On the syntax and semantics of virtual linguistic terms for information fusion in decision making

Information Fusion 34 (2017) 43–48

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Information Fusion journal homepage: www.elsevier.com/locate/inffus

On the syntax and semantics of virtual linguistic terms for information fusion in decision making Zeshui Xu a,b,∗, Hai Wang a a b

School of Economics and Management, Southeast University, Nanjing, Jiangsu 211189, China School of Computer and Software, Nanjing University of Information Science and Technology, Nanjing, Jiangsu 210044, China

a r t i c l e

i n f o

Article history: Received 1 June 2015 Revised 4 June 2016 Accepted 6 June 2016 Available online 7 June 2016 Keywords: Virtual linguistic terms (VLTs) Syntax Semantics Computing with words

a b s t r a c t The virtual linguistic model is a good technique for linguistic decision making and has been widely used in applications including linguistic information fusion. The main purpose of this paper is to define and specify the syntax and semantics of virtual linguistic terms (VLTs) in detail, and then to serve as the theoretical foundation of the computational models based on VLTs. The syntactical rule generates VLTs by a symbolic transformation, and then the semantic rule presents the semantics of VLTs by means of linguistic modifiers. Based on the syntax and semantics, VLTs could be a possible alternative for solving some current challenges of qualitative information fusion in decision making. © 2016 Elsevier B.V. All rights reserved.

1. Introduction Methodologies for Computing with Words (CWW) [1] are very useful for decision making problems with qualitative criteria and thus have been widely studied and applied in many practical areas. CWW manipulates natural and artificial linguistic expressions which are less precise than numbers but much closer to human’s brain mechanisms. All these linguistic expressions form the domain of possible values of a linguistic variable [2]. To reach a final decision, two scenarios of linguistic decision making models can be considered, which are from words to words and from words to numerical outputs/ranking [3]. The former outputs a linguistic representation of words, whereas the latter results in a ranking of alternatives based on numerical outputs. Till now, there are several famous linguistic decision making models, such as the membership function-based model [4], the type-2 fuzzy sets-based model [5], the ordinal scales-based model [6], the 2-tuple linguistic model [7] and the virtual linguistic model [8]. From a historical view, the virtual linguistic model can be considered as a variant of the 2tuple linguistic model. Both of them are very popular as they compute linguistic expressions without loss of information. Moreover, Herrera et al. [3] reported that the 2-tuple linguistic model follows the from words to words scenario while the virtual linguistic model falls into the other scenario.

When dealing with linguistic information by a certain computational model, the first and basic step is to choose linguistic term sets (LTSs) with syntax and semantics [9]. Although the virtual linguistic model has been widely applied in information fusion-driven decision making [10,11], its lack of clear representation of syntax and semantics has triggered off some discussions [12]. Recently, Liao et al. [13] started the discussion with a special case. When the virtual linguistic terms (VLTs) are balanced and uniformly distributed in the considered domain, they constructed a simple yet meaningful mapping between the VLTs and their semantics graphically. To build a sound foundation of the virtual linguistic model, in this paper, we mainly focus on the investigation of the syntax and semantics of VLTs in a general way. We begin our discussion with a predefined discrete LTS with syntax and semantics. The syntax of a VLT is generated by an algorithm based on proper linguistic modifiers. Then the semantic of a VLT can be derived by modifying the closest original linguistic term to a certain level. Finally, we reconstruct the computational model based on VLTs by some predefined operations following the classical computational models based on the ordered structure.

2. Preliminaries 2.1. Linguistic variables

∗

Corresponding author. E-mail addresses: [email protected] (Z. Xu), [email protected] (H. Wang).

http://dx.doi.org/10.1016/j.inffus.2016.06.002 1566-2535/© 2016 Elsevier B.V. All rights reserved.

