Linear Programming | Class 12 Mathematics Notes

Examples Illustrating Graphical Solution of Linear Programming Problems

This section presents several detailed examples demonstrating the graphical method and Corner Point Method for solving linear programming problems. Example 1 involves maximizing Z = 4x + y subject to constraints x + y ≤ 50, 3x + y ≤ 90, x ≥ 0, y ≥ 0. The feasible region is bounded with vertices O(0,0), A(30,0), B(20,30), and C(0,50). Evaluating Z at these points shows maximum Z = 120 at (30,0). Example 2 focuses on minimizing Z = 200x + 500y with constraints x + 2y ≥ 10, 3x + 4y ≤ 24, x,y ≥ 0. The feasible region is bounded with vertices A(0,5), B(4,3), C(0,6). Minimum Z = 2300 at (4,3). Example 3 involves minimizing and maximizing Z = 3x + 9y subject to x + 3y ≤ 60, x + y ≥ 10, x ≤ y, x,y ≥ 0. The feasible region is bounded with vertices A(0,10), B(5,5), C(15,15), D(0,20). Minimum Z = 60 at (5,5), maximum Z = 180 at both (15,15) and (0,20), illustrating multiple optimal solutions where every point on line segment joining these vertices also yields the same maximum. Example 4 deals with minimizing Z = -50x + 20y subject to inequalities 2x - y ≥ -5, 3x + y ≥ 3, 2x - 3y ≤ 12, x,y ≥ 0. The feasible region is unbounded. Evaluations at vertices show smallest Z = -300 at (6,0), but further analysis shows no minimum value exists due to unbounded region. Example 5 shows a problem with no feasible region, hence no solution. These examples reinforce the graphical method, Corner Point Method, and the importance of checking feasibility and boundedness.

📊 Diagram: See figure_3: Fig 12.2; See figure_4: Fig 12.3; See figure_5: Fig 12.4; See figure_6: Fig 12.5; See figure_7: Fig 12.6; See table_2: Table on page 7 (5×2); See table_3: Table on page 9 (5×2)

🧪 Activity: Graph the constraints and feasible regions for each example to visualize solutions.

🔗 Connection: Leads to Exercise 12.1 for practice of graphical solutions of linear programming problems.

Table on page 7 (5×2)

Table on page 9 (5×2)