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Symmetry 2018, 10, 383; doi:10.3390/sym10090383 www.mdpi.com/journal/symmetry
Article
Some Partitioned Maclaurin Symmetric Mean Based
on q-Rung Orthopair Fuzzy Information for Dealing
with Multi-Attribute Group Decision Making
Kaiyuan Bai 1, Xiaomin Zhu 1, Jun Wang 2 and Runtong Zhang 2,*
1 School of Mechanical, Electronic and Control Engineering, Beijing Jiaotong University,
Beijing 100044, China; kaiyuanbai@bjtu.edu.cn (K.B.); xmzhu@bjtu.edu.cn (X.Z.)
2 School of Economics and Management, Beijing Jiaotong University, Beijing 100044, China;
14113149@bjtu.edu.cn
* Correspondence: rtzhang@bjtu.edu.cn; Tel.: +86-010-5168-3854
Received: 20 August 2018; Accepted: 3 September 2018; Published: 5 September 2018
Abstract: In respect to the multi-attribute group decision making (MAGDM) problems in which the
evaluated value of each attribute is in the form of q-rung orthopair fuzzy numbers (q-ROFNs), a
new approach of MAGDM is developed. Firstly, a new aggregation operator, called the partitioned
Maclaurin symmetric mean (PMSM) operator, is proposed to deal with the situations where the
attributes are partitioned into different parts and there are interrelationships among multiple
attributes in same part whereas the attributes in different parts are not related. Some desirable
properties of PMSM are investigated. Then, in order to aggregate the q-rung orthopair fuzzy
information, the PMSM is extended to q-rung orthopair fuzzy sets (q-ROFSs) and two q-rung
orthopair fuzzy partitioned Maclaurin symmetric mean (q-ROFPMSM) operators are developed.
To eliminate the negative influence of unreasonable evaluation values of attributes on aggregated
result, we further propose two q-rung orthopair fuzzy power partitioned Maclaurin symmetric
mean (q-ROFPPMSM) operators, which combine the PMSM with the power average (PA) operator
within q-ROFSs. Finally, a numerical instance is provided to illustrate the proposed approach and a
comparative analysis is conducted to demonstrate the advantage of the proposed approach.
Keywords: partitioned Maclaurin symmetric mean; q-rung orthopair fuzzy set; q-rung orthopair
fuzzy partitioned Maclaurin symmetric mean; q-rung orthopair fuzzy power partitioned
Maclaurin symmetric; multi-attribute group decision making
1. Introduction
Multi-attribute group decision making (MAGDM) is one of the most important branches of
modern decision making theory. Generally speaking, MAGDM is an activity in which alternatives
are evaluated by a group of decision makers and the most suitable alternative is determined
accordingly. In MAGDM, one of critical problems is how to represent the information of attributes
given by decision makers, due to the appearance of fuzzy and uncertainty information. The other
critical problem is how to aggregate the attribute information and provide the ranking of
alternatives. For this problem, the aggregation operator is regarded as an effective tool to aggregate
decision information. A large number of studies on aggregation operator have been done and many
aggregation operators have been widely applied in MAGDM, such as the power average (PA)
operator [1], the Bonferroni mean (BM) operator [2], the Maclaurin symmetric mean (MSM)
operator [3], partitioned Bonferroni mean (PBM) operator [4], and so on. (A review of related
literature is listed in Section 2)
Symmetry 2018, 10, 383 2 of 32
The aforementioned aggregation approaches are used to capture various interrelationships of
attributes in MAGDM, but they ignore this situation in which the attributes are divided into several
parts and there are interrelationships among multiple attributes in each part. Thus, in this paper,
we extend the traditional Maclaurin symmetric mean (MSM) [3] and propose the partitioned
Maclaurin symmetric mean (PMSM) operator, which can model this circumstance in which
attributes are divided into several parts and multiple attributes in each part are interrelated. In
addition, as the complexity of MAGDM problems increase, we may encounter the following case:
the decision maker maybe evaluate the attributes in form of q-rung orthopair fuzzy number
(q-ROFN) and provide some unduly high or unduly low assessments owing to time shortage and a
lack of priori experience. These unreasonable assessments may negatively affect the finally decision
results.
In order to solve the above issues, we utilize PMSM to aggregate q-ROFNs. Meanwhile, we
combine the PMSM with PA in q-rung orthopair fuzzy set (q-ROFS) and propose q-rung orthopair
fuzzy power partitioned Maclaurin symmetric mean (q-ROFPPMSM) operator and the weighted
form of the q-ROFPPMSM operator. The q-ROFPPMSM not only reduces the negative influence of
unreasonable evaluations on the aggregating result, but also deals with this circumstance where
attributes are divided into several parts and multiple attributes in each part are interrelated.
We firstly define the PMSM operator and provide the mathematical formula. Some desirable
properties and special cases of PMSM are also investigated. It can be found that some existing
operators can be obtained from PMSM when the parameters of PMSM are assigned different values.
Further, we extend the PMSM in q-ROFS, and propose q-rung orthopair fuzzy partitioned
Maclaurin symmetric mean (q-ROFPMSM) operator and q-rung orthopair fuzzy weighted
partitioned Maclaurin symmetric mean (q-ROFWPMSM) operator to deal with q-rung orthopair
fuzzy information. In order to reduce the negative influence of unreasonable assessments on
decision result, we take advantage of PMSM and PA and propose q-ROFPPMSM and the weighted
form of q-ROFPPMSM, which is called q-rung orthopair fuzzy weighted power partitioned
Maclaurin symmetric mean (q-ROFWPPMSM) operator. Finally, a new approach based on the
q-ROFWPPMSM operator is introduced for solving the q-rung orthopair fuzzy MAGDM problems.
A numerical instance is also provided to illustrate the approach we proposed and a comparative
analysis is conducted to demonstrate the advantage of the proposed approach. The contributions of
this paper are as follows:
(1) We propose the PMSM operator, which can handle this situation where the attributes are
divided into several parts and there are interrelationships among multiple attributes in each
part.
(2) We extend the PMSM in q-ROFS for dealing with the q-rung orthopair fuzzy information.
(3) We combine PMSM and PA in q-ROFS and introduce the q-ROFPPMSM and the weighted
form of q-ROFPPMSM which not only take advantage of PMSM, but also reduce the negative
influence of unreasonable arguments on the aggregating result.
(4) We propose a new approach of MAGDM based on the proposed operator.
The rest of this paper is organized as follows: Section 2 provides a review of related literature.
Section 3 introduces some basic concepts. In Section 4, we define the PMSM, the q-ROFPMSM and
the q-ROFWPMSM. Meanwhile, we propose q-ROFPPMSM and q-ROFWPPMSM based on the PA
and PMSM operators. A new approach of q-rung orthopair fuzzy MAGDM based on
q-ROFWPPMSM is introduced in Section 5. Section 6 gives a numerical example to illustrate the
validity and advantages of the proposed approach and the last section summarizes the paper.
2. Literature Review
The application of fuzzy set theory in MAGDM and the application of aggregation operators in
MAGDM have been widely studied by researchers. In our review, we mainly focus on the literature
related to the q-rung orthopair fuzzy set (q-ROFS). In addition, we also concentrate on some
aggregation operators that are widely applied in fuzzy MAGDM problems. Due to the increasing
Symmetry 2018, 10, 383 3 of 32
complexity of real decision making problems, crisp numbers are insufficient and inadequate to
represent attribute values. Zadeh’s [5] fuzzy set (FS) theory is regarded to an effectively tool to deal
with impreciseness, and many works on MAGDM with fuzzy information has been done [6–8]. To
overcome the shortcomings of FS, Atanassov [9] proposed the concept of intuitionistic fuzzy set
(IFS), which has a membership degree and a non-membership degree simultaneously. Owing to its
great ability for handling fuzziness and uncertainty, IFSs have been widely applied in pattern
recognition [10,11], medical diagnosis [12,13], clustering analysis [14,15] and especially MAGDM
[16–18]. The constraint of IFS is that the sum of membership and non-membership degrees should
be less than or equal to one. Thus, Yager [19] generalized the IFS and proposed the Pythagorean
fuzzy set (PFS), whose constraint is that the square sum of membership and non-membership
degrees is less than or equal to one. Since its appearance, PFS has received much scholarly attention,
which has led to a wider range of applications [20–26].
More recently, Yager [27] introduced a new concept: the q-rung orthopair fuzzy set (q-ROFS),
which satisfies the condition that the sum of the qth power of the membership degree and the qth
power of the non-membership degree is bounded by one. This feature makes q-ROFS more powerful
than IFS and PFS in the aspect of dealing with the vagueness and fuzzy information. For instance,
when a decision maker provides 0.8 and 0.7 as the membership and non-membership degrees,
respectively, then the ordered pair (0.7, 0.8) is not valid for IFSs or PFSs, whereas it is valid for
q-ROFS. Many works on q-ROFS have been done to handle q-rung orthopair information. Peng [28]
defined new exponential operational laws of q-ROFNs in which the bases are positive real numbers
and the exponents are q-ROFNs and proposed a new score function for comparing two q-ROFNs.
Du [29] defined some Minkowski-type distance measures for q-ROFS and investigated the
application of the distance measure in decision making. Li et al. [30] combined the q-ROFS with a
picture fuzzy set and proposed a q-rung picture linguistic set. Liu and Wang [31] proposed a family
of simple weighted averaging and geometric operators for solving the q-rung orthopair fuzzy
MAGDM problems. Liu and Liu [32] and Wei et al. [33] respectively proposed some q-rung
orthopair fuzzy Bonferroni mean operators and some q-rung orthopair fuzzy Heronian mean
operators, which consider the interrelationship between any two q-ROFNs. Liu and Wang [34]
proposed some q-rung orthopair fuzzy Archimedean Bonferroni mean (q-ROFABM) operators,
which applied Bonferroni mean (BM) in the q-ROFS based on Archimedean T-norm and T-conorm.
Obviously, aggregation operators play an important role in MAGDM, especially the ones that
reflect the interrelationship among attributes. According to the type of relationship between
attributes, the aggregation operator can be divided into two groups. The one assumes each attribute
is related to the other attributes, such as the power average (PA) operator [1] and power geometric
(PG) operator [35], which allows the attributes to be aggregated to support and reinforce each other.
However, the PA and PG only capture the relationship by assigning the weight to each attribute
and they do not directly reflect the interrelationship structure among the attributes.
Thus, Yager [36] originally extended the BM [2] to capture the interrelationship between any
two attributes. Xia et al. [37] generalized the classical BM and proposed the generalized weighted BM
(GWBM) where the interrelationship among any three arguments can be measured. Zhang et al. [38]
also defined the dual generalized weighted BM (DGWBM) operator. To capture the interrelationship
among multiple attributes, Detemple and Robertson [39] explored the MSM [3] operator in MAGDM,
which assumes that each argument is related to other k-1 arguments and the parameter k can be
adjusted by decision maker. Owning to this flexibility of MSM, it has been used to deal with various
MAGDM problems [40–42].
The aforementioned operators are based on the assumption that each attribute is related with
the others in MAGDM. However, interrelationships do not usually exist among all attributes. Thus,
the second group operator mainly focuses on the circumstances in which parts of attributes are
related and others do not have any interrelationship. For such operators, the partitioned Bonferroni
mean (PBM) operator [4] is the representative. The PBM considers this situation where the
arguments are partitioned into several parts, and the argument in the same part is related to the
others. Similarly, Liu et al. [43] extended the Heronian mean (HM) to the partitioned Heronian
Symmetry 2018, 10, 383 4 of 32
mean (PHM). The PBM and PHM operators have been extensively applied in the process of
decision making [44,45]. Table 1 summarizes main characteristics of above aggregation operators.
