INTERNATIONAL JOURNAL OF COMPUTERS COMMUNICATIONS & CONTROL
ISSN 1841-9836, 12(2):254-264, April 2017.

A Multiple Attribute Group Decision Making Method Based on
2-D Uncertain Linguistic Weighted Heronian Mean Aggregation

Operator

W.H Liu, H.B. Liu, L.L. Li

WeiHua Liu*
School of Management Science and Engineering
Shandong University of Finance and Economics, Jinan 250014, China
the Second Ring Road No. 7366
Lixia District of Jinan, Shandong Province,China
*Corresponding author: WeihuaLiu@sdufe.edu.cn

HaiBo Liu
School of Chemistry and Chemical Engineering
QuFu Normal University, QuFu 273165, China
qflhb@163.com

LingLing Li
School of Management Science and Engineering
Shandong University of Finance and Economics, Jinan 250014, China
Lilingling@sdufe.edu.cn

Abstract: 2-Dimension uncertain linguistic variables can describe both subjective
evaluation result of attributes and reliability of the evaluation results in multiple
attribute decision making problems. However, it is difficult to aggregate these eval-
uation information and give comprehensive results. Heronian mean (HM) has the
characteristic of capturing the correlations between aggregated arguments and is ex-
tended to solve this problem. The 2-dimension uncertain linguistic weighted HM ag-
gregation(2DULWHMA) operator is employed in this paper. Firstly, the definition,
properties, expectations and the operational laws the 2-dimension uncertain linguistic
variables are investigated. Furthermore, the properties of the 2DULWHMA opera-
tors, such as commutativity, idempotency and monotonicity, etc. are studied. Some
special cases of the generalized parameters in these operators are analyzed. Finally,
an example is given to demonstrate the effectiveness and feasibility of the proposed
method.
Keywords: 2-Dimension (2-D) uncertain linguistic variables; Heronian mean; aggre-
gation operator.

1 Introduction

Multiple attribute decision making (MADM) refers to ranking and selecting the best alterna-
tives by utilizing the known information. It has been widely employed in economic, science and
technology etc. Since Churchman et al. [1] introduced the multi-attribute decision making and
employed it in enterprise investment decisions, the classical multi-attribute decision making has
attracted more and more research attention. However, owing to the complexity and uncertainty
of objective things and the fuzziness of human thought, a large number of MADM problems
are uncertain, which are called uncertain multiple attribute decision making problems. Those
uncertain multiple attribute decision making problems are difficult to evaluate alternatives for
decision makers using real numbers in many cases [25].It is more reasonable and natural expressed
by combining linguistic information (good, fair, poor) with fuzzy term (slightly, very, mightily,

Copyright © 2006-2017 by CCC Publications



A Multiple Attribute Group Decision Making Method Based on
2-D Uncertain Linguistic Weighted Heronian Mean Aggregation Operator 255

extremely, obviously) . The researches on the MADM based on linguistic variables have made
the fruitful achievements[614].

However, there exist a kind of the linguistic fuzzy MADM problems in practical decision-
making, such as review of the science technology project, blind evaluate of economic industry
system, etc. In these decision-making problems, decision makers not only assess all the indicators
of evaluation objects, but also estimate the familiarity with the given results. Therefore, Zhu
et al. [15] presented the concept of 2-dimension linguistic assessment information to solve this
kind of decision making problems. Liu et al. [16]extended 2-dimension linguistic information
to deal with the multiple attribute group decision making problems with unknown weight. 2-
dimension linguistic employs two class linguistic information to describe the judgment on the
object representing evaluation result and reliability of evaluation respectively. This can easily
distinguish indetermination between decision making problems and subjective understanding,
which is helpful to express opinions more accurately for decision makers. When 2-dimension
linguistic assessment information is described by uncertain linguistic variables, it is called as
2-dimension uncertain linguistic variables.

