Journal of Systems Engineering and Electronics

Vol. 23, No. 6, December 2012, pp.936–942

Importance measure of system reliability upgrade formulti-state consecutive k-out-of-n systems

Hongyan Dui, Shubin Si*, Zhiqiang Cai, Shudong Sun, and Yingfeng Zhang

School of Mechatronics, Northwestern Polytechnical University, Xi’an 710072, P. R. China

Abstract: Importance measures in reliability systems are used toidentify weak components in contributing to a proper function ofthe system. Traditional importance measures mainly concernedthe changing value of the system reliability caused by the changeof the reliability of the component, and seldom considered the jointeffect of the probability distribution, improvement rate of the objectcomponent. This paper studies the rate of the system reliabilityupgrading with an improvement of the component reliability for themulti-state consecutive k-out-of-n system. To verify the multi-stateconsecutive k-out-of-n system reliability upgrading by improvingone component based on its improvement rate, an increasing po-tential importance (IPI) and its physical meaning are describedat first. Secondly, the relationship between the IPI and Birnbaumimportance measures are discussed. And the IPI for some differ-ent improvement actions of the component is further discussed.Thirdly, the characteristics of the IPI are analyzed. Finally, an ap-plication to an oil pipeline system is given.

Keywords: system reliability, multi-state consecutive k-out-of-nsystem, improvement, Birnbaum importance.

DOI: 10.1109/JSEE.2012.00115

1. Introduction

The consecutive k-out-of-n system has attracted a greatdeal of attention in the field of the reliability. The con-secutive k-out-of-n system structure is a very popular typein engineering systems such as oil pipeline and telecom-munication systems [1], vacuum systems in accelera-tors [2], computer ring networks [3] and spacecraft relaystations [4].

In binary systems, a linear consecutive k-out-of-n:F (G) system consists of n linearly arranged componentssuch that the system fails (works) if and only if at leastk consecutive components fail (work). For the circular

Manuscript received November 2, 2011.*Corresponding author.This work was supported by the National Natural Science Foundation

of China (71271170; 71101116), the National High Technology Researchand Development Program of China (863 Progrom) (2012AA040914),and the Basic Research Foundation of Northwestern Polytechnical Uni-versity (JC20120228).

system, components are labeled clock wise from 1 to n

and the rules for the failure (operation) remain the sameas the linear system. Reference [1] firstly introduced thelinear consecutive k-out-of-n: F systems and describedthe telecommunication and pipeline systems which have asimilar structure. Reference [5] presented some stochas-tic ordering results for the reliability of the consecutivek-out-of-n: F system which consists of the independentand identical components. References [6,7] studied therelationship between n and k in a linear consecutive k-out-of-n: G system. Reference [8] studied the reliabilityproperty of consecutive k-out-of-n systems with arbitrarilydependent components. Reference [9] considered linearand circular consecutive k-out-of-n systems and the relia-bility of components of the systems which are assumed tobe independent but their probability distributions are non-identical.

In multi-state systems, both the system and componentsare in M +1 possible states, as {0, 1, . . . , M}, where M isthe perfect functioning state, while 0 is the complete fail-ure state. A few researchers have extended the definitionsof the binary consecutive k-out-of-n system to a multi-statecase [10,11]. Reference [12] extended the binary linearconsecutive k-out-of-n system to the multi-state case. Ref-erence [13] extended the binary circular consecutive k-out-of-n system to the multi-state case by allowing the systemto remain the binary, and its components with more thantwo possible states. Reference [14] proposed the definitionof the multi-state consecutive k-out-of-n: F and G sys-tems.

