Question

In an LPP, corner points of the feasible region determined by the system of linear constraints are \((1,1), (3,0), (0,3)\). If \(Z = ax + by\), where \(a>0, b>0\) is to be minimized, the condition on \(a\) and \(b\) so that the minimum of \(Z\) occurs at \((3, 0)\) and \((1, 1)\) will be:

• A. \(a = 2b\)
• B. \(a = \dfrac{b}{2}\)
• C. \(a = 3b\)
• D. \(a = b\)

Hint:

In linear programming problems, if the minimum or maximum occurs at more than one point, set the values of the objective function equal at those points to find the required condition.

Answer 1

The Correct Option is B

We are given the objective function: \[ Z = ax + by \] and it is to be minimized over the feasible region formed by the points: \[ A(1,1), \quad B(3,0), \quad C(0,3) \] Let us evaluate \(Z = ax + by\) at each of the corner points: At point \(A(1,1)\): \[ Z_1 = a(1) + b(1) = a + b \] At point \(B(3,0)\): \[ Z_2 = a(3) + b(0) = 3a \] At point \(C(0,3)\): \[ Z_3 = a(0) + b(3) = 3b \] We are told that the minimum of \(Z\) occurs at both \((3,0)\) and \((1,1)\). So we must have: \[ Z_1 = Z_2 \Rightarrow a + b = 3a \Rightarrow b = 2a \Rightarrow a = \frac{b}{2} \] \[ \boxed{a = \frac{b}{2}} \]