analyzing the behavior of an industrial system using fuzzy confidence interval based methodology

RESEARCH ARTICLE

Analyzing the Behavior of an Industrial System Using FuzzyConfidence Interval Based Methodology

Harish Garg

Received: 16 April 2013 / Revised: 14 August 2013 / Accepted: 24 October 2013 / Published online: 19 July 2014

� The National Academy of Sciences, India 2014

Abstract This paper presented a methodology, named as

confidence interval based lambda-tau, for analyzing the

behavior of complex repairable industrial systems by uti-

lizing vague, uncertain and imprecise data. In this, uncer-

tainties in the data related to each component of the system

are estimated with the help of fuzzy and statistical meth-

odology. Triangular fuzzy numbers are used for this pur-

pose as it allows expert opinions, operating conditions,

uncertainty and imprecision in reliability information.

Various reliability parameters are addressed for analyzing

the behavior of the system and their correspondingly

obtained results of the proposed approach are compared

with the existing fuzzy lambda-tau technique results. The

sensitivity as well as performance analysis has also been

performed to explore the effect of failure/repair rates of the

components on system availability. The approach has been

illustrated with an example of synthesis unit of a urea

fertilizer plant situated in Northern part of India. The

obtained results may be helpful for the plant personnel for

analyzing the systems’ behavior and to improve their per-

formance by adopting suitable maintenance strategies.

Keywords Uncertain system � Fuzzy reliability �Lambda-tau methodology � Fertilizer plant �Confidence interval

Introduction

Reliability and maintainability analysis is one of the

important engineering tasks in optimal design to study,

characterize and analyze the failure and repair of systems

in order to improve their operational use by increasing

their design life, eliminating or reducing the likelihood of

failures and safety risks, and reducing downtime, thereby

increasing available operating time. As a result, the job of

the reliability/ system analyst(s) has become more chal-

lenging as any unfortunate consequences of unreliable

behavior of systems or equipments have led to the desire

for reliability analysis [1, 2]. The conventional reliability

of a system is fully characterized in the context of

probability measures and is defined as the probability that

the system perform its function during a pre-defined

period under the defined conditions. Conventional meth-

ods assume that all the design information of the system

is precisely known which are based on the probabilistic

binary state model. But this precision is rarely true in the

real system because it is difficult to obtain a large

quantity of data from the system due to rare events of

components, human error and economic restraints. Even

if data is available, it is often inaccurate and thus exists

some uncertainty in the value of the survivor probabili-

ties. In such situations where sufficient information is not

available for defining a probability distribution, fuzzy set

theory can be used to represent the available data in an

analytical form [3]. Thus, the concept of fuzzy reliability

has been introduced and formulated either in the context

of the possibility measures or as a transition from fuzzy

success state to fuzzy failure state [4–6]. The main con-

tribution related to determine the component or system

reliability by using fuzzy set theory and fuzzy arithmetic

can be found in literature [7–11]. Yao et al. [12] applied a

H. Garg (&)

School of Mathematics and Computer Applications, Thapar

University Patiala, Patiala 147004, India

e-mail: [email protected]

URL: https://sites.google.com/site/harishg58iitr/

123

Natl. Acad. Sci. Lett. (July–August 2014) 37(4):359–370

DOI 10.1007/s40009-014-0239-2

statistical methodology in the fuzzy system reliability

analysis and got a fuzzy estimation of reliability. Jam-

khaneh et al. [13, 14] considered the fuzzy reliability of

both serial and parallel systems using fuzzy confidence

interval.

All the above-examined systems are non-repairable

ones and data used for their behavior analysis were col-

lected from historical records/logbooks/expert opinions

and taken as crisp data. Also, the traditional analytical

techniques need large amounts of data, which are difficult

to obtain because of various practical constraints such as

rare events of components, human errors, and economic

considerations for the estimation of failure/repair charac-

teristics of the system. In such circumstances, it is usually

not easy to analyze the behavior and performance of these

systems up to desired degree of accuracy by utilizing

available resources, data, and information. Furthermore, if

an analysis has been done by using some suitable tech-

niques listed above, then any reliability index alone is

inadequate to give a deeper idea about such type of sys-

tems’ behavior because a lot of factors exist which overall

influence the systems’ performance and consequently their

behavior. Thus, to analyze more closely the system’s

behavior, other reliability criteria should be included in the

traditional analysis and involved uncertainties must be

quantified. From the literature it is found that there are few

studies in which, using fuzzy set theory to account for

uncertainty in the analysis, system behavior in terms of

various reliability indices are analyzed [15–19]. In these

studies more emphasis is given to the evaluation of dif-

ferent reliability indices, reflecting system behavior, in the

form of fuzzy membership functions. Triangular fuzzy

numbers (TFNs) are used by using fuzzy possibility theory

for handling the uncertainty in the available/collected data.

