CHAPTER-4 NETWORK MODEL Techniques that enable complex projects to be scheduled.pdf

NETWORK MODEL CHAPTER FOUR
May 20, 2022 NETWORK MODEL 1

INTRODUCTION
Network model is a technique used for planning, and
scheduling large projects in the field of:
• Construction,
• Maintenance,
• Fabrication,
• Purchasing computer system,
• Distribution,
• Transportation,
• Facility allocation, and many others.

CONT’D
Reasons for the wide range applications of NM
• First visualize LP models with the context of
networks.
• Second network models of problems are often easier
to formulate.
• Third the pictorial representation of network models
are easy to understand.

CONT’D
Network diagram is a practical representation of the
various events and activities.
Network modeling: Techniques that enable complex
projects to be scheduled, taking into account the
precedence of each activity.

PROJECT MANAGEMENT APPLICATIONS
What is a project?
• Any unique endeavor with specific objective
• With multiple activities
• With defined precedence relationships
• With a specific time period for completion
Examples
• A major event like a wedding
• Any construction project
• Designing a political campaign

PHASES IN PM
1. Project Planning Phase
2. Project Scheduling Phase
3. Project Control Phase

CONT’D
There are two basic planning and control technique
that utilize network to complete predetermined project
or schedule. These are
Program Evaluation and Review Technique (PERT)
and
Critical Path Method.

NETWORK PLANNING TECHNIQUES
Program Evaluation & Review Technique (PERT):
Developed to manage the Polaris Ballistic Missile in
USA
In any new venture, uncertainties are bound to creep
in. PERT incorporated these uncertainties into a model,
which provides a reasonable answer to these
uncertainties.
Critical Path Method (CPM):
Developed to coordinate maintenance projects in the
chemical industry
A complex undertaking, but individual tasks are routine
(tasks’ duration = deterministic)

BOTH PERT AND CPM
▪Graphically display the precedence relationships &
sequence of activities
▪Estimate the project’s duration
▪Identify critical activities that cannot be delayed
without delaying the project
▪Estimate the amount of slack associated with non-
critical activities

Basic Difference Between PERT and CPM
PERT
In PERT analysis, a weighted average of the expected
completion time of each activity is calculated given
three time estimates of its completion. These time
estimates are derived from probability distribution of
completion times of an activity.
In PERT analysis emphasis is given on the completion
of a task rather than the activities required to be
performed to complete a task. Thus, PERT is also
called an event-oriented technique.
PERT is used for one time projects that involve
activities of non-repetitive nature, where completion
times are uncertain.
PERT helps in identifying critical areas in a project so
that necessary adjustments can be made to meet the
scheduled completion date of the project.
CPM
In CPM, the completion time of each
activity is known with certainty that
too unique.
CPM analysis explicitly estimate the
cost of the project in addition to the
completion time. Thus, this technique
is suitable for establishing a trade-off
for optimum balancing between
schedule time and cost of the project.
CPM is used for completing of
projects that involve activities of
repetitive nature.

Network Conventions
➢The project can be sub-divided into a set of predictable
independent activities each of which has a clear beginning and
ending.
➢Each activity can be sequenced as to its predecessors or
successors.
➢The network is not cyclical i.e each activity is executed once
and only once during the life of the project.
➢Activity times may be estimated either as a single point estimate
(CPM) or as a 3- point estimate (PERT)
▪CPM- Critical Path Method
▪PERT- Program Evaluation and Review Technique
➢The duration of the activities is independent of each other.

