t^
HD28
.M414
ALFRED
P.
WORKING PAPER
SLOAN SCHOOL OF MANAGEMENT
Sequential Screening in Semiconductor
Manufacturing, I: Exploiting Lot-to-Lot
Variability
Jihong
Ou
and Lawrence M. Wein
WP# 3451-92-MSA
July,
1992
MASSACHUSETTS
INSTITUTE OF TECHNOLOGY
50 MEMORIAL DRIVE
CAMBRIDGE, MASSACHUSETTS 02139
Sequential Screening in Semiconductor
Manufacturing, I: Exploiting Lot-to-Lot
Variability
Jihong
Ou
and Lawrence M. Wein
WP# 3451-92-MSA
July,
1992
MIT, LIBRARIES
\UG 2 7 1992-
REOclvED
SEQUENTIAL SCREENING IN SEMICONDUCTOR MANUFACTURING,
I:
EXPLOITING LOT-TO-LOT VARIABILITY
Jihong
Ou
Operations Research Center, M.I.T.
and
Lawrence M. Wein
Sloan School of Management, M.I.T.
Abstract
We
address a problem of simultaneous quality and quantity control motivated by
semiconductor manufacturing. After wafers are fabricated, they are probed, or electrically
tested, ajid in
turing process.
some
cases the probing facility
Under
this
is
the bottleneck for the entire IC manufac-
assumption, we consider the problem of choosing the optimal
start rate of lots of wafers into the fabrication facility
front of the probing facility to
and the optimal screening policy
maximize the expected
good chips minus the variable fabrication and probing
profit,
costs.
which
is
in
the revenue from
The screening
policy decides
which wafers to discard and which wafers to probe. These decisions axe subject to capacity constraints at
approach
random
is
both the wafer fabrication and probing
facihties.
employed: the number of bad chips on a wafer
variable,
ing to another
where the
gamma
scaie
parameter
distribution.
We
fit
is
unknown and
is
An
empirical Bayes
to be a
gamma
lot to lot
accord-
assumed
varies
from
the yield model to industrial data and test the
optimcd policy on this data.
July 1992
SEQUENTIAL SCREENING IN SEMICONDUCTOR MANUFACTURING,
EXPLOITING LOT-TO-LOT VARIABILITY
I:
Ou
Jihong
Operations Research Center. M.I.T.
and
Lawrence M. Wein
Sloan School of Management, M.I.T.
This paper and
management
will focus
companion (Longtin
main
1992) address a particular quality
et al.
issue in semiconductor manufacturing.
consists of four
we
its
The production
of integrated circuits
and
stages: wafer fabrication, probing, packaging
final testing,
on the interrelationship between wafer fabrication and probing.
and
In wafer
fabrication, disc-like wafers that contain hundreds of integrated circuits, or chips, are
produced
(in batches, or lots, of usually
20 to 50 wafers) by a very long and complex
procedure involving hundreds of operations.
on a wafer
good
chips.
is
After fabrication
completed, each chip
is
probed, or electrically tested, to distinguish between defective chips and
Each wafer
is
then separated into
its
respective chips, and nondefective
chips are covered in a protective plastic during packaging. In final testing, the chips are
functionally tested under a variety of environmental conditions before being shipped to
customers.
Problem Description. To
and the key material flow
issues
motivate our model formulation, the process economics
need to to be
briefly described. Building a wafer fabrica-
tion faciUty, or fab, costs hundreds of millions of dollars,
and consequently, fab managers
are very concerned with maintaining high utilization of the bottleneck equipment, and
one of the biggest operational decisions
rate of wafers into the fab.
fab (due primarily to
details),
for the fab
manager
Because of the huge amount of
random
yield, rework,
and
pushing the start rate beyond a certain
1
is
to determine the start
statistical variabihty in the
tool failures; see
level,
which we
Chen
call
et al.
1988 for
the fab's effective
capacity, will result in unacceptably high levels of work-in-process inventory and long
lead times.
Despite the well-documented congestion that occurs
where the probing (we
visited several facilities
rather than probing)
facility,
in
the more generic term testing
will often use
not the wafer fab,
is
wafer fabrication, we have
the bottleneck that determines the
production capacity of finished goods. There are several reasons for this phenomenon: the
testing
equipment
is
very expensive (machines can cost three to four million dollars) and
the testing procedure
testing capacity
is
is
a very time consuming and labor intensive process. Furthermore,
sometimes labor constrained because companies are either unable or
unwilling to hire and train full-time employees in the face of uncertain future demand.
Although
commodity
relative costs
or
custom
and revenues depend greatly on the type of market
chips)
and other
factors, the variable testing cost per wafer
typically only several percent of total variable production cost,
wafer of nondefective chips
is
constrained, they can
to
is
an increase
sell
in profit.
is
and the revenue from a
roughly ten times the variable production cost per wafer.
Also, the yield in wafer fabrication, which
very low and erratic. Since
(e.g.,
many
is
the fraction of chips that are good, can be
facilities are
capacity constrained rather than market
anything they make, and any increase
in yield leads directly
Consequently, yield dominates the economics of the process and
the primary concern of fab managers.
Semiconductor manufacturers typically use an exhaustive testing
chip of every wafer
is
tested and
is
deemed
it
undergoes testing (unless
from a custom order that has already been
grain of mainstream industry thinking.
is,
every
defective or nondefective. Indeed, the thought
of simply discarding a completed chip before
"leftovers"
policy; that
filled)
In contrast, our
goes very
represents
it
much
against the
paper and Longtin
based on the following simple premise that has also been put forth
in
et al.
axe
Goldratt and Cox
(1984): profitability can be increased by preventing bottleneck equipment from working
on products that are already defective. In particular,
under an exhaustive testing
policy,
if
testing
is
the bottleneck operation
then semiconductor manufacturers can increase their
profits
by simultaneously
(1)
employing a sequential screening procedure that adaplively
discards, rather than tests, portions of wafers (or entire wafers or even entire lots) that are
thought to have a sufficiently low proportion of nondefective chips, and
start rate of wafers.
in
Of
course,
if
the rate of wafer starts
is
(2) increasing the
increased, so
is
the congestion
the fab and the production costs, and these two factors need to be taken into account.
To
test this premise,
we consider the
following problem of simultaneous quality
quantity control: determine the start rate of
lots of
and
wafers into the fab and find a sequen-
screening policy for the testing facility to maximize the expected long run average
tial
revenue from nondefective chips minus the variable fabrication and testing costs of wafers.
The two
controls are subject to constraints on the average effective capacity of both the
fab and the testing station;
strictive
we assume
that the testing capacity constraint
than the fab constraint when an exhaustive testing policy
is
is
more
re-
To minimize
in use.
confusion, a screening procedure in isolation will be referred to as a policy and a screening
pohcy coupled with
a start rate will be referred to as a strategy.
