Intuitionistic fuzzy information aggregation under confidence levels

Applied Soft Computing 19 (2014) 147–160

Contents lists available at ScienceDirect

Applied Soft Computing journal homepage: www.elsevier.com/locate/asoc

Intuitionistic fuzzy information aggregation under confidence levels Dejian Yu ∗ School of Information, Zhejiang University of Finance and Economics, Hangzhou 310018, China

a r t i c l e

i n f o

Article history: Received 21 November 2012 Received in revised form 10 January 2014 Accepted 2 February 2014 Available online 21 February 2014 Keywords: Intuitionistic fuzzy sets Confidence levels Multi-criteria group decision making Aggregation operator

a b s t r a c t In actuality, for example, the review of the National Science Foundation and the blind peer review of doctoral dissertation in China, the evaluation experts are requested to provide two types of information such as the performance of the evaluation objects and the familiarity with the evaluation areas (called confidence levels). However, existing information aggregation research achievements cannot be used to fusion the two types information described above effectively. In this paper, we focus on the information aggregation issue in the situation where there are confidence levels of the aggregated arguments under intuitionistic fuzzy environment. Firstly, we develop some confidence intuitionistic fuzzy weighted aggregation operators, such as the confidence intuitionistic fuzzy weighted averaging (CIFWA) operator and the confidence intuitionistic fuzzy weighted geometric (CIFWG) operator. Then, based on the Einstein operations, we proposed the confidence intuitionistic fuzzy Einstein weighted averaging (CIFEWA) operator and the confidence intuitionistic fuzzy Einstein weighted geometric (CIFEWG) operator. Finally, a practical example about the review of the doctoral dissertation in Chinese universities is provided to illustrate the developed intuitionistic fuzzy information aggregation operators. © 2014 Elsevier B.V. All rights reserved.

1. Introduction Fuzzy set (FS) theory proposed by Zadeh [1], is a powerful tool and has been applied to various fields. However, FS theory only has a membership degree which is not perfect in expressing the fuzziness of the subjective world. The intuitionistic fuzzy set (IFS) theory is an extension of FS theory and it was developed by Atanassov [2]. IFS is characterized by a membership degree and a non-membership degree [2–5]. One of the research branches of IFS theory is intuitionistic fuzzy multi-criteria group decision making and have attracted many attentions from researchers [6–15]. To aggregate all the performance on attributes for alternatives is a very critical step in decision making problem and the aggregation operators play an important role during the information fusion process. Many scholars are interested in intuitionistic fuzzy information aggregation operators. The intuitionistic fuzzy weighted averaging (IFWA) operator and intuitionistic fuzzy weighted geometric (IFWG) operator are two widely cited operators and they were introduced by Xu [16] and Xu and Yager [17] respectively. Based on which, the IFOWA operator, IFOWG operator, intuitionistic fuzzy hybrid averaging (IFHA) operator, intuitionistic fuzzy hybrid geometric (IFHG) operator, generalized IFWA, generalized IFWG, generalized IFOWA, generalized IFOWG are proposed [18]. Combined with the Bonferroni mean (BM), intuitionistic fuzzy Bonferroni mean (IFBM) and weighted IFBM are introduced by Xu [19]. Later, Xia et al. [20] proposed a series of extended IFBM operators, such as the GIFBM and weighted GIFBM. Inspired by the power aggregation operators [21,22], some intuitionistic fuzzy power operators are proposed, such as IFPWA, IFPWG, IFPOWA and IFPOWG [23]. The intuitionistic fuzzy power operators and intuitionistic fuzzy Bonferroni operators are two types of correlated aggregate operators; they are able to describe the relationships quantitatively from the objective perspective. Choquet integral based intuitionistic fuzzy aggregation operators, such as the intuitionistic fuzzy Choquet average (IFCA) operator, the IFCG operator [24,25], the quasi intuitionistic fuzzy Choquet ordered averaging (QIFCOA) operator [26], and induced generalized intuitionistic fuzzy Choquet ordered averaging (I-GIFCOA) operator [27] are also able to depict the interrelations between the aggregated arguments. However, they are different from the intuitionistic fuzzy power operators and intuitionistic fuzzy Bonferroni operators, since they are depict the interrelations from the subjective perspective. In order to fusion the intuitionistic fuzzy preference information, Xia et al. [28] proposed some intuitionistic multiplicative preference information aggregation operators, such as intuitionistic multiplicative weighted averaging (IMWA) operator, intuitionistic multiplicative weighted geometric (IMWG) operator, generalized IMWA operator and generalized IMWG operator. Furthermore, Xia and Xu [29] proposed

∗ Tel.: +86 15336884185. E-mail address: [email protected] 1568-4946/$ – see front matter © 2014 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.asoc.2014.02.001

148

D. Yu / Applied Soft Computing 19 (2014) 147–160

Fig. 1. The development of the intuitionistic fuzzy aggregation operators.

some extended intuitionistic multiplicative preference information aggregation operators, such as extended intuitionistic multiplicative weighted averaging (EIMWA) operator, extended intuitionistic multiplicative power averaging (EIMPA) operator, extended intuitionistic multiplicative Choquet averaging (EIMCA) operator, extended intuitionistic multiplicative power ordered averaging (EIMPOA) operator and extended intuitionistic multiplicative Choquet ordered averaging (EIMCOA) operator. Fig. 1 shows the development and the relations of the above described aggregation operators. But despite these remarkable achievements, the intuitionistic fuzzy information aggregation method is far from an unmitigated perfection. Most of the existing aggregation operators not consider the confidence level of the aggregated arguments provided by the information providers. However, in many real decision making problems, such as the blind peer review of doctoral dissertation in China, the evaluation experts are requested to provide two types of information such as the performance of the evaluation objects and the familiarity with the evaluation areas (called confidence levels) [30,31]. In this paper, we focus on the intuitionistic fuzzy information aggregation issue in the situation where the confidences levels of the aggregated arguments are asked to be considered. The main research contents can be summarized as following five parts, (1) confidence intuitionistic fuzzy aggregation operator; (2) confidence intuitionistic fuzzy ordered aggregation operator; (3) confidence intuitionistic fuzzy Einstein aggregation operator; (4) confidence intuitionistic fuzzy Einstein ordered aggregation operator; (5) group decision making method based on the above operators. The logistic relationship of above main content can be vivid compared to the construction of two – story house which shown in Fig. 2. This paper focuses on intuitionistic fuzzy set and information aggregation theory and uses them as the theory basis and methodology, just as the foundation of the house. Then this paper studied confidence intuitionistic fuzzy aggregation operators and confidence intuitionistic fuzzy ordered aggregation operators, the two are two supporting pillars of the house’s first floor and presents the progressive relationship. Meanwhile, based on Intuitionistic fuzzy Einstein operations, this paper further studied confidence intuitionistic fuzzy Einstein aggregation operators and confidence intuitionistic fuzzy Einstein ordered aggregation operators, the two also have the progressive relationship and refer to the supporting pillars of the second floor. At the end, an approach to intuitionistic fuzzy group decision making under confidence levels is introduced based on the proposed aggregation operators, it can be compared to the roof of the house. 2. Some basic concepts The Bulgarian scholar Atanossv [2] extended the fuzzy set theory [1] and introduced the intuitionistic fuzzy set (IFS) theory. The Definition of IFS was defined as follows and has been cited thousands of times. It could be found in most of the research papers about IFS theory. Definition 1.

The concept of intuitionistic fuzzy set (IFS) A on X is defined as follows:



A = { < x, A (x), vA (x) > x ∈ X}

(1)

where the functions A (x) and vA (x) denote the degrees of membership and non-membership of the element x ∈ X to the set A, respectively. In Eq. (1), the A (x) and vA (x) are functions with the values between closed interval [0, 1]. In addition, the sum of A (x) and vA (x) are also values between closed interval [0, 1]. For convenience, Xu [16] named ˛ = (˛ , v˛ ) an intuitionistic fuzzy number (IFN).

D. Yu / Applied Soft Computing 19 (2014) 147–160

149

Fig. 2. The logistic relationship graph of the main contents.

In the following, we introduce the basic operations of IFNs which was proposed by Xu [16] and Xu and Yager [17]. Definition 2.

