Group Decision Making in Information
Systems Security Assessment Using
Dual Hesitant Fuzzy Set
Dejian Yu,1,∗ José M. Merigó,2,3 Yejun Xu4
1
School of Information, Zhejiang University of Finance and Economics,
Hangzhou, Zhejiang 310018, People’s Republic of China
2
Manchester Business School, University of Manchester, M15 6PB Manchester,
UK
3
Department of Management Control and Information Systems, School of
Economics and Business, University of Chile, 8330015 Santiago, Chile
4
Business School, Hohai University, Jiangning, Nanjing, Jiangsu 211100,
People’s Republic of China
Network information system security has become a global issue since it is related to the economic
development and national security. Information system security assessment plays an important role
in the development of security solutions. Aiming at this issue, a dual hesitant fuzzy (DHF) group
decision-making (GDM) method was proposed in this paper to assist the assessment of network
information system security. A systemic index containing four aspects was established including organization security, management security, technical security, and personnel management
security. The DHF group evaluation matrix was constructed based on the individual evaluation
information from each expert. Some power average operator–based DHF information aggregation operators are proposed and used to fusion the performance of each criterion for information
systems. The advantage of these operators is that they can describe the relationship between the
indexes quantitatively. Finally, a case study about information systems security assessment was
C 2016 Wiley Periodicals, Inc.
presented to verify the effectiveness of proposed GDM methods. 1.
INTRODUCTION
Since Torra and Narukawa1 and Torra2 proposed the hesitant fuzzy set (HFS),
many extensions have been appeared to describe the vagueness of our subjective
world. For example, Rodrı́guez et al.3 introduced the hesitant fuzzy linguistic term
set. Interval-valued HFS was first presented by Chen et al.,4 it is characterized by
the membership degrees were expressed by a set of possible intervals, is another
extension of HFS. Qian et al.5 proposed the concept of generalized HFS by combine
∗
Author to whom all correspondence should be addressed; [email protected]
INTERNATIONAL JOURNAL OF INTELLIGENT SYSTEMS, VOL. 31, 786–812 (2016)
C 2016 Wiley Periodicals, Inc.
View this article online at wileyonlinelibrary.com. • DOI 10.1002/int.21804
GROUP DECISION MAKING IN INFORMATION SYSTEMS SECURITY
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HFS with intuitionistic fuzzy set.6 Yu7 using the triangular fuzzy numbers rather than
determined numbers to express the membership degree and proposed the triangular
fuzzy HFS, based on which, a new decision-making method is proposed for a
teaching quality evaluation problem. Yang et al.8 introduced the hesitant fuzzy
rough set by combing the HFS and rough set.
Zhu et al.9 proposed the dual HFS (DHFS) by adding the nonmembership
degree to HFS. The DHFS was acknowledged as one of the effective mathematical
means for investigating vagueness, uncertain, and incomplete information.10 For
example, the review of a Ph.D. thesis in China is always taken by three experts
anonymously, and the experts cannot exchange ideas. Owing to the complexity of
reviewing a Ph.D. thesis, it is very hard for an expert to give exact evaluating values.
The first expert thinks that the possibility of the Ph.D. thesis meeting the requirement
is 0.6 and the possibility of the Ph.D. thesis does not meeting the requirement is
0.2. The second expert thinks that the possibility of the Ph.D. thesis meeting the
requirement is 0.7 and the possibility of the Ph.D. thesis does not meeting the
requirement is 0.2. The third expert thinks that the possibility of the Ph.D. thesis
meeting the requirement is 0.8 and the possibility of the PhD thesis does not meeting
the requirement is 0.1. In this situation, the evaluation information provided by
three experts for the Ph.D. thesis can be expressed by a dual hesitant fuzzy element
(DHFE){{0.6, 0.7, 0.8}, {0.1, 0.2}}. Therefore, it is far better for expressing the
above phenomenon using DHFS than the HFS.
DHFS has been received extensive attentions from scholars in recent years.
Wang et al.11 defined the correlation measures for DHFS, based on which a clustering algorithm was also studied for DHFS and applied it to the issue of diamond
evaluation and classification. Farhadinia12 studied the correlation coefficients of
DHFS and interval-valued DHFS. Ye13 studied the decision-making method based
on DHFS and correlation coefficient. Zhu and Xu (2013)14 developed typical DHFS
and introduced the operations to improve the DHFS theory.
DHF information aggregation is an interesting research area, which is also an
object of study of DHFS. Some aggregation operators have been proposed to aggregate DHF information, such as DHF-weighted averaging, DHF-weighted geometric,
DHF-ordered weighted averaging, DHF-ordered weighted geometric, DHFHA, and
DHFHG operators.15 Ju et al.16 proposed some aggregation operators for aggregating
interval-valued DHF information.
The above aggregation operators for DHF information and interval-valued
DHF information are with the assumption that the aggregated data are independent. However, in real situation, the aggregated data may correlate between each
other.7,17–20 Therefore, it is important to study some new approach which can deal
with this issue. The power average (PA)21 and the power geometric (PG)22 operators
are two aggregation operators, which can deal with the correlation phenomenon between aggregated arguments. The PA and PG operators have been studied by many
researchers and achieved many research results. The development of the power
operators is summarized in Figure 1. We could see that the research about power operator is quite extensive. However, despite current research achievements on power
operator, there are still some unsolved issues to be explored. To extend the power
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Figure 1. The main development of power operator.
operator to DHF and interval-valued DHF environment could enhance the research
about power operator theory. First of all, we develop some aggregation operators for
aggregating DHF and interval-valued DHF information based on power operator.
Furthermore, we pay our attention to the GDM method using DHF information. The
problem about information systems security assessment is investigated in this paper
based on the proposed GDM method.
The remainder of this paper is set as follows. In Section 2, we briefly review
basic concepts of DHFS, interval-valued dual DHFS, and power operators. In Section 3, we present some new power operators for aggregating DHF information.
Section 4 presents some new power operators for aggregating interval-valued DHF
information. In Section 6, we present a GDM method using DHF information; the
real case about information systems security assessment is presented in this section.
The conclusions are discussed in Section 6.
2.
SOME BASIC CONCEPTS
The DHFS was defined by Zhu et al.9 as follows:
DEFINITION 1. Suppose there is a fixed set X. A DHFS D on the fixed set X is
described as follows.
D = {x, h(x), g(x)|x ∈ X}.
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The h(x) and g(x) are expressed by several possible determined numbers and
indicated the membership degree and nonmembership degree, respectively.9
Zhu et al.9 introduced the comparative laws for any two DHFEs.
DEFINITION 2. Suppose there are two DHFEs d1 = (h1 , g1 ) and d2 = (h2 , g2 ). The
scores functions of the two DHFEs are defined as follows:
S(d1 ) =
1 1 γ1 −
η1 ,
#h1 γ ∈h
#g1 η ∈g
(2)
1 1 γ2 −
η2 .
