Lecture 14. Resource Allocation involving Continuous Variables (Linear Programming) 1.040/1.401/ESD.018 Project Management.

Transcription

1 1.040/1.401/ESD.018 Project Management Lecture 14 Resource Allocation involving Continuous Variables (Linear Programming) April 2, 2007 Samuel Labi and Fred Moavenzadeh Massachusetts Institute of Technology 1

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

3 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 3

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 4

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 W X Y W X Y W X Y Examples of W =f(x,y) response surfaces Figure by MIT OCW. 5

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) Output, W Resource Y Resource X 6

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 Local space Output, W Resource Y Resource X 7

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 Local space Output, W Resource Y Local maximum Resource X 8

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

10 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 multidimensional linear programming problem. 10

11 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 11

12 Sketch of the region: y > 2 y Critical Boundary x y = 2 12

13 Example 2: Sketch of the region: x -5 < 0 y x x = 5 (Critical Boundary) 13

14 Example 3: Sketch of the region: y > 2 y x y = 1 (Critical Boundary) 14

15 Example 4: Sketch of the region: 1 x 0 y x x = 1 (Critical Boundary) 15

16 Example 5: Sketch of the region: y > 0 y x axis, or y = 0 (Critical Boundary) 16

17 Example 6: Sketch of the region: y y y = 3 (Critical Boundary) x axis -2 17

18 Linear Mean Programming and Variance Example 7: Sketch of the region: x y x x = -1 (Critical Boundary) 18

19 Linear Mean Programming and Variance Example 8: Sketch of the region: 2 - x 0 y x x = -2 (Critical Boundary) 19

20 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) 20

21 y x Thus, the critical boundary is: y = x Example 9: Sketch of the region: y x y y = x x (Critical Boundary) 21

22 y < x Thus, the critical boundary is: y = x Example 10: Sketch of the region: y < x y y = x x (Critical Boundary) 22

23 Example 11: Sketch of the region: x y 0 x y 0 Making y the subject yields: y x Thus, the critical boundary is: y = x y Linear Programming y = x (Critical Boundary) x 23

24 Example 12: Sketch of the region: y > 2x + 1 y > 2x +1 y Linear Programming y = 2x+1 Thus, the critical boundary is: y = 2x+1 When x = 0, y = (Critical Boundary) x CB passes thru (0,-0.5) When y = 0, x = 1 CB passes thru (1,0) 24

25 Example 13: Sketch of the region: y < 4x - 3 y < 4x -3 Thus, the critical boundary is: y y = 4x- 3 (Critical Boundary) y = 4x - 3 When x = 0, y = x CB passes thru (0, -3) -3 When y = 0, x = 3/4 CB passes thru (0.75, 0) 25

26 Example 14: Sketch of the region: y -3.8x + 13 y < -3.8x + 3 y Thus, the critical boundary is: y = - 3.8x +3 When x = 0, y = 13 CB passes thru (0, 13) 13 13/3.8 y = 4x- 3 x When y = 0, x = 13/3.8 (Critical Boundary) CB passes thru (13/3.8, 0) 26

27 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 27

28 y > 0 Its critical boundary is: y = 0 Example 15: Sketch the region bounded (or constrained) by the following functions y > 0 x > 0 y < -3.5x + 5 y Linear Programming y=0 (Critical Boundary) 28

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

30 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) Example 15: Sketch the region bounded (or constrained) by the following functions y > 0 x > 0 y < -3x + 5 y Linear Programming y=0 (Critical Boundary) x=0 (Critical Boundary) y= -3x + 5 (Critical Boundary) 30

31 This is the FEASIBLE region. All points in this region satisfy all the three constraining functions. Example 15: Sketch the region bounded (or constrained) by the following functions y > 0 x > 0 y < -3x + 5 y Feasibl e Region Linear Programming y=0 (Critical Boundary) x=0 (Critical Boundary) y= -3x + 5 (Critical Boundary) 31

32 Example 16: Sketch the region bounded (or constrained) by the following functions y > 0 y> - 0.2x + 5 y < -0.5x + 5 y Linear Programming x 32

33 This is the FEASIBLE region. All points in this region satisfy all the three constraining functions. Example 16: Sketch the region bounded (or constrained) by the following functions y > 0 y> - 0.2x + 5 y < -0.5x + 5 y Feasibl e Region Linear Programming y=0 (Critical Boundary) y= -0.5x + 5 (Critical Boundary) y = -0.2x + 5 (Critical Boundary) 33

34 Example 17: Sketch the region bounded (or constrained) by the following functions y > 3 y < -2x + 6 y < x + 1 y Linear Programming x 34

