数据、模型与决策（第12版）课件 第9章多准则决策

MulticriteriaDecisionMakingChapter9Copyright©2016PearsonEducation,Inc.GoalProgrammingGraphicalInterpretationofGoalProgrammingComputerSolutionofGoalProgrammingProblemswithQMforWindowsandExcelTheAnalyticalHierarchyProcessScoringModelsChapterTopicsCopyright©2016PearsonEducation,Inc.Studyofproblemswithseveralcriteria,i.e.,multiplecriteria,insteadofa

singleobjectivewhenmakingadecision.Threetechniquesdiscussed:goalprogramming,the

analyticalhierarchyprocess

and

scoringmodels.Goalprogrammingisavariationoflinearprogrammingconsideringmorethanoneobjective(goals)intheobjectivefunction.Theanalyticalhierarchyprocessdevelopsascoreforeachdecisionalternativebasedoncomparisonsofeachunderdifferentcriteriareflectingthedecisionmakers’preferences.Scoringmodelsarebasedonarelativelysimpleweightedscoringtechnique.OverviewCopyright©2016PearsonEducation,Inc.BeaverCreekPotteryCompanyExample:MaximizeZ=$40x1+50x2subjectto: 1x1+2x2

40hoursoflabor 4x1+3x2

120poundsofclay x1,x2

0Where:x1=numberofbowlsproducedx2=numberofmugsproducedGoalProgrammingExampleProblemData(1of2)Copyright©2016PearsonEducation,Inc.Addingobjectives(goals)inorderofimportance,the

company……doesnotwanttousefewerthan40hoursoflaborperday.…wouldliketoachieveasatisfactoryprofitlevelof$1,600perday.…prefersnottokeepmorethan120poundsofclayonhandeachday.…wouldliketominimizetheamountofovertime.

GoalProgrammingExampleProblemData(2of2)Copyright©2016PearsonEducation,Inc.Allgoalconstraintsareequalitiesthatincludedeviationalvariablesd-andd+.Apositivedeviationalvariable(d+)istheamountbywhichagoallevelisexceeded.Anegativedeviationvariable(d-)istheamountbywhichagoallevelisunderachieved.Atleastoneorbothdeviationalvariablesinagoalconstraintmustequalzero.Theobjectivefunctionseekstominimizethedeviationfromtherespectivegoalsintheorderofthegoalpriorities.GoalProgrammingGoalConstraintRequirementsCopyright©2016PearsonEducation,Inc.Laborgoal: x1+2x2+d1--d1+=40(hours/day)Profitgoal: 40x1+50x2+d2

--d2

+=1,600($/day)Materialgoal: 4x1+3x2+d3

--d3

+=120(lbsofclay/day)GoalProgrammingModelFormulationGoalConstraints(1of3)Copyright©2016PearsonEducation,Inc.Laborgoalsconstraint(priority1-lessthan40hourslabor;priority4-minimumovertime):MinimizeP1d1-,P4d1+Addprofitgoalconstraint(priority2-achieveprofitof$1,600):MinimizeP1d1-,P2d2-,P4d1+Addmaterialgoalconstraint(priority3-avoidkeepingmorethan120poundsofclayonhand):MinimizeP1d1-,P2d2-,P3d3+,P4d1+GoalProgrammingModelFormulationObjectiveFunction(2of3)Copyright©2016PearsonEducation,Inc.CompleteGoalProgrammingModel:MinimizeP1d1-,P2d2-,P3d3+,P4d1+

subjectto: x1+2x2+d1--d1+=40 (labor) 40x1+50x2+d2--d2+=1,600 (profit) 4x1+3x2+d3--d3+=120 (clay)x1,x2,d1-,d1+,d2-,d2+,d3-,d3+