At present, research based on 2-dimension linguistic information mainly focus on the follow-
ing. Aggregation operators is a hot topic. 2-dimension uncertain linguistic power generalized
weighted aggregation operator is proposed and some properties are discussed[15]. 2-dimension
uncertain linguistic generalized hybrid weighted average operator is proposed and employed in
technological innovation ability evaluation[17]. Liu and Qi[18] proposed some generalized de-
pendent aggregation operators for 2-dimension linguistic information and developed a group
decision making method based on these operators. 2-dimension uncertain linguistic density
geometric aggregation operators and 2-dimension uncertain linguistic density generalized aggre-
gation operators is proposed and used in an example[19]. Another topic is operational rules,
the operational rules of 2-dimension uncertain linguistic variables are improved by transferring
it into intuitionistic fuzzy numbers to make the operations more accurate[20]. Yu et al. [21]
transformed 2-dimension linguistic information into generalized triangular fuzzy number, and
proposed 2-dimension linguistic weighted average (2DLWA) operator and 2-dimension linguistic
ordered weighted average (2DLOWA) operator.

Although the 2-dimension linguistic variables can reflect the evaluation of decision makers on
objects, some information to be aggregated can still be omitted easily, which makes the integrated
result cannot be better to express alternatives synthetically. However Heronian mean operator
can capture the correlations of the aggregated arguments, so we combine 2-dimension linguistic
variables with Heronian mean and propose some 2-dimension uncertain linguistic Heronian mean
aggregation operators and weighted Heronian mean aggregation operators, further apply them
to solve the MADM problems.

In order to do so, the remainder of this paper is shown as follows. In Sect. 2, we briefly
introduce the operational rules of 2-dimension linguistic variables and the Heronian mean. In
Sect. 3, some Heronian mean aggregation operators based on 2-dimension linguistic variables
are proposed and commutativity, idempotency and monotonicity are studied.In Sect.4 detailed
calculating steps are given to solve the group decision making problem with 2-dimension uncertain
linguistic information. In Sect. 5, we use an illustrate example to verify the efficiency of the
proposed method, and some special cases are also discussed. In the last section, the conclusions
are given.



256 W.H Liu, H.B. Liu, L.L. Li

2 2-Dimension uncertain linguistic variable and Heronian mean

2.1 Uncertain linguistic variable

When decision makers need to express the qualitative information, generally it is necessary
to set an appropriate linguistic assessment set in advance. Let the linguistic assessment set be
S = {sl |l = 1, 2, · · · ,L− 1} , where sl represents a linguistic variable, L is odd number. when
L = 7, it is represented as = (extremely poor, very poor, poor, medium, good, very good,
extremely good).

2.2 2-Dimension uncertain linguistic variable

In many cases, decision makers should give both evaluation conclusion of evaluation objects
and other similar the reliability of their evaluation. In order to reliably describe decision makers
judgment to the evaluated object, 2-dimension uncertain linguistic variable s̃ = (sF ,sT ) has been
proposed[22]. sF = {ṡi |i = 1, 2, · · · ,n− 1 }represents decision makers judgment to an evaluated
object and sT = {s̈i |i = 1, 2, · · · ,m− 1 } represents the subjective evaluation on the reliability
of their given results.

Definition 1. Let s̃ = ([ṡa, ṡb] , [s̈c, s̈d]) where ṡa, ṡb ∈ sF is I class uncertain linguistic in-
formation, s̈c, s̈d ∈ sT is II class uncertain linguistic information [21]. In order to minimize
the loss of linguistic information, the discrete linguistic assessment sets of 2-dimension un-
certain linguistic information are extended to continuous linguistic assessment sets, such that
ṡa, ṡb ∈ sF = {si |i ∈ [0, t]} and Let s̈c, s̈d ∈ sT = {sj |j ∈ [0, t]} be the set of all 2-dimension
uncertain linguistic variables.

Consider any three uncertain linguistic variables s̃ = ([ṡa, ṡb][s̈c, s̈d]),s̃i = ([ṡai, ṡbi][s̈ci, s̈di])
and s̃j = ([ṡaj, ṡbj ][s̈cj, s̈dj ]).let λ ≥ 0,then the operational rules are defined as follows[15][23˜24]:

s̃i ⊕ s̃j = ([ṡai, ṡbi][s̈ci, s̈di]) ⊕ ([ṡaj, ṡbj ][s̈cj, s̈dj ]) = ([ṡai+aj, ṡbi+bj ][s̈min(ci,cj), s̈min(di,dj)]) (1)

s̃i ⊗ s̃j = ([ṡai, ṡbi][s̈ci, s̈di]) ⊗ ([ṡaj, ṡbj ][s̈cj, s̈dj ]) = ([ṡai×aj, ṡbi×bj ][s̈min(ci,cj), s̈min(di,dj)]) (2)
λs̃ = λ([ṡa, ṡb][s̈c, s̈d]) = ([ṡλa, ṡλb][s̈c, s̈d]) (3)

s̃λ = ([ṡa, ṡb][s̈c, s̈d])
λ = ([ṡaλ, ṡbλ][s̈c, s̈d]) (4)

