Multi Q-fuzzy soft expert set and its application

Abstract

Molodtsov introduced the concept of soft set as a general mathematical tool for dealing with uncertainty. In this work,we first defined a multi Q-fuzzy soft expert set and its basic operations, namely complement, union, intersection, OR and AND. We then construct a decision-making method on multi Q-fuzzy soft expert set. We finally provide an example to show that the method can be successfully applied to problems which contain uncertainties.

Keywords

Multi Q-fuzzy set soft set soft expert set

1 Introduction

Most of our real life problems in medical sciences, engineering, management, environment and social sciences often involve data which are not necessarily crisp, precise and deterministic in character due to various uncertainties associated with these problems. Most of these problems were solved by fuzzy set provided by Zadeh [1]. In 1999, Molodtsov [2] introduced the concept of soft set theory, which was completely a new approach for modeling vagueness and uncertainties. Later on, Maji et al. [3] studied the theory of fuzzy soft set and intuitionistic fuzzy soft set. After presentation of the operations of soft sets, the properties and applications on the soft set theory have been studied increasingly. Cagman et al. [4, 5] studied fuzzy soft set theory and fuzzy soft matrix theory which was extended to neuro-fuzzy soft sets by Varnamkhasti and Hassan [6, 7]. Alkhazaleh et al. [8] introduced the concept of fuzzy parameterized interval-valued fuzzy soft set and gave its application in decision making. Alkhazaleh et al. [9] also defined the idea of soft multisets as a generalization of Molodtsov’s soft set, followed by the concept of fuzzy soft multiset [10] possibility fuzzy soft set [11], generalised interval-valued fuzzy soft set [12] and interval-valued vague soft sets [13 –15] as extensions of soft set. Bashir and Salleh [16] introduced the concept of the fuzzy parameterized soft expert set, while Hazaymeh et al. [17] defined the generalized fuzzy soft expert set as a combination between the soft expert set and the fuzzy set. Alhazaymeh et al. [18] introduced the concept of soft intuitionistic fuzzy sets, followed by studies on vague soft sets [19 –23], and extended to generalised vague soft expert sets [24 –27]. In this paper we extend the work on soft expert sets [28] and multi Q-fuzzy soft set [29 –34] by proposing the concept of a multi Q-fuzzy soft expert set and its operations, namely complement, union, intersection, OR and AND along with illustrative examples. Finally we provide an application of this new concept to solve a decision-making problem.

2 Preliminaries

In this section we review the basic definitions of a soft expert set theory, and a multi Q-fuzzy set required as preliminaries.

Let U be a universe, E a set of parameters, and X a set of experts (agents). Let O be a set of opinions, Z = E × X × O and A ⊆ Z.

Definition 2.1. (see [28]) A pair (F, A) is called a soft expert set (over U) where F is a mapping given by F : A → P (U) where P (U) denotes the power set of U.

Definition 2.2. (see [28]) Let E be a set of parameters and X a set of experts. The NOT set of Z = E × X × O denoted by $\overset{´}{I} Z,$ is defined by $\overset{´}{I} Z = {\overset{´}{I} e_{i}, x_{j}, o_{k}},$ ∀ i, j, k where $\overset{´}{I} e_{i}$ is not e_i.

Definition 2.3. (see [28] The complement of a soft expert set (F, A) is denoted by (F, A) ^c and is defined by $(F, A)^{c} = (F^{c}, \overset{´}{I} A)$ where $F^{c} : \overset{´}{I} A \to P (U)$ is a mapping given by $F^{c} (α) = U - F (\overset{´}{I} α), \forall α \in \overset{´}{I} A .$

Definition 2.4. (see [28]) For two soft expert sets (F, A) and (G, B) over U, (F, A) is called a soft expert subset of (G, B) if

A ⊆ B,

∀ɛ ∈ A, F (ɛ) ⊆ G (ɛ) .

This relationship is denoted by $(F, A) \tilde{\subseteq} (G, B)$ . In this case (G, B) is called a soft expert superset of (F, A).

Definition 2.5. (see [28]) An agree-soft expert set (F, A) ₁ over U is a soft expert subset of (F, A) defined as follows: (F, A) ₁ = {F₁ (α) : α ∈ E × X × {1}} .

Definition 2.6. (see [28]) A disagree-soft expert set (F, A) ₀ over U is a soft expert subset of (F, A) defined as follows: (F, A) ₀ = {F₀ (α) : α ∈ E × X × 0} .

Definition 2.7. (see [28]) Union of two soft expert sets of (F, A) and (G, B) over the common universe U is the soft set (H, C) , where C = A ∪ B,

and ∀e ∈ C $H (e) = {\begin{matrix} F (e), & if e \in A - B; \\ G (e), & if e \in B - A; \\ F (e) \cup G (e), & if e \in A \cap B . \end{matrix}$

We write $(F, A) \tilde{\cup} (G, B) = (H, C)$ .

Definition 2.8. (see [28]) Intersection of two soft expert sets (F, A) and (G, B) over a common universe set U denoted by $(F, A) \tilde{\cap} (G, B)$ is the soft set (H, C), where C = A ∩ B, and ∀e ∈ C,

$H (e) = {\begin{matrix} F (e), & if e \in A - B; \\ G (e), & if e \in B - A; \\ F (e) \cap G (e), & if e \in A \cap B . \end{matrix}$

Definition 2.9. (see [28]) If (F, A) and (G, B) are two soft expert sets then (F, A) AND (G, B) denoted by (F, A) ∧ (G, B) is defined by (F, A) ∧ (G, B) = (H, A × B) , where H (α, β) = F (α) ∩ G (β) , ∀ (α, β) ∈ A × B.

Definition 2.10. (see [28]) If (F, A) and (G, B) are two soft expert sets then (F, A) OR (G, B) denoted by (F, A) ∨ (G, B) is defined by (F, A) ∨ (G, B) = (H, A × B), where H (α, β) = F (α) ∪ G (β) , ∀ (α, β) ∈ A × B.

