Hesitant Trapezoid Fuzzy Hamacher Aggregation Operators and Their Application to Multiple Attribute Decision Making

Qian YU; Jun CAO; Ling TAN; Yubing ZHAI; Jiongyan LIU

doi:10.21078/JSSI-2020-524-25

PDF(265 KB)

Journal of Systems Science and Information ›› 2020, Vol. 8 ›› Issue (6) : 524-548. DOI: 10.21078/JSSI-2020-524-25

Hesitant Trapezoid Fuzzy Hamacher Aggregation Operators and Their Application to Multiple Attribute Decision Making

Author information +

History +

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) 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.

Key words

multiple attribute decision making (MADM) / hesitant trapezoid fuzzy set (HTrFS) / Hamacher operation / Choquet integral

Cite this article

EndNote

Ris (Procite)

Bibtex

Download Citations

Qian YU , Jun CAO , Ling TAN , Yubing ZHAI , Jiongyan LIU. Hesitant Trapezoid Fuzzy Hamacher Aggregation Operators and Their Application to Multiple Attribute Decision Making. Journal of Systems Science and Information, 2020, 8(6): 524-548 https://doi.org/10.21078/JSSI-2020-524-25

1 Introduction

Since the theory of fuzzy sets (FSs) was first proposed by Zadeh^[1], many extension forms have 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 intuitionistic fuzzy sets (IVIFSs)^[6]. Recently, the concept of hesitant fuzzy set (HFS) was introduced by Torra^[7] to enrich the fuzzy set. It permits the membership of an element to a given set having a few different values, which has been a suitable tool to describe the imprecise or uncertain decision information. As a new generalized type of fuzzy sets, HFS has received a considerable attention 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 operators and their application in decision making problems. Xu and Xia^[15] defined the distance and correlation 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 fuzzy information and applied them to MADM problems. Inspired by the idea of prioritized aggregation operators^{[19, 20]}, Wei^[21] defined some prioritized aggregation operators for aggregating hesitant fuzzy information, and then applied them to develop some models for hesitant fuzzy MADM 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 Choquet integral and proposed some Choquet hesitant fuzzy information aggregation operators: Hesitant fuzzy Choquet ordered averaging (HFCOA) operator, hesitant fuzzy Choquet ordered geometric (HFCOG) operator, the generalized hesitant fuzzy Choquet ordered averaging (GHFCOA) operator and generalized hesitant fuzzy Choquet ordered geometric (GHFCOG) operator. Motivated by the power aggregation operators^[23], Zhang^[24] proposed a family of hesitant fuzzy power aggregation operators and applied them to solve multiple attribute group decision making problems.

The above-mentioned approaches have already been proven effective and feasible for dealing with multiple attribute decision making problems. However, the current approaches for MADM problem may induce the information losing and cannot represent the real preference of decision maker 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 the concept of hesitant trapezoid fuzzy set. To do so, the remainder of this paper is set out as follows. In Section 2, a brief introduction to some basic notations of hesitant fuzzy set and Hamacher operations is reviewed. In Section 3, some basic concepts related to hesitant trapezoid 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 information are proposed. In Section 5, we propose an approach for multi-attribute decision making based on the hesitant trapezoid fuzzy Hamacher weighted average (HTrFHWA) operator and the hesitant trapezoid fuzzy Hamacher weighted geometric (HTrFHWG) operator under hesitant trapezoid fuzzy environment. In Section 6, an illustrative example is pointed out to verify the developed approach and to demonstrate its practicality and effectiveness. In Section 7, 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

X

is denoted by a function

h_{A} (x)

that returns a subset of

[0, 1]

when it is applied to

X

. In order to make it easier to understood, the HFS is expressed by a mathematical symbol

\begin{array}{rcl} A = {⟨ x, h_{A (x)} ⟩ | x \in X}, \end{array}

(1)

where

h_{A (x)}

is a set of some different values indicating all possible membership degrees of the element

x \in X

to the set A. For convenience,

h_{A (x)}

is called a hesitant fuzzy element (HFE), which is denoted by

h_{A (x)} = {γ | γ \in h_{A (x)}}

Let

h

h_{1}

and

h_{2}

be three HFEs, then some operations are presented as follows^[14]:

h^{c} = {1 - γ | γ \in h}

;

h^{λ} = {(γ)^{λ} | γ \in h}

;

λ h = {1 - (1 - γ)^{2} | γ \in h}

;

h_{1} \oplus h_{2} = {γ_{1} + γ_{2} - γ_{1} γ_{2} | γ_{1} \in h_{1}, γ_{2} \in h_{2}}

;

h_{1} \otimes h_{2} = {γ_{1} γ_{2} | γ_{1} \in h_{1}, γ_{2} \in h_{2}}

Definition 2 (see [14]) For an HFE

h = {γ | γ \in h}, s (h) = \frac{1}{l_{h}} \sum_{y \in h} γ

is called the score function of

h

, where

l_{h}

is the number of the values in

h

. For two HFEs

h_{1}

and

h_{2}

s (h_{1}) > s (h_{2}),

then

h_{1} > h_{2};

s (h_{1}) = s (h_{2}),

then

h_{1} = h_{2}

2.2 Hamacher Operations

T-norm and t-conorm are an important notion in fuzzy set theory, which are used to define a generalized union and intersection of fuzzy sets^[26]. Roychowdhury and Wang^[27] gave the definition 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 introduced by Deschrijver and Kerre^[28]. Further, Hamacher^[29] proposed a more generalized t-norm and t-conorm. Hamacher operations^[29] include the Hamacher product and Hamacher sum, which are examples of t-norms and t-conorms, respectively. They are defined as follows:

Hamacher product

\otimes

is a t-norm and Hamacher sum

\oplus

is a t-conorm, where

\begin{array}{rcl} T (a, b) = a \otimes b = \frac{a b}{γ + (1 - γ) (a + b - a b)}, \end{array}

(2)

\begin{array}{rcl} T^{*} (a, b) = a \oplus b = \frac{a + b - a b - (1 - γ) a b}{1 - (1 - γ) a b} . \end{array}

(3)

Especially, when

γ = 1

, then Hamacher t-norm and t-conorm will reduce to

\begin{array}{l} T (a, b) = a \otimes b = a b, \\ T^{*} (a, b) = a \oplus b = a + b - a b, \end{array}

which are the algebraic t-norm and t-conorm respectively; when

γ = 2

, then Hamacher t-norm and t-conorm will reduce to

\begin{array}{l} T (a, b) = a \otimes b = \frac{a b}{1 + (1 - a) (1 - b)}, \\ T^{*} (a, b) = a \oplus b = \frac{a + b}{1 + a b}, \end{array}

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 to trapezoid fuzzy numbers, and define the degree of possibility of two trapezoid fuzzy numbers.

Definition 3 (see [25]) A trapezoid fuzzy numbers

\tilde{n}

can be defined as

[n_{1}, n_{2}, n_{3}, n_{4}]

. The membership function

μ_{\tilde{n}} (x)

is defined as

μ_{n} (x) = {\begin{cases} 0, & x < n_{1}, \\ \frac{x - n_{1}}{n_{2} - n_{1}}, & n_{1} \leq x \leq n_{2}, \\ 1, & n_{2} \leq x \leq n_{3}, \\ \frac{x - n_{4}}{n_{3} - n_{4}}, & n_{3} \leq x \leq n_{4}, \\ 0, & x > n_{4} . \end{cases}

For a trapezoidal fuzzy number

\tilde{n} = [n_{1}, n_{2}, n_{3}, n_{4}]

, if

n_{2} = n_{3}

, then

\tilde{n}

is called a triangular fuzzy number.

Given any two trapezoidal fuzzy numbers,

\tilde{m} = [m_{1}, m_{2}, m_{3}, m_{4}]

\tilde{n} = [n_{1}, n_{2}, n_{3}, n_{4}]

, and

λ > 0

, some operations can be expressed as follows^[3]:

\begin{array}{l} \tilde{m} \oplus \tilde{n} = [m_{1} + n_{1}, m_{2} + n_{2}, m_{3} + n_{3}, m_{4} + n_{4}]; \\ \tilde{m} \otimes \tilde{n} = [m_{1} \times n_{1}, m_{2} \times n_{2}, m_{3} \times n_{3}, m_{4} \times n_{4}]; \\ λ \tilde{m} = λ [m_{1}, m_{2}, m_{3}, m_{4}] = [λ m_{1}, λ m_{2}, λ m_{3}, λ m_{4}]; \\ \tilde{m} = {[m_{1}, m_{2}, m_{3}, m_{4}]}^{λ} = [m_{1}^{λ}, m_{2}^{λ}, m_{3}^{λ}, m_{4}^{λ}] . \end{array}

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

Definition 4 Let

\tilde{m} = [m_{1}, m_{2}, m_{3}, m_{4}]

and

\tilde{n} = [n_{1}, n_{2}, n_{3}, n_{4}]

be two trapezoid fuzzy numbers. Then, the possibility degree of

\tilde{m} \geq \tilde{n}

is defined as follows:

\begin{array}{rcl} p (\tilde{m} \geq \tilde{n}) = min {max {\frac{(m_{3} + m_{4}) - (n_{1} + n_{2})}{(m_{3} + m_{4}) - (m_{1} + m_{2}) + (n_{3} + n_{4}) - (n_{1} + n_{2})}, 0}, 1} . \end{array}

(4)

Obviously, the possibility degree

p (\tilde{m} \geq \tilde{n})

satisfies the following properties:

0 \leq p (\tilde{m} \geq \tilde{n}) \leq 1, 0 \leq p (\tilde{n} \geq \tilde{m}) \leq 1

;

p (\tilde{m} \geq \tilde{n}) + p (\tilde{n} \geq \tilde{m}) = 1 .

Especially,

p (\tilde{m} \geq \tilde{n}) = p (\tilde{n} \geq \tilde{m}) = 0.5

3.2 Hesitant Trapezoid Fuzzy Set

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

Definition 5 Let

X

be a reference set, a HTrFS

A

X

is denoted by a function

{\tilde{h}}_{A} (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.

\begin{array}{rcl} A = {⟨ x, {\tilde{h}}_{A (x)} ⟩ | x \in X}, \end{array}

(5)

where

{\tilde{h}}_{A (x)}

is a set of some different trapezoid fuzzy numbers indicating all possible membership degrees of the element

x \in X

to the set

A

. For convenience,

{\tilde{h}}_{A (x)}

is called a hesitant trapezoid fuzzy element (HTrFE), which is denoted

by {\tilde{h}}_{A (x)} = \tilde{h} = {[a, b, c, d]}

Given three HTrFEs,

\tilde{h} = {[a, b, c, d]}, {\tilde{h}}_{1} = {[a_{1}, b_{1}, c_{1}, d_{1}]}

and

{\tilde{h}}_{2} = {[a_{2}, b_{2}, c_{2}, d_{2}]}

and

λ > 0,

the operations are defined as follows.

{\tilde{h}}_{1} \oplus {\tilde{h}}_{2} = \cup_{[a_{1}, b_{1}, c_{1}, d_{1}] \in {\tilde{h}}_{1}, [a_{2}, b_{2}, c_{2}, d_{2}] \in {\tilde{h}}_{2}} {[a_{1} + a_{2} - a_{1} a_{2}, b_{1} + b_{2} - b_{1} b_{2}, c_{1} + c_{2} - c_{1} c_{2},

d_{1} + d_{2} - d_{1} d_{2}]}

;

{\tilde{h}}_{1} \otimes {\tilde{h}}_{2} = \cup_{[a_{1}, b_{1}, c_{1}, d_{1}] \in {\tilde{h}}_{1}, [a_{2}, b_{2}, c_{2}, d_{2}] \in {\tilde{h}}_{2}} {[a_{1} a_{2}, b_{1} b_{2}, c_{1} c_{2}, d_{1} d_{2}]}

;

{\tilde{h}}^{λ} = \cup_{[a, b, c, d] \in \tilde{h}} {[a^{λ}, b^{λ}, c^{λ}, d^{λ}]}

;

λ \tilde{h} = \cup_{[a, b, c, d] \in \tilde{h}} {[λ a, λ b, λ c, λ d]}

Definition 6 For a HTrFE

\tilde{h}, s (\tilde{h}) = \frac{1}{l_{\tilde{h}}} \sum_{γ \in \tilde{h}} γ

is called the score function of

\tilde{h}

, where

l_{\tilde{h}}

is the number of the trapezoid fuzzy values in

\tilde{h}

, and

s (\tilde{h})

is a trapezoid fuzzy number in the range of

[0, 1]

. For two HTrFEs

{\tilde{h}}_{1}

and

{\tilde{h}}_{2}

, if

s ({\tilde{h}}_{1}) > s ({\tilde{h}}_{2})

, then

{\tilde{h}}_{1} > {\tilde{h}}_{2}

; if

s ({\tilde{h}}_{1}) = s ({\tilde{h}}_{2})

, then

{\tilde{h}}_{1} = {\tilde{h}}_{2}

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

4 Some Aggregating Operators with Hesitant Trapezoid Fuzzy Information

4.1 Hesitant Trapezoid Fuzzy Aggregation Operators

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

Definition 7 Let

{\tilde{h}}_{j} (j = 1, 2, \dots, n)

be a collection of HTrFEs on

X .