Table 1. The main characteristics of different aggregation operators.
Approaches
Captures
Two Attributes
Captures
Three Attributes
Captures
Multiple Attributes
Attributes Partitions
PA [1]
Yes
No
No
No
PG [35]
Yes
No
No
No
BM [2]
Yes
No
No
No
GWBM [37]
Yes
Yes
No
No
DGWBM [38]
Yes
Yes
Yes
No
MSM [3]
Yes
Yes
Yes
No
PBM [4]
Yes
No
No
Yes
PHM [43]
Yes
No
No
Yes
It is worthy to point that as PBM and PHM inherit the features of BM and HM respectively,
and they fail to capture the interrelationship among multiple arguments. That motivates us to
propose PMSM operator and extend it in q-ROFS to deal with heterogeneous among attributes and
capture the interrelationship among multiple attributes in that same partition. In addition, we take
advantage of PMSM and PA and propose q-ROFPPMSM and q-ROFWPPMSM. Finally, a new
approach based on q-ROFWPPMSM operator is introduced for solving the q-rung orthopair fuzzy
MAGDM problems.
3. Preliminaries
3.1. q-ROFS
Definition 1 [27]. Let X be a universe of discourse, a q-rung orthopair fuzzy set (q-ROFS) A defined on X is
given by
( ) ( )
,,
AA
A x u x v x x X=
(1)
where
( ) [0,1]
Ax
and
( ) [0,1]
A
vx
respectively represent the membership and non-membership degrees
of the element x to the set A satisfying
( ) ( ) 1
qq
AA
u x v x+
,
( 1)q
. The indeterminacy degree of the element
x to the set A is
( )
( )
( )
1
1 ( ) ( ) q
qq
A A A
x x v x
= − +
. For convenience, Liu and Wang [31] called the pair
( )
( ), ( )
AA
u x v x
as a q-rung orthopair fuzzy number (q-ROFN), which can be denoted by
( )
,
AA
A u v=
.
Definition 2 [31]. Let
( )
1 1 1
,a u v=
and
( )
2 2 2
,a u v=
be two q-ROFNs, and
be a positive real number,
then the operational laws of the q-ROFN are defined as follows:
1.
( )
1
1 2 1 2 1 2 1 2
,
q
q q q q
a a u u u u v v
= + −
,
2.
( )
1
1 2 1 2 1 2 1 2
,q
q q q q
a a u u v v v v
= + −
,
3.
( )
1
1 1 1
1 1 ,
q
q
a u v
= − −
,
4.
( )
1
1 1 1
, 1 1
q
q
a u v
= − −
.
Symmetry 2018, 10, 383 5 of 32
Definition 3 [31]. Let
( )
,
aa
av
=
be a q-ROFN, then the score function of
a
is defined as
( )
qq
aa
Sa
=−
and the accuracy function is defined as
( )
qq
aa
Ha
=+
. For any two q-ROFNs
1 1 1
( , )av
=
and
2 2 2
( , )av
=
, then
1. If
( ) ( )
12
S a S a
, then
12
aa
;
2. If
( ) ( )
12
S a S a=
, then
(1) If
( ) ( )
12
H a H a
, then
12
aa
;
(2) If
( ) ( )
12
H a H a=
, then
12
aa=
.
Distance measure, as an effective tool to comparing the fuzzy information, has been widely
used in decision making. Recently, a distance measure for q-ROFNs called as the Minkowski-type
distance measure was proposed by Du [29] for evaluating the fuzzy degree. The definition is
presented as follows:
Definition 4 [29]. Let
1 1 1
( , )av
=
and
2 2 2
( , )av
=
be any two q-ROFNs, then the Minkowski-type
distance between
1
a
and
2
a
is given by
( )
1
1 2 1 2 1 2
11
,22
p
pp
d a a v v
= − + −
( 1)p
(2)
Example 1. Assume that
( )
10.8,0.3a=
,
( )
20.6,0.4a=
be two q-ROFNs and the parameter p is equal to
three. Based on the Definition 4, we can obtain the Minkowski-type distance measure
( ) ( )
( )
13
33
11
0.8, 0.3 , 0.6, 0.4 0.8 0.6 0.3 0.4 0.1651
22
d
= − + − =
3.2. PA Operator and MSM Operator
The power average (PA), introduced by Yager [1], can assign lower weights for arguments by
calculating the support degree between arguments so that they can reduce the bad influence of the
unduly high or unduly low arguments on the aggregation result. The original form of PA is
presented as follows:
Definition 5 [1]. Let
( 1,2,..., )
i
a i n=
be a collection of non-negative real numbers, if
( )
( )
( )
( )
()
12 1
1
( , ,..., ) 1 1
nn
n i i j
j
i
PA a a a T a a T a
=
=
= + +
(3)
then the PA is called the power average operator, where
( )
1, ( , )
n
i i j
j j i
T a Sup a a
=
=
(4)
and the
( , )Sup a b
is denoted as the support degree for a from b, which satisfies following properties:
1.
( , ) [0, 1]Sup a b
;
2.
( , ) ( , )Sup a b Sup b a=
;
3.
( , ) ( , )Sup a b Sup x y
, if
| | | |a b x y− −
The Maclaurin symmetric mean (MSM) is firstly proposed by Maclaurin [3] and developed by
Detemple and Robertson [39]. It can depict the interrelationship among any arguments by setting
different values for parameter k. The mathematical form is defined as follows:
Symmetry 2018, 10, 383 6 of 32
Definition 6 [39]. Let
i
a
( 1, 2,..., )in=
be a collection of non-negative real numbers and
1,2,...,kn=
, if
1
()
12
11
( , ,..., ) j
ik
k
k
kk
n i n
i i n j
MSM a a a a C
=
=
(5)
where
12
( , ,..., )
k
i i i
traverses all the k-tuple combination of
(1,2,..., )n
and
! !( )!
k
n
C n k n k=−
is the
binomial coefficient. Then the
()k
MSM
is called the Maclaurin symmetric mean (MSM) operator.
4. Some q-Rung Orthopair Fuzzy Power Partitioned Maclaurin Symmetric Mean Operators
In this section, we firstly extend the traditional MSM and propose the PMSM operator to
handle this situation in which the input arguments are divided into several parts and there are
interrelationships among multiple arguments in each part. Then, we extend the PMSM in q-ROFS
and define two q-ROFPMSM operators to deal with the aggregation information in the form of
q-ROFNs. Finally, we introduce a q-ROFPPMSM operator and the weighted form of the
q-ROFPPMSM operator based on PMSM and PA, which not only take advantage of PMSM, but also
reduce the negative influence of unduly high or unduly low evaluating values of attributes on the
decision result.
4.1. PMSM Operator
In many practical MAGDM problems, we may encounter a situation where the input
arguments can be divided into several classes and there are interrelationships among multiple
arguments in each class, whereas the attributes in different classes are not related. These situations
can be mathematically depicted as follows:
Let
12
{ , ,..., }
n
T a a a=
be a collection of nonnegative real numbers that are corresponding to the
performance value of each attribute, respectively. On the basis of the aforementioned
interrelationship pattern, suppose that the arguments
i
a
( 1, 2,..., )in=
are divided into d different
classes
12
, ,..., d
P P P
, satisfying
ij
PP=
and
1
d
h
hPT
==
. Furthermore, suppose that there is an
interrelationship among any kh arguments in each class
h
P
( 1,2,..., )hd=
and there is no relationship
among arguments of classes Pi and Pj. Then the partitioned Maclaurin symmetric mean (PMSM)
operator, which can aggregate the input arguments with above relationship structure, is defined as
follows:
Definition 7. Let
i
a
( 1, 2,..., )in=
be a collection of nonnegative real numbers, which are divided into
different classes
12
, ,..., d
P P P
. For the parameter vector
12
, ,..., d
k k k
with
1,2,...,
hh
kP=
and the
h
P
being
the cardinality of
h
P
( 1,2,..., )hd=
, if
12
12
12
1
( , ,..., )
12
1 , , , 1
11
( , , ..., )
h
h
d
j
h
kh
h
h
kh
k
k
d
k k k
ni
k
h i i i P j
pi i i
PMSM a a a a
dC
=
=
=
(6)
then the
12
( , ,..., )
d
k k k
PMSM
is called the partitioned Maclaurin symmetric mean (PMSM) operator, where
12
( , ,..., )
h
k
i i i
traverses all the kh-tuple combination of
(1,2,..., )
h
P
and the
h
h
k
P
C
is the binomial coefficient
satisfying following formula:
( )
!
!!
h
h
h
k
P
h h h
P
Ck P k
=−
(7)
From Equation (6), we can know that the PMSM firstly models the interrelationship of
attributes belonged to class
h
P
( 1,2,..., )hd=
and provides the satisfaction degree of interrelated
Symmetry 2018, 10, 383 7 of 32
attributes of each class by the expression
12
12
,, 1
,1
(1 )
h
hh
hj
kh
kh
h
i i i P
i
k
kk
i
Pii j
Ca
=
, it is noted that the PMSM
can model this case where the relationship type of attributes belonged to class
i
P
and
j
P
are
different by setting different values for parameter
i
k
and
j
k
. Then, the
12
( , ,..., )
12
( , ,..., )
d
k k k
n
PMSM a a a
gives the average satisfaction degree of all attributes, which are
belonging to class
12
, ,..., d
P P P
. Therefore, the PMSM is a more reasonable method to solve this
situation, where the arguments are divided into several classes and there are interrelationships
among multiple arguments in each class.
For the sake of illustrating the calculation procedure of the PMSM operator, a numerical
example is provided and depicted as follows:
Example 2. Let
i
A
( 1,2,...,7)i=
represent a collection of attributes, which are divided into two classes
1 1 3 4 6
{ , , , }P A A A A=
and
2 2 5 7
{ , , }P A A A=
according to the attribute characteristic. Moreover, assume that
each attribute is interrelated to any other two attributes in class
1
P
and each attribute in class
2
P
is
interrelated to each other, that is to say, the parameter
13k=
and
22k=
. The actual value of arguments
i
a
=i( 1,2,...,7)
corresponding to the attributes is as follows: a1 = 0.4, a2 = 0.7, a3 = 0.5, a4 = 0.6, a5 = 0.3, a6 = 0.8
and a7 = 0.2.