According to above equation from (1) to (4),following relationship can be easily proved:

s̃i ⊕ s̃j = s̃j ⊕ s̃i (5)

s̃i ⊗ s̃j = s̃j ⊗ s̃i (6)
λ(s̃i ⊕ s̃j) = λs̃i ⊕λs̃j (7)
λ1s̃⊕λ2s̃ = (λ1 + λ2)s̃ (8)
s̃λ1 ⊕ s̃λ2 = s̃(λ1+λ2) (9)

Definition 2. Let s̃ = ([ṡa, ṡb][s̈c, s̈d]) be a 2-dimension uncertain linguistic variable, m is the
length of sF and n is the length of sT ,then the expectation E(s̃) of s̃ is defined as

E(s̃) =
a + b

2(m− 1) ×
c + d

2(n− 1) (10)

For any 2-dimension uncertain linguistic variable s̃i and s̃j, if E(s̃i) ≥ E(s̃j) then s̃i ≥ s̃j ,or
vice versa.



A Multiple Attribute Group Decision Making Method Based on
2-D Uncertain Linguistic Weighted Heronian Mean Aggregation Operator 257

2.3 Heronian mean

Heronian Mean is one of aggregation methods, which can reflect the interrelationship of the
input arguments[25][26].

Definition 3. Let si(i = 1, 2, · · · ,n) be a collection of nonnegative numbers, then HM is defined
as

HM(s1,s2, · · · ,sn) =
2

n(n + 1)

n∑
i=1

n∑
j=i

√
aiaj (11)

Definition 4. Let p,q ≥ 0 and p , q do not take the value 0 simultaneously, then generalized
Heronian mean(GHM) is defined

GHM(s1,s2, · · · ,sn) =


 2
n(n + 1)

n∑
i=1

n∑
j=i

a
p
ia
q
j




1
p+q

(12)

3 The 2-dimension uncertain linguistic weighted Heronian mean
aggregation operators

Definition 5. Let p,q ≥ 0 and p,q do not take the value 0 simultaneously. s̃i = ([ṡai, ṡbi][s̈ci, s̈di])
be a collection of 2-dimension uncertain linguistic variables. 2-dimension uncertain linguistic
generalized Heronian mean aggregation (2DULGHMA) operators :Ωn → Ω is defined as

2DULGHMA(s̃1, s̃2, · · · , s̃n) =


 2
n(n + 1)

n∑
i=1,j=i

s̃
p
i s̃
q
j




1
p+q

(13)

According to the operational rules of 2-dimension uncertain linguistic variables, formula(13)
can be transformed into the following formŁş

2DULGHMA(s̃
1
, s̃

2
, · · · , s̃n) =

(
2

n(n+1)

n∑
i=1,j=i

s̃
p
i s̃
q
j

) 1
p+q

=




ṡ

( 2
n(n+1)

n∑
i=1,j=i

a
p
i a
q
j)

1
p+q

, ṡ
( 2
n(n+1)

n∑
i=1,j=i

b
p
i b
q
j)

1
p+q


 ,[min

i
s̈ci, min

i
s̈di

]
(14)

Theorem 6. (Commutativity). Let (s̃′1, s̃
′
2, · · · , s̃′n) is any permutation of (s̃1, s̃2, · · · , s̃n) then

2DULGHMA(s̃1, s̃2, · · · , s̃n) = 2DULGHMA(s̃′1, s̃′2, · · · , s̃′n)

Proof: Since (s̃′1, s̃
′
2, · · · , s̃′n) is any permutation of (s̃1, s̃2, · · · , s̃n) ,then

2DULGHMA(s̃1, s̃2, · · · , s̃n) =
(

2
n(n+1)

n∑
i=1,j=i

s̃
p
i s̃
q
j

) 1
p+q

=

(
2

n(n+1)

n∑
i=1,j=i

s̃i
′ps̃j

′q
) 1

p+q

= 2DULGHMA(s̃′1, s̃
′
2, · · · , s̃′n)

