Use of Grey Numbers for Evaluating a System's Performance Under Fuzzy Conditions

Use of Grey Numbers for Evaluating a System's Performance Under Fuzzy Conditions

Michael Voskoglou
Copyright: © 2021 |Pages: 17
DOI: 10.4018/978-1-7998-3479-3.ch023
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In the present research, a method using Grey Numbers as tools is developed for assessing a system's mean performance, which is useful when utilizing qualitative grades and not numerical scores for this purpose. Although this new method is proved to be equivalent with an analogous method using Triangular Fuzzy Numbers as tools developed in an earlier work, it reduces the required computational burden, since it requires the calculation of two components only (instead of three in the case of the Triangular Fuzzy Numbers) for obtaining the mean value of the Grey Numbers involved. Examples are also presented on student and athlete assessment illustrating the new method and showing that the system's quality performance, calculated by the traditional GPA index, may lead to different assessment conclusions.
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In Management a system is understood to be a set of interacting components forming an integrated whole and working together for achieving a common target. A factory, a hospital or a bank, constitute standard examples of systems, as well as a physical or biological system, an abstract knowledge system, etc. As a multi- perspective domain systems’ theory serves as a bridge for an interdisciplinary dialogue between autonomous areas of study (Balley, 1994, Backlund, 2000). The assessment of a system’s performance constitutes a very important part of this theory, because it enables the designers of the system to correct its weaknesses and therefore to increase its effectiveness.

When the performance of a system’s components is evaluated by numerical scores, then the traditional way for assessing the system’s mean performance is the calculation of the average of those scores. However, in order to comfort the reviewer’s existing uncertainty about the exact value of the numerical scores corresponding to each of the system’s components, frequently in practice the assessment is made not by numerical scores but by linguistic grades, like excellent, very good, good, etc. This involves a degree of fuzziness and makes the calculation of the mean value of those grades impossible.

A popular in such cases method for evaluating the overall system’s performance is the calculation of the Grade Point Average (GPA) index (e.g. see Voskoglou, 2017b, Chapter 6, p.125). However, GPA is a weighted average in which greater coefficients (weights) are assigned to the higher grades, which means that it reflects not the mean, as we wish, but the quality performance of the system.

Analysis of the objectives and of the scope of the present work: In order to overcome the above difficulties, we have utilized in earlier works the system’s total uncertainty under fuzzy conditions (created by the qualitative assessment of its components) as a measure of its effectiveness (Voskoglou, 2017b, Chapter 5). This manipulation is based on a fundamental principle of the Information Theory according to which the reduction of a system’s uncertainty is connected to the increase of information obtained by a system’s activity. In other words, lower uncertainty indicates a greater amount of information and therefore a better system’s performance with respect to the corresponding activity. However, this method needs laborious calculations, cannot give a precise qualitative characterization of a system’s performance and, most importantly, it is applicable for comparing the performance of two different systems with respect to a common activity only under the assumption that the existing uncertainty is the same in the two systems before the activity.

For this reason, Fuzzy Numbers (FNs) have been also used in later works for assessing a system’s mean performance under fuzzy conditions (Voskoglou, 2017a). However it was observed that, although the calculation of three components is needed for expressing the mean value of the qualitative grades in the form of a Triangular FN (TFN), only the middle component is used for defuzzifying it. The above observation suggests the search for a method analogous to the use of TFNs that possibly reduces the required computational burden. This search led us to utilizing Grey Numbers (GNs) (Liu & Lin, 2010) as an alternative tool for assessing a system’s mean performance with qualitative grades (Voskoglou & Theodorou, 2017). Although this new method has been proved to be equivalent with the use of TFNs, it needs the calculation of two components only (instead of three) for expressing the mean value of the qualitative grades in the form of a GN.

Key Terms in this Chapter

Fuzzy Logic (FL): A logic based on the concept of fuzzy set that, in contrast to the classical bi-valued logic of Aristotle (yes – no), characterizes cases with multiple values.

Defuzzification: The fuzzy logic’s approach for a problem’s solution involves the fuzzification of the problem’s data by representing them with properly defined fuzzy sets, the evaluation of the fuzzy data in order to express the problem’s solution in the form of a unique fuzzy set and the defuzzification of this fuzzy set in order to translate the problem’s mathematical solution to the natural language for use with the original real-life problem. The most popular defuzzification method is the Centre of Gravity (COG) Technique in which the corresponding system’s fuzzy outputs are represented by the coordinates of the COG of the level’s section contained between the graph of the membership function expressing those outputs and the OX axis.

Whitenization of a gn: The process of determining the white number with the highest probability to be the representative real value of a GN A INSERT PICT [a, b], which is denoted by w(A). When the distribution of A is unknown, one usually takes w(A) = .

Grey System (GS): A system which lacks information, such as structure message, operation mechanism and behaviour document. Usually, on the grounds of existing grey relations and elements one can identify where “grey” means poor, incomplete, uncertain, etc.

Triangular Fuzzy Number (TFN): The simplest form of a FN, whose membership function’s graph forms a triangle with the OX-axis. The defuzzification of a TFN A= ( a. b, c ) may be obtained by considering as its representative crisp value the x-coordinate X (A) = of the COG of its membership function.

Grey Number GN): An indeterminate number whose probable range is known, but which has unknown position within its boundaries. If R denotes the set of real numbers, a GN, say A, can be expressed mathematically with the help of a real interval [ a, b ] in the form A INSERT PICT [ a, b ] = {x ? R : a INSERT PICT x INSERT PICT b} . If a = b , then A is called a white number and if A?(-8,+8), then it is called a black number. Compared with the interval.

Fuzzy Set (FS): A generalization of the concept of crisp set that gives a mathematical formulation to real situations in which certain definitions have not clear boundaries (e.g. the high mountains of a country).. A FS, say A, is characterized by its membership function y = m A (x) defined on the universal set of the discourse U and taking values in the interval [0, 1], thus assigning to each element x of U a membership degree m A (x) with respect to A. The closer is m A (x) to 1, the better x satisfies the characteristic property of A.

Grade Point Average (GPA): A weighted average assigning greater coefficients (weights) to the higher scores and therefore focusing on a system’s quality performance.

Fuzzy Number (FN): A fuzzy set A on the set R of the real numbers which is normal (i.e. there exists x in R such that m A (x) = 1) and convex (i.e. all its a -cuts A a = { x ?U: m A (x)= a }, a in [0, 1], are closed real intervals) and whose membership function y = m A ( x) is a piecewise continuous function.

System (in Management): A set of interacting components forming an integrated whole and working together for achieving a common target. A factory, a hospital, or a bank constitute common examples of systems.

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