N-soft rough sets and its applications

Abstract

In this paper, we propose a new hybrid model called N-soft rough sets, which can be seen as a combination of rough sets and N-soft sets. Moreover, approximation operators and some useful properties with respect to N-soft rough approximation space are introduced. Furthermore, we propose decision making procedures for N-soft rough sets, the approximation sets are utilized to handle problems involving multi-criteria decision-making(MCDM), aiming at electing the optional objects and the possible optional objects based on their attribute set. The algorithm addresses some limitations of the extended rough sets models in dealing with inconsistent decision problems. Finally, an application of N-soft rough sets in multi-criteria decision making is illustrated with a real life example.

Keywords

Rough sets Decision making analysis

1 Introduction

Rough set theory (RST) [33] proposed by Z. Pawlak in 1980s, is a systematic approach for the classification of objects. The model is based on the indiscernibility relation, extracted by separating from exemplary decisions certain and doubt knowledge to deal with the inconsistent problem. However, we may encounter cases that some attributes are ordinal in some applications. For example, popularity of food, evaluations of product quality, scores of movies, etc. can be regarded as ordinal attributes(criteria), which have been monotonically ordered as per decision makers’ preference [32, 34]. In order to better deal with information systems with preference relationships, Greco et al. presented the concept of dominance-based rough set approach(DRSA) [21 , 37], the innovation was mainly based on the substitution of the equivalence relation in RST [33] by a dominance relation. Many interesting models of the dominance-based rough set approach have been proposed see [13 , 28]. In recent years, many extensions of DRSA in different directions have been reported. For instance, Du et al. established the dominance-based rough fuzzy set model [16], the idea of fuzzy dominance relation be considered to deal with fuzziness in preference representation. After, Rehman, Ali et al. considered soft preference and soft dominance relation in an information system and proposed variable precision multi decision λ-soft dominance based rough sets [35]. The hybridization of dominance based rough set approach and other mathematical tools have applied in multiple-criteria decision-analysis see [35, 36].

In 1999, the concept of soft sets was first introduced by Molodtsov [29], as a mathematical tool to deal with uncertain and vagueness data. Maji and Ali et al. [8 , 31] investigated many useful properties and applications related to this model. By combining the theories of soft sets and rough sets, Feng et al. [18, 19] proposed soft rough sets model and discussed the relationships between soft rough sets and Pawlak’s rough sets. Researchers have studied the relevant theories of soft rough sets [5 , 40] and applied it to multiple-criteria decision-analysis [17, 41]. Furthermore, a partial relation are introduced in soft sets [9, 10] by Alshami et al. In 2019, Shaheen et al. launched the notion of dominance-based soft rough sets [36] by combining the soft rough sets with the dominance relations.

However, non-binary estimations are also expected in rating or ranking positions. Examples that are closer to our daily experience are movie ratings [20], ternary voting system [11], or evaluation in a high school [24] et al., which can be seen as the form of natural numbers. In 2009, Herawan et al. [26] posed n binary-valued information system in soft sets where each of parameter has its own rankings. In addition, Chen et al. [15] described rating orders compared with the method in [26]. Instead of ratings as assessment, Ali et al. [6] introduced the concepts of lattice(anti-lattice) ordered soft sets to organize rating system among the soft sets parameters. To describe the importance of ordered ranks in actual problems, Fatimah et al. [20] proposed an extended soft sets model N-soft sets, which introduced the parameterized characterizations of the universe of objects that depend on a finite number of grades. Furthermore, the extensions of N-soft sets were developed and applied to resolve real world multi-criteria decision-making problems in the types of fuzzy N-soft sets [2], hesitant N-soft sets [3] and hesitant fuzzy N-soft sets [4], intuitionistic fuzzy N-soft rough sets [1], these models account for the possibility that the parameterized characterization of the universe is fuzzy or the decision-maker have hesitancy in providing their multinary evaluations of the objects. In addition, Hüseyin et al. presented the bipolar N-soft set [27], to tackle various issues based on the bipolar-rating system.

Decision analysis arising in business, medical diagnosis, seasonal influenza and political issues, etc, which involves a large number of criteria which make it difficult to find optimal solutions. Many scholars studied inconsistance decision problems [7 , 38]. One of these decision making problems is the multi-criteria decision-making(MCDM) [2 , 34–36], whose attribute values are preference-ordered. Many scholars projected many valuable techniques and mathematical approaches to address such situations. Although these approaches present possible solutions, there are still certain limitations. In this paper, we propose the theory of N-soft rough sets, aiming at evaluating and finding optimal alternatives based on their criteria. The advantages of using this new method is that it can solve the inconsistency problems in decision analysis. Our approach is to select excellent elements into decision set based on a decision attribute, and the set is approximated by excellent element sets according to the attribute value of each condition attribute, which is a process of choosing the better objects among the good objects.

In this paper, we introduce N-soft rough sets and investigate some fundamental properties. Then, we pose an algorithm to find the optional alternatives and the possible optional alternatives in multi-criteria decision-making (MCDM) problems. Furthermore, we present an application of this model to decision-making situation whose optional alternatives are given by a real example. The rest of the paper is organized as follows. In Section 2, we recall some basic concepts of information system, rough sets and N-soft sets. We introduce basic operations and properties of our new model in Section 3. Moreover, we pose an algorithm to illustrate this novel concept with real life example in Section 4.

2 Preliminaries

In the following, we first recall necessary concepts and preliminaries required in the sequel of our work.

Definition 2.1. [33] Information systems can be formally expressed as a four-tuple $I = 〈 U, Q, V, φ 〉$ , where U = (u₁, u₂, …, u_n) is a non-empty finite set of objects and called the universe, Q = C ∪ D is a non-empty finite set of attributes in which C and D are respectively condition attributes and decision attributes and V is the domain of attributes and φ : U × Q → V assigning each attribute of each object with a value in V. In this article, the values of V are integers.

