PROJECT MANAGEMENT
Time Management*
Dr. L. K. Gaafar
The American University in Cairo
* This Presentation is Based on information from the PMBOK Guide 2000
L. K. Gaafar
4/7/2015
Critical Path Method (CPM)
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CPM is a project network analysis technique
used to predict total project duration
A critical path for a project is the series of
activities that determines the earliest time by
which the project can be completed
The critical path is the longest path through the
network diagram and has the least amount of
float
L. K. Gaafar
4/7/2015
Finding the Critical Path

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Develop a network diagram
Add the durations of all activities to the project
network diagram
Calculate the total duration of every possible
path from the beginning to the end of the project
The longest path is the critical path
Activities on the critical path have zero float
L. K. Gaafar
4/7/2015
Simple Example
Consider the following project network diagram.
Assume all times are in days.
A ctiv ity
A
B
C
D
E
F
L. K. Gaafar
IP A
--A
B
B
C
D
D u ra tio n (d a y s)
2
5
2
7
1
2
4/7/2015
Simple Example
C=2
s ta rt
1
A=2
2
B=5
4
E=1
3
6
D=7
5
fin is h
F=2
Activity-on-arrow network
a. 2 paths on this network: A-B-C-E, A-B-D-F.
b. Paths have lengths of 10, 16
c. The critical path is A-B-D-F
d. The shortest duration needed to complete this project is 16 days
L. K. Gaafar
4/7/2015
Time Management
Key
ES
EF
7
Slack Act Dur.
6
LS
LF
0
0
0
2
A
13
2
2
0
2
2
9
2
6
15
15
10
E
1
16
7
B
Activity-on-node network
Dummy
5
7
7
0
7
L. K. Gaafar
C
9
D
14
14
7
0
14
14
16
F
2
16
4/7/2015
A ctiv ity
A
B
C
D
E
F
D ays
2
5
2
7
1
2
L. K. Gaafar
C ost ($)
200
500
200
500
100
100
C ost/day
100
100
100
71.4
100
50
Cash Flow
4/7/2015
Cash Flow
Daily Expenses
180
D ays
2
5
2
7
1
2
C ost ($)
200
500
200
500
100
100
C ost/day
100
100
100
71.4
100
50
160
140
120
Cost ($)
A ctiv ity
A
B
C
D
E
F
100
80
60
40
A c tiv ity
A
A
B
B
B
B
B
C ,D
C ,D
D ,E
D
D
D
D
F
F
C o st o f d a y T o ta l c o st
100
100
100
200
100
300
100
400
100
500
100
600
100
700
1 7 1 .4
871
1 7 1 .4
1043
1 7 1 .4
1214
7 1 .4
1286
7 1 .4
1357
7 1 .4
1428
7 1 .4
1500
50
1550
50
1600
20
0
1
2
3
4
5
6
7
8
9
10
11
12
11
12
13
14
15
16
13
14
15
4/7/2015
16
Day
Cumulative Expenses
1800
1600
1400
1200
Cost ($)
D ay
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
1000
800
600
400
200
L. K. Gaafar
0
1
2
3
4
5
6
7
8
9
10
Determining the Critical Path for Project X
a. How many paths are on this network diagram?
b. How long is each path?
c. Which is the critical path?
d. What is the shortest duration needed to complete this project?
L. K. Gaafar
4/7/2015
Stochastic (non-deterministic)
Activity Durations
Project Evaluation and Review
Technique (PERT)
4/7/2015
Stochastic Times
Uniform
Triangular
Beta
L. K. Gaafar
4/7/2015
Important Distributions
L. K. Gaafar
4/7/2015
Stochastic Times
The Central Limit Theorem
The sum of n mutually independent random
variables is well-approximated by a normal
distribution if n is large enough.
L. K. Gaafar
4/7/2015
PERT: Finding the Critical Path
(Stochastic Times)
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Develop a network diagram
Calculate the mean duration and variance of each activity
Calculate the total mean duration and the variance of every
possible path from the beginning to the end of the project by
summing the mean duration and variances of all activities on the
path.
The path with the longest mean duration is the critical path
If more than one path have the longest mean duration, the critical
path is the one with the largest variance.
Calculate possible project durations using the normal distribution
L. K. Gaafar
4/7/2015
Example I
Activity
IPA
A
B
C
D
E
F
G
--A
-A,C
B,D
A,C
--
Duration (wks)
a
m
b
4
6
14
3
4
8
4
5
6
7
7
7
3
3
6
6
8
14
13 18 20
Assuming that all activities are beta distributed,
what is the probability that the project duration
will exceed 19 weeks?
4/7/2015
L. K. Gaafar
A ct iv it y
IP A
A
---
B
A
C
--
D
A ,C
E
B ,D
F
A ,C
G
--
A 7, 2.8
D u ratio n
a
m
b
4
3
4
7
3
6
13
6
4
5
7
3
8
18
14
8
6
7
6
14
20



7 .0 0
2 .7 8
4 .5 0
0 .6 9
5 .0 0
0 .1 1
7 .0 0
0 .0 0
3 .5 0
0 .2 5
8 .6 7
1 .7 8
1 7 .5 0
1 .3 6
7
B 4.5, 0.7
E 3.5, 0.25
14
C 5,0.1
D 7, 0
17.5
F8.7, 1.8
7
G 17.5, 1.36
L. K. Gaafar
4/7/2015
A 7, 2.8
7
B 4.5, 0.7
E 3.5, 0.25
14
C 5,0.1
D 7, 0
17.5
F8.7, 1.8
7
G 17.5, 1.36
L. K. Gaafar
Path


ADE
ABE
AF
CDE
CF
G
17.50
15.00
15.67
15.50
13.67
17.50
3.03
3.72
4.56
0.36
1.89
1.36
4/7/2015
Example II
Duration
Activity IPA Distribution
a
m
A
--- Uniform
4
NA
B
--- Triangular
3
4
C
--- Beta
4
5
D
C Beta
5
7
E
A Triangular
3
3
F
A, B Triangular
5
8
G
E, D Uniform
9
NA
b
8
5
6
12
6
8
9
Construct an activity-on-arrow network for the project above.
Provide a 95% confidence interval on the completion time of the project.
L. K. Gaafar
4/7/2015
Example II
Activity
A
B
C
D
E
F
G
IPA
------C
A
A, B
E, D
Distribution
Uniform
Triangular
Beta
Beta
Triangular
Triangular
Uniform
Duration
a
m
4
NA
3
4
4
5
5
7
3
3
5
8
9
NA
b
8
5
6
12
6
8
9
F
B
A
Start
Finish
E
G
C
D
L. K. Gaafar
4/7/2015
Example II
B (4, 0.17)
F (7, 0.5)
A (6, 1.33)
Start
C (5, 0.11)
Finish
E (4, 0.5)
G (9, 0.0)
D (7.5, 1.36)
Path


