Computers & Industrial Engineering 72 (2014) 255–260

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Computing marginal and joint Birnbaum, and Barlow–Proschanimportances in weighted-k-out-of-n:G systems q

http://dx.doi.org/10.1016/j.cie.2014.03.0250360-8352/� 2014 Elsevier Ltd. All rights reserved.

q This manuscript was processed by Area Editor Min Xie.⇑ Corresponding author. Tel.: +90 312 586 85 58; fax: +90 312 586 80 91.

E-mail address: seryilmaz@atilim.edu.tr (S. Eryilmaz).

Serkan Eryilmaz a,⇑, Ali Riza Bozbulut b

a Atilim University, Department of Industrial Engineering, Ankara, Turkeyb TMA, Defense Sciences Institute, Ankara, Turkey

a r t i c l e i n f o a b s t r a c t

Article history:Received 25 October 2013Received in revised form 27 March 2014Accepted 30 March 2014Available online 13 April 2014

Keywords:Universal generating functionReliability importanceWeighted-k-out-of-n:G system

A weighted-k-out-of-n:G system is a system that consists of n binary components, each with its ownpositive weight, and operates only when the total weight of working components is at least k. Such astructure is useful when the components have different contributions to the performance of the entiresystem. This paper is concerned with both marginal and joint Birnbaum, and Barlow–Proschan (BP)importances of the components in weighted- k-out-of-n:G systems. The method of universal generatingfunction is used for computing marginal and joint Birnbaum importances. The method for computingBP-importance is based on a direct probabilistic approach. Extensive numerical calculations arepresented. By the help of these calculations and illustrations, it is possible to observe how the marginaland joint importances change with respect to the weights of components.

� 2014 Elsevier Ltd. All rights reserved.

1. Introduction

Component importance measures are useful tools in design,improvement, and control of engineered systems. Various impor-tance measures have been defined and studied in reliabilityliterature. Birnbaum (1969) categorized importance measures intothree classes which are structure importance measures, reliabilityimportance measures, and lifetime importance measures. Struc-tural importance measures the relative importance of componentswith respect to their positions in a system. Thus such importancemeasures need knowledge only about the system’s structurefunction. Reliability importance measures depend on both thestructure of the system and the reliabilities of components. Simi-larly, lifetime importance measures depend on both the structureof the system and the component lifetime distributions.

In this paper, we study the well-known reliability and lifetimeimportance measures in weighted-k-out-of-n:G systems. Becausein such a system the positions of the components are irrelevant,it is not attractive to use structural importance measures. Inparticular, we focus on Birnbaum marginal and joint reliabilityimportances, and Barlow–Proschan lifetime importance measurefor the components in a weighted-k-out-of-n:G structure.These importance measures depend on both the structure and

reliabilities/lifetime distributions of the components. Thereforewe will be able to have a knowledge about the ranking of compo-nent importances with respect to their weights and reliabilities/lifetime distributions. One important reason for studying Birn-baum importance is that many importance measures introducedin the literature are either motivated by the Birnbaum measure,or closely related to it.

For a binary coherent system with n binary components, let Xi

denote the state of the ith component, where Xi ¼ 1 if the ith com-ponent is working, and Xi ¼ 0, if it has failed, i ¼ 1;2; . . . ;n. Let S bethe event that the system works. The Birnbaum marginal reliabilityimportance (MRI) of the ith component is defined by

MRIi ¼ P SjXi ¼ 1f g � P SjXi ¼ 0f g; ð1Þ

and it measures the effect of the ith component on the reliability ofthe system considering both the operation and the failure of thecomponent (Birnbaum, 1969). The joint reliability importance(JRI) of two components is a measure of the interaction of two com-ponents in a system for their contribution to the system reliability(Armstrong, 1995; Hagstrom, 1990; Hong & Lie, 1993). The JRI ofcomponents i and j is defined as

JRIði; jÞ ¼ P SjXi ¼ 1;Xj ¼ 1� �

� P SjXi ¼ 1;Xj ¼ 0� �

� P SjXi ¼ 0;Xj ¼ 1� �

þ P SjXi ¼ 0;Xj ¼ 0� �

: ð2Þ

Depending on the sign of JRI of two components, Hagstrom(1990) introduced the concepts of ‘‘reliability substitutes’’ and ‘‘reli-ability complements’’. Two components are said to be reliability

