Extended TOPSIS method based on the entropy measure and probabilistic hesitant fuzzy information and their application in decision support system

Abstract

Probabilistic hesitant fuzzy Set (PHFs) is the most powerful and comprehensive idea to support more complexity than developed fuzzy set (FS) frameworks. In this paper, it can explain a novel, improved TOPSIS-based method for multi-criteria group decision-making (MCGDM) problem through the Probabilistic hesitant fuzzy environment, in which the weights of both experts and criteria are completely unknown. Firstly, we discuss the concept of PHFs, score functions and the basic operating laws of PHFs. In fact, to compute the unknown weight information, the generalized distance measure for PHFs was defined based on the Probabilistic hesitant fuzzy entropy measure. Second, MCGDM will be presented with the PHF information-based decision-making process.

Full text

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Journal of Intelligent & Fuzzy Systems xx (20xx) x–xx
DOI:10.3233/JIFS-202700
IOS Press
1
Extended TOPSIS method based on the
entropy measure and probabilistic hesitant
fuzzy information and their application
in decision support system
1
2
3
4
Muhammad Naeema, Muhammad Ali Khanb,∗, Saleem Abdullahb, Muhammad Qiyasb
and Saifullah Khanb
5
6
aDeanship of Combined First Year, Umm Al-Qura University Makkah, KSA7
bDepartment of Mathematics Abdul Wali Khan University, Mardan, Pakistan8
Abstract. Probabilistic hesitant fuzzy Set (PHFs) is the most powerful and comprehensive idea to support more complexity
than developed fuzzy set (FS) frameworks. In this paper, it can explain a novel, improved TOPSIS-based method for multi-
criteria group decision-making (MCGDM) problem through the Probabilistic hesitant fuzzy environment, in which the
weights of both experts and criteria are completely unknown. Firstly, we discuss the concept of PHFs, score functions and
the basic operating laws of PHFs. In fact, to compute the unknown weight information, the generalized distance measure for
PHFs was defined based on the Probabilistic hesitant fuzzy entropy measure. Second, MCGDM will be presented with the
PHF information-based decision-making process.
9
10
11
12
13
14
15
Keywords: Probabilistic hesitant fuzzy Set, extended TOPSIS method, application in decision making16
1. Introduction17
Multi-criteria group decision-making (MCGDM)18
problems is the method of identifying the most19
acceptable solution to all possible alternatives for
20
problems in evaluation and selection that have been21
thoroughly Implemented in real-life environments. In
22
real life, with imprecise information, there are many23
low-precision decision-making (DM) techniques. In24
1970, Zadeh introduced MCGDM by using the the-25
ory of fuzzy sets (FSs) that was an effective method
26
of dealing with such a problem during the DM pro-27
cess known as fuzzy MCGDM. As a consequence,28
∗Corresponding author. Muhammad Ali Khan, Department of
Mathematics Abdul Wali Khan University, Mardan, Pakistan.
E-mail: aliwazir.edu@gmail.com.
several extensions of the fuzzy set were developed, 29
like as hesitant fuzzy fuzzy sets (HFSs), probabilis- 30
tic hesitant fuzzy sets (PHFSs), and type-2 fuzzy set, 31
etc. These methods are used to resolve the complexity 32
inherent in useful MAGDM problems. 33
In order to deal with unpredictable circumstances 34
in DM processes, Zadeh [40] has implemented fuzzy 35
sets. FSs are generalized by applying the feature func- 36
tion to the membership function of classical sets. 37
The values between 0 and 1 are used in a fuzzy set 38
to describe the membership function. The member- 39
ship function and the function of non-membership 40
are sums equal to 1. Such drawback points to the 41
detriment that in fact, fuzzy sets are unable to con- 42
tain all kinds of uncertainty. Representing the use of 43
the item in a numerical meaning that is the expert’s 44
role in terms of values. We express in fuzzy sets or 45
ISSN 1064-1246/$35.00 © 2021 – IOS Press. All rights reserved.

