The document provides information about an Operations Research course. It includes the objective of the course, which is to develop and analyze mathematical models for decision problems and their systematic solution. It also lists the various topics that will be covered in the course, including linear programming, transportation problems, game theory, and metaheuristics. The course aims to help students identify and solve real-world business problems by applying appropriate operations research techniques.

1. Subject Name:
OPERATIONS RESEARCH
Subject Code:
10CS661
Prepared By:
Sindhuja K
Department:
CSE
Date
12/6/2023

2. 12/6/2023
OBJECTIVE
• With the growth of technology, the World has seen a remarkable changes in
the size and complexity of organizations.
• An integral part of this had been the division of labour and segmentation of
management responsibilities in these organizations.
• The results have been remarkable but with this, increasing specialization has
created a new problem to meet out organizational challenges.
• The allocation of limited resources to various activities has gained significant
importance in the competitive market.
• These types of problems need immediate attention which is made possible by
the application of OR techniques.
• Operations Research (OR) is a science which deals with problem,
formulation, solutions and finally appropriate decision making.

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LEARNING OUTCOMES
• Identify and develop operational research models
from the verbal description of the real world
problems.
• Solve business problems and apply it's applications
by using mathematical analysis.
• Develop the ideas of developing and analyzing
mathematical models for decision problems, and
their systematic solution.
• Understand the mathematical models that are
needed to solve optimization problems.

4. Engineered for Tomorrow
12/6/2023
Text Books:
1. Frederick S. Hillier and Gerald J. Lieberman: Introduction to
Operations Research, 8th Edition, Tata McGraw Hill, 2005.
(Chapters: 1, 2, 3.1 to 3.4, 4.1 to
4.8, 5, 6.1 to 6.7, 7.1 to 7.3, 8, 13, 14, 15.1 to 15.4)
Reference Books:
1. Wayne L. Winston: Operations Research Applications and
Algorithms, 4th Edition, Thomson Course Technology, 2003.
2. Hamdy A Taha: Operations Research: An Introduction,
8th Edition, Prentice Hall India, 2007.

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COURSE TOPICS
UNIT 1 - Introduction, Linear Programming – 1
UNIT 2 - LP – 2, Simplex Method – 1
UNIT 3 - Simplex Method – 2
UNIT 4 - Simplex Method – 2, Duality Theory
UNIT 5 - Duality Theory and Sensitivity Analysis,
Other Algorithms for LP
UNIT 6 - Transportation and Assignment Problems
UNIT 7 - Game Theory, Decision Analysis
UNIT 8 - Metaheuristics

6. Unit –I
Introduction, Linear
Programming – 1
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7. 12/6/2023
Unit 1- TOPICS
• Introduction: The origin, nature and impact of OR;
• Phases of OR
– Defining the problem and gathering data;
– Formulating a mathematical model;
– Deriving solutions from the model;
– Testing the model;
– Preparing to apply the model;
– Implementation .
• Introduction to Linear Programming: Prototype
example;
• The linear programming (LP) model

8. INTRODUCTION
• Operations Research is a very important area of study,
which tracks its roots to business applications. It
combines the three broad disciplines of Mathematics,
Computer Science, and Business Applications.
• This course will formally develop the ideas of
developing, analyzing, and validating mathematical
models for decision problems, and their systematic
solution. The course will involve programming and
mathematical analysis
12/6/2023

9. • Operations research involves “research on operations.” Thus,
operations research is applied to problems that concern how to
conduct and coordinate the operations (i.e., the activities) within
an organization .
• The nature of the organization is essentially immaterial, and, in
fact, OR has been applied extensively in such diverse areas as
manufacturing, transportation, construction,
telecommunications, financial planning, health care, the
military, and public services, to name just a few. Therefore, the
breadth of application is unusually wide.
• The research part of the name means that operations research
uses an approach that resembles the way research is conducted
in established scientific fields. To a considerable extent, the
scientific method is used to investigate the problem of concern.
(In fact, the term management science sometimes is used as a
synonym for operations research.)
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10. 12/6/2023
WHAT IS OPERATIONS RESEARCH
• Operations Research ( OR) Is the application of scientific methods in
solving problems facing management and to help to taking decisions.
• A rose by any other name…
– Management Science
– Systems Engineering
– Industrial Engineering
– Operations Management
– Applied Mathematics
• Operations research (also known as management science) is
a collection of techniques based on mathematics and other
scientific approaches that finds solutions to your problems.