Definition 2.11. (see [29]) Let I be a unit interval and k be a positive integer. A multi Q-fuzzy set ${\tilde{A}}_{Q}$ in U and a non-empty set Q is a set of ordered sequences ${\tilde{A}}_{Q} = {(u, q), μ_{i} (u, q) : u \in U, q \in Q}$ where $μ_{i} : U \times Q \to I^{k}, i = 1, 2, . . ., k .$

The function (μ₁ (u, q) , μ₂ (u, q) , . . . , μ_k (u, q)) is called the membership function of multi Q-fuzzy set ${\tilde{A}}_{Q};$ and μ₁ (u, q) + μ₂ (u, q) + . . . + μ_k (u, q)) ≤1, k is called the dimension of ${\tilde{A}}_{Q}$ . The set of all multi Q-fuzzy sets of dimension k in U and Q is denoted by M^kQF (U).

3 Multi Q-fuzzy soft expert set

In this section we introduce the concept of a multi Q-fuzzy soft expert set and define some operations on a multi Q-fuzzy soft expert set, namely subset, equality, complement, union, intersection, AND and ORoperations.

We begin by proposing the definition of a multi Q-fuzzy soft expert set followed by an example.

Let U be a universal set, E be a set of parameters, X a set of expert set (agents), and O = {1 = agree, 0 = disagree} a set of opinions. Let Z = E × X × O and A ⊆ Z .

Definition 3.1. A pair $({\tilde{F}}_{Q}, A)$ is called a multi Q-fuzzy soft expert set over U where ${\tilde{F}}_{Q}$ is a mapping given by ${\tilde{F}}_{Q} : A \to M^{k} QF (U)$ such that M^kQF (U) denotes the set of all multi Q-fuzzy set over U .

Example 3.2. Suppose that a company wants to buy three types of products from two suppliers and wants to take the opinion of two experts about these products. Let U = {u₁, u₂, u₃} be a set of products, Q = {q₁, q₂} be a set of suppliers, E = {e₁ = easy to use, e₂ = quality} is a set of decision parameters. Let X = {p, q} be a set of experts. Suppose that

$\tilde{F_{Q}} (e_{1}, p, 1) = {\frac{(u_{1}, q_{1})}{(0.1, 0.3, 0.5)}, \frac{(u_{1}, q_{2})}{(0.3, 0.4, 0.2)}, \frac{(u_{2}, q_{1})}{(0, 0.8, 0.2)}, \frac{(u_{2}, q_{2})}{(0.4, 0.4, 0.1)}, \frac{(u_{3}, q_{1})}{(0.5, 0.1, 0.3)}, \frac{(u_{3}, q_{2})}{(0.2, 0.4, 0)}},$

$\tilde{F_{Q}} (e_{1}, q, 1) = {\frac{(u_{1}, q_{1})}{(0.3, 0.4, 0.5)}, \frac{(u_{1}, q_{2})}{(0.5, 0.1, 0.2)}, \frac{(u_{2}, q_{1})}{(0.1, 0.6, 0.2)}, \frac{(u_{2}, q_{2})}{(0.3, 0.5, 0)}, \frac{(u_{3}, q_{1})}{(0.4, 0.6, 0)}, \frac{(u_{3}, q_{2})}{(0.2, 0.5, 0.1)}},$

$\tilde{F_{Q}} (e_{2}, p, 1) = {\frac{(u_{1}, q_{1})}{(0.7, 0.1, 0.1)}, \frac{(u_{1}, q_{2})}{(0.6, 0.2, 0.2)}, \frac{(u_{2}, q_{1})}{(0.1, 0.2, 0.3)}, \frac{(u_{2}, q_{2})}{(0.3, 0.2, 0.4)}, \frac{(u_{3}, q_{1})}{(0.5, 0.1, 0.3)}, \frac{(u_{3}, q_{2})}{(0.2, 0.4, 0)}},$

$\tilde{F_{Q}} (e_{2}, q, 1) = {\frac{(u_{1}, q_{1})}{(0.8, 0.1, 0.1)}, \frac{(u_{1}, q_{2})}{(0.6, 0.2, 0.2)}, \frac{(u_{2}, q_{1})}{(0.1, 0.8, 0)}, \frac{(u_{2}, q_{2})}{(0.2, 0.2, 0.1)}, \frac{(u_{3}, q_{1})}{(0.2, 0.5, 0.1)}, \frac{(u_{3}, q_{2})}{(0.3, 0.6, 0.1)}},$

$\tilde{F_{Q}} (e_{1}, p, 0) = {\frac{(u_{1}, q_{1})}{(0.2, 0.3, 0.3)}, \frac{(u_{1}, q_{2})}{(0.3, 0.4, 0.2)}, \frac{(u_{2}, q_{1})}{(0, 0.5, 0.2)}, \frac{(u_{2}, q_{2})}{(0.4, 0.3, 0.1)}, \frac{(u_{3}, q_{1})}{(0.4, 0.2, 0.3)}, \frac{(u_{3}, q_{2})}{(0.2, 0.4, 0.2)}},$

$\tilde{F_{Q}} (e_{1}, q, 0) = {\frac{(u_{1}, q_{1})}{(0.2, 0.2, 0.5)}, \frac{(u_{1}, q_{2})}{(0.3, 0.5, 0.2)}, \frac{(u_{2}, q_{1})}{(0.1, 0.3, 0.2)}, \frac{(u_{2}, q_{2})}{(0.3, 0.3, 0.2)}, \frac{(u_{3}, q_{1})}{(0.4, 0.6, 0.1)}, \frac{(u_{3}, q_{2})}{(0.1, 0.2, 0.3)}},$

