Stressing the use of several software packages based on simplex method variations, this text teaches linear programming's four phases through actual practice. It shows how to decide whether LP models should be applied, set up appropriate models, use software to solve them, and examine solutions to a
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1 Introduction to Systems of Linear Equations (Linear Systems) and Related Properties of Matrices
The topics presented in this introductory chapter are used to formulate and analyze models for a wide variety of practical problems in subsequent chapters. If you have been exposed to systems of linear equations and matrix operations, you may wish merely to skim this chapter to pick up the notation and then proceed to Chapter 2, where linear programming is introduced. In this chapter we discuss solving linear systems by row reduction, matrix operations, rank, and linear independence. Two examples are used to motivate definitions and illustrate concepts.
1.1 Linear Systems
The discussion of linear systems begins with Example 1.1.
Example 1.1. Solve the linear system
2 x1 + 3 x2 = 3
−x1 + 2 x2 = 4.
Solve means to find all two-dimensional vectors
that satisfy both equations (1) and (2); here x is a pair x1, x2 of numbers written in column form. Except for the names x1 and x2 of the variables (unknowns), all the information that we need to solve the system is given by the parameters of the system; the parameters of the system are the coefficients of the variables and the numbers on the right-hand side of the equations. For Example 1.1, all of this information is contained in the augmented matrix
which is composed of the coefficient matrix
followed by the right-hand-side vector
. Examples 1.1 and 1.2 below illustrate how to solve the general m × n linear system:
Solving this linear system means finding all n-dimensional column vectors
that satisfy the equations (1), (2),…, (m). The ai,j’s and bi’s are (specific) numbers called parameters of the system. Again, except for the names x1, x2, …, xn o...
Table of contents
Cover
Half Title
Series Page
Title Page
Copyright Page
Dedication
Preface
Table of Contents
Original Half Title
Chapter 1. Introduction to Systems of Linear Equations (Linear Systems) and Related Properties of Matrices
Chapter 2. Introduction to Linear Programming
Chapter 3. Elementary Properties of the Feasible Set for an LP
