Neutrosophic Assessment and Decision Making
MICHAEL GR. VOSKOGLOU
School of Engineering - Department of Mathematical Sciences
University of Peloponnese (Ex Graduate TEI of Western Greece)
26334 Patras - GREECE
Abstract: - Smarandache’s theory of neutrosophy is a generalization of Zadeh’s fuzziness characterizing each
element of the universal set with the membership degree, as in fuzzy sets, and in addition with the degrees of
non-membership and indeterminacy with respect to the corresponding neutrosophic set. In the present work we
use neutrosophic sets as tools for assessment and decision making. This turns out to be very useful when one is
not sure about the correctness of the grades assigned to the elements of the universal set. Examples are also
presented illustrating our results.
Key-Words: - Fuzzy Set (FS), Neutrosophic Set (NS), Neutrosophic Triplet (NT), Soft Set (SS), Fuzzy
Assessment, Decision Making (DM).
Received: December 12, 2022. Revised: August 9, 2023. Accepted: September 5, 2023. Published: October 2, 2023.
1 Introduction
The theory of Neutrosophy introduced by
Smarandache [1] in 1995, is an extension of Zadeh’s
Fuzziness [2]. The term neutrosophy was formed by
the synthesis of the words neutral and the Greek
“sofia”, which means wisdom or complete
knowledge. As we will see in detail in the next
section, in a neutrosophic set (NS) all the elements of
the universal set of the discourse are characterized by
three parameters, which take values in the unit
interval [0, 1].
NSs have found important applications to practical
everyday life problems, in which the use of Zadeh’s
FSs has not been proved to be sufficient for obtaining
the required results. The purpose of this work is to
present applications of NSs to assessment [3] and
decision making (DM) [4] processes.
The rest of the paper is organized as follows:
Section 2 contains the mathematical background
about NSs and soft sets (SSs) [5], needed for the
understanding of the paper. The application of NSs
to assessment processes is developed in section 3,
whereas section 4 describes their application to DM.
The article closes with the final conclusions and
some hints for further research, contained in its last
section 5.
2 Mathematical Background
2.1 Fuzzy Sets
Zadeh, in order to tackle mathematically the
existing in real life partial truths, defined in 1965 the
concept of fuzzy set (FS) as follows [2]:
Definition 1: Let U be the universal set of the
discourse, then a FS Α in U is defined with the help
of its membership function m: U
[0,1] as the set of
the ordered pairs
A = {(x, m(x)): x
U} (1)
The real number m(x) is called the membership
degree of x in Α. The greater m(x), the more x
satisfies the characteristic property of Α.
There is not any exact rule for defining the
membership function of a FS. The methods used for
this are usually empirical or statistical and its
definition is not unique, depending on each
observer’s subjective criteria about the
corresponding situation. Defining, for example, the
FS of the tall men, one may consider all men with
heights greater than 1.90 m as tall and another one all
those with heights greater than 2 m. As a result, the
first observer will attach membership degree 1 to all
men between 1.90 m and 2 m, whereas the second
one will attach membership degrees <1. Analogous
differences will obviously appear to the membership
degrees of the men with heights <1.90 m.
Consequently, the only restriction in the definition
of the membership function is that it must be
compatible with common sense; otherwise the
resulting FS does not give a creditable description of
the corresponding real situation. This could happen,
for instance, if in the previous example men with
heights <1.60 m possessed membership degrees
≥0.5.
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In a later stage, when membership degrees were
reinterpreted as possibility distributions, FSs were
extensively used for managing the existing in real life
uncertainty, which is connected with incomplete or
vague information. Possibility is an alternative
mathematical theory for tackling the uncertainty [6].
Zadeh articulated the relationship between
possibility and probability by noticing that whatever
is probable must be primarily possible [7].
All notions and operations defined on crisp sets are
extended in a natural way to FSs For general facts on
FSs and the connected to them uncertainty we refer
to the book [8].
2.2 Neutrosophic Sets
Following the introduction of FSs, a series of
extensions and related theories have been developed
for tackling more effectively all the forms of the
existing uncertainties [9].
Atanassov in 1986, added to Zadeh’s membership
degree the degree of non-membership and defined
the concept of intuitionistic FS (IFS), as an extension
of FS [10].
Smarandache in 1995, motivated by the various
neutral situations appearing in real life situations -
like <positive, zero, negative>, <small, medium,
high>, <win, draw, defeat>, etc. – introduced, in
addition of the degrees of membership and non-
membership, the degree of indeterminacy and
extended the concept of IFS to the concept of NS [1].
In this work we will make use only of the simplest
form of a NS, termed as single valued NS (SVNS)
and defined as follows [11]:
Definition 2: A SVNS A in the universal set U is
defined as the set of the ordered tumbles
A = {(x,T(x),I(x),F(x)): x
U, T(x),I(x),F(x)
[0,1],
0
T(x)+I(x)+F(x)
3} (2) .
In (2) T(x), I(x), F(x) are the degrees of truth (or
membership), indeterminacy (or neutrality) and
falsity (or non-membership) of x in A respectively,
called the neutrosophic components of x. For
simplicity, we write A<T, I, F> and the elements of
A in the form (t, i, f) of neutrosophic triplets (NTs),
with t, i, f in [0, 1].
Example 1: Let U be the set of the players of a
football team and let A be the SVNS of the good
players of U. Then each player x of U is characterized
by a NT (t, i, f), with t, i, f in [0, 1]. For instance,
x(0.7, 0.1, 0.4) ∈ A means that there is a 70% belief
that x is a good player, a 10% doubt about it and at
the same time a 40% belief that x is not a good
player. In particular, x(0,1,0) ∈ A means that we do
not know absolutely nothing about x’s affiliation
with A.
When T(x)+I(x)+F(x)<1, it leaves room for
incomplete information, when T(x)+I(x)+F(x)=1 for
complete information, and when T(x)+I(x)+F(x)>1
for inconsistent information about x in A. A SVNS
may contain simultaneously elements corresponding
to all kinds of the previous information. All notions
and operations defined on FSs are extended in a
natural way to NSs [11]
Since the NTs of a SVNS A are ordered triplets,
one may define addition among them and scalar
multiplication of a positive number with a NT in the
usual way, as follows:
Definition 3: Let (t1, i1, f1), (t2, i2, f2) be in A and
let r be a positive number. Then:
The sum (t1, i1, f1) + (t2, i2, f2) = (t1+ t2, i1+
i2, f1+ f2) (3)
The scalar product r(t1, i1, f1) = (rt1, r i1, f1)
(4)
The sum and the scalar product of the NTs of a
SVNS A with respect to Definition 3 need not, be a
NT of A, since it may happen that (t1+ t2)+(i1+ i2)+(f1+
f2)>3 or rt1+ri1+ rf1>3. With the help of Definition 3,
however, one can define the mean value of a finite
number of NTs of A, which is always in A. as
follows:
Definition 4: Let A be a SVNS and let (t1, i1, f1),
(t2, i2, f2), …., (tk, ik, fk) be a finite number of elements
of A. Assume that (ti, ii, fi) appears ni times in an
application, i = 1,2,…., k. Set n = n1+n2+….+nk. Then
the mean value of all these elements of A is defined
to be the NT of A
(tm,im,fm) =
1
n
[n1(t1, i1,f1)+n2(t2,i2,f2)+….+nk(tk,ik, fk)]
(5)
2.3 Soft Sets
The disadvantage of FSs concerning the definition of
their membership function remains obviously the
same for all their extensions involving membership
degrees, like the IFSs, the NSs, etc. To overcome this
difficulty, the concept of interval-valued FS (IVFS)
was introduced in 1975. An IVFS is defined by
mapping the universe U to the set of the closed
subintervals of [0, 1] [12]. Other theories related to
FSs were also developed, in which the definition of a
membership function is either not necessary (grey
systems and numbers [13]), or it is overpassed by
using a pair of crisp sets giving the lower and upper
bound of the original set (rough sets) [14].
Molodstov introduced in 1999 the concept of SS
for tackling the uncertainty in a parametric manner,
which does not need the definition of a membership
function. Namely, a SS is defined as follows [5]:
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Definition 5: Let E be a set of parameters and let
f be a map from E into the power set P(U) of the
universe U. Then the SS (f, E) in U is defined as the
set of the ordered pairs
(f, E) = {(e, f(e)): e ∈ A} (6)
In other words, a SS is a parametrized family of
subsets of U. The name "soft" is due to the fact that
the form of (f, E) depends on the parameters of E.
Example 2: Let U= {C1, C2, C3} be a set of cars and
let E = {e1, e2, e3} be the set of parameters e1=cheap,
e2=elegant and e3= expensive. Let us further assume
that C1, C2 are cheap, C3 is expensive and C2, C3 are
elegant cars. Then, a map f: E
P(U) is defined by
f(e1)={C1, C2}, f(e2)={C2, C3} and f(e3)={C3}.