$\tilde{F_{Q}} (e_{2}, p, 0) = {\frac{(u_{1}, q_{1})}{(0.2, 0.4, 0.5)}, \frac{(u_{1}, q_{2})}{(0.1, 0.3, 0.3)}, \frac{(u_{2}, q_{1})}{(0.3, 0.3, 0.2)}, \frac{(u_{2}, q_{2})}{(0.4, 0.1, 0.1)}, \frac{(u_{3}, q_{1})}{(0.1, 0.2, 0.3)}, \frac{(u_{3}, q_{2})}{(0.1, 0.4, 0.1)}},$

$\tilde{F_{Q}} (e_{2}, q, 0) = {\frac{(u_{1}, q_{1})}{(0.2, 0.3, 0.4)}, \frac{(u_{1}, q_{2})}{(0.1, 0.2, 0.2)}, \frac{(u_{2}, q_{1})}{(0, 0.8, 0.2)}, \frac{(u_{2}, q_{2})}{(0.3, 0.4, 0.3)}, \frac{(u_{3}, q_{1})}{(0.5, 0.2, 0.2)}, \frac{(u_{3}, q_{2})}{(0.6, 0.4, 0)}} .$

Then we can view the multi Q-fuzzy soft expert set $({\tilde{F}}_{Q}, E)$ as consisting of the following collection of approximations:

$(\tilde{F_{Q}}, E) = {((e_{1}, p, 1), {\frac{(u_{1}, q_{1})}{(0.1, 0.3, 0.5)}, \frac{(u_{1}, q_{2})}{(0.3, 0.4, 0.2)}, \frac{(u_{2}, q_{1})}{(0, 0.8, 0.2)}, \frac{(u_{2}, q_{2})}{(0.4, 0.4, 0.1)}, \frac{(u_{3}, q_{1})}{(0.5, 0.1, 0.3)}, \frac{(u_{3}, q_{2})}{(0.2, 0.4, 0)}}),$

$((e_{1}, q, 1), {\frac{(u_{1}, q_{1})}{(0.3, 0.4, 0.5)}, \frac{(u_{1}, q_{2})}{(0.5, 0.1, 0.2)}, \frac{(u_{2}, q_{1})}{(0.1, 0.6, 0.2)}, \frac{(u_{2}, q_{2})}{(0.3, 0.5, 0)}, \frac{(u_{3}, q_{1})}{(0.4, 0.6, 0)}, \frac{(u_{3}, q_{2})}{(0.2, 0.5, 0.1)}}),$

$((e_{2}, p, 1), {\frac{(u_{1}, q_{1})}{(0.7, 0.1, 0.1)}, \frac{(u_{1}, q_{2})}{(0.6, 0.2, 0.2)}, \frac{(u_{2}, q_{1})}{(0.1, 0.2, 0.3)}, \frac{(u_{2}, q_{2})}{(0.3, 0.2, 0.4)}, \frac{(u_{3}, q_{1})}{(0.5, 0.1, 0.3)}, \frac{(u_{3}, q_{2})}{(0.2, 0.4, 0)}}),$

$((e_{2}, q, 1), {\frac{(u_{1}, q_{1})}{(0.8, 0.1, 0.1)}, \frac{(u_{1}, q_{2})}{(0.6, 0.2, 0.2)}, \frac{(u_{2}, q_{1})}{(0.1, 0.8, 0)}, \frac{(u_{2}, q_{2})}{(0.2, 0.2, 0.1)}, \frac{(u_{3}, q_{1})}{(0.2, 0.5, 0.1)}, \frac{(u_{3}, q_{2})}{(0.3, 0.6, 0.1)}}),$

$((e_{1}, p, 0), {\frac{(u_{1}, q_{1})}{(0.2, 0.3, 0.3)}, \frac{(u_{1}, q_{2})}{(0.3, 0.4, 0.2)}, \frac{(u_{2}, q_{1})}{(0, 0.5, 0.2)}, \frac{(u_{2}, q_{2})}{(0.4, 0.3, 0.1)}, \frac{(u_{3}, q_{1})}{(0.4, 0.2, 0.3)}, \frac{(u_{3}, q_{2})}{(0.2, 0.4, 0.2)}}),$

$((e_{1}, q, 0), {\frac{(u_{1}, q_{1})}{(0.2, 0.2, 0.5)}, \frac{(u_{1}, q_{2})}{(0.3, 0.5, 0.2)}, \frac{(u_{2}, q_{1})}{(0.1, 0.3, 0.2)}, \frac{(u_{2}, q_{2})}{(0.3, 0.3, 0.2)}, \frac{(u_{3}, q_{1})}{(0.4, 0.6, 0.1)}, \frac{(u_{3}, q_{2})}{(0.1, 0.2, 0.3)}}),$

$((e_{2}, p, 0), {\frac{(u_{1}, q_{1})}{(0.2, 0.4, 0.5)}, \frac{(u_{1}, q_{2})}{(0.1, 0.3, 0.3)}, \frac{(u_{2}, q_{1})}{(0.3, 0.3, 0.2)}, \frac{(u_{2}, q_{2})}{(0.4, 0.1, 0.1)}, \frac{(u_{3}, q_{1})}{(0.1, 0.2, 0.3)}, \frac{(u_{3}, q_{2})}{(0.1, 0.4, 0.1)}}),$

$((e_{2}, q, 0), {\frac{(u_{1}, q_{1})}{(0.2, 0.3, 0.4)}, \frac{(u_{1}, q_{2})}{(0.1, 0.2, 0.2)}, \frac{(u_{2}, q_{1})}{(0, 0.8, 0.2)}, \frac{(u_{2}, q_{2})}{(0.3, 0.4, 0.3)}, \frac{(u_{3}, q_{1})}{(0.5, 0.2, 0.2)}, \frac{(u_{3}, q_{2})}{(0.6, 0.4, 0)}})} .$

Each element of the multi Q-fuzzy soft expert set implies the opinion of each expert based on each parameter about the products with their own companies.

We establish the definitions of subsets, equal, agree and disagree-multi Q-fuzzy soft expert sets belowfollowed by an example.

Definition 3.3. If $({\tilde{F}}_{Q}, A)$ and $({\tilde{H}}_{Q}, B)$ are two multi Q-fuzzy soft expert sets then we say that $({\tilde{F}}_{Q}, A)$ is a multi Q-fuzzy soft expert subset of $({\tilde{H}}_{Q}, B)$ , denoted by $({\tilde{F}}_{Q}, A) \subseteq ({\tilde{H}}_{Q}, B)$ , if A ⊆ B and ${\tilde{F}}_{Q} (x) \subseteq {\tilde{H}}_{Q} (x)$ for all x ∈ A.

Definition 3.4. If $({\tilde{F}}_{Q}, A)$ and $({\tilde{H}}_{Q}, B)$ are two multi Q-fuzzy soft expert sets then we say that $({\tilde{F}}_{Q}, A)$ and $({\tilde{H}}_{Q}, B)$ are equal if $({\tilde{F}}_{Q}, A)$ is a multi Q-fuzzy soft expert subset of $({\tilde{H}}_{Q}, B)$ and $({\tilde{H}}_{Q}, B)$ is a multi Q-fuzzy soft expert subset of $({\tilde{F}}_{Q}, A) .$

Definition 3.5. An agree-multi Q-fuzzy soft expert set $({\tilde{F}}_{Q}, A)_{1}$ over U and Q is a multi Q-fuzzy soft expert subset of $({\bar{F}}_{Q}, A)$ defined as follows: $({\tilde{F}}_{Q}, A)_{1} = {{\tilde{F_{Q}}}_{1} (α) : α \in E \times X \times {1}} .$

Definition 3.6. A disagree-multi Q-fuzzy soft expert set $({\tilde{F}}_{Q}, A)_{0}$ over U and Q is a multi Q-fuzzy soft expert subset of $({\bar{F}}_{Q}, A)$ defined as follows: $({\tilde{F}}_{Q}, A)_{0} = {{\tilde{F_{Q}}}_{0} (α) : α \in E \times X \times {0}} .$

Example 3.7. Consider Example 3.2. The agree-multi Q-fuzzy soft expert set $({\tilde{F}}_{Q}, E)_{1}$ over U and Q is

and the disagree-multi Q-fuzzy soft expert set $({\tilde{F}}_{Q}, E)_{0}$ over U and Q is

In order to give a deeper insight into this issue, we propose the definition and a proposition on complement of multi Q-fuzzy soft expert set.

Definition 3.8. The complement of a multi Q-fuzzy soft expert set $({\tilde{F}}_{Q}, A)$ is denoted by $({\tilde{F}}_{Q}, A)^{c}$ and is defined by $({\tilde{F}}_{Q}, A)^{c} = ({\tilde{F}}_{Q}^{c}; ⌉ A)$ where ${\tilde{F}}_{Q}^{c} : ⌉ A \to M^{k} QF (U)$ is a mapping given by ${\tilde{F}}_{Q}^{c} (α) = c ({\tilde{F}}_{Q} (⌉ α)), \forall α \in ⌉ A$ such that c is a multi Q-fuzzy complement.

Proposition 3.9. If $({\tilde{F}}_{Q}, A)$ is a multi Q-fuzzy softexpert set then

$(({\tilde{F}}_{Q}, A)^{c})^{c} = ({\tilde{F}}_{Q}, A) .$

${({\tilde{F}}_{Q}, A)_{1}}^{c} = ({\tilde{F}}_{Q}, A)_{0}$

${({\tilde{F}}_{Q}, A)_{0}}^{c} = ({\tilde{F}}_{Q}, A)_{1}$

Proof. The proof is straightforward by using the properties of multi Q-fuzzy set.

4 Union and intersection of multi Q-fuzzy soft expert sets

In this section, we introduce the definitions of union and intersection of multi Q-fuzzy soft expert sets, derive their properties, and give some examples.

Definition 4.1. The union of two multi Q-fuzzy soft expert sets $({\tilde{F}}_{Q}, A)$ and $({\tilde{G}}_{Q}, B)$ over U and Q, denoted by $({\tilde{F}}_{Q}, A) \tilde{\cup} ({\tilde{G}}_{Q}, B)$ , is the multi Q-fuzzy soft expert set $({\tilde{H}}_{Q}, C)$ where C = A ∪ B, and ∀e ∈ C,

${\tilde{H}}_{Q} (e) = {\begin{matrix} {\tilde{F}}_{Q} (e), & if e \in A - B; \\ {\tilde{G}}_{Q} (e), & if e \in B - A; \\ {\tilde{F}}_{Q} (e) \cup {\tilde{G}}_{(} e), & if e \in A \cap B . \end{matrix}$

Example 4.2. Assume that U = {u₁, u₂, u₃, u₄, u₅} be a universal set, E = {e₁, e₂, e₃} be a set of parameters, and Q = {q₁, q₂} be a non-empty set. Let X = {p, q} be a set of experts, $({\tilde{F}}_{Q}, A)$ and $({\tilde{G}}_{Q}, B)$ are two multi fuzzy soft expert sets over U and Q such that

A = {(e₁, p, 1) , (e₂, p, 0) , (e₁, q, 1) , (e₂, q, 1)} ,

B = {(e₁, p, 1) , (e₁, q, 1) , (e₂, q, 1)} ,

$({\tilde{F}}_{Q}, A) = {((e_{1}, p, 1), {\frac{(u_{1}, q_{1})}{(0.1, 0.2, 0.5)}, \frac{(u_{1}, q_{2})}{(0.1, 0.4, 0.2)}, \frac{(u_{2}, q_{1})}{(0, 0.8, 0.3)}, \frac{(u_{2}, q_{2})}{(0.4, 0.4, 0.1)}, \frac{(u_{3}, q_{1})}{(0.1, 0.2, 0.2)}, \frac{(u_{3}, q_{2})}{(0.5, 0.2, 0)}}),$

