A method for multi-attribute decision making applying soft rough sets

Abstract

This paper investigates a method for multi-attribute decision making applying soft rough sets. Firstly, some new concepts such as soft decision systems, soft relative positive regions, soft relative parameter reduct of soft decision systems, dependent degree of decision partition soft sets and conditional significance relative to decision partition soft sets are proposed based on soft rough sets. Secondly, the multi-attribute decision rule applying soft rough sets is given. Thirdly, an algorithm of multi-attribute decision making applying soft rough sets is presented. Finally, an application for the component retrieval problem is given to show the validity of this method.

Keywords

Soft rough set partition soft set soft decision system soft relative reduct decision making decision rule

1 Introduction

To solve complicated problems in economics, engineering, environmental science and social science, methods in classical mathematics are not always successful because various types of uncertainties are present in these problems. While probability theory, fuzzy set theory [30], rough set theory [23], and other mathematical tools are well-known and useful approaches to describe uncertainty, each of these theories has its difficulties as pointed out in [17]. One major problem shared by those theories is their incompatibility with the parameterizations tools.

In 1999, Molodtsov [17] initiated soft set theory as a new mathematical tool for dealing with uncertainties which classical mathematical tools cannot handle. Recently, there has been a rapid growth of interest in soft set theory. Many efforts have been devoted to further generalizations and extensions of Molodtsov’s soft sets. Recently there has been a rapid growth of interest in soft set theory and its applications. Many efforts have been devoted to further generalizations and extensions of Molodtsov’s soft sets. Maji et al. [19] defined fuzzy soft sets, combining soft sets with fuzzy sets. This line of exploration was further investigated by several researchers [20 , 27]. Maji et al. [21] reported a detailed theoretical study on soft sets, with emphasis on the algebraic operations. Jiang et al. [9] extended soft sets with description logics. Aktas and Cağman [1] initiated the notion of soft groups, extending fuzzy groups. Jun et al. discussed the applications of soft sets to the study of BCK/BCI-algebras [7 , 10]. Feng et al. investigated the relationships among soft sets, rough sets and fuzzy sets, obtaining three types of hybrid models: rough soft sets, soft rough sets and soft-rough fuzzy sets [2, 5]. Li et al. [13, 14] considered roughness of fuzzy soft sets and obtained the relationship among soft sets, soft rough sets and topologies. Li et al. [16] studied parameter reductions of soft coverings.

Applications of soft sets in decision making problems were initiated in [18]. To address fuzzy soft set based decision making problems, Roy and Maji [24] presented a novel method of object recognition from an imprecise multi-observer data. Using level soft sets, Feng et al. [3] proposed an adjustable approach to decision making based on fuzzy soft sets. This approach was further investigated in [4, 11]. Li et al. [15] investigated decision making based on intuitionistic fuzzy soft sets. Although Molodtsov’s soft sets have been applied by several authors to the study of decision making under uncertainty, it seems that soft set based group decision making has not been discussed yet in the literature. Thus the present study can be seen as a first attempt toward the possible application of soft rough approximations in multi-attribute decision making problems under uncertainty.

The remaining part of this paper is organized as follows. In Section 2, we have presented some concepts related to rough sets, soft sets, bijective soft sets, soft rough sets and information systems centered around our problem. Section 3 introduces some new concepts such as soft decision systems, soft relative positive regions, soft relative attribute reduct of soft decision systems, dependent degree of decision partition soft sets, conditional significance relative to decision partition soft sets, the multi-attribute decision rule in soft decision systems and so on. At the same time, we simply discuss some propositions about them. An algorithm of multi-attribute decision making based on soft rough set is presented. In Section 4, to show the validity of this algorithm we give an application for a multi-attribute decision making problem. This paper concludes in Section 5.

2 Preliminaries

In this section, we briefly recall some basic concepts about rough sets, soft sets, bijective soft sets, soft rough sets and information systems.

Throughout this paper, U denotes an initial universe, E denotes the set of all possible attributes, 2^U denotes the family of all subsets of U and ∣·∣ is the cardinality of a set. We only consider the case where both U and E are finite sets.

2.1 Rough sets

Let R be an equivalence relation on U. Then the pair (U, R) is called a Pawlak approximation space. Based on (U, R), we can define the following two rough approximations: $\underline{R} X = {x \in U | [x]_{R} \subseteq X},$ $\bar{R} X = {x \in U | [x]_{R} \cap X \neq \emptyset} .$ Then $\underline{R} X$ and $\bar{R} X$ are called the Pawlak lower approximation and Pawlak upper approximation of X ∈ 2^U, respectively. Moreover, the sets ${Pos}_{R} X = \underline{R} X, {Neg}_{R} X = U - \bar{R} X, {Bnd}_{R} X = \bar{R} X - \underline{R} X$ are referred to as R-positive, R-negative and R-boundary region of X ∈ 2^U, respectively.

A set is Pawlak rough if its boundary region is not empty, that is, X is Pawlak rough if $\underline{R} X \neq \bar{R} X$ . Otherwise, the set is crisp.

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

$x \in {Pos}_{R} X = \underline{R} X$ means that x certainly has property P;

$x \in \bar{R} X$ means that x possibly has property P;

x ∈ Neg_RX means that definitely x does not have property P.

2.2 Soft sets

Definition 2.1. [17] Let A ⊆ E. A pair (f, A) is called a soft set over U, if f is a mapping defined by f : A → 2^U.

In other words, a soft set over U is a parameterized family of subsets of the U. For e ∈ A, f (e) may be considered as the set of e-approximate elements of (f, A).

To illustrate this idea, let us consider the following example.

