Interval valued intuitionistic fuzzy parameterized soft set theory and its decision making

Abstract

In this paper, we first define interval-valued intuitionistic fuzzy parameterized soft sets by combining the interval-valued intuitionistic fuzzy sets and the soft sets from parametrization point of view. By using soft level sets we then construct a parameter reduction method to obtain a better decision making. We finally give a numerical example to show the method working successfully for problems containing uncertain data.

Keywords

Soft sets interval-valued intuitionistic fuzzy sets threshold on parameter sets level sets parameter reductions decision making

1 Introduction

There are many complicated problems in real life that involve uncertain data. Therefore, to deal with uncertain data, some authors introduced theories, such as the probability theory, the rough set theory [25], the fuzzy set theory [38], the interval valued fuzzy set theory [18], the intuitionistic fuzzy set theory [2], the interval valued intuitionistic fuzzy set theory [3] etc. A theory, called “soft set theory”, was added to this theories by Molodtsov [23], which is a completely different approach for modeling vagueness and uncertainties. Then, the operations and properties of soft sets were studied by [1 , 27].

In recent years, some researchers on soft set theory have received much attention by combining soft set theory with other mathematical models. Maji et al. [22] introduced the concept of fuzzy soft sets, Çağman et al. [7] introduced fuzzy parameterized soft sets, Deli and Çağman [11] introduced intuitionistic fuzzy parameterized soft sets, Yang et al. [37] initiated interval valued fuzzy soft sets, Çağman et al. [6] gave intuitionistic fuzzy soft sets, Jiang et al. [19] defined interval valued intuitionistic fuzzy soft sets, Feng et al. [16, 17] presented soft rough sets. The theories have been applied to wide variety of fields such as soft decision making [40], fuzzy soft decision making [30], FP-soft decision making [8 , 12], relations of soft sets [28], relations of fuzzy soft sets [29], relations of intuitionistic fuzzy soft sets [13, 24] and so on.

For an adjustable approach, Jiang et al. [20] introduced a decision making method based on the level soft sets. Zhang et al. [39] presented a decision making method by using the interval-valued intuitionistic fuzzy soft sets. In this paper, we define a decision making method based on the interval valued intuitionistic fuzzy parameterized soft sets. Three major contributions of this paper are as follows. First, we define interval-valued intuitionistic fuzzy parameterized soft sets (ivifp-soft sets). Then, we give some operations and propositions on the ivifp-soft sets. Finaly, we introduce parameter reduction and algorithm to make better decisions by using soft level sets.

2 Preliminaries

In this section, we briefly recall the notions of intuitionistic fuzzy sets [2], interval-valued intuitionistic fuzzy sets [3], soft sets [10], fuzzy parameterized soft sets [7] and intuitionistic fuzzy parameterized soft sets [11].

Throughout this paper U, E and $P (U)$ denote initial universe, set of parameters and power set of U, respectively.

Definition 1. [2] Let E be a universe. An intuitionistic fuzzy set K on E can be defined as follows: $K = {< x, μ_{K} (x), γ_{K} (x) > : x \in E}$ where, μ_K : E → [0, 1] and γ_K : E → [0, 1] such that 0 ≤ μ_K (x) + γ_K (x) ≤1 for any x ∈ E.

Here, μ_K (x) and γ_K (x) is the degree of membership and degree of non-membership of the element x, respectively.

Definition 2. [3] An interval-valued intuitionistic fuzzy set A on E can be defined as follows: $A = {〈 x, [μ_{A}^{-} (x), μ_{A}^{+} (x)], [γ_{A}^{-} (x), γ_{A}^{+} (x)] 〉 : x \in E}$ where, $[μ_{A}^{-} (x), μ_{A}^{+} (x)] \subseteq [0, 1]$ and $[γ_{A}^{-} (x), γ_{A}^{+} (x)] \subseteq [0, 1]$ such that such that $0 \leq {mu}_{A}^{+} (x) + γ_{A}^{+} (x) \leq 1$ , $0 \leq μ_{A}^{-} (x) \leq μ_{A}^{+} (x) \leq 1$ and $0 \leq γ_{A}^{-} (x) \leq γ_{A}^{+} (x) \leq 1$ for any x ∈ E.

If A and B are two interval-valued fuzzy sets on E, then subset, equal, complement, union and intersection of the interval-valued fuzzy sets are defined as follows [3];

A ⊆ B if and only if $μ_{A}^{-} (x) \leq μ_{B}^{-} (x)$ , $μ_{A}^{+} (x) \leq μ_{B}^{+} (x)$ , $γ_{A}^{-} (x) \geq γ_{B}^{-} (x)$ and $γ_{A}^{+} (x) \geq γ_{B}^{+} (x)$ for ∀x ∈ E,

A = B if and only if A ⊆ B and B ⊆ A

$A^{c} = {〈 x, [γ_{A}^{-} (x), γ_{A}^{+} (x)], [μ_{A}^{-} (x), μ_{A}^{+} (x)] 〉 : x \in E}$ ,

Union of A and B is $\begin{matrix} A \cup B = & {〈 x, [sup {μ_{A}^{-} (x), μ_{B}^{-} (x)}, sup {μ_{A}^{+} (x), μ_{B}^{+} (x)}], \\ [inf {γ_{A}^{-} (x), γ_{B}^{-} (x)}, inf {γ_{A}^{+} (x), γ_{B}^{+} (x)}] 〉 : x \in E} \end{matrix}$

Intersection of A and B is $\begin{matrix} A \cap B = & {〈 x, [inf {μ_{A}^{-} (x), μ_{B}^{-} (x)}, inf {μ_{A}^{+} (x), μ_{B}^{+} (x)}], \\ [sup {γ_{A}^{-} (x), γ_{B}^{-} (x)}, sup {γ_{A}^{+} (x), γ_{B}^{+} (x)}] 〉 : x \in E} . \end{matrix}$

Definition 3. [10] A soft set F over U is a set valued function from E to $P (U)$ . It can be written a set of ordered pairs $F = {(x, F (x)) : x \in E} .$ Note that if F (x) =∅, then the element (x, F (x)) is not appeared in F. Set of all soft sets over U is denoted by S.

Example 1. Let U = {u₁, u₂, u₃, u₄, u₅, u₆}, E = {x₁, x₂, x₃, x₄, x₅} and

F = {(x₁, {u₁, u₂}) , (x₂, {u₁, u₄, u₅, u₆}) , (x₄, U) , (x₅, {u₁, u₂, u₃, u₄, u₅}) }. The tabular representation of the soft set F is as follow:

U	u₁	u₂	u₃	u₄	u₅	u₆
x₁	1	1	0	0	0	0
x₂	1	0	0	1	1	1
x₃	0	0	0	0	0	0
x₄	1	1	1	1	1	1
x₅	1	1	1	1	1	0

Table: The tabular representation of the soft set F.