Importance measures reflect the effect of the individ-ual component reliability on the system reliability. Theyhave been widely used for identifying system weaknessesand supporting system improvement activities. Importancemeasures were first introduced in [15]. The Birnbaum im-portance measure deals with the effects of changes in theunreliability of a given component. Reference [16] derivedthe Birnbaum importance for the consecutive k-out-of-n:

Hongyan Dui et al.: Importance measure of system reliability upgrade for multi-state consecutive k-out-of-n systems 937

F system. Reference [17] provided the structure impor-tance ordering for the consecutive 2-out-of-n system. Ref-erence [18] obtained a complete ordering of componentswith respect to their structure importance for some con-secutive k-out-of-n: F systems, and a partial ordering forother systems. Reference [19] claimed that the compari-son of Birnbaum importance between two components fora consecutive k-out-of-n: G system is the same as that forthe F system. Reference [20] pointed out that the com-parison in [19] is not the case and gave a correct relationbetween the two systems. Reference [21] solved the prob-lem that a new component and a consecutive k-out-of-n:F system were given, in which the component should bereplaced with the new one such that the resulting system re-liability is maximized based on the Birnbaum importance.Reference [22] studied the relationship between the con-secutive k-out-of-n: F system and the consecutive k-out-of-n: G system, developed theorems for such systems, andthe optimal configuration rule was suggested by using theBirnbaum importance index. Reference [23] used the fi-nite Markov chain imbedding approach to evaluate the sys-tem state distribution for generalized multi-state k-out-of-n systems. Reference [24] developed an interval universalgenerating function to estimate the interval-valued relia-bility of the multi-state system. Reference [25] evaluatedmean residual and past lifetime functions of multi-statek-out-of-n: G systems. References [26,27] introduced amulti-state system structure for the weighted multi-state k-out-of-n system. Reference [28] presented three decisiondiagrams for the multi-state system modeling.

Although significant contributions were made by aboveresearchers, traditional importance measures mainly con-cerned the changing value of the system reliability causedby the change of the reliability of the component, and sel-dom considered the joint effect of the probability distribu-tions, improvement rate of the object component. Further-more, the expected upgrading of the system reliability fromthe component improvement is related to the improvementrate of components and the effectiveness of the systemstructure. In this paper, a new importance measure in themulti-state consecutive k-out-of-n system is presented tostudy the rate of the system reliability upgrading.

The rest of the paper is organized as follows. An im-portance measure of the multi-state consecutive k-out-of-nsystem reliability upgrading is proposed in Section 2. Sec-tion 3 demonstrates the characteristics of the importancemeasure. An application to an oil pipeline system is givenin Section 4. Finally, Section 5 draws some conclusions.

2. Importance measure

In this paper, the state space of each component and sys-

tem is {0, 1, 2, . . . , M}, where 0 corresponds to a completefailure of the system or its components and M is the perfectfunctioning state of the system or components. The statesare ordered from 0 to M . All components are statisticallyindependent.

2.1 Definition of importance measure

Reference [17] proved that matching components of bothconsecutive k-out-of-n: F and G systems have the sameBirnbaum importance, and [22] studied the reliability re-lationship between the consecutive k-out-of-n: F systemand the consecutive k-out-of-n: G system. Therefore inthis section, we will discuss the multi-state consecutivek-out-of-n: F system based on the system performancelevel m (1 � m � M). m represents that the system isworking when the system structure function Φ(X) � m,where X represents the vector of states of the components(x1, x2, . . . , xn), and xi represents the state of the compo-nent i (i = 0, 1, 2, . . . , M).

Reference [14] proposed the definition of the multi-stateconsecutive k-out-of-n: F system: Φ(X) < j (j =1, 2, . . . , M) if at least kl consecutive components are instates below l for all l such that j � l � M . An n-component system with such a property is called a multi-state consecutive k-out-of-n system. When k1 � k2 �· · · � kM , the system is called a decreasing multi-stateconsecutive k-out-of-n system, and when k1 � k2 � · · · �kM , the system is called an increasing multi-state consec-utive k-out-of-n system [14]. When kj is a constant, i.e.,k1 = k2 = · · · = kM = k, the structure of the system isthe same for all the system state levels. So the multi-stateconsecutive k-out-of-n: F system based on the system per-formance level m is equivalent to: Φ(X) < m if and onlyif at least consecutive km components have xi < m.