(15)

2



258 W.H Liu, H.B. Liu, L.L. Li

Theorem 7. (Idempotency). Let s̃i = ([ṡai, ṡbi][s̈ci, s̈di])(i = 1, 2, · · · ,n) be a collection of
uncertainty linguistic variables. If all s̃i are equal, for all i ,s̃i = s̃ = ([ṡa, ṡb][s̈c, s̈d]) then
2DULGHMA(s̃

1
, s̃

2
, · · · , s̃n) = s̃

Proof:
2DULGHMA(s̃

1
, s̃

2
, · · · , s̃n) = 2DULGHMA(s̃, s̃, · · · , s̃)

=

(
2

n(n+1)

n∑
i=1,j=i

s̃ps̃q

) 1
p+q

=
(

2
n(n+1)

(n + (n− 1) + · · ·1)s̃p+q
) 1
p+q

= s̃

(16)

2

Theorem 8. (Monotonicity). Let s̃i = ([ṡai, ṡbi][s̈ci, s̈di]) and s̃
′
i = ([ṡ

′
ai
, ṡ′bi][s̈

′
ci
, s̈′di])(i =

1, 2, · · · ,n) be a collection of uncertainty linguistic variables. If all s̃i are 2-dimension uncertain
linguistic variables. For all i,if s̃i ≤ s̃i′ ,then 2DULGHMA(s̃1, s̃2, · · · , s̃n) ≤ 2DULGHMA(s̃′1, s̃′2, · · · , s̃′n)

Proof:
2DULGHMA(s̃1, s̃2, · · · , s̃n)

=

(
2

n(n+1)

n∑
i=1,j=i

s̃
p
i s̃
q
j

) 1
p+q

≤
(

2
n(n+1)

n∑
i=1,j=i

s̃i
′ps̃i
′q
) 1

p+q

= 2DULGHMA(s̃′1, s̃
′
2, · · · , s̃′n)

(17)

2

Theorem 9. (Boundedness).Let 2DULGHMA operator s̃i = ([ṡai, ṡbi][s̈ci, s̈di])(i = 1, 2, · · · ,n)
,then min

i
(s̃
i
) ≤ 2DULGHMA(s̃

1
, s̃

2
, · · · , s̃n) ≤ max

i
(s̃
i
)

Proof: Let s̃min = min(s̃1, s̃2, · · · , s̃n) s̃max = max(s̃1, s̃2, · · · , s̃n) ,since s̃min ≤ s̃i ≤ s̃max,then

(
2

n(n+1)

n∑
i=1,j=i

s̃
p
mins̃

q
min

) 1
p+q

≤
(

2
n(n+1)

n∑
i=1,j=i

s̃
p
i s̃
q
j

) 1
p+q

≤
(

2
n(n+1)

n∑
i=1,j=i

s̃
p
maxs̃

q
max

) 1
p+q

(18)
2

Eq.(14) assumes that all of arguments being aggregated are of equal importance. However, in
many real cases, the importance degrees are not equal. Thus, we need to assign different weights
for different arguments, and further define a new aggregation operator to process this case.

Because unfair evaluation information may be provided by some decision makers, the weight
of evaluation information are larger when they are consistent with other evaluation information
given by other decision makers. Otherwise, smaller weight will be set. As result, the influence
of outlier values can be reduced, and the decision results based on these operators will be more
reasonable and reliable.

Definition 10. Let p,q ≥ 0 and p,q do not take the value 0 simultaneously. s̃i = ([ṡai, ṡbi][s̈ci, s̈di])
be a collection of 2-dimension uncertain linguistic variables whose weight vector is

ω = (ω1,ω2, · · · ,ωn)T satisfying ωi ∈ [0, 1] and
n∑
i=1

ωj = 1 , then 2DULWHMA is called the

2-dimension uncertain linguistic weighted HM aggregation (2DULWHMA) operator: Ωn → Ω .



