Linear Programming?

Question

Consider the following Linear Programming problem developed at Jeff Spencer’s San Antonio optical scanning firm:
Maximize profit = $1x 1 + $1 x 2
Subject to: 2 x 1 + 1 x2 ≤ 100
1 x1+ 2 x2≤ 100
a) What is the optimal solution to this problem? Solve it graphically.
b) If a technical breakthrough occurred that raised the profit per unit of , would this affect the optimal solution?
c) Instead of an increase in the profit coefficient that profit was overestimated and should only have been $1.25. Does this change the optimal solution?

Answer 1

Hi,
There are seems to be some inconsistencies in your problem, but I’ll mention them as I go along.

Let’s do the graph. If I interpret your problem correctly you have this:
Maximize:
P= x1 +x2
Subject to:
2x1 +x2 ≤ 100 (Ineq #1)
x1+2x2 ≤ 100 (Ineq #2)

Write the inequalities as equalities and find two points for each resulting equation.
Eq #1
X1….|….X2
0….| 100
50...| 0

Eq #2

X1….|….X2
0….|…50
100.|….0

Now, graph that the two equations using those points, and you’ll find that the intersection is the point (33.3, 33.3).
Now, make a table of the profit function at various corner points: (The dots are just for spacing, ignore them.)

Corner……….|….P=X1 +X2
(0, 0)………...|……0
(50, 0)……….|……50
(0, 50)……….|……50
(33.33)(33.33)|…….66.67

b)If a technical breakthrough occurred that raised the profit per unit of , would this affect the optimal solution?
The question seems to be incomplete, (of which one or both?) but, of course, if the profit for only one of the products were to be raised to above $2, for example, the maximum profit would shift from the intersection to one of the other points.

c) Instead of an increase in the profit coefficient that profit was overestimated and should only have been $1.25. Does this change the optimal solution?
This question seems contradictory since both coefficients are one, and, as you state the problem, 1.25 would be an increase rather than a decrease. We would need to know what the coefficients originally were to decide this. But generally, if the profit of both are increased the same amount, the intersection remains the most profitable. If only one coefficient in this problem decreases, the intersection point becomes more profitable. If only one point is increased, the maximizing solution will shift to one of the other points if the ratio of the coefficients its greater than 2. That is assuming you have given the coefficients correctly.

FE