Taguchi Method and Robust Design: Tutorial and Guideline

Taguchi Method and Robust Design: Tutorial and Guideline CONTENT 1. Introduction 2. Microsoft Excel: graphing 3. Microsoft Excel: Regression 4. Microsoft Excel: Variance analysis 5. Robust Design: An Example Reference Appendix A 1

1. INTRODUCTION Taguchi method is also known as quality Engineering. The objective of quality engineering is to choose from all possible designs the one that can ensure the highest functional robustness of products at the lowest possible cost. Taguchi method involves a three-step approach: i.e., system design, parameter design, and tolerance design. System design is the process of applying basic scientific and engineering principles in order to develop a functional design. Parameter design is the investigation conducted in order to identify settings that minimize or reduce the performance variation in the product or process. Tolerance design is a method for determining tolerances that minimize the sum of product manufacturing and lifetime cost. If the parameter design cannot achieve the required performance variation, tolerance design can be used to reduce the variation by reducing the tolerances based on the quality loss function. Robust design is the operation of choosing settings for product or process parameters to reduce variation of that product or process s response from target. Because it involves determination of parameter settings, robust design is called parameter design. In order to design a system so that its performance is insensitive to uncontrollable (noise) variables, one needs to systematically investigate the relationship between appropriate control factors and noise variables, typically through off-line experiments, and judiciously choose the settings of the control factors to make the system robust to uncontrollable noise variation. Thus, the implementation of the robust design method includes the following operational steps: 1. state the problem and objective. 2. identify responses, control factors, and sources of noise. 2

3. plan an experiment to study the relationships between responses and control and noise factors. 4. run the experiment and collect the data. Analyze the data to determine the control factor settings that predict improvement on the product or process design. 5. run a small experiment to confirm if the control factor settings determined in step 4 actually improve the product or process design. If so, adopt the control factor settings and consider another iteration for further improvement. If not, correct or modify the assumptions and go back to step 2. This lab deals with relevant aspects in step 4, i.e., data analysis using Microsoft Excel. We will deal with graphing in Section 2. Section 3 deals with regression. Section 4 deals with analysis of variance. Finally, a numerical example is given in Section 5. 2. MICROSOFT EXCEL: GRAPHING Excel is a spreadsheet program. By clicking Microsoft Excel, a blank worksheet will appear on screen. A worksheet is a grid of columns and rows. The intersection of any column and row is called a cell. Each cell in a worksheet has a unique cell reference, the designation formed by combining the row and column headings. For example, B6 refers to the cell at the intersection of column B and row 6. The cell point is a white cross-shaped pointer that appears over cells in the worksheet. You use the cell pointer to select any cell in the worksheet. The selected cell is called the active cell. You always have at least one cell selected at all times. A range is a specified group of cells. A range can be a single cell, a column, a row, or any combination of cells, columns, and rows. Range coordinates identify a range. The first element in the range coordinates is the location of the upper left cell in the range; the second element is the location of the lower-right cell. A colon (:) separates these two elements. For example, the range B6:D8 includes the cells B6, B7, B8, C6, C7, C8, D6, D7, and D8. 3

With the Excel, you can create a chart based on data in a worksheet. The axes are the grid on which the data is plotted. On a 2D chart, the y-axis is the vertical axis on a chart (value axis), and the x-axis is the horizontal axis (category axis). A 3D chart has three (add a z- axis). Example 1: Draw x-y curvs of the following data: x =: 8, 9, 10, 12, 15, 18, 20, 25, 30, 35; y =: 25, 26.5, 28, 33, 36, 36.5, 36, 32.5, 26, 21. Step 1: Sequentially input x values to A1 through A10, and input y values to B1 through B10. Step 2: Select the range A1:B10. Step 3: Click Chart Wizard, and then hold the mouse s left button and move on screen from the upper-left to the lower-right. This forms the region of the chart. Click Next. Step 4: Select XY (Scatter) or what you want and click Next. Step 5: Select the format No. 2 or what you want and click Next. Step 6: Input Chart title, title of category (x) axis, and title of value (y) axis. Click Finish. Step 7: Save the file as Lab-1. The curve is shown in Figure 1. 4

