1. OPERATIONS RESEARCH
INTRODUCTION TO OPERATIONS RESEARCH
Operations Research is a new branch of Mathematics dealing in the
optimization problems in real-life situations. It is also a quantitative technique to
deal many management problems, In this discipline ,we study cost minimization of
various inventory problems, the minimization of transportation costs of sending
goods from various warehouses to different centers, the profit maximization or cost
minimization in linear programming models, the assignment of different person to
different jobs so that total time taken to perform the jobs is minimized, the
congestion problem in traffic places, airline counters, supermarket, to find out of
the waiting time of customers in the queue, the project completion time with
limited resources and many other similar problems.
ORIGIN OF OPERATIONS RESEARCH
The germination of the concept Operations Research occurred during
World War 1. In England in the year 1915, F.W. Lanchester attempted to treat
military operations quantitatively. He derived equations relating the outcome of a
battle to both the relative numerical strength of the combatants and their relative
manpower. He modeled a situation involving strategic choices and then tested that
a model against a known situation.
During the same period, Thomas Alva Edison in America was studying
the process of Anti-submarine warfare. He devised a war game to be used for
simulating problems of naval manoeuvre. In 1917, A.K. Erlang, a Danish
Mathematician has developed solutions for some waiting line problems. In 1915,
F.W.Harris had developed the first model on an inventory problem for economic
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2. lot size In 1930 W.Leontieff developed a linear programming model representing
the entire United status economy. Active research works were done during World
War II in Great Britain and United States of America.
DEFINITION OF OPERATIONS RESEARCH
Operations Research was initially a subject dealing with military
operations during the World War II. Later several techniques were developed to
suit much of humanities progress in science, technologies, business administration,
etc. A vast variety of fields is brought in the purview of this branch of science.
There are served definitions for O.R.
They only specify the applications of the discipline. None of the definitions is well
defined. We mention some of the definitions here.
DEFINITION BY D’CLARKE
Operations Research is defined as the art of winning wars without
actually fighting.
DEFINITION BY ACOFF, ANNOFF AND CHURCHMANN
O.R. is the application of scientific methods, techniques and tools to
problem involving the operations of system so as to provide those in control of the
operations with optimum solution to the problem.
DEFINITION BY T.L.SASTRY
O.R. is the art of giving bad answers to problems where otherwise
worse answers are given.
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3. DEFINITION BY JAMES LUNTRY
O.R. is the sophisticated name given to multidisciplinary problem-
oriented approach to the top management problem. It involves the applications of
scientific methods in situations where executives require description, prediction,
and comparison for the purpose of decision making.
DEFINITION BY AMERICAN SOCIETY OF O.R.
O.R. is an experimental and applied science devoted to observe,
understanding and predicting the behaviors of purposeful man-machine system,
many of these definitions only broad line the applications of O.R. to war, industry,
management and humanity progress.
APPLICATIONS OF OPERATIONS RESEARCH
Here we mention only some of the areas where O.R. techniques can be
applied. This science is applied widely in areas of accounting facilities, planning,
finance, manufacturing, marketing, purchasing and in organizations and
government and quasi government activities. We mention some of the applications
of O.R. in the above areas.
Cash flow planning, credit policy planning of delinquent account
strategy are some of the areas in accounting where O.R. techniques are used.
Warehouses locations, Transportation loading and unloading, factory
size and location, Hospitals planning are some of the areas of facilities planning .
In finance, it is a applied to qualitative study of investment analysis,
portfolio management, dividend policy, etc. In marketing, O.R. is applied to study
the selection of product-mix, prediction scheduling time, advertising allocation,
etc.
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4. In order to arrive at an optimal solution to the problems in O.R., first we
have to construct a model. Once a project is selected, we have to describe the
problem as a model. The model should describe all the features of the problem.
We have to express the description of the problem in a mathematical formulation.
This formulation has not be done satisfying all the assumptions of the problem.
MODELS AND MODELLING
Modelling a real life situation helps us to study the different behavior of
the problem corresponding to the description of the problem. Great efforts have
been taken by experts to model business situations, military operations, motion of
planets and stars, congestion in traffic places and so on.
A model is an abstraction on an idealized representation of a real life
problem. The object of a model is to provide means for analyzing the behavior of
the system for future improvement. A map of multiple activity chart, a project
network, the representation of the behavior of a queuing system, a model to
forecast the future, based on the past and the present factors of a time series, etc..
are all examples of models, A model can be a picture, map, a curve or an equation.
The reliability of the decision drawn from the model may depend upon the validity
of the model on the basic assumption on which the model is built.