Therefore, the SS (f, E) in U is the set of the ordered
pairs
(f, E) = {(e1, {C1, C2}), (e2, {C2, C3}, (e3, {C3}} (7)
Maji et al. [15] introduced a tabular representation
of a SS in the form of a binary matrix in order to be
stored easily in a computer’s memory. For instance,
the tabular representation of the soft set (f, E) of
Example 2 is given by Table 1.
Table 1. Tabular representation of the SS of
Example 2
e1
e2
e3
C1
1
0
0
C2
1
1
0
C3
0
1
1
A FS in U with membership function y = m(x) is
a SS in U of the form (f, [0, 1]), where f(α)={x
U:
m(x)
α} is the corresponding a-cut of the FS, for
each α in [0, 1]. Consequently the concept of SS is a
generalization of the concept of FS. All notions and
operations defined on FSs are extended in a natural
way to SSs [16].
3 Neutrosophic Assessment
The performance of the members of a group is
assessed frequently by using qualitative grades
(linguistic expressions) instead of numerical scores.
This happens either because the existing data about
their performance are not very clear, or for reasons
of elasticity (e.g. from teacher to students).
Obviously, in such cases the mean performance of
the group cannot be assessed by calculating the mean
value of the individual scores of its members. For
tackling this situation, we have used in earlier works
either triangular fuzzy numbers or grey numbers
(closed real intervals) and we have shown that these
two methods are equivalent [17] (sections 5.2 and
6.2).
Cases appear, however, in practice, in which one
is not sure about the accuracy of the qualitative
grades assigned to the objects under assessment (e.g.
students). In such cases, the use of NSs is possibly
the best way for evaluating a group’s overall
performance. The following example illustrates this
situation.
Example 3: The new teacher of a student class is
not sure yet about the quality of each of the students.
He characterized, therefore, the set of the very good
students by NTs as follows: s1(1, 0, 0), s2(0.9, 0.1,
0.1), s3(0.8, 0.2, 0.1), s4(0.4, 0.5, 0.8), s5(0.4, 0.5,
0.8), s6(0.3, 0.7, 0.8), s7(0.3, 0.7, 0.8), s8(0.2, 0.8,
0.9), s9(0.1, 0.9, 0.9), s10(0.1, 0.9, 0.9} and the
remaining 10 students of the class by (0, 0, 1). This
means that the teacher is absolutely sure that s1 is a
very good student, 90% sure that s2 is a very good
student too, but at the same time he has a 10% doubt
about it and a 10% belief that s2 is not a very good
student, etc. For the last 10 students the teacher is
absolutely sure that they are not very good students.
Evaluate the mean level of the student skills.
Solution: The mean level of the student skills can
be estimated by the mean value of the corresponding
NTs, i.e. by
1
20
[ (1, 0, 0)+(0.9, 0.1, 0.1)+(0.8, 0.2,
0.1)+2(0.4, 0.5, 0.8)+2(0.3, 0.7, 0.8)+(0.2, 0.8,
0.9)+2(0.1, 0.9, 0.9)+10(0, 0, 1)], which by equations
(3) and (4) is equal to
1
20
(4.5, 5.3, 16.3) = (0.225,
0.265, 0.815). This means that a random student of
the class has a 22.5 % probability to be a very good
student, however, there exists also a 26.5% doubt
about it and an 81.5% probability to be not a very
good student.
4 Neutrosophic Decision Making
Maji et al. [15] used the tabular form of a SS as a tool
for DM in a parametric manner. The following
example highlights their method:
Example 4: A person wants to buy one of the six
cars C1, C2, C3, C4, C5 and C6. His ideal preference is
a high-speed, automatic (gear-box), hybrid (petrol
and electric power) and cheap car. Assume that C1,
C2, C6 are the high-speed, C2, C3, C5, C6 are the
automatic, C3, C5 are the hybrid and C4 is the unique
cheap car. Which is the best choice for the candidate
buyer?
Solution: Set V = {C1, C2, C3, C4, C5, C6} and let
E = {e1, e2, e3, e4} be the set of the parameters
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e1=high-speed, e2=automatic, e3=hybrid and
e4=cheap. Consider the map g: E → Δ(V) defined by
g(e1) = { C1, C2, C6}, g(e2) = { C2, C3, C5, C6}, g(e3)
= { C3, C5}, g(e4) = { C4} and the corresponding SS
(g, E) = {(e1, {C1, C2, C6}), (e2, {C2, C3, C5, C6}),
(e3, {C3, C5}), (e4, {C4})} (7)
Table 2: Tabular representation of the SS of Example 4
e1
e2
e3
e4
C1
1
0
0
0
C2
1
1
0
0
C3
0
1
1
0
C4
0
0
0
1
C5
0
1
1
0
C6
1
1
0
0
Forming the tabular representation of the SS (g, P)
(Table 2) the choice value of each car is calculated by
adding the binary elements of the corresponding row
of it. The cars C1 and C4 have, therefore, choice value
1 and all the others have choice value 2.
