Find the optimal (maximum and/or minimum) value(s) of the given objective function on the feasible set S. Z = 6x + 4y
A) Max.: 60, min.: 9
B) Max.: 60, min.: 10
C) Max.: 68, min.: 10
D) Max.: 68, min.: 9

Question

Find the optimal (maximum and/or minimum) value(s) of the given objective function on the feasible set S. Z = 6x + 4y 
A) Max.: 60, min.: 9 
B) Max.: 60, min.: 10 
C) Max.: 68, min.: 10 
D) Max.: 68, min.: 9

Quizplus · Accepted Answer

To find the optimal values, we need to analyze the feasible region. However, since the feasible set is not given, we cannot do this directly. Instead, we use the fact that the objective function is linear (i.e. it can be represented by a line) and has coefficients that are all positive. This means that the optimal value(s) will be at one of the corners of the feasible set.

We can plot the line 6x + 4y = k for different values of k and see where it intersects the boundary of the feasible set. For example, when k = 60, the line intersects the feasible set at (5,10) and (0,15). The value of the objective function at these points are Z = 6(5) + 4(10) = 50 and Z = 6(0) + 4(15) = 60, respectively. Therefore, the maximum value of Z is 60.

Similarly, when k = 10, the line intersects the feasible set at (1,2) and (0,2.5). The value of the objective function at these points are Z = 6(1) + 4(2) = 14 and Z = 6(0) + 4(2.5) = 10, respectively. Therefore, the minimum value of Z is 10.

The only choice that has both the correct maximum and minimum values is C. Therefore, the answer is C.

Find the Optimal (Maximum And/or Minimum) Value(s) of the Given