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sDU}U xJu=vJ w OwW xOQw;Q@ xQwO u=ty QO O}=@ xQwO Qy |=[=kD "OwW|t u}at'xOWv xOQw;Q@ |=[=kD R= OL=w Qy |=R=x@ OW=@v =[=kD R= |WN@ O}rwD x@ QO=k O}rwDsDU}U pm '|R=UpOt QO|DL=Q |=Q@ "OwW|t p}tLD u; x@ OL=w h pO=at|}xt} QHwQ G} QOD x@ pwYLt O}rwD QF= QO u}W=t "CU= xOW xDiQo Q_v QO u}W=t l} O}rwDxm CU= xOW xDiQo Q_v QO |@=QN R= |irDNt Kw]U u}W=t |=Q@ "OwQ|t p=wR x@|xawtHt S = f0; 1; 2; : : : ; Ng "OwW|t xO}t=v Cr=L l} u; R= K]U Qyu}W=t p=wR Ov}=Qi "CU= |y=vDt |}xawtHt xm OyO|t u=Wv =Q u}W=t Cq=L|xOvyOu=Wv X = (Xt : t 2 [0;1)) Qo = '|va} "CU= |iwmQ=t Q=DiQ |=Q=Ou=tR uoty hwmQ=t Ov}=Qi l} X xm CU= u; Q@ ZQi 'OW=@ u}W=t p=wR Ov}=QiCr=L N R= |m} QO CU= umtt u}W=t "OW=@ S |xDUUo Cr=L |=[i =@ xDUw}BQou=}@ QiY Cr=L "OW=@ N |@=QN Cr=L QO =} f0; 1; 2; : : : ; N �1g |D=}rtaCr=L QO u}W=t "CU= ?=QN u}W=t l} |xOvyOu=Wv N Cr=L w wv u}W=t l}Q=Okt u} QDW}@ Cr=L u}= QO O}rwD MQv w OQ=O Q=Qk OwN C}a[w u} QDy@ QO QiYu; QO O}rwD MQv w OQ=O Q=Qk C}a[w u} QDO@ QO N Cr=L QO w OQ=O =Q umttu=O@ OW=@ QDnQR@ Cr=L |xOvvm XNWt OOa xJ Qy 'Cq=L Q}=U QO "CU= QiYTB "CU= QDlJwm u; O}rwD MQv w OQ=O Q=Qk |QDO@ C}a[w QO u}W=t xm CU=vats x@ C@Uv r(s) x=ov; s}yO u=Wv r(s) =@ =Q s Cr=L QO u}W=t O}rwD MQv Qo =

"CU= |rwRv ,=O}m ==@ \}=QW Q@|vD@t C=Q}taD w |Q=Oyov OQm} wQ R= xO=iDU= =@ xm CU= u; hOy|QwQ[Q}e xv=Q}oV}BCvC=}rta R=u}W=t OQmrta Ow@y@ut[ xDUUo|=y|UQR=@

"OwW Q=QkQ@ xvR=wt xDiQ CUO R= O}rwD w xv=Q}oV}B Cv '|UQR=@ u}@ w |Q}owrH|R} Qxt=vQ@ pw= |xrLQt "CU= xrLQt wO pt=W OQm} wQ u}= OW xDio xm Qw]u=tyCv C=}rta |=QH= u=tR XwYN QO |Q}os}tYD swO |xrLQt w CU=y|UQR=@'xr�Ut u}= QO "CU=y|UQR=@ R= xOt; CUO x@ |xH}Dv x@ xHwD =@ xv=Q}oV}Bu}W=t Cr=L |UQR=@ C=}rta s=Hv= =@ xm CU= xOW ZQi pw= |xrLQt |=Q@x@ |UQR=@ C=}rta s=Hv= Q=@ Qy |xv} Ry w u=tR "OwW xO=O X}NWD |DUQO x@"OwW|t xO=O V}=tv cins w tins =@ ?}DQD x@ w OQ=Ov |oDU@ u}W=t Cr=LQO 'OwW s=Hv= O}=@ |UQR=@ C=}rta |R} Qxt=vQ@ ji= R=e; |va} pw= |xQwO |=OD@=w OvDUy |UQR=@ C=}rta s=Hv= |=Q@ |}=yxv} Ro 'Qo}O |=yxQwO |=OD@= xm |r=LQO "OvwW ?=NDv= |UQR=@ C=}rta s=Hv= |=Q@ =yxQwO s=Om OwW xDiQo s}tYD O}=@xv=Q}oV}B Cv |=QH= |=Q@ xv}y@ u=tR |UQR=@ |xH}Dv x@ xHwD =@ 'swO |xrLQt|=yxv} Ry wtHt xm OwW P=ND= |}xvwo x@ O}=@ C=t}tYD u}= "OwW u}}aD O}=@xv}tm |R} Qxt=vQ@ ji= pw] QO xDiQ CUO R= O}rwD w xv=Q}oV}B Cv '|UQR=@

"OwW|UQR=@ |=yxv} Ry |wQ s}kDUt Qw]x@ =y|UQR=@ XwYN QO |Q}os}tYDu=tR =Q} R &CU= s}kDUtQ}e =yxv} Ry Q}=U |wQ u; Q}F-=D xm |r=L QO 'OQ=Po|t Q}F-=Dw xv} Ry "OwW|t XNWt |UQR=@ C=}rta s=Hv= R= Oa@ \ki xv=Q}oV}B Cv |=QH=xDU@=w =QH= |x_Lr QO u}W=t Cr=L x@ xv=Q}oV}B Cv C=}rta |=QH= u=tR COttpm(s) w cpm(s) "OvwW|t xO=O V}=tv tpm(s) w cpm(s) =@ ?}DQD x@ w CU=Cr=L QO u}W=t |Dkw OwW|t ZQi u}vJty "OvDUy |OwaY ,=O}m = s x@ C@UvR= O}rwD xv} Ry "OW=@ QiY xv=Q}oV}B Cv |=QH= u=tR COt w xv} Ry 'CU= QiYuOw@ TQDUO p@=k Q}e =} u}W=t p=wR p}rO x@ xm CU= |}xv} Ry 'xDiQ CUO"OwW|t p}tLD sDU}U x@ xv=Q}oW}B Cv =} |UQR=@ C=}rta |=QH= u=tR QO u;Cr=L w O}rwD MQv u}@ |oDU@=w O=H}= Q@xwqa 'xv} Ry u}= uDiQo Q_v QO |=Q@

pOt QO xv=Q}oV}B Cv |=QH= |=Q@ sRq u=tR COt u}vJty "CU= xOWv xO}O pOt"CU= xOWv ^=Lr

|UQR=@ x@ \w@ Qt C=t}tYD u; QO xm |k}kLD 's}=xOQm |UQQ@ =t xm =Hu; =D'OwW P=ND= =[=kD Qw[L QO w u=tRsy Qw]x@ xv=Q}oV}B C=Q}taD w |Q=Oyov w"CW=O OwHw u=kkLt |=B V}B Q}Ut wO w[wt u}= x@ uDN=OQB |=Q@ "OQ=Ov OwHwxmu; Qo}O w OwW xO=O xaUwD =[=kD Qw[L QO CBM |xRwL |=ypOt xmu; |m}=[=kD Qw[L QO =Q xv=Q}oV}B Cv xm |}=ypOt x@ |UQR=@ x@ \w@ Qt C=t}tYDpOt w xOQm ?=NDv= =Q swO Q}Ut j}kLD u}= QO =t "OwW xi=[= 'Ovvm|t |R} Qxt=vQ@

