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# Project Management Pert

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Submitted By Petemiller
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Chapter 13 Project Management

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.

What is Project Management
■ To

schedule and control the progress and cost of a project.

PERT/CPM:

Input:
– – –

Activities in a project; Precedence relationships among tasks; Expected performance times of tasks. 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; ...

Output:
– – – – –

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?

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 outgoing arc except the ‘finish’ node.

Example of Foundry Inc., p.523
Activity A B C D E F G H Immediate Predecessors A B C C D, E F, G

PERT Network for Foundry Inc. Example

Example of a Hospital Project:
Activity
A B C D E F G H I J K

Immediate Predecessor(s)
— — A B B A C D A E, G, H F, I, G

PERT Network for Hospital Project

Performance Time t of an Activity
§

t is calculated as follows: a + 4m + b t = 6 where a=optimistic time, b=pessimistic time, m=most likely time.

§

Note: t is also called the expected performance time of an activity.

Variance of Activity Time t

If a, m, and b are given for the optimistic, most likely, and pessimistic estimations of activity k, variance σk2 is calculated by the formula σk 2

⎛ b − a ⎞ = ⎜ ⎟ ⎝ 6 ⎠

2

Variance, a Measure of Variation
■ Variance

is a measure of variation of possible values around the expected value. ■ The larger the variance, the more spreadout the random values. ■ The square root of variance is called standard deviation.

Example, Foundry Inc., p.525
Activity a m b t variance

A B C D E F G H

1 2 1 2 1 1 3 1

2 3 2 4 4 2 4 2

3 4 3 6 7 9 11 3

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.

Time and Timings of an Activity:
• • • • • •

t=estimated performance time; ES=Earliest starting time; LS=Latest starting time; EF=Earliest finish time; LF=Latest finish time; s=Slack time of a task.

Uses of Time and Timings
■ Earliest

times (ES and EF) and latest times (LS and LF) show the timings of an activity’s “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.

Computing Earliest Times
Step 1. Mark “start” node: ES=EF=0. 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: EF=ES at the Finish node.

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 children’s are all marked with LF and LS, mark it as below:
• •

LF = Earliest LS of its immediate children, LS = LF – t

Note: LS=LF at Start node.

Computing Slack Times

For each activity:
– slack

= LS – ES = LF – EF

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.

Example, Foundry Inc.
Activity a m b t variance

A B C D E F G H

1 2 1 2 1 1 3 1

2 3 2 4 4 2 4 2

3 4 3 6 7 9 11 3

2 3 2 4 4 3 5 2

0.111 0.111 0.111 0.444 1 1.777 1.777 0.111

A
ES= LS=

2
EF= LF=

C
ES= LS=

2
EF= LF=

F
ES= LS=

3
EF= LF=

slack=

slack=

slack=

Start
ES=EF= LS=LF=

E
ES= LS=

4
EF= LF=

H
ES= LS=

2
EF= LF=

Finish
ES=EF= LS=LF=

slack= B
ES= LS=

slack= G
ES= LS=

3
EF= LF=

D
ES= LS=

4
EF= LF=

5
EF= LF=

slack=

slack=

slack=

Network for Foundry Inc.

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.

Example: A Hospital Project
Activity
A B C D E F G H I J K

Immediate Predecessor(s)
— — A B B A C D A E, G, H F, I, G

Performance time t (weeks)
12 9 10 10 24 10 35 40 15 4 6

F
ES=

10
EF= LF=

A
ES= LS=

12
EF= LF=

LS=

K
ES= LS=

6
EF= LF=

slack= I
ES= LS=

15
EF= LF=

slack=

slack=

slack= Start
ES=EF= LS=LF=

G 10
EF= LF= ES= LS=

35
EF= LF=

C
ES= LS=

Finish
ES=EF= LS=LF=

slack=

slack= D B
ES= LS=

10
EF= LF=

H
ES= LS=

40
EF= LF=

9
EF= LF=

ES= LS=

J
ES= LS=

4
EF= LF=

slack= E
ES= LS=

slack= 24
EF= LF=

slack=

slack=

slack=

A Hospital Project

Slack and the Critical Path
• •

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

Critical Path, Examples

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

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.

C
ES= LS=

4
EF= LF=

A
ES= LS=

2
EF= LF=

slack=

slack= Start
ES=EF= LS=LF=

D
ES= LS=

3
EF= LF=

Finish
ES=EF= LS=LF=

slack= B
ES= LS=

7
EF= LF=

E
ES= LS=

2
EF= LF=

slack=

slack=

Calculate the critical path

Example: Draw diagram and find critical path Activity A B C D E F G H I Predecessor B A C A,D E,G G t 5 3 6 4 8 12 7 6 5

Example: Draw diagram and find critical path Activity A B C D E F G H Predecessor A B A,B C D,E,F E,F t 3 4 6 5 8 2 4 5

Solved Problem 13-1&2, p.547-548

Calculate the Critical Path
Activity A B C D E F G a 1 2 4 8 2 4 1 m b 2 3 3 4 5 6 9 10 5 8 5 6 2 3 Immediate predecessor A B C, D B E

Steps for Solving 13-1&2
1. 2. 3. 4.

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

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.

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.

Project Completion Time and its Variance

The expected project completion time T: T = earliest completion time of the project. The variance of T, σT2 : σT2 = Σ(variances of activities on the critical path)

Example, Foundry Inc.
Activity a m b t variance

A B C D E F G H

1 2 1 2 1 1 3 1

2 3 2 4 4 2 4 2

3 4 3 6 7 9 11 3

2 3 2 4 4 3 5 2

0.111 0.111 0.111 0.444 1 1.777 1.777 0.111
Variance of T, σT2 =

Critical path: A-C-E-G-H Project completion time, T =

Solved Problem 13-1&2, p.547-548

Project completion time and variance
Activity A B C D E F G a 1 2 4 8 2 4 1 m 2 3 5 9 5 5 2 b 3 4 6 10 8 6 3 t 2 3 5 9 5 5 2 variance 0.111 0.111 0.111 0.111 1 0.111 0.111

Critical path: B-D-E-G Project completion time, T = Variance of T, σT2 =

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 σT2 .
2. Calculate Z = x −T

σT 2

3. Find probability from a normal distribution table (as on page 698).

The Idea of the Approach

The table on p.698 gives the probability P(z

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