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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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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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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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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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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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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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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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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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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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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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, TL, 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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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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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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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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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

Activity

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Finish eventStart event

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

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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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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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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

project. These are computed by reversing the method of calculation used for

earliest event times.

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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 te 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

Compute the standard normal deviate

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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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