Spring 2000 IE 361 Exam Solutions
Exam 1
1.
(a) A calibration experiment is done to determine how to correct for instrument bias, by
figuring out how to translate easily available measurements to estimated "gold standard"
measurements.
An R&R study is done to determine the precision of measurement equipment, specifically
to assess the size of "intrinsic"/repeatability measurement variation and the size of
"operator-to-operator" measurement variation.
2.
(a) Use B
s8€" „ >8•#
B
s8€" œ
"
," aC8€"
ÈQWI
k," k Ë"
• ,! b œ
&
É" €
"!'Þ& „ "Þ)'! lÞ**l
€
"
8
€
as
B8€" •B8 b#
! aB3 •B8 b#
8
.
3œ"
"
Þ** Ð"!& • Þ!"Ñ œ "!'Þ!&.
Ð"!'Þ!&•"!!Ñ#
"
.
€
"!
*Ð#!Ñ#
So limits are
3.
Þ!!$$
(a) 5
s repeatability œ .#V
Ð7Ñ œ "Þ"#) œ Þ!!#* inch. This is an estimate of the standard
deviation a single operator would experience measuring the same disk many times.
‰# • "# ˆ Þ!!$$
‰#
(b) Use display (2.11). 5
s reproducibility œ ÊmaxŠ!ß ˆ Þ!!*#
"Þ'*$
"Þ"#) ‹ œ Þ!!&! inch.
Yes, there is important operator-to-operator variability here. This standard deviation is
1.7 times as large as the repeatability standard deviation.
s œ ' 5s R&R œ 'É5s repeatability+5s reproducibility œ ÈÐÞ!!#*Ñ# €ÐÞ!!&!Ñ# œ Þ#). This is a pretty
(c) KGV
Y•P
#ÐÞ!'#&Ñ
#ÐÞ!'#&Ñ
big/unsatisfactory estimated KGV . We'd like a value on the order of Þ" to Þ!". This is at
least #Þ) times that size.
#
#
4.
(a) These limits don't allow for any real "process variation" but only for measurement
variation. They are set up as if one could produce consistently perfect diameters.
(b) NO. Doing things this way would be setting oneself up to have stratification hide
consistent differences between heads. The $ heads should be monitored separately.
(c) V œ ")Þ%!
"! œ ")Þ%. Use display (3.18). Y G PV œ #Þ""&Ð"Þ)%Ñ œ $Þ)* with no
PGPV . There are no "out-of-control" ranges. There is no evidence in the ranges of
physical change/instability in the process.
"Þ)%
(d) 5
œ Þ(* ‚ "!•# inch (Note, by the way, that consistent with
s œ .#VÐ8Ñ œ #Þ$#'
problems 3a and 4a, this is larger than 5
s repeatability, reflecting process variation in addition
to the measurement/repeatability variation.)
(e) Supposing the mean measured diameter to be at the ideal, the specifications are at
D œ Þ!'#&
Þ!!(* œ (Þ*"" and D œ • (Þ*"". Since T Òl^l • (Þ*""Ó ¸ " my projection for
measured diameters should be "Þ!!. Since the actual diameters should be less variable
than the measured diameters, if this weren't already essentially "Þ!! we could expect the
best possible fraction of actual diameters in spec.s to exceed the fraction for the measured
ones.
(f) Use display (3.7), i.e. Þ*'# „ Þ&((Ð"Þ)%Ñ. That is Y G PB œ #Þ!#% and
PGPB œ • Þ"!! in the units of the table. There are no B's outside these limits and
therefore no indication of physical instability.
5.
(a) Use display (3.33). When Y G Ps: œ Þ!!" € $É Þ!!"ÐÞ***Ñ
œ Þ!!(( and since
#!!
Þ!!" • $É Þ!!"ÐÞ***Ñ
• !, no lower control limit is appropriate.
#!!
(b) ; œ T Òthe first s: plots outside limitsÓ œ T Ò:
s ž Þ!!((Ó œ T Ò\ ž "Þ&%Ó for \ a
binomial random variable with 8 œ #!! and : œ Þ!!&. This is " • 0 Ð!Ñ • 0 Ð"Ñ œ Þ#'%Þ
So EVP œ "Î; œ "ÎÞ#'% œ $Þ().
Exam 2
Quality Culture Questions
1. Malcom Baldridge
2. ?Quality Function Deployment? and Robust Design
3. 6 Sigma-Motorola Corp., Malcom Baldridge-National Institute for Standards and
Technology, Quality Function Deployment-Yoji Akao, Robust Design-Genichi Taguchi,
ISO 9000-International Organization for Standards, Total Quality Management-W.
Edwards Deming
4. There is typically no way to provide a useful operational definition of a company
"using" a method. Does that mean that at least one person in the company attended a
seminar or what? There is similarly no obvious way to invent a valid measure of dollars
saved. How could I ever compare what happened to what might have happened? There
is also the likely problem of bias/inaccuracy in that people providing these figures (if
there really are any) are presumably disciples of those making the claims.
