Critical Path Analysis Module

TOPIC : CRITICAL PATH ANALYSIS 950 / 3

Compiled by: Goh PC 1

LEARNING OUTCOMES:

When you have completed this module, you should be able to

(Networks)

Identify activities on arcs and activities on nodes; construct an activity network for a project;

(Critical paths)

calculate the earliest and latest start and finish times for activities; calculate and interpret the total float for an activity; identify critical activities and critical paths; determine the minimum completion time of a project;

(Scheduling and crashing)

construct a Gantt chart and resource histogram for a project; determine the minimum number of workers to complete a project in a given time; determine the minimum time to complete a project for a given number of workers; determine the effect of adjusting the duration of an activity on the critical path and

completion time;

determine the effect of adjusting the number of workers on the critical path and completion time.

INTRODUCTION

Critical path analysis (CPA) is a technique used in project management for planning, scheduling,

and controlling a complex project. CPA involves the logical sequencing of the activities,

managing the time required for each activity, and determining the most efficient plan for

carrying out the various activities to ensure the whole scheme is completed in the minimum time.

NETWORK

A project defines a combination of interrelated activities that must be executed in a certain order

before the entire task can be completed. The sequence in which the activities in a project must be

carried out is summarized in a table known as the dependence table or precedence table.

An example of a precedence table is as follow.

Activity Preceding Activity

Designing Purchasing Designing

Cutting Purchasing

Sewing Cutting

A project network is a diagram that shows the sequence in which the activities of a project will

take place and their interdependencies. It is also known as a network diagram.

Two popular forms of project network are:

activity on arc (AOA) network diagram, and activity on node (AON) network diagram.



Activity on arc (AOA) network diagram

The basic elements in an AOA network diagram are activities and events.

An activity in a project is a job or task requiring both time and resources for its completion. For example, training of workers, laying tiles etc.

In an AOA network diagram, an activity is represented by an arrow (or a directed arc).

An event represents a point in time that signifies the completion of some activities and the beginning of new ones.

In an AOA diagram, an event is represented by a node (circle).

Rules in AOA network construction:

Network typically flows from left to right.

A complete network must have a single starting event and a single ending event.

Each activity must have a preceding event (tail event) and a succeeding event (head event).

The tail event has a smaller number than the head event.

There can be more than one activity having the same tail event or head event.

Each event has at least a preceding activity and succeeding activity except the start event and the end event.

tail head

(denotes the start

of an activity)

(denotes the completion

of an activity)

A

1

A

B C

D 1

2

3 4 Start event End event

(No preceding

activity) (No succeeding

activity)

Tail event

1 2

Tail event Head event

A

(Denotes the beginning of

Activity A)

(Denotes the end of

Activity A)

Head event



An activity cannot start until the tail event has occurred.

An event cannot occur until all the preceding activities are completed.

Arrows (or arcs) cannot cross each other.

Two activities CANNOT have the same tail event and head event.

Loop (a sequence of activities that starts and ends at the same event) is NOT allowed.

Dummy Activity

A dummy activity is an activity which does not consume time or resources. Dummy activities

are represented by broken arrows .

A dummy activity is used in the following situations.

When two activities are having the same tail event and head event.

To illustrate logical dependencies of activities.

For example:

Activities A and B precede activity C.

Activity B precedes activity D.

1 2

A

B

1 2

3

A

B C

1 2

A

B

1 3

A

B

2

Dummy (Not allowed)

(Use dummy

activity)

Dummy

A

B

C

D

(Use dummy

activity)

A

B

C

D

(Incorrect)



Activity on node (AON) network diagram

An AON network diagram uses boxes (or nodes) to denote activities. These boxes are connected

from beginning to end with arrows to depict a logical sequence and the interdependencies

between the schedule activities. The following diagram is an example of an AON network

diagram.

In the above AON network diagram, activity A precedes activities B and E, whereas activity E

succeeds activities A and D.

Rules in AON network construction:

Network typically flows from left to right.

An activity cannot begin until all the preceding activities are completed.

Arrows indicate precedence and flow and can cross over each other.

Looping is not allowed.

The network has a unique start node and a unique end node.

Each activity has at least one entering arrow and leaving arrow except the start and end

nodes.

The precedence table for a project is given as follow.


