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Applying cost-reliability analysis to improve systemreliabilityTien-Hui Chena & Chiao-Pin Baob

a Department of Travel Industry Management, Far-East University, No. 49, Zhonghua Rd.,Xinshi Dist., Tainan City 74448, ROCb Department of Industrial Engineering and Management, I-Shou University, No.1, Sec. 1,Syuecheng Rd., Dashu Dist., Kaohsiung City 84001, ROCPublished online: 13 Dec 2013.

To cite this article: Tien-Hui Chen & Chiao-Pin Bao (2013) Applying cost-reliability analysis to improve system reliability,Journal of Industrial and Production Engineering, 30:7, 467-472, DOI: 10.1080/21681015.2013.869511

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Applying cost-reliability analysis to improve system reliability

Tien-Hui Chena* and Chiao-Pin Baob

aDepartment of Travel Industry Management, Far-East University, No. 49, Zhonghua Rd., Xinshi Dist., Tainan City 74448, ROC;bDepartment of Industrial Engineering and Management, I-Shou University, No.1, Sec. 1, Syuecheng Rd., Dashu Dist., Kaohsiung

City 84001, ROC

(Received May 2013; revised September 2013; accepted November 2013)

A system is generally constructed using a set of individual components based on their own reliabilities and inter-precedenceconstraints, with such systems generally referred to as a stochastic one. Improving reliability is one of the key tasks ofmanagers, and since this is generally based on the reliability of individual system components, it is necessary to establishthe priority of various improvement actions based on these. This study proposes two techniques to achieve this: one is aprocess of cost-reliability analysis, that can help decision-makers in the selection of the appropriate policies to raise systemreliability, and the other is a way to identify the importance of various system components in order to ascertain the order inwhich they should be improved. In addition, this study shows that if the lifetime of a component satisfies an exponentialdistribution, then in order to maximize overall system reliability the optimal replacement time for this component is itshalf-lifetime. By considering both the results of a cost-reliability analysis and the importance of individual components,decision-makers can set appropriate policies to effectively improve system reliability.

Keywords: graphical evaluation and review technique; reliability; cost

1. Introduction

As stochastic systems (such as military equipment andweapons systems) have become more advanced andcomplicated, a key task of logistic engineers is toimprove system reliability, and thus their performance.Many factors must be considered to achieve this, andsince they are often inherently uncertain, scholars usuallyapply probabilistic approaches to decide the best way tomake improvements. However, such methods can onlyrepresent the randomness underlying the success or fail-ure of a particular action, and require complete data andpredetermined conditions. To overcome such problems,the graphical evaluation and review technique (GERT)[11,12] has been widely utilized in order to makeeffective improvements to stochastic systems.

For example, Whitehouse [15] applied GERT toimprove the reliability of a rocket recovery system, whileOhta [10] modeled and analyzed a single conveyorsystem through the same approach. Bao and Xia [2] usedGERT to evaluate the reliability of a two-unit coldstandby redundant system with random switching time,and Kuo and Bao [5] later applied GERT to measure thereliability of a generator set. In addition, Shankar andSahani [13] used GERT to analyze not only to a two-unitcold standby system, but also to a two-unit warmstandby system with repairs [14]. Cheng [4] proposed afuzzy GERT method to calculate fuzzy reliability basedon the membership functions of component reliabilityin order to improve the reliability of fuzzy systems.Kurihara and Nishiuchi [6] proposed a Monte Carlosimulation method to estimate the characteristics of a

GERT network and showed that it could achieve thisrapidly and accurately. Liu et al. [9] proposed three pro-cedures to develop fuzzy GERT and to overcome someof the weakness of GERT networks. Lin [7] employedGERT analysis to deal with the problem of designing acontinuous sampling plan with return cost.

More recently, Wang et al. [16] proposed a novelvague GERT for evaluating wafer manufacturing yieldand finishing time in the lithographic process, anddeveloped a simple decision support system to carry outthe complex calculation procedures in order to providemore information to managers. Lin et al. [8] adoptedapproximate fuzzy arithmetic operations based on theweakest t-norm arithmetic operations to evaluate fuzzyrepairable reliability models using GERT simulationtechnology. They concluded that the proposed approxi-mate fuzzy arithmetic operations could successfullycalculate the results of fuzzy arithmetic operations usinginterval arithmetic, and effectively reduce fuzzy spreads.However, none of these earlier works identify the rela-tive importance of a set of components in order to carryout more effective and efficient reliability improvements.

Cost is an important factor when considering systemreliability, since it is possible that a component with alower unit cost is less reliable. Therefore, this studyproposes a cost-reliability index that can help decision-makers to determine which reliability improvement policyshould be utilized for each component in order to increasethe effectiveness of such actions. In addition to cost, it isalso important to identify the relative importance of eachcomponent before deciding which actions should be taken

*Corresponding author. Email: thchen@cc.feu.edu.tw

© 2013 Chinese Institute of Industrial Engineers

Journal of Industrial and Production Engineering, 2013Vol. 30, No. 7, 467–472, http://dx.doi.org/10.1080/21681015.2013.869511

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and in what order to improve over system reliability in themost efficient and effective manner.

2. Literature review

Whitehouse [15] first proposed the graphical method(GERT) to resolve stochastic network problems, withnodes and directed paths being used to solve a boardrange of practical problems. The GERT approach com-bines Gantt charts, PERT, CPM, and the flowcharts, andis based on a network of nodes. In a stochastic GERTnetwork, the appropriate probability value (or reliability)and finishing time (or lifetime) is given to each compo-nent, and operating parameters are also added ifnecessary. This method has the following features:

(1) Each network consists of nodes denotingoperations and directed branches.

(2) Each branch has a probability that the activityassociated with it will be performed.

(3) Various parameters describe the activitiesrepresented by the branches, and these parametersare usually additive, such as time and/or cost.

2.1. The advantages of GERT

The component relationships in a stochastic networkinclude graphical portrays and the sequential relationsamong these. Tasks related to certain that must precedeor follow those related to other components are thenclearly identified, in time as well as function. Such a net-work is a powerful tool for planning and controlling asystem, and has the following advantages. First, it is aconsistent framework for planning, monitoring and con-trolling system reliability. Second, the GERT method canefficiently estimate the changes in key system character-istics caused by changes in branching probability, andalso estimate the sensitivity of the overall system to cer-tain network characteristics. Third, the GERT approachillustrates the interdependence of all components andfunction packages. GERT networks have been shown tobe very effective when used to improve the reliability ofcomplicated systems in various industries [1], helpingmanagers to make better decisions.

2.2. Notations

The GERT network includes three input nodes:“Exclusive-or” “Inclusive-or” and “And” nodes, and twooutput nodes: “Deterministic” and “Probabilistic” nodes.In this study, the needed notations of GERT method aredefined as follow:

Ck the unit cost of component kC0ik the total cost of component k when

utilizing policy iIk(t) the importance of component k at time tRk(t) the reliability of component k at time t

wiRkðtÞ the increased reliability of component kat time t when utilizing policy i

RonðRkðtÞÞ the system reliability when the reliabilityof component k is Rk(t) at time t, wherenodes o and n are the source and sinkingnodes in the GERT network, respectively

RonðRkðtÞ þ DÞ the system reliability when the reliabilityof component k is ½RkðtÞ þ D� at time t,where Δ is the increased reliability ofcomponent k.

3. The measurement of component importance

In order to effectively and efficiently improve the reli-ability of a system, decision-makers should consider theimportance of its individual components. This is charac-terized as the rate at which changes in the reliability of aspecific component affect the overall reliability of thesystem. The importance of component k at time t isdefined as the ratio of RonðRkðtÞ þ DÞ to RonðRkðtÞÞminus 1, i.e. RonðRkðtÞþDÞ

RonðRkðtÞÞ � 1� �

, and referred to as Ik(t)

[3]. Ik(t) can be interpreted as the influence of compo-nent k in a system based on how much it affects the unitreliability, Δ. In general, the larger the value of Ik(t), themore influence component k has on system reliability,and thus the more important it is.

4. Heuristic policies to improve system reliability

A system is constructed of components and each onegenerally has its own distribution function over its life-time. To improve system reliability, this study discussesthree common policies, as follows:

Policy 1: The decision-maker parallels a specifiedspare part for component k.Policy 2: The decision-maker increases the reliabilityof component k by using reliability engineeringtechniques.Policy 3: The decision-maker replaces component kbefore the end of its lifetime.