A Multiple Attribute Group Decision Making Method Based on
2-D Uncertain Linguistic Weighted Heronian Mean Aggregation Operator 259

If

2DULWHMA(s̃1, s̃2, · · · , s̃n)

=

(
2

n(n+1)

n∑
i=1,j=i

(ωis̃i)
p
(ωjs̃j)

q

) 1
p+q

=




ṡ(

2
n(n+1)

n∑
i=1,j=i

(ωiai)
p(ωjaj)

q

) 1p+q , ṡ(
2

n(n+1)

n∑
i=1,j=i

(ωibi)
p(ωjbj)

q

) 1
p+q


 ,
[
min
i
s̈ci, mini

s̈
di

]
(19)

Similar to Theorems 69,we can prove that the 2DULWHMA operator with the following
properties:

Theorem 11. (Commutativity) Let (s̃′1, s̃
′
2, · · · , s̃′2) is any permutation of (s̃1, s̃2, · · · , s̃n) then

2DULWHMA(s̃1, s̃2, · · · , s̃n) = 2DULWHMA(s̃′1, s̃′2, · · · , s̃′2)

Theorem 12. (Idempotency) Let s̃i = s̃(i = 1, 2, · · · ,n) then 2DULWHMA(s̃1, s̃2, · · · , s̃n) = s̃

Theorem 13. (Monotonicity) Let s̃i and s̃′i (i = 1, 2, · · · ,n) are 2-dimension uncertain linguistic
variables. For all i, if s̃i ≤ s̃′i ,then 2DULWHMA(s̃1, s̃2, · · · , s̃n) ≤ 2DULWHMA(s̃′1, s̃′2, · · · , s̃′n)

Theorem 14. (Boundedness) The operator lies between the max and min operators:
min(s̃1, s̃2, · · · , s̃n) ≤ 2DULWHMA(s̃1, s̃2, · · · , s̃n) ≤ max(s̃1, s̃2, · · · , s̃n)

4 A Method for group decision making based on Weighted HM
under 2-dimension uncertain linguistic environment

In this section, we consider a group decision making problem with 2-dimension uncertain
linguistic information. 2DULWHMA proposed in section 3 will be used to solve a group decision
making problem.
Let A = {A1,A2, · · · ,Am} be a discrete set of alternatives and C = {C1,C2, · · · ,Cn} be the set
of attributes, whose weighting vector is ω = {ω1,ω2, · · · ,ωn}T such that ωi ≥ 0 and

n∑
i=1

ωi = 1

. Let D = {D1,D, · · · ,Dt} be the set of decision makers, and λ = (λ1,λ2, · · · ,λt) is the
expert weight vector such that λk ≥ 0 and

t∑
k=1

λk = 1 ,The decision matrix Dk = [d̃kij]m×n

is 2-dimension uncertain linguistic variable about the attribute Cj for the alternative Ai and
d̃kij = ([ṡak

ij
, ṡak

ij
][s̈ck

ij
, s̈dk

ij
]) where ṡa, ṡb ∈ sF and s̈c, s̈d ∈ sT is uncertain linguistic information.

Then the ranking of alternatives is required. The method involves the following steps:
Step 1: Utilize the 2DULWHMA to aggregate the evaluation values of each expert (j = 1, 2, · · · ,n)



260 W.H Liu, H.B. Liu, L.L. Li

d̃ij = 2DULWHMA(d̃
1
ij
, d̃2

ij
, · · · , d̃t

ij
)

=

(
2

t(t+1)

t∑
k=1,l=k

(λk
ij
d̃k
ij

)
p
(λl

ij
d̃l
ij

)q

) 1
p+q

=




[ṡ(

2
t(t+1)

t∑
k=1,l=k

(λkija
k
ij

)
p
(λl
ij
al
ij

)q

) 1
p+q

, ṡ
 2
t(t+1)

t∑
k=1,l=k

(λk
ij
bk
ij

)
p
(λl
ij
bl
ij

)q




1
p+q


 ,
[
min
k
s̈cik , mink

s̈
dik

]
(20)

Step 2: Aggregate the evaluation information of each attribute by 2DULWHMA operator based
on the following formula (i = 1, 2, · · · ,m)

d̃i = 2DULWHMA(d̃i1, d̃i2, · · · , d̃im)

=

(
2

m(m+1)

m∑
j=1,k=j

(ωjd̃ij)
p
(ωkd̃ik)

q

) 1
p+q

=




ṡ 2

m(m+1)

m∑
j=1,k=j

(ωjaij)
p(ωkaik)

q




1
p+q

, ṡ
 2
m(m+1)

n∑
j=1,k=j

(ωjbij)
p(ωkbik)

q




1
p+q


 ,
[
min
j
s̈cij, min

j
s̈dij

]
(21)

Step 3: Calculate the expectation E(d̃i) of 2-dimension uncertain linguistic variable according
to Equation(10).
Step 4: Rank all the alternatives and select the best ones in accordance with the ranking of E(s̃)
.

