Multi-granularity belief interval-valued soft set

Abstract

A new model named multi-granularity belief interval-valued soft set is introduced in this paper. Some basic properties about it are presented and illustrated. The improved concepts of the soft belief value and soft belief degree are proposed, which provided an easier and better compared horizontally and vertically method among the different objects and different parameters. An algorithm for decision-making problems on multi-granularity belief interval-valued soft set is put forward and its validity is proved by the application of an example. Moreover, the newly proposed algorithm is compared with existing method to indicate its extensive application.

Keywords

Belief interval-valued soft set soft belief value soft belief degree decision making

1 Introduction

Molodsov [18] presented the soft set theory in 1999 as a novel mathematical tool to develop uncertainty theory. Maji et al. [19] made further efforts to study the soft set theory in the application of decision-making problems. Kong [14, 15] introduced normal parameter reduction of soft sets and further studied on normal parameter reduction in soft set based on particle swarm optimization algorithm. For the incomplete soft sets and fuzzy soft sets, Kong [16, 17] put forward an efficient approach of decision-making and a new parameter reduction respectively. A novel notion of soft set parameterization reduction was defined by Chen et al. [2]. Intutionistic fuzzy set theory was proposed by Atanassov [1]. Broad extensions of the basic soft set theory and fuzzy set theory were contributed such as fuzzy soft sets by Maji et al. [20], interval-valued fuzzy soft sets by Yang et al. [31], intutionistic fuzzy soft sets by Maji [21], interval-valued intutionistic fuzzy soft sets by Jiang et al. [10]. Roy and Maji [27] generalized a theoretic approach on fuzzy soft sets to deal with object recognition and decision-making problems. Feng et al. [6, 7], Dey et al. [3], Kamacı [12] and Fatimah et al. [8] have extended classical soft set theory to soft fuzzy sets, soft rough sets, soft fuzzy rough sets, soft rough fuzzy sets and N-soft sets and applied them in many directions. Kamacı [11] developed some new operations on N-soft sets and discussed the algebraic properties of N-soft sets in some algebraic structures. Then Kamacı and Petchimuthu [13] introduced a novel model of bipolar N-soft set and its related properties. They also provided two excellent algorithms to dispose the decision-making problems on the basis of bipolar N-soft sets and compared their approaches with some previous methods to illustrate the efficiencies of their algorithms. Hong et al. [9] used the interval representation to replace the former expression of intuitionistic fuzzy set. Jiang et al. [10] combined the interval-valued intuitionistic fuzzy set and soft set, then proposed interval-valued intuitionistic fuzzy soft set theory and some basic properties. Dymova and Sevastjanov [4, 5] interpreted intuitionistic fuzzy sets and defined some operations on them in the frame of Dempster–Shafer theory by the representation [9]. Nguyen [22] adapted interval-valued intuitionistic fuzzy sets via a new interval-valued knowledge measure and applied it in decision making. In 2019, Vijayabalaji et al. [28] combined belief interval-value and soft set and proposed the notion of belief interval-valued soft set theory. And then the operations were defined on belief interval-valued soft sets and applied them in multi-attribute decision making.

The fuzzy information granulation and granular computing were initiated by Zadeh [34, 35]. He considered an equivalence relation on the universe as a granulation, and a partition as a granulation space. Pawlak [23] initiated the rough set in 1982. Then he continued studying the rudiments and applications in [24, 25]. The rough set theory was deemed as a single granulation by Yao [32, 33]. That is to say, a equivalence relation was formed a single granulation space. The worth noticeably, a certain equivalence relation was aroused by an attribute set. The description of the object via two original granulations namely two attribute sets is less missed in uncertainty management, but it will destroy the structure of the classical granulation. In 2010, original single-granulation rough sets were developed to multi-granulation rough sets which were called multi-granulation rough sets by Qian et al. [26]. They introduced the set approximations via multiple equivalence relations on the universe. In 2013, two new types of multiple granulation rough set were put forward by Xu et al. [30]. Wang et al. [29] introduced multi-granularity soft rough set model to handle multi-attribute decision-making problems and constructed a multi-granularity soft decision system. And they defined some new properties and proposed a relative attribute reduction in it.

Rough set is a kind of granulation model based on equivalence relation, while soft set theory uses approximation set instead of equivalence class as information grain. In order to get more original information from two or more belief interval-valued soft sets without breaking the original structure, we will introduce multi-granularity belief interval-valued soft set(MGBIVSS), which combines multi-granularity and belief interval-valued soft set. We will propose some relative concepts of MGBIVSS and deal with decision-making problems. In Section 2, the basic concepts of soft set theory, intuitionistic fuzzy set theory, intuitionistic fuzzy soft set theory and belief interval-valued soft set theory are introduced. Section 3 is dedicated to the study of multi-granularity belief interval-valued soft set theory. This paper is organized as follows. In this section, the related concepts of the multi-granularity belief interval-valued soft sets are proposed. The intersection, the union and the complement operations and some properties are defined and proved on MGBIVSS. In Section 4, two new concepts of soft belief value and soft belief degree are proposed. Then we put forward an algorithm as application of the new concepts to solve decision-making problems and illustrate it by a corporate recruitment example. In section 5, it was demonstrated that the algorithm is more widely applied and more easily comparable by a previous example. At last, the concluding section summarizes the paper.

2 Preliminaries

In this section, we recall some concepts such as soft sets, intuitionistic fuzzy sets, intuitionistic fuzzy soft sets and belief interval-valued soft sets. Let U = {u_k : k = 1, 2, …, n} be a finite universe set, A = {a_i : i = 1, 2, …, l} be a subset of the parameter set E, i . e . A ⊆ E.

2.1 Intuitionistic fuzzy set

Definition 2.1. [1] An intuitionistic fuzzy set S can be represented as. $S = {< u_{k}, μ_{S} (u_{k}), ν_{S} (u_{k}) > | u_{k} \in U}$ subject to the condition 0 ≤ μ_S (u_k) + ν_S (u_k) ≤1 for every u_k ∈ U.