#h2 γ ∈h
#g2 η ∈g
(3)
1
S(d2 ) =
2
1
2
1
1
2
2
If S(d1 ) > S(d2 ), then d1 d2 . Example 1 illustrates the comparative laws
described above.
Example 1. Let d1 = {{0.2, 0.3}, {0.4, 0.7}}, d2 = {{0.2, 0.6}, {0.2}}, and d3 =
{{0.5}, {0.4, 0.5}}, be three DHFEs. We can get the scores of three DHFEs according
to Definition 2.
1
1
(0.2 + 0.3) − (0.4 + 0.7) = −0.3,
2
2
1
1
S(d2 ) = (0.2 + 0.6) − (0.2) = 0.2,
2
1
1
1
S(d3 ) = (0.5) − (0.4 + 0.5) = 0.05.
1
2
S(d1 ) =
Since
S(d2 ) > S(d3 ) > S(d1 ),
We have
d2 d3 d1 .
DEFINITION 3. Let d, d1 , and d2 be three DHFEs, Then
(1)
(2)
(3)
(4)
d1 ⊕ d2 = ∪γ1 ∈h1 ,γ2 ∈h2 ,η1 ∈g1 ,η2 ∈g2 {{γ1 + γ2 − γ1 γ2 }, {η1 η2 }},
d1 ⊗ d2 = ∪γ1 ∈h1 ,γ2 ∈h2 ,η1 ∈g1 ,η2 ∈g2 {{γ1 γ2 }, {η1 + η2 − η1 η2 }},
nd = ∪γ ∈h,η∈g {{1 − (1 − γ )n }, {ηn }},
d n = ∪γ ∈h,η∈g {{γ n }, {1 − (1 − η)n }}.
Farhadinia3 extended the DHFS to a more generalized form and introduced the
concept of interval-valued DHFS (IVDHFS). The definition of IVDHFS is defined
as follows:
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DEFINITION 4. Suppose there is a fixed set X. A IVDHFS on the fixed set X is
described as follows:
= {x, h̃(x), g̃(x)|x ∈ X}.
(4)
The h̃(x) and g̃(x) are expressed by several possible intervals of real numbers
and indicated the membership degree range and nonmembership degree range,
respectively.
DEFINITION 5. Suppose there are two IVDHFEs d˜1 = (h̃1 , g̃1 ) and d˜2 = (h̃2 , g̃2 ). The
scores functions of the two IVDHFEs are defined as follows:
S(d˜1 ) =
1 1 γ̃1 −
η̃1
#g̃1 η̃ ∈g̃
#h̃1
γ̃1 ∈h̃1
1
1
1 γ̃1L + γ̃1U
1 η̃1L + η̃1U
=
−
,
2
#g̃1 η̃ ∈g̃
2
#h̃1
γ̃1 ∈h̃1
S(d˜2 ) =
1
(5)
1
1 1 γ̃2 −
η̃2
#g̃2 η̃ ∈g̃
#h̃2
γ̃2 ∈h̃2
2
2
1 η̃2L + η̃2U
1 γ̃2L + γ̃2U
−
.
=
2
#g̃2 η̃ ∈g̃
2
#h̃2
γ̃2 ∈h̃2
2
(6)
2
If S(d˜1 ) > S(d˜2 ), then d˜1 d˜2 . Example 2 illustrates the comparative laws
described above. In Equations (5) and (6), L and U are the lower and upper bounds
of intervals of real numbers, respectively.
Example 2. Let d˜1 = {{[0.2, 0.3], [0.3, 0.4], [0.2, 0.5]}, {[0.1, 0.2], [0.2, 0.5]}},
and
d̃3 = {{[0.1, 0.2]},
d˜2 = {{[0.2, 0.3], [0.5, 0.6]}, {[0.1, 0.2], [0.2, 0.3]}},
{[0.3, 0.4], [0.1, 0.3]}}, be three IVDHFEs. We can get the scores of three
IVDHFEs according to Definition 5.
0.2 + 0.3 0.3 + 0.4 0.2 + 0.5
+
+
2
2
2
1 0.1 + 0.2 0.2 + 0.5
−
+
= 0.0667,
2
2
2
1 0.1 + 0.2 0.2 + 0.3
1 0.2 + 0.3 0.5 + 0.6
S(d˜2 ) =
+
−
+
= 0.2,
2
2
2
2
2
2
1 0.3 + 0.4 0.1 + 0.3
0.1 + 0.2
−
+
= −0.125.
S(d˜3 ) =
2
2
2
2
S(d˜1 ) =
1
3
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Since
S(d˜2 ) > S(d˜1 ) > S(d̃3 ),
We have
d˜2 d˜1 d̃3 .
In the following, we define the operations for IVDHFEs; it is very important
for the rest of this paper.
DEFINITION 6. Let d̃, d˜1 and d˜2 be three IVDHFEs, Then
(1) d˜1 ⊕ d˜2 = ∪γ̃1 ∈h̃1 ,γ̃2 ∈h̃2 ,η̃1 ∈g̃1 ,η̃2 ∈g̃2 {{[γ̃1L + γ̃2L − γ̃1L γ̃2L , γ̃1U + γ̃2U − γ̃1U γ̃2U ]}, {[η̃1L η̃2L ,
η̃1U η̃2U ]}},
(2) d˜1 ⊗ d˜2 = ∪γ̃1 ∈h̃1 ,γ̃2 ∈h̃2 ,η̃1 ∈g̃1 ,η̃2 ∈g̃2 {{[γ̃1L γ̃2L , γ̃1U γ̃2U ]}, {[η̃1L + η̃2L − η̃1L η̃2L , η̃1U + η̃2U −
η̃1U η̃2U ]}},
n
n
n
n
(3) nd̃ = ∪γ̃ ∈h̃,η̃∈g̃ {{[1 − (1 − γ̃ L ) , 1 − (1 − γ̃ U ) ]}, {[(η̃L ) , (η̃U ) ]}},
n
L n
U n
L n
U n
(4) d̃ = ∪γ̃ ∈h̃,η̃∈g̃ {{[(γ̃ ) , (γ̃ ) ]}, {[1 − (1 − η̃ ) , 1 − (1 − η̃ ) ]}}.
Yager21 introduced the PA operator to aggregate a collection of data ai (i =
1, 2, . . . , n), defined as follows:
n
(1 + T (ai ))ai
P A(a1 , a2 , . . . , an ) = i=1
n
i=1 (1 + T (ai ))
(7)
where T (ai ) = nj=1,j =i sup(ai , aj ) and sup(ai , aj ) is the support for ai from aj ,
which satisfies the following conditions:23
(1) Sup(ai , aj ) ∈ [0, 1];
(2) Sup(ai , aj ) = Sup(aj , ai );
(3) Sup(ai , aj ) ≥ Sup(as , at ), if d(ai , aj ) < d(as , at ).
Xu and Yager22 defined a PG operator as follows:.
P G(a1 , a2 , . . . , an ) =
n
ai
n
(1+T (ai ))
i=1
n (1+T (a ))
i
i=1
(8)
i=1
3.
DHF POWER AGGREGATION OPERATORS
In this section, we first propose the distance measures for the DHFEs. Then,
we propose the DHF power weighted average (DHFPWA) operator and the DHF
power–weighted geometric (DHFPWG) operator. Next, we suggest the generalized
DHF power–weighted average (GDHFPWA) operator and the generalized DHF
power–weighted geometric (GDHFPWG) operator.
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3.1.
Distance Measures for DHFES
Motivated by the distance measure of HFS studied by Xu and Xia24 and
distance measure of Intuitionistic fuzzy set (IFS), we define some distance measure
for DHFEs.
DEFINITION 7. For two DHFEs d1 = (h1 , g1 ) and d2 = (h2 , g2 ), the distance measure
between d1 and d2 , denoted as d(d1 , d2 ),
d(d1 , d2 ) =
l
2
1 ρ(i)
ρ(i) h1 − h2 l i=1
1/2
l
2
1 ρ(i)
ρ(i) +
g1 − g2 s i=1
1/2
(9)
.
In Equation (9), l(h1 ) means the number of elements in h1 and l =
max{l(h1 ), l(h2 )}. Similarly, s(g1 ) and s = max{s(g1 ), l(g2 )} have corresponding
meanings.
Example 3. Let d1 = (h1 , g1 ) = {{0.1, 0.2, 0.3}, {0.4, 0.5}} and d2 = (h2 , g2 ) =
{{0.3, 0.6}, {0.2}} be two DHFEs. In the following, we calculate the distance of
these two DHFEs was given as follows:
Since there are three elements in h1 and only two elements in h2 , therefore, we
add the minimum value in h2 to h2 , in other words, we add 0.3 to h2 . Similarly, we
add 0.2 to g2 .
According to Equation (9), we have
1/λ
1
2
2
2
(0.1 − 0.3) + (0.2 − 0.3) + (0.3 − 0.6)
d(d1 , d2 ) =
3
1/λ
1
2
2
(0.4 − 0.2) + (0.5 − 0.2)
+
= 0.4710.
2
3.2.
DHF Power Information Aggregation Operators
In the following, we study the power operators under DHF environment.
DEFINITION 8. Let dj (j = 1, 2, . . . , n) be a collection of DHFEs, and w =
(w1 , w2 , . . . , wn )T be the standardized weight vector of dj (j = 1, 2, . . . , n), then
the DHFPWA operator is defined as follows:
DHFPWA (d1 , d2 , . . . , dn )
=
(w1 (1 + T (d1 ))) d1 ⊕ (w2 (1 + T (d2 ))) d2 ⊕ · · · ⊕ (wn (1 + T (dn ))) dn
n
j =1 wj (1 + T (dj ))
(10)
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where
n
T (dj ) =
j = 1, 2, . . . , n
wi sup(di , dj ),
(11)
i=1,i=j
Sup(di , dj ) = 1 − d(di , dj ) is the support for di from dj , d is a distance measure.
Especially, if w = ( n1 , n1 , . . . , n1 )T , then the DHFPWA reduces to an DHF power
average (DHFPA) operator:
DHFPA (d1 , d2 , . . . , dn )
=
(1 + T (d1 )) d1 ⊕ (1 + T (d2 )) d2 ⊕ · · · ⊕ (1 + T (dn )) dn
n
j =1 (1 + T (dj ))
(12)
n
1
sup(di , dj ),
T (dj ) =
n
i=1,i=j
(13)
where
j = 1, 2, . . . , n
Motivated by the idea of the PG operator, we define the DHFPWG operator as
follows:
DEFINITION 9. Let dj (j = 1, 2, . . . , n) be a collection of DHFEs, and w =
(w1 , w2 , . . . , wn )T be the standardized weight vector of dj (j = 1, 2, . . . , n), then
the DHFPWG operator is defined as follows:
DHFPWG (d1 , d2 , . . . , dn )
= (d1 )
w (1+T (d1 ))
n 1
j =1 wj (1+T (dj ))
⊗ (d2 )
w (1+T (d2 ))
n 2
j =1 wj (1+T (dj ))
⊗ · · · ⊗ (dn )
nwn (1+T (dn ))
j =1 wj (1+T (dj ))
(14)
where
T (dj ) =
n
wi sup(di , dj ), j = 1, 2, . . . , n
(15)
i=1,i=j
Sup(di , dj ) = 1−d(di , dj ) is the support for di from dj , d is a distance measure.
Especially, if w = ( n1 , n1 , . . . , n1 )T , then the DHFPWG reduces to a DHF power
geometric (DHFPG) operator:
DHFPG (d1 , d2 , . . . , dn )
= (d1 )
1+T (d1 )
n
j =1 (1+T (dj ))
⊗ (d2 )
1+T (d2 )
n
j =1 (1+T (dj ))
⊗ · · · ⊗ (dn )
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where
T (dj ) =
n
1
sup(di , dj ),
n
i=1,i=j
j = 1, 2, . . . , n
(17)
We can drive Theorem 1 based on the operation laws of DHFEs and Definitions
8 and 9.
THEOREM 1. Let dj (j = 1, 2, . . . , n) be a collection of DHFEs and w =
(w1 , w2 , . . . , wn )T be the standardized weight vector of dj (j = 1, 2, . . . , n), then,
their aggregated value by using the DHFPWA or DHFPWG operator is also an
DHFE, and
DHFPWA A(d1 , d2 , . . . , dn )
w (1+T (dj ))
w (1+T (dj ))
n j
n j
n
n
wj (1+T (dj ))
wj (1+T (dj ))
j
=1
j
=1
1 − (1 − γj )
, (ηj )
= ∪γj ∈hj , ηj ∈gj
j =1
j =1
(18)
DHFPWG (d1 , d2 , . . . , dn )
w (1+T (dj ))
w (1+T (dj ))
n j
n j
n
n
w (1+T (dj ))
, 1 − (1 − ηj ) j =1 wj (1+T (dj ))
= ∪γj ∈hj , ηj ∈gj
(γj ) j =1 j
j =1
j =1
(19)
Example 4. Let d1 = (h1 , g1 ) = {{0.1, 0.2, 0.3}, {0.4, 0.5}}, d2 = (h2 , g2 ) =
{{0.3, 0.6}, {0.2}}, and d3 = {{0.5}, {0.3, 0.5}} be three DHFEs. and w =
(0.24, 0.45, 0.31)T be the standardized weight vector of dj (j = 1, 2, 3), In the following, we first calculate the distances between the three DHFEs.