The major drawback of the above techniques is that none

of them found the estimation of reliability index in the

fuzzy sense. Furthermore, they neither use fuzzy data nor

the robust and comprehensive fuzzy confidence interval. In

the evaluation of the reliability indices, data uncertainty is

one of the major challenges in the real world. The statis-

tical failure data (in the form of failure rates and repair

times) of systems are still sparse, which introduces

imprecision in the estimation of reliability parameters of

the system’s components. Moreover, due to uncertainties

in statistical data, it is necessary to estimate a range of a

reliable parameter rather than a single value. Thus current

failure data (crisp) are not sufficient to account the

involved uncertainties. Since, the population of reliability

parameters of the subsystem is unknown, therefore using

sampling, it is desirable to use the statistical confidence

interval for estimation of reliability parameters. To tackle

this problem, a new method for reliability evaluation of the

repairable industrial system by using both fuzzy data and

comprehensive fuzzy confidence interval has been pre-

sented here.

Thus, the main objective of the present paper is to

quantify the uncertainties of reliability data with the help of

both fuzzy numbers and fuzzy confidence intervals and to

develop a methodology named as confidence interval based

lambda-tau (CIBLT) for analyzing the behavior of the

complex repairable industrial systems by utilizing vague,

imprecise and uncertain data. The proposed methodology

involves qualitative modeling using PN and quantitative

analysis using lambda-tau method of solution with basic

events represented by fuzzy numbers of triangular mem-

bership functions through statistical estimation technique

and data records. To strengthen the analysis various reli-

ability indices such as system’s failure rate, repair time,

MTBF etc. are computed in the form of fuzzy membership

functions. Results obtained from CIBLT technique are

compared with the fuzzy lambda-tau (FLT) results. The

obtained results will help the management for reallocating

the resources to achieve the targeted goals of higher profit.

The synthesis unit of the urea fertilizer plant, situated in the

Northern part of India, producing approximately

1,500–2,000 metric tons of urea per day has been taken to

demonstrate the approach.

Basic Concepts of Fuzzy Set Theory

Zadeh [3] first introduced the fuzzy set theory and gener-

alize the mathematical concepts of the set to the fuzzy set.

He then theorized that if the available information is such

that the uncertain value can be located inside a closed

interval, which he called interval of confidence, then a

membership function that maps each element of the

interval of confidence to a value in the interval [0,1] can be

defined. In classical set theory, an element x in a universe

U is either a member of some crisp set A or not. This binary

issue of membership can be represented mathematically by

the characteristic function

vAðxÞ ¼1 if x 2 A

0 if x 62 A

�ð1Þ

Zadeh [3] extended the notion of binary membership to

accommodate various degrees of membership on the real

continuous interval [0,1]. The fuzzy set eA in the universe of

discourse U can be defined as a set of ordered pairs and is

given by,

eA ¼ fðx; leAðxÞÞ : x 2 Ug ð2Þ

where leAðxÞ is the degree of membership of element x in

fuzzy set eA and its value is given as leA 2 ½0; 1�.A fuzzy set eA in universe U is convex if and only if

membership functions of leAðxÞ of eA is fuzzy-convex i.e.

360 H. Garg

123

leAðkx1 þ ð1� kÞx2Þ� minðleAðx1Þ; leAðx2ÞÞ8x1; x2 2 U; 0� k� 1

ð3Þ

and is said to be normal if there exist at least one points

x 2 U such that leAðxÞ ¼ 1. A fuzzy subset eA ¼fðx; leAðxÞÞ j x 2 Rg of the real line R is called fuzzy

number if eA is convex, normal and bounded.

a-Cut

An a-cut of a fuzzy set eA is a crisp set which consists of

elements of eA having at least degree a. It is denoted by AðaÞ

and is defined mathematically as

AðaÞ ¼ fx 2 U : leAðxÞ� ag ð4Þ

where a is the parameter in the range 0� a� 1. The

concept of a-cut offers a method for resolving any fuzzy

sets in terms of constituent crisp sets. Every a-cut of a

fuzzy number is a closed interval and a family of such

intervals describes completely a fuzzy number under study.

Hence we have AðaÞ ¼ ½AðaÞL ;AðaÞU � where

AðaÞL ðaÞ ¼ inffx 2 R j leAðxÞ > ag

AðaÞU ðaÞ ¼ supfx 2 R j leAðxÞ > ag

Membership Functions and Interval Arithmetic

Membership function defines the fuzziness in a fuzzy set

irrespective of the elements in the set, which are discrete or

continuous. For a fuzzy set eA a membership function,

denoted by leAð�Þ maps U to the subset of the non-negative

real numbers [0, 1] i.e. leA : U ! ½0; 1�. Many membership

functions such as normal, triangular, trapezoidal can be

used to represent fuzzy numbers. However, triangular

membership functions (TMF) are widely used for calcu-

lating and interpreting reliability data because of their

simplicity and understandability [20, 16, 21].