Example 2.2. 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 which 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 houses” 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 : E ⟶ 2^U given by “houses(.)”, where (.) is to be filled in by one of the attributes. Suppose

f (e₁) = {h₁, h₂, h₅}, f (e₂) = {h₁, h₆}, f (e₃) = {h₃, h₄}, f (e₄) = {h₃, h₄, h₆} .

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

Definition 2.3. [21] Let (f, A) and (g, B) be two soft sets over U.

(f, A) is called a soft subset of (g, B), if A ⊆ B and f (e) = g (e) for each e ∈ A. We denote it by $(f, A) \tilde{\subset} (g, B)$ .

(f, A) and (g, B) are called soft equal, if A = B and f (e) = g (e) for each e ∈ A. We denote it by (f, A) = (g, B).

Obviously, (f, A) = (g, B) if and only if $(f, A) \tilde{\subset} (g, B)$ and $(f, A) \tilde{\supset} (g, B)$ .

Definition 2.4. [21] Let (f, A), (g, B) and (h, C) be three soft sets over U.

(h, C) is called the intersection of (f, A) and (g, B), if C = A ∩ B and h (e) = f (e) ∩ g (e) for each e ∈ C. We denote (h, C) by $(f, A) \tilde{\cap} (g, B)$ .

(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 $(f, A) \tilde{\cup} (g, B)$ .

Definition 2.5. [21] Let (f, A) and (g, B) be two soft sets over U. (f, A) AND (g, B) denoted by (f, A) ∧ (g, B) is defined by (f, A) ∧ (g, B) = (h, A × B), where h (a, b) = f (a) ∩ g (b) for each (a, b)∈ A × B.

Definition 2.6. [5] 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 Bijective soft sets

Definition 2.7. [6] Let A ⊆ E and (f, A) a soft set over U. (f, A) is called a bijective soft set, if f is a mapping f : A → 2^U such that

⋃_e∈Af (e) = U;

For e_i, e_j ∈ A and e_i ≠ e_j, f (e_i)∩ f (e_j) = ∅.

Suppose B = {f (e₁), f (e₂), . . ., f (e_n)} with e₁, e₂, …, e_n ∈ A. From Definition 2.7, the mapping f : A → 2^U can be transformed to the mapping f : A → B, which is a bijective function, namely, for every X ∈ B, there is exactly one attribute e ∈ A such that f (e) = X and no unmapped element remains in both A and B.

Proposition 2.8. Let (f, A) be a bijective soft set over U and (g, B) a null soft set over U. Then $(f, A) \tilde{\cup} (g, B)$ is a bijective soft set over U.

Proposition 2.9. Let (f, A) and (g, B) be two bijective soft sets over U. Then (f, A) ∧ (g, B) is also a bijective soft set over U.

2.4 Soft rough sets

Definition 2.10. [5] Let (f, A) be a soft set over U. Then the pair P = (U, (f, A)) is called a soft approximation space. Based on the soft approximation space P, we define the following two operations: $\begin{matrix} {\underline{apr}}_{P} X & = & {x \in U | \exists e \in A s . t . x \in f (e) \subseteq X}, \\ {\bar{apr}}_{P} X & = & {x \in U | \exists e \in A s . t . x \in f (e), f (e) \cap X \\ \neq & \emptyset} . \end{matrix}$

Then ${\underline{apr}}_{P} (X)$ and ${\bar{apr}}_{P} (X)$ are called the soft P-lower approximation and the soft P-upper approximation of X ∈ 2^U, respectively. 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 ∈ 2^U, respectively.

X is said to be a soft P-definable set if ${\underline{apr}}_{P} (X) = {\bar{apr}}_{P} (X)$ ; otherwise, X is called a soft P-rough set.

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

$x \in {Pos}_{P} (X) = {\underline{apr}}_{P} (X)$ 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: $\begin{matrix} {\underline{apr}}_{P} (X) & = & ⋃_{e \in A} {f (e) | f (e) \subseteq X}, \\ {\bar{apr}}_{P} (X) & = & ⋃_{e \in A} {f (e) | f (e) \cap X \neq \emptyset} . \end{matrix}$

2.5 Information systems

Definition 2.11. [31] Let U be a finite set of objects and let A be a finite set of attributes. The quadruple (U, A, V, g) is called an information system, if g is an information function from U × A to V = ⋃ _a∈AV_a where every V_a = {g (x, e) | e ∈ A and x ∈ U} is the value set of the attribute e. (U, A, V, g) is called a 2-valued information system, if V ⊆ {0, 1}.

If A = C ∪ D and C∩ D = ∅, then (U, A, V, g) is called a decision information system where C is called a condition attribute set and D is called a decision attribute set. Sometimes the decision information system (U, C ∪ D, V, g) denotes by (U, C, D, V, g).

Definition 2.12. Let S = (f, A) be a soft set over U. Then I_S = (U, A, V, g_s) is called a 2-value information system induced by S, if g_s : U × A → V is a mapping such that for any x ∈ U and e ∈ A $g_{s} (x, e) = {\begin{matrix} 1, x \in f (e), \\ 0, x \notin f (e) . \end{matrix}$

Definition 2.13. Let I = (U, A, V, g) be a 2-value information system. Then S_I = (f_I, A) is called a soft set over U induced by I, if f_I : A → 2^U is a mapping such that f_I (e) = {x ∈ U| g (x, e) =1} for any x ∈ U and e ∈ A.

Lemma 2.14. Let S = (f, A) be a soft set over U, I_S = (U, A, V, g_s) a 2-value information system induced by S and S_{I
_S} = (f_{I
_S}, A) a soft set over U induced by I_S. Then S = S_{I
_S}.

