A *Linear Programming* problem is a special case of a *
Mathematical Programming* problem. From an analytical perspective, a
mathematical program tries to identify an *extreme* (i.e., minimum
or maximum) point of a function , which
furthermore satisfies a set of constraints, e.g.,
. Linear programming is the
specialization of mathematical programming to the case where both,
function *f* - to be called the *objective function* - and the
problem constraints are *linear*.

From an applications perspective, mathematical (and therefore, linear)
programming is an *optimization* tool, which allows the
rationalization of many managerial and/or technological decisions
required by contemporary techno-socio-economic applications. An
important factor for the applicability of the mathematical programming
methodology in various application contexts, is the computational
tractability of the resulting analytical models. Under the advent of
modern computing technology, this tractability requirement translates
to the existence of effective and efficient algorithmic procedures
able to provide a systematic and fast solution to these models. For
Linear Programming problems, the *Simplex* algorithm, discussed
later in the text, provides a powerful computational tool, able to
provide fast solutions to very large-scale applications, sometimes
including hundreds of thousands of variables (i.e., decision
factors). In fact, the Simplex algorithm was one of the first
Mathematical Programming algorithms to be developed (George Dantzig,
1947), and its subsequent successful implementation in a series of
applications significantly contributed to the acceptance of the
broader field of *Operations Research* as a scientific approach to
decision making.

As it happens, however, with every modeling effort, the effective application of Linear Programming requires good understanding of the underlying modeling assumptions, and a pertinent interpretation of the obtained analytical solutions. Therefore, in this section we discuss the details of the LP modeling and its underlying assumptions, by means of the following example.

Fri Jun 20 15:03:05 CDT 1997