NTRODUCTION TO OPERATIONS RESEARCH

1. CHAPTER ONE: INTRODUCTION
Definition of Operation Research: Different authors have defined OR in different ways. In general, OR
can be defined as;
Operation research is defined as the application of scientific methods, techniques, and tools to problems
involving the operations of a system so as to provide those in control of the system with optimum solutions
to the problems.
Operation research is defined as the experimental and applied science devoted to observing, understanding,
and predicting the behavior of purposeful man-machine systems; and operations research workers are
actively engaged in applying this knowledge to practical problem in business, government and society.
Operation research is defined as the scientific method of providing executive departments with a quantitative
basis for decisions regarding operations under their control.
Morse & Kimball
Example: Suppose a company produces both interior and exterior paints from two raw materials 1 and 2.
The following table provides the basic data of the problem;
Exterior Paint from per
ton of Raw Material
Interior Paint from Per
Ton of Raw Material
Maximum Daily
Availability (tons)
Raw Material 1
Raw Material 2
6
1
4
2
24
6
Profit per ton (in 1000
USD)
5 4
A market survey restricts the maximum daily demand of interior paint to 2 tons. Additionally the daily
demand for interior paint cannot exceed that of exterior paint by more than 1 ton. Determine the best product
mix of interior and exterior paints for which the daily profit will be maximum level.
Now to solve this problem we apply the scientific methods, techniques and tools which is called the
operation research.
Here the OR model includes three basic elements
(1) Decision variables that we seek to determine
(2) Objective (goal) that we aim to maximize
(3) Constraints that we need to satisfy
Characteristics of Operation Research
The followings are the most important characteristics of OR
(i) It is system oriented
(ii) OR is the application of scientific method, techniques and tools
(iii) OR is used by interdisciplinary team in order to present complex functional relationships as
mathematical models
(iv) OR deals with uncovering new problems for quantitative analysis
(v) OR utilizes computer facilities like as software
(vi) OR deals with quantitative analysis
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2. (vii) OR also deals with human factor
Application of OR
(i) In order to take decision on scientific basis in the field of business and industrial management, OR team
will have to consider different OR techniques of producing goods and returns. In production management,
OR techniques will help to change the overall structure like installation of new machine, introduction of
more automation etc.
(ii) OR techniques are widely applicable in the field of industry for production, blending, product mix,
inventory control, demand forecast, sales, and purchase, transportation, repair, and maintenance, scheduling
and sequencing planning, scheduling and control of projects etc.
(iii) OR is widely applicable in business and society. For instance, it is widely applicable to decide the
premium rates of various policies. It is also extensively used in petroleum, paper, chemical, metal,
processing, aircraft, rubber, textile, mining, transport and distribution industries.
(iv) OR is widely applicable both in developing and developed economies for planning. In developing and
developed societies, economists, statisticians, mathematicians, econometricians, administrators, technicians,
politicians, researchers and agricultural experts workers together to solve the problem of poverty, hunger,
sustainable economic growth, infrastructure development, optimum allocation of resources to fulfill our
unlimited demands,
(v) OR approach is equally applicable both in big and small organizations. For example, whenever a
departmental store faces a problem like employing, additional sales, purchasing additional van etc,
techniques of OR can be applied to minimize cost and maximize benefit for each such decision.
Scope of OR in Modern Management:
OR is a scientific method which is widely applicable for solving and decision making about different
problems. It is a kit of scientific and programmable rules providing the management a quantitative basis for
decision regarding operations under its control. Some of the areas of management where OR techniques
have been successfully applied given below;
(1) Allocation and distribution
(i) Optimal allocation of limited resources such as agricultural land, water, gas, labor force, machines,
materials, time and money to fulfill our maximum demand.
(ii) Location and size of warehouses, distribution centers retail depots etc
(iii) distribution policy
(2) Production and Facility Planning
(i) Selection, location and design of production plants
(ii) Project scheduling and allocation of resources
(iii) Preparation of forecast for the various inventory items and computing economic order quantities and
reorder levels
(iv) Determination of the number and size of the items to be produced
((v) Maintenance policy
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3. (v) Scheduling and sequencing of production runs by proper allocation of machines
(3) Procurement
(i) What, how and when to purchase at the minimum procurement cost
(ii) Bidding and replacement policies
(4) Marketing
(i) Product selection, timing and competitive actions
(ii) Selection of advertising media
(iii) Demand forecasts and stock levels
(iv) Customer’s preference for size, colour, and packing various products
(5) Finance
(i) Capital requirements, cash-flow analysis
(ii) Credit policies and credit risks etc
(iii) Profit plan for the company
(iv) Determination of optimum replacement policies
(6) Personal
(i) Selection of personnel, determination of retirement age and skills
(ii) Recruitment policies and assignment of jobs
(6) Research and Development
(i) Determination of areas for research and development
(ii) Reliability and control of development projects
(iii) Selection of projects and preparation of their budgets
Thus it can be concluded that OR can be widely used in taking timely management decision and framing
corrective measures.
