Decision Making Theories and Methods Based on Interval-Valued Intuitionistic Fuzzy Sets

Preface
Contents
1 A Possibility Degree Method for Interval-Valued Intuitionistic Fuzzy Multi-attribute Group Decision Making
	Abstract
	1.1 Introduction
	1.2 A New Ranking Method of IVIFS from the Probability Viewpoint
	1.2 A New Ranking Method of IVIFS from the Probability Viewpoint
		1.2.1 New Ranking Method for Intervals from the Probability Viewpoint
	1.2.1 New Ranking Method for Intervals from the Probability Viewpoint
	1.2.1 New Ranking Method for Intervals from the Probability Viewpoint
		1.2.2 New Ranking Method for IVIFNs Based on the Possibility Degree
	1.2.2 New Ranking Method for IVIFNs Based on the Possibility Degree
	1.2.2 New Ranking Method for IVIFNs Based on the Possibility Degree
		1.2.3 Comparative Analysis with Score and Accuracy Functions for IVIFNs
	1.3 Ordered Weighted Average Operator and Hybrid Weighted Average Operator for IVIFNs
		1.3.1 Weighted Average Operator for IVIFNs
	1.3.1 Weighted Average Operator for IVIFNs
		1.3.2 Proposed Ordered Weighted Average Operator for IVIFNs
	1.3.2 Proposed Ordered Weighted Average Operator for IVIFNs
		1.3.3 Proposed Hybrid Weighted Average Operator for IVIFNs
	1.3.3 Proposed Hybrid Weighted Average Operator for IVIFNs
	1.3.3 Proposed Hybrid Weighted Average Operator for IVIFNs
	1.4 MAGDM Problem and Method with IVIFS
		1.4.1 Problem Description for MAGDM with IVIFS
		1.4.2 Determination of the Weights of DMs
	1.4.2 Determination of the Weights of DMs
		1.4.3 Group Decision Making Method
	1.5 An Air-Condition System Selection Example and Comparison Analysis
		1.5.1 An Air-Condition System Selection Problem and the Analysis Process
		1.5.2 Comparison Analysis of the Obtained Results
	1.6 Conclusions
	References
2 A New Method for Atanassov’s Interval-Valued Intuitionistic Fuzzy MAGDM with Incomplete Attribute Weight Information
	Abstract
	2.1 Introduction
	2.2 Preliminaries
		2.2.1 Interval Objective Programming
		2.2.2 Orderings of Intervals
		2.2.3 Atanassov’s Intuitionistic Fuzzy Set and Atanassov’s Interval-Valued Intuitionistic Fuzzy Set
	2.3 A Novel Method for MAGDM with AIVIFVs and Incomplete Attribute Weight Information
		2.3.1 Presentation of the Problems
		2.3.2 Determine the DMs’ Weights with Respect to Different Attributes
			2.3.2.1 Calculate the Similarity Degree Based on an Extended TOPSIS
			2.3.2.2 Calculate Proximity Degree Using the Distance Measure
			2.3.2.3 Obtain the Weights of DMs with Respect to Different Attributes
		2.3.3 Converting Individual Decision Matrices into a Collective Interval Matrix
		2.3.4 Construct Multi-objective Interval-Programming for Deriving Attribute Weights
		2.3.5 Decision Process and Algorithm for MAGDM with AIVIFVs
	2.4 A Real-World R & D Project Selection Example and Comparison Analyses
		2.4.1 A Real-World R & D Project Selection Problem and the Solution Process
		2.4.2 Comparison with the Extended TOPSIS Method
		2.4.3 Comparison with Barrenechea et al.’s Method
	2.5 Conclusions
	References
3 Interval-Valued Intuitionistic Fuzzy Mathematical Programming Method for Hybrid Multi-criteria Group Decision Making with Interval-Valued Intuitionistic Fuzzy Truth Degrees
	Abstract
	3.1 Introduction
	3.2 Basic Concepts
		3.2.1 Concepts of Interval-Valued Intuitionistic Fuzzy Sets and Distances
		3.2.2 Definition and Distance for Trapezoidal Fuzzy Numbers and Triangular Fuzzy Numbers
		3.2.3 Linguistic Variables
	3.3 Hybrid MCGDM Problems Considering Alternative Comparisons with IVIF Truth Degrees
		3.3.1 Hybrid MCGDM Problems with IVIF Truth Degrees and Incomplete Weight Information
		3.3.2 Normalization Method
