HD28
.M414
no.'64i4
-Tl
Inspection for Circuit Board Assembly
Philippe
B.
Chevalier
and
Lawrence M. Wein
WP#
3465-92-MSA
September, 1992
MASSACHUSETTS
INSTITUTE OF TECHNOLOGY
50 MEMORIAL DRIVE
CAMBRIDGE, MASSACHUSETTS 02139
Inspection for Circuit Board Assembly
Philippe
B.
Chevalier
and
Lawrence M. Wein
WP#
3465-92-MSA
September, 1992
MIT. LIBRARIES
'
CT 2
1992]
REUcivB;
INSPECTION FOR CIRCUIT BOARD ASSEMBLY
Phillipe B. Chevalier
Operations Research Center, M.I.T.
and
Lawrence M. Wein
Sloan School of Management, M.I.T.
Abstract
Several stages of tests are performed in circuit board assembly, and each test consists
of one or
more noisy measurements.
allocation of inspection
We
and the testing
consider the problem of jointly optimizing the
policy: that
at
is,
which stages should
a
board
be inspected, and at these stages, whether to accept or reject a board based on noisy
test
measurements.
The
objective
and defective items shipped
is
is
to
minimize the exp(^cted costs
very sensitive to some input data that
practice.
We
to customers.
is
observe that the optimal testing policy
extremely hard to estimate accurately
Hence, we do not pursue an optimal solution, although
with considerable
effort.
an optimal inspection allocation given
model
to
an industrial
significantly improves
it
and type
this testing policy.
facility that
upon the
I
is
using
it,
and
facility's historical policy.
September 1992
We
11
in
could be calculated
Rather, we derive a closed form testing policy that
optimal with respect to the expected number of type
of the
for testing, repair,
errors,
also describe
find that the
is
Pareto
and then
find
an application
proposed policy
INSPECTION FOR CIRCUIT BOARD ASSEMBLY
Philippe B. Chevalier
Operations Research Center, M.I.T.
and
Lawrence M. Wein
Sloan School of Management, M.I.T.
1.
Introduction
This paper considers the problem of inspection in a circuit board assemblj' plant, which
came
to our attention while
advances have given
working with a Hewlett-Packard
rise to circuit
facilitj-.
Recent technological
boards of increasing complexity, and consequentlj' testing
has become one of the most challenging aspects of circuit board manufacturing. Inspection
costs can account for over half of the total manufacturing cost,
utilization of inspection resources
The assembly
is
is
of circuit boards
and hence optimizing the
a crucial task.
is
performed
in
a single manufacturing stage, which
Assembled
followed by several successi\'e inspection stages.
inspected for manufacturing defects and defective components.
circuit
The
boards haA'e to be
inspection process at
each stage includes three different activities: testing, diagnosis and repair.
or
more measurements
reject a board.
The
are taken from a board,
defect
during repair. Diagnosis
of difficulty
often the
most
is
on
testing,
errors.
of that board.
The
The
its cost,
diagnostic power of a test
1
made whether
one
to accept or
and
is
corrected
and time consuming task and the degree
and henceforth repair
cost for performing a test
is
identified during diagnosis,
difficult
attractiveness of a test depends on
surement
is
depends on the board type and the type of
focus of this paper
The
is
on a rejected board
and a decision
In testing,
test that
is
performed.
The main
refers to diagnosis as well as repair.
diagnostic power, coverage and mea-
on a particular board depends on the type
is
the amount of information that
a
test
provides for diagnosis, and
strongly influences the time required and the cost for a repair.
it
Defects are considered to be of different types, and the coverage of a test refers to the range
of defects that can be detected,
in
Figure
2,
and the extent to which they can be detected. For example,
the defect types linked to defective assembly, such as opens and shorts, are very
well covered at the first inspection stage,
whereas defective components are much harder to
The outcome
detect and
some components can only be detected
of a test
one or more measurements, and the accuracy of these measurements determines
is
the prevalence of type
We
I
(false reject)
and type
The
consider two interrelated decisions.
inspect a board; this problem
is
known
At the stages where inspection
is
in
at the last inspection stage.
II (false
first
accept) errors.
decision
is
to decide at which stage(s) to
the literature as the inspection allocation problem.
performed, the testing policy decides whether to accept
The
joint
decision of inspection allocation and testing will be referred to as an inspection policy.
The
on the noisy measurements obtained from the
or reject each board based
optimization problem
which includes costs
is
to find an inspection policy to minimize the total expected cost,
for testing, repair
assume that every defective board
We make
is
and defective items shipped to customers. Lince we
repaired, no scrapping cost
one crucial assumption that allows
distribution of the true value of a
measurement
the inspection policy employed at previous stages.
is
trying to detect a type
inspections related to type
tests,
i
i
for a tractable anah'sis:
the probability
at a particular stage
independent of
If
The
measured
effect of
defects, as well as the use of tight acceptance intervals for these
of the true
measurement
is
value.
The exact nature
not measured at different
quantities measured at later stages are typically an aggregation of quantities
earlier,
and the nature
a defect. Even
be extremely
is
the measurement under consideration
dependency can be very complex since the same quantity
stages.
included.
is
one would expect that the presence of previous
defect, then
would lead to a narrower distribution
of this
test.
if
a
of this aggregation might either
model of
difficult. Typically,
this
dampen
dependency was developed, data
the only available data
2
is
or amplify the
collection
would
the noisy measurements under the
existing inspection policy; experiments would probably need to be performed under various
In our case study, the derivation of
inspection policies to gather the necessary data.
optimal testing policy was primarily an issue at
had binary
results),
which
assumption was not an
We encountered
lem:
the
is
first test
precision.
(many
of the other tests
undertaken, and consequently the independence
issue.
a formidable obstacle to implementing an optimal solution to this prob-
the optimal testing policy
measurement
in circuit testing
an
errors,
is
very sensitive to the probability distributions of the
which are exceedingly
difficult to
Hence, the considerable computational
effort required to derive the
spection pohcy did not appear to be worthwhile.
several important features:
(i)
estimate to the required degree of
Instead,
we found a
optimal
in-
testing policy with
the cutoff limits for acceptance and rejection that charac-
terize the testing policy are expressed in closed form; consequently, the test engineers at
Hewlett-Packard found the policy intuitively appealing, and easy to understand and use;
the policy
is
Pareto optimal with respect to the expected number of type
incurred; and
and type
I
the cutoff limits should be close to the optimal cutoff limits
(iii)
if
II
(ii)
errors
reasonably
accurate parameter estimates are used.
Given
culated.
optimal inspection allocation policy
this testing policy, the
Since the cardinahty of the action space
is
is
then easily
cal-
small and the dimensionality of the
continuous state space can be large, we employed an exhaustive search procedure instead of
a dynamic programming algorithm.
The paper
is
contains a short case study of the Hewlett-Packard
being applied.
The
case studj' includes a description of the
facility,
facility,
oui-
model
the methodology for
gathering the data, which has been disguised, and the proposed policy.
facility's historical
where
Relative to this
inspection policy, our limited numerical results suggest that the proposed
allocation policy can reduce total inspection costs by 10-20%,
and the proposed testing policy
can reduce costs by roughly 5%. The proposed inspection policy
implemented, and we are not yet
in a
is
in
the process of being
position to report on the cost reduction realized by
3
the
facility.