Figure 1: x-y curve y 40 35 30 25 20 15 10 5 Series1 0 0 10 20 30 40 x 3. REGRESSION Example 2: Fit the data in Example 1 into a function: y = f(x). Step 1: Select a fitting model. According to the shape of the curves shown in Figure 1, the following model may be appropriate: 2 y a bx cx e where the parameters a, b, and c are the constants to be determined by regression, and e is a random deviation which is assumed to be normally distributed with mean = 0 and standard deviation. Step 2: Linearize the above model by the following transformations: 5

Thus, the model can be rewritten as x x, x x 1 2 y a bx1 cx2 e 2 Step 3: Generate data. Now, open the file Lab-1. We use the column D as x 1, the column E as x 2 and the column F as y. Select D1 and type =A1 ; select E1 and type =A1^2 ; select F1 and type =B1. Select D1:F1 and hold the left button of the mouse and move to the row 10. This completes the input of the data. Step 4: Regression. Click Tools, click Data analysis and regression. Type F1:F10 to Input Y range and Type D1:E10 to Input X range. Press Enter. The result is shown in Table 1. For the convenience of understanding, we introduce the following definitions and notations: Residual Sum of Squares (SSr): ( y y ) 2, where y is observed value and y is predicted value. Total Sum of Squares (SSt): ( y y) 2, where y is the mean of the y observations in the sample. The coefficient of multiple determination R 2 : = 1 - SSr/SSt. Adjusted R 2 : = 1 - [(n-1)/(n-k)]ssr/sst. Random deviation variance: 2 SSr / ( n k ). Where n is sample size and k is the number of estimated constants in the model. 6

Table 1: Output of the regression SUMMARY OUTPUT Regression Statistics Multiple R 0.962769656 R Square 0.926925411 Adjusted R 0.906046956 Square Standard Error 1.671778115 Observations 10 ANOVA df SS MS F Significance F Regression 2 248.1611056 124.0805528 44.39626638 0.000105483 Residual 7 19.56389445 2.794842064 Total 9 267.725 Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Lower 95.0% Upper 95.0% Intercept 7.313500551 3.029630589 2.41399086 0.046500657 0.149567712 14.4774334 0.14956771 14.477433 X Variable 1 2.880994396 0.336804616 8.553904132 5.93012E-05 2.084578603 3.67741019 2.0845786 3.6774102 X Variable 2-0.072645789 0.007964304-9.12142385 3.90931E-05-0.091478361-0.0538132-0.0914784-0.0538132 Step 5: Translate the result. The Multiple R in regression statistics reflects how good the fitting is. The greater it is, the better. In this example, it equals 0.9628. It is close to 1. This implies that the fitting model is not too bad. The Coefficient corresponding to Intercept is a, the Coefficient corresponding to X Variable 1 is b, and the Coefficient corresponding to X Variable 2 is c. They are 7.31350, 2.88099, and -0.072646, respectively. Now, we draw this fitting curve and the data curve together. 7

Open the file Lab-1. Let A11 = 6, A12 = A11+1,, A41 = A40+1. Select C11 and type =7.3135+2.88099*A11-0.07265*A11^2. Select C11, hold the left button of the mouse and move to the row C41. This completes the calculation of y. Now, select A1:C41 and repeat those steps presented in Example 1. Figure 2 shows the fitting curve together with the data curve. As can be seen, the fitting is not very ideal. Therefore, we can try the following model: y ax b e cx where the parameters a, b, and c are the constants to be determined by regression. It can be linearized by the following transformations: Thus, the model can be rewritten as z ln( y), A ln( a), x ln( x), x x 1 2 z A bx 1 cx 2 Similarly, we have the regression result shown in Table 2: 8

Table 2: Output of the regression SUMMARY OUTPUT Regression Statistics Multiple R 0.993972276 R Square 0.987980886 Adjusted R 0.984546853 Square Standard Error 0.023425092 Observations 10 ANOVA df SS MS F Significance F Regression 2 0.315745171 0.157872585 287.7028229 1.9035E-07 Residual 7 0.003841144 0.000548735 Total 9 0.319586315 Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Lower 95.0% Upper 95.0% Intercept 0.341692322 0.13804421 2.475238351 0.042505533 0.015269869 0.6681148 0.0152699 0.6681148 X Variable 1 1.75652854 0.076387431 22.99499435 7.45711E-08 1.575901098 1.937156 1.5759011 1.937156 X Variable 2-0.101289717 0.004239343-23.89278486 5.72056E-08-0.111314163-0.091265-0.111314-0.091265 At this time, multiple R = 0.9940. This implies that this model is much better than the previous model. The model parameters are as follows: a = 1.40733, b=1.75653, c = -0.10129 Figure 3 shows the fitting curve together with the data curve. As can be seen, the fitting is very good. 9

Figure 3: Fitting curve y 40 35 30 25 20 15 10 5 Series1 Series2 0 0 10 20 30 40 x 4. ANALYSIS OF VARIANCE (ANOVA) Many investigations involve a comparison of several population means. A single-factor analysis of variance problem involves a comparison of k population means. The objective is to test if or not these means are equal. The analysis of variance method analyzes variability in the data to see how much can be attributed to between-group differences and how much is due to within-group variability. Notation: Sample size: n i Sample mean: x x / n i 2 2 Sample variance: si ( xij xi ) / ( ni 1) j ij j i 10

Total number of observations: N Grant mean: x xij / N j i n i i Example 3: Suppose that three different teaching methods are used to teach a course to three groups of students. Then we have their scores on the final examination. Based on the scores, we can analyze the effectiveness of each teaching method by Analysis of Variance technique. The scores are shown in Table 3: Table 3: Data for Example 3 Method 1 Method 2 method 3 Student 1 57 64 68 Student 2 75 71 75 Student 3 98 79 50 Student 4 61 45 53 Student 5 84 50 61 Student 6 40 74 Now, input the five data under Method 1 into A1 through A5; input the six data under Method 2 into B1 through B6; and input the six data under Method 3 into C1 through C6. Click Tools. Then click data analysis and select Anova: single factor. Input A1:C6 as Input range and then click ok. The result is shown in Table 4. It includes two parts: the first part is the summary and the second is ANOVA. Definition or notation: 2 Mean square (MS) for groups: MS1 ( xi x) ni / ( k 1). It reflects between-group variation. Df: degrees of freedom. SS: sum of squares. i P-value: it is the smallest level of significance at which the hypotheses can be rejected. 2 Mean square for error: MS2 si ( ni 1) / ( N k). It reflects within-group variation. i F ratio: F MS1 / MS2. If F > F critical value, then it implies that there are great differences between the means of the groups. One cannot think that the means are equal. 11

Where F critical value is taken from the F distribution table based on numerator and denominator degrees of freedom and a significance level. In our example, MS1= 396.8333, and MS2 = 206.7381, so F = 1.919498. F critical value = 3.73889. So, we cannot think that there are big differences between the means at a significance level of 0.05. Anova: Single Factor Table 4: Output of the ANOVA SUMMAR Y Groups Count Sum Average Variance Column 1 5 375 75 282.5 Column 2 6 349 58.16667 240.5667 Column 3 6 381 63.5 112.3 ANOVA Source of Variation Between Groups Within Groups SS df MS F P-value F crit 793.6667 2 396.8333 1.919498 0.18336 3.73889 2894.333 14 206.7381 Total 3688 16 Example 4: Two-factor ANOVA. An investigator will often be interested in assessing the effects of two different factors A and B on a response variable. There are several levels corresponding to each factor. Suppose that an experiment is carried out, resulting in a data set that contains some number of observations for each combination of the factor levels, see Table 5. Here, we can have 3+4 = 7 sample average (or variance) and 1 grant average. These are given in the first part of ANOVA, see Table 6. We can view each column as a group, then we can have a single-factor ANOVA. Similarly, we can view 12