A B A

C A

D B, C

E C

AOA network diagram:

AON network diagram:

Dummy

A

C

B 1 2

3

4 5 D

E

Start Finish

B

A

C E

D



Exercise 1

1. Base on the network diagram, construct a precedence table.

2. Base on the precedence table, draw an AOA network diagram.


A B C A

D B

E C

F D, E

3. Draw an AOA network and an AON network for each of the following projects.

(a)


A B A

C A

D A

E B, C

F D

(b)


A B C D A

E B

F C

G D, E

H F, G

(c)


A B C A

D A, B

E A, B

F C

G C, D

A

B

C

D 1 3

2

4 5 E



CRITICAL PATHS

A path in a network diagram is a continuous line leading from the first event (or start) and

connecting adjacent activities until the last event (or finish). A critical path is the path with the

longest duration in the network. It gives the shortest time in which the entire project can be

completed.

In this diagram, the paths are A-C-G, A-E-F and B-D-F.

Path Duration (weeks)

A-C-G 2 + 4 + 3 = 9

A-E-F 2 + 3 + 5 = 10

B-D-F 1 + 2 + 5 = 8

The critical path is A-E-F and the minimum time required to complete the project is 10 weeks.

The critical activities are A, E, and F.

In this diagram, the paths are A-C-E-F, B-E-F and B-D-F.

Path Duration (days)

A-C-E-F 2 + 1 + 5 + 3 = 11

B-E-F 3 + 5 + 3 = 11

B-D-F 3 + 7 + 3 = 13

The critical path is B-D-F and the minimum time required to complete the project is 13 days.

The critical activities are B, D and F.

Note:

A network may have more than one critical path. All the activities on the critical path are known as the critical activities. A delay in any of the critical activities will increase the project duration.

A

B

C

D 1

3

2

4

5

E 2

1

4

2

3

F

5

G 3

6

Act t

A 2

B 3

Start

C 1

E 5

D 7

Finish F 3



Exercise 2

1. Base on the following network diagram, list all the possible paths of the project and their

corresponding total duration (days). Hence, determine the critical path and the minimum time

required to complete the project.

2. Draw an AOA network diagram for each of the following projects, list all the possible

paths of the project and their corresponding total duration. Hence, determine the critical path and

the minimum time required to complete the project.

(a)

Activity Preceding Activity Duration (days)

A 2

B - 3

C A, B 4

D B 7

E C 5

(b)

Activity Preceding Activity Duration (weeks)

A 3

B A 1

C A 2

D B 6

E B, C 5

F E 2

3. Draw an AON network diagram for the following project, list all the possible paths of the

project and their corresponding total duration. Hence, determine the critical activities and the

minimum time required to complete the project.


A 3

B 1

C B 2

D A 3

E A 4

F C, D 7

G E 5

H F, G 2

A

B

C

D 1

3

2 4 5

E

2

3

4

6

F

4

3 Dummy

5



Calculation using AOA network diagram

Earliest and Latest Event Time

ie = the earliest time event i can occur

il = the latest time event i can occur

Calculation of the earliest event time (forward pass):

Begins from the start event and move to the end event.

The earliest time for the start event is 0.

One tail event:

22012 Atee

53223 Btee

86224 Ctee

More than one tail events:

1138

1055

4

3

E

D

te

te

11

11 ,10max.5

e

The earliest time of the end event is the shortest time taken to complete the entire project.

1541156 Ftee

i A B

il ie

B

1

0

A

2

Start

event

D

5

C

3 E

4

End

event F 3

6

3

4

5

6

2

2e

3e

4e

5e 6e



Calculation of the latest event time (backward pass):

Begins from the end event and move to the start event.

For the end event, the latest event time equals the earliest event time.

1566 el

One head event:

1141565 Ftll

831154 Etll

651153 Dtll

More than one head events:

336

268

3

4

B

C

tl

tl

2

3 ,2min.2

l

The latest time for the start event is 0.

The AOA network diagram showing the earliest and latest event times:

Earliest Start Time (EST), Earliest Finish Time (EFT), Latest Start Time (LST) and Latest Finish

Time (LFT) of An Activity

EST of activity A = ie

EFT of activity A = Atei

LST of activity A = Atl j

LFT of activity A = jl

B

1

0

A

2

Start

event

D

5

C

3 E

4

End

event F 3

6

3

4

5

6

2

2l

3l

4l

5l 6l 15 11 2

5

8

B

1

0

A

2

Start

event

D

5

C

3 E

4

End

event F 3

6

3

4

5

6

2

15 11 2

5

8

0 15 11

8

6

2

i A

il ie

j

je jl At



Calculation using AON network diagram

An AON network diagram can be constructed using the following notations.

Calculation of the earliest start time of an activity (forward pass):

Begins from the Start activity and move to the Finish activity.

The earliest start time for the start activity is 0 (the duration of start activity is 0).

One preceding activity:

EST of A = 0

EST of B = 0

EST of D = EST of B + Bt = 0 + 5 = 5

More than one preceding activities:

EST of A + At = 0 + 2 = 2 EST of C = max. (2, 5)

EST of B + Bt = 0 + 5 = 5 = 5

EST of C + Ct = 5 + 4 = 9 EST of E = max. (8, 9)

EST of D + Dt = 5 + 3 = 8 = 9

The earliest start time of the Finish activity is the shortest time taken to complete the

entire project.

EST of Finish = EST of E + Et = 9 + 3 = 12

Act t

EST LST

C 4

Finish

A 2

B 5

Start

0

D 3

E 3

Act

t

EST

LST

EFT

LFT Act

EST EFT t

LST T.F. LFT



Calculation of the latest start time of an activity (backward pass):

Begins from the Finish activity and move to the Start activity.

For the Finish activity, the latest start time equals the earliest start time.

LST of Finish = EST of Finish = 12

One succeeding activity:

LST of E = LST of Finish Et = 12 3 = 9

LST of D = LST of E Dt = 9 3 = 6

LST of C = LST of E Ct = 9 4 = 5

LST of A = LST of C At = 5 2 = 3

More than one succeeding activities:

LST of D Bt = 6 5 = 1 LST of B = min. (0, 1)

LST of C Bt = 5 5 = 0 = 0

The latest start time for the Start activity is 0.

Conditions for a critical activity:

In an AOA network diagram

A is a critical activity if and only if

ie = il

je = jl

Aijij tllee

In an AON network diagram

A is a critical activity if and only if

EST of A = LST of A

EST of B = LST of B

EST of B EST of A = duration of A

A 3

2 2

B 4

5 5

A i

ie

j

il je jl At



Float

Float is the spare time or slack by which an activity can be delayed without increasing the overall

project completion time.

An activity with a slack of zero is critical since it must be completed on time to avoid increasing

the completion time of the project.

Three varieties of float are distinguished:

Total float the delay possible for an activity if all preceding activities start as early as possible whilst all subsequent activities start as late as possible.

Free float the delay possible for an activity if all preceding activities start as early as possible whilst all subsequent activities start at their earliest time.

Independent float the delay possible for an activity if all preceding activities start as late as possible whilst all subsequent activities start at their earliest time.

All non-critical activities have positive total float. These activities can be delayed by the amount

of time equals to the total float without changing the minimum completion time of the project.

However, it affects the latest finish times of the preceding activities and the earliest start times of

the subsequent activities.

Calculations of the floats:

Total float = tel ij = 19 8 3 = 8 days

Free float = tee ij = 15 8 3 = 4 days

Independent float = tle ij = 15 10 3 = 2 days (if the value is negative, then the

independent float is considered zero).

i

8 10

j

15 19

A

3 days(t)

8 10 13 15 19

maximum time available

3 days

A Total float

A

3 days

Free float

A

3 days

Independent

float

ei ej li lj



Calculations of the total floats in an AOA network diagram

Activity Duration

(days)

Earliest Latest Total Float

Start Finish Start Finish

A 2 0 2 0 2 0

B 3 2 5 3 6 1

C 6 2 8 2 8 0

D 5 5 10 6 11 1

E 3 8 11 8 11 0

F 4 11 15 11 15 0

Only activities B and D have positive total float. So activities B and D are non-critical activities.

Both activities B and D can be delayed by one day without delaying the minimum completion

time of the project.

The critical activities are A, C, E and F and the minimum completion time of the project is 15

days.

Calculations of the total floats in an AON network diagram

Activity Duration

(days)

Earliest Latest Total Float

Start Finish Start Finish

A 2 0 2 3 5 3

B 5 0 5 0 5 0

C 4 5 9 5 9 0

D 3 5 8 6 9 1

E 3 9 12 9 12 0

Only activities A and D have positive total float. So activities A and D are non-critical activities.