In practice, the resulting increase in profit that
tive to the exhaustive testing strategy
an optimal strategy
commonly used
in
industry (that
will
is,
an exhaustive
testing policy with a start rate that keeps the testing facility working at
capacity) depends greatly on two factors that will be discussed below:
congestion levels of the fab and the testing
and
(2) the
facility
nature of the yield variability. Indeed,
than the testing
facility
achieve rela-
its
effective
(1) the relative
under the exhaustive testing
if
under an exhaustive testing
the fab was more highly congested
policy,
then this policy might be
optimal, and hence sequential screening would be of no value. However, testing
performed after various key operations
tion
in
policy,
is
also
the fab, and often (for example, see the simula-
models of Atherton and Dayhoff 1985, Glassey and Resende 1988 and Wein 1988) the
bottleneck workstation in the fab
make many (up
is
the photolithography workstation, to which wafers
to twenty) visits during their processing. Thus, the
framework presented
here can also be used to perform sequential screening at key tests in the fab. That
start rate of wafers
is,
the
can be increased and undesirable wafers or chips can be discarded at
in-fab tests so that the bottleneck
procedure
this
type
II
may
not be as effective
errors are apt to be
often visual and
is
equipment works on higher quality
in
the fab as
more prevalent
in
it
is
the fab.
at probe, because type
at
an in-fab
to the bottleneck workstation.
We
test
may become
will hereafter
defective before
is
at
90%
of
its
effective capacity
and
assume that testing
is
when the exhaustive
is
correctly
is
next
its
visit
the bottleneck
operation, and more specifically, our numerical studies here and in Longtin et
that the fab
I
In particular, in-fab testing
not as discerning as electrical testing, and a chip that
found to be nondefective
However,
chips.
assume
al.
testing strategy
is
employed.
Yield Modeling.
We now
in wafer fabrication occurs for a variety of reasons, including short
particulate contamination (see
Osburn
heavily on the probing machines.
effective
After
if
difficult to detect visually,
life
yield
cycles,
and
and the industry
Intuitively, sequential screening will only
dependencies and/or nonuniformities
all, if
product
1988), misalignment of operations,
et al.
chemical imbalances. Also, defective chips are
relies
Low
discuss the nature of the yield variability.
in yield
be
can be identified and exploited.
every chip processed by the fab had the same probability of being defective,
independently of
all
other chips, then sequential screening would be fruitless. However,
several types of dependencies do exist and, indeed, one of our primary goals in this pair of
papers
and
is
to analyze industrial data
and determine which dependencies are most prevalent
easiest to exploit.
Recall that chips are produced on wafers, and wafers travel through the fab
Dependencies
(1)
may be
present at
all
in lots.
three levels (lots, wafers, chips), including
dependence across consecutive
lots:
the yield of consecutive lots
may be
positively
correlated because of machines that go in and out of control, or batch operations, such
as diffusion or oxidation, that simultaneously process multiple lots;
(2)
nonuniformity
than others;
in
chip type:
some chip types may be inherently
easier to
produce
(3)
a lot
dependence of wafers within a
may be due
positive serial correlation of wafer yields within
lot:
one or more
to operations that simultaneously process
lots of wafers,
or to wafer- by- wafer operations that incur a joint set-up for an entire lot;
dependence of neighboring chip locations on a
(4)
found
(see Mallory et
in clusters
wafer,
defective chips are often
1983 for empirical data), which
al.
may be due
to
processing or particulate contamination;
(5) radial
nonuniformity on a wafer:
handling and processing can cause a donut-
shaped yield with more defective chips on the edge of the wafer and. to a
in
the center of the wafer (see Ferris-Prabhu et
al.
1987 for empirical data): and
dependence of a chip location across wafers within a
(6)
lesser extent,
lot:
mask
and batch
defects
operations can cause the yield of a chip location to be positively correlated across consecutive wafers.
(i)
Furthermore, sequential screening can be performed at
all
entire lots of wafers based on the yield
lots, (ii)
from previous
on the yield of previously tested wafers from the same
lot.
three levels:
or
(iii)
we can
wafers in a
We
dependency
wafer fabrication; since a wafer fab
not very prevalent
is
in
do not pursue screening of type
flow line operation, lots that are processed together in the
lot
based
chips on a wafer based
on the yield of previously tested chips.
(1)
discard
(i)
is
same oven during a
because
far
from a
particular
batch operation tend to go their separate ways and do not arrive together at the testing
facihty.
type,
Also,
all
the industrial data sets that we analyze contain lots of only one chip
and hence nonuniformity
(2) will not
be addressed. However, this factor could be
addressed in our framework by developing a different yield model
Our two
studies
dencies (3)-(6).
randomness
ity in
mean
The
employ sequential screening
factors underlying
of types
dependency
(3),
(ii)
and
for
each type of chip.
(iii)
to exploit depen-
coupled with the high degree of
in the
production process, lead to a significant amount of lot-to-lot variabil-
yield.
In this paper, sequential screening of type
lot-to-lot variabiUty.
Our
yield
(ii) is
model assumes that the number
employed
to exploit
of defective chips
on each
wafer in a given
lot
and scale parameter
(3 is
unknown and
an independent
is
fi.
An
varies
dently from a (different)
setting decides
when
empirical Bayes approach
from
lot to lot; for
gamma
each
lot,
prior distribution.
variable with shape parameter
is
used, where the scale parameter
the parameter 3
A
in this paper.
chosen indepen-
is
sequential screening policy
Bohn (1991)
is
used to estimate the parameters
Since the primitive empirical data
in
Bohn
is
wafer
and temporal dependencies described
maps
in this
Appendix
(see the
and model the chip
yield by a
in (4)-(6).
of that paper for
Markov random
Longtin
et al.
for the yield
number
the
defective chips on each wafer, this data cannot be used to analyze the
spatial
q
to discard the remaining wafers in a lot.
Industrial data from
model
gamma random
of non-
more detailed
analyze over 300
some e.xamples) from two wafer
field,
which
is
fabs.
a stochastic model that
allows the probability of a chip being nondefective to depend on the resulting yield of the
neighboring chips.
A
variety of sequential screening strategies of type
(iii)
are proposed
that discard individual chips on a wafer. In summary, the present paper employs sequential
screening at the wafer level to exploit lot-to-lot variability and Longtin et
al.
employ
sequential screening at the chip level to exploit detailed spatial dependencies within a
lot.
The two key
spatial
aspects of yield modeling that we focus on, lot-to-lot variablity and
dependence on and across wafers, have received very
modeling hterature.
