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DOI:10.24200/J65.2020.55246.2092

"""pQDvm

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Dk=|L=Q]|O=yvW}B pOt pL |=Q@ "OwW|t xv}W}@ 4(ARL1) pQDvm CLD \}=QW QO xr=@vO

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p�=Ut pL QO w [14]'OQ=O OQ@ Q=m 'OvQ=O |v=Um} h}=_w xm x@=Wt |=yOL=w |@Uv%Ovv=t CU= xDiQ Q=mx@ |oOQDUo x@ |O=YDk= w |D} Q}Ot |=yxv}tR QO xQ=}atOvJptL |=yCmQW u}t-=D |xQ}HvR |@=} RQ= w [15]|v=DUQ=t}@ C=tON OQmrta |@=} RQ=

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"OvOQm|=ysDU}U sytVN@l} x@|v}r=@ |=yOv}=Qi w|v=DUQ=t}@ |=y|HwQNV}=BOvQ}o|t Q=Qk u=tQO CLD =ysDU}U u}= QO xm |v=Q=t}@ "CU= xOW p}O@D CtqUQ=Wi 'C@=}O 'TvH 'uU Ovv=t Cw=iDt pta R= V}B |=y|oS} w uDW=O p}rO x@Qy nQt p=tDL= wQ u}= R= "OvyO|t p}mWD =Q |vwoty=v |xat=H 'xQ}e w uwNV}B |=y|oS} w x@ xmr@ 'K=QH CQ=yt u=R}t x@ =yvDxv |L=QH pta R= TB Q=t}@O}=@ |L=QH OQmrta |@=} RQ= w V}=B 'u}=Q@=v@ "OQ=O |oDU@ R}v Q=t}@ Qy pta R=xDmv u}= "OQ=O s=v �lU} Q p}OaD� Ov}=Qi u}= "OwW p}OaD =y|oS} w u}= T=U= Q@uwO@ |L=QH C}i}m uOw@ ?U=vt=v =} ?U=vt X}NWD xm CU= C}ty= R�=L'lU} Q p}OaD sOa CQwY QO w CU= umttQ}e lU} Q p}OaD uDiQo Q_v QO[10]"OvyO|tCUO R= =Q OwN|}=Q=mC}i}m Ow@y@ Q=R@=l} u=wva x@ pQDvm |=yQ=OwtvOwR Q}twnQt MQv QO Vy=m w V}=Ri= X}NWD |=Q@ 'xv}tR u}= QO |}xar=]t QOQ=Owtv R= |tmW CQw�; sU} Qwv; |oQ=B s}tQD |L=QH pta R= Oa@ u=Q=t}@ s=ovyxH}Dv w OW xO=iDU= 2 (RACUSUM) xOW lU} Q p}OaD |atHD `tH pQDvm[11]"CU= xDi=} Vy=m xOW |v}@V}B Q=Okt x@ C@Uv Q}twnQt MQv O=O u=Wv|atHD `tH Q=Owtv l} R= xO=iDU= =@ xm OW xO=O u=Wv Qo}O |}xar=]t QO u}vJtyxOW xDiQo Q_v QO ?rk |L=QH |=y|HwQN V}=B O}OH xOW lU} Q p}OaD

[12]"CU=|L=Q] uwvm =D xm OyO|t u=Wv C=}@O= |UQQ@ 'CU= XNWt xm =Hu; =DCtqU |=ysDU}UV}=B |=Q@ xOWlU} Q p}OaD pQDvm |=yQ=Owtv |Q=t; |O=YDk=l} VQDUo xr=kt u}= hOy 'u}=Q@=v@ "CU= xOWv xDiQo Q_v QO u=QoVywSB \UwDCtqU |=ysDU}U QO RACUSUM pQDvm Q=Owtv |xiOyOvJ |L=Q] pOtQ=Owtv |L=Q] |=yQDt=Q=B pOt u}= QO "CU= |Q=t; w |O=YDk= C=_Lqt =@pw] \UwDt 'u}vJty w Q=_Dv= |xv} Ry xm OwW|t u}}aD |}xvwox@ RACUSUM

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� = ��(1 + 1�

) (5)

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%CU=Vi = Min(0; Vi�1 �Wi); i = 1; 2; :::V0 = 0 (6)

x@U=Lt 7 |x]@=Q R= xO=iDU= =@ xm CU= CUSUM R=}Dt= Wi u; QO xm%OwW|t

Wi = log(L(yij!i1)L(yij!i0) ) (7)