The most well-known importance measure is the Birn-baum importance measure which gives the contribution ofone component reliability to the system reliability in bi-nary systems. Birnbaum first introduced the importance ofthe component i for binary systems as follows [15]:

I(BM)i =∂ Pr{Φ(X) = 1}

∂Pi1=

Pr{Φ(X) = 1|xi = 1} − Pr {Φ(X) = 1|xi = 0} (1)

where Pi1 = Pr{xi = 1}.Reference [29] generalized the Birnbaum importance

measure from the binary systems to the multi-state ones.For a multi-state system of m levels, the system workswhen Φ(X) � m and fails when Φ(X) < m. So theBirnbaum importance of the component i with the systemperformance level of multi-state systems is [29]

I(BM)i = Pr{Φ(X) � m|xi � m}−

938 Journal of Systems Engineering and Electronics Vol. 23, No. 6, December 2012

Pr{Φ(X) � m|xi < m}. (2)

Reference [16] derived the Birnbaum importance for theconsecutive k-out-of-n: F system in binary systems as fol-lows:

I(BM)i =∂R(n)∂Pi1

= R(n|xi = 1) − R(n|xi = 0) =

R(i − 1)R′(n − i) − R(n)1 − Pi1

(3)

where R(j) represents the reliability of the multi-stateconsecutive k-out-of-j: F subsystem consisting of com-ponents {1, 2, . . . , j} with the system performance level,R′(j) represents the reliability of the multi-state consec-utive k-out-of-j: F subsystem consisting of components{(n − j + 1), (n − j + 2), . . . , (n − 1), n} with thesystem performance level, R(n) represents the reliabilityof the multi-state consecutive k-out-of-n: F system andR(n|·i) = R(P1,m, . . . , Pi−1,m, ·, Pi+1,m, . . . , Pn,m).

Then according to (2), the Birnbaum importance for theconsecutive k-out-of-n: F system in multi-state systemsbased on the system performance level m is as follows:

I(BM)i =∂R(n)∂Pi,m

= R(n|xi � m) − R(n|xi < m) =

R(i − 1)R′(n − i) − R(n)1 − Pi,m

(4)

where Pi,m = Pr{xi � m}.In the multi-state consecutive k-out-of-n: F system

based on the system performance level m, according to (4),we have

R(n)=Pi,mR(n|xi �m)+(1−Pi,m)R(n|xi < m) =

Pi,m [R(n|xi �m)−R(n|xi < m)] + R(n|xi < m) =

Pi,mI(BM)i + R(n|xi < m). (5)

The improving component i is equivalent to an upgradeof Pi,m. Here suppose that P ′

i,m is the reliability for theimproved component i and R̄(n) is the system reliabilityafter the improvement on the component i. According to(5), the system reliability upgrading is

R̄(n) − R(n) = P ′i,mI(BM)i + R(n|xi < m)−

Pi,mI(BM)i − R(n|xi < m) =

(P ′i,m − Pi,m)I(BM)i. (6)

But it would make more sense to consider the rate of thesystem reliability upgrading with the improvement time ofthe component i rather than the value of the system re-liability upgrading for managers. When considering themean effect of the improving component i on the systemreliability upgrading per unit time, we need to introducethe improvement rate μi of the component i to (6). The

μi represents the expected times that the component i im-proves from Pi,m to P ′

i,m in an unit time. So the rate of thesystem reliability upgrading based on the improvement ofthe component i is

μi(R̄(n) − R(n)) = μi(P ′i,m − Pi,m)I(BM)i. (7)

Definition 1 The increasing potential importance (IPI)of the component i, I(IPI)i, is defined as the rate of thesystem reliability upgrading due to an improvement of thecomponent i in the multi-state consecutive k-out-of-n: F

system based on the system performance level m. That is

I(IPI)i = μi(P ′i,m − Pi,m)I(BM)i =

μi(P ′i,m − Pi,m)

R(i − 1)R′(n − i) − R(n)1 − Pi,m

. (8)

From Definition 1, the physical meaning of I(IPI)i isthe rate of the system reliability upgrading due to an im-provement of the component i. For a multi-state consec-utive k-out-of-n: F system consisting of n components,an effort should be given to the most important componentwhich represents the maximization of the rate of the sys-tem reliability upgrading based on I(IPI)i.

