Mb0048 Unit 01-Slm

Operations Research Unit 1

Sikkim Manipal University Page No. 1

Unit 1 Introduction to Operations Research

Structure:

1.1 Introduction

Objectives

1.2 Historical Background

Definitions of operations research

1.3 Scope of Operations Research

1.4 Features of Operations Research

1.5 Phases of Operations Research

1.6 Types of Operations Research Models

A broad classification of OR models

1.7 Operations Research Methodology

Definition

Construction

Solution

Validation

Implementation

1.8 Operations Research Techniques and Tools

1.9 Structure of the Mathematical Model

1.10 Limitations of Operations Research

1.11 Summary

1.12 Glossary

1.13 Terminal Questions

1.14 Answers

1.15 Case Study

1.1 Introduction

Welcome to the unit on operations research management. Operations

research management focuses on the mathematical scoring of

consequences of a decision aiming to optimise the use of time, effort, and

resources to avoid blunders. The act of obtaining best results under any

given circumstances is known as optimising. The key purpose of Operations

Research (OR) is to do preparative calculations that aid the decision-making

process.



Hence, decision-making is a key part of our daily life. The ultimate goal of all

decisions is to maximise benefits and to minimise effort and time. OR gives

decision makers the power to make effective decisions and improve day-to-

day operations. Decision makers consider all the available options, study

the outcomes, and estimate the risks.

In simple situations, common sense and judgement can be used to take

decisions. For example, if you are buying a microwave or washing machine,

the decision-making process is not very complicated. You can simply

compare the price, quality, and durability of the well-known brands and

models in the market and take a decision based on it.

However, in complex situations, although it is possible to take decisions

based on one’s common sense, a decision backed by mathematical

calculations reduces the risk factor and increases the probability of success.

Some such situations, where decision-makers have to depend on

mathematical scoring and reasoning, are finding an appropriate product mix

amidst competitor’s products or planning a public transportation network in a

city.

Objectives:

After studying this unit, you should be able to:

describe the historical background of OR

list the significant features of OR

describe the methodology of OR

define the structure of a mathematical model in OR

describe the significance of the function of OR

1.2 Historical Background

During the World War II, scientists from United Kingdom studied the

strategic and tactical problems associated with air and land defence of the

country. The aim of this study was to determine the effective utilisation of

limited military resources to win the battle. The technique was named

operations research. After World War II, operations research techniques

were developed and deployed in the decision-making process in

complicated situations in various fields, such as industrial, academic, and

government organisations.



1.2.1 Definitions of operations research

Churchman, Aackoff, and Aruoff defined operations research as “the

application of scientific methods, techniques and tools to the operation of a

system with optimum solutions to the problems” where 'optimum' refers to

the best possible alternative.

The objective of OR is to provide a scientific basis to the decision-makers

for solving problems involving interaction with various components of the

organisation. This can be achieved by employing a team of scientists from

different disciplines to work together for finding the best possible solution in

the interest of the organisation as a whole. The solution thus obtained is

known as an optimal decision.

You can also define operations research as “The use of scientific methods

to provide criteria for decisions regarding man, machine, and systems

involving repetitive operations.”

Self Assessment Questions

1. The main objective of OR is to provide a _______ ________ to the

decision-makers.

2. OR employs a team of _________ from _________ __________.

1.3 Scope of Operations Research

Any problem, either simple or complicated, can use OR techniques to find

the best possible solution. This section will explain the scope of OR by

analysing its application in various fields of everyday life.

In defence operations – In modern warfare, the three major military

components namely, Air Force, Army, and Navy carry out the defence

operations. The activities in each of these components can be further

divided in four sub-components - administration, intelligence, operations,

training and supply. The applications of modern warfare techniques in

each of the components of military organisations require expert

knowledge in respective fields. Furthermore, each component works to

drive maximum gains from its operations and there is always a

possibility that the strategy beneficial to one component may be

unfeasible for another component. Thus in defence operations, there is

a requirement to co-ordinate the activities of various components. This



gives maximum benefit to the organisation as a whole, having maximum

use of the individual components. A team of scientists from various

disciplines gets together to study the strategies of different components.

After appropriate analysis of the various courses of actions, the team

selects the best course of action, known as the ‘optimum strategy’.

In industry – The system of modern industries is so complex that an

individual cannot intuitively judge the optimum point of operation in its

various components. The business environment is always changing and

any decision useful at one time may not be suitable some time later.

