Question 1 Use the simplex method to solve the following simplex problems. a) Maximise 2 x1 + 2 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.