Three-way decisions based on semi-three-way decision spaces
Introduction
After three-way decisions (3WD) were proposed by Yao [36], [37], [38], the theory of three-way decisions obtained the rapid development both in theory and applications [44]. The researches on three-way decisions mainly focus on three aspects, background researches, theoretical framework researches and application researches.
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Background researches
The background researches on 3WD are extension researches of rough sets which is the root of theory on three-way decisions, such as decision-theoretic rough sets [6], [43], game-theoretic rough sets [1], [2], [3], interval-valued fuzzy rough sets [11], [14], [15], intuitionistic fuzzy decision-theoretic rough sets [21], triangular fuzzy decision-theoretic rough sets [24], interval-value fuzzy decision-theoretic rough sets [20], [46], [48], dominance-based fuzzy rough sets [7], covering-based rough sets [17], granular variable precision fuzzy rough sets [32] and fuzzy variable precision rough sets [47], multi-granulation rough sets [31], [42] and so on.
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Theoretical framework researches
Theoretical framework researches on 3WD mainly refer to choice of value domain, construction and interpretation of decision evaluation functions [9], [10], [16], the mode of three-way decisions [37], [38], [40], proximity structures of three-way decisions [30] and so on.
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Application researches
The researches of application on 3WD contain a lot of fields, such as incomplete information system [26], risk decision making [18], classification and cluster [45], investment [27], multi-agent [34], group decision making [25], face recognition [19], social networks [30] and cognitive networks [28], recommender systems [1], multi-granular mining [4], medicine [41] and so on.
In theoretical research of three-way decisions, the author introduced axiomatic definitions for decision measurement, decision condition and decision evaluation function and established three-way decision spaces (3WDS) [9], [10]. On this basis 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 [9], interval-valued fuzzy sets [9], type-2 fuzzy sets and interval-valued type-2 fuzzy sets [33], intuitionistic fuzzy sets [21], interval sets [35], shadowed sets [9], [29], hesitant fuzzy sets [10], [22], interval-valued hesitant fuzzy sets [10], random sets [9], [10] and rough sets [9], [39] etc.
Three-way decision spaces were established on a fuzzy lattice, i.e., a complete distributive lattice with an involutive negator [9]. Soon after that, three-way decision spaces were generalized to partially ordered sets in [10], in order to consider three-way decisions based on type-2 fuzzy sets [12], interval-valued type-2 fuzzy sets [13], hesitant fuzzy sets [10] and interval-valued hesitant fuzzy sets [8], [10].
In the formulations of three-way decisions, Yao suggested three refinements/interpretations while introducing the three-way decisions [40]. The first one is based on rough sets, the second one is based on evaluation of objects and the third one is based on trisecting and acting framework, which corresponding functions used in [40] are Pr(A|[x]), v(x) and τ(x), respectively. The notion of three-way decision spaces is a summary and abstraction of Yao's three interpretations. If we consider decision evaluation function E(A)(x) in three-way decision spaces as Pr(A|[x]), then it is a 3WD based on rough sets; if Yao's evaluation function v(x) is E(A)(x), then it is a 3WD based on evaluation-based 3WD; if Yao's three values function τ(x) is taken as E(A)(x), then it is a 3WD based on trisecting and acting framework. The following Fig. 1.1 shows a relationship between Hu's 3WDS and Yao's three interpretations on 3WD.
In three-way decision spaces, it is important to construct a decision evaluation function which satisfies three axioms, minimum element axiom, monotonicity axiom and complement axiom [9], [10]. As referred to [9], [10], background of these three axioms comes from the common properties of the large amount of evaluation functions in probabilistic rough sets and decision-theoretic rough sets, such as for a crisp subset A and an equivalence relation R over a finite universe. Many functions satisfy minimum element axiom and monotonicity axiom which are the most basic conditions. Complement axiom is important and is essential for three-way decisions, which is stressed in the following literatures.
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In [20], “Since , we simplify the rules based only on the probability P(C|[x]) and the losses”.
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In [21], “P(C|[x]) is the conditional probability of an object x belonging to C given that the object is described by its equivalence class [x]. Analogously, is the conditional probability of an object x belonging to , i.e., ”.
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In [23–25,36], “Since , we simplify the rules based only on the probability P(C|[x]) and the loss function(s)”.
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In [26], “Due to we find the rules are only depended on the conditional probability P(X|[x]) and the loss functions (s)”.
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In [37], “By following the same procedure of the two-way model, we can simplify the conditions in these rules based on the assumption (c1) and the relationship P(C|[x]) + P(Cc|[x]) = 1”.
- (6)
In [46], [48], author proved P(A|B) + P(Ac|B) = 1. (Proposition 2.5, [46]; Remark 2.5, [48]).
The above literatures are only part of the evidence on complement axiom and all show the importance of complement axiom. However, there are some useful functions which do not satisfy the complement axiom. For example, in [9,10,33], some evaluation functions must satisfy the special conditions, otherwise they do not satisfy the complement axiom. The examples are listed as follows.
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In Example 2.3 (1)–(4) of [9], if A is a fuzzy set or R is a fuzzy equivalence relation over finite universe U, then
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In Example 2.3 (5)–(7) of [9], if A is a fuzzy set (interval-valued fuzzy set) or R is an interval-valued fuzzy equivalence relation over finite universe U, then the considered functions do not necessarily satisfy the complement axiom.