256 S. Eryilmaz, A.R. Bozbulut / Computers & Industrial Engineering 72 (2014) 255–260

substitutes (complements) if their sign of JRI are non-positive (non-negative). The sign and the value of the JRI represent the type andthe degree of interactions between two components with respectto system reliability. If JRI > 0 then one component becomes moreimportant when the other is functioning (synergy); if JRI < 0 thenone component becomes less important when the other is function-ing (diminishing returns); and if JRI ¼ 0 then one component’simportance is unchanged by the functioning of the other(Armstrong, 1995).

Let T denote the failure time of a given n-component systemwhose components’ failure times are T1; . . . ; Tn. Barlow andProschan (1975) introduced the importance measure (hereafterBP importance)

IBPi ¼ P T ¼ Tif g

which represents the probability that the system fails upon the fail-ure of the ith component, i ¼ 1; . . . ;n.

A k-out-of-n:G system is a system that consists of n binary com-ponents, and operates only when at least k components function.MRI and BP importances in k-out-of-n:G systems have been stud-ied in various papers including Proschan and Boland (1983), Iyer(1992), and Boland and El-Neweihi (1995). Hong, Koo, and Lie(2002) and Gao, Cui, and Li (2007) studied JRI of components inbinary k-out-of-n:G systems. Joint reliability importance measuresin multi-state systems have been studied in Wu (2005) and Si, Dui,Cai, Sun, and Zhang (2012). Some recent works on reliabilityimportance are in Si, Cai, Sun, and Zhang (2010), Qi, Zuo, Zhang,and Yang (2011), Rani, Jain, and Dewan (2011), Kuo and Zhu(2012), Peng, Coit, and Feng (2012), Zhu, Yao, and Kuo (2012),Eryilmaz (2013a, 2013b), Marichal and Mathonet (2013), Wu andCoolen (2013), Zhu and Kuo (2014).

In a usual k-out-of-n:G system, the contribution of each compo-nent to the performance of the system is assumed to be identical.The case when the components have different impacts on the func-tioning of the system has formulated by Wu and Chen (1994) (seealso Higashiyama, 2001). According to their definitions, aweighted-k-out-of-n:G system is a system which functions if andonly if the total weight of working components is at least k. Sucha structure is useful for modeling many real life systems. For exam-ple, a lighting system may consist of a fixed number of light bulbswith different wattages. In this example, the weight of a compo-nent (light bulb) is considered to be its wattage. If all componentshave the same weight of 1, then the weighted-k-out-of-n:G systemis equivalent to the usual k-out-of-n:G system.

In the present paper, we study MRI, JRI and BP-importance mea-sures of the components in weighted-k-out-of-n:G system. Ourresults extend some of the results in Hong et al. (2002) and Gaoet al. (2007) from usual k-out-of-n:G systems to the weighted-k-out-of-n:G systems. In a usual k-out-of-n:G system, all componentsare assumed to have identical weights. Therefore, importance mea-sures of components are functions of their reliabilities. However, ina k-out-of-n:G system with weighted components the importanceof a component is expected to be dependent on both the reliabilityand the weight of the component. The main objective of the pres-ent paper is to examine how the importance of the componentschanges with respect to their reliabilities and weights. Our methodfor computing MRI and JRI is based on the well-known universalgenerating function (UGF) approach. The UGF method has beenfound to be very useful in reliability evaluations (Levitin, 2004,2005; Levitin & Lisnianski, 1999; Li & Zuo, 2008). The BP-impor-tance of the components is computed via classical probabilistictechniques.

The remainder of the paper is organized as follows. In Section 2,we present the method of UGF for computing MRI and JRI in aweighted-k-out-of-n:G system. Section 3 includes a probabilistic

approach for computing component BP-importance. Finally in Sec-tion 4, we present computational results to illustrate the methods.

