Monte Carlo Methods in Fuzzy Optimization (Studies in Fuzziness and Soft Computing, 222)

✍ Scribed by James J. Buckley, Leonard J. Jowers

• Publisher: Springer
• Year: 2008
• Tongue: English
• Leaves: 256
• Category: Library

✦ Synopsis

1. 1 Introduction The objective of this book is to introduce Monte Carlo methods to ?nd good approximate solutions to fuzzy optimization problems. Many crisp (nonfuzzy) optimization problems have algorithms to determine solutions. This is not true for fuzzy optimization. There are other things to discuss in fuzzy optimization, which we will do later onin the book, like? and

✦ Table of Contents

Title Page
Contents
Part I: Introduction
Introduction
Introduction
Part I
Part II
Part III
Part IV
Notation
Previous Research
MATLAB/C++ Programs
Fuzzy Sets
Introduction
Fuzzy Sets
Fuzzy Numbers
Alpha-Cuts
Inequalities
Discrete Fuzzy Sets
Fuzzy Arithmetic
Extension Principle
Interval Arithmetic
Fuzzy Arithmetic
Fuzzy Functions
Extension Principle
Alpha-Cuts and Interval Arithmetic
Differences
Min/Max of a Fuzzy Number
Ordering Fuzzy Numbers
Buckley's Method
Kerre's Method
Chen's Method
Breaking Ties
Undominated Fuzzy Vectors
Buckley's Method
Kerre's Method
Chen's Method
Crisp Random Numbers and Vectors
Introduction
Random Numbers
Quasi-random Sequences
Random Number Generator
Random Non-negative Integers
Random Vectors: Real Numbers
Random Vectors: Non-negative Integers
Random Fuzzy Numbers and Vectors
Introduction
Random Triangular/Trapezoidal Fuzzy Numbers
Random Quadratic Fuzzy Numbers
Generated from Implicit Quadratic Functions
Generated from Parametric Quadratic Functions, Bézier Fuzzy Numbers
Comparison of Random Fuzzy Vectors
Random Fuzzy Vectors
Tests for Randomness
Introduction
Random Fuzzy Numbers
Run Test
Frequency Test
Search Space
Search Space $[a,b]$ for QBGFNs
Search Space $\Omega$ for QBGFNs
Search Space $[a,b]$ for TFNs
Search Space $\Omega$ for TFNs
Other Search Spaces
Part II: Applications
Fuzzy Monte Carlo Method
Introduction
Crisp Linear Program
Fuzzy Linear Program
Kerre's Method
Chen's Method
Comparison of Solutions
Fully Fuzzified Linear Programming I
Introduction
Fully Fuzzified Linear Programming
Product Mix Problem
Fuzzy Monte Carlo Method
Kerre's Method
Chen's Method
Comparison of Solutions
Fully Fuzzified Linear Programming II
Introduction
Diet Problem
Fuzzy Monte Carlo Method
Comparison of Solutions
Fuzzy Multiobjective LP
Introduction
Multiobjective Fully Fuzzified Linear Programming
Example Problem
Fuzzy Monte Carlo Method
Compare Solutions
Solving Fuzzy Equations
Introduction
$\overline{A}$ $\overline{X}$+ $\overline{B}$= $\overline{C}$
Other Solutions
Fuzzy Monte Carlo Method
Fuzzy Quadratic Equation
Fuzzy Monte Carlo Method
Fuzzy Matrix Equation
Fuzzy Monte Carlo Method
Summary and Conclusions
Fuzzy Linear Regression I
Introduction
Random Fuzzy Vectors
Application
First Choice of Intervals
Second Choice of Intervals
Comparison of Solutions
Summary and Conclusions
Univariate Fuzzy Nonlinear Regression
Introduction
Univariate Fuzzy Nonlinear Regression
Evolutionary Algorithm (EA)
Fuzzy Monte Carlo Method
Fuzzy Quadratic
First Choice of Intervals
Second Choice of Intervals
Comparison of Solutions
Fuzzy Logarithmic
First Choice of Intervals
Second Choice of Intervals
Comparison of Solutions
Summary and Conclusions
Multivariate Nonlinear Regression
Introduction
Universal Approximator
Evolutionary Algorithm
Fuzzy Monte Carlo Method
Applications
First Application
Second Application
Third Application
Fourth Application
Summary and Conclusions
Fuzzy Linear Regression II
Introduction
Error Measures
Random Vectors
Example Problem
First Choice of Intervals
Second Choice of Intervals
Comparison of Solutions
Summary and Conclusions
MATLAB Program
Fuzzy Two-Person Zero-Sum Games
Introduction
Two-Person Zero-Sum Games
Fuzzy Two-Person Zero-Sum Games
Fuzzy Monte Carlo
Random Sequences of Fuzzy Mixed Strategies
Max/Min of Fuzzy Numbers
Fuzzy Monte Carlo Solution Method
Conclusions and Future Research
Fuzzy Queuing Models
Introduction
Queuing Model
Fuzzy Queuing Model
Fuzzy Monte Carlo Method
Random Sequence $\overline{V}_k$
Maximum of Fuzzy Profit
Fuzzy Monte Carlo Solution
Summary and Conclusions
Part III: Unfinished Business
Fuzzy Min-Cost Capacitated Network
Introduction
Min-Cost Capacitated Network
Fuzzy Monte Carlo Method
Fuzzy Shortest Path Problem
Introduction
Fuzzy Shortest Path Problem
Monte Carlo Method
Fuzzy Max-Flow Problem
Introduction
Max-Flow Problem
Fuzzy Max-Flow Problem
Fuzzy Monte Carlo Solution
Inventory Control: Known Demand
Introduction
Inventory Problem
Monte Carlo Method
Monte Carlo Solution
Inventory Control: Fuzzy Demand
Introduction
Inventory Model
Monte Carlo Solution
Inventory Control: Backordering
Introduction
Inventory Model
Monte Carlo Method
Fuzzy Transportation Problem
Introduction
Transportation Problem
Fuzzy Transportation Problem
Fuzzy Integer Programming
Introduction
Fuzzy Integers
An Integer Programming Problem
A Fuzzy Integer Programming Problem
Fuzzy Monte Carlo Solution
Fuzzy Dynamic Programming
Introduction
A Dynamic Programming Problem
A Fuzzy Dynamic Programming Problem
Fuzzy Monte Carlo Solution
Fuzzy Project Scheduling/PERT
Introduction
Job Times Fuzzy Numbers
Job Times Discrete Fuzzy Sets
Fuzzy Monte Carlo Method
Max/Min Fuzzy Function
Introduction
Max/Min $f(\overline{X})$
Max/Min $f(\overline{X},\overline{Y})$
Part IV: Summary, Conclusions, Future Research
Summary, Conclusions, Future Research
Summary
Future Research
Conclusions
Index
List of Figures
List of Tables