Performance Evaluation of Production System Using Lambda Tau Methodology -A Case from Paper Industry

Abstract

In this paper a methodology is applied which uses Petri nets and are evaluated for reliability indices to ascertain the performance of the system using fuzzy Lambda-Tau Method. Fuzzy set theory is used to represent the failure rate and repair time instead of classical (crisp) set theory because fuzzy numbers allow expert opinions, uncertainty and imprecision in the reliability information to be incorporated into the system model. Various Performance Parameters of Interest such as Failure Rate, Repair Time, Mean Time between Failures, Reliability, Unreliability, Availability and Unavailability are evaluated at different values of Degree of Membership Started from 0 to 1 and at increment of 0.1 to quantify the behavior in terms of fuzzy, crisp and defuzzified values. The centriod method of defuzzification is used to represent fuzzy output into crisp values at different spread i.e. 15%, 25% and 40%. The application of the described methodology is applied on a real system of a paper industry consists of Feeding system and Paper Production system.

Full text

3rd International Conference on Production and Industrial Engineering
CPIE-2013
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Performance Evaluation of Production System Using Lambda Tau
Methodology – A Case from Paper Industry
1Mr. Rajesh Babbar, 2Dr. Anish Sachdeva, 3Mr. Ashish Bhateja
1 Assistant Professor, P.D.M School of Technology & Management, Bahadurgarh, Haryana
(rajesh.babbar85@gmail.com)
2Associate Professor, National Institute of Technology Jalandhar, Punjab
3Assistant Professor, Gulzar Group of institutions Khanna, Punjab
Abstract: In this paper a methodology is applied
which uses Petri nets and are evaluated for
reliability indices to ascertain the performance of
the system using fuzzy Lambda – Tau Method.
Fuzzy set theory is used to represent the failure rate
and repair time instead of classical (crisp) set theory
because fuzzy numbers allow expert opinions,
uncertainty and imprecision in the reliability
information to be incorporated into the system
model. Various Performance Parameters of Interest
such as Failure Rate, Repair Time, Mean Time
between Failures, Reliability, Unreliability,
Availability and Unavailability are evaluated at
different values of Degree of Membership Started
from 0 to 1 and at increment of 0.1 to quantify the
behavior in terms of fuzzy, crisp and defuzzified
values. The centriod method of defuzzification is
used to represent fuzzy output into crisp values at
different spread i.e. 15%, 25% and 40%. The
application of the described methodology is applied
on a real system of a paper industry consists of
Feeding system and Paper Production system.
1. INTRODUCTION
Reliability is a popular concept that has been
celebrated for years as a commendable attribute of a
person or an artifact. The Oxford English Dictionary
defines it as „the quality of being reliable, that may be
relied upon; in which reliance or confidence may be
put; trustworthy, safe, sure. According to the
International Electro technical Vocabulary, failure is
defined as‟ the termination of the ability of an item to
perform the required function. It is nearly an
unavoidable phenomenon in mechanical
system/components. One can observe various kinds
of failure in past under various circumstances such as
nuclear explosions, 1986, Industrial plant leakages,
Union carbide plant, Bhopal 1984, Oil pipeline
leakage at Jesse Nigeria 1988 and electrical network
shutdown etc. which may be due to human error,
poor maintenance, inadequate testing/inspection.
1.1 The Use of Fuzzy Logic to Reliability Analysis: The
use of fuzzy logic and fuzzy arithmetic to determine
component or system reliability can be easily found.
It is well known fact that the most databases, on
which most of the reliability analysis depends, are
either out of data or collected under different
operating and environmental conditions. J. Knezevic
et al. [1] developed a methodology which uses Petri
nets instead of fault tree methodology and solve for
reliability indices utilising fuzzy Lambda Tau
method. Fuzzy set theory is used for representing the
failure rate and repair time instead of the classical
(crisp) set theory. Because fuzzy numbers gives

