The document presents a novel technique called particle swarm optimization-based Lambda-Tau (PSOBLT) for analyzing the stochastic behavior of complex repairable industrial systems using uncertain data. PSOBLT combines Lambda-Tau methodology and particle swarm optimization to model system interactions using Petri nets and optimize the membership functions of reliability indices like failure rate and repair time. The technique reduces uncertainty in behavior analysis results compared to existing methods. The document demonstrates PSOBLT on a paper mill feeding unit to analyze system performance and help managers improve profit through maintenance strategies.

1. Stochastic behavior analysis of complex repairable industrial systems
utilizing uncertain data
Harish Garg n
, S.P. Sharma
Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee 247667, Uttarakhand, India
a r t i c l e i n f o
Article history:
Received 13 November 2011
Received in revised form
26 June 2012
Accepted 26 June 2012
Available online 15 July 2012
Keywords:
Paper mill
Particle swarm optimization
Fuzzy logic
Lambda-Tau methodology
a b s t r a c t
The purpose of this paper is to present a novel technique for analyzing the behavior of an industrial
system stochastically by utilizing vague, imprecise, and uncertain data. In the present study two
important tools namely Lambda-Tau methodology and particle swarm optimization are combinedly
used to present a novel technique named as particle swarm optimization based Lambda-Tau (PSOBLT)
for analyzing the behavior of a complex repairable system stochastically up to a desired degree of
accuracy. Expressions of reliability indices like failure rate, repair time, mean time between failures
(MTBF), expected number of failures (ENOF), reliability and availability for the system are obtained by
using Lambda-Tau methodology and particle swarm optimization is used to construct their member-
ship function. The interaction among the working units of the system is modeled with the help of Petri
nets. The feeding unit of a paper mill situated in a northern part of India, producing approximately
200 ton of paper per day, has been considered to demonstrate the proposed approach. Sensitivity
analysis of system’s behavior has also been done. The behavior analysis results computed by PSOBLT
technique have a reduced region of prediction in comparison of existing technique region,
i.e. uncertainties involved in the analysis are reduced. Thus, it may be a more useful analysis tool to
assess the current system conditions and involved uncertainties.
& 2012 ISA. Published by Elsevier Ltd. All rights reserved.
1. Introduction
The industrial systems are generally repairable and consist of
several subsystems. Each subsystem is composed of various
complex components and the probability of system survival
depends directly on each of its constituent components. Industrial
systems are expected to be operational and available for the
maximum possible time so as to maximize the overall production
and hence profit. However, failure is an unavoidable phenomenon
in mechanical systems/process plants/components. These failures
may be the result of human error, poor maintenance, or inade-
quate testing and inspection. Therefore, the systems and compo-
nents undergo several failure–repair cycles that include logistic
delays while performing repair leads to the degradation of
systems’ overall performance. Behavior of these systems will help
to analyze the systems’ overall performance and to carry out
design modifications so that timely action may be initiated to
achieve the desired industrial goals.
But, the complexity of industrial systems and the non-linearity
of their behavior are such that explicit functions modeling of the
system behavior are not readily available. Due to these obstacles,
researchers gave attention to the systems’ behavior analysis [1–6].
Most of the above recorded works depended on available histor-
ical records, gathered from various sources and utilized traditional
analysis techniques like Markovian approach, fault tree analysis
(FTA), reliability block diagrams (RBD), Petri nets (PN), etc. to
model the systems’ behavior. They analyzed or optimized systems’
behavior in terms of some specific reliability indices like relia-
bility, availability or maintainability etc. at a time. For example, in
[1,3,4] they analyzed the behavior/performance of industrial
systems utilizing Markovian approach. Gupta et al. [5] used
numerical method for behavior analysis of a dairy plant. Aksu
et al. [2] proposed a methodology based on FTA and Markovian
approach for the reliability and availability assessment of a pod
propulsion system. Yuzgec [7] had optimized the feeding profile of
an industrial scale fed-batch baker’s yeast fermentation process
using four different differential evolution algorithms. Wu et al. [8]
proposed an improved particle swarm optimization algorithm for
solving the reliability problems. Additionally, there are some other
types of reliability problems developed by the researchers such as
process control [9] reliability, distribution system reliability [10],
reliability of dynamic systems [11] and so on.
All of them have used the historical data which are either out of
date or collected under different operating and environmental
conditions. Thus, the used data were vague, imprecise, and uncer-
tain, i.e. historical records can only represent the past behavior but
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ISA Transactions
0019-0578/$ - see front matter & 2012 ISA. Published by Elsevier Ltd. All rights reserved.
http://dx.doi.org/10.1016/j.isatra.2012.06.012
n
Corresponding author. Fax: þ91 9897599923.
E-mail address: harishg58iitr@gmail.com (H. Garg).
ISA Transactions 51 (2012) 752–762

2. may be unable to predict future behavior of the equipment.
Unfortunately, using historical database and rough (approximate)
estimates, estimated failure and repair rates (crisp) have some
uncertainties. Thus current failure and repair rates (crisp) are not
sufficient to account the involved uncertainties.
