Linear Programming
An Introduction to Linear Programming
Steven J. Miller∗ March 31, 2007
Mathematics Department Brown University 151 Thayer Street Providence, RI 02912
Abstract We describe Linear Programming, an important generalization of Linear Algebra. Linear Programming is used to successfully model numerous real world situations, ranging from scheduling airline routes to shipping oil from refineries to cities to finding inexpensive diets capable of meeting the minimum daily requirements. In many of these problems, the number of variables and constraints are so large that it is not enough to merely to know there is solution; we need some way of finding it (or at least a close approximation to it) in a reasonable amount of time. We describe the types of problems Linear Programming can handle and show how we can solve them using the simplex method. We discuss generalizations to Binary Integer Linear Programming (with an example of a manager of an activity hall), and conclude with an analysis of versatility of Linear Programming and the types of problems and constraints which can be handled linearly, as well as some brief comments about its generalizations (to handle situations with quadratic constraints).
Contents
1 Linear Programming 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 The Canonical Linear Programming Problem . . . . . . . . . . . . . . . . 1.2.1 Statement of the Diet Problem . . . . . . . . . . . . . . . . . . . . 1.2.2 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Solution to the Diet Problem . . . . . . . . . . . . . . . . . . . . . 1.3 Duality and Basic Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Basic Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Solving the Canonical Linear Programming Problem: The Simplex Method 1.4.1 Phase I of the Simplex Method . . . . . . .
Steven J. Miller∗ March 31, 2007
Mathematics Department Brown University 151 Thayer Street Providence, RI 02912
Abstract We describe Linear Programming, an important generalization of Linear Algebra. Linear Programming is used to successfully model numerous real world situations, ranging from scheduling airline routes to shipping oil from refineries to cities to finding inexpensive diets capable of meeting the minimum daily requirements. In many of these problems, the number of variables and constraints are so large that it is not enough to merely to know there is solution; we need some way of finding it (or at least a close approximation to it) in a reasonable amount of time. We describe the types of problems Linear Programming can handle and show how we can solve them using the simplex method. We discuss generalizations to Binary Integer Linear Programming (with an example of a manager of an activity hall), and conclude with an analysis of versatility of Linear Programming and the types of problems and constraints which can be handled linearly, as well as some brief comments about its generalizations (to handle situations with quadratic constraints).
Contents
1 Linear Programming 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 The Canonical Linear Programming Problem . . . . . . . . . . . . . . . . 1.2.1 Statement of the Diet Problem . . . . . . . . . . . . . . . . . . . . 1.2.2 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Solution to the Diet Problem . . . . . . . . . . . . . . . . . . . . . 1.3 Duality and Basic Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Basic Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Solving the Canonical Linear Programming Problem: The Simplex Method 1.4.1 Phase I of the Simplex Method . . . . . . .
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