For three IFNs ˛, ˛1 , ˛2 ∈ V,  > 0, some operations were given as follows:

1) ˛1 ⊕ ˛2 = (˛1 + ˛2 − ˛1 ˛2 , v˛1 v˛2 ) 2) ˛1 ⊗ ˛2 = (˛1 ˛2 , v˛1 + v˛2 − v˛1 v˛2 )

3) ˛ = (1 − (1 − ˛ ) , v˛ ) 4) ˛ = (˛ , 1 − (1 − v˛ ) )

Xu and Yager [17] introduced an effective approach to rank two IFNs ˛i = (˛i , v˛i ) (i = 1, 2). The main idea of this method is to rank any two IFNs according to their score functions and accuracy functions. There are two different cases when comparing any two IFNs. 1) If the score functions are different, the bigger the score functions of an IFN, the bigger IFN. 2) If two score functions are equal, then the accuracy function should be adopted and the ranking is depends on it totally. In other words, the bigger the accuracy functions of an IFN, the bigger IFN. 3. Intuitionistic fuzzy information aggregation operators under confidence levels In actuality, the evaluation experts are requested to provide two types of information such as the performance of the evaluation objects and the familiarity with the evaluation areas (called confidence levels) [30,31]. In this Section, we investigate the information aggregation methods under confidence levels in the context of intuitionistic fuzzy environment and proposed a series of aggregation operators. 3.1. Confidence intuitionistic fuzzy aggregation operators In the following, we propose the confidence intuitionistic fuzzy weighted averaging (CIFWA) operator and confidence intuitionistic fuzzy weighted geometric (CIFWG) operator and study the desirable properties of the proposed operators. The definition of the CIFWA operator was given as follows: Definition 3. Let (˛1 , ˛2 , . . ., ˛n ) be a collection of IFNs, lj be the confidence levels of IFN ˛j and 0 ≤ lj ≤ 1, w = (w1 , w2 , . . ., wn ) be the n weight vector of them, such that wj ∈ [0, 1] and w = 1. If j=1 j n

CIFWA(< l1 , ˛1 >, < l2 , ˛2 >, . . ., < ln , ˛n >) = ⊕ wj (lj ˛j ) = w1 (l1 ˛1 ) ⊕ w2 (l2 ˛2 ) ⊕ · · · ⊕ wn (ln ˛n )

(2)

j=1

Then CIFWA is called confidence intuitionistic fuzzy weighted averaging (CIFWA) operator. Especially, if l1 = l2 = · · · = ln = 1, then the CIFWA operator reduces to the intuitionistic fuzzy weighted averaging (IFWA) operator n

IFWA(˛1 , ˛2 , . . ., ˛n ) = ⊕ wj ˛j = w1 ˛1 ⊕ w2 ˛2 ⊕ · · · ⊕ wn ˛n j=1

which was proposed by Xu [16]. By Definitions 2 and 3, we can get the following Theorem.

(3)

150

D. Yu / Applied Soft Computing 19 (2014) 147–160

Theorem 1. Let (˛1 , ˛2 , . . ., ˛n ) be a collection of IFNs, lj be the confidence levels of IFN ˛j and 0 ≤ lj ≤ 1, then their aggregated value by using CIFWA operator is also an IFN and

⎛

CIFWA(< l1 , ˛1 >, < l2 , ˛2 >, . . ., < ln , ˛n >) = ⎝1 −

n 

n 

j=1

j=1

⎞

(vj )lj ×wj ⎠

(1 − j )lj ×wj ,

(4)

On the one hand,

Proof.

2 ˛ = (1 − (1 − ˛ )2 , v˛2 )

(5)

2

2 1

1 (1 − (1 − ˛ ) , v˛2 ) = (1 − (1 − 1 + (1 − ˛ ) ) , v˛1 2 ) = (1 − (1 − ˛ )

1 2

, v˛1 2 )

(6)

On the other hand, (1 2 )˛ = (1 − (1 − ˛ )1 2 , v˛1 2 )

(7)

Hence, 1 (2 ˛) = (1 2 )˛

(8)

Then, n

n

j=1

j=1

CIFWA(< l1 , ˛1 >, < l2 , ˛2 >, . . ., < ln , ˛n >) = ⊕ wj (lj ˛j ) = ⊕ (wj lj )˛j = (w1 l1 )˛1 ⊕ (w2 l2 )˛2 ⊕ · · · ⊕ (wn ln )˛n In the following, we prove

⎛ n

CIFWA(< l1 , ˛1 >, < l2 , ˛2 >, . . ., < ln , ˛n >) = ⊕ (wj lj )˛j = ⎝1 − j=1

n 

n 

j=1

j=1

(1 − j )lj ×wj ,

(9)

⎞

(vj )lj ×wj ⎠

(10)

Eq. (10) by using mathematical induction on n: 1) For n = 2: Since (w1 l1 )˛1 = (1 − (1 − 1 )w1 l1 , (v1 )w1 l1 ) (w2 l2 )˛2 = (1 − (1 − 2 )

w2 l2

, (v2 )

w2 l2

(11)

)

(12)

We have CIFWA(< l1 , ˛1 >, < l2 , ˛2 >) = (w1 l1 )˛1 ⊕ (w2 l2 )˛2 = (1 − (1 − 1 )w1 l1 (1 − 2 )w2 l2 , (v1 )w1 l1 (v2 )w2 l2 )

(13)

2) If Eq. (10) holds for n = k, that is

⎛ CIFWA(< l1 , ˛1 >, < l2 , ˛2 >, . . ., < lk , ˛k >) = (w1 l1 )˛1 ⊕ (w2 l2 )˛2 ⊕ · · · ⊕ (wk lk )˛k = ⎝1 −

k 

k 

j=1

j=1

(1 − j )wj lj ,

⎞

(vj )wj lj ⎠

(14)

then, when n = k + 1, by the operational laws described in Section 2, we have CIFWA(< l1 , ˛1 >, < l2 , ˛2 >, . . ., < lk+1 , ˛k+1 >) = (w1 l1 )˛1 ⊕ (w2 l2 )˛2 ⊕ · · · ⊕ (wk lk )˛k ⊕ (wk+1 lk+1 )˛k+1

⎛

= ⎝1 −

⎛ = ⎝1 −

⎛ = ⎝1 −

k 

k 

j=1

j=1

(1 − j )wj lj ,

⎞

⎛

 k



(vj )wj lj ⎠ ⊕ 1 − (1 − k+1 )wk+1 lk+1 , (vk+1 )wk+1 lk+1

(1 − j )wj lj + 1 − (1 − k+1 )wk+1 lk+1 − ⎝1 −

j=1





j=1

j=1

k+1

(1 − j )wj lj ,

k+1

⎞

k 

⎞





(1 − j )wj lj ⎠ 1 − (1 − k+1 )wk+1 lk+1

j=1

k+1



⎞

(vj )wj lj ⎠

,

j=1

(vj )wj lj ⎠

i.e. Eq. (10) holds for n = k + 1. Thus, Eq. (10) holds for all n. Then

⎛

CIFWA(< l1 , ˛1 >, < l2 , ˛2 >, . . ., < ln , ˛n >) = ⎝1 −

n 

n 

j=1

j=1

(1 − j )lj ×wj ,

⎞

(vj )lj ×wj ⎠

(15)

Obviously, the prove method of Theorem 1 is based on mathematical induction. This kind of proves method was widely used in existing research results, such as Xu [16], Zhao et al. [18].

D. Yu / Applied Soft Computing 19 (2014) 147–160

151

Combined the CIFWA operator with geometric mean, we study the confidence intuitionistic fuzzy information aggregation from the geometric perspective. Firstly, we give the definition of the confidence intuitionistic fuzzy weighted geometric (CIFWG) operator as follows: Definition 4.

Let (˛1 , ˛2 , . . ., ˛n ) be a collection of IFNs, lj be the confidence levels of IFN ˛j and 0 ≤ lj ≤ 1, w = (w1 , w2 , ..., wn ) be the

weight vector of them, such that wj ∈ [0, 1] and

n 

wj = 1. A confidence intuitionistic fuzzy weighted geometric (CIFWG) operator is a

j=1

mapping In → I such that:

n

CIFWG(< l1 , ˛1 >, < l2 , ˛2 >, . . ., < ln , ˛n >) = ⊗ (˛ljj )wj = (˛l11 )w1 ⊗ (˛l22 )w2 ⊗ · · · ⊗ (˛lnn )wn

(16)

j=1

Especially, if l1 = l2 = · · · = ln = 1, then the CIFWG operator reduces to the intuitionistic fuzzy weighted geometric (IFWG) operator n

j = ˛w1 ⊗ ˛w2 ⊗ · · · ⊗ ˛wn IFWG(˛1 , ˛2 , . . ., ˛n ) = ⊗ ˛w n 1 2 j

(17)

j=1

which was proposed by Xu and Yager [17]. Theorem 2. Let (˛1 , ˛2 , . . ., ˛n ) be a collection of IFNs, lj be the confidence levels of IFN ˛j and 0 ≤ lj ≤ 1, then their aggregated value by using CIFWG operator is also an IFN and

⎛

CIFWG(< l1 , ˛1 >, < l2 , ˛2 >, . . ., < ln , ˛n >) = ⎝

n 

n 

j=1

j=1

(j )lj ×wj , 1 −

⎞

(1 − vj )lj ×wj ⎠

(18)

where w = (w1 , w2 , . . ., wn )T is the weight vector of (j = 1, 2, . . ., n) with wj ∈ [0, 1] and Proof.

n j=1

wj = 1.

