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 . . . . . . . . . . . . . . . . . . . . 1.4.2 Phase II of the Simplex Method . . . . . . . . . . . . . . . . . . . ∗

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E-mail: sjmiller@math.brown.edu

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1.4.3 Summary of the Simplex Method . . . . . . . . . . . . . . . . . . . . Binary Integer Linear Programming . . . . . . . . . . . . . . . . . . . . . . .

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2 Versatility of Linear Programming 2.1 Binary Indicator Variables . . 2.2 Or Statements . . . . . . . . . 2.3 If-Then Statements . . . . . . 2.4 Truncation . . . . . . . . . . . 2.5 Minimums and Maximums . . 2.6 Absolute Values . . . . . . . .

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3 Removing Quadratic (and Higher Order) Terms 3.1 Problem Formulation . . . . . . . . . . . . . 3.2 Linearizing Certain Polynomials . . . . . . . 3.3 Example: Applications to Scheduling . . . . 3.4 Summary . . . . . . . . . . . . . . . . . . .

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1

Linear Programming

1.1 Introduction

We describe the ideas and applications of Linear Programming; our presentation is heavily inﬂuenced by Joel Franklin’s excellent book, Methods of Mathematical Economics [Fr]. We strongly recommend this book to anyone interested in a very readable presentation, replete with examples and references. Linear Programming is a generalization of Linear Algebra. It is capable of handling a variety of problems, ranging from ﬁnding...