Vague parameterized vague soft set theory and its decision making

Abstract

In this paper, we first define vague parameterized vague soft sets (vpvs-sets) and study some of their properties. We then introduce vpvs-aggregation operator to form vpvs-decision making method that allows constructing more efficient decision processes. Finally, we give a numerical example to show the method working successfully for problems containing uncertain data.

Keywords

vague set soft set vague soft set vague parameterized vague soft set decision making

1 Introduction

Researchers in economics, engineering, environment science, the social science, medical science, business, management, and many other fields deal daily with the complexities of modeling uncertain data. Classical methods are not always successful, because the uncertainties appearing in these domains may be of various types. Probability theory, fuzzy set theory [1], intuitionistic fuzzy set theory [2], vague set theory [3], interval mathematics [4], and other mathematical tools are well know and often useful approaches to describing uncertainty. However, all of these theories have their own difficulties which have been pointed out in [5]. Molodtsov suggested that one reason for these difficulties may be due to the inadequacy of the parametrization tools of these theories. To overcome these difficulties, Molodtsov [5] introduced the concept of soft sets as a new mathematical tool for dealing with uncertainties that is free from the difficulties that have troubled the usual theoretical approaches. Since then, many researches have investigated soft sets and have established some significant conclusions. For example, Jun and Park [6] proposed the notion of soft ideals and idealistic soft BCK/BCI-algebras, and constructed several examples. Ali et al. [7] corrected some errors in earlier studies and proposed some new operations on soft sets. Çağman and Enginoglu [8] redefined the operations of soft sets and constructed a uni-int decision making method which selected a set of optimum elements from the alternatives, they also defined soft matrices, a matrix representation of soft sets, and constructed a soft max-min decision making method [9]. Jiang et al. [10] proposed a novel approach to semantic decision making by using ontology-based soft sets and ontology reasoning. Qin and Hong [11] introduced the concept of soft equality and some related properties were derived, some equivalent conditions for soft sets being equal were also given. Herawan and Deris [12] presented an alternative approach for mining regular association rules and maximal association rules from transactional data sets using soft set theory.

It is worth noting that all of above works are built on the classical soft set theory. The generalizations of soft sets to environments in which uncertainty is a factor have become a rapidly progressing research area receiving much attention in recent years. Maji et al. [13] first introduced the concept of fuzzy soft sets by combining fuzzy sets and soft sets. Majumdar and Samanta [14] further generalized the concept of fuzzy soft sets and some of their properties were studied, and relations on generalized fuzzy soft sets were also discussed by them. Yang et al. [15] introduced the concept of interval-valued fuzzy soft set, which is a combination of interval-valued fuzzy sets and soft sets. Xiao et al. [16] introduced the notion of exclusive disjunctive soft sets and gave an application of these new sets. Maji et al. [17, 18] initiated the notion of intuitionistic fuzzy soft sets by integrating the intuitionistic fuzzy sets with soft sets. By combing the vague set and the soft set, Xu et al. [19] introduced the notion of vague soft sets, derived its basic properties and illustrated its potential applications. Jiang et al. [20] constructed a new soft set model called interval-valued intuitionistic fuzzy soft sets by integrating the interval-valued intuitionistic fuzzy sets and soft sets. Some interesting applications of these sets can be found in [21, 22]. By combing the related works and the soft sets from parametrization point of view, fuzzy parameterized soft set theory [23], fuzzy parameterized fuzzy soft set theory [24], fuzzy parameterized interval-valued fuzzy soft set theory [25], intuitionistic fuzzy parameterized soft set theory [26], interval valued intuitionistic fuzzy parameterized soft set theory [27], multi Q-fuzzy parameterized soft set theory [28] and the decision making methods based on these theories have also been introduced by some scholars.

By definition, a soft set is a parameterized family of subsets of the universal set. In other words, a soft set is a mapping from a set of parameters to the power set of an initial universe set. In the real world, the difficulty is that the objects in the universal set may not precisely satisfy the problems parameters, which usually represent some attributes, characteristics, or properties of the objects in the universal set. The concept of fuzzy soft sets proposed in [13] partially resolves this difficulty, but falls short in dealing with additional complexity - that is, the mapping may be too vague. It is, therefore, desirable to extend soft set theory and fuzzy soft set theory using the concept of vague set theory. Vague set theory is actually an extension of fuzzy set theory and vague sets are regards as a special case of context-dependent fuzzy sets. The basic concepts of vague set theory and its extensions can be found in [29 –33]. Vague soft set theory makes descriptions of the object world more realistic, practical and accurate, at least in some cases, making it a very promising tool. Since vague sets are equivalent to intuitionistic fuzzy sets [34], so vague soft sets are equivalent to intuitionistic fuzzy soft sets. Some scholars have studied intuitionistic fuzzy soft sets from different aspects. For example, Gunduz and Bayramov [35] introduced the concept of an intuitionistic fuzzy soft module and some operations on intuitionistic fuzzy soft sets were given, they also studied some of its basic properties. Jiang et al. [36] presented an adjustable approach to intuitionistic fuzzy soft sets based decision making by using level soft sets of intuitionistic fuzzy soft sets and gave some illustrative examples, the weighted intuitionistic fuzzy soft sets were introduced and its application to decision making was investigated. Zhang [37] proposed a novel approach to intuitionistic fuzzy soft set based decision making problems using rough set theory. Wang and Qu [38] introduced the definitions of entropy, similarity measure and distance measure of vague soft sets, the relations between these measures were discussed in detail. The extensions and some interesting applications of vague soft sets can be found in [39 –41]. However, there has been rather little work completed for vague parameterized vague soft set theory. The purpose of this paper is to further extend the concept of vague soft set theory proposed by Xu et al. in [19]. In this paper, we will present the definition of vague parameterized vague soft set and introduce a decision making method based on vpvs-sets. An example is provided illustrates the effectiveness of the method which is more practical.

The rest of this paper is organized as follows. Section 2 recalls some basic concepts of vague sets, soft sets and vague soft sets et al. In Section 3, we introduce the definition of vague parameterized vague soft sets and study some of their properties. In Section 4, we define vpvs-aggregation operator to form vpvs-decision making method and give an example which shows that the method can be successfully applied to problems that contain uncertainties. Concluding remarks and open questions for further investigation are provided in Section 5.

2 Preliminaries

In this section, we will recall several definitions and results which are necessary for our paper. They are stated as follows:

Definition 2.1. [3] A vague set X in the universe U = {u₁, u₂, . . . , u_n} can be expressed by the following notion, X = {(u_i, [t_X (u_i) , 1 - f_X (u_i)]) |u_i ∈ U}, i.e X (u_i) = [t_X (u_i) , 1 - f_X (u_i)] and the condition 0 ≤ t_X (u_i) ≤1 - f_X (u_i) should hold for any u_i ∈ U, where t_X (u_i) is called the membership degree (true membership) of element u_i to the vague set X, while f_X (u_i) is the degree of nonmembership (false membership) of the element u_i to the vague set X.

Definition 2.2. [3] Let A, B be two vague sets in the universe U = {u₁, u₂, . . . , u_n}, then the union, intersection and complement of vague sets are defined as follows: $\begin{matrix} A \cup B & = & {(u_{i}, [\max (t_{A} (u_{i}), t_{B} (u_{i})), \\ \max (1 - f_{A} (u_{i}), 1 - f_{B} (u_{i}))]) | u_{i} \in U}, \\ A \cap B & = & {(u_{i}, [\min (t_{A} (u_{i}), t_{B} (u_{i})), \\ \min (1 - f_{A} (u_{i}), 1 - f_{B} (u_{i}))]) | u_{i} \in U}, \\ A^{c} & = & {(u_{i}, [f_{A} (u_{i}), 1 - t_{A} (u_{i})]) | u_{i} \in U} . \end{matrix}$

Definition 2.3. [3] Let A, B be two vague sets in the universe U = {u₁, u₂, . . . , u_n}. If ∀u_i ∈ U, t_A (u_i) ≤ t_B (u_i) , 1 - f_A (u_i) ≤1 - f_B (u_i), then A is called a vague subset of B, denoted by A ⊆ B, where i = 1, 2, 3, . . . , n.

Definition 2.4. [5] Let U be an initial universe set, P (U) be the power set of U, E be the set of all parameters and A ⊆ E. Then, a soft set F_A over U is a set defined by a function f_A representing a mapping f_A : E → P (U) such that f_A (x) =∅ if x ∉ A . Here, f_A is called approximate function of the soft set F_A and the value f_A (x) is a set called x-element of the soft set for all x ∈ E. It is worth noting that the sets f_A (x) may be arbitrary. Some of them may be empty, some may have nonempty intersection. Thus, a soft set F_A over U can be represented by the set of ordered pairs $F_{A} = {(x, f_{A} (x)), x \in E, f_{A} (x) \in P (U)} .$

Note that, the set of all soft sets over U will be denoted by S (U).

Definition 2.5. [19] Let U be an initial universe set, V (U) be the set of all vague sets over U, E be the set of all parameters and A ⊆ E. Then, a vague soft set (VS-set) Γ_A over U is a set defined by a function γ_A representing a mapping γ_A : E → V (U) such that γ_A (x) =∅ if x ∉ A . Thus, a soft set Γ_A over U can be represented by the set of ordered pairs $Γ_{A} = {(x, γ_{A} (x)), x \in E, γ_{A} (x) \in V (U)} .$

The value γ_A (x) is a vague set over U. That is $γ_{A} (x) = {(u, [t_{A (x)} (u), 1 - f_{A (x)} (u)]), x \in E, u \in U}$ where t_A(x) (u) and f_A(x) (u) are the membership and non-membership degrees of u to the parameter x, respectively. Note that, the set of all vague soft sets over U is denoted by VS (U).

Definition 2.6. [19] Let Γ_A and Γ_B be two vague soft sets over a universe U. If A ⊆ B and ∀x ∈ A, γ_A (x) is a vague subset of γ_B (x), then Γ_A is called a vague soft subset of Γ_B. This relation is denoted by $Γ_{A} \tilde{\subseteq} Γ_{B}$ .

Definition 2.7. [19] Two vague soft sets Γ_A and Γ_B over a universe U are said to be vague soft equal, if Γ_A is a vague soft subset of Γ_B and Γ_B is a vague soft subset of Γ_A. This relation is denoted by Γ_A = Γ_B.

Definition 2.8. [19] A vague soft set Γ_A over U is said to be a null vague soft set denoted by Γ_∅, if γ_A (x) =∅ for all x ∈ E, that is γ_A (x) = {(u, [0, 0]) , x ∈ E, u ∈ U}.

Definition 2.9. [19] A vague soft set Γ_A over U is said to be a A-universal vague soft set denoted by $Γ_{\tilde{A}}$ , if γ_A (x) = {(u, [1, 1]) , x ∈ E, u ∈ U}.

