H-max distance measure of intuitionistic fuzzy sets in decision making
Graphical abstract
Introduction
Fuzzy set (FS) [53], [54], [55] proposed by Zadeh is the most successful mathematical framework for handling uncertain, imprecision and ambiguous information. A fuzzy set consists of a membership function whose values are in the unit interval [0,1]. Fuzzy set have been applied successfully to various areas both in theoretical and applicable aspects such as pattern recognition, data mining, economics, artificial intelligent, decision making problems [24], [26]. However, in reality, it may not be certain that the degree of non-membership of an element in a fuzzy set is equal to 1 minus the degree of membership which means the uncertainty is not fully grasped by the set. Indeed, Atanassov proposed the concept of intuitionistic fuzzy set (IFS) [1] which comprises the degrees of membership and non-membership both takes values in the unit interval with an extra condition that their sum does not exceed 1. It has been observed that IFS can better designate fuzziness and is a more generalized framework than the fuzzy set. In the practical application point of view, IFS gained much attention from the research community and has been successfully validated in the fields of modelling imprecision [14], [15], pattern recognition [49], [7], computational intelligence [10] and medical diagnosis [42], [43], [44], and decision making [9], [30], [8] and so on. Some literature on IFS can be found in [29], [38], [27], [6], [34], [18].
As an important content in fuzzy mathematics, distance measures, divergence measures and similarity measures [7], [13], [16], [5], [57] of IFSs have attracted many researchers from various fields. Szmidt and Kacprzyk [41] proposed four distance measures of IFSs, which were based on the geometric interpretation of IFSs, and have some good geometric properties. Later, Grzegorzewski [17], Szmidt and Kacprzyk [44] modified these distance measures. Further in this regard, Wang and Xin [50], Park et al. [36], Yang and Chiclana [52], Hatzimichailidis et al. [19] proposed several distance and similarity measures of IFS. Maheshwari and Srivastava [28] studied on divergence measures for IFS. The definition of divergence measure on IFS [28] relied on the basic characteristics of distance measures, except the condition of inclusion relation between two IFSs. We can consider divergence measure as a specific case of the distance measure. More discussion on measures of IFSs can be found in [2], [3], [4], [31], [32], [33], [45], [46], [47], [48], [22], [58].
However, the previous proposed distance measures and divergence measures are not effective in some cases that requires the establishment of inclusion relation between IFSs [44], [41], [17], [28]. Some authors modified the inclusion relation between IFSs [1] to obtain new distance measures such as Park et al. [36]. Nonetheless, the modified inclusion relation between IFSs is not a suitable way to approach mathematical logic reasoning as well as reasoning in reality. Further in [28], Maheshwari and Srivastava proved that their divergence measure satisfies basic properties of a distance measure. However, we notice that the divergence measure proposed by Maheshwari and Srivastava [28] is a metric of IFSs based on the modified inclusion relation between IFSs of Park et al. [36] instead of using the inclusion relation between IFSs of Atanassov as Maheshwari and Srivastava presented in [28]. In the other words, the inclusion relation between IFSs is an important factor for building the distance measure between IFSs. In some cases, the existing distance measures give the values which are not really convinced. For instance, the distance measure proposed by Wang and Xin [50] gives out the equal values while in reality the distinction is clearly observed. Therefore, it is necessary to study an effective distance measure between IFSs.
Moreover, the previous distance measures such as those of Szmidt and Kacprzyk [41], [44], Grzegorzewski [17], Wang and Xin [50], Park et al. [36], Maheshwari and Srivastava [28], have not thoroughly evaluated the intuitionistic fuzzy information because the cross-evaluation between the degrees of membership and non-membership has not been considered (see Definition 5). Hence, these measures were not really effective in the complex decision making problems. For example, medical diagnosis is not only based on current symptoms but also medical history of patients. In this situation, if a distance measure uses the cross-evaluation, it will be easy to evaluate all important degrees between the membership degree and the non-membership degree of a patient at present. It is able to measure those degrees of a patient in the past time as well as the cross-time between the past and present. Therefore, this shows the difference and novelty of the proposal in term of practical application. Evaluating intuitionistic fuzzy medical information fully through cross-evaluation will bring more information and accuracy of diagnosis for patients.
Motivated from the above mentioned drawbacks, in this paper, we propose a new distance measure called H-max of IFSs, where the cross-evaluation is considered. Besides the basic properties of distance measures, we also point out some new results on intuitionistic t-norms and intuitionistic t-conorms and evaluate distance measure between two IFSs which are structured from these operations. Further, we combine the classification of t-representable intuitionistic fuzzy t-norms and t-conorms with the proposed distance measure to study some interesting properties. Furthermore, we study De Morgan triplets of IFSs based on the proposed distance measure. Finally, we apply the proposed distance measure to medical diagnosis problem with numerical example and experimental validation on real-world datasets to check the applicability and effectiveness.
The novelty of this paper in comparison with the relevant researches is highlighted as follows. Firstly, we propose a new method to choose the inclusion relation for building distance measures between IFSs. Secondly, we also design a new distance measure with properties and values that are more realistic reasoning than the previous ones between IFSs. Finally, we investigate some interesting, meaningful and logically connection between the proposed distance measure and De Morgan triplets of IFSs.
The rest of the paper is organized as follows. Section 2 provides some fundamental concepts of the IFS. Section 3 proposes the H-max distance measure of IFS and points out its important properties. Section 4 shows some new results on intuitionistic t-norms and intuitionistic t-conorms and evaluate distance measure between IFSs structured from these operations. Section 5 presents an application of H-max distance measure to the medical diagnosis problem including a new algorithm and numerical example. Section 6 shows the experimental results on real-world datasets. Finally, conclusion is given in Section 7.
Section snippets
Preliminary
Firstly, let us consider the basis concepts of the intuitionistic fuzzy set. Definition 1 Zadeh [53]: Let X be a space of points and let x ∈ X. A fuzzy set A in X is characterized by a membership function μA with a range in [0, 1]. Fuzzy set can be represented in the following way: Definition 2 Atanassov [1]: Let X be a space of points and let x ∈ X. An intuitionistic fuzzy set A in X is characterized by a membership function μA and a non-membership function νA with a range in [0, 1] such that 0 ≤ μA + νA ≤ 1.
H-max distance measure of intuitionistic fuzzy sets
Firstly, we analyze the definition of “distance measure of IFSs”. Wang and Xin [50] defined this concept as in Definition 4 where the condition (iv) mentions the inclusion relation between IFSs. Some other researchers defined this concept similarly to Definition 4 but the condition (iv) is replaced with the triangle inequality condition (Papakostas et al. [35]), which is not related to the inclusion relation between IFSs. It is necessary that the inclusion relation should be established between
Distance measure of IFSs with the intuitionistic fuzzy t-norm and t-conorm
In this section, we combine the classification of t-representable intuitionistic fuzzy t-norms and t-conorms [25] with some properties of the proposed distance measures. Firstly, we recall some related definitions. Definition 6 ([25]) A fuzzy negation n is a nonincreasing function satisfies
For example: is an involutive fuzzy negation, called standard fuzzy negation. Definition 7 ([25]) A mapping is a fuzzy
An application of H-max distance measure for medical diagnosis
Medical diagnosis is the process of investigation of diseases from a patient's symptoms. Most decisions in medical science have substantial uncertainties which deal with imprecision and fuzziness. There are a lot of diseases in medical science which share some common symptom [28]. It is indeed difficult for (young) clinicians to give diagnosis for patients on particular diseases. The notion of distance measures under intuitionistic fuzzy setting plays a decisive role in tackling the problem. In
Experimental environments
Experimental tools: We compare the proposed methods (H-max, WX-H-max) against the methods of Szmidt & Kacprzyk [41] (SK1-1 (14), SK1-2 (15), SK1-3 (16), SK1-4 (17)), Szmidt & Kacprzyk [44] (SK2), Wang & Xin [50] (WX), Vlachos & Sergiadis [49] (VS), Zhang & Jiang [56] (ZJ), Wei et al. [51] & Hung [20] (W), Jujun et al. [23] (J) and Maheshwari & Srivastava [28] (SA) in the combination of Matlab 2015a programming language and R programming language.
Experimental datasets: The benchmark datasets
Conclusions
In this paper, we introduced a new distance measures called H-max of intuitionistics fuzzy sets (IFSs). We also studied some new results on intuitionistic t-norms, intuitionistic t-conorms and evaluated distance measure between two IFSs which basically structured from these operations. Further, we combined the classification of t-representable intuitionistic fuzzy t-norms and t-conorms with the proposed distance measure to study some interesting properties. Moreover, we studied De Morgan
Acknowledgment
The author (R.T. Ngan) would like to thank the Project 911 of VNU University of Science, Vietnam National University for supporting her work. The author (L.H. Son) would like to thank Sejong University, Korea for their approval of Visiting Professor position in 2018. This research is funded by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 102.01-2017.02.
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