samuel labi and fred moavenzadeh department of civil and environmental engineering

1.040/1.401 – Project Management

Probabilistic Planning Part I

Samuel Labi and Fred MoavenzadehSamuel Labi and Fred Moavenzadeh

Department of Civil and Environmental EngineeringDepartment of Civil and Environmental EngineeringMassachusetts Institute of TechnologyMassachusetts Institute of Technology

March 19, 2007

2

Project Management PhasesProject Management Phases

FEASIBILITY CLOSEOUTDEVELOPMENT OPERATIONS DESIGNDESIGNPLANNING PLANNING

Project Evaluation

Project Finance

Project Organization

Project Cost Estimation

Project Planning and Scheduling

Recall …

3

Outline for this Lecture

Deterministic systems decision-making – the general picture

Probabilistic systems decision-making – the general picture

– the special case of project planning

Why planning is never deterministic

Simplified examples of deterministic, probabilistic planning

- everyday life- project planning

Illustration of probabilistic project planning

PERT – Basics, terminology, advantages, disadvantages, example

4

Deterministic Systems Planning and Decision-making

DeterministicAnalysis of Engineering

System

Input variables and their FIXED VALUES

Examples: queuing systems, network systems, etc.

Examples: costs, time duration, quality, interest rates, etc.

FIXED VALUES of the Outputs (system performance criteria, etc.)

VALUE OF combined output (index, utility or value) representing multiple performance measures of the system

A SINGLE evaluation outcome

Alt. 1 Alt. k Alt. n …

…

…

Final evaluation result and decision

Influence of deterministic inputs on the outputs of engineering systems -- the general picture

X*

Y*

Z*

O1*

O2*

OM*

O*COMBO

5

But engineering systems are never deterministic!

Why?For sample, for Project Planning Systems … Variations in planning input parameters Beyond control of project manager Categories of the variation factors

● Natural (weather -- good and bad, natural disasters, etc.)

● Man-made (equipment breakdowns, strikes, new technology, worker morale, poor design, site problems, interest rates, etc.)

Combined effect of input factor variation is a variation in the outputs (costs, time, quality)

6

Why planning is never deterministic -- II

Probabilistic Analysis of Engineering

System

fX

Input variables and their probability distributions

Examples: queuing systems, network systems, etc.

Examples: costs, time durations, quality, interest rates, etc.

f(O1)

Outputs (system performance criteria, etc.) and their probability distributions

Probability distributions for individual outputs

Probability distribution for combined output (index, utility or value) representing multiple performance measures

Discrete probability distribution for evaluation outcome

PAlt. i is the probability that an alternative turns out to be most desirable or most critical

PAlt. i

Alt. 1 Alt. 2 Alt. n …

…

…

Variability of final evaluation result and

decision

Influence of stochastic inputs on the outputs of engineering systems -- the general picture

fY

fZ

f(O2)

f(OM)

f(OCOMBO)

7

Why planning is never deterministic -- III

f

Input variable and its probability distribution

Probability distribution?

What exactly do you mean by that?

See survey for dinner durations (times) of class members

8

Probability Distribution for your Dinner Times:

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0 2.5 7.5 12.5 17.5 22.5 27.5 32.5

Time Spent for Dinner (minutes)

Pro

bab

ility

9

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0 2.5 7.5 12.5 17.5 22.5 27.5 32.5


Pro

bab

ility

Probability that a randomly selected student spends less than T* minutes for dinner is:

… less than T* = P(T < T*) =

*)(*

ZZPTT

P

10

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0 2.5 7.5 12.5 17.5 22.5 27.5 32.5


Pro

bab

ility

Probability that a randomly selected student spends less than T* minutes for dinner is:

… less than T1 = P(T < T*) =

P(T)

T

f(Z)

Z

T*

Z*

Mean = μStd dev = σ

Mean = 0Std dev = 1*)(

*ZZP

TTP

μ = 0

μ0

Standard Normal Transformation

11

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0 2.5 7.5 12.5 17.5 22.5 27.5 32.5


Pro

bab

ility

Probability that a randomly selected student spends …

… less than T1 = P(T < T*) =

… more than T1 = P(T > T*) =

P(T)

T

f(Z)

Z

T*

Z*


Mean = 0Std dev = 1

*)(*

ZZPTT

P

μ = 0

μ0


*)(*

ZZPTT

P

12

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0 2.5 7.5 12.5 17.5 22.5 27.5 32.5


