23 CHAPTER 3 AN OVERVIEW OF DESIGN OF EXPERIMENTS AND RESPONSE SURFACE METHODOLOGY 3.1 DESIGN OF EXPERIMENTS Design of experiments is a systematic approach for investigation of a system or process. A series of structured tests are designed in which planned changes are made to the input variables of a process or system. The effects of these changes on a pre-defined output are then assessed. DOE is a technique to lay out experimental research studies plan in the most logical, economical and statistical way. Through this technique, researchers can determine the most desirable design of product, the best parameters combination for the required process, the most robust recipe for formulation, the most critical validation/durability test conditions and the most effective data collection plan. DOE consists of a set of experimental runs, in which each run is defined by the combination of each factor level (variables) and analysis of experiments. DOE helps to make product and processes more robust. It is a proven technique that continues to show the increasing usage in manufacturing, and chemical process industries especially for fast, cost saving and accurate results. Historically, DOE technique was first developed by Fisher in the 1920 s to study the effect of multiple variables simultaneously (Dowey and
24 Matthews, 1998). DOE technique now has become a very useful statistical tool to help us understand process characteristics and to investigate how inputs affect responses based on statistical backgrounds. In addition, it has been used to systematically determine the optimal process parameters with fewer testing trials (Park and Ahn 2004). The order of tasks using this tool starts with identifying the input variables and the response (output) that is to be measured. For each input variable, a number of levels are defined, that represent the range for which the effect of that variable is desired to be known. An experimental plan is prepared which describes the experimenter to set each test parameter for each run of the test. The response is then measured for each run. The method of analysis is carried out to look for the differences between response (output) readings for different groups of the input changes. These differences are then attributed to the input variables acting alone (called a single effect) or in combination with another input variable (called an interaction). DOE is team oriented and a variety of backgrounds (e.g. design, manufacturing, statistics and so on) should be involved, when identifying factors and levels and developing the matrix, as this is the most skilled part. Moreover, as this tool is used to answer specific questions, the team should have a clear understanding of the difference between control and noise factors. It is very important to get more information from each experiment performed. Well-designed experiments can produce significantly more information and often require fewer runs than haphazard or unplanned experiments. For example, if there is an interaction between two input variables, both variables should be included in the design rather than doing a ''one factor at a time'' experiment. An interaction occurs when the effect of one input variable is influenced by the level of another input variable. Designed experiments are often carried out in four phases: planning, screening, optimization, and verification.
25 3.1.1 Analysis of Variance Analysis of variance is a general method for studying sampled-data relationships (Clarke and Cooke 1998). This method gives the difference between two or more sample means to be analyzed, achieved by subdividing the total sum of squares. In one way, ANOVA is the simplest case. The purpose is to test significant differences between class means, and this is done by analyzing the variances. ANOVA is similar to regression, in that, it is used to investigate and model the relationship between a response variable and one or more independent variables. In effect, analysis of variance extends the twosample t-test for testing the equality of two population means to a more general null hypothesis (Draper and Smith 1982). 3.2 RESPONSE SURFACE METHODOLOGY Optimization tools are obviously beneficial to designers, but all such tools require the collection of large volumes of performance data. Simulations that provide a detailed analysis of a potential design s performance are time consuming and computationally intensive. Running a full scale simulation of every possible design in order to find the optimal one is completely infeasible when there are billions or trillions of potential component configurations. The modeling and simulation research community has developed the family of techniques called response surface methodology to address exactly such situations. Response surface methodology is a combination of experimental, regression analysis and statistical inferences. The concept of a response surface method involves a dependent variable "y" called the response variable and several independent variables x 1, x 2,,x k (Hicks 1993). If all of these
26 variables are assumed to be measurable, the response surface can be expressed as y =f (x 1,x 2,x 3,...x k ) (3.1) The goal is to optimize the response variable "y". The response or the dependent variable is assumed to be random variable. It is assumed that the independent variables (x 1, x 2 x k ) are continuous and controllable by the experimenter with negligible error. Hence the model can be written as y =f (x 1,x 2,...,x k)+ε (3.2) where "ε" is a random error. If the expected response is denoted by E(y) = η, the surface represented by η = f (x 1, x 2, x 3 ) is called as response surface. It is required to find a suitable approximation for the true functional relationship between "y" and the set of independent variables "x i " s. Usually a second order model is utilized in response surface methodology (Montgomery 1997), and is given by Equation (3.3). k k 2 0 i i ii i i j i j i= 1 i= 1 i j for i < j (3.3) y = β + β x + β x + β x x + ε of least squares. The "β" parameters of the polynomials are estimated by the method 3.2.1 Popular Designs of RSM Response surface method is created from factorial design and there are two categories of quadratic factorial designs, namely Box-Behnken design and central composite design.