Modelling is the essence of operations research building a model helps
us to convert the complexities and uncertainties of a decision making problem into
a concrete logical structure which is amenable to formed analysis. A model is a
vehicle for arriving a well structured problem of reality. A commentator of a
cricket match describes the play as a model to enable us to predict the future
course of events of the play. It is a descriptive model available for further analysis.
It is not always possible to analyse a situation only with the description of the
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5. situation we have to formulate the problem into concrete mathematical
representation in the form of a curve, graph or equations. Models, could be
classified as iconic model, analogue models and symbolic model.
ADVANTAGES
 A iconic model is concrete.
 It is easy to construct the model.
 It is easy study the model then the system itself.
DISADVANTAGES
 This model is not suited for further manipulation.
 It cannot be used to study the changes in the operation of
the system.
 It is not possible to make any modification of the model.
 Adjustment with changing situations cannot be done in
this model.
ANALOGUE MODEL
In an analogue model, one set of properties is used to represent another
set of properties. After analysing the model for decision making the results of the
analysis can be re-interpreted in terms of the original system. For example,
Contour lines on a map are analogues of elevation as they represent the size and
fall of heights, Graphs are analogues as distance is used to represent a wide variety
of variables such as time, percentage, weight, etc. It is easier to manipulate the
analogue model. But it is less specific and less concrete.
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6. SYMBOLIC MODEL
Symbolic models employ a set of mathematical symbols and functions
to represent the decision variables and their functions to describe the behavior of
the system. Almost all the models in O.R. are symbolic model.
ADVANTAGES
 These models are most abstract and most general.
 These models are amicable for experimental manipulation.
 They yield reasonably good results to the real life problem.
 A good model should have the following characteristics :
 It should be capable of taking into account new formulations with having
any significant change in its frame.
 The assumptions should be well defined and the number of assumptions
should be as small as possible.
 The assumptions should be simple and coherent.
 Only a limited number of variables should be used.
 It should be acceptable to parametric treatment.
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7. ADVANTAGE OF A MODEL
 It is a description of a physical problem.
 It gives a systematic approach to a problem and is subject to logical
treatment.
 It is easy to make decisions based on a model.
 If a model is built on a broad based assumption, it is easy to modify it
according to new situations.
 Model help us finding avenues for new research and improvement in a
system.
LIMITATION OF A MODEL
 Models are only an attempt in describing a system and should be taken to be
as absolute representation of a system.
 Model constructed is valid only if all the assumptions of the model are true
in the system for which the model is constructed.
 Validity of the model is subject to experimental testing.
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8. PERT
PROGRAM EVALUATION AND REVIEW
TECHNIQUE
INTRODUCTION
Network scheduling is a technique used for planning and scheduling
large projects in the fields of construction, maintenance, fabrication purchasing,
computer system installation, research and development designs etc. The technique
is a method of minimizing trouble spots, such as, production bottlenecks, delays
and interruptions by determining critical factors and coordinating various parts of
overall job.
There are two basic planning and control techniques that utilize a
network that to complete a pre determined project or scheduling. These are :
Program Evaluation and Review Technique(PERT) and the Critical Path
Method(CPM) several variations of these have also been developed. One such
important variation being the Review Analysis of Multiple Projects(RAMP) which
is useful for guiding the „activities‟ of several projects at one time.
NETWORK ANALYSIS
CPM was developed in 1957 by J.E. Kelly of Remington and M.R
Walker of Dupont to aid in the scheduling of routine plant overhaul, maintenance
and construction work. This method differentiates between planning and
scheduling. Planning refers to the determination of activities that must be
accomplished and the order in which such activities should be performed to
achieve the objectives of the project.
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9. PERT was developed in the late 1950‟s by the US Navy Special
Projects Office in operation with the management consulting firm of Booz , Allen
and Hamilton. The technique received substantial favourable publicity for its use in
the engineering and development program of the Polaris missile, a complicated
project that had 250 prime characters and over 9000 sub characters. But now this
technique is very popular in the hands of project planner and controller of various
departments in government and in industry. In PERT, we usually assume that the
time to perform the activity is uncertain and as such three time estimates are used.
METHODOLOGY OF PERT/CPM NETWORKS
The methodology involved in applying PERT for any project may be
split into the following steps:
 PROJECT PLANNING
The purpose of this is to identify all important events/activities
which are essential for completion as well as making up of the project and their
dependence upon one another is shown explicitly in the form of a network.
 TIME ESTIMATION
Estimates of the time required perform each of network activities
are made, the estimates are based upon manpower and equipment availability
and certain assumptions that may have been made in planning the project. By
incorporating the time required for completing each of the activities in the
network, the project duration as well as the criticality of the activities are found. At
this stage it is also possible to compute the probability of completing the project or
a part of the project by a specified time.
