Content uploaded by Pei Zhang
Author content
All content in this area was uploaded by Pei Zhang on Jul 24, 2022
Content may be subject to copyright.
Received: 10 April 2020
|
Revised: 4 September 2020
|
Accepted: 24 September 2020
DOI: 10.1002/int.22306
RESEARCH ARTICLE
A normal wiggly hesitant fuzzy MABAC
method based on CCSD and prospect theory for
multiple attribute decision making
Peide Liu |Pei Zhang
School of Management Science and
Engineering, Shandong University of
Finance and Economics, Jinan,
Shandong, China
Correspondence
Peide Liu, School of Management
Science and Engineering, Shandong
University of Finance and Economics,
Jinan, 250014 Shandong, China.
Email: liupd@sdufe.edu.cn
Funding information
National Natural Science Foundation of
China, Grant/Award Number: 71771140;
Special Funds of Taishan Scholars
Project of Shandong Province,
Grant/Award Number: ts201511045;
Major bidding projects of National Social
Science Fund of China,
Grant/Award Number: 19ZDA080
Abstract
Normal wiggly hesitant fuzzy set (NWHFS) is a new
fuzzy information form to help decision makers (DMs)
express their evaluations, which can further dig the
potential uncertain information hidden in the original
data given by the DMs. Firstly, we define a new dis-
tance measure and new operational laws of NWHFSs.
Then, for the situation where attribute weights are
completely unknown, we propose an extended CCSD
method to produce them objectively, which compre-
hensively uses standard deviation (SD) and correlation
coefficient (CC). What's more, we introduce the
MABAC (multiattributive border approximation area
comparison) method, which takes the distance be-
tween alternatives and the border approximation area
(BAA) into consideration for handling the complex and
uncertain decision‐making problems. Meanwhile, we
combine the MABAC method with prospect theory
(PT), which considers DMs' psychological behavior,
and propose a new NWHF‐CCSD‐PT‐MABAC method
to cope with the multi‐attribute decision making pro-
blems under normal wiggly hesitant fuzzy environ-
ment. Lastly, we illustrate the validity and advantages
of the proposed method through an example of college
book supplier selection.
KEYWORDS
CCSD, MABAC, MADM, normal wiggly hesitant fuzzy set,
prospect theory
Int J Intell Syst. 2020;1–31. wileyonlinelibrary.com/journal/int © 2020 Wiley Periodicals LLC
|
1
1|INTRODUCTION
Multi‐attribute decision making (MADM) problem refers to the problem that makes the best
choice from a group of alternatives based on multi‐attribute characteristics. This kind of pro-
blem is common in people's real life, so it is particularly important to skillfully apply
the MADM method to cope with real problem. For solving such problems, many scholars have
proposed some feasible multi‐attribute decision models and applied them to the practice
activity. Since Zadeh
1
first proposed the definition of fuzzy set (FSs) and used it to describe the
uncertainty and fuzziness, which indicates how much an element belongs to a collection by
using the value on [0,1], different extension forms of FSs have been put forward successively,
such as intuitionistic FSs,
2,3
type‐2 FSs,
4,5
and hesitant fuzzy sets (HFSs).
6,7
Chief of all, HFSs
proposed by Torra
6
allows a couple of possible values in [0,1] to indicate the degree to which an
element belongs to a set. It is more suitable for some particularly complex evaluation problems.
Therefore, compared with other FSs, HFSs have a broader application range.
However, no matter how complex the form of information is, it cannot completely and
accurately describe the detailed preference information from all DMs for an alternative. In
general, DMs are more concerned about how to express information in specific forms and give
detailed evaluations. As a result, the final evaluation information may be biased against their
actual preferences. To effectively describe such a complicated uncertain information, Ren et al.
8
proposed a concept of the NWHFS in which DMs give evaluation results based on the normal
wiggly range (NWR) and the real preference degree (RPD). At present, NWHFS has become a
form of information representation that plays an important role in the process of uncertain
decision making because it can dig deep into the uncertainty hidden in the initial assessment
results. In this paper, in accordance with the information form of NWHFS, we make full use of
its advantages and combine it with an effective MADM method to solve practical problems.
In the MADM process, the attribute weights are an important parameter. In general,
there are three main methods to determine attribute weights in MADM. One is the sub-
jective weight determination method from experts' judgment, the other is the objective
weight determination method form decision data, and the last one is the mixed method that
considers both subjective and objective weights. Experts play a significant role in de-
termining attributes in subjective methods. Based on the expert's preference information, the
weights of each attribute can be calculated. However, in practical application, to overcome
the subjectivity, the attribute weights can be obtained by the objective weight determination
method based on the decision matrix. This can reduce the influence of subjective preference
and personal preference of DMs, and improve the accuracy of evaluation as well. There are
many objective methods to obtain the attribute weights, the most common methods are
shown as follows: the CRITIC method,
9
the Entropy method,
10
the maximizing deviation
method,
11
the standard deviation method
11
and CCSD method.
12
CCSD is a new objective
weight determination method proposed recently. It comprehensively uses standard deviation
(SD) and correlation coefficient (CC) to determine attribute weights. In recent years, it has
been applied to assign more accurate weights to the attributes of financial performance
evaluation to eliminate the influence of subjective factors. Igoulalene et al.
13
and Singh
et al.
14
used CCSD method to solve practical problems in different application contexts,
respectively. The CCSD approach does not require a specific standardized method and
provides a more comprehensive and reliable weighting mechanism. This paper will make
full use of the advantages of the CCSD approach to obtain the weights.
2
|
LIU AND ZHANG
With the deepening of research, more and more methods are used to deal with MADM problem.
These approaches mainly include the QUALIFLEX approach,
15
the TOPSIS approach,
16
the BWM
approach,
17
the VIKOR approach,
18
the PROMETHEE approach,
19
the ELECTRE approach,
20
and
the MULTIMOORA approach,
21
which have been studied by innumerable investigators and have
been well applied in practical life. However, none of the approaches we just mentioned take the
distance between alternatives and BAA into consideration. To solve this problem, Pamucar and
Cirovic
22
proposed the MABAC method, which divides the alternatives into the border, upper and
lower approximation regions. Later, Peng and Yang
23
extended the MABAC method to the process
of R&D project selection to choose the ideal alternative by making full use of the Pythagorean FSs
and Choquet integral average operators. What's more, Xue et al.
24
put forward an extended form of
MABAC approach under the interval‐valued intuitionistic fuzzy (IVIF) environment to solve ma-
terial selection problems, in which the weights of attributes were unknown. Subsequently, Yu
et al.
25
applied the MABAC method to choose a hotel on the travel web under interval type‐2fuzzy
(IT2F) context. Then, Gigovic et al.
26
developed a GIS‐DANP‐MABAC model for the sake of
choosing the site of wind farms. In addition, Sun et al.
27
put the MABAC approach into hesitant
fuzzy linguistic context for the problem of Patients' prioritization. Recently, Peng and Dai
28
use the
MABAC approach to cope with the single‐valued neutrosophic MADM problems, Wang et al.
29
put
forward the MABAC method under q‐rung orthopair fuzzy environment. Furthermore, Ji et al.
30
made use of single‐valued neutrosophic linguistic sets to choose an outsourcing provider by
combing the MABAC approach with ELECTRE method. Although its calculation process is simple,
the program is systematic and logic is good, to date, MABAC has not been developed under normal
wiggly hesitant fuzzy environment. It will be a good research topic to apply MABAC method to
normal wiggly hesitant fuzzy environments.
In addition, the existing MADM methods with normal wiggly hesitant fuzzy information
have a common deficiency, that is, they do not think of the behavioral preference of DMs. In
most practical decisions, the DMs' behavior is a critical factor that affects the decision results.
Prospect theory (PT)
31
is an efficient and practical tool to deal with behavioral characteristics,
and it is widely applied in decision‐making under risks and uncertainties. PT claims that people
make decisions in accordance with the potential value of losses and gains, not end results. At
present, many scholars combine PT with different methods to cope with MADM problems in
practical. The aim is also to extend the approach with the advantage of PT to better solve these
problems. Fan et al.
32
calculated the total prospect value of each alternative by applying
the value function of PT and the simple additive weighting (SAW) approach to determine the
ranking of alternatives. Li and Chen
33
investigated a novel TOPSIS method in accordance with
PT for group decision making, which considered risk psychology of DMs and information
fuzziness. Therefore, inspired by MABAC method and PT, we propose an extended normal
wiggly hesitant fuzzy MABAC method based on PT to deal with MADM in NWHFSs.
Based on the above statements, the motivations of this paper are shown as follows.
(1) NWHFSs can obtain deeper uncertain information on the basis of keeping the original
hesitant fuzzy information and the MABAC method can take into account the potential value
of gains and losses and gets stable results, obviously, it is necessary to extend the MABAC
method to normal wiggly hesitant fuzzy set (NWHFS) to deal with this complex fuzzy
information; (2) the key to the combination of MABAC method and NWHFSs is how to
measure the degree of deviation between two NWHFSs, which can be solved by the distance
measure and operational laws between NWHFSs. (3) The CCSD is a new objective weight
determination method, which comprehensively uses SD and CC to determine attribute weights.
Obviously, it has a more comprehensive and reliable weighting mechanism that can be
LIU AND ZHANG
|
3
effectively combined with the NWHFSs to eliminate the subjective influence of DMs. (4) PT
considers the psychological behavior of DMs, while the MABAC method can obtain stable
results through simple calculation, so it can make the calculation process simple and easy to
understand. In summary, combining with NWHFSs and CCSD method, we propose normal
wiggly hesitant fuzzy projection‐based PT and MABAC method (NWHFS‐CCSD‐PT‐MABAC),
which can be better used to solve practical MADM problems.
The innovations of this study can be mainly manifested as follows:
(1) Define the distance measure and new operational laws of normal wiggly hesitant fuzzy
elements (NWHFEs) between any two NWHFEs to facilitate the comparison of two
NWHFEs.
(2) Take advantage of the CCSD approach to obtain the weights of all attributes for the MADM
problems with unknown weight information under normal wiggly hesitant fuzzy environment.
(3) Combine the MABAC method with prospect theory (PT‐MABAC) to obtain an extended
MABAC method that can consider the risk behavior of DMs.
(4) Put forward an extended NWHF‐CCSD‐PT‐MABAC method under normal wiggly hesitant
fuzzy environment, which fully considers the advantages of CCSD method, prospect theory
and MABAC method.
(5) A numerical example of library supplier selection is described to illustrate the applicability of
the proposed algorithm. A multi‐angle validity test and detailed comparative analysis and
discussion are provided to verify the effectiveness and superiority of the proposed algorithm.
The rest of this paper is structured as follows. In Section 2,webrieflyintroducesomedefinitions,
including the NWHFSs, the classical CCSD approach, PT and the classical MABAC approach. In
Section 3, we propose a distance measure and new operational rules of NWHFEs. In Section 4,we
propose an extension of the MABAC method based on PT and CCSD method to cope with the
MADM problems under normal wiggly hesitant fuzzy environment. In Section 5,throughan
example of university book supplier selection, the calculation process of the NWHF‐CCSD‐PT‐
MABAC method is illustrated, and compared with other methods, the effectiveness and super-
iorities of the proposed method are proved. In Section 6, we draw a conclusion.