5 Examples illustration and discussion

In this section, the method are illustrated through an application case of evaluation in extra-
efficient economic industry system of Shandong province. From this example, we explains the
actual application and the effectiveness of the proposed method. In order to evaluate industry
ecological level, four typical industries are selected and expressed by {a1,a2,a3,a4}. a1 is for
ecological agriculture,a2 is for environmentally friendly industry, a3 is for energy saving indus-
try, and a4 is for circular economy. Three experts {e1,e2,e3} were invited to evaluate these
projects by following indexes expressed by { c1,c2,c3,c4}. c1 expressed Industrial efficiency re-
flectting the quantity and quality of industry economic growth, and it could be evaluated by
the energy consumption, material consumption and water consumption, and etc. c2 expressed
industrial structure coordination reflecting industry coordination among the economic benefits,
environmental benefits and social benefits, and it could be evaluated by industrial spatial distri-
bution harmony, industrial structure harmony and R&D to investment industrial gross output
etc. c3 expressed environment benefits reflecting impacts on the industry environmental over
entire product life cycle ,it could be evaluated by cleaner production, pollution control, reclama-
tion of wastes and condition of work safety etc. c4 expressed performance indicator measuring
the industry development contribution for social progress ,it could be evaluate by total tax and
profit payment, total assets contribution and green GDP per person etc.

Supposed that λ = (0.243, 0.514, 0.243) is the weight vector of the five experts, and ω =
(0.25, 0.27, 0.25, 0.23) is the index weight. The index values given by the experts take the form
of 2- dimension uncertain linguistic variables, and they are shown in Tables 1-3.

The experts adopted I class linguistic set and the II class linguistic set.



A Multiple Attribute Group Decision Making Method Based on
2-D Uncertain Linguistic Weighted Heronian Mean Aggregation Operator 261

According to step 1- step 4, we got the following result as table 4.

The 2DULWHMA were utilized to aggregate the arguments, some different overall attribute
value d̃i of the alternatives ai (i = 1, 2, 3, 4) were list in Table 4. From this table, we could find
that the overall attribute values of each alternative depended on the choice of the parameters p
and q,but the ranking was kept unchanged. According to table 4,a1 has higher ecological level.

Conclusions and future works

The multiple attribute decision making problems based on 2-dimension uncertain linguistic
variables have been applied to a wide range of areas. Compared with the traditional uncertain
linguistic variables, 2-dimension uncertain linguistic variables add a subjective evaluation on the
reliability of the evaluation results given by decision makers, so they can better express fuzzy
information.

In addition, the HM can take all the decision arguments and their relationships into ac-
count. Based on HM and 2-dimension uncertain linguistic variables, 2DULWHMA operators
are developed in this paper. Some desirable properties, such as commutativity, idempotency,
monotonicity and boundedness are discussed. Moreover, different p,q cannot affect ranking of
alternatives , which proved that it is a flexible multiple attribute decision making method in that
the decision makers, and they can choose different values of the parameters p and q according
to their actual needs. Finally, to demonstrate the effectiveness and feasibility of the developed
method, an example about industry ecological level is given. In further research, the proposed
operators and methods can be extended to other fuzzy information.

Table 1: the index values of industry given by expert e1

industry attribute(c1) attribute (c2) attribute (c3) attribute (c4)
a1 ([ṡ4, ṡ5], [s̈2, s̈3]) ([ṡ2, ṡ3], [s̈3, s̈3]) ([ṡ3, ṡ5], [s̈4, s̈4]) ([ṡ4, ṡ5], [s̈1, s̈2])
a2 ([ṡ3, ṡ4], [s̈2, s̈3]) ([ṡ3, ṡ5], [s̈3, s̈3]) ([ṡ4, ṡ4], [s̈4, s̈4]) ([ṡ4, ṡ4], [s̈1, s̈2])
a3 ([ṡ2, ṡ3], [s̈2, s̈3]) ([ṡ3, ṡ4], [s̈3, s̈3]) ([ṡ3, ṡ4], [s̈4, s̈4]) ([ṡ4, ṡ5], [s̈1, s̈2])
a4 ([ṡ5, ṡ6], [s̈3, s̈4]) ([ṡ1, ṡ2], [s̈3, s̈3]) ([ṡ3, ṡ4], [s̈4, s̈4]) ([ṡ3, ṡ4], [s̈1, s̈2])