Given a nonempty domain U , a fuzzy set F on U is characterized by a membership function μF : U → [0, 1]. For each u of U, μF (x ) represents the membership degree of u in F . Generally, a fuzzy set

44

Z. Xu, H. Wang / Information Fusion 34 (2017) 43–48

F can be denoted by [14]:



F=

U

μF (u )/u

3. The syntax and semantics of VLTs (1)

The class of all fuzzy sets on U is denoted by F (U ). Furthermore, given A, B ∈ F (U ), A is a subset of B, denoted by A ⊆ B, which holds if and only if A(u ) ≤ B(u ) (for all u ∈ U). A linguistic variable, whose values are words or sentences in a natural or artificial language, serves as an approximation of characterization of phenomena that is too complex or too ill-defined to be described by a conventional numerical variable. Fuzzy sets are used to represent the restrictions associated with the values of a linguistic variable. The definition of linguistic variable is as follows: Definition 1. [2]. A linguistic variable is characterized by a quintuple (X, S(X ), U, G, M ), where X is the name of the variable; S(X ) (or simply S) denotes the term set of X with each term being a fuzzy variable denoted generically by s and ranging over the domain U which is associated with the base variable u; G is a syntactic rule for generating the names, s, of values of X; and M is a semantic rule for associating with each s its meaning, M (s ), which is a fuzzy set of U. Remark 1. As suggested by Zadeh [2], three denotations, i.e., the name s, its meaning (semantic) M (s ) and its restriction R(s ) will be used interchangeably to avoid a profusion of symbols. A particular s, a name generated by G, is called a term. We can denote it as S = {s}. An important facet of a linguistic variable is the following two rules: (1) A syntactic rule, having the form of a grammar, to generate the names of the values of the variable. (2) A semantic rule, to compute the meaning of each value. If the number of terms in S is infinite, it is necessary to use an algorithm, rather than a table look-up procedure, to generate the elements of S and compute their semantics. When generating terms in S, linguistic modifiers play an important role. Given an atomic term, composite terms can be generated by modifying the atomic term to certain levels. Generally, given U, a fuzzy modifier F M on U, is a mapping such that [15]:

F M : F (U ) → F (U ) s → F M (s, δ )

(2)

In order to generate VLTs, it is natural to begin with a predefined LTS associated with semantics having the form of Eq. (3). Note that the symbol st is used to represent both the name of the term and its semantic taking the form of a fuzzy set defined on the domain U. Each linguistic term st ∈ S is called an original linguistic term and considered as an atomic term. VLTs are generated by the proper linguistic modifiers based on the original linguistic terms. Roughly, we will generate a VLT by its closest original linguistic term. It is easy to generate a new VLT by a symbolic transformation as follows: Definition 2. (Syntactical generation of a VLT). Let S = {st |t = 0, 1, . . . , τ } be a LTS with the semantics defined on the domain U. For any t ∈ {0, 1, . . . , τ }, let

δ∈

⎧ ⎨[0, 0.5 ),

t=0

[−0.5, 0],

t=τ

[−0.5, 0.5 ),

else

⎩

(4)

then the pair (t, δ ) generates a VLT sα , with α = t + δ . The set of VLTs is denoted by S¯ = {sα |α ∈ [0, τ ]}. According to Definition 2, a VLT, sα , is generated by an atomic term st and a real number δ satisfying t = round (α ) and δ = α − t, where round is the classical round function. The original linguistic term can be viewed as a special VLT with δ = 0. Example 1. Given the following LTS (whose semantics are shown in Fig. 1):

S = {s0 = extremely poor, s1 = very poor, s2 = poor, s3 = sl ightl y poor, s4 = f air, s5 = sl ightl y good, s6 = good, s7 = very good, s8 = extremely good} Let t = 5 and δ = 0.4, a new VLT, named s5.4 , can be generated. Note that, different from original linguistic terms, it is hard to name a VLT by words or sentences exactly. For example, s5 can be named by slightly good, but s5.4 can not be endowed with a linguistic name. Till now, only the symbolic has been generated for a VLT. Their meanings and semantics should be assigned. This will be completed by linguistic modifiers defined below:

where s is a given term and δ is a real number representing the degree of modification. Two classes of famous modifiers are the power modifiers [16] and the shifting modifiers [17].

Definition 3. (The semantic rule). Let S = {st |t = 0, 1, . . . , τ } be a LTS with the semantics defined on the domain U. For any α ∈ [0, τ ], the semantic of the VLT sα generated by Definition 2 is given by

2.2. Virtual linguistic model

sα = F M (st , δ )

Generally, a LTS with the semantics defined on the domain U can be denoted by

where t = round (α ), δ = α − t and F M is a linguistic modifier on U.