2|PRELIMINARIES
In this section, as a foundation, we mainly review some existing concepts including HFSs,
NWHFSs, PT and MABAC method.
2.1 |HFSs
Definition 1 (Torra
6
). Suppose
7
Γ
is a set and a HFS HF on
Γ
can be defined by hτ(
)
HF
that returns a subset belonging to [0,1], which can be expressed as follows:
{}
τhττ
H
F= , ( ) | Γ
,
HF ∈(1)
4
|
LIU AND ZHANG
where hτ()
HF is a set of several computable values in [0,1]. As a matter of convenience, hτ()
HF
is simplified as
h
H
F
and is defined as a hesitant fuzzy element (HFE).
34
2.2 |NWHFS
NWHFS proposed first by Ren et al.
8
is a new information form, which uses a normal fluc-
tuation range to represent the deeper uncertainty when the DMs evaluate an object under a
certain criterion. It realizes the development of deeper uncertainty information behind the
assessments while maintaining the original hesitant fuzzy information.
Definition 2 (Ren et al.
8
). Given a HFE hδδ δ={ , , , }
h12 #
⋅⋅⋅ . Let hδ=hi
h
i
1
#=1
#
∑and
εδh=(−)
hhi
hi
1
#=1
#2
∑be the average and the standard deviation of all alternatives in h,
respectively. The function
g
represents the mapping from hto
ε
[
0, ]
h
, which is called the
normal wiggly range (NWR) of δi, and the function gδ(
)
i
can be described as follows:
gδεe
~()=
,
ih
δh
ε
−(−)
2
i
h
2
2
(2)
where h#denotes the association granularity of HFE h. Ren et al.
8
indicated that the function gδ(
)
i
is
a similar normal function on the basis of the mean and standard deviation of all arguments in h.
Definition 3 (Ren et al.
8
). Suppose hδδ δ={ , , , }
h12 #
⋅⋅⋅ is a HFE and hδ={ =
δsum h δh/()| }
ii
∈
represents the standardized HFE, where sum h δ()= i
hi
=1
#
∑is the sum
of all elements in h.TheRPDrpd g(
)
in hcan be estimated in line with the orness measure,
35
which could be expressed by
()
()
rpd g
δmean h
δmean h
mean h
(~)=
~,if ()<0.5;
1−~,if ()>0.5;
0.5, if ( ) = 0.5;
i
hihi
h
i
hihi
h
=1
#~#~−
#~−1
=1
#~#~−
#~−1
⎧
⎨
⎪
⎪
⎩
⎪
⎪
∑
∑(3)
where mean h(
)
is the average value in hand RPD can represent DMs' uncertainty in HFE.
Definition 4. Suppose
30
Γis a set and
HF τhττ= {( , ( ))| Γ}∈
is a HFS on
Γ
. Then, the
homologous
N
WH
F
on
Γ
could be described by:
{}
N
WHF τhτζhττ= , (), ( ()) | Γ∈(4)
where hτ(
)
is a hesitant fuzzy element in the hesitant fuzzy set.
˘˘ ˘
ζhτδδδ(())={, , , }
,
hτ12 #()
⋯
˘
δμμμ δgδrpd g δgδδ δgδ={ , , }={max( −(),0),(2 (())−1) ( ) + , min ( + ( )), 1
}
ii
L
i
M
i
Uii i ii ii
, and
δiis one of the values of hτ(
)
,gδ(
)
i
represents the normal wiggly parameter of δi, and rpd g δ(())
i
LIU AND ZHANG
|
5
represents the RPD of hτ(
)
,ζhτ(()
)
is called a normal wiggly element (NWE). As a matter of
convenience, the pairwise
hτζhτ<(), (())
>
is simplified as hζh<, ()
>
and could be described as
a normal wiggly hesitant fuzzy element (NWHFE).
Example 1. Given a reference set
τττΓ={ , , }
123
. A HFS
˘
Hcould be:
˘
{}
()()()
Hτττ= , 0.1, 0.2, 0.3 , , 0.7, 0.8, 0.9 , , 0.4, 0.5, 0.6
.
123
Based on the Definitions 3 and 4, a
˘
N
WHF Hcould be calculated:
˘
N
WHF
τ
τ
τ
=
, (0.1, 0.2, 0.3), {(0.0614, 0.0871, 0.1386), (0.1184, 0.1728, 0.2816), (0.2614, 0.2871, 0.3386) }
, (0.7, 0.8, 0.9), {(0.6614, 0.7032, 0.7386), (0.7184, 0.8068, 0.8816), (0.8614, 0.9032, 0.9386) }
, (0.4, 0.5, 0.6), {(0.3614, 0.4000, 0.4386), (0.4184, 0.5000, 0.5816), (0.6000, 0.6386, 0.0819) }
H
1
2
3
⎧
⎨
⎪
⎪
⎩
⎪
⎪
⎫
⎬
⎪
⎪
⎭
⎪
⎪
To better help you understand the NWHFS, we take the NWHFE (0.1, 0.2, 0.3)
,
{(0.0614, 0.0871, 0.1386), (0.1184, 0.1728, 0.2816), (0.2614, 0.2871, 0.3386)} as an example,
(
0.1, 0.2, 0.3) is the original hesitant fuzzy information given by DMs, and the potential
uncertain information mined from the original hesitant fuzzy information
(
0.1, 0.2, 0.3) is
{(0.0614, 0.0871, 0.1386), (0.1184, 0.1728, 0.2816), (0.2614, 0.2871, 0.3386)}
.
For the convenience of understanding, we give the graphic form of NWHFEs corresponding
to HFEs
(
0.1, 0.2, 0.3) and
(
0.7, 0.8, 0.9) to explain the specific meaning of NWHFS. From
Figures 1and 2, we can find that the normal wiggly range ζh(
)
of each value δiin
hδδ δ={ , , , }
h12 #
⋅⋅⋅ is a triangle area. Among them, δδ−g( )
ii
and δδ+g( )
ii
represent the ab-
scissa coordinates of the bottom end points of each triangle area, that is to say, δiswings around
the normal wiggly parameter gδ(
)
i
. As Ren et al.
8
said, people's uncertainty is like a normal
wiggly pendulum in a range centered on one value. Hence, the analogous normal function
g
is
constructed to obtain the normal wiggly parameter gδ(
)
i
of each value in HFE based on the
mean and the standard deviation. And we can further obtain the left and right wiggly ranges
δδ−g( )
ii
and δδ+g( )
ii
of each value, which is the uncertain wiggly range of the DMs.
FIGURE 1 The normal wiggly hesitant fuzzy element of the corresponding hesitant fuzzy element (0.1, 0.2, 0.3)
[Color figure can be viewed at wileyonlinelibrary.com]
6
|
LIU AND ZHANG
What's more, we think that the DMs will tend to give smaller values for evaluation when
he /she uses
(
0.1, 0.2, 0.3), and tend to give bigger values when he/she uses
(
0.7, 0.8.0.9
)
. Here,
the preference information is reflected by the RPD function rpd g δ(())
i
. In Figures 1and 2, the
abscissa coordinate of each triangle vertex is denoted by
rpd g δgδδ
(
2(())−1) ( ) +
iii
. It's not
difficult to find that the RPD rpd g δ(( ))
i
of the DMs determines the bias degree of the triangular
vertex. For example, the vertices of all triangles in Figure 1deviate to the left because the mean
of
(
0.1, 0.2, 0.3) is less than 0.5 and the vertices of all triangles in Figure 2deviate to the right
because the mean of
(
0.7, 0.8.0.9
)
is significantly larger than 0.5. Therefore, the NWHFE can
accurately describe the potential preference of the DMs.
According to the above analysis, through the analogous normal function
g
and the RPD
function rpd g δ(())
i
, the potential uncertain information of DMs hidden in the original hesitant
fuzzy information can be mined.
Definition 5 (Ren et al.
8
). Suppose hζh<, ()
>
is a NWHFE, his the average value and
ε
h
is the standard deviation value in h. Then, the scoring function hℕ(
)
of a NWHFE
hζh<, ()
>
is defined as
˘˘
()
hS hζhρhερ
hδεℕ()= (<, ()>)= ( −)+(1−)1
#−,
NW h
i
h
iδ
=1
#
i
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
∑(5)
where
˘
δ=
i
μμ μ++
3
i
L
i
M
i
Uand
˘
ε μ μ μ μμ μμ μ μ=()+( )+( )−−−
δi
L
i
M
i
U
i
L
i
M
i
L
i
U
i
M
i
U
222
i.
ρ
(0, 1
)
∈can be considered as DMs' confidence level to initial hesitant fuzzy evaluation results.
For ease of comparison and analysis, we set
ρ
=1/
2
in this paper to mean that the DMs
have a 50% confidence in its original hesitant fuzzy information. In practical decision problems,
the parameter
ρ
should be confirmed in line accordance with the knowledge level of the DMs
on relevant problems.
Example 2. Let h= (0.3, 0.5, 0.7
)
be a HFE,
h= 0.5000
is the average value, ε= 0.1633
h
is the standard deviation value of h. Based on Definition 4, the NWHFE can be get as
follows:
FIGURE 2 The normal wiggly hesitant fuzzy element of the corresponding hesitant fuzzy element (0.7, 0.8, 0.9)
[Color figure can be viewed at wileyonlinelibrary.com]
hζh,()
= (0.3, 0.5, 0.7), {(0.2229, 0.3000, 0.3771), (0.3367, 0.5000, 0.6633), (0.6229, 0.7000, 0.7771) }
LIU AND ZHANG
|
7
Based on Definition 5, we can get hℕ( ) = 0.326
7
.
Definition 6 (Ren et al.
8
). Let
h
1
and h
2
be two HFEs, hζh,()
11
and hζh,()
22
be two
diverse NWHFEs, hℕ()
1and hℕ(
)
2be their scoring function values.
If hhℕ()>ℕ()
12
, then hζh,()
11
is better than hζh,()
22
, which can be denoted
as hζhhζh<,()> < ,()
>
11 22
≻.If hhℕ()=ℕ(
)
12
, then hζh,()
11
and hζh,()
22
are indis-
tinguishable, which can be denoted as hζhhζh,()~ ,()
11 22
.If hhℕ()<ℕ()
12
, the hζh,()
11
is inferior to hζh,()
22
, which can be denoted as hζhhζh,() ,()
11 22
≻.
Example 3. Given two HFEs h= (0.2, 0.4, 0.6)
1and h= (0.3, 0.4, 0.5
)
2, based on
Definition 4, we can get
hζh
hζh
, ( ) = (0.2, 0.4, 0.6), {(0.1229, 0.1743, 0.2771), (0.2367, 0.3456, 0.5633), (0.5229, 0.5743, 0.6771)}
, ( ) = (0.3, 0.4, 0.5), { (0.2614, 0.2936, 0.3386), (0.3184, 0.3864, 0.4816), (0.4614, 0.4936, 0.5386) }
11
22
Based on Definition 5, let
ρ
=1/
2
, we can get hhℕ( ) = 0.2191, ℕ( ) = 0.311
7
12
.So
hhℕ()<ℕ()
12
, we think hζh,()
11
is inferior than hζh,()
22
.
2.3 |The classical CCSD method
The CCSD approach is an objective weight determination method that can determine the
SD and CC of data through statistical analysis to eliminate the subjectivity of expert opinions,
which was initially proposed by Wang and Luo.
