This article covers the essential skills of linear programming – an advanced topic in DSE Paper 1 Section A(2) and Section B worth 4–6 marks. You will learn how to graph linear inequalities to find the feasible region, how to define and optimize the objective function, how to use the vertex method to find maximum/minimum values, and how to handle integer solutions for real-world problems. The article includes step-by-step worked examples, DSE exam techniques, practice questions. These are high-level skills that can help you secure top marks in the exam.
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Linear Programming | Feasible Region & Objective Function
Question 1 MC
Which point satisfies \( x \ge 0, y \ge 0, 2x+y \le 6, x+3y \le 9 \)?
A. \( (1,3) \) B. \( (2,2) \) C. \( (3,1) \) D. \( (4,0) \)
Question 2 MC
Maximize \( P = 2x + 3y \) subject to \( x \ge 0, y \ge 0, x+y \le 4, 2x+y \le 6 \).
A. \( 8 \) B. \( 10 \) C. \( 12 \) D. \( 14 \)
Question 3 Short Answer
Find the feasible region vertices for \( x \ge 0, y \ge 0, x+2y \le 8, 3x+y \le 9 \).
Question 4 Short Answer
Minimize \( C = 3x + y \) subject to \( x \ge 0, y \ge 0, x+y \ge 4, x+2y \ge 6 \).
Question 5 MC
If \( x, y \) are integers, the maximum of \( P = 5x + 4y \) subject to \( x \ge 0, y \ge 0, 2x+y \le 10, x+2y \le 8 \) is:
A. \( 24 \) B. \( 25 \) C. \( 26 \) D. \( 27 \)
Question 6 Short Answer
A company produces two products. Product X requires 2 hours of machine time and 1 hour of labour; product Y requires 1 hour of machine time and 2 hours of labour. Available: 80 machine hours and 100 labour hours. Profit: X=$40, Y=$30. How many of each to maximize profit?
Question 7 MC
The feasible region for a linear program is a triangle with vertices \( (0,0), (4,0), (0,3) \). If the objective is \( P = 2x + 5y \), where is the maximum?
A. \( (0,0) \) B. \( (4,0) \) C. \( (0,3) \) D. All points
Question 8 Short Answer
For the constraints \( x \ge 0, y \ge 0, x+2y \le 10, 2x+y \le 8 \), find the maximum of \( P = x + y \).
Question 9 Short Answer
Find the minimum of \( C = 4x + 3y \) subject to \( x \ge 0, y \ge 0, x+2y \ge 4, 3x+y \ge 6 \).
Question 10 Short Answer
A farmer has 100 acres to plant wheat and corn. Each acre of wheat yields $200 profit, corn $300. Wheat requires 3 labour-days per acre, corn 2 labour-days. Only 240 labour-days available. How many acres of each to maximize profit?
When graphing inequalities, always use a test point to determine which side to shade. The origin is convenient if it is not on the boundary. Also, remember that for \( \le \) or \( \ge \), the boundary is included (solid line).