Crow Extended Confidence Bounds Example 2

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This example appears in the Reliability Growth and Repairable System Analysis Reference.


Calculate the 2-sided confidence bounds at the 90% confidence level on the demonstrated, projected and growth potential MTBF for the Test-Fix-Find-Test data in the Crow Extended Test-Fix-Find-Test Example.

Solution
For this example, there are A, BC and BD failure modes, so the estimated demonstrated failure intensity, λ^D(T), is simply the Crow-AMSAA model applied to all A, BC, and BD data.


λ^D(T)=λ^CA=λ^β^Tβ^1=0.12744


Therefore, the demonstrated MTBF is:


MTBFD=[λ^D(T)]1=7.84708


Based on this value, the Fisher Matrix confidence bounds for the demonstrated failure intensity at the 90% confidence level are:


[λD(T)]L=λ^CA(T)ezαVar(λ^CA(T))/λ^i(T)=0.09339


[λD(T)]U=λ^CA(T)ezαVar(λ^CA(T))/λ^i(T)=0.17390


The Fisher Matrix confidence bounds for the demonstrated MTBF at the 90% confidence level are:


MTBFDL=1[λD(T)]U=5.75054MTBFDU=1[λD(T)]L=10.70799


The Crow confidence bounds for the demonstrated MTBF at the 90% confidence level are:


MTBFDL=1[λD(T)]U=1λ^D(T)χ2(2N,α/2)2N=5.6325MTBFDU=1[λD(T)]L=1λ^D(T)χ2(2N,1α/2)2N=10.8779


The projected failure intensity is:


λ^P(T)=λ^CAλ^BD+i=1M(1di)NiT+d¯h^(T|BD)=0.0885


Based on this value, the Fisher Matrix confidence bounds at the 90% confidence level for the projected failure intensity are:


[λP(T)]L=λ^P(T)ezαVar(λ^P(T))/λ^P(T)=0.0681


[λP(T)]U=λ^P(T)ezαVar(λ^P(T))/λ^P(T)=0.1152


The Fisher Matrix confidence bounds for the projected MTBF at the 90% confidence level are:


MTBFPL=1[λP(T)]U=8.6818MTBFPU=1[λP(T)]L=14.6926


The Crow confidence bounds for the projected failure intensity are:


[λP(T)]L=λ^P(T)+C22λ^P(T) C2+C44=0.0672[λP(T)]U=λ^P(T)+C22+λ^P(T)C2+C44=0.1166


The Crow confidence bounds for the projected MTBF at the 90% confidence level are:


MTBFPL=1[λ^P(T)]U=8.5743MTBFPU=1[λ^P(T)]L=14.8769


The growth potential failure intensity is:


λ^GP=λ^CAλ^BD+i=1M(1di)NiT=0.0670


Based on this value, the Fisher Matrix and Crow confidence bounds at the 90% confidence level for the growth potential failure intensity are:


rL=r^GP+C22r^GPC2+C44=0.0488rU=r^GP+C22+r^GPC2+C44=0.0919


The Fisher Matrix and Crow confidence bounds for the growth potential MTBF at the 90% confidence level are:


MTBFGPL=1rU=10.8790MTBFGPU=1rL=20.4855


The figure below shows the Fisher Matrix confidence bounds at the 90% confidence level for the demonstrated, projected and growth potential MTBF.

Fisher Matrix confidence bounds on MTBF.

The next figure shows these bounds based on the Crow method.

Crow confidence bounds on MTBF.