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Objective Function Explicit Constraints

This document describes an optimization problem to minimize a linear objective function subject to several explicit and implicit constraints. The objective is to minimize the weighted sum of 10 decision variables (A-J) representing amounts of different products. There are 7 explicit linear constraints on combinations of the decision variables that must be satisfied. The problem is modified to include 7 slack variables (S1-S7) to convert inequality constraints into equality constraints. The problem is solved using the simplex method, with 7 tables showing the simplex iterations and optimal solution values for the decision variables.

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Sinteya Yeo
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40 views16 pages

Objective Function Explicit Constraints

This document describes an optimization problem to minimize a linear objective function subject to several explicit and implicit constraints. The objective is to minimize the weighted sum of 10 decision variables (A-J) representing amounts of different products. There are 7 explicit linear constraints on combinations of the decision variables that must be satisfied. The problem is modified to include 7 slack variables (S1-S7) to convert inequality constraints into equality constraints. The problem is solved using the simplex method, with 7 tables showing the simplex iterations and optimal solution values for the decision variables.

Uploaded by

Sinteya Yeo
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as XLSX, PDF, TXT or read online on Scribd
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OBJECTIVE FUNCTION Minimize: 0.18A + 0.22B + 0.10C + 0.12D + 0.10E + 0.09F + 0.40G + 0.16H + 0.50I + 0.

07J
EXPLICIT CONSTRAINTS 90A + 110B + 100C + 90D + 75E + 35F + 65G + 100H + 120I + 65J ≥ 420
2B + 2C + 2D + 5E + 3F + 4H + 1J ≤ 20
270E + 8F + 12H ≤ 30
6A + 4B + 2C + 3D + 1E + 1G + 1J ≥ 5
20A + 48B + 12C + 8D + 30E + 52G + 250H + 3I+ 26J ≥ 400
3A + 4B + 5C + 6D + 7E + 2F + 1G + 9H + 1I+ 3J ≥ 20
5A + 2B + 3C + 4D + 1G + 3J ≥ 12
IMPLICIT CONSTRAINTS A, B, C, D, E, F, G, H, I, J ≥ 0
MODIFICATIONS:
OBJECTIVE FUNCTION Minimize: 0.18A + 0.22B + 0.10C + 0.12D + 0.10E + 0.09F + 0.40G + 0.16H + 0.50I + 0.07J + 0S1
EXPLICIT CONSTRAINTS 90A + 110B + 100C + 90D + 75E + 35F + 65G + 100H + 120I + 65J - 1S1 + 1A1 = 420
2B + 2C + 2D + 5E + 3F + 4H + 1J + 1S2 = 20
270E + 8F + 12H + 1S3 = 30
6A + 4B + 2C + 3D + 1E + 1G + 1J -1S4 + IA2 = 5
20A + 48B + 12C + 8D + 30E + 52G + 250H + 3I+ 26J - 1S5 + 1A3 = 400
3A + 4B + 5C + 6D + 7E + 2F + 1G + 9H + 1I+ 3J - 1S6 + 1A4 = 20
5A + 2B + 3C + 4D + 1G + 3J - 1S7 + 1A5 = 12
TABLE 1
CJ --- --- 0.18 0.22 0.10 0.12 0.10 0.09 0.40
S VAR SVAL A B C D E F G
1000.00 A1 420.00 90.00 110.00 100.00 90.00 75.00 35.00 65.00
0.00 S2 20.00 0.00 2.00 2.00 2.00 5.00 3.00 0.00
0.00 S3 30.00 0.00 0.00 0.00 0.00 270.00 8.00 0.00
1000.00 A2 5.00 6.00 4.00 2.00 3.00 1.00 0.00 1.00
1000.00 A3 400.00 20.00 48.00 12.00 8.00 30.00 0.00 52.00
1000.00 A4 20.00 3.00 4.00 5.00 6.00 7.00 2.00 1.00
1000.00 A5 12.00 5.00 2.00 3.00 4.00 0.00 0.00 1.00
--- ZJ ### ### ### ### ### ### 37000.00 ###
--- CJ-ZJ ### ### ### ### ### ### ###

TABLE 2
CJ --- --- 0.18 0.22 0.10 0.12 0.10 0.09 0.40
S VAR SVAL A B C D E F G
1000.00 A1 260.00 82.00 90.80 95.20 86.80 63.00 35.00 44.20
0.00 S2 13.60 -0.32 1.23 1.81 1.87 4.52 3.00 -0.83
0.00 S3 10.80 -0.96 -2.30 -0.58 -0.38 268.56 8.00 -2.50
1000.00 A2 5.00 6.00 4.00 2.00 3.00 1.00 0.00 1.00
0.16 H 1.60 0.08 0.19 0.05 0.03 0.12 0.00 0.21
1000.00 A4 5.60 2.28 2.27 4.57 5.71 5.92 2.00 -0.87
1000.00 A5 12.00 5.00 2.00 3.00 4.00 0.00 0.00 1.00
--- ZJ ### 95280.01 99072.03 ### 99512.01 69920.02 37000.00 45328.03
--- CJ-ZJ ### ### ### ### ### ### ###