On the one hand,

˛2 = (˛2 , 1 − (1 − v˛ )2 ) (˛2 , 1 − (1 − v˛ )2 )1

=

(19)

(˛1 2 , 1 − (1 − 1 + (1 − v˛ )2 )1 )

=

(˛1 2 , 1 − (1 − v˛ )1 2 )

(20)

On the other hand, ˛1 2 = (˛1 2 , 1 − (1 − v˛ )1 2 )

(21)

Hence, (˛2 )1 = ˛1 2

(22)

Then, n

n

j=1

j=1

CIFWG(< l1 , ˛1 >, < l2 , ˛2 >, . . ., < ln , ˛n >) = ⊗ (˛ljj )wj = ⊗ (˛j )wj lj = (˛1 )w1 l1 ⊗ (˛2 )w2 l2 ⊗ · · · ⊗ (˛n )wn ln

(23)

In the following, we prove

⎛ n

CIFWG(< l1 , ˛1 >, < l2 , ˛2 >, . . ., < ln , ˛n >) = ⊗ (˛j )wj lj = ⎝ j=1

n 

n 

j=1

j=1

(j )lj ×wj , 1 −

⎞

(1 − vj )lj ×wj ⎠

(24)

Eq. (24) by using mathematical induction on n: 1) For n = 2: Since (˛1 )w1 l1 = ((1 )w1 l1 , 1 − (1 − v1 )w1 l1 ) (˛2 )

w2 l2

= ((2 )

w2 l2

, 1 − (1 − v2 )

w2 l2

(25) (26)

)

We have CIFWG(< l1 , ˛1 >, < l2 , ˛2 >) = (˛1 )w1 l1 ⊗ (˛2 )w2 l2 = ((1 )w1 l1 (2 )w2 l2 , 1 − (1 − v1 )w1 l1 (1 − v2 )w2 l2 )

(27)

2) If Eq. (24) holds for n = k, that is

⎛ CIFWA(< l1 , ˛1 >, < l2 , ˛2 >, . . ., < lk , ˛k >) = (˛1 )w1 l1 ⊗ (˛2 )w2 l2 ⊗ · · · ⊗ (˛k )wk lk = ⎝

k 

k 

j=1

j=1

(j )wj lj , 1 −

⎞

(1 − vj )wj lj ⎠

(28)

152

D. Yu / Applied Soft Computing 19 (2014) 147–160

then, when n = k + 1, by the operational laws described in Section 2, we have CIFWG(< l1 , ˛1 >, < l2 , ˛2 >, . . ., < lk+1 , ˛k+1 >) = (˛1 )w1 l1 ⊗ (˛2 )w2 l2 ⊗ · · · ⊗ (˛k )wk lk ⊗ (˛k+1 )wk+1 lk+1

⎛

=⎝

k 

(j )wj lj , 1 −

⎞

k 



(1 − vj )wj lj ⎠ ⊗ (k+1 )wk+1 lk+1 , 1 − (1 − vk+1 )wk+1 lk+1



j=1 ⎛ j=1 ⎛ ⎞ ⎞ k+1 k k

   =⎝ (j )wj lj , 1 − (1 − vj )wj lj + 1 − (1 − vk+1 )wk+1 lk+1 − ⎝1 − (1 − vj )wj lj ⎠ 1 − (1 − vk+1 )wk+1 lk+1 ⎠ j=1 ⎛ j=1 ⎞ k+1 k+1   =⎝ (j )wj lj , 1 − (1 − vj )wj lj ⎠ j=1

j=1

j=1

i.e. Eq. (24) holds for n = k + 1. Thus, Eq. (24) holds for all n. Then

⎛ ⎞ n n   l ×w l ×w CIFWG(< l1 , ˛1 >, < l2 , ˛2 >, . . ., < ln , ˛n >) = ⎝ (j ) j j , 1 − (1 − vj ) j j ⎠ j=1

(29)

j=1

Similar to Theorem 1, the proof of Theorem 2 borrows from Xu [16], Zhao et al. [18]. Let ˛j = (˛j , v˛j ) and ˇj = (ˇj , vˇj ) (j = 1, 2, . . ., n) be two collections of IFNs, lj (0 ≤ lj ≤ 1) and wj (wj ∈ [0, 1]) be the confidence levels

n

and weight of them, respectively, with w = 1. Then we can easily prove the CIFWA and CIFWG operators have the properties of j=1 j monotonicity, boundness and idempotency as follows: 1) (Monotonicity) If ˛j ≤ ˇj and v˛j ≥ vˇj , for all j, then CIFWA(< l1 , ˛1 >, < l2 , ˛2 >, . . ., < ln , ˛n >) ≤ CIFWA(< l1 , ˇ1 >, < l2 , ˇ2 >, . . ., < ln , ˇn >)

(30)

CIFWG(< l1 , ˛1 >, < l2 , ˛2 >, . . ., < ln , ˛n >) ≤ CIFWG(< l1 , ˇ1 >, < l2 , ˇ2 >, . . ., < ln , ˇn >)

(31)

2) (Boundness) min(lj ˛j ) ≤ CIFWA(< l1 , ˛1 >, < l2 , ˛2 >, . . ., < ln , ˛n >) ≤ max(lj ˛j ) j

(32)

j

(33)min(˛ljj ) ≤ CIFWG(< l1 , ˛1 >, < l2 , ˛2 >, . . ., < ln , ˛n >) ≤ max(˛ljj ) j

j

3) (idempotency) If ˛ = (˛ , v˛ ) be an IFN, ˛j = ˛ and v˛j = v˛ , for all j, then CIFWA(< l1 , ˛1 >, < l2 , ˛2 >, . . ., < ln , ˛n >) = l˛

(34)

CIFWG(< l1 , ˛1 >, < l2 , ˛2 >, . . ., < ln , ˛n >) = ˛l

(35)

3.2. Confidence ordered intuitionistic fuzzy aggregation operators In this section, we introduce the idea of OWA [32] into confidence intuitionistic fuzzy information aggregation problem and propose some aggregation operator correspondingly. Definition 5. Let (˛1 , ˛2 , . . ., ˛n ) be a collection of IFNs, lj be the confidence levels of IFN ˛j and 0 ≤ lj ≤ 1, a confidence intuitionistic fuzzy ordered weighted averaging (CIFOWA) operator is a mapping In → I: n

CIFOWA(< l1 , ˛1 >, < l2 , ˛2 >, . . ., < ln , ˛n >) = ⊕ ωj (lı(j) ˛ı(j) ) = ω1 (lı(1) ˛ı(1) ) ⊕ ω2 (lı(2) ˛ı(2) ) ⊕ · · · ⊕ ωn (lı(n) ˛ı(n) )

(36)

j=1

A confidence intuitionistic fuzzy ordered weighted geometric (CIFOWG) operator is a mapping In → I: n

ω ω ω ω ı(j) ) j = (˛lı(1) ) 1 ⊗ (˛lı(2) ) 2 ⊗ · · · ⊗ (˛lı(n) ) n CIFOWG(< l1 , ˛1 >, < l2 , ˛2 >, . . ., < ln , ˛n >) = ⊗ (˛lı(j) ı(1) ı(2) ı(n)

(37)

j=1

where ω = (ω1 , ω2 , . . ., ωn )T is the associated weight vector such that ωj ∈ [0, 1] and

n 

ı(j) is the jth ωj = 1. ı : (1, 2, . . ., n) → (1, 2, . . ., n), ˛lı(j)

j=1

largest of ˛ljj , lı(j) ˛ı(j) is the jth largest of lj ˛j . Especially, if l1 = l2 = · · · = ln = 1, then the CIFOWA operator reduces to the intuitionistic fuzzy ordered weighted averaging (IFOWA) operator n

IFOWA(˛1 , ˛2 , . . ., ˛n ) = ⊕ ωj ˛j = ω1 ˛1 ⊕ ω2 ˛2 ⊕ · · · ⊕ ωn ˛n j=1

which was proposed by Xu [16].

