Linear programing

LINEAR PROGRAMMING

By:-
Sankheerth P. Uma Maheshwar Rao
Aakansha Bajpai Abhishek Bose
Amit Kumar Das Aniruddh Tiwari
Ankit Sharma Archana Yadav
Arunava Saha Arvind Singh
Awinash Chandra Ashok Kumar Komineni

LINEAR PROGRAMMING

 What is LP ?
 The word linear means the relationship which can
be represented by a straight line .i.e the relation is of the
form
ax +by=c. In other words it is used to describe the
relationship between two or more variables which are
proportional to each other
The word “programming” is concerned with the
optimal allocation of limited resources.
Linear programming is a way to handle certain types of
optimization problems
Linear programming is a mathematical method for
determining a way to achieve the best outcome

DEFINITION OF LP
 LP is a mathematical modeling technique useful for
the allocation of “scarce or limited’’ resources such
as labor, material, machine ,time ,warehouse space
,etc…,to several competing activities such as
product ,service ,job, new
equipments, projects, etc...on the basis of a given
criteria of optimality

DEFINITION OF LPP
 A mathematical technique used to obtain an
optimum solution in resource allocation
problems, such as production planning.

 It is a mathematical model or technique for
efficient and effective utilization of limited recourses
to achieve organization objectives (Maximize profits
or Minimize cost).

 When solving a problem using linear programming
, the program is put into a number of linear
inequalities and then an attempt is made to
maximize (or minimize) the inputs

REQUIREMENTS
 There must be well defined objective function.

 There must be a constraint on the amount.

 There must be alternative course of action.

 The decision variables should be interrelated and
non negative.

 The resource must be limited in supply.

ASSUMPTIONS
 Proportionality

 Additivity

 Continuity

 Certainity

 Finite Choices

APPLICATION OF LINEAR PROGRAMMING
 Business

 Industrial

 Military

 Economic

 Marketing

 Distribution

AREAS OF APPLICATION OF LINEAR
PROGRAMMING
 Industrial Application
 Product Mix Problem
 Blending Problems
 Production Scheduling Problem
 Assembly Line Balancing
 Make-Or-Buy Problems
 Management Applications
 Media Selection Problems
 Portfolio Selection Problems
 Profit Planning Problems
 Transportation Problems
 Miscellaneous Applications
 Diet Problems
 Agriculture Problems
 Flight Scheduling Problems
 Facilities Location Problems

ADVANTAGES OF L.P.
 It helps in attaining optimum use of productive
factors.

 It improves the quality of the decisions.

 It provides better tools for meeting the changing
conditions.

 It highlights the bottleneck in the production
process.

LIMITATION OF L.P.
 For large problems the computational difficulties are
enormous.

 It may yield fractional value answers to decision
variables.

 It is applicable to only static situation.

 LP deals with the problems with single objective.

TYPES OF SOLUTIONS TO L.P. PROBLEM

 Graphical Method

 Simplex Method

FORMS OF L.P.
 The canonical form
 Objective function is of maximum type
 All decision variables are non negetive

 The Standard Form
 All variables are non negative
 The right hand side of each constraint is non negative.

 All constraints are expressed in equations.

 Objective function may be of maximization or minimization

type.

IMPORTANT DEFINITIONS IN L.P.
 Solution:
A set of variables [X1,X2,...,Xn+m] is called a
solution to L.P. Problem if it satisfies its constraints.
 Feasible Solution:
A set of variables [X1,X2,...,Xn+m] is called a
feasible solution to L.P. Problem if it satisfies its
constraints as well as non-negativity restrictions.
 Optimal Feasible Solution:
The basic feasible solution that optimises the
objective function.
 Unbounded Solution:
If the value of the objective function can be
increased or decreased indefinitely, the solution is called
an unbounded solution.

VARIABLES USED IN L.P.

 Slack Variable

 Surplus Variable

 Artificial Variable

Non-negative variables, Subtracted from the L.H.S of the
constraints to change the inequalities to equalities. Added when the
inequalities are of the type (>=). Also called as “negative slack”.

 Slack Variables

Non-negative variables, added to the L.H.S of the constraints to
change the inequalities to equalities. Added when the inequalities
are of the type (<=).

 Surplus Variables
In some L.P problems slack variables cannot provide a solution.
These problems are of the types (>=) or (=) . Artificial variables are
introduced in these problems to provide a solution.
Artificial variables are fictitious and have no physical meaning.

 Artificial Variables

DUALITY :
 For every L.P. problem there is a related unique L.P.
problem involving same data which also describes
the original problem.
 The primal programme is rewritten by transposing
the rows and columns of the algebraic statement of
the problem.
 The variables of the dual programme are known as
“Dual variables or Shadow prices” of the various
resources.
 The optimal solution of the dual problem gives
complete information about the optimal solution of
the primal problem and vice versa.

ADVANTAGES :
 By converting a primal problem into dual
, computation becomes easier , as the no. of
rows(constraints) reduces in comparison with the
no. of columns( variables).
 Gives additional information as to how the optimal
solution changes as a result of the changes in the
coefficients . This is the basis for sensitivity
analysis.
 Economic interpretation of dual helps the
management in making future decisions.
 Duality is used to solve L.P. problems in which the
initial solution in infeasible.

SENSITIVITY ANALYSIS :
(POST OPTIMALITY TEST)
Two situations:
 In formulation , it is assumed that the parameters
such as market demand, equipment
capacity, resource consumption, costs, profits
etc., do not change but in real time it is not
possible.

 After attaining the optimal solution, one may
discover that a wrong value of a cost coefficient
was used or a particular variable or constraint was
omitted etc.,

 Changes in the parameters of the problem may be
discrete or continuous.
 The study of effect of discrete changes in parameters on
the optimal solution is called as “Sensitivity analysis”.
 The study of effect of continuous changes in parameters
on the optimal solution is called as “Parametric
Programming.”
 The objective of the sensitivity analysis is to determine
how sensitive is the optimal solution to the changes in
the parameters.

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Linear programing

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