# Solving a Linear Programming Problem with Mixed Constraints Operation Research Minggu 3 Part 2.

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Solving a Linear Programming Problem with Mixed Constraints Operation Research Minggu 3 Part 2

Maximize Z = 5x 1 + x 2 subject to x 1  10 x 1 – 2x 2  3 x 1 + x 2 = 12 With x 1, x 2  0

 Ubah pertidaksamaan menjadi persamaan  Tambahkan variabel artifisial (A)  Tambahkan koefisien –M pada fungsi tujuan (karena fungsi maximasi)  Tambahkan slack variabel (S)

Aturan Koefisien BatasanPenyesuaianMaximasiMinimasi ≤Tambah variabel pengurang 00 =Tambah variabel artifisial -MM ≥Kurang variabel penambah 00 Dan tambah variabel artifisial -MM

Maximize Z = 5x 1 + x 2 + 0S 1 + 0S 2 – MA 1 – MA 2 subject to x 1 + S 1 = 10 x 1 – 2x 2 - S 2 + A 1 = 3 x 1 + x 2 + A 2 = 12 With x 1, x 2, S 1, S 2, A 1, A 2  0

Initial Simplex Tableau cjcj 5100-M cbcb BASISx1x1 x2x2 S1S1 S2S2 A1A1 A1A1 Solution 0 -M S1A1A2S1A1A2 111111 0 -2 1 100100 0 0 010010 001001 10 3 12 ZjZj -2MM0M-M -15M c j - Z j 5+2M1-M0-M00

Second Simplex Tableau cjcj 5100-M cbcb BASISx1x1 x2x2 S1S1 S2S2 A1A1 A1A1 Solution 0 5 -M S1x1A2S1x1A2 010010 2 -2 3 100100 1 1 1 001001 739739 ZjZj 5-10-3M0-5-M5+M-M15-9M c j - Z j 011+3M05+M-5-2M0

Third Simplex Tableau cjcj 5100-M cbcb BASISx1x1 x2x2 S1S1 S2S2 A1A1 A1A1 Solution 051051 S1x1x2S1x1x2 010010 001001 100100 1/3 -1/3 1/3 -1/3 1/3 -1/3 -2/3 2/3 1/3 193193 ZjZj 510-4/34/311/348 c j - Z j 0004/3-M-4/3-M-11/3

Modified Simplex Tableau for Artificial Variable Illustration cjcj 5100 cbcb BASISx1x1 x2x2 S1S1 S2S2 Solution 051051 S1x1x2S1x1x2 010010 001001 100100 1/3 -1/3 1/3 193193 ZjZj 510-4/348 c j - Z j 0004/3

Optimal Simplex Tableau for Artificial Variable Illustration cjcj 5100 cbcb BASISx1x1 x2x2 S1S1 S2S2 Solution 051051 S2x1x2S2x1x2 010010 001001 3 1 100100 3 10 2 ZjZj 514052 c j - Z j 00-40

Solving the Minimization Problem

 The pivot column : the nonbasic variable with the largest |c j – Z j | value, for c j – Z j < 0. (negatif terbesar)  Or choose the maximum value for zj-cj  The pivot row is as same as maximizing problem.

Minimize Z = 1200y 1 + 3000y 2 + 3600y 3 subject to y 1 + 2y 2 + y 3  3 y 1 + 3y 2 + 4y 3  4 with y 1, y 2, y 3  0

Minimize Z = 1200y 1 + 3000y 2 + 3600y 3 + MA 1 + MA 2 subject to y 1 + 2y 2 + y 3 – S 1 + A 1 = 3 y 1 + 3y 2 + 4y 3 - S 2 + A 2 = 4 with y 1, y 2, y 3, S 1, S 2, A 1, A 2  0

Initial Simplex Tableau for Minimization Problem cjcj 12003000360000MM cbcb BASISy1y1 y2y2 y3y3 S1S1 S2S2 A1A1 A2A2 Solution MMMM A1A2A1A2 1111 2323 1414 0 1010 0101 3434 ZjZj 2M5M -M MM7M c j - Z j 1200- 2M 3000- 5M 3600- 5M MM00

Second Simplex Tableau cjcj 12003000360000MM cbcb BASISy1y1 y2y2 y3y3 S1S1 S2S2 A1A1 A1A1 Solution M 3000 A1y2A1y2 1/3 0101 -5/3 4/3 0 2/3 -1/3 1010 -2/3 1/3 4/3 ZjZj (1/3M+ 1000) 3000 (-5/3M +4000) -M (2/3M - 1000) M (-2/3M +1000) 1/3M + 4000 c j - Z j (-1/3M +200) 0 (5/3M - 400) M (-2/3M +1000) 0 (5/3M - 1000)

Third Simplex Tableau cjcj 12003000360000MM cbcb BASISy1y1 y2y2 y3y3 S1S1 S2S2 A1A1 A1A1 Solution 0 3000 S1y2S1y2 ½½½½ 0101 -5/2 ½ -3/2 -1/2 1010 3/2 ½ 0 ½ 3/2 ZjZj 150030001500-15000150004500 c j - Z j -3000210015000M-1500M

Optimal Simplex Tableau for Minimization Problem cjcj 12003000360000MM cbcb BASISy1y1 y2y2 y3y3 S1S1 S2S2 A1A1 A1A1 Solution 1200 3000 y1y2y1y2 1010 0101 -5 3 -3 1 2 3 -2 1 1111 ZjZj 12003000-3000-600 600 4200 c j - Z j 00600 (M-600)

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