SOLUTION: Minimize the function f(x, y) = 0.5x + 0.4y in the region determined by the following constraints. 5x + y ≥ 12 5x + 4y ≥ 30 x ≥ 0 y ≥ 0

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Question 1114281: Minimize the function
f(x, y) = 0.5x + 0.4y
in the region determined by the following constraints.
5x + y ≥ 12
5x + 4y ≥ 30
x ≥ 0
y ≥ 0
Answer by greenestamps(13200)   (Show Source): You can put this solution on YOUR website!


The two oblique constraint lines intersect at (1.2,6). The corners of the feasibility region are (0,12), (1.2,6), and (6,0).

When you evaluate the objective function at those corners, it has the same minimum value of 3 at both (1.2,6) and (6,0). That means the objective function has the minimum value 3 at any point on the segment between (1.2,6) and (6,0).

Answer: The minimum value of the objective function is 3.

Note the usual method taught for maximizing or minimizing an objective function is to evaluate it at all the corners of the feasibility region.

In fact, that is not necessary. You can determine which corner of the feasibility region will give the maximum or minimum value of the objective function by comparing the slope of the objective function with the slopes of the constraint lines.

In this example, the slope of the objective function is the same as the slope of one of the constraint lines; that immediately tells us that the minimum value of the objective function will be obtained at any point on that constraint line that is part of the feasibility region.
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