment combination which gives the optimal response (Hinkelmann & Kempthorne, 2007). There are many important applications in the design, development, and formulation of new products, as well as the improvement of existing product designs (Myers & Montgomery, 2002). The RSM usually involves the following two steps. The first step is often to carry out a fractional factorial design to determine which factors significantly affect the response and their interactions as the preliminary screening. This can determine the fastest
rising or falling direction and the range of controlling variables for further analysis. In our study, the first step in the analysis showed that the relationship exhibits a non-linear change. Once in the vicinity of the optimum response, the experimenter needs to peruse a more elaborate model between the response and the factors. The second step of the RSM is to use special experimental designs, which include a central composite design (CCD) or BBD, to perform response surface regression. If a full factorial design is used, the midpoint of each factor in its range must be set to estimate the fitting error of the model. However, the 3k factorial usually results in many repeated testings and low accuracy when more influential factors are involved. The CCD, as an RSM design, is efficient for sequential experimentation and allows a reasonable amount of information to test the lack of fit, but does not involve a large number of design points. This design requires five levels for each factor and contains combinations in which all factors are simultaneously at their highest or lowest level. As the depth levels of the cup of different head sizes are difficult to set due to the specific ratio for factorial and axial points, this method is not adopted in this research. The BBD, as another RSM design, was proposed by using only three levels of each factor and at the same time with a ‘‘reasonable’’ number of experimental points (Hinkelmann & Kempthorne, 2007). The use of the BBD should be confined to a situation in which one is not interested in predicting extreme responses. Furthermore, this design is rotatable (or near rotatable) which means that the model would possess a reasonably stable distribution of scaled prediction variance throughout the experimental design region (Montgomery, 2005). It requires three levels of each factor instead of five as in the CCD, which results in fewer experimental trials to evaluate multiple variables and their interactions, and is more convenient and less expensive to run than the CCD with the same number of factors (k < 5) (Ragonese, Macka, Hughes, & Petocz, 2002). 3. Experimental work This research proposes the application of the BBD to optimize the process of foam cup molding, and efficiently determine the optimal molding temperatures and length of dwell time in accordance with the cup depths of mold heads and their interactions. The materials used were provided by a commercial bra cup manufacturer, including a flexible PU foam material, a series of different sized mold heads and their corresponding plastic cup templates for visual examination of cup depth and shape. 3.1. Material for foam cup molding Thermo-set foam cups are an integral part of molded and padded bra construction, made by a mold compression of flexible PU foam in accordance with temperature, pressure and time (Chong, 2010). The major advantage of using flexible foam is that it provides fit and support for brassieres with good softness and durability. Flexible PU foam possesses a unique ability that is not offered by many other materials (Yu, 1996). It is a complex assembly of elastomer and air. Through compression, it can be molded into different cup shapes of different sizes. PU foam has a cellular structure, which contains a large number of interconnected open and closed cells. The physical and thermal properties directly influence the enduse performance of a foam cup (Bicerano, 2002). The specifications of the flexible PU foam tested in this study are shown in Table 1. 3.2. Instruments and shape conformity A contour molding machine is used for the foam cup molding. This machine has a set of aluminum molding units that consist of
8061
L. Wu et al. / Expert Systems with Applications 39 (2012) 8059–8065 Table 1 Specifications of the flexible PU foam. Items
Values
Density Cell count (cells per 25 mm) Tensile stress at 8% strain Compression stress value at a compression of 40% Hardness Softening temperature Melting point
28.23 kg/m3 46.2 2.80 kPa 2.16 kPa 21.96 Shore A 169 °C 220–240 °C
a pair of top and bottom molds which compress each other during molding. A contour molding machine, New Pads DM-021HP4-2PR (USA), is used in the current study. A series of mold heads sized 34B, 36C and 36DD was also provided by the bra cup manufacturer. Their corresponding plastic cup templates, which are traditionally used as the quality standard for visual examinations of shape conformity of molded foam cups, were also prepared. Unfortunately, this is a subjective approach in that the quality and conformity of the molded cups are generally subjectively assessed based on the experience of individual quality inspectors. No quantitative information can be provided regardless of the effects of the molding conditions on foam cup quality. In this study, a desktop NextEngine 3D laser scanner (NextEngine, USA) with ScanStudio HD PRO software are employed to capture the outer surface of the molded foam cups in full color. The scanner has a maximal accuracy and resolution of 0.127 mm. The shape conformity of the molded foam cups was calculated from the deviation of the outside surfaces between each molded foam cup and its corresponding plastic cup template (Yick, Ng, Zhou, Yu, & Chan, 2008).
measured values of the cup depths are 56 mm, 76 mm and 96 mm, which respectively correspond to sizes 34B, 36C and 36DD. The bending force of the mold heads compresses the foam sheet to result in a foam cup. The space between the male and female molds defines the compression rate at the center and edge of a foam cup. As the gap is usually fixed by the male and female molds during the molding process, the pressure setting can not be adjusted on the molding machine which corresponds to the specific size of the aluminum mold. Thus, pressure is not included in the main factors. The three factors (molding temperature, dwell time and cup depth of mold head) chosen for this BBD experiment are designated as x1, x2, x3 and prescribed into three levels, and successively coded as +1, 0, 1 for high, intermediate and low values, respectively. Three test variables were coded according to the following equation:
Xi ¼
xi x0 Dx
i ¼ 1; 2; 3;
ð1Þ
where Xi is the coded value of an independent variable; xi is the actual value of an independent variable; x0 is the actual value of an independent variable at center point; and Dx is the step change value of an independent variable (Sharma, Singh, & Dilbaghi, 2009). Table 2 lists the process factors and factor levels of the bra cup molding conditions.Data from the BBD were analyzed by multiple regression to fit the following quadratic polynomial model:
Y ¼ a0 þ
3 X i¼1
ai X i þ
3 X
aii X 2i þ
i¼1
2 X 3 X i¼1
aij X i X j þ e;
ð2Þ
j¼2
where Y is the response variable; a0 is the model constant; ai represents the linear coefficient; aii denotes the quadratic coefficient; aij is the interaction coefficient; and e is the statistical error.
