A Hybrid Method for Assessing Student Mathematical Modelling Skills
under Fuzzy Conditions
MICHAEL GR. VOSKOGLOU
Mathematical Sciences, School of Technological Applications
University of Peloponnese (ex T.E.I. of Western Greece)
Meg. Alexandrou 1, 26334 Patras
GREECE
Abstract: - Mathematical modelling (MM) appears today as a valuable tool in mathematics education, which
connects mathematics with everyday life situations on the purpose of increasing the student interest on the
subject. In this paper a hybrid method is applied for the assessment of student groups’ MM skills with
qualitative grades (i.e. under fuzzy conditions). Namely, soft sets are used as tools for a parametric assessment
of a group’s performance, the calculation of the GPA index and the Rectangular Fuzzy Assessment Model are
applied for evaluating the group’s qualitative performance, grey numbers are used as tools for assessing the
group’s mean performance and neutrosophic sets are utilized when the teacher is not sure about the individual
grades assigned to some (or all) students of the group.
Key-Words: - Mathematical Modelling (MM), Fuzzy Logic (FL), Fuzzy Assessment Methods, GPA Index,
Rectangular Fuzzy Assessment Model (RFAM), Grey Number (GN), Neutrosophic Set (NS), Soft Set (SS).
Received: May 5, 2022. Revised: October 19, 2022. Accepted: November 27, 2022. Published: December 27, 2022.
1 Introduction
A model is understood to be a simplified
representation of a real system including only its
characteristics which are related to a certain
problem concerning the system (assumed real
system); e.g. maximizing the system’s productivity,
minimizing its functional costs, etc. The process of
modelling is a fundamental principle of the systems’
theory, since the experimentation on the real system
is usually difficult (or impossible sometimes)
requiring a lot of money and time. Modelling a
system involves a deep abstracting process, which is
graphically represented in Fig. 1 [1].
Fig. 1: Graphical representation of the modelling process
There are several types of models used according
to the form of the system and of the corresponding
problem to be solved [1]. In simple cases iconic
models may be used, like maps, bas-relief
representations, etc. Analogical models, such as
graphs, diagrams, etc., are frequently used when the
corresponding problem concerns the study of the
relationship between only two of the system’s
variables; e.g. speed and time, temperature and
pressure, etc. The mathematical or symbolic models
use mathematical symbols and representations
(functions, equations, inequalities, etc.) to describe
the system’s behavior. This is the most important
type of models, because they provide accurate and
general (i.e. holding even if the system’s parameters
are changed) solutions to the corresponding
problems. In case of complex systems, however,
like the biological ones, where the solution cannot
be expressed in solvable mathematical terms or the
mathematical solution requires laborious
calculations, simulation models are often used.
These models mimic the system’s behavior over a
period of time with the help of a well-organized set
of logical orders, usually expressed in the form of a
computer program. Also, heuristic models can be
utilized for improving already existing solutions,
International Journal of Computational and Applied Mathematics & Computer Science
DOI: 10.37394/232028.2022.2.15
Michael Gr. Voskoglou
E-ISSN: 2769-2477
106
Volume 2, 2022
obtained either empirically or by using other types
of models.
Until the middle of the 1970’s Mathematical
Modelling (MM) mainly used to be a tool at hands
of the scientists for solving problems related to their
disciplines. The failure of the introduction of the
“new mathematics” to school education [2],
however, turned the attention of the specialists to
problem-solving activities as a more effective way
for teaching and learning mathematics. MM in
particular, has been widely used for connecting
mathematics to everyday life situations, on the
purpose of increasing the student interest on the
subject.
Quality is a desirable characteristic of all human
activities. This makes assessment one of the most
important components connected to those activities.
Assessment takes place in two ways, either with the
help of numerical or with the help of qualitative
grades, like excellent, good, mediocre, etc.
When numerical grades are used, standard
methods are applied for the overall assessment of
the skills of a group of objects participating in a
certain activity, like the calculation of the mean
value of all the individual scores or the Grade Point
Average (GPA) index, a weighted average in which
greater coefficients are assigned to the higher
scores.
