Some power generalized aggregation operators based on the interval neutrosophic sets and their application to decision making

Abstract

The interval neutrosophic set (INS) can be better to express the incomplete, indeterminate and inconsistent information, and the power average (PA) operator can take all the decision arguments and their relationships into account, and the generalized weighted aggregation (GWA) operator can reflect the mentality of the decision-makers. In this paper, we combined PA and GWA operators to INS, and proposed some aggregation operators, include interval neutrosophic power generalized aggregation (INPGA) operator, interval neutrosophic power generalized weighted aggregation (INPGWA) operator and interval neutrosophic power generalized ordered weighted aggregation (INPGOWA) operator. Firstly, we presented some new operational laws for interval neutrosophic sets and studied their properties. Then we proposed INPGA, INPGWA and INPGOWA operators, and studied some properties and special cases of them. Further, we gave a decision making method based on these operators. Finally, an illustrative example is given to verify the developed approach and to demonstrate its practicality and effectiveness, and to illustrate the influence of the parameter on decision making results in INPGWA operator.

Keywords

Interval neutrosophic set power average generalized weighted aggregation (GWA) operator ordered weighted aggregation (OWA) operator multiple attribute decision making

1 Introduction

In real decision making, the decision information is often incomplete, indeterminate and inconsistent information. The fuzzy set theory proposed by Zadeh [1], is a good tool to process fuzzy information, however, it only has a membership, and cannot express non-membership. Atanassov [2, 3] proposed the intuitionistic fuzzy set (IFS) by adding a non-membership function, i.e., the intuitionistic fuzzy sets consider both membership (or called truth-membership) T_A (x) and non-membership (or called falsity-membership) F_A (x) and satisfy the conditions T_A (x) , F_A (x) ∈ [0, 1] and 0 ≤ T_A (x) + F_A (x) ≤1. Further, Atanassov and Gargov [4], Atanassov [5] proposed the interval-valued intuitionistic fuzzy set (IVIFS) in which the truth-membership function and falsity-membership function were extended to interval numbers. IFSs and IVIFSs can only handle incomplete information not the indeterminate information and inconsistent information. In IFSs, the indeterminacy is 1 - T_A (x) - F_A (x) by default. Further, Smarandache [6] proposed the neutrosophic set (NS) by adding an independent indeterminacy membership on the basis of IFS, which is a generalization of fuzzy set, interval valued fuzzy set, intuitionistic fuzzy set, and so on. In NS, the indeterminacy is quantified explicitly and truth-membership, indeterminacy membership, and false-membership are completely independent. For example, during a voting process, thirty percent vote “yes”, twenty percent vote “no”, ten percent give up and forty percent are undecided. Obviously, that is beyond the scope of IFS because the indeterminacy in IFS is 1-membership-non-membership by default, and in this example, it cannot distinguish the information between giving up and undecided. However, it is easily expressed by NS (0.3,0.4,0.2). Another example about inconsistent information, when we ask the opinion of an expert about a certain statement, he or she may think the possibility that the statement is right is 0.5 and the statement is false is 0.6 and the degree that he or she is not sure is 0.2 [7]. In this case, we can only express the inconsistent information by NS (0.5,0.2,0.6) and IFS cannot do.

Recently, NSs have attracted widely attention, and made some applications [6 –13]. Wang et al. [7] proposed a single valued neutrosophic set (SVNS), which is an instance of the neutrosophic set. Ye [9] proposed a simplified neutrosophic weighted arithmetic average operator and a simplified neutrosophic weighted geometric average operator. Based on the two aggregation operators, a multicriteria decision-making method is established in which the evaluation values of alternatives with respective to criteria are represented by the form of SNSs. Ye [10] proposed the single valued neutrosophic cross entropy, and developed a multicriteria decision-making method based on the entropy. Ye [11] proposed the correlation coefficient and weighted correlation coefficient of SVNSs, and proved that the cosine similarity degree is a special case of the correlation coefficient in SVNS. Wang et al. [12] defined interval neutrosophic sets (INSs) in which the truth-membership, indeterminacy-membership, and false-membership were extended to interval numbers, and discussed various properties of INSs. Ye [13] defined the similarity measures between INSs on the basis of the Hamming and Euclidean distances, and a multicriteria decision-making method based on the similarity degree is proposed. However, so far, there has been no research on aggregation operators for INSs.

The information aggregation operators are an [3] interesting and important research topic, which are receiving more and more attention [14 –33]. Yager [27] proposed a power-average (PA) operator and a power OWA (POWA) operator, which weighting vectors depend on the input data and allow values being fused to support and reinforce each other. Xu and Yager [28] developed an uncertain power ordered weighted geometric (UPOWG) operator. Xu and Wang [29] developed 2-tuple linguistic power average (2TLPA) operator, 2-tuple linguistic weighted power average (2TLWPA) operator, and 2-tuple linguisticpower-ordered-weighted average (2TLPOWA) operator. Xu [30] developed a series of PA operators for aggregating the intuitionistic fuzzy numbers. Zhou et al. [31] proposed a generalized intuitionistic fuzzy power averaging (GIFPA) operator and the generalized intuitionistic fuzzy power ordered weighted averaging (GIFPOWA) operator. Xu et al. [32] proposed the linguistic weighted PA operator, the LPOWA operator, the uncertain linguistic weighted PA operator, and the ULPOWA operator. Obviously, these PA operators cannot aggregate the INSs. In this paper, we will study some PA aggregation operators based on INSs, and discuss some special cases and properties of them. Further, we give a decision making method for multiple attribute decision making (MADM) problems based on these operators.

In order to do so, the remainder of this paper is shown as follows. In Section 2, we briefly review some basic concepts and operational rules of INS and propose PGA and PGOWA operators. In Section 3, we propose INPGA, aINPGWA and INPGOWA operators, and discuss some desirable properties and special cases. In Section 4, we propose a decision making method based on the INPGWA and INPGOWA operators for the multiple attribute decision making problems in which attribute values take the form of INSs. In Section 5, we give an example to illustrate the application of proposed method, and compare the developed method with the existing methods. In Section 6, we conclude the paper.

2 Preliminaries

2.1 The interval neutrosophic set

Definition 1. [6] Let X be a universe of discourse, with a generic element in X denoted by x. A neutrosophic set A in X is $A = {x (T_{A} (x), I_{A} (x), F_{A} (x)) | x \in X}$ (1) where, T_A is the truth-membership function, I_A is the indeterminacy-membership function, and F_A is the falsity-membership function. T_A (x) , I_A (x) and F_A (x) are real standard or nonstandard subsets of ] 0^-, 1⁺].

