laporan golden section

7/23/2019 Laporan Golden Section

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LAPORAN PRAKTIKUM KOMPUTASI PROSES

OPTIMASI SATU VARIABEL DENGAN CARA GOLDEN SECTION

Disusun Oleh :

NAMA : M.Fakhi!al A"#illah

NIM : $% &'$ ()'

KELAS : B

ASISTEN : $. *eni An++,,-ai

'. An#/ Se0ian

%. A+us Kunia-an

1. Khui/ai A23alina

LABORATORIUM KOMPUTASI PROSES

4URUSAN TEKNIK KIMIA

FAKULTAS TEKNOLOGI INDUSTRI

UNIVERSITAS ISLAM INDONESIA

'($&



BAB $

PENDA*ULUAN

OPTIMASI SATU VARIABEL DENGAN CARA GOLDEN SECTION

A.Tu5uan

A+a 3ahasis-a #a0a 3en/elesaikan "enuk 0en/elesaian #i66eensial ,#ine 5enis

iniial 7alue 0,"le3 3en++unakan 0en/elesaian nu3eik.

B. Dasa Te,i

Se8aa u3u3 ,0i3asi "eai 0en8aian nilai e"aik 93ini3u3 aau 3aksi3u3 #ai

"e"ea0a 6un+si /an+ #i"eikan 0a#a suau k,neks. O0i3asi 5u+a #a0a "eai u0a/a unuk

3enin+kakan kine5a sehin++a 3e30un/ai kualias /an+ "aik #an hasil ke5a /an+ in++i.

O0i3asi #a0a #iaikan se"a+ai suau 0,ses unuk 3en8ai k,n#isi /an+ ,0i3u3; #ala3

ai 0alin+ 3en+unun+kan; 5a#i #a0a "eu0a 3aksi3asi aau 3ini3asi. Bila kia "eha#a0an

#en+an 3asalah keunun+an; kea#aan ,0i3u3 a#alah kea#aan /an+ 3e3"eikan

keunun+an 3aksi3u3 93aksi3asi; se#an+kan "ila "eha#a0an #en+an 3asalah

0en+eluaan aau 0en+,"anan; kea#aan ,0i3u3 a#alah /an+ 3e3"eikan 0en+eluaan aau

0en+,"anan 3ini3u3 93ini3asi. Se8aa u3u3 6un+si /an+ akan #i3aksi3u3kan aau

#i3ini3u3kan #ise"u objective function; se#an+kan ha+a < ha+a /an+ "e0en+auh #an

"isa #i0ilih #ise"u 7aia"el 90eu"ah.

Se8aa analiis; nilai 3aksi3u3 aau 3ini3u3 #ai suau 0esa3aan :

y=f ( x)⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯(1 .1)

#a0a #i0e,leh 0a#a ha+a = /an+ 3e3enuhi

y' ( x )= f

' ( x )⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯(1 .2)

Na3un unuk "e"ea0a 6un+si; 0,ses 3en8ai uunan 6un+sin/a 8uku0 u3i

sehin++a 0,ses unuk 3enenukan iik ,0i3u3 haus 3en++unakan 3e,#e lain

3isaln/a 3e,#e G,l#en Se8i,n unuk sau 7aia"el; #an *,,ke 4ee7es unuk

3uli7aia"le.

Suau 6un+si #a0a 3e3iliki sau iik ,0i3u3 /an+ #ise"u se"a+ai 6un+si uni3,#al.

Be"ea0a 6un+si 5u+a #a0a 3e3iliki sau aau le"ih iik ,0i3u3; 6un+si se0ei ini

#ise"u se"a+ai 6un+si 3uli3,#al. Unuk 6un+si 3uli3,#al 3aka iik ,0i3u3 /an+



#i0e,leh san+a e+anun+ #ai iik a-al /an+ #i+unakan unuk 3ene"ak iik ,0i3u3

ese"u.

O0i3asi 6un+si 5u+a #a0a #iklasi6ikasikan "e#asakan ke0a#a 5u3lah 7aia"el

/an+ akan #i,0i3asi. Unuk sau 7aia"el; 3aka #a0a #ise"u ,0i3asi 6un+si sau

7aia"el se#an+kan unuk le"ih #ai sau 7aia"el 3aka #ise"u se"a+ai ,0i3asi 6un+si

3uli7aia"el. Me,#e G,l#en Se8i,n 3eu0akan 3e,#e unuk 3enenukan iik ,0i3u3

9iik 3aksi3u3 aau 3ini3u3 unuk sau 7aia"el.