Proof. By Definition 2.13, for any e ∈ A, f_{I
_S} (e) = {x ∈ U| g_s (x, e) =1} .

By Definition 2.12, for any x ∈ U and e ∈ A, $g_{s} (x, e) = {\begin{matrix} 1, x \in f (e), \\ 0, x \notin f (e) . \end{matrix}$

This implies that g_s (x, e) =1 ⇔ x ∈ f (e) . So for any x ∈ U, e ∈ A, f (e) = f_{I
_S} (e). Hence f_A = (f_{I
_S}, A). This implies S = S_{I
_S}. □

Lemma 2.15. Let I = (U, A, V, g) be a 2-value information system, S_I = (f_I, _A) a soft set over U induced by I and I_{S
_I} = (U, A, V, g_{s
_I}) a 2-value information system induced by S_I. Then I = I_{S
_I}.

Proof. By Definition 2.12, for any x ∈ U and e ∈ A, $g_{s_{I}} (x, e) = {\begin{matrix} 1, x \in f_{I} (e), \\ 0, x \notin f_{I} (e) . \end{matrix}$

For any x ∈ U and e ∈ A, by Definition 2.13 f_I (e) = {x ∈ U| g (x, e) =1}. Since I = (U, A, V, g) is a 2-value information system, g (x, e) =0 for x ∉ f_I (e), This implies $g (x, e) = {\begin{matrix} 1, x \in f_{I} (e), \\ 0, x \notin f_{I} (e) . \end{matrix}$

So for any x ∈ U and e ∈ A, g_{s
_I} (x, e) = g (x, e) . Hence g_{s
_I} = g . This implies I = I_{S
_I} . □

Theorem 2.16. Let Σ = {S| S = (f, A) is a soft set over U} and Γ = {I| I = (U, A, V, g) is a 2-valueinformation system} . Then there exists a one-to-one correspondence between Σ and Γ.

Proof. Two maps F : Σ → Γ and G : Γ → Σ are defined as follows:

F (S) = I_S, for S∈ Σ ; G (I) = S_I, for I ∈ Γ .

By Lemma 2.14, G ∘ F = i_Σ, where G ∘ F is the composition of F and G, and i_Σ is the identity map on Σ.

By Lemma 2.15, F ∘ G = i_Γ, where G ∘ F is the composition of G and F, and i_Γ is the identity map on Γ.

Hence F and G are both a one-to-one correspondence. This prove that there exists a one-to-one correspondence between Σ and Γ. □

To illustrate this idea, let us consider the following example.

Example 2.17. In Example 2.2, we consider the information system (U, A, V, g), which also describes “attractiveness of the houses”. For instance, f (h₁, e₁) =1 means “h₁ is beautiful” and its functional value is 1; “f (h₁, e₃) =0 ”means “h₁ is not cheap” and its functional value is 0. That is, the value = 1 whenever h_i ∈ f (e_j) (1 ⩽ i ⩽ 6, 1 ⩽ j ⩽ 4). Otherwise, the value = 0. Thus we described (U, A, V, g) as the following Table 1.

3 Soft decision systems

It is worth nothing that information systems and soft sets are closely related. Given the soft set (f, A) over U, (f, A) could induce an information system in a natural way. In the section, we mainly discuss soft decision systems.

Let (f_i, C_i) (i = 1, 2, ⋯, n) be bijective soft sets over U where C_i∩ C_j = ∅ for i ≠ j. Denote $(f, C) = {\tilde{\cup}}_{i = 1}^{n} (f_{i}, C_{i}), (φ, K) = ⋀_{i = 1}^{n} (f_{i}, C_{i}),$ where $C = ⋃_{i = 1}^{n} C_{i}$ and K = C₁ × C₂ × … × C_n .

3.1 Soft relative positive regions

Definition 3.1. Let (f, A) and (g, B) be two soft sets over U. Then the soft positive region of (f, A) to (g, B) is defined as follows: $\begin{matrix} {Pos}_{(f, A)} (g, B) & = & ⋃_{b \in B} {\underline{apr}}_{P} g (b) \\ = & ⋃_{b \in B} {x \in U | \exists e \in A \\ s . t . x \in f (e) \subseteq g (b)}, \end{matrix}$ where P = (U, (f, A)).

Example 3.2. Let U = {x₁, x₂, x₃, x₄, x₅, x₆, x₇} be a set. Let A = {e₁, e₂, e₃, e₄} and B = {b₁, b₂} be two attribute sets. Suppose that (f, A) and (g, B) are two soft sets over U.

(f, A) is given as follows:

f (e₁) = {x₁, x₂}, f (e₂) = {x₄, x₅, x₆}, f (e₃) = {x₃, x₇} .

(g, B) is given as follows::

g (b₁) = {x₁, x₂, x₃}, g (b₂) = {x₄, x₅, x₆, x₇}.

Then ${\underline{apr}}_{(U, (f, A))} g (b_{1}) = {x_{1}, x_{2}}$ ,

${\underline{apr}}_{(U, (f, A))} g (b_{2}) = {x_{4}, x_{5}, x_{6}} .$

Thus Pos_(f,A) (g, B) = {x₁, x₂, x₄, x₅, x₆} .

Definition 3.3. Let (f_i, C_i) (i = 1, 2, …, n) be bijective soft sets over U where C_i∩ C_j = ∅ for i ≠ j. Let (g, D) be a partition soft set over U where C∩ D = ∅. Then the triple (U, (f, C), (g, D)) is called a soft decision system, (f, C) is called the condition bijective soft set and (g, D) is called the decision partition soft set.

Accordingly, in a soft decision system (U, (f, C), (g, D)), we have $\begin{matrix} {Pos}_{(φ, K)} (g, D) & = & ⋃_{d \in D} {\underline{apr}}_{P} g (d) \\ = & ⋃_{d \in D} {x \in U | \exists e \in K \\ s . t . x \in φ (e) \subseteq g (d)}, \end{matrix}$ where K = C₁ × C₂ × … × C_n and P = (U, (φ, K)). We call it soft relative positive regions of soft decision systems.

For a given soft decision system, we always consider Pos_(φ,K) (g, D) ≠ ∅ .

Example 3.4. Suppose that U = {x₁, x₂, x₃, x₄, x₅, x₆} is a common universe, which is a set of six shops. $C = ⋃_{i = 1}^{3} C_{i}$ denotes the attribute set where C₁ stands for the empowerment of sales personnel, C₂ stands for the perceived quality of goods, and C₃ stands for the high traffic location, respectively. The value sets of these attributes are C₁ = {high, medium, low}, C₂ = {good, average} and C₃ = {no, yes}, respectively. And D = {profit, loss} describes shop profit or loss. Suppose that the six shops are characterized by the condition bijective soft set ${\tilde{\cup}}_{i = 1}^{3} (f_{i}, C_{i})$ , and the management benefit of shop is characterized by the decision partition soft set (g, D).

The mapping of each bijective soft set over U is defined as follows:

f₁ (high) = {x₁, x₆}, f₁ (medium) = {x₂, x₃, x₅}, f₁ (low) = {x₄},

f₂(good) = {x₁, x₂, x₃}, f₂(average) = {x₄, x₅, x₆},

f₃ (no) = {x₁, x₂, x₃, x₄}, f₃ (yes) = {x₅, x₆} .

The mapping of the decision partition soft set over U is defined as follows:

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

Then we can view each bijective soft set (f_i, C_i) as a collection of approximations as follows:

(f₁, C₁) = {high = {x₁, x₆}, medium = {x₂, x₃, x₅}, low = {x₄}},

(f₂, C₂) = {good = {x₁, x₂, x₃}, average = {x₄, x₅, x₆}},

(f₃, C₃) = {no = {x₁, x₂, x₃, x₄}, yes = {x₅, x₆}} .

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

Denote $(f, C) = {\tilde{\cup}}_{i = 1}^{3} (f_{i}, C_{i}), (φ, K) = ⋀_{i = 1}^{3} (f_{i}, C_{i}),$ where $C = ⋃_{i = 1}^{3} C_{i}$ and K = C₁ × C₂ × C₃ .

Let e_i ∈ K. Then

e₁ =high and good and no,

e₂ =medium and good and no,

e₃ =low and average and no,

e₄ =medium and average and yes,

e₅ =high and average and yes. $φ (e_{1}) = {x_{1}}, φ (e_{2}) = {x_{2}, x_{3}},$ $φ (e_{3}) = {x_{4}}, φ (e_{4}) = {x_{5}}, φ (e_{5}) = {x_{6}} .$

Besides, we have the tabular form of (φ, K) given in Table 2.

So (U, (f, C), (g, D)) is a soft decision system on how to choose profitable shops. Thus $\begin{matrix} {\underline{apr}}_{(U, (f, C))} g (profit) = {x_{1}, x_{6}}, \\ {\underline{apr}}_{(U, (f, C))} g (loss) = {x_{4}, x_{5}} . \end{matrix}$

Therefore Pos_(φ,K) (g, D) = {x₁, x₄, x₅, x₆} .

3.2 Soft relative attribute reduct of soft decision systems

Definition 3.5. Let (U, (f, C), (g, D)) be a soft decision system and let 1 ≤ j ≤ n. Then

(1) (f_j, C_j) is called a soft dispensable set of (f, C) relative to (g, D), if Pos_(φ,K) (g, D) = Pos_(ψ,Q) (g, D), where $(ψ, Q) = ⋀_{i = 1, i \neq j}^{n} (f_{i}, C_{i})$ . Otherwise, (f_j, C_j) is called a soft indispensable set of (f, C) relative to (g, D).

(2) (f, C) is called a soft independent set relative to (g, D), if every soft bijective set (f_i, C_i) of (f, C) is a soft indispensable set relative to (g, D). Otherwise, (f, C) is called a soft dependent set relative to (g, D) .

(3) The unit set of all the soft indispensable set of (f, C) relative to (g, D) is called the core of (f, C) relative to (g, D), denoted by core ((f, C), (g, D)).

Definition 3.6. Let (U, (f, C), (g, D)) be a soft decision system. For k ≤ m and 1 ≤ j_k ≤ n. Denote $\begin{matrix} (f^{'}, C^{'}) & = & {\tilde{\cup}}_{k = 1}^{m} (f_{j_{k}}, C_{j_{k}}) and (φ^{'}, K^{'}) \\ = & ⋀_{k = 1}^{m} (f_{j_{k}}, C_{j_{k}}) . \end{matrix}$

Then (f′, C′) is called a soft relative attribute reduct of (U, (f, C), (g, D)), if

(1) Pos_(φ,K) (g, D) = Pos_(φ′,K′) (g, D),

(2) (f′, C′) is a soft independent set relative to (g, D).

Example 3.7. In Example 3.4, denote

(φ₁, K₁) = (f₁, C₁) ∧ (f₂, C₂), (φ₂, K₂) = (f₁, C₁) ∧ (f₃, C₃), (φ₃, K₃) = (f₂, C₂) ∧ (f₃, C₃) .

We have

Pos_(φ₁,K₁) (g, D) = Pos_(φ₂,K₂) (g, D) = Pos_(φ,K) (g, D) = {x₁, x₄, x₅, x₆}, Pos_(φ₃,K₃) (g, D) = {x₄} .

