hesitant trapezoid fuzzy hamacher aggregation operators

Journal of Systems Science and Information

Dec., 2020, Vol. 8, No. 6, pp. 524–548

DOI: 10.21078/JSSI-2020-524-25

Hesitant Trapezoid Fuzzy Hamacher Aggregation Operators and

Their Application to Multiple Attribute Decision Making

Qian YUSchool of Business and Administration, Chongqing University of Science & Technology, Chongqing

401331, China

E-mail: [email protected]

Jun CAOSchool of Business and Administration, Chongqing University of Science & Technology, Chongqing

401331, China


Ling TANSchool of Business and Administration, Chongqing University of Science & Technology, Chongqing

401331, China


Yubing ZHAISchool of Management and Economics, Beijing Institute of Technology, Beijing 100081, China


Jiongyan LIUSchool of Business and Administration, Chongqing University of Science & Technology, Chongqing

401331, China


Abstract In this paper, we investigate the multiple attribute decision making (MADM) problems

in which the attribute values take the form of hesitant trapezoid fuzzy information. Firstly, inspired

by the idea of hesitant fuzzy sets and trapezoid fuzzy numbers, the definition of hesitant trapezoid

fuzzy set and some operational laws of hesitant trapezoid fuzzy elements are proposed. Then some

hesitant trapezoid fuzzy aggregation operators based on Hamacher operation are developed, such as

the hesitant trapezoid fuzzy Hamacher weighted average (HTrFHWA) operator, the hesitant trapezoid

fuzzy Hamacher weighted geometric (HTrFHWG) operator, the hesitant trapezoid fuzzy Hamacher

Choquet average (HTrFHCA), the hesitant trapezoid fuzzy Hamacher Choquet geometric (HTrFHCG),

etc. Furthermore, an approach based on the hesitant trapezoid fuzzy Hamacher weighted average

(HTrFHWA) operator and the hesitant trapezoid fuzzy Hamacher weighted geometric (HTrFHWG)

Received December 20, 2019, accepted May 20, 2020

Supported by the Science and Technology Research Project of Chongqing Municipal Education Commission

(KJQN201901505), the Key Project of Humanities and Social Sciences Research of Chongqing Education Com-

mission in 2019 (19SKGH181)

Hesitant Trapezoid Fuzzy Hamacher Aggregation Operators and ... 525

operator is proposed for MADM problems under hesitant trapezoid fuzzy environment. Finally, a

numerical example for supplier selection is given to illustrate the application of the proposed approach.

Keywords multiple attribute decision making (MADM); hesitant trapezoid fuzzy set (HTrFS); Hamacher

operation; Choquet integral

1 Introduction

Since the theory of fuzzy sets (FSs) was first proposed by Zadeh[1], many extension formshave been developed to generalize the FS theory, such as vague sets[2], type-2 fuzzy sets[3],interval-valued fuzzy sets[4] , intuitionistic fuzzy sets (IFSs)[5] and interval-valued intuitionisticfuzzy sets (IVIFSs)[6]. Recently, the concept of hesitant fuzzy set (HFS) was introduced byTorra[7] to enrich the fuzzy set. It permits the membership of an element to a given set havinga few different values, which has been a suitable tool to describe the imprecise or uncertaindecision information. As a new generalized type of fuzzy sets, HFS has received a considerableattention and has been applied to a various field of decision making[8−13].

Xia, et al.[14] gave an intensive study on hesitant fuzzy information aggregation operatorsand their application in decision making problems. Xu and Xia[15] defined the distance andcorrelation measures for hesitant fuzzy information and then discussed their properties in detail.Based on the Bonferroni mean (BM)[16] and geometric Bonferroni mean (GBM)[17] operators,Zhu, et al.[18] developed a series of Bonferroni mean aggregation operators for hesitant fuzzyinformation and applied them to MADM problems. Inspired by the idea of prioritized aggre-gation operators[19,20], Wei[21] defined some prioritized aggregation operators for aggregatinghesitant fuzzy information, and then applied them to develop some models for hesitant fuzzyMADM problems in which the attributes are in different priority levels. And Wei, et al.[22]

depicted the interactions phenomena among the aggregated arguments with the aid of Choquetintegral and proposed some Choquet hesitant fuzzy information aggregation operators: Hesitantfuzzy Choquet ordered averaging (HFCOA) operator, hesitant fuzzy Choquet ordered geomet-ric (HFCOG) operator, the generalized hesitant fuzzy Choquet ordered averaging (GHFCOA)operator and generalized hesitant fuzzy Choquet ordered geometric (GHFCOG) operator. Mo-tivated by the power aggregation operators[23], Zhang[24] proposed a family of hesitant fuzzypower aggregation operators and applied them to solve multiple attribute group decision makingproblems.

The above-mentioned approaches have already been proven effective and feasible for dealingwith multiple attribute decision making problems. However, the current approaches for MADMproblem may induce the information losing and cannot represent the real preference of decisionmaker precisely. In order to process uncertain and inaccuracy information as precise as possible,motivated by the idea of HFSs and trapezoid fuzzy numbers[25], in this paper, we propose theconcept of hesitant trapezoid fuzzy set. To do so, the remainder of this paper is set outas follows. In Section 2, a brief introduction to some basic notations of hesitant fuzzy setand Hamacher operations is reviewed. In Section 3, some basic concepts related to hesitanttrapezoid fuzzy sets and some operational laws of hesitant trapezoid fuzzy elements are defined.In Section 4, we develop a series of hesitant trapezoid fuzzy aggregation operators, furthermore,some aggregation operators based on the Hamacher operations with hesitant trapezoid fuzzy

526 YU Q, CAO J, TAN L, et al.

information are proposed. In Section 5, we propose an approach for multi-attribute decisionmaking based on the hesitant trapezoid fuzzy Hamacher weighted average (HTrFHWA) operatorand the hesitant trapezoid fuzzy Hamacher weighted geometric (HTrFHWG) operator underhesitant trapezoid fuzzy environment. In Section 6, an illustrative example is pointed out toverify the developed approach and to demonstrate its practicality and effectiveness. In Section7, we conclude the paper and give some remarks.

2 Preliminaries

2.1 Hesitant Fuzzy Set

Definition 1 (see [7, 14]) Let X be a reference set, a hesitant fuzzy set (HFS) A on X isdenoted by a function hA(x) that returns a subset of [0, 1] when it is applied to X . In order tomake it easier to understood, the HFS is expressed by a mathematical symbol

A ={⟨

x, hA(x)

⟩ |x ∈ X}

, (1)

where hA(x) is a set of some different values indicating all possible membership degrees of theelement x∈X to the set A. For convenience, hA(x) is called a hesitant fuzzy element (HFE),which is denoted by hA(x) =

{γ|γ ∈ hA(x)

}.

Let h, h1 and h2 be three HFEs, then some operations are presented as follows[14]:1) hc = {1 − γ|γ ∈ h};2) hλ =

{(γ)λ|γ ∈ h

};

3) λh ={1 − (1 − γ)2|γ ∈ h

};

4) h1 ⊕ h2 = {γ1 + γ2 − γ1γ2|γ1 ∈ h1, γ2 ∈ h2};5) h1 ⊗ h2 = {γ1γ2|γ1 ∈ h1, γ2 ∈ h2}.Definition 2 (see [14]) For an HFE h = {γ|γ ∈ h}, s(h) = 1

lh

∑y∈h γ is called the

score function of h, where lh is the number of the values in h. For two HFEs h1 and h2 ifs (h1) > s (h2) , then h1 > h2; if s (h1) = s (h2) , then h1 = h2.

2.2 Hamacher Operations

T-norm and t-conorm are an important notion in fuzzy set theory, which are used to definea generalized union and intersection of fuzzy sets[26]. Roychowdhury and Wang[27] gave thedefinition and conditions of t-norm and t-conorm. Based on a t-norm(T ) and t-conorm(T ∗),a generalized union and a generalized intersection of intuitionistic fuzzy sets were introducedby Deschrijver and Kerre[28]. Further, Hamacher[29] proposed a more generalized t-norm andt-conorm. Hamacher operations[29] include the Hamacher product and Hamacher sum, whichare examples of t-norms and t-conorms, respectively. They are defined as follows:

Hamacher product ⊗ is a t-norm and Hamacher sum ⊕ is a t-conorm, where

T (a, b) = a ⊗ b =ab

γ + (1 − γ)(a + b − ab), (2)

T ∗(a, b) = a ⊕ b =a + b − ab − (1 − γ)ab

1 − (1 − γ)ab. (3)


Especially, when γ = 1, then Hamacher t-norm and t-conorm will reduce to

T (a, b) = a ⊗ b = ab,

T ∗(a, b) = a ⊕ b = a + b − ab,

which are the algebraic t-norm and t-conorm respectively; when γ = 2, then Hamacher t-normand t-conorm will reduce to

T (a, b) = a ⊗ b =ab

1 + (1 − a)(1 − b),

T ∗(a, b) = a ⊕ b =a + b

1 + ab,

which are called the Einstein t-norm and t-conorm, respectively[30].

