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�˚ 4;cu$ e˝ 3˚gs S�k S>J˝ Aw˚* 4; [�˚5� �"� �8 � 4;c

u$� [�˚ S�@ z*� ~ ˝ K0 S˘ ˚jM( J˘Ih �/z˝ :Ie&

t0k(t)"0� (�˚5� $g* fH0 &QNRRL@M @MC 'DKOL@M�����˚˙ m={ K�

+_6(t)+vj

t0j(t)

I B� ˚jM( J@ �/* :Io˚ t0k(t)=02 U# 4;cu$@ [�

* 3˚gs S>J˝ _6(t)+ vj(k

t0j(t)( *�</�5�

=@ H0 �8 3˚gs& :Ie& ? 5s B� gs& :Ie&( +

˘˚ 7Neˇ˘) ˚jM( F˝ 4;cu$� EE&z˙ p@/d: );�

˝ j U:$ @5˘ * U:$� EEB3( ) / IG�e w5˝ =@˚

5� �"� ˚ =@˚ ~�z\ /˝5 ˛ QH ˚jM(0 *�"* :Ie&

( +˘˚ 7N� � 4;cu$� 5s 4; [�ˇ˘Ih EE˙ pk�e

z0˘ �8 1A ˚q˝ =@˚ � & ˚5� =@ I˝ ˚jM( J˘Ih

:Io˙ Yi�\ >*5 4;cu$˝ �8 1A � *˙ & P˝ =H&

z˙ T�˘ K\�\~ ˚jM( J� EE&z˙ B3/d˝ 4; S>J( [�

�e w5˝ =@˚5���˚

˚jM( J@ �/�˝ :Ie& t0k(t)@ 2i( @4˙ ~�� ˛ U6 L

2 B(* 5s ˚jM( I�`J˚@ �/�˝ :Ie& ��� �[� �<2

U#˛ �8 3˚gs S� C: � )� =@F˚ ~�;d *5� �"� �8

3˚gs˙ *n�: )�� Gk H =@ �* _7*e& =@˙ !;>

5�& 85�

vj

Gt0j(t0)+_G6(t0)=0

Gvk(t0)-Gt0k(t0)+v

jGt0

j(t0)+_G6(t0)=0

�"� )� L =@ˇ˘Ih 9/ ~�z˝ ˛ B(* ˚jM( J �;

�*˛ Gt0k(t0)=Gv0

k(t0) =@˚ (x;d *5˝ K0 ! <˝k ?&˚5� ˚

=@ˇ˘Ih J˚jM( (��˝ �u. Q20 4; T< �<�� $G

� =H T<v& 4tO _0�*� 4;cu$ e �/�˝ :Io &z˙

T</[d e˙ �'*5� �"� ˚ \9�˛2 U:� �8 3˚gs S�@

3QTSGETK -@RG �8ˇ˘ *nz: )* =@0 5�& 85�

Page 16: 4%.4˙ 3 vn4 LO

31D-LJK.ps, page #16, Fri Dec 27 14:58:23 2013

��� 40. &4˙ +* 3˚�vn4;& LO9� 8'. [�

�˚ t1" � �n$ �-˝ 3˚79 �M(~˚ 4;cu$ ��� 40. RX7˙ hJ�

˝ U#( �'*5� :' - 5� ˝ 40.ˇ˘ U6�˝ ˚jM(� 4'@ u:zN

t0=argmaxvj

vj(t0)+_6(t0) �˚

˚k e$ �8 :Ie&( *n; >$� 4;cu$ e˝ 3˚gs S�k S>J

˝ Aw˚* 4; [�˚0˘ �8 :Ie&˝ 4;cu$˘ ��o �8 3˚

gs S�( [��*� K*�˝ t0k(t)@ w5� �"� �8 :Ie& t0

k(t)˝

&QNRRL@M @MC 'DKOL@M�����˚� r ���˚ �" 4;cu$� X.e&˘Ih

~�z˝ G(t0, t0k, {t0

j}j(k)=G(t-k, 0, {t0j}j(k) =@ � *n� & P5�

�� #j% ˚jM( 4%& Lc3˚79� 8'. [�

? f �˝ 3˚�vn N� � Lc3˚79� ˚;Q7 �s 8'.

