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"�OW xQ=W=u; =@ Q_=vDt h=Oy= w Ckwt |v=tQO Rm =Qt x@ u}LwQHt |x}rND uDiQo Q_v QO "3

"|v=tQO Rm =Qt C}iQ_ C}OwOLt w hOy `@=wD QO|OvtC}=[Q |R=Uxv}W}@ u=wva CLD xv=DUwOQW@ hOy `@=D uDiQo Q_v QO "4xO}O?}U; O=Qi= |x}rND OYQO u} QDsm |R=Uxv}W}@ j} Q] R= xO}O?}U; \=kv

"�swO hOy `@=D� Q]N |xk]vt R=w hOy `@=D QO u; x@ \w@ Qt C=_Lqt R}v w x}rkv |xr}Uw uDiQo Q_v QO "5

"p}=Uw u}= C}iQ_ w O=OaD C}OwOLtl}DUHr |xr�Ut 'swO VN@ QO %CU= xOW |yOv=tR=U u}vJ xr=kt 'xt=O= QOVN@ QO "CU= xOW K} QWD |O=yvW}B |R=i |[=} Q |R} Qxt=vQ@ pOt w O=Ot=|OQwt |xar=]t 'sQ=yJ VN@ QO w xOW xOQw; |R=i |}xrLQtwO pL OQm} wQ 'swU'sHvB VN@ QO ,=D}=yv "CU= xOW x�=Q= u=Q}= QyW u}OvJ QO xwkr=@ |=yu=QL@ |=Q@

"CU= xOW x�=Q= |D; C=ar=]t |=Q@ C=O=yvW}B w |Q}oxH}Dv

sOa CLD O=Ot= l}DUHr |R} Qxt=vQ@ |R=UpOt "2C}a]k

|=yQ}eDt '=yQDt=Q=B '=yQo =U=vW 'C=}[Qi x=Qty x@|UQQ@ OQwt |xr�UtVN@u}= QO"OwW|t |iQat xr�Ut |[=} Q |R} Qxt=vQ@ pOt w

pOt C=}[Qi w xr�Ut u=}@ "1"2QO C=t=Ok= u} QDsyt R= u}twOYt |x}rND C=}rta w |O=Ot= O=Ot= `} RwD C=}rta=@ 'C=t=Ok= u}= ?U=vt w CUQO |R} Qxt=vQ@ "OwW|t ?wULt u=QL@ x@ MU=B R=iw |r=t w |v=H C=Q=UN Vy=m QO |}=RUx@ Q}F-=D 'C=v=mt= w `@=vt C}OwOLt x@ xHwDR=}v OQwt |QwQ[ sqk= O}=@ 'MU=B R=i QO "OQ=O |} woMU=B |WN@ QF= w |}=Q=m V}=Ri=p=UQ= xO}O?}U; j]=vt x@ O=Ot= `} RwD Rm =Qt R= w '|}=U=vW u}LwQHt w u}twOYtw |oO}UQ OQwt =Hv; QO w pkDvt |v=tQO Rm =Qt x@ xO}O?}U; O=Qi= u}vJty w OwW

"OvQ}o Q=Qk =w=Otpt=W 1 pmW j@=]t j}kLD u}= QO xar=]t OQwt O=Ot= l}DUHr |xm@W\=kv "3 &|O=Ot= |=yq=m `} RwD Rm =Qt "2 &u}t-=D \=kv "1 %CU= |rY= VN@ Q=yJ'O=Ot= |xQ}HvR QO |O=Ot= |=yq=m u=oOvvmu}t-=D "Ckwt |v=tQO Rm =Qt w xO}O?}U;\=kv QO xm |O=Qi= |=Q@ =Q R=}v OQwt |O=Ot= |=yq=m w Ovvm|t |R=@ |U=U= |Wkv

"Ovvm|t u}t-=D 'OvQ=O Q=Qk xO}O?}U;`} RwD Rm =Qt ?=NDv= w O}Ov=m \=kv |UQQ@ %R=CU= CQ=@a |O=yvW}B pOt hOyx@ |O=Ot= |=yq=m ?U=vt `} RwD w X}YND '?U=vt Ckwt |v=tQO Rm =Qt w O=Ot=