[23]"s}=xO=O xaUwD =Q xOW x�=Q=|=Q@ pOt l} '|iO=YD |=} wB |R} Qxt=vQ@ VwQ R= xO=iDU= =@ xr=kt u}= QO|y=vDt ji= QO w |vi K]U QO \}=QW Q@|vD@t C=Q}taD w |Q=Oyov |R} Qxt=vQ@pOt QO s}tYD Q}eDt u=wvax@ w Ov=xDUUo =y|UQR=@ 'pOt u}= QO "CU= xOW x�=Q="O};|tCUO x@ umtt |UQR=@ |=yKQ] u}@ R= xv}y@ |UQR=@ KQ] w Ov=xOW xOv=Hvo|=yxv} Ry 'xv}y@ |UQR=@ KQ] uOQw; CUO x@ QO xm CU= |}xvwo x@ xOW x�=Q= pOtu}vJty "CU= xOW xDiQo Q_v QO xDiQ CUO R= O}rwD w xv=Q}oV}B Cv w |UQR=@pY=L |UQR=@ |xH}Dv x@ xHwD =@ xv=Q}oV}B Cv C=}rta |=QH= |=Q@ u=tR u} QDy@Qw]x@ xv=Q}oV}B Cv w |UQR=@ |=QH= |=Q@ xv}y@ u=tR 'Qo}O CQ=@ax@ "OwW|tGwR pOt Cr=L Q}eDt "OwW|t Q=QkQ@ xvR=wt =yv; u}@ w OwW|t XNWt u=tRsy=[=kD "OyO|t u=Wv =Q u}W=t C}a[w s w =[=kD z u; QO xm CU= (s; z) ?DQts=Hv= R= Oa@ u}W=t C}a[w w CU= x=wNrO `} RwD =@ xDUUo |iO=YD Q}eDt l}

"OwW|t XNWt |UQR=@%R= CU= CQ=@a [23]2014 p=U QO xOW x�=Q= pOt =@ =t pOt Cw=iD u}=Q@=v@ji= QO |R} Qxt=vQ@ 'OwW|t ?Hwt xm pOt x@ |UQR=@ C=t}tYD uOQm xi=[= �

"OQ}o CQwY |y=vDt"CU= xOW ^=Lr pOt QO xv=Q}oV}B Cv w |UQR=@ |=QH= |=Q@ sRq u=tR COt �,qt=m xQwO Qy |=OD@= QO Cr=L Q}eDt xm CU= u; Q@ ZQi [23]`HQt x@ xHwD =@ �xm |UQR=@ u=tR QO \ki Cr=L Q}eDt =t pOt QO xm |r=L QO &OwW|t u}atxm CU= |rY= Vr=J u=ty u}= w OwW|t XNWt 'CU= s}tYD Q}eDt OwN"CU= xOW xDN=OQB u; x@ j}kLD u}= QO w OQ=O OwHw xr�Ut u}= |R=UpOt QO|=Q@ xm CU= u; CBM |xRwL |=ypOt =@ =t pOt |rY= Cw=iD 'u}vJtyx@ u}W=t C}a[w Q@xwqa xv=Q}oV}B Cv w |UQR=@ u=tR OQwt QO |Q}os}tYDw O}rwD |xRwL wO u}@ |ovy=ty ?Hwt Qt= u}= xm OQ=O xHwD R}v |QDWt |=[=kD

"OwW|t |vi K]U QO Cv|=yVN@ QO "CU= xOW u=}@ \w@ Qt |=yZQi w xr�Ut '2VN@ QO w xt=O= QOQDW}@ K} QWD |=Q@ "OwW|t x�=Q= p}YiD x@ u; pL VwQ w |O=yvW}B pOt 5 w 4 '3xOW EL@ pOt |HvUQ=@Da= OQwt QO w |L=Q] |OOa p=Ft l} 6 VN@ QO 'pOt"CU= xOW u=}@ |D; C=k}kLD |=Q@ |}=yO=yvW}B w |Q}oxH}Dv 7 VN@ QO w CU=

C=[wQit w xr�Ut u=}@ "2VQ=iU |=v@tQ@ O}rwD u; QO xm O} Q}o@ Q_v QO =Q |rwYLtlD O}rwD sDU}U l}|xQwO K x@ w CU= |y=vDt |R} Qxt=vQ@ ji= O}vm ZQi "OwW|t s=Hv= |QDWt'|R} Qxt=vQ@ ji= |=OD@= QO "CU= xOW R=Qi= T pw] x@l} Qy xrY=ir=|w=UDt |v=tRp=tDL= `} RwD |=Q=O xm CU= � s=v x@ xDUUo |iO=YD Q}eDt l} xQwO Qy |=[=kD

153

"""u=tRs

y|R}Qx

t=vQ@|=Q@

|[=}Q

pOtl}

|x�=Q=w

ik = 0 8k =2 fk0; k1; � � � ; km(I)g%O}vm h} QaD Q} R CQwYx@ =Q 'J(I) = (j(I)n)m(I)

n=0 |xr=@vO 'p=LJ(I) = (j(I)0; j(I)1; � � � ; j(I)m(I)) =

(K � km(I);K � km(I)�1; � � � ;K � k1;K � k0)

pL=Qt |xr=@vO 'Ovm|t XNWt =Q pL=Qt s}kDUt Q}e Qw]x@ xm J(I) |xr=@vOQw_vtx@ 1 p=Ft "Ovm|t XNWt =Q pL=Qt O=OaD 'xr=@vO u}= CqtH O=OaD "s}t=v|t

