Multi-granularity soft rough set and its application in multi-attribute decision making

Abstract

In this paper, new concepts such as multi-granularity soft rough sets and multi-granularity soft relative attribute reduction are introduced. Basic properties of multi-granularity soft rough approximations are presented and illustrated by examples. A multi-granularity soft decision system is constructed based on multi-granularity soft rough sets. Furthermore, an algorithm for multi-attribute decision-making problems is proposed. Finally, the validity of this method is proved by the application of the component retrieval problem.

Keywords

Soft rough set multi-granularity soft rough set multi-granularity soft decision system multi-granularity soft relative attribute reduction decision rule

1 Introduction

The soft set theory, initiated by Molodtsov [16] in 1999, and rough set theory, initiated by Pawlak [19] in 1982, are two different and effective tools to describe the phenomena and concepts of an ambiguous, undefined, vague, and imprecise meaning. Maji et al. [17] discussed the application of the soft set theory in a decision-making problem. Kong [11, 12] presented an efficient decision-making approach in an incomplete soft set and a new parameter reduction in fuzzy soft sets. The same author studied the normal parameter reduction of a soft set and its algorithm [9, 10]. Chen et al. [4] gave a new definition of soft set parametrization reduction. Many models have been proposed to generalize and extend Molodtsov’s soft sets. Maji et al. [14] extended classical soft sets to fuzzy soft sets by combining the interval-valued fuzzy set and soft set models. Yang et al. [28] further introduced interval-valued fuzzy soft sets. Roy and Maji [23] applied fuzzy soft sets to object recognition and decision making. Feng et al. [6, 7] proposed the relationships among soft sets, rough sets, and fuzzy sets, obtaining three types of hybrid models: soft rough sets, rough soft sets, and soft-rough fuzzy sets. The same authors also applied soft rough sets in multicriteria group decision making [5]. Zhan [33] reviewed some decision-making methods based on (fuzzy) soft sets in more detail. Yu [29] proposed an algorithm based on a soft rough set. Zhang et al. [32] further used soft rough sets in multi-attribute decision making and applied them to formulate component adaptation schemes.

Zadeh [30, 31] presented the fuzzy information granulation in 1979 and proposed granular computing in 1997. In the perspective of granular computing, an equivalence relation on the universe can be viewed as a granulation, and a partition on the universe can be viewed as a granulation space [13, 18]. Yao [26, 27] assumed that the classical rough set theory is based on a single granulation. In other words, the single granulation space results in only one equivalence relation. Most notably, an attribute set can induce a certain equivalence relation in an information system. In applications, we often describe the target concept by using a finer granulation (partition) formed by combining two known granulations (partitions) from two different attribute subsets. This method can generate a much finer granulation and more knowledge, but the combination or fining destroys the original granulation structure or partitions. Qian et al. [22] extended Pawlak’s single-granulation rough set model to a multi-granulation rough set model. The set approximations are defined by multiple equivalence relations on the universe. Xu et al. [25] defined two new types of multiple granulation rough set models.

The main purpose of this paper is to introduce multi-granularity soft rough sets, which are applied to describe multi-attribute decision-making problems. We propose some relative properties and a decision-making reduction algorithm. Our methods preserve the integrity of the information granulation space and extract the decision rules. The rest of the paper is organized as follows. In Section 2, the basic concepts of rough set, soft set, soft rough set, and decision information system are introduced. Section 3 is dedicated to the study of multi-granularity soft rough sets. In this section, we define the multi-granularity soft approximation space, multi-granularity soft rough approximations, and multi-granularity soft rough sets. The basic properties of multi-granularity soft rough approximations are established. In Section 4, we introduce some new concepts such as multi-granularity soft decision system, multi-granularity soft relative attribute reduction, dependent degree of decision partition soft set, condition significance relative to the decision partition soft set, and the decision rules. Based on the above study, we present an algorithm for the multi-attribute decision-making problem. In Section 5, we give an application for component adaptation schemes to validate this algorithm.

2 Preliminaries

In this section, we recall some concepts such as rough set, soft set, soft rough set, decision information system, etc.

2.1 Rough sets

Definition 2.1. [19] Let U be a nonempty finite set, X be a nonempty subset of U and R be an equivalence relation on U. If X can be written as the union of some equivalence classes, then we say X is definable, otherwise it is not definable. If X is not definable, then we can approximate it by two definable subsets, which are called lower approximation and upper approximation of X, respectively, as the following: $\begin{matrix} \underline{apr} (X) & = & ⋃ {[x]_{R} : [x]_{R} \subseteq X} \\ \bar{apr} (X) & = & ⋃ {[x]_{R} : [x]_{R} \cap X \neq \emptyset} \end{matrix}$ where [x] _R denotes the equivalence class determined by R containing the element x ∈ U. A rough set is the pair $(\underline{apr} (X), \bar{apr} (X))$ . The difference set $\bar{apr} (X) - \underline{apr} (X)$ is called boundary region. Clearly if $\underline{apr} (X) = \bar{apr} (X)$ then X is definable. It can be seen that, $\underline{apr} (X)$ is the greatest definable set contained in X, whereas $\bar{apr} (X)$ is the least definable set containing X. Moreover, the sets $Pos (X) = \underline{apr} (X)$ , $Neg (X) = U - \bar{apr} (X)$ , $Bnd (X) = \bar{apr} (X) - \underline{apr} (X)$ are referred to as positive, negative and boundary region of X, respectively.

If X ⊆ U is defined by a predicate Q and x ∈ U, we have the following interpretation:

x ∈ Pos (X) means that x certainly has property Q;

$x \in \bar{apr} (X)$ means that x possibly has property Q;

x ∈ Neg (X) means that x definitely does not have property Q.

2.2 Soft sets

Let U be a finite universe of objects and E_U (simply denoted by E) the set of certain parameters in relation to the objects in U. Parameters are often attributes, characteristics, or properties of the objects in U. Let P (U) denote the power set of U. The concept of soft sets is defined as follows.

Definition 2.2. [14] A pair S = (f, A) is called a soft set over U, where A ⊆ E and f : A → P (U) is a set-valued mapping.

In other words, a soft set over U is a parameterized family of subsets of the universe U. For ɛ ∈ A, f (ɛ) might be considered as the set of ɛ-approximate elements in the soft set S = (f, A). Notably, f (ɛ) might be arbitrary: some of them might be empty, and some might have nonempty intersection [16]. In the absence of restriction on the approximate description, the soft set theory is very convenient and easily applicable in practice. Actually, we can choice suitable words, sentences, real numbers, functions, mappings and so on as parameters.

To illustrate this idea, let us consider the classical example in soft set theory.

Example 2.3. Let U = {h₁, h₂, h₃, h₄, h₅, h₆} be a set of houses and A = {e₁, e₂, e₃, e₄} ⊆ E a set of status of houses, in which e₁, e₂, e₃, e₄ stand for the attributes “beautiful”, “modern”, “cheap” and “in the green surroundings”, respectively.

Now, we consider a soft set (f, A), which describes the attractiveness of the house that Mr. X is going to buy. In this case, to define the soft set (f, A) means to point out beautiful houses, modern houses and so on.

Consider the mapping f : A → P (U) given by $\begin{matrix} f (e_{1}) & = & {h_{1}, h_{2}, h_{5}}, f (e_{2}) = {h_{1}, h_{6}}, \\ f (e_{3}) & = & {h_{3}, h_{4}}, f (e_{4}) = {h_{3}, h_{4}, h_{6}} . \end{matrix}$

Then, f (e₁) means “the beautiful houses”, whose functional value is the {h₁, h₂, h₅}. Thus, we can view the soft set (f, A) as a collection of approximations as follows $(f, A) = {\begin{matrix} beautiful = {h_{1}, h_{2}, h_{5}}, \\ modern = {h_{1}, h_{6}}, \\ cheap = {h_{3}, h_{4}}, \\ in the green surroundings = {h_{3}, h_{4}, h_{6}} \end{matrix}}$

Definition 2.4. [15] A soft set (f, A) over U is called a partition soft set if {f (e) |e ∈ A} forms a partition of U.