3 Hesitant Trapezoid Fuzzy Set (HTrFS)

3.1 Trapezoid Fuzzy Numbers

In this section, we briefly describe some basic concepts and operational laws related totrapezoid fuzzy numbers, and define the degree of possibility of two trapezoid fuzzy numbers.

Definition 3 (see [25]) A trapezoid fuzzy numbers n can be defined as [n1, n2, n3, n4].The membership function μn (x) is defined as

μn(x) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

0, x < n1,x − n1

n2 − n1, n1 ≤ x ≤ n2,

1, n2 ≤ x ≤ n3,x − n4

n3 − n4, n3 ≤ x ≤ n4,

0, x > n4.

For a trapezoidal fuzzy number n = [n1, n2, n3, n4], if n2 = n3, then n is called a triangularfuzzy number.

Given any two trapezoidal fuzzy numbers, m = [m1, m2, m3, m4], n = [n1, n2, n3, n4], andλ > 0, some operations can be expressed as follows[3]:

m ⊕ n = [m1 + n1, m2 + n2, m3 + n3, m4 + n4] ;

m ⊗ n = [m1 × n1, m2 × n2, m3 × n3, m4 × n4] ;

λm = λ [m1, m2, m3, m4] = [λm1, λm2, λm3, λm4] ;

m = [m1, m2, m3, m4]λ =

[mλ

1 , mλ2 , mλ

3 , mλ4

].

Motivated by the degree of possibility of two trapezoid fuzzy linguistic variables[31], in thefollowing, we introduce a formula for comparing trapezoid fuzzy numbers.

Definition 4 Let m = [m1, m2, m3, m4] and n = [n1, n2, n3, n4] be two trapezoid fuzzynumbers. Then, the possibility degree of m ≥ n is defined as follows:

p(m ≥ n) = min{

max{

(m3 + m4) − (n1 + n2)(m3 + m4) − (m1 + m2) + (n3 + n4) − (n1 + n2)

, 0}

, 1}

. (4)


Obviously, the possibility degree p(m ≥ n) satisfies the following properties:1) 0 ≤ p(m ≥ n) ≤ 1, 0 ≤ p(n ≥ m) ≤ 1;2) p(m ≥ n) + p(n ≥ m) = 1. Especially, p(m ≥ n) = p(n ≥ m) = 0.5.

3.2 Hesitant Trapezoid Fuzzy Set

In many practical situations, it is relatively easy for decision makers to define the possiblevalues rather than a precise number. Therefore, the trapezoid fuzzy number is usually moreenough to describe real-life decision problems than crisp numbers. So, we propose the hesitanttrapezoid fuzzy set based on HFS and trapezoid fuzzy numbers.

Definition 5 Let X be a reference set, a HTrFS A on X is denoted by a function hA(x)that returns a subset of [0, 1] when it is applied to X. In order to make it easier to understood,the HTrFS is expressed by a mathematical symbol.

A ={⟨

x, hA(x)

⟩|x ∈ X

}, (5)

where hA(x) is a set of some different trapezoid fuzzy numbers indicating all possible membershipdegrees of the element x ∈ X to the set A. For convenience, hA(x) is called a hesitant trapezoidfuzzy element (HTrFE), which is denoted by hA(x) = h = {[a, b, c, d]}.

Given three HTrFEs, h = {[a, b, c, d]}, h1 = {[a1, b1, c1, d1]} and h2 = {[a2, b2, c2, d2]} andλ > 0, the operations are defined as follows.

1) h1 ⊕ h2 = ∪[a1,b1,c1,d1]∈h1,[a2,b2,c2,d2]∈h2{[a1 + a2 − a1a2, b1 + b2 − b1b2, c1 + c2 − c1c2,

d1 + d2 − d1d2]};2) h1 ⊗ h2 = ∪[a1,b1,c1,d1]∈h1,[a2,b2,c2,d2]∈h2

{[a1a2, b1b2, c1c2, d1d2]};3) hλ = ∪[a,b,c,d]∈h

{[aλ, bλ, cλ, dλ

]};

4) λh = ∪[a,b,c,d]∈h {[λa, λb, λc, λd]}.Definition 6 For a HTrFE h, s(h) = 1

lh

∑γ∈h γ is called the score function of h, where lh

is the number of the trapezoid fuzzy values in h, and s(h) is a trapezoid fuzzy number in therange of [0, 1]. For two HTrFEs h1 and h2, if s(h1) > s(h2), then h1 > h2; if s(h1) = s(h2),then h1 = h2.

So, we can utilize Equation (4) to compare two score functions and judge the magnitudesof two HTrFEs.

4 Some Aggregating Operators with Hesitant Trapezoid Fuzzy Infor-

mation

4.1 Hesitant Trapezoid Fuzzy Aggregation Operators

Based on the operational principle for HTrFEs, we shall develop the hesitant trapezoid fuzzyweighted average (HTrFWA) operator and the hesitant trapezoid fuzzy weighted geometric(HTrFWG) operator.

Definition 7 Let hj (j = 1, 2, · · · , n) be a collection of HTrFEs on X. A HTrFWA operatoris a mapping Qn → Q, and denoted by

HTrFWA(h1, h2, · · · , hn

)


=n⊕

j=1wj hj

=⋃

γ1∈h1,γ2∈h2,··· ,γn∈hn⎧⎨⎩⎡⎣1 −

n∏j=1

(1 − aj)wj , 1 −

n∏j=1

(1 − bj)wj , 1 −

n∏j=1

(1 − cj)wj , 1 −

n∏j=1

(1 − dj)wj

⎤⎦⎫⎬⎭ , (6)

where w=(w1, w2, · · · , wn) is the weight vector of hj (j = 1, 2, · · · , n), wj ≥0 and∑n

j=1 wj = 1.

Definition 8 Let hj (j = 1, 2, · · · , n) be a collection of HTrFEs on X. A HTrFWGoperator is a mapping Qn → Q, and denoted by

HTrFWG(h1, h2, · · · , hn

)=

n⊗j=1

hwj

j

=⋃

γ1∈h1,γ2∈h2,··· ,γn∈hn

⎧⎨⎩⎡⎣ n∏

j=1

awj

j ,

n∏j=1

bwj

j ,

n∏j=1

cwj

j ,

n∏j=1

dwj

j

⎤⎦⎫⎬⎭ , (7)

where w = (w1, w2, · · · , wn) is the weight vector of hj(j = 1, 2, · · · , n), wj ≥ 0 and∑n

j=1 wj = 1.

Considering the weights of the ordered positions of hesitant trapezoid fuzzy arguments,the hesitant trapezoid fuzzy ordered weighted average (HTrFOWA) operator and the hesitanttrapezoid fuzzy ordered weighted geometric (HTrFOWG) operator are defined as follows.

Definition 9 Let hj (j = 1, 2, · · · , n) be a collection of HTrFEs, then we define theHTrFOWA operator as follows

HTrFOWA(h1, h2, · · · , hn

)=

n⊕j=1

ωj hσ(j)

=⋃

γσ(j)∈hσ(j),j=1,2,··· ,n

⎧⎨⎩⎡⎣1 −

n∏j=1

(1 − aσ(j)

)ωj, 1 −

n∏j=1

(1 − bσ(j)

)ωj,

1 −n∏

j=1

(1 − cσ(j)

)ωj, 1 −

n∏j=1

(1 − dσ(j)

)ωj

⎤⎦⎫⎬⎭ , (8)

where (σ(1), σ(2), · · · , σ(n)) is a permutation of (1, 2, · · · , n), such that hσ(j−1) ≥ hσ(j) for allj = 2, 3, · · · , n, and ω = (ω1, ω2, · · · , ωn)T is the aggregation-associated weight vector such thatωj ≥ 0,

∑nj=1 ωj = 1.

Definition 10 Let hj (j = 1, 2, · · · , n) be a collection of HTrFEs, then we define theHTrFOWG operator as follows

HTrFOWG(h1, h2, · · · , hn

)=

n⊗j=1

hωj

σ(j)


= ∪ γσ(j)∈hσ(j),j=1,2,··· ,n

⎧⎨⎩⎡⎣ n∏

j=1

awj

σ(j),

n∏j=1

bwj

σ(j),

n∏j=1

cwj

σ(j),

n∏j=1

dwj

σ(j)

⎤⎦⎫⎬⎭ , (9)


∑nj=1 ωj = 1.

From the above-mentioned definitions, we know that the HTrFWA and HTrFWG operatorsweight the hesitant trapezoid fuzzy argument itself, while the HTrFOWA and HTrFOWG op-erators weight the ordered positions of hesitant trapezoid fuzzy arguments. However, neither ofthese operators can consider both the two aspects. To solve this drawback, in the following weshall propose the hesitant trapezoid fuzzy hybrid average (HTrFHA) operator and the hesitanttrapezoid fuzzy hybrid geometric (HTrFHG) operator.

Definition 11 Let hj (j = 1, 2, · · · , n) be a collection of HTrFEs, then the HTrFHAoperator is defined as follows

HTrFHA(h1, h2, · · · , hn

)=

n⊕j=1

wj

.

hσ(j)

=⋃

.