A HTrFWA operator is a mapping

Q^{n} \to Q,

and denoted by

\begin{array}{rcl} H T r F W A ({\tilde{h}}_{1}, {\tilde{h}}_{2}, \dots, {\tilde{h}}_{n}) \\ = & \oplus_{j = 1}^{n} w_{j} {\tilde{h}}_{j} \\ = & ⋃_{{\tilde{γ}}_{1} \in {\tilde{h}}_{1}, {\tilde{γ}}_{2} \in {\tilde{h}}_{2}, \dots, {\tilde{γ}}_{n} \in {\tilde{h}}_{n}} \\ {[1 - \prod_{j = 1}^{n} {(1 - a_{j})}^{w_{j}}, 1 - \prod_{j = 1}^{n} {(1 - b_{j})}^{w_{j}}, 1 - \prod_{j = 1}^{n} {(1 - c_{j})}^{w_{j}}, 1 - \prod_{j = 1}^{n} {(1 - d_{j})}^{w_{j}}]}, \end{array}

(6)

where

w = (w_{1}, w_{2}, \dots, w_{n})

is the weight vector of

{\tilde{h}}_{j} (j = 1, 2, \dots, n), w_{j} \geq 0

and

\sum_{j = 1}^{n} w_{j} = 1

Definition 8 Let

{\tilde{h}}_{j} (j = 1, 2, \dots, n)

be a collection of HTrFEs on

X .

A HTrFWG operator is a mapping

Q^{n} \to Q,

and denoted by

\begin{array}{rcl} H T r F W G ({\tilde{h}}_{1}, {\tilde{h}}_{2}, \dots, {\tilde{h}}_{n}) \\ = & \otimes_{j = 1}^{n} {\tilde{h}}_{j}^{w_{j}} \\ = & ⋃_{{\tilde{γ}}_{1} \in {\tilde{h}}_{1}, {\tilde{γ}}_{2} \in {\tilde{h}}_{2}, \dots, {\tilde{γ}}_{n} \in {\tilde{h}}_{n}} {[\prod_{j = 1}^{n} a_{j}^{w_{j}}, \prod_{j = 1}^{n} b_{j}^{w_{j}}, \prod_{j = 1}^{n} c_{j}^{w_{j}}, \prod_{j = 1}^{n} d_{j}^{w_{j}}]}, \end{array}

(7)

where

w = (w_{1}, w_{2}, \dots, w_{n})

is the weight vector of

{\tilde{h}}_{j} (j = 1, 2, \dots, n), w_{j} \geq 0

and

\sum_{j = 1}^{n} w_{j} = 1

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

Definition 9 Let

{\tilde{h}}_{j} (j = 1, 2, \dots, n)

be a collection of HTrFEs, then we define the HTrFOWA operator as follows

\begin{array}{rcl} H T r F O W A ({\tilde{h}}_{1}, {\tilde{h}}_{2}, \dots, {\tilde{h}}_{n}) \\ = & \oplus_{j = 1}^{n} ω_{j} {\tilde{h}}_{σ (j)} \\ = & ⋃_{\binom{{\tilde{γ}}_{σ (j)} \in {\tilde{h}}_{σ (j)},}{j = 1, 2, \dots, n}} {[1 - \prod_{j = 1}^{n} {(1 - a_{σ (j)})}^{ω_{j}}, 1 - \prod_{j = 1}^{n} {(1 - b_{σ (j)})}^{ω_{j}}, \\ 1 - \prod_{j = 1}^{n} {(1 - c_{σ (j)})}^{ω_{j}}, 1 - \prod_{j = 1}^{n} {(1 - d_{σ (j)})}^{ω_{j}}]}, \end{array}

(8)

where

(σ (1), σ (2), \dots, σ (n))

is a permutation of

(1, 2, \dots, n),

such that

{\tilde{h}}_{σ (j - 1)} \geq {\tilde{h}}_{σ (j)}

for all

j = 2, 3, \dots, n,

and

ω = {(ω_{1}, ω_{2}, \dots, ω_{n})}^{T}

is the aggregation-associated weight vector such that

ω_{j} \geq 0, \sum_{j = 1}^{n} ω_{j} = 1

Definition 10 Let

{\tilde{h}}_{j} (j = 1, 2, \dots, n)

be a collection of HTrFEs, then we define the HTrFOWG operator as follows

\begin{array}{rcl} H T r F O W G ({\tilde{h}}_{1}, {\tilde{h}}_{2}, \dots, {\tilde{h}}_{n}) \\ = & \otimes_{j = 1}^{n} {\tilde{h}}_{σ (j)}^{ω_{j}} \\ = & \cup_{\binom{{\tilde{γ}}_{σ (j)} \in {\tilde{h}}_{σ (j)},}{j = 1, 2, \dots, n}} {[\prod_{j = 1}^{n} a_{σ (j)}^{w_{j}}, \prod_{j = 1}^{n} b_{σ (j)}^{w_{j}}, \prod_{j = 1}^{n} c_{σ (j)}^{w_{j}}, \prod_{j = 1}^{n} d_{σ (j)}^{w_{j}}]}, \end{array}

(9)

where

(σ (1), σ (2), \dots, σ (n))

is a permutation of

(1, 2, \dots, n),

such that

{\tilde{h}}_{σ (j - 1)} \geq {\tilde{h}}_{σ (j)}

for all

j = 2, 3, \dots, n,

and

ω = {(ω_{1}, ω_{2}, \dots, ω_{n})}^{T}

is the aggregation-associated weight vector such that

ω_{j} \geq 0, \sum_{j = 1}^{n} ω_{j} = 1

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

Definition 11 Let

{\tilde{h}}_{j} (j = 1, 2, \dots, n)

be a collection of HTrFEs, then the HTrFHA operator is defined as follows

\begin{array}{rcl} H T r F H A ({\tilde{h}}_{1}, {\tilde{h}}_{2}, \dots, {\tilde{h}}_{n}) \\ = & \oplus_{j = 1}^{n} w_{j} {\tilde{h}}_{σ (j)}^{.} \\ = & ⋃_{_{\binom{{\tilde{γ}}_{σ (j)}^{.} \in {\tilde{h}}_{σ (j)}^{.},}{j = 1, 2, \dots, n}}} {[1 - \prod_{j = 1}^{n} {(1 - {\dot{a}}_{σ (j)})}^{w_{j}}, 1 - \prod_{j = 1}^{n} {(1 - {\dot{b}}_{σ (j)})}^{w_{j}}, \\ 1 - \prod_{j = 1}^{n} {(1 - {\dot{c}}_{σ (j)})}^{w_{j}}, 1 - \prod_{j = 1}^{n} {(1 - {\dot{d}}_{σ (j)})}^{w_{j}}]}, \end{array}

(10)

where

w = {(w_{1}, w_{2}, \dots, w_{n})}^{T}

is the associated weighting vector, with

w_{j} \geq 0, \sum_{j = 1}^{n} w_{j} = 1

and

{\tilde{h}}_{σ (j)}

is the

j

th largest element of the hesitant trapezoid fuzzy arguments

{\tilde{h}}_{σ (j)} = (n ω_{j}) {\tilde{h}}_{j} (j = 1, 2, \dots, n), ω = {(ω_{1}, ω_{2}, \dots, ω_{n})}^{T}

is the associated weighting vector, with

ω_{j} \geq 0, \sum_{j = 1}^{n} ω_{j} = 1,

and

n

is the balancing coefficient.

Definition 12 Let

{\tilde{h}}_{j} (j = 1, 2, \dots, n)

be a collection of HTrFEs, then the HTrFHG operator is defined as follows

\begin{array}{rcl} H T r F H G ({\tilde{h}}_{1}, {\tilde{h}}_{2}, \dots, {\tilde{h}}_{n}) & = & \otimes_{j = 1}^{n} {\dot{\tilde{h}}}_{σ (j)}^{w_{j}} \\ = & ⋃_{_{\binom{{\tilde{γ}}_{σ (j)} \in {\tilde{h}}_{σ (j)},}{j = 1, 2, \dots, n}}} {[\prod_{j = 1}^{n} {\dot{a}}_{σ (j)}^{w_{j}}, \prod_{j = 1}^{n} {\dot{b}}_{σ (j)}^{w_{j}}, \prod_{j = 1}^{n} {\dot{c}}_{σ (j)}^{w_{j}}, \prod_{j = 1}^{n} {\dot{d}}_{σ (j)}^{w_{j}}]}, \end{array}

(11)

where

w = {(w_{1}, w_{2}, \dots, w_{n})}^{T}

is the associated weighting vector, with

w_{j} \geq 0, \sum_{j = 1}^{n} w_{j} = 1

and

{\tilde{h}}_{σ (j)}

is the

j

th largest element of the hesitant trapezoid fuzzy arguments

{\tilde{h}}_{σ (j)} = (n ω_{j}) {\tilde{h}}_{j} (j = 1, 2, \dots, n), ω = {(ω_{1}, ω_{2}, \dots, ω_{n})}^{T}

is the associated weighting vector, with

ω_{j} \geq 0, \sum_{j = 1}^{n} ω_{j} = 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

\otimes

and t-conorm

T^{*}

be the Hamacher

sum \oplus,

then the generalized intersection and union on two HTrFEs

{\tilde{h}}_{1}

and

{\tilde{h}}_{2}

become the Hamacher product (denoted by

{\tilde{h}}_{1} \otimes {\tilde{h}}_{2}

) and Hamacher sum (denoted by

{\tilde{h}}_{1} \oplus {\tilde{h}}_{2}

) of two HTrFEs

{\tilde{h}}_{1}

and

{\tilde{h}}_{2},

respectively, as follows:

\begin{array}{rcl} {\tilde{h}}_{1} \oplus {\tilde{h}}_{2} & = & ⋃_{{\tilde{γ}}_{1} \in {\tilde{h}}_{1}, {\tilde{γ}}_{2} \in {\tilde{h}}_{2}} {[\frac{a_{1} + a_{2} - a_{1} a_{2} - (1 - γ) a_{1} a_{2}}{1 - (1 - γ) a_{1} a_{2}}, \frac{b_{1} + b_{2} - b_{1} b_{2} - (1 - γ) b_{1} b_{2}}{1 - (1 - γ) b_{1} b_{2}}, \\ \frac{c_{1} + c_{2} - c_{1} c_{2} - (1 - γ) c_{1} c_{2}}{1 - (1 - γ) c_{1} c_{2}}, \frac{d_{1} + d_{2} - d_{1} d_{2} - (1 - γ) d_{1} d_{2}}{1 - (1 - γ) d_{1} d_{2}}]}; \\ {\tilde{h}}_{1} \otimes {\tilde{h}}_{2} & = & ⋃_{{\tilde{γ}}_{1} \in {\tilde{h}}_{1}, {\tilde{γ}}_{2} \in {\tilde{h}}_{2}} {[\frac{a_{1} a_{2}}{γ + (1 - γ) (a_{1} + a_{2} - a_{1} a_{2})}, \frac{a_{1} a_{2}}{γ + (1 - γ) (a_{1} + a_{2} - a_{1} a_{2})}, \\ \frac{c_{1} c_{2}}{γ + (1 - γ) (c_{1} + c_{2} - c_{1} c_{2})}, \frac{d_{1} d_{2}}{γ + (1 - γ) (d_{1} + d_{2} - d_{1} d_{2})}]}; \\ λ {\tilde{h}}_{1} & = & ⋃_{{\tilde{γ}}_{1} \in {\tilde{h}}_{1}} {[\frac{{(1 + (γ - 1) a_{1})}^{λ} - {(1 - a_{1})}^{λ}}{{(1 + (γ - 1) a_{1})}^{λ} + (γ - 1) {(1 - a_{1})}^{λ}}, \frac{{(1 + (γ - 1) b_{1})}^{λ} - {(1 - b_{1})}^{λ}}{{(1 + (γ - 1) b_{1})}^{λ} + (γ - 1) {(1 - b_{1})}^{λ}}, \\ \frac{{(1 + (γ - 1) c_{1})}^{λ} - {(1 - c_{1})}^{λ}}{{(1 + (γ - 1) c_{1})}^{λ} + (γ - 1) {(1 - c_{1})}^{λ}}, \frac{{(1 + (γ - 1) d_{1})}^{λ} - {(1 - d_{1})}^{λ}}{{(1 + (γ - 1) d_{1})}^{λ} + (γ - 1) {(1 - d_{1})}^{λ}}]} λ > 0; \\ {\tilde{h}}_{1}^{λ} & = & ⋃_{{\tilde{γ}}_{1} \in {\tilde{h}}_{1}} {[\frac{γ {(a_{1})}^{λ}}{{(1 + (γ - 1) (1 - a_{1}))}^{λ} + (γ - 1) {(a_{1})}^{λ}}, \frac{γ {(b_{1})}^{λ}}{{(1 + (γ - 1) (1 - b_{1}))}^{λ} + (γ - 1) {(b_{1})}^{λ}}, \\ \frac{γ {(c_{1})}^{λ}}{{(1 + (γ - 1) (1 - c_{1}))}^{λ} + (γ - 1) {(c_{1})}^{λ}}, \frac{γ {(d_{1})}^{λ}}{{(1 + (γ - 1) (1 - d_{1}))}^{λ} + (γ - 1) {(d_{1})}^{λ}}]} λ > 0. \end{array}

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

Definition 13 Let

{\tilde{h}}_{j} (j = 1, 2, \dots, n)

be a collection of HTrFEs, then we define the HTrFHWA operator as follows:

\begin{array}{rcl} H T r F H W A ({\tilde{h}}_{1}, {\tilde{h}}_{2}, \dots, {\tilde{h}}_{n}) = \oplus_{j = 1}^{n} w_{j} {\tilde{h}}_{j}, \end{array}

(12)

where

w = {(w_{1}, w_{2}, \dots, w_{n})}^{T}

be the weight vector of HTrFEs, and

w_{j} \geq 0, \sum_{j = 1}^{n} w_{j} = 1

Based on the Hamacher operations of hesitant trapezoid fuzzy elements described in Section 3, we can drive the following theorem.

