# An Insight Into The Z-number Approach To CWW

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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.

Abstract: Real-world information is imperfect and is usually described in natural language (NL). Moreover, this information is often partially reliable and a degree of reliability is also expressed in NL. In view of this, L.A. Zadeh suggested the concept of a Z-number as a more adequate concept for description of real-world information. A Z-number is an ordered pair Z = ( A , B ) of fuzzy numbers A and B used to describe a value of a random variable X, where A is an imprecise estimation of a value of X and B is an imprecise estimation of reliability of A. The main critical problem that naturally arises in processing Z-numbers-based information is computation with Z-numbers. The general ideas underlying computation with continuous Z-numbers (Z-numbers with continuous components) is suggested by the author of the Z-number concept. However, as he mentions, "Problems involving computation with Z-numbers is easy to state but far from easy to solve". Nowadays there is no arithmetic of Z-numbers suggested in the existing literature. Taking into account the fact that real problems are characterized by linguistic information which is, as a rule, described by a discrete set of meaningful linguistic terms, in our study we consider discrete Z-numbers. We suggest theoretical aspects of such arithmetic operations over discrete Z-numbers as addition, subtraction, multiplication, division, square root of a Z-number and other operations. The validity of the suggested approach is demonstrated by a series of numerical examples.

180 citations

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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.

105 citations

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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.

98 citations

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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.

Abstract: In order to deal with imprecision and partial reliability of real-world information, Prof. Zadeh suggested the concept of a Z-number Z=(A, B), as an ordered pair of continuous fuzzy numbers A and B. The first describes a linguistic value, and the second one is the associated reliability. Unfortunately, up to day there is no works devoted to arithmetic of continuous Z-numbers in existence. An original formulation of operations over continuous Z-numbers proposed by Zadeh includes complex non-linear variational problems. We propose an alternative approach which has a better computational complexity and accuracy tradeoff. The proposed approach is based on linear programming and other simple optimization problems. We 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. Vast compendium of examples shows validity of the suggested approach.

94 citations

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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.

91 citations

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##### References

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TL;DR: WordNet1 provides a more effective combination of traditional lexicographic information and modern computing, and is an online lexical database designed for use under program control.

Abstract: Because meaningful sentences are composed of meaningful words, any system that hopes to process natural languages as people do must have information about words and their meanings. This information is traditionally provided through dictionaries, and machine-readable dictionaries are now widely available. But dictionary entries evolved for the convenience of human readers, not for machines. WordNet1 provides a more effective combination of traditional lexicographic information and modern computing. WordNet is an online lexical database designed for use under program control. English nouns, verbs, adjectives, and adverbs are organized into sets of synonyms, each representing a lexicalized concept. Semantic relations link the synonym sets [4].

13,247 citations

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TL;DR: The lexical database: nouns in WordNet, Katherine J. Miller a semantic network of English verbs, and applications of WordNet: building semantic concordances are presented.

Abstract: Part 1 The lexical database: nouns in WordNet, George A. Miller modifiers in WordNet, Katherine J. Miller a semantic network of English verbs, Christiane Fellbaum design and implementation of the WordNet lexical database and searching software, Randee I. Tengi. Part 2: automated discovery of WordNet relations, Marti A. Hearst representing verb alterations in WordNet, Karen T. Kohl et al the formalization of WordNet by methods of relational concept analysis, Uta E. Priss. Part 3 Applications of WordNet: building semantic concordances, Shari Landes et al performance and confidence in a semantic annotation task, Christiane Fellbaum et al WordNet and class-based probabilities, Philip Resnik combining local context and WordNet similarity for word sense identification, Claudia Leacock and Martin Chodorow using WordNet for text retrieval, Ellen M. Voorhees lexical chains as representations of context for the detection and correction of malapropisms, Graeme Hirst and David St-Onge temporal indexing through lexical chaining, Reem Al-Halimi and Rick Kazman COLOR-X - using knowledge from WordNet for conceptual modelling, J.F.M. Burg and R.P. van de Riet knowledge processing on an extended WordNet, Sanda M. Harabagiu and Dan I Moldovan appendix - obtaining and using WordNet.

12,607 citations

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TL;DR: By relying on the use of linguistic variables and fuzzy algorithms, the approach provides an approximate and yet effective means of describing the behavior of systems which are too complex or too ill-defined to admit of precise mathematical analysis.

Abstract: The approach described in this paper represents a substantive departure from the conventional quantitative techniques of system analysis. It has three main distinguishing features: 1) use of so-called ``linguistic'' variables in place of or in addition to numerical variables; 2) characterization of simple relations between variables by fuzzy conditional statements; and 3) characterization of complex relations by fuzzy algorithms. A linguistic variable is defined as a variable whose values are sentences in a natural or artificial language. Thus, if tall, not tall, very tall, very very tall, etc. are values of height, then height is a linguistic variable. Fuzzy conditional statements are expressions of the form IF A THEN B, where A and B have fuzzy meaning, e.g., IF x is small THEN y is large, where small and large are viewed as labels of fuzzy sets. A fuzzy algorithm is an ordered sequence of instructions which may contain fuzzy assignment and conditional statements, e.g., x = very small, IF x is small THEN Y is large. The execution of such instructions is governed by the compositional rule of inference and the rule of the preponderant alternative. By relying on the use of linguistic variables and fuzzy algorithms, the approach provides an approximate and yet effective means of describing the behavior of systems which are too complex or too ill-defined to admit of precise mathematical analysis.

8,223 citations

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TL;DR: If the meaning of the words “machine” and “think” are to be found by examining how they are commonly used it is difficult to escape the conclusion that the meaning and the answer to the question, “Can machines think?” is to be sought in a statistical survey such as a Gallup poll.

Abstract: I propose to consider the question, “Can machines think?”♣ This should begin with definitions of the meaning of the terms “machine” and “think”. The definitions might be framed so as to reflect so far as possible the normal use of the words, but this attitude is dangerous. If the meaning of the words “machine” and “think” are to be found by examining how they are commonly used it is difficult to escape the conclusion that the meaning and the answer to the question, “Can machines think?” is to be sought in a statistical survey such as a Gallup poll.

6,121 citations

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TL;DR: The point of this note is that fuzzy logic plays a pivotal role in CW and vice-versa and, as an approximation, fuzzy logic may be equated to CW.

Abstract: As its name suggests, computing with words (CW) is a methodology in which words are used in place of numbers for computing and reasoning. The point of this note is that fuzzy logic plays a pivotal role in CW and vice-versa. Thus, as an approximation, fuzzy logic may be equated to CW. There are two major imperatives for computing with words. First, computing with words is a necessity when the available information is too imprecise to justify the use of numbers, and second, when there is a tolerance for imprecision which can be exploited to achieve tractability, robustness, low solution cost, and better rapport with reality. Exploitation of the tolerance for imprecision is an issue of central importance in CW. In CW, a word is viewed as a label of a granule; that is, a fuzzy set of points drawn together by similarity, with the fuzzy set playing the role of a fuzzy constraint on a variable. The premises are assumed to be expressed as propositions in a natural language. In coming years, computing with words is likely to evolve into a basic methodology in its own right with wide-ranging ramifications on both basic and applied levels.

2,938 citations

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