In this approach, product and process designs are achieved by adjusting factor levels to reduce variability. The process follows two distinct steps, with the assumption that reduction of variability is more important than being on the target:
1. Reduce variability by adjusting the levels of factors deter- mined to be influential
2. Adjust performance mean to target by adjusting those fac- tors with less influence on variability
Dr. Taguchi recommends the two-step optimization strategy when multiple factors influence the outcome. The following ex- ample demonstrates how robust factor levels are determined when there is only one major noise factor.
Table 5-33. L4 with results and S/N ratios
COLUMN TRIAL 1 2 3 R1 R2 R3 S/N 1 1 1 1 5 6 7 15.316 2 1 2 2 3 4 5 11.47 3 2 1 2 7 8 9 17.92 4 2 2 2 4 5 6 13.62
The optimization using the above two steps can be achieved by three different independent types of analyses: (a) noise-to-control factor interaction study, (b) mean and standard deviation analysis, and (c) analyses using S/N ratios of results. While all three types give a deeper understanding of the optimization process, the S/N analysis is the recommended approach. The three analyses are explained in the example using the same experimental results.
Example 5-12: Experimental Study to Reduce Rejects Due to Short Shots in an Injection Molding Process
Objective: Reduce rejects due to short shots.
Quality characteristic: Percent of rejects with desirable per- formance—smaller is better.
Factors and levels: The top six of the list of 18 qualified and “Pa- retoized” factors were selected for the study. To keep the size of the experiment small and study as many factors as possible, all factors were studied at two extreme ranges of values (two levels). These factors and their levels are shown next as a long list of qualified factors in descending order of importance to the project team.
1. Injection pressure 2. Mold closing speed 3. Mold pressure 4. Backpressure 5. Screw speed 6. Spear temperature 7. Manifold temperature 8. Mold opening speed 9. Mold opening time 10. Forward screw speed 11. Nozzle heater-on time 12. Screw retract speed 13. Cooling time
14. Holding pressure time 15. Ejection speed
16. Coolant type (water/oil)
17. Room temperature (cold/warm) 18. Operator skill level (new/experienced)
Selected factors and their levels (six among 18 factors selected for the study) are shown in Table 5-34.
Interaction: Interaction between factors A and B (A × B) was identified but not studied.
Noise factors: Among the factors identified, the coolant type was considered uncontrollable, noise factor: Coolant type (water = N1 and oil = N2)
Scope of Experiment: Based on the number of factors (six two- level factors and one interaction), the experiment using an L-8 array and six samples, three in each noise condition, were tested in each trial condition.
The experiment design [Fig. 5-13(a)] shows how the six factors are assigned. The results collected after the tests exposing them to the two noise levels are shown in Figure 5-13(b). The three columns next to the results show the average values under each noise level (N1 and N2) and the average of all results in a trial.
(a) Two-Step Optimization Using Noise and Control Factor Interaction
The values of the trial averages [Fig. 5-13(b)] are used to calculate the combined factor and noise (A1N1, A2N1, and so on) effects and their plots as shown in Figure 5-13(c).
The combined effect of factor A and noise is calculated as:
Table 5-34. Selected factors and their levels—Example 5-12 (Six among 18 factors selected for the study)
NOTATION FACTOR DESCRIPTION LEVEL 1 LEVEL 2
A Injection pressure 1,800 psi 2,250 psi
B Mold closing speed Low
(not revealed)
Moderate
C Mold pressure 600 psi
(4.1 MPa)
950 psi
D Backpressure 950 psi 1,075 psi
E Screw speed 50 sec. 65 sec.