[�˙ 0*�: )�� F @3˚79� ˚jM( 40˛ G @3˚79&

3˚79˚ 40.ˇ˘ U6�˝ U# # ˚ �s LO9� 8'. [�˙

S$?5� JO˛ G� U# ˝ Lc3˚79˚ P$.ˇ˘ 40. RX7˙ h

J�˝ U#˛ @3˚79& 7�˝ U# # 3˚79& 7�˝ U#˘

�� NvB 0*� T˘Ih Lc3˚79� 8'. [�ˇ˘Ih ˝�

& P˝ 40U�,. e�( �/� $ *5�

��˚ @3˚79� ˚jM( 4%

4I� 3˚�vn N� ��� B� 3˚79* 40.2 RX7˙ hJ

�: )�� 4;cu$ e :Io˙ Yi�\ /˝5 ˛ ?4z˝ 3˚gs

0 JZEE &z˙ *�<�˝ &z � ?4� K˚ ˚ 7N 3˚�vn

4;0 /hz\ /˝ K˚ �[� [�o˙ a� S$>.5� ˚ ?&˝ 3

˚79 e @@ K˛* [�˚0˘ 3˚�vn( ['�˝ @3˚790 ?6

˚jM( 40 '3�e � K˚5� T|0˘ @3˚79~˚ $GF e K

˛�*� �vn4;˚ /h� & P*� ˚jM(˙ 8(�� 40. RX7

˙ hJ�˝ U# [�z˝ 3˚gs0 r �˚ �" 5�& 8˚ *n� &

P5�

t1=argmax_6(t)+X1v1(t)��˚

Page 17: 4%.4˙ 3 vn4 LO

31D-LJK.ps, page #17, Fri Dec 27 14:58:23 2013

˚k�˘MX ��

˚ U#˝ 3˚79 �M(~˚ 40. RX7˙ hJ�˝ JH( u/�0˘ ]A K*

u:z\ /.5� ~_ 3˚79 �M(& 3˚79 �M(˚ 40.ˇ˘ 2��� ˚jM

(˙ 8(�˝ U#˝ �˛�-˘ u:�:˘ *5� F* ˚F˚ 3˚79 �& 40.ˇ˘

U6*5 �˛�-�ˇ˘ u:�� ˚7 ?4w 3˚gs0 t1,2"3ˇ˘ �� u:�:˘ Y

i*5� T˛ 3˚79 �˛ � # �M(˚ bL P$.ˇ˘ 40. RX7˙ hJ�� 40/@ � ?4z˝ 3˚gs0 t1"2"3ˇ˘ u:�:˘ Yi*5�

t1"=CX1(E (nZ)-nZ

1)X1(ZE (nZ)-nZ

1)+(Z-1)_E (nZ)

�"�@3˚79~˚ 4;cu$ ���˚jM(40( *5 E (nZ)"

