operation research explanation

operation research explanation - OPERATIONS RESEARCH...

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OPERATIONS RESEARCH LECTURE NOTES Y. İ lker Topcu, Ph.D. Acknowledgements: We would like to acknowledge Prof. W.L. Winston's "Operations Research: Applications and Algorithms" and Prof. J.E. Beasley's lecture notes which greatly influence these notes. .. We retain responsibility for all errors and would love to hear from visitors of this site! Istanbul Technical University OR/MS team www.isl.itu.edu.tr/ya Dr. Y. İ lker Topcu (www.ilkertopcu.net)
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CONTENTS 1. INTRODUCTION TO OR. .................................................................................... 1 1.1 TERMINOLOGY ...................................................................................................... 1 1.2 THE METHODOLOGY OF OR ............................................................................. 1 1.3 HISTORY OF OR .................................................................................................... 2 2. BASIC OR CONCEPTS. ..................................................................................... 6 3. LINEAR PROGRAMMING. ............................................................................... 11 3.1 FORMULATING LP ............................................................................................... 13 3.1.1 Giapetto Example .......................................................................................... 13 3.1.2 Advertisement Example ................................................................................ 15 3.1.3 Diet Example .................................................................................................. 16 3.1.4 Post Office Example ...................................................................................... 17 3.1.5 Sailco Example .............................................................................................. 18 3.1.6 Customer Service Level Example ............................................................... 19 3.2 SOLVING LP .......................................................................................................... 20 3.2.1 LP Solutions: Four Cases ............................................................................. 20 3.2.2 The Graphical Solution (Minimization) ....................................................... 21 3.2.3 The Graphical Solution (Maximization) ...................................................... 22 3.2.4 The Simplex Algorithm .................................................................................. 23 3.2.5 Example illustrating Simplex (Dakota Furniture) ...................................... 24 3.2.6 Other Solution Cases by Simplex ............................................................... 29 3.2.7 The Big M Method ......................................................................................... 31 3.2.8 Example illustrating Big M (Oranj Juice) .................................................... 32 3.3 SENSITIVITY ANALYSIS ..................................................................................... 35 3.3.1 Reduced Cost ................................................................................................. 35 3.3.2 Shadow Price ................................................................................................. 35 3.3.3 Simple Example ............................................................................................. 36 3.3.4 Utilizing Lindo Output for Sensitivity ........................................................... 37 3.3.5 Additional Example (Utilizing Simplex for Sensitivity) .............................. 39 3.4 DUALITY ................................................................................................................. 42 3.4.1 The Dual Theorem ......................................................................................... 42 3.4.2 Finding the Dual of a Normal Max Problem .............................................. 42 3.4.3 Finding the Dual of a Normal Min Problem ............................................... 43 Y. İ lker Topcu, Ph.D. (www.ilkertopcu.net) i
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3.4.4 Finding the Dual of a Nonnormal Max Problem ........................................ 43 3.4.5 Finding the Dual of a Nonnormal Min Problem ......................................... 43 3.4.6 An Example for Economic Interpretation (Dakota Furniture) .................. 44 4. TRANSPORTATION PROBLEMS. ................................................................... 45 4.1 FORMULATING TRANSPORTATION PROBLEMS ....................................... 46 4.1.1 LP Representation ......................................................................................... 46 4.1.2 Powerco Example for Balanced Transportation Problem ....................... 46 4.1.3 Balancing an Unbalanced Transportation Problem ................................. 47 4.1.4 Modified Powerco Example for Excess Supply ........................................ 48 4.1.5 Modified Powerco Example for Unmet Demand ....................................... 49 4.2 FINDING BFS FOR TRANSPORT’N PROBLEMS .......................................... 50 4.2.1 Northwest Corner Method ............................................................................ 51 4.2.2 Minimum Cost Method .................................................................................. 53 4.2.3 Vogel’s Method .............................................................................................. 55 4.3 THE TRANSPORTATION SIMPLEX METHOD ............................................... 57 4.3.1 Steps of the Method ...................................................................................... 57 4.3.2 Solving Powerco Example by Transportation Simplex ............................ 58 4.4 TRANSSHIPMENT PROBLEMS ........................................................................ 61 4.4.1 Steps ................................................................................................................ 61 4.4.2 Kuruoglu Example ......................................................................................... 62 4.5 ASSIGNMENT PROBLEMS ................................................................................ 64 4.5.1 LP Representation ......................................................................................... 64 4.5.2 Hungarian Method ......................................................................................... 64 4.5.3 Flight Crew Example ..................................................................................... 66 5. INTEGER PROGRAMMING . ............................................................................ 68 5.1 FORMULATING IP ................................................................................................ 69 5.1.1 Budgeting Problems ...................................................................................... 69 5.1.2 Knapsack Problems ...................................................................................... 72 5.1.3 Fixed Charge Problems ................................................................................ 73 5.1.4 Set Covering Problems ................................................................................. 78 5.1.5 Either - Or Constraints .................................................................................. 80 5.1.6 If - Then Constraints ...................................................................................... 81 5.1.7 Traveling Salesperson Problems ................................................................ 82 Y. İ lker Topcu, Ph.D. (www.ilkertopcu.net) ii
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1. INTRODUCTION TO OR 1.1 TERMINOLOGY The British/Europeans refer to "operational research", the Americans to "operations research" - but both are often shortened to just "OR" (which is the term we will use).
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