$((e_{1}, q, 1), {\frac{(u_{1}, q_{1})}{(0.3, 0.4, 0.5)}, \frac{(u_{1}, q_{2})}{(0.5, 0.1, 0.2)}, \frac{(u_{2}, q_{1})}{(0.1, 0.3, 0.2)}, \frac{(u_{2}, q_{2})}{(0.2, 0.5, 0)}, \frac{(u_{3}, q_{1})}{(0.4, 0.6, 0)}, \frac{(u_{3}, q_{2})}{(0.2, 0.5, 0.1)}}),$

$((e_{2}, q, 1), {\frac{(u_{1}, q_{1})}{(0.2, 0.1, 0.1)}, \frac{(u_{1}, q_{2})}{(0.6, 0.3, 0)}, \frac{(u_{2}, q_{1})}{(0.1, 0.8, 0)}, \frac{(u_{2}, q_{2})}{(0.2, 0.2, 0.1)}, \frac{(u_{3}, q_{1})}{(0.2, 0.1, 0.1)}, \frac{(u_{3}, q_{2})}{(0.3, 0.2, 0.1)}}),$

$((e_{2}, p, 0), {\frac{(u_{1}, q_{1})}{(0.3, 0.4, 0.1)}, \frac{(u_{1}, q_{2})}{(0.1, 0.3, 0.2)}, \frac{(u_{2}, q_{1})}{(0.3, 0.3, 0.2)}, \frac{(u_{2}, q_{2})}{(0.4, 0.1, 0.1)}, \frac{(u_{3}, q_{1})}{(0.1, 0.1, 0.3)}, \frac{(u_{3}, q_{2})}{(0.1, 0.4, 0.2)}})},$

and $({\tilde{G}}_{Q}, B) = {((e_{1}, p, 1), {\frac{(u_{1}, q_{1})}{(0, 0.2, 0.5)}, \frac{(u_{1}, q_{2})}{(0.3, 0.2, 0.2)}, \frac{(u_{2}, q_{1})}{(0, 0.8, 0.1)}, \frac{(u_{2}, q_{2})}{(0.2, 0.2, 0.1)}, \frac{(u_{3}, q_{1})}{(0.2, 0.1, 0.4)}, \frac{(u_{3}, q_{2})}{(0.1, 0.3, 0)}}),$

$((e_{2}, q, 1), {\frac{(u_{1}, q_{1})}{(0.2, 0.5, 0.1)}, \frac{(u_{1}, q_{2})}{(0, 0.2, 0.4)}, \frac{(u_{2}, q_{1})}{(0.1, 0.2, 0)}, \frac{(u_{2}, q_{2})}{(0.1, 0.4, 0.1)}, \frac{(u_{3}, q_{1})}{(0.2, 0.5, 0.1)}, \frac{(u_{3}, q_{2})}{(0.3, 0.6, 0.1)}})} .$

The union of $({\tilde{F}}_{Q}, A)$ and $({\tilde{G}}_{Q}, B)$ is $({\tilde{F}}_{Q}, B)$ $\tilde{\cup} ({\tilde{G}}_{Q}, B) = ({\tilde{H}}_{Q}, C)$ , such that

$({\tilde{H}}_{Q}, C) = {((e_{1}, p, 1), {\frac{(u_{1}, q_{1})}{(0.1, 0.2, 0.5)}, \frac{(u_{1}, q_{2})}{(0.3, 0.4, 0.2)}, \frac{(u_{2}, q_{1})}{(0, 0.8, 0.3)}, \frac{(u_{2}, q_{2})}{(0.4, 0.4, 0.1)}, \frac{(u_{3}, q_{1})}{(0.2, 0.2, 0.4)}, \frac{(u_{3}, q_{2})}{(0.5, 0.3, 0)}}),$

$((e_{2}, p, 1), {\frac{(u_{1}, q_{1})}{(0.4, 0.1, 0.2)}, \frac{(u_{1}, q_{2})}{(0.6, 0.2, 0.3)}, \frac{(u_{2}, q_{1})}{(0.2, 0.2, 0.3)}, \frac{(u_{2}, q_{2})}{(0.3, 0.3, 0.4)}, \frac{(u_{3}, q_{1})}{(0.5, 0.1, 0.3)}, \frac{(u_{3}, q_{2})}{(0.2, 0.7, 0)}}),$

$((e_{2}, q, 1), {\frac{(u_{1}, q_{1})}{(0.2, 0.5, 0.1)}, \frac{(u_{1}, q_{2})}{(0.6, 0.3, 0.4)}, \frac{(u_{2}, q_{1})}{(0.1, 0.8, 0)}, \frac{(u_{2}, q_{2})}{(0.2, 0.4, 0.1)}, \frac{(u_{3}, q_{1})}{(0.2, 0.5, 0.1)}, \frac{(u_{3}, q_{2})}{(0.3, 0.6, 0.1)}}),$

Definition 4.3. The intersection of two multi Q-fuzzy soft expert sets $({\tilde{F}}_{Q}, A)$ and $({\tilde{G}}_{Q}, B)$ over U and Q, denoted by $({\tilde{F}}_{Q}, A) \tilde{\cap} ({\tilde{G}}_{Q}, B),$ is the multi Q-fuzzy soft expert set $({\tilde{H}}_{Q}, C)$ where C = A ∪ B, and ∀e ∈ C, ${\tilde{H}}_{Q} (e) = {\begin{matrix} {\tilde{F}}_{Q} (e), & if e \in A - B; \\ {\tilde{G}}_{Q} (e), & if e \in B - A; \\ {\tilde{F}}_{Q} (e) \cap {\tilde{G}}_{(} e), & if e \in A \cap B . \end{matrix}$

Example 4.4. From Example 4.2, we have $({\tilde{F}}_{Q}, A) \tilde{\cap} ({\tilde{G}}_{Q}, B) = ({\tilde{H}}_{Q}, C)$ where

$({\tilde{H}}_{Q}, C) = {((e_{1}, p, 1), {\frac{(u_{1}, q_{1})}{(0, 0.2, 0.5)}, \frac{(u_{1}, q_{2})}{(0.1, 0.2, 0.2)}, \frac{(u_{2}, q_{1})}{(0, 0.8, 0.1)}, \frac{(u_{2}, q_{2})}{(0.2, 0.2, 0.1)}, \frac{(u_{3}, q_{1})}{(0.1, 0.1, 0.2)}, \frac{(u_{3}, q_{2})}{(0.1, 0.2, 0)}}),$

$((e_{2}, p, 1), {\frac{(u_{1}, q_{1})}{(0.3, 0.1, 0.1)}, \frac{(u_{1}, q_{2})}{(0.3, 0.2, 0.2)}, \frac{(u_{2}, q_{1})}{(0.1, 0.2, 0.3)}, \frac{(u_{2}, q_{2})}{(0.1, 0.2, 0.4)}, \frac{(u_{3}, q_{1})}{(0.5, 0.1, 0.2)}, \frac{(u_{3}, q_{2})}{(0.1, 0.4, 0)}}),$

$((e_{2}, q, 1), {\frac{(u_{1}, q_{1})}{(0.2, 0.5, 0.1)}, \frac{(u_{1}, q_{2})}{(0, 0.2, 0)}, \frac{(u_{2}, q_{1})}{(0.1, 0.2, 0)}, \frac{(u_{2}, q_{2})}{(0.1, 0.2, 0.1)}, \frac{(u_{3}, q_{1})}{(0.2, 0.1, 0.1)}, \frac{(u_{3}, q_{2})}{(0.3, 0.2, 0.1)}}),$

The intersection operation implies the common sets of experts and their opinions with respect to the set of parameters about the objects using multi Q-fuzzy sets.

The following proposition explicitly characterizes the combined operations of union and intersection of multi Q-fuzzy soft expert sets.

Proposition 4.5. If $({\tilde{F}}_{Q}, A), ({\tilde{G}}_{Q}, B)$ and $({\tilde{H}}_{Q}, C)$ are three multi Q-fuzzy soft expert sets over U and Q then

$({\tilde{F}}_{Q}, A) \tilde{\cup} (({\tilde{G}}_{Q}, B) \tilde{\cup} ({\tilde{H}}_{Q}, C)) = (({\tilde{F}}_{Q}, A) \tilde{\cup} ({\tilde{G}}_{Q}, B)) \tilde{\cup} ({\tilde{H}}_{Q}, C) .$

$({\tilde{F}}_{Q}, A) \tilde{\cup} ({\tilde{F}}_{Q}, A) = ({\tilde{F}}_{Q}, A) .$

$({\tilde{F}}_{Q}, A) \tilde{\cap} (({\tilde{G}}_{Q}, B) \tilde{\cap} ({\tilde{H}}_{Q}, C)) = (({\tilde{F}}_{Q}, A) \tilde{\cap} ({\tilde{G}}_{Q}, B)) \tilde{\cap} ({\tilde{H}}_{Q}, C) .$

$({\tilde{F}}_{Q}, A) \tilde{\cap} ({\tilde{F}}_{Q}, A) = ({\tilde{F}}_{Q}, A) .$

$({\tilde{F}}_{Q}, A) \tilde{\cap} (({\tilde{G}}_{Q}, B) \tilde{\cup} ({\tilde{H}}_{Q}, C)) = (({\tilde{F}}_{Q}, A) \tilde{\cap} ({\tilde{G}}_{Q}, B)) \tilde{\cup} (({\tilde{F}}_{Q}, A) \cap ({\tilde{H}}_{Q}, C)) .$

$({\tilde{F}}_{Q}, A) \tilde{\cup} (({\tilde{G}}_{Q}, B) \tilde{\cap} ({\tilde{H}}_{Q}, C)) = (({\tilde{F}}_{Q}, A) \tilde{\cup} ({\tilde{G}}_{Q}, B)) \tilde{\cap} (({\tilde{F}}_{Q}, A) \cup ({\tilde{H}}_{Q}, C)) .$

Proof. The proof is straightforward by using the properties of multi Q-fuzzy sets as in Definition 2.11.

5 AND and OR operations

In this section, we introduce the definitions of AND and OR operations for multi Q-fuzzy soft expert sets, derive their properties, and give some examples.

Definition 5.1. If $({\tilde{F}}_{Q}, A)$ and $({\tilde{G}}_{Q}, B)$ are two multi Q-fuzzy soft expert sets over U and Q then $({\tilde{F}}_{Q}, A)$ AND $({\tilde{G}}_{Q}, B)$ denoted by $({\tilde{F}}_{Q}, A) \land ({\tilde{G}}_{Q}, B)$ is defined by $({\tilde{F}}_{Q}, A) \land ({\tilde{G}}_{Q}, B) = (\tilde{H}, A \times B)$ such that ${\tilde{H}}_{Q} (α, β) = ({\tilde{F}}_{Q} (α) \cap \tilde{G} (β)); \forall (α, β) \in A \times B .$

Example 5.2. Consider Example 4.2. If

A = {(e₁, p, 1) , (e₂, p, 0)} , and

B = {(e₁, p, 1) , (e₁, q, 1)} ,

Thus $({\tilde{F}}_{Q}, A)$ AND $({\tilde{G}}_{Q}, B) = {(((e_{1}, p, 1), (e_{1}, p, 1)), {\frac{(u_{1}, q_{1})}{(0, 0.2, 0.5)}, \frac{(u_{1}, q_{2})}{(0.1, 0.2, 0.2)}, \frac{(u_{2}, q_{1})}{(0, 0.8, 0.1)}, \frac{(u_{2}, q_{2})}{(0.2, 0.2, 0.1)}, \frac{(u_{3}, q_{1})}{(0.1, 0.1, 0.2)}, \frac{(u_{3}, q_{2})}{(0.1, 0.2, 0)}}),$

