Research Article Reliability Analysis of the …downloads.hindawi.com › archive › 2013 › 943972.pdfResearch Article Reliability Analysis of the Engineering Systems Using Intuitionistic

Hindawi Publishing CorporationJournal of Quality and Reliability EngineeringVolume 2013, Article ID 943972, 10 pageshttp://dx.doi.org/10.1155/2013/943972

Research ArticleReliability Analysis of the Engineering SystemsUsing Intuitionistic Fuzzy Set Theory

Harish Garg,1 Monica Rani,2 and S. P. Sharma2

1 School of Mathematics and Computer Applications, Thapar University Patiala, Punjab 147004, India2Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee, Uttarakhand 247667, India

Correspondence should be addressed to Harish Garg; [email protected]

Received 22 April 2013; Revised 22 July 2013; Accepted 29 July 2013

Academic Editor: Adiel Teixeira de Almeida

Copyright © 2013 Harish Garg et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The present paper investigates the reliability analysis of industrial systems by using vague lambda-tau methodology in whichinformation related to system components is uncertain and imprecise in nature. The uncertainties in the data are handled withthe help of intuitionistic fuzzy set (IFS) theory rather than fuzzy set theory. Various reliability parameters are addressed forstrengthening the analysis in terms of degree of acceptance and rejection of IFS. Performance as well as sensitivity analysis ofthe system parameter has been investigated for accessing the impact of taking wrong combinations on its performance. Finallyresults are compared with the existing traditional crisp and fuzzy methodologies results. The technique has been demonstratedthrough a case study of bleaching unit of a paper mill.

1. Introduction

Engineering systems have become complicating day by day,and rapidly increasing the cost of the equipment challengesthe plant personnel or job analyst that to maintain the systemperformance so that to produce the desirable profit undera predetermined time. However the failure is an inevitablephenomenon in an industrial system. Also the effects ofproduct failures range from those that cause minor nuisancesto catastrophic failures involving loss of life and property.Therefore it is difficult for the system analyst to maintainthe performance of the system for a longer period of time.For this, it is common knowledge that large amount of datais required in order to estimate more accurately the failure,error or repair rates. However, it is usually impossible toobtain such a large quantity of data from any particular plant.From this point of view, fuzzy reliability is a novel concept insystem engineering as fuzzy set can easily capture subjective,uncertain, and ambiguous information. Thus based on thatsystem reliability has been evaluated by using various fuzzyarithmetic and interval of confidence operations [1–10].

After the successful applications of the fuzzy set theorysince 1970, several researchers are engaged in their exten-sions. Out of existence of several extensions, intuitionistic

fuzzy set theory (IFS) defined by Atanassov [11] is one ofthe most successful extensions and has been found to bewell suited for dealing with problems concerning vague-ness. In fuzzy set theory, it is assumed that the acceptanceand rejection grades of membership are complementary innature. But during deciding the degree of membership ofan object there is always a degree of hesitation between themembership functions. This feature is highlighted in IFStheory by defining the membership and nonmembershipvalues of an object by means of a mathematical relation.Thus there exists some degree of the indeterministic sit-uation with respect to any object. Also an idea of fuzzyset theory has been extended to a vague set by Gau andBuehrer [12] while Bustince and Burillo [13] showed thatthere is an equivalence relation between vague sets andthe IFSs. Apart from that a lot of work has been done todevelop and enrich the IFS theory given in [14–19] and theircorresponding references in terms of reliability evaluationof series-parallel system, as almost all the above researchershave analyzed the system reliability for performing thesystem behavior. But it is quite understood that the otherreliability parameters such as failure rate, mean time betweenfailure, and availability are also affecting the system perfor-mance directly. Thus in order to access all these reliability

2 Journal of Quality and Reliability Engineering

parameters, Garg [20] proposed an approach for analyzingthe system behavior of various reliability parameters interms of degree of membership and nonmembership valuesat different levels of confidence. Recently, IFS theory hasbeen applied in different areas such as logic programming[21], decision making problems [22], in medical diagnosis[23], pattern recognitions [24], and reliability optimization[25].

In this paper, a structural framework has been pre-sented for analyzing the behavior of the bleaching unit ofa paper mill using intuitionistic/vague lambda-tau method-ology (VLTM) [20]. For this, data related to various com-ponents of the system are extracted in the form of com-ponent failure rates and repair times from the varioussources such as historical records, reliability databases, andsystem reliability expert opinion, and triangular intuitionisticfuzzy numbers (TIFNs) have been used in the analysisfor handling the uncertainty between the data. Sensitiv-ity on system MTBF as well as performance analysis ofavailability index has been conducted for different com-binations of other reliability parameters. Finally obtainedresults are compared to crisp as well as fuzzy techniqueresults. The rest of the paper is organized as follow.Section 2 discusses the basic concepts of the IFS theorythat have been used during the analysis. In Section 3, themethodology for conducting the analysis has been pre-sented through an illustrative example of the paper industry.Section 4 contains the sensitivity as well as performanceanalysis for finding the critical component of the system.Finally, some concrete conclusions have been presented inSection 5.

