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Meaning of linear programming

Mathematical technique used in computer modeling (simulation) to find the best possible solution in allocating limited resources (energy, machines, materials, money, personnel, space, time, etc.) to achieve maximum profit or minimum cost. However, it is applicable only where all relationships are linear (see linear relationship), and can accommodate only a limited class of cost functions. For problems involving more complex cost functions, another technique called 'mixed integer modeling' is employed. Developed by the Russian economist Leonid Kantorovich (1912-86) and the US economist C. Koopmans (1910-86), on the basis of the work of the Russian mathematician Andrei Nikolaevich Kolmogorov (1903-87).

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A pictorial representation of a simple linear program with two variables and six inequalities. The set of feasible solutions is depicted in light red and forms a 2-dimensional polytope. The linear cost function is represented by the red line and the arrow: The red line is a level set of the cost function, and the arrow indicates the direction in which we are optimizing.

Linear programming (LP, or linear optimization) 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....