Question

In a linear programming problem, the feasible region representing the maximization of the objective function \( Z = px + qy \), where \( p, q > 0 \), is a shaded polygonal region. If all points on the line segment joining the corner points \( A(0, 3) \) and \( B(3, 2) \) give the maximum value of \( Z \), then which of the following is true?

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

Hint:

When an objective function \(Z = px + qy\) achieves its maximum along an entire edge segment \(AB\), the slope of the objective function line must equal the slope of that boundary line segment \(AB\). \[ \text{Slope of } Z \text{ line} = -\frac{p}{q}, \quad \text{Slope of } AB = \frac{2 - 3}{3 - 0} = -\frac{1}{3} \implies -\frac{p}{q} = -\frac{1}{3} \implies q = 3p \]

Answer 1

The Correct Option is C

Concept: According to the fundamental theorem of Linear Programming Problems (LPP), if the maximum or minimum value of an objective function \( Z = px + qy \) occurs at more than one corner point of the feasible region, then it must occur at every single point lying along the line segment connecting those two corner points. Therefore, if all points on the segment \(AB\) yield the maximum value of \(Z\), the value of \(Z\) at point \(A\) must be perfectly equal to the value of \(Z\) at point \(B\): \[ Z_A = Z_B \]

Step 1: Computing the value of \(Z\) at corner point \(A(0, 3)\).
Substitute \(x = 0\) and \(y = 3\) into the given objective function equation \( Z = px + qy \): \[ Z_A = p(0) + q(3) = 3q \]

Step 2: Computing the value of \(Z\) at corner point \(B(3, 2)\).
Substitute \(x = 3\) and \(y = 2\) into the objective function equation \( Z = px + qy \): \[ Z_B = p(3) + q(2) = 3p + 2q \]

Step 3: Equating the two values to establish the relationship.
Since all points on segment \(AB\) give the maximum value, we set the two expressions equal to each other: \[ Z_A = Z_B \implies 3q = 3p + 2q \]

Step 4: Simplifying the equation.
Subtract \(2q\) from both sides of the equation to isolate the variables: \[ 3q - 2q = 3p \implies q = 3p \] Hence, the correct relation is \(q = 3p\), corresponding to option (C).