But

Pos_(f₁,C₁) (g, D) = {x₁, x₄, x₆} ≠ Pos_(φ,K) (g, D), Pos_(f₃,C₃) (g, D) = ∅ ≠ Pos_(φ,K) (g, D) .

Thus

(f₁, C₁) ∪ (f₂, C₂) and (f₁, C₁) ∪ (f₃, C₃) are both soft relative attribute reducts of (U, (f, C), (g, D)) .

3.3 Dependent degree of decision partition soft sets

Definition 3.8. Let (f, A) and (g, B) be two soft sets over U. (f, A) is said to depend on (g, B) to a degree k (0 ≤ k ≤ 1), denoted (f, A) ⇒ _k (g, B), if $k = γ ((f, A), (g, B)) = \frac{∣ {Pos}_{(f, A)} (g, B) ∣}{∣ U ∣} .$

Accordingly, in a soft decision system (U, (f, C), (g, D)), we have $k = γ ((φ, K), (g, D)) = \frac{∣ {Pos}_{(φ, K)} (g, D) ∣}{∣ U ∣} .$

We call it the dependent degree of decision partition soft sets upon condition bijective soft sets. It characters a degree of condition bijective soft sets in classifying decision partition soft sets. Obviously, we have 0 ≤ k ≤ 1.

If k = 1, then (g, D) is completely dependent on (f, C).

If k = 0, then (g, D) is completely independent on (f, C).

Example 3.9. In Example 3.4, the dependent degree of the decision partition soft set (g, D) upon the condition bijective soft set $(f, C) = {\tilde{\cup}}_{i = 1}^{3} (f_{i}, C_{i}) :$ $\begin{matrix} k & = & γ (⋀_{i = 1}^{3} (f_{i}, C_{i}), (g, D)) \\ = & \frac{| {x_{1}, x_{4}, x_{5}, x_{6}} |}{| U |} = \frac{4}{6} = \frac{2}{3} \end{matrix}$

Proposition 3.10. Let (U, (f, C), (g, D)) be a soft decision system. Then, for m, n ∈ N and m < n, $γ (⋀_{i = 1}^{m} (f_{i}, C_{i}), (g, D)) \leq γ (⋀_{i = 1}^{n} (f_{i}, C_{i}), (g, D)) .$

Proof. Since $\begin{matrix} γ ((φ, K), (g, D)) \\ = \frac{∣ {Pos}_{(φ, K)} (g, D) ∣}{∣ U ∣} \\ = \frac{∣ ⋃_{d \in D} {\underline{apr}}_{P} g (d) ∣}{∣ U ∣} \\ = \frac{∣ ⋃_{d \in D} {x \in U | \exists e \in K s . t . x \in φ (e) \subseteq g (d)} ∣}{∣ U ∣}, \\ γ ((φ^{'}, K^{'}), (g, D)) \\ = \frac{∣ {Pos}_{(φ^{'}, K^{'})} (g, D) ∣}{∣ U ∣} \\ = \frac{∣ ⋃_{d \in D} {\underline{apr}}_{P^{'}} g (d) ∣}{∣ U ∣} \\ = \frac{∣ ⋃_{d \in D} {x \in U | \exists e \in K^{'} s . t . x \in φ^{'} (e) \subseteq g (d)} ∣}{∣ U ∣}, \end{matrix}$ where $K = ⋀_{i = 1}^{n} C_{i}, K^{'} = ⋀_{i = 1}^{m} C_{i}, P = (U, (f, C))$ andP′ = (U, (f′, C′)), by Definition 2.6, for any (c₁, c₂, ⋯, c_n) ∈ C₁ × C₂ × ⋯ × C_n, we have $\begin{matrix} φ (c_{1}, c_{2}, \dots, c_{n}) = f_{1} (c_{1}) \cap f_{2} (c_{2}) \\ \cap \dots \cap f_{m} (c_{m}) \cap \dots \cap f_{n} (c_{n}) \end{matrix}$ and $φ^{'} (c_{1}, c_{2}, \dots, c_{m}) = f_{1} (c_{1}) \cap f_{2} (c_{2}) \cap \dots \cap f_{m} (c_{m}) .$

For m, n ∈ N and m < n, ${\underline{apr}}_{P^{'}} g (d) \subseteq {\underline{apr}}_{P} g (d) .$

So $⋃_{d \in D} {\underline{apr}}_{P^{'}} g (d) \subseteq ⋃_{d \in D} {\underline{apr}}_{P} g (d) .$

Hence $γ (⋀_{i = 1}^{m} (f_{i}, C_{i}), (g, D))} \leq γ (⋀_{i = 1}^{n} (f_{i}, C_{i}), (g, D)) . □$

In other words, the condition bijective soft set can explain the most detailed classification of decision partition soft sets. And deleting some condition bijective soft sets can lose some information about the decision partition soft set. Thus, more information (more condition bijective soft sets) can result in bigger dependent degree of the decision partition soft set.

3.4 Conditional significance relative to decision partition soft sets

Definition 3.11. Let (U, (f, C), (g, D)) be a soft decision system and 1 ≤ j ≤ n. The conditional significance of (f_j, C_j) in (f, C) relative to (g, D) is denoted and defined as follows $\begin{matrix} s ((f_{j}, C_{j}), (f, C), (g, D)) \\ = γ (⋀_{i = 1}^{n} (f_{i}, C_{i}), (g, D)) \\ - γ (⋀_{i = 1, i \neq j}^{n} (f_{i}, C_{i}), (g, D)) . \end{matrix}$

This definition indicates the decrease of the dependent degree of decision partition soft sets when deleting one bijective soft set (f_j, C_j) from (f, C). The following results are easily obtained from the above definitions.

