MA 3231

Linear Programming

Section E162

Assignment 4

Content: up to Section 4

1. Solve the following linear program using

a) the perturbation (lexicographic) method

b) Bland’s rule

max z = −

3

4

x1 + 150×2 −

1

50×3 + 6×4

subject to

1

4

x1 + 150×2 −

1

25×3 + 9×4 ≤ 0

1

2

x1 − 90×2 −

1

50×3 + 3×4 ≤ 0

x3 ≤ 1

x1, x2, x3, x4 ≥ 0.

2. Consider the following linear programming problem:

max z = x1 + 2×2

subject to

−2×1 − x2 + x3 ≤ 1

x1 + x2 ≤ 2

x1 + x3 ≤ 3

x1, x2, x3 ≥ 0

a) Solve the linear program.

b) Find the dual program.

c) Solve the dual program.

d) Compare the solutions of primal and dual program.

2

3. Consider the following linear programming problem:

max z = x1 + 2×2 + x3 + x4

subject to

2×1 + x2 + 5×3 + x4 ≤ 8

2×1 + 2×2 + 4×4 ≤ 12

3×1 + x2 + 2×3 ≤ 18

x1, x2, x3, x4 ≥ 0

You know that the final dictionary for this program is given by

z = 12.4 − 1.2×1 − 0.2×5 − 0.9×6 − 2.8×4

x2 = 6 − x1 − 0.5×6 − 2×4

x3 = 0.4 − 0.2×1 − 0.2×5 + 0.1×6 + 0.2×4

x7 = 11.2 − 1.6×1 + 0.4×5 + 0.3×6 + 1.6×4

(where x5, x6, x7 are slack variables)

a) What will be the optimal solution to the problem if the objective function is

changed to

3×1 + 2×2 + x3 + x4

b) What will be the optimal solution to the problem if the objective function is

changed to

x1 + 2×2 + 0.5×3 + x4

c) For each of the three objective functions above, find the range of values for which

the final dictionary will remain optimal.

4. Use the parametric self-dual simplex method to solve the following problem

max z = 3×1 − x2

subject to

x1 − x2 ≤ 1

−x1 + x2 ≤ −4

x1, x2 ≥ 0

12 points per problems

MA 3231

# Linear Programming Section E162 Assignment 4 SOLVED

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