CHAPTER NINE
Project Scheduling
Networks, Duration Estimation, and Critical Path
9.1 RULES FOR DEVELOPING ACTIVITY NETWORKS
1. Some determination of activity precedence ordering must be done
prior to creating the network.
2. Network diagrams usually flow from left to right.
3. An activity cannot begin until all preceding connected activities
have been completed.
4. Arrows on networks indicate precedence and logical flow.
Arrows can cross over each other, although it is helpful for
clarity’s sake to limit this effect when possible.
5. Each activity should have a unique identifier associated with it
(e.g., number, letter, code, etc.).
6. Looping, or recycling through activities, is not permitted.
7. Although not required, it is common to start a project from a
single beginning node. A single node point also is typically used as
a project end indicator.
9.2 LABELS FOR ACTIVITY NODE
Early
Start
ID Number
Activity
Float
Activity Descriptor
Late
Start
Activity
Duration
Early
Finish
Late
Finish
9.3 PROJECT ACTIVITIES LINKED IN SERIES
9.4 ACTIVITIES LINKED IN PARALLEL (CONCURRENT)
9.5 MERGE ACTIVITIES
Activity A
Activity B
Activity C
Activity D
9.6 BURST ACTIVITIES
Activity B
Activity A
Activity C
Activity D
9.7 EXAMPLE OF CREATING A PROJECT ACTIVITY
NETWORK
Information for Network Construction
Name: Project Delta
Activity
Description
Predecessors
A
Contract signing
None
B
Questionnaire design
A
C
Target market ID
A
D
Survey sample
B, C
E
Develop presentation
B
F
Analyze results
D
G
Demographic analysis
C
H
Presentation to client
E, F, G
9.8 ACTIVITY NETWORK FOR EXAMPLE
E
Dev. Present.
B
Design
A
Contract
D
Survey
C
Market ID
F
Analysis
G
Demog.
H
Present
9.9 ACTIVITY DURATION ESTIMATION – BETA
DISTRIBUTION
ESTIMATED TIME FORMULA
TE =
A + 4(B) + C
6
WHERE:
A = MOST OPTIMISTIC TIME
B = MOST LIKELY TIME
C = MOST PESSIMISTIC TIME
9.10 CONSTRUCTING THE CRITICAL PATH
INFORMATION FOR PROJECT DELTA
Activity
Description
Predecessors
A
Contract signing
None
5
B
Questionnaire design
A
5
C
Target market ID
A
6
D
Survey sample
B, C
13
E
Develop presentation
B
6
F
Analyze results
D
4
G
Demographic analysis C
9
H
Presentation to client
2
E, F, G
Estimated Duration
9.10 CONSTRUCTING THE CRITICAL PATH (Continued)
Partial Project Activity Network with Task Durations
B
Design
5
A
Contract
5
E
Dev. Present
6
D
Survey
13
C
Market ID
6
F
Analysis
4
G
Demog.
9
H
Present
2
9.11 RULES WHEN USING THE FORWARD PASS
1. Add all activity times along each path as we move through the
network (ES + Dur = EF).
2. Carry the EF time to the activity nodes immediately succeeding
the recently completed node. That EF becomes the ES of the next
node, unless the succeeding node is a merge point.
3. At a merge point, the largest preceding EF becomes the ES for
that node.
9.12 ACTIVITY NETWORK WITH FORWARD PASS
5
0
B 10
Design
5
A
5
Contract
5
10
C
11
Market ID
6
16
6
11
5
E
Dev. Present
D 24
Survey
13
24 F 28
Analysis
4
11
G 20
Demog.
9
28
H 30
Present
2
9.13 RULES FOR USING THE BACKWARD PASS
1. Subtract activity times along each path as you move through the
network (LF – Dur = LS).
2. Carry back the LS time to the activity nodes immediately
preceding the successor node. That LS becomes the LF of the next
node, unless the preceding node is a burst point.
3. In the case of a burst point, the smallest succeeding LS becomes
the LF for that node.
9.14 ACTIVITY NETWORK WITH BACKWARD PASS
5
6
0
0
B 10
Design
5
11
A
5
Contract
5
5
10
22
11
11
5
C
11
Market ID
5
6
11
E
16
Dev. Present
6
D 24
Survey
13 24
28
24
F
28
28
Analysis
24
11
G
20
Demograph.