γσ(j)

∈.

hσ(j)

,

j=1,2,··· ,n

⎧⎨⎩⎡⎣1 −

n∏j=1

(1 − .

aσ(j)

)wj, 1 −

n∏j=1

(1 − .

bσ(j)

)wj

,

1 −n∏

j=1

(1 − .

cσ(j)

)wj, 1 −

n∏j=1

(1 − .

dσ(j)

)wj

⎤⎦⎫⎬⎭ , (10)

where w = (w1, w2, · · · , wn)T is the associated weighting vector, with wj ≥ 0,∑n

j=1 wj = 1 andhσ(j) is the jth largest element of the hesitant trapezoid fuzzy arguments hσ(j) = (nωj) hj (j =1, 2, · · · , n), ω = (ω1, ω2, · · · , ωn)T is the associated weighting vector, with ωj ≥ 0,

∑nj=1 ωj =

1, and n is the balancing coefficient.

Definition 12 Let hj (j = 1, 2, · · · , n) be a collection of HTrFEs, then the HTrFHGoperator is defined as follows

HTrFHG(h1, h2, · · · , hn

)=

n⊗j=1

˙h

wj

σ(j)

=⋃

γσ(j)∈hσ(j),j=1,2,··· ,n

⎧⎨⎩⎡⎣ n∏

j=1

awj

σ(j),

n∏j=1

bwj

σ(j),

n∏j=1

cwj

σ(j),

n∏j=1

dwj

σ(j)

⎤⎦⎫⎬⎭ , (11)



∑nj=1 ωj =

1, and n is the balancing coefficient.


4.2 Hesitant Trapezoid Fuzzy Hamacher Aggregation Operators

Motivated by Equations (2) and (3), let a t-norm T be the Hamacher product ⊗ and t-conorm T ∗ be the Hamacher sum⊕, then the generalized intersection and union on two HTrFEsh1 and h2 become the Hamacher product (denoted by h1 ⊗ h2) and Hamacher sum (denotedby h1 ⊕ h2) of two HTrFEs h1 and h2, respectively, as follows:

h1 ⊕ h2 =⋃

γ1∈h1,γ2∈h2

{[a1 + a2 − a1a2 − (1 − γ) a1a2

1 − (1 − γ) a1a2,b1 + b2 − b1b2 − (1 − γ) b1b2

1 − (1 − γ) b1b2,

c1 + c2 − c1c2 − (1 − γ) c1c2

1 − (1 − γ) c1c2,d1 + d2 − d1d2 − (1 − γ) d1d2

1 − (1 − γ) d1d2

]};

h1 ⊗ h2 =⋃

γ1∈h1,γ2∈h2

{[a1a2

γ + (1 − γ) (a1 + a2 − a1a2),

a1a2

γ + (1 − γ) (a1 + a2 − a1a2),

c1c2

γ + (1 − γ) (c1 + c2 − c1c2),

d1d2

γ + (1 − γ) (d1 + d2 − d1d2)

]};

λh1 =⋃

γ1∈h1

{[(1 + (γ − 1) a1)

λ − (1 − a1)λ

(1 + (γ − 1) a1)λ + (γ − 1) (1 − a1)

λ,

(1 + (γ − 1) b1)λ − (1 − b1)

λ

(1 + (γ − 1) b1)λ + (γ − 1) (1 − b1)

λ,

(1 + (γ − 1) c1)λ − (1 − c1)

λ

(1 + (γ − 1) c1)λ + (γ − 1) (1 − c1)

λ,

(1 + (γ − 1) d1)λ − (1 − d1)

λ

(1 + (γ − 1) d1)λ + (γ − 1) (1 − d1)

λ

]}λ > 0;

hλ1 =

⋃γ1∈h1

{[γ(a1)

λ

(1 + (γ − 1) (1 − a1))λ + (γ − 1) (a1)

λ,

γ(b1)λ

(1+(γ−1) (1−b1))λ + (γ−1) (b1)

λ,

γ(c1)λ

(1 + (γ − 1) (1 − c1))λ + (γ − 1) (c1)

λ,

γ(d1)λ

(1+(γ−1) (1−d1))λ + (γ−1) (d1)

λ

]}λ > 0.

In the following, we shall develop some hesitant trapezoid fuzzy Hamacher aggregation op-erators, such as the hesitant trapezoid fuzzy Hamacher weighted average (HTrFHWA) operatorand the hesitant trapezoid fuzzy Hamacher weighted geometric (HTrFHWG) operator.

Definition 13 Let hj (j = 1, 2, · · · , n) be a collection of HTrFEs, then we define theHTrFHWA operator as follows:

HTrFHWA(h1, h2, · · · , hn

)=

n⊕j=1

wj hj , (12)

where w = (w1, w2, · · · , wn)T be the weight vector of HTrFEs, and wj ≥ 0,∑n

j=1 wj = 1.

Based on the Hamacher operations of hesitant trapezoid fuzzy elements described in Section3, we can drive the following theorem.

Theorem 1 Let hj (j = 1, 2, · · · , n) be a collection of HTrFEs, then their aggregatedvalue by using the HTrFHWA operator is also a HTrFE, and

HTrFHWA(h1, h2, · · · , hn

)=

n⊕j=1

wj hj


=⋃

γj∈hj,j=1,2,··· ,n

⎧⎪⎪⎨⎪⎪⎩⎡⎢⎢⎣

n∏j=1

(1 + (γ − 1) aj)wj −

n∏j=1

(1 − aj)wj

n∏j=1

(1 + (γ − 1) aj)wj + (γ − 1)

n∏j=1

(1 − aj)wj

,

n∏j=1

(1 + (γ − 1) bj)wj −

n∏j=1

(1 − bj)wj

n∏j=1

(1 + (γ − 1) bj)wj + (γ − 1)

n∏j=1

(1 − bj)wj

,

n∏j=1

(1 + (γ − 1) cj)wj −

n∏j=1

(1 − cj)wj

n∏j=1

(1 + (γ − 1) cj)wj + (γ − 1)

n∏j=1

(1 − cj)wj

,

n∏j=1

(1 + (γ − 1) dj)wj −

n∏j=1

(1 − dj)wj

n∏j=1

(1 + (γ − 1) dj)wj + (γ − 1)

n∏j=1

(1 − dj)wj

⎤⎥⎥⎦⎫⎪⎪⎬⎪⎪⎭ . (13)

In what follows, we give some special cases of the HTrFHWA operator with respect to theparameter γ.

When γ = 1, the HTrFHWA operator reduces to the HTrFWA operator.When γ = 2, the HTrFHWA operator reduces to the hesitant trapezoid fuzzy Einstein

weighted average (HTrFEWA) operator as follows:

HTrFEWA(h1, h2, · · · , hn

)=

n⊕j=1

wj hj

⋃γj∈hj ,j=1,2,··· ,n

⎧⎪⎪⎨⎪⎪⎩⎡⎢⎢⎣

n∏j=1

(1 + aj)wj −

n∏j=1

(1 − aj)wj

n∏j=1

(1 + aj)wj +

n∏j=1

(1 − aj)wj

,

n∏j=1

(1 + bj)wj −

n∏j=1

(1 − bj)wj

n∏j=1

(1 + bj)wj +

n∏j=1

(1 − bj)wj

,

n∏j=1

(1 + cj)wj −

n∏j=1

(1 − cj)wj

n∏j=1

(1 + cj)wj +

n∏j=1

(1 − cj)wj

,

n∏j=1

(1 + dj)wj −

n∏j=1

(1 − dj)wj

n∏j=1

(1 + dj)wj +

n∏j=1

(1 − dj)wj

⎤⎥⎥⎦⎫⎪⎪⎬⎪⎪⎭ .

Definition 14 Let hj (j = 1, 2, · · · , n) be a collection of HTrFEs, then we define theHTrFHWG operator as follows

HTrFHWG(h1, h2, · · · , hn

)=

n⊗j=1

hwj

j , (14)

where w = (w1, w2, · · · , wn)T be the weight vector of HTrFEs, and wj ≥ 0,∑n

j=1 wj = 1.

Based on the Hamacher operations of hesitant trapezoid fuzzy elements described in Section3, we can drive the following theorem.

Theorem 2 Let hj (j = 1, 2, · · · , n) be a collection of HTrFEs, then their aggregated


value by using the HTrFHWG operator is also a HTrFE, and

HTrFHWG(h1, h2, · · · , hn

)=

n⊗j=1

hwj

j

⋃γj∈hj ,j=1,2,··· ,n

⎧⎪⎪⎨⎪⎪⎩⎡⎢⎢⎣

γn∏

j=1

awj

j

n∏j=1

(1 + (γ − 1) (1 − aj))wj

+ (γ − 1)n∏

j=1

awj

j

,

γn∏

j=1

bwj

j

n∏j=1

(1 + (γ − 1) (1 − bj))wj

+ (γ − 1)n∏

j=1

bwj

j

,

γn∏

j=1

cwj

j

n∏j=1

(1 + (γ − 1) (1 − cj))wj

+ (γ − 1)n∏

j=1

cwj

j

,

γn∏

j=1

dwj

j

n∏j=1

(1 + (γ − 1) (1 − dj))wj

+ (γ − 1)n∏

j=1

dwj

j

⎤⎥⎥⎦⎫⎪⎪⎬⎪⎪⎭ . (15)

In the following, we can discuss some special cases of the HTrFHWG operator with respectto the parameter γ.

