Multivariate Cumulative Sum Control Chart for Dispersion Matrix

2 002 , V ol. 13, N o.2 p p . 21～2 9

M u ltiv ari at e Cu m u lativ e S u m Con tro l Ch art for D i s pe rs ion M atrix ^{1 )}

D uk - Jo o n Ch an g ^{2 )} , Jae - K y o un g S hin ^{3 )}

A b s trac t

S ev er al differ en t cont r ol st at istics t o sim ult an eou sly m onit or disper sion m atrix of s ev er al qu alit y v ariables ar e pre sen t ed sin ce differ en t cont r ol st at ist ics can b e u sed t o de scrib e v ariability . M ultiv ariar e cu m ulativ e su m (CU S UM ) con tr ol ch art s are pr oposed an d t h e perform an ces of t h e pr op os ed CU S U M ch art s ar e ev alu at ed in t erm s of av er ag e ru n len gt h (A RL ). M ultiv ariat e S h ew h art ch art s ar e also pr op osed t o com p ar e th e pr opert ies of th e pr opos ed CU S UM ch art s . T h e n um erical r esult s sh ow th at m u lt iv ariat e CU S UM ch art s ar e m ore efficien t t h an m ultiv ariat e S h ew h art ch art s for sm all or m oderat e shift s . A n d w e also foun d t h at sm all r efer en ce v alu e of t h e CU S U M ch art is m or e efficien t for sm all shift .

K e y w o rd s : cont r ol st atist ic , ex pect ed t im e t o sign al, false alarm

1 . In tro du c ti on

Con t rol ch art is a r ep etit iv e st at istical pr ocedu re for cont in u ou sly m on it orin g p ar am t er s of t h e pr odu ction pr oces s , an d is also u s ed t o quickly det ect w h en th e pr oces s ch an g es an d elim in at e a s sign able cau s es th at m ay pr odu ce an y det erior at ion in t h e qu alit y of t h e m anu fa ctu r ed pr odu ct s . T o m aint ain a cont r ol ch art , s am ple s of size n ar e t ak en at r eg ular sam plin g t im e int erv al of len gt h d . T h e ab ilit y of a cont r ol ch art is det erm in ed by t h e len g t h of t im e r equ ir ed for th e ch art t o sig n al w h en th e pr oce s s is ou t - of- cont r ol, an d th e r at e of false alarm

1. T his r es earch is fin ancially s upport ed by Changw on Nat ional Univ er sity in 2000.

2. Pr ofes s or , Dept . of St atist ics , Changw on National Univer sity , Changw on , 641- 773, Kor ea.

E - m ail : djchan g @s ar im .changw on .ac.kr

3. Ass ociat e Profess or , Dept . of St atistics , Changw on National University, Changw on, 641- 773,

Korea.

w h en th e proces s is in - cont r ol. T h er efor e a g ood ch art sh ou ld qu ick ly det ect ch an g es in t h e produ ct ion proces s w hile pr odu cin g few fals e alarm s .

T h e b a sic S h ew h art ch art , alth ou gh sim ple t o u n der st an d an d apply , u s es on ly t h e in form ation in t h e curr en t s am ples an d is th u s r elat iv ely in efficien t in det ect in g sm all ch an g es on t h e pr oces s par am et er s . M odificat ion s su ch a s t h e u s e of supplem ent ary ru n s ru le s m ay im pr ov e th e efficien cy of S h ew h art ch art s om ew h at , bu t ot h er appr oach es w hich t ak e full adv ant ag es of th e in form ation in p a st sam ples ar e n eeded .