TABLE 3
CJ --- --- 0.18 0.22 0.10 0.12 0.10 0.09 0.40
S VAR SVAL A B C D E F G
0.50 I 2.19 0.69 0.76 0.80 0.73 0.53 0.29 0.37
0.00 S2 13.71 -0.29 1.27 1.85 1.91 4.55 3.01 -0.81
0.00 S3 11.12 -0.86 -2.19 -0.46 -0.28 268.64 8.04 -2.44
1000.00 A2 5.00 6.00 4.00 2.00 3.00 1.00 0.00 1.00
0.16 H 1.57 0.07 0.18 0.04 0.02 0.11 0.00 0.20
1000.00 A4 3.65 1.66 1.59 3.85 5.06 5.45 1.74 -1.20
1000.00 A5 12.00 5.00 2.00 3.00 4.00 0.00 0.00 1.00
--- ZJ 20649.16 12664.67 7590.65 8853.61 12060.64 6447.25 1737.35 796.35
--- CJ-ZJ ### -7590.43 -8853.51 ### -6447.15 -1737.26 -795.95

TABLE 4
CJ --- --- 0.18 0.22 0.10 0.12 0.10 0.09 0.40
S VAR SVAL A B C D E F G
0.50 I 1.61 0.00 0.30 0.57 0.39 0.42 0.29 0.26
0.00 S2 13.94 0.00 1.46 1.94 2.05 4.59 3.01 -0.77
0.00 S3 11.83 0.00 -1.62 -0.17 0.15 268.78 8.04 -2.30
0.18 A 0.83 1.00 0.67 0.33 0.50 0.17 0.00 0.17
0.16 H 1.51 0.00 0.14 0.01 -0.01 0.10 0.00 0.19
1000.00 A4 2.26 0.00 0.48 3.30 4.23 5.17 1.74 -1.48
1000.00 A5 7.83 0.00 -1.33 1.33 1.50 -0.83 0.00 0.17
--- ZJ 10095.42 0.18 -852.34 4632.11 5728.40 4336.51 1737.35 -1314.40
--- CJ-ZJ 0.00 852.56 -4632.01 -5728.28 -4336.41 -1737.26 1314.80

TABLE 5
CJ --- --- 0.18 0.22 0.10 0.12 0.10 0.09 0.40
S VAR SVAL A B C D E F G
0.50 I 1.41 0.00 0.26 0.27 0.00 -0.06 0.14 0.39
0.00 S2 12.85 0.00 1.23 0.34 0.00 2.09 2.17 -0.05
0.00 S3 11.75 0.00 -1.64 -0.29 0.00 268.59 7.98 -2.25
0.18 A 0.57 1.00 0.61 -0.06 0.00 -0.44 -0.21 0.34
0.16 H 1.52 0.00 0.14 0.02 0.00 0.12 0.00 0.19
0.12 D 0.53 0.00 0.11 0.78 1.00 1.22 0.41 -0.35
1000.00 A5 7.03 0.00 -1.50 0.16 0.00 -2.67 -0.62 0.69
--- ZJ 7032.36 0.18 -1503.59 163.38 0.12 -2667.28 -616.22 692.41
--- CJ-ZJ 0.00 1503.81 -163.28 0.00 2667.38 616.31 -692.01
TABLE 6
CJ --- --- 0.18 0.22 0.10 0.12 0.10 0.09 0.40
S VAR SVAL A B C D E F G
0.50 I 1.05 0.00 0.18 -0.25 -0.67 -0.88 -0.14 0.63
0.00 S2 12.87 0.00 1.23 0.38 0.04 2.14 2.19 -0.06
0.00 S3 13.54 0.00 -1.26 2.31 3.34 272.68 9.35 -3.42
0.18 A 0.56 1.00 0.61 -0.07 -0.01 -0.46 -0.21 0.35
0.16 H 1.37 0.00 0.10 -0.19 -0.28 -0.22 -0.11 0.28
0.07 J 1.64 0.00 0.35 2.40 3.07 3.76 1.26 -1.08
1000.00 A5 4.28 0.00 -2.09 -3.86 -5.15 -8.97 -2.73 2.50
--- ZJ 4275.97 0.18 -2089.64 -3857.94 -5154.64 -8969.84 -2734.16 2498.31
--- CJ-ZJ 0.00 2089.86 3858.04 5154.76 8969.94 2734.25 -2497.91

TABLE 7
CJ --- --- 0.18 0.22 0.10 0.12 0.10 0.09 0.40
S VAR SVAL A B C D E F G
0.50 I 0.03 -1.81 -0.92 -0.13 -0.65 -0.05 0.24 0.00
0.00 S2 12.97 0.18 1.34 0.36 0.04 2.05 2.15 0.00
0.00 S3 19.06 9.87 4.75 1.66 3.22 268.15 7.28 0.00
0.40 G 1.62 2.89 1.76 -0.19 -0.03 -1.33 -0.61 1.00
0.16 H 0.91 -0.82 -0.40 -0.14 -0.27 0.15 0.06 0.00
0.07 J 3.38 3.11 2.24 2.19 3.03 2.33 0.61 0.00
1000.00 A5 0.23 -7.22 -6.48 -3.38 -5.07 -5.65 -1.22 0.00
--- ZJ 235.85 -7219.27 -6482.46 -3381.49 -5068.86 -5654.70 -1215.79 0.40
--- CJ-ZJ 7219.45 6482.68 3381.59 5068.98 5654.80 1215.88 0.00