(38)

D. Yu / Applied Soft Computing 19 (2014) 147–160

153

If l1 = l2 = · · · = ln = 1, then the CIFOWG operator reduces to the intuitionistic fuzzy ordered weighted geometric (IFOWG) operator n

1 ⊗ ˛ω2 ⊗ · · · ⊗ ˛ωn IFOWG(˛1 , ˛2 , . . ., ˛n ) = ⊗ ωj ˛(j) = ˛ω ı(1) ı(2) ı(n)

(39)

j=1

which was proposed by Xu and Yager [17]. Theorem 3. Let (˛1 , ˛2 , . . ., ˛n ) be a collection of IFNs, lj be the confidence levels of IFN ˛j and 0 ≤ lj ≤ 1, then their aggregated value by using CIFOWA operator is also an IFN and

⎛

CIFOWA(< l1 , ˛1 >, < l2 , ˛2 >, . . ., < ln , ˛n >) = ⎝1 −

n 

n 

j=1

j=1

(1 − (j) )l(j) ×wj ,

⎞

(v(j) )l(j) ×wj ⎠

their aggregated value by using CIFOWG operator is also an IFN and

⎛

CIFOWA(< l1 , ˛1 >, < l2 , ˛2 >, . . ., < ln , ˛n >) = ⎝

n 

n 

j=1

j=1

((j) )l(j) ×wj , 1 −

(40)

⎞

(1 − v(j) )l(j) ×wj ⎠

where ω = (ω1 , ω2 , ..., ωn )T is the associated weight vector such that ωj ∈ [0, 1] and

n 

(41)

ωj = 1.

j=1

Proof.

The proof of Theorem 3 is similar to the proof of Theorem 1 and Theorem 2, the description will not repeat here.

3.3. Confidence intuitionistic fuzzy weighted aggregation operators based on Einstein operations It should be noted that the confidence intuitionistic fuzzy information aggregation operators proposed in Sections 3.1 and 3.2 are based on Algebraic t-conorm and t-norm. However, Wang and Liu [33,34] pointed out that the Algebraic t-conorm and t-norm is not the only operations for intuitionistic fuzzy set and proposed the Einstein operations for IFNs as follows. For three IFNs ˛, ˛1 , ˛2 ∈ V, some Einstein operational laws were given as follows: 1) ˛1 ⊕ε ˛2 = 2) ˛1 ⊗ε ˛2 = 3) ˛ = 4) ˛ =





˛1 +˛2 1+˛1 ˛2



v˛1 v˛2

,

1+(1−v˛1 )(1−v˛2 )

˛1 ˛2 v˛ +v˛2 , 1 1+(1−˛1 )(1−˛2 ) 1+v˛1 v˛2

2 ˛ 



(2−˛ ) +˛

,



(1+v˛ ) −(1−v˛ )



(1+v˛ ) +(1−v˛ )

(1+˛ ) −(1−˛ )



(1+˛ ) +(1−˛ )

,

2v ˛



;





;

( > 0);

(2−v˛ ) +v ˛

( > 0).

Suppose ˛ is an IFN, the following equations are valid (Wang and Liu [33,34]) ˛1 2 = (˛1 )2 (1 > 0, 2 > 0)

(42)

1 (2 ˛) = (1 2 )˛(1 > 0, 2 > 0)

(43)

Based on the intuitionistic fuzzy Einstein operational laws, we develop some confidence operators for aggregating the intuitionistic fuzzy information. Definition 6. Let (˛1 , ˛2 , . . ., ˛n ) be a collection of IFNs, lj be the confidence levels of IFN ˛j and 0 ≤ lj ≤ 1, w = (w1 , w2 , ..., wn ) be the weight vector of them, such that wj ∈ [0, 1] and

n 

wj = 1. A confidence intuitionistic fuzzy Einstein weighted averaging (CIFEWA) operator is a

j=1

mapping In → I such that:

n

CIFEWA(< l1 , ˛1 >, < l2 , ˛2 >, . . ., < ln , ˛n >) = ⊕ wj (lj ˛j ) = w1 (l1 ˛1 ) ⊕ w2 (l2 ˛2 ) ⊕ · · · ⊕ wn (ln ˛n )

(44)

j=1

Especially, if l1 = l2 = · · · = ln = 1, then the CIFEWA operator reduces to the intuitionistic fuzzy Einstein weighted averaging (IFEWA) operator n

IFEWA(˛1 , ˛2 , . . ., ˛n ) = ⊕ wj ˛j = w1 ˛1 ⊕ w2 ˛2 ⊕ · · · ⊕ wn ˛n

(45)

j=1

which was proposed by Wang and Liu [34]. Theorem 4. Let (˛1 , ˛2 , . . ., ˛n ) be a collection of IFNs, lj be the confidence levels of IFN ˛j and 0 ≤ lj ≤ 1, then their aggregated value by using CIFEWA operator is also an IFN and

n

CIFEWA(< l1 , ˛1 >, < l2 , ˛2 >, . . ., < ln , ˛n >) =

(1 + j )lj wj j=1 n (1 + j )lj wj j=1

− (1 − j )lj wj + (1 − j )lj wj

,

n j=1

2

n

(2 − vj )



(vj )lj wj

j=1 lj wj

+

n j=1

(vj )lj wj

(46)

154

D. Yu / Applied Soft Computing 19 (2014) 147–160

where w = (w1 , w2 , ..., wn )T is the weight vector of (j = 1, 2, ..., n) with wj ∈ [0, 1] and Proof.

n j=1

wj = 1.

According to Eqs. (42) and (43), we can get n

n

j=1

j=1

CIFEWA(< l1 , ˛1 >, < l2 , ˛2 >, . . ., < ln , ˛n >) = ⊕ wj (lj ˛j ) = ⊕ (wj lj )˛j In the following, we prove

(47)

n

n

CIFEWA(< l1 , ˛1 >, < l2 , ˛2 >, . . ., < ln , ˛n >) = ⊕ (wj lj )˛j = j=1

(1 + j )lj wj j=1 n (1 + j )lj wj j=1

− (1 − j )lj wj + (1 − j )lj wj

,

2

n j=1

n



(vj )lj wj

j=1 lj wj

(2 − vj )

+

n j=1

(vj )lj wj (48)

by using mathematical induction on n: 1) For n = 2: Since



(l1 w1 )˛1 =

 (l2 w2 )˛2 =

(1 + 1 )l1 w1 − (1 − 1 )l1 w1



2(v1 )l1 w1

,

(1 + 1 )l1 w1 + (1 − 1 )l1 w1 (2 − v1 )l1 w1 + (v1 )l1 w1 (1 + 2 )l2 w2 − (1 − 2 )l2 w2

2(v2 )l2 w2

(49)



, (1 + 2 )l2 w2 + (1 − 2 )l2 w2 (2 − v2 )l2 w2 + (v2 )l2 w2

(50)

Then



 

l w1

2(v1 ) 1



1 + (1 + 1 ) l w1

/(2 − v1 ) 1

l w1

1 + 1 − (2(v1 ) 1

l w1

(1 + j )lj wj j=1 2 (1 + j )lj wj j=1

− +

⊕ (wj lj )˛j =

/(1 + 1 )

1 − (2(v2 ) 2

1)

2 /(2

2

l w1

+ (1 − 1 ) 1

l1 w1

+ (1 − 1 )

  l w l w l w 2(v2 ) 2 2 /(2 − v2 ) 2 2 + (v2 ) 2 2   l w l w l w

+ (v1 ) 1

(1 − j )lj wj j=1 2 (1 − j )lj wj j=1

k

j=1

− (1 − 1 )

l w1

/(1 + 1 ) 1

l1 w1

(1 + j )lj wj j=1 k (1 + j )lj wj j=1

− +

,

2

2 j=1

− v2 ) 2

2 j=1

(2 − vj )lj wj +

(1 − j )lj wj j=1 k (1 − j )lj wj j=1

j=1

l w2

k

2 j=1

j=1

j=1

⊕ (wj lj )˛j = ⊕ (wj lj )˛j ⊕ (wk+1 lk+1 )˛k+1 =



CIFEWA(< l1 , ˛1 >, < l2 , ˛2 >, . . ., < ln , ˛n >) =

,



(51)

(52)

j=1

− +



(vj )lj wj

k j=1

(53)

(vj )lj wj

k+1

(1 − j )lj wj j=1 k+1 (1 − j )lj wj j=1

k+1

,

k+1 j=1

2

(1 + j )lj wj j=1 n (1 + j )lj wj j=1

− (1 − j )lj wj + (1 − j )lj wj

j=1

n

,

n j=1

2

k+1 j=1

+

n j=1

(54)