Uncorrected Author Proof
2M. Naeem et al. / Extended TOPSIS method based on the entropy measure and PHF
their novel extensions for public servers. We pre-46
sented different problems in daily life to make the best
47
option in the selection decision. It is a difficult task48
to develop the decision-making technique using the49
uncertain information. Several scientists have been50
drawn to deal with confusion in decision-making
51
issues in order to figure out the best alternative52
according to certain criteria. In this respect, several53
decision-making techniques of multi attributes were54
formed within fuzzy sets and their extension [31,
55
34, 36, 43]. Torra [27] defined the idea of the hes-56
itant fuzzy set to develop the fuzzy set form, which57
has a set of values without having a single value in
58
the form of membership. The Hesitant fuzzy set is59
a powerful tool for solving, decision-making prob-
60
lems with uncertainty. Several scholars, such as Chen61
et al. [7], are motivated to contribute to the hesi-62
tant fuzzy set theory. The notion of multi-criteria
63
decision-making (MCGDM) was widely discussed64
by researchers motivated by the influence of Proba-65
bilistic hesitant fuzzy set [3, 11, 13, 15, 18, 35, 44]. To66
alleviate the problem, Xu & Zhou [35] put forward the67
PHFS associating the probability of incidence with68
each HF. It mitigates the problem of inaccuracy and69
imprecision in generating the probability values of
70
occurrence by providing multiple value preferences
71
as the probability of occurrence for each HF. There72
are many uncertainties and hesitations in real-world
73
applications which are ordered as stochastic and non-74
stochastic [20]. Stochastic hesitancy can usually be75
assumed precisely by probabilistic modeling [22].
76
In any case, the probabilistic models and the tradi-77
tional FS theory are only useful for the preparation
78
of one part of the specificity. It will also be useful
79
to incorporate probability theory into FS theory [17,80
42].81
TOPSIS (Order Value Technique by Similarity to82
Ideal Solution) approach was first time defined by83
Hwang & Yoon [12] to solve a problem MCGDM84
that focuses on selecting an alternative with the small-
85
est distance from the positive ideal solution (P•˜
Iˆ
S)
86
and the longest distance from the negative ideal solu-
87
tion (N˜
Iˆ
S). Then, based on the TOPSIS method [25,88
26, 29, 30, 32, 46] most scholars have indulged in89
MAGDM problem in recent years. In this century,90
several authors developed successful TOPSIS imple-91
mentations in several decision sciences fields [1, 6,92
24, 28] by applying TOPSIS in different fuzzy envi-93
ronments [4, 5]. In 2015, Mardani et al. [21] published
94
a new work on fuzzy MCGDM from 1994 −2014,
95
which included a list of research implement through
96
the TOPSIS method.
97
Entropy is a very important and efficient tool for 98
measuring uncertain details. First time Zadeh [41] 99
define the fuzzy entropy. Shannon [23] developed 100
an information theory based on cross-entropy mea- 101
sure. Kullback and Leibler [9] introduced a cross 102
entropic distance measure between the two probabil- 103
ity distributions. In flexible decision-making, Furtan 104
[10] analyzed entropy theory. Dhar et al. [8] moni- 105
tored entropy reduction future decision-making. The 106
decision-making process was explored by Yang and 107
Qiu [37] on the basis of future value and entropy. 108
To obtain criteria weights of completely unknown 109
weight information in solving MCGDM problems. 110
Determined by the above discussion, in order to 111
benefit from the advantages of the TOPSIS tech- 112
nique and hesitant fuzzy sets, we have to develop 113
a new extended TOPSIS method with the probabilis- 114
tic hesitant fuzzy information. As, the generalized 115
form of the present fuzzy set structure like as HF set, 116
PHF sets are the hesitant fuzzy set, so PHF set dis- 117
cuss more confusion compared to FS, and HF set. 118
Hence, a novel improved TOPSIS-based method is 119
developed in this paper to address unknown weight 120
information of both experts and criteria weights with 121
such circumstances and to solve the MCGDM prob- 122
lem after calculating all the weights. To solve the 123
DM problems, it is important to use the ideal opin- 124
ion that is better connected to each matrix of experts. 125
Ideal opinion is selected according to PHF average 126
method in the presented procedure. In order to find 127
differences between two PHFSs, generalized distance 128
measurement is developed. In the present probabilis- 129
tic hesitant fuzzy TOPSIS (PHF-TOPSIS) method 130
for solving MCGDM problems, generalized distance- 131
based entropy measure is applied to determine the 132
criteria weights utilized in this study under the PHF 133
setting. 134
The description of this study is arranged as fol- 135
lows. Sec. 2, presents some FSs, HFSs, PHFSs related 136
information. The methodological development of 137
hesitant probabilistic entropy measures was proposed 138
in Sec 3. Section 4 consists with the new TOPSIS- 139
based methodology in solving probabilistic hesitant 140
fuzzy MCGDM problems with completely unknown 141
weight information. Several numerical examples in 142
probabilistic hesitant settings are provided to demon- 143
strate the application transport of the developed 144
method in Section 5. Section 6 represents a compara- 145
tive discussion between the proposed TOPSIS-based 146
method and the other existing methods in solving 147
MCGDM problems in the probabilistic hesitant fuzzy 148
environment. Section 7 represents the conclusions.

Uncorrected Author Proof
M. Naeem et al. / Extended TOPSIS method based on the entropy measure and PHF 3
2. Preliminaries149
In this portion, we briefly examine some of150
the basic notions of fuzzy sets, hesitant fuzzy set,151
probabilistic hesitant fuzzy set, operational laws of
152
probabilistic hesitant fuzzy set, and probabilistic hes-153
itant fuzzy set score function.154
Definition 1. [40] Let ˇ
Tbe a univeral set. Then, a
fuzzy set is defined as;
γ=ˆg, Pγ(ˆg)/ˆg∈ˇ
T,(1)
where Pγ(ˆg)∈[0,1]represent the membership
155
grade of ˆg∈ˇ
Tin γ.156
Definition 2. [34] Let ˇ
Tbe a univeral set. Then, a
hesitant fuzzy set is defined as;
γ=ˆg, hγ(ˆg)/ˆg∈ˇ
T,(2)
where hγ(ˆg)in the form of the set, that contained
157
some possible values in [0,1],denoted the member-158
ship degree of ˆg∈ˇ
Tin γ.
159
Definition 3. [34] Suppose that h, h1and h2be any
160
three sets of HFNs. Then, the basic operations are
161
described as;162
(1) Complement
hc=
β∈h
{1−β}.(3)
(2) Sum
h1⊕h2=
β1∈h1,β2∈h2
{β1+β2−β1β2}.
(4)
(3) Multiplication
h1⊗h2=
β1∈h1,β2∈h2
{β1β2}.(5)
(4) Exponentiation
ζh =
β∈h1−(1−β)ζ,ζ ≥0.(6)
Definition 4. [16] Let ˇ
Tbe a universal set. Then, the
probabilistic hesitant fuzzy set (PHFS) is defined as;
γ=ˆg, hγ(ˆg/pı)|ˆg∈ˇ
T,(7)
where hγ(ˆg)∈[0,1]and hγ(ˆg/pı)denoted the
163
membership grade of ˆg∈ˇ
Tin γ, where pıdenoted
164
the palpabilities of hγ(ˆg)which satisfy ıpı≤1.
Definition 5. [16] Suppose that h(βı/pı),h
1(βj/pj)165
and h2(βk/pk) are three sets of PHFNs. Then, the 166
below operational laws are described as; 167
Complement
(h(βı|pı))c=
ı∈1,..., /=h(βı|pı)
{(1 −βı)|pı}.(8)
Sum 168
h1(βj|pj)⊕h2(βk|pk)=h1(βj|pj)⊕h2(βk|pk)169
=
j∈1,..., /=h1(βı|pı),k∈1,..., /=h2(βk|pk)
170
βj+βk−βjβk|pjpk.(9) 171
Multiplication 172
h1(βj|pj)⊗h2(βk|pk)=h1(βj|pj)⊗h2(βk|pk)173
=
j∈1,..., /=h1(βı|pı),k∈1,..., /=h2(βk|pk)
174
βjβk|pjpk.(10) 175
Exponentiation
ζh(βı|pı)=
ı∈1,..., /=h(βı|pı)
{1−(1 −βı)ζ|pı},ζ ≥0.
(11)
Definition 6. [45] For any PHFS ˆ
h(βı|pı),the score
function is defined as;
Sˆ
h(βı|pı)=
n

j=1
ˆ
h(βı|pı)
n,(12)
is said to be probabilistic score function of 176
ˆ
h(βı|pı),177
3. Methodological development of 178
probabilistic hesitant fuzzy entropy 179
measure 180
The generalized distance measures and weighted 181
distance measures for PHFs in this section are used 182
to determine the difference between two PHFs in the 183
same universe of expression. Then, a new entropy 184
measure for PHF is proposed based on generalized 185
distance measures to evaluate a PHFs. 186
3.1. Distance measure for PHFs 187
Definition 7. Let ˇ
Tbe be a univeral set and ˘
A=
˘
A1, ..., ˘
Anand B•=B•
1, ..., B•
nare any two