Two
by-products of our analysis are of significant practical value. First, we determine
the cost reduction that would be achieved by reducing the measurement noise of the test
This quantity can help circuit board manufacturers evaluate new test equip-
equipment.
ment, and can
assist test
equipment manufacturers focus their
equipment. Although inspection
ity
improvement
benefit
Many
serial
efforts are also vital to a
efforts in
and market
company's success.
We
These quantities can be used to focus quality
defects.
an economic fashion.
(i.e.,
no type
or type
I
Yum
allowed for imperfect inspection (Eppen and Hurst 1974,
A
in
multistage
Early studies on this problem (see, for example. White 1966 and
and Bishop 1967) assumed perfect inspection
Garci^-Diaz et
al.
their
also derive the marginal
papers have been published on the optimal allocation of inspection
systems.
repeated.
efforts
necessary in the context of circuit board assembly, qual-
is
from reducing various types of
improvement
R&D
1984), where cne determines the
number
errors). Later
II
al.
indsay
papers
and McDowell 1981 and
of times that a test should
survey of work published on inspection allocation can be found
Recently, Villalobos et
I
(1992) studied a dynamic version of the
in
be
Raz (1986).
same problem, where
inspection of an item at a particular stage can depend on the result of the inspection of that
item at previous stages. Our paper appears to be the
first
to address the presence of distinct
defect types, the joint optimization of inspection allocation and testing,
of a
model
to an industrial facility.
Section 2 presents a detailed formulation of the problem, which
Section
3.
A
case study
is
presented in Section
2.
A
and the application
4,
is
then analyzed in
and conclusions are drawn
in Section 5.
Problem Formulation
typical flow chart of the assembly of circuit boards
main manufacturing stage
is
circuit
board assembly, where
is
all
presented in Figure
1.
The
the components are soldered
onto the printed circuit boards. At the system assembly, the different boards are plugged
4
into the final product.
The
in circuit test takes
a measurement from each of the individual
components soldered on a board, and the functional
simulated working conditions. At the system
system.
Many
test,
test assesses
each board
is
variants of the configuration displayed in Figure
the response of a board to
tested as part of a complete
1
are possible. For example,
there could be multiple levels of each test, or the system assembly step could be performed
in several stages,
some
tests
and a
test could
be performed on each subassembly; on the other hand,
might not be present.
Board
Assembly
Circuit
Printed CircuN
In Circuit
Test
Boards
System
Test
Figure
As mentioned
assume that
earlier,
1
Flow chart of
.
circuit
board assemblj'.
an important characteristic of each
test
for each type of defect, the successive tests that are
is its
coverage.
performed on the
boards have an increasing coverage. Figure 2 illustrates this property, which we
chical test coverage, for the
successive tests tend to be
We
circuit
call hierar-
example introduced above. Since they cover more defect types,
more complex and more expensive
to perform.
However, as the
complexity of the test increases, the precision of the information obtained from the
decreases. Consequently, type
I
and type
longer and has to be performed by
in test
more
II
errors are
in traditional
test
more prevalent, and diagnosis takes
qualified personnel.
The
hierarchical assumption
coverage holds for the circuit board assembly systems we have encountered.
assumption also holds
will
manufacturing settings, where additional work
This
is
per-
formed between successive
tests,
and each
test
measures the cumulative functionahty of the
manufactured item.
Main Defect type
Captured
Inspection Stage
at inspection
Defective Assembly
In Circuit Test
•
Defeaive Component
•
Defeaive Component
•
Incompatible
Component
Figure
In this section, the
tion 3.3
A
we
problem
how
discuss
error
turing process.
type
will
I
is
The
II
for
a single board type
between
in isolation.
In Sec-
board types.
different
to different aspects of the overall functional
Most of these measurements are subject
to
some
noise,
and
this
quite sensitive to small changes in the board design or the manufactesting policy at a stage specifies an interval for each
The presence
and type
formulated
up to several thousand measurements. These measurements
such that a board will be accepted
otherwise.
Hierarcliical test coverage.
components on a board, or
performance of a board.
measurement
is
2.
Syiem Test
to account for the dependencies
test consists of a series of
relate to different
Functional Test
of
if
every measurement
measurement
errors
makes
it
lies
inside
its
interval,
measurement
and rejected
impossible to completely eliminate
errors. Larger intervals will lead to the
acceptance of more boards, which
reduce the number of good boards falsely rejected but increase the number of defective
boards falsely accepted. Similarly, smaller intervals
will
have the opposite
effect.
For tests that have binary results, only one testing policy needs to be considered. Ex-
amples of such situations are systemwide functional
For the purpose of generality,
we
tests
and
tests for
opens and shorts.
allow the model to have a choice of testing policies at each
6
However, the model easily accommodates the case where the
stage.
policies at
We
and n
i
defects,
where
is
reduced to a single policy.
assume that each measurement can detect only one type
=
stage n.
one or more stages
1,
.
,
.
.
and
TV, let A',„
let v^f.
set of possible testing
of defect. For
i
=
1,
.
.
.
,
/
be the number of measurements taken at stage n that detect type
be the value obtained from the
k*^^
measurement
for
type
i
defects at
Then
f'^^
is
the true measurement value and
that these two
random
the measurement error.
is
e,„fc
We
assume
and that the measurement errors are
variables are independent,
independent across different measurements. These independence assumptions appear to hold
in practice.
As mentioned
the distribution of
i^'^^.
is
the Introduction, we also
in
make
the simplifying assumption that
independent of the inspection pohcy at previous stages.
Let Pink{^) be the conditional probability that the true value
v]^). is
inside
Gmk. which
is
the interval in which the true value should be for the proper functioning of the board, given
the measurement value; that
is,
p^ir) = Pt[vI, e
The function
p,nk{-f)
can be expressed solely
/
^
_
in
G'.nfc
I
C, =
,r].
terms of the problem inputs as
-
y)^,nk(y) dy
J_oc e,„t(3--
yK:nk{y)dy
^G,„, g<nA-(j-
where ^inki^)
's
and e,„t(x)
the density function of the measurement error.
The
1,
.
.
.
,
is
the probability density function of the true value of the quantity measured,
testing policy for defect type
A',„},
where the
A-"^
i
measurement
at stage n
for
type
i
defined by T,n
=
defects at stage n
is
is
inside [L,„fc,t/,„t]. Define
Q,nk{L,nkJ^mk)
=
Pi"
''La-
€ G,nk and
V^^i^.
^
[i,nkA',nk
{(L,„i,-.U,nk)-k
accepted
if it
=
lies
.
and
l3,nk{L,nk.U,nk)
where
Q,nk{I^ink,f^'ink)
is
value of the component
is
=
Pr {l^k ^ G^nk and
V^^^
^ [L,nkJhnk]]
the probability that the measurement
is
rejected although the true
is
is
the probability that the measurement
is
bad.
good, and ^inkiL,nki U,nk)
accepted while the true value of the component
,
If
we
let
f,nk{^) be the density
function of the measured values, then
+
P,nk{x)frnk{x)dx
P,nk{x) f,nk{x) dx
(2)
JU,nk
-OO
and
=
/3ink{L,nk-lhnk)
Let Q,n(T,„) be the expected
testing policy T,„,
board at stage
??
and
number
lei /ij„(r,n)
-
(\
f^nki^-)) f>nk(x)
of false defects of type
= X!
(3)
per board at stage
be the expected number of defects of type
that are not detected at that stage.
Olin{T,n)
i
dx
under
present on a
follows that
It
+
Ihnk{-r).f,nk(-r)dx
?'
