kuliah or ii

7/21/2019 kuliah OR II

http://slidepdf.com/reader/full/kuliah-or-ii-56da2421a6271 1/56

susy susmartinioperations research II, 2006

OPERATION RESEARCH II

3 SKSMATERI KULIAH 1




DYNAMIC PROGRAMMING

KARAKTERISTIK DP :

PROBE! "APAT "IBA#I "AA! STAGE , "E$#A$ S%AT%POLICY DECISION ,

SETIAP STAGE TER"IRI "ARI SAT% ATA% EBI& STATE (POSSIBLE CONDITIONS)

PE$#AR%& POLICY DECISION PA"A SETIAP STAGE !E$#%BA& CURRENT STATE KE "AA! S%AT% STATE 'A$#BER&%B%$#A$ "E$#A$ NEXT STAGE

SOLUTION PROBLEM "I"ISAI$ %$T%K !E$"APATKA$OPTIMAL POLICY BA#I PROBE! SE(ARA KESE%R%&A$

OPTIMAL POLICY PA"A S%AT% STAGE BERSI)AT

INDEPENDENT "ARI POLICY PA"A STAGE SEBE%!$'A SOLUTION PROCEDURE "I!%AI "E$#A$ !E$E$T%KA$

OPTIMAL POLICY PA"A LAST STAGE

RECURSIVE RELATIONSHIP :

( ) ( ) }{ nnnnn x s f Min Max s f ,/* =




( ) ( ) }{ nnnnn x s f Min Max s f ,/* =

( )

( ) ( )**

*

,

dan,, pada

tujuanfungsiuntuk kontribusi:,

) pada(:

untuk :

untuk :

)......,,2,1(untuk label:

Jumlah:

nnnnn

nn

nnn

nnn

n

n

x s f s f

xdecisionn stage s state

n stage x s f

s xof valueoptimal x

n stagevariabledecision x

n stage statecurrent s

N n stagen

stage N

=

−

=




JENIS DYNAMIC PROGRAMMING

DETERMINISTIC DYNAMIC PROGRAMMING

PROBABILISTIC DYNAMIC PROGRAMMING

n s

( )nnn x s f ,

Stagen

State :1+n s

( )1

*

1 ++ nn s f

Stagen+1

n xof oncontributi

n sState :

( )nnn x s f ,

n xdecision

( )1*

1+n f

( ) s f n*

1+

( )2*

1+n f

1

2

s

2c

1c

sc

1 p

s p

2 p

probability

Contributionfrom stage n





(ontoh soa* +:$o o-

!e.ica*Teams

Thousan.s o- A..itiona*Person/'ears o- i-e

(ountry

+ 2 3

0

+

2

3

1

0

1

0

0

+01

+20

0

20

1

1

++0

+10

0

10

0

40

+00

+30




Penye*esaian :

)({max)(

2,1)()({max)(

)()(),(

),(max)(

)(max)(),(

!

)(

"",.....,1,#

""

1,.....,1,#

1

,.....,1,#

"

"

1

"

1

"

1

""

x p s f

n for x s f x p s f

x s f x p x s f

x s f s f

s x

x p x p x s f

integersnegativenonare x

xtoSubject

x p Maximize

s x

nnnnn s x

nn

nnnnnnnn

nnn s x

nn

ni

ni

ni

iinnnnn

i

i

i

i

ii

nn

nn

=

∗

∗+

=

∗

∗+

=

∗

=

+=

=

=

=

=−+=

−+=

=

=

+=

=

∑

∑

∑

∑




0

+

23

1

0

10

040

+00

+30

0

+

23

1

" s )( "" s f ∗ ∗" x

n = 3




0 + 2 3 1

0

+

2

3

1

0

10

0

40

+00+30

20

0

0

+00+20

1

1

++1+21

1

+21+1

++0+60 +10

0

10

0

1

+21+60

0

0

0 or +

2

3

2 x

2 s

)()(),( 22"22222 x s f x p x s f −+= ∗

)( 22 s f ∗ ∗

2 x

n = 2




0 + 2 3 1

1 +60 +0 +61 +60 +11 +20 +0 +

1 s

1 x )()(),( 11211111 x s f x p x s f −+= ∗

)( 11 s f

∗ ∗

1 x

n = +

Optim! S"!#ti"n :

###.1$#

1

1"%

"

%1!

1

"

"

2

2

1

=−

=

=−=→

=

=−=→

=

∗

∗

∗

lifeof years personadditional

x

s

x

s

x





(O$TO& SOA 2 :

T&E !I$I!%! E!PO'!E$T RE5%IRE!E$T

SEASO$ SPRI$# S%!!ER A%T%!$ I$TER SPRI$#

RE5%IRE!E$T 211 220 20 200 211

BIA'A KEEBI&A$ TE$A#AKER7A

8 2,000 9 OR# 9 !%SI!

