introduction to project risk analysisdorpjr/emse388/session 13...project risk analysis why is it...

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:


Lecture Notes by Instructor: Dr. J. Rene van Dorp Chapter 27 - 76 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



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.



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



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



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)



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



Calculating Critical Milestones and Critical Activities

D1

D2

D3

E1

E2

E3

Milestone Latest Start time

t = Min(Ei - Di)



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?



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



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



Density Function of Triangular Distribution

L M UL

2A1 A2(U-L)

• Mean

3][ UMLXE ++=

• Variance

18))(()(][

2 MULMULXVar −−−−=



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

PDF



• 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



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



WHERE IS UNCERTAINTY COMING FROM?

Rank Correlations by Activity

-1 -0.5 0 0.5 1


Install Windows

Hang Drywall

Do Plumbing

Paint and Clean Up

Put in Duct Work

Put Up Frame

Erect Outside Walls


Prepare Foundation

Dummy Activity

Crit

ical

ity In

dex

Activity



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!



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



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.



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.



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



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.)



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

PDF



• 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



WHAT ARE THE CRITICAL ACTIVITIES?

Criticality Index by Activity

0 0.2 0.4 0.6 0.8 1


Dummy Activity

Install Windows

Do Plumbing

Put in Duct Work

Prepare Foundation

Put Up Frame

Erect Outside Walls


Hang Drywall

Paint and Clean Up

Criti

calit

y In

dex

Activity



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


Install Windows

Hang Drywall

Put Up Frame

Prepare Foundation

Erect Outside Walls


Dummy Activity

Crit

ical

ity In

dex

Activity

introduction to project risk analysisdorpjr/emse388/session 13...project risk analysis why is it...

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