There is always a need to check the validity of decisions continuously

against the situations. The industrial revolution with increased division of

labour and introduction of management responsibilities has made each

component an independent unit having its own goals. For example,

production department minimises the cost of production but maximises

output. Marketing department maximises the output, but minimises cost

of unit sales. Finance department tries to optimise the capital investment

and personnel department appoints good people at minimum cost. Thus,

each department plans its own objectives and all these objectives of

various departments or components come to conflict with one another

and may not agree to the overall objectives of the organisation. The

application of OR techniques helps in overcoming this difficulty by

integrating the diversified activities of various components to efficiently

serve the interest of the organisation as a whole. OR methods in

industry can be applied in the fields of production, inventory controls and

marketing, purchasing, transportation, and competitive strategies.

Planning – In modern times, it has become necessary for every

government to carefully plan, for the economic development of the

country. OR techniques can be fruitfully applied to maximise the per

capita income, with minimum sacrifice and time. A government can thus

use OR for framing future economic and social policies.

Agriculture – With increase in population, there is a need to increase

agriculture output. However, this cannot be done arbitrarily. There are

several restrictions. Hence, the need to determine a course of action

that serves the best under the given restrictions. You can solve this

problem by applying OR techniques.



In hospitals – OR methods can solve waiting problems in outpatient

department of big hospitals and administrative problems of the hospital

organisations.

In transport – Different OR methods can be applied to regulate the

arrival of trains and processing times, to minimise the passengers

waiting time and reduce congestion, and to formulate suitable

transportation policy, thereby reducing the costs and time of trans-

shipment.

Research and development – OR methodologies can be applied in the

field of R&D for several purposes, such as to control and plan product

introductions.


3. A government can thus use OR for framing future ______ and _______

4. In hospital OR methods can solve waiting problems in ______

department of big hospitals and ______ problems of the hospital

organisations.

1.4 Features of Operation Research

Some key features of OR are as follows:

OR is system-oriented. OR scrutinises the problem from an

organisation’s perspective. The results can be optimal for one part of the

system, while the same can be unfavourable for another part of the

system.

OR imbibes an inter–disciplinary team approach. Since no single

individual can have a thorough knowledge of all the fast developing

scientific know-how, personalities from different scientific and

managerial cadre form a team to solve the problem.

OR uses scientific methods to solve problems.

OR increases effectiveness of the management’s decision-making

ability.

OR uses computers to solve large and complex problems.

OR offers a quantitative solution.

OR also takes into account the human factors.




5. OR ________ inter-disciplinary approach.

6. OR increases the effectiveness of ________ ability.

1.5 Phases of Operations Research

The scientific method in OR study generally involves three phases.

Figure 1.1 depicts the three phases of OR.

Fig. 1.1: Phases of Operations Research

Let us now study the phases in detail.

Judgment phase

This phase includes the following activities:

Determination of the operations

Establishment of objectives and values related to the operations

Determination of suitable measures of effectiveness

Formulation of problems relative to the objectives

Research phase

This phase utilises the following methodologies:

Operation and data collection for a better understanding of the problems

Formulation of hypothesis and model

Observation and experimentation to test the hypothesis on the basis of

additional data

Analysis of the available information and verification of the hypothesis

using pre-established measure of effectiveness

Prediction of various results and consideration of alternative methods



Action phase

This phase involves making recommendations for the decision process. The

recommendations can be made by those who identify and present the

problem or by anyone who influences the operation in which the problem

has occurred.


7. Action phase involves making recommendations for the decision

process. (True/False)

8. One of the OR phases is judgement phase. (True/False)

1.6 Types of Operations Research Models

A model is an idealised representation or abstraction of a real-life system.

The objective of a model is to identify significant factors that affect the real-

life system and their interrelationships. A model aids the decision-making

process as it provides a simplified description of complexities and

uncertainties of a problem in a logical structure. The most significant

advantage of a model is that it does not interfere with the real-life system.

1.6.1 A broad classification of OR models

You can broadly classify OR models into the types depicted in figure 1.2.

Fig. 1.2: Classification of Models



Let us now study the models in detail.

Physical models

These models include all forms of diagrams, graphs, and charts. They are

designed to tackle specific problems. They bring out significant factors and

interrelationships in pictorial form to facilitate analysis. There are two types

of physical models. They are:

Iconic models

Analogue models

Let us now study the two types of physical models in detail.