Pro

bab

ility

Probability that a randomly selected student spends …

… less than T* = P(T < T*) =

… more than T* = P(T > T*) =

… between T*1 and T*2 = P(T*1 < T < T*2) =

P(T)

T

f(Z)

Z

T*

Z*


Mean = 0Std dev = 1

*)(*

ZZPTT

P

μ = 0

μ0


)( *2

*1

*2

*1 ZZZP

TTTP

*)(*

ZZPTT

P

13

Probability is the area under the probability distribution/density curves)

Probability can be found using any one of three ways:

- coordinate geometry

- calculus

- statistical tables

14

Like-wise, we can build probability distributions for project planning parameters by …

- using historical data from past projects, OR

- computer simulation

And thus we can find the probability that project durations falls within a certain specified range

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0 2.5 7.5 12.5 17.5 22.5 27.5 32.5


Pro

bab

ility

15

Probabilistic planning of project

management systems can involve uncertainties in:● Need for an Activity (need vs. no need)● Durations

» Activity durations» Activity start-times and end-times

● Cost of activities● Quality of Workmanship and materials● Etc.

16

Probabilistic planning of project management systems can involve uncertainties in:● Need for an Activity (need vs. no need)● Durations

» Activity durations» Activity start-times and end-times

● Cost of activities● Quality of Workmanship and materials● Etc.

17

Probabilistic Analysis of

Project Planning

Input variable and its probability distribution

Project planning system

Project activity durations

Outputs (system performance criteria, etc.) and their probability distributions

Probability distributions for the output

In this case, …“Output” is the identification of the critical path for the project.

Frequency distribution or probability distribution for evaluation outcome

PAlt.i is the probability that a given path turns out to be the critical path

PAlt. i

Path 1 Path 2 Path n …

Variability of final evaluation result and

decision

Influence of stochastic inputs -- the specific picture of project planning

fXf(O1)

18

SAM Waking up and meditating

START = 7AM

Duration 1hr

FINISH =8AM

ActivityStart Time

Activity Duration Finish Time

KEY

SAM Bathing, Breakfast,,

Reading, Transport to MIT, etc.

START =8AM

Duration 5hrs FINISH = 1 PM

US Meeting in Class

For this LectureSTART= 1PM

Duration1.5hr FINISH= 2:30

YOU Preparing for Classes, etc.

