1. 1 Lecture 11 Resource Allocation Part1 (involving Continuous Variables- Linear Programming) 1.040/1.401/ESD.018 Project Management Samuel Labi and Fred Moavenzadeh Massachusetts Institute of Technology April 2, 2007

2. 2 Linear Programming This Lecture Part 1: Basics of Linear Programming Part 2: Methods for Linear Programming Part 3: Linear Programming Applications

3. 3 Linear Programming Part 1: Basics of Linear Programming - The link to resource allocation in project management - What is a “feasible region”? - How to sketch a feasible region on a 2-D Cartesian axis - Vertices of a feasible region - Some standard terminology

4. 4 The link to resource allocation in project management Project output = f(Resource 1, Resource 2, Resource 3, … Resource n) The goal is to determine the levels of each resource that would maximize project output. Assume only 1 resource variable: X Project output Amount of Resource X

5. 5 The link to resource allocation in project management Project output = f(Resource 1, Resource 2, Resource 3, … Resource n) The goal is to determine the levels of each resource that would maximize project output. Assume only 2 resources: X and Y Linear Programming W W W X X X Y Y Y Examples of W =f(X,Y) response surfaces

6. 6 The link to resource allocation in project management Project output = f(Resource 1, Resource 2, Resource 3, … Resource n) The goal is to determine the levels of each resource that would maximize project output. Assume only 2 resources: X and Y (consider simplified cross section of response surface) Resource Y Output, W Resource X Linear Programming

7. 7 The link to resource allocation in project management Project output = f(Resource 1, Resource 2, Resource 3, … Resource n) The goal is to determine the levels of each resource that would maximize project output. Assume only 2 resources: X and Y Resource Y Output, W Resource X Linear Programming Local space

8. 8 The link to resource allocation in project management Project output = f(Resource 1, Resource 2, Resource 3, … Resource n) The goal is to determine the levels of each resource that would maximize project output. Assume only 2 resources: X and Y Resource Y Output, W Resource X Linear Programming Local space Local maximum

9. 9 The link to resource allocation in project management Project output = f(Resource 1, Resource 2, Resource 3, … Resource n) The goal is to determine the levels of each resource that would maximize project output. Assume only 2 resources: X and Y Resource Y Output, W Resource X Linear Programming Global Space Local maximum Local space Global Maximum

10. 10 The link to resource allocation in project management Project output = f(Resource 1, Resource 2, Resource 3, … Resource n) The goal is to determine the levels of each resource that would maximize project output. Assume only 2 resources: X and Y Resource Y Output, W Resource X Linear Programming Local space Local maximum

11. 11 In the real world, there are more than 2 resource types (variables) - equipment types - labor types or crew types - money Therefore, in project management, resource allocation can be a multi- dimensional linear programming problem. Linear Programming

12. 12 Linear Programming Example 1: Sketch the following region: y – 2 > 0 Solution First, make y the subject Write the equation of the critical boundary Sketch the critical boundary Indicate the region of interest

13. 13 Linear Programming Sketch of the region: y > 2 2 - 1 1 3 4 5 - 2 y = 2 x y Critical Boundary

14. 14 Linear Programming Example 2: Sketch of the region: x - 5 < 0 21345 x = 5 x y (Critical Boundary)

15. 15 Linear Programming Example 3: Sketch of the region: y > 2 2 - 1 1 3 4 5 - 2 y = 1 x y (Critical Boundary)

16. 16 Linear Programming Example 4: Sketch of the region : 1 – x ≤ 0 21345 x = 1 x y (Critical Boundary)

17. 17 Linear Programming Example 5: Sketch of the region: y > 0 2 - 1 1 3 4 5 - 2 x axis, or y = 0 y (Critical Boundary)

18. 18 Linear Programming Example 6: Sketch of the region: y - 3 ≤ 0 2 - 1 1 3 4 5 - 2 x axis y (Critical Boundary) y = 3

19. 19 Mean and Variance Linear Programming Example 7: Sketch of the region : x + 1 ≤ 0 21-2-3 x = -1 x y (Critical Boundary) 3

20. 20 Linear Programming Mean and Variance Linear Programming Example 8: Sketch of the region : 2 - x ≤ 0 21-2-3 x = -2 x y (Critical Boundary) 3

21. 21 Linear Programming How to Sketch a Region whose Critical Boundary is a bi-variate Function First, make y the subject of the inequality Write the equation of the critical boundary Sketch the critical boundary (often a sloping line) Indicate the region of interest Note that … - the sign < means the region below the sloping line - the sign > means the region above the sloping line)

22. 22 Linear Programming Example 9: Sketch of the region : y ≤ x y = x x y (Critical Boundary) y x Thus, the critical boundary is: y = x

23. 23 Linear Programming Example 10: Sketch of the region : y < x y = x x y (Critical Boundary) y < x Thus, the critical boundary is: y = x

24. 24 Linear Programming Example 11: Sketch of the region : x – y ≤ 0 y = x x y (Critical Boundary) x – y ≤ 0 Making y the subject yields: y x Thus, the critical boundary is: y = x