2. Computation of MRI and JRI

If pi ¼ 1� qi is the reliability of component i and xi is theweight of component i; i ¼ 1; . . . ;n, then the UGF of component i is

UiðzÞ ¼ qiz0 þ piz

xi : ð3Þ

The UGF of a system when component i works can beformulated as

US;Xi¼1ðzÞ ¼X

m2 0;...;M1f gP1

m;izH1

m;i ; ð4Þ

where M1 is the largest possible state of the system, P1m;i is the prob-

ability of state m and H1m;i is the total performance associated with

the state m when component i is working. Similarly, we can define

US;Xi¼0ðzÞ ¼X

m2 0;...;M0f gP0

m;izH0

m;i ; ð5Þ

The reliability of the system when the component i is in aworking state can be computed from

P S;Xi ¼ 1f g ¼ dðUS;Xi¼1ðzÞ; kÞ ¼ dX

m2 0;...;M1f gP1

m;izH1

m;i ; k

!

¼X

m2 0;...;M1f gP1

m;iaðzH1

m;i � kÞ; ð6Þ

where aðxÞ ¼ 1 if x P 0 and aðxÞ ¼ 0 if x < 0. Similarly,

P S;Xi ¼ 0f g ¼X

m2 0;...;M0f gP0

m;iaðzH0

m;i � kÞ; ð7Þ

The MRI of the component i can be computed using (6) and (7) in (1).For an illustration, consider a weighted 5-out-of-3:G system

with x1 ¼ 3;x2 ¼ 5;x3 ¼ 2. The system works if and only if thetotal weight of working components is at least 5. The individualUGFs of the components are

U1ðzÞ ¼ q1z0 þ p1z3;

U2ðzÞ ¼ q2z0 þ p2z5;

U3ðzÞ ¼ q3z0 þ p3z2:

The UGF of the system when component ‘‘1’’ is working

US;X1¼1ðzÞ ¼ ðp1z3Þðq2z0 þ p2z5Þðq3z0 þ p3z2Þ¼ p1q2q3z3 þ p1q2p3z5 þ p1p2q3z8 þ p1p2p3z10:

Therefore

P S;X1 ¼ 1f g ¼ p1q2p3 þ p1p2q3 þ p1p2p3 ¼ p1p3 þ p1p2 � p1p2p3:

Similarly,

US;X1¼0ðzÞ ¼ ðq1z0Þðq2z0 þ p2z5Þðq3z0 þ p3z2Þ¼ q1q2q3z0 þ q1q2p3z2 þ q1p2q3z5 þ q1p2p3z7;

and

P S;X1 ¼ 0f g ¼ q1p2q3 þ q1p2p3 ¼ q1p2;

Using (1), the MRI of component ‘‘1’’ is found to be

MRI1 ¼ p3ð1� p2Þ

Repeating the similar arguments we obtain

MRI2 ¼ 1� p1p3 and MRI3 ¼ p1ð1� p2Þ:

Time dependent MRI of component i is defined as

MRIiðtÞ ¼ P T > tjXiðtÞ ¼ 1f g � P T > tjXiðtÞ ¼ 0f g; ð8Þ

Table 2CPU time for all JRI values in a weighted-k-out-of-n:Gsystem.

n CPU time (in s)

2 0.000943 0.000804 0.000635 0.00125

10 0.0114915 0.0304625 0.1432050 1.2666575 4.35989

100 10.73874

S. Eryilmaz, A.R. Bozbulut / Computers & Industrial Engineering 72 (2014) 255–260 257

for i ¼ 1; . . . ;n, where XiðtÞ denotes the state of the ith component attime t; i ¼ 1; . . . ; n, i.e.

XiðtÞ ¼1; if the ith component functions at time t

0; if the ith component has failed at time t:

�

If Ti denotes the lifetime of the ith component, thenXiðtÞ ¼ 1f g � Ti > tf g; i ¼ 1; . . . ; n.

