INTUITIONISTIC FUZZY INTERACTION MACLAURIN SYMMETRIC MEANS AND THEIR APPLICATION TO MULTIPLE-ATTRIBUTE DECISION-MAKING
1 − ui + vi
j
j
 
1≤i1 <<in ≤n j =1
 


∏



))  +

) ( (

))

∏(

(

))

wi

j


+



))

(29)

3. A group decision-making approach based on the IFWIMSM operator
In this section, we will apply the IFIWMSM operator to solve the MAGDM problems.
3.1. Description of the MAGDM problems
A MAGDM problem with IFNs is described as follows. Suppose { A1 , A2 ,, Am } is the set of alternatives and {C1 , C2 ,, Cn } is the set of attributes which weight vector is w = (w1 , w2 ,, wn )T
with w j ≥ 0, j =
1,2, , n, and

n

∑ w j = 1 . Further, suppose {D1 , D2 ,, Dt } is the set of decision
j =1

makers (DMs) and ω = (ω1 , ω2 ,, ωt ) be the weight vector of them with ωk ≥ 0(k =
1,2,, t ),
t

∑ ωk =1 . Let Rk = rijk m×n be the decision matrix of MAGDM problems, where
k=1
rijk = ( uijk , vijk ) is the evaluation information expressed by the IFN with respect to alternative
and

Ai for attribute Cj given by the DM Dk. Then, the goal of this decision problem is to rank
alternatives.
Based on the IFWIMSM operator proposed in section 2, we will give its application in
the MAGDM problems and establish the detailed decision-making process shown as follows.
3.2. The decision-making steps based on the IFWIMSM operator
Step 1. Normalize the attribute values.
 In real decision-making, the attribute values have two types, i.e., cost attribute and benefit
attribute. In order to eliminate the difference in types, we need convert them to the same

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Technological and Economic Development of Economy, 2018, 24(4): 1533–1559

(

)

type. In general, we need convert cost type to benefit type. If the attribute value rijk = uijk , vijk
is cost type, it can be transformed to benefit one shown as follows (suppose the transformed
attribute value is still expressed by rijk ):
rijk = vijk , uijk.
(30)

(

)

Step 2. Aggregate the evaluation information of individual DM to collective information by
IFWIMSM operator shown as follows:
rij = IFWIMSM ( k ) (rij1 , rij2 ,, rijt ).
(31)

Step 3. Aggregate the evaluation information of each attribute to the comprehensive evaluation value of each alternative by IFWIMSM operator shown as follows:
zi = IFWIMSM ( k ) (ri1 , ri 2 ,, rin ).
(32)

Step 4. Calculate the score function S(zi )(i = 1,2,, m) of the collective overall values
zi (i = 1,2,, m), and then rank all the alternatives { A1 , A2 ,, Am } .When two score functions S(zi ) and S(z j ) are equal, it is necessary to calculate their accuracy functions H (zi )
and H (z j ), then we can rank them by accuracy functions.
Step 5. Rank the alternatives.
Rank all the alternatives { A1 , A2 ,, Am } and choose the best one(s) according to score
function S(zi ) and accuracy function H (zi ).
Step 6. End.
 4. An illustrative example
In order to show the application of the proposed method, we will give an example about the
route selection for five possible distribution schemes { A1 , A2 , A3 , A4 , A5 } . There are three DMs
Dk (k = 1,2,3) to evaluate these five alternatives according to four attributes which are shown
as follows: the risk analysis (C1 ) , the cost of transportation analysis (C2 ) , the convenience
of transportation analysis (C3 ) and the environmental impact analysis (C4 ) . The evaluation
results are used to construct three decision matrices
=
R k =
rijk  ( k 1,2,3 ) listed in Tables
5×4
1–3, where rijk can be expressed as IFN uijk , vijk . Suppose the weight vector of three DMs is
T

(

)

ω =( 0.35,0.40,0.25 ) and the weight vector of the attributes is w = (0.2,0.1,0.3,0.4)T . The
goal of this MAGDM problem is to select the optimization route for distribution schemes.
Table 1. Intuitionistic fuzzy decision matrix R1 given by D1

A1
A2
A3
A4
A5

C1

C2

C3

C4

( 0.5,0.4 )
( 0.6,0.3)
( 0.5,0.4 )
( 0.6,0.2 )
( 0.4,0.3)

( 0.6,0.3)
( 0.6,0.3)
( 0.2,0.6 )
( 0.7,0.2 )
( 0.7,0.2 )