π_S (u_k) =1 - (μ_S (u_k) + ν_S (u_k)) is the hesitation degree of u_k ∈ U. Hong et al. [9] utilized [μ_S (u_k) , 1 - ν_S (u_k)] to express the intuitionistic fuzzy set S to substitute the previous representation. The advantage is that the representation [μ_S (u_k) , 1 - ν_S (u_k)] is a classical interval for the right boundary is bigger than the left boundary.

Definition 2.2. [4] An intuitionistic fuzzy set S can be represented as. $S = {< u_{k}, {Bl}_{S} (u_{k}) > | u_{k} \in U}$

where Bl_S (u_k) = [Bel_S (u_k) , Pl_S (u_k)] represents the belief interval. Bel_S (u_k) = μ_S (u_k) and Pl_S (u_k) = μ_S (u_k) + π_S (u_k) =1 - ν_S (u_k) represent the degrees of belief and plausibility respectively.

2.2 Intuitionistic fuzzy soft set

Definition 2.3. [18] A pair (F, A) is a soft set over U, where F is a mapping F : A → P (U) and P (U) represents the power set of U.

Definition 2.4. [27] A pair (G, A) is a fuzzy soft set over U, where G is a mapping $G : A \to P (U)$ and $P (U)$ represents the set of all fuzzy sets over U.

Definition 2.5. [21] A pair (I, A) is an intuitionistic fuzzy soft set over U, where I is a mapping $I : A \to P (U)$ and $P (U)$ is the set of all intuitionistic fuzzy sets defined on U.

Definition 2.6. [21] Let (I₁, A₁) and (I₂, A₂) be two intuitionistic fuzzy soft sets on U. (I₁, A₁) is an intuitionistic fuzzy soft subset of (I₂, A₂) if and only if

(1)A₁ ⊂ A₂ ; A₁, A₂ ⊆ E.

(2)∀A_i ∈ E, I₁ (a_i) ⊂ I₁ (a_i)

In this case, we write (I₁, A₁) ⊆ (I₂, A₂)

2.3 Belief interval-valued soft set

Definition 2.7. [28] $ℙ (U)$ is a set of all belief interval-valued subsets of U. A mapping $Y : A \to ℙ (U)$ is a belief interval-valued soft set and denoted by (Y, A) on U, represented as. $Y (a_{i}) = {< u_{k}, {Bl}_{Y (a_{i})} (u_{k}) > | u_{k} \in U},$ where Bl_{Y(a_i)} (u_k) = [Bel_{Y(a_i)} (u_k) , Pl_{Y(a_i)} (u_k)], ∀a_i ∈ A ⊆ E.

Vijayabalaji and Ramesh [28] introduced belief interval-valued soft set (BIVSS) by combining belief interval-value and soft set. They redefined the belief interval in terms of Dymova and Sevastjanov [4]. Bel_{Y(a_i)} (u_k) is the belief degree, and Pl_{Y(a_i)} (u_k) ∈ [0, 1] is the plausibility degree, where Bel_{Y(a_i)} (u_k) = μ_{Y(a_i)} (u_k) and Pl_{Y(a_i)} (u_k) = μ_{Y(a_i)} (u_k) + π_{Y(a_i)} (u_k) + ν_{Y(a_i)} (u_k) × π_{Y(a_i)} (u_k) =1 - ν_{Y(a_i)} (u_k) ².

Definition 2.8. [28] Let (Y₁, A₁) and (Y₂, A₂) be two belief interval-valued soft sets on the common set U = {u_k : k = 1, 2, …, n}. The intersection and union (Y₁, A₁) and (Y₂, A₂) are defined as follows.

(1)Intersection

(Y₁, A₁) ∩ (Y₂, A₂) = (L, O), where O = A₁ ∩ A₂, andA₁, A₂ ⊆ E. $L (a_{i}) = Y_{1} (a_{i}) \cap Y_{2} (a_{i})$

(2)Union

(Y₁, A₁) ∪ (Y₂, A₂) = (Q, T), where T = A₁ ∪ A₂, andA₁, A₂ ⊆ E.

$Q (a_{i}) = {\begin{matrix} Y_{1} (a_{i}) & a_{i} \in A_{1} - A_{2}, \\ Y_{2} (a_{i}) & a_{i} \in A_{2} - A_{1}, \\ Y_{1} (a_{i}) \cup Y_{2} (a_{i}) & a_{i} \in A_{1} \cap A_{2} . \end{matrix}$

Definition 2.9. [28] Let (Y, A) be a belief interval-valued soft set on the set U = {u_k : k = 1, 2, …, n}. (Y, A) ^c = (Y, A^c) is the complement of (Y, A), where ∀a_i ∈ A, A^c = [1 - Pl_{Y(a_i)} (u_k) , 1 - Bel_{Y(a_i)} (u_k)] and for $a_{i}^{c} \in A^{c}$

3 Multi-granularity belief interval-valued soft set

In the previous research work, there is only one parameter to each object, which we can call it single-granularity. Now we propose multi-granularity belief interval-valued soft set which contains two or more parameters to each object. We only introduce two-granularity belief interval-valued soft set. Let A = {a_i : i = 1, 2, …, l} ⊆ E and B = {b_j : j = 1, 2, …, m} ⊆ E.

Definition 3.1. Let (F, A) and (G, B) be two belief interval-valued soft sets over a common universe set U, (H, C) is called a multi-granularity belief interval-valued soft set. We define the following operation (H, C), where C = A × B, and ∀a_i ∈ A, b_j ∈ B, u_k ∈ U $\begin{matrix} H (a_{i}, b_{j}) \\ = {\frac{u_{k}}{{Bel}_{H (a_{i}, b_{j})} (u_{k}), {Pl}_{H (a_{i}, b_{j})} (u_{k})} : \forall u_{k} \in U} \end{matrix}$ where Bel_{H(a_i,b_j)} (u_k) = min (Bel_{F(a_i)} (u_k) , Bel_{G(b_j)} (u_k)) and Pl_{H(a_i,b_j)} (u_k) = max (Pl_{F(a_i)} (u_k) , Pl_{G(b_j)} (u_k)) for all u_k ∈ U.

