Homework Week 13
IE 5342 - Dr. Timothy I. Matis
Hunter Swerdloff
28 Nov 2021
Setup
Load Libraries Into Session
library(dplyr)
library(tidyr)
library(GAD)
library(DoE.base)
library(FrF2)## Warning: package 'FrF2' was built under R version 4.1.2Problem 8.2
Suppose that in Problem 6.15, only a one-half fraction of the \(2^4\) design could be run. Construct the design and perform the analysis, using the data from replicate I.
des.res8.2<-FrF2(nfactors=4,resolution=4,randomize=FALSE)
des.res8.2## A B C D
## 1 -1 -1 -1 -1
## 2 1 -1 -1 1
## 3 -1 1 -1 1
## 4 1 1 -1 -1
## 5 -1 -1 1 1
## 6 1 -1 1 -1
## 7 -1 1 1 -1
## 8 1 1 1 1
## class=design, type= FrF2From the table of pluses and minuses, we can determine which corner points/responses from Problem 6.15 to use:
(1), ad, bd, ab, cd, ac, bc, and abcd.
response8.2<-c(7.037,16.867,13.876,17.273,
11.846,4.368,9.360,15.653)
des.resp8.2<-add.response(des.res8.2,response8.2)
summary(des.resp8.2)## Call:
## FrF2(nfactors = 4, resolution = 4, randomize = FALSE)
##
## Experimental design of type FrF2
## 8 runs
##
## Factor settings (scale ends):
## A B C D
## 1 -1 -1 -1 -1
## 2 1 1 1 1
##
## Responses:
## [1] response8.2
##
## Design generating information:
## $legend
## [1] A=A B=B C=C D=D
##
## $generators
## [1] D=ABC
##
##
## Alias structure:
## $fi2
## [1] AB=CD AC=BD AD=BC
##
##
## The design itself:
## A B C D response8.2
## 1 -1 -1 -1 -1 7.037
## 2 1 -1 -1 1 16.867
## 3 -1 1 -1 1 13.876
## 4 1 1 -1 -1 17.273
## 5 -1 -1 1 1 11.846
## 6 1 -1 1 -1 4.368
## 7 -1 1 1 -1 9.360
## 8 1 1 1 1 15.653
## class=design, type= FrF2DanielPlot(des.resp8.2,half=TRUE)
The half normal plot shows that there are no significant main effects or interaction effects.
Problem 8.24
Construct a \(2^{5-1}\) design. Show how the design may be run in two blocks of eight observations each. Are any main effects or two-factor interactions confounded with blocks?
des.res8.24<-FrF2(nruns=16,nfactors=5,blocks=2,randomize=FALSE,alias.block.2fis = TRUE)
summary(des.res8.24)## Call:
## FrF2(nruns = 16, nfactors = 5, blocks = 2, randomize = FALSE,
## alias.block.2fis = TRUE)
##
## Experimental design of type FrF2.blocked
## 16 runs
## blocked design with 2 blocks of size 8
##
## Factor settings (scale ends):
## A B C D E
## 1 -1 -1 -1 -1 -1
## 2 1 1 1 1 1
##
## Design generating information:
## $legend
## [1] A=A B=B C=C D=D E=E
##
## $`generators for design itself`
## [1] E=ABCD
##
## $`block generators`
## [1] AB
##
##
## no aliasing of main effects or 2fis among experimental factors
##
## Aliased with block main effects:
## [1] AB
##
## The design itself:
## run.no run.no.std.rp Blocks A B C D E
## 1 1 5.1.1 1 -1 1 -1 -1 -1
## 2 2 6.1.2 1 -1 1 -1 1 1
## 3 3 7.1.3 1 -1 1 1 -1 1
## 4 4 8.1.4 1 -1 1 1 1 -1
## 5 5 9.1.5 1 1 -1 -1 -1 -1
## 6 6 10.1.6 1 1 -1 -1 1 1
## 7 7 11.1.7 1 1 -1 1 -1 1
## 8 8 12.1.8 1 1 -1 1 1 -1
## run.no run.no.std.rp Blocks A B C D E
## 9 9 1.2.1 2 -1 -1 -1 -1 1
## 10 10 2.2.2 2 -1 -1 -1 1 -1
## 11 11 3.2.3 2 -1 -1 1 -1 -1
## 12 12 4.2.4 2 -1 -1 1 1 1
## 13 13 13.2.5 2 1 1 -1 -1 1
## 14 14 14.2.6 2 1 1 -1 1 -1
## 15 15 15.2.7 2 1 1 1 -1 -1
## 16 16 16.2.8 2 1 1 1 1 1
## class=design, type= FrF2.blocked
## NOTE: columns run.no and run.no.std.rp are annotation,
## not part of the data frameNo main effects are confounded with blocks. The only two-factor interaction confounded with blocks is AB.