For policy 1, we suppose the unit cost and the reliabilityof component k at time t are Ck and Rk(t), respectively,and the increased reliability and the total cost for compo-nent k will be 2Rk � RkðtÞ � RkðtÞ2 ¼ RkðtÞ � RkðtÞ2 andC01k ¼ 2Ck , respectively.For policy 2, the new reliability and new unit cost of

the enhanced component k after applying the engineeringtechniques are defined as and, respectively.

For policy 3, we suppose the lifetime of componentk satisfies a Weibull distribution with a shape parametera and scale parameter k, as shown in Equation (1).

f ðt ; k; aÞ ¼ ak

tk

� �a�1e�

tkð Þa ; t� 0

0; t\0

�(1)

The reliability of component k at time t is Rk(t), t� Tk ,where Tk is the expected lifetime of component k.

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Because a is the shape parameter of the Weibull distribu-tion, it can be interpreted as follows:

(1) A value of a\1 indicates that the failure ratedecreases over time. This happens if there is sig-nificant “infant mortality,” or a high number ofdefective items fail early in the production pro-cess, and the failure rate then decreases overtime as the defective items are weeded out of thepopulation.

(2) A value of a ¼ 1 indicates that the failure rate isconstant over time, which suggests that randomexternal events are causing mortality, or failure.The Weibull distribution is related to the expo-nential distribution when a ¼ 1.

(3) A value of a[ 1 indicates that the failure rateincreases with time. This happens if there is an“aging” process, or parts that are more likely tofail as time goes on.

Because α = 1 indicates that the failure rate is con-stant over time, it is a meaningful value that decision-makers can use to calculate the optimal componentreplacement time in order to improve system reliability.As stated in Appendix 1, the optimal replacement time,t*, is the half-lifetime, i.e. t� ¼ Tk

2 , under the assumptionthat the lifetime of component k satisfies an exponentialdistribution. In other words, when the decision-makerreplaces component k at its half-lifetime, this willachieve the greatest increase in system reliability. Underthis policy, the total cost C0

3k for component k is equal to2Ck . To simplify the illustration of this, we assume thatthe lifetime of component k satisfies an exponential dis-tribution with parameter λ, i.e. α = 1 and thus Tk ¼ 1

k, asseen in Section 5.

Cost is an important factor when working to improvesystem reliability, since it is possible that a componentthat has a lower unit cost is less reliable. This study thusassigns a cost-reliability index to component k in orderto select the most appropriate policy from the three onesoutlined about, shown as Function (2).

minC01k � Ck

w1RkðtÞ ;C02k � Ck

w2RkðtÞ ;C03k � Ck

w3RkðtÞ� �

; (2)

where w1RkðtÞ and ðC0ik � CkÞ are the increased reliabil-

ity and the added cost of component k when utilizingpolicy i, respectively. We thus have w1RkðtÞ ¼ RkðtÞ�RkðtÞ2, w2RkðtÞ ¼ R0

kðtÞ � RkðtÞ2 and

w3RkðtÞ ¼ 0; t\ Tk2

Rk t � Tk2

� �� RkðtÞ; t� Tk2

�

5. An illustration

To demonstrate the proposed approaches, Figure 1presents a GERT-type system, and we assume that theunit costs of the components are Ca=500, Cb= 1000,Cc=1000, Ce=3000, Cf=3000, and Cg=4000.

Moreover, we assume that the lifetimes of componentsa, b, c, e, f, and g satisfy the exponential distributionwith parameters ka ¼ 1

10, kb ¼ 115, kc ¼ 1

15, andke ¼ kf ¼ kg ¼ 1

20. In other words, the mean time to fail-ure is 10 months for component a.

5.1. Cost-reliability analysis

In Figure 1, R2;�3;4 ¼ Re þ Rg � ReRg;R2;3;4 ¼ RfþfRe þ Rg � ReRgg � Rf Re � Rf Rg þ Rf ReRg.