START= 7 AM

Duration 1 hr FINISH= 8AM

YOU showing up at Other Classes

START= 8AM

Duration 5 hr FINISH=1 PM

YOU missing the lectureSTART=

1 PMDuration1.5hr FINISH

= 2:30

Perfectly deterministic

Probabilistic planning: An example in everyday life

19


LS = 7AM

ES = 6AM

Duration = 1 hour

EF = 7AMLF = 8AM

Activity

Latest Start

Earliest Start

Duration of

Activity

Earliest FinishLatest Finish

KEY


Reading, Transport to MIT, etc.LS =

8AM

ES =7AM

Duration = 5 hrs

EF = NOONLF = 1 PM

US Meeting in Class

For this LectureLS = 1 PM

ES = 12:55

Duration = 1.5 hrs

EF = 2:25LF = 2:30


LS = 7AM

ES = 6:45

Duration = 1 hr

EF = 7:45LF = 8AM


LS = 8AM

ES = 7:45

Duration = 5 hour

EF = 12:45LF = 1 PM

YOU missing the lecture

LS = 7AM

ES = 6AM

Duration = 1 hour

EF = 7AMLF = 8AM

Partly deterministic, Partly probabilistic


20


LS = 7AM

ES = 6AM

Duration = 1 hour

EF = 7AMLF = 8AM

Activity

Latest Start

Earliest Start

Duration of

Activity


KEY



8AM

ES =7AM

Duration = 5 hrs

EF = NOONLF = 1 PM

US Meeting in Class


ES = 12:55

Duration = 1.5 hrs

EF = 2:25LF = 2:30


LS = 7AM

ES = 6:45

Duration = 1 hr

EF = 7:45LF = 8AM


LS = 8AM

ES = 7:45

Duration = 5 hour

EF = 12:45LF = 1 PM


LS = 7AM

ES = 6AM

Duration = 1 hour

EF = 7AMLF = 8AM



21


LS = 7AM

ES = 6AM

Duration = 1 hour

EF = 7AMLF = 8AM

Activity

Latest Start

Earliest Start

Duration of

Activity


KEY



8AM

ES =7AM

Duration = 5 hrs

EF = NOONLF = 1 PM

US Meeting in Class


ES = 12:55

Duration = 1.5 hrs

EF = 2:25LF = 2:30


LS = 7AM

ES = 6:45

Duration = 1 hr

EF = 7:45LF = 8AM


LS = 8AM

ES = 7:45

Duration = 5 hour

EF = 12:45LF = 1 PM


LS = 7AM

ES = 6AM

Duration = 1 hour

EF = 7AMLF = 8AM



22


LS = 7AM

ES = 6AM

Duration = 1 hour

EF = 7AMLF = 8AM

Activity

Latest Start

Earliest Start

Duration of

Activity


KEY



8AM

ES =7AM

Duration = 5 hrs

EF = NOONLF = 1 PM

US Meeting in Class


ES = 12:55

Duration = 1.5 hrs

EF = 2:25LF = 2:30


LS = 7AM

ES = 6:45

Duration = 1 hr

EF = 7:45LF = 8AM


LS = 8AM

ES = 7:45

Duration = 5 hour

EF = 12:45LF = 1 PM


LS = 7AM

ES = 6AM

Duration = 1 hour

EF = 7AMLF = 8AM



23


LS = 7AM

ES = 6AM

Duration = 1 hour

EF = 7AMLF = 8AM

Activity

Latest Start

Earliest Start

Duration of

Activity


KEY



8AM

ES =7AM

Duration = 5 hrs

EF = NOONLF = 1 PM

US Meeting in Class


ES = 12:55

Duration = 1.5 hrs

EF = 2:25LF = 2:30


LS = 7AM

ES = 6:45

Duration = 1 hr

EF = 7:45LF = 8AM


LS = 8AM

ES = 7:45

Duration = 5 hour

EF = 12:45LF = 1 PM


LS = 7AM

ES = 6AM

Duration = 1 hour

EF = 7AMLF = 8AM



24


LS = 7AM

ES = 6AM

Duration = 1 hour

EF = 7AMLF = 8AM

Activity

Latest Start

Earliest Start

Duration of

Activity


KEY



8AM

ES =7AM

Duration = 5 hrs

EF = NOONLF = 1 PM

US Meeting in Class


ES = 12:55

Duration = 1.5 hrs

EF = 2:25LF = 2:30


LS = 7AM

ES = 6:45

Duration = 1 hr

EF = 7:45LF = 8AM


LS = 8AM

ES = 7:45

Duration = 5 hour

EF = 12:45LF = 1 PM


LS = 7AM

ES = 6AM

Duration = 1 hour

EF = 7AMLF = 8AM



25


LS = 7AM

ES = 6AM

Duration = 1 hour

EF = 7AMLF = 8AM

Activity

Latest Start

Earliest Start

Duration of

Activity


KEY



8AM

ES =7AM

Duration = 5 hrs

EF = NOONLF = 1 PM

US Meeting in Class


ES = 12:55

Duration = 1.5 hrs

EF = 2:25LF = 2:30


LS = 7AM

ES = 6:45

Duration = 1 hr

EF = 7:45LF = 8AM


LS = 8AM

ES = 7:45

Duration = 5 hour

EF = 12:45LF = 1 PM


LS = 7AM

ES = 6AM

Duration = 1 hour

EF = 7AMLF = 8AM



26


LS = 7AM

ES = 6AM

Duration = 1 hour

EF = 7AMLF = 8AM

Activity

Latest Start

Earliest Start

Duration of

Activity


KEY



8AM

ES =7AM

Duration = 5 hrs

EF = NOONLF = 1 PM

