or - ORMS 1020 Operations Research with GNU Linear...

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ORMS 1020 Operations Research with GNU Linear Programming Kit Tommi Sottinen [email protected] www.uwasa.fi/ tsottine/orms1020 August 27, 2009
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Contents I Introduction 5 1 On Operations Research 6 1.1 What is Operations Research . . . . . . . . . . . . . . . . . . . 6 1.2 History of Operations Research* . . . . . . . . . . . . . . . . . 8 1.3 Phases of Operations Research Study . . . . . . . . . . . . . . . 10 2 On Linear Programming 13 2.1 Example towards Linear Programming . . . . . . . . . . . . . . 13 2.2 Solving Linear Programs Graphically . . . . . . . . . . . . . . . 15 3 Linear Programming with GNU Linear Programming Kit 21 3.1 Overview of GNU Linear Programming Kit . . . . . . . . . . . 21 3.2 Getting and Installing GNU Linear Programming Kit . . . . . . 23 3.3 Using glpsol with GNU MathProg . . . . . . . . . . . . . . . . 24 3.4 Advanced MathProg and glpsol * . . . . . . . . . . . . . . . . . 32 II Theory of Linear Programming 39 4 Linear Algebra and Linear Systems 40 4.1 Matrix Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 4.2 Solving Linear Systems . . . . . . . . . . . . . . . . . . . . . . . 48 4.3 Matrices as Linear Functions* . . . . . . . . . . . . . . . . . . . 50 5 Linear Programs and Their Optima 55 5.1 Form of Linear Program . . . . . . . . . . . . . . . . . . . . . . 55 5.2 Location of Linear Programs’ Optima . . . . . . . . . . . . . . 61 5.3 Karush–Kuhn–Tucker Conditions* . . . . . . . . . . . . . . . . 64 5.4 Proofs* . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
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CONTENTS 2 6 Simplex Method 68 6.1 Towards Simplex Algorithm . . . . . . . . . . . . . . . . . . . . 68 6.2 Simplex Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 75 7 More on Simplex Method 87 7.1 Big M Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 87 7.2 Simplex Algorithm with Non-Unique Optima . . . . . . . . . . 94 7.3 Simplex/Big M Checklist . . . . . . . . . . . . . . . . . . . . . 102 8 Sensitivity and Duality 103 8.1 Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 103 8.2 Dual Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 8.3 Primal and Dual Sensitivity . . . . . . . . . . . . . . . . . . . . 136 III Applications of Linear Programming 137 9 Data Envelopment Analysis 138 9.1 Graphical Introduction to Data Envelopment Analysis . . . . . 138 9.2 Charnes–Cooper–Rhodes Model . . . . . . . . . . . . . . . . . . 152 9.3 Charnes–Cooper–Rhodes Model’s Dual . . . . . . . . . . . . . . 160 9.4 Strengths and Weaknesses of Data Envelopment Analysis . . . 167 10 Transportation Problems 168 10.1 Transportation Algorithm . . . . . . . . . . . . . . . . . . . . . 168 10.2 Assignment Problem . . . . . . . . . . . . . . . . . . . . . . . . 179 10.3 Transshipment Problem . . . . . . . . . . . . . . . . . . . . . . 184 IV Non-Linear Programming 190 11 Integer Programming 191 11.1 Integer Programming Terminology . . . . . . . . . . . . . . . . 191 11.2 Branch-And-Bound Method . . . . . . . . . . . . . . . . . . . . 192 11.3 Solving Integer Programs with GNU Linear Programming Kit . 199
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Preface These lecture notes are for the course ORMS 1020 “Operations Research” for fall 2009 in the University of Vaasa. The notes are a slightly modified version of the notes for the fall 2008 course ORMS 1020 in the University of Vaasa. The chapters, or sections of chapters, marked with an asterisk (*) may be omitted — or left for the students to read on their own time — if time is scarce. The author wishes to acknowledge that these lecture notes are collected from the references listed in Bibliography, and from many other sources the author has forgotten. The author claims no originality , and hopes not to be sued for plagiarizing or for violating the sacred c laws. Vaasa August 27, 2009 T. S.
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Bibliography [1] Rodrigo Ceron: The GNU Linear Programming Kit, Part 1: Introduction to linear optimization , Web Notes, 2006. http://www-128.ibm.com/developerworks/linux/library/l-glpk1/ . [2] Matti Laaksonen: TMA.101 Operaatioanalyysi , Lecture Notes, 2005. http://lipas.uwasa.fi/ mla/orms1020/oa.html .
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