Unuk 6un+si /an+ suka unuk #iuunkan aau 3e30un/ai uunan /an+ suka #i8ai

akan/a; 0,ses ,0i3asi #a0a #ilakukan se8aa nu3eis.

Golden section 3eu0akan salah sau 8aa aau 3e,#e ,0i3asi nu3eik /an+ #a0a

#iea0kan unuk 6un+si /an+ "esi6a uni3,#al. Ke#ua i0e ,0i3asi; /aiu 3aksi3asi #an3ini3asi #a0a #iselesaikan #en+an 8aa ini. G,l#en>se8i,n 9sea8h 3eh,# 3eu0akan

3e,#e ,0i3asi sau 7aia"el /an+ se#ehana; #an 3e30un/ai 0en#ekaan /an+ 3ii0

#en+an 3e,#e "ise8i,n #ala3 0enenuan aka 0esa3aan ak linie. Mii0 #en+an "ise8i,n;

i#e #asa 3e,#e ini a#alah 3e3an6aakan nilai /an+ la3a se"a+ai nilai /an+ "au; se8aa

ieai6. Se"a+ai aki"an/a; enan+? ine7al a-al 7aia"el /an+ #i0ilih se3akin la3a akan

se3akin 3en/e30i; kaena a#a se"a+ian su">ine7al 7aia"el /an+ #ieli3inasi; hin++a

#i0e,leh in+ka k,n7e+ensi /an+ #iin+inkan.

Golden section 3eu0akan salah sau 8aa ,0i3asi nu3eis /an+ "isa #i0akai unuk

6un+si /an+ "esi6a uni3,#al. Ke#ua i0e ,0i3asi; /aiu 3aksi3asi #an 3ini3asi #a0a

#iselesaikan #en+an 8aa ini. Misal akan #ilakukan 3aksi3asi eha#a0 0esa3aan 9$.$

#ala3 ine7al =A sa30ai =B.

/

l

l

@A @P @ @B @

Ga3"a $.$ Eli3inasi #en+an G,l#en Se8i,n



Misal akan #ilakukan 3aksi3asi / 0a#a ine7al =A =B. Di0ilih ' iik unuk

e7aluasi; 3isal = 0 #an =. 4ika 6un+si uni3,#al 9han/a 0un/a sau iik ekse3 3aka #en+an

"e#asa ha+a / 0a#a ' iik ese"u 3aka a#a se"a+ian ine7al /an+ #a0a #ieli3inasi.

Dihaa0kan 0ila "ah-a 0a#a e7aluasi lan+kah selan5un/a; salah sau iik la3a "isa #i0akai

la+i. 4a#i han/a #i0elukan $ iik "au. P,ses eli3inasi ine7al eliha se0ei 0a#a +a3"a

$.'.

/

$

l

=A =F = = 0 =

/

=A =F = = 0 =

Ga3"a ). '. P,ses Eli3inasi Ine7al

Disini a#a 0,"le3; #i3ana leak iik P #an a+a 0a#a ine7al "eikun/a salah

sau iikn/a 3asih "isa #i0akai. Misal iik P #an 3asin+ < 3asin+ "e5aak l = ine7al

a-al #ai iik B #an A. Dala3 hal ini ha+a l akan #i8ai.

Dai +a3"a $.$. eliha "ah-a :

( xQ− x p )lama=( x P− x A )baru

Selan5un/a :



{l−(1−l)} ( xB− x A )lama= (1−l ). ( x P− x A )baru

{2. l−(1−l)}( x B− x A )lama=(1−l ) . l . ( x P− x A )baru (2. l−1)=(l−l2) l

2+l−1=0

O0i3u3

=l = 0 = =ll

Fis ieai,n

l(

l$ l'

Se8,n# ieai,n

l$

l'

Se8aa 3ae3ais :

l1

l0=l2

l1

Kaena

l0=l

1+l

2



l1

l1+l2=l0

l11−

l1

l2=

1

R→

l1

l2= R 1+ R=

1

R R+ R2=1

R+ R2−1=0 R1,1=−b±√ b2

−4 ac2a

R1,1=−1±√ 12

−4.1 .(−1)2.1

R1,1=−1+√ 5

2=0,618

Ke3un+kinan < ke3un+kinan /an+ e5a#i 0a#a eli3inasi #en+an 8aa Golden Section a#alah :