Theorem 1 Let ${\tilde{h}}_{j} (j = 1, 2, \dots, n)$ be a collection of HTrFEs, then their aggregated value by using the HTrFHWA operator is also a HTrFE, and

\begin{array}{rcl} H T r F H W A ({\tilde{h}}_{1}, {\tilde{h}}_{2}, \dots, {\tilde{h}}_{n}) \\ = & \oplus_{j = 1}^{n} w_{j} {\tilde{h}}_{j} \\ = & ⋃_{\binom{{\tilde{γ}}_{j} \in {\tilde{h}}_{j},}{j = 1, 2, \dots, n}} {[\frac{\prod_{j = 1}^{n} {(1 + (γ - 1) a_{j})}^{w_{j}} - \prod_{j = 1}^{n} {(1 - a_{j})}^{w_{j}}}{\prod_{j = 1}^{n} {(1 + (γ - 1) a_{j})}^{w_{j}} + (γ - 1) \prod_{j = 1}^{n} {(1 - a_{j})}^{w_{j}}}, \\ \frac{\prod_{j = 1}^{n} {(1 + (γ - 1) b_{j})}^{w_{j}} - \prod_{j = 1}^{n} {(1 - b_{j})}^{w_{j}}}{\prod_{j = 1}^{n} {(1 + (γ - 1) b_{j})}^{w_{j}} + (γ - 1) \prod_{j = 1}^{n} {(1 - b_{j})}^{w_{j}}}, \\ \frac{\prod_{j = 1}^{n} {(1 + (γ - 1) c_{j})}^{w_{j}} - \prod_{j = 1}^{n} {(1 - c_{j})}^{w_{j}}}{\prod_{j = 1}^{n} {(1 + (γ - 1) c_{j})}^{w_{j}} + (γ - 1) \prod_{j = 1}^{n} {(1 - c_{j})}^{w_{j}}}, \\ \frac{\prod_{j = 1}^{n} {(1 + (γ - 1) d_{j})}^{w_{j}} - \prod_{j = 1}^{n} {(1 - d_{j})}^{w_{j}}}{\prod_{j = 1}^{n} {(1 + (γ - 1) d_{j})}^{w_{j}} + (γ - 1) \prod_{j = 1}^{n} {(1 - d_{j})}^{w_{j}}}]} . \end{array}

(13)

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

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:

\begin{array}{rcl} H T r F E W A ({\tilde{h}}_{1}, {\tilde{h}}_{2}, \dots, {\tilde{h}}_{n}) \\ = & \oplus_{j = 1}^{n} w_{j} {\tilde{h}}_{j} \\ ⋃_{\binom{{\tilde{γ}}_{j} \in {\tilde{h}}_{j},}{j = 1, 2, \dots, n}} {[\frac{\prod_{j = 1}^{n} {(1 + a_{j})}^{w_{j}} - \prod_{j = 1}^{n} {(1 - a_{j})}^{w_{j}}}{\prod_{j = 1}^{n} {(1 + a_{j})}^{w_{j}} + \prod_{j = 1}^{n} {(1 - a_{j})}^{w_{j}}}, \frac{\prod_{j = 1}^{n} {(1 + b_{j})}^{w_{j}} - \prod_{j = 1}^{n} {(1 - b_{j})}^{w_{j}}}{\prod_{j = 1}^{n} {(1 + b_{j})}^{w_{j}} + \prod_{j = 1}^{n} {(1 - b_{j})}^{w_{j}}}, \\ \frac{\prod_{j = 1}^{n} {(1 + c_{j})}^{w_{j}} - \prod_{j = 1}^{n} {(1 - c_{j})}^{w_{j}}}{\prod_{j = 1}^{n} {(1 + c_{j})}^{w_{j}} + \prod_{j = 1}^{n} {(1 - c_{j})}^{w_{j}}}, \frac{\prod_{j = 1}^{n} {(1 + d_{j})}^{w_{j}} - \prod_{j = 1}^{n} {(1 - d_{j})}^{w_{j}}}{\prod_{j = 1}^{n} {(1 + d_{j})}^{w_{j}} + \prod_{j = 1}^{n} {(1 - d_{j})}^{w_{j}}}]} . \end{array}

Definition 14 Let

{\tilde{h}}_{j} (j = 1, 2, \dots, n)

be a collection of HTrFEs, then we define the HTrFHWG operator as follows

\begin{array}{rcl} H T r F H W G ({\tilde{h}}_{1}, {\tilde{h}}_{2}, \dots, {\tilde{h}}_{n}) = \otimes_{j = 1}^{n} {\tilde{h}}_{j}^{w_{j}}, \end{array}

(14)

where

w = {(w_{1}, w_{2}, \dots, w_{n})}^{T}

be the weight vector of HTrFEs, and

w_{j} \geq 0, \sum_{j = 1}^{n} w_{j} = 1

Based on the Hamacher operations of hesitant trapezoid fuzzy elements described in Section 3, we can drive the following theorem.

Theorem 2 Let ${\tilde{h}}_{j} (j = 1, 2, \dots, n)$ be a collection of HTrFEs, then their aggregated value by using the HTrFHWG operator is also a HTrFE, and

\begin{array}{rcl} H T r F H W G ({\tilde{h}}_{1}, {\tilde{h}}_{2}, \dots, {\tilde{h}}_{n}) \\ = & \otimes_{j = 1}^{n} {\tilde{h}}_{j}^{w_{j}} \\ ⋃_{\binom{{\tilde{γ}}_{j} \in {\tilde{h}}_{j},}{j = 1, 2, \dots, n}} {[\frac{γ \prod_{j = 1}^{n} a_{j}^{w_{j}}}{\prod_{j = 1}^{n} {(1 + (γ - 1) (1 - a_{j}))}^{^{w_{j}}} + (γ - 1) \prod_{j = 1}^{n} a_{j}^{w_{j}}}, \\ \frac{γ \prod_{j = 1}^{n} b_{j}^{w_{j}}}{\prod_{j = 1}^{n} {(1 + (γ - 1) (1 - b_{j}))}^{^{w_{j}}} + (γ - 1) \prod_{j = 1}^{n} b_{j}^{w_{j}}}, \\ \frac{γ \prod_{j = 1}^{n} c_{j}^{w_{j}}}{\prod_{j = 1}^{n} {(1 + (γ - 1) (1 - c_{j}))}^{^{w_{j}}} + (γ - 1) \prod_{j = 1}^{n} c_{j}^{w_{j}}}, \\ \frac{γ \prod_{j = 1}^{n} d_{j}^{w_{j}}}{\prod_{j = 1}^{n} {(1 + (γ - 1) (1 - d_{j}))}^{^{w_{j}}} + (γ - 1) \prod_{j = 1}^{n} d_{j}^{w_{j}}}]} . \end{array}

(15)

In the following, we can discuss some special cases of the HTrFHWG operator with respect to 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:

\begin{array}{rcl} H T r F E W G ({\tilde{h}}_{1}, {\tilde{h}}_{2}, \dots, {\tilde{h}}_{n}) \\ = & \otimes_{j = 1}^{n} ({\tilde{h}}_{j}^{w_{j}}) \\ ⋃_{\binom{{\tilde{γ}}_{j} \in {\tilde{h}}_{j},}{j = 1, 2, \dots, n}} {[\frac{2 \prod_{j = 1}^{n} a_{j}^{w_{j}}}{\prod_{j = 1}^{n} {(2 - a_{j})}^{w_{j}} + \prod_{j = 1}^{n} a_{j}^{w_{j}}}, \frac{2 \prod_{j = 1}^{n} b_{j}^{w_{j}}}{\prod_{j = 1}^{n} {(2 - b_{j})}^{w_{j}} + \prod_{j = 1}^{n} b_{j}^{w_{j}}}, \\ \frac{2 \prod_{j = 1}^{n} c_{j}^{w_{j}}}{\prod_{j = 1}^{n} {(2 - c_{j})}^{w_{j}} + \prod_{j = 1}^{n} c_{j}^{w_{j}}}, \frac{2 \prod_{j = 1}^{n} d_{j}^{w_{j}}}{\prod_{j = 1}^{n} {(2 - d_{j})}^{w_{j}} + \prod_{j = 1}^{n} d_{j}^{w_{j}}}]} . \end{array}

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

Definition 15 Let

{\tilde{h}}_{j} (j = 1, 2, \dots, n)

be a collection of HTrFEs, then we define the HTrFHOWA operator as follows:

\begin{array}{rcl} H T r F H O W A ({\tilde{h}}_{1}, {\tilde{h}}_{2}, \dots, {\tilde{h}}_{n}) \\ = & \oplus_{j = 1}^{n} ω_{j} {\tilde{h}}_{σ (j)} \\ = & ⋃_{\binom{{\tilde{γ}}_{σ (j)} \in {\tilde{h}}_{σ (j)},}{j = 1, 2, \dots, n}} {[\frac{\prod_{j = 1}^{n} {(1 + (γ - 1) a_{σ (j)})}^{ω_{j}} - \prod_{j = 1}^{n} {(1 - a_{σ (j)})}^{ω_{j}}}{\prod_{j = 1}^{n} {(1 + (γ - 1) a_{σ (j)})}^{ω_{j}} + (γ - 1) \prod_{j = 1}^{n} {(1 - a_{σ (j)})}^{ω_{j}}}, \\ \frac{\prod_{j = 1}^{n} {(1 + (γ - 1) b_{σ (j)})}^{ω_{j}} - \prod_{j = 1}^{n} {(1 - b_{σ (j)})}^{ω_{j}}}{\prod_{j = 1}^{n} {(1 + (γ - 1) b_{σ (j)})}^{ω_{j}} + (γ - 1) \prod_{j = 1}^{n} {(1 - b_{σ (j)})}^{ω_{j}}}, \\ \frac{\prod_{j = 1}^{n} {(1 + (γ - 1) c_{σ (j)})}^{ω_{j}} - \prod_{j = 1}^{n} {(1 - c_{σ (j)})}^{ω_{j}}}{\prod_{j = 1}^{n} {(1 + (γ - 1) c_{σ (j)})}^{ω_{j}} + (γ - 1) \prod_{j = 1}^{n} {(1 - c_{σ (j)})}^{ω_{j}}}, \\ \frac{\prod_{j = 1}^{n} {(1 + (γ - 1) d_{σ (j)})}^{ω_{j}} - \prod_{j = 1}^{n} {(1 - d_{σ (j)})}^{ω_{j}}}{\prod_{j = 1}^{n} {(1 + (γ - 1) d_{σ (j)})}^{ω_{j}} + (γ - 1) \prod_{j = 1}^{n} {(1 - d_{σ (j)})}^{ω_{j}}}]}, \end{array}

(16)

where

(σ (1), σ (2), \dots, σ (n))

is a permutation of

(1, 2, \dots, n),

such that

{\tilde{h}}_{σ (j - 1)} \geq {\tilde{h}}_{σ (j)}

for all

j = 2, \dots, n,

and

ω = {(ω_{1}, ω_{2}, \dots, ω_{n})}^{T}

is the aggregation-associated weight vector such that

ω_{j} \geq 0, \sum_{j = 1}^{n} ω_{j} = 1

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

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:

\begin{array}{rcl} H T r F E O W A ({\tilde{h}}_{1}, {\tilde{h}}_{2}, \dots, {\tilde{h}}_{n}) \\ = & \oplus_{j = 1}^{n} w_{j} {\tilde{h}}_{σ (j)} \\ ⋃_{\binom{{\tilde{γ}}_{σ (j)} \in {\tilde{h}}_{σ (j)},}{j = 1, 2, \dots, n}} {[\frac{\prod_{j = 1}^{n} {(1 + a_{σ (j)})}^{w_{j}} - \prod_{j = 1}^{n} {(1 - a_{σ (j)})}^{w_{j}}}{\prod_{j = 1}^{n} {(1 + a_{σ (j)})}^{w_{j}} + \prod_{j = 1}^{n} {(1 - a_{σ (j)})}^{w_{j}}}, \frac{\prod_{j = 1}^{n} {(1 + b_{σ (j)})}^{w_{j}} - \prod_{j = 1}^{n} {(1 - b_{σ (j)})}^{w_{j}}}{\prod_{j = 1}^{n} {(1 + b_{σ (j)})}^{w_{j}} + \prod_{j = 1}^{n} {(1 - b_{σ (j)})}^{w_{j}}}, \\ \frac{\prod_{j = 1}^{n} {(1 + c_{σ (j)})}^{w_{j}} - \prod_{j = 1}^{n} {(1 - c_{σ (j)})}^{w_{j}}}{\prod_{j = 1}^{n} {(1 + c_{σ (j)})}^{w_{j}} + \prod_{j = 1}^{n} {(1 - c_{σ (j)})}^{w_{j}}}, \frac{\prod_{j = 1}^{n} {(1 + d_{σ (j)})}^{w_{j}} - \prod_{j = 1}^{n} {(1 - d_{σ (j)})}^{w_{j}}}{\prod_{j = 1}^{n} {(1 + d_{σ (j)})}^{w_{j}} + \prod_{j = 1}^{n} {(1 - d_{σ (j)})}^{w_{j}}}]} . \end{array}