Definition 4. [21] Let U = {u₁, u₂, …, u_k} be an initial universe of objects, E = {x₁, x₂, …, x_m} be a set of parameters and F be a soft set over U. For any x_j ∈ E, F (x_j) is a subset of U. Then, the choice value of an object u_i ∈ U is c_i, given by c_i = ∑_ju_ij, where a_ij are the entries in the table of the reduct-soft-set. That is, $a_{ij} = {\begin{matrix} 1, & u_{i} \in F (x_{j}) \\ 0, & u_{i} \notin F (x_{j}) \end{matrix}$

Example 2. Consider the above Example 1. Clearly, $c_{1} = \sum_{j = 1}^{5} a_{1 j} = 4,$ $c_{3} = c_{6} = \sum_{j = 1}^{5} a_{3 j} = \sum_{j = 1}^{5} a_{6 j} = 2,$ $c_{2} = c_{4} = c_{5} = \sum_{j = 1}^{5} a_{2 j} = \sum_{j = 1}^{5} a_{4 j} = \sum_{j = 1}^{5} a_{5 j} = 3 .$

Definition 5. [10] Let F, G ∈ S. Then,

If F (x) =∅ for all x ∈ E, F is said to be a null soft set, denoted by Φ.

If F (x) = U for all x ∈ E, F is said to be absolute soft set, denoted by $\hat{U}$ .

F is soft subset of G, denoted by $F \tilde{\subseteq} G$ , if F (x) ⊆ G (x) for all x ∈ E.

F = G, if $F \tilde{\subseteq} G$ and $G \tilde{\subseteq} F$ .

Soft union of F and G, denoted by $F \tilde{\cup} G$ , is a soft set over U and defined by $F \tilde{\cup} G : E \to P (U)$ such that $(F \tilde{\cup} G) (x) = F (x) \cup G (x)$ for all x ∈ E.

Soft intersection of F and G, denoted by $F \tilde{\cap} G$ , is a soft set over U and defined by $F \tilde{\cap} G : E \to P (U)$ such that $(F \tilde{\cap} G) (x) = F (x) \cap G (x)$ for all x ∈ E.

Soft complement of F is denoted by $F^{\tilde{c}}$ and defined by $F^{\tilde{c}} : E \to P (U)$ such that $F^{\tilde{c}} (x) = U \ F (x)$ for all x ∈ E.

Definition 6. [5] Let F (E) be a set of all fuzzy sets over E. Then a fuzzy parameterized soft set (FP-soft set) F_X on the universe U is defined as follows: $F_{X} = {(μ_{X} (x) / x, F_{X} (x)) : x \in E}$ where μ_X : E → [0, 1], and $F_{X} : F (E) \to P (U)$ such that F_X =∅ if μ_X (x) =0 for all x ∈ E.

Example 3. Assume that U = {u₁, u₂, u₃, u₄, u₅, u₆} is a universal set and E = {x₁, x₂, x₃, x₄} is a set of parameters. If X = {0.2/x₂, 0.5/x₃, 1/x₄} and F_X (x₂) = {u₂, u₃, u₄, u₅, u₆}, F_X (x₃) =∅, and F_X (x₄) = U, then

$\begin{matrix} F_{X} = & {(0.2 / x_{2}, {u_{2}, u_{3}, u_{4}, u_{5}, u_{6}}), (0.5 / x_{3}, \emptyset), \\ (1 / x_{4}, U)} . \end{matrix}$

Definition 7. [11] Let U be an initial universe, P (U) be the power set of U, E be a set of all parameters and K be an intuitionistic fuzzy set over E. An intuitionistic FP-soft sets ⨿_K over U is defined as follows: $\begin{matrix} ⨿_{K} = & {(< x, μ_{K} (x), γ_{K} (x) >, f_{K} (x)) : x \in E} \end{matrix}$ where μ_K : E → [0, 1], γ_K : E → [0, 1] and f_K : E → P (U) such that f_K (x) =∅ if μ_K (x) =0 and γ_K (x) =1.

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

Example 4. Assume that U = {u₁, u₂, u₃} is a universal set and E = {x₁, x₂} is a set of parameters. If $K = {< x_{1}, 0.2, 0.3 >, < x_{2}, 0.1, 0.8 >}$ and $f_{K} (x_{1}) = {u_{2}, u_{3}}, f_{K} (x_{2}) = U$ then an intuitionistic FP-soft set ⨿_K as given by $\begin{matrix} ⨿_{K} & = {(< x_{1}, 0.2, 0.3 >, {u_{2}, u_{3}}), \\ (< x_{2}, 0.1, 0.8 >, U)} \end{matrix}$

3 Interval-valued intuitionistic fuzzy parameterized soft sets

In this section, we define interval-valued intuitionistic fuzzy parameterized soft sets and their operations. Some of it is quoted from [10 , 39].

Definition 8. Let A be an interval-valued intuitionistic fuzzy set over E. Then an interval-valued intuitionistic fuzzy parameterized soft set (ivifp-soft set) F_A over U is defined as follows: $\begin{matrix} F_{A} = & {(〈 x, [μ_{A}^{-} (x), μ_{A}^{+} (x)], [γ_{A}^{-} (x), γ_{A}^{+} (x)] 〉, \\ F_{A} (x)) : x \in E} \end{matrix}$ where, $[μ_{A}^{-} (x), μ_{A}^{+} (x)] \subseteq [0, 1]$ and $[γ_{A}^{-} (x), γ_{A}^{+} (x)] \subseteq [0, 1]$ such that $0 \leq μ_{A}^{+} (x) + γ_{A}^{+} (x) \leq 1$ , $0 \leq μ_{A}^{-} (x) \leq μ_{A}^{+} (x) \leq 1$ and $0 \leq γ_{A}^{-} (x) \leq γ_{A}^{+} (x) \leq 1$ for any x ∈ E.

Here, $[μ_{A}^{-} (x), μ_{A}^{+} (x)]$ and $[γ_{A}^{-} (x), γ_{A}^{+} (x)]$ called membership function and non-membership of interval-valued intuitionistic fuzzy set, respectively. The value $[μ_{A}^{-} (x), μ_{A}^{+} (x)]$ and $[γ_{A}^{-} (x), γ_{A}^{+} (x)]$ are the degree of importance and unimportant of the parameter x. $μ_{A}^{-} (x)$ and $μ_{A}^{+} (x)$ are referred to as the lower and upper degrees of membership the element x, and $γ_{A}^{-} (x)$ and $γ_{A}^{+} (x)$ are referred to as the lower and upper degrees of membership the element x, respectively. The sets of all ivifp-soft sets over U will be denoted by IVIFPS.