The use of qualitative grades is usually preferred
when more elasticity is desirable (as it frequently
happens in case of student assessment), or when no
exact numerical data are available. In this case,
assessment methods based on principles of fuzzy
logic (FL) are frequently used.
The present author has developed in earlier works
several methods for the assessment of
human/machine performance under fuzzy conditions
including the measurement of uncertainty in fuzzy
systems, the use of the Center of Gravity (COG)
defuzzification technique, the use of fuzzy numbers
(FNs) or of grey numbers (GNs), etc. All these
methods are reviewed in [3]. Recently, Voskoglou
developed also assessment models using soft sets
(SSs) and neutrosophic sets (NSs) as tools [4, 5]
In this work a hybrid method is applied for the
assessment of student groups’ MM skills with
qualitative grades (i.e. under fuzzy conditions).
Namely, SSs are used for the parametric assessment
of a group’s performance, the calculation of the
GPA index and the Rectangular Fuzzy Assessment
Model (RFAM) are applied for evaluating the
group’s qualitative performance, GNs are used as
tools for assessing the group’s mean performance
and NSs are utilized when the teacher is not sure
about the individual grades assigned to some (or all)
students of the group. The paper closes with the
final conclusions and some hints for future research.
2. Mathematical Modelling in
Education
One of the first who proposed the use of MM as a
tool for teaching mathematics was H. O. Pollak
[6], who presented in 1976 during the ICME-3
Conference in Karlsruhe the scheme of Fig. 2,
known as the circle of modelling. In this scheme,
given a problem for solution from a topic
different from mathematics (other world), the
solver, following the direction of the arrows, is
transferred to the “universe” of mathematics.
There, the solver uses or creates suitable
mathematics for the solution of the problem and
then returns to the other world to check the
validity of the mathematical solution obtained. If
the verification of the solution is proved to be
non-compatible to the existing real conditions, the
same circle is repeated one or more times.
Fig. 2: The Pollak’s Circle of Modelling
Following the Pollak’s presentation, much
effort has been placed by mathematics education
researchers to study and analyze in detail the
process of MM on the purpose of using it for
teaching mathematics. Several models have been
developed towards this direction, a brief but
comprehensive account of which can be found in
[7], including the present author’s model (Fig. 3).
Fig. 3: Flow-diagram of Voskoglou’s model for the MM
process
In this model [8] Voskoglou described the MM
process in terms of a Markov chain introduced on its
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E-ISSN: 2769-2477
107
Volume 2, 2022
main steps, which are: S1 = analysis of the problem,
S2 = mathematization (formulation and construction
of the model), S3 = solution of the model, S4 =
validation of the solution and S5 = implementation
of the solution to the real system. When the MM
process is completed at step S5, it is assumed that a
new problem is given to the class, which implies
that the process restarts again from step S1.
Mathematization is the step of the MM process
with the greatest gravity, since it involves a deep
abstracting process, which is not always easy to be
achieved by a non-expert. A solver who has
obtained a mathematical solution of the model is
usually able to “translate” it in terms of the
corresponding real situation and to check its
validity. There are, however, sometimes problems in
which the validation of the model and/or the
implementation of the final mathematical results to
the real system hide surprises, which force solvers
to look back to the construction of the model and
make the necessary changes to it. A characteristic
example is presented in [9].
Models like Voskoglou’s (Fig. 3) are useful for
describing the solvers’ ideal behavior when tackling
MM problems. Relative researches [10-12],
however, report that the reality is not like that. In
fact, modelers follow individual routes related to
their learning styles and the level of their cognition.
Consequently, from the teachers’ part there exists an
uncertainty about the student way of thinking at
each step of the MM process. Those findings
inspired the present author to use principles of FL
for describing in a more realistic way the process of
MM in the classroom on the purpose of
understanding, and therefore treating better, the
student reactions during the MM process [13]. The
steps of the MM process in this model are
represented as fuzzy sets on a set of linguistic labels
characterizing the student performance in each step.