12
The detailed steps are as follows:
Step 1: Suppose all the alternatives are expressed by Ai m(=1,2, )
i⋯
and Cj n( = 1, 2,
)
j⋯are criterion. Let Hh=( )
ij mn be the original decision matrix,
where hi
j
denotes the value for alternative Aiunder the criterion Cjgiven by the DMs.
For convenience, let
M
mN n= (1, 2, ), = (1, 2,
)
⋯⋯
.
Step 2: Normalize all the elements in the original matrix Hand get the normalized matrix
Hh=( )
ij mn
∼
iMjN
(
,)∈∈
to eliminate the influence of various physical dimensions. The
specific normalization process can be realized by Equation (6):
hhC
hC
iMjN
~=, for benefit criterion
( ) , for cost criterion ,,
.
ij
ij j
ij cj
⎪
⎪
⎧
⎨
⎩
∈∈
(6)
Step 3: Let
W
ww w=( , , , )
n
T
12
⋯be the attribute weights, which satisfies
w
[0, 1
]
j∈and
w=
1
j
nj
=1
∑. According to SAW approach,
36
the overall evaluation of alternative Aicould
be counted by
ehwiM=,
.
i
k
n
ik k
=1
∑
∈
(7)
When Cjis removed, the overall evaluation of alternative Aicould be redefined as
8
|
LIU AND ZHANG
ehwiM=,
.
ij
kkj
n
ik k
=1,
∑∈
≠
(8)
Step 4: The CC
L
j
between Cjand the overall assessment value can be shown as
Lhhee
hh ee
jN=(~−~)( −)
(~−~)(−)
,
,
ji
mij j ij j
i
mij j i
mij j
=1
=1 2=1 2
∑
∑∑
∈(9)
where
hmhj N=1,
.
j
i
m
ij
=1
∑∈
(10)
emehwjN=1=,
.
j
i
m
ij
kkj
m
kk
=1 =1,
∑∑ ∈
≠
(11)
(1) If
L
j
is close to 1, their evaluations will have nearly the same numerical distribution
whether or not the Cjis included. In this case, removing the Cjhas little impact on deci-
sion, so it could be given a very small weight.
(2) If
L
j
is close to −1, their evaluations will have an almost opposite numerical distribution
whether or not Cjis included. In other words, adding Cjto the attribute set will have a
critical effect on decision. Now, the Cjcould be given a very high weight.
(3) If an attribute has the same utility on all the alternatives under consideration, it could be
removed without any effect on the decision. In other words, an attribute with a large SD
should receive more weight than an attribute with a small SD.
Step 5: Confirm the weights of attributes as
w
ιL
ιLjN=1−
1−,
,
j
jj
k
nkk
=1
∑∈
(12)
where
ι
mhhjN=1(−),
.
j
i
m
ij j
=1
2
∑∈
(13)
Here,
L
1
−
j
is the square of the root to reduce the difference between the maximum and
minimum weights. To solve Equation (12), we transform it into the nonlinear optimization model:
Jw
ιL
ιL
M
inimize = ‐1−
1−
,
j
n
j
jj
k
nkk
=1 =1
2
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
∑∑(14)
wwjN
S
ubject to = 1, 0,
.
j
n
jj
=1
∑≥∈ (15)
The final attribute weight is called CCSD weights.
LIU AND ZHANG
|
9
2.4 |Prospect theory (PT)
PT
31
proposed by Kahneman and Tversky can reflect the behavior of DMs under risks and
uncertainties. The decision process described by PT is divided into two stages. In the edit phase,
the results are sorted in accordance with reference points and utility functions. In the eva-
luation stage, a value function and a weight function are used to assess the edited prospect and
select the prospect with the highest value. PT assumed that DMs choose the optimal scheme in
line with the prospect value of all schemes, and the prospect value is jointly determined
through the value function and probability weight function, that is
vwpνx=()(Δ)
,
i
n
ii
=1
∑
(16)
where vis the prospect value, and νx(Δ)
iis a value function expressed by Equation (17), which
can be described by asymmetric s‐shaped functions, as shown in Figure 3.
xxx
Δ
=‐
ii0denotes
deviation between the existing achieve and the psychological equilibrium of DMs, where
x
0
is
the reference point, namely the psychological equilibrium of DMs.
vx xx
λxx
(Δ)= (Δ), Δ0
−(−Δ ), Δ<0
,
i
iαi
iβi
⎧
⎨
⎩
≥(17)
where
α
and
β
are the adjustable coefficient and convexity specifying concavity, respectively
satisfying the constraint conditions
αβ
0
,
1
≤≤
, indicating decreased sensitivity.
α
and
β
are
larger, and DMs are more risk‐oriented. The parameter
λ
describes loss aversion, which
satisfies λ> 1, indicating the DMs are loss averse.
The probability weighting function
w
p()
ishows that people respond too much to events
with small probabilities and not enough to events with medium and large probabilities. In 1992,
the most commonly used parameterized form of probability weighting function in empirical
and theoretical applications is put forward by Tversky and Kahneman
37
as follows:
()
w
p
x
x
()=
,Δ0
,Δ<0
,
i
p
pp
i
p
pp i
+(1−)
(+(1−))
i
γ
i
γiγγ
i
σ
σσ
σ
1
1
⎧
⎨
⎪
⎪
⎩
⎪
⎪
≥
(18)
FIGURE 3 A value function of prospect theory
vx(
)
[Color figure can be viewed at wileyonlinelibrary.com]
10
|
LIU AND ZHANG
where
p
is probability,
γ
and
σ
are risk‐benefit attitude coefficient and risk‐loss attitude
coefficient.
w
p()
is shown as Figure 4.
2.5 |The traditional MABAC approach
The traditional TODIM approach was originally discovered by Pamučar and Ćirović.
22
To use
the MABAC method, we must first define the distance between the criterion function of each
alternative and the boundary approximation region. Next, we give the specific steps of the
traditional MABAC approach as follows:
Step 1: Form the original decision matrix H, which is the same as the classical CCSD
approach.
Step 2: Normalize Hand get the normalized matrix Hh=( )
ij mn
∼
(
i
Mj N,∈∈
)by
Equation (6).
Step 3: Get weighting matrix Taccording to Equation (19):
t
wh=(
~+1)
.
ij j ij
⋅(19)
Step 4: Get the BAA matrix
G
gg g=[ , , ,
]
n12
⋯
according to Equation (20):
gv=
.
j
i
m
ij
m
=1
1/
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
∏(20)
Step 5: Compute the distance Dd=( )
ij mn between the matrix elements in the weighted
matrix
V
and the approximate region of the boundary
G
. That is to say:
DVG=−
.
(21)
FIGURE 4 The probability weight function
w
p(
)
LIU AND ZHANG
|
11
The belongings of Aito the lower (
G
˜
), border (
G
) and upper (
G+
) approximation area are
decided based on Equation (22), expressed by Figure 5:
A
Gifd
Gifd
Gifd
,=0
,=0
,=0
.
i
ij
ij
ij
+
−
⎧
⎨
⎪
⎩
⎪
∈(22)
Step 6: Rank the alternatives. The criterion function value of each alternative is calculated by
S
di Mj N=, ,
.
i
j
n
ij
=1
∑∈∈ (23)
3|DISTANCE MEASURE AND NEW OPERATIONAL
LAWS OF NWHFSS
Definition 7 (Xu and Zhang
38
). It should be noted that different HFEs may have various
cardinal numbers, and the values are generally out of orders, thus, we can arrange them
at will for convenience. Given two HFEs
h
1
and h
2
, and hh#,#
12
denotes the numbers
of values in
h
1
and h
2
, respectively. To ensure the correct operations between NWHFEs,
the shorter one should be added until
h
1
and h
2
have the same length when hh##
12
≠,
Based on the maximum terms
h+
and minimum terms h
−
, Xu and Zhang
38
introduce a
method which can use the parameter
λ
to add different values to the HFE by
hλhλh
*=+(1−)
,
+−(24)
where
λ
λ(0 1)≤≤ is a parameter to reflect the risk preference of DMs.
λ
takes 1/2 here, that
is, hhh
*=1/2( + )
+−.
3.1 |Distance measure of NWHFSs
Definition 8. Let
˘
hζhδδϒ=,()>=<{},{}
111 11
and
˘
hζhδδϒ=,()={},{}
222 22
be
two given NWHFEs. If
d
(ϒ,ϒ
)
NW 12
meets the following conditions:
FIGURE 5 Description of the
G
−
,
G
, and
G
+
12
|
LIU AND ZHANG
(1) d
0
(ϒ,ϒ)
1
NW 12
≤≤
,
(2)
d
(ϒ,ϒ)=
0
NW 12 if and only if
ϒ=ϒ
1
2
(3)
d
d(ϒ,ϒ)= (ϒ,ϒ
)
NW NW12 21
,
(4) Let
˘
hζhδδϒ=,()={},
{}
333 33be ∀NWHFEs, and
d
(ϒ,ϒ
)
NW 13
≤
d
(ϒ,ϒ
)
NW 12 d
+
(ϒ,ϒ
)
NW 23
.
Then,
d
(ϒ,ϒ
)
12
can be called distance measure (DME) between ϒ
1
and
ϒ
2
.
Definition 9. Let
˘
hζhδδϒ=,()={},{}
aaa aa
and
˘
hζhδδϒ=,()={},{}
bbb bb
be
two given NWHFEs, where hδlh={ =1,2,…, # }
aa
la,hδlh={ =1,2,…, # }
bb
lb,
ζh(
)
a
˘
δlh
=
{=1,2,…,#}
a
l
a
μμμ lh={( , , ) = 1, 2, …, #
}
aLl a
Ml aUl aand
˘
ζhδlhμμ μ( )={ =1,2,…, # }={( , ,
)
bb
l
bb
Ll
b
Ml bUl
lh=1,2,…, # }
b.
Suppose that HFEs in
h
a
and h
b
are arranged in an ascending order. Among them,
hh
H
#=#=
ab
should be held. Otherwise, jaorbϒ(= )
jwith fewer cardinalities needs to
be added to maintain the same length for
ϒ
a
and ϒ
b
, and then we can define the distance
measure with preference coefficient
ρ
as follows:
dρHδδ ρ
Hμμ μ μ
μμ
(ϒ,ϒ)= 1−+(1−)11
3(−+−
+−)
NW a b
l
H
a
lb
l
l
H
a
Ll
b
Ll
a
Ml
b
Ml
a
Ul
b
Ul
=1
2
=1
2
2
2
⎜
⎟
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
⎛
⎝
⎜
⎜⎛
⎝
⎞
⎠
⎞
⎠
⎟
⎟
∑∑
(25)
where
ρ
[0, 1]∈.
Proof.
First, we prove that Equation (25) satisfies the condition (1).
It is easy to observe that
()
δδ
0
−
1
Hl
H
a
lb
l
1
=1
2
≤∑≤
and
()
()
μμ μ μ μμ
0
(−+−+−)
1
Hl
H
a
Ll
b
Ll
a
Ml
b
Ml
a
Ul
b
Ul
1
=1
1
3
222
≤∑≤.
