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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: [email protected]
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
basic actual oduce ioners
hold ues to cesses. l and stakes
h the al data 1996; other
es to oyels,
d on a ndices
rmal
been normal
hod using
2.1. Hsicapvarinormmeathe andmetsplifromweiskewdistdistdiffpopdevof aequout thanestarespdistdiffthe esti
respobsthe simsamobs
=
Als
valu
my, P. Nagesh
. Weighted va
n-Hung Wu ability index iance control mal processasurement of pprocess data a
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
Median ed 0.135
Lp = x - sLp d 99.865 Up = x + s
410
CaEqsk
fopo
Caca
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,
skewness
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.
u,, D.R., Swam
using ositive
leave
rocess ions.
,
=Min
y used icated
n not values,
data t the
quality rmally n be
mily of XII
of the normal use of is that
on can he real g non-is that nction n may formal nge of ts of can be and k
nclude ormal,
Log-
Burthe probvari
f(
k≥ 0
Whkurtdistcumdist
F(
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
,k) = 0,if y <
rr-XII distribumate capabilitmate of the p
mmonly used Cen introduced aments method
arson curve pm with percenrr distribution (
.1 Procedure frr XII Distrib
rr method invo
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
Square rooroot transfo
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
d R chart
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
Perc
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
.
.
.
Lp
Up
M
C C
C
C
ble as shown ate of lambda (d in the transupper Confid
er Confidencemarked on ththis case study
hat correspondnsforming th
CIs. The figurenitab 14 statisti
Calculations
205.60 205.30 205.00
100 205.32 0.0405 0.39 0.21
0.384 3.13
2.5377 12.5234
-2.085 -0.082 3.595
205.2355
205.4655
205.3166
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.
StDe
v
SamStDeStDe
ALSL*Targ
LSL
USLSamStDeStDe
TargUSLSam
O bsePPMPPMPPM
5.3. Compmethod In this cadistribution
-5.0
175
150
125
100
75
50
Box
Figu
Process Data
mple N 100ev (Within) 66.9729ev (O v erall) 75.5519A fter Transformation* 0.380189get* ** 0.271154
mple Mean* 0.319472ev (Within)* 0.0184948ev (O v erall)* 0.0194258
100get *
500mple Mean 241.35
erv ed Performance < LSL 0.00 > USL 10000.00 Total 10000.00
ExPPP
Figure 6. Pr
putation of P
se study, theons like expon
Lamb0.0-2.5
Lower CL
x-Cox Plot of R
ure 5. Box Cox
00.28
USL*
xp. Within PerformancePPM > LSL* 513.62PPM < USL* 4494.34PPM Total 5007.96
Process CapUsing Box-Cox T
rocess capabili
PCIs using B
oretical non-nnential, weibul
bda2.50
Upper CL
Resistivity Da
x plot to estima
0.0.320.30
transformed dat
Exp. O v erall PerformPPM > LSL* 887.PPM < USL* 6436.PPM Total 7323.
pability of Resransformation Wi
ty Analysis usi
Burr’s
normal ll and
logn(resdist14 ias s
5.0
Limit
ata
ate optimal valu
0.30.3634
LSLta
ance.14.14.29
sistivity Dataith Lambda = -0.2
ing Box- Cox t
normal are ussistivity of stribution identiis used to comshown in the fig
Lambda(using 95.0% confide
Estimate -
Lower CL -Upper CL
Rounded Value
ue of
38
L*
Potential (Within) C
C C pk 0.98O v erall C apa
Pp 0.94PPL 1.04PPU 0.83Ppk
C p
0.83C pm *
0.98C PL 1.09C PU 0.87C pk 0.87
WithinO v erall
a21
transformation
sed to model silicon wafer)ification featur
mpare the fit ofgure 7.
417
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8bility
4433*
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the response ) .Individual re in Minitab f distributions
418
Per
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90
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10
10.1
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10
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0.1
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
Lognormal - 95
Weibull - 95%
Figur
Comparison ons with P-vce Interval)
distribution idal software (M
probability ns in order
omparison of An type
eme value
pability indicelognormal dis
e output of Ms shown in the
Y. Wooluru
ty500
ty1000
Probabil% C I
% C I
re 7. Probability
of alternvalues (For
dentification feinitab 14) is u
plots for to compare
Alternative Dis
es for the case stribution are
Minitab 14 statifigure 8.
u,, D.R., Swam
Per
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10.1
99.99050
10
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Per
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ity Plot for
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native
95%
feature sed to
said their
goosevethe lognin cp-va(0.0
stributions usinA
study found istical
6. Thecalc
my, P. Nagesh
Resistiv ity100.010.0.0
Resistiv ity20000
ResistivityE xponential - 95%
G amma - 95% C
individual dist
odness of fit wen distributionappropriate o
normal distribcomparison wialue (0.295) is
05).
ng output fromAD value
2.286 23.55 0.432 0.359 0.435 0.793 2.045
Results and
e following Tculation results
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
ned values ofhows that the
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
ibution Box Tran
0.981.090.870.87
ighted ethods PCIs data.
d with ethods ase of ed and d with thods.
s good ations.
gives d to od has better
s that ta are itional easure eading
ed to rmally .
ability of the g the
resiall vdispmeatarg241
Cletradaccobasi
Burdistmod1.5)
Ovesligdistmodexccase732specconlognone
ThewhiClehelpatteand
gsekul, P. (200Comparison ofC642-C665.
nd Classical MResults Cox
nsformation
istivity of silicvalues are lesspersion need an have to bget value of 2.79.
ments methodditional 6σ mount the possic data.
rr-based methtributions thderately from ).
erall performanghtly inferior tribution modedel exhibits theeding the spe of Box –C
23 parts percification lim
ncluded that normal distribu
e.
e estimates maich are highements’s methp quality conentive to and fod improvement.
08). Process capf Clements, Bu
Method
Lognormal Model m
0.97 21.01 20.93 20.93 2
con wafer is is than 1.33 soto be reduced
be shifted to c225 from exist
d is simple extmethod, whichsible non-norm
hod works hat depart
normality (Sk
nce of Box-Co than the
el, Lognormalhat 6179 partspecification limox transforma
r million exmits, hence modeling ution approach
ade using the Ber than those hod are good ntrol engineeocus on proces.
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
well under slightly or
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
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.
, & Lin G.H.cess with asym
nt, 16(5), 507-5
999). Computrative study.02/(sici)1099-1
ru ical
SwaJSS EduBanIndidrsw
u,, D.R., Swam
lity with asymation, 23(3), 6
f non-normal 587-593.
R.L. (1998). Trof Quality Tec
capability indon computer an
ility calculatio
to Statistical Q
An analysis ofological), 26(2)
.A., & Messimormal process
(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
apability indicliab. Engng.10)15:5<339::a
chnical
com
h
ances. Commu
bility indices
on-normal data133-141.
ed populations Engineering, 12
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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