If A = E, then the A-universal vague soft set is called universal vague soft set and denoted by $Γ_{\tilde{E}}$ .

Definition 2.10. [19] Let E = {e₁, e₂, . . . , e_n} be a parameter set. The not set of E denoted by ¬E is defined by ¬E = {¬ e₁, ¬ e₂, . . . , ¬ e_n}, where ¬e_i = not e_i.

Definition 2.11. [19] The complement of vague soft set Γ_A is denoted by $Γ_{A}^{c}$ and is defined by $Γ_{A}^{c} = {(x, γ_{A^{c}} (x)), x \in E}$ , where $γ_{A^{c}} (x) = γ_{A}^{c} (x)$ is the complement of vague set γ_A (x), defined by $γ_{A}^{c} (x) = {(u, [f_{A (x)} (u), 1 - t_{A (x)} (u)]), x \in E, u \in U} .$

Clearly $(Γ_{A}^{c})^{c} = Γ_{A}$ .

Definition 2.12. [19] Let Γ_A and Γ_B be two vague soft sets over a universe U, the union of two Γ_A and Γ_B, denoted by $Γ_{A} \tilde{\cup} Γ_{B}$ , and is defined by $Γ_{A} \tilde{\cup} Γ_{B} = {(x, γ_{A} (x) \cup γ_{B} (x)), x \in E}$ where γ_A (x) ∪ γ_B (x) = {(u, max (t_A(x) (u) , t_B(x) (u)) , max (1 - f_A(x) (u) , 1 - f_B(x) (u)) , x ∈ E, u ∈ U} .

Definition 2.13. [19] Let Γ_A and Γ_B be two vague soft sets over a universe U, the intersection of two Γ_A and Γ_B, denoted by $Γ_{A} \tilde{\cap} Γ_{B}$ , and is defined by $Γ_{A} \tilde{\cap} Γ_{B} = {(x, γ_{A} (x) \cap γ_{B} (x)), x \in E}$ where γ_A (x) ∩ γ_B (x) = {(u, min (t_A(x) (u) , t_B(x) (u)) , min (1 - f_A(x) (u) , 1 - f_B(x) (u)) , x ∈ E, u ∈ U} .

Definition 2.14. [23] Let U be an initial universe, P (U) be the power set of U, E be a set of all parameters and X be a fuzzy set over E. Then a FP-soft set (f_X, E) on the universe U is defined as follows: $(f_{X}, E) = {(u_{X} (x) / x, f_{X} (x)), x \in E}$ where u_X : E → [0, 1] and f_X : E → P (U) such that f_X (x) =∅ if u_X (x) =0.

Here f_X called approximate function and u_X called membership function of FP-soft sets.

Definition 2.15. [24] Let U be an initial universe, F (U) be the set of all fuzzy sets over U, E be a set of all parameters and X be a fuzzy set over E with the membership function u_X : E → [0, 1] and γ_X (x) be a fuzzy set over U for all x ∈ E. Then, a fpfs-set Γ_X over U is a set defined by a function γ_X representing a mapping γ_X : E → F (U) such that γ_X (x) =∅ if u_X (x) =0.

Here, γ_X is called fuzzy approximate function of the fpfs-set for all x ∈ E. Thus, a fpfs-set Γ_X over U can be represented by the set of ordered pairs $\begin{matrix} Γ_{X} & = & {(u_{X} (x) / x, γ_{X} (x)), x \in E, \\ γ_{X} (x) \in F (U), u_{X} (x) \in [0, 1]} . \end{matrix}$

It must be noted that the sets of all fpfs-sets over U will be denoted by FPFS (U).

Definition 2.16. [26] Let U be an initial universe, P (U) be the power set of U, E be a set of all parameters and X be an intuitionistic fuzzy set over E. An intuitionistic FP-soft sets Γ_X over U is defined as follows: $Γ_{X} = {((x, α_{X} (x), β_{X} (x)), f_{X} (x)), x \in E}$ where α_X : E → [0, 1] , β_X : E → [0, 1] and f_X : E → P (U) with the property f_X (x) =∅ if α_X (x) =0, β_X (x) =1.

Here the function α_X and β_X called membership function and non-membership of intuitionistic FP-soft set, respectively. The value α_X (x) and β_X (x) is the degree of importance and unimportance of the parameter x.

3 Vague parameterized vague soft sets

In this section, we define vague parameterized vague soft sets (vpvs-sets)and their operations.

Definition 3.1. Let U be an initial universe, E be the set of all parameters and X be a vague set over E with membership function t_X : E → [0, 1] and nonmembership function f_X : E → [0, 1], let η_X (x) be a vague set over U for all x ∈ E. Then, a vague parameterized vague soft set Ψ_X over U is a set defined by a function η_X representing a mapping $η_{X} : E \to V (U)$ such that η_X (x) =∅ if t_X (x) =0, f_X (x) =1. Here, η_X is called vague approximate function of the vpvs-set Ψ_X, and η_X (x) is a vague set called x-element of the vpvs-set for all x ∈ E. Thus, a vpvs-set Ψ_X over U can be represented by the set of ordered pairs $\begin{matrix} Ψ_{X} & = & {(\frac{[t_{X} (x), 1 - f_{X} (x)]}{x}, η_{X} (x)) : \\ x \in E, η_{X} (x) \in V (U)} . \end{matrix}$

It must be noted that the set of all vpvs-sets over U will be denoted by VPVS (U).