We know
of
little
attention in the IC yield
no models capturing the former aspect and Flack (1985)
appears to contain the only yield model that explicitly accounts for spatial dependence
of chips
on a wafer. Nearly
the existing yield literature (see
all
Cunningham 1990
for a
recent survey) calculates the proportion of nondefective chips on a wafer by considering
the chip axea and density of point defects on the wafer. These derivations lead to a two
parameter distribution (the negative binomial distribution, which describes a Poisson
random
variable mixed with a
fitted to the
mean and
gamma, appears
to
be the most
variance of the empirical data for the
chips per wafer. According to
Cunningham, the goal
of
effective) that
number
most of the chip
can be
of nondefective
yield
modeling
research has been to predict costs and actual yields, and to determine the appropriate
level of circuit integration.
appears to be the
to
first
Albin and Friedman's (1989) work on acceptance sampling
employ a
yield
model
parameter distribution (the N'eyman type A. which
model the number
they use a two
in a quality control context:
is
a Poisson
compounded
Poisson) to
on a wafer. Because they were interested
of defective chips
in
quality
control issues rather than circuit design issues, they directly modeled the yield without
resorting to the defect density and chip area, and
Summary
tially
The optimization problem addressed
of Results.
in this pair of papers.
in this
paper
essen-
is
an optimal stopping problem embedded within a mathematical program, and the
optimal solution
is
determined numerically by solving a
stopping problems. Since the optimal strategy
optimal fixed sample
after
we do the same
where a
size strategy,
fixed
which the controller either discards or
is
parameterized optimal
series of
difficult to calculate,
number
we
also find the
of wafers from each lot
tests all the
remaining wafers
is
tested,
Five
in a lot.
of Bohn's industrial data sets are used to estimate the parameters of the yield model,
and the two proposed
maximum
the
policies are derived for all five
data
sets.
For our parameter values,
possible profit increase that an optimal strategy can achieve relative to
the exhaustive strategy
commonly used
in
industry
is
between 11.1% and 12%; the exact
upper bound cannot be mentioned without revealing the true
The
fixed
sample
size strategy
profit increase over the five
randomly
sets, respectively.
also tested
on the actual data
in a
simulation study.
shuffling the wafers in a lot, 100 lots of wafers are generated
the five data
sets.
If
Bohn's wafers.
and the optimal strategy achieve a 2.2% and 2.5% average
data
These two strategies are
yield of
from each
By
lot in
the yield model underestimates the average number of discarded
wafers per lot in the simulation study, then the testing facility will be underutilized and a
suboptimal strategy
is
obtained.
If
discarded wafers, then the testing
the yield model overestimates the average
facility will
number
of
be overutilized, and an infeasible solution
can result. Under the fixed sample size strategy, the model accurately predicts the average
number
of discarded wafers per lot,
and an average
profit increase of
1.2%
is
achieved.
Under the optimal
model underestimates the average number
strategy, the
of discarded
wafers by an average of 2.5% over the five data sets, which results in an average profit
decrease of 0.7%.
In
since
summary, the
it
is
much
is
size strategy
easier to derive
analytical model,
We
sample
fixed
and appears
and
to be
to
may be
implement,
preferable to the optimal strategy,
performs nearly as well on the
it
more robust when faced with the actual data
sets.
believe that the discrepancy between the theoretical results and the simulation results
due primarily
parameter
q.
to the
assumption that
the same data set have the same shape
Hence, a relaxation of this assumption
more accurate estimate
lead to a
all lots in
more
of the average
effective
and
number
is
probably required to obtain a
of discarded chips per lot, which should
reliable strategy.
The
profit increases reported here are
relatively small and, in particular, are significantly smaller the the increases achieved by
screening at the chip level in Longtin et
a
1%
al.
However, readers should keep
in
mind that
increase in revenue minus variable cost can represent millions of dollars annually.
Also, since the fixed cost
would translate
into a
component
much
is
1% improvement
so large in this industry, a
larger percentage
improvement
in
here
a company's reported
profits.
The remainder
of the paper
is
organized as follows. In Section
I,
the yield model
is
described in detail, and the modeling assumptions are compared with the conclusions of
Bohn's empirical study. The stochastic optimization problem
The optimal
fixed
sample
size screening strategy
sequential screening strategy
Section
7 of
5.
is
is
derived in Section
Concluding remarks on
this
found
4.
is
formulated
in
Section
Section 3, and the optimal
in
Numerical results are reported
paper and Longtin et
2.
al.
can be found
in
in Section
the latter paper.
1.
The Yield Model and
Our yield model assumes
Industrial
that the
number
Data
of defective chips
on each wafer
in
a given
lot
is
J.
an independent
An
gamma random
empirical Bayes approach
for all lots
is
is
parameters a and
h.
gamma
chosen independently from a
The two gamma
lot
gamma
gamma
(a
+
is
the
prior distribution with
distributions form a conjugate pair,
are found to be defective, then 3 has a
1
scale parameter
same
to lot: for each lot. the value of
ia.b) distribution prior to testing a wafer,
Figure
q and
used, where the shape parameter q
but the scale parameter J varies from
the parameter J
J has a
variable with shape parameter
a,b
+
and
if .r
if
known
the parameter
chips on the wafer
x) posterior distribution.
Die Yields of 11 Batches from Factory
CI
of
good
on each wafer of each
chips, or die.
Bohn. which displays a summary of data
to a lot of wafers
lot.
In Figure
we reproduce Figure
1.
Cl. Each column in Figure
corresponds
and each point represents the number of good chips on
a particular
set
essentially contains 11 yield histograms, one for each lot.
1
Ccime to the following three conclusions concerning his 11 data sets:
of each lot varies considerably from lot to lot (e.g..
there
lot
is
variabihty
(i.e..
compare the
considering the
gamma-gamma
we performed our
pair,
binomial pair and the gamma-Poisson
intuitive appeal, since the
integer between zero
and
(iii)
compare the second
number
of
and the number
pair: the
specifically,
the five data
sets.
lot.
ratio over all 53 lots
was
is
less
than (equal
the controls derived from these two
\-ield
modeled as
ein
However, the binomial and
lot
variabihty of chip yield.
number
of the
7.6
30.3. In contrast, the corresponding \ariance- to-mean ratio
respectively) assumption
explicitly
and determined the variance- to- mean
The average
In fact, before
this effect.
is
on a wafer.
we calculated the mean and variance
each wafer of a given
yield.
beta-binomial model, in particular, has
bad chips per wafer
of chips
mean
entire analysis using the beta-
Poisson assumptions significantly underestimate the within
of
good chips on
ratio for each lot in
and the range was
1.8 to
under the binomial (Poisson.
to. respectively) one.
Consequently, when
models were tested on the actual data, too
wafers were discarded at the testing facihty. which led to a significant reduction
in overadl profit.
level of
within
the
high:
is
yield
lots): (ii)
certainly captures the lot-to-lot variabihty in
However, plenty of other conjugate pairs would also capture
mamy
two
mean
Bohn
lots).