nZ1 # `H� =@ �"� j˙ ˚˙ L˝ 3˚gs�t1""0˚˚ [�z˝˘˛ ˚

˝ @3˚79˚ ['�˝ 3˚gs& KJ* Kˇ˘ ˚ gs˚ [�z ˚

JZ � 3˚�vn@ ˚bB\e w5�

��˚ @3˚ * 3˚79� ˚jM( 4%& Lc3˚79� ˚jM( 4%

m,

@3˚79˚ $G e K˛* 3˚gs˚ ?4� & P*� 40. RX7

˙ hJ*5 3˚79� 3˚� _I@ @3˚79 e ˚8z˝ 3˚�v

n@ ˚bBR & Pˇ0˘˛ 3˚79 F* ˚ �<�� U6.ˇ˘ 4I

� 3˚gs ?4&4 Qc{ K˚5� ˚ U# 40. /@ � ?4z˝ 3

˚gs0 b� U:$F˚ 3QTSGETK �8 :I8'˙ JH�˝ U# 5�& 8

˚ ?4� K˚5�

t1"3=argmax_6(t)+X1v1(t)+X3v3(t) � ˚

r � ˚ � @3˚79 �& 3˚79 � � =*w ˚jM(F˚ 4;cu

$ e :Io Yi˙ +�� U6.ˇ˘ RX7˙ hJ�˝ U#˛ 5�& 8

˚ Lc3˚79ˇ˘ vNw Q2F� gs � � �� 693˚� 1�. &

z �" 40. �8ˇ˘ [�z˝ 3˚gs0 5�& 80 1˚* &z˙

>˚e � K˚5�

F E (nZ)&nZ2 U#˛ t1"3=0

G E (nZ)#nZ2 U#˛ t1"3=C

X2(nZ2-E (nZ))

X2(nZ2-ZE (nZ))+(_+1)(Z-1)E (nZ)

"0��˚

Page 18: 4%.4˙ 3 vn4 LO

31D-LJK.ps, page #18, Fri Dec 27 14:58:23 2013

�� 40. &4˙ +* 3˚�vn4;& LO9� 8'. [�

��˚ r � ˚� _7*e& =@ˇ˘Ih fI;@ (��˝ U# t1"3"0@ (x*5�

�:� YX;d { K0 E (nZ)#nZ22 U#˛ Lc3˚79˚ @@ ['�˝

�. 3˚gs0 S�D�˚\~ @3˚79& 3˚79� ˚j˙ �T�˝ ˚

jM(F� 40. U6 ��� T�˘ [�z˝ 3˚gs˚ t1"3"0˚ w5

˝ /˚5� ) E (nZ)&nZ2˚" ˛ Lc3˚79˚ @@ ['�˝ gs˚

t2=CE (nZ)-nZ

2

ZE (nZ)-nZ2"0˚\~˛ T�˘ @3˚79& 3˚79� ˚j˙ �T�

˝ ˚jM(F� 40. U6� ?&˛ [�z˝ 3˚gs0 t1"3=0˚ w5� @

3˚79& 3˚79� 40. U6 � � ?4z˝ 3˚gs0 B(* U

#˚�\ Lc3˚79� �. gs&˝ w˚@ E:e w5� �"� Lc3˚

790 Na\ L 3˚79� 40. U6˙ +�� ?4z˝ 3˚gs ��

� b� U# PB� EE&z˚ B3z0˘˛ ?6 3˚gs ?4&4 $

GF* 40.ˇ˘ Qc� $ �˝ K2˚ E/5� 5/ h� ˛ ˚jM( 4

0 ��� 3˚gs˚ ?4w5 b� 3˚790 �vn N� � $GF

� ˚j˙ >'�: )�� ˚ 40. U6 m�;d �˝ K˚ �[� [�

˚ w5�

b� U# PB� Lc3˚790 ˚jM( 40 m��/\~ T :V

0 8'. V � ˝zBd *5� +˛ Lc3˚790 8'. ˝ �"

F P$.ˇ˘ ˚jM(˙ 8(�N G �. 3˚gs ��� C�* ['

