4 Programa Linear Metode Dua Fasa (1)
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![Page 1: 4 Programa Linear Metode Dua Fasa (1)](https://reader036.fdokumen.com/reader036/viewer/2022062309/55cf973d550346d03390764a/html5/thumbnails/1.jpg)
PROGRAMA LINEARMETODE SIMPLEKS DUA FASASESI 4
STMIK MERCUSUAR
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BENTUK BAKU LP• Semua Kendala/contraint berupa persamaam dengan sisi kanan
Nonnegatif• Semua Variabel Nonnegatif• Fungsi tujuan dapat Maksimum maupun Minimum• Kendala– Bentuk <, ditambah Slack (S>0).
x1+x2<15 menjadi x1+x2+S=0– Bentuk >, ditambah Surplus (S) dan Artificial (A)
x1+x2>15 menjadi x1+x2-S+A=0– Bentuk =, ditambah Artificial (A)
x1+x2=15 menjadi x1+x2+A=0• Bila bentuk ketidaksamaan dikalikan dengan -1, tandanya akan
berbalik. Mis -x1+x2>-15 jadi x1-x2<15
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METODE SIMPLEKS DUA FASA• Untuk penyelesaian Programa linier yang memiliki minimal 1
(satu) fungsi pembatas dengan tanda (≥) atau tanda (=)• Tahap 1 untuk memperoleh niali Zj = 0, kemudian tahap 2 untuk
mendapatkan jawaban optimal
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• Prosedur hampir sama dengan Metode Simpleks biasa, kecuali ditambah variabel surplus dan variabel artificial serta 2 fasa penyelesaian.
Max Z= 250X1 + 200X2 - MX6
Pembatas 20X1 + 45X2 + X3 = 10.750
30X1 + 25X2 + X4 = 9.750
X1 - X5 + X6 = 100
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Contoh (pernah dibahas pada bab sebelumnya)
– Pabrik membuat meja dan kursi, harga meja Rp 250 ribu dan kursi Rp 200 ribu.
– Pembuatan Meja perlu 20 sat asembling dan 30 sat finishing– Pembuatan Kursi perlu 45 sat asembling dan 25 sat finishing– Kapasitas mesin asembling 10.750 sat asembling dan mesin
finishing 9.750 sat finishing– Produk minimal yang harus dibuat adalah 100 unit meja
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Z = 250X1 + 200X2
20X1 + 45X2 ≤ 10.750
30X1 + 25X2 ≤ 9.750
X1 ≥ 100
Z - 250X1 - 200X2 + MX6 = 0
20X1 + 45X2 + X3 = 10.750
30X1 + 25X2 + X4 = 9.750
X1 - X5 + X6 = 100
SLACK
SURPLUS
SLACK
"M" Koefisien fungsi tujuan
artificial
ARTFICIAL
Contoh Soal (methode 2 fasa)
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Basis x1 x2 x3 x4 x5 x6 Ruas Kanan
x3 20,00 45,00 1,00 0,00 0,00 0,00 10.750,00
x4 30,00 25,00 0,00 1,00 0,00 0,00 9.750,00
x6 1,00 0,00 0,00 0,00 -1,00 1,00 100,00
Zj-Cj -250,00 -200,00 0,00 0,00 0,00 M 0,00
Zj-Cj -250-M -200,00 0,00 0,00 M 0,00 -100M
Zj-Cj -250,00 -200,00 0,00 0,00 0,00 0,00 0,00
Zj-Cj -1 0,00 0,00 0,00 1 0,00 -100
Nilai M dijadikan Nol
(-M)x(1)+M(-M)x(0)+0(-M)x(0)+(-200)(-M)x(1)+(-250) (-M)x(100)+0(-M)x(0)+0 (-M)x(-1)+0
Komponen Zj-Cj tanpa M
Komponen Zj-Cj dengan M
