Linear Programming

Linear Programming is a method for making the best decision, such as maximising profit or minimising cost, when resources like money, labour, machine time or raw materials are limited. In this chapter you will learn to formulate a linear programming problem from a word problem: choosing the decision variables, writing the linear objective function Z = px + qy, and expressing the restrictions as linear inequalities called constraints, together with the non-negativity conditions x >= 0 and y >= 0. Typical examples include manufacturing, diet and allocation problems. You will solve these problems graphically. Each constraint is drawn as a line, the common region satisfying all constraints is the feasible region, and every point in it is a feasible solution. The corner point method states that the optimal value, if it exists, occurs at a corner of the feasible region. You will learn how to deal with bounded and unbounded feasible regions, how to check whether a minimum or maximum exists in an unbounded region, and what happens when the feasible region is empty.

Detailed Worksheet: Linear Programming Section A - Definitions (10 marks) 1. Define objective function and constraints in a linear programming problem. (2 marks) 2. What is a feasible region? What is an infeasible solution? (2 marks) 3. State the corner point theorem for a bounded feasible region. (2 marks) 4. Distinguish between a bounded and an unbounded feasible region. (2 marks) 5. What is an optimal solution? Can an LPP have more than one optimal solution? (2 marks) Section B - Calculations and Applications (15 marks) 6. Solve graphically: maximise Z = 3x + 4y subject to x + y <= 4, x >= 0, y >= 0. (3 marks) 7. Solve graphically: minimise Z = -3x + 4y subject to x + 2y <= 8, 3x + 2y <= 12, x >= 0, y >= 0. (3 marks) 8. Solve graphically: maximise Z = 5x + 3y subject to 3x + 5y <= 15, 5x + 2y <= 10, x >= 0, y >= 0. (3 marks) 9. A dealer wishes to buy table fans at Rs 360 each and sewing machines at Rs 240 each. He has Rs 5,760 and space for at most 20 items. His profit is Rs 22 per fan and Rs 18 per sewing machine. Formulate the LPP and find the number of each he should buy to maximise profit. (3 marks) 10. Minimise Z = 3x + 5y subject to x + 3y >= 3, x + y >= 2, x >= 0, y >= 0. Show that the minimum exists even though the region is unbounded. (3 marks) Section C - Diagrams (10 marks) 11. Draw the feasible region for x + 2y <= 10, 3x + y <= 15, x >= 0, y >= 0. Mark all corner points with coordinates and shade the region. (4 marks) 12. Draw the unbounded feasible region for x + 3y >= 3, x + y >= 2, x >= 0, y >= 0, mark its corner points, and draw the line 3x + 5y = 7 to show that the open half-plane 3x + 5y < 7 has no point in common with the region. (3 marks) 13. Draw a flowchart showing the steps of the corner point method: formulate, draw constraints, identify the feasible region, find corners, evaluate Z, and check unboundedness. (3 marks) Section D - Analysis and Higher-order Thinking (15 marks) 14. A diet must contain at least 8 units of vitamin A and 11 units of minerals. Food F1 contains 3 units of vitamin A and 5 units of minerals per kg and costs Rs 60 per kg; food F2 contains 4 units of vitamin A and 2 units of minerals per kg and costs Rs 80 per kg. Formulate and solve the LPP to minimise cost. Comment on the nature of the optimal solution. (5 marks) 15. Maximise Z = -x + 2y subject to x >= 3, x + y >= 5, x + 2y >= 6, y >= 0. Explain why Z has no maximum value even though its value at a corner point is largest. (5 marks) 16. A factory makes nuts and bolts. A packet of nuts needs 1 hour on machine A and 3 hours on machine B; a packet of bolts needs 3 hours on machine A and 1 hour on machine B. Each machine runs at most 12 hours a day. The profit is Rs 17.50 per packet of nuts and Rs 7 per packet of bolts. Formulate and solve the LPP to maximise profit. (5 marks) Instructions: Time allowed 2 hours. Attempt all sections. Draw graphs to scale on graph paper, shade the feasible region and tabulate values of Z at the corners in words.