Problem 8.25
Construct a \(2^{7-2}\) design. Show how the design may be run in four blocks of eight observations each. Are any main effects or two-factor interactions confounded with blocks?
des.res8.25<-FrF2(nruns=32,nfactors=7,blocks=4,randomize=FALSE,alias.block.2fis = TRUE)
summary(des.res8.25)## Call:
## FrF2(nruns = 32, nfactors = 7, blocks = 4, randomize = FALSE,
## alias.block.2fis = TRUE)
##
## Experimental design of type FrF2.blocked
## 32 runs
## blocked design with 4 blocks of size 8
##
## Factor settings (scale ends):
## A B C D E F G
## 1 -1 -1 -1 -1 -1 -1 -1
## 2 1 1 1 1 1 1 1
##
## Design generating information:
## $legend
## [1] A=A B=B C=C D=D E=E F=F G=G
##
## $`generators for design itself`
## [1] F=ABC G=ABDE
##
## $`block generators`
## [1] AB AC
##
##
## no aliasing of main effects or 2fis among experimental factors
##
## Aliased with block main effects:
## [1] AB AC AF BC BF CF
##
## The design itself:
## run.no run.no.std.rp Blocks A B C D E F G
## 1 1 13.1.1 1 -1 1 1 -1 -1 -1 -1
## 2 2 14.1.2 1 -1 1 1 -1 1 -1 1
## 3 3 15.1.3 1 -1 1 1 1 -1 -1 1
## 4 4 16.1.4 1 -1 1 1 1 1 -1 -1
## 5 5 17.1.5 1 1 -1 -1 -1 -1 1 -1
## 6 6 18.1.6 1 1 -1 -1 -1 1 1 1
## 7 7 19.1.7 1 1 -1 -1 1 -1 1 1
## 8 8 20.1.8 1 1 -1 -1 1 1 1 -1
## run.no run.no.std.rp Blocks A B C D E F G
## 9 9 9.2.1 2 -1 1 -1 -1 -1 1 -1
## 10 10 10.2.2 2 -1 1 -1 -1 1 1 1
## 11 11 11.2.3 2 -1 1 -1 1 -1 1 1
## 12 12 12.2.4 2 -1 1 -1 1 1 1 -1
## 13 13 21.2.5 2 1 -1 1 -1 -1 -1 -1
## 14 14 22.2.6 2 1 -1 1 -1 1 -1 1
## 15 15 23.2.7 2 1 -1 1 1 -1 -1 1
## 16 16 24.2.8 2 1 -1 1 1 1 -1 -1
## run.no run.no.std.rp Blocks A B C D E F G
## 17 17 5.3.1 3 -1 -1 1 -1 -1 1 1
## 18 18 6.3.2 3 -1 -1 1 -1 1 1 -1
## 19 19 7.3.3 3 -1 -1 1 1 -1 1 -1
## 20 20 8.3.4 3 -1 -1 1 1 1 1 1
## 21 21 25.3.5 3 1 1 -1 -1 -1 -1 1
## 22 22 26.3.6 3 1 1 -1 -1 1 -1 -1
## 23 23 27.3.7 3 1 1 -1 1 -1 -1 -1
## 24 24 28.3.8 3 1 1 -1 1 1 -1 1
## run.no run.no.std.rp Blocks A B C D E F G
## 25 25 1.4.1 4 -1 -1 -1 -1 -1 -1 1
## 26 26 2.4.2 4 -1 -1 -1 -1 1 -1 -1
## 27 27 3.4.3 4 -1 -1 -1 1 -1 -1 -1
## 28 28 4.4.4 4 -1 -1 -1 1 1 -1 1
## 29 29 29.4.5 4 1 1 1 -1 -1 1 1
## 30 30 30.4.6 4 1 1 1 -1 1 1 -1
## 31 31 31.4.7 4 1 1 1 1 -1 1 -1
## 32 32 32.4.8 4 1 1 1 1 1 1 1
## class=design, type= FrF2.blocked
## NOTE: columns run.no and run.no.std.rp are annotation,
## not part of the data frameNo main effects or two-factor interactions are confounded with blocks.
Problem 8.28
A 16-run experiment was performed in a semiconductor manufacturing plant to study the effects of six factors on the curvature or camber of the substrate devices produced. The six variables and their levels are shown in Table P8.2.
