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Introduction to
Operations Research
SUJIT SHEDAGE
email id – sujit.shedage@flame.edu.in
Introduction
Operations Research is a systematic and analytical approach
to decision making and problem solving.
OR is a branch of applied mathematics that uses techniques
and statistics to arrive at optimal solutions to solve complex
problems.
It is typically concerned with determining the maximum
profit, sale, output, yield and efficiency and minimum
losses, risks, cost, and time.
Problem Solving Process
Constructing a Model
Problem must be translated from
verbal, qualitative terms to logical and
quantitative terms.
A mathematical model is a collection
of functional relationships by which
allowable actions are delimited and
evaluated.
Solving a Mathematical Model
Many tools are available
Some lead to “optimal” solution
Others only evaluate candidates to
trial and error to find ”best” course of
action
Operations Research Models
Deterministic Model
Linear Programming
Network Optimization
Integer Programming
Non-Linear Programming
Stochastic/Probabilistic Model
• Discrete-Time Markov Chains
• Continuous-Time Markov Chains
• Queuing Theory
• Decision Analysis
Introduction to Linear Programming
Problem
A linear Programming model seeks to maximize or minimize a
linear function, subject to a set of linear constraints
A linear model consists of the following components:
◦ A set of decision variables
◦ An objective function
◦ A set of constraints
Process to formulate a LP problem
1. Identify the decision variables
2. Write the objective function
3. Mention the constraints
4. Explicitly state the non-negativity restrictions
For a problem to be a linear programming problem, the decision
variables, objective function and constraints all must be linear
functions.
Formulation of LPP
1. The decision variables (activities): These are activities for
which we want to determining a solution. These are usually
denoted by x1, x2, …, xn.
2. The objective function (goal): This is a function which is
expressed in terms of decision variables, and we want to
optimize (maximize or minimize) the function.
3. The constraints: These are limiting conditions on the use of
resources. The solution of LPP must satisfy all these
constraints.
Example 1
Consider a chocolate manufacturing company which produces only two types of chocolate –
A and B. Both the chocolates require Milk and Choco only.
To manufacture each unit of A and B, following quantities are required:
Each unit of A requires 1 unit of Milk and 3 units of Choco
Each unit of B requires 1 unit of Milk and 2 units of Choco
The company kitchen has a total of 5 units of Milk and 12 units of Choco. On each sale, the company
makes a profit of
Rs 6 and Rs 5 per unit sold of A and B respectively.
Now, the company wishes to maximize its profit. How many units of A and B should it produce
respectively
Example 1
Let, the total number of units of A and B be X and Y respectively.
Then,
Maximize Profit = Z = 6X +5Y
Subject to constraints:
X + Y ≤ 5
3X + 2Y ≤ 12
X ≥ 0 and Y ≥ 0
Example 2
A farmer has recently acquired a 110 hectares piece of land. He has decided to
grow Wheat and barley on that ;and. Due to the quality of the sun and the
region’s excellent climate. The entire production of wheat and barley can be sold.
He wants to know how to plant each variety in the 110 hectares given the costs,
net profit and labor requirements according to the data shown below. Given the
farmer has a budget of US$ 10,000 and an availability of 1200 man-days during
the planning horizon, find the optimal solution.
Variety Cost
(Price/Hec)
Net Profit
(Price/Hec)
Man-days/Hec
Wheat 100 50 10
Barley 200 120 30
Example 2
1. Decision variables:
The total are for growing Wheat = X (in
hectares)
The total are for growing Barley = Y (in
hectares)
2. Objective Function:
Max Z = 50X + 120Y
3. Constraints:
Subject to: 100X + 200Y ≤ 10,000
10X + 20Y ≤ 1200
X + Y ≤ 110
X ≥ 0 and Y ≥ 0

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session 3 OR_L1.pptx

  • 1. Introduction to Operations Research SUJIT SHEDAGE email id – sujit.shedage@flame.edu.in
  • 2. Introduction Operations Research is a systematic and analytical approach to decision making and problem solving. OR is a branch of applied mathematics that uses techniques and statistics to arrive at optimal solutions to solve complex problems. It is typically concerned with determining the maximum profit, sale, output, yield and efficiency and minimum losses, risks, cost, and time.
  • 4. Constructing a Model Problem must be translated from verbal, qualitative terms to logical and quantitative terms. A mathematical model is a collection of functional relationships by which allowable actions are delimited and evaluated.
  • 5. Solving a Mathematical Model Many tools are available Some lead to “optimal” solution Others only evaluate candidates to trial and error to find ”best” course of action
  • 6. Operations Research Models Deterministic Model Linear Programming Network Optimization Integer Programming Non-Linear Programming Stochastic/Probabilistic Model • Discrete-Time Markov Chains • Continuous-Time Markov Chains • Queuing Theory • Decision Analysis
  • 7. Introduction to Linear Programming Problem A linear Programming model seeks to maximize or minimize a linear function, subject to a set of linear constraints A linear model consists of the following components: ◦ A set of decision variables ◦ An objective function ◦ A set of constraints
  • 8. Process to formulate a LP problem 1. Identify the decision variables 2. Write the objective function 3. Mention the constraints 4. Explicitly state the non-negativity restrictions For a problem to be a linear programming problem, the decision variables, objective function and constraints all must be linear functions.
  • 9. Formulation of LPP 1. The decision variables (activities): These are activities for which we want to determining a solution. These are usually denoted by x1, x2, …, xn. 2. The objective function (goal): This is a function which is expressed in terms of decision variables, and we want to optimize (maximize or minimize) the function. 3. The constraints: These are limiting conditions on the use of resources. The solution of LPP must satisfy all these constraints.
  • 10. Example 1 Consider a chocolate manufacturing company which produces only two types of chocolate – A and B. Both the chocolates require Milk and Choco only. To manufacture each unit of A and B, following quantities are required: Each unit of A requires 1 unit of Milk and 3 units of Choco Each unit of B requires 1 unit of Milk and 2 units of Choco The company kitchen has a total of 5 units of Milk and 12 units of Choco. On each sale, the company makes a profit of Rs 6 and Rs 5 per unit sold of A and B respectively. Now, the company wishes to maximize its profit. How many units of A and B should it produce respectively
  • 11. Example 1 Let, the total number of units of A and B be X and Y respectively. Then, Maximize Profit = Z = 6X +5Y Subject to constraints: X + Y ≤ 5 3X + 2Y ≤ 12 X ≥ 0 and Y ≥ 0
  • 12. Example 2 A farmer has recently acquired a 110 hectares piece of land. He has decided to grow Wheat and barley on that ;and. Due to the quality of the sun and the region’s excellent climate. The entire production of wheat and barley can be sold. He wants to know how to plant each variety in the 110 hectares given the costs, net profit and labor requirements according to the data shown below. Given the farmer has a budget of US$ 10,000 and an availability of 1200 man-days during the planning horizon, find the optimal solution. Variety Cost (Price/Hec) Net Profit (Price/Hec) Man-days/Hec Wheat 100 50 10 Barley 200 120 30
  • 13. Example 2 1. Decision variables: The total are for growing Wheat = X (in hectares) The total are for growing Barley = Y (in hectares) 2. Objective Function: Max Z = 50X + 120Y 3. Constraints: Subject to: 100X + 200Y ≤ 10,000 10X + 20Y ≤ 1200 X + Y ≤ 110 X ≥ 0 and Y ≥ 0