Template:Data analysis fleet rsa: Difference between revisions

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===Fleet Data Analysis===
#REDIRECT [[Fleet Data Analysis]]
Once the accumulated timeline has been generated, it is then converted into grouped data. To accomplish this, a group interval is required. The group interval length should be chosen so that it is representative of the data.  Also note that the intervals do not have to be of equal length. Once the data points have been grouped, the parameters can be obtained using maximum likelihood estimation as described in the Grouped Data Analysis section of the [[Crow-AMSAA - NHPP|Crow-AMSAA (NHPP)]] chapter. The data from the table above can be grouped into 5 hour intervals. This interval length is sufficiently large to insure that there are failures within each interval. The grouped data set is given in the following table.
 
<br>
{|style= align="center" border="1"
|-
|colspan="2" style="text-align:center"|Grouped data
|-
!Failures in Interval
!Interval End Time
|-
|1|| 5
|-
|1|| 10
|-
|1|| 15
|-
|1|| 20
|-
|1|| 25
|}
 
The Crow-AMSAA model for Grouped Failure Times is used for the data in Table 13.3 and the parameters of the model are solved by satisfying the following maximum likelihood equations (Chapter 5).
 
 
::<math>\begin{matrix}
  \widehat{\lambda }=\frac{n}{T_{k}^{\widehat{\beta }}} \\
  \underset{i=1}{\overset{k}{\mathop \sum }}\,{{n}_{i}}\left[ \frac{T_{i}^{\widehat{\beta }}\ln {{T}_{i-1}}-T_{i-1}^{\widehat{\beta }}\ln {{T}_{i-1}}}{T_{i}^{\widehat{\beta }}-T_{i-1}^{\widehat{\beta }}}-\ln {{T}_{k}} \right]=0 \\
\end{matrix}</math>
 
 
===Fleet Analysis Example===
The following table presents data for a fleet of 27 systems. A cycle is a complete history from overhaul to overhaul. The failure history for the last completed cycle for each system is recorded. This is a random sample of data from the fleet. These systems are in the order in which they were selected. Suppose the intervals to group the current data are 10000, 20000, 30000, 40000 and the final interval is defined by the termination time. Conduct the fleet analysis.
<br>
{|style= align="center" border="1"
|-
|colspan="4" style="text-align:center"|Sample fleet data
|-
!System
!Cycle Time  <math>{{T}_{j}}</math>
!Number of failures  <math>{{N}_{j}}</math>
!Failure Time  <math>{{X}_{ij}}</math>
|-
|1|| 1396|| 1|| 1396
|-
|2|| 4497|| 1|| 4497
|-
|3|| 525|| 1|| 525
|-
|4|| 1232|| 1|| 1232
|-
|5|| 227|| 1|| 227
|-
|6|| 135|| 1|| 135
|-
|7|| 19|| 1|| 19
|-
|8|| 812|| 1|| 812
|-
|9|| 2024|| 1|| 2024
|-
|10|| 943|| 2|| 316, 943
|-
|11|| 60|| 1|| 60
|-
|12|| 4234|| 2|| 4233, 4234
|-
|13|| 2527|| 2|| 1877, 2527
|-
|14|| 2105|| 2|| 2074, 2105
|-
|15|| 5079|| 1|| 5079
|-
|16|| 577|| 2|| 546, 577
|-
|17|| 4085|| 2|| 453, 4085
|-
|18|| 1023|| 1|| 1023
|-
|19|| 161|| 1|| 161
|-
|20|| 4767|| 2|| 36, 4767
|-
|21|| 6228|| 3|| 3795, 4375, 6228
|-
|22|| 68|| 1|| 68
|-
|23|| 1830|| 1|| 1830
|-
|24|| 1241|| 1|| 1241
|-
|25|| 2573|| 2|| 871, 2573
|-
|26|| 3556|| 1|| 3556
|-
|27|| 186|| 1|| 186
|-
|Total||52110|| 37||
|}
 
'''Solution'''
<br>
The sample fleet data set can be grouped into 10000, 20000, 30000, 4000 and 52110 time intervals. The following table gives the grouped data.
 
 
{|style= align="center" border="2"
|-
|colspan="2" style="text-align:center"|Grouped data
|-
!Time
!Observed Failures
|-
|10000|| 8
|-
|20000|| 16
|-
|30000|| 22
|-
|40000|| 27
|-
|52110|| 37
|}
Based on the above time intervals, the maximum likelihood estimates of  <math>\widehat{\lambda }</math>  and  <math>\widehat{\beta }</math>  for this data set are then given by:
 
 
::<math>\begin{matrix}
  \widehat{\lambda }=0.00147 \\
  \widehat{\beta }=0.93328 \\
\end{matrix}</math>
 
 
The next figure shows the System Operation plot.
 
[[Image:rga13.7.png|thumb|center|300px|System Operation plot for fleet data.]]
<br>

Latest revision as of 06:39, 23 August 2012

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