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Pemrograman Linier
(Linear Programming - LP)
Metode Simpleks
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Kasus produksi panel kaca
Perusahaan memproduksi : Produk 1 = panel pintu kaca bingkai aluminium
Produk 2 = panel jendela kaca bingkai kayu
Dari 3 pabrik : Pabrik 1 = pekerjaan bingkai aluminium
Pabrik 2 = pekerjaan bingkai kayu
Pabrik 3 = pekerjaan pemasangan kaca
Data kapasitas pabrik dan keuntungan sbb.:
Kapasitas yangdapat digunakan
Produk 1 Produk 2
Pabrik 1 1 0
Pabrik 2 0 2 12
Pabrik 3 3 2 1!
Keuntungan perunit
" 3 " #
Kapasitas yang digunakan per unit
ukuran produksi
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Kasus Panel Kaca :
Metode Simpleks - Formulasi
X1 X2 S1 S2 S3 RHS
(Slack-1) (Slack-2) (Slack-3)
Maks Z = 3 X1 + 5 X2
e!"atas 1 # X1 + S1 = $
e!"atas 2 # 2 X2 + S2 = 12
e!"atas 3 # 3 X1
+ 2 X2
+ S3
= 1%
$aksimumkan : % = 3&1 ' #&2
Pembatas : &1 ≤
2&2≤ 12
3&1 ' 2&2 ≤ 1!
&1( &2 ≥ 0
Dirubah menjadi :
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Kasus Panel Kaca :
Metode Simpleks - Tabel
) j )1 )2 *.. )n 0 0 *.. 0
)b +asic,ariable
s
-1 -2 *.. -n 1 2 *.. m /uantity
0 1 a11 a12 *.. a1n 1 0 *.. 0 b1
0 2 a21 a22 *.. a2n 0 1 *.. 0 b2
: : :
0 m am1 am2 *.. amn 0 0 *.. 1 bm
j ∑)b.ai1 ∑)b.ai2 *.. ∑)b.ain 0 0 *.. 0
) j j )1 1
)2
2
*.. )n n
0 0 *.. 0
Bentuk Tabel Simpleks :
Kriteria opti!u! #&'ntuk persoalan tuuan !e!aksi!u!kan "ila se!ua * Z ≤ ,
&'ntuk persoalan tuuan !e!ini!u!kan "ila se!ua Z * ≤ ,
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Kasus Panel Kaca : Metode Simpleks - Iterasi
Cj 3 5 0 0 0
)b +asic,ariables
-1 -2 slack 1 slack 2 slack 3 /uantity
teration 1
0 slack 1 1 0 1 0 0
0 slack 2 0 2 0 1 0 12
0 slack 3 3 2 0 0 1 1!
%j 0 0 0 0 0 0
cj%j 3 # 0 0 0teration 2
0 slack 1 1 0 1 0 0
# -2 0 1 0 0.# 0 4
0 slack 3 3 0 0 1 1 4
%j 0 # 0 2.# 0 30
cj%j 3 0 0 2.# 0
teration 3
0 slack 1 0 0 1 0.3333 0.3333 2
# -2 0 1 0 0.# 0 4
3 -1 1 0 0 0.3333 0.3333 2
%j 3 # 0 1.# 1 34
cj%j 0 0 0 1.# 1
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Kasus Panel Kaca :
Metode Simpleks - Solusi
Variable Status Value
X1 asic 2
X2 asic .