2. Basic Concepts of IntuitionisticFuzzy Set Theory

The basic concepts and definition which are used for analyz-ing the behavior of the system are discussed in this section.

2.1. Intuitionistic Fuzzy Set [12]. A vague or IFS

𝐴 in theuniverse 𝑈 is characterized by a membership function 𝜇

𝐴

:

𝑈 → [0, 1] and a nonmembership function ]

𝐴

: 𝑈 → [0, 1]

with the condition 𝜇

𝐴

(𝑥) + ]

𝐴

(𝑥) ≤ 1, for all 𝑥 ∈ 𝑈, where𝜇

𝐴

(𝑥) is considered as the lower bound for the degree ofmembership of 𝑥 in

𝐴 (based on evidences) and ]

𝐴

(𝑥) isthe lower bound of the negation (derived from the evidenceagainst of 𝑥) of the membership of 𝑥 in

𝐴. Therefore, thedegree of membership of 𝑥 in the vague set 𝐴 is characterizedby the interval [𝜇

𝐴

(𝑥), 1 − ]

𝐴

(𝑥)]. A typical illustration ofvague set 𝐴 is shown in Figure 1.

2.2. 𝛼-Cut of the Vague Set. 𝛼-cut of an IFS, denoted by𝐴

(𝛼), contains all those elements of the universe at which itsmembership function is at least degree 𝛼 and is defined as

𝐴

(𝛼)

= {𝑥 ∈ 𝑈 : 𝜇

𝐴

(𝑥) ≥ 𝛼, 1 − ]

𝐴

(𝑥) ≥ 𝛼} ,(1)

1Againstregion Hesitation

region

Membershipvalues

Supportregion

x

𝜇A

(x)

1− �A

(x)

Figure 1: A vague set.

where 𝛼 is the parameter in the range 0 ≤ 𝛼 ≤ 1. Every 𝛼-cutof a fuzzy number is a closed interval and is represented as𝐴

(𝛼)

= [𝐴

(𝛼)

𝐿

, 𝐴

(𝛼)

𝑈

], where

𝐴

(𝛼)

𝐿

(𝛼) = inf {𝑥 ∈ 𝑅𝜇

𝐴

(𝑥) ⩾ 𝛼} ,

𝐴

(𝛼)

𝑈

(𝛼) = sup {𝑥 ∈ 𝑅 | 𝜇

𝐴

(𝑥) ⩾ 𝛼} .

(2)

2.3. Triangular Intuitionistic Fuzzy Number (TIFN). An IFSdenoted by

𝐴 = ⟨[(𝑎, 𝑏, 𝑐); 𝜇, ]]⟩, where 𝑎, 𝑏, 𝑐 ∈ R is said tobe TIFN if its membership function is given by

𝜇

𝐴

(𝑥) =

{

{

{

{

{

{

{

{

{

{

{

{

{

𝜇(

𝑥 − 𝑎

𝑏 − 𝑎

) , 𝑎 ≤ 𝑥 ≤ 𝑏

𝜇, 𝑥 = 𝑏

𝜇 (

𝑐 − 𝑥

𝑐 − 𝑏

) , 𝑏 ≤ 𝑥 ≤ 𝑐

0, otherwise,

1 − ]

𝐴

(𝑥) =

{

{

{

{

{

{

{

{

{

{

{

{

{

(1 − ]) (𝑥 − 𝑎

𝑏 − 𝑎

) , 𝑎 ≤ 𝑥 ≤ 𝑏

1 − ], 𝑥 = 𝑏

(1 − ]) (𝑐 − 𝑥

𝑐 − 𝑏

) , 𝑏 ≤ 𝑥 ≤ 𝑐

0, otherwise,

(3)

where the parameter 𝑏 gives the modal values of 𝐴, thatis, 𝜇

𝐴

(𝑏) = 1, and 𝑎, 𝑐 are the lower and upper bounds ofavailable area for the evaluation data. A triangular IFS definedby the triplet (𝑎, 𝑏, 𝑐) with 𝛼-cuts is defined below and showngraphically in Figure 2:

𝐴

(𝛼)

= [𝑎

(𝛼)