(vj )lj wj



(vj )lj wj

j=1 lj wj

(2 − vj )



(vj )lj wj

(2 − vj )lj wj +

i.e. Eq. (48) holds for n = k + 1. Thus, Eq. (48) holds for all n. Then

n

l w2

+ (1 − 2 ) 2

(vj )lj wj

k

(1 + j )lj wj j=1 k+1 (1 + j )lj wj j=1

/(1 + 2 )

l w2

+ (1 − 2 ) 2

l2 w2

)

(2 − vj )lj wj +

k+1

− (1 − 2 )

l w2

/(1 + 2 ) 2

l2 w2



then, when n = k + 1, by the operational laws of IFNs, we have k+1

(1 + 2 )

l w2

− (1 − 2 ) 2

l2 w2



+ (v2 ) 2

2

k

,



(vj )lj wj

k

2

l w2

+ (1 + 2 ) 2

l1 w1

l w1

If Eq. (48) holds for n = k, that is k

l w1

− (1 − 1 ) 1

l1 w1

+ (v1 ) 1

/(2 − v1 ) 1

2 =

l w1

(1 + 1 ) 1

(l1 w1 )˛1 ⊕ (l2 w2 )˛2 =

(vj )lj wj

(55)

It should be noted that the above proof are largely inspired by the idea of Xu [16], Zhao et al. [18], Wang and Liu [33,34]. Definition 7. Let (˛1 , ˛2 , . . ., ˛n ) be a collection of IFNs, lj be the confidence levels of IFN ˛j and 0 ≤ lj ≤ 1, w = (w1 , w2 , ..., wn ) be the weight vector of them, such that wj ∈ [0, 1] and

n 

wj = 1. A confidence intuitionistic fuzzy Einstein weighted geometric (CIFEWG) operator is a

j=1

mapping In → I such that:

n

CIFEWG(< l1 , ˛1 >, < l2 , ˛2 >, . . ., < ln , ˛n >) = ⊗ (˛ljj )wj = (˛1l1 )w1 ⊗ (˛l22 )w2 ⊗ · · · ⊗ (˛lnn )wn

(56)

j=1

Especially, if l1 = l2 = · · · = ln = 1, then the CIFEWG operator reduces to the intuitionistic fuzzy Einstein weighted geometric (IFEWG) operator n

j = ˛w1 ⊗ ˛w2 ⊗ · · · ⊗ ˛wn IFEWG(˛1 , ˛2 , . . ., ˛n ) = ⊗ ˛w n 1 2 j

(57)

j=1

which was proposed by Wang and Liu [33]. Theorem 5. Let (˛1 , ˛2 , . . ., ˛n ) be a collection of IFNs, lj be the confidence levels of IFN ˛j and 0 ≤ lj ≤ 1, then their aggregated value by using CIFEWG operator is also an IFN and

D. Yu / Applied Soft Computing 19 (2014) 147–160



n

2

n

CIFEWG(< l1 , ˛1 >, < l2 , ˛2 >, . . ., < ln , ˛n >) =

j=1

155

n

(j )lj wj

j=1 lj wj

(2 − j )

+

n j=1

(j )lj wj

,

where w = (w1 , w2 , ..., wn )T is the weight vector of (j = 1, 2, ..., n) with wj ∈ [0, 1] and Proof.

(1 + vj )lj wj j=1 n (1 + vj )lj wj j=1

n

j=1



− (1 − vj )lj wj

(58)

+ (1 − vj )lj wj

wj = 1.

According to Eqs. (42) and (43), we can get n

n

j=1

j=1

CIFEWG(< l1 , ˛1 >, < l2 , ˛2 >, . . ., < ln , ˛n >) = ⊗ (˛ljj )wj = ⊗ (˛j )lj wj In the following, we prove

(59)

n

CIFEWG(< l1 , ˛1 >, < l2 , ˛2 >, . . ., < ln , ˛n >) = ⊗ (˛j )

lj wj

n

=

j=1

n

2

j=1

n

(j )lj wj

j=1 lj wj

(2 − j )

+

n j=1

(j )lj wj

,

(1 + vj )lj wj j=1 n (1 + vj )lj wj j=1

− (1 − vj )lj wj



+ (1 − vj )lj wj (60)

by using mathematical induction on n: 1) For n = 2: Since



(˛1 )

l1 w1

(˛2 )

l2 w2

=

 =



(1 + v1 )l1 w1 − (1 − v1 )l1 w1

2(1 )l1 w1

, (2 − 1 )l1 w1 + (1 )l1 w1 (1 + v1 )l1 w1 + (1 − v1 )l1 w1 (1 + 2 )l2 w2 − (1 − 2 )l2 w2

(61)



2(v2 )l2 w2

, (1 + 2 )l2 w2 + (1 − 2 )l2 w2 (2 − v2 )l2 w2 + (v2 )l2 w2

Then



⎛

(˛1 )l1 w1 ⊗ (˛2 )l2 w2 = ⎝



(62)

2(1 )l1 w1 /(2 − 1 )l1 w1 + (1 )l1 w1



2(2 )l2 w2 /(2 − (2 ))l2 w2 + (2 )l2 w2



1 + (1 − (2(1 )l1 w1 /(2 − 1 )l1 w1 + (1 )l1 w1 )



,

1 − (2(2 )l2 w2 /(2 − 2 )l2 w2 + (2 )l2 w2 )

⎞ (1 + v1 )



l1 w1

1 + (1 + v1 )

− (1 − v1 )

l1 w1

l1 w1

− (1 − v1 )

(˛1 )

l1 w1

⊗ (˛2 )

l2 w2

=

/(1 + v1 )

l1 w1

/(1 + v1 )

2

2 j=1

l1 w1

2

j=1

+ (1 − v1 )

l1 w1

l1 w1

+ (1 − v1 )



(1 + v2 )

2 j=1

(j )lj wj

,

− (1 − v2 )

l2 w2

2

(j )lj wj

(2 − j )lj wj +

+ (1 + v2 )

l1 w1

l2 w2

(1 + vj )lj wj j=1 2 (1 + vj )lj wj j=1

− +

− (1 − v2 )



⊗ (˛j )

lj wj

j=1

k

k

=

j=1

2

j=1

k

(j )lj wj

(2 − j )lj wj +

k j=1

(j )lj wj

,

(1 + vj )lj wj j=1 k (1 + vj )lj wj j=1

− +

(1 − vj )lj wj j=1 k (1 − vj )lj wj j=1

l2 w2

/(1 + v2 )

(1 − vj )lj wj j=1 2 (1 − vj )lj wj j=1

k

/(1 + v2 )

l2 w2

2

If Eq. (60) holds for n = k, that is k

l2 w2

+ (1 − v2 )

l2 w2

l2 w2

+ (1 − v2 )l2 w2



⊗ (˛j )

j=1

lj wj

k

= ⊗ ˛j j=1

lj wj

⊗ (˛k+1 )

lk+1 wk+1

=

k+1

k+1 j=1

2

j=1

(64)

 (65)

k+1

(j )lj wj

(2 − j )lj wj +

k+1 j=1

(j )lj wj

,

(63)



then, when n = k + 1, by the operational laws of IFNs, we have k+1

⎠

(1 + vj )lj wj j=1 k+1 (1 + vj )lj wj j=1

− +

k+1

(1 − vj )lj wj j=1 k+1 (1 − vj )lj wj j=1

 (66)

i.e. Eq. (60) holds for n = k + 1. Thus, Eq. (60) holds for all n. Then

CIFEWG(< l1 , ˛1 >, < l2 , ˛2 >, . . ., < ln , ˛n >) =

n  lj wj

n j=1

2

j=1

j

(2 − j )lj wj +

n

n  lj wj , j=1

j

(1 + vj )lj wj j=1 n (1 + vj )lj wj j=1

− (1 − vj )lj wj



+ (1 − vj )lj wj

(67)

It should be noted that the above proof are also largely inspired by the idea of Xu [16], Zhao et al. [18], Wang and Liu [33,34]. 3.4. Confidence intuitionistic fuzzy ordered weighted aggregation operators based on Einstein operations Similar to the Section 3.2, in this section, we introduce the idea of OWA aggregation operator into confidence intuitionistic fuzzy information aggregation problem based on the Einstein operations and propose some aggregation operator correspondingly. Definition 8. Let (˛1 , ˛2 , . . ., ˛n ) be a collection of IFNs, lj be the confidence levels of IFN ˛j and 0 ≤ lj ≤ 1, a confidence intuitionistic fuzzy Einstein ordered weighted averaging (CIFEOWA) operator is a mapping In → I: n