BASIC TERMS
A. Network: it is the graphic representation of logically and
sequentially connected arrows and nods representing activities
and events of a project. Networks are also called arrow diagram.
B. Event: an event is a point in time that marks the beginning or
ending of activity. Events are commonly represented by circles
(nodes) in the network diagram.
❖Event divided in to two
1. Merge event
2. Burst event

MERGE EVENT
❖An event which represents the joint completion of more than
one activity is known as merge event.
Merge

BRUST EVENT
An event which represents the initiation (beginning) of
more than one activity is known as burst event.
Burst

C. Activities: Activity is a time-consuming job or task
that is a key sub-part of the total project.
Activities are identified by the numbers of their starting
(tail or initial) event and ending (head or terminals)
event.
i j
Activity
CONT’D

Activities
 Predecessor activity
 Successor activity
 Dummy activity
1 2 4
3
A
Activity
B
Activity
C
Activity
Dummy
Activity
D
Activity
• B is the predecessor of C and D
•B is the successor of A;C and D for B
CONT’D

NETWORK DIAGRAMS
Activity-on-Node (AON):
Uses nodes to represent the activity
Uses arrows to represent precedence relationships

NETWORK DIAGRAMS
Activity-on-Arrow (AOA) network
In this type of precedence network at the each end
of the activity arrow is a node (or circle).

AOA network AON network
1. Activity A
A
2. B must follow A A B
3 C must follow A
& B
B
C
A
4 C must follow A,
and D must follow
A&B
A C
B D
A
A B
A
B
C
A C
B D
CONT’D

Rules of Network Construction
✓In network diagram, arrows represent activities and circles the
events. The length of an arrow is of no significance.
✓Each activity should be represented by one arrow and must
start and end in a circle called event. The tail of an activity
represents the start and head the completion of work.
✓The event numbered 1 denotes start of the project and is
called initial event. All activities emerging (or taking off) from
event 1 should not be preceded by any other activity or
activities.
✓Event carrying the highest number denotes the completion
events. A network should have only one initial event and only
one terminal event.

✓The general rule of numbering the event is that the number at
an activity’s head should always be larger than that at its tail.
That is, events should be numbered such that for each activity (i,
j), i< j.
✓An activity must be uniquely identified by its starting and
completion event which implies that:
•An event number should not get repeated or duplicated.
•Two activities should not be identified by the same completion
event.
•Activities must be represented either by their symbols or by
the corresponding ordered pair of starting-completion events.

✓The logical sequence (or interrelationship) between activities must
follow the following rules :
•An event cannot occur until all the incoming activities into it have
been completed.
•An activity cannot start unless all the preceding activities, on which
it depends, have been completed.
•Though a dummy activity does not consume ether any resource or
time, even then it has to follow the rules 6(a) and 6(b).

ERRORS AND DUMMIES IN NETWORK
Looping and Dangling
✓Looping (cycling) and dangling are considered as faults in a
network. Therefore, these must be avoided.
✓A case of endless loop in a network which is also known as
looping is shown in figure below, where activities A, B, and C
form a cycle.
1
2
4
3
Looping
C
A
B

CONT’D
Dangling
A case of disconnect activity before the completion of all
activities which is also known as dangling.
B
2
1 4
3
A
C
Dangling

The following is one of the cases in which the use of dummy
activity may help in drawing the network correctly as per the
various rules.
When two or more parallel activities in a project have the same
head and tail events: i.e: two events are connected with more
than one arrow.
2
1
A
B
C
D
3
1
A B D
2
Dummy
C C
3
1
A
B
D
2
Dum
my
C
CONT’D

Example
An assembly is to be made from two parts X and Y. Both parts must be
turned on a lathe and Y need not be polished. The sequence of activities
together with their predecessors is given below:
Activity Description Predecessor Activity
A Open work order _
B Get material for X A
C Get material for Y A
D Turn X on Lathe B
E Turn Y on Lathe B, C
F Polish Y E
G Assemble X and Y D, F
H Pack G

SOLUTION
The network diagram for the project is shown below:
1 2
4
3
5
6 7 8
A
B D
C
E
F
G H
Dum
my

LISTED IN THE TABLE ARE THE ACTIVITIES AND
SEQUENCING NECESSARY FOR A MAINTENANCE JOB ON
THE HEAT EXCHANGERS IN A REFINERY.
Activity Description Predecessor Activity
A Dismantle pipe connections -
B Dismantle header, closure, and floating front A
C Remove tube bundle B
D Clean bolts B
E Clean header and floating head front B
F Clean tube bundle C
G Clean shell C
H Replace tube bundle F, G
I Prepare shell pressure test D, E, H
J Prepare tube pressure test and reassemble I
REQUIRED DRAW A NETWORK DIAGRAM