When (F, A) = (G, B), it is clear that multi-granularity belief interval-valued soft set is (F, A) or (G, B). When (F, A) ≠ (G, B), it is clear that multi-granularity belief interval-valued soft set in Definition 3.1 is a generalization of belief interval-valued soft set which is a more comprehensive summary of information of (F, A) and (G, B). For ∀u_k ∈ U, H (a_i, b_j) is a comprehensive description about belief and plausibility for parameter a_i and parameter b_j. For each u_k with parameter a_i and parameter b_j, the description of belief is lower than u_k with parameter a_i or parameter b_j, for each u_k with parameter a_i and parameter b_j, the description of plausibility is greater than u_k with parameter a_i or parameter b_j.

Remark 3.1. Symmetric: (H, C) is called a multi-granularity belief interval-valued soft set, C = A × B, and ∀a_i ∈ A, b_j ∈ B, u_k ∈ U, H (a_i, b_j) = H (b_j, a_i).

Example 3.1. Let U = {u₁, u₂, u₃, u₄, u₅}, A = {a₁, a₂} ⊂ E and B = {b₁, b₂, b₃} ⊂ E. Let (F, A) be a belief interval-valued soft set over U given by Table 1 and (G, B) be a belief interval-valued soft set over U given by Table 2.

Table 1
Table for belief interval-valued soft set (F, A)

u ₁ u ₂ u ₃ u ₄ u ₅

a ₁ [0.3, 0.5] [0.6, 0.7] [0.6, 0.7] [0.5, 0.8] [0.3, 0.7]

a ₂ [0.6, 0.8] [0.1, 0.3] [0.7, 0.9] [0.8, 0.9] [0.4, 0.9]

	u ₁	u ₂	u ₃	u ₄	u ₅
a ₁	[0.3, 0.5]	[0.6, 0.7]	[0.6, 0.7]	[0.5, 0.8]	[0.3, 0.7]
a ₂	[0.6, 0.8]	[0.1, 0.3]	[0.7, 0.9]	[0.8, 0.9]	[0.4, 0.9]

Table 2

Table for belief interval-valued soft set (G, B)

	u ₁	u ₂	u ₃	u ₄	u ₅
b ₁	[0.5, 0.7]	[0.3, 0.4]	[0.2, 0.8]	[0.4, 0.9]	[0.6, 0.7]
b ₂	[0.5, 0.6]	[0.5, 0.6]	[0.2, 0.4]	[0.5, 0.7]	[0.3, 0.7]
b ₃	[0.8, 0.9]	[0.5, 0.7]	[0.1, 0.3]	[0.2, 0.5]	[0.4, 0.7]

According to the Definition 3.1, the multi-granularity belief interval-valued soft set (H, C) is given by Table 3.

Table 3

Table for multi-granularity belief interval-valued soft set (H, C)

	u ₁	u ₂	u ₃	u ₄	u ₅
(a₁, b₁)	[0.3, 0.7]	[0.3, 0.7]	[0.2, 0.8]	[0.4, 0.9]	[0.3, 0.7]
(a₁, b₂)	[0.3, 0.6]	[0.5, 0.7]	[0.2, 0.7]	[0.5, 0.8]	[0.3, 0.7]
(a₁, b₃)	[0.3, 0.9]	[0.5, 0.7]	[0.1, 0.7]	[0.2, 0.8]	[0.3, 0.7]
(a₂, b₁)	[0.5, 0.8]	[0.1, 0.4]	[0.2, 0.9]	[0.4, 0.9]	[0.4, 0.7]
(a₂, b₂)	[0.5, 0.8]	[0.1, 0.6]	[0.2, 0.9]	[0.5, 0.9]	[0.3, 0.7]
(a₂, b₃)	[0.6, 0.9]	[0.1, 0.7]	[0.1, 0.9]	[0.2, 0.9]	[0.4, 0.7]

Definition 3.2. A multi-granularity belief interval-valued soft set (H, C) over U is said a NULL multi-granularity belief interval-valued soft set denoted by Φ, if ∀ (a_i, b_j) ∈ C, u_k ∈ U, $H (a_{i}, b_{j}) = {\frac{u_{k}}{[0, 0]} : \forall u_{k} \in U}$

Definition 3.3. A multi-granularity belief interval-valued soft set (H, C) over U is said an ABSOLUTE multi-granularity belief interval-valued soft set denoted by $\tilde{U}$ , if ∀ (a_i, b_j) ∈ C, u_k ∈ U, $H (a_{i}, b_{j}) = {\frac{u_{k}}{[1, 1]} : \forall u_{k} \in U}$

Definition 3.4. Let A = {a_i : i = 1, 2, …, l} and B = {b_j : j = 1, 2, …, m} be the subsets of parameter set E and (F, A) and (G, B) be two belief interval-valued soft sets on the common universe set U = {u_k : k = 1, 2, …, n}. Then (H, C) and (W, D) are two multi-granularity belief interval-valued soft sets derived from (F, A) and (G, B). The intersection, union and complement of two multi-granularity belief interval-valued soft sets (H, C) and (W, D) over U are defined as follows.

(1)Intersection

(H, C) ∩ (W, D) = (M, K), where K = C ∩ D

M (a_i, b_j) = H (a_i, b_j) ∩ W (a_i, b_j)

(2)Union

(H, C) ∪ (W, D) = (N, P), where K = C ∪ D

$\begin{matrix} N (a_{i}, b_{j}) \\ = {\begin{matrix} H (a_{i}, b_{j}) & (a_{i}, b_{j}) \in C - D, \\ W (a_{i}, b_{j}) & (a_{i}, b_{j}) \in D - C, \\ H (a_{i}, b_{j}) \cup W (a_{i}, b_{j}) & (a_{i}, b_{j}) \in C \cap D . \end{matrix} \end{matrix}$

(3) Complement

(H, C) ^c = (H^c, ⌝ C), where ∀ ¬ (a_i, b_j) ∈ ⌝ C,

H^c = [1 - Pl_{H(a_i,b_j)} (u_k) , 1 - Bel_{H(a_i,b_j)} (u_k)], ¬ (a_i, b_j) = not a_i and not b_j.

