Unit 1

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