On the basis of Definition 7, the aggregated result of the arguments in class P1 is given as
follows:
1
1
1
1 2 1 1 2 3 1
1
11 2 3
12 1
113
3
3
, , , , ,
11
4
11
jj
k
k
k
k
ii
ki i i P i i i P
jj
Pi i i
i i i
aa
C
C
==
=
( ) ( )
(
(
( ) ( )
)
)
13
3
4
1 0.4 0.5 0.6 0.4 0.5 0.8 0.4 0.6 0.8 0.5 0.6 0.8 0.5625C + + += =
Then, the aggregated result of arguments in class P2 is
2
2
2
1 2 2 1 2 2
2
212
12 2
112
2
2
, , , ,
11
3
11
jj
k
k
k
k
ii
ki i i P i i P
jj
Pii
i i i
aa
C
C
==
=
( ) ( ) ( )
( )
12
2
3
10.7 0.3 0.7 0.2 0.3 0.2 0.3697
C
= + + =
Finally, the degree of satisfaction over all arguments can be obtained
12
12
1
2
(3 2 )
1 2 7
1 , , , 1
11
( , ,..., ) 0.4661
2
h
h
j
h
kh
h
h
kh
k
k
i
k
h i i i P j
pi i i
PMSM a a a a
C
=
=
==
，
Meanwhile, the MSM operator is used to solve the aforementioned example and the
aggregated results under the condition of the parameter k taking two or three are obtained as
follows:
Symmetry 2018, 10, 383 8 of 32
12
12
2
17
1
2
1 2 7 2
7
( , , ..., ) 0.4933
j
i
ii j
a
MSM a a a C
=
==
,
1 2 3
13
3
17
1
3
1 2 7 3
7
( , , ..., ) 0.4863
j
i
i i i j
a
MSM a a a C
=
==
.
The calculation result obtained by the PMSM is different from the results of the MSM. This
difference is a result of the former partitioning the argument set into different classes and
considering various relationship types among the arguments in each class, whereas the later only
assumes that there is an interrelationship among any k arguments.
Some special cases with respect to the cardinality of class and the parameter vector of the
PMSM operator are investigated.
Remark 1. When all arguments belong to same class and the types of the interrelationship among arguments
are also the same, namely, the cardinality of
1
Pn=
and
11,2,...,k k n==
, then the PMSM reduces to the
MSM [3] operator as follows:
1
1
12
1
1
1 2 1
1
1
12 1
1
1
11
12
, , , 1
1
( , ,..., )
j
k
j
k
k
kk
k
i
ki i i n j
k
ni
kk
i i i P jn
Pi i i
a
PMSM a a a a C
C
=
=
==
(8)
Remark 2. In some practical decision making situations, the attributes can be divided into different classes
12
, ,..., d
P P P
and the type of relationship structure is consistent in each class
h
P
( 1,2,..., )hd=
, that is to say,
12 d
k k k k= = =
and
1,2,...,min{| |}
h
kP=
for
1,2,...,hd=
. Then Equation (6), can be modified as
follows:
( )
12
12
1
()
12
1 , , , 1
11
, ,..., j
kh
hk
k
k
d
k
ni
k
h i i i P j
Pi i i
PMSM a a a a
dC
=
=
=
(9)
Remark 3. It is noted that the PMSM can be reduced to a special case of the partitioned Bonferroni mean
operator [4], with the parameters s and t being equal to one, when the attributes can be divided into different
classes
12
, ,..., d
P P P
and there is an interrelationship between any two attributes in each class
h
P
( 1,2,..., )hd=
, that is to say,
12 2
d
k k k k= = = =
for
1,2,...,hd=
.
( )
(2 )
12
, , ..., n
PMSM a a a
( )
1 2 1 2
1 2 1 2
11
22
22
2
1 , 1 ,
11
1 1 1 2 1
2
1
jj
hh
h
dd
ii
h i i P h i i P
jj
Phh
i i i i
aa
dd
CPP
= =
==
= =
−
(10)
Symmetry 2018, 10, 383 9 of 32
( )
12
12
12
11
22
1 , 1
1 1 1 1 1
1
1h h h
dd
i i i j
h i i P h i P j P
hh
hh i i i j
a a a a
dd
PP
PP
= =
= =
−
−
(Let
1
ii=
and
2
ij=
)
1,1
12
( , ,..., )
n
PBM a a a=
Remark 4. In some practical decision making situations, it may happen that some attributes have no
relationship with any of the rest of the attributes, namely, they do not belong to any classes. In order to solve
this case, we can divide the attributes into two sets. Meanwhile, we put these attributes, which are not related
to any attributes in a single set denoted by
1
C
and put other attributes in another set denoted by
2
C
.
Assume that the attributes in
2
C
are divided based on a previous relationship structure. Equation (6) can be
modified as follows:
12
1 2 1
12
1
11
( , , , )
12
1 , , , 11
1 1 1
( , ,..., )
h
h
d
j
h
kh
h
h
kh
k
k
d
k k k
n i i
k
h i i i P i C
j
pi i i
n C C
PMSM a a a a a
n d n C
C
=
=
−
= +
(11)
In the following, some properties of PMSM operator are discussed as follows:
Theorem 1 (Idempotency). Let
i
a
( 1, 2,..., )in=
be a collection of nonnegative real numbers. For the
parameter vector
12
, ,..., d
k k k
with
1,2,...,
hh
kP=
and the
h
P
being the cardinality of
h
P
( 1,2,..., )hd=
,
if
12 n
a a a a= = = =
, then we can get
12
( , ,..., )
12
( , ,..., )
d
k k k
n
PMSM a a a a=
(12)
Proof. Based on the assumption that
i
a
are equal to a for all
1,2,...,in=
, then we can get
12
1 2 1 2
1 2 1 2
11
( , , , )
1 , , , 1 , , ,
1
1 1 1 1
( , , ..., )
hh
h
dh
hh
k h k h
hh
hh
kk
hh
kk
k
dd
k k k k
kk
h i i i P h i i i P
j
pp
i i i i i i
PMSM a a a a a
dd
CC
= =
=
==
( )
1
1
11 h
hk
h
h
h
k
dkk
P
k
hp
C a a
dC
=
= =
□
Theorem 2 (Monotonicity). Let
i
a
and
i
b
be two collections of nonnegative real numbers. For the
parameter vector
12
, ,..., d
k k k
with
1,2,...,
hh
kP=
and the
h
P
being the cardinality of
h
P
( 1,2,..., )hd=
,
if
ii
ab
for all
1,2,...,in=
, then
1 2 1 2
( , , , ) ( , , , )
1 2 1 2
( , ,..., ) ( , ,..., )
dd
k k k k k k
nn
PMSM a a a PMSM b b b
(13)
Symmetry 2018, 10, 383 10 of 32
Proof. Based on the assumption that
ii
ab
for all
1,2,...,in=
, then we can obtain
1 2 1 2
1 2 1 2
, , , , , ,
1 1 1 1
h h h h
j j j j
k h k h
hh
kk
hh
k k k k
i i i i
i i i P i i i P
j j j j
i i i i i i
a b a b
= = = =
1 2 1 2
1 2 1 2
11
, , , , , ,
11
11
hh
hh
jj
hh
k h k h
hh
hh
kk
hh
kk
kk
ii
kk
i i i P i i i P
jj
PP
i i i i i i
ab
CC
==
1 2 1 2
1 2 1 2
11
1 , , , 1 , , ,
11
1 1 1 1
hh
hh
jj
hh
k h k h
hh
hh
kk
hh
kk
kk
dd
ii
kk
h i i i P h i i i P
jj
pp
i i i i i i
ab
dd
CC
= =
==
□
Theorem 3 (Boundedness). Let
i
a
( 1, 2,..., )in=
be a collection of nonnegative real numbers. For the
parameter vector
12
, ,..., d
k k k
with
1,2,...,
hh
kP=
and the
h
P
being the cardinality of
h
P
( 1,2,..., )hd=
,
if
min{ }
i
i
aa
−=
and
max{ }
i
i
aa
+=
, then
12
( , , , )
12
( , ,..., )
d
k k k
n
a PMSM a a a a
−+
(14)
Proof. Based on the Theorem 2, we can obtain
1 2 1 2
( , ,..., ) ( , ,..., )
12
( , ,..., ) ( , ,..., )
dd
k k k k k k
n
PMSM a a a PMSM a a a
− − −
And
1 2 1 2
( , ,..., ) ( , ,..., )
12
( , ,..., ) ( , ,..., )
dd
k k k k k k + + +
n
PMSM a a a PMSM a a a
Furthermore, based on the Theorem 1, we can obtain
12
( , ,..., )( , ,..., )
d
k k k
PMSM a a a a
− − − −
=
and
12
( , ,..., )( , ,..., )
d
k k k
PMSM a a a a
+ + + +
=
Hence, we can obtain
12
( , , , )
12
( , ,..., )
d
k k k
n
a PMSM a a a a
−+
□
4.2. q-ROFPMSM Operator and q-ROFWPMSM Operator
The PMSM can only deal with evaluation values in the form of nonnegative real numbers, but
it is not valid to the information that is expressed by the q-ROFNs. In this section, we shall apply the
PMSM operator in q-rung orthopair fuzzy environment and propose the q-rung orthopair fuzzy
partitioned Maclaurin symmetric mean (q-ROFPMSM) operator and q-rung orthopair fuzzy
weighted partitioned Maclaurin symmetric mean (q-ROFPMSM) operator
Symmetry 2018, 10, 383 11 of 32
Definition 8. Let
i
a
( 1, 2,..., )in=
be a collection of q-ROFNs which are divided into d different classes
12
, ,..., d
P P P
. For parameter vector
12
, ,..., d
k k k
with
1,2,...,
hh
kP=
and
h
P
the being the cardinality of
h
P
( 1,2,..., )hd=
, if
12
12
2
1
( , , , )
12 1 , , , 1
11
( , ,..., )
h
h
d
j
hkh
h
hik
h
k
k
d
k k k
ni
k
h i i i P j
Pi i i
q-ROFPMSM a a a a
dC
= =
=
(15)
where the
12
( , ,..., )
h
k
i i i
traverses all the kh-tuple combination of
( 1,2,..., )
h
iP=
and
h
h
k
P
C
is the binomial
coefficient. Then the
12
( , , , )
d
k k k
q-ROFPMSM
is called the q-rung orthopair fuzzy partitioned Maclaurin
symmetric mean (q-ROFPMSM) operator.
Theorem 4. Let
i
a
( 1, 2,..., )in=
be a collection of q-ROFNs. For the parameter vector
12
, ,..., d
k k k
with
1,2,...,
hh
kP=
and the
h
P
being the cardinality of
h
P
( 1,2,..., )hd=
, then the aggregating result obtained
by Equation (15) is still a q-ROFN and presented as follows:
12
( , , , )
12
( , ,..., )
d
k k k
n
q-ROFPMSM a a a =
12
12
1
1
1
1
1 , , , 1
1 1 1 1
h
kh
P
h
h
j
kh
h
kh
q
d
k
C
q
k
d
i
h i i i P j
i i i
= =
− − − −
,
( )
12
12
1
1
1
1
1 , , , 1
1 1 1 1
h
kh
Ph
h
j
kh
h
kh
d
q
k
C
k
dq
i
h i i i P j
i i i
v
= =
− − − −
(16)
The proof of Theorem 4 is provided in Appendix A.
Considering the influence of the partition number of the argument set and the relationship
structure of the argument on q-ROFPMSM, some special cases of the q-ROFPMSM operator are put
as the remark below:
Remark 5. When all arguments belong to the same class and the types of the interrelationship among
arguments are also the same, that is to say, the number of the class
1d=
, the cardinality of
1
Pn=
and the
11,2,...,k k n==
, then the q-ROFPMSM reduces to the q-rung orthopair fuzzy Maclaurin symmetric mean
(q-ROFMSM) operator as follow:
Symmetry 2018, 10, 383 12 of 32
1
1
12
11 2 1
1
121
1
1
11
()
12 , , , 1
1
( , ,..., ) j
k
j
k
ik
kk
k
ki
i i i n j
k
ni
kk
i i i P j n
Pi i i
a
q-ROFPMSM a a a a C
C
=
=
= =
(17)
Remark 6. When there is no partition among argument sets and the types of the interrelationship among
arguments are same, namely, the cardinality
1
Pn=
and the parameter
11,2,...,k k n==
. Under the above
conditions, we further investigate some special cases of q-ROFPMSM with parameter k taking some
particular values.