Proposition 3.12. Let (U, (f, C), (g, D)) be a soft decision system and 1 ≤ j ≤ n. Then

(1) 0 ≤ s ((f_j, C_j), (f, C), (g, D)) ≤1 .

(2) (f_j, C_j) is a soft indispensable set of (f, C) to (g, D) if and only if $s ((f_{j}, C_{j}), (f, C), (g, D)) > 0 .$

(3) $core ((f, C), (g, D)) = \tilde{\cup} {(f_{j}, C_{j}) ∣ s ((f_{j}, C_{j}), (f, C), (g, D)) > 0, j \leq n}$ .

Theorem 3.13. Let (U, (f, C), (g, D)) be a soft decision system. Let k = 1, 2, …, m and 1 ≤ j_k ≤ n. Denote

$(f^{'}, C^{'}) = {\tilde{\cup}}_{k = 1}^{m} (f_{j_{k}}, C_{j_{k}}) and (φ^{'}, K^{'}) = ⋀_{k = 1}^{m} (f_{j_{k}}, C_{j_{k}}),$

where $C^{'} = ⋃_{j_{k} = 1}^{m} C_{j_{k}}$ and K′ = C_j1 × C_j2 × … × C_jm .

If γ ((φ′, K′), (g, D)) = γ ((φ, K), (g, D)) and s ((f_j, C_j), (f′, C′), (g, D)) >0, then (f′, C′) is a soft relative attribute reduct of (U, (f, C), (g, D)).

3.5 The multi-attribute decision rule in soft decision systems

Definition 3.14. Let (U, (f, C), (g, D)) be a soft decision system. Let e ∈ K = C₁ × C₂ × … × C_n, d ∈ D. The soft rough membership function of φ (e) relative to g (d) is denoted and defined as follows $ξ (φ (e), g (d)) = \frac{| φ (e) \cap g (d) |}{| φ (e) |} .$

Definition 3.15. Let (U, (f, C), (g, D)) be soft decision system. For k ≤ m and 1 ≤ j_k ≤ n, denote $(f^{'}, C^{'}) = {\tilde{\cup}}_{k = 1}^{m} (f_{j_{k}}, C_{j_{k}}), (φ^{'}, K^{'}) = ⋀_{k = 1}^{m} (f_{j_{k}}, C_{j_{k}}),$ where $C = ⋃_{j_{k} = 1}^{m} C_{j_{k}}$ and K′ = C_j1 × C_j2 × … × C_jm . Let (f′, C′) be a soft relative attribute reduct of (U, (f, C), (g, D)). We call $If e, then d (ξ (φ^{'} (e), g (d)))$ the multi-attribute decision rule by induced (f′, C′) in (U, (f, C), (g, D)), where e ∈ K′, d ∈ D and ξ (φ′ (e), g (d)) denotes the soft rough membership function of φ′ (e) relative to g (d), which expresses the support degree of rules.

3.6 An algorithm for the multi-attribute decision rule

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

An algorithm:

Step 1. Construct a soft decision system (U, (f, C), (g, D)).

Step 2. Calculate the dependent degree of (g, D) upon $⋀_{i = 1, i \neq j}^{n} (f_{i}, C_{i}) (j = 0, 1, 2, \dots, n)$ .

Step 3. Calculate each conditional significance of (f_j, C_j) in (f, C) relative to (g, D) by Definition 3.11.

Step 4. Find core ((f, C), (g, D)) by Proposition 3.12.

Step 5. Find soft relative attribute reducts of (U, (f, C), (g, D)) by Theorem 3.13.

(1) core ((f, C), (g, D)) is a soft relative attribute reduct of (U, (f, C), (g, D)) if γ (core ((f, C), (g, D)) = γ ((f, C), (g, D)). In this case, the process stops.

Otherwise, it continues (2).

(2) Denote $core ((f, C), (g, D)) = ⋀_{k = 1}^{m} (f_{j_{k}}, C_{j_{k}}),$ where k ≤ m and 1 ≤ j_k ≤ n.

(a) Calculate the conditional significance of each bijective soft set (f_i, C_i) (i ≠ j_k) about ${\tilde{\cup}}_{k = 1}^{m} (f_{j_{k}}, C_{j_{k}})$ relative to (g, D) by Definition 3.11.

(b) Select (f_i, C_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 $core ((f, C), (g, D)) \tilde{\cup} (f_{i}, C_{i})$ is a soft relative attribute reduct of (U, (f, C), (g, D)).

Step 6. Obtain decision rules by soft relative attribute reducts in the soft decision system (U, (f, C), (g, D)). (see Fig.1)

4 An application for a multi-attribute decision making problem

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

Firstly, we collect the feedback about component reuse information [26].

Let U = {x₁, x₂, x₃, x₄, x₅, x₆, x₇, x₈} be a set of eight components. Let $C = ⋃_{i = 1}^{7} C_{i}$ be a condition attribute set of eight components. C₁ describes the flexibility and interoperability. C₂ describes adaptable for reuse. C₃ describes specification match. C₄ describes the quality of components. C₅ describes criticality. C₆ describes the functional usage. C₇ describes resource utilization. The values of these attributes are as follows, respectively:

$\begin{matrix} C_{1} & = & {c_{11} (high), c_{12} (medium), c_{13} (low)}, \\ C_{2} & = & {c_{21} (high), c_{22} (medium), c_{23} (low)}, \\ C_{3} & = & {c_{31} (high), c_{32} (medium), c_{33} (low)}, \\ C_{4} & = & {c_{41} (high), c_{42} (medium)}, \\ C_{5} & = & {c_{51} (high), c_{52} (medium)}, \\ C_{6} & = & {c_{61} (high), c_{62} (medium)}, \\ C_{7} & = & {c_{71} (medium), c_{72} (low)} . \end{matrix}$

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

Now we use the feedback to form a decision table such as Table 3.