Definition 16 Let

{\tilde{h}}_{j} (j = 1, 2, \dots, n)

be a collection of HTrFEs, then we define the HTrFHOWG operator as follows:

\begin{array}{rcl} H T r F H O W G ({\tilde{h}}_{1}, {\tilde{h}}_{2}, \dots, {\tilde{h}}_{n}) \\ = & \otimes_{j = 1}^{n} {\tilde{h}}_{σ (j)}^{ω_{j}} \\ ⋃_{\binom{{\tilde{γ}}_{σ (j)} \in {\tilde{h}}_{σ (j)},}{j = 1, 2, \dots, n}} {[\frac{γ \prod_{j = 1}^{n} a_{σ (j)}^{ω_{j}}}{\prod_{j = 1}^{n} {(1 + (γ - 1) (1 - a_{σ (j)}))}^{ω_{j}} + (γ - 1) \prod_{j = 1}^{n} a_{σ (j)}^{ω_{j}}}, \\ \frac{γ \prod_{j = 1}^{n} b_{σ (j)}^{ω_{j}}}{\prod_{j = 1}^{n} {(1 + (γ - 1) (1 - b_{σ (j)}))}^{ω_{j}} + (γ - 1) \prod_{j = 1}^{n} b_{σ (j)}^{ω_{j}}}, \\ \frac{γ \prod_{j = 1}^{n} c_{σ (j)}^{ω_{j}}}{\prod_{j = 1}^{n} {(1 + (γ - 1) (1 - c_{σ (j)}))}^{ω_{j}} + (γ - 1) \prod_{j = 1}^{n} c_{σ (j)}^{ω_{j}}}, \\ \frac{γ \prod_{j = 1}^{n} d_{σ (j)}^{ω_{j}}}{\prod_{j = 1}^{n} {(1 + (γ - 1) (1 - d_{σ (j)}))}^{ω_{j}} + (γ - 1) \prod_{j = 1}^{n} d_{σ (j)}^{ω_{j}}}]}, \end{array}

(17)

where

(σ (1), σ (2), \dots, σ (n))

is a permutation of

(1, 2, \dots, n),

such that

{\tilde{h}}_{σ (j - 1)} \geq {\tilde{h}}_{σ (j)}

for all

j = 2, 3, \dots, n,

and

ω = {(ω_{1}, ω_{2}, \dots, ω_{n})}^{T}

is the aggregation-associated weight vector such that

ω_{j} \geq 0, \sum_{j = 1}^{n} ω_{j} = 1

Now, we can discuss some special cases of the HTrFHOWG operator with respect to the parameter

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:

\begin{array}{rcl} H T r F E O W G ({\tilde{h}}_{1}, {\tilde{h}}_{2}, \dots, {\tilde{h}}_{n}) \\ = & \otimes_{j = 1}^{n} ({\tilde{h}}_{σ (j)}^{w_{j}}) \\ ⋃_{\binom{{\tilde{γ}}_{σ (j)} \in {\tilde{h}}_{σ (j)},}{j = 1, 2, \dots, n}} {[\frac{2 \prod_{j = 1}^{n} a_{σ (j)}^{w_{j}}}{\prod_{j = 1}^{n} {(2 - a_{σ (j)})}^{w_{j}} + \prod_{j = 1}^{n} a_{σ (j)}^{w_{j}}}, \frac{2 \prod_{j = 1}^{n} b_{σ (j)}^{w_{j}}}{\prod_{j = 1}^{n} {(2 - b_{σ (j)})}^{w_{j}} + \prod_{j = 1}^{n} b_{σ (j)}^{w_{j}}}, \\ \frac{2 \prod_{j = 1}^{n} c_{σ (j)}^{w_{j}}}{\prod_{j = 1}^{n} {(2 - c_{σ (j)})}^{w_{j}} + \prod_{j = 1}^{n} c_{σ (j)}^{w_{j}}}, \frac{2 \prod_{j = 1}^{n} d_{σ (j)}^{w_{j}}}{\prod_{j = 1}^{n} {(2 - d_{σ (j)})}^{w_{j}} + \prod_{j = 1}^{n} d_{σ (j)}^{w_{j}}}]} . \end{array}

Similarly, from the above mentioned definitions, the HTrFHWA and HTrFHWG operators weight the hesitant trapezoid fuzzy argument itself, while the HTrFHOWA and HTrFHOWG operators weight the ordered positions of hesitant trapezoid fuzzy arguments. In order to consider both the two aspects, in the following we shall propose the hesitant trapezoid fuzzy Hamacher hybrid average (HTrFHHA) operator and the hesitant trapezoid fuzzy Hamacher hybrid geometric (HTrFHHG) operator.

Definition 17 Let

{\tilde{h}}_{j} (j = 1, 2, \dots, n)

be a collection of HTrFEs, then the HTrFHHA operator is defined as follows:

\begin{array}{rcl} \begin{array}{lll} H T r F H H A ({\tilde{h}}_{1}, {\tilde{h}}_{2}, \dots, {\tilde{h}}_{n}) \\ = & \oplus_{j = 1}^{n} w_{j} {\tilde{h}}_{σ (j)}^{.} = ⋃_{\binom{{\tilde{γ}}_{σ (j)}^{.} \in {\tilde{h}}_{σ (j)}^{.},}{j = 1, 2, \dots, n}} {[\frac{\prod_{j = 1}^{n} {(1 + (γ - 1) {\dot{a}}_{σ (j)})}^{w_{j}} - \prod_{j = 1}^{n} {(1 - {\dot{a}}_{σ (j)})}^{w_{j}}}{\prod_{j = 1}^{n} {(1 + (γ - 1) {\dot{a}}_{σ (j)})}^{w_{j}} + (γ - 1) \prod_{j = 1}^{n} {(1 - \dot{a_{σ (j)}})}^{w_{j}}}, \\ \frac{\prod_{j = 1}^{n} {(1 + (γ - 1) {\dot{b}}_{σ (j)})}^{w_{j}} - \prod_{j = 1}^{n} {(1 - {\dot{b}}_{σ (j)})}^{w_{j}}}{\prod_{j = 1}^{n} {(1 + (γ - 1) {\dot{b}}_{σ (j)})}^{w_{j}} + (γ - 1) \prod_{j = 1}^{n} {(1 - {\dot{b}}_{σ (j)})}^{w_{j}}}, \\ \frac{\prod_{j = 1}^{n} {(1 + (γ - 1) {\dot{c}}_{σ (j)})}^{w_{j}} - \prod_{j = 1}^{n} {(1 - {\dot{c}}_{σ (j)})}^{w_{j}}}{\prod_{j = 1}^{n} {(1 + (γ - 1) {\dot{c}}_{σ (j)})}^{w_{j}} + (γ - 1) \prod_{j = 1}^{n} {(1 - {\dot{c}}_{σ (j)})}^{w_{j}}}, \\ \frac{\prod_{j = 1}^{n} {(1 + (γ - 1) {\dot{d}}_{σ (j)})}^{w_{j}} - \prod_{j = 1}^{n} {(1 - {\dot{d}}_{σ (j)})}^{w_{j}}}{\prod_{j = 1}^{n} {(1 + (γ - 1) {\dot{d}}_{σ (j)})}^{w_{j}} + (γ - 1) \prod_{j = 1}^{n} {(1 - {\dot{d}}_{σ (j)})}^{w_{j}}}]}, \end{array} \end{array}

(18)

where

w = {(w_{1}, w_{2}, \dots, w_{n})}^{T}

is the associated weighting vector, with

w_{j} \geq 0

\sum_{j = 1}^{n} w_{j} = 1

and

{\tilde{h}}_{σ (j)}

is the

j

th largest element of the hesitant trapezoid fuzzy arguments

{\tilde{h}}_{σ (j)} = (n ω_{j}) {\tilde{h}}_{j} (j = 1, 2, \dots, n)

ω = {(ω_{1}, ω_{2}, \dots, ω_{n})}^{T}

is the associated weighting vector, with

ω_{j} \geq 0

\sum_{j = 1}^{n} ω_{j} = 1,

and

n

is the balancing coefficient.

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

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:

\begin{array}{rcl} H T r F E H A ({\tilde{h}}_{1}, {\tilde{h}}_{2}, \dots, {\tilde{h}}_{n}) \\ = & \oplus_{j = 1}^{n} w_{j} {\tilde{h}}_{σ (j)}^{.} \\ ⋃_{\binom{{\tilde{γ}}_{σ (j)}^{.} \in {\tilde{h}}_{σ (j)}^{.},}{j = 1, 2, \dots, n}} {[\frac{\prod_{j = 1}^{n} {(1 + {\dot{a}}_{σ (j)})}^{w_{j}} - \prod_{j = 1}^{n} {(1 - {\dot{a}}_{σ (j)})}^{w_{j}}}{\prod_{j = 1}^{n} {(1 + {\dot{a}}_{σ (j)})}^{w_{j}} + \prod_{j = 1}^{n} {(1 - {\dot{a}}_{σ (j)})}^{w_{j}}}, \frac{\prod_{j = 1}^{n} {(1 + {\dot{b}}_{σ (j)})}^{w_{j}} - \prod_{j = 1}^{n} {(1 - {\dot{b}}_{σ (j)})}^{w_{j}}}{\prod_{j = 1}^{n} {(1 + {\dot{b}}_{σ (j)})}^{w_{j}} + \prod_{j = 1}^{n} {(1 - {\dot{b}}_{σ (j)})}^{w_{j}}}, \\ \frac{\prod_{j = 1}^{n} {(1 + {\dot{c}}_{σ (j)})}^{w_{j}} - \prod_{j = 1}^{n} {(1 - {\dot{c}}_{σ (j)})}^{w_{j}}}{\prod_{j = 1}^{n} {(1 + {\dot{c}}_{σ (j)})}^{w_{j}} + \prod_{j = 1}^{n} {(1 - {\dot{c}}_{σ (j)})}^{w_{j}}}, \frac{\prod_{j = 1}^{n} {(1 + {\dot{d}}_{σ (j)})}^{w_{j}} - \prod_{j = 1}^{n} {(1 - {\dot{d}}_{σ (j)})}^{w_{j}}}{\prod_{j = 1}^{n} {(1 + {\dot{d}}_{σ (j)})}^{w_{j}} + \prod_{j = 1}^{n} {(1 - {\dot{d}}_{σ (j)})}^{w_{j}}}]} . \end{array}

Definition 18 Let

{\tilde{h}}_{j} (j = 1, 2, \dots, n)

be a collection of HTrFEs, then the HTrFHHG operator is defined as follows:

\begin{array}{rcl} H T r F H H G ({\tilde{h}}_{1}, {\tilde{h}}_{2}, \dots, {\tilde{h}}_{n}) \\ = & \otimes_{j = 1}^{n} {\dot{\tilde{h}}}_{σ (j)}^{w_{j}} \\ = & ⋃_{\binom{{\tilde{γ}}_{σ (j)}^{.} \in {\tilde{h}}_{σ (j)}^{.},}{j = 1, 2, \dots, n}} {[\frac{γ \prod_{j = 1}^{n} {\dot{a}}_{σ (j)}^{w_{j}}}{\prod_{j = 1}^{n} {(1 + (γ - 1) (1 - {\dot{a}}_{σ (j)}))}^{^{w_{j}}} + (γ - 1) \prod_{j = 1}^{n} {\dot{a}}_{σ (j)}^{w_{j}}}, \\ \frac{γ \prod_{j = 1}^{n} {\dot{b}}_{σ (j)}^{w_{j}}}{\prod_{j = 1}^{n} {(1 + (γ - 1) (1 - {\dot{b}}_{σ (j)}))}^{^{w_{j}}} + (γ - 1) \prod_{j = 1}^{n} {\dot{b}}_{σ (j)}^{w_{j}}}, \\ \frac{γ \prod_{j = 1}^{n} {\dot{c}}_{σ (j)}^{w_{j}}}{\prod_{j = 1}^{n} {(1 + (γ - 1) (1 - {\dot{c}}_{σ (j)}))}^{^{w_{j}}} + (γ - 1) \prod_{j = 1}^{n} {\dot{c}}_{σ (j)}^{w_{j}}}, \\ \frac{γ \prod_{j = 1}^{n} {\dot{d}}_{σ (j)}^{w_{j}}}{\prod_{j = 1}^{n} {(1 + (γ - 1) (1 - {\dot{d}}_{σ (j)}))}^{^{w_{j}}} + (γ - 1) \prod_{j = 1}^{n} {\dot{d}}_{σ (j)}^{w_{j}}}]}, \end{array}

(19)

where

w = {(w_{1}, w_{2}, \dots, w_{n})}^{T}

is the associated weighting vector, with

w_{j} \geq 0

\sum_{j = 1}^{n} w_{j} = 1

and

{\tilde{h}}_{σ (j)}

is the

j

th largest element of the hesitant trapezoid fuzzy arguments

{\tilde{h}}_{σ (j)} = (n ω_{j}) {\tilde{h}}_{j} (j = 1, 2, \dots, n)

ω = {(ω_{1}, ω_{2}, \dots, ω_{n})}^{T}

is the associated weighting vector, with

ω_{j} \geq 0

\sum_{j = 1}^{n} ω_{j} = 1,

and

n

is the balancing coefficient.