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expert opinions, linguistic variables, operating
conditions, uncertainties and imperfection in
reliability information to be incorporated into the
system model. T.S Liu & S.B. Chiou [2] presented a
Petri net approach to failure analysis instead of fault
tree analysis that has been widely applied to the
system failure analysis in reliability engineering. It is
essentially a graphical method for describing
relations between conditions and events. The use of
Petri net in failure analysis enables to replace logic
gate function in fault tree, efficiently obtain minimal
cut sets, and absorb models. Wang [3] investigated
that Centroid defuzzification and the maximizing set
and minimizing set methods are two commonly used
approaches to ranking fuzzy numbers and often
require membership functions to be known. In that
paper, the two methods are reinvestigated when
explicit membership functions are not known but
alpha level sets are available.
1.2 Releavent Concept of Fuzzy Set Theory to
Reliability analysis: Classical sets contain objects
that satisfy precise properties of membership. Fuzzy
sets on the other hand, contains objects that satisfy
imprecise properties of membership i.e. membership
of an object in a fuzzy set can be partial. For classical
sets, an element x in a universe U is either a member
of some crisp set A or it is not. This binary issue of
membership can be represented mathematically by
the indication factor: 1    
    
Notion of binary membership to accommodate
various degree of membership on the real continuous
interval, where the endpoints of 0 and 1 confirm
to no membership and full membership respectively
Just as the indicator function does for crisp sets, the
infinite number of values in between the end points
can represent various degree of membership for an
element x in some set of the universe U. The sets of
the universe U that can be accommodating degree of
membership were termed as fuzzy set. Hence a fuzzy
set can be represented by a functional mapping as
(x)  [0, 1], Where (x) is the degree of
membership of element x in fuzzy set or simply
membership function of , the value (x) is on the
unit interval that measures the degree to which x
belongs to fuzzy set
(x) = The degree to which x  . The larger (x)
is, the stronger the degree of belongingness for x in
A. In other words, a fuzzy subset of a universe of
discourse U = {x} is defined as a mapping by which
x is assigned a number in {0, 1}. This indicates the
extent to which x belongs to A membership
function (M.F) is a curve that defines how each point
in the input space is mapped to a membership value
(partial truth) between 0 and 1.There are various
types of membership functions such as triangular,
trapezoidal, gamma, rectangular that can be used for
reliability analysis. However triangular and
trapezoidal membership functions are widely used for
calculating and interpreting reliability data. [4, 5]. A
triangular M.F is defined by a triplet (a, b, c), where
the parameter b gives maximal grade of membership
i.e.
1)(
~b
A

with a and c being lower and upper
bounds is expressed as

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(x) = (x)
0 a a (

) b b(

To resolve fuzzy sets in terms of consistuent crisp
sets, the concept of alpha cut is used which we have
discussed earlier. They are indispensable in
performing arithmetic operations with fuzzy sets. The
α cut of a fuzzy set A denoted by , is the crisp set
comprised of all the elements x of universe of
discourse for which membership function of greater
than or equal to α that is,
With introduction of α cuts, A is defined as
The confidence level defined by
alpha cut is written as
2. DESCRIPTION OF THE SYSTEM
To illustrate the application of proposed approach a
case study from paper mill is taken [6, 7]. The
production of paper depends upon the availability of
pulp, failure free operation of the system, availability
of a repair facility to the system and reliability of
each piece of equipment. There are two main
functional units in the paper mill.
2.1 Feeding System: The paper mill has a chipping
house in which the wood is chipped and stored. The
chipped wood is carried to a digester for preparing
the pulp. The processing system, called the feeding
system, comprises three subsystems as follows:
 The blower (1), for pushing the wood chips
through the pipe by compressed air whose
failure causes failure of the feeding system;
 A standby unit (2) for carrying the chips by
compressed air from the store, but with less
capacity (this unit works either when there is
an extra demand for chips or there is a
sudden failure in the main subsystem);
 This main subsystem (SS1) consists of three
operating units in series – chain (3),
conveyor (4) and bucket conveyor (5) for
lifting the chips up to the height of the
digester (when there is a failure in this
system Unit B is switched on, which feeds
the digester slowly, causing a delay in the
digestion process and hence loos in
production.
1
3
4
SS1
5
Feeding
System
2

