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区间值q-Rung模糊Choquet积分算子及其在群决策中的应用_Interval-valued q-Rung Orthopai
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区间值q-Rung模糊Choquet积分算子及其在群决策中的应用_Interval-valued q-Rung Orthopair Fuzzy Choquet Integral Operators and Its Application in Group Decision Making.pdf
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Interval-valued q-Rung Orthopair Fuzzy Choquet Integral Operators
and Its Application in Group Decision Making
Benting Wan
1,
*, Juelin Huang
2
and Xi Chen
3
1
School of software and Internet of things Engineering, Jiangxi University of Finance and Economics, Nanchang 330013,
China; wanbenting@jxufe.edu.cn
2
School of software and Internet of things Engineering, Jiangxi University of Finance and Economics, Nanchang 330013,
China; 2202021682@stu.jxufe.edu.cn
3
School of software and Internet of things Engineering, Jiangxi University of Finance and Economics, Nanchang 330013,
China; 517060730@qq.com.
* Correspondence: e-mail: wanbenting@jxufe.edu.cn
Abstract: It is more flexible for decision makers to evaluate by interval-valued q-rung orthopair fuzzy set
(IVq-ROFS), which offers fuzzy decision-making more applicational space. Meanwhile, Choquet integral uses
non-additive set function (fuzzy measure) to describe the interaction between attributes directly. In particular,
there are a large number of practical issues that have relevance between attributes. Therefore, this paper pro-
poses the correlation operator and group decision-making method based on the interval-valued q-rung or-
thopair fuzzy set Choquet integral. First, interval-valued q-rung orthopair fuzzy Choquet integral average op-
erator (IVq-ROFCA) and interval-valued q-rung orthopair fuzzy Choquet integral geometric operator (IVq-
ROFCG) are investigated, and their basic properties are proved. Furthermore, several operators based on IVq-
ROFCA and IVq-ROFCG are developed. Then, a group decision-making method based on IVq-ROFCA is
developed, which can solve the decision-making problems with interaction between attributes. Finally,
through the implementation of the warning management system for hypertension, it is shown that the operator
and group decision-making method proposed in this paper can handle complex decision-making cases in re-
ality, and the decision result is consistent with the doctor’s diagnosis result. Moreover, the comparison with
the results of other operators shows that the proposed operators and group decision-making method are correct
and effective, and the decision result will not be affected by the change of q value.
Keywords: Choquet integral; Interval-valued q-rung orthopair fuzzy set; group decision-making
1. Introduction
The fuzzy set (FS) theory and method proposed by Zadeh [1] have been widely used in real
life, such as medical treatment, manufacturing, education, etc. With the increase of people’s aware-
ness of complex issues and uncertainty issues, the research on fuzzy theories and methods has re-
ceived great attention from researchers [2-6]. Since decision-makers in real life need to deal with
the possibilities of support, opposition, and neutrality, Atanassov further proposed the intuitionistic
fuzzy set (IFS) [7], that is, the sum of membership degree (u) and non-membership degree (v) of
each ordered pair is less than or equal to one: . In response to the sum of membership
degree and non-membership degree of each ordered pair being greater than one, Yager proposed
the Pythagorean fuzzy set (PFS) [8-9], the sum of the square of membership and membership is
less than or equal to one:
, which covers a wider range compared with intuitionistic
fuzzy. To deal with much more complicated problems, Yager proposed the situation that the degree
of membership and non-membership of the q power is less than or equal to one in 2016:
[10]. It not only combines IFS and PFS, but also expands the fuzzy set to a wider range
of applications based on the different values of q. Researchers have put forward numerous excellent
results in recent years: Yager [11-12] and Alajlan studied some properties of q-rung orthopair fuzzy
set (q-ROFS) [13]; Liu and Wang proposed q-rung orthopair fuzzy numbers (q-ROFNs) and their
properties, and developed q-rung orthopair fuzzy weighted average operators and Bonferroni mean
operator, which was used to deal with multi-attribute group decision-making(MAGDM) problems
[14-15]; Xing et al. developed a q-rung orthopair fuzzy point weighted aggregation operator and
applied it to multi-attribute decision-making [16]; Garg solved the MAGDM issues through the
investigated trigonometric operation and connection number based on q-ROFS[17-18]; Hussain et
al. also published many research results based on q-ROFNs [19].
However, the attributes are not independent of each other in decision-making, there are more
mutual influences and correlations that can be appropriately solved by Choquet integral [20]. The
Choquet integral based on fuzzy measures comes decision-making problems related to attributes in
handy [21-23], which is had conducted in-depth research: Tan defined the intuitionistic fuzzy Cho-
quet integral average operator and the interval-valued integral average operator [24-26]; Xu inves-
tigated the intuitionistic fuzzy Choquet integral average operator and geometric operator [27]. In-
spired by Tan and Xu, many scholars expanded the intuitionistic fuzzy Choquet integral operator.
