Computers in Industry 108 (2019) 45–52
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Computers in Industry
journal homepage: www.elsevier.com/locate/compind
Dynamic interval valued neutrosophic set: Modeling decision making
in dynamic environments
Nguyen Tho Thonga,b , Luu Quoc Datc , Le Hoang Sona,* , Nguyen Dinh Hoaa , Mumtaz Alid,
Florentin Smarandachee
a
VNU Information Technology Institute, Vietnam National University, Hanoi, Viet Nam
VNU University of Engineering and Technology, Vietnam National University, Hanoi, Viet Nam
VNU University of Economics and Business, Vietnam National University, Hanoi, Viet Nam
d
University of Southern Queensland, 4300, QLD, Australia
e
University of New Mexico, 705 Gurley Ave., Gallup, New Mexico 87301, USA
b
c
A R T I C L E I N F O
A B S T R A C T
Article history:
Received 24 October 2018
Received in revised form 31 January 2019
Accepted 10 February 2019
Available online xxx
Dynamic decision problems constrained by time are of highly-interested in many aspects of real life. This
paper proposes a new concept called the Dynamic Interval-valued Neutrosophic Set (DIVNS) for such the
dynamic decision-making applications. Firstly, we define the definitions and mathematical operations,
properties and correlations of DIVNSs. Next, we develop a new TOPSIS (Technique for Order of Preference
by Similarity to Ideal Solution) method based on the proposed DIVNS theory. Finally, a practical
application of the method for evaluating lecturers’ performance at the University of Languages and
International Studies, Vietnam National University, Hanoi (ULIS-VNU) is given to illustrate the efficiency
of our approach.
© 2019 Elsevier B.V. All rights reserved.
Keywords:
Dynamic environment
Interval valued set
Neutrosophic set
1. Introduction
Neutrosophic set (NS) [45] is able to handle indeterminacy
information [51,52,58]. NS and its extensions have become
widely applied in almost areas, such as decision-making
[1,12,20,21,33,34,41,42,49,58–62], clustering analysis [56,59], image
processing [27,28], etc. However, in some complex problems in reallife, data may be collected from different time intervals or multiperiods, which raises the need for dynamic decision making for such
the situations. The term ‘dynamic’ can be regarded in term of criteria
such as (a) a series of decisions required to reach a goal; (b) path
dependent decision; (c) the state of decision. This research considers
the ‘dynamic’ decision problems which are constrained by time, as
seen, for example, in emergency management and patient care.
Specifically, when the economic situation of a certain company is
investigated, the economic growth level of product series should be
investigated by changes of the trend of profit of all products through
the periods. Another example can be found in medical diagnosis
where clinicians have to exam patients by different time intervals.
* Corresponding author at: 144 Xuan Thuy, Cau Giay, Hanoi, Viet Nam.
Tel.: +84 904.171.284.
E-mail addresses: 17028003@vnu.edu.vn (N.T. Thong), datlq@vnu.edu.vn
(L.Q. Dat), sonlh@vnu.edu.vn (L.H. Son), hoand@vnu.edu.vn (N.D. Hoa),
Mumtaz.Ali@usq.edu.au (M. Ali), fsmarandache@gmail.com (F. Smarandache).
https://doi.org/10.1016/j.compind.2019.02.009
0166-3615/© 2019 Elsevier B.V. All rights reserved.
Recently, Yan et al. [53] developed a dynamic multiple attribute
decision making method with grey number (considering both
attribute value aggregation of all periods and their fluctuation
among periods) to calculate degree of every alternative. This model
was also used in [32] to manage linguistic bipolar scales using
transformation between bipolar and unipolar linguistic terms. Ye
[57] proposed a dynamic neutrosophic multiset. For decision
assistance in dynamic environments, some algorithms that used
TOPSIS under neutrosophic linguistic environments were presented
in [2,10,11,22,23,25,26,33–37,40,55]. There have been also some
works that applied the Interval-Valued Neutrosophic Set (IVNS) with
the TOPSIS method for decision making [6,11,29,33,49,54,62]. Other
relevant decision making methods can be retrieved in [3–5,7–9,13–
19]. However, the existing researches did not consider different time
intervals as the objective of this research aims. To the best of our
knowledge, fluctuation of alternative’s attribute values within
periods on NSs has not been examined. In many practical cases,
there is not enough available information to judge complicated
situations, indeed it often given approximate ranges.
In this paper, we propose a new TOPSIS method based on a new
extension of NS called the Dynamic Interval-valued Neutrosophic
Set (DIVSN) for dynamic decision-making problems. The main
contribution includes:
(a) We define definitions and mathematical operations, properties
and correlations of DIVNSs.
46
N.T. Thong et al. / Computers in Industry 108 (2019) 45–52
(b) We develop a new TOPSIS method based on the proposed
DIVNS theory.
(c) A practical application of the method for evaluating
lecturers’ performance at the Vietnam National University,
Hanoi (ULIS-VNU) is given to illustrate the efficiency of our
approach.