F Spear temperature 325°C 380°C
Average effect of A1N1 =
(11.5 + 8.70 + 11.7 + 12.70)/4 = 11.15 (Highlighted data) Average effect of A2N1 =
(13.7 + 13.4 + 12.5 + 12.0)/4 = 12.90
Similarly, all other combined effects [Fig. 5-13(c)] are calculated and are used to plot factor effects at each level using N1 and N2 along the x-axis. For example, A1N1 (11.5) and A1N2 (13.49) are
Figure 5-13(a). Experimental design for six two-level factors and noise exposure—Example 5-12
FACTOR RESULTS (y) AVERAGE
TRIAL A B – C D E F N1 N2 N1 N2 y 1 1 1 1 1 1 1 1 11.5 11.8 11.3 14.1 14.5 13.8 11.5 14.3 12.8 2 1 1 1 2 2 2 2 9.2 8.7 8.2 9.3 10.7 9.6 8.7 9.9 9.3 3 1 2 2 1 1 2 2 11.7 11.8 11.5 14.3 14.4 14.1 11.7 14.3 12.9 4 1 2 2 2 2 1 1 12.7 12.7 12.6 15.6 15.6 15.4 12.7 15.5 14.1 5 2 1 2 1 2 1 2 13.8 13.5 13.8 13.3 12.8 12.4 13.7 12.8 13.3 6 2 1 2 2 1 2 1 13.2 13.5 13.4 16.2 16.6 16.4 13.4 16.4 14.9 7 2 2 1 1 2 2 1 12.6 12.9 12.1 15.4 15.8 14.8 12.5 15.3 13.9 8 2 2 1 2 1 1 2 12.3 11.7 12 15.1 14.3 14.2 12.0 14.7 13.3 Grand averages => 12.0 14.1 13.1
Figure 5-13(b). Experimental results and calculated trial results averages— Example 5-12
FACTOR RESULTS
TRIAL A B – C D E F Noise N1 Noise N2
1 1 1 1 1 1 1 1
For each trial condition: 3 sample results were exposed to noise condition N1.
3 sample results were exposed to noise condition N2. 2 1 1 1 2 2 2 2 3 1 2 2 1 1 2 2 4 1 2 2 2 2 1 1 5 2 1 2 1 2 1 2 6 2 1 2 2 1 2 1 7 2 2 1 1 2 2 1 8 2 2 1 2 1 1 2
used to obtain plot of A1 line. Plots for all other factor level effects are obtained in the same manner.
Review of the noise and control factor interaction plots [Fig. 5-13(c)] shows that the plots for factors B, C, D, and F have more angle between the lines, indicating that there is significant in- teraction. Because, for robust design, the line with a shallower angle to horizontal is likely to produce less variation, levels B1, C1,
NOISE AND CONTROL FACTOR INTERACTION EFFECTS
A1 A2 B1 B2 C1 C2 D1 D2 E1 E2 F1 F2
Noise N1 11.15 12.90 11.82 12.21 12.36 11.68 12.14 11.90 12.48 11.57 12.52 11.52 Noise N2 13.49 14.82 13.31 14.97 14.15 14.13 14.88 13.39 14.30 13.98 15.35 12.93
Figure 5-13(c). Calculated noise and control factor interactions and plots (N×A, N×B, ..., N×F)—Example 5-12 15.50 15.00 14.50 14.00 13.50 13.00 12.50 12.00 11.50 11.0
N1 Noise N2 N1 Noise N2 N1 Noise N2
A1 A2 B1 B2 C1 C2 15.50 15.00 14.50 14.00 13.50 13.00 12.50 12.00 11.50 11.0
N1 Noise N2 N1 Noise N2 N1 Noise N2
D1 D2 E1 E2 F1 F2
FACTOR AVERAGE EFFECTS (MAIN EFFECTS)
A B – C D E F
Level 1 12.29 12.57 – 13.25 13.49 13.37 13.94
Level 2 13.84 13.57 – 12.88 12.65 12.78 12.20
Diff. L2 – L1 1.54 1.00 – –.37 –.84 –.60 –1.74
Figure 5-13(d). Calculated factor average effects and plots (Main effects of A, B, C, D, E, and F) 14.00 13.80 13.60 13.40 13.20 13.00 12.80 12.60 12.40 12.20
A1 Factor A2 B1 Factor B2 C1 Factor C2
14.00 13.80 13.60 13.40 13.20 13.00 12.80 12.60 12.40 12.20
D1 Factor D2 E1 Factor E2 F1 Factor F2
D2, and F2 are the choices for these factors. Factors A and E are considered to have interaction of lesser degree and are treated by analysis using the main effects of factors.