Theorem 3.2. Let C be a set of parameter binaries, if Φ is a NULL multi-granularity belief interval-valued soft set, $\tilde{U}$ is an ABSOLUTE multi-granularity belief interval-valued soft set, (H, C) is a multi-granularity belief interval-valued soft set over U, then

(1)(H, C) ∪ (H, C) = (H, C);

(2)(H, C) ∩ (H, C) = (H, C);

(3)(H, C) ∪ Φ = (H, C);

(4)(H, C) ∩ Φ = Φ;

(5) $(H, C) \cup \tilde{U} = \tilde{U}$ ;

(6) $(H, C) \cap \tilde{U} = (H, C)$ .

Proof. It is easily obtained from Definitions 3.1-3.4. □

Definition 3.5. Let A = {a_i : i = 1, 2, …, l} and B = {b_j : j = 1, 2, …, m} be the subsets of parameter set E and (F, A) and (G, B) be two belief interval-valued soft sets on the common set U = {u_k : k = 1, 2, …, n}. Then (H, C) and (W, D) are two multi-granularity belief interval-valued soft sets derived from (F, A) and (G, B), we say that (H, C) is a multi-granularity belief interval-valued soft subset of (W, D) if and only if

(1) C ⊆ D;

(2) ∀ (a_i, b_j) ∈ C, H (a_i, b_j) ⊆ W (a_i, b_j), that is, for all u_k ∈ U and ∀ (a_i, b_j) ∈ C, Bel_{H(a_i,b_j)} (u_k) ≤ Bel_{W(a_i,b_j)} (u_k) and Pl_{H(a_i,b_j)} (u_k) ≤ Pl_{W(a_i,b_j)} (u_k),

That is denoted by (H, C) ⊑ (W, D). Similarly, if (W, D) is a multi-granularity belief interval-valued soft subset of (H, C), that is denoted by (W, D) ⊑ (H, C).

Definition 3.6. Let A = {a_i : i = 1, 2, …, l} and B = {b_j : j = 1, 2, …, m} be the subsets of parameter set E and (F, A) and (G, B) be two belief interval-valued soft sets on the common set U = {u_k : k = 1, 2, …, n}. Then (H, C) and (W, D) are two multi-granularity belief interval-valued soft sets derived from (F, A) and (G, B). Then (H, C) and (W, D) are said to be multi-granularity belief interval-valued soft equal if and only if

(1)(H, C) is a multi-granularity belief interval-valued soft subset of (W, D);

(2)(W, D) is a multi-granularity belief interval-valued soft subset of (H, C).

We write (H, C) = (W, D).

Theorem 3.3. Let Φ be a NULL multi-granularity belief interval-valued soft set, $\tilde{U}$ be an ABSOLUTE multi-granularity belief interval-valued soft set. (H, C), (W, D) and (M, K) are three multi-granularity belief interval-valued soft sets over U, then

(1)Φ ⊑ (H, C);

(2) $(H, C) ⊑ \tilde{U}$ ;

(3)(H, C) ⊑ (H, C) ∪ (W, D), (W, D) ⊑ (H, C) ∪ (W, D);

(4)(H, C) ∩ (W, D) ⊑ (H, C), (H, C) ∩ (W, D) ⊑ (W, D);

(5)(H, C) ∪ (W, D) = (W, D) ∪ (H, C);

(6)(H, C) ∩ (W, D) = (W, D) ∩ (H, C);

(7)((H, C) ∪ (W, D)) ∪ (M, K)=(H, C) ∪ ((W, D) ∪ (M, K));

(8)((H, C) ∩ (W, D)) ∩ (M, K)=(H, C) ∩ ((W, D) ∩ (M, K)).

Proof. (1)-(8) It is easily obtained from Definitions 3.1-3.6. □

Theorem 3.4. For any three multi-granularity belief interval-valued soft sets over U, (H, C), (W, D) and (M, K), then we have the following properties:

(1)((H, C) ∪ (W, D)) ∩ (M, K)=((H, C)∩ (M, K)) ∪ ((W, D) ∩ (M, K)) ;

(2)((H, C) ∩ (W, D)) ∪ (M, K)=((H, C) ∪ (M, K)) ∩ ((W, D) ∪ (M, K)) .

Proof. (1)Suppose that (H, C) ∪ (W, D) = (N, P), where P = C ∪ D and ∀ (a_i, b_j) ∈ P, ∀u_k ∈ U, ${Bel}_{N (a_{i}, b_{j})} (u_{k}) = {\begin{matrix} {Bel}_{H (a_{i}, b_{j})} (u_{k}) & (a_{i}, b_{j}) \in C - D, \\ {Bel}_{W (a_{i}, b_{j})} (u_{k}) & (a_{i}, b_{j}) \in D - C, \\ \max ({Bel}_{H (a_{i}, b_{j})} (u_{k}), {Bel}_{W (a_{i}, b_{j})} (u_{k})) & (a_{i}, b_{j}) \in C \cap D . \end{matrix}$ ${Pl}_{N (a_{i}, b_{j})} (u_{k}) = {\begin{matrix} {Pl}_{H (a_{i}, b_{j})} (u_{k}) & (a_{i}, b_{j}) \in C - D, \\ {Pl}_{W (a_{i}, b_{j})} (u_{k}) & (a_{i}, b_{j}) \in D - C, \\ \max ({Pl}_{H (a_{i}, b_{j})} (u_{k}), {Pl}_{W (a_{i}, b_{j})} (u_{k})) & (a_{i}, b_{j}) \in C \cap D . \end{matrix}$