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

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:

\begin{array}{rcl} H T r F E H G ({\tilde{h}}_{1}, {\tilde{h}}_{2}, \dots, {\tilde{h}}_{n}) \\ = & \otimes_{j = 1}^{n} ({\dot{\tilde{h}}}_{σ (j)}^{w_{j}}) \\ = & ⋃_{\binom{{\tilde{γ}}_{σ (j)}^{.} \in {\tilde{h}}_{σ (j)}^{.},}{j = 1, 2, \dots, n}} {[\frac{2 \prod_{j = 1}^{n} {\dot{a}}_{σ (j)}^{w_{j}}}{\prod_{j = 1}^{n} {(2 - {\dot{a}}_{σ (j)})}^{w_{j}} + \prod_{j = 1}^{n} {\dot{a}}_{σ (j)}^{w_{j}}}, \frac{2 \prod_{j = 1}^{n} {\dot{b}}_{σ (j)}^{w_{j}}}{\prod_{j = 1}^{n} {(2 - {\dot{b}}_{σ (j)})}^{w_{j}} + \prod_{j = 1}^{n} {\dot{b}}_{σ (j)}^{w_{j}}}, \\ \frac{2 \prod_{j = 1}^{n} {\dot{c}}_{σ (j)}^{w_{j}}}{\prod_{j = 1}^{n} {(2 - {\dot{c}}_{σ (j)})}^{w_{j}} + \prod_{j = 1}^{n} {\dot{c}}_{σ (j)}^{w_{j}}}, \frac{2 \prod_{j = 1}^{n} {\dot{d}}_{σ (j)}^{w_{j}}}{\prod_{j = 1}^{n} {(2 - {\dot{d}}_{σ (j)})}^{w_{j}} + \prod_{j = 1}^{n} {\dot{d}}_{σ (j)}^{w_{j}}}]} . \end{array}

4.3 Hesitant Trapezoid Fuzzy Hamacher Choquet Aggregation Operators

The previous aggregation operators in MADM problem under hesitant trapezoid fuzzy environment 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 characteristics between attributes. In 1974, fuzzy measure was first introduced by Sugeno^[32] to deal with the phenomenon of mutual influence, and it has been applied in various fields^[33-36]. In real decision-making problems, fuzzy measures define a weight on not only each attribute but also each combination of attribute, and the sum of every weight does not equal to one.

Definition 19 (see [32]) Let

X = {1, 2, \dots, n}

be a universe of discourse,

P (X)

be the power set of

X .

A fuzzy measure on

X

is a set function

μ : P (X) \to [0, 1]

satisfying the following axioms:

μ (\emptyset) = 0, μ (X) = 1

;

2) If

A, B \in P (X)

and

A \subseteq B,

then

μ (A) \leq μ (B)

In multi-attribute decision making,

μ (A)

can be viewed as the importance of the attribute set

A

. Thus, in addition to the usual weights on attributes taken separately, weights on any combination of attributes also can be defined.

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

Definition 20 (see [41]) Let

f

be a positive real-valued function on

X,

and

μ

be a fuzzy measure on

X

. The discrete Choquet integral of

f

with respect to

μ

is defined by

\begin{array}{rcl} C_{μ} (f (x_{(1)}), f (x_{(2)}), \dots, f (x_{(n)})) = \sum_{i = 1}^{n} f (x_{(i)}) (μ (A_{(i)}) - μ (A_{(i + 1)})), \end{array}

(20)

where

(\cdot)

indicates a permutation of

(1, 2, \dots, n),

such that

f (x_{(1)}) \leq f (x_{(2)}) \leq \dots \leq f (x_{(n)})

and

A_{(i)} = {i, i + 1, \dots, n}

with

A_{(n + 1)} = \emptyset

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

Definition 21 Let

{\tilde{h}}_{j} (j = 1, 2, \dots, n)

be a collection of HTrFEs on

X,

and

μ

be a fuzzy measure on

X

. A HTrFHCA operator is a mapping

Q^{n} \to Q

, and

\begin{array}{rcl} H T r F H C A_{μ} ({\tilde{h}}_{1}, {\tilde{h}}_{2}, \dots, {\tilde{h}}_{n}) = \oplus_{j = 1}^{n} (μ (A_{σ (j)}) - μ (A_{σ (j - 1)})) {\tilde{h}}_{σ (j)}, \end{array}

(21)

where

(σ (1), σ (2), \dots, σ (n))

is a permutation of

(1, 2, \dots, n),

such that

{\tilde{h}}_{σ (j - 1)} \geq {\tilde{h}}_{σ (j)}

for all

j = 2, 3, \dots, n

A_{σ (k)} = {x_{σ (j)} | j \leq k},

for

k \geq 1,

and

A_{σ (0)} = \emptyset

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

Theorem 3 Let ${\tilde{h}}_{j} (j = 1, 2, \dots, n)$ be a collection of HTrFEs, then their aggregated value by using the HTrFHCA operator is also a HTrFE, and

\begin{array}{rcl} H T r F H C A_{μ} ({\tilde{h}}_{1}, {\tilde{h}}_{2}, \dots, {\tilde{h}}_{n}) \\ = & \oplus_{j = 1}^{n} (μ (A_{σ (j)}) - μ (A_{σ (j - 1)})) {\tilde{h}}_{σ (j)} \\ = & ⋃_{\binom{{\tilde{γ}}_{σ (j)} \in {\tilde{h}}_{σ (j)},}{j = 1, 2, \dots, n}} {[\frac{\prod_{j = 1}^{n} {(1 + (γ - 1) a_{σ (j)})}^{^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})}} - \prod_{j = 1}^{n} {(1 - a_{σ (j)})}^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})}}{\prod_{j = 1}^{n} {(1 + (γ - 1) a_{σ (j)})}^{^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})}} + (γ - 1) \prod_{j = 1}^{n} {(1 - a_{σ (j)})}^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})}}, \\ \frac{\prod_{j = 1}^{n} {(1 + (γ - 1) b_{σ (j)})}^{^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})}} - \prod_{j = 1}^{n} {(1 - b_{σ (j)})}^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})}}{\prod_{j = 1}^{n} {(1 + (γ - 1) b_{σ (j)})}^{^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})}} + (γ - 1) \prod_{j = 1}^{n} {(1 - b_{σ (j)})}^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})}}, \\ \frac{\prod_{j = 1}^{n} {(1 + (γ - 1) c_{σ (j)})}^{^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})}} - \prod_{j = 1}^{n} {(1 - c_{σ (j)})}^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})}}{\prod_{j = 1}^{n} {(1 + (γ - 1) c_{σ (j)})}^{^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})}} + (γ - 1) \prod_{j = 1}^{n} {(1 - c_{σ (j)})}^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})}}, \\ \frac{\prod_{j = 1}^{n} {(1 + (γ - 1) d_{σ (j)})}^{^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})}} - \prod_{j = 1}^{n} {(1 - d_{σ (j)})}^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})}}{\prod_{j = 1}^{n} {(1 + (γ - 1) d_{σ (j)})}^{^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})}} + (γ - 1) \prod_{j = 1}^{n} {(1 - d_{σ (j)})}^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})}}]}, \end{array}

(22)

where $(σ (1), σ (2), \dots, σ (n))$ is a permutation of $(1, 2, \dots, n),$ such that ${\tilde{h}}_{σ (j - 1)} \geq {\tilde{h}}_{σ (j)}$ for all $j = 2, 3, \dots, n$ , $A_{σ (k)} = {x_{σ (j)} | j \leq k},$ for $k \geq 1,$ and $A_{σ (0)} = \emptyset$ .

Now, we can discuss some special cases of the HTrFHCA operator with respect to the parameter

γ

When

γ = 1,

the HTrFHCA operator reduces to the hesitant trapezoid fuzzy Choquet average (HTrFCA) operator as follows:

\begin{array}{rcl} H T r F C A_{μ} ({\tilde{h}}_{1}, {\tilde{h}}_{2}, \dots, {\tilde{h}}_{n}) \\ = & \oplus_{j = 1}^{n} (μ (A_{σ (j)}) - μ (A_{σ (j - 1)})) {\tilde{h}}_{σ (j)} \\ = & ⋃_{\binom{{\tilde{γ}}_{σ (j)} \in {\tilde{h}}_{σ (j)},}{j = 1, 2, \dots, n}} {[1 - \prod_{j = 1}^{n} {(1 - a_{σ (j)})}^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})}, 1 - \prod_{j = 1}^{n} {(1 - b_{σ (j)})}^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})}, \\ 1 - \prod_{j = 1}^{n} {(1 - c_{σ (j)})}^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})}, 1 - \prod_{j = 1}^{n} {(1 - d_{σ (j)})}^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})}]} . \end{array}

When

γ = 2

, the HTrFHCA operator reduces to the hesitant trapezoid fuzzy Einstein Choquet average (HTrFECA) operator as follows:

\begin{array}{rcl} H T r F E C A_{μ} ({\tilde{h}}_{1}, {\tilde{h}}_{2}, \dots, {\tilde{h}}_{n}) \\ = & \oplus_{j = 1}^{n} (μ (A_{σ (j)}) - μ (A_{σ (j - 1)})) {\tilde{h}}_{σ (j)} \\ = & ⋃_{\binom{{\tilde{γ}}_{σ (j)} \in {\tilde{h}}_{σ (j)},}{j = 1, 2, \dots, n}} {[\frac{\prod_{j = 1}^{n} {(1 + a_{σ (j)})}^{^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})}} - \prod_{j = 1}^{n} {(1 - a_{σ (j)})}^{^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})}}}{\prod_{j = 1}^{n} {(1 + a_{σ (j)})}^{^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})}} + \prod_{j = 1}^{n} {(1 - a_{σ (j)})}^{^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})}}}, \\ \frac{\prod_{j = 1}^{n} {(1 + b_{σ (j)})}^{^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})}} - \prod_{j = 1}^{n} {(1 - b_{σ (j)})}^{^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})}}}{\prod_{j = 1}^{n} {(1 + b_{σ (j)})}^{^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})}} + \prod_{j = 1}^{n} {(1 - b_{σ (j)})}^{^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})}}}, \\ \frac{\prod_{j = 1}^{n} {(1 + c_{σ (j)})}^{^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})}} - \prod_{j = 1}^{n} {(1 - c_{σ (j)})}^{^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})}}}{\prod_{j = 1}^{n} {(1 + c_{σ (j)})}^{^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})}} + \prod_{j = 1}^{n} {(1 - c_{σ (j)})}^{^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})}}}, \\ \frac{\prod_{j = 1}^{n} {(1 + d_{σ (j)})}^{^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})}} - \prod_{j = 1}^{n} {(1 - d_{σ (j)})}^{^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})}}}{\prod_{j = 1}^{n} {(1 + d_{σ (j)})}^{^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})}} + \prod_{j = 1}^{n} {(1 - d_{σ (j)})}^{^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})}}}]} . \end{array}

Definition 22 Let

{\tilde{h}}_{j} (j = 1, 2, \dots, n)

be a collection of HTrFEs on

X,

and

μ

be a fuzzy measure on

X

. A HTrFHCG operator is a mapping

Q^{n} \to Q,

and

\begin{array}{rcl} H T r F H C G_{μ} ({\tilde{h}}_{1}, {\tilde{h}}_{2}, \dots, {\tilde{h}}_{n}) = \otimes_{j = 1}^{n} ({\tilde{h}}_{σ (j)}^{(μ (A_{σ (j)}) - μ (A_{σ (j - 1)}))}), \end{array}

(23)

where

(σ (1), σ (2), \dots, σ (n))

is a permutation of

(1, 2, \dots, n),

such that

{\tilde{h}}_{σ (j - 1)} \geq {\tilde{h}}_{σ (j)}

for all

j = 2, \dots, n, A_{σ (k)} = {x_{σ (j)} | j \leq k},

for

k \geq 1,

and

A_{σ (0)} = \emptyset

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

Theorem 4 Let ${\tilde{h}}_{j} (j = 1, 2, \dots, n)$ be a collection of HTrFEs, then their aggregated value by using the HTrFHCG operator is also a HTrFE, and

\begin{array}{rcl} H T r F H C G_{μ} ({\tilde{h}}_{1}, {\tilde{h}}_{2}, \dots, {\tilde{h}}_{n}) \\ = & \otimes_{j = 1}^{n} ({\tilde{h}}_{σ (j)}^{(μ (A_{σ (j)}) - μ (A_{σ (j - 1)}))}) \\ = & ⋃_{\binom{{\tilde{γ}}_{σ (j)} \in {\tilde{h}}_{σ (j)},}{j = 1, 2, \dots, n}} {[\frac{γ \prod_{j = 1}^{n} a_{σ (j)}^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})}}{\prod_{j = 1}^{n} {(1 + (γ - 1) (1 - a_{σ (j)}))}^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})} + (γ - 1) \prod_{j = 1}^{n} a_{σ (j)}^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})}}, \\ \frac{γ \prod_{j = 1}^{n} b_{σ (j)}^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})}}{\prod_{j = 1}^{n} {(1 + (γ - 1) (1 - b_{σ (j)}))}^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})} + (γ - 1) \prod_{j = 1}^{n} b_{σ (j)}^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})}}, \\ \frac{γ \prod_{j = 1}^{n} c_{σ (j)}^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})}}{\prod_{j = 1}^{n} {(1 + (γ - 1) (1 - c_{σ (j)}))}^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})} + (γ - 1) \prod_{j = 1}^{n} c_{σ (j)}^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})}}, \\ \frac{γ \prod_{j = 1}^{n} d_{σ (j)}^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})}}{\prod_{j = 1}^{n} {(1 + (γ - 1) (1 - d_{σ (j)}))}^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})} + (γ - 1) \prod_{j = 1}^{n} d_{σ (j)}^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})}}]}, \end{array}

(24)

where $(σ (1), σ (2), \dots, σ (n))$ is a permutation of $(1, 2, \dots, n),$ such that ${\tilde{h}}_{σ (j - 1)} \geq {\tilde{h}}_{σ (j)}$ for all $j = 2, \dots, n, A_{σ (k)} = {x_{σ (j)} | j \leq k},$ for $k \geq 1,$ and $A_{σ (0)} = \emptyset$ .

