The arithmetic of discrete Z-numbers
TL;DR: In this article, the main critical problem that naturally arises in processing Z-number-based information is computation with Z-numbers, which is a more adequate concept for description of real-world information.
About: This article is published in Information Sciences.The article was published on 2015-01-01. It has received 234 citations till now. The article focuses on the topics: Discrete system & Multiplication.
Citations
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TL;DR: A novel framework of dependence assessment in human reliability analysis is proposed based on Z
Abstract: Z -number is an effective model to describe uncertainty in the real world. Under the condition that uncertainty reasoning is an important issue to process information, how to achieve Z -valuation uncertainty reasoning is a problem. As a Z -number involves both fuzzy and probabilistic uncertainty, main difficulty in the problem to be solved is accomplishing both two uncertainties’ reasoning. In this paper, a novel Z -network model and its associated reasoning algorithm are proposed to overcome the difficulty. Structure of the proposed Z -network that contains three basic structures is directed acyclic graph, and this is similarly with Bayesian network (BN). Process of reasoning algorithm involves two parts: first, Bayesian reasoning is applied to establish an optimization model for probabilistic uncertainty reasoning in a Z -number; second, the arithmetic approach of discrete Z -number on if–then rule and maximum entropy approach are proposed for fuzzy uncertainty reasoning. Z -network is essentially an extended model on the basis of BN and properties of a Z -number for Z -valuation uncertainty reasoning. In application, a novel framework of dependence assessment in human reliability analysis is proposed based on Z -network, and a case study demonstrates its effectiveness.
186 citations
Cites background from "The arithmetic of discrete Z-number..."
...[8], [9] is proposed in this paper as follows:...
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...At first, some definitions and properties of Z-number based on existing research works [8], [14]–[19] are proposed, and with it, the arithmetic process of a discrete Znumber on the if–then rule and maximum entropy approach are proposed tomakepreparations forZ-network reasoning....
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...Definition 1: (A discrete Z-number [8]) A discrete Z-number is an ordered pair of discrete fuzzy sets denoted asX is (A,B), where A plays the role of the fuzzy restriction on values that a random variable X may take with a membership function...
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...Until now, varieties of mathematical models are proposed to express uncertainties, such as probability theory [2], Dempster–Shafer evidence theory [3]–[5], fuzzy mathematics [6], [7], Z-number [1], [8], [9], and so on....
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TL;DR: An extended TODIM method based on the Choquet integral for multi-criteria decision-making (MCDM) problems with linguistic Z-numbers is developed, which is a more comprehensive reflection of the decision-makers’ cognition but also is more in line with expression habits.
Abstract: Z-numbers are a new concept considering both the description of cognitive information and the reliability of information. Linguistic terms are useful tools to adequately and effectively model real-life cognitive information, as well as to characterize the randomness of events. However, a form of Z-numbers, in which their two components are in the form of linguistic terms, is rarely studied, although it is common in decision-making problems. In terms of Z-numbers and linguistic term sets, we provided the definition of linguistic Z-numbers as a form of Z-numbers or a subclass of Z-numbers. Then, we defined some operations of linguistic Z-numbers and proposed a comparison method based on the score and accuracy functions of linguistic Z-numbers. We also presented the distance measure of linguistic Z-numbers. Next, we developed an extended TODIM (an acronym in Portuguese of interactive and multi-criteria decision-making) method based on the Choquet integral for multi-criteria decision-making (MCDM) problems with linguistic Z-numbers. Finally, we provided an example concerning the selection of medical inquiry applications to demonstrate the feasibility of our proposed approach. We then verified the applicability and superiority of our approach through comparative analyses with other existing methods. Illustrative and comparative analyses indicated that the proposed approach was valid and feasible for different decision-makers and cognitive environments. Furthermore, the final ranking results of the proposed approach were closer to real decision-making processes. Linguistic Z-numbers can flexibly characterize real cognitive information as well as describe the reliability of information. This method not only is a more comprehensive reflection of the decision-makers’ cognition but also is more in line with expression habits. The proposed method inherited the merits of the classical TODIM method and considers the interactivity of criteria; therefore, the proposed method was effective for dealing with real-life MCDM problems. Consideration about bounded rational and the interactivity of criteria made final outcomes convincing and consistent with real decision-making.
144 citations
Cites background from "The arithmetic of discrete Z-number..."
...[19] and Bhanu [20] presented some operations for Z-numbers....
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...The first area focuses on the fundamental research on Z-numbers, including operations [19, 20], converting methods [21], and extension studies [22–27]....
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...In particular, the operations for Znumbers [18] are complex and include several variational problems [19]....
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TL;DR: This work developed basic arithmetic operations such as addition, subtraction, multiplication and division, and some algebraic operations as maximum, minimum, square and square root of continuous Z-numbers.
135 citations
TL;DR: Results show that the proposed framework can improve the previous methods with comparability considering the reliability of information using Z-numbers and is more flexible comparing with previous work.
