FLEXIBLE AND GENERALIZED UNCERTAINTY OPTIMIZATION : theory and approaches.

Preface
Contents
1 An Introduction to Generalized Uncertainty Optimization
	1.1 Introduction
	1.2 What Interval, Fuzzy, and Generalized Uncertainty Optimization …
		1.2.1 The Importance of More General Types of Optimization
		1.2.2 Fuzzy Set and Possibility Theory—Differences
	1.3 Fundamental Entities: Intervals, Fuzzy Intervals and Possibility Intervals
		1.3.1 Intervals
		1.3.2 Fuzzy Intervals
		1.3.3 Possibility Intervals
		1.3.4 Interval, Fuzzy Interval, Representations, Arithmetic
	1.4 A Taxonomy of Generalized Uncertainty in Optimization
	1.5 Summary
	1.6 Exercises
	References
2 Generalized Uncertainty Theory: A Language for Information Deficiency
	2.1 Introduction
	2.2 Generalized Uncertainty
		2.2.1 Intervals
		2.2.2 Fuzzy Interval
		2.2.3 Possibility/Necessity Measures and Distributions
		2.2.4 Interval-Valued Probability
		2.2.5 P-Boxes
		2.2.6 Cloud
		2.2.7 Kolmogorov-Smirnov Statistic
		2.2.8 Dempster-Shafer Belief/Plausibility Measures and Distributions
		2.2.9 Probability Interval
		2.2.10 Random Sets
	2.3 Equivalences Between Generalized Uncertainty Types
		2.3.1 Possibility Interval, Generalized Uncertainty Arithmetic
	2.4 Summary
	2.5 Exercises
	References
3 The Construction of Flexible and Generalized Uncertainty Optimization Input Data
	3.1 Introduction
	3.2 Construction Flexibility/Fuzzy Distributions
		3.2.1 Construction of Data for Coefficients in Right-Hand Side
		3.2.2 Construction of Data for Flexible/Soft Relationships
	3.3 Construction of Generalized Uncertainty Distributions
		3.3.1 Intervals and Fuzzy Intervals
		3.3.2 Possibility Pairs Distributions
		3.3.3 Interval-Valued Probabilities: Construction of Interval-Valued Probability Measures from Upper/Lower Cumulative Probabilities
		3.3.4 Construction from Kolmogorov–Smirnov Statistics
		3.3.5 Belief and Plausibility Construction
		3.3.6 Random Sets
		3.3.7 Probability Intervals and Random Sets
	3.4 Summary
	3.5 Exercises
	References
4 An Overview of Flexible and Generalized Uncertainty Optimization
	4.1 Introduction
	4.2 Flexibility and Generalized Uncertainty Translations
		4.2.1 Input Coefficients
		4.2.2 Soft Constraint Relationships
		4.2.3 Soft Objective Relationships
	4.3 Distinctions of Fuzzy and Possibility—Generalized Uncertainty in Optimization
		4.3.1 Basic Notions
		4.3.2 Relations: Flexible Versus Generalized Uncertainty in Constraint Sets
		4.3.3 Generalized Uncertainty in Constraint Sets
		4.3.4 Three Types of Constraints: Semantic Considerations
		4.3.5 Objective Function: Flexible Versus Generalized Uncertainty in Objective Functions
	4.4 Solution Methods
	4.5 Summary
	4.6 Exercises
	References
5 Flexible Optimization
	5.1 Introduction
	5.2 Flexible Optimization
		5.2.1 Fuzzy Components of a Flexible Optimization Model
		5.2.2 Translation of Components into Fuzzy Membership Functions
		5.2.3 Transformation of the Simultaneous Fuzzy Memberships into a Real-Valued Mathematical Programming Problem
	5.3 Three Generic Flexible Optimization Model Types
		5.3.1 Decision-Making in a Fuzzy Environment
		5.3.2 Flexible Optimization that Returns Fuzzy Interval Solutions
		5.3.3 Fuzzy Interval Right-Hand Sides
	5.4 Summary
	5.5 Exercises
	References
6 Generalized Uncertainty Optimization
	6.1 Introduction to Generalized Uncertainty Optimization
		6.1.1 Distributions
		6.1.2 Fuzzy Optimization Mapped to Fuzzy Banach Spaces
		6.1.3 Real Euclidean Space Approaches
		6.1.4 Minimum Maximum Regret
	6.2 Example—Investment Strategies
		6.2.1 MiniMax Regret Under Generalized Uncertainty
	6.3 Summary
	6.4 Exercises
	References