Uncorrected Author Proof
4M. Naeem et al. / Extended TOPSIS method based on the entropy measure and PHF
PHFSs, where ˘
Aj=˘g, ˆ
hkαj/pj/˘g∈ˇ
Tand
B•
j=˘g, ˆ
hkβj/pj/˘g∈ˇ
T,j =1, ..., n, gener-
alized distance measures between ˘
Aand βis defined
for ζ>0 as,
d(˘
A, B•)=1
2n
n

j=1
n

j=1
ˆ
hk(αj/pj)−
n

j=1
ˆ
hk(βj/pj)
ζ
1
ζ
.
(13)
188
Definition 8. Let ˘
A=˘
A1, ..., ˘
Anand
B•=B•
1, ..., B•
nbe any two PHFS sets,
where ˘
Aj=˘g, ˆ
hkαj/pj/˘g∈ˇ
Tand
B•
j=˘g, ˆ
hkβj/p/˘g∈ˇ
T,j =1, ..., n. Then,
the weighted generalized distance measures between
˘
Aand B•is defined for ζ>0 as,
d(˘
A, B•)=1
2n
n

j=1
wj
n

j=1
ˆ
hk(αj/pj)−
n

j=1
ˆ
hk(βj/pj)
ζ
1
ζ
,
(14)
where wj∈[0,1],with wj(j=1, ..., n) and
189
n
j=1wj=1190
(1) If we put ζ=1,then Equation (14) is called a191
weighted Hamming distance measure.
192
(2) If we put ζ=2,then Equation (14) is called a193
weighted Euclidean distance measure.194
(3) If we put ζ=+∞,then Equation (14) is called195
a weighted Chebychev distance measure.
196
Definition 9. Let ˘
Aj=˘g, ˆ
hkαj/pj/˘g∈ˇ
Tand
B•
j=˘g, ˆ
hkβj/p/˘g∈ˇ
T,be any two PHFS
sets. Then, the generalized distance measure is
defined as;
d˘
A, B•=1
2
ˆ
hkαj/Pj−ˆ
hkβj/pj
ζ1
ζ
,
(15)
then the distance measure defined in Equation (15),
197
satisfies the below properties;198
(1) 0 ≤d˘
A, B•≤1,
199
(2) d˘
A, B•=0,iff ˘
A=B•,
200
(3) d˘
A, B•=dB•,˘
A.
201
3.2. Entropy measure for PHFs
202
Definition 10. [14] Let ˘
A=˘
A1, ..., ˘
Anbe any
PHFSs, where ˘
Aı=ˆ
h(αı/pı)is a PHFN for ı=
1, ..., n. The entropy measure for PHFS by defined
as;
˘
E(˘
A)=1−2
n
n

j=11
k
k
ˆ
h(αı/pı)−1
2.(16)
Properties of entropy measure; 203
Let ˘
Aand B•are the PHFSs in universal set ˇ
T,204
the entropy measure ˘
E(˘
A) and ˘
E(B•) satisfies the 205
following properties; 206
(˘a1)˘
E(˘
A)=0,iff ˘
Ais a crisp set; 207
(˘a2)˘
E(˘
A)=1,iff ˘
Ais a most PHFs; 208
(˘a3)˘
E(˘
A)=˘
E(˘
Ac); 209
(˘a4)˘
E(˘
A)≤˘
E(B•),if ˆ
h(αı/pı)≤ˆ
h(βı/pı)≤210
1/2or1/2≤ˆ
h(βı/pı)≤ˆ
h(αı/pı).211
F(o). llows form [14]. 212
4. Probabilistic hesitant fuzzy TOPSIS 213
4.1. Probabilistic hesitant fuzzy MCGDM 214
problem 215
Let ˆ
S=ˆ
S1, ..., ˆ
Snbe the family of alterna-
tives, and let C={C1, ..., Cm}be the family of
attribute. Let there also be a number of experts,E
k
(k=1, ..., e), to express their opinion on n alterna-
tives with respect to the mattributes by using PHFs
ˆ
H(k)
ıj =ˆ
hk(βı/pı)ıj. The decision matrix of the kth
experts as.
ˆ
H(k)=ˆ
H(k)
ıj m×n=ˆ
hk(βı/pı)ıjm×n
,
where
ˆ
H(k)=
ˆ
S1
ˆ
S2
.
.
ˆ
Sn
⎡
⎢
⎢
⎢
⎢
⎢
⎣
C1C2.. Cn
ˆ
H(k)
11 ˆ
H(k)
12 . . ˆ
H(k)
1n
ˆ
H(k)
21 ˆ
H(k)
22 . . ˆ
H(k)
2n
.. . . .
. . . . .
ˆ
H(k)
m1ˆ
H(k)
m2. . ˆ
H(k)
mn
⎤
⎥
⎥
⎥
⎥
⎥
⎦
It is represent that the all the about given infor- 216
mation the weight of experts and attribute both is 217
unknown in the decision- making. 218
4.2. PHF-TOPSIS method 219
This section his main three steps. The first step, 220
TOPSIS-based aggregation method for finding the 221
weight of experts. The second step of the criteria 222
given weight utilizing the entropy weight. The third 223
step is a scoring process based on the grade of simi- 224
larity with P•˜
Iˆ
S(k)and N˜
Iˆ
Sto the ideal solution. The 225
many steps were provided to solve the probabilistic 226