slack 1 asic 2
slack 2 /0/asic ,slack 3 /0/asic ,
Optimal Value (Z) 36
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Kasus Panel Kaca :
Metode Simpleks - Sensitivitas
Variable Value Reduced Original Lower Upper
Cost Value Bound Bound
X1 2 , 3 , 5X2 . , 5 2 n4inity
Constraint Dual Slack/ Original Lower Upper
Value Surplus Value Bound Bound
*onstraint 1 , 2 $ 2 n4inity
*onstraint 2 15 , 12 . 1%
*onstraint 3 1 , 1% 12 2$
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Kasus Panel Kaca :
Metode Simpleks - asil
X1 X2 RHS Dual
Mai!i6e 3 5*onstraint 1 1 , ≤ $ 0
*onstraint 2 , 2 ≤ 12 1.5
*onstraint 3 3 2 ≤ 1% 1
Slutin 2 6 36
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Kasus Panel Kaca :
Metode Simpleks - Formulasi
X1 X2 S1 S2 S3 RHS
(Slack-1) (Slack-2) (Slack-3)
Maks Z = 3 X1 + 5 X2
e!"atas 1 # X1 + S1 = $
e!"atas 2 # 2 X2 + S2 = 12
e!"atas 3 # 3 X1 + 2 X2 + S3 = 1%
$aksimumkan : % = 3&1 ' #&2
Pembatas : &1 ≤
2&2≤ 12
3&1 ' 2&2 ≤ 1!
&1( &2 ≥ 0
Dirubah menjadi :
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!onto" Kasus Produksi Keramik
erusa7aan Kera!ik 8ea9er *reek ottery *o!pany: !e!produksi
!angkuk dan !ug Ke"utu7an su!"erdaya seperti pada ta"el "erikut#
Produk Sumberdaya Keuntungan($/unit)Buruh
(jam/unit)Lempung
(pounds/unit)
$angkuk 1 0
$ug 2 3 #0
'ntuk keperluan produksi 7arian tersedia $, a!;7ari untuk "uru7 dan
12, pound;7ari untuk "a7an le!pung uatla7 ru!usan pe!odelan
progra!a linier untuk !e!aksi!alkan keuntungan
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Kasus Produksi Keramik #
$umusan P%L
Maksi!u!kan # Z = $, X1 + 5, X2
e!"atas # X1 + 2 X2 ≤ $, (a!;7ari "uru7) $ X1 + 3 X2 ≤ 12, (pound;7ari "a7an le!pung)
X1 X2 ≥ ,
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soal1
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soal
2
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Kasus Produksi Keramik #
Iterasi Tabel Simpleks
)j 0 #0 0 0
)b +asic,ariables
-1 -2 slack 1 slack 2 /uantity
teration 1
0 slack 1 1 2 1 0 0
0 slack 2 3 0 1 120
%j 0 0 0 0 0
cj%j 0 #0 0 0
teration 2
#0 -2 0.# 1 0.# 0 20
0 slack 2 2.# 0 1.# 1 40
%j 2# #0 2# 0 1(000
cj%j 1# 0 2# 0
teration 3
#0 -2 0 1 0.! 0.2 !
0 -1 1 0 0.4 0. 2
%j 0 #0 14 4 1(340
cj%j 0 0 14 4
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Kasus Produksi Keramik #
asil Per"itungan
X1 X2 RHS Dual
Mai!i6e $, 5,
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M-&'S($ &I-M)
$etode impleks
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Metode Simpleks M-&esar
5ungsi 6ujuan dengan 78ariabel arti9sial;i000 -2 $.;1 $.;2 $.;3 $.;
$in = #000 -1 ' >000 -2 ' $.;1 ' $.;2 ' $.;3 ' $.;
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Metode Simpleks M-&esar (Lan*utan)
!"!R R#$#S!% %"S' $*!+!S $%,!D' RS!$!!% :
> ≤ > dita!"a7 slack → > + Si = >
> = > dita!"a7 arti4isial → > + ? i = >
> ≥ > dikurang "ilangan surplus dita!"a7 arti4isial → > - Si + ? i = >
-O%+OH :
X1 ≤ $ !enadi X1 + S1 = $
2X2 = 12 !enadi 2X2 + ? 2 = 12
3X1 + 2X2 ≥ 1% !enadi 3X1 + 2X2 - S3 + ? 3 = 1%
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!onto" Kasus Panel Kaca: Metode M-&esar
X1 X2 S1 R2 S3 R3 RHS
(Slack-1) (@rt-2) (Surplus-3) (@rt-3)
Min Z = 3 X1 + 5 X2 + M? 2 + M? 3
e!"atas 1 # X1 + S1 = $
e!"atas 2 # 2 X2 + ? 2 = 12
e!"atas 3 # 3 X1 + 2 X2 - S3 + ? 3 = 1%
$inimumkan : % = 3&1 ' #&2
Pembatas : &1 ≤
2&2= 12
3&1 ' 2&2 ≥ 1!
