Fuzziness

In this article it has been tried to show that fuzzy theory performs better than probability theory in monitoring the product quality.A method that uses statistical techniques to monitor and control product quality is called statistical process control (SPC), where control charts are test tools frequently used for monitoring the manufacturing process.In this study, statistical quality control and the fuzzy set theory are aimed to combine.As known, fuzzy sets and fuzzy logic are powerful mathematical tools for modeling uncertain systems in industry, nature and humanity; and facilitators for common-sense reasoning in decision making in the absence of complete and precise information.In this basis for a textile firm for monitoring the yarn quality, control charts according to fuzzy theory by considering the quality in terms of grades of conformance as opposed to absolute conformance and nonconformance.And then with the same data for a textile factory, the control chart based on probability theory is constructed.The results of control charts based on two different approaches are compared.It's seen that fuzzy theory performs better than probability theory in monitoring the product quality.

The technological progress in the field of artificial intelligence (AI) has brought new opportunities and challenges to the intelligent development and innovative research in fuzzy spatial relations, fuzzy theory is the basic theory of intelligent information processing, and the paper gives a brief analysis of fuzzy calculation method in spatial analysis and spatial relation calculation. In particular, the spatial relations of fuzzy objects are more difficult to express and describe accurately. This paper proposes a method to calculate the topological relations between fuzzy objects with 4-Intersection model and 9-Intersection model, in the analysis of spatial relations, the extension from deterministic entity to uncertain entity is realized with fuzzy set theory, the fuzzy space matrix is constructed, then fuzzy topological relations and fuzzy membership are calculated and output, the model simplifies many complex relations, and it can describe eight topological relations including disjoint, meet, overlap, covers, contains, covered-by, inside and equal. In practical application, the position of spatial objects can be controlled and adjusted according to topological relations. Through analysis of algorithm and fuzzy membership calculation, the model is effective in theory.

Image processing has to deal with many ambiguous situations. Fuzzy set theory is a useful mathematical tool for handling the ambiguity or uncertainty. In order to apply the fuzzy theory, selecting the fuzzy region of membership function is a fundamental and important task. In this paper, a new method of membership function based on fuzzy theory by PSO algorithm optimized was proposed by analyzing the deficiencies of traditional enhancement algorithm. A new entropy definition of a fuzzy set was proposed. The new entropy definition of a fuzzy set was not only related to the membership (fuzzy domain) but also related to the probability distribution (space domain), it can respond to the variety of image input information. In addition, by quoting a novel particle swarm optimization (PSO) algorithm to find the optimization parameters for membership. We have employed the new proposed approach to perform image enhancement and thresholding and obtained satisfactory results.

The ability to describe variables in a health risk model through probability theory enables us to estimate human health risk. These types of risk assessment are interpreted as probabilistic risk assessment (PRA). Generally, PRA requires specific estimate of the parameters of the probability density of the input variables. In all circumstances, such estimates of the parameters may not be available due to the lack of knowledge or information. Such types of variables are treated as uncertain variables. These types of information are often termed uncertainty which are interpreted through fuzzy theory. The ability to describe uncertainty through fuzzy set theory enables us to process both random variable and fuzzy variable in a single framework. The method of processing aleatory and epistemic uncertainties into a same framework is coined as hybrid method. In this paper, we are going to talk about such type of hybrid methodology for human health risk assessment. Risk assessment on human health through different pathways of exposure has been attempted many a times combining Monte Carlo analysis and extension principle of fuzzy set theory. The emergence of credibility theory enables transforming fuzzy variable into credibility distribution function which can be used in those hybrid analyses. Hence, an attempt, for the first time, has been made to combine probability theory and credibility theory to estimate risk in human health exposure. This method of risk assessment in the presence of credibility theory and probability theory is identified as probabilistic-credibility method (PCM). The results obtained are then interpreted through probability theory, unlike the other hybrid methodology where the results are interpreted in terms of possibility theory. The results obtained are then compared with probability-fuzzy risk assessment (PFRA) method. Generally, decision under hybrid methodology is made on the index of optimism. An optimistic decision maker estimates from the $$\alpha$$ -cut at 1, whereas a pessimistic decision maker estimates from the $$\alpha$$ -cut at 0. The PCM is an optimistic approach as the decision is always made at $$\alpha$$ =1.

Mathematical theories of uncertainty in which measures of uncertainty are now well established are fuzzy set theory, evidence theory, and possibility theory, in addition to classical set theory and probability theory. Some of these theories are connected. Classical set theory is subsumed under fuzzy set theory. Probability theory and possibility theory are branches of evidence theory, while evidence theory is, in turn, a branch of fuzzy measure theory. Although information aspects of fuzzy measure theory have not been investigated as yet, the theory is briefly introduced in this chapter because it represents a broad framework for future research.