When γ = 1, the HTrFHWG operator reduces to the HTrFWG operator.When γ = 2, the HTrFHWG operator reduces to the hesitant trapezoid fuzzy Einstein

weigthed geometric (HTrFEWG) operator as follows:

HTrFEWG(h1, h2, · · · , hn

)=

n⊗j=1

(h

wj

j

)⋃

γj∈hj,j=1,2,··· ,n

⎧⎪⎪⎨⎪⎪⎩⎡⎢⎢⎣

2n∏

j=1

awj

j

n∏j=1

(2 − aj)wj +

n∏j=1

awj

j

,

2n∏

j=1

bwj

j

n∏j=1

(2 − bj)wj +

n∏j=1

bwj

j

,

2n∏

j=1

cwj

j

n∏j=1

(2 − cj)wj +

n∏j=1

cwj

j

,

2n∏

j=1

dwj

j

n∏j=1

(2 − dj)wj +

n∏j=1

dwj

j

⎤⎥⎥⎦⎫⎪⎪⎬⎪⎪⎭ .

Considering the weights of the ordered positions of hesitant trapezoid fuzzy arguments, thehesitant trapezoid fuzzy Hamacher ordered weighted average (HTrFHOWA) operator and thehesitant trapezoid fuzzy Hamacher ordered weighted geometric (HTrFHOWG) operator aredefined as follows.

Definition 15 Let hj (j = 1, 2, · · · , n) be a collection of HTrFEs, then we define the


HTrFHOWA operator as follows:

HTrFHOWA(h1, h2, · · · , hn

)=

n⊕j=1

ωj hσ(j)

=⋃

γσ(j)∈hσ(j),j=1,2,··· ,n

⎧⎪⎪⎨⎪⎪⎩⎡⎢⎢⎣

n∏j=1

(1 + (γ − 1) aσ(j)

)ωj −n∏

j=1

(1 − aσ(j)

)ωj

n∏j=1

(1 + (γ − 1) aσ(j)

)ωj + (γ − 1)n∏

j=1

(1 − aσ(j)

)ωj

,

n∏j=1

(1 + (γ − 1) bσ(j)

)ωj −n∏

j=1

(1 − bσ(j)

)ωj

n∏j=1

(1 + (γ − 1) bσ(j)

)ωj + (γ − 1)n∏

j=1

(1 − bσ(j)

)ωj

,

n∏j=1

(1 + (γ − 1) cσ(j)

)ωj −n∏

j=1

(1 − cσ(j)

)ωj

n∏j=1

(1 + (γ − 1) cσ(j)

)ωj + (γ − 1)n∏

j=1

(1 − cσ(j)

)ωj

,

n∏j=1

(1 + (γ − 1) dσ(j)

)ωj −n∏

j=1

(1 − dσ(j)

)ωj

n∏j=1

(1 + (γ − 1) dσ(j)

)ωj + (γ − 1)n∏

j=1

(1 − dσ(j)

)ωj

⎤⎥⎥⎦⎫⎪⎪⎬⎪⎪⎭ , (16)

where (σ(1), σ(2), · · · , σ(n)) is a permutation of (1, 2, · · · , n), such that hσ(j−1) ≥ hσ(j) for allj = 2, · · · , n, and ω = (ω1, ω2, · · · , ωn)T is the aggregation-associated weight vector such thatωj ≥ 0,

∑nj=1 ωj = 1.

In what follows, we give some special cases of the HTrFHOWA operator with respect to theparameter γ.

When γ = 1, the HTrFHOWA operator reduces to the HTrFOWA operator.When γ = 2, the HTrFHOWA operator reduces to the hesitant trapezoid fuzzy Einstein

ordered weighted average (HTrFEOWA) operator as follows:

HTrFEOWA(h1, h2, · · · , hn

)=

n⊕j=1

wj hσ(j)

⋃γσ(j)∈hσ(j),j=1,2,··· ,n

⎧⎪⎪⎨⎪⎪⎩⎡⎢⎢⎣

n∏j=1

(1 + aσ(j)

)wj −n∏

j=1

(1 − aσ(j)

)wj

n∏j=1

(1 + aσ(j)

)wj +n∏

j=1

(1 − aσ(j)

)wj

,

n∏j=1

(1 + bσ(j)

)wj −n∏

j=1

(1 − bσ(j)

)wj

n∏j=1

(1 + bσ(j)

)wj +n∏

j=1

(1 − bσ(j)

)wj

,

n∏j=1

(1 + cσ(j)

)wj −n∏

j=1

(1 − cσ(j)

)wj

n∏j=1

(1 + cσ(j)

)wj +n∏

j=1

(1 − cσ(j)

)wj

,

n∏j=1

(1 + dσ(j)

)wj −n∏

j=1

(1 − dσ(j)

)wj

n∏j=1

(1 + dσ(j)

)wj +n∏

j=1

(1 − dσ(j)

)wj

⎤⎥⎥⎦⎫⎪⎪⎬⎪⎪⎭ .

Definition 16 Let hj (j = 1, 2, · · · , n) be a collection of HTrFEs, then we define the


HTrFHOWG operator as follows:

HTrFHOWG(h1, h2, · · · , hn

)=

n⊗j=1

hωj

σ(j)

⋃γσ(j)∈hσ(j),j=1,2,··· ,n

⎧⎪⎪⎨⎪⎪⎩⎡⎢⎢⎣

γn∏

j=1

aωj

σ(j)

n∏j=1

(1 + (γ − 1)

(1 − aσ(j)

))ωj + (γ − 1)n∏

j=1

aωj

σ(j)

,

γn∏

j=1

bωj

σ(j)

n∏j=1

(1 + (γ − 1)

(1 − bσ(j)

))ωj + (γ − 1)n∏

j=1

bωj

σ(j)

,

γn∏

j=1

cωj

σ(j)

n∏j=1

(1 + (γ − 1)

(1 − cσ(j)

))ωj + (γ − 1)n∏

j=1

cωj

σ(j)

,

γn∏

j=1

dωj

σ(j)

n∏j=1

(1 + (γ − 1)

(1 − dσ(j)

))ωj + (γ − 1)n∏

j=1

dωj

σ(j)

⎤⎥⎥⎦⎫⎪⎪⎬⎪⎪⎭ , (17)


∑nj=1 ωj = 1.

Now, we can discuss some special cases of the HTrFHOWG operator with respect to theparameter γ.

When γ = 1, the HTrFHOWG operator reduces to the HTrFOWG operator.When γ = 2, the HTrFHOWG operator reduces to the hesitant trapezoid fuzzy Einstein

ordered weigthed geometric (HTrFEOWG) operator as follows:

HTrFEOWG(h1, h2, · · · , hn

)=

n⊗j=1

(h

wj

σ(j)

)⋃

γσ(j)∈hσ(j),j=1,2,··· ,n

⎧⎪⎪⎨⎪⎪⎩⎡⎢⎢⎣

2n∏

j=1

awj

σ(j)

n∏j=1

(2 − aσ(j)

)wj +n∏

j=1

awj

σ(j)

,

2n∏

j=1

bwj

σ(j)

n∏j=1

(2 − bσ(j)

)wj +n∏

j=1

bwj

σ(j)

,

2n∏

j=1

cwj

σ(j)

n∏j=1

(2 − cσ(j)

)wj +n∏

j=1

cwj

σ(j)

,

2n∏

j=1

dwj

σ(j)

n∏j=1

(2 − dσ(j)

)wj +n∏

j=1

dwj

σ(j)

⎤⎥⎥⎦⎫⎪⎪⎬⎪⎪⎭ .

Similarly, from the above mentioned definitions, the HTrFHWA and HTrFHWG operatorsweight the hesitant trapezoid fuzzy argument itself, while the HTrFHOWA and HTrFHOWG


operators weight the ordered positions of hesitant trapezoid fuzzy arguments. In order toconsider both the two aspects, in the following we shall propose the hesitant trapezoid fuzzyHamacher hybrid average (HTrFHHA) operator and the hesitant trapezoid fuzzy Hamacherhybrid geometric (HTrFHHG) operator.

Definition 17 Let hj (j = 1, 2, · · · , n) be a collection of HTrFEs, then the HTrFHHAoperator is defined as follows:

HTrFHHA(h1, h2, · · · , hn

)

=n⊕

j=1wj

.

hσ(j)

=⋃

.

γσ(j)

∈.

hσ(j)

,

j=1,2,··· ,n

⎧⎪⎪⎨⎪⎪⎩⎡⎢⎢⎣

n∏j=1

(1 + (γ − 1)

.aσ(j)

)wj −n∏

j=1

(1 − .

aσ(j)

)wj

n∏j=1

(1 + (γ − 1)

.aσ(j)

)wj + (γ − 1)n∏

j=1

(1 − .

aσ(j)

)wj

,

n∏j=1

(1 + (γ − 1)

.

bσ(j)

)wj −n∏

j=1

(1 − .

bσ(j)

)wj

n∏j=1

(1 + (γ − 1)

.

bσ(j)

)wj

+ (γ − 1)n∏

j=1

(1 − .

bσ(j)

)wj,

n∏j=1

(1 + (γ − 1)

.cσ(j)

)wj −n∏

j=1

(1 − .

cσ(j)

)wj

n∏j=1

(1 + (γ − 1)

.cσ(j)

)wj + (γ − 1)n∏

j=1

(1 − .

cσ(j)

)wj

,

n∏j=1

(1 + (γ − 1)

.

dσ(j)

)wj −n∏

j=1

(1 − .

dσ(j)

)wj

n∏j=1

(1 + (γ − 1)

.

dσ(j)

)wj

+ (γ − 1)n∏

j=1

(1 − .

dσ(j)

)wj

⎤⎥⎥⎦⎫⎪⎪⎬⎪⎪⎭ , (18)



∑nj=1 ωj = 1,

and n is the balancing coefficient.