?i
/
pt„i,{x)f,„i,(x}dx
(4)
and
An(r,„)
= 2^
k=l
For technical reasons that
will
{I
-
Pznk{3-))f>nk(x)dx.
become apparent
are continuous unimodal functions for
In addition to deciding
/
upon a
(5)
•^^'"'•
all i,
later,
n and
we assume
that
Pmki-''^)
and /,„t(x)
k.
we
testing policy,
also need to choose the inspection
allocation policy at each stage, which specifies whether or not to test the boards at that stage.
Notice that additional stages can be
particular stages.
The
objective
is
to
artificially
added to consider partial testing options
minimize the total expected cost of testing,
repair,
at
and
defects leaving the plant; this total cost will often be referred to as the total inspection cost.
We
consider a per unit testing cost /„ at stage
test engineering,
??,
which typically includes operator time,
equipment depreciation and maintenance, and various overhead
possible benefit, or negative cost, of testing
is
that
it
will lead to
costs.
A
quicker learning and hence
we did not attempt
process improvements; however, in our case study,
Also, no fixed testing cost
benefit into the per unit testing cost.
The
repair cost r,„
board
cost
at stage n.
assume that
all
incurred, whether the defect
is
included in the model.
is
the total cost incurred to diagnose and repair a type
is
We
to incorporate this
defective boards are repaired,
is
on a
and that the same repair
The
a real defect or a false defect.
on a board that leaves the plant includes the cost of a
defect
i
field repair,
cost
/ of a
defect
the cost of the analysis
and repair of the defective board that comes back to the plant, and a cost measuring the
We
customer's loss of goodwill.
assume that the
system, which simplifies the analysis. However,
then
it
is
if
cost
is
there are two or
more
defects in a system,
unlikely that these defects would be detected by the customer at the
Moreover, since the number of defects leaving the plant
same system are very
cost
per defect and not per defective
is
We
also
time.
very low, multiple defects on the
would be very similar
unlikely; consequently, the results obtained
was incurred per defective system.
same
assume that the
cost
if
a
/ does not depend on
the type of defect on a board. Although the more general case can be easily accommodated,
this
assumption seems practical, since the costs of
dominate the other
costs,
As a consequence
of the hierarchical test coverage assumption,
appear per board at stage
board that leave stage
and
stages,
cf,„
and
6,„
goodwill and visiting the repair
n.
Let
n.
denote the average number of defects of type
at stage
stage
policies
earlier
n and testing policy
T,,,
6,^
this as
if,
on average,
is
/
!=1
(i.e.,
d,„
new
defects of type
depends on the inspection policy
The
distributions of the
policies
all
random
at
all
/
per
earlier
variables from
the costs in the model are
where the decision to inspect depends
on the board) are not considered.
is
become
6,n
are expected values.
upon what was observed
defects
model
Notice that
and dynamic allocation
more
We
which these expected values are derived do not matter because
linear
site
and are independent of the defect type.
detectable at each inspection stage.
i
lost
If
inspection
is
performed
used, then the expected cost of testing and repair at that
V
and the expected number of defects of type
On
the other hand,
stage.
if
no inspection
The expected number
The optimization problem
testing policy Tin to use,
if
is
per board leaving stage n
performed at stage
is
of type
?'
i
is
no cost
n, then
is
incurred at that
defects per board leaving stage n in this case
to decide whether to inspect at each stage,
and
if
is
so.
what
such a choice exists, to minimize the expected total inspection
cost,
N
/
\
I
Hi'" + X]
n=l
''" (^'."-1
+
d,n
In this section,
in
-
3.1.
+
/^
(6)
^tA'-
i=l
/
Analysis
we analyze the problem formulated
Section
in
Seciion 3.1 and the inspection allocation pohcy
Various extensions of the basic problem are described
3.2.
I
l3,n(Tin))
i-1
3.
addressed
+
0,n(T,n)
is
in
2.
The
testing
numerically derived
problem
in
is
Section
the last three subsections.
The Testing Policy
This subsection contains three main
results.
First,
we
derive a set of testing policies
that
is
optimal with respect to the tradeoff between the expected number of type
type
II
errors per board in the sense of Pareto optimahty
(i.e.,
it
is
one quantity without increasing the other). This result enables us to
to these policies
when searching
additional assumptions are
optimal testing policy
in practice.
is
made
for
an optimal testing
to sharpen the
polic\'.
first result.
1
and
impossible to reduce
restrict
our attention
Second, three rather weak
However, we observe that the
very sensitive to certain input data that are difficult to estimate
Hence, we conclude this subsection by proposing a closed form policy that
Pareto optimal and easy to implement and understand.
10
is
The minimum expected
is
denoted by Jn{^\n^^2ni-
Jn{pi,p2,---,Pi)
+
Jn + \{P\
/„
until exiting the plant,
which
the dynamic programming optimality equations
i^/n)i satisfies
•
= Min
from stage n
total inspection cost
dxn,p2
+
+
<^2n,-- -,/>/
<^/n),
+ min
(7)
«=1
+
J„ + l(/9ln(r.n),^2n(7^.n),
•
•
,
for
/^/nC^.n))
n
=
1
,
A'
and
Jn+i{PuP2,-
,Pi)
= IYIP'^
(8)
1=1
where
{p\,p2,-
are
,Pi)
dummy
The second minimization
variables.
in (7)
min|/!„(Qi„(r;„),a2„(r,„),
.
.
can be rewritten more concisely as
.,aj„{T,n))
+
5n(/3i„(r,„),/32„(r,n),
/3/„(r,„))}.
(9)
where
= J2 x,r,
/i„(.ri,.T2,...,.r/)
:io)
1=1
and
9n{x\,X2,.
.
.
,xi)
=
J„+i(,ri,a-2,.
.
.
,.17)
+
'^{p,
+
d,„
-
x,)r,„.
(ii:
l:=l
If Q',n(T,„)
and
l-^in{T,„) in (9)
are replaced with the expressions obtained in equations (4)-(5),
then the derivative of this function with respect to the upper acceptance limit
-
-7^(Q^ln,0'2nv
•
•
(',„;,.
is
^0:in)Pink{Uink)finkiU,nk)
r\
+
Tr^(/?ln,/?2n,-
where, to simplify notation, a,„ stands for Q,n(7',n) and
sion equals zero
•
•
,/^/n) (1
"
P,nk{l',nk)) f,nk{l'n,k)<
OX,
/?,„
stands for
f3,„{T,n).
This expres-
if
l^{01nJ2n,...,l3ln)
(12)
.
The second
P'inki^^'nk)
limit
derivative of the objective function (9) will be positive
<
0-
One would expect
at a point
is
Pink[^,nk}
>
the second order conditions to hold, since the upper
to
show that the optimal lower acceptance
- ^T-
X
da„,^
,
From the
limit satisfies
U<JJ
TT^'
n
derivative of the objective function (9) will be positive
0.
and
to a valid board are decreasing.
The same argument can be used
P'inki^^rik)
<
f'ni,iU,nk)
where both the frequency of measurement and the probability that a
measurement corresponds
The second
if
if
>
fink{L>tnk)
and
Again, the second order conditions are consistent with our intuition.
definition of
/;„
and g„
in (lO)-(ll),
-7^— (oin,02„,..
it
follows that
=
,a/„)
.
r,„
(14
and
^"^
dpi
^(An,/?2nv,/^/n) =
OX,
(/^In, /^2nv
•
•
,/^/n)
"
(15)
?m.
