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=Logistic=
#REDIRECT [[Logistic]]
The Logistic reliability growth model has an S-shaped curve and is given by [3]:
where  <math>b</math>  and  <math>k</math>  are parameters. Similar to the analysis given for the Gompertz curve (Chapter 7), the following may be concluded:
<br>
:1) The point of inflection is given by:
::<math>{{T}_{i}}=\frac{\ln (b)}{k}</math>
:2) When  <math>b>1</math>  then  <math>{{T}_{i}}>0</math>  and an S-shaped curve will be generated. However, w<math>%</math>hen  <math>0<b\le 1</math>  then  <math>{{T}_{i}}\le 0</math>  and the Logistic reliability growth model will not be described by an S-shaped curve.
<br>
:3) The value of  <math>R</math>  is equal to 0.5 at the inflection point.
<br>
{{parameter estimation logistic rga}}
 
{{confidence bounds logistic rga}}
 
==General Examples==
===Example 5===
Table 8.4 presents the reliabilities observed monthly for an automobile transmission that was tested for one year.
<br>
<br>
:1) Find a Logistic reliability growth curve that best represents the data.
:2) Plot it comparatively with the raw data.
:3) If design changes continue to be incorporated and the testing continues, when will the reliability goal of 99% be achieved?
:4) If design changes continue to be incorporated and the testing continues, what will be the attainable reliability at the end of January the following year?
 
 
{|style= align="center" border="1"
|+Table 8.4 - Reliability data
!Month
!Observed Reliability(%)
|-
|June|| 22
|-
|July|| 26
|-
|August|| 30
|-
|September|| 34
|-
|October|| 45
|-
|November|| 58
|-
|December|| 68
|-
|January|| 79
|-
|February|| 85
|-
|March|| 89
|-
|April|| 92
|-
|May|| 95
|}
 
<br>
[[Image:rga8.7.png|thumb|center|400px|Entered data and the estimated Logistic parameters.]]
<br>
<br>
[[Image:rga8.8.png|thumb|center|400px|Reliability vs. Time plot.]]
<br>
<br>
[[Image:rga8.9.png|thumb|center|400px|When the reliability goal of 99% will be achieved.]]
<br>
<br>
[[Image:rga8.10.png|thumb|center|400px|The reliability at the end of the following January (month=20)]]
<br>
 
====Solution to Example 5====
:1) Figure Loge21 shows the estimated parameters.
:2) Figure Loge22 displays the Reliability vs. Time plot.
:3) Using the QCP, Figure Loge23 displays when the reliability goal of 99% will be achieved.
:4) Figure Loge24 shows the reliability at the end of January the following year (i.e. after 20 months of testing and development).
<br>
<br>
 
===Example 6===
<br>
Table 8.5 presents the results for a missile launch test. The test consisted of 20 attempts. If the missile launched, it was recorded as a success. If not, it was recorded as a failure. Note that, at this development stage, the test did not consider whether or not the target was destroyed.
<br>
<br>
:1) Find a Logistic reliability growth curve that best represents the data.
:2) Plot it comparatively with the raw data.
:3) If design changes continue to be incorporated and the testing continues, when will the reliability goal of 99.5% with a 90% confidence level be achieved?
:4) If design changes continue to be incorporated and the testing continues, what will be the attainable reliability at the end of the 35th launch?
<br>
 
{|style= align="center" border="1"
|+Table 8.5 - Sequential success/failure data
!Launch Number
!Result
|-
|1|| F
|-
|2|| F
|-
|3|| S
|-
|4|| F
|-
|5|| F
|-
|6|| S
|-
|7|| S
|-
|8|| S
|-
|9|| F
|-
|10|| S
|-
|11|| F
|-
|12|| S
|-
|13|| S
|-
|14|| S
|-
|15|| S
|-
|16|| S
|-
|17|| S
|-
|18|| S
|-
|19|| S
|-
|20|| S
|}
 
====Solution to Example 6====
:1) Figure Loge31 shows the entered data and the estimated parameters.
:2) Figure Loge32 displays the Reliability vs. Time plot.
:3) Figure Loge33 displays when the reliability goal of 99.5% will be achieved with a 90% confidence level.
:4) Figure Loge34 displays the reliability achieved after the 35th launch.
 
[[Image:rga8.11.png|thumb|center|400px|Entered data and the estimated logistic parameters.]]
<br>
<br>
[[Image:rga8.12.png|thumb|center|400px|Reliability vs. Time plot.]]
<br>
<br>
[[Image:rga8.13.png|thumb|center|400px|When the reliability goal of 99.5% with a 90% confidence level will be achieved.]]
<br>
<br>
[[Image:rga8.14.png|thumb|center|400px|The reliability at the end of the 35th launch.]]
<br>
<br>
 
===Example 7===
<br>
Consider the data given for the missile launch test in Example 6. Now suppose that the engineers assigned failure modes to each failure and that the appropriate corrective actions were taken. <br>
Table 8.6 presents the data.
:1) Find the Logistic reliability growth curve that best represents the data.
:2) Plot it comparatively with the raw data.
:3) If design changes continue to be incorporated and the testing continues, when will the reliability goal of 99.50% be achieved?
:4) If design changes continue to be incorporated and the testing continues, what will be the attainable reliability at the end of the 35th launch?
<br>
 
{|style= align="center" border="1"
|+Table 8.6 - Sequential success/failure data with modes
!Launch Number
!Result
!Mode
|-
|1|| F|| 2
|-
|2|| F|| 1
|-
|3|| S||
|-
|4|| F|| 3
|-
|5|| F|| 3
|-
|6|| S||
|-
|7|| S||
|-
|8|| S||
|-
|9|| F|| 2
|-
|10|| S||
|-
|11|| F|| 1
|-
|12|| S
|-
|13|| S
|-
|14|| S
|-
|15|| S
|-
|16|| S
|-
|17|| S
|-
|18|| S
|-
|19|| S
|-
|20|| S
|}
 
====Solution to Example 7====
:1) Figure Loge51 shows the estimated parameters.
:2) Figure Loge52 displays the Reliability vs. Time plot.
:3) Figure Loge53 displays when the reliability goal of 99.5% will be achieved.
:4) Figure Loge54 displays the reliability after the 35th launch.
 
[[Image:rga8.15.png|thumb|center|400px|Entered data and the estimated Logistic parameters.]]
<br>
<br>
[[Image:rga8.16.png|thumb|center|400px|Reliability vs. Time plot.]]
<br>
<br>
[[Image:rga8.17.png|thumb|center|400px|Calculate when the reliability goal of 99.5% will be achieved.]]
<br>
<br>
[[Image:rga8.18.png|thumb|center|400px|Calculate the reliability at the end of the 35th launch.]]

Latest revision as of 03:05, 24 August 2012

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