PERT/CPM

Variability in the Project Completion Date

• In doing Critical Path calculations, estimated activity times are not certain.

• What is the effect of these uncertainties on calculated project completion date?

Critical Path determines this date

Critical Path is a sequence of activities

Variation in Critical Path activities can cause variation in the completion date

Variation in non-Critical Path activities will not usually cause variation in the completion date

But… if variation along a non-Critical Path uses all of the slack time, this path may become Critical

But if… if variation along the Critical Path results in an earlier completion date, a non-Critical Path may become Critical.

3 Estimates for each Activity

Optimistic time ( a )

Everything progresses in an ideal manner

Most probable time ( m )

Most likely under normal conditions

Pessimistic time ( b )

Encounter breakdowns and/or delays

Expected time

4

6

t

a m bt

( a ) ( m ) ( b ) Expected

Most Time

Activity Optimistic Probable Pessimistic (Weeks)

A 4 5 12 6

B 1 1.5 5 2

C 2 3 4 3

D 3 4 11 5

E 2 3 4 3

F 1.5 2 2.5 2

G 1.5 3 4.5 3

H 2.5 3.5 7.5 4

I 1.5 2 2.5 2

J 1 2 3 2

Expected time

4

6

t

a m bt

Case Study (1) (2) (3) (4)

Earliest Latest Earliest Latest

Activity Start Start Finish Finish

A 0 0 6 6

B 0 7 2 9

C 6 10 9 13

D 6 7 11 12

E 6 6 9 9

F 9 13 11 15

G 11 12 14 15

H 9 9 13 13

I 13 13 15 15

J 15 15 17 17

C F

A D G

J

E

B H I

(1) (2) (3) (4) (2) - (1) (4) - (3)

Earliest Latest Earliest Latest SLACK SLACK

Activity Start Start Finish Finish (LS - ES) (LF - EF)

A 0 0 6 6 0 0

B 0 7 2 9 7 7

C 6 10 9 13 4 4

D 6 7 11 12 1 1

E 6 6 9 9 0 0

F 9 13 11 15 4 4

G 11 12 14 15 1 1

H 9 9 13 13 0 0

I 13 13 15 15 0 0

J 15 15 17 17 0 0

C F

A D G

J

E

B H I

Project Duration

T stands for the project duration

The activities along the Critical Path are A – E – H – I – J

E H IA Jt t t t tT

Expected

Time

Activity (Weeks)

A 6

B 2

C 3

D 5

E 3

F 2

G 3

H 4

I 2

J 2

6 3 4 2 2 17T

Variance in Project Activities 2

Variance of an activity time = 6

b a

( a ) ( m ) ( b ) Expected

Most Time Variance

Activity Optimistic Probable Pessimistic (Weeks) σ2

A 4 5 12 6 1.78

B 1 1.5 5 2 0.44

C 2 3 4 3 0.11

D 3 4 11 5 1.78

E 2 3 4 3 0.11

F 1.5 2 2.5 2 0.03

G 1.5 3 4.5 3 0.25

H 2.5 3.5 7.5 4 0.69

I 1.5 2 2.5 2 0.03

J 1 2 3 2 0.11

Variance in Project Duration 2 2 2 2 22

E H IA J

2 1.78 0.11 0.69 0.03 0.11 2.72E H IA J

Variance

Activity σ2

A 1.78

B 0.44

C 0.11

D 1.78

E 0.11

F 0.03

G 0.25

H 0.69

I 0.03

J 0.11

Standard Deviation in Project Duration

2.72 1.65

Final Assumption

The distribution of project completion time T is

Normally distributed.

The next step:

• Given the Expected Completion time (mean of the Normal)

• Given the Expected Standard Deviation in the Expected

Completion time (standard deviation of the Normal)

• Convert this Normal Curve to an equivalent Z-Normal Curve

17

1.65

xz

This “says” that any time you want to know the

“what if” probability that another completion

time (x) might actually occur, it represents a

random variable associated with this normal

distribution … and can be converted to an

equivalent z normal value…

What if… example

What is the probability that the project would be

completed in 20 weeks or less?

This would be an x = 20 from this normal distribution.

Convert the x to a z

17

1.65

171.82

1.65

xz

xz

This “says” that a completion time

of 20 is 1.82 standard deviations

above the mean (17).

Here is where we use the Table Handout

Z = 0 Z = 1.82 µ = 17 x = 20

Z table gives the area to the left, but only to the mean line. You need to add in the 0.50 to get the probability.

171.82

1.65

xz

xz

Steps

1. Develop list of activities

2. Draw the network

3. Estimate expected activity time and variance for each

activity

4. Use expected activity time estimates to determine

earliest start & finish and latest finish and start for each

activity

5. Use the project completion time as the finishing time

6. Compute the slack for each activity & identify the

Critical Path

7. Use the variability in the project completion date to

compute probabilities of meeting a specified date