Section 2 defines the new concept of Dynamic Intervalvalued Neutrosophic Set (DIVSN). Section 3 presents the TOPSIS
method for DIVSN. Section 4 illustrates the proposed method in
a practical application. Finally, Section 5 summarizes the
findings.
xðhT x ðt1 Þ; Ix ðt1 Þ; F x ðt1 Þi; hT x ðt2 Þ; Ix ðt2 Þ; F x ðt2 Þi; :::; hT x ðtk Þ; Ix ðtk Þ; F x ðtk ÞiÞ
Example 2.1. A DIVNS in time sequence t ¼ ft1 ; t2 ; t3 g and a
universal NS ¼ fx1 ; x2 ; x3 g is given:
2. Dynamic interval-valued neutrosophic set
2.1. Set definition
We can also use the notation AðtÞ and xðtÞ, meaning that
each element x in A depends on t. Or T x ðtÞ; Ix ðtÞ; F x ðtÞ are interval –
valued functions (a particular case of neutrosophic function [1]).
The difference of the new definition against the existing one in
[57]:
We have extended Ye’s DSVNS [57] to DIVNS by considering a
time sequence: t ¼ ft1 ; t2 ; :::; tk g then at each time tl ; 1 l m, the
neutrosophic components of the generic element x 2 A change as
follow:
8
9
< hx1 ; ð½0:1; 0:25; ½0:15; 0:2; ½0:3; 0:6Þ; ð½0:45; 0:5; ½0:1; 0:3; ½0:2; 0:4Þ; ð½0:6; 0:7; ½0:52; 0:6; ½0:7; 0:9Þi;
=
A ¼ hx2 ; ð½0:38; 0:4; ½0:25; 0:4; ½0:12; 0:3Þ; ð½0:07; 0:1; ½0:1; 0:2; ½0:09; 0:1Þ; ð½0:22; 0:3; ½0:4; 0:5; ½0:3; 0:43Þi;
:
;
hx3 ; ð½0:7; 0:9; ½0:33; 0:45; ½0:59; 0:6Þ; ð½0:2; 0:22; ½0:5; 0:6; ½0:2; 0:3Þ; ð½0:8; 0:9; ½0:3; 0:41; ½0:3; 0:33Þi
Definition 2.1. [45]: Let U be a universe of discourse. A
neutrosophic set is:
2.2. Set theoretic operations of DIVNS
Let AðtÞ and BðtÞ be two DIVNSs included in U;
A ¼ fhx : T A ðxÞ; IA ðxÞ; F A ðxÞi; x 2 U g
T A ðxÞ; IA ðxÞ; F A ðxÞ 2 ½0; 1
where
0 supðT A ðxÞÞ þ supðIA ðxÞÞ þ supðF A ðxÞÞ 3.
and
Definition 2.2. [45]: A neutrosophic number is defined as
N ¼ a þ bI, where a and b are real numbers, and I is the
indeterminacy.
Definition 2.3. [57]: A Dynamic Single-Valued Neutrosophic Set
(DSVNS) is: A ¼ fx 2 U; xðT x ðtÞ; Ix ðtÞ; F x ðtÞÞg for all x 2 A:
E
o
n
D
xðtÞ; T Ax ðtl Þ; IAx ðtl Þ; F Ax ðtl Þ ; 8tl 2 t; x 2 U ; BðtÞ
AðtÞ ¼
n
D
E
o
xðtÞ; T Bx ðtl Þ; IBx ðtl Þ; F Bx ðtl Þ ; 8tl 2 t; x 2 U
¼
Definition 2.5. : DIVNS Intersection
AðtÞ \ BðtÞ ¼
D
n
E
xðtÞ; T Ax ðtl Þ ^ T Bx ðtl Þ; IAx ðtl Þ _ IBx ðtl Þ; F Ax ðtl Þ _ F Bx ðtl Þ ;
8tl 2 t; x 2 Ug
T x ; Ix ; F x : ½0; 1Þ ! ½0; 1
where T x ; Ix ; F x are continuous functions whose argument is
time ðtÞ.
Based on the definition of DSVNS above, we formulate the new
definition as below.
Definition 2.4. A Dynamic Interval-Valued Neutrosophic Set
(DIVNS) is in the form below:
h
i h
i h
i
L
U
L
U
x T Lx ðtÞ; T U
x ðt Þ ; I x ðt Þ; I x ðt Þ ; F x ðtÞ; F x ðt Þ
where t 0,
L
U
L
U
T Lx ðtÞ < T U
x ðt Þ; Ix ðtÞ < Ix ðtÞ; F x ðt Þ < F x ðt Þ
And
h
i h
i h
i
L
U
L
U
T Lx ðtÞ; T U
x ðt Þ ; I x ðt Þ; I x ðt Þ ; F x ðtÞ; F x ðt Þ ½0; 1
In other words, a DIVNS is a neutrosophic set whose elements’
neutrosophic components (membership, indeterminacy, nonmembership) are all intervals changing with respect to time.
For a simplified notation, we denote:
i
i
i
h
h
h
L
U
L
U
T x ðtÞ ¼ T Lx ðtÞ; T U
x ðt Þ ; I x ðt Þ ¼ Ix ðt Þ; I x ðt Þ ; F x ðt Þ ¼ F x ðt Þ; F x ðt Þ
where T x ðtÞ; Ix ðtÞ; F x ðtÞ : ½0; 1Þ ! Pð½0; 1Þ with Pð½0; 1Þ been the
power set of ½0; 1.