There is no restriction on the sum of T_A (x) , I_A (x) and F_A (x), so we can obtain 0^- ≤ T_A (x) + I_A (x) + F_A (x) ≤3⁺.

The neutrosophic set was presented from philosophical point of view. Obviously, it was difficult to apply in the real applications. Wang [7] further proposed the single valued neutrosophic set (SVNS) from scientific or engineering point of view, which was defined as follows.

Definition 2. [7] Let X be a universe of discourse, with a generic element in X denoted by x. A single valued neutrosophic set A in X is $A = {x (T_{A} (x), I_{A} (x), F_{A} (x)) | x \in X}$ (2)

Definition 3. [7] Let X be a universe of discourse, with a generic element in X denoted by x. A interval neutrosophic set A in X is $A = {x (T_{A} (x), I_{A} (x), F_{A} (x)) | x \in X}$ (3) where, T_A is the truth-membership function, I_A is the indeterminacy-membership function, and F_A is the falsity-membership function. For each point x in X, we have T_A (x) , I_A (x) , F_A (x) ⊆ [0, 1], and 0 ≤ sup(T_A (x)) + sup(I_A (x)) + sup(F_A (x)) ≤3.

For convenience, we can use x = ([T^L, T^U] , [I^L, I^U] , [F^L, F^U]) to represent an INS.

In the following, we will discuss the distance and similarity degree between two INSs.

Definition 4. [13] Let $x = ([T_{1}^{L}, T_{1}^{U}], [I_{1}^{L}, I_{1}^{U}],$ $[F_{1}^{L}, F_{1}^{U}])$ and $y = ([T_{2}^{L}, T_{2}^{U}], [I_{2}^{L}, I_{2}^{U}], [F_{2}^{L}, F_{2}^{U}])$ be two INSs, then the normalized Hamming distance between x and y is defined as follows: $\begin{matrix} d (x, y) = \frac{1}{6} (| T_{1}^{L} - T_{2}^{L} | + | T_{1}^{U} - T_{2}^{U} | + | I_{1}^{L} - I_{2}^{L} | \\ + | I_{1}^{U} - I_{2}^{U} | + | F_{1}^{L} - F_{2}^{L} | + | F_{1}^{U} - F_{2}^{U} |) \end{matrix}$ (4)

Definition 5. [13] Let $x = ([T_{1}^{L}, T_{1}^{U}], [I_{1}^{L}, I_{1}^{U}],$ $[F_{1}^{L}, F_{1}^{U}])$ and $y = ([T_{2}^{L}, T_{2}^{U}], [I_{2}^{L}, I_{2}^{U}], [F_{2}^{L}, F_{2}^{U}])$ be two INSs, then the cosine of included angle between x and y is $\begin{matrix} cos (x, y) \\ = \frac{T_{1}^{L} T_{2}^{L} + T_{1}^{U} T_{2}^{U} + I_{1}^{L} I_{2}^{L} + I_{1}^{U} I_{2}^{U} + F_{1}^{L} F_{2}^{L} + F_{1}^{U} F_{2}^{U}}{∥ x ∥ • ∥ y ∥} \end{matrix}$ (5) where $\begin{matrix} ∥ x ∥ = \sqrt{{(T_{1}^{L})}^{2} + {(T_{1}^{U})}^{2} + {(I_{1}^{L})}^{2} + {(I_{1}^{U})}^{2} + {(F_{1}^{L})}^{2} + {(F_{1}^{U})}^{2}}, \\ ∥ y ∥ = \sqrt{{(T_{2}^{L})}^{2} + {(T_{2}^{U})}^{2} + {(I_{2}^{L})}^{2} + {(I_{2}^{U})}^{2} + {(F_{2}^{L})}^{2} + {(F_{2}^{U})}^{2}} \end{matrix}$

Obviously, 0 < cos(x, y) ≤1. When y is the ideal solution I = ([1, 1] , [0, 0] , [0, 0]), the bigger the cos(x, I) between x and I is, the more consistent the direction between x and I is. In this condition, $\begin{matrix} {cos}^{*} (x, I) \\ = \frac{T_{1}^{L} + T_{1}^{U}}{\sqrt{2 ({(T_{1}^{L})}^{2} + {(T_{1}^{U})}^{2} + {(I_{1}^{L})}^{2} + {(I_{1}^{U})}^{2} + {(F_{1}^{L})}^{2} + {(F_{1}^{U})}^{2})}} \end{matrix}$ (6)

In order to compare two INSs, we can give the following definition.

Definition 6. [13] Let $x = ([T_{1}^{L}, T_{1}^{U}], [I_{1}^{L}, I_{1}^{U}],$ $[F_{1}^{L}, F_{1}^{U}])$ and $y = ([T_{2}^{L}, T_{2}^{U}], [I_{2}^{L}, I_{2}^{U}], [F_{2}^{L}, F_{2}^{U}])$ be two INSs, if cos ^* (x, I) < cos ^* (y, I) then x < y.

Definition 7. [12, 13] Let $x = ([T_{1}^{L}, T_{1}^{U}], [I_{1}^{L}, I_{1}^{U}],$ $[F_{1}^{L}, F_{1}^{U}])$ and $y = ([T_{2}^{L}, T_{2}^{U}], [I_{2}^{L}, I_{2}^{U}], [F_{2}^{L}, F_{2}^{U}])$ be two INSs, then the operational laws are defined as follows.

