Uncertain Computation-Based Decision Theory

The following sections are included:

• Dedication
• Preface
• Contents

The following sections are included:

• Certain environment
• Risk environments
  • Expected utility theory of von Neumann and Morgenstern
  • Prospect theory
• Uncertain environment
  • The Laplace criterion, the Hurwitz criterion and the minimax regret criterion
  • Subjective expected utility theory of Savage
  • Choquet expected utility
  • Multiple priors models
  • Cumulative prospect theory
• General theory of decisions

The following sections are included:

• A brief review of the main existing decision theories
• Comparison of existing decision theories

The following sections are included:

• Classical interval arithmetic
• Preference relations of interval values
• Relative distance measure based interval arithmetic
• Solving interval equations

The following sections are included:

• Operations on random variables
• Estimation of statistical characteristic of interval uncertainty
  • Statistical estimates
  • In general case when the distributions are not necessarily Gaussian
  • Complexity of the estimation calculation

The following sections are included:

• Type-1 fuzzy sets and numbers
• Type-2 fuzzy sets and numbers
• Arithmetic with continuous type-1 fuzzy numbers
  • Method based on the extension principle
  • Method based on interval arithmetic and α-cuts
• Arithmetic with discrete type-1 fuzzy numbers
• Arithmetic of type-2 fuzzy numbers
• Horizontal membership function based fuzzy arithmetic
• Adequacy of fuzzy models
• Fuzzy logic formalization for approximate reasoning

The following sections are included:

• Z-number, Z-information
• The arithmetic with continuous Z-numbers
  • State-of-the-art
  • General procedure of computation with Z-numbers
    • A general approach suggested by L. Zadeh
    • The suggested approach
  • Operations on continuous Z-numbers
    • Addition of continuous Z-numbers
    • Standard subtraction of continuous Z-numbers
    • Multiplication of continuous Z-numbers
    • Standard division of continuous Z-numbers
    • Square of a continuous Z-number
    • Square root of a continuous Z-number
    • Minimum and maximum of continuous Z-numbers
• The arithmetic of discrete Z-numbers
  • Discrete Z-numbers
    • Addition of discrete Z-numbers
    • Standard subtraction of discrete Z-numbers
    • Multiplication of discrete Z-numbers
    • Standard division of discrete Z-numbers
  • Power of a discrete Z-number
    • Square of a discrete Z-number
    • Square root of a discrete Z-number
• Z-numbers valued functions
  • Methodology for construction of functions of Z-numbers
  • Basic properties of functions of Z-numbers
• Reasoning with Z-information
  • State-of-the-art
  • Fuzzy and Z-number-valued If-Then rules
  • Approximate reasoning with Z-rules
    • Linear interpolation-based reasoning with Z-rules
• Ranking of Z-numbers
  • State-of-the-art
  • Ranking of Z-numbers
• Aggregation of Z-information
  • State-of-the-art
  • Aggregation operators for Z-numbers based information
    • Z-arithmetic mean
    • Z-geometric mean
    • Z-weighted arithmetic mean
    • Z-T-norm and Z-T-conorm operators
    • Z-ordered weighted averaging operators
• Z-numbers and type-2 fuzzy sets: A representation result
  • Z-numbers and type-2 fuzzy sets
  • Data processing to Z-numbers and type-2 fuzzy set

The following sections are included:

• Usuality and U-number concept
• A general approach to computation with U-numbers
• Approximate reasoning with usual information

In this chapter we present the fuzzy logic for formal geometric reasoning with extended objects. Based on the idea that extended objects may be seen as location constraints to coordinate points, the geometric primitives point, line, incidence and equality are interpreted as fuzzy predicates of a first order language. An additional predicate for the “distinctness” of point like objects is also used. Fuzzy Logic [28] is discussed as a reasoning system for geometry of extended objects. A fuzzification of the axioms of incidence geometry based on the proposed fuzzy logic is presented. In addition we discuss a special form of positional uncertainty, namely positional tolerance that arises from geometric constructions with extended primitives. We also address Euclid’s first postulate, which lays the foundation for consistent geometric reasoning in all classical geometries by taken into account extended primitives and gave a fuzzification of Euclid’s first postulate by using of our fuzzy logic. Fuzzy equivalence relation “Extended lines sameness” is introduced. For its approximation we use fuzzy conditional inference, which is based on proposed fuzzy “Degree of indiscernibility “and “Discernibility measure “of extended points…

The following sections are included:

• Decision making with interval probabilities
  • Consistency of interval probabilities
  • Statement of the problem and its solution
• Interval-based multi-attribute decision making
• Interval linear programming based decision making

The following sections are included:

• Fuzzy decision theory with imperfect information
  • State-of-the-art
  • Definitions
  • Statement of the problem
• Fuzzy optimality based decision making
  • State-of the-art
  • Some preliminary information
  • Statement of the problem
  • Method of solution
• Decision making with combined states
  • State-of the-art
  • Statement of the problem
  • Utility model axiomatization
  • Solution of the problem
• Fuzzy economics and decision making
• Type-2 fuzzy clustering
  • State-of-the-art
  • Type-2 fuzzy clustering and rule extraction
• Fuzzy regression analysis
  • State-of-the-art
  • Statement of problem
  • Genetic algorithm for defining parameters of fuzzy regression model
  • Simulations

The following sections are included:

• The Z-restriction based general decisions theory
  • Motivation
  • A unified decision model
    • Formal framework
  • Discussion of some special cases
  • Methodology for the general theory of decisions
• Z-number based linear programming
  • State-of-the art
  • Statement of a Z-number based linear programming problem
  • Solution method
• Fair price based decision making under interval, set-valued, fuzzy, and Z-number uncertainty
  • Case of interval uncertainty
  • Case of set-valued uncertainty
  • Case of Z-valued uncertainty
  • Case of fuzzy uncertainty
• Z-regression analysis
  • Z-regression model

The following sections are included:

• Decision making for well interventions under interval probability
  • Statement of the problem
  • Solution of the problem
  • A numerical solution
• Fuzzy production–distribution planning in supply chain management
  • State-of-the-art
  • Statement of the problem
  • A solution of the production–distribution problem
  • Experimental investigation
• Application of fuzzy logic to economics
  • Fuzzy language in modeling of an economic agent
  • Fuzzy stability
  • Stability and time path planning of national economy
  • Economic growth control problem
  • Fuzzy nonlinear model of a manufacture dynamics
  • Fuzzy approach to portfolio construction
• Decision making in a products mix problem
• Multi-attribute decision making for web services selection
  • State-of-the-art
  • Statement of the problem and its solution
  • Practical example
• Z-number modeling in psychological research
  • Set of problem
  • Z-modeling of educational achievement
• Optimal port choice under Z-information
  • State-of-the-art
  • Decision making on port selection under Z-number-valued information

The following sections are included:

• Bibliography
• Index