Rk is the reliability of component k, and �4 means thatcomponent 4 has failed. Therefore, node ð2; �3Þ indicatesthat component 2 is working but component 3 has failed.The probability of component k operating at time t isshown as Pðx� tÞ, where x is a random variable. If t = 8months, then the reliabilities of the components are asfollows:

Rað8Þ ¼ Pðx� 8Þ ¼ e�810 ¼ 0:4493;Rbð8Þ ¼ e

�815 ¼ 0:5867;

Rcð8Þ ¼ e�815 ¼ 0:5867; andReð8Þ

¼ e�820 ¼ 0:6703 ¼ Rf ð8Þ ¼ Rgð8Þ:

If a decision-maker utilizes policy 2, to acquire the valueof w2RkðtÞ, we assume that increasing the unit cost by30% can decrease the failure rate by 50% for componentk. If the decision-maker utilizes policy 3, based on theresults presented in Appendix 1, we obtain ta ¼ t � Tk

2 ¼8� 10

2 ¼ 3; tb ¼ tc ¼ 8� 152 ¼ 1

2 and te ¼ tf ¼ tg ¼ 0because of t\ Tk

2 ; k ¼ e; f ; g. For each component, thereliabilities, added costs and cost-reliability indices withdifferent policies are presented in Table 1.

The indices of all components by utilizing policy 2,presented in the ninth column of Table 1, attain the low-est values among these three policies. Therefore, thedecision-maker should utilize policy 2 to improve systemreliability under the assumption that each componentincreases unit cost by 30% to achieve a 50% fall inthe failure rate. The decision-maker can undertake asensitivity analysis under different assumptions and thenselect the appropriate policy to improve the reliability ofsystem.

The cost-reliability indices indicate which policyshould be utilized for each component. However, thepriority with which component should be improved israrely obvious, and thus it worth identifying the impact

Figure 1. A GERT-type system.

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that the various components have on the reliability of theoverall system.

5.2. The relative importance of various components

For the system shown in Figure 1, the reliability at timet is as follows:

R0;4ðtÞ ¼ fRaðtÞRbðtÞð1� RcðtÞÞ½ReðtÞ þ RgðtÞ � ReðtÞRgðtÞ�gþ fRaðtÞRbðtÞRcðtÞ½Rf ðtÞ þ ReðtÞ þ RgðtÞ � ReðtÞRgðtÞ�Rf ðtÞReðtÞ � Rf ðtÞRgðtÞ þ Rf ðtÞReðtÞRgðtÞ�gþfRaðtÞ½1� RbðtÞ�RcðtÞRf ðtÞg ð3Þ

As mentioned above, policy 2 achieves a greaterimprovement in system reliability under the assumptionthat the unit cost of each component increases by 30%and its failure rate decreases by 50%. However, if thereliability of a specific component is changed, then theoverall system reliability is also likely to vary. A sensitiv-ity analysis needs to be carried out to obtain the relativeimportance of the various components with differentnΔ’s, where nΔ is the increasing component reliability,and the results with t = 8 are presented in Table 2.

It is obvious that the importance of component a hasthe largest values under the various nΔ’s. In other words,component a has the highest priority with regard toimprovement, followed by components f, e, and g, whenfollowing policy 2. For example, if the decision-makerenhances the reliability of component a when the valueof nΔ = 0.005, then the increased reliability of compo-nent a is equal to 0.1324 ðR0

a0 ð8Þ � Rað8Þ ¼ 0:5867�0:4543 ¼ 0:1324Þ. The system reliability will thenincrease 0.4127 0:3196

0:4543 � ð0:4543þ 0:4127Þ� �with the

added cost of 150 ðC0a � CaÞ ¼ 500� 1:3� 500 ¼ 150.

6. Conclusions

In a stochastic system, it is possible that a componentwith a lower unit cost is associated with less reliability.Moreover, different policies to increase component reli-ability may have different impacts on system reliability,as well as different costs. Decision-makers should thusundertake a cost-reliability analysis for individual com-ponents to find out the most effective items to focus onbefore implementing any policies to improve componentreliability.