2.3 Soft rough sets

Definition 2.5. [7] Let S = (f, A) be a soft set over U. Then the pair P = (U, S) is called a soft approximation space. Based on the soft approximation space P, we define the following two operations: $\begin{matrix} {\underline{apr}}_{P} (X) & = & {u \in U : \exists a \in A such that u \in f (a) \subseteq X} \\ {\bar{apr}}_{P} (X) & = & {u \in U : \exists a \in A such that u \in f (a) \\ and f (a) \cap X \neq \emptyset} \end{matrix}$ assigning to every subset X ⊆ U two sets ${\underline{apr}}_{P} (X)$ and ${\bar{apr}}_{P} (X)$ , which are called the soft P-lower approximation and the soft P-upper approximation of X, respectively. In general, we refer to ${\underline{apr}}_{P} (X)$ and ${\bar{apr}}_{P} (X)$ as soft rough approximations of X with respect to P. Moreover, the sets $\begin{matrix} {Pos}_{P} (X) & = & {\underline{apr}}_{P} (X), \\ {Neg}_{P} (X) & = & U - {\bar{apr}}_{P} (X), \\ {Bnd}_{P} (X) & = & {\bar{apr}}_{P} (X) - {\underline{apr}}_{P} (X) \end{matrix}$ are called the soft P-positive region, the soft P-negative region and the soft P-boundary region of X, respectively. If ${\underline{apr}}_{P} (X) = {\bar{apr}}_{P} (X)$ , X is said to be soft P-definable; otherwise X is called a soft rough set with respect to P.

It is clear that ${\underline{apr}}_{P} (X) \subseteq X$ and ${\underline{apr}}_{P} \subseteq {\bar{apr}}_{P} (X)$ . But $X \subseteq {\bar{apr}}_{P} (X)$ does not hold in general [7].

From the analogy with Pawlak rough sets, we also have the following interpretation of above concepts.

x ∈ Pos_P (X) means that x surely belongs to X with respect to P;

$x \in {\bar{apr}}_{P} (X)$ means that x possibly belongs to X with respect to P;

x ∈ Neg_P (X) means that x surely does not belong to X with respect to P.

Clearly, ${\underline{apr}}_{P} (X)$ and ${\bar{apr}}_{P} (X)$ can be expressed equivalently as: $\begin{matrix} {\underline{apr}}_{P} (X) & = & ⋃_{e \in A} {f (e) | f (e) \subseteq X}, \\ {\bar{apr}}_{P} (X) & = & ⋃_{e \in A} {f (e) | f (e) \cap X \neq \emptyset} . \end{matrix}$

2.4 Decision information system

Definition 2.6. [8] A decision information system can be defined as S = (U, At = C ∪ D, {V_a|a ∈ At} , {f_a|a ∈ At}), where U is a nonempty set of objects, At is a nonempty set of attributes, C is a set of condition attributes, D is a decision attribute, V_a is a nonempty set of values for each attribute a ∈ At, and f_a : U → V_a is an information function for each attribute a ∈ At.

For a subset of attributes A ⊆ At, we define an equivalence relation R_A (or A for short) as follows: $R_{A} = {(x, y) \in U \times U | \forall a \in A, f_{a} (x) = f_{a} (y)}$

According to the above definition, two objects in U satisfy R_A if and only if they have the same values on all attributes of A. Consequently, we can form two equivalence relations by C and D respectively.

Definition 2.7. [8] Let U/D = {d₁, d₂, …, d_m} be the partition of the universe U defined by the decision attribute D. Then the positive region and boundary region of D with respect to C in S are as follows: $\begin{matrix} {Pos}_{C} (D) & = & \overset{m}{⋃_{i = 1}} \underline{C} (d_{i}) \\ {Bnd}_{C} (D) & = & U - {Pos}_{C} (D) \end{matrix}$ where $\underline{C} (d_{i}) = {x \in U | [x]_{C} \subseteq d_{i}}$ is the lower approximation of d_i in the approximation space produced by C.

A decision information system is consistent if each equivalence class defined by C leads to a unique decision. In other words, there is a unique d_i ∈ U/D and [x] _C ⊆ d_i. In this case, we have Pos_C (D) = U and Bnd_C (D) =∅. Otherwise the system is inconsistent.

Let B be a subset of C, an attribute a ∈ B is dispensable in B if Pos_B-{a} (D) = Pos_B (D); otherwise a is indispensable in B. The collection of all the indispensable attributes in C is called the core of C with respect to D. An attribute set B ⊆ C is a reduction of C with respect to D if it satisfies the following two conditions:

Pos_B (D) = Pos_C (D);

For any attribute a ∈ B, Pos_B-{a} (D) ≠ Pos_B (D).

The reduction of a decision information system is the minimal set of condition attributes to ensure the positive region unchanged. The aim of attribute reduction is to simplify a decision information system but not to reduce its classification abilities.

Since the attribute reduction is to underline the functional dependencies between condition attributes and decision attributes, a decision information system may also be seen as a set of decision rules. Classically, the decision rules are logic statements exactly as $If conditions, then decision .$

The “if” part of a decision rule is called condition part, and the “then” part is called decision part. Błaszczyński [3] pointed out that an object supports a decision rule if it matches (that is, verifies) both the condition and decision parts of the rule. Alternatively, an object is covered by a decision rule when it matches the condition part of the rule.

3 On multi-granularity soft rough sets

Definition 3.1. Let S₁ = (f, A) and S₂ = (g, B) be two soft sets over a common universe U and S = (S₁, S₂). Then the pair W = (U, S) is called a multi-granularity soft approximation space. Based on the W, we define the following two operations $\begin{matrix} {\underline{apr}}_{W} (X) \\ = {u \in U : \exists a \in A such that u \\ \in f (a) \subseteq X or \exists b \in B such that u \\ \in g (b) \subseteq X} \\ {\bar{apr}}_{W} (X) \\ = {u \in U : \exists a \in A such that u \in f (a) and \\ f (a) \cap X \neq \emptyset or \exists b \in B such that u \in g (b) \\ and g (b) \cap X \neq \emptyset} \end{matrix}$ assigning to every subset X ⊆ U two sets ${\underline{apr}}_{W} (X)$ and ${\bar{apr}}_{W} (X)$ , which are called the multi-granularity soft W-lower approximation and the multi-granularity soft W-upper approximation of X, respectively. In general, we refer to ${\underline{apr}}_{W} (X)$ and ${\bar{apr}}_{W} (X)$ as multi-granularity soft rough approximations of X with respect to W. Moreover, the sets $\begin{matrix} {Pos}_{W} (X) & = & {\underline{apr}}_{W} (X), \\ {Neg}_{W} (X) & = & U - {\bar{apr}}_{W} (X), \\ {Bnd}_{W} (X) & = & {\bar{apr}}_{W} (X) - {\underline{apr}}_{W} (X) . \end{matrix}$ are called the multi-granularity soft W-positive region, the multi-granularity soft W-negative region and the multi-granularity soft W-boundary region of X, respectively. If ${\underline{apr}}_{W} (X) = {\bar{apr}}_{W} (X)$ , X is said to be multi-granularity soft W-definable; otherwise X is called a multi-granularity soft rough set (MGSR-set) with respect to W.