each row as a group, then we can have another single-factor ANOVA. These are shown in the second part. Table 5: Data for Example 4 B1 B2 B3 B4 A1 9.2 12.43 12.9 10.8 A2 8.93 12.63 14.5 12.77 A3 16.3 18.1 19.93 18.17 Anova: Two-Factor Without Replication Table 6: Output of the ANOVA SUMMAR Y Count Sum Average Variance Row 1 4 45.33 11.3325 2.830892 Row 2 4 48.83 12.2075 5.497492 Row 3 4 72.5 18.125 2.1971 Column 1 3 34.43 11.47667 17.46663 Column 2 3 43.16 14.38667 10.35163 Column 3 3 47.33 15.77667 13.57763 Column 4 3 41.74 13.91333 14.55963 ANOVA Source of Variation SS df MS F P-value F crit Rows 109.2273 2 54.61366 122.0985 1.38E-05 5.143249 Columns 28.8927 3 9.6309 21.53159 0.001299 4.757055 Error 2.68375 6 0.447292 Total 140.8038 11 5. ROBUST DESIGN: AN EXAMPLE 13

Example 5: The example is taken from Reference [9]. In this example, there are 8 control factors, each at two levels. we denote the two levels as -1 and 1 although they correspond to different specific values for different factor. The control array is a 2 8 4 fractional factorial design, see Table 7., 16-run, Table 7: Data for Example 5-1 -1-1 -1-1 -1-1 -1 1-1 -1-1 -1 1 1 1-1 1-1 -1 1-1 1 1 1 1-1 -1 1 1-1 -1-1 -1 1-1 1 1 1-1 1-1 1-1 1-1 -1 1-1 1 1-1 -1 1-1 1 1 1 1-1 -1-1 1-1 -1-1 -1 1 1 1-1 1 1-1 -1 1 1-1 1-1 -1 1-1 1-1 1 1-1 1 1-1 1-1 -1-1 1-1 -1 1 1-1 -1 1 1 1-1 1 1-1 1-1 -1-1 1 1 1 1-1 -1-1 1 1 1 1 1 1 1 1 M1=100 M1=100 M2=200 M2=200 M3=300 M3=300 N1 N2 N1 N2 N1 N2 1 119.2 123.8 239.9 244.4 359.8 365.4 2 155.6 164.2 314.2 322.7 471.6 482.1 3 129 136.1 261.7 267.6 392.9 400.1 4 168.2 176.2 339.7 348 510.1 519.9 5 142.5 150.8 289.4 297 434.5 444 6 160 168.7 323.7 331.2 486.4 496 7 142.6 149.8 288.5 294.8 433 441 8 151.6 160.3 307 314.6 460.3 470.4 9 165.3 174.8 335.6 343.7 503.8 514.2 10 186 199.4 378.2 390.9 568.4 584.5 11 141.8 149.2 287.6 294.2 430.9 439.7 12 184.9 195.4 373.7 383.8 561.2 574.1 13 148.6 157.5 302.3 310.4 453.3 463.5 14 180.6 191 365.9 375.8 549.5 562 14

15 154.9 163.5 314.3 322 471.8 480.7 16 182.8 196.5 371.7 384.3 558.9 574.2 The second part of the table is the outer array which consists of 3 levels of a signal factor (M1 = 100, M2 = 200, M3 = 300) crossed with two levels of a noise factor for a total of 6 runs. Thus, we have 16 6 = 96 observations. Let Y ijk denote the observation corresponding to the i-th setting of the control factors, j-th setting of the signal factor, and k-th setting of the noise factors. Under the assumption of a linear ideal function with no intercept (i.e. no constant term), we have the model Y M ijk i j ijk where i, the sensitivity measure, and 2 var( ) both depend on the control factor setting, where i = 1, 2,, 16. i ijk Measure of robustness is so-called signal-to-noise ratio (SN ratio). The SN ratio for evaluating the stability of the product is defined as 2 2 2 2 /, or 10log ( / ) 1 2 10 1 where 1 and 2 are the standard deviation of the first part and the second part, respectively. Basically, the SN ratio indicates the degree of the predictable performance of the product in the presence of noise factors. 2 In the current example 1 i and 2 = i, so the SN ratio is defined as 2 2 i 10log 10( i / i ). For each given combination of the control array, we find i and i by regression. Input signal factor values 100, 100, 200, 200, 300, 300 into A1 through A6, and Y values in the first row 119.2, 123.8, 239.9, 244.4, 359.8, 365.4 -- into B1 through B6. Select regression. Input Y range B1:B6, input X range A1: A6, and click the item constant is zero. We have the result of regression shown in Table 8. Thus, i is given by 15