Now, we can discuss some special cases of the HTrFHCG operator with respect to the parameter

γ

When

γ = 1,

the HTrFHCG operator reduces to the hesitant trapezoid fuzzy Choquet geometric (HTrFCG) operator as follows:

\begin{array}{rcl} H T r F C G_{μ} ({\tilde{h}}_{1}, {\tilde{h}}_{2}, \dots, {\tilde{h}}_{n}) & = & \otimes_{j = 1}^{n} ({\tilde{h}}_{σ (j)}^{(μ (A_{σ (j)}) - μ (A_{σ (j - 1)}))}) \\ = & ⋃_{\binom{{\tilde{γ}}_{σ (j)} \in {\tilde{h}}_{σ (j)},}{j = 1, 2, \dots, n}} {[\prod_{j = 1}^{n} a_{σ (j)}^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})}, \prod_{j = 1}^{n} b_{σ (j)}^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})}, \\ \prod_{j = 1}^{n} c_{σ (j)}^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})}, \prod_{j = 1}^{n} d_{σ (j)}^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})}]} . \end{array}

When

γ = 2,

the HTrFHCG operator reduces to the hesitant trapezoid fuzzy Einstein Choquet geometric (HTrFECG) operator as follows:

\begin{array}{rcl} H T r F E C G_{μ} ({\tilde{h}}_{1}, {\tilde{h}}_{2}, \dots, {\tilde{h}}_{n}) \\ = & \otimes_{j = 1}^{n} ({\tilde{h}}_{σ (j)}^{(μ (A_{σ (j)}) - μ (A_{σ (j - 1)}))}) \\ = & \cup_{\binom{{\tilde{γ}}_{σ (j)} \in {\tilde{h}}_{σ (j)},}{j = 1, 2, \dots, n}} {[\frac{2 \prod_{j = 1}^{n} a_{σ (j)}^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})}}{\prod_{j = 1}^{n} {(2 - a_{σ (j)})}^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})} + \prod_{j = 1}^{n} a_{σ (j)}^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})}}, \\ \frac{2 \prod_{j = 1}^{n} b_{σ (j)}^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})}}{\prod_{j = 1}^{n} {(2 - b_{σ (j)})}^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})} + \prod_{j = 1}^{n} b_{σ (j)}^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})}}, \\ \frac{2 \prod_{j = 1}^{n} c_{σ (j)}^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})}}{\prod_{j = 1}^{n} {(2 - c_{σ (j)})}^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})} + \prod_{j = 1}^{n} c_{σ (j)}^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})}}, \\ \frac{2 \prod_{j = 1}^{n} d_{σ (j)}^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})}}{\prod_{j = 1}^{n} {(2 - d_{σ (j)})}^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})} + \prod_{j = 1}^{n} d_{σ (j)}^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})}}]} . \end{array}

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 to MADM problems under hesitant trapezoid fuzzy environment.

For a MADM problem, let

A = {A_{1}, A_{2}, \dots, A_{m}}

be a discrete set of alternatives, and

C = {C_{1}, C_{2}, \dots, C_{n}}

be a collection of attributes.

w = {(w_{1}, w_{2}, \dots, w_{n})}^{T}

is the weighting vector of the attribute

C_{j} (j = 1, 2, \dots, n),

with

w_{j} \geq 0

\sum_{j = 1}^{n} w_{j} = 1 .

Suppose that the decision matrix

H = ({\tilde{h}}_{\bar{y}})_{m \times n}

is the hesitant trapezoid fuzzy matrix, where

{\tilde{h}}_{i j}

is the evaluation value for alternative

A_{i} \in A (i = 1, 2, \dots, m)

with respect to the attribute

C_{j} \in C (j = 1, 2, \dots, n),

and takes the form of HTrFE.

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

Step 1 From the given decision matrix

H,

and we utilize the HTrFHWA operator

\begin{array}{rcl} {\tilde{h}}_{i} & = & H T r F H W A ({\tilde{h}}_{i 1}, {\tilde{h}}_{i 2}, \dots, {\tilde{h}}_{i n}) \\ = & {[\frac{\prod_{j = 1}^{n} {(1 + (γ - 1) a_{i j})}^{w_{j}} - \prod_{j = 1}^{n} {(1 - a_{i j})}^{w_{j}}}{\prod_{j = 1}^{n} {(1 + (γ - 1) a_{i j})}^{w_{j}} + (γ - 1) \prod_{j = 1}^{n} {(1 - a_{i j})}^{w_{j}}}, \frac{\prod_{j = 1}^{n} {(1 + (γ - 1) b_{i j})}^{w_{j}} - \prod_{j = 1}^{n} {(1 - b_{i j})}^{w_{j}}}{\prod_{j = 1}^{n} {(1 + (γ - 1) b_{i j})}^{w_{j}} + (γ - 1) \prod_{j = 1}^{n} {(1 - b_{i j})}^{w_{j}}}, \\ \frac{\prod_{j = 1}^{n} {(1 + (γ - 1) c_{i j})}^{w_{j}} - \prod_{j = 1}^{n} {(1 - c_{i j})}^{w_{j}}}{\prod_{j = 1}^{n} {(1 + (γ - 1) c_{i j})}^{w_{j}} + (γ - 1) \prod_{j = 1}^{n} {(1 - c_{i j})}^{w_{j}}}, \frac{\prod_{j = 1}^{n} {(1 + (γ - 1) d_{i j})}^{w_{j}} - \prod_{j = 1}^{n} {(1 - d_{i j})}^{w_{j}}}{\prod_{j = 1}^{n} {(1 + (γ - 1) d_{i j})}^{w_{j}} + (γ - 1) \prod_{j = 1}^{n} {(1 - d_{i j})}^{w_{j}}}]} \end{array}

to aggregate all the hesitant trapezoid fuzzy values

{\tilde{h}}_{i j} (i = 1, 2, \dots, m; j = 1, 2, \dots, n)

into the overall hesitant trapezoid fuzzy value

{\tilde{h}}_{i} (i = 1, 2, \dots, m)

of each alternative

A_{i} (i = 1, 2, \dots, m)

Step 2 Calculate the score values

s (h_{i}) (i = 1, 2, \dots, m)

of the overall hesitant trapezoid fuzzy values

{\tilde{h}}_{i} (i = 1, 2, \dots, m)

by Definition 6.

Step 3 To rank these score values

s ({\tilde{h}}_{i}) (i = 1, 2, \dots, m),

we first compare each

s ({\tilde{h}}_{i})

with all the

s ({\tilde{h}}_{i}) (i = 1, 2, \dots, m)

by using Equation (4). For simplicity, we let

p_{i j} = p (s ({\tilde{h}}_{i}) \geq s ({\tilde{h}}_{j}))

, then we develop a complementary matrix as

P = {(p_{i j})}_{m \times n}

, where

p_{i j} \geq 0, p_{i j} + p_{j i} = 1, p_{i i} = 0.5, i, j = 1, 2, \dots, m .

Summing all the elements in each line of matrix

P,

we have

p_{i} = \sum_{j = 1}^{m} p_{i j}, i = 1, 2, \dots, m .

Then we rank the score values

s ({\tilde{h}}_{i}) (i = 1, 2, \dots, m)

in descending order in accordance with the values of

p_{i} (i = 1, 2, \dots, m)

Step 4 Rank all the alternatives

A_{i} (i = 1, 2, \dots, m)

and select the best one(s) in accordance with

s (h_{i}) (i = 1, 2, \dots, m)

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

Step 1 $^{'}$ From the given decision matrix

H

, and we utilize the HTrFHWG operator

\begin{array}{rcl} {\tilde{h}}_{i} \\ = & H T r F H W G ({\tilde{h}}_{i 1}, {\tilde{h}}_{i 2}, \dots, {\tilde{h}}_{i n}) \\ = & {[\frac{γ \prod_{j = 1}^{n} {(a_{i j})}^{w_{j}}}{\prod_{j = 1}^{n} {(1 + (γ - 1) (1 - a_{i j}))}^{w_{j}} + (γ - 1) \prod_{j = 1}^{n} {(a_{i j})}^{w_{j}}}, \frac{γ \prod_{j = 1}^{n} {(b_{i j})}^{w_{j}}}{\prod_{j = 1}^{n} {(1 + (γ - 1) (1 - b_{i j}))}^{w_{j}} + (γ - 1) \prod_{j = 1}^{n} {(b_{i j})}^{w_{j}}}, \\ \frac{γ \prod_{j = 1}^{n} {(c_{i j})}^{w_{j}}}{\prod_{j = 1}^{n} {(1 + (γ - 1) (1 - c_{i j}))}^{w_{j}} + (γ - 1) \prod_{j = 1}^{n} {(c_{i j})}^{w_{j}}}, \frac{γ \prod_{j = 1}^{n} {(d_{i j})}^{w_{j}}}{\prod_{j = 1}^{n} {(1 + (γ - 1) (1 - d_{i j}))}^{w_{j}} + (γ - 1) \prod_{j = 1}^{n} {(d_{i j})}^{w_{j}}}]} \end{array}

to aggregate all the hesitant trapezoid fuzzy values

{\tilde{h}}_{i j} (i = 1, 2, \dots, m; j = 1, 2, \dots, n)

into the overall hesitant trapezoid fuzzy value

{\tilde{h}}_{i} (i = 1, 2, \dots, m)

of each alternative

A_{i} (i = 1, 2, \dots, m)

Step 2 $^{'}$ Calculate the score values

s (h_{i}) (i = 1, 2, \dots, m)

of the overall hesitant trapezoid fuzzy values

{\tilde{h}}_{i} (i = 1, 2, \dots, m)

by Definition 6.

Step 3 $^{'}$ To rank these score values

s ({\tilde{h}}_{i}) (i = 1, 2, \dots, m),

we first compare

each s ({\tilde{h}}_{i})

with all the

s ({\tilde{h}}_{i}) (i = 1, 2, \dots, m)

by using Equation (4). For simplicity, we let

p_{i j} = p (s ({\tilde{h}}_{i}) \geq s ({\tilde{h}}_{j})),

then we develop a complementary matrix as

P = (p_{i j})_{m \times n},

where

p_{i j} \geq 0, p_{i j} + p_{j i} = 1, p_{i j} = 0.5, i, j = 1, 2, \dots, m .

Summing all the elements in each line of matrix

P

, we have

p_{i} = \sum_{j = 1}^{m} p_{i j}, i = 1, 2, \dots, m .

Then we rank the score values

s ({\tilde{h}}_{i}) (i = 1, 2, \dots, m)

in descending order in accordance with the values of

p_{i} (i = 1, 2, \dots, m)

Step 4 $^{'}$ Rank all the alternatives

A_{i} (i = 1, 2, \dots, m)

and select the best one(s) in accordance with

s (h_{t}) (i = 1, 2, \dots, m)

6 Illustrative Example

Let us suppose that there is a manufacturing company, which wants to select the best global supplier according to the core competencies of suppliers (adapted from ^[42]). There are four suppliers

A_{i} (i = 1, 2, 3, 4)

to be evaluated under the following four attributes

C_{j} (j = 1, 2, 3, 4) : C_{1}

is the level of technology innovation;

C_{2}

is the control ability of flow;

C_{3}

is the ability of management;

C_{4}

is the level of service.

w = (0.2, 0.4, 0.1, 0.3)^{T}

is the weight vector of attributes. The four possible candidates

A_{i} (i = 1, 2, 3, 4)

are to be evaluated using the hesitant trapezoid fuzzy information by the decision maker under the above four attributes, and the hesitant trapezoid fuzzy decision matrix

H = ({\tilde{h}}_{i j})_{4 \times 4}

is shown in Table 1.