27 The Box-Behnken design is a spherical design with all points lying on a sphere of radius 2 shown in Figure 3.1. Also this does not contain any points at the vertices of the cubic region created by the upper and lower limits for each variable. This could be advantageous, when the points on the corners of the cube represent factor level combinations that are prohibitively expensive or impossible to test because of physical constraints. Figure 3.1 Box-Behnken design The central composite design is an efficient and the most popular class of design for fitting the second order model and it is shown in Figure 3.2. There are two parameters in the design that must be specified: the distance "α" of the axial runs from the design centre and the number of centre points. For a spherical region of interest, the best choice of α from a prediction variance viewpoint is to set α = k. Three to five runs are recommended for reasonably stable variance of predicted response. CCD contains an imbedded factorial or fractional factorial design with center points that is augmented with a group of ''star points'' that allow estimation of curvature. Due to the limited capability of orthogonal blocking of Box- Behnken design, central composite design is employed in this study.
28 Figure 3.2 Central composite design 3.2.2 Individual Terms Used in RSM The various terms for model estimation processes are listed below. predictive model. Factor: Experimental variables selected for inclusion in the Mean: The sum of squares associated with the mean of Y. It is presented since it is the amount used to adjust the other sums of squares. Model: The sum of squares associated with the model. of squares. Model (Adjusted): The model sum of squares minus the mean sum Error: The sum of the squared residuals. This is often called the sum of squares error. Total: The sum of the squared Y values.
29 Total (Adjusted): The sum of the squared Y values minus the mean sum of squares. Mean Square: The sum of squares divided by the degrees of freedom. The mean square for error is an estimate of the underlying variation in the data. Sum of Squares: Sum of the squared differences between the average values for the blocks and the overall mean. DF: Degrees of freedom attributed to the blocks, generally equal to one less than the number of blocks. Mean Square: Estimate of the block variance, calculated by the block sum of squares divided by block degrees of freedom. Block: Removes any variation attributed to the blocks prior to computing the ANOVA for the factor effects. F Value: Test for comparing model variance with residual (error) variance. If the variances are close to the same, the ratio will be close to one and it is less likely that any of the factors have a significant effect on the response. Prob > F: The F distribution itself is determined by the degrees of freedom associated with the variances being compared. If the Prob>F value is very small (less than 0.05) then the terms in the model have a significant effect on the response. Residual: Consists of terms used to estimate experimental error
30 Lack of Fit (LOF): This is the variation of the data around the fitted model. If the model does not fit the data well, this will be significant. design points. Pure Error: Amount of variation in the response in replicated Cor. Total: Totals of all information corrected for the mean. Std Dev: (Root MSE) Square root of the residual mean square. C.V.: Coefficient of Variation. The standard deviation expressed as percentage of the mean. PRESS: Predicted Residual Error Sum of Squares A measure of how the model fits each point in the design. Adj R-Squared: A measure of the amount of variation around the mean explained by the model, adjusted for the number of terms in the model. Pred R-Squared: A measure of the amount of variation in new data explained by the model. The predicted R-squared and the adjusted R- squared should be within 0.20 to each other. Otherwise there may be a problem with either the data or the model. Adequate Precision: This is a signal to noise ratio. It compares the range of the predicted values at the design points to the average prediction error. Ratios greater than 4 indicate adequate model discrimination. 95% CI High and Low: Represent the range that the true coefficient should be found in 95% of the time. If this range spans 0 (one limit is positive and the other negative) then the coefficient of 0 could be true, indicating that the factor has no effect.