19
9
28
4
28
H
30
Presentation
28
2
30
9.15 COMPLETED ACTIVITY NETWORK WITH CRITICAL PATH AND ACTIVITY
SLACK TIMES IDENTIFIED
Critical Path is indicated in bold.
5
1
6
0
0
0
B 10
Design
5
11
A
5
Contract
5
5
10
22
11
0
11
5
C
11
0 Market ID
5
6
11
E
16
12 Dev. Present
6
D 24
Survey
13 24
28
24
0
24
11
G
F
28
28
H
30
Analysis
0 Presentation
4
28
28
2
30
20
8 Demograph.
19
9
28
ES
ID
Slack
Task Name
LS
Duration
EF
LF
9.16 Probability of Project Completion
The Formula for Activity Variance is:
s2 = [1/6(b – a)]2, where b is the most pessimistic time and a is the most optimistic.
Example: Activity A: [(12 – 4)/6]2 = (8/6)2 = 64/36, or 1.78 days variance
The Formula for Project Variance is:
σp2 = Project variance = Σ (variances of activities on critical path)
Example: Suppose our critical path had 4 activities with the following variances:
Activity A: 1.78
Activity B: 4.00
Activity C: 0.69
Activity D: 1.00
Project Variance: 1.78 + 4.00 + 0.69 + 1.00 = 7.47 days
Project Standard Deviation = √7.47 = 2.73 days
Suppose our project has a critical path of 16 days. What is the probability that the
project will finish by day 18?
Solution: Using the Standard Normal Equation:
Z = (Due date – Expected date of completion)/ σp
Z = (18 – 16)/2.73, or 0.73
Normal Tables show that a Z value of 0.73 equates to a probability of .7673
There is a 76.7% chance of finishing the project by day 18.
How many extra days do we need to generate a 99% chance of finishing on time?
Solution:
Using the normal table, we find that a 99% likelihood equates to a Z score of 2.33,
therefore:
Due Date
= Expected date of completion + (Z * σp)
= 16 days + (2.33 * 2.73)
= 16 + 6.36
= 22.36 days
9.17 ACTIVITY NETWORK DEMONSTRATING LADDERING TECHNIQUE
A1
Design
A2
Design
A3
Design
A1
Coding
A2
Coding
A3
Coding
A1
Debugging
A2
Debugging
A3
Debugging
9.18 NETWORK DEMONSTRATING HAMMOCK ACTIVITY
5
13
18
0
0
0
A
5
5
5
B
9
4
22
5
9
14
12
10
22
C
12
7
21
5
D
11
0 user needs
5
6 11
21
9
31
12
9
21
11
0
11
5
G
E
25
Coding
14
A
31
Hammock
26
25
H
22
31
I
10
31
31
4
25
0
25
F
31
Debugging
6
31
35
0
35
9.19 STEPS TO REDUCE THE CRITICAL PATH
1. ELIMINATE TASKS ON THE CRITICAL PATH.
2. REPLAN SERIAL PATHS TO BE PARALLEL.
3. OVERLAP SEQUENTIAL TASKS.
4. SHORTEN THE DURATION ON CRITICAL PATH
ACTIVITIES.
5. SHORTEN EARLY TASKS.
6. SHORTEN LONGEST TASKS.
7. SHORTEN EASIEST TASKS.
8. SHORTEN TASKS THAT COST THE LEAST TO SPEED UP.
DISCUSSION QUESTIONS
9.1 Define the following terms:
a. Path: group of activities sequenced by relationship through project network logic
b. Activity: any piece of work that will be performed during the project which has an
expected time and cost for completion
c. Early start: the earliest possible date upon which an uncompleted activity or project can
start based on sequencing and scheduling constraints
d. Early finish: the earliest possible date upon which an uncompleted activity or project
can be completed
e. Late start: the latest date an activity may start without delaying other project milestones
or the project’s expected completion date
f. Late finish: the latest date an activity may end without delaying other project
milestones or the project’s expected completion date
g. Forward pass: a process that works forward though the project network to determine
the earliest start and earliest finish time for an activity
h. Backward pass: a process that works backwards through the project network to
calculate the latest finish time for an uncompleted activity
i. Node: a convergence point of dependent paths in a network
j. AON: Activity on Node; a method of logic that determines activity networks in which a
node depicts an activity and arrows indicate sequencing between nodes
k. Float or Slack: a calculation which determines the amount of time an activity can be
delayed from its earliest start date without delaying the project’s completion date
l. Critical path: the path through the project network having the least amount of float time
and the longest time duration
m. PERT: Project Evaluation and Review Technique; a network analysis system based on
events and probability used when activities and their duration are difficult to define
9.2 What are the main benefits of constructing a network diagram for a project? Why is it
a useful project management tool?
