Section A

1. (a) The linear programming problem

maximise z = 5x1 + 8x2

subject to: x1 + x2 ≤ 6

5x1 + 9x2 ≤ 45

x1, x2 ≥ 0

has the following solution (in the final Simplex tableau):

i. Perform sensitivity analysis on the right-hand side value of the first constraint of the LP problem.

ii. Find a solution to the following linear programming problem, without resolving it from scratch: 

maximise z = 5x1 + 8x2 + 2x3

subject to: x1 + x2 + x3 ≤ 6

5x1 + 9x2 + 2x3 ≤ 45

x1, x2, x3 ≥ 0.

Provide all necessary details and explanation.

(b) Find the optimal value of the objective function of the following LP problem by directly solving its dual problem:

maximise z = 3x1 − 14x2 − 2x3

subject to: −3x1 + 2x2 − 2x3 ≥ −20

x1, x2, x3 ≥ 0

Provide all the details.