Abstract
Atanassov [Intuitionistic fuzzy set, Fuzzy Sets and Systems
Introduction
Zadeh [1] defined the fuzzy set in 1965, which has been widely applied to amounts of areas, such as pattern recognition [2–4], decision making [5–8], medical diagnosis [9, 10], clustering analysis [11], etc. However, the fuzzy set is characterized by only a membership function, which is a single-valued function, when using it, we will overlook some uncertain information in many practical situations. In order to avoid such shortcomings of the fuzzy set, Atanassov [12, 13] developed the fuzzy set to Atanassov’s intuitionistic fuzzy set (A-IFS), each of which consists of a non-membership function, a membership function and a hesitancy function. In the past few decades, the A-IFS has received great attention and has been applied to various fields of modern life [13–15]. Several aspects for the intuitionistic fuzzy information have been researched widely, including aggregation techniques [15–20], clustering algorithms [21], distance measures [22–24], correlation measures [25, 26] and intuitionistic fuzzy calculus [27, 28]. Atanassov [12] and De et al. [29] introduced some basic operations on IFSs, including “intersection”, “union”, “supplement”, “power” and so on, which not only ensure that the operational results are also IFSs, but also are useful in the calculus of linguistic variables in an intuitionistic fuzzy environment. Based on the concept of A-IFS, Xu and Yager [16] defined the intuitionistic fuzzy number (IFN), which consists of a membership degree and a non-membership degree. Lots of work has been done by using IFNs. For example, Xu and Yager [16, 17] defined some basic operational laws of IFNs. In 2006, Atanassov [30] defined the subtraction and division operational laws based on A-IFSs, and inspired by which, Lei and Xu [27] developed the subtraction and division operations for IFNs. However, as the study of the IFS theory extends in both depth and scope, effective aggregation and handling of intuitionistic fuzzy information have become increasingly necessary and important. These basic operations on IFSs or IFNs have been far from meeting the actual needs. Therefore, lots of techniques for aggregating intuitionistic fuzzy information have been developed [15, 18], such as the intuitionistic fuzzy weighted averaging operator, the intuitionistic fuzzy ordered weighted averaging operator, the intuitionistic fuzzy weighted geometric operator and the intuitionistic fuzzy ordered weighted geometric operator, etc. When we make some decisions by using information aggregation methods which have been mentioned above, especially using the intuitionistic fuzzy weighting geometric operator, whose basic element consists of a positive real number and an intuitionistic fuzzy number, and we only know one well-known operational law [16, 17]. However, we are lack of another important operational law, and thus make us unable to deal with lots of calculations when using IFNs. Therefore, as a necessary supplement of the existing intuitionistic fuzzy aggregation techniques, in this paper, we define the exponential operational law of IFNs, in which the bases are positive real numbers and the exponents are IFNs, and then many properties of the operational law will be discussed.
In the process of multiple criteria decision making with intuitionistic fuzzy information, if the data collected in the decision matrix are given in exact real numbers, and the criteria weights provided by the experts are represented by intuitionistic fuzzy information, then the traditional intuitionistic fuzzy aggregation techniques are unsuitable for dealing with these cases. Although the intuitionistic fuzzy weighted averaging operator (IFWA) can be used to deal such a situation, this method will also show some limitations and there are a lot of unreasonable places. But the aggregation technique based on the exponential operational law developed in this paper can effectively solve the issue. To do that, the remainder of the paper is set out as follows: Section 2 introduces some basic knowledge related to A-IFSs and IFNs. Section 3 introduces the exponential operational laws of A-IFSs and IFNs, and develops the corresponding intuitionistic fuzzy information aggregation method. Section 4 gives an example to show how to utilize the exponential operational law in the process of information aggregation. The paper ends with some conclusions in Section 5.
Preliminaries
In this section, let’s recall some basic concepts related to the A-IFSs and IFNs.
which is characterized by a membership function:
and a non-membership function:
These two functions satisfy the condition: 0 ≤ μ A (x) + ν A (x) ≤ 1, for x ∈ X, where μ A (x) and ν A (x) represent the membership degree and the non-membership degree of x in A, respectively. Moreover, for each IFS A in X, if π A (x) = 1 - μ A (x) - ν A (x), for all x ∈ X, then π A (x) is called the indeterminacy degree of x to A. For convenience, Xu and Yager [16] called α = (μ α , ν α ) an intuitionistic fuzzy number (IFN) (or an intuitionistic fuzzy value (IFV)), where μ α ∈ [0, 1], ν α ∈ [0, 1] and 0 ≤ μ α + ν α ≤ 1.
