Novel basic operational laws for linguistic terms, hesitant fuzzy linguistic term sets and probabilistic linguistic term sets

Information Sciences 372 (2016) 407–427

Contents lists available at ScienceDirect

Information Sciences journal homepage: www.elsevier.com/locate/ins

Novel basic operational laws for linguistic terms, hesitant fuzzy linguistic term sets and probabilistic linguistic term sets Xunjie Gou a, Zeshui Xu a,b,∗ a b

Business School, Sichuan University, Chengdu 610064, China School of Computer and Software, Nanjing University of Information Science and Technology, Nanjing 210044, China

a r t i c l e

i n f o

Article history: Received 24 June 2016 Revised 2 August 2016 Accepted 11 August 2016 Available online 12 August 2016 Keywords: Operational laws Equivalent transformation functions Linguistic term set Hesitant fuzzy linguistic term set Probabilistic linguistic term set

a b s t r a c t In the process of decision making, people sometimes may feel more comfortable to express their preferences by linguistic terms instead of the quantitative form. However, as the basic premise of operations, the existing operational laws of linguistic terms and the extended linguistic term sets are very unreasonable. In order to overcome this issue, in this paper, we redefine some more logical operational laws for linguistic terms, hesitant fuzzy linguistic elements (HFLEs) and probabilistic linguistic term sets (PLTSs) based on two equivalent transformation functions. These novel operational laws can not only avoid the operation values exceeding the bounds of LTSs, but also keep the operation results more reasonable in decision making with linguistic information. Furthermore, the operational laws can keep the probability information complete when computing with PLTSs. Additionally, lots of properties of the operational laws are discussed, and some three-dimensional figures are drawn to show the regions of different operational laws of linguistic terms more vividly. © 2016 Elsevier Inc. All rights reserved.

1. Introduction Decision making is a common activity in our daily life. Over the past decades, a lot of decision making techniques have been developed, including the TOPSIS methods [2,5,21,23], the VIKOR method [12], the LINMAP method [16] and granular computing [1,3,4,14,17-19,22-24,26,31,37,38], etc. In the process of decision making, people sometimes may feel more comfortable to express their preferences by linguistic terms instead of the quantitative form. Therefore, the fuzzy linguistic approach [39] has received lots of scholars’ attention, and it is an effective way to model linguistic information. In this approach, an important step is to transform the linguistic information into a machine manipulative format, in which the computation can be carried out [12]. Therefore, Xu [34] proposed a subscript-symmetric additive linguistic term set (LTS), but it is discrete and sometimes inconvenient for calculation and analysis. In order to preserve all linguistic information, he further extended the discrete LTS to a continuous LTS (or called virtual LTS), and based on which, Liao et al. [9] established the mapping between virtual linguistic terms and their corresponding semantics as shown in Fig. 1. Later on, a series of extended LTSs have been put forward, such as hesitant fuzzy linguistic term set (HFLTS) [2,512,15,29,30,33,34,40-42], linguistic hesitant fuzzy set (LHFS) [16,20], probabilistic linguistic term set (PLTS) [21], etc. The HFLTS, combining the LTS and the hesitant fuzzy set (HFS), was introduced by Rodríguez et al. [25]. It is a more reasonable information expression form, which can be used to describe people’s subjective cognitions. Liao et al. [11] redefined the concept of HFLTS in term of mathematical representation, and the elements of a HFLTS were called hesitant fuzzy ∗

Corresponding author. Tel.: 88615850776685. E-mail addresses: [email protected] (X. Gou), [email protected] (Z. Xu).

http://dx.doi.org/10.1016/j.ins.2016.08.034 0020-0255/© 2016 Elsevier Inc. All rights reserved.

408

X. Gou, Z. Xu / Information Sciences 372 (2016) 407–427

s s

3

none

0 1 / 15

s

2

very low

1/ 6

s1.3

1.6

s

1

low

1/ 3

2/5

0

1

s2

s3

medium

high

very high

perfect

1 / 2 11 / 20 2 / 3

5 / 6 53 / 60

1

Fig. 1. Semantics of virtual linguistic terms.

linguistic elements (HFLEs). In recent years, a lot of scholars have studied HFLTSs from different angles, and developed the hesitant fuzzy linguistic information aggregation theory [7,30,33,40], the hesitant fuzzy linguistic measurement theory [6,8,9-11,28], the hesitant fuzzy linguistic preference relation theory [11,15,41,42], and the hesitant fuzzy linguistic decision making methodologies [2,5,12,28,29], etc. However, due to the lack of considering the weight information in most of the current studies about HFLTSs, we always give tacit consent to all the linguistic terms having the same importance or weight, but it rarely happens in reality. In fact, people may prefer some of the possible linguistic terms so that they may have different importance degrees. Therefore, Pang et al. [21] introduced PLTS to extend HFLTS through adding probabilities without loss of any original linguistic information. Additionally, they developed an extended TOPSIS method and an aggregation-based method respectively for multi-attribute group decision making (MAGDM) with probabilistic linguistic information. In general, we need to make some operations when dealing with all kinds of linguistic information in practical decision making problems. However, the existing operational laws of linguistic terms have some shortcomings as follows: (1) Suppose that S = {st |t = −3, . . . , −1, 0, 1, . . . , 3} is a LTS, s2 and s3 are two linguistic terms, and let λ = 2, then by the basic operational laws of linguistic terms given by Xu [35], we obtain s2  s3 = s5 and λs2 = s4 . Obviously, both these two results exceed the upper bound s3 . Considering that the operations of HFLEs are based on the operational laws of linguistic terms [35], they also have this drawback. (2) Let S be the LTS as defined above, and let L1 ( p) = {s1 (0.2 ), s2 (0.3 ), s3 (0.5 )} and L2 ( p) = {s2 (0.2 ), s3 (0.8 )} be two PLTSs, then by using the operational laws of PLTSs given by Pang et al. [21], we obtain L1 ( p)  L2 ( p) = {0.2s1  0s2 , 0.3s2  0.2s2 , 0.5s3  0.8s3} = {s0.2 , s1 , s3.9 }. Clearly, the result s3.9 not only exceeds the bound [s−3 , s3 ], but also loses the corresponding probability information. In order to avoid these issues, we need to define some novel operational laws for linguistic terms, HFLEs and PLTSs. Gou et al. [7] proposed two equivalent transformation functions g and g−1 , which can achieve the equivalent transformations between the HFLEs and the HFEs. Based on g and g−1 , in this paper we define some novel operational laws of linguistic terms, HFLEs and PLTSs, including addition, multiplication, power, subtraction, division and supplement. These operational laws not only can avoid the operation results exceeding the bounds of LTSs, but also can keep the probability information complete after operations. Fig. 2 describes the operation process of the novel operational laws clearly. The remainder of this paper is organized as follows: In Section 2, we review some concepts of LTSs, HFLTSs and PLTSs, and introduce two equivalent transformation functions g and g−1 . In Section 3, we define some novel operational laws for linguistic terms and discuss their properties. We also draw some three-dimensional figures to show the regions of different operational laws vividly. Additionally, some novel operational laws of HFLTSs are defined and their properties are also discussed in Section 4. Section 5 defines the operational laws of PLTSs considering the factors of linguistic terms and probability information. In Section 6, we use a practical example to show the novel operational laws of HFLEs and PLTSs for dealing with a multiple criteria decision making (MCDM) problem. We also summarize the advantages of these operational laws by comparing them with the existing ones. Finally, we draw some conclusions and point out the future research directions of the linguistic terms, HFLEs and PLTSs in Section 7.

2. Linguistic term set, hesitant fuzzy set and some extended forms In this section, we mainly recall some basic concepts and properties of LTSs, HFSs, HFLTSs and PLTSs:

X. Gou, Z. Xu / Information Sciences 372 (2016) 407–427

409

Fig. 2. The operation process of the novel operational laws.

2.1. Linguistic term set As we have discussed in the introduction, Xu [34] proposed the subscript-symmetric additive linguistic term set S =

{st |t = −τ , . . . , −1, 0, 1, . . . , τ}, where the mid linguistic label s0 expresses the assessment of “indifference”, and the rest linguistic labels are distributed symmetrically around it. τ is a positive integer, s−τ and sτ are respectively the lower and upper bounds of linguistic terms. For any two linguistic terms sα , sβ ∈ S˜, and λ, λ1 , λ2 ∈ [0, 1], Xu [35] introduced some operational laws as follows: (1) (2) (3) (4)

sα  sβ = sα +β ; λsα = sλα ; (λ1 + λ2 )sα = λ1 sα + λ2 sα ; λ(sα + sβ ) = λsα + λsβ .

2.2. Hesitant fuzzy set Torra [27] defined the concept of HFS, which permits the membership degree of an element to a reference set represented by several possible values. Definition 2.1. [27]. Let X be a fixed set. A HFS on X is in terms of a function that when applied to X returns a subset of [0,1]. To be easily understood, Xia and Xu [32,36] expressed the HFS by a mathematical symbol B = {< x, hB (x ) > |x ∈ X }, where hB (x ) is a set of some values in [0,1], denoting the possible membership degrees of the element x ∈ X to the set B. Additionally, they called h = hB (x ) a hesitant fuzzy element (HFE). Definition 2.2. [32]. Some operational laws about any three HFEs h, h1 and h2 are given as follows: (1) hλ = ∪γ ∈h {γ λ };

λ

(2) λh = ∪γ ∈h {1 − (1 − γ ) }; (3) h1  h2 = ∪γ1 ∈h1 ,γ2 ∈h2 {γ1 + γ2 − γ1 γ2 }; (4) h1  h2 = ∪γ1 ∈h1 ,γ2 ∈h2 {γ1 γ2 }. In addition, Liao [13] defined the subtraction and division operations for HFEs:   { γ11−−γγ22 }, if γ1 ≥ γ2 and γ2 = 1 (5) h1  h2 = γ1 ∈h1 ,γ2 ∈h2 ; otherwise 0,  { γγ12 }, if γ1 ≤ γ2 and γ2 = 0 (6) h1  h2 = γ1 ∈h1 ,γ2 ∈h2 . 0, otherwise

410

X. Gou, Z. Xu / Information Sciences 372 (2016) 407–427

2.3. Hesitant fuzzy linguistic term set Based on HFSs and fuzzy linguistic approach [39], Rodríguez et al. [25] defined the concept of HFLTS: Definition 2.3. [25]. Let S = {s0 , . . . , sτ } be a LTS. A HFLTS, HS , is an ordered finite subset of the consecutive linguistic terms of S. Obviously, in Definition 2.3, there is no any mathematical form for HFLTS. To overcome this issue, Liao [11] gave the definition of HFLTS mathematically: Definition 2.4. [11]. Let xi ∈ X (i = 1, 2, . . . , N ), be fixed and S = {st |t = −τ , . . . , −1, 0, 1, . . . , τ} be a LTS. A HFLTS on X, HS , is in mathematical form of HS = {< xi , hS (xi ) > |xi ∈ X }, where hS (xi ) is a set of some values in S and can be expressed as hS (xi ) = {sφl (xi )|sφl (xi ) ∈ S, l = 1, . . . , L} with L being the number of linguistic terms in hS (xi ). hS (xi ) denotes the possible degree of the linguistic variable xi to S. For convenience, hS (xi ) is called the HFLE and hS is the set of all HFLEs. As we know, there exist some relationships between the HFLE and the HFE, and they can be transformed each other equivalently based on two equivalent transformation functions [7]: Definition 2.5. [7]. Let S = {st |t = −τ , . . . , −1, 0, 1, . . . , τ} be a LTS, hS = {st |t ∈ [−τ , τ ]} be a HFLE, and hγ = {γ |γ ∈ [0, 1]} be a HFE. Then the linguistic variable st that expresses the equivalent information to the membership degree γ is obtained with the following function g:

g : [−τ , τ ] → [0, 1], g(st ) =

t 1 + =γ 2τ 2



g : [−τ , τ ] → [0, 1], g(hS ) = g(st ) =

(1)

t 1 + |t ∈ [−τ , τ ] 2τ 2



= hγ

(2)