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Figure 1 Petri net models of Feeding System
2.2 Paper Production System: In the paper industry
raw material (chips) is fed into a digester and pulp is
prepared. The pulp, free from all foreign particles, is
stored in a tank with the help of a pump, from where
it is fed to the head box of the paper machine to
adjust the thickness of the paper. The pulp runs over
a wire mat running on three rollers (Felt, Top and
Bottom Rollers) and water from the pulp is sucked
through two vacuum pumps arranged in parallel. The
wet water passes through heated rollers together with
a synthetic belt and is dried in drier. The paper
formation system consists of four subsystems:
 The wire mat (6), which provides width for
the paper and helps in removing the water.
Its failure cause complete failure of the
system.
 The rollers (SS2) consists of three rollers,
(Felt, Top and Bottom i.e. 8, 9, 10) in series,
which help the wire mat and the synthetic
belt which is rolling upon them. Failure of
any one cause the complete failure of the
system
 The synthetic Belt (7) which carries the
paper to the press and drier section. Its
failure cause complete failure of the system.
 The vacuum (SS3), which sucks water from
the pulp. Two vacuum pumps (11, 12) are
arranged in parallel. As proper vacuum is
required, the failure of any one is not
desirable. Failure of more than one pumps at
a time cause the complete failure of the
system
3
4
SS1
5
8
9
SS2
10
11
12
SS3
6
11
12
7
8
9
SS2
10
SS3
Paper
Production
System

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Figure 2 Petri Net of Paper Production System
Now Under the information extraction phase, the data
regarding the failure rate and repair time of the
units are required. In the present case, the data
regarding the Failure Rate and Repair Time for all the
components i.e. (i = 1 to 12) are collected and are
presented in Table 1
Table 1 Failure Rate and Repair Time Data of the Petri Net Models
Blower
Stand
By Unit
Main Sub System
(SS1)
Synthetic
Belt
Wire
Mat
Rollers
(SS2)
Vacuum
Pumps
(SS3)
i
1
2
3
4
5
6
7
8
9
10
11
12
Repair Time τ
(hrs.)
10
5
10
5
3.3
10
10
5
5
5
5
5
Failure Rate λ
(10-3 Failure/hr.)
2
3
3
4
5
1
1
1
1
1
2.5
2.5
3. RESULTS & DISCUSSIONS
For the system components, fuzzification is carried
out with the help of triangular membership function
[as shown in the Figure 3, for the first component i.e.
Blower, with ±15% spread on crisp value].
Figure 3 Fuzzification of Failure Rate & Repair Time of First Component (Blower)
After knowing the input fuzzy triangular numbers for
all the components shown in Petri net model Figure 1
and Figure 2, the corresponding fuzzy values of
for all the Components and Sub systems at different
confidence levels α ranging from 0 to 1 with an
increment of 0.1 are determined using fuzzy
transition expressions as presented here. The interval
expressions for the fuzzy triangular number of the

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basic places for the failure rate ( ) and repair time ( ) to AND transition expressions can be determined
from the following Equation 1 & Equation 2.
Similarly, The interval expressions for the fuzzy
triangular number of the basic places for the failure
rate ( ) and repair time ( ) to OR transition
expressions can be determined from the following
Equation 3 & Equation 4.
By using fuzzy transistion expression described
above Failure Rate (λ) and Repair Time (τ) for
Feeding system and Paper Produciton System are
computed and are presented here in Table 2.
Table 2 Failure Rate and Repair Time for Feeding System and paper production system
Failure Rate x 10-3 (hr-1)
Feeding System
Failure Rate x 10-3 (hr-1)
Paper Production System
Repair Time (hrs)
Feeding system
Repair Time (hrs)
Paper Production System
DOMF
L.S
R.S
L.S
R.S
L.S
R.S
L.S
R.S
1
2.380
2.380
5.062
5.062
8.824
8.824
6.944
6.944
0.9
2.328
2.434
4.985
5.140
8.326
9.352
6.550
7.362
0.8
2.276
2.489
4.907
5.2183
7.857
9.916
6.179
7.8056
0.7
2.226
2.545
4.8294
5.296
7.415
10.517
5.827
8.276
0.6
2.177
2.603
4.7519
5.374
6.997
11.160
5.495
8.775
0.5
2.129
2.662
4.674
5.453
6.601
11.850
5.182
9.306
0.4
2.081
2.723
4.597
5.531
6.227
12.593
4.885
9.869
0.3
2.035
2.786
4.519
5.609
5.871
13.393
4.605
10.469
0.2
1.990
2.851
4.442
5.688
5.534
14.260
4.340
11.108

