* Tel.: +86 312 5079660; fax: +86 312 5079660. E-mail address: zhimingzhang@ymail.com.

SDI Paper Template Version 1.6 Date 11.10.2012 1

Weighted hesitant fuzzy sets and their 2

application to multi-criteria decision making 3

4

Zhiming Zhang1,2*, Chong Wu2 5 6

1School of Management, Harbin Institute of Technology, Harbin 150001, Heilongjiang 7 Province, PR China 8

2College of Mathematics and Computer Science, Hebei University, Baoding 071002, Hebei 9 Province, PR China 10

11 12 13 .14 ABSTRACT 15 Aims: The aim of this paper is to investigate weighted hesitant fuzzy sets and their application to multi-criteria decision making. Study design: This paper puts forward the concept of a weighted hesitant fuzzy set (WHFS), in which several possible membership degrees of each element have different weights. Archimedean t-conorm and t-norm provide a generalization of a variety of other t-conorms and t-norms that include as special cases Algebraic, Einstein, Hamacher and Frank t-conorms and t-norms. Place and Duration of Study: Hesitant fuzzy set, permitting the membership degree of an element to be a set of several possible values, can be referred to as an efficient mathematical tool for modeling people’s hesitancy in daily life. It is noted that several possible membership degrees of each element in the hesitant fuzzy set are of equal importance, but in many practical problems, especially in multi-criteria decision making, the weights of several possible membership degrees of each element should be taken into account. Methodology: In this paper, based on Archimedean t-conorm and t-norm, we present some operations on weighted hesitant fuzzy sets (WHESs), and based on which, we develop two weighted hesitant fuzzy aggregation operators for aggregating weighted hesitant fuzzy information. Furthermore, some desired properties and special cases of the developed operators are discussed in detail. Results: We develop an approach for multi-criteria decision making under weighted hesitant fuzzy environment. Conclusion: An illustrative example is provided to show the effectiveness and practicality of the proposed operators and approach. 16 Keywords: Multi-criteria decision making; Hesitant fuzzy sets; Weighted hesitant fuzzy sets; 17 Archimedean t-conorm and t-norm; Weighted hesitant fuzzy aggregation operator. 18 19 1. INTRODUCTION 20 21 Due to the fact that when defining the membership degree of an element to a set, the 22 difficulty of establishing the membership degree is not because we have a margin of error 23 (as in intuitionstic fuzzy set [1], interval-valued fuzzy set [2], or interval-valued intuitionistic 24 fuzzy set [3]) or some possibility distribution (as in type-2 fuzzy set [4]) on the possible 25 values, but because we have some possible values, Torra [5] defined the hesitant fuzzy sets 26 (HFS) to permit the membership degree of an element to a set represented as several 27 possible values between 0 and 1. The HFS can be used to efficiently manage the situation 28 where people hesitate between several possible values to express their opinions. Since it 29

was introduced, HFS has attracted much attention. Torra and Narukawa [6] first applied 30 hesitant fuzzy sets (HFSs) to decision making. Xu and Xia [7,8] proposed a lot of distance 31 measures, similarity measures, and correlation measures for HFSs. Farhadinia [9] 32 investigated the relationship between the entropy, the similarity measure, and the distance 33 measure for HFSs and interval-valued hesitant fuzzy sets (IVHFSs) [10,11]. Peng et al. [12] 34 presented a generalized hesitant fuzzy synergetic weighted distance (GHFSWD) measure 35 based on the generalized hesitant fuzzy weighted distance (GHFWD) measure and the 36 generalized hesitant fuzzy ordered weighted distance (GHFOWD) measure proposed in [7]. 37 Qian et al. [13] extended hesitant fuzzy sets by intuitionistic fuzzy sets and referred to them 38 as generalized hesitant fuzzy sets. 39 It should be noted that only several possible values are involved in the classical HFS, but the 40 importance of each possible value is not emphasized. Nevertheless, in many practical 41 situations, especially in multi-criteria decision making, several possible values usually have 42 different importance and thus need to be assigned different weights. For example, to get a 43 reasonable decision result, ten decision makers who are very familiar with this area are 44 invited to estimate the degree that an alternative satisfies an attribute. Suppose there are 45 four cases, four decision makers provide 0.8 , three decision makers provide 0.7 , two 46 decision makers provide 0.6 , and one decision maker provides 0.5 , and these ten decision 47 makers cannot persuade each other to change their opinions. In [7,14,15,16,17], the authors 48 do not consider the importance of all of the possible values for an alternative under an 49 attribute and allow these values repeated many times appear only once. According to 50 [7,14,15,16,17], the degree that the alternative satisfies the attribute is represented by a HFS 51 {0.5, 0.6, 0.7, 0.8}, which is somewhat inconsistent with our intuition because these values 52 repeated many times at least denote strength of the decision makers’ preferences. 53 According to the strategy given in [7,14,15,16,17], more experts may not contribute to more 54 reasonable decision results. When we consider a multiple attribute group decision making 55 (MAGDM) problem, if two or more decision makers who are familiar with this area give the 56 same preferences, then their preferences will be close to group preference. In such cases, 57 the value repeated many times may be more important than the one repeated only one time. 58 Therefore, the importance of all of the possible membership degrees should be attached to 59 the construction of the HFS. To do it, in this paper, we introduce the concept of a weighted 60 hesitant fuzzy set (WHFS), which is a new generalization of the classical hesitant fuzzy set 61 by adding the weight information to the classical hesitant fuzzy set. In the WHFS, the 62 importance of all of the possible membership degrees is taken into account and the weight 63 information is associated with all of the possible membership degrees. Thus, the WHFS can 64 contain more information than the classical hesitant fuzzy set and can help the decision 65 makers get more accurate, reasonable, and reliable decision results than the classical 66 hesitant fuzzy set. In the previous example, the degree that the alternative satisfies the 67 attribute can be represented by a WHFS {(0.5, 0.1), (0.6, 0.2), (0.7, 0.3), (0.8, 0.4)}. 68 In order to aggregate hesitant fuzzy information, Xia and Xu [15] proposed some Algebraic t-69 conorm and t-norm based operational laws for HFSs, based on which, a variety of hesitant 70 fuzzy aggregation operators have been developed in recent years. For example, Xia and Xu 71 [15] developed the hesitant fuzzy weighted averaging (HFWA) operator, the hesitant fuzzy 72 weighted geometric (HFWG) operator, the generalized hesitant fuzzy weighted averaging 73 (GHFWA) operator, the generalized hesitant fuzzy weighted geometric (GHFWG) operator, 74 the hesitant fuzzy ordered weighted averaging (HFOWA) operator, the hesitant fuzzy 75 ordered weighted geometric (HFOWG) operator, the generalized hesitant fuzzy ordered 76 weighted averaging (GHFOWA) operator, the generalized hesitant fuzzy ordered weighted 77 geometric (GHFOWG) operator, the hesitant fuzzy hybrid averaging (HFHA) operator, the 78 hesitant fuzzy hybrid geometric (HFHG) operator, the generalized hesitant fuzzy hybrid 79 averaging (HFHA) operator, and the generalized hesitant fuzzy hybrid geometric (GHFHG) 80 operator. Xia et al. [18] proposed some new hesitant fuzzy aggregation operators, such as 81 the quasi hesitant fuzzy weighted aggregation (QHFWA) operator, the hesitant fuzzy 82

modular weighted averaging(QHFWA) operator, the hesitant fuzzy modular weighted 83 geometric (HFMWG) operator, the quasi hesitant fuzzy ordered weighted aggregation 84 (QHFOWA) operator, the hesitant fuzzy modular ordered weighted averaging (QHFOWA) 85 operator, the hesitant fuzzy modular ordered weighted geometric (HFMWG) operator, the 86 induced quasi hesitant fuzzy ordered weighted aggregation (IQHFOWA) operator, the 87 induced hesitant fuzzy modular ordered weighted averaging (IHFMOWA) operator, and the 88 induced hesitant fuzzy modular ordered weighted geometric (IHFMWG) operator. By 89 extending the Bonferroni mean (BM) [19] to hesitant fuzzy environments, Zhu and Xu [20] 90 developed the hesitant fuzzy Bonferroni means (HFBMs) and the weighted hesitant fuzzy 91 Bonferroni mean (WHFBM). By extending the geometric Bonferroni mean (BM) [21] to 92 hesitant fuzzy environments, Zhu et al. [22] proposed the hesitant fuzzy geometric 93 Bonferroni mean (HFGBM) and the weighted hesitant fuzzy Choquet geometric Bonferroni 94 mean (WHFCGBM). In order to consider the relationship between the hesitant fuzzy input 95 arguments, Zhang [23] developed several new hesitant fuzzy aggregation operators, 96 including the hesitant fuzzy power average (HFPA) operator, the hesitant fuzzy power 97 geometric (HFPG) operator, the generalized hesitant fuzzy power average (GHFPA) 98 operator, the generalized hesitant fuzzy power geometric (GHFPG)operator, the weighted 99 the generalized hesitant fuzzy power average (WGHFPA) operator, the weighted 100 generalized hesitant fuzzy power geometric (WGHFPG) operator, the hesitant fuzzy power 101 ordered weighted average (HFPOWA) operator, the hesitant fuzzy power ordered weighted 102 geometric(HFPOWG) operator, the generalized hesitant fuzzy power ordered weighted 103 average (GHFPOWA) operator, and the generalized hesitant fuzzy power ordered weighted 104 geometric (GHFPOWG) operator. Wei and Zhao [24] introduced some operations on 105 hesitant interval-valued fuzzy sets (HIVFSs) based on Einstein t-conorm and t-norm, and 106 based on which, developed some induced hesitant interval-valued fuzzy Einstein 107 aggregation operators for aggregating hesitant interval-valued fuzzy information. 108 Archimedean t-conorm and t-norm [25,26] are generalizations of lots of other t-conorms and 109 t-norms, such as Algebraic, Einstein, Hamacher and Frank t-conorms and t-norms [27]. 110 Based on Archimedean t-conorm and t-norm, Beliakov et al. [28] gave some operations 111 about intuitionistic fuzzy sets (IFSs). Xia et al. [27] further gave some other operations on 112 IFSs, and proposed some specific intuitionistic fuzzy aggregation operators. Motivated by 113 the work of Beliakov et al. [28] and Xia et al. [27], this paper proposes some Archimedean t-114 conorm and t-norm based operation laws on weighted hesitant fuzzy sets (WHFSs), 115 investigates their properties, and based on which, develops two Archimedean t-conorm and 116 t-norm based weighted hesitant fuzzy aggregation operators, including the Archimedean t-117 conorm and t-norm based weighted hesitant fuzzy weighted averaging (ATS-WHFWA) 118 operator and the Archimedean t-conorm and t-norm based weighted hesitant fuzzy weighted 119 geometric (ATS-WHFWG) operator. Moreover, we study some desired properties of the new 120 operators and give their special cases, such as the weighted hesitant fuzzy weighted 121 averaging (WHFWA) operator, the weighted hesitant fuzzy Einstein weighted averaging 122 (WHFEWA) operator, the weighted hesitant fuzzy Hammer weighted averaging (WHFHWA) 123 operator, the weighted hesitant fuzzy Frank weighted averaging (WHFFWA) operator, the 124 weighted hesitant fuzzy weighted geometric (WHFWG) operator, the weighted hesitant fuzzy 125 Einstein weighted geometric (WHFEWG) operator, the weighted hesitant fuzzy Hammer 126 weighted geometric (WHFHWG) operator, and the weighted hesitant fuzzy Frank weighted 127 geometric (WHFFWG) operator. Finally, we develop an approach for multi-criteria decision 128 making under weighted hesitant fuzzy environment, and provide a numerical example to 129 illustrate the proposed approach. 130 This paper is organized as follows. Section 2 introduces some basic concepts of hesitant 131 fuzzy sets and Archimedean t-conorm and t-norm. In Section 3, we define the concept of 132 weighted hesitant fuzzy sets (WHFSs) and introduce some operational laws for them based 133 on Archimedean t-conorm and t-norm. Section 4 proposes two Archimedean t-conorm and t-134 norm based weighted hesitant fuzzy aggregation operators for aggregation weighted 135