17

"""|=Q@|

R=i|xi

OyOvJ

l}DUH

rpOtl

}|x�=Q=X

i

Xirj =Xk

Yjrk; 8j; r (5)Xj

Xirj � ~Si; 8i; r (6)Xo

Qko � ~d2k; 8k (7)Xi

Xirj � ZjCazj ; 8j; r (8)Xk

Yjrk � ZjCazj ; 8j; r (9)Xk

Qko �WoCawo; 8o (10)

Xirj � CaprTirj ; 8i; r; j (11)

Yjrk � CaprVjrk; 8j; r; k (12)Xi;j

Tirj � Nm; 8r (13)Xj;k

Vjrk � Nm; 8r (14)

Xirj ; Yjrk � 0; (15)

Tirj ; Vjrk; Qko 2 Z; (16)

Wo; Zj 2 f0; 1g ; (17)

R= pkv w ptL |=yxv} Ry '|R=Ov=x=Q |=yxv} Ry wtHt uOQmxv}tm pw= hOy `@=D\=kv x@ O=Ot= `} RwD Rm =Qt R= pkv w ptL |=yxv} Ry 'O=Ot= `} RwD Rm =Qt x@ u}t-=D \=kv|=yxv} Ry '|v=tQO Rm =Qt x@ xO}O?}U; \=kv R= pkv w ptL |=yxv} Ry 'xO}O?}U;|=yxv} Ry w xO}O?}U; \=kv w `} RwD Rm =Qt 'u}t-=D \=kv u}@ x}rkvp}=Uw X=YDN=

"CU= =[=kD uOW xOQw;Q@ sOa |xt} QH|R=Uxv}W}@j} Q] R= xO}O?}U;\=kv |OvtC}=[Q|R=Uxv}W}@ swOhOy `@=D|]NQ}eKw[wx@ swOhOy "CU= xO}O?}U;\=kv QOVWwB OYQOu} QDsm wtHt

%CWwv u}vJ u=wD|t =Q pO=at |]N pOt "CU=MaxF2 = �; (18)

� �8<:PoQko

~d2k

9=; ; (19)

u; CU}=@|t xmr@ OvQ=O xov =Q q=m OvyO|tv xR=H= xm CU=vat u}= x@ 3 C}OwOLtu}t[D 4 C}OwOLt "OQm p=UQ= =Q OwHwt=v |q=m u=wD|tv u}vJty &Ovvm p=UQ= =Q"OwW |Q}owrH xO}O?}U; \=kv x@ |i=[= |O=Ot= |=yq=m p=UQ= R= xm Ovm|tO=Ot= `} RwD RmQt Qy QO q=m Qy |=Q@ u=} QH pO=aD |xrO=at Qou=Wv 5 C}OwOLtRmQt u; R= 'O=Ot= `} RwD RmQt l} x@ xOWOQ=w |=yq=m pm xm Ovm|t u}t[D w CU='u}t-=D \=kv QO TQDUOQO |=yq=m pm xm CU=vat u}O@ 6 C}OwOLt "OwW GQ=N|v=tQO Rm =Qt x@ O=Qi= QDW}@ p=UQ= R= 7 C}OwOLt "OvO=Ot= `} RwD Rm =Qt x@ p=UQ= p@=kGwQN w OwQw R= w Ov=C}iQ_ x@ \w@ Qt 10 =D 8 |=yC}OwOLt "Ovm|t |Q}owrHOv=xOWv |R=Ov=x=Q xm Ckwt |v=tQO Rm =Qt x@ O=Qi= OwQw w |O=Ot= `} RwD Rm =Qt x@ q=mO=OaD w x}rkv p}=Uw C}iQ_ x@ \w@ Qt 14 =D 11 |=yC}OwOLt "Ovvm|t |Q}owrHQiY w K}LY OOa 'uOw@ |ivtQ}e x@ \w@ Qt 17 =D 15 |=yC}OwOLt "CU=yv;

