International Journal for Quality Research 10(2) 407–420 ISSN 1800-6450

407

Yerriswamy Wooluru 1

Swamy D.R.

Nagesh P.

Article info:

Received 12.10.2014 Accepted 13.01.2015

UDC – 54.061

DOI – 10.18421/IJQR10.02-11

PROCESS CAPABILITY ESTIMATION FOR NON-NORMALLY DISTRIBUTED DATA

USING ROBUST METHODS - A COMPARATIVE STUDY

Abstract: Process capability indices are very important process quality assessment tools in automotive industries. The common process capability indices (PCIs) Cp, Cpk, Cpm are widely used in practice. The use of these PCIs based on the assumption that process is in control and its output is normally distributed. In practice, normality is not always fulfilled. Indices developed based on normality assumption are very sensitive to non- normal processes. When distribution of a product quality characteristic is non-normal, Cp and Cpk indices calculated using conventional methods often lead to erroneous interpretation of process capability. In the literature, various methods have been proposed for surrogate process capability indices under non normality but few literature sources offer their comprehensive evaluation and comparison of their ability to capture true capability in non-normal situation. In this paper, five methods have been reviewed and capability evaluation is carried out for the data pertaining to resistivity of silicon wafer. The final results revealed that the Burr based percentile method is better than Clements method. Modelling of non-normal data and Box-Cox transformation method using statistical software (Minitab 14) provides reasonably good result as they are very promising methods for non –normal and moderately skewed data (Skewness ≤ 1.5).

Keywords: Process capability indices, Non - normal process, Clements method, Box - Cox transformation, Burr distribution, probability plots

1. Introduction1

Process mean µ, Process standard deviation σ and product specifications are basic information used to evaluate process capability indices however, product

1 Corresponding author: Yerriswamy Wooluru

email: ysprabhu@gmail.com

specifications are different in different products (Pearn et al., 1995). A frontline manager of a process cannot evaluate process performance using µ and σ only. For this reason Dr. Juran combined process parameters with product specifications and introduces the concept of process capability indices (PCI).Since then ,the most common indices being applied by manufacturing industry are process capability index Cp and

408

process ratdefined as

Cp = –

Cpk Min Capability determine producing specificatiocapability depend onunderlying measuremedistributedassumptionpractice as non- normneed to vbefore depdetermine Some authinsightful iin interprmisapplica(Choi and Box and authors hahandle the1994) Tangcomparativdesigned fo 2. Surro

Distri Here the presented tdistribution

W Cl Bu Bo M

St

tio for off - cen

’

n,

indices arewhether a pro

items won limits or

indices Cp n an implicit a

quality ents are indepe. Howeverns are not

many physicamal data and qverify that theploying any the capability hors have prinformation regretation that tion of indicesBai, 1996; M

Cox, 1964). Aave introducee skewness ing and Than (1

ve analysis amor non-normal

ogate PCIs fibutions

following mto compute PCn.

Weighted varianlements Methourr Method ox-Cox Transf

Modelling nontatistical softwa

Y. Wooluru

ntre process Cp

widely useocess is capabwithin cus

not. The prand Cpk he

assumption thacharacte

endent and nor, these fulfilled in

al processes proquality practitie assumptions PCI techniquof their proc

rovided usefulgarding the mis

occur withs to non-normaMontgomery, Alternatively,

ed new indicn the data (B1999) reportedmong seven indistribution.

for Non-Nor

methods have CIs for non-n

nce Method od

formation Methn-normal data are

u,, D.R., Swam

pk are

(1)

(2)

ed to ble of tomer rocess eavily at the eristic rmally

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hod using

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. Weighted va

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d shows that tthod are based it a skewed om the mean. ighted variancwed distributribution from tributions whicferent standapulation with aviation of σ, tha total observatual to µ. Also,

of n total obsen µ.the two ablished by usipectively. Thtributions will ferent standard

estimation omated by ,i.ecan be estipectively. Stanervations whicvalue of ca

milar formula tomple standard

ervations (Ahm

=∑

2∑ 2

o, the sample observations wue of can be

2∑ 2

h

ariance method

Proposed a napplying th

charting methses to improcess perfor

are non-normalthe two weigh

d on the same por asymmetric The main

ce method is ution into tits mean to cre

ch have the samard deviationa mean of µ anhere are obstions which ar, there are ervations whicnew distributing and hat is, the have the samed deviations

of µ, and e.∑ / ,imated by ndard deviationch are less thaan be computeo that used to

d deviation med et al., 200

1

standard deviawhich are grecalculated as

1

d

new process he weighted hod for non–mprove the rmance when lly distributed hted variance philosophy to c distribution idea of the to divide a

two normal eate two new

ame mean but ns. For a nd a standard servations out re less than or

observations ch are greater tions can be observations

two new e mean µ, but

and .For ,µ can be

and and and

n with an or equal to ed by using calculate the for n total

08)