Q=Owtv xmCU=O}B 7 |x]@=Q x@ xHwD =@ "CU=|}=tvCUQO `@=D hQat x]@=Q u}= QOLQ=t; xm u}=ZLt x@ "CU= xOW |L=Q] !i1 x@ !i0 R= C=Q}}eD `} QUX}NWD |=Q@'Ovm Rw=HD|rQDvm OLl} =} X=N hOy Q=Oktl} R= Q=Owtv u}= |xOW|v=UQ RwQx@CUSUM Q=Owtv l} R= Q=DWwv u}= QO uwJ "OwW|t QO=Y pQDvm R= GQ=N Q=OWy'CU=k@ u=tR u}ov=}t QO|Wy=m C=Q}}eDhWm u; hOy w OwW|t xO=iDU= xiQ]l}pQDvm |v}}=B OL LCL '=DU=Q u}ty QO "s} Q}o@ Q_v QO =Q pQDvm u}}=B OL l} O}=@xJv=vJ 'u}=Q@=v@ "s}UQ@ ?wr]t ARL0 l} x@ xm OwW|t ?=NDv= |k} Q] x@ wCU="OyO|t u=Wv =Q pQDvm R= GQ=N Q=OWy l} CUSUM Q=Owtv x=ov; 'LCL > ViVwQ 'u; |Q}oQ=mx@ w RACUSUM pQDvm Q=Owtv |xaUwD Qw_vt x@ 'xt=O= QO|}=tvCUQO `@=D O=H}= |=Q@VwQu}= "OW xDiQo Q_v QO [17]u=Q=mty w woU|O=yvW}B\UwD xOW |v}@V}B |=k@ w |r=oJ `@=wD R= 'CUSUM pQDvm Q=Owtv R=}Dt= QOQO =Q |oOWQwUv=U C=Oy=Wt |=Q@ VwQ u}= 'u}vJty "Ovm|t xO=iDU= AFT pOtQwUv=U u=tR =} xOW u}}aD V}B R= u=tR l} =D u=Q=t}@ |=k@ u=tR |va} 'OQ}o|t Q_vOy=wN CQ=@a |}=tvCUQO `@=D Q=m u}= s=Hv= =@ "OwW|t xOy=Wt (c) CU=Q R= xOW

%R= Ow@L(zij!i; �i) = [f(zi)]�i [S(zi)]1��i

zi = Min(yi; c)

�i =

8<:1 if yi � c0 if yi > c

(8)

8 |x]@=Q QO 'AFT pOt Q@ |vD@t |=k@ w p=tDL= |r=oJ `@=wD |Q=Po}=H =@=@ u=wD|t 'TBU "OwW|t pY=L RACUSUM pQDvm Q=Owtv |}=tvCUQO `@=DRACUSUM pQDvm Q=Owtv R=}Dt= QO |}=tvCUQO C@Uv 7 |x]@=Q R= xO=iDU=R=}Dt= 'T=}kt QDt=Q=B QO � xR=Ov= x@ Q}}eD l} X}NWD |=Q@ 'u}=Q@=v@ "OQm O=H}= =Q

%R= CU= CQ=@a |}=tvCUQO sD} Q=or

CU= |Q}eDt �=k@ u=tR� xm OwW xHwD O}=@ "Ovm|t pta QDy@ w QDT=UL |r}Nu=v}t]= p@=k |=yxO=O X=N |=y|oS} w R= |m} "CU= u=v}t]= C}r@=k ,=D}y=t xmu}}aD V}B R= OL l} =D =k@ u=tR V}=B xm OyO|t MQ |v=tR w CU= |oOWQwUv=U|L=QH Ov}=Qi QO u=wD|t xv} Ry w u=tRVy=m Qw_vt x@ u}=Q@=v@ "OW=@ xHwD OQwt xOWu=v}t]= p@=k |=yxO=O 'u}= Q@ xwqa "OQm xO=iDU= CU=Q R= |oOWQwUv=U Q=mwR=U R=|R=UpOt T=}kt u=mt sD} Q=or w T=}kt u=mt |=y`} RwD R= |w[a |xr}Uwx@|=yC}akwt QO xm CU=y`} RwD u} QDOQ@ Q=mQB R= |m} pw@} w '=yv; u=}t QO "OvwW|tQ@ ZQi 'C=}rm uO=O CUO R= uwO@ 'xH}Dv QO [18]"CU= xOW `k=w O}it wvDt|r=oJ `@=wD "Ovm|t |wQ}B pw@} w `} RwD R= =k@ u=tR =} |HwQN Q}eDt xm CU= u}=%R= Ov=CQ=@a 'OwW|t xO=O u=Wv S w f =@ '?}DQD x@ xm `} RwD u}= |=k@ w p=tDL=

f(y) = �� ( y� )��1 exp(�( y� )�)

S(y) = exp(�( y� )�)(1)

'u}vJty "CU=k@ u=tR |va} xOW |Q}oxR=Ov= |HwQN R= xOWC@F VRQ= y u; QO xm"CU= pw@} w `} RwD T=}kt QDt=Q=B � > 0 w pmW QDt=Q=B � > 0

|xat=H QO OwHwt |vwoty=v w =k@ u=tR u=v}t]= C}r@=k C}y=t 'Qo}O hQ] R=xrtH R= "OR=U|t uwtvyQ =k@ p}rLD |vw}UQoQ |=ypOt R= xO=iDU= x@ =Q =t 'u=Q=t}@7(AFT) xDi=}`} QUD CUmW u=tR pOt x@ u=wD|t =k@ p}rLD uw}UQoQ |=ypOtQ=PoQ}F-=D |=yQ}eDt R= |a@=D |xr}Uwx@ =Q `} RwD QDt=Q=B OvJ =} l} xm OQm xQ=W=AFT pOt R= lU} Q p}OaD Qw_vt x@ Q=DWwv u}= QO [19]"Ovm|t u=}@ xOW |}=U=vWu=Wv 2 |x]@=Q =@ u=wD|t =Q x}=B `} RwD |=yQDt=Q=B 'u; T=U= Q@ w s}vm|t xO=iDU=

%O=O!i = q(�;Xi) (2)

|=yQDt=Q=B Q=OQ@ � '`} RwD |=yQDt=Q=B R= |Q=OQ@ !i 'uw}UQoQ pOt `@=D q u; QO xmi Q=t}@ |=Q@ =Q lU} Q pt=wa xm CU= Q=PoQ}F-=D |=yQ}eDt Q=OQ@ Xi w |vw}UQoQx@ =Q T=}kt QDt=Q=B 'pw@} w `} RwD |=Q@ AFT pOt R= xO=iDU= =@ "Ovm|t Tmavt