The MRI defined by (8) can be readily obtained replacingcomponent reliabilities in MRIi by the corresponding survivalfunctions. For the abovementioned example of n ¼ 3 components,we have

MRI1ðtÞ ¼ F3ðtÞF2ðtÞ;

MRI2ðtÞ ¼ 1� F1ðtÞF3ðtÞ;

MRI3ðtÞ ¼ F1ðtÞF2ðtÞ;

where FiðtÞ ¼ P Ti > tf g ¼ 1� FiðtÞ.Next, we compute the JRI between components i and j. If MRIi;�j

(MRIi;�j) denotes the marginal reliability importance of componenti when the component j is in a working (failed) state, then

JRIði; jÞ ¼MRIi;�j �MRIi;�j ð9Þ

(see, e.g. Hong & Lie, 1993). MRIi;�j (MRIi;�j) can be calculated by set-ting pj ¼ 1 (pj ¼ 0) in marginal reliability importance of component i.

For the example given above, using (9) and the marginal reliabil-ity importances, the JRI of components ‘‘1’’ and ‘‘2’’ is found to be

JRIð1;2Þ ¼MRI1;�2 �MRI1;�2 ¼ p3ð1� 1Þ � p3ð1� 0Þ ¼ �p3 < 0

which implies that the components ‘‘1’’ and ‘‘2’’ are reliabilitysubstitutes. Similarly, we obtain

JRIð1;3Þ ¼ 1� p2 > 0 and JRIð2;3Þ ¼ �p1 < 0

which respectively implies that the components ‘‘1’’ and ‘‘3’’ arereliability complements and the components ‘‘2’’ and ‘‘3’’ arereliability substitutes.

For general values of k and n, we have written a MATLAB codefor computing MRI and JRI in weighted-k-out-of-n:G system. Ouralgorithm is based on convolution and polynomial multiplicationin MATLAB. MATLAB codes are available upon request.

The method of UGF has been widely used for the systems withweighted components. An important advantage of the method liesin its ability to represent both component reliability and the asso-ciated weight in a very simple form. To examine the efficiency ofthe method, in Tables 1 2 we provide the CPU times required forcomputing all MRI and JRI values in weighted-k-out-of-n:G sys-tems. To run MATLAB programs we used Intel Core i5 (2.53 GHz)processor with 4 GB of RAM. Random numbers between 0 and 1were generated to represent component reliabilities and randominteger numbers between 1 and 14 were generated to representcomponent weights.

Table 1CPU time for all MRI values in a weighted-k-out-of-n:Gsystem.

n CPU time (in s)

2 0.000223 0.000274 0.000515 0.00087

10 0.0031715 0.0059125 0.0139150 0.0533175 0.12388

100 0.22695

3. BP importance

Let T denote the failure time of a weighted-k-out-of-n:G systemwith component failure times T1; . . . ; Tn. Denote by WnðtÞ the totalweight of the system at time t, i.e.

WnðtÞ ¼Xn

i¼1

xiXiðtÞ; ð10Þ

where XiðtÞ ¼ 1f g � Ti > tf g; i ¼ 1; . . . ;n. Then the time to failure(or lifetime) of the weighted-k-out-of-n:G system can be expressedas

T ¼ inf t : WnðtÞ < kf g:

Our aim is to compute the probability P T ¼ Tif g for i ¼ 1; . . . ;n.Let }r;n;i;j extend all permutations ði1; . . . ; in�2Þ of 1; . . . ; j� 1; jþf

1; . . . ; i� 1; iþ 1; . . . ;ng for which i1 < � � � < in�r andin�rþ1 < � � � < in�2 such that

k�xi 6Xn�r

m¼1

xim < k: ð11Þ

To illustrate the above definition of the class }r;n;i;j, letn ¼ 5; k ¼ 6;x1 ¼ 2;x2 ¼ 1;x3 ¼ 3;x4 ¼ 2;x5 ¼ 1. For j ¼ 2;i ¼ 4; r ¼ 3; }r;n;i;j includes all permutations ði1; i2; i3Þ of 1;3;5f gsuch that i1 < i2 and

4 6X2

m¼1

xim < 6:

The permutations ð1;3;5Þ and ð3;5;1Þ satisfy the last inequalitysince 4 6 x1 þx3 < 6 and 4 6 x3 þx5 < 6. Therefore for theweight vector ð2;1;3;2;1Þ; }3;5;4;2 ¼ ð1;3;5Þ; ð3;5;1Þf g.

