代写MATH2003J, OPTIMIZATION IN ECONOMICS, BDIC 2023/2024, SPRING Problem Sheet 4代做Java语言

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.