coefficient of X variable 1 1.2097, and i is given by standard error 2.7286. These values along with the corresponding SN ratio is given in Table 9. Similarly, we can find other i and i. In other words, we need to undertake such 16 regressions. SUMMARY OUTPUT Table 8: Find i and i by regression Regression Statistics Multiple R 0.999679939 R Square 0.99935998 Adjusted R 0.79935998 Square Standard Error 2.728631263 Observations 6 ANOVA df SS MS F Significance F Regression 1 58128.38119 58128.38119 7807.257921 9.83521E-08 Residual 5 37.22714286 7.445428571 Total 6 58165.60833 Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Lower 95.0% Upper 95.0% Intercept 0 #N/A #N/A #N/A #N/A #N/A #N/A #N/A X Variable 1 1.209714286 0.005156628 234.5940399 2.67076E-11 1.196458772 1.2229698 1.1964588 1.2229698 The results of these regressions are shown in Table 9. As can be seen from the table, the combination No. 1 gives the maximum SN ratio. Table 9: i, i and the SN ratio beta sigma SN ratio 1.2097 2.7286-7.06524 1.591 5.0998-10.1177 1.3224 3.7115-8.96373 1.7178 4.8004-8.9261 1.4649 4.6625-10.0562 16

1.6315 10.3545-16.0508 1.4607 4.4665-9.70813 1.5575 6.3244-12.1718 1.6974 5.145-9.63202 1.9223 7.7537-12.1138 1.4523 4.2123-9.24926 1.8933 6.1794-10.2745 1.5293 5.0048-10.2979 1.8534 6.0305-10.2477 1.5888 4.6269-9.28442 1.8895 7.6348-12.129 Under the assumption of a linear ideal function with no intercept, the sensitivity measurements (i.e. coefficients) and the robustness are obtained in the above. The next steps are: 1) Determine which control factors have the significant effects on the robustness (i.e. SN ratio) and their appropriate settings of the levels. 2) Identifying which control factors significantly affect the sensitivity measurements and their appropriate settings. 3) Determine the settings of these significant control factors such that the system is high in both robustness (SN ratio) and sensitivity. To identify the active dispersion effects (i.e., factors that are important for reducing variability), one considers a linear model to the estimated SN ratios as a function of the control factors. With the data in Table 7 for control factor A to H and the SN ratios in Table 9, we use SPSS and apply ANOVA to obtain Table 10. Dependent Variable:SN Table 10. Tests of Between-Subjects Effects Source Type III Sum of Squares Df Mean Square F Sig. Corrected Model 20.912 a 8 2.614 10.083.003 Intercept 1525.618 1 1525.618 5884.526.000 A 5.979 1 5.979 23.062.002 B.109 1.109.421.537 C.788 1.788 3.039.125 D 6.532 1 6.532 25.195.002 E 1.249 1 1.249 4.817.064 17

F.211 1.211.813.397 G 5.671 1 5.671 21.874.002 H.374 1.374 1.441.269 Error 1.815 7.259 Total 1548.345 16 Corrected Total 22.727 15 a. R Squared =.920 (Adjusted R Squared =.829) The significance of the each control factor to the SN ratio can be seen from the last column of Table 10. When the p-value (i.e. sig. value in last column) is less than 0.05, we say the corresponding factor is significant. From Table 10, we have the factors A, D and G are significant (cf. Fig. 2(a) in [9]). To further determine the level settings of these significant factors, the main effect of each factor is studied by plotting their profiles using SPSS, as shown in Figure 4, Figure 5, and Figure 6, respectively. Figure 4. Profile of factor A for SN ratio. 18