Definition 2.6. DIVNS Union
AðtÞ [ BðtÞ ¼
n
E
D
xðtÞ; T Ax ðtl Þ _ T Bx ðtl Þ; IAx ðtl Þ ^ IBx ðtl Þ; F Ax ðtl Þ ^ F Bx ðtl Þ ;
8tl 2 t; x 2 Ug
Definition 2.7. DIVNS Complement
E
o
n
D
AðtÞC ¼
xðtÞ; F Ax ðtl Þ; 1 IAx ðtl Þ; T Ax ðtl Þ ; 8tl 2 t; x 2 U
Definition 2.8. DIVNS inclusion
AðtÞ BðtÞ T Ax ðtl Þ T Bx ðtl Þ; IAx ðtl Þ IBx ðtl Þ and F Ax ðtl Þ F Bx ðtl Þ:
Definition 2.9. DIVNS Equality
AðtÞ ¼ BðtÞ,AðtÞ BðtÞ and AðtÞ BðtÞ:
In the above DIVNS aggregation operators by “^” we meant the
“t-norm” and by “_”
the t-conorm from the single–valued fuzzy sets
47
N.T. Thong et al. / Computers in Industry 108 (2019) 45–52
Definition 2.14. Correlation coefficient of DIVNSs
2.3. Operations on DIVNS numbers
Let
Let us consider two DIVNS numbers:
n
D
E
o
xðtÞ; T A ðx; tl Þ; IA ðx; tl Þ; F A ðx; tl Þ ; 8tl 2 t; x 2 U ;
E
o
n
D
xðtÞ; T B ðx; tl Þ; IB ðx; tl Þ; F B ðx; tl Þ ; 8tl 2 t; x 2 U
BðtÞ ¼
AðtÞ ¼
D
nD
E
Eo
T Ax ðt1 Þ; IAx ðt1 Þ; F Ax ðt1 Þ ; :::; T Ax ðtk Þ; IAx ðtk Þ; F Ax ðtk Þ
að t Þ ¼
nD
E
Eo
D
bðtÞ ¼
T Bx ðt1 Þ; IBx ðt1 Þ; F Bx ðt1 Þ ; :::; T Bx ðtk Þ; IBx ðtk Þ; F Bx ðtk Þ :
be two DIVNs in t ¼ ft1 ; t2 ; :::; tk g and U ¼ ðx1 ; x2 ; :::; xn Þ.
A correlation coefficient is:
1
0
infT A ðxi ; tl Þ infT B ðxi ; tl Þ þ supT A ðxi ; tl Þ supT B ðxi ; tl Þ
n
P
C
B
A
@ þinfIA ðtl Þ infIB ðxi ; tl Þ þ supIA ðxi ; tl Þ supIB ðxi ; tl Þ
i¼1
A
B
A
B
k
X
þinfF ðxi ; tl Þ infF ðxi ; tl Þ þ supF ðxi ; tl Þ supF ðxi ; tl Þ
1
ffi1
rðAðtÞ; BðtÞÞ ¼
0 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
2
2
2 3
k l¼1 u
u n
A
A
A
B uX6 infT ðxi ; tl Þ þ supT ðxi ; tl Þ þ infI ðxi ; tl Þ
7C
Bu 4
2
2
2 5 C
Bt
C
B i¼1 þ supIA ðxi ; tl Þ þ infF A ðxi ; tl Þ þ supF A ðxi ; tl Þ
C
B
C
B vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
C
ffi
B u 2
2
2
2 3 C
B u n
C
B
B
B
B uX6 infT ðxi ; tl Þ þ supT ðxi ; tl Þ þ infI ðxi ; tl Þ 7 C
B u 4
C
5
@ t
A
2
2
2
B
B
B
i¼1 þ supI ðxi ; tl Þ
ð
Þ
ð
Þ
þ inf xi ; tl
þ supF xi ; tl
Definition 2.10. Addition of DIVNS numbers
8*
+
>
T Ax ðt1 Þ þ T Bx ðt1 Þ T Ax ðt1 Þ T Bx ðt1 Þ;
>
>
;
>
>
< IAx ðt1 Þ IBx ðt1 Þ; F Ax ðt1 Þ F Bx ðt1 Þ
aðtÞ bðtÞ ¼ :::;
*
+
>
>
T Ax ðtk Þ þ T Bx ðtk Þ T Ax ðtk Þ T Bx ðtk Þ;
>
>
>
: IA ðt Þ IB ðt Þ; F A ðt Þ F B ðt Þ
x k
x k
x k
x k
9
>
>
>
>
>
=
Theorem 2.1. The correlation coefficient between A and B
satisfies:
ð1Þ
>
>
>
>
>
;
Definition 2.11. Multiplication of DIVNS numbers
aðtÞ
bðtÞ
8*
+
>
T Ax ðt1 Þ T Bx ðt1 Þ; IAx ðt1 Þ þ IBx ðt1 Þ IAx ðt1 Þ IBx ðt1 Þ;
>
>
;
>
>
A
B
A
B
< F x ðt1 Þ þ F x ðt1 Þ F x ðt1 Þ F x ðt1 Þ
¼ :::;
+
*
>
>
T Ax ðtk Þ T Bx ðtk Þ; IAx ðtk Þ þ IBx ðtk Þ IAx ðtk Þ IBx ðtk Þ;
>
>
>
: F A ðt Þ þ F B ðt Þ F A ðt Þ F B ðt Þ
x k
x k
x k
x k
9
>
>
>
>
>
=
ð2Þ
>
>
>
>
>
;
Definition 2.12. Scalar Multiplication of DIVNS numbers
nD
a
E
a aðtÞ ¼ 1
1 T Ax ðt1 Þ ; IAx ðt1 Þa ; F Ax ðt1 Þa ; :::;
D
1
1
T Ax ðtk Þ
a
E
; IAx ðtk Þa ; F Ax ðtk Þa g
ð5Þ
ð3Þ
ðPr1Þ 0 rðAðtÞ; BðtÞÞ 1;
ðPr2Þ rðAðtÞ; BðtÞÞ ¼ 1 if AðtÞ ¼ BðtÞ;
ðPr3ÞrðAðtÞ; BðtÞÞ ¼ rðBðtÞ; AðtÞÞ
Proof.