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10.  SCHEDULING
The scheduling computations give the earliest and the latest allowable
start and finish time for each activity, and as a by product, they identify the
critical path through the network, and indicate the amount of “slack” time
may associated with the non critical paths.
 TIME COST TRADE OFF’S
If the scheduled time to complete the project as determined in step 3
satisfactory, the project planning and scheduling may be complete
However, if one interested in determining the cost of reducing the project
completion time. Then time cost trade-offs of activity performs time must be
considered for those activities on the critical and near critical path(s).
 RESOURCE ALLOCATION
The feasibility of each schedule must be checked with respective
manpower and equipment requirements. Establishing complete feasibility of
a specific schedule may require replanning and rescheduling or time-cost
trade-offs. Hence a final solution may require the performance of a number
of cycles of steps 3, 4 and 5.
 PROJECT CONTROL
When the network plan and the schedule have been developed to a
satisfactory extent, they are repaired to final form for use in the field. The
project is controlled by checking progress against the schedule, assigning
and scheduling manpower and equipment, and analysing the effects of
delays.
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11. PROBABILTY CONSIDERATIONS IN PERT
The network methods discussed so far may be termed as deterministic,
since estimated activity times are assumed to be the expected values. But no
recognition is given to the fact that expected activity time is the mean of a
distribution of possible values which could occur.
Under the conditions of uncertainty, the estimate time for each activity
are PERT network is represented by a probability distribution. This
probability distribution of activity time is based upon three different time
estimates mode for each activity. These are as follows:
to = the optimistic time, is the shortest possible time to complete the
activity if all goes well.
tp = the pessimistic time, is the longest time that an activity could
take if everything goes wrong.
tm = the most likely time, is the estimate of the normal time an
activity would take. If only one time where available, this would
be it. Otherwise it is mode the probability distribution.
PROBABILITY OF MEETING THE SCHEDULE TIME
With PERT, it is possible to determine the probability of completing a
contract on schedule. The scheduled dates are expressed as number of time units
from the present time. Initially they may be the latest time, T L, for each event, but
after a project is started we shall know how far it has progressed at any given date,
and the scheduled time will be the latest time if the project is to be completed on its
original schedule.
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12. The probability distribution of times for completing an event can be
approximated by the normal distribution due to the control limit theorem. Thus the
probability of completing the project by schedule time(TS) is given by:
Prob (z<(Ts-Te)/σ)
the standard normal variate is given by,
Z=(Ts-Te)/σe
Where
Te = Expected completion time of the project.
σ e = Number of the standard deviations the scheduled time lies from the
expected time. (i.e) the standard deviations of the scheduled time.
Using the commutative normal distribution table, the corresponding value
of the standard normal variate is read off. This will give the require probability of
completing the project on schedule time.
RULES OF NETWORK CONSTRUCTION
For the construction of a network, generally, the following rules are
followed
 Each activity is represented by one and only one arrow.
 Each activity must be identified by its starting and end node which implies
that.
i. Two activities should not be identified by the same completion
events, and
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13. ii. Activities must be represented by their symbols are by the
corresponding ordered pair of starting completion events.
 Notes are numbered to identify an activity uniquely Tail node should be
lower than the head node, of an activity.
 Between any pair of nodes, there should be one and only one activity,
however more than one activity may emanate from and terminate to a node.
 Arrows should be kept straight and not curved or bent.
NUMBERING THE EVENTS
After the network is drawn in a logical sequence, every event is
assigned a number. The number sequenced must be such so as to reflect the flow of
the network. In event numbering, the following rules should be observed:
a. Event number should be unique.
b. Event numbering should be carried out on a sequential basis from left to
right.
c. The initial event which has all outgoing arrows with no incoming arrow is
numbered 0 or 1.
d. The head of an arrow should always bear a number higher than the one
assigned at the tail of the arrow.
e. Gaps should be left in the sequence of event numbering to accommodate
subsequent inclusion of activities, if necessary.
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14. BASIC CONCEPTS OF NETWORK ANALYSIS
A fundamental ingredient in both PERT and CPM is the use of
network system as a means of graphically depicting the current problems or
proposed project. Because of its importance to a basic understanding of both PERT
and CPM, the network concept will be examined. When a network is being
constructed, certain conventions are followed to represent a project graphically, for
it is essential that the relationship between activities and events are correctly
depicted. Before illustrating the network representation, it is necessary to define
some of the concepts.
ACTIVITY
All projects may be viewed as being composed of operations or tasks
called activities, which require the expenditure of time and resources for the
accomplishment. An activity as depicted by a single arrow ( ) on the project
network. The activity arrows are called arcs. The activity arrow is not scaled, the
length of the activity time is only a matter of convenience and clarity, and does not
represent important of time. The head of the arrow shows the sequence or flow of
activities. An activity cannot begin until the completion of the preceding activities.