( 0.3,0.6 )
( 0.6,0.2 )
( 0.6,0.2 )
( 0.5,0.4 )
( 0.4,0.5)

( 0.5,0.4 )
( 0.6,0.3)
( 0.4,0.4 )
( 0.4,0.4 )
( 0.4,0.5)

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P. Liu, W. Liu. Intuitionistic fuzzy interaction Maclaurin symmetric means and their application ...
 Table 2. Intuitionistic fuzzy decision matrix R2 given by D2

A1
A2
A3
A4
A5

C1

C2

C3

C4

( 0.4,0.2 )
( 0.5,0.3)
( 0.4,0.4 )
( 0.5,0.4 )
( 0.6,0.3)

( 0.6,0.2 )
( 0.6,0.2 )
( 0.3,0.5)
( 0.7,0.2 )
( 0.7,0.2 )

( 0.5,0.4 )
( 0.6,0.2 )
( 0.5,0.4 )
( 0.5,0.2 )
( 0.4,0.2 )

( 0.5,0.3)
( 0.5,0.4 )
( 0.4,0.2 )
( 0.7,0.2 )
( 0.6,0.2 )

Table 3. Intuitionistic fuzzy decision matrix R3 given by D3

A1
A2
A3
A4
A5

C1

C2

C3

C4

( 0.4,0.5)
( 0.5,0.4 )
( 0.4,0.5)
( 0.5,0.3)
( 0.6,0.2 )

( 0.5,0.2 )
( 0.5,0.3)
( 0.4,0.4 )
( 0.4,0.5)
( 0.5,0.3)

( 0.5,0.3)
( 0.6,0.2 )
( 0.5,0.3)
( 0.5,0.4 )
( 0.4,0.4 )

( 0.5,0.2 )
( 0.7,0.2 )
( 0.6,0.3)
( 0.5,0.3)
( 0.6,0.3)

4.1. The decision-making steps
To get the best alternative(s), the steps are shown as following.
Step 1. Normalize the attribute values.
All the measured values are the same type, so they do not need to do the normalization.
Step 2. Aggregate the evaluation information of individual DM to collective information by
IFIWMSM operator (without loss of generality, we suppose k = 3).
 r11 = ( 0.176,0.265 ) , r12 = ( 0.256,0.186 ), r13 = ( 0.177,0.331) , r14 = ( 0.212,0.230 ) ,

r21 = ( 0.227,0.264 ), r22 = ( 0.253,0.208 ) , r23 = ( 0.265,0.150 ) , r24 = ( 0.260,0.276 ) ,
r31 = ( 0.175,0.315 ) , r32 = ( 0.109,0.306 ) , r33 = ( 0.232,0.235 ) , r34 = ( 0.185,0.210 ) ,

r41 = ( 0.232,0.235 ) , r42 = ( 0.263,0.234 ) , r43 = ( 0.206,0.257 ) , r44 = ( 0.246,0.221) ,
r51 = ( 0.237,0.204 ) , r52 = ( 0.308,0.201) , r53 = ( 0.159,0.250 ) , r54 = ( 0.221,0.270 ).
Step 3. Aggregate the evaluation information of each attribute to the comprehensive evaluation value of each alternative by IFWIMSM operator (without loss of generality, we let k = 3).
z1 = ( 0.0530,0.7901) , z2 = ( 0.0679,0.7821), z3 = ( 0.0483,0.7885 ) , z4 = ( 0.0634,0.7857 ) ,
z5 = ( 0.0613,0.7837 ).
Step 4. Calculate the score function S(zi )(i = 1,2,,4) of the collective overall values
zi (i = 1,2,,5).
S(z1 ) = −0.7371 , S(z2 ) = −0.7141 , S(z3 ) = −0.7402 , S(z4 ) = −0.7223 , S(z5 ) = −0.7224.
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Step 5. Rank the alternatives.
On the basis of the score functions S(zi ) ( i = 1,2,3,4 ) , we can rank the alternatives
{A1 , A2 , A3 , A4 , A5 } shown as follows.
A2  A4  A5  A1  A3.
So, the best alternative is A2.
Step 6. End.
 4.2. The influence of the parameters k on decision-making result of this example
In order to clarify the influence of the parameter k on decision-making of this example, we
use the different values k in steps 2 and 3 to rank the alternatives. The ranking results are
shown in Table 4.
Table 4. Ordering of the alternatives by utilizing the different k
k

score function S(zi )

