Stochastic Modelling and Optimization using ...


 
 "!#$ %'& ()%+*-,#.%/10 !'(2#3 5476%"86!99:&;8=@?BAC?AEDFFD G5HCAC?JI5?@KLG )M>DN< GPOQ?7RTSU=WV+=NX.Y5RZ=NGGPQ[\5]N ^D_
`D ¨D_F`©¡GQ"GD" 1 Ÿ ªO G«X.Qž"ª G«
¦¬RTDD G G ­®°¯ GP"Q"°.±–
¦ ¥ D )MOG H ¥ D_ )M"O Gk=;±;QžG Q[YtQ"Q
N P¨D_"GQ[²³ "GO ´ ‰DOa;ށÊ/Ò;ÐTË°ÒØ+Ö;ËÍÕ-ʕÉ+Ø+ËÍÌ1ÅÇ)úyǤÕ+ÖÏ2úCÖWÏ2úüØ+ÖWËÍÕ-Ì1Ê/ÒûÈÇ)ÐÏÔÒÇÜÏ/ɫʎÕ+ËÍÎ5Ú;ÅÍÇkÇàÊ/Î5Ú;ÅÇ ÚWÉ+Ï/È;ÅÍÇÑÎøLýUÖÇ1É+ÇʕÉ+ÇhÖÏ2úyÇ1×/Ç1ÉÑÞWÕ.Ç1×/Ç1ɞÊ/ÅÕ+ËÍØ+ÓWʕØ+ËÏÔÒWÕmúUÖÇÑҚØ+ÖÇÒÏ ÐWÊ/Å;ÜÏ/ɞÎtÓWÅÍʕØ+ËÏÔҜÈÇÑÌÏÔÎ5ÇÑÕ Ê«ÅÍËÍÎkËØ+ʕØ+ËÏÔÒ>ø‚þUÇÑÊ/ÅÍËÍÕ.Ø+ËÍÌÕ.Ø.Ï ÌžÖ;Ê/Õ.Ø+ËÍÌ¡Ú;É+Ï/Æ/ɞÊ/ΫΫ˰ÒÆ-ÚWÉ+Ï/È;ÅÇÑÎkՃΫÊÑÙ¤ÖWÊ2×/Ç[×/Ç1É+ÙÅÍʕɞÆ/Ç[ÐÇ1Ø.Ç1É.Ý Î«ËÍÒWË°Õ.Ø+ËÍÌ®ÇÑÿ_ÓWËÍ×Ê/ÅÇÑÒ_Ø+Õ1ÞÊ/ÒWДË܁úyÇkÌ1Ê/ҐÊ2×/ÏÔËÍДËØúyÇkÐϤÒÏ/Øhú7ËÍÕh֖Ø.ÏËÍÒ;Õ.Ø+Ê/Ò'Ø+˰ʕØ.Ç5Ø+ÖÇÑÕ.Ç«ËÍÒ ÜÓ;ÅÍÅmʕثÊ/Ò_ÙTÚ ÏÔË°Ò'Ø-ÐWÓɞ˰ÒÆØ+ÖÇÎ5Ï ÐÇÑÅÍÅ°ËÍÒÆÚ;É+ÏÌÇÑ՞Õ1øÄeÅÍÕ.ϖô¡õ öÕ-Ø.ÇÑÒ;Ð Ø.ϐÖWÊÑ×/ǤʖÖWËÍÆÔÖWÅÙ É+Ç1ÚÇ1Ø+ËØ+Ë×/Ç-Õ.Ø.ɞÓWÌØ+ÓWÉ+Ç/Þ Ê/Ò;Ð ËÍØeËÍÕ7úyÏ/É+Ø+Ö Æ/ÏÔËÍÒƚØ.Ï)Õ+ÏÔÎ5Ç-Ç
Ï/É+ØØ.Ï)Çà Ú;ÅÏÔËØ7Ø+ÖWËÍÕeÜÏ/ÉÇ kÌ1ËÍÇÑÒ'Ø ÚWÉ+Ï/È;ÅÍÇÑ΅Æ/ÇÑÒÇ1ɞʕØ+ËÏÔÒ> ø LË°ÒWÊ/ÅÍÅ٘Ø+Ö Ç ™Ê/ÅÆ/Ï/ɞËØ+Ö; Î BÕÜÏ/ɞ Î ƒÏ/Ü5ʒÕ.Ø.Ï ÌžÖWÊ/Õ+Ø+ËÍ̖ÚWÉ+Ï/Æ/ÉÊ/ΫΫËÍÒÆ ÚWÉ+Ï/È;ÅÍÇÑÎ ËÍÕtÐWË
Ç1É+ÇÑÒ_Ø[Ø.ώØ+ÖWʕØtÏ/ÜUËØ+Õ-ÐÇ1Ø.Ç1ɞΫ˰ÒWËÍÕ.Ø+ËÍÌ-ÇÑÿÓWËוÊ/ÅÇÑÒ_Ø1ނÊ/ÒWÐØ+ÖǚΫÏÐÇÑÅ°ÅÍËÍÒƤÕ.Ù Õ Ý Ø.ÇÑÎÕ+ÖÏÔÓWÅ°ÐØ+Ê /Ç5Ø+ÖWË°ÕeË°Ò'Ø.ÏÊ/Ì1ÌÏÔÓWÒ_Ø1ø®ÄeÐWÐÉ+ÇÑ՞Õ+ËÍÒƜØ+ÖÇÑÕ.Ç-ÚÏÔËÍÒ'Ø+ÕÑÞ úyÇ5ÐÇÑÕ+ÌɞËÍÈ