BIA'A PER%BA&A$ 7%!A&TE$A#A KER7A

8 200 9 PERBE"AA$ 7%!A& TK;2




PENYELESAIAN :

• STAGE 1 : SUE!• STAGE 2 : AUTU"• STAGE # : $%"TE!

• STAGE & : S'!%"G

( ) ( )

2!!,1:

###,22##

2!!

2!!,2##,2%#,22#

pdkerjatenagaminimumkebutuhan

2!!

%,",2,1

, pdkerjatenaga jumlah

%#1

1

2

1

%"21

%

===→=

=

−+−=

≤≤⇒

====

=→

=→

=→

=→

−

−

x x sn Ketika

x sState

r x x xn stage for cost

xr

r r r r

n stager

x

n

n stage x

nn

nnnn

nn

n

n

State :

Stagen

Stagen+1

n s n x

( )nnn x s f ,

n x

( ) ( )nnnn r x s x −+− ###,22## 2

( )nn x f *

1+Value :




( ) ( )[ ]

%,",2,12!!:

###,22##

:%

1

2

1

=≤≤

−+−∑=

−

i for xr to subject

r x x x Minimize

functionObject

ii

iiiii

)easi<*e Possi<*e (ost

+ 220

2 20

3 200

211

nn x 1−= nn x snr

2!!22# 1 ≤≤ x

2!!2%# 2 ≤≤ x

2!!2## " ≤≤ x

2!!% = x

2!!1 = s

2!!22# 2 ≤≤ s

2!!2%# " ≤≤ s

2!!2## % ≤≤ s

( ) ( )22####,22!!2## 1

2

1 −+− x x

( ) ( )2%####,22## 2

2

12 −+− x x x

( ) ( )2#####,22## "2

2" −+− x x x

( ) 2

"2!!2## x−




Solution procedure

Stage & :n ( &

211

*

% x( )%

*

% s f % s

2!!2## % ≤≤ s ( ) 2%2!!2## s−

Stage # :n ( #

( ) ( ) ( ) ( ){ }

( ) ( ) ( ){ }

2!!2%#

2!!2##2#####,22##min

2#####,22##min

"

2

""

2

""2!!2##

"

*

%"

2

""

2!!2##

"

*

"

"

"

≤≤→

−+−+−=

+−+−=

≤≤

≤≤

svalues Possible

x x s x

x f x s x s f

x

x

( ) ( ) ( ) ( ){ }nnnnnn xr

nn x f r x s x s f

shiprelationrecursive

nn

*

1

2

2!!

* ###,22##min

:

+≤≤

+−+−=




( )

( ) ( ) ( )

( )

( )

−

+

+

+−+

−+=

+=→

= −−=

−−+−=∂∂

2##2

2!####,2

22!#2!!2##

22!#2##

:2

2!#

#2!#2%##

2!!%#####,2%##,

:,min

"

2

"

2

""

"*

"

"*"

""

""""""

"

"""

s

s s s s f

sehingga

s x

s x

x s x x s f x

x s f

" s ( )"*

" s f *" x

2!!2%# " ≤≤ s ( ) ( ) ( )1!####,12'#!#2!#!# "

2

"

2

" −+−+− s s s2

2!#" + s




Stage 2 :n ( 2

( ) ( ) ( ) ( )

( ) ( )( ) ( ) ( )1!####,12'#!#2!#!#

###,22##

###,22##,

"

2

"

2

"

22222

2

*

"22

2

22222

−+−+−+−+−=

+−+−=

s s s

r x s x

x f r x s x x s f

( ) ( )

( )( )

( ) ( )2!!2%#2!!22#

2!!2%#min

#'##,:

"

2%#2

,min

22

222

22222

2

22

222

2!!2%#

2*2

2

≤≤→≤≤

≤≤

≥=∂

∂

+=⇒

=≤≤

s jika feasiblehanya s

x feasible x jika xnilai

x s f x Karena

s x

x s f s f x




( )

( )

valueadalah x

xuntuk x s f x

s Ketika

sutk x s f

kondisi padadihitung tetap feasible yg xnilaiapi

min2%#

2!!2%#,#,

:2%#

2%#22#,

2

2222

2

2

2222

2

=→

≤≤>∂

∂<

<≤

2%#22# 2 ≤≤ s

2!!2%# 2 ≤≤ s

( ) ###,11!2%#2## 2

2 +− s

( ) ( ) ( )[ ]!$!""#2'!2!#2

2##2

2

2

2

2 −+−+− s s s

2%#

"

2%#2 2 + s

2 s ( )2

*

2 s f *

2 x




Stage 1 :n ( 1

( ) ( ) ( ) ( )

( )

( )

( ) ( ) ( )

( ) ( ) ( )

( ) ( )

≤≤

−+−

+−+−+−

≤≤

+−+−+−

=

+−+−=

2!!2%#

!$!""#2'!