(Pr1) It is obvious that rðAðtÞ; BðtÞÞ 0. From Cauchy–Schwarz
inequality, we have
1
0
infT A ðxi ; tl Þ infT B ðxi ; tl Þ þ supT A ðxi ; tl Þ supT B ðxi ; tl Þ
n
X
C
B
A
@ þinfIA ðtl Þ infIB ðxi ; tl Þ þ supIA ðxi ; tl Þ supIB ðxi ; tl Þ
i¼1
þinfF A ðxi ; tl Þ infF B ðxi ; tl Þ þ supF A ðxi ; tl Þ supF B ðxi ; tl Þ
v
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi1
0u 2
2
2
2 3
u n
A
A
A
B uX6 infT ðxi ; tl Þ þ supT ðxi ; tl Þ þ infI ðxi ; tl Þ
7C
Bu 4
2
2
2 5 C
Bt
C
A
A
A
B i¼1 þ supI ðxi ; tl Þ þ infF ðxi ; tl Þ þ supF ðxi ; tl Þ
C
B
C
B vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffiC
B u 2
2
2
2 3 C
B u n
C
B uX6 infT B ðxi ; tl Þ þ supT B ðxi ; tl Þ þ infIB ðxi ; tl Þ
7C
B u 4
2
2
2 5 C
@ t
A
B
B
i¼1 þ supI ðxi ; t l Þ
þ inf ðxi ; tl Þ þ supF B ðxi ; tl Þ
for each l 2 f1; 2; :; kg. Thus, 0 rðAðtÞ; BðtÞÞ 1.
(Pr2) AðtÞ ¼ BðtÞ.
B
A
8l 2 f1; 2; :::; kg.
B
We have
infT A ðxi ; tl Þ ¼
A
infT ðxi ; tl Þ; supT ðxi ; tl Þ ¼ supT ðxi ; tl Þ; infI ðxi ; tl Þ ¼ infIB ðxi ; tl Þ;
supIA ðxi ; tl Þ ¼ supIB ðxi ; tl Þ; infF A ðxi ; tl Þ ¼ infF B ðxi ; tl Þ; supF A ðxi ; tl Þ ¼
Definition 2.13. Power of the DIVNS numbers
8D
a
A
>
>
< T x ðt1 Þ ; 1
:::;
aðtÞa ¼ D
>
>
: T Ax ðtk Þa ; 1
1
1
a
IAx ðt1 Þ ; 1
a
IAx ðtk Þ ; 1
1
1
a E 9
>
;>
F Ax ðt1 Þ
=
a E >
>
;
F Ax ðtk Þ
supF B ðxi ; tl Þ; infT A ðxi ; tl Þ ¼ infT B ðxi ; tl Þ ) rðAðtÞ; BðtÞÞ ¼ 1
(Pr3) It is easily observed.
Definition 2.15. Weighted Correlation Coefficient of DIVNSs
Different weights for xi ði ¼ 1; :::; nÞ and tl ðl ¼ 1; :::; kÞ are
integrated as follows.
