On Some Structural Importance of System Components

In this note a new method of comparing component structural importance is introduced and compared to other existing ones. Especially, relationships of the new comparison method to the H-importance due to Hwang (2001,2005), the criticality ordering due to Boland et al. (1989) and Birnbaum importance are obtained. Illustrative examples are given.


Introduction
Consider a binary coherent system (C, φ) composed of n independent components, where C = {1, 2, . . ., n} denotes the index set of the n components, and φ, a mapping from {0, 1} n to {0, 1}, denotes the (nondecreasing) structure function of the system.Denote by p i the reliability of the ith component, and by h(p ¯) = h(p 1 , . . ., p n ) the system reliability function.A vector x ¯for which φ(x ¯) = 0 (1) is called a cut (path) vector.A minimal cut (path) vector is a cut (path) vector x ¯, and φ(y ¯) = 1 (0) for all y ¯> (<) x, where y ¯> x ¯means y i ≥ x i for all i and y i > x i for some i.For any cut (path) vector x ¯the index subset {i : x i = 0} ({i : x i = 1}) is the corresponding cut (path) set.If x ¯is a minimal cut (path) vector then {i : x i = 0} ({i : x i = 1}) is called a minimal cut (path) set for the system.See Barlow and Proschan (1981) for a full account of the theory of binary coherent systems.
Various measures (orderings) of component importance for binary coherent systems have been introduced in the literature.The most fundamental one is Birnbaum reliability importance measure, defined by The Birnbaum reliability importance of components may be used to evaluate the effect of an improvement in component reliability on system reliability (see Barlow and Proschan (1981)).In the case where p 1 = • • • = p n = 1/2, the Birnbaum reliability importance measure reduces to Birnbaum structural importance measure, denoted by I φ (i), where x ¯∈ {0, 1} n−1 , and | • | denotes set cardinality.
In the absence of information about component reliabilities, the Birnbaum structural importance measure provides a fair comparison of relative importance among system components.However, for systems with highly reliable components, using the Birnbaum structural importance may give a misleading picture of which components are most important.Butler (1979) developed an alternative structural ordering, namely the cut-importance ordering, and obtained a relationship between the cut-importance and the Birnbaum reliability importance measure for high values of p ¯. Since structural importance of a component represents importance of the node (position), we use the terms structural importance of component i or importance of node i alternatively in this note without ambiguity.
Definition 1.1: (Butler (1979)) For each component s, let d (s) ij denote the number of unions of i distinct minimal cut sets such that the union contains exactly j nodes and includes node s.Let b Theorem 1.2: (Butler (1979)) For p ¯= (p, p, • • • , p) and the scalar p is sufficiently close to one, the cut-importance ordering > c and the ordering induced by I h (i; p ¯) are identical.Later, Boland et al. (1989) introduced the following notion of criticality.Based on the structural criticality ordering, they developed a principle of pairwise rearrangement of components to improve system reliability.
and strict inequality holds for some x ¯.If equality holds for all x ¯, components i and j are said to be permutation equivalent, denoted by i = c j.
It is shown in Boland et al. (1989) that the criticality ordering is in some sense stronger than that of the Birnbaum structural importance: Theorem 1.4: (Boland et al. (1989) The relationships of the structural criticality and the cut-importance to the Birnbaum reliability importance are presented in Meng (1995): ( (III) It readily follows that i > c j =⇒ i > c j, from (I) and (II).
Recently, Hwang (2001Hwang ( ,2005) ) introduced the following two new indices of component structural importance, based on the cut sets (path sets) and minimal cut sets of a binary coherent system.
It is shown in Hwang (2001Hwang ( ,2005) ) that the notion of H-( H)-importance is between the criticality and the cut-importance in the sense that The above implications in (i) and (ii) are not reversible (see examples in section 3 of this note).

Results
First, we introduce a new method of comparing component relative importance, instead of using quantitative measures.

Definition 2.1:
That is, there exists a path set of size d which contains i but not j.
Lemma 2.2: The Birnbaum structural importance, From the above expression, we easily see that Next we show that the structural ordering in Definition 2.1 is essentially equivalent to the one induced by the number of cut sets (path sets) introduced in Hwang ( 2001).

Proof:
The sets C i (d) and C j (d) can be decomposed as unions of two disjoint sets.That is, , where C ij (d) denotes cut sets that contains both i and j; and C i j (d) (Cī j (d)) denotes cut sets that contains i (j) but not j (i).Thus, The conclusion follows.
The Birnbaum structure importance I φ (i) is identical to the Birnbaum reliability importance Hwang (2001) that if components i is more H-important than j, then i dominates j in the universal Birnbaum comparison, namely, I h (i; p ¯) ≥ I h (j; p ¯) for all p = (p, • • • , p), all p ∈ (0, 1).The following theorem slightly strengthens this result.

Theorem 2.4:
From the above expression, we see that From Lemma 2.3, Theorem 2.4 and result (II) in section 1, we conclude that more H-important implies more cut-important.This fact was proved, but the proof in Hwang ( 2001) is rather complicated.
The main result obtained in Boland et al. (1989) provides a principle of rearrangement of components in order to achieve better system reliability.This principle can be rephrased as: i > c j if and only if h(α i , β j ; p ¯) ≥ h(β i , α j ; p ¯) for all 1 ≥ α > β ≥ 0, all p ¯, with strict inequality holding for some 1 ¯> p ¯> 0 ¯.Under a weaker than criticality assumption on component importance, we obtain the following theorem pertaining to rearrangement of components.

Proof:
(see the proof of T heorem 2.4).

Illustrative examples
In this section examples are given to illustrate that the implication relationships obtained in Hwang (2001Hwang ( , 2005)), concerning criticality, cut importance, H-importance and H-importance, are not reversible.

Figure 2 :
Figure 2: A parallel-series system Node s is said to be more cut-important than node t,