Induced generalized hesitant fuzzy Shapley hybrid operators and their application in multi-attribute decision making

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Induced generalized hesitant fuzzy Shapley hybrid operators and their application in multi-attribute decision making

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Fanyong Meng a,b , Xiaohong Chen a,∗ , Qiang Zhang c a b c

School of Business, Central South University, Changsha 410083, China School of Management, Qingdao Technological University, Qingdao 266520, China School of Management and Economics, Beijing Institute of Technology, Beijing 100081, China

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a r t i c l e

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a b s t r a c t

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Article history: Received 25 March 2013 Received in revised form 23 August 2014 Accepted 18 November 2014 Available online xxx

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Keywords: Multi-attribute decision making Hesitant fuzzy set Grey relational analysis (GRA) method Shapley function

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1. Introduction

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In this study, two induced generalized hesitant fuzzy hybrid operators called the induced generalized hesitant fuzzy Shapley hybrid weighted averaging (IG-HFSHWA) operator and the induced generalized hesitant fuzzy Shapley hybrid geometric mean (IG-HFSHGM) operator are defined. The prominent characteristics of these two operators are that they do not only globally consider the importance of elements and their ordered positions, but also overall reflect their correlations. Furthermore, when the weight information of the attributes and the ordered positions is partly known, using grey relational analysis (GRA) method and the Shapley function models for the optimal fuzzy measures on an attribute set and on an ordered set are respectively established. Finally, an approach to hesitant fuzzy multi-attribute decision making with incomplete weight information and interactive conditions is developed, and an illustrative example is provided to show its practicality and effectivity. © 2014 Published by Elsevier B.V.

As an extension of fuzzy sets [1], hesitant fuzzy sets [2] permit the membership degree having a set of possible values. The popularity of hesitant fuzzy sets for solving decision-making problems is that they facilitate effectively representing inherent hesitancy and uncertainty in the human decision making process. Torra [2] discussed the relationship between hesitant fuzzy sets and intuitionistic fuzzy sets and showed that the envelope of a hesitant fuzzy set h = / Ø is an intuitionistic fuzzy set. Based on the relationship between hesitant fuzzy sets and intuitionistic fuzzy sets, Xia and Xu [3] defined some operational laws on hesitant fuzzy sets and presented some hesitant fuzzy aggregation operators. Based on the idea of quasi-arithmetic means, Xia et al. [4] developed a series of hesitant fuzzy aggregation operators and applied them to decision making. Motivated by the idea of the prioritized aggregation operator, Wei [5] developed the hesitant fuzzy prioritized weighted average (HFPWA) operator and the hesitant fuzzy prioritized weighted geometric (HFPWG) operator. Based on the Bonferroni mean (BM), Zhu et al. [6] introduced the weighted hesitant fuzzy geometric Bonferroni mean (WHFGBM) operator. Xu

∗ Corresponding author at: Yuelu District, Changsha 410083, Hunan Province, China. Tel.: +86 18254298903; fax: +86 053286875851. E-mail address: [email protected] (X. Chen).

and Xia [7,8] defined some distance measures for hesitant fuzzy sets and presented a distance and correlation measure for hesitant fuzzy information, whilst Xu and Xia [9] studied entropy and crossentropy of hesitant fuzzy sets and applied them to multi-attribute decision making. Liao and Xu [10] applied the VIKOR-based method to hesitate fuzzy multi-criteria decision making, whilst Chen et al. [11] developed an approach to multi-criteria decision making under hesitant fuzzy environment using the ELECTRE I. Furthermore, Chen et al. [12] studied some correlation coefficients of hesitant fuzzy sets and researched their application in clustering analysis. However, these measures are all defined under the assumption that the values in all HFEs are arranged in an increasing order, and two compared HFEs must have the same length. To cope with the situation where the elements in a set are correlative, some hesitant fuzzy aggregation operators based on fuzzy measures [13] are defined, such as the weighted hesitant fuzzy Choquet geometric Bonferroni mean (WHFCGBM) operator [6], the hesitant fuzzy Choquet geometric (HFCG) operator [14], the generalized hesitant fuzzy Choquet ordered averaging (GHFCOA) operator and the generalized hesitant fuzzy Choquet ordered geometric (GHFCOG) operator [15]. All these operators are based on the assumption that the fuzzy measure on an attribute set is completely known. However, because of various reasons, such as time pressure and the expert’s limited expertise about the problem domain, the weight information in the process of decision making may be partly known. As Meng et al. [16] noted, the Choquet integral only reflects

http://dx.doi.org/10.1016/j.asoc.2014.11.017 1568-4946/© 2014 Published by Elsevier B.V.

Please cite this article in press as: F. Meng, et al., Induced generalized hesitant fuzzy Shapley hybrid operators and their application in multi-attribute decision making, Appl. Soft Comput. J. (2014), http://dx.doi.org/10.1016/j.asoc.2014.11.017

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the interaction between two adjacent coalitions. Furthermore, the Choquet integral can either give the importance of elements or that of ordered positions, but it cannot both consider these two aspects. To cope with these issues, this study presents two induced generalized hesitant fuzzy hybrid Shapley operators, which do not only globally consider the importance of the elements in a set and their ordered positions, but also overall reflect their correlations. Furthermore, when the weight vectors on the attribute set and on the ordered set are partly known, models for the optimal fuzzy measures are respectively established, by which the weight vectors on them can be obtained. As a series of development, a method to hesitant fuzzy multi-attribute decision making with incomplete weight information and interaction conditions is developed. To do these, the rest parts of this paper are organized as follows: In Section 2, some basic concepts about hesitant fuzzy sets are briefly reviewed, and some existing hesitant fuzzy hybrid aggregation operators are introduced. In Section 3, the induced generalized hesitant fuzzy Shapley hybrid weighted averaging (IG-HFSHWA) operator and the induced generalized hesitant fuzzy Shapley hybrid geometric mean (IG-HFSHGM) operator are defined, and some important cases are examined. In Section 4, based on grey relational analysis (GRA) method and the Shapley function, models for the optimal fuzzy measures on the attribute set and on the ordered set are respectively established. In Section 5, an approach to hesitant fuzzy multi-attribute decision making with incomplete weight information and interaction conditions is developed, and a numerical example is provided to illustrate the developed procedure. The conclusion is made in the last section.