ð4Þ
48
N.T. Thong et al. / Computers in Industry 108 (2019) 45–52
1
infT A ðxi ; tl Þ infT B ðxi ; tl Þ þ supT A ðxi ; tl Þ supT B ðxi ; tl Þ
C
B
wi @ þinfIA ðtl Þ infIB ðxi ; tl Þ þ supIA ðxi ; tl Þ supIB ðxi ; tl Þ A
i¼1
A
B
A
B
k
X
þinfF ðxi ; tl Þ infF ðxi ; tl Þ þ supF ðxi ; tl Þ supF ðxi ; tl Þ
1
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi1
rW ðAðtÞ; BðtÞÞ ¼
v 0v
0
u
2
2
2 1
k l¼1 l
u n
A
A
A
X
infT
ð
x
;
t
Þ
þ
supT
ð
x
;
t
Þ
þ
infI
ð
x
;
t
Þ
Bu
i l
i l
i
CC
C
B u wðxi Þ B
@
Bt
2
2
2 A C
A
A
A
C
B i¼1
þ supI xi ; tj
þ infF xi ; tj
þ supF xi ; tj
C
B
C
B vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi
1
0
C
B u
2
2
2
C
B u n
B
B
B
B uX
ð
x
;
t
Þ
þ
supT
ð
x
;
t
Þ
þ
infI
ð
x
;
t
Þ
infT
i l
i l
i l
C C
B u wðx Þ B
@
2
2
2 A C
i
@ t
A
B
i¼1
þ supIB ðxi ; tl Þ þ inf ðxi ; tl Þ þ supF B ðxi ; tl Þ
n
P
0
where
w ¼ ðw1 ; w2 ; :::; wn ÞT and v ¼ ðv1 ; v2 ; :::; vm ÞTn are
P
weighting
vectors of xi ði ¼ 1; :::; nÞ and tl ðl ¼ 1; :::; kÞ with
wi ¼
k
P
i¼1
1 and
vl ¼ 1.
l¼1
wi ¼ 1 =n ; i ¼ 1; :::; n and vl ¼ 1 =k ; l ¼ 1; :::; m, Eq. (6) turns
When
to (5).
The weighted correlation coefficient between A and B also
satisfies the properties as in Theorem 2.1.
3. A topsis method for divns
Assume A ¼ fA1 ; A2 ; :::; Av g and C ¼ fC 1 ; C 2 ; :::; C n g and D ¼
fD1 ; D2 ; :::; Dh g are sets of alternatives, attributes, and decision
makers. For a decision maker Dq ; q ¼ 1; :::; h; the evaluation
characteristic of an alternatives Am ; m ¼ 1; :::; v; on an attribute
C p ; p ¼ 1; :::; n; in time sequence t ¼ ft1 ; t2 ; :::; tk g is represented by
q
; l ¼ 1; 2; :::; k: where
the decision matrix Dq ðtl Þ ¼ dmp ðtÞ
vn
D
q
q
q
E
q
q
dmp ðtÞ¼ xdmp ðtÞ; T dmp ; t ; I dmp ; t ; F dmp ; t
; t ¼ ft1 ; t2 ; :::; tk g
1
wp ¼
hk
ð6Þ
3
1 1 0
1 1
h
h
6 X
hk X
h k7
7
6
A
Imp ðxÞ ¼ 6@ ILpmq xtl A
; @ IU
7
pmq xtl
5
4 q¼1
q¼1
2
0
2
3
0
1 1 0
1 1
h
h
6 X
hk X
h k7
6
7
A
; @ FU
F mp ðxÞ ¼ 6@ F Lpmq xtl A
7
pmq xtl
4 q¼1
5
q¼1
3.2. Importance weight aggregation
nh
i h L U i
; Ipq xtl ; Ipq xtl ; ½ F Lpq xtl ;
Let xpq ðtl Þ ¼ T Lpq xtl ; T U
pq xtl
FU
pq xtl g be weight of Dq to criterion C p in time sequence t l ;
where: p ¼ 1; :::; n; q ¼ 1; :::; h; l ¼ 1; :::; k: The average weight
h
h
nh
o
L
U
L
U
wp ¼ T Lp ðxÞ; T U
p ðxÞ; I p ðxÞ; I p ðxÞ; F p ðxÞ; F p ðxÞ can be evaluated
as:
* nh L U i h L U i h L U io
+
þ :::þ
T p1 xt1 ; T p1 xt1 ; Ip1 xt1 ; Ip1 xt1 ; F p1 xt1 ; F p1 xt1
nh
i h
i h
io
;
; ILph xth ; IU
; F Lph xth ; F U
T Lph xth ; T U
ph xth
ph xth
ph xth
ð8Þ
taken by DIVNSs evaluated by decision maker Dq.
