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Unformatted text preview: ORMS1020 Operations Research with GNU Linear Programming Kit Tommi Sottinen tommi.sottinen@uwasa.fi www.uwasa.fi/ ∼ tsottine/orms1020 August 27, 2009 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 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...
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