A triangular fuzzy number (TFN) is defined by the

ordered triplet eA ¼ ða; b; cÞ representing, respectively, the

lower value, the modal value, and the upper value of a

triangular fuzzy membership function. Its membership

function l ~A : R �! ½0; 1�; is defined as:

leAðxÞ ¼x�ab�a

; a� x� b

1; x ¼ bc�xc�b

; b� x� c

0; otherwise

8>>><>>>:

ð5Þ

and their corresponding a-cut is defined as

AðaÞ ¼ ½aðaÞ; cðaÞ� ¼ ½ðb� aÞaþ a; c� aðc� bÞ� ð6Þ

The two-sided 100(1� c)% confidence interval for the a-

cut of the fuzzy set ~A is

AðaÞ ¼�

aþ aðb� aÞ � rffiffiffikp tk�1

c2

� �;

c� aðc� bÞ þ rffiffiffikp tk�1

c2

� �� ð7Þ

where r be estimation of the population standard deviation

of b. Let T be a t distributed random variable with k � 1

degree of freedom then tk�1ðc2Þ satisfies

P T � tk�1

c2

� �� ¼ c

2ð8Þ

The basic arithmetic operations, i.e., addition, subtraction,

multiplication and division, of fuzzy numbers depends

upon the arithmetic of the interval of confidence. The four

main arithmetic operation on two triangular fuzzy sets eA ¼\ða1; b1; c1Þ[ and eB ¼\ða2; b2; c2Þ[ described by

the a-cuts are given below for the following intervals:

AðaÞ ¼ ½AðaÞ1 ;AðaÞ3 � and BðaÞ ¼ ½BðaÞ1 ;B

ðaÞ3 �; a 2 ½0; 1�

(i) Addition : ~Aþ ~B ¼ ½AðaÞ1 þ BðaÞ1 ;A

ðaÞ3 þ B

ðaÞ3 �

(ii) Subtraction : ~A� ~B ¼ ½AðaÞ1 � BðaÞ3 ;A

ðaÞ3 � B

ðaÞ1 �

(iii) Multiplication : ~A � ~B ¼ ½PðaÞ;QðaÞ�where PðaÞ ¼ minðAðaÞ1 � B

ðaÞ1 ;A

ðaÞ1 � B

ðaÞ3 ;A

ðaÞ3 � B

ðaÞ1 ;

AðaÞ3 � B

ðaÞ3 Þ

and QðaÞ ¼ maxðAðaÞ1 � BðaÞ1 ;A

ðaÞ1 � B

ðaÞ3 ;A

ðaÞ3 � B

ðaÞ1 ;

AðaÞ3 � B

ðaÞ3 Þ

(iv) Division : ~A� ~B ¼ ~A � 1~B

if 0 62 ~B

It is clear that the multiplication and division of two TFNs

is not again a TFN with linear sides but it is a new fuzzy

number with parabolic sides.

Confidence Interval Based Lambda-Tau Methodology

Lambda-tau methodology is a traditional method in which

fault tree is used to model the system. The constant failure

rate model is adopted in this method and the basic

expressions used to evaluate the system’s failure rate ðkÞand repair time (s) associated with the logical AND-gates

and OR-gates are summarized in Table 1.

Knezevic and Odoom [18] and Garg [22] extended this

idea by coupling it with PN and fuzzy and vague set theory

respectively. But they did not find the estimation of reli-

ability of the system using statistical confidence interval.

Keeping this in mind, Jamkhaneh et al. [14] has analyzed

the fuzzy system reliability, for non-repairable system,

using the statistical confidence interval for estimation of

reliability. But any reliability index alone is inadequate to

give a deeper idea about such type of systems’ behavior

because a lot of factors exist which overall influence the

systems’ performance and consequently their behavior. To

overcome this, the present paper presented a methodology,

named as confidence interval based lambda-tau (CIBLT)

for analyzing the behavior of a repairable system. The

Analyzing the Behavior of an Industrial System 361

123

constant failure rate model is adopted in this technique

because most of the technical systems exhibit constant

failure and repair rates (i.e. exponentially distributed) after

initial burn-in-period in bathtub curve.

The basic assumptions used in this methodology are:

– Component failures and repair rates are statistically

independent, constant, very small and obey exponential

distribution function;

– After repairs, the repaired component is considered as

good as new;

– There are no simultaneous failures among the

subsystems.

– Separate maintenance facility is available for each com-

ponent. The repair process begins soon after a unit fails.

The details of the methodology are given as below:

The technique starts from the information extraction

phase in which data related to various components of the

systems are collected from the various resources in the

form of their failure rate (ki) and repair time (si). As mostly

the collected data are imprecise in nature due to various

constraints and hence fuzzy set theory has been used for

handling the uncertainties in the data. For more specfically,

triangular fuzzy numbers with equal spread, say �15 %, in

both the directions (left and right to the middle) are used

for converting the crisp data into fuzzy numbers and are

shown in Fig. 1 where, ~ki is a fuzzy failure rate and ~si is a

fuzzy repair time of ith component in the form of triangular

fuzzy numbers.