(3)

4

ation with eater than the

5

The two process camodified method as

Cp (WV) =

Cpk (WV)

Cpk (WV)

2.2. Cleme For non-n(which i“populationcharacterisproposed percentiles and proceprocessC

Fig ProcedureClements m

ObLSch

EsgiMSk

commonly usapability indice

using the follows

=

index can be e

= min

ents method

normal pearsincludes a ns” withtics) Clemea novel methto calculate p

ess capabilityindices bas

gure 1. probabi

e for calculamethod (Boylebtain specificaSL for a haracteristic stimate sampleiven sample da

Mean, Standard kewness, Kurto

sed normally-es are p, pweighted var

expressed as:

,

sonian distribwide class

h non-nnts (1989)

hod of non-nprocess capabily for off csed on the m

ility distributio

ating PCIs es, 1994):

ation limits USgiven q

e statistics for tata: Sample size

deviation, osis

-based pk are riance

(6)

7

bution s of normal

has normal lityC centre mean,

stanUndparadisttablfuncreplequ

C

Whthe meatwo−LPCIdist

C

on curve for a n

using

L and quality

the e,

ndard deviatioder the assuameters determtribution curvle of the famiction of skewnlaced 6σby

uation (6).

here, U is the 0.135 percen

an μ is estimao 3 are estim

respectivelyI are obtaintributed quality

min USU

non- normal da

Look upercentil

Look upercentil

Look up Calculate

percentil Calculate

percentilUp

on, skewness aumption that mine the type ove, Clements ily of Pearsonness and kurtosU −L ) in

99.865 percenntile. For C ,ated by medianmated by U –y, Figure 1 de

ned for a ny attribute. SL MU M

,MM

ata with spec. L

up standarde, up standardie standardized M

e estimatee using Eqn. Le estimatee using Eqn.

409

and kurtosis. these four

of the Pearson utilized the

n curves as a sis. Clements n the below

(8)

ntile andL is , the process n M, and the – M) and (M epicts how a non-normally

LSLL

9

Limits

dized 0.135

ized 99.865

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Lp = x - sLp d 99.865 Up = x + s

410

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fopo

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C

an

[C

2.3. Burr d Although Cin industrythat the accurately especially distributionprocess capcharacterisdistributedmodified bprobabilitydistributionestimates process datthe Burr Xthe two pabe used to world and normal prothe direct uinstead of aavoid the integrationthe skewnevarious procovered by.Such probmost knowGamma, normal and

alculate estimqn. M = x +kewness revers

or negative ositive

alculate nonapability indice

= , C

nd C

C , C ]

distribution

Clements’s mey today, Wu et

Clements’s measure thewhen the

n is skewed.pability analystic data is, Clements’s

by replacing thy curves win to improve tof the indiceta. Two reason

XII distributionarameter Burr-Xdescribe data especially tho

ocesses. The seuse of a fitted a probability dneed for a n. It is found thess and kurtoobability densiy different combability densitywn functions, Beta, Weibul

d other function

Y. Wooluru

mated median + sM , for poe sign,

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n-normal pres using Equati

C =

,C

ethod is widelyal. (1999) indimethod can

e nominal vunderlying

. To conducsis when the qs non- nor

method canhe Pearson famith a Burr the accuracy oes for non-nns justify the u. First reason iXII distributiothat arise in th

ose concerningecond reason icumulative fun

density functionnumerical or fhat a wide ransis coefficientity functions c

mbinations of c y functions in

including noll, Logistic, ns.

