Enfoscados, guarnecidos y enlucidos Descripción

0 0.2 0.4 0.6 0.8 1 alpha 28 28.5 29 29.5 x

Figure 3.5: Fuzzy Estimatorµin Example 3.6.1

3.7

Summary

We saw in this chapter that our fuzzy estimators can be triangular fuzzy numbers; or triangular shaped fuzzy numbers where we complete the base by drawing short vertical line segments from the horizontal axis up to the graph and the base represents a 99% confidence interval. In the rest of this book, for simplicity, all our fuzzy estimators will be triangular fuzzy numbers.

3.8

References

1. J.J.Buckley: Fuzzy Statistics, Springer, Heidelberg, Germany, 2004. 2. J.J.Buckley: Fuzzy Probabilities and Fuzzy Sets for Web Planning,

Springer, Heidelberg, Germany, 2004.

3. J.J.Buckley: Simulating Fuzzy Systems, Springer, Heidelberg, Ger- many, 2005.

4. Maple 9, Waterloo Maple Inc., Waterloo, Canada.

5. M.Olinick: An Introduction to Mathematical Models in the Social and Life Sciences, Addison-Wesley, Reading, MA, 1978.

32 CHAPTER 3. FUZZY ESTIMATION 7. R.V.Hogg and E.A.Tanis: Probability and Statistical Inference, Sixth

Chapter 4

Fuzzy Systems

4.1

Introduction

We have argued before that many crisp systems become fuzzy systems be- cause the values of some of the parameters in the crisp system are not known precisely and need to be estimated. Too often point estimators for these para- meters are obtained and used in the models. But the point estimators do not contain any uncertainty. We proposed incorporating uncertainty into these estimators by employing fuzzy estimators discussed in Chapter 3. Fuzzy es- timators are fuzzy numbers so some of the parameters in the model become fuzzy numbers making it into a fuzzy system. We will now discuss this trans- formation from crisp system into a fuzzy system in more detail for continuous systems.

This book is concerned with using crisp continuous simulation to estimate the evolution of continuous fuzzy systems. The continuous fuzzy systems we will look at are all governed by fuzzy differential equations. So, let us first look at our previous work in this area of using crisp simulation to study fuzzy systems([1],[2],[6]-[9]).

We started with studying what are called discrete event (fuzzy) systems. These systems can usually be described as queuing networks. Items (transac- tions) 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 net- works, 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 transac- tions leaving the system per unit time. We argued that these point estimators

34 CHAPTER 4. FUZZY SYSTEMS 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 (Chapter 3). 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.

Continuous systems are usually described by a system of ordinary dif- ferential equations (ODEs). Many parameters in the system of ODEs are not known precisely and must be estimated. To show the uncertainty in these parameter values we will use fuzzy number estimators. Fuzzy number parameter values produce a system of fuzzy ODEs to solve and we have a continuous fuzzy system. Solution trajectories become fuzzy trajectories. We plan to use crisp continuous simulation to estimate these fuzzy trajectories.

We will start with a continuous crisp system whose description in time depends on crisp ordinary differential equations. Let us consider an example of a predator/prey model, also discussed in detail in Chapter 7. This is adopted from an example in ([10],[11]). The system of differential equations is

˙

x=−ax+bxy, (4.1)

˙

y=dy−cxy, (4.2)

for constantsa, b, c, dall positive and initial conditionsx(0) =x₀,y(0) =y₀. We write the time derivative ofx(y) as ˙x( ˙y). Non-trivial solutions to this system can not be obtained in terms of elementary functions. So we would need to employ some software to obtain the graphs ofx(t) andy(t). How do we get values fora, b, c, d? We considered two cases in in Chapter 3: (1) their values are estimated by expert opinion; and (2) their values are estimated from data by placing confidence intervals one on top of another. In either case the estimators become fuzzy, or fuzzy numbers.

In the predator/prey model above we could have some, or all the parame- ters, fuzzy. Therefore,a, b, c, d, x₀ andy₀ may all be fuzzy. Then we have a system of fuzzy nonlinear differential equations to solve. The trajectories forx(t) andy(t), t ≥0, will be fuzzy which means for each value of t x(t) and y(t) will be fuzzy numbers. We have considered solving fuzzy differ- ential equations before ([3]-[5]). However, in those papers/book we almost always allowed for only fuzzy initial conditions because the fuzzy solution became too difficult to obtain when more parameters became fuzzy. Now we may fuzzify more parameters because we are not finding a formula for the mathematical solution but instead we will use simulation.