The gamma-gamma model
More
last
lot variability (e.g..
the
(i)
the vertical spread of points in each column)
a high variation between lots of within
and seventh
of
1
wafer; hence. Figure
within
1
within
lot
.Although the
lot variabihty.
variabihty.
scale peirameters are
it
gamma-gamma
is
conjugate pair captures the substantial
unable to capture the high variation between
Perhaps a gamma- gamma model
unknown would capture
this effect:
prevented us from pursuing this avenue. In summary, the
the effects in conclusion
(i)
and
(ii),
in
which both the shape and
computationad considerations
gamma-gamma model
but does not capture the
10
lots of
effect in
captures
conclusion
(iii).
To
further investigate the validity of our model,
data sets
This
is
may
on the actual
note that the number of wafers
in
a lot
is
not constant in Figure
due to the scrapping of entire wafers during fabrication. Hence, we also assume
that each wafer in a
has a certain probabihty of being scrapped during fabrication,
lot
so that the size of a lot exiting the fab
2.
test the derived policies
in Section 5.
Finally, readers
1.
we
a binomial
is
random
variable.
Problem Formulation
In this section,
we mathematically formulate the optimization problem described
the Introduction, and pictured in Figure
and each wafer consists of
with probability
q.
M
chips.
and hence the
variable with parameters L and
when the
lot arrives to
1
Each wafer
lot size
— q.
Each
2.
/
lot
the testing faciHty.
entering the fab contains I wafers
in a lot
is
scrapped during fabrication
of a wafer exiting the fab
Since the exact
it is
in
number
is
a binomial
of wafers in a lot
random
is
known
natural to use this information to develop
an optimal screening policy. However, this would require us to derive a different optimal
screening pohcy for every possible value of
more
difficult to
compute and harder
our screening policy to
more than
/
differ
this
which makes the optimal solution much
implement
lot to lot.
wafers can be tested from a
about 5-10% of their wafers,
in
from
to
/.
lot
in practice.
Instead,
we do not allow
except for the obvious constraint that no
with
/
wafers. Since
most fabs
t\-pically scrap
assumption should not lead to significant degradations
performance.
untested wafer?
scrapped
qlA wafers week
'
good wafers
TESTING
(l-q)LX wafers«'weeK
X lots/week
Mj wafers. week
bad wafers
Figure
2.
The semiconductor manufacturing
11
faciHty.
For a typical
=
wafer, for n
the
u„
u
=
/.
1
and
let .s^
=
let
Xn be the number of defective chips on the nth
Ill=\
be the total number of defective chips on
J^k
=
n wafers. Let the decision variable u„
first
=
exiting the fab.
lot
the nth wafer
if
(ui,U2,
....
where u„
the nth wafer
be discarded. A screening policy
=
for
n
>
=
For n
/.
1
/,
g(xn,u„)
=
=
and
defined by the vector
is
0.
The
^in
depends on
profit
generated
if
u„
=
0,
if
u„
=
1,
(i:
<
r{M r
to be tested
is
''
where
is
the decision
only through the sufficient statistic 5„_i, where ^o
(xi, ...,Xn_i)
bv wafer n
ul).
to
is
if
1
-
ir,)
ct
the revenue received from a good chip and cj
is
is
the variable testing cost per
wafer. For a given policy u. the expected profit from testing one lot of wafers
is
L
V{u)
=
E[Y,g(^n.Ur,)l
(2)
n= \
and the expected number
of wafers tested per lot
is
N{u) = E['tunl
(3)
n=l
where both expectations are over the random variables
definition of
and
//„.
The problem
decision variable
A,
which
the
is
of finding a screening policy u that
of simultaneous quality
We
its effective
levels will
maximizes
(2)
is
is
embedded
in
the
number
of lots introduced into the fab per week.
assume that
if
let
an optimal stopping
and quantity control involves one more
and two extra constraints. The decision variable
capacity be ftp lots per week and
per week.
which
(xi,...,j().
Our problem
problem.
/,
is
the
lot start
Let the fab's effective
the testing facility's effective capacity be
the rate of work entering either of these
capacity, then unacceptably high lead times
rate
and work
in
hj wafers
facilities
exceeds
process inventory
be incurred. Hence, the two constraints are
and
XN{U) <
12
fiT.
(5)
Our optimization problem
to choose the start rate A
is
and a screening pohcy u
to
maximize
\{V(u)-cr)
subject to constraints (4) and (5), where Cf
We
conclude
any testing
is
this section
is
the variable fabrication cost per
lot.
with some assumptions on the problem parameters. Before
performed, the a priori expected number of defective chips on a wafer
E[Xn]
=
To ensure that
this
quantity
positive,
is
—
(7)
I
we need
to
assume that
the parameter estimates obtained from Bohn's data
exhaustive testing policy by u
V'(u^)
,
is
-.
a
for
(6)
Section
in
a
>
5.
1,
If
which holds
we denote the
then
=
(1
- q)L[r{M -
- CtI
-^)
a —
(8)
[
and
N{u^) = {\-q)L.
We
(9)
assume
^^
(1
so that the testing facility
is
(101
-q)L
the bottleneck under the exhaustive testing policy, and
r(A/-^)-cx>-^.
— q)L
a —
so that exhaustive testing
3.
is
(11)
(1
I
profitable.
The Optimal Fixed Sample
Size Screening Policy
Since the optimal solution (A,u) to problem (4)-(6)
is
difficult to obtain,
ourselves in this section to a fixed sample size screening policy, which
Under
this strategy, the
number
of wafers tested
13
from a
lot is
is
we
restrict
denoted by
min{n,/}.
If
u"-
.
the total
number
of defective chips found in these wafers
remaining wafers
the
in
than or equal to B. then the
less
is
are tested; otherwise, the remaining wafers are discarded.
lot
Standard calculations show that
6-^7'^"^
+ (n-l)a)
r(a
,
..
'""' ^^'-'^ r(a)r((n-l)a)(6 + .„_,r<"-"-'
is
the probability density function for the number of bad chips on the
tested,
first
n
—
1
'
wafers
and
=—+
^x„|s„_i
a
is
'
'
the expected
number
chips are found on the
testing facihty has
/
(n
—
i)Q
—
(13)
1
of defective chips on the nth wafer, given that Sn-i defective
first
wafers
—
n
is
I
wafers.
.Also,
the probability that a
lot
entering the
given by
\
L
=
H{1)
I'-'iL-l)".
(14)
"
/
Hence, the expected profit per
V^K"^)
lot of
wafers
is
= J2HU){rl{M-E[x^])-lcT}
1=0
+
X^ //(/)r{n(M-£[x„]) + (/-n) /
-
^
//(/)cT{n
+ (/-n)
-
£[x„+,|5„])/(.sjrf.s,}
(15)
f{sn)dsr,}
/
-
l=n+l
and the expected number
iV(t/"-^)
(A/
of wafers tested per lot
= f^
//(/)/+
is
X^ //(0{n + (/-n) /
f(Sn]dsn}.