In what follows, we give some special cases of the HTrFHHA operator with respect to theparameter γ.

When γ = 1, the HTrFHHA operator reduces to the HTrFHA operator.When γ = 2, the HTrFHHA operator reduces to the hesitant trapezoid fuzzy Einstein

hybrid average (HTrFEHA) operator as follows:

HTrFEHA(h1, h2, · · · , hn

)=

n⊕j=1

wj

.

hσ(j)

⋃.

γσ(j)

∈.

hσ(j)

,

j=1,2,··· ,n

⎧⎪⎪⎨⎪⎪⎩⎡⎢⎢⎣

n∏j=1

(1 +

.aσ(j)

)wj −n∏

j=1

(1 − .

aσ(j)

)wj

n∏j=1

(1 +

.aσ(j)

)wj +n∏

j=1

(1 − .

aσ(j)

)wj

,

n∏j=1

(1 +

.

bσ(j)

)wj −n∏

j=1

(1 − .

bσ(j)

)wj

n∏j=1

(1 +

.

bσ(j)

)wj

+n∏

j=1

(1 − .

bσ(j)

)wj,


n∏j=1

(1 +

.cσ(j)

)wj −n∏

j=1

(1 − .

cσ(j)

)wj

n∏j=1

(1 +

.cσ(j)

)wj +n∏

j=1

(1 − .

cσ(j)

)wj

,

n∏j=1

(1 +

.

dσ(j)

)wj −n∏

j=1

(1 − .

dσ(j)

)wj

n∏j=1

(1 +

.

dσ(j)

)wj

+n∏

j=1

(1 − .

dσ(j)

)wj

⎤⎥⎥⎦⎫⎪⎪⎬⎪⎪⎭ .

Definition 18 Let hj (j = 1, 2, · · · , n) be a collection of HTrFEs, then the HTrFHHGoperator is defined as follows:

HTrFHHG(h1, h2, · · · , hn

)=

n⊗j=1

˙h

wj

σ(j)

=⋃

.

γσ(j)

∈.

hσ(j)

,

j=1,2,··· ,n

⎧⎪⎪⎨⎪⎪⎩⎡⎢⎢⎣

γn∏

j=1

awj

σ(j)

n∏j=1

(1 + (γ − 1)

(1 − .

aσ(j)

))wj

+ (γ − 1)n∏

j=1

awj

σ(j)

,

γn∏

j=1

bwj

σ(j)

n∏j=1

(1 + (γ − 1)

(1 − .

bσ(j)

))wj

+ (γ − 1)n∏

j=1

bwj

σ(j)

,

γn∏

j=1

cwj

σ(j)

n∏j=1

(1 + (γ − 1)

(1 − .

cσ(j)

))wj

+ (γ − 1)n∏

j=1

cwj

σ(j)

,

γn∏

j=1

dwj

σ(j)

n∏j=1

(1 + (γ − 1)

(1 − .

dσ(j)

))wj

+ (γ − 1)n∏

j=1

dwj

σ(j)

⎤⎥⎥⎥⎦⎫⎪⎪⎪⎬⎪⎪⎪⎭ , (19)



∑nj=1 ωj = 1,

and n is the balancing coefficient.

In what follows, we give some special cases of the HTrFHHG operator with respect to theparameter γ.

When γ = 1, the HTrFHHG operator reduces to the HTrFHG operator.When γ = 2, the HTrFHHG operator reduces to the hesitant trapezoid fuzzy Einstein

hybrid geometric (HTrFEHG) operator as follows:

HTrFEHG(h1, h2, · · · , hn

)=

n⊗j=1

( .

hσ(j)

wj)


=⋃

.

γσ(j)

∈.

hσ(j)

,

j=1,2,··· ,n

⎧⎪⎪⎨⎪⎪⎩⎡⎢⎢⎣

2n∏

j=1

awj

σ(j)

n∏j=1

(2 − .

aσ(j)

)wj +n∏

j=1

awj

σ(j)

,

2n∏

j=1

bwj

σ(j)

n∏j=1

(2 − .

bσ(j)

)wj

+n∏

j=1

bwj

σ(j)

,

2n∏

j=1

cwj

σ(j)

n∏j=1

(2 − .

cσ(j)

)wj +n∏

j=1

cwj

σ(j)

,

2n∏

j=1

dwj

σ(j)

n∏j=1

(2 − .

dσ(j)

)wj

+n∏

j=1

dwj

σ(j)

⎤⎥⎥⎦⎫⎪⎪⎬⎪⎪⎭ .

4.3 Hesitant Trapezoid Fuzzy Hamacher Choquet Aggregation Operators

The previous aggregation operators in MADM problem under hesitant trapezoid fuzzy en-vironment are based on the assumption that the attributes are independent of one another.However, in real decision-making problems, there exist some degree of inter-dependent charac-teristics between attributes. In 1974, fuzzy measure was first introduced by Sugeno[32] to dealwith the phenomenon of mutual influence, and it has been applied in various fields[33−36]. Inreal decision-making problems, fuzzy measures define a weight on not only each attribute butalso each combination of attribute, and the sum of every weight does not equal to one.

Definition 19 (see [32]) Let X = {1, 2, · · · , n} be a universe of discourse, P (X) be thepower set of X. A fuzzy measure on X is a set function μ : P (X) → [0, 1] satisfying the followingaxioms:

1) μ(∅) = 0, μ(X) = 1;2) If A, B ∈ P (X) and A ⊆ B, then μ(A) ≤ μ(B).

In multi-attribute decision making, μ(A) can be viewed as the importance of the attributeset A. Thus, in addition to the usual weights on attributes taken separately, weights on anycombination of attributes also can be defined.

Fuzzy integrals, as important aggregation operators for uncertain information, have beenstudied by many researchers [36−40]. One of the most important fuzzy integrals is the Choquetintegral proposed by Grabisch[41]. The concept of the Choquet integral on discrete sets isdefined as follows.

Definition 20 (see [41]) Let f be a positive real-valued function on X, and μ be a fuzzymeasure on X . The discrete Choquet integral of f with respect to μ is defined by

Cμ

(f(x(1)

), f

(x(2)

), · · · , f

(x(n)

))=

n∑i=1

f(x(i)

) (μ(A(i)

)− μ(A(i+1)

)), (20)

where (·) indicates a permutation of (1, 2, · · · , n), such that f(x(1)

) ≤ f(x(2)

) ≤ · · · ≤ f(x(n)

)and A(i) = {i, i + 1, · · · , n} with A(n+1) = ∅.

Based on the aggregation principle of hesitant trapezoid fuzzy elements and Choquet in-tegral, in the following, we shall develop the hesitant trapezoid fuzzy Hamacher Choquet av-erage (HTrFHCA) operator and the hesitant trapezoid fuzzy Hamacher Choquet geometric(HTrFHCG) operator.

Definition 21 Let hj (j = 1, 2, · · · , n) be a collection of HTrFEs on X, and μ be a fuzzy


measure on X . A HTrFHCA operator is a mapping Qn → Q, and

HTrFHCAμ

(h1, h2, · · · , hn

)=

n⊕j=1

(μ(Aσ(j)

)− μ(Aσ(j−1)

))hσ(j), (21)

where (σ(1), σ(2), · · · , σ(n)) is a permutation of (1, 2, · · · , n), such that hσ(j−1) ≥ hσ(j) for allj = 2, 3, · · · , n, Aσ(k) =

{xσ(j)|j ≤ k

}, for k ≥ 1, and Aσ(0) = ∅.

Based on the Hamacher operations of HTrFEs described in Section 3, we can drive thefollowing theorem.