S = {st |t = 0, 1, . . . , τ }

(3)

where τ is a positive integer. In the computational process of the membership function- based model and the ordinal scales-based model, dealing with these discrete linguistic terms may lead to the loss of information. Thus, Xu [8] extended Eq. (3) to a continuous form S¯ = {sα |α ∈ [0, τ ]}. Given sα ∈ S¯, if sα ∈ S, then it is an original linguistic term (atomic term); otherwise, it is a VLT. Due to the lack of syntax and semantics, Xu [8] had to state that the VLTs can only appear in operations. Based on some simple operational laws, the virtual linguistic model is convenient for information fusion for decision making without loss of information. However, the output VLTs limit the interpretability of this kind of decision making methods.

(5)

Remark 2. This definition only supplies a strategy to obtain the semantics of VLTs. The linguistic modifier should be determined according to the type of original terms distributed on the domain. We will specify the choice of linguistic modifiers in the coming subsections. Remark 3. Similar to the 2-tuple linguistic model [7], the syntax and semantic of a VLT should be clarified by two parameters. There is a pair of symbolic transformation functions in 2-tuple linguistic model. One is used to transform a 2-tuple linguistic term to a real number β ∈ [0, τ ] which represents the equivalent linguistic information and the other is used for inverse transformation. However, there are some different details. In the syntactical aspect, a 2-tuple linguistic term is generated by a real number β , whereas a VLT

Z. Xu, H. Wang / Information Fusion 34 (2017) 43–48

45

s1.7

s5.4

extremely poor

very poor

poor

0

0.125

0.25

0.0875

slightly poor

fair

0.375

0.5

0.3375

slightly good

good

0.625

0.75

0.55

very good

0.80

extremely good

0.875

1

Fig. 1. Semantics of original linguistic terms and VLTs.

is generated by an original linguistic term and a real number δ . Moreover, in the semantic aspect, a 2-tuple linguistic term takes use of a real number β to represent its information, but we obtain the semantic of a VLT by means of a proper linguistic modifier. The use of linguistic modifiers is more complex than the semantic representation of 2-tuple terms, but we can fully use the semantics of original linguistic terms. As will be seen in the coming subsections, Definitions 2 and 3 enable us to deal with VLTs whose semantics are either inclusive interpretations or non-inclusive interpretations.

In this context, one or more terms act as atomic terms and other terms are generated by means of modifiers. This kind of LTSs is often used in approximate reasoning [2]. Usually, the following order is assumed for the purpose of CWW:

We can see that the semantic of sα is defined by modifying its closest original linguistic term st to a certain level which is subject to another parameter δ . Especially, in the virtual linguistic model, each original linguistic term st keeps its original semantic defined in S. As mentioned in Section 2, VLTs are not interpretable without semantics. Based on Definition 3, a VLT sα can be interpreted by the two parameters t and δ . In fact, t implies the original linguistic term st which is closest to sα , and |δ| expresses the degree that sα is close to st . For instance, let s5.4 be the VLT shown in Fig. 1, it is close to the original linguistic term s5 (slightly good), and δ = 5.4 − 5 = 0.4 can serve as a similarity measure of s5.4 and s5 . Based on Definitions 2 and 3, an expert’s evaluation can also be expressed by VLTs. In fact, if the expert can express his/her opinion by an original linguistic term accurately, then the original linguistic term is suitable; if there does not exist st ∈ S for expressing the opinion accurately, then: (1) an original linguistic term st which is closest to the opinion, can be determined, (2) a number δ (described in Eq. (4)) can be provided to measure the degree that the objective term sα is close to st (More specifically, |δ| represents the degree of closeness, and δ > 0 if sα is greater than st , otherwise δ ≤ 0), and (3) a VLT sα , where α = t + δ , can be used for the representation of the opinion. In the coming part of this section, we specify Definition 3 according to the graphical shapes of membership functions of original linguistic terms in S.

Example 2. Given a domain U = [0, 100] representing human’s age, a linguistic term, denoted by s1 , named old, of the linguistic variable Age is defined by

(7)

Notice that st (t = 1, 2, . . . , τ ) in Eq. (6) represents the fuzzy sets corresponding to their semantics, whereas the symbols in Eq. (7) are just the names of linguistic terms.

 s1 =

100

50

−1

(1 + ( (u − 50 )/5 )2 ) /u

Then another two terms, s0 = more or less old and s2 = very old can be defined by

 s0 =

100 50

(μs1 (u ))1/2 /u, s2 =

(6)

