Project Management
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Introduction to project management
Critical path method (CPM)
Crashing
Uncertain activity durations
Reading: Page 760 – 794
Project
• Project:
– A special type of process / transformation
– A series of related jobs directed toward some
major output
– Require time and resources
– Examples
• MS Windows 2000, Fund raising, Preparation for a
private party, Building a new airport…
The Campus Wedding
Reserve
Church
(1)
Decorate
Church
(3)
2
Choose
Cake (2)
Get Gifts
(1)
Start
Preparatio
ns
1
Church
Notice
(17)
3
Jack’s
Catering
(10)
Rehearsal
Dinner (1)
Wedding
(April 22)
Choose
Pattern
(3)
5
Travel
from
Guatemala
(10)
Receive
Lace (8)
Choose
Invitation
s (3)
7
Prepare
Guest List
(4)
Invitation
s (12)
4
Fit Dress
(2)
Sew
Dress (11)
Address
Inv. (4)
Clean &
Press (2)
6
Inv. To
Post
Office (1)
Invitation
Lead Time
(10)
Project Management
• Project management
Planning, directing, and controlling resources (people,
equipment, material) to meet the technical, cost, and time
constraints of the project.
• Different aspects of PM
– Team work
– Leadership
– We focus on technical aspect
Project Manager
Responsible for:
Work
Human Resources
Communications
Quality
Time
Costs
Project Management
• Statement of work (SOW)
Objectives, work to be done, a proposed
schedule, performance measures (cost, time,
quality)
• Work breakdown structure
– Break project down into manageable
pieces
Work Breakdown Structure
Level 1
Level 2
Level 3
Level 4
Project
Questions about a Project
• How long does it take to complete a project?
• When should each activity start?
• Which activities might delay the project? In other
words, which activities should the project
manager pay particular attention to?
• How to expedite the completion of a project?
• Time-cost trade off
9
Example 1:
Clear Ocean Bay
Motion Pictures Ltd
Activity
Immediately
Predecessor
Durations
(weeks)
A Hire actors, actresses
-
6
B Hire film directors
-
11
C Hire crew members
B
8
D Cast announcement
A,C
9
E Stage setting
C
8
F Music Recording
C
7.5
D,E,F
7
G Film shooting
Question
• How long will it take to complete the
project?
• When should each activity start?
• If we wish to reduce the completion
time, which activities should be
crashed (expedited)?
Answer: Critical path method (CPM)
Basic Four Steps
Step
1: Construct the Network Diagram
2: Compute the Earliest Start time &
Earliest Finish time
(determine the earliest project
completion time)
3: Compute the Latest Start time &
Latest Finish time
(determine the Slack time)
4: Determine the Critical Activities and the
Critical Paths.
12
Critical Path Method
Step 1. Construct the Network Diagram.
AON: Activity-on-node
Each node represents an activity.
Arrows indicate sequencing
requirements.
A
D
Example: A immediately precedes D.
13
Another Representation
• AOA: Activity-on-arrow
Activity A immediately precedes D.
In our class, we use AON more frequently
A
D
1
The Network Diagram
D,9
A,6
Start
C,8
E,8
B,11
F,7.5
A path is a sequence from Start to Finish.
The “Start” and “finish” nodes may be omitted.
G,7
Finish
Critical Path Method
Step 2: Compute the Earliest Start time ES and
Earliest finish time EF for each activity.
ES = The earliest start time of an activity
EF = The earliest finish time of an activity
An activity may start when all its predecessors are finished:
ES(an activity) = maximum(EF’s of all its predecessors)
EF = ES + activity duration
16
Compute ES’s and EF’s
ES EF
A,6
19 28
0 6
D,9
A,6
0
0
Start
0 11
B,11
11 19
19 27
28 35
C,8
E,8
G,7
19 26.5
F,7.5
The project can be completed in 35 weeks.
Question
• How long will it take to complete
the project?
• When should each activity start?
• If we wish to reduce the
completion time, which activities
should be crashed (expedited)?
Slack Time and Critical Path
• Slack time of an activity: the amount of time by
which the activity may be delayed without
delaying the project, given that all other activities
remain unchanged.
• Critical activity: an activity with zero slack time.
• Critical path: a sequence of activities with zero
slack time. It is the longest path from Start to
Finish.
• There can be more than one critical path.
Step 3: Compute the Latest Finish and
Start Time.
