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Keywords: Z-number, a continuous fuzzy number, a random variable, convolution, a probability measure, arithmetic operations
1.    Introduction
Imperfect information naturally related to real-world phenomena is often characterized by
two main aspects. From the one side, values of interest are often not measured, but are estimated
based on perception and knowledge of a human being. In such cases, natural language (NL)
based estimations are used which can be formally described by fuzzy numbers. From the other
side, we should account for the fact that estimations are subjective, and therefore, are not fully
reliable. The reliability is, as a rule, partial - it is to an extent of a confidence of a human being.
This partial reliability, stemming from perceptual and mental sources can be considered as
imprecise probabilistic belief for modeling of which fuzzy value of probability measure can be
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used.
In order to ground a formal basis for dealing with real-world information, L.A. Zadeh
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suggested the concept of a Z-number [35] as an ordered pair Z = (A, B) of continuous fuzzy numbers used to describe a value of a random variable X, where A is a fuzzy constraint on values of X and B is a fuzzy reliability of A, and is considered as a value of probability measure of A. The author suggests a general approach for computations over Z -numbers on the base of the Zadeh’s extension principle. The approach is used to solve a problem of computation of a Z-number-valued function of Z-number-valued arguments.
Let us note that the Z-number concept is not the first attempt to model real-world uncertainty which is too complex to be captured by interval or fuzzy number-based representations. In fuzzy numbers, uncertainty is described by a numerical membership function. This means that they do not take into account inferred uncertainty interval. The first attempt to deal with such uncertainty
intervals was made in the theory of type-2 fuzzy sets [9,10,11,22]. However, in contrast to a
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type-2 fuzzy set, a Z-number explicitly represents reliability described in NL, and is a more structured formal construct. Hence, processing of Z-information requires to develop a new theory, new approaches and methodologies of computation with Z-numbers.
A general framework to computation with Z-numbers is suggested in [35]. In particular, implementation of arithmetic operations over two Z-numbers is expressed. However, practical realization of the suggested framework is computationally very complex, it includes several variational problems. As Zadeh claims, “Problems involving computation with Z-numbers is easy to state but far from easy to solve”, and there is no detailed and effective methods in existence to develop arithmetic, algebraic and other operations of Z-numbers. The author also raises important issues of computations with Z-numbers including ranking of Z-numbers and calculus of IF-THEN rules with Z-number valued components. In general, paper [35] opens a door for a lot of potential investigations of computation with Z-numbers.
Let us overview the existing works devoted handling of Z-numbers and its practical applications.
In [16] they suggest an approach to dealing with Z-numbers which naturally arise in the areas of control, decision making, modeling and others. The approach is based on converting a Z- number to a fuzzy number on the base of an expectation of a fuzzy set. However, converting Z- numbers to fuzzy numbers [18,20] leads to loss of original information reducing the benefit of using original Z-number-based information.
In [17] they considered an approach to multi-criteria decision making with Z -numbers on the base of the approach given in [16]. In the suggested framework, criteria weights and criteria values of alternatives are given as Z-numbers. However, the resulting overall performance evaluations of alternatives are computed as real numbers. This significant loss of information contained in Z-numbers is likely to lead to an inadequate multicriteria choice.
Papers [33,34] are devoted to new approaches of processing Z-numbers in various important
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fields. It is suggested to deal with Z-number Z = (A, B)in terms of a possibility distribution G(p) of probability distributions p which underlie Z = (A, B). Based on such representation, the author suggests manipulations over Z-numbers and their applications to reasoning, decision making and answering questions. Several detailed examples on computation with Z-information are provided to illustrate usefulness of the suggested approach. Another important potential impact of the suggested research is application to formalization of linguistic summaries. The author also suggests an alternative formulation of Z -information in terms of a Dempster-Shafer belief structure [28] which involves type-2 fuzzy sets [23,27,41]. In the paper, it is also suggested to rank Z-numbers by proceeding to the corresponding fuzzy numbers. The resulting fuzzy numbers are then compared on the base of their defuzzified values. However, this comparison is based on reducing a Z-number to a numeric value which is naturally characterized by a sufficient loss of information.
The work [39] is devoted to computation over continuous Z-numbers and several important practical problems in control, decision making and other areas. The suggested investigation is based on the use of normal probability density functions for modeling random variables. A special emphasis in some of the examples is made on calculus of Z-numbers based IF-THEN rules. A series of illustrative examples is provided on problems with Z-information in economics, social sphere, engineering, everyday activity and other fields.
