Aggregation distance measure and its induced similarity measure between intuitionistic fuzzy sets
Introduction
Since the seminal work of Zadeh, the fuzzy set (FS) theory characterized by a membership function between zero and one has become a useful tool to handle with imprecision and uncertainty [1]. In real-life situations, taking the hesitation or uncertainty about the membership degree into consideration, the degree of non-membership is not always equal to one minus the degree of membership, which is treated as reasonable in FS theory. To address this issue, Atanassov introduced the notion of intuitionistic fuzzy sets (IFSs) as an extension of FS, in which not only the degree of membership is given, but also the degree of non-membership degree [2].
Ever since IFSs’ appearance, many authors have paid great attention to the measures of distance and similarity between IFSs. Szmidt and Kacprzyk proposed four distance measures between IFSs, which were in some extent based on the geometric interpretation of intuitionistic fuzzy sets [3]. Li and Cheng proposed similarity measures of IFSs based on an axiomatic approach and applied these measures to pattern recognition [4]. But it was later pointed out that Li and Cheng’s measures are not always effective in some cases [5], [6], [7]. Hung and Yang proposed several similarity measures of IFSs based on Hausdorff distance and LP metric which can effectively be used with linguistic variables [8], [9]. Hatzimichailidis et al. introduced distance metric between IFSs which makes use of matrix norms and fuzzy implications [10]. Farhadinia presented a new similarity measure for IFSs by using the convex combination of endpoints and also focusing on the property of min and max operators [11]. Iancu also introduced two families of similarity measures based on Frank t-norms [12]. Most of these aforementioned measures in different formats are originated from axiomatic definitions. Recently, Baccour et al. and Xu and Chen gave a comprehensive overview of distance and similarity measures of IFSs [13], [14].
Measuring the distance and similarity between IFSs is now being extensively applied in many research fields, such as pattern recognition, fuzzy clustering and decision making. However, there may exist inconsistent results if we adopt different distance and/or similarity measures in practical applications as exemplified in Section 4, which will certainly get the decision makers into trouble. This situation can arise in a decision-making problem. How to make a decision to choose the optimal alternative from the conflict conclusions is still a problem to be solved. Li et al. gave a comparative analysis of the existing similarity measures for IFSs to benefit selection of similarity measures [15].
In this paper we attempt to deal with this problem from another viewpoint. Taken n distances we interested in as inputs, it is expected to produce a reasonable output, based on which the final decision is made. It is within the domain of the theory of aggregation functions. In this paper, the distance measures are aggregated as an aggregation distance by using aggregation functions. The weighted average of the existing distance measures is accepted as the overall evaluation. We propose two approaches to set the weights based on mathematical foundations. (I) We select the optimal solution of the shortest distance between a moving point and a vector with all its components degrees of distance we care about as the expectation. (II) We choose the normalized associated eigenvector of spectral radius of a non-negative symmetric matrix as an assignment and the weighted average of the degrees of distance the aggregation distance. Moreover, similarity measures generated by non-filling fuzzy negations for a given distance measure are provided to meet the axiomatic definition.
The rest of this paper is organized as follows: In Section 2, we recall basic concepts of intuitionistic fuzzy sets and some commonly used distance and similarity measures between IFSs. In Section 3, the aggregation distance measure for IFSs aggregated by an aggregation function without zero divisors is provided, as well as its induced similarity measure generated from a given distance measure with respect to a non-filling fuzzy negation. And their applications in pattern recognition, fuzzy clustering and decision making are given in Section 4. Finally, Section 5 concludes the present paper.
Section snippets
Preliminaries
In this section, we briefly recall some basic concepts relating to IFSs and some popular distance and similarity measures between IFSs.
Definition 2.1 Let a (crisp) set E be fixed. An Atanassov’s intuitionistic fuzzy set A in E is an object of the form where functions define the degree of membership and the degree of non-membership of the element x ∈ E to A, respectively, and for every x ∈ E, [2]
The function πA(x): E → [0, 1], given by
Aggregation distance measure and its induced similarity measure of intuitionistic fuzzy sets
In this section, the aggregation distance measure of IFSs on the basis of the existing distance measures is presented, so are the similarity measures generated from a given distance measure with respect to non-filling fuzzy negations.
Definition 3.1 Let a mapping f: [0, 1]n → [0, 1] (n > 1) satisfy the following properties.
f is idempotent at and i.e., and ; f is monotonic increasing in each of its components, i.e., if xi ≤ yi, then [23]
Numerical examples for pattern recognition
In this section, numbers of examples are provided from pattern recognition, fuzzy clustering, and decision making to validate applications of the proposed measures.
Liang and Shi use the principle of the maximum degree of similarity between IFSs to solve the problem of pattern recognition [5]. Similarly, Wang and Xin apply the principle of minimum degree of difference between IFSs to solve this problem because of the duality of similarity and distance [17]. That is to say, the less the
Conclusions
The research on distances and similarities between IFSs is a hot topic in the IFS theory. In this paper we start from the properties of distance measures and introduce the concept of aggregation distance measure, aggregation of the existing distance measures by using aggregation functions in the IFSs setting. Then we investigate the relationship between the distance measure and similarity measure and propose the generated similarity measure with respect to a non-filling fuzzy negation in
Acknowledgments
The authors are greatly thankful to the editor and anonymous reviewers for sharing their valuable comments that significantly improved the quality of the paper. This research was supported by the National Natural Science Foundation of China (Grantno. 61179038).
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This paper has been recommended for acceptance by N. Sladoje.