An Introduction to Operations

Research

Shivam Kushwaha
Assistant Professor (POM & QT)
Institute of Rural Management Anand
General Steps of Problem Solving

Steps of Problem Solving

1. Define the problem.

2. Determine the set of alternative solutions.
3. Determine the criteria for evaluating alternatives.
4. Evaluate the alternatives.
5. Choose an alternative (make a decision).
6. Implement the selected alternative.
7. Evaluate the results.

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Analysis Phase of Decision-Making Process

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Analysis Phase of Decision-Making Process

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Analysis Phase of Decision-Making Process

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Quantitative Analysis

Quantitative Analysis Process

▪Model Development
▪Data Preparation
▪Model Solution
▪Report Generation

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Example: Flight Simulator

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Essential features

Simplification: It abstracts the key components of flying—controls, navigation,

weather conditions, and physics—while omitting unnecessary complexities.

Representation: It replicates the environment of real-world flying, including

cockpit instruments, weather effects, and flight dynamics.

Purpose: It is designed to achieve specific objectives like pilot training, testing

responses to emergencies, or studying aircraft behavior in a controlled, risk-free
environment.

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Can you guess which model best explains the image
below?

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Model Development

What are models?

▪ A model is a simplified representation or abstraction of a real-world
system, process, or phenomenon.

Why to build models?

▪ To analyze and manipulating the model gives in- sight into how the real
system behaves under various conditions.

▪ Improve our understanding and communication

▪ Experimentation
▪ Standardization for analysis

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In this course, we will learn and apply quantitative
models to take better decision for real-world
problems

FOCUS: Linear Programming

Introduction to Operations Research

Operations Research
▪ Early roots of OR can be traced to World War II.
▪ The first formal use of OR was by the British military in 1937 to
improve radar detection systems.
▪ In 1947, George Dantzig made significant contribution to the
field by developing simplex method for solving linear
programming problems.
▪ It has been applied to various business and industries.
▪ Transportation, Banking and Finance, Agriculture

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Optimization

What is Optimization ?
▪ Optimization is the process of selecting the best among a set of
alternatives based on certain criteria/conditions/resources.
Let’s understand through an example:
Task Assignment
▪ 5 employees have 5 tasks to complete.
▪ Each employee is qualified to carry out one task.
▪ Each employee can only be assigned one task.
▪ Minimize the cost of getting all task done by employees.
TASK-1 TASK-2 TASK-3 TASK-4 TASK-5
Employee-1 138 129 132 105 116
Employee-2 132 116 123 110 111
Employee-3 138 102 109 137 131
Employee-4 150 145 105 114 136
Employee-5 123 113 108 150 130

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Optimization

Task Assignment
▪ What about 20 employees have 20 tasks to complete.
▪ The cost of assignment of each employee to each tasks is given.
▪ Do you really want to enumerate the 20!

2432902008176640000

Optimization problem does not always require iterating through and

assessing every possible solution.

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 Recap

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Case Study-I: A Case of ABC Glass Company

Product I: a glass door

with aluminium framing
▪ Two Products Manufactured by ABC
Glass Company
▪ Three plants;
▪ Aluminium frames and hardware in plant I,
▪ Wood frames in plant 2
▪ Glass production and assembly in plant 3
▪ Product I requires some of the prod Product II: a foot double hung
capacity of plants 1 and 3 while product 2 wood framed window
requires only that of plants 2 and 3.
▪ Each unit of Product 1 generates 3000 of
profit in the market while each unit of
Product 2 generates 5000 of profit.

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Constraints

Production time
Product I Product II available per week
Plant (Requirement) (Requirement) (hours)
1 1 0 4
2 0 2 12
3 3 2 18

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A Case of ABC Glass Company contd..

GOAL:

Determination of Profit Maximizing Output

Alternatively, we can ask

What should be production of two products to maximize their

total profit, subject to restrictions of limited production capacities
of plants.

