Fuzzy Random Theory

Chapter 2 introduces some key literatures and their comments on fields related to the research and emphasizes some related thoughts and the theories and technical methods that the researches are based on, including supply chain management and its uncertainties, random theory, fiizzy theory, fuzzy random theory, tendency and characteristics of supply chain planning research, neural network algorithm and genetic algorithm. 

Uncertainty theory is also referred to as probability theory, credibility theory, or reliability theory and includes fuzzy random theory, random fuzzy theory, double stochastic theory, double fiizzy theory, the dual rough theory, fiizzy rough theory, random rough theory, and rough stochastic theory. This section focuses on the probability theory and fiizzy set theory, including probability spaces, random variables, probability spaces, credibility measurement, fuzzy variable and its expected value operator, and so on. 

Data collected by modern analytical instalments are usually presented by the multidimensional arrays. To perform the detection/identification of the supposed component or to verify the authenticity of a product, it is necessary to estimate the similarity of the analyte to the reference. The similarity is commonly estimated with the use of the distance between the multidimensional arrays corresponding to the compared objects. To exclude within the limits of the possible the influence of the random errors and the nonreproductivity of the experimental conditions and to make the comparison of samples more robust, it is possible to handle the arrays with the use of the fuzzy set theory apparatus. 

Considering the fuzzy stochastic customer demand in the three-level supply chain, we apply the fuzzy expectation theory to build a supply chain logistics planning fuzzy random expected value programming model with the restriction of manufacturing capacity, storage capacity, and transportation capacity. 

Objective function (6.8) and its restrictions might not have strict mathematical sense as the models includes fuzzy random variables. We transform the model by using the theory in Sect. 6.1 to the following fuzzy stochastic expected value programming model. 

Certainly, no microscope would let you see the twists and turns of an individual molecule s path. However, the Einstein-Smoluchowski theory tells us how to spot the difference between a fuzzy line which consists of a great number of tiny random kinks, and an ordinary smooth curve, even though we cannot discern the individual kinks. (We do not always need to see everything, e.g. we can happily tell water from alcohol even though the individual molecules are invisible ) In the same way, a poljrmer chain looks nothing like a shape stretched in a certain direction. And the path of a man in a forest would depend quite noticeably on whether he is equipped with a compass or not ... 

The effective-medium approach is valid only for the random-dispersion structure including the cases in which phase B disperses in matrix phase A and phase A conversely disperses in matrix phase B. However, for the percolationlike structure, in which the identification of dispersion phase and matrix phase is difficult to determine, the effective-medium theory cannot be used directly. To deal with such a transition area, a newly developed type of fuzzy logic [19, 20] may be useful for describing the complex microstructure and thermophysical properties. 

Usually, the demand is described with random variables under uncertain environment. When we describe the demand with variables, we need a great amount of empirical statistics to get the distribution function. However, these data might be hard to get in some cases. The fussy sets theory is then a commonly used and elfeclive method which can quantihably describe the uncertain demand. The membership function of fuzzy numbers can be determined by the decision-makers when using fuzzy numbers to describe the demand, which is much easier to determine the membership function than the determination of distribution functions of random variable. 

ABSTRACT The paper proposes an alternative approach to the failure risk analysis of water supply network which includes random emergency events inaccuracy, diversity and a small amount of data. The presented method is based on the so-called theory of facts (Shafer) based on the concept of inaccurate probabilities and possibilities (Zadech), which combines the so-called distribution capabilities of the belonging function of fuzzy set. The main aim of this paper is to present the concept of failure risk analysis of water supply network. The proposed model was used for failure risk analysis of water supply system serving 63 thousand people in the east of Poland. 

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