Additionally, the membership degree γ that expresses the equivalent information to the linguistic variable st is obtained with the following function g−1 :

g−1 : [0, 1] → [−τ , τ ], g−1 (γ ) = s(2γ −1)τ = st

 

(3)





g−1 : [0, 1] → [−τ , τ ], g−1 hγ = g−1 (γ ) = s(2γ −1)τ |γ ∈ [0, 1] = hS

(4)

2.4. Probabilistic linguistic term set Pang et al. [21] introduced the concept of PLTS, which extends HFLTS through adding probability information without loss of any original linguistic information: Definition 2.6. [21]. Let S = {st |t = −τ , . . . , −1, 0, 1, . . . , τ} be a LTS. Then a PLTS can be defined as:

L( p) = {L(k ) ( p(k ) )|L(k ) ∈ S, p(k ) ≥ 0, k = 1, 2, ..., #L( p),

#L( p) k=1

p(k ) ≤ 1}

(5)

where L(k ) ( p(k ) ) is the linguistic term L(k ) associated with the probability p(k ) , and #L( p) is the number of all different linguistic terms in L( p). 3. Some novel operational laws for linguistic terms Over the past few decades, we always use the basic operational laws expressed in Section 2.1 to make operations on the linguistic terms by calculating their subscripts. However, one very obvious shortcoming of these operational laws is that the calculation results may exceed the bounds of the LTSs. To solve this issue, it is necessary to put forward some new operational laws for linguistic terms. Fig. 1 shows the mapping between the virtual linguistic terms and the numerical value (here we regard it as fuzzy number) included in [0, 1]. Two mapping (or equivalent transformation) functions (Eqs. (2) and (4)) have been introduced in Definition 2.5 [7], based on which, we can define some novel operational laws for linguistic terms, including addition, multiplication, power, subtraction, division and supplement: Definition 3.1. Let S = {st |t = −τ , . . . , −1, 0, 1, . . . , τ} be a LTS, sα and sβ be two linguistic terms, g and g−1 be the equivalent transformation functions of linguistic terms, and λ be a real number. Then (1) sα  sβ = g−1 (g(sα ) + g(sβ ) − g(sα )g(sβ )); (2) sα  sβ = g−1 (g(sα )g(sβ )); λ

(3) λsα = g−1 (1 − (1 − g(sα ) ) ); λ (4) (sα )λ = g−1 ( (g(sα ) ) );

X. Gou, Z. Xu / Information Sciences 372 (2016) 407–427

411

Fig. 3. The region of the addition operation sα  sβ .



g−1 (

(5) sα  sβ =

 (6) sα  sβ =

0,

g(sα )−g(sβ ) ), 1−g(sβ )

if g(sα ) ≥ g(sβ ) and g(sβ ) = 1

;

otherwise

g−1 ( gg((ssα )) ), if g(sα ) ≤ g(sβ ) and g(sβ ) = 0 β

0, (7) sα = g−1 (1 − g(sα ) ).

otherwise

;

Remark 1. In Definition 3.1, these two mapping functions g and g−1 are the positive function and the inverse function, respectively, and they provide two equivalent transformations between linguistic terms and fuzzy numbers. So we can obtain the basic operations of linguistic terms by the corresponding operations of fuzzy numbers. Furthermore, the operation results are also limited in [s−τ , sτ ] and we can prove it by taking the addition operation as example: We establish one binary function f (x, y ) = x + y − xy, x ∈ [0, 1], y ∈ [0, 1], and calculate the partial derivatives of x and y, respectively. Then ∂∂ xf = 1 − y ≥ 0 and ∂∂ yf = 1 − x ≥ 0, so we can obtain that the binary function f (x, y ) is an increasing function for both x and y. Moreover, both the extreme values are obtained at the endpoint. So the extreme values are also the maximum values, and the operation results are limited in [s−τ , sτ ]. Furthermore, the values of other operational laws are also limited in [s−τ , sτ ]. In order to understand these operational laws of linguistic terms better, we draw six three-dimensional figures (Figs. 3–8) and a plane figure (Fig. 9) to show the region of each operation: Remark 2. In Figs. 3–8, there exists a curve surface drawn by blue color in each three-dimensional figure, respectively. In particular, we denote λ ∈ [0, 1] in Figs. 5 and 6 for describing these figures better and intuitively because we cannot draw them clearly when λ ∈ [0, +∞). An example can be set up to show these operational laws: Example 3.1. Let S = {st |t = −3, . . . , −1, 0, 1, . . . , 3} be a LTS, s−3 , s2 and s3 be three linguistic terms, and λ = 2. Then (1) s2  s3 = g−1 (5/6 + 1 − (5/6 ) × 1) = s(2×1−1)×3 = s3 ; s2  s−3 = g−1 (5/6 + 0 − (5/6 ) × 0) = s(2× 5 −1)×3 = s2 . (2) s2  s3 = g−1 ( (5/6 ) × 1 ) = s(2× 5 −1)×3 = s2 ; s2  s−3 = g−1 ( (5/6 ) × 0 ) = s(2×0−1)×3 = s−3 . 6

(3) λs2 = g−1 (1 − (1 − 5/6) ) = s(2× 35 −1)×3 = s2.83 , 2

36

2

λs3 = g−1 (1 − (1 − 1 )2 ) = s(2×1−1)×3 = s3

6

and

λs−3 = g−1 (1 −

(1 − 0 ) ) = s(2×0−1)×3 = s−3 . 2 2 (4) (s2 )λ = g−1 ( (5/6 ) ) = s(2× 25 −1)×3 = s1.17 and (s3 )λ = g−1 ( (1 ) ) = s(2×1−1)×3 = s3 . 36

/6 −1 5/6−0 (5) s3  s2 = g−1 ( 11−5 −5/6 ) = s(2×1−1)×3 = s3 and s2  s−3 = g ( 1−0 ) = s(2× 5 −1 )×3 = s2 . 6

(6) s2  s3 = g−1 ( 51/6 ) = s(2× 5 −1)×3 = s2 . 6

(7) s−3 = g−1 (1 − 0 ) = s(2×1−1)×3 = s3 , s2 = g−1 (1 − 5/6) = s(2× 1 −1)×3 = s−2 , s3 = g−1 (1 − 1 ) = s(2×0−1)×3 = s−3 . 6

412

X. Gou, Z. Xu / Information Sciences 372 (2016) 407–427

Fig. 4. The region of the multiplication operation sα  sβ .

Fig. 5. The region of the multiplication operation λsα .

Remark 3. In this example, we can obtain some special properties about these operational laws of linguistic terms. Let S = {st |t = −τ , . . . , −1, 0, 1, . . . , τ} be a LTS, then (1) The result must be sτ no matter what linguistic term adds the maximum sτ ; Similarly, the result must be itself no matter what linguistic term adds the minimum s−τ . (2) The result must be itself no matter what linguistic term multiplies the maximum sτ , and the result must be s−τ no matter what linguistic term multiplies the minimum s−τ . (3) The result must be sτ no matter what positive real number multiplies the maximum sτ , and the result must be s−3 no matter what positive real number multiplies the minimum s−3 . (4) The exponentiation operation of the maximum sτ must be sτ no matter what the positive real number be. (5) The result must be sτ when the maximum sτ minus any linguistic term, and the result must be itself when any linguistic term minus s−τ . (6) The result must be itself when any linguistic divide the maximum sτ . (7) In fact, the supplement of any linguistic term can be obtained by transforming the subscript into its opposite number. All of these seven properties conform to the common sense of people, and these operational laws of linguistic terms can be reasonable and effective when computing with linguistic terms.

X. Gou, Z. Xu / Information Sciences 372 (2016) 407–427

413

Fig. 6. The region of the power operation (sα )λ .

Fig. 7. The region of the subtraction operation sα  sβ .

In what follows, we investigate some important properties for the novel operational laws: Theorem 3.1. Let S = {st |t = −τ , . . . , −1, 0, 1, . . . , τ} be a LTS, sα and sβ be two linguistic terms, λ be a positive real number. Then (1) (2) (3) (4) (5)

sα  sβ = sβ  sα ; sα  sβ = sβ  sα ; (sα  sβ )  sβ = sα , if sα ≥ sβ and sβ = 1; (sα  sβ )  sβ = sα , if sα ≤ sβ and sβ = 0; λ(sα  sβ ) = λsβ  λsα and λ(sα  sβ ) = λsα  λsβ ;

(6) (sα  sβ )λ = (sα )λ  (sβ )λ and (sα  sβ )λ = (sα )λ  (sβ )λ ; (7) λ1 sα  λ2 sα = (λ1 + λ2 )sα and λ1 sα  λ2 sα = (λ1 − λ2 )sα ; (8) (sα )λ1  (sα )λ2 = (sα )λ1 +λ2 and (sα )λ1  (sα )λ2 = (sα )λ1 −λ2 ; Proof. (1) and (2) are obvious, so we omit the proofs of them here.

414

X. Gou, Z. Xu / Information Sciences 372 (2016) 407–427

Fig. 8. The region of the division operation sα  sβ .

Fig. 9. The region of the supplement operation sα .

