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Advances in Fuzzy Mathematics.
ISSN 0973-533X Volume 12, Number 3 (2017), pp. 747-762
© Research India Publications
http://www.ripublication.com
Application of FEWMA Control Chart for
Monitoring Yarn Process in the Textile Industry
S. Subbulakshmi1, A. Kachimohideen2 and R. Sasikumar3
1 Department of Statistics, Dr. MGR Janaki College of Arts and Science for Women,
Chennai – 28, Tamil Nadu, India.
2Department of Statistics, Periyar E.V.R. College, Tiruchirappalli, Tamil Nadu, India.
3 Department of Statistics, Manonmaniam Sundaranar University, Tirunelveli, Tamil Nadu, India.
Abstract
A process that uses statistical techniques to observe and control product
quality is called Statistical Quality Control (SQC), where control charts are
test tools commonly used for monitoring the manufacturing process.
Statistical Process Control (SPC) is a process to get better the quality of
products and lessen rework and scrap, so that the quality and productivity
prospect can be met. Control charting is the most importantpart ofSPC. One
of themost important control charts is Exponentially Weighted Moving
Average (EWMA) to detect the small shifts. But uncertainty in human
cognitive processes makes the traditional control charts not be appropriate,
since they give only particular information. Thus, Fuzzy Exponentially
Weighted Moving Average (FEWMA) control chart which is accustomed to
use when the statistical data under concern are unsure or hazy.In this paper,
FEWMA control chart is applied tomonitor the yarn process in the textile
industry.
Keywords: Statistical Quality Control, Statistical Process Control,
Exponentially Weighted Moving Average, Control Chart, Fuzzy
Transformation Technique.
748 S. Subbulakshmi, A. Kachimohideen and R. Sasikumar
1. INTRODUCTION
Statistical Quality Control (SQC) and development is a branch of industrial statistics
which includes primarily the areas of acceptance sampling, statistical process
monitoring and control (SPC), design of experiments and capability analysis. Most
SQC research has focussed on precision.The manufacturingsector plays a very
important role in promoting the economic development and pushing the development
forward. So the manufacturing sector for making the right decision needs to have a
scientific way for increasing the qualification of the production processes. One way to
get this is by applying Quality Control.SPC is employed to monitor production
processes over time to detect changes in process performance. The basic
fundamentals of SPC and control charting were proposed by Walter A Shewhart in
the 1920’s and 1930’s. Until the mid to late 1970’s there were many important
advances but relatively few individuals conducting research in the area compared to
other areas of applied statistics. Research activity has greatly increased since about
1980 onwards. Much of the increase in interest was due to the quality revolution,
which was caused by an increasingly competitive international market place.
Improvements in quality were required for survival in many industries. SPC methods
are developed and applied largely in the discrete parts industries.Montgomery
provides an excellent discussion about SPC procedure in the manufacturing
industry.A control chart is a valuable statistical tool that aids practitioners in
statistically controlling and monitoring one or more variables when the quality of the
product or the quality of the process is characterised by certain values of these
variables. In general a control chart is very easy to be implemented in any type of
process. Thus control charts areextensively used in manufacturing area nowadays
preserving the quality of the process or the final product. A control chart is a chart
thatshows whether a sample of data falls inside the common or normal range of
variation.A control chart has upper and lower control limits that divide common
fromassignable causes of variation. The general range of variation is defined by the
utilize ofcontrol chart limits. A process is out of controlwhen a plot of data revealsthat
one or more samples fall outside the control limits.In this paper, we usedFEWMA
control chart for monitoring the yarn process. The EWMA utilizesall previous
observations, but the weight close to data is exponentiallywaning as the observations
get older and older. By changing the parameterof the EWMA statistic the `memory' of
the EWMA control chartcan be inclined.