Case 1: If
1k=
, then Equation (16) reduces to q-rung orthopair fuzzy average mean (q-ROFA)
operator as follows:
(1)
12
( , ,..., )
n
q-ROFPMSM a a a
( )
1 2 1 2
1
11
1
11
1 1 1 1
1 1 , 1 1 1 1
kk
nn
jj
kk
q
kq
k
CC
q
kk
q
ii
i i i n j i i i n j
v
= =
= − − − − − −
,
( )
11
11
11
11
11
1 1 1 1
1 1 , 1 1
nn
jj
qq
CC
q
q
ii
i n j i n j
v
= =
= − − − −
,
( )
( )
1
11
11
11
11
11
i
qq
nn
nn
qq
ii
ii
v
==
= − −
，
Let
1
ii=
( )
()
1
11
11
11
q
nn
nn
q
ii
ii
v
==
= − −
，
(18)
which is a special case of the q-rung orthopair fuzzy weighted average mean (q-ROFWA) operator
defined by Liu and Wang [31].
Case 2: If
2k=
, then Equation (16) reduces to the q-rung orthopair fuzzy Bonferroni mean
(q-ROFBM) operator introduced by Liu and Liu [32].
(2 )
12
( , ,..., )
n
q-ROFMSM a a a
( )
1 2 1 2
1
1 2 1
2 ( 1) 1 2
2 ( 1)
22
1 1 1 1
1 1 , 1 1 1 1
jj
qq
nn nn
q
q
ii
i i n j i i n j
v
−−
= =
= − − − − − −
( )
1 2 1 2
1 2 1 2
1
1 2 1 2
22
1 2 1
( 1) ( 1)
22
, 1 1 , 1 1
1 1 , 1 1 1 1
jj
q
n n n n
q
nn
q
ii
i i j i i j
i i i i
v
−−
= = = =
− − − − − −
=
1
q
(19)
Symmetry 2018, 10, 383 13 of 32
( )
1 2 1 2
1 2 1 2
11
1 2 1 2
1 ( 1) 1 ( 1)
22
, 1 1 , 1 1
1 1 , 1 1 1 1
jj
qq
n n n n
q
nn
q
ii
i i j i i j
i i i i
v
−−
= = = =
= − − − − − −
( ) ( )
1 2 1 2 1 2
1 2 1 2
1 2 1 2
11
1 2 1 2
1 ( 1) 1 ( 1)
, 1 , 1
1 1 , 1 1
qq
n n n n
nn
qq q q q
i i i i i i
i i i i
i i i i
v v v v
−−
==
− − − − += −
which is a special case of the q-ROFBM operator with the parameters s and t being equal to 1.
Case 3: If
kn=
, then Equation (16) reduces to the q-rung orthopair fuzzy geometric (q-ROFG)
operator as follows:
()
12
( , ,..., )
n
n
q-ROFPMSM a a a
( )
1 2 1 2
1
11
11
1
1 1 1 1
1 1 , 1 1 1 1
kk
nn
jj
kk
q
kq
k
qC
kk
C
q
ii
i i i n j i i i n j
v
= =
=
− − − − − −
( )
1 2 1 2
1
11
1
1 1 1 1
1 1 , 1 1 1 1
jj
nk
q
nq
n
q
nn
q
ii
i i i n j i i i n j
v
= =
=
− − − − − −
( )
1
11
11
, 1 1
jj
q
nn
nn
q
ii
jj
v
==
= − −
(20)
which is a special case of the q-rung orthopair fuzzy weighted geometric (q-ROFWG) operator
proposed by Liu and Wang [31].
Theorem 5 (Idempotency). Let
( , )
i i i
av
=
( 1, 2,..., )in=
be a collection of q-ROFNs. For the parameter
vector
12
, ,..., d
k k k
with
1,2,...,
hh
kP=
and
h
P
the being the cardinality of
h
P
( 1,2,..., )hd=
, if
( , )
i
a a v
==
for all
1,2,...,in=
, then
12
( , , , )
12
( , ,..., )
d
k k k
n
q-ROFPMSM a a a a=
(21)
Theorem 6 (Monotonicity). Let be
( , )
ii
i a a
av
=
and
( , )
ii
ibb
bv
=
two collections of q-ROFNs. For the
parameter vector
12
, ,..., d
k k k
with
1,2,...,
hh
kP=
and
h
P
the being the cardinality of
h
P
( 1,2,..., )hd=
,
if
ii
ab
and
ii
ab
vv
, then
1 2 1 2
( , , , ) ( , , , )
1 2 1 2
( , ,..., ) ( , ,..., )
dd
k k k k k k
nn
q-ROFPMSM a a a q-ROFPMSM b b b
(22)
Theorem 7 (Boundedness). Let
( , )
i i i
av
=
( 1, 2,..., )in=
be a collection of q-ROFNs. For the
parameter vector
12
, ,..., d
k k k
with
1,2,...,
hh
kP=
and
h
P
the being the cardinality of
h
P
( 1,2,..., )hd=
,
if
min{ }
i
i
aa
−=
and
max{ }
i
i
aa
+=
, then
Symmetry 2018, 10, 383 14 of 32
12
( , , , )
12
( , ,..., )
d
k k k +
n
a q-ROFPMSM a a a a
−
(23)
The proof of Theorems 5–7 are provided in Appendix A.
Note that the argument weights can produce a great impact on aggregated results, so we take
into account the importance of the argument itself and propose the q-ROFWPMSM operator to
overcome the drawbacks of q-ROFPMSM.
Definition 9. Let
( , )
i i i
av
=
( 1, 2,..., )in=
be a collection of q-ROFNs which are divided into d different
classes
12
, ,..., d
P P P
. For parameter vector
12
, ,..., d
k k k
with
1,2,...,
hh
kP=
and the
h
P
being the
cardinality of
h
P
( 1,2,..., )hd=
, if
( )
12
12
12
1
( , , , )
12 1 , , , 1
11
( , ,..., )
h
h
d
jj
hkh
h
hkh
k
k
d
k k k
n i i
k
h i i i P j
Pi i i
q-ROFWPMSM a a a w a
dC
= =
=
(24)
where the
12
( , ,..., )
h
k
i i i
traverses all the kh-tuple combination of
( 1,2,..., )
h
iP=
and
h
h
k
P
C
is the binomial
coefficient. The
i
w
denotes the weight information of
i
a
with
[0,1]
i
w
( 1, 2,..., )in=
and
11
n
i
iw
==
.
Then the
12
( , , , )
d
k k k
q-ROFWPMSM
is called the q-rung orthopair fuzzy weighted partitioned Maclaurin
symmetric mean (q-ROFPMSM) operator.
Theorem 8. Let
( , )
i i i
av
=
( 1, 2,..., )in=
be a collection of q-ROFNs and
i
w
denote the weight
information of
i
a
with
[0,1]
i
w
( 1, 2,..., )in=
and
11
n
i
iw
==
. For the parameter vector
12
, ,..., d
k k k
with
1,2,...,
hh
kP=
and
h
P
the being the cardinality of
h
P
( 1,2,..., )hd=
, then the aggregating result
obtained by Equation (24) is still a q-ROFN and presented as follows:
12
( , , , )
12
( , ,..., )
d
k k k
n
q-ROFWPMSM a a a =
( )
12
12
1
1
1
1
1 , , , 1
1 1 1 1 1 1
h
kh
hijPh
j
kh
h
kh
q
d
k
k
dwC
q
i
h i i i P j
i i i
= =
= − − − − − −
,
( )
12
12
1
1
1
1
1 , , , 1
1 1 1
h
kh
hijPh
j
kh
h
kh
d
q
k
k
dqw C
i
h i i i P j
i i i
v
= =
− − −
1-
(25)
The proof of this theorem is similar to Theorem 4, so it is omitted here.
Meanwhile, it is easily proved that the q-ROFWPMSM satisfies the Monotonicity and
Boundedness properties.
Remark 7. When the arguments can be divided into d different class
12
, ,..., d
P P P
and each member of class
h
P
( 1,2,..., )hd=
is interrelated to each other, namely,
2
h
kk==
for all
1,2,...,hd=
. Then the
Symmetry 2018, 10, 383 15 of 32
q-ROFWPMSM reduces to a special case of q-rung orthopair fuzzy weighted partitioned Bonferroni mean
(q-ROFWPBM) operator with the parameters s and t being equal to one.
(2 ,2 , ,2)
12
( , ,..., )
n
q-ROFWPMSM a a a
( )
( )
( )
1 2 1 2
1 2 1 2
11
22
22
2
1 , 1 1 , 1
1 1 1 2 1
2
1
j j j j
hh
h
dd
i i i i
h i i P j h i i P j
Pi i i i
hh
w a w a
dd
CPP
= = = =
= =
−
( )
( )
( )
( )
1 1 2 2
12
12
11
22
1 , 1 ,
1 1 1 1
11
hh
dd
i i i i i i j j
h i i P h i j P
i i i j
h h h h
w a w a w a w a
dd
P P P P
= =
= =
−−
(26)
4.3. q-ROFPPMSM Operator and q-ROFWPPMSM Operator
In a practical decision making process, the decision maker may provide unduly high or unduly
low evaluation values for attributes due to the lack of time and the difference of knowledge. The PA
can reduce the bad influence of unreasonable argument on aggregation result by calculating the
support measure between arguments. Thus, we propose the q-rung orthopair fuzzy power
partitioned Maclaurin symmetric mean (q-ROFPPMSM) and the q-rung orthopair fuzzy weighted
power partitioned Maclaurin symmetric mean (q-ROFPPMSM) operators that take advantage of
PMSM and PA.
Definition 10. Let
i
a
( 1, 2,..., )in=
be a collection of q-ROFNs which are divided into d different classes
12
, ,..., d
P P P
. For parameter vector
12
, ,..., d
k k k
with
1,2,...,
hh
kP=
and the
h
P
being the cardinality of
h
P
, if
( )
( )
= =
=
+
=
+
，
h
hj
d
j
hkh
h
hkh
k
k
di
k k k
ni
kn
h i i i P j
Pl
i i i l
n T a
q-ROFPPMSM a a a a
dCTa
12
12
12
1
( , , , )
12 1 , , , 1
1
1 ( )
11
( , ,..., )
1 ( )
(27)
then
12
( , , , )
d
k k k
q-ROFPPMSM
the is called the q-rung orthopair fuzzy power partitioned Maclaurin
symmetric mean (q-ROFPPMSM) operator, where the
12
( , ,..., )
h
k
i i i
traverses all the kh-tuple combination of
( )
1,2,..., h
iP=
and
h
h
k
P
C
is the binomial coefficient. Meanwhile, the
1,
( ) ( , )
n
i i l
l l i
T p Sup a a
=
=
and
( , )
il
Sup a a
is the support for
i
a
and
l
a
which satisfies following properties:
(1)
( , ) [0,1]
il
Sup a a
;
(2)
( , ) ( , )
i j r l
Sup a a Sup a a
;
(3) if
( , ) ( , )
i j r l
d a a d a a
, the
( , )
ij
d a a
is the distance of q-ROFNs
In order to simplify Equation (27), we define
( ) ( )
1
1 ( ) 1 ( )
n
i i l
l
T a T a
=
= + +
(28)
And
12
( , ,..., )
n
=
. The
is called as the power weighting vector which satisfies
[0,1]
i
and
11
n
i
i
==
. Therefore Equation (27) can be expressed as follows:
Symmetry 2018, 10, 383 16 of 32
( )
12
12
2
1
( , , , )
12 1 , , , 1
11
( , , ..., )
h
h
d
jj
hkh
h
hik
h
k
k
d
k k k
n i i
k
h i i i P j
Pi i i
q-ROFPPMSM a a a n a
dC
= =
=
(29)
Theorem 9. Let
( , )
i i i
av
=
( 1, 2,..., )in=
be a collection of q-ROFNs. For the parameter vector
12
, ,..., d
k k k
with
1,2,...,
hh
kP=
and the
h
P
being the cardinality of
h
P
( 1,2,..., )hd=
, then the
aggregating result obtained by Equation (29) is still a q-ROFN and is presented as follows:
12
( , , , )
12
( , ,..., )
d
k k k
n
q-ROFPPMSM a a a
( )
12
12
1
1
1
1
1 , , , 1
1 1 1 1 1 1
h
kh
hijPh
j
kh
h
kh
q
d
k
k
dnC
q
i
h i i i P j
i i i
= =
= − − − − − −
，
( )
12
12
1
1
1
1
1 , , , 1
1 1 1
h
kh
hijPh
j
kh
h
kh
d
q
k
k
dqn C
i
h i i i P j
i i i
v
= =
− − −
1-
(30)
The proof of this theorem is similar to Theorem 4, so it is omitted here.
Theorem 10 (Idempotency). Let
( , )
i i i
av
=
( 1, 2,..., )in=
be a collection of q-ROFNs. For the
parameter vector
12
, ,..., d
k k k
with
1,2,...,
hh
kP=
and
h
P
the being the cardinality of
h
P
( 1,2,..., )hd=
,
if
( , )
i
a a v
==
for all
1,2,...,in=
, then
12
( , , , )
12
( , ,..., )
d
k k k
n
q-ROFPPMSM a a a a=
(31)
Theorem 11 (Boundedness). Let
( , )
i i i
av
=
( 1, 2,..., )in=
be a collection of q-ROFNs. For the
parameter vector
12
, ,..., d
k k k
with
1,2,...,
hh
kP=
and the
h
P
being the cardinality of
h
P
( 1,2,..., )hd=
, if
min{ }
i
i
aa
−=
and
max{ }
i
i
aa
+=
, then
12
( , , , )
12
( , ,..., )
d
k k k
n
x q-ROFPPMSM a a a y
(32)
where
( )
( )
12
12
1
1
1
1
1 , , , 1
1 1 1 1 1 1 ,
h
kh
Ph
hij
kh
h
kh
q
d
k
C
kn
dq
h i i i P j
i i i
x
−
= =
= − − − − − −
Symmetry 2018, 10, 383 17 of 32
( )
( )
12
12
1
1
1
1
1 , , , 1
1 1 1
h
kh
Ph
hij
kh
h
kh
d
q
k
C
k
dqnw
h i i i P j
i i i
v−
= =
− − −
1-
and
( )
( )
12
12
1
1
1
1
1 , , , 1
1 1 1 1 1 1 ,
h
kh
Ph
hij
kh
h
kh
q
d
k
C
kn
dq
+
h i i i P j
i i i
y
= =
= − − − − − −
( )
( )
12
12
1
1
1
1
1 , , , 1
1 1 1
h
kh
Ph
hij
kh
h
kh
d
q
k
C
k
dqnw
+
h i i i P j
i i i
v
= =
− − −
1-
The proof of these theorems is provided in Appendix B.
In the following, we provide the weighted form of q-ROFPPMSM operator.
Definition 11. Let
( , )
i i i
av
=
( 1, 2,..., )in=
be a collection of q-ROFNs which are divided into d
different class
12
, ,..., d
P P P
and the
h
P
represent the cardinality of
h
P
. The
12
( , , , )T
n
w w w w=
is the
weighted vector with
[0,1]
i
w
( 1, 2,..., )in=
and
11
n
i
iw
==
. For the parameter vector r
12
, ,..., d
k k k
with
1,2,...,
hh
kP=
for all
1,2,...,hd=
, if
12
12
12
1
( , , , )
12 1 , , , 1
1
(1 ( ))
11
( , ,..., )
(1 ( ))
h
hjj
d
j
hkh
h
hkh
k
k
dii
k k k
ni
kn
h i i i P j
Pll
i i i l
nw T a
q-ROFWPPMSM a a a a
dCw T a
= =
=
+
=
+
(33)
then the
12
( , , , )
d
k k k
q-ROFWPPMSM
is called the q-rung orthopair fuzzy weighted power partitioned
Maclaurin symmetric mean (q-ROFWPPMSM) operator, where the
12
( , ,..., )
h
k
i i i
traverses all the kh-tuple
combination of
( )
1,2,..., h
iP=
and
h
h
k
P
C
is the binomial coefficient. Meanwhile, the
1,
( ) ( , )
n
i i l
l l i
T p Sup a a
=
=
and
( , )
il
Sup a a
is the support for
i
a
and
l
a
which satisfies following properties:
(1)
( , ) [0,1]
il
Sup a a
;
(2)
( , ) ( , )
i l l i
Sup a a Sup a a=
;
(3)
( , ) ( , )
i j r l
Sup a a Sup a a
, the
( , )
ij
d a a
is the distance of q-ROFNs
In order to simplify Equation (33), we define
( ) ( )
1
1 ( ) 1 ( )
n
i i l
l
T a T a
=
= + +
(34)
and
12
( , ,..., )
n
=
. The
is called as the power weighting vector which satisfies
[0,1]
i
and
11
n
i
i
==
. Therefore Equation (33) can be expressed as follows:
Symmetry 2018, 10, 383 18 of 32
12
12
12
1
( , , , )
12 1 , , , 1
1
11
( , , ..., )
h
hjj
d
j
hkh
h
hkh
k
k
dii
k k k
ni
kn
h i i i P j
Pll
i i i l
nw
q-ROFWPPMSM a a a a
dCw
= =
=
=
(35)
Theorem 12. Let
( , )
i i i
av
=
( 1, 2,..., )in=
be a collection of q-ROFNs and
i
w
denote the weight
information of
i
a
with
[0,1]
i
w
( 1, 2,..., )in=
and
11
n
i
iw
==
. For parameter vector
12
, ,..., d
k k k
with
1,2,...,
hh
kP=
and
h
P
the being the cardinality of
h
P
( 1,2,..., )hd=
, then the aggregating result
obtained by Equation (35) is still a q-ROFN and presented as follows:
12
( , , , )
12
( , ,..., )
d
k k k
n
q-ROFWPPMSM a a a
( )
1
12
12
1
1
1
1
1 , , , 1
1 1 1 1 1 1 ,
h
kh
P
h
ii
jj
hn
ll
l
j
kh
h
kh
q
d
k
C
nw
k
dqw
i
h i i i P j
i i i
=
= =
= − − − − − −
( )
1
12
12
1
1
1
1
1 , , , 1
1 1 1
h
kh
P
h
ii
jj
hn
ll
l
j
kh
h
kh
d
q
k
C
nw
k
dqw
i
h i i i P j
i i i
v
=
= =
− − −
1-
(36)
The proof of the theorem is similar to the Theorem 4, which is omitted here.
5. A Novel Approach of MAGDM Based on q-ROFWPPMSM Operator
In order to solve MAGDM problems, a new approach based on a q-ROFWPPMSM operator is
proposed.
A typical MAGDM is the process that the most desirable alternative is selected from a set of
alternatives
12
{ , ,..., }
m
X X X X=
based on a collection of attributes
12
{ , ,..., }
n
C C C C=
. The process is
carried out by a group of decision makers
12
{ , ,..., }
t
D D D D=
whose weight vector is
12
( , ,..., )
t
=
satisfying
[0,1]
k
( 1, 2,..., )kt=
and
11
t
k
k
==
. Meanwhile, the attribute
weight vector
12
( , ,..., )
n
w w w w=
, which satisfies
[0,1]
i
w
( 1, 2,..., )in=
and
11
n
i
iw
==
,
represents the importance of attribute
j
A
( 1,2,..., )jn=
in the decision making process. Suppose
that the attributes are divided into d different classes
12
, ,..., d
P P P
and there is an interrelationship
among any kh attributes in each class
h
P
( 1,2,..., )hd=
whereas the attributes in different classes are
not related. Due to the existence of uncertainty in a MAGDM problem, the performance value of
alternative
i
X
with respect to the attribute
j
A
given by decision maker
k
D
is provided in the
form of q-ROFN and is summarized in the decision matrix
[]
k
k ij m n
Rr
=
.
For the sake of select the best alternative, an algorithm based on q-ROFWPPMSM operator is
provided and the key steps of the algorithm are given as follows:
Step 1: To ensure the consistence of the type of each attribute, we transform the given decision
matrix
[]
k
k ij m n
Rr
=
into normalized q-rung orthopair fuzzy decision matrix
[]
k
k ij m n
Rr
=
by the
following method:
Symmetry 2018, 10, 383 19 of 32
, for benefit attribute of
( ) , for cost attribute of
k
ij j
k
ij kc
ij j
rC
rrC
=
(37)
where the
( ) ( , )
k c k k
ij ij ij
r
=
.
Step 2: Calculate the support between the q-ROFN
k
ij
r
with other q-ROFNs
l
ij
r
( , 1,2,..., )k l t=
.
( , ) 1 ( , )
k l k l
ij ij ij ij
Sup r r d r r=−
( 1,2,..., ; 1, 2, ..., )i m j n==
(38)
where the
( , )
kl
ij ij
d r r
is the distance of q-ROFNs based on Definition 4.
Step 3: Calculate the
()
k
ij
Tr
of the q-ROFN
k
ij
r
( 1, 2,..., )kt=
.
1
( ) ( , )
t
k k l
l
ij ij ij
lk
T r Sup r r
=
=
( 1,2,..., ; 1, 2, ..., )i m j n==
(39)
Step 4: Calculate the power weights
k
ij
corresponding to the q-ROFNs
k
ij
r
( 1, 2,..., )kt=
.
( )
( )
( )
( )
1
11
t
k k l
ij ij ij
l
T r T r
=
= + +
( 1,2,..., ; 1,2,..., )i m j n==
(40)
Step 5: For the alternative Xi, aggregate the evaluation of attributes
j
A
provided by decision
makers
k
D
( 1, 2,..., )kt=
based on q-ROFWPPMSM operator.
12
( , ,..., )
t
ij ij ij ij
r q-ROFWPPMSM r r r=
( 1,2,..., ; 1,2,..., )i m j n==
(41)
and obtain the comprehensive decision matrix
[]
ij m n
Rr
=
.
Step 6: Calculate the supports
( , )
ij il
Sup r r
( , 1,2,..., )j l n=
.
( , ) 1 ( , )
ij il ij il
Sup r r d r r=−
,
1,2, ...,im=
(42)
Step 7: Calculate the
()
ij
Tr
( 1,2,..., ; 1,2,..., )i m j n==
.
1,
( ) ( , )
n
ij ij il
l j l
T r Sup r r
=
=
(43)
Step 8: Calculate the power weights
ij
which are corresponded to attributes
j
A
( 1,2,..., )jn=
, respectively.