TABLE 8
CJ --- --- 0.18 0.22 0.10 0.12 0.10 0.09 0.40
S VAR SVAL A B C D E F G
0.00 S4 0.08 -4.44 -2.25 -0.33 -1.59 -0.11 0.59 0.00
0.00 S2 12.99 -0.78 0.86 0.29 -0.30 2.03 2.28 0.00
0.00 S3 19.21 1.39 0.45 1.03 0.18 267.93 8.40 0.00
0.40 G 1.66 0.32 0.46 -0.38 -0.96 -1.39 -0.27 1.00
0.16 H 0.90 -0.12 -0.04 -0.09 -0.01 0.17 -0.03 0.00
0.07 J 3.42 1.24 1.29 2.05 2.36 2.28 0.86 0.00
1000.00 A5 0.09 0.96 -2.34 -2.78 -2.13 -5.45 -2.30 0.00
--- ZJ 90.94 955.25 -2336.81 -2775.30 -2135.05 -5449.81 -2303.42 0.40
--- CJ-ZJ -955.07 2337.03 2775.40 2135.17 5449.91 2303.51 0.00

TABLE 9
CJ --- --- 0.18 0.22 0.10 0.12 0.10 0.09 0.40
S VAR SVAL A B C D E F G
0.00 S4 0.50 0.00 -13.11 -13.22 -11.51 -25.43 -10.11 0.00
0.00 S2 13.06 0.00 -1.04 -1.96 -2.04 -2.40 0.41 0.00
0.00 S3 19.08 0.00 3.86 5.08 3.29 275.88 11.76 0.00
0.40 G 1.63 0.00 1.25 0.56 -0.24 0.45 0.51 1.00
0.16 H 0.91 0.00 -0.32 -0.42 -0.27 -0.49 -0.31 0.00
0.07 J 3.30 0.00 4.33 5.66 5.14 9.36 3.85 0.00
0.18 A 0.09 1.00 -2.45 -2.91 -2.24 -5.71 -2.41 0.00
--- ZJ 1.05 0.18 0.31 0.03 -0.18 -0.27 -0.01 0.40
--- CJ-ZJ 0.00 -0.09 0.07 0.30 0.37 0.10 0.00

TABLE 10
CJ --- --- 0.18 0.22 0.10 0.12 0.10 0.09 0.40
S VAR SVAL A B C D E F G
0.00 S4 9.35 0.00 -1.48 1.97 2.29 -0.29 0.23 0.00
0.00 S2 14.56 0.00 0.93 0.61 0.30 1.86 2.16 0.00
0.00 S3 13.69 0.00 -3.22 -4.16 -5.10 0.00 5.48 0.00
0.40 G 0.23 0.00 -0.60 -1.86 -2.43 -3.54 -1.13 1.00
0.16 H 1.36 0.00 0.27 0.35 0.42 0.78 0.21 0.00
0.50 I 0.48 0.00 0.63 0.82 0.75 1.36 0.56 0.00
0.18 A 2.35 1.00 0.52 0.97 1.29 0.71 0.23 0.00
--- ZJ 0.97 0.18 0.21 -0.10 -0.30 -0.48 -0.10 0.40
--- CJ-ZJ 0.00 0.01 0.20 0.42 0.58 0.19 0.00

TABLE 11
CJ --- --- 0.18 0.22 0.10 0.12 0.10 0.09 0.40
S VAR SVAL A B C D E F G
0.00 S4 9.40 0.00 -1.60 1.60 1.80 -1.00 0.00 0.20
0.00 S2 14.39 0.00 1.38 2.02 2.14 4.55 3.01 -0.76
0.00 S3 13.18 0.00 -1.85 0.06 0.41 8.05 8.04 -2.27
0.00 S6 0.35 0.00 -0.92 -2.85 -3.73 -5.45 -1.74 1.54
0.16 H 1.40 0.00 0.15 0.00 -0.03 0.11 0.00 0.19
0.50 I 0.53 0.00 0.49 0.39 0.18 0.53 0.29 0.23
0.18 A 2.40 1.00 0.40 0.60 0.80 0.00 0.00 0.20
--- ZJ 0.92 0.18 0.34 0.30 0.23 0.28 0.15 0.18
--- CJ-ZJ 0.00 -0.12 -0.20 -0.11 -0.18 -0.06 0.22