100 50

(μs1 (u ))2 /u

Given a VLT in this context, its semantic can be specified based on the power modifier. Let S = {st |t = 0, 1, . . . , τ } be a LTS with inclusive interpretation. Suppose that the semantics of st (t ∈  {0, 1, . . . , τ }) are defined by st = U μst (u )/u, then the semantic of sα is obtained by

sα = F M (st , δ ) =



U

δ

(μst (u ))2 /u

(8)

where t = round (α ) and δ = α − t. Example 3. Let S = {s0 , s1 , s2 } be the LTS defined in Example 2. Then the semantic of s0.5 is (as shown in Fig. 2):



In the case when linguistic terms are with inclusive interpretations, it is assumed that the semantic entailment always holds. That is, given S = {st |t = 0, 1, . . . , τ } defined on the domain U, it follows that



Thus, a LTS S = {s0 , s1 , s2 } with inclusive interpretation is formed. The semantics of which are shown in Fig. 2.

s0.5 = F M (s1 , −0.5 ) =

3.1. The semantics of VLTs with inclusive interpretations

sτ ⊆ sτ −1 ⊆ · · · ⊆ s0

s0 ≤ s1 ≤ · · · ≤ sτ

=

100 50



U

(μs1 (u ))2

−0.5

√ 2 −1/ 2

(1 + ((u − 50 )/5 ) )

/u

/u

Motivated by the relationship between Eqs. (6) and (7), it is natural to define the order of sα1 and sα2 according to the inclusive relation of their semantics, where sα1 , sα2 ∈ S¯. Then it is easy to prove the following theorem:

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Z. Xu, H. Wang / Information Fusion 34 (2017) 43–48

1

s0.5

s1

s0 s2 0

100

50 Fig. 2. An example of semantics of VLTs with inclusive interpretation.

Theorem 1. Given two VLTs sα1 , sα2 ∈ S¯ with the semantics defined by Eq. (8), then

t =

sα1 ≤ sα2 ⇔ α1 ≤ α2

The inclusion is not satisfied in LTSs with non-inclusive interpretations, as shown in Fig. 1. Most of the recent literature focuses on this kind of semantics. The predefined LTS is usually constructed by a few atomic terms and other terms are generated by shifting modifiers. Let’s begin the semantics of VLTs in this context with a simply case. If original terms are uniformly distributed in the domain U = [lU , rU ], as shown in Fig. 1, then the semantic of sα can be defined by specifying Eq. (5) as:

sα =

μst (u − δ · l /τ )/u

U

sα = (max{lU , a + δ · l/τ }, b + δ · l/τ , c (10)

Example 4. Let S be the LTS of Example 1, whose semantics are shown in Fig. 1, then τ = 8, U = [0, 1] and l = 1. Moreover, by using triangular fuzzy numbers, we have s2 = (1/8, 2/8, 3/8) and s5 = (4/8, 5/8, 6/8). According to Eq. (9), the semantics of VLTs s1.7 and s5.4 are:



s1. 7 =

s5. 4

U

δ<0 δ=0 δ>0

respectively. Then Eq. (5) can be specified by a shifting modifier which shifts from st towards st to a certain extent. Different from Eq. (9), this step includes an affine transformation rather than simply translation. If the semantics of original terms can be expressed by trapezoidal fuzzy numbers, that is, st = (a, b, c, d ) and st = (a , b , c , d ), then the semantic of sα can be obtained by

sα = (max{lU , a˜}, b˜ , c˜, min{rU , d˜} )

(11)

where a˜ = (1 − |δ| )a + |δ|a , b˜ = (1 − |δ| )b + |δ|b , c˜ = (1 − |δ| )c + |δ|c and d˜ = (1 − |δ| )d +|δ|d . More generally, suppose s = 

U

μst (u )/u and st =



U

t

μst (u )/u, which are defined on a non-

uniformly distributed domain U, then the semantic of sα can be derived by three steps as shown in Fig. 3:

(9)

where l = rU − lU is the length of the domain U. Graphically, Eq. (9) shifts the membership function of st to a certain extent. As can be seen in most cases, trapezoidal fuzzy numbers are used to represent the piecewise linear membership functions. Thus, for any VLT sα (α ∈ [0, τ ]), suppose st = (a, b, c, d ) (where t = round (α )), then Eq. (9) can be rewritten as:

+ δ · l/τ , min{rU , d + δ · l/τ } )

t,

⎩

t + 1,

3.2. The semantics of VLTs with non-inclusive interpretations



⎧ ⎨t − 1,

μs2 (u − (−0.3 )/8 )/u

= (1/8 − 0.3/8, 2/8 − 0.3/8, 3/8 − 0.3/8 )  = μs5 (u − (0.4 )/8 )/u

Step 1. Construct a transformation function f to transform the non-uniformly distributed domain U into a uniformly distributed version U˜ . The semantics of original terms can be transformed simultaneously. Step 2. Obtain the semantic of sα on U˜ by Eq. (9). Step 3. Transform the obtained function of semantics into the domain U by f −1 . Example 5. Let S = {st |t = 0, 1, . . . , 6} be the non-uniformly distributed LTS defined on U = [0, 1] proposed by Xu [18]. The semantics of original terms are shown in Fig. 4. According to Eq. (11), the semantic of the VLT s1.3 can be derived by s1 = (0, 5/18, 5/12 ) and s2 = (5/18, 5/12, 1/2 ) as follows:

s1.3 = (0.7 × 0 + 0.3 × 5/18, 0.7 × 5/18 + 0.3 × 5/12, 0.7 × 5/12 + 0.3 × 1/2 ) = (0.08, 0.32, 0.44 ) Moreover, we can calculate the semantics by the idea of Fig. 3, i.e., we construct the transformation function f : U = [0, 1] → U˜ = [0, 1], where

U

= (4/8 + 0.4/8, 5/8 − 0.4/8, 6/8 − 0.4/8 ) which are also shown in Fig. 1. However, if original terms are non-uniformly distributed in the considered domain, it is hard to determine the semantics of VLTs by a simply shifting modifier. Given S = {st |t = 0, 1, . . . , τ } with non-inclusive interpretation, for any VLT sα (α ∈ [0, τ ]), two original terms st and st can be fixed by t = round (α ) and

u˜ = f (u ) =

⎧ 3u/5, ⎪ ⎪ ⎪ ⎪ ⎨6u/5 − 1/6,

0 ≤ u < 5/18 5/18 ≤ u < 5/12

2u − 1/2,

5/12 ≤ u < 7/12

3u/5 + 2/5,

7/12 ≤ u ≤ 1

⎪ ⎪ ⎪ 6u/5 − 1/30, ⎪ ⎩

7/12 ≤ u < 13/18

then the semantic of an original term st can be transformed  into s˜t = U˜ μst (u )/u˜. For example, let s˜5 = (4/6, 5/6, 1 ), then according to Eq. (9), s˜4.7 = (4/6 − 0.3/6, 5/6 − 0.3/6, 1 − 0.3/6 ). Using the inverse of f , the semantic of s4.7 on the domain U is s4.7 = (0.56, 0.78, 0.92 ).

Z. Xu, H. Wang / Information Fusion 34 (2017) 43–48

47

Fig. 3. The steps of obtaining the semantics of VLTs on non-uniform domain.

s0

0

s1 s1.3

0.28

s2

s3

s4

0.42 0.5 0.58

s4.7 s5

s6

0.72

1

Fig. 4. Examples of the semantics of VLTs on non-uniform domain.

4. Computational model of VLTs The computational model of VLTs plays an important role of fusing a collection of linguistic information and making decisions. In this section, we reconstruct the computational models based on the semantics of VTLs. We have extended the original LTS S = {st |t = 0, 1, . . . , τ } to a continuous version S¯ = {sα |α ∈ [0, τ ]} based on the abovementioned syntax and semantics. The computational model based on VLTs can be expressed as: Ind C → [0, τ ] − → S¯ S¯n −→ [0, τ ]n −

(1) Total order of VLTs. According to the discussion in Section 3, for any sα , sβ ∈ S¯, it is rational to define the total order as: (12)

where t1 = round (α ), δ1 = α − t1 and t2 = round (β ), δ2 = β − t2 . It is clear that Eq. (12) is equivalent to [8]:

sα ≤ sβ ⇔ α ≤ β

If the linguistic terms are uniformly distributed, in other words, they are equally informative, then the negation Neg becomes a simply case which has been widely used in the literature. In this case, ∀sα ∈ S¯, its negation can be defined as [8]:

Neg(sα ) = sτ −α

where Ind returns the indices of the input VLTs, and C is an aggregation operator. Obviously, this manipulation shows the same process of computing with linguistic 2-tuple. The first and third procedures correspond to the pair of transformation functions of the 2-tuple linguistic model. Some researchers have reported that the computational models of both the 2-tuple linguistic model and the virtual linguistic model are equivalent [19]. However, it should be noted that, in the virtual linguistic model, the above manipulation enables us to compute linguistic information according to the indices of VLTs, rather than computing by three discrete steps. To operate with VLTs, the following issues should be figured out at first:

sα ≤ sβ ⇔ (st1 < st2 ) ∨ ((st1 = st2 ) ∧ (δ1 ≤ δ2 ))

