Question

The corner points of the feasible region determined by the system of linear constraints are (0,3), (1,1) and (3,0). If the objective function is \( Z = px + qy \), \( p, q > 0 \), then the condition on \( p \) and \( q \) so that the minimum of \( Z \) occurs at (3,0) and (1,1) is:

• A. \( p = 3q \)
• B. \( 3p = q \)
• C. \( p = \frac{q}{2} \)
• D. \( p = 2q \)

Hint:

When solving linear programming problems, always check for the necessary conditions at the corner points using the constraints and objective function.

Answer 1

The Correct Option is C

Step 1: Write the objective function for the given points.
At the given corner points, we can write the value of the objective function as:
\[ Z_1 = 3q, \quad Z_2 = p + q, \quad Z_3 = 3p \]

Step 2: Find the conditions for minimum.
For the objective function to be minimized at \( (3, 0) \) and \( (1, 1) \), we require: \[ Z_3 = 3p \leq Z_1 = 3q, \quad Z_2 = p + q \geq Z_1 = 3q \quad \text{and} \quad Z_2 = p + q \geq Z_3 = 3p \]

Step 3: Solve the inequalities.
Solving these inequalities:
\[ p \leq q \quad \text{and} \quad p \geq 2q \]

Step 4: Conclusion.
Thus, \( p = \frac{q}{2} \). The correct answer is option (C).