When (f, A) = (g, B), it is clear that multi-granularity soft W-upper approximation and multi-granularity soft W-lower approximation of X in Definition 3.1 are exact the soft P-upper approximation and P-lower approximation in Definition 2.5 (P = W = (U, S)), respectively. When (f, A) ≠ (g, B), it is clear that MGSR-set in Definition 3.1 might not a soft rough set in Definition 2.5. Thus it can be seen as a generalization of the soft rough set model which is introduced in [7]. It is clear that ${\underline{apr}}_{W} (X) \subseteq X$ and ${\underline{apr}}_{W} (X) \subseteq {\bar{apr}}_{W} (X)$ for all X ⊆ U. The following example shows that $X \subseteq {\bar{apr}}_{W} (X)$ does not hold in general.

Example 3.2. Let U = {u₁, u₂, u₃, u₄, u₅}, E = {e₁, e₂, e₃, e₄}, A = {e₁, e₂} ⊆ E and B = {e₃, e₄} ⊆ E. Let S₁ = (f, A) be a soft set over U given by Table 1 and the S₂ = (g, B) be a soft set over U given by Table 2.

Table 1
Table for soft set (f, A)

u ₁ u ₂ u ₃ u ₄ u ₅

e ₁ 0 0 1 1 0

e ₂ 0 0 0 1 0

	u ₁	u ₂	u ₃	u ₄	u ₅
e ₁	0	0	1	1	0
e ₂	0	0	0	1	0

Table 2

Table for soft set (g, B)

	u ₁	u ₂	u ₃	u ₄	u ₅
e ₃	0	0	0	1	1
e ₄	0	0	0	0	1

For X = {u₁, u₂, u₃, u₄} ⊆ U, we have ${\underline{apr}}_{W} (X) = {u_{3}, u_{4}}$ , ${\bar{apr}}_{W} (X) = {u_{3}, u_{4}, u_{5}}$ . Thus $X ⊈ {\bar{apr}}_{W} (X)$ and X is a MGSR-set.

In the following of this section, let S₁ = (f, A) and S₂ = (g, B) be two soft sets over U, S = (S₁, S₂) and W = (U, S) be the corresponding multi-granularity soft approximation space.

The following results can be easily established from the definition of multi-granularity soft rough approximations.

Proposition 3.3.For eachX ⊆ U, $\begin{matrix} {\underline{apr}}_{W} (X) & = & ⋃_{a \in A} {f (a) : f (a) \subseteq X} \\ \cup ⋃_{b \in B} {g (b) : g (b) \subseteq X}, \\ {\bar{apr}}_{W} (X) & = & ⋃_{a \in A} {f (a) : f (a) \cap X \neq \emptyset} \\ \cup ⋃_{b \in B} {g (b) : g (b) \cap X \neq \emptyset} . \end{matrix}$

One can easily verify the following properties: $\begin{matrix} {\underline{apr}}_{W} (\emptyset) = {\bar{apr}}_{W} (\emptyset) = \emptyset; \\ {\underline{apr}}_{W} (U) = {\bar{apr}}_{W} (U) = ⋃_{a \in A} f (a) \cup ⋃_{b \in B} g (b); \\ X \subseteq Y \Rightarrow {\underline{apr}}_{W} (X) \subseteq {\underline{apr}}_{W} (Y); \\ X \subseteq Y \Rightarrow {\bar{apr}}_{W} (X) \subseteq {\bar{apr}}_{W} (Y); \\ {\underline{apr}}_{W} (X \cap Y) \subseteq {\underline{apr}}_{W} (X) \cap {\underline{apr}}_{W} (Y); \\ {\underline{apr}}_{W} (X \cup Y) \supseteq {\underline{apr}}_{W} (X) \cup {\underline{apr}}_{W} (Y); \\ {\bar{apr}}_{W} (X \cap Y) \subseteq {\bar{apr}}_{W} (X) \cap {\bar{apr}}_{W} (Y); \\ {\bar{apr}}_{W} (X \cup Y) = {\bar{apr}}_{W} (X) \cup {\bar{apr}}_{W} (Y) . \end{matrix}$

Proposition 3.4. For any X ⊆ U, X is multi-granularity soft W-definable if and only if ${\bar{apr}}_{W} (X) \subseteq X$ .

Proof. Note that if X is multi-granularity soft W-definable, then ${\underline{apr}}_{W} (X) = {\bar{apr}}_{W} (X)$ . By ${\underline{apr}}_{W} (X) \subseteq X$ , we have ${\bar{apr}}_{W} (X) \subseteq X$ .

Conversely, suppose that ${\bar{apr}}_{W} (X) \subseteq X$ . To show that X is multi-granularity soft W-definable, we only need to prove that ${\bar{apr}}_{W} (X) \subseteq {\underline{apr}}_{W} (X)$ . Indeed, for each $u \in {\bar{apr}}_{W} (X)$ . Then u∈ f (a) , f (a) ∩ X ≠ ∅ for some a ∈ A or u∈ g (b) , g (b) ∩ X ≠ ∅ for some b ∈ B. Thus $u \in f (a) \subseteq {\bar{apr}}_{W} (X) \subseteq X$ or $u \in g (b) \subseteq {\bar{apr}}_{W} (X) \subseteq X$ . It follows that $u \in {\underline{apr}}_{W} (X)$ , and so ${\bar{apr}}_{W} (X) = {\underline{apr}}_{W} (X)$ as required. □

Theorem 3.5. For any X ⊆ U, we have

${\underline{apr}}_{W} ({\underline{apr}}_{W} (X)) = {\underline{apr}}_{W} (X)$ ;

${\underline{apr}}_{W} ({\bar{apr}}_{W} (X)) = {\bar{apr}}_{W} (X)$ ;

${\bar{apr}}_{W} ({\underline{apr}}_{W} (X)) \supseteq {\underline{apr}}_{W} (X)$ ;

${\bar{apr}}_{W} ({\bar{apr}}_{W} (X)) \supseteq {\bar{apr}}_{W} (X)$ ;

Proof. (1) Let $Y = {\underline{apr}}_{W} (X)$ . By ${\underline{apr}}_{W} (Y) \subseteq Y$ , we only need to show $Y \subseteq {\underline{apr}}_{W} (Y)$ . Let y ∈ Y. Then there is a ∈ A such that y ∈ f (a) ⊆ X or b ∈ B such that y ⊆ g (b) ⊆ X. Hence y ∈ f (a) ⊆ Y or y ∈ g (b) ⊆ Y. Then $y \in {\underline{apr}}_{W} (Y)$ and so $Y \subseteq {\underline{apr}}_{W} (Y)$ .

(2) The proof is similar to (1).

(3) It is obvious from ${\underline{apr}}_{W} (X) \subseteq {\bar{apr}}_{W} (X)$ and (1).

(4) It is obvious from ${\underline{apr}}_{W} (X) \subseteq {\bar{apr}}_{W} (X)$ and (2). □

The following example shows that the inclusion relations in Theorem 3 might be strict.

Example 3.6. Let U = {u₁, u₂, u₃, u₄, u₅, u₆}, E = {e₁, e₂, e₃, e₄}, A = {e₁, e₂, e₄} ⊆ E and B = {e₂, e₃} ⊆ E. Let S₁ = (f, A) be a soft set over U given by Table 3 and the S₂ = (g, B) be a soft set over U given by Table 4. Take X = {u₁, u₃, u₆} ⊆ U.