, 𝑐

(𝛼)

] , 𝐴

(1−𝛼)

= [𝑎

(1−𝛼)

, 𝑐

(1−𝛼)

] , (4)

Journal of Quality and Reliability Engineering 3

1

0a b c

𝛼

𝜇

1− �

A

a(1−𝛼)

a(𝛼)

c(𝛼)c(1−𝛼)

𝜇A(x) 1− �

A(x)

Figure 2: 𝛼-cut of IFS

𝐴.

where

𝑎

(𝛼)

= 𝑎 +

𝛼

𝜇

𝑖

(𝑏 − 𝑎) ,

𝑎

(1−𝛼)

= 𝑎 +

𝛼

1 − ]𝑖

(𝑏 − 𝑎) ,

𝑐

(𝛼)

= 𝑐 −

𝛼

𝜇

𝑖

(𝑐 − 𝑏) ,

𝑐

(1−𝛼)

= 𝑐 −

𝛼

1 − ]𝑖

(𝑐 − 𝑏) .

(5)

The four basic arithmetic operations that is, addition,subtraction, multiplication, and division, on two triangularvague sets 𝐴 = ⟨(𝑎

1

, 𝑏

1

, 𝑐

1

); 𝜇

1

, ]1

⟩ and

𝐵 = ⟨(𝑎

2

, 𝑏

2

, 𝑐

2

); 𝜇

2

, ]2

⟩

with 𝜇 = min(𝜇1

, 𝜇

2

) and ] = min(]1

, ]2

) are given below:

(i) addition: 𝐴 +

𝐵 = [𝑎

1

+ 𝑎

2

, 𝑏

1

+ 𝑏

2

, 𝑐

1

+ 𝑐

2

],

(ii) subtraction: 𝐴 −

𝐵 = [𝑎

1

− 𝑐

2

, 𝑏

1

− 𝑏

2

, 𝑐

1

− 𝑎

2

],

(iii) multiplication: 𝐴 ⋅

𝐵 = [𝑎

1

𝑎

2

, 𝑏

1

𝑏

2

, 𝑐

1

𝑐

2

],

(iv) division: 𝐴 ÷

𝐵 = [𝑎

1

/𝑐

2

, 𝑏

1

/𝑏

2

, 𝑐

1

/𝑎

2

], if 0 ∉

𝐵.

3. System Description and Methodology

The present work is based on the evaluation of the bleachingunit of a paper mill, a complex repairable industrial sys-tem. The system consists of three components/subsystems,namely, tank, filter, and washer. The function of the tankis to store the washed pulp from the washing system, andchlorine is passed to it for brightening the pulp. On the otherhand, filter and washer have two units arranged in parallelin which entrapped gases during the washing of the pulpare removed from filter component and chlorine is from thewasher component after washing the fibers. The interactionsamong the various working unit of the system are shownthrough its reliability block diagram (RBD) in Figure 3. Thedescribed study is valid for the steady state period only, that is,during which the failure rate of the system can be consideredas constant, that is, exponentially distributed. Exponential

Filters Washers

Bleaching tank

B1

A

C1

B2 C2

Figure 3: Reliability block diagram of bleaching system.

Table 1: Failure rate and repair time of the system.

Components Bleaching tank Filter Washer(𝑖 = 1) (𝑖 = 2, 3) (𝑖 = 4, 5)

Failure rate (𝜆𝑖

hrs−1) 0.0000832 0.007 0.009Repair time (𝜏

𝑖

hrs) 2.5 2.0 3.0

distribution is easy to understand and implement and hasbeen demonstrated in the literature to provide good approx-imations to machine failure distributions [26]. The followingassumptions were taken into account for modeling:

(i) both failure rates (𝜆𝑖

) and repair times (𝜏𝑖

) of eachcomponent are constant and obey exponent distribu-tions;

(ii) separate maintenance facility is available for eachcomponent;

(iii) after repairs, the repaired component is considered asgood as new;

(iv) repair or replacement is carried out in case of failuresonly;

(v) subsystem either in the working or in failed state, thatis, operating or failed state.

The procedural step of the methodology is given below.

Step 1 (information extraction phase). The methodologystarts with the information extraction phase in which infor-mation in the form of system components’ failure rate (𝜆

𝑖

’s)and repair times (𝜏

𝑖

’s) is extracted from various sourcessuch as historical records, reliability databases, and systemreliability expert opinion and is integrated with the help ofplant personnel. This information is given in Table 1. Themission time for analysing the reliability parameter is takento be 10 hours.