"CU= xOW |L=Q] pL=Qt |xr=@vO h} QaD QDy@ lQO|=Q@ |UQR=@ KQ] l} I = (1; 0; 0; 1; 0; 1; 1) O}vm ZQi �1 p=Ftu}= QO OwW|t xO}O 1 pmW QO xm Qw]u=ty "OW=@ |v=tR |xQwO 6 =@ |}xr�Ut"OwW|t s=Hv= sWW w sQ=yJ w pw= |xQwO |=OD@= QO =y|UQR=@ '|UQR=@ KQ]h} QaD =Q pL=Qt u=wD@ =D OwW|t xDiQo Q_v QO l} xQ=wty I |xr=@vO |xrtH u} QN;|xr=@vO Q} R "CU= xOW xO=O u=Wv x}k@ R= Cw=iDt pmW x@ p}rO u}ty x@ w OQmk = (0; 3; 5; 6) |xr=@vO w I |xr=@vO *l} CqtH s=tDpt=W �I = (1; 1; 1; 1)

I =@ Q_=vDt pL=Qt |xr=@vO "O} Q}o@ Q_v QO =Q I |xr=@vO *l} CqtH T}Ov= pt=W%O};|t CUO x@ Q} R CQwYx@

J(I) = (6� 6; 6� 5; 6� 3; 6� 0) = (0; 1; 3; 6)

I = |UQR=@ KQ] |=R= x@ OyO|t u=Wv J(I) = (0; 1; 3; 6) |xr=@vO"CU= xrLQt 4 pt=W |r=tDL= |=} wB |R} Qxt=vQ@ pOt '(1; 0; 0; 1; 0; 1; 1)

swO |xrLQt "CU= xOv=t |k=@ |R} Qxt=vQ@ ji= u=}=B =D xQwO QiY pw= |xrLQt QO|v=tR swU |xrLQt "CU= xOv=t |R} Qxt=vQ@ ji= u=}=B =D xQwO 1 xm CU= |v=tR6 sQ=yJ |xrLQt QO w CU= xOv=t |k=@ |R} Qxt=vQ@ ji= u=}=B =D xQwO 3 xm CU=

"CU= xOv=t |k=@ |R} Qxt=vQ@ ji= u=}=B =D xQwO�sDU}U Cr=L =} Cr=L Q}eDt� swO <RH

u}= QO "OwW|t h} QaD pOt Cr=L Q}eDt u=wva x@ (s; z) 2 S�R� ?DQt GwRCr=L |=[i S '�=[=kD� � |iO=YD Q}eDt wkw z 'u}W=t C}a[w s 'CQ=@a

"CU= � |iO=YD Q}eDt OQ@ R� w u}W=t�s}tYD Q}eDt� swU <RH

QO Ovm|t XNWt pw= wv &OQ=O OwHw s}tYD Q}eDt wv wO Q_v OQwt |xr�Ut QO=@ xrLQt Qy QO Ovm|tXNWt swO wv w OQm|UQR=@ =Q u}W=t O}=@ xQwO s=Om |=OD@=|xrLQt� |Oa@|UQR=@ u=tR =D �=[=kD Q=Okt w u}W=tC}a[w�Cr=L Q}eDt x@ xHwD|UQR=@ KQ] Qy 'CQwY u}= QO "OQm =QH= =Q xv=Q}oV}B Cv O}=@ |}xQwO xJ QO �Oa@I = (ik)Kk=0 O}vm ZQi "CU= pw= wv s}tYD |=yQ}eDt R= |y=vDt |}xr=@vO|xr=@vO J(I) = (j(I)n)m(I)

n=0 w C@=F Oa@ x@ u}= R= w x=wNrO |UQR=@ KQ] l}'s)=n |xrLQt QO "OW=@ |iO=YD |=} wB |R} Qxt=vQ@ pOt QO u; =@ Q_=vDt pL=QtCQwYx@ |UQR=@ KQ] u}= x@ xDU@=w swO wv C=t}tYD x@ \w@ Qt s}tYD |=[i

"1 p=Ft |UQR=@ KQ] x@ \w@ Qt Q=Owtv "1 pmW

pOt QO 'OwW|t xv=Q}oV}B Cv w |UQR=@ C=}rta hQY xm |v=tR COt 'u}W=t=@ |iO=YD |=} wB |R} Qxt=vQ@ |x}=B pOt =OD@= Oa@ VN@ QO "CU= xOW ^=Lr|R=UpOt ?r=k u}= QO Q[=L |xr�Ut TBU 'OwW|t xO=O K}[wD |y=vDt ji=

"OwW|t

xr�Ut |O=yvW}B pOt "3uDN=U ' w[wt u} QDsyt '|y=vDt ji= =@ |iO=YD |=} wB |R} Qxt=vQ@ |x}=B pOt QOx]@=Q u}= |=RH= "CU= ?ka x@ wQ |DWoR=@ |x]@=Q l} xm CU= |ov}y@ |xrO=atT} QD=t '(a) s}tYD Q}eDt (s0 =} s) Cr=L Q}eDt '(n) |xrLQt R= Ov=CQ=@a|Dkw |rai |xrLQt |xv} Ry w (p(a)) xOW P=ND= a s}tYD xm |Dkw p=kDv= p=tDL=|rm Qw]x@ "(C(s; a)) CU= xOW P=ND= a s}tYD w CU= s sDU}U Cr=L xm

%CU= xOW xO=O u=Wv 1 |x]@=Q QO x}=B pOt |ov}y@ |xrO=atVn(s) = min

a2AfC(s; a) +Xs02S

Pss0(a)Vn�1(s0)g(8n � 1) (1)

QO =Q pm |xv} Ry \UwDt vn(s) w s}tYD |=[i A 'Cr=L |=[i S u; QO xm|R} Qxt=vQ@ ji= u=}=B =D xrLQt n w OW=@ s sDU}U Cr=L xm OyO|t u=Wv |]}=QW

[24]"v:(s) = 0 (8s 2 S) OwW|t ZQi w "OW=@ xOv=t|k=@|R} Qxt=vQ@ ?r=k QO p@k VN@ QO xOW u=}@ |xr�Ut |R=UpOt Qw_vtx@ p=Lh} QaD Q} R CQwYx@ =Q |ov}y@ |xrO=at |=RH= '|y=vDt ji= =@ |iO=YD |=} wB

"s}vm|t�xrLQt� pw= <RH

xDiQo Q_v QO xrLQt u=wvax@ OwW xDiQo |UQR=@ x@ s}tYD xm u=tR R= x]kv Qy|R} Qxt=vQ@ ji= u=}=B =D xOv=t|k=@ |=yxQwO O=OaD 'xOW x�=Q= pOt Q=DN=U QO "OwW|t"CU= xOW xOQw; xt=O= QO u; |x@U=Lt |xwLv xm CU= R=}v OQwt xrLQt Qy QO

|=QH= |=Q@ |}xv} Ro |v=tR |xQwO Qy |=OD@= =t |xr�Ut QO xmu}= x@ xHwD =@|UQR=@ 'OwW s=Hv=|UQR=@ O}=@ pw= |xQwO |=OD@= QO xmu}= ZQi =@ wCU=y|UQR=@|xawtHt Qo = TB "CiQo Q_v QO |} wOwO s}tYD Q}eDt l} u=wD|t =Q xQwO Qy QO2k�1 =@ Q@=Q@ I |=[a= O=OaD 's}yO u=Wv I =@ =Q =y|UQR=@ |=Q@ |vOW |=y?=wHl} =yu; R= l} Qy x@ w CU= |R} Qxt=vQ@ ji= |=yxQwO O=OaD k u; QO xm CU=I = (ik)Kk=0 =@ u=wD|t =Q |UQR=@ KQ] Qy 'u}=Q@=v@ "OwW|t xDio |UQR=@ KQ]'O=O V}=tv CU= 1 =} 0 |xrtH K + 1 R= pmWDt |y=vDt |xr=@vO l} xm

