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Problems in The Simplex Method Minggu 3 Part 3. Solusi yang tidak fisibel (infeasible solution) Masalah yang tidak terbatas (The unbounded linear programming)

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Presentasi berjudul: "Problems in The Simplex Method Minggu 3 Part 3. Solusi yang tidak fisibel (infeasible solution) Masalah yang tidak terbatas (The unbounded linear programming)"— 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 = 2x 1 + 4x 2 subject to 2.5x 1 + 3x 2  300 5x 1 + 2x 2  400 2x 2  150 x 1  60 x 2  60 with x 1, x 2  0

4 Initial Simplex Tableau for Infeasibility Problem cjcj M 0 cbcb BASISx1x1 x2x2 S1S1 S2S2 S3S3 S4S4 A1A1 S5S5 A2A2 Solution 0 -M S1S2S3A1A2S1S2S3A1A ZjZj -M 000M M -120M c j - Z j 2+M4+M000-M0 0

5 Second Simplex Tableau for Infeasibility Problem cjcj M 0 cbcb BASISx1x1 x2x2 S1S1 S2S2 S3S3 S4S4 A1A1 S5S5 A2A2 Solution 0 -M 4 S1S2S3A1x2S1S2S3A1x ZjZj -M 000M M c j - Z j 2+MM000-M04-M-4

6 Third Simplex Tableau for Infeasibility Problem cjcj M 0 cbcb BASISx1x1 x2x2 S1S1 S2S2 S3S3 S4S4 A1A1 S5S5 A2A2 Solutio n 2 0 -M 4 x1S2S3A1x2x1S2S3A1x ZjZj 24 ( M) 00M-M ( M) ( M M c j - Z j 00 ( M) 00-M0 ( M ( M)

7 Final Simplex Tableau for Infeasibility Problem cjcj M 0 cbcb BASISx1x1 x2x2 S1S1 S2S2 S3S3 S4S4 A1A1 S5S5 A2A2 Solution 2 -M 0 -M 4 x1A2S3A1x2x1A2S3A1x ZjZj 24 ( M) ( M) 0M-MM 320 – 10M c j - Z j 00 ( M) ( M) 0-M0 0

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 = 2x 1 + 3x 2 subject to x 1 - x 2  2 -3x 1 + x 2  4 with x 1, x 2  0

10 Initial Simplex Tableau for Unbounded Illustration cjcj 2300 cbcb BASISx1x1 x2x2 S1S1 S2S2 Solution 0000 S1S2S1S ZjZj c j - Z j 2300

11 Second Simplex Tableau for Unbounded Illustration cjcj 3400 cbcb BASISx1x1 x2x2 S1S1 S2S2 Solution 0303 S1x2S1x ZjZj c j - Z j

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 = 10/3x 1 + 4x 2 subject to 2.5x 1 + 3x 2  300 5x 1 + 2x 2  400 2x 2  150 with x 1, x 2  0

14 Initial Simplex Tableau for Alternate Optimal Solution cjcj 10/34000 cbcb BASISx1x1 x2x2 S1S1 S2S2 S3S3 Solution S1S2S3S1S2S ZjZj c j - Z j 10/34000 Second Simplex Tableau for Alternate Optimal Solution cjcj 10/34000 cbcb BASISx1x1 x2x2 S1S1 S2S2 S3S3 Solution S1S2x2S1S2x ZjZj c j - Z j 10/3000-2

15 Fourth Simplex Tableau for Alternate Optimal Solution cjcj 10/34000 cbcb BASISx1x1 x2x2 S1S1 S2S2 S3S3 Solution 10/3 0 4 x1S3x2x1S3x ZjZj 10/344/ c j - Z j 00-4/300 Third Simplex Tableau for Alternate Optimal Solution cjcj 10/34000 cbcb BASISx1x1 x2x2 S1S1 S2S2 S3S3 Solution 10/3 0 4 x1S2x2x1S2x ZjZj 10/344/ c j - Z j 00-4/300

16 Jika melakukan iterasi lanjut pada tabel simplex yang sudah memenuhi syarat optimal Diindikasikan oleh nilai 0 pada baris c j -z j 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 = 4x 1 + 3x 2 subject to x 1 - x 2  2 2x 1 + x 3  4 x 1 + x 2 + x 3  3 with x 1, x 2, x 3  0

18 Initial Simplex Tableau for Degeneracy Illustration cjcj cbcb BASISx1x1 x2x2 x3x3 S1S1 S2S2 S3S3 Solution Ratio S1S2S3S1S2S /1 = 2 4/2 = 2 3/1 = 3 ZjZj c j - Z j Second Simplex Tableau for Degeneracy Illustration cjcj cbcb BASISx1x1 x2x2 x3x3 S1S1 S2S2 S3S3 Solution Ratio x1S2S3x1S2S /2 1/2 ZjZj c j - Z j

19 Third Simplex Tableau for Degeneracy Illustration cjcj cbcb BASISx1x1 x2x2 x3x3 S1S1 S2S2 S3S3 Solution Ratio x1x2S3x1x2S /0.5 0/0.5 - ZjZj c j - Z j Fourth Simplex Tableau for Degeneracy Illustration cjcj cbcb BASISx1x1 x2x2 x3x3 S1S1 S2S2 S3S3 Solution Ratio x1x3S3x1x3S /1 - 1/1 ZjZj c j - Z j

20 Fifth Simplex Tableau for Degeneracy Illustration cjcj cbcb BASISx1x1 x2x2 x3x3 S1S1 S2S2 S3S3 Solution Ratio x1x3S1x1x3S ZjZj c j - Z j

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


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