Each run was replicated four times, and a camber measurement was taken on the substrate. The data are shown in Table P8.3.
Part (a)
What type of design the the experimenters use?
des.res8.28a<-FrF2(nruns=16,nfactors=6,randomize=FALSE)
summary(des.res8.28a)## Call:
## FrF2(nruns = 16, nfactors = 6, randomize = FALSE)
##
## Experimental design of type FrF2
## 16 runs
##
## Factor settings (scale ends):
## A B C D E F
## 1 -1 -1 -1 -1 -1 -1
## 2 1 1 1 1 1 1
##
## Design generating information:
## $legend
## [1] A=A B=B C=C D=D E=E F=F
##
## $generators
## [1] E=ABC F=ABD
##
##
## Alias structure:
## $fi2
## [1] AB=CE=DF AC=BE AD=BF AE=BC AF=BD CD=EF CF=DE
##
##
## The design itself:
## A B C D E F
## 1 -1 -1 -1 -1 -1 -1
## 2 1 -1 -1 -1 1 1
## 3 -1 1 -1 -1 1 1
## 4 1 1 -1 -1 -1 -1
## 5 -1 -1 1 -1 1 -1
## 6 1 -1 1 -1 -1 1
## 7 -1 1 1 -1 -1 1
## 8 1 1 1 -1 1 -1
## 9 -1 -1 -1 1 -1 1
## 10 1 -1 -1 1 1 -1
## 11 -1 1 -1 1 1 -1
## 12 1 1 -1 1 -1 1
## 13 -1 -1 1 1 1 1
## 14 1 -1 1 1 -1 -1
## 15 -1 1 1 1 -1 -1
## 16 1 1 1 1 1 1
## class=design, type= FrF2None of the main effects are aliased with other main effects or two-factor interactions, so this is a \(2^{6-2}\) resolution IV design
Part (b)
What are the alias relationships in this design?
The alias relationships from the design output in Part (a) are as follows:
A=BCE=CDF=ABDEF
B=ACE=DEF=ABCDF
C=ABE=ADF=BCDEF
D=ACF=BEF=ABCDE
E=ABC=BDF=ACDEF
F=ACD=BDE=ABCEF
AB=CE=BCDF=ADEF
AC=BE=DF=ABCDEF
AD=CF=ABEF=BCDE
AE=BC=ABDF=CDEF
AF=CD=ABDE=BCEF
BD=EF=ACDE=ABCF
BF=DE=ACEF=ABCD
Part (c)
Do any of the process variables affect average camber?
dat8.28c <- read.csv("~/Grad School/IE 5342/Homework/8.28cData.csv")
model8.28c <- lm(dat8.28c$Camber~dat8.28c$LamTemp+
dat8.28c$LamTime+dat8.28c$LamPressure+
dat8.28c$FiringTemp+dat8.28c$FiringCycleTime+
dat8.28c$FiringDewPoint,data=dat8.28c)
summary(model8.28c)##
## Call:
## lm.default(formula = dat8.28c$Camber ~ dat8.28c$LamTemp + dat8.28c$LamTime +
## dat8.28c$LamPressure + dat8.28c$FiringTemp + dat8.28c$FiringCycleTime +
## dat8.28c$FiringDewPoint, data = dat8.28c)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.0067672 -0.0022703 -0.0003875 0.0028797 0.0081328
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.0107016 0.0004793 22.328 < 2e-16 ***
## dat8.28c$LamTemp 0.0019453 0.0004793 4.059 0.000152 ***
## dat8.28c$LamTime 0.0002891 0.0004793 0.603 0.548823
## dat8.28c$LamPressure 0.0028016 0.0004793 5.845 2.57e-07 ***
## dat8.28c$FiringTemp -0.0007109 0.0004793 -1.483 0.143492
## dat8.28c$FiringCycleTime -0.0017234 0.0004793 -3.596 0.000676 ***
## dat8.28c$FiringDewPoint -0.0038734 0.0004793 -8.082 5.03e-11 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.003834 on 57 degrees of freedom
## Multiple R-squared: 0.6975, Adjusted R-squared: 0.6657
## F-statistic: 21.91 on 6 and 57 DF, p-value: 3.566e-13halfnormal(model8.28c)## Warning in halfnormal.lm(model8.28c): halfnormal not recommended for models with
## more residual df than model df##
## Significant effects (alpha=0.05, Lenth method):## [1] dat8.28c$FiringDewPoint dat8.28c$LamPressure dat8.28c$LamTemp
##
## [4] dat8.28c$FiringCycleTime lof9 lof1
##
## [7] lof7 e28
From the model summary and the half normal plot, we can see that Lamination Temperature, Lamination Pressure, Firing Cycle Time, and Firing Dew Point affect average camber.