Iconic models are primarily images of objects or systems, represented on a

smaller scale. These models can simulate the actual performance of a

product. Analogue models are small physical systems having characteristics

similar to the objects they represent, such as toys.

Mathematical or symbolic models

These models employ a set of mathematical symbols to represent the

decision variable of the system. The variables are related by mathematical

systems. Some examples of mathematical models are allocation,

sequencing, and replacement models.

By nature of environment

These models can be further classified as follows:

Deterministic models - These are the models in which everything is

defined and the results are certain, such as an EOQ model.

Probabilistic models - These are the models in which the input and

output variables follow a defined probability distribution, such as the

games theory.

By the extent of generality

These models can be further classified as follows:

General models – These are the models which you can apply in general

to any problem. For example, linear programming.

Specific models - These are the models that you can apply only under

specific conditions. For example, you can use the sales response curve

or equation as a function in the marketing function.




9. Diagram belongs to physical models. (True/False)

10. Allocation problems are represented by iconic models. (True/False)

1.7 Operations Research Methodology

The basic dominant characteristic feature of operations research is that it

employs mathematical representations or models to analyse problems. This

distinct approach represents an adaptation of the scientific methodology

used by the physical sciences. The scientific method translates a given

problem into a mathematical representation which is solved and

retransformed into the original context. Figure 1.3 depicts the OR approach

to problem solving.

Fig. 1.3: Steps in the OR methodology

As shown in figure 1.3, OR methodology consists of five steps. They are -

defining the problem, constructing the model, solving the model, validating

the model, and implementing the result.

Let us now study the steps in detail.

1.7.1 Definition

The first and the most important step in the OR approach of problem solving

is to define the problem. One needs to ensure that the problem is identified

properly because this problem statement will indicate the following three

major aspects:

Description of the goal or the objective of the study

Identification of the decision alternative to the system



Recognition of the limitations, restrictions, and requirements of the

system

1.7.2 Construction

Based on the problem definition, you need to identify and select the most

appropriate model to represent the system. While selecting a model, you

need to ensure that the model specifies quantitative expressions for the

objective and the constraints of the problem in terms of its decision

variables. A model gives a perspective picture of the whole problem and

helps in tackling it in a well-organised manner. Therefore, if the resulting

model fits into one of the common mathematical models, you can obtain a

convenient solution by using mathematical techniques. If the mathematical

relationships of the model are too complex to allow analytic solutions, a

simulation model may be more appropriate. Hence, appropriate models can

be constructed.

1.7.3 Solution

After deciding on an appropriate model, you need to develop a solution for

the model and interpret the solution in the context of the given problem. A

solution to a model implies determination of a specific set of decision

variables that would yield an optimum solution. An optimum solution is one

which maximises or minimises the performance of any measure in a model

subject to the conditions and constraints imposed on the model.

1.7.4 Validation

A model is a good representation of a system. However, the optimal solution

must work towards improving the system’s performance. You can test the

validity of a model by comparing its performance with some past data

available from the actual system. If under similar conditions of inputs, your

model can reproduce the past performance of the system, then you can be

sure that your model is valid. However, you will still have no assurance that

future performance will continue to duplicate the past behaviour. Secondly,

since the model is based on careful examination of past data, the

comparison should always reveal favourable results. In some instances, this

problem may be overcome by using data from trial runs of the system. One

must note that such validation methods are not appropriate for non-existent

systems because data will not be available for comparison.



1.7.5 Implementation

You need to apply the optimal solution obtained from the model to the

system and note the improvement in the performance of the system. You

need to validate this performance check under changing conditions. To do

so, you need to translate these results into detailed operating instructions

issued in an understandable form to the individuals who will administer and

operate the recommended system. The interaction between the operations

research team and the operating personnel reaches its peak in this phase.

1.8 Operations Research Techniques and Tools

The different techniques and tools used in OR are as follows:

Linear programming – You can use linear programming to find a

solution for optimising a given objective. The objective may be to

maximise profit or to minimise cost. You need to ensure that both the

objective function and the constraints can be expressed as linear

expressions of decision variables. You will learn about the various uses

of linear programming in Unit 2.

Inventory control methods – The production, purchasing, and material

managers are always confronted with questions, such as when to buy,

how much to buy, and how much to keep in stock. The inventory model

aims at optimising these inventory levels.

