Discussion on Weighted Similarity Measure under Intuitionistic Fuzzy Sets Environment
Abstract
We analytically show that the findings of pattern recognition problems with weighted similarity measures under intuitionistic fuzzy sets environment that is dominated by relative weights of elements in the universe of discourse for the discrete case and the weighted function for the continuous case. In the past, researchers focus on constructing new similarity measures or developing new algorithms applying their similarity measures. Hence, previous results depended on a special weight to decide the pattern of the sample that may be required further considerations. How to select a proper weight will be an important issue for researchers in the future when deal with pattern recognition problems.
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Introduction
More and more business had being developed in the real world, so it is hard for a company to get accurate market data. Therefore, fuzzy set data may be a kind of data that can be obtained more easily to explain the market and managerial situation. If the data is a traditional fuzzy set, there are some methods to be applied to solve the fuzzy problem. Recently, many similarity measures have been proposed, for examples, Hung and Lin [1], Julian et al. [2], Hung and Lin [3], Tung et al. [4], Yen et al. [5], Hung and Wang [6], Chu and Guo [7], Hung et al. [8], and Tung and Hopscotch [9], for measuring the degree of similarity between fuzzy sets, under a kind of special fuzzy sets, which is called Intuitionistic Fuzzy Sets (IFSs) that were initiated presented by Atanassov [10, 11]. Since Atanassov originated the idea of IFSs, many different similarity measures between IFSs have been proposed in the literature. Atanassov and Rangasamy[12] and Kuppannan et al. [13] provided practical applications for IFSs. The importance of suitable distance measures between IFSs takes place because they play an important role in the theoretical development and implication problem. Two existing similarity measures for IFSs were proposed by Li and Cheng [14] to indicate the dominated factor for the selection of pattern recognition problems. This paper is a detailed analysis for the similarity measure of intuitionistic fuzzy sets (IFSs) in Li and Cheng [14]. Mitchell [15] already provided an example to demonstrate that the similarity measure proposed by Li and Cheng [14] may lead to counterintuitive result. However, there are 373 papers continuously referred to Li and Cheng [14]. Owing to the high citation, it deserves a detailed study of their paper. To be compatible with previous results, we directly study the examples of Li and Cheng [14] to show that their weighted similarity measures contained inherent problems that is their results are dependent on weights of elements in the universe of discourse for the discrete case and the weighted function for the continuous case. Hence, how to derive the weights or the weighted function should be the crucial issue in the future research. Our consideration will offer a patch work to enhance the operational development of similarity measure for pattern recognition under IFSs. Recently, there is a trend to improve published papers, for example, Hung et al. [16], Lin et al. [17], Tung [18], and Chao et al. [19]. Following this trend, we will provide improvements for Li and Cheng [14]. In this paper, based on the same numerical examples of Li and Cheng [14], we will demonstrate that the proposed measures of Li and Cheng [14] performs dependent on the relative weight in pattern recognition. Our findings presented here could arouse attention to take care of the decision of relative weights in the selection and applications of similarity measures for IFSs and vague sets in practice.
Conclusion
A similarity measure is a useful tool for determining the similarity of two objects. Based on the same numerical examples of Li and Cheng [14], we demonstrated that their proposed similarity measures are dominated by the relative weight of the domain for IFS in pattern recognition problems. In the past, researchers focus on developing new similarities to replace previous established similarity measures, moreover, Yen et al. [5], Hung et al. [8], Chu et al. [21] and Chou [22] constructed algorithms that is related to the size of universe of discourse for the discrete cases to repeatedly applied their proposed similarity measures. However, Yen et al. [4], Hung et al. [8], Chu et al. [21] and Chou [22] did not pay attention to how to decide relative weights for elements in the universe of discourse. Based on our discussion, we show that applying the same similarity measure with different relative weights will result in different finding for pattern recognition problems. Consequently, we point out their proposed measures to analyze the behavior of decision making that should be put more attention to the relative weight of the domain for an IFS.