Activity A can be delayed by three days while activity D can be delayed by 1 day without

delaying the minimum completion time of the project.

The critical activities are B, C and E and the minimum completion time of the project is 12 days.

B

1

0

A

2

Start

event

D

5

C

3 E

4

End

event F 3

6

3

4

5

6

2

15 11 2

5

8

0 15 11

8

6

2

C 4

5 5

Finish

12 12

A 2

0 3

B 5

0 0

Start

0 0

D 3

5 6

E 3

9 9

Act t

EST LST



The following activity network shows the earliest start time, earliest finish time, latest start time,

latest finish time and the total float for each activity in the project.

Activity Preceding Activities Duration (days)

A - 2

B - 3

C A, B 1

D A 5

E C, D 2

Exercise 3

1. Based on the following network diagram, construct a table showing the earliest start time

(EST), earliest finish time (EFT), latest start time (LST) and latest finish time (LFT) for each

activity.

2. Determine the earliest and latest time for each event in the following network diagram.

Hence, construct a table showing the earliest start time (EST), earliest finish time (EFT), latest

start time (LST) and latest finish time (LFT) and the total float for each activity.

0 2 2

0 0 2

A

0 3 3

3 3 6

B

2 5 7

2 0 7

D

3 1 4

6 3 7

C

7 2 9

7 0 9

E

9

9

FINISH

3

1 1

5

7 8

E 2 2

2 3

A

3

1

0 0

6

10 10

4

6 6 B

C

D

F

G

5

2

5

4

1

Act

EST EFT t

LST T.F. LFT

A

B

C

D

E

1

1

0

3

2

5

6

5

10

3

4

4

F

2 Dummy



3. The following table shows the activities for a project.


A - 2

B - 1

C A 3

D B 5

E C, D 2

F B 4

G E, F 3

H B 7

(a) Draw an activity network for the project showing the earliest and latest start time for each activity.

(b) Determine the critical path and the minimum time required to complete the project.



A - 1

B - 1

C - 3

D A 2

E B 3

F B 5

G D, E 5

H C, F 4

(a) Draw an AOA network for the project. (b) Construct a table showing the earliest start time, earliest finish time, latest start

time, latest finish time and the total float for each activity.

(c) Hence, determine the critical activities and the minimum completion time for the project.


Activity Preceding Activities Duration (weeks)

A - 3

B A 5

C A 6

D B 4

E C 8

F B 2

G D, E 11

(a) Draw an AOA activity network for the project. (b) Construct a table showing the duration, earliest start time, earliest finish time,

latest start time, latest finish time and total float for each activity.

(c) Hence, state the critical path and the minimum time for the project to be completed.

(d) If the duration of activity B has to be extended for 3 weeks, determine whether the project will be delayed.



6. The following table shows the activities involved in a project.

Activity Preceding Activities Duration (Hours)

A - 6

B A 4

C - 5

D B, C 1

E D 1

F A 7

G E, F 4

(a) Draw an activity network for the project showing the earliest start time and the latest start time for each activity.

(b) Determine the total float for each activity. (c) Determine the critical path and the minimum time required to complete the

project.

(d) If activity B is extended to 7 hours, determine the number of hours the project will be delayed.

7. The following table shows the activities, their durations and their preceding activities for

a project.

Activity Duration (weeks) Preceding activities

A 6 -

B 4 -

C 2 A

D 3 A

E 7 D

F 10 B, C

G 5 C

H 1 E, G

(a) Draw an AOA network for the project.

(b) Construct a table showing the earliest start time, earliest finish time, latest start

time and latest finish time for each activity. Hence, determine the critical activities and find the

minimum time needed to complete the project.

(c) Determine the minimum time needed to complete the project,

(i) if the duration for activity E is reduced by a week,

(ii) if the duration for activity F is reduced by a week.



SCHEDULING AND CRASHING

Scheduling

The objective of critical path analysis is to obtain a schedule or time chart that determines the

start and finish dates of each activity in the project.

Gantt chart

A Gantt chart is a horizontal-bar chart, which provides a graphical illustration of a schedule

showing activity start, duration and completion. It is constructed with a horizontal axis

representing the total time span of the project, broken down into increments (days, weeks or

months) and a vertical axis representing the activities (or tasks) that make up the project.