(1) The complement of x is $\bar{x} = ([F_{1}^{L}, F_{1}^{U}], [1 - I_{1}^{U}, 1 - I_{1}^{L}], [T_{1}^{L}, T_{1}^{U}])$ (7) $\begin{matrix} (2) x \oplus y & = ([T_{1}^{L} + T_{2}^{L} - T_{1}^{L} T_{2}^{L}, T_{1}^{U} + T_{2}^{U} - T_{1}^{U} T_{2}^{U}], \\ [I_{1}^{L} I_{2}^{L}, I_{1}^{U} I_{2}^{U}], [F_{1}^{L} F_{2}^{L}, F_{1}^{U} F_{2}^{U}]) \end{matrix}$ (8) $\begin{matrix} (3) x \otimes y & = ([T_{1}^{L} T_{2}^{L}, T_{1}^{U} T_{2}^{U}], \\ [I_{1}^{L} + I_{2}^{L} - I_{1}^{L} I_{2}^{L}, I_{1}^{U} + I_{2}^{U} - I_{1}^{U} I_{2}^{U}], \\ [F_{1}^{L} + F_{2}^{L} - F_{1}^{L} F_{2}^{L}, F_{1}^{U} + F_{2}^{U} - F_{1}^{U} F_{2}^{U}]) \end{matrix}$ (9) $\begin{matrix} (4) nx & = ([1 - (1 - T_{1}^{L})^{n}, 1 - (1 - T_{1}^{U})^{n}], \\ [(I_{1}^{L})^{n}, (I_{1}^{U})^{n}], [(F_{1}^{L})^{n}, (F_{1}^{U})^{n}]) n > 0 \end{matrix}$ (10) $\begin{matrix} (5) x^{n} & = ([(T_{1}^{L})^{n}, (T_{1}^{U})^{n}], [1 - (1 - I_{1}^{L})^{n}, 1 - (1 - I_{1}^{U})^{n}], \\ [1 - (1 - F_{1}^{L})^{n}, 1 - (1 - F_{1}^{U})^{n}]) n > 0 \end{matrix}$ (11)

Theorem 1.Let $x = ([T_{1}^{L}, T_{1}^{U}], [I_{1}^{L}, I_{1}^{U}], [F_{1}^{L}, F_{1}^{U}])$ and $y = ([T_{2}^{L}, T_{2}^{U}], [I_{2}^{L}, I_{2}^{U}], [F_{2}^{L}, F_{2}^{U}])$ be two INSs, andη, η₁, η₂ > 0, then we have $(1) x \oplus y = y \oplus x$ (12) $(2) x \otimes y = y \otimes x$ (13) $(3) η (x \oplus y) = η x \oplus η y$ (14) $(4) η_{1} x \oplus η_{2} x = (η_{1} + η_{2}) x$ (15) $(5) x^{η} \otimes y^{η} = (x \otimes y)^{η}$ (16) $(6) x^{η_{1}} \otimes x^{η_{2}} = x^{η_{1} + η_{2}}$ (17)

2.2 GWA operator

Definition 8. [18, 31]. A GWA operator of dimension n is a mapping GWA : Rⁿ → R. Such that, $GWA (a_{1}, a_{2}, \dots, a_{n}) = {(\sum_{j = 1}^{n} w_{j} a_{j}^{λ})}^{1 / λ}$ (18) where W = (w₁, w₂, ⋯ , w_n) ^T is a weight vector of (a₁, a₂, ⋯ , a_n), and satisfying w_j ∈ [0, 1] (j = 1, 2, ⋯ , n) and $\sum_{j = 1}^{n} w_{j} = 1$ , then GWA is called the generalized weighted aggregation (GWA) operator. λ is a parameter such that λ ∈ (- ∞ , 0) ∪ (0, + ∞).

2.3 The power generalized aggregation operators

Definition 9. [27] A power average (PA) operator proposed by Yager [27] can be defined as follows: $PA (a_{1}, a_{2}, \dots, a_{n}) = \sum_{i = 1}^{n} (1 + T (a_{i})) a_{i} / \sum_{i = 1}^{n} (1 + T (a_{i}))$ (19) where $T (a_{i}) = \overset{n}{\underset{j \neq i}{\sum_{j = 1}}} Sup (a_{i}, a_{j})$ (20) and Sup (a, b) is regarded as the support for a from b, which meets the following properties.

Sup (a, b) ∈ [0, 1].

Sup (a, b) = Sup (b, a).

Sup (a, b) ≥ Sup (x, y), if |a - b| < |x - y|

Further, Yager [27] also defined a power ordered weighted average (POWA) operator.

Definition 10. [27] A POWA operator of dimension n is a mapping POWA : Rⁿ → R. Such that, $POWA (a_{1}, a_{2}, \dots, a_{n}) = \sum_{i = 1}^{n} u_{i} a_{index (i)}$ (21) where $u_{i} = g (\frac{R_{i}}{TV}) - g (\frac{R_{i - 1}}{TV}), R_{i} = \sum_{j = 1}^{i} V_{index (j)}$

g : [0, 1] → [0, 1] is a basic unit-interval monotonic (BUM) function which satisfies the following properties.

1) g (0) =0; 2) g (1) =1;

3) g (x) ≥ g (y), if x > y. $TV = \sum_{i = 1}^{n} V_{index (i)}, V_{index (i)} = 1 + T (a_{index (i)})$ (22)

T (a_index(i)) denotes the support of the ith largest argument by all the other arguments, i.e., $T (a_{index (i)}) = \overset{n}{\underset{j \neq i}{\sum_{j = 1}}} Sup (a_{index (i)}, a_{index (j)})$ (23) where Sup (a_index(i), a_index(j)) indicates the support of jth largest argument for the ith largest argument.

In the following, we will combine the PA and POWA operators with GWA operator respectively, and propose power generalized aggregation (PGA) operator and power generalized ordered weighted aggregation (PGOWA) operator which can generalize PA and POWA operators.

Definition 11. A PGA operator of dimension n is a mapping PGA : Rⁿ → R. Such that, $PGA (a_{1}, a_{2}, \dots, a_{n}) = {(\sum_{j = 1}^{n} w_{j} a_{j}^{λ})}^{1 / λ}$ (24) where $w_{i} = V_{i} / \sum_{i = 1}^{n} V_{i}$ , V_i = 1 + T (a_i) and $T (a_{i}) = {\sum_{j = 1}^{n}}_{j \neq i} Sup (a_{i}, a_{j})$ , then PGA is called the power generalized aggregation (PGA) operator. In addition, λ is a parameter such that λ ∈ (- ∞ , 0) ∪ (0, + ∞).

It’s easy to prove that the PGA operator has commutativity, idempotency, and boundedness. Limited to the space, the proofs of these properties are omitted here.

Definition 12. A PGOWA operator of dimension n is a mapping PGA : Rⁿ → R. Such that, $PGOWA (a_{1}, a_{2}, \dots, a_{n}) = {(\sum_{j = 1}^{n} ζ_{j} a_{index (j)}^{λ})}^{1 / λ}$ (25) where index is an indexing function such that a_index(j) is the jth largest of the a_j (j = 1, 2 ⋯ , n). ζ_j (j = 1, 2 ⋯ , n) is a collection of weights such that $ζ_{j} = g (\frac{R_{j}}{TV}) - g (\frac{R_{j - 1}}{TV})$ (26) where, $R_{j} = \sum_{i = 1}^{j} V_{index (i)}$ , $TV = \sum_{i = 1}^{n} V_{index (i)}$ , V_index(i) = 1 + T (a_index(i))

g : [0, 1] → [0, 1] is a basic unit-interval monotonic (BUM) function.

T (a_index(i)) indicates the support of the ith largest argument by all the other arguments, i.e., $T (a_{index (i)}) = \overset{n}{\underset{k \neq i}{\sum_{k = 1}}} Sup (a_{index (i)}, a_{index (k)})$ (27)

λ is a parameter such that λ ∈ (- ∞ , 0) ∪ (0, + ∞).

Similar to the PGA operator, PGOWA operator has commutativity, idempotency, and boundedness.

3 The interval neutrosophic power generalized aggregation operators

The prominent characteristic of PGA and PGOWA operators is that they can take all the decision arguments and their relationships into account. In the following, we will extend PGA and PGOWA operators to the interval neutrosophic sets (INSs), and propose an interval neutrosophic power generalized aggregation (INPGA) operator, an interval neutrosophic power generalized weighted aggregation (INPGWA) operator and an interval neutrosophic power generalized ordered weighted aggregation (INPGOWA) operator.

Definition 13. Let $x_{j} = ([T_{j}^{L}, T_{j}^{U}], [I_{j}^{L}, I_{j}^{U}], [F_{j}^{L},$ $F_{j}^{U}]) (j = 1, 2 \dots, n)$ be a collection of the INSs, and INPGA : Ωⁿ → Ω, if $INPGA (x_{1}, x_{2}, \dots, x_{n}) = {(\frac{\sum_{j = 1}^{n} (1 + T (x_{j})) x_{j}^{λ}}{\sum_{j = 1}^{n} (1 + T (x_{j}))})}^{1 / λ}$ (28) where Ω is the set of all INSs, and $T (x_{j}) = {\sum_{i = 1}^{n}}_{i \neq j} Sup (x_{j}, x_{i})$ , λ is a parameter such that λ ∈ (0, + ∞) (because we cannot define x^λ when λ < 0), then INPGA is called the interval neutrosophic power generalized aggregation (INPGA) operator.

Theorem 2.Let $x_{j} = ([T_{j}^{L}, T_{j}^{U}], [I_{j}^{L}, I_{j}^{U}], [F_{j}^{L}, F_{j}^{U}])$ (j = 1, 2 ⋯ , n) be a collection of the INSs, then, the result aggregated from Definition 13 is still an INS, and even $\begin{matrix} INPGA (x_{1}, x_{2}, \dots, x_{n}) \\ = ([\begin{matrix} {(1 - \prod_{j = 1}^{n} (1 - (T_{j}^{L})^{λ})^{\frac{(1 + T (x_{j}))}{\sum_{j = 1}^{n} (1 + T (x_{j}))}})}^{1 / λ}, \\ {(1 - \prod_{j = 1}^{n} (1 - (T_{j}^{U})^{λ})^{\frac{(1 + T (x_{j}))}{\sum_{j = 1}^{n} (1 + T (x_{j}))}})}^{1 / λ} \end{matrix}], \\ [\begin{matrix} 1 - {(1 - \prod_{j = 1}^{n} (1 - (1 - I_{j}^{L})^{λ})^{\frac{(1 + T (x_{j}))}{\sum_{j = 1}^{n} (1 + T (x_{j}))}})}^{1 / λ}, \\ 1 - {(1 - \prod_{j = 1}^{n} (1 - (1 - I_{j}^{U})^{λ})^{\frac{(1 + T (x_{j}))}{\sum_{j = 1}^{n} (1 + T (x_{j}))}})}^{1 / λ} \end{matrix}], \\ [\begin{matrix} 1 - {(1 - \prod_{j = 1}^{n} (1 - (1 - F_{j}^{L})^{λ})^{\frac{(1 + T (x_{j}))}{\sum_{j = 1}^{n} (1 + T (x_{j}))}})}^{1 / λ}, \\ 1 - {(1 - \prod_{j = 1}^{n} (1 - (1 - F_{j}^{U})^{λ})^{\frac{(1 + T (x_{j}))}{\sum_{j = 1}^{n} (1 + T (x_{j}))}})}^{1 / λ} \end{matrix}]) \end{matrix}$ (29)

We can use mathematical induction to prove this theorem, and it is omitted here because of the constraints of space.

The INPGA operator satisfies the following properties:

(1) Theorem 3 (Commutativity). Let $(x_{1}^{'}, x_{2}^{'}, \dots, x_{n}^{'})$ be any permutation of (x₁, x₂, ⋯ , x_n), then $INPGA (x_{1}^{'}, x_{2}^{'}, \dots, x_{n}^{'}) = INPGA (x_{1}, x_{2}, \dots, x_{n})$ (30)

(2) Theorem 4 (Idempotency). Letx_j = x, j = 1, 2, ⋯ , n, then $INPGA (x_{1}, x_{2}, \dots, x_{n}) = x$ (31)(3) Theorem 5 (Boundedness). TheINPGAoperator lies between the max and min operators: $\begin{matrix} min (x_{1}, x_{2}, \dots, x_{n}) \leq INPGA (x_{1}, x_{2}, \dots, x_{n}) \\ \leq max (x_{1}, x_{2}, \dots, x_{n}) \end{matrix}$ (32)

Limited to the space, the proofs of these properties are omitted here

In the following, we will discuss some cases of the INPGA operator.

(1) When λ → 0, $\begin{matrix} INPGA (x_{1}, x_{2}, \dots, x_{n}) \\ = ([\prod_{j = 1}^{n} (T_{j}^{L})^{\frac{(1 + T (x_{j}))}{\sum_{j = 1}^{n} (1 + T (x_{j}))}}, \prod_{j = 1}^{n} (T_{j}^{U})^{\frac{(1 + T (x_{j}))}{\sum_{j = 1}^{n} (1 + T (x_{j}))}}], \\ [1 - \prod_{j = 1}^{n} (1 - I_{j}^{L})^{\frac{(1 + T (x_{j}))}{\sum_{j = 1}^{n} (1 + T (x_{j}))}}, 1 - \prod_{j = 1}^{n} (1 - I_{j}^{U})^{\frac{(1 + T (x_{j}))}{\sum_{j = 1}^{n} (1 + T (x_{j}))}}], \\ [1 - \prod_{j = 1}^{n} (1 - F_{j}^{L})^{\frac{(1 + T (x_{j}))}{\sum_{j = 1}^{n} (1 + T (x_{j}))}}, 1 - \prod_{j = 1}^{n} (1 - F_{j}^{U})^{\frac{(1 + T (x_{j}))}{\sum_{j = 1}^{n} (1 + T (x_{j}))}}]) \end{matrix}$ (33)

The INPGA operator can reduce to the interval neutrosophic power geometric (INPG) operator.

