Risk Matrix: Rating Scheme Design and Risk Aggregation

Contents
1 Risk Matrix: Foundations and Overview
	1.1 What Is Risk Matrix?
		1.1.1 What Is Risk?
		1.1.2 What Is Risk Measure
		1.1.3 Objective Risk Measure Versus Subjective Risk Measure
	1.2 Theoretical Flaws and the Topics in This Book
		1.2.1 Some Theoretical Flaws of Risk Matrix
		1.2.2 The Topics in This Book
	References
2 Different Types of Risk Matrices and Typical Applications
	2.1 Origin and Fundamental Model
		2.1.1 The Origin of the Risk Matrix
		2.1.2 Fundamental Mathematical Model of Risk Matrix
	2.2 Structure and the Design of Different Kinds of Risk Matrix
		2.2.1 Qualitative Risk Matrix
		2.2.2 Semi-quantitative Risk Matrices
		2.2.3 Quantitative Risk Matrices
	2.3 Applications in Different Fields
		2.3.1 Food Safety Risk Assessment
		2.3.2 Risk Assessment of Public Infrastructure Operation
		2.3.3 Medical and Health Risk Assessment
	References
3 Rating Scheme Design Methods
	3.1 Summary of Some Unwritten Rules
		3.1.1 Input Classification
		3.1.2 Measure of Risks
		3.1.3 Classification of Different Risks
	3.2 Usage of Utility Functions to Design Qualitative Risk Matrix Design
	3.3 Cox’s Risk Matrix Design Axioms
	3.4 Sequential Updating Approach
		3.4.1 Principles of the SUA to Design Rating Schemes for Risk Matrices
		3.4.2 The Uniqueness Principle of the Sequential Updating Approach
		3.4.3 A Global Rating Algorithm
		3.4.4 Application: Design the Rating Scheme of a 4 times4 Risk Matrix Using SUA
	3.5 Summary and Comparison of the Rating Scheme Design Methods
		3.5.1 Summary
		3.5.2 Comparison
	References
4 Risk Perceptions in Risk Matrix: Sources and Impact to Risk Matrix Design
	4.1 Introduction
	4.2 Identifying Risk Perception in Risk Matrices
		4.2.1 Perception of the Scaling of Inputs
		4.2.2 Perception of the Location of Inputs
		4.2.3 Perception of Input Category Membership
		4.2.4 Perception of the Measure of Risks
		4.2.5 Perception of Attitudes Towards Risks
	4.3 A Sequential Updating Approach Used to Integrate Different Perception
		4.3.1 Review of the SUA
		4.3.2 Integrating Different Types of Risk Perception
	4.4 Risk Matrices Integrating Different Risk Perceptions
		4.4.1 Performance of Integrating Risk Perception on the Scaling of Inputs
		4.4.2 Performance of Integrating Risk Perception on the Location of Inputs
		4.4.3 Performance of Integrating Risk Perception on Input Category Membership
		4.4.4 Performance of Integrating Risk Perception on the Measure of Risks
		4.4.5 Performance of Integrating Risk Perception on Attitudes Towards Risks
	4.5 Conclusion and Discussions
	References
5 Risk Matrix Design Assessment: Criteria and Quantitative Indicators
	5.1 Criteria Used to Assess Risk Matrix
		5.1.1 Monotonicity of Risks
		5.1.2 Consistency with the Risk Measure
		5.1.3 Effective Resolution of Risk Ratings
	5.2 Quantitative Indicators of the Criteria
		5.2.1 Proportion of Wrong Risk Pairs (PWRP)
		5.2.2 The Volatility of Risk Measures (VRM)
		5.2.3 Probability of a Correct Decision (PCD)
	5.3 Applications of the Criteria
		5.3.1 Instruction for Practitioners Designing Risk Matrices
		5.3.2 Determine the Risk Rating of Some Cells
		5.3.3 Assessing Some of the Risk Matrices Used in the Literature
	References
6 Risk Matrix Aggregation: A General Framework
	6.1 Term Explanation of Risk Matrix Aggregation
	6.2 Normal Framework to Aggregate Risk Matrices
		6.2.1 Assess Individual Risk According to the Normalized Quantitative Risk Matrices
		6.2.2 Find the Appropriate Expressions of the Ratings
		6.2.3 Aggregate the Individual Risks by Composition Methods
		6.2.4 Transform the Result of Aggregation into a Specific Value
	References
7 Risk Matrix Aggregation Methods: Introduction and Comparative Analysis
	7.1 Fuzzy Set-Based Method
		7.1.1 Similarities Between Risk Matrices and Fuzzy Sets
		7.1.2 Fuzzy Membership in Risk Matrices
	7.2 Interval Number-Based Method
	7.3 Probability Density Function-Based Method
	7.4 Comparison of Different Methods to Aggregate Risk Matrices
		7.4.1 An Illustrative Example
		7.4.2 Robustness of Different Methods
		7.4.3 Comparison of the Three Aggregation Methods
	References
8 Three-Dimensional Risk Matrix: Theoretical Basis and Construction
	8.1 Background of the Three-Dimensional Risk Matrix
	8.2 Three-Dimensional Risk Measure Considering the Strength of Knowledge
	8.3 Impact Factors of the Strength of Knowledge
	8.4 Measure of the Strength of Knowledge Based on Fuzzy MCDM
		8.4.1 Multi-Attribute Decision-Making Method Based on Fuzzy Measure and Choquet Integral
		8.4.2 Knowledge Strength Under Multi-Layer Factors
		8.4.3 Knowledge Strength Under Multiple Decision Makers
	8.5 Construction of Three-Dimensional Risk Matrices
	References
9 Conclusions and Future Research
	9.1 Conclusions
	9.2 Future Research