T h e m ultiv ariat e procedur e t o qu ality cont r ol w a s fir st in t rodu ced by H ot ellin g (1947 ) an d b ecam e popu lar in r ecen t y ear s . J ack son (1959 ) an d Gh ar e an d T or g er sen (1968) pr esen t ed m ultiv ariat e S h ew h art ch art b a sed on H ot ellin g ' s T

st atist ic. W oodall an d N cu b e (1985 ) ex t en ded t h e un iv ariat e CU S UM pr ocedur e t o t h e m ult iv ariat e ca se for m on it orin g m ean v ect or of qu ality v ariable s . T h ey oper at ed p t w o - sided un iv ariat e CU S UM sch em es sim ult an eou sly , an d ev alu at ed t h e p erform an ce of t h e collect ion of t h e s ch em e. Cr osier (1988 ) an d P ign atiello an d Run g er (1990) con sider ed n ew m u lt iv ariat e CU S UM pr ocedur es t h at accu m ulat e pa st s am ple in form at ion for each p ar am et er an d t h en form a u niv ariat e CU S UM st atist ic fr om t h e m ultiv ariat e dat a for m on it orin g t h e m ean v ect or .

Up t o th e pr esen t , m ultiv ariat e cont r ol pr ocedur es h av e b een w idely u sed for m onit orin g pr oces s m ean v ect or . But , r elat iv ely lit t le at t ent ion h a s b een giv en t o t h e u se of CU S UM ch art s for m onit orin g disper sion m at rix . In t his paper , w e pr opose sev er al differ en t cont r ol st at ist ics an d CU S UM ch art s for m on it orin g t h e disper sion m at rix of correlat ed qu ality v ariab les w h er e th e t ar g et pr oce s s m ean v ect or r em ain ed k n ow n con st an t .

2 . E v alu atin g Con trol S t ati s ti c s

T h e qu alit y of a pr odu ct is oft en ch ara ct erized by j oint lev els of sev er al qu alit y ch ar act eristics . B ecau se it is in appr opriat e t o u se in div idu al ch art s t o det ect any ch an g es of each pr oces s p ar am et er s in th is ca s e, a m ultiv ariat e qu ality con t rol pr ocedu re for sim ult an eou sly m onit orin g correlat ed v ariables is n eeded . S in ce th e a s sign able cau ses w h ich pr odu ce r an dom v ariat ion of th e qu alit y ar e t r eat ed a s inh er ent t o th e pr oces s in it s curr en t st at e, t h e se qu ality ch ar act erist ics ar e r an dom v ariables .

A s sum e t h at t h e pr odu ct ion pr oces s of int er est h a s p ( p 2) correlat ed qu alit y

v ariables r epr esent ed by t h e r an dom v ect or X = ( X

, X

, X

, , X

) ' an d w e

obt ain an in depen dent sequ en ce of ran dom v ect or s X

, X

, X

, , w h er e th e

v ect or X

= ( X '

, , X '

) ' is a s am ple of ob serv at ion s at each sam plin g t im e

i an d X

= ( X

, X

, , X

) ' . T h e u n derly in g prob abilit y dist ribu t ion of th e p

qu alit y v ariables is a s sum ed t o b e m ultiv ariat e n orm al distribut ion w it h m ean

v ect or an d disp er sion m at rix .

In pr act ice , it m ay b e n eces s ary in m any ca s es t o e st im at e b oth an d from t h e pa st dat a , but for sim plicity w e a s su m e th at

an d

ar e kn ow n . Let

0

= (

,

) b e th e k n ow n t arg et v alu es for th e pr oces s par am et er s of p qu alit y ch ar act eristics an d

is r epr esen t ed a s

0

=

10 20

p0

an d

=

2

10 120 10 20 1p0 10 p0

120 10 20 2

20 2p0 20 p0

1p0 10 p0 2p0 20 p0

2 p0

,

w h er e t h e t arg et cov arian ce com p on ent of X

an d X

is

=

for r , s = 1, 2 , , p .

In u niv ariat e ca se , t h e pr oce s s disp er sion can b e m on it ored by S

ch art or r an g e ch art , w h er e S

den ot e an u nbia sed sam ple v arian ce for a r an dom sam ple of size n fr om a pr oces s . T h e S

ch art sig n als for lar g e v alu es of S

or equ iv alen t ly for lar g e v alu es of T

= ( n - 1) S

/

w h er e

is t ar g et v alu e of t h e pr oces s disper sion

an d S

is obt ain ed at sam plin g t im e i . W h en t h e pr oces s is in - con t rol, th e st atist ic T

h a s a ch i- squ ar ed dist rib ut ion w it h ( n - 1) deg r ees of fr eedom .