2. REVIEW OF LITERATURE
Statistical theory began to be function effectively to quality control in the year 1920
and in1924 Shewhart prepared the first sketch of a fresh control chart. His attempt
was later developed by Deming and the before time work of Shewhart, Deming,
Dodge and Roming constitutes a great deal of what today comprises the theory of
Application of FEWMA Control Chart for Monitoring Yarn Process.. 749
SPC. In 1931 Walter A. Shewhart, print a book about the fundamental concept of
statistical control and sketch out five economic advantagesaccessible through
statistical control of worth manufactured product.The main impetus for implementing
SPC in manufacturing processes is need for higher and stable quality. Garvin (1987)
defined quality in various dimensions: level of performance, reliability, durability,
serviceability, aesthetic, features, perceived quality and conformance to standards.
Montgomery (2009) elucidated an important modern definition that
“Qualityimprovement is the reduction of variability in process and products. In this
view, SPC is seen as a mechanism for controlling variables.Wang and Raz (1990)
exemplify two approaches for constructing variable control charts based on linguistic
data. Roberts (1959) introduced the control chart based on the EWMA. Most current
references include Hunter (1986), Crowder (1987) and Lucas and Saccucci
(1990).The survivalof fuzzy uncertainty in manufacturing system is an
indisputabletruth. In a global market, companies must deal with a high rate of
changes in business environment. The parameters, variables and restrictions of the
production system are naturallyvagueness. Such ahonestidentificationcertainly
broughtfuzzy mathematics initiated by Zadeh(1965)defined a fuzzy set as a class of
objects with grades membership.Raz and Wnag (1990) proposed an approach based
on the fuzzy set theory by assigning a fuzzy set to each linguistic term. El-Shal and
Morris (2002) described a research to make use of of fuzzy logic to alter SPC rules,
with the plan of sinking the cohort of false alarms.
3. FUZZY TRANSFORMATION TECHNIQUES
With reference to Wang and Raz (1990) fuzzy transformation techniques are worn to
convert the fuzzy numbers into crunchy values. The four fuzzy measures of central
tendency, fuzzy mode, α- level fuzzy midrange, fuzzy median and fuzzy average
which famous in descriptive statistics, are given below:
The fuzzymode(fmode) :The membership function of the value of the base variable of a
fuzzy set equals 1 in the fuzzy mode. That is fmode = {x|μF(x) =1}, ∀x∈F. Also, Fuzzy
mode is unique if the membership function is unimodal.
The α-level fuzzy midrange(mrf α
): F α, the average of the end points of a α-cut is
a non fuzzy subset of the base variable x containing all the values with membership
function values greater than or equal to α. Thus F α={𝑥|
Fx αμ }. If aα
and
cα are end points of α – cut F α
such that aα=Min { F α
} and cα=Max { F α
},
then, mrf α
= 1
2( aα
+ cα)
750 S. Subbulakshmi, A. Kachimohideen and R. Sasikumar
The fuzzy median(fmed):Fuzzy median is the point which divides the curve of the
fuzzy set in to two equal regions under the membership function satisfying the
following equations:
med
F
f
a
x dx μ = med
F
c
f
x dx μ =1
2
F
c
a
x dxμ Where a and c are the points in the base
variable of the fuzzy set F such that a is less than c.
The fuzzy average(favg): According to Zadeh, the fuzzy average is
favg=Av(x;F)=
1
0
1
0
F
F
x dx
x dx
x
α
α
μ
μ
It should be noted that there is no hypothetical basis following any one particularly or
the choiceamong them. In general, when the membership function is nonlinear the
first two methods are easier to calculate than the last two. The fuzzy mode may go
ahead to biased results when the membership function is tremendously asymmetrical.
The fuzzy midrange is stretchier because one can choose different levels of
membership (α) of interest. If the area under the membership function is measured to
be an appropriate measure of fuzziness, the fuzzy median is appropriate.