Then,
()
ρδδρ
0
−
Hl
H
a
lb
l
1
=1
2
≤∑≤and
ρHμμ μ μ μμ ρ
0
(1‐)11
3(−+−+−)1‐
.
l
H
a
Ll
b
Ll
a
Ml
b
Ml
a
Ul
b
Ul
=1
222
⎜⎟
⎛
⎝
⎜
⎜⎛
⎝
⎞
⎠
⎞
⎠
⎟
⎟
∑
≤≤
Thus,
()
() ( )
ρδδ ρ μμμμ μμ ρρ
0
−+(1−)(−+−+−)+(1‐)=
1
Hl
H
a
lb
l
Hl
H
a
Ll
b
Ll
a
Ml
b
Ml
a
Ul
b
Ul
1
=1
21
=1
1
3
222
≤∑∑ ≤
d
0
(ϒ,ϒ)1
.
NW a b
≤≤
Second, we prove that Equation (25) satisfies the condition (2).
If
d
(ϒ,ϒ)=
0
NW a b ,
Then,
()
() ( )
ρ
δδ ρ μ μ μ μ μ μ−+(1−)(−+−+−)=
0
Hl
H
a
lb
l
Hl
H
a
Ll
b
Ll
a
Ml
b
Ml
a
Ul
b
Ul
1
=1
21
=1
1
3
222
∑∑
LIU AND ZHANG
|
13
Thus,
()
ρδδ−=0
Hl
H
a
lb
l
1
=1
2
∑
and
()
()
ρμμμμμμ(1 −)(−+−+−)=
0
Hl
H
a
Ll
b
Ll
a
Ml
b
Ml
a
Ul
b
Ul
1
=1
1
3
222
∑,
Thus, δδ−=0
l
H
a
lb
l
=1
2
∑and μμ μ μ μμ
(−+−+−)=
0
a
Ll
b
Ll
a
Ml
b
Ml
a
Ul
b
Ul
1
3
222
,
Thus, δδ−=0
a
lb
l2and μμ μ μ μμ−+−+−=
0
a
Ll
b
Ll
a
Ml
b
Ml
a
Ul
b
Ul
222
,
Thus, δδ−=
0
a
lb
land μμ−=
0
a
Ll
b
Ll ,μμ−=
0
a
Ml
b
Ml ,μμ−=0
a
Ul
b
Ul ,
ϒ=ϒ
1
2
.
If
ϒ=ϒ
1
2
,δδ−=
0
a
lb
land μμ−=
0
a
Ll
b
Ll ,μμ−=
0
a
Ml
b
Ml ,μμ−=0
a
Ul
b
Ul ,
Thus, δδ−=0
l
H
a
lb
l
=1
2
∑and μμ μ μ μμ
(−+−+−)=
0
a
Ll
b
Ll
a
Ml
b
Ml
a
Ul
b
Ul
1
3
222
,
Thus,
()
() ( )
ρ
δδ ρ μ μ μ μ μ μ−+(1−)(−+−+−)=
0
Hl
H
a
lb
l
Hl
H
a
Ll
b
Ll
a
Ml
b
Ml
a
Ul
b
Ul
1
=1
21
=1
1
3
222
∑∑ ,
d
(ϒ,ϒ)=
0
NW a b
Third, we prove that Equation (25) satisfies the condition (3).
dρHδδ ρ
Hμμ μ μ
μμ ρ
Hδδ
ρHμμ μ μ μμ
d
(ϒ,ϒ)= 1−+(1−)11
3(−+−
+−)=1−
+(1−)11
3(−+−+−)
=(ϒ,ϒ)
NW b a
l
H
a
lb
l
l
H
a
Ll
b
Ll
a
Ml
b
Ml
a
Ul
b
Ul
l
H
a
lb
l
l
H
a
Ll
b
Ll
a
Ml
b
Ml
a
Ul
b
Ul
NW a b
=1
2
=1
2
2
2
=1
2
=1
222
⎜
⎟
⎜⎟
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
⎛
⎝
⎜
⎜⎛
⎝
⎞
⎠
⎞
⎠
⎟
⎟
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
⎛
⎝
⎜
⎜⎛
⎝
⎞
⎠
⎞
⎠
⎟
⎟
∑∑
∑
∑
Finally, we prove that Equation (25) satisfies the condition (4), that is,
d
dd(ϒ,ϒ)(ϒ,ϒ)+ (ϒ,ϒ)
.
NW a c NW a b NW b c
≤
Let
˘
hζhδδϒ=,()={},{}
ccc cc
, in which hδlh={ =1,2,…, # }
cc
lc,
˘
ζhδlhμcμμ lh( )={ =1,2,…, # }={( , , ) =1,2,…, # }
cc
l
cLl c
Ml
c
Ul c, then
dρHδδ ρ
Hμμ μ μ
μμ
(ϒ,ϒ)= 1−+(1−)11
3(−+−
+−),
NW a c
l
H
a
lc
l
l
H
a
Ll
c
Ll
a
Ml
c
Ml
a
Ul
c
Ul
=1
2
=1
2
2
2
⎜
⎟
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
⎛
⎝
⎜
⎜⎛
⎝
⎞
⎠
⎞
⎠
⎟
⎟
∑∑
dρHδδ ρ
Hμμ μ μ
μμ
(ϒ,ϒ)= 1−+(1−)11
3(−+−
+−),
NW a b
l
H
a
lb
l
l
H
a
Ll
b
Ll
a
Ml
b
Ml
a
Ul
b
Ul
=1
2
=1
2
2
2
⎜
⎟
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
⎛
⎝
⎜
⎜⎛
⎝
⎞
⎠
⎞
⎠
⎟
⎟
∑∑
14
|
LIU AND ZHANG
dρHδδ ρ
Hμμ μ μ
μμ
(ϒ,ϒ)= 1−+(1−)11
3(−+−
+−)
NW b c
l
H
b
lc
l
l
H
b
Ll
c
Ll
b
Ml
c
Ml
b
Ul
c
Ul
=1
2
=1
2
2
2
⎜
⎟
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
⎛
⎝
⎜
⎜⎛
⎝
⎞
⎠
⎞
⎠
⎟
⎟
∑∑
Since the mathematical theorem ac a c++≤holds, then ab bc(−)+( −)
ab bc−+−≤, i.e., ac ab bc−−+−≤.
Hence, δδ δδ δδ μ μ μ μ μ μ−−+−,−−+−
a
lc
la
lb
lb
lc
l
a
Ll
c
Ll
a
Ll
b
Ll
b
Ll
c
Ll
≤≤
,
μμ μμ μ μ−−+−
a
Ml
c
Ml
a
Ml
b
Ml
b
Ml
c
Ml
≤and μμ μμ μμ−−+−
a
Ul
c
Ul
a
Ul
b
Ul
b
Ul
c
Ul
≤.
Further,
d
dd(ϒ,ϒ)(ϒ,ϒ)+ (ϒ,ϒ
)
NW a c NW a b NW b c
≤is established.
The proof is finished.
Example 4. Given two NWHFSs
ϒ= {0.2, 0.4, 0.6}, {(0.1229, 0.1743, 0.2771), (0.2367, 0.3456, 0.5633), (0.5229, 0.5743, 0.6771) }
ϒ= (0.3, 0.5, 0.7), {(0.2229, 0.3000, 0.3771), (0.3367, 0.5000, 0.6633), (0.6229, 0.7000, 0.7771)}
1
2
and
ρ
=0.5
, so the distance between ϒ
1
and
ϒ
2
is
d
(ϒ,ϒ) = 0.011
4
NW 12 .
3.2 |New operational laws of NWHFSs
Definition 10 (Ren et al.
8
). Let hζh,()
11
and hζh,()
22
be two different NWHFEs, then
(1)
˘˘
˘˘
hζhhζhδδδδ δ δ,() ,( )= + −,
δhδhδζhδζh
11 22 ,1212 (), ()
12
1122 1122
⊕∪ ∪⊕
∈∈ ∈∈
(2)
˘˘
˘˘
hζhhζhδδ δ δ,() ,( )= ,
δhδhδζhδζh
11 22 ,12 (), ( )
12
1122 112 2
⊗∪∪⊗
∈∈ ∈∈
(3)
˘
˘
hζhδδλ
(
,()
)=, ,>0
λ
δh
λ
δζh
λ
11 1 ()
1
11 11
⎧
⎨
⎩
⎫
⎬
⎭
∪∪
∈∈
(4)
˘
˘
λ
hζhδλδλ
(,()
)=1‐(1‐), , >0
δh
λ
δζh
11 1 () 1
11 11
⎧
⎨
⎩
⎫
⎬
⎭
∪∪
∈∈
After calculated by Definition 10, the number of derived HFE values will be increased,
which makes the calculation extremely complicated. Liao et al.
39
proposed several new
methods to solve this problem by modified the operational rules, Ren et al.
8
proposed the
operational rules of NWHFEs based on hesitancy fuzzy number and triangular fuzzy number.
Inspired by Liao et al.,
39
we put forward the new operational laws of NWHFEs as follows:
Definition 11. Let hζhj n,()|=1,2, ,
jj ⋯be a set of NWHFEs, and
λ
be an
integer, then
(1)
˘˘
hζhhζhδδδδδ δlh,() ^,()= + −,=1,2,…,#
ql ql ql ql ql ql
11 2 2 1() 2() 1() 2() 1
()
2
()
⊕⊕
LIU AND ZHANG
|
15
(2)
˘˘
hζhhζhδδ δ δ lh,() ^,()= , =1,2,…,#
ql ql ql ql
11 22 1() 2() 1
()
2
()
⊗⊗
(3)
()
˘
()
hζhδδlhλ
(
, ( ) ) ={ , | =1,2,…, # }, >0
λql λql λ
11 1() 1
()
(4)
˘
λ
hζhδλδlhλ(,())={1‐(1‐), ( )| =1,2,…, # }, >0
ql λql
11 1() 1
()
(5)
()
˘
()
hζhδδlh
^,()=1‐1‐,=1,2,…,#
j
n
jj j
n
j
ql j
nj
ql
=1 =1
() =1
()
⊕
∏⊕
(6)
()
˘
hζhδδlh
^,()= , =1,2,…,#
j
n
jj j
n
j
ql j
nj
ql
=1 =1
() =1
()
⊗∏⊗
where δj
ql(
)
is the
l
th smallest value in hj,
˘
δj
ql(
)
is the
l
th smallest value in ζh()
j, and
˘
δj
ql(
)
can be considered as triangular fuzzy numbers.
⊕
and ⊗are the operations of triangular fuzzy
numbers.
40
From the new operational laws proposed above, we can see that the results are also
NWHFEs. The operations of HFEs and TFHFEs are combined in the operational laws of
NWHFEs. Thus, the rationality of the proposed operational laws is clear.
4|EXTENDED MABAC METHOD BASED ON THE CCSD
AND PROSPECT THEORY UNDER NORMAL WIGGLY
HESITANT FUZZY ENVIRONMENT
In this section, we put forward an extension of the MABAC method in accordance with CCSD
and PT to cope with the MADM problems under normal wiggly hesitant fuzzy environment.
4.1 |The description of the MADM problems under normal wiggly
hesitant fuzzy environment
Suppose AAA A=( , , , )
m12
⋯is a set including all the alternatives and CCC C=(,,,
)
n12
⋯is a
set which includes the attributes. Suppose the hesitant fuzzy decision matrix is Hh=( )
ij m n×,in
which the hesitant fuzzy number hi
j
is regarded as the evaluation of alternative Ai M(
)
i∈
about the attribute Cj N()
j∈, and shown in Table 1.