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DEFINITIONS
• It is an Act of winning wars without actually
fighting.
– Aurther Clark
• It is a Scientific Approach to problem solving for
executive management.
– H.M. Wagner
• It is Art of giving bad answers to problem which
otherwise have worse answers.
– T.L. Saaty

12. AIMS OF OR
Maxima of ;
Profit
Assembly line performance
Crop yield
Bandwidth
Minima of;
Loss
Risks

13. ORIGIN OF OPERATIONS
RESEARCH
• The formal activities of Operations Research (OR) were
initiated in England during World War II by McClosky and
Trefthen in 1940. A team of British scientists set out to make
decisions regarding the best utilization of war material.
• Following the end of the war, the ideas advanced in military
operations were adapted to improve efficiency and
productivity in the civilian sector.
• Today, OR is a dominant decision making tool, that seeks the
optimum state in all conditions and thus provides optimum
solution to organizational problems
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14. THE NATURE OF OR
• OR is applied to problems that concern how to conduct and
coordinate the operation within a organization.
• The steps followed are
– Construct a model that attempts to abstract the essence of the real problem
– Conduct suitable tests on the model, modify it as needed and verify some.
• OR frequently attempts to find a best solution (optimal
solution) for the problem under consideration(“Search for
optimality”).
• OR gives bad answers to the problems where worse could be
given(it cannot give perfect answers to the problem. Thus OR
improves only the quality of the solution)
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15. IMPACT OF OR
• Operations Research (OR) applies scientific method to the management of
organized systems in business, industry, government and other enterprises.
• OR can play a significant role in bringing a balance among different inter
disciplinary people to managerial problem.
• OR is regularly applied in areas such as:
– supply chain management
– marketing and revenue management systems
– manufacturing plants
– financial engineering
– telecommunication networks
– healthcare management
– transportation networks
– energy and the environment
– service systems
– web commerce
– military defense
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16. PHASES OF AN OR STUDY
• As a decision-making tool, OR is both a science and an art.
• The principal phases for implementing OR in practice
includes:
– Definition of the problem.
– Construction of the model.
– Solution of the model.
– Validation of the model.
– Implementation of the solution.

17. Operation Research 17
Defining the problem and gathering data
Formulating of the model
To obtain solution from model
Testing of the model and the solution
Implementation of the solution
PHASES OF OR

18. DEFINING THE PROBLEM AND
GATHERING DATA
• In formulating a problem for O.R. study analysis must be
made of the following major components:
(i) The environment
(ii) The objectives
(iii) The decision maker
(iv) The alternative courses of action and constraints.
• Out of the above four components environment is most
comprehensive as it provides a setting for the remaining three.
• The operation researcher shall attend conferences, pay visits,
send observation and perform research work thus succeeds in
getting sufficient data to formulate the problems.
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19. FORMULATING OF THE MODEL
The operation researcher can now construct the model to show
the relations and interrelations between a cause and effect or
between an action and a reaction. The following steps are used
in linear programming models
– Specify solution objectives
– Specify decision variables and stipulate constraints on the
solution
– Identify or construct an appropriate model for solution
development
– Determine and obtain required data
– Develop solutions using analytical model
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20. TO OBTAİN SOLUTİON FROM
MODEL
• A solution may be extracted form a model either by
conducting experiments on it i.e. by simulation or by
mathematical analysis.
• It and even use standard algorithms.”Optimizing is the science
of the ultimate; satisficing is the art of the feasible”
• OR team uses Heuristic procedures and post optimality
analysis(“what-if analysis”).
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21. TESTİNG OF THE MODEL AND
THE SOLUTİON
• Before the application of the model solution, the validity of
model and reliability of the solution should be tested.
• Validity of the model can be decided by comparing its outputs
with the results of past.
• Retrospective test: If past behavior is repeated when provided
similar inputs (a stable solution is being obtained), the model
will be valid.
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22. PREPARING TO APPLY THE
MODEL
• Well documented installation
• Computer based systems
• Interactive decision support systems
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23. IMPLEMENTATİON
• Implementation of the solution obtained from a validated
model is a reliable solution to the the real-life problems.
• Implementation of the solution is the duty of operation
research team.
12/6/2023

24. SOCIETIES
• Operational Research Society,Turkey
• The International Federation of Operational Research
Societies(IFORS)
• The Association of European Operational Research
Societies(EURO)
12/6/2023

25. OR SOFTWARE PACKAGES
• A very popular approach now is to use today’s premier
spreadsheet package, Microsoft Excel, to formulate small OR
models in a spreadsheet format. The Excel Solver then is used
to solve the models.
• LINDO (and its companion modeling language LINGO)
continues to be a dominant OR software package. Student
versions of LINDO and LINGO now can be downloaded free
from the Web.
• CPLEX is a state-of-the-art software package that is widely
used for solving large and challenging OR problems
• MPL is a user-friendly modeling system that uses CPLEX as
its main solver.
12/6/2023