hesitant fuzzy information. Some desired properties and special cases of the proposed 136 operators are also investigated in this section. In the sequel, Section 5 develops an 137 approach to multi-criteria decision making under weighted hesitant fuzzy environment and 138 gives a practical example to illustrate the developed approach. The final section offers some 139 concluding remarks. 140 2. PRELIMINARIES 141 142 In this section, we will give a brief introduction of hesitant fuzzy sets [5] and Archimedean t-143 conorm and t-norm [25,26]. 144 Definition 2.1 [5]. Let X be a fixed set, a hesitant fuzzy set (HFS) on X is in terms of a 145 function that when applied to X returns a subset of [ ]0,1 . 146

Xia and Xu [15] expressed a HFS by the following form: 147

( ){ }, EE x h x x X= Î (1) 148

where ( )Eh x is a set of some values in [ ]0,1 , denoting the possible membership degrees of 149

the element x XÎ to the set E . For convenience, Xia and Xu [15] called ( )Eh h x= a 150

hesitant fuzzy element (HFE) and H the set of all hesitant fuzzy elements (HFEs). 151 Assume three HFEs represented by h , 1h and 2h , Torra [5] defined some operations on 152 them, which can be described as: 153 (1) { }1ch hg g= - Î ; (2) 154

(2) { }1 2 1 2 1 1 2 2,h h h hU g g g g= Ú Î Î ; (3) 155

(3) { }1 2 1 2 1 1 2 2,h h h hI g g g g= Ù Î Î . (4) 156

Definition 2.2 [25,26]. A function [ ] [ ] [ ]: 0,1 0,1 0,1T ´ ® is called a t-norm if it satisfies the 157

following four conditions: 158 (1) ( )1,T a a= , for all [ ]0,1aÎ . 159

(2) ( ) ( ), ,T a b T b a= , for all [ ], 0,1a bÎ . 160

(3) ( )( ) ( )( ), , , ,T a T b c T T a b c= , for all [ ], , 0,1a b cÎ . 161

(4) If a a¢£ and b b¢£ for all [ ], , , 0,1a a b b¢ ¢Î , then ( ) ( ), ,T a b T a b¢ ¢£ . 162

Definition 2.3 [25,26]. A function [ ] [ ] [ ]: 0,1 0,1 0,1S ´ ® is called a t-conorm if it satisfies the 163

following four conditions: 164 (1) ( )0,S a a= , for all [ ]0,1aÎ . 165

(2) ( ) ( ), ,S a b S b a= , for all [ ], 0,1a bÎ . 166

(3) ( )( ) ( )( ), , , ,S a S b c S S a b c= , for all [ ], , 0,1a b cÎ . 167

(4) If a a¢£ and b b¢£ for all [ ], , , 0,1a a b b¢ ¢Î , then ( ) ( ), ,S a b S a b¢ ¢£ . 168

Definition 2.4 [25,26]. A t-norm function ( ),T a b is called Archimedean t-norm if it is 169

continuous and ( ),T a a a< for all ( )0,1aÎ . An Archimedean t-norm is called strictly 170

Archimedean t-norm if it is strictly increasing in each variable for ( ), 0,1a bÎ . 171

Definition 2.5 [25,26]. A t-conorm function ( ),S a b is called Archimedean t-conorm if it is 172

continuous and ( ),S a a a> for all ( )0,1aÎ . An Archimedean t-conorm is called strictly 173

Archimedean t-conorm if it is strictly increasing in each variable for ( ), 0,1a bÎ . 174

It is well known [29] that a strict Archimedean t-norm ( ),T a b is expressed via its additive 175

generator g as ( ) ( ) ( )( )1,T a b g g a g b-= + , where [ ] [ ]: 0,1 0,g ® +¥ is a strictly decreasing 176

function such that ( )1 0g = . A dual Archimedean t-conorm ( ),S a b is expressed as 177

( ) ( ) ( )( )1,S a b f f a f b-= + with ( ) ( )1f t g t= - . 178

179 3. WEIGHTED HESITANT FUZZY SETS (WHFSS) AND WEIGHTED HESITANT 180 FUZZY ELEMENTS (WHFES) 181 182 Considering that the classical hesitant fuzzy set does not involve the importance of all of the 183 possible membership degrees of each element, in this section, we will propose a new 184 concept of weighted hesitant fuzzy set by assigning a weight vector to all of the possible 185 membership degrees of each element. 186 Definition 3.1. Let X be a reference set, a weighted hesitant fuzzy set (WHFS) on X is 187 defined as: 188

( ){ } ( ){ }( ){ }, , ,A

xA h xA x h x x X x w x Xgg

gÎ

= Î = Î%%%% U (5) 189

where ( )Ah x% is a set of some different values in [ ]0,1 , denoting all possible membership 190

degrees of the element x XÎ to the set A% , xw g is the weight of g , [ ]0,1xw g Î , and 191

( )1

A

xh x

w ggÎ

=å%

for any x XÎ . 192

For convenience, we call ( ){ },h

h wggg

Î=% U a weighted hesitant fuzzy element (WHFE), 193

where h is a set of some different membership degrees in [ ]0,1 , wg is the weight of g , 194

[ ]0,1wg Î for any hg Î , and 1h

wggÎ

=å . Let H% denote the set of all weighted hesitant 195

fuzzy elements (WHFEs). 196

Let ( ){ } ( ){ }( ){ }, , ,A

xA h xA x h x x X x w x Xgg

gÎ

= Î = Î%%%% U be a WHFS. If for any x XÎ 197

and ( )Ah xg Î % ,

( )1

x

A

wh xg = # %

( ( )Ah x# % is the number of the elements in ( )A

h x% ), then A% 198

reduces to a HFS. Let ( ){ },h

h wggg

Î=% U be a WHFE. If

1w

hg = #, then h% reduces to a 199

HFE. 200 By Definition 3.1, the WHFS extends the HFS to contain several membership degrees and 201 their corresponding weights. The difference between the HFS and WHFS is that the former 202 assumes that the possible membership degrees of each element are of equal importance, 203 while the latter assigns different weights to different membership degrees. Thus, compared 204 with the HFS, the WHFS can depict human uncertainty more objectively and precisely. 205 In the following, we illustrate how to construct a WHFE. Suppose that l experts are required 206

to evaluate the membership degree of the element x in the set A% . 1l experts provide 1g , 2l 207

experts provide 2g , L , and kl experts provide kg , where 1

k

ki

l l=

=å . Assume that these l 208

experts cannot persuade each other to change their opinions. In such cases, the 209

membership degree of the element x in the set A% has k possible values 1g , 2g , L , and 210

kg . The weights of ig ( 1,2, ,i k= L ) are i

ilwlg = ( 1,2, ,i k= L ). Thus, the membership 211

degree of the element x in the set A% can be represented by a WHFE 212

1 21 2, , , , , , k

k

ll lh

l l lg g g

ì üæ öæ ö æ ö= í ýç ÷ ç ÷ ç ÷è ø è ø è øî þ

% L . Based on the above analysis, we can see that 213

constructing of a WHFE consists of two steps: (1) collecting different possible membership 214 degree values into a HFE; (2) assigning the weights to these different membership degree 215 values. 216 The WHFS is an efficient tool to represent situations in which several different membership 217 functions for a fuzzy set are possible and different membership functions have different 218 weights. It is particularly suitable to address the hesitancy and uncertainty that are quite 219 usual in real world decision making problems. 220 Example 3.1. Let { }1 2 3, ,X x x x= , 221

( ) ( ){ } ( ) ( ) ( ){ } ( ) ( ){ }{ }1 2 3, 0.5,0.3 , 0.6,0.7 , , 0.1,0.5 , 0.3,0.2 , 0.4,0.3 , , 0.7,0.5 , 0.9,0.5%A x x x= , 222

and ( ) ( ) ( ){ }0.1,0.5 , 0.3,0.2 , 0.4,0.3h =% . Then, A% is a WHFS on X and h% is a WHFE. 223

To compare two WHFEs, we define the following comparison laws: 224

Definition 3.2. For a WHFE ( ){ },h

h wggg

Î=% U , ( ) ( )

h

s h wgg

gÎ

= ×å% is called the score 225

function of h% . For two WHFEs 1h% and 2h% , if ( ) ( )1 2s h s h>% % , then 1 2h h>% % ; if ( ) ( )1 2s h s h=% % , 226

then 1 2h h=% % . 227

Given three WHFEs represented by ( ){ },h

h wggg

Î=% U , ( ){ }11 1

1 1 1,h

h w ggg

Î=% U , and 228

( ){ }22 22 2 2,

hh w gg

gÎ

=% U , we define some basic operations on them as below: 229

(1) ( ){ }1 ,c

hh wgg

gÎ

= -% U ; (6) 230

(2) ( ){ }1 21 1 2 21 2 1 2 1 2,

,h h

h h w wg gg gg g

Î Î= Ú ×% %U U ; (7) 231

(3) ( ){ }1 21 1 2 21 2 1 2 1 2,

,h h

h h w wg gg gg g

Î Î= Ù ×% %I U . (8) 232

Theorem 3.1. Let h% , 1h% , and 2h% be three WHFEs. Then, ch% , 1 2h h% %U , and 1 2h h% %I are also 233 WHFEs. 234

Proof. It is clear that ch% is a WHFE. 235 According to Definition 3.1, we have 236

( ) ( )1 2 1 2 1

1 1 2 2 1 1 2 2 1 1

1 2 1 2 1,

1 1h h h h h

w w w w wg g g g gg g g g gÎ Î Î Î Î

æ öæ ö× = × = × =ç ÷ç ÷ç ÷ç ÷è øè ø

å å å å 237

which shows that 1 2h h% %U is a WHFE. 238

Similarly, we can conclude that 1 2h h% %I is also a WHFE. 239 This completes the proof of Theorem 3.1. o 240 In the above operations, we allow the membership degrees repeated many times appear 241 only once, whose weight is a sum of the weights of the membership degrees repeated many 242 times. 243