Another prominent shortcoming of existing methodologies is
that traditional analytical techniques need large amounts of data,
which are difficult to obtain because of various practical con-
straints such as rare events of components, human errors, and
economic considerations for the estimation of failure/repair
characteristics 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 analysis has
been done by using some suitable techniques listed above, then
any reliability index alone is inadequate to give deeper idea about
such a type of systems’ behavior because a lot of factors exist
which overall influence the systems’ performance and conse-
quently 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 quanti-
fied. The inclusion of various reliability indices as criteria helps
the management to understand the effect of increasing/decreas-
ing the failure and repair rates of a particular component or
subsystem upon the overall performance of the system and
quantification of uncertainties provide results closer to the real
situational environment’s results.
Knezevic and Odoom [12] highlighted these ideas and
analyzed the behavior of a general repairable system by introdu-
cing the concept of fuzzy Lambda-Tau technique coupled with PN
in terms of various reliability indices utilizing quantified data. In
their approach, PN is used to model the system while fuzzy set
theory is used to quantify the uncertain, vague, and imprecise
data. They used fuzzy triangular numbers to quantify the involved
uncertainties in the failure/repair data because it is easy for
preparation, evaluation, and interpretation of engineering data.
In their analysis several reliability indices are used such as failure
rate, repair time, mean time between failures (MTBF), expected
number of failures (ENOF), and availability and reliability of the
system which gave more sound idea about the system’s behavior.
Komal et al. [13] used this approach for behavior analysis of press
unit using FTA instead of PN in a paper mill while the authors in
[14–17] have analyzed the behavior of some complex repairable
industrial system by using PN and fuzzy approach.
It has been analysed from these studies that when this app-
roach has been applied on system whose structure become more
complex or number of components in the system increases, then
the computed reliability indices in the form of fuzzy membership
function have wide spread due to various arithmetic operations
involved in the calculations and thus cannot give the precise idea
about the behavior of the system [18]. To reduce the uncertainty
level in the analysis, spread for each reliability index must be
reduced up to a desired degree of accuracy so that plant personnel
may use these indices to analyze the system’s behavior more
closely and take more sound decisions to improve the performance
of the plant. Mon and Cheng [19] suggested a way to optimize the
spread of fuzzy membership function, of a nonrepairable system,
using some available software packages GINO. Also in literature
variety of methods and algorithms exist for optimization and have
been applied in various technological fields, during the last three
decades [20–22]. Particle swarm optimization (PSO) is one of such
type of widely used algorithm and hence can be used to optimize
the spread of fuzzy membership function to reduce the uncertain-
ties up to a desired degree of accuracy.
Thus, the main objective of this paper is to quantify the
uncertainties with the help of fuzzy numbers and to develop a
technique to analyze the system’s behavior more closely and to
make the decisions more realistic and generic for further applica-
tion. In this paper, a technique named as particle swarm optimi-
zation-based Lambda-Tau (PSOBLT) has been developed for
analyzing the behavior of complex repairable industrial systems.
Thus, it is observed from the study that using uncertain and
limited data for complex repairable industrial system, stochastic
behavior can be analyzed up to a desired degree of accuracy. Plant
personnel may use the results and can give guidelines to improve
the system’s performance by adopting suitable maintenance
strategies. An example of the feeding unit in a paper mill is taken
into account to demonstrate the proposed technique. Results
obtained from PSOBLT technique are compared with the existing
Lambda-Tau and genetic algorithms-based Lambda-Tau (GABLT)
techniques result. The obtained results will help the management
for reallocating the resources to achieve the targeted goal of
higher profit.
2. Petri net theory
Petri nets (PN), developed by Carl Petri [23], are a useful tool
for analyzing and modelling the dynamic behaviour of complex
systems with concurrent discrete events [24]. Mathematically,
Petri net is a 5-tuple, PN ¼ ðP,T,F,W,M0Þ, where P ¼ fp1,p2 . . . pmg is
a finite set of places, T ¼ ft1,t2 . . . tng is a finite set of transitions,
F DðP Â TÞ [ ðT Â PÞ is a set of arcs, W : F-f1; 2,3 . . .g is a weight
associated with the arcs in F, M0 : P-f0; 1,2, Á Á Ág is the initial
marking, P T ¼ f and P [ T af.
The PN in its simplest form is a directed bipartite graph, where
the two types of disjoint nodes are known as places (drawn as
circles) and transitions (drawn as boxes or bars). For building a
Petri net model, the events and their conditions and conse-
quences in a system are first defined and then represented by
transitions and places in a Petri net model. In modeling [24], using
the concept of conditions and events, places represent conditions,
and transitions represent events. A transition has a certain
number of input places and output places representing the
preconditions and post-conditions of an event. The places are
connected to the transitions by input and output arcs. A directed
arc (F) from a transition to a place is said to be input arc and the
one from place to transition is called an output arc, with respect
to the place and vice versa with respect to transition.
Similar to fault tree model, PN also represents graphically the
cause and effect relationship and interaction among the working
units of a system to be modeled [25]. As obtaining minimal cut
sets in a fault tree model is a tedious process due to the large
number of gates and basic events. Contrary to fault trees,
Petrinets can more efficiently derive the minimal cut and path
sets simultaneously [12,25]. PN has a static part as well as
dynamic part. The static part consists of places, transitions, and
arrows. Meanwhile the dynamic part is related with marking of
graph by tokens, which are present, not present or evolves
dynamically on firing of valid transitions. In this study, only the
static part of PN is used to model the quantitative behavior of
system, i.e. the tokens are omitted and it is assumed that
transitions are not timed, i.e. they are immediate transitions.
