OR PERT CPM AND JOB SCHEDULING

2640003 - Operations Research (OR)
Unit - 5
1By: Jignesh Kariya, NJSMTI VERAVAL

Contents
Project Management and Scheduling
1. Project Management (CPM & PERT)
• Network concepts
• Components
• Rules for network construction
• Critical path method (CPM)
• Project evaluation and Review Techniques (PERT)
2. Production scheduling (job sequencing)
• Introduction
• Johnson’s algorithm for n jobs 2 machines
• Johnson’s algorithm for N jobs m machines
• 2 jobs m machines using graphical method.

Project Management
and Scheduling

Introduction
• A project is an endeavor to create a unique product or service. It is
specific, timely, usually multi-disciplinary and always conflict-ridden.
Projects are a part of an overall programme and are broken down into
well-defined set of tasks (Jobs), subtasks and further if desired, all of
which must be completed within a specified time, along with minimum
cost.
• Examples of projects include, construction of a bridge, highway, power
plant, repair and maintenance of an oil refinery or an air plane; design.
development and marketing of a new product, research and
development work, etc. Such projects involve a large number of
interrelated activities (or tasks) that must be completed in a specified
time, in a specified sequence (or order) and require resources such as
personnel, money, materials, facilities and/or space.
• The main objective before starting any project is to schedule the
required activities in an efficient manner so as to complete it on or
before a specified time limit at a minimum cost of its completion

Introduction
• Hence, before starting any project, it is necessary to prepare a plan for
scheduling and controlling the various activities (or tasks) involved in
the given project. The techniques of operations research used for
planning, scheduling and controlling large and complex projects are
often referred to as network analysis, network planning or network
planning and scheduling techniques.
• All these techniques are based on the representation of the project as
a network of activities. A network is a graphical plan consisting of a
certain configuration of arrows and nodes for showing the logical
sequence of various activities to be performed to achieve project
objectives.

Introduction
• PERT and CPM are two methods where (Programme Evaluation and
Review Technique) was developed in 1956-58 by a research team to
help in the planning and scheduling of the US Navy's Polaris Nuclear
Submarine Missile project, which involved thousands of activities. The
objective of the team was to efficiently plan and develop the Polaris
missile system. Since 1958, this technique has proved to be useful for
all jobs or projects that have an element of uncertainty in the
estimation of duration, as is the case with new types of projects the
likes of which have never been taken up before.
• CPM (Critical Path Method) was developed by E.L. DuPont company
along with Remington Rand Corporation. The aim behind its
development was to provide a technique for the control of the
maintenance of company's chemical plants. In course of time, the use
of CPM was extended to the field of cost and resource allocation.

Basic Difference Between PERT & CPM
Basis PERT CPM
Stands for PERT stands for Programme Evaluation
and Review Technique
CPM stands for Critical Path Method.
Model It is a probabilistic model under which the
result estimated in a manner of
probability.
It is a deterministic model under which
the result is ascertained in a manner of
certainty.
Time IT deals with the activities of uncertain
time.
It deals with the activities of precise well
known time.
Jobs It is used for onetime projects that
involve activities of non-repetitive nature
It is used for completing of projects that
involve activities of repetitive nature.
Orientation It is activity oriented in as much as its
result is calculated on the basis of the
activities.
It is even oriented, in as much as its
results are calculated on the basis of the
events.
Dummy
Activities
It does not make use of dummy activities. It makes use of dummy activities to
represent the proper sequencing of the
activities.
Cost It has nothing to do with cost of a project.
It deals with the cost of a project
schedules and their minimization.
Estimation
It finds out expected time of each activity
on the basis of three types of estimates.
Its calculation is based on one type of
time estimation that is precisely known.
Time PERT is restricted to time variable. CPM includes time-cost trade off.

• There are no essential differences between PERT and CPM as such. Both of
them share in common the determination of a critical path and are based on
the network representation of activities and their scheduling that determines
the most critical activities to be controlled so as to meet the completion date of
a project. However. The following are the some of their other major
differences.
PERT : Project Evolution And Review Technique
1. In PERT, three estimates are used to form a weighted average of the expected
completion time of each activity, based on the probability distribution of
completion times.
2. It is basically a tool for planning and control of time.
3. To perform PERT analysis on a project, the emphasis is on the completion of a
task rather than the activities required to be performed to reach a particular
event or task. Thus, it is also called an event oriented technique.
4. It is used for onetime projects that involve activities of non-repetitive nature
(i.e. activities that may never have been performed before), in which time
estimates are uncertain, such as redesigning an assembly line or installing a
new information system. 8By: Jignesh Kariya, NJSMTI VERAVAL

5. It 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 : Critical Path Method
1. In CPM, there is only one estimate of completion time of each activity. This
technique was developed in connection with a construction and
maintenance project in which the duration of each activity was known
with certainty.
2. It allows an explicit estimate of costs in addition to time. Thus, it is suitable
for establishing a trade off for optimum balancing between schedule time
and cost of the project.
3. It is used for completing of projects that involve activities of repetitive
nature.