US Meeting in Class


ES = 12:55

Duration = 1.5 hrs

EF = 2:25LF = 2:30


LS = 7AM

ES = 6:45

Duration = 1 hr

EF = 7:45LF = 8AM


LS = 8AM

ES = 7:45

Duration = 5 hour

EF = 12:45LF = 1 PM


LS = 7AM

ES = 6AM

Duration = 1 hour

EF = 7AMLF = 8AM



27


LS = 7AM

ES = 6AM

Duration = 1 hour

EF = 7AMLF = 8AM

Activity

Latest Start

Earliest Start

Duration of

Activity


KEY



8AM

ES =7AM

Duration = 5 hrs

EF = NOONLF = 1 PM

US Meeting in Class


ES = 12:55

Duration = 1.5 hrs

EF = 2:25LF = 2:30


LS = 7AM

ES = 6:45

Duration = 1 hr

EF = 7:45LF = 8AM


LS = 8AM

ES = 7:45

Duration = 5 hour

EF = 12:45LF = 1 PM


LS = 7AM

ES = 6AM

Duration = 1 hour

EF = 7AMLF = 8AM



28


LS = 7AM

ES = 6AM

Duration = 1 hour

EF = 7AMLF = 8AM

Activity

Latest Start

Earliest Start

Duration of

Activity


KEY



8AM

ES =7AM

Duration = 5 hrs

EF = NOONLF = 1 PM

US Meeting in Class


ES = 12:55

Duration = 1.5 hrs

EF = 2:25LF = 2:30


LS = 7AM

ES = 6:45

Duration = 1 hr

EF = 7:45LF = 8AM


LS = 8AM

ES = 7:45

Duration = 5 hour

EF = 12:45LF = 1 PM


LS = 7AM

ES = 6AM

Duration = 1 hour

EF = 7AMLF = 8AM



29


LS = 7AM

ES = 6AM

Duration = 1 hour

EF = 7AMLF = 8AM

Activity

Latest Start

Earliest Start

Duration of

Activity


KEY



8AM

ES =7AM

Duration = 5 hrs

EF = NOONLF = 1 PM

US Meeting in Class


ES = 12:55

Duration = 1.5 hrs

EF = 2:25LF = 2:30


LS = 7AM

ES = 6:45

Duration = 1 hr

EF = 7:45LF = 8AM


LS = 8AM

ES = 7:45

Duration = 5 hour

EF = 12:45LF = 1 PM


LS = 7AM

ES = 6AM

Duration = 1 hour

EF = 7AMLF = 8AM



30


LS = 7AM

ES = 6AM

Duration EF = 7AMLF = 8AM

Activity

Latest Start

Earliest Start

Duration of

Activity


KEY



8AM

ES =7AM

Duration EF = NOONLF = 1 PM

US Meeting in Class

For this Lecture

LS = 1 PM

ES = 12:55

Duration EF = 2:25LF = 2:30


LS = 7AM

ES = 6:45

Duration

EF = 7:45LF = 8AM


LS = 8AM

ES = 7:45

Duration

EF = 12:45LF = 1 PM


LS = 7AM

ES = 6AM


Fully probabilistic

μ = 1hrσ = 0.25hr

μ = 5 hrσ = 0.6hr

μ =1.5 hrσ = 0.31hr

μ = 1 hrσ = 0.15 hr

μ = 5 hrσ = 0.35 hr

μ = 1hrσ = 0.2 hr


31


LS = 7AM

ES = 6AM


Activity

Latest Start

Earliest Start

Duration of

Activity


KEY



8AM

ES =7AM

Duration EF = NOONLF = 1 PM

US Meeting in Class

For this Lecture

LS = 1 PM

ES = 12:55

Duration EF = 2:25LF = 2:30


LS = 7AM

ES = 6:55

Duration

EF = 7:55LF = 8AM


LS = 8AM

ES = 7:55

Duration

EF = 12:55LF = 1 PM


LS = 7AM

ES = 6AM


Fully probabilistic

μ = 1hrσ = 0.25hr

μ = 5 hrσ = 0.6hr

μ =1.5 hrσ = 0.31hr

μ = 1 hrσ = 0.15 hr

μ = 5 hrσ = 0.35 hr

μ = 1hrσ = 0.2 hr


32

Probabilistic planning: An example in Project Management

Activity NameEarly StartLate Start

DurationEarly FinishLate Finish

KEY

Activity RMonth 0 4

months Month 4

Month 0

Month 4

Activity GMonth 4

3 months

Month 10

Month 7

Month 7

Activity JMonth 10

5 months

Month 15

Month 10

Month 15

Activity WMonth 15

2 months

Month 17

Month 15

Month 17

Activity AMonth 4 4

months Month 15

Month 11

Month 8

Activity CMonth 4 6

months Month 10

Month 4

Month 10

Activity MMonth 10 3

months Month 15

Month 12

Month 13

33

Activity NameEarly StartLate Start

Duration(O-M-P)

Early FinishLate Finish

KEY

Activity RMonth 0

4 Months(3-4-8) Mon

th 4Month 0

Month 4

Activity GMonth 4

3 Months(2-3-5) Mont

h 10Month 7

Month 7

Activity JMonth 10


h 15Month 10

Month 15

Activity WMonth 15


h 17Month 15

Month 17

Activity AMonth 4


h 15Month 11

Month 8

Activity CMonth 4


h 10Month 4

Month 10

Activity MMonth 10

3Months(1-3-5) Mont

h 15Month 12

Month 13

O: Optimistic (earliest time)M: Most probable timeP: Pessimistic (latest time)