For more details, refer to [26]. Fig. 1(a) and (b) illustrate the
equivalent PN models, corresponding to the logical basic AND and
OR gates.
3. Basic notation on fuzzy approach
The section presents only those basic concepts related to fuzzy
set theory, which are helpful for analyzing system behavior.
H. Garg, S.P. Sharma / ISA Transactions 51 (2012) 752–762 753

3. 3.1. Crisp versus fuzzy set
Crisp(classical) sets contain objects that satisfy precise proper-
ties of membership functions. Only two possibilities exist – an
element belongs to, or does not belong to a set. This binary issue
of membership can be represented mathematically by the indi-
cator function as,
XAðxÞ ¼
1 if xAA
0 if x =2 A
(
ð1Þ
On the other hand fuzzy sets contain objects that satisfy
imprecise properties of membership functions i.e. membership
of an object in a fuzzy set can be partial [27]. Contrary to classical
sets, fuzzy sets accommodate various degrees of membership on
the real continuous interval ½0; 1Š where ‘0’ conforms to no
membership and ‘1’ conforms to full membership. Mathemati-
cally, a fuzzy set ~A is defined by its m~A ðxÞ that satisfies
m~A ðxÞA½0; 1Š ð2Þ
where m~A ðxÞ is the degree of membership of element x in fuzzy set ~A.
3.2. Extension principle
The extension principle was developed by [27,28] and later
elaborated by [29] to enable the extension of the domain of a
function on fuzzy sets. It plays a fundamental role in translating
the set-based concepts to fuzzy set counterparts. A principle for
fuzzifying crisp functions (or possibly crisp relations) is called
extension principle [30].
A crisp function f : X-Y, defined on two universes of dis-
course X and Y, is fuzzified when it is extended to act on fuzzy sets
~FðXÞ and ~FðYÞ. The corresponding fuzzified function f has the form,
f : ~FðXÞ-~FðYÞ.
3.3. a-cuts
The a-cut of a fuzzy set ~A, denoted by Aa, is a crisp set
consisting of elements of ~A having the degree of membership at
least a and is mathematically defined as
Aa ¼ fxAX : m~A ðxÞZag ð3Þ
where a is the parameter in the range of 0rar1 and X is the
universe of discourse. The concept of a-cut offers a method for
resolving any fuzzy sets in terms of constituent crisp sets.
3.4. Membership functions
The concept of membership function is an most important
aspect in the fuzzy set theory. They are used to represent various
fuzzy sets. Many membership functions such as normal, triangular,
trapezoidal can be used to represent fuzzy numbers. However,
triangular membership functions (TMF) are widely used for
calculating and interpreting reliability data because of their
simplicity and understandability [31,32]. The decision of selecting
triangular fuzzy numbers (TFNs) lies in their ease to represent the
membership function effectively and to incorporate the judgement
distribution of multiple experts. This is not true for complex
membership functions, such as trapezoidal one, etc. For instance,
imprecise or incomplete information such as low/high failure rate
i.e. about 4 or between 5 and 7 is well represented by TMF. In the
present paper triangular membership function is used as it not
only conveys the behavior of system parameters but also reflect
the dispersion of the data adequately. The dispersion takes care of
inherent variation in human performance, vagueness in system
performance due to age and adverse operating conditions. Thus it
becomes intuitive for the engineers to arrive at right decisions.
A triangular fuzzy number (TFN) is defined by the ordered
triplet ~A ¼ ða,b,cÞ representing, respectively, the lower value, the
modal value, and the upper value of a triangular fuzzy member-
ship function. Its membership function m~A : RÀ!½0; 1Š is defined as
m~A ðxÞ ¼
xÀa
bÀa
if arxrb
1 if x ¼ b
cÀx
cÀb
if brxrc
8
>>><
>>>:
ð4Þ
The a-cut of fuzzy number ða,b,cÞ is defined below and shown
graphically in Fig. 2
Aa ¼ ½aðaÞ
,cðaÞ
Š ð5Þ
The interval of confidence defined by a-cuts can be written as
Aa ¼ ½ðbÀaÞaþa,ÀðcÀbÞaþcŠ ð6Þ
The basic arithmetic operations, i.e. addition, subtraction,
multiplication and division based on two fuzzy sets ~A and ~B, are
shown in Table 1 for the following intervals: Aa ¼ ½AðaÞ
1 ,AðaÞ
3 Š,
Ba ¼ ½BðaÞ
1 ,BðaÞ
3 Š, aA½0; 1Š.
It is clear that the multiplication and division of two TFNs are
not again a TFN with linear sides but it is a new fuzzy number
with parabolic sides.
P1
P2
P3
P1
P2
P3
AND OR
Fig. 1. Petri Net model of Logical- AND and OR operations.
1
0 b ca
Fig. 2. Triangular fuzzy number of fuzzy set ~A.
Table 1
Basic operations on fuzzy numbers.