( )
( )
( )
1
1 ( ) 1
n
ij ij il
l
T r T r
=
= + +
,
1,2,...,in=
(44)
Step 9: Calculate the overall performance value of alternatives
i
X
( 1,2,..., )im=
over all
attributes.
12
( , ,..., )
i i i in
r q-ROFWPPMSM r r r=
(45)
Step 10: Calculate the score function of alternatives and rank the alternatives based on the
comparison rule presented in Definition 3
6. Numerical Instance
In this section, a MAGDM problem about company location selection is provided to illustrate
the application of the proposed approach (cited from Liu et al. [12]).
Example 3. A corporation needs to select a best location to build new company building from five
alternatives denoted by
i
X
( 1,2,...,5)i=
. Considering the company’s strategic benefits, the company decides
to evaluate the alternatives based on the following four factors, including: the cost of rent C1, the convenience
Symmetry 2018, 10, 383 20 of 32
of transportation C2, the cost of labor C3, and the influence of surrounding environment C4. The
corresponding attribute weighting vector is
(0.25,0.1,0.3,0.35)w=
. Assume that the attributes are divided
into two classes
1 1 3
{ , }P C C=
and
24
{ , }P C C=
, there is interrelationship between any two attributes in
each class, that is to say, the
12
2kk==
. Three experts
l
D
( 1,2, 3)l=
, whose weight vector is
(0.35,0.45,0.2)
=
, are invited to evaluate five alternatives by taking form of q-ROFNs according to the
above four attributes and the decision matrices
54
[]
ll
ij
Rr
=
( 1,2, 3)l=
are presented in Tables 2–4.
Table 2. The q-rung orthopair fuzzy decision matrix R1 provided by D1.
C1
C2
C3
C4
X1
(0.5,0.4)
(0.5,0.3)
(0.2,0.6)
(0.5,0.4)
X2
(0.6,0.2)
(0.6,0.3)
(0.6,0.2)
(0.6,0.3)
X3
(0.5,0.4)
(0.2,0.6)
(0.6,0.2)
(0.4,0.4)
X4
(0.6,0.2)
(0.6,0.2)
(0.4,0.2)
(0.3,0.6)
X5
(0.4,0.3)
(0.7,0.2)
(0.4,0.5)
(0.4,0.5)
Table 3. The q-rung orthopair fuzzy decision matrix R2 provided by D2.
C1
C2
C3
C4
X1
(0.4,0.2)
(0.6,0.2)
(0.4,0.4)
(0.5,0.3)
X2
(0.5,0.3)
(0.6,0.2)
(0.6,0.2)
(0.5,0.4)
X3
(0.4,0.4)
(0.3,0.5)
(0.5,0.3)
(0.7,0.2)
X4
(0.5,0.4)
(0.7,0.2)
(0.5,0.2)
(0.7,0.2)
X5
(0.6,0.3)
(0.7,0.2)
(0.4,0.2)
(0.4,0.2)
Table 4. The q-rung orthopair fuzzy decision matrix R3 provided by D3.
C1
C2
C3
C4
X1
(0.4,0.5)
(0.5,0.2)
(0.5,0.3)
(0.5,0.2)
X2
(0.5,0.4)
(0.5,0.3)
(0.6,0.2)
(0.7,0.2)
X3
(0.4,0.5)
(0.3,0.4)
(0.4,0.3)
(0.3,0.3)
X4
(0.5,0.3)
(0.5,0.3)
(0.3,0.5)
(0.5,0.2)
X5
(0.6,0.2)
(0.6,0.3)
(0.4,0.4)
(0.6,0.3)
6.1. The Decision-Making Process
Step 1: It is noted that the same type of each attribute is consistent, then we can get the
kk
ij ij
rr=
( 1,2,3)k=
based on Equation (37) and the normalized the decision matrix
54
[]
k
k k ij
R R r
==
;
Step 2: Calculate the support
( , )
kk
ij ij
Sup r r
( 1,2,...,5; 1,2,3,4; , 1,2,3)i j k l= = =
based on
Equation (38). To simplify,
( )
54
( , )
kl
ij ij
Sup r r
is denoted as
kl
Sup
and presented as follows:
12 21
0.8349 0.9000 0.8000 0.9206
0.9000 0.9206 1 0.9000
0.9206 0.9000 0.9000 0.7404
0.8349 0.9206 0.9206 0.6000
0.8413 1 0.7619 0.7619
Sup Sup
==
;
13 31
0.9000 0.9206 0.7000 0.8413
0.8349 0.9206 1 0.9000
0.9000 0.8349 0.8349 0.9000
0.9000 0.9000 0.7590 0.6698
0.8349 0.9000 0.9026 0.8000
Sup Sup
==
;
Symmetry 2018, 10, 383 21 of 32
23 32
0.7619 0.9206 0.9000 0.9206
0.9206 0.9000 1 0.8000
0.9206 0.9206 0.9206 0.6809
0.9206 0.8349 0.7404 0.8413
0.9206 0.9000 0.8413 0.8349
Sup Sup
==
;
Step 3: Calculate the
()
k
ij
Tr
( 1,2,..., 5; 1,2, 3, 4; 1,2,3)i j k= = =
based on Equation (39). For
simplify,
( )
54
()
k
ij
Tr
is denoted as
k
T
and presented as follows:
1
1.7349 1.8206 1.5000 1.7619
1.7349 1.8413 2 1.8000
1,8206 1.7349 1.7349 1.6404
1.7349 1.8206 1.6796 1.2698
1.6762 1.9000 1.6825 1.5619
T
=
;
2
1.5968 1.8206 1.7000 1.8413
1.8206 1.8206 2 1.7000
1.8413 1.8206 1.8206 1.4213
1.7555 1.7555 1.6610 1.4413
1.7619 1.9000 1.6031 1.5968
T
=
;
3
1.6619 1.8413 1.6000 1.7619
1.7555 1.8206 2 1.7000
1.8206 1.7555 1.7555 1.5809
1.8206 1.7349 1.4994 1.5111
1.7555 1.8000 1.7619 1.6349
T
=
;
Step 4: Calculate the power weights
k
ij
( 1,2,..., 5; 1,2, 3, 4; 1,2,3)i j k= = =
based on Equation
(40). For simplify,
( )
54
k
ij
is denoted as
k
W
and presented as follows:
1
0.3601 0.3495 0.3352 0.3455
0.3446 0.3517 0.3500 0.3583
0.3489 0.3446 0.3446 0.3653
0.3467 0.3559 0.3559 0.3317
0.3430 0.3524 0.3526 0.3459
W
=
;
2
0.4396 0.4493 0.4655 0.4570
0.4570 0.4489 0.4500 0.4442
0.4518 0.4570 0.4570 0.4307
0.4491 0.4470 0.4544 0.4587
0.4552 0.4531 0.4399 0.4508
W
=
;
3
0.2003 0.2012 0.1992 0.1974
0.1984 0.1995 0.200 0.1974
0.1993 0.1984 0.1984 0.2040
0.2043 0.1972 0.1897 0.2097
0.2018 0.1944 0.2075 0.2033
W
=
Step 5: For alternative Xi, we aggregate the evaluation of attributes Aj given by decision makers
Dk (k = 1, 2, 3) based on Equation (41) and the comprehensive decision matrix is presented in Table 5
(Suppose k1 = 1).
Table 5. Comprehensive q-rung orthopair fuzzy decision matrix.
C1
C2
C3
C4
A1
(0.3096, 0.6756)
(0.3896, 0.6131)
(0.2707, 0.7564)
(0.3518, 0.6738)
A2
(0.3815, 0.6512)
(0.4144, 0.6300)
(0.4271, 0.5848)
(0.4191, 0.6802)
A3
(0.3088, 0.7478)
(0.1902, 0.7986)
(0.3716, 0.6390)
(0.4078, 0.6541)
A4
(0.3817, 0.6669)
(0.4569, 0.6006)
(0.3085, 0.6197)
(0.4193, 0.6603)
A5
(0.3894, 0.6514)
(0.4941, 0.6004)
(0.2794, 0.6833)
(0.3223, 0.6681)
Step 6: Calculate the
( , )
ij il
Sup r r
( 1,2, ...,5; , 1,2,3, 4)i j l==
based on Equation (42). To simplify,
( )
54
( , )
ij il
Sup r r
is denoted as
jl
Sup
and obtain
Symmetry 2018, 10, 383 22 of 32
12 21
0.9277
0.9717
0.9035
0.9290
0.9138
Sup Sup
==
;
13 31
0.9336
0.9421
0.9084
0.9371
0.9120
Sup Sup
==
;
14 41
0.9662
0.9433
0.9036
0.9701
0.9465
Sup Sup
==
;
23 32
0.8678
0.9638
0.8288
0.8822
0.8263
Sup Sup
==
;
24 42
0.9482
0.9602
0.8118
0.9489
0.8609
Sup Sup
==
;
34 43
0.9181
0.9243
0.9706
0.9106
0.9654
Sup Sup
==
;
Step 7: Calculate the
()
ij
Tr
( 1,2,..., 5; 1,2, 3,4)ij==
based on Equation (43). To simplify,
( )
54
()
ij
Tr
is denoted as T and obtain
2.8278 2.7437 2.7195 2.8329
2.8800 2.8958 2.8302 2.8506
2.7154 2.5441 2.7078 2.6860
2.8362 2.7601 2.7299 2.8296
2.7722 2.6010 2.7038 2.7728
T
=
Step 8: Calculate the power weight vector of alternative Xi
( 1,2,...,5)i=
with respect to the
attributes Aj
( 1,2,3, 4)j=
based on Equation (44) and obtain
1=(0.2526,0.0988,0.2945,0.3541)
;
2(0.2515,0.1010,0.2980,0.3495)=
;
3=(0.2520,0.0962,0.3018,0.3500)
;
4=(0.2518,0.0991,0.2949,0.3533)
;
5(0.2525,0.0964,0.2975,0.3536)
=
;
Step 9: Calculate the overall performance of alternative Xi
( 1,2,...,5)i=
over all attributes based
on Equation (45).
1 2 3
(0.2015,0.9181); (0.2532,0.9009); (0.1953,0.9263)r r r= = =
45
(0.2394,0.9029); (0.2228,0.9080)rr==
Step 10: Calculate the score function of alternative Xi
( 1,2,...,5)i=
based on Definition 4.
1 2 3 4 5
( ) 0.7657; ( ) 0.7150; ( ) 0.7874; ( ) 0.7223; ( ) 0.7377S r S r S r S r S r= − = − = − = − = −
and on the basis the value of the score function of alternative, we rank the alternatives by using the
comparison and get
2 4 5 1 3
X X X X X
6.2. The Influence of the Parameters on the Results
It is noted that the parameter q and the parameter vector (k1, k2) have great impacts on the
aggregated result of alternatives in Example 3. Firstly, the influence of parameter q on aggregation
results of alternatives is investigated by calculating the score functions of alternatives under the
condition of parameter q taking different values. The results are presented in Figure 1.
Symmetry 2018, 10, 383 23 of 32
Figure 1. Score values of the alternatives Xi
( 1,2,...,5)i=
when
[1,10],q
.
