The PERT Approach for ProjectRisk Assessment• The program evaluation and review technique (PERT) was

developed by the late 1950’s. The objective was to evaluatethe risk in meeting the time goals of the execution ofprojects whose activities had some uncertainty in theirduration estimates. To represent the uncertainty induration estimates, the PERT technique recognizes theprobabilistic, rather than deterministic, nature of theoperations involved in high-risk activities. Accordingly, thePERT technique incorporates three durations for eachactivity into its methodology. The 3 estimates are: 1

The PERT Approach• Optimistic duration (a): estimated time (comparatively

short) of executing the activity under very favorableworking conditions. The probability of attaining thisduration is about 0.01;

• Pessimistic duration (b): estimated time (comparativelylong) of executing the activity under very unfavorableworking conditions. The probability of attaining thisduration is also about 0.01; and

• Most Likely duration (m): estimated time of executing theactivity that is closest to the actual duration. Thisestimates lies in between the above two extremes. 2

The PERT Approach• In PERT, the given estimates of times and the likelihood of

occurrence are represented by a beta curve, as shownbelow. However, with the three estimates of time for eachactivity, we cannot perform traditional CPM analysis todetermine project duration. Therefore, we need to get asingle weighted average duration for each activity. Theformulas for the expected duration, called expected elapsedtime (te) are as follows:

Beta-Distribution Curve

a

m

b

0.5

ActivityDurationte

3

The PERT ApproachAnalysis Steps:

• Step 1: Individual Activity Durations• a = Optimistic duration = Minimum duration• m = Most Frequent duration (most likely)• b = Pessimistic duration = Maximum duration• te = activity expected duration = (a + 4 m + b) / 6• te

2 = activity duration variance = [(b - a) / 6]2

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The PERT Approach• Step 2: CPM Calculations

• Using the activities’ te durations, CPM calculations are performedfollowing the forward and backward passes to determine the projectduration (TE). Activity floats and also calculated and critical activitiesidentified.

• Step 3: Distribution of Project Duration• Since the probability is 0.5 that each activity will finish at its te

durations, there is a probability of 0.5 for the entire project beingfinished at time TE. However, the expected project duration does notfollow a beta curve as did the activities comprising the project.Assuming that the project is executed a large number of times, theresulting population of project durations may be assumed normallydistributed.

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The PERT Approach• The normal distribution of project duration is defined by its mean ()

and standard deviation () values, determined as follows:• TE = TE = te of critical activities;• TE = te

2 of critical activities

• Step 4: Analysis of Project Completion Probabilities• Using the project normal distribution, it is possible now to find the

probability values associated with specific project duration. By scalingthe project distribution to the standard normal distribution, we canobtain probabilities from standard probability tables and makeconclusions, as follows: Z = Desired Completion Date - TE

TE

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34.1% 34.1%

13.6%13.6%2.1% 2.1%

-3ơ -2ơ -ơ ơ 2ơ 3ơTe ơ = 68.2% of the area under the curve+ -

Te 2ơ = 95.4% of the area under the curve+ -

Te 3ơ = 99.7% of the area under the curve+ -

activity2= ơvariance)(activityV

Ơ Total = (Ơ)2A + (Ơ)2

B + (Ơ)2C + ……

(for the critical path) (activities in the critical path)

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Example

• Consider a project whose critical path consists of 4activities A,B,C,D. Each activity has three time estimates asshown:

A

2 6 8

B

3 9 12

C

2 8 10

D

4 6 9

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Solution

• The Expected Time and Standard Deviation for each criticalActivity

6.16

7.33

8.5

5.66

Te

0.835964D

1.3381082C

1.591293B

16862A

Activity standarddeviation

(Tp – To) / 6

Tp - ToTpTmToActivity

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• Expected time for the project =5.66 + 8.5 +7.33 + 6.16 = 27.65

Ơ Total = (1)2 + (1.5)2 + (1.33)2 + (0.83)2

= 2.39

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• To calculate the probability of finishing within any given time,you calculate Z where:

Where T is the time requiredAnd from the normal curve table we find the probability

T - Te

Ơ T

Z =

11

12

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The PERT Approach (Example)PessimisticTime (b)

Most Prob.Time (m)

OptimisticTime (a)

PredecessorsActivity

852-------A1296AB876AC741B,CD888AE17145D,EF21123CG963F,GH1185HI

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Variance[(b-a)/6]2

Standard Deviation[(b-a)/6]

Expected timeTe =

Activity

115A119B

1/91/37C114D008E4213F9312G116H118I

a + 4m + b6

The PERT Approach (Example)

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0 5

0 5 5A

5 14

5 9 14B

5 12

7 7 14C

5 13

10 8 18E

14 18

14 4 18D

12 24

19 12 31G

18 31

18 13 31F

31 37

31 6 37H

37 45

37 8 45I

Ơ Total = Ơ2A + Ơ2

B + Ơ2D + Ơ2

F + Ơ2H + Ơ2

I

Ơ Total = (1)2 + (1)2 + (1)2 + (2)2 + (1)2 + (1)2

= 9 = 316

• What is the probability of completing the project in 50days? And what is the probability of completing theproject in 4 days less than the expected duration.