"CU= uOw@ 1 w

x@ r |x}rkv |xr}Uw \UwD i u}t-=D |x]kv R= q=m OL=w Qy p=kDv= |xv} Ry %~cirj&j O=Ot= `} RwD RmQt

r |x}rkv |xr}Uw \UwD j O=Ot= `} RwD RmQt R= q=m OL=w Qy p=kDv= |xv} Ry %~cjrk&k ?}U; |x]kv x@

&o |v=tQO RmQt x@ k ?}U; |x]kv R= OQi Qy p=kDv= |xv} Ry %~cko&k ?}U; |x]kv QO q=m OL=w Qy Ow@tm |xv} Ry %b1k

&k xO}O?}U; |x]kv QO u}twOYt p=kDv= sOa |xv} Ry %b2k&k xO}O?}U; |x]kv QO |O=Ot= |q=m |=[=kD u=R}t % ~d1k

&k ?}U; |x]kv �u}twOYt O=OaD� |v=tQO |=[=kD u=R}t % ~d2k&i u}t-=D |x]kv u}t-=D u=R}t % ~Si

&j `} RwD RmQt C}iQ_ %Cazj&o |v=tQO RmQt C}iQ_ %Cawo

&r wv TQDUO QO x}rkv p}=Uw O=OaD %Nr"r wv |x}rkv |xr}Uw C}iQ_ %Capr

s}tYD |=yQ}eDt "4"2RmQt x@ r |x}rkv |xr}Uw \UwD i u}t-=D |x]kv R= xOW pkDvt |q=m u=R}t %Xirj

&j `} RwD|x]kv x@ r |x}rkv |xr}Uw \UwD j `} RwD RmQt R= xOW pkDvt |q=m Q=Okt %Yjrk

&k |xO}O?}U;&o |v=tQO RmQt x@ k |xO}O?}U; |x]kv R= xOW pkDvt O=Qi= O=OaD %Qko

CQwYu}= Q}e QO w 1 Q@=Q@ OwW ?=NDv= uOWp=ai CyH j O=Ot= `} RwD RmQt Qo = %Zj&QiY Q@=Q@

&QiY CQwYu}= Q}e QO w 1 OwW ?=NDv= uOWp=ai CyH o |v=tQO RmQt Qo = %Wo

&j `} RwD RmQt w i u}t-=D |x]kv u}@ r |x}rkv p}=Uw O=OaD %Tirj"k |xO}O?}U; |x]kv w j `} RwD RmQt u}@ r |x}rkv p}=Uw O=OaD %Vjrk

=yC}OwOLt w hOy `@=wD "5"2Min ~F1 =

Xj

~fjZj +Xo

~hoWo+Xi;r;j

~arTirj +Xj;r;k

~arVjrk

+Xi;r;j

~cirjXirj +Xj;r;k

~cjrkYjrk +Xk;o

~ckoQko

+Xk

b1k

~d1k �Xj

Yjrk

!+Xk

b2k

~d2k �Xo

Qko

!; (1)

MaxF2 = mink

8<:PoQko

~d2k

9=; ; (2)

S:t :Xj;k

Yjrk � min

(Xi

~Si;Xk

~d1k); (3)X

j

Yjrk � ~d1k; 8r; k (4)

18

|L=Diu

}tQ;O}U

w|Q}t=

|oQR@|

ra

optimal|x@U=Lt =@ =Q=m |xawtHt |wQ hOy `@=D Qy |xOwOLt u}}aD "swO s=ow Xoptimal

1 � 'a� optimal |x@U=Lt |=Q@ "hOy `@=D Qy |=Q@ nadir wO}=@ hOy `@=DQy |=Q@ j}kO pOt '�W optimal

2 w Xoptimal2 � w �W optimal

126 w 25 \@=wQ j} Q] R= hOy `@=D Qy |=Q@ nadirTBU w OwW pL xv=o =OH

%CU= u}tND p@=kWnadir2 = max

nW2jW1 �W optimal

1 &X 2 f(x)o; (25)

Wnadir2 = minnW1jW2 �W optimal

2 &X 2 f(x)o; (26)