	3.4 IVIF Mathematical Programming Method for Hybrid MCGDM
		3.4.1 IVIFS-Type Consistency and Inconsistency Measurements
		3.4.2 Bi-Objective IVIF Mathematical Programming Model
		3.4.3 Goal Programming Approach to Solving Bi-Objective IVIF Mathematical Programming Model
		3.4.4 GDM Process and Steps for Solving Hybrid MCGDM
	3.5 Empirical Example of Critical Infrastructure
		3.5.1 A Critical Infrastructure Evaluation Example and the Analysis Process
		3.5.2 Comparison Analysis with the Existing LINMAP Methods
	3.6 Conclusions
	Appendix 1
	Appendix 2
	References
4 A Selection Method Based on MAGDM with Interval-Valued Intuitionistic Fuzzy Sets
	Abstract
	4.1 Introduction
	4.2 Preliminaries
		4.2.1 Interval-Valued Intuitionistic Fuzzy Set
		4.2.2 Gray Relation Analysis
	4.3 A Novel Method for MAGDM with IVIFSs and Incomplete Attribute Weight Information
		4.3.1 Determine the Weights of Experts by the Extended GRA Method
		4.3.2 Integrate Individual Decision Matrices into a Collective Matrix
		4.3.3 Identify the Attribute Weights by a New Multi-objective Linear Programming Model
		4.3.4 Decision Process and Algorithm for MAGDM Problems with IVIFSs
		4.3.5 Decision Support System Framework Based on MAGDM with IVIFSs
	4.4 A Cloud Service Selection Problem and Comparison Analysis
		4.4.1 A Cloud Service Provider Selection Problem and the Solution Process
		4.4.2 Sensitivity Analysis for Parameter
		4.4.3 Comparison Analysis with the Method Using the Score Function
	4.5 Conclusions
	References
5 Aggregating Decision Information into Interval-Valued Intuitionistic Fuzzy Numbers for Heterogeneous Multi-attribute Group Decision Making
	Abstract
	5.1 Introduction
	5.2 Some Basic Concepts
		5.2.1 Interval-Valued Intuitionistic Fuzzy Set
		5.2.2 Definitions and Distances for Real Number, Interval Number, TFN and TrFN
	5.3 Aggregating Heterogeneous Decision Information into IVIFNs
		5.3.1 Presentation of Heterogeneous MAGDM Problem
		5.3.2 A General Method for Aggregating Heterogeneous Decision Information into IVIFNs
			5.3.2.1 Compute the Qsd, Qdd and Qud
			5.3.2.2 Calculate Qsi, Qdi and Qui
			5.3.2.3 Induce an IVIFN
		5.3.3 Concrete Computation Formulas for Aggregating Heterogeneous Information into IVIFNs
			5.3.3.1 For Trapezoidal Fuzzy Numbers
			5.3.3.2 For Triangular Fuzzy Numbers
			5.3.3.3 For Interval Numbers
			5.3.3.4 For Real Numbers
	5.4 A Novel Approach for Heterogeneous MAGDM Problems
		5.4.1 Construct an Intuitionistic Fuzzy Programming Model to Determine the Attribute Weights
		5.4.2 Algorithm for Solving Heterogeneous MAGDM
	5.5 Comparison Analysis with Existing Methods
	5.6 Illustrative Examples
		5.6.1 An IT Outsourcing Service Provider Evaluation Example
			5.6.1.1 Decision Process Using the Proposed Method
			5.6.1.2 Comparison with Extended TOPSIS Method
		5.6.2 A Supplier Selection Example
	5.7 Conclusions
	References
6 A Novel Method for Group Decision Making with Interval-Valued Atanassov Intuitionistic Fuzzy Preference Relations
	Abstract
	6.1 Introduction
	6.2 Multiplicative Consistency of Atanassov Intuitionistic Fuzzy Preference Relations
		6.2.1 A New Multiplicative Consistency Index of Atanassov Intuitionistic Fuzzy Preference Relations
	6.2.1 A New Multiplicative Consistency Index of Atanassov Intuitionistic Fuzzy Preference Relations
	6.2.1 A New Multiplicative Consistency Index of Atanassov Intuitionistic Fuzzy Preference Relations
	6.2.1 A New Multiplicative Consistency Index of Atanassov Intuitionistic Fuzzy Preference Relations
	6.2.1 A New Multiplicative Consistency Index of Atanassov Intuitionistic Fuzzy Preference Relations