1/λ
1
2
2
2
(0.1 − 0.3) + (0.2 − 0.3) + (0.3 − 0.6)
d(d1 , d2 ) =
3
1/λ
1
(0.4 − 0.2)2 + (0.5 − 0.2)2
+
= 0.4710,
2
1/λ
1
(0.1 − 0.5)2 + (0.2 − 0.5)2 + (0.3 − 0.5)2
d(d1 , d3 ) =
3
1/λ
1
(0.4 − 0.3)2 + (0.5 − 0.5)2
+
= 0.3816,
2
1/λ
1
(0.3 − 0.5)2 + (0.6 − 0.5)2
d(d2 , d3 ) =
2
1/λ
1
(0.2 − 0.3)2 + (0.2 − 0.5)2
+
= 0.2288.
2
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Therefore,
sup(d1 , d2 ) = 0.5290, sup(d1 , d3 ) = 0.6184, sup(d2 , d3 ) = 0.7712.
and
T (d1 ) = 0.4298, T (d2 ) = 0.3660, T (d3 ) = 0.4954.
w1 (1 + T (d1 ))
n
= 0.2414,
j =1 wj (1 + T (dj ))
w2 (1 + T (d2 ))
n
= 0.4325,
j =1 wj (1 + T (dj ))
w3 (1 + T (d3 ))
n
= 0.3261.
j =1 wj (1 + T (dj ))
DHFPWA (d1 , d2 , . . . , dn )
= {{0.3335, 0.4768, 0.3522, 0.4915, 0.3727, 0.5076},
{0.2698, 0.3188, 0.2848, 0.3364}},
DHFPWG (d1 , d2 , . . . , dn )
= {{0.2718, 0.3668, 0.3213, 0.4337, 0.3544, 0.4783},
{0.2855, 0.3597, 0.3162, 0.3873}}.
3.3.
Generalized DHF Power Information Aggregation Operators
Based on the OWA operator,25 Yager26 proposed the generalized OWA (GOWA)
operator. Based on the GOWA operator, some generalized power operators have
been proposed.27 In this section, we study the generalized form of DHFPWA and
DHFPWG operators and propose the GDHFPWA and GDHFPWG operators.
DEFINITION 10. Let dj (j = 1, 2, . . . , n) be a collection of DHFEs, and w =
(w1 , w2 , . . . , wn )T be the standardized weight vector of dj (j = 1, 2, . . . , n), then
the GDHFPWA operator and the GDHFPWG operator are defined as follows:
⎞1/λ
⎛
n
wj 1 + T (dj )
n
djλ ⎠ ,
GDHFPWA (d1 , d2 , . . . , dn ) = ⎝
w
(1
+
T
(d
))
j
j
j =1
j =1
⎛
⎞
w 1+T (d )
n
1 ⎝ (nj=1j (wj (1+Tj (d))j )) ⎠
λdj
GDHFPWA (d1 , d2 , . . . , dn ) =
,
λ j =1
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where λ ∈ (0, +∞), and
T (dj ) =
n
wi sup(di , dj ),
j = 1, 2, . . . , n
(22)
i=1,i=j
Sup(d, dj ) = 1−d(di , dj ) is the support for di from dj , d is a distance measure.
Especially, if λ = 1, then the GDHFPWA reduces to a DHFPWA operator
proposed in Definition 8, and the GDHFPWG reduces to a DHFPWG operator
proposed in Definition 9.
THEOREM 2. Let dj (j = 1, 2, . . . , n) be a collection of DHFEs, and w =
(w1 , w2 , . . . , wn )T be the standardized weight vector of dj (j = 1, 2, . . . , n), then
their aggregated value by using the GDHFPWA operator or the GDHFPWG operator are also a DHFE, and
GHFPWA (d1 , d2 , . . . , dn )
1/λ w (1+T (dj ))
j
n
λ nj=1 wj (1+T (dj ))
= ∪γj ∈hj , ηj ∈gj
1 − (1 − γj )
,
j =1
1/λ (dj ))
nwj (1+T
n w
(1+T
(d
))
λ
j
j
.
1− 1− 1−(1−ηj ) j =1
j =1
(23)
GHFPWG (d1 , d2 , . . . , dn )
1/λ (dj ))
nwj (1+T
n w
(1+T
(d
))
λ
j
= ∪γj ∈hj , ηj ∈gj
1− 1− 1−(1−γj ) j =1 j
,
j =1
n
1 − (1 −
j =1
ηjλ )
w (1+T (dj ))
n j
j =1 wj (1+T (dj ))
1/λ .
(24)
Example 5. Let d1 = (h1 , g1 ) = {{0.3, 0.4}, {0.1, 0.2, 0.4}}, d2 = (h2 , g2 ) =
{{0.3, 0.4, 0.5}, {0.2}}, and d3 = {{0.4}, {0.1, 0.3}} be three DHFEs. and w =
(0.31, 0.38, 0.31)T be the standardized weight vector of dj (j = 1, 2, 3), Then,
1/2
1
(0.3 − 0.3)2 + (0.3 − 0.4)2 + (0.4 − 0.5)2
3
1/2
1
2
2
2
(0.1 − 0.2) + (0.2 − 0.2) + (0.4 − 0.2)
+
= 0.2107
3
1/2
1
(0.3 − 0.4)2 + (0.4 − 0.4)2
d(d1 , d3 ) =
2
d(d1 , d2 ) =
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797
1/2
1
(0.1 − 0.1)2 + (0.2 − 0.1)2 + (0.4 − 0.3)2
3
1/2
1
(0.3 − 0.4)2 + (0.4 − 0.4)2 + (0.5 − 0.4)2
d(d2 , d3 ) =
3
1/2
1
2
2
(0.2 − 0.1) + (0.2 − 0.3)
+
= 0.1816.
2
+
= 0.1524,
Therefore,
sup(d1 , d2 ) = 0.7893, sup(d1 , d3 ) = 0.8476, sup(d2 , d3 ) = 0.8184.
and
T (d1 ) = 0.6179, T (d2 ) = 0.4431, T (d3 ) = 0.5717.
w1 (1 + T (d1 ))
w2 (1 + T (d2 ))
n
= 0.3263, n
= 0.3567,
j =1 wj (1 + T (dj ))
j =1 wj (1 + T (dj ))
w3 (1 + T (d3 ))
n
= 0.3170.
j =1 wj (1 + T (dj ))
When λ = 1, then
GDHFPWA (d1 , d2 , d3 ) = {{0.3334, 0.3690, 0.4088, 0.3661, 0.4000, 0.4378},
{0.1280, 0.1814, 0.1605, 0.2274, 0.2013, 0.2852}},
and its score equals 0.1885.
GDHFPWG (d1 , d2 , d3 ) = {{0.3286, 0.3642, 0.3943, 0.3610, 0.4000, 0.4331},
{0.1370, 0.2031, 0.1696, 0.2332, 0.2440, 0.3019}},
and its score equals 0.1654.
When λ = 5, then
GDHFPWA (d1 , d2 , d3 ) = {{0.3453, 0.3778, 0.4332, 0.3755, 0.4000, 0.4467},
{0.1256, 0.1724, 0.1571, 0.2244, 0.1834, 0.2734}},
and its score equals 0.2070.
