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.

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;– ...

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 out-going arc except the ‘finish’ node.

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

PERT Network for Foundry Inc. Example

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

PERT Network for Hospital Project

Performance Time t of an Activity

t is calculated as follows:

where a=optimistic time,b=pessimistic time,m=most likely time.

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

ta m b

4

6

Variance of Activity Time t

• If a, m, and b are given for the optimistic, most likely, and pessimistic estimations of activity k, variance k

2 is calculated by the formula

22

6

ab

k

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.

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

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 1 2 3 2 0.111B 2 3 4 3 0.111C 1 2 3 2 0.111D 2 4 6 4 0.444E 1 4 7 4 1F 1 2 9 3 1.777G 3 4 11 5 1.777H 1 2 3 2 0.111

H 2

ES=

LS=

EF=

LF=

A 2

ES=

LS=

EF=

LF=

B 3

ES=

LS=

EF=

LF=

F 3

ES=

LS=

EF=

LF=

G 5

ES=

LS=

EF=

LF=

ES=EF=

LS=LF=

ES=EF=

LS=LF=

Start Finish

slack= slack=

slack= slack=

slack=

Network for Foundry Inc.

D 4

ES=

LS=

EF=

LF=

slack=

C 2

ES=

LS=

EF=

LF=

slack=

E 4

ES=

LS=

EF=

LF=

slack=

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 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

J 4

ES=

LS=

EF=

LF=

A 12

ES=

LS=

EF=

LF=

B 9

ES=

LS=

EF=

LF=

G 35

ES=

LS=

EF=

LF=

H 40

ES=

LS=

EF=

LF=

ES=EF=

LS=LF=

ES=EF=

LS=LF=

Start Finish

slack=

slack=

slack=

slack=

slack=

A Hospital Project

E 24

ES=

LS=

EF=

LF=

slack=

F 10

ES=

LS=

EF=

LF=

slack=

D 10

ES=

LS=

EF=

LF=

slack=

I 15

ES=

LS=

EF=

LF=

slack=

C 10

ES=

LS=

EF=

LF=

slack=

K 6

ES=

LS=

EF=

LF=

slack=

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.

D 3

ES=

LS=

EF=

LF=

A 2

ES=

LS=

EF=

LF=

B 7

ES=

LS=

EF=

LF=

C 4

ES=

LS=

EF=

LF=

E 2

ES=

LS=

EF=

LF=

ES=EF=

LS=LF=ES=EF=

LS=LF=

StartFinish

slack= slack=

slack=

slack=

slack=

Calculate the critical path

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

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

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

Calculate 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

Steps for Solving 13-1&2

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.

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, T2 :

T2 = (variances of activities on the critical path)

Example, Foundry Inc.Activity a m b t variance

A 1 2 3 2 0.111B 2 3 4 3 0.111C 1 2 3 2 0.111D 2 4 6 4 0.444E 1 4 7 4 1F 1 2 9 3 1.777G 3 4 11 5 1.777H 1 2 3 2 0.111

Critical path: A-C-E-G-H

Project completion time, T = Variance of T, T2 =

Solved Problem 13-1&2, p.547-548Project completion time and varianceActivity 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-GProject completion time, T = Variance of T, T

2 =

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).

2.2

T

TxZCalculate

The Idea of the Approach

• The table on p.698 gives the probability P(z<=Z) where z is a random variable with standard normal distribution, i.e. zN(0,1); Z is a specific value.

• P(project finishes within x days)

)(2

ZzPTx

zP

T

Notes (1)

• P(project is finished within x days)

= P(z<=Z)• P(project is not finished within x days)

= 1P(project finishes within x days)

= 1P(z<=Z)

Notes (2)

• If x<T, 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(z<=1.5) = P(z>=1.5) = 1P(z<=1.5)

Example of Foundry Inc. p.530-531

• Project completion time T=15 weeks.• Variance of project time, T

2=3.111.

• We want to find the probability that project is finished within 16 weeks. Here, x=16, and

• So, P(project is finished within 16 weeks)

= P(z<=Z) = P(z<=0.57) = 0.71566.

57.076.1

1

111.3

15162

T

TxZ

Examples of probability analysis

• If a project’s expected completing time is T=246 days with its variance T

2=25, 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?

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?

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

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

D 3

ES=

LS=

EF=

LF=

A 2

ES=

LS=

EF=

LF=

B 7

ES=

LS=

EF=

LF=

C 4

ES=

LS=

EF=

LF=

E 2

ES=

LS=

EF=

LF=

ES=EF=

LS=LF=ES=EF=

LS=LF=

StartFinish

slack= slack=

slack=

slack=

slack=

Calculate the critical path

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.