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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 : $. *eni An++,,-ai
'. An#/ Se0ian
%. A+us Kunia-an
1. Khui/ai A23alina
LABORATORIUM KOMPUTASI PROSES
4URUSAN TEKNIK KIMIA
FAKULTAS TEKNOLOGI INDUSTRI
UNIVERSITAS ISLAM INDONESIA
'($&
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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+
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#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
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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 :
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{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
7/23/2019 Laporan Golden Section
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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
7/23/2019 Laporan Golden Section
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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,. $>' :
7/23/2019 Laporan Golden Section
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N,.$
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
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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
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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>$(
L (;$
7/23/2019 Laporan Golden Section
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@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 -
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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
$. O0i3asi #iaikan se"a+ai suau 0,ses unuk 3en8ai k,n#isi /an+
,0i3u3;#ala3 ai 0alin+ 3en+unun+kan.
'. O0i3asi #a0a "eu0a 3aksi3asi aau 3ini3asi /an+ "ekaian #en+an
keunun+an.
%. Golden section 3eu0akan salah sau 8aa aau 3e,#e ,0i3asi nu3eik /an+
#a0a #iea0kan unuk 6un+si /an+ "esi6a uni3,#al.
1. Me,#e golden section #i+unakan unuk 3en8ai nilai ,0i3u3 #ai suau 6un+si.
Pa#a 0akiku3 ini nilai ,0i3u3 #i0e,leh #ala3 "enuk k,,#ina 9=,0; /,0.
&. Me,#e golden section 3eu0akan 3e,#e nu3eis; 3e,#e lain /an+ #a0a
#i+unakan unuk ,0i3asi a#alah 3e,#e +a6is #en+an 3en++unakan 0enuunan
6un+si.
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Kuantitatif :
Latihan No.1
Pa#a s,alMen8ai ha+a3ini3asi
@a >$
@" $
,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
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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 $;$)$
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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?&$J&&?BAB)GOLDENSECTION 9'$.% ;)>$'>'($&
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