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EMGT 388 – Quantitative Methods in Cost Engineering
Lecture Notes by Instructor: Dr. J. Rene van Dorp Chapter 27 - Page 275 Source: Financial Models Using Simulation and Optimization by Wayne Winston
PROJECT RISK ANALYSIS
WHY IS IT NEEDED?
• Large Project (e.g. construction projects)
generally fail to meet company deadlines
• Large Projects generally exceed the
calculated budget.
WHY DOES THIS HAPPEN?
• At least partially because scheduling and budgeting are based on point
estimates for activity completion times and activity cost.
PROPOSED ALTERNATIVE?
Perform Scheduling and Budgeting under Uncertainty:
EMGT 388 – Quantitative Methods in Cost Engineering
Lecture Notes by Instructor: Dr. J. Rene van Dorp Chapter 27 - Page 276 Source: Financial Models Using Simulation and Optimization by Wayne Winston
STEP 1: Model Activity/Cost Uncertainty
STEP 2: Identify Risk Factors
STEP 3: Calculate Project Uncertainty in Completion Time and Cost
STEP 4: Schedule and Budget using Companies Risk Averseness.
THIS LECTURE WILL FOCUSS PRIMARILY ON SCHEDULE UNCERTIANTY
EMGT 388 – Quantitative Methods in Cost Engineering
Lecture Notes by Instructor: Dr. J. Rene van Dorp Chapter 27 - Page 277 Source: Financial Models Using Simulation and Optimization by Wayne Winston
INTERMEZZO: Modeling a Project via Activities and Milestones
Tom Lingley, an independent contractor, has agreed to build a new room on an
existing house. He plans to begin work on Monday Morning, June 1. The main
question is when will he complete his work. The owner of the house is particularly
hopeful that the room will be ready by Tuesday Morning, June 29 , that is, in 21
or fewer working days. The work proceeds in Stages (or Activities) labeled A
through J.
EMGT 388 – Quantitative Methods in Cost Engineering
Lecture Notes by Instructor: Dr. J. Rene van Dorp Chapter 27 - Page 278 Source: Financial Models Using Simulation and Optimization by Wayne Winston
Index Duration PREDECESSORS
A Prepare Foundation 4.5 None
B Put Up Frame 4 A
C Order Custom Windows 12 None
D Erect Outside Walls 3.5 B
E Do electrical Wiring 4.5 D
F Do Plumbing 3.5 D
G Put in Duct Work 4 D
H Hang Drywall 3 E,F,G
I Install Windows 1 B,C
J Paint and Clean Up 2 H
EMGT 388 – Quantitative Methods in Cost Engineering
Lecture Notes by Instructor: Dr. J. Rene van Dorp Chapter 27 - Page 279 Source: Financial Models Using Simulation and Optimization by Wayne Winston
1
2
4
3 5
A-4.5
B-4
C-12
Dummy-0
D-3.56
7
8
E-4.5
F-3.5
G-4
H-3
I-1
J-2
Calculating Project Completion Time
EMGT 388 – Quantitative Methods in Cost Engineering
Lecture Notes by Instructor: Dr. J. Rene van Dorp Chapter 27 - Page 280 Source: Financial Models Using Simulation and Optimization by Wayne Winston
D1
D2
D3
S1
S2
S3
Milestone Earliest Start time
t = Max(Si+ Di)
D1
D2
D3
S1
S2
S3
Milestone Earliest Start time
t = Max(Si+ Di)
EMGT 388 – Quantitative Methods in Cost Engineering
Lecture Notes by Instructor: Dr. J. Rene van Dorp Chapter 27 - Page 281 Source: Financial Models Using Simulation and Optimization by Wayne Winston
1
2
4
3 5
A-4.5
B-4
C-12
Dummy-0
D-3.56
7
8
E-4.5
F-3.5
G-4
H-3
I-1
J-2
?4.5?4.5
?0?0
?8.5?8.5
?12?12
?16.5?16.5 ?
19.5?19.5
?21.5?21.5
?12?12
EMGT 388 – Quantitative Methods in Cost Engineering
Lecture Notes by Instructor: Dr. J. Rene van Dorp Chapter 27 - Page 282 Source: Financial Models Using Simulation and Optimization by Wayne Winston
Calculating Critical Milestones and Critical Activities
D1
D2
D3
E1
E2
E3
Milestone Latest Start time
t = Min(Ei - Di)
EMGT 388 – Quantitative Methods in Cost Engineering
Lecture Notes by Instructor: Dr. J. Rene van Dorp Chapter 27 - Page 283 Source: Financial Models Using Simulation and Optimization by Wayne Winston
1
2
4
3 5
A-4.5
B-4
C-12
Dummy-0
D-3.56
7
8
E-4.5
F-3.5
G-4
H-3
I-1
J-2
4.54.54.54.5
0000
8.58.58.58.5
12121212
16.516.516.516.5 19.5
19.519.519.5
21.521.521.521.5
20.51220.512
COMPLICATING FACTORS: Activity Durations are uncertain?