Table 3

Table for soft set (f, A)

	u ₁	u ₂	u ₃	u ₄	u ₅	u ₆
e ₁	1	0	1	0	0	1
e ₂	0	1	1	0	1	0
e ₄	0	1	0	1	1	0

Table 4

Table for soft set (g, B)

	u ₁	u ₂	u ₃	u ₄	u ₅	u ₆
e ₂	0	1	0	1	1	0
e ₃	1	0	1	0	0	1

Then ${\underline{apr}}_{W} (X) = {u_{1}, u_{3}, u_{6}} = X$ and ${\bar{apr}}_{W} (X) = {u_{1}, u_{2}, u_{3}, u_{5}, u_{6}}$ . We have that ${\bar{apr}}_{W} ({\underline{apr}}_{W} (X)) ⫌ {\underline{apr}}_{W} (X)$ which suggests that the inclusion (3) in Theorem 3.5 might hold strictly. Moreover, we have that ${\bar{apr}}_{W} ({\bar{apr}}_{W} (X)) = U$ . It follows that ${\bar{apr}}_{W} ({\bar{apr}}_{W} (X)) ⫌ {\bar{apr}}_{W} (X)$ which shows that the inclusion (4) in Theorem 3.5 might be strict.

Comparing with full soft set we introduce the concept of jointly full soft sets.

Definition 3.7. The set U is called full soft universe if ⋃_a∈Af (a) ∪ ⋃ _b∈Bg (b) = U, meanwhile (f, A) and (g, B) are called jointly full soft sets.

The following results are related with jointly full soft sets.

Theorem 3.8. The following statements are equivalent:

U is a full soft universe;

${\underline{apr}}_{W} (U) = U$ ;

${\bar{apr}}_{W} (U) = U$ ;

$X \subseteq {\bar{apr}}_{W} (X)$ for all X ⊆ U;

${\bar{apr}}_{W} ({u}) \neq \emptyset$ for all u ∈ U.

Proof. (1)⇒(2) Let U be a full soft universe, (f, A) and (g, B) be jointly full soft sets. Then, for each u ∈ U, there is a ∈ A such that u ∈ f (a) ⊆ U or b ∈ B such that u ∈ g (b) ⊆ U that is $u \in {\underline{apr}}_{W} (U)$ . Thus $U \subseteq {\underline{apr}}_{W} (U)$ which implies that ${\underline{apr}}_{W} (U) = U$ .

(2)⇒(3) Clear.

(3)⇒(1) By ${\bar{apr}}_{W} (U) = U$ and Proposition 3.3, we have U = ⋃ _a∈A {f (a) : f (a) ∩ U ≠ ∅} ∪ ⋃ _b∈B {g (b) : g (b) ∩ U ≠ ∅}. Hence ⋃_a∈Af (a) ∪ ⋃ _b∈Bg (b) = U, that is U is a full soft universe.

(1)⇒(4) Let U be a full soft universe, (f, A) and (g, B) be jointly full soft sets and X ⊆ U. Then, for each x ∈ X, there exist a ∈ A such that x ∈ f (a) and f (a)∩ X ≠ ∅ or b ∈ B such that x ∈ g (b) and g (b)∩ X ≠ ∅. It implies that $X \subseteq {\bar{apr}}_{W} (X)$ .

(4)⇒(5) Clear.

(5)⇒(1) Suppose that ${\bar{apr}}_{W} ({u}) \neq \emptyset$ for all u ∈ U. Then there exist a ∈ A such that u ∈ f (a) or b ∈ B such that u ∈ g (b). Thus U is full soft universe. □

Proposition 3.9. If U is a full soft universe, then for any X ⊆ U we have the following:

$U - {\underline{apr}}_{W} (X) \subseteq {\bar{apr}}_{W} (U - X)$ ,

$U - {\bar{apr}}_{W} (X) \subseteq {\underline{apr}}_{W} (U - X)$ .

Proof. (1) Let $u \in U - {\underline{apr}}_{W} (X)$ . Since U is a full soft universe, then there are jointly full soft sets, namely (f, A) and (g, B) such that ⋃_a∈Af (a) ∪ ⋃ _b∈Bg (b) = U. So there is a ∈ A such that u ∈ f (a) or b ∈ B such that u ∈ g (b). By $u \notin {\underline{apr}}_{W} (X)$ , we have f (a) ⊈ X and g (b) ⊈ X. It implies that f (a)∩ (U - X) ≠ ∅ and g (b)∩ (U - X) ≠ ∅. Hence, $u \in {\bar{apr}}_{W} (U - X)$ .

(2) Let $u \in U - {\bar{apr}}_{W} (X)$ . Since U is a full soft universe, then there are jointly full soft sets, namely (f, A) and (g, B) such that ⋃_a∈Af (a) ∪ ⋃ _b∈Bg (b) = U. So there is a ∈ A such that u ∈ f (a) or b ∈ B such that u ∈ g (b). By $u \notin {\bar{apr}}_{W} (X)$ , we have f (a)∩ X = ∅ and g (b)∩ X = ∅. It implies that f (a) ⊆ U - X and g (b) ⊆ U - X. Hence, $u \in {\underline{apr}}_{W} (U - X)$ . □

Definition 3.10. For any X, Y ⊆ U, we define $\begin{matrix} X {\underset{̇}{∼}}_{W} Y & \Leftrightarrow & {\underline{apr}}_{W} (X) = {\underline{apr}}_{W} (Y), \\ X {\dot{\sim}}_{W} Y & \Leftrightarrow & {\bar{apr}}_{W} (X) = {\bar{apr}}_{W} (Y), \end{matrix}$ and $X \approx_{W} Y \Leftrightarrow X {\underset{̇}{∼}}_{W} Y and X {\dot{\sim}}_{W} Y .$

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

It is easy to verify that the above relations are all equivalence relations over P (U).

Theorem 3.11. For any X ⊆ U, we have the following results:

${\underline{apr}}_{W} (X) = ⋂ {Y \subseteq U : X {\underset{̇}{∼}}_{W} Y}$ ;

If U is a full soft universe, then ${\bar{apr}}_{W} (X) \supseteq ⋃ {Y \subseteq U : X {\dot{\sim}}_{W} Y}$ .

Proof. (1) Let $u \in {\underline{apr}}_{W} (X)$ . If $X {\underset{̇}{∼}}_{W} Y$ , then ${\underline{apr}}_{W} (X) = {\underline{apr}}_{W} (Y)$ . For any $Y \subseteq U, {\underline{apr}}_{W} (Y) \subseteq Y$ . Thus $u \in {\underline{apr}}_{W} (X) = {\underline{apr}}_{W} (Y) \subseteq Y$ . It implies that $u \in ⋂ {Y \subseteq U : X {\underset{̇}{∼}}_{W} Y}$ , and so ${\underline{apr}}_{W} (X) \subseteq ⋂ {Y \subseteq U : X {\underset{̇}{∼}}_{W} Y}$ . On the other hand, let $u \in ⋂ {Y \subseteq U : X {\underset{̇}{∼}}_{W} Y}$ . By Theorem 3.5, ${\underline{apr}}_{W} ({\underline{apr}}_{W} (X)) = {\underline{apr}}_{W} (X)$ . Thus $X {\underset{̇}{∼}}_{W} {\underline{apr}}_{W} (X)$ . It follows that $u \in {\underline{apr}}_{W} (X)$ and this implies that $⋂ {Y \subseteq U : X {\underset{̇}{∼}}_{W} Y} \subseteq {\underline{apr}}_{W} (X)$ . It follows that ${\underline{apr}}_{W} (X) = ⋂ {Y \subseteq U : X {\underset{̇}{∼}}_{W} Y}$ .

