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Joint Integrated Importance Measure for Multi-StateTransition SystemsShubin Si a , Hongyan Dui a , Zhiqiang Cai a , Shudong Sun a & Yingfeng Zhang aa Ministry of Education Key Laboratory of Contemporary Design and IntegratedManufacturing Technology, School of Mechatronics , Northwestern Polytechnical University ,Shaanxi , P. R. ChinaPublished online: 10 Sep 2012.

To cite this article: Shubin Si , Hongyan Dui , Zhiqiang Cai , Shudong Sun & Yingfeng Zhang (2012) Joint Integrated ImportanceMeasure for Multi-State Transition Systems, Communications in Statistics - Theory and Methods, 41:21, 3846-3862, DOI:10.1080/03610926.2012.688158

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Communications in Statistics—Theory and Methods, 41: 3846–3862, 2012Copyright © Taylor & Francis Group, LLCISSN: 0361-0926 print/1532-415X onlineDOI: 10.1080/03610926.2012.688158

Joint Integrated ImportanceMeasureforMulti-State Transition Systems

SHUBIN SI, HONGYAN DUI, ZHIQIANG CAI,SHUDONG SUN, AND YINGFENG ZHANG

Ministry of Education Key Laboratory of Contemporary Design andIntegrated Manufacturing Technology, School of Mechatronics,Northwestern Polytechnical University, Shaanxi, P. R. China

Joint reliability importance (JRI) evaluates the interaction of two components incontributing to the system reliability in a system. Traditional JRI measures mainlyconcern the change of the system reliability caused by the interactive change of thereliabilities of the two components and seldom consider the probability distributions,transition rates of the object component states, and system performance. This articleextends the JRI concept of two components from multi-state systems to multi-statetransition systems and mainly focuses on the joint integrated importance measure(JIIM) which considers the transition rates of component states. Firstly, the conceptand physical meaning of JIIM in binary systems are described. Secondly, the JIIMfor deterioration process (JIIMDP) and the JIIM for maintenance process (JIIMMP)in multi-state systems are studied respectively. The corresponding characteristics ofJIIMDP and JIIMMP for series and parallel systems are also analyzed. Finally, anapplication to an offshore electrical power generation system is given to demonstratethe proposed JIIM.

Keywords Deterioration process; Joint reliability importance; Maintenanceprocess; Performance utility; Reliability.

Mathematics Subject Classification 60K10; 90B25.

1. Introduction

Acronyms

JRI Joint reliability importanceIIM Integrated importance measureJIIM Joint integrated importance measure

Received October 20, 2011; Accepted April 20, 2012Address correspondence to Shubin Si, Ministry of Education Key Laboratory of

Contemporary Design and Integrated Manufacturing Technology, School of Mechatronics,Northwestern Polytechnical University, P.O. Box 554 Xi’an, Shaanxi 710072, P. R. China;E-mail: sbsi1974@163.com

3846

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Joint Integrated Importance Measure 3847

Notation.

n number of components in the systemi index for component i, i = 1� 2� � � � � nxi state of component i, xi = 0� 1� 2� � � � �MX �x1� x2� � � � � xn�: state vector of the components��X� system structure function with domain �0� 1� � � � �M�n

and range �0� 1� � � � �M� where M and n are positive integers,��X� = ��x1� x2� � � � � xn�

aj the performance corresponding to state j of the systemcj the maintenance cost corresponding to state j of the system�i the failure rate of component i in a binary systemi the repair rate of component i in a binary systemh�P� the reliability function of the binary system

�h�P� = Pr���X� = 1��Pi the reliability of components i in a binary system�im�q the transition rate of component i from state m to state q �q < m�

in a multi-state systemim�q the transition rate of component i from state m to state q �q > m�

in a multi-state systemPim Pr�xi = m�im im = Pr�xi ≥ m� = Pim + Pi�m+1� + · · · + PiMi

Qim Qim = Pr�xi ≤ m� = Pi0 + · · · + Pim

JRI�i� j� joint reliability importance of components i and jI�BM�i Birnbaum importance of component i in binary systemsI�IIM�i IIM value of component i in binary systemsI�Gr�mi Griffith importance of state m of component i

Assumptions.

1) The systems are coherent.2) The state space of each component and system is �0� 1� � � � �M�, where 0

corresponds to complete failure of the system or components and M is the perfectfunctioning state of the system or components. The states are ordered from 0 toM , and the transition rate between the states of the component is constant.

Importance measures are used to quantify the criticality of a particularcomponent within a system design, and they have been widely used for identifyingsystem weaknesses and supporting system improvement activities. With theimportance values of all components, proper actions can be applied on the weakestcomponent to improve system performance at the minimum cost or effort. Inaddition, the valuable information can also be provided for the safety and operationof a system.