SOLUTION
2 3
7
1
4
8
A B
C
E
D
D1
5
6
D2
G
F
H
6
10
I J

EXERCISE 1
The following table gives the activities in a construction project:
May 20, 2022 NETWORK MODEL
30
Activity Immediate Predecessor
A -
B -
C -
D A
E C
F A
G D, B, E
Required: Draw an arrow diagram for the project

Critical Path Analysis
The objective of critical path analysis is to estimate the
total project duration & to assign starting & finishing
times to all activities involved in the project.
This helps in checking actual progress against the
scheduled duration of the project.
Path: a path is unbroken chain of activities that connects
from the start node to the end node (terminal node).
Certain activities in a network diagram of a project are
called critical activities because delay in their execution
will cause further delay in the project completion time.

CONT’D
Critical path is the continuous chain of critical activities
in a network diagram. It is the longest path starting
from first to the last event & is shown by the thick line
or double lines in the network diagram most of the
time.

CONT’D
The following factors should be known
to prepare project scheduling
• Total completion time of the project
• Earliest & latest start time of each activity
• Float for each activity

CRITICAL ACTIVITIES & CRITICAL PATH
Notation for the purpose of calculating various times
of event & activities.
𝐸𝑖 =Earliest occurrence time of an event, i. It is the
earliest time at which an event can occur without
affecting the total project time.
𝐿𝑖 =Latest occurrence time of event i. It is the latest
time at which an event can occur without affecting
the total project time 𝐸𝑆𝑖𝑗 = Earliest start time for
activity (i, j). It is the time at which the activity can
start without affecting the total project time

CONT’D
𝐿𝑆𝑖𝑗 =Latest start time for activity (i, j). It is the latest
possible time by which an activity must start without
affecting the total project time
𝐸𝐹𝑖𝑗 =Earliest finish time for activity (i, j ). It is the earliest
possible time at which an activity can finish without
affecting the total project time.
𝐿𝐹 𝑖𝑗=Latest finish time for activity (i, j). It is the latest
time by which an activity must get completed without
delaying the project completion.
𝑡𝑖𝑗 Duration of activity ( i, j ).

APBON company is bringing a new product on line to be
manufactured in their current facility in some existing
space. The owners have identified 11 activities and their
precedence relationships. Develop an AON for the
project.
Activity Description
Immediate
Predecessor
Duration
(weeks)
A Develop product specifications None 4
B Design manufacturing process A 6
C Source & purchase materials A 3
D Source & purchase tooling & equipment B 6
E Receive & install tooling & equipment D 14
F Receive materials C 5
G Pilot production run E & F 2
H Evaluate product design G 2
I Evaluate process performance G 3
J Write documentation report H & I 4
K Transition to manufacturing J 2
CONT’D

DIAGRAM THE NETWORK FOR NAPBON COMPANY

ADD DETERMINISTIC TIME ESTIMATES AND
CONNECTED PATHS

CALCULATE THE PATH COMPLETION TIMES
The longest path (ABDEGIJK) limits the project’s
duration (project cannot finish in less time than its
longest path)
ABDEGIJK is the project’s critical path
Paths Path duration
ABDEGHJK 40
ABDEGIJK 41
ACFGHJK 22
ACFGIJK 23

SOME NETWORK DEFINITIONS
All activities on the critical path have zero slack
Slack defines how long non-critical activities can be delayed
without delaying the project
Slack = the activity’s late finish minus its early finish (or its late
start minus its early start)
Earliest Start (ES) = the earliest finish of the immediately
preceding activity
Earliest Finish (EF) = is the ES plus the activity time
Latest Start (LS) and Latest Finish (LF) = the latest time an activity
can start (LS) or finish (LF) without delaying the project
completion

ES, EF NETWORK

LS, LF NETWORK

Float (or slack) of an Activity and Event
The float of an activity is the amount of time by which it is
possible to delay its completion time without affecting the total
project completion time.
1. Event Float
The float (also sometimes called ‘slack’) of an event is the
difference between its latest time (Li) and its earliest time
(𝑬𝒊).That is
It is a measure of how much later than expected a particular
event could occur without delaying the completion of the entire
project.
𝑬𝒗𝒆𝒏𝒕 𝑭𝒍𝒐𝒂𝒕 = 𝑳𝒊
− 𝑬𝒊