Figure 1: Triangular fuzzy numbers
4. FUZZY EXPONENTIALLY WEIGHTED MOVING AVERAGE CONTROL
CHART
The process was monitored by the control charts. Fuzzy control charts are commonly
used when data are covered by ambiguity and vagueness for providing litheness on
control limits to put off false alarms. The EWMA is chosen to perceive small shifts in
the process. The traditional EWMA control chart was introduced by Roberts and
Hunter details of theses charts are as below:
Application of FEWMA Control Chart for Monitoring Yarn Process.. 751
Zt= λtX + 1 Zt-1where Ztis the tthexponentially weighted moving average,
tX
denotes the tth sample average, 0< λ≤1 is a constant, X is the overall mean, m is
thesamplenumber and t=1, 2,... m.Z0= X , If iX are independent random variables
with variance σ2/n (σ is the population standard deviation and known), then the
variance of Zt isσzt2=
2
n
2
[
21 1
t ] where n is the sample size.As t
increases, σzt2 increases to a limiting value:σz=
2
2n
. If the sample number t is
moderately large, the traditional EWMA control chart is given as follows: UCL
EWMA= X +3 n
2
, CLEWMA= X , LCLEWMA= X -3
n
2
. For small t,
the traditional EWMA control chart is given as follows:
UCL EWMA= X +3 n
2
[1 1 ]2
t
CLEWMA= X
LCLEWMA= X -3 n
2
[1 1 ]2
t
If σ is estimated from sample, R is used for constructing tradition EWMA control
chart like following:
UCL EWMA= X +A2 R2
CLEWMA= X
LCLEWMA= X - A2 R2
where R is the average of the Ri’s while Ri is a rangefor each sample.
4.1 FEWMA Control Charts When (σa, σb, σc) are Known
First calculate aR , bR , cR , the arithmetic means of the least possible values, the
most possible values and the largest possible values of an elongation of a cotton in a
yarn process respectively.(Ra1, Rb1 , Rc1) and( aR , bR , cR ) are obtained as
follows:Firstly, (Raj, Rbj , Rcj ) values are calculated, where Raj=max{Xaj}-min{Xcj},
752 S. Subbulakshmi, A. Kachimohideen and R. Sasikumar
Rbj =max{Xbj}-min{Xbj}, Rcj=max{Xcj}-min{Xaj}and max{Xij} is the maximum of
fuzzy numbers in the sample and min{Xij} is the minimum of fuzzy numbers in the
sample.
Table 1: Fuzzy Number Representation of the samples
SAMPLE xa xb Xc R
1
1 10.21 10.22 8.16
Ra1=1.79
Rb1=1.81
Rc1=1.85
2 10.94 10.95 9.12
3 9.8 9.81 9.93
4 10.24 10.26 9.42
5 9.34 9.36 11.34
6 10 10.01 10.83
7 9.78 9.79 10.16
8 9.26 9.28 10.25
9 9.65 9.66 9.71
10 9.12 9.14 9.99
1aX =9.834
1bX =9.85
1cX =9.86
2
1 10.21 9.75 9.56 Ra2=2.45
Rb2=2.47
Rc2=2.5 2 10.94 8.62 9.68
3 9.8 9.09 10.35
4 10.24 9.63 10.5
5 9.34 10.23 9.56
6 10 10.53 9.75
7 9.78 9.28 9.63
8 9.26 11.09 11.37
9 9.65 9.71 11.13
10 9.12 9.52 10.69
2aX =9.729
2bX =9.75
2cX =9.76
...
... ... ... ...
1 9.56 9.57 9.58
Ra25=1.41
2 10.22 10.24 10.25
3 9.65 9.68 9.69
4 9.26 9.27 9.28
Application of FEWMA Control Chart for Monitoring Yarn Process.. 753
3 5 9.49 9.5 9.51 Rb25=1.43
Rc25=1.45
6 10.05 10.06 10.07
7 8.93 8.94 8.95
8 10.36 10.37 10.38
9 10.14 10.15 10.16
10 9.48 9.49 9.51
25aX =9.714
25bX =9.73
25CX =9.74
aR =2.1
bR =2.13
cR =2.13 aX =9.78
bX =9.80
cX =9.81
When detecting the small shift in process with fuzzy observations,the FEWMA
control chart should be used to evaluate the process. Fuzzy observations (Xa1,Xb1,Xc1)
are collected from process when the fuzzy observations are represented by triangular
membership function with sample size n. (atX ,
btX ,ctX )represents the fuzzy
average of rth sample(Table 2).
