Project Management

1
Chapter 13

• Project Management

2
Characteristics of a project

• A project is unique (not routine),
• A project is composed of interrelated
  sub-projects/activities,
• It is associated woth a large investment.

3
What is Project Management

• To schedule and control the progress and cost of
  a project.

4
PERT/CPM

• Input
• Activities in a project
• Precedence relationships among tasks
• Expected performance times of tasks.
• Output
• The earliest finish time of the project
• The critical path of the project
• The required starting time and finish time of
  each task
• Probabilities of finishing project on a certain
  date
• ...

5
PERT/CPM is supposed to answer questions such as

• How long does the project take?
• What are the bottle-neck tasks of the project?
• What is the time for a task ready to start?
• What is the probability that the project is
  finished by some date?
• How additional resources are allocated among the
  tasks?

6
PERT Network

• It is a directed network.
• Each activity is represented by a node.
• An arc from task X to task Y if task Y follows
  task X.
• A start node and a finish node are added to
  show project start and project finish.
• Every node must have at least one out-going arc
  except the finish node.

7
Example of Foundry Inc., p.523
Activity Immediate Predecessors
A -
B -
C A
D B
E C
F C
G D, E
H F, G
8
PERT Network for Foundry Inc. Example
9
Example of a Hospital Project
Activity Immediate Predecessor(s)
A
B
C A
D B
E B
F A
G C
H D
I A
J E, G, H
K F, I, G
10
PERT Network for Hospital Project
11
Performance Time t of an Activity

• t is calculated as follows
• where
• aoptimistic time,
• bpessimistic time,
• mmost likely time.
• Note t is also called the expected performance
  time of an activity.

12
Variance of Activity Time t

• If a, m, and b are given for the optimistic, most
  likely, and pessimistic estimations of activity
  k, variance ?k2 is calculated by the formula

13
Variance, a Measure of Variation

• Variance is a measure of variation of possible
  values around the expected value.
• The larger the variance, the more spread-out the
  random values.
• The square root of variance is called standard
  deviation.

14
Example, Foundry Inc., p.525
Activity a m b t variance
A 1 2 3
B 2 3 4
C 1 2 3
D 2 4 6
E 1 4 7
F 1 2 9
G 3 4 11
H 1 2 3
15
Critical Path

• It is the longest path in the PERT network from
  the start to the end.
• It determines the duration of the project.
• It is the bottle-neck of the project.

16
Time and Timings of an Activity

• testimated performance time
• ESEarliest starting time
• LSLatest starting time
• EFEarliest finish time
• LFLatest finish time
• sSlack time of a task.

17
Uses of Time and Timings

• Earliest times (ES and EF) and latest times (LS
  and LF) show the timings of an activitys
  in/out of project.
• ES and LS of an activity tell the time when the
  preparations for that activity must be done.
• For calculating the critical path.

18
Computing Earliest Times

• Step 1. Mark start node ESEF0.
• Step 2. Repeatedly do this until finishing all
  nodes
• For a node whose immediate predecessors are all
  marked, mark it as below
• ES Latest EF of its immediate predecessors,
• EF ES t
• Note EFES at the Finish node.

19
Computing Latest Times

• Step 1. Mark Finish node
• LF LS EF of Finish node.
• Step 2. Repeatedly do this until finishing all
  nodes
• For a node whose immediate childrens are all
  marked with LF and LS, mark it as below
• LF Earliest LS of its immediate children,
• LS LF t
• Note LSLF at Start node.

20
Computing Slack Times

• For each activity
• slack LS ES LF EF

21
Foundry Inc. Example

• Calculate ES, EF, LS, LF, and slack for each
  activity of the Foundry Inc. example on its PERT
  network, given the data about the project as in
  the next slide.

22
Example, Foundry Inc.
Activity a m b t variance
A 1 2 3 2 0.111
B 2 3 4 3 0.111
C 1 2 3 2 0.111
D 2 4 6 4 0.444
E 1 4 7 4 1
F 1 2 9 3 1.777
G 3 4 11 5 1.777
H 1 2 3 2 0.111
23
F
3
A
2
C
2
EF
ES
EF
ES
ES
EF
LF
LS
LF
LS
LS
LF
slack
slack
slack
H
2
Start
Finish
E
4
ES
EF
ESEF
ESEF
ES
EF
LS
LF
LSLF
LSLF
LS
LF
slack
slack
G
5
D
4
B
3
EF
EF
ES
EF
ES
ES
LF
LF
LS
LF
LS
LS
slack
slack
slack
Network for Foundry Inc.
24
Example of Hospital Project

• Calculate ES, EF, LF, LS and slack of each
  activity in this project on its PERT network,
  given the data about the project as in the next
  slide.

25
Example A Hospital Project
Activity Immediate Predecessor(s) Performance time t (weeks)
A 12
B 9
C A 10
D B 10
E B 24
F A 10
G C 35
H D 40
I A 15
J E, G, H 4
K F, I, G 6
26
F
10
ES
EF
K
6
LS
LF
A
12
EF
ES
slack
ES
EF
LF
LS
I
15
LS
LF
slack
slack
ES
EF
LS
LF
slack
G
35
Start
Finish
ES
EF
C
10
ESEF
ESEF
LS
LF
ES
EF
LSLF
LSLF
slack
LS
LF
slack
H
40
D
10
EF
ES
ES
EF
B
9
J
4
LF
LS
LS
LF
ES
EF
ES
EF
slack
slack
LS
LF
LS
LF
slack
slack
E
24
ES
EF
LS
LF
A Hospital Project
slack
27
Slack and the Critical Path

• The slack of any activity on the critical path is
  zero.
• If an activitys slack time is zero, then it is
  must be on the critical path.

28
Critical Path, Examples

• What is the critical path in the Foundry Inc.
  example?
• What is the critical path in the Hospital project
  example?

29
Calculate the Critical Path

• Step 1. Mark earliest times (ES, EF) on all
  nodes, forward
• Step 2. Mark latest times (LF, LS) on all nodes,
  backward
• Step 3. Calculate slack of each activity
• Step 4. Identify the critical path that contain
  the activities with zero slack.

30
C
4
ES
EF
LS
LF
A
2
slack
ES
EF
LS
LF
slack
D
3
Finish
Start
ES
EF
ESEF
ESEF
LS
LF
LSLF
LSLF
slack
E
2
B
7
ES
EF
ES
EF
LS
LF
LS
LF
slack
slack
Calculate the critical path
31
Example Draw diagram and find critical path

• Activity Predecessor t
• A - 5
• B - 3
• C - 6
• D B 4
• E A 8
• F C 12
• G A,D 7
• H E,G 6
• I G 5

32
Example Draw diagram and find critical path

• Activity Predecessor t
• A - 3
• B - 4
• C A 6
• D B 5
• E A,B 8
• F C 2
• G D,E,F 4
• H E,F 5

33
Solved Problem 13-12, p.547-548Calculate the
Critical Path
Activity a m b Immediate predecessor
A 1 2 3 -
B 2 3 4 -
C 4 5 6 A
D 8 9 10 B
E 2 5 8 C, D
F 4 5 6 B
G 1 2 3 E
34
Steps for Solving 13-12

1. Calculate activity performance time t for each
   activity
2. Draw the PERT network
3. Calculate ES, EF, LS, LF and slack of each
   activity on PERT network
4. Identify the critical path.

35
Probabilities in PERT

• Since the performance time t of an activity is
  from estimations, its actual performance time may
  deviate from t
• And the actual project completion time may vary,
  therefore.

36
Probabilistic Information for Management

• The expected project finish time and the variance
  of project finish time
• Probability the project is finished by a certain
  date.

37
Project Completion Time and its Variance

• The expected project completion time T
• T earliest completion time of the project.
• The variance of T, ?T2
• ?T2 ?(variances of activities on the critical
  path)

38
Example, Foundry Inc.
Activity a m b t variance
A 1 2 3 2 0.111
B 2 3 4 3 0.111
C 1 2 3 2 0.111
D 2 4 6 4 0.444
E 1 4 7 4 1
F 1 2 9 3 1.777
G 3 4 11 5 1.777
H 1 2 3 2 0.111
Critical path A-C-E-G-H
Variance of T, ?T2
Project completion time, T
39
Solved Problem 13-12, p.547-548Project
completion time and variance
Activity a m b t variance
A 1 2 3 2 0.111
B 2 3 4 3 0.111
C 4 5 6 5 0.111
D 8 9 10 9 0.111
E 2 5 8 5 1
F 4 5 6 5 0.111
G 1 2 3 2 0.111
Critical path B-D-E-G
Project completion time, T
Variance of T, ?T2
40
Probability Analysis

• To find probability of completing project within
  a particular time x
• 1. Find the critical path, expected project
  completion time T and its variance ?T2 .
• 3. Find probability from a normal distribution
  table (as on page 698).

41
The Idea of the Approach

• The table on p.698 gives the probability P(zltZ)
  where z is a random variable with standard normal
  distribution, i.e. z?N(0,1) Z is a specific
  value.
• P(project finishes within x days)

42
Notes (1)

• P(project is finished within x days)
• P(zltZ)
• P(project is not finished within x days)
• 1?P(project finishes within x days)
• 1?P(zltZ)

43
Notes (2)

• If xltT, then Z is a negative number.
• But the table on p.698 is only for positive Z
  values.
• For example, Z ?1.5, per to the symmetry feature
  of the normal curve,
• P(zlt?1.5) P(zgt1.5) 1?P(zlt1.5)

44
Example of Foundry Inc. p.530-531

• Project completion time T15 weeks.
• Variance of project time, ?T23.111.
• We want to find the probability that project is
  finished within 16 weeks. Here, x16, and
• So, P(project is finished within 16 weeks)
• P(zltZ) P(zlt0.57) 0.71566.

45
Examples of probability analysis

• If a projects expected completing time is T246
  days with its variance ?T225, then what is the
  probability that the project
• is actually completed within 246 days?
• is actually completed within 240 days?
• is actually completed within 256 days?
• is not completed by the 256th day?

46
A Comprehensive Example

• Given the data of a project as in the next slide,
  answer the following questions
• What is PERT network like for this project?
• What is the critical path?
• Activity E will be subcontracted out. What is
  earliest time it can be started? What is time it
  must start so that it will not delay the project?
• What is probability that the project can be
  finished within 10 weeks?
• What is the probability that the project is not
  yet finished after 12 weeks?

47
Data of One-more-example

• Activity Predecessor a m b
• A - 1 2 3
• B - 5.5 7 8.5
• C A 3.5 4 4.5
• D A 2 3 4
• E B,D 0 2 4

48
Example (cont.)

• Activity Predecessor t v
• A - 2 0.111
• B - 7 0.250
• C A 4 0.028
• D A 3 0.111
• E B,D 2 0.444

49
C
4
ES
EF
LS
LF
A
2
slack
ES
EF
LS
LF
slack
D
3
Finish
Start
ES
EF
ESEF
ESEF
LS
LF
LSLF
LSLF
slack
E
2
B
7
ES
EF
ES
EF
LS
LF
LS
LF
slack
slack
Calculate the critical path
50
Solving on QM

• Critical path, ES, LS, EF, LF, and slack can be
  calculated by QM for Windows. We need to enter
  activities times and immediate predecessors.
• But QM does not provide the network.