Step 2 (formulate the membership and nonmembershipfunctions). The fact that the extracted database on whichreliability analysis depends is either out of date or collectedunder different operating and environmental conditionsleads to the problem of uncertainty in the current failurerates and repair times. So to account for the uncertaintiesin the analysis, the obtained data are converted into thetriangular intuitionistic fuzzy numbers with some knownspread say ±15% (also at ±25% and ±50%) suggested bydecision maker/design maintenance expert/system reliabilityanalyst, as depicted in Figure 4. Based on these triangularinput data, the expressions of their systems’ failure rate


1

𝜇

1− �

𝛼-cuts

𝜇��𝑖

(x)

��i

𝜆i2 𝜆i2− 𝜆i2 × 0.15 𝜆i2 + 𝜆i2 × 0.15

(a) Membership functions of 𝜆𝑖

1

𝜇

1− �

𝛼-cuts

𝜇��𝑖 (x)

��i

𝜏i2 𝜏i2− 𝜏i2 × 0.15 𝜏i2 + 𝜏i2 × 0.15

(b) Membership functions of 𝜏𝑖

Figure 4: Input for the 𝑖th component of the system.

(𝜆𝑠

) and repair time (𝜏𝑠

) corresponding to membership andnonmembership functions are obtained for AND- and OR-transitions at different 𝛼-cuts using interval arithmetic oper-ations on TIFNs. These interval expressions are given below.

For Membership FunctionsExpressions for AND-Transitions. Consider thefollowing:

𝜆

(𝛼𝜇)

𝑠

=

[

[

[

𝑛

∏

𝑖=1

{(𝜆

𝑖2

− 𝜆

𝑖1

)

𝛼

𝜇

𝜇

𝑖

+ 𝜆

𝑖1

}

⋅

𝑛

∑

𝑗=1

[

[

[

𝑛

∏

𝑖=1

𝑖 = 𝑗

{(𝜏

𝑖2

− 𝜏

𝑖1

)

𝛼

𝜇

𝜇

𝑖

+ 𝜏

𝑖1

}

]

]

]

,

𝑛

∏

𝑖=1

{− (𝜆

𝑖3

− 𝜆

𝑖2

)

𝛼

𝜇

𝜇

𝑖

+ 𝜆

𝑖3

}

⋅

𝑛

∑

𝑗=1

[

[

[

𝑛

∏

𝑖=1

𝑖 = 𝑗

{− (𝜏

𝑖3

− 𝜏

𝑖2

)

𝛼

𝜇

𝜇

𝑖

+ 𝜏

𝑖3

}

]

]

]

]

]

]

,

𝜏

(𝛼𝜇)

𝑠

=

[

[

[

[

∏

𝑛

𝑖=1

{(𝜏

𝑖2

− 𝜏

𝑖1

) (𝛼

𝜇

/𝜇

𝑖

) + 𝜏

𝑖1

}

∑

𝑛

𝑗=1

[∏

𝑛

𝑖=1

𝑖 = 𝑗

{− (𝜏

𝑖3

− 𝜏

𝑖2

) (𝛼

𝜇

/𝜇

𝑖

) + 𝜏

𝑖3

}]

,

∏

𝑛

𝑖=1

{− (𝜏

𝑖3

− 𝜏

𝑖2

) (𝛼

𝜇

/𝜇

𝑖

) + 𝜏

𝑖3

}

∑

𝑛

𝑗=1

[∏

𝑛

𝑖=1

𝑖 = 𝑗

{(𝜏

𝑖2

− 𝜏

𝑖1

) (𝛼

𝜇

/𝜇

𝑖

) + 𝜏

𝑖1

}]

]

]

]

]

.

(6)

Expressions for OR-Transitions. Consider thefollowing:

𝜆

(𝛼𝜇)

𝑠

= [

𝑛

∑

𝑖=1

{(𝜆

𝑖2

− 𝜆

𝑖1

)

𝛼

𝜇

𝜇

𝑖

+ 𝜆

𝑖1

} ,

𝑛

∑

𝑖=1

{− (𝜆

𝑖3

− 𝜆

𝑖2

)

𝛼

𝜇

𝜇

𝑖

+ 𝜆

𝑖3

}] ,

𝜏

(𝛼𝜇)

𝑠

=

[

[

(

𝑛

∑

𝑖=1

[{(𝜆

𝑖2

− 𝜆

𝑖1

)

𝛼

𝜇

𝜇

𝑖

+ 𝜆

𝑖1

}

⋅ {(𝜏

𝑖2

− 𝜏

𝑖1

)

𝛼

𝜇

𝜇

𝑖

+ 𝜏

𝑖1

}])

