OPERATION RESEARCH
Hemal Rajyaguru
Introduction


One has to take decisions for himself and for
others.
(E.g.-students which specialization they
should take.)

Everyone has to develop their talents in such
a way so that they can take correct decision
at a proper time.

(E.g. selection of proper engg.
branch,Investments in proper property,etc.)
Decision Dependency

An effective decision depends on
following factors:
Economic,
Social,
Political

Starting of a new factory
Economic factors would be construction cost,
labour cost, raw material cost, transportation
cost, taxes, energy, etc.
Social factors would be number of persons
available for labour, engineers, experts, etc.
Political factors would be government policies,
rules & regulations
(E.g. Gujarat is the best)

Decision maker has to faced with…..
(i)large number of interacting variables
as discussed in factors
(ii)actions of the other competitors,
over which we don’t have control
People make decisions…..

Based on their past experience and
intuitions(understanding) but they are not
always enough.

Some formal system is needed to
determine an effective course of actions.
Operation Research
Quantitative methods,
 Management science,
 Decision science
provides scientific approach to the
executives for making better decisions for
operation under their control.


Operation research is not decision making

It hopefully improve the decisions made
by the management in rational, logical,
systematic and scientific way.
Operations Research

Operations research is “a scientific approach to decision making
which seeks to determine how best to design and operate a
system, under conditions requiring the allocation of scarce
resources.”

Provides a set of algorithms that act as tools for effective
problem solving and decision making

Extensive applications in engineering, business and public systems.

Used extensively by manufacturing and service industries in
decision making that’s why also useful in IT industry.
Operations Research

Origin during World War II when the British military asked
scientists to analyze military problems

The application of mathematics and scientific method to military
applications was called Operations Research.

Operational analysis

System analysis

Cost benefit analysis

Management science

Decision Science
Purpose

Helps in determining the best(optimum)
solution(course of action) to problems
where decision has to be taken under the
restriction of limited resources.

It is possible to convert any real life problem
into a mathematical model.

Basic feature of OR is to formulate a real
world problem as a mathematical model.

Any industry want to lower their labour or
production costs to achieve higher profits
and OR can implied that.
OR=problem solving technique

=science+art
where, science is using mathematical
techniques for solving decision problems
Art is the ability to develop good report
with good supplying information

Quantitative approach to decision-making: or/ms

Decision under certainty or uncertainty

Static decision: Decisions for one time period only
come under this.



E.g.: (1)Toss result in match
(II)Match result
Dynamic decision: Decisions over several time periods
comes under this.

E.g.: Bowling, Batting, Fielding strategies at different stage
of match.
Methodology of Operations Research
Methodology of Operations Research

Problem Formulation

Model Building
◦
Physical Models
◦
Mathematical Models

Obtaining input Data

Solution of Model
◦
Feasible and Infeasible Solutions
◦
Optimal and Non-optimal Solutions
◦
Unique and Multiple Solutions
Fundamentals of Operations
Research
Linear Programming – Formulations
 Linear Programming – Solution
 Duality and Sensitivity Analysis
 Transportation Problem
 Assignment Problem
 Dynamic Programming
 Deterministic Inventory Models

Linear Programming
First conceived by George B Dantzig around
1947.
 The work of Kantorovich (1939) was
published in 1959
 Dantzig’s first paper was titled “Programming
in Linear Structure”
 Koopmans
coined the term “Linear
Programming” in 1948
 Simplex method was published in 1949 by
Dantzig

LINEAR PROGRAMMING
FORMULATIONS
Example 1 – Maximisation Problem
(Product Mix Problem)






Consider a small manufacturer making two products
A and B.
Two resources R1 and R2 are require to make these
products.
Each unit of product A requires 1 unit of R1 and 3
units of R2.
Each unit of product B requires 1 unit of R1 and 2
units of R2.
The manufacturer has 5 units of R1 and 12 units of R2
available,
The manufacturer also makes a profit of
◦ Rs. 6 per unit of product A sold and
◦ Rs. 5 per unit of product B sold.
Product
Resource 1
Resource 2
Profit
A
1
3
Rs. 6
B
1
2
Rs. 5
TOTAL
5
12
Solution

The manufacturer has to decide on the number of units of products A and B
to produce.

It is acceptable that the manufacturer would like to make as much profit as
possible and would decide on the production quantities accordingly.

The manufacturer has to ensure that the resources needed to make the
products are available.

Before we attempt to find out the decisions of the manufacturer, let us
redefine the problem in an algebraic form.

The manufacturer has to decide on the production quantities. Let us call them
x and y and define

Let x be the number of units of product A made

Let y be the number of units of product B made.

The profit associated with x units of product A and y units of product B is 6x + 5y. The manufacturer
would determine x and y such that this function has a maximum value.