It will be seen $(〈 x, [μ_{A}^{-} (x), μ_{A}^{+} (x)], [γ_{A}^{-} (x), γ_{A}^{+} (x)] 〉, F_{A} (x))$ as an element of F_A.

Remark 1. Note that the following statements

If $μ_{A}^{-} (x) = μ_{A}^{+} (x) = 0$ , then it must be F_A (x) =∅. Thus, this statement is not written in the ivifp-soft set.

If $μ_{A}^{-} (x) = μ_{A}^{+} (x) = 1$ , then it must not be F_A (x) =∅.

If F_A (x) =∅ for x ∈ E, then it means that x is not suitable parameter. In this statement, the ivifp-soft set F_A do not include such elements.

Definition 9. If F_A and G_B are two ivifp-soft sets on U, then ivifp-soft subset, ivifp-soft equal, ivifp-soft complement, ivifp-soft union and ivifp-soft intersection of the interval-valued fuzzy sets are defined as follows, respectively.

If F_A (x) =∅, $μ_{A}^{-} (x) = μ_{A}^{+} (x) = 0$ and $γ_{A}^{-} (x) = γ_{A}^{+} (x) = 1$ for all x ∈ E, then F_A is called a empty ivifp-soft set, denoted by $\tilde{Φ}$ .

If F_A (x) = U, $μ_{A}^{-} (x) = μ_{A}^{+} (x) = 1$ and $γ_{A}^{-} (x) = γ_{A}^{+} (x) = 0$ for all x ∈ E, then F_A is called a universal ivifp-soft set, denoted by U.

F_A ⊑ G_B if and only if $μ_{A}^{-} (x) \leq μ_{B}^{-} (x)$ , $μ_{A}^{+} (x) \leq μ_{B}^{+} (x)$ , $γ_{A}^{-} (x) \geq γ_{B}^{-} (x)$ , $γ_{A}^{+} (x) \geq γ_{B}^{+} (x)$ and F_A (x) ⊆ G_B (x) for all x ∈ E,

F_A = G_B if and only if F_A ⊑ G_B and G_B ⊑ F_A

$F_{A}^{\tilde{c}} = {(〈 x, [γ_{A}^{-} (x), γ_{A}^{+} (x)], [μ_{A}^{-} (x), μ_{A}^{+} (x)] 〉, U \ F_{A} (x)) : x \in E}$

ivifp-union of F_A and G_B is $\begin{matrix} F_{A} ⊔ G_{B} & = & {(〈 x, [sup {μ_{A}^{-} (x), μ_{B}^{-} (x)}, \\ sup {μ_{A}^{+} (x), μ_{B}^{+} (x)}], \\ [inf {γ_{A}^{-} (x), γ_{B}^{-} (x)}, \\ inf {γ_{A}^{+} (x), γ_{B}^{+} (x)}] 〉, \\ F_{A} (x) \cup G_{B} (x)) : x \in E} \end{matrix}$

ivifp-intersection of F_A and G_B is $\begin{matrix} F_{A} ⊓ G_{B} & = & {(〈 x, [inf {μ_{A}^{-} (x), μ_{B}^{-} (x)}, \\ inf {μ_{A}^{+} (x), μ_{B}^{+} (x)}], \\ [sup {γ_{A}^{-} (x), γ_{B}^{-} (x)}, \\ sup {γ_{A}^{+} (x), γ_{B}^{+} (x)}] 〉, \\ F_{A} (x) \cap G_{B} (x)) : x \in E} \end{matrix}$

Example 1. Let U = {u₁, u₂, u₃, u₄}, E = {x₁, x₂, x₃}, F_A and G_B be ivifp-sets such that $\begin{matrix} F_{A} & = & {(〈 x_{1}, [0.1, 0.5], [0.4, 0.5] 〉, {u_{1}, u_{2}}), \\ (〈 x_{2}, [0.2, 0.3], [0.5, 0.7] 〉, {u_{2}, u_{3}})} \\ G_{B} & = & {(〈 x_{2}, [0.1, 0.6], [0.2, 0.3] 〉, {u_{3}, u_{4}}), \\ (〈 x_{3}, [0.4, 0.7], [0.1, 0.2] 〉, {u_{3}})} . \end{matrix}$ Then, $\begin{matrix} F_{A} ⊔ G_{B} & = & {(〈 x_{1}, [0.1, 0.5], [0.4, 0.5] 〉, {u_{1}, u_{2}}), \\ (〈 x_{2}, [0.2, 0.6], [0.2, 0.3] 〉, {u_{2}, u_{3}, u_{4}}), \\ (〈 x_{3}, [0.4, 0.7], [0.1, 0.2] 〉, {u_{3}})} \\ F_{A} ⊓ G_{B} & = & {(〈 x_{2}, [0.1, 0.3], [0.5, 0.7] 〉, {u_{3}})} \\ F_{A}^{\tilde{c}} & = & {(〈 x_{1}, [0.4, 0.5], [0.1, 0.5] 〉, {u_{3}, u_{4}}), \\ (〈 x_{2}, [0.5, 0.7], [0.2, 0.3] 〉, {u_{2}, u_{3}})} . \end{matrix}$

Remark 2.F_A ⊑ G_B does not imply that every element of F_A is an element of G_B as in the definition of classical subset. For example, ivifp-soft sets F_A and G_B are defined as $\begin{matrix} F_{A} & = & {(〈 x_{1}, [0.1, 0.3], [0.5, 0.5] 〉 {u_{1}, u_{2}}), \\ (〈 x_{2}, [0.3, 0.4], [0.4, 0.5] 〉), {u_{2}}} \\ G_{B} & = & {(〈 x_{1}, [0.2, 0.6], [0.3, 0.4] 〉, U), \\ (〈 x_{2}, [0.4, 0.7], [0.1, 0.3] 〉, {u_{1}, u_{4}})} . \end{matrix}$ It can be seen that F_A ⊑ G_B, but every element of F_A is not an element of G_B.

Proposition 1.LetF_A, G_B ∈ IVIFPS. Then

$F_{A} ⊑ \tilde{U}$

$\tilde{Φ} ⊑ F_{A}$

F_A ⊑ F_A

Proof. It is clear from Definition 9 and Definitions 8.