CIFEOWA(< l1 , ˛1 >, < l2 , ˛2 >, . . ., < ln , ˛n >) = ⊕ ωj (lı(j) ˛ı(j) ) = ω1 (lı(1) ˛ı(1) ) ⊕ ω2 (lı(2) ˛ı(2) ) ⊕ · · · ⊕ ωn (lı(n) ˛ı(n) ) j=1

(68)

156

D. Yu / Applied Soft Computing 19 (2014) 147–160

A confidence intuitionistic fuzzy Einstein ordered weighted geometric (CIFEOWG) operator is a mapping In → I: n

ω ω ω ω ı(j) ) j = (˛lı(1) ) 1 ⊗ (˛lı(2) ) 2 ⊗ · · · ⊗ (˛lı(n) ) n CIFEOWG(< l1 , ˛1 >, < l2 , ˛2 >, . . ., < ln , ˛n >) = ⊗ (˛lı(j) ı(1) ı(2) ı(n)

(69)

j=1

where ω = (ω1 , ω2 , ..., ωn )T is the associated weight vector such that ωj ∈ [0, 1] and

n 

lı(j) is the jth ωj = 1. ı : (1, 2, ..., n) → (1, 2, . . . n), ˛ı(j)

j=1

largest of ˛ljj , lı(j) ˛ı(j) is the jth largest of lj ˛j . Especially, if l1 = l2 = · · · = ln = 1, then the CIFEOWA operator reduces to the intuitionistic fuzzy ordered weighted averaging (IFEOWA) operator n

IFEOWA(˛1 , ˛2 , . . ., ˛n ) = ⊕ ωj ˛j = ω1 ˛1 ⊕ ω2 ˛2 ⊕ · · · ⊕ ωn ˛n

(70)

j=1

If l1 = l2 = · · · = ln = 1, then the CIFEOWG operator reduces to the intuitionistic fuzzy ordered weighted geometric (IFEOWG) operator n

1 ⊗ ˛ω2 ⊗ · · · ⊗ ˛ωn IFOWG(˛1 , ˛2 , . . ., ˛n ) = ⊗ ωj ˛(j) = ˛ω ı(1) ı(2) ı(n)

(71)

j=1

Theorem 6. Let (˛1 , ˛2 , ..., ˛n ) be a collection of IFNs, lj be the confidence levels of IFN ˛j and 0 ≤ lj ≤ 1, then their aggregated value by using CIFEOWA operator is also an IFN and CIFEOWA(< l1 , ˛1 >, < l2 , ˛2 >, . . ., < ln , ˛n >)

n

=

(1 + ı(j) )lı(j) ×ωj j=1 n (1 + ı(j) )lı(j) ×ωj j=1

− (1 − ı(j) )lı(j) ×ωj + (1 − ı(j) )lı(j) ×ωj

,

n 

n j=1

2

j=1

vı(j)



lı(j) ×ωj

(2 − vı(j) )lı(j) ×ωj +

n  j=1

vı(j)

lı(j) ×ωj

(72)

their aggregated value by using CIFEOWG operator is also an IFN and

CIFEOWG =

n

n j=1

2

(ı(j) ) j=1 lı(j) ×ωj

(2 − ı(j) )

+

n

lı(j) ×ωj

n j=1

(ı(j) )lı(j) ×ωj

,

(1 + vı(j) )lı(j) ×ωj j=1 n (1 + vı(j) )lı(j) ×ωj j=1

− (1 − vı(j) )lı(j) ×ωj



+ (1 − vı(j) )lı(j) ×ωj n 

where ω = (ω1 , ω2 , ..., ωn )T is the associated weight vector such that ωj ∈ [0, 1] and

(73)

ωj = 1.

j=1

4. Group decision making under confidence levels in the context of intuitionistic fuzzy environment The idea of group decision making is a widely used and has been applied to many areas, such as natural disasters [37–39], energy [40], personnel evaluation [41,42]. During the process of group decision making, the aggregation operators play an important role in aggregating the experts’ evaluations to a comprehensive one [43–46]. Motivated by the existing research idea, we use the confidence intuitionistic fuzzy aggregation operators to group decision making problem. 4.1. An approach to group decision making under intuitionistic fuzzy environment In order to solve the group decision making problems with confidence levels in the context of intuitionistic fuzzy environment, we introduce a decision analysis approach in this section. Firstly, decision makers are required to give their evaluation on alternatives using IFNs, together with the confidence levels of the evaluation subject. Secondly, the aggregation operators proposed in Section 3 are applied to aggregate the each decision maker’s evaluation to collective evaluation on each alternative. Thirdly, we aggregate the performance of each attribute for each alternative and get the comprehensive IFNs. Finally, the ranking of the alternatives can be got by calculating their score function and accuracy function of the comprehensive intuitionistic fuzzy numbers. Based on the ranking, the optimal alternative can be singled out and we will describe the detailed decision making method in the remainder of this section. For a multi-criteria decision making problem, let X = {X1 , X2 , .., Xm } be a set of m alternatives, C = {c1 , c2 , .., cn } a set of n criteria, whose weight vector is w = (w1 , w2 , ..., wn )T , satisfying wj > 0, j = 1, 2, ..., n and

n 

wj = 1, and E = {e1 , e2 , · · · , ep } is the set of decision makers

i=1

and their weight vector is ω = (ω1 , ω2 , ..., ωp )T , satisfying ωq > 0, q = 1, 2, ..., n and

p 

ωq = 1.

q=1 (q)

Step 1: Let A(q) = (˛ij )

(q)

m×n

(q)

(q)

be an intuitionistic fuzzy decision matrix, and ˛ij = (ij , vij ) is an attribute value provided by the decision

maker eq , which is expressed in an IFN. Simultaneously, the decision makers provide the degrees that they are familiar with the research topics and give out the confidence levels lq (0 ≤ lq ≤ 1).

D. Yu / Applied Soft Computing 19 (2014) 147–160

157

Table 1 Review form for doctoral dissertation. No.

Criteria

Weights

1

Topic selection and literature review

0.15

Content

Evaluations

Belong to the leading edge of subject or the hot research point, has important theoretic significance and applied value; familiar with the research status and process for subject. Innovation 0.3 Have theoretical breakthrough; have positive influence and impact on the development for social economy and culture; creativity points. 0.2 Theory basis and Solid and broad theoretical foundation, also have the specialized special knowledge knowledge for the subject and related area. Capacity of scientific 0.2 Independently scientific research ability; informative citing research information; subject to be explored in depth. Theses writing 0.15 Clear concept and logistics, smooth sentences, format specification, good school ethos Degrees of familiarity with the research content (use the number between 0 and 1 to identify, 0 refer to not familiar at all while 1 refer to totally familiar)

2 3 4 5 Confidence levels

Step 2: Utilize the CIFWA operator

⎛

n 

˛ij = CIFWA(˛ij , ˛ij , ..., ˛ij ) = ⎝1 − (1)

(2)

(p)

(q) lq ×ωq

(1 − ij )

,

j=1

or the CIFWG operator:

⎛

˛ij = CIFWG(˛ij , ˛ij , ..., ˛ij ) = ⎝ (1)

(2)

(p)

n 

n 

⎞ (q) lq ×ωq

(vij )

⎠

(74)

j=1

(q) lq ×ωq

(ij )

,1 −

j=1

⎞

n 

(q) lq ×ωq

(1 − vij )

⎠

(75)

j=1 (q)

to aggregate all the individual intuitionistic fuzzy decision matrix A(q) = (˛ij )

m×n

(q = 1, 2, ..., p) into the collective intuitionistic fuzzy

decision matrix A = (˛ij )m×n , i = 1, 2, ..., m; j = 1, 2, ..., n. Step 3: Aggregate the intuitionistic fuzzy numbers ˛ij for each alternative Xi by the IFWA (or IFWG) operator:



˛i = IFWA (˛i1 , ˛i2 , ..., ˛in ) =

wj

n

1 − ˘ (1 − ij ) , ˘ (vij )

 ˛i = IFWG (˛i1 , ˛i2 , ..., ˛in ) =

n

j=1



wj

j=1

n

n

j=1

j=1

w w ˘ (ij ) j , 1 − ˘ (1 − vij ) j

i = 1, 2, ..., m

(76)

i = 1, 2, ..., m

(77)