PROBLEM THAT NEEDS TO BE SOLVED: What is the uncertainty in the Project
Completion Time given the Uncertainty in the Activity Durations?
EMGT 388 – Quantitative Methods in Cost Engineering
Lecture Notes by Instructor: Dr. J. Rene van Dorp Chapter 27 - Page 284 Source: Financial Models Using Simulation and Optimization by Wayne Winston
Index Min Most Likely Max Mean
A Prepare Foundation 1.5 4.5 8.5 4.5
B Put Up Frame 3 4 5 4
C Order Custom Windows 7 12 19 12
Dummy Activity 0 0 0 0
D Erect Outside Walls 2 3.5 6 3.5
E Do electrical Wiring 3 4.5 7 4.5
F Do Plumbing 2 3.5 6 3.5
G Put in Duct Work 2 4 6 4
H Hang Drywall 2.5 3 3.5 3
I Install Windows 0.5 1 1.5 1
J Paint and Clean Up 1.5 2 2.5 2
EMGT 388 – Quantitative Methods in Cost Engineering
Lecture Notes by Instructor: Dr. J. Rene van Dorp Chapter 27 - Page 285 Source: Financial Models Using Simulation and Optimization by Wayne Winston
APPROACH STEP 1:
MODEL THE UNCERTAINTY IN ACTIVITY DURATION USING TRIANGULAR DISTRIBUTIONS
INTERMEZZO – TRIANGULAR DISTRIBUTION
Three Parameters: Lower Bound: L Upper Bound: U Most Likely (or Best Guess): M
EMGT 388 – Quantitative Methods in Cost Engineering
Lecture Notes by Instructor: Dr. J. Rene van Dorp Chapter 27 - Page 286 Source: Financial Models Using Simulation and Optimization by Wayne Winston
Density Function of Triangular Distribution
L M UL
2A1 A2(U-L)
• Mean
3][ UMLXE ++=
• Variance
18))(()(][
2 MULMULXVar −−−−=
EMGT 388 – Quantitative Methods in Cost Engineering
Lecture Notes by Instructor: Dr. J. Rene van Dorp Chapter 27 - Page 287 Source: Financial Models Using Simulation and Optimization by Wayne Winston
RESULTS UNCERTAINTY ANALYSIS
Distribution Earliest Start Time Milestone 8
00.020.040.060.080.1
0.120.140.160.180.2
0 10 20 30
Earliest Event Time
EMGT 388 – Quantitative Methods in Cost Engineering
Lecture Notes by Instructor: Dr. J. Rene van Dorp Chapter 27 - Page 288 Source: Financial Models Using Simulation and Optimization by Wayne Winston
• There is a 10% chance that the project will finish in 20.7 days. • You are 90% certain that the project will finish within 26.6 days.
Comparison Distributions Earliest and Latest Start Time Milestone 8
0.00%
10.00%
20.00%
30.00%
40.00%
50.00%
60.00%
70.00%
80.00%
90.00%
100.00%
0 5 10 15 20 25 30 35 40 45
Cum
ulat
ive
Perc
enta
ge
Earliest Start Time Latest Start Time
EMGT 388 – Quantitative Methods in Cost Engineering
Lecture Notes by Instructor: Dr. J. Rene van Dorp Chapter 27 - Page 289 Source: Financial Models Using Simulation and Optimization by Wayne Winston
WHAT ARE THE CRITICAL ACTIVITIES?
Criticality Index by Activity
0 0.2 0.4 0.6 0.8 1
Order Custom Windows
Dummy Activity
Install Windows
Do Plumbing
Put in Duct Work
Erect Outside Walls
Do electrical Wiring
Prepare Foundation
Put Up Frame
Hang Drywall
Paint and Clean Up
Criti
calit
y In
dex
Activity
EMGT 388 – Quantitative Methods in Cost Engineering
Lecture Notes by Instructor: Dr. J. Rene van Dorp Chapter 27 - Page 290 Source: Financial Models Using Simulation and Optimization by Wayne Winston
WHERE IS UNCERTAINTY COMING FROM?
Rank Correlations by Activity
-1 -0.5 0 0.5 1
Order Custom Windows
Install Windows
Hang Drywall
Do Plumbing
Paint and Clean Up
Put in Duct Work
Put Up Frame
Erect Outside Walls
Do electrical Wiring
Prepare Foundation
Dummy Activity
Crit
ical
ity In
dex
Activity
EMGT 388 – Quantitative Methods in Cost Engineering
Lecture Notes by Instructor: Dr. J. Rene van Dorp Chapter 27 - Page 291 Source: Financial Models Using Simulation and Optimization by Wayne Winston
IS THE ASSUMPTION OF INDEPENDENCE OF ACTIVITY DURATIONS
REASONABLE?
ANSWER:
• No, with a booming economy people invest in their homes through home
improvement projects. With a limited of contractors available you typically,
see over commitment of their resources, resulting in overall activity delays.
• No, some of the activities occur under the open sky and these activities will
all be delayed when weather is bad.
CONCLUSION:
Activity Duration exhibit (strong?) positive dependence!