Where z᷉0= (za0, zb0, zc0) = (aX ,
bX , cX )While knowing the fuzzy standard
deviations, the fuzzy averages, fuzzy standard deviations and λ are used to construct
the fuzzy EWMA control chart.If the sample number t is moderately large, the fuzzy
EWMA control chart is given as follows:
U ~
C LEWMA= ( aX ,
bX , cX ) + 3
n(σa,σb, σc)
2
=
aX + 3 a
n
2
,
bX + 3 b
n
2
,
cX + 3 c
n
2
~
C LEWMA= ( aX ,
bX , cX )
L ~
C LEWMA= (aX ,
bX , cX ) _ 3
n(σa,σb, σc)
2
=
aX _ 3 a
n
2
,
bX _ 3 b
n
2
,
cX _
3 c
n
2
754 S. Subbulakshmi, A. Kachimohideen and R. Sasikumar
If the sample number t is small following equations are obtained:
U ~
C LEWMA= ( aX ,
bX , cX ) + 3
n(σa,σb, σc)
2
[
21 1
t ]=
aX + 3 a
n
2
[
21 1
t ],
bX + 3 b
n
2
[
21 1
t ],
cX + 3 c
n
2
[
21 1
t ]
~
C LEWMA= (aX ,
bX ,cX )
L~
C L= ( aX ,
bX , cX ) _ 3
n(σa,σb, σc)
2
[
21 1
t ]=
aX _ 3 a
n
2
[
2
1 1t
], bX _ 3 b
n
2
[
21 1
t ],
cX _ 3 c
n
2
[
21 1
t ]
Table 2: Fuzzy averages and fuzzy exponentially weighted moving averages
T X t
~
Z t
1
2
3
...
M
(1aX ,
1bX ,1cX )
(2aX ,
2bX , 2cX )
(3 ,aX 3bX ,
3 cX )
...
( amX , bmX , cmX )
~
1Z =λ(1aX ,
1bX ,1cX )+(1-λ) (
aX ,bX ,
cX )
~
2Z =λ(2aX ,
2bX , 2cX )+(1-λ)
~
1Z
~
3Z =λ(3 ,aX 3bX ,
3 cX )(1-λ) ~
2Z
... ~
mZ =λ( amX , bmX , cmX )+(1-λ) 1
~
mZ
4.2 α-cuts FEWMA control charts when (σa, σb, σc) are known
An α-cuts is a restricted fuzzy set which includes the elements whose membership
degrees are greater than equal to α.After applying the α-cuts on means and standard
deviations, the α-cuts fuzzy overall means and α-cuts fuzzy standard deviation are
calculated as follows, respectively:
aX
=aX + α ( bX - aX ),
cX
= cX - α ( cX - bX ) and a
= a + α( b – a ),
c
= c - α ( c - b )
Application of FEWMA Control Chart for Monitoring Yarn Process.. 755
If the sample number t is moderately large, the α-cuts fuzzy EWMA control chart is
given as follows:EWMAUCL
= (aX
, bX ,cX
)+ 3
n(
a
, b , c
)2
=
aX
+ 3 a
n
2
,
bX +3 b
n
2
,
cX
+3 c
n
2
EWMACL
= (aX
, bX ,cX
)
EWMALCL
= (aX
, bX ,cX
)- 3
n(
a
, b , c
)2
=
aX
- 3 a
n
2
, bX -
3 b
n
2
,
cX
-3 c
n
2
If the sample number t is small, the α-cuts FEWMA control chart is obtained by the
following equations:
EWMAUCL
=(aX
, bX ,cX
)+ 3
n(
a
, b , c
) 2
[1 ( )]2
1t
=aX
+ 3 a
n
2[1 ( )]
21
t
, bX +3 b
n 2
[1 ( )]2
1t
,cX
+3 c
n
2[1 ( )]
21
t
EWMACL
= (aX
, bX ,cX
)
EWMALCL
= (aX
, bX ,cX
)- 3
n(
a
, b , c
)2
[1 ( )]2
1t
=aX
-3 a
n
2[1 ( )]
21
t
, bX -3 b
n 2
[1 ( )]2
1t
, cX
-3 c
n
2[1 ( )]
21
t
756 S. Subbulakshmi, A. Kachimohideen and R. Sasikumar
4.3 α-level fuzzy median for α-cuts FEWMA control chart for (σa, σb, σc) are
known
The α-level fuzzy median transformation techniques is applied on α-cuts FEWMA
control charts for obtaining the crisp values of control limits. The α-level fuzzy
median for α-cuts FEWMA control chart is obtained for the sample number t is
moderately large and t is small as follows, respectively;
med EWMAUCL
=
med EWMACL
+
1
n(
a
, b , c
)2
med EWMACL
=
1
3(
aX
, bX ,cX
)
med EWMALCL
=
med EWMACL
-
1
n(
a
, b , c
)2
med EWMAUCL
=
med EWMACL
+
1
n(
a
, b , c
)2
[1 ( )]2
1t
med EWMACL
=
1
3(
aX
, bX ,cX
)
med EWMALCL
=
med EWMACL
-
1
n(
a
, b , c
)2
[1 ( )]2
1t
While evaluating
the sample with FEWMA control chart, we can calculate the α-level fuzzy median
value.