LF = The latest time at which an activity may be
finished without delaying the project beyond its
earliest completion date
LS = The latest time at which an activity may start
without delaying the project beyond its earliest
completion date
LF (an activity) = minimum(LS’s of all its
successors)
LS = LF – activity duration
Compute LF’s and LS’s
ES EF
19 28
A,6
0 6
LS LF
0
A,6
13
0
19 28
19 27
11 19
13 19
Start
0
0
D,9
0
0 11
0
Critical path
C,8
B,11
0
0
11 19
28 35
1
E,8
20 28
19 26.5
0 11
F,7.5 1.5
Slack time = LF – EF = LS – ES
20.5
28
G,7
28 35
0
Step 4: Compute the Slack Time.
Activity
Slack Time = LF – EF = LS - ES
A
13
B
0
C
0
D
0
E
1
F
1.5
G
0
Critical Path Method (CPM):
Four Steps
1. Construct the network diagram.
2. Compute the earliest start and finish times of
each activity. Determine the earliest
completion time of the project.
3. Compute the latest finish and start times of
each activity. Determine the slacks.
4. Determine critical activities and critical paths.
CPM applies if activity times are known with
certainty.
The Gantt Chart
Project Completion Time = 35 weeks
Critical Paths: B-C-D-G
A
C
B
D
E
F
G
11
19
28
35
Time
Gantt Chart is a useful tool for scheduling and
describing activities in a project. It can be used with or
without CPM.
Exercise 1: Writing a Term Paper
Activity
Immediately
Predecessor
Durations (days)
A Computer programming
-
7
B Implementation
A
3
C Drafting
-
6
D Proof reading
C
3
E Final write-up
B,D
2
How long does it take? Which activities are critical?
Critical Path and Slacks
A,7
B,3
7 + 3 + 2 = 12
E,2
Start
C,6
1
D,3
The Project takes 12 days.
Activities A, B and E are critical.
1
Problem
• If we need to reduce the completion time of a
project, which activities should be crashed? by
how many units of time?
– Partial crash is allowed : we can use crashing
program for any part of an activity
– Partial crash is not allowed: an activity should
be carried out either with normal program, or
with crashing program, but not both.
The case when Partial crash is not allowed will
not be covered here.
Example 2: Writing a Term Paper: How
to Finish Sooner
In exercise 1, suppose the due date is 10 days from now. How
would you shorten the paper writing time? Which activities would
you crash and by how many weeks?
Activity
Normal Time
Max Crashing Time
Cost/day to expedite
A
7
1
$100
B
3
1
$150
C
6
2
$200
D
3
2
$150
E
2
1
$250
Idea: start from the cheapest critical activity.
Critical Path and Slacks
A,7
B,3
E,2
Start
C,6
1
D,3
The Project takes 12 days.
Activities A, B and E are critical.
1
Crashing -- Step 1
A,6
B,3
E,2
Start
C,6
0
D,3
Shorten A by 1 day, cost: $100.
Project completed in 11 days.
All activities are critical.
0
Crashing -- Step 2
A,5
B,3
E,2
Start
C,6
D,2
Shorten A and D by 1 day, cost: $250.
(Alternatively, shorten E by 1 day at the same cost).
Project completed in 10 days.
Summary: Writing a Term Paper
1. Critical path: A-B-E. A is the cheapest to crash. Crash A
for 1 day.
• Now there are two critical paths: A-B-E, and C-D-E,
with completion time 11 days
2. To complete it in 10 days, you can crash 1 day in A-B-E
and 1 day in C-D-E
• Option 1: crash E for 1 day, with a cost of $250
• Option 2: crash B for 1 day, and D for 1 day, with a
cost of $300.
• Option 3: crash B for 1 day, and C for 1 day, with a
cost of $350.
Option 1 is the best.
In summary, we should crash A for 1 day, and E for 1 day,
with a total cost of $350
Exercise 2:
Clear Ocean Bay Motion Pictures Ltd
In Example 1, we have
found that the project
takes 35 weeks.
Now for some reason, the
project needs to be
completed in 32 weeks.
Which activities would you
crash and by how many
weeks?
Idea: start from the
cheapest critical activity.
Activity
Normal
Time
Max
Crashing
Time
Cost/week
to expedite
A
6
2
40
B
11
2
300
C
8
1
200
D
9
2
80
E
8
1
60
F
7.5
0.5
100
G
7
1
500
Time-Cost Trade-Off:
To What Extent Should a Project be Crashed?
Total
cost
Expected indirect costs
Shorten
Cumulative
cost of crashing
CRASH
Shorten
Optimum
Time/Cost Tradeoff
• Goal: To reduce the project duration by crashing
on selected activities as long as the benefit is
greater than the cost.
• Crashing priorities:
– critical activities
• more economical
• on more than more critical path
Example 3
•Suppose a project consists of activities A,B,C,D,E
and F. The activities crashing can take place over a
wide range of reduction time. The cost per day of
reduction is as follows:
Activity Immed. Pred. Normal time (days) Cost/day to crash Low Time Limit
A
None
3
$450
1
B
None
4
$300
2
C
A
3
$200
1
D
A,B
2
$500
1
E
D
2
$100
1
F
C,E
4
$140
3
Suppose the project is due in 9 days and a penalty of
$150 per day is imposed after the 9th day. How
should we schedule the project to minimize the total
cost?
Example 3
A,3
C,3
F,4
Finish
Start
B,4
D,2
E,2
Completion time: 12 days. The project time should
be reduced if crashing cost is less than $150.
Example 3
1. Critical path: B-D-E-F. Crash E for one day,
with cost $100. –--new completion time: 11
days
2. Critical path: B-D-E-F. Crashing F for one day,
with cost $140. ---- new completion time 10
days
3. Critical path: B-D-E-F. Cost for crashing one
day is at least $300, which is greater than
$150. Stop here.
Summary: Crash E and F for one day
respectively. The project will be finished in 10
days, with a crashing cost of $240.
Project Management in Practice
• In practice, there are many software for
project planning. i.e., CPM-GOMS
• People find that in reality, such software
always slightly underestimate the project
completion time
• Why?
Uncertain Activity Durations
• Because the duration of each activity is usually
stochastic, instead of deterministic; while in
those software input, we simply use mean
activity duration
• Consider the following example
A
B
tA = 4 w.p. 0.5; =5 w.p. 0.5
F
S
tB = 5
C
tC = 9 w.p. 0.5; = 10 w.p. 0.5
Example
• If we use mean activity duration, we have
E[tA] = 4.5
4.5
5
A
B
E[tB] = 5
F
9.5
S
E[tC] = 9.5
C
• The two path (S, A, B, F), (S, C, F) are all critical
paths, those software gives us the project
completion time = max{4.5+5, 9.5}=9.5
Example
• In practice, there are four possibilities: corresponding to
these cases, the project completion times are 9, 10, 10,
10 respectively
• The real mean project completion time should be 9.75
4
5
4
5
A
B
A
B
F
9
S
C
C
S
F
10
S
5
5
5
5
A
B
A
B
9
C
F
S
10
C
F
Project Management in Practice
• From the example, we can see that the real
critical path is stochastic, using mean activity
duration as input generates a static critical path,
thus we over-simplify the problem
• In practice if we want to estimate the project
completion time precisely, we should
– Estimate the activity duration distribution based on
historical data, and then run simulation to determine
the project completion time (to be discussed later)
– Use some approximation, as in the following
CPM with Three Activity Time Estimates
Immediate
Task Predecesors Optimistic Most Likely Pessimistic
A
None
3
6
15
B
None
2
4
14
C
A
6
12
30
D
A
2
5
8
E
C
5
11
17
F
D
3
6
15
G
B
3
9
27
H
E,F
1
4
7
I
G,H
4
19
28
Expected Time Calculations
ET(A)= 3+4(6)+15
Task
A
B
C
D
E
F
G
H
I
Immediate Expected
Predecesors
Time
None
7
None
5.333
A
14
A
5
C
11
D
7
B
11
E,F
4
G,H
18
6
ET(A)=42/6=7
Opt. Time + 4(Most Likely Time) + Pess. Time
Expected Time =
6
Network
Duration = 54 Days
C(14)
E(11)
H(4)
A(7)
D(5)
F(7)
I(18)
B
(5.333)
G(11)
Probability Exercise
What is the probability of finishing this project in
less than 53 days?
P(t < D)
D=53
t
TE = 54
Z =
D - TE
2
σ
∑ cp
Pessim. - Optim. 2
Activity variance, σ = (
)
6
2
Task
A
B
C
D
E
F
G
H
I
Optimistic Most Likely Pessimistic Variance
3
6
15
4
2
4
14
6
12
30
16
2
5
8
5
11
17
4
3
6
15
3
9
27
1
4
7
1
4
19
28
16
(Sum the variance along the critical path.)
2
σ
∑ = 41
P(t < D)
TE = 54
D=53
Z =
D - TE
2
σ
∑ cp
t
53- 54
=
= -.156
41
P(Z < -.156) = .44, or 44 % (Appendix B)
There is an approximate 44% probability that this
project will be completed in less than 53 weeks.
What is the probability that the project duration
will exceed 56 weeks?
P(t < D)
TE = 54
Z =
D - TE
2
σ
∑ cp
t
D=56
56 - 54
=
= .312
41
Approximation: P(Z >.312) = 1-0.62 = .38, or 38 %
Review Problems
• Page 802 to 807, problems 1.b, 2, 5.b, 14,
15, 16, 17.