Ç >Þ Ø+ÖÇkÕ.Ø.Ï ÌžÖ;Ê/Õ.Ø+ËÍÌkΫÏÐÇÑÅ°ÅÍËÍÒÆ)ÌÏÔÎ5ÚÏÔÒÇÑÒ_Ø¡Ï/ܬØ+Ö Ç !#"%$'&()*+&-, Õ+ÙÕ.Ø.ÇÑΤÞ@úUÖ;ËÍ̞֎úyÏ/.É

Õ[ËÍÒ ÌÏÔ0Ò /+ÓWÒWÌØ+ËÏÔҎúUËÍØ+֎ÇÑËØ+ÖÇ1É[2 Ä 14 ö 3’Ï/6 É 5ö8 þ 7yõWõ9Ý 1ö Ê/ÒWЖÊ/ÅÍÅÏPúUÕhØ+ÖÇ5Ç kÌ1ËÍÇÑÒ'Ø¡Æ/ÇÑÒWÇ1ɞʕØ+ËÏÔÒ Ï/ܳÅÍʕÉ+Æ/ÇÝ$Õ+Ì1Ê/ÅÇtÕ.Ø.Ï ÌžÖWÊ/Õ.Ø+Ë°ÌhÚWÉ+Ï/Æ/ɞÊ/ÎkΫËÍÒÆ5ÚWɞÏ/È;ÅÇÑΫÕ1; ø :–ÇtÆÔË×/Ç[Õ+ÏÔÎ5Ç[ÐÇ1Ø+Ê/ËÍŰՃÏ/ܳÕ+ÓW̞ÖÚWÉ+Ï/ÈÝ ÅÇÑΫÕ@Ê/ÒWЮÐWÇÑÕ+ÌɞËÈ< Ç =>?>ÞPÊ/ÒtËÍÎ5Ú;ÅÍÇÑÎ5ÇÑÒ'Ø+ʕØ+ËÍÏÔÒ¡Ï/ÜWÒÇÑÕ.Ø.ÇÑA Ð @yÇÑÒWÐÇ1ÉÕÐÇÑÌÏÔÎ5ÚÏÔÕ+ËÍØ+ËÏÔÒ¡úUÖ;ËÍÌžÖ úmÏ/BÉ ÕmÇÑËØ+ÖÇ1ɬËÍÒWÐÇ1ÚÇÑÒWÐWÇÑÒ'Ø+ÅÙ®Ï/Ü>Ï/ÉyË°ÒkÌÏÔ0Ò /+ÓWÒWÌØ+ËÏÔÒkúUËØ+C Ö 5Ê/Ò;МÖWÊ/ÕmÈÇ1ÇÑÒkÓWÕ+ÇÑЫØ.Ï Õ.ÏÔÅ×/Ç-ʚ×ʕɞËÍÇ1Ø$Ù¤Ï/܂ɞÇÑÊ/Å@úyÏ/ɞÅÍЉÚWÉ+Ï/ÈNÅÇÑΫÕ18 ø :–ÇtØ+ÖWÇÑÒ Ð;ËÍÕ+Ì1ÓWÕ+Õ7ÖÏ2ú}× ËÍÕ+ÓWÊ/Å°EË DÑʕØ+ËÏÔÒ¤Ø.ÏÏÔÅÍÕÌ1Ê/Ò ÈÇ-Ó;Õ.ÇÑЎØ.ÏÊ/ËÍЎØ+ÖÇ«ô¡õ ö Î5ÏÐWÇÑÅÍÅÍËÍÒƚÚWÉ+Ï ÌÇÑÕ+Õ1ÞÊ/Ò;АÕ.Ç1ØhÏÔÓØhØ+ÖÇ5Ú;É+Ï/Æ/É+ÇÑÕ+ÕhúyÇ«ÖWÊ2×/ÇkΫÊ/ÐÇ Ø.Ï2ú¬Ê•ÉžÐWÕ5Ê/Ò ËÍÒ_Ø.Ç1Æ/ɞʕØ.ÇÑÐûÕ.Ø.Ï ÌžÖWÊ/Õ+Ø+ËÍ̚ÚWÉ+Ï/Æ/ɞÊ/ΫÎkËÍÒÆ ÇÑÒ'× ËÉ+ÏÔÒWΫÇÑÒ'Ø-ËÍÒØ+ÖǚÐWÇ1×/ÇÑÅÏ/Ú;Î5ÇÑÒ_Ø-Ï/Ü '"$!&()*+&F,Uø Á ML¿N GIH>JKAL 0

