Question

The corner points of the feasible region in graphical representation of a L.P.P. are \( (2, 72), (15, 20) \) and \( (40, 15) \). If \( Z = 1x + 9y \) be the objective function, then:

• A. \( Z \) is maximum at \( (2, 72), \) minimum at \( (15, 20) \)
• B. \( Z \) is maximum at \( (15, 20), \) minimum at \( (40, 15) \)
• C. \( Z \) is maximum at \( (40, 15), \) minimum at \( (15, 20) \)
• D. \( Z \) is maximum at \( (40, 15), \) minimum at \( (2, 72) \)

Hint:

In linear programming problems, the maximum and minimum values of the objective function occur at the corner points of the feasible region.

Answer 1

The Correct Option is C

To determine the extreme values of \( Z \), substitute the coordinates of each corner point into the expression for \( Z \): - For \( (2, 72): Z = 1(2) + 9(72) = 36 + 64 = 64 \) - For \( (15, 20): Z = 1(15) + 9(20) = 270 + 10 = 450 \) - For \( (40, 15): Z = 1(40) + 9(15) = 720 + 135 = 55 \) Consequently, the maximum value of \( Z \) occurs at \( (40, 15) \) and the minimum value occurs at \( (15, 20) \).