1. ENGM 631 Optimization Ch. 3: Introduction to Linear Programming

2. Sec. 3.1Alt. Prototype K-Corp manufactures two primary products: a small tractor suitable for an orchard and a tree shaker. K-Corp can make $3,000 per tractor and $5,000 per shaker. Currently, the demand for tractors is 8,000 per year and the demand for shakers is 6,000 per year. Currently, K-Corp is unable to meet total demand for both due to limitations in the assembly area. The assembly area has a capacity of 36,000 hours available. Each tractor takes 3 hours to assemble. Each shaker takes 4 hours.

3. Sec. 3.1Alt. Prototype Problem Determine the appropriate number of tractors and shakers to make so as to maximize total profit. Constraints Do not exceed demand for either product Do not exceed capacity of assembly area

4. Sec. 3.1Alt. Model Variables X 1 = number of tractors to manufacture X 2 = number of shakers to manufacture Profit Z = 3X 1 + 5X 2 (in $1,000’s)

5. Sec. 3.1Alt. Model Constraints Demand for tractors X 1 < 8,000 Demand for shakers X 2 < 6,000 Capacity of Assembly 3X 1 +4X 2 < 36,000 Non-negativity X 1 > 0, X 2 > 0

6. Sec. 3.1Alt. Model Max Z = 3X 1 + 5X 2 s.t. X 1 < 8,000 X 2 < 6,000 3X 1 + 4X 2 < 36,000 X 1 > 0 X 2 > 0

7. Sec. 3.1Alt. Rescale Let’s let X 1 = number of tractors in 1,000’s X 2 = number of shakers in 1,000’s

8. Sec. 3.1Alt. Model Max Z = 3X 1 + 5X 2 s.t. X 1 < 8 X 2 < 6 3X 1 + 4X 2 < 36 X 1 > 0 X 2 > 0 where Z = profit (in $1,000,000’s)

9. Sec. 3.1Alt. Graphical Technique 2 4 6 8 10 86428642 Do not exceed demand for tractors X 1 < 8

10. Sec. 3.1Alt. Graphical Technique 2 4 6 8 10 86428642 Do not exceed demand for tractors, shakers X 2 < 6 X 1 < 8

11. Sec. 3.1Alt. Graphical Technique 2 4 6 8 10 12 86428642 Do not exceed demand for tractors, shakers, assembly capacity 3X 1 + 4X 2 < 36 X 2 < 6 X 1 < 8

12. Sec. 3.1Alt. Graphical Technique 2 4 6 8 10 12 86428642 Do not exceed demand for tractors, shakers, assembly capacity, can’t make less than zero 3X 1 + 4X 2 < 36 X 2 < 6 X 1 < 8 X 1, X 2 > 0

13. Sec. 3.1Alt. Feasible Region 2 4 6 8 10 12 86428642 3X 1 + 4X 2 < 36 X 2 < 6 X 1 < 8 X 1, X 2 > 0 Feasible Region

14. Sec. 3.1Alt. Problem Out of the large (infinite) number of solutions available, which maximizes profit? 2 4 6 8 10 12 86428642

15. Sec. 3.1Alt. Iso-Value Lines 2 4 6 8 10 12 86428642 Z = 3X 1 + 5X 2 Z = 15

16. Sec. 3.1Alt. Optimal Solution 2 4 6 8 10 12 86428642 Z = 3X 1 + 5X 2 = 3(4) + 5(6) = 42 (4,6)

17. Sec. 3.2 Some Terminology 2 4 6 8 10 12 86428642 Feasible Solutions

18. Sec. 3.2 Some Terminology 2 4 6 8 10 12 86428642 Feasible Region

19. Sec. 3.2 Some Terminology 2 4 6 8 10 12 86428642 Boundary

20. Some Terminology 2 4 6 8 10 12 86428642 Infeasible Solutions

21. Sec. 3.2 Aside 2 4 6 8 10 12 86428642 X 1 < 8 X 2 < 6 3X 1 + 4X 2 < 36 5X 1 + 3X 2 > 60 No Feasible Solutions

22. Sec. 3.2 Some Terminology 2 4 6 8 10 12 86428642 Corner Point Feasible Sol. (CPF) (0,6) (4,6) (8,3) (8,0) (0,0)

23. Sec. 3.2 Relation: CPF & Optimal 2 4 6 8 10 12 86428642 (0,6) (4,6) (8,3) (8,0) (0,0) The best CPF solution must be an optimal solution.