4.2 |The extended normal wiggly hesitant fuzzy CCSD‐PT‐MABAC
approach
In this subsection, an extended CCSD‐PT‐MABAC method is presented for normal wiggly
hesitant fuzzy information. First, we get the normal wiggly hesitant fuzzy decision matrix by
computing the NWE. Second, we calculate the criteria weights
W
ww=( , , )
m
T
1⋯by using the
CCSD approach, which satisfy
w
[0, 1]∈and w=
1
j
nj
=1
∑. Thirdly, in accordance with the PT,
31
we get the prospect decision matrix. Finally, we rank the alternatives in line with the MABAC
method by getting the BAA matrix and computing the distance between the matrix elements
and the approximate region of the boundary. The steps of the proposed NWHF‐CCSD‐PT‐
MABAC are shown as follows.
16
|
LIU AND ZHANG
4.2.1 |Normalize the original decision matrix
Step 1: Form the original hesitant decision matrix Hh=( )
ij mn and normalize it to get the
normalized Hh=( )
ij mn
∼
by Equation (26), shown in Table 2:
hhC
hC
iMjN
~=, for benefit criterion
( ) for cost criterion ,,
.
ij
ij j
ij cj
⎪
⎪
⎧
⎨
⎩
∈∈
(26)
Step 2: Compute the NWE ζh
()
ij
and we could get the NWHFE
˘
η
hζhδδ=~,(~)={}
,{}
ij ij ij ij ij .
Then
˘
N
ηhζhδδ=( ) = ~,(~)={~},~
ij mn ij ij mn ij
ql
ij
ql
mn
() ()
⎧
⎨
⎩
⎫
⎬
⎭is determined, where h=
ij
{}
δlh=1,2,…, #
ij
ql
ij
()
,
˘
{}
()
ζhδlhμμμ lh
()
==1,2,…,#=,, =1,2,…,#
ij ij
ql
ij ijLq l ij
Mq l ijUq l ij
() () () ()
⎧
⎨
⎩
⎫
⎬
⎭
∼∼∼
∼
,
shown in Table 3.
The above matrix can be simplified as
N
A
A
ηηη
ηηη
=
m
jn
mmjmn
CCC
111 1 1
1
jn1
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
⋮
⋯
⋮⋱⋮
⋯
⋯
(27)
TABLE 1 Original hesitant decision matrix H
C1
C2
C
n−1
C
n
A
1
h
11
h
1
2
⋯
h
n1, −1
h
n1
A
2
h
2
1
h
2
2
⋯
h
n2, −
1
h
n
2
⋯
⋯
⋯
⋯
⋯
⋯
A
m−1
h
m−1,1
h
m−1,
2
⋯
h
mn−1, −
1
h
mn−1,
A
4
h
m
1
h
m
2
⋯
h
mn,−
1
h
mn
TABLE 2 Normalized hesitant decision matrix H
∼
C1
C2
C
n−1
C
n
A
1
h
~
11
h
˜1
2
⋯
h
˜n1, −1
h
˜n1
A
2
h
˜2
1
h
˜2
2
⋯
h
˜n2, −
1
h
˜
n2
⋯
⋯
⋯
⋯
⋯
⋯
A
m−1
h
˜m−1,1
h
˜m−1,
2
⋯
h
˜mn−1, −
1
h
˜mn−1,
A
m
h
˜
m
1
h
˜m
2
⋯
h
˜mn,−
1
h
˜mn
LIU AND ZHANG
|
17
4.2.2 |Determine the criteria weights
Step 3: Let
W
ww=( , , )
m
T
1⋯be attribute weights, which satisfies
wwj n=1, 0, =1, ,
.
i
mjj
=1
∑≥⋯
Remove the attribute Cj, the overall evaluation value of
each alternative can be redefined as
eηwiM=^,
.
ij
n
kkj
ik k
=1,
⊕∈
≠
(28)
Step 4: The CC
L
jbetween Cjand the overall evaluation value can be expressed as
Ldηηdee
dηη dee
jN=(,) (,)
((,)) ((,))
,
,
ji
mNW ij j NW ij j
i
mNW ij j i
mNW ij j
=1
=1 2=1 2
∑
∑∑ ∈(29)
where
η
mηjN=1^,
,
j
m
iij
=1
⊕∈ (30)
emeηwjN=1^=^,.
j
m
iij
n
kkj
kk
=1 =1,
⊕⊕ ∈
≠
(31)
Step 5: Count the attribute weights by
w
ιL
ιLjN=1−
1−,
.
j
jj
k
nkk
=1
∑∈(32)
where
ι
mdηη jN=1((
~,~)) ,
.
j
i
m
NW ij j
=1
2
∑
∈(33)
TABLE 3 The normal wiggly hesitant decision matrix
N
C1
C2
C
n−1
C
n
A
1
hζh,( )
11 11
∼∼
hζh,( )
12 12
∼∼
⋯
hζh,( )
nn1, −11,−1
∼∼
hζh,( )
nn1, 1,
∼∼
A
2
hζh,( )
21 21
∼∼
hζh,( )
22 22
∼∼
⋯
hζh,( )
nn2, −12,−1
∼∼
hζh,( )
nn2, 2,
∼∼
⋯
⋯
⋯
⋯
⋯
⋯
A
m−1
hζh,( )
mm−1, 1 −1,1
∼∼
hζh,( )
mm−1,2 −1, 2
∼∼
⋯
hζh,( )
mn mn−1, −1−1, −1
∼∼
hζh,( )
mn mn−1, −1,
∼∼
A
m
hζh,( )
mm11
∼∼
hζh,( )
mm22
∼∼
⋯
hζh,( )
mn mn,−1,−1
∼∼
hζh,( )
mn mn
∼∼
18
|
LIU AND ZHANG
To solve Equation (32), we transform it into the nonlinear optimization model:
Jw
ιL
ιL
M
inimize = ‐1−
1−
,
j
n
j
jj
k
nkk
=1 =1
2
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
∑∑
wwjN
S
ubject to = 1, 0,
.
j
n
jj
=1
∑≥∈ (34)
4.2.3 |Rank the alternatives
Step 6: Determine the normal wiggly hesitant fuzzy prospect decision matrix
V
v=( )
ij mn.
Give a reference point
η
j, which is the average of each alternative Aifor each attribute Cj.
vπpνη=()()
,
ij jij (35)
where
()
()
πp
ηη
ηη
()=
,ifℕ()−ℕ() 0,
,ifℕ()−ℕ()<0,
j
p
pp ij j
p
pp ij j
+(1−)
+(1−)
j
γ
j
γjγγ
j
σ
j
σjσσ
1
1
⎧
⎨
⎪
⎪
⎩
⎪
⎪
≥
(36)
νη dηη η η
λdηη η η
()= ((,)),ifℕ()−ℕ() 0,
−((,)),ifℕ()−ℕ()<0.
ij
NW ij j αij j
NW ij j βij j
⎪
⎪
⎧
⎨
⎩
≥
(37)
where vij denote the prospect value of
η
ij
,
η
jdenotes the reference point value under the
attribute Cjin the PT. ηℕ(
)
ij and ηℕ()
jdenotes the score function values of
η
ij
,
η
j.
d
ηη(,
)
NW ij j
denotes the distance between
η
ij
and
η
j. Tversky and Kahneman
37
gave
α
βλγσ= = 0.88, = 2.25, = 0.61, = 0.69 through experimental verification.
Step 7: Compute all the elements from the weighting matrix Tby
t
wv=
,
ij j ij (38)
Step 8: Get the BAA matrix
G
gg g=[ , , ,
]
n12
⋯
by
gt=
.
j
i
m
ij
m
=1
1/
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
∏(39)
LIU AND ZHANG
|
19
Step 9: Compute the distance Dd=( )
ij mn between the matrix elements in the prospect
decision matrix
V
and the approximate region of the boundary
G
by
d
tg=‐
ij ij j
.
(40)
Step 10: Rank the alternatives. The criterion function value of each alternative is calculated
as the sum of the distance between the alternative and the boundary approximation
region by
S
diMjN=,,
.
i
j
n
ij
=1
∑
∈∈
(41)
5|AN ILLUSTRATIVE EXAMPLE
In this section, we adopt the proposed method to solve the supplier selection problem in the
book procurement bidding of a university library. By analyzing the parameters and comparing
with other methods, the effectiveness and advantages of the proposed method are verified.
5.1 |Problem description
With the opening of China's book market, it is particularly important for libraries to select
suppliers conditionally. The problem of library supplier selection is a typical MADM problem.
Some scholars have studied the bidding work of university library books. Wang
41
analyzed the
feasibility and superiority of book bidding procurement in university library, and compared
with various methods of document bidding procurement. Xie
42
put forward the selection cri-
teria and evaluation methods of suppliers according to the traits of book procurement bidding
and the influence of the supplier selection on the quality of library collection construction.
There are many attributes which are considered to select book suppliers in university
libraries, this paper will adopt the five attributes put forward by Xie
42
to evaluate and analyze
the book suppliers in university libraries. These five attributes include Good reputation and
comprehensive strength (C
1
), Standardized interview and catalog data (C
2
), Standardized
document processing services (C3), Efficient logistics distribution system (C
4
), Quality perso-
nalized service (C5). The details of these five attributes are described in Table 4. There are four
alternative book suppliers which are represented as AAAA,,,
123
4
. The DMs give evaluation
results after comprehensive analysis and consideration of relevant attribute standards, and
express their preference information in the form of HFE, shown in Table 5.
5.2 |The steps of decision making
We can solve this MADM problem by the proposed NWHF‐CCSD‐PT‐MABAC method, which
involves the following seven steps.
20
|
LIU AND ZHANG
5.2.1 |Normalize the original decision matrix
Step 1: Form the original hesitant decision matrix Hh=( )
ij mn and normalize it to get
Hh=( )
ij mn
∼
according to Equation (16). Since Cjare all benefit types, thus
Hh Hh=( ) = =( )
ij mn ij mn
∼
, as shown in Table 6.
Step 2: Compute the NEW ζh(
)
ij
and we could get the NWHFE
η
hζh=~,(
~)
ij ij ij and the
NWHF decision matrix
N
ηhζh=( ) = ~,(
~)
ij mn ij ij mn described in Table 7.
5.2.2 |Determine the criteria weights
Step 3–5: Solve the CCSD model by Equation (34) using Microsoft Excel Solver. The optimal
attribute weights can be obtained as
W
wwwww= ( , , , , ) = (0.0363, 0.5031, 0.1511, 0.1370, 0.1724)
.