TABLE 12
CJ --- --- 0.18 0.22 0.10 0.12 0.10 0.09 0.40
S VAR SVAL A B C D E F G
0.00 S4 7.20 0.00 -3.62 0.00 1.06 -3.19 -1.22 -0.77
0.00 S2 11.62 0.00 -1.16 0.00 1.21 1.78 1.48 -1.98
0.00 S3 13.10 0.00 -1.92 0.00 0.38 7.97 8.00 -2.30
0.00 S6 4.27 0.00 2.67 0.00 -2.41 -1.54 0.43 3.26
0.16 H 1.41 0.00 0.16 0.00 -0.03 0.12 0.00 0.19
0.10 C 1.37 0.00 1.26 1.00 0.46 1.37 0.76 0.60
0.18 A 1.58 1.00 -0.36 0.00 0.52 -0.82 -0.46 -0.16
--- ZJ 0.65 0.18 0.09 0.10 0.14 0.01 -0.01 0.06
--- CJ-ZJ 0.00 0.13 0.00 -0.02 0.09 0.10 0.34

TABLE 13
CJ --- --- 0.18 0.22 0.10 0.12 0.10 0.09 0.40
S VAR SVAL A B C D E F G
0.00 S4 0.02 -4.56 -1.99 0.00 -1.32 0.55 0.86 -0.03
0.00 S2 10.01 -1.02 -0.80 0.00 0.67 2.62 1.94 -1.81
0.00 S3 15.10 1.27 -2.37 0.00 1.05 6.93 7.42 -2.51
0.00 S6 5.22 0.60 2.46 0.00 -2.10 -2.03 0.16 3.16
0.16 H 1.24 -0.11 0.20 0.00 -0.09 0.21 0.05 0.21
0.10 C 1.02 -0.22 1.34 1.00 0.34 1.55 0.86 0.64
0.07 J 2.98 1.89 -0.67 0.00 0.99 -1.55 -0.86 -0.31
--- ZJ 0.51 0.09 0.12 0.10 0.09 0.08 0.03 0.08
--- CJ-ZJ 0.09 0.10 0.00 0.03 0.02 0.06 0.32

C = 1.-2 cups of oatmeal


H = 1.24 cups of milk
J = 2.98 slices of wheat toast
Minimum cost = 0.51
G + 0.16H + 0.50I + 0.07J
120I + 65J ≥ 420

G + 0.16H + 0.50I + 0.07J + 0S1 + 0S2 + 0S3 + 0S4 + 0S5 + 0S6 + 0S7 + 1000A1 + 1000A2 + 1000A3 + 1000A4 + 1000A5
- 1S1 + 1A1 = 420

0.16 0.50 0.07 0.00 0.00 0.00 0.00 0.00 0.00 0.00
H I J S1 S2 S3 S4 S5 S6 S7
100.00 120.00 65.00 -1.00 0.00 0.00 0.00 0.00 0.00 0.00
4.00 0.00 1.00 0.00 1.00 0.00 0.00 0.00 0.00 0.00
12.00 0.00 0.00 0.00 0.00 1.00 0.00 0.00 0.00 0.00
0.00 0.00 1.00 0.00 0.00 0.00 -1.00 0.00 0.00 0.00
250.00 3.00 26.00 0.00 0.00 0.00 0.00 -1.00 0.00 0.00
9.00 1.00 3.00 0.00 0.00 0.00 0.00 0.00 -1.00 0.00
0.00 0.00 3.00 0.00 0.00 0.00 0.00 0.00 0.00 -1.00
### ### 98000.00 -1000.00 0.00 0.00 -1000.00 -1000.00 -1000.00 -1000.00
### ### -97999.93 1000.00 0.00 0.00 1000.00 1000.00 1000.00 1000.00

0.16 0.50 0.07 0.00 0.00 0.00 0.00 0.00 0.00 0.00
H I J S1 S2 S3 S4 S5 S6 S7
0.00 118.80 54.60 -1.00 0.00 0.00 0.00 0.40 0.00 0.00
0.00 -0.05 0.58 0.00 1.00 0.00 0.00 0.02 0.00 0.00
0.00 -0.14 -1.25 0.00 0.00 1.00 0.00 0.05 0.00 0.00
0.00 0.00 1.00 0.00 0.00 0.00 -1.00 0.05 0.00 0.00
1.00 0.01 0.10 0.00 0.00 0.00 0.00 0.00 0.00 0.00
0.00 0.89 2.06 0.00 0.00 0.00 0.00 0.04 -1.00 0.00
0.00 0.00 3.00 0.00 0.00 0.00 0.00 0.00 0.00 -1.00
0.16 ### 60664.02 -1000.00 0.00 0.00 -1000.00 484.00 -1000.00 -1000.00
0.00 ### -60663.95 1000.00 0.00 0.00 1000.00 -484.00 1000.00 1000.00

0.16 0.50 0.07 0.00 0.00 0.00 0.00 0.00 0.00 0.00
H I J S1 S2 S3 S4 S5 S6 S7
0.00 1.00 0.46 -0.01 0.00 0.00 0.00 0.00 0.00 0.00
0.00 0.00 0.61 0.00 1.00 0.00 0.00 0.02 0.00 0.00
0.00 0.00 -1.18 0.00 0.00 1.00 0.00 0.05 0.00 0.00
0.00 0.00 1.00 0.00 0.00 0.00 -1.00 0.05 0.00 0.00
1.00 0.00 0.10 0.00 0.00 0.00 0.00 0.00 0.00 0.00
0.00 0.00 1.65 0.01 0.00 0.00 0.00 0.04 -1.00 0.89
0.00 0.00 3.00 0.00 0.00 0.00 0.00 0.00 0.00 -1.00
0.16 0.50 5654.29 7.50 0.00 0.00 -1000.00 84.00 -1000.00 -108.00
0.00 0.00 -5654.22 -7.50 0.00 0.00 1000.00 -84.00 1000.00 108.00