(2) When λ = 1, $\begin{matrix} INPGA (x_{1}, x_{2}, \dots, x_{n}) \\ = ([1 - (\prod_{j = 1}^{n} {(1 - T_{j}^{L})}^{\frac{(1 - T (x_{j}))}{\sum_{j = 1}^{n} (1 + T (x_{j}))}}), \\ 1 - (\prod_{j = 1}^{n} {(1 - T_{j}^{U})}^{\frac{(1 + T (x_{j}))}{\sum_{j = 1}^{n} (1 + T (x_{j}))}})], \\ [\prod_{j = 1}^{n} (I_{j}^{L})^{\frac{(1 + T (x_{j}))}{\sum_{j = 1}^{n} (1 + T (x_{j}))}}, \prod_{j = 1}^{n} (I_{j}^{U})^{\frac{(1 + T (x_{j}))}{\sum_{j = 1}^{n} (1 + T (x_{j}))}}], \end{matrix}$ (34) $[\prod_{j = 1}^{n} (F_{j}^{L})^{\frac{(1 + T (x_{j}))}{\sum_{j = 1}^{n} (1 + T (x_{j}))}}, \prod_{j = 1}^{n} (F_{j}^{U})^{\frac{(1 + T (x_{j}))}{\sum_{j = 1}^{n} (1 + T (x_{j}))}}])$

The INPGA operator can reduce to the interval neutrosophic power aggregation (INPA)operator.

In Definition 13, we assumed that all of the objects (x₁, x₂, ⋯ , x_n) being aggregated were of equal importance. However, in many real cases, the importance degrees are not equal; thus, we can assign different weights for different objects, and further define a new aggregation operator to process thiscase.

Definition 14. Le $x_{j} = ([T_{j}^{L}, T_{j}^{U}], [I_{j}^{L}, I_{j}^{U}], [F_{j}^{L},$ $F_{j}^{U}]) (j = 1, 2 \dots, n)$ be a collection of the INSs, and INPGWA : Ωⁿ → Ω, if $INPGWA (x_{1}, x_{2}, \dots, x_{n}) = {(\frac{\sum_{j = 1}^{n} ω_{j} (1 + T (x_{j})) x_{j}^{λ}}{\sum_{j = 1}^{n} ω_{j} (1 + T (x_{j}))})}^{1 / λ}$ (35) where Ω is the set of all INSs, and ω = (ω₁, ω₂, ⋯ , ω_n) ^T is the weight vector of x_j (j = 1, 2, ⋯ , n), $ω_{j} \in [0, 1], \sum_{j = 1}^{n} ω_{j} = 1$ . T (x_j) = ∑_i=1_i≠jⁿSup (x_j, x_i), λ is a parameter such that λ ∈ (0, + ∞). Specially, if $ω = {(\frac{1}{n}, \frac{1}{n}, \dots, \frac{1}{n})}^{T}$ , INPGWA operator should be INPGA operator.

Theorem 6.Let $x_{j} = ([T_{j}^{L}, T_{j}^{U}], [I_{j}^{L}, I_{j}^{U}], [F_{j}^{L}, F_{j}^{U}])$ (j = 1, 2 ⋯ , n) be a collection of the INSs, then, the result aggregated from Definition 14 is still an INS, and even

$\begin{matrix} INPGWA (x_{1}, x_{2}, \dots, x_{n}) = ([{(1 - \prod_{j = 1}^{n} (1 - (T_{j}^{L})^{λ})^{{\bar{ω}}_{j}})}^{1 / λ}, {(1 - \prod_{j = 1}^{n} (1 - (T_{j}^{U})^{λ})^{{\bar{ω}}_{j}})}^{1 / λ}], \\ [1 - {(1 - \prod_{j = 1}^{n} (1 - (1 - I_{j}^{L})^{λ})^{{\bar{ω}}_{j}})}^{1 / λ}, 1 - {(1 - \prod_{j = 1}^{n} (1 - (1 - I_{j}^{U})^{λ})^{{\bar{ω}}_{j}})}^{1 / λ}], \\ [1 - {(1 - \prod_{j = 1}^{n} (1 - (1 - F_{j}^{L})^{λ})^{{\bar{ω}}_{j}})}^{1 / λ}, 1 - {(1 - \prod_{j = 1}^{n} (1 - (1 - F_{j}^{U})^{λ})^{{\bar{ω}}_{j}})}^{1 / λ}]) \end{matrix}$ (36) where ${\bar{ω}}_{j} = \frac{ω_{j} (1 + T (x_{j}))}{\sum_{j = 1}^{n} ω_{j} (1 + T (x_{j}))}$

The proof of this theorem is similar with Theorem 2, it’s omitted here.

In the following, we will discuss some properties of the INPGWA operator. Similar to Theorems 4 and 5, it can be easily proved that the INPGWA operator has the following properties

(1) Theorem 7 (Idempotency).Letx_j = x, j = 1, 2, ⋯ , n, then $INPGWA (x_{1}, x_{2}, \dots, x_{n}) = x$ (37)

(2) Theorem 8 (Boundedness).The INPGWA operator lies between the max and min operators:

$\begin{matrix} min (x_{1}, x_{2}, \dots, x_{n}) \leq INPGWA (x_{1}, x_{2}, \dots, x_{n}) \\ \leq max (x_{1}, x_{2}, \dots, x_{n}) \end{matrix}$ (38)

Note that The INPGWA operator has not the commutativity.

In the following, we will discuss some cases of the INPGWA operator.