Main effects of factor are plotted from the calculated average effects using the trial result averages [last column in Fig. 5-13(b)], as shown in Figure 5-13(d). The levels of the remaining two fac-
tors, A and E, now can be identified from the lower values (QC = smaller is better) of the factor average effects as A1 and E2.
Optimization Step Summary
1. Reduce variability by identifying the factors that interact with noise.
• Factors with strong interaction: B, C, D, F [A and E are found to have less interaction with N; see interaction plot Figure 5-13(c)].
• Levels for least variability: B1, C1, D2, and F2
2. Adjust mean by selecting factors with least interaction with noise.
• Factors with lesser interaction: A and E
• Levels for mean closer to target: A1 and E2 (smaller is better QC, see plots of main effect above)
(b) Two-Step Optimization Using Mean and Standard Deviation
This type of analysis uses standard deviation (S) of trial re- sults for selecting robust factor levels along with main effects for adjusting mean response. Using the same trial results, standard
FACTOR RESULTS (y) AVG SD
TRIAL A B – C D E F N1 N2 y S 1 1 1 1 1 1 1 1 11.5 11.8 11.3 14.1 14.5 13.8 12.8 1.45 2 1 1 1 2 2 2 2 9.2 8.7 8.2 9.3 10.7 9.6 9.3 0.85 3 1 2 2 1 1 2 2 11.7 11.8 11.5 14.3 14.4 14.1 12.9 1.43 4 1 2 2 2 2 1 1 12.7 12.7 12.6 15.6 15.6 15.4 14.1 1.57 5 2 1 2 1 2 1 2 13.8 13.5 13.8 13.3 12.8 12.4 13.3 0.56 6 2 1 2 2 1 2 1 13.2 13.5 13.4 16.2 16.6 16.4 14.9 1.67 7 2 2 1 1 2 2 1 12.6 12.9 12.1 15.4 15.8 14.8 13.9 1.59 8 2 2 1 2 1 1 2 12.3 11.7 12 15.1 14.3 14.2 13.3 1.43 Grand averages=> 13.1 1.32
Figure 5-13(e). Experimental results and calculated standard deviation of trial results—Example 5-12
deviation and averages are calculated [Fig. 5-13(e)]. Figure 5-13(f) shows the average effects of factor on S and the plots. Because vari- ability is never desirable, regardless of the quality characteristic of the process under study, a smaller value (smaller is better) of S becomes the levels for robust design. Based on the variability [Fig.
FACTOR EFFECTS ON STANDARD DEVIATION OF RESULTS
A B – C D E F
Level 1 1.33 1.13 – 1.26 1.50 1.26 1.57
Level 2 1.31 1.51 – 1.38 1.14 1.38 1.07
Diff. L2 – L1 –.01 .37 – .12 .35 .13 –.50
Figure 5-13(f). Calculated average factor effects on standard deviation and plots—Example 5-12 1.50 1.45 1.40 1.35 1.30 1.25 1.20 1.15 1.10 1.05
A1 Factor A2 B1 Factor B2 C1 Factor C2
1.50 1.45 1.40 1.35 1.30 1.25 1.20 1.15 1.10 1.05
5-13(f)], factors B, D, and F are found significant and their levels for robust design are B1, D2, and F2. The levels of the remaining factors are selected based on the main effects as before [A1, C2, and E2 from Fig. 5-13(d)].
Optimization Step Summary
1. Reduce variability by identifying factors with significant effects of standard deviation of results.
• Significant factors: B, D, and F (A and E are found to have less interaction with N, see interaction plots above) • Levels for least variability: B1, D2, and F2
