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Understanding Linear Programming: Maximizing Profits with Constraints

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This guide explores the fundamentals of linear programming, including definitions of key terms such as objective function and feasible region. It details how to find maximum and minimum values for profit equations while adhering to constraints in real-world scenarios, such as production limits and available working hours. Learn to apply these concepts through examples like maximizing profits from H-Dub T-shirts and Hats production, ensuring efficient use of resources and adherence to time restrictions.

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Understanding Linear Programming: Maximizing Profits with Constraints

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  1. Unit 1 Linear programming

  2. Linear programming • Define: LINEAR PROGRAMMING – is a method for finding a minimum or maximum value of some quantity, given a set of constraints (Limits). • Define: OBJECTIVE FUNCTION – the quantity you are trying to maximize or minimize • Define: FEASIBLE REGION – The area created by the system of inequalities (constraints)

  3. Find the Maximum for P = 5x -y Substitute each of the coordinates (A-D)into the equation above to see which gives the maximum value. A: P = 5(0) – 0 P = 0 B: P = 5(6) – 0 P = 30 C: P = 5(2) – 4 P = 6 D: P = 5(0) – 4 P = – 4

  4. Find the coordinates and then substitute each (A-F)into the equation above to see which gives the maximum and minimum values. • Find the Maximum & Minimum for P = - .04x + 3.2y A: P = –0.04(0) + 3.2(2) P = 6.4 C (1, 6) B: P = –0.04(0) + 3.2(5) P = 16 Maximum B (0, 5) C: P = –0.04(1) + 3.2(6) P = 19.16 D: P = –0.04(5) + 3.2(2) P = 6.2 D (5, 2) Minimum A (0, 2) E: P = –0.04(5) + 3.2(0) P = – 0.2 E (5, 0) F: P = –0.04(5) + 3.2(0) P = – 0.16 F (4, 0)

  5. Solving linear programming (2, 2) (4, 1) (2, 1) Minimum for: C = 3x + 4y (2, 1) (4, 1) (2, 2)

  6. Profit: P = 6x + 20y • (0, 0) • P = 6(0) + 20(0) • P = 0 • (50, 0) • P = 6(50) + 20(0) • P = 300 • (15, 35) • P = 6(15) + 20(35) • P = 790 • (0, 4) • P = 6(0) + 20(4) • P = 80 • You are making H-Dub T-shirts & Hats to sell for homecoming and under the following constraints. • You have at most 20 hours to work • You want to have at most 50 items to sell H-DUB T-SHIRT Takes 30 minutes to make Supplies cost $20 Profit $20 • Constraints: Time = 10x + 30y ≤ 1200 Amount= x + y < 50 Real life= x ≥ 0 & y ≥ 0 150 – 100 – 50– H-DUB HAT Takes 10 minutes to make Supplies cost $4 Profit $6 • Objective function: Profit: P = 6x + 20y (15, 35) (0, 40) (50, 0) | | | 50 100 150 (0, 0)

  7. CLASSWORK • P. 160

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