Additionally, for any IFN α = (μ
α
, ν
α
), its score function [5] has the form below:
where s (α) ∈ [0, 1], and s (α) is the score of α. However, in order to distinguish any two IFNs, Hong and Choi [6] defined an accuracy function of α, which is denoted by h, and h (α) = μ α + ν α .
Based on the two functions s (α) and h (α), for the purpose of comparing and ranking any two IFNs, Xu and Yager [16] developed the following method:
(1) If s (α1) < s (α2), then α1 is smaller than α2, denoted by α1 < α2;
(2) If s (α1) = s (α2), then
(a) If h (α1) = h (α2), then α1 is equal to α2, denoted by α1 = α2;
(b) If h (α1) < h (α2), then α1 is smaller than α2, denoted by α1 < α2.
Next, let us take a look at some basic operational laws of IFNs:
; α1 ∩ α2 = (min {μ
α
1
, μ
α
2
} , max {ν
α
1
, ν
α
2
}); α1 ∪ α2 = (max {μ
α
1
, μ
α
2
} , min {ν
α
1
, ν
α
2
}); α1 ⊕ α2 = (μ
α
1
+ μ
α
2
- μ
α
1
μ
α
2
, ν
α
1
ν
α
2
); α1 ⊗ α2 = (μ
α
1
μ
α
2
, ν
α
1
+ ν
α
2
- ν
α
1
ν
α
2
);
;
.
(1) A⊖ B = { x, μA⊖B (x) , νA⊖B (x) |x ∈ X }, where
(2) A ø B = {〈x, μAøB (x) , νAøB (x) 〉|x ∈ X}, where
Motivated by the subtraction and division operations of IFSs, Lei and Xu [20] developed the subtraction and division operations for IFNs as follows:
(1) α ! β = (μα!β, να!β), where
and
(2) α % β = (μα%β, να%β), where
and
Based on the basic operational laws in Definition 2.3, Xu and Yager [16, 17] introduced an intuitionistic fuzzy weighted geometric (IFWG) operator:
where ω = (ω1, ω2, ⋯ , ω
m
)
T
is the weight vector of α
i
(i = 1, 2, ⋯ , m), with ω
i
∈ [0, 1] (i = 1, 2, ⋯ , m) and . The aggregated value by using the IFWA operator is also an IFN, and
where ω = (ω1, ω2, ⋯ , ω
m
)
T
is the weight vector of α
i
(i = 1, 2, ⋯ , m), with ω
i
∈ [0, 1] (i = 1, 2, ⋯ , m) and . The aggregated value by using the IFWG operator is also an IFN, and
The definitions of exponential operational laws of IFSs and IFNs
We have already introduced some operational laws and aggregation techniques of IFSs and IFNs in Section 2. As we know from Definitions 2.3 and 2.6 that the weights of all the exponential operational laws and the corresponding aggregation methods for IFNs are non-negative real numbers, which cannot be used to deal with some decision making problems in actual applications as described in the introduction. As a supplement, in the following, we define the new exponential operational laws about IFSs and IFNs respectively, in which the bases are positive real numbers and the exponents are IFSs or IFNs:
We can prove that λ
A
is also an IFS: In fact, by the definition of IFS, we can get that the membership function and the non-membership function of A satisfy:
and 0 ≤ μ A (x) + ν A (x) ≤ 1, x ∈ X, so 0 ≤ 1 - μ A (x) ≤ 1 and 1 - μ A (x) ≥ ν A (x), x ∈ X. Now we discuss the following two cases:
(1) If λ ∈ (0, 1), we get the membership function:
and
So λ A ={ 〈 x, λ1-μ A (x), 1 - λν A (x) 〉 |x ∈ X } is an IFS.
(2) If λ ≥ 1, then 0 ≤ 1/ λ ≤ 1, we can also get
It is different with the operational law A λ , in Definition 3.1, we exchange the positions of the intuitionistic fuzzy set A and the real number λ, and get λ A , which can be called an exponential operational law based on intuitionistic fuzzy set (EOL-IFS). Obviously, it can be known from Definition 3.1 that when λ ≥ 1, in order to ensure that λ A is still an IFS, we let 1/ λ instead of λ.
