THE COMPETITIVE SEMICONDUCTOR MANUFACTURING HUMAN
RESOURCES PROJECT:
Second Interim Report
CSM32
Clair Brown, Editor
12.3 A Model of Learning by Doing
The learning by doing model used here is based on that developed
by Hatch and Reichelstein (1994). The basic idea is that cost is
a function of yields which are a function of learning by doing activities.
Let x denote the gross number of dice manufactured, q denote
the number of functional dice, and y denote the net yield (line
and die yield). Then by definition q =x·y.
Now assume that the unit variable cost of manufacturing a chip is
a constant, c, whether the die functions or not. Since
the saleable output must bear the cost of scrapped dice and wafers,
c·x = v·q, where v is the average variable
cost of production of a functioning device. We can now restate the
average variable cost as
where LY and DY refer
to line yield and die yield respectively. Hence, to estimate the
learning curve for semiconductors it is sufficient to estimate the
line and die yield improvement curves as functions of the determinants
of learning by doing. Due to data limitations, only improvements
in die yield will be analyzed in this study.
The earlier discussion emphasizes that better process control results
in higher yields. In this model, process control is hypothesized
to improve as fabspecific knowled,e of the manufacturing environment
and processes grow. The rate at which this knowledge accumulates
depends on quantity and quality of resources devoted to yield improvement.
Thus, the rate of learning by doing is determined not only by the
level of manufacturing experience, but also by investments in physical
and human capital that facilitate knowledge acquisition.
The accumulation of manufacturing experience is summarized by the
cumulative number of wafers fabricated and by the cumulative number
of engineering hours devoted to the process. Let CV_{t}
denote the number of wafers produced for a given process between
time zero (the date of the first observation) and time t.
Similarly, the cumulative number of engineering hours for a process
is denoted bv CE_{t}. Obviously, the level
of processspecific knowledge is unobservable, but we can make use
of it through the learning index L_{t}.
Ignoring the effect of learning capital, the learning index is defined
by the level of manufacturing experience:
where L_{o} is the existing level
of knowledge or experience in the first period. L_{o} embodies
the learning that has been accumulated prior to the first observation
of the process. 5
As was explained above, one determinant of die yield is the die
size. It is common to normalize die yields for die size by focusing
on the density of fatal defects per CM2 of silicon. To do this,
defect density values are derived from reported die yields and used
as the object of analysis. There are a variety of defect densitv
models available, for this study the "Murphy model" is
used. The relationship between die yield and the average number
of fatal defects per CM2 is given by:
where A is the die area and DD is the
defect density parameter.'
The objective now is to identify and evaluate the factors that reduce
the defect density parameter. To incorporate learning by doing into
reductions in the defect density parameter, we first assume that
the defect density curve is additively separable into a dynamic
(learning by doing) component and a static component. The learning
by doing component depends on the learning index.
where L_{t} is the unobservable
"learning index" defined in equation (12.2).
The second term in equation (12.4) includes the influence of the
variables that do not directly affect the rate of learning by doing
but exercise a constant influence over improvements in the defect
density. These variables include the cleanliness of the clean room
CR (measured by the number of particles of per cubic foot),
and the number of mask layers ML. The number of particles
per cubic foot in the clean room directly influences the incidence
of fatal defects due to particulate contamination. The number of
mask layers is a measure of the total number of steps in the processmore
process steps increase the probability of a fatal defect on the
die. The final static input to the level of defect density is the
equipment vintage. The variable refers to the date at which the
machine was manufactured and/or purchased. Newer equipment, with
a "higher" vintage value, generally allows greater process
control, but the vintage of the equipment does not change the amount
of process specific knowledge in the fab and is not included in
the learning index. As new equipment is installed, the average vintage
of equipment in the fab rises, overall process control is enhanced,
and the density of defects falls.
With the exception of defect density models discussed earlier, neither
economics nor engineering have many insights into the appropriate
choice of functional form for h_{1}(L_{t})
and h_{2}(·). Following Hatch
and Reichelstein (1994), a negative exponential functional form
is chosen for h_{1}(L_{t})
while h_{2}(·) is assumed to be linear.