Probabilistic planning: An example in Project Management

34

Probabilistic planning involving activity durations An Illustration

Activity RMonth 0

4 Months(3-4-8) Mon

th 4Month 0

Month 4

Activity GMonth 4


h 10Month 7

Month 7

Activity JMonth 10


h 15Month 10

Month 15

Activity WMonth 15


h 17Month 15

Month 17

Activity AMonth 4


h 15Month 11

Month 8

Activity CMonth 4


h 10Month 4

Month 10

Activity MMonth 10

3Months(1-3-5) Mont

h 15Month 12

Month 13

Activity

Op

tim

isti

c

Mo

st

Lik

ely

Pes

sim

isti

c

Mean Variance

R 3 4 8 4.50 0.833 0.69444A 3 4 5 4.00 0.333 0.11111G 2 3 5 3.17 0.500 0.25000C 3 6 7 5.67 0.667 0.44444J 2 5 6 4.67 0.667 0.44444M 1 3 5 3.00 0.667 0.44444W 1 2 4 2.17 0.500 0.25000

35


Activity RMonth 0

4 Months(3-4-8) Mon

th 4Month 0

Month 4

Activity GMonth 4


h 10Month 7

Month 7

Activity JMonth 10


h 15Month 10

Month 15

Activity WMonth 15


h 17Month 15

Month 17

Activity AMonth 4


h 15Month 11

Month 8

Activity CMonth 4


h 10Month 4

Month 10

Activity MMonth 10

3Months(1-3-5) Mont

h 15Month 12

Month 13

Activity

Op

tim

isti

c

Mo

st

Lik

ely

Pes

sim

isti

c

Mean Variance

R 3 4 8 4.50 0.833 0.69444A 3 4 5 4.00 0.333 0.11111G 2 3 5 3.17 0.500 0.25000C 3 6 7 5.67 0.667 0.44444J 2 5 6 4.67 0.667 0.44444M 1 3 5 3.00 0.667 0.44444W 1 2 4 2.17 0.500 0.25000

Let’s say we have lots of data on the durations of each activity. Such data is typically from:

- Historical records (previous projects)

- Computer simulation (Monte Carlo)

36


Activity RMonth 0

4 Months(3-4-8) Mon

th 4Month 0

Month 4

Activity GMonth 4


h 10Month 7

Month 7

Activity JMonth 10


h 15Month 10

Month 15

Activity WMonth 15


h 17Month 15

Month 17

Activity AMonth 4


h 15Month 11

Month 8

Activity CMonth 4


h 10Month 4

Month 10

Activity MMonth 10

3Months(1-3-5) Mont

h 15Month 12

Month 13

Activity

Op

tim

isti

c

Mo

st

Lik

ely

Pes

sim

isti

c

MeanStandard

Dev.Variance

R 3 4 8 4.50 0.833 0.69444A 3 4 5 4.00 0.333 0.11111G 2 3 5 3.17 0.500 0.25000C 3 6 7 5.67 0.667 0.44444J 2 5 6 4.67 0.667 0.44444M 1 3 5 3.00 0.667 0.44444W 1 2 4 2.17 0.500 0.25000

37

Calculate the Expected Duration of each path, and Identify the Critical Path on the basis of the mean only:

Path:Deterministic

Duration:Probabilistic

Duration:R-A-W 10 10.67R-G-J-W 14 14.50R-C-J-W 17 17.00 Critical PathR-C-M-W 15 15.33

Activity

Op

tim

isti

c

Mo

st

Lik

ely

Pe

ss

imis

tic

MeanStandard

Dev.Variance

R 3 4 8 4.50 0.833 0.69444A 3 4 5 4.00 0.333 0.11111G 2 3 5 3.17 0.500 0.25000

C 3 6 7 5.67 0.667 0.44444J 2 5 6 4.67 0.667 0.44444M 1 3 5 3.00 0.667 0.44444W 1 2 4 2.17 0.500 0.25000

38

Calculate the Expected Duration of each path, and Identify the Critical Path on the basis of both the mean and the std dev:

Activity

Op

tim

isti

c

Mo

st

Lik

ely

Pes

sim

isti

c

MeanStandard

Dev.Variance

R 3 4 8 4.50 0.833 0.69444A 3 4 5 4.00 0.333 0.11111G 2 3 5 3.17 0.500 0.25000C 3 6 7 5.67 0.667 0.44444J 2 5 6 4.67 0.667 0.44444M 1 3 5 3.00 0.667 0.44444W 1 2 4 2.17 0.500 0.25000