(3) If sα ≥ sβ and sβ = 1, then





sα  sβ  sβ = g−1

α +τ

− β2+ττ 1 − β2+ττ

2τ

 2ατ −βτ −τ 2 τ −β

= g−1

 sβ = g−1

+τ

2τ

α−β τ −β

β +τ + − 2τ

 sβ = s 2ατ −βτ −τ 2  sβ τ −β

2ατ −βτ −τ 2 τ −β

2τ

(4) If sα ≤ sβ and sβ = 0, then





−1

sα  sβ  sβ = g

α +τ 2τ β +τ 2τ

= λg−1

 −1

=g

= s 

β +τ × 2τ

 2ατ −βτ +τ 2 β +τ

−1

 sβ = s 2ατ −βτ +τ 2  sβ = g

α+τ β +τ α+τ β +τ + − × 2τ 2τ 2τ 2τ

 = g−1

β +τ × 2τ

⎛



= λs τ 2 +ατ +βτ −αβ = g−1 ⎝1 − 2τ

α + τ 

+τ

2τ

β +τ

(5) λ(sα  sβ )



+τ

2τ

 = s(2× α+τ −1 )τ = sα 2τ

τ 2 +ατ +βτ −αβ +τ 2τ 1−



λ

λ   τ − α λ  τ − α λ τ −β τ −β 1− +1− − 1− 1− 2τ 2τ 2τ 2τ 

  s

λ 2× 1− τ2−τα −1 τ

(

)





2× 1− τ2−τβ

λ

−1 τ

= s(2× α+τ −1 )τ = sα 2τ

2τ

λ ⎞ ⎠

X. Gou, Z. Xu / Information Sciences 372 (2016) 407–427



α + τ λ 1− 1− 2τ

= g−1





 g−1

λ 



β +τ 1− 1− 2τ

415

= λsβ  λsα

  λ sα  sβ = λg

−1

α +τ

− β2+ττ 1 − β2+ττ

2τ



= λs

 =g

−β 2× ατ − β −1 τ

−1



α−β 1− 1− τ −β

= s



(

2× 1− τ2−τα

(6) (sα  sβ )λ





−1

=

g

 = g−1



sα  sβ

)

 λ

  s  λ = g−1 −1 τ 2× 1− τ −β −1 τ 2τ

α+τ β +τ × 2τ 2τ

=



 −1

=g

2× (α +τ )(2β +τ ) −1 τ

2τ

 = g−1

=g

α + τ λ 1− 1− 2τ

λ

s

 −1



4τ

 α + τ λ β + τ λ 2τ



λ

λ 



α + τ λ 2τ

  g−1

  τ −α  λ  τ −β λ − 2τ 2τ  τ − β λ 2τ



−1

g

β +τ 1− 1− 2τ

(α + τ ) (β + τ ) 4τ 2

β +τ 2τ

λ 



λ 

= λsα  λsβ

λ 

 λ

= ( s α )λ  s β

λ

 

 α + τ λ  β + τ λ α + τ λ −1 = g =g =g β +τ β +τ 2τ 2τ   

λ  λ α + τ λ β +τ = g−1  g−1 = (sα )λ  sβ 2τ 2τ 

−1

 α + τ λ

(7) λ1 sα  λ2 sα



−1





 α + τ λ1 α + τ λ2 −1   =g 1− 1− g 1− 1− = s  λ 2× 1− ( τ2−τα ) 1 −1 τ 2τ 2τ

 α + τ λ1 +λ2 −1 =g 1− 1− = (λ1 + λ2 )sα 2τ 

−1

 s







λ 2× 1− τ2−τα 2 −1 τ

(

)

λ1 sα  λ2 sα



 α + τ λ1 α + τ λ2    s     g−1 1 − 1 − = s  λ λ 2× 1− ( τ2−τα ) 1 −1 τ 2× 1− ( τ2−τα ) 2 −1 τ 2τ 2τ     

τ −α λ2 − τ −α λ1  τ − α λ1 −λ2  α + τ λ1 −λ2 2τ 2τ −1 −1 −1 =g = g 1 − = g 1 − 1 − = (λ1 − λ2 )sα  τ −α λ2 2τ 2τ 

= g−1 1 − 1 −

2τ

(8) (sα )λ1  (sα )λ2 −1

=g



α + τ λ1 2τ

−1

g



α + τ λ2 2τ

= s

  s  = g−1 λ λ 2× ( α2+ττ ) 1 −1 τ 2× ( α2+ττ ) 2 −1 τ



α + τ λ1 +λ2 2τ



α + τ λ1 −λ2 2τ

= (sα )λ1 +λ2

(sα )λ1  (sα )λ2 = g−1



α + τ λ1 2τ

 g−1



α + τ λ2 2τ

= s

(

2× α2+ττ

λ1

)

  s  = g−1 λ −1 τ 2× ( α2+ττ ) 2 −1 τ

Theorem 3.2. Let S = {st |t = −τ , . . . , −1, 0, 1, . . . , τ} be a LTS, sα , sβ and sδ be three linguistic terms, then

= (sα )λ1 −λ2

416

X. Gou, Z. Xu / Information Sciences 372 (2016) 407–427

2 (1) sα  sβ  sδ = sα  sδ  sβ = sα  (sβ  sδ ), if α ≥ τ +βτ2+τδτ −βδ and g(s τ 2 +βτ +δτ −βδ ) = 1. 2τ

δ +τ ) δ +τ ) (2) sα  sβ  sδ = sα  sδ  sβ = sα  (sβ  sδ ), if α ≤ (2 × (β +τ4)( − 1 )τ and g( (2 × (β +τ4)( − 1 )τ ) = 0. τ2 τ2

Proof. 2 (1) If α ≥ τ +βτ2+τδτ −βδ and g(s τ 2 +βτ +δτ −βδ ) = 1, then 2τ

sα  sβ  sδ =g−1

−1

sα  sδ  sβ =g



α +τ

− β2+ττ 2τ 1 − β2+ττ

α +τ

− δ2+ττ 1 − δ2+ττ

2τ





−1

sα  sβ  sδ = sα  g = s

⎛

 sδ =s

  s = g−1 ⎝ δ

−β 2× ατ − β −1 τ



−1

 sβ =s(2× α−δ −1 )τ  sγ = g τ −δ

2τ

1 − δ2+ττ

− δ2+ττ

(2× ατ −−δδ −1 )τ +τ − 2τ 1 − β2+ττ

β +τ 2τ

⎞ ⎠ = s

= s

 −1

= sα  s τ 2 +βτ +δτ −βδ = g 2τ





−τ +βδ 2× 2ατ −(τδτ−−ββτ −1 τ )(τ −δ ) 2



β +τ δ+τ β +τ δ+τ + − × 2τ 2τ 2τ 2τ

−τ 2 +βδ 2× 2ατ −(τδτ−−ββτ )(τ −δ )



−β 2× ατ − β −1 τ +τ



−τ +βδ 2× 2ατ −(τδτ−−ββτ −1 τ )(τ −δ ) 2

α +τ − 2τ



τ 2 +βτ +δτ −βδ +τ 2τ

2τ

τ 2 +βτ +δτ −βδ +τ 2τ

1−

2τ

−1 τ

Thus, we obtain sα  sβ  sδ = sα  sδ  sβ = sα  (sβ  sδ ). δ +τ ) δ +τ ) (2) If α ≤ (2 × (β +τ4)( − 1 )τ and g( (2 × (β +τ4)( − 1 )τ ) = 0, then τ2 τ2

sα  sβ  sδ = g−1 sα  sδ  sβ = g−1

α + τ  β +τ

α + τ 



 sδ = s

  s = g−1 δ τ −1 τ 2× βα + +τ

2τ ( α + τ ) (β + τ ) (δ + τ )

 sβ = s(2× α+τ −1 )τ  sβ = g−1 δ +τ

2τ ( α + τ ) (β + τ ) (δ + τ )

= s



= s



2× (β2+τ τ(α)(+δτ+)τ ) −1 τ

2× (β2+ττ(α)(+δτ+)τ ) −1 τ δ+τ

  δ+τ 2τ ( α + τ ) −1 β + τ −1    sα  sβ  sδ = sα  g × = sα  s =g = s 2× (β +τ )(2δ +τ ) −1 τ 2× (β2+τ τ(α)(+δτ+)τ ) −1 τ 2τ 2τ (β + τ ) (δ + τ ) 4τ

and then, we get sα  sβ  sδ = sα  sδ  sβ = sα  (sβ  sδ ). 4. Some novel operational laws of HFLEs From Definition 2.5, we know that there exist two equivalent transformation functions between the HFLE and the HFE. However, there is a problem to be solved: Suppose that S = {s−3 = worst, s−2 = very bad, s−1 = bad, s0 = medium, s1 = good, s2 = very good, s3 = best } is a LTS, then some linguistic expressions can be shown as: l l1 = morethangood, l l2 = at least bad, and l l3 = betweenmediumand verygood. Meanwhile, we can also use HFLEs to express these linguistic expressions as hS1 = {s1 , s2 , s3 }, hS2 = {s−1 , s0 , s1 , s2 , s3 } and hS3 = {s0 , s1 , s2 }. Superficially, we find that these three HFLEs are discrete, but in fact, based on these linguistic expressions, the elements included in these HFLEs are continuous. Therefore, when we transform the HFLEs into the HFEs, the elements included in HFEs must be continuous. Once we understand this problem clearly, the operational laws of the HFLEs can be defined as follows: Definition

4.1. Let

S = {st |t = −τ , . . . , −1, 0, 1, . . . , τ}

S, l = 1, 2, . . . , #hS1 } and hS2 =

{s2

φl

be

|s2

a

LTS,

hS = {sφl |sφl ∈ S, l = 1, 2, . . . , #hS},

hS1 =

{s1φ |s1φ ∈ l

l

φl ∈ S, l = 1, 2, . . . , #hS2 } be three HFLEs (#hS , #hS1 and #hS2 are the numbers of

linguistic terms included in three HFLEs, respectively), g and g−1 be the equivalent transformation functions of HFLEs and HFEs, and λ be a real number. Then



(1) hS1  hS2 = g−1

 (3) λhS = g−1

γ1 ∈g(hS1 ),γ2 ∈g(hS2 )



{γ1 + γ2 − γ1 γ2 } ; 

{γ1 γ2 } ;

 {1 − (1 − γ )λ } ;

g−1

(2) hS1  hS2 =





γ1 ∈g(hS ),γ2 ∈g(hS )

γ ∈ g( h S )

1

2

X. Gou, Z. Xu / Information Sciences 372 (2016) 407–427

(4) (hS )λ = g−1



 λ {γ } ; γ ∈ g( h S ) 



417

 γ1 −γ2

, if γ1 ≥ γ2 and γ2 = 1 {θ } , where θ = 1−γ2 ; 0 , otherwise γ ∈ g( h ) , γ ∈ g( h )  1 S1 2 S2   γ1  , if γ1 ≤ γ2 and γ2 = 0 (6) hS1  hS2 = g−1 {θ } , where θ = γ2 ; 1, otherwise γ1 ∈g(hS1 ),γ2 ∈g(hS2 )

 (7) hS = g−1 {1 − γ } . (5) hS1  hS2 =



g−1

γ ∈ g( h S )

Remark 4. From (1) to (7), we can first transform the HFLEs into HFEs equivalently based on the function g, and then calculate these HFEs by using the operational laws of HFEs. Finally, by using the other function g−1 , we can transform these calculation values into the HFLEs equivalently. Additionally, we rank the calculation values in increasing order and omit the extra repeated elements. In the following, we give an example to verify the operational laws given in Definition 4.1: Example 4.1. Let S = {st |t = −3, . . . , −1, 0, 1, . . . , 3} be a LTS, hS1 = {s1 , s2 , s3 } and hS2 = {s−1 , s0 } be two HFLEs, λ = 2. Then, based on the function g, we obtain g(hS1 ) = { 23 , 56 , 1} and g(hS2 ) = { 13 , 12 }. In addition, we get



(1) hS1  hS2 =

g−1





 γ1 ∈g(hS1 ),γ2 ∈g(hS2 )



