Optimization in Practice with MATLAB® For Engineering Students and Professionals

Overview

Linear programming (LP) is a technique for optimizing a linear objective function, subject to linear equality and linear inequality constraints. Linear programming is an important field of optimization for several reasons (Refs. [1, 2]). Many practical problems in engineering and operations research can be expressed as linear programming problems. Linear programming problems can be solved in an easy, fast, and reliable manner.

Linear programming was first used in the field of economics, and still remains popular in the areas of management, economics, finance, and engineering. During World War II, George Dantzig of the U.S. Air Force used LP techniques for planning problems. He invented the Simplex method, which is one of the most popular methods for solving LP problems.

Linear programming is a well developed subject in operations research, and several references that focus on linear programming alone are available [3, 4, 5, 6]. The reader is encouraged to consult these references for a more detailed discussion of linear programming.

In this chapter, the basics of linear programming problems will be studied. First, an introduction of the basic terminology and important concepts in LP problems will be presented in Sec. 11.2. In Sec. 11.3, the graphical approach for solving LP problems will be discussed, and four types of possible solutions for an LP problem will be introduced. Details on the Matlab linear programming solver are presented in Sec. 11.4. Sections 11.5 and 11.6 discuss the various concepts involved in the Simplex method. In Sec. 11.7, the notion of duality and interior point methods are briefly discussed. Section 11.8 concludes the chapter with a summary.

Basics of Linear Programming

In this section, some basic terminology and definitions in linear programming will be discussed.

A generic linear programming problem consisting of linear equality and linear inequality constraints is given as where z is the linear objective function to be minimized;