Theorem 3 Let hj (j = 1, 2, · · · , n) be a collection of HTrFEs, then their aggregatedvalue by using the HTrFHCA operator is also a HTrFE, and

HTrFHCAμ

(h1, h2, · · · , hn

)=

n⊕j=1

(μ(Aσ(j)

)− μ(Aσ(j−1)

))hσ(j)

=⋃

γσ(j)∈hσ(j),j=1,2,··· ,n

⎧⎪⎪⎪⎨⎪⎪⎪⎩⎡⎢⎢⎢⎣

n∏j=1

(1+(γ−1) aσ(j)

)μ(Aσ(j))−μ(Aσ(j−1))−n∏

j=1

(1−aσ(j)

)μ(Aσ(j))−μ(Aσ(j−1))

n∏j=1

(1+(γ−1) aσ(j)

)μ(Aσ(j))−μ(Aσ(j−1))+(γ−1)

n∏j=1

(1−aσ(j)

)μ(Aσ(j))−μ(Aσ(j−1)),

n∏j=1

(1 + (γ − 1) bσ(j)

)μ(Aσ(j))−μ(Aσ(j−1)) −n∏

j=1

(1 − bσ(j)


n∏j=1

(1 + (γ − 1) bσ(j)

)μ(Aσ(j))−μ(Aσ(j−1))+ (γ − 1)

n∏j=1

(1 − bσ(j)

)μ(Aσ(j))−μ(Aσ(j−1)),

n∏j=1

(1 + (γ − 1) cσ(j)

)μ(Aσ(j))−μ(Aσ(j−1)) −n∏

j=1

(1 − cσ(j)


n∏j=1

(1 + (γ − 1) cσ(j)

)μ(Aσ(j))−μ(Aσ(j−1))+ (γ − 1)

n∏j=1

(1 − cσ(j)

)μ(Aσ(j))−μ(Aσ(j−1)),

n∏j=1

(1 + (γ − 1) dσ(j)

)μ(Aσ(j))−μ(Aσ(j−1)) −n∏

j=1

(1 − dσ(j)


n∏j=1

(1+(γ − 1) dσ(j)

)μ(Aσ(j))−μ(Aσ(j−1))+ (γ − 1)

n∏j=1

(1−dσ(j)


⎤⎥⎥⎥⎦⎫⎪⎪⎪⎬⎪⎪⎪⎭ , (22)

where (σ(1), σ(2), · · · , σ(n)) is a permutation of (1, 2, · · · , n), such that hσ(j−1) ≥ hσ(j) for allj = 2, 3, · · · , n, Aσ(k) =

{xσ(j)|j ≤ k

}, for k ≥ 1, and Aσ(0) = ∅.

Now, we can discuss some special cases of the HTrFHCA operator with respect to theparameter γ.

When γ = 1, the HTrFHCA operator reduces to the hesitant trapezoid fuzzy Choquetaverage (HTrFCA) operator as follows:

HTrFCAμ

(h1, h2, · · · , hn

)=

n⊕j=1

(μ(Aσ(j)

)− μ(Aσ(j−1)

))hσ(j)


=⋃

γσ(j)∈hσ(j),j=1,2,··· ,n

⎧⎨⎩⎡⎣1 −

n∏j=1

(1 − aσ(j)

)μ(Aσ(j))−μ(Aσ(j−1)) , 1 −n∏

j=1

(1 − bσ(j)

)μ(Aσ(j))−μ(Aσ(j−1)),

1 −n∏

j=1

(1 − cσ(j)

)μ(Aσ(j))−μ(Aσ(j−1)), 1 −n∏

j=1

(1 − dσ(j)

)μ(Aσ(j))−μ(Aσ(j−1))⎤⎦⎫⎬⎭ .

When γ = 2, the HTrFHCA operator reduces to the hesitant trapezoid fuzzy EinsteinChoquet average (HTrFECA) operator as follows:

HTrFECAμ

(h1, h2, · · · , hn

)=

n⊕j=1

(μ(Aσ(j)

)− μ(Aσ(j−1)

))hσ(j)

=⋃

γσ(j)∈hσ(j),j=1,2,··· ,n

⎧⎪⎪⎪⎨⎪⎪⎪⎩⎡⎢⎢⎢⎣

n∏j=1

(1 + aσ(j)

)μ(Aσ(j))−μ(Aσ(j−1)) −n∏

j=1

(1 − aσ(j)


n∏j=1

(1 + aσ(j)

)μ(Aσ(j))−μ(Aσ(j−1))+

n∏j=1

(1 − aσ(j)

)μ(Aσ(j))−μ(Aσ(j−1)),

n∏j=1

(1 + bσ(j)

)μ(Aσ(j))−μ(Aσ(j−1)) −n∏

j=1

(1 − bσ(j)


n∏j=1

(1 + bσ(j)

)μ(Aσ(j))−μ(Aσ(j−1))+

n∏j=1

(1 − bσ(j)

)μ(Aσ(j))−μ(Aσ(j−1)),

n∏j=1

(1 + cσ(j)

)μ(Aσ(j))−μ(Aσ(j−1)) −n∏

j=1

(1 − cσ(j)


n∏j=1

(1 + cσ(j)

)μ(Aσ(j))−μ(Aσ(j−1))+

n∏j=1

(1 − cσ(j)

)μ(Aσ(j))−μ(Aσ(j−1)),

n∏j=1

(1 + dσ(j)

)μ(Aσ(j))−μ(Aσ(j−1)) −n∏

j=1

(1 − dσ(j)


n∏j=1

(1 + dσ(j)

)μ(Aσ(j))−μ(Aσ(j−1))+

n∏j=1

(1 − dσ(j)


⎤⎥⎥⎥⎦⎫⎪⎪⎪⎬⎪⎪⎪⎭ .

Definition 22 Let hj (j = 1, 2, · · · , n) be a collection of HTrFEs on X, and μ be a fuzzymeasure on X . A HTrFHCG operator is a mapping Qn → Q, and

HTrFHCGμ

(h1, h2, · · · , hn

)=

n⊗j=1

(h(μ(Aσ(j))−μ(Aσ(j−1)))σ(j)

), (23)

where (σ(1), σ(2), · · · , σ(n)) is a permutation of (1, 2, · · · , n), such that hσ(j−1) ≥ hσ(j) for allj = 2, · · · , n, Aσ(k) =

{xσ(j)|j ≤ k

}, for k ≥ 1, and Aσ(0) = ∅.

Based on the Hamacher operations of HTrFEs described in Section 3, we can drive thefollowing theorem.

Theorem 4 Let hj (j = 1, 2, · · · , n) be a collection of HTrFEs, then their aggregatedvalue by using the HTrFHCG operator is also a HTrFE, and

HTrFHCGμ

(h1, h2, · · · , hn

)


=n⊗

j=1


)

=⋃

γσ(j)∈hσ(j),j=1,2,··· ,n

⎧⎪⎪⎨⎪⎪⎩⎡⎢⎢⎣

γn∏

j=1

aμ(Aσ(j))−μ(Aσ(j−1))σ(j)

n∏j=1

(1 + (γ−1)

(1−aσ(j)

))μ(Aσ(j))−μ(Aσ(j−1)) + (γ−1)n∏

j=1


,

γn∏

j=1

bμ(Aσ(j))−μ(Aσ(j−1))σ(j)

n∏j=1

(1 + (γ − 1)

(1 − bσ(j)

))μ(Aσ(j))−μ(Aσ(j−1)) + (γ − 1)n∏

j=1


,

γn∏

j=1

cμ(Aσ(j))−μ(Aσ(j−1))σ(j)

n∏j=1

(1 + (γ − 1)

(1 − cσ(j)

))μ(Aσ(j))−μ(Aσ(j−1)) + (γ − 1)n∏

j=1


,

γn∏

j=1

dμ(Aσ(j))−μ(Aσ(j−1))σ(j)

n∏j=1

(1 + (γ − 1)

(1 − dσ(j)

))μ(Aσ(j))−μ(Aσ(j−1)) + (γ − 1)n∏

j=1


⎤⎥⎥⎦⎫⎪⎪⎬⎪⎪⎭ , (24)

where (σ(1), σ(2), · · · , σ(n)) is a permutation of (1, 2, · · · , n), such that hσ(j−1) ≥ hσ(j) for allj = 2, · · · , n, Aσ(k) =

{xσ(j)|j ≤ k

}, for k ≥ 1, and Aσ(0) = ∅.

Now, we can discuss some special cases of the HTrFHCG operator with respect to theparameter γ.

When γ = 1, the HTrFHCG operator reduces to the hesitant trapezoid fuzzy Choquetgeometric (HTrFCG) operator as follows:

HTrFCGμ

(h1, h2, · · · , hn

)=

n⊗j=1


)

=⋃

γσ(j)∈hσ(j),j=1,2,··· ,n

⎧⎨⎩⎡⎣ n∏

j=1

aμ(Aσ(j))−μ(Aσ(j−1))σ(j) ,

n∏j=1

bμ(Aσ(j))−μ(Aσ(j−1))σ(j) ,

n∏j=1

cμ(Aσ(j))−μ(Aσ(j−1))σ(j) ,

n∏j=1


⎤⎦⎫⎬⎭ .

When γ = 2, the HTrFHCG operator reduces to the hesitant trapezoid fuzzy EinsteinChoquet geometric (HTrFECG) operator as follows:

HTrFECGμ

(h1, h2, · · · , hn

)=

n⊗j=1


)

= ∪ γσ(j)∈hσ(j),j=1,2,··· ,n

⎧⎪⎪⎨⎪⎪⎩⎡⎢⎢⎣

2n∏

j=1


n∏j=1

(2 − aσ(j)

)μ(Aσ(j))−μ(Aσ(j−1)) +n∏

j=1


,


2n∏

j=1


n∏j=1

(2 − bσ(j)

)μ(Aσ(j))−μ(Aσ(j−1)) +n∏

j=1


,

2n∏

j=1


n∏j=1

(2 − cσ(j)

)μ(Aσ(j))−μ(Aσ(j−1)) +n∏

j=1


,

2n∏

j=1


n∏j=1

(2 − dσ(j)

)μ(Aσ(j))−μ(Aσ(j−1)) +n∏

j=1


⎤⎥⎥⎦⎫⎪⎪⎬⎪⎪⎭ .