Abstract: Environmental assessment and decision making is complex leading to uncertainty due to multiple criteria involved with uncertain information. Uncertainty is an unavoidable and inevitable element of any environmental evaluation process. The published literatures rarely include the studies on uncertain data with variable fuzzy reliabilities. This research has proposed an environmental evaluation framework based on Dempster–Shafer theory and Z-numbers. Of which a new notion of the utility of fuzzy number is proposed to generate the basic probability assignment of Z-numbers. The framework can effectively aggregate uncertain data with different fuzzy reliabilities to obtain a comprehensive evaluation measure. The proposed model has been applied to two case studies to illustrate the proposed framework and show its effectiveness in environmental evaluations. Results show that the proposed framework can improve the previous methods with comparability considering the reliability of information using Z-numbers. The proposed method is more flexible comparing with previous work.
133 citations
TL;DR: An innovative method for addressing multicriteria group decision-making (MCGDM) problems with Z-numbers under the condition that the weight information is completely unknown is developed.
Abstract: Z -number is the general representation of real-life information with reliability, and it has adequate description power from the point of view of human perception. This study develops an innovative method for addressing multicriteria group decision-making (MCGDM) problems with Z -numbers under the condition that the weight information is completely unknown. Processing Z -numbers requires effective support of reliable tools. Then, the normal cloud model can be employed to analyze the Z -number construct. First, the potential information involved in Z -numbers is invoked, and a novel concept of normal Z +-value is proposed with the aid of the normal cloud model. The operations, distance measurement, and power aggregation operators of normal Z +-values are defined. Moreover, an MCGDM method is developed by incorporating the defined distance measurement and power aggregation operators into the MultiObjective Optimization by Ratio Analysis plus the Full Multiplicative Form. Finally, an illustrative example concerning air pollution potential evaluation is provided to demonstrate the proposed method. Its feasibility and validity are further verified by a sensitivity analysis and comparison with other existing methods.
129 citations
Cites background or methods from "The arithmetic of discrete Z-number..."
...such as Gaussian distribution, are employed based on a reasonable hypothesis [14]....
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...in [8] are very complex because several nonlinear variational problems need to be solved in the computation process [14]....
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..., membership function and probability density function) need to be handled [14]....
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...In addition, the sum and product operations of Z-valuations were studied in [13], some operations of discrete Znumbers were defined in [14], and the arithmetic of continuous Z-numbers was introduced in [15]....
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References
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Book•
[...]
01 Aug 1996
TL;DR: A separation theorem for convex fuzzy sets is proved without requiring that the fuzzy sets be disjoint.
Abstract: A fuzzy set is a class of objects with a continuum of grades of membership. Such a set is characterized by a membership (characteristic) function which assigns to each object a grade of membership ranging between zero and one. The notions of inclusion, union, intersection, complement, relation, convexity, etc., are extended to such sets, and various properties of these notions in the context of fuzzy sets are established. In particular, a separation theorem for convex fuzzy sets is proved without requiring that the fuzzy sets be disjoint.
52,705 citations
Book•
01 Jan 1995TL;DR: Fuzzy Sets and Fuzzy Logic is a true magnum opus; it addresses practically every significant topic in the broad expanse of the union of fuzzy set theory and fuzzy logic.
Abstract: Fuzzy Sets and Fuzzy Logic is a true magnum opus. An enlargement of Fuzzy Sets, Uncertainty,
and Information—an earlier work of Professor Klir and Tina Folger—Fuzzy Sets and Fuzzy Logic
addresses practically every significant topic in the broad expanse of the union of fuzzy set theory
and fuzzy logic. To me Fuzzy Sets and Fuzzy Logic is a remarkable achievement; it covers its vast
territory with impeccable authority, deep insight and a meticulous attention to detail.
To view Fuzzy Sets and Fuzzy Logic in a proper perspective, it is necessary to clarify a point
of semantics which relates to the meanings of fuzzy sets and fuzzy logic.
A frequent source of misunderstanding fias to do with the interpretation of fuzzy logic. The
problem is that the term fuzzy logic has two different meanings. More specifically, in a narrow
sense, fuzzy logic, FLn, is a logical system which may be viewed as an extension and generalization
of classical multivalued logics. But in a wider sense, fuzzy logic, FL^ is almost synonymous
with the theory of fuzzy sets. In this context, what is important to recognize is that: (a) FLW is much
broader than FLn and subsumes FLn as one of its branches; (b) the agenda of FLn is very different
from the agendas of classical multivalued logics; and (c) at this juncture, the term fuzzy logic is
usually used in its wide rather than narrow sense, effectively equating fuzzy logic with FLW
In Fuzzy Sets and Fuzzy Logic, fuzzy logic is interpreted in a sense that is close to FLW. However,
to avoid misunderstanding, the title refers to both fuzzy sets and fuzzy logic.