A complete methodology for teaching
mathematics on the basis of MM has been
eventually developed, which is usually referred as
the application-oriented teaching of mathematics
[14]. However, as the present author underlines in
[9], teachers must be careful, because the extensive
use of the application-oriented teaching as a general
method for teaching mathematics could lead to far-
fetched situations, in which more attention is given
to the choice of the applications rather than to the
mathematical content!
More details about MM from the viewpoint of
Education and representative examples can be found
in earlier works of the author [9, 14].
3. Mathematical Background
3.1 Fuzzy Sets and Logic
Zadeh, in order to deal with partial truths,
introduced in 1965 the concept of fuzzy set (FS) as
follows [15]:
Definition 1: Let U be the universe, then a FS F
in U is of the form
F = {(x, m(x)): x
U} (1)
In equation (1) m: U
[0,1] is the membership
function of F and m(x) is called the membership
degree of x in F. The greater m(x), the more x
satisfies the characteristic property of F. A crisp
subset F of U is a FS in U with membership
function such that m(x)=1 if x belongs to F and 0
otherwise.
Based on the concept of FS Zadeh developed the
infinite-valued FL [16], in which truth values are
modelled by numbers in the unit interval [0, 1]. FL
is an extension of the classical bivalent logic (BL) of
Aristotle embodying the Lukasiewicz’s “Principle
of Valence” [17]. In contrast to the Aristotle’s
principle of the “Excluded Middle”, Lukasiewicz’s
principle states that propositions are not only either
true or false, but they can have intermediate truth-
values too.
It was only in a second moment that FS theory
and FL were used to embrace uncertainty modelling
[18, 19]. This happened when membership functions
were reinterpreted as possibility distributions.
Possibility theory is an uncertainty theory devoted
to the handling of incomplete information [20].
Zadeh [18] articulated the relationship between
possibility and probability, noticing that what is
probable must preliminarily be possible. For general
facts on FSs and the connected to them uncertainty
we refer to the book [21].
3.2 Neutrosophic Sets
Following the introduction of FSs, various
generalizations and other related to FSs theories
have been proposed enabling a more effective
management of all types of the existing in real
world uncertainty. A brief description of the main
among those generalizations and theories can be
found in [22].
Atanassov added in 1986 to Zadeh’s membership
degree the degree of non-membership and
introduced the concept of intuitionistic fuzzy set
(IFS) [23] as the set of the ordered triples
A = {(x, m(x), n(x)): x
U, 0
m(x) + n(x)
1} (2)
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Smarandache, motivated by the various neutral
situations appearing in real life - like <friend,
neutral, enemy>, <positive, zero, negative>, <small,
medium, high>, <male, transgender, female>, <win,
draw, defeat>, etc. – introduced in 1995 the degree
of indeterminacy/neutrality of the elements of the
universal set U in a subset of U and defined the
concept of neutrosophic set (NS) [24]. The term
neuttrosophic is the result of the synthesis of the
words “neutral” and “sophia” which means in Greek
language “wisdom”. In this work we need only the
simplest version of the concept of NS, which is
defined as follows:
Definition 2: A single valued NS (SVNS) A in U
is of the form
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} (3)
In (3) T(x), I(x), F(x) are the degrees of truth (or
membership), indeterminacy and falsity (or non-
membership) of x in A respectively, called the
neutrosophic components of x. For simplicity, we
write A<T, I, F>.
For example, let U be the set of the players of a
basketball team and let A be the SVNS of the good
players of U. Then each player x of U is
characterized by a neutrosophic triplet (t, i, f) with
respect to A, 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 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.
In an IFS the indeterminacy coincides by default
to 1- T(x) – F(x). Also, in a FS is I(x)=0 and F(x) =
1 – T(x), whereas in a crisp set is T(x)=1 (or 0) and
F(x)= 0 (or 1). In other words, crisp sets, FSs and
IFSs are special cases of SVNSs.
