Question

Solve the following linear programming problem graphically: Maximise \( Z = x + 2y \) Subject to the constraints: \[ x - y \geq 0 \] \[ x - 2y \geq -2 \] \[ x \geq 0, \, y \geq 0 \]

Hint:

To solve linear programming problems graphically, plot the constraints to form the feasible region and evaluate the objective function at each vertex of this region.

Answer 1

The process involves graphing the constraints to identify the feasible region. The constraints are: - \( x \geq y \) derived from \( x - y \geq 0 \). - \( x \geq 2y - 2 \) derived from \( x - 2y \geq -2 \). - \( x \geq 0 \) and \( y \geq 0 \), confining the region to the first quadrant. Following the plotting of these constraints, the feasible region is established. The objective function \( Z = x + 2y \) is then evaluated at the vertices of this region. The vertex points identified are (0, 0), (4, 2), and (5, 3). The evaluation of \( Z \) at these points yields: - At (0, 0), \( Z = 0 + 2(0) = 0 \). - At (4, 2), \( Z = 4 + 2(2) = 8 \). - At (5, 3), \( Z = 5 + 2(3) = 11 \). Consequently, the maximum value of \( Z \) is 8 at the point (4, 2).