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SEN301 - SEN301 OPERATIONS RESEARCH I LECTURE...

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Y. İ lker Topcu, Ph.D. (www.ilkertopcu.info) SEN301 OPERATIONS RESEARCH I LECTURE NOTES (2011-2012) 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.yoneylem.itu.edu.tr
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2011-2012 Y. İ lker Topcu, Ph.D. (www.ilkertopcu.info) i 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 ................................................................................ 14 3.1.3 Diet Example .................................................................................................. 15 3.1.4 Post Office Example ...................................................................................... 16 3.1.5 Sailco Example .............................................................................................. 16 3.1.6 Customer Service Level Example ............................................................... 17 3.2 SOLVING LP .......................................................................................................... 19 3.2.1 LP Solutions: Four Cases ............................................................................. 19 3.2.2 The Graphical Solution ................................................................................. 19 3.2.3 The Simplex Algorithm .................................................................................. 24 3.2.4 The Big M Method ......................................................................................... 30 3.3 DUALITY ................................................................................................................. 33 3.3.1 Primal – Dual .................................................................................................. 33 3.3.2 Finding the Dual of an LP ............................................................................. 33 3.3.3 The Dual Theorem ......................................................................................... 35 3.3.4 Economic Interpretation ................................................................................ 35 3.4 SENSITIVITY ANALYSIS ..................................................................................... 37 3.4.1 Reduced Cost ................................................................................................. 37 3.4.2 Shadow Price ................................................................................................. 37 3.4.3 Conceptualization .......................................................................................... 37 3.4.4 Utilizing Lindo Output for Sensitivity ........................................................... 38 3.4.5 Some important equations ........................................................................... 39 3.4.6 Utilizing Simplex for Sensitivity .................................................................... 40 3.4.7 Utilizing Graphical Solution for Sensitivity ................................................. 42 3.4.8 Duality and Sensitivity Analysis ................................................................... 42 3.4.9 The %100 Rule .............................................................................................. 42
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2011-2012 Y. İ lker Topcu, Ph.D. (www.ilkertopcu.info) ii 3.5 DUAL SIMPLEX ALGORITHM ............................................................................ 43 3.6 REVISED SIMPLEX ALGORITHM ..................................................................... 44 3.7 COMPLEMENTARY SLACKNESS THEOREM ............................................... 51 4. TRANSPORTATION PROBLEMS .................................................................... 53 4.1 FORMULATING TRANSPORTATION PROBLEMS ....................................... 53 4.1.1 Formulating Balanced Transportation Problem ........................................ 54 4.1.2 Balancing an Unbalanced Transportation Problem ................................. 55 4.2 FINDING BFS FOR TRANSPORT’N PROBLEMS .......................................... 56 4.2.1 Northwest Corner Method ............................................................................ 57 4.2.2 Minimum Cost Method .................................................................................. 59 4.2.3 Vogel’s Method .............................................................................................. 60 4.3 THE TRANSPORTATION SIMPLEX METHOD ............................................... 61 4.4 SENSITIVITY ANALYSIS FOR TRANSPORTATION PROBLEMS .............. 65 4.5 TRANSSHIPMENT PROBLEMS ........................................................................ 68 4.6 ASSIGNMENT PROBLEMS ................................................................................ 70 4.6.1 LP Representation ......................................................................................... 71 4.6.2 Hungarian Method ......................................................................................... 71
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2011-2012 Y. İ lker Topcu, Ph.D. (www.ilkertopcu.info) 1 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). Another term which is used for this field is "management science" ("MS"). The Americans sometimes combine the terms OR and MS together and say " OR/MS " or "ORMS". Yet other terms sometimes used are "industrial engineering" ("IE"), "decision science" ("DS"), and “problem solving”. In recent years there has been a move towards a standardization upon a single term for the field, namely the term "OR". “Operations Research (Management Science) is a scientific approach to decision making that seeks to best design and operate a system, usually under conditions requiring the allocation of scarce resources.” A system is an organization of interdependent components that work together to accomplish the goal of the system.
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