A Gantt chart can be created from an activity network:

Start by scheduling critical activities. Since a critical activity doesnt have spare time or float, its start date is fixed at the earliest start time (EST) and finish date is fixed at the

latest finish time (LFT).

Next, scheduling the non-critical activities. For each non-critical activity, the duration between the earliest start time and the latest finish time is determined. Non-critical

activities can be scheduled anywhere within this duration as long as it doesnt affect the precedence relationships.

o If the total float equals the free float, the activity can be scheduled anywhere within this duration without affecting the precedence relationships.

o If the free float is less than the total float, the start time of the activity can be delayed from the earliest start time by not more than the free float, without

affecting the precedence relationships.

Two commonly used schedules are: o Earliest schedule schedule all non-critical activities as earliest as possible. o Latest schedule schedule all non-critical activities as late as possible.

Scheduling must take into consideration: o Parallelism tasks can be undertaken simultaneously. o Dependency task has an effect on subsequent tasks.



A project involves 10 activities. The following network diagram shows the dependency

relationships between activities and the duration (days) for each activity.

The table below shows the number of workers needed for each of the activities.

Activity Number of workers Activity Number of workers

A 3 F 2

B 2 G 4

C 4 H 6

D 3 I 3

E 5 J 4

The Gantt chart below shows the schedule (the earliest and the latest) for each activity in the

project.

Time

1 2 3 4 5 6 7 8 9 10 11 12 13 14

A(3)

C(4)

H(6)

J(4)

B(2)

D(3)

E(5)

F(2)

G(4)

I(3)

5 5 11 12 12 12 4 15 15 13 8 8 8 4

3 5 5 12 12 12 12 6 12 12 8 11 11 11

Row (i) shows the number of workers needed each day if each activity starts as early as possible.

Row (ii) shows the number of workers needed each day if each activity starts as late as possible.

3

2 3

4

7 7

E

3

2

3 3

A

3

1

0 0

7

14 14

5

7 7

B

C

D

F G

4

2 6

4

2 4

4

6

10 10

3 H

I

J Dummy

(i)

(ii)



Resource histogram

The number of workers needed in a project for each time period may be illustrated in a resource

histogram as follow. The non critical activities, activities which are not on the critical path, do

not have fixed starting and finishing times but are constrained by the earliest and latest starting

and finishing times. This situation offers the planner chance for adjusting the demand for

resources.

(i) Resource histogram showing the number of workers needed when activities are scheduled on their earliest start time.

No. of workers Time

1 2 3 4 5 6 7 8 9 10 11 12 13 14

15

14

13

12

11

10

9

8

7

6

5

4

3

2

1

5 5 11 12 12 12 4 15 15 13 8 8 8 4

(ii) Resource histogram showing the number of workers needed when activities are scheduled on their latest start time.

No. of workers Time

1 2 3 4 5 6 7 8 9 10 11 12 13 14

12

11

10

9

8

7

6

5

4

3

2

1

3 5 5 12 12 12 12 6 12 12 8 11 11 11

The peaks and valleys in the above resource histograms indicate high day-to-day variation in the

resource demand. Resource leveling shifts non-critical activities within their float times so as to

move resources from the peak periods (high usage) to the valley periods (low usage), without

delaying the project.



Scheduling limited resource

A resource conflict occurs when at any point in the schedule several activities are in parallel and

the total amount of required resource(s) exceeds the availability limit. The solution to this

situation is to give the resource to higher priority activities and delay the others until the earliest

time the resource become available again. Resource is assigned to the activities having:

the least total float; and the earliest latest start time.

The following network diagram shows a simple project involving three activities A, B and C.

Assuming activity A needs 2 workers, B needs 4 workers and C needs 2 workers and the

resource available is 6 workers per day.

No. of workers Time (day)

1 2 3 4 5

8 A

7

6 C

5

4

B 3

2

1

8 8 6 4 4

The above resource histogram shows resource usage if all activities are scheduled on their

earliest start time. Note that activities A, B & C require more than 6 workers at time period 1 & 2.

No. of workers Time (day)

1 2 3 4 5

6 A C

5

4

B 3

2

1

6 6 6 6 6

The above resource histogram shows resource usage if activity C is delayed by 2 days, its total

slack.