× (

𝑛

∑

𝑖=1

{− (𝜆

𝑖3

− 𝜆

𝑖2

)

𝛼

𝜇

𝜇

𝑖

+ 𝜆

𝑖3

})

−1

,

(

𝑛

∑

𝑖=1

[{− (𝜆

𝑖3

− 𝜆

𝑖2

)

𝛼

𝜇

𝜇

𝑖

+ 𝜆

𝑖3

}

⋅ {− (𝜏

𝑖3

− 𝜏

𝑖2

)

𝛼

𝜇

𝜇

𝑖

+ 𝜏

𝑖3

}])

× (

𝑛

∑

𝑖=1

{(𝜆

𝑖2

− 𝜆

𝑖1

)

𝛼

𝜇

𝜇

𝑖

+ 𝜆

𝑖1

})

−1

]

]

.

(7)


For Nonmembership Functions

Expressions for AND-Transitions. Consider thefollowing:

𝜆

(𝛼1−])

𝑠

=

[

[

[

𝑛

∏

𝑖=1

{(𝜆

𝑖2

− 𝜆

𝑖1

)

𝛼]

1 − ]𝑖

+ 𝜆

𝑖1

}

⋅

𝑛

∑

𝑗=1

[

[

[

𝑛

∏

𝑖=1

𝑖 = 𝑗

{(𝜏

𝑖2

− 𝜏

𝑖1

)

𝛼]

1 − ]𝑖

+ 𝜏

𝑖1

}

]

]

]

,

𝑛

∏

𝑖=1

{− (𝜆

𝑖3

− 𝜆

𝑖2

)

𝛼]

1 − ]𝑖

+ 𝜆

𝑖3

}

⋅

𝑛

∑

𝑗=1

[

[

[

𝑛

∏

𝑖=1

𝑖 = 𝑗

{− (𝜏

𝑖3

− 𝜏

𝑖2

)

𝛼]

1 − ]𝑖

+ 𝜏

𝑖3

}

]

]

]

]

]

]

,

𝜏

(𝛼1−])

𝑠

=

[

[

[

[

∏

𝑛

𝑖=1

{(𝜏

𝑖2

− 𝜏

𝑖1

) (𝛼]/ (1 − ]𝑖

)) + 𝜏

𝑖1

}

∑

𝑛

𝑗=1

[∏

𝑛

𝑖=1

𝑖 = 𝑗

{− (𝜏

𝑖3

− 𝜏

𝑖2

) (𝛼]/ (1 − ]𝑖

)) + 𝜏

𝑖3

}]

,

∏

𝑛

𝑖=1

{− (𝜏

𝑖3

− 𝜏

𝑖2

) (𝛼]/ (1 − ]𝑖

)) + 𝜏

𝑖3

}

∑

𝑛

𝑗=1

[∏

𝑛

𝑖=1

𝑖 = 𝑗

{(𝜏

𝑖2

− 𝜏

𝑖1

) (𝛼]/ (1 − ]𝑖

)) + 𝜏

𝑖1

}]

]

]

]

]

.

(8)

Expressions for OR-Transitions. Consider thefollowing:

𝜆

(𝛼1−])

𝑠

= [

𝑛

∑

𝑖=1

{(𝜆

𝑖2

− 𝜆

𝑖1

)

𝛼]

1 − ]𝑖

+ 𝜆

𝑖1

} ,

𝑛

∑

𝑖=1

{− (𝜆

𝑖3

− 𝜆

𝑖2

)

𝛼]

1 − ]𝑖

+ 𝜆

𝑖3

}] ,

𝜏

(𝛼1−])

𝑠

=

[

[

(

𝑛

∑

𝑖=1

[{(𝜆

𝑖2

− 𝜆

𝑖1

)

𝛼]

1 − ]𝑖

+ 𝜆

𝑖1

}

⋅ {(𝜏

𝑖2

− 𝜏

𝑖1

)

𝛼]

1 − ]𝑖

+ 𝜏

𝑖1

}])

× (

𝑛

∑

𝑖=1

{− (𝜆

𝑖3

− 𝜆

𝑖2

)

𝛼]

1 − ]𝑖

+ 𝜆

𝑖3

})

−1

,

(

𝑛

∑

𝑖=1

[{− (𝜆

𝑖3

− 𝜆

𝑖2

)

𝛼]

1 − ]𝑖

+ 𝜆

𝑖3

}

Table 2: Some reliability parameters.