Since (H, C) ∪ (W, D) = (N, P), We suppose that (N, P) ∩ (M . K) = (I, S), where S = P ∪ K = (C ∪ D) ∩ K, Then, ∀ (a_i, b_j) ∈ S, ∀u_k ∈ U, ${Bel}_{I (a_{i}, b_{j})} (u_{k}) = {\begin{matrix} \min ({Bel}_{H (a_{i}, b_{j})} (u_{k}), {Bel}_{M (a_{i}, b_{j})} (u_{k})) & (a_{i}, b_{j}) \in (C - D) \cap K, \\ \min ({Bel}_{W (a_{i}, b_{j})} (u_{k}), {Bel}_{M (a_{i}, b_{j})} (u_{k})) & (a_{i}, b_{j}) \in (D - C) \cap K, \\ \min (\max ({Bel}_{H (a_{i}, b_{j})} (u_{k}), {Bel}_{W (a_{i}, b_{j})} (u_{k})), {Bel}_{M (a_{i}, b_{j})} (u_{k})) & (a_{i}, b_{j}) \in C \cap D \cap K . \end{matrix}$ ${Pl}_{I (a_{i}, b_{j})} (u_{k}) = {\begin{matrix} \min ({Pl}_{H (a_{i}, b_{j})} (u_{k}), {Pl}_{M (a_{i}, b_{j})} (u_{k})) & (a_{i}, b_{j}) \in (C - D) \cap K, \\ \min ({Pl}_{W (a_{i}, b_{j})} (u_{k}), {Pl}_{M (a_{i}, b_{j})} (u_{k})) & (a_{i}, b_{j}) \in (D - C) \cap K, \\ \min (\max ({Pl}_{H (a_{i}, b_{j})} (u_{k}), {Pl}_{W (a_{i}, b_{j})} (u_{k})), {Pl}_{M (a_{i}, b_{j})} (u_{k})) & (a_{i}, b_{j}) \in C \cap D \cap K . \end{matrix}$ Assume that ((H, C) ∩ (M, K)) ∪ ((W, D) ∩ (M, K)) = (J . V), where V = (C ∩ K) ∪ (D ∩ K) = (C ∪ D) ∩ K, Then, ∀ (a_i, b_j) ∈ S, ∀u_k ∈ U, ${Bel}_{J (a_{i}, b_{j})} (u_{k}) = {\begin{matrix} \min ({Bel}_{H (a_{i}, b_{j})} (u_{k}), {Bel}_{M (a_{i}, b_{j})} (u_{k})) & (a_{i}, b_{j}) \in (C - D) \cap K, \\ \min ({Bel}_{W (a_{i}, b_{j})} (u_{k}), {Bel}_{M (a_{i}, b_{j})} (u_{k})) & (a_{i}, b_{j}) \in (D - C) \cap K, \\ \max (\min ({Bel}_{H (a_{i}, b_{j})} (u_{k}), {Bel}_{M (a_{i}, b_{j})} (u_{k})), \\ \min ({Bel}_{W (a_{i}, b_{j})} (u_{k}), {Bel}_{M (a_{i}, b_{j})} (u_{k}))) & (a_{i}, b_{j}) \in C \cap D \cap K . \end{matrix}$ ${Pl}_{J (a_{i}, b_{j})} (u_{k}) = {\begin{matrix} \min ({Pl}_{H (a_{i}, b_{j})} (u_{k}), {Pl}_{M (a_{i}, b_{j})} (u_{k})) & (a_{i}, b_{j}) \in (C - D) \cap K, \\ \min ({Pl}_{W (a_{i}, b_{j})} (u_{k}), {Pl}_{M (a_{i}, b_{j})} (u_{k})) & (a_{i}, b_{j}) \in (D - C) \cap K, \\ \max (\min ({Pl}_{H (a_{i}, b_{j})} (u_{k}), {Pl}_{M (a_{i}, b_{j})} (u_{k})), \\ \min ({Pl}_{W (a_{i}, b_{j})} (u_{k}), {Pl}_{M (a_{i}, b_{j})} (u_{k}))) & (a_{i}, b_{j}) \in C \cap D \cap K . \end{matrix}$ Since min (max (Bel_{H(a_i,b_j)} (u_k) , Bel_{W(a_i,b_j)} (u_k)) , Bel_{M(a_i,b_j)} (u_k) )=max (min (Bel_{H(a_i,b_j)} ( u_k) , Bel_{M(a_i,b_j)} (u_k)) , min (Bel_{W(a_i,b_j)} (u_k) , Bel_{M(a_i,b_j)} (u_k))) and min (max (Pl_{H(a_i,b_j)} (u_k) , Pl_{W(a_i,b_j)} (u_k)) , Pl_{M(a_i,b_j)} (u_k))= max (min (Pl_{H(a_i,b_j)} (u_k) , Pl_{M(a_i,b_j)} (u_k)) , min (Pl_{W(a_i,b_j)} (u_k) , Pl_{M(a_i,b_j)} (u_k) ) ), Therefore, Bel_{I(a_i,b_j)} (u_k)= Bel_{J(a_i,b_j)} (u_k), Pl_{I(a_i,b_j)} (u_k)=Pl_{J(a_i,b_j)} (u_k) for ∀ (a_i, b_j) ∈ (C ∩ D) ∪ (D ∩ K), ∀u_k ∈ U.

Thus ((H, C) ∪ (W, D)) ∩ (M, K)=((H, C) ∩ (M, K)) ∪ ((W, D) ∩ (M, K))

(2)The proof is similar to that of (1).□

Theorem 3.5. For any two multi-granularity belief interval-valued soft sets over U, (H, C) and (W, C) then we have the following properties:

(1)((H, C) ∩ (W, C)) ^c=(H, C) ^c∪ (W, C) ^c ;

(2)((H, C) ∪ (W, C)) ^c=(H, C) ^c ∩ (W, C) ^c .

