Question 1 Use the simplex method to solve the following simplex problems. a) Maximise 2 x1 + 2 x2 + 3 x3 subject to x1 + 3 x2 + 2 x3 6, 3 x1 + x2 +...

Question 1 Use the simplex method to solve the following simplex problems. a) Maximise 2 x1 + 2 x2 + 3 x3 subject to x1 + 3 x2 + 2 x3 ≤ 6, 3 x1 + x2 + 3 x3 ≤ 10 x1 + x2 + x3 ≤ 4, x1, x2, x3 ≥ 0. b) Maximise −3 x1 − x2 − 5 x3 subject to −x1 + 4 x2 + 2 x3 ≤ 0, −2 x1 + x2 − 3 x3 ≤ −2 7 x2 + x3 ≤ 1, x1, x2, x3 ≥ 0.

b) Maximise 4 x1 + 3 x2 + 5 x3 + x4 subject to x1 + x2 + x3 + x4 ≤ 6, 3 x1 + x3 = 6 4 x2 − x3 + 2 x4 = 1, x1 unrestricted, x2, x3, x4 ≥ 0. (15 marks)

Question 2 Consider the primal linear programming problem below.

Maximise 3 x1 + 4 x2 + 6 x3, subject to 2 x1 + 3 x2 + 5 x3 ≤ 15, 3 x1 + 2 x2 + 5 x3 ≤ 12, x1, x2, x3 ≥ 0. a) Construct the dual problem and solve it graphically. b) Find a solution to the primal problem, stating clearly any results that you use.