Applied Statistics and Probability for Engineers

14-3 FACTORIAL EXPERIMENTS 509

0.5, 1.0, 1.5, 2.0, and 2.5 hours. The results of this series of runs are shown in Fig. 14-5. This

figure indicates that maximum yield is achieved at about 1.7 hours of reaction time. To opti-

mize temperature, the engineer then fixes time at 1.7 hours (the apparent optimum) and per-

forms five runs at different temperatures, say, 140, 150, 160, 170, and 180F. The results of this

set of runs are plotted in Fig. 14-6. Maximum yield occurs at about 155F. Therefore, we would

conclude that running the process at 155F and 1.7 hours is the best set of operating conditions,

resulting in yields of around 75%.

Figure 14-7 displays the contour plot of actual process yield as a function of temperature

and time with the one-factor-at-a-time experiments superimposed on the contours. Clearly,

this one-factor-at-a-time approach has failed dramatically here, as the true optimum is at least

20 yield points higher and occurs at much lower reaction times and higher temperatures. The

failure to discover the importance of the shorter reaction times is particularly important be-

cause this could have significant impact on production volume or capacity, production plan-

ning, manufacturing cost, and total productivity.

The one-factor-at-a-time approach has failed here because it cannot detect the interac-

tion between temperature and time. Factorial experiments are the only way to detect inter-

actions. Furthermore, the one-factor-at-a-time method is inefficient. It will require more

Figure 14-5 Yield versus reaction time with

temperature constant at 155 F.

0.5

50

1.0 1.5 2.0 2.5

60

70

80

Yield (%)

Time (hr)

140

50

150 160 170 180

60

70

80

Yield (%)

Temperature (°F)

Figure 14-6 Yield versus temperature with reaction

time constant at 1.7 hours.

• 0.2 0.2 0.6 1

–1 – 0.6

–15

–5

5

15

25

35

45

A

B

y

–1

• 0.6
  
  
  • 0.20.2
  
  
  
  

0.6

1

• 1 – 0.6
  
  
  • 0.2^0
  
  
  
  

0.20.6

0.2 0.6 1

• 0.6 –0.2


• 1

10

15

20

25

30

35

40

A

y

B

1

Figure 14-3 Three-dimensional surface plot of the data from

Table 14-1, showing main effects of the two factors Aand B.

Figure 14-4 Three-dimensional surface plot of the data from

Table 14-2 showing the effect of the Aand Binteraction.

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