Equations (12) and (13) now bpcom^"
j^
P,nk{L,nk)
=
P,nkif-',nk)
=
1
"
dj ^,
,
-,
T
;—T
^if^mjhn
(16)
/i/n)
This expression has an intuitive meaning: the probability that a component
acceptance and rejection cutoff points should be equal to the marginal cost of a
is
of (16)
is
accept parts, and the acceptance region
a constant for each measurement k
=
1,
.
of Pareto optimal testing policies can be generated
The
.
/\r,„,
becomes
it
which we denote by Ctn- The
by letting C,„ vary from
/,nt(-T)
the
increased. Notice that the right side
is
existence of lower and upper limits L,nk and
the continuity and unimodality assumptions of
will
.
at
false rejection
divided by the marginal cost of a false acceptance. As this cost ratio increases,
less costly to
bad
to
set
1.
satisfying (16)
is
guaranteed by
and p,nk{^) stated
in
Section
Umk
2.
We
not pursue weaker necessary and sufficient conditions for the existence of these optimal
upper and lower hmits, since
in
our case stud}' these functions
density functions that satisfy the necessary conditions.
12
will
be modeled
b}'
Gaussian
We can
compute
(16) to
<^tn{C,n)
and
in
advance
/S,„(C',n).
As a
all
functions.
result, the
The nonlinear curve
minimization
in
k^
dynamic programming recursion
Section 4.2 shows an example of the tradeoff
the curve was derived by aggregating
in circuit test that detects defective
We now turn to a special
explicitly described.
for this testing stage
1.
several
and defect type, which
components.
more
This special case requires three further assumptions that are satisfied
in practice.
First,
we assume that the measurement
and the true value of the component being measured are normally distributed.
can be simplified to
Pj„/:(.r)
all
to
case where the set of Pareto optimal testing policies can be
our case study and often hold
that
and
and generate the functions
of type II errors per board (/?,„(C,„)) obtained by letting C,„ vary from
hundred measurements (one per component)
in
in the
(1), (4), (5)
errors per board (a,n(C,n)) versus the expected
I
Since these quantities are independent of
was the
and use equations
possible testing policies, but only these two
all
Figure 7
in
curve of the expected number of type
number
1,
possible optimal testing policies,
does not have to explicitly consider
(7)
and
the right side of (16) vary between
let
subscripts
(all
,nk
"
<
It
noise
follows
being dropped for better readability)
p{x)
]_/
AGu -
.
2V'^
+
+ fta,a^^2[al + al)
/<^)<^e
JGu
r)(rh
'
AGt -
__
""'
^
+
{G,
+
n,
a,a^^2[al
+
a})
fi^)aj
-
.r)rr|
.
^
(17)
where:
/if
and
Gf.
are the expected value and the standard deviation, respectively, of the mea-
surement noise;
/i^
and
cr^
are the expected value
values of the
and the standard deviation,
component being measured; we
will also refer to
/i^
respectively, of the true
as the
nominal value of
the component;
Gi and G'u are the lower and upper limits respectively of the interval
measured component
is
good
if
its
true value
13
lies
inside
it;
G
such that the
erf
is
the error function, erf(r)
= -^ Jq
Figure
We
also
e
The
3.
which
dt,
'
is
plotted in Figure
3.
error function.
assume that
—
|<7u
Gi
>
(18)
1
and
CT^
In
is
19)
3(7e.
quaUty management terminology, inequality (18) states that the machine capability index
greater than
1;
that
ification range, IGj,
(i.e.,
\Gu
—
Gi\
>
—
is,
Gi\.
the natural process range, 6a^,
Since
By
1
many companies
is
smaller than the product spec-
are currently striving for "Ga"' capability
often holds in practice. Inequality (19) requires that the test
12cr^), (18)
measurements be reasonably
to
>
reliable.
conditions (18)-(19), one term of the right side of equation (17) will always be equal
or -1, implying that p{L)
=
p{U)
L =
for
//^
any L and
+
//e
U
such that
+
(Gl
-
fi^)z
(20)
+
(Gn
-
/i^)-,
(21)
and
U=
//.^
+
ft,
14
A
1
1
1
1
1
various components in our study and found that p(t) takes on values between 0.2 and 0.4.
Unfortunately, despite gathering a large amount of data
that the
95%
in
the case study, we will see later
confidence intervals on the estimates used for p{x) are
11.1% overestimate. For
this reason,
it
extremely
is
difficult to
much wider than
this
obtain an optimal testing
policy that provides reliable performance.
Consequently,
it
does not seem worth the considerable effort required to derive the
optimal testing policy. Instead, we propose to simply set the lower and upper cutoff limits
and descent of p{x)
in
These points, denoted by
x/
to the middle points of the ascent
points such that p{x)
=
0.5.
argument of either error function
r/
=
in (17)
+
/'^
Figure
and
.r^,
that
4;
is,
we
find the
two
are derived by setting the
equal to zero, which yields
/^
f
^,,
+
(r;/-//4)(^^^)
(22)
and
.r,
.Although the value
in
=
//,.
+
(G,
- //^)(^^^).
(23)
the right side of (16) will often be closer to 0.9 than 0.' in
practice, the corresponding difference in the cutoff points will typically be dwarfed by the
inaccuracies in the measurement errors. Moreover, the cutoff points x/ and Xu have several
desirable features.
is
still
Even
if
Pareto optimal.
to the optimal policy
Also, this policy
if
since the testing policy
the parameter estimates are inaccurate, the testing policy (.r/,Xu)
is
easy to find and can be expected to he close
the estimates used for p{x) are reasonably accurate.
is
derived in closed form,
understood and implemented by the
intuitively appealing,
it is
test engineers.
More
have the following intuitive meaning (consult Figure
shifted
4):
and
Furthermore,
is
very easily
specifically, expressions (22)-(2.3)
the acceptance interval should be
from the tolerance interval by the expected value of the measurement
tolerance on both sides should be multiplied by a factor that
is
noise; the
the ratio of the
sum
of
variance of the true values of the component and the measurement noise over the variance of
the true values of the component. This ratio shows verj' clearly
16
how the acceptance
interval
is
affected
by the measurement
noise.
For these reasons,
and Xu are used to define the
xi
testing policy in the case study described in the next section.
The Inspection Allocation Policy
3.2.
The Pareto optimal curves
for a,„(T',n)
and
characterized by a,>i(C,„) and ^in{Cin) can be substituted
^,n(Tin), respectively, into the
dynamic programming optimahty equation
and the optimal inspection allocation policy can be derived by solving the resulting
(7),
dynamic program. However, the optimal solution becomes very tedious
to calculate, since
the functions a,„(C„) and fi,n[C,n) and the /—dimensional state space are continuous, and
/
is
policy
has
moderate
of
is
More
difficult to reliably
many
(22)-(23)
size.
importantl}', as pointed out in Section 3.1, the optimal testing
implement
desirable characteristics.
and then
this proposal, the
-^n(/'l,/52v,/'/)
in practice,
and the testing policy derived
in (22)-(23)
Hence, we propose to use the testing policy defined
Under
find the optimal allocation policy given this fixed testing pohcy.
dynamic programming equation
=Min
in
(7) simplifies to
J„ + \{p\-\- dxn,p2-\- din
,Pl
+
dl„),
(24)
I
1
!
where q,„ and
to xi
/3,„
are
^
=1
computed from equations
(4)-(5)
by setting the cutoff hmits L and
('
and x^-
Solving (8) and (24) numerically requires a discretization of the /—dimensional state
space of incoming defects, and can be cumbersome in
and
exhaustive search procedure over
all 2^^ policies,
Let
TTn
cases. In contrast,
as described below.
represent the inspection allocation policy at stage n, and
denote the allocation policy
for the line
up
to stage
m. For
i
=
1,
let
H^ =
.
