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The first think to do is to explain what is our “new approach” and how it fits into the area of uncertain probabilities. We first consider a very simple example using interval probabilities. Let X = {x1, x2, x3} be a finite set and let P be a probability function defined on all subsets of X with P({xi}) = a...
In this chapter we have collected together the basic ideas from fuzzy sets and fuzzy functions needed for the book. Any reader familiar with fuzzy sets, fuzzy numbers, the extension principle, α-cuts, interval arithmetic, and fuzzy functions may go on and have a look at Sections 2.5 through 2.10. In Section 2.5 we discuss our method of handling the maximun/minimum of a fuzzy set to be used in Chapter...
This chapter is based on [1]. However, many new results have been added mostly on fuzzy conditional probability (Section 3.3), fuzzy independence (Section 3.4), fuzzy Bayes’ formula (Section 3.5) and the applications in Section 3.6.
The results in Section 5.3 come from [1]. However, the applications in Sections 5.4 were adapted from [2]. We first review some basic results from regular, finite, Markov chains (also in the next Chapter) applied to queuing theory and then show in Section 5.3 that we also obtain steady state fuzzy probabilities for fuzzy regular, finite, Markov chains. Then in Section 5.4 we discuss two applications.
This Chapter continues our research into fuzzy Markov chains. In [4] we employed possibility distributions in finite Markov chains. The rows in a transition matrix were possibility distributions, instead of discrete probability distributions. Using possibilities we went on to look at regular, and absorbing, Markov chains and Markov decision processes.
A decision problem under risk is characterized by three things: (1) the decision maker must choose an action ai from a finite set of actions A = {a1,..., am}; (2) a finite set of chance events (also called the states of nature) ε = {e1,..., en} over which we have no control;...
This Chapter is based on [1]. New material includes the applications in Section 8.5. We consider the fuzzy uniform in Section 8.2, the fuzzy normal is in Section 8.3, followed by the fuzzy negative exponential in Section 8.4. In each case of a fuzzy density function we first discuss how they are used to compute fuzzy probabilities and then we find their fuzzy mean and their fuzzy variance. We always...
Most of the applications of fuzzy sets to inventory control have been in the area of the EOQ, or the economic order quantity problem. See ([4]-[12],[14]-[16]), and the references in these papers, for a survey of this literature. What was usually done by these authors is to fuzzify some of the, difficult to estimate, parameters in the model. We are going to do something different in this Chapter in...
This Chapter generalizes Chapters 4 and 8 to multivariable fuzzy probability distributions. Since the discrete case and the continuous case are similar, just interchange summation and integrals symbols, we only present the continuous case in Section 10.2. Applications are in the next Chapter. The continuous case is based on [1]. For simplicity we will consider only the joint fuzzy probability distributions...
This Chapter presents two applications of joint fuzzy probability distributions discussed in Chapter 10. In the next section we have an application of a joint fuzzy discrete probability distribution followed by a discussion of fuzzy reliability theory which is an application of joint fuzzy continuous probability distribution.
In this chapter we will look at a few examples of functions of a single discrete (or continuous) fuzzy random variable. In the next chapter we consider functions of continuous fuzzy random variables (more than one). We start with discrete fuzzy random variables and then we discuss continuous fuzzy random variables.
We first discuss some theoretical results and then look at applications in the next two sections. The theoretical results are stated in terms of continuous fuzzy random variables.
Let X1, ..., Xn be a random sample from N(μ, σ2) and define $$ % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4vaiabg2...
We first summarize Chapters 3 through 15, without discussing any of the applications contained within a chapter, and then present our suggestions for future research. Our conclusions are at the end of the Chapter.
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