Definition 2.2. [24] For a given information system $I = 〈 U, Q, V, φ 〉$ , if there is a partial order relation “ ≥ _q " on the value range of an attribute q ∈ Q, then we call q as a criterion, and u_i ≥ _qu_j represents that u_i is at least as good as u_j under the criterion q, where u_i, u_j ∈ U, i.e., u_i is superior than u_j on q. When all attributes in the information system are criteria, the information system is called an ordered information system.

For a subset of attributes A ⊆ Q, u_i ≥ _Au_j means that u_i dominates u_j on all the criteria in A.

In general, we denote an ordered information system with a single decision attribute d by $I_{d}^{≽} = {U, Q, V, φ}$ , where Q = C ∪ {d}. For simplicity, we only consider attributes with increasing preference in the following. In an ordered information system $I_{d}^{≽}$ , we say that u_i dominates u_j with respect to A ⊆ C if u_i ≽ _Au_j, denoted by $u_{i} R_{A}^{≽} u_{j}$ , and u_i dominates u_j with respect to d if u_i ≽ _du_j, denoted by $u_{i} R_{d}^{≽} u_{j}$ .

Denote $\begin{matrix} R_{A}^{≽} & = {(u_{i}, u_{j}) \in U \times U | u_{i} ≽_{A} u_{j}} = {(u_{i}, u_{j}) \in U \times U | φ (u_{i}, q) \geq φ (u_{j}, q), \forall q \in A}, \\ R_{d}^{≽} & = {(u_{i}, u_{j}) \in U \times U | u_{i} ≽_{d} u_{j}} = {(u_{i}, u_{j}) \in U \times U | φ (u_{i}, d) \geq φ (u_{j}, d)}, \end{matrix}$

$\begin{matrix} [u_{i}] [u_{i}]_{A}^{≽} & = {u_{j} \in U | (u_{j}, u_{i}) \in R_{A}^{≽}} = {u_{j} \in U | φ (u_{j}, q) \geq φ (u_{i}, q), \forall q \in A}, \\ [u_{i}]_{d}^{≽} & = {u_{j} \in U | (u_{j}, u_{i}) \in R_{d}^{≽}} = {u_{j} \in U | φ (u_{j}, d) \geq φ (u_{i}, d)} . \end{matrix}$

Where $R_{A}^{≽}$ and $R_{d}^{≽}$ are called dominance relation and dominance class [21, 24] in ordered information system $I_{d}^{≽} .$

Definition 2.3. [39] Let $I_{d}^{≽} = {U, Q, V, φ}$ be an ordered information system with a single decision attribute, where Q = C ∪ {d}. If $R_{C}^{≽} ⊈ R_{d}^{≽}$ , which is defined by $\forall u \in U, [u]_{C}^{≽} ⊈ [u]_{d}^{≽}$ , then the ordered information system is an inconsistent ordered information system.

Next, we recall some basic concepts regarding rough sets and N-soft sets.

Definition 2.4. [33] Let U be a non-empty finite universe and R an equivalence relation on U. The pair (U, R) is called a Pawlak approximation space. Using the equivalence relation R, one can define the following two operations:

${\underline{X}}_{R} = {x \in U : [x]_{R} \subseteq X}$ ,

${\bar{X}}_{R} = {x \in U : [x]_{R} \cap X \neq \emptyset}$ assigning to every subset X ⊆ U, two sets ${\underline{X}}_{R}$ and ${\bar{X}}_{R}$ called the R - lower and the R - upper rough approximation of X, respectively.

Definition 2.5. [20] Let U be a universe of objects and A the set of attributes, A ⊆ Q. Let G = {0, 1, …, N - 1} be a set of ordered grades where N ∈ {2, 3, …}. A triple (F, A, N) is called a N-soft set on U if F : A→ 2^U ×G, with the property that for each q ∈ A, there exists a unique (u, g_q) ∈ U × G such that (u, g_q) ∈ F (q) , u ∈ U, g_q ∈ G.

Let U = {u_i, i = 1, 2, …, m} and A = {q_j, j = 1, 2, …, n} be finite sets. The N-soft set can be presented by Table 1, where r_ij means (u_i, r_ij) ∈ F (q_j) or F (q_j) (u_i) = r_ij.

Table 1
Table for (F, A, N)

(F,A,N) q ₁ q ₂ … q _n

u ₁ r ₁₁ r ₁₂ ⋯ r _1n

u ₂ r ₂₁ r ₂₂ ⋯ r _2n

⋯ ⋯ ⋯ ⋯ ⋯

u _m r _m1 r _m2 ⋯ r _mn

(F,A,N)	q ₁	q ₂	…	q _n
u ₁	r ₁₁	r ₁₂	⋯	r _1n
u ₂	r ₂₁	r ₂₂	⋯	r _2n
⋯	⋯	⋯	⋯	⋯
u _m	r _m1	r _m2	⋯	r _mn

3 N-soft rough approximations and N-soft rough sets

In this section, we introduce N-soft rough approximation space and N-soft rough sets. Let U be the initial universe and X ⊆ U. The complement of X is denoted by -X.

Definition 3.1. Let S = (F, A, N) be a N-soft set over U and 0 ≤ t < N a threshold. For every q ∈ A, the set associated with (F, A, N) and t denoted by f (q, t^≥), where f : q → 2^U, and it is defined by the expression f (q, t^≥) = {u_i ∈ U|F (q) (u_i) ≥ t} .

From the perspective of complement, for every q ∈ A, we can also define

f (q, t^≤) = - f (q, (t + 1) ^≥), where 0 ≤ t < N,

f (q, [t, k]) = f (q, t^≥) ⋂ f (q, k^≤),where 0 ≤ t ≤ k < N.