Example 3.2. Given two WHFEs 1h% and 2h% as follows: 244

( ) ( ) ( ){ }1 0.1,0.5 , 0.3,0.2 , 0.4,0.3h =% , ( ) ( ){ }2 0.7,0.6 , 0.8,0.4h =% . 245

Then, we have 246

( ) ( ) ( ){ }1 0.9,0.5 , 0.7,0.2 , 0.6,0.3ch =% 247

( ) ( ) ( )( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ){ }( ) ( ){ }

1 2

0.1 0.7,0.5 0.6 , 0.1 0.8,0.5 0.4 , 0.3 0.7,0.2 0.6 ,

0.3 0.8,0.2 0.4 , 0.4 0.7,0.3 0.6 , 0.4 0.8,0.3 0.4

0.7,0.3 , 0.8,0.2 , 0.7,0.12 , 0.8,0.08 , 0.7,0.18 , 0.8,0.12

0.7,0.6 , 0.8,0.4

h hì üÚ × Ú × Ú ×ï ï= í ý

Ú × Ú × Ú ×ï ïî þ=

=

% %U

248

( ) ( ) ( )( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ){ }( ) ( ) ( ){ }

2

0.1 0.7,0.5 0.6 , 0.1 0.8,0.5 0.4 , 0.3 0.7,0.2 0.6 ,

0.3 0.8,0.2 0.4 , 0.4 0.7,0.3 0.6 , 0.4 0.8,0.3 0.4

0.1,0.3 , 0.1,0.2 , 0.3,0.12 , 0.3,0.08 , 0.4,0.18 , 0.4,0.12

0.1,0.5 , 0.3,0.2 , 0.4,0.3

h hì üÙ × Ù × Ù ×ï ï= í ý

Ù × Ù × Ù ×ï ïî þ=

=

% %I

249

In order to aggregate weighted hesitant fuzzy information, we defined some new operations 250

on the WHFEs h% , 1h% and 2h% : 251

(1) ( )( ){ } ( ) ( )( )( ){ }1 2 1 21 1 2 2 1 1 2 2

11 2 1 2 1 2 1 2 1 2, ,

, , ,% % U Uh h h hh h S w w f f f w wg g g gg g g g

g g g g-

Î Î Î ÎÅ = × = + × ; (9) 252

(2) ( )( ){ } ( ) ( )( )( ){ }1 2 1 21 1 2 2 1 1 2 2

11 2 1 2 1 2 1 2 1 2, ,

, , ,% % U Uh h h hh h T w w g g g w wg g g gg g g g

g g g g-

Î Î Î ÎÄ = × = + × ; (10) 253

(3) ( )( )( ){ }1 ,h

h f f wggl l g-

Î=% U , 0l > ; (11) 254

(4) ( )( )( ){ }1 ,h

h g g wlgg

l g-Î

=% U , 0l > . (12) 255

Theorem 3.2. For three WHFEs h% , 1h% , and 2h% , we have the following properties: 256

(1) 1 2 2 1h h h hÅ = Å% % % % ; 257

(2) 1 2 2 1h h h hÄ = Ä% % % % ; 258

(3) ( )1 2 1 2h h h hl l lÅ = Å% % % % , 0l > ; 259

(4) ( )1 2 1 2h h h hl l lÄ = Ä% % % % , 0l > ; 260

(5) ( )1 2 1 2h h hl l l lÅ = +% % % , 1 2, 0l l > ; 261

(6) 1 2 1 2h h hl l l l+Ä =% % % , 1 2, 0l l > ; 262

(7) ( ) ( )1 2 1 2% % % % % %h h h h h hÅ Å = Å Å ; 263

(8) ( ) ( )1 2 1 2% % % % % %h h h h h hÄ Ä = Ä Ä ; 264

(9) ( )1 2 1 2% % % %U Ic c ch h h h= ; 265

(10) ( )1 2 1 2% % % %I Uc c ch h h h= ; 266

(11) ( ) ( )% % cch hl

l= ; 267

(12) ( ) ( )% % cch hll = ; 268

(13) ( )1 2 1 2% % % %c c ch h h hÅ = Ä ; 269

(14) ( )1 2 1 2% % % %c c ch h h hÄ = Å . 270

271 4. AGGREGATION OPERATORS FOR WEIGHTED HESITANT FUZZY 272 INFORMATION 273 In the current section, we will propose several operators for aggregating the weighted 274 hesitant fuzzy information and investigate some properties of these operators. 275

Definition 4.1. Let ih% ( 1,2, ,i n= L ) be a collection of WHFEs, and let ( )1 2, , ,T

nw w w w= L 276

be the weight vector of ih% ( 1,2, ,i n= L ) with [ ]0,1iw Î and 1

1n

ii

w=

=å . Then, an 277

Archimedean t-conorm and t-norm based weighted hesitant fuzzy weighted averaging (ATS-278 WHFWA) operator is a mapping nH H®% % , where 279

( ) ( )1 21

ATS-WHFWA , , ,n

n i ii

h h h hw=

= Å% % % %L (13) 280

Theorem 4.1. Let ( ){ },ii i

i i ihh w gg

gÎ

=% U ( 1,2, ,i n= L ) be a collection of WHFEs, and 281

( )1 2, , ,T

nw w w w= L be the weight vector of ih% ( 1,2, ,i n= L ), where iw indicates the 282

importance degree of ih% , satisfying [ ]0,1iw Î and 1

1n

ii

w=

=å , then the aggregated value by 283

using the ATS-WHFWA operator is also a WHFE, and 284

( ) ( )1 1 2 2

11 2 , , ,

1 1

ATS-WHFWA , , , ,in n

nn

n i i ih h hi i

h h h f f w gg g gw g-

Î Î Î= =

ì üæ öæ öï ï= í ýç ÷ç ÷è øï ïè øî þå ÕL

% % %L U (14) 285

Proof. By using mathematical induction on n : For 2n = , since 286

( )( )( ){ }11 1

11 1 1 1 1,

hh f f w gg

w w g-Î

=% U 287

( )( )( ){ }22 2

12 2 2 2 2,

hh f f w gg

w w g-Î

=% U 288

we have 289

( )( )( ){ }( ) ( )( )( ){ }( )( )( )( ) ( )( )( )( )( ){ }

( ) ( )( )( ){ }

1 21 1 2 2

1 21 1 2 2

1 21 1 2 2

1 11 1 2 2 1 1 1 2 2 2

1 1 11 1 2 2 1 2,

11 1 2 2 1 2,

, ,

,

,

h h

h h

h h

h h f f w f f w

f f f f f f f w w

f f f w w

g gg g

g gg g

g gg g

w w w g w g

w g w g

w g w g

- -Î Î

- - -Î Î

-Î Î

Å = Å

= + ×

= + ×

% % U UUU

290

That is, the Eq. (14) holds for 2n = . Suppose that the Eq. (14) holds for n k= , i.e., 291

( ) ( )1 1 2 2

1

, , ,11 1

,ik k

kkk

i i i i ih h hii i

h f f w gg g gw w g-

Î Î Î== =

ì üæ öæ öï ïÅ = í ýç ÷ç ÷è øï ïè øî þå ÕL

% U 292

then, when 1n k= + , we have 293

( ) ( ) ( )

( ) ( )( )( ){ }( )

( ) ( )

11 1 2 2 1 1

1

1 11 1

1 11 1 1, , ,

1 1

11

1 11 1

, ,

,

L

% % %

U Ui kk k k k

i

k k

i i i i k ki i

kk

i i i k k kh h h hi i

kk

i i k k ii i

h h h

f f w f f w

f f f w

g gg g g g

g

w w w

w g w g

w g w g

++ +

+

+ += =

- -+ + +Î Î Î Î

= =

+-

+ += =

æ öÅ = Å Åç ÷è ø

æ öì üæ öæ öï ï= Åç ÷í ýç ÷ç ÷ç ÷è øï ïè øî þè øì üæ öæ öï= +í ýç ÷ç ÷

è øïè øî

å Õ

å Õ

( )

1 1 2 2 1 1

1 1 2 2 1 1

, , , ,

111

, , , ,1 1

,

L

L

U

Uk k k k

ik k k k

h h h h

kk

i i ih h h hi i

f f w

g g g g

gg g g gw g

+ +

+ +

Î Î Î Î

++-

Î Î Î Î= =

ï

ïþì üæ öæ öï ï= í ýç ÷ç ÷

è øï ïè øî þå Õ

294

i.e., Eq. (14) holds for 1n k= + . Thus Eq. (14) holds for all n . 295

In addition, because [ ] [ ]: 0,1 0,g ® +¥ is a strictly decreasing function and ( ) ( )1f t g t= - , 296

[ ] [ ]: 0,1 0,f ® +¥ is a strictly increasing function, which implies that 297

( )1

1

0 1n

i ii

f fw g-

=

æ ö£ £ç ÷

è øå 298

Furthermore, we have 299

( )

1 1 2 2 1 1 2 2 1 1

1 2

1 1 2 2 1 1 1 1 2 2

1

1 1

1

, , , , , ,1 1

1

1 2, , , 1

1

i i n

n n n n n n

i

n n

n n

i i nh h h h h h hi i

n

ih h h h hi

h

w w w

w w w

w

g g gg g g g g g g

g g gg g g g g

gg

- -

- -

-

Î Î Î Î Î Î Î= =

-

Î Î Î Î Î=

Î

æ öæ öæ öæ öç ÷= ×ç ÷ç ÷ç ÷ ç ÷ç ÷ç ÷è ø è øè øè ø

æ öæ öæ ö= = = ×ç ÷ç ÷ç ÷ ç ÷ç ÷è ø è øè ø

= =

å å åÕ Õ

å å åÕ

å

L L

LL

1

300

This completes the proof of Theorem 4.1. o 301 In the following, let’s study some desirable properties of the ATS-WHFWA operator. 302

Theorem 4.2. Let ih% ( 1,2, ,i n= L ) be a collection of WHFEs, ( )1 2, , ,T

nw w w w= L be their 303

weight vector with [ ]0,1iw Î ( 1,2, ,i n= L ) and 1

1n

ii

w=

=å , if 0r > , then 304

( ) ( )1 2 1 2ATS-WHFWA , , , ATS-WHFWA , , ,n nrh rh rh r h h h=% % % % % %L L (15) 305

Proof. Since for any 1,2, ,i n= L , 306

( )( )( ){ }1 ,ii i

i i ihrh f rf w gg

g-

Î=% U 307

Based on Theorem 4.1, we have 308

( )

( )( )( )

( )

1 1 2 2

1 1 2 2

1 2

1 1

, , ,1 1

1

, , ,1 1

ATS-WHFWA , , ,

,

,

in n

in n

n

nn

i i ih h hi i

nn

i i ih h hi i

rh rh rh

f f f rf w

f r f w

gg g g

gg g g

w g

w g

- -Î Î Î

= =

-Î Î Î

= =

ì üæ öæ öï ï= í ýç ÷ç ÷è øï ïè øî þ

ì üæ öæ öï ï= í ýç ÷ç ÷è øï ïè øî þ

å Õ

å Õ

L

L

% % %L

U

U

309

According to Eq. (11), we can get 310

( )

( )

( )

( )

