METODE SIMPLEKS

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Metode simpleks digunakan untuk menyelesaikan masalah optimalisasi linear dengan dua variabel keputusan (X1 dan X2) dan empat variabel slack (S1, S2, S3, S4). Masalah ini mengalami tiga iterasi untuk mencapai solusi optimal dengan nilai fungsi tujuan Z = 6.

Education

Operation Research
Metode Simpleks
SOEDARMANTO
JANUARI 2013

Pertanyaan

DIKETAHUI
FUNGSI TUJUAN
Z = 2X₁ + 3X₂
FUNGSI KENDALA
5X ₁ + 6X₂ ≤ 60
X ₁ +2X₂ ≤ 16
X₁ ≤ 10
X₂ ≤6

Menyisipkan Slack

2X ₁ + 3X ₂ + OS ₁ + OS ₂ + OS ₃ + OS₄ =0
5X ₁ + 6X ₂ + S ₁ = 60
X ₁ + 2X ₂ +S₂ = 16
X₁ +S₃ = 10
X₂ + S₄ =6

Iterasi 0

Ci Vb X ₁ X₂ S₁ S₂ S₃ S₄
bi
0 Cj ₁ 2
S 5 36 01 00 00 00
60
0 S₂ 1 2 0 1 0 0
16
0 S₃ 1 0 0 0 1 0
10
0 S₄ 0 1 0 0 0 1
6
Zj 0 0 0 0 0 0

Langkah - Langkah

PENENTUAN KOLOM KUNCI
Konsep Opportunity cost (Cj-Zj) terbesar
PENENTUAN BARIS KUNCI
Rasio= bi/aij (pilih nilai positif terkecil)
PENGISIAN ELEMEN-ELEMEN YG KOSONG DENGAN ASB
(ATURAN SUDUT YANG BERLAWANAN)

Aturan Sudut Berlawanan
Elemen Sudut Elemen Lama
yang
Berlawanan
E₂ EX Elemen Baru

Elemen Sudut
Elemen Kunci
EK E₁ yang
Berlawanan

E₁ xE₂
EX (baru) = EX (lama) - -------------
EK

Iterasi 0

Ci Vb X ₁ X₂ S₁ S₂ S₃ S₄
bi
0Cj ₁ 25
S 36 01 00 00 00
60
0 S₂ 1 2 0 1 0 0
16
0 S₃ 1 0 0 0 1 0
10
0 S₄ 0 1 0 0 0 1
6
Zj 0 0 0 0 0 0
0

Iterasi 1

Ci Vb 2 X ₁ 3 X ₂ 0 S ₁ 0 S ₂ 0 S ₃ 0 S₄
Cj
bi
S₁ 0
S₂ 0
S₃ 0
X₂ 0 1 0 0 0 1
6
0 Zj
Cj-Zj

Iterasi 1
Ci Vb X ₁ X₂ S₁ S₂ S₃ S₄
bi
0 S ₁ 2 5 30
Cj 01 00 00 0-6
24
0S ₂ 1 0 0 1 0 -2
4
0S ₃ 1 0 0 0 1 0
10
3X ₂ 0 1 0 0 0 1
6
0 Zj 0 3 0 0 0 3

Iterasi 2

Ci Vb 2 X ₁ 3 X ₂ 0 S ₁ 0 S ₂ 0 S ₃ 0 S₄
Cj
bi
S₁ 0
X₁ 1 0 0 1 0 -2
4
S₃ 0
X₂ 0
Zj
Cj-Zj

Iterasi 2
Ci Vb X ₁ X₂ S₁ S₂ S₃ S₄
bi
0 S ₁ 20
Cj 30 01 0-5 00 04
4
2X ₁ 1 0 0 1 0 -2
4
0S ₃ 0 0 0 -1 1 2
6
3X ₂ 0 1 0 0 0 1
6
Zj 2 3 0 2 0 -1

Iterasi 3
Ci Vb X ₁ X ₂ S ₁ S ₂ S ₃ S₄
Cjbi 2 3 0 0 0 0
0S ₁ 0 0 1 -5 0 1
4
2X ₁ 0

0S ₃ 0

3X ₂ 0

Zj

Iterasi 3
Ci Vb X ₁ X₂ S₁ S₂ S₃ S₄
bi
0 S ₁ 20
Cj 30 01 0-5 00 01
4
2X ₁ 1 0 ½ -1½ 0 0
6
0S ₃ 0 0 -½ 1½ 1 0
4
3X ₂ 0 1 -¼ 1¼ 0 0
5
Zj 2 3 ¼ ¾ 0 0

