Question

Minimize $Z = -50x + 20y$ subject to the constraints: \[ 2x - y \geq -5, \quad 3x + y \geq 3, \quad 2x - 3y \leq 12, \quad x \geq 0, \quad y \geq 0. \] Then which of the following is/are true:
(A) Feasible region is unbounded.
(B) $Z$ has no minimum value.
(C) The minimum value of $Z$ is 100.
(D) The minimum value of $Z$ is -300.

• A. (A), (C) and (D) only
• B. (C) and (D) only
• C. (A) and (C) only
• D. (A) and (D) only

Answer 1

The Correct Option is A

The objective is to minimize \( Z = -50x + 20y \) subject to the following constraints:

• \( 2x - y \geq -5 \)
• \( 3x + y \geq 3 \)
• \( 2x - 3y \leq 12 \)
• \( x \geq 0 \)
• \( y \geq 0 \)

The process involves defining the feasible region and then determining the minimum value of \( Z \).

Step 1: Define the Feasible Region

Graph each constraint to identify the feasible region.

• Constraint 1: \( y \leq 2x + 5 \)
• Constraint 2: \( y \geq -3x + 3 \)
• Constraint 3: \( y \geq \frac{2}{3}x - 4 \)
• Non-negativity Constraints: \( x \geq 0 \), \( y \geq 0 \)

The feasible region is the area where all these constraints intersect.

Step 2: Assess Region Boundedness

The graph indicates that the feasible region extends infinitely in the positive \( y \)-direction and to the right, meaning it is unbounded.

Step 3: Calculate Vertex Points

The vertices are found by solving the systems of equations formed by the constraint boundaries:

Intersection of Lines	Coordinates of Vertex
\(2x-y=-5\) and \(3x+y=3\)	\((1,0)\)
\(3x+y=3\) and \(2x-3y=12\)	\((3, -6)\)
\(2x-y=-5\) and \(y=\frac{2}{3}x-4\)	\((0,5)\)

Step 4: Evaluate \( Z \) at Vertex Points

• For vertex \((1,0)\): \( Z = -50 \times 1 + 20 \times 0 = -50 \)
• For vertex \((0,5)\): \( Z = -50 \times 0 + 20 \times 5 = 100 \)
• Vertex \((3,-6)\) is invalid as it violates the \( y \geq 0 \) constraint.

Step 5: Interpretation of Findings

(A) The feasible region is unbounded.

(B) Due to the unbounded nature of the feasible region, \( Z = -50x + 20y \) can approach negative infinity as \( x \) increases indefinitely. Therefore, no minimum value exists.

(C) The value of \( Z \) at vertex \((0,5)\) is 100. This is not the minimum value.

(D) A value of -300 can be attained if \( x \) and \( y \) tend towards infinity in a manner that further decreases \( Z \).

The conclusions that are correct are (A), (C), and (D).