Table 2: the index values of industry given by expert e2

industry attribute(c1) attribute (c2) attribute (c3) attribute (c4)
a1 ([ṡ4, ṡ4], [s̈3, s̈4]) ([ṡ3, ṡ4], [s̈2, s̈3]) ([ṡ4, ṡ4], [s̈3, s̈3]) ([ṡ5, ṡ6], [s̈3, s̈4])
a2 ([ṡ4, ṡ5], [s̈2, s̈3]) ([ṡ3, ṡ3], [s̈2, s̈3]) ([ṡ4, ṡ5], [s̈3, s̈3]) ([ṡ4, ṡ4], [s̈3, s̈3])
a3 ([ṡ3, ṡ4], [s̈3, s̈3]) ([ṡ4, ṡ5], [s̈2, s̈3]) ([ṡ2, ṡ3], [s̈3, s̈3]) ([ṡ3, ṡ4], [s̈3, s̈4])
a4 ([ṡ5, ṡ5], [s̈3, s̈4]) ([ṡ4, ṡ5], [s̈2, s̈3]) ([ṡ2, ṡ2], [s̈3, s̈3]) ([ṡ3, ṡ4], [s̈3, s̈4])

Table 3: the index values of industry given by expert e3

industry attribute(c1) attribute (c2) attribute (c3) attribute (c4)
a1 ([ṡ5, ṡ5], [s̈2, s̈3]) ([ṡ3, ṡ3], [s̈2, s̈2]) ([ṡ4, ṡ4], [s̈3, s̈4]) ([ṡ4, ṡ5], [s̈1, s̈1])
a2 ([ṡ4, ṡ4], [s̈2, s̈3]) ([ṡ4, ṡ5], [s̈2, s̈2]) ([ṡ1, ṡ2], [s̈3, s̈4]) ([ṡ3, ṡ3], [s̈1, s̈1])
a3 ([ṡ3, ṡ4], [s̈2, s̈3]) ([ṡ5, ṡ5], [s̈2, s̈2]) ([ṡ2, ṡ3], [s̈3, s̈4]) ([ṡ4, ṡ4], [s̈1, s̈1])
a4 ([ṡ3, ṡ4], [s̈2, s̈3]) ([ṡ2, ṡ3], [s̈2, s̈2]) ([ṡ4, ṡ5], [s̈3, s̈4]) ([ṡ3, ṡ5], [s̈1, s̈1])



262 W.H Liu, H.B. Liu, L.L. Li

Table 4: aggregate attribute values by the 2DULWHMA and the rankings of the industry

p,q industry d̃i(i = 1, 2, 3, 4) E(d̃) Ranking

p=0.5,q=0.5
a1 ([ṡ0.188, ṡ0.336], [s̈1, s̈1]) 0.01091

a1 > a2 > a4 > a3a2 ([ṡ0.131, ṡ0.240], [s̈1, s̈1]) 0.007745
a3 ([ṡ0.091, ṡ0.229], [s̈1, s̈1]) 0.006672
a4 ([ṡ0.115, ṡ0.248], [s̈1, s̈1]) 0.007554

p=1,q=1
a1 ([ṡ0.324, ṡ0.375], [s̈1, s̈1]) 0.014547

a1 > a2 > a4 > a3a2 ([ṡ0.300, ṡ0.349], [s̈1, s̈1]) 0.013526
a3 ([ṡ0.267, ṡ0.342], [s̈1, s̈1]) 0.012687
a4 ([ṡ0.282, ṡ0.347], [s̈1, s̈1]) 0.013122

p=5,q=5
a1 ([ṡ0.765, ṡ0.770], [s̈1, s̈1]) 0.0319827

a1 > a2 > a4 > a3a2 ([ṡ0.763, ṡ0.768], [s̈1, s̈1]) 0.031904
a3 ([ṡ0.759, ṡ0.767], [s̈1, s̈1]) 0.031798
a4 ([ṡ0.761, ṡ0.768], [s̈1, s̈1]) 0.031843

