Chapter 5 Component Importance

Transcription

1 Chapter 5 Component Importance Marvin Rausand Department of Production and Quality Engineering Norwegian University of Science and Technology marvin.rausand@ntnu.no Marvin Rausand, March 19, 2004 System Reliability Theory (2nd ed.), Wiley, 2003 p. 1/19

2 Measures Covered Measures Covered Importance Depends On The following component importance measures are defined and discussed in this chapter: Birnbaum s measure (and some variants) The improvement potential measure (and some variants) Risk achievement worth Risk reduction worth The criticality importance measure Fussell-Vesely s measure Marvin Rausand, March 19, 2004 System Reliability Theory (2nd ed.), Wiley, 2003 p. 2/19

3 Importance Depends On Measures Covered Importance Depends On The various measures are based on slightly different interpretations of the concept component importance. Intuitively, the importance of a component should depend on two factors: The location of the component in the system The reliability of the component in question and, perhaps, also the uncertainty in our estimate of the component reliability. Marvin Rausand, March 19, 2004 System Reliability Theory (2nd ed.), Wiley, 2003 p. 3/19

4 Marvin Rausand, March 19, 2004 System Reliability Theory (2nd ed.), Wiley, 2003 p. 4/19

5 Birnbaum s Measure (1) Birnbaum (1969) proposed the following measure of the reliability importance of a component: Birnbaum s measure of importance of component i at time t is I B (i t) = h(p(t)) p i (t) Birnbaum s measure is thus obtained by partial differentiation of the system reliability with respect to p i (t). This approach is well known from classical sensitivity analysis. If I B (i t) is large, a small change in the reliability of component i will result in a comparatively large change in the system reliability at time t Marvin Rausand, March 19, 2004 System Reliability Theory (2nd ed.), Wiley, 2003 p. 5/19

6 Birnbaum s Measure (2) By fault tree notation, Birnbaum s measure may be written as where I B (i t) = Q 0(t) q i (t) q i (t) = 1 p i (t) Q 0 (t) = 1 p S (t) = 1 h(p(t)) Birnbaum s measure is named after the Hungarian-American professor Zygmund William Birnbaum ( ) Marvin Rausand, March 19, 2004 System Reliability Theory (2nd ed.), Wiley, 2003 p. 6/19

7 Birnbaum s Measure (3) By pivotal decomposition we have h(p(t)) = p i (t) h(1 i,p(t)) + (1 p i (t)) h(0 i,p(t)) = p i (t) [h(1 i,p(t)) h(0 i,p(t))] h(0 i,p(t)) Birnbaum s measure can therefore we written as I B (i t) = h(p(t)) p i (t) = h(1 i,p(t)) h(0 i,p(t)) Note that Birnbaum s measure I B (i t) of component i only depends on the structure of the system and the reliabilities of the other components. I B (i t) is independent of the actual reliability p i (t) of component i. This may be regarded as a weakness of Birnbaum s measure Marvin Rausand, March 19, 2004 System Reliability Theory (2nd ed.), Wiley, 2003 p. 7/19

8 Birnbaum s Measure (4) Since h( i,p(t)) = E[φ( i,x(t)] we can write I B (i t) = E[φ(1 i,x(t)] E[φ(0 i,x(t)] = E[φ(1 i,x(t) φ(0 i,x(t)] When the structure is coherent [φ(1 i,x(t)) φ(0 i,x(t))] can only take on the values 0 and 1. Therefore I B (i t) = Pr(φ(1 i,x(t)) φ(0 i,x(t)) = 1) This is to say that I B (i t) is equal to the probability that (1 i,x(t)) is a critical path vector for component i at time t Birnbaum s measure is therefore the probability that the system is such a state at time t that component i is critical for the system Marvin Rausand, March 19, 2004 System Reliability Theory (2nd ed.), Wiley, 2003 p. 8/19

9 Birnbaum s Measure (5) Assume that component i has failure rate λ i. In some situations we may be interested in measuring how much the system reliability will change by making a small change to λ i. The sensitivity of the system reliability with respect to changes in λ i can obviously be measured by h(p(t)) λ i = h(p(t)) p i (t) p i(t) λ i = I B (i t) p i(t) λ i A similar measure can be used for all parameters related to the component reliability p i (t), for i = 1, 2,..., n. In some cases, several components in a system will have the same failure rate λ. To find the sensitivity of the system reliability with respect to changes in λ, we can still use h(p(t))/ λ Marvin Rausand, March 19, 2004 System Reliability Theory (2nd ed.), Wiley, 2003 p. 9/19

10 Birnbaum s Measure (6) Consider a system where component i has reliability p i (t) that is a function of a parameter θ i. The parameter θ i may be the failure rate, the repair rate, or the test frequency, of component i. To improve the system reliability, we may want to change the parameter θ i (by buying a higher quality component or changing the maintenance strategy). Assume that we are able to determine the cost of the improvement as a function of θ i, that is, c i = c(θ i ), and that this function is strictly increasing or decreasing such that we can find its inverse function. The effect of an extra investment related to component i may now be measured by h(p(t)) c i = h(p(t)) θ i θ i c i = I B (i t) p i(t) θ i θ i c i Marvin Rausand, March 19, 2004 System Reliability Theory (2nd ed.), Wiley, 2003 p. 10/19