Maksi3asi : / 0 / =A = 0

=P =

=B =B

= Di8ai

/ 0 / =A =A

=B =

= =P

=P Di8ai

Mini3asi : / 0 / =A =A

=B =

= =P

=P Di8ai

/ 0 / =A = 0

=P =

=B =B

= Di8ai



Algoritma :

$. Menenukan 0esa3aan /69=

'. Menenukan nilai @a;@";T,leansi 9@">@a; R aau L (;$%. Men8ai nilai

x P= xa+[ (1− L ) ( xb− xa ) ] xQ= xa+[ ( L ) ( xb− xa ) ]

1. Men8ai nilai /P #an / ke 0esa3aan / 69=

&. Men8ai nilai / 0 < /

. Meliha nilai /P #an / unuk 3e3asikan 3en++unakan ke3un+kinan eli3inasi /an+

#i0akai.

). Men8ai nilai =,0 #an /,0

xopt =

( xa+ xb)

2

/,0 3ensu"siusikan =,0 ke 0esa3aan / 69=

BAB II

PERSOALAN DAN PENELESAIAN

Laihan #i la" N,. $>' :



N,.$

8ailah ha+a minimasi unuk :

@a >$

@" $

,leansi 9@">@a(.((($

L'HL>$(

L (;$

@a @" 9@">@a @0 @ 0 0>

-1,00001,000

0 2,0000 -0,2360 0,2360 -0,2361 -0,8345 0,5983

-0,2360 1,0000 1,2360 0,2360 0,5278 -0,8345 -1,3964 0,5619

0,23601,000

0 0,7640 0,5278 0,7082 -1,3964 -1,8704 0,4740

0,52781,000

0 0,4722 0,7082 0,8196 -1,8704 -2,2360 0,3656

0,70821,000

0 0,2918 0,8196 0,8885 -2,2360 -2,4985 0,2625

0,81961,000

0 0,1804 0,8885 0,9311 -2,4985 -2,6777 0,1792

0,88851,000

0 0,1115 0,9311 0,9574 -2,6777 -2,7958 0,1180

0,93111,000

0 0,0689 0,9574 0,9737 -2,7958 -2,8718 0,0760

0,95741,000

0 0,0426 0,9737 0,9837 -2,8718 -2,9200 0,0482

0,97371,000

0 0,0263 0,9837 0,9899 -2,9200 -2,9502 0,0303

0,98371,000

0 0,0163 0,9899 0,9938 -2,9502 -2,9691 0,0189

0,98991,000

0 0,0101 0,9938 0,9962 -2,9691 -2,9809 0,0118

0,9938

1,000

0 0,0062 0,9962 0,9976 -2,9809 -2,9882 0,0073

0,99621,000

0 0,0038 0,9976 0,9985 -2,9882 -2,9927 0,0045

0,99761,000

0 0,0024 0,9985 0,9991 -2,9927 -2,9955 0,0028

0,99851,000

0 0,0015 0,9991 0,9994 -2,9955 -2,9972 0,0017

0,99911,000

0 0,0009 0,9994 0,9997 -2,9972 -2,9983 0,0011

0,99941,000

0 0,0006 0,9997 0,9998 -2,9983 -2,9989 0,0007

0,99971,000

0 0,0003 0,9998 0,9999 -2,9989 -2,9993 0,00040,9998 1,000 0,0002 0,9999 0,9999 -2,9993 -2,9996 0,0003



0

0,99991,000

0 0,0001 0,9999 0,9999 -2,9996 -2,9997 0,0002

Xopt=

0,999

9

Yopt=

-2,999

7

jadi ,didapatkan nilai masing-masingyaitu

Xa=0,999

9

X!=1,000

0

Xopt=0,999

9

Yopt=

-2,999

7

N,.'