Definition 3.2. Let S = (F, A, N) be a N-soft set over U and 0 ≤ t < N a threshold. We denoted that f (A, t^≥) = ⋂ _q∈Af (q, t^≥) and f^∗ (A, t^≥) = ⋃ _q∈Af (q, t^≥).

Definition 3.3. Let S = (F, A, N) be a N-soft set over U. Then R = (U, S, N) is called a N-soft rough approximation space. For each subset X ⊆ U, we define the following two approximate operations

${\underline{L}}_{R} (X, A, t) = {u \in U : \exists q \in A, [u \in f (q, t^{\geq}), u \in f (q, t^{\geq}) \subseteq X]},$

${\bar{L}}_{R} (X, A, t) = {u \in U : \exists q \in A, [u \in f (q, t^{\geq}), u \in f (q, t^{\geq}) \cap X \neq \emptyset]} .$

The two sets ${\underline{L}}_{R} (X, A, t)$ and ${\bar{L}}_{R} (X, A, t)$ are called the N-soft rough R-lower approximation and the N-soft rough R-upper approximation of X, respectively.

In general, we use ${\underline{L}}_{R} (X, A, t)$ and ${\bar{L}}_{R} (X, A, t)$ as N-soft rough approximations of X with respect to R. Moreover, the sets

${Pos}_{R} (X, A, t) = {\underline{L}}_{R} (X, A, t)$

${Neg}_{R} (X, A, t) = - {\bar{L}}_{R} (X, A, t)$

${Bnd}_{R} (X, A, t) = {\bar{L}}_{R} (X, A, t) - {\underline{L}}_{R} (X, A, t)$

are called N-soft rough R-positive region, the N-soft rough R-negative region and the N-soft rough R-boundary region of X, respectively. If ${\underline{L}}_{R} (X, A, t) = {\bar{L}}_{R} (X, A, t)$ , that is Bnd_R (X, A, t) =∅, then X is said to be a N-soft rough R-definable set. If ${\underline{L}}_{R} (X, A, t) \neq {\bar{L}}_{R} (X, A, t)$ , that is Bnd_R (X, A, t)≠ ∅, then X is said to be a N-soft R-rough set. It is clear that N-soft rough approximation space is a generalized soft rough approximation space.

For every X ⊆ U and 0 ≤ t < N be a threshold, we can know that ${\underline{L}}_{R} (X, A, t) \subseteq X$ and ${\underline{L}}_{R} (X, A, t) \subseteq {\bar{L}}_{R} (X, A, t)$ . However, we notice that $X \subseteq {\bar{L}}_{R} (X, A, t)$ does not hold in general, which can be explained by the following example.

Example 3.4. Let U = {u₁, u₂, u₃, u₄, u₅, u₆} is universe and Q = C ⋃ {d}, where A = {q₁, q₂, q₃, q₄, d} ⊆ Q. Let S = (F, A, 4) be a 4-soft set over U and R = (U, S, 4) a 4-soft rough approximation space. The related concepts are explained by the following example.

From Table 2, we can obtain that f (q₁, 1^≥) = {u₁, u₅}, f (q₂, 1^≥) = {u₃}, f (q₃, 1^≥) =∅, f (q₄, 1^≥) = {u₁, u₂, u₄}.

Table 2
Table for (F, A, 4)

q ₁ q ₂ q ₃ q ₄ d

u ₁ 1 0 0 2 3

u ₂ 0 0 0 3 3

u ₃ 0 1 0 0 1

u ₄ 0 0 0 2 2

u ₅ 2 0 0 0 3

u ₆ 0 0 0 0 1

	q ₁	q ₂	q ₄	d
u ₁	1	0	2	3
u ₂	0	0	3	3
u ₃	0	1	0	1
u ₄	0	0	2	2
u ₅	2	0	0	3
u ₆	0	0	0	1

Let X₁ = f (d, 1^≤) = {u₃, u₆}, we have ${\underline{L}}_{R} (X_{1}, A, 1) = {\bar{L}}_{R} (X_{1}, A, 1) = {u_{3}}$ by Definition 3 and Definition 3, therefore X₁ is a 4-soft rough R-definable set.

Let X₂ = f (d, 2^≤) = {u₃, u₄, u₆}, then we have ${\underline{L}}_{R} (X_{2}, A, 1) = {u_{3}}$ and ${\bar{L}}_{R} (X_{2}, A, 1) = {u_{1}, u_{2}, u_{3}, u_{4}}$ . ${\underline{L}}_{R} (X_{2}, A, 1) \neq {\bar{L}}_{R} (X_{2}, A, 1)$ , X₂ is a 4-soft rough set.

It is clear that {u₃, u₄, u₆} ⊈ {u₁, u₂, u₃, u₄}. It implies that $X_{2} ⊈ {\bar{L}}_{R} (X_{2}, A, 2) .$

Remark 3.5. In this paper, we apply the approach of N-soft rough sets to deal with MCDM problem with a single decision attribute, this method can be also used to address the problem with multi-decision attributes in D, where D = {d₁, d₂, ⋯ , d_l}. Then we can find the set of optimal elements X = f (D, t^≥), and use a series of f (q_i, t^≥) to approximate X, where q_i ∈ C is a criteria and 0 ≤ t < N a threshold.

Proposition 3.6. Let S = (F, A, N) be a N-soft set over U, 0 ≤ t < N a threshold and R = (U, S, N) a N-soft rough approximation space. Then we have the following two approximations.

${\underline{L}}_{R} (X, A, t) = ⋃_{q \in A} {f (q, t^{\geq}) : f (q, t^{\geq}) \subseteq X}$

and

${\bar{L}}_{R} (X, A, t) = ⋃_{q \in A} {f (q, t^{\geq}) : f (q, t^{\geq}) \cap X \neq \emptyset} .$

They are called the t th lower approximation and the t th upper approximation of X, respectively.