This study proposes two technique that can be usedto achieve this: one is a form of cost-reliability analysis,which can assist the decision-maker in the selection ofthe most appropriate policy to improve system reliability,and the other is a way to identify the relative importanceof the various components in order to decide which onesto focus on first. Here, the greater the effect that a com-ponent has on overall system reliability, the more impor-tant it is. By utilizing the proposed techniques andperforming a sensitivity analysis, decision-makers canthus improve system reliability in an effective and effi-cient manner. In future works, scholars can apply thesetechniques to real world cases to examine how effectivethey are in practice.

Notes on contributorsTien-Hui Chen is an associate professor at the Department ofTravel Industry Management, Far-East University, Taiwan. Hisresearch interests include performance evaluation and workloadassessment. He has published papers in International Journal ofProduction Research, Journal of the Operational ResearchSociety, International Journal of Hospitality Management,Cornell Hospitality Quarterly, Tourism Economics, etc.

Chiao-Pin Bao is a professor at the Department of IndustrialEngineering and Management, I-Shou University, Taiwan. Hisresearch interests include performance evaluation and statisticsanalysis. He has published papers in Journal of the OperationalResearch Society, Journal of the Chinese Institute of IndustrialEngineers, etc.

References

[1] Adbi, M., H. R. Ghasemzadeh, S. Abdollahpour, M.Sabzeparvar, A. D. M. Nasab, “Modeling and analysis ofmechanization project of wheat production by GERTnetworks,” Agricultural Sciences in China, 9, 1078–1083(2010).

Table 1. The reliabilities, added cost and indices of each component at t = 8.

Component

Policy 1 Policy 2 Policy 3C01k�Ck

w1Rk ð8ÞC02k�Ck

w2Rkð8ÞC03k�Ck

w3Rkð8ÞReliability Added cost Reliability Added cost Reliability Added cost

a 0.248 500 0.132 150 0.292 500 2021 1092 3430b 0.243 1000 0.114 300 0.382 1000 4123 2627 2628c 0.243 1000 0.114 300 0.382 1000 4123 2627 2628e 0.221 3000 0.096 900 0 3000 13,574 9414 ∞f 0.221 3000 0.096 900 0 3000 13,574 9414 ∞g 0.221 4000 0.096 1200 0 4000 18,099 12,552 ∞

Table 2. Relative importance of the components with differentnΔ’s.

Component

Ik (8)

nΔ =0.001

nΔ =0.005

nΔ =0.01

nΔ =0.05

nΔ =0.1

a 0.3203 0.3228 0.3264 0.3548 0.3903b 0.3195 0.3205 0.3217 0.3314 0.3435c 0.3194 0.3200 0.3207 0.3264 0.3336e 0.3194 0.3196 0.3198 0.3219 0.3246f 0.3196 0.3199 0.3209 0.3255 0.3318g 0.3194 0.3196 0.3198 0.3219 0.3246

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[2] Bao, C. P. and W. G. Xia, “The reliability of a two-unitcold standby redundant and system with random switchingtime,” Journal of Technology, 8, 51–56 (1993).

[3] Chen, T. H., M. F. Yang, C. J. Hsu, W. J. Lin and C. P.Bao, “Crashing analysis of a stochastic project,” Journalof Far East University, 25, 301–306 (2008).

[4] Cheng, C. H., “Fuzzy repairable reliability based on fuzzyGERT,” Microelectronics Reliability, 36, 1557–1563 (1996).

[5] Kuo, C. Y. and C. P. Bao, “Reliability calculation forgenerating units via GERT,” Journal of KaohsiungPolytechnic Institute, 1, 1–12 (1994).

[6] Kurihara, K. and N. Nishiuchi, “Efficient Monte Carlosimulation method of GERT-type network for projectmanagement,” Computer & Industrial Engineering, 42,521–531 (2002).

[7] Lin, T. Y., “Optimal inspection policy for CSP-T withreturn cost using GERT,” Journal of the Chinese Instituteof Industrial Engineers, 25, 446–456 (2008).

[8] Lin, K. P., W. Wen, C. C. Chou, C. H. Jen and K. C.Hung, “Applying fuzzy GERT with approximate fuzzyarithmetic based on the weakest t-norm operations toevaluate repairable reliability,” Applied mathematicalModelling, 35, 5314–5325 (2011).