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0.1
1.945
2.917
4.365
5.766
5.214
15.200
4.090
11.787
0
1.901
2.985
4.2883
5.845
4.910
16.226
3.853
12.512
3.2 Determination of Reliability
Parameters for Feeding system and paper
production system: After defuzzifying failure
rate and repair time values for the top place of the
feeding system and Paper Production System as
presented in Table 2, they are used to determining a
number of quantifiable parameters such as Mean
Time Between Failure, Reliability, Unreliability,
Availability, Unavailability were determined for the
system‟s TOP place using the mission time 10 hrs.
Various parameter of interest such as system
availability, Unavailability, System Reliability,
Unreliability and Mean time between failures are
computed at different alpha cuts to analyze the
behaviour of system in quantitative terms. The
summary of the fuzzy reliability parameters, for each
confidence factor i.e. α level ranging from 0 to 1 with
increments of 0.1, are presented in Table 3 and Table
4 with left and right spreads. Depending upon the
value of α, the analyst predicts the measures for the
system.
Table 3 Computed Parameters of Mean Time between Failure, Reliability, Unreliability, Availability and
Unavailability for Feeding Sustem
Mean Time
Between Failure x
102(hr)
Reliability
Un Reliability
Availability
Unavailability
DOMF
L.S
R.S
L.S
R.S
L.S
R.S
L.S
R.S
L.S
R.S
1
4.289
4.289
0.976481
0.976481
0.023519
0.023519
0.979846
0.979846
0.020154
0.020154
0.9
4.192
4.389
0.975956
0.976994
0.023006
0.024043
0.978178
0.980526
0.019474
0.021822
0.8
4.097
4.492
0.97542
0.977495
0.022505
0.024580
0.976368
0.981181
0.018819
0.023632
0.7
4.003
4.597
0.97487
0.977986
0.022014
0.025130
0.974401
0.981814
0.018186
0.025599
0.6
3.912
4.705
0.974306
0.978467
0.021533
0.025694
0.972261
0.982427
0.017573
0.027739
0.5
3.822
4.816
0.973728
0.978938
0.021062
0.026272
0.969928
0.983022
0.016978
0.030072
0.4
3.734
4.930
0.973134
0.979400
0.020600
0.026866
0.967378
0.983599
0.016401
0.032622
0.3
3.648
5.047
0.972524
0.979854
0.020146
0.027476
0.964587
0.984161
0.015839
0.035413
0.2
3.563
5.168
0.971897
0.980299
0.019701
0.028103
0.961524
0.984707
0.015293
0.038476
0.1
3.481
5.293
0.971252
0.980736
0.019264
0.028748
0.958154
0.985240
0.014760
0.041846
0
3.398
5.422
0.970588
0.981166
0.018834
0..029412
0.954436
0.985760
0.014240
0.045564

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Table 4 Computed Parameters of Mean Time between Failure, Reliability, Unreliability, Availability and
Unavailability for Paper Produciton System
Mean Time
Between Failure
x 102(hr)
Reliability
Un Reliability
Availability
Unavailability
DOMF
L.S
R.S
L.S
R.S
L.S
R.S
L.S
R.S
L.S
R.S
1
2.045
2.045
0.95063
0.950635
0.049365
0.049365
0.96715
0.96715
0.032847
0.03284
0.9
2.011
2.080
0.94989
0.951375
0.048625
0.050104
0.96468
0.96845
0.031546
0.03532
0.8
1.978
2.116
0.94915
0.952114
0.047886
0.050845
0.96204
0.96971
0.030287
0.03796
0.7
1.946
2.153
0.94841
0.952853
0.047146
0.051585
0.95921
0.97093
0.029067
0.04079
0.6
1.916
2.192
0.94767
0.953592
0.046408
0.052326
0.95620
0.97211
0.027887
0.04380
0.5
1.886
2.232
0.94693
0.954331
0.045669
0.053067
0.95297
0.97326
0.026743
0.04703
0.4
1.857
2.274
0.94619
0.95507
0.044930
0.053808
0.94953
0.97436
0.025636
0.05047
0.3
1.829
2.317
0.94545
0.955808
0.044192
0.054549
0.94585
0.97544
0.024564
0.05414
0.2
1.802
2.362
0.94471
0.956546
0.043453
0.055291
0.94192
0.97647
0.023527
0.05808
0.1
1.775
2.409
0.94397
0.957285
0.042715
0.056033
0.93773
0.97748
0.022522
0.06227
0
1.749
2.457
0.94323
0.958023
0.041977
0.056775
0.93325
0.97845
0.02155
0.06675