2.2 Relationships between Birnbaum importance andIPI for some different improvement actions

Based on Definition 1, we have I(IPI)i = μi(P ′i,m −

Pi,m)I(BM)i. And according to some different improve-ment actions, we can get the following points.

Firstly, if the improvement action is to upgrade the com-ponent reliability by the same amount, P ′

i,m − Pi,m =A, A is a constant, for all components i, then we haveI(IPI)i = AμiI(BM)i. If μ1 = μ2 = · · · = μn = μ,then I(IPI)i = AμI(BM)i. So I(IPI)i has the samecharacteristic as the one of I(BM)i in the multi-state con-secutive k-out-of-n: F system [18−22].

Nextly, if we improve the component i to the perfectfunctioning, then P ′

i,m = 1. We can get

I(IPI)i = μi(1 − Pi,m)I(BM)i =

μi(1 − Pi,m)R(i − 1)R′(n − i) − R(n)

1 − Pi,m=

μi[R(i − 1)R′(n − i) − R(n)]. (9)

Firally, if we use a parallel redundancy to represent theimprovement of the component reliability, then P ′

i,m =1 − (1 − Pi,m)2. So we can get

I(IPI)i = μi

(P ′

i,m − Pi,m

)I(BM)i =

μi

(1 − (1 − Pi,m)2 − Pi,m

)I(BM)i =

μi (1 − Pi,m)Pi,mI(BM)i =

Hongyan Dui et al.: Importance measure of system reliability upgrade for multi-state consecutive k-out-of-n systems 939

μiPi,m [R(i − 1)R′(n − i) − R(n)] . (10)

3. Characteristics of IPI

Theorem 1 The optimal improvement of the systemreliability per unit time is to improve the component i withthe maximal I(IPI)i for a multi-state consecutive k-out-of-n: F system.

Proof Immediate from (7) and (8). �We use RL(j), R′

L(j), RL(n) to represent R(j),R′(j), R(n) of the linear system. We can get the followingTheorem.

Theorem 2 For a multi-state linear consecutive k-out-of-n: F system, we can get that I(IPI)i = μi(P ′

i,m −

Pi,m)RL(i − 1)R′

L(n − i) − RL(n)1 − Pi,m

.

Proof According to (4), we have

I(BM)i =∂R(n)∂Pi,m

= R(n|xi � m) − R(n|xi < m) =

R(i − 1)R′(n − i) − R(n)1 − Pi,m

.

For a multi-state linear consecutive k-out-of-n: F sys-tem, we can get R(i − 1) = RL(i − 1), R′(n − i) =R′

L(n − i) and R(n) = RL(n). So according to (8), wecan get that

I(IPI)i =μi(P ′i,m−Pi,m)

RL(i−1)R′L(n−i)−RL(n)

1−Pi,m.

�Corollary 1 If all components are independent identi-

cally distributed (i.i.d), and P1,m = P2,m = · · · = Pn,m =P , P ′

1,m = P ′2,m = · · · = P ′

n,m = P ′, then we can get that

I(IPI)i = μi(P ′ − P )RL(i − 1)RL(n − i) − RL(n)

1 − P

for a multi-state linear consecutive k-out-of-n: F system.Proof If all components are i.i.d then we can get that

R′L(n − i) = RL(Pi+1,m, . . . , Pn,m) = RL(P, . . . , P︸ ︷︷ ︸

n−i

) =

RL(n − i). According to Theorem 2, we have that

I(IPI)i = μi(P ′ − P )RL(i − 1)RL(n − i) − RL(n)

1 − P.

