Consider a linear programming problem with the following constraints and objective function, f(x,y).
y≥5
y≤x
y≤
–
1
3
x+8
f(x,y)=–6x–11y
The constraints are graphed below.
(5,5)
(6,6)
(9,5)
What are the values of x and y that minimize the objective function for the feasible region?
To find the values of x and y that minimize the objective function for the feasible region, we need to find the point in the feasible region where the objective function is at its lowest value.
From the graph, we can see that the feasible region is the region below the lines: y ≥ 5, y ≤ x, and y ≤ (1/3)x + 8.
To minimize the objective function -6x - 11y, we need to find the point that gives the lowest value when substituted into the objective function.
From the graph, it seems that the point (6, 6) lies within the feasible region and is the closest to the origin. Let's test this point:
f(6, 6) = -6(6) - 11(6) = -36 - 66 = -102
Checking the other points:
f(5, 5) = -6(5) - 11(5) = -30 - 55 = -85
f(9, 5) = -6(9) - 11(5) = -54 - 55 = -109
Therefore, the values of x and y that minimize the objective function for the feasible region are x = 6 and y = 6.