%|va} s}vm|t xDU@=w xr=kt u}= QO Q=PoQ}F-=D Q}eDt =yvD� = exp(�0 + �1x) (3)

%OwW|t |U} wvR=@ u}vJ =k@ w p=tDL= |r=oJ `@=D 'wQu}= R=

f(yjx) = �exp(�0+�1x) ( y

exp(�0+�1x) )��1

exp(�( yexp(�0+�1x) )�)

S(yjx) = exp(�( yexp(�0+�1x) )�)

(4)

CtUk Vkv '3 |x]@=Q QO "OvDUy |vw}UQoQ pOt |=yQDt=Q=B �1 w �0 u; QO xm4 |x]@=Q x@ xHwD =@ 'xH}Dv QO "CU= u=Um} T=}kt QDt=Q=B =@ exp(�0) C@=Fw � pmW QDt=Q=B |=Q=O pw@} w `} RwD |=Q@ AFT pOt xm O=O u=Wv u=wD|tpQDvm CLD |=yQDt=Q=B Q=OQ@ 'u}=Q@=v@ [19]"CU= � exp(�1xi) T=}kt QDt=Q=Bxm CU= u}= Q@ ZQi "OwW|t XNWt !i0 = (�; � exp(�1xi)) CQwYx@u; QO xm OyO|t u=Wv =Q pQDvm R= GQ=N \}=QW !i1 = (�; �� exp(�1xi))X}NWD |=Q@ CUSUM pQDvm Q=Owtv xm CU= xOW u}}aD V}B R= Q}eDt l} �u}ov=}t w T=}kt QDt=Q=B u}@ s}kDUt |x]@=Q p}rO x@ "CU= xOW |L=Q] u; |xv}y@"CU= =k@ u=tR u}ov=}t QO Q}}eD =@ pO=at T=}kt QDt=Q=B QO |Q}}eD Qy 'pw@} w `} RwD

89

"""pQDvm

Q=Owtvl

}|Q=t;

�|O=Y

Dk=|L=Q]

Wi = log[ ��� exp(�1xi) ( zi

�� exp(�1xi) )��1 exp(�( zi�� exp(�1xi) )�)]

�i [exp(�( zi�� exp(�1xi) )�)]1��i

[ �� exp(�1xi) ( zi

� exp(�1xi) )��1 exp(�( zi� exp(�1xi) )�)]

�i [exp(�( zi� exp(�1xi) )�)]1��i

(9)

%OwW|t xrwtQi u}vJ xQwO l} Q=_Dv= pm |xv} Ry 'u}vJtyEC =

A:1�+[(h:ARL1)�� ]+ 1:TF+ 2:TD

h +

CO:ARL1 + (CF + CD)

(12)

%R= Ov=CQ=@a `@=wD u}= |=yQDt=Q=B&pQDvm CLD \}=QW QO xr=@vO pw] \UwDt %ARL0 �

&pQDvm R= GQ=N \}=QW QO xr=@vO pw] \UwDt %ARL1 �ltm x@ xm 'i + 1 w i |=yxvwtv u}@ p}rO =@ h=QLv= wkw Q=_Dv= u=tR %� �

%O};|t CUO x@ 13 |x]@=Q1� (1 + �:h):e��:h

�:(1� e��:h)(13)

"CU= xOt; CUw}B �?� CtUk QO x]@=Q u}= C=}�RH&p}rO =@ h=QLv= |}=U=vW w wHwCUH |=Q@ u=tR \UwDt %TF �

&|LqY= C=t=Ok= |=QH= |=Q@ u=tR \UwDt %TD �&Q=t}@ Qy |=Q@ |Q}oxvwtv |=yxv} Ry %A �

Qy |=Q@ pQDvm R= GQ=N |L=QH p}rO x@ u=DUQ=t}@ Q@ xOW p}tLD |xv} Ry %CO �&p}rO =@ h=QLv= wkw R= |W=v Q=t}@

&p}rO =@ h=QLv= |}=U=vW w wHwCUH |xv} Ry %CF �&p}rO =@ h=QLv= `iQ |=Q@ |LqY= C=t=Ok= |xv} Ry %CD �

p}rO =@ h=QLv= |}=U=vW w wHwCUH u=tR pw] QO Ov}=Qi Qo = 'CU= 1 % 1 �&CU= QiY pO=at CQwY u}= Q}e QO 'OwWv hkwDt

'OwWv hkwDt p}rO =@ h=QLv= hPL w `iQ u=tR pw] QO Ov}=Qi Qo = 'CU= 1 % 2 �"CU= QiY pO=at CQwY u}= Q}e QO

|xv} Ry s}UkD =@ 'Ov}=Qi Q@ xOW p}tLD |v=tR OL=w QO Q=_Dv= |xv} Ry 'C}=yv QOx@U=Lt 14 |x]@=Q j@=]t w xQwO l} Q=_Dv= pw] Q@ xQwO l} pw] QO Q=_Dv= pm

%OwW|tEA = EC

ET(14)

QO RACUSUM pQDvm Q=Owtv |L=Q] xiOyOvJ pOt "3"2p}rO =@ h=QLv= l} Qw[L

|=yQ=Owtv |O=YDk= |L=Q] |rY= pmWt 'OW u=}@ �xtOkt� QO xm u=vJQ@ x@re Qw_vt x@ 'u}=Q@=v@ "CU= ?wr]t |Q=t; X=wN x@ xHwD sOa 'pQDvml} 'RACUSUM pQDvm Q=Owtv |O=YDk= |L=Q] QO |Q=t; X=wN ha[C=_Lqt wO Qy u=tRsy Qw] x@ xm OwW|t O=yvW}B xiOyOvJ |R} Qxt=vQ@ pOt