3.3. Determination of control factors and levels
3.4. Steps for process parameter optimization
After contour molding, the foam cup shape is always affected by process parameters, such as the molding temperature, dwell time and pressure. The molding temperature is the dominant parameter among the three factors (Chong, 2010). The molding temperature setting not only provides the foam microstructure with sufficient energy to restructure, but also satisfies the appearance and color requirements. If the molding temperature is not high enough, it will be difficult for the molded cup to form the required stability in profile and thickness. On the contrary, if the molding temperature is too high, the cup will become brittle and be easily torn, and the polymeric material will stick to the molds (Yu, 1996). In this research, the molding temperature range is set between 180 °C and 210 °C based on the first step of the analysis in the fractional factorial design. Dwell time is another dominant parameter. In fact, temperature and dwell time are inter-related in order to reach a balance for a molded profile that fits the overall quality requirements (Chong, 2010). Excessive dwell time would certainly cause foam aging because the PU elastomer will be damaged by too much heat. Therefore, the dwell time must be compatible with the molding temperature for quality in each individual piece of foam and its resultant cup shapes and sizes. The dwell time was designated from 60 s to 180 s based on the first step of the analysis. Different sized mold heads have a direct effect on the molded cup shape and shape conformity. As the size of the mold head cannot be put into a regression formula due to the nominal variable, the cup depth that corresponded to each head size was applied to the DOE as a numerical variable. The cup depth is defined as the vertical height from the top plane to the lowest point identified at the concave mold head by reverse engineering software. The
The following steps are followed for process optimization and a flow chart diagram is presented in Fig. 1. In this process of optimization, the molding temperature, dwell time and cup depth of the mold head were entered as the explanatory variables, and the optimal shape conformity as the response variable. Step 1. Make aluminum mold heads based on specified material and plastic cup template according to the client’s requirements. Step 2. Carry out BBD arrays and scan the molded foam cups with the NextEngine 3D scanner. Step 3. Calculate the shape conformities between each molded cup and its corresponding plastic cup template. In this research, 15-run trials assigned by the statistical software Design-Expert were conducted to determine their optimum levels. The BBD matrix consisted of 12 different level combinations of the independent variables as well as three center point runs used to fit a second-order response surface and provide a measure of process stability and inherent variability (Dong, Xie, Wang, Zhan, & Yao, 2009). The BBD design matrix along with the experimental values of the responses are shown in Table 3 below (in terms of coded factors).
Table 2 Levels and codes chosen for the BBD. Variables
Molding temperature (°C) Dwell time (s) Cup depth (mm)
Symbols
Uncoded levels
Uncoded
Coded
1
0
1
x1 x2 x3
X1 X2 X3
180 60 56
195 120 76
210 180 96
8062
L. Wu et al. / Expert Systems with Applications 39 (2012) 8059–8065
Step 6. Adjust the final shape of the aluminum molds in accordance with the foam cup quality (viz., handfeel, wrinkles after washing, yellowing, etc.) to evaluate the fatigue states of the struts against thermal deformation. Step 7. Verify the simulations and experiment results. Step 8. Duplicate the standard aluminum mold for mass production once the foam cup shape and the aluminum mold shape are approved by client. 4. Results and discussion 4.1. ANOVA results and regression model Whether a quadratic model is significant or not could be determined through ANOVA. As seen in Table 4, ANOVA shows that this quadratic regression model is highly significant. This is evident from the Fisher’s F-value of 6.48 and a low probability value (P = 0.027 < 0.05). The results of the goodness-of-fit of the models are also summarized in Table 4. The model adequacy was further confirmed by a satisfactory value of the determination coefficient, which was calculated to be 0.9210. The value suggests that the model could predict 92.10% of the variability in the response. It is adequate for this regression model within the range of experimental variables. Furthermore, there is a very strong agreement between the predicted values of the shape conformity and the experimental data, as seen in Table 4. The regression coefficient values of the shape conformity model are listed in Table 4. The p-values of each coefficient were used to examine the significance level, which also indicate the interaction effects between each independent variable. The p-value of the Ftest for lack of fit is 0.056 which implies that there is marginally no significant lack-of-fit at a = 0.05 significance level and the model is acceptable. The second-order polynomial equation in an uncoded form illustrates the relationship of the three variables, which was established to explain shape conformity. The polynomial model for the shape conformity of the molded foam cups is regressed in the following formula (in terms of uncoded factors) Fig. 1. Flow chart of the optimization design.
Y ¼ 2106:33 þ 21:30x1 þ 2:38x2 2:40x3 0:05x21 0:01x23 0:01x1 x2 þ 0:02x1 x3 þ 0:0003x2 x3 :
Step 4. Conduct RSM simulation, including second-order regression and analysis of variance (ANOVA). Step 5.Trace the optimal conditions of the different cup depths of the mold heads based on the contour and the surface plots of the RSM simulation.
Table 3 The BBD with experimental values of shape conformity of the molded foam cups. Run no.
X1
X2
X3
Response: shape conformity (%)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
1 1 1 1 1 1 1 1 0 0 0 0 0 0 0
1 1 1 1 0 0 0 0 1 1 1 1 0 0 0
0 0 0 0 1 1 1 1 1 1 1 1 0 0 0
35.18 70.68 74.27 73.94 64.15 72.35 45.33 76.89 74.59 76.03 65.78 68.89 82.91 79.15 81.11
ð3Þ
4.2. Optimization of molding process parameters RSM plays a very critical role in efficiently exploring the optimal values of explanatory variables. As a function of two factors, three dimensional response surfaces and their corresponding contour plots are more helpful in understanding both the main and the interaction effects of these two factors, maintaining all other factors at fixed levels (Adinarayana & Ellaiah, 2002). They can be used to describe and examine the regression equations in a visualized way to reflect the effects of experimental variables on the required response (Wu, Zhou, & Li, 2009). In this research, the shape conformities of molded cups could be simulated with different sizes of mold heads along with two continuous experimental variables, which are molding temperature and dwell time. The maximum predicted values for shape conformity as well as the optimal values of the experimental values could be located by the equation from the response surface. Fig. 2 indicates the changing response surface at a cup depth of 56 mm. The shape conformity shows an upward trend when the temperature is increased from 180 °C to around 194.3 °C and dwell time to around 152.6 s. If the temperature and time exceed the above values simultaneously, there will emerge a decline in the de-
8063
L. Wu et al. / Expert Systems with Applications 39 (2012) 8059–8065 Table 4 ANOVA results of the quadratic regression model for optimization of the shape conformity of molded foam cups. Source of variations
Sum of squares
Model x1-Molding temperature (°C) x2-Dwell time (s) x3-Cup depth (mm) x1 x2 x1 x3 x2 x3 x21
2199.01 701.81 274.95 114.23 320.95 136.42 0.70 539.77
x22 x23 Residual Lack-of-fit Pure error Total error *
Degree of freedom 9 1 1 1 1 1 1 1
Coefficient estimate
Mean square
F-value
p-value (Prob > F)
Determination coefficient (R2)
Adjusted determination coefficient (Adj R2)
244.33 701.81 274.95 114.23 320.95 136.42 0.70 539.77
6.48 18.60 7.29 3.03 8.51 3.62 0.018 14.31
0.0267* 0.0076 0.0428 0.1424 0.0331 0.1156 0.8972 0.0129
92.10%
77.88%
21.30 2.38 2.40 0.01 0.02 0.0003 0.05
109.60
1
0.00
109.60
2.90
0.1490
67.82
1
0.01
67.82
1.80
0.2377
188.66 181.59 7.07 2387.67
5 3 2 14
37.732 60.529 3.537
17.12
0.056
Significant at the 5% level.