Table 1 Hesitant trapezoid fuzzy decision matrix

	$C_{1}$	$C_{2}$	$C_{3}$	$C_{4}$
$A_{1}$	{[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]}
$A_{1}$	[0.6, 0.7, 0.8, 0.9]}	${[0.2, 0.3, 0.4, 0.5]}$	[0.1, 0.2, 0.3, 0.4]}
$A_{2}$	{[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]
$A_{2}$		[0.3, 0.4, 0.5, 0.6]}		[0.5, 0.6, 0.7, 0.8]}
$A_{3}$	{[0.3, 0.4, 0.5, 0.6]		{[0.4, 0.5, 0.6, 0.7]	{[0.2, 0.3, 0.4, 0.5]}
$A_{3}$	[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]}
$A_{4}$	${[0.1, 0.2, 0.3, 0.4]}$	{[0.4, 0.5, 0.6, 0.7]	{[0.3, 0.4, 0.5, 0.6]}	{[0.1, 0.2, 0.3, 0.4]
$A_{4}$	${[0.1, 0.2, 0.3, 0.4]}$	[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 fuzzy Hamacher weighted average (HTrFHWA) operator and the hesitant trapezoid fuzzy Hamacher weighted geometric (HTrFHWG) operator to develop an approach to multiple attribute decision making problems with hesitant trapezoid fuzzy information, which can be described as following:

Step 1 We utilize the HTrFHWA operator to aggregate all the hesitant trapezoid fuzzy information

{\tilde{h}}_{i j} (i = 1, 2, 3, 4; j = 1, 2, 3, 4)

into the overall hesitant trapezoid fuzzy values

{\tilde{h}}_{i} (i = 1, 2, 3, 4) .

Take

{\tilde{h}}_{1}

for example, here let

γ = 1, 2

, we have

\begin{array}{rcl} h_{1} & = & H T r F H W A_{1} ({\tilde{h}}_{11}, {\tilde{h}}_{12}, {\tilde{h}}_{13}, {\tilde{h}}_{14}) \\ = & H T r F W A ({\tilde{h}}_{11}, {\tilde{h}}_{12}, {\tilde{h}}_{13}, {\tilde{h}}_{14}) = \oplus_{j = 1}^{4} w_{j} {\tilde{h}}_{1 j} \\ = & ⋃_{\binom{{\tilde{γ}}_{1 j} \in {\tilde{h}}_{1 j},}{j = 1, 2, 3, 4}} {[1 - \prod_{j = 1}^{4} {(1 - a_{1 j})}^{w_{j}}, 1 - \prod_{j = 1}^{4} {(1 - b_{1 j})}^{w_{j}}, 1 - \prod_{j = 1}^{4} {(1 - c_{1 j})}^{w_{j}}, 1 - \prod_{j = 1}^{4} {(1 - d_{1 j})}^{w_{j}}]} \\ = & {[0.3408, 0.4429, 0.5458, 0.6508] [0.3241, 0.4266, 0.5303, 0.6363] \\ [0.3696, 0.4740, 0.5812, 0.6960] [0.3536 0.4587 0.5669 0.6834]}, \\ h_{1} & = & H T r F H W A_{2} ({\tilde{h}}_{11}, {\tilde{h}}_{12}, {\tilde{h}}_{13}, {\tilde{h}}_{14}) = H T r F E W A ({\tilde{h}}_{11}, {\tilde{h}}_{12}, {\tilde{h}}_{13}, {\tilde{h}}_{14}) = \oplus_{j = 1}^{4} w_{j} {\tilde{h}}_{1 j} \\ = & ⋃_{\binom{{\tilde{γ}}_{1 j} \in {\tilde{h}}_{1 j},}{j = 1, 2, 3, 4}} {[\frac{\prod_{j = 1}^{4} {(1 + a_{1 j})}^{w_{j}} - \prod_{j = 1}^{4} {(1 - a_{1 j})}^{w_{j}}}{\prod_{j = 1}^{4} {(1 + a_{1 j})}^{w_{j}} + \prod_{j = 1}^{4} {(1 - a_{1 j})}^{w_{j}}}, \frac{\prod_{j = 1}^{4} {(1 + b_{1 j})}^{w_{j}} - \prod_{j = 1}^{4} {(1 - b_{1 j})}^{w_{j}}}{\prod_{j = 1}^{4} {(1 + b_{1 j})}^{w_{j}} + \prod_{j = 1}^{4} {(1 - b_{1 j})}^{w_{j}}}, \\ \frac{\prod_{j = 1}^{4} {(1 + c_{1 j})}^{w_{j}} - \prod_{j = 1}^{4} {(1 - c_{1 j})}^{w_{j}}}{\prod_{j = 1}^{4} {(1 + c_{1 j})}^{w_{j}} + \prod_{j = 1}^{4} {(1 - c_{1 j})}^{w_{j}}}, \frac{\prod_{j = 1}^{4} {(1 + d_{1 j})}^{w_{j}} - \prod_{j = 1}^{4} {(1 - d_{1 j})}^{w_{j}}}{\prod_{j = 1}^{4} {(1 + d_{1 j})}^{w_{j}} + \prod_{j = 1}^{4} {(1 - d_{1 j})}^{w_{j}}}]} \\ = & {[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]} . \end{array}

Step 2a) When

γ = 1,

calculate the score values

s ({\tilde{h}}_{i}) (i = 1, 2, 3, 4)

of the overall hesitant trapezoid fuzzy preference values

{\tilde{h}}_{i} (i = 1, 2, 3, 4)

\begin{array}{rcl} s ({\tilde{h}}_{1}) = {[0.3470, 04505, 0.5561, 0.6666]}, s ({\tilde{h}}_{2}) = {[0.3105, 0.4124, 0.5150, 0.6193]}, \\ s ({\tilde{h}}_{3}) = {[0.2603, 0.3613, 0.4627, 0.5649]}, s ({\tilde{h}}_{4}) = {[0.3212, 0.4240, 0.5279, 0.6340]} . \end{array}

Step 2b) When

γ = 2

, calculate the score values

s ({\tilde{h}}_{i}) (i = 1, 2, 3, 4)

of the overall hesitant trapezoid fuzzy preference values

{\tilde{h}}_{i} (i = 1, 2, 3, 4)

\begin{array}{rcl} s ({\tilde{h}}_{1}) = {[0.3389, 0.4429, 0.5487, 0.6594]}, s ({\tilde{h}}_{2}) = {[0.3049, 0.4071, 0.5101, 0.6147]}, \\ s ({\tilde{h}}_{3}) = {[0.2173, 0.3177, 0.4181, 0.5187]}, s ({\tilde{h}}_{4}) = {[0.3125, 0.4159, 0.5205, 0.6272]} . \end{array}

Step 3 Rank all the suppliers

A_{i} (i = 1, 2, 3, 4)

in accordance with the score value

s ({\tilde{h}}_{i})

of the overall fuzzy preference value

{\tilde{h}}_{i} : A_{1} ≻ A_{4} ≻ A_{2} ≻ A_{3} .

And thus, the most desirable supplier is

A_{1}

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

Step 1 $^{'}$ We apply the HTrFHWG operator to obtain the overall hesitant trapezoid fuzzy values

{\tilde{h}}_{i}

. Take

{\tilde{h}}_{1}

for example, here let

γ = 1, 2,

we have

\begin{array}{rcl} h_{1} & = & H T r F H W G_{1} ({\tilde{h}}_{11}, {\tilde{h}}_{12}, {\tilde{h}}_{13}, {\tilde{h}}_{14}) \\ = & H T r F W G ({\tilde{h}}_{11}, {\tilde{h}}_{12}, {\tilde{h}}_{13}, {\tilde{h}}_{14}) = \otimes_{j = 1}^{4} {\tilde{h}}_{1 j}^{w_{j}} \\ = & ⋃_{\binom{{\tilde{γ}}_{1 j} \in {\tilde{h}}_{1 j},}{j = 1, 2, 3, 4}} {[\prod_{j = 1}^{4} a_{1 j}^{w_{j}}, \prod_{j = 1}^{4} b_{1 j}^{w_{j}}, \prod_{j = 1}^{4} c_{1 j}^{w_{j}}, \prod_{j = 1}^{4} d_{1 j}^{w_{j}}]} \\ = & {[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]}, \\ {\tilde{h}}_{1} & = & H T r F H W G_{2} ({\tilde{h}}_{11}, {\tilde{h}}_{12}, {\tilde{h}}_{13}, {\tilde{h}}_{14}) = H T r F E W G ({\tilde{h}}_{11}, {\tilde{h}}_{12}, {\tilde{h}}_{13}, {\tilde{h}}_{14}) = \otimes_{j = 1}^{4} ({\tilde{h}}_{1 j}^{w_{j}}) \\ = & ⋃_{\binom{{\tilde{γ}}_{1 j} \in {\tilde{h}}_{1 j},}{j = 1, 2, 3, 4}} {[\frac{2 \prod_{j = 1}^{4} a_{1 j}^{w_{j}}}{\prod_{j = 1}^{4} {(2 - a_{1 j})}^{w_{j}} + \prod_{j = 1}^{4} a_{1 j}^{w_{j}}}, \frac{2 \prod_{j = 1}^{4} b_{1 j}^{w_{j}}}{\prod_{j = 1}^{4} {(2 - b_{1 j})}^{w_{j}} + \prod_{j = 1}^{4} b_{1 j}^{w_{j}}}, \\ \frac{2 \prod_{j = 1}^{4} c_{1 j}^{w_{j}}}{\prod_{j = 1}^{4} {(2 - c_{1 j})}^{w_{j}} + \prod_{j = 1}^{4} c_{1 j}^{w_{j}}}, \frac{2 \prod_{j = 1}^{4} d_{1 j}^{w_{j}}}{\prod_{j = 1}^{4} {(2 - d_{1 j})}^{w_{j}} + \prod_{j = 1}^{4} d_{1 j}^{w_{j}}}]} \\ = & {[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]} . \end{array}

Step 2a $^{'}$ ) When

γ = 1

, calculate the score values

s ({\tilde{h}}_{i}) (i = 1, 2, 3, 4

) of the overall hesitant trapezoid fuzzy preference values

{\tilde{h}}_{i} (i = 1, 2, 3, 4)

\begin{array}{l} s ({\tilde{h}}_{1}) = {[0.2974, 0.4058, 0.5106, 0.5480]}, s ({\tilde{h}}_{2}) = {[0.2717, 0.3803, 0.4847, 0.5875]}, \\ s ({\tilde{h}}_{3}) = {[0.2117, 0.3132, 0.4140, 0.5145]}, s ({\tilde{h}}_{4}) = {[0.2558, 0.3717, 0.4795, 0.5843]} . \end{array}

Step 2b $^{'}$ ) When

γ = 2

, calculate the score values

s ({\tilde{h}}_{i}) (i = 1, 2, 3, 4)

of the overall hesitant trapezoid fuzzy preference values

{\tilde{h}}_{i} (i = 1, 2, 3, 4)

\begin{array}{l} s ({\tilde{h}}_{1}) = {[0.3033, 0.4122, 0.5175, 0.5676]}, s ({\tilde{h}}_{2}) = {[0.2757, 0.3847, 0.4895, 0.5926]}, \\ s ({\tilde{h}}_{3}) = {[0.2123, 0.3139, 0.4147, 0.5153]}, s ({\tilde{h}}_{4}) = {[0.2630, 0.3793, 0.4875, 0.5927]} . \end{array}

Step 3 $^{'}$ Rank all the suppliers

A_{i} (i = 1, 2, 3, 4)

in accordance with the score value

s ({\tilde{h}}_{i})

of the overall fuzzy preference value

{\tilde{h}}_{i} : A_{1} ≻ A_{2} ≻ A_{4} ≻ A_{3} .

And thus the most desirable supplier is

A_{1}

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

A_{1}

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
${H T r F H W A}_{1}$	$A_{1} ≻ A_{4} ≻ A_{2} ≻ A_{3}$
${H T r F H W A}_{2}$	$A_{1} ≻ A_{4} ≻ A_{2} ≻ A_{3}$
${H T r F H W G}_{1}$	$A_{1} ≻ A_{2} ≻ A_{4} ≻ A_{3}$
${H T r F H W G}_{2}$	$A_{1} ≻ A_{2} ≻ A_{4} ≻ A_{3}$

Through the above analysis, the advantages of our method based on the HTrFHWA operator and HTrFHWG operator can be summarized as follow. First, we studied the MADM problems in which the attribute values take the form of HTrFSs, which can flexibly describe the uncertainty in the reality. Secondly, since there are attribute-related associations, compared the conventional operators, our proposed operators consider the information about the relationship among arguments being aggregated. Thirdly, the proposed method based on the HTrFHWA operator and HTrFHWG operator has different parameters, DM can choose different parameters 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 is more effective and powerful for solving MADM problems.