Parameters Expressions

Failure rate MTTF𝑠

=

1

𝜆

𝑠

Repair time MTTR𝑠

=

1

𝜇

𝑠

= 𝜏

𝑠

MTBF MTBF𝑠

= MTTF𝑠

+MTTR𝑠

ENOF 𝑊

𝑠

(0, 𝑡) =

𝜆

𝑠

𝜇

𝑠

𝑡

𝜆

𝑠

+ 𝜇

𝑠

+

𝜆

2

𝑠

(𝜆

𝑠

+ 𝜇

𝑠

)

2

[1 − 𝑒

−(𝜆𝑠+𝜇𝑠)𝑡]

Reliability 𝑅

𝑠

= 𝑒

−𝜆𝑠𝑡

Availability 𝐴

𝑠

=

𝜇

𝑠

𝜆

𝑠

+ 𝜇

𝑠

+

𝜆

𝑠

𝜆

𝑠

+ 𝜇

𝑠

𝑒

−(𝜆𝑠+𝜇𝑠)𝑡

⋅ {− (𝜏

𝑖3

− 𝜏

𝑖2

)

𝛼]

1 − ]𝑖

+ 𝜏

𝑖3

}])

× (

𝑛

∑

𝑖=1

{(𝜆

𝑖2

− 𝜆

𝑖1

)

𝛼]

1 − ]𝑖

+ 𝜆

𝑖1

})

−1

]

]

.

(9)

Step 3 (calculate various reliability parameters). Based onthese expressions of the system 𝜆

𝑠

and 𝜏

𝑠

, the various relia-bility parameters of the system, which depicts the behavior ofthe system, are analyzed at various membership grades alongwith left and right spreads. The expression of these reliabilityparameters is given in Table 2. The results obtained corre-sponding to these reliability indices are shown inFigure 5 thatcorresponds to ±15% spread along with the fuzzy lambda-tau (FLT) and traditional (crisp)methodologies results whichindicate that results obtained from VLTM are in betweenthem. That is, VLTM technique acts as a bridge between theMarkov process (crisp values) and FLT technique. From theseplots the following conclusions are drawn.

(i) The values of all reliability indices computed by usingtraditional methods (crisp) are independent of thedegree of confidence level (𝛼). It shows that whileobtaining the results by these methods attention hasnot been paid to the uncertainties in the data. Thusthis methodology is not practically sound, and hencetheir results will be suitable only for a system withprecise data.

(ii) The results computed by the FLT technique arepresented in figure with FLT legend. In their tech-nique there is a zero degree of hesitation betweenthe membership functions. Moreover the domain ofconfidence level is taken to be one, that is, 𝛼 = 1.Therefore the results computed by FLT methodologyare not that practical.

(iii) The proposed approach provides improvement overthe above shortcoming by considering 0.2 degreeof hesitation between the degree of membershipand nonmembership functions shown by solid and


4 6 8 10 120

0.2

0.4

0.6

0.8

1D

egre

e of m

embe

rshi

p

Failure rate (hrs−1)×10

−4

(a)

0 1 2 3 4 50

0.2

0.4

0.6

0.8

1

Repair time (hrs)

Deg

ree o

f mem

bers

hip

(b)

800 1000 1200 1400 1600 1800 2000 22000

0.2

0.4

0.6

0.8

1

Mean time between failures (hrs)

Deg

ree o

f mem

bers

hip

(c)

4 6 8 10 120

0.2

0.4

0.6

0.8

1

Expected number of failures

Deg

ree o

f mem

bers

hip

×10−3

(d)

0.988 0.99 0.992 0.994 0.9960

0.2

0.4

0.6

0.8

1

Reliability

Deg

ree o

f mem

bers

hip

FLTMembership

NonmembershipCrisp

(e)

0.994 0.995 0.996 0.997 0.998 0.999 10

0.2

0.4

0.6

0.8

1

Availability

Deg

ree o

f mem

bers

hip

FLTMembership

NonmembershipCrisp

(f)

Figure 5: Reliability indices plot at ±15% spread.


Table 3: Defuzzified values of the reliability parameters.

Spread Defuzzified values of parameters atFailure rate Repair time MTBF ENOF Reliability Availability

±0% Crisp 0.0007652 1.4806586 1308.3285 0.00076446 0.9923772 0.9988695

±15%I 0.00078845 2.1162556 1387.3031 0.00078748 0.9921490 0.9980299II 0.00078836 2.1129933 1386.9562 0.00078738 0.9921499 0.9980344III 0.00078832 2.1114761 1386.7954 0.00078734 0.9921503 0.9980365

±25%I 0.00082994 3.7328473 1545.0429 0.00082843 0.9917425 0.9956702II 0.00082968 3.7161699 1543.8254 0.00082817 0.9917449 0.9956992III 0.00082956 3.7083396 1543.2595 0.00082805 0.9917461 0.9957130

