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Simulating Continuous Fuzzy Systems (Studies in Fuzziness and Soft Computing)

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This book is the companion text to Simulating Fuzzy Systems which investigated discrete fuzzy systems through crisp discrete simulation. The current book studies continuous fuzzy dynamical systems using crisp continuous simulation. We start with a crisp continuous dynamical system whose evolution depends on a system of ordinary differential equations (ODEs). The system of ODEs contains parameters many of which have uncertain values. Usually point estimators for these uncertain parameters are used, but the resulting system will not display any uncertainty associated with these estimators. Instead we employ fuzzy number estimators, constructed from expert opinion or from data, for the uncertain parameters. Fuzzy number estimators produces a system of fuzzy ODEs to solve whose solution will be fuzzy trajectories for the variables. We use crisp continuous simulation to estimate the trajectories of the support and core of these fuzzy numbers in a variety of twenty applications of fuzzy dynamical systems. The applications range from Bungee jumping to the AIDS epidemic to dynamical models in economics.

This book is written in two major parts. The first part includes the introductory
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 system
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 finally exits the system. Examples considered included machine
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 confidence 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. In general, computing these fuzzy numbers
can be difficult. We showed how crisp discrete event simulation can be used
to estimate the fuzzy numbers used to describe system behavior.
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