A

B

C

5

1

0 0

3

2 5

2

3 5

3

4

5 5

2

Resource

available 6

workers/day



The activities of a project along with their durations, predecessors and resource used are given in

the following table.

Activity Preceding Activities Duration (days) Resource (men/day)

A - 6 3

B - 2 6

C A 10 4

D A, B 16 4

E B 6 2

F C 8 3

G D, E 10 5

H F 6 2

The AON network is drawn and the project completion time is 32 days without considering the

resource limits.

Time (day) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32

A(3) D(4) G(5)

B(6)

C(4)

E(2)

F(3)

H(2)

9 9 5 5 5 5 10 10 8 8 8 8 8 8 8 8 7 7 7 7 7 7 8 8 7 7 7 7 7 7 5 5

The Gantt chart above shows the activities scheduled at their earliest time and the resource

histogram shows the resource usage per day. The project can be completed in 32 days with a

requirement of at least 10 workers per day.

Act t

EST LST

C 10

6 8

Finish

32 32

A 6

0 0

B 2

0 4

Start

0

D 16

6 6

G 10

22 22

E 6

2 16

F 8

16 18

H 6

24 26

10

9

8

7

6

5

4

3

2

1



If the resource is limited to 8 men per day, determine the activities schedule start and finish times

so that the daily resource usage does not exceed the resource limits.

Current

time

Eligible

activities

Resource Duration ELS Decision Finish time

0 A

B

3

6

6

2

0

4

Start

Delay

6

-

6 B

C

6

4

2

10

4

8

Start

Delay

8

-

8 D

C

E

4

4

2

16

10

6

6

8

16

Start

Start

Delay

24

18

-

18 D

E

F

4

2

3

16

6

8

-

16

18

Continue

Start

Delay

24

24

-

24 F

G

3

5

8

10

18

22

Start

Start

32

34

32 G

H

5

2

10

6

-

26

Continue

Start

34

38

Therefore, the project completion time is 38 days.

Time (day) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38

A(3) B(6) D(4) G(5)

C(4) E(2) F(3) H(2)

3 3 3 3 3 3 6 6 8 8 8 8 8 8 8 8 8 8 6 6 6 6 6 6 8 8 8 8 8 8 8 8 7 7 2 2 2 2

Resource (men/day)

8

7

6

5

4

3

2

1



Crashing

Crashing an activity refers to the speeding up or shortening of the duration of an activity by

using additional resources. These include overtime, hiring temporary staff, renting more efficient

equipment, and other measures. Project crashing refers to the process of shortening the duration

of the project by crashing the duration of a number of activities. Since it generally results in an

increase of the overall project costs, the challenge faced by the project manager is to identify the

activities to crash and the duration reduction for each activity such that the project crashing is

done in the least expensive manner possible.

The key to project crashing is to attain maximum reduction in schedule time with minimum cost.

The time to stop crashing is when it no longer becomes cost effective.

A simple guideline for schedule crashing is:

Crash only activities that are critical. Crash from the least expensive to the most expensive. Crash an activity only until

o It reaches its maximum time reduction. o It causes another path to also become critical. o It becomes more expensive to crash than not to crash.

The AOA network diagram of a project along with the durations (days) of the activities is given

as follow. The minimum completion time of the project is 21 days.

The critical activities are A, F and H and the minimum completion time for the project is 21 days.

The duration of the project can be reduced by crashing one of the activities A, F or H.

First, calculate the free floats for all the non-critical activities and identify those activities with

free float more than zero. These activities are C, D and G.

Activity Duration (days) Free float

A 7 0

B 4 0

C 5 2

D 10 1

E 5 0

F 8 0

G 7 2

H 6 0

3

7 7

7

21 21 6 4

2

4 5

A

5

1

0 0

4

7 7

B

C

D

E

G

10

7

5

8

7

5

12 14

6

15 15

F H

X



To crash activity A: o Reduce the duration of activity A by 1 day and recalculate the free floats for

activities C, D and G.

o The free floats for activities C and D have reduced by 1 day each. So activity A can only be reduced by 1 day (minimum of the original FF of C and D).

o In fact, by reducing the duration of activity A by 1 day has caused the path B-D-H to become critical path. So, activity A can be crashed by at most 1 day.

o The project completion time has been reduced by 1 day to 20 days.