Significance of Using PERT / CPM
1. A network diagram helps to translate highly complex project into a set of
simple and logical arranged activities and therefore,
• Helps in the clarity of thoughts and actions.
• Helps in clear and unambiguous communication developing from top to
bottom and vice versa among the people responsible for executing the
project.
2. Detailed analysis of a network helps project incharge to peep into the
future because
• Difficulties and problems that can be reasonably expected to crop up
during the course of execution can be foreseen well ahead of its actual
execution.
• Delays and holdups during course of execution arc minimized. Corrective
action can also be taken well in time. 10By: Jignesh Kariya, NJSMTI VERAVAL

3. Isolates activities that control the project completion and therefore, results
in expeditious completion of the project.
4. Helps in the division of responsibilities and therefore, enhance effective
coordination among different departments/agencies involved.
5. Helps in timely allocation of resources to various activities in order to
achieve optimal utilization of resources.

Phases of Project Management
In general, project management consists of three phases.
1. Project planning phase
2. Scheduling phase
3. Project control phase
1. Project Planning Phase :
• In order to visualize the sequencing or precedence requirements of the
activities in a project, it is helpful to draw a network diagram.
For this the following tips are adopted:
(a) Identify various activities (task or work elements) to be performed in
the project, that is, develop a breakdown structure (WBS).
(b) Determine the requirement of resources such as men, materials,
machines, money, etc., for carrying out activities listed above.

(c) Assign responsibility for each work package. The work packages
corresponds to the smallest work efforts defined in a project and forms the set
of elemental tasks that are the basis for planning, scheduling and controlling
the project.
(d) Allocate resources to work packages.
(e) Estimate cost and time at various levels of project completion.
(I) Develop work performance criteria.
(g) Establish control channels for project personnel.

2. Scheduling Phase :
Once all work packages (i.e. tasks) have been identified and given unique names
or identifiers, scheduling the project, i.e. when each of the activities are
required to be performed, is taken up. It prepares an analysis/estimate of the
likelihood of the project to be completed on or before the specified time.
The various steps involved during this phase are listed below:
(a) Identify all people who will be responsible for each task.
(b) Estimate the expected duration(s) of each activity, taking into consideration
the resources required for their execution in the most economic manner.
(c) Specify the interrelationship (i.e. precedence relationship) among various
activities.

2. Scheduling Phase cont..
(d) Develop a network diagram, showing the sequential interrelationship
between various activities. For this, tips such as; what is required to be done;
why it must be done, can it be dispensed with how to carry out the job; what
must precede it; what has to follow; what can be done concurrently, may be
followed.
(e) Based on these time estimates, calculate the total project duration, identify
critical path; calculate floats; carry out resources smoothing (or levelling)
exercise for critical (or search) resources, taking into account the resource
constraints (if any).
3. Project Control Phase
Project control refers to the evaluation of the actual progress (status) against
the plan. If significant differences are observed, then the scheduling and
resource allocation decisions are changed in order to update and revise the
uncompleted pan of the project. In other words, remedial (modifying planning)
or reallocation of resources (cost minimization) measures are adopted in such
cases. 15By: Jignesh Kariya, NJSMTI VERAVAL

16
Relationship between phases of Project Management
By: Jignesh Kariya, NJSMTI VERAVAL

PERT/CPM Network Components & Precedence Relationships
17
• PERT/CPM network consists of two major components.
1. Events
2. Activities
Events
• Events in the network diagram represent project milestones, such
as the start or the completion of an activity (task) or activities, and
occur at a particular instant of time at which some specific part of
the project has been or is to be achieved. Events are commonly
represented by circles (nodes) in the network diagram.
• The events can be further classified into the following two
categories:
1. Merge Events
2. Burst Events

18
• Merge Event :
An event which represents the joint completion of more than one
activity is known as a merge event.
• Burst Event :
An event that has represents the initiation (beginning) of more than
one activity is known as burst event.
Events
Events