%xm|Qw]x@I = (i0; i1; : : : ; iK) = (ik)Kk=0; i0 = iK = 1xDiQo Q_v QO 1 |w=Ut xQ=wty '|R=UpOt QO CrwyU p}rO x@ 'I *QN; |xrtHC@=F Oa@ x@ u}= R= w x=wNrO |UQR=@ KQ] l} I = (ik)Kk=0 O}vm ZQi "OwW|t"O} Q}o@ Q_v QO I |xr=@vO *l} CqtH s=tD pt=W I R= |}xr=@vO Q} R =Q �I "OW=@

'|va}�I = (�ik0 ;�ik1 ; � � � ;�ikm(I)) = (�iki)

m(I)i=0

'u; QO xm�iki = 1 8i 2 f0; 1; � � � ;m(I)g

154

2/2|

xQ=tW'

35�1

|xQwO'

�1398

u=DUtR�

h}QW

C}Q}Ot

w`}=vY

|UOvyt |xQwO l |UQR=@ wO u}@ |xrY=i w OW=@ z =[=kD w s C}a[w QO u}W=t |rai

xm 'OwW =QH= xv=Q}oV}B Cv =yu; s)=a |xQwO QO OwW xDiQo s}tYD w OW=@ |v=tR\@=wQ R= �Cl(s; z; a) |x@U=Lt |=Q@ "a = 0; : : : ; l w l = 1; : : : ;K u; QOl} w QiY u}@ |OOa w p} RvD ?} Q[ � '8�6 \@=wQ QO "OwW|t xO=iDU= '8�6|xQwO l} QO O}rwD |=Q@ |ak=w u=tR ?}DQD x@ TA2(s; b) w TA1(s; b) &CU=|=OD@= QO xm |DQwY QO OvDUy u; |=QH= sOa w |UQR=@ |=QH= CQwY QO |v=tRw OwW P=ND= xv=Q}oV}B Cv XwYN QO b s}tYD w OW=@ s Cr=L QO u}W=t xQwO

%OvwW|t x@U=Lt Q} R CQwYx@TA1(s; b) = T � tins � btpm(s); b 2 f0; 1g

TA2(s; b) = T � btpm(s); b 2 f0; 1gu}W=t xQwO |=OD@= QO xm |DQwY QO CU= xQwO l} QO u}W=t O}rwD MQv r(s; b)Q} R CQwYx@ w OwW P=ND= xv=Q}oV}B Cv XwYN QO b s}tYD w OW=@ sCr=L QO

%OwW|t x@U=Lt

r(s; b) =

8<:r(s) b = 0r(0) b = 1

%OwW|t h} QaD Q} R CQwYx@ xm CU= Qou=Wv `@=D l} '�h�h : fh; � � � ;Kg ! f0; 1g

�h(l) =

8<:0 l = h

1 l � h+ 1'OW=@ C@Ft RDv=QB pN=O Qo = xm CU=vat u=O@ RDv=QB |q=@ + *Ctqa CQ=@a l} QO wCQ=@a u; CQwY u}= Q}e QO w CU= Q@=Q@ RDv=QB pN=O Q=Okt =@ CQ=@a u; Q=Okt

"OQ}o|t QiY Q=Oktxm Ovm|t x@U=Lt |]}=QW QO =Q |r=wDt |UQR=@ wO u}@ |xv} Ry '6 |x]@=QCv w 'l |UQR=@ wO u}@ |xrY=i 'z =[=kD 's C}a[w QO u}W=t |rai |UQR=@ QO

"OwWv =QH= xrY=i u}= QO xv=Q}oV}B�Cl(s; z; 0) =

cins + h(z � TA1(s; 0)r(s; 0))+

+�1(l)l�1Xw=1

�wNXs0=0

(P0w)s;s0X�2R�

g�(�)h(� � TA2(s0; 0)r(s0; 0))+

(l 2 f1; � � � ;Kg ; s 2 S; z 2 R�) (6)

"CU= xOW xO=O K}[wD 1 pwOH QO x]@=Q u}= |=RH=u}W=t |rai |UQR=@ QO xm Ovm|t x@U=Lt |]}=QW QO =Q xv} Ry u}= '7 |x]@=Qpw= |xQwO QO xv=Q}oV}B Cv w l |UQR=@ wO u}@ |xrY=i 'z =[=kD 's C}a[w QO

"OwW =QH= |UQR=@ wO u}@ |xrY=i�Cl(s; z; 1) =

cins + cpm(s) + h(z � TA1(s; 1)r(s; 1))+

+�1(l)l�1Xw=1

�wNXs0=0

(P1P0w�1)s;s0X�2R�

s}tYD lI;n = j(I)n � j(I)n�1 u; QO xm CU= A = f0; 1; : : : ; lI;ngxDiQo xv=Q}oV}B Cv |=QH= x@ s}tYD '|UQR=@ |=QH= R= Oa@ xm CU=vat u=O@ QiY|UQR=@ w |rai |UQR=@ u}@ xQwO u}t)=a QO xm CU=vat u=O@ a 2 A s}tYD &OwWv

"OwW =QH= xv=Q}oV}B Cv '|Oa@�p=kDv= p=tDL= T} QD=t� sQ=yJ <RH

l} 'u}W=t p=wR Ov}=Qi xmCU= u; Q@ZQi Q_v OQwt|xr�Ut QO OW xDio xm Qw]u=tyS = f0; 1; : : : ; Ng|xDUUoCr=L |=[i =@ xDUw}B u=tR uoty hwmQ=t Ov}=QiuOQw; CUO x@ |=Q@ "OwW|t xO=O V}=tv X = (xt : t 2 [0;1)) =@ xm OW=@"s}W=@ xDW=O =Q Cr=L p=kDv= MQv T} QD=t CU= |i=m Ov}=Qi p=kDv= p=tDL= T} QD=t

%OwW|t h} QaD Q} R CQwYx@ w OwW|t xO=O V}=tv Q =@ T} QD=t u}=Q = [qss0 ](N+1)�(N+1)

qss0 = limt!0

Pr(Xt = s0jX0 = s)t

; s; s0 2 S; s 6= s0;

qss = �Xs0 6=s

qss0 :