0.16 0.50 0.07 0.00 0.00 0.00 0.00 0.00 0.00 0.00
H I J S1 S2 S3 S4 S5 S6 S7
0.00 1.00 0.34 -0.01 0.00 0.00 0.12 -0.01 0.00 0.00
0.00 0.00 0.65 0.00 1.00 0.00 -0.05 0.02 0.00 0.00
0.00 0.00 -1.04 0.00 0.00 1.00 -0.14 0.05 0.00 0.00
0.00 0.00 0.17 0.00 0.00 0.00 -0.17 0.01 0.00 0.00
1.00 0.00 0.09 0.00 0.00 0.00 0.01 0.00 0.00 0.00
0.00 0.00 1.38 0.01 0.00 0.00 0.28 0.02 -1.00 0.89
0.00 0.00 2.17 0.00 0.00 0.00 0.83 -0.04 0.00 -1.00
0.16 0.50 3543.54 7.50 0.00 0.00 1110.75 -17.32 -1000.00 -108.00
0.00 0.00 -3543.47 -7.50 0.00 0.00 -1110.75 17.32 1000.00 108.00

0.16 0.50 0.07 0.00 0.00 0.00 0.00 0.00 0.00 0.00
H I J S1 S2 S3 S4 S5 S6 S7
0.00 1.00 0.22 -0.01 0.00 0.00 0.09 -0.01 0.09 -0.08
0.00 0.00 -0.01 0.00 1.00 0.00 -0.18 0.01 0.48 -0.43
0.00 0.00 -1.09 0.00 0.00 1.00 -0.15 0.05 0.04 -0.03
0.00 0.00 0.00 0.00 0.00 0.00 -0.20 0.01 0.12 -0.11
1.00 0.00 0.09 0.00 0.00 0.00 0.01 0.00 0.00 0.00
0.00 0.00 0.33 0.00 0.00 0.00 0.07 0.01 -0.24 0.21
0.00 0.00 1.68 0.00 0.00 0.00 0.73 -0.05 0.35 -1.32
0.16 0.50 1678.44 -2.67 0.00 0.00 734.94 -48.05 354.81 -1316.49
0.00 0.00 -1678.37 2.67 0.00 0.00 -734.94 48.05 -354.81 1316.49
0.16 0.50 0.07 0.00 0.00 0.00 0.00 0.00 0.00 0.00
H I J S1 S2 S3 S4 S5 S6 S7
0.00 1.00 0.00 -0.01 0.00 0.00 0.05 -0.01 0.25 -0.22
0.00 0.00 0.00 0.00 1.00 0.00 -0.18 0.01 0.47 -0.42
0.00 0.00 0.00 0.01 0.00 1.00 0.07 0.07 -0.75 0.67
0.00 0.00 0.00 0.00 0.00 0.00 -0.20 0.01 0.12 -0.11
1.00 0.00 0.00 0.00 0.00 0.00 -0.01 -0.01 0.06 -0.06
0.00 0.00 1.00 0.01 0.00 0.00 0.20 0.02 -0.73 0.65
0.00 0.00 0.00 -0.01 0.00 0.00 0.40 -0.08 1.57 -2.40
0.16 0.50 0.07 -11.82 0.00 0.00 396.77 -75.71 1573.97 -2403.98
0.00 0.00 0.00 11.82 0.00 0.00 -396.77 75.71 -1573.97 2403.98

0.16 0.50 0.07 0.00 0.00 0.00 0.00 0.00 0.00 0.00
H I J S1 S2 S3 S4 S5 S6 S7
0.00 1.00 0.00 -0.01 0.00 0.00 0.41 -0.02 0.03 -0.03
0.00 0.00 0.00 0.00 1.00 0.00 -0.22 0.01 0.50 -0.44
0.00 0.00 0.00 0.00 0.00 1.00 -1.91 0.12 0.44 -0.39
0.00 0.00 0.00 0.00 0.00 0.00 -0.58 0.02 0.35 -0.31
1.00 0.00 0.00 0.00 0.00 0.00 0.16 -0.01 -0.04 0.03
0.00 0.00 1.00 0.00 0.00 0.00 -0.42 0.03 -0.35 0.31
0.00 0.00 0.00 -0.01 0.00 0.00 1.84 -0.11 0.70 -1.62
0.16 0.50 0.07 -5.26 0.00 0.00 1842.45 -113.64 699.94 -1624.34
0.00 0.00 0.00 5.26 0.00 0.00 -1842.45 113.64 -699.94 1624.34

