1.040/1.401/ESD.018 Project Management Spring 2007 Lecture 14 Project Monitoring
1 Lecture 11 Resource Allocation Part1 (involving Continuous Variables- Linear Programming)...
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Lecture 11 Resource Allocation Part1
(involving Continuous Variables-
Linear Programming)
1.040/1.401/ESD.018Project Management
Samuel Labi and Fred Moavenzadeh
Massachusetts Institute of Technology
April 2, 2007
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Linear Programming
This Lecture
Part 1: Basics of Linear Programming
Part 2: Methods for Linear Programming
Part 3: Linear Programming Applications
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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
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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
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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
WW
W
X
XX
YY
Y
Examples of W =f(X,Y) response surfaces
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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
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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
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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
![Page 9: 1 Lecture 11 Resource Allocation Part1 (involving Continuous Variables- Linear Programming) 1.040/1.401/ESD.018 Project Management Samuel Labi and Fred.](https://reader036.fdocuments.in/reader036/viewer/2022062714/56649d545503460f94a309fc/html5/thumbnails/9.jpg)
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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
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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 YOutput, W
Resource X
Linear Programming
Local space
Local maximum
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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
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Linear Programming
Example 1: Sketch the following region:y – 2 > 0
SolutionFirst, make y the subjectWrite the equation of the critical boundarySketch the critical boundaryIndicate the region of interest
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Linear Programming
Sketch of the region: y > 2
2
- 1
1
3
4
5
- 2
y = 2
x
y
Critical Boundary
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Linear Programming
Example 2: Sketch of the region: x - 5 < 0
21 3 4 5
x = 5
x
y
(Critical Boundary)
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Linear Programming
Linear Programming
Example 3: Sketch of the region: y > 2
2
- 1
1
3
4
5
- 2
y = 1
x
y
(Critical Boundary)
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Linear Programming
Example 4: Sketch of the region: 1 – x ≤ 0
21 3 4 5
x = 1
x
y
(Critical Boundary)
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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)
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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
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Mean and Variance
Linear Programming
Example 7: Sketch of the region: x + 1 ≤ 0
21-1-2-3
x = -1
x
y
(Critical Boundary)
3
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Linear Programming
Mean and Variance
Linear Programming
Example 8: Sketch of the region: 2 - x ≤ 0
21-1-2-3
x = -2
x
y
(Critical Boundary)
3
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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)
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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
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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
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Linear Programming
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
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Linear Programming
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
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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
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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
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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
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Linear Programming
Example 15: Sketch the region bounded (or constrained) by the following functions
y > 0x > 0y < -3.5x + 5
y=0
y
(Critical Boundary)
y > 0
Its critical boundary is:
y = 0
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Linear Programming
Example 15: Sketch the region bounded (or constrained) by the following functions
y > 0x > 0y < -3.5x + 5
y=0
y
(Critical Boundary)
x > 0
Its critical boundary is:
x = 0
x=0
(Critical Boundary)
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Linear Programming
Example 15: Sketch the region bounded (or constrained) by the following functions
y > 0x > 0y < -3x + 5
y=0
y
(Critical Boundary)
y < -3x + 5
Thus, the critical boundary is:y = -3x + 5When x = 0, y = 5CB passes thru (0, 5)
When y = 0, x = 5/3CB passes thru (5/3, 0)
x=0
(Critical Boundary)
(Critical Boundary)
y= -3x + 5
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Linear Programming
Example 15: Sketch the region bounded (or constrained) by the following functions
y > 0x > 0y < -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)
(Critical Boundary)
y= -3x + 5
Feasible Region
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Linear Programming
Example 16: Sketch the region bounded (or constrained) by the following functions
y > 0y> - 0.2x + 5y < -0.5x + 5
x
y
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Linear Programming
Example 16: Sketch the region bounded (or constrained) by the following functions
y > 0y> - 0.2x + 5y < -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 + 5y= -0.5x + 5
(Critical Boundary)
Feasible Region
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Linear Programming
Example 17: Sketch the region bounded (or constrained) by the following functions
y > 3y < -2x + 6y < x + 1
y
x
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Linear Programming
Example 17: Sketch the region bounded (or constrained) by the following functions
y > 3y < -2x + 6y < 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
x3-1
6
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Linear Programming
Example 18: Sketch the region bounded (or constrained) by the following functions
y > 0x > 0y < -x + 5y < x+2
y
x
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Linear Programming
Example 18: Sketch the region bounded (or constrained) by the following functions
y > 0x > 0y < -x + 5y < 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=03-2
5
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Linear Programming
Example 19: Sketch the region bounded (or constrained) by the following functions
y > 0x > 0y < -0.33x + 1y > 2x - 5
y
x
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Linear Programming
Example 19: Sketch the region bounded (or constrained) by the following functions
y > 0x > 0y < -0.33x + 1y > 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)
(Critical Boundary)
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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
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Linear Programming
Example 19: Sketch the region bounded (or constrained) by the following functions
y > 0x > 0y < -0.33x + 1y > 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)
(Critical Boundary)
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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.
Linear Programming
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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
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Symbols for decision variables
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.
x1
x2
x3
x2
x1
Linear Programming
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Dimensionality of Optimization Problems
An optimization problem with n decision variables n-dimensional
Linear Programming
W=f(x1)
1 Decision Variable
x1
1-dimensional
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Dimensionality of Optimization Problems
An optimization problem with n decision variables n-dimensional
x1
x2
2-dimensional
2 Decision Variables
Intersecting lines yield vertices (problem solutions)
Linear Programming
W=f(x1 , x2)W=f(x1)
1 Decision Variable
x1
1-dimensional
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Dimensionality of Optimization Problems
An optimization problem with n decision variables n-dimensional
x1
x2
x2
x1
2-dimensional 3-dimensional
2 Decision Variables 3 Decision Variables
Intersecting lines yield vertices (problem solutions)
Intersecting planes yield vertices (problem solutions)
x3
Linear Programming
W=f(x1 , x2)W=f(x1) W=f(x1 , x2, x3)
1 Decision Variable
x1
1-dimensional
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Dimensionality of Optimization Problems
An optimization problem with n decision variables n-dimensional
x1
x2
x2
x1
2-dimensional 3-dimensional
2 Decision Variables 3 Decision Variables
n-dimensional
n Decision Variables
Sorry! Cannot
be visualize
d
Intersecting lines yield vertices (problem solutions)
Intersecting planes yield vertices (problem solutions)
Intersecting objects yield vertices (problem solutions)
x3
Linear Programming
W=f(x1 , x2)W=f(x1) W=f(x1 , x2, x3) W=f(x1 , x2, …, xn)
1 Decision Variable
x1
1-dimensional
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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
Linear Programming
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Solution
The objective function is: W = 8x + 5y
The constraints are:y > 0x > 0y < -0.33x + 1y < 2x – 5
The control values are x and y.
Linear Programming
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Linear Programming
y=3
y
(Critical Boundary)
(Critical Boundary)
y = 2x - 5y= 0.33x + 1
(Critical Boundary)
Feasible Region
x
(0, 1)
(3.6, 2.2)
(0, 0)(2.5, 0)
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) 2.5 0 = 8(2.5) + 5(0) = 20
(3.6, 2.2) 3.6 2.2 = 8(3.6) + 5(2.2) = 36
Solution (cont’d)
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Solution (continued)
Therefore, the maximum value of W is 36,And this happens when x = 3.6 and y = 2.2
That is: Wopt = 36 units
yopt = 3.6 units
xopt = 2.2 units
This set of answers represents the “optimal solution”.
Linear Programming
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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
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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
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Questions?
Linear Programming