p=10,q=10
a1 ([ṡ0.873, ṡ0.874], [s̈1, s̈1]) 0.036389

a1 > a2 > a4 > a3a2 ([ṡ0.872, ṡ0.873], [s̈1, s̈1]) 0.036367
a3 ([ṡ0.871, ṡ0.873], [s̈1, s̈1]) 0.036336
a4 ([ṡ0.872, ṡ0.873], [s̈1, s̈1]) 0.036349

p=1,q=0.5
a1 ([ṡ0.252, ṡ0.326], [s̈1, s̈1]) 0.012046

a1 > a2 > a4 > a3a2 ([ṡ0.218, ṡ0.286], [s̈1, s̈1]) 0.010498
a3 ([ṡ0.180, ṡ0.277], [s̈1, s̈1]) 0.009517
a4 ([ṡ0.199, ṡ0.285], [s̈1, s̈1]) 0.010091

p=1,q=2
a1 ([ṡ0.444, ṡ0.474], [s̈1, s̈1]) 0.019121

a1 > a2 > a4 > a3a2 ([ṡ0.430, ṡ0.460], [s̈1, s̈1]) 0.018551
a3 ([ṡ0.407, ṡ0.455], [s̈1, s̈1]) 0.017963
a4 ([ṡ0.417, ṡ0.458], [s̈1, s̈1]) 0.01824

p=1,q=5
a1 ([ṡ0.647, ṡ0.658], [s̈1, s̈1]) 0.0272

a1 > a2 > a4 > a3a2 ([ṡ0.643, ṡ0.654], [s̈1, s̈1]) 0.02701
a3 ([ṡ0.633, ṡ0.652], [s̈1, s̈1]) 0.02677
a4 ([ṡ0.637, ṡ0.653], [s̈1, s̈1]) 0.026875

p=1,q=10
a1 ([ṡ0.783, ṡ0.787], [s̈1, s̈1]) 0.032726

a1 > a2 > a4 > a3a2 ([ṡ0.782, ṡ0.786], [s̈1, s̈1]) 0.03266
a3 ([ṡ0.778, ṡ0.785], [s̈1, s̈1]) 0.03257
a4 ([ṡ0.780, ṡ0.785], [s̈1, s̈1]) 0.032608

p=0.5,q=1
a1 ([ṡ0.252, ṡ0.786], [s̈1, s̈1]) 0.012046

a1 > a2 > a4 > a3a2 ([ṡ0.218, ṡ0.286], [s̈1, s̈1]) 0.010498
a3 ([ṡ0.180, ṡ0.277], [s̈1, s̈1]) 0.009517
a4 ([ṡ0.199, ṡ0.285], [s̈1, s̈1]) 0.010091

p=2,q=1
a1 ([ṡ1.233, ṡ1.234], [s̈1, s̈1]) 0.051394

a1 > a2 > a4 > a3a2 ([ṡ1.106, ṡ1.160], [s̈1, s̈1]) 0.047219
a3 ([ṡ0.917, ṡ1.106], [s̈1, s̈1]) 0.042142
a4 ([ṡ1.039, ṡ1.164], [s̈1, s̈1]) 0.045891

p=5,q=1
a1 ([ṡ1.108, ṡ1.108], [s̈1, s̈1]) 0.046167

a1 > a2 > a4 > a3a2 ([ṡ1.050, ṡ1.074], [s̈1, s̈1]) 0.044247
a3 ([ṡ0.958, ṡ1.050], [s̈1, s̈1]) 0.041829
a4 ([ṡ1.016, ṡ1.075], [s̈1, s̈1]) 0.043543

p=10,q=1
a1 ([ṡ1.057, ṡ1.057], [s̈1, s̈1]) 0.044041

a1 > a2 > a4 > a3a2 ([ṡ1.027, ṡ1.039], [s̈1, s̈1]) 0.043031
a3 ([ṡ0.977, ṡ1.027], [s̈1, s̈1]) 0.041738
a4 ([ṡ1.008, ṡ1.039], [s̈1, s̈1]) 0.04264



A Multiple Attribute Group Decision Making Method Based on
2-D Uncertain Linguistic Weighted Heronian Mean Aggregation Operator 263

Acknowledgment

This paper is supported by Social Sciences Planning Project of Shandong Province of China
(13CGLZ02).

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