Lecture Introduction to Management Science with Spreadsheets: Chapter 12 - Stevenson, Ozgur

Chapter 12 Decision theory, after completing this chapter, you should be able to: Outline the characteristics of a decision theory approach to decision making; describe and give examples of decisions under certainty, risk, and complete uncertainty; make decisions using maximin, maximax, minimax regret, Hurwicz, equally likely, and expected value criteria and use Excel to solve problems involving these techniques;...

Introduction to Management Science with Spreadsheets Stevenson and Ozgur First Edition Part Probabilistic Decision Models Chapter 12 Markov Analysis McGraw­Hill/Irwin Copyright © 2007 by The McGraw­Hill Companies, Inc. All rights reserved Learning Objectives After completing this chapter, you should be able to: Give examples of systems that may lend themselves to be analyzed by a Markov model Explain the meaning of transition probabilities Describe the kinds of system behaviors that Markov analysis pertains to Use a tree diagram to analyze system behavior Use matrix multiplication to analyze system behavior Use an algebraic method to solve for steady-state probabilities Copyright © 2007 The McGraw­Hill  McGraw­ Companies. All rights reserved.   Hill/Irwin  12–2 Learning Objectives (cont’d) After completing this chapter, you should be able to: Analyze absorbing states, namely accounts receivable, using a Markov model List the assumptions of a Markov model Use Excel to solve various problems pertaining to a Markov model Copyright © 2007 The McGraw­Hill  Companies. All rights reserved.   McGraw­ Hill/Irwin  12–3 Characteristics Characteristics of of aa Markov Markov System System It will operate or exist for a number of periods In each period, the system can assume one of a number of states or conditions The states are both mutually exclusive and collectively exhaustive System changes between states from period to period can be described by transition probabilities, which remain constant The probability of the system being in a given state in a particular period depends only on its state in the preceding period and the transition probabilities It is independent of all earlier periods Copyright © 2007 The McGraw­Hill  McGraw­ Companies. All rights reserved.   Hill/Irwin  12–4 Markov Markov Analysis: Analysis: Assumptions Assumptions • Markov Analysis Assumptions – The probability that an item in the system either will change from one state (e.g., Airport A) to another or remain in its current state is a function of the transition probabilities only – The transition probabilities remain constant – The system is a closed one; there will be no arrivals to the system or exits from the system Copyright © 2007 The McGraw­Hill  Companies. All rights reserved.   McGraw­ Hill/Irwin  12–5 Table Table12–1 12–1 Examples ExamplesofofSystems SystemsThat ThatMay MayBe BeDescribed Describedas asMarkov Markov Copyright © 2007 The McGraw­Hill  Companies. All rights reserved.   McGraw­ Hill/Irwin  12–6 Table Table12–2 12–2 Transition TransitionProbabilities Probabilitiesfor forCar CarRental RentalExample Example Copyright © 2007 The McGraw­Hill  Companies. All rights reserved.   McGraw­ Hill/Irwin  12–7 System System Behavior Behavior • Both the long-term behavior and the short-term behavior of a system are completely determined by the system’s transition probabilities • Short-term behavior is solely dependent on the system’s state in the current period and the transition probabilities • The long-run proportions are referred to as the steady-state proportions, or probabilities, of the system Copyright © 2007 The McGraw­Hill  Companies. All rights reserved.   McGraw­ Hill/Irwin  12–8 Figure Figure12–1 12–1 Expected ExpectedProportion ProportionofofPeriod Period00Rentals RentalsReturned ReturnedtotoAirport AirportAA Copyright © 2007 The McGraw­Hill  Companies. All rights reserved.   McGraw­ Hill/Irwin  12–9 Methods Methods of of System System Behavior Behavior Analysis Analysis • Tree Diagram – A visual portrayal of a system’s transitions composed of a series of branches, which represent the possible choices at each stage (period) and the conditional probabilities of each choice being selected • Matrix Multiplication – Assumes that “current” state proportions are equal to the product of the proportions in the preceding period multiplied by the matrix of transition probabilities – Involves the multiplication of the “current” proportions, which is referred to as a probability vector, by the transition matrix Copyright © 2007 The McGraw­Hill  Companies. All rights reserved.   McGraw­ Hill/Irwin  12– 10 Figure Figure12–8 12–8 Tree TreeDiagram Diagramfor forExample Example12-5, 12-5,Starting Startingfrom fromYY (Initial (InitialState State=Y) =Y) Copyright © 2007 The McGraw­Hill  Companies. All rights reserved.   