%s} Q=O u; |xDi=}Vy=m CQwY |=Q@ xmWi = [(1� ���)( zi

� exp(�1xi))�]� �i� log � (10)

"CU= xOW QmP CUw}B �hr=� CtUk QO 10 |x]@=Q x@ 9 |x]@=Q p}O@D |xwLvxO=iDU= =@ u=wD|t 'RACUSUM pQDvm Q=Owtv R=}Dt= x@ |@=}CUO =@ 'C}=yv QO

"Ci=} CUO u; |xQ=t; x@ |oO=Ux@ 6 |x]@=Q R=

RACUSUM pQDvm Q=Owtv |O=YDk= |L=Q] |xv} Ry `@=D "2"2p}rO =@ h=QLv= l} Qw[L QO

|xv} Ry `@=D l} |R=Uxv}tm =@ RACUSUM pQDvm Q=Owtv |=yQDt=Q=B u}}aDxrY=i� h '�xvwtv xR=Ov=� n =yQDt=Q=B u}= "OwW|t xO}t=v |O=YDk= |L=Q] '?U=vt|=Q@ CUSUM QO xOW |L=Q] ?} Q[� � w �pQDvm u}}=B OL� LCL '�|Q}oxvwtv|=yQDt=Q=B |xv}y@ |=yVRQ= u}}aD |=Q@ Q=DWwv u}= QO "OvDUy �Q}}eD |xv}y@ hWmu}= [3]"s} Q}o|t ltm Tvw w uRvQwr |xv} Ry `@=D R= RACUSUM pQDvm Q=Owtv|xv} Ry `@=D "CU= xOW xDiQo Q_v QO Ov}=Qi QO p}rO =@ h=QLv= l} Qw[L |=Q@ `@=D

%OwW|t x} RHD |rY= CtUk xU x@ |O=yvW}B&|Q}oxvwtv |=yxv} Ry �hr=

|L=QH \}=QW QO �CtqU |=ysDU}U� u=DUQ=t}@ Q@ xOW p}tLD |=yxv} Ry �?&Q=t}@ Qy |=Q@ pQDvm R= GQ=N Cr=L QO

"p}rO =@ h=QLv= |=Q@ |LqY= s=Ok= w |}=U=vW |=yxv} Ry �GOwW|t R=e; pQDvm CLD Cr=L l} R= 'xQwO l} QO Ov}=Qi xm CU= u}= Q@ ZQiu}ov=}t =@ |}=tv `} RwD l} R= p}rO =@ h=QLv= l} O=ONQ =D u=tR 'Cr=L u}= QO wpQDvm R= GQ=N Cr=L x@ Ov}=Qi 'p}rO =@ h=QLv= O=ONQ R= TB "Ovm|t |wQ}B 1=�|xQwO TBU w O@=}|t xt=O= p}rO =@ h=QLv= KqY= w uOQm =O}B u=tR =D w OwQ|tQ=_Dv= pm |xv} Ry s}UkD =@ '|v=tR OL=w QO Q=_Dv= |xv} Ry "OwW|t R=e; |O}OHQ=_Dv= pw] 'u}vJty "OwW|t x@U=Lt 'xQwO l} Q=_Dv= pw] Q@ xQwO l} pw] QO

%CU= CtUk Q=yJ pt=W xQwO l}'Qo}O CQ=@a x@ "Ov=t|t |k=@ pQDvm CLD Cr=L QO Ov}=Qi xm |v=tR COt �hr=

&CU= 1=� u}ov=}t =@ |}=tv |iO=YD Q}eDt l} u=tR COt u}=&CU= pQDvm R= GQ=N Ov}=Qi xm |v=tR COt �?

&p}rO =@ h=QLv= X}NWD w |}=U=vW |=Q@ u=tR COt �G"p}rO =@ h=QLv= KqY= CyH C=t=Ok= |=Q@ u=tR COt �O

CU= Q@=Q@ xQwO l} Q=_Dv= pw] '|v=tR CtUk Q=yJ uDiQo Q=Qk sy Q=vm =@ 'u}=Q@=v@%=@

ET = 1�

+ [(h:ARL1)� � ] + TF + TD (11)

90

2/2|

xQ=tW'

36�1

|xQwO'

�1399

u=DUtR�

h}QW

C}Q}Ot

w`}=vY

|UOvyt =@ |Q}os}tYD |=yOL=w R= uwoty |}xWwN |@Uv |}=Q=m xm |[=} Q |R} Qxt=vQ@

|xawtHt Qy 'Q=DWwv u}= QO [22]"OQ}o|t xR=Ov= =Q xv=oOvJ |=y|HwQN w |=y|OwQwxwqa "OwW|t ZQi |Q}os}tYD OL=w l} u=wvax@ |L=Q] |=yQDt=Q=B R= umtt|Q=t; hOy wO '|O=yvW}B |xiOyOvJ pOt |=Q@ DEA swyit x@ xHwD =@ 'u}= Q@'=y|HwQN u=wvax@ OvDUy |R=Uxv}W}@ CQwYx@ xm =Q 1=ARL1 w ARL0 |va}Q_v QO |OwQw =yvD u=wvax@ CU= |R=Uxv}tm CQwYx@ xm =Q (EA) xv} Ry hOy w|=yuRw CU= sRq |Q}os}tYD |=yOL=w |@Uv |}=Q=m |x@U=Lt |=Q@ "s} Q}o|t'hOy u}= x@ uO}UQ |=Q@ "OvwW xv}y@ =y|HwQN w =y|OwQw x@ xOW xO=O X=YDN=w CCR |=ypOt QO "OQm xQ=W= |atH w 9 BCC 'CCR |=ypOt x@ u=wD|tpOt QO =t= 'CU= |QwQ[ QwLt|HwQN w QwLt|OwQw |=yCr=L R}=tD BCC

w u}rw= R= |atH pOt "OwW|t x�=Q= OL=w pOt l} CQwYx@ Cr=L wO Qy |atH'pOt u}= OwN xm CU= DEA QO |}=Q=m u}}aD |=Q@ xOW x�=Q= |=ypOt u} QDsyt|=Q@ xr=kt u}= QO 'xH}Dv QO "CU= Qo}O |=ypOt R= |Q=}U@ h} QaD -=Wvt w x}=B|=yQDt=Q=B R= umtt xawtHt l} u=wvax@ |Q}os}tYD OL=w Qy |}=Q=m |x@U=Ltpt=m pOt u}= C}ty= p}qO Qo}O R= "s}vm|t xO=iDU= |atH pOt pL R= '|L=Q]|=yQ}eDt |R=Uxv}tm x@ ,=t}kDUt pOt =Q} R 'CU= u;\UwD xOW x@U=Lt |}=Q=m uOw@QO |mtm |=yQ}eDt OwHw X}NWD =ypOt Qo}O QO xm |r=L QO "OR=OQB|t |mtmswO |xrLQt u=wvax@ w |atH pOt x}@W Qo}O pOt l} |xr}Uwx@ ,=twta '|}=Q=mxm CU= u}= |atH pOt |=yC} Rt R= 'u}=Q@=v@ "OwW|t s=Hv= |}=Q=m |Q}oxR=Ov=CUO x@ pOt l} pL =@ OL=w |}=Q=m |@=} RQ= w CU}v |}xrLQtwO VwQ x@ |R=}v"OQ=O OwHw |HwQN b w |OwQw a =@ |Q}os}tYD OL=w m OwW|t ZQi "O};|tx@ '(k = 1; 2; : : : ;m)k |Q}os}tYD OL=w |=Q@ |HwQN w |OwQw |=yVRQ="OwW|t xO=O u=Wv (r = 1; 2; : : : ; b)qrk w (j = 1; 2; : : : ; a)pjk =@ ?}DQDk |Q}os}tYD OL=w |}=Q=m |x@U=Lt |=Q@ |atH pOt |[=} Q sQi 'p=L u}= =@

%R= CU= CQ=@aMax Ek(D) =Pbr=1 urqrk(D)�Pa

j=1 ejpjk(D)� Ts:t:Pbr=1 urqrk(D)�Pa

j=1 ejpjk(D)� T � 0;ur; ej � 1;T is free:

(17)

xOR=@ wv T Ctqa w OvDUy =y|HwQN w =y|OwQw u=Rw=?}DQD x@ ur w ej u; QO xmR= |@}mQD =} |Q}os}tYD OL=w Qy |=Q@ O}=@ 17 |x]@=Q "OyO|t u=Wv =Q T=}kt x@OL=w |}=Q=m |R=Uxv}W}@ |=Q@ u=Rw= R= |}xawtHt =D OwW xrwtQi |L=Q] |=yQDt=Q=B|Q}os}tYD OL=w x=ou; E�k = 1 Qo = "O}; CUO x@ 'xOW xDiQo Q_v QO |Q}os}tYD|O=yvW}B OQm} wQ QO '=Pr "CU=Q=m =v CQwY u}= Q}e QO w CU= |@Uv |=Q=m kuDi=} Qw_vtx@ xiOyOvJ pOt wDQ=B K]U |}=Q=m VHvU |=Q@ DEA R= s} Q=O OYku}= |=Q@ "OwW xO=iDU= RACUSUM pQDvm Q=Owtv |xv}y@ |L=Q] |=yQDt=Q=B|Oa@ EL@t QO xm CU= xOW xDiQo Q_v QO |}xrLQtOvJ pL sD} Qwor= l} Qw_vt

"OwW|t xO=O K}[wD

pL sD} Qwor= "3"3|xiOyOvJ pOt |R=Uxv}y@ |=Q@ |}xrLQtGvB pL sD} Qwor= l} VN@ Q} R u}= QOumtt |xawtHt Qy |=Q@ O}=@ pOt u}= QO xm OwW xHwD "OwW|t |iQat |O=yvW}B

pt=W |O=YDk= hOy l} |=Q=O pOt u}= "OQ}o|t Q_v QO =Q |Q=t; w |O=YDk=hOy wO w Ov}=Qi V}=B |=Q@ (EA) |v=tR OL=w Qy Q=_Dv= |xv} Ry |R=Uxv}tm(ARL0) pQDvm CLD \}=QW QO xr=@vO pw] \UwDt |R=Uxv}W}@ pt=W |Q=t;pw] \UwDt Twmat |R=Uxv}W}@ w |y=@DW= Q=OWy |=yO=ONQ Vy=m Qw_vt x@\}=QW `} QU X}NWD Qw_vtx@ (1=ARL1) pQDvm R= GQ=N \}=QW QO xr=@vOCU= CQ=@a |O=yvW}B |xiOyOvJ pOt |D=}[=} Q pmW "CU= pQDvm R= GQ=N

%R=Min EA(D)

Max ARL0(D)

Max 1ARL1(D)

s:t:

ARL0(D) � ARLLARL1(D) � ARLU

(15)

|xOvyOu=Wv ?}DQD x@ '|Q=t; C}OwOLt wO QO ARLU w ARLL u; QO xmR= umtt ?}mQD l} D 'u}vJty "OvDUy ARL1 |q=@ OL w ARL0 u}}=B OL?=NDv= Qw_vt x@ 'u}=Q@=v@ "CU= � w LCL 'h 'n pt=W |L=Q] |=yQDt=Q=B|=yVRQ= =@ RACUSUM pQDvm Q=Owtv |L=Q] |=yQDt=Q=B R= ?}mQD l}l} |Oa@ EL@t QO "OQm pL =Q 15 |x]@=Q |xiOyOvJ pOt u=wD|t 'xv}y@p}rLD |=yVwQ w wDQ=B pY= |R=UxJQ=Bm} T=U= Q@ |}xrLQtOvJ pL sD} Qwor=xO=O KQW 15 |x]@=Q |xiOyOvJ pOt |R=Uxv}y@ |=Q@ (DEA) =yxO=O |WWwB

"OwW|t

pL OQm} wQ "3wDQ=B pY= "1"3

CQwYx@re swyit R= xO=iDU= =@ =y?wret=v T=U= Q@ =ypLx=Q |Ov@x@DQ wDQ=B pY= QOx=Q Qy '=y?wret=v Kw]U T=U= Q@ =ypLx=Q uOQm ?DQt |=Q@ |rm Qw] x@ "OQ}o|t=} uOw@ ?wret =D 'xOW xU}=kt C}atH QO OwHwt Qo}O |=ypL x=Q s=tD =@ pLs=Om I}y xm OQ=O OwHw =ypLx=Q R= |}xawtHt C}=yv QO "OwW XNWt u; uOw@v"OvyO|t p}mWD =Q wDQ=B K]U =ypL x=Q u}= =Pr "OvwW|tv sy ?wret w ?r=eo pLx=Q x@ |DQwY QO g pLx=Q 'hOy `@=D Q =@ |R=Uxv}tm |xr�Ut l} |=Q@

%xm Ovm|t x@reg < oj8t : ft(g) � ft(o)and

9t : ft(g) < ft(o)

(16)

"OvDUy o w g |=ypL x=Q =@ \@DQt s= t h=Oy= Q}O=kt ft(o) w ft(g) u; QO xm|w=Ut =} QDlJwm g x@ \w@ Qt h=Oy= |xty xm CU= swyit u}= x@ q=@ CQ=@ag w OW=@v g R= QDy@ Q_v I}y R= o |va} "OvW=@ o x@ \w@ Qt Q_=vDt hOy `@=DK]U pL x=Q� =Q ?wret=v |=ypL x=Q "OW=@ o R= QDy@ ,=O}m = Q_v l} R= pk=OL

[21w20]"Ovt=v|t �wDQ=B

(DEA) =yxO=O |WWwB p}rLD "2"3T=U= Q@CU= |WwQ w OW |iQat ROwQ w QB wm 'RvQ=J\UwD Q=@u}rw= 8 CCR VwQ

91

"""pQDvm

Q=Owtvl

}|Q=t;

�|O=Y

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10000 |R=Ux}@W '=y=]N |R=Uxv}tm Qw_vt x@ xm CU= QmP x@ sRq "Ot;QO =QH= x@QO |Q}}eD uDiQo Q_v QO uwO@ '|L=Q] |=yQDt=Q=B R= ?}mQD Qy |=Q@ "OW Q=QmD Q=@OwHw ?@U x@ xm |v=tR |=Q@ 'u}vJty w OW x@U=Lt ARL0 |=yVRQ= '=yxO=OARL1 |=yVRQ= 'CW=O OwHw =k@ u=tR QO Vy=m �95 |v=Uv= p}rO =@ h=QLv=|@=}CUO w |y=@DW= Q=OWy |q=@ |=yO=ONQ MQv R= |Q}owrH Qw_vt x@ "OW C@FARL1 |q=@ OL w 20 =@ Q@=Q@ ARL0 u}}=B OL 'Q}}eD hWm pw@k p@=k CQOk x@xm OW p=ta= |L=Q] QDt=Q=B wO |wQ |OwOL 'u}= Q@ xwqa "OW xDiQo Q_v QO 5 =@ Q@=Q@Q=_Dv= |xv} Ry x@ |@=}CUO 'p=L u}= =@ "OW xO=O KQW pL sD} Qwor= QO u; \}=QW|L=QH RmQt QO RACUSUM pQDvm Q=Owtv R= xO=iDU= x@ \w@ Qt |v=tR OL=w Qy|Q=t; � |O=YDk= pOt |R=Uxv}y@ Qw_vt x@ 'C}=yv QO "OW Oy=wN umtt ?rkR= umtt ?}mQD u} QDy@ x@ |@=}CUO w RACUSUM pQDvm Q=Owtv |xiOyOvJC=@U=Lt |t=tD 'CU= QmP x@ sRq "OW =QH= pL sD} Qwor= '|L=Q] |=yQDt=Q=BR= TB "OW p}yUD 2016 xNUv MATLAB \}Lt QO xOW Om |=yxt=vQ@ CLDxO=iDU= =@ 'OW XNWt (LCL; �) ?}mQD CQwYx@ xwkr=@ pL x=Q 1000 xm u}=u=}t R= 'TBU "OW xDW=Po Q=vm umttQ}e |=ypL x=Q |Q=t; |=yC}OwOLt R=wDQ=B K]U u=wvax@ pLx=Q 36 wDQ=B pY= R= xO=iDU= =@ 'xOv=t|k=@ |=ypL x=Q'?wret=v |=ypL x=Q uOW XNWt R= TB "OW ?=NDv= ?wret=v |=ypL x=Q =}|=Q@ |L=Q] |=yQDt=Q=B u=ty =} pL x=Q u} QD=Q=m ?=NDv= w =yv; |Ov@C} wrw= |=Q@'DEA QO "OW xO=iDU= DEA |=yVwQ R= RACUSUM pQDvm Q=Owtv Q=QkDU=|Q}os}tYD OL=w l} u=wvax@ '(LCL; �) |va} |L=Q] |=yQDt=Q=B R= ?}mQD QyQ=Okt u} QDsm |=Q=O *|Q}os}tYD OL=w 'DEA |=yVwQ uwJ "OW xDiQo Q_v QOxar=]t u}= QO 'Ovm|t ?=NDv= =Q=m OL=w u=wvax@ =Q |HwQN Q=Okt u} QDW}@ w |OwQw

