RELIABILITY IMPROVEMENT THROUGH ROBUST DESIGN
5.11 DESIGN OPTIMIZATION
5.11.4 Graphical Response Analysis
The purpose of graphical response analysis is to identify the factors and inter-actions that significantly affect the response, and determine the combination of factor levels to achieve the most desirable response. This graphical method is intuitive, simple, and powerful, and often is a good choice for engineers. The analysis has been computerized; commercial software packages such as Minitab provide the capability for graphical analysis.
To better understand graphical analysis, let’s look at an example.
Example 5.7 The attaching clips in automobiles cause audible noise while vehicles are operating. The two variables that may affect the audible noise level are length of clip and type of material. Interaction between these two variables is possible. The noise factors that influence the audible noise level include vehicle
speed and temperature. The levels of the control factors and noise factors are as follows:
ž Control factor A: length; A0= 25 cm, A1 = 15 cm
ž Control factor B: material; B0= plastic, B1= metal
ž Noise condition z: driving speed and temperature;z1= 40 miles per hour and 15◦C,z2= 85 miles per hour and 30◦C
L4(23) is used as an inner array to accommodate the control factors. The outer array contains two columns each for a noise condition. The experimental layout is shown in Table 5.10. Then the experiments are conducted according to the cross array. The audible noise data (in dB) were collected after each vehicle accumulated 1500 miles for this test purpose. The data are summarized in Table 5.10.
The audible noise level is a smaller-the-better characteristic. The signal-to-noise ratio is calculated from (5.34) and summarized in Table 5.10. For example, the value of the ratio for the first run is
ˆη1 = −10 log1
2(152+ 192)
= −24.7.
Then the average responses at levels 0 and 1 of factors A and B are computed:
Level A B
0 −29.1 −27.4
1 −29.5 −31.2
For example, the average response at level 0 of factor B is B0= −24.7 − 30.2
2 = −27.4.
Next, a two-way table is constructed for the average response of the interaction between factors A and B:
TABLE 5.10 Experimental Layout for the Clip Design
Run A B A× B z1 z2 ˆη
1 0 0 0 15 19 −24.7
2 0 1 1 47 49 −33.6
3 1 0 1 28 36 −30.2
4 1 1 0 26 29 −28.8
DESIGN OPTIMIZATION 163
A0 A1
B0 −24.7 −30.2 B1 −33.6 −28.8
Having calculated the average response at each level of factors and interac-tions, we need to determine significant factors and interactions and then select optimal levels of the factors. The work can be accomplished using graphical response analysis.
Graphical response analysis is to plot the average response for factor and interaction levels and determine the significant factors and their optimal levels from the graphs. The average response for a level of a factor is the sum of the observations corresponding to the level divided by the total number of obser-vations. Example 5.7 shows the calculation of average response at B0. Plot the average responses on a chart in which the x-axis is the level of a factor and y-axis is the response. Then connect the dots on the chart. This graph is known as a main effect plot. Figure 5.23a shows the main effect plots for factors A and B of Example 5.7. The average response of an interaction between two factors is usually obtained using a two-way table in which a cross entry is the average response, corresponding to the combination of the levels of the two factors (see the two-way table for Example 5.7). Plot the tabulated average responses on a chart where thex-axis is the level of a factor. The chart has more than one line segment, each representing a level of the other factor, and is called an interaction plot. The interaction plot for Example 5.7 is shown in Figure 5.23b.
The significance of factors and interactions can be assessed by viewing the graphs. A steep line segment in the main effect plot indicates a strong effect of the factor. The factor is insignificant if the line segment is flat. In an interaction plot, parallel line segments indicate no interaction between the two factors. Otherwise an interaction is existent. Let’s revisit Example 5.7. Figure 5.23a indicates that factor B has a strong effect on the response because the line segment has a steep slope. Factor A has little influence on the response because the corresponding line segment is practically horizontal. The interaction plot in Figure 5.23b indi-cates the lack of parallelism of the two line segments. Therefore, the interaction between factors A and B is significant.