Fig. 2. Response surface and contour plots that represent the effect of molding temperature, dwell time, and their interaction on shape conformity when the cup depth is 56 mm.
Fig. 4. Response surface and contour plots that represent the effect of molding temperature and dwell time, and their interaction on shape conformity when the cup depth is 96 mm.
Table 5 The final settings for the optimal process parameters via soft computing.
Fig. 3. Response surface and contour plots that represent the effect of molding temperature, dwell time, and their interaction on shape conformity when the cup depth is 76 mm.
gree of conformity. The pattern of the elliptic contours reveals that an apparent interactive effect exists between the molding temperature and dwell time. Therefore, the optimal molding temperature
Molding temperature (°C)
Dwell time (s)
Cup depth (mm)
Shape conformity (%)
194.3 198.8 204.0
152.6 140.8 125.4
56 76 96
81.93 83.24 77.80
and dwell time are 194.3 °C and 152.6 s, respectively, when the cup depth is 56 mm. Fig. 3 demonstrates the variation trend of the shape conformity vs. temperature and time when the cup depth is 76 mm. The surface shape is more or less the same as the cup depth of 56 mm. The optimal values of the molding temperature and dwell time are 198.8 °C and 140.8 s, respectively. As indicated by the contours, conformity reaches a maximum value at 83.24%. Optimal shape conformity can never be reached with a longer time even with a higher or lower temperature. For a cup depth of 96 mm, the effect of the molding temperature and dwell time on shape conformity is presented in Fig. 4. According to the surface and contours, it can be concluded that the optimal molding temperature and dwell time for this cup
8064
L. Wu et al. / Expert Systems with Applications 39 (2012) 8059–8065
the shape conformity for the same cup depth. The interaction between temperature and time also has a significant effect on shape conformity. It can be seen from the optimal conditions with different cup depths that for a larger cup depth, foam cup molding needs to be increased in temperature and cannot be implemented for a long time; on the contrary, for a small cup depth, a lower temperature and longer time are comparatively required; and 4. The changing trends and optimal parameters of shape conformity are statistically discussed. Although the properties of the material were not taken into account on the effect on foam cup molding, the method can be enforced in present design development and production by bra cup manufacturers. Based on the simulation and experimental results, this approach tries to strike a balance amongst low cost, high efficiency and good accuracy. It has proven not only to be scientific and reasonable, but also quick and efficient. Future work will involve the molecular structure of materials and thermal transfer mechanisms during the molding process. Fig. 5. Normal probability plot of residuals for shape conformity of molded foam cups.
Acknowledgement depth are 204.0 °C and 125.4 s, respectively, with a corresponding degree of shape conformity at 77.80%. The final settings for the optimal process parameters are shown in Table 5. These settings correspond to the maximum shape conformity at a given cup depth in the mold heads. 5. Verification test A normal probability plot of the residuals is depicted in Fig. 5, which reveals that the residuals generally fall on a least-square line which is used to estimate the cumulative distribution function for the population. As evident from the figure, the errors are normally distributed and there are almost no serious violations of the assumptions that underlie the analysis (Kilickap, 2010). By displaying a satisfactory normal distribution, the normality assumptions made earlier could be confirmed and the predictive regression model has extracted all information available from the experimental data (Joseph Davidson, Balasubramanian, & Tagore, 2008). 6. Conclusions Determination of the optimal process parameter settings is critical work in the manufacturing industry which influences productivity, quality, production costs and time for delivery. By applying the BBD method, an effective approach for process parameter optimization in foam cup molding with a minimal number of experimental runs has been presented in this paper. The conclusions of the present study are as follows: 1. This study demonstrates the optimization of molding conditions (viz., molding temperature and length of dwell time) for corresponding foam material and cup depths of mold heads, providing an objective evaluation approach that not only optimizes the foam cup molding process, but also facilitates communication between manufacturers and clients; 2. A Box–Behnken experimental design is utilized for optimization, which replaces the traditional DOE. The optimum sets of molding temperatures and dwell times are graphically obtained at different cup depths of mold heads. The experimental shape conformity closely agrees with the predicted values; 3. The ANOVA results reveal that molding temperature is the main parameter, which has a greater influence than the dwell time on
The authors would like to thank the Research Grant Council for funding this research through project account PolyU RPGV.
References Adinarayana, K., & Ellaiah, P. (2002). Response surface optimization of the critical medium components for the production of alkaline protease by a newly isolated Bacillus sp.. Journal of Pharmacy and Pharmaceutical Sciences, 5(3), 272–276. Beauregard, M. R., Mikulak, R. J., & Olson, B. A. (2000). Experimenting for breakthrough improvement: The step-by-step guide for harnessing the power of experimental design. Waitsfield, VT: Quality Training Portal. Bicerano, J. (2002). Prediction of polymer properties (3rd ed.). New York: Marcel Dekker Inc. Box, G. E. P., & Wilson, K. B. (1951). On the experimental attainment of optimum conditions. Journal of the Royal Statistical Society, 13(1), 1–45. Chang, H. W. (2008). A data mining approach to dynamic multiple responses in Taguchi experimental design. Expert Systems with Applications, 35(3), 1095–1103. Chong, R. (2010). Molding & lamination technology handbook for lingerie industry. Hong Kong: Hong Kong Productivity Council. Dong, C. H., Xie, X. Q., Wang, X. L., Zhan, Y., & Yao, Y. J. (2009). Application of Box– Behnken design in optimisation for polysaccharides extraction from cultured mycelium of Cordyceps sinensis. Food and Bioproducts Processing, 87(2), 139–144. Ferreira, S. L. C., Bruns, R. E., Ferreira, H. S., Matos, G. D., David, J. M., Brandão, G. C., et al. (2007). Box–Behnken design: An alternative for the optimization of analytical methods. Analytica Chimica Acta, 597(2), 179–186. Fowlkes, W. Y., & Creveling, C. M. (1995). Engineering methods for robust product design: Using Taguchi methods in technology and product development. New Jersey: Prentice Hall. Haycock, C. E. (1978). Breast support and protection in female athlete. In AAHPERD symposium papers (Vol. 1(2), pp. 50–53). Hinkelmann, K., & Kempthorne, O. (2007). Design and analysis of experiments: Introduction to experimental design (2nd ed.). New Jersey: John Wiley & Sons, Inc. Hou, T. H., Su, C. H., & Liu, W. L. (2007). Parameters optimization of a nano-particle wet milling process using the Taguchi method, response surface method and genetic algorithm. Powder Technology, 173(3), 153–162. Joseph Davidson, M., Balasubramanian, K., & Tagore, G. R. N. (2008). Surface roughness prediction of flow-formed AA6061 alloy by design of experiments. Journal of Materials Processing Technology, 202(1–3), 41–46. Khuri, A. I. (2006). Response surface methodology and related topics. New Jersey: World Scientific Publishing Co. Pte. Ltd. Kilickap, E. (2010). Optimization of cutting parameters on delamination based on Taguchi method during drilling of GFRP composite. Expert Systems with Applications, 37(8), 6116–6122. Montgomery, D. C. (2005). Design and analysis of experiments (6th ed.). USA: John wiley & Sons, Inc. Myers, R. H., & Montgomery, D. C. (2002). Response surface methodology: Process and product optimization using designed experiments (2nd ed.). USA: John wiley & Sons, Inc. Ragonese, R., Macka, M., Hughes, J., & Petocz, P. (2002). The use of the Box–Behnken experimental design in the optimisation and robustness testing of a capillary electrophoresis method for the analysis of ethambutol hydrochloride in a