5 An Approach to Multiple Attribute Decision Making with Hesitant

Trapezoid Fuzzy Information

In this section, we utilize the proposed aggregation operators to develop an approach toMADM problems under hesitant trapezoid fuzzy environment.

For a MADM problem, let A = {A1, A2, · · · , Am} be a discrete set of alternatives, andC = {C1, C2, · · · , Cn} be a collection of attributes. w = (w1, w2, · · · , wn)T is the weightingvector of the attribute Cj (j = 1, 2, · · · , n), with wj ≥ 0,

∑nj=1 wj = 1. Suppose that the decision

matrix H = (hy)m×n is the hesitant trapezoid fuzzy matrix, where hij is the evaluation valuefor alternative Ai ∈ A (i = 1, 2, · · · , m) with respect to the attribute Cj ∈ C (j = 1, 2, · · · , n),and takes the form of HTrFE.

In the following, we apply the HTrFHWA operator operator to deal with multi-attributedecision making problems. The main steps are summarized as follows.

Step 1 From the given decision matrix H, and we utilize the HTrFHWA operator

hi = HTrFHWA(hi1, hi2, · · · , hin

)

=

⎧⎪⎪⎨⎪⎪⎩⎡⎢⎢⎣

n∏j=1

(1+(γ−1) aij)wj −

n∏j=1

(1−aij)wj

n∏j=1

(1+(γ−1) aij)wj +(γ−1)

n∏j=1

(1−aij)wj

,

n∏j=1

(1+(γ−1) bij)wj −

n∏j=1

(1−bij)wj

n∏j=1

(1+(γ−1) bij)wj +(γ−1)

n∏j=1

(1−bij)wj

,

n∏j=1

(1+(γ−1) cij)wj −

n∏j=1

(1−cij)wj

n∏j=1

(1+(γ−1) cij)wj +(γ−1)

n∏j=1

(1−cij)wj

,

n∏j=1

(1+(γ−1) dij)wj −

n∏j=1

(1−dij)wj

n∏j=1

(1+(γ−1) dij)wj +(γ−1)

n∏j=1

(1−dij)wj

⎤⎥⎥⎦⎫⎪⎪⎬⎪⎪⎭

to aggregate all the hesitant trapezoid fuzzy values hij (i = 1, 2, · · · , m; j = 1, 2, · · · , n) intothe overall hesitant trapezoid fuzzy value hi (i = 1, 2, · · · , m) of each alternative Ai (i =1, 2, · · · , m).

Step 2 Calculate the score values s (hi) (i = 1, 2, · · · , m) of the overall hesitant trapezoidfuzzy values hi (i = 1, 2, · · · , m) by Definition 6.

Step 3 To rank these score values s(hi) (i = 1, 2, · · · , m), we first compare each s(hi) withall the s(hi) (i = 1, 2, · · · , m) by using Equation (4). For simplicity, we let pij = p(s(hi) ≥


s(hj)), then we develop a complementary matrix as P = (pij)m×n, where

pij ≥ 0, pij + pji = 1, pii = 0.5, i, j = 1, 2, · · · , m.

Summing all the elements in each line of matrix P, we have

pi =m∑

j=1

pij , i = 1, 2, · · · , m.

Then we rank the score values s(hi) (i = 1, 2, · · · , m) in descending order in accordancewith the values of pi (i = 1, 2, · · · , m).

Step 4 Rank all the alternatives Ai (i = 1, 2, · · · , m) and select the best one(s) in accor-dance with s (hi) (i = 1, 2, · · · , m).

In the following, we apply the HTrFHWG operator to deal with multi-attribute decisionmaking problems. The main steps are summarized as follows.

Step 1′ From the given decision matrix H , and we utilize the HTrFHWG operator

hi

= HTrFHWG(hi1, hi2, · · · , hin

)

=

⎧⎪⎪⎨⎪⎪⎩⎡⎢⎢⎣

γn∏

j=1

(aij)wj

n∏j=1

(1+(γ−1) (1−aij))wj +(γ−1)

n∏j=1

(aij)wj

,

γn∏

j=1

(bij)wj

n∏j=1

(1+(γ−1) (1−bij))wj +(γ−1)

n∏j=1

(bij)wj

,

γn∏

j=1

(cij)wj

n∏j=1

(1+(γ−1) (1−cij))wj +(γ−1)

n∏j=1

(cij)wj

,

γn∏

j=1

(dij)wj

n∏j=1

(1+(γ−1) (1−dij))wj +(γ−1)

n∏j=1

(dij)wj

⎤⎥⎥⎦⎫⎪⎪⎬⎪⎪⎭

to aggregate all the hesitant trapezoid fuzzy values hij (i = 1, 2, · · · , m; j = 1, 2, · · · , n) intothe overall hesitant trapezoid fuzzy value hi (i = 1, 2, · · · , m) of each alternative Ai (i =1, 2, · · · , m).

Step 2′ Calculate the score values s (hi) (i = 1, 2, · · · , m) of the overall hesitant trapezoidfuzzy values hi (i = 1, 2, · · · , m) by Definition 6.

Step 3′ To rank these score values s(hi) (i = 1, 2, · · · , m), we first compare each s(hi) withall the s(hi) (i = 1, 2, · · · , m) by using Equation (4). For simplicity, we let pij = p(s(hi) ≥s(hj)), then we develop a complementary matrix as P = (pij)m×n, where

pij ≥ 0, pij + pji = 1, pij = 0.5, i, j = 1, 2, · · · , m.

Summing all the elements in each line of matrix P , we have

pi =m∑

j=1

pij , i = 1, 2, · · · , m.

Then we rank the score values s(hi) (i = 1, 2, · · · , m) in descending order in accordancewith the values of pi (i = 1, 2, · · · , m).

Step 4′ Rank all the alternatives Ai (i = 1, 2, · · · , m) and select the best one(s) in accor-dance with s (ht) (i = 1, 2, · · · , m).


6 Illustrative Example

Let us suppose that there is a manufacturing company, which wants to select the bestglobal supplier according to the core competencies of suppliers (adapted from [42]). There arefour suppliers Ai (i = 1, 2, 3, 4) to be evaluated under the following four attributes Cj (j =1, 2, 3, 4) : C1 is the level of technology innovation; C2 is the control ability of flow; C3 is theability of management; C4 is the level of service. w = (0.2, 0.4, 0.1, 0.3)T is the weight vector ofattributes. The four possible candidates Ai (i = 1, 2, 3, 4) are to be evaluated using the hesitanttrapezoid fuzzy information by the decision maker under the above four attributes, and thehesitant trapezoid fuzzy decision matrix H = (hij)4×4 is shown in Table 1.

Table 1 Hesitant trapezoid fuzzy decision matrix

C1 C2 C3 C4

A1

{[0.5,0.6,0.7,0.8] {[0.2, 0.3, 0.4, 0.5]} {[0.3,0.4,0.5,0.6] {[0.4,0.5,0.6,0.7]}[0.6,0.7,0.8,0.9]} [0.1,0.2,0.3,0.4]}

A2

{[0.4,0.5,0.6,0.7]} {[0.1,0.2,0.3,0.4] {[0.2,0.3,0.4,0.5]} {[0.3,0.4,0.5,0.6]

[0.3,0.4,0.5,0.6]} [0.5,0.6,0.7,0.8]}

A3

{[0.3,0.4,0.5,0.6] {[0.4,0.5,0.6,0.7] {[0.2,0.3,0.4,0.5]}[0.2,0.3,0.4,0.5]} {[0.2,0.3,0.4,0.5]} [0.1,0.2,0.3,0.4]

[0.3,0.4,0.5,0.6]}

A4 {[0.1, 0.2, 0.3, 0.4]} {[0.4,0.5,0.6,0.7] {[0.3,0.4,0.5,0.6]} {[0.1,0.2,0.3,0.4]

[0.5,0.6,0.7,0.8]} [0.4,0.5,0.6,0.7]}

In order to select the most desirable supplier, we utilize the hesitant trapezoid fuzzyHamacher weighted average (HTrFHWA) operator and the hesitant trapezoid fuzzy Hamacherweighted geometric (HTrFHWG) operator to develop an approach to multiple attribute deci-sion making problems with hesitant trapezoid fuzzy information, which can be described asfollowing:

Step 1 We utilize the HTrFHWA operator to aggregate all the hesitant trapezoid fuzzyinformation hij (i = 1, 2, 3, 4; j = 1, 2, 3, 4) into the overall hesitant trapezoid fuzzy valueshi (i = 1, 2, 3, 4). Take h1 for example, here let γ = 1, 2, we have

h1 = HTrFHWA1

(h11, h12, h13, h14

)= HTrFWA

(h11, h12, h13, h14

)=

4⊕j=1

wj h1j

=⋃

γ1j∈h1j,j=1,2,3,4

⎧⎨⎩⎡⎣1 −

4∏j=1

(1 − a1j)wj , 1 −

4∏j=1

(1 − b1j)wj , 1 −

4∏j=1

(1 − c1j)wj , 1 −

4∏j=1

(1 − d1j)wj

⎤⎦⎫⎬⎭

= {[0.3408, 0.4429, 0.5458, 0.6508] [0.3241, 0.4266, 0.5303, 0.6363]