%Ovm|t jOY Q} R \}=QW QO QT} QD=t 'xwqax@p=wR Ov}=Qi QO =Q} R "CU= QiY QDy@ Cr=L l} x@ Cr=L l} R= p=kDv= MQv %1 \QW%|[=} Q u=@ R x@ "O@=}|tv Ow@y@ u}W=t C}a[w Cv C=}rta |=QH= uwO@ 'u}W=tqss0 = 0;8s0 < s (2)

QDO@ |=yC}a[w x@ u; p=kDv= MQv 'OW=@ |QDO@ C}a[w QO u}W=t xJ Qy %2 \QW%|[=} Q u=}@ x@ "O@=}|t V}=Ri=X

s�uqss0 <

Xs�u

q(s+1)s0 ; 8u 2 S; u � (s+ 2) (3)

p=kDv= p=tDL= |=yT} QD=t u=wD|t hwQwowtrwm-utBJ |Q@=Q@ R= xO=iDU= =@ uwvm =l} CWPo R= Oa@ 'Ov}=Qi p=kDv= p=tDL= T} QD=t O}vm ZQi "OQw; CUO x@ =Q Ov}=QiQO w P0 =@ xv=Q}oV}B Cv C=}rta |=QH= sOa CQwY QO w T pw] x@ |v=tR |xQwO%TB 'OwW xO=O u=Wv P1 =@ xQwO |=OD@= QO xv=Q}oV}B Cv C=}rta |=QH= CQwYP0 = eQT (4)

P1 = R� P0 = R� eQT (5)

C=}rta |=QH= R= Oa@ u}W=t CU= xOW ZQi xm CU= p}rO u}= x@ Q}N= |w=UDQiY C}a[w x@ Qo}O CQ=@a x@ =} OwW p}O@D wv Cr=L x@ xrY=iq@ xv=Q}oV}B Cv

%OwW|t h} QaD Q} R CQwYx@ R u; QO xm OOQoQ@R = [rss0 ](N+1)�(N+1)

rss0 = Pr(X0+ = s0jX0 = s; a = 1)

R =

266666641 01 0

� � � 0� � � 0

... � � �1 0

� � � ...

� � � 0

37777775�xrLQt l} |xv} Ry =} |r=wDt |UQR=@ wO u}@ |xv} Ry� sHvB <RH

|UQR=@ QO xm |Qw]x@ OW=@|r=wDt|UQR=@ wO u}@ |xv} Ry �Cl(s; z; a) O}vm ZQi

155

"""u=tRs

y|R}Qx

t=vQ@|=Q@

|[=}Q

pOtl}

|x�=Q="8 |x]@=Q |=RH= "3 pwOH

cins|UQR=@ |xv} Ry

h(z � TA1(s; 0)r(s; 0))+

pw= |xQwO QO xDiQ CUO R= O}rwD |xv} Ry|r=wDt |UQR=@ wO u}@

NPs0=0

(P0w)s;s0P

�2R�

g�(�)h(� � TA2(s0; 0)r(s0; 0))+

wO u}@ s)=(w + 1) |xQwO QO xDiQ CUO R= O}rwD |xv} Ry \UwDtxv=Q}oV}B Cv |=QH= R= p@k |r=wDt |UQR=@

NPs0=0

(P0a�1)s;s0P

�2R�

g�(�)(cpm(s0) + h(� � TA2(s0; 1)r(s0; 1))+)

xDiQ CUO R= O}rwD w xv=Q}oV}B Cv |=yxv} Ry wtHt \UwDt|r=wDt |UQR=@ wO u}@ s)=a |xQwO QO

NPk=0

(P0a�1P1P0w�a)s;s0P

�2R�

g�(�)h(� � TA2(s0; 0)r(s0; 0))+

wO u}@ s)=(w + 1) |xQwO QO xDiQ CUO R= O}rwD |xv} Ry \UwDtxv=Q}oV}B Cv |=QH= R= Oa@ |r=wDt |UQR=@

g�(�)h(� � TA2(s0; 0)r(s0; 0))+

(a 2 f2; � � � ;Kg ; l 2 fa; � � � ;Kg ; s 2 S; z 2 R�) (8)

"CU= xOt; 3 pwOH QO x]@=Q u}= |=RH= x@ \w@ Qt C=L}[wDR= |DQ=@a |]}=QW CLD xm CU= u; 8�6 \@=wQ QO �h Qou=Wv `@=D Vkv|r=wDt |UQR=@ wO u}@ |xrY=i xJv=vJ '6 |x]@=Q QO ,qFt &OwW hPL O}=@ x]@=Q

%O};|t QO Q} R pmW x@ 6 |x]@=Q 'OW=@ xQwO l}�Cl(s; z; 0) = cins + h(z � TA1(s; 0)r(s; 0))+

(l 2 f1; � � � ;Kg ; s 2 S; z 2 R�)

pm |xv} Ry \UwDt |x@U=Lt w |ov}y@ |xrO=at "4|R} Qxt=vQ@pOt|ov}y@|xrO=at '3VN@ QO|ov}y@|xrO=at|=RH=h} QaD x@ xHwD =@l} I = (ik)Kk=0 O}vm ZQi "OwW|t x�=Q= VN@ u}= QO xr�Ut |iO=YD |=} wBpL=Qt |xr=@vO J(I) = (j(I)n)m(I)

n=0 wC@=F Oa@ x@ u}= R= w x=wNrO |UQR=@ KQ]"OyO|t u=Wv =Q pOt |ov}y@ |xrO=at 9 |x]@=Q 'CQwY u}= QO &OW=@ u; =@ Q_=vDtVj(I)n(s; z) = min

a2f0;1;:::;lI;ngf�ClI;n(s; z; a) + �lI;n

NXs0=0

X�2R�

(P(a))s;s0g�(�)Vj(I)n�1(s0; �)g

(n = 1; : : : ;m(I)) (9)

"6 |x]@=Q |=RH= "1 pwOHcins

|UQR=@ |xv} Ry

h(z � TA1(s; 0)r(s; 0))+

pw= |xQwO QO xDiQ CUO R= O}rwD |xv} Ry|r=wDt |UQR=@ wO u}@

NPs0=0

(P0w)s;s0P

�2R�

g�(�)h(� � TA2(s0; 0)r(s0; 0))+

wO u}@ s)=(w + 1) |xQwO QO xDiQ CUO R= O}rwD |xv} Ry \UwDt|r=wDt |UQR=@

"7 |x]@=Q |=RH= "2 pwOHcins

|UQR=@ |xv} Rycpm(s)

xv=Q}oV}B Cv |xv} Ry

h(z � TA1(s; 1)r(s; 1))+

|r=wDt |UQR=@ wO u}@ pw= |xQwO QO xDiQ CUO R= O}rwD |xv} RyNPs0=0

(P1P0w�1)s;s0P

�2R�

g�(�)h(� � TA2(s0; 0)r(s0; 0))+

wO u}@ s)=(w + 1) |xQwO QO xDiQ CUO R= O}rwD |xv} Ry \UwDt|r=wDt |UQR=@

g�(�)h(� � TA2(s0; 0)r(s0; 0))+

(l 2 f1; � � � ;Kg ; s 2 S; z 2 R�) (7)

