A pictorial representation of a simple linear program linear programming in operations research pdf two variables and six inequalities. The set of feasible solutions is depicted in yellow and forms a polygon, 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. A closed feasible region of a problem with three variables is a convex polyhedron. The linear programming problem is to find a point on the polyhedron that is on the plane with the highest possible value.
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 polytope, which is a set defined as the intersection of finitely many half spaces, each of which is defined by a linear inequality. 0 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 it can be said that the first vector is less-than or equal-to the second vector. Linear programming can be applied to various fields of study.
It is widely used in mathematics, and to a lesser extent in business, economics, and for some engineering problems. Industries that use linear programming models include transportation, energy, telecommunications, and manufacturing. It has proven useful in modeling diverse types of problems in planning, routing, scheduling, assignment, and design. In 1939 a linear programming formulation of a problem that is equivalent to the general linear programming problem was given by the Soviet economist Leonid Kantorovich, who also proposed a method for solving it.
It is a way he developed, during World War II, to plan expenditures and returns in order to reduce costs of the army and to increase losses incurred to the enemy. Kantorovich’s work was initially neglected in the USSR.
About the same time as Kantorovich, the Dutch-American economist T. Koopmans formulated classical economic problems as linear programs. Kantorovich and Koopmans later shared the 1975 Nobel prize in economics.