McGraw­ Hill/Irwin  12– 23 Exhibit Exhibit12–3 12–3 Worksheet Worksheetfor forthe theMarkov MarkovAnalysis Analysisofofthe theAcorn AcornUniversity University Problem Problem Copyright © 2007 The McGraw­Hill  Companies. All rights reserved.   McGraw­ Hill/Irwin  12– 24 Exhibit Exhibit12–4 12–4 Second SecondWorksheet Worksheetfor forthe theMarkov MarkovAnalysis Analysisofofthe theAcorn Acorn University UniversityProblem Problem Copyright © 2007 The McGraw­Hill  Companies. All rights reserved.   McGraw­ Hill/Irwin  12– 25 Exhibit Exhibit12–5 12–5 Third ThirdWorksheet Worksheetfor forthe theMarkov MarkovAnalysis Analysisand andSteady-State Steady-State Probabilities Probabilitiesofofthe theAcorn AcornUniversity UniversityProblem Problem Copyright © 2007 The McGraw­Hill  Companies. All rights reserved.   McGraw­ Hill/Irwin  12– 26 Exhibit Exhibit12–6 12–6 Parameters ParametersSpecification SpecificationScreen Screenfor forthe theAcorn AcornUniversity University Problem Problem Copyright © 2007 The McGraw­Hill  Companies. All rights reserved.   McGraw­ Hill/Irwin  12– 27 Cyclical, Cyclical, Transient, Transient, and and Absorbing Absorbing Systems Systems • Cyclical system – A system that has a tendency to move from state to state in a definite pattern or cycle • Transient system – A system in which there is at least one state—the transient state—where once a system leaves it, the system will never return to it • Absorbing system – A system that gravitates to one or more states—once a member of a system enters an absorbing state, it becomes trapped and can never exit that state McGraw­ Copyright © 2007 The McGraw­Hill  Companies. All rights reserved.   Hill/Irwin  12– 28 Table Table12–6 12–6 An AnExample ExampleofofaaCyclical CyclicalSystem System Table Table12 12–7 –7 An AnExample ExampleofofSystem Systemwith withaaTransient TransientState State Table Table12 12–8 –8 An AnExample ExampleofofaaSystem Systemwith withAbsorbing AbsorbingStates States Copyright © 2007 The McGraw­Hill  Companies. All rights reserved.   McGraw­ Hill/Irwin  12– 29 Figure Figure12–9 12–9 Probability ProbabilityTransition TransitionDiagrams Diagramsfor forthe theTransition TransitionMatrices Matrices Given GivenininTables Tables12-6, 12-6,12-7, 12-7,and and12-8 12-8 Copyright © 2007 The McGraw­Hill  Companies. All rights reserved.   McGraw­ Hill/Irwin  12– 30 Exhibit Exhibit12–7 12–7 Excel ExcelWorksheet Worksheetfor forExample Example12-10: 12-10:The TheAcorn AcornHospital Hospital Absorbing AbsorbingState StateProblem Problem Copyright © 2007 The McGraw­Hill  Companies. All rights reserved.   McGraw­ Hill/Irwin  12– 31 Table Table12–9 12–9 Answers AnswerstotoExample Example12-10, 12-10,Part Part11aathrough throughf f Copyright © 2007 The McGraw­Hill  Companies. All rights reserved.   McGraw­ Hill/Irwin  12– 32 Exhibit Exhibit12–8 12–8 Worksheet Worksheetfor forThe TheMarkov MarkovAnalysis AnalysisofofSolved SolvedProblem Problem44 Copyright © 2007 The McGraw­Hill  Companies. All rights reserved.   McGraw­ Hill/Irwin  12– 33 Exhibit Exhibit12–9 12–9 Solver SolverParameters ParametersSpecification SpecificationScreen Screenfor forthe theSteady-State Steady-State Calculations Calculationsfor forSolved SolvedProblem Problem44 Copyright © 2007 The McGraw­Hill  Companies. All rights reserved.   McGraw­ Hill/Irwin  12– 34 Exhibit Exhibit12–10 12–10 Worksheet Worksheetfor forthe theSteady-State Steady-StateCalculations CalculationsofofSolved Solved Problem Problem55 Copyright © 2007 The McGraw­Hill  Companies. All rights reserved.   McGraw­ Hill/Irwin  12– 35 Exhibit Exhibit12–11 12–11 Solver SolverParameters ParametersSpecification SpecificationScreen Screenfor forthe theSteady-State Steady-State Calculations Calculationsfor forSolved SolvedProblem Problem55 Copyright © 2007 The McGraw­Hill  Companies. All rights reserved.   McGraw­ Hill/Irwin  12– 36 Exhibit Exhibit12–12 12–12 Excel ExcelWorksheet Worksheetfor forSolved SolvedProblem Problem6:6:Accounts AccountsReceivable Receivable —Absorbing —AbsorbingState StateProblem Problem Copyright © 2007 The McGraw­Hill  Companies. All rights reserved.   McGraw­ Hill/Irwin  12– 37 ... Companies. All rights reserved.   McGraw­ Hill/Irwin ? ?12? ?? 20 Table Table12–5 12? ??5 Transition TransitionMatrix Matrixfor forExamples Examples1 2-5 , 1 2- 5 ,1 2- 6, 1 2- 6,and and1 2-7 1 2- 7 Copyright © 2007 The McGraw­Hill ... Figure12–9 12? ??9 Probability ProbabilityTransition TransitionDiagrams Diagramsfor forthe theTransition TransitionMatrices Matrices Given GivenininTables Tables1 2-6 , 1 2- 6 ,1 2- 7, 1 2- 7,and and1 2-8 1 2- 8... McGraw­ Hill/Irwin ? ?12? ?? 13 Table Table12–3 12? ??3 Period-by-Period Period-by-PeriodProportions Proportionsfor forthe theRental RentalExample, Example,and andthe the Steady-State Steady-StateProportions