ôeÙÒ;Ê/ΫËÍÌhÕ.Ø.Ï ÌžÖWÊ/Õ+Ø+ËÍÌ¡ÚWɞÏ/Æ/ɞÊ/ΫΫËÍÒWÆÞ Î«ÏÐÇÑÅ°ÅÍËÍÒÆ-ÅÍÊ/ÒÆÔÓWʕÆ/ÇÑÕÑÞ;ÐÇÑÌÏÔÎ5ÚÏÔÕ+ËØ+ËÍÏÔÒ>ø

O

PRQTSUWV

XY

Z4[]\^Y`_UaZ4[bXYdcfeb\hgCebUWV

V

XSg

IhG« •D_
 •DF¬#jie%>k (2# !/8ml no_&!9 ®(2#+%/10 !'(2#$ 6%"86!99plAq3«= Ssr7·¡"F G•D7 
Qž"DQ¡ D MG«·¡G  tu

QP?™?†…

v?w

qyx

{}|Z ‰ qyx q | r„ | r |€ vƒ‚ ‚ t~B

r w

z

x w‡w | w … q ‡r x ‚-w x b„ w w2‹ z

x

… q| r ‚

‹

‚‡‚

q | r‡x

b

q| r ‚

ˆ ˆ

‰0w | q r ‰‚

qž¸r

!Š°(0Š !Š°0( Š

·¡GP"GŒx DQ" h]
N PGQQ¬·¡
Q" G n!# !  !#:0¢ ( ‘ ·¡FF³M@GD_]]"
›N ‰D_"GOTM±TC D £> GœWª )MGt
_¦ß]>D_ Q®¦§"
 qDOQ GPGœ GkQ
; 1>DQž Q" )ªFD_
`G">DŽ"G->D_"ªD_Fy#$:p 9 l l26Ñ 0% n q$G?™?ßDš·7GGP»
k ‰
W"%  r-
¦e"GŽD_]]F •D_"
L?‰IhGP"G‰ D•±ûMG QGP¯ GPDFUQ"ª 1E ‰G¤]G"
NO QkMGP.·7GGP  nml12(2 %'& (2# !/m8 l1(
¦‚"G« ‰
;OGFLDO–Wª¤ Q Ÿ "G] "GQGQ[Q" )ªFD_GPO–Q1D'GQ¡
¦‚"G«ªOGP"F±;O>D_D¤] "
; GQQ QD_ ‰]FGO D_¤ G”QG P
O Qž1DG”OD_G_¥ ? q$>
 ‰
"G
 Q¤OQž P
 MGP.·7GG  GQ •DFGQ O ±;>D  5Qž
; 1>DQž «]
 "D ‰  (2#3  l-¨0N’ ©'!* l2&>#@]"
MFGP
` ¦ q+ƒ¸ rU·¡ 1` •DF ªFD_œ"G5OQ" GP"G5G ›N]@GP PD_
 G ›N]F PF°± ‚