We have written a MATLAB code to find the elements of theclass }r;n;i;j. For finding the elements in the class, we first choosen� r elements from ði1; . . . ; in�2Þ. This can be done in

n� 2n� r

� �¼ n� 2

r � 2

� �ways and hence the corresponding computa-

tional complexity for this operation is Oðnr�2Þ. For each selectedgroup of elements and the other remaining r � 2 elements weapply quick sort algorithm. The computational complexity for sort-ing m elements is Oðm2Þ. Finally, after adding the comparison in(11) the overall computational complexity is obtained as OðnrÞ.

For the weighted k-out-of-n:G system consisting of indepen-dent components, the BP-importance of the ith component is com-puted from (see Appendix A for the proof of the result)

IBPi ¼ P T ¼ Tif g ¼ I

Xj–i

xj < k

!Z 1

0

Ym–i

FmðtÞdFiðtÞ

þXn

r¼2

Xj–i

X}r;n;i;j

Z 1

0

Z 1

s

Yn�r

m¼1

Fim ðtÞYn�2

m¼n�rþ1

Fim ðsÞdFiðtÞdFjðsÞ; ð12Þ

258 S. Eryilmaz, A.R. Bozbulut / Computers & Industrial Engineering 72 (2014) 255–260

where IðAÞ ¼ 1 if A occurs, and IðAÞ ¼ 0, otherwise, andQb

i¼axi � 1for a > b.

Thus computation of the BP-importance of the component isbased on the computation of the integralsZ 1

0

Z 1

s

Yn�r

m¼1

Fim ðtÞYn�2

m¼n�rþ1

Fim ðsÞdFiðtÞdFjðsÞ ð13Þ

for some ði1; . . . ; in�2Þ satisfying a certain property which dependson the weights of the components. For an illustration, below wecompute BP-importances of all components in a weighted-5-out-of-3:G system when x1 ¼ 3;x2 ¼ 5;x3 ¼ 2.

Using (12), becauseP

j–1xj ¼ 7 P k ¼ 5

IBP1 ¼ P T ¼ T1f g¼

X3

r¼2

Xj–1

X}r;3;1;j

Z 1

0

Z 1

s

Y3�r

m¼1

Fim ðtÞY1

m¼4�r

Fim ðsÞdF1ðtÞdFjðsÞ

¼X3

j¼2

X}2;3;1;j

Z 1

0

Z 1

sFi1 ðtÞdF1ðtÞdFjðsÞ

þX3

j¼2

X}3;3;1;j

Z 1

0

Z 1

sFi1 ðsÞdF1ðtÞdFjðsÞ: ð14Þ

}2;3;1;2 includes all permutations ði1Þ of 3f g such that

k�x1 6 xi1 ¼ x3 < k:

which is satisfied. }2;3;1;3 includes all permutations ði1Þ of 2f g suchthat

k�x1 6 xi1 ¼ x2 < k

which is not satisfied. That is,}2;3;1;3 ¼£. Thus, the first sum in (14) isZ 1

0

Z 1

sF3ðtÞdF1ðtÞdF2ðsÞ:

On the other hand, }3;3;1;2 ¼£ and }3;3;1;3 ¼£. Thus the contribu-tion of the second sum in (14) is zero. Therefore

IBP1 ¼ P T ¼ T1f g ¼

Z 1

0

Z 1

sF3ðtÞdF1ðtÞdF2ðsÞ: ð15Þ

Similarly,

IBP2 ¼ P T ¼ T2f g¼

Z 1

0

Z 1

sF3ðtÞdF2ðtÞdF1ðsÞþ

Z 1

0

Z 1

sF1ðtÞdF2ðtÞdF3ðsÞ

þZ 1

0

Z 1

sF3ðsÞdF2ðtÞdF1ðsÞþ

Z 1

0

Z 1

sF1ðsÞdF2ðtÞdF3ðsÞ; ð16Þ

and

IBP3 ¼ P T ¼ T3f g ¼

Z 1

0

Z 1

sF1ðtÞdF3ðtÞdF2ðsÞ: ð17Þ

If in particular, the components have exponential lifetimedistributions with FiðtÞ ¼ e�ki t; t P 0; i ¼ 1;2;3, then