When the sum T(x) + I(x) + F(x) of the
neutrosophic components of x ∈ U in a SVNS A on
U is <1, then x leaves room for incomplete
information, when is equal to 1 for complete
information and when is greater than 1 for
paraconsistent (i.e. contradiction tolerant)
information. A SVNS may contain simultaneously
elements leaving room to all the previous types of
information. For general facts on SVNSs we refer to
[25].
Summation of neutrosophic triplets is equivalent
to the neutrosophic union of sets. That is why the
neutrosophic summation and implicitly its extension
to neutrosophic scalar multiplication can be defined
in many ways, equivalently to the known in the
literature neutrosophic union operators [26]. Here,
writing the elements of a SVNS A in the form of
neutrosophic triplets we define addition and scalar
product in A as follows:
Let (t1, i1, f1), (t2, i2, f2) be in A and let k be a
positive number. Then:
The sum (t1, i1, f1) + (t2, i2, f2) = (t1+ t2, i1+
i2, f1+ f2) (4)
The scalar product k(t1, i1, f1) = (kt1, k i1,
kf1) (5)
3.3 Soft Sets
A disadvantage connected to the concept of FS is
that there is not any exact rule for defining properly
the membership function. The methods used for this
are usually empirical or statistical and the definition
of the membership function is not unique depending
on the “signals” that each observer receives from the
environment, which are different from person to
person. For example, defining the FS of “tall men”
one may consider as tall all men having heights
more than 1.90 meters and another all those having
heights more than 2 meters. As a result, the first
observer will assign membership degree 1 to men of
heights between 1.90 and 2 meters, in contrast to the
second one, who will assign membership degrees
<1. Consequently, analogous differences is logical
to appear for all the other heights. The only
restriction, therefore, for the definition of the
membership function is to be compatible to the
common sense; otherwise the resulting FS does not
give a reliable description of the corresponding real
situation. This could happen for instance, if in the
FS of “tall men”, men with heights less than 1.60
meters have membership degrees ≥0.5.
The same difficulty appears to all generalizations
of FSs in which membership functions are involved
(e.g. IFSs, NSs, etc.). For this reason, the concept of
interval-valued FS (IVFS) [27] was introduced in
1975, in which the membership degrees are replaced
by sub-intervals of the unit interval [0, 1].
Alternative to FS theories were also proposed, in
which the definition of a membership function is
either not necessary (grey systems/GNs [28]), or it is
overpassed by considering a pair of sets which give
the lower and the upper approximation of the
original crisp set (rough sets [29]).
Molodstov, in order to tackle the uncertainty in
a parametric manner, initiated in 1999 the concept
of soft set (SS) as follows [30]:
Definition 3: Let E be a set of parameters, let A
be a subset of E, and let f be a map from A into the
power set P(U) of all subsets of the universe U.
Then the SS (f, A) in U is defined to be the set of
the ordered pairs
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(f, A) = {(e, f(e)): e ∈ A} (6)
The term "soft" is due to the fact that the form of
(f, A) depends on the parameters of A. For example,
let U= {C1, C2, C3} be a set of cars and let E = {e1,
e2, e3} be the set of the parameters e1=cheap,
e2=hybrid (petrol and electric power) and e3=
expensive. Let us further assume that the cars C1, C2
are cheap, C3 is expensive and C2, C3 are hybrid
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}}.
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 α – cut of the FS, for
each α in [0, 1]. For general facts on SSs we refer
to [31].
Obviously, an important advantage of SSs is that,
by using the parameters, they pass through the need
of defining membership functions. The theory of
SSs has found many and important applications to
several sectors of human activity like decision
making, parameter reduction, data clustering and
data dealing with incompleteness, etc. One of the
most important steps for the theory of SSs was to
define mappings on SSs, which was achieved by A.