q’x

q’‘`r‡x ‚

‚

q’‘`r(„F‘`r

q’‘`r

ˆ

‰0w ˆ

‰‚

q˜‘`r

a

‘

‘





Ÿ ŸA„

q  a r

·¡GP"G q’‘`r«OG
GPQ¤"G–]
M>DMF°.±ŒDQ"Q
N P¨D_"GO ·¡" GŽG2¯ GPWh‘pªO G¤Q"
GŽ]
M>DM F°.± G•DQª"± GT²¤? Ih G¦§
" )ªFD_
 ¥q3a?ršQ¤»W
/·¡}D_Q¤"G (2# !&³n!6Fn´q$
T1%'9t !;2#*rœ¦§
 OGP"G+V Q ‰G œWª¯_DFG•?’›[G"GŽ·7G–>D•¯ G
 G ¯ GP P
)¯D_"¨D_MFGhx ·¡ 1 QµQ"D"GPO+¶’M@G2.·eGPG DFF "GDF^•D_"
 Q?¡YtF°G>D_"¯ GPF±·7G) •D ¦§
" )ªFD_G5"G)OG2G ‰w Qž œGœ;ª ¯_DFGP-ªQ–D¯D_"¨D_MFG x q’‘`r¡¦§
-G•D_ 1f  ‘ ŸpDOû"GŒDOO &L%'&!&>#$ 2¼ !#3’ ©/:j# oš P
QžD_A Q x q˜‘`r x q˜‘©¡
N 1»'D_¦§GPFF¨D_DOA˜GPQP©[ª Q"^ P^P¤  ± Á Q"»W3kž!&;8 N°!6ߦ§
" 
"GQ]@
O "
"G«]"
M;V FGP¡ ęQeON±ND ‰ tQž"ª Pª "G?ßY®QhD–DF°GP">D_"¯G®
šQ"
F°¯Nš"GtOG2GP" Q ®G œWª¯_DFG•N6(l7¦§ª P"
r ÉWqyx

 € ¦ ° q’‘`r qyx ’q ‘`r„B‘`r vƒ‚ ‚ t ~ ˬ ¬®0¯ ¬m®0¯4± Q?B•?̅ q’‘`r‡x ’q ‘`r q˜‘`r4͗… q˜‘`r.x ‚‡‚ ‚ ˆ¥‰‚ -‚ w w r

w

ˆ

qjÏr ‘ 

ŸAÎ

X  Q G•DQž± G
ª  
Q"
Ô·q>D'ÐÉ{QD P
¯ G(› ¦§ª P"

¦Ñx ?ÒÅeGO G"Q”OGP 
]@
Q""
 ]
N PGGO Q[M± MªFO )ª]”D)]GP G2·¡Q"G5FGDeF
/·eGP¡D]]
›N D_"
‰w
¦6É M±Ž"±;¤D)QG0œWªGP G €
¦hO Ô Ó@G"GPœ6%1>%/(!¼(Õx ¦Ö ¦§
œ·¡ 1 Oª>D_F7Q
Fª "
 Q× ¦Ö € q’‘`r«DG
M D GO`?IhGPQ"GDG ªQGO 
¦§
" %1#$:9!:j# o” ;#§(2D•¯G‰ G‰ DQ"GÆÉ q’x w

