6: II 7: III
5.9 Exercises
1. Run all the code examples in the chapter as you read.
2. Johnson and McNeilly[45] describe the use of a factorial experiment to quantify the impact of process inputs on outputs so that they can meet the customer specifications. The experiments were conducted at Metalor Technologies to understand its silver powder production process. High- purity silver powder is used in conductive pastes by the microelectronics industry. Physical properties of the powder (density and surface area in particular) are critical to its performance. The experiments set out to dis- cover how three key input variables: (1) reaction temperature, (2) %am- monium, and (3) stir rate effect the density and surface area. The results of their experiments (in standard order) are shown in the following Tables.
Two replicates were run at each of the eight treatment combinations in random order.
Factors Level (−) Level (+)
A=Ammonium 2% 30%
B=Stir Rate 100 rpm 150 rpm
C=Temperature 8°C 40°C
XA XB XC Density Surface Area
− − − 14.68 15.18 0.40 0.43
+ − − 15.12 17.48 0.42 0.41
− + − 7.54 6.66 0.69 0.67
+ + − 12.46 12.62 0.52 0.36
− − + 10.95 17.68 0.58 0.43
+ − + 12.65 15.96 0.57 0.54
− + + 8.03 8.84 0.68 0.75
+ + + 14.96 14.96 0.41 0.41
(a) Analyze the density response. Draw conclusions about the effects of the factors. Are there any significant interactions?
(b) Is the fitted model adequate?
(c) Make any plots that help you explain and interpret the results.
(d) Analyze the surface area response. Draw conclusions about the effects of the factors. Are there any significant interactions?
(e) Is the fitted model adequate?
(f) Make any plots that help you explain and interpret the results.
(g) If the specifications require surface area to be between 0.3 and 0.6 cm2/g and density, must be less than 14 g/cm3, find operating con- ditions in terms of (1) reaction temperature, (2)
3. Create the minimum aberration maximum resolution (25−2) design.
4. Create a (25−2) design using the generators D=ABC, E=AC.
(a) What is the defining relation for this design?
(b) what is the alias pattern for this design?
(c) Not knowing what factors will be represented by A-D, can you say whether the design you created in this exercise is any better or worse than the minimum aberration maximum resolution design you cre- ated in exercise 2?
5. Hill and Demier[41] showed an example of the use of the (25−2) design you created in exercise 3 for improving the yield of dyestuffs to make the product economically competitive. The product formation was thought to take place by the reaction steps shown.
Where F is the basic starting material that reacts in the presence of a solvent with material C to give intermediates and then the final product.
The second reaction is a condensation step, and the third reaction is a ring- closing step. It wasn’t certain what the role of materials B and E played in reactions 2 and 3. Even though there were at least these three separate reactions, preliminary laboratory runs indicated that the reactants could be charged all at once. Therefore, the reaction procedure was to run for t1 hr at T1°C, corresponding to the steps needed for condensation, and then the temperature raised to T2°C for t2 more hours to complete the ring- closure. Preliminary experiments indicated that the time and temperature of the condensation deserved additional study, because this step appeared more critical and less robust than the ring-closing step. Other variables thought to affect the reaction were the amount of B, E, and the solvent.
Therefore, these five variables were studied in an initial screening design to determine which had significant effects on product yield. The factors and levels were:
Factors Level (−) Level (+) A=Condensation temp 90 °C 110 °C B=Amount of B 29.3 cc 39.1 cc C=Solvent volume 125 cc 175 cc D=Condensation time 16 h 24 h
E=Amount of E 29 cc 43 cc
(a) Given that the observed yields and filtration times for the (25−2) experiment (in standard order) were:
Yields = 23.2,16.9,16.8,15.5,23.8,23.4,16.2,18 Filtration Time = 32,20,25,21,30,8,17,28
determine what factors and or interactions had significant effects on these responses, and interpret or explain them.
(b) The researchers believed that factor D=Condensation Temperature would have a large effect on both Yield and Filtration Time. The fact that it did not in these experiments, led them to believe that it may have a quadratic effect that was missed in the two-level experiments.
What additional experiments would you recommend to check the possibility of a quadratic effect?
6. Compare the following two designs for studying the main effects and two- factor interactions among 6 factors. A=the minimum aberration maximum resolution 26−1and B=an Alternative Screening design for 6 factors. Cre- ate a two-by-two table with column headings for design A. and design B., and row headings Advantages and Disadvantages. Make entries in the four cells of the table showing the advantages and disadvantages for using each of the two designs for stated the purpose.
7. Compare the following two designs for studing the main effects and two- factor interactions among 7 factors. A=the minimum aberration maximum resolution 27−3and B=an Alternative Screening design for 7 factors. Cre- ate a two-by-two table with column headings for design A and design B, and row headings Advantages and Disadvantages. Make entries in the four cells of the table showing the advantages and disadvantages for using each of the two designs for stated the purpose.
(a) Which of the two designs would you recommend for the comparisom in question 6?
(b) Which of the two designs would you recommend for the comparisom in question 7?
8. Create a Definitive Screening Design for 6 three-level main effects that includes 3 additional center points.
(a) Create the design in standardized and randomized order.
(b) Assuming experiments had been conducted using the design you created in (a) and the responses in standard order were:
10.95,7.83,11.26,11.31,9.81,19.94,16.33,4.90,17.36,14.57,27.30,5.65, 9.27,9.30,8.77,11.11,
use theFitDefSc() function to find a model for the data.
(c) Does the model fit the data well? (i.e., is there any significant lack of fit?)
(d) If there are insignificant terms in the model you found, refit the model using the lm() eliminating one term at a time until all remaining terms are significant.
(e) Make the diagnostic plots to check the least-squares model assump- tions for your final model.
Time Weighted Control Charts in Phase II
Shewhart control charts like the X −R charts discussed in Chapter 4 are very useful in Phase I. They can quickly detect a large shift in the mean or variance, and the patterns on the chart defined by the Western Electric rules mentioned inSection 4.2.3, are helpful in hypothesizing the cause of out-of- control conditions. If it can be demonstrated that a process is in "in-control"
with a capability index of 1.5 or higher in Phase I, then there will be no more than 1,350 ppm (or 0.135%) nonconforming output in future production (as long as the process mean does not shift by more that 1.5 standard deviations).
Therefore, to maintain a high quality level it is important that small shifts in the mean on the order of one standard deviation are quickly detected and corrected in during Phase II monitoring.
However, Shewhart charts are not sensitive to small shifts in the mean when used for Phase II process monitoring. When the only out-of-control sig- nal used on a Shewhart chart is a point beyond the control limits, the OC may be too high and the ARL too long to quickly detect a small shift in the process mean. If additional sensitizing rules (like the Western Electric rules) are used in Phase II to decrease the out-of-control ARL, they also dramatically de- crease the in-control ARL. This could cause unnecessary process adjustments or searches for causes. The solution is to use time weighted control charts.
These charts plot the cumulative or weighted sums of all past observations, and are more likely to detect a small shift in the process mean.