Proposition 2.LetF_A ∈ IVIFPS. Then

F_A = G_BandG_B = H_C ⇔ F_A = H_C

F_A ⊑ G_BandG_B ⊑ F_A ⇔ F_A = G_B

F_A ⊑ G_BandG_B ⊑ H_C ⇒ F_A ⊑ H_C

Proof. It can be proved by Definition 9.

Proposition 3.LetF_A ∈ IVIFPS. Then

$(F_{A}^{\tilde{c}})^{\tilde{c}} = F_{A}$

${\tilde{Φ}}^{\tilde{c}} = \tilde{U}$

${\tilde{U}}^{\tilde{c}} = \tilde{Φ}$

Proof. It is trivial.

Proposition 4.LetF_A, G_B, H_C ∈ IVIFPS. Then

F_A ⊔ F_A = F_A

$F_{A} ⊔ \tilde{Φ} = F_{A}$

$F_{A} ⊔ \tilde{U} = \tilde{U}$

F_A ⊔ G_B = G_B ⊔ F_A

(F_A ⊔ G_B) ⊔ H_C = F_A ⊔ (G_B ⊔ H_C)

Proof. It is clear.

Proposition 5.LetF_A, G_B, H_C ∈ IVIFPS. Then

F_A ⊓ F_A = F_A

$F_{A} ⊓ \tilde{Φ} = \tilde{Φ}$

$F_{A} ⊓ \tilde{U} = F_{A}$

F_A ⊓ G_B = G_B ⊓ F_A

(F_A ⊓ G_B) ⊓ H_C = F_A ⊓ (G_B ⊓ H_C)

Proof. It is clear.

Remark 3. Let F_A ∈ IVIFPS. If $F_{A} \neq \tilde{Φ}$ or $F_{A} \neq \tilde{U}$ , then $F_{A} ⊔ F_{A}^{\tilde{c}} \neq \tilde{U}$ and $F_{A} ⊓ F_{A}^{\tilde{c}} \neq \tilde{Φ}$ . It can be seen clearly in the Example 5.

Proposition 6.LetF_A, G_B, H_C ∈ IVIFPS. Then

F_A ⊔ (G_B ⊓ H_C) = (F_A ⊔ G_B) ⊓ (F_A ⊔ H_C)

F_A ⊓ (G_B ⊔ H_C) = (F_A ⊓ G_B) ⊔ (F_A ⊓ H_C)

Proof. It can be proved easily from Definition 9.

Proposition 7.LetF_A, G_B ∈ IVIFPS. Then

$(F_{A} ⊔ G_{B})^{\tilde{c}} = F_{A}^{\tilde{c}} ⊓ G_{B}^{\tilde{c}}$

$(F_{A} ⊓ G_{B})^{\tilde{c}} = F_{A}^{\tilde{c}} ⊔ G_{B}^{\tilde{c}}$

Proof. Let F_A, G_B ∈ IVIFPS such that $\begin{matrix} F_{A} & = & {(〈 x, [μ_{A}^{-} (x), μ_{A}^{+} (x)], \\ [γ_{A}^{-} (x), γ_{A}^{+} (x)] 〉, F_{A} (x)) : x \in E} \\ G_{B} & = & {(〈 x, [μ_{B}^{-} (x), μ_{B}^{+} (x)], \\ [γ_{B}^{-} (x), γ_{B}^{+} (x)] 〉, G_{B} (x)) : x \in E} . \end{matrix}$

Then, $\begin{matrix} (F_{A} ⊔ G_{B})^{\tilde{c}} & = & {(〈 x, [inf {γ_{A}^{-} (x), γ_{B}^{-} (x)}, \\ inf {γ_{A}^{+} (x), γ_{B}^{+} (x)}], \\ [sup {μ_{A}^{-} (x), μ_{B}^{-} (x)}, \\ sup {μ_{A}^{+} (x), μ_{B}^{+} (x)}] 〉, \\ U \ (F_{A} (x) \cup G_{B} (x))) : x \in E} \\ = & {(〈 x, [inf {γ_{A}^{-} (x), γ_{B}^{-} (x)}, \\ inf {γ_{A}^{+} (x), γ_{B}^{+} (x)}], \\ [sup {μ_{A}^{-} (x) μ_{B}^{-} (x)}, sup {μ_{A}^{+} (x), μ_{B}^{+} (x)}] 〉, \\ (U \ F_{A} (x)) \cap (U \ G_{B} (x))) : x \in E} \\ = & F_{A}^{\tilde{c}} ⊓ G_{B}^{\tilde{c}} . \end{matrix}$

It can be proved similar way (i.).

Definition 10.F_A, G_B ∈ IVIFPS. The

OR-product of F_A and G_B, denoted by F_A ∨ G_B, is defined as following $\begin{matrix} F_{A} \lor G_{B} (x, y) & = & {(〈 (x, y), [sup {μ_{A}^{-} (x), μ_{B}^{-} (y)}, \\ sup {μ_{A}^{+} (x), μ_{B}^{+} (y)}], \\ [inf {γ_{A}^{-} (x), γ_{B}^{-} (y)}, \\ inf {γ_{A}^{+} (x), γ_{B}^{+} (y)}] 〉, \\ F_{A} \lor G_{B} (x, y)) : x, y \in E} . \end{matrix}$ where (F_A ∨ G_B) (x, y) = F_A (x) ∪ G_B (y).

AND-product of F_A and G_B, denoted by F_A ∧ G_B, is defined as following $\begin{matrix} (F_{A} \land G_{B}) (x, y) & = & {(〈 (x, y), [inf {μ_{A}^{-} (x), μ_{B}^{-} (y)}, \\ inf {μ_{A}^{+} (x), μ_{B}^{+} (y)}], \\ [sup {γ_{A}^{-} (x), γ_{B}^{-} (y)}, \\ sup {γ_{A}^{+} (x), γ_{B}^{+} (y)}] 〉, \\ (F_{A} \land G_{B}) (x, y)) : x, y \in E} . \end{matrix}$ where (F_A ∧ G_B) (x) = F_A (x) ∩ G_B (x).

