Problems in The Simplex Method Minggu 3 Part 3

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)

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 x1  60 x2  60 with x1 , x2  0

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

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

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 (0.8 + 0.4M) M (-1.6 + 1.2M) (1.6 - 1.2M 336 -12M cj - Zj (-0.8 - 0.4M) (-1.6 + 0.2M)

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 (1.6 + 0.3M) (-0.4 + 0.05M) M 320 – 10M cj - Zj (-1.6 - 0.3M) (0.4 - 0.05M)

Infeasible LP Ketidaklayakan solusi Kesalahan memformulasi program linear

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

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

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

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

The alternate optimal solution If two or more solutions yield the optimal objective value. Maximize Z = 10/3x1 + 4x2 subject to 2.5x1 + 3x2  300 5x1 + 2x2  400 2x2  150 with x1 , x2  0

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

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

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

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 - x2  2 2x1 + x3  4 x1 + x2 + x3  3 with x1 , x2 , x3  0

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

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

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

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