# Mathematical Modeling using Fuzzy Logic:Applications to Sustainability

Rule-Based Fuzzy Logic SystemsNew Concepts and Their Historical BackgroundFundamental Diamond RequirementThe Flow of UncertaintiesExisting Literature on Type 2 Fuzzy SetsCoverageApplicability Outside of Rule-Based FLSComputationPrimer on Fuzzy SetsFuzzy SetsFrom Fuzzy Sets to Crisp SetsLinguistic VariablesMembership FunctionsSome TerminologySet-Theoretic Operations on Crisp SetsSet-Theoretic Operations for Fuzzy SetsCrisp Relations and Compositions on the Same Product SpaceRelations and CompositionsHedgesExpansion PrincipleFL PrimerCrisp LogicFrom Crisp Logic to FLRemarksExerciseReferencesSources of UncertaintyUncertaintyUncertainty: General DiscussionUncertainty at FLSWords Mean Different Things to Different PeopleExerciseReferencesMembership Functions and UncertaintyIntroductionType 1 Membership FunctionThe Concept of a Type 2 Fuzzy SetDefinition of Type 2 Fuzzy Sets and Related ConceptsType 2 Fuzzy Sets and Examples of FOUUpper and Lower Membership FunctionsA Type 1 Purge Set Represented by a Type 2 Fuzzy Setand 1 Membership of Type 2 Fuzzy SetBack to the Language LabelExerciseReferencesCase StudiesIntroductionTime Series PredictionExtracting Rules from DataClassic Time Series Forecasting MethodAutoregression (AR)Moving Average (MA)Autoregressive Movement Average Type (ARMA)Autoregressive Integrated Moving Average (ARIMA)Seasonal Autoregressive Integrated Moving Average (SARIMA)Vector Autoregression (VAR)Vector Autoregressive Moving Average (VARMA)Vector Autoregression Moving-Average with Exogenous Regressors (VARMAX)Simple Exponential Smoothing (SES)Holt–Winters Exponential Smoothing (HWES)Knowledge Mining Using SurveysKnowledge Mining MethodologyExerciseReferencesSingleton Type 1 Fuzzy Logic Systems: No UncertaintiesIntroductionRulesFuzzy Inference EngineFuzzification and Its Effect on ReasoningFuzzifierDefuzzificationCentroid DefuzzifierBisecting DefuzzifierWeighted Average DefuzzifierMidpoint of Maximum DefuzzifierLargest of Maximum DefuzzifierSmallest Maximum DefuzzifierFLS DesignBack-Propagation (the Steepest Descent) MethodSVD-QR MethodRepetitive Diamond MethodSample Study: Time Series PredictionCase Study: Data MiningExerciseReferencesCentroid of a Type 2 Fuzzy Set: Type ReductionIntroductionUnspecified Consequences for the CenterGeneralization Center for Interval Type 2 Fuzzy SetInterval Type 2 Center of Fuzzy SetType Reduction: Unspecified ConsequencesCenter Type ReductionHeight Type ReductionSet Center Type ReductionComputational Complexity of Type ReductionConclusionExerciseReferencesModeling of SustainabilityThe Meaning of SustainabilityIntroduction to Sustainability by Fuzzy Assessment (SAFE) ModelSAFE Model OverviewKey Indicators of Sustainable DevelopmentMeasuring SustainabilityFuzzy AssessmentSensitivity AnalysisAdvantages and Disadvantages of the SAFE ModelSample Study for the SAFE ModelSAFE for Energy SustainabilityConclusionExerciseReferencesEpilogueIntroductionType 2 vs. Type 1 FLSApplication for Type 2 FLSRule-Based Nomenclature for Video TrafficSelected FunctionFOU on FunctionRulesFOU for MeasurementFL RBC’s ParametersCalculation Formula for Type 1 FL RBCCalculation Formula for Type 2 FL RBCOptimization of Rule Design ParametersFL RBC TestResults and ConclusionsEqualization of Time-Varying Nonlinear Digital ContactsPreparation for Equalization of Water SupplyWhy Type 2 FAFs Are Needed?FAF DesignSimulation and ConclusionLiaison System with ISI and CCIConnection Ticket Rental for ATM NetworksSurvey-Based CAC Using Type 2 FLS: OverviewExtraction of Knowledge for CACSurvey ProcessCAC Visualization Boundaries and ResultsExerciseReferencesAppendix A: Join, Meet, and Negation Operations for Non-Interval Type 2 Fuzzy SetsAppendix B: Properties of Type 1 and Type 2 Fuzzy SetsAppendix C: ComputationAppendix D: Medical Diagnosis by Fuzzy LogicAppendix E: Fuzzy Logic System OptimizedAppendix F: Heart Disease DemoAppendix G: Linear Tip Demo, Mamdani Tip Demo, Sugeno Tip DemoAppendix H: Miscellaneous