1 1 2 2

1 1 2 2

1 2

1

, , ,1 1

1 1

, , ,1 1

1

1

ATS-WHFWA , , ,

,

,

in n

in n

n

nn

i i ih h hi i

nn

i i ih h hi i

n

i ii

r h h h

r f f w

f rf f f w

f r f

gg g g

gg g g

w g

w g

w g

-Î Î Î

= =

- -Î Î Î

= =

-

=

æ öì üæ öæ öï ï= ç ÷í ýç ÷ç ÷ç ÷è øï ïè øî þè øì üæ öæ öæ öæ öï ï= ç ÷ç ÷í ýç ÷ç ÷ç ÷ç ÷è øè øè øï ïè øî þ

=

å Õ

å Õ

å

L

L

% % %L

U

U

1 1 2 2, , ,1

,in n

n

ih h hi

w gg g gÎ Î Î=

ì üæ öæ öï ïí ýç ÷ç ÷

è øï ïè øî þÕLU

311

This completes the proof of Theorem 4.2. o 312

Theorem 4.3. Let ( ){ },ii i

i i ihh w gg

gÎ

=% U and ( ){ },ii i

i i ill xx

x vÎ

=% U ( 1,2, ,i n= L ) be two 313

collections of WHFEs, ( )1 2, , ,T

nw w w w= L be their weight vector with [ ]0,1iw Î 314

( 1,2, ,i n= L ) and 1

1n

ii

w=

=å , then 315

( )( ) ( )1 1 2 2

1 2 1 2

ATS-WHFWA , , ,

ATS-WHFWA , , , ATS-WHFWA , , ,

% % % % % %L% % % % % %L L

n n

n n

h l h l h l

h h h l l l

Å Å Å

= Å (16) 316

Proof. According to Eq. (9), we have 317

( ) ( )( )( ){ }1

,,

i ii i i ii i i i i ih l

h l f f f w g xg xg x v-

Î ÎÅ = + ×% % U 318

According to Theorem 4.1, we have 319

( )

( ) ( )( )( ) ( )

( ) ( ) ( )

1 1 1 1

1 1 1

1 1 2 2

1 1

, , , , ,1 1

1

, , ,1 1 1

ATS-WHFWA , , ,

,

,

i in n n n

i in n

n n

nn

i i i i ih h l li i

nn n

i i i i i ih hi i i

h l h l h l

f f f f f w

f f f w

g xg g x x

g xg g x

w g x v

w g w x v

- -Î Î Î Î

= =

-Î Î Î

= = =

Å Å Å

ì üæ öæ öï ï= + ×í ýç ÷ç ÷è øï ïè øî þ

ì üæ öæ öï ï= + ×í ýç ÷ç ÷è øï ïè øî þ

å Õ

å å Õ

L L

L

% % % % % %L

U

1 , , n nl lx ÎLU

320

On other hand, according to Theorem 4.1 and Eq. (9), we have 321

( ) ( )

( ) ( )1 1 2 2 1 1 2 2

1 2 1 2

1 1

, , , , , ,1 11 1

1

ATS-WHFWA , , , ATS-WHFWA , , ,

, ,L L

% % % % % %L L

U Ui in n n n

n n

n nn n

i i i i i ih h h l l li ii i

h h h l l l

f f w f f

f f f

g xg g g x x xw g w x v- -

Î Î Î Î Î Î= == =

- -

Å

æ ö æ öì ü ì üæ ö æ öæ ö æ öï ï ï ï= Åç ÷ ç ÷í ý í ýç ÷ ç ÷ç ÷ ç ÷ç ÷ ç ÷è ø è øï ï ï ïè ø è øî þ î þè ø è ø

=

å åÕ Õ

( ) ( )

( ) ( ) ( )

1 1 1 1

1 1

1 1

, , , , ,1 1 1 1

1

, , ,1 1 1

,

,

L L

L

U i in n n n

i in n

n nn n

i i i i i ih h l li i i i

nn n

i i i i i ih hi i i

f f f f w

f f f w

g xg g x x

g xg g

w g w x v

w g w x v

-Î Î Î Î

= = = =

-Î Î

= = =

ì üæ öæ öæ ö æ öæ ö æ öï ï+ ×ç ÷ç ÷í ýç ÷ ç ÷ç ÷ ç ÷ç ÷ç ÷è ø è øè ø è øï è ø ïè øî þì üæ öæ öï ï= + ×í ýç ÷ç ÷

è øï ïè øî þ

å å Õ Õ

å å Õ1 1 , ,LU

n nl lx xÎ Î

322

which completes the proof of Theorem 4.3. o 323 If the additive generator g is assigned different forms, then some specific weighted hesitant 324

fuzzy aggregation operators can be obtained as follows: 325 Case 1. If ( ) ( )logg t t= - , then the ATS-WHFWA operator reduces to the following form: 326

( ) ( )1 1 2 2

1 2 , , ,1 1

WHFWA , , , 1 1 ,i

in n

n n

n i ih h hi i

h h h ww

gg g gg

Î Î Î= =

ì üæ ö= - -í ýç ÷

è øî þÕ ÕL

% % %L U (17) 327

which is the weighted hesitant fuzzy weighted averaging (WHFWA) operator. 328 In fact, if ( ) ( )logg t t= - , then ( ) ( ) ( )1 log 1f t g t t= - = - - and ( )1 1 tf t e- -= - . Thus, 329

( ) ( )

( )

( )

1 1 2 2

1

1 1 2 2

1

11 2 , , ,

1 1

log 1

, , ,1

log 1

1

WHFWA , , , ,

1 ,

1 ,

in n

n

i ii

in n

ni

ii

i

nn

n i i ih h hi i

n

ih h hi

n

ii

h h h f f w

e w

e ww

gg g g

w g

gg g g

g

g

w g

=

=

-Î Î Î

= =

-

Î Î Î=

æ öç ÷-ç ÷è ø

=

ì üæ öæ öï ï= í ýç ÷ç ÷è øï ïè øî þ

ì üæ öåï ïç ÷= -í ýç ÷ï ïè øî þìæ öÕïç ÷= -íç ÷ç ÷è øî

å Õ

Õ

Õ

L

L

% % %L U

U

( )

1 1 2 2

1 1 2 2

, , ,

, , ,1 1

1 1 ,

n n

i

in n

h h h

n n

i ih h hi i

w

g g g

wgg g g

g

Î Î Î

Î Î Î= =

üïý

ï ïþì üæ ö

= - -í ýç ÷è øî þÕ Õ

L

L

U

U

330

Furthermore, if 1

iii

whg =#

for any 1,2, ,i n= L , where ih# is the number of the elements in 331

ih , then the Eq. (17) is transformed to 332

( ) ( )1 1 2 2

1 2 , , ,1

1

1HFWA , , , 1 1 ,i

n n

n

n i nh h hi

ii

h h hh

w

g g gg

Î Î Î=

=

ì üæ öï ïç ÷ï ïç ÷= - -í ýç ÷ï ï#ç ÷ï ïè øî þ

ÕÕ

L% % %L U (18) 333

which is the hesitant fuzzy weighted average (HFWA) operator proposed by Xia and Xu [15]. 334

Case 2. If ( ) 2log

tg t

t-æ ö= ç ÷

è ø, then the ATS-WHFWA operator reduces to the following form: 335

( )( ) ( )

( ) ( )1 1 2 2

1 11 2 , , ,

1

1 1

1 1WHFEWA , , , ,

1 1

i i

in ni i

n n

i i ni i

n in nh h hi

i ii i

h h h w

w w

gg g g w w

g g

g g

= =Î Î Î

=

= =

ì üæ ö+ - -ï ïç ÷ï ïç ÷= í ý

ç ÷ï ï+ + -ç ÷ï ïè øî þ

Õ ÕÕ

Õ ÕL

% % %L U (19) 336

which is the weighted hesitant fuzzy Einstein weighted averaging (WHFEWA) operator. 337

In fact, if ( ) 2log

tg t

t-æ ö= ç ÷

è ø, then ( ) ( ) 1

1 log1

tf t g t

t+æ ö= - = ç ÷-è ø

and ( )1 11

t

t

ef t

e- -

=+

. Thus, 338

( ) ( )1 1 2 2

1

1 1 2 2

1

11 2 , , ,

1 1

1log

1

, , , 1log 1

1

WHFEWA , , , ,

1,

1

in n

ni

iii

n in n ii

ii

nn

n i i ih h hi i

n

ih h hi

h h h f f w

ew

e

gg g g

gw

g

gg g g gw

g

w g

=

=

-Î Î Î

= =

æ ö+ç ÷ç ÷-è ø

Î Î Î æ ö+ç ÷ =ç ÷-è ø

ì üæ öæ öï ï= í ýç ÷ç ÷è øï ïè øî þ

ì üæ öåï ïç ÷-ï ï= ç ÷í ýåç ÷ï ïç ÷+ï ïè øî þ

å Õ

Õ

L

L

% % %L U

U

1

1 1 2 2

1

1log

1

, , , 1 1log1

1

1

1

1,

1

11

1,

11

1

ini

ii

in in n i

ii

i

ii

n

ih h hi

ni

ni i

in i

i

i i

ew

e

w

w

w

gg

gg g g gg

w

gw

gg

gg

=

=

æ öæ ö+ç ÷ç ÷ç ÷ç ÷-è øè ø

æ öÎ Î Î æ ö+ç ÷ =ç ÷ç ÷ç ÷-è øè ø

=

=

=

ì üæ öÕï ïç ÷-ï ïç ÷= í ýç ÷ï ïÕç ÷ï ï+è øî þ

æ öæ ö+ç ÷-ç ÷-ç ÷è ø= ç ÷æ ö+ç +ç ÷ç -è øè ø

Õ

ÕÕ

Õ

LU

( ) ( )

( ) ( )

1 1 2 2

1 1 2 2

, , ,

1 1, , ,

1

1 1

1 1,

1 1

n n

i i

in ni i

h h h

n n

i i ni i

in nh h hi

i ii i

w

g g g

w w

gg g g w w

g g

g g

Î Î Î

= =Î Î Î

=

= =

ì üï ïï ïí ýï ï÷ï ï÷î þì üæ ö

+ - -ï ïç ÷ï ïç ÷= í ýç ÷ï ï+ + -ç ÷ï ïè øî þ

Õ ÕÕ

Õ Õ

L

L

U

U 339

Furthermore, if 1

iii

whg =#

for any 1,2, ,i n= L , then the Eq. (19) is transformed to 340

( )( ) ( )

( ) ( )1 1 2 2

1 11 2 , , ,

1 1 1

1 11

HFEWA , , , ,1 1

i i

n ni i

n n

i ii i

n n n nh h h

i i ii i i

h h hh

w w

g g g w w

g g

g g

= =Î Î Î

= = =

ì üæ ö+ - -ï ïç ÷ï ïç ÷= í ý

ç ÷ï ï+ + - #ç ÷ï ïè øî þ

Õ Õ

Õ Õ ÕL

% % %L U (20) 341

which is the hesitant fuzzy Einstein weighted average (HFEWA) operator given by Wei and 342 Zhao [24]. 343