2!#2

2##22####,22##

2%#22#

###,11!2%#2##22####,22##

,

:

###,22##,

1

2

2

2

2

1

1

2

11

1

2

11

2

11

111

1*

2

1

*

211

2

11111

xutk

s s

x x s x

xutk

x x s x

x s f

x f pada Mengacu

x f r x s x x s f

( ) ( )

( ) ( )

2%#22#min2%#

,

2%#,#2%!##,2!!

2"!2%##,

:2%#22#

11

11111

1

1

11111

1

1

≤≤=

≤∀<−=∂∂

→=

−−=∂∂

≤≤

xuntuk valueadalah x

itukarenaOleh

x x x s f x

s

s x x s f x

x!ntuk




211 +41,000 21

( )1

*

1 s f 1 s *

1 x21 21 21 211

*

1 x *

2 x *

" x *

% x

( ) ( )

( )

( )

( ) ( )

###,1!)2!!(

2!!22#min!2%$

!2%$,2%#,2!!2%#min!2%$

2%#22#min2%#

2!!2%#!.2%$2!!

%

22!"#,

:

#,

:

22!"%"

%##,

,2!!2%#

*

1

11

1111

11

11

111

11111

1

11112

1

2

11111

1

1

=

≤≤=→

>→

≤≤=

≤≤=

≤≤=→=

+=→=

∂∂

∀>∂∂

−−=∂∂

≤≤

f

xutk valueadalah" x

" s f s f xutk value" x

xutk value x

xutk x s Karena

s x x s f

x

untuk maka

x x s f x

Karena

s x x s f x

x Ketika




PROBABILISTIC DYNAMIC PROGRAMMING

(O$TO& :

Se<uah perusahaan menerima or.er .=n >etentuan s<< : Keputusan pro.u> .iterima9.ito*a> .i tan=an CUSTOMER CUSTOMER hanya mem<utuh>an SATU PRO"%K SA7A Perusahaan mempunyai >esempatan hanya 3 KALI PRODUCTION RUN 7i>a pa.a a>hir p$"%#&ti"n $#n y= KE 3 BE%! A"A pro.u> y= .apat

.iterima o*eh &#'t"m$ , perusahaan a>an men.apat>an PENALTY COST

se<esar 8 +,600

Perusahaan men=estimasi>an : Pe*uan= pro.u> "ITERI!A ? "ITOAK, masin=2 : @ SETUP COST %i 'tip ! PRODUCTION RUN : 8 300

PRODUCTION COST : 8 +00 per ITE!Berapa um*ah pro.u> pa.a masin=/masin= PRODUCTION RUN , a=ar tota*

on=>os pro.u>si minima*




PENYELESAIAN :

( )

n

s n x

nnn

n

n

stage beginninghen

+ero)or(oneneededstillitems aeptableof number:-tate stageforsi+elot:

%,",2,1run prodution:-tage =

( )

( ) ( )

( ) ##

:imana

,min:

isdeisionimmediatetheand

,stageatstateinstartssstemtheif

,stageforostexpetedtotal:,:-ehingga

*

,.....1,#*

=

=

n

nnn x

nn

n

n

nnn

f

x s f s f

x

nS

n x s f

n




( )

( )

>

==+

+

#,"

#,#here:0tau

, produksiJumlah1##"##

:stageatostrodution

n

nn xif

xif K x K

n

( ) ( ) ( ) ( ) ( )

( ) ( )

( ) 1'1

:imana

12

1

#2

1112

1,1

,1untuk -ehingga

*

%

*

1

*

1

*

1

=

++=

−+++=

=

+

++

f

f x K

f f x K x f

s

n

x

n

n

x

n

x

nnn

n

n

nn




State : 1 n x

( )nn x f ,1

#

1

( ) ##*1 =+n f

( )1*

1+n f

decision ( )

n x

2

11−

( ) n x

21

n x K +

n x K +

( ) ( )121 *

1+++= n

x

n f x K

n

Value

:

"=n

0 + 2 3 1

0

+

0

+6 +2 4 4 41

0

4

0

3 or

( ) ( )1'2

1 "

"

x

x K ++( ) ="" ,1 x f

( )"