0.16 0.50 0.07 0.00 0.00 0.00 0.00 0.00 0.00 0.00
H I J S1 S2 S3 S4 S5 S6 S7
0.00 2.45 0.00 -0.02 0.00 0.00 1.00 -0.05 0.07 -0.07
0.00 0.53 0.00 -0.01 1.00 0.00 0.00 0.00 0.51 -0.46
0.00 4.67 0.00 -0.04 0.00 1.00 0.00 0.03 0.58 -0.52
0.00 1.42 0.00 -0.01 0.00 0.00 0.00 -0.01 0.39 -0.35
1.00 -0.39 0.00 0.00 0.00 0.00 0.00 0.00 -0.05 0.04
0.00 1.03 1.00 -0.01 0.00 0.00 0.00 0.01 -0.32 0.28
0.00 -4.51 0.00 0.03 0.00 0.00 0.00 -0.02 0.56 -1.50
0.16 -4505.04 0.07 33.70 0.00 0.00 0.00 -20.23 561.92 -1501.24
0.00 4505.54 0.00 -33.70 0.00 0.00 0.00 20.23 -561.92 1501.24

0.16 0.50 0.07 0.00 0.00 0.00 0.00 0.00 0.00 0.00
H I J S1 S2 S3 S4 S5 S6 S7
0.00 -18.49 0.00 0.14 0.00 0.00 1.00 -0.14 2.68 -7.04
0.00 -3.13 0.00 0.02 1.00 0.00 0.00 -0.02 0.97 -1.68
0.00 11.25 0.00 -0.09 0.00 1.00 0.00 0.06 -0.24 1.67
0.00 2.94 0.00 -0.03 0.00 0.00 0.00 -0.01 0.20 0.16
1.00 -0.94 0.00 0.01 0.00 0.00 0.00 0.00 0.02 -0.14
0.00 6.88 1.00 -0.05 0.00 0.00 0.00 0.04 -1.05 2.23
0.00 -4.72 0.00 0.04 0.00 0.00 0.00 -0.02 0.59 -1.57
0.16 0.66 0.07 -0.01 0.00 0.00 0.00 0.00 0.12 -0.09
0.00 -0.16 0.00 0.01 0.00 0.00 0.00 0.00 -0.12 0.09

0.16 0.50 0.07 0.00 0.00 0.00 0.00 0.00 0.00 0.00
H I J S1 S2 S3 S4 S5 S6 S7
0.00 0.00 2.69 0.00 0.00 0.00 1.00 -0.04 -0.13 -1.04
0.00 0.00 0.46 0.00 1.00 0.00 0.00 0.00 0.49 -0.66
0.00 0.00 -1.63 -0.01 0.00 1.00 0.00 -0.01 1.48 -1.98
0.00 0.00 -0.43 0.00 0.00 0.00 0.00 -0.02 0.65 -0.80
1.00 0.00 0.14 0.00 0.00 0.00 0.00 0.00 -0.12 0.17
0.00 1.00 0.15 -0.01 0.00 0.00 0.00 0.01 -0.15 0.32
0.00 0.00 0.69 0.00 0.00 0.00 0.00 0.00 -0.13 -0.04
0.16 0.50 0.05 -0.01 0.00 0.00 0.00 -0.01 0.14 -0.14
0.00 0.00 0.02 0.01 0.00 0.00 0.00 0.01 -0.14 0.14

0.16 0.50 0.07 0.00 0.00 0.00 0.00 0.00 0.00 0.00
H I J S1 S2 S3 S4 S5 S6 S7
0.00 0.00 2.60 0.00 0.00 0.00 1.00 -0.05 0.00 -1.20
0.00 0.00 0.78 0.00 1.00 0.00 0.00 0.02 0.00 -0.06
0.00 0.00 -0.67 0.00 0.00 1.00 0.00 0.05 0.00 0.05
0.00 0.00 -0.66 -0.01 0.00 0.00 0.00 -0.04 1.00 -1.22
1.00 0.00 0.06 0.00 0.00 0.00 0.00 0.00 0.00 0.01
0.00 1.00 0.05 -0.01 0.00 0.00 0.00 0.00 0.00 0.14
0.00 0.00 0.60 0.00 0.00 0.00 0.00 0.00 0.00 -0.20
0.16 0.50 0.14 0.00 0.00 0.00 0.00 0.00 0.00 0.04
0.00 0.00 -0.07 0.00 0.00 0.00 0.00 0.00 0.00 -0.04

0.16 0.50 0.07 0.00 0.00 0.00 0.00 0.00 0.00 0.00
H I J S1 S2 S3 S4 S5 S6 S7
0.00 -4.13 2.41 0.03 0.00 0.00 1.00 -0.05 0.00 -1.77
0.00 -5.21 0.54 0.04 1.00 0.00 0.00 0.02 0.00 -0.78
0.00 -0.14 -0.67 0.00 0.00 1.00 0.00 0.05 0.00 0.03
0.00 7.37 -0.32 -0.07 0.00 0.00 0.00 -0.04 1.00 -0.21
1.00 0.01 0.06 0.00 0.00 0.00 0.00 0.00 0.00 0.02
0.00 2.58 0.12 -0.02 0.00 0.00 0.00 0.00 0.00 0.36
0.00 -1.55 0.53 0.01 0.00 0.00 0.00 0.00 0.00 -0.41
0.16 -0.02 0.12 0.00 0.00 0.00 0.00 0.00 0.00 -0.04
0.00 0.52 -0.05 0.00 0.00 0.00 0.00 0.00 0.00 0.04