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OQwt u=Q=t}@ |=k@ u=tR u}ov=}t QO |Wy=m C=Q}}eD hWm |=Q@ xm CU= xiQ]l}|xr�Ut Q@ x@re w lU} Q p}OaD |=Q@ Q=Owtv u}= QO 'u}vJty "CiQo Q=Qk xHwD|=ylU} Q u}@ \=@DQ= xm OW xO=iDU= pw@} w AFT pOt R= u=Q=t}@ u=}t |vwoty=v'xr=kt u}= |rY= hOy "OQm|t XNWt =Q Q=t}@ Qy |=k@ u=tR |wQ pta R= p@kw |O=YDk= X=wN xm |}xvwox@ Ow@ RACUSUM pQDvm Q=Owtv |=yQDt=Q=B u}}aD|xrY=i w xvwtv xR=Ov= uOw@ C@=F x@ xHwD =@ "Ovm xOQw;Q@ u=tRty Qw] x@ =Q |Q=t;|xrO=@t u} QDy@ xm OvOW u}}aD |}xvwox@ (LCL; �) |L=Q] |=yQDt=Q=B '|Q}oxvwtvuOWXNWt |=Q@ "ODi}@ j=iD= pOt |Q=t; w |O=YDk= hOy xU u}@ umtt pO=aDt|R=Uxv}y@ pY= T=U=Q@|}xrLQtOvJ pL sD} Qwor=l} R= xv}y@|L=Q] |=yQDt=Q=Bu=wva x@ =yQDt=Q=B R= ?}mQD Qy DEA QO "OW xO=iDU= DEA |=yVwQ w wDQ=BxiOyOvJ pOt |=yQDt=Q=B |xv}y@ ?}mQD w OW xDiQo Q_v QO |Q}os}tYD OL=w l}R= |O=yvW}B OQm} wQ OQ@ Q=m 'C}=yv QO "OW XNWt `]=kDt |}=Q=m |Q}oxR=Ov= =@|L=Q] TBU w OW xO=O u=Wv ?rk |L=QH RmQt QO |OQwt |xar=]t l} j} Q]u=Wv xrY=L G}=Dv "CiQo Q=Qk xU}=kt OQwt |O=YDk= Xr=N |L=Q] =@ xiOyOvJ"OQ=O |QDy@ |Q=t; OQmrta 'xv} Ry Q=Okt QO |tm V}=Ri= =@ xiOyOvJ |L=Q] O=OQw] x@ Ov=wD|t RACUSUM pQDvm Q=Owtv |L=Q] QO |O=yvW}B OQm} wQ 'pm QOQ=Qk xO=iDU= OQwt u=Q=t}@ |=k@ u=tR COt QO |Wy=m C=Q}}eD hWm Qw_vt x@ QF-wt

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xU}=kt QO 'p=} Q 1457098 46 Q=Okt =@ xiOyOvJ |Q=t; |O=YDk= |L=Q] |v=tR1438149 52 Q=Okt =@ |O=YDk= Xr=N |L=Q] |v=tR OL=w Qy Q=_Dv= |xv} Ry =@xv} Ry QO V}=Ri= u}= 'p=L u}= =@ "CU= xDi=} V}=Ri= OYQO 1 3 u=R}t x@ 'p=} Q49 391 Q=Okt =@ xiOyOvJ |Q=t; |O=YDk= |L=Q] ARL0 xm OwW|t Ea=@,=@} QkD u=R}t x@ '24 041 Q=Okt =@ |O=YDk= Xr=N |L=Q] ARL0 =@ xU}=kt QO|L=Q] wv wO Qy |=Q@ ARL1 u}@ \}=QW u}= QO 'u}vJty "O@=} V}=Ri= �100,=D@Uv V}=Ri= =@ xm OyO|t u=Wv xU}=kt u}= |xH}Dv "OQ=Ov OwHw |Q}osWJ Cw=iDV}=Ri= |}x_Lqt p@=k Qw] x@ ARL0 '|v=tR OL=w Qy Q=_Dv= |xv} Ry QO |tmxiOyOvJ |Q=t; |O=YDk= |L=Q] R= xO=iDU= =@ xm CU=vat u=O@ Qt= u}= "CU= xDi=}Q=Owtv |Q=t; OQmrta '|v=tR OL=w Qy Q=_Dv= |xv} Ry QO |tm V}=Ri= =@ u=wD|t|}x_Lqt p@=k Qw]x@ =Q p}rO =@ h=QLv= l} Qw[L QO RACUSUM pQDvm