L. Wu et al. / Expert Systems with Applications 39 (2012) 8059–8065 pharmaceutical formulation. Journal of Pharmaceutical and Biomedical Analysis, 27(6), 995–1007. Sharma, P., Singh, L., & Dilbaghi, N. (2009). Optimization of process variables for decolorization of Disperse Yellow 211 by Bacillus subtilis using Box–Behnken design. Journal of Hazardous Materials, 164(2–3), 1024–1029. Sun, Y., Li, T., Yan, J., & Liu, J. (2010). Technology optimization for polysaccharides (POP) extraction from the fruiting bodies of Pleurotus ostreatus by Box– Behnken statistical design. Carbohydrate Polymers, 80(1), 242–247. Tong, L. I., Chang, Y. C., & Lin, S. H. (2011). Determining the optimal re-sampling strategy for a classification model with imbalanced data using design of experiments and response surface methodologies. Expert Systems with Applications, 38(4), 4222–4227. Wu, D., Zhou, J., & Li, Y. (2009). Effect of the sulfidation process on the mechanical properties of a CoMoP/Al2O3 hydrotreating catalyst. Chemical Engineering Science, 64(2), 198–206.
8065
Yick, K. L., Ng, S., Zhou, X., Yu, W., & Chan, D. (2008). Wire frame representation of 3D moulded bra cup and its application to example-based design. Fibers and Polymers, 9(5), 653–658. Yick, K. L., Wu, L., Yip, J., Ng, S.-p., & Yu, W. (2010). Study of thermal-mechanical properties of polyurethane foam and the three-dimensional shape of molded bra cups. Journal of Materials Processing Technology, 210(1), 116–121. Yu, W. (1996). The effects of polyurethane foam properties and moulding conditions on the shape characteristics of brassiere cups. UK: The University of Leeds. Yu, W., Fan, J., Harlock, S. C., & Ng, S. P. (2006). Innovation and technology of women’s intimate apparel. England: Woodhead Publishing Ltd. Yu, W. M., Yeung, K. W., Harlock, S. C., & Leaf, G. A. V. (1998). Predicting energy requirements for moulding flexible polyurethane foam sheets. Textile Research Journal, 68(4), 289–295.
Contents lists available at SciVerse ScienceDirect
Expert Systems with Applications journal homepage: www.elsevier.com/locate/eswa
Application of the Box–Behnken design to the optimization of process parameters in foam cup molding Long Wu a, Kit-lun Yick a,⇑, Sun-pui Ng b, Joanne Yip a a b
Institute of Textiles & Clothing, The Hong Kong Polytechnic University, Hong Kong Hong Kong Community College, The Hong Kong Polytechnic University, Hong Kong
a r t i c l e
i n f o
Keywords: Box–Behnken Response surface methodology Optimization Shape conformity Foam
a b s t r a c t Currently, foam molding technologies are widely adopted for most bra styles, which demonstrate the incomparable advantages in the contemporary intimate apparel industry. The determination of proper molding conditions, such as molding temperatures and length of time on the basis of cup sizes and styles, is crucial in achieving the required cup shape with high stability, which is regarded as the most challenging part of the molded bra making process. To determine the optimal process parameter settings, numerous process trials are generally required to evaluate the molding variables and their interactions. This study proposes a novel systematic methodology to identify the optimal molding process parameters based on design of experiment (DOE) and a parameterization-based remesh method to evaluate the 3D shape conformity of molded cups. By solving the regression equation obtained from a Box–Behnken design (BBD) and analyzing the response surface plots, the results prove that molding temperature has greater influence than the length of the dwell time on the 3D shape conformity of molded cups. The optimal molding conditions can be determined for the cup depths of different sized mold heads, which are validated by the experimental results. Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction The optimization of process parameters is routinely performed in the manufacturing industry so as to solve the challenges in product quality and/or optimize the cost effectiveness of manufacturing processes (Chang, 2008). In the intimate apparel industry, foam cup molding is a critical process in the course of making seamless molded bras. Due to the smooth surface and natural configuration under the outer garment, seamless molded bras nowadays represent almost 80–90% of the overall bra market (Haycock, 1978). Nevertheless, the determination of optimal molding conditions is a long standing problem in controlling the process of foam cup molding. The process has been carried out on a flat polyurethane (PU) foam sheet with a certain thickness by a hot male mold head press which is actuated by means of compressed air, and maintained for a certain dwell time before the male mold is automatically released back (Yu, Fan, Harlock, & Ng, 2006). The PU foam is then softened by heat and forms the required shape and thickness that are defined by a cavity between the pair of molds (Yick, Wu, Yip, Ng, & Yu, 2010). The process and the ultimate quality of the molded cups are greatly influenced by the applied molding temperature, length of time and clamping pressure. Factors such as cup size, style, and foam physical ⇑ Corresponding author. Tel.: +852 27666551; fax: +852 27731432. E-mail address: [email protected] (K.-l. Yick). 0957-4174/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.eswa.2012.01.137
and thermal mechanical properties can also increase difficulties in the determination of optimal molding conditions for the required quality and shape conformity. For over 40 years since garment molding was first introduced in the 1970s, the molding conditions are roughly determined by the experience of the molder on a trial-and-error basis. As the molded cups are soft and readily deform, the quality and shape of the molded cups are visually inspected through comparisons with a plastic cup template. The judgment depends on the experience of the inspectors. Problems have further increased in the new century due to the complex design of mold cups, the extreme size and softness requirements of bras, and a vast number of foam materials introduced into the market. As a consequence of the increasing demand of seamless molded bras and the short cycle time in manufacturing, the optimization of the foam cup molding process has become a very important subject for researchers in order to provide accurate and reliable procedures that will assure molding quality and improve production efficiency (Yu, Yeung, Harlock, & Leaf, 1998). The optimization approach provided by the Box–Behnken design (BBD), which is a response surface methodology (RSM) that uses Design-Expert software (Version 7.0.0, Stat-Ease Inc., Minneapolis, USA), is proposed in this study. As a collection of statistical and mathematical techniques for developing, improving, and optimizing processes, RSM is specifically applied in the industrial world, particularly in situations where several input variables potentially influence a performance measure or
8060
L. Wu et al. / Expert Systems with Applications 39 (2012) 8059–8065
quality characteristic of the product or process (Myers & Montgomery, 2002). On the basis of the BBD, the process parameters (molding temperatures, length of dwell time and cup depths of mold heads) in the foam cup molding process can be optimized with a minimum number of experimental runs and achieve competitive advantages in product quality and costs.