Proof. (1) Suppose (H, C) ∩ (W, C) = (M, C), where (M, C) ^c = (M^C, ⌝ C), for ∀ ¬ (a_i, b_j) ∈ ⌝ C, ∀u_k ∈ U, the belief measure is 1 - min (Pl_{H(a_i,b_j)} (u_k) , Pl_{W(a_i,b_j)} (u_k)) and the plausibility measure is 1 - min (Bel_{H(a_i,b_j)} (u_k) , Bel_{W(a_i,b_j)} (u_k)). When (H, C) ^c ∪ (W, C) ^c, ∀ ¬ (a_i, b_j) ∈ ⌝ C, ∀u_k ∈ U, the belief measure is max (1 - Pl_{H(a_i,b_j)} (u_k) , 1 - Pl_{W(a_i,b_j)} (u_k)) and the plausibility measure is max (1 - Bel_{H(a_i,b_j)} (u_k) , 1 - Bel_{W(a_i,b_j)} (u_k)). Since 1 - min (Pl_{H(a_i,b_j)} (u_k) , Pl_{W(a_i,b_j)} (u_k)) =max (1 - Pl_{H(a_i,b_j)} (u_k) , 1 - Pl_{W(a_i,b_j)} (u_k)), 1 - min (Bel_{H(a_i,b_j)} (u_k) , Bel_{W(a_i,b_j)} (u_k)) =max (1 - Bel_{H(a_i,b_j)} (u_k) , 1 - Bel_{W(a_i,b_j)} (u_k)) Thus ((H, C) ∩ (W, C)) ^c=(H, C) ^c ∪ (W, C) ^c. (2) The proof is similar to that of (1). □

4 Soft belief value and soft belief degree

In the paper of Vijayabalaji et al. [28], the belief interval was redefined based on Dymova et al. [4] and the definition of belief interval-valued soft set was introduced. Now, we present two new concepts to process belief interval based on Vijayabalaji et al. [28] on MGBIVSS.

Definition 4.1. Let U = {u_k : k = 1, 2, …, n} be the universe set and (F, A) and (G, B) be two belief interval-valued soft sets over a common universe U, where A = {a_i : i = 1, 2, …, l} and B = {b_j : j = 1, 2, …, m}. (H, C) is a multi-granularity belief interval-valued soft set(MGBIVSS) derived from (F, A) and (G, B). The soft belief value of u_k for (a_i, b_j), (k = 1, 2, …, n, i = 1, 2, …, l, j = 1, 2, …, m) on (H, C) is defined by

$\begin{matrix} {SBV}_{H (a_{i}, b_{j})} (u_{k}) = {Bel}_{H (a_{i}, b_{j})} (u_{k}) \\ + \frac{{Bel}_{H (a_{i}, b_{j})} (u_{k})}{{Bel}_{H (a_{i}, b_{j})} (u_{k}) + \sqrt[2]{1 - {Pl}_{H (a_{i}, b_{j})} (u_{k})}} \\ ({Pl}_{H (a_{i}, b_{j})} (u_{k}) - {Bel}_{H (a_{i}, b_{j})} (u_{k})) \end{matrix}$ the soft belief degree of u_k, k = 1, 2, …, n on (H, C) is defined by ${SBD}_{H} (u_{k}) = \frac{1}{m \times l} \sum_{i = 1}^{l} (\sum_{j = 1}^{m} {SBV}_{H (a_{i}, b_{j})} (u_{k}))$

(∀ (a_i, b_j) ∈ C, ∀ u_k ∈ U)

The following is an algorithm to solve decision making on MGBIVSS and an example is to illustrate it.

Algorithm 1:

Decision-making based on MGBIVSS

BEGIN

1.Input two belief interval-valued soft sets (F, A) and (G, B).

2.Obtain the multi-granularity belief interval-valued soft set

(H, C).

3.Calculate soft belief degree of u_k (k = 1, 2, …, n).

4.Rank the alternatives u_k (k = 1, 2, …, n) according to

SBD_H (u_k).

END

Example 4.1. Suppose a company recruits algorithm engineers, there are multiple candidates competing. After the first round of academic background screening, let U = {u₁, u₂, u₃, u₄, u₅} be a set of five candidates. The company wants to choose two most suitable candidates from two aspects which are working ability and personal information. Let A = {a₁, a₂, a₃} be the working ability set, a₁, a₂, a₃ describe “experience”, “computer knowledge”, “skilled foreign languages”, B = {b₁, b₂, b₃} be the personal information set, b₁, b₂, b₃ describe “young age”, “marriage status”, “good health”. There is a decision maker from the department of human resources, who considers the parameters above to evaluate the five candidates from working ability and personal information indicated by BIVSSs (F, A) and (G, B).

Step 1. Input BIVSSs of (F, A) and (G, B) represented by Tables 5.

Table 4

Table for belief interval-valued soft set (F, A)

	u ₁	u ₂	u ₃	u ₄	u ₅
a ₁	[0.8, 0.9]	[0.4, 0.8]	[0.7, 0.8]	[0.6, 0.8]	[0.5, 0.7]
a ₂	[0.6, 0.8]	[0.6, 0.9]	[0.4, 0.9]	[0.6, 0.9]	[0.5, 0.6]
a ₃	[0.7, 0.8]	[0.5, 0.6]	[0.6, 0.8]	[0.7, 0.9]	[0.6, 0.8]

Table 5

Table for belief interval-valued soft set (G, B)

	u ₁	u ₂	u ₃	u ₄	u ₅
b ₁	[0.7, 0.8]	[0.4, 0.6]	[0.6, 0.7]	[0.5, 0.7]	[0.4, 0.7]
b ₂	[0.5, 0.6]	[0.3, 0.7]	[0.6, 0.9]	[0.6, 0.8]	[0.7, 0.9]
b ₃	[0.6, 0.7]	[0.5, 0.8]	[0.5, 0.6]	[0.7, 0.8]	[0.5, 0.6]

Step 2. Obtain the MGBIVSS (H, C) represented by Table 6.

Table 6

Table for MGBIVSS (H, C)

	u ₁	u ₂	u ₃	u ₄	u ₅
(a₁, b₁)	[0.7, 0.9]	[0.4, 0.8]	[0.6, 0.8]	[0.5, 0.8]	[0.4, 0.7]
(a₁, b₂)	[0.5, 0.9]	[0.3, 0.8]	[0.6, 0.9]	[0.6, 0.8]	[0.5, 0.9]
(a₁, b₃)	[0.6, 0.9]	[0.4, 0.8]	[0.5, 0.8]	[0.6, 0.8]	[0.5, 0.7]
(a₂, b₁)	[0.6, 0.8]	[0.4, 0.9]	[0.4, 0.9]	[0.5, 0.9]	[0.4, 0.7]
(a₂, b₂)	[0.5, 0.8]	[0.3, 0.9]	[0.4, 0.9]	[0.6, 0.9]	[0.5, 0.9]
(a₂, b₃)	[0.6, 0.8]	[0.5, 0.9]	[0.4, 0.9]	[0.6, 0.9]	[0.5, 0.6]
(a₃, b₁)	[0.7, 0.8]	[0.4, 0.6]	[0.6, 0.8]	[0.5, 0.9]	[0.4, 0.8]
(a₃, b₂)	[0.5, 0.8]	[0.3, 0.7]	[0.6, 0.9]	[0.6, 0.9]	[0.6, 0.9]
(a₃, b₃)	[0.6, 0.8]	[0.5, 0.8]	[0.5, 0.8]	[0.7, 0.9]	[0.5, 0.8]

Step 3. Calculate soft belief degree of u_k (k = 1, 2, 3, 4, 5) in Table 7.