Ranking

k=1

S(z1 ) =
−0.0383, S(z2 ) =
−0.0125
S(z3 ) =
−0.0329, S(z4 ) =
−0.019

S(z5 ) = −0.0251

A2  A4  A5  A3  A1

k=2

S(z1 ) =
−0.7337, S(z2 ) =
−0.7075
S(z3 ) =
−0.7332, S(z4 ) =
−0.7178
S(z5 ) = −0.7211

A2  A4  A5  A3  A1

k=3

S(z1 ) =
−0.7371, S(z2 ) =
−0.7141
S(z3 ) =
−0.7402, S(z4 ) =
−0.7223
S(z5 ) = −0.7224

A2  A4  A5  A1  A3

As we can see from Table 4, the ranking results with the different parameter k are slightly
different, but the best choice is always A2. In addition, we can also find that the score functions on the basis of the IFWIMSM operator became smaller as the parameter k increases, so
we can regard parameter k as the DM’s risk preference. With the increase of the parameter
k, the decision-making results will be changed from the optimism to pessimism. In real
practical decision-making situations, DMs can choose the appropriate value in accordance
with their risk preferences.
 4.3. The verification of the validity
To prove the effectiveness of the developed method in this paper, we solve the same illustrative example by using the three existing MAGDM methods including the weighted intuitionistic fuzzy Maclaurin symmetric mean (WIFMSM) operator proposed by Qin and Liu
(2014), the intuitionistic fuzzy weighted average (IFWA) operator proposed by Xu (2007).
For convenience, we let p= q= 1, k= 3 , then the final ranking orders of the alternatives
obtained by the above three methods are described in Table 5.
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Table 5. Ranking results by different method
Method

Aggregation
operator

Score values S(Zi)
S(z1 )= 0.1489, S(z2 )= 0.3164,

Xu (2007)

IFWA

S(z3 )= 0.1377, S(z4 )= 0.2586,
S(z5 ) = 0.2211
S(z1 ) =
−0.3352, S(z2 ) =
−0.2678,

Qin and Liu
(2014)

WIFMSM

Proposed method

IFWIMSM

S(z3 ) =
−0.3704, S(z4 ) =
−0.2864,
S(z5 ) = −0.2944

Ranking

A2  A4  A5  A1  A3

A2  A4  A5  A1  A3

S(z1 ) =
−0.7371, S(z2 ) =
−0.7141
A2  A4  A5  A1  A3
S(z3 ) =
−0.7402, S(z4 ) =
−0.7223
S(z5 ) = −0.7224

From Table 5, we can find that there are the same ranking results by using three methods.
So the method in this paper is effective and feasible.
 4.4. Further compared with other methods
From the above subsections, we have testified the validity of our proposed method. However,
we find that they have the same ranking results, so it is difficult to illustrate the advantages
of our method and the drawbacks of the existing methods in some situations. So we give
two examples to show that our method is more extensive. In example 4, we will show the
advantages of MSM operator of IFNs comparing with BM operator of IFNs, and example
5 will illustrate the advantages of interaction operational rules of IFNs comparing with the
traditional rules.
Example 4. We solve the same illustrative example by the MAGDM methods based on the
weighted intuitionistic fuzzy interaction BM (WIFIBM) operator proposed by Y. D. He,
Z. He, and Chen (2015) and the proposed method in this paper because these two methods
adopt the same operational rules of IFNs. For easily comparing, we let k = 2 and k = 3. Then
we can get the results shown in Table 6.
From table 6, we can see that the ranking results based on WIFIBM operator (p = q = 1)
and IFWIMSM operator (k = 2) are same. Obviously, this conclusion can be easily explained
that these two methods consider interrelationship only for two attributes, even, we can also
know that when k = 2, the IFWIMSM operator will reduce to the WIFIBM (p = q = 1).
 However, when k = 3, the method based on IFWIMSM operator can consider interrelationship for three attributes, and the ranking result is different from the method proposed by He
et al. (2015) because it is only for two attributes. In addition, we also know that the WIFIBM
operator proposed by He et al. (2015) is a special case of the IFWIMSM operator proposed
in this paper. So the IFWIMSM is more flexible than WIFIBM operator.
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Table 6. Ranking results by different methods and parameters
Method and operators parameters

Score values S(Zi)

Ranking

Method by He et al. p = q = 1
(2015) based on WIFIBM

S(z1 ) =
−0.0423, S(z2 ) =
−0.0191
S(z3 ) =
−0.0372, S(z4 ) =
−0.0257
A2  A4  A5  A3  A1
S(z5 ) = −0.0311