*

" s f

*

" x

" x

" s




0 + 2 3

0+

04 4 1

0

02 or 3

n =

2 ( ) =22 ,1 x f ( ) ( )1

21 *

"2

2

f x K x

++2 x

2 s ( )2

*

2 s f *

2 x

n = 1

0 + 2 3

+ 1 6 39 6 94 9+6 6 39 2

( )1

*

1 s f *

1 x

( ) =11 ,1 x f ( ) ( )12

1 *

21

1

f x K x

++1 x

1 s

KESIMPULAN

'$!dengan

%atau"

diterima/gadatidak jika,"atau2

diterima/gadatidak jika,2

*

"

*

2

*

1

cost expected total

x

x

x

=

=

=




CONTOH

:

2/"sebesarmenang peluangmempunaiia baha

akiniaitu,3ntuktersebut.n teruhanmemenangkadapatiaagar

405,setiap padakandipertaruhharusang678- jumlah

ngkanmemperhituharustersebutstatistikahlidemikian,engan

kandipertaruhdapat405setiap padaadag678-setiapdan678-,"dengandimulai90;,aal pada jika

!,milikiiag678-405,"dengan90;akhir pada

: baha bertaruh,merekaitu,3ntuk

temanna.<oleh temandiperaai but tidak baru tersesistem =amun>egas.4asdi populerang/90; permainansuatu

nmemenangkauntuksistemruimemperbahastatistikahli-eorang

→→

<→




( )

( )

( ) ( )

( ) ( ) ( )nnnnnnnnn

nnn s xnn

n

n

nnn

n

n

th

x s f x s f x s f

x s f s f

xdecisionimmediatethemakesand s stateinn stage

startsan statisticithethat givenchipsleast at #ith playsthreethe finishing of y probabilit x s f

n stagebegintohand inchipsof number sStaten stageat bet tochipsof number x

nnnStage

nn

++−=

=

++

=

*

1"2*

1"1

.....,,1,#

*

,

,max:

,,!

:,

::

",2,1:

:anenelesai




0

+

23

C1

0

0

0293

293

+

/

/

/2 or more;

+ or more;

0or D s3 1;

n =

3

n =

2

0 + 2 3

0

+

2

3

C1

0

0

0

293

293

+

0

9

9

49

9

293

293

293

293 293

0

0

9

293

49

+

/

/

+ or 2

0, 2 or 3

+

0orDs2/1;

( )"

*

" s f " s *

" x

2 s 2 x ( ) ( ) ( )22

*

""2

22

*

""1

222 , x s f x s f x s f ++−= ( )2

*

2 s f *

2 x




0 + 2 3

3 293 2092 293 293 2092 +

n = 1

1 s( ) ( ) ( )11

*

2"2

11

*

2"1

111 , x s f x s f x s f ++−=( )1

*

1 s f *

1 x

( )( )

=

===

=

==

=

kalahn taruhankeseluruhaseara,kalah

2%atau",2,1

1"atau2 menang,2atau1kalah,

"atau2kalah,

#menang,1menang,

1

:?esimpulan

*2

*

2*"*2

*"

*"*

2

1

x

x x x

x

x x

x$




MARKOV PROCESSES

"E)I$ISI :

!ARKOF PRO(ESS !O"ES a.a*ah suatu proses sto>asti> untu>memper>ira>an >ea.aan .i masa men.atan=, .en=an hanyamempertim<an=>an >ea.aan tepat se<e*umnya Sehin==a untu> suatu!ar>oG Process, .en=an p$'nt 'tt yan= .i>etahui, ma>a &"n%iti"n!

p$"**i!it+ >ea.aan <eri>utnya <ersi-at in.epen.en .ari >ea.aan

se>aran=

Suatu 't"&'ti& p$"&'' .en=an suatu -init "$

&"#nt*! 'tt 'p&, .i>ata>an mempunyai suatu M$."/ &in

't$#&t#$ i>a

{ }...,2,1,#, =n % n

{ }{ }

....,1,#untukdan,semuauntuk

,.....,,,

1

1111##1

=

==

======

+

−−+

n ji

i % j % P

i % i % i % i % j % P

nn

nnnn




( ) ( )

( ) proses.terhadap

hanadanlampau,masakeadaanterhadapangdan......,,

untuk ),(suatu

pernataanterhadapeki@alen

11##

1

i % state present dependent

t independen

i % state present i % i %

state past j % state &uture y probabilit

l conditiona Property Markovian

n

nnn

n

=

===

=

−−

+

{ }

ies probabilit transition

i % j % P

ies probabilit l 'onditiona

nn

sebagai jugadisebut

1 ==+

∑=

==

≤≤≤≤ N

j

ij

ij

N i p

N ji p

1

.......,2,1,1

,11#




State + 2 HH $

P

+

2

$

P ++

p2+

0

0

0

p$+

p+2

p2+

p$2

P +$

p2$

0

0

0

p$$

{ } ij

NN N N

N

N

ij p Probabiliyransition

p p p

p p p

p p p

p P

Matrixransition stepOne

:

.....

