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Monte Carlo Methods in Fuzzy Optimization is a clear and
didactic book about Monte Carlo methods using random fuzzy numbers
to obtain approximate solutions to fuzzy optimization problems. The
book includes various solved problems such as fuzzy linear
programming, fuzzy regression, fuzzy inventory control, fuzzy game
theory, and fuzzy queuing theory. The book will appeal to
engineers, researchers, and students in Fuzziness and applied
mathematics.
1. 1 Introduction This book is written in four major divisions. The
first part is the introductory chapters consisting of Chapters 1
and 2. In part two, Chapters 3-11, we develop fuzzy estimation. For
example, in Chapter 3 we construct a fuzzy estimator for the mean
of a normal distribution assuming the variance is known. More
details on fuzzy estimation are in Chapter 3 and then after Chapter
3, Chapters 4-11 can be read independently. Part three, Chapters
12- 20, are on fuzzy hypothesis testing. For example, in Chapter 12
we consider the test Ho : /1 = /10 verses HI : /1 f=- /10 where /1
is the mean of a normal distribution with known variance, but we
use a fuzzy number (from Chapter 3) estimator of /1 in the test
statistic. More details on fuzzy hypothesis testing are in Chapter
12 and then after Chapter 12 Chapters 13-20 may be read
independently. Part four, Chapters 21-27, are on fuzzy regression
and fuzzy prediction. We start with fuzzy correlation in Chapter
21. Simple linear regression is the topic in Chapters 22-24 and
Chapters 25-27 concentrate on multiple linear regression. Part two
(fuzzy estimation) is used in Chapters 22 and 25; and part 3 (fuzzy
hypothesis testing) is employed in Chapters 24 and 27. Fuzzy
prediction is contained in Chapters 23 and 26. A most important
part of our models in fuzzy statistics is that we always start with
a random sample producing crisp (non-fuzzy) data.
Provides a broad and deep survey of Roman Catholic life and
thought, updated and expanded throughout The Wiley Blackwell
Companion to Catholicism provides an authoritative overview of the
history, doctrine, practices, and expansion of Catholicism. Written
by a group of distinguished scholars, this comprehensive reference
work offers an illuminating account of the global, historical, and
cultural phenomena of Catholicism. Accessible chapters address
central topics in the practice of Catholic theology and the
development of doctrine, including God and Jesus Christ, creation
and Church, the Virgin Mary, the sacraments, moral theology,
eschatology, and more. Throughout the text, the authors illustrate
the unity and diversity of Catholic life and thought while
highlighting the ways Catholicism overlaps with, and transforms,
other ways of living and thinking. Now in its second edition, The
Wiley Blackwell Companion to Catholicism is fully updated to
include recent developments in the study of Catholicism.
Extensively revised and expanded chapters, many of which written by
new authors, address contemporary issues such as theology and
politics, environmentalism, and the clerical sexual abuse crisis.
Entirely new chapters cover the early modern Church, the Bible in
Catholic theology, the Eastern Catholic churches, liturgy, care for
creation, the consecrated life, challenges for the Catholic Church,
and more. An informed and engaging intellectual journey through the
past and present of Roman Catholicism, The Wiley Blackwell
Companion to Catholicism: Illustrates the diversity of modern
Catholic life and thought Describes Catholics in different lands,
including the Holy Land, India, Africa, Europe, the British Isles,
Asia, Oceania, and the Americas Surveys spirituality and ecumenism,
inter-religious dialog, Catholic schools and hospitals, art and the
sciences, the Holy See, and other central Catholic institutions and
practices Covers major eras in Catholic history, from the
Scriptures and the early Church to Post-Modernity Features new
material on diverse practices of Catholicism across cultures, the
global dimensions of the Catholic Church, race and ethnicity, and
Eastern Catholic Churches The Wiley Blackwell Companion to
Catholicism, Second Edition, is the ideal textbook for surveys
classes on Catholicism and Catholic theology in Catholic,
Protestant, and non-confessional colleges and universities. It is
also an invaluable resource for scholars and general readers
interested in broadening their knowledge of Catholicism.