This gives
where CR denotes the clean room grade,
ML is the number of mask layers, and Vz'n is the average equipment
vintage.
Since we cannot observe the learning index directly, it is necessary
to solve for Lo in terms of the initial defect density value, DDO,
and substitute the result. In the initial period, i.e., when t 
0, the defect density curve is
Solving for L_{o} gives
Finally, substituting the previous expression
into (12.2) and (12.5) gives us the defect density curve with the
observable DDO:
Hatch and Mowery (1994) have shown that some of the knowledge acquired
through learning by doing is specific to the equipment in the manufacturing
environment. Initially, the traits of the new equipment are unknown
and its performance with existing manufacturing processes will likely
differ from existing, equipment. For example, the process specification
limits for the temperature in a given step may need to be adjusted
for the new equipment to accommodate its differences from existing
equipment. Because the new machine is not fully understood, the
number of defects will often rise until the specific characteristics
of the new machine are learned. In other words, the lack of understanding
of new equipment is equivalent to a loss in process specific information
which continues until the traits of the new machine are learned.
The result is a temporary yield excursion corresponding to the installation
of new equipment. To test this hypothesis, the influence of the
installation of new equipment is included in the learning index.
The new learning index is
where Install is a dummy variable that
takes on a value of one in periods that new equipment is installed.'
Based on this revised learning index and the inclusion of the equipment
vintage in h2 ('),the defect density curve becomes
where Vin_{t}
is the average equipment vintage in the fab at time t and Vin_{o}
is the average equipment vintage in the initial period.
This hypothesized relationship between equipment installations and
the learning curve is shown in Figure 2,
where the defect density at time t, denoted by DDt(L_{t})
is falling over time as a result of learning by doing. Let L_{t
}denote the level of learning, then if new equipment is
installed at time Z', the defect density may temporarily spike
'The installation variable does not interact with the cumulative
volume and cumulative engineering variables because the disruption
caused is assumed to last only one period. In other words, it is
expected to only require one month to learn the characteristics
of the new equipment.
up to the level DDi(L_{i}+Install_{i}).
Due to the greater process control of the new equipment, the defect
density should quickly fall to a lower level than previously obtained.
It is not straightforward how to reduce the variety and intensity
of human capital investments into a few measurable variables. Each
fab has a wide range of human resource practices in hiring, developing
skills, and performing manufacturing operations. While it is difficult
to measure the amount of human capital available, its role in enhancing
the rate of learning, by doing is relatively clear. When operators
acquire sufficient human capital to participate actively in problem
solving activities, more information is available for analysis and
decision making and engineering resources are freed to focus on
more difficult projects. I have chosen the degree of operator involvement
in improvement teams as the most appropriate measure of operator
human capital influencing learning by doing activities. Virtually
all fabs use teams of some form to improve performance. The proportion
of operators involved in teams is a reasonable (but not the only)
measure of the amount of problem solving that operators do. It also
provides an estimate of the human capital in the fab because operators
must have a minimum level of skills to participate meaningfully
in teams. Team participation is also an important training opportunity
where operators can learn technical and problem solving skills from
engineers and technicians. To incorporate the influence of operator
problem solving, the proportion of operators involved in teams is
assumed to be a component of the learning index and is allowed to
interact with the basic learning drivers. The defect density curve
becomes
where Team gives the degree of involvement
of operators in problem solving teams.
While it is difficult to quantify the amount and nature of human
capital in any organization, it is not difficult to measure the
impact of losing it by studying the effect of operator turnover.
Turnover is particularly harmful in two respects. Some knowledge
is lost to the fab with every operator that leaves and the operators
who replace them typically have less training and experience. The
new operators are more likely to make mistakes that result in yield
losses; they are not able to continue in the yield improvement activities
of their predecessors, and they actually become part of the problem.
Another result is that engineering resources must be diverted from
other improvement activities to training operators and preventing
their mistakes until they get sufficient experience to become contributors
rather than detractors. Inserting the rate, of operator turnover
into the learnin, index results in the following defect density
curve:
where Turn is the annual proportion of
operators who leave the fab. Turnover is expected to increase the
level of defect density and reduce the rate of learning.
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