Our proposed method k = 2
Based on IFWIMSM

S(z1 ) =
−0.7337, S(z2 ) =
−0.7075
S(z3 ) =
−0.7332, S(z4 ) =
−0.7178
A2  A4  A5  A3  A1
S(z5 ) = −0.7211

Our proposed method k = 3
Based on IFWIMSM

S(z1 ) =
−0.7371, S(z2 ) =
−0.7141
S(z3 ) =
−0.7402, S(z4 ) =
−0.7223
A2  A4  A5  A1  A3
S(z5 ) = −0.7224

Example 5. Assume that a company wants to choose a new salary plan and there are five choices
{A1 , A2 , A3 , A4 , A5 }, and four attributes (let their weight vector be w = (0.25,0.25,0.25,0.25)T )
are shown as follows: the employee satisfaction (C1), the feasibility analysis (C2), the influence
of surrounding environment (C3) and the influence of social-politic (C4).
 The DM D evaluates the plans Ai (i = 1, 2, 3, 4, 5) with respect to the attributes Cj (j = 1,
2, 3, 4) by the IFNs and the decision matrix R is listed in Table 7, where rij can be expressed
as uij , vij.
(

)

Table 7. Intuitionistic fuzzy decision matrix R given by D

A1
A2
A3
A4
A5

C1

C2

C3

C4

( 0.65,0.25)
( 0.62,0.33)
( 0.65,0.32 )
( 0.66,0.23)
( 0.58,0.23)

( 0.71,0.23)
( 0.64,0.30 )
( 0.53,0 )
( 0.77,0.18 )
( 0.62,0.24 )

( 0.53,0.32 )
( 0.56,0.34 )
( 0.57,0.34 )
( 0.48,0.24 )
( 0.72,0.12 )

( 0.62,0 )
( 0.68,0.2 )
( 0.71,0.25)
( 0.57,0.37 )
( 0.64,0.15)

The aggregation results for the different methods are shown in Table 8.
From Table 8, we obtain that the best choice based on the IFWA operator proposed by Xu
(2007) is A1 and it is same with WIFMSM operator (Qin & Liu, 2014). While the best choices
produced by the WIFIBM operator proposed by He et al. (2015) and IFWIMSM operator
proposed in this paper are both A5. Because the methods proposed by Xu (2007) and by Qin
and Liu (2014) are all based on traditional operations which do not consider the interactions between membership function and non-membership function of different IFNs, and
there exist some weaknesses in operations of IFNs when one of non-memberships is zero, it
is possible to produce the unreasonable results. In this example, because non-memberships
in the attributes r14 and r32 are zero, the aggregation results for the choices A1 and A3 are

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P. Liu, W. Liu. Intuitionistic fuzzy interaction Maclaurin symmetric means and their application ...
 unreasonable, so the ranking results for five choices are not reasonable. However, the methods proposed by He et al. (2015) and in this paper can consider the interactions between
membership function and non-membership function of different IFNs, and they can relieve
the weakness when the non-membership function of any one IFN is zero. So the ranking
results are reasonable, i.e., A5 is the best choice and not A1.
 Table 8. Ranking results by different methods
Method and adopted aggregation
Operators

Score values S(Zi)

S(z1 ) 0.6331,
=
S(z2 ) 0.3410
Method by Xu (2007) based =
=
S(z3 ) 0.6216,
=
S(z4 ) 0.3901
on
IFWA operator
S(z5 ) = 0.4663
Method by Qin and Liu (2014)
based on WIFMSM operator

S(z1 ) =
−0.3774, S(z2 ) =
−0.5166
S(z3 ) =
−0.4077, S(z4 ) =
−0.4923
S(z5 ) = −0.4271

S(z1 ) 0.9868,
=
S(z2 ) 0.9896
Method by He et al. (2015) =
=
S(z3 ) 0.9839,
=
S(z4 ) 0.9865
based on WIFIBM (p = q = 1)
operator
S(z5 ) = 0.9902
Our proposed method based
on IFWIMSM (k = 3) operator

S(z1 ) =
−0.5022, S(z2 ) =
−0.5421
S(z3 ) =
−0.5032, S(z4 ) =
−0.5203
S(z5 ) = −0.5002