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2. Some basic concepts

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To cope with the situation where the membership degree of an element has several possible values, Torra [2] introduced the concept of hesitant fuzzy sets. Definition 1. [2] Let X = {x1 , x2 , . . ., xn }be a finite set, a hesitant fuzzy set (HFS) on X is in terms of a function that when applied to X returns a subset of [0,1]. To be easily understood, the HFS is expressed by a mathematical symbol E = (< xi , hE (xi ) > |xi ∈ X), where hE (xi ) is a set of some values in [0,1] denoting the possible membership degrees of the element xi ∈ X to the set E. For convenience, Xia and Xu [3] called h = hE (xi ) a hesitant fuzzy element (HFE). Let H be the set of all HFEs. Given three HFEs represented by h, h1 and h2 , Torra [2] defined some operations on them, which can be described as follows: (1) hc =∪ r∈h {1 − r} ; (2) h1 ∪ h2 = ∪rr1 ∈ h1 ,r2 ∈ h2 max{r1 , r2 }; (3) h1 ∩ h2 = ∪r1 ∈ h1 ,r2 ∈ h2 min{r1 , r 2 }. Based on the relationship between HFEs and intuitionistic fuzzy values (IFVs), Xia and Xu [3] defined the following new operations on the HFEs h, h1 and h2 : (1) (2) (3) (4)

h =∪ r∈h {r }  > 0 ; h =∪ r∈h {1 − (1 − r) }  > 0 ; h1 ⊕ h2 = ∪r1 ∈ h1 ,r2 ∈ h2 {r1 + r2 − r1 r2 }; h1 ⊗ h2 = ∪r1 ∈ h1 ,r2 ∈ h2 {r1 r2 }.

Similar to the order relationship between intuitionistic fuzzy elements, Xia and Xu [3] defined the following score function to rank HFEs.



Definition 2. [3] For a HFE h, S(h) = (1/#h)

r is called the score

125

r ∈h

function of h, where #h is the number of the elements in h. For two HFEs h1 and h2 , if S(h1 ) > S(h2 ), then h1 > h2 ; if S(h1 ) = S(h2 ), then h1 = h2 . In some cases, the score function fails to distinguish between two distinct HFEs. For example, let h1 = {0.1, 0.9} and h2 = {0.3, 0.5, 0.7} be two HFEs, then their scores are both equal to 0.5. According to Definition 2, it has h1 = h2 . But they are obviously different. For any HFE h, the averaging deviation function is defined by 1 D(h) = #h

 r ∈h



1  1  (r − S(h)) = r− r #h #h

,

128 129 130 131 132 133

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r ∈h

where #h is the number of possible values in h. In order to increase the distinction of HFEs, the order relationship, for any two HFEs h1 and h2 , is defined by If S(h1 ) < S(h2 ), then h 1 < h2 . D(h1 ) > D(h2 ), h1 < h2 If S(h1 ) < S(h2 ), then . D(h1 ) = D(h2 ), h1 = h2

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In above example, if the improvement method is used to rank h1 = {0.1, 0.9} and h2 = {0.3, 0.5, 0.7}, then h1 < h2 for S(h1 ) < S(h2 ) and D(h1 ) > D(h2 ). To both consider the importance of elements and their ordered positions, Xia and Xu [3] defined two generalized hesitant fuzzy hybrid operators called the generalized hesitant fuzzy hybrid averaging (HFHA) operator and the generalized hesitant fuzzy hybrid geometric (GHFHG) operator, given as in the following definition. Definition 3. [3] Let hi (i = 1, 2, . . ., n) be a collection of HFEs in H, w = (w1 , w2 , . . ., wn )T be the weight vector on {hi }i = 1, 2, . . . , n with n 

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2

2

r ∈h

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wi = 1, and ω = (ω1 , ω2 , . . ., ωn )T be the associated

150

weight vector on the ordered set N = {1, 2, . . ., n} with ωi ∈ [0,1] and

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wi ∈ [0,1] and

i=1

n 

ωi = 1.

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i=1

(1) The generalized hesitant fuzzy hybrid averaging (GHFHA) operator is defined by



(1)

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j=1

⎛ = ∪r

154

1/

n

⊕ ωj h˙ (j)

GHFHA (h1 , h2 , . . ., hn ) =

153

∈ h˙ (1) ,r(2) ∈ h˙ (2) ,...,r(n) ∈ h˙ (n)

⎝1 −

⎞1/

n

 ωj (1 − r(j) )

⎠

,

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j=1

where  > 0, (·) is a permutation on the weighted HFEs nwi hi (i = 1, 2, . . ., n) with h˙ (j) = nw(j) h(j) being the jth largest value of nwi hi (i = 1, 2, . . ., n), and n is the balancing coefficient. (2) The generalized hesitant fuzzy hybrid geometric (GHFHG) operator is defined by GHFHG (h1 , h2 , . . ., hn ) = = ∪r



×

(1)

1 



n

ωj ⊗ (h˙ (j) )



n

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 162

j=1

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∈ h˙ (1) ,r(2) ∈ h˙ (2) ,...,r(n) ∈ h˙ (n)

1−

157

1 − ˘ (1 − (1 − r(j) ) )

ωj

1/  ,

j=1

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nw

where  > 0, (·) is a permutation on the weighted HFEs hi i (i = 1, 2, nwi (j) being the jth largest value of h . . ., n) with h˙ (j) = hnw (i = 1, 2, i (j) . . ., n), and n is the balancing coefficient. Although the GHFHA and GHFHG operators both consider the importance of elements and their ordered positions, they are based on the assumption that the elements in a set are independent. In the practical decision-making problems, this assumption is usually violated, so it is unsuitable to aggregate their values using additive measures. In 1974, Sugeno [13] introduced the following concept of fuzzy measures. Definition 4. [13] A fuzzy measure on finite set N = {1, 2, . . ., n} is a set function  : P(N) → [0, 1] satisfying

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(1) (∅) =0, (N) = 1; (2) For all A, B ∈ P(N) with A ⊆ B, (A) ≤ (B),

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where P(N) is the power set of N.

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To reflect the interactive characteristics between elements, Wei et al. [15] defined the following generalized hesitant fuzzy Choquet integral operators. Definition 5. [15] Let hi (i = 1, 2, . . ., n) be a collection of HFEs in H, and  be a fuzzy measure on N = {1, 2, . . ., n}. (1) The generalized hesitant fuzzy Choquet ordered averaging (GHFCOA) operator is defined by

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1/

n

⊕ ((A(i) ) − (A(i+1) ))h(i) i=1

GHFCOA (h1 , h2 , . . ., hn ) =

(2) The generalized hesitant fuzzy Choquet ordered geometric (GHFCOG) operator is defined by GHFCOG(h1 , h2 , ..., hn ) =

1 



n

⊗ (h(i) )



(A(i) )−(A(i+1) )

;

where  > 0, (·) indicates a permutation on N such that h(1) ≤ h(2) ≤ . . . ≤ h(n) , and A(i) = {i, ..., n} with A(n+1) =∅.

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3. New induced generalized hesitant fuzzy operators

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 (n − s − 1)!s! n!