,
.
define
^.
n
=
l^im
if
I'm
—
inspection,
d„n
if
TT,,,
=
no inspection.
"
<^,.n„,_,
+
an inspection
be found relatively easily by employing an
allocation policy that solves (8)
(24) can
many
.
/
(7i"i, 7r2,
and
m —
.
1,
.
.
.
.
,
.
tt^)
,
A',
which
77?
=
is
the expected
1,...,A'^,
number
the cost
of defects of type
that leave stage
i
of allocation policy
cn,„
m
under poHcy
n„
is
Q,m
" Am)
if
T^m
crim-i
if
^m — no
\\,„-
For
calculated from the cost cn„,_j of
allocation policy Tlm-i by
/
=
CUrr,
where
=
cx\^
=
(f,_no
entire line, which
3.3.
+tm + Yl '^•mi^t.Urr^-i +
CUm-l
+
d,m
=
InSpection,
<
0.
The
search procedure calculates the total inspection cost over the
given by cn^
is
inspection,
+ /IlLi
^xS\n^ f°r
^ 2^
policies.
Dependencies Across Different Board Types
we have been considering only one type
Until now,
types were completely independent, then
measure several board
tests actually
all
of circuit board.
board
would carry ovor. However, some
of our results
tj'pes together, in
If different
which case either
the board types
all
are tested or none are tested. Thus, the inspection allocation decisions taken independently
might yield an infeasible solution. For shared
results), this
where the
tests that are
dependence can be dealt with by employing
total cost of the test
split
is
among
with the cost
t
two board types
split
the allocation of
t
is
feasible
the cost for the board type that
types,
and
if
the solutions
solutions agree.
Due
and hence we are not
algorithm.
test,
We run
still
is
us suppose that the total cost
If
If
the decisions for the two types
the solutions do not agree, then modify
board type that
The problem
is
is
tested
and decreasing
solved again for both board
do not agree, then we repeat the entire procedure
to lack of data, this
in
for the
not tested.
let
the model for the board types independently
and optimal.
by increasing the cost
have l)inary
the different board types such that the solutions
between the two types.
arbitrarily
agree, then the solution
is i.
(i.e.,
Lagrangian relaxation approach,
a
found independently coincide. To illustrate this approach,
of a test that covers
nonparametric
procedure could not be applied
in
until the
our case study,
a position to report on the efficiency of any particular cost allocation
Notice that this approach becomes almost impossible to use with a parametric
because the Lagrange multiplier must be
18
a function of
the testing policy that
is
used.
under study, the
In the plant
comprehensive system
tests,
measured more than one board type were the more
tests that
and hence were nonparametric.
Congestion also leads to dependencies across board types:
are tested at the
Moreover,
if
same
many
stage, then the testing load
may
if
many board
too
types
be undesirably high at that stage.
of these board types are not tested at the previous stages, then
repairs will be required at that stage, which leads to additional work.
many
At considerable com-
putational expense, queueing effects could be incorporated here by a Lagrangian relaxation
method
similar to that described above, or by directly incorporating queueing costs (derived
from a product form queueing network,
for
example) into the objective function.
We
did
not pursue this avenue because the congestion effects at the Hewlett-Packard facility were
negligible.
Readers are referred to Tang (1991)
for
an inspection allocation problem with
queueing costs.
3.4.
Improved Testing Equipment
Our model
explicitly takes into account the effect of
measurement
noise,
us to determine the value of improved testing equipment that possesses lower
noise.
it
This issue
is
very important, since testing equipment
how
not clear, a priori,
is
is
which enables
measurement
extremely expensive and
a reduction in measurement errors affects the overall cost of
inspection.
To evaluate improved
parameters
/Zg
testing
equipment
in the
context of our model, the
and a^ are substituted into equations
procedure described
in
Section 3.2
is
carried out.
(22)-('23)
The optimal
We
to illustrate the
more
precise equipment.
noise
whose standard deviation was
test that detects defective
total inspection cost for the
for the old
tester
had
half of that of the current tester.
in
equipment.
magnitude of the savings achievable with
assumed that a hypothetical new
components
noise
and the exhaustive search
new equipment can then be compared with the corresponding value
An experiment was performed
new
a
measurement
For the
in circuit
our case study, the tradeoff curve for the new and
19
original test
equipment
is
pictured in Figure
5,
which shows that both types of errors could
simultaneously be divided by at least two using the new equipment; the upper curve
figure
identical to the curve in Figure
is
7.
If
components on the board, then the reduction
we assume
in total
in this
similar error reductions for
inspection cost
is
all
on the order of 5
percent.
0.05
0.02
0.01
0.03
0.04
0.05
0.06
0.07
0.03
0.09
0.1
a
New
Test Equipment
Figure
3.5.
5.
Reduction
A Original Test Equipment
in testing errors.
Allocation of Quality Improvement Efforts
Engineering resources to work on quality improvements are limited, and consequentlj'
is
important to dedicate resources to projects that have the
different context, Albin
project
chart
is
is
maximum
it
impact. In a slightly
and Friedman (1991) show that the choice of the most valuable
not always straightforward. In particular, they show that the traditional Pareto
not an adequate tool
Chart would be misleading
when
for the
defects are clustered.
It
is
also possible that a Pareto
problem considered here, since
different types of defects
are detected in different propoitions at different stages and hence the cost of different types
of defects can
be very
different.
20
If
the inspection policy for
respect to
its
i*'^
argument
cost of having one
more
is
a constant for each
defect of type
probability that a defect of type
that
all
defects of type
stages following n
all
i
i
is
m
and
m
arriving at stage
reaching stage
then the derivative of
This constant
i.
leave stage n
i
fixed,
is
is
is
with
Jj,+i
the marginal expected
easily calculated. Let
detected at stage m.
If
be the
q,rn
we assume
are equally likely to be detected, then
9.tm
—1
*'i,Tn
The
probability that a defect of type
As a
result,
leaving stage n
i
n
(
"r O-im
'?''j(l
repaired at stage
is
,o-(-)=
N
m-1
E
n
Hence, under any fixed inspection
tained from reducing type
i
> n
is
-Qtm)-
the expected total inspection cost created by this defect
dJn
m
is
A^
(i-9™)r™ +
9=/
policj', this
defects at stage
quantity
is
n
<?''/
the marginal cost reduction ob-
7i.
Recall that the hierarchical test coverage assumption implies that
more
defects
detectable at each inspection stage; this was modeled by assuming that, on average.
type
i
i
defects appeared per board at stage
of type
i
i
Z^n=l
2.^71
4.
plant has
result,
of type
i
3 inspection stages as
new
working on the root cause of type
defects appearing at every stage.
defects
If
dp,
=1
\
is
'"'"
"m
Case Study
describe the application of this model to the Hewlett-Packard
N=
d,„
defects across the different stages does not change,
then the marginal cost from reducing type
We now
As a
mmiber
defects will simultaneously reduce the
we assume that the proportion
??.
become
shown
in
21
Figure
1.