$(((e_{1}, p, 1), (e_{1}, q, 1)), {\frac{(u_{1}, q_{1})}{(0.1, 0.2, 0.5)}, \frac{(u_{1}, q_{2})}{(0.1, 0.1, 0.2)}, \frac{(u_{2}, q_{1})}{(0, 0.6, 0.2)}, \frac{(u_{2}, q_{2})}{(0.3, 0.4, 0)}, \frac{(u_{3}, q_{1})}{(0.4, 0.2, 0.0)}, \frac{(u_{3}, q_{2})}{(0.1, 0.2, 0)}}),$

$(((e_{2}, p, 0), (e_{1}, p, 1)), {\frac{(u_{1}, q_{1})}{(0, 0.2, 0.1)}, \frac{(u_{1}, q_{2})}{(0.1, 0.2, 0.2)}, \frac{(u_{2}, q_{1})}{(0, 0.3, 0.1)}, \frac{(u_{2}, q_{2})}{(0.2, 0.1, 0.1)}, \frac{(u_{3}, q_{1})}{(0.1, 0.1, 0.3)}, \frac{(u_{3}, q_{2})}{(0.1, 0.3, 0)}}),$

$(((e_{2}, p, 0), (e_{1}, q, 1)), {\frac{(u_{1}, q_{1})}{(0.3, 0.4, 0.1)}, \frac{(u_{1}, q_{2})}{(0.1, 0.1, 0.2)}, \frac{(u_{2}, q_{1})}{(0.1, 0.3, 0.1)}, \frac{(u_{2}, q_{2})}{(0.3, 0.1, 0)}, \frac{(u_{3}, q_{1})}{(0.1, 0.1, 0)}, \frac{(u_{3}, q_{2})}{(0.1, 0.4, 0.1)}})} .$

Definition 5.3. If $({\tilde{F}}_{Q}, A)$ and $({\tilde{G}}_{Q}, B)$ are two multi Q-fuzzy soft expert sets over U and Q then $({\tilde{F}}_{Q}, A)$ OR $({\tilde{G}}_{Q}, B)$ denoted by $({\tilde{F}}_{Q}, A) \lor ({\tilde{G}}_{Q}, B)$ is defined by $({\tilde{F}}_{Q}, A) \lor ({\tilde{G}}_{Q}, B) = (\tilde{H}, A \times B)$ such that ${\tilde{H}}_{Q} (α, β) = ({\tilde{F}}_{Q} (α) \cup \tilde{G} (β)); \forall (α, β) \in A \times B .$

Example 5.4. Consider Example 5.2. $({\tilde{F}}_{Q}, A)$ OR $({\tilde{G}}_{Q}, B)$ is

$({\tilde{H}}_{Q}, C) = {(((e_{1}, p, 1), (e_{1}, p, 1)), {\frac{(u_{1}, q_{1})}{(0.1, 0.2, 0.5)}, \frac{(u_{1}, q_{2})}{(0.1, 0.4, 0.2)}, \frac{(u_{2}, q_{1})}{(0, 0.8, 0.3)}, \frac{(u_{2}, q_{2})}{(0.4, 0.4, 0.1)}, \frac{(u_{3}, q_{1})}{(0.2, 0.2, 0.4)}, \frac{(u_{3}, q_{2})}{(0.5, 0.3, 0)}}),$

${(((e_{1}, p, 1), (e_{1}, q, 1)), {\frac{(u_{1}, q_{1})}{(0.3, 0.4, 0.5)}, \frac{(u_{1}, q_{2})}{(0.5, 0.4, 0.2)}, \frac{(u_{2}, q_{1})}{(0.1, 0.8, 0.3)}, \frac{(u_{2}, q_{2})}{(0.4, 0.5, 0.1)}, \frac{(u_{3}, q_{1})}{(0.4, 0.6, 0.3)}, \frac{(u_{3}, q_{2})}{(0.5, 0.5, 0.1)}}),$

${(((e_{2}, p, 0), (e_{1}, q, 1)), {\frac{(u_{1}, q_{1})}{(0.3, 0.4, 0.5)}, \frac{(u_{1}, q_{2})}{(0.3, 0.3, 0.2)}, \frac{(u_{2}, q_{1})}{(0.3, 0.8, 0.2)}, \frac{(u_{2}, q_{2})}{(0.4, 0.2, 0.1)}, \frac{(u_{3}, q_{1})}{(0.2, 0.1, 0.4)}, \frac{(u_{3}, q_{2})}{(0.1, 0.4, 0.2)}}),$

${(((e_{2}, p, 0), (e_{1}, q, 1)), {\frac{(u_{1}, q_{1})}{(0.3, 0.4, 0.5)}, \frac{(u_{1}, q_{2})}{(0.5, 0.3, 0.2)}, \frac{(u_{2}, q_{1})}{(0.3, 0.6, 0.2)}, \frac{(u_{2}, q_{2})}{(0.4, 0.5, 0.1)}, \frac{(u_{3}, q_{1})}{(0.4, 0.6, 0.3)}, \frac{(u_{3}, q_{2})}{(0.2, 0.5, 0.2)}})} .$

6 An application of multi Q-Fuzzy soft expert set

In this section, we present an application of multi Q-fuzzy soft expert set theory in a decision making problem by using the mean of each multi Q-fuzzy set.

Assume that a hospital wants to fill a position to be chosen by an expert committee. There are three candidates U = {u₁, u₂, u₃} with two types of qualifications Q = {q₁ = masters, q₂ = doctorate} and the hiring committee considers a set of parameters, E = {e₁, e₂, e₃} representing experience, computer knowledge and language fluency respectively. Let X = {p, q} be the set of two expert committee members. After a serious deliberation the committee constructs the following multi Q-fuzzy softexpert set.

