Quantitative Management Forfurthervolumes: http://www.springer.com/series/10621 Gang Kou • Daji Ergu (cid:129) Yi Peng (cid:129) Yong Shi Data Processing for the AHP/ANP 123 GangKou DajiErgu SchoolofManagementandEconomics CollegeofElectricalandInformation UniversityofElectronicScience Engineering andTechnologyofChina SouthwestUniversityforNationalities NorthJiansheRoad YihuanRoad Chengdu Chengdu China,People’sRepublic China,People’sRepublic YiPeng YongShi SchoolofManagmentandEconomics UniversityofNebraskaatOmahaIS&T UniversityofElectronicScience DodgeStreet andTechnologyofChina OmahaNebraska NorthJiansheRoad USA Chengdu China,People’sRepublic ISSN2194-086X ISSN2194-0878 (electronic) ISBN978-3-642-29212-5 ISBN978-3-642-29213-2(eBook) DOI10.1007/978-3-642-29213-2 SpringerHeidelbergNewYorkDordrechtLondon LibraryofCongressControlNumber:2012944253 ©Springer-VerlagBerlinHeidelberg2013 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped.Exemptedfromthislegalreservationarebriefexcerptsinconnection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. 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Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) Abstract The positive reciprocal pairwise comparison matrix (PCM) is a well established techniqueandwidelyusedinmultiplecriteriadecisionmaking(MCDM)methods toperformpairwisecomparisonsandderivetheweightvectorsofbeingcompared items, especially in the analytical hierarchy (network) process (AHP/ANP). The PCM is also used to quantifythe qualitativeand/orintangibleattributes into mea- surablequantities.Consistencytest,inconsistentdataidentificationandadjustment, missing or uncertain data estimation, and sensitivity analysis of rank reversal are importantresearchissuesinthisfield.Althoughtheseissueshavebeenextensively studied, there is no universally accepted, simple and generalized data processing model to handle the above mentioned issues simultaneously. In this book, the maximum eigenvalue threshold method is proposed as the new consistency index forthe AHP/ANP. An inducedbias matrixmodel(IBMM) is proposedto identify andadjusttheinconsistentdata,andestimatethemissingoruncertaindata.Based on the IBMM, several questionnaire design improvement formats are proposed to quickly collect the data to make rapid and efficient decision making. Besides, the IBMM is further used to analyze the sensitivity of rank reversal issue when addingnewcriteriaoralternativesordeletingoldcriteriaoralternatives.Finally,two applicationsofIBMM,taskschedulingandresourceallocationincloudcomputing environment, and risk assessment and decision analysis, are used to illustrate the proposedIBMM. As anextendedmodelofIBMM, aninducedarithmeticaverage biasmatrix(IAABM)isdescribedindetailinfinalChapter. v Keywords The analytical hierarchy (network) process (AHP/ANP), Induced bias matrix model(IBMM),Dataprocessing,Consistencytest,Maximumeigenvaluethreshold, Inconsistencyidentificationandadjustment,Uncertainormissingvaluesestimation, Rankreversal,Inducedarithmeticaveragebiasmatrix(IAABM). vii Contents 1 Introduction .................................................................. 1 References..................................................................... 5 2 ANewConsistencyTestIndexfortheDataintheAHP/ANP .......... 11 2.1 BasicsoftheAHP/ANP ................................................ 11 2.1.1 ThePositiveReciprocalPairwiseComparisonMatrix........ 11 2.1.2 BasicsoftheAHP .............................................. 13 2.1.3 BasicsoftheANP .............................................. 14 2.2 ConsistencyTestIssueintheAHP/ANP............................... 16 2.2.1 AnalysisoftheConsistencyRatio(CR)Method.............. 16 2.2.2 TheIssuesofConsistencyTestintheAHP/ANP............. 17 2.3 TheNew ConsistencyIndex:MaximumEigenvalue ThresholdfortheAHP/ANP ........................................... 18 2.3.1 The Advantagesof Maximum Eigenvalue ThresholdforAHP/ANP....................................... 20 2.4 TheProcessesofDataConsistencyTestinAHP/ANP................ 21 2.5 IllustrativeExample..................................................... 25 References..................................................................... 27 3 IBMMforInconsistentDataIdentificationandAdjustment intheAHP/ANP.............................................................. 29 3.1 TheTheoremsofInducedBiasMatrixModel(IBMM) .............. 29 3.1.1 TheTheoreticalProofsofIBMM.............................. 31 3.2 IBMMforInconsistentDataIdentificationandAdjustment.......... 40 3.2.1 TheBasicsoftheInconsistencyIdentification andAdjustmentMethod........................................ 40 3.2.2 TheProcessesofInconsistencyIdentification andAdjustmentMethod........................................ 41 3.2.3 FastInconsistencyIdentificationandAdjustmentMethod ... 44 3.3 IllustrativeExamples.................................................... 50 3.3.1 IllustrativeExamplesforGeneralInconsistency IdentificationandAdjustmentMethod ........................ 50 ix x Contents 3.3.2 IllustrativeExamplesfor FastInconsistency IdentificationandAdjustmentMethod ........................ 62 References..................................................................... 64 4 IBMMforMissingDataEstimation....................................... 65 4.1 BasicsoftheIBMMforMissingDataEstimation .................... 65 4.2 TheProcessesofEstimatingMissingDatabytheIBMM ............ 66 4.3 ProofsoftheIBMMforIPCMinOrderThree........................ 68 4.4 IllustrativeExamples.................................................... 71 4.4.1 IllustrativeExamplesinOrderThree.......................... 71 4.4.2 IllustrativeExamplesinOrderFour ........................... 73 References..................................................................... 75 5 IBMMforQuestionnaireDesignImprovement .......................... 77 5.1 MotivationoftheResearch............................................. 77 5.2 ThePrinciplesofImprovingtheQuestionnaireDesign............... 79 5.3 IllustrativeExample..................................................... 83 References..................................................................... 91 6 IBMMforRankReversal................................................... 93 6.1 RankReversalIssueintheAHP/ANP................................. 93 6.2 SensitivityAnalysisofRankReversalbytheIBMM ................. 94 6.3 IllustrativeExamples.................................................... 94 References..................................................................... 105 7 ApplicationsofIBMM....................................................... 107 7.1 Task SchedulingandResourceAllocationin Cloud ComputingEnvironmentbytheIBMM................................ 107 7.1.1 ResourceAllocationinCloudComputing..................... 108 7.1.2 Task-OrientedResourceAllocationinCloudComputing.... 109 7.1.3 IllustrativeExamples ........................................... 110 7.2 RiskAssessmentandDecisionAnalysisbytheIBMM............... 116 7.2.1 BackgroundofRiskAssessmentandDecisionAnalysis..... 116 7.2.2 IllustrativeExamples ........................................... 116 References..................................................................... 126 8 InducedArithmeticAverageBiasMatrixModel(IAABMM).......... 129 8.1 TheTheoremofIAABMM............................................. 129 8.2 TheInconsistencyIdentificationProcessesofIAABMM............. 130 8.3 TheEstimatingFormulaofInconsistencyAdjustment................ 132 8.4 IllustrativeExamples.................................................... 133 References..................................................................... 135 Index............................................................................... 137