%j} Q] R= hOy `@=D Qy |=Q@ |]N C} w[a `@=D u}}aD "swU s=o

�1(X) =

8>>>><>>>>:1; if W1 �W optimal

1Wnadir1 �W1

Wnadir1 �Woptimal1

; if W optimal1 �W1 �Wnadir1

0; if W1 �W optimal1

(27)

�2(X) =

8>>>><>>>>:1; if W1 �W optimal

1W1�Wnadir1

Woptimal1 �Wnadir1

; if Wnadir1 �W1 �W optimal1

0; if W1 �Wnadir1(28)

VwQ T=U=Q@ xiOy lD pOt x@ xiOy OvJ |a]k pOt p}O@D "sQ=yJ s=o"" C}OwOLt

Max�1(X)

s:t :

�2(X) � "; X 2 f(x); " 2 [0; 1] ; (29)

"hrDNt |xv}y@ |=y?=wH O}rwD |=Q@ " Q=Okt Q}}eD "sHvB s=o=Q u; 'OW=@ ji=wt xOWO}rwD |=y?=wH R= |m} =@ xOvQ}os}tYD Qo = "sWW s=oQO "OQm Oy=wN ?=NDv= QN; s}tYD u=wvax@ =Q xOW ZQi pL w Ovm|t pw@k"OOQo|tR=@ 5 s=o x@ w OyO|t Q}}eD =Q OwN pw@k OQwt |xOwOLt CQwYu}=Q}e

j}kO=v |=yC}OwOLt w j}kO=v pm |xv} Ry hOy `@=D p}O@D "3"3|a]k |xr�Ut l} x@ 21 |x]@=Q R= xO=iDU= =@ 'xr�Ut pw= hOy `@=D xrLQt u}= QO

"OwW|t p}O@D

MinF1 =Xj

�fuj + 2fmj + fpj4

�Zj

+Xo

�huo + 2hmo + hpo

4�Wo

+Xi;r;j

�aur + 2amr + apr

4�Tirj

|}xrLQt wO pL OQm} wQ "3"CU= xOW x�=Q= |}xrLQtwO OQm} wQ l} '|O=yvW}B |R=i |xiOywO pOt pL |=Q@xO=iDU= |a]k pOt x@ |a]k Q}e pOt p}O@D |=Q@ [30]Rvt}H VwQ R= pw= |xrLQt QOuDi=} |=Q@ xOW KqY= uwr}UB= C}OwOLt VwQ R= swO |xrLQt QO "CU= xOW

"CU= xOW xO=iDU= |}=yv ?=wH

1 |xrLQt "1"3u}at w |a]k pOt x@ |R=i |a]kQ}e |R} Qxt=vQ@ |xr�Ut l} pw= |xrLQt QOOQwt |xrY=i p}@k R= |wk |[=} Q swyit Q@ |vD@t Rvt}H VwQ "OwW|t p}O@D|=Q@ |tD} Qwor= Q_v R= VwQ u}= u}vJty "CU= |R=i O=Oa= |[=} Q O}t= w Q=_Dv=u}a QO w Ovm ^iL =Q =yv; |Q}PB|]N Ov=wD|t xm CU=Q=m |R=i |]N p�=Ut pLl} ~c xm OwW|t ZQi "O}=Ri}v |w=Ut=v |=yC}OwOLt w hOy `@=D O=OaD Q@ p=Lxv=v}@O@ 'ptDLt ?}DQDx@ |x]kv xU R}v cp w cu 'cm w CU= |FrFt |R=i OOaw Q=_Dv= OQwt |xrY=i |xOvyOu=Wv 21 w 20 |xrO=at wO "OvDUy xv=v}@VwN w

"CU= ~c |FrFt OOa |[=} Q O}t=EI(~c) = [Ec1 ; Ec2 ] =

�12 (cu + cm); 12 (cm + cp)

�; (20)

EV (~c) = Ec1 + Ec22 = cu + 2cm + cp

4 ; (21)

OOa u; |=yQDt=Q=B xm OwW|t xDiQo Q_v QO |R=i |[=} Q |R} Qxt=vQ@ pOt p=L"OW=@ |FrFt |R=i