Example 6. Let consider the Example 5, again. Then we have $\begin{matrix} F_{A} \lor G_{B} & = & {(〈 (x_{1}, x_{2}), [0.1, 0.6], [0.2, 0.3] 〉, U), \\ (〈 (x_{1}, x_{3}), [0.4, 0.7], [0.1, 0.2] 〉, {u_{1}, u_{2}, u_{3}}), \\ (〈 (x_{2}, x_{2}), [0.2, 0.6], [0.2, 0.3] 〉, {u_{2}, u_{3}, u_{4}}), \\ (〈 (x_{2}, x_{3}), [0.4, 0.7], [0.1, 0.2] 〉, {u_{2}, u_{3}})} \end{matrix}$ and $\begin{matrix} F_{A} \land G_{B} & = & {(〈 (x_{2}, x_{2}), [0.1, 0.3], [0.5, 0.7] 〉, {u_{3}}), \\ (〈 (x_{2}, x_{3}), [0.2, 0.3], [0.5, 0.7] 〉, {u_{3}})} \end{matrix}$

Proposition 8.LetF_A, G_B, H_C ∈ IVIFPS. Then

$F_{A} \land \tilde{Φ} = \tilde{Φ}$

$F_{A} \land \tilde{U} = F_{A}$

F_A ∧ G_B = G_B ∧ F_A

F_A ∨ G_B = G_B ∨ F_A

(F_A ∧ G_B) ∧ H_C = F_A ∧ (G_B ∧ H_C)

(F_A ∨ G_B) ∨ H_C = F_A ∨ (G_B ∨ H_C)

Proof. It can be proved easily from Definition 10.

4 Parameter reduction method

In this section, we first define a level set for ivifp-soft sets. We then define a parameter reduction method of an ivifp-soft set that produce a soft set from an ivifp-soft set. This concept present an adjustable approach to the ivifp-soft sets based decision making problems. Some of it is quoted from [19–21 , 39].

In the following, some definitions and operations on intuitionistic fuzzy soft sets and interval valued intuitionistic fuzzy soft sets defined in [20, 39], we extend these definitions and operations to ivifp-soft sets.

Definition 11. Let F_A ∈ IVIFPS. Then. For s = [s^-, s⁺], t = [t^-, t⁺] ⊆ [0, 1], the (s, t)-level soft set of F_A is a crisp soft set, denoted by (F_A ; (s, t)) and defined by $(F_{A}; (s, t)) = {(x_{i}, F_{A} (x_{i})) : x_{i} \in E}$ where, $s^{-} \leq μ_{A}^{-} (x_{i})$ , $s^{+} \leq μ_{A}^{+} (x_{i})$ , $t^{-} \geq γ_{A}^{-} (x_{i}),$ and $t^{+} \geq γ_{A}^{+} (x_{i})$ .

Remark 4. In Definition 11, s ⊆ [0, 1] can be viewed as a given least threshold on degrees of membership and t ⊆ [0, 1] can be viewed as a given greatest threshold on degrees of non-membership. If $s^{-} \leq μ_{A}^{-} (x_{i})$ , $s^{+} \leq μ_{A}^{+} (x_{i})$ , $t^{-} \geq γ_{A}^{-} (x_{i})$ and $t^{+} \geq γ_{A}^{+} (x_{i})$ , it shows that the degree of the membership of x with respect to the u is not less than s and the degree of the nonmembership of u with respect to the parameter x is not more than t. In practical applications of ivifp-soft sets, the thresholds s, t ⊆ [0, 1] is pre-established by decision makers and reflect decision makers' requirements on “membership levels” and “non-membership levels”.

Example 7. Consider the above Example 5. Clearly, for s = [0.2, 0.5] and t = [0.6, 0.9] the (s, t)-level soft set of F_A as follows $(F_{A}; (s, t)) = {(x_{2}, {u_{2}, u_{3}})} .$

Theorem 1.LetF_A, G_B ∈ IVIFPS.

If $s_{2}^{-} \leq s_{1}^{-}$ , $s_{2}^{+} \leq s_{1}^{+}$ , $t_{2}^{-} \geq t_{1}^{-}$ and $t_{2}^{+} \geq t_{1}^{+}$ then we have (F_A ; (s₁, t₁)) ⊑ (F_A ; (s₂, t₂)), where (F_A ; (s₁, t₁)) and (F_A ; (s₂, t₂)) are (s₁, t₁)-level soft set and (s₂, t₂)-level soft sets of F_A, respectively.

If F_A ⊑ G_B, then we have $(F_{A}; (s, t)) \tilde{\subseteq}$ (G_B ; (s, t)), where (F_A ; (s, t)) and (G_B ; (s, t)) are (s, t)-level soft sets F_A and G_B, respectively.

Proof. It can be proved easily by using Definition 9 (iii.) and Definition 11.

Remark 5. In some practical applications the thresholds s, t ⊆ [0, 1] decision makers need to impose different thresholds. To cope with such problems, we replace a constant value the thresholds by a function as the thresholds membership values.

Definition 12. Let F_A ∈ IVIFPS and $s_{mid} = [s_{mid}^{-}, s_{mid}^{+}]$ , $t_{mid} = [t_{mid}^{-}, t_{mid}^{+}] \subseteq [0, 1]$ which is called a threshold of ivifp-soft set. The level soft set of F_A with respect to (s_mid, t_mid) is a crisp soft set, denoted by (F_A ; (s_mid, t_mid)), defined by $(F_{A}; (s_{mid}, t_{mid})) = {((x_{i}, F_{A} (x_{i})) : x_{i} \in E}$ where, $s_{mid}^{-} \leq μ_{A}^{-} (x_{i})$ , $s_{mid}^{+} \leq μ_{A}^{+} (x_{i})$ , $t_{mid}^{-} \geq γ_{A}^{-} (x_{i})$ , and $t_{mid}^{+} \geq γ_{A}^{+} (x_{i})$ such that, $\begin{matrix} s_{mid}^{-} & = & \sum_{x_{i} \in E^{*}} μ_{A}^{-} (x_{i}) / | E^{*} |, \\ s_{mid}^{+} & = & \sum_{x_{i} \in E^{*}} μ_{A}^{+} (x_{i}) / | E^{*} |, \\ t_{mid}^{-} & = & \sum_{x_{i} \in E^{*}} γ_{A}^{-} (x_{i}) / | E^{*} |, \\ t_{mid}^{+} & = & \sum_{x_{i} \in E^{*}} γ_{A}^{+} (x_{i}) / | E^{*} | \end{matrix}$ for all x_i ∈ E^*, where if x_j ∈ E \ E^*, then F_A (x_j) =∅.

The (s_mid, t_mid) is called the mid-threshold of the ivifp-soft set F_A. In the following discussions, the mid-level decision rule will mean using the mid-threshold and considering the mid-level soft set in ivifp-soft sets based decision making.