2. Response surface methodology and Box–Behnken design In recent years, design of experiment (DOE) has been frequently applied to optimize analytical methods due to its advantages, such as a reduction in the number of experiments that need to be executed, which results in lower consumption and considerably less laboratory work (Ferreira et al., 2007). Although many experimental design methods are available to optimize product quality and the manufacturing process, there are two mainstreams in the view of development. One is the Taguchi method conceived and developed by Dr. Genichi Taguchi, a Japanese quality control expert, in the 1950s. The other is presented from the classical western method which consists of factorial design and RSM. Both methods realize robust design and process optimization of products by using statistical methods, but the procedures and principles are different. The Taguchi experimental design is considered a highly effective method to determine the optimal values for various parameters involved in a given manufacturing system (Fowlkes & Creveling, 1995). It entails system, parameter and tolerance design processes to achieve a robust design and result for the best product quality. The purpose of system design is to determine the suitable working levels of design factors (Hou, Su, & Liu, 2007). Parameter design determines the factor levels while tolerance design is used to fine-tune the results of parameter design to seek the least combination of fluctuation. This method can recognize significant factors that warrant attention at the next stage to improve product quality and obtain optimal collocation in designated levels, so it is generally used for screening tests (Beauregard, Mikulak, & Olson, 2000). Box and Wilson (1951) used RSM to explore the relationships between explanatory and response variables in statistics. When many factors and interactions affect the desired response, RSM is an effective tool for optimizing the process (Sun, Li, Yan, & Liu, 2010). The RSM is a form of visualization that allows engineers to better comprehend the characteristics of a process and the law, providing support to track suboptimal points (Khuri, 2006). In most RSM problems, the relationship between the response and the independent variables is unknown. The objective of the RSM is to find a suitable approximation for the true functional relationship between them (Montgomery, 2005). Since the quality characteristics of a production process may not be linear with the input variables, the polynomial of the second order is employed in some regions of the independent variables if there is curvature in the system. A great deal of empirical evidence has shown that the quadratic model (or very occasionally, some higher-order polynomial) is usually sufficient for the optimum region (Myers & Montgomery, 2002). The response surface can be graphically used to make judgments about the relationship between explanatory and response variables (Tong, Chang, & Lin, 2011). Knowledge of this relationship is important to find the treatment combination which gives the optimal response (Hinkelmann & Kempthorne, 2007). There are many important applications in the design, development, and formulation of new products, as well as the improvement of existing product designs (Myers & Montgomery, 2002). The RSM usually involves the following two steps. The first step is often to carry out a fractional factorial design to determine which factors significantly affect the response and their interactions as the preliminary screening. This can determine the fastest
rising or falling direction and the range of controlling variables for further analysis. In our study, the first step in the analysis showed that the relationship exhibits a non-linear change. Once in the vicinity of the optimum response, the experimenter needs to peruse a more elaborate model between the response and the factors. The second step of the RSM is to use special experimental designs, which include a central composite design (CCD) or BBD, to perform response surface regression. If a full factorial design is used, the midpoint of each factor in its range must be set to estimate the fitting error of the model. However, the 3k factorial usually results in many repeated testings and low accuracy when more influential factors are involved. The CCD, as an RSM design, is efficient for sequential experimentation and allows a reasonable amount of information to test the lack of fit, but does not involve a large number of design points. This design requires five levels for each factor and contains combinations in which all factors are simultaneously at their highest or lowest level. As the depth levels of the cup of different head sizes are difficult to set due to the specific ratio for factorial and axial points, this method is not adopted in this research. The BBD, as another RSM design, was proposed by using only three levels of each factor and at the same time with a ‘‘reasonable’’ number of experimental points (Hinkelmann & Kempthorne, 2007). The use of the BBD should be confined to a situation in which one is not interested in predicting extreme responses. Furthermore, this design is rotatable (or near rotatable) which means that the model would possess a reasonably stable distribution of scaled prediction variance throughout the experimental design region (Montgomery, 2005). It requires three levels of each factor instead of five as in the CCD, which results in fewer experimental trials to evaluate multiple variables and their interactions, and is more convenient and less expensive to run than the CCD with the same number of factors (k < 5) (Ragonese, Macka, Hughes, & Petocz, 2002). 3. Experimental work This research proposes the application of the BBD to optimize the process of foam cup molding, and efficiently determine the optimal molding temperatures and length of dwell time in accordance with the cup depths of mold heads and their interactions. The materials used were provided by a commercial bra cup manufacturer, including a flexible PU foam material, a series of different sized mold heads and their corresponding plastic cup templates for visual examination of cup depth and shape. 3.1. Material for foam cup molding Thermo-set foam cups are an integral part of molded and padded bra construction, made by a mold compression of flexible PU foam in accordance with temperature, pressure and time (Chong, 2010). The major advantage of using flexible foam is that it provides fit and support for brassieres with good softness and durability. Flexible PU foam possesses a unique ability that is not offered by many other materials (Yu, 1996). It is a complex assembly of elastomer and air. Through compression, it can be molded into different cup shapes of different sizes. PU foam has a cellular structure, which contains a large number of interconnected open and closed cells. The physical and thermal properties directly influence the enduse performance of a foam cup (Bicerano, 2002). The specifications of the flexible PU foam tested in this study are shown in Table 1. 3.2. Instruments and shape conformity A contour molding machine is used for the foam cup molding. This machine has a set of aluminum molding units that consist of
8061
L. Wu et al. / Expert Systems with Applications 39 (2012) 8059–8065 Table 1 Specifications of the flexible PU foam. Items
Values
Density Cell count (cells per 25 mm) Tensile stress at 8% strain Compression stress value at a compression of 40% Hardness Softening temperature Melting point
28.23 kg/m3 46.2 2.80 kPa 2.16 kPa 21.96 Shore A 169 °C 220–240 °C
a pair of top and bottom molds which compress each other during molding. A contour molding machine, New Pads DM-021HP4-2PR (USA), is used in the current study. A series of mold heads sized 34B, 36C and 36DD was also provided by the bra cup manufacturer. Their corresponding plastic cup templates, which are traditionally used as the quality standard for visual examinations of shape conformity of molded foam cups, were also prepared. Unfortunately, this is a subjective approach in that the quality and conformity of the molded cups are generally subjectively assessed based on the experience of individual quality inspectors. No quantitative information can be provided regardless of the effects of the molding conditions on foam cup quality. In this study, a desktop NextEngine 3D laser scanner (NextEngine, USA) with ScanStudio HD PRO software are employed to capture the outer surface of the molded foam cups in full color. The scanner has a maximal accuracy and resolution of 0.127 mm. The shape conformity of the molded foam cups was calculated from the deviation of the outside surfaces between each molded foam cup and its corresponding plastic cup template (Yick, Ng, Zhou, Yu, & Chan, 2008).