IBP1 ¼

k1k2

ðk1 þ k3Þðk1 þ k2 þ k3Þ; ð18Þ

IBP2 ¼

k1 þ k3

k1 þ k2 þ k3; ð19Þ

IBP3 ¼

k2k3

ðk1 þ k3Þðk1 þ k2 þ k3Þ: ð20Þ

4. Numerical illustrations

In Table 3, we present the relation between MRIs of the compo-nents in a system with n ¼ 5 components when the reliabilityvector of the components is

p ¼ ð0:95;0:93;0:96;0:95;0:97Þ: ð21Þ

The relation between the marginal reliability importance of the com-ponents depends not only on the individual weights and reliabilitiesof the components but also on the value of k. This can be observedfrom the last two lines of Table 3. For x ¼ ð2;1;5;2;4Þ; althoughthe component ‘‘3’’ is less important than component ‘‘5’’ for k ¼ 7,the reverse is true when k ¼ 9. That is, component ‘‘3’’ becomes moreimportant when we need a higher capacity. Indeed for all k P 9, themost important component is the component ‘‘3’’ whenx ¼ ð2;1;5;2;4Þ and p ¼ ð0:95;0:93;0:96;0:95;0:97Þ even if thecomponent ‘‘5’’ has a larger reliability.

If all components had the same weight, then we would haveMRI2 < MRI1 ¼MRI4 < MRI3 < MRI5 for all k. However, in a k -out-of-n:G system with weighted components, the rankingbetween MRIs depends on p;x and the value of k. This is clearlyobserved from our numerical calculations.

In Tables 4,5, we compute the JRI between all pairs of n ¼ 5components for the reliability vector given by (21). As observedfrom Tables 4,5, the weight vector x has an effect on the signand the value of JRI. For the weighted-9-out-of-5:G system inTable 5, while the components ‘‘3’’ and ‘‘5’’ are reliability comple-ments for x3 ¼ 3 and x5 ¼ 4, they are reliability substitutes whenx3 ¼ 5 and x5 ¼ 4. The change in x3 also affects the value of JRIbetween the components ‘‘4’’ and ‘‘5’’ when k ¼ 7, and the signand the value of JRI between the components ‘‘4’’ and ‘‘5’’ whenk ¼ 9.

In Tables 6–8, we compute BP-importances when the systemconsists of two different types of components such that the weightin the first (second) group of components is x(x�) and the survivalfunction of the first (second) group of components is F1ðtÞ ¼ e�k1t

(F2ðtÞ ¼ e�k2t), t P 0. That is, we assume

xi ¼x; i ¼ 1; . . . ;m

x�; i ¼ mþ 1; . . . ;n

�

and

FiðtÞ ¼e�k1t ; i ¼ 1; . . . ;m

e�k2t ; i ¼ mþ 1; . . . ;n;

�

where m is the number of components in the first group.We may make the following comments from the Tables 6–8.If the survival probability of the components with weight x is

greater than the survival probability of the components withweight x� (the case in Table 6), and x > x� then IBP

i;x > IBPi;x� .

If the survival probabilities of the components are same (thecase in Table 7), then IBP

i;x P IBPi;x� when x > x�.

If the survival probability of the components with weight x issmaller than the survival probability of the components withweight x�, and x > x� the relation between IBP

i;x and IBPi;x� depends

on the number of components with weight x. See the lines 2and 3 of Table 8.