26. INTRODUCTION TO LINEAR
PROGRAMMING (LP)
12/6/2023
Real-World
Problem
Recognition and
Definition of the
Problem
Formulation and
Construction of the
Mathematical
Model
Solution
of the Model
Interpretation
Validation and
Sensitivity Analysis
of the Model
Implementation

27. LINEAR PROGRAMMING
• In mathematics, linear programming (LP) is a technique for
optimization of a linear objective function, subject to linear
equality and linear inequality constraints.
• Linear programming determines the way to achieve the best
outcome (such as maximum profit or lowest cost) in a given
mathematical model and given some list of requirements
represented as linear equations.
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28. MATHEMATICAL FORMULATION OF LINEAR PROGRAMMING MODEL:
• Step 1
• - Study the given situation
• - Find the key decision to be made
• Identify the decision variables of the problem
• Step 2
• Formulate the objective function to be optimized
• Step 3
• Formulate the constraints of the problem
• Step 4
• - Add non-negativity restrictions or constraints
• The objective function , the set of constraints and the non-
negativity
• restrictions together form an LP model.
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29. • The Wyndor Glass Co. produces high-quality glass products, including windows and
glass doors. It has three plants.
•
• Plant 1 produces Aluminum frames
• Plant 2 produces wood frames
• Plant 3 produces the glass and assembles the products.
• The company has decided to produce two new products.
• Product 1: An 8-foot glass door with aluminum framing
• Product 2: A 4x6 foot double-hung wood framed window
• Each product will be produced in batches of 20. The production rate is defined as the
number of batches produced per week.
• The company wants to know what the production rate should be in order to maximize
their total profit, subject to the restriction imposed by the limited production capacities
available in the 3 plants.
PROTOTYPE EXAMPLE

30. • To get the answer, we need to collect the following data.
• (a) Number of hours of production time available per week in each plant for these two
new products. (Most of the time in the 3 plants is already committed to current products,
so the available capacity for the 2 new products is quite limited).
• Number of hours of production time available per week in Plant 1 for the new products: 4
• Number of hours of production time available per week in Plant 2 for the new products:
12
• Number of hours of production time available per week in Plant 3 for the new products:
18

31. 12/6/2023
(b) Number of hours of production time used in each plant for each batch produced
of each new product (Product 1 requires some of the production capacity in
Plants 1 and 3, but none in Plant 2. Product 2 needs only Plants 2 and 3).
Number of hours of production time used in Plant 1 for each batch produced of
Product 1: 1
Number of hours of production time used in Plant 2 for each batch produced of
Product 1: 0
Number of hours of production time used in Plant 3 for each batch produced of
Product 1: 3
Number of hours of production time used in Plant 1 for each batch produced of
Product 2: 0
Number of hours of production time used in Plant 2 for each batch produced of
Product 2: 2
Number of hours of production time used in Plant 3 for each batch produced of
Product 2: 2

32. (c) Profit per batch produced of each new product.
Profit per batch produced of Product 1: $3,000
Profit per batch produced of Product 2: $5,000

33. Formulation as a Linear Programming Problem
To formulate the LP model for this problem, let
x1 = number of batches of product 1 produced per week
x2 = number of batches of product 2 produced per week
Z = total profit ( in thousands of dollars) from producing the two new products.
Thus, x1 and x2 are the decision variables for the model. Using the data of Table , we obtain

34. HOW TO SOLVE LP PROBLEMS
• Graphical Solution
• Simplex Method for Standard form LP
– Geometric Concepts
– Setting up and Algebra
– Algebraic solution of Simplex

35. (4) Graphical Solution
The Wyndor Glass Co. example is used to illustrate the graphical solution.
Shaded area shows values of (x1, x2)
allowed by x1 ≥ 0, x2 ≥ 0, x1 ≤ 4
Shaded area shows values of (x1, x2) ,
called feasible region

36. The value of (x1, x2) that maximize 3x1 + 5x2 is (2, 6)

37. Common Terminology for LP Model
Objective Function:
The function being maximized or minimized is called the objective
function.
Constraint:
The restrictions of LP Model are referred to as constraints.
The first m constraints in the previous model are sometimes
called functional constraints.
The restrictions xj >= 0 are called nonnegativity constraints.

38. 12/6/2023
Feasible Region:
The feasible region is the
collection of all feasible
solutions.
Feasible Solution:
A feasible solution is a solution
for which all the constraints are
satisfied.
Infeasible Solution:
An infeasible solution is a solution
for which at least one constraint is
violated.
A feasible solution is
located in the feasible
region. An infeasible
solution is outside the
feasible region.