Operation Crisp Fuzzy
Addition AþB ~A þ ~B ¼ ½AðaÞ
1 þBðaÞ
1 ,AðaÞ
3 þBðaÞ
3 Š
Subtraction AÀB ~AÀ ~B ¼ ½AðaÞ
1 ÀBðaÞ
3 ,AðaÞ
3 ÀBðaÞ
1 Š
Multiplication A Á B ~A Á ~B ¼ ½AðaÞ
1 Á BðaÞ
1 ,AðaÞ
3 Á BðaÞ
3 Š
Division ACB ~AC ~B ¼ ½AðaÞ
1 CBðaÞ
3 ,AðaÞ
3 CBðaÞ
1 Š, if 0 =2 ½BðaÞ
1 ,BðaÞ
3 Š
H. Garg, S.P. Sharma / ISA Transactions 51 (2012) 752–762754

4. 4. Methodology for behavior analysis
The motive of the study is to analyze the behavior of the
system by utilizing quantified vague, imprecise and uncertain
information/data.
4.1. 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 ðlÞ and repair time ðtÞ
associated with the logical AND- and OR-gates are summarized
in Table 2 [12,33]. But, Knezevic and Odoom [12] extended this
idea by coupling it with PN and fuzzy set theory and have
analysed the various reliability parameters (indices) in the form
of fuzzy membership functions for a repairable system. Their
approach is based on qualitative modeling using PN and quanti-
tative analysis using Lambda-Tau method of solution with basic
events represented by fuzzy numbers of triangular membership
functions.
But disadvantage of this methodology is that as the number of
components increases or system structure become more complex,
results in the form of fuzzy membership function have wide
spread due to various fuzzy arithmetic operations used in the
calculations [18]. So to analyze the stochastic behavior of complex
industrial system up to a desired degree of accuracy, an effective
and advanced technique is needed. For this PSOBLT technique is
included in this paper and is described herein.
4.2. PSOBLT technique
In PSOBLT technique, two important tools, namely Lambda-
Tau methodology and PSO are combinely used. This technique
utilizes ordinary arithmetic and optimization techniques instead
of fuzzy arithmetic for the computation of system’s fuzzy relia-
bility indices.
The main assumptions used in this technique are given below:
component failures and repair rates are statistically indepen-
dent, constant, very small and obey exponential distribution
function;
l5t and their product is small.
after repairs, the repaired component is considered as good
as new.
the standby units are of same nature and capacity as the
active units.
system structure is precisely known.
Strategy followed through this approach is shown by flow
chart in Fig. 3 and details are given hereafter.
First step in this technique is the information extraction phase.
In this, information in the form of failure rates (l’s) and repair
times (t’s) of each component of the system is extracted from the
available historical data/logbooks etc. which is imprecise in
nature due to the reasons already stated above.
In the next step, the obtained crisp data is converted into fuzzy
numbers, for accounting the uncertainties in the analysis, as it
allow experts opinion, linguistic variables, operating conditions,
uncertainty and imprecision in reliability information. Triangular
fuzzy number (TFN) is used for this purpose because it is easy for
presentation, evaluation and interpretation of engineering data
[31,32]. Thus, more specifically extracted crisp failure rates and
repair times are converted into triangular fuzzy numbers having
known spread (support) suggested by decision maker (DM)/design
maintenance expert/system reliability analyst. An input data for
failure rate li and repair time ti of the ith component of a system in
the form of TFNs with equal spread 715% in both the directions
(left and right to the middle) are shown in Fig. 4.
Table 2
Basic expressions of Lambda-Tau methodology.
Gate lAND tAND lOR tOR
Expression
Qn
j ¼ 1
lj
Pn
i ¼ 1
Qn
j ¼ 1
i a j
tj
2
4
3
5
Qn
i ¼ 1 ti
Pn
j ¼ 1
Qn
i ¼ 1
i a j
ti
h i
Pn
i ¼ 1
li
Pn
i ¼ 1 liti
Pn
i ¼ 1 li
Information
extraction in
the form of
parameters
of failure
rate and
repair time
Historical records
system reliability analyst
reliability database
gnisuybreifizzuF
triangular fuzzy
numbers
Obtain
reliability
indices using
ledomsNP
Construct fuzzy
reliability indices
membership function
using PSO
ybreifizzufeD
COG method
R
E
L
I
A
B
I
L
I
T
Y
P
A
R
A
M
E
T
E
R
S
System
behavior
analysis
Fuzzy
Crisp
Defuzzifed
fuzzy
output
crisp
input
fuzzy
data
Step 2
Step 3
Step 4
Step 1
Fig. 3. Flow chart of PSOBLT technique.
H. Garg, S.P. Sharma / ISA Transactions 51 (2012) 752–762 755

5. In the next step of the technique, system is modeled with the
help of Petri nets by finding its minimum-cut sets. Based on these
cut sets, expressions of various reliability indices of interest such
as system’s failure rate, repair time, MTBF, ENOF, availability and
reliability are obtained using Lambda-Tau methodology i.e. by
using Tables 2 and 3 results [12,33].
As the expressions of the obtained reliability index are highly
complex or non-linear in nature which contains high level of the
uncertainties. But, in order to take more appropriate decision for
improving the performance of the system, it is necessary that
spread for each reliability index must be reduced up to a desired
degree of accuracy. For this, membership functions of each relia-
bility index is constructed by formulating a non-linear program-
ming problem, at each cut level a, by utilizing the quantified fuzzy
l’s and t’s. In this optimization problem, expression of reliability
indices are obtained by using an ordinary arithmetic unlike of the
fuzzy arithmetic operations.