Phases of OR or OR Approach or How OR Works
Like all other scientific researches, OR is based on scientific methodology, which proceeds the following
lines
(i) Formulating the problem
(ii) Construction a model to represent the system under study
(iii) Deriving the solution from the model
(iv) Testing the model and the solution derived from it
(v) Establishing controls over the solution
(vi) Putting the solution to work i.e. implementation
Example:
Example: Suppose a company produces both interior and exterior paints from two raw materials 1 and 2.
The following table provides the basic data of the problem;
Exterior Paint from per
ton of Raw Material
Interior Paint from Per
Ton of Raw Material
Maximum Daily
Availability (tons)
Raw Material 1
Raw Material 2
6
1
4
2
24
6
Profit per ton (in 1000
USD)
5 4
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4. A market survey restricts the maximum daily demand of interior paint to 2 tons. Additionally the daily
demand for interior paint cannot exceed that of exterior paint by more than 1 ton. Determine the best product
mix of interior and exterior paints for which the daily profit will be maximum level.
Now to solve this problem we apply the scientific methods, techniques and tools which is called the
operation research.
Solution: Let us define
The variable 1x indicates the daily production in tons of exterior paint
The variable 2x indicates the daily production in tons of interior paint
The total daily profit is given by;
1 2z = 5x +4x
The objective of the company is to maximize 1 2z = 5x +4x
The last element of the model deals with the constraints that restrict raw materials usage and demand.
From the data of the problem we have
Usage of raw material 1 = 1 26x +4x
Usage of raw material 2 = 1 2x +2x
Now from the problem it is clear to us that the daily availabilities of raw materials 1 and 2 are limited to 24 tons and
6 tons respectively, the associated restrictions are given by;
1 26x +4x 24�
1 2x +2x 6�
There are two types of demand restrictions (i) maximum daily demand of interior paint is limited to 2 tons
and (2) excess of daily production of interior paint over that of exterior paint is at most 1 ton. The first
restriction is very straightforward which expressed as
2x 2�
The second restriction can be translated to state that the difference between the daily production of interior and
exterior paints does not exceed 1. Thus we can write that
2 1x -x 1�
An implicit (or understood-to-be) restriction on the model is that the variables 1x and 2x must be positive.
That is
1x 0�
2x 0�
The complete model of the problem can be written as;
Maximize 1 2z = 5x +4x
Subject to;
1 26x +4x 24�
1 2x +2x 6�
2x 2�
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5. 2 1x -x 1�
1x 0�
2x 0�
Feasible Solution: Any solution that satisfies all the constraints of the model is called a feasible solution. For
example 1x = 3 and 1x = 1 is a feasible solution because it does not violate any of the constraints.
Thus the maximum profit will be z = 5 3+4 = 19� (thousand USD)
Formulating the Problems
In formulating a problem for OR study, analysis must be made of the four major components
(i) the environment
(ii) the decision maker
(iii) the objectives
(iv) alternative courses of action and constraints
Environment: Environment is defines as the framework within which a system of organized activity is
directed to attain the prescribed objectives or goals. It involves physical, social, and economic factors which
may affect the problem under consideration.
Decision Maker: Decision maker is the second component of the problem. Decision maker, or researcher or
system operator is the person who is in actual control the operations under study.
The operation researcher (operation analyst) must study the decision maker and his/her relationship to the
problem
Objectives: Objectives are the third component of the problem to which analysis must be made. Objectives
should be defined by taking into account the system as a whole. A common error is to identify the
objectives, considering a portion of the entire system. Under such conditions, what is considered best for the
portion of the system, may actually prove harmful for the entire system. OR tries to takes into account as
broad scope of objectives as possible.
Alternative Courses of Action and Constraints: Alternatives are the final components of the problem. The
research problem is to determine which alternative course of action is most effective to achieve a certain set
of objectives. Other affected by the decisions under study should be identified. There must be complete
agreement on these points between the persons initiating the OR study and the persons performing these
operations.
Once the modified list of objectives, alternative courses of actions and counteractions is ready and the
operations research team has studied the environment and identified the decision maker, the problem can be
taken up for further research.
Model Construction:
After formulating the problem, the next step is to construct a model for the system under study. In OR study,
model construction simply means the translating the problem into a mathematical relationships or
mathematical model. A mathematical model consists of a set of equations which describe the system or
problem. These equations represent (i) the effectiveness function and (ii) constraints
The effectiveness function usually called the objective function is a mathematical expression of the
objectives i.e. mathematical expression of cost, output or profit of the operation.
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6. Constraints or restrictions are mathematical expressions of the limitations on the fulfillment of the
objectives.
The objective function and constraints are function of two types of variables, (i) controllable variables
(called decision variables) (ii) uncontrollable variables
Controllable Variable: A variable that is directly under the control of the operations analyst is called
controllable variable; the values of these variables are to be determined
Uncontrollable Variable: A variable that is not under the control of operations analyst is called
uncontrollable variable.