5. Much of the available material comes from consultants wishing to sell their
advice/services. One should expect their claims to be overblown and self-serving.
Information from academics is sometimes (not always) more dispassionate/"objective."
(Academic payoff is more often in terms of ego/reputation than dollars, but the impulse to
be self-serving remains.)
6. If their methodology is truly effective, it constitutes a "strategic advantage" and the
company is likely to consider it proprietary, not something to be "given away" fairly
cheaply on the open market.
Technical Questions
1.
(a) .D ¸ .B • $cotÐ.) Ñ œ # • $Ð!Ñ œ # and
`D ‰# #
•$
#
#
‰# #
5D# ¸ ˆ `B
5B € ˆ `D
` ) 5) œ "ÐÞ!"Ñ € Š Ð!•"Ñ ‹ ÐÞ!#Ñ œ Þ!!$( so that
5D ¸ ÈÞ!!$( œ Þ!'" cm.
#
(b) (I got the correlation wrong originally. It should be 3 œ Þ"'%.) Use a multivariate
chart with 8 œ $ß .B œ .D œ #ß 5B œ Þ!"ß 5D œ Þ!'" and 3 œ Þ"'%. Plot
\ # œ $ÐB • #ß D • #ÑŒ
ÐÞ!"Ñ#
Þ"'%ÐÞ!"ÑÐÞ!'"Ñ
with control limit Y G P\ # œ # € $È#Ð#Ñ œ )
Þ"'%ÐÞ!"ÑÐÞ!'"Ñ
B•#
•
Œ
#
D•#•
ÐÞ!'"Ñ
•"
2. SPC is process watching for purposes of change-detection. EFC is on-line process
knob-turning for purposes of process guidance/creation of process stability. These are
different but complimentary activities that both can contribute to reduction in overall
process variation.
3.
(a) This plot is remarkably (almost unbelievably!) linear. It indicates that a normal
distribution model for coating weight is a sensible one.
$!•"
$!•"
ß 'ÐÞ!"%)ÑÉ "(Þ(!)
(b) Use equation (5.5). Here this gives the interval Š6(.0148)É %#Þ&&(
‹
i.e. ÐÞ!($ß Þ""%Ñ.
s :5 œ
(c) Use equations (5.9) and (5.10). Here this is G
Þ$&!!•Þ$#"!
$ÐÞ!"%)Ñ
ÐÞ'&$Ñ
"
confidence bound is then Þ'&$ • "Þ'%&É *Ð$!Ñ
€ #Ð$!Ñ•#
œ Þ%)!.
œ Þ'&$ and the lower
#
(d) Use equation (5.13). Here this is Þ$#"! „ $Þ$&&ÐÞ!"%)Ñ i.e. Þ$#"! „ Þ!%*(
opt
Þ!!(%
(e) (CUSUM) 5"opt œ Þ$#"! € Þ!!(%
# œ Þ$#%( and 5# œ Þ$#"! • # œ Þ$"($ (using
equations (4.12)). Use Y! œ P! œ !. Then apply display (4.13) to enter Table 4.6.
ÐÞ!!(%ÑÎ#
Þ!"%)
^ œ ÐÞ!"%)ÑÎ
È% œ Þ& and using display (4.14) 2 œ Ð%Þ((Ñ È% œ Þ!$&$.
(EWMA) Using display (4.4) =230 > œ
Þ!!(%
Þ!"%)ÎÈ%
œ " and -opt œ Þ"% Round this to Þ" so
we can use Table 4.3 to get ^ œ #Þ(!. Then use
Þ"
- œ Þ"ß I [ Q E! œ Þ$#"!ß Y GPI [ Q E œ Þ$#"! € #Þ(! Þ!"%)
È% É Ð#•Þ"Ñ œ Þ$#"! € Þ!!%&)
and PGPI [ Q E œ Þ$#"! • Þ!!%&).
lÞ$"!!•Þ$#"!l
œ "Þ" from display (4.18). ^‡
Þ!#!!ÎÈ%
Þ!$&$
[‡ œ Þ!#!!Î
È% œ $Þ&$ (from display (4.15)).
#ÐÞ!!(%ÑÎ#
Þ!#!!ÎÈ%
(f) (CUSUM) H‡ œ
œ
display (4.19)) and
Now go to Table A.5.
?EVP ¸ "!?.
(EWMA) H‡ œ
lÞ$"!!•Þ$#"!l
Þ!#!!ÎÈ%
œ "Þ" from display (4.6). ^‡ œ
(from display (4.7). Now go to Table A.3. ?EVP ¸ '?.
œ Þ(% (from
#ÐÞ!!%&)Ñ
É Ð#•Þ"Ñ
Þ"
#ÐÞ!#!!ÑÎÈ%
œ "Þ**'
Exam 3
1. Here C"Þ œ #Þ&)!#'( and C#Þ œ #Þ&(##'( (these are needed in part (f)).
(a) No , I am not surprised. I expect to capture both measurement variation and copy to
copy variation. The preliminary study captured only measurement variation.
