Chapter 2

1. Chapter 2 An Introduction to Linear Programming MT 235

2. George Dantzig • Courtesy of NPR: • “The Mathematician Who Solved Major Problems” • http://www.npr.org/dmg/dmg.php?prgCode=WESAT&showDate=21-May-2005&segNum=14& MT 235

3. General Form of an LP Model MT 235

4. General Form of an LP Model • where the c’s, a’s and b’s are constants determined from the problem and the x’s are the decision variables MT 235

5. Components of Linear Programming • An objective • Decision variables • Constraints • Parameters MT 235

6. Assumptions of the LP Model • Divisibility - basic units of x’s are divisible • Proportionality - a’s and c’s are strictly proportional to the x’s • Additivity - each term in the objective function and constraints contains only one variable • Deterministic - all c’s, a’s and b’s are known and measured without error • Non-Negativity (caveat) MT 235

7. Sherwood Furniture Company Recently, Sherwood Furniture Company has been interested in developing a new line of stereo speaker cabinets. In the coming month, Sherwood expects to have excess capacity in its Assembly and Finishing departments and would like to experiment with two new models. One model is the Standard, a large, high-quality cabinet in a traditional design that can be sold in virtually unlimited quantities to several manufacturers of audio equipment. The other model is the Custom, a small, inexpensive cabinet in a novel design that a single buyer will purchase on an exclusive basis. Under the tentative terms of this agreement, the buyer will purchase as many Customs as Sherwood produces, up to 32 units. The Standard requires 4 hours in the Assembly Department and 8 hours in the Finishing Department, and each unit contributes $20 to profit. The Custom requires 3 hours in Assembly and 2 hours in Finishing, and each unit contributes $10 to profit. Current plans call for 120 hours to be available next month in Assembly and 160 hours in Finishing for cabinet production, and Sherwood desires to allocate this capacity in the most economical way. MT 235

8. Sherwood Furniture Company – Linear Equations MT 235

9. Sherwood Furniture Company – Graph Solution MT 235

10. Sherwood Furniture Company – Graph Solution Constraint 1 MT 235

11. Sherwood Furniture Company – Graph Solution Constraint 1 MT 235

12. Sherwood Furniture Company – Graph Solution Constraint 2 MT 235

13. Sherwood Furniture Company – Graph Solution Constraint 1 & 2 MT 235

14. Sherwood Furniture Company – Graph Solution Constraint 3 MT 235

15. Sherwood Furniture Company – Graph Solution Constraint 1, 2 & 3 MT 235

16. Sherwood Furniture Company – Graph Solution MT 235

17. Sherwood Furniture Company – Graph Solution MT 235

18. Sherwood Furniture Company – Solve Linear Equations MT 235

19. Sherwood Furniture Company – Solve Linear Equations MT 235

20. Sherwood Furniture Company – Solve Linear Equations MT 235

21. Sherwood Furniture Company – Graph Solution Optimal Point (15, 20) MT 235

22. Sherwood Furniture Company – Slack Calculation MT 235

23. Sherwood Furniture Company - Slack Variables Max 20x1 + 10x2 + 0S1 + 0S2 + 0S3 s.t. 4x1 + 3x2 + 1S1 = 120 8x1 + 2x2 + 1S2 = 160 x2 + 1S3 = 32 x1, x2, S1 ,S2 ,S3 >= 0 MT 235

24. Sherwood Furniture Company – Graph Solution 3 1 2 MT 235

25. Sherwood Furniture Company – Slack Calculation Point 1 Point 1 MT 235

26. Sherwood Furniture Company – Graph Solution 3 1 2 MT 235

27. Sherwood Furniture Company – Slack Calculation Point 2 Point 2 MT 235

28. Sherwood Furniture Company – Graph Solution 3 1 2 MT 235

29. Sherwood Furniture Company – Slack Calculation Point 3 Point 3 MT 235

30. Sherwood Furniture Company – Slack Calculation Points 1, 2 & 3 Point 1 Point 2 Point 3 MT 235

31. Sherwood Furniture Company – Slack Variables • For each ≤ constraint the difference between the RHS and LHS (RHS-LHS). It is the amount of resource left over. • Constraint 1; S1 = 0 hrs. • Constraint 2; S2 = 0 hrs. • Constraint 3; S3 = 12 Custom MT 235

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38. Pet Food Company A pet food company wants to find the optimal mix of ingredients, which will minimize the cost of a batch of food, subject to constraints on nutritional content. There are two ingredients, P1 and P2. P1 costs $5/lb. and P2 costs $8/lb. A batch of food must contain no more than 400 lbs. of P1 and must contain at least 200 lbs. of P2. A batch must contain a total of at least 500 lbs. What is the optimal (minimal cost) mix for a single batch of food? MT 235

39. Pet Food Company – Linear Equations MT 235

40. Pet Food Company – Graph Solution MT 235

41. Pet Food Company – Graph Solution Constraint 1 MT 235

42. Pet Food Company – Graph Solution Constraint 1 MT 235

43. Pet Food Company – Graph Solution Constraint 2 MT 235

44. Pet Food Company – Graph Solution Constraint 1 & 2 MT 235

45. Pet Food Company – Graph Solution Constraint 3 MT 235

46. Pet Food Company – Graph Solution Constraint 1, 2 & 3 MT 235

47. Pet Food Company – Solve Linear Equations MT 235

48. Pet Food Company – Graph Solution MT 235

49. Pet Food Company – Solve Linear Equations MT 235

50. Pet Food Company – Solve Linear Equations MT 235