Definition 3.2. Let Ψ_X ∈ VPVS (U). If η_X (x) =∅ for all x ∈ E, then Ψ_X is called a X-empty vpvs-set, denoted by Ψ_{∅_X}. If X =∅, then the X-empty vpvs-set Ψ_{∅_X} is called an empty vpvs-set, denoted by Ψ_∅.

Definition 3.3. Let Ψ_X ∈ VPVS (U). If t_X (x) =1, f_X (x) =0 and η_X (x) = U for all x ∈ E, then Ψ_X is called a X-universal vpvs-set, denoted by $Ψ_{\tilde{X}}$ . If X = E, then the X-universal vpvs-set $Ψ_{\tilde{X}}$ is called a universal vpvs-set, denoted by $Ψ_{\tilde{E}}$ .

Example 3.1. Assume that U = {u₁, u₂, u₃, u₄} is a universe set and E = {x₁, x₂, x₃} is a set of parameters. If $X = {\frac{[0.2, 0.6]}{x_{1}}, \frac{[0.5, 0.5]}{x_{2}}, \frac{[1, 1]}{x_{3}}}$ and η_X (x) is defined as follows: $\begin{matrix} η_{X} (x_{1}) & = & {\frac{[0.2, 0.7]}{u_{1}}, \frac{[0.6, 0.9]}{u_{2}}, \frac{[0.1, 0.6]}{u_{4}}}, \\ η_{X} (x_{2}) = \emptyset, η_{X} (x_{3}) = U, \end{matrix}$ then a vpvs-set Ψ_X is written by $Ψ_{X} = {(\frac{[0.2, 0.6]}{x_{1}}, {\frac{[0.2, 0.7]}{u_{1}}, \frac{[0.6, 0.9]}{u_{2}}, \frac{[0.1, 0.6]}{u_{4}}}), (\frac{[0.5, 0.5]}{x_{2}}, \emptyset), (\frac{[1, 1]}{x_{3}}, U)} .$

If $Y = {\frac{[0.1, 0.8]}{x_{1}}, \frac{[0.2, 0.6]}{x_{3}}}$ and η_Y (x₁) =∅ , η_Y (x₃) = ∅, then the vpvs-set Ψ_Y is a Y-empty vague parameterized vague soft set, i.e. Ψ_Y = Ψ_{∅_Y}.

If $L = {\frac{[0, 0]}{x_{1}}, \frac{[0, 0]}{x_{2}}, \frac{[0, 0]}{x_{3}}}$ , then the vpvs-set Ψ_L is an empty vague parameterized vague soft set.

If $Z = {\frac{[1, 1]}{x_{2}}, \frac{[1, 1]}{x_{3}}}$ and η_Z (x₂) = U, η_Z (x₃) = U, then the vpvs-set Ψ_Z is a Z-universal vpvs-set, i.e. $Ψ_{Z} = Ψ_{\tilde{Z}}$ .

If $M = {\frac{[1, 1]}{x_{1}}, \frac{[1, 1]}{x_{2}}, \frac{[1, 1]}{x_{3}}}$ and η_M (x₁) = U, η_M (x₂) = U, η_M (x₃) = U, then the vpvs-set Ψ_M is a universal vague parameterized vague soft set, i.e. $Ψ_{M} = Ψ_{\tilde{E}}$ .

Definition 3.4. Let Ψ_X, Ψ_Y ∈ VPVS (U). Then Ψ_X is a vague parameterized vague soft subset of Ψ_Y, denoted by $Ψ_{X} \tilde{\subseteq} Ψ_{Y}$ , if and only if t_X (x) ≤ t_Y (x) , f_X (x) ≥ f_Y (x) and η_X (x) ⊆ η_Y (x) for all x ∈ E.

Remark 3.1. $Ψ_{X} \tilde{\subseteq} Ψ_{Y}$ does not imply that every element of Ψ_X is an element of Ψ_Y as in the definition of classical subset.

Example 3.2. Assume that U = {u₁, u₂, u₃, u₄} is a universal set of objects and E = {x₁, x₂, x₃} is a set of all parameters. If $X = {\frac{[0.2, 0.6]}{x_{1}}}$ and $Y = {\frac{[0.5, 0.7]}{x_{1}}, \frac{[0.4, 0.8]}{x_{3}}}$ , and $\begin{matrix} Ψ_{X} & = & {(\frac{[0.2, 0.6]}{x_{1}}, {\frac{[0.3, 0.7]}{u_{2}}, \frac{[0.4, 0.8]}{u_{4}}})}, \\ Ψ_{Y} & = & {(\frac{[0.5, 0.7]}{x_{1}}, {\frac{[0.2, 0.6]}{u_{1}}, \frac{[0.4, 0.9]}{u_{2}}, \frac{[0.5, 0.9]}{u_{4}}}), (\frac{[0.4, 0.8]}{x_{3}}, {\frac{[0.4, 0.5]}{u_{1}}, \frac{[0.2, 0.4]}{u_{3}}})}, \end{matrix}$ then for all x ∈ E, t_X (x) ≤ t_Y (x) , f_X (x) ≥ f_Y (x) and η_X (x) ⊆ η_Y (x) is valid. Hence $Ψ_{X} \tilde{\subseteq} Ψ_{Y}$ . It is clear that $(\frac{[0.2, 0.6]}{x_{1}}, {\frac{[0.3, 0.7]}{u_{2}}, \frac{[0.4, 0.8]}{u_{4}}}) \in Ψ_{X}$ but $(\frac{[0.2, 0.6]}{x_{1}}, {\frac{[0.3, 0.7]}{u_{2}}, \frac{[0.4, 0.8]}{u_{4}}}) \notin Ψ_{Y}$ .

