Project Scheduling

Probabilistic PERT

PERT Probability Approach to

Project Scheduling• Activity completion times are seldom known with

cetainty.

• PERT is a technique that treats activity completion times as random variables.

• Completion time estimates can be estimated using the Three Time Estimate approach. In this approach, three time estimates are required for each activity:

– Results from statistical studies

– Subjective best estimates

a = an optimistic time to perform the activity P(Finish < a) < .01

m = the most likely time to perform the activity (mode)

b = a pessimistic time to perform the activity P(Finish > b) < .01

3-Time Estimate Approach

Probability Distribution• With three time estimates, the activity completion time

can be approximated by a Beta distribution.

• Beta distributions can come in a variety of shapes:

a m b ba mm a b

Mean and Standard Deviation for

Activity Completion Times

• The best estimate for the mean is a weighted

average of the three time estimates with weights

1/6, 4/6, and 1/6 respectively on a, m, and b.

• Since most of the area is with the range from a to

b (b-a), and since most of the area lies 3 standard

deviations on either side of the mean (6 standard

deviations total), then the standard deviation is

approximated by Range/6.

6

a-b=deviation standard the=

6

b+4m+a= timecompletionmean the=

• Assumption 2– There are enough activities on the critical path so that

the distribution of the overall project completion time can be approximated by the normal distribution.

PERT Assumptions

• Assumption 1– A critical path can be determined by using the mean

completion times for the activities.

– The project mean completion time is determined solely by the completion time of the activities on the critical path.

• Assumption 3– The time to complete one activity is independent of the

completion time of any other activity.

The three assumptions imply that the

overall project completion time is normally

distributed, with:

The Project Completion Time

Distribution

= Sum of the ’s on the critical path

2 = Sum of the 2 ’s on the critical path

Activity Optimistic Most Likely Pessimistic

A 76 86 120

B 12 15 18

C 4 5 6

D 15 18 33

E 18 21 24

F 16 26 30

G 10 13 22

H 24 28 32

I 22 27 50

J 38 43 60

The Probability Approach(76 + 4(86) +120)/6 (120-76)/6

2

90 7.33 53.73

15 1.00 1.00

5 0.33 0.11

20 3.00 9.00

21 1.00 1.00

25 2.33 5.43

14 2.00 4.00

28 1.33 1.77

30 4.67 21.81

45 3.67 13.47

2

90 7.33 53.73

15 1.00 1.00

5 0.33 0.11

20 3.00 9.00

21 1.00 1.00

25 2.33 5.43

14 2.00 4.00

28 1.33 1.77

30 4.67 21.81

45 3.67 13.47

2

90 7.33 53.73

15 1.00 1.00

5 0.33 0.11

20 3.00 9.00

21 1.00 1.00

25 2.33 5.43

14 2.00 4.00

28 1.33 1.77

30 4.67 21.81

45 3.67 13.47

(7.33)2

2

90 7.33 53.73

15 1.00 1.00

5 0.33 0.11

20 3.00 9.00

21 1.00 1.00

25 2.33 5.43

14 2.00 4.00

28 1.33 1.77

30 4.67 21.81

45 3.67 13.47

Distribution For Klone Computers

• The project has a normal distribution.

• The critical path is A-F-G-D-J.

45 20 14 25 90

μμμμμμ JDGFA

194

13.44 9 4 5.44 53.78

σσσσσσ 2

J

2

D

2

G

2

F

2

A

2

85.66

85.66σσ 2

9.255

Standard Probability Questions1. What is the probability the project will be finished

within 194 days? • P(X < 194)

2. Give an interval within which we are 95% sure of completing the project.

• X values, xL, the lower confidnce limit, and xU, the upper confidnce limit, such that P(X<xL) = .025 and P(X>xU) = .025

3. What is the probability the project will be completed within 180 days?

• P(X < 180)

4. What is the probability the project will take longer than 210 days.

• P(X > 210)

5. By what time are we 99% sure of completing the project?

• X value such that P(X < x) = .99

Excel Solutions

NORMDIST(194, 194, 9.255, TRUE)

NORMINV(.025, 194, 9.255)

NORMINV(.975, 194, 9.255)

NORMDIST(180, 194, 9.255, TRUE)

1 - NORMDIST(210, 194, 9.255, TRUE)

NORMINV(.99, 194, 9.255)

Using the PERT-CPM Template for

Probabilistic Models

• Instead of calculating µ and by hand,

the Excel template may be used.