0

5

10

15

20

25

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

Mea

n D

urat

ion

Path R-A-W

Path R-G-J-

W

Path R-C-J-

W

Path R-C-M-W

39

0

5

10

15

20

25

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

Mea

n D

urat

ion

Path R-A-W

Path R-G-J-

W

Path R-C-J-W Path

R-C-M-W

Critical Path

40

0

5

10

15

20

25

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

Mea

n D

urat

ion

Critical path here can be considered as that with:- Longest duration (mean)- Greatest variation (stdev)

41

0

5

10

15

20

25

30

35

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

Mea

n D

urat

ion

Consider the following hypothetical project paths:

Path P-Q-R

Path P-F-W-

R

Path P-H-W-

RPath P-H-M-R

On the basis of mean duration only, Path P-H-W-R is the critical path

On the basis of the variance of durations only, Path P-F-W-R is the critical path

How would you decide the critical path on the basis of both mean duration and variance of durations?

42

Better way to identify critical path is using the amount of slack in each path (see later slides)

43

Another Example of Probabilistic Project Scheduling

- Monte Carlo Simulation- Similar activity structure as before, but Start and End activities are dummies (zero durations).

See Excel Sheet Attached

44

Benefits of Probabilistic Project Planning

Helps identify likely critical paths in situations where there is great uncertainty

Helps ascertain the likelihood (probability) that overall project duration will fall within a given range

Helps establish a scale of “criticality” among the project activities

Discussed in previous slides

45

Benefits of Probabilistic Project Planning

Helps identify likely critical paths in situations where there is great uncertainty

Helps ascertain the likelihood (probability) that overall project duration will fall within a given range

Helps establish a scale of “criticality” among the project activities Discussed in

subsequent slides

46

Probabilistic planning …

Is it ever used in real-life project management?

47

A Tool for Stochastic Planning: PERT

Program Evaluation and Review Technique (PERT)

- Need for PERT arose during the Space Race, in the late fifties- Developed by Booz-Allen Hamilton for US Navy, and Lockheed Corporation

- Polaris Missile/Submarine Project- R&D Projects- Time Oriented- Probabilistic Times- Assumes that activity durations are

Beta distributed

48

PERT Parameters

Optimistic duration a

Most Likely duration m

Pessimistic duration b

Expected duration

Standard deviation

Variance

6

4

2

12

3

1_ bmabamd

v s 2

sb a

6

49

Steps in PERT Analysis

Obtain a, m and b for each activity

Compute Expected Activity Duration d=te

Compute Variance v=s2

Compute Expected Project Duration D=Te

Compute Project Variance V=S2 as Sum of Critical Path Activity Variance

In Case of Multiple Critical Path Use the One with the Largest Variance

Calculate Probability of Completing the Project

50

PERT Example

Activity Predecessor a m b d vA - 1 2 4 2.17 0.25B - 5 6 7 6.00 0.11C - 2 4 5 3.83 0.25D A 1 3 4 2.83 0.25E C 4 5 7 5.17 0.25F A 3 4 5 4.00 0.11G B,D,E 1 2 3 2.00 0.11

51

PERT Example

Finding the Standard Deviation of the duration of a given path comprising Activities C, E, and G.

T

S V C V E V G

S

e

11

0 25 0 25 01111

0 6111

0 6111

0 7817

2 [ ] [ ] [ ]

. . .

.

.

.

52

PERT AnalysisFinding probability that project duration is less than some valueExample: probability that project ends before 10 months

P T T P T

P zT

S

P z

P z

P z

d

e

( )

.

.

.

.

.

10

10

10 11

0 7817

12793

1 12793

1 08997

01003

10%

53

Probability that the project will end before 13 months

P T P z

P z

1313 11

0 7817

2 5585

0 9948

.

.

.

54

Probability that the project will have a duration between 9 and 11.5 months

P T T T P T

P T P T

P z P z

P z P z

P z P z

L U

9 15

115 9

115 11

0 7817

9 11

0 7817

0 6396 2 5585

0 6396 1 2 5585

0 7389 1 0 9948

0 7389 0 0052

0 7337

.

.

. .

. .

. .

. .

. .

.