�We use RC(j), R′

C(j), RC(n) to represent R(j),R′(j), R(n)of the circular system. We can get the follow-ing Theorem.

Theorem 3 For a multi-state circular consecutive k-out-of-n: F system, we can get that I(IPI)i = μi(P ′

i,m −

Pi,m)RL(i + 1, i − 1) − RC(n)

1 − Pi,m, where RL(i+1, i−1) =

RL(Pi+1,m, . . . , Pn,m, P1,m, . . . , Pi−1,m).Proof According to (4), we have

I(BM)i =∂R(n)∂Pi,m

= R(n|xi � m) − R(n|xi < m) =

R(i − 1)R′(n − i) − R(n)1 − Pi,m

.

For a multi-state circular consecutive k-out-of-n: F sys-tem, we can get that R(n) = RC(n) and

R(i − 1)R′(n − i) = RC(i − 1)R′C(n − i) =

RC(P1,m, . . . , Pi−1,m, xi � m, Pi+1,m, . . . , Pn,m) =

RL(Pi+1,m, . . . , Pn,m, P1,m, . . . , Pi−1,m) =

RL(i + 1, i − 1).

Hence according to (8), we can get I(IPI)i =

μi(P ′i,m − Pi,m)

RL(i + 1, i − 1) − RC(n)1 − Pi,m

. �Corollary 2 If all components are i.i.d, and P1,m =

P2,m = · · · = Pn,m = P , P ′1,m = P ′

2,m = · · · =P ′

n,m = P ′, then we can get that I(IPI)i = μi(P ′ −

P )RL(n − 1) − RC(n)

1 − Pfor a multi-state circular consec-

utive k-out-of-n: F system.Proof If all components are i.i.d, then we can get that

RL(i + 1, i − 1) =

RL(Pi+1,m, . . . , Pn,m, P1,m, . . . , Pi−1,m) =

RL(P, . . . , P︸ ︷︷ ︸n−1

) = RL(n − 1).

According to Theorem 3, we have that I(IPI)i =

μi(P ′ − P )RL(n − 1) − RC(n)

1 − P. �

Corollary 3 Suppose all components are i.i.d, andP1,m = P2,m = · · · = Pn,m = P, P ′

1,m = P ′2,m =

· · · = P ′n,m = P ′. If μi � μj , then I(IPI)i � I(IPI)j

for a multi-state circular consecutive k-out-of-n: F system.Proof Immediate from Corollary 2. �Theorem 4 In the multi-state linear consecutive k-

out-of-n: F system based on the system performance levelm, if all components are i.i.d, and P1,m = P2,m = · · · =Pn,m = P , P ′

1,m = P ′2,m = · · · = P ′

n,m = P ′, then wehave the following results.

(i) If μi � μi+1, then I(IPI)i � I(IPI)i+1 forn − km + 1 � i < km.

(ii) Whenn

2< km < n, if μi � μi+1, then I(IPI)i <

I(IPI)i+1 for i + 1 � n − km + 1.

(iii) When 2 < km � n

2, if μi � μi+1, then I(IPI)i >

I(IPI)i+1 for i > n − km.

940 Journal of Systems Engineering and Electronics Vol. 23, No. 6, December 2012

(iv) When 2 < km � n

2, if μi � μi+1, then I(IPI)i <

I(IPI)i+1 for i < km.

Proof According to Corollary 1, we have that

I(IPI)i = μi(P ′ − P )RL(i − 1)RL(n − i) − RL(n)

1 − P

and

I(IPI)i+1 =μi+1(P ′−P )RL(i)RL(n−i−1)− RL(n)

1 − P.