Figure 5. Profile of factor D for SN ratio Figure 6. Profile of factor G for SN ratio Figures 4 to 6 (cf. Figure 3 in [9]) show that the level setting for A, D and G should all be -1. Similarly, with the data in Table 7 and sensitivity in Table 9, we apply the ANOVA to find the significant factors for sensitivity. The results from SPSS are shown in Table 11. Dependent Variable:BETA Table 11. Tests of Between-Subjects Effects Source Type III Sum of Squares df Mean Square F Sig. Corrected Model.617 a 8.077 13.296.001 19

Intercept 41.537 1 41.537 7157.844.000 A.341 1.341 58.740.000 B 6.202E-5 1 6.202E-5.011.921 C.002 1.002.302.600 D.219 1.219 37.766.000 E.031 1.031 5.303.055 F.014 1.014 2.356.169 G.007 1.007 1.184.313 H.004 1.004.708.428 Error.041 7.006 Total 42.195 16 Corrected Total.658 15 a. R Squared =.938 (Adjusted R Squared =.868) From Table 11, we can see that only factor A and D are significant. Hence, only factors A and D will affect the sensitivity of the system. In order to determine the levels of these two factors, the profiles of factors A and D with respect to sensitivity are plotted in Figure 7 and Figure 8, respectively. Figure 7. Profile of factor A for sensitivity 20

Figure 8. Profile of factor D for sensitivity From figures 7 and 8, we can see that the setting for factor A and D should all be +1. The results are consistent with those in [9] (cf. Figure 4 in [9]). One can conclude that the factors A, D and G have significant effects on the dispersion, and from the profile we know that A, D and G should choose the first level (-1). From the sensitivity analysis, in order to make the sensitivity measure as large as possible, the significant factors A and D should choose the second level (+1). This conflicts with the setting in the earlier choice of the first level of reducing the variability. As mentioned in [9], a compromise is required. The results may be one of the following situations: G is set the first level (-1). Either first or second level may be chosen for A and D. From [9], we know, the optimal choice is, -1 for G, +1 for both A and D. It is noted that in Appendix A, there is another approach to determine the control factor to trade off between the dispersion (SN ratio) and sensitivity (beta). REFERENCES: 1. Christine H. Muller (1997), Robust planning and analysis of experiments. 2. Nancy D. Warner (1999), Easy Microsoft Excel 2000. 3. Ron Person (1997), Using Microsoft Excel 97. 21

4. Genichi Taguchi (translated by Shih-Chung Tsai, 1993), Taguchi on robust technology development: bringing quality engineering upstream. 5. N. Logothetis (1992), Managing for total quality, From Deming to Taguchi and SPC. 6. Thomas J. Lorenzen & Virgil L. Anderson (1993), Design of experiments. 7. Secial issue on Taguchi methods, Quality and Reliability Engineering International, Vol. 4, No. 2, 1988. 8. Jay Devore & Roxy Peck (1993), Statistics: The Exploration and Analysis of Data, Duxbury Press. 9. Mahesh Lunani etc. (1997), Graphical methods for robust design with dynamic characteristics, Journal of Quality Technology, Vol. 29, No. 3, 327-338. 22

Appendix A: To identify the active dispersion effects (i.e., factors that are important for reducing variability), one fits a linear model to the estimated SN ratios as a function of the control factors: a a A a H 0 1 8 The fitting result is shown in Table 10. The greater the coefficient (the absolute value) is, the more important the corresponding factor. One would conclude from this analysis that the variables 1, 3, 8, and 5 (i.e., A, C, H, and E) are the important factors in terms of reducing variability. Thus, the control factors which are important for reducing variability are determined and their appropriate settings are chosen maximize the SN ratio. SUMMARY OUTPUT Table 10: Output of the regression Regression Statistics Multiple R 0.856664144 R Square 0.733873456 Adjusted R 0.429728834 Square Standard Error 1.513310723 Observations 16 ANOVA df SS MS F Significance F Regression 8 44.20661329 5.525826662 2.412909529 0.131419679 Residual 7 16.03076542 2.290109345 Total 15 60.23737871 Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Lower 95.0% Upper 95.0% 23