There are a number of reasons why a network diagram is useful for a project. The
network allows project managers to clearly illustrate all activities associates with the
project and demonstrate any interdependencies between tasks. Another benefit is being
able to identify any ‘critical’ activities once the calculation has been performed, therefore
highlighting the path in the project which is most crucial to meeting time constraints. The
network also helps with identifying organizational resources.
Overall, project managers will find this a useful tool as it helps to time schedule tasks
within a project, identify the project duration, and helps the project manager co-ordinate
and plan the project accordingly.
9.3 List three methods for deriving duration estimates for project activities. What are the
strengths and weaknesses associated with each method?
One method for deriving time estimates is past experience. This method is beneficial in
that it is easy and uses past examples of similar activities to predict future time estimates.
However, it is limited in that estimates can be distorted by extenuating circumstances,
changes in time and conditions, and information obsolescence. Another method uses
expert opinion. Again, the approach is simple to use and draws on experience and
knowledge of experts. The shortcomings here involve potential inadequacy of staff (at
least relative to the expert giving the opinion) and project-specific complications. A third
method employs mathematical derivations. This approach is more objective and allows
multiple estimates based on best, most likely, and worst-case analysis. The weaknesses of
this method are that it is slightly more difficult to use and it disregards past failures (a.k.a.
lessons learned).
9.4 In your opinion, what are the chief benefits and drawbacks of using beta distribution
calculations (based on PERT techniques) to derive activity duration estimates?
Beta distribution allows for the likelihood that optimistic and pessimistic times will not
be symmetrical. By including realistic estimates of pessimistic and optimistic durations,
beta distribution creates a more accurate distribution of alternative duration times. One
drawback to this method is that it relies on estimates of pessimistic and optimistic time
estimates, which may not be reliable. There has also been some debate related to how the
time estimates in this method should be calculated and/or interpreted.
9.5 What options are available to reduce the critical path on a project? What
considerations would you need to make in choosing an appropriate option?
Project managers can select a number of different options in helping to reduce the critical
path within the network diagram. These are:

Eliminate tasks

Re-plan some serial activities to run concurrently

Overlap sequential tasks

Shorten the duration of critical path tasks

Shorten early tasks / longest tasks / easiest tasks

Shorten tasks that cost the least to speed up
Project managers will need to consider which of these options to apply depending on
which project constraints are fixed, what the project client expects from the project,
whether there are additional resources available to the project, and what acceptable
practice within the organization is.
9.6 The float associated with each project task can only be derived following the
completion of the forward and backward passes. Explain why this is true.
The forward pass establishes the earliest time that activities in the network can begin and
end. The backward pass determines the latest time activities in the network can begin and
end. Float time is the difference between the task’s latest and earliest end time (or the
task’s latest and earliest start time). Hence, float cannot be calculated until the forward
and backward passes have been completed.
====================================
PROBLEMS
9.1 You are a project manager for a medium-sized charitable organization. You are
required to plan an outdoor fundraiser that needs to be publicized, with a target of
attracting 300 people to the event.
a. Create a table to illustrate the activities associated with delivering this project and
identifying which tasks have predecessors.
b. Draw out the activity diagram illustrating burst and merge activities.
9.2 What is the time estimate of an activity in which the optimistic estimate is 2 days,
pessimistic is 12 days, and most likely is 4 days? Show your work.
9.3 What is the time estimate of an activity in which the pessimistic time is 68 hours,
optimistic time is 24 hours, and likely time is 48 hours? Show your work.