	6.2.1 A New Multiplicative Consistency Index of Atanassov Intuitionistic Fuzzy Preference Relations
	6.2.1 A New Multiplicative Consistency Index of Atanassov Intuitionistic Fuzzy Preference Relations
	6.2.1 A New Multiplicative Consistency Index of Atanassov Intuitionistic Fuzzy Preference Relations
	6.2.1 A New Multiplicative Consistency Index of Atanassov Intuitionistic Fuzzy Preference Relations
		6.2.2 An Iterative Algorithm for Repairing the Consistency of AIFPRs
	6.2.2 An Iterative Algorithm for Repairing the Consistency of AIFPRs
	6.2.2 An Iterative Algorithm for Repairing the Consistency of AIFPRs
	6.3 Multiplicative Consistency of IV-AIFPRs
		6.3.1 Define and Check Multiplicative Consistency of IV-AIFPRs
	6.3.1 Define and Check Multiplicative Consistency of IV-AIFPRs
	6.3.1 Define and Check Multiplicative Consistency of IV-AIFPRs
	6.3.1 Define and Check Multiplicative Consistency of IV-AIFPRs
	6.3.1 Define and Check Multiplicative Consistency of IV-AIFPRs
	6.3.1 Define and Check Multiplicative Consistency of IV-AIFPRs
	6.3.1 Define and Check Multiplicative Consistency of IV-AIFPRs
	6.3.1 Define and Check Multiplicative Consistency of IV-AIFPRs
	6.3.1 Define and Check Multiplicative Consistency of IV-AIFPRs
		6.3.2 Repair Multiplicative Consistency of IV-AIFPRs
	6.3.2 Repair Multiplicative Consistency of IV-AIFPRs
	6.4 A Novel Method for Group Decision Making with IV-AIFPRs
		6.4.1 Determine DMs’ Weights Objectively and Integrate Individual IV-AIFPRs
	6.4.1 Determine DMs’ Weights Objectively and Integrate Individual IV-AIFPRs
		6.4.2 Derive IVAIF Priority Weights and Rank Alternatives
			6.4.2.1 Derive IVAIF Priority Weights of Alternatives
			6.4.2.2 A TOPSIS Based Approach to Ranking IVAIF Priority Weights
	6.4.2.2 A TOPSIS Based Approach to Ranking IVAIF Priority Weights
		6.4.3 A Novel Method for Solving GDM Problems with IV-AIFPRs
	6.5 A Practical Example of a Virtual Enterprise Partner Selection and Comparative Analyses
		6.5.1 A Practical Example of a Virtual Enterprise Partner Selection
		6.5.2 Comparative Analyses
			6.5.2.1 Comparison with Liao’s Method
			6.5.2.2 Comparison with Other Existing Group Decision Making Methods
	6.6 Conclusions
	Appendix 1
	References
7 Additive Consistent Interval-Valued Atanassov Intuitionistic Fuzzy Preference Relation and Likelihood Comparison Algorithm Based Group Decision Making
	Abstract
	7.1 Introduction
	7.2 Preliminaries
	7.2 Preliminaries
	7.2 Preliminaries
	7.2 Preliminaries
	7.2 Preliminaries
	7.2 Preliminaries
	7.3 A New Likelihood Comparison Algorithm of IVAIFVs
	7.3 A New Likelihood Comparison Algorithm of IVAIFVs
	7.4 Additive Consistency Analyses for IVAIFPR
		7.4.1 Additive Consistency Definition of IVAIFPR
	7.4.1 Additive Consistency Definition of IVAIFPR
	7.4.1 Additive Consistency Definition of IVAIFPR
		7.4.2 Derive the IVAIF Priority Weights from IVAIFPR
	7.5 Method for Solving the Group Decision Making Problems with IVAIFPRs
		7.5.1 Description for GDM Problems with IVAIFPRs
		7.5.2 Determination of DMs’ Weights
	7.5.2 Determination of DMs’ Weights
		7.5.3 Method for Group Decision Making with IVAIFPRs
	7.5.3 Method for Group Decision Making with IVAIFPRs
	7.5.3 Method for Group Decision Making with IVAIFPRs
	7.6 An Example of ERP System Selection and Comparative Analysis
		7.6.1 A Practical Example of ERP System Selection
		7.6.2 Comparative Analysis with Existing GDM Methods
	7.7 Conclusions
	Appendix 1
	References
8 A Three-Phase Method for Group Decision Making with Interval-Valued Intuitionistic Fuzzy Preference Relations
	Abstract
	8.1 Introduction
	8.2 Preliminaries
		8.2.1 Some Related Concepts on IFPR
	8.2.1 Some Related Concepts on IFPR