€ šÖ r

q˜‘`r.x

‚-w

ˆ

zÑÞ

w

Ø

z

Ø q§

ˆ ˆ

‰0w °

q’‘`r‡× šÖ q˜‘`r

¬®¯ ±

?™G? ¦§
kQ"
 ‰G¤]
]@
QDFQx ¶

qÝÜr €

‰‚

w

¦Ö

q˜‘`rÚÙ €

ˆ

¸?„ÎÎ΄ÛÎ

 G "GP 
ª "Q"G¦§ª 2
˜Q

 N¦§G•DQMFGr2·7G •DT GGPD_"Gt*l"!'(2jº::#joN#:(®M±TªQ OG P"
 Q®
¦ßG PGQ"Q
û
M 1DGO¦§"
 Gßq§ªWM@
ª OGOrOªDF«Q"
Fª 
Q? >
 ‰D± ]D_ P •DFt]"
M FG QTÅeG OGQTOGP 
]@
Q""
 >DQ®M@GPGûQ
/·¡ 
ŽMGœ¦$DQžGP®D Q"
F¯;GœOGP"G ‰Qž «G œ;ª ¯_DFGP-]
MFGP ‡DQtªQª>DFF±
F°±D‰¦§GP·pGPQ-
¦ß Pª Q«DGš0G œWªGO "
D  PªD_"GF°±TD_]]"
›N ‰D_"T G ɜD"DFFGF3D_FF±[ GeQ"
D_ Ge0G œWªG GQm
%¦ ÅeGPOGP"Q OGP 
]@
Q""
– P
NO GQ[D"G5
" ‰DFF°± )ª 1”Q" ‰DFFGe">D G GD_F°V.] ª"]
Q"G« GP"
NO Q¡D]]FGOŽ"
 GkO GPGP" Q" -G œ;ª ¯_DFGP¡]"
M FGÚ q ?a rÑ? IhGOGDQ
¦¡ GŽ.·e
ŒQD GŽ¦§
 )ªF¨D'
EG ›;"GO "GDOF°±’
ûGû 9 ;#$(2# !/m8 lš •DQGDFF±D_Qß" Gh]"
 "GPQ"Q"GPQ?ßI
k¦§
 D_FF±k(G › ] "GQQ7"Qƒ¦§
UÕ D â¡V QD Gh]
MFGP < ·7G[ªQ"G®D ã #$6"!#$ %'I & äæå ´ ä§ç]åè ¸ „ÎÎ0Î0„-⫹WF°D'
Q¡G ›N]F °F±< ·7Gš GPG"DFF°±–»–GP" Q®
§ ¦ l©l2&>#e#$6 ll1(2< ·¡ 1pD/¯G D 
NOG˜¦§
ûG•D 1CQGP’Ê  éWç.„ìè ¸ „ÎÎ΄-â DOpDpD" EM@G2.·eGPG£GD 1p
NO G  ˆ "GP]"GPQ"GPW"“  … · éWç˜ê D_ O … éWç.? =N
D•¯ G "GŽGP¯G¤GGŽQ
/·¡  y ªGT¸ ? Ih GŽªQ"ªDF7""GG‰G ‰
F
±ìq$D GPQ"
DMF.± "
G•D 1 FG•D_¦Iq§
ŽG0œWª°¯_DFGPW"F± 
GD 1 Q" PG>D_"
r2D )M ª
ª Q"F± OG £G5G«]
M>DMF°.±
¦yDFF`
;OGQ¡–"G«GP¯ GP®""GG_? X UQ¬
_ßQ 2F±šG PGQQDž±>D_ƒ"G¡]>Dž"
 QyMG 
 ‰2 G £> GUD_ßG•D_ 1‰]G
;O`Dw Q" OGP Q"
 ]"
; GPQ"QÕx4q | rÕå q’x „-x q | r„ÎÎ0΄-x ò q |Aò r-r[¦§
®·¡ 1û·7GªQ"G G 
D_
Æ  x ç q | ç r å qyx „ÎÎ0Î0„-xçBq | çjr-r-
”"GP]ˆ "GPQ"Gw )"‚ G ‚ ] "
; GQQšª]Œ
TDOE FªOQ1D_ GCèÑ? A˜Gk D”
/· G ›N]ˆ GQQ® w GkFGD"F°± 
Qž"DGPO˜ 9 ;#$(2# !/m8 lano'&!9 -(2#.%/10 !'(2#3 6%"86!9p 9 lDQ  

tu

w

x



zÐ{ | w

‰| 

~

€

t~F

~ vƒ‚

qyx ‚

q| ‚

r„ | ‚

r¢z—{ |  | ~ z

Q?B•?ʅ

{ |  |

| 

t 

  u 

x

w‡w | w q r.x ‚ ‚-w | í w … í q .r x w w …

… ò w

q|

ò

r.x

q|

z

… z

… í

??