Underlying the organization of Fuzzy Sets and Fuzzy Logic is a fundamental fact, namely,
that any field X and any theory Y can be fuzzified by replacing the concept of a crisp set in X and Y
by that of a fuzzy set. In application to basic fields such as arithmetic, topology, graph theory, probability
theory and logic, fuzzification leads to fuzzy arithmetic, fuzzy topology, fuzzy graph theory,
fuzzy probability theory and FLn. Similarly, hi application to applied fields such as neural network
theory, stability theory, pattern recognition and mathematical programming, fuzzification leads to
fuzzy neural network theory, fuzzy stability theory, fuzzy pattern recognition and fuzzy mathematical
programming. What is gained through fuzzification is greater generality, higher expressive
power, an enhanced ability to model real-world problems and, most importantly, a methodology for
exploiting the tolerance for imprecision—a methodology which serves to achieve tractability,
7,131 citations
"The arithmetic of discrete Z-number..." refers background in this paper
...However, it should be mentioned that converting Z-numbers to classical fuzzy numbers[17,19] leads to loss of original information....
[...]
...The second aspect is the fact that computation with discrete fuzzy numbers[7,8,25] and discrete probability distributions are characterized by a significantly lower computational complexity than that with continuous fuzzy numbers[17,19] and density functions....
[...]
TL;DR: In probability theory, an event, A, is a member of a a-field, CY, of subsets of a sample space ~2, where CY is any collection of disjoint events.
2,396 citations
"The arithmetic of discrete Z-number..." refers background in this paper
...Definition 4 ([37] Probability measure of a discrete fuzzy number)....
[...]
TL;DR: This book introduced many novel mathematical operations based on this concept of level of confidence and have presented many generalizations, and presented several operations and functions of fuzzy numbers, such as integer modulo operations, trigonometric functions, and hyperbolic functions.
Abstract: We were rather pleased to read the review of our book, Introduction to Fuzzy Arithmetic: Theory and Applications. This review was done quite carefully by Caroline M. Eastman of the University of South Carolina, and we are grateful to her for pointing out many interesting, positive aspects as well as some shortcomings of our book. As members of the fuzzy community, we are concerned with studies and developments of concepts and techniques basic to the analysis of uncertainty arising from human perception, thinking, and reasoning processes. In this book we present such concepts and some novel tools for dealing with uncertainties. We start our introduction with the definition for the interval of confidence [al, a2], where al and a2 represent, respectively, the lower and upper bounds of our (subjective) confidence. Next, we introduce some arithmetic operations on these numbers. We then introduce the level of presumption ue [13, 1] and, using it, introduce the uncertain or fuzzy number that is so pervasive in our reasoning process. The reviewer has rightly pointed out that in certain situations, interval arithmetic can be considered a subset of fuzzy arithmetic, the main topic of our book. However, we intentionally did not want to confuse the issue by introducing interval arithmetic and then giving a generalization. We liked our approach, as have many other researchers and students who have used the book. In our approach, we have been guided throughout by a desire to lay a firm foundation for the definition of fuzzy numbers using the basic concept of level of confidence. We have introduced many novel mathematical operations based on this concept and have presented many generalizations. In addition, we have presented several operations and functions of fuzzy numbers, such as integer modulo operations, trigonometric functions, and hyperbolic functions. These studies have been included for students as well as researchers who wish to have an extended view of the theory. We have attempted to give a thorough exposition of fuzzy numbers; this exposition is illustrated by about 115 worked-out examples, 150 diagrams, and 90 tables. We did not include problems or exercises, which would have put this book in the category of a textbook. The subtitle of the book is \"Theory and Applications,\" but as is rightly
2,238 citations
"The arithmetic of discrete Z-number..." refers background in this paper
...However, it should be mentioned that converting Z-numbers to classical fuzzy numbers[17,19] leads to loss of original information....
[...]
...The second aspect is the fact that computation with discrete fuzzy numbers[7,8,25] and discrete probability distributions are characterized by a significantly lower computational complexity than that with continuous fuzzy numbers[17,19] and density functions....
[...]
Book•
01 Jan 1986TL;DR: The text has tried to strike a balance between simplicity in exposition and sophistication in analytical reasoning, and ensure that the mathematically oriented reader will find here a smooth development without major gaps.
Abstract: The course is attended by a large number of undergraduate and graduate students with diverse backgrounds. Acccordingly, we have tried to strike a balance between simplicity in exposition and sophistication in analytical reasoning. Some of the more mathematically rigorous analysis has been just sketched or intuitively explained in the text, so that complex proofs do not stand in the way of an otherwise simple exposition. At the same time, some of this analysis and the necessary mathematical results are developed (at the level of advanced calculus) in theoretical problems, which are included at the end of the corresponding chapter. The theoretical problems (marked by *) constitute an important component of the text, and ensure that the mathematically oriented reader will find here a smooth development without major gaps.
1,326 citations