Problems in The Simplex Method

Presentasi berjudul: "Problems in The Simplex Method"— Transcript presentasi:

1 Problems in The Simplex Method
Minggu 3 Part 3

2 Solusi yang tidak fisibel (infeasible solution)
Masalah yang tidak terbatas (The unbounded linear programming) Solusi optimal majemuk (The alternate optimal solution) Pivot row yang seri (degeneracy)

3 The infeasible linear programming problem
No solution that satisfies the constraints and non-negativity conditions for the problem. Maximize Z = 2x1 + 4x2 subject to 2.5x1 + 3x2  300 5x1 + 2x2  400 2x2  150 x  60 x2  60 with x1 , x2  0

4 Initial Simplex Tableau for Infeasibility Problem
cj 2 4 -M cb BASIS x1 x2 S1 S2 S3 S4 A1 S5 A2 Solution 2.5 5 1 3 -1 300 400 150 60 Zj M -120M cj - Zj 2+M 4+M

5 Second Simplex Tableau for Infeasibility Problem
cj 2 4 -M cb BASIS x1 x2 S1 S2 S3 S4 A1 S5 A2 Solution 2.5 5 1 -1 3 -3 -2 120 280 30 60 Zj M -4 240-6M cj - Zj 2+M -M-4

6 Third Simplex Tableau for Infeasibility Problem
cj 2 4 -M cb BASIS x1 x2 S1 S2 S3 S4 A1 S5 A2 Solution 1 0.4 -2 -0.4 -1 1.2 -4 -1.2 48 40 30 12 60 Zj ( M) M ( M) ( M M cj - Zj ( M) ( M)

7 Final Simplex Tableau for Infeasibility Problem
cj 2 4 -M cb BASIS x1 x2 S1 S2 S3 S4 A1 S5 A2 Solution 1 -0.2 -0.5 0.2 0.5 0.3 0.25 -0.3 -0.25 -1 1.2 -4 -1.2 -2 120 280 30 60 Zj ( M) ( M) M 320 – 10M cj - Zj ( M) ( M)

8 Infeasible LP Ketidaklayakan solusi
Kesalahan memformulasi program linear

9 The unbounded linear programming
If the objective function can be made infinitely large without violating any of the constraints. Maximize Z = 2x1 + 3x2 subject to x1 - x2  2 -3x1 + x2  4 with x1 , x2  0

10 Initial Simplex Tableau for Unbounded Illustration
cj 2 3 cb BASIS x1 x2 S1 S2 Solution 1 -3 -1 4 Zj cj - Zj

11 Second Simplex Tableau for Unbounded Illustration
cj 3 4 cb BASIS x1 x2 S1 S2 Solution -2 -3 1 6 Zj -9 12 cj - Zj 11

12 Suatu masalah yang tidak terbatas diidentifikasikan dalam simplex pada saat pemilihan pivot row tidak mungkin dilakukan  saat nilai pivot row negatif atau tak terhingga

13 The alternate optimal solution
If two or more solutions yield the optimal objective value. Maximize Z = 10/3x1 + 4x2 subject to 2.5x1 + 3x2  x1 + 2x2  400 2x2  150 with x1 , x2  0

14 Initial Simplex Tableau for Alternate Optimal Solution
cj 10/3 4 cb BASIS x1 x2 S1 S2 S3 Solution 2.5 5 3 2 1 300 400 150 Zj cj - Zj Second Simplex Tableau for Alternate Optimal Solution cj 10/3 4 cb BASIS x1 x2 S1 S2 S3 Solution 2.5 5 1 -1.5 -1 0.5 75 250 Zj 2 cj - Zj -2

15 Third Simplex Tableau for Alternate Optimal Solution
cj 10/3 4 cb BASIS x1 x2 S1 S2 S3 Solution 1 0.4 -2 -0.6 2 0.5 30 100 75 Zj 4/3 400 cj - Zj -4/3 Fourth Simplex Tableau for Alternate Optimal Solution cj 10/3 4 cb BASIS x1 x2 S1 S2 S3 Solution 1 -0.2 -1 0.5 0.3 -0.25 60 50 Zj 4/3 400 cj - Zj -4/3

16 Jika melakukan iterasi lanjut pada tabel simplex yang sudah memenuhi syarat optimal
Diindikasikan oleh nilai 0 pada baris cj-zj untuk variabel bukan dasar Memberi keleluasaan pada perusahaan untuk memilih kombinasi produk

17 The concept of degeneracy in linear programming
If one or more of the basic variables has a value of zero. Occurs whenever two rows satisfy the criterion of selection as pivot row. Maximize Z = 4x1 + 3x2 subject to x1 - x  2 2x x3  4 x1 + x2 + x3  3 with x1 , x2 , x3  0

18 Initial Simplex Tableau for Degeneracy Illustration
cj 4 3 cb BASIS x1 x2 x3 S1 S2 S3 Solution Ratio 1 2 -1 2/1 = 2 4/2 = 2 3/1 = 3 Zj cj - Zj Second Simplex Tableau for Degeneracy Illustration cj 4 3 cb BASIS x1 x2 x3 S1 S2 S3 Solution Ratio 1 -1 2 -2 - 0/2 1/2 Zj -4 8 cj - Zj

19 Third Simplex Tableau for Degeneracy Illustration
cj 4 3 cb BASIS x1 x2 x3 S1 S2 S3 Solution Ratio 1 0.5 -1 2 2/0.5 0/0.5 - Zj 8 cj - Zj -2 Fourth Simplex Tableau for Degeneracy Illustration cj 4 3 cb BASIS x1 x2 x3 S1 S2 S3 Solution Ratio 1 -1 2 -2 2/1 - 1/1 Zj 8 cj - Zj -3

20 Fifth Simplex Tableau for Degeneracy Illustration
cj 4 3 cb BASIS x1 x2 x3 S1 S2 S3 Solution Ratio 1 -1 2 Zj 10 cj - Zj -2

21 Degenerasi muncul ketika terdapat pivot row yang seri
Pilih salah satu pivot row secara acak dan lakukan iterasi lanjut secara normal