Case 3. If ( ) ( )1log

tg t

t

q qæ + - ö= ç ÷

è ø, 0q > , then the ATS-WHFWA operator reduces to the 344

following form: 345

( )( )( ) ( )

( )( ) ( ) ( )1 1 2 2

1 11 2 , , ,

1

1 1

1 1 1WHFHWA , , , ,

1 1 1 1L

% % %L Ui i

in ni i

n n

i i ni i

n in nh h hi

i ii i

h h h w

w w

gg g g w w

q g g

q g q g

= =Î Î Î

=

= =

ì üæ ö+ - - -ï ïç ÷ï ïç ÷= í ý

ç ÷ï ï+ - + - -ç ÷ï ïè øî þ

Õ ÕÕ

Õ Õ(21) 346

which is the weighted hesitant fuzzy Hammer weighted averaging (WHFHWA) operator. 347 Especially, if 1q = , then the WHFHWA operator reduces to the WHFWA operator; if 2q = , 348 then the WHFHWA operator reduces to the WHFEWA operator. 349

In fact, if ( ) ( )1log

tg t

t

q qæ + - ö= ç ÷

è ø, then ( ) ( ) ( )1 1

1 log1

tf t g t

t

qæ + - ö= - = ç ÷-è ø

and 350

( )1 11

t

t

ef t

eq- -

=- -

. Thus, 351

( ) ( )

( )

( )

1 1 2 2

1

1

11 2 , , ,

1 1

1 1log

1

1 1 1log1

WHFHWA , , , ,

1,

1

L% % %L U in n

ini

ii

in ii

ii

nn

n i i ih h hi i

n

ii

h h h f f w

ew

e

w

w

gg g g

q gg

gq gg

w g

q

=

=

-

Î Î Î= =

æ ö+ -æ öç ÷ç ÷ç ÷ç ÷-è øè ø

æ ö+ -æ öç ÷ =ç ÷ç ÷ç ÷-è øè ø

ì üæ öæ öï ï= í ýç ÷ç ÷è øï ïè øî þ

ì üæ öÕï ïç ÷-ïç ÷= í ýç ÷ï Õç ÷ï - -è øî

å Õ

Õ

( )( ) ( )

( )( ) ( ) ( )

1 1 2 2

1 1 2 2

, , ,

1 1

, , ,1

1 1

1 1 1,

1 1 1 1

L

L

U

U

n n

i i

in ni i

h h h

n n

i i ni i

in nh h hi

i ii i

w

g g g

w w

gg g g w w

q g g

q g q g

Î Î Î

= =Î Î Î

=

= =

ï

ïïþ

ì üæ ö+ - - -ï ïç ÷ï ïç ÷= í ý

ç ÷ï ï+ - + - -ç ÷ï ïè øî þ

Õ ÕÕ

Õ Õ

352

Furthermore, if 1

iii

whg =#

for any 1,2, ,i n= L , then the Eq. (21) is transformed to 353

( )( )( ) ( )

( )( ) ( ) ( )1 1 2 2

1 11 2 , , ,

1 1 1

1 1 11

HFHWA , , , ,1 1 1 1

L% % %L U

i i

n ni i

n n

i ii i

n n n nh h h

i i ii i i

h h hh

w w

g g g w w

q g g

q g q g

= =Î Î Î

= = =

ì üæ ö+ - - -ï ïç ÷ï ïç ÷= í ý

ç ÷ï ï+ - + - - #ç ÷ï ïè øî þ

Õ Õ

Õ Õ Õ (22) 354

which is the hesitant fuzzy Hammer weighted average (HFHWA) operator. Especially, if 355 1q = , then the HFHWA operator reduces to the HFWA operator; if 2q = , then the HFHWA 356

operator reduces to the HFEWA operator. 357

Case 4. If ( ) 1log

1tg t

qq-æ ö= ç ÷-è ø

, 0q > , then the ATS-WHFWA operator reduces to the 358

following form: 359 360

( ) ( )1 1 2 2

11 2 , , ,

1 1

WHFFWA , , , 1 log 1 1 ,i

i

in n

n n

n ih h hi i

h h h wwg

q gg g gq -

Î Î Î= =

ì üæ öæ öï ï= - + -í ýç ÷ç ÷è øï ïè øî þÕ ÕL

% % %L U (23) 361

which is the weighted hesitant fuzzy Frank weighted averaging (WHFFWA) operator. 362 Especially, if 1q ® , then the WHFFWA operator reduces to the WHFWA operator. 363

In fact, if ( ) 1log

1tg t

qq-æ ö= ç ÷-è ø

, then ( ) ( ) 1

11 log

1tf t g t

qq -

-æ ö= - = ç ÷-è ø and 364

( )1 11 log

t

t

ef t

eqq- æ ö- +

= - ç ÷è ø

. Thus, 365

( ) ( )1 1 2 2

11

11

11 2 , , ,

1 1

1log

1

11log

1

WHFFWA , , , ,

11 log ,

L% % %L U in n

n i

ii

n i i

ii

nn

n i i ih h hi i

n

ii

h h h f f w

ew

e

w

g

w

g

gg g g

qq

q gqq

w g

q-

=

-=

-

Î Î Î= =

æ ö-æ öç ÷ç ÷ç ÷-è øè ø

æ ö-æ öç ÷ =ç ÷ç ÷-è øè ø

ì üæ öæ öï ï= í ýç ÷ç ÷è øï ïè øî þ

æ öæ öÕç ÷ç ÷- +ç ÷ç ÷= -ç ÷ç ÷Õç ÷ç ÷

è øè ø

å Õ

Õ

( )

1 1 2 2

1 1 2 2

, , ,

11

, , ,1

11

1

1 1

11

11 log ,

11

1 log 1 1 ,

L

L

U

U

n n

i

i

iin n

i

ii

i

h h h

n

ni

ih h h ni

i

n

ii i

w

w

g g g

w

g

q gwg g g

g

wgq g

qqq

qq

q

Î Î Î

-=

Î Î Î=

-=

-

= =

ì üï ïï ïí ýï ïï ïî þì üæ öæ ö-æ öï ï- +ç ÷ç ÷ç ÷-ï ïè øç ÷ç ÷= -í ýç ÷ç ÷-æ öï ïç ÷ç ÷ç ÷ç ÷ç ÷ï ï-è øè øè øî þ

æ ö= - + -ç ÷

è ø

ÕÕ

Õ

Õ1 1 2 2, , ,LU

n n

n

h h hg g gÎ Î Î

ì üæ öï ïí ýç ÷ï ïè øî þ

Õ

366

Furthermore, if 1

iii

whg =#

for any 1,2, ,i n= L , then the Eq. (23) is transformed to 367

( ) ( )1 1 2 2

11 2 , , ,

1

1

1HFFWA , , , 1 log 1 1 ,

ii

n n

n

n nh h hi

ii

h h hh

wgqg g g

q -

Î Î Î=

=

ì üæ öï ïç ÷æ öï ïç ÷= - + -í ýç ÷ç ÷è øï ï#ç ÷ï ïè øî þ

ÕÕ

L% % %L U (24) 368

which is the hesitant fuzzy Frank weighted average (HFFWA) operator. 369 Based on the ATS-WHFWA operator and the geometric mean, here we define an 370 Archimedean t-conorm and t-norm based weighted hesitant fuzzy weighted geometric (ATS-371 WHFWG) operator: 372

Definition 4.2. Let ih% ( 1,2, ,i n= L ) be a collection of WHFEs, and let ( )1 2, , ,T

nw w w w= L 373

be the weight vector of ih% ( 1,2, ,i n= L ) with [ ]0,1iw Î and 1

1n

ii

w=

=å . Then, an 374

Archimedean t-conorm and t-norm based weighted hesitant fuzzy weighted geometric (ATS-375 WHFWG) operator is a mapping nH H®% % , where 376

( ) ( )1 21

ATS-WHFWG , , , in

n ii

h h h hw=

= Ä% % % %L (25) 377

Theorem 4.4. Let ( ){ },ii i

i i ihh w gg

gÎ

=% U ( 1,2, ,i n= L ) be a collection of WHFEs, and 378

( )1 2, , ,T

nw w w w= L be the weight vector of ih% ( 1,2, ,i n= L ), where iw indicates the 379

importance degree of ih% , satisfying [ ]0,1iw Î and 1

1n

ii

w=

=å , then the aggregated value by 380

using the ATS-WHFWG operator is also a WHFE, and 381

( ) ( )1 1 2 2

11 2 , , ,

1 1

ATS-WHFWG , , , ,in n

nn

n i i ih h hi i

h h h g g w gg g gw g-

Î Î Î= =

ì üæ öæ öï ï= í ýç ÷ç ÷è øï ïè øî þå ÕL

% % %L U (26) 382

Theorem 4.5. Let ih% ( 1,2, ,i n= L ) be a collection of WHFEs, ( )1 2, , ,T

nw w w w= L be their 383

weight vector with [ ]0,1iw Î ( 1,2, ,i n= L ) and 1

1n

ii

w=

=å , if 0r > , then 384

( ) ( )( )1 2 1 2ATS-WHFWG , , , ATS-WHFWG , , ,r

r r rn nh h h h h h=% % % % % %L L (27) 385

Theorem 4.6. Let ( ){ },ii i

i i ihh w gg

gÎ

=% U and ( ){ },ii i

i i ill xx

x vÎ

=% U ( 1,2, ,i n= L ) be two 386

collections of WHFEs, ( )1 2, , ,T

nw w w w= L be their weight vector with [ ]0,1iw Î 387

( 1,2, ,i n= L ) and 1

1n

ii

w=

=å , then 388

( )( ) ( )1 1 2 2

1 2 1 2

ATS-WHFWG , , ,

ATS-WHFWG , , , ATS-WHFWG , , ,

% % % % % %L% % % % % %L L

n n

n n

h l h l h l

h h h l l l

Ä Ä Ä

= Ä (28) 389

In what follows, we will investigate the relationship between ATS-WHFWA operator and 390 ATS-WHFWG operator. 391

Theorem 4.7. Let ih% ( 1,2, ,i n= L ) be a collection of WHFEs, ( )1 2, , ,T

nw w w w= L be their 392

weight vector with [ ]0,1iw Î ( 1,2, ,i n= L ) and 1

1n

ii

w=

=å , then we have 393

(1) ( ) ( )( )1 2 1 2ATS-WHFWA , , , ATS-WHFWG , , ,c

c c cn nh h h h h h=% % % % % %L L (29) 394

(2) ( ) ( )( )1 2 1 2ATS-WHFWG , , , ATS-WHFWA , , ,c

c c cn nh h h h h h=% % % % % %L L (30) 395

Proof. (1) According to Eqs. (6), (14), and (26), we can get 396

( ) ( )

( )

( )

