1. THE PROBLEM – Method 1 Linear Programming : Introductory Example Let x represent number of litres of energy drink Let y represent number of litres of refresher drink Maximise x + 0.8y Subject to: x + y  1000 2x + y  1500 3x + 2y  2400

2. First constraint: x + y  1000 GRAPHING

3. First constraint x + y  1000 Second constraint: 2x + y  1500 GRAPHING

4. First constraint: x + y  1000 Second constraint: 2x + y  1500 Third constraint: 3x + 2y  2400 GRAPHING

5. First constraint: x + y  1000 Second constraint: 2x + y  1500 Third constraint: 3x + 2y  2400 Adding: x  0 and y  0 GRAPHING

6. Feasible region is the unshaded area and satisfies: x + y  1000 2x + y  1500 3x + 2y  2400 x  0 and y  0 GRAPHING

7. Evaluate the objective function x + 0.8y at vertices of the feasible region: O: 0 + 0 = 0 A: 0 + 0.8 x 1000 = 800 B: 400 + 0.8 x 600 = 880 C: 600 + 0.8 x 300 = 840 D: 750 + 0 = 750 O A B C D Maximum income = £880 at (400, 600) SOLUTION

8. THE PROBLEM – Method 2 Linear Programming : Introductory Example Let x represent number of litres of energy drink Let y represent number of litres of refresher drink Maximise x + 0.8y Subject to: x + y  1000 2x + y  1500 3x + 2y  2400

9. All three constraints: First: x + y  1000 SOLUTION

10. All three constraints: First: x + y  1000 Second: 2x + y  1500 SOLUTION

11. All three constraints: First: x + y  1000 Second: 2x + y  1500 Third: 3x + 2y  2400 SOLUTION

12. All three constraints: First: x + y  1000 Second: 2x + y  1500 Third: 3x + 2y  2400 Adding: x  0 and y  0 SOLUTION

13. Feasible region is the unshaded area and satisfies: x + y  1000 2x + y  1500 3x + 2y  2400 x  0 and y  0 SOLUTION

14. Draw a straight line x + 0.8y = k. O A B C D Maximum income = £880 at (400, 600) SOLUTION Move a ruler parallel to this line until it reaches the edge of the feasible region. The furthest point you can move it to is point B. At B (400, 600) the value of the objective function is 880.

15. P86 – Ex7A

16. A factory produces two types of drink, an ‘energy’ drink and a ‘refresher’ drink. The day’s output is to be planned. Each drink requires syrup, vitamin supplement and concentrated flavouring, as shown in the table. The last row in the table shows how much of each ingredient is available for the day’s production. How can the factory manager decide how much of each drink to make? THE PROBLEM - Blending Linear Programming : Introductory Example

17. Syrup Vitamin supplement Concentrated flavouring 5 litres of energy drink 1.25 litres2 units30 cc 5 litres of refresher drink 1.25 litres1 unit20 cc Availabilities250 litres300 units4.8 litres Energy drink sells at £1 per litre Refresher drink sells at 80 p per litre THE PROBLEM

18. Syrup constraint: Let x represent number of litres of energy drink Let y represent number of litres of refresher drink 0.25x + 0.25y  250  x + y  1000 FORMULATION

19. Vitamin supplement constraint: Let x represent number of litres of energy drink Let y represent number of litres of refresher drink 0.4x + 0.2y  300  2x + y  1500 FORMULATION

20. Concentrated flavouring constraint: Let x represent number of litres of energy drink Let y represent number of litres of refresher drink 6x + 4y  4800  3x + 2y  2400 FORMULATION

21. Objective function: Let x represent number of litres of energy drink Energy drink sells for £1 per litre Let y represent number of litres of refresher drink Refresher drink sells for 80 pence per litre Maximise x + 0.8y FORMULATION

22. First constraint: x + y  1000 GRAPHING

23. First constraint x + y  1000 Second constraint: 2x + y  1500 GRAPHING

24. First constraint: x + y  1000 Second constraint: 2x + y  1500 Third constraint: 3x + 2y  2400 GRAPHING

25. First constraint: x + y  1000 Second constraint: 2x + y  1500 Third constraint: 3x + 2y  2400 Adding: x  0 and y  0 GRAPHING

26. Feasible region is the unshaded area and satisfies: x + y  1000 2x + y  1500 3x + 2y  2400 x  0 and y  0 GRAPHING

27. Evaluate the objective function x + 0.8y at vertices of the feasible region: O: 0 + 0 = 0 A: 0 + 0.8 x 1000 = 800 B: 400 + 0.8 x 600 = 880 C: 600 + 0.8 x 300 = 840 D: 750 + 0 = 750 O A B C D Maximum income = £880 at (400, 600) SOLUTION

28. Formulating Constraints

29. x = number of milk chocolates y = number of dark chocolates There must be at least 30 milk chocolates in a batch.

30. x = number of milk chocolates y = number of dark chocolates There must be more milk chocolates than dark ones in a batch.

31. x = number of milk chocolates y = number of dark chocolates The total number of chocolates must not exceed 50.

32. x = number of milk chocolates y = number of dark chocolates The number of dark chocolates must not be more than 25.

33. x = number of milk chocolates y = number of dark chocolates There must be more than 50 milk chocolates in a batch.

34. x = number of milk chocolates y = number of dark chocolates There must be no more than 20 milk chocolates in a batch.

35. x = number of milk chocolates y = number of dark chocolates There must be the same number of milk chocolates as dark chocolates in a batch.

36. x = number of milk chocolates y = number of dark chocolates There must be at least 40 chocolates in total in a batch.

37. x = number of milk chocolates y = number of dark chocolates There must be exactly 15 dark chocolates in a batch.

38. x = number of milk chocolates y = number of dark chocolates There must be more dark chocolates than milk chocolates in a batch.

39. x = number of milk chocolates y = number of dark chocolates Milk chocolates cost 2p to make and dark chocolates cost 3p. The cost of making them must be less than £5.

40. x = number of milk chocolates y = number of dark chocolates Each milk chocolate takes 2 minutes to make and each dark chocolate takes 4 minutes. There is 4 hours available to make all the sweets.

41. x = number of milk chocolates y = number of dark chocolates Milk chocolates each require 5cm² of wrapping and dark chocolates each require 4cm². 900cm² of wrapping paper is available for use.

42. x = number of milk chocolates y = number of dark chocolates The profit on the milk chocolates is 3p and the profit on the dark chocolates is 2p. The firm needs to make a profit of at least £4 on each batch.

43. x = number of milk chocolates y = number of dark chocolates There must be at least twice as many dark chocolates as milk chocolates.

44. x = number of milk chocolates y = number of dark chocolates The milk chocolates must account for at least 10% of the total batch of chocolates.

45. x = number of milk chocolates y = number of dark chocolates The dark chocolates must be less than 40% of the total number of chocolates.

46. x = number of milk chocolates y = number of dark chocolates The number of dark chocolates must be less than half the number of milk chocolates.

47. P92 – Ex7B – Q1,4,6,7,9