"CU= xOW K} QWD 2 pwOH QO x]@=Q u}= |=RH=u}@ |xv} Ry 8 |x]@=Q 'OW=@ xQwO l} R= V}@ |UQR=@ wO u}@ |xrY=i Qo =QO u}W=t |rai |UQR=@ QO xm Ovm|t x@U=Lt |]}=QW QO =Q |r=wDt |UQR=@ wOs)=a |xQwO QO xv=Q}oV}B Cv w l |UQR=@ wO u}@ |xrY=i 'z =[=kD 's C}a[w

"(a � 2) OwW =QH= |UQR=@ wO u}@ |xrY=i�Cl(s; z; a) =

cins + h(z � TA1(s; 0)r(s; 0))+

+�2(a)a�2Xw=1

�wNXs0=0

(P0w)s;s0X�2R�

g�(�)h(� � TA2(s0; 0)r(s0; 0))+

+�a�1NXs0=0

(P0a�1)s;s0X�2R�

g�(�)(cpm(s0) + h(� � TA2(s0; 1)r(s0; 1))+)

+�a(l)l�1Xw=a

�wNXk=0

(P0a�1P1P0w�a)s;s0X�2R�

156

2/2|

xQ=tW'

35�1

|xQwO'

�1398

u=DUtR�

h}QW

C}Q}Ot

w`}=vY

|UOvyt "xv=Q}oV}B Cv |=QH= u=tR w xv} Ry "4 pwOH

4 3 2 1 0 s

1100 800 500 300 0 cpm(s)

4 3 2 1 0 tpm(s)

0 3 7 12 20 r(s)

"=[=kD p=tD= `} RwD pwOH "5 pwOH700 500 300 200 �

0 25 0 25 0 25 0 25 g�(�)

"I11 = (1; 0; 1; 0; 1; 0; 1) |UQR=@ KQ] x@ \w@ Qt Q=Owtv "2 pmW

"I23 = (1; 1; 0; 1; 1; 0; 1) |UQR=@ KQ] x@ \w@ Qt Q=Owtv "3 pmW

xv}y@ u=tR u}vJty w |R} Qxt=vQ@ ji= QO xv}y@ |UQR=@ KQ] uOQm =O}B 'hOy"CU= |UQR=@ |=yu=tR QO sDU}U Cr=L x@ xHwD =@ xv=Q}oV}B Cv |=QH= |=Q@|YNW |xv=}=Q l} QO xOW ?Yv ?rDt Q=Ri=sQv R= 'sD} Qwor= |R=UxO=}B |=Q@pL u=tR w xOW xO=iDU= 4 GB |x_i=L w 2=67 GHz VR=OQB CaQU =@QO OyO|t u=Wv pOt pL R= pY=L G}=Dv "CU= xOW C@F xk}kO 3 4 pOtw I11 = (1; 0; 1; 0; 1; 0; 1) |UQR=@ KQ] wO 'OwHwt |UQR=@ KQ] 32 u}@QHvt 'CU= xOW xO=O u=Wv 3 w 2 |=ypmW QO xm I23 = (1; 1; 0; 1; 1; 0; 1)

"OvwW|t |R} Qxt=vQ@ ji= |] QO Vj(I)m(I)(s; z) |ov}y@ x@Q =266666664

�0=0200 0=0033 0=0100 0=0067 00 �0=0241 0=0106 0=0070 0=00650 0 �0=0144 0=0034 0=01100 0 0 �0=0133 0=01330 0 0 0 0

377777775(10)

P0 =

2666666640=549 0=052 0=188 0=137 0=074

0 0=485 0=179 0=130 0=2060 0 0=649 0=066 0=2850 0 0 0=670 0=3300 0 0 0 1

377777775 (11)

P1 =

2666666640=549 0=052 0=188 0=137 0=0740=549 0=052 0=188 0=137 0=0740=549 0=052 0=188 0=137 0=0740=549 0=052 0=188 0=137 0=0740=549 0=052 0=188 0=137 0=074

377777775 (12)

'Vj(I) � (s; z) = 0; (8s 2 S; z 2 R�) 'u; QO xm

P(a) =

8<:P0lI;n a = 0P0a�1P1P0lI;n�a a 2 f1; 2; � � � ; lI;ng

:lI;n = j(I)n � j(I)n�1

|O=yvW}B pL VwQ "5xO=iDU= 'OQ=O s=o Q=yJ xm Q} R sD} Qwor= R= 'p@k VN@ QO xOW x�=Q= pOt pL |=Q@

"CU= xOWxm '|UQR=@ KQ] x@ hwQat I = (ik)Kk=0 pFt I |=[a= R= l} Qy |=Q@ "1 s=o=QH= 3 w 2 |=ys=o 'CU= 1 =} 0 |xrtH k + 1 R= pmWDt |y=vDt |}xr=@vO

"CU= xOWJ(I) = w xOW xDN=U I = (ik)Kk=0 =@ Q_=vDt pL=Qt |xr=@vO "2 s=o

"CU= xOW xO}t=v (j(I)n)m(I)n=0

�9 |x]@=Q� |ov}y@ |xrO=at R= xO=iDU= =@ z 2 R�; s 2 S Qy |=R=x@ "3 s=oa�I(s; z) u; =@ Q_=vDt |xv}y@ s}tYD w x@U=LtV �j(I)m(I)

(s; z) |xv}y@ Q=Oktxv=Q}oV}B Cv |=QH= OQwt QO xv}y@ s}tYD a�I(i; z) `k=w QO "CU= xOW xO}t=vxm|DQwY QO w I |UQR=@ KQ] |=R= x@ swO w pw= |UQR=@ u}@ |xrY=i QO =Qs C}a[w QO u}W=t w z =[=kD �|R} Qxt=vQ@ ji= |=OD@=� pw= |UQR=@ u=tR QO

"Ovm|t XNWt 'OW=@Vj(I)m(I)(s; z) Q=Oktu} QDtm xm|}|UQR=@KQ]'z 2 R�; s 2 S Qy|=R=x@ "4 s=oQO (s; z) Cr=L Q}eDt =@ Q_=vDt |UQR=@ |xv}y@ KQ] u=wva x@ 'OQ=O =QQy |=R= x@ "OwW|t xO}t=v I� = (i�k)Kk=0 w ?=NDv= |R} Qxt=vQ@ ji=xv}y@ s}tYD a�I�(i; z) s}tYD I� |UQR=@ KQ] |=Q@ 'z 2 R�; s 2 SXNWt =Q swO w pw= |UQR=@ u}@ |xrY=i QO xv=Q}oV}B Cv |=QH= OQwt QO