25. 25 Linear Programming Example 12: Sketch of the region : y > 2x + 1 y = 2x+1 x y (Critical Boundary) y > 2x +1 Thus, the critical boundary is: y = 2x+1 When x = 0, y = -0.5 CB passes thru ( 0, - 0.5 ) When y = 0, x = 1 CB passes thru ( 1,0 ) 1 -3

26. 26 Linear Programming Example 13: Sketch of the region : y < 4x - 3 y = 4x- 3 x y (Critical Boundary) y < 4x - 3 Thus, the critical boundary is: y = 4x - 3 When x = 0, y = -3 CB passes thru (0, -3) When y = 0, x = 3/4 CB passes thru (0.75, 0) 0.75 -3

27. 27 Linear Programming Example 14: Sketch of the region : y ≤ -3.8x + 13 y = 4x- 3 x y (Critical Boundary) y < -3.8x + 3 Thus, the critical boundary is: y = - 3.8x +3 When x = 0, y = 13 CB passes thru (0, 13) When y = 0, x = 13/3.8 CB passes thru (13/3.8, 0) 13/3.8 13

28. 28 How to sketch a region bounded by two or more critical boundaries First make y the subject of each inequality Write the equation of the critical boundary Sketch the critical boundaries for each inequality Indicate the overlapping region of interest Linear Programming

29. 29 Linear Programming Example 15: Sketch the region bounded (or constrained) by the following functions y > 0 x > 0 y < -3.5x + 5 y=0 y (Critical Boundary) y > 0 Its critical boundary is: y = 0

30. 30 Linear Programming Example 15: Sketch the region bounded (or constrained) by the following functions y > 0 x > 0 y < -3.5x + 5 y=0 y (Critical Boundary) x > 0 Its critical boundary is: x = 0 (Critical Boundary)

31. 31 Linear Programming Example 15: Sketch the region bounded (or constrained) by the following functions y > 0 x > 0 y < -3x + 5 y=0 y (Critical Boundary) y < -3x + 5 Thus, the critical boundary is: y = -3x + 5 When x = 0, y = 5 CB passes thru (0, 5) When y = 0, x = 5/3 CB passes thru (5/3, 0) x=0 (Critical Boundary) y= -3x + 5

32. 32 Linear Programming Example 15: Sketch the region bounded (or constrained) by the following functions y > 0 x > 0 y < -3x + 5 y=0 y (Critical Boundary) This is the FEASIBLE region. All points in this region satisfy all the three constraining functions. x=0 (Critical Boundary) y= -3x + 5 Feasible Region

33. 33 Linear Programming Example 16: Sketch the region bounded (or constrained) by the following functions y > 0 y> - 0.2x + 5 y < -0.5x + 5 x y

34. 34 Linear Programming Example 16: Sketch the region bounded (or constrained) by the following functions y > 0 y> - 0.2x + 5 y < -0.5x + 5 y=0 y (Critical Boundary) This is the FEASIBLE region. All points in this region satisfy all the three constraining functions. (Critical Boundary) y = -0.2x + 5 y= -0.5x + 5 (Critical Boundary) Feasible Region

35. 35 Linear Programming Example 17: Sketch the region bounded (or constrained) by the following functions y > 3 y < -2x + 6 y < x + 1 y x

36. 36 Linear Programming Example 17: Sketch the region bounded (or constrained) by the following functions y > 3 y < -2x + 6 y < x + 1 y=3 y (Critical Boundary) This is the FEASIBLE region. All points in this region satisfy all the three constraining functions. (Critical Boundary) y = -0.2x + 6 y= x + 1 (Critical Boundary) Feasible Region x 3 6

37. 37 Linear Programming Example 18: Sketch the region bounded (or constrained) by the following functions y > 0 x > 0 y < -x + 5 y < x+2 y x

38. 38 Linear Programming Example 18: Sketch the region bounded (or constrained) by the following functions y > 0 x > 0 y < -x + 5 y < x+2 y=3 x=0 (Critical Boundary) This is the FEASIBLE region. All points in this region satisfy all the three constraining functions. (Critical Boundary) y = -x + 5y= x + 2 (Critical Boundary) Feasible Region y=0 3 -2 5

39. 39 Linear Programming Example 19: Sketch the region bounded (or constrained) by the following functions y > 0 x > 0 y < -0.33x + 1 y > 2x - 5 y x

40. 40 Linear Programming Example 19: Sketch the region bounded (or constrained) by the following functions y > 0 x > 0 y < -0.33x + 1 y > 2x - 5 y=3 y (Critical Boundary) This is the FEASIBLE region. All points in this region satisfy all the three constraining functions. (Critical Boundary) y = 2x - 5 y= 0.33x + 1 (Critical Boundary) Feasible Region x 5/2 1 (Critical Boundary)

41. 41 What are the “vertices” of a feasible region? Simply refers to the corner points How do we determine the vertices of a feasible region? - Plot the boundary conditions carefully on a graph sheet and read off the values at the corners, OR - Solve the equations simultaneously Linear Programming