Suppose that (f_i, C_i) (i = 1, …, 7) and (g, D) are soft sets over U. The mapping of (f_i, C_i) (i = 1, …, 7) and (g, D) over U are described asfollows:

f₁ (c₁₁) = {x₂, x₄, x₆, x₇}, f₁ (c₁₂) = {x₅, x₈}, f₁ (c₁₃) = {x₁, x₃},

f₂ (c₂₁) = {x₁, x₃, x₅, x₈}, f₂ (c₂₂) = {x₄, x₆, x₇}, f₂ (c₂₃) = {x₂},

f₃ (c₃₁) = {x₂, x₄, x₆, x₇}, f₃ (c₃₂) = {x₅, x₈}, f₃ (c₃₃) = {x₁, x₃},

f₄ (c₄₁) = {x₂, x₄, x₆, x₇}, f₄ (c₄₂) = {x₁, x₃, x₅, x₈},

f₅ (c₅₁) = {x₅, x₈}, f₅ (c₅₂) = {x₁, x₂, x₃, x₄, x₆, x₇},

f₆ (c₆₁) = {x₁, x₂, x₃, x₄, x₆, x₇}, f₆ (c₆₂) = {x₅, x₈},

f₇ (c₇₁) = {x₁, x₃}, f₇ (c₇₂) = {x₂, x₄, x₅, x₆, x₇, x₈},

g (d₁) = {x₃, x₅, x₈}, g (d₂) = {x₁, x₆, x₇}, g (d₃) = {x₄}, g (d₄) = {x₂} .

Then we view each soft set (f_i, C_i) and (g, D) as collections of approximations as follows:

$\begin{matrix} (f_{1}, C_{1}) & = & {{x_{2}, x_{4}, x_{6}, x_{7}}, {x_{5}, x_{8}}, {x_{1}, x_{3}}}, \\ (f_{2}, C_{2}) & = & {{x_{1}, x_{3}, x_{5}, x_{8}}, {x_{4}, x_{6}, x_{7}}, {x_{2}}}, \\ (f_{3}, C_{3}) & = & {{x_{2}, x_{4}, x_{6}, x_{7}}, {x_{5}, x_{8}}, {x_{1}, x_{3}}}, \\ (f_{4}, C_{4}) & = & {{x_{2}, x_{4}, x_{6}, x_{7}}, {x_{1}, x_{3}, x_{5}, x_{8}}}, \\ (f_{5}, C_{5}) & = & {{x_{5}, x_{8}}, {x_{1}, x_{2}, x_{3}, x_{4}, x_{6}, x_{7}}}, \\ (f_{6}, C_{6}) & = & {{x_{1}, x_{2}, x_{3}, x_{4}, x_{6}, x_{7}}, {x_{5}, x_{8}}}, \\ (f_{7}, C_{7}) & = & {{x_{1}, x_{3},}, {x_{2}, x_{4}, x_{5}, x_{6}, x_{7}, x_{8}}}, \\ (g, D) & = & {{x_{3}, x_{5}, x_{8}}, {x_{1}, x_{6}, x_{7}}, {x_{4}}, {x_{2}}} . \end{matrix}$

Denote $(f, C) = {\tilde{\cup}}_{i = 1}^{7} (f_{i}, C_{i}), (φ, K) = ⋀_{i = 1}^{7} (f_{i}, C_{i}),$ where $C = ⋃_{i = 1}^{7} C_{i}$ and K = C₁ × C₂ × … × C₇ .

Step 1. We construct a soft decision system (U, (f, C), (g, D)) on how much is the possibility of adapt decision schemes chosen by users. Here the condition bijective soft set (f, C) describes “the character of components” and the decision partition soft set (g, D) describes “the adapt decision of components”.

Then e_i ∈ K respectively stand for as follows: $\begin{matrix} e_{1} & = & (c_{11}, c_{22}, c_{31}, c_{41}, c_{52}, c_{61}, c_{72}), \\ e_{2} & = & (c_{11}, c_{23}, c_{31}, c_{41}, c_{52}, c_{61}, c_{72}), \\ e_{3} & = & (c_{12}, c_{21}, c_{32}, c_{42}, c_{51}, c_{62}, c_{72}), \\ e_{4} & = & (c_{13}, c_{21}, c_{33}, c_{42}, c_{52}, c_{61}, c_{71}) . \end{matrix}$

So φ (e₁) = {x₄, x₆, x₇}, φ (e₂) = {x₂}, φ (e₃) = {x₅, x₈}, φ (e₄) = {x₁, x₃} .

The tabular form of (φ, K) is given in Table 4.

Step 2. For

${\underline{apr}}_{(U, (f, C))} g (d_{1}) = {x_{5}, x_{8}}, {\underline{apr}}_{(U, (f, C))} g (d_{2}) = {\underline{apr}}_{(U, (f, C))} g (d_{3}) = \emptyset,$

${\underline{apr}}_{(U, (f, C))} g (d_{4}) = {x_{2}},$

we have Pos_(φ,K) (g, D) = {x₂, x₅, x₈} .

So $k = γ (⋀_{i = 1}^{7} (f_{i}, C_{i}), (g, D)) = \frac{| {x_{2}, x_{5}, x_{8}} |}{| U |} = \frac{3}{8} .$

Similarly, $⋀_{i = 1, i \neq j}^{7} (f_{i}, C_{i}) (j = 1, 2, \dots, 7)$ are also expressed as following tabular forms (Tables 5–11), respectively.