It is important that activities be defined so that beginning and end of each activity
can be identified clearly.
PREDECESSOR ACTIVITY
Activities that must be completed immediately prior to the start of
another activity are called predecessor activities.
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15. SUCCESSOR ACTIVITY
Activities that cannot be started until one or more of the other
activities are completed, but immediately succeed them are called successor
activities.
CONCURRENT ACTIVITY
Activities which can be accomplished concurrently are known as
concurrent activities. It may be noted that an activity can be a predecessor or
successor to an event or it may be concurrent with one or more of the other
Activities.
EVENT
An Event represent a specific accomplishment in the project and takes
place at a particular instant of time, and does not, therefore, consume time or
resources. An event in a network is a time oriented reference point that signifies
the end of the activity and the beginning of another. Events are usually represented
in the project network by circles (o). The event circles are called nodes. Therefore,
a major difference between activities and events is that activities represent the
passage of time where as events are point in time. All activity arrows must begin
and end with event nodes as shown below
Start Finish
event Activity event
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16. MERGE EVENT
Where more than one activity comes and joins, the event is known as
merge event
BURST EVENT
When more than one activity leaves an event, the event is known as
burst event.
MERGE AND BURST EVENT
An activity may be a merge and burst event simultaneously as with
respect some activities it can be merge event and with respect to some other
activities it may be burst event
.
Merge event Burst Event Merge & Burst
DUMMY ACTIVITY
In most projects many activities can be performed concurrently or
simultaneously. It is possible that two activities could be drawn by the same
beginning and end events, In situations where two are more activities can be
performed concurrently, the concept of dummy activity is introduced to solve this
problem. Therefore there will be only one activity between two events.
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17. An activity which does not consume either any resource or time is
known as dummy activity. A dummy activity represented by dotted line in the
network diagram.
Predecessor Successor
Activity Activity
Dummy activity
PERT SYSTEM OF THREE TIME ESTIMATE
The traditional single estimate of duration of any activity is replaced by
three time estimates in PERT system an optimistic, a pessimistic, and a most likely
time.
OPTIMISTIC TIME (a or to)
The time estimate of an activity when everything is assumed to go well
as per plan. In other words, it is the estimate of the minimum possible time, which
an activity takes in completion under ideal conditions. However no provisions are
made for breakdown, delays, etc
MOST LIKELY TIME (m or tm)
The time which the activity will take most frequently performed a
number of times the model value.
PESSIMISTIC TIME (tp)
The unlikely but possible performance time if whatever could go
wrong, goes wrong in series. In other words it is the longest time the activity can
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18. conceivably take. This however does not include major catastrophies like labour
strikes, acts of God unrest, etc.
EST- It means Earliest start time for an activity represent the time at which an
activity begins at the earliest.
EFT- EFT means Earliest finish time of an activity is it earliest start time „+‟
(plus) the required time to perform the activity.
LFT- LFT means latest finish time. Latest finish time of an activity represent the
latest by which an activity must be completed without delaying the completion of
project.
LST- Latest start time for an activity is the Latest finish time „-‟(minus) the
activity duration methods.
FORWARD PASS METHOD (For Earliest Event Time)
Based on fixed occurrence time of the initial network event, the
forward pass computation yields the earliest start and earliest finish times for each
activity and indirectly the earliest expected occurrence time for each event.
BACKWORD PASS METHOD (For latest allowable time)
The latest occurrence event time (L) specifies the time by which all
activities entering into that event must completed, without delaying the total
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19. project. These are computed by reversing the method of calculation used for
earliest event times.
CRITICAL PATH METHOD
The longest path is called the critical path. An activity is said be critical
if the delay in its start will delay the project completion time.
PERT-ALGORITHM
The various step involved in developing PERT network for analyzing
any project are summarized below
 Make a list of activities that make up the project including immediate
Predecessors.
 Making use of step1 sketch the required network.
 Denote the most likely time by tm, the optimistic time to and
pessimistic time by tp.
 Using beta distribution for each activity duration the expected time t e for
te = (to+tm+tp)/6
 Tabulated various time (i.e) expected activity times, earliest and latest
times and mark the EST and LFT on the arrow diagram.
 Determine the total float for each activity by taking the difference
between EST & LFT.
 Identify the critical activities and connect them with the beginning node
and the ending node in the network diagram by double line arrows. The
critical path and expected date of completion of the project
 Using the values of tp and to to compute the variance (σ2) of each
activity . This is done with the following formula,
σ2 = [(tp-to)/6]2
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20.  Compute the standard normal deviate
Due date –Expected date of completion
Zo =
√project variance
 Use standard normal tables to find the probability p(z ≤ zo) of
completing the project within the scheduled time, where
Z ~ N(0,1).
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