Research and Applications in Economics Volume 3, 2016 doi: 10.14355/rae.2016.03.001
www.seipub.org/rae
Ranking Fuzzy Numbers with Goodness Criteria and Its Applications in Stock Performance Evaluation Li Zhang*1, Zhenyuan Wang2 Department of Economics, University of Nebraska at Omaha, 6001 Dodge ST Omaha, NE, USA
*1
Department of Mathematics, University of Nebraska at Omaha, 6001 Dodge ST Omaha, NE, USA
2
lwestman@unomaha.edu; 2zhenyuanwang@unomaha.edu
*1
Abstract This work summarizes some ranking methods for fuzzy numbers. A point view of reference system is proposed as a general way to establish ranking methods on sets of fuzzy numbers. Four goodness criteria of ranking methods are established, which can be applied further to decision-making on environment with uncertainty. The idea of ranking fuzzy numbers can also be used for ranking stock performance. As an application, a set of real NASDAQ data is adopted in an example to show the details of the ranking method by lexicography that is a reference system as well. Keywords Rankings; Fuzzy Numbers; Ranking Criteria; Reference Systems; Stock Evaluation
Introduction Fuzzy number [9, 20] is the most important mathematical quantities concerning fuzziness and vagueness which is hard to be ordered directly and intuitively. In literature, the concepts of ranking and ordering are frequently confused. As a matter of fact,ranking and ordering can be clearly distinguished under a mathematical concept, called relation. Ranking, as a relation on nonempty sets, can be established on a set of fuzzy numbers. Rankings on sets of fuzzy numbers play a critical role in data analysis with uncertainty. To be clear and intuitive, some reference systems,whose rankings have been already defined, are required. The most common reference system is the set of all real numbers with the natural ordering. In literature, there are various ranking techniques widely used in many areas such as decision-making and artificial intelligence, however, many of them do not lead to the unique commonly-recognized results [1-8, 10-14, 16-19, 21]. To eliminate the unreasonable ranking methods and reduce the candidates of effective ranking method, a number of criteria to judge the goodness of ranking techniques are necessary. In this work, the methods of reference systems are established and discussed for constructing ranking methods based on fuzzy number rankings established by lexicography as one of reference systems. Four proposed criteria are used to judge various examples of ranking methods. In fact, some ranking indexes of fuzzy numbers based on the centroid of area between the curve of their membership function and horizontal axis seem problematic under the judgement of those criteria. It is well-known that the stock’s daily performance is measured by a few numerical indexes, just as rankings of fuzzy numbers are determined by their parameters. Thus, we attempt to use a reasonable ranking method for fuzzy numbers to evaluate the daily performance of stock, such as real NASDAQ data. After the introduction, this paper is arranged as follows. Section 2 provides the fundamental knowledge of relations on nonempty sets and concepts of fuzzy numbers. In section 3, establishing rankings on given sets of fuzzy numbers is presented with some examples. Section 4 newly introduces the criteria for goodness of the rankings. In section 5, a fuzzy number ranking method is shown torank the daily stock performances by lexicography. Finally, conclusions are summarized in Section 6.
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