( @\ P˝ 5s ˚jM(& 2�( +�� 40.2 RX7˙ hJ�˝

:V˙ .!�e � K˚5� �"� 5� �� �˝ Lc3˚79� ˚jM(

m� I/˙ @9B P$.2 ˚jM( 40 m��˝ U#( S$? E˛

˚F� 693˚ &z˚ E (nZ)&nZ2 U#k T )�2 U#˘ NvB ˚F e

�[� 8'. [�˙ v;*5�

�˚ Lc3˚79� P$.� ˚jM( 4% m

@3˚79& 3˚79˚ ˚jM( 40 m�� P˝ 1M � Lc3

˚79 F* P$.ˇ˘ ˚jM( 40 m�*5 40. �8ˇ˘�� 3

˚gs0 b� ˚jM(F˚ 3QTSGETK 8' �" U6.ˇ˘ :I"9˙ �˝

U# r �˚ �" 5�& 8˚ ?4� K˚5�

t1"2"3=argmax_6(t)+X1v1(t)+X2v2(t)+X3v3(t) ��˚

Page 19: 4%.4˙ 3 vn4 LO

31D-LJK.ps, page #19, Fri Dec 27 14:58:24 2013

˚k�˘MX ���

��˚ r ���˚ � JZEEe&˝ 6<(t)=(X1+X2)[(1-b)v1(t)+bv2(t)]+X3v3(t)ˇ˘ 4�w

5� ˚ U# JZEE˙ *�</d˝ 3˚gs0 �˚ ]y5�

t1"2"3=�

˚˝ r ��˚ � S$? JZ.ˇ˘ �.� gs�t6D�˚& 9_* ?&( N

$f PB� b� 3˚79˚ P$.ˇ˘ ˚jM( 40 m��˝ U#˛

3QTSGETK 8' [� �" =s.2 3˚gs˚ *n� & P�˙ < & P5�

�˚ E (nZ)&nZ2 U#� @3˚79�� 4%. 7

Lc3˚79 3iw Q2F� EO(˚ E (nZ)&nZ2 )0 P˝ U#˛ ˚

F˚ @3˚79& 3˚79� 40. U6 m��\ /˝5 4I� 4

; [� t1"3=0 * �e � K˚5� T|N E (nZ)&nZ22 U# Lc3˚7

9˚ @@ ['�˝ 3˚gs0 S�H�@ w5� �"� ˚7 Lc3˚790 3

˚�vn4; PB� $G� EE ,T˙ �3</d: )�� 40.ˇ˘

@3˚79& 40.ˇ˘ 7{ K2˚ E:e w5� ? �� �˝ @3˚7

9& Lc3˚79� 7 �" 8(w +˘5 M(� �2> {j� V:˝

!(]F e 9_�5 @4*5� �"� T V:˝ L M(� �2> {jF

� h�?>�BNMUDW BNLAHM@SHNM˚ˇ˘ N$� & P5�

40. 7ˇ˘ 8(w ˚jM(� �2> {j˙ ��ˇh˚U��S˚˝hU��S˚˛ h˚��˛ �˚ˇ˘ 4�{ & P5� �"� L M(� 40. 7ˇ˘ 8(w +˘5

˚jM(� :Ie&˝ 3QTSGETK :I8' �" ˚|* h�?>ˇ˘ N$N

˝ �2> 4;['e&@ 4;cu$� 4; [� / )Rz*� 7N� K˚

5� ˚ U# ˚jM( 40˘ 2; ?4z˝ 3˚gs0 5�& 8˚ ?4w5�

t1,"2"3=argmax_6<(t)+(X1+X2)[(1-b)v1+bv2(t)]+X3v3(t) ���˚

r ���˚0 3QTSGETK :I8' �" � ˚jM(F˚ 4;cu$ e RX7

˙ hJ�˝ U# 40. �8 � ?4z˝ �8 3˚gs˙ N$f P5��˚

�:� @3˚79& Lc3˚79˚ +˘5 ˚jM(˙ 8(� � 9_* &

z� �2> {j˙ b� !(]F e >@*5 ˛ T 7 @D* �2> ��{

j0 vmax1,2 ,C

X1

X1+X2v1(t)+C

X2

X1+X2v2(t)˚0˘ vmax

1.2 &(1-b)v1(t)+bv2(t)� =

Page 20: 4%.4˙ 3 vn4 LO

31D-LJK.ps, page #20, Fri Dec 27 14:58:24 2013

��� 40. &4˙ +* 3˚�vn4;& LO9� 8'. [�

��˚ T|N ~_ L M( c� 40. 2� 8( GCH˚ (�*5 1-b&CX1

X1+X2=

@~ (x�� r ���˚� �. gs0 f1 �>5 1 K˚5�

@˙ ~���d *5� ˚ =@˙ c(O 4˛� 1-b&CX1

X1+X2@ w5��˚ ˚

( ˝{ 7˛ r ���˚� _7'v =@˙ ˚H�� 40. 7 �s �.