The requirements of two resources are x + y for R1 and 3x + 2y for R2 and the manufacturer has to
ensure that these are available.

The problem is to find x and y such that 6x + 5y is maximized and

x + y ≤ 5 and 3x + 2y ≤ 12 are satisfied. Also it is necessary that x, y ≥ 0 so that only nonnegative
quantities are produced.

If we redefine the production quantities as x1 and x2 (for reasons of uniformity and consistency) then
problem is to
Maximize Z = 6x1 + 5x2
Subject to
x1 + x2 ≤ 5
3x1 + 2x2 ≤ 12
x1, x2 ≥ 0
Terminology

The problem variables x1 and x2 are called decision variables and they
represent the solution or the output decision from the problem.

The profit function that the manufacturer wishes to increase, represents
the objective of making the decisions on the production quantities and is
called the objective function.

The conditions matching the resource availability and resource
requirement are called constraints.
◦ These usually limit (or restrict) the values the decision variables can take.

We have also explicitly stated that the decision variable should take non
negative values.
◦ This is true for all Linear Programming problems.
◦ This is called non negativity restriction

The problem that we have written down in algebraic form represents the
mathematical model of the given system and is called the Problem
Formulation.

The problem formulation has the following steps.
◦
◦
◦
◦

Identifying the decision variables.
Writing the objective function
Writing the constraints.
Writing the non negativity restrictions.
In the above formulation, the objective function and
the constraints are linear.
◦ Therefore the model that we formulated is a linear
programming problem.

A Linear Programming problem has
◦ A linear objective function,
◦ linear constraints and
◦ the non negativity constraints on all the decision variables.
Example – 2 Minimization Problem

The Agricultural Research institute suggested to a farmer to spread out at
least 4800 kg of a special phosphate fertilizer and not less than 7200 kg of
a special nitrogen fertilizer to raise productivity of crops in his fields.

There are two sources two obtaining these-mixtures A and B. Both of
these are available in bag weighing 100 kg each and they cost Rs 40 and Rs
24 respectively.

Mixture A contains phosphate and nitrogen equivalent of 20 kg and 80 kg
respectively.

While mixture B contains these ingredients equivalent of 50 kg each.

Write this as a linear programming problem to determine how many bags
of each type the farmer should buy in order to obtain the required
fertilizer at minimum cost.
Bag Type
(100 Kg)
Price per
Bag
Phosphate/
Bag
Nitrogen /
Bag
A
40
20
80
B
24
50
50
4800
7200
Min
Requirement
Solution
The objective function
 The constraints
 Non-negativity condition

Minimise
Z = 40x1 + 24x2
Cost
Subject to
20x1 + 50x2 ≥ 4800
Phosphate requirement
80x1 + 50x2 ≥ 7200
Nitrogen requirement
x 1, x 2
≥0
Non-negativity restriction
Examples

A firm is engaged in producing two products, A and B. Each
unit of product A requires 2 kg of raw material and 4 labour
hours for processing, whereas each unit of product B
requires 3 kg of raw material and 3 hours of labour, of the
same type. Every week, the firm has an availability of 60 kg of
raw material and 96 labour hours. One unit of product A
sold yields Rs 40 and one unit of product B sold gives Rs 35
as profit.

Formulate this problem as a linear programming problem to
determine as to how many units of each of the products
should be produced per week so that the firm can earn the
maximum profit. Assume that there is no marketing
constraint so that all that is produced can be sold.
Product
Raw Material
Labour
Profit
A
2
4
40
B
3
3
35
Total
60
96
A company is manufacturing two different types
of products, A and B. Each product has to be
processed in 3 different departments – casting,
machining and finally inspection. The capacity of 3
departments is limited to 35 hrs., 32 hrs and 24
hrs. per week respectively. Product A requires 7
hrs. in casting department, 8 hrs. in machining
shop and 4 hrs. in inspection, whereas product B
requires 5 hrs., 4 hrs. and 6 hrs. respectively in
respective shop. The profit contribution for a unit
product of A and B is Rs. 30 and Rs. 40
respectively.
 Formulate the problem.

Department
Product
A
B
Capacity
Per week
Casting
7
5
35
Machining
8
4
32
Inspection
4
6
24
Profit
contribution
Per unit
Rs. 30 /-
Rs. 40 /-


A24-hour supermarket has the following minimal
requirements for cashiers:
Period:
1
2
3
4
5
6
Time of day
(24-hour
clock):
3-7
7-11
11-15
15-19
19-23
23-3
Minimum
number
required:
7
20
14
20
10
5
Period 1 follows immediately after period 6. A cashier
works eight consecutive hours, starting at the beginning
of one of the six time periods. Determine a daily
employee Worksheet which satisfies the requirements
with the least number of personnel. Formulate the
problem as a linear programming problem.