F or m u lt iv ariat e ca s e, on e pos sible m u lt iv ariat e v er sion of T

is V

=

n

j = 1

( X

- X

) '

( X

- X

) = tr ( A

) (2.1)

w h er e A

=

n

j = 1

( X

- X

) ( X

- X

) ' an d t h e p p s am ple disper sion m at rix

S

is A

/ ( n - 1) . W h en th e pr oces s is in - cont r ol, th e disp er sion m atrix is

an d th e cont r ol st atist ic V

h a s a ch i- squ ar ed dist ribu tion w it h ( n - 1) p degr ees of freedom . H ot ellin g (1947 ) pr op osed th at th e st at ist ic V

can b e u sed t o m on it or t h e pr oces s disp er sion m at rix of p qu alit y v ariables .

T h e g en er al m ultiv ariat e st atist ical qu ality cont r ol ch art can b e con sider ed a s a r ep et it iv e t est s of sign ifican ce w h er e each qu alit y ch ar act erist ic is defin ed by p qu alit y v ariables X

, X

, , X

. T h er efor e, w e can obt ain an ot h er ch art st at istic for m om it orin g b y u sin g th e lik elih ood r atio t e st (LRT ) st atist ic for t est in g H

: =

v s H

:

w h er e t ar g et m ean v ect or of th e qu alit y v ariables

is k n ow n . T h e r eg ion s ab ov e t h e U CL corr esp on ds t o t h e LRT r ej ection r eg ion .

F or t h e i t h s am ple , lik elih ood r at io can b e ex pr es sed a s

= n

- np

2

|A

- 1

0

|

n

2

ex p [ ^- 2 ¹ tr (

A

) + 1 2 np ] ^.

Let T V

b e - 2 ln . T h en T V

= tr ( A

- 1

0

) - n ln |A

| + n ln |

| + np ln n - np . (2.2) H en ce, t h e st atist ic T V

can b e u s ed a s th e sam ple st at ist ics for m on it orin g . A lt (1982) pr op os ed t h e u se of sam ple g en er alized v arian ce | S

| t o m onit or disper sion m at rix . A n d H ui (1980) also u s ed th e sam ple g en er alized v arian ces for m onit orin g t h e pr oces s disper sion b y u sin g t h e follow in g st at istic n - 1 ( | S

| / |

| - 1 ) . T his st at istic is a sy m pt ot ically n orm ally dist ribu t ed w it h m ean 0 an d v arian ce 2p .

If t h e st at istic V

or T V

plot s ab ov e t h e upper con tr ol lim it s , t h e pr oces s disper sion m at rix is deem ed ou t - of- cont r ol st at e an d a s sig n able cau ses ar e s ou ght . T h e st atist ic V

h a s a chi - s qu ar ed distribut ion w it h ( n - 1) p deg r ees of fr eedom an d th e per cen t a g e poin t of t h e V

can b e obt ain ed from chi - s qu ar e distribut ion w h en th e proces s is in - cont r ol. But , if th e pr oces s sh ift s fr om

th en it is difficult t o obt ain t h e ex act dist ribu t ion of V

. T hu s in or der t o ob t ain th e p er cen t ag e p oint s of V

w h en t h e pr oes s is ou t - of - con tr ol st at e, it is n eces sary t o u se com pu t er sim ulat ion s . A n d , it is difficu lt t o obt ain t h e ex act dist ribu t ion of T V

w h en t h e pr oces s is in - con t rol or ou t - of- cont r ol st at es . T hu s , in or der t o ev alu at e t h e p erform an ce s of t h e ch art s b a sed on t h e st atist ic s V

or T V

for it is n eces s ary t o carry out com put er sim u lation s .