Importance measures were first introduced in binary systems by Birnbaum(1969). The Birnbaum importance gives the contribution of the componentreliability to the system reliability. Vasseur and Llory (1999) considered ReliabilityAchievement Worth, Reliability Reduction Worth, Fussell-Vesely, and Birnbaumimportance measures as the most valuable importance measures in probabilisticsafety assessment and probabilistic risk assessment. Currently, most reliability workhas focused on multi-state systems. In multi-state systems, the system and its

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3848 Si et al.

components have multiple states or their performance levels vary from perfectoperation to complete failure. For example, such systems appear in communicationnetworks, production systems, manufacturing systems, power generation systems,computer systems, sensor networks, logic circuits, transportation of oil and gas etc.(Lisnianski and Levitin, 2003; Levitin, 2005; Natvig, 2011). The research on multi-state systems began in the 1970s. El-Neweihi et al. (1978) analyzed the theoreticalrelationships between the multi-state system reliability behavior and componentperformance. Barlow and Wu (1978) defined a system state function for coherentsystems with multi-state components and investigated its properties. Lisnianski andLevitin (2003) wrote the first book on the multi-state system reliability analysisand optimization which provides basic methods for multi-state system reliabilityassessment and presented many applications to a variety of technical problems.Guo et al. (2006) and Zhao et al. (2006) presented the system reliability formulaewith product of matrices by means of finite Markov chain imbedding approach.Mu et al. (2008) studied inferences for reliability functions of the system havingtwo components connected in series. Eryilmaz (2008) considered the reliability andits estimation of consecutive k-out-of-n: G system. Eryilmaz and Iscioglu (2011)classified the states of the system and evaluated the reliability of multi-state system.

For multi-state systems, the importance measure has been used as a toolfor system performance improvement. Griffith (1980) formalized the concept ofsystem performance, proposed the importance vectors of component states, andstudied the effect of component improvement on system performance using ageneralization of Birnbaum importance. Wu and Chan (2003) defined a newperformance importance of a state of a component in multi-state systems to measurewhich component affects it the most, or which state of a certain componentcontributes the most, and analyzed the relationships between the Wu importancemeasure and Griffith importance measure. Zio and Podofillini (2003) generalizedthe Birnbaum importance measure with the system performance level from binarysystems to multi-state systems.

The Birnbaum importance characterizes the rate at which the system reliabilitychanges with respect to changes in the reliability of a given component and itis also defined as marginal reliability importance. Improvements in reliability ofcomponents with the highest marginal reliability importance cause the greatestincrease in system reliability. However, marginal reliability importance does notprovide all information on how components affect the system reliability. The jointreliability importance (JRI) is used in this kind of situation, since it indicates theeffect of interaction between components on system reliability. Hong and Lie (1993)first introduced the JRI of two edges in an undirected network, concepts of jointfailure importance, and marginal failure importance. Armstrong (1995) removedthe statistical independence restriction of the JRI by showing that similar resultshold in a more general case where component failures is statistically dependent.Hong et al. (2000) investigated the JRI of two gate events in a fault tree. Honget al. (2002) analyzed the JRI for k-out-of-n systems with independently identicallydistributed components, non identical and independent components, and pair-wisedependent components. Wu (2005) extended the JRI of two components from thebinary system case to the multi-state system case. Gao et al. (2007) extended aconcept of the JRI of two components to multi-components and investigated theconcept of conditional reliability importance. Si et al. (2010) defined the integratedimportance measure (IIM) of coherent systems and applied it to practical head

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Joint Integrated Importance Measure 3849

up display system analysis. Si et al. (2011, 2012) developed IIM of componentstates based on the performance in multi-state systems for deterioration process andmaintenance process.

Traditional JRI measures mainly concern the change of the system reliabilitycaused by the interactive change of the reliabilities of the two components. Althoughsignificant contributions are made by the above researchers, they seldom considerthe probability distributions, transition rates of the object component states, andsystem performance.

In this article, in order to analyze how components interact in determiningsystem performance, we investigate the definitions and characteristics of the jointintegrated importance measure (JIIM) of component states and components. TheJIIM is appropriate for the system whose behavior is described by its states andthe possible transitions among these states. It is used to analyze tradeoff betweencomponents for design engineers since it indicates how components interact indetermining system utility.

The rest of the article is organized as follows. The JIIM of binary systemsis described in Sec. 2. Section 3 discusses the JIIM of multi-state systems fordeterioration process. Section 4 analyzes the JIIM of multi-state systems formaintenance process. Section 5 demonstrates an application to an offshore electricalpower generation system for JIIM. The conclusions are given in Sec. 6.