CONT’D
2. Activity Float
It is the float (or slack) in the activity time estimates.
Types of activity floats:
a. Total Float
It is the difference between the time available to perform the
activity (measured from the earliest start time to the latest finish
time) and the expected completion time of the activity.
𝑻𝒐𝒕𝒂𝒍 𝑭𝒍𝒐𝒂𝒕 (𝑻𝑭𝒊𝒋
) = (𝑳𝒋
− 𝑬𝒊
) – 𝒕𝒊𝒋
= 𝑳𝑺𝒊𝒋
– 𝑬𝑺𝒊𝒋
= 𝑳𝑭𝒊𝒋
– 𝑬𝑭𝒊𝒋

CONT’D
If total float value is
a) Negative (i.e. L-E<0),-resources are not adequate
and activity may not finish in time
b) Zero (i.e. L-E=0) -Resources are just sufficient to
complete the activity. Any delay in activity
execution will necessarily increase the project cost.
c) Positive (i.e. L-E>0) Resources are surplus -Surplus
resources can be deployed elsewhere or execution
of the activity can be delayed.

CALCULATING SLACK
Activity
Late
Finish
Early
Finish
Slack
(weeks)
A 4 4 0
B 10 10 0
C 25 7 18
D 16 16 0
E 30 30 0
F 30 12 18
G 32 32 0
H 35 34 1
I 35 35 0
J 39 39 0
K 41 41 0

PERT NETWORKS
Program Evaluation and Review Technique (PERT) - is a network
analysis technique used to estimate project duration when there
is a high degree of uncertainty about the individual activity
duration estimates.
The three time estimates are:
1) Optimistic time (𝑡𝑜 𝑜𝑟 𝑎): This is the shortest possible time to
perform an activity, assuming that everything goes well.
2) Pessimistic time (𝑡𝑝 𝑜𝑟 𝑏): This is the maximum (longest) time
that is required to perform an activity, under extremely bad
conditions.
3) Most likely time (𝑡𝑚 𝑜𝑟 𝑚): This is the most realistic time to
complete the activity.

CONT’D
Ħ Expected activity time (𝑡𝑒) =
(𝑡𝑜+ 𝑡𝑝)/2+2𝑡𝑚
3
=
𝑡𝑜+ 4𝑡𝑚+𝑡𝑝
6
Ħ Variance of activity time, 𝜎2 =
1
6
(𝑡𝑝 − 𝑡𝑜)
2
Ħ Standard deviation, σ = Variance
Ħ Estimation of project completion time
Ħ Calculate the probability of actually meeting the scheduled
time to the project as well as activities time of the project

The probability of completing the project by schedule time (𝑇𝑠) is
given by: Prob 𝑍 =
𝑡𝑠− 𝑡𝑒
𝜎𝑒
Where 𝑡𝑒= expected completion
Example
A project is represented by a network shown below & has the following data
Task A B C D E F G H I
Immediate predecessor - - - A A B C D E&F
Optimistic time 5 18 26 16 15 6 7 7 3
Pessimistic time: 10 22 40 20 25 12 12 9 5
Most Likely time: 8 20 33 18 20 9 10 8 4
CONT’D

Determine the following
a) Draw the arrow diagram
b) Expected task time & their variance
c) The earliest & latest expected times to reach each
event
d) The critical path
e) The probability of an event occurring at the
expected completion date if the original scheduled
time of completing the project is 41.5 weeks.
f) The duration of the project that will have 95%
chance of being completed.
CONT’D

SOLUTION
a.
H
G
C
E
D
I
F
B
A
1
3
4
2
6 7
5

Activity 𝑡𝑜 𝑡𝑝 𝑡𝑚 𝑡𝑒=
(𝑡𝑜+ 4𝑡𝑚)+𝑡𝑝
6 𝜎2
=
1
6
(𝑡𝑝 − 𝑡𝑜)
2
1-2
1-3
1-4
2-5
2-6
3-6
4-7
5-7
6-7
5
18
26
16
15
6
7
7
3
10
22
40
20
25
12
12
9
5
8
20
33
18
20
9
10
8
4
7.8
20.0
33.0
18.0
20.0
9.0
9.8
8.0
4.0
0.694
0.444
5.444
0.444
2.778
1.000
0.694
0.111
0.111
CONT’D

C&D
1
E2=7.8
L2=16.
8
E5=25.
8
L5=34.
8
E4=33
L4=33
E6=29
L6=38.
8
E1=0
L1=0
E3=20
L3=29.
8
B (20) H (8)
G (9.8)
C (33)
E (20)
D (18)
I (4)
F (9)
A (7.8)
1
3
4
2
6 7
5
E7=42.
8
L7=42.
8
Critical path 1-4-7

E
Given
1. Expected length of critical path (Te)= 33+9.8=42.8
2. Variance of critical path length = 5.429+0.694=6.123-----
𝜎𝑒 = 6.123 = 2.474
3. Scheduled time (Ts)=41.5,
Therefore probability of meeting the scheduled time is given by
Prob 𝑍 ≤
𝑡𝑠− 𝑡𝑒
𝜎𝑒
=prob (Z≤-0.5) =0.3 from normal distribution
table
Thus, the probability that the project can be completed in less
than or equal to 41.5 weeks is 0.30. In other words, the
probability that the project will gate delayed beyond 41.5
weeks is 70%

F
Given that Prob 𝑍 ≤
𝑡𝑠−𝑡𝑒
𝜎𝑒
=0.95
But Z0.95=1.64, from normal distribution table. Thus 1.64= (Ts-
42.8)/2.47)or Ts= 1.64 × 2.474 + 42.8 = 46.85 weeks.

EXERCISE REVISITING AAA COMPANY USING
PROBABILISTIC TIME ESTIMATES
Activity Description
Optimistic
time
Most likely
time
Pessimistic
time
A Develop product specifications 2 4 6
B Design manufacturing process 3 7 10
C Source & purchase materials 2 3 5
D Source & purchase tooling & equipment 4 7 9
E Receive & install tooling & equipment 12 16 20
F Receive materials 2 5 8
G Pilot production run 2 2 2
H Evaluate product design 2 3 4
I Evaluate process performance 2 3 5
J Write documentation report 2 4 6
K Transition to manufacturing 2 2 2

REQUIRED
Determine the following
a) Draw the arrow diagram
b) Expected task time & their variance
c) The earliest & latest expected times to reach each event
d) The critical path
e) The probability of an event occurring at the expected
completion date if the original scheduled time of completing
the project is 48 and 40 weeks. Z=1.52=93.57%(table
value) and (Z=-2.12=48.3%)
f) The duration of the project that will have 90% chance of
being completed. (Z0.9=1.28)

PROJECT TIME-COST TRADE-OFF
Crashing is employed to reduce the project completion
time by spending extra resource (cost). However, as shown
in the figure in the next slide beyond point A, cost increases
more quickly when time is reduced. Similarly, beyond point
B, the time increases while the cost decrease. Since for
technical reasons time may not be reduced indefinitely,
therefore, we call this limit as crash point. There is also a
cost efficient duration called normal point. Thus extending
the activity duration beyond normal point may increase
costs.

CONT’D
Crashing is employed to reduce the project completion time by
spending extra resource (cost). The decision maker is interested
in the central region of the curve between A and B.

CONT’D
The project completion time can be reduced by reducing
(crashing) the normal completion time of critical activities.
The reduction in normal time of completion will increase
the total budget of the project.
However, the decision maker always looks trade-off
between total cost of project and total time required to
complete it.
Project crashing is employed to reduce the project
completion time by spending extra resource (cost).

REDUCING PROJECT COMPLETION TIME
Project completion times may need to be shortened
because of
Different deadlines
Penalty clauses
Need to put resources on a new project
Promised completion dates
Reduced project completion time is “crashing”

CONT’D
Crashing a project needs to balance
• Shorten a project duration
• Cost to shorten the project duration
Crashing a project requires you to know
• Crash time of each activity
• Crash cost of each activity

STEPS IN CRASHING
The method of establishing time-cost trade-off for the
completion of a project can be summarized as follows:
Step 1: Determine the normal project completion time and
associated critical path.
Step 2: Identify critical activity and compute the cost slope for
each of these by using the relationship
The values of cost slope for critical activities indicate the direct
extra cost required to execute an activity per unit of time.

CONT’D
Step 3: For reducing the total project completion time,
identify and crash an activity time on the critical path
with lowest cost slope value to the point where
i. another path in the network becomes critical, or
ii. the activity has been crash to its lowest possible time.

CONT’D
Step 4: If the critical path under crashing is still critical,
return to Step 3. However, if due to crashing of an
activity time in Step 3, other path(s) in the network also
become critical, then identify and crash the activity(s)
on the critical path(s) with the minimum joint cost slope.
Step 5: Terminate the procedure when each critical
activity has been crashed to its lowest possible time.
Determine total project cost (indirect cost plus direct
cost) corresponding to different project durations.

EXAMPLE
Activities Normal Crash
Time (Weeks) Cost (Birr) Time (Weeks) Cost (Birr)
1-2 3 300 2 400
2-3 3 30 3 30
2-4 7 420 5 580
2-5 9 720 7 810
3-5 5 250 4 300
4-5 0 0 0 0
5-6 6 320 4 410
6-7 4 400 3 470
6-8 13 780 10 900
7-8 10 1000 9 1200

CONT’D
Indirect cost is Birr 50 per week.
a) Draw the network diagram for the project and
identify the critical path.
b) What are the normal project duration and
associated cost?
c) Find out the total float associated with non-critical
activities.
d) Crash the relevant activities and determine the
optimal project completion time and cost.

6
4
1
0
5
3
E6=1
8
L6=17
E5=1
2
L5=11
E4=1
0
L4=11
E1=0
L1=0
E3=6
L3=6
13
7
9
0
E2=3
L2=3
3
1 2
4
6
5
E7=2
2
L7=21
3
7
8
E8=3
2
L8=32
CONT’D

May 20, 2022 NETWORK MODEL 69 69
6
4
1
0
5
3
E6=1
7
L6=17
E5=1
1
L5=11
E4=1
0
L4=11
E1=0
L1=0
E3=6
L3=6
13
7
8
0
E2=3
L2=3
3
1 2
4
6
5
E7=2
1
L7=21
3
7
8
E8=3
1
L8=31
×
New Total cost=Total Direct normal cost + increased direct cost
due to crashing of activity (2-5)

4
4
1
0
5
3
E6=1
5
L6=15
E5=1
1
L5=11
E4=1
0
L4=11
E1=0
L1=0
E3=6
L3=6
13
7
8
0
E2=3
L2=3
3
1 2
4
6
5
E7=1
9
L7=19
3
7
8
E8=2
9
L8=29
×
Critical activities Crash cost per week (Birr)
1-2 100
2-3 0 (Crashing is not required)
2-5 X Crashed
3-5 50
5-6 45
6-7 70
7-8 200

New Total cost=Total Direct normal cost + increased direct cost
due to crashing of 5-6 + indirect cost for 29 weeks
=(4,220+1×45+2×45)+29×50 = birr 5805
Critical activities Crash cost per week (Birr)
1-2 100
2-3 0 (Crashing is not required)
2-5 X Crashed
3-5 50
5-6 X crashed
6-7 70
CONT’D

CONT’D
Crashing 6-7 from 4 weeks to 3 weeks will results in
increased direct cost than the gain due to reduction in
project time. Consequently, here we must stop further
crashing.

CONT’D

EXERCISE
The data on the normal time and cost along with crashed time
and cost associated with a project are shown in the table below:

CHAPTER-4 NETWORK MODEL Techniques that enable complex projects to be scheduled.pdf

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CHAPTER-4 NETWORK MODEL Techniques that enable complex projects to be scheduled.pdf