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using ositive

leave

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,

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nclude ormal,

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k≥ 0

Whkurtdistcumdist

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F(

BurestiesticomCheClePeathemBur 2.3.Bur Bur

my, P. Nagesh

rr XII distributrequired perc

bability densitiate Y is

,k) =

0, f ( ,k) = 0

here c and k rtosis coefficitribution respmulative distribtribution is deri

,k) = 1

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rr-XII distribumate capabilitmate of the p

mmonly used Cen introduced aments method

arson curve pm with percenrr distribution (

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Estimatestandard dekurtosis of th Calculateskewness (αgiven sample

is mean ofthe standard d

where n is thethe data. Use the select the appand k.Then u ) / s = (Q

h

ution can be uscentiles of varty function of

if y ≥ 0; c ≥

0 if y < 0

represent the sients of thepectively .Thbution functionived as:

if y ≥ 0

0

ution can bety indices to pprocess capabiClements metha modification

d, whereby instpercentiles, thntiles from an(Castagliola, 1

for calculatingution method

lves following

s the sample meviation, skee original sampe standardized) and kurtosis size n, as follo

∑

f the observatideviation.

∑

-

e number of ob

values of propriate Burr

use the standardQ - µ)/σ, wh

sed to obtain ariate X .The f a Burr XII

≥ 0 (10)

(11)

skewness and e Burr XII herefore, the n of the Burr

(12)

(13)

e applied to provide better ility than the hod. Liu and

n based on the tead of using hey replaced n appropriate 996).

g PCIs using d

g steps:

mean, sample ewness and ple data.

d moments of s (α ) for the ows:

, where,

ions and s is

,

bservations in

and to parameters c

dized Z = (x -here x is the

randomthe secorrespdeviatistandaand kcollectfound and (tables,and up Causing Bupper s .

. Causing e

C =

2.4. Box-C Box and Cpower trannormalize the need transformaIt transformdata on thvariable X

=

= ln This continparameter λvalues. Tincorporatetransforma

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Fourth roolog transfor

Reciprocal0.50, Recip

No transfoproduces re

m variate of thelected Burr vponding meaion respective

ard deviations akurtosis coeffiction of Burrin the tables o

(Castaglioa, 1 the standardi

pper percentilesalculate estiBurr table for percentiles as

, M= +

alculate procesequations prese

, C

C = Min [C

Cox power tran

Cox (1964) prnsformations ta particular v

to randomtions to determm non-normahe necessarilyas shown in th

For ≠ 0

For 0 nuous family dλ, it can on an

This family es mantions like:

ot transformatiormation, λ=0.

ot transformatiormation, λ =0.0

square root procal transform

rmation needeesults identical

he original datavariate,µ and an and staely. The meanas well as skewcients, for a r distributionsf Burr (Chou, 996). From ized lower, ms are obtained.mated percelower, medianfollows: Lp =s . , Up =

ss capability inented below.

C , C ]

nsformation

rovides a famthat will opti

variable, eliminmly try difmine the best ol data into n

y positive reshe below equati

depends on a

n infinite numbof transforma

ny tradi

ion, λ= 0.50, 33.

on, λ=0.25, N00

transformationmation, λ=-1.0

ed, when λ= 1.l to original dat

a. Q is σ its

andard n and wness large

s are 1996) these

median

entiles n, and = + + s

ndices

C =

ily of imally nating fferent option. normal

ponse ion

(14)

(15) single

bers of ations itional

Cube

Natural

n, λ=-00,

.00, it ta.

Moposskewtranfor 2.5.Sta Quaaskecapfolldemcapnormdo resube trandistnormsoftappa gdistnormrespdistexpdiststatproccapcha 3. Met

st common itive skew buw unless the v

nsformation. Bit (Box and Co

. Modellingtistical softwa

ality control ed to evaluaability for keyow non-norma

monstrating ability requimally distributnot follow th

ults generated incorrect. W

nsform data tribution or idmal distributware’s can be

propriate non-ngood approactribution that mal distributioponse, but if atribution is gponential, lotributions usuaistical softwarcess stabilityability for racteristics.

Methodolog

thodology invo Understa

process cnormal d

Data Col Calculate

case stud Validate Estimatio

using noclassical

transformatiut may exacerbvariable is refleox-Cox elimin

ox, 1964).