S⊆N\i 207 208 209

((S ∪ i) − (S)) ∀i ∈ N,

(1)

where s and n respectively denote the cardinalities of S and N. From Eq. (1), we know that psn = ((n − s − 1)!s!)/n! is a probability weight with



psn = 1 and psn > 0 for each S ⊆ N. The Shapley

S⊆N\i 210 211 212

n 

ϕi (, N) = 1. It means that {ϕi (, N)}i ∈ N

213

i=1

is a weight vector. When the fuzzy measure  is defined on an attribute set, (A) can be viewed as the importance of the attribute set A. Thus, in addition to the usual weights on the attribute set taken separately, weights on any combination of attributes are also defined. If we apply the attributes’ Shapley values as their weights, then it does not only consider their importance, but also reflect the influence from the other attributes. It is worth pointing out that if there is no interaction between attributes, then their Shapley values are equal to the importance of their own. Thus, the Shapley value can be seen as an extension of additive weights. In the multi-attribute decision making, if we consider the interactions between attributes, then the Shapley function may be a good choice. The induced generalized aggregation operator is an important kind of aggregation operators that has been researched by many scholars, such as the generalized OWA (GOWA) operator [18], the induced generalized OWA (IGOWA) operator [19], the uncertain induced quasi-arithmetic OWA (UIQAOWA) operator [20], the induced generalized hybrid averaging (IGHA) operator [21], the induced generalized intuitionistic fuzzy aggregation (IGIFA) operator [22], the induced interval-valued intuitionistic fuzzy hybrid aggregation (I-IVIFHA) operator [23], the induced 2-tuple linguistic generalized OWA (I-2TLGOWA) operator [24], and the induced uncertain linguistic OWA (I-ULOWA) operator [25]. Based on the Shapely function, we define the following two induced generalized hesitant fuzzy Shapley hybrid operators.

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Definition 6. An IG-HFSHWA operator of dimension n is a mapping IG-HFSHWA: Hn → H on the set of second arguments of two tuples u1 , h1  , u2 , h2  , . . ., un , hn  with a set of order-inducing variables ui (i = 1, 2, . . ., n) and a parameter  such that  ∈ (0, + ∞), denoted by

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⎛

n





⊕ ϕj (, N)ϕh (v, Q )h(j)

⎜ j=1 = ⎝ n

(j)



ϕ (, N)ϕh (v, Q ) j=1 j

245

⎞1/ ⎟ ⎠

,

246

(j)

where (·) is a permutation on ui (i = 1, 2, . . ., n) with u(j) being the jth largest value of ui (i = 1, 2, . . ., n), ϕj (, N) is the Shapley value with respect to (w.r.t.) the associated fuzzy measure  on N = {1, 2, . . ., n} for the jth ordered position, and ϕh (v, Q ) is the Shapley

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j

To eliminate the mentioned issues, this section presents two new hesitant fuzzy operators that globally consider the importance of the elements and their ordered positions as well as reflect their interactions. The Shapley function [17] as one of the most important payoff indices in game theory receives considerable attentions. When it is calculated with respect to the fuzzy measure  on finite set N = {1, 2, . . ., n}, it has ϕi (, N) =

N) ≥ 0 for any i ∈ N and

IG-HFSHWA ( u1 , h1 , u2 , h2 , . . ., un , hn )

i=1

From Definition 5, we know that the GHFCOA and GHFCOG operators consider the importance of their ordered positions and reflect their correlations. However, they neither consider the importance of the elements nor reflect their correlations.

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3

function is actually an expectation value of the marginal contribution between the element i and any coalition S ⊆ N\i. From the definition of fuzzy measures, it is not difficult to know that ϕi (,

value w.r.t. the fuzzy measure v on Q = {hj }j=1,...,n for hj (j = 1, 2, . . ., n).

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Remark 1. If  and v are both additive measures, then the IGHFSHWA operator degenerates to the induced generalized hesitant fuzzy hybrid weighted averaging (IG-HFHWA) operator

 n IG-HFHWA ( u1 , h1 , u2 , h2 , . . ., un , hn )=



j=1

n



wj ωh h(j)

,



wj ωh(j)

where wj = (j) and ωhj = v(hj ) for each j = 1, 2, . . ., n.

n





⊕ ϕj (, N)ϕh (v, Q )h(j)

⎜ j=1

G-HFSHOWA (h1 , h2 , . . ., hn ) == ⎝ n

(j)



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/ j, then the IGRemark 2. If ui = uj for all i, j = 1, 2, . . ., n with i = HFSHWA operator degenerates to the generalized hesitant fuzzy Shapley hybrid OWA (G-HFSHOWA) operator

⎛

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1/

(j)

j=1

253

ϕ (, N)ϕh (v, Q ) j=1 j

258 259 260

⎞1/ ⎟ ⎠

,

(j)

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where ϕh(j) (v, Q )h(j) is the jth largest value of the Shapley weighted HFEs ϕh (v, Q )hi (i = 1, 2, . . ., n). i

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Remark 3. If  = 1, then the IG-HFSHWA operator degenerates to the induced hesitant fuzzy Shapley hybrid weighted averaging (I-HFSHWA) operator

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I-HFSHWA ( u1 , h1 , u2 , h2 , . . ., un , hn )

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=

n

ϕ (, N)ϕh(j) (v, Q ) j=1 j

.

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Remark 4. If  = 2, then the IG-HFSHWA operator degenerates to the induced hesitant fuzzy Shapley hybrid quadratic weighted averaging (I-HFSHQWA) operator

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I-HFSHQWA ( u1 , h1 , u2 , h2 , . . ., un , hn )

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⎛

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⎜

n

⊕ ϕj (, N)ϕh2 (v, Q )h2(j) (j) j=1 n ϕ (, N)ϕh2 (v, Q ) j=1 j (j)

= ⎝

⎞1/2 ⎟ ⎠

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IG-HFSOWA ( u1 , h1 , u2 , h2 , . . ., un , hn )

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=

1/

n



⊕ ϕj (, N)h(j)

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Furthermore, if  → 0+ , then the IG-HFSHWA operator degenerates to the induced hesitant fuzzy Shapley ordered geometric mean (I-HFSOGM) operator n

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j (,N) . I-HFSOGM ( u1 , h1 , u2 , h2 , . . ., un , hn ) = ⊗ hϕ (j)

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Remark 6. If (j) = 1/n for each j = 1, 2, . . ., n, then the IG-HFSHWA operator degenerates to the induced generalized hesitant fuzzy Shapley weighted averaging (IG-HFSWA) operator

⎛

290

n





⊕ ϕh (v, Q )h(j)

⎜ j=1

= ⎝ n

(j)

 ϕ (v, Q ) j=1 h(j)

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Theorem 1. Let u1 , h1  , u2 , h2  , . . ., un , hn  be a set of two tuples with a set of order-inducing variables ui (i = 1, 2, . . ., n) and hi (i = 1, 2, . . ., n) being a collection of HFEs in H, let  be a fuzzy measure on the ordered set N = {1, 2, . . ., n}, and let v be the associated fuzzy measure on Q = {hi }i∈N . Then, their aggregated value using the IG-HFSHWA operator is also a HFE, denoted by IG − HFSHWA ( u1 , h1 , u2 , h2 , . . ., un , hn )