55

PERT Advantages

Includes Variance

Assessment of Probability of Achieving a Goal

56

PERT Disadvantages

Data intensive - Very Time Consuming

Validity of Beta Distribution for Activity Durations

Only one Critical Path considered

Assumes independence between activity durations

57

PERT Monte Carlo Simulation

Determine the Criticality Index of an Activity

Used 10,000 Simulations, Now from 1000 to 400 Have Been Reported as Giving Good Results

58

PERT Monte Carlo Simulation Process

Set the Duration Distribution for Each Activity

Generate Random Duration from Distribution

Determine Critical Path and Duration Using CPM

Record Results

59

Example Network

FF44

BB66

GG22 EndEndStartStart

AA2.172.17

CC3.833.83

DD2.832.83

EE5.175.17

AA

FFBB

EE

CC

DD GGEndEndStartStart

60

Monte Carlo Simulation Example

Statistics for Example Activities

Activity

OptimisticTime,

a

Most LikelyTime,

m

PessimisticTime,

b

ExpectedValue,

d

StandardDeviation,

sA 2 5 8 5 1B 1 3 5 3 0.66C 7 8 9 8 0.33D 4 7 10 7 1E 6 7 8 7 0.33F 2 4 6 4 0.66G 4 5 6 5 0.33

61


Activity DurationSummary of Simulation Runs for Example Project

RunNumber A B C D E F G

CriticalPath

CompletionTime

1 6.3 2.2 8.8 6.6 7.6 5.7 4.6 A-C-F-G 25.42 2.1 1.8 7.4 8.0 6.6 2.7 4.6 A-D-F-G 17.43 7.8 4.9 8.8 7.0 6.7 5.0 4.9 A-C-F-G 26.54 5.3 2.3 8.9 9.5 6.2 4.8 5.4 A-D-F-G 25.05 4.5 2.6 7.6 7.2 7.2 5.3 5.6 A-C-F-G 23.06 7.1 0.4 7.2 5.8 6.1 2.8 5.2 A-C-F-G 22.37 5.2 4.7 8.9 6.6 7.3 4.6 5.5 A-C-F-G 24.28 6.2 4.4 8.9 4.0 6.7 3.0 4.0 A-C-F-G 22.19 2.7 1.1 7.4 5.9 7.9 2.9 5.9 A-C-F-G 18.910 4.0 3.6 8.3 4.3 7.1 3.1 4.3 A-C-F-G 19.7

62

Project Duration Distribution

1717

101099

88

77

66

55

44

33

2211

001188

1199

2200

2211

2222

2233

2244

2255

2266

2277

2288

2929

Frequ

ency

Frequ

ency

Project LengthProject Length

63

Probability Computations from Monte Carlo results

*TTPNumber of Times Project Finished in Less Than or Equal to T* unitsTotal Number of Replications

*TTPNumber of Times Project Finished in More Than or Equal to T* unitsTotal Number of Replications

ETC.

64

Criticality Index for an Activity

Definition: Proportion of Runs in which the Activity is in the Critical Path

100%

0%

80%

20%

0%

100% 100%

0%

20%

40%

60%

80%

100%

A B C D E F G

Activity

Criticality

65

Criticality Index for a Path

Definition I (“Naïve Definition): Proportion of Runs in which the Activity is in the Critical Path (see Slide # 60)

66


Definition I (“Naïve Definition): Proportion of Runs in which the Activity is in the Critical Path (see Slide # 60)


Activity DurationSummary of Simulation Runs for Example Project


CriticalPath

CompletionTime

Summary of Simulation Runs for Example Project


CriticalPath

CompletionTime

1 6.3 2.2 8.8 6.6 7.6 5.7 4.6 A-C-F-G 25.42 2.1 1.8 7.4 8.0 6.6 2.7 4.6 A-D-F-G 17.41 6.3 2.2 8.8 6.6 7.6 5.7 4.6 A-C-F-G 25.42 2.1 1.8 7.4 8.0 6.6 2.7 4.6 A-D-F-G 17.43 7.8 4.9 8.8 7.0 6.7 5.0 4.9 A-C-F-G 26.54 5.3 2.3 8.9 9.5 6.2 4.8 5.4 A-D-F-G 25.03 7.8 4.9 8.8 7.0 6.7 5.0 4.9 A-C-F-G 26.54 5.3 2.3 8.9 9.5 6.2 4.8 5.4 A-D-F-G 25.05 4.5 2.6 7.6 7.2 7.2 5.3 5.6 A-C-F-G 23.06 7.1 0.4 7.2 5.8 6.1 2.8 5.2 A-C-F-G 22.35 4.5 2.6 7.6 7.2 7.2 5.3 5.6 A-C-F-G 23.06 7.1 0.4 7.2 5.8 6.1 2.8 5.2 A-C-F-G 22.37 5.2 4.7 8.9 6.6 7.3 4.6 5.5 A-C-F-G 24.28 6.2 4.4 8.9 4.0 6.7 3.0 4.0 A-C-F-G 22.17 5.2 4.7 8.9 6.6 7.3 4.6 5.5 A-C-F-G 24.28 6.2 4.4 8.9 4.0 6.7 3.0 4.0 A-C-F-G 22.19 2.7 1.1 7.4 5.9 7.9 2.9 5.9 A-C-F-G 18.910 4.0 3.6 8.3 4.3 7.1 3.1 4.3 A-C-F-G 19.79 2.7 1.1 7.4 5.9 7.9 2.9 5.9 A-C-F-G 18.910 4.0 3.6 8.3 4.3 7.1 3.1 4.3 A-C-F-G 19.7