	8.2.1 Some Related Concepts on IFPR
	8.2.1 Some Related Concepts on IFPR
	8.2.1 Some Related Concepts on IFPR
	8.2.1 Some Related Concepts on IFPR
	8.2.1 Some Related Concepts on IFPR
	8.2.1 Some Related Concepts on IFPR
		8.2.2 Additive Consistency of IVIFPR
	8.2.2 Additive Consistency of IVIFPR
	8.2.2 Additive Consistency of IVIFPR
	8.2.2 Additive Consistency of IVIFPR
	8.2.2 Additive Consistency of IVIFPR
	8.2.2 Additive Consistency of IVIFPR
	8.3 Determination of the Intuitionistic Fuzzy Priority Weights from an IVIFPR
		8.3.1 Extracting a Risk Attitudinal-Based Consistent IFPR from an IVIFPR
	8.3.1 Extracting a Risk Attitudinal-Based Consistent IFPR from an IVIFPR
		8.3.2 Deriving the Intuitionistic Fuzzy Priority Weights from the Extractive IFPR
	8.3.2 Deriving the Intuitionistic Fuzzy Priority Weights from the Extractive IFPR
	8.4 A Novel Three-Phase Method for Solving GDM with IVIFPRs
		8.4.1 Presentation of Problem for GDM with IVIFPRs
		8.4.2 Integrating Individual IVIFPRs to a Collective One
	8.4.2 Integrating Individual IVIFPRs to a Collective One
	8.4.2 Integrating Individual IVIFPRs to a Collective One
		8.4.3 A Three-Phase Method for GDM with IVIFPRs
	8.5 An Example of Network System Selection and Comparison Analyses
		8.5.1 A Network System Selection Example and the Analysis Process
		8.5.2 Comparison Analysis with Xu’s Method
		8.5.3 Comparison Analysis with Wang’s Method
	8.6 Conclusions
	References
9 A Group Decision-Making Method Considering Both the Group Consensus and Multiplicative Consistency of Interval-Valued Intuitionistic Fuzzy Preference Relations
	Abstract
	9.1 Introduction
	9.2 Preliminaries
	9.2 Preliminaries
	9.2 Preliminaries
	9.2 Preliminaries
	9.3 A New Ranking Method of IVIFVs
		9.3.1 A New Ranking Method of IVIFVs
	9.3.1 A New Ranking Method of IVIFVs
		9.3.2 Comparison with Existing Ranking Methods of IVIFVs
	9.3.2 Comparison with Existing Ranking Methods of IVIFVs
	9.3.2 Comparison with Existing Ranking Methods of IVIFVs
	9.3.2 Comparison with Existing Ranking Methods of IVIFVs
	9.4 Analyses of Group Consensus in GDM
		9.4.1 Problem Description for GDM with IVIFPRs
		9.4.2 Group Consensus Analysis of GDM
	9.4.2 Group Consensus Analysis of GDM
	9.4.2 Group Consensus Analysis of GDM
		9.4.3 An Iteration Algorithm in Group Consensus Improving Process
	9.4.3 An Iteration Algorithm in Group Consensus Improving Process
	9.4.3 An Iteration Algorithm in Group Consensus Improving Process
	9.4.3 An Iteration Algorithm in Group Consensus Improving Process
	9.4.3 An Iteration Algorithm in Group Consensus Improving Process
		9.4.4 Statistical Comparative Study on Group Consensus
	9.5 Multiplicative Consistency of IVIFPR
		9.5.1 Multiplicative Consistency Concept of IVIFPR
	9.5.1 Multiplicative Consistency Concept of IVIFPR
	9.5.1 Multiplicative Consistency Concept of IVIFPR
		9.5.2 Determine the Priority Weights from an IVIFPR
	9.6 Method for GDM with IVIFPRs
		9.6.1 Determination of Expert Weights Based on the Markov Model
	9.6.1 Determination of Expert Weights Based on the Markov Model
		9.6.2 Method for GDM with IVIFPRs
	9.7 Two Case Studies
		9.7.1 Case 1: An Example of a Virtual Enterprise Partner Selection
			9.7.1.1 A Practical Example of a Virtual Enterprise Partner Selection
			9.7.1.2 Comparative Analyses and Spearman’s Rank-Correlation Test
			9.7.1.3 Fitted Error Analysis of the Obtained Results
		9.7.2 Case 2: A Practical Example of an Enterprise Resource Planningsystem Selection
			9.7.2.1 A Practical Example of an Enterprise Resource Planning System Selection
			9.7.2.2 Comparative Analysis with the Method of Liao et al
	9.8 Conclusions
	Appendix 1
	Appendix 2
	References