=

∪

n

n

ϕj (,N)ϕh (v,Q )/ r (j) (1−r(j) )

1−

r(1) ∈ h(1) ,r(2) ∈ h(2) ,...,r(n) ∈ h(n)

j=1

1/ ϕj (,N)ϕh

(j)

(v,Q )

j=1

303 304 305

j=1

wj ωh

315 316

318 319 320

(j)

322

j=1

323



n



⊗ h(j)

n



ϕh

(j)

(v,Q ) ϕj (,N)/

ϕ (,N)ϕh (v,Q ) j=1 j (j)

324 325 326 327

 ,

328

j=1

h (v,Q ) is the jth largest value of the Shapley weighted where hϕ (j) (j)

329

HFEs hi hi

330

ϕ (v,Q )

(i = 1, 2, . . ., n).

I-HFSHGM ( u1 , h1 , u2 , h2 , . . ., un , hn ) ϕj (,N)ϕh

n

(j)

(v,Q )/

331 332 333 334

n

ϕ (,N)ϕh (v,Q ) j=1 j (j)

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Remark 10. If  = 2, then the IG-HFSHGM operator degenerates to the induced hesitant fuzzy Shapley hybrid quadratic geometric mean (I-HFSHQGM) operator

.

301 302

w j /

314

321



I-HFSHQGM ( u1 , h1 , u2 , h2 , . . ., un , hn )



300



h ⊗ hω (j) (j)

313

336

298 299

310

j=1

291 292

n

n

Remark 8. If ui = uj for all i, j = 1, 2, . . ., n with i = / j, then the IGHFHSGM operator degenerates to the generalized hesitant fuzzy Shapley hybrid ordered geometric mean (G-HFSHOGM) operator

⎞1/ ⎟ ⎠



= ⊗ h(j)

IG-HFSWA ( u1 , h1 , u2 , h2 , . . ., un , hn )

309

317

where wj = (j) and ωhj = v(hj ) for each j = 1, 2, . . ., n.

1 

308

312

Remark 9. If  = 1, then the IG-HFSHGM operator degenerates to the induced hesitant fuzzy Shapley hybrid geometric mean (IHFSHGM) operator

j=1

286

,

(j)

1 

307

311



measure v on Q = {hj }j=1,...,n for hj (j = 1, 2, . . ., n).

=

281 282

ϕ (,N)ϕh (v,Q ) j=1 j (j)

306

j=1

G-HFSHOGM (h1 , h2 , . . ., hn )

.

j=1

h ⊗ hϕ (j) (j)

n  (v,Q ) ϕj (,N)/

where (·) is a permutation on ui (i = 1, 2, . . ., n) with u(j) being the jth largest value of ui (i = 1, 2, . . ., n), ϕj (, N) is the Shapley value w.r.t. the associated fuzzy measure  on N = {1, 2, . . ., n} for the jth ordered position, and ϕh (v, Q ) is the Shapley value w.r.t. the fuzzy

=

278



n



IG-HFHGM ( u1 , h1 , u2 , h2 , . . ., un , hn )

.

Remark 5. If v(hi ) = 1/n for each i = 1, 2, . . ., n, then the IGHFSHWA operator degenerates to the induced generalized hesitant fuzzy Shapley OWA (IG-HFSOWA) operator

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1 



Remark 7. If  and v are both additive measures, then the IGHFSHGM operator degenerates to the induced generalized hesitant fuzzy hybrid geometric mean (IG-HFHGM) operator

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IG-HFSHGM ( u1 , h1 , u2 , h2 , . . ., un , hn ) =

n

⊕ ϕj (, N)ϕh(j) (v, Q )h(j)

j=1

Definition 7. An IG-HFSHGM operator of dimension n is a mapping IG-HFSHGM: Hn → H on the set of second arguments of two tuples u1 , h1  , u2 , h2  , . . ., un , hn  with a set of order-inducing variables ui (i = 1, 2, . . ., n) and a parameter  such that  ∈ (0, + ∞), denoted by

where the notations as shown in Definition 6. In a similar way to the IG-HFSHWA operator, we define the following induced generalized hesitant fuzzy Shapley hybrid geometric mean (IG-HFSHGM) operator.

=

1 2



n

ϕh (v,Q ) ⊗ (2h(j) ) (j)

n

ϕj (,N)/

ϕ (,N)ϕh (v,Q ) j=1 j (j)

337 338 339 340



341

j=1

342

Remark 11. If (j) = 1/n for each j = 1, 2, . . ., n, then the IG-HFSHGM operator degenerates to the induced generalized hesitant fuzzy Shapley geometric mean (IG-HFSGM) operator 1 IG-HFSGM ( u1 , h1 , u2 , h2 , . . ., un , hn ) = 



n

343 344 345



ϕh (v,Q ) ⊗ h(j) (j) j=1

Theorem 2. Let u1 , h1  , u2 , h2  , . . ., un , hn  be a set of two tuples with a set of order-inducing variables ui (i = 1, 2, . . ., n) and hi (i = 1, 2, . . ., n) being a collection of HFEs in H, let  be a fuzzy measure on the ordered set N = {1, 2, . . ., n}, and let v be the associated fuzzy measure on Q = {hi }i∈N . Then, their aggregated value using the IG-HFSHGM operator is also a HFE, denoted by

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For the hesitant fuzzy decision matrix H = (hij )m×n , let

353 354

IG − HFSHGM ( u1 , h1 , u2 , h2 , . . ., un , hn ) =

⎛ 355



⎝1 −

×

1−

∪

r(1) ∈ h(1) ,r(2) ∈ h(2) ,...,r(n) ∈ h(n)

1/ ⎞ ϕj (,N)ϕh (v,Q ) (j) j=1 ⎠

n

n



ϕh (v,Q ) (1 − (1 − r(j) ) ) (j)

ϕj (,N)/

j=1

356 357

where the notations as shown in Definition 7.

361

From Theorems 1 and 2, it is not difficult to know that the IG-HFSHWA and IG-HFSHGM operators satisfy commutativity, monotonicity and boundary. Here, we on longer discuss them in detail.