(2) For each $u \in ⋃ {Y \subseteq U : X {\dot{\sim}}_{W} Y}$ , there is Y₀ ⊆ U such that u ∈ Y₀ and $X {\dot{\sim}}_{W} Y_{0}$ . Since U is full soft universe. By Theorem 3.8, we have $Y_{0} \subseteq {\bar{apr}}_{W} (Y_{0}) = {\bar{apr}}_{W} (X)$ . It follows that $u \in {\bar{apr}}_{W} (X)$ . Hence, we have ${\bar{apr}}_{W} (X) \supseteq ⋃ {Y \subseteq U : X {\dot{\sim}}_{W} Y}$ . □

4 Decision making method based on MGSR-Sets

4.1 Multi-granularity soft decision systems

Definition 4.1. Let S_i = (f_i, C_i) (i = 1, 2, …, n) and (g, D) be soft sets over U, value set C_i = {c_i1, c_i2, …, c_ij(i)} (j (i) is a positive integer). Denote W⁽ⁿ⁾ = (U, S) as a multi-granularity soft approximation space, where S = (S₁, S₂, ⋯ , S_n). Then the multi-granularity soft positive region of S to (g, D) is defined as follows: $\begin{matrix} {Pos}_{S} (g, D) \\ = ⋃_{d \in D} {\underline{apr}}_{W^{(n)}} g (d) \\ = ⋃_{d \in D} {x \in U : \exists 1 \leq n_{0} \leq n, \\ 1 \leq j_{n_{0}} \leq j (n_{0}) such that c_{n_{0} j_{n_{0}}} \in C_{n_{0}}, \\ x \in f_{n_{0}} (c_{n_{0} j_{n_{0}}}) \subseteq g (d)} . \end{matrix}$

Definition 4.2. Let S_i = (f_i, C_i) (i = 1, 2, …, n) be soft sets over U, where C_i is a attribute set and C_i⋂ C_i′ = ∅ for i ≠ i′, value set C_i = {c_i1, c_i2, …, c_ij(i)} (j (i) is a positive integer). Let (g, D) be a partition soft set over U, where C_i ⋂ D = ∅ (i = 1, 2, …, n). Then the triple (U, CS, DS) is called a multi-granularity soft decision system (MGSDS), where CS = (S₁, S₂, …, S_n) is the multi-granularity condition soft set, DS = (g, D) is the decision partition soft set and CW⁽ⁿ⁾ = (U, CS) is the multi-granularity soft approximation space.

Accordingly, in a multi-granularity soft decision system, we call ${Pos}_{CS} (DS) = ⋃_{d \in D} {\underline{apr}}_{{CW}^{(n)}} g (d)$ the multi-granularity soft relative positive region of MGSDS.

For a given MGSDS, we always consider Pos_CS (DS)≠ ∅.

Example 4.3. Suppose that U = {x₁, x₂, x₃, x₄, x₅, x₆} is a common universe, which is a set of six shops. C₁, C₂, C₃ denote the attribute sets, and C₁, C₂, C₃ stand for the empowerment of sales personnel, the perceived quality of goods, the high traffic location, respectively. The value sets of these attributes are C₁ = {high, medium, low}, C₂ = {good, average}, C₃ = {no, yes}. D = {profit, loss} describes shop profit or loss.

The mapping of each soft set over U is defined as follows: $\begin{matrix} f_{1} (high) & = & {x_{1}, x_{6}}, f_{1} (medium) = {x_{2}, x_{3}, x_{5}}, \\ f_{1} (low) & = & {x_{4}}, \\ f_{2} (good) & = & {x_{1}, x_{2}, x_{3}}, f_{2} (average) = {x_{4}, x_{5}, x_{6}}, \\ f_{3} (no) & = & {x_{1}, x_{2}, x_{3}, x_{4}}, f_{3} (yes) = {x_{5}, x_{6}} . \end{matrix}$

The mapping of the decision partition soft set over U is defined as follows: $g (profit) = {x_{1}, x_{3}, x_{6}}, g (loss) = {x_{2}, x_{4}, x_{5}} .$

Then we can regard each soft set (f_i, C_i) (i = 1, 2, 3) as a collection of approximations as follows: $\begin{matrix} (f_{1}, C_{1}) & = & {high = {x_{1}, x_{6}}, medium = {x_{2}, x_{3}, x_{5}}, \\ low = {x_{4}}}, \\ (f_{2}, C_{2}) & = & {good = {x_{1}, x_{2}, x_{3}}, \\ average = {x_{4}, x_{5}, x_{6}}}, \\ (f_{3}, C_{3}) & = & {no = {x_{1}, x_{2}, x_{3}, x_{4}}, yes = {x_{5}, x_{6}}} . \end{matrix}$

Similarly, (g, D) = {profit = {x₁, x₃, x₆} , loss = {x₂, x₄, x₅}}.

So (U, CS, DS) is a multi-granularity soft decision system on how to choose profitable shops. Thus, $\begin{matrix} {\underline{apr}}_{{CW}^{(3)}} g (profit) = {x_{1}, x_{6}}, \\ {\underline{apr}}_{{CW}^{(3)}} g (loss) = {x_{4}} . \end{matrix}$

Therefore Pos_CS (DS) = {x₁, x₄, x₆}.

4.2 Multi-granularity soft relative attribute reduction of MGSDS

Definition 4.4. Let (U, CS, DS) be a MGSDS.

For some 1 ≤ i ≤ n, S_i = (f_i, C_i) is called a soft dispensable set of CS relative to DS, if Pos_CS (DS) = Pos_{CS-S_i} (DS), where CS - S_i = (S₁, S₂, ⋯ , S_i-1, S_i+1, ⋯ , S_n). Otherwise, S_i = (f_i, C_i) is called a soft indispensable set of CS relative to DS.

CS is called a multi-granularity soft independent set relative to DS, if every soft set S_i of CS is a soft indispensable set relative to DS. Otherwise, CS is called a multi-granularity soft dependent set relative to DS.

The union set of all the soft indispensable set of CS relative to DS is called the core of CS relative to DS, denoted by core (CS, DS).

Definition 4.5. Let (U, CS, DS) be a MGSDS, (S_{i
₁}, S_{i
₂}, ⋯ , S_{i
_t}), denoted by ${CS}_{re}^{(t)}$ , be a subset of CS.

Then ${CS}_{re}^{(t)}$ is called a multi-granularity soft relative attribute reduction of (U, CS, DS), if

${Pos}_{CS} (DS) = {Pos}_{{CS}_{re}^{(t)}} (DS)$

${CS}_{re}^{(t)}$ is a multi-granularity soft independent set relative to DS.

Example 4.6. In Example 4.3, denote ${CS}_{{re}_{1}}^{(2)} = (S_{1}, S_{2})$ , ${CS}_{{re}_{2}}^{(2)} = (S_{1}, S_{3})$ , ${CS}_{{re}_{3}}^{(2)} = (S_{2}, S_{3})$ . We have $\begin{matrix} {Pos}_{{CS}_{{re}_{3}}^{(2)}} (DS) & = & \emptyset, {Pos}_{{CS}_{{re}_{1}}^{(2)}} (DS) = {Pos}_{{CS}_{{re}_{2}}^{(2)}} (DS) \\ = & {Pos}_{CS} (DS) = {x_{1}, x_{4}, x_{6}} . \end{matrix}$

Since ${Pos}_{S_{2}} (DS) = \emptyset \neq {Pos}_{{CS}_{{re}_{1}}^{(2)}} (DS)$ , so S₁ is a soft indispensable set of ${CS}_{{re}_{1}}^{(2)}$ relative to DS. For ${Pos}_{S_{1}} (DS) = {Pos}_{{CS}_{{re}_{1}}^{(2)}} (DS)$ , so S₂ is a soft dispensable set of ${CS}_{{re}_{1}}^{(2)}$ relative to DS. Hence, ${CS}_{{re}_{1}}^{(2)}$ is a multi-granularity soft dependent set relative to DS. Similarly, we know the ${CS}_{{re}_{2}}^{(2)}$ is a multi-granularity soft dependent set relative to DS.