measured values of the cup depths are 56 mm, 76 mm and 96 mm, which respectively correspond to sizes 34B, 36C and 36DD. The bending force of the mold heads compresses the foam sheet to result in a foam cup. The space between the male and female molds defines the compression rate at the center and edge of a foam cup. As the gap is usually fixed by the male and female molds during the molding process, the pressure setting can not be adjusted on the molding machine which corresponds to the specific size of the aluminum mold. Thus, pressure is not included in the main factors. The three factors (molding temperature, dwell time and cup depth of mold head) chosen for this BBD experiment are designated as x1, x2, x3 and prescribed into three levels, and successively coded as +1, 0, 1 for high, intermediate and low values, respectively. Three test variables were coded according to the following equation:
Xi ¼
xi x0 Dx
i ¼ 1; 2; 3;
ð1Þ
where Xi is the coded value of an independent variable; xi is the actual value of an independent variable; x0 is the actual value of an independent variable at center point; and Dx is the step change value of an independent variable (Sharma, Singh, & Dilbaghi, 2009). Table 2 lists the process factors and factor levels of the bra cup molding conditions.Data from the BBD were analyzed by multiple regression to fit the following quadratic polynomial model:
Y ¼ a0 þ
3 X i¼1
ai X i þ
3 X
aii X 2i þ
i¼1
2 X 3 X i¼1
aij X i X j þ e;
ð2Þ
j¼2
where Y is the response variable; a0 is the model constant; ai represents the linear coefficient; aii denotes the quadratic coefficient; aij is the interaction coefficient; and e is the statistical error.
3.3. Determination of control factors and levels
3.4. Steps for process parameter optimization
After contour molding, the foam cup shape is always affected by process parameters, such as the molding temperature, dwell time and pressure. The molding temperature is the dominant parameter among the three factors (Chong, 2010). The molding temperature setting not only provides the foam microstructure with sufficient energy to restructure, but also satisfies the appearance and color requirements. If the molding temperature is not high enough, it will be difficult for the molded cup to form the required stability in profile and thickness. On the contrary, if the molding temperature is too high, the cup will become brittle and be easily torn, and the polymeric material will stick to the molds (Yu, 1996). In this research, the molding temperature range is set between 180 °C and 210 °C based on the first step of the analysis in the fractional factorial design. Dwell time is another dominant parameter. In fact, temperature and dwell time are inter-related in order to reach a balance for a molded profile that fits the overall quality requirements (Chong, 2010). Excessive dwell time would certainly cause foam aging because the PU elastomer will be damaged by too much heat. Therefore, the dwell time must be compatible with the molding temperature for quality in each individual piece of foam and its resultant cup shapes and sizes. The dwell time was designated from 60 s to 180 s based on the first step of the analysis. Different sized mold heads have a direct effect on the molded cup shape and shape conformity. As the size of the mold head cannot be put into a regression formula due to the nominal variable, the cup depth that corresponded to each head size was applied to the DOE as a numerical variable. The cup depth is defined as the vertical height from the top plane to the lowest point identified at the concave mold head by reverse engineering software. The
The following steps are followed for process optimization and a flow chart diagram is presented in Fig. 1. In this process of optimization, the molding temperature, dwell time and cup depth of the mold head were entered as the explanatory variables, and the optimal shape conformity as the response variable. Step 1. Make aluminum mold heads based on specified material and plastic cup template according to the client’s requirements. Step 2. Carry out BBD arrays and scan the molded foam cups with the NextEngine 3D scanner. Step 3. Calculate the shape conformities between each molded cup and its corresponding plastic cup template. In this research, 15-run trials assigned by the statistical software Design-Expert were conducted to determine their optimum levels. The BBD matrix consisted of 12 different level combinations of the independent variables as well as three center point runs used to fit a second-order response surface and provide a measure of process stability and inherent variability (Dong, Xie, Wang, Zhan, & Yao, 2009). The BBD design matrix along with the experimental values of the responses are shown in Table 3 below (in terms of coded factors).
Table 2 Levels and codes chosen for the BBD. Variables
Molding temperature (°C) Dwell time (s) Cup depth (mm)
Symbols
Uncoded levels
Uncoded
Coded
1
0
1
x1 x2 x3
X1 X2 X3
180 60 56
195 120 76
210 180 96
8062
L. Wu et al. / Expert Systems with Applications 39 (2012) 8059–8065
Step 6. Adjust the final shape of the aluminum molds in accordance with the foam cup quality (viz., handfeel, wrinkles after washing, yellowing, etc.) to evaluate the fatigue states of the struts against thermal deformation. Step 7. Verify the simulations and experiment results. Step 8. Duplicate the standard aluminum mold for mass production once the foam cup shape and the aluminum mold shape are approved by client. 4. Results and discussion 4.1. ANOVA results and regression model Whether a quadratic model is significant or not could be determined through ANOVA. As seen in Table 4, ANOVA shows that this quadratic regression model is highly significant. This is evident from the Fisher’s F-value of 6.48 and a low probability value (P = 0.027 < 0.05). The results of the goodness-of-fit of the models are also summarized in Table 4. The model adequacy was further confirmed by a satisfactory value of the determination coefficient, which was calculated to be 0.9210. The value suggests that the model could predict 92.10% of the variability in the response. It is adequate for this regression model within the range of experimental variables. Furthermore, there is a very strong agreement between the predicted values of the shape conformity and the experimental data, as seen in Table 4. The regression coefficient values of the shape conformity model are listed in Table 4. The p-values of each coefficient were used to examine the significance level, which also indicate the interaction effects between each independent variable. The p-value of the Ftest for lack of fit is 0.056 which implies that there is marginally no significant lack-of-fit at a = 0.05 significance level and the model is acceptable. The second-order polynomial equation in an uncoded form illustrates the relationship of the three variables, which was established to explain shape conformity. The polynomial model for the shape conformity of the molded foam cups is regressed in the following formula (in terms of uncoded factors) Fig. 1. Flow chart of the optimization design.