Proposition 3.1. Let Ψ_X, Ψ_Y, Ψ_Z ∈ VPVS (U). Then,

$Ψ_{X} \tilde{\subseteq} Ψ_{\tilde{E}}$

$Ψ_{\emptyset_{X}} \tilde{\subseteq} Ψ_{X}$

$Ψ_{\emptyset} \tilde{\subseteq} Ψ_{X}$

$Ψ_{X} \tilde{\subseteq} Ψ_{X}$

$Ψ_{X} \tilde{\subseteq} Ψ_{Y}$ and $Ψ_{Y} \tilde{\subseteq} Ψ_{Z} \Rightarrow Ψ_{X} \tilde{\subseteq} Ψ_{Z}$

Proof. The above properties of $\tilde{\subseteq}$ trivally follow from the above definitions. □

Definition 3.5. Let Ψ_X, Ψ_Y ∈ VPVS (U). Then Ψ_X and Ψ_Y are vague parameterized vague soft equal, written by Ψ_X = Ψ_Y, if and only if t_X (x) = t_Y (x) , f_X (x) = f_Y (x) and η_X (x) = η_Y (x) for all x ∈ E.

Proposition 3.2. Let Ψ_X, Ψ_Y, Ψ_Z ∈ VPVS (U). Then,

Ψ_X = Ψ_YandΨ_Y = Ψ_Z ⇒ Ψ_X = Ψ_Z

$Ψ_{X} \tilde{\subseteq} Ψ_{Y}$ and $Ψ_{Y} \tilde{\subseteq} Ψ_{X} \Rightarrow Ψ_{X} = Ψ_{Y}$

Proof. The above properties of = and $\tilde{\subseteq}$ trivially follow from the above Definitions 3.4 and 3.5. □

Definition 3.6. Let Ψ_X ∈ VPVS (U). Then complement of Ψ_X, denoted by $Ψ_{X}^{c}$ , is a vague parameterized vague soft set defined by $\begin{matrix} Ψ_{X}^{c} = {(\frac{[f_{X} (x), 1 - t_{X} (x)]}{x}, η_{X}^{c} (x)) : x \in E} \end{matrix}$ where $η_{X}^{c} (x)$ is complement of the vague set η_X (x), that is $η_{X}^{c} (x) = η_{X^{c}} (x)$ for every x ∈ E.

Proposition 3.3. Let Ψ_X ∈ VPVS (U). Then,

$Ψ_{\tilde{E}}^{c} = Ψ_{\emptyset}$

$(Ψ_{X}^{c})^{c} = Ψ_{X}$

$Ψ_{\emptyset}^{c} = Ψ_{\tilde{E}}$

Proof. Let $Ψ_{\tilde{E}} = {(\frac{[1, 1]}{x}, U), \forall x \in E}$ .

Then, from Definition 3.6, we have: $Ψ_{\tilde{E}}^{c} = {(\frac{[0, 0]}{x}, \emptyset), \forall x \in E} = Ψ_{\emptyset} .$

Similarity (2) and (3) easily can be made. □

Definition 3.7. Let Ψ_X, Ψ_Y ∈ VPVS (U). Then union of Ψ_X and Ψ_Y, denoted by $Ψ_{X} \tilde{\cup} Ψ_{Y}$ , is a vpvs-set defined by $\begin{matrix} Ψ_{X} \tilde{\cup} Ψ_{Y} \\ = {\frac{[\max (t_{X} (x), t_{Y} (x)), \max (1 - f_{X} (x), 1 - f_{Y} (x))]}{x}, \\ η_{X \tilde{\cup} Y} (x) : x \in E} \end{matrix}$ where $η_{X \tilde{\cup} Y} (x) = η_{X} (x) \cup η_{Y} (x)$ .

Proposition 3.4. Let Ψ_X, Ψ_Y, Ψ_Z ∈ VPVS (U). Then,

$Ψ_{X} \tilde{\cup} Ψ_{X} = Ψ_{X}$

$Ψ_{X} \tilde{\cup} Ψ_{\emptyset} = Ψ_{X}$

$Ψ_{X} \tilde{\cup} Ψ_{\tilde{E}} = Ψ_{\tilde{E}}$

$Ψ_{X} \tilde{\cup} Ψ_{Y} = Ψ_{Y} \tilde{\cup} Ψ_{X}$

$(Ψ_{X} \tilde{\cup} Ψ_{Y}) \tilde{\cup} Ψ_{Z} = Ψ_{X} \tilde{\cup} (Ψ_{Y} \tilde{\cup} Ψ_{Z})$

Proof. The proofs can be easily obtained from Definition 3.7. □

Definition 3.8. Let Ψ_X, Ψ_Y ∈ VPVS (U). Then intersection of Ψ_X and Ψ_Y, denoted by $Ψ_{X} \tilde{\cap} Ψ_{Y}$ , is a vpvs-set defined by $\begin{matrix} Ψ_{X} \tilde{\cap} Ψ_{Y} \\ = {\frac{[\min (t_{X} (x), t_{Y} (x)), \min (1 - f_{X} (x), 1 - f_{Y} (x))]}{x}, \\ η_{X \tilde{\cap} Y} (x) : x \in E} \end{matrix}$ where $η_{X \tilde{\cap} Y} (x) = η_{X} (x) \cap η_{Y} (x)$ .