GDHFPWG (d1 , d2 , d3 ) = {{0.3247, 0.3590, 0.3788, 0.3557, 0.4000, 0.4276},
{0.1645, 0.2453, 0.1859, 0.2507, 0.3222, 0.3353}},
and its score equals 0.1237.
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Figure 2. Comparison between GDHFPWA and GDHFPWG (λ ∈ (0, 10)).
When λ = 10, then
GDHFPWA (d1 , d2 , d3 ) = {{0.3607, 0.3855, 0.4554, 0.3839, 0.4000, 0.4591},
{0.1223, 0.1611, 0.1520, 0.2203, 0.1657, 0.2585}},
and its score equals 0.2274.
GDHFPWG (d1 , d2 , d3 ) = {{0.3196, 0.3506, 0.3599, 0.3473, 0.4000, 0.4209},
{0.1804, 0.2680, 0.1925, 0.2684, 0.3577, 0.3596}},
and its score equals 0.0953.
Taking λ ∈ (0, 10), the comparison between GDHFPWA operator and GDHFPWG operator is shown in Figure 2.
4.
INTERVAL-VALUED DHF POWER AGGREGATION OPERATORS
Farhadinia3 extended the DHFS to a more generalized form and introduced
the interval-valued DHFS (IVDHFS). In this section, we focus on the intervalvalued DHF information aggregation problem based on power operators. First of
all, we give the definition of the distance measures for the IVDHFEs. Second, we
propose some power operators for aggregating interval-valued DHF information;
some generalized forms of these operators are also investigated.
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4.1.
Distance Measures for IVDHFEs
In Section 3.1, we studied the distance measures of DHFS, based on which, we
investigate the distance measures of IVDHFS.
DEFINITION 11. For two IVDHFEs d˜1 = (h̃1 , g̃1 ) and d˜2 = (h̃2 , g̃2 ), the distance
measure between d˜1 and d˜2 , is denoted as d(d˜1 , d˜2 ),
d(d˜1 , d˜2 ) =
l
2
1 ρ(i)
ρ(i) −
h̃
h̃
2 l i=1 1
1/2
l
2
1 ρ(i)
ρ(i) +
−
g̃
g̃
2 s i=1 1
1/2
(25)
In Equation (25), l(h̃1 ) means the number of elements in h̃1 and l =
max{l(h̃1 ), l(h̃2 )}. If l(h̃1 ) < l(h̃2 ), then h̃1 should be extended by adding the minimum value in it until it has the same length with h̃2 .28 s(g̃1 ) and s = max{s(g̃1 ), l(g̃2 )}
have corresponding meanings.
Example 6. Let d˜1 = {{[0.1, 0.3], [0.3, 0.4], [0.2, 0.5]}, {[0.1, 0.2], [0.2, 0.5]}},
d˜2 = {{[0.2, 0.3], [0.5, 0.6]}, {[0.1, 0.2], [0.2, 0.3]}} be two IVDHFEs. In the following, we calculate the distance of these two IVDHFEs.
According to Equation (25), we have
1/2
1
2
2
2
(0.1 − 0.2) + (0.2 − 0.2) + (0.3 − 0.5)
d(d̃1 , d̃2 ) =
3
1/2
1
(0.3 − 0.3)2 + (0.4 − 0.3)2 + (0.5 − 0.6)2
+
3
1/2
1
2
2
(0.1 − 0.1) + (0.2 − 0.2)
+
2
1/2
1
2
2
(0.2 − 0.2) + (0.5 − 0.3)
+
= 0.3522.
2
4.2. Interval-Valued DHF Power Information Aggregation Operators
In this section, we study the power operators using interval-valued DHF
information.
DEFINITION 12. Let d˜j (j = 1, 2, . . . , n) be a collection of DHFEs, and w =
(w1 , w2 , . . . , wn )T be the standardized weight vector of d˜j (j = 1, 2, . . . , n), then
the interval-valued DHF power weighted average (IVDHFPWA) operator and
interval-valued DHF power weighted geometric (IVDHFPWG) operator were
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defined as follows:
IVDHFPWA (d˜1 , d˜2 , . . . , d̃n )
w1 1 + T (d˜1 ) d˜1 ⊕ w2 1 + T (d̃2 ) d̃2 ⊕ · · · ⊕ wn 1 + T (d̃n ) d̃n
,
=
n
˜
j =1 wj (1 + T (d j ))
(26)
IVDHFPWG (d˜1 , d˜2 , . . . , d˜n
= (d˜1 )
w (1+T (d̃1 ))
n 1
˜
j =1 wj (1+T (d j ))
⊗ (d˜2 )
w (1+T (d̃2 ))
n 2
˜
j =1 wj (1+T (d j ))
⊗ · · · ⊗ (d˜n )
w (1+T (d̃n ))
n n
˜
j =1 wj (1+T (d j ))
.
(27)
where
T (d˜j ) =
n
j = 1, 2, . . . , n.
wi sup(d˜i , d˜j ),
(28)
i=1,i=j
Sup(d˜i , d˜j ) = 1 − d(d˜i , d˜j ) is the support for d˜i from d˜j , d is a distance measure.
Especially, if w = ( n1 , n1 , . . . , n1 )T , then the IVDHFPWA reduces to an intervalvalued DHF power average (IVDHFPA) operator:
IVDHFPA (d˜1 , d˜2 , . . . , d˜n )
1 + T (d̃1 ) d̃1 ⊕ 1 + T (d̃2 ) d̃2 ⊕ · · · ⊕ 1 + T (d̃n ) d̃n
.
=
n
˜
j =1 (1 + T (d j ))
(29)
The IVDHFPWG reduces to an interval-valued DHF power geometric (IVDHFPG) operator:
IVDHFPG (d˜1 , d˜2 , . . . , d˜n )
= (d˜1 )
1+T (d̃1 )
n
˜
j =1 (1+T (d j ))
⊗ (d˜2 )
1+T (d̃2 )
n
˜
j =1 (1+T (d j ))
⊗ · · · ⊗ (d˜n )
n 1+T (d̃n )
˜
j =1 (1+T (d j ))
,
(30)
where
T (d˜j ) =
n
1
sup(d˜i , d˜j ),
n
i=1,i=j
j = 1, 2, . . . , n.