Uncorrected Author Proof
M. Naeem et al. / Extended TOPSIS method based on the entropy measure and PHF 5
hesitant MAGDM problem utilizing TOPSIS-based227
method:
228
Step 1. In the first place, the information presented229
by the group of experts is structured, and we have a230
decision matrix in the profit parameters used only.231
ˆ
H(k)
ıj =ˆ
hk(βı/pı)ıj ,
where k=1, ..., e, ı =1, ..., m, j =1, ..., n,
232
The normalized decision matrices (NEs) are there-
fore presented as follows:
ˇ
D(k)=ˆ
H(k)
ıj =
ˆ
S1
ˆ
S2
.
.
ˆ
Sn
⎡
⎢
⎢
⎢
⎢
⎣
C1C2..C
n
ˆ
H(k)
11 ˆ
H(k)
12 ..ˆ
H(k)
1n
ˆ
H(k)
21 ˆ
H(k)
22 ..ˆ
H(k)
2n
. . ...
. . ...
ˆ
H(k)
m1ˆ
H(k)
m2..ˆ
H(k)
mn
⎤
⎥
⎥
⎥
⎥
⎦
for k=1, ..., e.
233
Step 2. If the weights of expert are completely234
unknown, the final decision result can not be exe-235
cuted. The weights of the expert are therefore
236
calculated as follows at first:237
I. The expert opinion is similar to the group opin-
ion (or ˜
IO
ıj), so the best ˜
IO
ıj should be obtained
by an aggregate of all expert opinions. Thus, ˜
IO
ıj
is obtained by taking probabilistic hesitant fuzzy
weighted average of the alternatives’ decision values
corresponding to the attribute given by the experts,
considering the same weights of experts at the initial
stage;
˜
IO=⎡
⎣
˜
IO
11 ˜
IO
12 ... ˜
IO
1n
˜
IO
21 ˜
IO
21 ... ˜
IO
2n
....
˜
IO
m1˜
IO
m2... IO
mn
⎤
⎦
where,
˜
IO
ıj =
e

k=11
e
ˆ
H(k)
ıj =1−
e

k=1 1−ˆ
H(k)
ıj !1
e"
II. Calculate the right ideal opinion ( ˇ
R˜
IO◦) and the
left ideal opinion ( ´
L˜
IO•). We have,
ˇ
R˜
IO◦
ıj =⎡
⎢
⎣
ˇ
R˜
IO◦
11 ˇ
R˜
IO◦
12 ... ˇ
R˜
IO◦
1n
ˇ
R˜
IO◦
21 ˇ
R˜
IO◦
22 ... ˇ
R˜
IO◦
2n
. . ... .
ˇ
R˜
IO◦
m1ˇ
R˜
IO◦
m2... ˇ
R˜
IO◦
mn
⎤
⎥
⎦
and
´
L˜
IO•=⎡
⎢
⎣
´
L˜
IO•
11 ´
L˜
IO•
12 ... ´
L˜
IO•
1n
´
L˜
IO•
21 ´
L˜
IO•
22 ... ´
L˜
IO•
2n
. . ... .
´
L˜
IO•
m1´
L˜
IO•
m2... ´
L˜
IO•
mn
⎤
⎥
⎦
where ˇ
R˜
IO◦
ıj =#ˆ
H(k)
ıj : maxk$Sˆ
H(k)
ıj %&238
and ´
L˜
IO•
ıj =#ˆ
H(k)
ıj : mink$Sˆ
H(k)
ıj %&for 239
ı=1, ..., m, j =1, ..., n, 240
III. With Equation (13), determined the
distances of each personal decision matrix
ˇ
D(k)from ˜
IO, ˇ
R˜
IO◦, and ´
L˜
IO•, identified as
ˇ
D˜
IO(k),ˇ
Dˇ
R˜
IO◦(k), and ˇ
D´
L˜
IO•(k), respectively,
we have
ˇ
D˜
IO(k)
ı=⎡
⎣
1
2n
n

j=1⎧
⎨
⎩
n

j=1
ˆ
H(k)
ıj −
n

j=1
˜
IO
ıj
ζ⎫
⎬
⎭⎤
⎦
1
ζ
ˇ
Dˇ
R˜
IO◦(k)
ı=1
2n
n

j=1
n

j=1
ˆ
H(k)
ıj −
n

j=1
ˇ
R˜
IO◦
ıj
ζ"
1
ζ
ˇ
D´
L˜
IO•(k)
ı=1
2n
n

j=1
n

j=1
ˆ
H(k)
ıj −
n

j=1
´
L˜
IO•
ıj
ζ"
1
ζ
for ı=1, ..., m and k=1, ..., e. 241
IV . The closeness indices ( ˇ
C˜
Is) are used as below:
ˇ
C˜
Ik=
m
ı=1ˇ
Dˇ
R˜
IO◦(k)
ı+m
ı=1ˇ
D´
L˜
IO•(k)
ı
m
ı=1ˇ
D˜
IO(k)
ı+m
ı=1ˇ
Dˇ
R˜
IO◦(k)
ı+m
ı=1ˇ
D´
L˜
IO•(k)
ı
for k=1, ..., e
V. The experts weights are calculated as follows:
η∗
k=
ˇ
C˜
Ik
e
k=1ˇ
C˜
Ik
242
Step 3. The attributes weights are obtain by using 243
the entropy measure as follows: 244
I. The revised ideal opinion (REV ˜
IO) are calcu-
late as;
REV ˜
IO
ıj =
e
-
k=1η∗
k.ˆ
H(k)
ıj 
=1−
e

k=11−ˆ
H(k)
ıj η∗
k"
II. The entropy measure relation to each attributes
are obtain by which using Equation (16) which as;
EMj=EREV ˜
IO
ıj, ..., REV ˜
IO
mj,j =1, ..., n.

Uncorrected Author Proof
6M. Naeem et al. / Extended TOPSIS method based on the entropy measure and PHF
III. Using the entropy weighted, this the weight of
attributes as follows;
ˆ
Wˇ
Cj=1−EMj
n−n
j=1EMj
Thus, we have the following weights of the criteria245
as ˆ
Wˇ
C=ˆ
Wˇ
C1,..., ˆ
Wˇ
CnT
246
Step 4. Again, the TOPSIS method is utilized to247
classify the alternatives to each expert as follows
248
through the evaluation of the attribute weights. We249
have250
I. Obtained weighted NEk
ıj by the weight vector
of the attributes calculated as:
NE(k)
ıj =ˆ
Wˇ
Cj.ˆ
H(k)
ıj =#1−1−ˆ
H(k)
ıj ˆ
Wˇ
Cj&
for each k=1, ..., e
II.P•˜
Iˆ
S(k)and N˜
IS(k)are calculated for each NE(k),
251
weighted for each E(k)(k=1, ..., e)in the following
252
method:253
P•˜
Iˆ
S(k)= P•˜
Iˆ
S(k)
j!e×n=
NE(k)
ıj : max
ısNE(k)
ıj ! for j=1, ..., n
N˜
Iˆ
S(k)= N˜
Iˆ
S(k)
j!e×n=
NE(k)
ıj : min
ısNE(k)
ıj ! for j=1, ..., n
254
III. Determined weights of the attributes are
255
taken into consideration. Utilizing Equation (9), the
256
weighted distances NE(k)
ıj from P•˜
Iˆ
S(k)and N˜
Iˆ
S(k)
257
are shown by ˇ
D˜
I´
S
ıand ˇ
D˜
I´
S−
ırespectively, and are
258
determined for each Ek, as follows: 259
ˇ
D˜
I´
S(k)
ı=1
2n
n