11 {γ1 + γ2 − γ1 γ2 } = g−1 ( 79 , 56 , 89 , 12 , 1) = {s1.67 , s2 , s2.33 , s2.5 , s3 }; 

5 1 5 1 {γ1 γ2 } = g−1 ( 29 , 18 , 3 , 12 , 2 ) = {s−1.67 , s−1.33 , s−1 , s−0.5 s0 };

 2 2 (3) λhS1 = g−1 {1 − (1 − γ )λ } = g(1 − (1 − 23 ) , 1 − (1 − 56 ) , 1 − (1 − 1 )2 ) = {s2.33 , s2.83 , s3 }; γ ∈ g( h S )

 2 2 (4) (hS )λ = g−1 {γ λ } = g{( 32 ) , ( 56 ) , (1 )2 } = {s−0.33 , s1.67 , s3 }; γ ∈ g( h S )  

(2) hS1  hS2 = g−1

γ1 ∈g(hS ),γ2 ∈g(hS ) 1

(5) hS1  hS2 = g−1





{ γ11−−γγ22 } = g−1 { 13 , 12 , 23 , 34 , 1} = {s−1 , s0 , s1 , s1.5 , s3 }; γ ∈ g( h ) , γ ∈ g( h )  1 S1 2 S2 

(6) hS2  hS1 = g−1 (7) hS1 = g−1

2



γ1 ∈g(hS1 ),γ2 ∈g(hS2 )





γ1 ∈g(hS )

{ γγ21 } = g−1 { 13 , 25 , 12 , 35 , 34 } = {s−1 , s−0.6 , s0 , s0.6 , s1.5 };

{1 − γ1 } = g−1 {1 − 23 , 1 − 56 , 1 − 1} = g−1 {0, 16 , 13 } = {s−3 , s−2 , s−1 };

1

= g−1 {1 − 13 , 1 − 12 } = g−1 { 12 , 23 } = {s0 , s1 }.

 hS2 = g−1



 γ2 ∈g(hS )

{1 − γ2 }

2

Furthermore, we investigate some properties of these operational laws below: Theorem 4.1. Let S = {st |t = −τ , . . . , −1, 0, 1, . . . , τ} be a LTS, hS , hS1 and hS2 be three HFLEs, and λ, λ1 and λ2 be three positive real numbers. Then (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12)

hS1  hS2 = hS2  hS1 ; hS1  hS2 = hS2  hS1 ; (hS1  hS2 )  hS2 = hS1 , if γ1 ≥ γ2 and γ2 = 1, for γ1 ∈ g(hS1 ), γ2 ∈ g(hS2 ); (hS1  hS2 )  hS2 = hS1 , if γ1 ≤ γ2 and γ2 = 0, for γ1 ∈ g(hS1 ), γ2 ∈ g(hS2 ); λ(hS1  hS2 ) = λhS1  λhS2 ; λ(hS1  hS2 ) = λhS1  λhS2 , if γ1 ≥ γ2 and γ2 = 1, for γ1 ∈ g(hS1 ), γ2 ∈ g(hS2 ); ( hS1  hS2 )λ = ( hS1 )λ  ( hS2 )λ ; (hS1  hS2 )λ = (hS1 )λ  (hS2 )λ ; if γ1 ≤ γ2 and γ2 = 0, for γ1 ∈ g(hS1 ), γ2 ∈ g(hS2 ); λ1 hS  λ2 hS = (λ1 + λ2 )hS ; λ1 hS  λ2 hS = (λ1 − λ2 )hS , if λ1 ≥ λ2 ; ( h S ) λ1  ( h S ) λ2 = ( h S ) λ1 + λ 2 ; (hS )λ1  (hS )λ2 = (hS )λ1 −λ2 , if λ1 ≥ λ2 ;

Proof. (1) and (2) are obvious, and we can omit the proofs of them here. (3) If γ1 ≥ γ2 and γ2 = 1, for γ1 ∈ g(hS1 ), γ2 ∈ g(hS2 ), then

( hS1  hS2 )  hS2

418

X. Gou, Z. Xu / Information Sciences 372 (2016) 407–427

⎛ = g−1 ⎝

⎛



 γ1 ∈g(hS1 ),γ2 ∈g(hS2 )

⎛ ⎞  γ1 − γ2 ⎠  hS2 = g−1 ⎝ 1 − γ2

γ1 ∈g(hS1 ),γ2 ∈g(hS2 )

⎞



= g−1 ⎝

γ1 ∈ g ( h S 1 )





⎞  γ1 − γ2 γ1 − γ2 + γ2 − × γ2 ⎠ 1 − γ2 1 − γ2

{γ1 }⎠ = hS1

(4) If γ1 ≤ γ2 and γ2 = 0, for γ1 ∈ g(hS1 ), γ2 ∈ g(hS2 ), then

⎛





(hS1  hS2 )  hS2 = g−1 ⎝

γ1 ∈g(hS1 ),γ2 ∈g(hS2 )

⎛

⎞



= g−1 ⎝

γ1 ∈ g ( h S 1 )

(5) λ(hS1  hS2 ) = λg−1 (

γ1 ∈g(hS ),γ2 ∈g(hS )





= g−1 ⎝

γ1 ∈ g ( h S 1 )



 γ1 ∈g(hS1 ),γ2 ∈g(hS2 )

⎞  γ1 × γ2 ⎠ γ2

{γ1 }⎠ = hS1

 1

⎛

⎛ ⎞  γ1 ⎠  hS2 = g−1 ⎝ γ2

{γ1 + γ2 − γ1 γ2 } ) = g−1 (

2

1 − (1 − γ1 )λ



 γ1 ∈g(hS ),γ2 ∈g(hS ) 1

⎛

⎞





⎠  g−1 ⎝

γ2 ∈ g ( h S 2 )

{1 − (1 − γ1 )λ (1 − γ2 )λ })

2

1 − (1 − γ2 )λ



⎞ ⎠ = λhS1  λhS2

(6) If γ1 ≥ γ2 and γ2 = 1, for γ1 ∈ g(hS1 ), γ2 ∈ g(hS2 ), then

⎛







⎛

⎞









1 − (1 − γ1 )λ − 1 − (1 − γ2 )λ

γ1 − γ2 ⎠ −1 ⎝ =g   1 − γ2 1 − 1 − (1 − γ2 )λ γ1 ∈g(hS1 ),γ2 ∈g(hS2 ) γ1 ∈g(hS1 ),γ2 ∈g(hS2 ) ⎞ ⎞ ⎛ ⎛     = g−1 ⎝ 1 − (1 − γ1 )λ ⎠  g−1 ⎝ 1 − (1 − γ2 )λ ⎠ = λhS1  λhS2 γ1 ∈ g ( h S 1 ) γ2 ∈ g ( h S 2 )

λ(hS1  hS2 ) = λg−1 ⎝

 ⎞ ⎠

(7) (hS1  hS2 )λ

⎛

⎛

⎞⎞λ



= ⎝g−1 ⎝

γ1 ∈g(hS1 ),γ2 ∈g(hS2 )

⎛ 

{γ1 γ2 }⎠⎠ = g−1 ⎝

γ1 ∈ g ( h S 1 )



λ

⎞

⎛

γ1 ⎠  g−1 ⎝





γ2 ∈ g ( h S 2 )

λ

⎞

γ2 ⎠ = (hS1 )λ  (hS2 )λ

(8) If γ1 ≤ γ2 and γ2 = 0, for γ1 ∈ g(hS1 ), γ2 ∈ g(hS2 ), then

⎞ ⎞ ⎛ ⎛ ⎞⎞λ      γ 1 ⎠⎠ = g−1 ⎝ γ1λ ⎠  g−1 ⎝ γ2λ ⎠ = (hS1 )λ  (hS2 )λ (hS1  hS2 ) = ⎝g−1 ⎝ γ2 γ1 ∈g(hS1 ),γ2 ∈g(hS2 ) γ1 ∈ g ( h S 1 ) γ2 ∈ g ( h S 2 ) ⎛

⎛



λ

(9) λ1 hS  λ2 hS = g−1 (

 −1

=g

 = g−1



 γ ∈ g( h S )



γ ∈g ( hS )

 γ ∈g ( hS )





{1 − (1 − γ )λ1 } )  g−1 (

λ1

1 − (1 − γ )

+ 1 − (1 − γ )

 γ ∈ g( h S )

λ2



{1 − (1 − γ )λ2 })

− 1 − (1 − γ )

 1 − (1 − γ )λ1 +λ2 = (λ1 + λ2 )hS

λ1



1 − (1 − γ )

λ2





X. Gou, Z. Xu / Information Sciences 372 (2016) 407–427

(10) If λ1 ≥ λ2 , then



λ1 hS  λ2 hS = g−1 





1 − (1 − γ )





=g

(11) (hS )λ1  (hS )λ2 = g−1 (12) If λ1 ≥ λ2 , then

( hS )  ( hS )



1 − (1 − γ )





=g

 = g−1

γ λ1











= (λ1 − λ2 )hS





−1

g

γ ∈g ( hS )





γ ∈g ( hS )

γ ∈ g( h S )





1 − (1 − γ )

λ2





  {γ λ1 }  g−1 {γ λ2 } = g−1 { γ λ1 + λ 2 } = ( h S ) λ1 + λ 2



γ ∈ g( h S )

−1







 g−1

λ1 −λ2

γ ∈g ( hS )

λ2



γ ∈g ( hS )

−1

λ1



λ1

419

γ λ1 −λ2





γ ∈ g( h S )



γ λ2





 −1

=g

γ ∈g ( hS )

 γ ∈g ( hS )



γ λ1 γ λ2



λ1 −λ2

= ( hS )

γ ∈g ( hS )

Theorem 4.2. Let S = {st |t = −τ , . . . , −1, 0, 1, . . . , τ} be a LTS, hS1 , hS2 and hS3 be three HFLEs, and λ be a positive real number. Then (1) hS1  hS2  hS3 = hS1  hS3  hS2 = hS1  (hS2  hS3 ), if γ1 ≥ γ2 + γ3 − γ2 γ3 and γ2 + γ3 − γ2 γ3 = 1 for γ1 ∈ g(hS1 ), γ2 ∈ g(hS2 ), and γ3 ∈ g(hS3 ). (2) hS1  hS2  hS3 = hS1  hS3  hS2 = hS1  (hS2  hS3 ), if γ1 ≤ γ2 γ3 and γ2 γ3 = 0 for γ1 ∈ g(hS1 ), γ2 ∈ g(hS2 ), and γ3 ∈ g( h S 3 ) . Proof. (1) If γ1 ≥ γ2 + γ3 − γ2 γ3 and γ2 + γ3 − γ2 γ3 = 1 for γ1 ∈ g(hS1 ), γ2 ∈ g(hS2 ), and γ3 ∈ g(hS3 ), then

⎛





hS1  hS2  hS3 = g−1 ⎝

γ1 ∈g(hS1 ),γ2 ∈g(hS2 )