TT
12345
TABLE 4 The explanation of attributes
Attribute Explanation
C
1
: Good reputation and
comprehensive strength
Business scale, capital situation, company personnel quality,
network management degree, number of local customers,
local sales, book varieties coverage, group purchase ability,
order delivery speed and delivery rate
C
2: Standardized interview and
catalog data
Timely and accurately report book information to the library,
including both printed books and digital books; Already had
the reservation bibliography of prospective new book; should
reflect the spot bibliography of stock more
C
3
: Standardized document processing
services
Complete the simple procedures such as adding the collection
seal, adding antitheft magnetic strip, book marking and bar
code, greatly reducing the labor of the personnel of acceptance
and cataloging
C
4
: Efficient logistics distribution
system
The process operation time is short, the cost is low, the efficiency
is high, is advantageous to the book and the reader to meet
in time
C
5
: Quality personalized service Free door‐to‐door delivery, legitimate return of goods, targeted to
make personalized subscription data submitted to the
corresponding library
TABLE 5 The Original hesitant decision matrix H
C1
C2
C3
C4
C
5
A
1
{0.2,0.4,0.6} {0.2,0.3,0.6} {0.3,0.4,0.5} {0.4,0.5,0.6} {0.2,0.3,0.6}
A
2
{0.3,0.5,0.7} {0.1,0.3,0.4} {0.1,0.3,0.4} {0.3,0.4,0.5} {0.4,0.6,0.7}
A
3
{0.2,0.4,0.5} {0.4,0.5,0.7} {0.2,0.4,0.6} {0.2,0.3,0.4} {0.2,0.4,0.5}
A
4
{04,0.5,0.6} {0.3,0.5,0.7} {0.4,0.5,0.7} {0.3,0.5,0.6} {0.3,0.5,0.6}
LIU AND ZHANG
|
21
5.2.3 |Rank the alternatives
Step 6: Give the reference point
η
j, here, we set
α
βλγσ= = 0.88, = 2.25, = 0.61, = 0.69 and
p= (0.0363, 0.5031, 0.1511, 0.1370, 0.1724
)
. Then, we compute vij according to the Equations
(35)–(37), and determine the prospect decision matrix
V
v=( )
ij mn, as shown in Table 8.
Step 7: Compute the weighting matrix Tshown in Table 9. According to Equation (38), we
can get
W
wwwww= ( , , , , ) = (0.0363, 0.5031, 0.1511, 0.1370, 0.1724)
TT
12345 .
TABLE 6 The normalized hesitant decision matrix H
˜
C1
C2
C3
C4
C
5
A
1
{0.2,0.4,0.6} {0.2,0.3,0.6} {0.3,0.4,0.5} {0.4,0.5,0.6} {0.2,0.3,0.6}
A
2
{0.3,0.5,0.7} {0.1,0.3,0.4} {0.1,0.3,0.4} {0.3,0.4,0.5} {0.4,0.6,0.7}
A
3
{0.2,0.4,0.5} {0.4,0.5,0.7} {0.2,0.4,0.6} {0.2,0.3,0.4} {0.2,0.4,0.5}
A
4
{04,0.5,0.6} {0.3,0.5,0.7} {0.4,0.5,0.7} {0.3,0.5,0.6} {0.3,0.5,0.6}
TABLE 7 The normal wiggly hesitant decision matrix
N
C
1
A
1
η
= {0.2, 0.4, 0.6}, {(0.1229, 0.1743, 0.2771), (0.2367, 0.3456, 0.5633), (0.5229, 0.5743, 0.6771)}
11
A
2
η
= (0.3, 0.5, 0.7), {(0.2229, 0.3000, 0.3771), (0.3367, 0.5000, 0.6633), (0.6229, 0.7000, 0.7771)}
21
A
3
η
= {0.2, 0.4, 0.5}, {(0.1489, 0.1861, 0.2511), (0.2797, 0.3672, 0.5203), (0.4296, 0.4808, 0.5704)}
31
A
4
η
= {0.4, 0.5, 0.6}, {(0.3614, 0.4000, 04386), (0.4184, 0.5000, 0.5816), (0.5614, 0.6000, 0.6386)}
41
C
2
A
1
η
= {0.2, 0.3, 0.6}, {(0.0949, 0.1618, 0.3051), (0.1426, 0.2428, 0.4574), (0.5338, 0.5759, 0.6662)}
12
A
2
η
= {0.1, 0.3, 0.4}, {(0.0489, 0.0808, 0.1511), (0.1797, 0.2549, 0.4203), (0.3296, 0.3736, 0.4704)}
22
A
3
η
= {0.4, 0.5, 0.7}, {(0.3296, 0.4132, 0.4704), (0.3797, 0.5226, 0.6203), (0.6489, 0.7096, 0.7511)}
32
A
4
η
= {0.3, 0.5, 0.7}, {(0.2229, 0.3000, 0.3771), (0.3367, 0.5000, 0.6633), (0.6229, 0.7000, 0.7771)}
42
C
3
A
1
η
= {0.3, 0.4, 0.5}, {(0.2614, 0.2936, 0.3386), (0.3184, 0.3864, 0.4816), (0.4614, 0.4936, 0.5386)}
13
A
2
η
= {0.1, 0.3, 0.4}, {(0.0489, 0.0808, 0.1511), (0.1797, 0.2549, 0.4203), (0.3296, 0.3736, 0.4704)}
23
A
3
η
= {0.2, 0.4, 0.6}, {(0.1229, 0.1743, 0.2771), (0.2367, 0.3456, 0.5633), (0.5229, 0.5743, 0.6771)}
33
A
4
η
= {0.4, 0.5, 0.7}, {(0.3296, 0.4132, 0.4704), (0.3797, 0.5226, 0.6203), (0.6489, 0.7096, 0.7511)}
43
C
4
A
1
η
= {0.4, 0.5, 0.6}, {(0.3614, 0.4000, 04386), (0.4184, 0.5000, 0.5816), (0.5614, 0.6000, 0.6386)}
14
A
2
η
= {0.3, 0.4, 0.5}, {(0.2614, 0.2936, 0.3386), (0.3184, 0.3864, 0.4816), (0.4614, 0.4936, 0.5386)}
24
A
3
η
= {0.2, 0.3, 0.4}, {(0.1614, 0.1914, 0.2386), (0.2184, 0.2819, 0.3816), (0.3614, 0.3914, 0.4386)}
34
A
4
η
= {0.3, 0.5, 0.6}, {(0.2489, 0.2891, 0.3511), (0.3797, 0.4742, 0.6203), (0.5296, 0.5849, 0.6704)}
44
C
5
A
1
η
= {0.2, 0.3, 0.6}, {(0.0949, 0.1618, 0.3051), (0.1426, 0.2428, 0.4574), (0.5338, 0.5759, 0.6662)}
15
A
2
η
= {0.4, 0.6, 0.7}, {(0.3489, 0.4090, 0.4511), (0.4797, 0.6212, 0.7203), (0.6296, 0.7124, 0.7704)}
25
A
3
η
= {0.2, 0.4, 0.5}, {(0.1489, 0.1861, 0.2511), (0.2797, 0.3672, 0.5203), (0.4296, 0.4808, 0.5704)}
35
A
4
η
= {0.3, 0.5, 0.6}, {(0.2489, 0.2891, 0.3511), (0.3797, 0.4742, 0.6203), (0.5296, 0.5849, 0.6704)}
45
22
|
LIU AND ZHANG
Step 8: Get the BAA matrix
G
by Equation (39), we can get:
G
gggg= ( , , , ) = (0.0001, 0.0063, 0.0006, 0.0003, 0.0007)
.
1234
Step 9: Compute the distance Dd=( )
ij mn between the matrix elements and the approximate
region of the boundary by t Equation (40), as shown in Table 10.
Step 10: Rank the alternatives according to Equation (42), we can get:
S
ASASASA()=−0.0145, ( ) = ‐0.0307, ( ) = −0.0059, ( ) = −0.0038
.
123 4
Thus,
AAAA
431
2
≻≻≻
.
5.3 |Parameter sensitivity analysis
In Section 4, we combine the PT with the MABAC approach to describe the behavioral pre-
ference of DMs. The sensitivity analysis of parameters can help us better control the function of
TABLE 8 The prospect decision matrix
V
C1
C2
C3
C4
C
5
A
1
−0.0014 −0.0105 0.0009 0.0027 −0.0102
A
2
0.0010 −0.0427 −0.0157 −0.0006 0.0066
A
3
−0.0025 0.0101 −0.0021 −0.0112 −0.0065
A
4
0.0013 0.0054 0.0068 0.0015 0.0005
TABLE 9 The weighting matrix T
C1
C2
C3
C4
C
5
A
1
−0.0001 −0.0053 0.0001 0.0004 −0.0018
A
2
0.0000 −0.0215 −0.0024 −0.0001 0.0011
A
3
−0.0001 0.0051 −0.0003 −0.0015 −0.0011
A
4
0.0000 0.0027 0.0010 0.0002 0.0001
TABLE 10 The matrix D
C1
C2
C3
C4
C
5
A
1
−0.0001 −0.0116 −0.0004 0.0001 −0.0024
A
2
0.0000 −0.0278 −0.0030 −0.0004 0.0005
A
3
−0.0001 −0.0012 −0.0009 −0.0018 −0.0018
A
4
0.0000 −0.0036 0.0005 −0.0001 −0.0006
LIU AND ZHANG
|
23
parameters in MADM. Now, we adjust the parameters
λ
to discuss the effects of the parameters
on the resultant ranking results. Table 11 demonstrates the numerical results and final ranking
results under different values of
λ
.
Here, we set
α
βγσ= = 0.88, = 0.61, = 0.69 according to Tversky and Kahneman.
37
And
λ
varies from 1.0 to 30.0. To more clearly represent the effect of parameters, we present the
results in Table 11 as a graph, as shown in Figure 6, and the final ranking results obtained for
different
λ
is shown in Figure 7.
As seen from Figure 6, the numerical results of each alternative decrease with the increase
of
λ
. In the range of 1.0 to 30.0, the ranking of alternatives remains unchanged, while the
optimal solution remains unchanged, however, the gap between alternatives
A
1
and
A
2
tends to
widen. According to the PT, with the increase of the
λ
, the loss value increases, the risk
aversion degree of DM increases, and the optimal choice of alternatives tends to minimize the
loss. As seen from Figure 6, there is no change in the optimal alternative. When the parameter
increases significantly, the ranking hasn't been changed, which means that the
λ
is insensitive
to the behavior of DMs. It can also be seen from Figure 7that there is no change in the ranking
in accordance with the overall results of numerical results and ranking results.
TABLE 11 Sort the results of cases using different parameters
λ
Alternatives
λ=1
λ=1.25
λ=1.5
λ=1.75
λ=
2
S
Order
S
Order
S
Order
S
Order
S
Order
A
1
−0.0082 3 −0.0096 3 −0.0010 3 −0.0124 3 −0.0075 3
A
2
−0.0147 4 −0.0180 4 −0.0212 4 −0.0244 4 0.0046 4
A
3
−0.0013 2 −0.0022 2 −0.0031 2 −0.0039 2 −0.0407 2
A
4
−0.0012 1 −0.0018 1 −0.0024 1 −0.0029 1 −0.0121 1
λ=2.25
λ=2.5
λ=2.75
λ=3
λ=30
Alternatives
S
Order
S
Order
S
Order
S
Order
S
Order
A
1
−0.0149 3 −0.0088 3 −0.0174 3 −0.0186 3 −0.1273 3
A
2
−0.0307 4 0.0033 4 −0.0386 4 −0.0399 4 −0.3470 4
A
3
−0.0054 2 −0.0499 2 −0.0069 2 −0.0076 2 −0.0603 2
A
4
−0.0038 1 −0.0144 1 −0.0046 1 −0.0050 1 −0.0248 1
FIGURE 6 Numerical results obtained for different
λ
[Color figure can be viewed at wileyonlinelibrary.com]
24
|
LIU AND ZHANG
5.4 |Validity analysis of the proposed method
To prove the validity of the proposed method in this paper, we use three existing methods
under normal wiggly hesitant fuzzy environment to solve the problem of library supplier
selection. The existing methods under normal wiggly hesitant fuzzy environment we used to
compare include the normal wiggly hesitant fuzzy weighted averaging (NWHFWA) operator,
8
the normal wiggly hesitant fuzzy weighted geometric (NWHFWG) operator
8
and the normal
wiggly hesitant fuzzy TODIM (NWHF‐TODIM) method.