j=1
ˆ
Wˇ
Cj
n

j=1
NE(k)
ıj −
n

j=1
P•˜
Iˆ
Sk
ζ"
1
ζ
260
ˇ
D˜
I´
S−(k)
ı=1
2n
n

j=1
ˆ
Wˇ
Cj
n

j=1
NE(k)
ıj −
n

j=1
N˜
Iˆ
Sk
ζ"
1
ζ
261
IV . The following the revised closeness indices
(ˇ
Rˇ
C˜
I) of the alternatives are calculated for each Ek,
as
ˇ
Rˇ
C˜
Ik
ı=
ˇ
D˜
I´
S−(k)
ı
ˇ
D˜
I´
S−(k)
ı+ˇ
D˜
I´
S(k)
ı
Step 5. Using the calculated weights of experts η∗
k,262
ˇ
Rˇ
C˜
Is the final revised closeness index Fˇ
Rˇ
C˜
Iıfor 263
each alternative is aggregated as; 264
Fˇ
Rˇ
C˜
Iı=
e

k=1
η∗
k.ˇ
Rˇ
C˜
Ik
ı
The greater Fˇ
Rˇ
C˜
Iıis the most suitable alternative. 265
5. Numerical example 266
First of all, this section uses a numerical analysis of 267
the selection of air transport to illustrate the planned 268
MAGDM method. The present DM procedures and 269
the existing DM techniques are then compared using 270
PHF information to illustrate the features and advan- 271
tages of the proposed technique. 272
Air transport is important for the continuation of 273
airline services, selecting the best airport code is 274
considered a MCGDM problem. Four airport codes 275
are identified which are used as alternatives in this
Table 1
PNE1
ˆ
H(k)ˆ
C1ˆ
C2ˆ
C3
E1ˆ
S1{0.4/0.3,0.3/0.2,0.6/0.5}{
0.5/0.2,0.5/0.5,0.3/0.3}{
0.2/0.5,0.4/0.3,0.0/0.2}
ˆ
S2{0.2/0.5,0.4/0.3,0.4/0.2}{
0.4/0.3,0.5/0.3,0.5/0.4}{
0.3/0.2,0.6/0.4,0.5/0.4}
ˆ
S3{0.5/0.2,0.3/0.5,0.6/0.3}{
0.3/0.4,0.4/0.3,0.5/0.3}{
0.7/0.2,0.6/0.5,0.6/0.3}
ˆ
S4{0.6/0.3,0.4/0.4,0.3/0.3}{
0.7/0.4,0.9/0.2,0.3/0.4}{
0.3/0.3,0.4/0.3,0.8/0.4}
Table 2
PNE2
E2ˆ
C1ˆ
C2ˆ
C3
ˆ
S1{0.6/0.3,0.7/0.4,0.4/0.3}{
0.7/0.2,0.8/0.4,0.2/0.4}{
0.7/0.4,0.7/0.2,0.2/0.4}
ˆ
S2{0.3/0.6,0.4/0.2,0.3/0.2}{
0.2/0.7,0.4/0.1,0.8/0.2}{
0.5/0.5,0.2/0.4,0.0/0.1}
ˆ
S3{0.6/0.6,0.4/0.3,0.0/0.1}{
0.4/0.1,0.2/0.6,0.3/0.3}{
0.5/0.4,0.4/0.2,0.2/0.4}
ˆ
S4{0.3/0.5,0.4/0.2,0.4/0.3}{
0.4/0.5,0.6/0.3,0.0/0.2}{
0.6/0.3,0.6/0.4,0.7/0.3}