8ailah ha+a maksimasi unuk :

@a $

@" %

,leansi 9@">@a(.(($&

L'HL>$(

L (;$

@a @" 9@">@a @0 @ 0 0>

1,00003,000

0 2,0000 1,7640 2,2360 2,3218 1,3604 0,96140919

1,00002,236

0 1,2360 1,4722 1,7640 2,5520 2,3218 0,23012621

1,00001,764

0 0,7640 1,2918 1,4722 2,5503 2,5520 -0,00170710

1,29181,764

0 0,4722 1,4722 1,5836 2,5520 2,4982 0,05379865

1,29181,583

6 0,2918 1,4033 1,4722 2,5642 2,5520 0,01219539

1,29181,472

2 0,1803 1,3607 1,4033 2,5637 2,5642 -0,00043131

1,3607

1,472

2 0,1114 1,4033 1,4296 2,5642 2,5614 0,002769041,3607 1,429 0,0689 1,3870 1,4033 2,5647 2,5642 0,00055199



6

1,36071,403

3 0,0426 1,3770 1,3870 2,5646 2,5647 -0,00010228

1,37701,403

3 0,0263 1,3870 1,3933 2,5647 2,5646 0,00010615

1,37701,393

3 0,0163 1,3832 1,3870 2,5647 2,5647 0,00000079

1,37701,387

0 0,0100 1,3808 1,3832 2,5647 2,5647 -0,00002421

1,38081,387

0 0,0062 1,3832 1,3847 2,5647 2,5647 -0,00000547

1,38321,387

0 0,0038 1,3847 1,3856 2,5647 2,5647 0,00000018

1,38321,385

6 0,0024 1,3841 1,3847 2,5647 2,5647 -0,00000124

1,38411,385

6 0,0015 1,3847 1,3850 2,5647 2,5647 -0,00000026

Xopt=1,384

8

Yopt=2,564

7

jadi ,didapatkan nilai masing-masingyaitu

Xa=1,384

1

X!=

1,385

6

Xopt=1,384

8

Yopt=2,564

7

TUGAS

N,.$

8ailah ha+a minimum

unuk :

@a >'

@" $

,leansi 9@">@a(.(((

L'HL>$(

L (;$



@a @" 9@">@a @0 @ 0 0>-

2,0000

1,0000 3,0000 -0,8540

-0,146

01,926

13,916

7-

1,99069068-

2,0000

-

0,1460 1,8540 -1,2918

-

0,8540

1,1876

1,9261

-0,73850568

-2,000

0

-0,854

0 1,1460 -1,5622

-1,291

82,100

41,187

6 0,91289251-

1,5622

-0,854

0 0,7082 -1,2918

-1,124

51,187

61,251

9-

0,06431425-

1,5622

-1,124

5 0,4377 -1,3950

-1,291

81,360

41,187

6 0,17280040-

1,3950

-1,124

5 0,2705 -1,2918

-1,227

91,187

61,168

0 0,01954792-

1,2918

-1,124

5 0,1672 -1,2279

-1,188

41,168

01,184

6-

0,01656438-

1,2918

-1,188

4 0,1033 -1,2523

-1,227

91,168

41,168

0 0,00036311-

1,2523

-1,188

4 0,0639 -1,2279

-1,212

81,168

01,171

9-

0,00389764

-1,252

3

-1,212

8 0,0395 -1,2372

-1,227

91,167

11,168

0-

0,00085749-

1,2523

-1,227

9 0,0244 -1,2430

-1,237

21,167

21,167

1 0,00008168-

1,2430

-1,227

9 0,0151 -1,2372

-1,233

61,167

11,167

3-

0,00018460-

1,2430

-1,233

6 0,0093 -1,2394

-1,237

21,167

11,167

1-

0,00002410-

1,2430

-1,237

2 0,0057 -1,2408

-1,239

41,167

11,167

1 0,00001927-

1,2408

-1,237

2 0,0036 -1,2394

-1,238

61,167

11,167

1-

0,00000120-

1,2408

-1,238

6 0,0022 -1,2399

-1,239

41,167

11,167

1 0,00000431-

1,239

9

-1,238

6 0,0014 -1,2394

-1,239

1

1,167

1

1,167

1 0,00000069- - 0,0008 -1,2391 - 1,167 1,167 -



1,2394

1,2386

1,2389 1 1 0,00000028

Xopt

=

-1,239

0

Yopt=1,167

1

jadi ,didapatkan nilai masing-masing yaitu

Xa=

-1,239

4

X!=

-1,238

6

Xopt=

-1,239

0

Yopt=1,167

1

BAB III

PENUTUP

A. Kesimpulan

$. O0i3asi #iaikan se"a+ai suau 0,ses unuk 3en8ai k,n#isi /an+

,0i3u3;#ala3 ai 0alin+ 3en+unun+kan.