0GoalProgrammingModelFormulationCompleteModel(3of3)Copyright©2016PearsonEducation,Inc.Changingfourth-prioritygoal“limitovertimeto10hours”insteadofminimizingovertime:d1-+d4--d4+=10minimizeP1d1-,P2d2-,P3d3+,P4d4+Additionofafifth-prioritygoal-“importanttoachievethegoalformugs”:x1+d5-=30bowlsx2+d6-=20mugsminimizeP1d1-,P2d2-,P3d3+,P4d4+,4P5d5-+5P5d6-GoalProgrammingAlternativeFormsofGoalConstraints(1of2)Twoormoregoalsatthesameprioritylevelcanbeassignedweightstoindicatetheirrelativeimportance.Copyright©2016PearsonEducation,Inc.GoalProgrammingAlternativeFormsofGoalConstraints(2of2)CompleteModelwithAddedNewGoals:MinimizeP1d1-,P2d2-,P3d3+,P4d4+,4P5d5-+5P5d6-subjectto: x1+2x2+d1--d1+=40 40x1+50x2+d2--d2+=1,600 4x1+3x2+d3--d3+=120 d1++d4--d4+=10 x1+d5-=30 x2+d6-=20 x1,x2,d1-,d1+,d2-,d2+,d3-,d3+,d4-,d4+,d5-,d6-

0Copyright©2016PearsonEducation,Inc.Figure9.1Goalconstraints

GoalProgrammingGraphicalInterpretation(1of6)MinimizeP1d1-,P2d2-,P3d3+,P4d1+

subjectto:x1+2x2+d1--d1+=4040x1+50x2+d2--d2+=1,6004x1+3x2+d3--d3+=120x1,x2,d1-,d1+,d2-,d2+,d3-,d3+

0Copyright©2016PearsonEducation,Inc.Figure9.2Thefirst-prioritygoal:minimized1-MinimizeP1d1-,P2d2-,P3d3+,P4d1+

subjectto:x1+2x2+d1--d1+=4040x1+50x2+d2--d2+=1,6004x1+3x2+d3--d3+=120x1,x2,d1-,d1+,d2-,d2+,d3-,d3+

0GoalProgrammingGraphicalInterpretation(2of6)Copyright©2016PearsonEducation,Inc.Figure9.3Thesecond-prioritygoal:Minimized2-GoalProgrammingGraphicalInterpretation(3of6)MinimizeP1d1-,P2d2-,P3d3+,P4d1+

subjectto:x1+2x2+d1--d1+=4040x1+50x2+d2--d2+=1,6004x1+3x2+d3--d3+=120x1,x2,d1-,d1+,d2-,d2+,d3-,d3+

0Copyright©2016PearsonEducation,Inc.Figure9.4Thethird-prioritygoal:minimized3+GoalProgrammingGraphicalInterpretation(4of6)MinimizeP1d1-,P2d2-,P3d3+,P4d1+

subjectto:x1+2x2+d1--d1+=4040x1+50x2+d2--d2+=1,6004x1+3x2+d3--d3+=120x1,x2,d1-,d1+,d2-,d2+,d3-,d3+

0Copyright©2016PearsonEducation,Inc.Figure9.5TheFourth-PriorityGoal:Minimized1+GoalProgrammingGraphicalInterpretation(5of6)MinimizeP1d1-,P2d2-,P3d3+,P4d1+

subjectto:x1+2x2+d1--d1+=4040x1+50x2+d2--d2+=1,6004x1+3x2+d3--d3+=120x1,x2,d1-,d1+,d2-,d2+,d3-,d3+

0Copyright©2016PearsonEducation,Inc.Goalprogrammingsolutionsdonotalwaysachieveallgoalsandtheyarenotoptimal;

theyachievethebestormostsatisfactorysolutionpossible.MinimizeP1d1-,P2d2-,P3d3+,P4d1+

subjectto: x1+2x2+d1--d1+=40 40x1+50x2+d2--d2+=1,600 4x1+3x2+d3--d3+=120 x1,x2,d1-,d1+,d2-,d2+,d3-,d3+

0

Solution: x1=15bowls x2=20mugs d1-=15hoursGoalProgrammingGraphicalInterpretation(6of6)Copyright©2016PearsonEducation,Inc.Exhibit9.1MinimizeP1d1-,P2d2-,P3d3+,P4d1+

subjectto:x1+2x2+d1--d1+=4040x1+50x2+d2--d2+=1,6004x1+3x2+d3--d3+=120x1,x2,d1-,d1+,d2-,d2+,d3-,d3+