0.16 0.50 0.07 0.00 0.00 0.00 0.00 0.00 0.00 0.00
H I J S1 S2 S3 S4 S5 S6 S7
0.00 2.93 0.00 -0.02 0.00 0.00 1.00 -0.05 0.00 0.11
0.00 -3.63 0.00 0.03 1.00 0.00 0.00 0.02 0.00 -0.35
0.00 -2.11 0.00 0.02 0.00 1.00 0.00 0.05 0.00 -0.50
0.00 6.43 0.00 -0.06 0.00 0.00 0.00 -0.04 1.00 -0.46
1.00 0.18 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.06
0.00 2.93 0.00 -0.02 0.00 0.00 0.00 0.00 0.00 0.45
0.00 -2.93 1.00 0.02 0.00 0.00 0.00 0.00 0.00 -0.78
0.16 0.12 0.07 0.00 0.00 0.00 0.00 0.00 0.00 0.00
0.00 0.38 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
1000.00 1000.00 1000.00 1000.00 1000.00
A1 A2 A3 A4 A5
1.00 0.00 0.00 0.00 0.00
0.00 0.00 0.00 0.00 0.00
0.00 0.00 0.00 0.00 0.00
0.00 1.00 0.00 0.00 0.00
0.00 0.00 1.00 0.00 0.00
0.00 0.00 0.00 1.00 0.00
0.00 0.00 0.00 0.00 1.00
1000.00 1000.00 1000.00 1000.00 1000.00
0.00 0.00 0.00 0.00 0.00

1000.00 1000.00 1000.00 1000.00 1000.00


A1 A2 A3 A4 A5
1.00 0.00 -0.40 0.00 0.00
0.00 0.00 -0.02 0.00 0.00
0.00 0.00 -0.05 -12.00 0.00
0.00 1.00 0.00 0.00 0.00
0.00 0.00 0.00 0.00 0.00
0.00 0.00 -0.04 1.00 0.00
0.00 0.00 0.00 0.00 1.00
1000.00 1000.00 -436.00 1000.00 1000.00
0.00 0.00 1436.00 0.00 0.00

1000.00 1000.00 1000.00 1000.00 1000.00


A1 A2 A3 A4 A5
0.01 0.00 0.00 0.00 0.00
0.00 0.00 -0.02 0.00 0.00
0.00 0.00 -0.05 -12.00 0.00
0.00 1.00 0.00 0.00 0.00
0.00 0.00 0.00 0.00 0.00
0.00 0.00 -0.04 1.00 0.00
0.00 0.00 0.00 0.00 1.00
0.00 1000.00 -36.00 1000.00 0.00
-1000.00 0.00 1036.00 0.00 1000.00

1000.00 1000.00 1000.00 1000.00 1000.00


A1 A2 A3 A4 A5
0.01 -0.12 0.00 0.00 0.00
0.00 0.05 -0.02 0.00 0.00
0.00 0.14 -0.05 -12.00 0.00
0.00 0.17 0.00 0.00 0.00
0.00 -0.01 0.00 0.00 0.00
0.00 -0.28 -0.04 1.00 0.00
0.00 -0.83 0.00 0.00 1.00
0.00 -1110.75 -36.00 1000.00 1000.00
1000.00 2110.75 1036.00 0.00 0.00

1000.00 1000.00 1000.00 1000.00 1000.00


A1 A2 A3 A4 A5
0.01 -0.09 0.00 -0.09 0.00
0.00 0.18 0.00 -0.48 0.00
0.00 0.15 -0.05 -12.04 0.00
0.00 0.20 0.00 -0.12 0.00
0.00 -0.01 0.00 0.00 0.00
0.00 -0.07 -0.01 0.24 0.00
0.00 -0.73 0.01 -0.35 1.00
0.00 -734.94 12.77 -354.81 1000.00
1000.00 1734.94 987.23 1354.81 0.00
1000.00 1000.00 1000.00 1000.00 1000.00
A1 A2 A3 A4 A5
0.01 -0.05 0.01 -0.25 0.00
0.00 0.18 0.00 -0.47 0.00
0.00 -0.07 -0.08 -11.25 0.00
0.00 0.20 0.00 -0.12 0.00
0.00 0.01 0.00 -0.06 0.00
0.00 -0.20 -0.03 0.73 0.00
0.00 -0.40 0.06 -1.57 1.00
0.00 -396.77 56.66 -1573.97 1000.00
1000.00 1396.77 943.34 2573.97 0.00

1000.00 1000.00 1000.00 1000.00 1000.00


A1 A2 A3 A4 A5
0.01 -0.41 0.00 -0.03 0.00
0.00 0.22 0.00 -0.50 0.00
0.00 1.91 -0.03 -12.44 0.00
0.00 0.58 0.01 -0.35 0.00
0.00 -0.16 0.00 0.04 0.00
0.00 0.42 -0.01 0.35 0.00
0.00 -1.84 0.03 -0.70 1.00
0.00 -1842.45 25.20 -699.94 1000.00
1000.00 2842.45 974.80 1699.94 0.00