w

… ò x

ò

‚

q|

r‡x

b

‚

w ‹ ç ¢ xç-q | r ‹

q| ‚

t



€

|  

íƒq’x%í>q | í(r(„ | ír;zÎ0ÎÎ v



€ v

ò q’x ò q |Õò r„ | ò rÎÎ0Î  ˆ

r

‚

‚‡‚ | í ‚ q r.x q | r ‚ ‚ ‚

? z

q|

r.x

‚



ˆ

í í …§í‡í>q | r‡x%í>q | r

z

?? ?

rRz

… òò q |

Î0ÎÎz

b è

‰0w | q r ‚ ‰‚ | í í>q r

ò

???

?? ˆ

r.x ò q |

ò r

‰

ˆ

? ‰

í>q |

ò r

a#„Î0Î΄Fâ6„ ˆ

qݿr

·¡GP"GŽD_DŒDFFƒ 
 Q"D"Qœ
FOŒDF 
Qž«Q"ªGF±?–›®GP"Gh{ | | u qB™Er-OGP
"GQš P
O°
>DF¬G(›WV ]G PD_
T·¡° GQ"]G 25
Ž"G) ª ""GP«Q1D'G)
¦ƒ"GO>D_D ]
N PGQ"Q | P
O°
GO
 "Q-]>DQž Q
± ?’Y®DLDQM
 Qž1DOD"O D O Q"]Fš¯_D¨DM FGOGP"G"  Q" ‰G0œWª¯_D'V FGP[¦§
" )ªFD_
A Q q§·¡GP"G«¯D_"¨D_MFGPQG•D_ 1TQD G¤?¸ „Î0ÎÎ0„-â Í ¸k D•±ŽMG«Q"]F r7·¡ 1·eGk
 GP"G ¦§
ŽM "GP¯;°.±  ? Åe
 Q GP>D"
OG P
]@
Q"
 D O ÅeG OGQ–OG P
‰]
Q
}M>D_Q"GO Q
FªNV 
E GP"
;OQš D MG G ›;"GOGPO 
TG   GPF±˜>D OFG¤G )ªF"Qž1D G¤ DQ"G_?TIhGŽ šªF"QD G ÅeGPOGQhOG P
]@
Q"
 DF
"" Q7»W
/·¡D_Q-! & l1(2.# l-nÕÅeGPOGP"QhOGP 
]@
Q""
  q9Åe" G_D_®Q® 
;OGF³GD 1T ‰G D’
MQGž¯D'
ŒQ5 D_OG«Q-¦§
FF
Ô·7GOû ‰ GO ¨D_"GF±”M±’D–OGP Q
L GœD'¡G•D 1
;OG5Ž"G5GP¯G["GPG-G 1% l
! 2Ý l2&># 6%/ l1(1# ( "0çFq | ç r¢å qy…§ç q | ç r„ÎÎ0Î0„ž…¤çEçFq | ç r„ çBq.™*„ | ç r(„ çFq | ç rFreD"GP]"GPQ"GD_
QªDMFG-¦§
Q"
'V FªN
šG°GPy
¦>ˆ "GOG2w GP" Q ß0G œWª¯_DFv Gƒ]
MF‰ G CM±)Dt P
¯ GPW"
DFOG2G ‰Qž UQ
F¯GP<
® Q
‰Gœ"GP]"GPQ"GD_
ûQ"ª°1DM FG«¦§
tQ
Fª "
M± DŽ«=NS¬V Q"]G PÔ £> )Q
F¯GW  q§¦§
®G › D ]FG_FG®¦§
" ‰D_  qjÅe G-GP[D_F ?°