The Z-Number Enigma: A Study through an Experiment

The arithmetic of discrete Z-numbers

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.

Multi-Criteria Decision-Making Method Based on Distance Measure and Choquet Integral for Linguistic Z-Numbers

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.

The arithmetic 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.

Approximate Reasoning on a Basis of Z -Number-Valued If–Then Rules

Abstract: Approximate reasoning is about reasoning with imperfect information. Nowadays, a large diversity of approaches to approximate reasoning with fuzzy information and fuzzy type-2 information exists. It should be stressed, however, that real-world imperfect information is characterized by combination of fuzzy and probabilistic uncertainties, which is referred to as bimodal information. In view of this, Zadeh introduced the concept of a Z -number regarded as an ordered pair Z = ( A , B ) of fuzzy numbers A and B , where A is a linguistic value of a variable of interest, and B is a linguistic value of probability measure of A , playing a role of its reliability. Unfortunately, up to day, there is no research on approximate reasoning realized on the basis of if–then rules with Z -number-valued antecedents and consequents, briefly, Z- rules. Zadeh addressed this problem as related to an uncharted territory. In this paper, a new approach is developed to study approximate reasoning with Z- rules on a basis of linear interpolation. We provide an application of the approach to job satisfaction evaluation and to students’ educational achievement evaluation problems related to psychological and perceptual issues naturally characterized by imperfect information. The obtained results show applicability and validity of the proposed approach.

Outline of a New Approach to the Analysis of Complex Systems and Decision Processes

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.

Computing Machinery and Intelligence

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.

Fuzzy logic = computing with words

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.

A Note on Z-numbers

Abstract: Decisions are based on information. To be useful, information must be reliable. Basically, the concept of a Z-number relates to the issue of reliability of information. A Z-number, Z, has two components, Z=(A,B). The first component, A, is a restriction (constraint) on the values which a real-valued uncertain variable, X, is allowed to take. The second component, B, is a measure of reliability (certainty) of the first component. Typically, A and B are described in a natural language. Example: (about 45min, very sure). An important issue relates to computation with Z-numbers. Examples: What is the sum of (about 45min, very sure) and (about 30min, sure)? What is the square root of (approximately 100, likely)? Computation with Z-numbers falls within the province of Computing with Words (CW or CWW). In this note, the concept of a Z-number is introduced and methods of computation with Z-numbers are outlined. The concept of a Z-number has a potential for many applications, especially in the realms of economics, decision analysis, risk assessment, prediction, anticipation and rule-based characterization of imprecise functions and relations.