This book presents important applications of soft computing and fuzziness to the growing field of web planning. A new method of using fuzzy numbers to model uncertain probabilities and how these can be used to model a fuzzy queuing system is demonstrated, as well as a method of modeling fuzzy queuing systems employing fuzzy arrival rates and fuzzy service rates. All the computations needed to get to the fuzzy numbers for system performance are described starting for the one server case to more than three servers. A variety of optimization models are discussed with applications to the average response times, server utilization, server and queue costs, as well as to phenomena identified with web sites such as "burstiness" and "long tailed distributions".
1. 1 Introduction This book is written in two major parts. The ?rst
part includes the int- ductory chapters consisting of Chapters 1
through 6. In part two, Chapters 7-26, we present the applications.
This book continues our research into simulating fuzzy systems. We
started with investigating simulating discrete event fuzzy systems
([7],[13],[14]). These systems can usually be described as queuing
networks. Items (transactions) arrive at various points in the s-
tem and go into a queue waiting for service. The service stations,
preceded by a queue, are connected forming a network of queues and
service, until the transaction ?nally exits the system. Examples
considered included - chine shops, emergency rooms, project
networks, bus routes, etc. Analysis of all of these systems depends
on parameters like arrival rates and service rates. These
parameters are usually estimated from historical data. These
estimators are generally point estimators. The point estimators are
put into the model to compute system descriptors like mean time an
item spends in the system, or the expected number of transactions
leaving the system per unit time. We argued that these point
estimators contain uncertainty not shown in the calculations. Our
estimators of these parameters become fuzzy numbers, constructed by
placing a set of con?dence intervals one on top of another. Using
fuzzy number parameters in the model makes it into a fuzzy system.
The system descriptors we want (time in system, number leaving per
unit time) will be fuzzy numbers.
The primary purpose of this book is to present information about
selected topics on the interactions and applications of fuzzy +
neural. Most of the discussion centers around our own research in
these areas. Fuzzy + neural can mean many things: (1)
approximations between fuzzy systems and neu ral nets (Chapter 4);
(2) building hybrid neural nets to equal fuzzy systems (Chapter 5);
(3) using neura.l nets to solve fuzzy problems (Chapter 6); (4)
approximations between fuzzy neural nets and other fuzzy systems
(Chap ter 8); (5) constructing hybrid fuzzy neural nets for certain
fuzzy systems (Chapters 9, 10); or (6) computing with words
(Chapter 11). This book is not intend to be used primarily as a
text book for a course in fuzzy + neural because we have not
included problems at the end of each chapter, we have omitted most
proofs (given in the references), and we have given very few
references. We wanted to keep the mathematical prerequisites to a
minimum so all longer, involved, proofs were omitted. Elementary
dif ferential calculus is the only prerequisite needed since we do
mention partial derivatives once or twice."
The book aims at surveying results in the application of fuzzy sets
and fuzzy logic to economics and engineering. New results include
fuzzy non-linear regression, fully fuzzified linear programming,
fuzzy multi-period control, fuzzy network analysis, each using an
evolutionary algorithm; fuzzy queuing decision analysis using
possibility theory; fuzzy differential equations; fuzzy difference
equations; fuzzy partial differential equations; fuzzy eigenvalues
based on an evolutionary algorithm; fuzzy hierarchical analysis
using an evolutionary algorithm; fuzzy integral equations. Other
important topics covered are fuzzy input-output analysis; fuzzy
mathematics of finance; fuzzy PERT (project evaluation and review
technique). No previous knowledge of fuzzy sets is needed. The
mathematical background is assumed to be elementary calculus.