19
• Activities
• Activities in network diagram represent project operations or tasks
to be conducted. As such each activity except dummy activates
requires resources and takes a certain amount of time for
completion. An arrow is commonly used to represent an activity
with its head indicating the directions of progress in the project.
• Activities are identified by the numbers of their starting (tail or
initial) event and ending (head or terminal) event.
Activities can be further classified into the following three
categories.
1. Predecessor Activity
2. Successor Activity
3. Dummy Activity

20
1. Predecessor Activity :
An activity which must be completed before one or more other
activities start is known as predecessor activity.
2. Successor activity
An activity which starts immediately after one or more of other
activities are completed is known as successor activity.
3. Dummy activity :
An activity which does not consume either any resources and/or time is
known as dummy activity.
A dummy activity in the network id added only to establish the given
precedence relationship among activities of the project. It is needed
when
> Two or more parallel activates in a project have same head and tail
events.
> Two or more activities have some (but not all) of their immediate
predecessor activities in common.

22
Network Shows the sequential relationships among activities using
nodes and arrows. Network models use the following two types of
precedence network to show precedence requirements of the activities
in the project.
AON (Activity–on-Node) Network :
In this type of precedence network each node (or circle) represents a
specific task while the arcs represent the ordering between tasks. AON
network diagrams place the activities within the nodes, and the arrows
are used to indicate sequencing requirements. Generally these diagrams
have no particular starting and ending nodes for the whole project . The
lack of dummy activities in these diagrams always make them easier to
draw and interpret.
AOA (Activity–on-Arrow) Network :
In this type of precedence network at each end of the activity arrow is a
node. These nodes represent points in times or instants, when an
activity is starting or ending.By: Jignesh Kariya, NJSMTI VERAVAL

23
AOA Network Cont…
The arrows itself represents the passage of time required for that
activity to be performed.
These diagrams have a single beginning node from which all activities
with no predecessors may start. The diagram then works its way from
left to right, ending with a single ending node, where all activities with
no followers come together.
Three important advantages of using AOA are as follows :
1. Many computer programs are based on AOA network.
2. AOA diagrams can be superimposed on a time scale with the arrows
drawn, the correct length to indicate the time requirement.
3. AOA diagrams give a better sense of the flow of time throughout a
project.

24
Rules for AOA Network Construction

25
Rules for AOA Network Construction cont..
1. In network diagram, arrows represent activities and circles the events.
The length of an arrow is of no significance.
2. Each activity should be represented only 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.
3. The event numbered 1 denotes the 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 . An event carrying
the highest number denotes the completion event. A network should
have only one initial event and only one terminal event.
4. The general rule for numbering the event is that the head event should
always be numbered such that for each activity (i,j), i < j.

26
Rules for AOA Network Construction cont..
5. An activity must be uniquely identified by its starting and completion
event, which implies that:
(a) An event number should not get repeated or duplicated.
(b) Two activities should not be identified by the same completion event.
(c) Activities must be represented either by their symbols or by the
corresponding ordered pair of Starting - completion events.
6. The logical sequence (or interrelationship) between activities must follow
these rules:
(a) An event cannot occur until all its incoming activities have been
completed.
(b) An activity cannot start unless all the preceding activities, on which it
depends, have been completed.
(c) Though a dummy activity does not consume either any resource of time,
even then it has to follow the rules 6(a) and (b).By: Jignesh Kariya, NJSMTI VERAVAL

27
Errors and Dummies in Network :
Looping and Dangling :
• Looping
• Looping (cycling) and dangling are considered as faults in a network.
Therefore, these must be avoided.
• A case of endless loop in a network diagram, which is also known as
looping, where activities A, Band C form a cycle.
• Due to precedence relationships, it
appears from Fig, that every activity in
looping (or cycle) is a predecessor of itself.
In this case it is difficult to number three
Events associated with activity A, B and C
So as to satisfy rule 6 of constructing the
network.

28
Errors and Dummies in Network cont..
• Dangling
• A case of disconnect activity before the completion of all activities, which
is also known as dangling, is shown in Fig.
• In this case, activity C does not give any result as per the rules of the
network The dangling may be avoided by adopting rule S of constructing
the network.

29
Dummy (or Redundant) Activity:
• The following are the two cases in which the use of dummy activity may
help in drawing the network correctly, as per the various rules.
(i) 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.
(ii) When two chains of activities have a common event, yet are wholly or
partly independent of each other, as shown in Fig. (a). A dummy which is
used in such a case, to establish proper logical relationships, is also known
as logic dummy activity.