5. Concluding remarks

In this work, both marginal and joint Birnbaum importancemeasures and Barlow–Proschan importance measure have beencomputed in a weighed-k-out-of-n:G system that consists of binarycomponents. The findings of this paper enable us to observe howthe corresponding importance measures change with respect tothe weights of the components. In other words, we were able toobserve the effect of the contribution made by the component onits importance in the system. Since the UGF method can alsobe used to solve multi-state weighted-k-out-of-n:G systems

Table 3Relation between MRIs.

x k Relation

(2,1,3,2,4) 7 MRI2 < MRI1 ¼MRI4 < MRI3 < MRI5

9 MRI2 < MRI1 ¼MRI4 < MRI3 < MRI5

(2,1,5,2,4) 7 MRI2 < MRI1 ¼MRI4 < MRI3 < MRI5

9 MRI2 < MRI1 ¼MRI4 < MRI5 < MRI3

Table 4JRI between all pairs when k ¼ 7.

x j ¼ 1 j ¼ 2 j ¼ 3 j ¼ 4 j ¼ 5

(2,1,3,2,4) i ¼ 1 �0.0349 �0.0811 �0.0046 �0.9074i ¼ 2 �0.0921 �0.0349 0.0038i ¼ 3 �0.0811 �0.8933i ¼ 4 �0.9074

(2,1,5,2,4) i ¼ 1 �0.0349 �0.1081 �0.0621 �0.0434i ¼ 2 �0.0921 �0.0349 0.0038i ¼ 3 �0.1081 �0.9883i ¼ 4 �0.0434

Table 5JRI between all pairs when k ¼ 9.

x j ¼ 1 j ¼ 2 j ¼ 3 j ¼ 4 j ¼ 5

(2,1,3,2,4) i ¼ 1 0.0368 �0.8084 �0.8951 0.0833i ¼ 2 �0.8754 0.0368 0.0361i ¼ 3 �0.8084 0.1582i ¼ 4 0.0833

(2,1,5,2,4) i ¼ 1 0.0368 �0.8284 0.0648 �0.8767i ¼ 2 �0.8754 0.0368 0.0361i ¼ 3 �0.8284 �0.7418i ¼ 4 �0.8767

Table 6BP-importances when k1 ¼ 0:2 and k2 ¼ 0:25.

n m k x x� IBPi;x IBP

i;x�

10 3 10 3 2 0.1428 0.081710 5 10 3 2 0.1195 0.080510 8 10 3 2 0.1009 0.096610 5 15 3 2 0.1165 0.083510 5 20 3 2 0.1152 0.084810 5 10 3 1 0.1560 0.044020 5 20 3 2 0.0652 0.044920 15 20 3 2 0.0565 0.0306

Table 7BP-importances when k1 ¼ 0:2 and k2 ¼ 0:2.

n m k x x� IBPi;x IBP

i;x�

10 3 10 3 2 0.1417 0.082110 5 10 3 2 0.1238 0.076210 8 10 3 2 0.1000 0.100010 5 15 3 2 0.1246 0.075410 5 20 3 2 0.1278 0.072210 5 10 3 1 0.1563 0.043720 5 20 3 2 0.0674 0.044220 15 20 3 2 0.0558 0.0325

Table 8BP-importances when k1 ¼ 0:25 and k2 ¼ 0:2.

n m k x x� IBPi;x IBP

i;x�

10 3 10 3 2 0.1372 0.084110 5 10 3 2 0.1278 0.072210 8 10 3 2 0.0999 0.100510 5 15 3 2 0.1326 0.067410 5 20 3 2 0.1400 0.060010 5 10 3 1 0.1555 0.044520 5 20 3 2 0.0694 0.043520 15 20 3 2 0.0551 0.0346

S. Eryilmaz, A.R. Bozbulut / Computers & Industrial Engineering 72 (2014) 255–260 259

(Faghih-Roohi, Xie, Ng, & Yam, 2014; Li & Zuo, 2008) extensionof the results to the importance measures of the multi-statecomponents will be among our future research problems.

For a usual k-out-of-n:G system, Proschan and Boland (1983)showed that if

pi Pk� 1n� 1

for each i; ð22Þ

and

pi P pj; ð23Þ

than MRIi P MRIj;1 6 i; j 6 n.In this context, one important problem can be the investigation

of sufficient conditions similar to (22) and (23) for the relationMRIi P MRIj between two components i and j in weighted-k-out-of-n:G systems. Based on our computational results, we can con-clude that these sufficient conditions must involve the weight vec-tor x. Furthermore, we have also observed that the sign and thevalue of the JRI between components i and j are sensitive to thechanges not only on the individual weights xi and xj of the com-ponents but also on the weights of the other components. Determi-nation of sufficient conditions on x; p and k for the sign of JRI of thepair of components i and j is currently a challenging problem sincewe do not have an explicit expression for JRIði; jÞ. Therefore thecomputational results presented in this paper are useful to havea knowledge about importance measures for given reliabilitiesand weight vectors.