(i) Since i < km, we can get RL(i) = 1 and RL(i −1) = 1. Similarly, since n − km + 1 � i, we haveRL(n − i) = 1 and RL(n − i − 1) = 1. Then we

can get I(IPI)i = μi(P ′ − P )1 − RL(n)

1 − P, I(IPI)i+1 =

μi+1(P ′ − P )1 − RL(n)

1 − P. So if μi � μi+1, then

I(IPI)i � I(IPI)i+1.

(ii) Whenn

2< km < n, we have n − km < km. Since

i+1 � n−km +1, we can get i+1 � n−km +1 � km.Then we have RL(i) = 1 and RL(i − 1) = 1. So

we can get I(IPI)i = μi(P ′ − P )RL(n − i) − RL(n)

1 − P

and I(IPI)i+1 = μi+1(P ′−P )RL(n − i − 1) − RL(n)

1 − P.

Since RL(n − i) < RL(n − i − 1), when μi � μi+1, wecan get I(IPI)i < I(IPI)i+1.

(iii) When 2 < km � n

2, we have km � n − km.

Since i > n − km, we can get km � n − km < i. Thenwe have RL(n − i) = 1 and RL(n − i − 1) = 1. So

we can get I(IPI)i = μi(P ′ − P )RL(i − 1) − RL(n)

1 − P

and I(IPI)i+1 = μi+1(P ′ − P )RL(i) − RL(n)

1 − P. Since

RL(i) < RL(i − 1), when μi � μi+1, we can getI(IPI)i > I(IPI)i+1.

(iv) When 2 < km � n

2, we have km � n − km.

Since i < km, we can get i < km � n − km. Thenwe have RL(i) = 1 and RL(i − 1) = 1. So we can

get that I(IPI)i = μi(P ′ − P )RL(n − i) − RL(n)

1 − Pand

I(IPI)i+1 = μi+1(P ′ − P )RL(n − i − 1) − RL(n)

1 − P.

Since RL(n − i) < RL(n − i − 1), when μi � μi+1,we can get I(IPI)i < I(IPI)i+1. �

From Corollary 3, we can get the IPI order of all com-ponents in the circular consecutive k-out-of-n: F system,and from Theorem 4 we can get the IPI order of the ad-

jacent components in the linear consecutive k-out-of-n: F

system.For decreasing or increasing multi-state consecutive k-

out-of-n: F systems, I(IPI)i,m represents the IPI of thecomponent i based on the system performance level m,μi,m represents the expected times that the component i

improves from Pi,m to P ′i,m in an unit time based on the

system performance level m, Rm(j) represents the relia-bility of the multi-state consecutive k-out-of-j: F subsys-tem consisting of components {1, 2, . . . , j} based on thesystem performance level m, and R′

m(j) represents the re-liability of the multi-state consecutive k-out-of-j: F sub-system consisting of components {(n − j + 1), (n − j +2), . . . , (n− 1), n} based on the system performance levelm. We can get the following theorems.

Theorem 5 For a decreasing multi-state consecutivek-out-of-n: F system based on two different systems per-formance levels r, s (r < s), if P ′

i,r −Pi,r = P ′i,s−Pi,s =

Δ and μi,r = μi,s = μ, then we can get that I(IPI)i,r �I(IPI)i,s.

Proof According to (8), we have

I(IPI)i,r = μΔRr(i − 1)R′

r(n − i) − R(n)1 − Pi,r

I(IPI)i,s = μΔRs(i − 1)R′

s(n − i) − R(n)1 − Pi,s

.