43
What's more, To ensure that the
decision result is not affected by the weights, we uniformly use the CCSD weight used in
calculating the case of university book supplier selection problem. That is
W
= (0.0363, 0.5031, 0.1511, 0.1370, 0.1724)
T
. Detailed evaluation results of three different
MADM methods are shown in Table 12.
As shown in Table 12, when we use the above three methods to calculate the problem of
selection of college book suppliers, the proposed method in this paper gives the same ranking
results as the NWHFWA method,
8
NWHFWG method,
8
and NWHF‐TODIM
43
method. As we
all know, the NWHFWA method and WNHFWG method are two basic method in the normal
wiggly hesitant fuzzy environment, the same ranking results
AAAA
413
2
≻≻≻
between the
above methods shows that our method is effective. In addition, since the TODIM method is
proposed based on PT, both of the NWHF‐PT‐MABAC method and NWHF‐TODIM approach
consider the psychological behavior of DMs, so these two methods get the same ranking results,
that is,
AAAA
413
2
≻≻≻
, which can prove the effectiveness of the proposed method as well.
FIGURE 7 Final ranking results obtained for different
λ
[Color figure can be viewed at
wileyonlinelibrary.com]
TABLE 12 Results obtained by different Multi‐attribute decision making methods
Methods Evaluation values Ranking
NWHFWA method
8
S
A( ) = 0.2348
,
1
S
A( ) = 0.2194
,
2
S
A( ) = 0.3170
,
3
S
A( ) = 0.3438
4
AAAA
431
2
≻≻≻
NWHFWG method
8
S
A( ) = 0.2185
,
1
S
A( ) = 0.1832
,
2
S
A( ) = 0.2997
,
3
S
A( ) = 0.341
6
4
AAAA
431
2
≻≻≻
NWHF‐TODIM
method
θ=1
5
43
S
A( ) = 0.0081
,
1
S
A( ) = 0.0000
,
2
S
A( ) = 0.1244
,
3
S
A( ) = 1.0000
4
AAAA
431
2
≻≻≻
The proposed method
S
A()=−0.0145
,
1
S
A()=−0.0307
,
2
S
A()=−0.0059
,
3
S
A( ) = 0.0038
4
AAAA
431
2
≻≻≻
LIU AND ZHANG
|
25
According to the above analysis results, it is obvious that we can prove that the proposed
method is effective for solving and processing MADM problems.
5.5 |Superiority analysis of the proposed method
In the following, we will further discuss the superiorities of the proposed method further from
four aspects. First, we compare with the methods under the hesitant fuzzy environment, such
as HF‐TODIM method,
44
to prove the advantages of NWHFS in digging uncertainty in depth.
Second, we will compare with the method that does not use CCSD method to determine the
weight to prove the advantages of CCSD method. Thirdly, we compared the methods of the
NWHF‐TODIM
43
method to demonstrate the advantages of the PT‐MABAC method when
consider the DMs' behavior. Fourthly, we will compare it with other method such as NWHF‐
MULTIMOORA method
45
to prove the advantages of the method proposed in this paper.
To ensure that the decision result is not affected by the weights, we uniformly use the CCSD
weight used in calculating the case of university book supplier selection problem. That is
W
= (0.0363, 0.5031, 0.1511, 0.1370, 0.1724)
T
. Then we will use the following methods to
calculate the case of university book supplier selection.
(1) Compared with HF‐TODIM
44
method
In this paper, the evaluation information of DMs is described by NWHFS and we will prove
the advantages of the NWHFS compared with the HFE. In this section, we will continue to deal
with the case of university book supplier selection and set
ρ
=0.5
to reflect the role of NWHF
information. Table 13 shows the detailed evaluation results for both methods.
As shown in Table 13, the ranking results obtained by the proposed method and the
HF‐TODIM method
44
are inconsistent. For this result, we can make the following analysis. As
we can see, the proposed method combines PT and MABAC method. The PT takes the psy-
chological behavior of DMs into account, and the method of HF‐TODIM
44
also has the ad-
vantage of considering the psychological behavior of DMs because the TODIM is a highly
effective method to capture the psychological behavior based on PT. As MABAC is a simple
MADM method, it can be seen from the above analysis that the two methods are similar
without considering the information form.
Therefore, the fundamental reason for the different ranking results of the two methods lies
in the form of information. As we all know, NWHFS cannot only maintain hesitant fuzzy
information effectively, but also dig out the potential uncertainty evaluation information of
DMs from a deeper level. As some uncertain information may be elusive in the actual situation,
even DMs cannot give a detailed description. If we do not dig deeper into the uncertainty
TABLE 13 Results obtained by different multi‐attribute decision making methods
Methods Score values Ranking
HF‐TODIM method
44
S
A( ) = 0.3022
,
1
S
A( ) = 0.2981
,
2
S
A( ) = 0.0000
,
3
S
A( ) = 1.0000
4
AAAA
4123
≻≻≻
The proposed
method ρ
(
=0.5)
S
A()=−0.0145
,
1
S
A()=−0.0307
,
2
S
A()=−0.0059
,
3
S
A()=−0.003
8
4
AAAA
4312
≻≻≻
26
|
LIU AND ZHANG
information of DMs, it is likely to lead to distortion of the results, which can be proved by the
difference between the above two sequencing results. Through the above analysis, we can
demonstrate the advantages of using the information form of NWHFS.
(2) Compared with the proposed method without using CCSD.
To prove the advantages of CCSD, we set the weight of all attributes to 0.2 to calculate the
case of university book supplier selection without considering the influence of weights. The
evaluation results are shown in Table 14.
As shown in Table 14, we find that the ranking results without considering the influence of
attribute weights are inconsistent with the attribute weights given by the CCSD method, which
fully demonstrate the benefits of using the CCSD method. The CCSD method combines the
standard deviation of each attribute and its correlation coefficient to determine the weight of
the attribute. It is a very objective method to determine the weight of the attribute, which can
effectively eliminate the influence of the DMs' subjective emotion on the decision. Moreover,
the process of CCSD method is simple and easy to understand, which also shows the
advantages of CCSD weighting method.
(3) Compared with the NWHF‐TODIM
43
method
To further illustrate the advantages of our method, we will compare it with the NWHF‐
TODIM
43
method by calculating the case of university book supplier selection. Since both the
TODIM method and PT consider the psychological behavior of DMs, we selected two identical
parameters respectively and analyzed the ranking results. The evaluation results are shown in
Table 15.
TABLE 14 Results obtained by different multi‐attribute decision making methods
Methods Score values Ranking
The proposed method without
using CCSD
S
A()=−0.0078
,
1
S
A()=−0.0116
,
2
S
A()=−0.0094
,
3
S
A()=−0.001
2
4
AAAA
4132
≻≻≻
The proposed method
S
A()=−0.0145
,
1
S
A()=−0.0307
,
2
S
A()=−0.0059
,
3
S
A()=−0.003
8
4
AAAA
4312
≻≻≻
TABLE 15 Results obtained by different Multi‐attribute decision making methods
Methods Score values Ranking
NWHF‐TODIM method
θ
(
=1
)
43
S
A( ) = 0.1233
,
1
S
A()=−0.2207
,
2
S
A( ) = 0.0000
,
3
S
A( ) = 1.0000
4
AAAA
4213
≻≻≻
NWHF‐TODIM method
θ
(
=15
)
43
S
A( ) = 0.0081
,
1
S
A( ) = 0.0000
,
2
S
A( ) = 0.1244
,
3
S
A( ) = 1.0000
4
AAAA
4312
≻≻≻
The proposed method
λ
(
=1
)
S
A()=−0.0079
,
1
S
A()=−0.0147
,
2
S
A()=−0.0016
,
3
S
A()=−0.001
2
4
AAAA
4312
≻≻≻
The proposed
method λ
(
=15
)
S
A()=−0.0672
,
1
S
A()=−0.1789
,
2
S
A()=−0.0358
,
3
S
A()=−0.0163
4
AAAA
4312
≻≻≻
LIU AND ZHANG
|
27
As shown in Table 15, the ranking results obtained by NWHF‐TODIM
43
method are in-
consistent when
θ=
1
and
θ=15
, while the ranking results of the method proposed in this
paper are consistent regardless of the parameters. In addition, we can also see that when
θ
is
large enough, the ranking result is consistent with the method proposed in this paper, that is
AAAA
431
2
≻≻≻
. Thus, although both the TODIM method and the PT consider the psychological
behavior of DMs, we can know from the sensitivity analysis that when the PT and the MABAC
method are combined, the change of their parameters has no influence on the ranking results,
but the TODIM method is greatly affected by parameters. Therefore, the method proposed in
this paper is more stable.
(4) Compared with the NWHF‐MULTIMOORA
45
method
Then, we will compare our method against alternative method such as NWHF‐
MULTIMOORA method
45
by calculating the case of university book supplier selection to make
the method proposed in this paper easier to understand. MULTIMOORA is based on three
subordinate methods, namely, the ratio system (RS), the reference point (RP), and the full
multiplicative form (FMF). It also uses the dominance theory to compute the final ranking.
Next, we will prove the advantages of this method by comparing and analyzing the ranking
results of the two methods. The evaluation results are shown in Table 16.
As shown in Table 16, the ranking results obtained by the NWHF‐MULTIMOORA
method
45
and the proposed method in this paper are inconsistent. Although the two methods
have their own advantages, the psychological behavior of DMs is very important in dealing with
the selection of university library suppliers. Therefore, the combination of the method in this
paper and PT is more reasonable. In addition, MABAC divides the alternatives into boundary,
upper and lower approximation areas, while MULTIMOORA method does not consider the
distance between the alternatives and BAA, which has certain limitations. Therefore, the
method proposed in this paper has more advantages.
6|CONCLUSION
In the decision‐making process, to make DMs more appropriate and reasonable to express its
cognitive preference, many complex information representation forms have been proposed.
However, the limited thinking ability of DMs cannot make them clearly express complex
psychological preferences, which also increases the psychological burden and time cost of DMs.
Therefore, to help DMs to reduce the difficulty of evaluation, we try to use an effective method
to excavate the uncertainty information hidden in the original evaluation information given by
DMs. In this paper, we use the NWHFS to automatically dig for deep uncertainty, which cannot
TABLE 16 Results obtained by different Multi‐attribute decision making methods
Methods Score values Ranking
NWHF‐MULTIMOORA
method
45
S
A( ) = 77.1851
,
1
S
A( ) = 1.5994
,
2
S
A( ) = 2.0948
,
3
S
A( ) = 0.826
4
4
AAAA
132
4
≻≻≻
The proposed method
S
A()=−0.0086
,
1
S
A()=−0.0116
,
2
S
A()=−0.0087
,
3
S
A()=−0.001
2
4
AAAA
4132
≻≻≻
28
|
LIU AND ZHANG
only retain the original hesitant fuzzy information of DMs, but also more deeply excavates the
uncertain information given by DMs.