3 . M ultiv ari at e S h e w h art Ch art

S h ew h art ch art is w idely u sed t o display s am ple dat a fr om a pr oces s for t h e purpose of det erm inin g w h et h er a pr odu ct ion proces s is in - cont r ol, for brin gin g an out - of - cont r ol pr oce s s int o in - cont r ol, an d for m on it orin g a pr oces s t o m ak e sur e t h at it st ay s in - cont r ol. A S h ew h art ch art h a s a g ood ab ilit y t o det ect larg e ch an g es in m onit or ed p ar am et er quickly an d is ea sy t o im plem en t t h e pr oces s . H ow ev er , th e S h ew h art ch art is slow t o sign al sm all or m oderat e ch an g es in t h e pr oces s par am et er s .

T h e ru n len gt h N is defin ed a s t h e r an dom n um b er of s am ple s r equ ir ed for t h e ch art t o sign al an d th e t im e r equir ed t o sig n al T is d N w h ere d is t h e len g th of th e sam plin g in t erv al. T h e av er ag e run len g th (A RL ) is E ( N ) an d t h e ex pect ed t im e t o sig n al E ( T ) is sim ply t h e pr odu ct of t h e A RL an d d . T h er efor e, th e A RL can b e t h ou g ht of a s t h e ex pect ed t im e t o sign al.

Let q b e th e pr ob abilit y th at a ch art st atist ic falls out - of - con t rol lim it s , t h en N

is g eom etrically dist ribu t ed w it h p ar am et er q w h en t h e pr oces s is in - cont r ol. T h e ex p ect ed tim e t o sign al E ( T ) an d t h e v arian ce of t h e t im e t o sig n al V ( T ) can b e repr esen t ed a s

E ( T ) = d E ( N ) = d

q an d V ( T ) = d

( 1 - q) q

.

S in ce t h e cont r ol lim it s for a m ultiv ariat e S h ew h art ch art b a s ed on t h e sam ple st atist ic V

w ould b e s et a s { ^{0 ,}

[ ( n - 1)p ] } , a S h ew h art ch art b a s ed on V

sig n als w h en ev er

V

[ ( n - 1) p ] . (3.1)

A n d b ecau se t h e cont r ol lim it s for a m u lt iv ariat e S h ew h art ch art b a sed on t h e st atist ic T V

w ou ld b e set by u sin g p er cent ag e poin t of T V

, a S h ew h art ch art b a sed on T V

sig n als w h en ev er

T V

h

(3.2)

w h er e h

can b e obt ain ed t o sat isfy a specified in - con tr ol A RL by sim ulat ion . [R esu lt 3.1] Let X

= ( X

, X

, X

) ' b e dist ribu t ed a s N

(

,

) an d X

's b e in depen dent w h en t h e pr oces s is in - cont r ol. A s su m e t h at m u lt iv ariat e S h ew h art ch art b a s ed on th e st at ist ic V

in (2.1) is u sed a s st at ed ab ov e. If th e pr oces s par am et er s of th e dist ribu tion shift ed a s N

(

, c

) w h er e c is a con st ant , th en

A R L = 1

1 - F ( h

/ c ) (3.3)

w h er e h

=

[ ( n - 1)p ] is th e con tr ol lim it of t h e ch art in (3.1) an d F ( ) is a chi- s qu ar ed dist ribu tion fu n ction w it h ( n - 1) p deg rees of freedom .

4 . M u ltiv ari at e CU S U M Ch art

T h e CU S U M ch art is a g ood alt ern at iv e t o t h e S h ew h art ch art an d is oft en u sed in st ea d of st an dar d S h ew h art ch art w h en det ect ion of sm all sh ift s in a pr odu ct ion pr oces s is im p ort ant . A CU S UM ch art dir ect ly in corp or at es all of th e inform at ion in th e sequ en ce of sam ple v alu e s by plot t in g t h e cum ulat iv e sum of t h e dev iat ion of t h e sam ple v alu es from th e t ar g et v alu e.