Proposition 3.7. Suppose that S = (F, A, N) is a N-soft set over U, 0 ≤ t < N a threshold and R = (U, S, N) is the corresponding N-soft rough approximation space. For X ⊆ U, the following properties can be easily established.

(1) ${\underline{L}}_{R} (\emptyset, A, t) = {\bar{L}}_{R} (\emptyset, A, t) = \emptyset$ ;

(2) ${\underline{L}}_{R} (U, A, t) = ⋃_{t \in R} {\bar{L}}_{R} (U, A, t) = ⋃_{q \in A} f (q, t^{\geq})$ ;

(3) ${\underline{L}}_{R} (X \cap Y, A, t) \subseteq {\underline{L}}_{R} (X, A, t) \cap {\underline{L}}_{R} (Y, A, t)$ ;

(4) ${\underline{L}}_{R} (X \cup Y, A, t) \supseteq {\underline{L}}_{R} (X, A, t) \cup {\underline{L}}_{R} (Y, A, t)$ ;

(5) ${\bar{L}}_{R} (X \cup Y, A, t) = {\bar{L}}_{R} (X, A, t) \cup {\bar{L}}_{R} (Y, A, t)$ ;

(6) ${\bar{L}}_{R} (X \cap Y, A, t) \subseteq {\bar{L}}_{R} (X, A, t) \cap {\bar{L}}_{R} (Y, A, t)$ ;

(7) $X \subseteq Y \Rightarrow {\underline{L}}_{R} (X, A, t) \subseteq {\underline{L}}_{R} (Y, A, t)$ ;

(8) $X \subseteq Y \Rightarrow {\bar{L}}_{R} (X, A, t) \subseteq {\bar{L}}_{R} (Y, A, t)$ ;

(9) $k \geq t \Rightarrow {\underline{L}}_{R} (X, A, k) \subseteq {\underline{L}}_{R} (X, A, t)$ ;

(10) $k \geq t \Rightarrow {\bar{L}}_{R} (X, A, k) \subseteq {\bar{L}}_{R} (X, A, t)$ .

Proposition 3.8. Let S = (F, A, N) be a N-soft set over U, 0 ≤ t < N a threshold and R = (U, S, N) is the corresponding N-soft rough approximation space. For any X ⊆ U, X is N-soft rough R-definable set if and only if ${\bar{L}}_{R} (X, A, t) \subseteq X$ .

Proof. Note that X is N-soft rough R-definable set, then ${\underline{L}}_{R} (X, A, t) = {\bar{L}}_{R} (X, A, t) \subseteq X$ . Conversely, suppose that ${\bar{L}}_{R} (X, A, t) \subseteq X$ for X ⊆ U. To prove that X is N-soft rough R-definable set, we only need to prove that ${\bar{L}}_{R} (X, A, t) \subseteq {\underline{L}}_{R} (X, A, t)$ . Let $u \in {\bar{L}}_{R} (X, A, t)$ . Then u ∈ f (q, t^≥) and f (q, t^≥)∩ X ≠ ∅ for some q ∈ A. It follows that $u \in f (q, t^{\geq}) \subseteq {\bar{L}}_{R} (X, A, t) \subseteq X$ . Hence $u \in {\underline{L}}_{R} (X, A, t)$ , and so ${\bar{L}}_{R} (X, A, t) \subseteq {\underline{L}}_{R} (X, A, t)$ as required. □

Theorem 3.9. Let S = (F, A, N) be a N-soft set over U, 0 ≤ t < N a threshold and R = (U, S, N) a N-soft rough approximation space. For any X ⊆ U, the following properties are available.

(1) ${\underline{L}}_{R} ({\bar{L}}_{R} (X, A, t), A, t) = {\bar{L}}_{R} (X, A, t)$ ;

(2) ${\bar{L}}_{R} ({\underline{L}}_{R} (X, A, t), A, t) \supseteq {\underline{L}}_{R} (X, A, t)$ ;

(3) ${\underline{L}}_{R} ({\underline{L}}_{R} (X, A, t), A, t) = {\underline{L}}_{R} (X, A, t)$ ;

(4) ${\bar{L}}_{R} ({\bar{L}}_{R} (X, A, t), A, t) \supseteq {\bar{L}}_{R} (X, A, t)$ ;

Proof.

(1) Let $Y = {\bar{L}}_{R} (X, A, t)$ and u ∈ Y. Then u ∈ f (q, t^≥) and f (q, t^≥)∩ X ≠ ∅ for some q ∈ A. By Proposition 3, $Y = {\bar{L}}_{R} (X, A, t) = ⋃_{q \in A} {f (q, t^{\geq}) : f (q, t^{\geq}) \cap X \neq \emptyset}$ , which means u ∈ f (q, t^≥) ⊆ Y for some q ∈ A. Therefore, we have $u \in {\underline{L}}_{R} (Y, A, t)$ and $Y \subseteq {\underline{L}}_{R} (Y, A, t)$ . On the other hand, it is obvious that ${\underline{L}}_{R} (Y, A, t) \subseteq Y$ holds for any Y ⊆ U. Thus, we can conclude that $Y = {\underline{L}}_{R} (Y, A, t)$ .

(2) Let $Y = {\underline{L}}_{R} (X, A, t)$ and u ∈ Y. Then u ∈ f (q, t^≥) ⊆ X for some q ∈ A. By proposition 3, $Y = {\underline{L}}_{R} (X, A, t) = ⋃_{q \in A} {f (q, t^{\geq}) : f (q, t^{\geq}) \subseteq X}$ , we deduce that u ∈ f (q, t^≥) and f (q, t^≥)∩ Y = f (q, t^≥) ≠ ∅. Hence, we have that $u \in {\bar{L}}_{R} (Y, A, t)$ and so $Y \subseteq {\bar{L}}_{R} (Y, A, t)$ as required.