.

.

.....

.....

:)(

21

22221

11211

==

−

N x N ransition

Probability

Matrix P




6ontoh 1:(urrent

se*ectionSe*ection $eJt ee>

Piaria A Piaria B Piaria (

Piaria A

Piaria B

Piaria (

01

0

03

03

02

03

02

0

0

=

=

%.#".#".#

%.#2.#%.#

2.#".#!.#

"""2"1

2"2221

1"1211

p p p

p p p

p p p

P ( )%.#2.#%.#1

2 =(



susy susmartini

operations research II, 2006

( )[ ]

iniiii

n

i

n

i

n

i

n

i

p p p P ( (

P ( P P ( P ( (

.....21

#1

11211

==→

=== −−−

( )%.#2.#%.#.1

2

2

2 == P ( (

%.#".#".#%.#2.#%.#

2.#".#!.#

[ ]"2.#2.#%#.#=

Se*ection Time



susy susmartini


(urrent

Se*ection,Paria B n0 ;

Piaria A

Piaria B

Piaria (

Piaria (

Piaria B

Piaria A 0; 01; 020

Piaria (

Piaria B

Piaria A 0; 03; 0+2

Piaria (

Piaria B

Piaria A 02; 0; 004

p03

p02

p02

p0

p01

p0

p03

p0

p0

p03

p02

p0

Se*ection$eJt Time, n +

Se*ection Time A-ter $eJt, n 2



susy susmartini


The Chapman-Kolmoo!o" E#$a%&on'

nnnn

M

k

r nkj

r ik

nij

n

ijij

P P P P P P P

nr n ji p p p

p y probabilit transition stepn p y probabilit ransition

====

≤≤=

−→

−−

=

−

∑

...........

#dan,,semuautk ,

:sebagaiditulisapat

::

)2()1()(

#

)(



susy susmartini


( )

( )

( ) ( )"1%.#2$".#%1".#

"%.#2$.#"(.#

"2.#2).#%#.#

"#.#2$.#%".#

2.#".#!.#

%.#".#".#

%.#2.#%.#

2.#".#!.#

%.#".#".#

%.#2.#%.#

2.#".#!.#

.2.#".#!.#

.2.#".#!.# 2

21

1

"

1

=

=

=

=

=

P

P ( (

( ) ( ) ( ) ( ) ( )

=

=

===

"1#.#2$2$.#%#".#

"1%.#2$2.#%#.#

"1$%.#2$2$.#%#.#

"%.#2$.#".#

"2.#2.#%#.#

"#.#2$.#%".#

"%.#2$.#".#

"2.#2.#%#.#

"#.#2$.#%".#

.... 222"% P P P P P P P P



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The LON(-RUN )EHAVIOR O* MARKOV PROCESSE+

M$."/ Cin mempunyai suatu si-at tertentu, yaitu <ahLa sete*ahme*a*ui operasi untu> perio.e La>tu yan= cu>up *ama <e<erapa'tp;, ma>a M$."/ p$"&'' a>an mencapai suatu 't%+1'tt

&"n%iti"n'0

Suatu M$."/ &in mencapai 't%+1'tt &"n%iti"n', ma>a ran=>aian&in; pasti $2"%i&0

Suatu $2"%i& M$."/ Cin mempunyai si-at yan= memun=>in>an 9 p"''i*!, per=era>an .ari suatu 'tt menuu >e 'tt yan= *ain,tanpa memperhati>an p$'nt 'tt0



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(ontoh 2 :)uture States

+ 2 3

PresentState

+ +93 +93 0 +93

2 0 +92 +9 +93 +9 0 29 +9

0 0 +93 293

"ari S%a%e 1 .apat *an=sun= >e semua 'tt >ecua*i >e 'tt 3 %ntu>

menuu >e 'tt 3, harus .i*a>u>an me*a*ui 'tt 2 ter*e<ih.ahu*u, <aru >e 'tt 3

"ari S%a%e , .apat *an=sun= >e semua 'tt >ecua*i >e 'tt+ %ntu>menuu >e 'tt +,harus .i*a>u>an me*a*ui 'tt 3 atau ter*e<ih .ahu*u, .ari state 3 <aru >e 'tt + Atau .ari 'tt

>e 'tt 3, <aru >e 'tt +"st pa.a prinsipnya, .ari suatu 'tt .apat menuu >e 'tt

*ainnya

T$n'iti"n Mt$i3 ts< suatu e!o&. .ha&n



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7enis $2"%i& &in yan= pentin= .i>etahui a.a*ah Re$la! Cha&n

