PSTS 04 LP Simpleks.ppt

8/15/2019 PSTS 04 LP Simpleks.ppt

1/27

Pemrograman Linier

(Linear Programming - LP)

Metode Simpleks


2/27

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


3/27

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 :


4/27

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 * ≤ ,


5/27

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


6/27

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


7/27

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$


8/27

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


9/27

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 :


10/27

!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


11/27

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 ≥ ,


12/27

soal1


13/27

soal

2


14/27


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


15/27


asil Per"itungan

X1 X2 RHS Dual

Mai!i6e $, 5,


16/27

M-&'S($ &I-M)

$etode impleks


17/27

Metode Simpleks M-&esar

5ungsi 6ujuan dengan 78ariabel arti9sial;i000 -2 $.;1 $.;2 $.;3 $.;

$in = #000 -1 ' >000 -2 ' $.;1 ' $.;2 ' $.;3 ' $.;


18/27

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%


19/27

!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 :


20/27

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)


21/27

!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


22/27

!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


23/27

!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$


24/27

!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


25/27

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


26/27

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


27/27

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

Documents

PSTS 04 LP Simpleks.ppt