Question

2. Solve the following LP problem graphically; confirm your results using Solver in MS Excel.

Maximize profit = 20x1 + 10x2

Subject to: 5x1 + 4x2 ≤ 250

2x1 + 5x2 ≤ 150

x1, x2 ≥ 0

Answer 1

Here the 1st constraint is 5x1+4x2<=250

Take x2 = 0 and so 5x1 = 250 or x1 = 50.

Now take x1 = 0 and so 4x2 = 250 or x2 = 62.5

Thus for the 1st constraint the line will have points - 50 and 62.5 for x1 and x2

The 2nd constraint is 2x1+5x2<=150

Take x2 = 0 and so 2x1 = 150 and x1 = 75

Now take x1 = 0 and so 5x2 = 150 and x2 = 30.

Thus for the 2nd constraint the line will have points - 75 and 30 for x1 and x2.

We plot both lines on a graph and then determine the feasible region. The feasible region is shaded and the corner points are - A, B, C and D

A = (0,0), B = (0,30), C = (40,15) and D = (50,0)

At A value of objective function = 20*0+10*0 = 0. At B it is 20*0+10*30 = 300. At C it is 20*40 + 10*15 = 950 and at D it is 20*50+10*0 = 1000

Thus profit (or the objective function) is maximized at x1 = 50 and x2 = 0 and the value of the maximized profit (objective function) is 1000.

The graph is attached below: