Linear Programming

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					                      Linear Programming
Linear Programming

Linear programming is a mathematical method for determining a way to achieve
the best outcome (such as maximum profit or lowest cost) in a given
mathematical model for some list of requirements represented as linear
relationships. Linear programming is a specific case of mathematical programming
(mathematical optimization).

More formally, linear programming is a technique for the optimization of a linear
objective function, subject to linear equality and linear inequality constraints.

Its feasible region is a convex polyhedron, which is a set defined as the
intersection of finitely many half spaces, each of which is defined by a linear
inequality.

Its objective function is a real-valued affine function defined on this polyhedron. A
linear programming algorithm finds a point in the polyhedron where this function
has the smallest (or largest) value if such point exists. Linear programs are
problems that can be expressed in canonical form:

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where x represents the vector of variables (to be determined), c and b are vectors
of (known) coefficients, A is a (known) matrix of coefficients, and is the matrix
transpose. The expression to be maximized or minimized is called the objective
function (cTx in this case).

The inequalities Ax ≤ b are the constraints which specify a convex polytope over
which the objective function is to be optimized. In this context, two vectors are
comparable when they have the same dimensions.

If every entry in the first is less-than or equal-to the corresponding entry in the
second then we can say the first vector is less-than or equal-to the second vector.

Linear programming can be applied to various fields of study. It is used in business
and economics, but can also be utilized for some engineering problems. Industries
that use linear programming models include transportation, energy,
telecommunications, and manufacturing. It has proved useful in modeling diverse
types of problems in planning, routing, scheduling, assignment, and design.

Uses :- Linear programming is a considerable field of optimization for several
reasons. Many practical problems in operations research can be expressed as
linear programming problems. Certain special cases of linear programming, such
as network flow problems and multicommodity flow problems are considered
important enough to have generated much research on specialized algorithms for
their solution.

A number of algorithms for other types of optimization problems work by solving
LP problems as sub-problems. Historically, ideas from linear programming have
inspired many of the central concepts of optimization theory, such as duality,
decomposition, and the importance of convexity and its generalizations.


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Likewise, linear programming is heavily used in microeconomics and company
management, such as planning, production, transportation, technology and other
issues. Although the modern management issues are ever-changing, most
companies would like to maximize profits or minimize costs with limited resources.
Therefore, many issues can be characterized as linear programming problems.

Firstly we will find out the satisfiable values of 'x' and 'y' for each equation and
create a graph which involves all three equations as part of it. These three
equations form a common region or may be uncommon region which satisfies the
cost for a set of inequalities.

This region if it is common then optimized maximum profit is and if regions are not
common then the cost found will be minimum. Consider the following graph, in
this way we have a set of two inequalities say: x + y < 2 and x = 2. One is an
inequality and other one is an equation. The region thus obtained considering
these two lines and cost is as follows:




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posted:8/28/2012
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