&1( &2 ≥ 0
Dirubah menjadi :
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Cj 3 5 0 M 0 M
)b +asic,ariables
-1 -2 1 ;2 3 ;3 /uantity
teration1
0 1 1 0 1 0 0 0
$ ;2 0 2 0 1 0 0 12
$ ;3 3 2 0 0 1 1 1!
j 3$ $ 0 $ $ $ 30 $
j )j 3$ 3 $ # 0 0 $ 0
teration2
0 1 1 0 1 0 0 0
# -2 0 1 0 0.# 0 0 4
$ ;3 3 0 0 1 1 1 4
j 3$ # 0 $ ' 2.# $ $ 4$ ' 30
j )j 3$ 3 0 0 2$ '2.#
$ 0
teration3
0 1 0 0 1 0.3333 0.3333 0.3333 2
# -2 0 1 0 0.# 0 0 4
3 -1 1 0 0 0.3333 0.3333 0.3333 2
!onto" Kasus Panel Kaca: Metode M-&esar (lan*utan)
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!onto" Kasus Panel Kaca:
+ersi ,M or .indo/s
)j +asic ,ari. 3 -1 # -2 0 slack 1 0 art?cl 2 0 art?cl 3 0 surplus 3 /uantity
teration 1
0 slack 1 1 0 1 0 0 0
0 art?cl 2 0 2 0 1 0 0 12
0 art?cl 3 3 2 0 0 1 1 1!
%j 0 1 0 0 0 1 30
cj%j 3 0 0 0 1
teration 2
0 slack 1 1 0 1 0 0 0
# -2 0 1 0 0.# 0 0 4
0 art?cl 3 3 0 0 1 1 1 4
%j 0 # 0 2 0 1 4
cj%j 3 0 0 2 0 1
teration 3
0 slack 1 0 0 1 0.3333 0.3333 0.3333 2
# -2 0 1 0 0.# 0 0 4
3 -1 1 0 0 0.3333 0.3333 0.3333 2
%j 3 # 0 1 1 0 0
cj%j 0 0 0 1 1 0
teration
0 slack 1 0 0 1 0.3333 0.3333 0.3333 2
# -2 0 1 0 0.# 0 0 4
3 -1 1 0 0 0.3333 0.3333 0.3333 2
%j 3 # 0 1.# 1 1 34
cj%j 0 0 0 1.# 1 1
teration #
0 surplus 3 0 0 3 1 1 1 4
# -2 0 1 0 0.# 0 0 4
3 -1 1 0 1.0 0 0 0 .0
%j 3 # 3 2.# 0 0 2
cj%j 0 0 3 2.# 0 0
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!onto" Kasus : Metode M-&esar - Solusi
Variable Status Value
X1 asic 2
X2 asic .
slack 1 asic 2
art4cl 2 /0/asic ,
surplus 3 /0/asic ,
Optimal Value (Z) 36
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!onto" Kasus : Metode M-&esar -
Sensitivitas
Var. Value Redu/. Ori0. er #pper
-st Value *und *und
X1 2 , 3 , n4inityX2 . , 5 -n4inity n4inity
-nstr. Dual Sla/4 Ori0. er #pper
Value Surpl. Value *und *und
*onstraint 1 , 2 $ 2 n4inity
*onstraint 2 -15 , 12 . 1%
*onstraint 3 -1 , 1% 12 2$
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!onto" Kasus : Metode M-&esar - asil
X1 X2 RHS Dual
Mini!i6e 3 5*onstraint 1 1 , ≤ $ 0
*onstraint 2 , 2 = 12 1.5
*onstraint 3 3 2 ≥ 1% 1
Slutin 2 6 36
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X1 X2 S1 S2 S3 R3 RHS
(Slack-1) (Slack-2) (Surplus-3) (@rti4icial-3)Maks Z = 25,,,,, X1 + $5,,,,, X2 - M? 3e!"atas 1 # X1 + X2 + S1 = ,,,