11 Birnbaum s Measure (7) In a practical reliability study of a complex system, one of the most time-consuming tasks is to find adequate estimates for the input parameters (failure rates, repair rates, etc.). In some cases, we may start with rather rough estimates, calculate Birnbaum s measure of importance for the various components, or the parameter sensitivities, and then spend the most time finding high-quality data for the most important components. Components with a very low value of Birnbaum s measure will have a negligible effect on the system reliability, and extra efforts finding high-quality data for such components may be considered a waste of time Marvin Rausand, March 19, 2004 System Reliability Theory (2nd ed.), Wiley, 2003 p. 11/19

12 Improvement Potential (1) The improvement potential of component i at time t is I IP (i t) = h(1 i,p(t)) h(p(t)) The improvement potential may be expressed as I IP (i t) = I B (i t) (1 p i (t)) or, by using the fault tree notation I IP (i t) = I B (i t) q i (t) Marvin Rausand, March 19, 2004 System Reliability Theory (2nd ed.), Wiley, 2003 p. 12/19

13 Improvement Potential (2) I IP (i t) is the difference between the system reliability with a perfect component i, and the system reliability with the actual component i. In practice, it is not possible to improve the reliability p i (t) of component i to 100% reliability. Let us assume that it is possible to improve p i (t) to new value (t) representing, for example, the state of the art for this type of components. We may then calculate the realistic or credible improvement potential (CIP) of component i at time t, defined by p (n) i I CIP (i t) = h(p (n) i (t),p(t)) h(p(t)) where h(p (n) i (t), p(t)) denotes the system reliability when component i is replaced by a new component with reliability p (n) i (t). Marvin Rausand, March 19, 2004 System Reliability Theory (2nd ed.), Wiley, 2003 p. 13/19

14 Risk Achievement Worth The importance measure risk achievement worth (RAW) of component i at time t is I RAW (i t) = 1 h(0 i,p(t)) 1 h(p(t)) The RAW is the ratio of the (conditional) system unreliability if component i is not present (or if component i is always failed) with the actual system unreliability. The RAW presents a measure of the worth of component i in achieving the present level of system reliability and indicates the importance of maintaining the current level of reliability for the component. Marvin Rausand, March 19, 2004 System Reliability Theory (2nd ed.), Wiley, 2003 p. 14/19

15 Risk Reduction Worth The importance measure risk reduction worth (RRW) of component i at time t is I RRW (i t) = 1 h(p(t)) 1 h(1 i,p(t)) The RRW is the ratio of the actual system unreliability with the (conditional) system unreliability if component i is replaced by a perfect component with p i (t) 1. In some applications, failure of a component may be an operator error or some external event. If such components can be removed from the system, for example, by canceling an operator intervention, this may be regarded as replacement with a perfect component. Marvin Rausand, March 19, 2004 System Reliability Theory (2nd ed.), Wiley, 2003 p. 15/19

16 Criticality Importance (1) The component importance measure criticality importance I CR (i t) of component i at time t is the probability that component i is critical for the system and is failed at time t, when we know that the system is failed at time t. I CR (i t) = IB (i t) (1 p i (t)) 1 h(p(t)) By using the fault tree notation, I CR (i t) may be written as I CR (i t) = IB (i t) q i (t) Q 0 (t) Marvin Rausand, March 19, 2004 System Reliability Theory (2nd ed.), Wiley, 2003 p. 16/19

17 Criticality Importance (2) Let C(1 i,x(t)) denote the event that the system at time t is in a state where component i is critical. We know that Pr(C(1 i,x(t))) = I B (i t) The probability that component i is critical for the system and at the same time is failed at time t is Pr(C(1 i,x(t)) (X i (t) = 0)) = I B (i t) (1 p i (t)) When we know that the system is in a failed state at time t, then Pr(C(1 i,x(t)) (X i (t) = 0) φ(x(t)) = 0) Marvin Rausand, March 19, 2004 System Reliability Theory (2nd ed.), Wiley, 2003 p. 17/19

18 Criticality Importance (3) Since the event C(1 i,x(t)) (X(t) = 0) implies that φ(x(t)) = 0), we get Pr(C(1 i,x(t)) (X i (t) = 0)) Pr(φ(X(t)) = 0) = IB (i t) (1 p i (t)) 1 h(p(t)) I CR (i t) is therefore the probability that component i has caused system failure, when we know that the system is failed at time t. For component i to cause system failure, component i must be critical, and then fail. When component i is repaired, the system will start functioning again. This is why the criticality importance measure may be used to prioritize maintenance actions in complex systems Marvin Rausand, March 19, 2004 System Reliability Theory (2nd ed.), Wiley, 2003 p. 18/19

19 Fussell-Vesely s Measure Fussell-Vesely s measure of importance, I FV (i t) is the probability that at least one minimal cut set that contains component i is failed at time t, given that the system is failed at time t. Fussell-Vesely s measure can be approximated by I FV (i t) 1 m i j=1 (1 ( ˇQ i j (t)) Q 0 (t) mi j=1 ˇQ i j (t) Q 0 (t) where ˇQ i j (t)) denotes the probability that minimal cut set j among those containing component i is failed at time t Marvin Rausand, March 19, 2004 System Reliability Theory (2nd ed.), Wiley, 2003 p. 19/19