⎛





= g−1 ⎝

γ1 ∈g(hS1 ),γ2 ∈g(hS2 ),γ3 ∈g(hS3 ),

⎛ hS1  hS3  hS2 = g−1 ⎝

⎞  γ1 − γ2 − γ3 + γ2 γ3 ⎠ = (1 − γ2 )(1 − γ3 )





γ1 ∈g(hS1 ),γ3 ∈g(hS3 )

⎛ 

= g−1 ⎝

γ1 ∈g(hS1 ),γ2 ∈g(hS2 ),γ3 ∈g(hS3 ),



hS1  hS2  hS3 = hS1  g−1 ⎝ 

 γ1 − γ2

− γ3 1 − γ2 1 − γ3

γ1 ∈g(hS1 ),γ2 ∈g(hS2 ),γ3 ∈g(hS3 ),





s

γ1 ∈g(hS1 ),γ2 ∈g(hS2 ),γ3 ∈g(hS3 ),



 γ −γ −γ +γ γ 2× 11−γ2 31−γ2 3 −1 τ ( 2 )( 3 )

 γ1 − γ3

− γ2 1 − γ3 1 − γ2

γ1 ∈g(hS1 ),γ2 ∈g(hS2 ),γ3 ∈g(hS3 ),



⎞  ⎠

⎞  ⎠

 s

γ1 ∈g(hS1 ),γ2 ∈g(hS2 ),γ3 ∈g(hS3 ),

  γ −γ −γ +γ γ 2× 11−γ2 31−γ2 3 −1 τ ( 2 )( 3 )

⎞ 

γ2 ∈ g ( h S 2 ) , γ3 ∈ g ( h S 3 )

⎛



γ1 ∈g(hS1 ),γ2 ∈g(hS2 ),γ3 ∈g(hS3 )

⎛ = g−1 ⎝

⎞   γ1 − γ2 − γ3 + γ2 γ3 ⎠ = (1 − γ2 )(1 − γ3 )





⎛ ⎞  γ1 − γ3 ⎠  hS2 = g−1 ⎝ 1 − γ3

⎛



= g−1 ⎝

⎛ ⎞  γ1 − γ2 ⎠  hS3 = g−1 ⎝ 1 − γ2



γ1 ∈g(hS1 ),γ2 ∈g(hS2 ),γ3 ∈g(hS3 ),

{γ2 + γ3 − γ2 γ3 }⎠ 

⎞

γ1 − (γ2 + γ3 − γ2 γ3 ) ⎠ 1 − (γ2 + γ3 − γ2 γ3 )



⎞  γ1 − γ2 − γ3 + γ2 γ3 ⎠ = (1 − γ2 )(1 − γ3 )

 γ1 ∈g(hS1 ),γ2 ∈g(hS2 ),γ3 ∈g(hS3 ),

 s

  γ −γ −γ +γ γ 2× 11−γ2 31−γ2 3 −1 τ ( 2 )( 3 )

and thus, we obtain hS1  hS2  hS3 = hS1  hS3  hS2 = hS1  (hS2  hS3 ) when γ1 ≥ γ2 + γ3 − γ2 γ3 and γ2 + γ3 − γ2 γ3 = 1 for γ1 ∈ g(hS1 ), γ2 ∈ g(hS2 ), and γ3 ∈ g(hS3 ).

420

X. Gou, Z. Xu / Information Sciences 372 (2016) 407–427

(2) If γ1 ≤ γ2 γ3 , γ2 γ3 = 0 for γ1 ∈ g(hS1 ), γ2 ∈ g(hS2 ), and γ3 ∈ g(hS3 ), then

hS1  hS2  hS3

⎛

⎛ ⎞ ⎞     γ γ 1 ⎠ 1 ⎠ = g−1 ⎝  sγ = g−1 ⎝ γ2 γ2 γ3 γ1 ∈g(hS1 ),γ2 ∈g(hS2 ) γ1 ∈g(hS1 ),γ2 ∈g(hS2 ),γ3 ∈g(hS3 )   





=

s

γ1 ∈g(hS1 ),γ2 ∈g(hS2 ),γ3 ∈g(hS3 )

γ



2× γ 1γ −1 τ 2 3

hS1  hS3  hS2

⎛ = g−1 ⎝







⎛

⎞



γ1 ⎠  hS2 = g−1 ⎝ γ3 γ1 ∈g(hS1 ),γ3 ∈g(hS3 ) γ1 ∈g(hS1 ),γ2 ∈g(hS2 ),γ3 ∈g(hS3 )   

=

s

γ1 ∈g(hS1 ),γ2 ∈g(hS2 ),γ3 ∈g(hS3 )



hS1  hS2  hS3

γ

⎞  γ1 ⎠ γ2 γ3



2× γ 1γ −1 τ 2 3

 ⎛

⎞ 

= hS1  g−1 ⎝

γ2 ∈g(hS2 ),γ3 ∈g(hS3 )

⎛

s

γ1 ∈g(hS1 ),γ2 ∈g(hS2 ),γ3 ∈g(hS3 )



{γ2 γ3 }⎠ = g−1 ⎝





=



 γ

γ1 ∈g(hS1 ),γ2 ∈g(hS2 ),γ3 ∈g(hS3 )



⎞  γ1 ⎠ γ2 γ3



2× γ 1γ −1 τ 2 3

Therefore, we get hS1  hS2  hS3 = hS1  hS3  hS2 = hS1  (hS2  hS3 ) when γ1 ≤ γ2 γ3 and γ2 γ3 = 0 for γ1 ∈ g(hS1 ), γ2 ∈ g(hS2 ), and γ3 ∈ g(hS3 ). 5. Some novel operational laws of PLTSs As we have discussed previously, the operation results by using the operational laws of Pang et al. [21] may exceed the bounds of LTSs, or the corresponding probability information may be lost after operations. To avoid this issue, in the following we propose some novel operational laws for PLTSs based on the equivalent transformation functions g and g−1 : Definition 5.1. Let S = {st |t = −τ , . . . , −1, 0, 1, . . . , τ} be a LTS, L( p), L1 ( p) and L2 ( p) be three PLTSs, and λ be a positive real numbers. Then (1) L1 ( p)  L2 ( p)

⎛





= g−1 ⎝

η1(i) + η2( j ) − η1(i) η2( j )

η1(i) ∈g(L1 ),η2( j ) ∈g(L2 )

⎛

(2) L1 ( p)  L2 ( p) = g−1 ⎝

η1(i ) ∈g(L1 ),η2( j ) ∈g(L2 )

 (3) λL( p) =



g−1

(4) (L( p) )λ = g−1



( i ) ∈ g( L )

η 



⎞  i j p(1 ) p(2 ) ⎠, i = 1, 2, . . . , #L1 ( p), j = 1, 2, . . . , #L2 ( p);

⎞    η1(i) η2( j ) p(1i) p(2j ) ⎠, i = 1, 2, . . . , #L1 ( p), j = 1, 2, . . . , #L2 ( p); 

λ {(1 − (1 − η (i) ) )( p(i) )} , i = 1, 2, . . . , #L( p);

 η ( i ) ∈ g( L )

 λ {(η (i) ) ( p(i) )} , i = 1, 2, . . . , #L( p);

X. Gou, Z. Xu / Information Sciences 372 (2016) 407–427

⎛ (5) L1 ( p)  L2 ( p) = g−1 ⎝

 =

( j)

η1 −η2 , j 1−η2( )

( j)

0,

i

⎛

j

j

otherwise

⎞

 ( j)

(i )

η1 ∈g(L1 ),η2 ∈g(L2 ) η1(i )

η2( j )

( j)

1,



(7) L( p) =

where

=

;

otherwise





g−1

{ ( p(1i) p(2j ) )}⎠, i = 1, 2, . . . , #L1 ( p), j = 1, 2, . . . , #L2 ( p),

( j)

if η1(i ) ≤ η2 and η2 = 0

,

{ ( p(1i) p(2j ) )}⎠, i = 1, 2, . . . , #L1 ( p), j = 1, 2, . . . , #L2 ( p), where

if η1( ) ≥ η2( ) and η2( ) = 1;

(6) L1 ( p)  L2 ( p) = g−1 ⎝



 (i )

η1 ∈g(L1 ),η2 ∈g(L2 ) (i )

421

⎞

{(1 − η (i) )( p(i) )} , i = 1, 2, . . . , #L( p).

η ( i ) ∈ g( L )

Remark 5. From the above seven operational laws of PLTSs, we can derive some conclusions as follows: (1) Combining the same LTSs together. For example, if there are two operation values s2 (0.1 ) and s2 (0.2 ), then we can combine them into s2 (0.3 ) by calculating the sum of two probabilities. (2) In the final results, we rank all the LTSs in increasing order. (3) In the final results, the probabilities information cannot be lost. (4) The equivalent transformation functions g−1 and g1 are only related to the LTSs, and have nothing to do with the probabilities. An example can be given to show the operational laws of PLTSs: Example 5.1. Let S = {st |t = −3, . . . , −1, 0, 1, . . . , 3} be a LTS, L1 ( p) = {s1 (0.3 ), s2 (0.2 ), s3 (0.5 )} and L2 ( p) = 0.2 0.3 {s−1 (0.2), s0 (0.3)} be two PLTSs, and λ = 2. Firstly, we need to normalize L2 ( p), and get L2 ( p) = {s−1 ( 0.2+0 .3 ), s0 ( 0.2+0.3 )} = {s−1 (0.4), s0 (0.6)}, then

⎛

(1) L1 ( p)  L2 ( p) = g−1 ⎝

= g−1

 ( j)

(i )

η1 ∈g(L1 ),η2 ∈g(L2 )

7

{(η1(i) + η2( j ) − η1(i) η2( j ) )( p(1i) p(2j ) )}⎠, i = 1, 2, 3, j = 1, 2

 5 8 11 (0.12), (0.18), (0.08), (0.12), 1(0.5 ) = {s1.67 (0.12), s2 (0.18), s2.33 (0.08), s2.5 (0.12), s3 (0.5)} 9 6 9 12 ⎛ ⎞

(2) L1 ( p)  L2 ( p) = g−1 ⎝

= g−1

⎞

 ( j)

(i )

η1 ∈g(L1 ),η2 ∈g(L2 )

2

{(η1(i) η2( j ) )( p(1i) p(2j ) )}⎠, i = 1, 2, 3, j = 1, 2

 5 1 5 1 (0.12), (0.08), (0.38), (0.12), (0.3) = {s−1.67 (0.12), s−1.33 (0.18), s−1 (0.08), s−0.5 (0.12), s0 (0.5)} 9 18 3 12 2   

(3) 2L1 ( p) = g−1

= g−1

2

η ( i ) ∈ g( L

8

1)

{(1 − (1 − η (i) ) )( p(i) )} , i = 1, 2, 3

 35 (0.3), (0.2 ), 1(0.5) = {s2.33 (0.3), s2.83 (0.2), s3 (0.5)} 9 36  

(4) (L1 ( p) )2 = g−1



2

η ( i ) ∈ g( L 1 )

−1

=g

( j)

(5) Since η1(i ) ≥ η2

  2 2 3

{(η (i) ) ( p(i) )} , i = 1, 2, 3

( 0 . 3 ),

 5 2 6

( j)