Since r < s, we get Pi,r � Pi,s ⇒ 1−Pi,r � 1−Pi,s. Fora decreasing multi-state consecutive k-out-of-n: F system,we have kr � ks, so Rr(i − 1) � Rs(i − 1), R′

r(n− i) �R′

s(n − i). Then we can get I(IPI)i,r � I(IPI)i,s. �Corollary 4 For a decreasing (increasing) multi-state

consecutive k-out-of-n: F system based on two differentsystems performance levels r, s (r < s), if we improvethe component i to the perfect functioning and μi,r =μi,s = μ, then we an get that I(IPI)i,r � I(IPI)i,s

(I(IPI)i,r � I(IPI)i,s).Proof According to (9), we have I(IPI)i,r =

μ[Rr(i−1)R′r(n− i)−R(n)] and I(IPI)i,s = μ[Rs(i−

1)R′s(n−i)−R(n)]. The rest proof is similar to Theorem 5.

�

4. Application to an oil pipeline system

An oil pipeline system with 14 pump stations which failswhen at least km = 7 consecutive pump stations are downis shown as in Fig. 1. The structure of the system is shownin Fig. 1. Components 1 to 14 are pump stations whichhave the same reliabilities (Pi,m = 0.5) as [22] and differ-ent improvement rates as

μ1 = 0.56, μ2 = 0.6, μ3 = 0.25, μ4 = 0.5, μ5 = 0.18

μ6 = 0.135, μ7 = 0.015, μ8 = 0.040, μ9 = 0.145

Hongyan Dui et al.: Importance measure of system reliability upgrade for multi-state consecutive k-out-of-n systems 941

μ10 = 0.237, μ11 = 0.32, μ12 = 0.48

μ13 = 0.225, μ14 = 0.7.

Fig. 1 Model of an oil pipeline system

Here assume that we improve the component i to theperfect function, P ′

i,m = 1. Then I(IPI)i = μi[R(i −1)R′(n− i)−R(n)]. Table 1 shows I(BM)i and I(IPI)i

of the component i based on (4) and (9).

Table 1 I(BM)i and I(IPI)i of component i

Component I(BM)i Rank I(IP I)i Rank1 0.007 812 50 13 0.002 187 5 112 0.015 625 00 11 0.004 687 5 43 0.023 437 50 9 0.002 929 687 5 94 0.031 250 00 7 0.007 812 5 15 0.039 062 50 5 0.003 515 625 66 0.046 875 00 3 0.003 164 062 5 87 0.054 687 50 1 0.000 410 156 25 148 0.054 687 50 1 0.001 093 75 139 0.046 875 00 3 0.003 398 437 5 710 0.039 062 50 5 0.004 628 906 25 511 0.031 250 00 7 0.005 312 0.023 437 50 9 0.005 625 213 0.015 625 00 11 0.001 757 812 5 1214 0.007 812 50 13 0.002 734 375 10

In Table 1, I(IPI)i is obtained based on (9). From Ta-ble 1, I(BM)i of all components are known as a symmet-ric distribution when the reliabilities of all components arethe same, while I(IPI)i of all components are sporadicconsidering the improvement rates of all components. Al-though components 7 and 8 have the largest Birnbaum im-portance when considering the change of the system relia-bility with the change of component reliability, component7 has the smallest IPI and component 4 has the largest IPIwhen considering the rate of the system reliability upgrad-ing due to an improvement of objective component.

Obviously, as Table 1 is obtained based on reliabilitiesand improvement rates of components, any change of re-liabilities and improvement rates of the components mayresult in the change of the IPI. Hence, the choice of the ex-ample and parameters [22] may be difficult to justify em-pirically for IPI. However, our aim is to illustrate differentranks of components between IPI and the Birnbaum im-portance.

The IPI can hence be an objective function subjected toreliabilities and improvement rates, which can lead to anoptimal design considering the rate of the system reliabil-ity upgrading due to an improvement of one component.

5. Conclusions

This paper discusses the IPI in multi-state consecutive k-out-of-n systems. Its main works can be summarized as

follows. Firstly, the IPI of the objective component is therate of the system reliability upgrading due to an improve-ment of the objective component. Nextly, the optimal im-provement of the system reliability per unit time is de-termined by the maximal IPI of all components. Finally,even if a component produces a better Birnbaum impor-tance than another component, its IPI may be lower. IPIis a good measure for engineers to determine that whichcomponent is the most responsible for the rate of the sys-tem reliability upgrading.