Proposition 3.5. Let Ψ_X, Ψ_Y, Ψ_Z ∈ VPVS (U). Then,

$Ψ_{X} \tilde{\cap} Ψ_{X} = Ψ_{X}$

$Ψ_{X} \tilde{\cap} Ψ_{\emptyset} = Ψ_{\emptyset}$

$Ψ_{X} \tilde{\cap} Ψ_{\tilde{E}} = Ψ_{X}$

$Ψ_{X} \tilde{\cap} Ψ_{Y} = Ψ_{Y} \tilde{\cap} Ψ_{X}$

$(Ψ_{X} \tilde{\cap} Ψ_{Y}) \tilde{\cap} Ψ_{Z} = Ψ_{X} \tilde{\cap} (Ψ_{Y} \tilde{\cap} Ψ_{Z})$

Proof. The proofs can be easily obtained from Definition 3.8. □

Remark 3.2. Let Ψ_X ∈ VPVS (U), If Ψ_X ≠ Ψ_∅ or $Ψ_{X} \neq Ψ_{\tilde{E}}$ , then $Ψ_{X} \tilde{\cup} Ψ_{X}^{c} \neq Ψ_{\tilde{E}}$ and $Ψ_{X} \tilde{\cap} Ψ_{X}^{c} \neq Ψ_{\emptyset}$ .

Example 3.3. Assume that U = {u₁, u₂, u₃, u₄} is a universal set of objects and E = {x₁, x₂} is a set of all parameters. If $X = {\frac{[0.2, 0.6]}{x_{1}}, \frac{[0.5, 0.5]}{x_{2}}}$ , and

$Ψ_{X} = {(\frac{[0.2, 0.6]}{x_{1}}, {\frac{[0.3, 0.6]}{u_{1}}, \frac{[0.4, 0.5]}{u_{2}}, \frac{[0.6, 0.8]}{u_{4}}}), (\frac{[0.5, 0.5]}{x_{2}}, {\frac{[0.3, 0.5]}{u_{1}}, \frac{[0.5, 0.8]}{u_{3}}})},$ then $X^{c} = {\frac{[0.4, 0.8]}{x_{1}}, \frac{[0.5, 0.5]}{x_{2}}}$ , and $Ψ_{X}^{c} = {(\frac{[0.4, 0.8]}{x_{1}}, {\frac{[0.4, 0.7]}{u_{1}}, \frac{[0.5, 0.6]}{u_{2}}, \frac{[0.2, 0.4]}{u_{4}}}), (\frac{[0.5, 0.5]}{x_{2}}, {\frac{[0.5, 0.7]}{u_{1}}, \frac{[0.2, 0.5]}{u_{3}}})},$

Therefore, $Ψ_{X} \tilde{\cup} Ψ_{X}^{c} = {(\frac{[0.4, 0.8]}{x_{1}}, {\frac{[0.4, 0.7]}{u_{1}}, \frac{[0.5, 0.6]}{u_{2}}, \frac{[0.6, 0.8]}{u_{4}}}), (\frac{[0.5, 0.5]}{x_{2}}, {\frac{[0.5, 0.7]}{u_{1}}, \frac{[0.5, 0.8]}{u_{3}}})} \neq Ψ_{\tilde{E}},$

and $Ψ_{X} \tilde{\cap} Ψ_{X}^{c} = {(\frac{[0.2, 0.6]}{x_{1}}, {\frac{[0.3, 0.6]}{u_{1}}, \frac{[0.4, 0.5]}{u_{2}}, \frac{[0.2, 0.4]}{u_{4}}}), (\frac{[0.5, 0.5]}{x_{2}}, {\frac{[0.3, 0.5]}{u_{1}}, \frac{[0.2, 0.5]}{u_{3}}})} \neq Ψ_{\emptyset} .$

Proposition 3.6. Let Ψ_X, Ψ_Y ∈ VPVS (U). Then, the following De Morgan’s types of results are true:

$(Ψ_{X} \tilde{\cup} Ψ_{Y})^{c} = Ψ_{X}^{c} \tilde{\cap} Ψ_{Y}^{c}$

$(Ψ_{X} \tilde{\cap} Ψ_{Y})^{c} = Ψ_{X}^{c} \tilde{\cup} Ψ_{Y}^{c}$

Proof. (1) For all x ∈ E, $\begin{matrix} η_{(X \tilde{\cup} Y)^{c}} (x) & = & η_{X \tilde{\cup} Y}^{c} (x) = (η_{X} (x) \cup η_{Y} (x))^{c} \\ = & (η_{X} (x))^{c} \cap (η_{Y} (x))^{c} = η_{X}^{c} (x) \cap η_{Y}^{c} (x) \\ = & η_{X^{c}} (x) \cap η_{Y^{c}} (x) = η_{X^{c} \tilde{\cap} Y^{c}} (x) . \end{matrix}$ and $\begin{matrix} Ψ_{X} \tilde{\cup} Ψ_{Y} & = & {\frac{[\max (t_{X} (x), t_{Y} (x)), \max (1 - f_{X} (x), 1 - f_{Y} (x))]}{x}, η_{X \tilde{\cup} Y} (x) : x \in E} \\ = & {\frac{[\max (t_{X} (x), t_{Y} (x)), 1 - \min (f_{X} (x), f_{Y} (x))]}{x}, η_{X \tilde{\cup} Y} (x) : x \in E} \end{matrix}$