[0.3696, 0.4740, 0.5812, 0.6960] [0.35360.45870.56690.6834]},h1 = HTrFHWA2

(h11, h12, h13, h14

)= HTrFEWA

(h11, h12, h13, h14

)=

4⊕j=1

wj h1j


=⋃

γ1j∈h1j,j=1,2,3,4

⎧⎪⎪⎪⎨⎪⎪⎪⎩⎡⎢⎢⎢⎣

4∏j=1

(1 + a1j)wj −

4∏j=1

(1 − a1j)wj

4∏j=1

(1 + a1j)wj +

4∏j=1

(1 − a1j)wj

,

4∏j=1

(1 + b1j)wj −

4∏j=1

(1 − b1j)wj

4∏j=1

(1 + b1j)wj +

4∏j=1

(1 − b1j)wj

,

4∏j=1

(1 + c1j)wj −

4∏j=1

(1 − c1j)wj

4∏j=1

(1 + c1j)wj +

4∏j=1

(1 − c1j)wj

,

4∏j=1

(1 + d1j)wj −

4∏j=1

(1 − d1j)wj

4∏j=1

(1 + d1j)wj +

4∏j=1

(1 − d1j)wj

⎤⎥⎥⎥⎦⎫⎪⎪⎪⎬⎪⎪⎪⎭

= {[0.3355, 0.4379, 0.5412, 0.6463] [0.3168, 0.4198, 0.5240, 0.6304]

[0.3608, 0.4656, 0.5731, 0.6877] [0.3424, 0.4481, 0.5567, 0.6733]} .

Step 2a) When γ = 1, calculate the score values s(hi) (i = 1, 2, 3, 4) of the overall hesitanttrapezoid fuzzy preference values hi (i = 1, 2, 3, 4).

s(h1) = {[0.3470, 04505, 0.5561, 0.6666]}, s(h2) = {[0.3105, 0.4124, 0.5150, 0.6193]},s(h3) = {[0.2603, 0.3613, 0.4627, 0.5649]}, s(h4) = {[0.3212, 0.4240, 0.5279, 0.6340]}.

Step 2b) When γ = 2, calculate the score values s(hi) (i = 1, 2, 3, 4) of the overall hesitanttrapezoid fuzzy preference values hi (i = 1, 2, 3, 4).

s(h1) = {[0.3389, 0.4429, 0.5487, 0.6594]}, s(h2) = {[0.3049, 0.4071, 0.5101, 0.6147]},s(h3) = {[0.2173, 0.3177, 0.4181, 0.5187]}, s(h4) = {[0.3125, 0.4159, 0.5205, 0.6272]}.

Step 3 Rank all the suppliers Ai (i = 1, 2, 3, 4) in accordance with the score value s(hi)of the overall fuzzy preference value hi : A1 � A4 � A2 � A3. And thus, the most desirablesupplier is A1.

Furthermore, the HTrFHWG operator is used to calculate comprehensive hesitant trapezoidfuzzy information shown as follows.

Step 1′ We apply the HTrFHWG operator to obtain the overall hesitant trapezoid fuzzyvalues hi. Take h1 for example, here let γ = 1, 2, we have

h1 = HTrFHWG1

(h11, h12, h13, h14

)= HTrFWG

(h11, h12, h13, h14

)=

4⊗j=1

hwj

1j

=⋃

γ1j∈h1j ,j=1,2,3,4

⎧⎨⎩⎡⎣ 4∏

j=1

awj

1j ,

4∏j=1

bwj

1j ,

4∏j=1

cwj

1j ,

4∏j=1

dwj

1j

⎤⎦⎫⎬⎭

= {[0.3080, 0.4134, 0.5166, 0.6188] [0.2759, 0.3857, 0.4909, 0.3314]

[0.3194, 0.4263, 0.5306, 0.6336] [0.2862, 0.3978, 0.5042, 0.6084]},h1 = HTrFHWG2

(h11, h12, h13, h14

)= HTrFEWG

(h11, h12, h13, h14

)=

4⊗j=1

(h

wj

1j

)

=⋃

γ1j∈h1j ,j=1,2,3,4

⎧⎪⎪⎪⎨⎪⎪⎪⎩⎡⎢⎢⎢⎣

24∏

j=1

awj

1j

4∏j=1

(2 − a1j)wj +

4∏j=1

awj

1j

,

24∏

j=1

bwj

1j

4∏j=1

(2 − b1j)wj +

4∏j=1

bwj

1j

,


24∏

j=1

cwj

1j

4∏j=1

(2 − c1j)wj +

4∏j=1

cwj

1j

,

24∏

j=1

dwj

1j

4∏j=1

(2 − d1j)wj +

4∏j=1

dwj

1j

⎤⎥⎥⎥⎦⎫⎪⎪⎪⎬⎪⎪⎪⎭

= {[0.3121, 0.4178, 0.5214, 0.6239] [0.2815, 0.3916, 0.4972, 0.3866]

[0.3256, 0.4331, 0.5380, 0.6416] [0.2939, 0.4062, 0.5134, 0.6184]}.

Step 2a′) When γ = 1, calculate the score values s(hi) (i = 1, 2, 3, 4) of the overall hesitanttrapezoid fuzzy preference values hi (i = 1, 2, 3, 4).

s(h1) = {[0.2974, 0.4058, 0.5106, 0.5480]}, s(h2) = {[0.2717, 0.3803, 0.4847, 0.5875]},s(h3) = {[0.2117, 0.3132, 0.4140, 0.5145]}, s(h4) = {[0.2558, 0.3717, 0.4795, 0.5843]}.

Step 2b′) When γ = 2, calculate the score values s(hi) (i = 1, 2, 3, 4) of the overall hesitanttrapezoid fuzzy preference values hi (i = 1, 2, 3, 4).

s(h1) = {[0.3033, 0.4122, 0.5175, 0.5676]}, s(h2) = {[0.2757, 0.3847, 0.4895, 0.5926]},s(h3) = {[0.2123, 0.3139, 0.4147, 0.5153]}, s(h4) = {[0.2630, 0.3793, 0.4875, 0.5927]}.

Step 3′ Rank all the suppliers Ai (i = 1, 2, 3, 4) in accordance with the score value s(hi)of the overall fuzzy preference value hi : A1 � A2 � A4 � A3. And thus the most desirablesupplier is A1.

From the above analysis, it can be easily seen that although the ranking orders of thesuppliers are slightly different, the most desirable supplier in supply chain management is A1.

The ranking results based on the different operators are shown in Table 2.

Table 2 Ranking results based on the different operators

Aggregation operator Ranking order

HTrFHWA1 A1 � A4 � A2 � A3

HTrFHWA2 A1 � A4 � A2 � A3

HTrFHWG1 A1 � A2 � A4 � A3

HTrFHWG2 A1 � A2 � A4 � A3

Through the above analysis, the advantages of our method based on the HTrFHWA op-erator and HTrFHWG operator can be summarized as follow. First, we studied the MADMproblems in which the attribute values take the form of HTrFSs, which can flexibly describethe uncertainty in the reality. Secondly, since there are attribute-related associations, com-pared the conventional operators, our proposed operators consider the information about therelationship among arguments being aggregated. Thirdly, the proposed method based on theHTrFHWA operator and HTrFHWG operator has different parameters, DM can choose differentparameters based on their attitude in order to choose the most optimal alternative reasonably.Furthermore, the proposed operators provide a new approach to aggregate HTrFSs, which ismore effective and powerful for solving MADM problems.


7 Conclusion

In this paper, we investigate the multiple attribute decision making (MADM) problemsin which the attribute values take the form of hesitant trapezoid fuzzy information. Firstly,the concept, the operational laws and the score function of hesitant trapezoid fuzzy elements(HTrFE) are proposed to measure the uncertain information difficult to express with crispnumbers. Then some aggregation operators based on the Hamacher operation for aggregat-ing hesitant trapezoid fuzzy information are defined, such as the hesitant trapezoid fuzzyHamacher weighted average (HTrFHWA) operator, the hesitant trapezoid fuzzy Hamacherordered weighted average (HTrFHOWA) operator, the hesitant trapezoid fuzzy Hamacherweighted geometric (HTrFHWG) operator, the hesitant trapezoid fuzzy Hamacher orderedweighted geometric (HTrFHOWG) operator, the hesitant trapezoid fuzzy Hamacher hybridaverage (HTrFHHA) operator and the hesitant trapezoid fuzzy Hamacher hybrid geometric(HTrFHHG) operator. Furthermore, we utilize the hesitant trapezoid fuzzy Hamacher weightedaverage (HTrFHWA) operator and the hesitant trapezoid fuzzy Hamacher weighted geomet-ric (HTrFHWG) operator to develop an approach to solve multiple attribute decision making(MADM) problems under hesitant trapezoid fuzzy environments. Finally, a practical exampleabout supplier selection is given to verify the practicality and validity of the proposed method.

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