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FASA 1
Basis x1 x2 x3 x4 x5 x6 Rs kanan Rasio
x3 20,00 45,00 1,00 0,00 0,00 0,00 10.750,00 537,50
x4 30,00 25,00 0,00 1,00 0,00 0,00 9.750,00 325,00
x6 1,00 0,00 0,00 0,00 -1,00 1,00 100,00 100,00
Zj-Cj -1,00 0,00 0,00 0,00 1,00 0,00 -100,00
Komponen Zj-Cj terkecil
Komponen Ruas kanan terkecil
Komponen Zj-Cj dengan M
PIVOT
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Basis x1 x2 x3 x4 x5 x6 Rs kanan
x3 0,00 45,00 1,00 0,00 20,00 -20,00 8.750,00
x4 0,00 25,00 0,00 1,00 30,00 -30,00 6.750,00
x1 1,00 0,00 0,00 0,00 -1,00 1,00 100,00
Zj-Cj 0,00 0,00 0,00 0,00 0,00 1,00 0,00
HASIL ITERASI FASA 1
Akhir Fasa 1, Komponen Zj-Cj di kolom ruas kanan sama dengan 0
Pada FASA 2, kolom x6 (artificial dihilangkan
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FASE 2
Basis x1 x2 x3 x4 x5 Rs kanan
x3 0,00 45,00 1,00 0,00 20,00 8.750,00
x4 0,00 25,00 0,00 1,00 30,00 6.750,00
x1 1,00 0,00 0,00 0,00 -1,00 100,00
Zj-Cj -250,00 -200,00 0,00 0,00 0,00 0,00
0,00 -200,00 0,00 0,00 -250,00 25.000,00
Komponen Zj-Cj tanpa M
Nilai Zj-Cj pada kolom x1 dijadikan 0
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Basis x1 x2 x3 x4 x5 Rs kanan Rasio
x3 0,00 45,00 1,00 0,00 20,00 8.750,00 437,50
x4 0,00 25,00 0,00 1,00 30,00 6.750,00 225,00
x1 1,00 0,00 0,00 0,00 -1,00 100,00 -100,00
Zj-Cj 0,00 -200,00 0,00 0,00 -250,00 25.000,00
Rasio non negatif terkecilNilai Zj-Cj terkecilPIVOT
(Nilai nya dijadikan 1
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Basis x1 x2 x3 x4 x5 Rs kanan
x3 0,00 28,33 1,00 -0,67 0,00 4.250,00
x5 0,00 0,83 0,00 0,03 1,00 225,00
x1 1,00 0,83 0,00 0,03 0,00 325,00
Zj-Cj 0,00 8,33 0,00 8,33 0,00 81.250,00
HASIL ITERASI FASA 2
Semua komponen Zj-Cj sdh NOL atau Positih
berarti sdh Optimal
Hasil tg diperoleh : x1 = 325 x2 = 0 x3 = 4.250 x4 = 0 x5 = 225 Z = 81.250
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Model Programa Linier (PL)
Z = 15X1 + 12X2
3X1 + 8X2 ≤ 39
10X1 + 4X2 ≤ 62
X1 ≥ 3
X2 ≥ 2
Bentuk Standar
Max Z - 15X1 - 12X2 + MX7 + MX8 = 0
Pembatas 3X1 + 8X2 + X3 = 39
10X1 + 4X2 + X4 = 62
X1 - X5 + X7 = 3
X2 - X6 + X8 = 2
Contoh ke 2
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Basis x1 x2 x3 x4 x5 x6 x7 x8 Rs kanan
x3 3 8 1 0 0 0 0 0 39,00
x4 10 8 0 1 0 0 0 0 62,00
x7 1 0 0 0 -1 0 1 0 3,00
x8 0 1 0 0 0 -1 0 1 2,00
Zj-Cj -15 -12 0 0 0 0 M M 0,00
Zj-Cj -15-M -12 0 0 M 0 0 M -3M
Zj-Cj -15-M -12-M 0 0 M M 0 0 -5M
Zj-Cj -15 -12 0 0 0 0 0 0 0,00
Zj-Cj -1 -1 0 0 1 1 0 0 -5,00
PERSIAPAN FASA 1
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Basis x1 x2 x3 x4 x5 x6 x7 x8 Rs kanan Rasio
x3 3 8 1 0 0 0 0 0 39,00 13,00
x4 10 4 0 1 0 0 0 0 62,00 6,20
x7 1 0 0 0 -1 0 1 0 3,00 3,00
x8 0 1 0 0 0 -1 0 1 2,00 #DIV/0!