(1) When λ → 0, $\begin{matrix} INPGWA (x_{1}, x_{2}, \dots, x_{n}) \\ = ([\prod_{j = 1}^{n} (T_{j}^{L})^{{\bar{ω}}_{j}}, \prod_{j = 1}^{n} (T_{j}^{U})^{{\bar{ω}}_{j}}], \\ [1 - \prod_{j = 1}^{n} (1 - I_{j}^{L})^{{\bar{ω}}_{j}}, 1 - \prod_{j = 1}^{n} (1 - I_{j}^{U})^{{\bar{ω}}_{j}}], \\ [1 - \prod_{j = 1}^{n} (1 - F_{j}^{L})^{{\bar{ω}}_{j}}, 1 - \prod_{j = 1}^{n} (1 - F_{j}^{U})^{{\bar{ω}}_{j}}]) \end{matrix}$ (39)

(2) When λ = 1, $\begin{matrix} INPGWA (x_{1}, x_{2}, \dots, x_{n}) \\ = ([1 - (\prod_{j = 1}^{n} (1 - T_{j}^{L})^{{\bar{ω}}_{j}}), 1 - (\prod_{j = 1}^{n} (1 - T_{j}^{U})^{{\bar{ω}}_{j}})], \\ [\prod_{j = 1}^{n} (I_{j}^{L})^{{\bar{ω}}_{j}}, \prod_{j = 1}^{n} (I_{j}^{U})^{{\bar{ω}}_{j}}], [\prod_{j = 1}^{n} (F_{j}^{L})^{{\bar{ω}}_{j}}, \prod_{j = 1}^{n} (F_{j}^{U})^{{\bar{ω}}_{j}}]) \end{matrix}$ (40)

Theorem 9.LettingSup (x_i, x_j) = k, for alli ≠ j, then $\begin{matrix} INPGWA (x_{1}, x_{2}, \dots, x_{n}) = {(\frac{\sum_{j = 1}^{n} ω_{j} (1 + T (x_{j})) x_{j}^{λ}}{\sum_{j = 1}^{n} ω_{j} (1 + T (x_{j}))})}^{1 / λ} \\ = {(\sum_{j = 1}^{n} ω_{j} x_{j}^{λ})}^{1 / λ} \end{matrix}$ (41) which indicates that when all the supports are the same, the INPGWA operator becomes an interval neutrosophic generalized weighted average (INGWA) operator.

The INPGWA operator can weight all the given INSs themselves, and the weighting vectors depend on the input arguments. However, in many decision making problems, we need to consider the weight of the argument positions, for example, diving games in Olympic Games. Base on the PGOWA operator, we define a new aggregation operator called an interval neutrosophic power generalized ordered weighted averaging (INPGOWA) operator.

Definition 15. Let $x_{j} = ([T_{j}^{L}, T_{j}^{U}], [I_{j}^{L}, I_{j}^{U}], [F_{j}^{L},$ $F_{j}^{U}]) (j = 1, 2 \dots, n)$ be a collection of the INSs, and INPGOWA : Ωⁿ → Ω, if $INPGOWA (x_{1}, x_{2}, \dots, x_{n}) = {(\sum_{j = 1}^{n} ζ_{j} x_{index (j)}^{λ})}^{1 / λ}$ (42) where Ω is the set of all INSs, and index is an indexing function such that x_index(j) is the jth largest of the INSs x_j (j = 1, 2 ⋯ , n). ζ_j (j = 1, 2 ⋯ , n) is a collection of weights such that $ς_{j} = g (\frac{R_{j}}{TV}) - g (\frac{R_{j - 1}}{TV}), R_{j} = \sum_{i = 1}^{j} V_{index (i)}$ (43) where, $TV = \sum_{i = 1}^{n} V_{index (i)}$ , V_index(i) = 1 + T (x_index(i))g : [0, 1] → [0, 1] is a basic unit-interval monotonic (BUM) function.

T (x_index(i)) indicates the support of the ith largest argument by all the other arguments, i.e., $T (x_{index (i)}) = \overset{n}{\underset{k \neq i}{\sum_{k = 1}}} Sup (x_{index (i)}, x_{index (k)})$ (44)

λ is a parameter such that λ ∈ (0, + ∞).

Theorem 10.Let $x_{j} = ([T_{j}^{L}, T_{j}^{U}], [I_{j}^{L}, I_{j}^{U}], [F_{j}^{L}, F_{j}^{U}])$ (j = 1, 2 ⋯ , n) be a collection of the INSs, then, the result aggregated from Definition 15 is still an INS, and even $\begin{matrix} INPGOWA (x_{1}, x_{2}, \dots, x_{n}) \\ = ([\begin{matrix} {(1 - (\prod_{j = 1}^{n} {(1 - {(T_{index (j)}^{L})}^{λ})}^{ς_{j}}))}^{1 / λ}, \\ {(1 - (\prod_{j = 1}^{n} {(1 - {(T_{index (j)}^{U})}^{λ})}^{ς_{j}}))}^{1 / λ} \end{matrix}], \\ [\begin{matrix} 1 - {(1 - \prod_{j = 1}^{n} {(1 - {(1 - I_{index (j)}^{L})}^{λ})}^{ς_{j}})}^{1 / λ}, \\ 1 - {(1 - \prod_{j = 1}^{n} {(1 - {(1 - I_{index (j)}^{U})}^{λ})}^{ς_{j}})}^{1 / λ} \end{matrix}], \\ [\begin{matrix} 1 - {(1 - \prod_{j = 1}^{n} {(1 - {(1 - F_{index (j)}^{L})}^{λ})}^{ς_{j}})}^{1 / λ}, \\ 1 - {(1 - \prod_{j = 1}^{n} {(1 - {(1 - F_{index (j)}^{U})}^{λ})}^{ς_{j}})}^{1 / λ} \end{matrix}] \end{matrix}$ (45)

The proof of this theorem is similar with Theorem 2, it’s omitted here.

Theorem 11.Especially, ifg (x) = x, then the INPGOWA operator can reduce to the INPGA operator. So, INPGA operator is a special case of INPGOWA operator.

Theorem 12.LettingSup (x_i, x_j) = k, for alli ≠ j, then $ILPGOWA (x_{1}, x_{2}, \dots, x_{n}) = {(\frac{1}{n} \sum_{j = 1}^{n} x_{j}^{λ})}^{1 / λ}$ (46) which indicates that when all the supports are the same, the INPGOWA operator becomes an interval neutrosophic generalized average (INGA) operator.

Similar to Theorems 3, 4 and 5, it can be easily proved that the INPGOWA operator has commutativity, idempotency, and boundedness.

Similarly, some special cases of the INPGOWA operators are shown as follows.