Table 7

Table for soft belief degree of u_k on MGBIVSS (H, C)

	u ₁	u ₂	u ₃	u ₄	u ₅
SBV _H(a₁,b₁)	0.8378	0.5889	0.7146	0.6584	0.5266
SBV _H(a₁,b₂)	0.7450	0.5007	0.7965	0.7146	0.7450
SBV _H(a₁,b₃)	0.7965	0.5889	0.6584	0.7146	0.5954
SBV _H(a₂,b₁)	0.7146	0.6792	0.6792	0.7450	0.5266
SBV _H(a₂,b₂)	0.6584	0.5921	0.6792	0.7965	0.7450
SBV _H(a₂,b₃)	0.7146	0.7450	0.6792	0.7965	0.5442
SBV _H(a₃,b₁)	0.7610	0.4775	0.7146	0.7450	0.5889
SBV _H(a₃,b₂)	0.6584	0.4416	0.7965	0.7965	0.7965
SBV _H(a₃,b₃)	0.7146	0.6584	0.6584	0.8378	0.6584
SBD _H	0.5809	0.5858	0.7085	0.7562	0.6363

Step 4. Rank the alternatives u_k (k = 1, 2, 3, 4, 5) according to SBD_H (u_k). From Table 7, we have decision u₄ > u₃ > u₅ > u₂ > u₁.

5 Comparison with existing work

We can also obtain soft belief value and soft belief degree based on belief interval-valued soft set as special case of MGBIVSS in single granularity space. That is to say, the algorithm above is equally applicable on belief interval-valued soft set. Next, we compare the presented approach with existing approach through a previous example to highlight the superiority of the new algorithm.

Example 5.1. Let U = {u₁, u₂, u₃, u₄, u₅, u₆} be the set of invalids, and E = {N, P, H} be the set of complementary and alternative medicine treatments; N = {n₁, n₂, n₃, n₄, n₅, n₆} indicates natural therapies projects, where n₁= no negative impact, n₂= non-poisonous, n₃= apply natural raw materials, n₄= enhance immunity, n₅= get the body able to cure for itself, and n₆= improve natural capacity; P = {p₁, p₂, p₃} is priority participation in healthy therapies projects, where p₁= equal companions, p₂= sufferers should be positive, and p₃= sufferers decide on their owns; H = {h₁, h₂, h₃, h₄} is for overall health programs, where h₁= coordinate your corporeity, heart and spirit, h₂= concentrate on people’holistic happiness, h₃= the body possesses an innate capability, and h₄= apply contemporary science and technique.

Step 1. Input the BIVSSs decision matrices M_k = (Y_k, E_k) (k = 1, 2, 3,), where E₁ = N, E₂ = P and E₃ = H. $\begin{matrix} M_{1} & = [\begin{matrix} [0.30, 0.70] & [0.40, 0.80] & [0.40, 0.70] & [0.50, 0.90] & [0.20, 0.60] & [0.20, 0.70] \\ [0.40, 0.70] & [0.40, 0.80] & [0.50, 0.80] & [0.40, 0.90] & [0.30, 0.70] & [0.50, 0.80] \\ [0.20, 0.50] & [0.30, 0.70] & [0.40, 0.80] & [0.60, 1.00] & [0.50, 0.70] & [0.40, 0.60] \\ [0.20, 0.60] & [0.30, 0.60] & [0.50, 0.90] & [0.60, 0.90] & [0.40, 0.50] & [0.30, 0.60] \\ [0.40, 0.60] & [0.40, 0.80] & [0.50, 0.80] & [0.50, 0.90] & [0.20, 0.50] & [0.30, 0.50] \\ [0.20, 0.50] & [0.40, 0.90] & [0.60, 0.80] & [0.70, 0.90] & [0.10, 0.50] & [0.20, 0.50] \end{matrix}] \\ M_{2} & = [\begin{matrix} [0.40, 0.70] & [0.30, 0.60] & [0.50, 0.70] & [0.20, 0.60] & [0.30, 0.50] & [0.40, 0.70] \\ [0.20, 0.50] & [0.40, 0.80] & [0.40, 0.60] & [0.50, 0.90] & [0.40, 0.70] & [0.30, 0.60] \\ [0.30, 0.60] & [0.40, 0.70] & [0.30, 0.60] & [0.60, 1.00] & [0.20, 0.50] & [0.30, 0.60] \end{matrix}] \\ M_{3} & = [\begin{matrix} [0.20, 0.50] & [0.10, 0.40] & [0.40, 0.70] & [0.50, 0.80] & [0.30, 0.60] & [0.20, 0.50] \\ [0.30, 0.70] & [0.50, 0.80] & [0.40, 0.50] & [0.60, 0.90] & [0.30, 0.70] & [0.40, 0.70] \\ [0.40, 0.60] & [0.40, 0.70] & [0.20, 0.60] & [0.40, 0.70] & [0.20, 0.40] & [0.50, 0.80] \\ [0.30, 0.60] & [0.70, 1.00] & [0.50, 0.70] & [0.30, 0.80] & [0.10, 0.50] & [0.40, 0.90] \end{matrix}] \end{matrix}$

Step 2. Calculate the soft belief value and soft belief degree according to the Definition 4.1 in the case of single granularity space, and they are expressed in Tables 8 –10.