- (3)
In Section 4.7.3 of [9], if A = [A–, A+] is an interval-valued fuzzy set of finite universe U and R = [R–, R+] is an interval-valued fuzzy equivalence relation of U, then
- (4)
In Theorem 4.6 of [10], for a hesitant fuzzy set H, is a decision evaluation function, but (a ∈ [0, 0.5)∪(0.5, 1] ) is not.
- (5)
In Theorems 3.5, Theorem 3.11 and 3.11 of [33], the authors also considered for type-2 fuzzy set A and for type-2 fuzzy relation R in order to satisfy the complement axiom.
Whether there is a way to construct a decision evaluation function from a function which satisfies only the minimum element axiom and monotonicity axiom? To address this problem, in this paper, firstly, the concept of semi-decision evaluation functions are introduced which only need to satisfy the minimum element axiom and monotonicity axiom. Secondly, this paper presents a transformation method from semi-decision evaluation functions to decision evaluation functions, which makes the applications of three-way decisions through decision evaluation functions more widely available. Thirdly, this paper discusses three-way decisions based on semi-decision evaluation functions for fuzzy sets, interval-valued fuzzy sets, intuionistic fuzzy sets, hesitant fuzzy sets and type-2 fuzzy sets, respectively. Finally, this paper considers three-way decisions over multiple semi-three-way decision spaces.
The rest of this paper is structured as follows. Section 2, as preliminaries, recalls the decision evaluation function axioms and three-way decision spaces based on partially ordered sets. In Section 3, on partially ordered sets, the notion of semi-decision evaluation functions is introduced and semi-three-way decision spaces are established. Taken into account this, we present a transformation method from semi-decision evaluation functions to decision evaluation functions. As special cases of semi-three-way decision spaces based on partially ordered sets, Section 4 discusses three-way decisions based on fuzzy sets, interval-valued fuzzy sets, intuionistic fuzzy sets, hesitant fuzzy sets and type-2 fuzzy sets. Section 5 discusses multiple semi-three-way decision spaces. Section 6 gives an illustrative example to demonstrate the significance of new notions and the transformation methods. Finally, Section 7 concludes this paper.
Section snippets
Preliminaries
The basic concepts, notations and results of bounded partially ordered sets, decision evaluation function and three-way decision spaces [9], [10] are briefly reviewed in this section.
In this paper (P, ≤P ) is a bounded partially ordered set (poset, for short) with an involutive negator NP, the minimum 0P and maximum 1P which is written as (P, ≤P, NP, 0P, 1P), where involutive negator is inverse order (x ≤ y⇔N(y) ≤P N(x), ∀x, y ∈ P) and involutive (NP(NP(x)) = x, ∀x ∈ P). The following examples
Semi-three-way decision spaces
This section introduces the notion of semi-decision evaluation functions and presents a transformation method from semi-decision evaluation functions to decision evaluation functions.
Definition 3.1 Let U be a decision universe and V be a condition universe. Then a mapping E: Map(V, PC) → Map(U, PD) is called a semi-decision evaluation function of U, if it satisfies the Minimum element axiom (E1) and Monotonicity axiom (E2).
In Definition 3.1, term “semi-” is in contrast to Definition 2.1. In Definition 2.1,
Three-way decisions based on semi-three-way decision spaces
This section discusses three-way decisions based on semi-three-way decision spaces for all kinds of decision conditions such as fuzzy sets, interval-valued fuzzy sets, intuitionistic fuzzy sets, random sets, hesitant fuzzy sets and type-2 fuzzy sets.
For the convenience of expression, unless confusion, decision measurement domain PD is denoted by (PD, ≤, N, 0, 1).
The aggregation of multiple semi-three-way decision spaces
Let Ei: Map(V, PC) → Map(U, PD) (i=1, 2,…, n) be n decision evaluation functions of U. Then and are semi-decision evaluation functions of U which were discussed in [9,10]. If we consider PD = [0, 1], then decision evaluation function constructed by Theorem 3.3 from or is .
This shows that the compromise method of multi-granulation rough sets is more reasonable than two extreme methods, optimistic and pessimistic
An illustrative example
Here, we use an example [9] to illustrate some notions of the semi-three-way decision space.
Let us consider an evaluation problem of credit card applicants. Suppose that U={x1, x2, …, x9} is a set of nine applicants. AT = {education, salary} is a set of two condition attributes. The values of attribute ‘‘education’’ are {best, better, good}. And the values of attribute ‘‘salary’’ are {high, middle, low}. We make three specialists I, II, III evaluate the attribute values for these applicants. It
Conclusions
This paper introduces a new notion of semi-decision evaluation functions in order to the theory of three-way decisions are more widely used and finds an effective transformation method from a semi-decision evaluation function to a decision evaluation function on decision domain [0,1], I(2) and I2. Main conclusions in this paper and continuous work to do are listed as follows.
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More functions can be applied in three-way decisions, especially in three-way decisions based on fuzzy sets,
Acknowledgments
The authors are extremely grateful to the anonymous referees and professor Witold Pedrycz, Editor-in-Chief, for their critical suggestions to improve the quality 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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