Through the above calculation, we have Pos_{S
₁} (DS) = Pos_CS (DS) and S₁ is a soft indispensable set of CS relative to DS. Hence, ${CS}_{re}^{(1)} = S_{1}$ is a multi-granularity soft relative attribute reduction of (U, CS, DS).

4.3 Dependent degree of decision partition soft sets

Definition 4.7. Let (U, CS, DS) be a MGSDS, we call $k = γ (CS, DS) = \frac{| {Pos}_{CS} (DS) |}{| U |}$ the dependent degree of DS upon CS, where | · | is the cardinal number of a set. It characters a degree of multi-granularity condition soft sets in classifying decision partition soft sets. Obviously, we have 0 ≤ k ≤ 1.

If k = 1, then DS is completely dependent on CS.

If k = 0, then DS is completely independent on CS.

Example 4.8. In Example 4.3, the dependent degree k of the DS upon the CS. $k = γ (CS, DS) = \frac{| {x_{1}, x_{4}, x_{6}} |}{| U |} = \frac{3}{6} = \frac{1}{2} .$

Proposition 4.9.Let (U, CS, DS) be a MGSDS. Then, for m < n, $γ ({CS}^{(m)}, DS) \leq γ (CS, DS) .$ whereCS^(m) = {S₁, S₂, …, S_m}.

Proof. Since $\begin{matrix} γ ({CS}^{(m)}, DS) \\ = \frac{| {Pos}_{{CS}^{(m)}} (DS) |}{| U |} = \frac{| ⋃_{d \in D} {\underline{apr}}_{{CW}^{(m)}} g (d) |}{| U |}, \end{matrix}$ where CW^(m) = (U, CS^(m)).

For any d ∈ D, let $x \in {\underline{apr}}_{{CW}^{(m)}} g (d)$ . Then there is 1 ≤ n₀ ≤ m and 1 ≤ j_{n
₀} ≤ j (n₀) such that c_{n
₀
j
_{n
₀}} ∈ C_{n
₀} and x ∈ f_{n
₀} (c_{n
₀
j
_{n
₀}}) ⊆ g (d).

By m < n we have 1 ≤ n₀ ≤ n. So $x \in {\underline{apr}}_{{CW}^{(n)}} g (d)$ . Then ${\underline{apr}}_{{CW}^{(m)}} g (d) \subseteq {\underline{apr}}_{{CW}^{(n)}} g (d) .$

From the arbitrary of d, we obtain $⋃_{d \in D} {\underline{apr}}_{{CW}^{(m)}} g (d) \subseteq ⋃_{d \in D} {\underline{apr}}_{{CW}^{(n)}} g (d)$ that is Pos_{CW
^(m)} ⊆ Pos_{CW
⁽ⁿ⁾} (DS). Hence $γ ({CS}^{(m)}, DS) \leq γ (CS, DS) . □$

Alternatively, the multi-granularity condition soft sets can explain in detail the classification of decision partition soft sets. We will lose some information about the decision partition soft set when deleting some condition soft sets. Essentially, the more information (more condition soft sets)can result in larger dependent degree of the decision partition soft set.

4.4 Condition significance relative to decision partition soft set

Definition 4.10. Let (U, CS, DS) be a MGSDS and 1 ≤ i ≤ n. The condition significance of S_i in CS relative to DS is defined as follows $sig (S_{i}, CS, DS) = γ (CS, DS) - γ (CS - S_{i}, DS) .$

This definition indicates the decrease case of the dependent degree of decision partition soft set when deleting one soft set S_i = (f_i, C_i) from CS.

We review the union of soft sets defined as follows:

Definition 4.11. [15] Let (f, A), (g, B) and (h, C) be three soft sets over U. (h, C) is called the union of (f, A) and (g, B), if C = A ∪ B and $h (e) = {\begin{matrix} f (e), & e \in A - B, \\ f (e) \cup g (e), & e \in A \cap B, \\ g (e), & e \in B - A . \end{matrix}$ we denote (h, C) by $(f, A) \tilde{\cup} (g, B)$ .

The following results are easily obtained from the above definitions.

Proposition 4.12. Let (U, CS, DS) be a MGSDS and 1 ≤ i ≤ n. Then

0 ≤ sig (S_i, CS, DS) ≤1.

S_i is a soft indispensable set of CS to DS if and only if $sig (S_{i}, CS, DS) > 0 .$

$core (CS, DS) = ⋃^{\sim} {S_{i} : sig (S_{i}, CS, DS) > 0, 1 \leq i \leq n}$ .

Theorem 4.13. Let (U, CS, DS) be a MGSDS. ${CS}_{re}^{(t)} = (S_{i_{1}}, S_{i_{2}}, \dots, S_{i_{t}})$ . If $γ ({CS}_{re}^{(t)}, DS) = γ (CS, DS)$ and $sig (S_{i_{k}}, {CS}_{re}^{(t)}, DS) > 0$ , (1 ≤ k ≤ t), then ${CS}_{re}^{(t)}$ is a multi-granularity soft relative attribute reduction of (U, CS, DS).

4.5 Decision rules in MGSDS

Definition 4.14. Let (U, CS, DS) be a MGSDS. For any c_ij ∈ C_i, d_r ∈ D, we define $δ_{ij} (d_{r}) = \frac{| f_{i} (c_{ij}) \cap g (d_{r}) |}{| g (d_{r}) |} .$ which is called soft rough coverage degree of c_ij relative to d_r.

The soft rough coverage degree expresses the correlation degree between c_ij and d_r. The larger value of δ_ij (d_r), the more closely connected between c_ij and d_r. The soft rough coverage degree is used for evaluating the quality of decision rules.

Definition 4.15. Let ${CS}_{re}^{(t)}$ be a multi-granularity soft relative attribute reduction of MGSDS (U, CS, DS). We might as well let ${CS}_{re}^{(t)} = (S_{1}, S_{2}, \dots, S_{t})$ . For any C_i (1 ≤ i ≤ t), d_r ∈ D, we define maximal coverage degree $δ_{i}^{r} = max_{1 \leq j \leq j (i)} δ_{ij} (d_{r}) .$

Let $δ_{i}^{r} = δ_{{ij}_{i}} (d_{r})$ , that is, $δ_{i}^{r}$ is the soft rough coverage degree of c_{ij
_i} relative to d_r. We call $If c_{{ij}_{i}} \in C_{i}, then σ (δ_{i}^{r})$ the decision rule induced by ${CS}_{re}^{(t)}$ in (U, CS, DS).

4.6 Algorithm for the multi-attribute decision problem

Based on above definitions and results, we will give an algorithm for the multi-attribute decision problem.

Step 1. Construct a multi-granularity soft decision system (U, CS, DS).

Step 2. Calculate the dependent degree of DS upon CS - S_i (i = 1, 2, ⋯ , n).

Step 3. Calculate each conditional significance sig (S_i, CS, DS) of S_i = (f_i, C_i) in CS relative to DS by Definition 4.10.

Step 4. Find core (CS, DS) by Proposition 4.12.

Step 5. Find multi-granularity soft relative attribute reduction of (U, CS, DS) by Theorem 4.13.

(1) core (CS, DS) is a multi-granularity soft relative attribute reduction of (U, CS, DS) if γ (core (CS, DS)) = γ (CS, DS). In this case, the process stops.

Otherwise, it continues (2).

(2) Denote $core (CS, DS) = {CS}_{re}^{(t)} = (S_{i_{1}}, S_{i_{2}}, \dots, S_{i_{t}}) .$

(a) Calculate the condition significance $sig (S_{i}, ({CS}_{re}^{(t)}, S_{i}), DS)$ of each soft set $S_{i} \in CS - {CS}_{re}^{(t)}$ about $({CS}_{re}^{(t)}, S_{i})$ relative to DS by Definition 4.10.