Part (d)
Do any of the process variables affect the variability in camber measurements?
dat8.28d <- read.csv("~/Grad School/IE 5342/Homework/8.28dData.csv")
A <- dat8.28d$LamTemp
B <- dat8.28d$LamTime
C <- dat8.28d$LamPressure
D <- dat8.28d$FiringTemp
E <- dat8.28d$FiringCycleTime
F <- dat8.28d$FiringDewPoint
model8.28d <- lm(dat8.28d$CamberStdDev~A+B+C+D+E+F+A*B+A*C+A*D+A*E+A*F+B*C+B*D+B*E+B*F+C*D+C*E+C*F+D*E+D*F+E*F,data=dat8.28d)
summary(model8.28d)##
## Call:
## lm.default(formula = dat8.28d$CamberStdDev ~ A + B + C + D +
## E + F + A * B + A * C + A * D + A * E + A * F + B * C + B *
## D + B * E + B * F + C * D + C * E + C * F + D * E + D * F +
## E * F, data = dat8.28d)
##
## Residuals:
## 1 2 3 4 5 6 7 8 9 10 11
## -1.358 -2.040 1.358 2.040 2.040 1.358 -2.040 -1.358 1.358 2.040 -1.358
## 12 13 14 15 16
## -2.040 -2.040 -1.358 2.040 1.358
##
## Coefficients: (8 not defined because of singularities)
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 25.35375 1.22537 20.691 0.00233 **
## A 7.95175 1.22537 6.489 0.02293 *
## B -8.28862 1.22537 -6.764 0.02116 *
## C 2.93725 1.22537 2.397 0.13872
## D -1.64625 1.22537 -1.343 0.31126
## E -1.16862 1.22537 -0.954 0.44089
## F -4.62800 1.22537 -3.777 0.06350 .
## A:B 0.47763 1.22537 0.390 0.73429
## A:C 1.26200 1.22537 1.030 0.41131
## A:D -2.31325 1.22537 -1.888 0.19967
## A:E -0.09012 1.22537 -0.074 0.94806
## A:F -5.43725 1.22537 -4.437 0.04722 *
## B:C NA NA NA NA
## B:D -2.42787 1.22537 -1.981 0.18607
## B:E NA NA NA NA
## B:F 4.10913 1.22537 3.353 0.07859 .
## C:D NA NA NA NA
## C:E NA NA NA NA
## C:F NA NA NA NA
## D:E NA NA NA NA
## D:F NA NA NA NA
## E:F NA NA NA NA
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 4.901 on 2 degrees of freedom
## Multiple R-squared: 0.9869, Adjusted R-squared: 0.9015
## F-statistic: 11.56 on 13 and 2 DF, p-value: 0.08237halfnormal(model8.28d)##
## The following effects are completely aliased:## [1] B:C B:E C:D C:E C:F D:E D:F E:F##
## Significant effects (alpha=0.05, Lenth method):## [1] B A
The model summary and half normal plot show that Lamination Temperature and Lamination Time affect the variability in camber measurement at \(\alpha=0.05\). The model summary shows (but the half normal plot does not show) that the interaction effect between Lamination Temperature and Firing Dew Point also affects the variability in camber measurement at \(\alpha=0.05\).
Part (e)
If it is important to reduce camber as much as possible, what recommendations would you make?
From part (c), we saw that Lamination Temperature, Lamination Pressure, Firing Cycle Time, and Firing Dew Point affect average camber, so we will look deeper into these and determine recommendations based off the variable’s effect on the mean camber. The average camber is lower when Lamination Temperature is low, Lamination Pressure is low, Firing Cycle Time is high, and Firing Dew Point is high. To summarize, I recommend running these 4 variables at the following levels:
Lamination Temperature: 55\(^\circ\)C
Lamination Pressure: 5 tn
Firing Cycle Time: 29 hours
Firing Dew Point: 26\(^\circ\)C
Problem 8.40
Consider the following experiment and answer the following questions.
Part (a)
How many factors did this experiment investigate?
4 factors
Part (b)
What is the resolution of this design?