35 This is the FEASIBLE region. All points in this region satisfy all the three constraining functions. Example 17: Sketch the region bounded (or constrained) by the following functions y > 3 y < -2x + 6 y < x + 1 y y = -0.2x + 6 (Critical Boundary) y= x Feasibl e Region Linear Programming 3 y=3 (Critical Boundary) x (Critical Boundary) 35

36 Example 18: Sketch the region bounded (or constrained) by the following functions y > 0 x > 0 y < -x + 5 y < x+2 y Linear Programming x 36

37 This is the FEASIBLE region. All points in this region satisfy all the three constraining functions. Example 18: Sketch the region bounded (or constrained) by the following functions y > 0 x > 0 y < -x + 5 y < x+2 x=0 y= x + 2 (Critical Boundary) 5-2 Feasibl e Region Linear Programming 3 y=3 (Critical Boundary) y=0 y = -x + 5 (Critical Boundary) 37

38 Example 19: Sketch the region bounded (or constrained) by the following functions y > 0 x > 0 y < -0.33x + 1 y > 2x - 5 y Linear Programming x 38

39 This is the FEASIBLE region. Example 19: Sketch the region bounded (or constrained) by the following functions y > 0 x > 0 y < -0.33x + 1 y > 2x - 5 y (Critical Boundary) Linear Programming y= 0.33x + 1 (Critical Boundary) All points in this region satisfy all the three constraining functions. 1 5/2 Feasibl e Region y=3 (Critical Boundary) x (Critical Boundary) y = 2x - 5 (Critical Boundary) 39

40 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 40

41 This is the FEASIBLE region. Example 19: Sketch the region bounded (or constrained) by the following functions y > 0 x > 0 y < -0.33x + 1 y > 2x - 5 y (Critical Boundary) (3.6, 2.2) Linear Programming y= 0.33x + 1 (Critical Boundary) All points in this region satisfy all the three constraining functions. (0, 1) (0, 0) (2.5, 0) Feasibl e Region y=3 (Critical Boundary) x (Critical Boundary) y = 2x - 5 (Critical Boundary) 41

42 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 = x 2 + y W = x y2 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. 42

43 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 43

44 Symbols for decision variables x 2 x 2 x 1 x 1 x 3 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. 44

45 Dimensionality of Optimization Problems An optimization problem with n decision variables n-dimensional x 1 1-dimensional 1 Decision Variable W=f(x 1 ) 45

46 Dimensionality of Optimization Problems An optimization problem with n decision variables n-dimensional x 2 1-dimensional 1 Decision Variable x 1 2-dimensional x 1 2 Decision Variables W=f(x 1 ) Intersecting lines yield vertices (problem solutions) W=f(x 1, x 2 ) 46

47 Dimensionality of Optimization Problems An optimization problem with n decision variables n-dimensional x 2 x 2 1-dimensional x 1 2-dimensional x 1 x 1 3-dimensional x 3 1 Decision Variable 2 Decision Variables 3 Decision Variables Intersecting lines yield vertices (problem solutions) Intersecting planes yield vertices (problem solutions) W=f(x 1 ) W=f(x 1, x 2 ) W=f(x 1, x 2, x 3 ) 47

48 Dimensionality of Optimization Problems 1-dimensional 1 Decision Variable An optimization problem with n decision variables n-dimensional x 1 x 2 x 2 2-dimensional x 1 x 1 3-dimensional 2 Decision Variables 3 Decision Variables Intersecting lines yield vertices (problem solutions) Intersecting planes yield vertices (problem solutions) x 3 Sorry! Cannot be visualized n-dimensional n Decision Variables Intersecting objects yield vertices (problem solutions) W=f(x 1 ) W=f(x 1, x 2 ) W=f(x 1, x 2, x 3 ) W=f(x 1, x 2,, x n ) 48

49 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

50 Solution The objective function is: W = 8x + 5y The constraints are: y > 0 x > 0 y < -0.33x + 1 y < 2x 5 The control values are x and y. 50

51 Solution (cont d) y y = 2x - 5 (Critical Boundary) y= 0.33x + 1 (3.6, 2.2) (Critical Boundary) (0, 1) Feasibl e Region y=3 (Critical Boundary) (0, 0) (2.5, 0) x Vertices of Feasible Region x y W = 8x+5y (0, 0) 0 0 = 8(0) + 5(0) = 0 (0, 1) 0 1 = 8(0) + 5(1) = 5 (2.5, 0) = 8(2.5) + 5(0) = 20 (3.6, 2.2) = 8(3.6) + 5(2.2) = 36 51

52 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. 52

53 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 53

54 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 54

55 Questions? 55