2. JIIM of Binary Systems

2.1. Definition of the JIIM

Hong and Lie (1993) introduced the JRI of two components in binary systems asfollows:

JRI�i� j� = �2h�P�

�Pi�Pj

� (1)

where h�P� is the reliability function of the system with h�P� = Pr���X� = 1�, Pi

and Pj are reliabilities of components i and j respectively, and P = �P1� P2� � � � � Pn�is the reliability vector of all components.

Armstrong (1995) generalized Eq. (1) as follows:

JRI�i� j� = �2h�P�

�Pi�Pj

= h�1i� 1j� P�+ h�0i� 0j� P�− h�1i� 0j� P�− h�0i� 1j� P�

= �h�1j� P�

�Pi

− �h�0j� P�

�Pi

(2)

Equation (2) suggests that JRI of two components describes the relativeimportance of one component when the other is functioning. JRI indicates that acomponent is more �JRI > 0� or less �JRI < 0� important or has the same �JRI = 0�importance when the other is functioning. The Birnbaum importance of componenti is defined as

I�BM�i =�h�P�

�Pi

= h�1i� P�− h�0i� P�� (3)

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so JRI�i� j� can also be interpreted as the change of the Birnbaum importance ofcomponent i with component j’s deteriorating from state 1 to state 0 as follows:

JRI�i� j� = �I�BM�i�Pj

= ��h�1i� P�− h�0i� P���Pj

= h�1i� 1j� P�+ h�0i� 0j� P�− h�1i� 0j� P�− h�0i� 1j� P�� (4)

The IIM of component i in binary systems is as follows (Si et al., 2012):

I�IIM�i = Pi × �i × I�BM�i = Pi × �i ×�h�P�

�Pi

= �h�1i� P�− h�0i� P��× Pi × �i� (5)

where �i is the failure rate of component i. So according to Eqs. (1)–(5), we can giveDefinition 2.1 as follows.

Definition 2.1. The JIIM of components i and j in a binary system is

JIIM�i� j� = PiPj�i�j�2h�P�

�Pi�Pj

= Pj�j�I�IIM�i

�Pj

= PiPj�i�j� �h�1i� P�− h�0i� P��

�Pj

= PiPj�i�j(h�1i� 1j� P�+ h�0i� 0j� P�− h�1i� 0j� P�− h�0i� 1j� P�

)�

From Definition 2.1, JIIM�i� j� can be interpreted as the change of the IIMvalue of component i caused by component j’s deteriorating from state 1 to state 0based on the reliabilities and failure rates of components i and j.

Definition 2.1 shows that JIIM can be represented in terms of IIM of eachcomponent in a modified system where components are guaranteed to be working orfailed. JIIM of two components describes the relative IIM value of one componentwhen the other is functioning.

2.2. Characteristics of the JIIM

Theorem 2.1. The JIIM of components i and j in a binary series system consisting ofn components is JIIM�i� j� = �i�j

∏nk=1 Pk ≥ 0.

Proof. In a series system, we have h�P� = P1P2 � � � Pn. So we can get h�0i� 0j� P� =h�1i� 0j� P� = h�0i� 1j� P� = 0. According to Definition 2.1, we have

JIIM�i� j� = PiPj�i�jh�1i� 1j� P� = PiPj�i�j

n∏k=1�k �=i�j

Pk = �i�j

n∏k=1

Pk ≥ 0�

This completes the proof of Theorem 2.1. �

Theorem 2.2. In a binary series system consisting of n components, if �i�j ≥ �k�m, then

JIIM�i� j� ≥ JIIM�k�m��

Proof. Immediate from Theorem 2.1.

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Joint Integrated Importance Measure 3851

Theorem 2.3. The JIIM of components i and j in a binary parallel system consistingof n components is JIIM�i� j� = −PiPj�i�j

∏nk=1�k �=i�j �1− Pk� ≤ 0.