(1) When λ → 0, $\begin{matrix} INPGOWA (x_{1}, x_{2}, \dots, x_{n}) = \prod_{j = 1}^{n} x_{index (j)}^{ζ_{j}} \\ = ([\prod_{j = 1}^{n} (T_{index (j)}^{L})^{ζ j}, \prod_{j = 1}^{n} (T_{index (j)}^{U})^{ζ j}], \\ [1 - \prod_{j = 1}^{n} (1 - I_{index (j)}^{L})^{ζ_{j}}, 1 - \prod_{j = 1}^{n} (1 - I_{index (j)}^{U})^{ζ_{j}}], \\ [1 - \prod_{j = 1}^{n} (1 - F_{index (j)}^{L})^{ζ_{j}}, 1 - \prod_{j = 1}^{n} (1 - F_{index (j)}^{U})^{ζ_{j}}]) \end{matrix}$ (47)

(2) When λ = 1, $\begin{matrix} INPGOWA (x_{1}, x_{2}, \dots, x_{n}) = \sum_{j = 1}^{n} ζ_{j} x_{index (j)} \\ = ([1 - \prod_{j = 1}^{n} (1 - F_{index (j)}^{L})^{ζ_{j}}, 1 - \prod_{j = 1}^{n} (1 - F_{index (j)}^{U})^{ζ_{j}}], \\ [\prod_{j = 1}^{n} (I_{index (j)}^{L})^{ζ_{j}}, \prod_{j = 1}^{n} (I_{index (j)}^{U})^{ζ j}], \\ [\prod_{j = 1}^{n} (F_{index (j)}^{L})^{ζ j}, \prod_{j = 1}^{n} (F_{index (j)}^{U})^{ζ j}]) \end{matrix}$ (48)

Clearly, both the INPGWA and INPGOWA operators can consider the given arguments and their relationships; the difference between two operators is that the ILPGWA operator emphasizes their own importance of these arguments, while the ILPGOWA operator stresses their ordered position importance.

4 A multiple attribute decision making method based on these operators

In this section, we will use INPGWA or INPGOWA operators to the multiple attribute decision making problems in which attribute values take the form of INSs, and propose a decision making method.

For a multiple attribute decision making problem, let A ={ A₁, A₂, ⋯ , A_m } be the collection of alternatives, and C ={ C₁, C₂, ⋯ , C_n } be the collection of attributes. Suppose that $x_{ij} = ([T_{ij}^{L}, T_{ij}^{U}], [I_{ij}^{L}, I_{ij}^{U}], [F_{ij}^{L}, F_{ij}^{U}])$ is the evaluation information of the alternative A_i on the criteria C_j which is represented by the form of INSs. where $T_{ij}^{L}, T_{ij}^{U}, I_{ij}^{L}, I_{ij}^{U}, F_{ij}^{L}, F_{ij}^{U} \in [0, 1]$ and $T_{ij}^{U} + I_{ij}^{U} + F_{ij}^{U} \leq 3$ . Then we can rank the order of the alternatives.

If the attribute weight is known, we can use the INPGWA operator to aggregate all attribute values, otherwise, we can use INPGA operator or INPGOWA operator aggregate them. When the attribute weight is known, we can suppose weight vector of attribute set C is w = (w₁, w₂, ⋯ , w_n), and $w_{j} \in [0, 1], \sum_{j = 1}^{n}$ w_j = 1. So, we can only introduce the decision making method based on the INPGWA operator.

Then the decision steps are shown as follows.

Step 1. Convert cost type to benefit type by the complement function in Equation (7). If C_j is a cost type, then we can replace x_ij with ${\bar{x}}_{ij}$ for all i.

Step 2. Calculate the supports.

$\begin{matrix} Sup (x_{ij}, x_{il}) & = 1 - d (x_{ij}, x_{il}), i = 1, 2, \dots, m; \\ j, l & = 1, 2, \dots, n \end{matrix}$ (49) where d (x_ij, x_il) is defined by formula (4).

Step 3. Calculate T (x_ij)

$\begin{matrix} T (x_{ij}) & = \overset{n}{\underset{l \neq j}{\sum_{l = 1}}} Sup (x_{ij}, x_{il}) i = 1, 2, \dots, m; \\ j & = 1, 2, \dots, n \end{matrix}$ (50)

Step 4. Calculate the weight ϖ_ij associated with the interval neutrosophic set x_ij

$\begin{matrix} ϖ_{ij} & = \frac{w_{j} (1 + T (x_{ij}))}{\sum_{j = 1}^{n} w_{j} (1 + T (x_{ij}))} i = 1, 2, \dots, m; \\ j & = 1, 2, \dots \end{matrix}$ (51)

Step 5. Calculate the comprehensive evaluation value of each alternative $\begin{matrix} x_{i} = INPGWA (x_{i 1}, x_{i 2}, \dots, x_{in}) \\ = ([\begin{matrix} {(1 - \prod_{j = 1}^{n} (1 - (T_{ij}^{L})^{λ})^{ϖ ij})}^{1 / λ}, \\ {(1 - \prod_{j = 1}^{n} (1 - (T_{ij}^{U})^{λ})^{ϖ ij})}^{1 / λ} \end{matrix}], \\ [\begin{matrix} 1 - {(1 - \prod_{j = 1}^{n} (1 - (1 - I_{ij}^{L})^{λ})^{ϖ ij})}^{1 / λ}, \\ 1 - {(1 - \prod_{j = 1}^{n} (1 - (1 - I_{ij}^{U})^{λ})^{ϖ ij})}^{1 / λ} \end{matrix}], \\ [\begin{matrix} 1 - {(1 - \prod_{j = 1}^{n} (1 - (1 - F_{ij}^{L})^{λ})^{ϖ ij})}^{1 / λ}, \\ 1 - {(1 - \prod_{j = 1}^{n} (1 - (1 - F_{ij}^{U})^{λ})^{ϖ ij})}^{1 / λ} \end{matrix}]) \end{matrix}$ (52) where i = 1, 2, ⋯ , m.

Step 6: Calculate the cos ^* (x_i, I), where I = ([1, 1] , [0, 0] , [0, 0]) by Equation (6).

Step 7: Rank all the alternatives {A₁, A₂, ⋯ , A_m } and select the best one(s) by the cosine similarity degree in Definition 6.

Step 8: End.