Intercept -10.39301976 0.378327681-27.47094724 2.17362E-08-11.28762193-9.498418-11.28762-9.498418 X Variable 1-1.11090713 0.378327681-2.936362275 0.021825858-2.005509299-0.216305-2.005509-0.216305 X Variable 2 0.304643465 0.378327681 0.805237048 0.447167182-0.589958704 1.199246-0.589959 1.199246 X Variable 3-0.850231131 0.378327681-2.247340531 0.059430809-1.7448333 0.044371-1.744833 0.044371 X Variable 4-0.01055197 0.378327681-0.027891085 0.978527523-0.905154139 0.88405-0.905154 0.88405 X Variable 5-0.501492822 0.378327681-1.325551494 0.226610264-1.396094992 0.393109-1.396095 0.393109 X Variable 6 0.384759664 0.378327681 1.017001091 0.343008356-0.509842505 1.279362-0.509843 1.279362 X Variable 7-0.244398544 0.378327681-0.645996992 0.53887405-1.139000713 0.650204-1.139001 0.650204 X Variable 8-0.503707277 0.378327681-1.331404765 0.22477671-1.398309446 0.390895-1.398309 0.390895 To identify the active sensitivity effects, one fits a linear model to the i s as a function of the control factors: Table 11 shows the result of the regression. b b A b H 0 1 8 SUMMARY OUTPUT Table 11: Output of the regression Regression Statistics Multiple R 0.967355084 R Square 0.935775858 Adjusted R 0.862376838 Square Standard Error 0.077585904 Observations 16 ANOVA df SS MS F Significance F Regression 8 0.61395595 0.076744494 12.74916014 0.001544208 Residual 7 0.042137008 0.006019573 Total 15 0.656092958 Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Lower 95.0% Upper 95.0% 24

Intercept 1.6113625 0.019396476 83.07501321 9.64079E-12 1.565497155 1.657228 1.565497 1.6572278 X Variable 1 0.145675 0.019396476 7.510384876 0.00013611 0.099809655 0.19154 0.09981 0.1915403 X Variable 2-0.001075 0.019396476-0.055422439 0.957350621-0.046940345 0.04479-0.04694 0.0447903 X Variable 3 0.0105875 0.019396476 0.545846575 0.602124582-0.035277845 0.056453-0.035278 0.0564528 X Variable 4 0.116925 0.019396476 6.028156867 0.000527269 0.071059655 0.16279 0.07106 0.1627903 X Variable 5 0.0429625 0.019396476 2.214964203 0.062338831-0.002902845 0.088828-0.002903 0.0888278 X Variable 6 0.0295125 0.019396476 1.521539273 0.17193473-0.016352845 0.075378-0.016353 0.0753778 X Variable 7-0.0202125 0.019396476-1.042070735 0.332025967-0.066077845 0.025653-0.066078 0.0256528 X Variable 8 0.015525 0.019396476 0.800403125 0.449783991-0.030340345 0.06139-0.03034 0.0613903 One would conclude from this analysis that the variables 1 and 4 (i.e., A and D) are the most important factors in affecting sensitivity. Once the important sensitivity and dispersion effects have been identified, we can choose the appropriate settings of the factors to reduce variability and get close to the desired sensitivity. To intuitively find parameter settings, we can fit the SN ratio as a function of individual control variable by regressing the following model: a b x, j j j j 1, 2,..., 8. The regression straight lines are displayed in Figure 4 which indicates the magnitudes of the effects of the factors. To make the SN ratio large, we have to choose the 1 st level (with -1 value) of A, C, D, E, G, and H, and the 2 nd level (with +1 value) of B and F. 25

-9-9.5-10 -10.5-11 -11.5-12 A B C D E F G H Figure 4: Regression lines: the SN ratio as the control factors 26