PART V. Practices and Emerging Applications
3.4 Component Importance and the Component
Replacement Problem
Component importance measures the relative importance of a component, or sometimes the position of a component in the system, with respect to system reliability. The importance index can be used to assist the allocation of redundant components or replacement by giving priority to the more important positions. The most popular importance index is the Birnbaum importance, but very few results have been obtained. Weaker versions have been introduced to obtain more results.
44 System Reliability and Optimization
3.4.1 The Birnbaum Importance
The Birnbaum (reliability) importance of compo- nentiin systemx∈ {L, C}is defined as
Ix(i)=∂Rx(p1, p2, . . . , pn)
∂pi
It is the rate at which the system reliability grows when the reliability of componentigrows.
When necessary, we use Ix(i, n) to denote Ix(i) with n fixed. Independently, Griffith and Govindarajulu [13] and Papastavridis [39] first studied the Birnbaum importance for consecutive lines. Their results can be quoted as follows.
Theorem 20.
IL(i)= [RL(1, i−1)RL(i+1, n)−RL(1, n)]/qi and
IC(i)= [RL(pi+1, pi+2, . . . , pi−1)
−RC(1, n)]/qi.
It can be shown thatIL(i)observes the same recursive equations as observed byL(1, n). Thus similar to Theorem 1, the following theorem holds.
Theorem 21. For an IID model
1. IL(i, n)=IL(i, n−1)−pqkIL(i, n−k−1) ifn−i≥k+1
2. IL(i, n)=IL(i−1, n−1)−pqkIL(i−k−1, n−k−1)ifi−1≥k+1.
The comparison of Birnbaum importance under the IND model is valid only for the underlying set (p1, p2, . . . , pn). On the other hand, the IID model is the suitable one if the focus is on comparing the positions by neutralizing the differences inpi. Even for the IID model, Hwang et al. [40] showed that the comparison is not independent ofp.
The following theorem is given by Changet al.
[30].
Theorem 22. Consider the consecutive-2 line un- der the IID model. Then
IL(2i) > IL(2i−1) for2i≤(n+1)/2 IL(2i) < IL(2i−2) for2i≤(n+1)/2 IL(2i+1) > IL(2i−1) for2i+1≤(n+1)/2
For generalk, not much is known. Kuoet al.
[35] first observed the following result.
Theorem 23. For the consecutive-kline under the IID model
IL(1) < IL(2) <ã ã ã< IL(k) ifn≥2k IL(n−k+1)=IL(n−k+2)
= ã ã ã =IL(k) ifn <2k The following theorem was proved (partially) by Zuo [41] and (partially) by Zakariaet al.[42].
Theorem 24. Consider the consecutive-k line un- der the IID model. ThenIL(1)≤IL(i)for alli≤ n/2andIL(k) > IL(k+1)forn >2k.
Changet al.[43, 44] gave a method to compare IL(i) with IL(i+1), and derived the following results.
Theorem 25.
IL(2k+1) < IL(2k), IL(k+1) < IL(k+2) and
IL(2k−1,4k−1) < IL(2k,4k−1).
Recently, Changet al.[45] extended the results in Theorem 25 as follows:
Theorem 26.
(i) IL((t−2)k−1, tk−1) < IL[(t−2)k, tk−1]fort≥3
(ii) IL(3k+1,6k+1) < IL(3k,6k+1).
Hwang [46] defined a new importance mea- sure. In Hwang’s definition, component i is said to be more important than component j, writ- ten asH (i) > H (j ), if for everyd=k, k+1, k+ 2, . . . , n, |CSi,d|, the number of d-cutsets con- tainingiis never fewer than|CSj,d|. Hwang [46]
proved thatHmore importance implies Birnbaum more importance. He gave the following theorem.
Theorem 27. H (i)≥H (j ) implies IL(i)≥IL(j ) under the IID model for allp.
Note that one cannot use computation to prove IL(i)≥IL(j ) for all p since there is an infinite number of them. But for any finite system, we
Reliabilities of Consecutive-kSystems 45 can verifyH (i)≥H (j )sinced is bounded byn.
OnceH (i)≥H (j )is verified, then the previously impossible-to-verify relationIL(i)≥IL(j )is also verified. Changet al.[43] also proved that:
Theorem 28. H (k)≥H (i)for alli≤(n+1)/2.
3.4.2 Partial Birnbaum Importance
Chang et al. [45] proposed the half-line impor- tance Ih, which requires Ih(i) > Ih(j ) only for all p≥1/2for a comparison. They justified this half-line condition by noting that, in most practi- cal cases,p≥1/2. For the consecutive-k-out-of-n line, they were able to establish:
Theorem 29.
Ih(1) < Ih(2) <ã ã ã< Ih(k−1)
< Ih(k+1) < Ih(i) < Ih(2k) < Ih(k) for alli > k+1andi=2k.
The Birnbaum importance for the special case p=1/2 is known as the “structure importance”
in the literature. Chang et al. [30] suggested calling it the “combinatorial importance” so that the term “structure importance” can be reserved for general use (there are other importance indices depending on structure only). Denote the combinatorial importance by IC(i); Lin et al.
[47] found an interesting correspondence between IC(i)andfk,n, the Fibonacci numbers of orderk, which is defined by
fk,n=
0 if1≤n≤k−1
1 ifn=k
k i=1
fk,n−i ifn≥k+1 They proved:
Theorem 30. Forp=1/2,
RL(n)=(1/2)nfk,n+k+1.
Thus fk,n+k+1 can be interpreted as the number of working consecutive-k-out-of-nlines.
Theorem 31.
IC(i)=(1/2)n−1(2fk,i+kfk,n−i+k−1−fk,n+k+1).