From Figure 1, we can know that the aggregation results depend on the parameter q, the score
values of alternatives
i
X
( 1,2,...,5)i=
become greater as the parameter q increases. However, it is
noted that the ranking result of alternatives is still
2 4 5 1 3
X X X X X
no matter what values
the parameter q takes. That means the parameter q is robust. The parameter q represents the space of
acceptable orthopairs, that is to say, the decision maker can set an appropriate value of parameter q
to model the uncertainty and fuzzy information in decision making.
In the following, under the condition of parameter vector (k1, k2) taking some special values, the
score functions of alternatives are calculated and the calculation results are showed in Table 6.
Table 6. Score values of alternatives with different values of parameter vector.
(k1, k2)
S(r1)
S(r2)
S(r3)
S(r4)
S(r5)
Ranking
(1,1)
−0.7492
−0.6960
−0.7407
−0.7036
−0.7219
2 4 5 3 1
X X X X X
(2,1)
−0.7497
−0.6996
−0.7473
−0.7058
−0.7224
2 4 5 3 1
X X X X X
(1,2)
−0.7651
−0.7113
−0.7806
−0.7201
−0.7372
2 4 5 1 3
X X X X X
(2,2)
−0.7657
−0.7150
−0.7874
−0.7223
−0.7377
2 4 5 1 3
X X X X X
It is known from Table 6 that the ranking order of X2, X4, and, X5 remain unchanged no matter
what values the parameter vector takes, whereas the ranking order of alternatives X3 and X1
is
31
XX
when the parameter vector takes (1,1) or (2,1) and the ranking order of alternatives X3
and X1 is
13
XX
when the parameter vector takes (1,2) or (2,2). The difference is due to the
relationship structure of the attributes has changed when the parameter vector takes different
values. The parameter vector models the types of interrelationships among attributes, therefore, a
decision maker can set the appropriate values of a parameter vector to model any kind of
interrelationship among attributes in decision making. Meanwhile, we can observe that the more
interrelationships of attributes in each class we consider, the smaller the score values will become.
6.3. Comparative Analysis
In the following, some comparisons of the proposed approach with existing approaches are
conducted to illustrate the validity and advantage of the q-ROFWPPMSM operator. We select
following approaches to solve aforementioned example, including: the approach proposed by Wei
and Lu [26] based on the Pythagorean fuzzy power weighted averaging (PFPWA) operators, the
Symmetry 2018, 10, 383 24 of 32
approach introduced by Wei and Lu [40] based on Pythagorean fuzzy weighted Maclaurin
symmetric mean (PFWMSM) operator, and the approach defined by Liu et al. [16] based on
intuitionistic fuzzy weighted interaction partitioned Bonferroni mean (IFWIPBM) operator. The
aggregated results of the alternatives obtained by the reference approaches and the proposed
approach are presented in Table 7.
Table 7. Score values and ranking results by different approaches in Example 3.
Operator
Score Values of ri (i = 1, 2, 3, 4)
Ranking Result
PFPWA [26]
S(r1) = 0.5510, S(r2) = 0.6347, S(r3) = 0.5170,
S(r4) = 0.6065, S(r5) = 0.5829;
2 4 5 3 1
X X X X X
PFWMSM [40]
(Suppose k = 2)
S(r1) = 0.0914, S(r2) = 0.1135, S(r3) = 0.0919,
S(r4) = 0.1091, S(r5) = 0.1029;
2 4 5 3 1
X X X X X
IFWIPBM [16]
(Suppose p = q = 2)
S(r1) = −0.0294, S(r2) = −0.0046, S(r3) = −0.0213
S(r4) = −0.0115, S(r5) = −0.0252;
2 4 3 5 1
X X X X X
q-ROFWPPMSM
(Supposek1 = k2 = 2)
S(r1) = −0.7657, S(r2) = −0.7150, S(r3) = −0.7874
S(r4) = −0.7223, S(r5) = −0.7377;
2 4 5 1 3
X X X X X
It is can be observed from Table 7 that the alternatives X2 and X4 are respectively identified as
the best alternative and second best alternative by all approaches, though the ranking orders of the
rest of alternatives X1, X3, and, X5 are slightly different. Thus, the validity of the proposed approach
is verified.
In Example 3, the attributes are divided into two classes and the attributes of each class are
interrelated to each other. In order to further demonstrate the advantage of the proposed approach,
a new example with more complicated scenarios is provided and it is depicted as follows:
Example 4. A corporation wants to select a new investment area from four alternatives
i
X
( 1,2,3, 4)i=
. After
preliminary screening, there are the following five factors denoted by
j
C
( 1,2,...,5)j=
, which are selected as
evaluation attributes, including: C1: the risk of losing capital sum, C2: the amount of interest received, C3: the
vulnerability of capital sum to modification by inflation, C4: the market potential and, C5: the growth potential.
The corresponding attribute weight vector is
(0.3,0.1,0.25,0.15,0.2)w=
. Considering the attribute
characteristics, the attributes are divided into two class P1 = {C1, C2, C3} and P2 = {C4, C5}. Moreover, there is
interrelationship among any three attributes in
1
P
and there is interrelationship between any two attributes
in
2
P
, that is to say, the
13k=
and
22k=
. The evaluating values of alternatives
i
X
( 1,2,3, 4)i=
with
respect to attributes Cj are given in form of q-ROFNs and the decision matrix is present in Table 8
Table 8. The q-rung orthopair fuzzy decision matrix.
C1
C2
C3
C4
C5
X1
(0.3, 0.6)
(0.7, 0.2)
(0.2, 0.8)
(0.8, 0.1)
(0.7, 0.3)
X2
(0.1, 0.8)
(0.8, 0.2)
(0.2, 0.6)
(0.7, 0.2)
(0.8, 0.1)
X3
(0.1, 0.85)
(0.6, 0.2)
(0.2, 0.75)
(0.8, 0.2)
(0.8, 0.2)
X4
(0.2, 0.7)
(0.9, 0.1)
(0.2, 0.7)
(0.6, 0.3)
(0.8, 0.1)
We utilize the aforementioned approaches to solve Example 4. The score value of alternatives
over all attributes and ranking order obtained by different approaches are presented in Table 9.
Symmetry 2018, 10, 383 25 of 32
Table 9. Score values and ranking results by different approaches in Example 4.
Method
Score Values
Ranking Result
PFPWA [26]
S(r1) = 0.6708; S(r2) = 0.6949; S(r3) = 0.6710; S(r4) =
0.7086;
4 2 3 1
X X X X
PFWMSM [40]
(Suppose k = 2)
S(r1) = 0.2201; S(r2) = 0.2315; S(r3) = 0.2061; S(r4) =
0.2384;
4 2 1 3
X X X X
IFWIPBM [16]
(Suppose p = q= 1)
S(r1) = −0.5967; S(r2) = −0.4690; S(r3) = −0.3541; S(r4) =
−0.3835;
3 4 2 1
X X X X
q-ROFWPPMSM
(Suppose k1 = 3, k2 = 2)
S(r1) = −0.4101; S(r2) = −0.3609; S(r3) = −0.4052; S(r4) =
−0.4189;
2 3 1 4
X X X X
It is known from Table 9 that the ranking order obtained by the proposed approach is
significantly different from the results given by the three existing approaches. The difference is due
to none of the above three approaches can exactly model the relationship structure where attributes
are divided into several classes and there is interrelationship among the arguments of each class.
In the following, we compare the differences of the ranking orders of alternatives obtained by
the aforementioned approaches in detail and analyze the principal cause of the above discrepancy
from the perspective of the model structure.
(1) Comparing the approach introduced by Wei and Lu [26] with the proposed approach, the
alternative X4 is respectively identified as the best alternative and worst alternative and the
alternative X2 is respectively identified as the second best alternative and the best alternative
by the approach of Wei and Lu [26] and the proposed approach. The difference is due to the
former only using the power aggregation operator, which can calculate the support degree
between arguments whereas the later not only includes the power aggregation operation, but
also considers the interrelationship among arguments. In Example 4, it is obvious that the
interrelationship exists among attributes, so the proposed approach may be more reasonable
than Wei and Lu’s approach [22].
(2) Similar to the ranking order of alternatives given by the PFPWA operator, the approach of Wei
and Lu [40] also respectively identified the alternatives X4 and X2 as the best alternative and
the second best alternative, whereas the proposed approach identifies the X4 and X2 as the
worst alternative and best alternative, respectively. The ranking order of the rest alternatives
X1 and X3 obtained by Wei and Lu’s [40] approach and the proposed approach is
13
XX
and
31
XX
, respectively. The difference is due to the approach of Wei and Lu [40],
which can capture the interrelationships among attributes by using a MSM operator, which
can calculate the average of the sum of satisfaction among any k attributes. However, the
proposed approach considers this situation where the attributes can be divided into different
classes and there is an interrelationship among any attribute in each class, whereas there is no
interrelationship among attributes of any two classes.
(3) The ranking orders obtained by the approach of Liu et al. [16] and the proposed approach are
significantly different. The alternative X3 and the alternative X2 are respectively identified as
the best alternative by the approach of Liu et al. [16] and the proposed approach. The approach
of Liu et al. [16], based on the IFIPBM operator, divides attributes into different classes and
assumes attributes in each class are interrelated to each other. But in Example 4, each attribute
is interrelated to any other two attributes in class P1 and each attribute in class P2 is interrelated
to each other, that is to say, the type of interrelationship of attributes in each class are different.
It is obvious that the proposed approach can model the above situation better than the
approach of Liu et al. [16]. Therefore the proposed approach may be more reasonable than Liu
et al. [16].
Symmetry 2018, 10, 383 26 of 32
7. Conclusions
In this paper, a new approach is proposed for dealing with q-rung orthopair fuzzy MAGDM
problems. The contribution of this paper includes three phases. Firstly, a new aggregation operator,
which is called the partitioned PMSM operator, is proposed for dealing with a situation where the
attributes are divided into several parts and there is interrelationship among any attributes in each
part whereas the attributes in different parts are not related. The mathematical form of the PMSM is
introduced and some special cases and desirable properties of a PMSM operator are also
investigated. Secondly, in order to aggregate the q-rung orthopair fuzzy information, the PMSM
operator is extended in a q-ROFS and q-rung orthopair fuzzy partitioned Maclaurin symmetric
mean (q-ROFPMSM) and q-rung orthopair fuzzy weighted partitioned Maclaurin symmetric mean
(q-ROFWPMSM) operators are proposed. Finally, to eliminate the negative effects of unreasonable
assessment values obtained by the decision maker on the decision results, we take advantage of the
PA operator and propose a q-rung orthopair fuzzy power partitioned Maclaurin symmetric mean
(q-ROFPPMSM) and q-rung orthopair fuzzy weighted power partitioned Maclaurin symmetric
mean (q-ROFWPPMSM) operators, which combine the advantages of PMSM and PA operators. A
new approach based on the q-ROFWPPMSM operator is proposed for solving q-rung orthopair
fuzzy MAGDM problems. A numerical example and some comparative analysis are also
conducted.
Based on the results of the comparative analysis, the main advantages of the proposed
approach include: (1) the proposed PMSM can reflect the relationship structure of attributes that
attributes are partitioned into several parts, and there is interrelationship among any attributes in
each part; (2) the proposed PMSM can reduce MSM or PBM operator by adjusting the cardinality of
set and setting different values of parameter vector; (3) the q-ROFPPMSM operator can reduce the
influence of the unduly high and low arguments on ranking results. In future works, we will apply
the proposed approach in other practical decision making problems, such as low carbon supplier
selection, risk management, medical diagnosis, and resource evaluation, etc.