( j)

and η2 = 1 for η1(i ) ∈ g(L1 ) and η2 ∈ g(L2 ), i = 1, 2, 3, j = 1, 2, then

⎛

1 2





L1 ( p)  L2 ( p) = g−1 ⎝ = g−1

 (0.2 ), (1 ) (0.5) = {s−0.33 (0.3 ), s1.17 (0.2), s3 (0.5)} 2

(i ) ∈g 1

η

( j) ∈g 2

(L1 ),η

( L2 )

(i )

( j ) 

η1 − η2 j 1 − η2( )

p(1 ) p(2 ) i

j



⎞

⎠, i = 1, 2, 3, j = 1, 2

 1 3 2 (0.12), (0.18), (0.08), (0.12), 1(0.5) = {s−1 (0.18), s0 (0.12), s1 (0.12), s1.5 (0.08), s3 (0.5)} 3

4

3

422

X. Gou, Z. Xu / Information Sciences 372 (2016) 407–427

( j)

(6) Since η1(i ) ≥ η2

( j)

( j)

and η1 = 0 for η1(i ) ∈ g(L1 ) and η2 ∈ g(L2 ), i = 1, 2, 3, j = 1, 2, then

⎛





L2 ( p)  L1 ( p) = g−1 ⎝

η1(i) ∈g(L1 ),η2( j ) ∈g(L2 )

= g−1 (7) L1 ( p) = g−1 (

= g−1



1 3

 2 1 3 3 (0.2), (0.08), (0.42), (0.12), (0.18) = {s−1 (0.2 ), s−0.6 (0. ), s0 (0.42), s0.6 (0.12), s1.5 (0.18)} 5



2

(i )

η1(i ) ∈g(L1 )

1−

⎞    η2 i j p(1 ) p(2 ) ⎠, i = 1, 2, 3, j = 1, 2 η1( j ) (i )

5

4

(i )

{(1 − η1 )( p1 )}), i = 1, 2, 3













2 5 1 1 ( 0 . 3 ), 1 − (0.2 ), (1 − 1 )(0.5) = 0(0.5), (0.2 ), (0.3) = {s−3 (0.5 ), s−2 (0.2), s−1 (0.3 )} 3 6 6 3

Theorem 5.1. Let S = {st |t = −τ , . . . , −1, 0, 1, . . . , τ} be a LTS, hS , hS1 and hS2 be three HFLEs, and λ, λ1 and λ2 be three positive real numbers. Then (1) (2) (3) (4) (5) (6) (7) (8) (9) (10)

L1 ( p )  L2 ( p ) = L2 ( p )  L1 ( p ); L1 ( p )  L2 ( p ) = L2 ( p )  L1 ( p ); λ(L1 ( p)  L2 ( p) ) = λL2 ( p)  λL1 ( p); λ(L1 ( p)  L2 ( p) ) = λL2 ( p)  λL1 ( p), if η1(i) ≥ η2( j ) and η2( j ) = 1 for η1(i) ∈ g(L1 ) and η2( j ) ∈ g(L2 ), i = 1, 2, . . . , #L1 ( p), j = 1, 2, . . . , #L2 ( p ); ( L1 ( p )  L2 ( p ) )λ = ( L1 ( p ) )λ  ( L2 ( p ) )λ ; (L1 ( p)  L2 ( p) )λ = (L1 ( p) )λ  (L2 ( p) )λ ; if η1(i) ≤ η2( j ) and η2( j ) = 0 for η1(i) ∈ g(L1 ) and η2( j ) ∈ g(L2 ), i = 1, 2, . . . , #L1 ( p ), j = 1, 2, . . . , #L2 ( p ); λ1 L( p)  λ2 L( p) = (λ1 + λ2 )L( p), if i = j for η (i) , η ( j ) ∈ g(L ) and i = 1, 2, . . . , #L( p); λ1 L( p)  λ2 L( p) = (λ1 − λ2 )L( p), if λ1 ≥ λ2 and i = j for η (i) , η ( j ) ∈ g(L ) and i = 1, 2, . . . , #L( p); (L( p))λ1  (L( p))λ2 = (L( p))λ1 +λ2 , if i = j for η (i) , η ( j ) ∈ g(L ) and i = 1, 2, . . . , #L( p); (L( p))λ1  (L( p))λ2 = (L( p))λ1 −λ2 , if λ1 ≥ λ2 and i = j for η (i) , η ( j ) ∈ g(L ) and i = 1, 2, . . . , #L( p).

Proof. (1) and (2) are obvious and we omit the proofs of them here. (3) λ(L1 ( p)  L2 ( p) )

⎛

=

λg−1 ⎝ (i )

η1(i) + η2( j ) − η1(i) η2( j )



⎞  i j p(1 ) p(2 ) ⎠

( j)

η1 ∈g(L1 ),η2 ∈g(L2 )

⎛





= g−1 ⎝ (i ) ∈g 1

η

⎛





= g−1 ⎝

1 − 1 − η1( )

( j) ∈g 2

(L1 ),η







i

λ 

1 − η2( ) j

( L2 )



1 − 1 − η1( ) i

λ 

p(1 ) i



⎞

( j)

p(1 ) p(2 ) i

⎛

⎠  g−1 ⎝

η1(i) ∈g(L1 )

(4) If η1(i ) ≥ η2

λ 



j





⎞ ⎠ 

1 − 1 − η2( ) j

λ 

p(2 ) j



⎞ ⎠ = λL2 ( p)  λL1 ( p)

η2( j ) ∈g(L2 ) ( j)

( j)

and η2 = 1 for η1(i ) ∈ g(L1 ) and η2 ∈ g(L2 ), i = 1, 2, . . . , #L1 ( p), j = 1, 2, . . . , #L2 ( p), then

λ ( L1 ( p )  L2 ( p ) ) ⎞  (i ) ( j )  η − η i j 1 2 = λg−1 ⎝ p(1 ) p(2 ) ⎠ j 1 − η2( ) (i ) ( j) η1 ∈g(L1 ),η2 ∈g(L2 ) ⎧ ⎫⎞ ⎛ λ  λ ( j) (i ) ⎪ ⎪ ⎨ ⎬   1 − η − 1 − η  2 1 ⎜ ⎟ i j = g−1 ⎝ p(1 ) p(2 ) ⎠   λ ⎪ ⎪ ( j) (i ) ( j) ⎩ ⎭ η1 ∈g(L1 ),η2 ∈g(L2 ) 1 − η2 ⎞ ⎞ ⎛ ⎛    λ    λ     i i i i = g−1 ⎝ 1 − 1 − η1( ) p(1 ) ⎠  g−1 ⎝ 1 − 1 − η2( ) p(2 ) ⎠ = λL2 ( p)  λL1 ( p) ⎛



η1(i) ∈g(L1 )



η2(i) ∈g(L2 )

X. Gou, Z. Xu / Information Sciences 372 (2016) 407–427

(5) (L1 ( p)  L2 ( p) )λ

⎛

=

⎛

⎝g−1 ⎝

η1(i) ∈g(L1 ),η2( j ) ∈g(L2 )

⎛ =





⎛





⎝g−1 ⎝

η1(i)



η1(i) η2( j )



⎞⎞λ ⎛  i j p(1 ) p(2 ) ⎠⎠ = g−1 ⎝

( j)

p(1 ) p(2 ) i

j

( j)



η2( j )



⎛





η1(i) ∈g(L1 ),η2( j ) ∈g(L2 )

⎛ ⎞⎞λ     η1 ⎜ i j p(1 ) p(2 ) ⎠⎠ = g−1 ⎝ ( j) η2 η (i) ∈g(L ),η ( j ) ∈g(L

(i )

(7) λ1 L( p)  λ2 L( p) =





g−1

η ( i ) ∈ g( L )





−1

=g

1−

λ1  λ2 1 − η (i ) 1 − η( j)

η (i) ∈g(L ),η ( j ) ∈g(L )









η2( j )

λ 

2

⎞  j p(2 ) ⎠ = (L1 ( p) )λ  (L2 ( p) )λ

η12 ∈g(L2 )



λ {(1 − (1 − η (i) ) 1 )( p(i) )}



1

1

⎫⎞ ⎧  λ ⎪ ⎨ η1(i)  ⎬ ⎪ ⎟ (i ) ( j )  λ p1 p2 ⎪⎠ ⎪ ⎩ η (i ) ⎭ 2) 2

( j)

η1 ∈g(L1 )



1− 1−η

  (i ) λ1 +λ2

 (i )



 g−1



p(i ) p( j )



 η ( j ) ∈ g( L )



λ {(1 − (1 − η ( j ) ) 2 )( p( j ) )}



 = (λ1 + λ2 )L( p)

p

η (i) ∈g(L ),i= j

(8) If λ1 ≥ λ2 and i = j for η (i ) , η ( j ) ∈ g(L ) and i = 1, 2, . . . , #L( p);



λ1 L( p)  λ2 L( p) = g

−1





1− 1−η

η ( i ) ∈g ( L )

 



=g







−1

 

(i ) λ1

1 − η( j)



η (i) ∈g(L ),η ( j ) ∈g(L )





−1

=g

1−



(i )

p

λ2







g

λ1 +λ2 1 − η (i )





λ1 

λ2

p(i ) p( j )



(L( p))λ1  (L( p))λ2

 = g−1

 = g−1





λ   η (i) 1 p(i)



 −1

g

η ( i ) ∈g ( L )





λ   η ( j ) 2 p( j )



η ( j ) ∈g ( L )





η

  (i ) λ1

η (i) ∈g(L ),η ( j ) ∈g(L )







η

  (i ) λ1 +λ2

η

  ( j ) λ2

   (i )

p

   ( j)

p(i ) p

= (L( p) )λ1 +λ2

η (i) ∈g(L ),i= j

(10) If λ1 ≥ λ2 , and i = j for η (i ) , η ( j ) ∈ g(L ) and i = 1, 2, . . . , #L( p), then

(L( p))λ1  (L( p))λ2



1− 1−η



p(i ) p( j )







(9) If i = j for η (i ) , η ( j ) ∈ g(L ) and i = 1, 2, . . . , #L( p), then





η ( j ) ∈g ( L )

− 1 − η (i )

1 − η( j)



−1

η (i) ∈g(L ),i= j

=g

⎠

⎞⎞λ  j p(2 ) ⎠⎠ = (L1 ( p) )λ  (L2 ( p) )λ

(i )

⎞ ⎛   λ     i = g−1 ⎝ p(1 ) ⎠  g−1 ⎝ η1(i)



⎞

( j)

⎛

−1



and η2 = 0 for η1(i ) ∈ g(L1 ) and η2 ∈ g(L2 ), i = 1, 2, . . . , #L1 ( p), j = 1, 2, . . . , #L2 ( p), then

(L1 ( p)  L2 ( p) )λ = ⎝g−1 ⎝

= g−1

η2( j )

λ 

η2 ∈g(L2 )

⎛



η1(i)

λ 

( j)

η1 ∈g(L1 )

(6) If η1(i ) ≤ η2



 η1(i) ∈g(L1 ),η2( j ) ∈g(L2 )

⎞⎞λ ⎛ ⎛   i p(1 ) ⎠⎠  ⎝g−1 ⎝

(i )

423

= (λ1 − λ2 )L( p)

 

( j ) λ2

( j)

p





424

X. Gou, Z. Xu / Information Sciences 372 (2016) 407–427

 =g





λ   η (i) 1 p(i)



−1



 g

η ( i ) ∈g ( L )



λ   η ( j ) 2 p( j )







η

  (i ) λ1 −λ2

   (i )

p





−1

=g

η ( j ) ∈g ( L )



= g−1







−1

η (i) ∈g(L ),η ( j ) ∈g(L )



η (i )

λ1

η( j)

λ2



p(i ) p( j )





= (L( p) )λ1 −λ2

η (i) ∈g(L ),i= j

Theorem 5.2. Let S = {st |t = −τ , . . . , −1, 0, 1, . . . , τ} be a LTS, hS1 , hS1 and hS2 be three HFLEs, and λ be a positive real number. Then ( j)

( j)

(1) L1 ( p)  L2 ( p)  L3 ( p) = L1 ( p)  L3 ( p)  L2 ( p) = L1 ( p)  (L2 ( p)  L3 ( p) ), if η1(i ) ≥ η2 + η3(k ) − η2 η3(k ) (k )

( j ) (k )

( j)

(i )

(k )

η3 − η2 η3 = 1 for η1 ∈ g(L1 ), η2 ∈ g(L2 ) and η3 ∈ g(L3 ).