Then, the upper boundary values of reliability indices are
computed at cut level a by solving the optimization problems
Maximize : ~Fðl1,l2, . . . ,ln,t1,t2, . . . ,tmÞ or
~Fðt=l1,l2, . . . ,ln,t1,t2, . . . ,tmÞ
Subject to : mli
ðxÞZa,
mtj
ðxÞZa,
0rar1,
i ¼ 1; 2, . . . ,n, j ¼ 1; 2, . . . ,m: ð7Þ
The obtained maximum value of F is denoted by Fmax.
The lower boundary value of reliability indices are computed
at cut level a by solving the optimization problem (8)
Minimize : ~Fðl1,l2, . . . ,ln,t1,t2, . . . ,tmÞ or
~Fðt=l1,l2, . . . ,ln,t1,t2, . . . ,tmÞ
Subject to : mli
ðxÞZa,
mtj
ðxÞZa,
0rar1,
i ¼ 1; 2, . . . ,n, j ¼ 1; 2, . . . ,m: ð8Þ
The obtained minimum value of F is denoted by Fmin.
The membership function values of ~F at Fmax and Fmin are both
a that is:
m~F ðFmaxÞ ¼ m~F ðFminÞ ¼ a
where ~Fðl1,l2, . . . ,ln,t1,t2, . . . ,tmÞ and ~Fðt=l1,l2, . . . ,ln,t1,
t2, . . . ,tmÞ are time independent and dependent fuzzy reliability
indices.
Since the problem is non-linear in nature, it needs some effective
techniques and tools for its global solution. Out of the existing variety
of methods and algorithms, evolutionary algorithmic (EAs) techni-
ques are widely used to determine the global optimal solution of
nonlinear optimization problems without any pre-assumptions such
as continuity and differentiability. PSO is one of the family of EAs
which is basically a random search technique and has been applied
effectively to many different problems like system reliability/avail-
ability/optimization [8,22,34,35]. Thus in the light of applicability, this
paper use PSO as a tool to solve the optimization problems (7) and (8)
in the process of determining the fuzzy membership function of each
reliability index which has optimized spread. The description of the
PSO algorithm is given below.
4.3. Particle swarm optimization
Particle Swarm Optimization (PSO), first introduced by Ken-
nedy and Eberhart [34], is a stochastic global optimization
technique inspired by social behavior of bird flocking or fish
schooling. It simulated the feature of bird flocking and fish
schooling to configure the heuristic learning mechanism.
The algorithm works by initializing a flock of birds randomly over
the searching space, where every bird is called as a ‘‘particle’’.
These ‘‘particles’’ fly with a certain velocity and find the global best
position after some iteration. At each iteration, each particle can
adjust its velocity vector, based on its momentum and the
influence of its best position (pbest) as well as the best position
of its neighbors (gbest), and then compute a new position that the
1 1
Triangular Membership functions of Triangular Membership functions of
Fig. 4. Input Triangular Fuzzy Numbers for the ith component of the system.
Table 3
Some reliability parameters.
Parameters Expressions
Failure rate
MTTFs ¼
1
ls
Repair time
MTTRs ¼
1
ms
¼ ts
MTBF MTBFs ¼ MTTFs þMTTRs
ENOF
Wsð0,tÞ ¼
lsmst
ls þms
þ
l2
s
ðls þmsÞ2
½1ÀeÀðls þ ms Þt
Š
Reliability Rs ¼ eÀls t
Availability
As ¼ ms
ls þms
þ
ls
ls þms
eÀðls þ msÞt
H. Garg, S.P. Sharma / ISA Transactions 51 (2012) 752–762756

6. ‘‘particle’’ is to fly to. Suppose the dimension for a searching space
is D, the total number of particles is n, the position of the ith
particle can be expressed as vector xi ¼ ½xi1,xi2, . . . ,xiDŠ the best
position of the ith particle is denoted as pbesti ¼ ½pbesti1,
pbesti2, . . . ,pbestiDŠ, and the best position of the total particle
swarm is denoted as vector gbest ¼ ½gbest1,gbest2, . . . ,gbestDŠ, the
velocity of the ith particle is represented as vector vi ¼ ½vi1,
vi2, . . . ,viDŠ. Then the position and velocity of the particle are
updated by the following relations:
viðtþ1Þ ¼ wnviðtÞþc1nr1nðpbestiðtÞÀxiðtÞÞþc2nr2nðgbestðtÞÀxiðtÞÞ
ð9Þ
xiðtþ1Þ ¼ xiðtÞþviðtþ1Þ ð10Þ
where c1 and c2 are constants, r1 and r2 are random variable with
uniform distribution between 0 and 1, w is inertia weight, which
shows that the effect of previous velocity vector on the new vector.
The pseudo code of the algorithm is described in Algorithm 1.
Algorithm 1. Pseudo code of Particle swarm optimization (PSO).
1: Objective function: fðxÞ, x ¼ ðx1,x2, . . . ,xK Þ;
2: For each particle:
Initialize particle position and velocity
3: Do:
4: For each particle:
(a) Calculate fitness value
(b) If the fitness value is better than the best fitness value
(pbest) in history.
(c) Set current value as the new pbest.
5: End for
6: For each particle:
(a) Find in the particle neighborhood, the particle with the
best fitness.
(b) Calculate particle velocity according to the velocity
equation (9).