Step 3. By Definition 3.11, we can calculate the conditional significance of (f_j, C_j) in (f, C) relative to (g, D) $\begin{matrix} s ((f_{j}, C_{j}), (f, C), (g, D)) \\ = \frac{3}{8} - \frac{3}{8} = 0 (j = 1, 3, 4, 5, 6, 7), \\ s ((f_{2}, C_{2}), (f, C), (g, D)) \\ = \frac{3}{8} - \frac{1}{4} = \frac{1}{8} . \end{matrix}$

Step 4. By Proposition 3.12 we have $core ((f, C), (g, D)) = (f_{2}, C_{2}) .$

Step 5. Since Pos_(f₂,C₂) (g, D) = {x₂}, we have

$γ ((f_{2}, C_{2}), (g, D)) = \frac{| {x_{2}} |}{| U |} = \frac{1}{8}$

$\neq γ (⋀_{i = 1, i \neq j}^{7} (f_{i}, C_{i}), (g, D)) (j = 1, 3, 4, 5, 6, 7) .$

By Theorem 3.13, (f₂, C₂) is not a soft relative attribute reduct of (U, (f, C), (g, D)). Next we find soft relative attribute reducts of (U, (f, C), (g, D)) . We addict (f_j, C_j) (j = 1, 3, 4, 5, 6, 7) to (f₂, C₂). Then by Definition 3.11, we calculate the conditional significance of each bijective soft set (f_j, C_j) in (f₂, C₂) ∪ (f_j, C_j) relative to (g, D):

$\begin{matrix} s ((f_{j}, C_{j}), (f_{2}, C_{2}) \cup (f_{j}, C_{j}), (g, D)) \\ = \frac{3}{8} - \frac{1}{8} = \frac{1}{4} (j = 1, 3, 5, 6, 7), \\ s ((f_{4}, C_{4}), (f_{2}, C_{2}) \cup (f_{4}, C_{4}), (g, D)) \\ = \frac{1}{8} - \frac{1}{8} = 0 . \end{matrix}$

Now we select the sets with the most elements from C_j (j = 1, 3, 5, 6, 7), and C₁ and C₃ is the sets with the most elements. Since

$\begin{matrix} γ (⋀_{i = 1, i \neq j}^{7} (f_{i}, C_{i}), (g, D)) \\ = γ ((f_{1}, C_{1}) \land (f_{2}, C_{2}), (g, D)) \\ = γ ((f_{2}, C_{2}) \land (f_{3}, C_{3}), (g, D)) = \frac{3}{8} . \end{matrix}$

We know that (f₁, C₁) ∪ (f₂, C₂) and (f₂, C₂) ∪ (f₃, C₃) are two soft relative reducts of (U, (f, C), (g, D)) by Theorem 3.13.

Step 6.

If (f₁, C₁) ∪ (f₂, C₂) is a soft relative reductions of (U, (f, C), (g, D)), then $e_{i}^{'} \in C_{1} \times C_{2}$ respectively stand for as follows:

$e_{1}^{'} = (c_{11}$ , c₂₂), $e_{2}^{'} = (c_{11}, c_{23})$ , $e_{3}^{'} = (c_{12}$ , c₂₁), $e_{4}^{'} = (c_{13}, c_{21}) .$

We can induce a decision rule of (U, (f, C), (g, D)) as follows:

(1) If the flexibility and interoperability of components is high and the adaptable for reuse is middle, then d₂( $\frac{2}{3}$ ).

(2) If the flexibility and interoperability of components is high and the adaptable for reuse is middle, then d₃( $\frac{1}{3}$ ).

(3) If the flexibility and interoperability of components is high and the adaptable for reuse is low, then d₄(1).

(4) If the flexibility and interoperability of components is middle and the adaptable for reuse is high, then d₁(1).

(5) If the flexibility and interoperability of components is low and the adaptable for reuse is high, then d₁( $\frac{1}{2}$ ).

(6) If the flexibility and interoperability of components is low and the adaptable for reuse is high, then d₂( $\frac{1}{2}$ ).

Similarly, we can also obtain another decision rule of (U, (f, C), (g, D)) if (f₂, C₂) ∪ (f₃, C₃) is a soft relative reduct of (U, (f, C), (g, D)). There we omit.

This method is based on cases of library history data analysis. We can find the useful information. The multi-attribute decision rule and the support degree of rules provides scientific objective basis. This method reduces the search domain and is more efficient than the existing methods. Therefore, this method can help users to decide the component adapter scheme and reduce pressure and subjectivity in the component reuse process adapter decision-making.

5 Conclusions

In this paper, we have proposed a method based on soft rough sets for supporting multi-attribute decision making problems. Through constructing a soft decision system, we introduce the concepts such as the dependency, the relative reduct of soft decision systems and decision rule and so on. Besides, we give an algorithm for the multi-attribute decision. We have analysed this method by the problem of component adaptation scheme. This method is proved to be practical and rational. In the future, we will consider to apply rough approximations of interval-valued fuzzy soft set to multi-attribute decision makingproblems.

Acknowledgments

The authors would like to thank the editors and the anonymous reviewers for their valuable comments and suggestions which have helped immensely in improving the quality of the paper. This work is supported by the National Natural Science Foundation of China (11461005, 11201490, 11401052), the Natural Science Foundation of Guangxi (2014GXNSFAA118001), Guangxi University Science and Technology Research Project (KY2015YB075, KY2015YB081), China Postdoctoral Science Foundation (2013M542558), Special Funds of Guangxi Distinguished Experts Construction Engineering and Key Laboratory of Optimization Control and Engineering Calculation in Department of Guangxi Education.

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