Paper [5] is the first work devoted to decision making under uncertainty when probabilities of states of nature and outcomes of alternatives are described by Z-numbers. The suggested decision analysis is based on two main stages. At the first stage, Z-numbers are reduced to fuzzy numbers on the base of the approach suggested in [16]. At the second stage, values of fuzzy utility function for alternatives are computed to choose the best one. The main disadvantage is related to the loss of information resulting from converting Z-numbers to fuzzy numbers.
Paper [7] is devoted to potential contributions of application of the Z-number concept to
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development of computing with words (CWW) methodology. The authors suggest an approach
to CWW using Z-numbers and provide a real-life illustrating example.
In [25] they suggest an outline of the general principles, challenges and perspectives of
CWW in light of the Z-number concept and consider issues of integration of CWW and Natural
Language Processing technology.
The work [24] is also devoted to Z-numbers based approach to CWW. The authors suggest
basis for a system of processing sentences in NL by using Bayes’ approach and Shannon’s
entropy theorem.
In [19] they suggest an enhanced inference engine toolkit for implementation of CWW
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technologies in a general scope including combinations of fuzzy IF-THEN rules, fuzzy
arithmetic and fuzzy probabilities. The authors mention that the suggested toolkit can be further
developed to apply for computation with Z-numbers, without involvement into problems with
high computational complexity.
In [6] they consider an application of the AHP approach under Z-information. The suggested procedure is based on the approach proposed in [16]. Despite that in the suggested procedure alternatives are described in the realm of Z-information, they are compared on the basis of
the AHP ap
numeric overall utilities. Unfortunately, this significantly reduces benefits of using Z¬/V
information.
In [21] decision making under interval, set-valued, fuzzy, and Z-number uncertainty are
consid ered. The decision analysis technique suggested by the authors is based on the fair price approach.
In [30] they suggest several approaches of approximate evaluation of a Z-number in order to reduce computational complexity. One of the suggested approaches is based on approximation of a fuzzy set of probability densities by means of fuzzy IF-THEN rules.
In [2,3,4] a general and computationally effective approach to computation with discrete Z-
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numbers is suggested. The authors provide strong motivation of the use of discrete Z-numbers as an alternative to the continuous counterparts. In particular, the motivation is based on the fact that NL-based information has a discrete framework. From the other side, in a discrete framework it is not required to decide upon a reasonable assumption to use some type of probability distributions. The suggested arithmetic of discrete Z-numbers includes basic arithmetic operations and important algebraic operations. The proposed approach allows dealing with Z-numbers directly without conversion to fuzzy numbers.
Let us mention, that despite of wide applicability of computation of discrete Z-numbers, in a lot of real-world problems relevant information comes in a continuous framework which may require computing with continuous Z-numbers. We can conclude that today there is no general and computationally effective approach to computations with continuous Z-numbers. In the existing papers, information described by continuous Z-numbers is reduced to fuzzy numbers or crisp numbers that always leads to loss and distortion of information. However, processing of original Z-numbers-based information is very important for solving a large variety of real-world problems.
A new approach should be developed to introduce basic arithmetic, algebraic and other important operations for Z-numbers, which are worth to consider from theoretical and practical perspectives. This approach needs to be relatively easily applied for problems in realms control, decision analysis, optimization, forecasting and other areas. Computation with Z-numbers is characterized by propagation of combinations of possibilistic-probabilistic restrictions, that is, involves restriction-based computation.
Nowadays, existing literature devoted to computation and reasoning with restrictions includes well-developed approaches and theories to deal with pure probabilistic or pure possibilistic restrictions.
For computation with probabilistic restrictions as probability distributions the well-known
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probabilistic arithmetic is used [26,31,32]. Fuzzy arithmetic [18,20] deals with possibilistic constraints, which describe objects as classes with “unsharp” boundaries.
In this paper we continue our research on computation with Z-numbers which was initiated in [2,3,4]. We suggest a new approach for basic arithmetic and some important algebraic operations over continuous Z-numbers. The suggested approach allows to relatively easy deal with complex problems of propagation of possibilistic-probabilistic restrictions underlying computation with Z-numbers. A series of examples are provided to show the validity of the suggested approach.
The paper is structured as follows. Section II includes the necessary prerequisite material such as arithmetic and algebraic operations over random variables, some operations over fuzzy numbers, etc. In section III we outline the general framework of computation with Z-numbers suggested by Zadeh for understanding the involved optimization and variational problems, and provide a general description of the new suggested approach which can be effectively applied to conduct arithmetic and algebraic operations over Z-numbers. In section IV we explain how the suggested approach is applied for each of the considered operations. In Section V we provide a series of examples on the considered operations to explain in details applicability and efficiency of the suggested approach. In Section VI some conclusions are given.
2.    Preliminaries
Operations over continuous Z-numbers is a synergy of the probabilistic arithmetic and the fuzzy arithmetic. In this section we provide the related definitions.
 

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