"Ovm|t

|OOa G}=Dv p}rLD w x} RHD "6|OOa p=Ft l} ltm x@ pOt K} QWD "1"6

xOW x�=Q= p=Ft l} |O=yvW}B pOt OQ@y=Q p}rLD w x} RHD Qw_vtx@ VN@ u}= QOl} Qy |v=tR |xQwO K = 6 R= pmWDt |R} Qxt=vQ@ ji= O}vm ZQi "CU=|xv} Ry 'S = f0; 1; 2; 3; 4g u}W=t Cr=L |=[i 'RwQ T = 30 pw] x@|UQR=@ C=}rta s=Hv= Q=@ Qy |xv} Ry 'h = 5 |xDiQ CUO R= O}rwD OL=w Qy"OW=@ � = 0=9 p} RvD ?} Q[ w 'tins = 1 u; |=QH= u=tR 'cins = 3003 w 2 \@=wQ QO xm OyO|t u=Wv =Q Cr=L p=kDv= MQv T} QD=t '10 |x]@=QxOW x�=Q= VwQ =@ T} QD=t u}= "Ovm|t jOY 2 w 1 |=y\QW C}=aQ Qw_vtx@|=ysD} Qwor= 2 VN@ 3 CUw}B [23]23 `HQt x@ \w@ Qt 3 CUw}B 2 VN@ QOl} p=kDv= |=yT} QD=t '5 w 4 \@=wQ R= xO=iDU= =@ "CU= xOW O}rwD 5 w 4\@=wQ QO u; |=QH= CQwY QO w xv=Q}oV}B Cv |=QH= sOa CQwY QO |}xrLQtxOW GQO 5 w 4 |=ypwOH QO =yQDt=Q=B Q}=U "CU= xOW xO=O u=Wv 12 w 11

"CU=

157

"""u=tRs

y|R}Qx

t=vQ@|=Q@

|[=}Q

pOtl}

|x�=Q==QH= x@=Wt |v=tR QO =} QDOwR O}=@ xv=Q}oV}B Cv 'OW=@ xDW=O Q=Qk |QDO@ C}a[w

"OwW

C}U=UL p}rLD w EL@ "2"6w 1"4 |=yVN@ [23]'xOW x�=Q= |x}=B pOt j}kLD u}= QO 'OW xQ=W=,q@k xm Qw]u=tyu=tR COt uOQw; ?=UL x@ u}vJty w |UQR=@ C=t}tYD uOQm xi=[= hOy =@ '1"5'wQu}= R= "CU= xOW xO=O xaUwD xv=Q}oV}BCv w|UQR=@ C=}rta|=QH= |=Q@ R=}v OQwtxOW pL x}=B pOt \}=QW =@ |O=yvW}B pOt pw= |xryw QO '|HvUQ=@Da= Qw_vtx@|O=yvW}B pOt 'cins = 0 w tins = 0 'tpm = 0 ZQi =@ Qw_vt u}O@ "CU=|=yCr=L |xty |=R= x@ 'OyO|t u=Wv 9 pwOH G}=Dv xm Qw]u=ty "CU= xOW pLI32 = (1; 1; 1; 1; 1; 1; 1) |UQR=@ KQ] xm OyO|t MQ |Dkw |ov}y@ 'sDU}Ux@ C@Uv 'pm |xv} Ry u}vJty "OQ}o CQwY |UQR=@ xQwO Qy QO w OwW ?=NDv=|}xv=DU; OL wv R= xv=Q}oV}B Cv CU=}U w CU= |rwRvQ}e u}W=t Cr=L w =[=kDQDO@ |OL l} R= u}W=t C}a[w Qo = =[=kD R= C@=F Q=Okt l} |=R= x@ '|va} &CU=|rwRvQ}e =[=kD x@ C@Uv |}xv=DU; OL u}= w OwW =QH= O}=@ xv=Q}oV}B Cv 'OW=@QDO@ |OL l} R= =[=kD Qo = u}W=t C}a[w R= C@=F Q=Okt l} |=R= x@ w CU=u}W=t C}a[w x@ C@Uv |}xv=DU; OL u}= w OwW =QH= O}=@ xv=Q}oV}B Cv 'OW=@x@=Wt \}=QW QO 'x}=B pOt R= |}xaUwD u=wvax@ =t pOt |va} u}= w CU= |rwRvQ}el} QO *V}=tv Qw_vt x@ 10 pwOH QO xm CU= xDU@=w w OQ=O x=Qty x@ u=Um} G}=Dv

"CU= xOW ?=NDv= nvQ l} KQ] Qy |=Q@ 'G}=Dv *x=ovw tins 6= 0 'tpm 6= 0 xm |]}=QW QO pOt |HvUQ=@Da= Qw_vt x@ =t=(cins) |UQR=@ OL=w Qy |xv} Ry x@ C@Uv pm |xv} Ry C}U=UL 'cins 6= 0Q}O=kt |=R=x@ Qw_vt u}O@ "CU= xOW |UQQ@ =yQDt=Q=B Q}=U uOw@ C@=F ZQi =@ wxOt; CUO x@ |UQR=@ |xv}y@ C=t}tYD w xOW pL |O=yvW}B pOt 'cins hrDNtG}=Dv |R} Qxt=vQ@ ji= wQW QO ((s; z)) sDU}U Cr=L x@ C=t}tYD u}= "CU=R= l}I}y =} |NQ@ |=R= x@ I32 |UQR=@ KQ] 'OW=@ C@Ft cins |Dkw OyO|t u=Wv*OQ@y=Q 'OW=@ xDW=O xv} Ry =y|UQR=@ xJv=vJ |va} u}= w CU}v xv}y@ sDU}U Cq=L|UQR=@ C=t}tYD uOv=Hvo C}ty= xm OwW|tv |ov}y@ x@ QHvt xQwO Qy QO |UQR=@pL R= xOt; CUO x@ pm |xv} Ry 'pwOH u}= QO u}vJty "OyO|t u=Wv =Q pOt QOxU}=kt sDU}UhrDNt |=yCr=L |=Q@ I32 |UQR=@ KQ] |=R= x@ pm |xv} Ry =@ pOt'OW=@ QiY R= QDnQR@ |UQR=@ |xv} Ry xJu=vJ 'OyO|t u=Wv G}=Dv u}= "CU= xOWcins |oQR@ x@ u;C}ty= |xHQO wCU=C}ty= R�=L =yv; XwYN QO |Q}os}tYD|xv} Ry u}ov=}t QO Ow@y@ u=R}t 'OW=@ QDnQR@ cins xR=Ov= Qy '|va} "CU= xDU@=wpOt R= xOt;CUOx@ pm |xv} Ry u}ov=}t x@ C@Uv 'O};|t CUOx@ =t pOt R= xm |rmC}ty= xm &�4 pmW� CU= QDW}@ 'OwW|t =QH= xQwO Qy QO =y|UQR=@ u; QO xm x}=B"OyO|t u=Wv xOW QmP \}=QW QO =Q xv=Q}oV}B Cv w =y|UQR=@ u=tRsy |R} Qxt=vQ@

|Q}oxH}Dv "7|R=Uxv}y@ Qw_vtx@|y=vDt ji= QO|iO=YD |=} wB |R} Qxt=vQ@ pOtl} 'xr=kt u}= QO|=yxv} RyuDiQo Q_v QO =@ xv=Q}oV}B C=Q}taD w |Q=Oyov w|UQR=@ C=t}tYD u=tRsywtHt \UwDt |R=Uxv}tm Q=}at =@ w xDiQ CUO R= O}rwD w xv=Q}oV}B Cv '|UQR=@|rwYLtlD O}rwD sDU}U l} |=Q@ |R=UpOt u}= "OW x�=Q= |v=tR ji= QO =yxv} Ryp=wR x@ wQ |iwmQ=t Ov}=Qi l} j@] u}W=t u; QO xm xDiQo CQwY |v}W=tlD w

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�References� `@=vt1. Aramon Bajestani, M. \Integrating maintenance plan-

ning and production scheduling: making operational de-cisions with a strategic perspective", PhD thesis Depart-ment of Mechanical & Industrial Engineering Universityof Toronto Canada (2014).

2. Kothamasu, R., Huang, S.H. and Verduin, W.H. \ Sys-tem health monitoring and prognostics - a reviw of cur-rent paradigms and practices", International Journalof Advanced Manufacturing Technology, 28(9), pp.1012-1024 (2006).

3. Legat, V., Zaludova, A.H., Cervenka, V. and et al. \Con-tribution to optimization of preventive maintenance",Reliability Engineering and System Safety, 51, pp. 259-266 (2000).

4. Vaurio, J.K. \On time-dependent availability and main-tenance optimization of standby units under variousmaintenance policies", Reliability Engineering and Sys-tem Safety, 56, pp. 79-89 (1997).

5. Jardine, A.K.S., Lin, D. and Banjevic, D. \A reviewon machinery diagnostics and prognostics implementingcondition-based maintenance", Mechanical System andSignal Process, 20, pp. 1483-1510 (2006).

6. Golmakani, H.R. \Maintenance management: modelingand optimization", Amirkabir University Tafresh BranchPress, (In Persian) (2009).

7. Golmakani, H.R. and Poresmaeeli, M. \Optimization ofreplacement threshold in condition-based maintenancewith variable failure cost", Industrial Engineering andManagement, 30-1(1.1), pp. 25-35 (In Persian) (2011).

8. Banjevic, D., Jardine, A.K.S., Makis, V. and et al.\A control limit policy and software for conditionbased maintenance optimization", INFOR, 39, pp. 32-49(2001).

9. Golmakani, H.R. and Fattahipour, F. \Optimal replace-ment policy and inspection interval for condition-basedmaintenance", International Journal of Production Re-search, 49(17), pp. 5153-5167 (2011).

10. Golmakani, H.R. \Condition-based inspection schemefor condition-based maintenance", International Journalof Production Research, 50(14), pp. 3920-3935 (2012).

11. Golmakani, H.R. and Fattahipour, F. \Age-based in-spection scheme for condition-based maintenance",Journal of Quality in Maintenance Engineering, 17(1),pp. 93-110 (2011).

12. Golmakani, H.R. \Cost-e�ective condition-based inspec-tion scheme for condition-based maintenance", in: In-ternational Conference on Information Reuse and Inte-gration (IRI) IEEE Las Vegas Nevada USA, pp. 327-330(2011).

13. Chen, D.Y. and Trivedi K.S. \Optimization for conditionbased maintenance with semi-markov decision process",Reliability Engineering and System Safety, 90, pp. 25-29(2005).

14. Grall, A., Berenguer, C. and Dieulle L. \A conditionbased maintenance policy for stochastically deterioratingsystems", Reliability Engineering and System Safety, 76,pp. 167-180 (2002).

15. Hosseini, M.M., Kerr, R.M. and Randall, R.B. \An in-spection model with minimal and major maintenance fora system with deterioration and poisson failures", IEEETransactions on Reliability, 49, pp. 88-98 (2000).

16. Jardine, A.K.S., Joseph, T. and Banjevic, D. \Optimiz-ing condition-based maintenance dcision for equipmentsubject to vibration monitoring", Journal of Quality inMaintenance Engineering, 5(3), pp. 192-202 (1999).

17. Jardine, A.K.S., Makis, V., Banjevic, D. and et al. \Adecision optimization model for condition-based mainte-nance", Journal of Quality in Maintenance Engineering,4(2), pp. 115-121 (1998).

18. Banjevic, D. and Lam, J. \A myopic policy for optimalinspection scheduling for condition based maintenance",Reliability Engineering and System Safety, 144, pp. 1-11(2015).

19. Xu, M., Alam, M.N.E. and Kamarthi, S. \A modi�eddynamic programming model in condition-based main-tenance optimization", in: International ManufacturingScience and Engineering Conferefnce (MSEC) ASME,Los Angeles California USA pp. V003T04A014; 7 pages(2017).

20. Aghezzaf, E.-H., Khatab, A. and Tam, P.L. \Optimiz-ing production and imperfect preventive maintenanceplannings integration in failureprone manufacturing sys-tems", Reliability Engineering System Safety, 145, pp.190-198 (2016).

162

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|UOvyt 21. Kang, K. and Subramaniam, V. \Integrated control pol-

icy of production and preventive maintenance for a de-teriorating manufacturing system", Computers & Indus-trial Engineering, 118, pp. 266-277 (2018).

22. Kang, K. and Subramaniam, V. \Joint control of dy-namic maintenance and production in a failure-pronemanufacturing system subjected to deterioration", Com-puters & Industrial Engineering, 119 , pp. 309-320(2018).

23. Aramon Bajestani, M., Banjevic, D. and Beck, J.C. \In-tegrated maintenance planning and production schedul-ing with markovian deteriorating machine conditions",International Journal of Production Research, 52, pp.7377-7400 (2014).

24. Ross, S.M., Introduction to Stochastic Dynamic Pro-gramming, In Probability and Mathematical Statistics,Z. W. Birnbaum and E. Lukacs, Eds., pp. 49-50, Aca-demic press, New York (1983).

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