[9] Liu, S. Y., S. C. Liu and J. W. Lin, “Model formulation anddevelopment of fuzzy GERT networks,” Journal of the Chi-nese Institute of Industrial Engineers, 21, 156–166 (2004).

[10] Ohta, H., “GERT analysis of a single conveyor system,”International Journal of Production Research, 17,405–410 (1979).

[11] Pritsker, A. A. B. and W. W. Happ, “GERT: Graphicalevaluation and review technique: Part I fundamental,” TheJournal of Industrial Engineering, 5–7, 267–274 (1966).

[12] Pritsker, A. A. B. and G. E. Whitehouse, “GERT: Graphi-cal evaluation and review technique: Part II probabilisticand industrial engineering application,” The Journal ofIndustrial Engineering, 5–7, 293–301 (1966).

[13] Shankar, G. and V. Sahani, “GERT analysis of a two-unitcold standby system with repair,” MicroelectronicsReliability, 35, 837–840 (1995).

[14] Shankar, G. and V. Sahani, “GERT analysis of a two-unitwarm standby system with repair,” MicroelectronicsReliability, 36, 1275–1278 (1996).

[15] Whitehouse, G. E., “GERT, a useful technique for analyzingreliability problems,” Technometrics, 12, 33–48 (1970).

[16] Wang, C. N., G. K. Yang, K. C. Chang and P. Chu,“Evaluating the manufacturing capacity of a lithographicarea by using a novel vague GERT,” Expert Systems withApplications, 38, 923–932 (2011).

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Appendix 1Theorem 1: If the lifetime of component k satisfies an exponen-tial distribution with a mean of Tk, to maximize the reliabilityof component k, the optimal replacement time isTk/2.Proof: Suppose the lifetime of component k satisfies an expo-nential distribution with parameter λ, i.e. 1/λ = Tk. If the deci-sion maker replaces component k at time t and t ≤ Tk then thereliability of component k can be shown as Rk (t)

RkðtÞ ¼ t

Tk

Z 1

tke�kxdxþ Tk � t

Tk

Z 1

Tk�tke�kxdx

¼ t

Tke�kt þ Tk � t

Tke�kðTk�tÞ:

Hence,

dðRkðtÞÞdt

¼ R0kðtÞ ¼

1

Tke�kt � kt

Tke�kt � 1

Tke�kðTk�tÞ

þ kðTk � tÞTk

e�kðTk�tÞ ¼ 1

Tke�kt½1� kt�

� 1

Tke�kðTk�tÞ½1� kðTk � tÞ�:

Let R0kðtÞ ¼ 0, i.e. 1

Tke�kt½1� kt� � 1

Tke�kðTk�tÞ½1� kðTk � tÞ�

¼ 0;

e�kt½1� kt� ¼ e�kðTk�tÞ½1� kðTk � tÞ�;

e�kðTk�2tÞ ¼ 1� kt1� kðTk � tÞ

�kðTk � 2tÞ ¼ ‘n1� kt

1� kðTk � tÞ�

Tk � 2t ¼ �1

k‘n

1� kt1� kðTk � tÞ

�

t ¼ 1

2Tk þ 1

k‘n

1� kt1� kðTk � tÞ

� � �(a)

Based on Equation (a), when t ¼ 12Tk , we have R0

kðTk2 Þ ¼ 0. Fur-

thermore,dðR0

k ðtÞÞdt2 ¼ R00

k ðtÞ ¼ �2kTke�kt þ k2t

Tke�kt � k

Tke�kTk ekt þ k2

e�kTk ekt � kTke�kTk ekt � k2t

Tke�kTk ekt:

If t ¼ Tk2 , then

R00Tk2

ðkÞ ¼ �2kTk

e�kTk2 þ k2

2e�

kTk2 � k

Tke�

kTk2

þ k2e�kTk2 � k

Tke�

kTk2 � k2

2e�

kTk2

¼ e�kTk2

� � �2kTk

þ k2

2� kTk

þ k2 � kTk

� k2

2

�

¼ e�kTk2

� � k2Tk � 4kTk

�

Since 1/λ = Tk, we have λ2Tk – 4λ < 0. Therefore, R00k

Tk2

� �< 0.

Hence, when the replacement time is Tk2 , the reliability of com-

ponent k attains the maximum value.

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