Author Contributions: The idea of the whole thesis was put forward by K.B. He also wrote the paper. X.Z.
analyzed the existing work and J.W. provided the numerical instance. The computation of the paper was
conducted by R.Z.
Funding: This research was funded by National Natural Science Foundation of China (Grant number 71532002),
the Fundamental Fund for Humanities and Social Sciences of Beijing Jiaotong University (Grant number
2016JBZD01) and a key project of Beijing Social Science Foundation Research Base with grant number of
18JDGLA017.
Acknowledgments: We would like to thank the anonymous referees and editors for their careful reading on
the manuscript and providing constructive comments
Conflicts of Interest: The authors declare no conflict of interest.
Appendix A
Proof of Theorem 4. Based on the operational laws of q-ROFNs described in Definition 2, we can
get
( )
1
111
, 1 1
hh
h
j j j
q
kk
k
q
i i i
jjj
av
===
= − −
( )
12 1 2 1 2
12 1 2 1 2
1
1
, , , 1 , , , 1 , , , 1
1 1 , 1 1
hh
h
j j j
kh
hk h k h
hh
khkk
hh
q
qq
kk
k
q
i i i
i i i P j i i i P j i i i P j
i i i i i i i i i
av
= = =
= − − − −
Furthermore, we can obtain
Symmetry 2018, 10, 383 27 of 32
( )
kh
P
h
hh
h
j j j
hkh
hk h k h
hh
hkhkk
hh
q
C
qq
kk
k
q
i i i
ki i i P j i i i P j i i i P j
Pi i i i i i i i i
av
C12 1 2 1 2
12 1 2 1 2
1
1
1
, , , 1 , , , 1 , , , 1
1= 1 1 , 1 1
= = =
− − − −
kh
Ph
C1
h
h
j
hkh
h
hkh
k
k
d
i
k
h i i i P j
Pi i i
a
C12
12
1
1 , , , 1
1
= =
=
12
12
1
1
1
1 , , , 1
,1 1 1
h
kh
Ph
h
j
kh
h
kh
q
k
C
q
k
d
i
h i i i P j
i i i
= =
− − −
1-
( )
12
12
1
1
1
1 , , , 1
1 1 1 1
h
kh
Ph
h
j
kh
h
kh
q
k
C
k
dq
i
h i i i P j
i i i
v
= =
− − − −
,
Finally, we can obtain
12
12
1
1 , , , 1
11 h
h
j
hkh
h
hkh
k
k
d
i
k
h i i i P j
Pi i i
a
dC
= =
=
12
12
1
1
1
1
1 , , , 1
1 1 1 1
h
kh
P
h
h
j
kh
h
kh
q
d
k
C
q
k
d
i
h i i i P j
i i i
= =
− − − −
,
( )
12
12
1
1
1
1
1 , , , 1
1 1 1 1
h
kh
Ph
h
j
kh
h
kh
d
q
k
C
k
dq
i
h i i i P j
i i i
v
= =
− − − −
.
thus, the proof of Theorem 4 is completed. □
Proof of Theorem 5. Since q-ROFNs
i
a
are equal to
a
for all
1,2,...,in=
, then we can get
12
( , , , ) ( , ,..., )
d
k k k
q-ROFPMSM a a a
Symmetry 2018, 10, 383 28 of 32
12
12
1
1
1
1
1 , , , 1
1 1 1 1 ,
h
kh
P
h
h
kh
h
kh
q
d
k
C
q
k
d
h i i i P j
i i i
= =
= − − − −
( )
12
12
1
1
1
1
1 , , , 1
1 1 1 1
h
kh
P
h
h
kh
h
kh
d
q
k
C
k
dq
h i i i P j
i i i
v
= =
− − − −
( )
( )
1
11
11
11
1
11
1 1 1 1 1 1 1 1
h
hkk
hh
kk
hhPP
h
hh
PP
h
hh
qd
dk
q
k
C
dd
C
Ck
C
qk q
hh
v
==
= − − − − − − − −
，
( )
( )
1
11
11
1
11
1 1 1 1
h
hh
h
d
qq
dk
dd
kk
qk q
hh
v
==
= − − − −
，
( ) ( )
( )
1
11
1
11
1 1 ,
q
dd
dd
q
qq
hh
vv
==
= − − =
，
thus, the proof of Theorem 5 is completed. □
Proof of Theorem 6.
1 1 1 1
11
h h h h
ii
ii
jj
jj
qq
k k k k
aa
bb
j j j j
= = = =
− −
1 2 1 2
1 2 1 2
11
, , , 1 , , , 1
1 1 1 1
kk
hh
PP
hh
hh
ii
jj
k h k h
hh
kk
hh
CC
qq
kk
aa
i i i P j i i i P j
i i i i i i
= =
− − − −
12
12
1
1
1
1
1 , , , 1
1 1 1 1 ( )
h
kh
P
h
h
j
kh
h
kh
q
d
k
C
q
k
d
i
h i i i P j
i i i
a
= =
− − − −
,
12
12
1
1
1
1
1 , , , 1
1 1 1 1 ( )
h
kh
P
h
h
j
kh
h
kh
q
d
k
C
q
k
d
i
h i i i P j
i i i
b
= =
− − − −
which means the
ab
. So we can obtain
qq
ab
. Similar, we can obtain
ab
and
qq
ab
.
If
qq
ab
and
qq
ab
, then
1 2 1 2
( , , ) ( , , )
1 2 1 2
( , ,..., ) ( , ,..., )
dd
k k k k k k
nn
q-ROFPMSM a a a q-ROFPMSM b b b
Symmetry 2018, 10, 383 29 of 32
If
qq
ab
=
and
qq
ab
, then
1 2 1 2
( , , ) ( , , )
1 2 1 2
( , ,..., ) ( , ,..., )
dd
k k k k k k
nn
q-ROFPMSM a a a q-ROFPMSM b b b
If
qq
ab
=
and
qq
ab
=
, then
1 2 1 2
( , , ) ( , , )
1 2 1 2
( , ,..., ) ( , ,..., )
dd
k k k k k k
nn
q-ROFPMSM a a a q-ROFPMSM b b b=
thus, the proof of Theorem 6 is completed. □
Proof of Theorem 7. Based on the Theorem 5 and Theorem 6, we can obtain that
1 2 1 2
( , , , ) ( , , , )
12
( , ,..., ) ( , ,..., )
dd
k k k k k k
n
q-ROFPMSM a a a q-ROFPMSM a a a a
− − − −
=
and
1 2 1 2
( , , , ) ( , , , )
12
( , ,..., ) ( , ,..., )
dd
k k k k k k + + + +
n
q-ROFPMSM a a a q-ROFPMSM a a a a=
Thus, the proof of Theorem 7 is completed. □
Appendix B
Proof of Theorem 10. Since all q-ROFNs
i
a
( 1, 2,..., )in=
are equal to
( , )av
=
, we can get
( , ) 1
ij
Sup a a =
for
, 1,2,...,i j n=
. Based on Equation (28), we can get
1
in
=
( 1, 2,..., )in=
, then
12
( , , , ) ( , ,..., )
d
k k k
q-ROFPPMSM a a a
( ) ( )
11
11
11
11
11
1 1 1 1 , 1 1 1
hh
kk
hh
PP
hh
kk
hh
hh
PP
hh
qd
dq
kk
CC
dd
kk
CC
qq
hh
v
==
=
− − − − − − −
1-
( ) ( )
11
11
11
11
1 1 , 1 1
hh
hh
qd
dq
kk
dd
kk
qq
hh
v
==
=
− − − −
( ) ( )
( )
11
11
11
1 1 , ,
qd
dd
dq
qq
hh
vv
==
= − − =
thus, the proof of Theorem 7 is completed. □
Proof of Theorem 11.
( )
( )
( )
( )
1
1
1 , 1 ,
ij
iji
ij
j
j j j j
q
qn
nqn
n
q
i i i i
n a v v
−−
= − −
1- 1-
( )
1
h
jj
k
ii
jna
=
=
( )
( )
( )
( )
( )
( )
11
1
1
1 1 1 1
1 , 1 1 1 , 1 1
h h h h
ij
iji
ij
j
jj
qq
q
q
k k k k
n
nq qn
qn
q
ii
j j j j
vv
−−
= = = =
− − − − − −
1- 1-
Further, we can obtain
Symmetry 2018, 10, 383 30 of 32
( ) ( )
( )
12 1 2 1 2
12 1 2 1 2
1
1
, , , 1 , , , 1 , , , 1
1 1 1 , 1 1
hh
hijij
j j j j
kh
hk h k h
hh
khkk
hh
q
q
kk
knqn
q
i i i i
i i i P j i i i P j i i i P j
i i i i i i i i i
n a v
= = =
= − − − − −
1-
( )
( )
( )
( )
1 2 1 2
1 2 1 2
1
1
, , , 1 , , , 1
1 1 1 , 1 1
hh
ijij
k h k h
hh
kk
hh
q
q
kk
n
q qn
i i i P j i i i P j
i i i i i i
v
−−
= =
− − − − −
1-
( )
12
12
1
, , , 1
1
h
h
jj
hkh
h
hkh
k
k
ii
ki i i P j
Pi i i
na
C
=
( )
( )
1 2 1 2
1 2 1 2
1
11
11
, , , 1 , , , 1
1 1 1 1 1 1
hh
kk
hh
PP
hh
hh
ijij
jj
k h k h
hh
kk
hh
q
qk k
CC
kk
nqn
q
ii
i i i P j i i i P j
i i i i i i
v
= =
= − − − − − −
1- ，1-
( )
12
12
1
1
, , , 1
1 1 1 ( ) ,
h
kh
P
h
hij
kh
h
kh
qk
C
kn
q
i i i P j
i i i
−
=
− − −
1-
( )
12
12
1
1
1
, , , 1
1 1 1 ( )
h
kh
P
h
hij
kh
h
kh
q
k
C
kqn
i i i P j
i i i
v
−
=
− − −
1-
Finally, we can get
( )
12
12
1
1 , , , 1
11
h
h
jj
hkh
h
hkh
k
k
d
ii
k
h i i i P j
Pi i i
na
dC
= =
=
( )
12
12
1
1
1
1
1 , , , 1
1 1 1 1 1
h
kh
hijPh
j
kh
h
kh
q
d
k
k
dnC
q
i
h i i i P j
i i i
= =
− − − − −
1-
( )
12
12
1
1
1
1
1 , , , 1
1 1 1
h
kh
hiP
jh
j
kh
h
kh
d
q
k
k
dC
qn
i
h i i i P j
i i i
v
= =
− − −
1-
( )
( )
12
12
1
1
1
1
1 , , , 1
1 1 1 1 1
h
kh
Ph
hij
kh
h
kh
q
d
k
C
kn
dq
h i i i P j
i i i
−
= =
− − − − −
1-
Symmetry 2018, 10, 383 31 of 32
( )
( )
12
12
1
1
1
1
1 , , , 1
1 1 1
h
kh
Ph
hij
kh
h
kh
d
q
k
C
k
dqn
h i i i P j
i i i
v
−
= =
− − −
1-
Similarly, we can easily prove that
12
( , , , )
12
( , ,..., )
d
k k k
n
q-ROFPPMSM a a a y
.
Therefore, we can obtain
12
( , , , )
12
( , ,..., )
d
k k k
n
x q-ROFPPMSM a a a y
. □
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