0GoalProgrammingComputerSolutionUsingQMforWindows(1of3)Copyright©2016PearsonEducation,Inc.Exhibit9.2GoalProgrammingComputerSolutionUsingQMforWindows(2of3)Copyright©2016PearsonEducation,Inc.Exhibit9.3GoalProgrammingComputerSolutionUsingQMforWindows(3of3)Copyright©2016PearsonEducation,Inc.Exhibit9.4GoalProgrammingComputerSolutionUsingExcel(1of3)GoalconstraintforcellG5Decisionvariables—B10:B11Deviationalvariables—E5:F7Copyright©2016PearsonEducation,Inc.Exhibit9.5GoalProgrammingComputerSolutionUsingExcel(2of3)Minimizedeviationalvariableforfirst-prioritygoalinE5DecisionanddeviationalvariablesGoalconstraintsCopyright©2016PearsonEducation,Inc.Exhibit9.6GoalProgrammingComputerSolutionUsingExcel(3of3)Copyright©2016PearsonEducation,Inc.MinimizeP1d1-,P2d2-,P3d3+,P4d4+,4P5d5-+5P5d6-subjectto: x1+2x2+d1--d1+=40 40x1+50x2+d2--d2+=1,600 4x1+3x2+d3--d3+=120 d1++d4--d4+=10 x1+d5-=30 x2+d6-=20 x1,x2,d1-,d1+,d2-,d2+,d3-,d3+,d4-,d4+,d5-,d6-

0GoalProgrammingSolutionforAlteredProblemUsingExcel(1of6)Thismodelincludesgoalsforovertimeandmaximumstoragelevelsforbowlsandmugs.Copyright©2016PearsonEducation,Inc.Exhibit9.7GoalProgrammingSolutionforAlteredProblemUsingExcel(2of6)GoalconstraintforlaborGoalconstraintforovertime;=F5+E8-F8=C9*B13+E9Exhibit9.8GoalProgrammingSolutionforAlteredProblemUsingExcel(3of6)Copyright©2016PearsonEducation,Inc.GoalProgrammingSolutionforAlteredProblemUsingExcel(4of6)Exhibit9.9Firsttwoprioritygoals,minimizingand,achievedThird-prioritygoaltominimizenotachievedCopyright©2016PearsonEducation,Inc.Exhibit9.10GoalProgrammingSolutionforAlteredProblemUsingExcel(5of6)First-andsecond-prioritygoalsachieved;addE5=0andE6=0Copyright©2016PearsonEducation,Inc.Exhibit9.11GoalProgrammingSolutionforAlteredProblemUsingExcel(6of6)Fourth-prioritygoaltominimizeovertime,,notachievedCopyright©2016PearsonEducation,Inc.Methodforrankingseveraldecisionalternativesandselectingthebestonewhenthedecisionmakerhasmultipleobjectives,orcriteria,onwhichtobasethedecision.Thedecisionmakermakesadecisionbasedonhowthealternativescompareaccordingtoseveralcriteria.Thedecisionmakerwillselectthealternativethatbestmeetsthedecisioncriteria.Aprocessfordevelopinganumericalscoretorankeachdecisionalternativebasedonhowwellthealternativemeetsthedecisionmaker’scriteria.AnalyticalHierarchyProcess(AHP)OverviewCopyright©2016PearsonEducation,Inc.SouthcorpDevelopmentCompanyshoppingmallsiteselection.Threepotentialsites:AtlantaBirminghamCharlotte.Criteriaforsitecomparisons:Customermarketbase.IncomelevelInfrastructureAnalyticalHierarchyProcessExampleProblemStatementCopyright©2016PearsonEducation,Inc.Topofthehierarchy:theobjective(selectthebestsite).Secondlevel:howthefourcriteriacontributetotheobjective.Thirdlevel:howeachofthethreealternativescontributestoeachofthefourcriteria.AnalyticalHierarchyProcessHierarchyStructureCopyright©2016PearsonEducation,Inc.Mathematicallydeterminepreferencesforsiteswithrespecttoeachcriterion.Mathematicallydeterminepreferencesforcriteria(rankorderofimportance).Combinethesetwosetsofpreferencestomathematicallyderiveacompositescoreforeachsite.Selectthesitewiththehighestscore.AnalyticalHierarchyProcessGeneralMathematicalProcessCopyright©2016PearsonEducation,Inc.Inapairwisecomparison,twoalternativesarecomparedaccordingtoacriterionandoneispreferred.Apreferencescaleassignsnumericalvaluestodifferentlevelsofperformance.AnalyticalHierarchyProcessPairwiseComparisons(1of2)Copyright©2016PearsonEducation,Inc.Table9.1PreferencescaleforpairwisecomparisonsAnalyticalHierarchyProcessPairwiseComparisons(2of2)Copyright©2016PearsonEducation,Inc.