Approximately
-50
facility.
different
This
board
types are manufactured at this
facility,
The number
line of final products.
of boards
from 1,000 to 10,000 per year. Hence,
different
board types.
grouped the many
and these boards go into various models of a
produced of each type
must be
tests
is
single
relatively low, ranging
flexible in order to
accommodate many
Figure 2 illustrates the basic structure of the tests performed.
different categories of defects that are
based on their similarity
used internally into 7 broad types,
These seven categories are open and
for testing purposes.
We
shorts,
missing or wrong components, defective components, two different soldering defect types
(one
last
much harder
is
to detect than the other), miscellaneous,
category consists entirely of false defects, and occurs when a defect
detected, but no problem can be found.
measurements
for
The nature
on the
circuit
number
of
ranges from about 20 to several hundred.
this facility cur-
all
stages.
and the policy
is
The
all
50 board types. Instead,
its
effectiveness,
and
left
facility uses
no consistent procedure
for
highly dependent on the particular engineer
determining
in
charge of
at this facility estimate that the total cost of inspection represents
would be an enormous task
cost.
to collect data
and derive the proposed inspection policy
we applied our model
in a
very limited
manner
in
The data
collection procedure
is
for
order to exhibit
the appropriate software with the Hewlett-Packard engineers,
are carrying out the implementation.
4.1.
for this last defect tj'pe, the
boards and on their components, which partially explains why
The managers
and our
thought to be
strict tolerances
about half of the total manufacturing
It
A',„,
is
which cannot be revealed, requires very
of the final product,
testing policies,
the test.
Except
each stage and defect type,
rently tests all boards at
4.1
and testing problems. This
who
described in Section
results are given in Section 4.2.
Data Collection
Our model
requires three different types of data.
and the second concerns the occurrence of
inspection stage.
The
last
defects,
and perhaps most
first
type
is
the cost parameters,
and the coverage and
difficult
22
The
reliability of
each
type of data to gather pertains to the
testing process.
Testing and repair are two activities that are closely linked, since they are performed
by the same people
in
the same work area. Their cost includes technician time, floor space,
equipment and overhead
we use estimates
activities.
management
for
To
time.
allocate cost
of the time that different engineers
Both of these
costs vary for each
complexity and design of a board.
of the available data.
board type;
in particular,
number
to estimate these cost parameters
volume
overall
of
of boards processed.
depended on the
level of detail
For some stages, we obtained estimates of each of the different costs
how much
or repair (and diagnosis).
total time the boards
they depend on the
These costs are proportional to the
(engineering, equipment, floor space, etc.) individually. In these cases,
costs according to
activities,
and technicians spend on each of these
inspection that takes place at a stage, and hence to the
The procedure we used
between these two
we
split
each of the
could be allocated either to testing (and testing maintenance)
The
total cost for each of these activities
underwent testing and
was then divided by the
repair, respectively, to obtain a cost rate for
each activity. Finally, multiplying the cost rate by the average time
a particular board type gave the corresponding cost estimate.
it
took to
In the cases
test or repair
where detailed
overhead costs could not be obtained, we multiplied the average time to test and repair a
particular board type by a
study, the repair cost was
type.
Although
this
is
flat
overhead rate that included
assumed
all
the above costs. In the case
to be identical for each defect type for a particular board
an approximation of
reality,
we
did not have enough data to estimate
these costs for each individual defect type. However, the test and repair costs did vary widely
across board types and across inspection stages.
The
analysis
cost of a defect
on a board leaving the plant includes the cost of an on
and repair of the defective board
of the cost of lost goodwill.
defective unit,
and then
We
at the plant,
site repair,
and the quality department's estimate
estimated the expected number of
lost sales
induced by a
set the goodwill cost equal to this quantity times the profit
by one unit.
23
generated
To estimate
n,
we used
cf,„,
number
the expected
historical data
about
tlie total
at the different stages of inspection,
new type
of
number
defects per board appearing at stage
i
of defects per board of each type detected
Then the engineers
<!),„.
in
charge of each test were asked
to estimate the proportion a,n of defects of each type that they felt their test should detect,
and the proportion
boards
of defects of each type detected by the test that were actually
b,n
Some
(false defects).
of these estimates were based on experiments, whereas others
were based on the experience of the engineers.
which
the number of type
is
good
failures per
i
We
also used historical data to estimate
7;,,
board that occurred during the warranty period
of the equipment.
From
this data,
at each stage n,
we
which
a.n
(
Yl
m=\
of defects per board that
become detectable
is
N
=
din
number
calculated the
n-1
'5^""
(
-b,m)
^
+
V>
-
H
for
^tm)
=
?:
1
,
.
.
,
.
and n
/
=
1
,
.
.
.
,
jV,
(25)
m=l
since a failure during the warranty period was considered to be caused by an undetected
defect. For
of type
'tf-ct
type
i
—
errors per board
I
1,
.
.
.
,
/
and stage n —
.
.
.
,
A^,
the historical expected number
is
Om =
and the
1,
historical expected
number
of type
(26)
4>xn^in,
II
errors per board
is
n
An = X]
Tn
y-iTix
-
fpimCi-
-
km))-
(27)
=l
These
historical estimates will be used later. Notice that the procedure culminating in (26)-
(27)
much
is
simpler and probably more reliable than attempting to estimate these historical
quantities via (4)-(5).
To derive the
measured
testing policy,
at each stage. For a
measured should be
in the
we need the input data G,
e(.r)
and
^(.r) for
each quantity
system to function properly, the true value of each quantity
corresponding interval Gink-
It is
often very hard to determine
the exact limit values on a component that will ensure the proper functioning of the board.
24
This problem was avoided here by using the specifications to which the component was
bought. Our justification
is
that even
if
a
component
is
bought with tighter specifications
than actually needed, a component that does not meet these specifications signals some kind
of abnormality, such as physical
after the
equipment has been
To estimate the
damage
utilized for
distribution of the
true value of the quantity measured
or a poor soldering, that
some
of the true value of the
measurement noise e(j) and the distribution
(,(x),
we only have the
is
most components used
are extremely small and fragile.
is
being measured,
to try to isolate a
how
The
mount components, which
depend on many things, such
component from the
is
rest of the circuit
also a delicate task
is
as the type of
guarded (guarding
is
component
the ^-"'hnique used
board) and the topolog\- of the
board. As a result, the measurement noise can only be determined
different
\-ia
experiments
each
for
board type.
At our request, a controlled experiment was performed at the Hewlett-Packard
to study the
measurement noise
testers, also called testheads, are
available testhead.
noise
is
of 79
measurements
different
the
is
true values are almost impossible
at this facility are surface
the measurement
measured
noise. Unfortunately, neither of
Estimating the measurement noise
since the distribution of this noise will
that
distribution of the
of the
recorded by the testing device,
component and the measurement
these two quantities can be estimated independently.
to measure, since
cause a real defect
time.
values at our disposal. This measured value, which
sum
may
at the in circuit test level.
At
facility
this facility, several in circuit
used in parallel, and boards are typically tested on the
first
Consequently, we wanted to find out what portion of the measurement
attributable to variations across different testheads.
A'
=
10 times consecutively on //
=
-3
The experiment repealed each
different testheads for /?
boards of the same type, and generated nearly 24,000 data points.
25
The measurement
10
One measurement
was taken from each of the 79 components on each board, which consisted of 28
capacitors, 13 diodes, 12 transistors and 4 inductors.
=
resistors.