A m ult iv ariat e CU S UM ch art b a sed on th e st at ist ic V

in (2.1) is g iv en by

Y

= m ax { Y

, 0 } + ( V

- k

) (4.1)

w h er e Y

=

(

0) an d r efer en ce v alu e k

0 . T his ch art for disp er sion m at rix sign als w h en ev er Y

h

.

W h en th e pr oces s par am et er s ar e on - t ar g et , decision int erv al h

can b e ev alu at ed by th e M ark ov ch ain or in t egr al equ at ion appr oach t o s at isfy a specified in - cont r ol A RL . A n d w h en t h e proces s par am et er s in h av e ch an g ed, th e p erform an ce s an d pr opert ies of t his ch art can b e ev alu at ed by sim ulation .

T h e CU S UM pr ocedu re can b e con sider ed a s a sequ en ce of in depen dent t est s w h er e each t est is act u ally equ iv alen t t o a sequ ent ial pr ob ability r at io t est (S P RT ) for t estin g H

: =

v s H

:

. T h is sequ en ce of S P RT ' s is equiv alent t o u sin g th e CU S UM st atist ic

Y

= m ax { Y

, 0 } + ( L

- k ) (4.2)

w h er e t h e lik elih ood r at io L

= max {L ( , ) } / m ax {L ( ,

) } an d L ( , ) is lik elih ood fu n ction of r an dom v ect or X

= ( X '

, X '

, , X '

) ' an d k 0 . T his ch art sign als w h en ev er Y

> c . T h er efore , a CU S UM procedur e b a s ed on t h e st atist ic T V

in (2.2) can also b e con st ru ct ed a s

Y

= m ax { Y

, 0 } + ( T V

- k

) (4.3)

w h er e Y

=

(

0) an d k

0 . T his ch art sign als w h en ev er Y

h

.

S in ce it is difficult t o obt ain t h e ex act perform an ces of m u lt iv ariat e CU S UM s ch em e b a sed on T V

, t h e per cent ag e p oint an d pr opert ies of t his ch art can b e ev alu at ed b y sim u lat ion u n der th e pr oces s p ar am et er s of th e pr oce s s ar e on - t ar g et or ch an g ed .

5 . N u m e ric al P e rf orm an c e s an d Con c lu din g R em ark s

T h e desig n of a CU S UM ch art r equ ir es t h e specification of t h e sam ple size n , t h e sam plin g int erv al d , an d t h e ch art par am et er s h an d k . A g ood ch oice for t h e ch art par am et er s depen ds on th e n um b er of qu alit y v ariables in t h e pr op os ed con tr ol sch em e an d t h e size of shift on in t ere st in g .

In or der t o ev alu at e th e perform an ces an d com pare t h e pr oposed m u lt iv ariat e

CU S U M an d S h ew h art ch art s fairly , it is n ece s sary t o calibr at e ea ch s ch em es so

t h at on - t ar g et A RL E ( N |

,

) b e t h e sam e for all t h e pr oposed sch em e s . In

our com put at ion , each sch em e w a s calibr at ed so t h at t h e on - t ar g et A RL w a s

appr ox im at ely equ al t o 370.4 an d t h e s am ple size for each ch ar act erist ic w a s fiv e

for p = 3 an d p = 4 . F or conv en ien ce, w e let th at t h e s am plin g int erv al of un it

t im e d = 1 an d k n ow n t ar g et m ean v ect or

= 0 . T h e perform an ce of th e

ch art s for m onit orin g a disper sion m at rix depen d s on t h e com pon en t s of . F or com pu t ation al sim plicity in our com pu t ation , w e a s sum e th at

= 1,

= 0 . 3 for

r , s = 1, 2 , , p .

S in ce it is n ot pos sible t o inv estig at e all of t h e differ en t w ay s in w h ich cou ld ch an g e, w e con sider t h e follow in g ty pical ty p es of shift s for com p arison in t h e pr oces s par am et er s :

(1) V

:

of

is in cr ea sed t o [ 1 + ( 4 i - 3) / 10 ] .