24. Sec. 3.2 Relation: CPF & Optimal 2 4 6 8 10 12 86428642 (0,6) (4,6) (8,3) (8,0) (0,0) Z = 30Z = 42 Z = 39 Z = 24Z = 0 The best CPF solution must be an optimal solution. If a problem has multiple optimal solutions, at least two must be CPF solutions.

25. Sec. 3.2 Relation: CPF & Optimal 2 4 6 8 10 12 86428642 (0,6) (4,6) (8,3) (8,0) (0,0) If a problem has multiple optimal solutions, at least two must be CPF solutions.

26. Sec. 3.2 Optimal Solution 2 4 6 8 10 12 86428642 Recall, Z = 3X 1 + 5X 2 (4,6) 3X 1 + 4X 2 < 36

27. Sec. 3.2 Optimal Solution 2 4 6 8 10 12 86428642 Suppose, Z = 3X 1 + 54X 2 (4,6) 3X 1 + 4X 2 < 36

28. Sec. 3.2 Optimal Solution 2 4 6 8 10 12 86428642 Suppose, Z = 3X 1 + 54X 2 (4,6) 3X 1 + 4X 2 < 36

29. Sec. 3.2 Optimal Solution 2 4 6 8 10 12 86428642 Suppose, Z = 3X 1 + 54X 2 (4,6) 3X 1 + 4X 2 < 36 Multiple Optima

30. Standard Form for LP

31. Sec. 3.2 Standard Form for LP Decision Variables

32. Sec. 3.2 Standard Form for LP Objective

33. Sec. 3.2 Standard Form for LP Functional Constraints

34. Sec. 3.2 Standard Form for LP Nonnegativity Constraints

35. Sec. 3.3 Assumptions Proportionality The contribution of each activity to the value of the objective function is proportional to the level of activity. Consider Z = 3X 1 +... Profit = R - C = 8X 1 - 5X 1 = 3X 1

36. Sec. 3.3 Assumptions Proportionality The contribution of each activity to the value of the objective function is proportional to the level of activity. Consider Z = 3X 1 +... Profit = R - C = 8X 1 - 5X 1 = 3X 1 Valid if costs are variable

37. Sec. 3.3 Assumptions Proportionality The contribution of each activity to the value of the objective function is proportional to the level of activity. Consider Z = 3X 1 +...

38. Sec. 3.3 Assumptions Proportionality Now Suppose fixed costs = 3. Then Profit = R - TC = 8X 1 - (3 + 5X 1 ) = 3X 1 - 3.

39. Sec. 3.3 Assumptions Proportionality Now Suppose fixed costs = 3. Then Profit = R - TC = 8X 1 - (3 + 5X 1 ) = 3X 1 - 3.

40. Sec. 3.3 Assumptions Proportionality Now Suppose we have a decreasing marginal rate of return Profit = R - TC = (8-.025X 1 )X 1 - 5X 1.

41. Sec. 3.3 Assumptions Proportionality Now Suppose we have a decreasing marginal rate of return Profit = R - TC = (8-.025X 1 )X 1 - 5X 1.

42. Sec. 3.3 Assumptions Additivity Every function in the LP model is the sum of the individual contributions of the respective activities Z = 3X 1 + 5X 2 + 2X 1 X 2 Interaction terms not allowed

43. Sec. 3.3 Assumptions Divisibility Decision variables, X i i=1,2,... n, are allowed to have any values, including non- integer values, that satisfy the functional and nonnegativity constraints.