EMGT 388 – Quantitative Methods in Cost Engineering
Lecture Notes by Instructor: Dr. J. Rene van Dorp Chapter 27 - Page 292 Source: Financial Models Using Simulation and Optimization by Wayne Winston
HomeImprovement
Market
Prepare Foundation
Put up Frame
Order custom Windows
Erect outside walls
Do electrical wiring
Do plumbing
Put in duct work
Install Windows
Paint and clean up
0.80
0.80
0.80
0.80
0.80
0.800.80
0.80
0.80
EMGT 388 – Quantitative Methods in Cost Engineering
Lecture Notes by Instructor: Dr. J. Rene van Dorp Chapter 27 - Page 293 Source: Financial Models Using Simulation and Optimization by Wayne Winston
INTERMEZZO: Modeling Dependence Between
Two Random Variables
• Suppose random variable X has cumulative distribution function F(.)
• Suppose random variable Y has cumulative distribution function G(.)
• Random Variable UX = F(X) is Uniform random variable on [0,1].
• Random Variable UY = F(Y) is Uniform random variable on [0,1].
CONCLUSION: Instead of modeling the positive dependence
between X and Y, we may equivalently model
positive dependence between UX and UY.
EMGT 388 – Quantitative Methods in Cost Engineering
Lecture Notes by Instructor: Dr. J. Rene van Dorp Chapter 27 - Page 294 Source: Financial Models Using Simulation and Optimization by Wayne Winston
APPROACH: Dependence between UX and UY is completely specified when a two-imensional
distribution on the unit square is defined with uniform marginal distributions
DEFINITION: A COPULA A copula is a bivariate distribution with
uniform marginal distribution on the unit square.
EMGT 388 – Quantitative Methods in Cost Engineering
Lecture Notes by Instructor: Dr. J. Rene van Dorp Chapter 27 - Page 295 Source: Financial Models Using Simulation and Optimization by Wayne Winston
Example Copula: Diagonal Band Distribution
1
UX - axis1- θ 10
v = u
- 1 + θ
v = u
+ 1 - θ
a
b
u
Area 3
Area 1
Area 5
Area 2
Area 4
UY - axis
EMGT 388 – Quantitative Methods in Cost Engineering
Lecture Notes by Instructor: Dr. J. Rene van Dorp Chapter 27 - Page 296 Source: Financial Models Using Simulation and Optimization by Wayne Winston
SAMPLING ALGORITHM:
Step 1: Sample uX from a uniform on [0,1]
Step 2: Sample uY from a uniform on [a,b]
Step 3: If uY < 0, uY:= - uY.
Step 4: If uY > 1, uY:= 1-(1- uY.)
EMGT 388 – Quantitative Methods in Cost Engineering
Lecture Notes by Instructor: Dr. J. Rene van Dorp Chapter 27 - Page 297 Source: Financial Models Using Simulation and Optimization by Wayne Winston
RESULTS WITH DEPENDENCE
Distribution Earliest Start Time Milestone 8
0
0.02
0.04
0.06
0.08
0.1
0.12
0 10 20 30
Earliest Event Time
EMGT 388 – Quantitative Methods in Cost Engineering
Lecture Notes by Instructor: Dr. J. Rene van Dorp Chapter 27 - Page 298 Source: Financial Models Using Simulation and Optimization by Wayne Winston
• There is a 25% chance (instead of 10%) that the project will finish in 20 days.
• You are 74% certain (instead of 10%) that the project will finish within 26.6
days.
Comparison Distributions Earliest and Latest Start Time Milestone 8
0.00%
10.00%
20.00%
30.00%
40.00%
50.00%
60.00%
70.00%
80.00%
90.00%
100.00%
0 5 10 15 20 25 30 35 40 45
Cum
ulat
ive
Per
cent
age
Earliest Start Time Latest Start Time
EMGT 388 – Quantitative Methods in Cost Engineering
Lecture Notes by Instructor: Dr. J. Rene van Dorp Chapter 27 - Page 299 Source: Financial Models Using Simulation and Optimization by Wayne Winston
WHAT ARE THE CRITICAL ACTIVITIES?
Criticality Index by Activity
0 0.2 0.4 0.6 0.8 1
Order Custom Windows
Dummy Activity
Install Windows
Do Plumbing
Put in Duct Work
Prepare Foundation
Put Up Frame
Erect Outside Walls
Do electrical Wiring
Hang Drywall
Paint and Clean Up
Criti
calit
y In
dex
Activity
EMGT 388 – Quantitative Methods in Cost Engineering
Lecture Notes by Instructor: Dr. J. Rene van Dorp Chapter 27 - Page 300 Source: Financial Models Using Simulation and Optimization by Wayne Winston
WHERE IS UNCERTAINTY COMING FROM?
Rank Correlations by Activity
-1 -0.5 0 0.5 1
Put in Duct Work
Do Plumbing
Paint and Clean Up
Order Custom Windows
Install Windows
Hang Drywall
Put Up Frame
Prepare Foundation
Erect Outside Walls
Do electrical Wiring
Dummy Activity
Crit
ical
ity In
dex
Activity
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