Remark 6. In Definition 12, s_mid ⊆ [0, 1] can be viewed as a given least threshold on degrees of membership and t_mid ⊆ [0, 1] can be viewed as a given least threshold on degrees of non-membership with respect to the u.

If $s_{mid}^{-} \leq μ_{A}^{-} (x_{i})$ , $s_{mid}^{+} \leq μ_{A}^{+} (x_{i})$ , $t_{mid}^{-} \geq γ_{A}^{-} (x_{i})$ and $t_{mid}^{+} \geq γ_{A}^{+} (x_{i})$ it shows that the degree of the membership of x with respect to the u is not less than s_mid and the degree of the nonmembership of u with respect to the parameter x is not more than t_mid.

Definition 13. Let F_A ∈ IVIFPS, $s_{\max} = [s_{\max}^{-}, s_{\max}^{+}]$ and $t_{\min} = [t_{\min}^{-}, t_{\min}^{+}] \subseteq [0, 1]$ which is called a threshold of ivifp-soft set. The level soft set of F_A with respect to (s_max, t_min) is a crisp soft set, denoted by (F_A ; (s_max, t_min)), defined by; $(F_{A}; (s_{\max}, t_{\min})) = {((x_{i}, F_{A} (x_{i})) : x_{i} \in E}$ where, $s_{\max}^{-} \leq μ_{A}^{-} (x_{i})$ , $s_{\max}^{+} \leq μ_{A}^{+} (x_{i})$ , $t_{\min}^{-} \geq γ_{A}^{-} (x_{i})$ and $t_{\min}^{+} \geq γ_{A}^{+} (x_{i})$ such that, $\begin{matrix} s_{\max}^{-} & = & sup {μ_{A}^{-} (x_{i}) : x_{i} \in E}, \\ s_{\max}^{+} & = & sup {μ_{A}^{+} (x_{i}) : x_{i} \in E}, \\ t_{\min}^{-} & = & inf {γ_{A}^{-} (x_{i}) : x_{i} \in E}, \\ t_{\min}^{+} & = & inf {γ_{A}^{+} (x_{i}) : x_{i} \in E} . \end{matrix}$ The (s_max, t_min) is called the Mm-threshold of the ivifp-soft set F_A. In the following discussions, the Mm-level decision rule will mean using the Mm-threshold and considering the Mm-level soft set in ivifp-soft sets based decision making.

Remark 7. In Definition 13, s_max [0, 1] can be viewed as a given least threshold on degrees of membership and t_min ⊆ [0, 1] can be viewed as a given least threshold on degrees of non-membership with respect to the u.

If $s_{\max}^{-} \leq μ_{A}^{-} (x_{i})$ , $s_{\max}^{+} \leq μ_{A}^{+} (x_{i})$ , $t_{\min}^{-} \geq γ_{A}^{-} (x_{i})$ and $t_{\min}^{+} \geq γ_{A}^{+} (x_{i})$ it shows that the degree of the membership of x with respect to the u is not less than s_max and the degree of the nonmembership of u with respect to the parameter x is not more than t_min.

Definition 14. Let F_A ∈ IVIFPS and an $s_{\min} = [s_{\min}^{-}, s_{\min}^{+}]$ , $t_{\min} = [t_{\min}^{-}, t_{\min}^{+}] \subseteq [0, 1]$ which is called a threshold of ivifp-soft set. The level soft set of F_A with respect to (s_min, t_min) is a crisp soft set, denoted by (F_A ; (s_min, t_min)), defined by; $(F_{A}; (s_{\min}, t_{\min})) = {((x_{i}, F_{A} (x_{i})) : x_{i} \in E}$ where, $s_{\min}^{-} \leq μ_{A}^{-} (x_{i})$ , $s_{\min}^{+} \leq μ_{A}^{+} (x_{i})$ , $t_{\min}^{-} \geq γ_{A}^{-} (x_{i})$ and $t_{\min}^{+} \geq γ_{A}^{+} (x_{i})$ such that $\begin{matrix} s_{\min}^{-} & = & inf {μ_{A}^{-} (x_{i}) : x_{i} \in E}, \\ s_{\min}^{+} & = & inf {μ_{A}^{+} (x_{i}) : x_{i} \in E}, \\ t_{\min}^{-} & = & inf {γ_{A}^{-} (x_{i}) : x_{i} \in E}, \\ t_{\min}^{+} & = & inf {γ_{A}^{+} (x_{i}) : x_{i} \in E} . \end{matrix}$

The (s_min, t_min) is called the mm-threshold of the ivifp-soft set F_A. In the following discussions, the mm-level decision rule will mean using the mm-threshold and considering the mm-level soft set in ivifp-soft sets based decision making.

Remark 8. In Definition 14, s_min [0, 1] can be viewed as a given least threshold on degrees of membership and t_min ⊆ [0, 1] can be viewed as a given least threshold on degrees of non-membership with respect to the u.

If $s_{\min}^{-} \leq μ_{A}^{-} (x_{i})$ , $s_{\min}^{+} \leq μ_{A}^{+} (x_{i})$ , $t_{\min}^{-} \geq γ_{A}^{-} (x_{i})$ and $t_{\min}^{+} \geq γ_{A}^{+} (x_{i})$ it shows that the degree of the membership of x with respect to the u is not less than s_min and the degree of the nonmembership of u with respect to the parameter x is not more than t_min.

Example 8. Let be E = {x₁, x₂, x₃}, U = {u₁, u₂, u₃} and F_A ∈ IVIFPS such that $\begin{matrix} F_{A} & = & {(〈 x_{1}, [0.1, 0.5], [0.3, 0.4] 〉, {u_{1}, u_{2}}), \\ (〈 x_{2}, [0.3, 0.4], [0.4, 0.5] 〉, {u_{2}, u_{3}}), \\ (〈 x_{3}, [0.5, 0.6], [0.2, 0.3] 〉, {u_{1}, u_{2}, u_{3}})} . \end{matrix}$ Then $\begin{matrix} s_{mid} = [0.3, 0.5], & t_{mid} = [0.3, 0.4] \\ s_{\max} = [0.5, 0.6], & t_{\min} = [0.2, 0.3] \\ s_{\min} = [0.1, 0.4], & t_{\min} = [0.2, 0.3] \end{matrix}$ Thus, we have $\begin{matrix} (F_{A}; (s_{mid}, t_{mid})) & = & {(x_{3}, {u_{1}, u_{2}, u_{3}})} \\ (F_{A}; (s_{\max}, t_{\min})) & = & {(x_{3}, {u_{1}, u_{2}, u_{3}})} \\ (F_{A}; (s_{\min}, t_{\min})) & = & {(x_{3}, {u_{1}, u_{2}, u_{3}})} \end{matrix}$

Theorem 2.LetF_A ∈ IVIFPS, (F_A ; (s_mid, t_mid)), (F_A ; (s_max, t_max)) and (F_A ; (s_min, t_min)) be the mid-level soft set, max-level soft set and min-level soft set ofF_A, respectively. Then,

$(F_{A}; (s_{\max}, t_{\max})) \tilde{\subseteq} (F_{A}; (s_{mid}, t_{mid}))$

$(F_{A}; (s_{\max}, t_{\max})) \tilde{\subseteq} (F_{A}; (s_{\min}, t_{\min}))$ .

