Question

In the Linear Programming Problem for objective function \( Z = 18x + 10y \) subject to constraints \[ 4x + y \geq 20 \] \[ 2x + 3y \geq 30 \] \[ x, y \geq 0 \] Find the minimum value of \( Z \).

Hint:

Quick Tip: In linear programming problems, solving the system of equations obtained by converting inequalities into equalities helps find the corner points. The objective function value at these points will give the minimum or maximum value depending on the problem.

Answer 1

The objective function is \( Z = 18x + 10y \), subject to the constraints: \[ 4x + y \geq 20 \quad \text{(1)} \] \[ 2x + 3y \geq 30 \quad \text{(2)} \] \[ x, y \geq 0 \quad \text{(3)} \] To find corner points, convert inequalities to equalities:
- Equation (1): \( 4x + y = 20 \)
- Equation (2): \( 2x + 3y = 30 \)
Solve the system:
Multiply equation (1) by 3: \[ 12x + 3y = 60 \] Subtract equation (2): \[ (12x + 3y) - (2x + 3y) = 60 - 30 \] \[ 10x = 30 \quad \Rightarrow \quad x = 3 \] Substitute \( x = 3 \) into equation (1): \[ 4(3) + y = 20 \quad \Rightarrow \quad 12 + y = 20 \quad \Rightarrow \quad y = 8 \] The corner point is \( (3, 8) \). Evaluate \( Z = 18x + 10y \) at \( (3, 8) \): \[ Z = 18(3) + 10(8) = 54 + 80 = 134 \] The minimum value of \( Z \) is \( \boxed{134} \).