(16)
Thus, problem (4)-(6) reduces to
max
subject to
A(V(u"'^)
A
<
c/r)
(1")
(IS)
/zf
AAr(u"'^)
14
-
<
MT,
(19)
which
is
equivalent to
max
mm{fiF.fiT/y{u''-'')}{Viu'''")
-
cp).
(20)
Since a closed form solution to (20) appears to be unattainable, we exhaustively
enumerate over the integer values {n.B
solution.
The
V(u-^)
:
<
n
< 1:0 < B < n.M)
to find the optimal
calculations are considerably streamlined by observing that
=
V{u--^-')+ Y. H{l)T{l-n)l
L
-
.B
f{Sr.)ds^
(21'
= .V(u-s-i)+ Y. H[l)[l-n)r f{sjds„.
(22)
Y, H{l)cT[l-n)
/
{M-E[x^^,\sr.])f{sr,)ds^
/
=n+l
and
A'(u-S)
Hence,
for
each B. only /g., £'[j„+i|5n]/(5„)(is„ and fg_^ f(Sr,)dsn have to be calculated.
The Optimal
4.
Solution
In this section, a computational procedure
is
essentially
First
is
developed to solve problem
(4)-(6).
which
an optimal stopping problem embedded within a mathematical program.
we reformulate the problem
into the equivalent two-step maximization
problem
max \{V\-cp)
(23)
0<A<MF
where
Vx
=
maxV(u)
subject to iV(u)
Proposition
only
if
u'
is
1.
(A*.u*)
is
(24)
<
^.
(25)
an optimal solution to problem problem (4)-(6)
an optimal solution to problem (24)-(25) with A
15
=
A* in (25),
if
and
and
A*
is
(he optimal solution to (23).
The optimal objective function value
is
the
same
for
both
problems.
Proof.
with A
=
(X\u')
If
A' and, for
an optimal solution to problem
is
any screening policy u satisfying
(4)-(6), then u' satisfies (25)
this condition,
we have
X'{V{u-)-cp-)>\-{V(u)-cr)
(26)
or
V{in >
Hence, u'
is
V{u).
(27)
an optimal solution to problem (24)-(25) with A
=
A' in (25),
and
V'(u")
=
Vx'.
Observe that
X-{V(u')-Cf)>\{V[u)-cr)
for all A
and
u satisfying (4)
and
(5).
(28)
Fixing A and maximizing over u subject to (25)
yields
^'{Vx'-cf)>\(Vx-cf)
for all A
<
such that
A
<
optimal objective function value
Conversely,
u'
if
is
Therefore, A*
f.if.
is
is
(29)
an optimal solution to (23) and the
the same for both problems.
an optimal solution to (24)-(25) and A*
(23), then they jointly satisfy constraints (4)
and
(5).
is
an optimal solution to
For any other feasible solution
(A,u) to (4)-(6), we have
XiViu) - of) < A(Vx - cr] < X'{Vx' - cp)
which implies that (A*,u*)
function value
is
the
same
is
for
=
X'(V{u')
-
c/r),
(30)
an optimal solution to (4)-(6), and the optimal objective
both problems.
I
Let u° be the screening policy that maximizes the function V(u) defined
Maximizing V(u)
is
an optimal stopping problem and
section.
16
will
be discussed
in
(2).
later in this
Proposition
and
\'
=
<
2. If .\'{u°)
the optimal solution to (23)-(25)
ht/i-I-f- f/^en
is
u'
=
fj.f.
Proof. Since the screening strategy «° maximizes V{u) with no side constraints,
also
maximizes (24)-(25)
for all A
that u° optimizes (24)-(25) for
profitable,
and hence
I
(ti°)
Thus, when .V(u°)
<
We now
/ir/.V(u°)].
[0.
G
A
all
>
6
[0,/i/r].
and setting
A
By
=
Since N(ii°)
<
^ij/nr-
^ip
all
A
€
[0.
is
I
not used to
is
it
follows
optimizes (23).
its
full
effective
obtained by solving a single optimal stopping
is
consider the more interesting situation where N(u^)
u° optimizes (24)-(25) for
it
(11). the exhaustive testing policy
jij/np. the probing facility
capacity, and the solution to (4)-(6)
problem.
u°
//T/^'(»°)]. (23) can be replaced
max
\{\\-cf).
>
j.ljI i.Lf.
Since
by
(31)
mt/.V("°)<a<mf
If
problem (24)-(25) can be solved
efficiently for a given A,
over A G [/ir/.V(u°)./i/r] for the largest value of A(Va
to our original problem.
Since (24)-(25)
is
—
then a one-dimensional search
cp) will yield an optimal solution
a constrained optimal stopping problem,
we
solve this problem by employing a Lagrangian approach. Let 7 be the Lagrange multiplier
for constraint (25)
and define
u„
=
0.
ifu„
=
l.
if
5"'(x„,{in)
=
(32)
<
r(iV/-x„)-CT-7
Notice that 7 plays the role of an additional testing cost, so that the total testing cost
per wafer
is
cj
+
7.
Define the Lagrangian function
Viu)
=
V{u)-tN{u)
=
£[^^(x„,u„)l-7^(u)
L
n=l
=
E[X;g-(x„,u„)],
(33)
n= l
and consider the Lagrangian problem
maxV^(u).
U
17
(34)
Proposition
7
>
0,
Lagrangian problenn for some
3. If the screening policy u'{f) solves the
and
vV(u-(7))
then u'{f)
is
= y.
(35)
the optimal solution to problem ('24)-(25).
Proof. For any screening strategy u satisfying
V(u)
(25),
<
V'(u)-7iV(u)
<
V{u'it))-fN{u'{f)) +
=
V'(u-(7)).
+
7^
f^
(36)
I
Since 7 enters the Lagrangian problem as an additional testing cost,
to
show that the optimal objective function value
function of 7.
The
in (34)
proof of the following proposition
that the optimal expected
nonincreasing function of
number
on
this conjecture has
not hard
is
a continuous nonincreasing
this fact
of wafers tested per lot N{u'(f))
Although
7.
relies
is
it
and the conjecture
is
also a continuous,
been borne out
in
our numerical
study and seems as intuitively obvious as the continuity and monotonicity of the optimal
awkward expression
objective function value, the
from providing a rigorous
Proposition
4. If
N(u°) > nxl I-lf- ^^^" there
As 7 increases from
the optimal solution to (34)
conjecture
Since
By
<
is
is
to
Proposition
3,
<
exists a
=
to discard
all
is
that u'(^)
wafers
decreases from V{u°) to
when 7 = rM.
from iV(u°) to
N{u°), there must be a 7 € (0,rM]
u'{^)
7G (O.rM] such
is
^f.
rM, V^(u*(7))
correct, A'^(u'(7)) decreases
/xt/mf
has prevented us
proof.
an optimal solution to (24)-(25) with start rate \
Proof.
for N{u'{'y)) in (52)
Similarly,
as 7 increases
for
0, since
from
an optimal solution to (24)-(25) with start rate A
to
=
which N(u'(^))
if
=
our
rM.