• Instead of entering data in the µ and

columns, input the estimates for a, m ,

and b into columns C, D, and E.

– The template does all the required

calculations

– After the problem has been solved,

probability analyses may be performed.

Enter a, m, b instead of

Call Solver

Click Solve

Go to PERT OUTPUT worksheet

Call Solver

Click Solve

To get a cumulative

probability, enter

a number here

P(Project is completed in less than 180 days)

Cost Analysis Using the

Expected Value Approach

• Spending extra money, in general

should decrease project duration.

• But is this operation cost effective?

• The expected value criterion can be

used as a guide for answering this

question.

Suppose an analysis of the competition

indicated:

– If the project is completed within 180 days,

this would yields an additional profit of $1

million.

– If the project is completed in 180 days to

200 days, this would yield an additional

profit of $400,000.

Cost Analyses Using Probabilities

• Completion time reduction can be achieved by

additional training.

• Two possible activities are being considered.

– Sales personnel training: (Activity H)

• Cost $200,000;

• New time estimates are a = 19, m= 21, and b = 23 days.

– Technical staff training: (Activity F)

• Cost $250,000;

• New time estimates are a = 12, m = 14, and b = 16.

• Which, if either option, should be pursued?

KLONE COMPUTERS -Cost analysis using probabilities

Sales personnel training (Activity H) is

not a critical activity.• Thus any reduction in Activity H will not affect

the critical path and hence the distribution of

the project completion time.

Analysis of Additional

Sales Personnel Training

This option should not be

pursued at any cost.

Analysis of Additional

Technical Staff Training

• Technical Staff Training (Activity F) is on

the critical path so this option should be

analyzed.

• One of three things will happen:

– The project will finish within 180 days:

• Klonepalm will net an additional $1 million

– The project will finish in the period from 180

to 200 days

• Klonepalm will net an additional $400,000

– The project will take longer than 200 days

• Klonepalm will not make any additional profit.

The Expected Value Approach

• Find the P(X < 180), P(180 < X < 200), and P(X > 200) under the scenarios that– No additional staff training is done

– Additional staff is done

• For each scenario find the expected profit:

• Subtract the two expected values. If the difference is less than the cost of the

training, do not perform the additional training.– Caution: These are expected values (long run average

values). But this approach serves as a good indicator for the decision maker to consider.

Expected Additional Profit

1000000(P(X<180)) + 400000(P(180<X<200)) + 0(P(X>200))

The Calculations• The PERT-CPM template can be used to

calculate the probabilities.No Additional

TrainingAdditional

Training

µ = 194

= 9.255

µ = 189

= 9.0185

.065192P(X < 180)

P(180 <X < 200) .676398

P(X > 200) .258410

X $1000000

$ 0

$ 65,192

$270,559X $400000

X $0

Total = $335,751

$159,152

$ 0

$291,824

.159152

.729561

.111287

Total = $450,976

Net increase = $450,976-$335,751 = $115,225

This is less than the $250,000 required for training.

Do not perform the

additional training!

Review• 3-Time Estimate Approach for PERT

– Each activity has a Beta distribution

– Calculation of Mean of each activity

– Calculation Variance and Standard Deviation for each activity

• Assumptions for using PERT approach

• Distribution of Project CompletionTime

– Normal

– Mean = Sum of means on critical path

– Variance = Sum of variances on critical path

• Using the PERT-CPM template

• Using PERT in cost analyses