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Second Interim Report
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 fab-specific 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 CVt 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 CEt. Obviously, the level of process-specific knowledge is unobservable, but we can make use of it through the learning index Lt. Ignoring the effect of learning capital, the learning index is defined by the level of manufacturing experience:

where Lo is the existing level of knowledge or experience in the first period. Lo 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 Lt 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 process-more 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 h1(Lt) and h2(·). Following Hatch and Reichelstein (1994), a negative exponential functional form is chosen for h1(Lt) while h2(·) 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 Lo 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 Vint is the average equipment vintage in the fab at time t and Vino 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(Lt) is falling over time as a result of learning by doing. Let Lt 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(Li+Installi). 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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