Proof. Let F_A ∈ IVIFPS. From Definition 12, Definition 13 and Definition 14, it can be seen that $s_{\min}^{-} \leq s_{mid}^{-} \leq s_{\max}^{-}$ and $t_{\min}^{-} \leq t_{mid}^{-} \leq t_{\max}^{-}$ . Thus,

For all x_j ∈ E which providing the inequalities $s_{mid}^{-} \leq μ_{A}^{-} (x_{j}) < s_{\max}^{-}$ , (x_i, F_A (x_i)) ∉ (F_A ; (s_max, t_max)). So, $(F_{A}; (s_{\max}, t_{\max})) \tilde{\subseteq} (F_{A}; (s_{mid}, t_{mid}))$

It can be proved similar way.

Now, we construct an ivifp-soft set decision making method by the following algorithm;

Algorithm:

Input the ivifp-soft set F_A,

Input a threshold (s_mid, t_mid) (or (s_max, t_min) , (s_min, t_min)) by using mid-level decision rule (or Mm-level decision rule, mm-level decision rule) for decision making.

Compute mid-level soft set (F_A ; (s_mid, t_mid)) (or Mm-level soft set ((F_A ; (s_max, t_min)), mm-level soft set (F_A ; (s_min, t_min))

Present the level soft set (F_A ; (s_min, t_mid)) (or the level soft set ((F_A ; (s_max, t_min)), the level soft set (F_A ; (s_min, t_min)) in tabular form.

Compute the choice value c_i of u_i for any u_i ∈ U,

The optimal decision is to select u_k if $c_{k} = max_{u_{i} \in U} c_{i} .$

Remark 9. If k has more than one value then any one of u_k may be chosen. If there are too many optimal choices in Step 6, we may go back to the second step and change the threshold (or decision rule) such that only one optimal choice remains in the end.

Example 9. Let us suppose that some one goes to the real estate office to buy a house. Moreover, assume that there are eight alternatives U = {u₁, u₂, u₃, u₄, u₅, u₆, u₇, u₈}. There may be for parameters E = {x₁, x₂, x₃, x₄, x₅, x₆} to evaluate the houses. The parameters x_i (i = 1, 2, 3, 4, 5, 6) stand for “with parking”, “spacious”, “expensive”, “green surrounding”, “mo dern” and “near to city”, respectively. He/She considers x₁, x₂, x₃ and x₄ parameters to buy a suitable house.

Step 1. After thinking thoroughly, he/she evaluates the alternative according to choosing parameters and constructs an ivifp-soft set F_A as following

\begin{matrix} F_{A} & = & {(〈 x_{1}, [0.6, 0.8], [0.1, 0.2] 〉, {u_{1}, u_{4}, u_{5}, u_{7}}), \\ (〈 x_{2}, [0.5, 0.6], [0.3, 0.4] 〉, {u_{2}, u_{5}, u_{6}, u_{8}}), \\ (〈 x_{3}, [0.4, 0.5], [0.3, 0.4] 〉, {u_{2}, u_{3}, u_{6}, u_{7}}), \\ (〈 x_{4}, [0.1, 0.2], [0.4, 0.8] 〉, {u_{1}, u_{4}, u_{5}, u_{8}}) \\ (〈 x_{5}, [0.4, 0.5], [0.2, 0.5] 〉, {u_{3}, u_{6}, u_{7}}), \\ (〈 x_{6}, [0.7, 0.7], [0.1, 0.3] 〉, {u_{2}, u_{4}, u_{8}})} . \end{matrix}

Step 2. Then, we have

\begin{matrix} s_{mid} = [0.54, 0.66], & t_{mid} = [0.28, 0.52] \\ s_{\max} = [0.7, 0.8], & t_{\min} = [0.1, 0.2] \\ s_{\min} = [0.1, 0.2], & t_{\min} = [0.1, 0.2] . \end{matrix}

Step 3. Thus, the (s_mid, t_mid)-level soft set of F_A is (After the necessary calculations, they can be seen that (s_max, t_min)-level soft set and (s_max, t_min)-level soft set of F_A are not suitable for decision making in this problem.)

\begin{matrix} (F_{A}; (s_{mid}, t_{mid})) = & {(x_{1}, {u_{1}, u_{4}, u_{5}, u_{7}}), \\ (x_{6}, {u_{2}, u_{4}, u_{8}})} . \end{matrix}

Step 4. Tabular form of

(F_{A}; (S_{m i d,} t_{m i d}))

U	u₁	u₂	u₃	u₄	u₅	u₆	u₇	u₈
x₁	1	0	0	1	1	0	1	0
x₆	0	1	0	1	0	0	0	1

Step 5. Then, we have the choice value c_i for i = 1, 2 ,..., 8

c₁=1, c₂=1, c₃=0, c₄=2,

c₅=1, c₆=0, c₇=1, c₈=1.

Step 6. So, the optimal decision is u₄.

5 Conclusion

In this paper, we first proposed intuitionistic fuzzy parameterized soft sets and some results. Then, we applied the level soft sets approach introduced by Zhang et al. [39] from parametrization point of view. These approaches are desirable to further apply level soft sets of intuitionistic fuzzy parameterized soft sets to other some practical theories that contain uncertainties. This decision making method can easily be applied for group decision making with the help of the Definition 10.

References

Ali

M.I.

, Feng

, Liu

, Min

W.K.

and Shabir

, On some new operations in soft set theory, Computers and Mathematics with Applications57 (2009), 1547–1553.

Atanassov

, Intuitionistic fuzzy sets, Fuzzy Sets and Systems20 (1986), 87–96.

Atanassov

and Garov

, Interval valued intuitionistic fuzzy sets, Fuzzy Sets and Systems31 (1989), 343–349.

Çağman

and Enginoğlu

, Soft set theory and uni-int decision making, European Journal of Operational Research207 (2010), 848–855.