,med EWMA jS
=
, , ,
1( )
3 a j b j c jX X X
FEWMA control chart for unknown standard deviations are calculated for yarn
process data as follows:
U~
C LEWMA= aX + A2 aR
2
,
bX +A2 bR2
,
cX + A2 cR2
aX + A2 aR2
= 9.78+0.577*2.1 0.2
2 0.2=10.18
bX +A2 bR2
= 9.8+0.577*2.13 0.2
2 0.2=10.21
cX + A2 cR2
= 9.81+0.577*2.13 0.2
2 0.2=10.22
Application of FEWMA Control Chart for Monitoring Yarn Process.. 757
U~
C LEWMA=(10.18, 10.21, 10.22)
C�̃�EWMA =(aX ,
bX , cX ) = (9.78, 9.80, 9.81)
C�̃�EWMA = (9.78, 9.80, 9.81)
L~
C LEWMA = aX - A2
cR2
,
bX - A2bR 2
,
cX - A2aR
2
aX - A2cR
2
= 9.78 -0.577*2.13 0.2
2 0.2=9.37
bX - A2bR 2
=9.80 -0.577*2.13 0.2
2 0.2=9.39
cX - A2aR
2
= 9.81 -0.577*2.1 0.2
2 0.2=9.41
L~
C LEWMA = (9.37, 9.39, 9.41)
where λ=0.2due to general approach in production process.
aX
, cX
and aR, cR
are calculated by using the following equations, where
α=0.65. Because there are quite a lot of applications in literature in which α-cuts is
preferred 0.65 for the manufacturing process.
aX
=aX + α( bX - aX ) =9.78+0.65(9.80-9.78) = 9.79
cX
= cX - α( cX - bX ) = 9.81-0.65(9.81-9.80) = 9.80
aR =aR +α (
bR -aR ) =2.10 + 0.65 (2.13 – 2.10) = 2.12
cR =cR -α(
cR -bR ) =2.13 - 0.65 (2.13 – 2.13) = 2.13
The limits of α-cuts FEWMA control chart are given as follows for yarn process:
EWMAUCL
= aX
+ A2aR
2
,
bX +A2 bR2
,
cX
+ A2 cR
2
aX
+ A2aR
2
= 9.79 + 0.577 * 2.12 0.2
2 0.2= 10.20
bX +A2 bR2
= 9.80+0.577*2.13 0.2
2 0.2 =10.21
758 S. Subbulakshmi, A. Kachimohideen and R. Sasikumar
cX
+ A2 cR
2
=9.80+0.577*2.13 0.2
2 0.2 = 10.21
EWMAUCL
= (10.20, 10.21, 10.21)
EWMACL
= (aX
, bX ,cX
) =(9.79, 9.80, 9.80)
EWMACL
= (9.79, 9.80, 9.80)
EWMALCL
=aX
- A2 cR
2
,
bX -A2 bR2
,
cX
- A2 aR
2
aX
- A2 cR
2
=9.79-0.577*2.13 0.2
2 0.2 = 9.38
bX - A2 bR2
=9.80-0.577*2.13 0.2
2 0.2 = 9.39
cX
- A2 aR
2
=9.80-0.577*2.12 0.2
2 0.2 = 9.39
EWMALCL
= (9.38, 9.39, 9.39)
Fuzzy median transformation technique is integrated to the α-level fuzzy median for
α-cutFEWMA control chart as follows:
med EWMAUCL
= med EWMACL
+
1
3A2(
aR + bR +cR )
2
=9.80+1
3*0.577(2.12+2.13+2.13) 0.2
2 0.2 = 10.21
med EWMACL
=
1
3(
aX
, bX ,cX
) =1
3(9.79+9.80+9.80)=9.80
med EWMALCL
= med EWMACL
-
1
n( a
, b , c
)2
=9.80-1
3*0.577(2.12+2.13+2.13) 0.2
2 0.2= 9.39
For each sample, α-level fuzzy median value (,med EWMA jS
) is calculated. α –cuts for
averages for the 25 samples are calculated as follows:
Application of FEWMA Control Chart for Monitoring Yarn Process.. 759
65
,1aX = ,1aX +0.65(
,1bX -,1aX ) =9.83+0.65(9.85-9.83) = 9.84
65
,2aX = ,2aX +0.65(
,2bX -,2aX )=9.73+0.65(9.75-9.73) =9.74
...