42. 42 Linear Programming Example 19: Sketch the region bounded (or constrained) by the following functions y > 0 x > 0 y < -0.33x + 1 y > 2x - 5 y=3 y (Critical Boundary) This is the FEASIBLE region. All points in this region satisfy all the three constraining functions. (Critical Boundary) y = 2x - 5 y= 0.33x + 1 (Critical Boundary) Feasible Region x (0, 1) (3.6, 2.2) (0, 0) (2.5, 0) (Critical Boundary)

43. 43 Why are vertices important? They often represent points at which certain combinations of X and Y is either a maximum or minimum. Certain combination … ? Yes! For example: W = x + y W = 2 x + 3y W = x 2 + y W = x 0.5 + 3y2 W = ( x + y) 2 etc., etc. So we typically seek to optimize (maximize or minimize) the value of W. In other words, W is the objective function. Linear Programming

44. 44 W is also referred to as the OBJECTIVE FUNCTION or project performance output. (It is our objective to maximize or minimize W x and y can be referred to as Project CONTROL VARIABLES or DECISION VARIABLES Linear Programming

45. 45 Symbols for decision variables In some books, (x 1, x 2 ) is used instead of (x,y) (x 1, x 2, x 3 ) is used instead of (x, y, z) (x 1, x 2, x 3, x 4 ) is used instead of (x, y, z, v) etc. x1x1 x2x2 x3x3 x2x2 x1x1 Linear Programming

46. 46 Dimensionality of Optimization Problems An optimization problem with n decision variables  n -dimensional Linear Programming W=f(x 1 ) 1 Decision Variable x1x1 1-dimensional

47. 47 Dimensionality of Optimization Problems An optimization problem with n decision variables  n -dimensional x1x1 x2x2 2-dimensional 2 Decision Variables Intersecting lines yield vertices (problem solutions) Linear Programming W=f(x 1, x 2 )W=f(x 1 ) 1 Decision Variable x1x1 1-dimensional

48. 48 Dimensionality of Optimization Problems An optimization problem with n decision variables  n -dimensional x1x1 x2x2 x2x2 x1x1 2-dimensional 3-dimensional 2 Decision Variables3 Decision Variables Intersecting lines yield vertices (problem solutions) Intersecting planes yield vertices (problem solutions) x3x3 Linear Programming W=f(x 1, x 2 )W=f(x 1 )W=f(x 1, x 2, x 3 ) 1 Decision Variable x1x1 1-dimensional

49. 49 Dimensionality of Optimization Problems An optimization problem with n decision variables  n -dimensional x1x1 x2x2 x2x2 x1x1 2-dimensional 3-dimensional 2 Decision Variables3 Decision Variables n-dimensional n Decision Variables Sorry! Cannot be visualized Intersecting lines yield vertices (problem solutions) Intersecting planes yield vertices (problem solutions) Intersecting objects yield vertices (problem solutions) x3x3 Linear Programming W=f(x 1, x 2 )W=f(x 1 )W=f(x 1, x 2, x 3 )W=f(x 1, x 2, …, x n ) 1 Decision Variable x1x1 1-dimensional

50. 50 Example of 2-dimensional problem Given that W = 8 x + 5 y Find the maximum value of Z subject to the following: y > 0 x > 0 y < -0.33x + 1 y < 2x - 5 Linear Programming

51. 51 Solution The objective function is: W = 8 x + 5 y The constraints are: y > 0 x > 0 y < -0.33x + 1 y < 2x – 5 The control values are x and y. Linear Programming

52. 52 Linear Programming y=3 y (Critical Boundary) y = 2x - 5 y= 0.33x + 1 (Critical Boundary) Feasible Region x (0, 1) (3.6, 2.2) (0, 0) (2.5, 0) Vertices of Feasible Region xy W = 8 x +5 y (0, 0)00 = 8(0) + 5(0) = 0 (0, 1)01 = 8(0) + 5(1) = 5 (2.5, 0)2.50 = 8(2.5) + 5(0) = 20 (3.6, 2.2)3.62.2 = 8(3.6) + 5(2.2) = 36 Solution (cont’d)

53. 53 Solution (continued) Therefore, the maximum value of W is 36, And this happens when x = 3.6 and y = 2.2 That is: W opt = 36 units y opt = 3.6 units x opt = 2.2 units This set of answers represents the “optimal solution”. Linear Programming

54. 54 What if there are several variables and constraints? - In project management resource allocation, a typical problem may have tens, hundreds, or even thousands of variables and several constraints. - Solutions methods - Graphical method - Simultaneous equations - Vector algebra (matrices) - Software packages Linear Programming

55. 55 Next Lecture Common Methods for Solving Linear Programming Problems Graphical Methods - The “Z-substitution” Method - The “Z-vector” Method Various Software Programs: - GAMS - CPLEX - SOLVER Linear Programming

56. 56 Questions? Linear Programming