bi4;~i~
" "
;;
February
24, 2004
K'~Y
IE 361 Exam 1
Prof. Vardeman
IE 361 students Brown, Freyer, and Parayno worked on the placement of "center" buttons on some circular
disks. They quantified the goodness of button placement by measuring
y = disk radial run-Qut
(in inches). The first step in the students' work was a measurement study.
a) One student took a single disk, placed it in the measuring device and measured y .She then removed the
disk, placed it a second time in the device, and measured y.
She repeated this (with the same disk) a total of 10
times. The values she obtained had R = .0045 in, s = .0012 in, and y = .0350 in .
~
i) The values R and s measure what kind of variation? (Circle exactly one of the following):
~~
~ ~\S
~
reproducibility variation
both repeatability and reproducibility
ii) Here, R is several times s .This should be no surprise. Explain v.:hy in 20 words or less.
Fo't"""
l(\:.jD
)A~=J2.rr
= "3.078a\,/\Ih\lA- jJ-$=c~ff'".:..S72.7G"'.
~
10£
';J-
~~
~
-h>
bR-
abp\A..T
("3,078/~727)
t'>~s
s.
iii) Use the value of s above and find 90% confidence limits for the "population" standard deviation, 0" ,
corresponding to s. (Plug correct numbers into a'correct formula, but don't bother to simplify.)
( S .~
I)~
..~
I'~
V~
(0012.
~
\.1
)
:s rg-
V ~)
S \
J .{)OIZIE
V 'G9
¥.3:32S
{.1!.)
( .OOc>~)
.002.0'
)
b) Each of 15 company technicians individually placed a second disk (all used the same one) in the measuring
device and measured y. The sample standard deviation of the values they obtained was .0050 in .
6As
i) In terms of the variances O"~' O"~' O"~P'and 0"2 that are parametersof the two-way random effects model
-:::.s:.;-
often used to describe a standard gauge R&R study, the standard deviation above is an empirical
approximation of what? (Give an expression involving some or all of these that .0050 in estimates.)
~
-I
R~-
IOP.g-ii)
~
-2.
-,
cr(!> + cr;~
~
.-'2..
+
(f-
If engineering specifications on run-out are 0 to .0600 in and one takes the .0050 in value as indicating
"gauge R&R variability," what is an estimated gauge capability ratio (or "precision to tolerance ratio")
here? Is it a "good" one? (Give a number and say whether or not it is favorable.)
~
:=.
!:-~
=-
~~~
lA -L
~s
tJ I\ol. M
i.s
A
.f/(P()()
te-rn
; 5;.
=.
/tio
hLt.
t;( i
( ~~'~-J<.
.t't,.. -rtl;r
"
-o
)
-I
-S
C::rt:. K
P-t
~ o v ~/.
.'N~
'(.t( ~
~~ T
1
;;:'~o
§~
c "
!i",
At a later date, run-out values were measured once (by a fixed technician) on each of20 other disks produced
over a very short period. These values had a sample standard deviation of .0040 in .
10r;t!; c) ~ased on the. 0040 in figu~e ~nd the results of th~ measurement.study'. give an es~imate of the actual disk !..:.1:- to-dlsk short term standard devIatIon of run-out for thIs process (not IncludIng measurement error).
; z.
Jo ~
~
=
J I!tM
( 0)
sj
-5~
.=
/(DO*
=
.C;O3.8
)"Z.. -(-;;;;)-;:
,.~
d) Your answer to c) is itself subject to uncertainty, that can be quantified in terms of a standard error (an
estimated standard deviation). Give this standard error for your answer to c). (Plug in, but you need not
simplify .)
m-
-L
~~
2-
-1l.oi$,5
-12.
=
.()f)O
)~
&)(~~~)1
-.$
5\
(
7
-s~
4
M=I
-1-
~0"tO)L-
r\-\
)
( ."t'IZ/
)
4-~
f (.()()IZ.)+
\ --10 ~ I
+
{.OP~
-Z:O=T
,
IY\
Supposenow that it is desirable to set up a process monitoring scheme for disk run-out. Samples of n = 5 disks
will be taken hourly from production and run-out measured. Under a scenario in which there is a single
technician that does all the measuring, a "stable process" model (stable production process and stable measuring
process model) with " 0- " of about .0040 in is perhaps appropriate. Suppose further that a standard mean runout of J.t= .0350 in is realistic.
I~
e) Find appropriate standards-given co,ntrol chart limits for x and s under this scenario.
~
-:;:: ~1\.,Y"r;
t:..6v\fM..
---~t~~
L!..L-~
s
e:~~
:
.c..e4--G.t,
--l;(C.L.s
L..c Ls
UCL =
x
LCL
=
x
.~4f;4
.O 2 ~b
/IM.
~T
M:.
= )J..~ 3~
=
=
=
I,~
a1-
:
"Bb cr-
=
B.s rr
-03S0
-03;$(:J+
I. ~b'f(
.!:::!:!::.