362

4. Models for the optimal fuzzy measures

358 359 360

363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378

379 380

381

Consider a decision-making problem, Let A = {a1 , a2 , . . ., am } be the set of alternatives, and C = {c1 , c2 , . . ., cn } be the set of attributes. The decision makers are allowed to make their individual preferences for alternatives w.r.t. each attribute. If the decision makers provide several values for the alternative ai w.r.t. the attribute cj , this preference can be considered as a HFE hij . By H = (hij )m×n , we denote the hesitant fuzzy decision matrix given by the decision makers. If the weight information about attributes and their ordered positions are completely known, then we can use some aggregation operator to calculate the comprehensive attribute values. Otherwise, we first need to obtain the weights of the attributes and their ordered positions. Grey relational analysis (GRA) method [26] as an important multi-attribute decision making method has been researched by many scholars [27–33]. To define the grey relational coefficient for HFEs, we first introduce the following distance measure on HFEs. Definition 8. Let h1 and h2 be any two HFEs in H, then the distance between h1 and h2 is defined by

d(h1 , h2 ) =

1 #h1 + #h2

⎧ ⎨ ⎩

min |r1 − r2 | +

r1 ∈ h1

r2 ∈ h2

 r2 ∈ h2

⎫ ⎬

min |r2 − r1 |

r1 ∈ h1

384

385 386

387 388 389

⎭

,

(2)

382

383

5

H+ =

{h+ , h+ , . . ., h+ n} 1 2

H− =

{h− , h− , . . ., h− n }, 1 2

(P1) 0 ≤ d(h1 , h2 ) ≤ 1; (P2) d(h1 , h2 ) = 0 if and only if h1 = h2 ; (P3) d(h1 , h2 ) = d(h2 , h1 ).

401

where h+ = h(m)j and h− = h(1)j for each j = 1, 2, . . ., n, and (·) is a j j permutation on M = {1, 2, . . ., m} such that h(1)j ≤ h(2)j ≤ . . . ≤ h(m)j . Similar to the grey relational coefficients of intuitionistic fuzzy sets given by Wei [29,30], we define the grey relational coefficients from the positive-ideal solution (PIS) and negative-ideal solution (NIS) as follows: ij+ =

ij−

=

) +  max max d(hij , h+ ) min min d(hij , h+ j j 1≤i≤m 1≤j≤n 1≤i≤m 1≤j≤n , d(hij , h+ ) +  max max d(hij , h+ ) j j 1≤i≤m 1≤j≤n

392

0≤



r1 ∈ h1 393 394 395 396 397 398 399

min |r1 − r2 | +

r2 ∈ h2



r2 ∈ h2

min |r2 − r1 | ≤ #h1 + #h2 .

r1 ∈ h1

Thus, 0 ≤ d(h1 , h2 ) ≤ 1. For (P2): If h1 = h2 , then min |r1 − r2 | = min |r2 − r1 | = 0 for all r2 ∈ h2

r1 ∈ h1

r1 ∈ h1 and all r2 ∈ h2 . Thus, d(h1 , h2 ) = 0. On the other hand, if d(h1 , h2 ) = 0, then min |r1 − r2 | = min |r2 − r1 | = 0. It means that there r2 ∈ h2

r1 ∈ h1

exists r2 ∈ h2 for any r1 ∈ h1 such that r1 = r2 as well as there exists r1 ∈ h1 for any r2 ∈ h2 such that r1 = r2 . Thus, h1 = h2 . For (P3): From (2), it is easy to get the conclusion.

404 405 406 407

408

(4)

409

) +  max max d(hij , h− ) min min d(hij , h− j j

1≤i≤m 1≤j≤n

1≤i≤m 1≤j≤n

d(hij , h− ) +  max max d(hij , h− ) j j

for each pair (i, j) (i = 1, 2, . . ., m; j = 1, 2, . . ., n), where  is usually equal to 0.5. If the weight information of the attributes is partly known, we build the following model for the optimal fuzzy measure v on the attribute set C w.r.t. the alternative ai (i = 1, 2, . . ., m). min

n 

+ j=1 ij

ij−

+ ij−

410 411 412 413 414

ϕcj (v, C)

⎧ v(C) = 1 ⎪ ⎨

(5)

415

where ϕc (v, C) is the Shapley value of the attribute cj w.r.t. the fuzzy

416

s.t.

v(S) ≤ v(T ) ∀S, T ⊆ C s.t.S ⊆ T ⎪ ⎩ v(cj ) ∈ Ucj ,

,

j = 1, 2, . . ., n

v(cj )≥0,

j

measure v, and U cj is the known weight information (j = 1, 2, . . ., n). Since all alternatives are non inferior, we further build the following model for the optimal fuzzy measure v on the attribute set C. min

n m  

+ i=1 j=1 ij

⎧ v(C) = 1 ⎪ ⎨ s.t.

ij−

+ ij−

417 418 419 420

ϕcj (v, C) (6)

421

v(S) ≤ v(T ) ∀S, T ⊆ C s.t.S ⊆ T ⎪ ⎩ v(cj ) ∈ Ucj , v(cj )≥0,

j = 1, 2, . . ., n

Now, we consider the optimal fuzzy measure on the ordered set N. For the hesitant fuzzy decision matrix H = (hij )m×n , let

Proof. For (P1): Since 0 ≤ |r1 − r2 |≤1 for all r1 ∈ h1 and all r2 ∈ h2 , we have

403

(3)

¯ = {h¯ 1 , h¯ 2 , . . ., h¯ n }, H 390 391

402

1≤i≤m 1≤j≤n

where #h1 and #h2 respectively denote the numbers of elements in h1 and h2 . Theorem 3. Let h1 and h2 be any two HFEs in H, then the distance between h1 and h2 defined by Eq. (2) satisfies the following properties:

400

422 423

424

m

where h¯ j = (1/m) ⊕ hij for each j = 1, 2, . . ., n.

425

i=1

We define the grey relational coefficient of hij and h¯ j as follows: ¯ ij =

426

min min d(hij , h¯ j ) +  max max d(hij , h¯ j )

1≤i≤m 1≤j≤n

1≤i≤m 1≤j≤n

d(hij , h¯ j ) +  max max d(hij , h¯ j )

.

(7)

427

1≤i≤m 1≤j≤n

for each pair (i, j) (i = 1, 2, . . ., m; j = 1, 2, . . ., n), where  is usually equal to 0.5. For each i = 1, 2, . . ., m, reorder ¯ ij (j = 1, 2, . . ., n) such that ¯ i(j) being the jth largest value of ¯ (j = 1, 2, . . ., n). If the weight informaij

tion of the ordered positions is partly known, we build the following

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model for the optimal fuzzy measure  on the ordered set N w.r.t. the alternative ai (i = 1, 2, . . ., m). min

s.t.

437 438 439 440 441 442

⎩

s.t.

