11/02/09 Chapter 7-Proje ct Planning 1 Elements of Project Planning Divide project into tasks, tasks into subtasks, subtasks into ... Estimate duration of each task, subtask, ... Estimate resource requirements for each task, subtask, ...(budget, personnel, facilities) Identify precedence relations among tasks
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11/02/09 Chapter 7-Project Planning
1
Elements of Project Planning
Divide project into tasks, tasks into subtasks, subtasks into ...
Estimate duration of each task, subtask, ...
Estimate resource requirements for each task, subtask, ...(budget, personnel, facilities)
Tasks for Designing and Building an Electric Substation
Table E 7.3.1
Task Duration
(days)
A. Design substation 10
B. Select site 7
C. Prepare site 3
D. Order transformers and related electrical equipment 2
E. Order concrete, fencing, and related construction
supplies
2
F. Excavate for foundation 3
G. Pour and cure concrete foundation 10
H. Erect structural frames 2
I. Install electrical equipment 3
J. Erect fence 2
K. Test electric equipment 2
L. Connect to grid 2
M. Clean up site 2
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Gantt Chart for Substation Design and Construction
Fig. E7.3.1(b)
A
B
C
D
E
F
G
H
I
J
K
L
M
M T W Th F
Week 1
Task M T W Th F
Week 2
M T W Th F
Week 3
M T W Th F
Week 4
M T W Th F
Week 5
M T W Th F
Week 6
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Critical Path Method (CPM)
Uses a network flow diagram to depict the precedence relations among activities (tasks)
Elements of diagram are directed line segments and nodes
Facilitates identification of activities whose timely completion are “critical” to timely completion of the project
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CPM Notation and Conventions
An activity is an ongoing effort on a project task (directed line segment).
Every activity has an initiating event and a
closing event (nodes).
Events consume no time. Their primary role in
CPM diagrams is to separate activities.
Consecutive activities must be separated by events.
Fig. 7.6
A3
(b ) E v en ts(a) A c tiv ities
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CPM Notation and Conventions (cont.)
No pair of events can be directly connected by more than one activity with no intervening events. If an activity R must immediately precede S and T, the relationship is depicted as follows
Fig. 7.7(a)
If activities R and S must both immediately precede T, the relationship is depicted as follows
Fig. 7.7(b)
All networks must begin with a single Start event and end with a single Finish event.
S
T
R
S
R
T
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Dummy Activities Sometimes precedence relationships require the use of a
dummy activity (depicted by a dashed line) to indicate the appropriate relationships. Dummy activities do not take up any time.
Dummy activity is needed to correctly depict that P and
Q must precede S, and P must precede R.
Dummy activity needed when several activities have same Start and End events. Activities R and S share the same Start and End events.
Fig. 7.10
Dummy activities should be be included only when needed to display the precedence relation.
S
P R
Q
S
R
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Precedence Relations for Several Activities of Bumper Project
Table 7.2
Activity Preceded by
Design bracket
Build bracket Design bracket
Build bumper
Drill holes in bumper Build bumper, Design bracket
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Using Dummy Activity for Bumper Project
Fig. 7.8
(a)Shows proper precedence relations for drilling the
holes but does not include building the bracket
drill holein bumper
(c)Inclusion of a dummy activity allows
proper relations to be depicted
(b)Improperly shows that building the bracket
requires the bumper to be built first
build
brac
ket
buildbumper
design
bracket
buildbumper
drill holein bumper
buildbracket
designbracket
drill holein bumper
buildbumper
design
bracket
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Multiple Dummy Activities
A precedes D A and B precede E B and C precede F
Fig. 7.9
D
C F
A
B E
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Project Activities and their Precedence Relations
Table 7.1
Activity Duration Preceded by
A 3
B 3 A
C 4
D 1 C
E 3 B, D
F 2 A, B, D
G 2 C, F
H 4 G
I 1 C
J 3 E, G
K 5 F, H, I
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Constructing a Network Diagram
Fig. 7.12
Fig. 7.11
Start
A3
B3
I1
D1
C4
Start
A3
Finish
B3
E3
H 4
G2
I1
J3
K5
F2
D1
C4
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Example of Constructing a Network Diagram
Table E7.4.1
Fig. E7.4.1
Activity Preceded by
A
B
C
D A,B
E C
F A
G D, E, F
H C
I H
J D, E
K I, J
L G
M L, K
A
F G
EC
LB D
H
J
I K M
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Activity Duration Preceded by A 4 — B 5 — C 4 A D 5 A E 6 A F 4 D,C G 7 F,B H 4 G,E
Another Example of Constructing a Network Diagram*
Start
Finish
A
4C 4
F 4
H4
G 7
D 5
E 6
B
5
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The Critical Path
The critical path is the path of activities from the start event to the finish event for which delay in any activity along that path will delay the project finish.