Proof. In a parallel system, we have h�P� = 1−∏nk=1 �1− Pk�. So we can get

h�1i� 1j� P� = h�1i� 0j� P� = h�0i� 1j� P� = 1� According to Definition 1, we have

JIIM�i� j� = PiPj�i�j(h�0i� 0j� P�− 1

) = PiPj�i�j

[(1−

n∏k=1�k �=i�j

�1− Pk�

)− 1

]

= −PiPj�i�j

n∏k=1�k �=i�j

�1− Pk� ≤ 0�

This completes the proof of Theorem 2.3. �

If failure and repair times of all components are exponentially distributed, thenthe stochastic process will have a Markov property and can be represented by aMarkov model (Trivedi, 2002). The following algebraic linear equation can be usedto evaluate the steady state distribution for component i as follows: Pi · �i − �1−Pi� · i = 0, where i is the repair rate of component i. So when components i andj have a Markov property, the JIIM of components i and j in a binary parallelsystem consisting of n components is

JIIM�i� j� = −PiPj�i�j

n∏k=1�k �=i�j

�1− Pk� = −ij

n∏k=1

�1− Pk�� (6)

Theorem 2.4. In a binary parallel system consisting of n components, when allcomponents have a Markov property, if ij ≥ km, then JIIM�i� j� ≤ JIIM�k�m�.

Proof. Immediate from Eq. (6).Theorems 2.1 and 2.3 give the expressions of the JIIM of components i and j in

binary series and parallel systems, respectively. We can compare the JIIM values ofdifferent components in binary series and parallel systems based on Theorems 2.2and 2.4.

3. JIIM of Multi-state Systems for Deterioration Process

3.1. Definition of the JIIM for Deterioration Process

Let 0 = a0 ≤ a1 ≤ · · · ≤ aM represent the performance levels corresponding to thestate space �0� 1� � � � �M� of the system. Griffith (1980) gave the following expectedperformance function of a multi-state system:

U =M∑j=1

ajPr���X� = j� =M∑j=1

ajPr���x1� x2� � � � � xn� = j�� (7)

and proposed the Griffith importance of state m of component i in multi-statesystems as

I�Gr�mi =M∑l=1

�al − al−1��Pr���mi� X� ≥ l�− Pr����m− 1�i� X� ≥ l� � (8)

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Wu (2005) extended the JRI of two components from the binary system to themulti-state system based on the performance in which components and the systemhave M+1 states: 0� 1� � � � �M as follows (Wu, 2005).

The JRI of state m of component i and state k of component j in a multi-statesystem (shortly, JPIM) is

JPIM�i� j�m� k� = �2U

�im�jk

=M∑l=1

�al − al−1��Pr���mi� kj� X� ≥ l

− Pr���mi� �k− 1�j� X� ≥ l − Pr����m− 1�i� kj� X� ≥ l

+ Pr����m− 1�i� �k− 1�j� X� ≥ l �� (9)

where im = Pim + Pi�m+1� + PiM and Pim = Pr�xi = m��m = 0� 1� � � � �M .In Eq. (9), JPIM�i� j�m� k� can be interpreted as the change of I�Gr�mi caused

by component j’s deteriorating from state k to state k− 1.The I�Gr�mi can be interpreted as the change of the system performance when

component i deteriorates from state m to state m− 1. However, the situation thata component deteriorates from state m to any one of states �m− 1�m− 2� � � � � 0�is more reasonable in practice. The transition rate matrix of component i fordeterioration process is listed in Eq. (10):

� =

�iM�M−1 �iM�M−2 · · · �iM�1 �iM�0

0 �iM−1�M−2 · · · �iM−1�1 �iM−1�0

������

� � ����

���0 0 · · · 0 �i1�0

� (10)

where the �im�q represents the transition rate of component i from state m to state q

(q < m�.According to this situation, Si et al. (2012) proposed the IIM concept for

deterioration process in the multi-state system based on the performance as follows:

I�IIM�m�qi = Pim · �im�q

m∑k=q+1

I�Gr�ki

= Pim · �im�q

M∑l=1

�al − al−1��Pr���mi� X� ≥ l�− Pr���qi� X� ≥ l�

= Pim · �im�q

M∑l=1

al�Pr���mi� X� = l�− Pr���qi� X� = l� � (11)

where q is any state less than m of component i, q < m.Just as the definition of the JRI for a binary system was introduced by

Armstrong (1995), one can propose the following definition to measure the JIIMof two components for deterioration process (shortly, JIIMDP) in a multi-statesystem.

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Joint Integrated Importance Measure 3853

Definition 3.1. The JIIM of state m deteriorating to state q of component i andstate k deteriorating to state o of component j for deterioration process in amulti-state system is

JIIMDP�i� j�m� q� k� o� = Pjk · Pim · �ik�o · �im�q

M∑l=1

�al − al−1��Pr���mi� kj� X� ≥ l�

− Pr���mi� oj� X� ≥ l�− Pr���qi� kj� X� ≥ l�

+ Pr���qi� oj� X� ≥ l��

= Pjk · Pim · �ik�o · �im�q

M∑l=1

al�Pr���mi� kj� X� = l�

− Pr���mi� oj� X� = l�− Pr���qi� kj� X� = l�

+ Pr���qi� oj� X� = l��� (12)

where o is any state less than k of component j, o < k.

From Definition 3.1, JIIMDP�i� j�m� q� k� o� can be interpreted as the changeof I�IIM�

m�qi caused by component j’s deteriorating from state k to state o.

Definition 3.2. The JIIM of state m of component i and state k of component j fordeterioration process in a multi-state system is

JIIMDP�i� j�m� k� =m−1∑q=0

k−1∑o=0

JIIMDP�i� j�m� q� k� o�� (13)

Definition 3.3. The JIIM of components i and j for deterioration process in amulti-state system is

JIIMDP�i� j� =M∑k=1

M∑m=1

JIIMDP�i� j�m� k�� (14)

In multi-state systems for deterioration process, Definition 3.1 describes theJIIM of transition of two different states of different components, Definition 3.2describes the JIIM of different states of different components and Definition 3.3describes the JIIM of different components.

3.2. Characteristics of the JIIM for Deterioration Process

When q = o = 0, we have (Si et al., 2012)

I�IIM�m�0i = Pim · �im�0

M∑l=1

�al − al−1��Pr���mi� X� ≥ l�− Pr���0i� X� ≥ l�

= Pim · �im�0

M∑l=1

al�Pr���mi� X� = l�− Pr���0i� X� = l� � (15)

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3854 Si et al.

The corresponding JIIMDP of Eq. (15) is as follows:

JIIMDP�i� j�m� 0� k� 0�

= Pjk · Pim · �ik�0 · �im�0

M∑l=1

�al − al−1��Pr���mi� kj� X� ≥ l�

− Pr���mi� 0j� X� ≥ l�− Pr���0i� 0j� X� ≥ l�+ Pr���0i� 0j� X� ≥ l��� (16)

From Eq. (16), JIIMDP�i� j�m� 0� k� 0� can be interpreted as the change ofI�IIM�m�0

i caused by component j’s deteriorating from state k to complete failure.

Theorem 3.1. In a multi-state series system consisting of n components,

JIIMDP�i� j�m� 0� k� 0� =

Pjk · Pim · �ik�0 · �im�0

k∑l=1

�al − al−1�n∏

x=1�x �=i�j

xl� m ≥ k�

Pjk · Pim · �ik�0 · �im�0

m∑l=1

�al − al−1�n∏

x=1�x �=i�j

xl� m < k�

�

Proof. By the definition of a series system, we have ��X� = min�x1� x2� � � � � xn�.So we have Pr���mi� 0j� X� ≥ l� = Pr���0i� 0j� X� ≥ l� = Pr���0i� 0j� X� ≥ l� = 0.Then,

JIIMDP�i� j�m� 0� k� 0� = Pjk · Pim · �ik�0 · �im�0

M∑l=1

�al − al−1�Pr���mi� kj� X� ≥ l��

(17)

If m ≥ k, then Pr���mi� kj� X� ≥ l� = 0� l = k+ 1� � � � �M . So we have

M∑l=1

�al − al−1�Pr���mi� kj� X� ≥ l�

=k∑

l=1

�al − al−1�Pr���mi� kj� X� ≥ l�+M∑

l=k+1

�al − al−1�Pr���mi� kj� X� ≥ l�

=k∑

l=1

�al − al−1�Pr���mi� kj� X� ≥ l� =k∑

l=1

�al − al−1�n∏

x=1�x �=i�j

xl�

If m < k, then Pr���mi� kj� X� ≥ l� = 0� l = m+ 1� � � � �M . Similarly, we can have

M∑l=1

�al − al−1�Pr���mi� kj� X� ≥ l� =m∑l=1

�al − al−1�n∏

x=1�x �=i�j

xl�

Based on Eq. (17), we can get

JIIMDP�i� j�m� 0� k� 0� =

Pjk · Pim · �ik�0 · �im�0

k∑l=1

�al − al−1�n∏

x=1�x �=i�j

xl� m ≥ k

Pjk · Pim · �ik�0 · �im�0

m∑l=1

�al − al−1�n∏

x=1�x �=i�j

xl� m < k

�

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Joint Integrated Importance Measure 3855

This completes the proof of Theorem 3.1. �

Similarly, the following theorem can be proved.

Theorem 3.2. In a multi-state parallel system consisting of n components,

JIIMDP�i� j�m� 0� k� 0� =

Pjk · Pim · �ik�0 · �im�0

k∑l=1

�al − al−1�n∏

x=1�x �=i�j

�1− xl�� m ≥ k

Pjk · Pim · �ik�0 · �im�0

m∑l=1

�al − al−1�n∏

x=1�x �=i�j

�1− xl�� m < k�

Theorems 3.1 and 3.2 give the expressions of the JIIMDP of components i andj in multi-state series and parallel systems for deterioration process, respectively.