/ir/zT-
fip-
I
Propositions 3 and Proposition 4 can be combined to develop a search procedure
for solving
problem
(4)-(6).
that 7 for which ;V(u'(7))
For fixed A € [ij.t/N{u°), hf], we solve (34) and search for
= ^iT/^
Proposition 4 guarantees the existence of such a
18
7
in
the interval [O.7]. Then, we evaluate the objective function
let
A
=
ij.t/N{u' {-))).
By Proposition
3,
"'(7)
and the objective function
this value of A.
For each 7 G [0,7]. we solve
7.
(31) can be evaluated.
over 7 G [O.7]
is
~
<7)-^-/'f] is
to (24)-(25).
7 G
all
/ij/-^-
Thus, one search
are searched.
[0,*/]
=
))
optimal start rate A' to (31) and the optimal
sufficient to find the
screening policy u'
end of
searched as
and
Since for every
A G [^t/-V(u°).^f] there exists a 7 in the interval [0,7] such that .V(u'(-
every A G [^r/(l
('i-i)
the optimal solution to (24)-(25) with
is
in
for
However, the search over A can be
a A' that has the largest objective function value.
accomplished simultaneously as we search over
and search
in (31)
Readers can find an outline of
this
algorithm
at the
this section.
We now
focus on solving the Lagrangian problem (34). Let
1
be the probability that a
n
^
I
-^(""1^''-'
lot
has more than
/
r(a + na)
= r(a)r(a + (n-l)a)
denote the posterior probability density
(6
on the
profit
first
obtained from wafers n
+
for the
+
3„-i)"^'"-"-(r„)-^
+
(6
given that Sn-i bad chips are found on the
expected
wafers, and let
+ xj-'^^
.„.,
number
-
n
first
1
""
'
"
^^^^
^'
bad chips on the nth wafer.
of
wafers.
1,...,I. given that Sn
If V'^*'(5n)
represents the
bad chips were detected
n wafers, then u'[-)) and V"'(u*(7)) can be found by solving the dynamic
programming equations
V^{sl)
=
V„''(5„)
= max{0,G(n)
(39)
/•oo
/
[r{M - x^+i) - ct - f
+
l^+.isn
+
Xn+i)\f{Xn+i\sn)d.r^+,}.
Jo
n
= L-l
and
1.
(40)
/•oo
V^(u)
=
^(7(0)
= max{0,G(0)
After n wafers have been tested,
If
/
[r{M -
i^)
-
ct
-
1
+
(41)
V^''{xi)]f(xi)dxi}.
we can discard the remaining wafers and obtain no
the lot has more than n wafers, then we can continue testing;
j„+i bad chips, then the immediate profit
is
19
r{M -
J„+i)
if
wafer n
-07-7
+
1
profit.
contains
and the expected
future profit
Vn+ii^n
is
+
-^n+i)-
These equations
optimal solution to (34), which are discussed
Proposition
The optimal
5.
u\ =
=
u-„
n
is
(ui(7),
...,
u^(-,
is
))
(42)
if5„_i >B;:_i,
(43)
— l,n =
where the stopping boundary B^ >
=
5„_i <B;:_i,
if
\
the two propositions below.
in
policy u'{f)
also reveal structural properties of the
and 5^_i
1,...,L,
= —1
indicates that wafer
not tested under any circumstances.
We
Proof.
only need to show that
a backward induction on
and consider the
n.
It
nonincreasing
V'„"'(5n) is
+
difference V^isn
1)
—
=
which
is
done by
+
1,
^ni^n)- In order to prove that this quantity
is
trivially true for
is
in Sn,
n
I.
Suppose
it
is
true for n
nonpositive, the following properties of the conditional density /(jn+i|5„) are required.
For n
=
1,
.
.
.
I —
,
1
and
>
.s„
there exists Xn+i
0,
>
such that
1
/(x„+i
-
1|5„
+
1)
>
/(j„+i|5„), for Xn+i
>
Xn+i, and
(44)
/(j^+i
-
l|.s„
+
1)
<
/(x„+i|5„), for Xn+i
<
x„+i.
(45)
These inequalities can be
verified using (38).
[(r(M - x„+i)
/
-CT-1 +
By
(40),
V;:+,{sn
+
it
1
+
suflfices to
consider the difference
X„+,)/(x„+i|.^„
+
l)(fx,+i]
Jo
/•CO
-
[{r{M
/
-
X„+i)
-
Cj
-
7
+
V;\i(5„
+
Xn+l)f{^n+l\^n)dXn+i]
Jo
=
+ na —
a
-+
Jo
1
K\i(5„+
1
+Xn + i)/(x„+i|s„+
l)(fXn + l
roo
Jo
roo
<
/
[KVl(^n
+
1
+
-C„+l)
- KT+JSn +
- /"[V„'Vi(5„+x„+i)-V;'Vi(5„ +
X„+i)]/(x„+i|5„
+
l)c?X„+i
x„+i)]/(x„+,|5„)(fx„+i.
(46)
Jo
Changing the integration variable
in
the
first
integral
20
from x„+i to x„+i + l and combining
the two integrals
/'
[Vn + liSn
-/
we
in (46),
+
[Vn'+l('Sn
-
^n+l)
+
-Tri
get
V;\
-
+ l)
5,
, (
+
^n''+l('^"
X„ + i )] [/(x„ +
+
- l|s„+
i
-
1)
/( X„+ ,
|.S
J](fx, +
,
(47)
•^n+l)]/(-rn + l|-«n)^-rn + l-
.'0
The two terms
inside the
hypothesis; hence, the
first
integral have opposite signs by (44)-(45)
integral
first
is
nonpositive.
because, by the induction assumption, V„\i(.s„
1
<
Xn+\- Therefore, (47)
The
and
is
is
The second
+ j-„^,) >
is
nonnegative
integral
is
+ x„+i
for
V'„\i(5„
nonpositive, and the induction
and the induction
)
<
verified.
|
following proposition establishes monotonicity of the optimal stopping boundary,
used to streamline the dynamic programming algorithm. The proof
the proof of Proposition
Proposition
6.
5,
and
is
similar to
is
omitted.
The optimal stopping boundary
satisfies
Bo<B]<...<B2_j.
variables.