Y ¼ 2106:33 þ 21:30x1 þ 2:38x2 2:40x3 0:05x21 0:01x23 0:01x1 x2 þ 0:02x1 x3 þ 0:0003x2 x3 :
Step 4. Conduct RSM simulation, including second-order regression and analysis of variance (ANOVA). Step 5.Trace the optimal conditions of the different cup depths of the mold heads based on the contour and the surface plots of the RSM simulation.
Table 3 The BBD with experimental values of shape conformity of the molded foam cups. Run no.
X1
X2
X3
Response: shape conformity (%)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
1 1 1 1 1 1 1 1 0 0 0 0 0 0 0
1 1 1 1 0 0 0 0 1 1 1 1 0 0 0
0 0 0 0 1 1 1 1 1 1 1 1 0 0 0
35.18 70.68 74.27 73.94 64.15 72.35 45.33 76.89 74.59 76.03 65.78 68.89 82.91 79.15 81.11
ð3Þ
4.2. Optimization of molding process parameters RSM plays a very critical role in efficiently exploring the optimal values of explanatory variables. As a function of two factors, three dimensional response surfaces and their corresponding contour plots are more helpful in understanding both the main and the interaction effects of these two factors, maintaining all other factors at fixed levels (Adinarayana & Ellaiah, 2002). They can be used to describe and examine the regression equations in a visualized way to reflect the effects of experimental variables on the required response (Wu, Zhou, & Li, 2009). In this research, the shape conformities of molded cups could be simulated with different sizes of mold heads along with two continuous experimental variables, which are molding temperature and dwell time. The maximum predicted values for shape conformity as well as the optimal values of the experimental values could be located by the equation from the response surface. Fig. 2 indicates the changing response surface at a cup depth of 56 mm. The shape conformity shows an upward trend when the temperature is increased from 180 °C to around 194.3 °C and dwell time to around 152.6 s. If the temperature and time exceed the above values simultaneously, there will emerge a decline in the de-
8063
L. Wu et al. / Expert Systems with Applications 39 (2012) 8059–8065 Table 4 ANOVA results of the quadratic regression model for optimization of the shape conformity of molded foam cups. Source of variations
Sum of squares
Model x1-Molding temperature (°C) x2-Dwell time (s) x3-Cup depth (mm) x1 x2 x1 x3 x2 x3 x21
2199.01 701.81 274.95 114.23 320.95 136.42 0.70 539.77
x22 x23 Residual Lack-of-fit Pure error Total error *
Degree of freedom 9 1 1 1 1 1 1 1
Coefficient estimate
Mean square
F-value
p-value (Prob > F)
Determination coefficient (R2)
Adjusted determination coefficient (Adj R2)
244.33 701.81 274.95 114.23 320.95 136.42 0.70 539.77
6.48 18.60 7.29 3.03 8.51 3.62 0.018 14.31
0.0267* 0.0076 0.0428 0.1424 0.0331 0.1156 0.8972 0.0129
92.10%
77.88%
21.30 2.38 2.40 0.01 0.02 0.0003 0.05
109.60
1
0.00
109.60
2.90
0.1490
67.82
1
0.01
67.82
1.80
0.2377
188.66 181.59 7.07 2387.67
5 3 2 14
37.732 60.529 3.537
17.12
0.056
Significant at the 5% level.
Fig. 2. Response surface and contour plots that represent the effect of molding temperature, dwell time, and their interaction on shape conformity when the cup depth is 56 mm.
Fig. 4. Response surface and contour plots that represent the effect of molding temperature and dwell time, and their interaction on shape conformity when the cup depth is 96 mm.
Table 5 The final settings for the optimal process parameters via soft computing.
Fig. 3. Response surface and contour plots that represent the effect of molding temperature, dwell time, and their interaction on shape conformity when the cup depth is 76 mm.
gree of conformity. The pattern of the elliptic contours reveals that an apparent interactive effect exists between the molding temperature and dwell time. Therefore, the optimal molding temperature
Molding temperature (°C)
Dwell time (s)
Cup depth (mm)
Shape conformity (%)
194.3 198.8 204.0
152.6 140.8 125.4
56 76 96
81.93 83.24 77.80
and dwell time are 194.3 °C and 152.6 s, respectively, when the cup depth is 56 mm. Fig. 3 demonstrates the variation trend of the shape conformity vs. temperature and time when the cup depth is 76 mm. The surface shape is more or less the same as the cup depth of 56 mm. The optimal values of the molding temperature and dwell time are 198.8 °C and 140.8 s, respectively. As indicated by the contours, conformity reaches a maximum value at 83.24%. Optimal shape conformity can never be reached with a longer time even with a higher or lower temperature. For a cup depth of 96 mm, the effect of the molding temperature and dwell time on shape conformity is presented in Fig. 4. According to the surface and contours, it can be concluded that the optimal molding temperature and dwell time for this cup
8064
L. Wu et al. / Expert Systems with Applications 39 (2012) 8059–8065
the shape conformity for the same cup depth. The interaction between temperature and time also has a significant effect on shape conformity. It can be seen from the optimal conditions with different cup depths that for a larger cup depth, foam cup molding needs to be increased in temperature and cannot be implemented for a long time; on the contrary, for a small cup depth, a lower temperature and longer time are comparatively required; and 4. The changing trends and optimal parameters of shape conformity are statistically discussed. Although the properties of the material were not taken into account on the effect on foam cup molding, the method can be enforced in present design development and production by bra cup manufacturers. Based on the simulation and experimental results, this approach tries to strike a balance amongst low cost, high efficiency and good accuracy. It has proven not only to be scientific and reasonable, but also quick and efficient. Future work will involve the molecular structure of materials and thermal transfer mechanisms during the molding process. Fig. 5. Normal probability plot of residuals for shape conformity of molded foam cups.