IncomeLevelInfrastructureTransportationABCAnalyticalHierarchyProcessPairwiseComparisonMatrix Apairwisecomparisonmatrixsummarizesthepairwisecomparisonsforacriteria.Copyright©2016PearsonEducation,Inc.AnalyticalHierarchyProcessDevelopingPreferencesWithinCriteria(1of3)Insynthesization,decisionalternativesareprioritizedwithineachcriterionCopyright©2016PearsonEducation,Inc.Table9.2ThenormalizedmatrixwithrowaveragesAnalyticalHierarchyProcessDevelopingPreferencesWithinCriteria(2of3)TherowaveragevaluesrepresentthepreferencevectorCopyright©2016PearsonEducation,Inc.Table9.3CriteriapreferencematrixAnalyticalHierarchyProcessDevelopingPreferencesWithinCriteria(3of3)Preferencevectorsforothercriteriaarecomputedsimilarly,resultinginthepreferencematrixCopyright©2016PearsonEducation,Inc.Table9.4NormalizedmatrixforcriteriawithrowaveragesAnalyticalHierarchyProcessRankingtheCriteria(1of2)PairwiseComparisonMatrix:Copyright©2016PearsonEducation,Inc.AnalyticalHierarchyProcessRankingtheCriteria(2of2)PreferenceVectorforCriteria:

Market Income Infrastructure TransportationCopyright©2016PearsonEducation,Inc.OverallScore:AnalyticalHierarchyProcessDevelopinganOverallRankingSiteAscore=.1993(.5012)+.6535(.2819)+.0860(.1790)+.0612(.1561)=.3091SiteBscore=.1993(.1185)+.6535(.0598)+.0860(.6850)+.0612(.6196)=.1595SiteCscore=.1993(.3803)+.6535(.6583)+.0860(.1360)+.0612(.2243)=.5314OverallRanking:Copyright©2016PearsonEducation,Inc.AnalyticalHierarchyProcessSummaryofMathematicalStepsDevelopapairwisecomparisonmatrixforeachdecisionalternativeforeachcriteria.SynthesizationSumeachcolumnvalueofthepairwisecomparisonmatrices.Divideeachvalueineachcolumnbyitscolumnsum.Averagethevaluesineachrowofthenormalizedmatrices.Combinethevectorsofpreferencesforeachcriterion.Developapairwisecomparisonmatrixforthecriteria.Computethenormalizedmatrix.Developthepreferencevector.ComputeanoverallscoreforeachdecisionalternativeRankthedecisionalternatives.Copyright©2016PearsonEducation,Inc.AnalyticalHierarchyProcess:Consistency(1of3)ConsistencyIndex(CI):CheckforconsistencyandvalidityofmultiplepairwisecomparisonsExample:Southcorp’sconsistencyinthepairwisecomparisonsofthe4 siteselectioncriteriaX(1)(0.1993)+(1/5)(0.6535)+(3)(0.0860)+(4)(0.0612)=0.8328(5)(0.1993)+(1)(0.6535)+(9)(0.0860)+(7)(0.0612)=2.8524(1/3)(0.1993)+(1/9)(0.6535)+(1)(0.0860)+(2)(0.0612)=0.3474(¼)(0.1993)+(1/7)(0.6535)+(½)(0.0860)+(1)(0.0612)=0.2473MarketIncomeInfrastructureTransportationCriteriaMarket15-Jan340.1993Income51970.6535Infrastructure3-Jan9-Jan120.086Transportation4-Jan7-Jan2-Jan10.0612Copyright©2016PearsonEducation,Inc.AnalyticalHierarchyProcess:Consistency(2of3)Step2:Divideeachvaluebythecorrespondingweightfromthe preferencevectorandcomputetheaverage 0.8328/0.1993=4.1786 2.8524/0.6535=4.3648 0.3474/0.0860=4.0401 0.2473/0.0612=4.0422