Appendix A

Proof of (12): Let Tr:n denote the rth smallest among the life-times T1; . . . ; Tn. Then, by the law of total probability

P T ¼ Tif g ¼Xn

r¼1

P T ¼ Ti; Ti ¼ Tr:nf g:

If the failure time of the system corresponds to the rth smallestcomponent’s failure time Tr:n, then we have the following relation

T ¼ Tr:nf g () WnðTr�1:nÞP k;WnðTr:nÞ < kf g:

Therefore

P T ¼ Tif g ¼Xn

r¼1

P WnðTr�1:nÞP k;WnðTr:nÞ < k; Ti ¼ Tr:nf g

¼ P WnðT1:nÞ < k; Ti ¼ T1:nf g

þXn

r¼2

P WnðTr�1:nÞP k;WnðTr:nÞ < k; Ti ¼ Tr:nf g ðA:1Þ

Clearly,

P WnðT1:nÞ<k;Ti¼T1:nf g¼P Ti <T1;. . . ;Ti <Ti�1;Ti <Tiþ1; . . .;Ti <Tnf g; if

Xj–i

xj <k

0 otherwise

8<:

¼ IXj–i

xj <k

!Z 1

0

Ym–i

FmðtÞdFiðtÞ: ðA:2Þ

For r P 2,

260 S. Eryilmaz, A.R. Bozbulut / Computers & Industrial Engineering 72 (2014) 255–260

P WnðTr�1:nÞP k;WnðTr:nÞ< k;Ti¼ Tr:nf g¼Xj–i

P WnðTjÞ�

P k;WnðTiÞ< k;Ti¼ Tr:n;Tj¼ Tr�1:n�:

¼Xj–i

P WnðTjÞP k;WnðTiÞ< k;Ti > Tj;r�

�2 of T1; . . . ;Tj�1;Tjþ1; . . . ;Ti�1;Tiþ1; . . . ;Tn is less than Tj; and n

�r of T1; . . . ;Tj�1;Tjþ1; . . . ;Ti�1;Tiþ1; . . . ;Tn is greater than Ti�:

Let Aij be the event that {Ti > Tj, r � 2 of T1, . . . , Tj�1, Tj+1, . . . , Ti�1,Ti+1, . . . , Tn is less than Tj and n � r of T1, . . . , Tj�1, Tj+1, . . . ,Ti�1, Ti+1, . . . , Tn is greater than Ti}.

When Aij occurs

WnðTjÞ ¼Xn

m¼1

xmIðTm > TjÞ ¼ xi þXn�r

m¼1

xim ;

and

WnðTiÞ ¼Xn

m¼1

xmIðTm > TiÞ ¼Xn�r

m¼1

xim

for a permutation ði1; . . . ; in�2Þ of 1; . . . ; j� 1; jþ 1; . . . ; i� 1;fiþ 1; . . . ;ng. Therefore

P WnðTr�1:nÞP k;WnðTr:nÞ< k;Ti¼ Tr:nf g

¼Xj–i

X}r;n;i;j

Z Zt>s

P r�2 of T1; . . . ;Tj�1;Tjþ1; . . . ;Ti�1;Tiþ1; . . . ;�

Tn is less than s n� r of T1; . . . ;Tj�1;Tjþ1; . . . ;Ti�1;Tiþ1; . . . ;Tn is greater than t�

dFiðtÞdFjðsÞ ¼Xj–i

X}r;n;i;j

Z 1

0

Z 1

s

Yn�r

m¼1

Fim ðtÞYn�2

m¼n�rþ1

Fim ðsÞdFiðtÞdFjðsÞ: ðA:3Þ

Thus the proof of (12) follows substituting A.2 and A.3 in Eq. A.1.

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