(b) Select S_i with maximal conditional significance one by one. If there are many soft sets with the same maximal significant, we choose the attribute set containing the most elements. So (core (CS, DS) , S_i) is a multi-granularity soft relative attribute reduction of (U, CS, DS).

Step 6. Obtain decision rules by multi-granularity soft relative attribute reduction in the multi-granularity soft decision system (U, CS, DS). (see Fig. 1).

Fig.1

The flow diagram of the algorithm.

5 Application for a multi-attribute decision making problem

Now we apply the above approach to make component adaptation schemes in process of software reuse as a multi-attribute decision making problem.

Firstly, we collect the feedback about component reuse information.

Let U = {x₁, x₂, x₃, x₄, x₅, x₆, x₇, x₈} be a set of eight components. Let C = (C₁, C₂, C₃, C₄, C₅, C₆, C₇) be a condition attribute set of eight components and C₁, C₂, C₃, C₄, C₅, C₆, C₇ describe the flexibility and inter operability, adaptable for reuse, specification match, the quality of components, the criticality, the functional usage, resource utilization, respectively. The values of these attributes are as follows: $\begin{matrix} C_{1} = {c_{11} (high), c_{12} (medium), c_{13} (low)}, \\ C_{2} = {c_{21} (high), c_{22} (medium), c_{23} (low)}, \\ C_{3} = {c_{31} (high), c_{32} (medium), c_{33} (low)}, \\ C_{4} = {c_{41} (high), c_{42} (medium)}, \\ C_{5} = {c_{51} (high), c_{52} (medium)}, \\ C_{6} = {c_{61} (high), c_{62} (medium)}, \\ C_{7} = {c_{71} (high), c_{72} (medium)} . \end{matrix}$

Let D = {d₁, d₂, d₃, d₄} be the decision attribute set, which describes component adaptation schemes. d₁, d₂, d₃, d₄ describe “Reused with no additional effort to adapt”, “Reused with change in code”, “Reused by changing the requirement specification”, “Develop afresh”, respectively.

Now we use the feedback to form a decision table such as Table 5.

Table 5
Table for feedback

U C ₁ C ₂ C ₃ C ₄ C ₅ C ₆ C ₇ D

x ₁ L H L M M H M d ₂

x ₂ H L H H M H L d ₄

x ₃ L H L M M H M d ₁

x ₄ H M H H M H L d ₃

x ₅ M H M M H M L d ₁

x ₆ H M H H M H L d ₂

x ₇ H M H H M H L d ₂

x ₈ M H M M H M L d ₁

U	C ₁	C ₂	C ₃	C ₄	C ₅	C ₆	C ₇	D
x ₁	L	H	L	M	M	H	M	d ₂
x ₂	H	L	H	H	M	H	L	d ₄
x ₃	L	H	L	M	M	H	M	d ₁
x ₄	H	M	H	H	M	H	L	d ₃
x ₅	M	H	M	M	H	M	L	d ₁
x ₆	H	M	H	H	M	H	L	d ₂
x ₇	H	M	H	H	M	H	L	d ₂
x ₈	M	H	M	M	H	M	L	d ₁

where L, H and M mean Low, High and Medium, respectively.

The mapping in S_i = (f_i, C_i) (i = 1, 2, ⋯ , 7) and (g, D) over U are described as follows: $\begin{matrix} f_{1} (c_{11}) & = & {x_{2}, x_{4}, x_{6}, x_{7}}, f_{1} (c_{12}) = {x_{5}, x_{8}}, \\ f_{1} (c_{13}) & = & {x_{1}, x_{3}}, \\ f_{2} (c_{21}) & = & {x_{1}, x_{3}, x_{5}, x_{8}}, f_{2} (c_{22}) = {x_{4}, x_{6}, x_{7}}, \\ f_{2} (c_{23}) & = & {x_{2}}, \\ f_{3} (c_{31}) & = & {x_{2}, x_{4}, x_{6}, x_{7}}, f_{3} (c_{32}) = {x_{5}, x_{8}}, \\ f_{3} (c_{33}) & = & {x_{1}, x_{3}}, \\ f_{4} (c_{41}) & = & {x_{2}, x_{4}, x_{6}, x_{7}}, f_{4} (c_{42}) = {x_{1}, x_{3}, x_{5}, x_{8}}, \\ f_{5} (c_{51}) & = & {x_{5}, x_{8}}, f_{5} (c_{52}) = {x_{1}, x_{2}, x_{3}, x_{4}, x_{6}, x_{7}}, \\ f_{6} (c_{61}) & = & {x_{1}, x_{2}, x_{3}, x_{4}, x_{6}, x_{7}}, f_{6} (c_{62}) = {x_{5}, x_{8}}, \\ f_{7} (c_{71}) & = & {x_{1}, x_{3}}, f_{7} (c_{72}) = {x_{2}, x_{4}, x_{5}, x_{6}, x_{7}, x_{8}}, \\ g (d_{1}) & = & {x_{3}, x_{5}, x_{8}}, g (d_{2}) = {x_{1}, x_{6}, x_{7}}, \\ g (d_{3}) & = & {x_{4}}, g (d_{4}) = {x_{2}} . \end{matrix}$

Then we view each soft set S_i = (f_i, C_i) and (g, D) as collections of approximations as follows: $\begin{matrix} S_{1} = (f_{1}, C_{1}) = {{x_{2}, x_{4}, x_{6}, x_{7}}, {x_{5}, x_{8}}, {x_{1}, x_{3}}}, \\ S_{2} = (f_{2}, C_{2}) = {{x_{1}, x_{3}, x_{5}, x_{8}}, {x_{4}, x_{6}, x_{7}}, {x_{2}}}, \\ S_{3} = (f_{3}, C_{3}) = {{x_{2}, x_{4}, x_{6}, x_{7}}, {x_{5}, x_{8}}, {x_{1}, x_{3}}}, \\ S_{4} = (f_{4}, C_{4}) = {{x_{2}, x_{4}, x_{6}, x_{7}}, {x_{1}, x_{3}, x_{5}, x_{8}}}, \\ S_{5} = (f_{5}, C_{5}) = {{x_{5}, x_{8}}, {x_{1}, x_{2}, x_{3}, x_{4}, x_{6}, x_{7}}}, \\ S_{6} = (f_{6}, C_{6}) = {{x_{1}, x_{2}, x_{3}, x_{4}, x_{6}, x_{7}}, {x_{5}, x_{8}}}, \\ S_{7} = (f_{7}, C_{7}) = {{x_{1}, x_{3}}, {x_{2}, x_{4}, x_{5}, x_{6}, x_{7}, x_{8}}}, \\ (g, D) = {{x_{3}, x_{5}, x_{8}}, {x_{1}, x_{6}, x_{7}}, {x_{4}}, {x_{2}}} . \end{matrix}$

Denote CS = {S₁, S₂, …, S₇}, DS = (g, D) and CW⁽⁷⁾ = (U, CS).

Step 1. Construct a multi-granularity soft decision system (U, CS, DS) on how much is the possibility of adapt decision schemes chosen by users. Here the multi-granularity soft set CS describes “the character of components” and the decision partition soft set DS describes “the adapt decision of components”.