446 447 448 449 450

451 452

453 454 455 456 457

458 459 460 461 462 463 464

(S) ≤ (T ) ∀S, T ⊆ N s.t.S ⊆ T (j) ∈ Uj ,

n M  

⎩

466 467 468 469 470 471 472

,

(j) ∈ Uj ,

(j)≥0,

j = 1, 2, . . ., n

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

hi = IG-HFSHWA ( u1 , h1 , u2 , h2 , . . ., un , hn ) =

473

∪

474

ri(1) ∈ hi(1) ,ri(2) ∈ hi(2) ,...,ri(n) ∈ hi(n)

⎛ ⎝1 −

n

n

(1 − ri(j)

 ϕj (,N)ϕc(j) (v,C)/ )

j=1

⎞1/ ϕj (,N)ϕc

(j)

(v,C)

⎠

475

j=1

for benefit attribute cj for cost attribute cj

(i=1, 2, . . ., m;

j=1, 2, . . ., n)

= ∪r ∈ hij {1 − r}. with Step 3: Utilize model (6) to solve the optimal fuzzy measure v on the attribute set C, and calculate their Shapley values. Step 4: Utilize model (9) to solve the optimal fuzzy measure  on the ordered set N, and calculate their Shapley values. Step 5: Let uj = ¯ ij (j = 1, 2, . . ., n) for each i = 1, 2, . . ., m, utilize the induced generalized hesitant fuzzy Shapley hybrid weighted averaging (IG-HFSHWA) operator

477 478 479

=

m×n

hcij

{0.7, 0.8} {0.2, 0.4, 0.6, 0.7} {0.8, 0.9} {0.6, 0.8} {0.2, 0.6}

hi = IG-HFSHGM ( u1 , h1 , u2 , h2 , . . ., un , hn )

Step 1: Suppose that there exist m alternatives A = {a1 , a2 , . . ., am } to be evaluated according to n attributes C = {c1 , c2 , . . ., cn } to form the hesitant fuzzy decision matrix H = (hij )m×n , where hij is a HFE. Step 2: If all attributes cj (j = 1, 2, . . ., n) are benefits (i.e., the bigger the better), then the attribute values do not need normalization. Otherwise, we need to normalize the hesitant fuzzy decision matrix H = (hij )m×n into H  = (hij ) , where

hcij

c4

{0.2, 0.5, 0.6} {0.3, 0.6} {0.2, 0.4} {0.1, 0.2, 0.3, 0.6} {0.4, 0.5, 0.6}

(9)

Based on the introduced aggregation operators and models for the optimal fuzzy measures, this section presents a method to hesitant fuzzy multi-attribute decision making with incomplete weight information and interactive conditions. The decision procedure can be described as follows:

hij

c3

{0.2, 0.3} {0.2, 0.4, 0.5} {0.5, 0.6, 0.7} {0.3, 0.7} {0.2, 0.3,0.7, 0.9}

476

5. An approach to hesitant fuzzy multi-attribute decision making



c2

a1 a2 a3 a4 a5

or the induced generalized hesitant fuzzy Shapley hybrid geometric mean (IG-HFSHGM) operator

¯ i(j) ϕj (, N)

Remark 12. In models (6) and (9), we use the elements’ Shapley values as their weights, which globally consider their importance as well as overall reflect their correlations. If there are no interactive characteristics between attributes and between their ordered positions, then models (6) and (9) respectively degenerate to models for the optimal additive weight vectors on the attribute set and on the ordered set.

hij =

c1

j = 1, 2, . . ., n

(S) ≤ (T ) ∀S, T ⊆ N s.t.S ⊆ T

 465

(j)≥0,

⎧I=1 j=1 ⎨ (N) = 1

443

445

(8)

where ϕj (v, N) is the Shapley value of the jth ordered position w.r.t. the fuzzy measure , and Uj is known weight information (j = 1, 2, . . ., n). Since there is no difference between fuzzy measures on the ordered set N w.r.t. different alternatives, we further build the following model for the optimal fuzzy measure  on the ordered set N. min

444

¯ i(j) ϕj (, N)

⎧j=1 ⎨ (N) = 1

435

436

n 

Table 1 Hesitant fuzzy decision matrix.

∪

481

ri(1) ∈ hi(1) ,ri(2) ∈ hi(2) ,...,ri(n) ∈ hi(n)

⎛ ×

480

⎝1 −

 1−

n

1/ ⎞ ϕj (,N)ϕc (v,C) (j) j=1 ⎠

n



(1 − (1 − ri(j) ϕc(j) (v,C) ) )

ϕj (,N)/

482

j=1

483

to calculate the comprehensive HFEs hi (i = 1, 2, . . ., m) of the alternatives ai (i = 1, 2, . . ., m). Step 6: According to the comprehensive HFEs hi (i = 1, 2, . . ., m), calculate the score value S(hi ) and the averaging deviation value D(hi ). Then, to rank the comprehensive HFEs hi (i = 1, 2, . . ., m), and select the best alternative(s). Step 7: End.

Example 1. There is a panel with five possible emerging technology enterprises ai (i = 1–5) to select (adapted from Ref. [15]). The experts select four attributes to evaluate the five possible emerging technology enterprises: c1 is the technical advancement; c2 is the potential market; c3 is the industrialization infrastructure, human resources and financial conditions; c4 is the employment creation and the development of science and technology. To avoid influence each other, the decision makers are required to evaluate the five possible alternatives ai (i = 1–5) under the above four attributes in anonymity and the hesitant fuzzy decision matrix H = (hij )m×n is presented in Table 1, where hij (i = 1–5; j = 1–4) are in the form of HFEs. Because of time pressure and the expert’s limited expertise about the problem domain, the weight information of the attributes is partly known. They are respectively given by U c1 = [0.2, 0.4], Uc2 = [0.25, 0.45], Uc3 = [0.2, 0.4] and Uc4 = [0.15, 0.3]. Furthermore, the importance of the ordered positions is respectively defined by U1 = [0.1, 0.2], U2 = [0.2, 0.3], U3 = [0.3, 0.4] and U4 = [0.4, 0.5]. To effectively solve this problem, the above proposed decision procedure is followed for determining the most desirable alternative(s). Step 1: Since all attributes are benefits, there is on need to normalize the hesitant fuzzy decision matrix H, namely, H = H .

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491 492 493 494 495 496 497 498 499 500 501 502

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Table 2 The fuzzy measure v on the attribute set C. Coalition

Fuzzy measure

Coalition

Fuzzy measure

Coalition

Fuzzy measure

{c1 } {c2 } {c3 } {c4 } {c1 , c2 }

0.2 0.25 0.25 0.3 0.25

{c1 , c3 } {c1 , c4 } {c2 , c3 } {c2 , c4 } {c3 , c4 }

0.25 1 0.25 0.3 1

{c1 , c2 , c3 } {c1 , c2 , c4 } {c1 , c3 , c4 } {c2 , c3 , c4 } {c1 , c2 , c3 , c4 }

0.25 1 1 1 1

Table 3 The fuzzy measure  on the ordered set N.

