With respect to the subassembly, there are i mechanical specifications such as dimensional tolerances, stress requirements, and thermal limits. In addition, there are j electrical specifications such as response time and circuit gain. There are also k other characteristics which must conform to specification such as quality characteristics, like solder joint wetting and paint finish. There are therefore i + j number of performance parameters and k number of inspection characteristics. The latter classification is arbitrary.

#### The Role of Parameter Capability Data

Suppose that a data base of process, material, and component capability data has been formed as the result of extensive characterization studies, and the data is of two varieties short-term and long-term. The computer analyses and simulations were undertaken for each of the i and j parameters using the characterization data as a basis of simulation. Obviously, such an undertaking would produce more realistic short-term and long-term first time parameter yield estimates because of working with known process, component, and material capabilities.

#### The Role of Mechanical Engineering

To more fully understand the implications of the latter discussion, refer to the radio example. Suppose any one of the assembly gaps associated with the subassembly as related to subassembly K, for example, can be selected. If the process capabilities of the related parts are known or capable of rational estimation, it would be possible to compute the probability of assembly using a statistically based methodology.

If there were a total of x = 6 independent assembly gaps related to subassembly K, and the long-term, first-time yield (Y_{FT}) projection for each gap was 99.73%, it could be seen that the rolled-throughput yield (Y_{RT}) for the product subassembly over many manufacturing intervals would be Y_{RT}= Y_{FT}^{X} = .9973^{6} =
.9839, or 98.39%. In other words, there would be a 98.39% likelihood that any given product subassembly could be put together without
encountering an interference fit.

Using the same methodology, it is possible to estimate the joint
yield or simultaneous probability of zero irregularities with respect to all
other mechanical specifications; i.e., thermal and stress requirements. In
turn, an overall rolled-throughput yield could be derived which describes all
mechanical aspects.

#### The Role of Electrical Engineering

In the context of electrical circuit
performance, refer to the printed circuit board (PCB) illustrated in Figure
4-4. Within this particular PCB are several circuits - only one circuit will
be considered. Suppose that circuit V was intended to output a certain
electrical response such as some level of voltage within specified limits.
Also assume that it was possible to simulate the response (such as voltage)
using a computer simulation program. If the known or rationally premised
statistical characteristics of each part related to circuit V into the
simulation were input, it would be possible to study the effect of component
variation on voltage. From that, the projected first time test yield of
circuit voltage could be estimated. Again, if there were a total of j = 6
independent electrical response parameters associated with the PCB and the
long- term first time yield (Y_{FT}) projection for each of the six
parameters was 99.73%, the rolled-throughput yield (Y_{RT}) for the
PCB over many manufacturing intervals would be Y_{RT} =
Y_{FT}^{m} = 9973< SUP
> 6
<
/SUP > = .9839, or 98.39%. In
other words, there would be a 98.39% likelihood that any PCB would pass the
required electrical tests.

Notice that all of the independent circuit response yields
related to the subassembly could be treated in the same manner, thereby
deriving the rolled- throughput yield for all j electrical parameters.
However, for this example, assume that there is only one PCB per product
subassembly and j = 6 circuit responses per PCB.

#### The Role of Manufacturing and Quality Engineering

Assume that the subassembly is subject to k
inspection characteristics. If the historical defects-per-unit (dpu) data is
available, or can be rationally postulated for each of the k characteristics,
the rolled-throughput inspection yield (using the Poisson relation) could be
estimated. For example, consider that the historical soldering performance (as
related to solder joint wetting) was such that dpu = .000001. Notice that in
this instance a unit is defined as an individual solder joint. Given this, the
expected first time wetting yield (per solder joint) would be
e^{-d/u}= 2.71828^{-.000001} = .999999, or 99.9999%. If there were
a total of w= 2,750 solder joints in the subassembly, the wetting
rolled-through-put yield would be Y_{RT}= Y_{FT}^{W}= .999999^{2750} = .9973, or 99.73%. If the same
rolled-throughput yield applied to each of the k=6 inspection categories, the
long-term probability of zero defects as related to any given subassembly
would be Y_{RT} = Y_{FT}^{k} = .9973^{6} =
.9839, or 98.39%. This would hold
true since there is by definition only one PCB per subassembly.

####
Estimating the Composite Long-Term Design
Producibility

The overall long-term first time yield for all i, j, and
k product subassembly parameters could be given by .9839x.9839x.9839= .9525.
There would be 95.25% confidence of specification conformance over agreat many
production intervals as related to the electrical mechanical, and inspection
aspects of the product subassembly.

If the same estimate applied to each of the
t=6 subassemblies, the product rolled-through put yield would be Y_{RT
}=Y_{FT}^{t} =.9525^{6}=.7467, or
74.67%. Therefore, there is now a relative benchmark to gauge the
producibility of the composite product design. It should be noted that
manufacturing cycle time as well as other manufacturing and business issues
can be studied at this point in the analysis.

#### Comparison of Design Alternatives and the Baselining
Process

The normalized rolled-throughput yield
Y_{RT;N} can now be addressed. Essentially such a
yield represents the throughput-per-opportunity, where an opportunity is
arbitrarily defined: i.e., part, process step, etc. In this sense, it may be
thought of as the first time yield of a "typical" defect opportunity.
Normalization has the effect of essentially neutralizing the influence of
complexity. Because of this feature, it is possible to compare the
producibility of one design alternative to another so long as the definition
of an opportunity remains consistent. To illustrate how this works extend the
previous example. If the long-term product level rolled-throughput yield for
design alternative A is .7467 and there are m =
5000 opportunities for a detect,
then the normalized long-term rolled-throughput yield per part would be
Y_{RT;N}^{1/m} =.7467^{1/5000}= .999942, or 99.9942%. The interactive
effect of manufacturing capability and product (or process) complexity on
rolled- throughput yield can be better understood by the graphs presented in
Figure 4-5 and Figure 4-6.

####