±50%I 0.00102659 30.128487 2712.3958 0.00102094 0.9898218 0.9442947II 0.00102556 29.505969 2706.8065 0.00101995 0.9898318 0.9452983III 0.00102508 29.196251 2699.9013 0.00101949 0.9898364 0.9457843

I: membership function, II: nonmembership function, and III: FLT.

dotted lines respectively. In the proposed approachthe domain of confidence level is clearly 𝛼 ≤ 0.8.In this, if 𝛾

1

is the degree of membership functionfor some reliability index and 𝛾

2

is the degree for thecorresponding nonmembership function then thereis 1−𝛾

1

−𝛾

2

degree of hesitation between the degree ofmembership functions. For instance, correspondingto availability value 0.9975027, there is 0.3 degree ofinterminancy between the membership functions ofthe availability 0.9975027. Thus, the proposed tech-nique is beneficial for the system analyst for analyzingthe behavior of the system in more flexible ways inlesser time than the other existing techniques.

Step 4 (defuzzified values of the reliability parameters). Inorder to implement these results by the system analyst orplant personnel then it is necessary to convert these fuzzifiedresults into the crisp or binary nature. For this, defuzzificationmethods, which convert fuzzified output into crisp value, areused, depending on the application. Here center of gravitymethod has been used for defuzzification because it givesthe mean of the data [27]. Based on that, the crispness anddefuzzified values of the reliability parameters at differentspreads (±15%,±25%, and±50%) are computed and tabulatedin Table 3. It follows from the tables that results computedby proposing an approach follows the same trend as the FLTresults, and hence results are conservative in nature.

4. Sensitivity and Performance Analyses

To analyze the effect of different reliability parameters on itsperformance, the following analysis have been done.

4.1. Sensitivity Analysis. To analyze the impact of changein values of reliability indices on to the system’s behavior,behavioral plots have been plotted for different combinationsof reliability indices and are shown in Figure 6. Throughoutthe combinations, ranges of repair time and ENOF are fixedand have been varied, along the 𝑥- and𝑦-axes, respectively, inthe range computed by their membership functions (Figures5(b) and 5(d)) at cut level 𝛼 = 0. The effects on MTBF by

taking different combinations of the remaining parameters(reliability, failure rate, and availability) are computed andhave been shown along the 𝑧-axis. For instance, in the firstthree combinations in these plots, reliability and availabilityparameters are set to be 0.9917 and 0.9965, respectively,while failure rate changes from 0.548 × 10

−3 to 0.765 ×

10

−3 and further to 0.834 × 10

−3. It is observed for thiscombination that the prediction ranges of MTBF shown inFigures 6(a)–6(c) are in the range 1781.8688, 1277.4099, and1172.0125, respectively. The corresponding plots show that asthe failure rate of the system increases then, for the prescribedranges and values of the other indices, MTBF of the systemdecreases exponentially as shown in Table 4. This suggeststhat a slight change in system’s failure rate may changeits MTBF largely and consequently behavior of the system.Similarly, for other combinations, behavior has been analyzedand corresponding results have been shown in Figures 6(a)–6(i). These plots will be beneficial for a plant maintenanceengineer for preserving particular index and to analyze theimpact of other indices on to the system’s behavior. Thus,based on the behavioral plots and corresponding table, thesystem manager can analyze the critical behavior of thesystem and plan for suitable maintenance.

4.2. Performance Analysis. As the performance of the systemis decreasing with the passage of time so it is necessary for thedecisionmakers or system analyst that initial condition of thesystem should be changed for maintaining the performanceof the system. For this, suitable maintenance strategy shouldbe adopted for it. But the problem to the system analyst is thatit is difficult to find the component measures of the systemon which more attention should be given to enhance theperformance of the system. For this, an analysis has beendoneon the availability index by varying the system componentparameters, that is, failure rate and repair time. In their anal-ysis, failure rate and repair time of each component of the sys-tem are varied simultaneously, and keeping the other param-eter remain fixed, the corresponding effect on availabilityindex has been shown graphically in Figure 7.This figure con-tains three subplots corresponding to threemain components


03

6

48

12

1000

2250

3500

MTB

F

Repair timeENOF

Reliability = 0.9917, failure rate = 0.000548,0.9965

×10−3

and availability =

(a)

03

6

48

12

500

1500

2500

MTB

F

Repair timeENOF


×10−3

and availability =

(b)

03

6

48

12

500

1500

2500

MTB

F

Repair timeENOF


×10−3

and availability =

(c)

03

6

48

12

1000

2000

3000

Repair timeENOF

MTB

F


×10−3

and availability =

(d)