514 515

Coalition

Fuzzy measure

Coalition

Fuzzy measure

Coalition

Fuzzy measure

{1} {2} {3} {4} {1, 2}

0.1 0.2 0.3 0.5 0.2

{1, 3} {1, 4} {2, 3} {2, 4} {3, 4}

0.3 0.5 0.3 1 1

{1, 2, 3} {1, 2, 4} {1, 3, 4} {2, 3, 4} {1, 2, 3, 4}

0.3 1 1 1 1

Step 2: Let  = 0.5, from Eqs. (3) and (4), we get the grey relational coefficient of each alternative from PIS and NIS as follows:

⎛ 516

 + = (ij+ )

5×4

0.3672

⎜ 0.582 ⎜

0.709

0.7647

0.3514

0.7303

0.3786

0.619

⎝ 0.619

0.619

1

0.52

0.587

=⎜ ⎜

1

0.5321

1

526

Step 3: From Eq. (7), it gets the following grey relational coefficient matrix

⎞

⎟ ⎟ ⎟, ⎟ 0.6842 0.5372 ⎠ 0.3333

⎛

0.1928

⎜ 0.0538 ⎜ ¯ = (¯ ij ) =⎜ 5×4 ⎜ 0.0535

1

517

 − = (ij− )

5×4

1

0.7911

0.4194

⎜ 0.6701 0.7647 0.7646 0.4037 ⎟ ⎜ ⎟ 0.3333 0.619 ⎟ ⎟. ⎝ 0.5652 1 0.52 0.619 ⎠

518 519

1

0.2453

⎧ v(C) = 1 ⎪ ⎪ ⎪ ⎨ v(S) ≤ v(T ) ∀S, T ⊆ {c1 , c2 , c3 , c4 }

521 522 523

524 525

(3) ∈ [0.3, 0.4],

0.0595 ⎟ ⎟.

0.034

0.0941

0.0714

⎟

530 531

(4) ∈ [0.4, 0.5]

533 534 535

v(c4 ) ∈ [0.15, 0.3]

ϕ1 (, N) = 0.025,

ϕ2 (, N) = 0.1417,

ϕ3 (, N) = 0.1917,

536

ϕ4 (, N) = 0.6417.

537

538

Step 4: Utilize the IG-HFSHWA operator to calculate the comprehensive HFEs hi (i = 1–5) for the emerging technology enterprises

ϕc4 (v, C) = 0.5833.

Table 4

Q4 Ranking results w.r.t.  = 1.

The GHFCOA operator The GHFCOG operator The IG-HFSHWA operator The IG-HFSHGM operator

529

532

Solve the above linear programming, the optimal fuzzy measure  on the ordered set N are obtained as shown in Table 3. According to Table 3, it derives the following Shapley values

ϕc2 (v, C) = 0.0667,

ϕc3 (v, C) = 0.1833,

0.1088

.

Solve the above linear programming, the optimal fuzzy measure v on the attribute set C are obtained as shown in Table 2. According to Table 2, it derives the following Shapley values ϕc1 (v, C) = 0.1667,

0.0516

0.0667 ⎟

⎧ (N) = 1 ⎪ ⎪ ⎨ (S) ≤ (T ) ∀S, T ⊆ {1, 2, 3, 4} . s.t. (1) ∈ [0.1, 0.2], (2) ∈ [0.2, 0.3] ⎪ ⎪ ⎩

v(c1 ) ∈ [0.2, 0.4], v(c2 ) ∈ [0.25, 0.45] ⎪ ⎪ ⎪ ⎩ v(c3 ) ∈ [0.2, 0.4],

0.0338

min 0.0732((1) − (2, 3, 4)) + 0.0145((2) − (1, 3, 4)) −0.0223((3) − (1, 2, 4)) − 0.0654((4) − (1, 2, 3)) +0.0439((1, 2) − (3, 4)) + 0.0254((1, 3) − (2, 4)) +0.0039((1, 4) − (2, 3)) + 0.4122

min 0.005(v(c1 ) − v(c2 , c3 , c4 )) + 0.0256(v(c2 ) − v(c1 , c3 , c4 )) −0.0079(v(c3 ) − v(c1 , c2 , c4 )) − 0.0227(v(c4 ) − v(c1 , c2 , c3 )) +0.0153(v(c1 , c2 ) − v(c3 , c4 )) − 0.0014(v(c1 , c3 ) − v(c2 , c4 )) −0.0088(v(c1 , c4 ) − v(c2 , c3 )) + 2.4786

s.t.

0.0931

According to model (9), it gets the following linear programming for the optimal fuzzy measure  on the ordered set N.

According to model (6), it gets the following linear programming for the optimal fuzzy measure v on the attribute set C.

520

0.0344

⎞

1

=⎜ ⎜ 0.3672 0.6611 0.4936 0.5871

528

⎞

0.076

⎝ 0.1186 0.0931 0.0623 0.0905 ⎠ 0.1186

⎛

0.1424

527

a1

a2

a3

a4

a5

Ranking orders

0.4433 0.3368 0.6007 0.4608

0.5512 0.5088 0.5525 0.4925

0.6398 0.5636 0.5089 0.4463

0.5283 0.477 0.5958 0.518

0.5877 0.5407 0.5944 0.555

h3 > h5 > h2 > h4 > h1 h3 > h5 > h2 > h4 > h1 h1 > h4 > h5 > h2 > h3 h5 > h4 > h2 > h1 > h2

Please cite this article in press as: F. Meng, et al., Induced generalized hesitant fuzzy Shapley hybrid operators and their application in multi-attribute decision making, Appl. Soft Comput. J. (2014), http://dx.doi.org/10.1016/j.asoc.2014.11.017

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8 Table 5 Ranking results w.r.t.  = 2.

The GHFCOA operator The GHFCOG operator The IG-HFSHWA operator The IG-HFSHGM operator

a1

a2

a3

a4

0.4909 0.3237 0.6283 0.5205

0.5661 0.5 0.58 0.6602

0.6615 0.5483 0.5279 0.6782

0.5501 0.4685 0.6135 0.6081

a5

a1

a2

a3

a4

0.5861 0.3004 0.6755 0.4904

0.6011 0.481 0.6094 0.7319

0.7171 0.5253 0.5843 0.6714

0.6028 0.4518 0.6484 0.6129

a1

a2

a3

a4

0.6549 0.2859 0.7069 0.4704

0.6307 0.4647 0.6382 0.7399

0.7659 0.5134 0.658 0.6645

0.6451 0.4386 0.6742 0.6055

Ranking orders

0.6051 0.5272 0.6025 0.7297

h3 > h5 > h2 > h4 > h1 h3 > h5 > h2 > h4 > h1 h1 > h4 > h5 > h2 > h3 h5 > h3 > h2 > h4 > h1

Table 6 Ranking results w.r.t.  = 5.

The GHFCOA operator The GHFCOG operator The IG-HFSHWA operator The IG-HFSHGM operator

a5

Ranking orders

0.6477 0.4978 0.6205 0.7603

h3 > h5 > h4 > h2 > h1 h3 > h5 > h2 > h4 > h1 h1 > h4 > h5 > h2 > h3 h5 > h2 > h3 > h4 > h1

Table 7 Ranking results w.r.t.  = 10.