Ranking

A1  A3  A5  A4  A2

A1  A3  A5  A4  A2

A5  A2  A1  A4  A3

A5  A1  A3  A4  A2

In addition, although the best choice for the methods proposed by He et al. (2015) and in
this paper is the same, the ranking for these two methods are different. The reason is that the
method proposed by He et al. (2015) is based on the interrelationship only for two attributes
and the method in this paper is based on interrelationship for three attributes.
In a word, our method can overcome the weakness of the some existing methods proposed by Xu (2007), Qin and Liu (2014) which are based on the traditional operational rules.
In addition, our method is more general because it can consider interrelationship from two
attributes to n attributes.
 In the following, we will give some comparisons of the three methods and our proposed
method, which are listed in Table 9.
According to the above analysis, the comparisons between our proposed method and the
existing three methods can be described as follows.
(1) Compared with the method based on the IFWA operator, we can find that the method
proposed by Xu (2007) can describe fuzzy information easier. However, this method is based
on the assumption that the attributes are independent and it doesn’t consider the interrelationship between them. The improved operator in this paper not only considers the interrelationship between two attributes but also can take interrelationship among multi-attributes
into account.
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Table 9. The comparisons of different methods
whether captures
interrelationship
of two attributes

whether captures
interrelationship of
multiple attributes

whether considerate the interaction
between the membership function
and non-membership function

Xu (2007)

No

No

No

Qin and Liu (2014)

Yes

Yes

No

He et al. (2015)

Yes

No

Yes

Proposed method

Yes

Yes

Yes

Methods

Moreover, in aggregation operations, the method proposed by Xu (2007) uses the traditional operational rules, while our method makes use of the interaction operational rules. As
we all know, the traditional operations do not consider the interactions between membership function and non-membership function of different IFNs, so they can get unreasonable
results in some special cases, especially when one of the non-memberships is zero. However,
the interaction operational rules can consider the interactions between membership function
and non-membership function sufficiently, so the method proposed in this paper is more
reasonable to produce the ranking result because it can overcome the weakness when one of
the non-memberships is zero.
 (2) Compared with the method proposed by Qin and Liu (2014) based on the WIFMSM
operator, obviously, these two methods are all based on MSM operator which can consider
the interrelationship among multiple attributes. However, the method proposed by Qin and
Liu (2014) is based on the traditional operational rules which do not consider the interactions between membership function and non-membership function of different IFSs, and it
can produce the unreasonable result when one of the non-memberships is zero, while the
method proposed in this paper can take account of the interactions between membership
function and non-membership function, and overcomes some existing problems by the traditional operational rules.
(3) Compared with the method proposed by He et al. (2015) based on the WIFIBM
operator. Obviously, these two methods are all based on the interaction operational rules
of IFNs which can overcome the existing problems when one of the non-memberships is
zero. However, the method proposed by He et al. (2015) adopts the BM operator which can
only consider the interrelationship between two attributes while the method proposed in
this paper not only considers the interrelationship between two attributes but also can take
interrelationship among multi-attributes into account. In addition, we can also know that
the WIFIBM operator is a special case of the WIFIMSM operator when k = 2 . Obviously,
the method based on the WIFIMSM operator is more general than the method based on
the WIFIBM operator.
According to the comparisons and analysis above, the IFWIMSM operator developed in
this paper is better than the existing other methods for aggregating the IFNs. Therefore, the
IFWIMSM operator is more suitable to deal with the problem of MAGDM in the intuitionistic fuzzy environment.
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 Conclusions
In this paper, we extended the MSM operator to IFNs based on the interaction operations
for the IFNs and proposed some interaction MSM operators for IFNs, such as the IFIMSM
and IFWIMSM operators, then we discussed some desirable characteristics of them, such as
idempotency and commutativity. Further we analyzed some special cases of these operators,
and proposed a method for the MAGDM problems based on the IFWIMSM operator. Comparing with the existing methods, the proposed method is more general than some existing
methods. The significant advantaged are that they can capture the interrelationship among
the multi-input arguments which have the flexibility by the parameters k, and they can also
consider interactions between membership function and non-membership function of IFNs
which can overcome some existing problems when one of the non-memberships is zero.
In further research, it is necessary and significant to take the applications of these operators to solve the real decision-making problems, such as evaluations on population resources
and environment (Zha & Kavuri, 2016; Zhang, He & Pan, 2017; Zhang & Shi, 2016; Zhu,
2017) or Chinese culture (Hou, 2016), or the proposed operators are extended to some new
fuzzy information (Liu & Chen, 2018; P. D. Liu, Zhang, X. Liu, & Wang, 2016).
Acknowledgments
This paper is supported by the National Natural Science Foundation of China
(Nos. 71771140, 71471172), the Special Funds of Taishan Scholars Project of Shandong Province (No. ts201511045), Shandong Provincial Social Science Planning Project
(Nos. 17BGLJ04,16CGLJ31 and 16CKJJ27), and Key research and development program of
Shandong Province (No. 2016GNC110016).
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