linear programming

Math.any of several methods for finding where a given linear function of several nonnegative variables assumes an extreme value and for determining the extreme value, the variable usually being subjected to constraints in the form of linear equalities or inequalities.[194550]
* * *
Mathematical modeling technique useful for guiding quantitative decisions in business, industrial engineering, and to a lesser extent the social and physical sciences.Solving a linear programming problem can be reduced to finding the optimum value (see optimization) of a linear equation (called an objective function), subject to a set of constraints expressed as inequalities. The number of inequalities and variables depends on the complexity of the problem, whose solution is found by solving the system of inequalities like a system of equations. The extensive use of linear programming during World War II to deal with transportation, scheduling and allocations of resources under constraints like cost and priority gave the subject an impetus that carried it into the postwar era. The number of equations and variables needed to model reallife situations accurately is large, and the solution process can be timeconsuming even with computers. See also simplex method.* * *
mathematical modeling technique useful for guiding quantitative decisions in business planning, industrial engineering, and—to a lesser extent—in the social and physical sciences.Applications of the method of linear programming were first seriously attempted in the late 1930s by the Soviet mathematician Leonid Kantorovich and by the American economist Wassily Leontief in the areas of manufacturing schedules and of economics, respectively, but their work was ignored for decades. During World War II, linear programming was used extensively to deal with transportation, scheduling, and allocation of resources subject to certain restrictions such as costs and availability. These applications did much to establish the acceptability of this method, which gained further impetus in 1947 with the introduction of the American mathematician George Dantzig's simplex method, which greatly simplified the solution of linear programming problems. However, as increasingly more complex problems involving more variables were attempted, the number of necessary operations expanded exponentially and exceeded the computational capacity of even the most powerful computers. Then, in 1979, the Russian mathematician Leonid Khachian discovered a polynomialtime algorithm—i.e., the number of computational steps grows as a power of the number of variables, rather than exponentially—thereby allowing the solution of hitherto inaccessible problems.The solution of a linearprogramming problem reduces to finding the optimum value (largest or smallest, depending on the problem) of the linear expression (called the objective function):subject to a set of constraints expressed as inequalities:The a's, b's, and c's are constants determined by the capacities, needs, costs, profits, and other requirements and restrictions of the problem. The basic assumption in the application of this method is that the various relationships between demand and availability are linear; that is, none of the x_{i} is raised to a power other than 1. In order to obtain the solution to this problem it is necessary to find the solution of the system of linear inequalities (that is, the set of n values of the variables x_{i} that simultaneously satisfies all the inequalities). The objective function is then evaluated by substituting the values of the x_{i} in the equation that defines f .* * *
Universalium. 2010.
Look at other dictionaries:
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.… … Wikipedia
linear programming — n. Math. a procedure for minimizing or maximizing a linear function of several variables, subject to a finite number of linear restrictions on these variables … English World dictionary
Linear programming — Dieser Artikel oder Abschnitt ist nicht hinreichend mit Belegen (Literatur, Webseiten oder Einzelnachweisen) versehen. Die fraglichen Angaben werden daher möglicherweise demnächst gelöscht. Hilf Wikipedia, indem du die Angaben recherchierst und… … Deutsch Wikipedia
linear programming — LP A method for the optimal allocation of scarce resources to alternative activities. The aim of the decision making process (termed the objective function ) and related constraints are expressed in mathematical terms, and may be plotted… … Auditor's dictionary
linear programming — tiesinis programavimas statusas T sritis automatika atitikmenys: angl. linear programming vok. lineare Programmierung, f rus. линейное программирование, n pranc. programmation linéaire, f … Automatikos terminų žodynas
linear programming — noun Date: 1949 a mathematical method of solving practical problems (as the allocation of resources) by means of linear functions where the variables involved are subject to constraints … New Collegiate Dictionary
linear programming — noun the branch of mathematics concerned with the minimization or maximization of a linear function of several variables and inequalities; used in many branches of industry to minimize costs or maximize production … Wiktionary
Linear programming — Technique for finding the maximum value of some equation subject to stated linear constraints. The New York Times Financial Glossary … Financial and business terms
linear programming — Technique for finding the maximum value of some equation, subject to stated linear constraints. Bloomberg Financial Dictionary … Financial and business terms
linear programming — noun a mathematical technique for maximizing or minimizing a linear function of several variables … English new terms dictionary