Repair timeENOF

03

6

48

12

500

1.500

2.500

MTB

F


×10−3

and availability =

(e)

Repair timeENOF

03

6

4

8

12

500

1250

2000

MTB

F


×10−3

and availability =

(f)

Repair timeENOF

03

6

48

12

500

1,500

2,500

MTB

F


×10−3

and availability =

(g)

Repair timeENOF

03

6

48

12

500

1000

1500

MTB

F

Reliability = 0.9948, failure rate = 0.000765,and availability = 0.999

×10−3

(h)

Repair timeENOF

03

6

48

12

400

900

1400

MTB

F


×10−3

and availability =

(i)

Figure 6: Effect of the variation of reliability parameters on system MTBF.

of the system while their corresponding ranges are summa-rized in Table 5. From this analysis, it has been seen thatbleaching tank components produce amarginal effect on sys-tem performance while washer produces significantly. Thusit has been concluded from their analysis and correspondingtable that, for increasing the performance of the system, it isnecessary that special attention should be done on a washercomponent than filter and bleaching tank. Therefore impor-tance measures for maintaining the system components andfor saving money, manpower, and time are given as perpreferential order, washer, filter, and bleaching tank.

5. Conclusion

The objective of this paper is to predict the behavior of anindustrial system under imprecise and vague environment.To handle the uncertainty in the data, IFS theory has beenused rather than fuzzy set theory as it offers additionalinformation about the degree of hesitation. The conversionof the input data into triangular intuitionistic fuzzy numberswill greatly increase the relevance of the study. Six reliabilityparameters of the system have been computed in the termsof membership functions and their results with crisp and


Table 4: Variation of the MTBF parameter by changing other reliability parameters.

Figure (Reliability, failure rate, and availability) Mean time between failuresMin Max Range

Figure 6(a) [0.9917, 0.000548, 0.9965] 1342.8829 3124.7517 1781.8688Figure 6(b) [0.9917, 0.000765, 0.9965] 961.9998 2239.4097 1277.4099Figure 6(c) [0.9917, 0.000834, 0.9965] 882.4213 2054.4338 1172.0125Figure 6(d) [0.9923, 0.000548, 0.9988] 1245.3500 2895.8775 1650.5275Figure 6(e) [0.9923, 0.000765, 0.9988] 892.1073 2074.7841 1182.6768Figure 6(f) [0.9923, 0.000834, 0.9988] 818.3038 1903.2318 1084.9280Figure 6(g) [0.9948, 0.000548, 0.9990] 839.9668 1953.3982 1113.4314Figure 6(h) [0.9948, 0.000765, 0.9990] 601.7129 1399.5899 797.8770Figure 6(i) [0.9948, 0.000834, 0.9990] 551.9342 1283.8820 731.9478

1.5

2.5

3.5

4

8

12

0.9997

0.9999

1

Repair time

Bleaching tank

Failure rate

Avai

labi

lity

×10−5

(a)

Repair time

Failure rate

1

2

3

4

7

10

0.97

0.985

1

Filter

Avai

labi

lity

×10−3

(b)

Repair time

Failure rate

1

3

5

0.005

0.01

0.015

0.95

0.975

1

Washer

Avai

labi

lity

(c)

Figure 7: Effect of varying components parameters on its performance.


Table 5: Effect of simultaneous variations of system’s components’ failure and repair times on its availability index.

Component Range of failure rate Range of repair time Availability index𝜆 (hrs −1) 𝜏 (hrs) Min Max

Bleaching Tank 0.00005408–0.00011232 1.6250–3.3750 0.99982662 0.99991230Filter 0.004550–0.009450 1.3000–2.7000 0.97486818 0.99405300Washer 0.005850–0.012150 1.9500–4.0500 0.95358060 0.98852416

FLT results, and it was found that the proposed results actas a bridge between them. The crisp and defuzzified valuesof the indices at different levels of uncertainties will help themaintenance engineer for establishing a suitablemaintenanceaction. Sensitivity as well as performance analysis has beendone in analyzing the most critical component of the system;based on that system analyst or plant personnel may changetheir target goals and maintenance actions. From the resultsit has been concluded that, for saving money and time,necessary actions should be done in the components washer,filter, and bleaching tank as per preferential order. Apartfrom that system analyst may conduct cost benefit analysisfrom the computed results by considering the manufacturingand repairing cost for achieving the goals of higher profit atreasonable cost.

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Research Article Reliability Analysis of the …downloads.hindawi.com › archive › 2013 › 943972.pdfResearch Article Reliability Analysis of the Engineering Systems Using Intuitionistic