(2) C

:

an d

of

ar e ch an g ed t o [ 0 . 3 + ( 2 i - 1) / 10 ] (3 ) ( V

, C

) for i = 1, 2 , 3 .

(4 ) S

:

is ch an g ed t o c

w h ere c

= [ 1 + ( 3 i - 2) / 10 ]

.

A ft er t h e r efer en ce v alu e of t h e proposed CU S UM ch art b a sed on t h e cont r ol st atist ic V

, decision in t erv al h

w a s calculat ed by M ark ov ch ain m et h od w it h t h e nu m b er of t r an sien t st at es r = 100 . A n d t h e p ar am et er s h

b a sed on T V

for CU S U M sch em e , an d th e A RL v alu es for all t h e pr oposed t y pes of shift s for S h ew h art an d CU S UM ch art s b a sed on V

or T V

w er e obt ain ed by sim ulation w it h 10,000 it er at ion s .

< T able 1> A RL v alu es for disp er sion m at rix ( p = 3 )

t y pes S h ew h art CU S UM

of shift s ^V

T V

b a s ed on V

b a sed on T V

n o shift 370.4 370.4 370.4 370.4 370.4 370.3 370.3 370.3

V

177.7 340.6 86.1 91.5 99.5 302.2 311.9 318.0

V

15.8 53.9 12.6 10.7 9.8 24.5 25.2 26.4

V

4.5 8.8 6.2 5.0 4.5 7.2 6.6 6.3

C

419.7 354.4 623.3 564.2 530.6 321.6 329.1 334.2

C

403.8 244.7 1663.7 1121.5 880.8 103.5 119.9 136.4

C

309.0 101.3 3019.5 1502.3 996.9 18.5 19.1 20.9

( V

, C

) 203.8 328.2 116.3 123.9 133.6 274.5 288.6 296.8

( V

, C

) 19.2 46.3 15.6 13.3 12.4 20.3 20.4 21.3

( V

, C

) 5.6 6.9 7.5 6.2 5.5 5.7 5.2 4.9

S

65.3 299.6 26.6 24.5 24.7 213.8 229.0 242.0

S

4.5 27.2 5.8 4.7 4.1 12.3 11.9 11.9

S

1.7 4.3 3.1 2.6 2.0 4.2 3.8 3.5

S

1.2 1.8 2.2 1.8 1.6 2.4 2.1 2.0

k_V= 12 . 5 k_V= 13 k_V= 13 . 5 k_{T V}= 9 k_{T V}= 9 . 5 k_V= 10

T h e perform an ces of t h e pr oposed m u lt iv ariat e S h ew h art an d CU S UM ch art ar e g iv en in T able 1 an d 2. By v ariou s n um erical com put at ion , w e foun d t h e follow in g pr opert ies . W h en a shift in v arian ce com p on ent s h av e occur ed , m ultiv ariat e sch em e b a sed on t h e con tr ol st atist ic V

is efficien t . A n d a shift in corr elation coefficien t s h av e occu r ed, con tr ol procedur e b a sed on t h e T V

w ill b e r ecom m en ded. A sh ift for b oth v arian ces an d corr elation coefficient s in h a s occurr ed , t h e m ultiv ariat e CU S U M pr ocedu re b a sed on T V

w ill b e r ecom m en ded.

W h en sm all or m oder at e ch an g es in t h e pr odu ction pr oces s h av e occu rr ed , CU S U M procedur es are m ore efficien t t h an S h ew h art ch art s in t erm s of A RL . N um erical r esu lt s for v ariou s r efer en ce v alu es of CU S UM sch em es sh ow th at sm all r efer en ce v alu es ar e m or e efficient in det ect in g sm all sh ift s an d lar g e r efer en ce v alu es ar e m or e efficient for lar g e shift s .

H en ce, w e r ecom m en d CU S UM ch art b a sed on th e con tr ol st atist ic T V

t o sim u lt an eou sly m onit or b oth v arian ce s an d corr elation coefficient s of in th e m u lt iv ariat e n orm al pr oces s w h en sm all or m oder at e sh ift h a s occu r ed in t h e pr odu ct ion pr oces s .