As soon as TFNs corresponding to each of the compo-

nents are known, the corresponding fuzzy value for the top

place (system fails) can be obtained using the extension

principle coupled with a-cut and interval arithmetic oper-

ations on triangular fuzzy numbers. The interval expression

for the triangular fuzzy number, for the failure rate ~k and

repair time ~s, for AND/OR-transitions are as follows:

Expressions for AND-Transitions

kðaÞ ¼Yn

i¼1

ðki2 � ki1Þaþ ki1 �riffiffiffi

kp tk�1

c2

� ��

"

Xn

j¼1

Yi¼1i6¼j

n

ðsi2 � si1Þaþ si1 �riffiffiffi

kp tk�1

c2

� �� 2664

3775;

Yn

i¼1

�ðki3 � ki2Þaþ ki3 þriffiffiffi

kp tk�1

c2

� ��

Xn

j¼1

Yi¼1i6¼j

n

�ðsi3 � si2Þaþ si3 þriffiffiffi

kp tk�1

c2

� �� 2664

37753775 ð9Þ

sðaÞ ¼

Qni¼1

ðsi2 � si1Þaþ si1 � riffiffikp tk�1

c2

n o

Pnj¼1

Qi¼1i 6¼j

n

�ðsi3 � si2Þaþ si3 þ riffiffikp tk�1

c2

n o264

375;

2666666664

Qni¼1

�ðsi3 � si2Þaþ si3 þ riffiffikp tk�1

c2

n o

Pnj¼1

Qi¼1i 6¼j

n

ðsi2 � si1Þaþ si1 � riffiffikp tk�1

c2

n o264

375

3777777775

ð10Þ

1 1

(a) Triangular Membership functions of (b) Triangular Membership functions of

Fig. 1 Input triangular fuzzy

number for the ith component of

the system

Table 1 Basic expressions of lambda tau methodology

Gate kAND sAND kOR sOR

Expression

Qnj¼1

kj

Pni¼1

Qj¼1

i 6¼j

n

sj

2664

3775

Qni¼1

si

Pn

j¼1

Qi¼1i 6¼j

n

si

24

35

Pni¼1

ki

Pn

i¼1

kisi

Pn

i¼1

ki

362 H. Garg

123

Expressions for OR-Transitions

kðaÞ ¼Xn

i¼1

ki1 þ aðki2 � ki1Þ �riffiffiffi

kp tk�1

c2

� �� ;

"

Xn

i¼1

ki3 � aðki3 � ki2Þ þriffiffiffi

kp tk�1

c2

� �� # ð11Þ

By using these systems expression (9)–(12), the reli-

ability parameters are analyzed, with left and right spreads,

at various degrees of membership functions with the

increment of 0.1 confidence level a. The expression of

these reliability parameters is summarized in Table 2. As

the obtained results are fuzzy in nature but the decision

maker or system analyst always wants a crisp or binary

nature value for implementing it into their system. In this

center of gravity [23] has been used for defuzzification due

to the property of their equivalent to the mean of the data.

Mathematically centroid or center of gravity (COG)

method is represented as an Eq (13)

�x ¼R

xx � l ~BðxÞdxRxl ~BðxÞdx

ð13Þ

where eB is the output fuzzy set, and l ~B is the membership

function.

Illustrative Example

To illustrate, a fertilizer plant situated in the northern

part of India and producing approximately 1,500–2,000

metric tons per day has been considered as a main sys-

tem [24]. The fertilizer plant is large, complex and

repairable engineering unit which is a combination of

two dependent systems namely ammonia production

system and the urea production system. The urea plant is

composed of synthesis, decomposition, crystallization and

prilling system, arranged in predetermined configuration.

Among these, urea synthesis is one of the most important

and vital functional processes which is the subject of our

discussion. The process of the system is briefly described

below.

System Description

In this process, an ammonia production plant releases

carbon dioxide gas (CO2) as a by-product which is pumped

through a CO2 booster compressor and a CO2 high-pres-

sure compressor to the synthesis reactor after getting it

passed through a methanol absorber. Liquid ammonia from

the first tank (raised to 250 atm pressure by two liquid

ammonia feed pumps arranged in parallel) is passed

through two ammonia preheaters (arranged in series) to

raise its temperature to 82:3 C then it is fed into the urea

synthesis reactor, maintained at 190 C and 250 atm. In

addition to this about 40 %, ammonium carbonate (con-

sisting of a CO2; NH3; CO2, Biuret mixture) recovered

from the recovery section is fed into the urea synthesis

reactor through a multistage centrifugal pump. This allows

CO2 and ammonia to react at 190 C and 250 atm in the

synthesis reactor to form urea. Naturally the failure of any

intermediate equipment in this process will stop the for-

mulation of the urea. So the behavior analysis of each part

of the equipment is necessary for the design modification

for the system.

In brief, the various subsystems and the components

associated with them are defined as below [24]:

– Subsystem 1 has one unit CO2 booster compressor (A),

a centrifugal type pump, which raise the pressure of

Table 2 Some reliability prarameters

Parameters Expressions

Mean time to failure MTTFs ¼ 1ks

Mean time to repair MTTRs ¼ 1ls¼ ss

ENOF Wsð0; tÞ ¼ ksls tksþls

þ k2s

ðksþlsÞ2½1� e�ðksþlsÞt�

MTBF MTBFs ¼ MTTFs þMTTRs

Reliability Rs ¼ e�ks t

Availability As ¼ ls

ksþlsþ ks

ksþlse�ðksþlsÞt

sðaÞ ¼

Pni¼1

ðki2 � ki1Þaþ ki1 � riffiffikp tk�1

c2

n o� ðsi2 � si1Þaþ si1 � riffiffi

kp tk�1

c2

n oh iPni¼1

�ðki3 � ki2Þaþ ki3 þ riffiffikp tk�1

c2

n o ;

2664

Pni¼1

�ðki3 � ki2Þaþ ki3 þ riffiffikp tk�1

c2

n o� �ðsi3 � si2Þaþ si3 þ riffiffi

kp tk�1

c2

n oh iPni¼1

ðki2 � ki1Þaþ ki1 � riffiffikp tk�1

c2

n o3775

ð12Þ


123

CO2 from 0.1 to 29.5 atm. Its failure causes complete

failure of the system.

– Subsystem 2 has one unit CO2 high-pressure compres-

sor (B), a reciprocating type pump, which raises the

pressure of CO2 from 29.5 to 250 atm. Its failure causes

complete failure of the system.

– Subsystem 3 contains the liquid ammonia feed pumps

(E) which are of reciprocating type that raise the

ammonia pressure from 16.5 to 250 atm. Two pumps

are in operation simultaneously and two remain in cold

standby. The system fails when three pumps fail.

– Subsystem 4 has two ammonia pre-heaters (D)

arranged in series. The first one raises the temperature

of gas up to 53:2 C and the second heat the gas to

82:3 C. Failure of either causes complete failure of the

system.

– Subsystem 5 has the recycle isolution feed pump (F), a

multistage centrifugal pump, to raise the pressure of

ammonia carbonate from 17 to 250 atm. It has one unit

in standby. The system fails only when both units fail.

The schematic diagram and the Petri Net model of this

system are shown in Fig. 2, where ‘‘top’’ in Fig. 2b

represents the system failure of the synthesis unit (Fig. 2).

Behavior Analysis

The procedural steps used for conducting the analysis by

using CIBLT technique are given as below.

Step 1: Under the information extraction phase, the data

related to failure rates (k’s) and repair times (s’s)

of the main components of the system are

collected from the historical/present records such

as historical records, reliability databases, system

reliability expert opinion etc and is integrated

with expertise of maintenance personnel as

presented in Table 3 [24].

Step 2: Since the extracted database on which reliability

analysis depends is either out of date or collected

under different operations and environmental

conditions and hence contains an imprecise or

uncertain data. So to handle these impreciseness

or vagueness, the obtained/collected data are

fuzzified into the triangular fuzzy numbers with

some known spread �15; �25 and �50 % as

suggested by the decision makers/system analyst

and two-sided significance level of confidence

interval is 95 %. After obtaining the fuzzified

data of the basic events of the system, the top

place event of the system is obtained by using the

extension principle coupled with the a-cuts along

with the interval expression of the system failure

rate and repair times as given in equation (9)–

(12) respectively for their membership functions.

Step 3: Based on their PN model, the top event of the

system failure events based on the basic events

are calculated by using the expressions of the

systems’ failure rate (ks) and repair time (ss)

listed in Table 1. Using these expressions of ks

and ss, various reliability parameters for the

mission time t = 10(h) with left and right spread,

for each level (a-level), ranging from 0.1 to 1,

with increments of 0.1, are obtained and shown

graphically in Fig. 3 for �15; �25 and �50 %spreads along with FLT results. From the Fig. 3,

it has been concluded that the membership

values, of the various reliability indices, obtained

by using the traditional (crisp) methodology are

CO2 BoosterCompressor

A

MethanolAbsorber

CO2Compressor

B

deefnoitulosIpump

F

LiquidAmmonia Feed

pumpsE

1 2

CO2fromtank

Mixture (urea + NH 3 + CO2 H+ 2 +teriuB+OCH3 )HO to Decomposer

Required AmmoniaCarbonate

NH3fromtank

Preheaters (D)