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Objective for ABC company

Assume
Total profit of ABC is denoted as (Z)
𝑥1 and 𝑥2 are number of quantities of Products I and II.
Given per unit Profit on products I and II to be Rs. 3,000 and Rs. 5,000
respectively.
To maximize profit, one need to choose 𝑥1 and 𝑥2 such that that maximize
total profit (Z).
Representing mathematically

Max 𝑍; where Z = 3000𝑥1 + 5000𝑥2

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Formulating a Mathematical Model

▪ Optimize the Objective Function Subject to Constraints

▪ Gives a Concise Description of the Problem
▪ Forms a Bridge Between Mathematical Techniques and Computers
to Analyze the Problem

Model for the ABC

𝑀𝑎𝑥 𝑧 = 3000𝑥1 + 5000𝑥2

𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜:
𝑥1 ≤ 4 [Plant 1]
2𝑥2 ≤ 12 [Plant 2]
3𝑥1 + 2𝑥2 ≤ 18 [Plant 3]
𝑥1 ≥ 0, 𝑥2 ≥ 0

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Linear Programming

Linear programming is a mathematical modelling technique that

involves maximizing or minimizing a linear function while taking
into account various constraints.

Key Components:
▪ Decision Variables
▪ Objective Function
▪ Constraints
▪ Feasible Region
▪ Objective Function Coefficients
▪ Non-Negativity Constraints

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General Form of Linear Programming

𝑥𝑗 values are decision variables and 𝑐𝑗, 𝑎𝑖𝑗 , and 𝑏𝑖 values are
constants, called parameters or coefficients

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Case Study-I: A problem of farmer

A farmer has two types of fertilizers, Fertilizer A and Fertilizer B,

to use on their crops.
▪ Nutrient Requirements:
▪ At least 80 units of nitrogen.
▪ At least 100 units of phosphorus.
▪ Fertilizer Content:
▪ Fertilizer A contains 4 units of nitrogen and 5 units of phosphorus per kilogram.
▪ Fertilizer B contains 2 units of nitrogen and 8 units of phosphorus per kilogram.
▪ Cost:
▪ Fertilizer A costs ₹30 per kilogram.
▪ Fertilizer B costs ₹20 per kilogram.

The farmer wants to minimize the total cost of purchasing fertilizers while
ensuring the crops get the minimum required nutrients.

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Case Study-I: A problem of farmer

Let us assume, 𝑥1 and 𝑥2 are the quantities of Fertilizer A and Fertilizer B

being purchased.

Total Cost of fertilizer = 30𝑥1 + 20𝑥2

Goal: Minimization of the fertilizer cost while satisfying the requirement of

Nitrogen and Phosphorus.

Linear Programming Model

𝑀𝑖𝑛 30𝑥1 + 20𝑥2
Subject to
4𝑥1 + 2𝑥2 ≥ 80
5𝑥1 + 8𝑥2 ≥ 100
𝑥1 , 𝑥2 ≥ 0

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Problem 1: Product-Mix

▪ A company manufactures two products, P1 and P2, using two

resources: Machine Hours and Labor Hours.
▪ Product Information:
• Product P1 generates ₹50 profit per unit.
• Product P2 generates ₹40 profit per unit.
▪ Resource Usage:
• Machine Hours : P1 uses 2 hours per unit. P2 uses 1 hour per unit.
• Labor Hours:P1 uses 1 hour per unit. P2 uses 2 hours per unit
▪ Resource Availability:
• Total available Machine Hours: 100 hours.
• Total available Labor Hours: 80 hours.

The company wants to maximize its profit while considering

resource constraints.

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Problem 2: Man-Power Requirement

The nurses start work at the beginning of the shift (8 am, 12 noon etc) and
work for 8 continuous hours.

What is the minimum number of nurses required to meet the daily demand?

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Problem 3: Production Planning

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Tasks for the next class

1. Try to Solve Problems 1-3

2. Linear Programming Formulation of Merton Truck.

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 Thank you

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