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Biographies

Hongyan Dui was born in China in 1982. He re-ceived his B.S. degree (2007) in applied mathemat-ics from Henna University, and M.S. degree (2009)from the major of applied mathematics, Northwest-ern Polytechnical University (NPU), China. Heis currently a Ph. D. candidate of the School ofMechatronics, NPU. In 2011, he went to the Uni-versity of Alberta as a visit collaborator for twomonths. His research interests include system re-

liability and importance analysis.E-mail: duihongyan@mail.nwpu.edu.cn

Shubin Si was born in China in 1974. Currently heis a professor in the School of Mechatronics, North-western Polytechnical University (NPU), China. Hereceived his B.S. degree in 1997 and received hisM.S. degree in 2002 at NPU, majored in mechatron-ics engineering. He also received his Ph.D. degreein 2006 at NPU, majored in management scienceand engineering. In 2007, with the support of ChinaScholarship Council, he went to the University of

Vassa, Finland as a visit scholar for one year. He has published over60 academic papers and articles in journals and conferences in the past 5years. He also headed and participated in 5 government supported foun-dations and more than 10 enterprise supported projects. His research top-ics include system reliability theory, importance measure theory, mainte-nance management systems, and decision support systems.E-mail: sisb@nwpu.edu.cn

Zhiqiang Cai was born in China in 1981. He iscurrently an associate professor in the School ofMechatronics, Northwestern Polytechnical Univer-sity (NPU), China. He is also a member of IEEEand IEEE RS. He received his B.S. degree in 2003at NPU with the title of “Excellent Undergraduateof Shaanxi Province”. He received his M.S. degreein 2006 at NPU with the award of “Excellent The-sis of NPU”. In 2007, with the support of China

Scholarship Council, he went to the Ecole Centrale Paris, France asa co-supervised Ph.D. student for one year. In 2010, he received hisPh.D. degree at NPU, majored in management science and engineering.His research topics include maintenance management, failure prediction,Bayesian network modeling and decision making support. He has pub-lished over 20 academic papers and articles in journals and conferences inthe past 5 years. He is an associate professor of the School of Mechatron-ics, Northwestern Polytechnical University. His research interests includesystem modeling, analysis and optimization.E-mail: caizhiqiang@nwpu.edu.cn

Shudong Sun was born in China in 1963. He isa professor of the School of Mechatronics, North-western Polytechnical University, China. He re-ceived his Ph. D. (1989) from the School of Mechan-ical Engineering, Nanjing University of Aeronauticaland Astronautical, China. He is currently a manag-ing director of manufacturing automation research,mechatronics and robotics research, and manufactur-ing technology and machine tool in National Univer-

sity, and a deputy director of Industrial Engineering Branch of Mechani-cal Engineering Society in Shannxi, China. He is also a member of IEEE.He went to the University of Sheffield, Strathclyde, and Malta, UK as avisit scholar for one year in 1994, 2004, and 2007, respectively. He re-ceived the outstanding young teacher award of ministry of education, andaviation youth technology award. His research interests include produc-tion management, maintenance management and robotics.E-mail: sdsun@nwpu.edu.cn

Yingfeng Zhang was born in China in 1979. He isa professor of the School of Mechatronics, North-western Polytechnical University, China. He re-ceived the B.S. M.S. and Ph. D. degrees in Mechan-ical Engineering from Xi’an Jiaotong University,China in 1999, 2002, and 2005, respectively. Heheld post doctoral research fellowships from 2005to 2007 in the University of Hong Kong. He is adirector of Industrial Engineering Branch of Me-

chanical Engineering Society in Shannxi, China. His research interestsinclude advanced manufacturing technology, networked manufacturing,intelligent manufacturing, and RFID technologies.E-mail: zhangyf@nwpu.edu.cn