The Linear Programming Problem

Abstract

Since the time it was first proposed by one of the authors (George B. Dantzig) in 1947 as a way for planners to set general objectives and arrive at a detailed schedule to meet these goals, linear programming has come into wide use. It has many nonlinear and integer extensions collectively known as the mathematical pro-gramming field, such as integer programming, nonlinear programming, stochastic programming, combinatorial optimization, and network flow maximization; these are presented in subsequent volumes. Here then is a formal definition of the field that has become an important branch of study in mathematics, economics, computer science, and decision science (i.e., operations research and management science): Mathematical programming (or optimization theory) is that branch of mathematics dealing with techniques for maximizing or minimizing an objective function subject to linear, nonlinear, and integer constraints on the variables. The special case, linear programming, has a special relationship to this more general mathematical programming field. It plays a role analogous to that of partial derivatives to a function in calculus—it is the first-order approximation. Linear programming is concerned with the maximization or minimiza-tion of a linear objective function in many variables subject to linear equality and inequality constraints. For many applications, the solution of the mathematical system can be interpreted as a program, namely, a statement of the time and quantity of actions to be per-formed by the system so that it may move from its given status towards some defined objective. 1 2 THE LINEAR PROGRAMMING PROBLEM Linear programming problems vary from small to large: The number of con-straints less than 1,000 is considered " small, " between 1,000 and 2,000 is consid-ered " medium, " and greater than 2,000 is considered " large. " Linear programming models can be very large in practice; some have many thousands of constraints and variables. To solve large systems requires special software that has taken years to develop. Other special tools, called matrix generators, are often used to help orga-nize the formulation of the model and direct the generation of the coefficients from basic data files. As the size of models that can be solved has grown, so has evolved the art of model management. These include, on the input side, model formulation and model updating, and, on the output side, summarizing of the detailed solution output in the form of graphs and other displays (so that the results may be more easily understood and implemented by decision makers).