Uncorrected Author Proof
M. Naeem et al. / Extended TOPSIS method based on the entropy measure and PHF 7
Table 3
PNE3
E3ˆ
C1ˆ
C2ˆ
C3
ˆ
S1{0.4/0.5,0.5/0.2,0.4/0.3}{
0.5/0.2,0.6/0.2,0.6/0.6}{
0.7/0.3,0.4/0.6,0.0/0.1}
ˆ
S2{0.3/0.2,0.6/0.4,0.6/0.4}{
0.5/0.7,0.7/0.2,0.0/0.1}{
0.4/0.1,0.5/0.2,0.6/0.7}
ˆ
S3{0.3/0.4,0.5/0.4,0.0/0.2}{
0.5/0.1,0.6/0.8,0.0/0.1}{
0.4/0.2,0.2/0.6,0.5/0.2}
ˆ
S4{0.3/0.2,0.4/0.4,0.5/0.4}{
0.4/0.5,0.6/0.3,0.4/0.2}{
0.5/0.6,0.3/0.3,0.0/0.1}
Abbreviation: E, expert.
Table 4
Ideal opinion
˜
IO
ıj ˆ
C1ˆ
C2ˆ
C3
ˆ
S1{0.4755/0.045,0.5278/0.016,0.4755/0.045}{0.5779/0.008,0.6576/0.04,0.3923/0.072}{0.5836/0.06,0.5234/0.036,0.0716/0.008}
ˆ
S2{0.2679/0.06,0.4755/0.024,0.4478/0.016}{0.3782/0.147,0.5515/0.006,0.5354/0.008}{0.4052/0.01,0.4567/0.032,0.4148/0.028}
ˆ
S3{0.4804/0.048,0.4052/0.06,0.2629/0.006}{0.4052/0.004,0.4227/0.144,0.2950/0.009}{0.5515/0.016,0.4227/0.06,0.5064/0.024}
ˆ
S4{0.4804/0.03,0.3996/0.032,0.4052/0.036}{0.5234/0.1,0.7476/0.018,0.2508/0.016}{0.4804/0.054,0.4478/0.036,0.6081/0.012}
Abbreviation: ˜
IO, Ideal opinion.
Table 5
Right ideal opinion
ˇ
R˜
IO◦ˆ
C1ˆ
C2ˆ
C3
ˆ
S1{0.6/0.3,0.7/0.4,0.4/0.3}{
0.5/0.2,0.6/0.2,0.6/0.6}{
0.7/0.4,0.7/0.2,0.2/0.4}
ˆ
S2{0.3/0.2,0.6/0.4,0.6/0.4}{
0.5/0.7,0.7/0.2,0.0/0.1}{
0.4/0.1,0.5/0.2,0.6/0.7}
ˆ
S3{0.6/0.6,0.4/0.3,0.0/0.1}{
0.5/0.1,0.6/0.8,0.0/0.1}{
0.4/0.1,0.5/0.2,0.6/0.7}
ˆ
S4{0.6/0.3,0.4/0.4,0.3/0.3}{
0.7/0.4,0.9/0.2,0.3/0.4}{
0.6/0.3,0.6/0.4,0.7/0.3}
Abbreviation: ˇ
R˜
IO◦, right ideal opinion.
Table 6
Lift ideal opinion
´
L˜
IO•ˆ
C1ˆ
C2ˆ
C3
ˆ
S1{0.4/0.5,0.5/0.2,0.4/0.3}{
0.5/0.2,0.5/0.5,0.3/0.3}{
0.2/0.5,0.4/0.3,0.0/0.2}
ˆ
S2{0.2/0.5,0.4/0.3,0.4/0.2}{
0.2/0.7,0.4/0.1,0.8/0.2}{
0.5/0.5,0.2/0.4,0.0/0.1}
ˆ
S3{0.3/0.4,0.5/0.4,0.0/0.2}{
0.4/0.1,0.2/0.6,0.3/0.3}{
0.4/0.2,0.2/0.6,0.5/0.2}
ˆ
S4{0.3/0.5,0.4/0.2,0.4/0.3}{
0.4/0.5,0.6/0.3,0.0/0.2}{
0.5/0.6,0.3/0.3,0.0/0.1}
Abbreviation: ´
L˜
IO•, lift ideal opinion.
decision, ˆ
S1is the airport code ABD,ˆ
S2is the air-
276
port code ADU,ˆ
S3is the airport code OMH,ˆ
S4is277
the airport code IFN. The following criteria are iden-278
tified when evaluating the airports: ˆ
C1represent total279
passengers, ˆ
C2represent total freight, ˆ
C3represent280
the movement of aircraft.281
The evaluation value of the alternatives (airports)
282
for each criterion provided by the experts is given by
283
PHFNs as shown in the probabilistic hesitant matrix
284
of decisions given in Tables 1–3. The following calcu-285
lations are performed to solve the MAGDM problem
286
by developed method.287
2.Many steps are taken to measure the expert’s288
weights:289
I.˜
IOis estimated and shown in Table 4.290
II.ˇ
R˜
IO◦and ´
L˜
IO•are estimated and shown in 291
Tables 5, 6 respectively. 292
III. Distances of each every decision matrix from 293
˜
IO,ˇ
R˜
IO◦, and ´
L˜
IO•are obtained using Equation 294
(13) for ζ=2 and displayed, correspondingly, in 295
Tables 7–9. 296
IV .ˇ
C˜
Iof expert are measured and shown in Table 297
10. 298
V. The expert weights (η∗
k) are now measured and 299
displayed with the current expert weights measured 300
as shown in Table 11. 301
3. The following methods are used to measure the 302
weights of the attributes. 303
I. The updated ideal opinion is obtained by taking 304
the PHF weighted average of the alternative decision 305

Uncorrected Author Proof
8M. Naeem et al. / Extended TOPSIS method based on the entropy measure and PHF
Table 7
Distance of ideal opinion from each decision matrix
ˇ
D˜
IO•ˆ
S1ˆ
S2ˆ
S3ˆ
S4
E10.4510 0.6307 0.7029 0.7187
E20.6901 0.4410 0.4049 0.6741
E30.6717 0.7160 0.5458 0.5500
Abbreviation: ˇ
D˜
IO,distance of ideal opinion; E, expert.
Table 8
Distance of right ideal opinion from each individual decision
matrix
ˇ
Dˇ
R˜
IO◦ˆ
S1ˆ
S2ˆ
S3ˆ
S4
E10.5213 0.5958 0.6109 0.32
E20.52 0.8238 0.6161 0.2080
E30.2629 0 0.3735 0.5036
Abbreviation: E, expert; ˇ
Dˇ
R˜
IO◦distance of right ideal opinion.
Table 9
Distance of right ideal opinion from each individual decision
matrix
ˇ
D´
L˜
IO•ˆ
S1ˆ
S2ˆ
S3ˆ
S4
E10.1224 0.5756 0.3905 0.5973
E20.3153 0.0571 0.2224 0.48
E30.4672 0.8062 0.4611 0.1059
Abbreviation: ˇ
D´
L˜
IO•, distance of left ideal opinion; E, expert.
Table 10
Closeness index
ˇ
C˜
I1ˇ
C˜
I2ˇ
C˜
I3
0.5987 0.5946 0.5654
Abbreviation: ˇ
C˜
Icloseness index.
values utilizing the measured expert weights as given306
in Table 11.307
II. Utilizing the entropy measurements respec-308
tively to each criteria, the weights of the attribute are309
measured and shown in Table 13.310
The weighted normalized decision matrices are
311
computed in Tables 14–16 as follows:
312
4.The following steps shall be taken to achieve a
313
judgement corresponding to each expert separately.
314
Table 11
Weight of experts
η∗
1η∗
2η∗
3
0.344 0.342 0.314
Abbreviation: Eexpert.
I. The weighted NE for each expert shall be estab- 315
lished as given in Table 13. 316
II. P•˜
Iˆ
Sand N˜
Iˆ
Sare defined by weighted NES for 317
each expert as shown in Tables 17, 18, respectively. 318
III. The ˇ
Rˇ
C˜
Is for each expert shall be determined 319
as shown in Table 19. 320
5.The final revised closeness indices (Fˇ
Rˇ
C˜
I)by 321
using the DMs weights are computed in Table 20, as 322
follows: 323
Hence, ˆ
S1is the best alternative according to given 324
attributes. 325
6. Comparison section 326
In this section, a comparison of the characteris- 327
tics of these proposed improved TOPSIS method and 328
the designed MAGDM method is made to show the 329
advantages of the designed technique. This compar- 330
ison is carried out by comparing the characteristics 331
of the different decision-making technique presents 332
in literature. In the method of [19], TOPSIS method 333
for probabilistic hesitant fuzzy information is pre- 334
sented. The Normalized DMs information are shown 335
in Tables 21–23: 336
Weight of expert are computed as follows
η∗
1η∗
2η∗
3
0.333 0.344 0.324
Weight of attributes are computed as follows;
ˆ
Wˇ
C1ˆ
Wˇ
C2ˆ
Wˇ
C3
0.342 0.332 0.326
The final revised closeness indices (Fˇ
Rˇ
C˜
Iı) by using 337
the DMs weights are computed in Table 24 as follows: 338
Hence, ˆ
S1is the best alternative according to given 339
Attributes. 340
Table 12
Revised ideal opinion
ˆ
C1ˆ
C2C3
ˆ
S1{0.4776/0.045,0.5286/0.016,0.4781/0.045}{
0.5804/0.008,0.6592/0.04,0.3883/0.072}{
0.5796/0.06,0.5266/0.036,0.0734/0.008}
ˆ
S2{0.2670/0.06,0.4717/0.024,0.4431/0.016}{
0.3748/0.147,0.5466/0.006,0.5456/0.008}{
0.4055/0.01,0.4561/0.032,0.4091/0.028}
ˆ
S3{0.4851/0.048,0.4025/0.06,0.2703/0.006}{
0.4025/0.004,0.4171/0.144,0.3026/0.009}{
0.5515/0.016,0.4287/0.06,0.5074/0.024}
ˆ
S4{0.4225/0.03,0.4/0.032,0.4025/0.036}{
0.5272/0.1,0.7517/0.018,0.0246/0.016}{
0.4804/0.054,0.4517/0.036,0.6191/0.012}
Abbreviation: REV ˜
IO, revised ideal opinion.