2. Adjust mean by selecting factors with less interaction with noise.
• Factors with lesser effects on standard deviation: A, C, and E
• Factor levels: A1, C2, and E2 (smaller is better QC, see plots of main effect above)
(c) Two-Step Optimization Using S/N Ratios
S/N of the trial results [Fig. 5-13(g)], which is directly related to deviation of results from the target, is used for computing fac-
RESULTS (y) Tr A B – C D E F N1 N2 S/N RATIO 1 1 1 1 1 1 1 1 11.5 11.8 11.3 14.1 14.5 13.8 –22.21 2 1 1 1 2 2 2 2 9.2 8.7 8.2 9.3 10.7 9.6 –19.38 3 1 2 2 1 1 2 2 11.7 11.8 11.5 14.3 14.4 14.1 –22.30 4 1 2 2 2 2 1 1 12.7 12.7 12.6 15.6 15.6 15.4 –23.03 5 2 1 2 1 2 1 2 13.8 13.5 13.8 13.3 12.8 12.4 –22.46 6 2 1 2 2 1 2 1 13.2 13.5 13.4 16.2 16.6 16.4 –23.50 7 2 2 1 1 2 2 1 12.6 12.9 12.1 15.4 15.8 14.8 –22.93 8 2 2 1 2 1 1 2 12.3 11.7 12 15.1 14.3 14.2 –22.50 Grand averages => –22.29
Figure 5-13(g). Experimental results and calculated S/N ratios—Example 5-12
tor average effects as shown in Figure 5-13(h). By definition, no matter the quality characteristic of the original results, larger values of S/N always represent lower variation.
Based on the plot above, the factor levels that shows the high- est S/N values are: A1, B1, C2, D2, E2, and F2. All significant factors now can be identified from the rest by performing ANOVA shown
FACTOR AVERAGE EFFECTS BASED ON S/N RATIOS
A B – C D E F
Level 1 –21.73 –21.89 – –22.48 –22.63 –22.55 –22.92
Level 2 –22.847 –22.69 – –22.10 –21.95 –22.03 –21.66
Diff. L2 – L1 –1.12 –.80 – .37 .68 .52 1.26
Figure 5-13(h). Plot of average effects of factors (S/N effects of A, B, C, D, E, and F)—Example 5-12 –21.20 –21.40 –21.60 –21.80 –22.00 –22.20 –22.40 –22.60 –22.80 –23.00
A1 Factor A2 B1 Factor B2 C1 Factor C2
–21.20 –21.40 –21.60 –21.80 –22.00 –22.20 –22.40 –22.60 –22.80 –23.00
in Figure 5-13(i). Calculation of ANOVA terms and its use will be discussed later (Chapter 6). Our immediate attention is directed to only the last column of ANOVA, which represents the relative percent influence of the factors to the variability of results in statistical and discrete terms.
From ANOVA, factors C and E are found to be insignificant (pooled) and are ignored in the estimation of the expected per- formance at optimum conditions below. Like in the previous two types of analyses, the levels of factors C and E are determined from the factor average effect plot [Fig. 5-13(d)] using the smaller is better quality characteristic.
Optimum condition: A1B1D2F2 (factors C and E are pooled)
Yopt
22 29 21 73 22 29 21 89 22 29 21 95 22 29 . . . . . . . 221 66 22 29 22 29 0 56 0 4 0 34 0 63 22 29 1 93 20 . . . . . . . . .
..36 S/N
(which translates to 10.4 in the original units of results)
# FACTOR AND INTERACTION DOF SS V F S P (%) 1 A: Injection pressure 1 2.486 2.486 6.031 2.073 18.96 2 B: Mold closing sp. 1 1.277 1.277 3.099 .865 7.91 3 Interaction A × B 1 2.277 2.277 5.525 1.865 17.05 4 C: Mold pressure (1) (.278) Pooled
5 D: Back pressure 1 .915 .915 2.222 .503 4.60 6 E: Screw speed (1) (.545) Pooled
7 F: Spear temperature 1 3.156 3.156 7.657 2.743 25.09
Other/Error 2 .824 .412 26.38
Total 7 10.937 100%
Optimization Step Summary
1. Reduce variability by identifying factors with significant effects on S/N.
• Significant factors: A, B, D, and F
• Factor levels: A1, B1, D2, and F2 (larger S/N)
2. Adjust mean by selecting factors with lesser effect on S/N. • Factors with lesser effects on S/N: C and E
• Factor levels: C2 and E2 (smaller is better, main effect)