30
In fig (a) : If head event of C and D do not depend on the completion of
activities A and B then the network can be redrawn, as in fig (b), otherwise
fig (a) must be followed.

Examples
31
Ex : 1

Examples
32
Solution : 1
ACTIVITY Predecessor
Activity
A -
B A
C A
D B
E B,C
F E
G D,F
H G

Examples
33
Ex : 2

Examples
34
Solution : 2
Activity
A -
B A
C B
D B
E B
F C
G C
H F, G
I D, E ,H
J I

Examples
35
Ex : 3

Examples
36
Solution : 3
Activity
A -
B A
C A
D C
E B
F C
G D,E
H G
I H
J F
K I,J
L K
M L
N J
O M,N By: Jignesh Kariya, NJSMTI VERAVAL

Examples

Critical Path Analysis
39
 Critical path is the longest path through the project network, the activities on the
path are the critical activities therefore any delay in their completion must be
avoided to prevent delay in project completion.
 The objective of critical path analysis is to estimate the total project duration and
to assign starting and finishing times to all activities involved in the project. This
helps to check the actual progress against the scheduled duration of the project.
The duration of individual activities may be uniquely determined (in case of CPM)
or may involve the three time estimates (in case of PERT), out of which the
expected duration of an activity is computed.
 Having done this, the following factors should be known in order to prepare the
project scheduling.
(i) Total completion time of the project.
(ii) Earlier and latest start time of each activity.
(iii) Critical activities and critical path.
(iv) Float for each activity, i.e. the amount of time by which the completion of a
non-critical activity can be delaying without delaying the total project completion
time.

41
For calculating the earliest occurrence and latest allowable times of
events, the following two methods are there
1. Forward pass method
We use forward and backward pas in CPM / PERT to find Earliest and
Latest occurrence times of events.
• Forward pass calculations are used to find Earliest occurrence times
of events (Ei). Forward pass calculations are from Left to Right. Start
value is equal to zero. (Earliest occurrence time of 1st event = 0).
• When two or more activities merge in an event, the maximum value
is taken as the Earliest occurrence time for that event.
• Forward pass time= Earliest time of Tail event + Activity time

42
2. Backward pass method
 Backward pass calculations are from Right to Left. Backward pass
calculations are used to find Latest occurrence times of events (Li).
For the last event, Latest time = Earliest time.
 If there is more than one activity coming back in an event, in
backward pass we take minimum value.
 Backward pass time= Latest time of Head event – Activity time

43
1. Forward Pass Method (For Earliest Event Time)

44
2. Backward Pass Method (For Latest Allowable Event Time)

45
Float (slack) of an activity and Event

46
Float (slack) of an activity and Event

49
Example :1

50
Example :1 conti…

51
Solution : 1
1. Forward Pass method

52
Solution : 1 cont…
2. Network Diagram

53
Solution : 1 cont..
3. Backward Pass Method

54
Solution : 1 cont..
4. Total And Free Float

55
Example :2

56
Solution : 2

57
Solution : 2 cont...

58
Solution : 2 cont...

Project Scheduling With Uncertain Activity Times
59
PERT was developed to handle projects where the time duration for each activity
is not known with certainty but is a random variable that is characterized by β
Beta distribution. To estimate the parameters : mean and variance, of the β –
distribution three time estimates for each activity are required to calculate its
expected completion time.
The three time estimates that required are as under :
Optimistic time (t0 or a) The shortest possible time (duration) in which an activity,
can be performed assuming that everything goes well.
Pessimistic time (tp or b) The longest possible time required to perform an
activity under extremely bad conditions. However, such conditions do not include
natural calamities like earthquakes, flood, etc.
Most likely time ( tm or m) The time that would occur most often to complete an
activity, if the activity was repeated under exactly the same conditions many
times. Obviously, it is the completion time that would occur most frequently (I.e.
model value).

60
The β – distribution is not necessarily symmetric, the degree of skewness depends on
the location of tm to t0 and tp.
The range of optimistic time t0 and pessimistic time tp is assumed to enclose every
possible duration of the activity. The most likely completion time for an activity may
not be equal to the midpoint (t0 + tp )/2 and may occur to its left or to its right.