Suatu Re$la! Cha&n .i.e-inisi>an se<a=ai &in yan= mempunyaisuatu t$n'iti"n mt$i3 P .imana poLer P hanya ter.iri .ari p"'iti/

p$"**i!it+ /!#'0

"en=an per>ataan *ain :

Semua R2#!$ Cin pasti suatu e!o&. .ha&n, tetapi %&a/ 'em$a $2"%i& &in merupa>an Re$la! Cha&n0

(ontoh :

( ) ( ) ( )

'hain

)egular chainergodicelements* y probabilit positive

matrixmatrixtransition

% % % %

% % % %

% % % %

% % % %

P

% % %

% % % %

% % % %

% % % %

P

% %

% % %

% % %

% % %

P

merupakantersebutsehingga

dariterdirihanaang menjadi telah",ke poerada

###

#

#

#

"21

=→

=→

=



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Con%oh 3 :

( ) ( )

( ) ( )

)+,!-.) +),O/0' Matrixransition

% %

% % %

% % %

% % %

% %

P

% % %

% %

% %

% %

% % %

P

% %

% % %

% % %

% % %

% %

P

% % %

% %

% %

% %

% % %

P

A3?0=B08 adalah

###

##

##

##

###

##

###

###

###

##

###

##

##

##

###

##

###

###

###

##

%"

21

⇒

=→

=

→

=→

=



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+ETERMINATION O* STEA+Y-STATE CON+ITIONS

=

==

"1".#2$2$.#%##.#

"1".#2$2$.#%##.#

"1".#2$2$.#%##.#

"1#.#2$2$.#%#".#

"1%.#2$2.#%#.#

"1$%.#2$2$.#%#.#

"1#.#2$2$.#%#".#

"1%.#2$2.#%#.#

"1$%.#2$2$.#%#.#

. %% P P P

( )

∑

∑

=

=

∞→

==

=

>

=

−

M

i

iji j

M

j

j

j

j

n

n

ij

j

M j p

p

j* stateconditions state steady

1

1

.,...,1untuk ,

1

#

lim

:makauntuk adalahJika

π π

π

π

π

π



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Con%oh 2 P&a!&a 4 :

#.1

#."1" padamenjadiakan""1".#

#.2$2$ padamenjadiakan22$2$.##.%## padamenjadiakan%##.#

#".#.#".#

#".#%.#!.#

#.1%.#%.#2.#

".#2.#".#

".#%.#!.#

1

"21

"

2

1

"21

"21

"21

"21"

"212

"211

"21

=++→

⇒=⇒= ⇒=→

=+−=++−

=++ ++=

++=++=→

++=

π π π

π

π

π

π π π

π π π

π π π

π π π π

π π π π

π π π π

π π π

te steady sta state

te steady sta statete steady sta state 1

(urrentse*ection

Se*ection $eJt ee>


Piaria A

Piaria BPiaria (

01

003

03

0203

02

00

S SS ( S



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*IRST PASSA(E TIMES

)IRST PASSA#E TI!ES T ij ; a.a*ah La>tu 9 um*ah transisi 9 proses yan=

.iper*u>an untu> me*a>u>an transisi .ari suatu state i k e state j pe!%ama

/al&07i>a j2i* ma>a 4i$'t P''2 Tim .ise<ut se<a=ai Re.$!!en.e T&me a>tu

Kam<uh9Beru*an=;, yaitu La>tu 9 um*ah transisi yan= .iper*u>an untu>

>em<a*i >e initial state i0

Con%oh 5 :Beri>ut ini a.a*ah .ata perse.iaan <aran= .a*am 4 min==u :

I 5 4

I 6 6

I 7 1I 8

I 9 1

I 4

I ; 6I < 2

4i$'t P''2 Tim .ari 'tt 4 >e 'tt 1 a.a*ah 2min==u, .ari .ari 'tt 4 >e 'tt a.a*ah 3 min==u,sementara

R&#$$n& Tim .ari 'tt 4 a.a*ah 1 min==u, .anR&#$$n& Tim .ari 'tt 1 a.a*ah 2 min==u

( )



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( )

( ) ( )

( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( ) jj

n

ij

n

jjij

n

jjij

n

ij

n

ij

jjijijij

ijijij

n

ij

p g p g p g p g

p g p g

p p g

n

j statei state g

12211

122

11

............

.

.