Prob. (T ≤ 50) = Prob. (Z1 ≤ 1.67)From table Prob. (T ≤ 50) = 0.9525 = 95.25%

T - Te

Ơ T

=1Z50 _ 45

3=

= 1.67

The PERT Approach (Example)

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Prob. (T ≤ 41) = Prob. (Z2 ≤ -1.33)From table Prob. (T ≤ 41) = 0.0918 = 9.18%

T - Te

Ơ T

=2Z 41 _ 45

3=

= - 1.33

The PERT Approach (Example)

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• What will be the total duration if you want to be97.5% confident that the project will not exceedit?From table Z = 1.96

duration = 1.96 (ơ) + 45= 51 days

you are adding 6 days (contingency) to be 97.5%confident that the project will not exceed theduration.

The PERT Approach (Example)

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Range Estimating• Range Estimating is similar to the PERT

approach, where the estimators are asked tosupply three estimates for each activity: lowest(L), highest (H) and most likely (M).

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• E = Expected Value =

• σ = Standard Deviation =For each individual activity

Standard deviation for the project

Where n is the total number of activities in theproject

L + 4M +H

6

H - L

6

= (σ1)2 + (σ2)2 + (σ3)2 + ...+ (σn)2

Range Estimating

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ExampleStandarddeviation

ExpectedValue

HighestMostLikely

LowestDescriptionAct. No.

1,3508,12010,4029,0052,300A10

3092,9803,6543,1061,800B20

61910,30813,2009,7969,485C30

6425,2346,4505,5892,600D40

2,76022,50027,76124,00911,200E50

2,10820,51525,74621,06113,100F60

4,35044,87554,20146,73828,100G70 22

• Expected value of the project = $114,532

• Range Estimate = $114,532 +/- 5804.9(we are 84.1% confident that the project will notexceed $120,336.9)

σ Project = (σ)210 + (σ)2

20 + ……+ (σ)270

= $5804.9

Example

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• How much should we add as a contingency to be 98%confident that the project will not exceed the estimatedvalue.

• From the tableP (98%) Z = 2.06 = (C – Ce)/ σ

Total value = 2.06 * 5,804.9 + 114,532= $ 126,490

contingency added = $11,958

Example

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The PERT Approach (Critique)• Criticisms to PERT Technique• Requires three estimated durations for each activity.• Assumes continuous not discrete distribution for

durations.• Beta distribution is debatable.• It focuses on a single critical path and ignores close-to-

critical paths.• It assumes independent activity durations.• It ignores the risk that occurs at path convergence points.

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Simulation• Simulation is an analytical method meant to imitate a real

– life problem / system especially when other analyses aretoo mathematically complex or too difficult to reproduce.

• Monte Carlo simulation is a form of simulation thatrandomly generates values for uncertain variables overand over to simulate a model.

• Spread sheet risk analysis uses both a model andsimulation to automatically analyze the effect of varyinginput on outputs of the modeled system / problem.

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• Simulation is a 4 step process1. Identify the uncertain cells in the model.2. Implement appropriate random number generators

(RNGs) for each uncertain cell.3. Replicate the model n times, and record the value of

the bottom – line performance measures.4. Analyze the sample values collected on the

performance measure.

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Monte Carlo SimulationStep-By-Step

1.Determine the duration (or cost, …) distribution of eachactivity. It is possible to use discrete values or to use thesimplified assumption of a triangular distribution;

Triangular Distribution

a

m

b ActivityDuration

ActivityDuration

Discrete Distribution

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Monte Carlo Simulation2. Generate one project scenario by randomly generating one

possible duration (or cost) for each activity in the project(based on its distribution). Perform CPM calculations (orcost) for this scenario and determine the project duration (orcost);

3. Repeat step 2 for the number of desired simulations(scenarios) and then tabulate the results;

4. Project Duration Distribution: Calculate the mean () and ()values for the resulting project durations (total cost); and

5. Using the () and () values, determine the probability ofthe project being completed on or before any given date, orwithin any estimated total cost.

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