4. JIIM of Multi-State Systems for Maintenance Process

4.1. Definition of the JIIM for Maintenance Process

Let 0 = cM ≤ cM−1 ≤ · · · ≤ c0 represent the maintenance cost levels attached to thestate space �0� 1� � � � �M� of the system. The maintenance cost levels are decreasingwith the increase of the state. Si et al. (2011) gave the following the maintenancecost function of a multi-state system:

C =M−1∑j=0

cjP���X� = j�� (18)

The transition rates matrix of component i for maintenance process is listed inEq. (19):

� =

i0�1 i

0�2 · · · i0�M−1 i

0�M

0 i1�2 · · · i

1�M−1 i1�M

������

� � ����

���0 0 · · · 0 i

M−1�M

� (19)

where the im�q represents the transition rate of component i from state m to state q

�q > m�.Si et al. (2011) proposed the IIM concept for maintenance process in the multi-

state system based on the maintenance cost as follows:

I�IIM�m�qi = Pim · i

m�q

M−1∑l=0

�cl − cl+1��P���mi� X� ≤ l�− P���qi� X� ≤ l�

= Pim · im�q

M−1∑l=0

cl�P���mi� X� = l�− P���qi� X� = l� � (20)

where q is any state more than m of component i, q > m.

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Just as the definition of the JRI for a binary system was introduced by Armstrong(1995), so too can one propose the following definition to measure the JIIM of twocomponents for maintenance process (shortly, JIIMMP) in a multi-state system.

Definition 4.1. The JIIM of state m maintaining to state q of component i and statek maintaining to state o of component j for maintenance process in a multi-statesystem is

JIIMMP�i� j�m� q� k� o�

= Pjk · Pim · ik�o · i

m�q

M−1∑l=0

�cl − cl+1��Pr���mi� kj� X� ≤ l�

− Pr���mi� oj� X� ≤ l�− Pr���qi� kj� X� ≤ l�+ Pr���qi� oj� X� ≤ l��

= Pjk · Pim · ik�o · i

m�q

M−1∑l=0

cl�Pr���mi� kj� X� = l�

− Pr���mi� oj� X� = l�− Pr���qi� kj� X� = l�+ Pr���qi� oj� X� = l��� (21)

where o is any state more than k of component j, o > k.

From Definition 4.1, JIIMMP�i� j�m� q� k� o� can be interpreted as the changeof I�IIM�

m�qi caused by component j’s maintaining from state k to state o.

Definition 4.2. The JIIM of state m of component i and state k of component j formaintenance process in a multi-state system is

JIIMMP�i� j�m� k� =M∑

q=m+1

M∑o=k+1

JIIMMP(i� j�m� q� k� o�� (22)

Definition 4.3. The JIIM of components i and j for maintenance process in a multi-state system is

JIIMMP�i� j� =M−1∑k=0

M−1∑m=0

JIIMMP(i� j�m� k�� (23)

In multi-state systems for maintenance process, Definition 4.1 describes theJIIM of transition of two different states of different components, Definition 4.2describes the JIIM of different states of different components and Definition 4.3describes the JIIM of different components.

4.2. Characteristics of the JIIM for Maintenance Process

When q = o = M , we have

I�IIM�m�Mi = Pim · i

m�M

M−1∑l=0

�cl − cl+1� �P���mi� X� ≤ l�− P���Mi� X� ≤ l�

= Pim · im�M

M−1∑l=0

cl �P���mi� X� = l�− P���Mi� X� = l� � (24)

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The corresponding JIIMMP of Eq. (24) is as follows:

JIIMMP�i� j�m�M� k�M�

= Pjk · Pim · ik�M · i

m�M

M−1∑l=0

�cl − cl+1��Pr���mi� kj� X� ≤ l�

− Pr���mi�Mj� X� ≤ l�− Pr���Mi� kj� X� ≤ l�+ Pr���Mi�Mj� X� ≤ l��

= Pjk · Pim · ik�M · i

m�M

M−1∑l=0

cl�Pr���mi� kj� X� = l�

− Pr���mi�Mj� X� = l�− Pr���Mi� kj� X� = l�+ Pr���Mi�Mj� X� = l��� (25)

From Eq. (25), JIIMMP�i� j�m�M� k�M� can be interpreted as the change ofI�IIM�m�M

i caused by component j’s maintaining from state k to perfect functioningstate.