For projects with a small number of alternative paths, the critical path can be most efficiently identified by finding the longest of the alternative paths.
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Alternative Paths and their Length
Path Length
A-B-E-J 12 A-B-F-G-dummy-J 13
A-B-F-G-H-K 19 C-D-E-J 11
C-D-F-G-dummy-J 12 C-D-F-G-H-K 18
C-I-K 10
For the project depicted in Fig. 7.12
Table 7.3
Thus, A-B-F-G-H-K is the critical path
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Critical Path Depicted on Network Diagram
Fig. 7.13
Start
A3
Finish
B3
E3
H 4
G2
I1
J3
K5
F2
D1
C4
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Example Problem of Finding the Critical Path*
Path Duration
A-E-H 14
A-C-F-G-H 23
A-D-dummy-F-G-H 24
B-G-H 16
Start
Finish
A
4C 4
F 4
H4
G 7
D 5
E 6
B
5
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Alternate Method for Determining Critical Path
This approach is more efficient for larger networks
– Use forward sweep to find earliest start (ES) for each activity
– Use backward sweep to find latest start (LS) for each activity
– Calculate total float (TF) for each activity
TF = LS - ES
– Critical Path consists of Activities with TF = 0
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Earliest Start*
Earliest Start (ES) is the earliest time an activity can start. It is found by tracing forward (from tail to head of each activity arrow) from the project Start event to the tail of the selected activity. When several paths are possible, use the longest path as determined by the sum of the activity durations on that path.
For activity F, ES = 9
Start
Finish
A
4C 4
F 4
H4
G 7
D 5
E 6
B
5
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Project Duration* Continue until we have ES for all activities that
terminate in the project Finish event. When duration of each of those activities are added to their respective ES times, the largest of the resulting sums is defined as the Project Duration.
For activity H, ES=20
Project Duration =20+4=24
Start
Finish
A
4C 4
F 4
H4
G 7
D 5
E 6
B
5
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Latest Start* Latest Start (LS) is the latest time an activity can start
and still have the project completed within the Project Duration time.
LS is found by tracing backwards (from head to tail of each activity) from the project Finish event to the tail of selected activity. Make sure you reach the tail of the selected activity via the head of that activity. When several paths are possible, use the longest path as determined by the sum of the activity durations on that path. The Project Duration minus the length of this longest path is the LS for the selected activity.
For activity A, LS=24-24=0
Start
Finish
A
4C 4
F 4
H4
G 7
D 5
E 6
B
5
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Total Float Total Float for each activity is the
difference between the latest start and the earliest start.
TF = LS - ES
The activities for which TF=0 define the critical path.
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Summary of Total Float Calculations*
Critical Path consists of A-D-F-G-H
Activity Duration ES LS TF
A 4 0 0 0
B 5 0 8 8
C 4 4 5 1
D 5 4 4 0
E 6 4 14 10
F 4 9 9 0
G 7 13 13 0
H 4 20 20 0
Project Duration 24 24
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Total Float Calculations
Activity Duration Earliest Start Latest Start Total 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
Project Duration 19
Table 7.4
For project shown in Fig. 7.12
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Example of Critical Path Determination
Fig. E7.4.2(a)
A1
B2
D 4
H5
E6
G1
L3
M4
O7
F7
J 2K 5I
10 N6
C3
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Summary of Example Float Calculations
Table E7.4.2
Activity ES LS TF
A 0 0 0
B 1 1 0
C 1 4 3
D 3 3 0
E 7 7 0
F 7 7 0
G 13 13 0
H 3 8 5 I 3 12 9 J 14 15 1
K 16 17 1
L 14 14 0
M 17 17 0
N 21 22 1
O 21 21 0 Project
Duration 28
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Critical Path for Example Problem
Fig. E7.4.2(b)
A1
B2
D 4
H5
E6
G1
L3
M4
O7
F7
J 2K 5I
10 N6
C3
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Program Evaluation and Review Technique (PERT)
Based on Critical Path Method
Replaces single estimate of activity
duration by a probability distribution
Allows estimate of probability of
completing project by a specified time
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Typical Beta Distributions for Activity Durations
to-optimistic estimate; the shortest time within which this activity can be completed assuming everything goes right. This is the left terminus of the pdf.
tm-the most likely time required to complete the activity. This is the mode of the pdf.
tp- pessimistic estimate; the longest time it will take this activity to be completed assuming everything goes wrong. This is the right terminus of the pdf.
