Introduction to Linear Programming
Linear programming was developed during World War II, when a system with
which to maximize the efficiency of resources was of utmost importance. New
war-related projects demanded attention and spread resources thin. “Programming” was a military term that referred to activities such as planning schedules
efficiently or deploying men optimally. George Dantzig, a member of the U.S.
Air Force, developed the Simplex method of optimization in 1947 in order to
provide an efficient algorithm for solving programming problems that had linear
structures. Since then, experts from a variety of fields, especially mathematics
and economics, have developed the theory behind “linear programming” and
explored its applications .
This paper will cover the main concepts in linear programming, including
examples when appropriate. First, in Section 1 we will explore simple properties, basic definitions and theories of linear programs. In order to illustrate
some applications of linear programming, we will explain simplified “real-world”
examples in Section 2. Section 3 presents more definitions, concluding with the
statement of the General Representation Theorem (GRT). In Section 4, we explore an outline of the proof of the GRT and in Section 5 we work through a
few examples related to the GRT.
After learning the theory behind linear programs, we will focus methods
of solving them. Section 6 introduces concepts necessary for introducing the
Simplex algorithm, which we explain in Section 7. In Section 8, we explore
the Simplex further and learn how to deal with no initial basis in the Simplex
tableau. Next, Section 9 discusses cycling in Simplex tableaux and ways to
counter this phenomenon. We present an overview of sensitivity analysis in
Section 10. Finally, we put all of these concepts together in an extensive case
study in Section 11.
What is a linear program?
We can reduce the structure that characterizes linear programming problems
(perhaps after several manipulations) into the following form: