Template:Gompertz models (standard and modified): Difference between revisions

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{{standard model overview gompz}}
REDIRECT [[Gompertz_Models]]
 
{{parameter estimation using least squares in nonlinear regression}}
 
{{cumulative reliability gumpz}}
 
{{modified gompertz model}}
 
==General Examples==
===Example 4===
<br>
A new design is put through a reliability growth test. The requirement is that after the ninth stage the design will exhibit an 85% reliability with a 90% confidence level. Given the data in Table 7.5, do the following:
<br>
:1) Estimate the parameters of the Standard Gompertz model.
:2) What is the initial reliability at  <math>T=0</math> ?
:3) Determine the reliability at the end of the ninth stage and check to see if the goal has been met.
<br>
{|style= align="center" border="1"
|+'''Table 7.5 - Grouped per configuration data for Example 4'''
!Stage
!Number of Units
!Number of Failures
|-
|1|| 10|| 5
|-
|2|| 8|| 3
|-
|3|| 9|| 3
|-
|4|| 9|| 2
|-
|5|| 10|| 2
|-
|6|| 10|| 1
|-
|7|| 10|| 1
|-
|8|| 10|| 1
|-
|9|| 10|| 1
|}
<br>
====Solution to Example 4====
:1) The data is entered in cumulative format and the estimated Standard Gompertz parameters are shown in Figure Gompex4a.
 
[[Image:rga7.6.png|thumb|center|400px|Entered data and the estimated Standard Gompertz parameters.]]
:2) The initial reliability at  <math>T=0</math>  is equal to:
 
::<math>\begin{align}
  & {{R}_{T=0}}= & a\cdot b \\
& = & 0.9497\cdot 0.5249 \\
& = & 0.4985 
\end{align}</math>
 
:3) The reliability at the ninth stage can be calculated using the Quick Calculation Pad (QCP) as shown in Figure Gompex4b.
 
[[Image:rga7.7.png|thumb|center|400px|Calculate the reliability at the end of the ninth stage with 90% confidence bounds.]]
<br>
The estimated reliability at the end of the ninth stage is equal to 91.92%. However, the lower limit at the 90% 1-sided confidence bound is equal to 82.15%. Therefore, the required goal of 85% reliability at a 90% confidence level has not been met.
 
===Example 5===
Using the data in Table 7.6, determine whether the Standard Gompertz or Modified Gompertz would be better suited for analyzing the given data.
<br>
{|style= align="center" border="1"
|+'''Table 7.6 - Reliability data for Example 5'''
!Stage
!Reliability (%)
|-
|0|| 36
|-
|1|| 38
|-
|2|| 46
|-
|3|| 58
|-
|4|| 71
|-
|5|| 80
|-
|6|| 86
|-
|7|| 88
|-
|8|| 90
|-
|9|| 91
|}
 
====Solution to Example 5====
The Standard Gompertz Reliability vs. Time plot is shown in Figure Ex5Std.
The Standard Gompertz seems to do a fairly good job of modeling the data. However, it appears that it is having difficulty modeling the S-shape of the data. The Modified Gompertz Reliability vs. Time plot is shown in Figure Ex5Mod.
 
The Modified Gompertz, as expected, does a much better job of handling the S-shape presented by the data and provides a better fit for this data.
 
[[Image:rga7.8.png|thumb|center|400px|Standard Gompertz Reliability vs. Time plot]]
<br>
<br>
[[Image:rga7.9.png|thumb|center|400px|Modified Gompertz Reliability vs. Time plot.]]

Revision as of 02:09, 27 August 2012

REDIRECT Gompertz_Models

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