$\begin{matrix} (Ψ_{X} \tilde{\cup} Ψ_{Y})^{c} & = & {\frac{[\min (f_{X} (x), f_{Y} (x)), 1 - \max (t_{X} (x), t_{Y} (x))]}{x}, η_{(X \tilde{\cup} Y)^{c}} (x) : x \in E} \\ = & {\frac{[\min (f_{X} (x), f_{Y} (x)), \min (1 - t_{X} (x), 1 - t_{Y} (x))]}{x}, η_{X^{c} \tilde{\cap} Y^{c}} (x) : x \in E} \\ = & {\frac{[(f_{X} (x), 1 - t_{X} (x)]}{x}, η_{X^{c}} (x) : x \in E} \tilde{\cap} {\frac{[(f_{Y} (x), 1 - t_{Y} (x)]}{x}, η_{Y^{c}} (x) : x \in E} = Ψ_{X}^{c} \tilde{\cap} Ψ_{Y}^{c} \end{matrix}$

The proof of (2) can be made similarly. □

Proposition 3.7. Let Ψ_X, Ψ_Y, Ψ_Z ∈ VPVS (U). Then,

$Ψ_{X} \tilde{\cup} (Ψ_{Y} \tilde{\cap} Ψ_{Z}) = (Ψ_{X} \tilde{\cup} Ψ_{Y}) \tilde{\cap} (Ψ_{X} \tilde{\cup} Ψ_{Z})$

$Ψ_{X} \tilde{\cap} (Ψ_{Y} \tilde{\cup} Ψ_{Z}) = (Ψ_{X} \tilde{\cap} Ψ_{Y}) \tilde{\cup} (Ψ_{X} \tilde{\cap} Ψ_{Z})$

Proof. (1) For all x ∈ E, $\begin{matrix} t_{X \tilde{\cup} (Y \tilde{\cap} Z)} (x) & = & \max (t_{X} (x), t_{Y \tilde{\cap} Z} (x)) = \max (t_{X} (x), \min (t_{Y} (x), t_{Z} (x))) \\ = & \min (\max (t_{X} (x), t_{Y} (x)), \max (t_{X} (x), t_{Z} (x))) = \min (t_{X \tilde{\cup} Y} (x), t_{X \tilde{\cup} Z} (x)) = t_{(X \tilde{\cup} Y) \tilde{\cap} (X \tilde{\cup} Z)} (x) \\ f_{X \tilde{\cup} (Y \tilde{\cap} Z)} (x) & = & \min (f_{X} (x), f_{Y \tilde{\cap} Z} (x)) = \min (f_{X} (x), \max (f_{Y} (x), f_{Z} (x))) \\ = & \max (\min (f_{X} (x), f_{Y} (x)), \min (f_{X} (x), f_{Z} (x))) = \max (f_{X \tilde{\cup} Y} (x), f_{X \tilde{\cup} Z} (x)) = f_{(X \tilde{\cup} Y) \tilde{\cap} (X \tilde{\cup} Z)} (x) \end{matrix}$ and $\begin{matrix} η_{X \tilde{\cup} (Y \tilde{\cap} Z)} (x) & = & η_{X} (x) \cup η_{Y \tilde{\cap} Z} (x) = η_{X} (x) \cup (η_{Y} (x) \cap η_{Z} (x)) \\ = & (η_{X} (x) \cup η_{Y} (x)) \cap (η_{X} (x) \cup η_{Z} (x)) = η_{X \tilde{\cup} Y} (x) \cap η_{X \tilde{\cup} Z} (x) = η_{(X \tilde{\cup} Y) \tilde{\cap} (X \tilde{\cup} Z)} (x) . \end{matrix}$

The proof of (2) can be made similarly. □

4 Vpvs-aggregation operator

In this section, we define an aggregate vague set of a vague parameterized vague soft set, we also define vpvs-aggregation operator that produced an aggregate vague set from a vpvs-set and its vague parameter set. Also we give an application of this operator in decision making problem.

Definition 4.1. Let Ψ_X ∈ VPVS (U). Then a vpvs-aggregation operator, denoted by VPVS_agg, is defined by $\begin{matrix} {VPVS}_{agg} : V (E) \times VPVS (U) \\ \to V (U), {VPVS}_{agg} (X, Ψ_{X}) = Ψ_{X}^{*} \end{matrix}$ where $\begin{matrix} Ψ_{X}^{*} = {\frac{[t_{Ψ_{X}^{*}} (u), 1 - f_{Ψ_{X}^{*}} (u)]}{u} : u \in U} \end{matrix}$ which is a vague set over U. The value $Ψ_{X}^{*}$ is called aggregate vague set of the Ψ_X. Here, the membership degree $t_{Ψ_{X}^{*}} (u)$ and nonmembership degree $f_{Ψ_{X}^{*}} (u)$ of u is defined as follows: $\begin{matrix} t_{Ψ_{X}^{*}} (u) = \frac{1}{| E |} \sum_{x \in E} t_{X} (x) t_{η_{X} (x)} (u) \end{matrix}$

and $\begin{matrix} f_{Ψ_{X}^{*}} (u) = \frac{1}{| E |} \sum_{x \in E} f_{X} (x) f_{η_{X} (x)} (u) \end{matrix}$ where |E| is the cardinality of E and t_{η_X(x)} (u) is membership degree and f_{η_X(x)} (u) is nonmembership degree of u ∈ U in the vague set η_X (x).

Now, we construct a vague parameterized vague soft set decision making method by the following algorithm:

Step 1. Constructs a feasible vague subset X over the parameters set E based on a decision maker (DM) which is expert.

Step 2. Constructs a vague parameterized vague soft set Ψ_X over the alternatives set U based on a DM.

Step 3. Computes the aggregate vague set $Ψ_{X}^{*}$ of vague parameterized vague soft set Ψ_X.