e!"atas 2 # 52,, X1 + 1.,, X2 + S2 = 1%,,,,,,
e!"atas 3 # X1 - 2 X2 - S3 + ? 3 = ,
Dirubah menjadi :
KS0S P1L T2M P3 3'$ I$ISI
$aksimumkan = 2.#00.000 -1 '
.#00.000 -2
Kendala :
-1 ' -2 @
>.000
#.200 -1 ' 1.400 -2 @1!.000.000
-1 2-2 A 0
-1 ( -2 A 0
$aksimumkan 2 #00 000 - ' #00 000
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Cj Basi !ariab"es 2#%%#%%% &' #%%#%%% &2 % S' % S2 *+3 % S3 ,uantity
-terasi '
0 1 1 1 1 0 0 0 >.000
0 2 #.200 1.400 0 1 0 0 1!.000.000
$ ;3 1 2 0 0 1 1 0
%j $ 2$ 0 0 $ $ 0
cj%j 2.#00.000'$ .#00.0002$ 0 0 0 $-terasi 2
0 1 0 3 1 0 1 1 >.000
0 2 0 12.000 0 1 #.200 #.200 1!.000.000
2.#00.000 -1 1 2 0 0 1 1 0
%j 2.#00.000 #.000.000 0 0 2.#00.000 2.#00.000 0
cj%j 0 B.#00.000 0 0 2.#00.000$ 2.#00.000
-terasi 3
0 1 0 0 1 0.0002# 0(3 0(3 2.#00
#00000 &2&2 0 1 0 0 0.333 0.333 '#%%'#%%
2#00000 &'&' 1 0 0 0.00014> 0.1333 0.1333 3#%%%3#%%%
.j.j 2.#00.000 .#00.000 0 14(444> 1.414.44> 1.414.44> '#%%#%%%#%%%'#%%#%%%#%%%
cj%j 0 0 0 14(444> 1.414.44>$ 1.414.44>
KS0S P1L T2M
P3 3'$ I$ISI
$aksimumkan = 2.#00.000 -1 ' .#00.000
-2
Kendala :
-1 ' -2 @ >.000
#.200 -1 ' 1.400 -2 @ 1!.000.000
-1 2-2 A 0
-1 ( -2 A 0
$aksimumkan = 2 #00 000 - ' #00 000
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Cj Basi !ariab"es 2%%%%% &' %%%%% &2 % S' % S2 % +3 % S3 ,uantity
-teration '
0 1 1 1 1 0 0 0 >(000
0 2 #(200 1(400 0 1 0 0 1!(000(000
0 ;3 1 2 0 0 1 1 0
%j 2#00000 #00000 0 0 0 1 0
cj%j 1 2 0 0 0 1
-teration 20 slack 1 0 3 1 0 1 1 >(000
0 slack 2 0 12(000 0 1 #(200 #(200 1!(000(000
2#00000 -1 1 2 0 0 1 1 0
%j 2#00000 #00000 0 0 1 0 0
cj%j 0 0 0 0 1 0
-teration 3
0 slack 1 0 3 1 0 1 1 >(000
0 slack 2 0 12(000 0 1 #(200 #(200 1!(000(000
2#00000 -1 1 2 0 0 1 1 0%j 2#00000 #000000 0 0 2#00000 2#00000 0
cj%j 0 B(#00(000 0 0 2(#00(000 2(#00(000
-teration
0 slack 1 0 0 1 0.0003 0.3 0.3 2(#00
#00000 &2&2 0 1 0 0.0001 0.333 0.333 '%%'%%
2#00000 &'&' 1 0 0 0.0002 0.1333 0.1333 3%%%3%%%
.j.j 2#00000 #00000 0 >B1.4> 141444>.0 141444>.0 '2%%%%%%%'2%%%%%%%
cj%j 0 0 0 >B1.444> 1(414(444.444> 1(414(444.444>
KS0S P1L T2M
P3 3'$ I$ISI+'$SI ,M F1$ .I231.S
$aksimumkan = 2.#00.000 -1 ' .#00.000
-2
Kendala :
-1 ' -2 @ >.000
#.200 -1 ' 1.400 -2 @ 1!.000.000
-1 2-2 A 0
-1 ( -2 A 0