(31)
THEOREM 3. Let d˜j (j = 1, 2, . . . , n) be a collection of IVDHFEs, and w =
(w1 , w2 , . . . , wn )T be the standardized weight vector of d˜j (j = 1, 2, . . . , n), then
their aggregated value by using the IVDHFPWA or IVDHFPWG operator is also
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an IVDHFE, and
IVDHFPWA (d˜1 , d˜2 , . . . , d˜n )
= ∪γ̃j ∈h̃j , η̃j ∈g̃j
n
×
1 − (1 − γ̃jL )
w (1+T (d˜j ))
n j
˜
j =1 wj (1+T (d j ))
j =1
n
j =1
(η̃Lj )
w (1+T (d˜j ))
n j
˜
j =1 wj (1+T (d j ))
n
,
j =1
IVDHFPWG (d˜1 , d˜2 , . . . , d˜n )
= ∪γ̃j ∈h̃j , η̃j ∈g̃j
n
j =1
n
1 − (1 − η̃Lj )
(γ̃jL )
(η̃Uj )
,
n
1 − (1 − γ̃jU )
w (1+T (d˜j ))
n j
˜
j =1 wj (1+T (d j ))
n
, j =1
j =1
j =1
w (1+T (d˜j ))
n j
˜
j =1 wj (1+T (d j ))
w (1+T (d˜j ))
n j
˜
j =1 wj (1+T (d j ))
w (1+T (d˜j ))
n j
˜
j =1 wj (1+T (d j ))
n
,
(γ̃jU )
(32)
w (1+T (d˜j ))
n j
˜
j =1 wj (1+T (d j ))
, 1 − (1 − η̃Uj )
w (1+T (d˜j ))
n j
˜
j =1 wj (1+T (d j ))
,
j =1
(33)
4.3. Generalized Interval-Valued DHF Power Information Aggregation
Operators
In this section, we extend the IVDHFPWA and IVDHFPWG operators and
propose the generalized IVDHFPWA and generalized IVDHFPWG operators.
DEFINITION 13. Let d˜j (j = 1, 2, . . . , n) be a collection of IVDHFEs and w =
(w1 , w2 , . . . , wn )T be the standardized weight vector of d˜j (j = 1, 2, . . . , n), then
the generalized interval-valued DHF power weighted average (GIVDHFPWA)
operator and the generalized interval-valued DHF power weighted geometric
(GIVDHFPWG) operator are defined as follows:
⎛
GIVDHFPWA (d˜1 , d˜2 , . . . , d˜n ) = ⎝
n
j =1
⎞1/λ
˜
wj 1 + T (d j )
d˜λ ⎠ , (34)
n
wj (1 + T (d˜j )) j
j =1
⎛
⎞
(d˜j )))
n
(nwj (w1+T
1
˜
GIVDHFPWG (d˜1 , d˜2 , . . . , d˜n ) = ⎝
λd˜j j =1 j (1+T (d j )) ⎠ ,
λ j =1
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where λ ∈ (0, +∞), and
n
T (d˜j ) =
j = 1, 2, . . . , n.
wi sup(d˜i , d˜j ),
(36)
i=1,i=j
Sup(d̃, d˜j ) = 1−d(d˜i , d˜j ) is the support for d˜i from d˜j , d is a distance measure.
THEOREM 4. Let d˜j (j = 1, 2, . . . , n) be a collection of IVDHFEs and w =
(w1 , w2 , . . . , wn )T be the standardized weight vector of d˜j (j = 1, 2, . . . , n), then
their aggregated value by using the GIVDHFPWA operator or the GIVDHFPWG
operator are also an IVDHFE, and
GIVHFPWA (d˜1 , d˜2 , . . . , d˜n )
⎧⎧⎡
1/λ
w (1+T (d˜ ))
⎨⎨
L λ nj=1j wj (1+Tj (d˜j ))
n ⎣ 1 − 1 − γ̃j
,
= ∪γ̃j ∈h̃j , η̃j ∈g̃j
⎩⎩
j =1
w (1+T (d˜ ))
λ nj=1j wj (1+Tj (d˜j ))
1 − 1 − γ̃jU
n
j =1
1/λ ⎤⎫
⎬
⎦ ,
⎭
⎧⎡ 1/λ
(d˜j ))
⎨
nwj (1+T
n ˜ ))
w
(1+T
(
d
L
λ
j
⎣1− 1− 1 − (1 − η̃j ) j =1 j
,
⎩
j =1
nwj (1+T (d˜j ))˜
n
U λ
j =1 wj (1+T (d j ))
1 − 1− 1−(1−η̃j )
j =1
1/λ ⎤⎫⎫
⎬⎬
⎦
.
⎭⎭
(37)
GIVHFPWG (d̃1 , d̃2 , . . . , d̃n )
⎧⎧⎡
1/λ
⎨⎨
wj (1+T (d˜j ))
n λ nj=1 wj (1+T (d˜j ))
L
⎣1 − 1− 1−(1−γ̃j )
,
= ∪γ̃j ∈h̃j , η̃j ∈g̃j
⎩⎩
j =1
(d j ))
nwj (1+T
wj (1+T (d˜j ))
U λ
j
=1
1 − 1− 1 − (1 − γ̃j )
˜
n
j =1
1/λ ⎤⎫
⎬
⎦ ,
⎭
⎧⎡
1/λ
w (1+T (d˜ ))
⎨
L λ nj=1j wj (1+Tj (d˜j ))
n ⎣ 1 − 1 − η̃j
,
⎩
j =1
w (1+T (d˜ ))
U λ nj=1j wj (1+Tj (d˜j ))
n 1 − 1 − η̃j
j =1
1/λ ⎤⎫⎫
⎬⎬
⎦
.
⎭⎭
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803
Example 7. Let d˜1 = {{[0.2, 0.3], [0.3, 0.4], [0.2, 0.5]}, {[0.1, 0.2], [0.2, 0.5]}},
d˜2 = {{[0.2, 0.3], [0.5, 0.6]}, {[0.1, 0.2], [0.2, 0.3]}}, d̃3 = {{[0.1, 0.2]},
{[0.3, 0.4], [0.1, 0.3]}}, be three IVDHFEs. and w = (0.31, 0.38, 0.31)T be
the standardized weight vector of d˜j (j = 1, 2, 3), Then,
d(d1 , d2 ) = 0.3385, d(d1 , d3 ) = 0.5282, d(d2 , d3 ) = 0.7538.
Therefore,
sup(d1 , d2 ) = 0.6615, sup(d1 , d3 ) = 0.4718, sup(d2 , d3 ) = 0.2462.
and
T (d1 ) = 0.3976, T (d2 ) = 0.2814, T (d3 ) = 0.2398.
w1 (1 + T (d1 ))
n
= 0.3321,
j =1 wj (1 + T (dj ))
w2 (1 + T (d2 ))
n
= 0.3733,
j =1 wj (1 + T (dj ))
w3 (1 + T (d3 ))
n
= 0.2946.
j =1 wj (1 + T (dj ))
When λ = 1, then
GDHFPWA (d1 , d2 , d3 )
= {{[0.1718, 0.2719], [0.3050, 0.4092],
[0.1718, 0.3082], [0.3050, 0.4387], [0.2077, 0.3489], [0.3352, 0.4716]},
{[0.1000, 0.2254], [0.1382, 0.2453], [0.1295, 0.2622], [0.1790, 0.2854],
[0.1259, 0.3055], [0.1740, 0.3326], [0.1631, 0.3555], [0.2254, 0.3869]}}.
and its score equals 0.1506.