< T able 2> A RL v alu es for disp er sion m at rix ( p = 4 )

t y pes S h ew h art CU S UM

of shift s ^V

T V

b a s ed on V

b a sed on T V

n o shift 370.4 370.4 370.4 370.4 370.4 370.4 370.4 370.4

V

199.9 357.3 97.9 103.6 111.9 316.3 322.8 325.8

V

19.3 118.7 14.9 12.6 11.6 43.5 41.8 42.8

V

5.3 19.3 7.2 5.9 5.2 13.8 12.0 11.0

C

397.6 359.7 540.7 506.5 488.8 331.8 334.7 339.9

C

374.3 291.6 1107.9 859.3 721.2 130.8 140.9 153.6

C

291.8 166.4 1795.4 1106.6 815.8 33.8 32.1 32.8

( V

, C

) 2191.1 349.6 123.6 130.5 139.9 290.7 298.0 305.9

( V

, C

) 22.5 100.7 17.7 15.1 14.0 35.7 33.6 33.9

( V

, C

) 6.2 14.2 8.5 7.0 6.2 10.7 9.2 8.4

S

55.3 326.6 23.2 20.7 20.0 215.1 226.6 237.2

S

3.4 47.9 5.2 4.2 3.7 17.9 15.7 14.7

S

1.4 5.7 2.9 2.4 2.1 6.3 5.3 4.8

S

1.1 2.0 2.0 1.7 1.5 3.5 2.9 2.6

k_V= 16 . 5 k_V= 17 k_V= 17 . 5 k_{T V}= 16 kT V= 16 . 5 k_V= 17

R e f e re n c e s

1. Alt , F .B. (1982). Multiv ariat e Qu ality Cont rol in T he E ncy clop ed ia of S ta tis tical S ciences , eds. S . Kot z and John son , J ohn W iley , New York . 2. Arnold, J .C. and Reynolds, M.R. (2001). CUSUM Control Chart s with Variable

S am ple S izes an d S am plin g Int erv als, J ournal of Quality T echn ology , V ol.

33, 66- 81.

3. Ch an g , T .C. an d Gan , F .F . (1995). A Cum ulativ e S um Control Ch art for M onit orin g P rocee V arian ce, J ournal of Quality T echn ology , V ol. 27, 109- 119.

4. Crosier , R.B. (1988). Multiv ariat e Generalization of Cumulativ e Sum Qu ality - Control S chem e, T echn om etrics, V ol. 30, 291- 303.

5. Gh ar e, P .H . an d T or ger sen , P .E . (1968). T he Multicharact eristic Control Ch art , J ournal of I nd us trial E ng in eering , Vol. 19, 269- 272.

6. Hot ellin g , H .(1947). Multiv ariat e Quality Control, T echniques of S tatis tical A naly s is , McGraw - Hill, New York , 111- 184.

7. Jack son , J .S .(1959). Qu ality Cont rol M et hods for S ev eral Relat ed Variables , T echn om e trics , V ol. 1, 359- 377.

8. Pign at iello, J .J ., Jr . an d Run g er , G.C. (1990). Com parison s of Multiv ariat e CUS UM Chart s , J ournal of Quality T echn ology , V ol. 22, 173- 186.

9. Pr abhu , S .S . and Run g er , G.C.(1997). Designin g a Multiv ariat e EW MA Cont rol Ch art , J ournal of Quality T echn ology , V ol. 29, 8- 15.

10. Reynolds , M .R., Jr . and Ghosh , B.K . (1981). Designing Control Ch art s for Mean s an d V arian ces , A S Q C Quality Cong ress T ransactions , S an F ran cisco, 400- 407.

11. W oodall, W .H . an d N cube, M .M . (1985). Multiv ariat e CUSUM Qu ality Control P rocedure, T echn om e trics , V ol. 27, 285- 292.

[ 2002년 8월 접수, 2002년 10월 채택 ]