Reliability Importance and Optimized Reliability
Allocation (Analytical)Reliability Importance and Optimized Reliability
Allocation (Analytical)This chapter includes the following sections:
Once the reliability of a system has been determined, engineers are often faced with the task of identifying the least reliable component(s) in the system in order to improve the design. For example, it was observed in the RBDs and Analytical System Reliability section of this on-line reference that the least reliable component in a series system has the biggest effect on the system reliability. In this case, if the reliability of the system is to be improved, then the efforts can best be concentrated on improving the reliability of that component first. (Note: At this point, the cost of improving the reliability of the component is not considered. Cost of improvement is covered in the Reliability Allocation section of this on-line reference.) In simple systems such as a series system, it is easy to identify the weak components. However, in more complex systems this becomes quite a difficult task. For complex systems, the analyst needs a mathematical approach that will provide the means of identifying and quantifying the importance of each component in the system.
Using reliability importance measures is one method of identifying the relative importance of each component in a system with respect to the overall reliability of the system. The reliability importance, IR, of component i in a system of n components is given by Leemis [17]:
(1)
Where:
Rs is the system reliability.
Ri is the component reliability.
The value of the reliability importance given by Eqn. (1) depends both on the reliability of a component and its corresponding position in the system. In the RBDs and Analytical System Reliability section of this on-line reference, we observed that for a simple series system (three components in series with reliabilities of 0.7, 0.8 and 0.9) the rate of increase of the system reliability was greatest when the least reliable component was improved. In other words, it was observed that Component 1 had the largest reliability importance in the system relative to the other two components (see Figure 6.1). The same conclusion can be drawn by using Eqn. (1) and obtaining the reliability importance in terms of a value for each component.
Figure 6.1: Rate of change of system reliability when increasing the reliability of each component.
Using BlockSim, the reliability importance values for these components can be calculated with Eqn. (1). Using the plot option and selecting the "Static Reliability Importance" plot type, Figure 6.2 can be obtained. Note that the time input required to create this plot is irrelevant for this example because the components are static.
Figure 6.2: Static Reliability Importance plot.
The values shown in Figure 6.2 for each component were obtained using Eqn. (1). The reliability equation for this series system was given by:
(2)
Taking the partial derivative of Eqn. (2) with respect to R1 yields:
Thus, the reliability importance of Component 1 is
0.72.
The reliability importance values for Components 2 and 3 are obtained
in a similar manner.
The same concept applies if the components have a time varying reliability.
That is, if Rs(t) = R1(t) • R2(t) • R3(t),
then one could compute 
at any time x
or 
This is quantified in Eqn. (3).
(3)
In turn, this can be viewed as either a static plot (at a given time)
or as time varying plot, as illustrated in the next figures. Specifically,
Figures 6.3, 6.4 and 6.5 present the analysis for three components configured
reliability-wise in series following a Weibull distribution with
β = 3 and η1 = 1,000,
η2 = 2,000
and
η3 = 3,000.
Figure 6.3 shows a bar chart of while Figure 6.4 shows
the in BlockSim's "tableau chart" format. In
this chart, the area of the square is . Lastly, Figure 6.5
shows the (t) vs. time.
Figure 6.3: Static Reliability Importance plot at t = 1,000.
Figure 6.4: Static Reliability Importance tableau plot at t = 1,000.
Figure 6.5: Reliability Importance vs. time plot.
Assume that a system has failure modes A, B, C, D, E and F. Furthermore, assume that failure of the entire system will occur if:
Mode B, C or F occurs.
Modes A and E, A and D or E and D occur.
In addition, assume the following failure probabilities for each mode.
Modes A and D have a mean time to occurrence of 1,000 hours (i.e. exponential with MTTF = 1,000).
Mode E has a mean time to occurrence of 100 hours (i.e. exponential with MTTF = 100).
Modes B, C and F have a mean time to occurrence of 700,000, 1,000,000 and 2,000,000 hours respectively (i.e. exponential with MTTFB = 700,000, MTTFC = 1,000,000 and MTTFF = 2,000,000).
Examine the mode importance for operating times of 100 and 500 hours.
The RBD for this example is (from Example 18 in the RBDs and Analytical System Reliability section of this on-line reference):
Figure 6.6 illustrates (t
= 100). It can be seen that even though B,
C and F
have a much rarer rate of occurrence, they are much more significant at
100 hours. By 500 hours, (t
= 500), the effects of the lower reliability components become greatly
pronounced and thus they become more important, as it can be seen in Figure
6.7. Finally, the behavior of (t)
can be observed in Figure 6.8. Note that not all lines are plainly visible
in Figure 6.8 due to overlap.
Figure 6.6: Plot of
(t
= 100) for Example 1.
Figure 6.7: Plot of
(t
= 500) for Example 1.
Figure 6.8: Plot of
(t)
for Example 1.
See Also:
Reliability Importance and Optimized Reliability Allocation (Analytical)
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