None of the main effects are aliased with other main effects or two-factor interactions, so this is a resolution IV design
Part (c)
Calculate the estimates of the main effects.
des.res8.40c<-FrF2(nruns=8,nfactors=4,randomize=FALSE)
response8.40c<-c(8,10,12,7,13,6,5,11)
des.resp8.40c<-add.response(des.res8.40c,response8.40c)
des.resp8.40c## A B C D response8.40c
## 1 -1 -1 -1 -1 8
## 2 1 -1 -1 1 10
## 3 -1 1 -1 1 12
## 4 1 1 -1 -1 7
## 5 -1 -1 1 1 13
## 6 1 -1 1 -1 6
## 7 -1 1 1 -1 5
## 8 1 1 1 1 11
## class=design, type= FrF2A8.40 <- (((10+7+6+11)/4)-((8+12+13+5)/4))/2
B8.40 <- (((12+7+5+11)/4)-((8+10+13+6)/4))/2
C8.40 <- (((13+6+5+11)/4)-((8+10+12+7)/4))/2
D8.40 <- (((10+12+13+11)/4)-((8+7+6+5)/4))/2Estimate of main effect A: -0.5
Estimate of main effect B: -0.25
Estimate of main effect C: -0.25
Estimate of main effect D: 2.5
Part (d)
What is the complete defining relation for this design?
des.res8.40d<-FrF2(nruns=8,nfactors=4,randomize=FALSE)
summary(des.res8.40d)## Call:
## FrF2(nruns = 8, nfactors = 4, randomize = FALSE)
##
## Experimental design of type FrF2
## 8 runs
##
## Factor settings (scale ends):
## A B C D
## 1 -1 -1 -1 -1
## 2 1 1 1 1
##
## Design generating information:
## $legend
## [1] A=A B=B C=C D=D
##
## $generators
## [1] D=ABC
##
##
## Alias structure:
## $fi2
## [1] AB=CD AC=BD AD=BC
##
##
## The design itself:
## A B C D
## 1 -1 -1 -1 -1
## 2 1 -1 -1 1
## 3 -1 1 -1 1
## 4 1 1 -1 -1
## 5 -1 -1 1 1
## 6 1 -1 1 -1
## 7 -1 1 1 -1
## 8 1 1 1 1
## class=design, type= FrF2The complete defining relation for this design is I=ABCD.
Problem 8.48
Consider the following design.
Part (a)
What is the generator for column D?
D=-ABC
Part (b)
What is the generator for column E?
E=BC
Part (c)
If this design were folded over, what is the resolution of the combined design?
Resolution IV design
Problem 8.60
Consider a partial fold over for the \(2^{7-4}\) resolution III design. Suppose that the partial fold over of this design is constructed using column A ( + signs only). Determine the alias relationships in the combined design.
des.res8.60<-FrF2(nfactors=7,resolution=3,randomize=FALSE)
folddes8.60 <- fold.design(des.res8.60,column=1)
summary(folddes8.60)## Multi-step-call:
## [[1]]
## FrF2(nfactors = 7, resolution = 3, randomize = FALSE)
##
## $fold
## [1] 1
##
##
## Experimental design of type FrF2.folded
## 16 runs
##
## Factor settings (scale ends):
## A B C fold D E F G
## 1 -1 -1 -1 original -1 -1 -1 -1
## 2 1 1 1 mirror 1 1 1 1
##
## Design generating information:
## $legend
## [1] A=A B=B C=C D=fold E=D F=E G=F H=G
##
##
## Alias structure:
## $main
## [1] B=CG=FH C=BG=EH E=CH=FG F=BH=EG G=BC=EF H=BF=CE
##
## $fi2
## [1] AB=-DE AC=-DF AD=-BE=-CF=-GH AE=-BD AF=-CD
## [6] AG=-DH AH=-DG
##
##
## The design itself:
## A B C fold D E F G
## 1 -1 -1 -1 original 1 1 1 -1
## 2 1 -1 -1 original -1 -1 1 1
## 3 -1 1 -1 original -1 1 -1 1
## 4 1 1 -1 original 1 -1 -1 -1
## 5 -1 -1 1 original 1 -1 -1 1
## 6 1 -1 1 original -1 1 -1 -1
## 7 -1 1 1 original -1 -1 1 -1
## 8 1 1 1 original 1 1 1 1
## 9 1 -1 -1 mirror 1 1 1 -1
## 10 -1 -1 -1 mirror -1 -1 1 1
## 11 1 1 -1 mirror -1 1 -1 1
## 12 -1 1 -1 mirror 1 -1 -1 -1
## 13 1 -1 1 mirror 1 -1 -1 1
## 14 -1 -1 1 mirror -1 1 -1 -1
## 15 1 1 1 mirror -1 -1 1 -1
## 16 -1 1 1 mirror 1 1 1 1
## class=design, type= FrF2.foldedThe alias relationships are described above.