'. O0i3asi #a0a "eu0a 3aksi3asi aau 3ini3asi /an+ "ekaian #en+an

keunun+an.

%. Golden section 3eu0akan salah sau 8aa aau 3e,#e ,0i3asi nu3eik /an+

#a0a #iea0kan unuk 6un+si /an+ "esi6a uni3,#al.

1. Me,#e golden section #i+unakan unuk 3en8ai nilai ,0i3u3 #ai suau 6un+si.

Pa#a 0akiku3 ini nilai ,0i3u3 #i0e,leh #ala3 "enuk k,,#ina 9=,0; /,0.

&. Me,#e golden section 3eu0akan 3e,#e nu3eis; 3e,#e lain /an+ #a0a

#i+unakan unuk ,0i3asi a#alah 3e,#e +a6is #en+an 3en++unakan 0enuunan

6un+si.



Kuantitatif :

Latihan No.1

Pa#a s,alMen8ai ha+a3ini3asi

@a >$

@" $

,leansi 9@">@a(.((($

L'HL>$(

L (;$

5a#i ;#i#a0akan nilai 3asin+>3asin+ /aiu :

@a (;JJJJ

@" $;((((

@,0 (;JJJJ

,0 >';JJJ)

No.2

Pa#a s,al

8ailah ha+a maksimasi unuk :

@a $

@" %

,leansi 9@">@a(.(($&

L'HL>$(

L (;$

jadi ,didapatkan nilai masing-masing yaitu

Xa = 1,3841

X! = 1,3856

Xopt= 1,3848

Yopt= 2,5647



TUGAS

N,.$

Pa#a s,al

8ailah ha+a minimum unuk :

@a >'

@" $

,leansi 9@">@a(.(((

L'HL>$(

L (;$

5a#i ;#i#a0akan nilai 3asin+>3asin+ /aiu :

@a >$;'%J1

@" >$;'%

@,0 >$;'%J(

,0 $;$)$



B. Saran

$. Ti#ak 3elakukan 8,0/ 0ase hasil 0eke5aan ,an+ lain.

'. Telii #ala3 3e3asukan u3us #i 6,3ula Ms.e=8el ;salah se#iki hasiln/a akan

"e"e#a.

%. Ban/ak "elaih #ala3 3en+,0easikan Ms.e=8el ;a+a i#ak ke"in+un+an .1. Me30ehaikan asisen 0a#a saa 3en5elaskan 3aei #i#e0an.

&. Ti#ak i"u #ikelas ;su0a/a i#ak 3en++an++u e3an /an+ lain.

. Dala3 3en+e5akan ,0i3asi sau 7aia"el #en+an 8aa +,l#en >se8i,n haus elii

;kaena kia sein+ salah #ala3 3e3asukan nilai =A ;=P; =B; = #ala3 0esa3aan

unuk 3aksi3asi aau 3ini3asi 9#en+an s/aa / 0 / aau / 0 /

DAFTAR PUSTAKA

h0s:??i#.s8i"#.8,3?#,8?'%'&JJ$?>Me,#e>O0i3asi>G,l#en>Se8i,n>Dan>T,,l",=>

O0i3asi>Mala"#,-nl,a# 9'(.'J ;)>$'>'($&

h0s:??i#.s8i"#.8,3?#,8?'&()(J$%%?1>'>G,l#en>Se8i,n#,-nl,a# 9'(.1 ;)>$'>'($&

h0:??---.#is,#,8.8,3?%J&>,0i3asi>nu3eik>sau>7aia"el>#en+an>3e,#e>+,l#en>

se8i,n 9'$.( ;)>$'>'($&

h0s:??---.a8a#e3ia.e#u?&$J&&?BAB)GOLDENSECTION 9'$.% ;)>$'>'($&

https://id.scribd.com/doc/232859916/6-Metode-Optimasi-Golden-Section-Dan-Toolbox-Optimasi-Matlab#download



https://id.scribd.com/doc/250709133/4-2-Golden-Section#download

http://www.distrodoc.com/3965-optimasi-numerik-satu-variabel-dengan-metode-golden-section


https://www.academia.edu/5616955/BAB_7_GOLDEN_SECTION

https://id.scribd.com/doc/250709133/4-2-Golden-Section#download



https://www.academia.edu/5616955/BAB_7_GOLDEN_SECTION



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