Note for Computational Method - CM by Abhishek Apoorv

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Linear Programming: Theory and Applications Catherine Lewis May 11, 2008 1

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Contents 1 Introduction to Linear Programming 1.1 What is a linear program? . . . . . . . . . . . . . . . . . . . . . . 3 3 1.2 1.3 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Manipulating a Linear Programming Problem . . . . . . . . . . . 5 6 1.4 1.5 The Linear Algebra of Linear Programming . . . . . . . . . . . . Convex Sets and Directions . . . . . . . . . . . . . . . . . . . . . 7 8 2 Examples from Bazaraa et. al. and Winston 11 2.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 The Theory Behind Linear Programming 16 17 3.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.2 The General Representation Theorem . . . . . . . . . . . . . . . 19 4 An Outline of the Proof 20 5 Examples With Convex Sets and Extreme Points From Bazaara et. al. 22 6 Tools for Solving Linear Programs 23 6.1 Important Precursors to the Simplex Method . . . . . . . . . . . 23 7 The Simplex Method In Practice 25 8 What if there is no initial basis in the Simplex tableau? 28 8.1 8.2 The Two-Phase Method . . . . . . . . . . . . . . . . . . . . . . . The Big-M Method . . . . . . . . . . . . . . . . . . . . . . . . . . 29 31 9 Cycling 33 9.1 The Lexicographic Method . . . . . . . . . . . . . . . . . . . . . 34 9.2 Bland’s Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 9.3 9.4 Theorem from [2] . . . . . . . . . . . . . . . . . . . . . . . . . . . Which Rule to Use? . . . . . . . . . . . . . . . . . . . . . . . . . 37 39 10 Sensitivity Analysis 39 10.1 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 10.1.1 Sensitivity Analysis for a cost coefficient . . . . . . . . . . 1 40

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10.1.2 Sensitivity Analysis for a right-hand-side value . . . . . . 41 11 Case Study: Busing Children to School 41 11.1 The Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 11.2 The Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.1 Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 42 11.2.2 Objective Function . . . . . . . . . . . . . . . . . . . . . . 11.2.3 Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . 43 43 11.3 The Complete Program . . . . . . . . . . . . . . . . . . . . . . . 11.4 Road Construction and Portables . . . . . . . . . . . . . . . . . . 47 49 11.4.1 Construction . . . . . . . . . . . . . . . . . . . . . . . . . 11.4.2 Portable Classrooms . . . . . . . . . . . . . . . . . . . . . 49 50 11.5 Keeping Neighborhoods Together . . . . . . . . . . . . . . . . . . 11.6 The Case Revisited . . . . . . . . . . . . . . . . . . . . . . . . . . 55 56 11.6.1 Shadow Prices . . . . . . . . . . . . . . . . . . . . . . . . 56 11.6.2 The New Result . . . . . . . . . . . . . . . . . . . . . . . 57 12 Conclusion 57 2

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1 Introduction to Linear Programming Linear programming was developed during World War II, when a system with which to maximize the efficiency of resources was of utmost importance. New war-related projects demanded attention and spread resources thin. “Programming” was a military term that referred to activities such as planning schedules efficiently or deploying men optimally. George Dantzig, a member of the U.S. Air Force, developed the Simplex method of optimization in 1947 in order to provide an efficient algorithm for solving programming problems that had linear structures. Since then, experts from a variety of fields, especially mathematics and economics, have developed the theory behind “linear programming” and explored its applications [1]. This paper will cover the main concepts in linear programming, including examples when appropriate. First, in Section 1 we will explore simple properties, basic definitions and theories of linear programs. In order to illustrate some applications of linear programming, we will explain simplified “real-world” examples in Section 2. Section 3 presents more definitions, concluding with the statement of the General Representation Theorem (GRT). In Section 4, we explore an outline of the proof of the GRT and in Section 5 we work through a few examples related to the GRT. After learning the theory behind linear programs, we will focus methods of solving them. Section 6 introduces concepts necessary for introducing the Simplex algorithm, which we explain in Section 7. In Section 8, we explore the Simplex further and learn how to deal with no initial basis in the Simplex tableau. Next, Section 9 discusses cycling in Simplex tableaux and ways to counter this phenomenon. We present an overview of sensitivity analysis in Section 10. Finally, we put all of these concepts together in an extensive case study in Section 11. 1.1 What is a linear program? We can reduce the structure that characterizes linear programming problems (perhaps after several manipulations) into the following form: 3