Non-Normal are

engineers arate process sy quality charaal distributionsprocess sta

ire the assuuted data. Howhe normal dist

under this assWhether it is

to follow dentify an apprution modele used. Identifnormal distribuch to find a

fits the data.on can be usedan alternative tgoing to be

ognormal, anally works w

re can be used y and estim

non-norm

gy

olves followinganding the basicapability analydata llection e required stady data

the critical asson of Cp, Cpon normal mmethod

411

ions reduce bate negative ected prior to

nates the need

data using

re frequently stability and cteristics that s. In the past, ability and umption of

wever, if data tribution, the sumption will

decided to the normal

ropriate non-l statistical fication of an ution model is

non-normal Many non-d to model a to the normal

viable, the nd weibull

well. Minitab to verify the

mate process al quality

g steps: ic concepts of ysis for non-

atistics of the

sumptions. pu, Cpl, Cpk methods and

412

Comm

3.1. Data c In order tmethods tothe data simDouglas Mstatistical consideredconsecutiveof Silicon w

Mean: 205Skewness; 0.09785 Table 1. D

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

omparison of Pmethods with method

collection

o discuss ando deal with nomilar to an exMontgomery Quality Contrin this paper

e measuremenwafers. Descrip

5.32; Standard 0.39; Kurto

Data of bore diaX1 X

205.324 20

205.302 20

205.346 20

205.326 20

205.330 20

205.333 20

205.282 20

205.297 20

205.409 20

205.342 20

205.368 20

205.389 20

205.356 20

205.252 20

205.326 20

205.334 20

205.287 20

205.333 20

205.325 20

205.369 20

Y. Wooluru

PCIs of non-nPCIs of cla

d compare theon-normality ixample presentin introductiorol, fifth editir. Table 1 pre

nts on the resisptive statistics:

deviation: 0.osis; 0.21; R

ameter using boX2 X3

05.275 205.

05.310 205.

05.280 205.

05.438 205.

05.397 205.

05.316 205.

05.396 205.

05.354 205.

05.313 205.

05.397 205.

05.397 205.

05.301 205.

05.298 205.

05.273 205.

05.297 205.

05.234 205.

05.262 205.

05.328 205.

05.297 205.

05.283 205.

u,, D.R., Swam

normal assical

e five ssues, ted by on to ion is esents stivity :

0405; Range:

4.

4.1.asse In tappcleaprodconveridispthe con

oring operationX4

.356 205.34

.312 205.26

.336 205.31

.288 205.42

.305 205.36

.271 205.31

.306 205.34

.329 205.33

.269 205.32

.265 205.30

.295 205.26

.316 205.31

.356 205.27

.350 205.24

.377 205.37

.318 205.30

.316 205.38

.259 205.33

.320 205.33

.336 205.30

my, P. Nagesh

ConstructioNormal prohistogram fstability anassumption

. Constructioess the stabilit

this study, in plicability of thar decision abduction pro

nstructed usingify stability oplays that the

mean and rantrol limits on t

n X5

49 205.343

60 205.300

5 205.346

29 205.299

68 205.354

4 205.318

48 205.297

30 205.324

23 205.319

05 205.303

62 205.315

9 205.353

70 205.294

41 205.361

71 205.316

03 205.342

83 205.312

36 205.396

35 205.285

06 205.336

h

on of Controbability plofor validatin

nd normalityn.

on of Controty of the proce

order to demhe method anbout the capacess, -R g Minitab 14 of the procesprocess is in c

ange values arthe both charts

X bar

205.329

205.297

205.325

205.356

205.351

205.310

205.326

205.327

205.327

205.322

205.327

205.336

205.315

205.295

205.337

205.306

205.312

205.330

205.312

205.326

205.323

rol chart, ot and ng the y

ol chart to ess

monstrate the nd to make a ability of the

chart are software to

ss. Figure 2 control as all re within the

R

0.081

0.052

0.066

0.150

0.092

0.062

0.114

0.057

0.140

0.132

0.135

0.088

0.086

0.120

0.080

0.108

0.121

0.137

0.050

0.086

0.09785

Sam

ple

Mea

nSa

mpl

eR

ange

4.2. Connormal normality Graphical and normal

F

31

205.38

205.35

205.32

205.29

205.26

Sam

ple

Ran

ge

31

0.20

0.15

0.10

0.05

0.00

struction ofprobability of the data

methods inclul probability pl

Freq

uenc

y

205.24

25

20

15

10

5

0

75

75

Xbar

F

f histogram plot to c

uding the histolot are used to

205205.28

His

Figure 3. H

Sample1119

Sample1119

r-R Chart of he

Figure 2. X and

and check

ogram check

the the normhistnorm

height-1205.365.32

stogram of heigNormal

Histogram for c

171513

171513

eight-1

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normality of histogram an

mal probabilitytogram for sammal.

205205.40

ght-1

ase study data

19

__X=205

UC L=2

LC L=20

19

_R=0.09

UC L=0

LC L=0

the data. Figund Figure 4 y plot for the

mple data appea

5.44

Mean 20StDev 0.04N

413

.3233

205.3796

05.2671

975

0.2061

ure 3 display display the

data set. The ars to be non-

205.34050100

414

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ent

99.

9

99

87654321

0.

The validiusing Andediameter dpass norma0.005),greais done byresult of tes 5. Comp

For case methods:

W Cl Bu Bo

5.1. Weigh The statistiStd. deviatMedian =205.00

Total numbn = 100

Number of

205.20

9

9

50

000000005

1

1

ty of non-norerson – Darling

data is considerality test becauater than criticy using Minitast is shown in F

putation of P

study data u

Weighted varianlements Methourr Method ox-Cox Transf

hted variance M

ics for the obtation =0.0405, M= 205.32, U

ber of observa

f observations l

Y. Wooluru

h205.30205.25

Probab

Figure 4

rmality is testeg test (AD).Thered as normaluse, the P-valucal value (0.05ab 14 softwarFigure 3and 4.

PCI’s

sing the follo

nce Method od

formation Meth

Method

ained sample daMean = 205.3

USL=205.60,

ation in the dat

less than or eq

u,, D.R., Swam

height-120205.350

bility Plot ofNormal - 95% C

4. Normal Prob

ed by e hole l as it

ue is(> ).This e ,the

owing

hod

ata: 2 and LSL=

ta set,

qual to

the

Nummea

Theobsof

=

Als

valu

=

Theprocmodmet

my, P. Nagesh

205.4505.40

height-1CI

bability Plot

mean value in

mber of obsean value in the

e sample standervations whic can be calcul

∑

= 0.0700

o, the sample observations wue of can be

∑

= 0.0288

e two commocess capabilidified using thod as follows

h

205.50

MeanStDevNADP-Value

the data set,ervations grea data set, =

dard deviationch are lower thlated as:

standard deviawhich are grecalculated as:

only used noity indices

the weights:

0.237

205.30.04050

1000.474

e

= 52

ater than the 48

n with han the value

(16)

ation with eater than the

(17)

ormally-based Cp,Cpk are

ted variance

Cp (WV) =

.

. =

Cpk (WV)

(WV) = mi

Table 2. PrStep No.

1

2

3

4

5

6

7

8

9

=

=

= 3.92

) index can be

in ,

rocess capabiliProcedure

SpecificationUpper speciTarget resistLower speciEstimate samSample sizeMean Standard devSkewness Kurtosis Look up stan Look up stan Look up stan Calculate using Eqn. L

Calculate eusing Calculate esEqn. M =

Calculate noindices using

C =

C

.

. .

e expressed as

ity calculation

ns : fication Limittivity ification Limitmple statistics:

viation

ndardized 0.13

ndardized 99.8

ndardized Me

estimated 0.1

Lp = - s

estimated 99. Eqn. Up =

stimated media + s

on-normal prog Equations.

, C =

,C = M

, Cpk

= m

= m Commet

procedure usin

35 percentile

865 percentile

dian in table 2

135 percentil

865 percentil + s

an using

ocess capabilit

,

Min [C , C ]

min .

. ,

.

min [3.24, 4.81]

mputation ofthod (Table 2)

ng the ClementNotations

USLSpec. M

LSL

N

S SkKu

Lp

Up

le Lp

le Up

M

ty

CCCC

.

] =3.24

f PCIs usin).

ts’s percentile mCalcu

L Mean

205.60205.30205.00

2

2

2

415

ng Clements

method ulations

0 0 0

100 205.32 0.0405 0.39 0.21

2.4676

3.5037

0.0652

205.220

205.461

205.322

2.489 3.156 2.00 2.00

416

Computatmethod (T Table 3. PrStep No.

