Type-2 fuzzy multigranulation rough sets
Introduction
Rough set theory [29] is a useful mathematical tool that enables experts to process various uncertainties. Afterwards, in order to enrich the application range of it, many efforts have been made to generalize the concept of rough sets [2], [5], [55], [62], [63]. Particularly, Qian et al. [30], [31] extended classic rough sets from a new perspective, and they discussed rough approximations based on several different granular structures and eventually proposed the MGRS model. To be specific, two decision strategies [30] are included in MGRSs, i.e., one is “seeking common ground while reserving differences” and another one is “seeking common ground while eliminating differences”. The above-stated two strategies correspond to optimistic MGRSs and pessimistic MGRSs respectively. The MGRS theory has attracted plenty of attentions since its establishment. Properties of MGRSs are discussed [32], [34] and several extensions are proposed, such as MGRSs based on tolerance relations [43], multigranulation with different grades rough sets [48], [49], multigranulation covering rough sets [17], etc. Recently, Ma and Mi [22] compared different kinds of MGRSs and proposed several updated uncertainty measures. By virtue of the grey system theory, Kang et al. [8] discussed the notion of variable precision grey MGRSs and applied it to attribute reductions.
In particular, MGRSs have been extended to fuzzy and intuitionistic fuzzy contexts. Yang et al. [47] and Xu et al. [42] discussed the fuzzy MGRS theory respectively. Among them, the former one is based on a family of fuzzy T-similarity relations, whereas the latter one is based on fuzzy tolerance relations. Then, fuzzy MGRSs are further studied [10], [12], [18], [28], [36], [38], [39] and extended by combining with other meaningful concepts. For instance, Zhan et al. [52], [53], [54], [64] discussed covering-based fuzzy MGRSs and intuitionistic fuzzy MGRSs, and further applied those models to multi-criteria or multi-attribute group decision-making problems. Zhang et al. [56], [57], [58], [60] proposed hesitant fuzzy MGRSs and discussed hesitant fuzzy multigranulation three-way decisions. Huang et al. [3], [4] studied intuitionistic fuzzy MGRSs. Xue et al. [45] explored multigranulation covering rough intuitionistic fuzzy sets. The MGRS model and its extensions have been applied in many areas as well, such as knowledge discovery [15], [41], attribute reductions [7], [14], [16], [46], information fusion [1], [59], medical diagnosis [11], [13], and conflict analysis [33].
In some practical applications, if the data contains more kinds of uncertainties and even the membership function of fuzzy sets and intuitionistic fuzzy sets cannot be determined, the utilization of type-2 fuzzy sets [50] is imperative. To be specific, there are three reasons that can explain why type-2 fuzzy sets are meaningful in practice [25]: (1) when satisfactory system performances cannot be achieved via type-1 and interval type-2 fuzzy systems, general type-2 fuzzy sets act as the promising logical fuzzy set model that should be considered; (2) it is accepted that type-2 fuzzy sets are more flexible than interval type-2 fuzzy sets; (3) type-2 fuzzy sets can cope with situations that are semantically incompatible. However, type-2 fuzzy sets are somewhat complex that make them difficult to use. Hence, three representations are further proposed to address the above-stated difficulties, i.e., the vertical-slice representation [27], the wavy-slice representation [26] and the horizontal-slice representation [19]. In addition, properties of type-2 fuzzy sets have been discussed in many literatures, including algebraic structures [27], centroid [9] and similarity measures [23], etc. Moreover, type-2 fuzzy sets are wildly used in a variety of realistic situations, such as computing with words [6], type-2 fuzzy control [35], [61] and medical diagnosis [51].
Existing MGRS models, including various fuzzy MGRSs and intuitionistic MGRSs, are able to address those problems described by symbolic, real-valued and interval-valued attributes. It is rare to see investigations on MGRS models dealing with diverse uncertainties, this motivates the study of the present paper. Based on the type-2 fuzzy rough set model on single granular structures [21], [40], [65], this paper aims to explore the T2FMGRS model that can generalize MGRSs to the type-2 fuzzy context.
The rest of the paper is organized as follows. Section 2 reviews several fundamental definitions and properties of MGRSs, type-2 fuzzy sets and type-2 fuzzy rough sets. In Section 3, the notion of T2FMGRSs is put forward, and properties of type-2 fuzzy multigranulation lower and upper approximations are discussed. The rough measure of T2FMGRSs is explored in Section 4. In Section 5, T2FMGRSs over two universes are established and some examples for decision-making problems are given to illustrate the applicability of T2FMGRSs. Section 6 sums up several conclusions and future study options.
Section snippets
Preliminaries
Qian et al. [30], [31] proposed MGRSs that enable experts to discuss a concept via multiple granular structures.
Definition 1 [30], [31] For a nonempty and finite universe X and a family of equivalence relations , the optimistic and pessimistic multigranulation lower and upper approximations of can be defined as follows:
Classic MGRSs have been extended to the fuzzy
T2FMGRSs
In this section, the concept of MGRSs is extended to the type-2 fuzzy context. Different from classic rough sets, the “equivalence class” in type-2 fuzzy rough sets is quite complicated, so it cannot be used in the definition of T2FMGRSs. Relationships between multigranulation approximations and single granulation approximations that have been proved in crisp and fuzzy MGRSs inspire us to put forward the following definitions.
Definition 5 Given a nonempty and finite universe X and a family of type-2 fuzzy
Rough measure of T2FMGRSs
Rough measure is one of the most important numerical characteristics of rough sets and it can be used to describe various uncertainties. In order to discuss the rough measure of T2FMGRSs, the “cardinality” of type-2 fuzzy sets should be considered at first. Probability measures proposed in Literature [20] will be used in this section.
Given a nonempty and finite universe X along with a probability space , the probability measure of an interval type-2 fuzzy set A is
T2FMGRSs over two universes
In this section, T2FMGRSs over two universes are discussed and two examples are given to illustrate the applicability of T2FMGRSs. Definition 7 Suppose that X and Y are two nonempty and finite universes and is a family of type-2 fuzzy relations from X to Y, that is, . is called a type-2 fuzzy multigranulation approximation space over two universes. The optimistic type-2 fuzzy multigranulation lower and upper approximations with respect to for a type-2 fuzzy set
Conclusions
In this paper, a model for T2FMGRSs is proposed by combining type-2 fuzzy sets with MGRSs. At first, the definition of T2FMGRSs is put forward and the properties of type-2 fuzzy multigranulation lower and upper approximations are discussed in detail. Then, the rough measure of the newly proposed T2FMGRSs is explored based on the probability measure of type-2 fuzzy sets. Finally, T2FMGRSs over two universes are discussed and a problem of subway line scheme selections along with a medical
Declaration of Competing Interest
The authors declared that they have no known conflicts of interest associated with the paper.
Acknowledgements
The author thanks the editor and the anonymous referees for their critical and insightful comments that have led to an improved version of this paper. The author also thanks Dr. Chao Zhang for his constructive suggestions and support. The works of this paper are supported by the National Natural Science Foundation of China (61672331, 61806116), the Key R&D program of Shanxi Province (International Cooperation, 201903D421041), the Natural Science Foundation of Shanxi Province (201801D221175),
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