Fig. 7.14
totm tp
(a) Skewed-left Beta distribution
to tm tp
(b) Skewed-right Beta distribution
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Network Diagram for PERT Example Problem
Fig. 7.15
Start
A1-3
-3
Finish
B2-3-4
E2-3-6
G1-2-4
I0-1-3
J1-3-6
K3-
5-12
F1-2-7
D1-
1-5
C3-4-10
H3-4-6
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PERT Procedure
Calculate the expected duration te (mean)
of each activity
Calculate the variance 2 of each activity
Use te to determine the expected project
duration Te and identify the critical path
6
4 pmoe
tttt
2
2
6
op tt
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Determination of Critical Path for PERT Example Calculation
Activity Expected Time
Variance Earliest Start
Latest Start
Total Float
A 3.00 0.44 0.00 0.50 0.50 B 3.00 0.11 3.00 3.50 0.50 C 4.83 1.36 0.00 0.00 0.00 D 1.67 0.44 4.83 4.83 0.00 E 3.33 0.44 6.50 14.84 8.34 F 2.67 1.00 6.50 6.50 0.00 G 2.17 0.25 9.17 9.17 0.00 H 4.17 0.25 11.34 11.34 0.00 I 1.17 0.25 4.83 14.34 9.51 J 3.17 0.69 11.34 18.17 6.83 K 5.83 2.25 15.51 15.51 0.00
Project Duration
21.34
Table 7.5
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PERT Critical Path
Fig. 7.16
Start
A1-3
-3
Finish
B2-3-4
E2-3-6
G1-2-4
I0-1-3
J1-3-6
K3-
5-12
F1-2-7
D1-
1-5
C3-4-10
H3-4-6
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Probability of Completing Project by Specified time Ts
Calculate the variance of the project duration as the sum of the variances of the activities on the critical path
Use the standard normal variable z to find the probability of completing the project in a specified time Ts
For Ts=20
From Table 5.1
Pr (z < -0.57) = 0.285
T
esS
TTz
362
001250250001440361
222222
.
......
)pathcritical(
T
KHGFDCT
570362
342120.
.
.TTz
T
esS
11/02/09 Chapter 7-Project Planning
43
Space Station PERT Problem
Fig. E7.5.1(a)
This image is not yet available
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Activities for Space Station PERT Problem
Table E7.5.1(a)
Activity Description
A Construct shell of module
B Order life support system and scientific experimentation package from same supplier
C Order components of control and navigational system
D Wire module
E Assemble control and navigational system
F Preliminary test of life support system
G Install life support in module
H Install scientific experimentation package in module I Preliminary test of control and navigational system in
module J Install control and navigational system in module
K Final testing and debugging
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PERT Diagram for Space Station Project
Fig. E7.5.1(b)
Start
A25
-30-
45
F1-1-1
G4-5-7
E5-7-12
C20-25-35
B10-15-20
D3-3-5
H2-2-3
J8-
10-1
4
I4-4-6
K6-8-15
Finish
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Summary of PERT Calculations for Space Station Project
Table E7.5.1(b)
Activity te 2 ES LS TF
A 31.67 11.11 0.00 0.50 0.50
B 15.00 2.78 0.00 19.50 19.50
C 25.83 6.25 0.00 0.00 0.00
D 3.33 0.11 31.67 32.17 0.50
E 7.50 1.36 25.83 25.83 0.00
F 1.00 0.00 15.00 34.50 19.50
G 5.17 0.25 35.00 42.83 7.83
H 2.17 0.03 35.00 35.50 0.50 I 4.33 0.11 33.33 33.33 0.00
J 10.33 1.00 37.67 37.67 0.00
K 8.83 2.25 48.00 48.00 0.00
Project Duration 56.83
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Another PERT Example Problem*
Activity te 2 ES LS TF A 4.32 1.00 0 0 0 B 5.40 1.56 0 8.64 8.64 C 4.32 1.00 4.32 5.40 1.08 D 5.40 1.56 4.32 4.32 0 E 6.48 2.25 4.32 15.12 10.80 F 4.32 1.00 9.72 9.72 0 G 7.56 3.06 14.04 14.04 0 H 4.32 1.00 21.60 21.60 0
Project Duration 25.92
Start
Finish
A
2-4-8
C2-4
-8
F2-
4-8
H2-4-8
G 3.5-7-14
D2.
5-5-
10
E 3-6-12
B
2.5-5-10
11/02/09 Chapter 7-Project Planning
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Probability of Completing Project by Specified time Ts*
Calculate the variance of the project duration as the sum of the variances of the activities on the critical path
Use the standard normal variable z to find the probability of completing the project in a specified time Ts
For Ts=30
From Table 5.1
Pr (z < 1.48) = 1 - Pr (z < -1.48) = 1 - 0.067 = 0.932