Secondary process

Main Process

A B

SS3

SS1

SS2

SS4 SS

5

F1

F2

Top

E2

E1

E3

E4

D1

D2

Fig. 2 a Systematic diagram and b Petri net model of the synthesis unit

364 H. Garg

123

Table 3 Data for failure rate and repair time

Subsystems (SS) ! A B E D F

Failure rate (ki 10�3)(h�1) 3.8730 5.3950 2.7590 4.4750 6.6910

ri (10�4) 4.7539 6.9386 3.1886 4.9943 7.7710

Repair time (si)(h) 4.5640 2.8380 4.7360 3.7270 4.3840

ri 0.4888 0.3469 0.5109 0.3869 0.4936

0.016 0.018 0.02 0.0220

0.2

0.4

0.6

0.8

1

Failure rate (hrs−1)

Deg

ree

of m

embe

rshi

pFuzzy failure rate

CIBLTFLTCrisp

(a)

1.5 2.2 2.9 3.6 4.3 5 5.7 6.40

0.2

0.4

0.6

0.8

1

Repair time (hrs)

Deg

ree

of m

embe

rshi

p

Fuzzy repair time

CIBLTFLTCrisp

(b)

43 48 53 58 63 68 730

0.2

0.4

0.6

0.8

1

MTBF (hrs)

Deg

ree

of m

embe

rshi

p

Fuzzy Mean Time Between Failures

CIBLTFLTCrisp

(c)

0.14 0.16 0.18 0.2 0.220

0.2

0.4

0.6

0.8

1

ENOF

Deg

ree

of m

embe

rshi

pFuzzy Expected Number of Failures

CIBLTFLTCrisp

(d)

0.8 0.81 0.82 0.83 0.84 0.85 0.860

0.2

0.4

0.6

0.8

1

Reliability

Deg

ree

of m

embe

rshi

p

Fuzzy Reliability

CIBLTFLTCrisp

(e)

0.9 0.92 0.94 0.960

0.2

0.4

0.6

0.8

1

Availability

Deg

ree

of m

embe

rshi

p

Fuzzy Availability

CIBLTFLTCrisp

(f)

Fig. 3 Various reliability plots

of the system at �15 % spread

along with FLT results


123

constant at all values of a. It means that they do

not consider the uncertainties in the data. Thus

the method is suitable only for a system whose

data are precise. On the other hand, results

proposed by the proposed technique have

reduced region and smaller spread than the FLT

results. This suggests that DM have smaller and

more sensitive region to make more sound and

effective decision in lesser time. Based on the

results shown in Fig. 3, decrease in the spread

from FLT to CIBLT results have been computed

and tabulated in Table 4. It shows that the largest

and the smallest decrease in spread occurs

corresponding to MTBF and ENOF, respectively

which means that the prediction range of reli-

ability indices decreased. The maintenance engi-

neer/expert may use this information to get

higher system reliability and/or availability to

achieve the goals of maximum profit.

Step 4: In order to take a decision related to these plots, it

is essential that resultant output should be in crisp

form. Thus defuzzification is essential for the

system analyst and hence the center of gravity

method [23] is used because it has the advantage

of being taken the whole membership function

into account for this transformation. From Fig. 3,

it is clear that sides of membership functions of

reliability parameters are parabolic, not linear as

were taken initially. The crisp and defuzzified

values for various reliability parameters at

�15; �25 and �50 % spreads are calculated

and depicted in Table 5, which reflects that the

crisp values do not change irrespective of the

spread chosen. However, defuzzified values of

various reliability parameters change with change

of spreads. From these results it has been clearly

shown that the results given by the proposed

technique acts as a bridge between the FLT and

Markovian (crisp) results. Also it has been

observed from Table 5 that when uncertainty

levels in the form of spread increases, defuzzified

values of reliability indices have almost the same

trend (increase or decrease) as shown by lambda-

tau. This suggests that values obtained through

proposed approach are conservative in nature,

which may be beneficial for plant personnel and

have some idea about the behavior of the system.