Step 2. Calculate the dependent degree of DS upon CS - S_i, (i = 1, 2, …, 7). $\begin{matrix} {\underline{apr}}_{{CW}^{(7)}} g (d_{1}) = {x_{5}, x_{8}}, {\underline{apr}}_{{CW}^{(7)}} g (d_{2}) = \\ {\underline{apr}}_{{CW}^{(7)}} g (d_{3}) = \emptyset, {\underline{apr}}_{{CW}^{(7)}} g (d_{4}) = {x_{2}} . \end{matrix}$

We have Pos_CS (DS) = {x₂, x₅, x₈}. So $\begin{matrix} k = γ (CS, DS) = \frac{| {x_{2}, x_{5}, x_{8}} |}{| U |} = \frac{3}{8} . \\ k = γ (CS - S_{i}, DS) = \frac{3}{8}, (i = 1, 3, 4, 5, 6, 7), \\ k = γ (CS - S_{2}, DS) = \frac{1}{4} . \end{matrix}$

Step 3. Calculate the conditional significance of (f_i, C_i) in CS relative to DS. $\begin{matrix} sig (S_{i}, CS, DS) = \frac{3}{8} - \frac{3}{8} = 0, (i = 1, 3, 4, 5, 6, 7), \\ sig (S_{2}, CS, DS) = \frac{3}{8} - \frac{1}{4} = \frac{1}{8} . \end{matrix}$

Step 4. Find core (CS, DS) $core (CS, DS) = S_{2} = (f_{2}, C_{2}) .$

Step 5. Since Pos_{S
₂} (DS) = {x₂}, we have $\begin{matrix} γ (S_{2}, DS) & = & \frac{| {x_{2}} |}{| U |} = \frac{1}{8} \neq γ (CS - S_{i}, DS), \\ i = (1, 3, 4, 5, 6, 7) . \end{matrix}$

So S₂ = (f₂, C₂) is not a multi-granularity soft relative attribute reduction of (U, CS, DS). Next we add some S_i = (f_i, C_i) (i = 1, 3, 4, 5, 6, 7) in (S₂, S_i) in turn. Then we calculate the conditional significance of each soft set S_i in (S₂, S_i) relative to DS: $\begin{matrix} sig (S_{i}, (S_{2}, S_{i}), DS) & = & \frac{3}{8} - \frac{1}{8} = \frac{1}{4}, (i = 1, 3, 5, 6), \\ sig (S_{4}, (S_{2}, S_{4}), DS) & = & sig (S_{7}, (S_{2}, S_{7}), DS) \\ = & \frac{1}{8} - \frac{1}{8} = 0 . \end{matrix}$

Now we select the sets C₁ and C₃ with the most elements from C_i (i = 1, 3, 5, 6). Since $γ ((S_{1}, S_{2}), DS) = γ ((S_{2}, S_{3}), DS) = \frac{3}{8},$ we know that (S₁, S₂) and (S₂, S₃) are two multi-granularity soft relative reductions of (U, CS, DS).

Step 6. As (S₁, S₂) is a multi-granularity soft relative reduction of (U, CS, DS), then for S₁ = (f₁, C₁) and S₂ = (f₂, C₂), we calculate the coverage degree of each attribute to d_r (r = 1, 2, 3, 4) in C₁ and C₂: $\begin{matrix} δ_{11} (d_{1}) & = & 0 δ_{12} (d_{1}) = \frac{2}{3} δ_{13} (d_{1}) = \frac{1}{3}, \\ δ_{11} (d_{2}) & = & \frac{2}{3} δ_{12} (d_{2}) = 0 δ_{13} (d_{2}) = \frac{1}{3}, \\ δ_{11} (d_{3}) & = & 1 δ_{12} (d_{3}) = 0 δ_{13} (d_{3}) = 0, \\ δ_{11} (d_{4}) & = & 1 δ_{12} (d_{4}) = 0 δ_{13} (d_{4}) = 0, \end{matrix}$

Similarly, for S₂ = (f₂, C₂), we have: $\begin{matrix} δ_{21} (d_{1}) & = & 1 δ_{22} (d_{1}) = 0 δ_{23} (d_{1}) = 0, \\ δ_{21} (d_{2}) & = & \frac{2}{3} δ_{22} (d_{2}) = \frac{2}{3} δ_{23} (d_{2}) = 0, \\ δ_{21} (d_{3}) & = & 0 δ_{22} (d_{3}) = 1 δ_{23} (d_{3}) = 0, \\ δ_{21} (d_{4}) & = & 0 δ_{22} (d_{4}) = 0 δ_{23} (d_{4}) = 1 . \end{matrix}$

For d₁, $δ_{12} = \frac{2}{3}$ and δ₂₁ = 1. So, if the flexibility and inter operability is medium, then $σ (\frac{2}{3})$ ; if the adaptable for reuse is high, then σ (1).

For d₂, $δ_{11} = \frac{2}{3}$ and $δ_{21} = δ_{22} = \frac{2}{3}$ . So, if the flexibility and inter operability is high, then $σ (\frac{2}{3})$ ; if the adaptable for reuse is high or medium, then $σ_{1} (\frac{2}{3})$ .

For d₃, δ₁₁ = 1 and δ₂₂ = 1. So, if the flexibility and inter operability is high, then σ (1); if the adaptable for reuse is medium, then σ (1).

For d₄, δ₁₁ = 1 and δ₂₃ = 1. So, if the flexibility and inter operability is high, then σ (1); if the adaptable for reuse is high or low, then σ (1).

Similarly, we can also obtain another decision rule of (U, CS, DS) if (S₂, S₃) is a multi-granularity soft relative reduction of (U, CS, DS).

This algorithm is based on cases of library history data analysis. The multi-attribute decision rule and the coverage degree of rules provide objective foundation for decision making problem. Moreover, this method can help users to choose the component adapter scheme and reduce subjectivity in the process of decision making.

6 Conclusion

In this paper, we have proposed a new concept of multi-granularity soft rough sets, which can be viewed as a generalized soft rough set model. We present some properties of multi-granularity soft rough approximations based on multi-granularity soft approximation spaces. We also study the multi-granularity soft decision system, which is applied to multi-attribute decision-making problems. The decision rule and an algorithm for the decision rule are proposed. As future work, connections between multi-granularity soft rough sets and various types of generalized rough set models could be explored.

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Multi-granularity soft rough set and its application in multi-attribute decision making

Abstract

Keywords

1 Introduction

2 Preliminaries

2.1 Rough sets

2.2 Soft sets

2.3 Soft rough sets

2.4 Decision information system

3 On multi-granularity soft rough sets

Table 1 Table for soft set (f, A) u 1 u 2 u 3 u 4 u 5 e 1 0 0 1 1 0 e 2 0 0 0 1 0

4.1 Multi-granularity soft decision systems

4.2 Multi-granularity soft relative attribute reduction of MGSDS

4.3 Dependent degree of decision partition soft sets

4.4 Condition significance relative to decision partition soft set

4.5 Decision rules in MGSDS

4.6 Algorithm for the multi-attribute decision problem

Table 5 Table for feedback U C 1 C 2 C 3 C 4 C 5 C 6 C 7 D x 1 L H L M M H M d 2 x 2 H L H H M H L d 4 x 3 L H L M M H M d 1 x 4 H M H H M H L d 3 x 5 M H M M H M L d 1 x 6 H M H H M H L d 2 x 7 H M H H M H L d 2 x 8 M H M M H M L d 1

References

Table 1
Table for soft set (f, A)

u ₁ u ₂ u ₃ u ₄ u ₅

e ₁ 0 0 1 1 0

e ₂ 0 0 0 1 0

Table 5
Table for feedback

U C ₁ C ₂ C ₃ C ₄ C ₅ C ₆ C ₇ D

x ₁ L H L M M H M d ₂

x ₂ H L H H M H L d ₄

x ₃ L H L M M H M d ₁

x ₄ H M H H M H L d ₃

x ₅ M H M M H M L d ₁

x ₆ H M H H M H L d ₂

x ₇ H M H H M H L d ₂

x ₈ M H M M H M L d ₁