GDHFPWG (d1 , d2 , d3 )
= {{[0.1631, 0.2662], [0.2296, 0.3448], [0.1631, 0.2929],
[0.2296, 0.3794], [0.1866, 0.3154], [0.2627, 0.4086]}, {[0.1000, 0.2309],
[0.1642, 0.2650], [0.1387, 0.2683], [0.2002, 0.3007], [0.1345, 0.3420],
[0.1963, 0.3712], [0.1718, 0.3740], [0.2309, 0.4018]}}.
and its score equals 0.0540.
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Figure 3. Comparison between GIVDHFPWA and GIVDHFPWG (λ ∈ (0, 10)).
Taking λ ∈ (0, 10), the comparison between GIVDHFPWA operator and
GIVDHFPWG operator is shown in Figure 3.
5.
INFORMATION SYSTEMS SECURITY ASSESSMENT USING THE
DHF GDM METHOD
With the rapid development of information and computer technology, more
and more people are on the information systems. At the same time, people have
paid more and more attention to the security of information system. The safety of
information system is very critical to an organization since any defects on privacy,
integrity, and some other aspects may cause negative impacts. Information security assessment is the base and prerequisite of information system security, and it
provides a fundamental basis for risk control and management.
Owing to the complex nature of information security assessment, it often happened under uncertainty environment and responsible by a group of experts. Based
on the existing research achievements,29,30 we adopt the following four criteria to
evaluate the security of information systems including (1) organization securityc1 ,
(2) management securityc2 , (3) technical security c3 , and (4) personnel management security c4 . Generally speaking, the four criteria have different importance
to the security of information systems; however, the study on weight information
of the different criteria is beyond the scope of this study. Therefore, we suppose
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Table I. Review table for expert e1 .
No.
1
2
3
4
Criteria
Weight
Organization
security (c1 )
Management
security (c2 )
Technical security
(c3 )
Personnel
management
security (c4 )
0.15
0.25
0.40
0.20
Description
Evaluation results
The satisfaction degree of four
information systems regarding
organization security
The dissatisfaction degree of four
information systems regarding
organization security
The satisfaction degree of four
information systems regarding
management security
The dissatisfaction degree of four
information systems regarding
management security
The satisfaction degree of four
information systems regarding
technical security
The dissatisfaction degree of four
information systems regarding
technical security
The satisfaction degree of four
information systems regarding
personnel management security
The dissatisfaction degree of four
information systems regarding
personnel management security
a1
a2
0.7
0.3
a3
a4
0.3
0.3
a1
a2
0.1
0.1
a3
a4
0.6
0.3
a1
a2
0.2
0.1
a3
a4
0.3
0.3
a1
a2
0.5
0.6
a3
a4
0.5
0.5
a1
a2
0.6
0.5
a3
a4
0.5
0.3
a1
a2
0.4
0.3
a3
a4
0.1
0.5
a1
a2
0.3
0.1
a3
a4
0.3
0.1
a1
a2
0.4
0.6
a3
a4
0.4
0.4
the weight vector of the four criteria is (0.15, 0.25, 0.40, 0.20)T . Suppose there
are four information systems (x1 , x2 , x3 , x4 ) need to be assessed and this assessment is taken by a group of three experts (e1 , e2 , e3 ). To obtain the information
given by the expert, the Tables I–III are designed and assigned to the three experts correspondingly. The experts are requested to select one number from the
set = {0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0} to fill the table. The
goal of this research is to evaluate the performance of the four information systems
regarding their security. To do this, the following steps are adopted.
Step 1. Collect information from experts and transform it to DHF information:
Tables I–III are received from the three experts, based on which, the DHF
decision-making matrix can be obtained and is shown in Table IV.
Take the assessment information of the first information system a1 regarding
organization security, for example, three experts all regard the satisfaction degree is
0.7. However, expert e1 think the dissatisfaction degree is 0.1, expert e2 believe the
dissatisfaction degree is 0.3 and the expert e3 think the dissatisfaction degree is 0.2.
In this situation, the comprehensive assessment information can be expressed by a
DHFE {{0.7}, {0.1, 0.2, 0.3}}. Likewise, all the evaluation results from the three
experts can be converted to DHF information and is shown in Table IV.
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Table II. Review table for expert e2 .
No.
1
2
3
4
Criteria
Weight
Organization
security (c1 )
0.15
Management
security (c2 )
0.25
Technical security
(c3 )
Personnel
management
security (c4 )
0.40
0.20
Description
Evaluation results
The satisfaction degree of four
information systems regarding
organization security
The dissatisfaction degree of four
information systems regarding
organization security
The satisfaction degree of four
information systems regarding
management security
The dissatisfaction degree of four
information systems regarding
management security
The satisfaction degree of four
information systems regarding
technical security
The dissatisfaction degree of four
information systems regarding
technical security
The satisfaction degree of four
information systems regarding
personnel management security
The dissatisfaction degree of four
information systems regarding
personnel management security
a1
a2
0.7
0.5
a3
a4
0.2
0.4
a1
a2
0.3
0.1
a3
a4
0.4
0.2
a1
a2
0.2
0.1
a3
a4
0.3
0.3
a1
0.5
0.7
a3
a4
0.5
0.6
a1
a2
0.6
0.5
a3
a4
0.6
0.4
a1
a2
0.4
0.3
a3
a4
0.2
0.6
a1
a2
0.2
0.1
a3
a4
0.4
0.1
a1
a2
0.6
0.7
a3
a4
0.3
0.5
Step 2. Calculate the T (DHFEij ) based on the DHF decision matrix:
⎡
0.3772
⎢0.2398
T (dij ) = ⎣
0.4868
0.5067
0.4109
0.3224
0.4778
0.5667
0.3293
0.2223
0.3129
0.4496
⎤
0.4096
0.3724⎥
0.5790⎦
0.5101
Take the first information system, for example, the T (DHFE11 ), T (DHFE12 ),
T (DHFE13 ), and T (DHFE14 ) were calculated as follows. The rest can be calculated
in the same way.
Since
DHFE1 = {{0.7}, {0.1, 0.2, 0.3}}, DHFE2 = {{0.2, 0.3}, {0.5}},
DHFE3 = {{0.6}, {0.4}}, DHFE4 = {{0.1, 0.2, 0.3}, {0.4, 0.6}}.
Then, the distance measures can be obtained based on Definition 7.
d12 = 0.7637, d13 = 0.3160, d14 = 0.7774,
d23 = 0.4536, d24 = 0.2158, d34 = 0.5497.
International Journal of Intelligent Systems
DOI 10.1002/int
GROUP DECISION MAKING IN INFORMATION SYSTEMS SECURITY
807
Table III. Review table for Expert e3 .
No.