Similarly, we can also propose an operational law for IFNs, and divide λ α into two parts based on the different values of λ.
We can prove that λ α is also an IFN. Firstly we let λ ∈ (0, 1), by the definition of IFN, we can get that α satisfies 0 ≤ μ α ≤ 1, 0 ≤ ν α ≤ 1 and 0 ≤ μ α + 1ν α ≤ 1. They can be changed by 0 ≤ 1 - μ α ≤ 1, 1 - μ α ≥ ν α , when λ ∈ (0, 1), we can get 0 ≤ λ1-μ α ≤ 1, 0 ≤ 1 - λ ν α ≤ 1 and 0 ≤ λ1-μ α + 1 - λ ν α ≤ 1. So λ α = (λ1-μ α , 1 - λ ν α ) is an IFN. When λ ≥ 1 and 0 < 1/ λ ≤ 1, we can also get λ α = ((1/ λ) 1-μ α , 1 - (1/ λ) ν α ) is an IFN. Just like EOL-IFS, we call λ α an exponential operational law based on intuitionistic fuzzy number (EOL-IFN).
Obviously, from Definition 3.2, we know that when λ ∈ (0, 1), λ α increases along with the increase of λ and α; and when λ ≥ 1, in order to ensure that λ α is still an IFN, we let 1/ λ instead of λ, in this case, the larger the value of λ, the smaller the value of λ α .
Next we give an example to show how the EOL-IFN works:
The properties of the exponential operational law of IFNs
In what follows, we discuss the properties of EOL-IFNs. When λ ∈ (0, 1), the properties of λ α are almost same as λ ≥ 1, and the operation processes and forms are simple. So here we only discuss the case when λ ∈ (0, 1). We first prove that the EOL-IFNs satisfy the commutative law and the associative law, respectively:
λ
α
1
⊕ λ
α
2
= λ
α
2
⊕ λ
α
1
; λ
α
1
⊗ λ
α
2
= λ
α
2
⊗ λ
α
1
.
(λ
α
1
⊕ λ
α
2
) ⊕ λ
α
3
= λ
α
1
⊕ (λ
α
2
⊕ λ
α
3
); (λ
α
1
⊗ λ
α
2
) ⊗ λ
α
3
= λ
α
1
⊗ (λ
α
2
⊗ λ
α
3
).
Obviously, Theorem 3.1 and Theorem 3.2 are reasonable and we omit the proofs of them.
k (λ
α
1
⊕ λ
α
2
) = kλ
α
1
⊕ kλ
α
2
; (λ
α
1
⊗ λ
α
2
)
k
= (λ
α
1
)
k
⊗ (λ
α
2
)
k
.
In Theorem 3.3, we only have two EOL-IFNs λ
α
1
and λ
α
2
, we can easily extend (1) and (2) of Theorem 3.3 to the general forms with a collection of n EOL-IFNs λ
α
1
, λ
α
2
, ⋯ , λ
α
n
, i.e.,
k1λ
α
⊕ k2λ
α
= (k1 + k2) λ
α
; (λ
α
)
k
1
⊗ (λ
α
)
k
2
= (λ
α
) k1+k2.
Similar to Theorem 3.3, we can further extend (1) and (2) of Theorem 3.4 to the following general forms:
(1) If μ α 1 ≥ μ α 2 , ν α 1 ≤ ν α 2 , ν α 2 ≥ 0, and ν α 1 π α 2 ≤ π α 1 ν α 2 , then (λ α 1 ⊖ λ α 2 ) ⊕ λ α 2 = λ α 1 ;
(2) If μ α 1 ≤ μ α 2 , ν α 1 ≥ ν α 2 , μ α 2 ≥ 0, and μ α 1 π α 2 ≤ π α 1 μ α 2 , then (λ α 1 ø λ α 2 ) ⊗ λ α 2 = λ α 1 .
(1) λ α 1 ⊖ λ α 2 ⊖ λ α 3 = λ α 1 ⊖ (λ α 2 ⊕ λ α 3 )
where μ λ α 1 ≥ μ λ α 2 , ν λ α 1 ≤ ν λ α 2 , ν λ α 2 ≥ 0, ν λ α 1 π λ α 2 ≤ π λ α 1 ν λ α 2 , μλ α 1 ⊖λ α 2 ≥ μ λ α 3 , νλ α 1 ⊖λ α 2 ≤ ν λ α 3 , ν λ α 3 ≥ 0 and νλ α 1 ⊖λ α 2 π λ α 3 ≤ πλ α 1 ⊖λ α 2 ν λ α 3 .
(2) λ α 1 ø λ α 2 ø λ α 3 = λ α 1 ø (λ α 2 ⊗ λ α 3 )
where μ λ α 1 ≤ μ λ α 2 , ν λ α 1 ≥ ν λ α 2 , μ λ α 2 ≥ 0, μ λ α 1 π λ α 2 ≤ π λ α 1 μ λ α 2 , μλ α 1 øλ α 2 ≤ μ λ α 3 , νλ α 1 øλ α 2 ≥ ν λ α 3 , μ λ α 3 ≥ 0 and μλ α 1 øλ α 2 π λ α 3 ≤ πλ α 1 øλ α 2 μ λ α 3 .
(1) kλ α 1 ⊖ kλ α 2 = k (λ α 1 ⊖ λ α 2 )
where μ α 1 ≥ μ α 2 , ν α 1 ≤ ν α 2 , ν α 2 ≥ 0 and ν α 1 π α 2 ≤ π α 1 ν α 2 .
(2) (λ α 1 ) k ø (λ α 2 ) k = (λ α 1 ø λ α 2 ) k
where μ α 1 ≤ μ α 2 , ν α 1 ≥ ν α 2 , μ α 2 ≥ 0 and μ α 1 π α 2 ≤ π α 1 μ α 2 .
In general, if we have n EOL-IFNs, and they satisfy the requirements of the subtraction and division of EOL-IFNs, then we can extend (1) and (2) of Theorem 3.7 to the following forms respectively:
(λ1)
α
⊗ (λ2)
α
= (λ1λ2)
α
;
, if λ1 ≤ λ2.
(1) (λ1) α ⊗ (λ2) α = ((λ1λ2) 1-μ α , 1 - (λ1λ2) ν α ) = (λ1λ2) α
(2)
k1λ
α
⊖ k2λ
α
= (k1 - k2) λ
α
; (λ
α
)
k
1
ø (λ
α
)
k
2
= (λ
α
) (k1-k2).
Besides what we have discussed above, there is a property that when we take different values of λ (λ ∈ (0, 1) or λ ≥ 1), we can get some different results:
If λ1 ≥ λ2, then we can get (λ1) 1-μ α ≥ (λ2) 1-μ α and 1 - (λ1) ν α ≤ 1 - (λ2) ν α easily. Let s ((λ1) α ) = (λ1) 1-μ α - (1 - (λ1) ν α ) and s ((λ2) α ) = (λ2) 1-μ α - (1 - (λ2) ν α ) be the scores of (λ1) α and (λ2) α , respectively. Then
(1) If s ((λ1) α ) > s ((λ2) α ), then (λ1) α > (λ2) α .
(2) If s ((λ1) α ) = s ((λ2) α ), then we can only obtain (λ1) 1-μ α = (λ2) 1-μ α and 1 - (λ1) ν α = 1 - (λ2) ν α , i.e., (λ1) α = (λ2) α . Based on these two cases, we can derive (λ1) α ≥ (λ2) α . However, when λ1, λ2 ≥ 1 and λ1 ≥ λ2, we can know 0 < 1/ λ1 ≤ 1/ λ2 ≤ 1, just like what we have discussed above, we can get (λ1) α ≤ (λ2) α . This completes the theorem.
In what follows, let’s take a look at some special values of λ α :
If λ = 1, then λ α = (λ1-μ α , 1 - λ ν α ) = (1, 0);
If α = (1, 0), then λ α = (λ1-μ α , 1 - λ ν α ) = (1, 0);
If α = (0, 1), then λ α = (λ1-μ α , 1 - λ ν α ) = (λ, 1 - λ).
For (1), because that the value of λ has a limitation, namely, 0 ≤ λ ≤ 1, and the result of exponential operation is increasing when we increase the value of λ. So if we let λ = 1, then no matter what we take the value of the IFN α, we always obtain the biggest IFN λ α = (λ1-μ α , 1 - λ ν α ) = (1, 0).
Similarly, if we take the biggest IFN α = (1, 0), then no matter what we take the real number λ, the result of exponential operation is also the biggest IFN λ α = (λ1-μ α , 1 - λ ν α ) = (1, 0).
However, if we take the smallest IFN α = (0, 1), then λ α = (λ1-μ α , 1 - λ ν α ) = (λ, 1 - λ). The result shows that when α = (0, 1), the value of the exponential operation λ α depends on the value of λ. The bigger the value of λ, the greater the value of λ α .
The aggregation technique of EOL-IFNs
Unlike the traditional IFWG operator [17], below we utilize the EOL-IFN (given in Definition 3.2) to develop an intuitionistic fuzzy weighted exponential aggregation operator, in which the bases are a collection of positive real numbers λ i (i = 1, 2, ⋯ , n) and the exponents are a collection of IFNs α i = (μ α i , ν α i ) (i = 1, 2, ⋯ , n).
(1) When n = 2, we have .
By Definition 3.2, we can see that both and are IFNs, and the value of is also an IFN. We also have and . Then
(2) Suppose that when n = k, Equation (3.1) holds, then
Combining (1) and (2), we can get that Equation (3.1) holds for all n. The proof is completed.
Obviously, in some uncertain situations, we can express the weights as the IFNs or the interval numbers, and these two kinds of numbers have a mathematical correspondence, that is, we can translate one form (IFN) into another form (interval number). In this paper, we express the weights as the IFNs. As we know, we usually take the weights in the form of the real numbers (such as a i (i = 1, 2, ⋯ , n)), and all these real numbers satisfy 0 ≤ a i ≤ 1 (i = 1, 2, ⋯ , n) and . However, here we take the IFNs α i = (μ α i , ν α i ) (i = 1, 2, ⋯ , n) as the weights, and we do not require to normalize them. Because if , then we must let at least one weight be (1, 0). However, in actual applications, we usually do not consider that one criterion is absolutely important. So the weights should lie between (0, 1) and (1, 0). In addition, we need to explain the meaning of the weight α = (μ α , ν α , π α ). The membership degree μ α is expressed as the degree that a criterion is preferred, the non-membership degree ν α is expressed as the degree that one attribute is not preferred, and π α = 1 - (μ α + ν α ) is expressed as the hesitancy degree. Actually, using the IFNs to express the weight information is comprehensive and reasonable.
Through this information aggregation operator, we can solve some decision making problems in actual applications, and we will give an example to illustrate how to use the IFWEA operator in Section 4. Next we investigate several properties of the IFWEA operator:
Let IFWEA (α1, α2, ⋯ , α
n
) = α = (μ
α
, ν
α
), α- = (μ
α
-
, ν
α
-
), and α + = (μ
α
+
, ν
α
+
), then based on the score function, where
In what follows, we discuss three cases:
(1) If s (α) < s (α+) and s (α) > s (α-), then we can get that Equation (3.2) holds obviously.
(2) If s (α) = s (α+), then μ
α
- ν
α
= μ
α
+
- ν
α
+
, we also have μ
α
= μ
α
+
, 1ν
α
= ν
α
+
. Hence
In this case, according to Definition 2.2, we have
(3) If s (α) = s (α-), then μ
α
- ν
α
= μ
α
-
- ν
α
-
, we also have μ
α
= μ
α
-
, 1ν
α
= ν
α
-
. Hence
so we have IFWEA (α1, α2, ⋯ , α n ) = α-.
Therefore, by (1)–(3), we can see that Theorem 3.12 holds. This completes the proof.
Specially, when normalizing these weights, namely, if we sum up all the weights, then the result must be the biggest IFN (1, 0). Thus,
Since and , for any i, then we have . Moreover,
So based on the score function, we have s (α) ≤ s (α*), and
(1) If s (α) < s (α*), then we can get
(2) If s (α) = s (α*), then
so .
Based on (1) and (2), we can see that Theorem 3.13 also holds.
In order to understand the new operational law better, we will show how the applicability of the information aggregation of EOL-IFNs works, below we introduce the steps of the information aggregation method:
In the following, we adapt an example from Ref. [31] to show the IFWEA operator and show its practicality in actual application:
G1: Booking and ticketing service, which involves convenience of booking or buying ticket, promptness of booking or buying ticket, courtesy of booking or buying ticket;
G2: Check-in and boarding process, which consists of convenience check-in, efficient check-in, courtesy of employee, clarity of announcement and so on;
G3: Cabin service, which can be divided into cabin safety demonstration, variety of newspapers and magazines, courtesy of flight attendants, flight attendant willing to help, clean and comfortable interior, in-flight facilities, and captain’s announcement;
G4: Responsiveness, which consists of fair waiting-list call, handing of delayed flight, complaint handing, and missing baggage handling.
There is no doubt that finding the best practice in each of the four main criteria and then calling all companies to learn from them respectively is better than determining the best company as a whole and trying to make the others follow all its practices, due to the fact that some of them would be inferior to the practice of some of the “followers”. However, selecting the best airline as a whole over all criteria is also very important especially for the passengers. Meanwhile, it is also very useful for the company to achieve brand effect.
Assume that the characteristics (criteria values) of the alternatives Y i (i = 1, 2, 3, 4) with respect to the criteria G j (j = 1, 2, 3, 4) are represented by some real numbers as shown in Table 1, which range from 0 and 1. Let ω = ((0.2, 0.6) , (0.4, 0.4) , (0.8, 0.1) , (0.6, 0.3)) (0.6, 0.3)) be the weighting vector of the criteria G j (j = 1, 2, 3, 4).
When i = 1, we can get
In a similar way, we can get
Obviously, s (d1) is the biggest one, and thus, UNI Air (Y1) will be the best one.
In this example, we know that Y1 is the best alternative. However, there is a problem that the weight α3 is the biggest weight and the characteristic value of the alternative Y2 with respect to the criterion G3 is also the biggest one, but Y2 is the third one in the final ranking. So we think that this ranking result is wrong and counter-intuitive, and explain this situation as follows:
In the process of information aggregation, both of the weights and the criteria values are important factors for calculating the results. In addition, we should pay close attention to the whole data rather than a part of them. Y2 has the biggest score with respect to the most important criterion, but the data with respect to the rest three criteria are very small. So when we aggregate all the data, the first one is not Y2. On the other hand, we can also realize that the IFWEA operator is very reasonable when aggregating the information.
In addition, below we utilize the intuitionistic fuzzy weighted averaging (IFWA) operator to aggregate the information in Table 1:
When i = 1, we can get
Similarly, we can calculate the overall criteria values of the rest airlines:
Obviously, s (d1) is also the biggest one. So UNI Air (Y1) is the best alternative.
Through the comparison between the IFWEA operator and the IFWA operator, we can make some analyses as follows:
1. For Step 1 using the second operator in Example 4.1, the aggregation process does not completely follow the IFWA operator. The information aggregation technique used for the UNI Air is:
As we know, α j (j = 1, 2, 3, 4) are the weights and λ1j (j = 1, 2, 3, 4) are the characteristics of the alternatives Y1 with respect to the criteria G j (j = 1, 2, 3, 4). If we use the IFWA operator, it needs to exchange the roles of α j and λ1j (j = 1, 2, 3, 4), namely, α j (j = 1, 2, 3, 4) are the characteristics of the alternatives Y1 with respect to the criteria G j (j = 1, 2, 3, 4) and λ1j (j = 1, 2, 3, 4) are the weights, then .
Based on the analysis above, we know that it is unreasonable to use the IFWA operator in such a situation. But we can utilize the IFWEA operator to deal with this situation, in the process of the information aggregation by using the IFWEA operator, we do not change the meanings and the positions of the weights and the criteria values.
2. The ranking results derived by the IFWEA operator and the IFWA operator are obviously different, which is mainly because that the positions and meanings of the weights are exchanged respectively with those of the criteria values when we use the IFWA operator, which may result in unreasonable decision.
Conclusions
This paper has given the exponential operational laws (EOL-IFNs) of IFSs and IFNs, which are the useful supplement of the existing intuitionistic fuzzy aggregation techniques. Then, we have investigated a series of properties of these EOL-IFNs. Next, we have developed the aggregation method using the EOL-IFNs, and also discussed some favorable properties of the aggregation method. Finally we have discussed the applicability of the EOL-IFNs and the corresponding aggregation method through an illustrative example. In the future, the derived results can be further considered in interval-valued intuitionistic fuzzy situations, and applied to some other fields, such as information retrieval, medical diagnosis, and clustering analysis, etc.
Acknowledgments
This research was funded by the National Natural Science Foundation of China (No. 61273209), and the Central University Basic Scientific Research Business Expenses Project (No. skgt201501).