Three-way decision spaces based on partially ordered sets and three-way decisions based on hesitant fuzzy sets
Introduction
The theory of three-way decisions (3WD) was proposed by Yao [57], [58], [59], whose basic ideas come from Pawlak's rough sets [35] and probabilistic rough sets [55], [56], [60]. In a few years, three-way decisions obtained the rapid development both in theory and applications. In theory, the researches on three-way decisions mainly focus on two aspects, the background researches and theoretical framework researches.
The background researches on three-way decisions mainly contain decision-theoretic rough sets (DTRS) [6], [33], [55], [56], [60], variable precision rough sets (VPRS) [66], Bayesian rough sets (BRS) [43], game-theoretic rough sets (GTRS) [13], fuzzy rough sets/rough fuzzy sets (FRS/RFS) [8], interval-valued fuzzy rough sets (IVFRS) [15], [19], [20], [44], intuitionistic fuzzy decision-theoretic rough sets [24], triangular fuzzy decision-theoretic rough sets [25], interval-value fuzzy decision-theoretic rough sets [23], dominance-based fuzzy rough sets [7], multi-granulation rough sets (MGRS) [38], [39], multi-granulation decision-theoretic rough sets [41], multi-granulation rough sets based on coving [28], neighborhood-based multi-granulation rough sets (NMGRS) [29] and so on. The common characteristics of these rough sets have their own evaluation functions or lower/upper approximations to determine the positive, negative and boundary regions of rough sets as three decision outcomes: acceptance, rejection, and uncertainty respectively.
The theoretical framework researches on three-way decisions mainly contain the value domain of evaluation functions [59], construction and interpretation of evaluation functions [57], [58], [59], the mode of three-way decisions [59] and the theory of three-way decision spaces [14] etc.
After the theory of three-way decisions was proposed, scholars began with the study of application. Existing application researches mainly contain incomplete information system [30], risk decision making [21], classification [32], investment [34], multi-agent [52], group decision making [26], face recognition [22], social networks [37] and so on.
In theoretical research of three-way decisions, Hu systematically studied all kinds of rough sets and probabilistic rough sets, introduced axiomatic definitions for decision measurement, decision condition and decision evaluation function and established three-way decision spaces [14]. Based on it the author gave a variety of three-way decisions on three-way decision spaces so that existing three-way decisions are the special examples of three-way decision spaces, such as three-way decisions based on fuzzy sets, interval-valued fuzzy sets, interval sets, shadowed sets, random sets and rough sets etc. At the same time, multi-granulation three-way decision spaces and their corresponding multi-granulation three-way decisions were also established in [14]. The author also introduced novel dynamic two-way decisions and dynamic three-way decisions based on three-way decision spaces and three-way decisions based on bi-evaluation functions in [14].
Three-way decision spaces were established on fuzzy lattice, i.e. complete distributive lattice with an involutive negator [14]. There are many popular fuzzy lattices, such as fuzzy sets [62], interval-valued fuzzy sets [15], [19], [20], [63], intuitionistic fuzzy sets [1], interval-valued intuitionistic fuzzy sets [2], interval sets [53], [54], [65], shadowed sets [36] and so on. However, some of focus structures do not constitute fuzzy lattices or constitute fuzzy lattices under special conditions, for example, type-2 fuzzy sets [16], interval-valued type-2 fuzzy sets [17], hesitant fuzzy sets [9], [11], [45] and interval-valued hesitant fuzzy sets [4], which are useful generalizations of fuzzy sets. It limits the wide applications of the theory of three-way decision spaces. Moreover, Yao in [59] first used a partially ordered set L and divided it into two disjoint nonempty sets and , where is the rejection region and is the acceptance region. So this paper attempts to generalize measurement on decision conclusion (decision domain) in three-way decision spaces from fuzzy lattices to partially ordered sets.
The rest of this paper is organized as follows. In Section 2, measurement of three-way decisions is specified in partially ordered sets, evaluation function axioms are given and three-way decision spaces based on partially ordered sets are established. Section 3 establishes three-way decisions theory on three-way decision spaces based on partially ordered sets, which contains general three-way decisions, the lower and upper approximations induced by three-way decisions and multi-granulation three-way decisions etc. In Section 4, as special cases of three-way decision spaces based on partially ordered sets, three-way decisions based on hesitant fuzzy sets are discussed and many decision evaluation functions are given. Likewise, as special cases of three-way decision spaces based on partially ordered sets, Section 5 discusses three-way decisions based on interval-valued hesitant fuzzy sets and gives many decision evaluation functions. Finally, in Section 6, this paper is concluded.
Section snippets
Three-way decision space based on partially ordered sets
In this section three-way decision spaces (3WDS) are established on partially ordered sets.
In [0, 1], operator (x ∈ [0, 1]) is applied. Similar operator can be discussed in general partially ordered set. For the convenience of research, the concept of involutive negator is defined as follows.
Definition 2.1 Let (X, ≤) be a partially ordered set. Then mapping N: X → X is called an involutive negator or inverse order and involutive operator on X, if it satisfies ∀x, y ∈ X,
x ≤ y⇒N(y) ≤ N(x) (inverse[14], [17]
Three-way decisions based on partially ordered sets
For the convenience of expression, unless confusion, decision measurement domain PD is denoted by (PD, ≤, N, 0, 1).
Three-way decisions based on hesitant fuzzy sets
In the section, we discuss a special partially ordered set which is constituted by hesitant fuzzy sets and discuss three-way decisions based on hesitant fuzzy sets.
Three-way decisions based on interval-valued hesitant fuzzy sets
Now hesitant fuzzy sets were extended to interval-valued hesitant fuzzy sets [4]. This section discusses another special partially ordered set which is constituted by interval-valued hesitant fuzzy sets and discusses three-way decisions based on interval-valued hesitant fuzzy sets.
Conclusions
In this paper, three-way decision spaces and three-way decisions based on partially ordered sets have been established and as a special example of partially ordered sets, , we systematically research three-way decision spaces and three-way decisions based on hesitant fuzzy sets. Main conclusions in this paper and continuous work to do are listed as follows.
- (1)
The measurement on decision conclusion (decision domain) in three-way decision spaces is generalized to partially ordered sets so
Acknowledgments
The authors are extremely grateful to the anonymous referees for their constructive comments and critical suggestions that led to an improved version of this paper. The work described in this paper was supported by grants from the National Natural Science Foundation of China (Grant nos. 11571010 and 61179038).
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