1 1 2 2

1 1 2 2

11 2 , , ,

1 1

1

, , ,1 1

1

1 1

ATS-WHFWA , , , 1 ,

1 ,

,

in n

in n

i

nnc c c

n i i ih h hi i

nn

i i ih h hi i

nn

i i ii i

h h h f f w

g g w

g g w

gg g g

gg g g

g

w g

w g

w g

-Î Î Î

= =

-Î Î Î

= =

-

= =

ì üæ öæ öï ï= -í ýç ÷ç ÷è øï ïè øî þ

ì üæ öæ öï ï= -í ýç ÷ç ÷è øï ïè øî þ

ìæ öæ öï= íç ÷ç ÷è øè ø

å Õ

å Õ

å Õ

L

L

% % %L U

U

( )( )1 1 2 2, , ,

1 2 ATS-WHFWG , , ,

n n

c

h h h

c

nh h h

g g gÎ Î Î

æ öüïç ÷ýç ÷ï ïî þè ø

=

L

% % %L

U 397

(2) According to Eqs. (6), (14), and (26), we have 398

( ) ( )

( )

( )

1 1 2 2

1 1 2 2

11 2 , , ,

1 1

1

, , ,1 1

1

1 1

ATS-WHFWG , , , 1 ,

1 ,

,

in n

in n

i

nnc c c

n i i ih h hi i

nn

i i ih h hi i

nn

i i ii i

h h h g g w

f f w

f f w

gg g g

gg g g

g

w g

w g

w g

-Î Î Î

= =

-Î Î Î

= =

-

= =

ì üæ öæ öï ï= -í ýç ÷ç ÷è øï ïè øî þ

ì üæ öæ öï ï= -í ýç ÷ç ÷è øï ïè øî þ

ìæ öæ öï= íç ÷ç ÷è øè ø

å Õ

å Õ

å Õ

L

L

% % %L U

U

( )( )1 1 2 2, , ,

1 2ATS-WHFWA , , ,

n n

c

h h h

c

nh h h

g g gÎ Î Î

æ öüïç ÷ýç ÷ï ïî þè ø

=

L

% % %L

U 399

This completes the proof of Theorem 4.7. 400

Theorem 4.8. Let ih% ( 1,2, ,i n= L ) be a collection of WHFEs, ( )1 2, , ,T

nw w w w= L be their 401

weight vector with [ ]0,1iw Î ( 1,2, ,i n= L ) and 1

1n

ii

w=

=å , if % %ih h= for all i , then 402

( )1 2ATS-WHFWA , , ,% % % %L nh h h h= 403

Theorem 4.9. Let ih% ( 1,2, ,i n= L ) be a collection of WHFEs, ( )1 2, , ,T

nw w w w= L be their 404

weight vector with [ ]0,1iw Î ( 1,2, ,i n= L ) and 1

1n

ii

w=

=å , if %h is a WHFE, then 405

( ) ( )1 2 1 2ATS-WHFWA , , , ATS-WHFWA , , ,% % % % % % % % % %L Ln nh h h h h h h h h hÅ Å Å = Å 406

Theorem 4.10. Let ih% ( 1,2, ,i n= L ) be a collection of WHFEs, ( )1 2, , ,T

nw w w w= L be their 407

weight vector with [ ]0,1iw Î ( 1,2, ,i n= L ) and 1

1n

ii

w=

=å , if 0r > and %h is a WHFE, then 408

( ) ( )1 2 1 2ATS-WHFWA , , , ATS-WHFWA , , ,% % % % % % % % % %L Ln nrh h rh h rh h r h h h hÅ Å Å = Å 409

Theorem 4.11. Let ih% ( 1,2, ,i n= L ) be a collection of WHFEs, ( )1 2, , ,T

nw w w w= L be their 410

weight vector with [ ]0,1iw Î ( 1,2, ,i n= L ) and 1

1n

ii

w=

=å , if % %ih h= for all i , then 411

( )1 2ATS-WHFWG , , ,% % % %L nh h h h= 412

Theorem 4.12. Let ih% ( 1,2, ,i n= L ) be a collection of WHFEs, ( )1 2, , ,T

nw w w w= L be their 413

weight vector with [ ]0,1iw Î ( 1,2, ,i n= L ) and 1

1n

ii

w=

=å , if %h is a WHFE, then 414

( ) ( )1 2 1 2ATS-WHFWG , , , ATS-WHFWG , , ,% % % % % % % % % %L Ln nh h h h h h h h h hÄ Ä Ä = Ä 415

Theorem 4.13. Let ih% ( 1,2, ,i n= L ) be a collection of WHFEs, ( )1 2, , ,T

nw w w w= L be their 416

weight vector with [ ]0,1iw Î ( 1,2, ,i n= L ) and 1

1n

ii

w=

=å , if 0r > and %h is a WHFE, then 417

( ) ( )( )1 2 1 2ATS-WHFWA , , , ATS-WHFWA , , ,% % % % % % % % % %L Lr

r r rn nh h h h h h h h h hÄ Ä Ä = Ä 418

If the additive generator g is assigned different forms, then some specific ATS-WHFWG 419

operators can be obtained as follows: 420 Case 1. If ( ) ( )logg t t= - , then the ATS-WHFWG operator reduces to the following form: 421

( )1 1 2 2

1 2 , , ,1 1

WHFWG , , , ,i

in n

n n

n i ih h hi i

h h h wwgg g g

gÎ Î Î

= =

ì üæ ö= í ýç ÷

è øî þÕ ÕL

% % %L U (31) 422

which is the weighted hesitant fuzzy weighted geometric (WHFWG) operator. Furthermore, if 423 1

iii

whg =#

for any 1,2, ,i n= L , where ih# is the number of the elements in ih , then the Eq. 424

(31) is transformed to 425

( )1 1 2 2

1 2 , , ,1

1

1HFWG , , , ,i

n n

n

n i nh h hi

ii

h h hh

wl g g g

gÎ Î Î

=

=

ì üæ öï ïç ÷ï ïç ÷= í ýç ÷ï ï#ç ÷ï ïè øî þ

ÕÕ

L% % %L U (32) 426

which is the hesitant fuzzy weighted geometric (HFWG) operator proposed by Xia and Xu 427 [15]. 428

Case 2. If ( ) 2log

tg t

t-æ ö= ç ÷

è ø, then the ATS-WHFWG operator reduces to the following form: 429

( )( )

1 1 2 2

11 2 , , ,

1

1 1

2WHFEWG , , , ,

2

i

in ni i

n

i ni

n in nh h hi

i ii i

h h h w

w

gg g g w w

g

g g

=Î Î Î

=

= =

ì üæ öï ïç ÷ï ïç ÷= í ýç ÷ï ï- +ç ÷ï ïè øî þ

ÕÕ

Õ ÕL

% % %L U (33) 430

which is the weighted hesitant fuzzy Einstein weighted geometric (WHFEWG) operator. 431

Furthermore, if 1

iii

whg =#

for any 1,2, ,i n= L , then the Eq. (33) is transformed to 432

( )( )

1 1 2 2

11 2 , , ,

1 1 1

21

HFEWG , , , ,2

i

n ni i

n

ii

n n n nh h h

i i ii i i

h h hh

w

l g g g w w

g

g g

=Î Î Î

= = =

ì üæ öï ïç ÷ï ïç ÷= í ýç ÷ï ï- + #ç ÷ï ïè øî þ

Õ

Õ Õ ÕL

% % %L U (34) 433

which is the hesitant fuzzy Einstein weighted geometric (HFEWG) operator given by Wei and 434 Zhao [24]. 435

Case 3. If ( ) ( )1log

tg t

t

q qæ + - ö= ç ÷

è ø, 0q > , then the ATS-WHFWG operator reduces to the 436

following form: 437 438

( )( )( )( ) ( )

1 1 2 2

11 2 , , ,

1

1 1

WHFHWG , , , ,1 1 1 1

L% % %L U

i

in ni i

n

i ni

n in nh h hi

i ii i

h h h w

w

gg g g w w

q g

q g q g

=Î Î Î

=

= =

ì üæ öï ïç ÷ï ïç ÷= í ýç ÷ï ï+ - - + -ç ÷ï ïè øî þ

ÕÕ

Õ Õ (35) 439

which is the weighted hesitant fuzzy Hammer weighted geometric (WHFHWG) operator. 440 Especially, if 1q = , then the WHFHWG operator reduces to the WHFWG operator; if 2q = , 441 then the WHFHWG operator reduces to the WHFEWG operator. 442

Furthermore, if 1

iii

whg =#

for any 1,2, ,i n= L , then the Eq. (35) is transformed to 443

( )( ) ( )( ) ( )

1 1 2 2

11 2 , , ,

1 1 1

1HFHWG , , , ,

1 1 1 1L

% % %L Ui

n ni i

n

ii

n n n nh h h

i i ii i i

h h hh

w

g g g w w

q g

q g q g

=Î Î Î

= = =

ì üæ öï ïç ÷ï ïç ÷= í ýç ÷ï ï+ - - + - #ç ÷ï ïè øî þ

Õ

Õ Õ Õ (36) 444

which is the hesitant fuzzy Hammer weighted geometric (HFHWG) operator. Especially, if 445 1q = , then the HFHWG operator reduces to the HFWG operator; if 2q = , then the HFHWG 446

operator reduces to the HFEWG operator. 447

Case 4. If ( ) 1log

1tg t

qq-æ ö= ç ÷-è ø

, 0q > , then the ATS-WHFWG operator reduces to the 448

following form: 449 450

( ) ( )1 1 2 2

1 2 , , ,1 1

WHFFWG , , , log 1 1 ,i

i

in n

n n

n ih h hi i

h h h wwg

q gg g gq

Î Î Î= =

ì üæ öæ öï ï= + -í ýç ÷ç ÷è øï ïè øî þÕ ÕL

% % %L U (37) 451

which is the weighted hesitant fuzzy Frank weighted geometric (WHFFWG) operator. 452

Furthermore, if 1

iii

whg =#

for any 1,2, ,i n= L , then the Eq. (37) is transformed to 453

( ) ( )1 1 2 2

1 2 , , ,1

1

1HFFWG , , , log 1 1 ,

ii

n n

n

n nh h hi

ii

h h hh

wgqg g g

qÎ Î Î

=

=

ì üæ öï ïç ÷æ öï ïç ÷= + -í ýç ÷ç ÷è øï ï#ç ÷ï ïè øî þ

ÕÕ

L% % %L U (38) 454

which is the hesitant fuzzy Frank weighted geometric (HFFWG) operator. 455 456 5. AN APPROACH TO MULTI-CRITERIA DECISION MAKING WITH WEIGHTED 457 HESITANT FUZZY INFORMATION 458 459 In this section, we shall utilize the proposed operators to develop an approach to multi-460 criteria decision making (MCDM) with weighted hesitant fuzzy information. For a MCDM 461 problem, let { }1 2, , , mY Y Y Y= L be a set of m alternatives, { }1 2, , , nG G G G= K be a 462

collection of n criteria, whose weight vector is ( )1 2, , ,T

nw w w w= L , with [ ]0,1jw Î , 463

1,2, ,j n= L , and 1

1n

jj

w=

=å , where jw denotes the importance degree of the criterion jG . 464

The decision makers provide all the possible values with their corresponding weights for the 465 alternative iY with respect to the criterion jG represented by a WHFE 466