22
noise was modeled
by expressing the measurements
ybhk
=
fi
+
Tb
+
0h
+
+
4'bh
ybhk as
for
tbhk
6=1,..., 5,
/;
=
!,...,//,
k=l
A',
(28)
board, and has zero
mean
where:
fi
is
the reference value for the component being measured;
Tj,
is
the average deviation of the measurements taken on the
and standard deviation
6h
is
6'^
cr^;
the average deviation of measurements taken on the h
head, and has zero
mean and
standard deviation ag;
4'bh
is
the average deviation of measurements taken on the 6'^ head and the
has zero
tbkk
is
mean and standard
deviation
6''*
board, and
cr^,;
the residual variation of a measurement that cannot be explained by the testhead. the
board, or the interaction between the testhead and the board, and has zero
stand a
'1
deviation a.
This model incorporates the implicit assumption that
variance on
mean and
all
testheads.
The data
th*^ r'^sidiial
noise
e
has the same
indicated that this assumption did not hold for
components, which suggests that the three testheads were not
all
ecjually calibrated wlien the
data was gathered. Without this assumption, the estimation of the model parameters would
have been greatly complicated.
Moreover, by only gathering data from three testheads,
a good estimation of the distribution of the variance of the residual noise across different
testheads was not possible.
We
is
estimate
/i
by
y,
which
is
a'
=
the average of
all
measurements taken. The variance
of
e
estimated by
which
is
Ylb=i T,h=\ T,k=iiyt>hk
-
ybh)
BH(K-l)
the variance of successive measurements of the very
same component on the same
testhead, and represents the precision limitation of the tester for the component.
26
The
vari-
ance of
^
is
estimated by
2
^"^
and the variance
of 6
K
{B-1){H-1)
estimated by
is
.2
On
= T.L,{yh-y?
H -\
<
<r'
B
BK
These two quantities represent the amount of variation that
One might
tween testheads.
is
Hnked
to the variations be-
think that a fixed effects model would be
number
for the testheads, since the facility uses only a fixed
more appropriate
of different testheads.
But the
variation between testheads evolves over time as a result of usage, maintenance, calibration,
etc.,
and thus the random
The average
of all
effects
model seems more appropriate.
measurements taken on the
6'^
board equals
/i
+
r;,,
and
Tf,
sidered to be the variation in the measured values associated with the individual
on the
6'^
board.
The
variance of r
.2
<7^
Finally, the estimated
is
can be con-
components
estimated by
= Y:txifh-yf
HK
H
B-\
parameters from the
^'
'^l
statistical
model
in
equation (28) are used to
estimate the parameters of the distributions of the measurement noise and the component
values.
We
assume that
resistor has a
;/^,
the nominal value of the component,
nominal value of 100 ohms), and
al
is
known
(e.g., a
100
ohm
let
/7,
=
y-//^,
=
a^
+
a].
al
=
ol.
+
(29)
a]
(30)
and
(31;
For each of the five component types, the Appendix contains tables of the estimates
that were derived from this experiment.
The
results
27
from
this
experiment were used to
test
assumptions (18)-(19), which were required to derive the proposed testing policy,
By
substituting a^ for a^
than one for
all
Appendix. For
noise, a^,
is
in (18),
79 components.
all
we found
To
.tj
and
x^-
that the machine capability index was greater
assess the validity of (19), readers are referred to the
four inductors in this sample, the estimated variance of the
measurement
higher than the estimated variance of the true values of these components,
o-|.
Hence, these measurements cannot distinguish between good and defective components. The
estimated variance of the measurement noise
the same order of magnitude as the estimated
is
variance of the true values for the 13 diodes, and none of the inductors or diodes satisfy (19).
In contrast, for the other three
component
types, 50 of the 62 components,
and hence 50 of
For these other three component types, most of the comjionents
79 in total, satisfy (19).
have an estimated noise variance much smaller than the estimated variance of their true
values.
These components can be tested with good accuracy. This analysis has important
practical implications:
some components should not be
tested (or, alternatively, a
new
test
should be devised for them).
An
important question
for the design of future expen.aients
is
whether the variance of
the noise associated with the different testheads can be predicted.
It
run experiments on a single testhead than on several testheads, because
experiments are much more
this question, a regression
coefficients of correlation
r^
=
difficult to
is
much
in
the latter case,
schedule into the production process.
was run to predict
a^.
To address
from a and the absolute value of
were consistently high; for example,
r^
=
easier to
fi^.
0.92 for the resistors
The
and
0.98 for the capacitors.
We
also gathered data
from another board type during an actual production run. and
used this data for two purposes.
The data was
first
used to investigate the normalitj^ as-
sumptions on the measurement error and true measurement value, which were also required
to derive xi
and
x^.
These assumptions are
tions are not at our disposal.
distribution, which
is
difficult to validate
because the two distribu-
However, we did assess the normality of the measurement
the convolution of the two distributions in question. Notice that the
28
results
from the controlled experiment are not appropriate
the experiment contains
many
components, we applied the goodness-of-fit
in
assumption, since
The production data
repeated measurements.
measured value from each of 350 components on 80
for testing this
different boards.
test associated
consists of a
For each of the 350
with the kurtosis measurement
equation (27.6a) of Duncan (1986) to the 80 measurement values. For more than 250 of
the components, the normality assumption was not rejected at the
The
sources,
controlled experiments
and the
facility's
95%
significance level.
compete with the regular production runs
emphasis on product output makes
it
for scarce re-
impractical to run a similar
controlled experiment for each of the 50 board types. Hence, the production data was also
used to assess the qualitative similarity between the two board types. More specifically, the
production data enabled us to estimate the total variance of the measurements, a^
and Appendix B of Chevalier (1992) displays the distribution of the
tion for the different
and the corresponding
resistors
We
components of each type on the board.
results in our
Appendix are
have a total standard de\ lation near 0.3%
and
this
earlier
is
For example, most
These
both cases.
in
product, but a more thorough investigation of this issue
experiment described
standard devia-
both appendices, and most capacitors
encouraging, particularly since the two board types plug into
final
a^,
found that these results
strikingly similar.
in
have a total standard deviation between 2 and 4%!
total
+
difi'erent
is still
results are very
subsystems of the
needed.
The
controlled
required to estimate the parameters a^ and a^ individually,
experiment should be repeated on some other board types before concluding that
the parameters do not vary significantly by board type.
Unfortunately, a
much
larger experiment than the
one performed here
is
required to
obtain reliable estimates for a^ and a?; the expected value of the measurement noise was
easier to estimate.
estimate could be
The 95%
off
confidence interval for a typical component was such that the
by about 50%
a 11.1% overestimation described
in
in either direction.
Section 3.1,
we
Comparing
example
of
see that these estimates are not reliable
enough to use with any degree of confidence. To get out
29
this with the
of this quandary,
we attempted
to group
we used
components
in subsets that
for aggregating
had very similar properties.
The key
characteristic
was that the standard deviation of the measurement noise and
the standard deviation of the component values represented nearly the
the nominal value of the component.
Not only
will
components increase the precision of the estimates
same percentage
of
the grouping of operations of different
and
for a^
<t|,
but
it
will also
make
the
tradeoff curve in Figures 5 and 7 less tedious to calculate. Although our aggregation led to
tighter confidence intervals for a^
optimal testing policy
4.2.
and
in a reliable
(T?,
the intervals were
still
too large to implement the
manner.
Results
As mentioned above, we did not apply the model
The model was
first
used to find the optimal inspection allocation policy given the
historical testing policy.
than the types analyzed
board
facility's
For this purpose, three types of circuit boards were chosen that
were representative of the
digital nature of a
to the entire Hewlett-Packard facility.