Step 4. Rank all the alternatives. 4.2. The anonymous review of the doctoral dissertation in Chinese universities In many Chinese universities, the doctoral dissertation will be reviewed by three experts anonymously and suppose they have same importance during this review process. And they will review dissertation according to five criteria, Including topic selection and literature review, innovation, theory basis and special knowledge, capacity of scientific research and theses writing. Different weights are given to different criteria and the standards for those principles can be referred to Table 1. Step 1: Three decision makers evaluate the five doctoral dissertation Xi (i = 1, 2, 3, 4, 5) with respect to the attributes Cj (j = 1, 2, . . ., 5), and the decision makers provide the degrees that they are familiar with the research topics and give out the confidence levels lq (0 ≤ lq ≤ 1) (q)

simultaneously. Then, we construct the following three intuitionistic fuzzy decision matrix D(q) = (dij )

5×4

(q = 1, 2, 3) (see Tables 2–4). (q)

Step 2: Utilize the CIFWA operator (Eq. (75)) to aggregate the three individual intuitionistic fuzzy decision matrix A(q) = (˛ij )

5×5

(q = 1, 2, 3) into the collective intuitionistic fuzzy decision matrix A = (˛ij )5×5 (see Table 5). Step 3: Utilize the IFWA operator (Eq. (77)) to aggregate all the preference values ˛ij (i = 1, 2, 3, 4, 5) in the ith line of A, and get the overall preference values ˛i . ˛1 = (0.5055, 0.3958) , ˛2 = (0.4219, 0.4582) ˛3 = (0.4949, 0.4020) , ˛4 = (0.4602, 0.4703) , ˛5 = (0.4781, 0.4923) Table 2 Intuitionistic fuzzy decision matrix R(1) .

x1 x2 x3 x4 x5

C1 0.15

C2 0.3

C3 0.2

C4 0.2

C5 0.15

<0.7, (0.8, 0.1)> <0.7, (0.6, 0.1)> <0.7, (0.9, 0.1)> <0.7, (0.8, 0.2)> <0.7, (0.7, 0.3)>

<0.7, (0.8, 0.1)> <0.7, (0.5, 0.4)> <0.7, (0.5, 0.5)> <0.7, (0.6, 0.4)> <0.7, (0.5, 0.5)>

<0.7, (0.8, 0.1)> <0.7, (0.7, 0.2)> <0.7, (0.9, 0.1)> <0.7, (0.7, 0.1)> <0.7, (0.6, 0.4)>

<0.7, (0.7, 0.3)> <0.7, (0.8, 0.1)> <0.7, (0.7, 0.2)> <0.7, (0.7, 0.3)> <0.7, (0.4, 0.5)>

<0.7, (0.5, 0.4)> <0.7, (0.9, 0.1)> <0.7, (0.8, 0.2)> <0.7, (0.8, 0.2)> <0.7, (0.8, 0.1)>

158

D. Yu / Applied Soft Computing 19 (2014) 147–160

Table 3 Intuitionistic fuzzy decision matrix R(2) .

x1 x2 x3 x4 x5

C1

C2

C3

C4

C5

<0.9, (0.7, 0.2)> <0.9, (0.8, 0.1)> <0.9, (0.8, 0.2)> <0.9, (0.8, 0.1)> <0.9, (0.7, 0.2)>

<0.9, (0.5, 0.5)> <0.9, (0.5, 0.2)> <0.9, (0.6, 0.3)> <0.9, (0.4, 0.6)> <0.9, (0.6, 0.4)>

<0.9, (0.8, 0.1)> <0.9, (0.8, 0.1)> <0.9, (0.7, 0.1)> <0.9, (0.8, 0.1)> <0.9, (0.9, 0.1)>

<0.9, (0.8, 0.1)> <0.9, (0.7, 0.1)> <0.9, (0.9, 0.1)> <0.9, (0.8, 0.2)> <0.9, (0.6, 0.4)>

<0.9, (0.9, 0.1)> <0.9, (0.5, 0.5)> <0.9, (0.7, 0.1)> <0.9, (0.8, 0.1)> <0.9, (0.8, 0.2)>

C1

C2

C3

C4

C5

<0.8, (0.6, 0.1)> <0.8, (0.7, 0.2)> <0.8, (0.6, 0.3)> <0.8, (0.4, 0.6)> <0.8, (0.8, 0.1)>

<0.8, (0.8, 0.1)> <0.8, (0.5, 0.5)> <0.8, (0.7, 0.1)> <0.8, (0.8, 0.1)> <0.8, (0.9, 0.1)>

<0.8, (0.9, 0.1)> <0.8, (0.9, 0.1)> <0.8, (0.7, 0.2)> <0.8, (0.8, 0.2)> <0.8, (0.6, 0.3)>

<0.8, (0.5, 0.4)> <0.8, (0.4, 0.5)> <0.8, (0.8, 0.1)> <0.8, (0.6, 0.4)> <0.8, (0.5, 0.5)>

<0.8, (0.6, 0.3)> <0.8, (0.4, 0.6)> <0.8, (0.8, 0.1)> <0.8, (0.5, 0.4)> <0.8, (0.8, 0.1)>

Table 4 Intuitionistic fuzzy decision matrix R(3) .

x1 x2 x3 x4 x5

Step 4: Calculate the scores of ˛i (i = 1, 2, 3, 4, 5) respectively: S1 = 0.1097, S2 = −0.0363, S3 = 0.0930, S4 = −0.0101, S5 = −0.0143

Since S1 > S3 > S4 > S5 > S2 we have X1 X3 X4 X5 X2 Based on the CIFWG operator, the main steps are as follows: Step 1 : See step 1 (q) Step 2 : Utilize the CIFWG operator (Eq. (76)) to aggregate the three individual intuitionistic fuzzy decision matrix A(q) = (˛ij )





5×5

(q = 1, 2, 3) into the collective intuitionistic fuzzy decision matrix A = ˛ ij 5×5 (see Table 6). Step 3 : Utilize the IFWG operator (Eq. (77)) to aggregate all the preference values ˛ ij (i = 1, 2, 3, 4, 5) in the ith line of A , and get the overall preference values ˛ i ˛ 1 = (0.8835,0.0847) , ˛ 2 = (0.8269,0.1143) , ˛ 3 = (0.8579,0.1011) , ˛ 4 = (0.8418,0.1393) , ˛ 5 = (0.8659,0.1256) Step 4 : Calculate the scores of ˛ ij (i = 1, 2, 3, 4, 5), respectively: S 1 = 0.7988, S 2 = 0.7126, S 3 = 0.7567, S 4 = 0.7025, S 5 = 0.7403 Since S 1 > S 3 > S 5 > S 2 > S 4

Table 5 Comprehensive intuitionistic fuzzy decision matrix A.

x1 x2 x3 x4 x5

C1

C2

C3

C4

C5

(0.3569, 0.4793) (0.3674, 0.4744) (0.4339, 0.5469) (0.3608, 0.5821) (0.3825, 0.5379)

(0.5980, 0.2943) (0.3929, 0.4523) (0.4944, 0.3594) (0.5116, 0.4136) (0.6115, 0.3885)

(0.5866, 0.3311) (0.5625, 0.3649) (0.5189, 0.3700) (0.5112, 0.3700) (0.4981, 0.4793)

(0.4340, 0.4821) (0.4077, 0.4284) (0.5685, 0.3649) (0.4538, 0.5462) (0.2934, 0.6888)

(0.3896, 0.5761) (0.3274, 0.6726) (0.4082, 0.4695) (0.3746, 0.5544) (0.4398, 0.4793)

C1

C2

C3

C4

C5

(0.8756, 0.0524) (0.8811, 0.0507) (0.9026, 0.0806) (0.8492, 0.1372) (0.8937, 0.0771)

(0.7501, 0.2091) (0.6071, 0.2839) (0.6914, 0.2344) (0.6648, 0.3161) (0.7343, 0.2657)

(0.9155, 0.0493) (0.8986, 0.0649) (0.8728, 0.0671) (0.8818, 0.0671) (0.8418, 0.1372)

(0.8179, 0.1398) (0.7850, 0.1346) (0.9007, 0.0649) (0.8421, 0.1579) (0.7181, 0.2591)

(0.8622, 0.1048) (0.8069, 0.1931) (0.9063, 0.0491) (0.8722, 0.0942) (0.9228, 0.0524)

Table 6 Intuitionistic fuzzy decision matrix A .

