MATH2003J, OPTIMIZATION IN ECONOMICS,
BDIC 2023/2024, SPRING
Problem Sheet 4
Question 1:
In each of the following, transform. the constraint set of the linear programming problem into the standard form, incorporating all supplemental variables.
(a).
3x1 + x2 ≤ 25
x1 + 5x2 ≤ 12
x1, x2 ≥ 0.
(b).
x1 ≤ 2
2x2 ≤ 13
x1 + x2 ≥ 3
x1, x2 ≥ 0.
(c).
x1 + x2 ≤ 5
x1 − x2 ≥ −2
3x2 ≤ 8
x1, x2 ≥ 0.
(d).
3x1 + 5x2 + 8x3 ≤ 100
x1 − 2x2 + 5x3 ≥ −5
−2x1 + x2 − 6x3 ≤ −2
x1, x2, x3 ≥ 0.
(e).
−3x1 + 3x2 − x3 ≤ 80
x1 − x2 + 4x3 ≥ 30
x1 + x2 − x3 ≤ 20
x1 ≤ 0, x2, x3 ≥ 0.
(f).
−x1 + 5x2 − x3 ≤ 50
x1 − x2 + 6x3 ≥ 22
x1 − x2 + x3 ≤ 18
x1, x3 ≥ 0.
For (a) and (c), find the maximum of f(x1, x2) = x1 + 2x2 both with the Simplex and the Graphical Methods.