7 Conclusion

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, 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 crisp numbers. Then some aggregation operators based on the Hamacher operation for aggregating hesitant trapezoid fuzzy information are defined, such as the hesitant trapezoid fuzzy Hamacher weighted average (HTrFHWA) operator, the hesitant trapezoid fuzzy Hamacher ordered weighted average (HTrFHOWA) operator, the hesitant trapezoid fuzzy Hamacher weighted geometric (HTrFHWG) operator, the hesitant trapezoid fuzzy Hamacher ordered weighted geometric (HTrFHOWG) operator, the hesitant trapezoid fuzzy Hamacher hybrid average (HTrFHHA) operator and the hesitant trapezoid fuzzy Hamacher hybrid geometric (HTrFHHG) operator. Furthermore, we utilize the hesitant trapezoid fuzzy Hamacher weighted average (HTrFHWA) operator and the hesitant trapezoid fuzzy Hamacher weighted geometric (HTrFHWG) operator to develop an approach to solve multiple attribute decision making (MADM) problems under hesitant trapezoid fuzzy environments. Finally, a practical example about supplier selection is given to verify the practicality and validity of the proposed method.

References

Publishing order | Descend order by publishing year | Descend order by cited within

1	Zadeh L A. Fuzzy sets. Information and Control, 1965, 8(3): 338- 353. https://doi.org/10.1016/S0019-9958(65)90241-X Cited in this article [1]

2	Gau W L, Buehrer D J. Vague sets. IEEE Transactions on Systems, Man, and Cybernetics, 1993, 23(2): 610- 614. https://doi.org/10.1109/21.229476 Cited in this article [1]

3	Dubois D J. Fuzzy sets and systems: Theory and applications. Academic Press, 1980. Cited in this article [2]

4	Turksen I B. Interval valued fuzzy-sets based on normal forms. Fuzzy Sets and Systems, 1986, 20(2): 191- 210. https://doi.org/10.1016/0165-0114(86)90077-1 Cited in this article [1]

5	Atanassov K T. Intuitionistic fuzzy sets. Fuzzy sets and Systems, 1986, 20(1): 87- 96. https://doi.org/10.1016/S0165-0114(86)80034-3 Cited in this article [1]

6	Atanassov K, Gargov G. Interval valued intuitionistic fuzzy-sets. Fuzzy Sets and Systems, 1989, 31(3): 343- 349. https://doi.org/10.1016/0165-0114(89)90205-4 Cited in this article [1]

7	Torra V. Hesitant fuzzy sets. International Journal of Intelligent Systems, 2010, 25(6): 529- 539. Cited in this article [2]

8	Xu Z, Zhang X. Hesitant fuzzy multi-attribute decision making based on TOPSIS with incomplete weight information. Knowledge-Based Systems, 2013, 52, 53- 64. https://doi.org/10.1016/j.knosys.2013.05.011 Cited in this article [1]

9	Yu D, Zhang W, Xu Y. Group decision making under hesitant fuzzy environment with application to personnel evaluation. Knowledge-Based Systems, 2013, 52, 1- 10. https://doi.org/10.1016/j.knosys.2013.04.010

10	Zhang N, Wei G. Extension of VIKOR method for decision making problem based on hesitant fuzzy set. Applied Mathematical Modelling, 2013, 37(7): 4938- 4947. https://doi.org/10.1016/j.apm.2012.10.002

11	Liu H, Rodriguez R M. A fuzzy envelope for hesitant fuzzy linguistic term set and its application to multicriteria decision making. Information Sciences, 2014, 258, 220- 238. https://doi.org/10.1016/j.ins.2013.07.027

12	Qian G, Wang H, Feng X. Generalized hesitant fuzzy sets and their application in decision support system. Knowledge-Based Systems, 2013, 37, 357- 365. https://doi.org/10.1016/j.knosys.2012.08.019

13	Zhang X, Xu Z. Hesitant fuzzy QUALIFLEX approach with a signed distance-based comparison method for multiple criteria decision analysis. Expert Systems with Applications, 2015, 42(2): 873- 884. https://doi.org/10.1016/j.eswa.2014.08.056 Cited in this article [1]

14	Xia M, Xu Z. Hesitant fuzzy information aggregation in decision making. International Journal of Approximate Reasoning, 2011, 52(3): 395- 407. https://doi.org/10.1016/j.ijar.2010.09.002 Cited in this article [4]

15	Xu Z, Xia M. Distance and similarity measures for hesitant fuzzy sets. Information Sciences, 2011, 181(11): 2128- 2138. https://doi.org/10.1016/j.ins.2011.01.028 Cited in this article [1]

16	Bonferroni C. Sulle medie multiple di potenze. Bollettino dell'Unione Matematica Italiana, 1950, 5(3-4): 267- 270. Cited in this article [1]

17	Xia M, Xu Z, Zhu B. Geometric Bonferroni means with their application in multi-criteria decision making. Knowledge-Based Systems, 2013, 40, 88- 100. https://doi.org/10.1016/j.knosys.2012.11.013 Cited in this article [1]

18	Zhu B, Xu Z, Xia M. Hesitant fuzzy geometric Bonferroni means. Information Sciences, 2012, 205, 72- 85. https://doi.org/10.1016/j.ins.2012.01.048 Cited in this article [1]

19	Yager R R. Prioritized aggregation operators. International Journal of Approximate Reasoning, 2008, 48(1): 263- 274. https://doi.org/10.1016/j.ijar.2007.08.009 Cited in this article [1]

20	Yager R R. Prioritized OWA aggregation. Fuzzy Optimization and Decision Making, 2009, 8(3): 245- 262. https://doi.org/10.1007/s10700-009-9063-4 Cited in this article [1]

21	Wei G. Hesitant fuzzy prioritized operators and their application to multiple attribute decision making. Knowledge-Based Systems, 2012, 31, 176- 182. https://doi.org/10.1016/j.knosys.2012.03.011 Cited in this article [1]

22	Wei G, Zhao X, Wang H, et al. Hesitant fuzzy Choquet integral aggregation operators and their applications to multiple attribute decision making. Information — An International Interdisciplinary Journal, 2012, 15(2): 441- 448. Cited in this article [1]

23	Yager R R. The power average operator. IEEE Transactions on Systems, Man, and Cybernetics, Part A: Systems and Humans, 2001, 31(6): 724- 731. https://doi.org/10.1109/3468.983429 Cited in this article [1]

24	Zhang Z. Hesitant fuzzy power aggregation operators and their application to multiple attribute group decision making. Information Sciences, 2013, 234, 150- 181. https://doi.org/10.1016/j.ins.2013.01.002 Cited in this article [1]

25	Kaufmann A, Gupta M M, Kaufmann A. Introduction to fuzzy arithmetic: Theory and applications. Van Nostrand Reinhold Company New York, 1985. Cited in this article [2]

26	Deschrijver G, Cornelis C, Kerre E E. On the representation of intuitionistic fuzzy t-norms and t-conorms. IEEE Transactions on Fuzzy Systems, 2004, 12(1): 45- 61. https://doi.org/10.1109/TFUZZ.2003.822678 Cited in this article [1]

27	Roychowdhury S, Wang B H. On generalized Hamacher families of triangular operators. International Journal of Approximate Reasoning, 1998, 19(3-4): 419- 439. https://doi.org/10.1016/S0888-613X(98)10018-X Cited in this article [1]

28	Deschrijver G, Kerre E E. A generalization of operators on intuitionistic fuzzy sets using triangular norms and conorms. Notes on IFS, 2002, 8(1): 19- 27. Cited in this article [1]

29	Hamacher H. Uber logische verknunpfungenn unssharfer Aussagen und deren Zugenhorige Bewertungsfunktione Trappl. Progress in Cybernatics and Systems Research, 1978, 3, 276- 288. Cited in this article [2]

30	Wang W, Liu X. Intuitionistic fuzzy geometric aggregation operators based on einstein operations. International Journal of Intelligent Systems, 2011, 26(11): 1049- 1075. https://doi.org/10.1002/int.20498 Cited in this article [1]

31	Liang X C, Chen S F. Multiple attribute decision making method based on trapezoid fuzzy linguistic variables. Journal of Southeast University (English Edition), 2008, 24(4): 478- 481. https://doi.org/10.3969/j.issn.1003-7985.2008.04.016 Cited in this article [1]

32	Sugeno M. Theory of fuzzy integrals and its applications. Tokyo Institute of Technology, 1974. Cited in this article [2]

Meng F, Zhang Q, Cheng H. Approaches to multiple-criteria group decision making based on interval-valued intuitionistic fuzzy Choquet integral with respect to the generalized

λ

-Shapley index. Knowledge-Based Systems, 2013, 37, 237- 249.

https://doi.org/10.1016/j.knosys.2012.08.007

Cited in this article [1]

34	Xu Z. Choquet integrals of weighted intuitionistic fuzzy information. Information Sciences, 2010, 180(5): 726- 736. https://doi.org/10.1016/j.ins.2009.11.011

35	Zhang S, Yu D. Some geometric Choquet aggregation operators using Einstein operations under intuitionistic fuzzy environment. Journal of Intelligent & Fuzzy Systems, 2014, 26(1): 491- 500.

36	Hu Y. Chebyshev type inequalities for general fuzzy integrals. Information Sciences, 2014, 278, 822- 825. https://doi.org/10.1016/j.ins.2014.03.095 Cited in this article [2]

37	Liou J J H, Chuang Y C, Tzeng G H. A fuzzy integral-based model for supplier evaluation and improvement. Information Sciences, 2014, 266, 199- 217. https://doi.org/10.1016/j.ins.2013.09.025

38	Bustince H, Galar M, Bedregal B, et al. A new approach to interval-valued Choquet integrals and the problem of ordering in interval-valued fuzzy set applications. IEEE Transactions on Fuzzy Systems, 2013, 21(6): 1150- 1162. https://doi.org/10.1109/TFUZZ.2013.2265090

39	Jang L C. A note on the interval-valued generalized fuzzy integral by means of an interval-representable pseudo-multiplication and their convergence properties. Fuzzy Sets and Systems, 2013, 222, 45- 57. https://doi.org/10.1016/j.fss.2012.11.016

40	Ko Y C, Fujita H, Tzeng G H. A fuzzy integral fusion approach in analyzing competitiveness patterns from WCY2010. Knowledge-Based Systems, 2013, 49, 1- 9. https://doi.org/10.1016/j.knosys.2013.04.001 Cited in this article [1]

41	Grabisch M. Fuzzy integral in multicriteria decision-making. Fuzzy Sets and Systems, 1985, 69(3): 279- 298. Cited in this article [2]

42	Tan C, Wu D D, Ma B. Group decision making with linguistic preference relations with application to supplier selection. Expert Systems with Applications, 2011, 38(12): 14382- 14389. https://doi.org/10.1016/j.eswa.2011.04.036 Cited in this article [1]

Funding

the Science and Technology Research Project of Chongqing Municipal Education Commission(KJQN201901505)

the Key Project of Humanities and Social Sciences Research of Chongqing Education Commission in 2019(19SKGH181)

PDF(265 KB)

180

Accesses

Citation

Detail

Sections

Recommended

Received	Accepted	Published
2019-12-20	2020-05-20	2020-12-25
Issue Date
2020-12-25

Please choose a citation manager

Content to export

Abstract

Key words

Cite this article

1 Introduction

2 Preliminaries

2.1 Hesitant Fuzzy Set

2.2 Hamacher Operations

3 Hesitant Trapezoid Fuzzy Set (HTrFS)

3.1 Trapezoid Fuzzy Numbers

3.2 Hesitant Trapezoid Fuzzy Set

4 Some Aggregating Operators with Hesitant Trapezoid Fuzzy Information

4.1 Hesitant Trapezoid Fuzzy Aggregation Operators

4.2 Hesitant Trapezoid Fuzzy Hamacher Aggregation Operators

4.3 Hesitant Trapezoid Fuzzy Hamacher Choquet Aggregation Operators

5 An Approach to Multiple Attribute Decision Making with Hesitant Trapezoid Fuzzy Information

6 Illustrative Example

Table 1 Hesitant trapezoid fuzzy decision matrix

Table 2 Ranking results based on the different operators

7 Conclusion

{{custom_sec.title}}

{{custom_sec.title}}

References

{{custom_fnGroup.title_en}}

Footnotes

Funding

Share

模态框（Modal）标题

Please choose a citation manager

Content to export

Abstract

Key words

Cite this article

1 Introduction

2 Preliminaries

2.1 Hesitant Fuzzy Set

2.2 Hamacher Operations

3 Hesitant Trapezoid Fuzzy Set (HTrFS)

3.1 Trapezoid Fuzzy Numbers

3.2 Hesitant Trapezoid Fuzzy Set

4 Some Aggregating Operators with Hesitant Trapezoid Fuzzy Information

4.1 Hesitant Trapezoid Fuzzy Aggregation Operators

4.2 Hesitant Trapezoid Fuzzy Hamacher Aggregation Operators

4.3 Hesitant Trapezoid Fuzzy Hamacher Choquet Aggregation Operators

5 An Approach to Multiple Attribute Decision Making with Hesitant Trapezoid Fuzzy Information

6 Illustrative Example

Table 1 Hesitant trapezoid fuzzy decision matrix

Table 2 Ranking results based on the different operators

7 Conclusion

{{custom_sec.title}}

{{custom_sec.title}}

References

{{custom_fnGroup.title_en}}

Footnotes

Funding