3.1. Aggregate ratings
h
i h L U i
; Impq xtl ; Impq xtl ;
xmpq ðtl Þ ¼ f
T Lmpq xtl ; T U
Let
mpq xtl
i
h
F Lmpq xtl ; F U
g be the suitability rating of alternative Am
mpq xtl
for criterion C p by decision-maker Dq in time sequence tl ; where:
m ¼ 1; :::; v; p ¼ 1; :::; n; q ¼ 1; :::; h; l ¼ 1; :::; k. The averaged suitnh
h
L
U
ability
rating
xmp ¼ T Lmp ðxÞ; T U
mp ðxÞ; I mp ðxÞ; I mp ðxÞ;
h
F Lmp ðxÞ; F U
mp ðxÞg can be evaluated as:
xmp
1
¼
hk
+
* nh L U i h L U i h L U io
þ :::þ
T mpq xt1 ; T mpq xt1 ; Impq xt1 ; Impq xt1 ; F mpq xt1 ; F mpq xt1
h
h
nh
i
i
i
o
;
; ILmpq xtk ; IU
; F Lmpq xtk ; F U
T Lmpq xtk ; T U
mpq xtk
mpq xtk
mpq xtk
where,
2
6
6*
6
6
T mp ðxÞ ¼ 6 1
6
6
4
8
>
>
>
<
1
>
>
>
:
0
@1
1
9
1
>
1 >k + *
>
h
h=
X
T Lpmq xtl A
; 1
>
>
q¼1
>
;
8
>
>
>
<
1
>
>
>
:
0
@1
1 3
9
k 7
11 >
>
> +7
h
7
h=
X
7
A
TU
7
pmq xtl
>
7
>
q¼1
>
; 7
5
ð7Þ
N.T. Thong et al. / Computers in Industry 108 (2019) 45–52
49
where,
2
6
6*
6
6
T p ð xÞ ¼ 6 1
6
6
4
8
>
>
>
<
>
>
>
:
1
1
9
k+ *
11 >
>
=
h
h>
X
T Lpq xtl A
; 1
>
>
q¼1
>
;
0
@1
8
>
>
>
<
>
>
>
:
1
0
@1
1 3
9
1
7
1 >
> k +7
= 7
h
h>
X
7
A
TU
7
pq xtl
>
7
>
q¼1
>
; 7
5
3
2
0
1 1 0
1 1
h
h
7
6 X
h
h
k
k
X
7
6
A
Ip ðxÞ ¼ 6@ ILpq xtl A
; @ IU
7
pq xtl
5
4 q¼1
q¼1
3
1 1 0
1 1
h
h
6 X
hk X
h k7
6
7
A
F p ðxÞ ¼ 6@ F Lpq xtl A
; @ FU
7
pq xtl
4 q¼1
5
q¼1
2
0
3.3. Compute the average weighted ratings
Average weighted ratings of alternatives in tl ; are:
Gm ¼
n
1X
xmp wp ; m ¼ 1; :::; v; p ¼ 1; :::; n;
n p¼1
ð9Þ
þ
3.4. Determination of Aþ ; A ; di and di
Interval neutrosophic positive and negative ideal solutions
namely (PIS, Aþ ) and (NIS, A ) are:
Aþ ¼ fx; ð½1; 1; ½0; 0; ½0; 0Þg
ð10Þ
A ¼ fx; ð½0; 0; ½1; 1; ½1; 1Þg
ð11Þ
The distances of each alternative Am ; m ¼ 1; . . . ; t from Aþ and
A in time sequence tl ; are calculated as:
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
þ
dm ¼
Gm Aþ
ð12Þ
dm
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
¼
Gm A
ð13Þ
þ
where dm and dm represents the shortest and farthest distances of
Am :
3.4. Obtain best coefficient
The best coefficient in time sequence tl ; is shown below where
high value indicates closer to interval neutrosophic PIS and farther
from interval neutrosophic NIS:
CC m ¼
dm
þ
dm þ dm
ð14Þ
and three decision makers, i.e. D1 ; . . . ; D3 ; are chosen. Ratings of
five lecturers are done by criteria as total of publications ðC 1 Þ;
teaching student evaluations ðC 2 Þ; personality characteristics ðC 3 Þ;
professional society ðC 4 Þ; teaching experience ðC 5 Þ; fluency of
foreign language ðC 6 Þ.
4.1. Aggregate ratings
Suitability ratings S= {Ve_Po,Po, Me, Go, Ve_Go} in t = {t1, t2, t3} is,
Ve_Po = Very_Poor = ([0.1, 0.2], [0.6, 0.7], [0.7, 0.8]),
Po = Poor = ([0.2, 0.3], [0.5, 0.6], [0.6, 0.7]),
Me = Medium = ([0.3, 0.5], [0.4, 0.6], [0.4, 0.5]),
Go = Good = ([0.5, 0.6], [0.4, 0.5], [0.3, 0.4]),
Ve_Go = Very_Good = ([0.6, 0.7], [0.2, 0.3], [0.2, 0.3]),
Table 1 presents suitability ratings where the aggregated
ratings of lecturers versus criteria are shown at the last column of
Table 1.
4.2. Importance weight aggregation
The importance V = {U_ IPA, O_ IPA, IPA, V_ IPA, A_IPA} in t = {t1,
t2, t3} is:
U_ IPA = ([0.1, 0.2], [0.4, 0.5], [0.6, 0.7]) = Unimportant,
O_IPA = ([0.2, 0.4], [0.5, 0.6], [0.4, 0.5]) = Ordinary_Important,
IPA = ([0.4, 0.6], [0.4, 0.5], [0.3, 0.4]) = Important,
V_IPA = ([0.6, 0.8], [0.3, 0.4], [0.2, 0.3]) = Very_Important,
A_IPA = ([0.7, 0.9], [0.2, 0.3], [0.1, 0.2]) = Absolutely_Important
(Tables 2–4),
4.3. Weighted ratings
þ
Aþ ; A ; di and di
4.4. Determine the lecturer
Table 5 shows the ranking order is A2 A3 A4 A1 A5 :
Thus, the best lecturer is A2 :
4. Applications
5. Comparison
This section applies the proposed method to evaluate lecturers’
performance in the case study of ULIS-VNU having 11 Faculties, 11
Departments, 09 Functional departments, 05 Centers and 01
Foreign Language Specializing High School with over 700 lecturers
and 8000 high school, undergraduate and graduate students.