(2) Negation operator of a VLT. Given S¯, the negation operator Neg is a mapping from S¯ to a subset of S¯ satisfying [20]: (i) Neg is not empty and convex; (ii) if sα < sβ , then Neg(sβ ) ≤ Neg(sα ), for any sα , sβ ∈ S¯; (iii) if sβ ∈ Neg(sα ), then sα ∈ Neg(sβ ), where sα , sβ ∈ S¯.

(13)

(14)

(3) Aggregation of VLTs. We begin the aggregation with two VLTs. For any sα , sβ ∈ S¯, their weights are w1 and w2 respectively, where w1 + w2 = 1, 0 ≤ w1 ≤ 1, and 0 ≤ w2 ≤ 1. Based on the semantics of VLTs, the weighted arithmetical averaging operator and the weighted geometric averaging operator can be defined as:

w1 sα  w2 sβ = F M (st1 , δ1 )

(15)

w1 sα  w2 sβ = F M (st2 , δ2 )

(16)

where t1 = round (w1 α + w2 β ), δ1 = w1 α + w2 β − t1 , t2 = round (α w1 · β w2 ), δ2 = α w1 · β w2 − t2 , and F M is the fuzzy modifier specified in Section 3. It is clear that the operation is conducted by two steps. An original term st1 (or st2 ) is determined at first. Then a modifier is used to move st1 (or st2 ) to a VLT based on the parameter δ1 (or δ2 ). It should be noted that we use Eqs. (15) and (16) to illustrate the semantics of the results of aggregation. For application, they can be rewritten as:

which means that the comparison of VLTs can be done by their indices. Thus, we have the following operations:

w1 sα  w2 sβ = sw1 α +w2 β

(17)

max{sα , sβ } = sβ ⇔ sα ≤ sβ , min{sα , sβ } = sα ⇔ sα ≤ sβ

w 1 s α  w 2 s β = s α w 1 ·β w 2

(18)

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Z. Xu, H. Wang / Information Fusion 34 (2017) 43–48

Eqs. (17) and (18) serve as the basic operation of VLTs as well. To aggregate n (n ≥ 2) VLTs, a series of aggregation operators based on these operations can be found in Refs. [8,21,22]. 5. Concluding remarks The virtual linguistic model is a good technique for CWW without loss of information. In this paper, we have focused on the syntactical rule and the semantic rule of VLTs. The former generates VLTs and the latter assigns semantics to them. These rules serve as the base of the computational model based on VLTs. Also based on these rules, VLTs can be used for the representation of the experts’ linguistic opinions and the output VLTs are interpretable. Therefore the paper can be seen as the theoretical foundation of the VLTsbased computational models. It had been demonstrated that VLTs play an important role in information fusion in decision making. Based on the syntax and semantics, VLTs could be an alternative way to solve the challenges in hesitant fuzzy information fusion [23] (especially in qualitative hesitant setting [24]), such as fusing hesitancy and defining the orders of hesitant linguistic information [25]. VLTs can also be used to implement the useful fuzzy measures in qualitative setting [26] or fuse linguistic preferences from distinct perspectives [27]. Detailed discussions about the connections to other similar techniques, such as the 2-tuple linguistic model, are very interesting for future investigation. For instance, a position paper can be contributed to help readers understand the role of each model and develop new models. Acknowledgements The authors would like to thank Editor-in-Chief, Area Editor and three anonymous reviewers for their insightful and constructive commendations that have led to an improved version of this paper. The work was supported by the National Natural Science Foundation of China (Nos. 61273209, 71571123), the Scientific Research Foundation of Graduate School of Southeast University (No. YBJJ1528) and the Scientific Innovation Research of College Graduates in Jiangsu Province (KYLX15-0191). References [1] L.A. Zadeh, Fuzzy logic = computing with words, IEEE Trans. Fuzzy Syst. 4 (2) (1996) 103–111. [2] L.A. Zadeh, The concept of a linguistic variable and its application to approximate reasoning-II, Inf. Sci. 8 (3) (1975) 301–357.

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