Proc

1 SpecUppeTargLow

2 EstimSamp MeaStandSkewKurt

3 Estimusing

4 Basevalue

5 Withthe tdeter

6 CalcEqn.

7 CalcEqn.

8 Calc Eqn

9 Calc

C =

C

5.2. Box-C The Box-Cis estimatedand correspare determCox transffrom normhaving to each distrib

ion of PCTable 3).

rocess capabiliedure

cifications : er specification L

get resistivity er specification

mate sample statple size an of sample datdard deviation (

wness osis

mate standard mg Sk and Ku valu

ed on and es using the Burr

h reference to thtable of standarrmine standardiz

ulate estimated 0 Lp = + s .

ulate estimated 9 Up = + s .

ulate estimated n. M = + sulate non-norma

= , C =

,C

Cox Transform

Cox transformd by Minitab14ponding proces

mined. The accformation is r

mal and it avosearch for a s

bution encount

Y. Wooluru

Is using B

ity calculation

Limit

Limit

tistics:

a ( overall)

moments of skewues from step 2.

from step 3 ,ser-XII distributio

e parameters c ardized tails of tzed lower, media

0.135 percentile

99.865 percentil

median using

.

al process capabi

= ,

= Min [C , C

mation

ation paramete4 statistical sofss capability incuracy of the obust to depaoids the troubsuitable methotered in practic

u,, D.R., Swam

Burr’s

using the Burr

wness ( ) and K

elect the parameon table

and k obtained ithe Burr XII dian and upper per

using

le using

ility indices usin

C ]

er (λ) ftware ndices

Box-artures ble of od for ce.

Theconis talso(0.40.95vertlamutilicalcoutp

my, P. Nagesh

r percentile met

Kurtosis ( )

eters c and k

n step 4, use istribution to rcentiles.

ng equations.

e Lambda tabntains an estimathe value usedo includes the 46) and lowe5),which are tical lines .In t

mbda value thized for tranculation of PCput of the Min

h

thod Notations

USL

Spec. Mean LSL

N

S Sk Ku

c k

.

.

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Calculations

205.60 205.30 205.00

100 205.32 0.0405 0.39 0.21

0.384 3.13

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205.2355

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2.6080 3.9038 1.9032 1.9032

in figure 5 (-0.21) which sformation. It dence Interval e Interval (-he graph by y, an optimal

ds to -0.21 is he data and e 6 shows the ical software.

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5.3. Compmethod In this cadistribution

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175

150

125

100

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mple N 100ev (Within) 66.9729ev (O v erall) 75.5519A fter Transformation* 0.380189get* ** 0.271154

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100get *

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erv ed Performance < LSL 0.00 > USL 10000.00 Total 10000.00

ExPPP

Figure 6. Pr

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Lamb0.0-2.5

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00.28

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Process CapUsing Box-Cox T

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pability of Resransformation Wi

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Lower CL -Upper CL

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38

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C C pk 0.98O v erall C apa

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5.3.1 CdistributioConfidenc Individual in statisticaconstruct distribution Table 5. CDistributionWeibull ExponentialLog logistic Largest extrLognormal Gamma Normal

Process capdata using through thsoftware as

Resistiv it200100

Resistiv it100

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Comparison ons with P-vce Interval)

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2.286 23.55 0.432 0.359 0.435 0.793 2.045

Results and

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h

1000.00

500

G o

P -V

G aA DP -V

LogA DP -V

E xpA DP -V

WeA D

C I

I

tribution

with the data. Ins are consideone that fits thbution providesith other distribs greater than

m probability pl P-value < 0.010 < 0.003 0.242 > 0.250 0.295 0.042 < 0.005

d Discussion

Table 6 presens of different m

oodness of F it T est

V alue < 0.010

ammaD = 0.793 V alue = 0.042

gnormalD = 0.435 V alue = 0.295

ponentialD = 23.551 V alue < 0.003

eibullD = 2.286

In this study, ered to select he data. The s the best fit butions as its critical value

lot

n

nts the PCIs methods.

Table 6. NPCIs

Wevarme

Cp 3.9Cpl 4.8Cpu 3.2Cpk 3.2 In this papvariance, Bare reviewefor non-nPCIs of Clthe PCIs consideredclassical mCpl is undePCIs of Weighted vresult but Box- Coxreasonably classical mbeen used results com 7. Concl

In practiceyields noninevitable, process ccapability oresults.