9.4 Using the following information, develop an activity network for Project Alpha.
Activity
Preceding Activities
A
---
B
A
C
A
D
B, C
E
B
F
D
G
C
H
E, F, G
9.5 Construct a network activity diagram based on the following information:
Activity
Preceding Activities
A
---
B
A
C
A
D
B, C
E
D
F
E
a. Which task is the burst activity?
b. Which task is the merge activity?
9.6 Consider the tasks in problem 5. Assume that the project scope has been changed and
as a result of this, new activities have been added to the project. Estimated durations for
each of the tasks have also been added. As the project manager, you have been asked to
re-develop your network activity diagram to include the following information:
Activity
Predecessors
Duration (Days)
A
_
1
B
A
2
C
A
5
D
B, C
1
E
D
4
F
E
5
G
D
2
H
D
1
I
G, H
3
J
F, I
1
a. Identify all burst activities and merge activities.
b. Calculate the forward and backward pass on the network diagram. What is the
estimated total duration for the project?
c. Which is the critical path for the project?
d. Which activities have slack time?
9.7 Consider the following project tasks and their identified best, likely, and worst-case
estimates of task duration. Assume the organization you work for computes TE based on
the standard beta distribution formula. Calculate the TE for each of the following tasks
(round to the nearest integer).
Activity
Best
Likely
Worst
A
5
5
20
B
3
5
9
C
7
21
26
D
4
4
4
E
10
20
44
F
3
15
15
G
6
9
11
H
32
44
75
I
12
17
31
J
2
8
10
TE
9.8 Consider the following project tasks and their identified best, likely, and worst-case
estimates of task duration. Assume the organization you work for computes TE based on
the standard beta distribution formula. Calculate the TE for each of the following tasks
(round to the nearest integer).
Activity
Best
Likely
Worst
A
4
5
10
B
4
6
9
C
2
5
8
D
5
8
10
E
12
16
20
F
6
10
12
G
5
9
14
H
14
16
22
I
10
14
20
J
1
2
5
TE
9.9 Using the information from the following table, create an AON network activity
diagram.
a. Calculate each activity TE (rounding to the nearest integer); the total duration of
the project; its early start, early finish, late start and late finish times; and slack for
each activity. Finally, show the project’s critical path.
Activity
Preceding activities
Best
Likely
Worst
A
-
12
15
25
B
A
4
6
11
C
-
12
12
30
D
B, C
8
15
20
E
A
7
12
15
F
E
9
9
42
G
D, E
13
17
19
H
F
5
10
15
I
G
11
13
20
J
G, H
2
3
6
K
J, I
8
12
22
TE
b. Now, assume that activity E has taken 10 days past its anticipated duration to
complete. What happens to the project’s schedule? Has the duration changed? Is there
a new critical path? Show your conclusions.
9.10 An advertising project manager has developed a program for a new advertising
campaign. In addition, the manager has gathered the time information for each
activity, as shown in the table below.
a. Calculate the expected activity times (round to nearest integer).
b. Calculate the activity slacks. What is the total project length? Make sure you
fully label all nodes in the network.
c. Identify the critical path. What are the alternative paths and how much slack time
is associated with each non-critical path?
d. Identify the burst activities and the merge activities.
e. Given the activity variances, what is the likelihood of the project finishing on
week 24?
f. Suppose you wanted to have a 99% confidence in the project finishing on time.
How many additional weeks would your project team need to negotiate for in order
to gain this 99% likelihood?
Time Estimates (week)
Activity
Optimistic
Most Likely
Pessimistic
Immediate Predecessor(s)
A
1
4
7
—
B
2
6
10
—
C
3
3
9
B
D
6
13
14
A
E
4
6
14
A, C
F
6
8
16
B
G
2
5
8
D, E, F
9.11
Consider a project with the following information:
Activity
Duration
A
3
--
B
5
A
C
7
A
D
3
B, C
E
5
B
F
4
D
G
2
C
H
5
E, F, G
Activity
Duration
A
Predecessors
ES
EF
LS
LF
Slack
3
0
3
0
3
--
B
5
3
8
5
10
2
C
7
3
10
3
10
--
D
3
10
13
10
13
--
E
5
8
13
12
17
4
F
4
13
17
13
17
--
G
2
10
12
15
17
5
H
5
17
22
17
22
--
a. Construct the project activity network using AON methodology and label each
node.
b. Identify the critical path and other paths through the network.