Table 8
Table for soft belief degree of u_k on (Y₁, E₁)

u ₁ u ₂ u ₃ u ₄ u ₅ u ₅

n ₁ 0.4416 0.5889 0.5266 0.7450 0.2721 0.3337

n ₂ 0.5267 0.5889 0.6584 0.6793 0.4416 0.6584

n ₃ 0.2662 0.4416 0.5889 1.0000 0.5954 0.4775

n ₄ 0.2962 0.3965 0.7450 0.7965 0.4361 0.3965

n ₅ 0.4775 0.5889 0.6584 0.7450 0.2661 0.3596

n ₆ 0.2661 0.6793 0.7146 0.8378 0.1496 0.2661

SBD_N (u_k) 0.3791 0.5474 0.6487 0.8006 0.3602 0.4153

	u ₁	u ₂	u ₃	u ₄	u ₅	u ₅
n ₁	0.4416	0.5889	0.5266	0.7450	0.2721	0.3337
n ₂	0.5267	0.5889	0.6584	0.6793	0.4416	0.6584
n ₃	0.2662	0.4416	0.5889	1.0000	0.5954	0.4775
n ₄	0.2962	0.3965	0.7450	0.7965	0.4361	0.3965
n ₅	0.4775	0.5889	0.6584	0.7450	0.2661	0.3596
n ₆	0.2661	0.6793	0.7146	0.8378	0.1496	0.2661
SBD_N (u_k)	0.3791	0.5474	0.6487	0.8006	0.3602	0.4153

Table 9

Table for soft belief degree of u_k on (Y₂, E₂)

	u ₁	u ₂	u ₃	u ₄	u ₅	u ₅
p ₁	0.5266	0.3965	0.5954	0.2961	0.3596	0.5266
p ₂	0.2661	0.5889	0.4775	0.7450	0.5266	0.3965
p ₃	0.3965	0.5266	0.3965	1.0000	0.2661	0.3596
SBD_P (u_k)	0.3964	0.5054	0.4898	0.6804	0.3841	0.4276

Table 10

Table for soft belief degree of u_k on (Y₃, E₃)

	u ₁	u ₂	u ₃	u ₄	u ₅	u ₅
h ₁	0.2661	0.1343	0.5266	0.6584	0.3965	0.2661
h ₂	0.4416	0.6584	0.4361	0.7965	0.4416	0.5266
h ₃	0.4775	0.5266	0.2961	0.5266	0.2410	0.6584
h ₄	0.3965	1.0000	0.5954	0.5007	0.1496	0.6793
SBD_H (u_k)	0.3954	0.5798	0.4636	0.6206	0.3072	0.5326

Step 3. In this example of multi-parameters decision making, we can use the averages of soft belief degree of alternatives for all categories parameters to obtain the optimal decision. That is to say, calculate the averages of soft belief degree of u_k for three categories parameters expressed by SBD_M (u_k). ${SBD}_{M} (u_{1}) = \frac{{SBD}_{N} (u_{1}) + {SBD}_{P} (u_{1}) + {SBD}_{H} (u_{1})}{3} \approx 0.3905$ , so similarly, SBD_M (u₂) ≈ 0.5437, SBD_M (u₃) ≈ 0.5340, SBD_M (u₄) ≈ 0.7005, SBD_M (u₅) ≈ 0.3505, SBD_M (u₆) ≈ 0.4585. Then, the ranking of the alternatives is shown as following. SBD_M (u₅) < SBD_M (u₁) < SBD_M (u₆) < SBD_M (u₃) < SBD_M (u₂) < SBD_M (u₄), so u₄ is the optimal choice. The comparison of our approach and the one of Vijayabalaji et al. [28] is showed in Table 11.

Table 11

Table for comparison of two methods

Method	sequence	optimal choice
Vijayabalaji et al.	u₄ < u₂ < u₃ < u₆ < u₁ < u₅	u ₄
our approach	u₅ < u₁ < u₆ < u₃ < u₂ < u₄	u ₄

Compared with the method of Vijayabalaji et al. [28], our approach is easier to calculate and understand. It is obvious that our method can compare any alternatives for each parameter or several parameters according to soft belief value and soft belief degree while the method in Vijayabalaji et al. [28] cann’t get the comparison of separate parameter. Thus, our approach has a wide range of applications and flexibility.

6 Conclusion and future work

We have introduced MGBIVSS to get more information and less destructive structures than simply combining two or more belief interval-valued soft sets. Some basic operations are defined and some properties are proved on MGBIVSS and several examples are provided to illustrate them. We have proposed two concepts of soft belief value and soft belief degree on MGBIVSS which are more easier to compare horizontally and vertically among the different objects and different parameters and more widely available than before. Then an algorithm is presented to deal with decision-making problems by using the concepts and the corresponding example is provided to explain the new algorithm. Finally, a comparison between our approach and existing work is expressed to explain the advantages.

This study contributes to analyze and describe the problems with regard to MGBIVSS. This method also provides new perspectives for research and applications of BIVSS in many areas containing uncertain data in the real-world situation. Next the reduction method of MGBIVSS can be developed according to the soft belief value and soft belief degree in the future work.

References

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Multi-granularity belief interval-valued soft set

Abstract

Keywords

1 Introduction

2 Preliminaries

2.1 Intuitionistic fuzzy set

2.2 Intuitionistic fuzzy soft set

2.3 Belief interval-valued soft set

3 Multi-granularity belief interval-valued soft set

Table 1 Table for belief interval-valued soft set (F, A) u 1 u 2 u 3 u 4 u 5 a 1 [0.3, 0.5] [0.6, 0.7] [0.6, 0.7] [0.5, 0.8] [0.3, 0.7] a 2 [0.6, 0.8] [0.1, 0.3] [0.7, 0.9] [0.8, 0.9] [0.4, 0.9]

References

Table 1
Table for belief interval-valued soft set (F, A)

u ₁ u ₂ u ₃ u ₄ u ₅

a ₁ [0.3, 0.5] [0.6, 0.7] [0.6, 0.7] [0.5, 0.8] [0.3, 0.7]

a ₂ [0.6, 0.8] [0.1, 0.3] [0.7, 0.9] [0.8, 0.9] [0.4, 0.9]