The general form of a mathematical model is given by;
i jE=f(x ,y )
E = Effectiveness function
ix
= ith controllable variable
jy
= jth uncontrollable variable
f = functional relationship between E, and ix
, jy
In OR, the mathematical model which is used is an approximation of the reality. Hence the system may not
include all the variables
Example of Linear Programming Problem:
Machine Type Products
1 2 3 4
Total Time Available Per
Week
A
B
C
1.5 1 2.4 1
1 5 1 3.5
1.5 3 3.5 1
2000
8000
5000
Unit Profits 5.24 7.30 8.34 4.18
Suppose three types of machines A, B and C turns out four products 1, 2, 3, 4. The above table shows
(i) the hours required on each machine type to produce per unit of each product (ii) total available
machine hours per week, and (iii) per unit profit on sale of each of the product.
Suppose jx (j = 1, 2, 3, 4) is the no. of units of product j produced per week. So we have the following
linear constraints;
1 2 3 4
1 2 3 4
1 2 3 4
1.5x +x +2.4x +x 2000
x +5x +x +3.5x 8000
1.5x +3x +3.5x +x 5000
�
�
�
Since the amount of production cannot be negative so, jx 0 (j = 1, 2, 3, 4)� . The weekly profit is
given by 1 2 3 4z= 5.24x +7.3x +8.34x +4.18x . Now we wish to determine the values of the variables jx 's
for which (i), (ii), (iii) and (iv) will be satisfied and (v) will be maximized
Models in OR
A model as used in operations research is defined as an idealized representation of the real life situation. It
represents one or a few aspects of reality. Diverse items such s a map, a multiple activity chart, breakeven
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7. equation, balance sheet, regression equation etc are all models because one of them represents a few aspects
of the real life situation. A map represents physical boundaries but normally ignores the heights of the
various places above from the sea level. The objective of the model is to provide a means for analyzing the
behavior of the system for the purpose of improving its performance.
TYPES OF MATHEMATICAL MODEL
Many OR models have been developed and applied to problems in business and industry. Some of these
models are given below:
(i) Mathematical techniques
(ii) Statistical techniques
(iii) Allocation models
(iv) Inventory models
(v) Sequencing models
(vi) Routing models
(vii) Competitive models
(viii) Queuing models
(ix) Dynamic programming models
(x) Simulation techniques
(xi) Decision theory
(xii) Replacement models
(xiii) Goal programming
(xiv) Markov analysis
(xv) Combined methods
Mathematical Techniques: Any mathematical technique can become a useful tool for operations analyst.
The most commonly used mathematical techniques are as (i) Differentiation, (ii) Integration (iii) Vector, (iv)
Matrix, (v) Determinant etc.
Example: For cost minimization, profit maximization, output maximization we use the concept of
differentiation. For solving different business and economic problems we use the concept of integration,
matrix, determinant etc.
Statistical Techniques
Some of the most commonly applied techniques come from probability theory and statistics. To deal with
the business and economic problems for decision making in uncertain situations we use the statistical
techniques. Statistical techniques include the discrete, continuous probability, renewal theory, Markov
process and stochastic process.
Example Suppose we want to know what is the expected no. of students of this class whose obtaining points
lies between 80 to 95, to deal with this problem we use the statistical technique.
Allocation Models
Allocation models are used to solve problems in which (i) there are a number of jobs to be performed and
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8. there are alternative ways of doing them, (ii) resources or facilities are limited.
In allocation model, the objective is to allocate the resources to the jobs in such a way as to optimize the
overall effectiveness ( i.e. minimize the total cost or maximize the total profit). This is called the
mathematical programming. When the constraints are expressed as linear equalities/inequalities, this is
called linear programming. If any of the constraints are non-linear, this is called the non-linear
programming.
The simplest type of allocation model involves the association of a number of jobs to the same number of
resources (men). This is called the assignment model. The assignment problem becomes more complex, if
some of the jobs require more than one resource, if the same resources are used for more than one job. Such
a problem is called transportation problem.
Classification of Models
The various schemes by which models can be classified are give as
(i) By degree of abstraction
(ii) By function
(iii) By structure
(iv) By nature of the environment
(v) By the extent of generality
(vi) By the time horizon
BY DEGREE OF ABSTRACTION
Mathematical models are the most abstract type since it requires not only the mathematical knowledge but
also great concentration to get the idea of the real-life situation they represent.
Example: Demand function, Production function
Language Models (cricket, football, baseball, hockey match commentary) are also abstract type.
Concrete Models ( model of earth, dam, building or plane or road) are the least abstract since they suggest
the shape or characteristics of the modeled entity.
BY FUNCTION
LIMITATION OF OPERATION RESEARCH
(1) OR deals with the mathematical models of quantitative variables but do not take into account of
qualitative factors or emotional factors which are quite real.
Example: OR deals with the mathematical model of cost minimization, profit maximization, output
maximization but cannot deal with the qualitative factors like as honesty, sincerity, political impact, etc.
(2) Mathematical models are applicable to only specific categories of problems
(3) Management which has to implement the advised proposals may itself offer a lot of resistance due to
conventional thinking.
(4) The experts must deal with the problems in OR for decision making which has to be implemented, but in
our country are not available.
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