IE 361 Homework Set #1 Fall 2000
1. Do problem 2.5 of the text using the range-based formulas. Then redo it using the
ANOVA-based formulas on page 27 of the text.
2. Below is a data set from a real calibration study (taken from a paper by John Mandel
of NBS/NIST). Unfortunately, the units are not given in the paper. For sake of argument,
suppose that they are ppm (parts per million) of some contaminant. B is the "truth"/goldstandard-measurement. C is the local laboratory measurement. All on 8 œ "% different
specimens.
B
C
'%(
'!&
(#)
'(&
"!$*
*'&
"!*&
**&
"""' "!")
""*% """(
"&&( "%##
"&*% "%(!
")*' "('#
"*)$ "($*
#"$' "*")
#"*# "*)$
###% #!!)
##%% #!"!
a) Fit a simple linear regression model to these data. For a fixed B/specimen, what do
you estimate "5measurement " to be at the local laboratory? What "conversion formula" do
you recommend for translating "local lab measurements" to estimated "gold standard
measurements"?
b) A new specimen is measured as C œ #!!! at the local laboratory. Give an
approximate 95% confidence interval for the "true"/gold standard value for this specimen.
IE 361 Homework Set #2 Fall 2000
Problems 3.18, 3.22, 3.27 and 3.30 of the text.
IE 361 Homework Set #3 Fall 2000
1. Do problem 4.31.
2. Use Minitab and generate 100 standard normal random numbers, placing them in
column 1. Put the integers 1 through 100 into column 2.
(a) Plot column 1 vs column 2 as an example of "stable process" behavior. Then use
Minitab to find the sample standard deviation of the normal random values. This should
be reasonably close to the "process short term standard deviation" 5 œ "Þ
(b) Use the "moving range chart" function in Minitab to calculate the average moving
range Q V for the data in column 1. How does Q V /1.128 serve as an estimate of the
"process short term standard deviation" 5 œ "?
(c) Add column 2 to column 1 and place it in column 1. Plot this new column 1 vs
column 2 as an example of a situation where one has samples of size 8 œ " and the
process mean is not constant (in fact .3 œ 3). Use Minitab to find the raw standard
deviation of the values now in column 1. How does this value serve as an estimate of the
"process short term standard deviation" 5 œ "?
(d) Repeat part (b) using the new values in column 1. Which estimate of 5 œ " do you
like best? The one in (c) or the one here?
(e) What is the moral told in parts (a)-(d) of this problem?
3. (a) For the data of part c) of Problem 4.2, give the best possible guess at 5 (the
process short term standard deviation) in this 8 œ " situation.
(b) If future monitoring of individual flatness values is contemplated and a mean flatness
of #Þ) is "standard," exactly how do you suggest monitoring these flatness values for an
"all-OK" ARL of 370?
4. Do Problems 5.2; 5.3b,c; 5.17a,b; 5.18; 5.28
5. Optional (5 exam points extra credit for a complete/correct solution)
This problem refers to the car door problem/figure on panel 3 of the class handouts over
Section 5.4.
Manufacturers wish to produce autos with attractive "fit and finish," part of which
consists of uniform (and small) gaps between adjacent pieces of sheet metal (like, e.g.,
doors and their corresponding frames). The attached figure is an idealized schematic of a
situation of this kind, where we (at least temporarily) assume that edges of both a door
and its frame are linear. (The coordinate system on this diagram is pictured as if its axes
are "vertical" and "horizontal." But the line on the body need not be an exactly "vertical"
line, and whatever this line's intended orientation relative to the ground, it is used to
establish the coordinate system as indicated on the diagram.)
On the figure, we are concerned with gaps 1" and 1# . The first is at the level of the top
hinge of the door and the second is . units "below" that level in the body coordinate
system (. units "down" the door frame line from the initial measurement). People
manufacturing the car body are responsible for the dimension A. People stamping the
doors are responsible for the angles )" and )# and the dimension C. People welding the
top door hinge to the door are responsible for the dimension B. And people hanging the
door on the car are responsible for the angle 9. The quantities Bß Cß Aß 9ß )" and )# are
measurable and can be used in manufacturing to verify that the various folks are "doing
their jobs." A door design engineer has to set nominal values for and produce tolerances
for variation in these quantities. This problem is concerned with how the propagation of
errors method might help in this tolerancing enterprise, through an analysis of how
variation in Bß Cß Aß 9ß )" and )# propagates to 1" ß 1# and 1" • 1# Þ
If I have correctly done my geometry/trigonometry, the following relationships hold for
labeled points on the diagram:
: œ Ð • Bsin9ß Bcos9Ñ
; œ : € ÐCcosŠ9 € Š)" •
1
1
‹‹ß CsinŠ9 € Š)" • ‹‹
#
#
= œ Ð;" € ;# tana9 € )" € )# • 1bß !Ñ
and
? œ Ð;" € Ð;# € .Ñtana9 € )" € )# • 1bß • .Ñ Þ
Then for the idealized problem here (with perfectly linear edges) we have
1" œ A • = "
and
1# œ A • ? "
.
Actually, in an attempt to allow for the notion of "form error" in the ideally linear edges,
one might propose that at a given distance "below" the origin of the body coordinate
system the realized edge of a real geometry is its nominal position plus a "form error."
Then instead of dealing with 1" and 1# , one might consider the gaps
1‡" œ 1" € %" • %#
and
1‡# œ 1# € %$ • %% ß
for body form errors %" and %$ and door form errors %# and %% . (The interpretation of
additive "form errors" around the line of the body door frame is perhaps fairly clear, since
"the error" at a given level is measured perpendicular to the "body line" and is thus welldefined for a given realized body geometry. The interpretation of an additive error on the
right side "door line" is not so clear, since in general one will not be measuring
perpendicular to the line of the door, or even at any consistent angle with it. So for a
realized geometry, what "form error" to associate with a given point on the ideal line or
exactly how to model it is not completely clear. We'll ignore this logical problem and
proceed using the models above.)
We'll use . œ %! cm, and below are two possible sets of nominal values for the
parameters of the door assembly:
Design A
B œ #! cm
C œ *! cm
A œ *!Þ% cm
9œ!
)" œ 1#
)# œ 1#
Design B
B œ #! cm
C œ *! cm
1
A œ Ð*!cos "!
€ Þ%Ñ cm
1
9 œ "!
)" œ 1#
)# œ %"!1
Partial derivatives of 1" and 1# (evaluated at the design nominal values of
Bß Cß Aß 9ß )" and )# ) are:
Design A
`1"
`B œ !
`1"
`C œ • "
Design B
`1"
`B œ Þ$!*
`1"
`C œ • Þ*&"
`1"
`A
`1"
`9
`1"
` )"
`1"
` )#
œ"
œ!
`1"
`A
`1"
`9
`1"
` )"
`1"
` )#
`1#
`B
`1#
`C
`1#
`A
`1#
`9
`1#
` )"
`1#
` )#
œ!
œ •"
œ • #!
œ • #!
œ"
œ • %!
œ • '!
œ • '!
`1#
`B
`1#
`C
`1#
`A
`1#
`9
`1#
` )"
`1#
` )#
œ"
œ!
œ • "*Þ!#"
œ • %'Þ)$$
œ Þ$!*
œ • Þ*&"
œ"
œ • %!
œ • &*Þ!#
œ • )'Þ)$$
(a) Suppose that a door engineer must eventually produce tolerances for Bß Cß Aß 9ß )" and
)# that are consistent with " „ Þ" cm" tolerances on 1" and 1# . If we interpret " „ Þ" cm"
tolerances to mean 51" and 512 are no more than Þ!$$ cm, consider the set of "sigmas"
and
5B œ Þ!" cm
5C œ Þ!" cm
5A œ Þ!" cm
59 œ Þ!!" rad
5)" œ Þ!!" rad
5)# œ Þ!!" rad .
First for Design A and then for Design B, investigate whether this set of "sigmas" is
consistent with the necessary final tolerances on 1" and 1# in two different ways. Make
propagation of error approximations to 51" and 51" . Then simulate "!! values of both
1" and 1# using independent normal random variables Bß Cß Aß 9ß )" and )# with means
equal to the design nominals and these standard deviations. (Compute the sample
standard deviations of the simulated values and compare to the Þ!$$ cm target.)
(b) One of the assumptions standing behind the propagation of error approximations is
the independence of the input random variables. Briefly discuss why independence of the
variables )" and )# may not be such a great model assumption in this problem.
(c) Notice that for Design A the propagation of error formula predicts that variation on
the dimension B will not much affect the gaps presently of interest, 1" and 1# , while the
situation is different for Design B. Argue, based on the nominal geometries, that this
makes perfectly good sense. For Design A, one might say that the gaps 1" and 1# are
"robust" to variation in B. For this design, do you think that the entire "fit" of the door to
the body of the car is going to be "robust to variation in B"? Explain.
"
(Note, by the way, that the fact that `1
` 9 œ ! for Design A also makes this design look
completely "robust to variation in 9" in terms of the gap 1" , at least by standards of the
propagation of error formula. But the situation for this variable is somewhat different
than for B. This partial derivative is equal to ! because for Cß Aß 9ß )" and )# at their
nominal values, 1" considered as a function of 9 alone has a local minimum at 9 œ !.
This is different from 1" being constant in 9. A more refined "second order" propagation
of error analysis of this problem, that essentially begins from a quadratic approximation
to 1" instead of a linear one, would distinguish between these two possibilities. But the
"first order" analysis done on the basis of formula (5.27) of the text is often helpful and
adequate for practical purposes.)
(d) What does the propagation of error formula predict for variation in the difference
1" • 1# , first for Design A, and then for Design B?
(e) Suppose that one desires to take into account the possibility of "form errors" affecting
the gaps, and thus considers analysis of 1‡" and 1‡# instead of 1" and 1# . If standard
deviations for the variables % are all Þ!!" cm, what does the propagation of error analysis
predict for variability in 1‡" and 1‡# for Design A?