x1 x2 x3 x4 x5

D. Yu / Applied Soft Computing 19 (2014) 147–160

159

we have X1 X3 X5 X2 X4 If we do not consider the confidence levels factor, in other words, if all the decision makers are marked by sure familiar with the evaluated objects, then our proposed operators are reduced to the existing intuitionistic fuzzy aggregation operators. However, the three experts are not familiar with the doctoral dissertation absolutely. To deal with such situations, the confidence intuitionistic fuzzy aggregation operators proposed in this paper are useful tools. From the above analysis, the main advantages over the traditional intuitionistic fuzzy operators are not only due to the fact that our operators accommodate the intuitionistic fuzzy environment but also due to the consideration of the confidence levels among the decision makers, which makes it more feasible and practical. 5. Concluding remarks We have developed a series of confidence intuitionistic fuzzy aggregation operators, such as CIFWA, CIFWG, CIFEWA and CIFWG operators. The main characteristic of these confidence aggregation operators is that they not only take into account the evaluation information of the decision makers but also consider the degrees that they are familiar with the research topics. Then, we have utilized the confidence operators to multiple attribute group decision making problems with intuitionistic fuzzy information. Furthermore, the confidence operators based on the intuitionistic fuzzy Einstein operations have also been studied, and a blind peer review of doctoral dissertation evaluation problem in China’s university has been used to verify the validity of our results and illustrate the application process. Acknowledgments The authors wish to thank the anonymous reviewers for their constructive comments on this study. This paper is supported by the National Natural Science Foundation of China (No. 71301142) and Zhejiang province Natural Science Foundation of China (No. LQ13G010004). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [37] [38] [39] [40] [41]

L.A. Zadeh, Fuzzy sets, Information and Control 8 (1965) 3388–4353. K.T. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems 20 (1986) 87–96. D.F. Li, Decision and Game Theory in Management with Intuitionistic Fuzzy Sets, Springer-Verlag, Heidelberg, Germany, 2014. Z.S. Xu, Intuitionistic Fuzzy Aggregation and Clustering, Springer-Verlag, Berlin, Heidelberg, 2013. Z.S. Xu, Uncertain Multiple Attribute Decision Making: Methods and Applications, Springer-Verlag, Berlin, Heidelberg, 2014. D.J. Yu, Group decision making based on generalized intuitionistic fuzzy prioritized geometric operator, International Journal of Intelligent Systems 27 (2012) 635–661. D.J. Yu, Y.Y. Wu, T. Lu, Interval-valued intuitionistic fuzzy prioritized operators and their application in group decision making, Knowledge-Based Systems 30 (2012) 57–66. Y.J. Xu, H.M. Wang, The induced generalized aggregation operators for intuitionistic fuzzy sets and their application in group decision making, Applied Soft Computing 12 (2012) 1168–1179. M.M. Xia, Z.S. Xu, Entropy/cross entropy-based group decision making under intuitionistic fuzzy environment, Information Fusion 13 (2012) 31–47. G.W. Wei, X.F. Zhao, Some induced correlated aggregating operators with intuitionistic fuzzy information and their application to multiple attribute group decision making, Expert Systems with Applications 39 (2012) 2026–2034. B. Huang, Y.L. Zhuang, H.X. Li, Information granulation and uncertainty measures in interval-valued intuitionistic fuzzy information systems, European Journal of Operational Research 231 (2013) 162–170. Y.C. Jiang, Y. Tang, H. Liu, Entropy on intuitionistic fuzzy soft sets and on interval-valued fuzzy soft sets, Information Sciences 240 (2013) 95–114. M. Agarwal, K.K. Biswas, M. Hanmandlu, Generalized intuitionistic fuzzy soft sets with applications in decision-making, Applied Soft Computing 13 (2013) 3552–3566. S. Chakrabortty, M. Pal, P.K. Nayak, Intuitionistic fuzzy optimization technique for Pareto optimal solution of manufacturing inventory models with shortages, European Journal of Operational Research 228 (2013) 381–387. S.M. Chen, T.S. Li, Evaluating students’ answer scripts based on interval-valued intuitionistic fuzzy sets, Information Sciences 235 (2013) 308–322. Z.S. Xu, Intuitionistic fuzzy aggregation operations, IEEE Transactions on Fuzzy Systems 15 (2007) 1179–1187. Z.S. Xu, R.R. Yager, Some geometric aggregation operators based on intuitionistic fuzzy sets, International Journal of General System 35 (2006) 417–433. H. Zhao, Z.S. Xu, M. Ni, S. Liu, Generalized aggregation operators for intuitionistic fuzzy sets, International Journal of Intelligent Systems 25 (2010) 1–30. Z.S. Xu, Intuitionistic fuzzy Bonferroni means, IEEE transactions on Systems Man and Cybernetics 41 (2011) 568–578. M.M. Xia, Z.S. Xu, B. Zhu, Generalized intuitionistic fuzzy Bonferroni means, International Journal of Intelligent Systems 27 (2012) 23–47. R.R. Yager, The power average operator, IEEE transactions on Systems, Man and Cybernetic 31 (2001) 724–731. Z.S. Xu, R.R. Yager, Power-geometric operators and their use in group decision making, IEEE Transactions on Fuzzy Systems 18 (2010) 94–105. Z.S. Xu, Approaches to multiple attribute group decision making based on intuitionistic fuzzy power aggregation operators, Knowledge-Based Systems 24 (2011) 749–760. Z.S. Xu, Choquet integrals of weighted intuitionistic fuzzy information, Information Sciences 180 (2010) 726–736. C.Q. Tan, X.H. Chen, Intuitionistic fuzzy Choquet integral operator for multi-criteria decision making, Expert Systems with Applications 37 (2010) 149–157. W. Yang, Z.P. Chen, The quasi-arithmetic intuitionistic fuzzy OWA operators, Knowledge-Based Systems 27 (2012) 219–233. Z.S. Xu, M.M. Xia, Induced generalized intuitionistic fuzzy operators, Knowledge-Based Systems 24 (2011) 197–209. M.M. Xia, Z.S. Xu, H.C. Liao, Preference relations based on intuitionistic multiplicative information, IEEE Transactions on Fuzzy Systems 21 (2013) 113–133. M.M. Xia, Z.S. Xu, Group decision making based on intuitionistic multiplicative aggregation operators, Applied Mathematical Modelling 37 (2013) 5120–5133. M.M. Xia, Z.S. Xu, N. Chen, Induced aggregation under confidence levels, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 19 (2011) 201–227. D.J. Yu, Intuitionistic fuzzy information aggregation and its application on multi-criteria decision-making, Journal of Industrial and Production Engineering 30 (2013) 281–290. R.R. Yager, On ordered weighted averaging aggregation operators in multi-criteria decision making, IEEE transactions on Systems Man and Cybernetics 18 (1988) 183–190. W.Z. Wang, X.W. Liu, Intuitionistic fuzzy geometric aggregation operators based on Einstein operations, International Journal of Intelligent Systems 26 (2011) 1049–1075. W.Z. Wang, X.W. Liu, Intuitionistic fuzzy information aggregation using Einstein operations, IEEE transactions on Fuzzy Systems 20 (2012) 923–938. C.P. Tseng, C.W. Chen, Y.P. Tu, A new viewpoint on risk control decision models for natural disasters, Natural Hazards 59 (2011) 1715–1733. J.W. Lin, C.W. Chen, C.Y. Peng, Kalman filter decision systems for debris flow hazard assessment, Natural Hazards 60 (2012) 1255–1266. C.W. Chen, C.P. Tseng, Default risk-based probabilistic decision model for risk management and control, Natural Hazards 63 (2012) 659–671. J.S. Chou, C.S. Ongkowijoyo, Risk-based group decision making regarding renewable energy schemes using a stochastic graphical matrix model, Automation in Construction 37 (2014) 98–109. D.J. Yu, W.Y. Zhang, Y.J. Xu, Group decision making under hesitant fuzzy environment with application to personnel evaluation, Knowledge-Based Systems 52 (2013) 1–10.

160

D. Yu / Applied Soft Computing 19 (2014) 147–160

[42] S.F. Zhang, S.Y. Liu, A GRA-based intuitionistic fuzzy multi-criteria group decision making method for personnel selection, Expert Systems with Applications 38 (2011) 11401–11405. [43] A. Hatami-Marbinia, M. Tavanab, V. Hajipourc, F. Kangic, A. Kazemic, An extended compromise ratio method for fuzzy group multi-attribute decision making with SWOT analysis, Applied Soft Computing 13 (2013) 3459–3472. [44] Z.L. Yue, Y.Y. Jia, A method to aggregate crisp values into interval-valued intuitionistic fuzzy information for group decision making, Applied Soft Computing 13 (2013) 2304–2317. [45] Y.H. Chang, C.H. Yeh, Y.W. Chang, A new method selection approach for fuzzy group multi-criteria decision making, Applied Soft Computing 13 (2013) 2179–2187. [46] Y. Jiang, Z.S. Xu, X.H. Yu, Compatibility measures and consensus models for group decision making with intuitionistic multiplicative preference relations, Applied Soft Computing 13 (2013) 2075–2086.