(3) Let $Y = {\underline{L}}_{R} (X, A, t)$ and u ∈ Y. Then u ∈ f (q, t^≥) ⊆ X for some q ∈ A. But $Y = {\underline{L}}_{R} (X, A, t) = ⋃_{q \in A} {f (q, t^{\geq}) : f (q, t^{\geq}) \subseteq X}$ , we deduce that u ∈ f (q, t^≥) ⊆ Y for q ∈ A. Thus we can receive $u \in {\underline{L}}_{R} (Y, A, t)$ , and that is to say $Y \subseteq {\underline{L}}_{R} (Y, A, t)$ . Since ${\underline{L}}_{R} (Y, A, t) \subseteq Y$ always holds for any Y ⊆ U. We have that $Y = {\underline{L}}_{R} (Y, A, t)$ .

(4) Let $Y = {\bar{L}}_{R} (X, A, t)$ and u ∈ Y. Then u ∈ f (q, t^≥) and f (q, t^≥)∩ X ≠ ∅ for some q ∈ A. But $Y = {\bar{L}}_{R} (X, A, t) = ⋃_{q \in A} {f (q, t^{\geq}) : f (q, t^{\geq}) \cap X \neq \emptyset}$ , it follows that u ∈ f (q, t^≥) and f (q, t^≥)∩ Y = f (q, t^≥) ≠ ∅. Hence $u \in {\bar{L}}_{R} (Y, A, t)$ , and so $Y \subseteq {\bar{L}}_{R} (Y, A, t)$ . □

Definition 3.10. Let S = (F, A, N) be a N-soft set over U, 0 ≤ t < N a threshold and R = (U, S, N) a N-soft rough approximation space. For any X, Y ⊆ U, we define

$X ⌢_{R} Y \Leftrightarrow {\underline{L}}_{R} (X, A, t) = {\underline{L}}_{R} (Y, A, t)$

$X ⌣_{R} Y \Leftrightarrow {\bar{L}}_{R} (X, A, t) = {\bar{L}}_{R} (Y, A, t)$

and

X ≍ _RY ⇔ X ⌢ _RY and X ⌣ _RY.

These binary relations are called the lower N-soft rough equal relation, the upper N-soft rough equal relation, and the N-soft rough equal relation, respectively.

Proposition 3.11. Let S = (F, A, N) be a N-soft set over U, 0 ≤ t < N a threshold and R = (U, S, N) a N-soft rough approximation space. Then for X, X₁, Y, Y₁ ⊆ U, we have

(1) X ⌣ _RX₁, X₁ ⌣ _RY ⇒ X ⌣ _RY;

(2) X ⌣ _RY ⇔ X ⌣ _R (X ∪ Y) ⌣ _RY;

(3) X ⌣ _RX₁, Y ⌣ _RY₁ ⇒ (X ∪ Y) ⌣ _R (X₁ ∪ Y₁);

(4) X ⌣ _RY ⇒ X ∪ (- Y) ⌣ _RU;

(5) X⊆ Y, Y ⌣ _R ∅ ⇒ X ⌣ _R ∅;

(6) X ⊆ Y, X ⌣ _RU ⇒ Y ⌣ _RU.

Proof. (1) Let X ⌣ _RX₁, X₁ ⌣ _RY. Then we have ${\bar{L}}_{R} (X, A, t) = {\bar{L}}_{R} (X_{1}, A, t)$ and ${\bar{L}}_{R} (X_{1}, A, t) = {\bar{L}}_{R} (Y, A, t)$ by Definition 3. Therefore, we have that ${\bar{L}}_{R} (X, A, t) = {\bar{L}}_{R} (Y, A, t)$ . That is X ⌣ _RY.

(2) Let X ⌣ _RY. Then ${\bar{L}}_{R} (X, A, t) = {\bar{L}}_{R} (Y, A, t)$ . From Proposition 3, we can obtain that ${\bar{L}}_{R} (X \cup Y, A, t) = {\bar{L}}_{R} (X, A, t) \cup {\bar{L}}_{R} (Y, A, t)$ . It implies ${\bar{L}}_{R} (X \cup Y, A, t) = {\bar{L}}_{R} (X, A, t) = {\bar{L}}_{R} (Y, A, t)$ . Therefore X ⌣ _R (X ∪ Y) ⌣ _RY. Conversely, if X ⌣ _R (X ∪ Y) ⌣ _RY, we immediately have that X ⌣ _RY by using the transitivity of the relation ⌣_R,

(3) Assume that X ⌣ _RX₁ and Y ⌣ _RY₁. Then by Definition 3, we know that ${\bar{L}}_{R} (X, A, t) = {\bar{L}}_{R} (X_{1}, A, t)$ and ${\bar{L}}_{R} (Y, A, t) = {\bar{L}}_{R} (Y_{1}, A, t)$ . From Proposition 3, ${\bar{L}}_{R} (X \cup Y, A, t) = {\bar{L}}_{R} (X, A, t) \cup {\bar{L}}_{R} (Y, A, t)$ and ${\bar{L}}_{R} (X_{1} \cup Y_{1}, A, t) = {\bar{L}}_{R} (X_{1}, A, t) \cup {\bar{L}}_{R} (Y_{1}, A, t)$ . Thus, we can obtain that ${\bar{L}}_{R} (X \cup Y, A, t) = {\bar{L}}_{R} (X_{1}, A, t) \cup {\bar{L}}_{R} (Y_{1}, A, t)$ . That is, X ⌣ _RX₁, Y ⌣ _RY₁ imply (X ∪ Y) ⌣ _R (X₁ ∪ Y₁).

