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1.040/1.401/ESD.018 Project Management

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

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1.040/1.401/ESD.018 Project Management

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

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

  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. 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. Linear Programming 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 W W W Y X Y Y X X Examples of W =f(X,Y) response surfaces

  6. Linear Programming 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) Output, W Resource Y Resource X

  7. Linear Programming 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 Output, W Resource Y Local space Resource X

  8. Linear Programming 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 Output, W Resource Y Local space Local maximum Resource X

  9. Linear Programming 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 Global Maximum Global Space Output, W Resource Y Local space Local maximum Resource X

  10. Linear Programming 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 Output, W Resource Y Local space Local maximum Resource X

  11. Linear Programming 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.

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

  14. Linear Programming Example 2: Sketch of the region: x - 5 < 0 y x 1 2 3 4 5 x = 5 (Critical Boundary)

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

  16. Linear Programming Example 4: Sketch of the region: 1 – x ≤ 0 y x 1 2 3 4 5 x = 1 (Critical Boundary)

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

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

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

  20. Mean and Variance Linear Programming Linear Programming Example 8: Sketch of the region: 2 - x ≤ 0 y x -3 -2 -1 1 2 3 x = -2 (Critical Boundary)

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

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

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

  25. Linear Programming Linear Programming Example 12: Sketch of the region: y > 2x + 1 y 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) y = 2x+1 (Critical Boundary) 1 x -3

  26. Linear Programming Example 13: Sketch of the region: y < 4x - 3 y y = 4x- 3 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) (Critical Boundary) x 0.75 -3

  27. Linear Programming Example 14: Sketch of the region: y ≤ -3.8x + 13 y 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 x 13/3.8 y = 4x- 3 (Critical Boundary)

  28. Linear Programming 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

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

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

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

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

  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 y x

  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 This is the FEASIBLE region. All points in this region satisfy all the three constraining functions. Feasible Region y=0 (Critical Boundary) y = -0.2x + 5 y= -0.5x + 5 (Critical Boundary) (Critical Boundary)

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

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

  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. 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 This is the FEASIBLE region. All points in this region satisfy all the three constraining functions. (Critical Boundary) y= 0.33x + 1 (Critical Boundary) Feasible Region 1 y=3 (Critical Boundary) x (Critical Boundary) 5/2 y = 2x - 5 (Critical Boundary)

  41. Linear Programming 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

  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 This is the FEASIBLE region. All points in this region satisfy all the three constraining functions. (Critical Boundary) y= 0.33x + 1 (3.6, 2.2) (Critical Boundary) (0, 1) Feasible Region y=3 (Critical Boundary) x (0, 0) (Critical Boundary) (2.5, 0) y = 2x - 5 (Critical Boundary)

  43. Linear Programming 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 = 2x + 3y W = x2 + y W = x0.5 + 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.

  44. Linear Programming 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

  45. Linear Programming Symbols for decision variables x2 In some books, (x1, x2) is used instead of (x,y) (x1, x2, x3) is used instead of (x, y, z) (x1, x2, x3 , x4) is used instead of (x, y, z, v) etc. x2 x1 x3 x1

  46. Linear Programming Dimensionality of Optimization Problems An optimization problem with n decision variables  n-dimensional x1 1-dimensional 1 Decision Variable W=f(x1)

  47. Linear Programming Dimensionality of Optimization Problems An optimization problem with n decision variables  n-dimensional x2 x1 x1 2-dimensional 1-dimensional 1 Decision Variable 2 Decision Variables Intersecting lines yield vertices (problem solutions) W=f(x1) W=f(x1 , x2)

  48. Linear Programming Dimensionality of Optimization Problems An optimization problem with n decision variables  n-dimensional x2 x2 x1 x3 x1 x1 3-dimensional 2-dimensional 1-dimensional 1 Decision Variable 2 Decision Variables 3 Decision Variables Intersecting planes yield vertices (problem solutions) Intersecting lines yield vertices (problem solutions) W=f(x1) W=f(x1 , x2) W=f(x1 , x2, x3)

  49. Linear Programming Dimensionality of Optimization Problems An optimization problem with n decision variables  n-dimensional x2 x2 Sorry! Cannot be visualized x1 x3 x1 x1 n-dimensional 3-dimensional 2-dimensional 1-dimensional n Decision Variables 1 Decision Variable 2 Decision Variables 3 Decision Variables Intersecting planes yield vertices (problem solutions) Intersecting objects yield vertices (problem solutions) Intersecting lines yield vertices (problem solutions) W=f(x1) W=f(x1 , x2) W=f(x1 , x2, x3) W=f(x1 , x2, …, xn)

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

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