(b) #Þ!"% • #Þ!"&) œ • Þ!!")ß #Þ!#! • #Þ!"&) œ Þ!!%#ß #Þ!"( • #Þ!"&) œ Þ!!"#ß
#Þ!"& • #Þ!"&) œ • Þ!!!) and #Þ!"$ • #Þ!"&) œ • Þ!!#)
(c) That normal plot is very linear. It indicates no obvious problems with the "normal
distributions with a common standard deviation" model here. There are no obvious
problems with using the standard analysis.
(d) Use ? œ > È=8P 34 œ #Þ!#)" Þ!!"(
È& œ Þ!!"&
(e) Use C34 • C3w 4w „ >=P É 8"34 €
or Þ!!#% „ Þ!!##.
"
83w 4w
This is #Þ!"&) • #Þ!"$% „ #Þ!#)"ÐÞ!!"(ÑÉ "& €
"
&
#
(f) Note that +" • +# œ C"Þ • C#Þ So use C"Þ • C#Þ „ >=P É 7N
This is
#
Ð#Þ&)!#'( • #Þ&(##'(Ñ „ #Þ!#)"ÐÞ!!"(ÑÉ &Ð$Ñ
or Þ!!)! „ Þ!!"$.
(g) Fitted interactions add to 0 across rows and down columns, so they are:
• Þ!!#*' • Þ!!"%#
Þ!!%$)
Þ!!#'%
Þ!!!$) • Þ!!$!#
Þ!!!$#
Þ!!"!% • Þ!!"$'
Confidence intervals for the !"34 are +,34 „ >=P É ÐM•"ÑÐN•"Ñ
So for 95% confidence the
7MN
%
"plus or minus part" of this is #Þ!#)"ÐÞ!!"(ÑÉ %&
œ Þ!!"!#). So only !"## and !"$" fail
to be detectable. YES there are detectable interactions.
2.
(a) Use the Yates algorithm.
CÞÞÞ œ "")Þ"#&ß +# œ ""Þ$(&ß ,# œ • "%Þ'#&ß +,## œ • 'Þ)(&ß -# œ &&Þ"#&ß
+-## œ • $Þ"#&ß ,-## œ "Þ$(&ß +,-### œ %Þ'#&
combination
(1)
a
b
ab
c
ac
bc
abc
C
&$
"!&
%%
&!
"('
"*(
"&%
"''
cycle #1
"&)
*%
$($
$#!
&#
'
#"
"#
cycle #2
#&#
'*$
&)
$$
• '%
• &$
• %'
•*
cycle #3
*%&
*"
• ""(
• &&
%%"
• #&
""
$(
cycle #3Î)
"")Þ"#&
""Þ$(&
• "%Þ'#&
• 'Þ)(&
&&Þ"#&
• $Þ"#&
"Þ$(&
%Þ'#&
s „ >=P "$ É ) and >Ð(Þ#Ñ "$ É ) œ Þ*>. For
(b) Confidence limits for effects are I
#
7
#
8
reasonable confidence levels, > is between, say, "Þ& and #Þ&Þ By this standard all effects
except the BC 2 factor interactions are detectable.
(c) A high, B low and C high will make all of the fitted main effects and the fitted AB
interaction positive. sC œ "")Þ"#& € ""Þ$(& € "%Þ'#& € &&Þ"#& € 'Þ)(& œ #!'Þ"#&.
(Note that this is substantially larger than the estimated mean given in the table.)
3.
(a) # possible œ #* œ &"#ß # run œ "'ß fraction œ
(b)
A
+
+
B
+
•
C
•
•
D
•
+
BC
E
•
+
BD
F
•
•
"
$#
ACD
G
+
•
AD
H
•
+
AB
J
+
•
(c) $"
(d) For example, from the last 3 generators, the CDG 3 factor interaction, the DH 2
factor interaction and the BJ 2 factor interaction.
(e) Note that • 'Þ$) estimates the D main effect plus aliases and that • "!Þ"$ estimates
the AD 2 factor interaction plus aliases. From the generators, the simplest possible
interpretation of • "!Þ"$ is as an H main effect (the negative sign indicating that that
"Existing" is best) and the simplest possible interpretation of • 'Þ$) is as a D main effect
(the negative sign indicating that "75°C" is best).
(f) Use any setting for A, B, C, E, F, G, J and the settings of D and H indicated above.
Then sC œ *# € Ð • "!Þ"$Ñ € Ð • 'Þ$)Ñ œ (&Þ%*.
(g) Depending upon how one "eye-balls" a line on this plot, either no effects are
statistically detectable, or perhaps the ( • "!Þ"$) and ( • 'Þ$)) effects are "bigger than
noise."