( ){ },ijij ij

ij ij ijrr w gg

gÎ

=% U . All ijr% ( 1,2, ,i m= L ; 1,2, ,j n= L ) construct the weighted hesitant 467

fuzzy decision matrix ( )ij m nR r

´= % (see Table 1). 468

Table 1. The weighted hesitant fuzzy decision matrix R 469

470 In general, there are benefit attributes (i.e., the bigger the attribute values the better) and 471 cost attributes (i.e., the smaller the attribute values the better) in a MCDM problem. In such 472 cases, we need transform the attribute values of cost type into the attribute values of benefit 473

1G L jG L nG

1Y 11r% L 1 jr% L 1nr%

L L L L L L

iY 1ir% L ijr% L inr%

L L L L L L

mY 1mr% L mjr% L mnr%

type, i.e., transform the weighted hesitant fuzzy decision matrix ( )ij m nR r

´= % into a 474

normalized weighted hesitant fuzzy decision matrix ( )ij m nA a

´= % by the method given by Xu 475

and Hu [30], where 476 , for benefit attribute

, for cost attribute

ij j

ij cij j

r Ga

r G

ìï= íïî

%% % , 1,2, ,i m= L , 1,2, ,j n= L , (39) 477

where cijr% is the complement of ijr% such that ( ){ }1 ,

ijij ij

cij ij ijrr w gg

gÎ

= -% U . 478

Step 1. Transform the weighted hesitant fuzzy decision matrix ( )ij m nR r

´= % into the 479

normalized weighted hesitant fuzzy decision matrix ( )ij m nA a

´= % based on Eq. (39). 480

Step 2. Utilize the ATS-WHFWA operator (Eq. (14)) 481

( ) ( )1 1 2 2

11 2 , , ,

1 1

ATS-WHFWA , , , ,iji i i i in in

nn

i i i in j ij ija a aj j

a a a a f f w gg g gw g-

Î Î Î= =

ì üæ öæ öï ï= = ç ÷í ýç ÷ç ÷è øï ïè øî þå ÕL% % % %L U (40) 482

or the ATS-WHFWG operator (Eq. (26)) 483

( ) ( )1 1 2 2

11 2 , , ,

1 1

ATS-WHFWG , , , ,iji i i i in in

nn

i i i in j ij ija a aj j

a a a a g g w gg g gw g-

Î Î Î= =

ì üæ öæ öï ï= = ç ÷í ýç ÷ç ÷è øï ïè øî þå ÕL% % % %L U (41) 484

to aggregate all the performance values ija% ( 1,2, ,j n= L ) in the ith line of A , and then 485

derive the overall performance value ia% ( 1,2, ,i m= L ) of the alternative iY ( 1,2, ,i m= L ). 486

Step 3. According to Definition 3.2, calculate the scores ( )is a% ( 1,2, ,i m= L ) of ia% 487

( 1,2, ,i m= L ) and rank all the alternatives iY ( 1,2, ,i m= L ) according to ( )is a% in 488

descending order. 489 Step 4. End. 490 In the following, we use a numerical example adapted from [20,22] to illustrate our approach. 491 Example 5.1 [20,22]. Suppose that a factory intends to select a new site for new buildings. 492 Assume that there are three possible alternatives iY ( 1,2,3i = ) and three criteria are 493

considered to decide which site to choose: (1) 1G (price); (2) 2G (location); and (3) 3G 494

(environment). The weight vector of three criteria jG ( )1, 2,3j = is ( )0.3,0.2,0.5Tw = . 495

Suppose that the characteristics of the alternatives iY ( 1,2,3i = ) with respect to the criteria 496

jG ( 1,2,3j = ) are denoted by the WHFE ( ){ },ijij ij

ij ij ijrr w gg

gÎ

=% U , where ijg indicates the 497

possible degree to which the alternative iY satisfies the criterion jG and ijijw g is the weight 498

of ijg . All ijr% ( 1,2,3i = ; 1,2,3j = ) are contained in the weighted hesitant fuzzy decision 499

matrix ( )ij m nR r

´= % (see Table 2). 500

501 502 503 504 505 506 507 508

Table 2. The weighted hesitant fuzzy decision matrix R 509

510 Step 1. Because all of the criteria jG ( 1,2,3j = ) are of the benefit type, the performance 511

values of the alternatives iY ( 1,2,3i = ) do not require normalization. 512

Step 2. Utilize the WHFHWA operator (Eq. (21)) (suppose that 1q = ) to aggregate all the 513 preference values ijr% ( 1,2,3j = ) in the ith line of R , and then derive the overall 514

performance value ir% ( 1,2,3i = ) of the alternative iY ( 1,2,3i = ): 515

( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( )

1

0.5528,0.0720 , 0.4708,0.1680 , 0.5150,0.0180 , 0.4262,0.0420 ,

0.5218,0.0720 , 0.4342,0.1680 , 0.4814,0.0180 , 0.3864,0.0420 ,

0.4949,0.0960 , 0.4024,0.2240 , 0.4523,0.0240 , 0.3519,0.0560

r

ì üï ï

= í ýï ïî þ

% 516

( ) ( ) ( )( ) ( ) ( )2

0.5184,0.1200 , 0.4798,0.1800 , 0.4438,0.3000 ,

0.4956,0.0800 , 0.4551,0.1200 , 0.4175,0.2000r

ì üï ï= í ýï ïî þ

% 517

( ) ( ) ( )( ) ( ) ( )3

0.5662,0.0500 , 0.5399,0.0500 , 0.5405,0.1500 ,

0.5126,0.1500 , 0.5196,0.3000 , 0.4904,0.3000r

ì üï ï= í ýï ïî þ

% 518

Step 3. According to Definition 3.2, we calculate the score values ( )is r% ( 1,2,3i = ) of ir% 519

( 1,2,3i = ) as: 520

( )1 0.4497s r =% , ( )2 0.4595s r =% , ( )3 0.5163s r =% . 521

Step 4. Since ( ) ( ) ( )3 2 1s r s r s r> >% % % , then we get the ranking of the alternatives iY ( )1,2,3i = 522

as 3 2 1Y Y Y> > . Thus, the best alternative is 3Y . 523

In the following, we will analyze how different values of the parameter q change the 524 aggregation results. As q is assigned different values between 0 and 30, the score functions 525 of the alternatives obtained by the WHFHWA operator are shown in Fig. 1. 526 527 528 529

1G 2G 3G

1Y {(0.6, 0.3), (0.5, 0.3), (0.4, 0.4)}

{(0.6, 0.8), (0.4, 0.2)} {(0.5, 0.3), (0.3, 0.7)}

2Y {(0.4, 0.6), (0.3, 0.4)} {(0.8, 1)} {(0.4, 0.2), (0.3, 0.3),

(0.2, 0.5)}

3Y {(0.8, 1)} {(0.7, 0.1), (0.6, 0.3),

(0.5, 0.6)} {(0.2, 0.5), (0.1, 0.5)}

0 5 10 15 20 25 300.4

0.45

0.5

0.55

0.6

0.65

thet

Sco

re fu

nctio

nss(Y1)s(Y2)s(Y3)thet=1.5087

s(Y1)=s(Y2)=0.4472

thet=6.2170s(Y1)=s(Y3)=0.4403

530 Fig. 1. Score functions for alternatives obtained by the WHFHWA operator 531 Fig. 1 demonstrates that all the score functions decrease as q increases from 0 to 30, from 532 which we can find that 533 (1) when ( ]0,1.5087q Î , the ranking of the four alternatives is 3 2 1Y Y Y> > and the best choice 534

is 3Y . 535

(2) when ( ]1.5087,6.2170q Î , the ranking of the four alternatives is 3 1 2Y Y Y> > and the best 536

choice is 3Y . 537

(3) when ( ]6.2170,30q Î , the ranking of the four alternatives is 1 3 2Y Y Y> > and the best 538

choice is 1Y . 539 In the above example, if we use the WHFHWG operator instead of the WHFHWA operator to 540 aggregate the values of the alternatives, then the score functions of the alternatives are 541 shown in Fig. 2. From Fig. 2, we can see that all the score functions obtained by the 542 WHFHWG operator increase as the parameter q increases from 0 to 30 and the 543 aggregation arguments are kept fixed. From Fig. 2, we can also see that as q increases 544 from 0 to 30, the ranking of alternatives is always 1 2 3Y Y Y> > and the best choice is always 545

1Y . 546 547

0 5 10 15 20 25 300.2

0.25

0.3

0.35

0.4

0.45

thet

Sco

re fu

nctio

ns

s(Y1)s(Y2)s(Y3)

548 Fig. 2. Score functions for alternatives obtained by the WHFHWG operator 549

0 5 10 15 20 25 300

0.1

0.2

0.3

0.4

thet

Dev

iatio

n va

lues

Y1Y2Y3

550 Fig. 3. Deviation values for alternatives between the WHFHWA and WHFHWG 551 operators 552 Fig. 3 illustrates the deviation values between the score functions obtained by the WHFHWA 553 operator and the ones obtained by the WHFHWG operator, from which we can find that the 554 values obtained by the WHFHWA operator are greater than the ones obtained by the 555 WHFHWG operator for the same value of the parameter q and the same aggregation 556 values, and the deviation values decrease as the value of the parameter q increases. 557 Fig. 3 indicates that the WHFHWA operator can obtain more favorable (or optimistic) 558 expectations, and therefore can be considered as an optimistic operator, while the 559 WHFHWG operator has more unfavorable (or pessimistic) expectations, and therefore can 560 be considered as a pessimistic operator. The values of the parameter q can be considered 561 as the optimistic or pessimistic levels. According to Figs. 1, 2, and 3, we can conclude that 562

the decision makers who take a gloomy view of the prospects could use the WHFHWG 563 operator and choose the smaller values of the parameter q , while the decision makers who 564 are optimistic could use the WHFHWA operator and choose the smaller values of the 565 parameter q . 566 If we use the WHFFWA (or WHFFWG) operator instead of the WHFHWA (or WHFHWG) 567 operator to aggregate the attribute values of alternatives, then the score functions of 568 alternatives are given in Figs. 4 and 5, respectively. Fig. 4 shows that all the score functions 569 obtained by the WHFFWA operator decrease as the parameter q increases from 0 to 30, 570 from which we can get that 571 (1) when ( ]0,2.3659q Î , the ranking of the four alternatives is 3 2 1Y Y Y> > and the best choice 572

is 3Y . 573

(2) when ( ]2.3659,30q Î , the ranking of the four alternatives is 3 1 2Y Y Y> > and the best 574

choice is 3Y . 575

0 5 10 15 20 25 300.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

thet

Sco

re fu

nctio

ns

s(Y1)s(Y2)s(Y3)

thet=2.3659s(Y1)=s(Y2)=0.4473

576 Fig. 4. Score functions for alternatives obtained by the WHFFWA operator 577

0 10 20 300.2

0.25

0.3

0.35

0.4

0.45

thet

Sco

re fu

nctio

ns

s(Y1)s(Y2)s(Y3)

578 Fig. 5. Score functions for alternatives obtained by the WHFFWG operator 579 Fig. 5 illustrates that all the score functions obtained by the WHFFWG operator increase as 580 the parameter q increases from 0 to 30, from which we can see that as q increases from 0 581 to 30, the ranking of alternatives is always 1 2 3Y Y Y> > and the best choice is always 1Y . 582