gst1,2"30 5�& 8˚ *nw5�

t1,2"3=mC

(X2-X1b-X2b)(nZ2-nZ

1)(X2-X1b-X2b)(nZ

2-nZ1)+(Z-1)E (nZ)

"0, i j1-b"CX1

X1+X2

0 , i j1-b=CX1

X1+X2

���˚

)� ?&( S$> ˛ 1-b"CX1

X1+X22 U# ˝ �. gs0 t1,2"3"0@ z

0˘ Lc3˚79˚ @3˚79& 7{ K2˚ (��\~˛ 1-b=CX1

X1+X22

U# ˝ r ���˚& r ��˚@ 9_�e zB t1,2"3=t1"2"3=0˚ z0˘ Lc3˚

79 e˝ @3˚79& 7{ K2˚ (��\ /˝5�

*{˛ Lc3˚79�Q2F e˝ r ���˚ �/w �8 3˚gs�t1,2"3"�˚

˚ [�z˝ K˚ S�� gs � � �vn ˚˚ $GF� 693˚ˇ˘~ 3

C�˝ K>5 x N0 [�2\ S$>$� Lc3˚79˚ @3˚79& 40

. 2�( *5 ˛ ˚ �s �2> 4;['e&˝ r ���˚ �" v1,2(t1,2"3)D

(1-b)v1(t1,2"3)+bv2(t1,2"3)˘ N$� & P5� TB5 3QTSGETK 8' �s

�2> :Ie&˝ 5�& 80 V:( L˝5�

t1,2(t1,2"3, B 1,2)=max[0, v1,2(t1,2"3)-B 1,2]

�:� !�˛�˝ Lc3˚79˚ @3˚79& 2��� 4;cu$ e RX

7˙ hJ�: )�� :I�˝ U# ˚ �s l{j�F˝ l>&˚� V:˚

5� �"� @3˚79& Lc3˚79� N�˝ ˚ l{j� V:( *�<�

˝ K˚" h & Pˇ�˛ ˚ X.˙ P(�: );�˝ 4;cu$� X. F

* 40.2 � 79F˚ 3QTSGETK 8' �" U6.ˇ˘ :I( �u�˝ U

# *�<� & PBd *5˝ K˚5� �"� ˚ L 79˚ 40.ˇ˘ 7

�˝ U# 5� =@˚ ~�zBd *5�

Page 21: 4%.4˙ 3 vn4 LO

31D-LJK.ps, page #21, Fri Dec 27 14:58:24 2013

˚k�˘MX ���

maxB 1,2, s.t. G(t1,2"3, B 1,2)&G(t, B 1,2)

�:� &�S˛ !�˛�˚D�^�˝^�˚z�˛��S˛ !�˛�˚˝^�z��S˛ B 31,2"3)+_6(t)� 8'.ˇ

˘ =Q7 P˝ ˚ L 79� N�˝ U6Q7 P˝ 3˚79˚

3QTSGETK �88' �" �2> �8 l{j� V:( B 31,2"3˘ ?4* U#

$GF F* 40. :I �s l{j� V:( ?4�e � K˚5� �"�

˚ L 79� 40. 2�˘ ˚bBk ˚jM(0 4;cu$@ �84;