39. Common Terminology for LP Model
No Feasible Solutions:
It is possible for a problem to have no feasible solutions.
An Example
he Wyndor Glass Co.
problem would have no
feasible solutions if the
constraint 3x1 + 5x2 ≤
50 were added to the
problem.
In this case, there is
no feasible region

40. Common Terminology for LP Model
Optimal Solution:
An optimal solution is a feasible solution that has the maximum or
minimum of the objective function.
Multiple Optimal
Solutions:
It is possible to have
more than one optimal
solution.
An Example
The Wyndor Glass Co. problem
would have multiple optimal
solutions if the objective function
were changed to Z = 3x1 + 2x2

41. Common Terminology for LP Model
Unbounded Objective:
If the constraints do not
prevent improving the
value of the objective
function indefinitely in
the favorable direction,
the LP model is called
having an unbounded
objective.
An Example
The Wyndor Glass Co. problem would have no optimal
solutions if the only functional constrait were x1 ≤ 4, because
x2 then could be increased indefinitely in the feasible region
without ever reaching the maximum value of Z = 3x1 + 2x2

42. Common Terminology for LP Model
Corner-Point Feasible (CPF) Solution:
A corner-point feasible (CPF) is a solution that lies at a corner of
the feasible region.
The five dots are the five
CPF solutions for the
Wyndor Glass Co. problem

43. Common Terminology for LP Model
Relationship between optimal solutions and CPF solutions :
Consider any linear programming problem with feasible solutions and a bounded feasible
region. The problem must posses CPF solutions and at least one optimal solution.
Furthermore, the best CPF solution must be an optimal solution. Therefore, if a problem
has exactly one optimal solution, it must be a CPF solution. If the problem has multiple
optimal solutions, at least two must be CPF solutions.
The prototype model has exactly one
optimal solution, (x1, x2)=(2,6), which
is a CPF solution
(2,6)
(4,3)
The modified problem has multiple
optimal solution, two of these optimal
solutions , (2,6) and (4,3), are CPF
solutions.

44. Matrix Standard Form of an LP Model

45. Tabular Standard Form of an LP Model

46. ASSIGNMENT 1
• A Company manufactures FM radios and calculators.The radios
contribute Rs 100 per unit and calculator Rs 150 per unit as a profit.Each
radio requires 4 diodes,4 registers while each calculator requires 10 diodes
and 2 registers.A radio takes 12 minutes and calculator takes 9.6 minutes
on the company electronic testing machine and the product manager
estimates that 160 hours of test time is available. The firm has 8000 diodes
and 3000 resisters in the stock . Formulate the problem as LPP.
• A computer company manufactures laptop and desktop that fetches a profit
of Rs.700 and Rs.500 unit respectively . Eachunit of laptops takes 4 hours
of assembly time and 2 hours of testing time while each unit desktop
requires 3 hours of assembly time and 1 hour for testing. In a given month
the total number of hours available for assembly is 210 hours and for
inspection is 90 hours . Formulate the problem as LPP to maximize the
profit.
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47. • Old hens can be bought at Rs.50 each but young one cost Rs.100 each . The
old hens lay 3 eggs/week and young hens 5eggs/week . Each egg cost Rs.2
,A hen cost Rs.5/week to feed. If a person has only Rs.2000 to spend for
hens . Formulate the problem to decide how many each kind of hen should
he buy. Assume he can not house more than 40 hens
• A toy company manufactures two types of dolls A and B . Each doll of type
B takes twice as long to produce as one of type A and the company would
have time to make maximum of 2000 dolls/day. The supply of plastic is
sufficient to produce 1500 dolls/day(both A and B combined). The doll B
requires a fancy dress of which there a only 600/day available. If the
company makes a profit of Rs.10 and Rs.18 /doll on doll A and B
respectively. Formulate the problem as LPP to maximize the profit.
• The standard of a special purpose brick is 5kg and it contains 2 ingredients
B1 and B2 costs Rs.5 and Rs.8/kg respectively. The brick contains not
more than 4kg of B1 and a minimum of 2kg of B2. Since the demand for
the profit is likely to be related to the price of brick . Formulate the problem
as LPP.
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48. • Solve the following LPP by graphical method
Minimize Z = 20x 1+ 10x2
STC x 1+ 2x2≤40
3x 1+ x2≥30
4x 1+ 3x2≥60
X1, x2 ≥0
maximize Z = 2x 1+ 3x2
STC x 1+ 2x2≤4
x 1+ x2=3
X1, x2 ≥0
maximize Z = 3x 1+ 2x2
STC 5x 1+ x2 ≥10
x 1+ x2≥6
x 1+ 4x2≥12 and X1, x2 ≥0
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49. maximize Z = 6x 1+ 4x2
STC x 1+ 2x2 ≥4
x 1+ 2x2≤2 and X1, x2 ≥0
maximize Z = x 1+ x2 /2
STC3 x 1+ 2x2≤12
5x 1≤10
x 1+ x2≤18
-x 1+ x2 ≥4
And X1, x2 ≥0
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