Box-Cox transform distributed

The obtainindices shproduction Referenc

Ahmed, S.normal qANZIAM

Numerical resul

eighted riance thod

CleMe

92 2.41 3.1

24 2.024 2.0

per, the ClemenBox-Cox transed and used to

normal quality assical methodof all the noin the case

method, Cpu is er estimated, w

other nonvariance (WV)it requires m

x transformatigood resu

method. Burr peeffectively a

mpared to Clem

lusions

e, manufacturn- normally d

therefore thecapability indof such proces

method is suthe non-normadata and estim

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process fo

ces:

, Abdollanian, quality charac

M Journal, 49(E

lts for PCIs of N

ements ethod

BurrDistri

8 2.605 3.900 1.900 1.90

nts, Burr, Weisformation meo estimate the characteristic

d are comparedon-normal me

study. In caover estimate

when comparedn-normal met) method givesmanual calculaion method

ults comparedercentile methoand it shows ments method.

ring processesdistributed date use of tradidices to messes give misle

uccessfully usal data to nor

mated the PCIs.

f process capae capability oor controlling

M., Zeephongcteristics: A CEMAC 2007), C

Non-normal anObtained R

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gives d to od has better

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nd Classical MResults Cox

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istivity of silicvalues are lesspersion need an have to bget value of 2.79.

ments methodditional 6σ mount the possic data.

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Method

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con wafer is is than 1.33 soto be reduced

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pability estimaurr and Box-C

419

Classical method (Normality assumption) 2.37 2.50 2.19 2.19

inadequate as o ,the process d and process closer to the ting mean of

ension of the h takes into mality of the

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kewness ≤

ox method is e lognormal l distribution s per million mits but in ation method xceeding the

it can be of data to

h is accurate

Burr method, made using indicator to

ers be more ss adjustment

ation for non-Cox method.

420

Boyles, RStatistics

Castaglioladistributi

Chou, Y., Pin statisti

Choi, I.S., of the 20

Clements, Progress

MontgomeYork.

Box, G.E.PStatistica

Wu, H.H.,capabilitInternati

Pearn, W.Lnormal pReliabili

Tang, L. &review http://dx

YerriswJSS AcaEducatioBangaloIndia ysprabhu

.A. (1994). Ps: Simulation a

a, P. (1996). ion. Quality En

Polansky, A.Mical process co

& Bai, D.S. (10 th Internation

J.A, (1989). Ps, 22, 95-100.

ery, D. (1996).

P., & Cox, Dal Society. Seri

, Swain, J.J., ty index for Gional, 15, 397-

L., Chen, K.S.pearsonian proity Managemen

& Than, S. (1and compar

.doi.org/10.100

wamy Woolurademy of Technion, ore

u@gmail.com

Y. Wooluru

rocess capabiand Communic

Evaluation ofngineering, 8, 5

M., & Mason, Rontrol. Journal

1996). Processnal conference

Process capabi

Introduction

D.R. (1964). Aies B (Methodo

Farrington, PGeneral non-no402.

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nt, 16(5), 507-5

999). Computrative study.02/(sici)1099-1

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SwaJSS EduBanIndidrsw

u,, D.R., Swam

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(1995). A genmmetric toleran521.

ting process c. Qual. Rel1638(199909/1

amy D.R. Academy of Te

ucation, ngalore

a wamydr@gmail.

my, P. Nagesh

mmetric tolera15-643

process capab

ransforming nochnology, 30, 1

dices for skewend Industrial E

ons for non-no

Quality Contro

f transformatio), 211-252.

mer, L.S. (19ses. Quality an

neralization of nces. Internati

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chnical

com

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ances. Commu

bility indices

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ormal distribut

ol. 5th edition.

ons. Journal

999). A weighnd Reliability

Clements metional journal of

ces for non-no Int., 15(5

aid-qre259>3.0

Nagesh P. JSS Centre for Mstudies, Mysore India pnagesh1973@r

unications in

using Burr’s

a to normality

.Proceedings 211-1214.

tions. Quality

. Wiley, New

of the Royal

hted variance Engineering

thod for non-of quality and

ormal data: a 5), 339-353. 0.co;2-a.

Management

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