In the numerical computations,
approximate the integrals by
amount
of computation.
finite
(48)
(39)-(41) involve L functions of continuous
The dynamic programming equations
the
<
Xn+\
we
summations.
discretize the continuous variables
Two observations
are helpful in reducing
boundary point can be
First, the final
and
explicitly derived,
and equals
^2-1 =
Also, since Ki(5„)
if
Sn
is
is
(a
+ (L-l)Q-l)(A/-(7 + CT)/r)/a-6.
nonincreasing
found such that K^l^n)
=
0,
in 5„,
is
then we set V;(x)
poUcy u'{f) cannot be optimal
number
the expected
Notice that the optimal boundary point 5o
=
calculate V^{sn) starting from 5„
we
=
for all
After the optimal solution u'(f) to the Lagrangian problem
determine N{u'{f)), which
(49)
= -1
for the original
or
0.
problem
If
21
is
derived,
we need
BJ = -1, then the
(4)-(6),
n
and
x € (sn,nM].
of wafers that are tested per
"
f{Sn-Sn-xK-^)..-f{si)dSn...ds„
Cn= /'.../
Js„=0
Jai=0
0,
Bq
lot.
screening
=
by
(11).
=
!,...,! -1,
If
to
0.
then
(50)
and the probability that
testing ceases after the nth wafer
=
Tn
where Co
=
1
C„_i
-Cn,
Then the expected number
•
N{n
n
= 1,...,I-
1.
of wafers tested per lot
(51)
is
= Y.^T^^L{\-Y^T^).
{-,))
n=l
We
is
(52)
n=l
conclude this section with an outline of the algorithm that solves the original
optimization problem (4)-(6).
Algorithm:
Step
1.
Step
2.
Let 7
=
and
=
set 7'
Find the optimal solution "'(7) to the Lagrangian problem (34) and the
optimal objective function value
stop.
The optimal
start rate
the optimal boundary Bq
and the optimal screening policy
Calculate N{u'{~i)) using (52), and define A^
Step
4.
Compute
(5,
is
the
where
maximum
6
is
=
is
—
—1, then
u'(7').
^itIN[u'{-i)).
the objective function value for the original problem,
over
= y[V\u{',)) +
7vV(u-(7))
a small step variable, and go to step
we could prove that P^
is
-
cf\.
(53)
P^'s calculated thus far, then let 7'
all
Notice that the algorithm
If
A^*
is
If
3.
If P-,
+
V~'{u'{~^)).
Step
P,
7
0.
concave
in 7,
7.
Change 7
to
2.
guaranteed to terminate, since Bq
is
=
= —\ when
7
— rM.
then a binary search, rather than an exhaustive
search, over 7 € [0,rM] could be employed, thereby saving a considerable
amount
of
indeed concave
computation. For
all five
with respect to
Although we have been able to prove that the optimal value function
V'^{u'{-^))
P-yl
is
7.
data
sets considered in the next section, P-,
is
decreasing and concave in 7, we have been unable to prove the concavity of
our obstacle
is
again the expression for N{u'{'^)) in (52).
22
5
Numerical Results
5.
we
In this section,
test
the optimal fixed sample size screening strategy and the
optimal sequential screening strategy on
10 lots
and each
lot
has
less
five sets of yield
than 25 wafers. The data
data, where each set has about
sets,
denoted by Cl,Cl -5,02.02.
and C3, were obtained by Bohn from the same factory producing the same product
five different
is
time periods. For each of the
used to obtain values of the
following procedure
maximum
is
gamma
(Qi,...,Qm).
followed for each data set.
on each wafer
in
on each wafer
maximum
b are
3^.
sets,
is
If
maximum
,3m) are
More
6.
likelihood parameter estimates a
to be the
(ai, ...,am); the profitability results for this case
lots,
then the
the median of
is
variable with
and
b.
known shape
(/^i, ..., Jrn)
are used
These parameter estimates
We
not reported here for reasons of confidentiality.
a
m
3m) by assuming that the number
(i3i
Finally, the revised estimates
identical estimation procedure, but chose
specifically, the
obtained from the number
estimate a by d, which
gamma random
a
likelihood estimation
a data set contains
{,3i
We
the set.
in lot k
parameter a and scale parameter
and
and
and then recompute the estimates
of defective chips
a. a
data
parameters a, a and
likelihood estimates (qi, ....q^)
of defective chips
to obtain
five
in
also
performed the
mean, rather than the median,
of
were quite similar to the results obtained
from the original procedure and are omitted.
As mentioned
earlier,
we assume
that
when the the probing
effective capacity
under the exhaustive testing
effective capacity.
The wafer
scrap rate
is
policy, the fab
facility
is
is
working
working at
its
90%
of
its
3%
of
at
5%, the variable probing cost per wafer
the variable fabrication cost per wafer and the revenue from a wafer containing
chips
is
is
all
good
10 times the wafer's variable production cost. These parameter values are based
on discussions with a variety
of
semiconductor managers and engineers, and are used to
derive the optimal fixed sample size strategy and the optimal sequential screening strategy
for
each of the
five
data
sets.
Both screening
policies vary little over the five
Figure 3 illustrates the two policies for data set Cl.
the optimal sequential screening policy
is
The stopping boundary
data
sets,
and
characterizing
nearly linear, but slightly convex, for every data
23
set,
and the slope increases with increased
the acceptable yield
size policy
is
in
the
lot.
)r
yield.
lot is
that the former only accepts lots of expected higher yield
<
a
o
O
<
fixed
sample
and requires a
it
is
not surprising
level.
Sequential
J
--
.^
.^
3500 -
-^
-^
STOP
3000 -
-^
-<
-^
z
o
required
considered acceptable
while the sequential screening strategy monitors yield continuously,
O 4000
is
than the optimal sequential policy. Since
the fixed sample size policy stops monitoring yield after a
c
3
determines
yield level over
The optimal
6 wafers from every lot in each data set,
slightly higher yield level to continue testing
4500
testing
The average acceptable
13.7% lower than the overall average
samples either
more
the lot-to-lot variability increases,
level, as
before discarding the remaining wafers
the five data sets
lot-to- lot variation. Since the slope
2500
-<
--
y
2000
.<
1500
CONTINUE
1000
^m^ffFixed Sample Size
500
5
Number
Cumulative
Figure
3.
Recall that the strategy
and to choose the
Optimal screening
commonly used
20
15
10
in
industry
Wafer
data
policies for
start rate so that the testing facility
24
of
is
25
set
Cl.
to perform exhaustive testing
works
at its effective capacity.
Before reporting our profit results,
is
it
increase that can be achieved relative to this straw strategy.
Let A^
denote the arrival rate under the exhaustive testing strategy, and
denote the average incoming
for
any strategy
yield.
let
(54,
the relative profit increase
is
X'-\^
__
A^
\^(V(u')-V{u^)\
\E\ V{u^)-cf
j
A^
\^\rL[\
MF-
^
A^
we assumed
effective capacity
that Lcj
=
0.0.3cf
.