Besides, Xu proposed intuitionistic fuzzy weighted average and weighted geometric operators [28-
29], Zhao et al. proposed the q-rung intuitionistic fuzzy operators [30], and Tan presented the q-
rung intuitionistic fuzzy geometric operators [31], which provide the theoretical basis for the sub-
sequent study of q-ROFSs. Many experts and scholars have conducted a lot of studies on fuzzy
integral (including Choquet integral) group decision-making methods [32-33], and found that the
group decision-making methods can effectively process decision data of multiple attributes and
multiple experts. Xing et al. [34] studied Choquet integral based on q-rung orthopair and proposed
related MAGDM methods; Keikha et al. [35] combined the Choquet integral and TOPSIS method
to solve the problem; Teng et al. [36] developed the generalized Shapley probabilistic linguistic
Choquet average (GS-PLCA) operator and investigated a method that can deal with large group
decision-making (LGDM) issues. Then, extending the Choquet integral and combining it with the
group decision-making method for in-depth research is of practical significance for solving many
complex and uncertain problems in real decision-making.
Inspired by Choquet integral operator and q-ROFS, this paper analyses the MAGDM problem
based on IVq-ROFS, proposes IVq-ROFCA, IVq-ROFCG, then proves their properties and devel-
ops some weighted operators based them. Combined with the group decision-making method pro-
posed in this paper, the case of warning management system for hypertension is verified and com-
pared. The research results of this paper are as follows.
(1) Based on the Choquet integral and q-ROFNs, this paper proposes IVq-ROFCA, IVq-
ROFCG, including some properties and extended operators of them. In addition, these operators
are verified, and idempotency, boundedness, commutativity and monotonicity of them are proved;
(2) A group decision-making method is proposed on the basis of IVq-ROFCA. In this method,
two aggregations of experts and attributes are carried out by IVq-ROFCA respectively, which col-
lects IVq-ROFNs s corresponding to each alternative. The ranking result of alternatives is obtained
via the score function and exact function of IVq-ROFNs.
(3) We also apply the proposed method into the case of warning management system for hy-
pertension, and the decision result correctly satisfies the diagnosis of doctor. It is not only found
that the decision results are discussed with methods proposed by other papers, but when q takes the
values 2, 3, 4, 5, same decision results can be also derived.
The organization of this paper is as follows: Section 2 reviews the concept of IVq-ROFS and
Choquet integral operator; Section 3 proposes the IVq-ROFCA, IVq-ROFCG; Moreover, Section
4 presents the group decision-making method based on given operators; Section 5 applies solves
practical cases by proposed group decision-making method and provides comparative analysis.
Section 6 concludes this paper.
2. Preliminaries
In this section, we make a brief review of IVq-ROFS and Choquet integral operator.
Definition 1 [11]. Let
be a fixed set,
is a q-
ROFS, where
,
, satisfy Equation (1):
(1)
Where, ,and for all
,
are the degree of membership;
is the degree
of non-membership, and the degree of hesitation
can be expressed as Equation (2):
(
(2)
Definition 2 [37]. Given a fixed Set
, IVq-ROFS on is defined as shown
in Equation (3):
(3)
The membership is represented by interval values, satisfies:
,
the non-membership satisfies:
, and also satisfied:
. The degree of hesitation of is shown as Equation (4).
(4)
In particular, when , IVq-ROFS reduces to interval-valued intuitionistic fuzzy set
(IVIFS).
Definition 3 [38]. Let
and
are interval val-
ued q-rung orthopair numbers (IVq-ROFNs), ,Then the Equation (5), (6), (7) and (8) are
established.
(5)
(6)
(7)
(8)
Definition 4 [38]. For any IVq-ROFNs
, the score function is shown as
Equation (9).
(9)
Definition 5 [38]. For any IVq-ROFNs
, the exact function is shown as
Equation (10).
(10)
Definition 6 [38]. For any two IVq-ROFNs
and
, the comparison law is as follows:
If
, then
;
If
, then
;
If
:
if
, then
;
if
, then
;
if
, then
.
Definition 7 [20]. Let
be a universe of discourse, be a positive real-valued
function and be the fuzzy measure on . Then the discrete Choquet integral of on fuzzy meas-
ure is defined as Equation (11).
(11)
Where
is a permutation of
that satisfies
,
.
3. Interval-valued q-rung orthopair fuzzy Choquet integral operators
In this section, we investigate IVq-ROFCA and IVq-ROFCG operators, then extend their
weighted operators and some properties.
3.1. IVq-ROFCA
Definition 8. Let be the fuzzy measure on nonempty finite set
(
),
and
,, , be IVq-ROFNs; then
the IVq-ROFCA can be defined as Equation (12).
(12)
Where
represements the Choquet integral,
is a permuta-
tion of
that satisfies
,
Theorem 1. If there are fuzzy measure (
) and IVq-ROFNs
,, ,. Then the IVq-ROFCA can be derived
as Equation (13):
,
(13)
Proof of Theorem 1: We prove Equation (13) by mathematical induction.
If n=2,
IVq-ROFCA
If n=k,
IVq-ROFCA
If n=k+1, the results of IVq-ROFCA are as follows:
IVq-ROFCA
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