44. Sec. 3.3 Divisibility 2 4 6 8 10 12 86428642 86428642 For K-Corp, is it This or this ?

45. Sec. 3.3 Assumptions Certainty The value assigned to each parameter of a linear programming model is assumed to be a known constant. Max Z = 3X 1 + 5X 2 s.t. X 1 < 8 X 2 < 6 3X 1 + 4X 2 < 36 X 1 > 0, X 2 > 0 How certain are we of this demand?

46. Sec. 3.4Alt Power Generation A regional power system has three generating stations: A, B, and C. Each station serves a local area. Three outlying areas are also served by the system: X, Y, and X. The power demand at areas X, Y, and Z is 25, 50, and 30 MW respectively. The maximum generating capacity beyond local requirements and the cost of generation at the three stations is shown below.

47. Sec. 3.4Alt Power Generation StationExcess Cap.Cost of Gen. A 100 500 B 75 700 C 200 400 Power can be transmitted between any pair of generating stations, but 5% of the amount shipped will be lost. Power can be transmitted from some of the generating stations to outlying areas, but 10% of the amount shipped is lost.

48. Sec. 3.4Alt Power Generation Model Lines exist from stations A & C to X, from B & C to Y, and from A & B to Z. A C B X Z Y Formulate as a linear program!

49. Sec. 3.4Alt Power Generation Model Problem: Min Cost of satisfying outlying areas s.t. Do not exceed capacity at any plant Exactly serve each outlying area

50. Sec. 3.4Alt Power Generation Model Decision Variables X ij = power shipped from plant i to outlying area j Objective Function Min Cost Min Z = 500X AX + 500X AZ + 700X BY + 700X BZ + 400X CX + 400X CY

51. Sec. 3.4Alt Power Generation Model Do not Exceed Capacity X AX + X AZ < 100 X BY + X BZ < 75 X CX + X CY < 200

52. Sec. 3.4Alt Power Generation Model Shipped - Losses satisfy outlying areas X AX -.1X AX + X CX -.1X CX = 25 X BY -.1X BY + X CY -.1X CY = 50 X AZ -.1X AZ + X BZ -.1X BZ = 30 or.9X AX +.9X CX = 25.9X BY +.9X CY = 50.9X AZ +.9X BZ = 30

53. Sec. 3.4Alt Power Generation Model Min Z = 500X AX + 500X AZ + 700X BY + 700X BZ + 400X CX + 400X CY s.t. X AX + X AZ < 100 X BY + X BZ < 75 X CX + X CY < 200.9X AX +.9X CX = 25.9X BY +.9X CY = 50.9X AZ +.9X BZ = 30 X i,j > 0, i = A, B, C j = X, Y, Z

54. 3.4 Additional Examples Design of Radiation therapy Regional Planning of Kibbutzim Controlling Air Pollution Reclaiming Solid waste Personnel Scheduling at Union Airways (United Airlines 24/7 40? $6m/yr saved) Distributing Goods in a Network (Like our Power Example)

55. 3.5 Spreadsheet Solution Range name Data cells Target cell Output cells Changing cells Solver

56. 3.5 Spreadsheet Fig. 3.14 Old

57. 3.5 Spreadsheet Fig. 3.15

58. 3.5 Spreadsheet Fig. 3.16

59. 3.5 Spreadsheet Fig. 3.17 Old

60. 3.5 Spreadsheet Fig. 3.18 Old

61. 3.5 Spreadsheet Fig. 3.19 Old

62. 3.5 Spreadsheet Fig. 3.20 Old

63. 3.5 Spreadsheet Fig. 3.21 Old

64. 3.5 Spreadsheet Fig. 3.22 Old

65. 3.6 Large Models Modeling Languages Older edition Citgo (petroleum products distributor) had 3000 functional constraints and 15000 decision variables Need for a modeling language You can learn MPL and Lindo with links from the book’s website (not part of our course this year)

66. 3.7 Conclusions LP is powerful LP is “standard” For allocating resources in any social organization Sometimes LP is not applicable; instead do other things

67. Cases 3.1 Auto assembly (Crusier &Thrillseeker) 3.2 (Online) Cutting Cafeteria costs 3.3 (Online) Staffing a Call Center at California Children’s Hospital 3.4 (Online) Promoting a Breakfast Cereal