5 An application example

In order to demonstrate the application of the proposed method, we will cite an example about the investment selection of a company (adapted from [13]). There is a company, which wants to invest a sum of money to an industry. There are 4 alternatives which can be considered by a panel, including: (1) A1 is a car company; (2) A2 is a food company; (3) A3 is a computer company; (4) A4 is an arms company. The evaluation on the alternatives is based on three criteria: (1) C1 is the risk; (2) C2 is the growth; (3) C3 is the environmental impact. where C1 and C2 are benefit criteria, and C3 is a cost criterion. The weight vector of the criteria is given by W=(0.35, 0.25, 0.4). The final decision information can be obtained by the INSs, and shown in Table 1.

We adopt the proposed method to rank the alternatives.

To get the best alternative(s), the following steps are involved:

(1) According to Equation (7), convert the cost criterion to benefit criterion.

(2) Calculate the supports Sup (x_ij, x_il) j, l = 1, 2, 3 . i = 1, 2, 3, 4 by formula (49) (for simplicity, we denote Sup (x_ij, x_il) with S_ij,il). We can get $S_{11, 12} = S_{12, 11} = 0.950, S_{11, 13} = S_{13, 11} = 0.683 .$

Limited to the space, the values of the other S_ij,il are not to be shown in detail.

(3) Calculate $T (x_{ij}) (i, = 1, 2, 3, 4 . j = 1, 2, 3)$ by formula (50) $T (x_{11}) = 1.633, T (x_{12}) = 1.583, T (x_{13}) = 1.317$

Limited to the space, the calculations of the other T (x_ij) are omitted here.

(4) Calculate the weights ϖ_ij (i, j = 1, 2, 3, 4) by formula (51) $ϖ_{11} = 0.370, ϖ_{12} = 0.259, ϖ_{1} 3 = 0.372$

Limited to the space, the calculations of the other ϖ_ij are omitted here.

(5) Calculate the comprehensive evaluation value of each alternative by formula (52), suppose λ = 1 $\begin{matrix} x_{1} & = & ([0 . 400, 0 . 528], [0 . 266, 0 . 432], [0 . 370, 0 . 541]) \\ x_{2} & = & ([0 . 691, 0 . 801], [0 . 278, 0 . 446], [0 . 239, 0 . 432]) \\ x_{3} & = & ([0 . 534, 0 . 763], [0 . 384, 0 . 564], [0 . 340, 0 . 440]) \\ x_{4} & = & ([0 . 721, 0 . 828], [0 . 307, 0 . 419], [0 . 333, 0 . 462]) \end{matrix}$

(6) Calculate the cos ^* (x_i, I) by formula (6), we can get $\begin{matrix} {cos}^{*} (x_{1}, I) = 0 . 619, {cos}^{*} (x_{2}, I) = 0 . 824, \\ {cos}^{*} (x_{3}, I) = 0 . 716, {cos}^{*} (x_{4}, I) = 0.817 \end{matrix}$

(7) Rank the alternatives

According to the cos ^* (x_i, I), the ranking is A₂ ≻ A₄ ≻ A₃ ≻ A₁.

In order to illustrate the influence of the parameter λ on decision making results of this example, we use the different value λ in INPGWA operator in step 5 to rank the alternatives. The ranking results are shown in Table 2.

As we can see from Table 2, the ordering of the alternatives may be different for the different value λ in INPGWA operator.

(1) When λ ≤ 0.7, the ordering of the alternatives is A₄ ≻ A₂ ≻ A₃ ≻ A₁ and the best alternative is A₄.

(2) When 0.8 ≤ λ ≤ 16.9, the best alternative is A₂, but the ordering of the alternatives is different with respect to different λ. When 0.8 ≤ λ ≤ 5.2, the ordering of the alternatives is A₂ ≻ A₄ ≻ A₃ ≻ A₁. When 5.3 ≤ λ ≤ 6.5, the ordering of the alternatives is A₂ ≻ A₄ ≻ A₁ ≻ A₃. When 6.6 ≤ λ ≤ 10.2, the ordering of the alternatives is A₂ ≻ A₁ ≻ A₄ ≻ A₃. When 10.3 ≤ λ ≤ 16.9, the ordering of the alternatives is A₂ ≻ A₁ ≻ A₃ ≻ A₄.

(3) When λ ≥ 17.0, the ordering of the alternatives is A₁ ≻ A₂ ≻ A₃ ≻ A₄ and the best alternative is A₁.

So, λ can be used to express the mentality of the decision-makers, the more the λ is, the more optimistic decision-makers are. On the contrary, the smaller the λ is, the more pessimistic decision-makers are. So, the organization or individual can properly select the desirable alternative according to his/her interest and the actual needs. Generally speaking, if they don’t have special preference, we can consider the rankings of some special values when λ gets 0, 1 and 2 because these values have some special meanings. In this example, we can think the best alternative is A₂.

In order to verify the effectiveness of the proposed method in this paper, we can compare with the method proposed by Ye [13]. Firstly, the same ranking results were produced by these methods. Secondly, the method proposed by Ye [13] was based on similarity measure, it cannot realize the information aggregation. The method proposed in this paper was based on the aggregation operators, and it can provide the more general and more flexible features as λ is assigned different values.

6 Conclusion

In real decision making, the decison information is often incomplete, indeterminate and inconsistent information, and the interval neutrosophic set (INS) can be better to express this kind of information. The traditional power average operators and the generalized aggregation operators are generally suitable for aggregating the crisp numbers, and yet they will fail in dealing with interval neutrosophic sets (INSs). In this paper, we have given some aggregation operators based on INSs, include INPGA operator, INPGWA operator and INPGOWA operator, and studied some properties and some special cases of them. Further, we have given a decision making method based on these operators. Finally, an illustrative example is given to verify the developed approach and to demonstrate its practicality and effectiveness, and to illustrate the influence of the parameter λ on decision making results in INPGWA operator. The prominent characteristic of the developed approaches is that they can take all the decision arguments and their relationships into account, and can reflect the mentality of the decision-makers, and the proposed method are more scientific to do decision making. In further research, it is necessary and meaningful to study some applications of these operators such as pattern recognition, medical diagnosis and inconsistent analysis, etc.

Footnotes

Acknowledgments

This paper is supported by the National Natural Science Foundation of China (Nos. 71471172 and 71271124), the Humanities and Social Sciences Research Project of Ministry of Education of China (No. 13YJC630104), Shandong Provincial Social Science Planning Project (No.13BGLJ10), the national soft science research project (2014GXQ4D192), and Graduate education innovation projects in Shandong Province (SDYY12065). The authors also would like to express appreciation to the anonymous reviewers and Editors for their very helpful comments that improved the paper.

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