Goal programming – In linear programming, you take a single objective

function and consider all other factors as constraints. However, in real

life there may be a number of important objective functions. Goal

programming has several objective functions, each having a target

value. Programming models are developed to minimise deviations from

these targets.

Queuing model – The queuing theory is based on the concept of

probability. It indicates the capability of a given system and the changes

possible in the system when you modify the system. In formulating a

queuing model, you need not take into account all the constraints. There

is no maximisation or minimisation of an objective function. Therefore,

the application of queuing theory cannot be viewed as an optimisation

process. You can use the queuing theory to estimate the required

balance between customer waiting time and the service capability of the



system. You need to first consider several alternatives, evaluate them

through queuing models, study their effect on the system, and then

make a choice. The criteria for evaluation will be measures of efficiency

of the system, such as the average length of a queue, expected waiting

time of a customer, and the average time spent by the customer in the

system. In this approach, your success primarily depends on the

alternatives considered and not much on the queuing models

developed.

Transportation model – The transportation model is an important class

of linear programs. The model studies the minimisation of the cost of

transporting a commodity from a number of sources to several

destinations. The supply at each source and the demand at each

destination are known. The objective of the model is to develop an

integral transportation schedule that meets all the demands from the

inventory at a minimum total transportation cost.

The transportation problem involves m sources, each of which has ai (i =

1, 2, …..,m) units of homogeneous product and n destinations available,

and each of which requires bj (j = 1, 2…., n) units of products. Here ai

and bj are positive integers. The cost cij of transporting one unit of the

product from the ith source to the jth destination is given for each

i and j. It is assumed that the total supply and the total demand are

equal.

n

1j

m

1i

i bja (1)

Condition (1) is guaranteed by creating either a fictitious destination with

a demand equal to the surplus if total demand is less than the total

supply or a (dummy) source with a supply equal to the shortage if total

demand exceeds total supply. The cost of transportation from the

fictitious destination to all sources and from all destinations to the

fictitious sources are assumed to be zero so that the total cost of

transportation will remain the same.

In addition to the above, there are tools such as the sequence model,

the assignment model, and network analysis, which you will learn in

detail in the later units.




11. OR methodology consists of definition, solution, and validation only.

(True/False)

12. The interaction between the OR team and management reaches peak

level in the implementation phase. (True/False)

1.9 Structure of the Mathematical Model

Many industrial and business situations are concerned with planning

activities. In each case of planning, there are limited sources, such as men,

machines, material, and capital at the disposal of the planner. One has to

take decisions regarding these resources to maximise production, minimise

the cost of production, or maximise the profit. These problems are referred

to as the problems of constrained optimisation.

Linear programming is a technique for determining an optimal schedule of

interdependent activities, for the given resources. Therefore, you can say

that programming refers to planning and the process of decision-making

about a particular plan of action from a given set of alternatives.

Any business activity or production activity to be formulated as a

mathematical model can best be discussed through its parts, which are as

follows:

Decision variables

Objective function

Constraints

Let us now study the parts in detail.

Decision variables

Decision variables are the unknowns, which you need to determine from the

solution of the model. The parameters represent the controlled variables of

the system.

Objective function

The objective function defines the measure of effectiveness of the system

as a mathematical function of its decision variables. The optimal solution to

the model is obtained when the corresponding values of the decision

variable yield the best value of the objective function whilst satisfying all

constraints. Therefore, you can say that the objective function acts as an

indicator for the achievement of the optimal solution.



While formulating a problem, the desire of the decision-maker is expressed

as a function of ‘n’ decision variables. (This function is a linear programming

problem, that is, each of its items will have only one variable raised to the

power one). Some of the objective functions in practice are:

Maximisation of contribution or profit

Minimisation of cost

Maximisation of production rate or minimisation of production time

Minimisation of labour turnover

Minimisation of overtime

Maximisation of resource utilisation

Minimisation of risk to environment or factory

Constraints

To account for the physical limitations of the system, you need to ensure

that the model includes constraints, which limit the decision variables to their

feasible range or permissible values. These are expressed as constraining

mathematical functions.

For example, in chemical industries, there are restrictions from the

government regarding the releasing of gases into the environment.

Restrictions from sales department about the marketability of some products

are also treated as constraints. Thus, a linear programming problem has a

set of constraints in practice.

The mathematical models in OR may be viewed generally as determining

the values of the decision variables x J, where J = 1, 2, 3, ------ n, which will

optimise Z = f (x 1, x 2, ---- x n).