1
2
3
4
Criteria
Weight
Organization
security (c1 )
Management
security (c2 )
Technical security
(c3 )
Personnel
management
security (c4 )
0.15
0.25
0.40
0.20
Description
Evaluation results
The satisfaction degree of four
information systems regarding
organization security
The dissatisfaction degree of four
information systems regarding
organization security
The satisfaction degree of four
information systems regarding
management security
The dissatisfaction degree of four
information systems regarding
management security
The satisfaction degree of four
information systems regarding
technical security
The dissatisfaction degree of four
information systems regarding
technical security
The satisfaction degree of four
information systems regarding
personnel management security
The dissatisfaction degree of four
information systems regarding
personnel management security
a1
a2
0.7
0.7
a3
a4
0.1
0.4
a1
a2
0.2
0.1
a3
a4
0.6
0.3
a1
a2
0.3
0.1
a3
a4
0.4
0.3
a1
a2
0.5
0.6
a3
a4
0.5
0.7
a1
a2
0.6
0.5
a3
a4
0.6
0.3
a1
a2
0.4
0.4
a3
a4
0.3
0.6
a1
a2
0.1
0.1
a3
a4
0.4
0.1
a1
a2
0.4
0.7
a3
a4
0.4
0.7
Table IV. DHF decision matrix.
a1
a2
a3
a4
c1
c2
c3
c4
{0.7, 0.1,0.2, 0.3}
{0.3,0.5,0.7, 0.1}
{0.1,0.2,0.3, 0.4,0.6}
{0.3,0.4, 0.2,0.3}
{0.2,0.3, 0.5}
{0.1, 0.6,0.7}
{0.3,0.4, 0.5}
{0.3, 0.5,0.6,0.7}
{0.6, 0.4}
{0.5, 0.3,0.4}
{0.5,0.6, 0.1,0.2,0.3}
{0.3,0.4, 0.5,0.6}
{0.1,0.2,0.3, 0.4,0.6}
{0.1, 0.6,0.7}
{0.3,0.4, 0.3,0.4}
{0.1, 0.4,0.5, 0.7}
According to Equation (11) and weight vector of the four criteria
(0.15, 0.25, 0.40, 0.20)T , we have,
T (DHFE11 ) = w2 × sup(DHFE1 , DHFE2 ) + w3 × sup(DHFE1 , DHFE3 )
+ w4 × sup(DHFE1 , DHFE4 ) = 0.3772
T (DHFE12 ) = w1 × sup(DHFE2 , DHFE1 ) + w3 × sup(DHFE2 , DHFE3 )
+ w4 × sup(DHFE2 , DHFE4 ) = 0.4109
T (DHFE13 ) = w1 × sup(DHFE3 , DHFE1 ) + w2 × sup(DHFE3 , DHFE2 )
+ w4 × sup(DHFE3 , DHFE4 ) = 0.3293
T (DHFE14 ) = w1 × sup(DHFE4 , DHFE1 ) + w2 × sup(DHFE4 , DHFE2 )
+ w3 × sup(DHFE4 , DHFE3 ) = 0.4096
International Journal of Intelligent Systems
DOI 10.1002/int
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YU, MERIGÓ, AND XU
Table V. Score values obtained by the GDHFPWA operator and the rankings of alternatives.
Scores
λ
0.1
1
5
10
a1
a2
a3
a4
–0.2913
0.0018
0.1970
0.2867
–0.5127
–0.1414
0.1415
0.2559
–0.2761
–0.0220
0.1581
0.2338
–0.4005
–0.2322
–0.1033
–0.0227
Ranking
a1
a1
a1
a1
a3
a3
a3
a2
a4
a2
a2
a3
a2
a4
a4
a4
Table VI. Score values obtained by the GDHFPWG operator and the rankings of alternatives.
Scores
λ
0.1
1
5
10
a1
a2
a3
a4
Ranking
0.4886
0.2060
–0.0053
–0.0998
0.4367
0.0609
–0.2335
–0.3586
0.3905
0.1327
–0.0747
–0.1697
0.0729
–0.0976
–0.1984
–0.2367
a1 a2 a3 a4
a1 a3 a2 a4
a1 a3 a4 a2
a1 a3 a4 a2
Step 3. Using the GDHFPWA operator to aggregate the performance of four criteria for each
information systems and ranking them:
When we set the different values to parameter in the GDHFPWA operator, different values for comprehensive performance can be obtained for each information
systems. We just express the scores and omit the DHFEs due to a large number
of data. Score values obtained by the GDHFPWA and GDHFPWG operators and
corresponding ranking results are shown in Table V and VI respectively.
Figure 4 shows the scores of the four information systems obtained by the
GDHFPWA operator as the parameter λ is set to different values; we can find
that the scores for four information systems increase as the value of the parameter
λ increases from 1 to 10.
Furthermore, we find that
(1) when λ ∈ [1, 6.777], the ranking of the four information systems is a1 a3 a2 a4 .
(2) when λ ∈ [6.777, 10], the ranking of the four information systems is a1 a2 a3 a4 .
Figure 5 shows the scores of the four information systems obtained by the
GDHFPWG operator as the parameter λ is set to different values; we can find that
the scores for four information systems decrease as the value of the parameter λ
increases from 1 to 10.
From Figure 5, we find that
(1) when λ ∈ [1, 3.854], the ranking of the four information systems is a1 a3 a2 a4 .
(2) when λ ∈ [3.854, 10], the ranking of the four information systems is a1 a3 a4 a2 .
From the above research, we find that the safest information system is the a1 .
International Journal of Intelligent Systems
DOI 10.1002/int
GROUP DECISION MAKING IN INFORMATION SYSTEMS SECURITY
Figure 4. Scores for alternatives obtained by the GDHFPWA operator (λ ∈ (0, 10]).
Figure 5. Scores for alternatives obtained by the GDHFPWG operator (λ ∈ (0, 10]).
International Journal of Intelligent Systems
DOI 10.1002/int
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YU, MERIGÓ, AND XU
6.
CONCLUDING REMARKS
In this paper, we have extended the PA and PG operators to DHF environment and interval-valued DHF environment. Some aggregation operators have been
proposed, such as DHFPWA, DHFPWG, GDHFPWA, and GDHFPWG operators.
An application of the proposed operators to GDM was given, and a real example
about the assessment of security for information systems is provided to illustrate
our proposed methods. From the numerical example, we found that the scores for
alternatives obtained by the GDHFPWG operator decrease as the value of the parameter λ increases whereas the scores for alternatives obtained by the GDHFPWA
operator increase as the value of the parameter λ increases.
Acknowledgments
Dejian Yu would like to acknowledge the support of National Natural Science Foundation
of China grant (no. 71301142), Zhejiang Science & Technology Plan of China (2015C33024),
Zhejiang Provincial Natural Science Foundation of China (no. LQ13G010004), Project Funded by
China Postdoctoral Science Foundation (no. 2014M550353), the National Education Information
Technology Research (no. 146242069), and Education Department Planning Project of Zhejiang
Province (2015SCG204). Yejun Xu would like to acknowledge the support of National Natural
Science Foundation of China grant (no. 71471056).
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