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METODE SIMPLEKS

1. Operation Research Metode Simpleks SOEDARMANTO JANUARI 2013

2. Pertanyaan DIKETAHUI FUNGSI TUJUAN Z = 2X₁ + 3X₂ FUNGSI KENDALA 5X ₁ + 6X₂ ≤ 60 X ₁ +2X₂ ≤ 16 X₁ ≤ 10 X₂ ≤6

3. Menyisipkan Slack 2X ₁ + 3X ₂ + OS ₁ + OS ₂ + OS ₃ + OS₄ =0 5X ₁ + 6X ₂ + S ₁ = 60 X ₁ + 2X ₂ +S₂ = 16 X₁ +S₃ = 10 X₂ + S₄ =6

4. Iterasi 0 Ci Vb X ₁ X₂ S₁ S₂ S₃ S₄ bi 0 Cj ₁ 2 S 5 36 01 00 00 00 60 0 S₂ 1 2 0 1 0 0 16 0 S₃ 1 0 0 0 1 0 10 0 S₄ 0 1 0 0 0 1 6 Zj 0 0 0 0 0 0

5. Langkah - Langkah PENENTUAN KOLOM KUNCI Konsep Opportunity cost (Cj-Zj) terbesar PENENTUAN BARIS KUNCI Rasio= bi/aij (pilih nilai positif terkecil) PENGISIAN ELEMEN-ELEMEN YG KOSONG DENGAN ASB (ATURAN SUDUT YANG BERLAWANAN)

6. Aturan Sudut Berlawanan Elemen Sudut Elemen Lama yang Berlawanan E₂ EX Elemen Baru Elemen Sudut Elemen Kunci EK E₁ yang Berlawanan E₁ xE₂ EX (baru) = EX (lama) - ------------- EK

7. Iterasi 0 Ci Vb X ₁ X₂ S₁ S₂ S₃ S₄ bi 0Cj ₁ 25 S 36 01 00 00 00 60 0 S₂ 1 2 0 1 0 0 16 0 S₃ 1 0 0 0 1 0 10 0 S₄ 0 1 0 0 0 1 6 Zj 0 0 0 0 0 0 0

8. Iterasi 1 Ci Vb 2 X ₁ 3 X ₂ 0 S ₁ 0 S ₂ 0 S ₃ 0 S₄ Cj bi S₁ 0 S₂ 0 S₃ 0 X₂ 0 1 0 0 0 1 6 0 Zj Cj-Zj

9. Iterasi 1 Ci Vb X ₁ X₂ S₁ S₂ S₃ S₄ bi 0 S ₁ 2 5 30 Cj 01 00 00 0-6 24 0S ₂ 1 0 0 1 0 -2 4 0S ₃ 1 0 0 0 1 0 10 3X ₂ 0 1 0 0 0 1 6 0 Zj 0 3 0 0 0 3

10. Iterasi 2 Ci Vb 2 X ₁ 3 X ₂ 0 S ₁ 0 S ₂ 0 S ₃ 0 S₄ Cj bi S₁ 0 X₁ 1 0 0 1 0 -2 4 S₃ 0 X₂ 0 Zj Cj-Zj

11. Iterasi 2 Ci Vb X ₁ X₂ S₁ S₂ S₃ S₄ bi 0 S ₁ 20 Cj 30 01 0-5 00 04 4 2X ₁ 1 0 0 1 0 -2 4 0S ₃ 0 0 0 -1 1 2 6 3X ₂ 0 1 0 0 0 1 6 Zj 2 3 0 2 0 -1

12. Iterasi 3 Ci Vb X ₁ X ₂ S ₁ S ₂ S ₃ S₄ Cjbi 2 3 0 0 0 0 0S ₁ 0 0 1 -5 0 1 4 2X ₁ 0 0S ₃ 0 3X ₂ 0 Zj

13. Iterasi 3 Ci Vb X ₁ X₂ S₁ S₂ S₃ S₄ bi 0 S ₁ 20 Cj 30 01 0-5 00 01 4 2X ₁ 1 0 ½ -1½ 0 0 6 0S ₃ 0 0 -½ 1½ 1 0 4 3X ₂ 0 1 -¼ 1¼ 0 0 5 Zj 2 3 ¼ ¾ 0 0

14. Terimakasih BAHAN AJAR

METODE SIMPLEKS

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