Question

The corner points of the bounded feasible region of an LPP are O(0, 0), A(250, 0), B(200, 50) and C(0, 175). If the maximum value of the objective function $Z = 2ax + by$ occurs at the points A(250, 0) and B(200, 50), then the relation between a and b is:

• A. $2a = b$
• B. $2a = 3b$
• C. $a = b$
• D. $a = 2b$

Hint:

For LPP maximum, equate $Z$ at both corner points where maximum occurs.

Answer 1

The Correct Option is B

The maximum of the objective function occurs at both points A and B. So, \[ Z_A = Z_B. \] At A(250, 0): \[ Z_A = 2a(250) + b(0) = 500a. \] At B(200, 50): \[ Z_B = 2a(200) + b(50) = 400a + 50b. \] Since $Z_A = Z_B$ for maximum: \[ 500a = 400a + 50b \implies 100a = 50b \implies 2a = b. \] Oops! Wait — but the question says $Z = 2ax + by$, so the coefficient of $x$ is $2a$, so the calculation is correct. So, \[ 500a = 400a + 50b \implies 100a = 50b \implies 2a = b. \] So the answer is (A) not (B).
Correct option is (A).