0 5 10 15 20 25 300

0.1

0.2

0.3

0.4

0.5

thet

Dev

iatio

n va

lues

Y1Y2Y3

583 Fig. 6. Deviation values for alternatives between the WHFFWA and WHFFWG 584 operators 585 Fig. 6 illustrates the deviation values between the score functions obtained by the WHFFWA 586 operator and the ones obtained by the WHFFWG operator, from which we can find that the 587

values obtained by the WHFFWA operator are greater than the ones obtained by the 588 WHFFWG operator for the same value of the parameter q and the same aggregation 589 values, and the deviation values decrease as the value of the parameter q increases. 590 Fig. 6 indicates that the WHFFWA operator can obtain more favorable (or optimistic) 591 expectations, and therefore can be considered as an optimistic operator, while the 592 WHFFWG operator has more unfavorable (or pessimistic) expectations, and therefore can 593 be considered as a pessimistic operator. The values of the parameter q can be considered 594 as the optimistic or pessimistic levels. According to Figs. 4, 5, and 6, we can conclude that 595 the decision makers who take a gloomy view of the prospects could use the WHFFWG 596 operator and choose the smaller values of the parameter q , while the decision makers who 597 are optimistic could use the WHFFWA operator and choose the smaller values of the 598 parameter q . 599 Based on the above analysis, we can see that the parameter q reflects the decision makers’ 600 preferences and the decision makers can choose the proper values of q according to their 601 preferences. By choosing different values of the parameter q , we can derive different score 602 functions, and then derive the different rankings of the alternatives and the different optimal 603 alternatives. That is, the final optimal decisions based on different values of the parameter q 604 could be different. Therefore, the developed aggregation operators with the parameters can 605 provide us with more choices and more flexibility than the existing ones due to the fact they 606 allow us to choose different values of the parameter in the light of the different practical 607 situations. 608 Example 5.2 (Continued with Example 5.1). In Example 5.1, if we do not consider the 609 importance of all of the possible values for an alternative under an attribute and only use the 610 HFEs to represent the performance values of an alternative under an attribute, then the 611 weighted hesitant fuzzy decision matrix R reduces to the hesitant fuzzy decision matrix 612

( )3 3

%ijR r´

¢ ¢= (see Table 3). 613

Table 3. The hesitant fuzzy decision matrix R¢ 614 615 616 617 618 619 620 621 We utilize the WHFHWA operator (Eq. (21)) (suppose that 1q = ) to aggregate all the 622 preference values %ijr¢ ( 1,2,3j = ) in the ith line of R¢ , and then derive the overall 623

performance value %ir¢ ( 1,2,3i = ) of the alternative iY ( 1,2,3i = ): 624

1

0.5528,0.4708,0.5150,0.4262,0.5218,0.4342,

0.4814,0.3864,0.4949,0.4024,0.4523,0.3519%r ì ü¢= í ý

î þ 625

{ }2 0.5184,0.4798,0.4438,0.4956,0.4551,0.4175%r¢ = 626

{ }3 0.5662,0.5399,0.5405,0.5126,0.5196,0.4904%r¢ = 627

According to Definition 3.2, we calculate the score values ( )%is r¢ ( 1,2,3i = ) of %ir¢ ( 1,2,3i = ) 628

as: 629

( )1 0.5282%s r¢ = , ( )2 0.4575%s r¢ = , ( )3 0.4684%s r¢ = . 630

Since ( ) ( ) ( )1 3 2% % %s r s r s r¢ ¢ ¢> > , then we get the ranking of the alternatives iY ( )1,2,3i = as 631

1 3 2Y Y Y> > . Thus, the best alternative is 1Y . 632

1G 2G 3G

1Y {0.6, 0.5, 0.4} {0.6, 0.4} {0.5, 0.3}

2Y {0.4, 0.3} {0.8} {0.4, 0.3, 0.2}

3Y {0.8} {0.7, 0.6, 0.5} {0.2, 0.1}

From the results of calculations, one can find a difference in the ranking results derived in 633 Examples 5.1 and 5.2. The reason is that Example 5.1 uses the WHFEs to represent the 634 preference values, which consider the importance of all of the possible values for an 635 alternative under an attribute; while, Example 5.2 uses the HFEs to represent the preference 636 values, which do not consider the importance of all of the possible values for an alternative 637 under an attribute. 638 In fact, suppose that ten decision makers are required to anonymously provide their 639 evaluations about the alternatives with regard to the attributes. For the evaluations about the 640 alternative 1Y with regard to the attribute 1G , assume that three decision makers provide 0.6, 641 three decision makers provide 0.5, the remaining four decision makers provide 0.4, and 642 these ten decision makers cannot persuade each other to change their opinions. When we 643 consider a MAGDM problem, if two or more decision makers who are familiar with this area 644 give the same preferences, then their preferences will be close to group preference. In such 645 cases, the value repeated many times may be more important than the one repeated only 646 one time. Therefore, the most likely evaluation about the alternative 1Y with regard to the 647

attribute 1G should be 0.4. For the evaluations about the alternative 1Y with regard to the 648

attribute 2G , assume that eight decision makers provide 0.6, the remaining two decision 649 makers provide 0.4, and these ten decision makers cannot persuade each other to change 650 their opinions. Then, the most likely evaluation about the alternative 1Y with regard to the 651

attribute 2G should be 0.6. For the evaluations about the alternative 1Y with regard to the 652

attribute 3G , assume that three decision makers provide 0.5, the remaining seven decision 653 makers provide 0.3, and these ten decision makers cannot persuade each other to change 654 their opinions. Then, the most likely evaluation about the alternative 1Y with regard to the 655

attribute 3G should be 0.3. If we use the WHFHWA operator (Eq. (21)) (suppose that 1q = ) 656 to aggregate 0.4, 0.6, and 0.3, then the most likely overall performance value of the 657 alternative 1Y should be 0.4024: 658

We can easily see that the score value ( )1 0.4497s r =% of the alternative 1Y obtained in 659

Example 5.1 is much closer to the most likely overall performance value 0.4024 than the 660 score value ( )1 0.5282%s r¢ = of the alternative 1Y obtained in Example 5.2. As a result, the 661

WHFEs are more reasonable and reliable than the HFEs in some practical applications. 662 663 6. CONCLUSIONS 664 665 Considering that the classical hesitant fuzzy set does not consider the importance of several 666 possible membership degrees of each element, in this paper, we have proposed a new 667 generalization of the classical hesitant fuzzy set, which we call the WHFS. The WHFS adds 668 a weight vector to several possible membership degrees of each element of the classical 669 hesitant fuzzy set, which denotes the importance of several possible membership degrees. 670 Then, based on Archimedean t-conorm and t-norm, we have defined some operational laws 671 for WHFEs and studied their properties, based on which, we have developed two weighted 672 hesitant fuzzy aggregation operators, including the ATS-WHFWA and ATS-WHFWG 673 operators, and investigated some desired properties of two new operators. Furthermore, 674 when the additive generator g is assigned different forms, some special cases of two new 675

operators have been obtained, such as the WHFWA, WHFEWA, WHFHWA, WHFFWA, 676 WHFWG, WHFEWG, WHFHWG, and WHFFWG operators. Finally, we have developed an 677 approach based on the proposed operators for multi-criteria decision making with weighted 678 hesitant fuzzy information, and the proposed operators and approach have been illustrated 679

by a practical example involving a detailed analysis of the variation trend of the score 680 functions and rankings of the alternatives with respect to the parameter q . 681 682 ACKNOWLEDGEMENTS 683 684 The authors thank the anonymous referees for their valuable suggestions in improving this 685 paper. This work is supported by the National Natural Science Foundation of China (Grant 686 Nos. 71271070 and 61375075) and the Natural Science Foundation of Hebei Province of 687 China (Grant Nos. F2012201020 and A2012201033). 688 689 COMPETING INTERESTS 690 691 AUTHORS’ CONTRIBUTIONS 692 693 ‘Zhiming Zhang’ designed the study, performed the statistical analysis, wrote the protocol, 694 and wrote the first draft of the manuscript. ‘Chong Wu’ managed the analyses of the study 695 and the literature searches. All authors read and approved the final manuscript. 696 697 REFERENCES 698 699 1. Atanassov K. Intuitionistic fuzzy sets. Fuzzy Sets and Systems. 1986;20:87-96. 700 2. Zadeh L. The concept of a linguistic variable and its application to approximate reasoning, 701 Part 1. Information Sciences. 1975;8:199-249. 702 3. Atanassov K, Gargov G. Interval valued intuitionistic fuzzy sets. Fuzzy Sets and Systems. 703 1989;31:343-349. 704 4. Mendel JM. Uncertain Rule-Based Fuzzy Logic Systems: Introduction and New Directions. 705 Prentice-Hall PTR; 2001. 706 5. Torra V. Hesitant fuzzy sets. International Journal of Intelligent Systems. 2010;25:529-539. 707 6. Torra V, Narukawa Y. On hesitant fuzzy sets and decision, in: The 18th IEEE International 708 Conference on Fuzzy Systems, Jeju Island, Korea, 2009. pp. 1378-1382. 709 7. Xu ZS, Xia MM. Distance and similarity measures for hesitant fuzzy sets. Information 710 Sciences. 2011;181:2128-2138. 711 8. Xu ZS, Xia MM. On distance and correlation measures of hesitant fuzzy information. 712 International Journal of Intelligent Systems. 2011;26:410-425. 713 9. Farhadinia B. Information measures for hesitant fuzzy sets and interval-valued hesitant 714 fuzzy sets. Information Sciences. 2013;240:129-144. 715 10. Chen N, Xu ZS, Xia MM. Correlation coefficients of hesitant fuzzy sets and their 716 applications to clustering analysis. Applied Mathematical Modelling. 2013;37(4):2197-2211. 717 11. Chen N, Xu ZS, Xia MM. Interval-valued hesitant preference relations and their 718 applications to group decision making. Knowledge-Based Systems. 2013;37:528-540. 719 12. Peng DH, Gao CY, Gao ZF. Generalized hesitant fuzzy synergetic weighted distance 720 measures and their application to multiple criteria decision-making. Applied Mathematical 721 Modelling. 2013;37:5837-5850. 722 13. Qian G, Wang H, Feng X. Generalized hesitant fuzzy sets and their application in 723 decision support system. Knowledge Based Systems. 2013;37:357-365. 724 14. Wei GW. Hesitant fuzzy prioritized operators and their application to multiple attribute 725 decision making. Knowledge-Based Systems. 2012;31:176-182. 726 15. Xia MM, Xu ZS. Hesitant fuzzy information aggregation in decision making. International 727 Journal of Approximate Reasoning. 2011;52:395-407. 728 16. Xu ZS, Zhang XL. Hesitant fuzzy multi-attribute decision making based on TOPSIS with 729 incomplete weight information. Knowledge-Based Systems. 2013;52 53-64. 730 17. Yu DJ, Zhang WY. Group decision making under hesitant fuzzy environment with 731 application to personnel evaluation. Knowledge-Based Systems. 2013;52:1-10. 732

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