Uncorrected Author Proof
M. Naeem et al. / Extended TOPSIS method based on the entropy measure and PHF 9
Table 13
Weight of attributes
ˆ
Wˇ
C1ˆ
Wˇ
C2ˆ
Wˇ
C3
0.335 0.330 0.335
Abbreviation: ˆ
Wˇ
C, Weight of attributes.
7. Result and discussion
341
The decision maker gives the information in the
342
form of probabilistic hesitant fuzzy sets. In compari-343
son section, we consider the neutral term equal to zero344
Table 14
Weighted normalized E1information
NE(k)ˆ
C1ˆ
C2ˆ
C3
ˆ
S1{0.1572/0.3,0.1126/0.2,0.2643/0.5}{
0.2072/0.2,0.2072/0.5,0.1126/0.3}{
0.0720/0.5,0.1572/0.3,0000/0.2}
ˆ
S2{0.0720/0.5,0.1572/0.3,0.1572/0.2}{
0.1572/0.3,0.2072/0.3,0.2072/0.4}{
0.1126/0.2,0.2643/0.4,0.2672/0.4}
ˆ
S3{0.2072/0.2,0.1126/0.5,0.2643/0.3}{
0.1126/0.4,0.1572/0.3,0.2072/0.3}{
0.3319/0.2,0.2643/0.5,0.3310/0.3}
ˆ
S4{0.2643/0.3,0.1572/0.4,0.1126/0.3}{
0.3319/0.4,0.5376/0.2,0.1126/0.4}{
0.1126/0.2,0.1572/0.3,0.4167/0.4}
Table 15
Weighted normalized E2information
ˆ
C1ˆ
C2ˆ
C3
ˆ
S1{0.2609/0.3,0.3278/0.4,0.1551/0.3}{
0.3272/0.2,0.4120/0.4,0.0709/0.4}{
0.3272/0.4,0.3278/0.2,0.0709/0.4}
ˆ
S2{0.1110/0.6,0.1551/0.2,0.1110/0.2}{
0.0709/0.7,0.1551/0.1,0.4120/0.2}{
0.2044/0.5,0.0709/0.4,0.000/0.1}
ˆ
S3{0.2609/0.6.0.1551/0.3,0.000/0.1}{
0.1551/0.1,0.0709/0.6,0.1110/0.3}{
0.2044/0.4,0.1551/0.2,0.0709/0.4}
ˆ
S4{0.1110/0.5,0.1551/0.2,0.1551/0.3}{
0.1551/0.5,0.2609/0.3,0.0/0.2}{
0.2609/0.3,0.2609/0.4,0.3278/0.3}
Table 16
Weighted normalized E3information
ˆ
C1ˆ
C2ˆ
C3
ˆ
S1{0.1572/0.5,0.2072/0.2,0.1572/0.3}{
0.2072/0.2,0.2643/0.2,0.2643/0.6}{
0.3319/0.3,0.1572/0.6,0.000/0.1}
ˆ
S2{0.1126/0., 0.2643/0.4,0.2643/0.4}{
0.2072/0.7,0.3319/0.2,0.000/0.1}{
0.1572/0.1,0.2072/0.2,0.2643/0.7}
ˆ
S3{0.1126/0.4,0.2072/0.4,0.000/0.2}{
0.2072/0.1,0.2643/0.8,0.000/0.1}{
0.1572/0.2,0.0720/0.6,0.2072/0.2}
ˆ
S4{0.1126/0.2,0.1572/0.4,0.2072/0.4}{
0.1572/0.5,0.2643/0.3,0.1572/0.2}{
0.2072/0.6,0.1126/0.3,0.000/0.1}
Abbreviation: E, expert; NE, normalized decision matrix.
Abbreviation: E, expert; P•˜
Iˆ
S, positive ideal solution.
Table 17
Positive ideal solution for each experts
ˆ
C1ˆ
C2ˆ
C3
P•˜
Iˆ
S1{0.1572/0.5,0.2072/0.2,0.1572/0.3}{
0.3319/0.4,0.5376/0.2,0.1126/0.4}{
0.3319/0.2,0.2643/0.5,0.3310/0.3}
P•˜
Iˆ
S2{0.2609/0.3,0.3278/0.4,0.1551/0.3}{
0.3278/0.2,0.4125/0.4,0.0709/0.4}{
0.2609/0.3,0.2609/0.4,0.3278/0.3}
P•˜
Iˆ
S3{0.1126/0.2,0.2643/0.4,0.2643/0.4}{
0.2072/0.2,0.2643/0.2,0.2643/0.4}{
0.1572/0.1,0.2072/0.2,0.2643/0.7}
Table 18
Negative ideal solution for each experts
ˆ
C1ˆ
C2ˆ
C3
N˜
Iˆ
S1{0.0720/0.5,0.1572/0.3,0.1572/0.2}{
0.2072/0.2,0.2072/0.5,0.1126/0.3}{
0.0720/0.5,0.1572/0.3,0.000/0.2}
N˜
Iˆ
S2{0.1110/0.6,0.1551/0.2,0.1110/0.2}{
0.1551/0.1,0.0709/0.6,0.1110/0.3}{
0.2644/0.5,0.2609/0.4,0.000/0.1}
N˜
Iˆ
S3{0.1126/0.4,0.2072/0.4,0.000/0.2}{
0.2072/0.7,0.2643/0.2,0.000/0.1}{
0.2072/0.6,0.1126/0.3,0.000/0.1}
Abbreviation: E, expert; N˜
Iˆ
S, negative ideal solution.
and used the proposed spherical improved TOPSIS 345
technique to solve the information. As in the obtain- 346
ing results, ˆ
S1are the best alternative which is same 347
as the given in the [19]. Here, we gave some compar- 348
ison of previously presented TOPSIS techniques and 349
proposed improved TOPSIS technique. 350
8. Conclusions 351
PHFS is a modern and powerful simplified idea 352
that has been selected as a strategic method to resolve 353