BF
AF
AEG
CDG
11
13
19
21.5
0.67
1.83
1.83
1.47
L. K. Gaafar
4/7/2015
Time Management: Crashing
Consider the following project network diagram.
Assume all times are in days.
A ctiv ity
A
B
C
D
E
F
L. K. Gaafar
IP A
--A
B
B
C
D
D u ra tio n (d a y s)
N o rm a l
C ra sh
2
2
5
3
2
1
7
4
1
1
2
1
T o ta l co st ($ )
N o rm a l
C ra sh
200
200
500
700
200
250
500
650
100
100
100
350
4/7/2015
Time Management
C(2,1,50)
A(2,2,0)
B(5,3,100)
D(7,4,50)
A ction
N o C rash in g
C rash D b y 1
C rash D b y 1
C rash D b y 1
C rash B by 1
C rash B by 1
C rash F b y 1
L. K. Gaafar
E(1,1,0)
F(2,1,250)
C ritical Path
D u ration (d ays)
T otal C ost ($ )
A -B -D -F
A -B -D -F
A -B -D -F
A -B -D -F
A -B -D -F
A -B -D -F
A -B -D -F
16
15
14
13
12
11
10
1600
1650
1700
1750
1850
1950
2200
4/7/2015
Duration/Cost Decision
Support Curve
2300
2200
2100
T o tal C o st ($)
2000
1900
1800
1700
1600
1500
10
11
L. K. Gaafar
12
13
D u ra tio n (d a y s )
14
15
16
4/7/2015
Time Management
C(2,1,50)
A(2,2,0)
E(1,1,0)
B(3,3,100)
D(4,4,50)
F(1,1,250)
Shortest Possible duration with crashing is 10 days.
Critical path is not changed.
L. K. Gaafar
4/7/2015
Example Problem
A ctivity
A
B
C
D
E
F
G
L. K. Gaafar
IPA
--A
-A ,C
B ,D
A ,C
--
D uration (days)
N orm al
C rash
6
4
4
3
5
4
7
7
4
2
8
6
18
13
T otal cost ($)
N orm al
C rash
100
120
80
93
95
110
115
115
64
106
75
99
228
318
4/7/2015
Project Network
A6
B4
E4
C5
D7
F8
G 18
P ath
A -B -E
A -D -E
A -F
C -D -E
C -F
G
D u ration
L. K. Gaafar
14
17
14
16
13
18
Shortest possible normal duration is 18 at a cost of $757
4/7/2015
Time Management
0
1
A
1
6
6
6
4
7
10
10
B
4
14
13
1
0
Dummy
L. K. Gaafar
5
6
5
1
2
7
7
14
0
18
6
14
0
G 18
4
0
18
2
C
10
13
D
F
17
E
14
4
18
7
8
Dummy
18
4/7/2015
Crashing
A(4,4,10)
B(4,3,13)
E(2,2,21)
D(7,7,0)
C(4,4,15)
F(8,6,12)
G(13,13,18)
A ction
N o C rashin g
C rash G
C rash G & A
C rash G & E
C rash G & E
C rash G ,A & C
L. K. Gaafar
A BE
14
14
13
12
11
10
ADE
17
17
16
15
14
13
D uration
AF
CDE
14
16
14
16
13
16
13
15
13
14
12
13
CF
13
13
13
13
13
12
G
18
17
16
15
14
13
E xtra
C ost
-18
28
39
39
43
T otal
C ost
757
775
803
842
881
924
4/7/2015
Final Crashed Network
A(4,4,10)
B(4,3,13)
E(2,2,21)
D(7,7,0)
C(4,4,15)
F(8,6,12)
G(13,13,18)
The shortest crashed project duration is 13 days at a minimum total cost of $924.
Further crashing of B or F is useless
L. K. Gaafar
4/7/2015
Using Critical Path Analysis to Make
Schedule Trade-offs
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Knowing the critical path helps you make
schedule trade-offs
Free slack or free float is the amount of time an
activity can be delayed without delaying the
early start of any immediately following activities
Total slack or total float is the amount of time an
activity may be delayed from its early start
without delaying the planned project finish date
This part is from a presentation by Kathy Schwalbe, schwalbe@augsburg.edu
L. K. Gaafar
http://www.augsburg.edu/depts/infotech/
4/7/2015
Techniques for Shortening a Project
Schedule



Shortening durations of critical tasks by adding
more resources or changing their scope
Crashing tasks by obtaining the greatest amount
of schedule compression for the least
incremental cost
Fast tracking tasks by doing them in parallel or
overlapping them
This part is from a presentation by Kathy Schwalbe, schwalbe@augsburg.edu
L. K. Gaafar
http://www.augsburg.edu/depts/infotech/
4/7/2015
Shortening Project Schedules
Original
schedule
Shortened
duration
Overlapped
tasks
This part is from a presentation by Kathy Schwalbe, schwalbe@augsburg.edu
L. K. Gaafar
http://www.augsburg.edu/depts/infotech/
4/7/2015
L. K. Gaafar
4/7/2015
Activity Definition
L. K. Gaafar
Activity Sequencing
4/7/2015
Duration Estimation
L. K. Gaafar
Schedule Development
4/7/2015
Schedule Control
L. K. Gaafar
4/7/2015