(c) Update particle position according to the position
equation (10).
(d) Apply the position constriction.
7: End for
8: While maximum iterations or minimum error criteria is not
attained
5. Illustrative example
The above mentioned technique, PSOBLT, for analyzing the
behavior of complex repairable system is illustrated through the
behavior of feeding system of a paper mill (situated in the
Northern part of India). The brief description of the system (paper
mill) is given below.
5.1. System description
For the production of paper, the raw material (softwood,
hardwood, bamboo, etc.) is chopped into small pieces of approxi-
mately uniform in size and transported for temporarily storage
through compressed air. Conveyor in the feeding system carry the
chips from the store to the digesters, whenever required. These
chips are cooked in the digester by using white liqueur
(NaOHþNa2S) with steam at a pressure of 8:5 kg=cm2
(around
180 1C temperature). The chips when cooked are referred to as
‘pulp’. The pulp is then transported to the storage tanks and
stirred continuously. After that it is further processed through
fiberlizer and refiner. The pulp is then filtered and washed
(in stages) with water to remove knots and chemicals. The final
washed pulp is stored in a surge tank. The next stages of
processing are bleaching and screening. For the production of
white paper, pulp is bleached by passing chlorine gas through the
pulp stored in the tank. For the production of brown pulp, used
for packaging purpose, pulp is screened directly. The white pulp
so obtained is passed through screeners to separate odd and
oversized particles. The pulp is then made to pass through
cleaners which separate heavy material from the pulp. Then, pulp
is fed to the head box of the paper machine comprising three
sections viz. forming, press and dryer. In the forming section of
the paper machine, the suction box (having six pumps) de-waters
the pulp by vacuum action. The paper in the form of sheets
produced by rolling presses is sent to press and dryer section to
reduce the moisture content by means of heat and vapour
transfer and to smooth out any irregularities. Finally, the rolled-
dried sheet of the paper (in the form of rolls) is sent for packaging.
Wood Chips
Blower for
pushing the
wood chips
Feeder unit for
carrying the
chips
Store of
wood chips
Chain
Conveyor
Belt
Conveyor
Bucket
Conveyor
Digester
Compressed
Air
Pipe filled by
compressed air
FSF
A F
E G
B C D
Fig. 5. (a) Systematic diagram and (b) PN model of feeding system.
Table 4
Failure rate and repair time data for feeding system.
Component Failure rate (li) Repair time (ti)
(Failures/h) (h)
A: Blower (i¼1) 2 Â 10À3 10
B: Chain Conveyor (i¼2) 3 Â 10À2 10
C: Belt Conveyor (i¼3) 4 Â 10À2 5.0
D: Butter Conveyor (i¼4) 5 Â 10À2 3.5
E: Feeder (i¼5) 2 Â 10À2 5.0
H. Garg, S.P. Sharma / ISA Transactions 51 (2012) 752–762 757

7. 5.2. Feeding system
Feeding system [15,36,37] is the first functioning part of a
paper mill and has a dominant role in the production of paper.
The function of feeding unit is to feed the chipped wood to the
digester for preparing the pulp from the chipping house where
the wood is chipped and stored. The system comprised the
following subsystems:
Blower (A): It is used for pushing the wooden chips through
the pipe by compressed air whose failure will cause complete
failure of the feeding system.
0.01 0.02 0.03 0.04 0.05
0
0.2
0.4
0.6
0.8
1
Failure rate
DegreeofMembership
Lambda−Tau
GABLT
PSOBLT
0 2 4 6 8 10 12 14
0
0.2
0.4
0.6
0.8
1
Repair time
DegreeofMembership
Lambda−Tau
GABLT
PSOBLT
20 30 40 50 60 70 80
0
0.2
0.4
0.6
0.8
1
MTBF
DegreeofMembership
Lambda−Tau
GABLT
PSOBLT
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
0
0.2
0.4
0.6
0.8
1
ENOF
DegreeofMembership
Lambda−Tau
GABLT
PSOBLT
0.6 0.65 0.7 0.75 0.8 0.85 0.9
0
0.2
0.4
0.6
0.8
1
Reliability
DegreeofMembership
Lambda−Tau
GABLT
PSOBLT
0.65 0.7 0.75 0.8 0.85 0.9 0.95 1
0
0.2
0.4
0.6
0.8
1
Availability
DegreeofMembership
Lambda−Tau
GABLT
PSOBLT
Fig. 6. Fuzzy reliability indices plots for feeding system. (a) failure rate; (b) repair time; (c) MTBF; (d) ENOF; (e) reliability; (f) availability.
H. Garg, S.P. Sharma / ISA Transactions 51 (2012) 752–762758

8. Conveyor Subsystem: It consists of three operating units in
series, namely Chain conveyor (B), Belt conveyor (C) and
Bucket conveyor (D) for lifting the chips up to the hight of
digester. When there is a failure in any of the subsystem,
standby unit (E) is switched on which feeds the digester slowly
causing delay in digesting process and hence loss in further
production.
Feeder (E): It acts as a standby unit with the conveyor
subsystem for carrying the chips by compressed air from the
store to the digester with less capacity. This unit works either
when there is an extra demand of chips or there is a sudden
failure in conveyor subsystem.