( j)

( j)

and η2 +

( j)

(2) L1 ( p)  L2 ( p)  L3 ( p) = L1 ( p)  L3 ( p)  L2 ( p) = L1 ( p)  (L2 ( p)  L3 ( p) ), if η1(i ) ≤ η2 η3(k ) and η2 η3(k ) = 0 for η1(i ) ∈ ( j)

(k )

g(L1 ), η2 ∈ g(L2 ) and η3 ( j)

∈ g( L 3 ) .

( j)

( j)

( j)

( j)

Proof. (1) If η1(i ) ≥ η2 + η3(k ) − η2 η3(k ) and η2 + η3(k ) − η2 η3(k ) = 1 for η1(i ) ∈ g(L1 ), η2 ∈ g(L2 ) and η3(k ) ∈ g(L3 ), then

⎛

L1 ( p)  L2 ( p)  L3 ( p) = g−1 ⎝





η1(i) ∈g(L1 ),η2( j ) ∈g(L2 )

⎛

⎞  η1(i) − η2( j )  (i) ( j )  ⎠ p1 p2  L3 ( p ) j 1 − η2( ) 



= g−1 ⎝

η1(i) ∈g(L1 ),η2( j ) ∈g(L2 ),η3(k) ∈g(L3 )

⎛ L1 ( p)  L3 ( p)  L2 ( p) = g−1 ⎝





η1(i) ∈g(L1 ),η3(k) ∈g(L3 )

⎛

⎞  η1(i) − η3(k)  (i) (k)  ⎠ p1 p3  L2 ( p ) k 1 − η3( ) 



= g−1 ⎝

⎞  η1( j ) − η2( j ) − η3(k) + η2( j ) η3(k)  (i) ( j ) (k)  ⎠ p1 p2 p3 j k j k 1 − η2( ) − η3( ) + η2( ) η3( )

η1(i) ∈g(L1 ),η2( j ) ∈g(L2 ),η3(k) ∈g(L3 )

⎞  η1( j ) − η2( j ) − η3(k) + η2( j ) η3(k)  (i) ( j ) (k)  ⎠ p1 p2 p3 j k j k 1 − η2( ) − η3( ) + η2( ) η3( )

⎛





L1 ( p)  (L2 ( p)  L3 ( p) ) = L1 ( p)  g−1 ⎝

η

η2( j ) ∈g(L2 ),η3(k) ∈g(L3 )

⎛





= g−1 ⎝

η1(i) ∈g(L1 ),η2( j ) ∈g(L2 ),η3(k) ∈g(L3 )

( j) 2

+η

(k ) 3

−η

( j ) (k ) 2 3

η



j k p(2 ) p(3 )

⎞  ⎠

⎞ 

η1( j ) − η2( j ) − η3(k) + η2( j ) η3(k)  (i) ( j ) (k)  ⎠ p1 p2 p3 j k j k 1 − η2( ) − η3( ) + η2( ) η3( ) ( j)

and thus, we obtain L1 ( p)  L2 ( p)  L3 ( p) = L1 ( p)  L3 ( p)  L2 ( p) = L1 ( p)  (L2 ( p)  L3 ( p) ) when η1(i ) ≥ η2 + η3(k ) − ( j ) (k )

( j)

( j ) (k )

(k )

(i )

( j)

(k )

η2 η3 and η2 + η3 − η2 η3 = 1 for η1 ∈ g(L1 ), η2 ∈ g(L2 ) and η3 ∈ g(L3 ). ( j)

( j)

( j)

(2) If η1(i ) ≤ η2 η3(k ) and η2 η3(k ) = 0 for η1(i ) ∈ g(L1 ), η2 ∈ g(L2 ) and η3(k ) ∈ g(L3 ), then

L1 ( p )  L2 ( p )  L3 ( p )

⎛

= g−1 ⎝





η1(i) ∈g(L1 ),η2( j ) ∈g(L2 )

L1 ( p )  L3 ( p )  L2 ( p )

⎛

= g−1 ⎝



η1(i) ∈g(L1 ),η3(k) ∈g(L3 )

⎞ ⎛  η1(i)  (i) ( j )  ⎠ p1 p2  L3 ( p) = g−1 ⎝ η2( j ) η ( i ) ∈g ( L 1



 1

⎞ ⎛    η1 i k p(1 ) p(3 ) ⎠  L2 ( p) = g−1 ⎝ η3(k) η ( i ) ∈g ( L

),η2( j ) ∈g(L2 ),η3( j ) ∈g(L3 )

(i )

1



 1

),η2( j ) ∈g(L2 ),η3(k) ∈g(L3 )



⎞  η1(i)  (i) ( j ) (k)  ⎠ p1 p2 p3 η2( j ) η3(k)

(i )

η1 η2( j ) η3(k)



⎞  i j k p(1 ) p(2 ) p(3 ) ⎠

X. Gou, Z. Xu / Information Sciences 372 (2016) 407–427

425

Table 1 The evaluation values of four hospitals.

A1 A2 A3 A4

C1

C2

C3

{s0 , s1 } {s2 , s3 } {s1 } {s2 , s3 }

{s2 } {s0 } {s1 , s2 } {s−2 , s−1 , s0 , s1 }

{s−1 , s0 } {s1 , s2 , s3 } {s2 , s3 } {s1 }

Table 2 The final results based two operational laws. Alternatives

HF LWA(hS1 , hS2 , . . . , hSn )

Ranking order

Utilize the existing operational laws

A1 A2 A3 A4

{s−0.5 , s−0.3 , s0.2 , s0.4 } { s1.1 , s1.3 , s1.8 , s2 , s2.5 , s2.7 } { s1.7 , s1.8 , s2.4 , s2.5 } { s0.9 , s1 , s1.1 , s1.2 , s1.3 , s1.4 }

A3 A2 A4 A1

A3

Utilize the novel operational laws

A1 A2 A3 A4

{s−0.52 , s−0.25 , s0.12 , s0.34 } {s1.19 , s1.88 , s3 } {s1.75 , s1.85 , s3 } {s−1.03 , s−0.91 , s−0.76 , s1.09 , s3 }

A3 A2 A4 A1

A3

⎛ L1 ( p)  (L2 ( p)  L3 ( p) ) = L1 ( p)  g−1 ⎝

⎛





η2 η3

η2( j ) ∈g(L2 ),η3(k) ∈g(L3 )



= g−1 ⎝

( j ) (k )



η1(i) ∈g(L1 ),η2( j ) ∈g(L2 ),η3(k) ∈g(L3 )



The optimal alternative

⎞  ⎠ p2 p3 ( j ) (k )

⎞

  η1(i) i j k p(1 ) p(2 ) p(3 ) ⎠ η2( j ) η3(k) ( j)

Therefore, we get L1 ( p)  L2 ( p)  L3 ( p) = L1 ( p)  L3 ( p)  L2 ( p) = L1 ( p)  (L2 ( p)  L3 ( p) ) when η1(i ) ≤ η2 η3(k ) and

( j ) (k )

(i )

( j)

(k )

η2 η3 = 0 for η1 ∈ g(L1 ), η2 ∈ g(L2 ) and η3 ∈ g(L3 ). 6. A case study

In this section, we use an example (adapted from Ref. [7]) to show the novel operational laws of HFLEs and PLTSs based on the aggregation operator and the TOPSIS methods [21], respectively. Example 6.1. [7]. The environmental pollution in China has become more and more serious because of some extreme weather conditions, such as haze, etc. With the limited medical resources, China has to face a tough problem, i.e., how to optimize the resource allocation and enhance the benefit about resource input and output. In recent years, several domestic hospitals in China have started to do some work on it. In order to evaluate these hospitals, we should consider three main criteria: (1) the environmental factor of medical and health service (C1 ); (2) personalized diagnosis and treatment optimization (C2 ); and (3) social resource allocation optimization under the pattern of wisdom medical and health services (C3 ). The weight vector of these three criteria is w = (0.2, 0.1, 0.7)T . There are four hospitals to be evaluated, i.e., the West China Hospital of Sichuan University (A1 ), the Huashan Hospital of Fudan University (A2 ), the Union Medical College Hospital (A3 ) and the Chinese PLA General Hospital (A4 ). Based on the three main criteria and the LTS S = {st |t = −3, −2, −1, 0, 1, −1, 3}, the invited experts use the HFLEs to provide their evaluations for these hospitals, and then construct the hesitant fuzzy linguistic decision matrix h = (hSi j )m×n as shown in Table 1. Based on the existing operational laws [35] and the novel operational laws of HFLEs, respectively, this problem can be solved by using the hesitant fuzzy linguistic weighted averaging (HFLWA) operator:

HF LWA(hS1 , hS2 , . . . , hSn ) =

n

j=1

w j hS j =





n

k1 ∈hS1 ,k2 ∈hS2 ,...,kn ∈hSn



w jk j

j=1

and the final results are shown in Table 2. From the numerical results in Table 2, we can make some analyses as follows: (1) We can obtain the same ranking and optimal alternative. However, the aggregated values of each alternative by using two kinds of operational laws are totally different. For example, when using the existing operational laws, we have

426

X. Gou, Z. Xu / Information Sciences 372 (2016) 407–427 Table 3 The evaluation values of four hospitals with PLTSs.

A1 A2 A3 A4

C1

C2

C3

{s0 (0.4), s1 (0.6)} {s2 (0.3), s3 (0.7)} { s 1 ( 1 )} {s2 (0.4), s3 (0.4)}

{ s 2 ( 1 )} {s0 (0.8)} {s1 (0.5), s2 (0.5)} {s−2 (0.4), s−1 (0.1), s0 (0.2), s1 (0.3)}