Assume that ULIS-VNU needs to evaluate the lecturers’ performance. After preliminary screening, five lecturers, i.e. A1 ; . . . ; A5 ;
This section compares the proposed TOPSIS method for DIVSN
with the similarity measures between INSs proposed by Ye [62] to
illustrate the advantages and applicability of the proposed method.
Using Ye’s [62] method and the data in Table 3, the score function,
the accuracy function and the certainty function of the lecturers
are shown in Table 6.
50
N.T. Thong et al. / Computers in Industry 108 (2019) 45–52
Table 1
Aggregated ratings.
Criteria
Lecturers
Decision makers
Aggregated ratings
t1
C1
A1
A2
A3
A4
A5
A1
A2
A3
A4
A5
A1
A2
A3
A4
A5
A1
A2
A3
A4
A5
A1
A2
A3
A4
A5
A1
A2
A3
A4
A5
C2
C3
C4
C5
C6
t2
t3
D1
D2
D3
D1
D2
D3
D1
D2
D3
Me
Go
Me
Go
Me
Go
Ve_Go
Ve_Go
Go
Ve_Go
Ve_Go
Go
Go
Go
Ve_Go
Me
Go
Go
Me
Me
Me
Go
Go
Ve_Go
Go
Ve_Go
Go
Ve_Go
Go
Go
Go
Go
Go
Me
Go
Go
Go
Go
Go
Go
Ve_Go
Ve_Go
Ve_Go
Go
Go
Go
Me
Go
Po
Me
Go
Ve_Go
Go
Go
Go
Go
Go
Go
Ve_Go
Go
Go
Ve_Go
Go
Go
Me
Go
Ve_Go
Go
Go
Go
Go
Go
Ve_Go
Go
Go
Me
Go
Go
Me
Po
Me
Go
Me
Go
Go
Go
Go
Ve_Go
Go
Go
Go
Ve_Go
Go
Go
Go
Ve_Go
Me
Go
Go
Go
Go
Ve_Go
Go
Ve_Go
Go
Go
Go
Go
Go
Me
Me
Ve_Go
Go
Ve_Go
Go
Ve_Go
Go
Ve_Go
Go
Ve_Go
Go
Go
Go
Go
Go
Go
Go
Me
Ve_Go
Ve_Go
Ve_Go
Go
Go
Go
Ve_Go
Go
Me
Go
Me
Me
Go
Go
Go
Go
Go
Go
Ve_Go
Go
Ve_Go
Go
Go
Ve_Go
Go
Go
Me
Go
Go
Go
Go
Go
Go
Ve_Go
Go
Go
Go
Me
Go
Me
Me
Me
Go
Go
Go
Go
Go
Ve_Go
Ge
Ve_Go
Go
Go
Go
Ve_Go
Go
Go
Go
Go
Ve_Go
Go
Go
Go
Go
Go
Go
Ve_Go
Go
Me
Go
Go
Go
Me
Go
Go
Go
Ve_Go
Go
Ve_Go
Go
Ve_Go
Go
Go
Ve_Go
Go
Go
Go
Go
Go
Go
Me
Go
Go
Me
Go
Ve_Go
Go
Go
Go
Me
Go
Go
Go
Me
V_G
Ve_Go
Go
Ve_Go
Go
Go
Go
Go
Ve_Go
Go
Ve_Go
Ve_Go
Go
Go
Go
Go
Go
Ve_Go
Me
Go
Ve_Go
Go
Go
Go
Me
Go
Ve_Go
Me
Me
Go
Go
Go
Go
Go
Ve_Go
Ve_Go
Ve_Go
Go
Go
([0.494, 0.603], [0.370, 0.5], [0.296, 0.4])
([0.558, 0.659], [0.272, 0.4], [0.239, 0.3])
([0.494, 0.603], [0.370, 0.5], [0.296, 0.4])
([0.481, 0.590], [0.400, 0.5], [0.310, 0.4])
([0.441, 0.569], [0.400, 0.5], [0.330, 0.4])
([0.512, 0.613], [0.370, 0.5], [0.287, 0.4])
([0.518, 0.627], [0.317, 0.4], [0.271, 0.4])
([0.474, 0.593], [0.370, 0.5], [0.306, 0.4])
([0.524, 0.625], [0.343, 0.4], [0.274, 0.4])
([0.506, 0.615], [0.343, 0.5], [0.283, 0.4])
([0.518, 0.627], [0.317, 0.4], [0.271, 0.4])
([0.547, 0.648], [0.294, 0.4], [0.251, 0.4])
([0.536, 0.637], [0.317, 0.4], [0.262, 0.4])
([0.524, 0.625], [0.343, 0.4], [0.274, 0.4])
([0.524, 0.625], [0.343, 0.4], [0.274, 0.4])
([0.397, 0.547], [0.400, 0.6], [0.352, 0.5])
([0.441, 0.569], [0.400, 0.5], [0.330, 0.4])
([0.494, 0.603], [0.370, 0.5], [0.296, 0.4])
([0.365, 0.518], [0.410, 0.6], [0.380, 0.5])
([0.316, 0.494], [0.410, 0.6], [0.405, 0.5])
([0.419, 0.558], [0.400, 0.5], [0.341, 0.4])
([0.536, 0.637], [0.317, 0.4], [0.262, 0.4])
([0.494, 0.603], [0.370, 0.5], [0.296, 0.4])
([0.536, 0.637], [0.317, 0.4], [0.262, 0.4])
([0.512, 0.613], [0.370, 0.5], [0.287, 0.4])
([0.558, 0.659], [0.272, 0.4], [0.239, 0.3])
([0.524, 0.625], [0.343, 0.4], [0.274, 0.4])
([0.569, 0.670], [0.252, 0.4], [0.229, 0.3])
([0.524, 0.625], [0.343, 0.4], [0.274, 0.4])
([0.524, 0.625], [0.343, 0.4], [0.274, 0.4])
Table 2
Aggregated weights.