Sensitivity Analysis

To analyze the impact of change in values of reliability

indices on the system’s behavior, behavioral plots have

been plotted for different combination of reliability,

availability and failure rate and the effects on MTBF are

Table 5 Defuzzified values of reliability parameters at different spreads

Spread (%) Method Reliability parameters at different spreads

Failure rate Repair time MTBF ENOF Reliability Availability

�0 Crisp 0.01889894 3.59212763 56.50513321 0.180809689 0.827795237 0.939680436

FLT 0.01892200 3.78455030 57.33118301 0.180763669 0.827796316 0.936292670

�15 CIBLT 0.01893238 3.68203939 56.76929755 0.180994924 0.827593279 0.938108116

FLT 0.01896301 4.15293800 58.88753679 0.180684419 0.827799104 0.929824443

�25 CIBLT 0.01899705 4.02558685 57.87044233 0.181148284 0.827316521 0.932413998

FLT 0.01915550 6.51000963 68.17447552 0.180483438 0.827827024 0.892024500

�50 CIBLT 0.01941744 12.3448105 69.89778872 0.180140644 0.825407465 0.859895864

Table 4 Data related to spread of reliability indices

Failure rate Repair time MTBF ENOF Reliability Availability

Spread related to reliability parameters

I 0.00608284 3.53943475 20.93593113 0.06773362 0.05033814 0.06726437

II 0.00368437 2.13131996 12.50205931 0.04103464 0.03048880 0.04056775

Decrease in spread from I to II

39.43010172 39.78360640 40.28419738 39.41761860 39.43200921 39.68909543

I: FLT II: CIBLT

366 H. Garg

123

computed and are shown in Fig. 4. Throughout the nine

combinations, ranges of repair time and ENOF are fixed

and computed by their membership functions (Fig. 3b, d)

at cut levels a ¼ 0 i.e. repair time are varied from 2.207237

to 5.746671 and 2.672436 to 4.803756 for FLT and CIBLT

techniques respectively. Similarly the computed range of

ENOF varies from 0.146851 to 0.214585 and 0.160455 to

0.201489 for respective techniques. The computed range of

MTBF for all combinations as well as of both techniques

are tabulated in Table 6. For instance, for the first

combination of Table 6, the selected value of reliability

and availability are 0.7854 and 0.9189 respectively while

the failure rate are changes from 1:4561 10�2 to

1:8898 10�2 and further to 2:5489 10�2. In this com-

bination, the computed range of MTBF is

78.144635–116.142677 and 83.410683–105.819301 for

FLT and CIBLT respectively. It may be observed that for

this combination the predicted range of MTBF is reduced

almost by 41.027 % from fuzzy lambda-tau when CIBLT

technique is applied. This observation infers that if system

2.2

3.6

50.16

0.19

0.2280

95

110

Repair time

Reliability = 0.7854, Failure rate = 0.014561,

Availability = 0.9189

ENOF

MT

BF

(a)

2.2

3.6

50.16

0.19

0.2260

72.5

85

Repair time



ENOF

MT

BF

(b)

2.2

3.6

50.16

0.19

0.2245

54

63

Repair time



ENOF

MT

BF

(c)

2.2

3.6

50.16

0.19

0.2265

75

85

Repair time



ENOF

MT

BF

(d)

2.2

3.6

50.16

0.19

0.2250

57.5

65

Repair time



ENOF

MT

BF

(e)

2.2

3.6

50.16

0.19

0.2235

42.5

50

Repair time



ENOF

MT

BF

(f)

2.2

3.6

50.16

0.19

0.2245

55

65

Repair time



ENOF

MT

BF

(g)

2.2

3.6

50.16

0.19

0.2235

42.5

50

Repair time



ENOF

MT

BF

(h)

2.2

3.6

50.16

0.19

0.2226

32

38

Repair time



ENOF

MT

BF

(i)

Fig. 4 Behavior of MTBF for different combinations of reliability parameters


123

analysts use CIBLT results, then they may have less range

of prediction which finally leads to more sound decisions.

Similar kind of reductions have been notified for other

combinations too. Thus, based on the behavioral and sen-

sitivity analysis plots and corresponding tables, the system

manager can analyze the critical behavior of the system

and plan for suitable maintenance.

Performance Analysis

As the performance of the system directly depends on each

of the constituent components. So to increase the perfor-

mance of the system, more attention should be given to

their corresponding subsystem for the effectiveness of the

maintenance program. In order to find the most critical

component, as per preferential order, of the system, an

investigation has been done on system availability by

varying their failure rate and repair time simultaneously

and fixing the failure rate and repair time of other com-

ponents’ at the same time. The results thus obtained are

shown graphically in Fig. 5 which contains five subplots

corresponding to five main components of the system.

It has been observed from Fig. 5a that the variation in

the failure rate and repair time of the booster compressor

component shows the significant impact on the availability

of the system i.e. an increase in their failure rate from

2:90475 10�3 to 4:84125 10�3 and repair time from

3.4230 to 5.7050 reduce the system availability up to 3.465

percent. The variation in the failure rate (5:01825 10�3

to 8:36375 10�3) and repair time (3.288–5.480) of the

isolution feed pump component have a large effect on

system availability (up to 7.616 %) as shown in Fig. 5e.

Similar effect on the system availability by the variation of

the other component failure rates and repair times are

analyzed from the Fig. 5. The magnitude of the effect of

variation in failure rates and repair times of various sub-

systems of the system on its performance is summarized in

Table 7. From the results, it can be analyzed that for

improving the performance of the system, more attention

should be given to the components as per the preferential

order; isolution feed pump, booster compressor, pre-heat-

ers, high-pressure compressor and feed pumps.