Uncorrected Author Proof
10 M. Naeem et al. / Extended TOPSIS method based on the entropy measure and PHF
Table 19
Revised closeness indices for each experts
ˇ
Rˇ
C˜
Ikˆ
S1ˆ
S2ˆ
S3ˆ
S4
ˇ
Rˇ
C˜
I10.9949 0.9856 0.9975 0.9969
ˇ
Rˇ
C˜
I20.4118 0.1024 0.2666 0.500
ˇ
Rˇ
C˜
I30.7238 0.6248 0.4036 0.4999
Abbreviation: E, expert; ˇ
Rˇ
C˜
I, revised closeness indices.
Table 20
Last the revised closeness indices
Attributes ˆ
S1ˆ
S2ˆ
S3ˆ
S4
Fˇ
Rˇ
C˜
Iı0.7004 0.5603 0.5510 0.6609
Table 21
PNE1
E1ˆ
H(k)ˆ
C1ˆ
C2ˆ
C3
ˆ
S1{.55/.3081,.68/.3501,.73/.3418}{
0.6/.4994,.66/.5006}{
.62/.4985,.68/.5005}
ˆ
S2{.62/.4936,.77/.5064}{
.68/.4987,.77/.5013 }{
.6/.3026,.73/.3570,.85/.3404}
ˆ
S3{.63/.3246,.71/.3412,.77/.3342}{
.66/.4997,.71/.5003}{
.68/.4996,0.74/.5004}
ˆ
S4{.67/.4998,.72/.5002}{
.62/.4992,.69/.5008}{
.67/.4999,0.71/.5001}
Table 22
PNE2
E2ˆ
H(k)ˆ
C1ˆ
C2ˆ
C3
ˆ
S1{.55/.1763,.68/.5097,.73/.3140}{
0.6/.4990,.66/.5010}{
.62/.4991,.68/.5009}
ˆ
S2{.62/.4899,.77/.5101}{
.68/.4980,.77/.5020}{
.6/.1749,.73/.5082,.85/.3169}
ˆ
S3{.63/.1856,.71/.5189,.77/.2955}{
.66/.4996,.71/.5004}{
.68/.4993,.74/.5007}
ˆ
S4{.67/.4996,.72/.5004}{
.62/.4987,.69/.5013}{
.67/.4998,.71/.5002}
Table 23
PNE3
E3ˆ
H(k)ˆ
C1ˆ
C2ˆ
C3
ˆ
S1{0.55/0.22,0.68/0.51,0.73/0.27}{
0.6/0.61,0.66/0.39}{
0.62/0.77,0.68/0.23,0.68/0}
ˆ
S2{0.62/0.31,0.77/0.69,0.77/0}{
0.68/0.29,0.77/0.71}{
0.6/0.18,0.73/0.21,0.85/0.61}
ˆ
S3{0.63/0.35,0.71/0.52,0.77/0.13}{
0.66/0.48,0.71/0.52}{
0.68/0.65,0.74/0.35,0.74/0}
ˆ
S4{0.67/0.53,0.72/0.47,0.72/0}{
0.62/0.55,0.69/0.45}{
0.67/0.7,0.71/0.3,0.71/0}
the complexity and vagueness of the data associated354
with MCGDM issues and thus experts feel secure355
with their decision to use HF data instead of PHFS.356
In this study, a novel improved DM approach based357
on TOPSIS is developed to resolve MCGDM issues358
in the sense of the PHF environment, with completely359
unknown weights of the experts and criteria. General-360
ized distance measure using the novel concept of PHF361
entropy measure is implemented to find information 362
on PHF entropy weights in the PHF setting. In order 363
to remove the combined loss of information during 364
the process, AOs are carried out in the last steps by uti- 365
lizing the computed weights of the experts to obtain 366
the final alternative standard. Finally, a numerical 367
examples are described to demonstrate the appli- 368
cability and benefits of the implemented method. 369
The developed technique can also be extended to 370
future research utilizing other fuzzy types in DM and 371
using them to solve various MCGDM problems with 372
unknown expert weights and criteria. 373
In the future, we will extend our proposed con- 374
cept to; Consensus is reaching for MAGDM with 375
multi-granular hesitant fuzzy linguistic term sets; 376
Table 24
Final revised closeness indices
E3ˆ
H(k)ˆ
C1ˆ
C2ˆ
C3
Attributes ˆ
S1ˆ
S2ˆ
S3ˆ
S4
Fˇ
Rˇ
C˜
Iı0.6655 0.5999 0.6121 0.6641

Uncorrected Author Proof
M. Naeem et al. / Extended TOPSIS method based on the entropy measure and PHF 11
Table 25
Comparative study table
Authors Uncertainty approach Modeling approach Unknown weights
FSs HFs PHFs Group TOPSIS Decision Attributes
DM Method Maker
Beg & Rashid [2] yes no no no yes no no
Yue [38] yes no no yes no yes no
Wu et al. [32] yes yes no yes yes yes yes
Proposed approach yes yes yes yes yes yes yes
Consensus is reaching for social network group deci-377
sion making by considering leadership and bounded
378
confidence.
379
Conflicts of interest380
The authors declare that they have no conflicts of381
interest.
382
Acknowledgments383
The authors would like to thank the Deanship of384
Scientific Research at Umm Al-Qura University for
385
supporting this work by grant number 19-SCI-101-
386
0056.
387
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