1.1 Introduction This book is written in the following divisions:
(1) the introductory chapters consisting of Chapters 1 and 2; (2)
introduction to fuzzy probability in Ch- ters3-5;
(3)introductiontofuzzyestimationinChapters6-11; (4)fuzzy/crisp
estimatorsofprobabilitydensity(mass)functionsbasedonafuzzymaximum
entropyprincipleinChapters12-14;
(5)introductiontofuzzyhypothesiste- ing in Chapters 15-18; (6)
fuzzy correlation and regression in Chapters 19-25; (7) Chapters 26
and 27 are about a fuzzy ANOVA model; (8) a fuzzy esti- tor of the
median in nonparametric statistics in Chapter 28; and (9) random
fuzzy numbers with applications to Monte Carlo studies in Chapter
29. First we need to be familiar with fuzzy sets. All you need to
know about fuzzy sets for this book comprises Chapter 2. For a
beginning introduction to fuzzysetsandfuzzylogicsee 8].
Oneotheritemrelatingtofuzzysets, needed infuzzyhypothesistesting,
isalsoinChapter2: howwewilldeterminewhich of the following three
possibilities is trueM N or M? N, for two fuzzy numbers M, N.
TheintroductiontofuzzyprobabilityinChapters3-5isbasedonthebook 1]
and the reader is referred to that book for more information,
especially applications. Whatisnewhereis:
(1)usinganonlinearoptimizationprogram in Maple 13] to solve certain
optimization problems in fuzzy probability, where previously we
used a graphical method; and (2) a new algorithm, suitable for
using only pencil and paper, for solving some restricted fuzzy
arithmetic problems. The introduction to fuzzy estimation is based
on the book 3] and we refer the interested reader to that book for
more about fuzzy estimators.
In probability and statistics we often have to estimate
probabilities and parameters in probability distributions using a
random sample. Instead of using a point estimate calculated from
the data we propose using fuzzy numbers which are constructed from
a set of confidence intervals. In probability calculations we apply
constrained fuzzy arithmetic because probabilities must add to one.
Fuzzy random variables have fuzzy distributions. A fuzzy normal
random variable has the normal distribution with fuzzy number mean
and variance. Applications are to queuing theory, Markov chains,
inventory control, decision theory and reliability theory.
Simulating Fuzzy Systems demonstrates how many systems naturally
become fuzzy systems and shows how regular (crisp) simulation can
be used to estimate the alpha-cuts of the fuzzy numbers used to
analyze the behavior of the fuzzy system. This monograph presents a
concise introduction to fuzzy sets, fuzzy logic, fuzzy estimation,
fuzzy probabilities, fuzzy systems theory, and fuzzy computation.
It also presents a wide selection of simulation applications
ranging from emergency rooms to machine shops to project
scheduling, showing the varieties of fuzzy systems.
How can Christians committed to the classical Christian tradition
address the issues raised by contemporary Islam? Before a
much-needed dialogue between Christians and Muslims is established,
Christians need to ask themselves how their Scriptures and
traditions might come to bear on such a dialogue. Do the divisions
among Catholic and Evangelical Christians fracture the classical
Christian tradition in ways that undercut Christian-Muslim dialogue
before it has even begun? Or could the classical tradition provide
invaluable resources for resolving divisions between Catholic and
Evangelical Christians in ways that would prepare them for
meaningful conversation with Muslim brothers and sisters? And what
does it have to teach us about what Christians can and must learn
from Muslims about their own traditions? The scholarly essays
compiled in Christian Theology and Islam consider these and further
questions, offering valuable insight for concerned Christians and
academics in the fields of theology and religion.
Simulating Fuzzy Systems demonstrates how many systems naturally
become fuzzy systems and shows how regular (crisp) simulation can
be used to estimate the alpha-cuts of the fuzzy numbers used to
analyze the behavior of the fuzzy system. This monograph presents a
concise introduction to fuzzy sets, fuzzy logic, fuzzy estimation,
fuzzy probabilities, fuzzy systems theory, and fuzzy computation.