Theorem 4.1. In a multi-state parallel system consisting of n components,

JIIMMP�i� j�m�M� k�M� =

Pjk · Pim · i

k�M · im�M

M−1∑l=m

�cl − cl+1�n∏

x=1�x �=i�j

Qxl� m ≥ k

Pjk · Pim · ik�M · i

m�M

M−1∑l=k

�cl − cl+1�n∏

x=1�x �=i�j

Qxl� m < k

�

Proof. By the definition of a parallel system,��X� = max�x1� x2� � � � � xn�. So, wehave Pr���mi�Mj� X� ≤ l = Pr���Mi� kj� X� ≤ l� = Pr���Mi�Mj� X� ≤ l� = 0. Then,

JIIMMP�i� j�m�M� k�M� = Pjk · Pim · ik�M · i

m�M

M−1∑l=0

�cl − cl+1�Pr���mi� kj� X� ≤ l��

(26)

If m ≥ k, then Pr���mi� kj� X� ≤ l� = 0� l = 0� 1� � � � � m− 1. So we have

M−1∑l=0

�cl − cl+1�Pr���mi� kj� X� ≤ l�

=m−1∑l=0

�cl − cl+1�Pr���mi� kj� X� ≤ l�+M−1∑l=m

�cl − cl+1�Pr���mi� kj� X� ≤ l�

=M−1∑l=m

�cl − cl+1�Pr���mi� kj� X� ≤ l� =M−1∑l=m

�cl − cl+1�n∏

x=1�x �=i�j

Qxl�

If m < k, then Pr���mi� kj� X� ≥ l� = 0� l = 0� 1� � � � � k− 1. Similarly, we canhave

M−1∑l=0

�cl − cl+1�Pr���mi� kj� X� ≤ l� =M−1∑l=k

�cl − cl+1�n∏

x=1�x �=i�j

Qxl�

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Based on Eq. (26), we can get

JIIMMP�i� j�m�M� k�M� =

Pjk · Pim · i

k�M · im�M

M−1∑l=m

�cl − cl+1�n∏

x=1�x �=i�j

Qxl� m ≥ k

Pjk · Pim · ik�M · i

m�M

M−1∑l=k

�cl − cl+1�n∏

x=1�x �=i�j

Qxl� m < k

�

This completes the proof of Theorem 4.1. �

Similarly, the following theorem can be proved.

Theorem 4.2. In a multi-state series system consisting of n components,

JIIMMP�i� j�m�M� k�M�

=

Pjk · Pim · i

k�M · im�M

M−1∑l=m

�cl − cl+1�n∏

x=1�x �=i�j

�1−Qxl�� m ≥ k

Pjk · Pim · ik�M · i

m�M

M−1∑l=k

�cl − cl+1�n∏

x=1�x �=i�j

�1−Qxl�� m < k

�

Theorems 4.1 and 4.2 give the expressions of the JIIMMP of components i andj in multi-state parallel and series systems for maintenance process, respectively.

5. A Case Study

We now look at an offshore electrical power generation system (Natvig et al., 1986;Wu, 2005). The amount of power that can be supplied to oil rig 1 depends on threecomponents: control unit U , generator A1, and standby generator A2, each has threestates �0� 1� 2�.

The reliabilities of the components and the system performance are in Table 1.For our use, assume that c0 = 5000� c1 = 3000� c2 = 0, and

�11�0 = 0�0192� �12�1 = 0�0374� �12�0 = 0�0262

�21�0 = 0�0453� �22�1 = 0�0556� �22�0 = 0�0392

�31�0 = 0�0453� �32�1 = 0�0556� �32�0 = 0�0392

�

10�1 = 0�5480� 1

0�2 = 0�4384� 11�2 = 0�6576

20�1 = 0�3358� 2

0�2 = 0�3976� 21�2 = 0�4862

30�1 = 0�3358� 3

0�2 = 0�3976� 31�2 = 0�4862

�

We have results shown in Tables 2–5. The transition rates of component states aresmall, so the JIIM values are small.

According to Eq. (13), we have the JRI of component states for deteriorationprocess as in Table 2.