t1,2"3 ˚? 5s B� �u˙ [��\ >�*� :Ie&( ˝�˝ @5

˘ $GF� l{j� V:@ *�<� & P*� 67{ K˚5� T|0˘

!DQMGDHL @MC 6GHMRSNM��� ,˚� 3GDNQDL � �" L 79� 2�˘ ˚bBk

˚jM(� :Ie&( ˝; >$� #[ ˚ ˚jM(ˇ˘Ih� :I@ ˚ˇ

� ˚F& 40. U6Q7 P˝ 3˚79~ 3QTSGETK 8' �" :I(

*5 4;cu$˝ t1,2"3 ˚? 3˚79˚ x ['{ & P˝ 4;˙ [

�{ & P˝˘˛ ˚( 5�& 8˚ 4�� & P5�

t"3=argmaxX3t3(t, B 1,2"33 )+_6(t) ���˚

r ���˚ �" t"3˚ 4�w5 ˛ ˚ L 79� 2�˘ 8(w ˚jM(ˇ˘

Ih 4;cu$ e �/z˝ 40. :Ie&˝˛ ˚jM(F˚ bL 3QTSGETK

-@RG �8 :I8'˙ [��˝ U# &QNRRL@M @MC 'DKOL@M�����˚ ���

5�& 8˚ 7N� K˚5�

(X1+X2)t1,2(t1,2"3, B 1,2"31,2 )+X3t3(t1,2"3, B 1,2"3

3 )+_6<(t1,2"3)

=X3t3(t"3, B 1,2"33 )+_6(t"3)

) =@˙ 3QTSGETK :I8' �s �2> :Ie&( ˝�� 8Q; >

5�& 85�

(X1+X2)(v1,2(t1,2"3)-B 1,2"31,2 )+X3(v3(t1,2"3)-B1,2"3

3 )+_6<(t1,2"3)

=X3v3(t"3)-X3B1,2"33 +_6(t"3) ���˚

Page 22: 4%.4˙ 3 vn4 LO

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��˚ $g* fH0 &QNRRL@M @MC 'DKOL@M�����˚( m={ K�

T=˘ ~_ 4;cu$� 4; [�˚ t"3 gs˚" ˛ 3QTSGETK -@RG �8

� �˝ L 79� 2�˘ wEw ˚jM(ˇ˘Ih� 40. :I˝ >2O

� &z˚ � K˚ ˛ ˚ U# v1,2(t"3)'B 1,2"31,2 � =@˚ `!w5���˚ ˚ e�(

˚H�� ) 8Qr˙ 4˛� 5�& 85�

(X1+X2)v1,2(t1,2"3)+X3v3(t1,2"3)+_6<(t1,2"3)

&X3v3(t"3)+_6(t"3)+(X1+X2)v1,2(t"3)

) Q7 IWr˙ 4� �" 8Q; > 5�& 85�

(_+1)[(X1+X2)(1-b)v1(t1,2"3)+(X1+X2)bv2(t1,2"3)+X3v3(t1,2"3)]

&[(X1+X2)(1-b)+X1_]v1(t"3)+[(X1+X2)b+X2_]v2(t"3)+X3v3(t"3)���˚

T=˘ r ���˚ˇ˘Ih 79 c 7ˇ˘Ih 8(w ˚jM(� 40. :

I@ j� ˚˙ L˝5 ˚F� l{j& QH�� 5� fWr˙ 8Q; h

& P5�

(_+1)[(X1+X2)(1-b)v1(t1,2"3)+(X1+X2)bv2(t1,2"3)+X3v3(t1,2"3)]

-(_+1)[X1v1(t"3)+X2v2(t"3)+X2v3(t"3)]

=(X1+X2)B 1,2"31,2 X1v1(t"3)-X2v2(t"3)

���˚

r ���˚� YT0 r ���˚� IWr˙ ˝; > �>5 H ˚o˙ < & P

5� ˚ ?&˘Ih 5�& 80 ?&@ *nw5�

(X1+X2)B 1,2"31,2 X1v1(t"3)-X2v2(t"3)&0

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