Being classic K-means++ clustering algorithm only for static data, dynamically incremental K-means++ clustering algorithm (DK-Means++) is presented based on fuzzy rough set theory in this paper. Firstly, in DK-Means++ clustering algorithm, the formula of similar degree is improved by weights computed by using of the important degree of attributes which are reduced on the basis of rough fuzzy set theory. Secondly, new data only need match granular which was clustered by K-means++ algorithm or seldom new data is clustered by classic K-means++ algorithm in global data. In this way, that all data is re-clustered each time in dynamic data set is avoided, so the efficiency of clustering is improved. Throughout our experiments showing, DK-Means++ algorithm can objectively and efficiently deal with clustering problem of dynamically incremental data.

Multiple variable modeling of purchasing performance is very important. Due to many fuzzy concepts which are not clearly defined or difficult to quantify in operation, we introduce the fuzzy theory in the neural network model to establish a fuzzy neural network to model the multiple variable purchasing performances. Fuzzy theory can explain the fuzzy phenomenon that is unique to human language while the neural network can automatically dig out the complex relationship between indicators. This paper analyzes the feasibility of the scheme.

Within the linguistic field, the phenomenon found by Fuzzy Set Theory application is called “fuzziness”. Linguistic study on fuzziness has lagged behind the multitudinous literature on vagueness and ambiguity. However, as an unconventional form of semantics and the entity of imprecision and rationality, the study of fuzziness remains a valuable area to explore. As fuzziness is a phenomenon existing in real world, as well as in natural language, the fuzzy set theory lighted up by linguistic method can serve as a tool to study language in turn. The close kinship created by linguistics and the birth of fuzzy sets in turn help to fetch the essence of fuzziness and resume the study of linguistics along the same line. Since there hasn't been generally agreed approach to Fuzzy Linguistics up till now, the linguistic application of Fuzzy Theory to natural language is clarified in this paper, and a working definition of linguistic fuzziness is provided.

A quantum mechanical approach to a fuzzy theory

With the increase of XLPE cable we used in power network, the corresponding technology of on-line monitoring becomes increasingly important.For improving the accuracy of on-line monitoring assessment of XLPE cable, we researched an assessment method and device of XLPE cable's insulation state by Fuzzy Theory.According to legitimately distribute between Fuzzy set and membership function in Fuzzy Theory, we solved the problem that how to distribute the membership of XLPE cable's insulation parameters and accomplished the assessment of XPLE cable's insulation state.The experimental operation results showed that the device can operate stably and interactwith background monitoring platform in real-time, which accomplished the on-line monitoring of XLPE cable and assessment of its insulation state.It can effectively ensure the safety of power network's operation.

In this chapter, some basic definitions and operations on the concepts of fuzzy set, fuzzy number, intuitionistic fuzzy set, single-valued neutrosophic set, single-valued neutrosophic number (SVN-number) are presented. Secondly, two centroid point are called 1. and 2. centroid point for single-valued trapezoidal neutrosophic number (SVTN-number) and single-valued triangular neutrosophic number (SVTrN-number) are presented. Then, some desired properties of 1. and 2. centroid point of SVTN-numbers and SVTrN-numbers studied. Also, based on concept of 1. and 2. centroid point of SVTrN-numbers, a new single-valued neutrosophic multiple-attribute decision-making method is proposed. Moreover, a numerical example is introduced to illustrate the availability and practicability of the proposed method. Finally, since centroid points of normalized SNTN-numbers or SNTrN-numbers are fuzzy values, all definitions and properties of fuzzy graph theory can applied to SNTN-numbers or SNTrN-numbers. For example, definition of fuzzy graph theory based on centroid points of normalized SVTN-numbers and SVTrN-numbers is given.

Information theory was introduced by Shannon in 1948 and it includes the study of uncertainty measures. Analogous to information theory which is based on probability theory fuzzy set theory was developed by Zadeh in 1965 which is a mechanism to manage uncertainty. The pedestal for present study is information theory and fuzzy set theory which is termed as fuzzy information theory. In this paper a new fuzzy entropy measure is proposed for two fuzzy sets A which is based upon fuzzy entropy given by Pal and Pal [7] with some of its properties.

The notion of a fuzzy graph was initially introduced by Kauffman (Introduction to the theory of fuzzy sets, Academic Press Inc., Orlando) in [91]. However, the development of fuzzy graph theory is due to the ground setting papers of Rosenfeld (Fuzzy sets and their applications, Academic Press, New York) [154] and Yeh and Bang (Fuzzy sets and their applications, Academic Press, New York) [186]. In Rosenfeld’s paper, basic structural and connectivity concepts were presented while Yeh and Bang introduced different connectivity parameters and discussed their application. Rosenfeld obtained the fuzzy analogs of several graph-theoretic concepts like bridges, paths, cycles, trees, and connectedness. Most of the theoretical development of fuzzy graph theory is based on Rosenfeld’s initial work.

Characterizations of endograph metric and Γ-convergence on fuzzy sets

Computation scheme for the general purpose VLSI fuzzy inference engine as expert system