Three-way decisions space and three-way decisions
Introduction
Theory of three-way decisions (3WD) is an extension of classic two-way decisions (2WD) [50], [51], [55], [56], whose basic ideas come from Pawlak rough sets [34], [35] and probability rough sets [7], [19], [29], [51], [52], [53], [54], [55], [56], [57], [58] and whose main purpose is to interpret the positive, negative and boundary regions of rough sets as three decisions outcomes, acceptance, rejection, and uncertainty (or deferment) in a ternary classification respectively. In addition to rough sets as delegates of the three-way decisions, there are other uncertainties, such as fuzzy uncertainty and random uncertainty. Table 1.1 lists typical delegates of three-way decisions.
It can be shown that, under certain conditions, probabilistic three-way decisions are superior to both Palwak three-way decisions and two-way (i.e., binary) decisions [56]. Many recent studies further investigated extensions and applications of three-way decisions [21], [26], [27], [28], [46], [51], [55], [56]. The researches on three-way decisions mainly focus on the following two aspects.
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The first aspect is the background researches on three-way decisions. It mainly contains the extension researches of rough sets. The first class is the extension from Pawlak rough sets to probability rough sets, such as decision-theoretic rough sets (DTRS) [7], [55], [56], [57], [58], variable precision rough sets (VPRS) [18], [62], Bayesian rough sets (BRS) [42], game-theoretic rough sets (GTRS) [13], fuzzy rough sets/rough fuzzy sets (FRS/RFS) [9], interval-valued fuzzy rough sets (IVFRS) [12], [15], [43] and Dominance-based fuzzy rough sets [8], [10]. The second class is the extension from one granular to multi-grualation, such as multi-granulation rough sets (MGRS) [38], [39], [40], multi-granulation decision-theoretic rough sets [41], multi-granulation rough sets based covering [23], Neighborhood-based multi-granulation rough sets (NMGRS) [24] and so on.
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The second aspect is theoretical framework researches on three-way decisions. It mainly contains value domain of evaluation functions [51], construction and interpretation of evaluation functions [51], [55], [56] and the mode of tree-way decisions [51].
These researches, however, are premature in theory and there are some problems about three-way decisions.
The first one is a measurement problem on decision conclusions (decision domain). More popular now is a linear order or totally ordered set whose typical representative is [0, 1]. The so-called linear order set (L, ⪯) means that ⪯ denotes a linear order relation (or total order relation) on L, i.e., it satisfies the following conditions.
- (1)
Reflexivity x ⪯ x.
- (2)
Anti-symmetry x ⪯ y, y ⪯ x ⇒ x = y.
- (3)
Transitivity x ⪯ y, y ⪯ z ⇒ x ⪯ z.
- (4)
Comparability x, y ∈ L ⇒ x ⪯ y or y ⪯ x.
But some problems may not be solved by a linear order for decision-making. Yao in [51] used a partially ordered set L and divided L into two nonempty sets, i.e. L = L− ∪ L+ (L− ∩ L+ = Ø), where L− is used to reject and L+ is used to accept. Indeed the problems of three-way decisions come down to the partitions, but he did not give any methods to divide. This paper uses a complete distributive lattice with an inverse order and involutive operator (for short, fuzzy lattice) as a measurement tool, so that its applications are much more comprehensive.
The second one is a decision condition problem (decision condition domain). Current common conditions used to three-way decisions are subsets of universe, fuzzy sets [59] or shadowed sets [36], [37]. It is unified to mappings from universe to fuzzy lattice in this paper.
The third one is an evaluation function problem. Evaluation functions are a key to decision-making. Different evaluation functions determine different decision results. Popular evaluation functions are associated with the conditional probability formulae. For example, in probabilistic rough sets [58], there are many models such as decision-theoretic rough sets based on Bayesian risk analysis (DTRS) [17], [26], [27], [51], [52], [53], [54], [55], [56], [57], [58], variable precision rough sets (VPRS) [18], [62], Bayesian rough sets (BRS) [42], [61], [63] and fuzzy probabilistic rough sets [16], [22], [45]. In these models, probability Pr (C∣[x]R) or (R is an equivalence relation) or (R is a fuzzy equivalence relation) are used as evaluation functions. This paper unifies the evaluation functions through their common properties.
The rest of this paper is organized as follows. In Section 2, the measures of three-way decisions are specified in fuzzy lattice represented by [0, 1], evaluation function axioms are given and three-way decisions spaces are established. Section 3 establishes three-way decisions theory on three-way decisions space, which contains general three-way decisions, lower and upper approximations induced by three-way decisions and multi-granulation three-way decisions. In Section 4, the existing three-way decisions come down to special cases of three-way decisions spaces, including three-way decisions based on fuzzy sets, interval-valued fuzzy sets, shadowed sets, interval sets, random sets and probability sets. At the same time, their corresponding multi-granulation three-way decisions are introduced. Section 5 gives novel dynamic two-way decisions and dynamic three-way decisions based on three-way decisions spaces. Section 6 presents three-way decisions with a pair of evaluation functions based on three-way decisions spaces. Finally this paper is concluded and two questions are discussed.
Section snippets
Three-way decisions space
In this section three-way decisions space (3WDS) is established through unifying decision measurement, decision conditions and evaluation functions of three-way decisions.
Three-way decisions
For the convenience of expression, unless confusion, decision measurement domain LD is denoted by (LD, ∧, ∨, N, 0, 1) and its order relation is shown by ⩽.
Several typical types of three-way decisions
For special cases of three-way decisions spaces, several typical types of three-way decisions are discussed, such as three-way decisions based on fuzzy sets, interval-valued fuzzy sets, fuzzy relations, shadowed sets, interval sets, random sets and probability rough sets. Particularly it is the first time to discuss interval-valued fuzzy decision rough sets in Section 4.7.3.
Dynamic two-way decisions based on three-way decisions spaces
Lots of practical decision-making are not just a one-time three-way decisions. That is to say, three-way decisions may be not made in the first place. But they are made up of multiple two-way decisions. Take a look at the following examples first. Example 5.1 New admissions decisions
New admissions decisions must go through more than one process such as verification of the qualification and multi-stage interview. In these decision-making sessions there exists a common feature to determine rejection region
Three-way decisions with a pair of evaluation functions
Depending on number of evaluation functions, Yao gave two modes of three-way decisions, which are the single evaluation function and dual evaluation functions [51]. The above three-way decisions are based on a single evaluation function. Multi-granulation uses more than one evaluation function, but the idea is based on a single evaluation function mode. Three-way decisions with a pair of evaluation evaluation functions are given below. Definition 6.1 Let (U, Map(V, LC), LD, Ea) and (U, Map(V, LC), LD, Eb) be
Conclusions
Three-way decisions space (3WDS) and three-way decisions theory (3WDT) have been established so that popular three-ways decisions are particular cases of 3WDSs advanced in this paper. At last we discuss the following two questions.
Question 1 Can definition of three-way decisions be changed to the following?
Let (U, Map(V, LC), LD, E) be a three-way decisions spaces, A ∈ Map(V, LC), α, β ∈ LD and 0 ⩽ β ⩽ α ⩽ 1. Then three-way decisions are defined as follows.
- (1)
Acceptance region: ACP(α,β)(E, A) = {x ∈ U∣E(A)(x) ⩾ α
Acknowledgments
The authors are extremely grateful to the anonymous referees and Professor Witold Pedrycz, Editor-in-Chief, for their critical suggestions for improvements. The work described in this paper was supported by grant from the National Natural Science Foundation of China (Grant No. 61179038).
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