Decision-makers
Criteria
Aggregated weights
t1
C1
C2
C3
C4
C5
C6
t2
t3
D1
D2
D3
D1
D2
D3
D1
D2
D3
IPA
V_IPA
IPA
IPA
IPA
V_IPA
IPA
V_IPA
IPA
V_IPA
IPA
V_IPA
IPA
IPA
V_IPA
IPA
IPA
IPA
IPA
V_IPA
IPA
IPA
V_IPA
IPA
V_IPA
V_IPA
IPA
O_IPA
IPA
IPA
IPA
V_IPA
V_IPA
IPA
V_IPA
IPA
V_IPA
A_IPA
V_IPA
IPA
IPA
V_IPA
IPA
V_IPA
IPA
IPA
IPA
V_IPA
V_IPA
V_IPA
V_IPA
IPA
IPA
IPA
Table 5
Closeness coefficient.
Table 3
Weighted ratings.
Lecturers
Aggregated weights
([0.170, 0.397], [0.648, 0.8], [0.545, 0.6])
([0.190, 0.436], [0.617, 0.7], [0.519, 0.6])
([0.187, 0.419], [0.642, 0.8], [0.535, 0.6])
([0.178, 0.400], [0.643, 0.8], [0.538, 0.6])
([0.173, 0.395], [0.649, 0.8], [0.549, 0.6])
A1
A2
A3
A4
A5
([0.476, 0.683], [0.363, 0.5], [0.262, 0.4])
([0.595, 0.800], [0.296, 0.4], [0.194, 0.3])
([0.499, 0.706], [0.352, 0.5], [0.251, 0.4])
([0.408, 0.613], [0.397, 0.5], [0.296, 0.4])
([0.452, 0.657], [0.375, 0.5], [0.274, 0.4])
([0.499, 0.706], [0.352, 0.5], [0.251, 0.4])
Table 4
The distance of each lecturer from Aþ and A .
Lecturers
Closeness coefficient
Ranking
A1
A2
A3
A4
A5
0.339
0.367
0.351
0.345
0.338
4
1
2
3
5
Table 6
Modified score, accuracy and certainty function of each lecturer.
Lecturers
d
d
Lecturers
Score function Accuracy function Certainty function Ranking
A1
A2
A3
A4
A5
0.346
0.375
0.359
0.352
0.345
0.675
0.647
0.662
0.668
0.676
A1
A2
A3
A4
A5
0,332
0,361
0,345
0,339
0,331
þ
0,297
0,241
0,267
0,284
0,300
0,283
0,313
0,303
0,289
0,284
4
1
2
3
5
N.T. Thong et al. / Computers in Industry 108 (2019) 45–52
Table 6 shows that the ranking order of the five lecturers is
A2 A3 A4 A1 A5 : Thus, the best lecturer is A2 : The result is
the same as that of the proposed method. This means that our
method in the simplest form can procedure the results of the best
method for this problem. Moreover, it is more generalized and
flexible than the Ye’s [62] method in dynamic environments.
6. Conclusion
This paper proposed a new concept of Dynamic Interval Valued
Neutrosophic Set (DIVNS) where all the factors in DIVNSs such as
truth, indeterminacy and falsity degrees are in different ranges of
time. Mathematical operations associated with DIVNSs and
correlation coefficients have also been defined. In addition, we
have proposed a new TOPSIS method based on the DIVNSs and
their application to evaluate lecturers' performance in the ULISVNU. This shows the feasibility and applications of Neutrosophic
Theory in Industry.
In the future, we will use DIVNSs as well as the TOPSIS method
to express dynamic information, and develop additional extention
theories for DIVNSs such as operators, similarity measure. In
addition, we extended this method to predictive problems such as
in [24, 30, 31, 38, 39, 43, 44,46, 47, 48, 50, 63–92].
Acknowledgement
This research is funded by Vietnam National Foundation for
Science and Technology Development (NAFOSTED) under grant
number 502.01-2015.16.
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Nguyen Dinh Hoa