(4) Suppose that X ⌣ _RY. Then by Definition 3, we know that ${\bar{L}}_{R} (X, A, t) = {\bar{L}}_{R} (Y, A, t)$ . Since ${\bar{L}}_{R} (X \cup (- Y), A, t) = {\bar{L}}_{R} (X, A, t) \cup {\bar{L}}_{R} (- Y, A, t)$ and ${\bar{L}}_{R} (U, A, t) = {\bar{L}}_{R} (Y, A, t) \cup {\bar{L}}_{R} (- Y, A, t)$ , it follows that ${\bar{L}}_{R} (X \cup (- Y), A, t) = {\bar{L}}_{R} (U, A, t)$ . Hence X ∪ (- Y) ⌣ _RU as required.

(5) Since X ⊆ Y and Y⌣ _R ∅. Then we deduce ${\bar{L}}_{R} (X, A, t) \subseteq {\bar{L}}_{R} (Y, A, t) = {\bar{L}}_{R} (\emptyset, A, t) = \emptyset$ . Hence ${\bar{L}}_{R} (X, A, t) = \emptyset = {\bar{L}}_{R} (\emptyset, A, t)$ , and so we have that X⌣ _R ∅.

(6) Suppose that X ⊆ Y and X ⌣ _RU. Then we have ${\bar{L}}_{R} (X, A, t) = {\bar{L}}_{R} (U, A, t) \subseteq {\bar{L}}_{R} (Y, A, t)$ . Since Y ⊆ U, it obvious that ${\bar{L}}_{R} (Y, A, t) \subseteq {\bar{L}}_{R} (U, A, t)$ . So we can obtain ${\bar{L}}_{R} (Y, A, t) = {\bar{L}}_{R} (U, A, t)$ . That is Y ⌣ _RU. □

4 Decision analysis using N-soft rough set

In this section, we aim at solving the problem of multi-criteria decision making(MCDM) with a single decision attribute. Algorithm 1 provides an approach to distribution reduction in inconsistent information systems. Attribute reduction based on proposed technique is applied to find the essential attribute required for classification. Based on the N-soft rough lower approximation ${\underline{L}}_{R} (X, A, t)$ and upper approximation ${\bar{L}}_{R} (X, A, t)$ , optimal objects and possible optimal objects will be chosen according to the following decision methodology.

4.1 Proposed methodology

Algorithm 1

Step 1: Input the original ordered information system $I_{d}^{≽} = 〈 U, C \cup {d}, V, φ 〉$ to apply N-soft rough set.

{Step 2: A single decision attribute {d} induces a partition of objects U into a finite number of decision classes Cl = {Cl_t, t ∈ T} , T = {0, . . . , N - 1} such that each u ∈ U belongs to one and only one class Cl_t ∈ Cl, and for all u_i ∈ Cl_t, they have the same grade t, where t ∈ T.

Step 3: Let C = C₁ ∪ C₂ ∪ ⋯ ∪ C_n, where C_i ∩ C_j = ∅ (1 ≤ i, j ≤ n) and the criteria of C_i select the same t_i^≥ (0 ≤ t_i < N).

Step 4: Find reduct sets. If there exists RED_Cl ⊆ C_i, then C_i = RED_Cl.

Intuitively speaking, a relative reduct of a information system is a subset of attributes that has the same or similar classification property as that of the entire set of condition attributes. There are the steps of distribution reduction for inconsistent ordered information system.

Substep 4.1: For A ⊆ C, we calculate the distribution function $μ_{A}^{≽} (u)$ about criterion set A.

; Let $R_{A}^{≽}$ , $R_{d}^{≽}$ be dominance relations derived from criterion set A and decision attribute d, respectively. For u_i, u_j ∈ U. Denote $U / R_{A}^{≽} = {_{A}^{≽} | u_{i} \in U},$ $U / R_{d}^{≽} = {D_{0}, D_{1}, D_{2}, \dots, D_{N - 1}},$ $μ_{A}^{≽} (u_{i}) = (\frac{| D_{0} \cap [u_{i}]_{A}^{≽} |}{| U |}, \frac{| D_{1} \cap [u_{i}]_{A}^{≽} |}{| U |}, \frac{| D_{2} \cap [u_{i}]_{A}^{≽} |}{| U |}, \dots, \frac{| D_{N - 1} \cap [u_{i}]_{A}^{≽} |}{| U |}),$ where $[u_{i}]_{A}^{≽} = {u_{j} \in U | (u_{i}, u_{j}) \in R_{A}^{≽}}$ , $D_{i} = {[u_{i}]_{d}^{≽} | u_{i} \in U}$ and |U| is the number of elements in the inconsistent ordered information system.

Substep 4.2: Based on the distribution function $μ_{A}^{≽} (u_{i})$ in Substep 4.1, we denote $D_{μ^{≽}}^{*} = {(u_{i}, u_{j}) : μ_{A}^{≽} (u_{i}) ≰ μ_{A}^{≽} (u_{j})},$ and $\begin{matrix} D_{μ^{≽}} (u_{i}, u_{j}) = \\ {\begin{matrix} {q \in A : F (q) (u_{i}) > F (q) (u_{j})}, & (u_{i}, u_{j}) \in D_{μ^{≽}}^{*}, \\ \emptyset, & (u_{i}, u_{j}) \notin D_{μ^{≽}}^{*} . \end{matrix} \end{matrix}$

Substep 4.3: Let M_{μ
^≽} = (D_{μ
^≽} (u_i, u_j) , u_i, u_j ∈ U) be distribution discernibility matrix. Denote $\begin{matrix} F_{μ}^{≽} & = \land {\lor {q_{k} : q_{k} \in D_{μ^{≽}} (u_{i}, u_{j})}, u_{i}, u_{j} \in U} \\ = \land {\lor {q_{k} : q_{k} \in D_{μ^{≽}} (u_{i}, u_{j})}, u_{i}, u_{j} \in D_{μ^{≽}}^{*}}}, \end{matrix}$ where $F_{μ}^{≽}$ is called discernibility formula of distribution discernibility.