0.65
,25aX = ,25aX + 0.65 (
,25bX -,25aX ) +9.75+0.65(9.73-9.71)=9.72
0.65
,1cX = ,1cX + 0.65 (
,1cX -,1bX ) =9.86+0.65 (9.86 – 9.85) = 9.87
0.65
,2cX = ,2cX + 0.65 (
,2cX - ,2bX ) = 9.76+0.65(9.76-9.75)=9.77
...
0.65
,25cX = ,25cX +0.65(
,25cX -,25bX )=9.74 + 0.65 (9.74 - 9.73) = 9.75
After calculating fuzzy number with α-cuts for averages for the first sample, α-level
fuzzy median value is obtained. Since this value is between control limits, the first
sample is in control. Also the condition of process control for each sample is defined
by using the following:
Process control = ,,
,
med EWMA med EWMA j med EWMAin control for
out ofcontrol otherwiseLCL S UCL
and
given in table.
,1med EWMAS
=
,1 ,1 ,1
1( )
3 a b cX X X
=1
3(9.84+9.85+9.87) = 9.85
,2med EWMAS
=
,2 ,2 ,2
1( )
3 a b cX X X
=1
3(9.74 + 9.75 + 9.77) =9.75
...
,25med EWMAS
=
,25 ,25 ,25
1( )
3 a b cX X X
=1
3(9.72 + 9.73 + 9.75)=9.73
760 S. Subbulakshmi, A. Kachimohideen and R. Sasikumar
Table 3: Control limits of FEWMA, α-levelfuzzy median value and the process
conditions.
sample ,med EWMA jS
9.39<Sα<10.21
1 9.85 In control
2 9.75 In control
3 9.55 In control
4 9.82 In control
5 9.80 In control
6 9.75 In control
7 9.66 In control
8 9.89 In control
9 10.22 Out of control
10 10.07 In control
11 9.85 In control
12 10.00 In control
13 9.88 In control
14 9.82 In control
15 9.81 In control
16 9.84 In control
17 9.72 In control
18 9.71 In control
19 9.60 In control
20 9.30 Out of control
21 9.71 In control
22 10.08 In control
23 9.8 In control
24 9.91 In control
25 9.73 In control
Application of FEWMA Control Chart for Monitoring Yarn Process.. 761
As seen in Table 3, the yarn process in a textile industry is “out of control” due to the
nineth and twentieth sample. Even though twenty three samples reveal an under
control process, two samples indicates an assignable causes. So, the production
process is out of control. The assignable causes for this shift should be searched and
after removing this cause, the process can run again.
5. CONCLUSION
In various real-world exertions, data are hazy and vague; the hypothesis of crunchy
profiles for processes is not sensible. Fuzzy set theory is acompetent tool to tackle this
limitation. In this paper, a new system of identifying small changes of process profile
has been planned in which profile parameters were assumed hazy and vague. For this
purpose, we have used FEWMA control charts and discussed the competence of yarn
process in a textile industry. The results have shown that this method was highly
capable in discovering assignable causes in profiles.
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