$({\tilde{G}}_{Q}, B) = {((e_{1}, p, 1), {\frac{(u_{1}, q_{1})}{(0, 0.1, 0.5)}, \frac{(u_{1}, q_{2})}{(0.2, 0.2, 0.2)}, \frac{(u_{2}, q_{1})}{(0.1, 0.8, 0.1)}, \frac{(u_{2}, q_{2})}{(0.2, 0.2, 0.1)}, \frac{(u_{3}, q_{1})}{(0.1, 0.1, 0.3)}, \frac{(u_{3}, q_{2})}{(0.3, 0.3, 0.1)}}),$

$((e_{1}, q, 1), {\frac{(u_{1}, q_{1})}{(0.3, 0.4, 0.1)}, \frac{(u_{1}, q_{2})}{(0.6, 0.1, 0.2)}, \frac{(u_{2}, q_{1})}{(0.1, 0.6, 0.2)}, \frac{(u_{2}, q_{2})}{(0.3, 0.5, 0)}, \frac{(u_{3}, q_{1})}{(0.4, 0.6, 0)}, \frac{(u_{3}, q_{2})}{(0.2, 0.5, 0.1)}}),$

$((e_{2}, p, 1), {\frac{(u_{1}, q_{1})}{(0.4, 0.3, 0.2)}, \frac{(u_{1}, q_{2})}{(0.2, 0.1, 0.2)}, \frac{(u_{2}, q_{1})}{(0.1, 0.2, 0.2)}, \frac{(u_{2}, q_{2})}{(0.2, 0.1, 0.4)}, \frac{(u_{3}, q_{1})}{(0.3, 0.1, 0.3)}, \frac{(u_{3}, q_{2})}{(0.4, 0.7, 0.1)}}),$

$((e_{2}, q, 1), {\frac{(u_{1}, q_{1})}{(0.3, 0, 0.1)}, \frac{(u_{1}, q_{2})}{(0.2, 0, 0)}, \frac{(u_{2}, q_{1})}{(0.1, 0.2, 0.3)}, \frac{(u_{2}, q_{2})}{(0.1, 0.4, 0.6)}, \frac{(u_{3}, q_{1})}{(0.2, 0.5, 0.1)}, \frac{(u_{3}, q_{2})}{(0, 0, 0.3)}}),$

$((e_{1}, p, 0), {\frac{(u_{1}, q_{1})}{(0.5, 0.1, 0.3)}, \frac{(u_{1}, q_{2})}{(0.4, 0.2, 0.2)}, \frac{(u_{2}, q_{1})}{(0.1, 0.4, 0.2)}, \frac{(u_{2}, q_{2})}{(0.3, 0.2, 0.1)}, \frac{(u_{3}, q_{1})}{(0.3, 0.1, 0.1)}, \frac{(u_{3}, q_{2})}{(0.1, 0.5, 0.2)}}),$

$((e_{1}, q, 0), {\frac{(u_{1}, q_{1})}{(0.7, 0.1, 0.1)}, \frac{(u_{1}, q_{2})}{(0.1, 0, 0.2)}, \frac{(u_{2}, q_{1})}{(0.1, 0.3, 0.2)}, \frac{(u_{2}, q_{2})}{(0.3, 0.3, 0.2)}, \frac{(u_{3}, q_{1})}{(0.4, 0.3, 0)}, \frac{(u_{3}, q_{2})}{(0.4, 0.2, 0.3)}}),$

$((e_{2}, p, 0), {\frac{(u_{1}, q_{1})}{(0.1, 0.5, 0.3)}, \frac{(u_{1}, q_{2})}{(0.1, 0, 0.1)}, \frac{(u_{2}, q_{1})}{(0.2, 0.3, 0.2)}, \frac{(u_{2}, q_{2})}{(0.2, 0.3, 0)}, \frac{(u_{3}, q_{1})}{(0.1, 0.2, 0.3)}, \frac{(u_{3}, q_{2})}{(0.1, 0.4, 0.1)}}),$

$((e_{2}, q, 0), {\frac{(u_{1}, q_{1})}{(0.4, 0.5, 0.1)}, \frac{(u_{1}, q_{2})}{(0.3, 0.2, 0.1)}, \frac{(u_{2}, q_{1})}{(0.4, 0.3, 0.2)}, \frac{(u_{2}, q_{2})}{(0.1, 0.1, 0.3)}, \frac{(u_{3}, q_{1})}{(0.3, 0.3, 0.1)}, \frac{(u_{3}, q_{2})}{(0.2, 0.3, 0.3)}})} .$

The following algorithm may be followed by the hospital to fill the position.

Input the multi Q-fuzzy soft expert set $({\tilde{F}}_{Q}, Z)$ .

Find the mean of each multi Q-fuzzy set.

Find an agree-multi Q-fuzzy soft expert set and a disagree-multi Q-fuzzy soft expert set.

Find C_j = ∑_i (u, q) _ij for agree- multi Q-fuzzy soft expert set.

Find K_j = ∑_i (u, q) _ij for disagree- multi Q-fuzzy soft expert set.

Find S_j = c_j - k_j.

Find r, for which s_r = maxs_j, where s_r is the optimal choice object. If r has more than one value, then any one of them could be chosen by the hospital using its option.

Tables 1 and 2 present the agree-multi Q-fuzzy soft expert set and disagree-multi Q-fuzzy soft expert set respectively by using the mean of each multi Q-fuzzy set.

From Tables 1 and 2 we are able to compute the values of S_j = c_j - k_j as in Table 3.

Since max S_j = 0.17, hence the committee will choose candidate u₂ with a masters degree for the job.

7 Conclusion

In this paper we introduced the concept of a multi Q-fuzzy soft expert set and its operations, which are equality, union, intersection and subset, OR, and AND. We illustrate application of this novel concept in a decision making process and it is expected that the approach will be useful to handle other realistic uncertain problems.

Acknowledgments

The authors would like to acknowledge the financial support received from Universiti Kebangsaan Malaysia under the research grant IP-2014-071.

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