Frequency as

PATH a Critical Path %A-C-F-G 8 80%A-D-F-G 2 20%E-F-G 0 0%

10

67


Frequency as

PATH a Critical Path %A-C-F-G 8 80%A-D-F-G 2 20%E-F-G 0 0%

10

80%

20%

0%0%

20%

40%

60%

80%

100%

A-C-F-G A-D-F-G E-F-G

Path

Frequency a

s a C

ritical P

ath

Definition I: Proportion of Runs in which the Activity is in the Critical Path (see Slide # 60)

68

Criticality Index for a PathDefinition II: How much slack exists in that path.

Less Slack Higher criticalityMore Slack Lower criticality

= minimum total float

= maximum total float

= total float or slack in current path

Using the index, we can rank project paths from most critical to least critical

100%mm inax

max

minaxm

Ranges from 0% to 100%

69

Criticality Index for a PathDefinition II: See Example In Excel File

(Path Criticality Slide)

70

References Kerzner, Harold, “Project Management: A Systems Approach to Planning, Scheduling, and

Controlling”, John Wiley and Sons, New York, 2000,7th Edition. Meredith, Jack, R., and Samuel J. Mantel, Jr., “Project Management: A Managerial Approach,”

John Wiley and Sons, New York, 2000, 4th Edition. Halpin, Daniel, W., and Woodhead, Ronald, W., “Construction Management’, John Wiley and

Sons, New York, 2000, 2nd Edition. Schtub, Avraham, Bard, Johnson and Globerson, Schlomo. “Project Management:

Engineering Technology, and Implementation”, Prentice Hall, New Jersey, 1994. Callahan, Quackenbush, and Rowings. “Construction Project Scheduling,” McGraw-Hill, New

York 1992. Pritsker, A. Alan, C. Elliot Sigal, “Management decision Making,” Prentice Hall, Englewood

Cliffs, NJ, 1983. Thamhain, Hans, “Engineering Management,” John Wiley and Sons, New York, 1992. Software Designed for Project Control Class 1.432, MIT by Roger Evje, Dan Lindholm, S. Rony

Mukhopadhyay, & Chun-Yi Wang. Fall Semester Term Project, 1996. Albano (1992), An Axiomatic Approach to Performance-Based Design, Doctoral Thesis, MIT Eppinger, S.D (1994). A Model-Based Method for Organizing Tasks in Product Development.

Research in Engineering Design 6: 1-13, Springer-Verlag London Limited. Eppinger, S.D. (1997) Three Concurrent Engineering Problems in Product Development

Seminar, MIT Sloan School of Management. Moder, Phillips and Davis (1983). Project Management with CPM, PERT and Precedence

Diagramming, Third Edition. Van Nostrand Reinhold Company. Naylor, T (1995). Construction Project Management: Planning and Scheduling. Delmar

Publishers, New York. Suh N.P. (1990). The Principle of Design. Oxford University Press, New York. Badiru and Pulat (1995) Comprehensive Project Management, Prentice Hall, New

Jersey. Hendrickson and Au (1989) Project Management for Construction, Prentice Hall, New

Jersey. Pritsker, Sigal, and Hammesfahr (1994). SLAM II: Network Models for Decision

Support Taylor and Moore (1980). R&D Project Planning with Q-GERT Network Modeling and

Simulation, Institute of Management Sciences. http://www.enr.com/

samuel labi and fred moavenzadeh department of civil and environmental engineering

Documents

outputs of engineering

queuing systems

network systems

dinner times

time durations

outputs costs

time minutes

value of combined output