Question

Optimize $Z = 3x + 9y$ subject to the constraints: \[ x + 3y \leq 60, \quad x + y \geq 10, \quad x \leq y, \quad x \geq 0, \quad y \geq 0. \]

• A. Maximum value of Z occurs at the point (15, 15) only.
• B. Maximum value of Z occurs at the point (0, 20) only.
• C. Maximum value of Z occurs exactly at two points (15, 15) and (0, 20).
• D. Maximum value of Z occurs at all the points on the line segment joining (15, 15) and(0, 20).

Answer 1

The Correct Option is D

To find the maximum value of \( Z = 3x + 9y \), we graph the constraints: \( x + 3y \leq 60 \) represents a half-plane below the line \( x + 3y = 60 \). \( x + y \geq 10 \) represents a half-plane above the line \( x + y = 10 \). \( x \leq y \) represents the region below the line \( x = y \). \( x \geq 0 \) and \( y \geq 0 \) restrict the solution to the first quadrant.

The feasible region is a polygon bounded by these lines. Evaluating \( Z \) at the corner points of this polygon: At \( (15, 15) \),

\[ Z = 3(15) + 9(15) = 180. \]

At \( (0, 20) \),

\[ Z = 3(0) + 9(20) = 180. \]

Since \( Z \) is linear and the line segment joining \( (15, 15) \) and \( (0, 20) \) lies within the feasible region, the maximum value of \( Z \) occurs at all points on this line segment.