To crash activity F: o Reduce the duration of activity F by 1 day and recalculate the free floats for


o The free floats for activities D and G have reduced by 1 day each. So activity A can only be reduced by 1 day (minimum of the original FF of D and G).

o In fact, by reducing the duration of activity A by 1 day has caused the path B-D-H to become critical path. So, activity A can be crashed by at most 1 day.

o The project completion time has been reduced by 1 day to 20 days.

3

6 6

7

20 20 6 4

2

4 4

A

5

1

0 0

4

6 6

B

C

D

E

G

10

7

5

8

6

5

11 13

6

14 14

F H

X

3

7 7

7

20 20 6 4

2

4 4

A

5

1

0 0

4

7 7

B

C

D

E

G

10

7

5

7

7

5

12 13

6

14 14

F H

X



To crash activity H: o Reduce the duration of activity H by 1 day and recalculate the free floats for


o The free floats for activity G has reduced by 1 day. So activity A can be reduced by 2 day (original FF of G).

o By reducing the duration of activity H by 2 days will cause the path A-E-G to become critical path. So, activity A can be crashed by 2 days.

o The project completion time has been reduced by 2 days to 19 days.

3

7 7

7

20 20 5 4

2

4 5

A

5

1

0 0

4

7 7

B

C

D

E

G

10

7

5

8

7

5

12 13

6

15 15

F H

X

3

7 7

7

19 19 4 4

2

4 5

A

5

1

0 0

4

7 7

B

C

D

E

G

10

7

5

8

7

5

12 12

6

15 15

F H

X



Exercise 4

1. The following table shows the activities for a project, their preceding activities, durations

and resource usage for each activity.


A - 5 3

B - 3 2

C B 6 4

D A 2 1

E A 3 6

F C, D 5 2

G E 7 5

H F 4 3

(a) Draw an AOA network for the project. (b) Determine the critical path and the minimum time to complete the project. (c) Draw a Gantt chart if all activities are scheduled on their earliest start time. (d) Based on your Gantt chart in (c), draw a resource histogram showing the day-to-

day resource usage. State the minimum number of workers required at any given

time.

(e) If the resource is limited to 8 men per day, draw a Gantt chart and a resource histogram to show the schedule start and finish times and resource allocation for

all the activities so that the daily resource usage does not exceed the resource

limits. State the minimum project completion time.

2. The following table shows the activities for a project, their preceding activities, durations

and resource used.


A - 4 3

B - 6 6

C - 2 4

D A 8 1

E D 4 4

F B 10 1

G B 16 4

H F 8 2

I E, H 6 4

J C 6 5

K G, J 10 2

(a) Draw an AON network for the project. (b) Determine the critical path and the minimum time to complete the project. (c) Draw a Gantt chart and a resource histogram if all activities are scheduled on their

earliest start time.

(d) Determine the minimum number of workers required at any given time by shifting the non-critical activities within their total float times without delaying

the project. Illustrate the resource allocation using a resource histogram.

(e) Determine the maximum number of days activity K can be crashed without affecting the dependency relationships between the activities. State the project

completion time after crashing activity K.



STPM 2012

A company is involved in construction projects. One of the projects awarded to the company

contains seven activities. The activities, the preceding activities and the duration required for

each activity are shown in table below.


A

B

C

D

E

F

G

-

-

A

A

B

B

C, D, E

3

9

4

10

2

15

8

(a) Draw an activity network for the project. [3]

(b) Determine the critical activities of the project, and find the minimum number of weeks required to complete the project. [4]

(c) Shortly before the company starts to implement the project, a technical assistant points out that the duration required to undertake activity F could be shortened to 11 weeks

with a new innovative approach. Determine whether the new approach adopted by the

company for activity F would affect your answer in (b). [3]

STPM 2011

The network of the activities on nodes of a project is shown below.

(d) Determine the values of r and s. [4]

(e) State the critical path, and determine the time required to complete the project. [3]

(f) Calculate the total floats for activities A and J. [2]

(g) If the duration of activity J is extended to four weeks, determine whether the project will be delayed. [2]

C 6

3 6

A 0

6 0

B 0

4 5

D 6

5 15

E 6

1 s Start

F 9

4 9

G r

2 16

I 13

5 13

H 13

6 14

J 19

3 20

Finish

K 18

5 18

Activity

Duration (weeks)

Earliest start time

Latest start time



STPM 2010

A group of students are involved in an orientation programme for new form six students. Nine

activities are required in order to organise the programme. The activities and the duration for

each activity are shown in the table below.