{s−1 (0.2), s0 (0.8)} {s1 (0.2 ), s2 (0.4), s3 (0.4)} {s2 (0.6), s3 (0.4)} {s1 (0.9)}

Table 4 The final results based two operational laws. Alternatives

d ( Ai , L ( p )+ )

d ( Ai , L ( p )− )

CI (xi )

Ranking order

Utilize the existing operational laws

A1 A2 A3 A4

0.1643 0.0759 0.1937 0.1794

0.0750 0.1797 0.0465 0.0841

−1.7490 0 −2.2947 −1.8969

A2 A1 A4 A3

A2

Utilize the novel operational laws

A1 A2 A3 A4

0.3851 0.0964 0.1503 0.2993

0.0890 0.3642 0.3618 0.1776

−3.7524 0 −0.5666 −2.6185

A2 A3 A4 A1

A2

The optimal alternative

0.1s2  0.7s3 = s0.2  s2.1 = s2.3 , but when using the novel operational laws, we get 0.1s2  0.7s3 = s−2.02  s3 = s3 . There are several unreasonable aspects about the existing operational laws. Firstly, for the multiplication between the real numbers and the LTSs, the aggregated values are always over s0 . In fact, if the real numbers reduce gradually, then the aggregated values decrease and approach to the lower bound s−3 . Secondly, based on the operational laws of HFSs, the aggregated values are always 1 no matter what real numbers multiply or add the HFS {1}. Therefore, the aggregated values must be s3 no matter what real numbers multiply or add the HFLE {s3 } considering that the HFS {1} and the HFLE {s3 } are equivalent. So the novel operational laws are much more reasonable than the existing ones, and fit for people’s habits. (2) When we deal with the problem by using the HFLWA operator, we can reduce the number of LTSs of the final aggregated values by using the novel operational laws. If we need to deal with the problem with amounts of HFLEs or lots of LTSs included in the HFLEs, the novel operational laws can also effectively reduce the computational complexity. Besides, If the experts provide their evaluations expressed as PLTSs (see Table 3), then we can utilize the TOPSIS method proposed in Ref. [21] to deal with this problem by using two different operational laws, and the final results are shown in Table 4. Obviously, we obtain different ranking orders but have the same optimal alternative. The reason is that the existing operational laws have some unreasonable aspects when calculating the multiplications of real numbers and LTSs. Additionally, in the process of calculation, the computational work is much smaller by using the novel operational laws. 7. Conclusions In this paper, we have used two equivalent transformation functions to define some novel operational laws for linguistic terms, HFLEs and PLTSs. These operational laws can not only avoid the operation values exceeding the bounds of LTSs, but also keep the probability information complete after operations. Then, we have verified the novel operational laws of HFLEs and PLTSs through a practical example, which involves the evaluations of some hospitals in China. We have also summarized the advantages of the novel operational laws by comparing them with the existing ones. In the future, we can also define the operational laws for some other extended LTSs, such as interval-valued HFLTS, 2-tuple LTS and unbalanced LTSs, and apply them to the fields of medical diagnosis, pattern recognition, and supply chain management, etc. Acknowledgments The authors would like to thank the editors and the anonymous referees for their insightful and constructive comments and suggestions that have led to this improved version of the paper. The work was supported in part by the National Natural Science Foundation of China (Nos. 71502235, 61273209 and 71532007), and the Central University Basic Scientific Research Business Expenses Project (No. skgt201501). References [1] M. Antonelli, P. Ducange, B. Lazzerini, F. Marcelloni, Multi-objective evolutionary design of granular rule-based classifiers, Granul. Comput. 1 (1) (2016) 37–58.

X. Gou, Z. Xu / Information Sciences 372 (2016) 407–427 [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] [35] [36] [37] [38] [39] [40] [41] [42]

427

I. Beg, T. Rashid, TOPSIS for hesitant fuzzy linguistic term sets, Int. J. Intell. Syst. 28 (2013) 1162–1171. D. Ciucci, Orthopairs and granular computing, Granul. Comput. 1 (3) (2016) 159–170. D. Dubois, H. Prade, Bridging gaps between several forms of granular computing, Granul. Comput. 1 (2) (2016) 115–126. F.J. Estrella, R.M. Rodriguez, L. Martinez, A hesitant linguistic fuzzy TOPSIS approach integrated into FLINTSTONES, in: 16th World Congress of the International-Fuzzy-Systems-Association (IFSA)/9th Conference of the European-Society-for-Fuzzy-Logic-and-Technology (EUSFLAT), 89, Gijon, SPAIN, 2015, pp. 799–806. B. Farhadinia, Multiple criteria decision-making methods with completely unknown weights in hesitant fuzzy linguistic term setting, Knowl. Based Syst. 93 (2016) 135–144. X.J. Gou, Z.S. Xu, H.C. Liao, Multi-criteria decision making based on bonferroni means with hesitant fuzzy linguistic information, Soft Comput. (2016), doi:10.10 07/s0 050 0- 016- 2211- 1. G. Hesamian, M. Shams, Measuring similarity and ordering based on hesitant fuzzy linguistic term sets, J. Intell. Fuzzy Syst. 28 (2) (2015) 983–990. H.C. Liao, Z.S. Xu, X.J. Zeng, Distance and similarity measures for hesitant fuzzy linguistic term sets and their application in multi-criteria decision making, Inf. Sci. 271 (2014) 125–142. H.C. Liao, Z.S. Xu, Approaches to manage hesitant fuzzy linguistic information based on the cosine distance and similarity measures for HFLTSs and their application in qualitative decision making, Expert Syst. Appl. 42 (2015) 5328–5336. H.C. Liao, Z.S. Xu, X.J. Zeng, J.M. Merigó, Qualitative decision making with correlation coefficients of hesitant fuzzy linguistic term sets, Knowl. Based Syst. 76 (2015) 127–138. H.C. Liao, Z.S. Xu, X.J. Zeng, Hesitant fuzzy linguistic VIKOR method and its application in qualitative multiple criteria decision making, IEEE Trans. Fuzzy Syst. 23 (5) (2015) 1343–1355. H.C. Liao, Z.S. Xu, Subtraction and division operations over hesitant fuzzy sets, J. Intell. Fuzzy Syst. 27 (2014) 65–72. P. Lingras, F. Haider, M. Triff, Granular meta-clustering based on hierarchical, network, and temporal connections, Granul. Comput. 1 (1) (2016) 71–92. H.B. Liu, J.F. Cai, L. Jiang, On improving the additive consistency of the fuzzy preference relations based on comparative linguistic expressions, Int. J. Int. Syst. 29 (2014) 544–559. Y.Z. Liu, Z.P. Fan, G.X. Gao, An extended LINMAP method for MAGDM under linguistic hesitant fuzzy environment, J. Intell. Fuzzy Syst. 30 (5) (2016) 2689–2703. L. Livi, A. Sadeghian, Granular computing, computational intelligence, and the analysis of non- geometric input spaces, Granul. Comput. 1 (1) (2016) 13–20. V. Loia, G. D’Aniello, A. Gaeta, F. Orciuoli, Enforcing situation awareness with granular computing: a systematic overview and new perspectives, Granul. Comput. 1 (2) (2016) 127–143. J.M. Mendel, A comparison of three approaches for estimating (synthesizing) an interval type-2 fuzzy set model of a linguistic term for computing with words, Granul. Comput. 1 (1) (2016) 59–69. F.Y. Meng, X.H. Chen, Q. Zhang, Multi-attribute decision analysis under a linguistic hesitant fuzzy environment, Inf. Sci. 267 (13) (2014) 287–305. Q. Pang, H. Wang, Z.S. Xu, Probabilistic linguistic term sets in multi-attribute group decision making, Inf. Sci. (2016), doi:10.1016/j.ins.2016.06.021. W. Pedrycz, S.M. Chen, Granular Computing and Decision-Making: Interactive and Iterative Approaches (2015). Z. Pei, A note on the TOPSIS method in MADM problems with linguistic evaluations, Appl. Soft Comput. 36 (2015) 24–35. G. Peters, R. Weber, DCC: a framework for dynamic granular clustering, Granul. Comput. 1 (1) (2016) 1–11. R.M. Rodríguez, L. Martínez, F. Herrera, Hesitant fuzzy linguistic terms sets for decision making, IEEE Trans. Fuzzy Syst. 20 (2012) 109–119. A. Skowron, A. Jankowski, S. Dutta, Interactive granular computing, Granul. Comput. 1 (2) (2016) 95–113. V. Torra, Hesitant fuzzy sets, Int. J. Intell. Syst. 25 (2010) 529–539. J.Q. Wang, J.T. Wu, J. Wang, H.Y. Zhang, X.H. Chen, Multi-criteria decision-making methods based on the Hausdorff distance of hesitant fuzzy linguistic numbers, Soft Comput. 20 (4) (2016) 1621–1633. C.P. Wei, Z.L. Ren, R.M. Rodriguez, A hesitant fuzzy linguistic TODIM method based on a score function, Int. J. Comput. Intell. Syst. 8 (4) (2015) 701–712. C.P. Wei, N. Zhao, X.J. Tang, Operators and Comparisons of Hesitant Fuzzy Linguistic Term Sets, IEEE Trans. Fuzzy Syst. 22 (3) (2014) 575–585. G. Wilke, E. Portmann, Granular computing as a basis of human-data interaction: a cognitive cities use case, Granul. Comput. 1 (3) (2016) 181–197. M.M. Xia, Z.S. Xu, Hesitant fuzzy information aggregation in decision making, Int. J. Approx. Reason. 52 (2011) 395–407. Y.J. Xu, A.W. Xu, J.M. Merigó, H.M. Wang, Hesitant fuzzy linguistic ordered weighted distance operators for group decision making, J. Appl. Math Comput. 49 (1-2) (2015) 1–24. Z.S. Xu, Deviation measures of linguistic preference relations in group decision making, Omega- Int. J. Manag. 33 (2005) 249–254. Z.S. Xu, Uncertain Multiple Attribute Decision Making: Methods and Applications (2004). Z.S. Xu, M.M. Xia, Distance and similarity measures for hesitant fuzzy sets, Inf. Sci. 181 (2011) 2128–2138. Z.S. Xu, H. Wang, Managing multi-granularity linguistic information in qualitative group decision making: an overview, Granul. Comput. 1 (1) (2016) 21–35. Y. Yao, A triarchic theory of granular computing, Granul. Comput. 1 (2) (2016) 145–157. L.A. Zadeh, The concept of a linguistic variable and its applications to approximate reasoning-Part I, Inf. Sci. 8 (1975) 199–249. Y.J. Zhang, Y.Z. Wang, J.P. Wang, Hesitant fuzzy linguistic multiple attribute decision making, in: 16th International Conference on Information Fusion Istanbul, Turkey, 2013, pp. 1421–1426. Z.M. Zhang, C. Wu, On the use of multiplicative consistency in hesitant fuzzy linguistic preference relations, Knowl. Based Syst. 72 (2014) 13–27. B. Zhu, Z.S. Xu, Consistency measures for hesitant fuzzy linguistic preference relations, IEEE Trans. Fuzzy Syst. 22 (1) (2014) 35–45.