Graphical Method and Corner Point Evaluation

Convert inequalities to boundary equations. Draw each line using intercepts or two points. Shade the side satisfying each inequality. Identify the common feasible region. List all corner points. Substitute each point in Z = ax + by. Choose the largest value for maximization or smallest value for minimization, then write the point and value together.

Draw x-axis and y-axis, plot each boundary line accurately, shade the common feasible region, label all vertices, and mark the direction of the feasible side for each constraint if needed.

Evaluating Z at intercepts of all lines instead of only the actual vertices of the feasible region leads to invalid answers.

The graphical method solves a two-variable LPP by drawing the constraint lines, shading the common feasible region, finding its corner points, and evaluating the objective function at those points.

The graphical method solves a two-variable LPP by drawing the constraint lines, shading the common feasible region, finding its corner points, and evaluating the objective function at those points. Corner point principle: For a bounded feasible region, maximum or minimum of Z = ax + by occurs at one or more vertices of the feasible region. Boundary line method: replace each inequality by equality to draw its line, then choose the valid half-plane. For constraints x + y <= 6, x <= 4, y <= 5, x >= 0, y >= 0, the feasible region lies in the first quadrant below or on each boundary line. The objective value is tested at all corner points of the region.

Each linear inequality represents a half-plane. The feasible region is the common part satisfying all constraints. If the feasible region is bounded, the maximum or minimum of a linear objective function occurs at a corner point. For an unbounded region, corner point values alone may not be enough to decide the answer unless the direction of improvement is checked. Corner point principle: For a bounded feasible region, maximum or minimum of Z = ax + by occurs at one or more vertices of the feasible region. Boundary line method: replace each inequality by equality to draw its line, then choose the valid half-plane. Maximize Z = 3x + 2y subject to x + y <= 4, x <= 3, y <= 2, x >= 0, y >= 0. Corner points are (0,0), (3,0), (3,1), (2,2), and (0,2). Evaluate: Z(0,0)=0, Z(3,0)=9, Z(3,1)=11, Z(2,2)=10, Z(0,2)=4. The maximum value is 11 at (3,1). Often asked as a complete long-answer problem requiring graph, feasible region, corner point table, optimum value, and final interpretation. Evaluating Z at intercepts of all lines instead of only the actual vertices of the feasible region leads to invalid answers.