A

B

C

D

E

F

G

H

I

-

-

A

A, B

A

C

C

E, F

D, G

3

3

4

2

5

2

6

4

2

(h) Draw an activity network for the programme. [3]

(i) Construct a table which shows the earliest start time, latest finish time and the total float for each activity. Hence, determine the critical path and the minimum number of

days needed to complete the programme. [8]

(j) If the duration of activity B has to be extended to four days, determine whether the programme will be extended or not. Give a reason for your answer. [2]

STPM 2009

A company wishes to develop a theme park on a 120-acre land. The major activities of the

project are listed in the table below.

Activity Duration

(month) Preceding activities

A Project application and approval 10 -

B Project design 6 A

C Project design approval 3 B

D Land clearing 2 A

E Machinery and equipment purchase 6 C

F Building construction 8 C, D

G Landscaping 6 C, D

H Park construction 10 C, D

I Testing 3 E, F, G, H

J Opening ceremony 1 I


(b) (i) List all the possible paths of the project and their corresponding total duration. [4]

(ii) Determine the critical path. [1]

(iii) Find the minimum time required to complete the project. [1]



STPM 2008

The following table shows the activities for a project and their preceding activities and duration.


A - 3

B - 4

C A 2

D B, C 5

E B 2

F D, E 3

(a) Draw an activity network for the project showing the earliest start time and the latest

start time for each activity. [7]

(b) State the critical activities of the project and the minimum time required to complete

the project. [2]

(c) If the duration of activity E has to be extended for a week, determine whether the

project will be delayed. [3]

STPM 2007

The following table shows the activities for a project and their preceding activities and duration.


A - 11

B - 5

C A 9

D B 6

E B 8

F E 6

G C, D, F 8

H E, G 7

I H 9


(b) Construct a table showing the duration, earliest start time, latest finish time and total

float for each activity. Hence, determine the critical path and the minimum duration of the

project. [9]

STPM 2006

The following table shows the activities, their durations and their preceding activities for a

project.

Activity Duration (weeks) Preceding activities

A 2 -

B 1 -

C 3 A

D 2 B

E 3 C, D

F 2 E

G 1 F


(b) Construct a table showing the earliest start time, earliest finish time, latest start time

and latest finish time for each activity. Hence, determine the critical activities and find the

minimum time needed to complete the project. [8]

(c) If the durations for activities C and D are each reduced by a week, determine whether

the project can be completed within 10 weeks. [2]



STPM 2005

The following table shows the activities, their preceding activities and their durations for a

project.


A - 7

B A 3

C A 3

D B, C 4

E B 5

F A 3

G D, E, F 6


(b) Construct a table which shows the earliest start time, earliest finish time, latest start

time, latest finish time, total float, free float and independent float for each activity. [7]

(c) Determine the critical path and the minimum time required to complete the project. [2]

(d) If the duration of activity D has to be extended to 8 weeks, determine the number of

weeks the project will be delayed. [3]

STPM 2004

The following table shows the activities involved in a particular project.

Activity Preceding

activities

Duration

(days)

Earliest start Latest finish

A - 5 0 5

B - 1 0 5

C B 2 1 7

D A, C 4 5 11

E A 6 5 11

F D, E 3 11 14


(b) Calculate the total float and free float of each activity. Hence, determine the critical

path of the project. [7]



STPM 2003

A training programme for young man managers involves seven activities. The activities and the

duration for each activity are shown in the table below.


A - 2

B - 5

C - 1

D B 10

E A, D 3

F C 6

G E, F 8

(a) Draw the network diagram for the training programme. [3]

(b) Determine the critical activities, and find the minimum time needed to complete the

training programme. [8]

STPM 2002

A project on setting up a student-registration system of a college involves seven activities. The

activities and their duration times (in days) are listed as follows:


A - 4

B - 2

C - 3

D A 8

E B 6

F C 3

G D, E 4

(a) Draw a network diagram for the project. [3]

(b) Determine the minimum duration for the project to be completed. [5]

(c) Calculate the total float for each activity and state the critical path of the project. [3]

Critical Path Analysis Module

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