61
In Beta Distribution the midpoint (t0 + tp )/2 is given half way weightage than
that of most likely point tm . Thus, the expected or mean ( te or µ ) time of
an activity, that is also the weighted average of three estimates, is computed
as arithmetic mean of
( t0 + tp ) /2 and 2tm .
If the duration of activities associated with project is uncertain, then variance
describes the dispersion (variation) in the activity time values.
The calculations are based on the concept of normal distribution where 99%
of area under normal curve falls within + 3σ from the mean or fall within the
range approximately 6 standard deviation in length.
Therefore, the interval (t0, tp ) or range (tp - t0) is assumed to enclose about 6
standard deviations of a symmetric distribution.
Thus if σ1 is the standard deviation of the duration of activity i

63
Estimation of Project Completion Time

64
Example : 1

65
Solution:

67
E values and L values are shown in below figure :
c :The critical path is : 1 -4 -7 and expected completion time for the
project is 42.8 weeks.

68
D

Extra Sum
69
Example - 1

Extra Sum
70
Solution - 1

Production Scheduling
(Job Sequencing)

Introduction
• The optimal order (sequence) shows the minimum time in which jobs,
equipment, people, materials, facilities and all other resources are
arranged to support the production schedules to give low costs and high
utilization.
• Other objective of calculating optimal production schedule are minimizing
customers waiting time for a product or service, meeting promised
delivery dates, keeping stock levels low providing preferred working
pattern and so on.
• The general scheduling or sequencing problem may be described as:
Let there be n jobs to be performed, one at a time, on each of m
machines. The sequence (order) of the machines in which each job should
be performed is given. The actual or expected time required by the jobs
on each of the machines is also given. The general sequencing problem,
therefore, is to find the sequence out of (n !)m possible sequences, which
minimize the total elapsed time between the start of the job in the first
machine and the completion of the last job on the last machine. 72By: Jignesh Kariya, NJSMTI VERAVAL

Introduction
• In particular, if there are n=3 jobs to be performed and m=3 machines are
to be used, then the total number of possible sequences will be (3 !)3
=216. Theoretically, it may be possible to find the optimum sequence but
this would require a lot of computational time. Thus, one should adopt the
sequencing technique.
• To find the optimum sequence, we first need to calculate the total elapsed
time for each of the possible sequences.
As stated earlier, even if the values of m and n are very small, it is difficult
to get the desired sequence with the total minimum elapsed time.
However, due to certain rules designed by Johnson, the task of
determining an optimum sequence has become quite easy.

Notations, Terminology and Assumptions
• Notations :
tij = Processing time (time required) for job i on machine j.
T = Total elapsed time for processing all the jobs. This includes idle time, if
any.
Iij= Idle time on machine j from the end of job (i - 1) to the start of job i.
• Terminology:
Number of Machines:
The number of machines refer to the number o f service facilities through
which a job must pass before it is assumed to be completed.
Processing Time : This is the time required by a job on each machine.
Processing Order : This refers to the order(sequence) in which machines
are required for completing the job.

Idle Time on a Machine :
This is the time during which a machine does not have a job to process,
i.e. idle time from the end of job (i - 1) to the start of job i.
Total Elapsed Time :
This is the time interval between starting the first job and completing the
last job, including the idle time (if any), in a particular order by the given
set of machines.
No Passing Rule :
This refers to the rule of maintaining the order in which jobs are to be
processed on given machines. for example, if n jobs are to be processed
on two machines, M1 and M2 in the order M1M2, then each job should go
first to machine M1 and then to M2.

Assumptions:
1. The processing time on different machines are exactly known and are
independent of the order of the jobs in which they are to be processed.
2. The time taken by the job in moving from one machine to another is negligible.
3. Once a job has begun on a machine, it must be completed before another job can
begin on the same machine.
4. All jobs are known and are ready for processing before the period under
consideration begins.
5. Only one job can be processed on a given machine at a time.
6. Machines to be used are of different types.
7. The order of completion of jobs are independent of the sequence of jobs. 76By: Jignesh Kariya, NJSMTI VERAVAL

Processing n Jobs Through Two Machines
77
Example -1
Solution -1

79
Example -2
Solution -2

81
Example -3
Solution -3

Processing n Jobs Through Three Machines
82
Example -4
Processing n Jobs Through
Three Machines

Processing n Jobs Through Three Machines
83
Solution -4

Processing n Jobs Through m Machines
84
Example -5
Processing n Jobs Through
m Machines

85
Solution - 5

86
Solution – 5 cont..

87
Example -6
Solution - 6

88
Solution – 6 cont..

89
Example - 7

90
Solution – 7

91
Questions ???

OR PERT CPM AND JOB SCHEDULING

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