From Table 2, we can get that the JRI of states of components JPIM�U�A1� 1� 1� is the largest and positive, but because the transition rates of statesof component A2 is larger than the ones of states of component U , the JIIMvalue JIIMDP�A1� A2� 1� 1� is the largest and positive. The JPIM and JIIMDP value

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Joint Integrated Importance Measure 3859

Table 1The reliabilities of the components and the system

performance (Wu, 2005)

State 0 State 1 State 2

Component 1: U 0.182 0.572 0.246Component 2: A1 0.138 0.808 0.054Component 3: A2 0.138 0.808 0.054System 0.266 0.516 0.218ai 0 1000 2000

Table 2The comparison between JPIM�i� j�m� k� (Wu, 2005) and

JIIMDP�i� j�m� k�

JPIM�i� J�m� k� Rank JIIMDP�i� j�m� k� Rank

�U�A1� 1� 1� 460�56 1 0�1851 2�U�A1� 1� 2� −399�00 8 −0�0177 8�U�A1� 2� 2� 55�46 5 0�00799 4�U�A2� 1� 1� 349�82 3 0�1406 3�U�A2� 1� 2� −393�68 7 −0�01517 7�U�A2� 2� 2� 50�14 6 0�00173 6�U�A1� A2� 1� 1� 388�45 2 0�5208 1�U�A1� A2� 1� 2� −425�08 9 −0�0507 9�U�A1� A2� 2� 2� 153�72 4 0�0035 5

Table 3The comparison between JPIM�i� j� (Wu, 2005) and

JIIMDP�i� j�

JPIM�i� J� Rank JIIMDP�i� j� Rank

�U�A1� 117�02 2 0�17539 2�U�A2� 6�28 3 0�12716 3�A1� A2� 117�39 1 0�4736 1

Table 4The JIIMMP�i� j�m� k� of component states

JIIMMP�i� j�m� k� Rank

�U�A1� 1� 1� 31�6984 1�U�A1� 0� 1� −2�526 9�U�A1� 0� 0� 0�6684 4�U�A2� 1� 1� 7�5452 3�U�A2� 0� 1� −0�5623 7�U�A2� 0� 0� 0�0877 6�U�A2� 1� 1� 27�9485 2�U�A2� 0� 1� −1�8791 8�U�A2� 0� 0� 0�1775 5

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Table 5The JIIMMP�i� j� of component

JIIMMP�i� j� Rank

�U�A1� 29�8408 1�U�A2� 7�0706 3�A1� A2� 26�2469 2

of �A1� A2� 1� 2� are all the smallest. JIIMDP concerns not only the probabilitydistributions and transition rates of the object component states, but also the changein the system performance with the change from state m to any deterioratingstate q�q < m� of component i. But JPIM concerns only the change in thesystem performance with the change from state m to deteriorating state m− 1 ofcomponent i.

Based on Eq. (14), we have the JRI of components for deterioration process asin Table 3.

From Table 3, we can get that any component is more important when anotheris functioning based on the JPIM�i� j� and JIIMDP�i� j�.

According to Eq. (22), we have the JRI of component states for maintenanceprocess as in Table 4.

From Table 4, we can get that the JIIM value JIIMMP�U�A1� 1� 1� is the largestand positive. The JIIMMP value of �U�A1� 0� 1� are the smallest. JIIMMP concernsnot only the probability distributions and transition rates of the object componentstates, but also the change in the system performance with the change from state mto any maintaining state q�q > m� of component i.

Based on Eq. (23), we have the JRI of components for maintenance process asin Table 5.

From Table 5, we can get that any component is more important when anotheris functioning based on the JIIMMP�i� j�.

From Tables 2–5, JIIM concerns not only the probability distributions andtransition rates of the object component states, but also the change in the systemperformance with the change from state m to any state q�m �= q� of component i.But Wu’s JPIM (Wu, 2005) concerns only the change in the system performancewith the change from state m to deteriorating state m− 1 of component i.

6. Conclusions

This article discussed the JIIM of component states and components in binary andmulti-state systems. Its main contributions can be summarized as follows.

(1) JIIM of two components gives how the component reliabilities affect each other,and can be used to describe the relative IIM value of one component when theother is functioning.

(2) JIIMDP concerns not only the probability distributions and transition rates ofthe object component states, but also the change in the system performance withthe change from state m to any deteriorating state q �q < m� of component i.But JPIM (Wu, 2005) concerns only the change in the system performance withthe change from state m to deteriorating state m− 1 of component i.

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Joint Integrated Importance Measure 3861

(3) JIIMMP concerns not only the probability distributions and transition rates ofthe object component states, but also the change in the system performance withthe change from state m to any maintaining state q�q > m� of component i.

(4) The JIIM proposed in this article is appropriate for the system whose behavioris described by its states and the possible transitions among these states. It canbe used to analyze tradeoff between components for design engineers since itindicates how components interact in determining system utility.

Acknowledgment

The authors gratefully acknowledge the financial support for this research from the973 Program of China (Grant No. 2010CB328000), the National Natural ScienceFoundation of China (Grant No. 71101116), the Science and Technology Projectof Shaanxi Province (Grant No. 2010K8-11), and the Grant of NorthwesternPolytechnical University (No. 11GH0134).

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