Subject to the constraints:

g i (x 1, x 2 ----- x n) b i, i = 1, 2, ---- m

And xJ 0 j = 1, 2, 3 ---- n where is , , or =.

The function f is called the objective function, where xj bi, represent the ith

constraint for i = 1, 2, 3 ---- m where bi is a known constant. The constraint xj

0 is called the non-negativity condition, which restricts the variables to

zero or positive values only.



Example, Diet problem

Formulate the mathematical model for the following:

Vitamin–A and Vitamin–B are found in food–1 and food–2.

One unit of food–1 contains 5 units of vitamin–A and 2 units of vitamin–B.

One unit of food–2 contains 6 units of vitamin–A and 3 units of vitamin–B.

The minimum daily requirement of a person is 60 units of vitamin–A and 80

units of Vitamin–B.

The cost per one unit of food–1 is Rs.5 and one unit of food–2 is Rs.6.

Assume that any excess units of vitamins are not harmful. Find the minimum

cost of the mixture (food–1 and food–2) which meets the daily minimum

requirements of vitamins.

Mathematical model of the diet problem:

Suppose, x1 = the number of units of food–1 in the mixture

x2 = the number of units of food–2 in the mixture.

Let us formulate the constraint related to vitamin-A. Since each unit of food–

1 contains 5 units of vitamin– A, we have that x1 units of food–1 contains 5x1

units of vitamin– A. Since each unit of food– 2 contains 6 units of vitamin–A,

we have that x2 units of food–2 contains 6x2 units of vitamin–A. Therefore,

the mixture contains 5x1 + 6x2 units of vitamin-A. Since the minimum

requirement of vitamin– A is 60 units, you can say that 5x1 + 6x2 60.

Now let’s formulate the constraint related to vitamin–B. Since each unit of

food–1 contains 2 units of vitamin–B we have that x1 units of food–1

contains 2x1 units of vitamin-B. Since each unit of food–2 contains 3 units of

vitamin–B, we have that x2 units of food–2 contains 3x2 units of vitamin–B.

Therefore the mixture contains 2x1 + 3x2 units of vitamin–B. Since the

minimum requirement of vitamin–B is 80 units, you can say that

2x2 + 3x2 80

Next let’s formulate the cost function. Given that the cost of one unit of

food–1 is Rs.5 and one unit of food–2 is Rs.6. Therefore, x1 units of food–1

costs Rs.5x1, and x2 units of food–2 costs Rs.6x2.

Therefore, the cost of the mixture is given by cost = 5x1 + 6x2.

If we write z for the cost function, then you can write z = 5x1 + 6x2.

Since cost has to be minimised, you can write min z = 5x1 + 6x2.



Since the number of units (x1 or x2) are always non-negative, you have x1

0, x2 0.

Therefore, the mathematical model is:

5x1 + 6x2 60

2x1 + 3x2 80

x1 0, x2 0, min z = 5x1 + 6x2.

1.10 Limitations of Operations Research

The limitations are more related to the problems of model building, time, and

money factors. The limitations are:

Magnitude of computation – Modern problems involve a large number

of variables. The magnitude of computation makes it difficult to find the

interrelationship.

Intangible factors – Non–quantitative factors and human emotional

factors cannot be taken into account.

Communication gap – There is a wide gap between the expectations of

managers and the aim of research professionals.

Time and money factors – When you subject the basic data to frequent

changes then incorporating them into OR models becomes a costly

affair.

Human factor – Implementation of decisions involves human relations

and behaviour.


13. OR imbibes _________ team approach.

14. Linear programming is tool of _______.

15. The three phases of OR are ________.

16. To solve any problem through OR approach, the first step is _______.

17. _________ represents a real life system.

18. _________ represents the controlled variables of the system.



1.11 Summary

Let us recapitulate the important concepts discussed in this unit:

The objective of OR is to provide a scientific basis to the decision-

makers for solving problems involving interaction with various

components of the organisation

The scope of OR is in various fields such as defence, industry,

government, agriculture, hospitals, transport and research and

development.

Some key features of OR are OR is system oriented, it imbibes inter–

disciplinary team approach and uses scientific methods to solve

problems.

The three phases of OR are Judgement phase, research phase, and

action phase.

You can broadly classify OR models according to physical models,

mathematical models , models by nature of environment and models by

extent of generality.

