This document discusses several project scheduling techniques including PERT, CPM, and GERT. It provides background on the development of PERT for the Polaris missile program and CPM for chemical plant construction. Key aspects of each technique are summarized, such as how PERT uses three time estimates to model uncertainty and calculate activity mean and variance. The document also covers topics like crashing, resource constraints, and limitations of PERT/CPM that GERT addresses.

1. Symbiosis Institute of
Management Studies (SIMS)
Project Management
PERT,CPM, Resource Allocation
and GERT
Manuja Goenka, E-12
Raj Jyoti Das-E-13
August 2013

2. Tools for Scheduling
• Two commonly used network methods for
planning and scheduling are:
1. Program Evaluation and Review Techniques
(PERT)
2. Critical path Method (CPM)
*Both PERT & CPM are termed critical path methods

3. History of PERT
• Project Evaluation and Review Technique (PERT)
– U S Navy (1958) for the POLARIS missile program
– Multiple task time estimates (probabilistic nature)

4. PERT
• PERT is based on the assumption that an activity’s
duration follows a probability distribution instead of
being a single value
• Three time estimates are required to compute the
parameters of an activity’s duration distribution:
– pessimistic time (a) - the time the activity would take
if things did not go well
– most likely time (m ) - the consensus best estimate of
the activity’s duration
– optimistic time (b) - the time the activity would take if
things did go well
a + 4m + b
Mean (expected time):te =
Variance: V =
6
b- a
6
2

5. Three Time Estimates of PERT
PERT uses three time estimates to address uncertainty of
project duration-
•Optimistic
•Most Likely
•Pessimistic

6. Mean or expected Time
It is the time where there is 50-50 chances that the activity will be
completed earlier or later than it.
For this case :

7. Variance
Variance is measure of variability in the activity completion
period.
The larger V, the less reliable te

8. The Expected Duration Of The Project
The expected duration of the project (Te) is the
sum of the expected activity times along the
critical path
Te = ∑ te
Where te are expected times of the activities on
the critical path

9. The Variation In The Project Duration
The variation in the project duration
distribution is computed as the sum of the
variances of the activity durations along the
critical path:
Vp = ∑ V
Where V is the variance of critical path

10. Near Critical Path
Path
(events)
Te = ∑ te
Vp = ∑ V
1-2-6-8
28**
6.34
1-7-8
20
17.00
1-2-5-7-8
Te=29*
Vp=6.00
1-4-5-7-8
18
3.89
1-3-4-5-78
27**
12.00

11. Probability of Finishing a Project
Let us assume,
Expected completion duration of a project = 29 weeks
Variance of the project duration = 6
Then what will be the probability of finishing the project by 27
weeks can be calculated by the formula:
Probability
Therefore,
Z=(27-29)/2.449
=-0.82
Prob. Of finishing the
project by 27 weeks
is app. 21%
Z
0.207
27
darla/smbs/vit
29
Time
11

12. History of Critical Path Method (CPM)
• E I Du Pont de Nemours & Co. (1957) for construction
of new chemical plant and maintenance shut-down
• CPM is a “Deterministic” approach
• CPM includes mathematical procedure for estimating
the trade off between project duration and project
cost
• CPM emphasis on applying additional resources to
particular key activities

13. Time-Cost Relationship
Normal Time, Tn: It is the time taken by an activity
under normal work conditions
Normal Cost, Cn: The cost incurred in doing an activity
in normal time.

14. Crashing
An activity is said to be crashed when maximum effort is
applied to finish that activity in the shortest possible
time.

15. Cost Slope
• The cost slope shows
by how much the cost
of job would change if
activities were speed
up or slowed down.
In this case,
Cost Slope= (18-9)/(5-8) = $3 K/week

16. CPM calculation
• Path
– A connected sequence of activities leading from
the starting event to the ending event
• Critical Path
– The longest path (time); determines the project
duration
• Critical Activities
– All of the activities that make up the critical path

17. CPM calculation
Forward Pass
• Earliest Start Time (ES)
– earliest time an activity can start
– ES = maximum EF of immediate predecessors
• Earliest finish time (EF)
– earliest time an activity can finish
– earliest start time plus activity time (EF= ES + t)
Backward Pass
Latest Start Time (LS)
Latest time an activity can start without delaying critical path time
LS= LF - t
Latest finish time (LF)
latest time an activity can be completed without delaying critical path time
LS = minimum LS of immediate predecessors

18. Project Crashing
• Crashing
– reducing project time by expending additional resources
• Crash time
– an amount of time an activity is reduced
• Crash cost
– cost of reducing activity time
• Goal
– reduce project duration at minimum cost

19. Time-Cost Relationship
 Crashing costs increase as project duration decreases
 Indirect costs increase as project duration increases
 Reduce project length as long as crashing costs are less than
indirect costs
Time-Cost Tradeoff
Total project cost
Indirect cost
Direct cost
time

20. Crashing Example
9
9
C
5
A
0
F
5
9
0
17
14
D
8
B
22
G
E
8
22
7
5
17
Activi
ty
17
Normal
(Wks)
Crush
(Wks)
10
Tn
Cn
A
8
Tc
Cost
Slope
(K$)
Cc
9
10 6
16 2
B
8
9
5
18 3
C
5
7
4
8
D
8
9
6
19 5
E
7
7
3
15 2
F
5
5
5
5
G
5
8
2
23 5
1
-

21. 9
9
C
5
A
0
9
0
17
14
F
5
D
8
22
G
B
E
8
7
8
5
17
22
17
Critical Path A-D-G=22wks
10

22. Types of Project Constraints
• Technical or Logic Constraints
– Constraints related to the networked sequence in
which project activities must occur.
• Physical Constraints
– Activities that cannot occur in parallel or are affected
by contractual or environmental conditions.
• Resource Constraints
– The absence, shortage, or unique interrelationship
and interaction characteristics of resources that
require a particular sequencing of project activities.