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3.3Defuzzification of Fuzzy Parameters: In
order to make decisions with respect to maintenance
actions it is necessary to convert fuzzy output into a
crisp value. Among the various techniques for
defuzzification such as centroid, bisector, middle of
the max, weighted average available in literature [10,
14] the centroid method [3] is used because of its
plausibility (lie approximately in the middle of the
area) and computational simplicity. To quantify the
system behaviour, the defuzzified values for the
respective system parameters are calculated at ±15%,
±25% spread, ±40% spread. The crisp value remains

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same irrespective of change in spread. Table 5
presents both crisp and defuzzified values for the unit
at different spreads for Feeding System and Paper
Production System respectively.
Table 5 Crisp and Defuzzified Values for Feeding system and Paper Production System
Feeding System
Paper Production System
System Parameter
Crisp
Value
Defuzzified
Value [15
% spread
Defuzzified
Value [25
% spread
Defuzzified
Value [40
% spread
Crisp
Value
Defuzzified
Value [15
% spread
Defuzzified
Value [25
% spread
Defuzzified
Value [40
% spread
Failure Rate x 10-3
(hr-1)
2.380
2.412
2.471
2.638
5.0625
5.0642
5.0684
5.0775
Repair Time (hr)
8.824
9.683
11.672
21.395
6.9444
7.5634
8.7976
12.7255
MTBF x 102(hr)
4.289
4.349
4.465
4.820
2.0447
2.0739
2.1294
2.8713
Availability x 10-1
9.7984
9.7418
9.6510
9.2623
9.6715
9.6056
9.5165
9.2723
Unavailability x 10-2
2.0154
2.2581
3.4896
7.3761
3.285
3.9440
4.8345
7.27704
Reliability x 10-1
9.7648
9.7617
9.7561
9.7402
9.50635
9.50629
9.50619
9.50594
Unreliability x 10-2
2.3519
2.3820
2.4388
2.5975
4.9365
4.9371
4.9380
4.9405
4 Behaviour Analysis of the system: From the
Table 5 it is evident that defuzzified value changes
with change in percentage-spread. For instance,
Repair Time first increase to 16.32% for Paper
Production system and 20.54% for Feeding System
when spread changes from ±15% to ±25% and
further to 44.65% for Paper Production System and
83.3 % for feeding system when spread changes from
±25% to ±40% with increase in spread. Similarly for
failure rate & Mean Time Between Failure increase
in defuzzified values, is observed. On the other hand,
Availability and Reliability decrease in defuzzified
values with increase in spread is observed. Thus from
the above discussions it is inferred that the
maintenance action for the system should be based on
defuzzified Reliability rather than on crisp value
because with the reduced Reliability values a safe
interval between maintenance actions can be
established and inspections (continuous or periodic)
can be conducted to monitor the condition or status of
various equipments constituting the system before it
reaches the crisp value. It can also be observed that
with increase in repair time for the unit, availability
goes on decreasing.
5 Conclusion: The fuzzy methodology can be
successfully used to evaluate and access the system
behaviour analysis. The estimation of various
parameters in terms of fuzzy, defuzzified and crisp
results not only helps the maintenance managers to
understand the behavioural dynamic of the respective
units but also depending upon the value of
confidence factor (alpha), the analyst can predict the
reliability measure for the system and take necessary

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steps to improve system performance.Thus, from the
study it is concluded that the application of an unified
approach presented in this paper help the
system/reliability analysts in the following manners.
 To analyze failure behaviour of industrial
systems in more realistic manner as they
often makes use of subjective judgments,
uncertain data and approximate system
models.
 The ability to model and deal with highly
complex systems (because fuzzy sets can
deal easily with approximations)
 The methodology so proposed in this paper
can be applied to any repairable technical
system or plant.
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