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"OQ=O

|Q}oxH}Dv "5V}=B |=Q@ RACUSUM pQDvm Q=Owtv xiOyOvJ |L=Q] pOt 'xr=kt u}= QO=@ h=QLv= l} Q}F-=D CLD 'pOt u}= "OW O=yvW}B CtqU |=ysDU}U QO =y|HwQN|Wy=m C=Q}}eD O=H}= Ea=@ w xOw@ |v=Uv= |=y=]N R= |W=v xm CW=O Q=Qk p}rOQ=Owtv l} xOW |L=Q] pQDvm Q=Owtv 'wQ u}= R= "OW|t u=Q=t}@ |=k@ u=tR QO

=yCWwv=B1. cumulative sum (CUSUM)2. risk-adjusted cumulative sum3. in-control average run length (ARL0)4. out-of-control average run length (ARL1)5. pareto principle

6. data envelopment analysis7. accelerated failure time (AFT)8. Charnes,Cooper& Rhodes (CCR)9. Banker, Charnes & Cooper (BCC)10. coronary artery bypass grafting11. morbidity12. mortality

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�References� `@=vt1. Woodall, W.H., Fogel, S.L. and Steiner, S.H. \The mon-

itoring and improvement of surgical-outcome quality",J. Qual. Technol, 47(4), pp. 383-399 (2015).

2. Duncan, A.J. \The economic design of �X charts usedto maintain current control of a process", J. Am. Stat.Assoc, 51(274), pp. 228-242 (1956).

3. Lorenzen, T.J. and Vance, L.C. \The economic designof control charts: a uni�ed approach", Technometrics,28(1), pp. 3-10 (1986).

4. Jiao, J.R. and Helo, P.T. \Optimization design of aCUSUM control chart based on Taguchi's loss func-tion", Int. J. Adv. Manuf. Tech, 35(11-12), pp. 1234-1243 (2008).

5. Lee, M.H. \Economic design of cumulative sum controlcharts for monitoring a process with correlated samples",Commun. Stat-Simul. C, 39(10), pp. 1909-1922 (2010).

6. Celano, G., Castagliola, P. and Trovato, E. \The eco-nomic performance of a CUSUM t control chart for moni-toring short production runs", Qual. Technol. Quant. M,9(4), pp. 329-354 (2012).

7. Saniga, E.M. \Economic statistical control chart designswith an application to �X and R charts", Technometrics,31(3), pp. 313-320 (1989).

8. Asadzadeh, S. and Khoshalhan, F. \Multiple-objectivedesign of an �X control chart with multiple assignablecauses", Int. J. Adv. Manuf. Tech, 43(3-4), pp. 312-322(2009).

9. Niaki, S.T.A., Malaki, M. and Ershadi, M.J. \A par-ticle swarm optimization approach on economic andeconomic-statistical designs of MEWMA control charts",Sci. Iran, 18(6), pp. 1529-1536 (2011).

10. Rasmussen, T.B, Ulrichsen, S.P. and N�rgaard, M. \Useof risk-adjusted CUSUM charts to monitor 30-day mor-tality in Danish hospital", J. Clin. Epidemiol, 10, pp.445-456 (2018).

11. Harris J.R., Forbes, T.L., Steiner, S.H. and et al. \Risk-adjusted analysis of early mortality after ruptured ab-dominal aortic aneurysm repair", J. Vasc. Surg, 42(3),pp. 387-391 (2005).

12. Keefe, M.J., Loda, J.B., Elhabashy, A.E. and et al. \Im-proved implementation of the risk-adjusted BernoulliCUSUM chart to monitor surgical outcome quality", Int.J. Qual. Health. C, 29(3), pp. 343-348 (2017).

13. Tzeng, G.H. and Huang, J.J. \Fuzzy multiple objectivedecision making", CRC Press (2013).

14. Liu, H.H., Song, Y.Y. and Yang, G.L. \Cross-e�ciencyevaluation in data envelopment analysis based onprospect theory", Eur. J. Oper. Res, 273(1), pp. 364-375 (2019).

15. Chowdhury, H. and Zelenyuk, V. \Performance of hos-pital services in Ontario: DEA with truncated regressionapproach", Omega, 63, pp. 111-122 (2016).

16. Omrani, H., Keshavarz, M. and Ghaderi, S.F. \Eval-uation of supply chain of a shipping company in Iranby a fuzzy relational network data envelopment analysismodel", Sci. Iran, 25(2), pp. 868-890 (2018).

17. Sego, L.H., Reynolds, M.R. and Woodall, W.H. \Risk-adjusted monitoring of survival times", Stat. Med, 28(9),pp. 1386-1401 (2009).

18. Asadzadeh, S. and Baghaei, A. \Robust AFT-basedmonitoring procedures for reliability data", Qual. Tech-nol. Quant. M, 17 (1), pp. 1-15 (2020).

19. Asadzadeh, S. and Aghaie, A. \Improving the productreliability in multistage manufacturing and service op-erations", Qual. Reliab. Eng. Int, 28(4), pp. 397-407(2012).

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21. Noghin, V.D., Reduction of the Pareto Set. An Ax-iomatic Approach, Springer Int. Publishing AG (2018).

22. Charnes, A., Cooper, W.W. and Rhodes, E. \Measuringthe e�ciency of decision making units", Eur. J. Oper.Res, 2 (6), pp. 429-444 (1978).

23. Bernstein, A.D. and Parsonnet, V. \Bedside estimationof risk as an aid for decision-making in cardiac surgery",Ann. Thorac. Surg, 69(3), pp. 823-828 (2000).

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