-q)My-
Lcj - cf
- y]
+ A^ \rL{l - q)My - Lcj - Cf *'"
f
Lct(1
fiF
(
\E
'
rLAf = lOcf
,
q
=
0.0.5
and the fab
is
at
90%
of
its
under the exhaustive strategy, the upper bound equals
1
lO( OM{l-y) \
,
^''^
'^^--9^Y[ 9.5y-im )This quantity increases from 1/9 when yield
level of
— q)L
^.ltI{\
^
-V^
Then an upper bound on
X'{V{u')-CF)-XEiV{n^)-CF)
\^{V{u^)-cf)
Since
=
profit
M-
=
y
bound on the
useful to determine an upper
10.58% required
is
100%
for profitability in (11).
be revealed, the yield was greater than
36%
to oo as yield approaches the critical
Although the exact average
for all five
data
yield cannot
and hence AF^ax
sets,
is
between 11.1% and 12.0%.
Under the heading "Theoretical Calculations", Table
I
relative to the exhaustive testing strategy obtained by the
all five
data
sets.
We
also display
pF
=
^Ip^f and pj
the effective capacity utilization of the fab and testing
seen that for every data
set,
both
facilities
work
3%
increase
policies
is
is
two proposed strategies
= {\N(u)/ pj),
for
which represent
facility, respectively.
at their effective capacity.
profit increcLses are rather small: out of a potential
to
reports the profit increases
It
can be
However, the
11.1% to 12.0% increase, only a
2%
achieved. Also, the difference in performance between the two proposed
relatively small; the fixed
sample
size policy averages a
2.24%
over the five data sets, compared to 2.51% for the optimal strategy.
25
profit increase
Table
Data Set
1.
Numerical
results.
wafers tested per
start rates
were recorded.
lot
These quantities and the theoretically calculated
were then used to calculate the
profit increases that are reported in
Table
I
under the heading "Simulation Results".
When
occur.
yield
the yield model underestimates the
If
facility
the derived policies are tested on the actual data, two undesirable things can
number
of discarded wafers, then the testing
underutilized and a feasible, but suboptimal, strategy
is
model overestimates the number
overutilized,
and an
is
obtained.
If
the
of discarded wafers, then the testing facility
infeasible strategy can result.
Referring to Table
I,
we
is
see that the
yield
model correctly predicts the average number
of discarded wafers per lot under the
fixed
sample
sets,
sets
C2 and
data
size policy for three of the five
C2.5. However, the yield model
is
and
is
accurate under the sequential policy,
less
underestimating the average number of discarded wafers per
five
data
testing.
sets. In
Both
these cases, the resulting profit
policies overestimated the
the resulting strategy
is
number
is
feasibility.
was achieved by reducing the
and pr
sample
by .3-5%
less
in three of the
than under exhaustive
and
not feasible: hence, the profit increases reported for this data set
of 0.75 (0.90, respectively)
sets for the fixed
sometimes
lot
of discarded wafers in data set C2.5
correspond to a reduced start rate that maintains
(0.993, respectively)
by about 1% on data
off
—
1.000.
size strategy
That
is,
the profit increase
start rate so that
pf
=
0.991
The average
profit increase over the five
1.23%
simulation study, about
1% below
The
sequential
is
the corresponding improvement achieved
in
in the
the theoretical calculations.
data
strategy averages a 0.70% profit decrease relative to the straw strategy, because of the
underutilization of the testing facility in cases Cl.5 and C2. Hence, in addition to being
eaisier to
derive
and
to
implement than the sequential strategy, the fixed sample
size
strategy performs nearly as well in the analytical calculations, and appears to be more
robust in our limited simulation study.
As a point of
number
yield
of
model
reference,
we
also considered the beta-binomial yield model,
bad chips on each wafer
is
modeled
as a binomial
significantly overestimated the average
27
number
random
where the
This
variable.
of wafers tested per
lot:
the
average value over the
five
data sets of pT vinder the sequential strategy
in
the simulation
study was only 0.861, which led to an average profit decrease of 11.68%.
Acknowledgment
We
are deeply indebted to Roger Bohn,
research
is
who generously gave
supported by a grant from the Leaders
for
and National Science Foundation grant DDM-9057297.
28
us his yield data. This
Manufacturing Program
at
MIT
References
Albin, S.
and D.
J.
Friedman. 1989. The Impact of Clustered Defect Distribution
Management Science
Fabrication.
W. and
Atherton, R.
J.
in
IC
35, 1066-1078.
E. Dayhoff.
Introduction to Fab Graph Structures.
1985.
ECS
Abstracts.
Bohn, R.
1991.
E.
Noise and Learning
in
Semiconductor Manufacturing.
Center
for
Technology Policy and Industrial Development, MIT, Cambridge, M.A.
Chen, H, M.,
J.
Harrison, A.
Mandelbaum, A. van
.Ackere,
and
L.
M.
VVein.
1988.
Empirical Evaluation of a Queueing Network Model for Semiconductor Wafer Fabrication.
Operations Research 36, 202-215.
Cunningham,
J. .A.
Manufacturing.
1990.
IEEE
The Use and Evaluation
of Yield
Variations in Semiconductor Wafers.
IEEE
M. G. C. Resende.
Circuit Manufacturing.
Goldratt, E. M. and
IEEE
Circuits
1988.
J.
K. Paulsen. 1987. Radial Yield
and Devices Magazine, March,
Closed-Loop Release Control
Trans, on Semiconductor Manufacturing
Cox. 1984. The Goal North River Press,
J.
in Integrated Circuit
Trans, on Semiconductor Manufacturing 3, 60-72.
Ferris-Prabhu, A. V., L. D. Smith, H. A. Bonges, and
Glassey, R. C. and
Models
Inc.,
1,
for
42-47.
VLSI
36-46.
Croton-on-Hudson,
NY.
Longtin, M., L.
Manufacturing,
Cambridge,
M. Wein, and
II:
R. E. Welsch. 1992. Sequential Screening in Semiconductor
Exploiting Spatial Dependence. Sloan School of Management. MIT,
MA.
Mallory, C. L., D. S. Perloff, T. F. Hasan, and R. N. Stanley. 1983. Spatial Yield Analysis
in Integrated Circuit
Manufacturing. Solid State Technology, November, 121-127.
29
Osburn, C. M., H. Berger. R.
P.
Donovan, and G. W. Jones.
1988.
The
Effects of
Contaniination on Semiconductor Manufacturing Yield. The Journal of Environmenta!
Sciences, March/.A.pril, 45-57.
VVein, L.
M.
1988. Scheduling Semiconductor Wafer Fabrication.
conductor Manufacturing
1,
115-130.
30
IEEE
Trans, on Senai-
Date Due
AU8.
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