31 VIF: Variance Inflation Factor Measures how much the variance of the model is inflated by the lack of orthogonality in the design. If the factor is orthogonal to all other factors in the model, the VIF is one. 3.2.3 Procedural Steps of Design Expert Software Design expert software is used to analyze the data collected. The procedural steps are given below: (i) A transformation is chosen, if desired. Otherwise, the option is "None". (ii) The appropriate model to be used is selected. The Fit Summary button displays the sequential F-tests, lack-of-fit tests and other adequacy measures that could be used to assist in selecting the appropriate model. (iii) The analysis of variance, post-anova analysis of individual model coefficients and case statistics for analysis of residuals are to be performed. (iv) Various diagnostic plots are inspected to statistically validate the model. (v) If the model looks good, model graphs are generated, i.e. the contour and 3D graphs, for interpretation. The analysis and inspection performed in steps (3) and (4) show, whether the model is good or otherwise. (vi) Optimization is carried out both numerically and graphically. (vii) The propagation of error method is applied to find the settings that minimize variation in the response. (viii) The fraction of the design space is plotted. (ix) Confirmation experiments are run.
32 3.2.3.1 RSM analysis A good model must be significant and the lack-of-fit must be insignificant. The various coefficient of determination, R 2 values should be close to 1. The diagnostic plots should also exhibit trends associated with a good model and these are elaborated subsequently. After analyzing each response, multiple response optimization is performed, either by inspecting the interpretation plots, or with the graphical and numerical tools provided for this purpose. It is mentioned previously that RSM designs also help in quantifying the relationships between one or more measured responses and the vital input factors. In order to determine that there exist a relationship between the factors and the response variables investigated, the data collected must be analyzed in a statistically sound manner using regression. A regression is performed in order to describe the data collected, whereby an observed, empirical variable (response) is approximated based on a functional relationship between the estimated variable, "Y est " and one or more regressor or input variable x 1, x 2,..., x i. In case, there exist a non-linear relationship between a particular response and three input variables, a quadratic Equation (3.4) Y = b + b x + b x + b x + b x x + b x x + b x x + b x + b x + b x + error 2 2 2 est 0 1 1 2 2 3 3 4 1 2 5 1 3 6 2 3 7 1 8 2 9 3 (3.4) may be used to describe the functional relationship between the estimated variable, "Y est " and the input variables x 1, x 2 and x 3. The least square technique is being used to fit a model equation containing the said regressors or input variables by minimizing the residual error measured by the sum of square of deviations between the actual and the estimated responses. This involves the calculation of estimates for the regression coefficients, i.e. the coefficients of the model variables including the intercept or constant term. The calculated
33 coefficients or the model equation need to however be tested for statistical significance. In this respect, the following tests are performed. (a) Test for significance of the regression model This test is performed as an ANOVA procedure by calculating the F-ratio. This ratio is used to measure the significance of the model under investigation with respect to the variance of all the terms included in the error term at the desired significance level, "α". A significant model is desired. (b) Test for significance on individual model coefficients This test forms the basis for model optimization by adding or deleting coefficients through backward elimination, forward addition or stepwise elimination/addition/exchange. It involves the determination of the P-value or probability value, usually relating the risk of falsely rejecting a given hypothesis. The "Prob. > F" value determined can be compared with the desired probability or α-level. In general, the lowest order polynomial would be chosen to adequately describe the system. (c) Test for lack-of-fit As replicate measurements are available, a test indicating the significance of the replicate error in comparison to the model dependent error can be performed. This test splits the residual or error sum of squares into two portions, one is due to pure error, which is based on the replicate measurements and the other is due to lack-of-fit based on the model performance. The test statistic for lack-of-fit is the ratio between the lack-offit mean square and the pure error mean square. As previously, this F-test statistic can be used to determine whether the lack-of-fit error is significant or otherwise at the desired significance level, "α". Insignificant lack-of-fit is
34 desired as significant lack-of-fit indicates that there might be contributions in the regressor response relationship that are not accounted for by the model. Additionally, checks need to be made in order to determine whether the model actually describes the experimental data. The checks performed here include the determination of the various coefficient, "R 2 ". These "R 2 " coefficients have values between 0 and 1. In addition to the above, the adequacy of the model is also investigated by the examination of residuals. The residuals, the difference between the respective, observed responses and the predicted responses are examined using the normal probability plots of the residuals and the plots of the residuals versus the predicted response. If the model is adequate, the points on the normal probability plots of the residuals should form a straight line. 3.2.4 Numerical Optimization Numerical optimization will optimize any combination of one or more goals. The goals may apply to either factors or responses. The desired goal for each factor and response is selected from the menu. The possible goals are: maximize, minimize, target, within range, none (for responses only) and set to an exact value (factors only.) If a response is transformed, the optimization will use either the original or transformed scale, as chosen by the current setting of the display options menu. A minimum and a maximum level must be provided for each parameter included. A weight can be assigned to each goal to adjust the shape of its particular desirability function. The "importance" of each goal can be changed in relation to the other goals. The default is for all goals to be equally important at a setting of 3 pluses (+++). The goals are combined into an overall desirability function. The program seeks to maximize this function. The goal seeking begins at a random starting point and proceeds up the
