Linear Programming

Linear programming is the solution of a mathematical problem concerning maximum and minimum values of a first-degree (linear) algebraic expression, with variables subject to certain stated conditions (restraints). A linear program (LP) can be expressed in two basic ways, the general form and the common form.
The General Form
minimize cx
subject to Ax = b
x > 0

( x = to the vector of variables to be solved for)
( A = a matrix of known coefficients)
( c and b = vectors of known coefficients)

Common Form
1 < x < u
( 1 and u = vectors of known lower and upper bounds)
( 1 and u can be positive or negative and they are infinite )
Linear programming has brought great advances to our economy as far as airline scheduling and the routing of information over complex communications networks are concerned. Through linear programming we are now able to schedule airline crews at a more mutable rate. This means that now it is possible to more easily cover shifts that have been changed. Also linear programming has made the routing of information over complex communications networks much more efficient. The use of linear programming also requires two basic methods, the simplex and ellipsoid, that once mastered make linear program a piece of cake to grasp.
The simplex method is a systematic procedure for generating and testing candidate vertex solutions to a linear program. A brilliant Russian mathematician named George Dantzig in 1947 introduced the simplex method. It visits “basic” solutions computed by fixing enough of the variables at their bounds to reduce the constraints Ax = b to a square system, which can be solved for unique values of the remaining variables. The simplex method can be viewed as moving from one point ( such as Ax = b , x > 0) to another along the edges of the boundary. Since it was first introduced it has greatly evolved. Thanks to the help of Dr. Narendra Karmakar, a mathematician at Bell Laboratories, we are now able to save time and money which is required for network planning in communications do to her development of the Karmakar algorithm.
The ellipsoid method is a slow, inefficient, but robust method of the cutting-plane method. Yudin, Nimerovsky, and Shor originally developed the ellipsoid method for convex nonlinear programming, but it became famous when Leonid Khachiyan, a Russian born mathematician who is currently teaching Computer Science at Rutgers University, used it to obtain a polynomial-time algorithm for linear programming.


Linear Programming affects us in all we do, whether through travel or communication. Our society and economy have greatly benefited from the scientific breakthrough of linear programming, for linear programming surrounds all arts of our life.

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