Once the significant control factors have been identified, the optimal setting of these factors should be determined. If interaction is important, the optimal levels of the factors involved are selected on the basis of the factor-level com-bination that results in the most desirable response. For factors not involved in an interaction, the optimal setting is the combination of factor levels at which the most desirable average response is achieved. When the interaction between two factors is strong, the main effects of the factors involved do not have much meaning. The levels determined through interaction analysis should override the levels selected from main effect plots. In Example 5.7, the interaction plot shows that interaction between factors A and B is important. The levels of A and B should be dictated by the interaction plot. From Figure 5.23b it is seen that A0B0
B
A
0 1
Level (a)
−23
−25
−27
−29
−31
−33
−35
h∧
A0
A1
B0 B1
−23
−25
−27
−29
−31
−33
−35
(b)
h∧
FIGURE 5.23 (a) Main effect plots; (b) interaction plot
produces the largest signal-to-noise ratio. This level combination must be used in design, although the main effect plot indicates that factor A is an insignificant variable whose level may be chosen for other considerations (e.g., using a shorter clip to save material cost).
If the experimental response is a nominal-the-best characteristic, we should generate the main effect and interaction plots for both signal-to-noise ratio and mean response. If a factor is identified to be both a dispersion factor and a mean adjustment factor, it is treated as a dispersion factor. Its level is selected to maximize the signal-to-noise ratio by using the strategy described above. To determine the optimal levels of the mean adjustment factors, we enumerate the average response at each combination of mean adjustment factor levels. The average response at a level combination is usually obtained by the prediction method described below. Then the combination is chosen to bring the average response on target.
Once the optimal levels of factors have been selected, the mean response at the optimal setting should be predicted for the following reasons. First, the prediction
DESIGN OPTIMIZATION 165
indicates how much improvement the robust design will potentially make. If the gain is not sufficient, additional improvement using other techniques, such as the tolerance design, may be required. Second, a subsequent confirmation test should be conducted and the result compared against the predicted value to verify the optimality of the design. The prediction is made based on an estimation of the effects of significant factors and interactions. For convenience, we denote by T and T the total of responses and the average of responses, respectively. Then we have
T =
N i=1
yi, T = T N,
where yi represents ˆηi oryi as shown in, for example, Table 5.9.
The average response predicted at optimal levels of significant factors is ˆy = T +
i∈MET
(Fi − T )+
j >i∈INT
[(Fij − T ) − (Fi− T ) − (Fj− T )], (5.37)
where MET is a set of significant main effects, INT a set of significant inter-actions, Fi the average response of factor Fi at the optimal level, and Fij the average response of the interaction between factors Fi and Fj at the optimal levels. Because the effect of an interaction includes the main effects of the fac-tors involved, the main effects should be subtracted from the interaction effect as shown in the second term of (5.37). If the response is a nominal-the-best characteristic, (5.37) should include all significant dispersion factors and mean adjustment factors and interactions. Then apply the equation to estimate the signal-to-noise ratio and the mean response.
In Example 5.7, B and A× B are significant and A0B0 is the optimal setting.
The grand average response isT = −29.3. The signal-to-noise ratio predicted at the optimal setting is obtained from (5.37) as
ˆη = T + (B0− T ) + [(A0B0− T ) − (A0− T ) − (B0− T )] = A0B0− A0+ T
= −24.7 + 29.1 − 29.3 = −24.9,
which is close to−24.7, the signal-to-noise ratio calculated from the experimental data at A0B0 and shown in Table 5.10.
In general, a confirmation experiment should be conducted before implemen-tation of the optimal setting in production. The optimality of the setting is verified if the confirmation result is close to the value predicted. A statistical hypothesis test may be needed to arrive at a statistically valid conclusion.