The GHFCOA operator The GHFCOG operator The IG-HFSHWA operator The IG-HFSHGM operator

541 542

ai (i = 1–5). Take the comprehensive hesitant fuzzy element h1 for an example, and let  = 1, it has h1 = G-HFSHWA ( u1 , h11 , u2 , h12 , u3 , h13 , u4 , h14 )



=

∪

1 − (1 − r11 )

r11 ∈ h11 ,r14 ∈ h14 ,r12 ∈ h12 ,r13 ∈ h13

4

543

×(1 − r13 )

ϕ4 (,N)ϕc3 (v,C)/

j=1

ϕj (,N)ϕc

(j)

a5

Ranking orders

0.6798 0.4778 0.6402 0.7604

h3 > h5 > h1 > h4 > h2 h3 > h5 > h2 > h4 > h1 h1 > h4 > h3 > h5 > h2 h5 > h2 > h3 > h4 > h1

6. Conclusions

569

We have developed an approach to hesitant fuzzy multiattribute decision making with incomplete weight information and

4

ϕ1 (,N)ϕc1 (v,C)/

j=1

ϕj (,N)ϕc



(j)

(v,C)

4

(1 − r14 )

ϕ2 (,N)ϕc4 (v,C)/

j=1

ϕj (,N)ϕc

(j)

(v,C)

4

(1 − r12 )

ϕ3 (,N)ϕc2 (v,C)/

j=1

ϕj (,N)ϕc

(j)

570 571

(v,C)

(v,C)

= {0.5783, 0.5528, 0.6049, 0.5079, 0.6409, 0.6614, 0.5214, 0.565, 0.5898, 0.6157, 0.6057, 0.6706, 0.5276, 0.5707, 0.5952, 0.6207, 0.6553, 0.6749, 0.5092, 0.554, 0.5794, 0.6059, 0.6419, 0.6623, 0.5226, 0.5661, 0.5909, 0.6167, 0.6516, 0.6715, 0.5288, 0.5718, 0.5962, 0.6217, 0.6562, 0.6758, }

544

where

4 

ϕj (, N)ϕc(j) (v, C) = ϕ1 (, N)ϕc1 (v, C) +

j=1

547

ϕ2 (, N)ϕc4 (v, C) + ϕ3 (, N)ϕc2 (v, C) + ϕ4 (, N)ϕc3 (v, C). Step 5: According to the comprehensive HFEs hi (i = 1–5), the score values S(hi ) (i = 1–5) are obtained as follows:

548

S(h1 ) = 0.6007,

545 546

549

= 0.5092,

S(h2 ) = 0.5525, S(h4 ) = 0.5958,

S(h3 ) S(h5 ) = 0.5944.

550

551 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567 568

Thus, the emerging technology enterprise a3 is the best choice. In the above example, we only use the IG-HFSHWA operator with  = 1 to get the best choice. For the convenience of comparison, ranking results w.r.t. the different aggregation operators and the different values of  are obtained as shown in Tables 4–7. The ranking results show that the different optimal alternatives may be yielded by using the different aggregation operators. Although the ranking results are different, the same optimal alternative is obtained by using the same operator and the different values of . Thus, the decision maker can properly select the desirable alternative according to his interest and the actual needs. However, if there is no special explanation that the attributes and the ordered positions are respectively independent, we suggest the experts use the aggregation operators based on fuzzy measures. Furthermore, since the hybrid aggregation operator considers more information than the OWA operator and the weighted averaging operator, we advise the experts to apply the hybrid aggregation operator.

interactive conditions. In order to get the comprehensive attribute values, two new induced generalized hesitant fuzzy aggregation operators are defined, which do not only consider the importance of the elements and their ordered positions but also reflect their interactions. In many practical decision-making problems, because of various reasons, the weight vectors on the attribute set and on the ordered set are usually partly known. To obtain their optimal weight vectors, the corresponding models are established by using grey relational analysis (GRA) method and the Shapley function. Although the fuzzy measures can well cope with the situations where the elements in a set are interactive, they are defined on the power set. It makes the problem exponentially complex. Thus, it will be interesting to research hesitant fuzzy multi-attribute decision making by using some special kinds of fuzzy measures, such as -fuzzy measures [13], k-additive measures [34] and p-symmetry measures [35]. In this paper, we have only studied HFSs, and it will be interesting to research other kinds of hesitant fuzzy sets using fuzzy measures, such as dual hesitant fuzzy sets [36], interval-valued hesitant fuzzy sets [37], interval-valued intuitionistic hesitant fuzzy sets [38], and hesitant fuzzy linguistic term sets [39]. Besides the hybrid aggregation operator, many researchers developed some other kinds of important aggregation operators. For example, Torra [40] introduced the weighted OWA (WOWA) operator using a monotonic function. Yager et al. [41] introduced the concept of immediate weights that both consider the importance of the elements and the ordered positions, whilst Merigó and Gil-Lafuente [42] defined some immediate weighted distance operators and researched their application to multi-person decision-making in production management. Furthermore, Yager [43] discussed the importance weights in the OWA operator using

Please cite this article in press as: F. Meng, et al., Induced generalized hesitant fuzzy Shapley hybrid operators and their application in multi-attribute decision making, Appl. Soft Comput. J. (2014), http://dx.doi.org/10.1016/j.asoc.2014.11.017

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ARTICLE IN PRESS F. Meng et al. / Applied Soft Computing xxx (2014) xxx–xxx

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transformation functions, and Merigó [44] defined the weights using weighted average of the weights of the elements and that of the ordered positions and presented the induced ordered weighted averaging-weighted average (IOWAWA) operator. It is one of our future research works to consider these operators in the setting of hesitant fuzzy environment based on fuzzy measures.

609

Acknowledgments

603 604 605 606 607

The authors first gratefully thank the managing editor professor Bas van Vlijmen and two anonymous referees for their valuable 611 and constructive comments, which have greatly improved the 612 Q3 paper. This work was supported by the Funds for Creative Research 613 614 Groups of China (No. 71221061), the Projects of Major International 615 Cooperation NSFC (No. 71210003), the Natural Science Foundation 616 Project of China (Nos. 71201089, 71071018 and 71271217), the 617 Natural Science Foundation Youth Project of Shandong Province, 618 China (ZR2012GQ005), and the Specialized Research Fund for the 619 Doctoral Programme of Higher Education (No. 20111101110036). 610

620

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Please cite this article in press as: F. Meng, et al., Induced generalized hesitant fuzzy Shapley hybrid operators and their application in multi-attribute decision making, Appl. Soft Comput. J. (2014), http://dx.doi.org/10.1016/j.asoc.2014.11.017

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