ME 380ME 380

Project Planning

Critical Path Method (CPM)Critical Path Method (CPM)

Elements: Activities & Events

Feature: Precedence relations

Activity Duration Precedence

A 4 -

B 5 -

C 3 A

D 3 A

E 2 B, C

Activities Table

Critical Path Method (CPM)Critical Path Method (CPM)

Graphical representation :

Activities :(edges)

Events :(vertices)

CT(Time reqd. for activity)

Critical Path Method (CPM)Critical Path Method (CPM)

Precedence:

Activities B & C precede Activity E

C

BE

This “Event” cannot occur before both activities B & C have been completed

Critical Path MethodCritical Path MethodExampleExample

Activity Duration Precedence

A 4 -

B 5 -

C 3 A

D 3 A

E 2 B, C

Critical Path Method (CPM)Critical Path Method (CPM)

The project sequence graph is constructed:

C

B

E

A

D

Now what ???ProjectStart

ProjectEnd

Critical Path Method (CPM)Critical Path Method (CPM) Events are consolidated to provide the

specified precedence. “Dummy” activities are added if necessary.

CB

E

A

D

ProjectStart

ProjectEnd

Dummy Activity ExampleDummy Activity ExampleTo be able to bolt a bracket to a panel, the operations

required are :

Design bracket A - Build bracket B A Build panel C - Drill holes in panel D A,C

C

BAD

A A

C CD D

B

Critical Path Method (CPM)Critical Path Method (CPM)

Activity times (duration) are added next :

CB

E

A

D

ProjectStart

ProjectEnd5

3 3

2

4

Critical Path Method (CPM)Critical Path Method (CPM)The CRITICAL PATH is the path through the project on which any

delay will cause the completion of the entire project to be delayed:

CB

E

A

D

ProjectStart

ProjectEnd5

3 3

2

4

Critical Path Method (CPM)Critical Path Method (CPM)

For fairly simple projects, the critical path is usually the longest path through the project.

For projects with several parallel and interlinked activities, this may not always be the case.

For more complicated projects, the critical path can be determined with an ‘earliest time’ forward sweep through the diagram followed by a ‘latest time’ reverse sweep.

Critical Path Method (CPM)Critical Path Method (CPM)The EARLIEST starting time of each activity is associated with the events.

It corresponds to the longest time of any path from any previous event.

CB

E

A

D

ProjectStart

ProjectEnd

3 3

2

40

4

975

Critical Path Method (CPM)Critical Path Method (CPM)The LATEST starting time of each activity is also associated with the events. It

corresponds to the longest time of any path from any subsequent event.

CB

E

A

D

ProjectStart

ProjectEnd

3 3

2

40

4

975

0

4

7 9

Critical Path Method (CPM)Critical Path Method (CPM)The CRITICAL PATH is the path along which the earliest time and latest time are

the same for all events, and the early start time plus activity time for any activity equals the early start time of the next activity.

CB

E

A

D

ProjectStart

ProjectEnd

3 3

2

40

4

975

0

4

7 9

Critical Path Method (CPM)Critical Path Method (CPM) This project cannot be completed in less than 9 weeks given the expected duration of the activities. However, activities B & D could be delayed or extended by up to 2 weeks each without affecting

the minimum project completion time. This is termed ‘float’ or ‘slack’ time.

CB

E

A

D

ProjectStart

ProjectEnd

3 3

2

40

4

975

0

4

7 9

ExampleExample Activity Duration Precedence

A 3 -B 3 AC 4 -D 1 CE 3 B, DF 2 A, B, DG 2 C, FH 4 GI 1 CJ 3 E, GK 5 F, H, I

ExampleExample

C

B EA

DProjectStart

ProjectEnd

HK

JF

I

G

ExampleExampleActivity Duration Earliest

StartLatest Start

Float

A 3 0 0 0

B 3 3 3 0

C 4 0 1 1

D 1 4 5 1

E 3 6 13 7

F 2 6 6 0

G 2 8 8 0

H 4 10 10 0

I 1 4 13 9

J 3 10 16 6

K 5 14 14 0

Summary: CPM StepsSummary: CPM Steps List all activities and expected durations.

Construct CPM diagram for activities list.

Determine EARLIEST start time for each event (working forward from project start).

Determine LATEST start time for each event (working backwards from project end).

Identify the CRITICAL PATH (and the ‘float’ time for any non-critical activities).

Using Estimates of Activity timesUsing Estimates of Activity times

The estimated duration of any activity is just that – an estimate.

There is usually an optimistic time (shortest time, TS) associated with any activity – 1 in 100 chance of taking less time than this.

There is also usually a pessimistic time (longest time, TL) associated with any activity – 1 in 100 chance of taking longer than this.

If TM is the most likely time for a specific activity, then a mean and variance for the activity can be calculated, assuming that TS, TL and TM are the parameters describing a Beta distribution.

Using Estimates of Activity timesUsing Estimates of Activity times

The estimated time TEST is calculated as:

TEST = (TS + 4.TM + TL)/6

and the variance of this is:

2 = (TL – TS)2/36

from the previous Examplefrom the previous ExampleActivity Duration

TM

TS TL TESTEarliest

StartLatest Start

Float

A 3 1 5 3.0 0 0.5 0.5

B 3 2 4 3.0 3 3.5 0.5

C 4 3 10 4.83 0 0 0

D 1 1 5 1.67 4.83 4.83 0

E 3 2 6 3.33 6.50 14.84 8.34

F 2 1 7 2.67 6.50 6.5 0

G 2 1 4 2.17 9.17 9.17 0

H 4 3 6 4.17 11.34 11.34 0

I 1 0 3 1.17 4.83 14.34 9.51

J 3 1 6 3.17 11.34 18.17 6.83

K 5 3 12 5.83 15.51 15.51 0

PERT/CPMPERT/CPM

The critical path has now become C-D-F-G-H-K

with a total estimated time of 21.3 days

(i.e. 15.51 + 5.83)

The std. deviation along the critical path is the square root of the sum of the individual variances:

CP = C2 + D

2 + F2 + G

2 + H2 + K

2

which for this data is 2.36 days