Step 5: Using Definition 3, we can get the sets of the lower approximation ${\underline{L}}_{R} (X, C_{i}, t_{i})$ and the upper approximation ${\bar{L}}_{R} (X, C_{i}, t_{i})$ , respectively.

Step 6: The set of optimal elements $O = ⋃_{C_{i} \subseteq C, t_{i} \in [0, N)} {\underline{L}}_{R} (X, C_{i}, t_{i})$ and possible optimal elements $P = ⋃_{C_{i} \subseteq C, t_{i} \in [0, N)} {\bar{L}}_{R} (X, C_{i}, t_{i})$ .

4.2 Explanation of proposed methodology

Example 4.1. Let U = {u₁, u₂, u₃, u₄, u₅, u₆} be a set of houses and C = {q₁, q₂, q₃, q₄, q₅} ⊆ Q a set of features of houses given by the purchaser, in which q₁, q₂, q₃, q₄, q₅ represent "emergency configuration", "community property service quality", "in the green surroundings", "price" and "geographical location", respectively. The attribute values are given by expert in Table 3, for the same attribute, the greater the value represents the level is higher. Now, we consider using N-soft rough sets to select the optimal houses for purchaser.

Specific steps are as follows:

Step 1: Input the original ordered information system $I_{d}^{≽}$ .

Step 2: Based on the decision attribute d, U can be divided into Cl₁, Cl₂, Cl₃ by different grade t, we choose $X = {Cl}_{2}^{\geq} = D_{2} = {u_{1}, u_{2}, u_{4}, u_{5}}$ .

Step 3: Divide criterions C into 3 partions. Let C = C₁ ∪ C₂ ∪ C₃ and C₁ = {q₁} , C₂ = {q₂, q₄, q₅} , C₃ = {q₃}. For q ∈ C₁, we choose the elements with grade 4^≥, for q ∈ C₂, we choose the elements with grade 3^≥ and for q ∈ C₃, we choose the elements with grade 2^≥.

Step 4: Find reduct sets.

From Table 3, we have $\begin{matrix} [u_{1}] [u_{1}]_{C}^{≽} & = {u_{1}, u_{2}, u_{5}, u_{6}}; & [u_{2}]_{C}^{≽} & = {u_{2}, u_{5}, u_{6}}; \\ [u_{3}]_{C}^{≽} & = {u_{2}, u_{3}, u_{4}, u_{5}, u_{6}}; & [u_{4}]_{C}^{≽} & = {u_{4}}; \\ [u_{5}]_{C}^{≽} & = {u_{5}}; & [u_{6}]_{C}^{≽} & = {u_{6}}; \end{matrix}$ and $\begin{matrix} D_{0} = [u_{1}]_{d}^{≽} = [u_{5}]_{d}^{≽} = {u_{1}, u_{5}}; \\ D_{1} = [u_{2}]_{d}^{≽} = [u_{4}]_{d}^{≽} = {u_{1}, u_{2}, u_{4}, u_{5}}; \\ D_{2} = [u_{3}]_{d}^{≽} = [u_{6}]_{d}^{≽} = {u_{1}, u_{2}, u_{3}, u_{4}, u_{5}, u_{6}} . \end{matrix}$

And we can caculate $\begin{matrix} μ_{C}^{≽} (u_{1}) & = (\frac{1}{3}, \frac{1}{2}, \frac{2}{3}); & μ_{C}^{≽} (u_{2}) & = (\frac{1}{6}, \frac{1}{3}, \frac{1}{2}); \\ μ_{C}^{≽} (u_{3}) & = (\frac{1}{6}, \frac{1}{2}, \frac{5}{6}); & μ_{C}^{≽} (u_{4}) & = (0, \frac{1}{6}, \frac{1}{6}); \\ μ_{C}^{≽} (u_{5}) & = (\frac{1}{6}, \frac{1}{6}, \frac{1}{6}); & μ_{C}^{≽} (u_{6}) & = (0, 0, \frac{1}{6}) . \end{matrix}$ In additional, $\begin{matrix} D_{μ^{≽}}^{*} = { & (u_{1}, u_{2}), (u_{1}, u_{3}), (u_{1}, u_{4}), (u_{1}, u_{5}), (u_{1}, u_{6}), (u_{2}, u_{4}), (u_{2}, u_{5}), (u_{2}, u_{6}), \\ (u_{3}, u_{1}) (u_{3}, u_{2}) (u_{3}, u_{4}), (u_{3}, u_{5}), (u_{3}, u_{6}), (u_{4}, u_{6}), (u_{5}, u_{4}), (u_{5}, u_{6})} . \end{matrix}$ The Table 4 is the distribution discernibility matrix of Table 3.

Table 3
Illustrative table

U q ₁ q ₂ q ₃ q ₄ q ₅ d

u ₁ 2 2 1 1 2 3

u ₂ 4 2 2 2 2 2

u ₃ 2 1 2 2 1 1

u ₄ 3 1 3 4 3 2

u ₅ 4 3 2 3 3 3

u ₆ 4 2 3 4 3 1

U	q ₁	q ₂	q ₃	q ₄	q ₅	d
u ₁	2	2	1	1	2	3
u ₂	4	2	2	2	2	2
u ₃	2	1	2	2	1	1
u ₄	3	1	3	4	3	2
u ₅	4	3	2	3	3	3
u ₆	4	2	3	4	3	1

Table 4

Distribution discernibility matrix

U	u ₁	u ₂	u ₃	u ₄	u ₅	u ₆
u ₁	∅	∅	q₂, q₅	q ₂	∅	∅
u ₂	∅	∅	∅	q₁, q₂	∅	∅
u ₃	q₃, q₄	∅	∅	∅	∅	∅
u ₄	∅	∅	∅	∅	∅	∅
u ₅	∅	∅	∅	q₁, q₂	∅	q ₂
u ₆	∅	∅	∅	∅	∅	∅

Consequently, we have $\begin{matrix} F_{μ}^{≽} = (q_{2} \lor q_{5}) \land q_{2} \land (q_{1} \lor q_{2}) \land (q_{3} \lor q_{4}) \\ = (q_{2} \land q_{3}) \lor (q_{2} \land q_{4}) . \end{matrix}$ There are two reducts for the criterion set C: ${RED}_{Cl}^{1} = {q_{2}, q_{3}};$ ${RED}_{Cl}^{2} = {q_{2}, q_{4}} .$ It is obvious that ${RED}_{Cl}^{2} = {q_{2}, q_{4}} \subseteq C_{2}$ . Then C₂ = {q₂, q₄} .

