This document presents an overview of linear programming, including: - Linear programming involves choosing a course of action when the mathematical model contains only linear functions. - The objective is to maximize or minimize some quantity subject to constraints. A feasible solution satisfies all constraints while an optimal solution results in the largest/smallest objective value. - Problem formulation involves translating a verbal problem statement into mathematical terms by defining decision variables and writing the objective and constraints in terms of these variables. - An example problem is presented to maximize profit by determining the optimal number of products A and B to manufacture, given constraints on money invested and labor hours. The objective and constraints are written mathematically to formulate the problem as a linear program.

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Presented by: Akshat Mishra
BBA (3rd
Sem)
Submitted to:
Mr. Durgesh Batra

2.  Linear Programming.
 Linear Programming Problem.
 Problem formulation.
 Guidelines for model formulations.
 Solved Example.
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3.  The use of the word “programming” hereThe use of the word “programming” here
means “choosing a course of action.”means “choosing a course of action.”
 Linear programming involves choosing aLinear programming involves choosing a
course of action when the mathematical modelcourse of action when the mathematical model
of the problem contains only linear functions.of the problem contains only linear functions.
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4.  The maximization or minimization of some quantity is the
objective in all linear programming problems.
 All LP problems have constraints that limit the degree to
which the objective can be pursued.
 A feasible solution satisfies all the problem's constraints.
 An optimal solution is a feasible solution that results in
the largest possible objective function value when
maximizing (or smallest when minimizing).
 A graphical solution method can be used to solve a linear
program with two variables.
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5.  If both the objective function and the constraints are
linear, the problem is referred to as a linear programming
problem.
 Linear functions are functions in which each variable
appears in a separate term raised to the first power and is
multiplied by a constant (which could be 0).
 Linear constraints are linear functions that are restricted to
be "less than or equal to", "equal to", or "greater than or
equal to" a constant.
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6.  Problem formulation or modeling is the process of
translating a verbal statement of a problem into a
mathematical statement.
 Formulating models is an art that can only be
mastered with practice and experience.
 Every LP problems has some unique features, but
most problems also have common features.
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7.  Understand the problem thoroughly.
 Describe the objective.
 Describe each constraint.
 Define the decision variables.
 Write the objective in terms of the decision variables.
 Write the constraints in terms of the decision variables.
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8.  XYZ ltd. can invest Rs40,000 in production and use
85 hours of labor. To manufacture one unit of product
“A” requires 15 minutes of labor, and to manufacture
one unit of product “B” requires 9 minutes of labor.
The company wants to maximize its profit. How many
units of product “A” and product “B” should it
manufacture? What is the maximized profit?
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9.  Since the profit to be maximized depend on the
number of product “A” and “B”, our decision
variables are:
x1 = number of product “A” produced;
x2 = number of product “B” produced;
 We want to maximize profit:
i.e. 30x1 + 20x2
Subject to the constraints:
Money: 40x1 + 60x2 ≤ 40,000
labor: 15x1 + 9x2 ≤ 5,100
Non-negativity: x1,x2 ≥ 0
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10.  Note the last constraint: x1,x2 ≥ 0 of product “B”
produced:.The unknowns x1 and x2 are called
decision variables.The function 30x1+20x2 to
be maximized is called the objective function.
 What we have now is a Linear Program.
maximum z = 30x1 + 20x2
40x1 + 60x2 ≤ 40;000
15x1 + 9x2 ≤ 5;100
x1; x2 ≥ 0
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11. 
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