Conclusion

The paper presents the methodology named as confidence

interval based lambda-tau (CIBLT) for analyzing the

behavior of the complex repairable industrial systems by

utilizing uncertain data. Major advantages of the proposed

technique are that it uses the fuzzy confidence interval for

estimating the uncertainties in the data by using triangular

fuzzy numbers. The technique has been demonstrated

through a case study of synthesis unit of a fertilizer plant

Table 6 Effect of various parameters on system MTBF

S. no. [Reliability, failure rate, availability] Mean time between failures

FLT CIBLT

Fig. 4a [0:7854; 1:4561 10�2; 0:9189] Min: 78.144635 83.410683

Max: 116.142677 105.819301

Fig. 4b [0:7854; 1:8898 10�2; 0:9189] Min: 60.402263 64.515192

Max: 90.216829 82.091494

Fig. 4c [0:7854; 2:5489 10�2; 0:9189] Min: 44.999026 48.110862

Max: 67.709018 61.491937

Fig. 4d [0:8276; 1:4561 10�2; 0:9396] Min: 61.181650 65.297441

Max: 90.856786 82.798839

Fig. 4e [0:8276; 1:8898 10�2; 0:9396] Min: 47.283336 50.495843

Max: 70.548028 64.211894

Fig. 4f [0:8276; 2:5489 10�2; 0:9396] Min: 35.217362 37.645673

Max: 52.916754 48.075433

Fig. 4g [0:8650; 1:4561 10�2; 0:9585] Min: 46.841453 49.981605

Max: 69.446801 63.315097

Fig. 4h [0:8650; 1:8898 10�2; 0:9585] Min: 36.189531 38.637388

Max: 53.881794 49.069721

Fig. 4i [0:8650; 2:5489 10�2; 0:9585] Min: 26.941946 28.788781

Max: 40.368863 36.702441

368 H. Garg

123

Table 7 Effect of varying failure rate and repair time on system availability

Components Range of failure rate Range of repair time System availability

(10�3 per hours) (h) Min Max

A: Booster compressor 2.90475–4.84125 3.4230–5.7050 0.93939503 0.97195047

B: High-pressure compressor 4.04625–6.74375 2.1285–3.5475 0.94157098 0.97465331

E: Feed pumps 2.06925–3.44875 3.5520–5.9200 0.95084559 0.97935308

D: Pre-heaters 3.35625–5.59375 2.7952–4.6587 0.93954719 0.97287215

F: Isolution feed pump 5.01825–8.36375 3.2880–5.4800 0.88594676 0.95341870

3

4.5

6

2.5

3.7

4.9

x 10−3

0.93

0.9525

0.975

Repair Time

Booster compressor

Failure rate

Ava

ilabi

lity

(a)

2

3

4

4

5.5

7

x 10−3

0.93

0.955

0.98

Repair Time

High−pressure compressor

Failure rate

Ava

ilabi

lity

(b)

3.4

4.7

6

2

2.75

3.5

x 10−3

0.945

0.9675

0.99

Repair Time

Feed pumps

Failure rate

Ava

ilabi

lity

(c)

2.5

3.7

4.9

3

4.3

5.6

x 10−3

0.93

0.955

0.98

Repair Time

Pre−heaters

Failure rateA

vaila

bilit

y

(d)

3.2

4.4

5.6

5

7

9

x 10−3

0.88

0.93

0.98

Repair Time

Isolution feed pump

Failure rate

Ava

ilabi

lity

(e)

Fig. 5 Effect on system

availability when failure rate

and repair time vary for a

particular component and fixing

these for remaining components


123

situated in the northern part of India, a complex repairable

industrial system. Various reliability parameters of the

system have been computed in terms of fuzzy membership

functions. The computed results are compared with the

existing FLT technique and concluded that the proposed

technique has less range of uncertainty which indicates that

they have a higher sensitivity zone and thus may be useful

for the reliability engineers/experts to make more sound

decisions. The sensitivity analysis has also been performed

to explore the effect of failure/repair rates of the units on

system performance. It is concluded that in order to

improve the availability and reliability aspects, it is nec-

essary to enhance the maintainability requirement of the

system. Thus, it will facilitate the management in reallo-

cating the resources, making maintenance decisions,

achieving long run availability of the system, and

enhancing the overall productivity of the industry. From

the results, it has been analyzed that maintenance should be

given as per the preferential order; isolution feed pump,

booster compressor, pre-heaters, high-pressure compressor

and feed pumps for improving the performance and hence

increasing the productivity of the system.

In nutshell, the important managerial implications

drawn using the discussed techniques are to:

– Predict the behavior of systems in more consistent

manner and in higher sensitivity zone;

– Analyze failure behavior of industrial systems in more

realistic manner as they often make use of imprecise

data;

– Determine reliability indices such as MTBF, MTTR

which are important for planning the maintenance need

of the systems; and

– Plan suitable maintenance strategies to improve system

performance and to reduce operation and maintenance

costs.

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analyzing the behavior of an industrial system using fuzzy confidence interval based methodology

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