Step 4. Find $\max (t) = \max {t_{Ψ_{X}^{*}} (u) : u \in U}$ and $\max (1 - f) = \max {1 - f_{Ψ_{X}^{*}} (v) : v \in U}$ .

Step 5. Find α ∈ [0, 1] such that $\frac{[\max (t), α]}{u} \in Ψ_{X}^{*}$ and β ∈ [0, 1] such that $\frac{[β, \max (1 - f)]}{v} \in Ψ_{X}^{*}$ .

Step 6. Computes $\frac{\max (t)}{\max (t) + (1 - α)} = α'$ and $\frac{β}{β + (1 - \max (1 - f))} = β'$ .

Step 7. If α′ > β′, the optimal decision is u, if α′ < β′, the optimal decision is v.

Example 4.1. Suppose that a workplace wants to fill a position. There are five candidates who fill in a form in order to apply formally for the position. There is a decision maker (DM), that is from the department of human resources.

He wants to interview the candidates, but it is very difficult to make it all of them. Therefore, by using the vpvs-set decision making method, the number of candidates are reduced to a suitable one. Assume that the set of candidates U = {u₁, u₂, u₃, u₄, u₅} which may be characterized by a set of parameters E = {x₁, x₂, x₃, x₄} which is ^″x₁ = experience ^″, ^″x₂ = technical information ^″, ^″x₃ = good speaking ^″, ^″x₄ = young age ^″. Now, we can apply the method as follows:

Step 1. Assume that DM constructs a feasible vague subset X over the parameters set E as follows: $\begin{matrix} X = {\frac{[0.8, 0.9]}{x_{1}}, \frac{[0.7, 0.8]}{x_{2}}, \frac{[0.5, 0.5]}{x_{3}}, \frac{[0.6, 0.7]}{x_{4}}} \end{matrix}$

Step 2. DM constructs a vague parameterized vague soft set Ψ_X over the alternatives set U as follows: $\begin{matrix} Ψ_{X} & = & {(\frac{[0.8, 0.9]}{x_{1}}, {\frac{[0.4, 0.7]}{u_{1}}, \frac{[0.6, 0.6]}{u_{2}}, \frac{[0.7, 0.8]}{u_{3}}, \frac{[0.2, 0.6]}{u_{4}}, \frac{[0.8, 0.9]}{u_{5}}}), \\ (\frac{[0.7, 0.8]}{x_{2}}, {\frac{[0.3, 0.8]}{u_{1}}, \frac{[0.5, 0.6]}{u_{2}}, \frac{[0.6, 0.9]}{u_{3}}, \frac{[0.3, 0.7]}{u_{4}}}), \\ (\frac{[0.5, 0.5]}{x_{3}}, {\frac{[0.5, 0.7]}{u_{1}}, \frac{[0.6, 0.8]}{u_{3}}, \frac{[0.7, 0.8]}{u_{5}}}), \\ (\frac{[0.6, 0.7]}{x_{4}}, {\frac{[0.6, 0.9]}{u_{1}}, \frac{[0.4, 0.8]}{u_{2}}, \frac{[0.3, 0.6]}{u_{4}}, \frac{[0.6, 0.6]}{u_{5}}})} \end{matrix}$

Step 3. DM computes the aggregate vague set $Ψ_{X}^{*}$ of vague parameterized vague soft set Ψ_X as: $\begin{matrix} Ψ_{X}^{*} = {\frac{[0.285, 0.9375]}{u_{1}}, \frac{[0.2675, 0.955]}{u_{2}}, \frac{[0.32, 0.965]}{u_{3}}, \frac{[0.1375, 0.945]}{u_{4}}, \frac{[0.3375, 0.9425]}{u_{5}}} \end{matrix}$

Step 4.max (t) =0.3375 and max (1 - f) =0.965.

Step 5. $\frac{[0.3375, 0.9425]}{u_{5}} \in Ψ_{X}^{*}, \frac{[0.32, 0.965]}{u_{3}} \in Ψ_{X}^{*}$ .

Step 6. $α^{'} = \frac{0.3375}{0.3375 + (1 - 0.9425)} = 0.8544, β^{'} = \frac{0.32}{0.32 + (1 - 0.965)} = 0.9014$ .

Step 7. Since α′ < β′, the optimal decision is u₃.

Note that, although membership degree of u₅ is bigger than u₃, opportune element of U is u₃. This example show how the effect on decision making of non-membership degrees of elements.

5 Conclusion

In this paper, we first defined vague parameterized vague soft set and their various operations. Then, we introduced the method of decision making on the vpvs-set theory. We also gave an example that demonstrated that the decision making method can successfully work. These conclusions can be extensively applied in many fields such as pattern recognition, image processing, approximate reasoning, and fuzzy control. For further study, we will study algebraic structure of vpvs-sets and extend our work to other decision models and applications for modeling vagueness and uncertainty.

Conflict of interests

The author declare that there is no conflict of interests regarding the publication of this paper.

Footnotes

Acknowledgments

The author would like to thank the anonymous referees for their constructive comments as well as helpful suggestions from the associate editor which helped in improving this paper significantly. The author is also grateful to Professor Tom Archibald for his help to improve the linguistic quality of this paper. The works described in this paper are supported by the National Natural Science Foundation of China under Grant nos. 11501444, 11726019; the Postdoctoral Science Foundation of China under Grant nos. 2013M532079, 2014T70932; the Science Research Foundation of Education Department of Shaanxi Provincial Government under Grant no. 15JK1735.

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