16.257

Average=16.257/4 =4.1564Step3:CalculatetheConsistencyIndex(CI) CI=(Average–n)/(n-1),wherenisnumberofitemscomparedCI=(4.1564-4)/(4-1)=0.0521

(CI=0indicatesperfectconsistency)Copyright©2016PearsonEducation,Inc.AnalyticalHierarchyProcess:Consistency(3of3)Step4:ComputetheRatioCI/RI whereRIisarandomindexvalueobtainedfromTable9.5Table9.5RandomIndexValuesfornItemsBeingCompared CI/RI=0.0521/0.90=0.0580

Note:DegreeofconsistencyissatisfactoryifCI/RI<0.10n2345678910RI00.580.901.121.241.321.411.451.51Copyright©2016PearsonEducation,Inc.Exhibit9.12AnalyticalHierarchyProcessExcelSpreadsheets(1of4)Clickon“Format,”then“Cells,”then“Fraction”toenterfractionsRowaverageformulaforF14=SUM(B5:B7)=B5/B8Copyright©2016PearsonEducation,Inc.Exhibit9.13AnalyticalHierarchyProcessExcelSpreadsheets(2of4)=SUM(B14:E14)/4=SUM(B5:B8)=B5/B9Copyright©2016PearsonEducation,Inc.Exhibit9.14AnalyticalHierarchyProcess

ExcelSpreadsheets(3of4)AtlantascoreincellC12CellsG14:G17fromExhibit9.13CellsF14:F16fromExhibit9.12Copyright©2016PearsonEducation,Inc.Exhibit9.15AnalyticalHierarchyProcessExcelSpreadsheets(4of4)=I5/G5=SUM(B11:B14)=(4.1564-4)/3=G12/0.90Copyright©2016PearsonEducation,Inc. Eachdecisionalternativeisgradedintermsofhowwellitsatisfiesthecriterionaccordingtofollowingformula:Si=

gijwj where: wj=aweightbetween0and1.00assignedtocriterionj; 1.00important,0unimportant; sumoftotalweightsequalsone. gij=agradebetween0and100indicatinghowwellalternativei satisfiescriteriaj; 100indicateshighsatisfaction,0lowsatisfaction.ScoringModelOverviewCopyright©2016PearsonEducation,Inc.Mallselectionwithfouralternativesandfivecriteria: S1=(.30)(40)+(.25)(75)+(.25)(60)+(.10)(90)+(.10)(80)=62.75 S2=(.30)(60)+(.25)(80)+(.25)(90)+(.10)(100)+(.10)(30)=73.50 S3=(.30)(90)+(.25)(65)+(.25)(79)+(.10)(80)+(.10)(50)=76.00 S4=(.30)(60)+(.25)(90)+(.25)(85)+(.10)(90)+(.10)(70)=77.75Mall4preferredbecauseofhighestscore,followedbymalls3,2,1.ScoringModelExampleProblemCopyright©2016PearsonEducation,Inc.Exhibit9.16ScoringModelExcelSolutionCopyright©2016PearsonEducation,Inc.GoalProgrammingExampleProblemProblemStatement Publicrelationsfirmsurveyinterviewerstaffingrequirementsdetermination.Onepersoncanconduct80telephoneinterviewsor40personalinterviewsperday.$50/dayfortelephoneinterviewer;$70/dayforpersonalinterviewer.Goals(inpriorityorder):Atleast3,000totalinterviewsconducted.Interviewerconductsonlyonetypeofintervieweachday;maintaindailybudgetof$2,500.Atleast1,000interviewsshouldbebytelephone. FormulateandsolveagoalprogrammingmodeltodeterminenumberofinterviewerstohireinordertosatisfythegoalsCopyright©2016PearsonEducation,Inc.Step1:ModelFormulation:MinimizeP1d1-,P2d2+,P3d3-subjectto: 80x1+40x2+d1--d1+=3,000interviews 50x1+70x2+d2--d2+=$2,500budget 80x1+d3--d3+=1,000telephoneinterviewswhere:x1=numberofte