The systematic diagram and interactions among the working
components of the system are modeled using PN and are shown
in Fig. 5(a) and (b) respectively, where FSF denotes the feeding
system top failure event. Under the information extraction phase,
the data related to failure rates ðli’sÞ and repair times ðti’sÞ of the
main components of the feeding system are collected from the
present/historical records of the paper mill. The collected data is
integrated with expertise of maintenance personnel and are given
in the Table 4 [15,36,37].
6. Computational results
6.1. Parameter setting
The optimization method has been implemented in Matlab
(MathWorks) and the program has been run on a T6400 @ 2 GHz
Intel Core(TM) 2 Duo processor with 2 GB of Random Access
Memory (RAM). In order to eliminate stochastic discrepancy,
25 independent runs have been made that involves 25 different
initial trial solutions with swarm size is 20 Â (no. of variables), the
acceleration coefficients parameters c1 and c2 are taken as
c1 ¼ c2 ¼ 1:5 while inertia weight w is defined as w ¼ ðw1Àw2Þ
ððitermaxÀiterÞ=itermaxÞþw2. Here w1 ¼0.9 and w2 ¼0.4 are the
initial and final values of inertia weight respectively, itermax
represents the maximum generation number (¼150) and ‘iter’
is used generation number. The termination criterion has been set
either limited to a maximum number of generations or to the
order of relative error equal to 10À6
, whichever is achieved first.
6.2. Result and discussion
The data, given in Table 4, as collected from historical records
and opinion of field experts are imprecise and vague so are
represented in the form of triangular fuzzy numbers with
715% spread suggested by systems expertise. Based on PN
model, the minimal cut sets, as obtained by using matrix method,
are fAg, fB,Eg, fC,Eg and fD,Eg. Using these minimal cut sets,
expression for systems’ failure rate ðlsÞ and repair time ðtsÞ are
obtained using the results given in Table 2, as follows:
ls ¼ l1 þl2l5ðt2 þt5Þþl3l5ðt3 þt5Þþl4l5ðt4 þt5Þ
ts ¼
l1t1 þl2l5t2t5 þl3l5t3t5 þl4l5t4t5
ls
Using these expressions of ls and ts, the basic steps of the
PSOBLT technique have been followed, at various membership
grades, for computing the fuzzy reliability indices for the mission
time t¼10 h with left and right spread. The computed results of
the PSOBLT technique are depicted graphically in Fig. 6 for 715%
spreads along with Lambda-Tau and GABLT techniques result.
Figure shows that PSOBLT results have reduced region and small
spread in comparison of existing results. The reason behind is that
PSO gives near to the optimal solution. This suggests that DM has
smaller and more sensitive region to make more sound and
effective decision in lesser time.
6.2.1. Analysis with different spreads
For defuzzification, center of gravity method [38] is used
because it has the advantage of being whole membership function
into account for this transformation. The crisp and defuzzified
values of reliability indices at 715%, 725% and 760% spreads
are computed and compared with Lambda-Tau and GABLT results
are shown in Table 5. It shows that when uncertainty level in the
form of spread increases from 715% to 725% and further 760%,
the variation in defuzzified values for almost all the reliability
indices is not so much as shown by results of Lambda-Tau and
GABLT techniques. From Table 5, it is evident that defuzzified
values change with change in spread. For example the failure rate
Table 5
Crisp and defuzzified values for feeding system.
Reliability indices Crisp Defuzzified values at (spread)
715% 725% 760%
Failure rate 0.0275000 Lambda-Tau: 0.0296364 0.0337164 0.0809759
GABLT: 0.0270031 0.0276512 0.0316168
PSOBLT: 0.0275748 0.0276272 0.0270348
Repair time 3.1818181 Lambda-Tau: 4.8672386 9.3490486 89.1941971
GABLT: 3.1890921 3.1787921 3.5311456
PSOBLT: 3.1881448 3.1980623 3.2053274
MTBF 39.5454545 Lambda-Tau: 44.6454958 55.8693987 101.5435504
GABLT: 42.6605303 55.5885769 58.8732983
PSOBLT: 39.4447996 39.7245899 42.29707718
ENOF 0.2591351 Lambda-Tau: 0.3072978 0.5114377 70.8981951
GABLT: 0.2536513 0.2581375 0.2915676
PSOBLT: 0.2596373 0.2590009 0.2582328
Availability 0.9221779 Lambda-Tau: 0.8713598 0.7824887 0.5861624
GABLT: 0.9243923 0.9232263 0.9082149
PSOBLT: 0.9220529 0.9226205 0.9258657
Reliability 0.7595721 Lambda-Tau: 0.7486219 0.7296919 0.6209493
GABLT: 0.7607065 0.7554107 0.7477982
PSOBLT: 0.7585523 0.7589268 0.7673566
H. Garg, S.P. Sharma / ISA Transactions 51 (2012) 752–762 759

9. of the system increases by 7.768%, 1.806%, 0.272% for fuzzy
Lambda-Tau, GABLT and PSOBLT respectively, when spread
changes from 715% to 725%, and it further increases by
13.766%, 2.401%, 0.190%, when spread changes from 725% to
760%. Based on results shown in Fig. 6, changes in defuzzified
value for all the techniques from crisp results have been com-
puted and given in Table 6 and concluded that variation in the
PSOBLT technique is smaller than existing Lambda-Tau and
GABLT techniques. Due to their reduced region of prediction,
the value obtained through PSOBLT technique may be beneficial
for system expert/analyst for future course of action, i.e. now the
maintenance will be based on the defuzzified values rather than
crisp values.
6.2.2. Behavior analysis
To analyze the impact of change in values of reliability indices
on to the system’s behavior, behavioral plots have been plotted
for different combination of reliability indices and are shown in
Fig. 7. Throughout the combinations, ranges of repair time and
ENOF are fixed and have been varied, along the x- and y-axes
0
5
10
15
0
0.2
0.4
0.6
0.8
0
100
200
300
400
Repair time
Reliability = 0.64, Failure rate = 0.017,
Availability = 0.74
ENOF
MTBF
2.5
3
3.5
4
0
0.2
0.4
50
100
150
200
Repair Time
Reliability=0.64, Failure Rate=0.017, Availability=0.74
ENOF
MTBF
2
2.5
3
3.5
0.2
0.22
0.24
0.26
80
90
100
110
Repair time
Reliability = 0.64, Failure rate=0.017,
Availability = 0.74
ENOF
MTBF
0
5
10
15
0
0.2
0.4
0.6
0.8
0
50
100
150
200
Repair time
Reliability = 0.64, Failure rate = 0.029,
Availability = 0.74
ENOF
MTBF
2.5
3
3.5
4
0
0.2
0.4
40
60
80
100
Repair Time
Reliability=0.64, Failure Rate=0.029, Availability=0.74
ENOF
MTBF
2
2.5
3
3.5
0.2
0.22
0.24
0.26
50
55
60
65
Repair time
Reliability = 0.64, Failure rate = 0.029,
Availability = 0.74
ENOF
MTBF
0
5
10
15
0
0.2
0.4
0.6
0.8
0
50
100
150
200
Repair time
Reliability = 0.64, Failure rate = 0.041,
Availability = 0.74
ENOF
MTBF
2.5
3
3.5
4
0
0.2
0.4
20
40
60
80
Repair Time
Reliability=0.64, Failure Rate=0.041, Availability=0.74
ENOF
MTBF
2
2.5
3
3.5
0.2
0.22
0.24
0.26
0.28
40
42
44
46
48
50
Repair time
Reliability = 0.64, Failure rate = 0.041,
Availability = 0.74
ENOF
MTBF
Fig. 7. Plots feeding unit behavior analysis: (a) Lambda-Tau; (b) GABLT; (c) PSOBLT.
Table 6
Change in defuzzified values of reliability indices.
Change in value (in %) from crisp to Reliability indices
Failure rate Repair time MTBF ENOF Availability Reliability
Lambda-Tau 7.768727 52.970359 12.896656 18.585942 5.510661 1.441627
GABLT 1.806909 0.228611 7.877203 2.116193 0.240127 0.149347
PSOBLT 0.272000 0.198839 0.254529 0.193798 0.013554 0.134259
H. Garg, S.P. Sharma / ISA Transactions 51 (2012) 752–762760

10. respectively, in the range computed by their membership func-
tions (Fig. 6(b) and (d)) at cut levels a ¼ 0. The effects on MTBF by
taking different combinations of the remaining parameters (relia-
bility, failure rate and availability) are computed and have been
shown along the z-axis. For instance, in the first three plots, the
reliability and availability are fixed to 0.64 and 0.74 respectively
while the failure rate changes from 0.017 to 0.029 and further to
0.041. The corresponding effects on MTBF for all the techniques
are shown graphically in Fig. 7.
It may be observed that for this combination the prediction
range of MTBF is reduced almost by 69.8064% and 94.0707% from
fuzzy Lambda-Tau when GABLT and PSOBLT techniques are
applied respectively while reduced by 80.3622% from GABLT
when PSOBLT technique is applied. The computed range of MTBF
for all the combinations as well as for all the techniques are
tabulated in Table 7. The plots show that as the failure rate of the
system increases then for the prescribed ranges and values of the
other indices, MTBF of the system decreases exponentially as
shown in Table 7. This observation infers that if system analysts
use PSOBLT results for the system then they may have less range
of prediction which finally leads to more sound decisions. Thus,
based on the behavioral plots and corresponding table, the system
manager can analyze the critical behavior of the system and plan
for suitable maintenance.
7. Conclusion
This paper presents a novel technique named as PSOBLT for
determining the membership function of the reliability indices of
complex repairable industrial system having lesser uncertainty.
Major advantage of the proposed technique is that it gives
compressed search space for each computed reliability index by
utilizing available information and uncertain data. The technique
has been demonstrated through an example of feeding unit of a
paper mill. This technique optimize the spread of the reliability
indices indicating higher sensitivity zone and thus may be useful
for the reliability engineers/experts to make more sound deci-
sions. Also, it is observed from the analysis that PSOBLT performs
consistently well in comparison to other existing techniques.
If system analysts use PSOBLT results then they may predict the
system behavior with more confidence. Thus, it will facilitate the
management in reallocating the resources, making maintenance
decisions, achieving long run availability of the system, and
enhancing the overall productivity of the paper industry.
In nutshell, the important managerial implications drawn
using the discussed techniques are to:
model and predict the behavior of industrial systems in more
consistent manner;
to analyze the behavior of the system 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 per-
formance and to reduce operation and maintenance costs.
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