.= .~+04=.02~b
t:.tQ'- =
=-
5 .~
~
.~o(.
oc>40)
.otJ40
.,. ~
)
s
UCL. =
"
LcL
=
=
=
-ao
s~
f u
.()
.OtJ3S
7~
s 1"t.e
o7~
~
s
2
Here are x and s values for run-outs from 10 consecutive periods.
8
.03
.0077
I~
~
.0050
.0028
.0030
.0016
=,3615
Is
= .0401
1) What do the limits of part e) applied to the values above indic~te about the stability of the (production plus
measuring) process?
~
~reev\.~~
IO~
~
.0044
Ix
:5:
~t
I'o~
A
~~/I\
:;;::
t:.6""M..( ~~
~f-
v;..llJ.l~
.~
?1~{lA,(
I"
l'(i~~
?/,lAV""
~,/i1..'V\.
7M
fYPtlS..S'
.
g) Do retrospective x chart control limits produce a much different outcome from that in part 1)? (Compute
the limits and say whether there is a.fundamental change in conclusion.)
\
V{~
tAtL z
= ~ ~ A.3 s: =
.()3bIS-+
(J.4'l.1 )(. 004-bI)
-= .01/9
Lt.Lx
.=c
1k
UCL =
.t
.o1/~
IV\
LC4
-() 3()t
J'(\
=
=- .0301-
ttvt:
vl'-ll)1~ "
rt./fl\-""i"~
~iJrl
3
IltrD -A-
'11iL
~,..J(It.t.t'\~,
M'~r0t4.
C#"r\cf'yz,{ 1\
~&
~
~ ~;~
1m=B h) As a matter of fact, a different technician made the measurementsin each of the 10 periods represented in the
:r:;-- values in the table. How might this help explain the results in parts 1) and g)?
r~~tt\~1"VI-~
t.DU~
~,')
k
.(/(It
-+et~V\\'"z.I~
~
T~~;c.t.(
I~ ~
~
d~~~
if\
sdV\y""~~f
VA{I;\.l~
M1.~<;CAr"-t!~f
1k
~t'~i~
"/;t?k-pfLdhfu(#(
.
IO~
i) The ideal is to eliminate technician differences in measuring technique. But if it is impossible to do so, one
!..::;r;..- can take them into account in setting up process monitoring schemes. If " a " representsstable "production
process plus single-technician measuring process" variability, and a reproducibility
representstechnician-totechnician variability, a sensible standard deviation for y becomes ~ar~producibility
+ a2j~
.In light of this, what
limits for an x chart would use in place of limits from part e) if areproducibility
= .0049 in ?
~.;~
MiL
;;:
LC.L-;
=
--~
M + 3 ~;£
=
-03So
+
~
.05"6
;~
.()
i~
I~t
V(
OD~)Z
1- ~
3
KEy
IE361Exam2
Apri112,2004
Prof. Vardeman
'9{)J9 .A case study in Creating Quality by Kolarik concerns the monitoring of a die-casting process for a 35~m
camera body. Over 20 shifts ofproduction, a total of2735 camera bodies were made and inspected. A
total of 215 nonconformities were identified on those 2735 bodies, and 97 of the bodies were judged to be
nonconforming. A retrospective analysis of the data from the 20 shifts found no evidence of process
instability over that period, and so these historical rates are taken as standard values.
a) In a subsequentshift, 150 camera bodies are produced. A total of 12 nonconformities are identified on
these, and 9 of the bodies are judged to be nonconforming. Do either of these counts provide clear evidence
of process change? (Compare the two counts to appropriate control limits and interpret your result.)
12 nonconformities
~ = ~
""!:
~
' ~ ::: ~
= .o,e,
3 rr~fS;;
':. ~ O I e,b
1}.~t
(.~.
"M~s.
I
I
:
,..("~
v p..\w.~
I Ir:::~ 31@I Iw
1$'0
.o7S't
.()be7
I
I
"'Sf t{~
~
t\ 0 c;,~.v
fyio(
= &
=
IF cJ1.A.r1i.t.
.o
s 5 f;.
11'MItS A ~
.t).g~St3'~~~
~95~
.Os5's.:t~o4-c;~
:
I~
t
-= .ob
3 ~
VTS(S
~
tt '(~
F = ~
.07S"'t
is' M~ ot.tT /)f l6Y\~(
Interpretation:
9 nonconforming
:.rp
(~
~.vld ~
( \ 'f\I\lts
ttJ
~f
~t
.~f
t""frD
rV1J t:t So~ c ~
t
1
<. .~
o~
.
Ilb.,ts b) Supposethat at some point in the future, the standard nonconformity rate is reduced, production is
~increased
to 200 bodies per shift, and a process monitoring system is adopted that signals "out of
control"/"intervention warranted" if there are 3 or more nonconformities observed on a shift. Evaluate the
ARL of this monitoring system if in fact the nonconformity rate is. 01 per camera body. Then interpret the
value in 15 words or less in the context of the application. (In case either is relevant, the binomial pmf is
(
J(x) = n ) px (l-p )n-xandthe Poissonpmfis J(x) =~.)
x
I1DfAt~
t6V\1» ( It
I"
= "'f>[ X ~ ~J ~ 1- p[X' ~2..J -=- 1- t(o)
~
~(.o\)
'IV.t""'~ 2
\A~ ~1~~d't\...
