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General Examples Using the Gompertz Models |
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The approximate reliability confidence bounds under the Gompertz model can be obtained with nonlinear regression. Additionally, the reliability is always between 0 and 1. In order to keep the endpoints of the confidence interval, the logit transformation is used to obtain the confidence bounds on reliability.
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where p is the total number of groups (in this case 3) and n is the total number of items in each group.
A device is required to have a reliability of 92% at the end of a 12-month design and development period. Table 7.1 gives the data obtained for the first five moths.
Having completed Steps 1 through 4 by preparing Table 7.1 and calculating the last column of the table to find S1, S2 and S3, proceed as follows:
Find c from Eqn. (16).

Find a from Eqn. (17).

This is the upper limit for the reliability
as
.
Find b from Eqn. (18).

Now, since the initial values have been determined, the Gauss-Newton method can be used. Therefore, substituting Yi = Ri, g1(0) = 94.16, g2(0) = 0.615, g3(0) = 0.731, Y(0), D(0), ν(0) become:



The estimate of the parameters ν(0) is given by:

The revised estimated regression coefficients in matrix form are:

If the Gauss-Newton method works effectively, then the relationship Q(k+1)< Q(k) has to hold, meaning that g(k+1) gives better estimates than g(k), after k. With the starting coefficients, g(0), Q is:

And with the coefficients at the end of the first iteration, g(1), Q is:

Therefore, it can be justified that the Gauss-Newton method works in the right direction. The iterations are continued until the relationship of Eqn. (6) is satisfied. Note that RGA uses a different analysis method called the Levenberg-Marquardt. This method utilizes the best features of the Gauss-Newton method and the method of the steepest descent, and occupies a middle ground between these two methods. The estimated parameters using RGA are shown in Figure 7.2. They are:


Figure 7.2: Estimated Standard Gompertz parameters |
The Gompertz reliability growth curve is:
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The required reliability is 92%. Consequently, from the previous result, this requirement will barely be met. Every effort should therefore be expended to implement the reliability program plan fully, and perhaps augment it slightly to assure that the reliability goal will be met.
| Table 7.2 - Comparison of the predicted reliabilities with the actual data | ||||||||||||||||||||||||||||||||||||||||||
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Figure 7.3: Plot of the reliability growth curve for Example 1 |
Calculate the parameters of the Gompertz model using the sequential data in Table 7.3.
Table 7.3 - Sequential data for Example 2 |
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Using RGA, the parameter estimates are shown in Figure 7.4.

Figure 7.4: Estimated Standard Gompertz parameters for Example 2 |