Chang and Hwang [48] considered the case that ptends to zero. The importance index, denoted by IR(i), actually measures the number of minimum pathsets (a subset of components whose collective successes induce a system success) containing component i. Since, in practice, p is not likely to approach zero, IR(i)is not of interest per se.
However, it could be a useful tool for the comparison of Birnbaum importance. While it is not easy to establish I (i) > I (j ), sometimes it is also difficult to establish the falsity of it. By proving IR(i) < IR(j ), we automatically establish the above falsity. Further, if we have provedIh(i) > Ih(j ), then provingIR(i)≥IR(j ) would add a lot of credibility to the conjecture that I (i) > I (j ) since IR(i)≥IR(j ) provides evidence from the other end of thepspectrum.
Represent n as n=qk+r with 0≤r < k.
Then q is the minimum number of working components for a pathset to exist. Let psq(k, n) denote the number of pathsets with q working components and let psi,q(k, n) the number of those containing componenti. Chang and Hwang [48] proved:
Theorem 32.
psq(k, n)=
q+k−r−1 k−r−1
Representiasi=uk+vwith0< v≤k: Theorem 33.
psi,q(k, n)=
u+k−v k−v
q−u+v−r−2 v−r−1
Theorem 34. IR(uk+v)≤IR[(u+1)k+v] for 1≤v≤(k+1)/2and(u+1)k+v≤(n+1)/2.
IR(uk+1) < IR(uk) foruk+1≤(n+1)/2 IR(uk+1)≤IR(j ) foruk+1≤j≤(n+1/2)
3.4.3 The Optimal Component Replacement
Consider the problem: “When a new extra component is given to replace a component in
46 System Reliability and Optimization
a linear consecutive-k-out-of-n:F system in order to raise the system reliability, which component should be replaced such that the resulting system reliability is maximized?” When a component is replaced, the change of system reliability is not only dependent on the working probabilities of the removed component and the new component, but also on the working probabilities of all other components. A straightforward algorithm is first re-computing the resulting system reliabilities of all possible replacements and then selecting the best position to replace a component. Even using the most efficient O(n) reliability algorithm for linear consecutive-k-out-of-n:F systems, the computational complexity of the straightforward component replacement algorithm isO(n2).
Chang et al. [49] proposed an O(n)-time al- gorithm for the component replacement problem based on the Birnbaum importance. They first observed the following results.
Lemma 2. IL(i)is independent ofpi.
Let the reliability of the new extra component bep∗. Changet al.[49] derived that:
Theorem 35.
RL(p1, . . . , pi−1, p∗, pi+1, . . . , pn)
−RL(p1, . . . , pi−1, pi, pi+1, . . . , pn)
=IL(i)(p∗−pi)
They provided an algorithm to find the optimal location where the component should be replaced.
The algorithm is quoted as follows.
Algorithm 1. (Linear component replacement algorithm)
1. Compute RL(1,1), RL(1,2), . . . , RL(1, n) inO(n)time.
2. Compute RL(n, n), RL(n−1, n), . . . , RL
(2, n)inO(n)time.
3. Compute IL(i) for i=1,2, . . . , n, with the equation given in Theorem 14.
4. ComputeIL(i)(p∗−pi)fori=1,2, . . . , n.
5. Choose the i in step 4 with the largest IL(i)(p∗−pi) value. Then replace com- ponent i with the new extra component.
The reliability of the resulting system is IL(i)(p∗−pi)+RL(1, n).
In Algorithm 1, each of the five steps takes at most O(n) time. Therefore, the total computational complexity isO(n).
Consider the component replacement problem for the circular system. As with the linear case, a straightforward algorithm can be designed as first re-computing the resulting system reliabilities of all possible replacements and then selecting the best position to replace a component. However, even using the most efficient O(kn) reliability algorithm for the circular systems, the computa- tional complexity of the straightforward algorithm is stillO(kn2). Changet al.[49] also proposed a similar algorithm for the circular component re- placement problem. The computational complex- ity isO(n2).
If the circular consecutive-k-out-of-n:F system contains some components with zero working probabilities such that the whole system reliability is zero, then the computational complexity of the circular component replacement algorithm can be further improved. The following theorem for the special case was given by Changet al.[49].
Theorem 36. If RC(1, n)=0 and there is an i in {1,2, . . . , n} such that IC(i) >0, then the following three conditions must be satisfied.
1. C(1, n) has just one run of at least k consecutive components, where the working probability of each component is 0.
2. The run that mentioned in condition 1 contains fewer than2kcomponents.
3. If the run that mentioned in condition 1 contains components 1,2, . . . , m, where k≤ m <2k, then:
IC(i) >0
for alli∈ {m−k+1, m−k+2, . . . , k}, IC(i)=0
for alli /∈ {m−k+1, m−k+2, . . . , k}. Based on Theorem 36, the circular compo- nent replacement algorithm can be modified as follows.
Reliabilities of Consecutive-kSystems 47 Algorithm 2. (Modified circular component re-
placement algorithm forRC(1, n)=0)
1. Find the largest run of consecutive compo- nents consisting of components with 0 work- ing probabilities 0. Then re-index the compo- nents such that this run contains components 1,2, . . . , m (m≥k).
2. Ifm≥2k, then any replacement is optimal.
STOP.
3. Compute RL(i+1, n+i−1) for i∈ {m− k+1, m−k+2, . . . , k}.
4. Choose theiin step 3 with the largestRL(i+ 1, n+i−1)value. Then replace componenti with the new extra component. The reliability of the resulting system is RL(i+1, n+i− 1)p∗. STOP.
In the modified algorithm, step 1 takes O(n) time, step 2 takesO(1)time, step 3 takes at most O(kn)time, and step 4 takes at mostO(n)time.
The total computational complexity is thusO(kn).