35 steepest slope to a maximum. There may be two or more maximums because of curvature in the response surfaces and their combination into the desirability function. 3.2.5 Graphical Optimization With multiple responses, it is required to find the regions where requirements simultaneously meet the critical properties, the "sweet spot". By superimposing or overlaying critical response contours on a contour plot, the best compromise can be visualized. Graphical optimization displays the area of feasible response values in the factor space. Regions that do not fit the optimization criteria are shaded. Any "window" that is NOT shaded satisfies the multiple constraints of the responses. The area that satisfies the constraints will be light grey, while the area that does NOT meet the criteria is dark grey. The graphical optimization allows visual selection of the optimum machining conditions according to certain criterion. The result of the graphical optimization is the overlay plot. These types of plots are extremely practical for quick technical use in the workshop to choose the values of the machining parameters that would achieve a certain response value for this type of materials. 3.2.6 Desirability Approach Myers and Montgomery (2003) describe a multiple response method called desirability. The desirability function approach is one of the most widely used methods in industry for the optimization of multiple response processes. It is based on the idea that the "quality" of a product or process that has multiple quality characteristics, with one of them outside of some "desired" limits, is completely unacceptable. The method finds operating conditions "x" that provide the "most desirable" response values.
36 The goal of optimization is to find a good set of conditions that will meet all the goals, not to get a desirability value of 1.0. For each response Y i (x), a desirability function di (Y i ) assigns numbers between 0 and 1 to the possible values of "Y i ", with d i (Y i ) = 0 representing a completely undesirable value of "Y i " and d i (Y i ) = 1 representing a completely desirable or ideal response value (Myers et al 2004). The individual desirability is then combined using the geometric mean, which gives the overall desirability D in Equation (3.5) 1/n n 1/n D = ( d1 d 2...d n ) = di i 1 (3.5) where "n" is the number of responses in the measure. If any of the responses or factors falls outside their desirability range, the overall function becomes zero. Depending on whether a particular response "Y i " is to be maximized, minimized, or assigned a target value, different desirability functions d i (Y i ) can be used. The shape of the desirability for each goal can be changed by the "weight" field. Weights are used to give added emphasis to the upper/lower bounds, or to emphasize the target value. With a weight of 1 the d i will vary from 0 to 1 in a linear fashion. Weights greater than 1 (maximum weight is 10), give more emphasis to the goal. Weights less than 1 (minimum weight is 0.1), give less emphasis to the goal. In the desirability objective function D(X), each response can be assigned with an important relative to the other responses. Importance (r i ) varies from the least important (+) a value of 1, to the most important (+++++) a value of 5. 3.2.7 Point Prediction The final step in any experiment is to predict the response at the optimal settings. Confirmation runs are done to verify the prediction. Point
37 prediction allows various levels for each factor or component into the current model. The software then calculates the expected responses and associated confidence intervals based on the prediction equation, which is shown in the ANOVA output. The predicted values are updated as the levels are changed. The prediction interval will be larger (a wider spread) than the confidence interval since more scatter in individual values than in averages are expected. 3.2.8 Propagation of Error The propagation of error method finds settings that minimize variation in the response. It makes the process or product more robust to variations in input factors. In essence, the POE method involves application of partial derivatives to locate flat areas on the response surface, preferably high plateaus. The POE technique provides more accurate results when the factors have lower standard deviation than when they have higher standard deviations. After the construction of response surface models for the responses, The standard deviations for one or more input factors are entered. These standard deviations can be entered on the column information sheet located under the view menu. Using the factor standard deviation information, it constructs a response surface map of the factor variation transmitted to the response. Finally, the multiple response optimizations are used to find factor settings that get the response on target with minimal variation (setting the goal for POE to minimize).
38 Lower the POE, the response is better. Less error in control factors will be transmitted to the selected responses. This results in a more robust process. However, POE will only work when the response surface is nonlinear. When the surface is linear, the error will be transmitted equally throughout the region. 3.3 SUMMARY In this chapter, design of experiments and response surface methodology are discussed. The various methods used for optimization are reported. The modeling, optimization and selection techniques have been discussed as an overview. Experimental methodology is explained in detail in the next chapter.