It also presents a wide selection of simulation applications
ranging from emergency rooms to machine shops to project
scheduling, showing the varieties of fuzzy systems.
In probability and statistics we often have to estimate
probabilities and parameters in probability distributions using a
random sample. Instead of using a point estimate calculated from
the data we propose using fuzzy numbers which are constructed from
a set of confidence intervals. In probability calculations we apply
constrained fuzzy arithmetic because probabilities must add to one.
Fuzzy random variables have fuzzy distributions. A fuzzy normal
random variable has the normal distribution with fuzzy number mean
and variance. Applications are to queuing theory, Markov chains,
inventory control, decision theory and reliability theory.
The primary purpose of this book is to present information about
selected topics on the interactions and applications of fuzzy +
neural. Most of the discussion centers around our own research in
these areas. Fuzzy + neural can mean many things: (1)
approximations between fuzzy systems and neu ral nets (Chapter 4);
(2) building hybrid neural nets to equal fuzzy systems (Chapter 5);
(3) using neura.l nets to solve fuzzy problems (Chapter 6); (4)
approximations between fuzzy neural nets and other fuzzy systems
(Chap ter 8); (5) constructing hybrid fuzzy neural nets for certain
fuzzy systems (Chapters 9, 10); or (6) computing with words
(Chapter 11). This book is not intend to be used primarily as a
text book for a course in fuzzy + neural because we have not
included problems at the end of each chapter, we have omitted most
proofs (given in the references), and we have given very few
references. We wanted to keep the mathematical prerequisites to a
minimum so all longer, involved, proofs were omitted. Elementary
dif ferential calculus is the only prerequisite needed since we do
mention partial derivatives once or twice."
In probability and statistics we often have to estimate
probabilities and parameters in probability distributions using a
random sample. Instead of using a point estimate calculated from
the data we propose using fuzzy numbers which are constructed from
a set of confidence intervals. In probability calculations we apply
constrained fuzzy arithmetic because probabilities must add to one.
Fuzzy random variables have fuzzy distributions. A fuzzy normal
random variable has the normal distribution with fuzzy number mean
and variance. Applications are to queuing theory, Markov chains,
inventory control, decision theory and reliability theory.
1.1 Introduction This book is written in the following divisions:
(1) the introductory chapters consisting of Chapters 1 and 2; (2)
introduction to fuzzy probability in Ch- ters3-5;
(3)introductiontofuzzyestimationinChapters6-11; (4)fuzzy/crisp
estimatorsofprobabilitydensity(mass)functionsbasedonafuzzymaximum
entropyprincipleinChapters12-14;
(5)introductiontofuzzyhypothesiste- ing in Chapters 15-18; (6)
fuzzy correlation and regression in Chapters 19-25; (7) Chapters 26
and 27 are about a fuzzy ANOVA model; (8) a fuzzy esti- tor of the
median in nonparametric statistics in Chapter 28; and (9) random
fuzzy numbers with applications to Monte Carlo studies in Chapter
29. First we need to be familiar with fuzzy sets. All you need to
know about fuzzy sets for this book comprises Chapter 2. For a
beginning introduction to fuzzysetsandfuzzylogicsee 8].
Oneotheritemrelatingtofuzzysets, needed infuzzyhypothesistesting,
isalsoinChapter2: howwewilldeterminewhich of the following three
possibilities is trueM N or M? N, for two fuzzy numbers M, N.
TheintroductiontofuzzyprobabilityinChapters3-5isbasedonthebook 1]
and the reader is referred to that book for more information,
especially applications. Whatisnewhereis:
(1)usinganonlinearoptimizationprogram in Maple 13] to solve certain
optimization problems in fuzzy probability, where previously we
used a graphical method; and (2) a new algorithm, suitable for
using only pencil and paper, for solving some restricted fuzzy
arithmetic problems. The introduction to fuzzy estimation is based
on the book 3] and we refer the interested reader to that book for
more about fuzzy estimators.
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