Zj-Cj -1 -1 0 0 1 1 0 0 -5,00
FASA 1 AWAL
PIVOT
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Basis x1 x2 x3 x4 x5 x6 x7 x8 Rs kanan
x3 0 8 1 0 3 0 -3 0 30,00
x4 0 4 0 1 10 0 -10 0 32,00
x1 1 0 0 0 -1 0 1 0 3,00
x8 0 1 0 0 0 -1 0 1 2,00
Zj-Cj 0 -1 0 0 0 1 1 0 -2,00
MASUK X1 KELUAR X7
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Basis x1 x2 x3 x4 x5 x6 x7 x8 Rs kanan Rasio
x3 0 8 1 0 3 0 -3 0 30,00 3,75
x4 0 4 0 1 10 0 -10 0 32,00 8,00
x1 1 0 0 0 -1 0 1 0 3,00 #DIV/0!
x8 0 1 0 0 0 -1 0 1 2,00 2,00
Zj-Cj 0 -1 0 0 0 1 1 0 -2,00
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Basis x1 x2 x3 x4 x5 x6 x7 x8 Rs kanan
x3 0 0 1 0 3 8 -3 -8 14,00
x4 0 0 0 1 10 4 -10 -4 24,00
x1 1 0 0 0 -1 0 1 0 3,00
x2 0 1 0 0 0 -1 0 1 2,00
Zj-Cj 0 0 0 0 0 0 1 1 0,00
MASUK X2 KELUAR X8
Komponen Zj-Cj pada ruas Kanan sdh “0”
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Basis x1 x2 x3 x4 x5 x6 Rs kanan
x3 0 0 1 0 3 8 14,00
x4 0 0 0 1 10 4 24,00
x1 1 0 0 0 -1 0 3,00
x2 0 1 0 0 0 -1 2,00
Zj-Cj -15 -12 0 0 0 0 0,00
Zj-Cj 0 -12 0 0 -15 0 45,00
Zj-Cj 0 0 0 0 -15 -12 69,00
FASA 2 (PERSIAPAN)
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Basis x1 x2 x3 x4 x5 x6 Rs kanan Rasio
x3 0 0 1 0 3 8 14,00 4,667
x4 0 0 0 1 10 4 24,00 2,400
x1 1 0 0 0 -1 0 3,00 -3,000
x2 0 1 0 0 0 -1 2,00 #DIV/0!
Zj-Cj 0 0 0 0 -15 -12 69,00
ITERASI KE1
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Basis x1 x2 x3 x4 x5 x6 Rs kanan
x3 0 0 1 0 3 8 14,00
x5 0 0 0 0,10 1 0 2
x1 1 0 0 0 -1 0 3,00
x2 0 1 0 0 0 -1 2,00
Zj-Cj 0 0 0 0 -15 -12 69,00
MASUK X5 KELUAR X4
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Basis x1 x2 x3 x4 x5 x6 Rs kanan
x3 0,00 0,00 1,00 -0,30 0,00 6,80 6,80
x5 0,00 0,00 0,00 0,10 1,00 0,40 2,40
x1 1,00 0,00 0,00 0,10 0,00 0,40 5,40
x2 0,00 1,00 0,00 0,00 0,00 -1,00 2,00
Zj-Cj 0,00 0,00 0,00 1,50 0,00 -6,00 105,00
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Basis x1 x2 x3 x4 x5 x6 Rs kanan Rasio
x3 0,00 0,00 1,00 -0,30 0,00 6,80 6,80 1,000
x5 0,00 0,00 0,00 0,10 1,00 0,40 2,40 6,000
x1 1,00 0,00 0,00 0,10 0,00 0,40 5,40 13,500
x2 0,00 1,00 0,00 0,00 0,00 -1,00 2,00 -2,000
Zj-Cj 0,00 0,00 0,00 1,50 0,00 -6,00 105,00
ITERASI KE2
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Basis x1 x2 x3 x4 x5 x6 Rs kanan
x6 0,00 0,00 0,15 -0,04 0,00 1,00 1,00
x5 0,00 0,00 0,00 0,10 1,00 0,40 2,40
x1 1,00 0,00 0,00 0,10 0,00 0,40 5,40
x2 0,00 1,00 0,00 0,00 0,00 -1,00 2,00
Zj-Cj 0,00 0,00 0,00 1,50 0,00 -6,00 105,00
MASUK X6 KELUAR X4
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Basis x1 x2 x3 x4 x5 x6 Rs kanan
x6 0,00 0,00 0,15 -0,04 0,00 1,00 1,00
x5 0,00 0,00 -0,06 0,12 1,00 0,00 2,00
x1 1,00 0,00 -0,06 0,12 0,00 0,00 5,00
x2 0,00 1,00 0,15 -0,04 0,00 0,00 3,00
Zj-Cj 0,00 0,00 0,88 1,24 0,00 0,00 111,00
HASIL AKHIR
Komponen Zj-Cj tidak ada yang negatif