Consequently, the buyer must choose one of the cars
C2, C3, C5 or C6.
The previous decision, obtained by applying the
method of Maji et al. [15], is not very helpful for the
candidate buyer, since it excludes only two among
the six available cars. This is due to the fact that in
the tabular form of the corresponding SS the
characterizations of the elements of the universal set
(cars in our example) by the corresponding
parameters are replaced with the binary elements
(truth values) 0, 1. In other words, although the
method starts from a fuzzy basis (SS), it uses bivalent
logic for making the required decision, e.g. cheap or
not cheap! Consequently, this methodology could
lead to a wrong decision, if some (or all) of the
parameters have not a bivalent texture; e.g. the
parameter “hybrid” has a bivalent texture, but not the
parameter “cheap”.
In order to tackle this problem, we have used in
earlier works grey numbers, instead of the binary
elements 0, 1, in the tabular representation of the
corresponding SS [18]. DM situations, however,
appear frequently in reality, in which the decision
maker has doubts about the correctness of the
qualitative (fuzzy) parameters assigned to some or all
of the elements of the set of the discourse. In such
cases, the best way to perform the DM process is to
use NSs. This is illustrated by the following example.
Example 5: Reconsider Example 4 and assume
that the candidate buyer, being not sure about the
correctness of the qualitative parameters p1 and p4
assigned to each of the six cars, decided to proceed
by replacing the parameters by NTs. As a result, the
tabular matrix of the DM process takes the form
shown in Table 3. Which is the best decision for the
candidate buyer in this case?
Table 3: Tabular representation of the SS of Example 5
e1
e2
e3
e4
C1
(1, 0,0)
(0,0,1)
(0,0,1)
(0.6,0.3,0.1)
C2
(1,0,0)
(1,0,0)
(0,0,1)
(0.2,0.2,0.6)
C3
(0.5,0.4, 0.1)
(1,0,0)
(1,0,0)
(0.6, 02,0.2)
C4
(0.5, 0.2, 0.3)
(0,0,1)
(0,0,1)
(1, 0, 0)
C5
(0.5, 0.1, 0.4)
(1,0,0)
(1,0,0)
(0.6,0.3,0.1)
C6
(1, 0, 0)
(1,0,0)
(0,0,1)
(0.4,0.4,0.2)
Solution: The choice value of each car in this case
is defined to be the mean value of the NTs of the line
of Table 3 in which he belongs. Thus, by equation
(5), the choice value of C1 is equal to
1
4
[(1, 0, 0)+2(0,
0, 1)+(0.6, 0.3, 0.1)] =
1
4
(1.6, 0.3, 2.1) = (0.4, 0.075,
0.525). In the same way one finds that the choice
values of C2, C3, C4, C5 and C6 are (0.55, 0.005, 0.4),
(0.775, 0.15, 0.075), (0.375, 0.05, 0.575), (0.775, 0.1,
0.125) and (0.6, 0.1, 0.3) respectively.
Now the candidate buyer may use either an
optimistic criterion by choosing the car with the
greatest truth degree, or a conservative criterion by
choosing the car with the lower falsity degree.
Consequently, using the optimistic criterion he must
choose one of the cars C3 and C5, whereas using the
conservative criterion he must choose the car C3. A
combination, therefore, of the two criteria leads to the
final choice of the car C3. Observe, however, that,
since the indeterminacy degree of C3 is 0.15 and of C5
is 0.1, there is a slightly greater doubt about the
suitability of C3 with respect to C5. In other words,
the choice of C3 is connected with a slightly greater
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risk. In final analysis, therefore, all the neutrosophic
components assigned to each car give useful
information about its suitability.
4. Discussion and Conclusions
In this work NSs were used as tools in assessment and
DM processes. This is a very useful approach when
one is not sure about the correctness of the parameters
/ qualitative grades assigned to the elements of the
universal set.
Our DM method in particular, was obtained by
adapting a parametric DM method of Maji et al. [15]
using SSs as tools. In our method the binary elements
0, 1 of the tabular representation of the corresponding
SS are replaced by NTs.
The combination of two or more of the extensions
of FSs that have been developed for tackling
efficiently the various forms of the existing in real
world uncertainty (SSs and NSs in this paper),
appears in general to be an effective way for
obtaining better results, not only for DM, but also for
assessment and for various other human activities.
Consequently this is a promising area for further
research.
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