Application of FEWMA Control Chart for Monitoring Yarn Process in the Textile … · 2017-06-17 · Application of FEWMA Control Chart for Monitoring Yarn Process in the Textile Industry

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)


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:


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},


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


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


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 )


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


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


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


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:


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


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


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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