\-ariety of
in
is
boards manufactured; these board types are different
the Appendix or Chevalier's Appendix B. Since iL^ analog or
one of
its
distinguishing features,
we chose one board type with
mostly digital components, one with mostly analog components, and one with a mixture of
components. These board types were produced
and
their yield ranged
in
volumes that were typical
for the facility,
from relatively low to relatively high. Also, these three board types
did not share any test and thus can be considered independently.
The optimal
testing policy
was derived
components, using the board type analyzed
calculated,
for
in
the in circuit test that detects defective
the Appendix. Hence, 79 (x/,a"u) pairs were
and the resulting expected number of type
to the corresponding quantities
Optimal allocation
under the
policy.
facility's
I
and type
H
errors were
compared
current testing policy.
Figure 6 displays the frequency of the different defect
types detected at each stage for the three board types under consideration. Each defect t}-pe
is
represented by the same pattern on
all
three charts.
30
The
values on each chart are arbitrary
in order to disguise the data, but the relative values across the three charts are
correct; these values are proportional to the quantities
<^,„
approximately
defined in Section 4.1.
shows that the number of defects and the predominant types of defects vary
type
defects as type
1
1
figure
significantly
For example, type 3 boards have roughly three times as
across the different board types.
many
The
board types, and the medium gray defect type
is
predominant on
boards but hardly present on type 2 boards.
70t
Board Type
Figu'c
The
costs
6.
Board Type 2
1
Board Type
Frequency of defects detected at each stage
and the quantities
d,„, Q,n
and
fS,„
in (25)-(27)
type, and also varied considerably across board types.
(27) represent the expected
number
These estimates were used
policy.
of errors per
in
I.
As expected, the optimal
to Figure 6
and Table
I,
we
board type.
were estimated
for
each lioard
Recall that the quantities in (26)-
board under the
facility's historical testing
the exhaustive search procedure described
3.2 to calculate the optimal inspection allocation for each
Table
for each
3
board type, which
is
in
Section
displayed in
allocation varies for the different board types. Referring
see that the system test
is
only bypassed by board type
has the lowest frequency of defects detected at system
test.
2,
which
Since the total inspection cost
represents about half of the total manufacturing cost, the savings realized by the optimal
policy in the different cases are significant.
Testing Policy. The
for
in circuit test for
the board type considered in the Appendix tests
the six different defect types listed earlier.
We
31
focus on the test for defective components,
Table
Board
Type
I.
Optimal
policies
and savings.
55%
Hewlett-Packard engineers have observed that about
the in circuit test are good components.
of our proposed testing policy, Figure 7 also
somewhat
55%
how
different estimates for a^
and
of the
if
still
same order
shows a straight
a^, then the
the tradeoff curve in the neighborhood of
pohcy would
at
performance compares to that
this
line that represents the set of
components are good components.
of the rejected
components
Further analysis found that this percentage did
not vary significantly by board type. To illustrate
policies such that
of the rejected
a,
and a
If
we had obtained
proposed policy would have been on
significant
be achieved. The percentage improvement
of magnitude, which represents about a
improvement over the current
in
Figures 5 and 7 appear to be
5%
reduction in inspection costs
implemented systemwide.
5.
The model presented here
measurement
is
Concluding Remarks
on two key features: hierarchical
built
errors in the testing process.
Although
this
model was developed
work of circuit board assembly, these features are present
in
many
when inspection
is
performed
test
coverage
is
a
ver\-
natural property
manufacturing process; the increasing
test
coverage
is
test
coverage and
in
the frame-
other settings. Hierarchical
at different stages in a
then a direct consequence of the fact
that potential defects are added between inspection stages by the intermediate manufacturing operations. Also, test measurements are typically subject to
some
noise, particularly in
the manufacturing of high precision products.
For the problem considered here, an optimal inspection policy could be pursued for a
modest
size
problem by discretizing the /—dimensional state space of defect types and the
curves that can be generated by the Pareto optimal policies
study reveals that the required precision
in
As a
result,
Howe\er, the case
the parameter estimates are very difficult to
achieve, even after performing controlled experiments
types.
in (20)-(21).
and aggregating similar component
we focus on developing a simple but systematic procedure
the cutoff limits that characterize the testing policy.
33
for deriving
For three representative board types
policy achieves a
10%
20%
to
the case study, the optimal inspection allocation
reduction in expected inspection costs relative to the
under the
historical inspection policy,
in
facility's historical testing policy.
facility's
For the in circuit test
that detects defective components, the proposed testing policy significantly outperforms the
facility's historical testing policy
in inspection costs.
manufacturing
on one board type, representing roughly a
Moreover, both policies are relatively
in practice.
The Hewlett-Packard
investigation of
facility
measurement
is
errors
Since a thorough
just beginning to apply our model.
had never been undertaken
at this facility,
perhaps the
We
encountered
biggest contribution thus far has been the statistical analysis of the data.
many
cases, previously
error
was an order of magnitude larger than the variance of the measurement
tables in the
some quick
unbeknownst
Appendix
for
to the facility's engineers,
some examples. These
measurement noise
fairly tight cutoff limits
is
where the mean measurement
error: see the
tables are extremely useful for developing
insights into the optimal testing policy.
R170 and R317 should perhaps not be
since the
reduction
Since the cost of inspection represents about half of the total direct
cost, these cost savings are significant.
ea^y to implement
5%
For example, resistors such as Rl-58,
tested (or at least ha^'e extremely slack cutoff limits),
so large relative to the true
component
can be set for most of the resistors, where o^ja^
noise.
is
In contrast,
very small.
has begun implementing the proposed testing policy, but has not yet implemented
facility
the optimal inspection allocation policy. Although they plan to implement the latter
the different inspection stages are
still
The
managed by
different people,
polic}-,
and organizational barriers
need to be overcome.
We
started from a real problem at a specific industrial facility, and developed and ana-
lyzed a mathematical
crude. Nevertheless,
els for
the large
model
it is
number
higher performance that
to address this problem; however, the
hoped that
this
work
will
model
is still
help future researchers find better
of problems of this nature that are beginning to emerge.
is
sought from manufactured products,
34
relatively
it
seems very
mod-
With the
likely that
inspection will
respect to
become an
increasingly important aspect of production
model refinement, we believe that a key shortcoming
in this
independence between the distribution of measurement values
and the inspection policy used at previous
dependence
is
to
stages.
A
natural
assume that the true measurement value
policy at previous stages only through 6,n_], which
is
the quantities a,„ and
policy T,„.
v\^^
l3,„
in (7)
All that remains
is
model
is
the assumed
at a given inspection stage
first
step towards modeling this
iij^f.
depends on the inspection
the expected number of type
present on a board at stage n. Under this assumption, the dynamic
and the derivation of the Pareto optimal testing
management. With
policies
still
i
programming framework
hold; the only difference
and (16) are now functions of
defects
(5,,„_i,
is
that
as well as the testing
the formidable task of estimating the influence of S^n-i on
from available data.
Acknowledgment
We
are grateful to the people at the Hewlett-Packard plant
advice and their tremendous effort
We
also
in
Andover. .MA
for their
helping us obtain the necessary data for our model.
thank Arnie Barnett and Steve Pollock
supported by a grant from the Leaders
Science Foundation Grant
in
for
for helpful discussions.
Manufacturing program at
This research
MIT and
is
National
Award No. DDM-9057297.
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36
in
Table
Component
II.
Resistors.
Table IV. Capacitors.
Component
Table VI. Diodes.
Component
Date Due
MIT
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