Çağman

, Çitak

and Enginoğlu

, FP-soft set theory and its applications, Annals of Fuzzy Mathematics and Informatics2/2 (2011), 219–226.

Çağman

and Karataş

, Intuitionistic fuzzy soft set theory and its decision making, Journal of Intelligent and Fuzzy Systems24/4 (2013), 829–836.

Çağman

, Erdoğan

and Enginoğlu

, FP-soft set theory and its applications, Annals of Fuzzy Mathematics and informatics2/2 (2011), 219–226.

Çağman

and Deli

, Means of FP-soft sets and its applications, Hacettepe Journal of Mathematics and Statistics41(5) (2012), 615–625.

Çağman

and Deli

, Product of FP-soft sets and its applications, Hacettepe Journal of Mathematics and Statistics41(3) (2012), 365–374.

10.

Çağman

, Contributions to the theory of soft sets, Journal of New Results in Science (4) (2014), 33–41.

11.

Deli

and Çağman

, Intuitionistic fuzzy parameterized soft set theory and its decision making, Applied Soft Computing28 (2015), 109–113.

12.

Deli

and Çağman

, Relations on FP-soft sets applied to decision making problems, Journal of New Theory3 (2015), 98–107.

13.

Dinda

and Samanta

T.K.

, Relations on intuitionistic fuzzy soft sets, Gen Math Notes1/2 (2010), 74–83.

14.

Feng

, Jun

Y.B.

and Zhao

, Soft semirings, Computers and Mathematics with Applications56/10 (2008), 2621–2628.

15.

Feng

, Jun

Y.B.

, Liu

and Li

, An adjustable approach to fuzzy soft sets based decision making, Journal of Computational and Applied Mathematics234 (2010), 10–20.

16.

Feng

, Liu

, Fotea

V.L.

and Jun

Y.B.

, Soft sets and soft rough sets, Information Sciences181 (2011), 1125–1137.

17.

Feng

, Li

, Davvaz

and Irfan Ali

, Soft sets combined with fuzzy sets and rough sets: A tentative approach, Soft Computing14 (2010), 899–911.

18.

Gorzalczany

M.B.

, A method of inference in approximate reasoning based on interval valued fuzzy sets, Fuzzy Sets and Systems21 (1987), 1–17.

19.

Jiang

, Tang

, Chen

, Liu

and Tang

, Interval-valued intuitionistic fuzzy soft sets and their properties, Computers and Mathematics with Applications60 (2010), 906–918.

20.

Jiang

, Tang

and Chen

, An adjustable approach to intuitionistic fuzzy soft sets based decision making, Applied Mathematical Modelling35 (2011), 824–836.

21.

Maji

P.K.

and Roy

A.R.

, An application of soft sets in a decision making problem, Computers and Mathematics with Applications44 (2002), 1077–1083.

22.

Maji

P.K.

, Biswas

and Roy

A.R.

, Fuzzy soft sets, Journal of Fuzzy Mathematics9(3) (2001), 589–602.

23.

Molodtsov

D.A.

, Soft set theory-first results, Computers and Mathematics with Applications37 (1999), 19–31.

24.

Mukherjee

and Chakraborty

S.B.

, On Intuitionistic fuzzy soft relations, Bulletin of Kerala Mathematics Association5/1 (2008), 35–42.

25.

Pawlak

, Rough sets, International Journal of Information and Computer Sciences11 (1982), 341–356.

26.

Peng

J.J.

, Wang

J.Q.

, Wu

X.H.

, Zhang

H.Y.

and Chen

X.H.

, The Fuzzy Cross-entropy for Intuitionistic Hesitant Fuzzy Sets and its Application in Multi-criteria Decision making, International Journal of Systems Science (2015). DOI: 10.1080/00207721.2014.993744

27.

Sezgin

and Atagün

A.O.

, On operations of soft sets, Computers and Mathematics with Applications61/5 (2011), 1457–1467.

28.

Som

, On the theory of soft sets, soft relation and fuzzy soft relation, Proc of the National Conference on Uncertainty: A Mathematical Approach, UAMA-06, Burdwan, 2006, pp. 1–9.

29.

Sut

D.K.

, An application of fuzzy soft relation in decision making problems, International Journal of Mathematics Trends and Technology3/2 (2012), 51–54.

30.

Xiao

, Gong

and Zou

, A combined forecasting approach based on fuzzy soft sets, J Comput Appl Math228 (2009), 326–333.

31.

Wang

J.Q.

and Zhang

H.Y.

, Multi-criteria decision-making approach based on Atanassov’s intuitionistic fuzzy sets with incomplete certain information on weights, IEEE Transactions on Fuzzy Systems21(3) (2013), 510–515.

32.

Wang

J.Q.

, Nie

R.R.

, Zhang

H.Y.

and Chen

X.H.

, New operators on triangular intuitionistic fuzzy numbers and their applications in system fault analysis, Information Sciences251 (2013), 79–95.

33.

Wang

J.Q.

, Nie

R.R.

, Zhang

H.Y.

and Chen

X.H.

, Intuitionistic fuzzy multi-criteria decision-making method based on evidential reasoning, Applied Soft Computing13(4) (2013), 1823–1831.

34.

Wang

J.Q.

, Li

K.J.

, Zhang

H.Y.

and Chen

X.H.

, A score function based on relative entropy and its application in intuitionistic normal fuzzy multiple criteria decision making, Journal of Intelligent and Fuzzy Systems25(3) (2013), 567–576.

35.

Wang

J.Q.

, Zhou

, Li

K.J.

, Zhang

H.Y.

and Chen

X.H.

, Multicriteria decision-making method based on normal intuitionistic fuzzy-induced generalized aggregation operator, TOP22 (2014), 1103–1122.

36.

Wang

X.F.

, Wang

J.Q.

and Yang

W.E.

, Multi-criteria group decision making method based on intuitionistic linguistic aggregation operators, Journal of Intelligent and Fuzzy Systems26(1) (2014), 115–125.

37.

Yang

, Lin

T.Y.

, Yang

, Li

and Yu

, Combination of interval-valued fuzzy set and soft set, Comput Math Appl58 (2009), 521–527.

38.

Zadeh

L.A.

, Fuzzy Sets, Inform and Control8 (1965), 338–353.

39.

Zhang

, Wang

, Tian

and Li

, A novel approach to interval-valued intuitionistic fuzzy soft set based decision making, Applied Mathematical Modellingxxx (2013) xxx (in press).

40.

Zou

and Xiao

, Data analysis approaches of soft sets under incomplete information, Knowl Base Syst21 (2008), 941–945.