MinZ = ~ctx

S:t ~aix � ~bi; i = 1; :::; k (22)

%CWwv u}vJ u=wD|t =Q q=@ pOt C}OwOLt Rvt}H VwQ j@]h(1� �)Eai2 + �Eai1

ix � �Ebi2 + (1� �)Ebi1 ; (23)

xOvQ}os}tYD OQi \UwD xm CU= VQ}PB pw@k p@=k |xHQO u} QDsm � u; QO xmO=Oa= |[=} Q O}t= w R=}v OQwt |xrY=i h} Q=aD ltmx@ xH}DvQO "OwW|t u}}aD

%OvU} wv|t u}vJ =Q � QDt=Q=B =@ 22 |a]k pOt '|R=i |FrFtMinEV (~c)x

s:t :

[(1� �)Eai2 + �Eai1 ]x � �Ebi2 + (1� �)Ebi1 ;

i = 1; :::; k; x � 0 (24)

|rt=aD |R=i pL OQm}wQ %2 |xrLQt "2"3xm OW Oy=wN pta uwr}UB= C}OwOLt VwQ T=U=Q@ '|R=i pL VwQ |xt=O= QOTBU w Ovm O=H}= xOvQ}os}tYD |=Q@ |} wDQ=B |=y?=wH R= |@U=vt Q} wYDCU= QO=k|O=yvW}B VwQ |=ys=o "Ow@ Oy=wN ?=wH u} QDy@ ?=NDv= x@ QO=k xOvQ}os}tYD

%R= CU= CQ=@a Q=wxYqN"Rvt}H VwQ \UwD 'u}at w |a]k pOt x@ |R=i |xiOyOvJ pOt p}O@D "pw= s=o

19

"""|=Q@|

R=i|xi

OyOvJ

l}DUH

rpOtl

}|x�=Q=xr�Ut |HwQN 'pOt T=UL |=yQDt=Q=B Q}}eD w |O=yvW}B pL OQm} wQ |Q}oQ=mx@

"CU= xDiQo Q=Qk |UQQ@ OQwt

|OQwt |xar=]t K} QWD "1"4'u=vtU 'u} wRk '|Q=U 'u=oQo � u}t-=D |x]kv VW 'xOW s=Hv= |=y|UQQ@ j@]|x]kvCiy 'O=Ot= `} RwD Rm =Qt |R=Ov=x=Q |=Q@ O}Ov=m |x]kvVW '�u=yiY= w l =Q=xDiQo Q_v QO Ckwt |v=tQO Rm =Qt |R=Ov=x=Q |=Q@ O}Ov=m |x]kv VW w Q}PB?}U;

"CU= xOW'u} wRk '|Q=U 'u=oQo |=yQyW O=Ot= `} RwD Rm =Qt O}Ov=m |x]kv VW C}iQ_Rm =Qt |=Q@ "CU= xOW ZQi 550000 Q@=Q@ w u=Um} u=W=m w ?Qe u=QyD 'u=vtUC}iQ_ u}= 'GQm w sk '?Qe u=QyD 'CWQ 'p@=@ '|Q=U |=yQyW QO Ckwt |v=tQOl} Qy QO OwHwt |q=m Q=Okt |xOvyOu=Wv 1 pwOH "CU= xOW ZQi 2000 Q@=Q@|=[=kD u=R}t w =yq=m |=[=kD u=R}t ?}DQDx@ '3 w 2 pwOH QO "CU= u}t-=D \=kv R=

"CU= xOW x�=Q= xO}O?}U; |x]kv Qy |=Q@ |v=tQO|FrFt |R=i OOa Q@=Q@ ?}DQDx@ |v=tQO Rm =Qt w `} RwD Rm =Qt |R=Ov=x=Q |xv} Rywv wO "CU= xOW ZQi OL=w Q=RyOY ?ULQ@ �18,20,24� w �9,10,12�

X=YDN= C@=F |=yxv} Ry ?}DQDx@ xm xOW xDiQo Q_v QO �lD w Qw=N� x}rkv |xr}UwQ=RyOY?ULQ@ �15,20,25� w �9,10,12� |FrFt |R=i OOa Q@=Q@ 2 w 1 |xr}UwQ@=Q@ 2 wv w OOa 1000 Q@=Q@ 1 wv |x}rkv |xr}Uw O=OaD u}vJty "CU= OL=w"CU= xOW ZQi 10000 w 3000 ?}DQDx@ s=OmQy C}iQ_ &CU= OOa 400w x]kv wO u}@ |xrY=i Q@ CU= |vD@t x]kv wO u}@ pkv w ptL |=yxv} Ry OQw;Q@|xv} Ry w 0 35 Q}PB?}U; \=kv x@ |Q=Q][= |=yq=m uO}UQv |xt} QH |xv} RyC}ty= p}rOx@ |v=tQO Rm =Qt x@ Q}PB?}U; |xk]vt R= O=Qi= p=UQ= sOa |xt} QH"CU= xOW xDiQo Q_v QO �5000 pO=at� nQR@ |OOa 'O=Qi= u=H ^iL w[wt

=yxDi=} w G}=Dv "2"4=@ Q_v OQwt xr�Ut "CU= xOW x�=Q= =yxDi=} w |D=@U=Lt G}=Dv 'VN@ u}= QO4GB RAM C=YNWt =@ |}xv=}=Q |wQGAMS 23 7 Q=Ri=sQv R= xO=iDU=xv} Ro xU u}@ QO VQ}PB |xHQO u} QDsm "CU= xOW =QH= Core i7 CPU and

"CU= xOW |UQQ@ 0 9 w 0 7 '0 5"CU= xOW x�=Q= |UQQ@ OQwt |xr�Ut |=Q@ |} wDQ=B |=y?=wH '4 pwOH QO&CU= xOW QDW}@ swO w pw= hOy `@=D Q=Okt �V}=Ri= =@ 'OwW|t xOy=Wt xm u=vJ

"(�1000) u}t-=D \=kv QO OwHwt |q=m u=R}t "1 pwOHu}t-=D \=kv S(Sp; Sm; Su)

u=oQo �340; 400; 500�|Q=U �382; 450; 562�u} wRk �510; 600; 750�u=vtU �425; 500; 625�l =Q= �595; 700; 875�

u=yiY= �595; 700; 875�

+Xj;r;k

�aur + 2amr + apr

4�Vjrk

+Xi;r;j

� cuirj + 2cmirj + cpirj4

�Xirj

+Xk

b1k �

d1uk + 2d1mk + d1pk4��X

j

Yjrk

!(30)

x@ 23 \@=wQ ltm =@ R}v �14 '7 '6 '4 '3 '2 |=yC}OwOLt� pOt |R=i CqO=at"OwW|t p}O@D |a]k Cr=LX

j;k

Yjrk �min�X

i

��Spi + Smi

2�

+ (1� �)�Smi + Sui

2�;X

i

��d1pk + d1mk

2�

+ (1� �)�d1mk + d1uk

2��

;

(31)Xj;k

Yjrk � ��d1pk + d1mk

2�

+ (1� �)�d1mk + d1uk

2�; (32)X

j

Xirj � ��Spi + Smi

2�

+ (1� �)�Smi + Sui

2�; (33)X

o

Qko � ��d2pk + d2mk

2�

+ (1� �)�d2mk + d2uk

2�; (34)

� �PoQko

(1� �)�d2mk +d2uk2

�+ �

�d2pk+d2mk2

� ; (35)

"Ovv=t|t |k=@ CQwY u=tyx@ x}=B pOt |R=iQ}e |=yC}OwOLt

|OQwt |xar=]t "4pL x=Q w pOt |WN@ QF= w |}=Q=m uO=Ou=Wv |=Q@ |OQwt |xar=]t VN@ u}= QO'u}t-=D \=kv pt=W 2 pmW QO xar=]t OQwt |xk]vt "CU= xOW x�=Q= |O=yvW}B=@ "CU= xOW xDiQo Q_v QO xO}O?}U; \=kv w Ckwt |v=tQO Rm =Qt '`} RwD Rm =Qt

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