The steps in OR methodology are problem definition, model

construction, model solution, model validation and result

implementation.

Linear programming, inventory control methods, goal programming,

queuing model and transportation model are the different tools and

techniques used in OR.

Any business activity or production activity has to be formulated as a

mathematical model.

Some of the limitations of OR are magnitude of computation, intangible

factors, communication gap, time and money factors and human factors

1.12 Glossary

Probability: possible outcomes of an event

Hypothesis: unproved theory

Network analysis: a mathematical representation of the problem by lines

and nodes joined to form a network



1.13 Terminal Questions

1. Define OR.

2. What are the features of OR?

3. What is a model in OR? Discuss the different models available in OR.

4. Write short notes on the different phases of OR.

5. What are the limitations of OR?

1.14 Answers


1. Scientific basis

2. Scientists, different disciplines

3. economic , social policies

4. Outpatient, administrative

5. Imbibes

6. Decision making

7. True

8. True

9. True

10. False

11. False

12. False

13. Inter-disciplinary

14. OR

15. Judgement phase, research phase, and action phase

16. Define the problem

17. Model

18. Parameters

Terminal Questions

1. OR is defined as the application of scientific methods, techniques and

tools to the operation of a system with optimum solutions to the

problems - refer 1.2.1

2. Some key features of OR are OR is system oriented, it imbibes inter–

disciplinary team approach and uses scientific methods to solve

problems - refer 1.4

3. Types of operations research models - refer 1.6



4. Phases of operations research are judgement, research and action

phase - refer 1.5

5. Limitations of OR - refer 1.10

1.15 Case Study

IMB Optimiser System

IBM was considering integrating its national network of spare parts

inventories to improve service support for their customers. They developed

a model for their inventory system that improved customer service while

reducing the value of IBM’s inventories by $ 250 million and saving an

additional $20 million per year through improved operational efficiency. A

particularly interesting aspect of the model validation phase of this study

was the way the future users of the inventory system had been incorporated

into the testing process. Because these future users were sceptical about

the system being developed, representatives were appointed to a user team

to serve as advisors to the OR team. After a preliminary version of the new

system had been developed, a pre-implementation test of the system was

conducted. Extensive feedback from the user team led to major

improvements in the proposed system.

Large computer system was used to apply this model. The system

developed was called ‘optimiser.’ It provided optimal control of service levels

and spare-parts inventories throughout IBM’s U.S. parts distribution

network, which included two central automated warehouses, dozens of field

distribution centres, and many thousands of outstation locations. The parts

inventory maintained in this network is valued in billions of dollars. Optimiser

consists of four major modules.

A forecasting system module contains a few programs for estimating the

failure rates of individual types of parts.

A data delivery system module consists of approximately 100 programs

that process over 15 gigabytes of data to provide the input for the model.

A decision system module then solves the model on a weekly basis to

optimise control of the inventories.

The fourth module includes six programs that integrate the optimiser into

IBM’s Parts Inventory Management System (PIMS). PIMS is a



sophisticated information and control system that contains millions of

lines of code.

Careful planning was required to implement the complex optimiser system

for controlling IBM’s national network of spare-parts inventories. Three

factors proved to be especially important in achieving a successful

implementation. By the time implementation phase was reached,

operational managers had a strong sense of ownership and had become

ardent supporters for installing optimiser in their functional areas. A second

success factor was a very extensive user acceptance test whereby users

could identify problem areas that needed to be rectified prior to

implementation. The third key was that a new system was phased in

gradually, with careful testing at each phase, so the major bugs could be

eliminated before the system went live nationally.

Discussion Questions:

1. Analyse the need for OR team at IBM.

2. Explain optimiser with its modules.

3. How did this new system help and what were the reasons for its

success.

Reference:

Kapoor V. K. (2005). Operations Research. Sultan Chand and Sons.

Sharma J. K. (2006). Operations Research. Macmillan India Limited.

Taha H. Operations Research. Prentice Hall.

Kanti Swarup & Gupta P. K., & Hira D. S., & Manmohan (2004).

Operation Research. Sultan Chand and Sons.

Cohen M., & Kamesan P. V., & Kleindorfer P., & Lee H., & Tekerian A.

(Jan-Feb. 1990). “Optimizer: IBM’s Multi-Echelon Invenentory System

for Managing Service Logistics”.

E- References:

newagepublishers.com.

http://www.newagepublishers.com/samplechapter/001012.pdf.

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