1000.00 1000.00 1000.00 1000.00 1000.00


A1 A2 A3 A4 A5
0.02 -1.00 0.00 -0.07 0.00
0.00 0.00 0.00 -0.51 0.00
0.04 0.00 -0.03 -12.58 0.00
0.01 0.00 0.01 -0.39 0.00
0.00 0.00 0.00 0.05 0.00
0.01 0.00 -0.01 0.32 0.00
-0.04 0.00 0.02 -0.56 1.00
-37.92 0.00 20.23 -561.92 1000.00
1037.92 1000.00 979.77 1561.92 0.00

1000.00 1000.00 1000.00 1000.00 1000.00


A1 A2 A3 A4 A5
-0.16 -1.00 0.10 -2.68 4.65
-0.03 0.00 0.02 -0.97 0.81
0.09 0.00 -0.06 -11.76 -1.46
0.02 0.00 0.01 -0.20 -0.34
-0.01 0.00 0.00 -0.02 0.12
0.06 0.00 -0.04 1.05 -1.30
-0.04 0.00 0.02 -0.59 1.05
0.01 0.00 0.00 -0.12 -0.02
999.99 1000.00 1000.00 1000.12 1000.02

1000.00 1000.00 1000.00 1000.00 1000.00


A1 A2 A3 A4 A5
0.00 -1.00 0.00 0.13 1.16
0.00 0.00 0.00 -0.49 0.22
0.00 0.00 0.01 -13.48 0.66
0.00 0.00 0.02 -0.65 0.22
0.00 0.00 0.00 0.12 -0.06
0.01 0.00 -0.01 0.15 -0.19
0.00 0.00 0.00 0.13 0.16
0.00 0.00 0.01 -0.14 0.01
1000.00 1000.00 999.99 1000.14 999.99

1000.00 1000.00 1000.00 1000.00 1000.00


A1 A2 A3 A4 A5
0.00 -1.00 0.00 0.00 1.20
0.00 0.00 -0.02 0.00 0.06
0.00 0.00 -0.05 -12.00 0.17
0.00 0.00 0.04 -1.00 0.33
0.00 0.00 0.00 0.00 -0.01
0.01 0.00 0.00 0.00 -0.14
0.00 0.00 0.00 0.00 0.20
0.00 0.00 0.00 0.00 -0.04
1000.00 1000.00 1000.00 1000.00 1000.04

1000.00 1000.00 1000.00 1000.00 1000.00


A1 A2 A3 A4 A5
-0.03 -1.00 0.00 0.00 1.77
-0.04 -0.04 -0.02 0.00 0.78
0.00 0.00 -0.05 -12.00 0.19
0.06 0.00 0.04 -1.00 -0.68
0.00 0.00 0.00 0.00 -0.02
0.02 0.00 0.00 0.00 -0.36
-0.01 0.00 0.00 0.00 0.41
0.00 0.00 0.00 0.00 0.04
1000.00 1000.00 1000.00 1000.00 999.96

1000.00 1000.00 1000.00 1000.00 1000.00


A1 A2 A3 A4 A5
0.02 -1.00 0.00 0.00 -0.11
-0.03 -0.04 -0.02 0.00 0.35
-0.02 0.00 -0.05 -12.00 0.72
0.05 0.00 0.04 -1.00 -0.43
0.00 0.00 0.00 0.00 -0.06
0.02 0.00 0.00 0.00 -0.45
-0.02 0.00 0.00 0.00 0.78
0.00 0.00 0.00 0.00 0.00
1000.00 1000.00 1000.00 1000.00 1000.00
Original objective function: Maximize: 9X + 7Y
Original Explicit constraints: 12X + 4Y ≤ 60
4X + 8Y ≤ 40
Original Implicit Constraint: X,Y ≥ 0 ???

MODIFICATIONS
Objective Function Maximize: 9x + 7y + 0S1 + 0S2
Explicit constraints 12x + 4Y+ 1S1 = 60
4X + 8Y + 1S2= 40
Implicit constraints X, Y ≥ 0

TABLE 1 (Initial Feasible Solution)


CJ _ _ 9.00 7.00 0.00 0.00
S VAR S VAL X Y S1 S2
0.00 S1 60.00 12.00 4.00 1.00 0.00 5
0.00 S2 40.00 4.00 8.00 0.00 1.00 10
ZJ 0.00 0.00 0.00 0.00 0.00
CJ-ZJ 9.00 7.00 0.00 0.00

CJ _ _ 9.00 7.00 0.00 0.00


S VAR S VAL X Y S1 S2
9.00 X 5.00 1.00 0.33 0.08 0.00 15
0.00 S2 20.00 0.00 6.67 -0.33 1.00 3
ZJ 45.00 9.00 3.00 0.75 0.00
CJ-ZJ 0.00 4.00 -0.75 0.00

CJ _ _ 9.00 7.00 0.00 0.00


S VAR S VAL X Y S1 S2
9.00 X 4.00 1.00 0.00 0.10 -0.05
7.00 Y 3.00 0.00 1.00 -0.05 0.15
ZJ 57.00 9.00 7.00 0.55 0.60
CJ-ZJ 0.00 0.00 -0.55 -0.60

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