In this paper, the basic theoretical knowledge of NWHFS has been elaborated in detail, and
the contribution of this paper can be mainly manifested by the following three ways. Firstly, we
define the distance measure and new operational laws of NWHFEs to facilitate the comparison
of two NWHFEs. Then, we apply the CCSD method to determine the attribute weights to
eliminate the subjective influence under the normal wiggly hesitant fuzzy environment. Fur-
thermore, considering the simplicity of MABAC calculation process, the stability of the results
and the feasibility of combining MABAC method with other methods, we develop a new
NWHF–CCSD‐PT‐MABAC method to solve the MADM problems by combining PT and
MABAC method, which can also consider the risk behavior of DMs.
In future research, we can apply the NWHF‐CCSD‐PT‐MABAC method proposed in this
paper to solve some real decision problems, such as fault diagnosis, time series prediction, and
medical diagnosis. Moreover, we can combine NWHFS with other MADM methods, such as
ORESTE method and PROMETHEE method. At the same time, we can combine it with
information aggregation operators, such as Maclaurin symmetric mean,
46,47
Bonferroni
mean,
48–50
and Heronian mean
51
operators in the future.
ACKNOWLEDGMENTS
This paper is supported by the National Natural Science Foundation of China (No. 71771140),
Project of cultural masters and "the four kinds of a batch" talents, the Special Funds of Taishan
Scholars Project of Shandong Province (No. ts201511045), Major bidding projects of National
Social Science Fund of China (19ZDA080).
REFERENCES
1. Zadeh LA. Fuzzy sets. Inf Control. 1965;8(3):338–356.
2. Atanassov KT. Intuitionistic fuzzy sets. Fuzzy Sets Syst. 1986;20(1):87‐96.
3. Castillo O, Kutlu F, Atan Ö. Intuitionistic fuzzy control of twin rotor multiple input multiple output
systems. J Intell Fuzzy Syst. 2020;38(1): 821‐833.
4. Karnik NN, Mendel JM. Operations on type‐2 fuzzy sets. Fuzzy Sets Syst. 2001;122(2): 327‐348.
5. Moreno JE, Sanchez MA, Mendoza O, et al. Design of an interval Type‐2 fuzzy model with justifiable
uncertainty. Inf Sci. 2020;513:206‐221.
6. Torra V. Hesitant fuzzy sets. Int J Intell Syst. 2010; 25(6):529‐539.
7. Vij S, Jain A, Tayal DK, Castillo O. Scientometric inspection of research progression in hesitant fuzzy sets.
J Intell Fuzzy Syst. 2020;38(1): 619‐626.
8. Ren Z, Xu Z, Wang H. Normal wiggly hesitant fuzzy sets and their application to environmental quality
evaluation. Knowl‐Based Syst. 2018;159:286‐297.
9. Diakoulaki D, Mavrotas G, Papayannakis L. Determining objective weights in multiple criteria problems:
the critic method. Comput Oper Res. 1995;22(7):763‐770.
10. Deng H, Yeh C, Willis RJ. Inter‐company comparison using modified TOPSIS with objective weights.
Comput Oper Res. 2000;27(10):963‐973.
11. Wu Z, Chen Y. The maximizing deviation method for group multiple attribute decision making under
linguistic environment. Fuzzy Sets Syst. 2007;158(14):1608‐1617.
12. Wang Y, Luo Y. Integration of correlations with standard deviations for determining attribute weights in
multiple attribute decision making. Math Comput Model Dyn Syst. 2010;51(1):1‐12.
13. Igoulalene I, Benyoucef L, Tiwari MK. Novel fuzzy hybrid multi‐criteria group decision making approaches
for the strategic supplier selection problem. Expert Syst Appl. 2015; 42(7):3342‐3356.
14. Singh RK, Benyoucef L. A consensus based group decision making methodology for strategic selection
problems of supply chain coordination. Eng Appl Artif Intell. 2013;26(1):122‐134.
LIU AND ZHANG
|
29
15. Li J, Wang J. An extended QUALIFLEX method under probability hesitant fuzzy environment for selecting
green suppliers. Int J Fuzzy Syst. 2017;19(6):1866‐1879.
16. Chen C. Extensions of the TOPSIS for group decision‐making under fuzzy environment. Fuzzy Sets Syst.
2000;114(1):1‐9.
17. Guo S, Zhao H. Fuzzy best‐worst multi‐criteria decision‐making method and its applications. Knowl‐Based
Syst. 2017;121:23‐31.
18. Wu Z, Ahmad J, Xu J. A group decision making framework based on fuzzy VIKOR approach for machine
tool selection with linguistic information. Appl Soft Comput. 2016;42:314‐324.
19. Deshmukh SC. Preference ranking organization method of enrichment evaluation (PROMETHEE). Int
J Eng Sci Inv. 2013;2(11):28‐34.
20. Chen N, Xu Z. Hesitant fuzzy ELECTRE II approach: a new way to handle multi‐criteria decision making
problems. Inf Sci. 2015;292:175‐197.
21. Hafezalkotob A, Hafezalkotob A, Liao H, Herrera F. An overview of MULTIMOORA for multi‐criteria
decision‐making: theory, developments, applications, and challenges. Inf Fusion. 2019;51:145‐177.
22. Pamucar D, Cirovic G. The selection of transport and handling resources in logistics centers using multi‐
attributive border approximation area comparison (MABAC). Expert Syst Appl. 2015;42(6):3016‐3028.
23. Peng X, Yang Y. Pythagorean fuzzy Choquet integral based MABAC method for multiple attribute group
decision making. Int J Intell Syst. 2016;31(10): 989‐1020.
24. Xue Y, You J, Lai X, Liu H. An interval‐valued intuitionistic fuzzy MABAC approach for material selection
with incomplete weight information. Appl Soft Comput. 2016;38:703‐713.
25. Yu S, Wang J, Wang J. An interval type‐2 fuzzy likelihood‐based MABAC approach and its application in
selecting hotels on a tourism website. Int J Fuzzy Syst. 2017,19(1):47‐61.
26. Gigovic L, Pamucar D, Božanic D, Ljubojevic S. Application of the GIS‐DANP‐MABAC multi‐criteria
model for selecting the location of wind farms: a case study of Vojvodina, Serbia. Renew Energy. 2017;103:
501‐521.
27. Sun R, Hu J, Zhou J, Chen X. A hesitant fuzzy linguistic projection‐based MABAC method for patients'
prioritization. Int J Fuzzy Syst. 2018;20(7):2144‐2160.
28. Peng X, Dai J. Approaches to single‐valued neutrosophic MADM based on MABAC, TOPSIS and new
similarity measure with score function. Neural Comput Appl. 2018;29(10):939‐954.
29. Wang J, Wei G, Wei C, Wei Y. MABAC method for multiple attribute group decision making under q‐rung
orthopair fuzzy environment. Def Technol. 2019; https://doi.org/10.1016/j.dt.2019.06.019
30. Ji P, Zhang H, Wang J. Selecting an outsourcing provider based on the combined MABAC–ELECTRE
method using single‐valued neutrosophic linguistic sets. Comput Ind Eng. 2018;120:429‐441.
31. Kahneman D, Tversky A. Prospect theory: an analysis of decision under risk. Econometrica 1979;47(2):
263‐291.
32. Fan Z, Zhang X, Chen F. Multiple attribute decision making considering aspiration‐levels: A method based
on prospect theory. Comput Ind Eng. 2013;65(2):341‐350.
33. Li X, Chen X. Extension of the TOPSIS method based on prospect theory and trapezoidal intuitionistic
fuzzy numbers for group decision making. J Syst Sci Syst Eng. 2014;3(2):231‐247.
34. Xia M, Xu Z. Hesitant fuzzy information aggregation in decision making. Int J Approx Reason. 2011;52(3):
395‐407.
35. Yager RR. On ordered weighted averaging aggregation operators in multicriteria decision making. IEEE
Trans Syst Man Cybern1988;18(1):183–190.
36. Tzeng GH, Huang JJ. Multiple attribute decision making: methods and applications. Boca Raton, FL:
CRC press. 2011.
37. Tversky A, Kahneman D. Advances in prospect theory: cumulative representation of uncertainty. J Risk
Uncertain. 1992;5(4):297‐323.
38. Xu Z, Zhang X. Hesitant fuzzy multi‐attribute decision making based on TOPSIS with incomplete weight
information. Knowl‐Based Syst. 2013;52:53‐64.
39. Liao H, Xu Z, Xia M. Multiplicative consistency of hesitant fuzzy preference relation and its application in
group decision making. Int J Inf Technol Decis Mak. 2014;13(1):47‐76.
40. Yu D. Triangular hesitant fuzzy set and its application to teaching quality evaluation. J Inf Comput Sci.
2013,10(7):1925‐1934.
30
|
LIU AND ZHANG
41. Wang Q. The bids and purchases for books in university Library. Library Tribune. 2005;(3):120‐122.
42. XLE X. Analyzing the suppliers selection on the public bidding of books procurement of university library.
Library Tribune. 2006;3.
43. Liu P, Zhang P. Normal wiggly hesitant fuzzy TODIM approach for multiple attribute decision making.
J Intell Fuzzy Syst.2020;39(1):627‐644.
44. Zhang X, Xu Z. The TODIM analysis approach based on novel measured functions under hesitant fuzzy
environment. Knowl‐Based Syst. 2014;61(1):48‐58.
45. Yang C, Wang Q, Peng W, Zhu J. A multi‐criteria group decision‐making approach based on improved
BWM and MULTIMOORA with normal wiggly hesitant fuzzy information. Int J Comput Int Sys. 2020;
13(1):366‐381.
46. Liu P, Chen S, Wang P. Multiple‐attribute group decision‐making based on q‐rung orthopair fuzzy power
maclaurin symmetric mean operators. IEEE Trans Syst Man Cybern B Cybern. 2019:1‐16.
47. Liu P, Li Y, Zhang M. Some Maclaurin symmetric mean aggregation operators based on two‐dimensional
uncertain linguistic information and their application to decision making. Neural Compute Appl. 2019;
31(8):4305‐4318.
48. Liu P, Gao H, Ma J. Novel green supplier selection method by combining quality function deployment with
partitioned Bonferroni mean operator in interval type‐2 fuzzy environment. Inf Sci. 2019;490:292‐316.
49. Liu P, Li H. Multiple attribute decision‐making method based on some normal neutrosophic Bonferroni
mean operators. Neural Comput Appl. 2017;28(1):179‐194.
50. Liu P, Wang P. Multiple‐attribute decision‐making based on Archimedean Bonferroni operators of q ‐Rung
orthopair fuzzy numbers. IEEE Trans Fuzzy Syst. 2019;27(5):834‐848.
51. Liu P, Shi L. Some neutrosophic uncertain linguistic number Heronian mean operators and their
application to multi‐attribute group decision making. Neural Comput Appl. 2017;28(5):1079‐1093.
How to cite this article: LiuP, ZhangP. A normal wiggly hesitant fuzzy MABAC
method based on CCSD and prospect theory for multiple attribute decision making. Int J
Intell Syst. 2020;1–31. https://doi.org/10.1002/int.22306
LIU AND ZHANG
|
31