ARL =
o~
~ ~
xI
l:t1'e;f
X = ~ }to't\~~YM
G
1-
-kl)
e-2.(
~ '3
-t-(.2.
1+- ~+-
)
~)
= .3Z3.S
A ~'V\
AR L ~ t
:.
Interpretation:
3. o;;r
Ttwj(
~
(
~
t>t\ ~V-l'(t£.
r",te
dY\
i'
0{
f
~r- bc"JLI -1k.tI(~
MA
.1-.I..
.L ( ,~~-I (
l'df,(1 ~r- Ul11YO
I'!'-(~ V',..,
e:V!~
3.t)~
"nl~
.1
2. A hole with a circular cross section of diameter D is to be drilled in a rectangular metal block of
thickness T .While the hole axis is supposedto be perpendicular to the top face of the block, it may in fact
make an angle of B (different from 7l'/2 ) with the faces of the block. This is pictured below.
I.)
rn
/1-
f
e
r
The amount of metal removed in drilling is then
( 7l'1 D2T
R=~4)~
In drilling, all of D,T~ and B are subject to variation. Supposethat means and standard deviations are
JlD = 3 and O'D= .3, JlT = 50 and O'T= .5, while Jlo = % and 0'0 = .017l' (the linear measuresare in mm and
the angle is in radians). Find an approximate standard deviation of R .
( .E. )
~..=.
~T
"1-
..-e::-
$/h@
)
~
{ J(\'b~r
~--
~-
{1I.
~b-
\2... s,Y\9
~
~1~2.e
de-tttt)
~t
:::.. r;;.4~
~
1)~~Z.4~
= 7°.77
yy\~3
3. The same air supply runs both heads ofa pneumatic wrench used to simultaneously tighten two nominally
identical bolts in the assembly of a hydraulic pump. Processstandardsfqr the torque required to loosen
either of those bolts are Jl = 40 ft lbs and 0' = 2 ft lb , and the standard for correlation between the two
torques is p = .8. A check of a single newly assembledpump produces a pair of measuredtorques of
37 ft lbs and 43 ft lbs .Does this clearly indicate a change from process standard values? (Showappropriate
calculations.)
,
X'3-=
.=
'r1 ( ~-~
)'
I (
4~-"kJ)
37-"'fO,
)
=
(-3
I
3
~
V-
(
(~-~
(
.e(2J(2.)
I.LL )
/
lJ/i)l/-.V1/
)
2-2.. ..e(2)(Z)
14
-3.Z
(-3.2.
1
2...".
)-,
(
) (-i )
37--40
1-.3 -40
)
.:::. -L
5.7b
( 4(-3)2 +-2(-3.2..)l-3)("3,)
+
'T I:? ~
~f~
e.i.t~~lft"..t
tV~~
evt~G.t.
lttt L,\('2.
Df
pr-o~~
=
2+
3r2:?;2.T
"~1..C..
= 22. S
-= S ~
4-(~)2-)
we.
~"c.
2
4. Lengths of steel shafts are measured by a laser gauge that produces a coded voltage proportional to shaft
length (above some reference length). In the units produced by the gauge, shafts ofa particular type have
length specifications 2300 :t 20. Below are measured lengths of n = 10 such shafts.
2298,2301,2298,2289,2291,2290,2290,2287
,2280, 2289
Assume that these shafts were produced by a physically stable process.
a) How confident are you that an 11th shaft produced by this process would measure at least 2280 ? (Give a
number. )
"...". /I, ,
J 1/.
(."Q'I1'l~A.tL'
Ie'v'~
I
-.!!--
-10
-t1+-/
I
-II
-===
~-
P/
I~
Henceforth assume that it is sensible to model shaft length as normally distributed.
above, x = 2291.3 and s = 6.2.
For the lengths listed
b) Give 90% confidence limits for a process capability index that measures process potential.
U~
(1~~
(5.7
)
(~~
)
l?S
)
~~)
1.4-1 )
c) Give two-sided limits that you are 95% sure contain at least 99% of all shaft lengths(;{ ~
I '~l+5...
{5.
/3
229(.3
)
I
22~1.3
1.437(6.2-
-:1::
27.
)
S
d) Give a 95% lower confidence bolmd for a process capability index that measures current process
performance.
1
~
IAs~
I,~rr
-Z
-~
2. 91- 3-
-.6075
( 5-ID)
.foo75-
l'. .e:,
Z z8()
(b.2. )-
.31
/.(P4-.$
5
~r--I"~~L~
)
T- %(i;;F2:
3
IE 361 Exam 3 Spring 2004 Key
Form A
1. e
2. d
3. d
4. e
5. b
6. c
7. e
8. b
9. e
10. d
11. d
12. c
13. d
14. e
15. b
16. e
17. c
18. d
19. e
20. b
Form B
1. d
2. e
3. e
4. d
5. c
6. c
7. a
8. d
9. a
10. a
11. b
12. c
13. b
14. a
15. b
16. d
17. b
18. e
19. d
20. d