Step 5: Find the sets of lower approximation ${\underline{L}}_{R} (X, C_{i}, t_{i})$ and the upper approximation ${\bar{L}}_{R} (X, C_{i}, t_{i})$ , respectively. $\begin{matrix} f (q_{1}, 4^{\geq}) & = {u_{2}, u_{5}, u_{6}}; & f (q_{2}, 3^{\geq}) & = {u_{5}}; \\ f (q_{3}, 2^{\geq}) & = {u_{2}, u_{3}, u_{4}, u_{5}, u_{6}}; & f (q_{4}, 3^{\geq}) & = {u_{4}, u_{5}, u_{6}} . \end{matrix}$ Then we get the sets of under approximation $\begin{matrix} {\underline{L}}_{R} (X, C_{1}, 4) = \emptyset; {\underline{L}}_{R} (X, C_{2}, 3) = {u_{5}}; \\ {\underline{L}}_{R} (X, C_{3}, 2) = \emptyset \end{matrix}$ and $\begin{matrix} {\bar{L}}_{R} (X, C_{1}, 4) = {u_{2}, u_{5}, u_{6}}; {\bar{L}}_{R} (X, C_{2}, 3) \\ = {u_{4}, u_{5}, u_{6}}; {\bar{L}}_{R} (X, C_{3}, 2) = {u_{2}, u_{3}, u_{4}, u_{5}, u_{6}} . \end{matrix}$

Step 6: The set of optimal elements $O = {\underline{L}}_{R} (X, C_{1}, 4) \cup {\underline{L}}_{R} (X, C_{2}, 3) \cup {\underline{L}}_{R} (X, C_{3}, 2) = {u_{5}}$ and the set of possible optimal elements $\begin{matrix} P = {\bar{L}}_{R} (X, C_{1}, 4) \cup {\bar{L}}_{R} (X, C_{2}, 3) \\ \cup {\bar{L}}_{R} (X, C_{3}, 2) = {u_{2}, u_{3}, u_{4}, u_{5}, u_{6}} . \end{matrix}$

5 Conclusions

In this article, we pose a new concept called N-soft rough sets to studied practical problems with ordered ranks and preference relations among the objects. We also give the N-soft rough approximation operators and present some related properties based on N-soft rough approximation space. Different from soft rough sets [19], the method of N-soft rough sets allow the practitioner to introduce subjectivity in order to account for personal preferences. The theory could be used to address the problem of multi-criterion decision making(MCDM). Moreover, in the decision-making process, we give the decision making procedures which contains a reduction based on distribution matrix. Furthermore, we get the set of optimal elements through a series of lower approximation union sets and the set of possible optimal elements through a series of upper approximation union sets. This approach could provide simpler and more efficient assessment, so as to help decision makers to find the solution of practical problems.

Footnotes

Acknowledgments

The authors would like to express their sincere appreciation to the referees for their careful reading of the paper and valuable comments.

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N-soft rough sets and its applications

Abstract

Keywords

1 Introduction

2 Preliminaries

Table 1 Table for (F, A, N) (F,A,N) q 1 q 2 … q n u 1 r 11 r 12 ⋯ r 1n u 2 r 21 r 22 ⋯ r 2n ⋯ ⋯ ⋯ ⋯ ⋯ u m r m1 r m2 ⋯ r mn

Table 2 Table for (F, A, 4) q 1 q 2 q 3 q 4 d u 1 1 0 0 2 3 u 2 0 0 0 3 3 u 3 0 1 0 0 1 u 4 0 0 0 2 2 u 5 2 0 0 0 3 u 6 0 0 0 0 1

4.1 Proposed methodology

4.2 Explanation of proposed methodology

Table 3 Illustrative table U q 1 q 2 q 3 q 4 q 5 d u 1 2 2 1 1 2 3 u 2 4 2 2 2 2 2 u 3 2 1 2 2 1 1 u 4 3 1 3 4 3 2 u 5 4 3 2 3 3 3 u 6 4 2 3 4 3 1

Footnotes

Acknowledgments

References

Table 1
Table for (F, A, N)

(F,A,N) q ₁ q ₂ … q _n

u ₁ r ₁₁ r ₁₂ ⋯ r _1n

u ₂ r ₂₁ r ₂₂ ⋯ r _2n

⋯ ⋯ ⋯ ⋯ ⋯

u _m r _m1 r _m2 ⋯ r _mn

Table 2
Table for (F, A, 4)

q ₁ q ₂ q ₃ q ₄ d

u ₁ 1 0 0 2 3

u ₂ 0 0 0 3 3

u ₃ 0 1 0 0 1

u ₄ 0 0 0 2 2

u ₅ 2 0 0 0 3

u ₆ 0 0 0 0 1

Table 3
Illustrative table

U q ₁ q ₂ q ₃ q ₄ q ₅ d

u ₁ 2 2 1 1 2 3

u ₂ 4 2 2 2 2 2

u ₃ 2 1 2 2 1 1

u ₄ 3 1 3 4 3 2

u ₅ 4 3 2 3 3 3

u ₆ 4 2 3 4 3 1