23. The Resource Problem
• Resources and Priorities
– Project network times are not a schedule until
resources have been assigned.
• The implicit assumption is that resources will be available in
the required amounts when needed.
• Adding new projects requires making realistic judgments of
resource availability and project durations.
• Resource-Constrained Scheduling
– Resource leveling (or smoothing) involves attempting
to even out demands on resources by using slack
(delaying noncritical activities) to manage resource
utilization.

25. Kinds of Resource Constraints
• People
• Materials
• Equipment
• Working Capital

26. Classification of A Scheduling
Problem
• Time Constrained Project
– A project that must be completed by an imposed
date.
• Time is fixed, resources are flexible: additional
resources are required to ensure project meets
schedule.
• Resource Constrained Project
– A project in which the level of resources available
cannot be exceeded.
• Resources are fixed, time is flexible: inadequate
resources will delay the project.

27. Example :
• Without resource constraints relatively easy
• With resource constraints very complex:
when jobs share resources with limited
availability, these jobs cannot be processed
simultaneously
Jobs
p(j)
R(1,j)
R(2,j)
1
8
2
3
2
4
1
0
3
6
3
4
4
4
1
0
5
4
2
3
Resource
Available
R1
4
R2
8
1
4
2
5
3
27

28. S 'j  earliest possible starting time of job j
C 'j  earliest possible completion time of job j
C ''  latest possible completion time of job j
j
slack j  C ''  p j  S 'j
j
28

29. Resource constraints
• Suppose jobs require a resource:
Job p(j) Predecessors S' C'' R(1,j)
1 2
0 3
3
2 3
0 3
1
3 1
0 6
2
4 4
1,2
3 7
2
5 2
2,3
3 8
3
6 1
4
7 8
3
6
5
3
4
3
2
5
2
1
resource requirements
1
6
4
1
2
3
4
5
6
7
8
29

30. Resource constraints (cont.)
• Suppose R 1  4 :
6
5
4
3
2
2
1
1
3
1
2
5
4
3
4
5
6
7
6
8
9
10
Cmax increases by 2
30

31. Resource-Constrained Project
Scheduling Problem (RCPSP)
•
•
•
•
•
•
n jobs j=1,…,n
N resources i=1,…,N
Rk:availability of resource k
pj: duration of job j
Rkj:requirement of resource k for job j
Pj: (immediate) predecessors of job j
31

32. RCPSP
• Goal: minimize makespan: Cmax  max C
j
• Restrictions:
'
j
– no job may start before T=0
– precedence relations
– finite resource capacity
32

33. RCPSP
example
Job p(j) P(j) S' C'' R(1,j) R(2,j)
1
2
- 0 3
3
2
2
3
- 0 3
1
1
3
1
- 0 6
2
1
4
4 1,2 3 7
2
1
5
2 2,3 3 8
3
2
6
1
4 7 8
3
1
4
R1  4
2
2
1
0
2
2
R2  2
4
3
2
1
0
6 5
3 4
2
6
8
4
4
10
6
6
8
5
10
12
33

34. Loading And Leveling
• Loading- amount of a resource necessary to
conduct a project
– Depends on the requirements of individual
activities.
– Changes throughout a project
• Resource Leveling- process of scheduling
activities so that the amount of a certain
required resource is balanced throughout the
resource.

35. Multiproject Resource Schedules
• Multiproject Scheduling Problems
– Overall project slippage
• Delay on one project create delays for other projects
– Inefficient resource application
• The peaks and valleys of resource demands create
scheduling problems and delays for projects.
– Resource bottlenecks
• Shortages of critical resources required for multiple
projects cause delays and schedule extensions.

36. Multiproject Resource Schedules
• Managing Multiproject Scheduling
– Create project offices or departments to oversee
the scheduling of resources across projects.
– Use a project priority queuing system: first come,
first served for resources.
– Centralize project management: treat all projects
as a part of a “megaproject.”
– Outsource projects to reduce the number of
projects handled internally.

37. Limitations of PERT/CPM
• All immediate predecessor activities must be
completed before a given activity can be started.
• No activity can be repeated and no “looping
back”
• Duration time for an activity is restricted to Beta
Distribution PERT and a single estimate in CPM.
• Critical Path is always considered the longest.
• There is only one terminal event and the only
way to reach it is by completing all activities in
the project

38. GERT
• A network analysis technique used in project
management.
• It allows probabilistic treatment of both network logic
and activity duration estimated.
• The technique was first described in 1966 by Dr. Alan
B. Pritsker of Purdue University and WW Happ.
• Compared to other techniques, GERT is an only rarely
used scheduling technique.

39. Contd..
• Utilizes probabilistic and branching nodes
• It represents the node will be reached if any m
of its p immediate predecessors are
completed.
m
n
p

40. Contd..
• It represents a probabilistic output where any
of q outputs are possible
• Each branch has an assigned probability
• When no probability is given, the probability is
assumed to be one for each branch.
1
2
q

41. Example

43. Thank YOu