:maka , ke periode pada

keaterjadinkali pertama peluangadalahJika

−−− −−−=

−=

==

=

=

%.#".#".#

%.#2.#%.#

2.#".#!.#

"""2"1

2"2221

1"1211

p p p

p p p

p p p

P

Con%oh 6 Piaria; :

( ) ( )

( ) ( ) ( )

( )

. 22.#

%.#2.#".#

2.#

""

1

1"

2

1"

2

1"

1

1"

1

1"

=

−=−===

p g p g

p g

=

"%.#2$.#"(.#

"2.#2).#%#.#

"#.#2$.#%".#2

P



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( )

( )

( ) ( )

( )

i" statei state state*recurrent i state

g

j*i

variablerandom ime Passage &irst

""""3*1*n g e4uality strict

j" statei state *ine4uality strict

g

g ji

n

n

ii

ij

n

ij

n

n

ij

n

ij

kekembaliakan pastisaatsuatudisebutkemudian

1

makaJika,

untukas probabilitdistribusimerupakanmaka,merupakansebut jumlah ter jikaBapi

menapai pernahakantidakmakamerupakansebut jumlah ter Jika

1

:nonnegatif nilaiadalahmakatetap,angdan3ntuk

1

1

∑

∑

∞

=

∞

=

=

=

=

≤

State.bsorbing dalamkhususkasussuatuadalah



susy susmartini


( )

( ) ( )

( )

( )kj

jk

ik ij

n

n

ijij

n

n

ijij

n

n

ij

n

n

ijij

ij

n

nii

ii

u pung u

variablerandomtime passage first the

g u

g ng u

j state i state

time passage first theof valueexpected u

i statei state

g Stateransient

p probabilty

transition stepone staterecurrent

State .bsorbing

∑∑

∑

∑∑

∑

≠

∞

=

∞

=

∞

=

∞

=

∞

=

+==

<∞=

==

<

=−

1 :sehingga

maka ,adalah jikaumumn/a,ada

1 jika

1 jika

:maka ,kedari

adalahJika

kekembali pernahakantidakini,kasus pada

1:situasiadalah

.1

dimana,suatu

dalamkhususkasussuatuadalah

1

1

11

1

(urrent Se*ection $eJt ee>



susy susmartini


(ontoh :>asus Piaria;

(urrentse*ection

Se*ection $eJt ee>


Piaria A

Piaria BPiaria (

01

003

03

0203

02

00

#eeksu

M ju

time recurrencethe state

time passage first expected i j

#eeksu

#eeksu

uu

u pu pu

uu

u pu pu

u pu

j

jj

j

kj

jk

ik ij

1%.""1".#

11

:-ehingga

.......,,2,1untuk 1

:denganterbalik berbandingsuatu

maka , Jika

21."

2."

2.#%.#1

1

".#!.#1

1

1

""

2"

1"

2"1"

2"221"212"

2"1"

2"121"111"

===

==

=

=

=

++=

++=

++=

++=

+= ∑≠

π

π

A l & 8 A)SOR)IN( MARKOV CHAINS



susy susmartini


Anal7'&' o8 A)SOR)IN( MARKOV CHAINS

Suatu !ar>oG (hain mena.i suatu A9'o!9&n Ma!/o" Cha&n i>a :

Se.i>itnya, ter.apat satu Absorbing State

Per=era>an 'tt .imun=>in>an .ari setiap nonabsorbing state >e pa*in=ti.a> satu absorbing state pa.a *an=>ah 9 'tp tertentu

Con%oh :

Stt + : Cmpi"n'ip t"#$nmn &!i*$

Stt 2 : >'"#t? @ 'it& t" n"t$ 'p"$t Stt 3 : Di!+ in't$#&ti"n n% p$&ti& n%%

Stt : Ti& %i!+ in't$#&ti"n n% p$&ti& n%%

#eek follo#ing thein j state

moving #eek oneini statea

in student tennisaof y probabilit p

P ij

=

=

%.#2.#2.#2.#

1.#!.#1.#".#

##1#

###1

###1



susy susmartini


lainnasuatu

kesuatudari peluangmatriks

lainang suatukesuatudari peluangmatriks

kesuatudari peluangmatriks,

lainna

kesuatudari peluangmatriks

%.#2.#

1.#!.#

##

##

2.#2.#

1.#".#

1#

#1

%.#2.#2.#2.#

1.#!.#1.#".#

##1#

###1

stateng nonabsorbi

ng statenonabsorbi s*by s N

stateabsorbing statengnonabsorbir*by s .

ng statenonabsorbi

stateabsorbing sbyr O

stateabsorbing

stateabsorbingr*byr 0

P

N

O

.