Acknowledgement depth are 204.0 °C and 125.4 s, respectively, with a corresponding degree of shape conformity at 77.80%. The final settings for the optimal process parameters are shown in Table 5. These settings correspond to the maximum shape conformity at a given cup depth in the mold heads. 5. Verification test A normal probability plot of the residuals is depicted in Fig. 5, which reveals that the residuals generally fall on a least-square line which is used to estimate the cumulative distribution function for the population. As evident from the figure, the errors are normally distributed and there are almost no serious violations of the assumptions that underlie the analysis (Kilickap, 2010). By displaying a satisfactory normal distribution, the normality assumptions made earlier could be confirmed and the predictive regression model has extracted all information available from the experimental data (Joseph Davidson, Balasubramanian, & Tagore, 2008). 6. Conclusions Determination of the optimal process parameter settings is critical work in the manufacturing industry which influences productivity, quality, production costs and time for delivery. By applying the BBD method, an effective approach for process parameter optimization in foam cup molding with a minimal number of experimental runs has been presented in this paper. The conclusions of the present study are as follows: 1. This study demonstrates the optimization of molding conditions (viz., molding temperature and length of dwell time) for corresponding foam material and cup depths of mold heads, providing an objective evaluation approach that not only optimizes the foam cup molding process, but also facilitates communication between manufacturers and clients; 2. A Box–Behnken experimental design is utilized for optimization, which replaces the traditional DOE. The optimum sets of molding temperatures and dwell times are graphically obtained at different cup depths of mold heads. The experimental shape conformity closely agrees with the predicted values; 3. The ANOVA results reveal that molding temperature is the main parameter, which has a greater influence than the dwell time on
The authors would like to thank the Research Grant Council for funding this research through project account PolyU RPGV.
References Adinarayana, K., & Ellaiah, P. (2002). Response surface optimization of the critical medium components for the production of alkaline protease by a newly isolated Bacillus sp.. Journal of Pharmacy and Pharmaceutical Sciences, 5(3), 272–276. Beauregard, M. R., Mikulak, R. J., & Olson, B. A. (2000). Experimenting for breakthrough improvement: The step-by-step guide for harnessing the power of experimental design. Waitsfield, VT: Quality Training Portal. Bicerano, J. (2002). Prediction of polymer properties (3rd ed.). New York: Marcel Dekker Inc. Box, G. E. P., & Wilson, K. B. (1951). On the experimental attainment of optimum conditions. Journal of the Royal Statistical Society, 13(1), 1–45. Chang, H. W. (2008). A data mining approach to dynamic multiple responses in Taguchi experimental design. Expert Systems with Applications, 35(3), 1095–1103. Chong, R. (2010). Molding & lamination technology handbook for lingerie industry. Hong Kong: Hong Kong Productivity Council. Dong, C. H., Xie, X. Q., Wang, X. L., Zhan, Y., & Yao, Y. J. (2009). Application of Box– Behnken design in optimisation for polysaccharides extraction from cultured mycelium of Cordyceps sinensis. Food and Bioproducts Processing, 87(2), 139–144. Ferreira, S. L. C., Bruns, R. E., Ferreira, H. S., Matos, G. D., David, J. M., Brandão, G. C., et al. (2007). Box–Behnken design: An alternative for the optimization of analytical methods. Analytica Chimica Acta, 597(2), 179–186. Fowlkes, W. Y., & Creveling, C. M. (1995). Engineering methods for robust product design: Using Taguchi methods in technology and product development. New Jersey: Prentice Hall. Haycock, C. E. (1978). Breast support and protection in female athlete. In AAHPERD symposium papers (Vol. 1(2), pp. 50–53). Hinkelmann, K., & Kempthorne, O. (2007). Design and analysis of experiments: Introduction to experimental design (2nd ed.). New Jersey: John Wiley & Sons, Inc. Hou, T. H., Su, C. H., & Liu, W. L. (2007). Parameters optimization of a nano-particle wet milling process using the Taguchi method, response surface method and genetic algorithm. Powder Technology, 173(3), 153–162. Joseph Davidson, M., Balasubramanian, K., & Tagore, G. R. N. (2008). Surface roughness prediction of flow-formed AA6061 alloy by design of experiments. Journal of Materials Processing Technology, 202(1–3), 41–46. Khuri, A. I. (2006). Response surface methodology and related topics. New Jersey: World Scientific Publishing Co. Pte. Ltd. Kilickap, E. (2010). Optimization of cutting parameters on delamination based on Taguchi method during drilling of GFRP composite. Expert Systems with Applications, 37(8), 6116–6122. Montgomery, D. C. (2005). Design and analysis of experiments (6th ed.). USA: John wiley & Sons, Inc. Myers, R. H., & Montgomery, D. C. (2002). Response surface methodology: Process and product optimization using designed experiments (2nd ed.). USA: John wiley & Sons, Inc. Ragonese, R., Macka, M., Hughes, J., & Petocz, P. (2002). The use of the Box–Behnken experimental design in the optimisation and robustness testing of a capillary electrophoresis method for the analysis of ethambutol hydrochloride in a
L. Wu et al. / Expert Systems with Applications 39 (2012) 8059–8065 pharmaceutical formulation. Journal of Pharmaceutical and Biomedical Analysis, 27(6), 995–1007. Sharma, P., Singh, L., & Dilbaghi, N. (2009). Optimization of process variables for decolorization of Disperse Yellow 211 by Bacillus subtilis using Box–Behnken design. Journal of Hazardous Materials, 164(2–3), 1024–1029. Sun, Y., Li, T., Yan, J., & Liu, J. (2010). Technology optimization for polysaccharides (POP) extraction from the fruiting bodies of Pleurotus ostreatus by Box– Behnken statistical design. Carbohydrate Polymers, 80(1), 242–247. Tong, L. I., Chang, Y. C., & Lin, S. H. (2011). Determining the optimal re-sampling strategy for a classification model with imbalanced data using design of experiments and response surface methodologies. Expert Systems with Applications, 38(4), 4222–4227. Wu, D., Zhou, J., & Li, Y. (2009). Effect of the sulfidation process on the mechanical properties of a CoMoP/Al2O3 hydrotreating catalyst. Chemical Engineering Science, 64(2), 198–206.
8065
Yick, K. L., Ng, S., Zhou, X., Yu, W., & Chan, D. (2008). Wire frame representation of 3D moulded bra cup and its application to example-based design. Fibers and Polymers, 9(5), 653–658. Yick, K. L., Wu, L., Yip, J., Ng, S.-p., & Yu, W. (2010). Study of thermal-mechanical properties of polyurethane foam and the three-dimensional shape of molded bra cups. Journal of Materials Processing Technology, 210(1), 116–121. Yu, W. (1996). The effects of polyurethane foam properties and moulding conditions on the shape characteristics of brassiere cups. UK: The University of Leeds. Yu, W., Fan, J., Harlock, S. C., & Ng, S. P. (2006). Innovation and technology of women’s intimate apparel. England: Woodhead Publishing Ltd. Yu, W. M., Yeung, K. W., Harlock, S. C., & Leaf, G. A. V. (1998). Predicting energy requirements for moulding flexible polyurethane foam sheets. Textile Research Journal, 68(4), 289–295.