0 P P

−−=

−−=

−−=

−−=

=→

=→

=

:matrixlfundamentaa



susy susmartini


( )

( )

( )

=−=

−

−=

−

=−

−=

−

−

$.1$1.#

"'.#1%.2

'.#2.#

1.#!.#

%.#2.#

1.#!.#

1#

#1

:

:

:

1

1

N 0 &

N 0

stateoneexactlyin stateng nonabsorbiother anyto state

ng nonabsorbiany from going of ies probabilit the N absorbed isit before stateng nonabsorbi

eachinis processatimesof number expected the &

N 0 &

matrixl fundamentaa

B2innin2 Stt E3p&t% 'tp' *-"$ A*'"$pti"n

S3

S

2+ M 036 210

0+ M + 210

: .bsorptionof y Probabilit he



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( )

( )

( )

( )

orang"$atau

%."$sebanak 2menapaiakanang jumlahsementaraorang,

'"atau'2.'sebanak1menapaidapatakanang jumlah0rtina,

%."$'.'2

%".#!$.#

2.#$1.#'#%#C

C

:adalahdanmenapaiakanang jumlahmaka

,'#dan%#dariterdiriang,1##terdapatJika

%".#!$.#2.#$1.#

2.#2.#

1.#".#

$.1$1.#

"'.#1%.2

%1

%

1

state students

state students

studentstheof ncompositio present of vector absorptionof y probabilit

S S students

S students students S students

. N 0 &.

pfy

5

=

×=

=×=

=

=

−= −

APPLICATION O* MARKOV PROCESS MO+ELS



susy susmartini


APPLICATION O* MARKOV PROCESS MO+ELS

Con%oh ;:State (on.ition o- Printer Output

+

2

3

EJce*ent

Accepta<*e, <ut o- mar=ina* Nua*ity

%naccepta<*e, <*urry an. unrea.a<*e

)romState

To State

+ 2 3

+

2

3

0

0

0

94

39

0

+94

+9

+

State EJpecte. (ost+

2

3

8 0

8 +000 cost o- i**e=i<*e reports;

8 1000 cost o- i**e=i<*e reports, p*us cost o- repairin= printer;

Transition matri o- maintenance po*ic



susy susmartini


Transition matriJ o- maintenance po*icy :

)romstate

To state

+ 2 3

+

2

3

0

0

+

94

39

0

+94

+9

0

###.1

11.#

'"'%.#

11.#1

1

"21

112

"

11$

2

112

1

2%1

11

"

2%"1$2

"1

"21

=++

==

==

==

+=+=

=

++=

π π π

π

π

π

π π π

π π π

π π

π π π



susy susmartini


Artinya :

%ntu> La>tu yan= cu>up *ama, printer a>an <era.a pa.a :

Stt + +4+4 "- t tim

Stt 2 636 "- t tim

Stt 3 +4+4 "- t tim

Sehin==a :

%ntu> La>tu yan= cu>up *ama, .iper>ira>an rata/rata <iaya untu>mintnn& p"!i&+ :

( ) ( ) ( ) %#.1!%!11.#!###'"'%.#1###11.##

!###1#### "21

=++=++ π π π

Con%oh 1< :



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Con%oh 1< :Suatu &$%it &$% &"mpn+ mem<a=i Stt#' "- A&&"#nt' R&i/*!

mena.i >ate=ori, yaitu :

Beri>ut ini a.a*ah t$n'iti"n mt$i3 perio.e min==uan; yan= <erhasi* .i<uat o*eh&"mpn+ <er.asar>an pen=amatan :

Accounts ReceiGa<*e(ate=ory states;

Status o- AccountsReceiGa<*e

+

2

3

Pai. in -u**

Ba. .e<t

0/30 .ays *ate

3+/+20 .ays *ate

To AR (ate=ory state;

+ 2 3

P

)rom

AR(ate=orystate;

+ + 0 0 0

2 0 + 0 0

3 0 02 02 02

03 03 03 0+

%ntu> mem<uat &$%it1

&"nt$"! p"!i&i' yan=*e<ih e--e>ti-, antara *ain.iper*u>an suatu=am<aran tentan= p$"**i!it+ "- *'"$pti"n

Pen7ele'a&an :



susy susmartini


Pen7ele'a&an :

( )

( )

=

=

−==

=

−−

=

−=

−=

=

=

−

−−

−

%!%.#!%'.#

"'".#'"$.#

".#".#

2.#%.#

212.1%!!.#

"#".#"'!.1.

212.1%!!.#

"#".#"'!.1

.#".#

2.#.#

1.#".#

2.#2.#

1#

#1

1.#".#2.#2.#

".#".#2.#%.#

##

##

1#

#1

1

11

1

. N 0 &.absorptionof y Probabilit

N 0 &

N

O

.

0 P



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( )

( )

( )

###,!#sebesar% dan###,1#sebesar1 menjadi berubah

dapattersebut%dan"dari baha berharapdapat0rtina,

###,!####,1#

%!%.#!%'.#

"'".#'"$.#

###,!#####,###,1

:-ehingga

###,!#####,###,1

:maka!##,###,dan1,###,###adalah

masing<masing%dan"komposisiaalkondisi padaJika

C

state state

state statecompany

vector

state state

=

=

=

kuliah or ii

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