Kharal and B. Ahmad and was applied to the
problem of medical diagnosis in medical expert
systems [32]. But fuzzy mathematics has also
significantly developed at the theoretical level
providing important insights even into branches of
classical mathematics like algebra, analysis,
geometry, topology etc.
3.4 Grey Numbers
Approximate data are frequently used nowadays in
many problems of everyday life, science and
engineering, because many constantly changing
factors are usually involved in large and complex
systems. Deng introduced in 1982 the grey system
(GS) theory as an alternative to the theory of FSs for
tackling such kind of data [27]. A GS is understood
to be a system that lacks information such as
structure message, operation mechanism and/or
behaviour document. The GS theory, which has
been mainly developed in China, has recently found
many important applications [33].
An interesting application of the closed intervals
of real numbers is their use in the GS theory for
handling approximate data. In fact, a numerical
interval I = [x, y], with x, y real numbers, x<y, can
be considered as representing a real number with
known range, whose exact value is unknown. The
closer x to y, the better I approximates the
corresponding real number. When no other
information is given about this number, it looks
logical to consider as its representative
approximation the real value
V(I) =
x+y
2
(7)
Moore et al. [34] introduced the basic arithmetic
operations on closed real intervals. In the present
work we shall make use only of the addition and
scalar product defined as follows: Let I1 = [x1, y1]
and I2 = [x2, y2] be closed intervals, then their sum I1
+ I2 is the closed interval
I1 + I2 = [x1+ x2, y1+ y2] (8)
Further, if k is a positive number then the scalar
product kI1 is the closed interval
kI1 = [kx1, ky1] (9)
When the closed real intervals are used for
handling approximate data, are usually referred as
grey numbers (GNs). A GN [x, y], however, may
also be connected to a whitenization function f: [x,
y] → [0, 1], such that, ∀ a ∈ [x, y], the closer f(a) to
1, the better a approximates the unknown number
represented by [x, y].
We close this subsection with the following
definition, which will be used in the assessment
method that will be presented later in this work.
Definition 4: Let I1, I2,…., In be a finite number
of GNs, n≥2, then the mean value of these GNs is
defined to be the GN
I =
1
n
(I1 + I2+….+ Ik) (10)
3.5 GPA Index and the Rectangular Fuzzy
Assessment Model
The calculation of the Grade Point Average (GPA)
Index is a classical method, very popular in the USA
and other western countries, for evaluating a group’s
qualitative performance, where greater coefficients
are assigned to the higher grades. For this, let n be
the total number of the objects of the group under
assessment and let nX be the number of the group’s
objects obtaining the grade X, X =A, B, C, D, F,
where A=excellent, B=very good, C=good,
D=mediocre and F=unsatisfactory. Then, the GPA
index is calculated by the formula
GPA =
F D C B A
0n +n +2n +3n +4n
n
(11)
[35] (Chapter 6, p. 125)
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Volume 2, 2022
In the worst case (n=nF) equation (11) gives that
GPA=0, whereas in the best case (n=nA) it gives that
GPA=4. We have in general, therefore, that
0≤GPA≤4, which means that values of GPA≥2
indicate a satisfactory qualitative performance.
Setting y1 =
F
n
n
, y2 =
D
n
n
, y3 =
C
n
n
, y4 =
B
n
n
and y5 =
A
n
n
, equation (11) can be written as
GPA = y2 + 2y3 + 3y4 + 4y5 (12)
Voskoglou developed a fuzzy model for
representing mathematically the process of learning
a subject matter in the classroom [36]. Later,
considering a student class as a fuzzy system, he
calculated the existing in it total possibilistic
uncertainty for assessing the student mean
performance [37]. Subbotin et al., based on
Voskoglou’s model, adapted properly the Center of
Gravity (COG) defuzzification technique for use as
an assessment method of student learning skills
[38]. Since then, Subbotin and Voskoglou applied,
jointly or separately, the COG technique, termed by
them as the Rectangular Fuzzy Assessment Model
(RFAM), in many other types of assessment
problems; e.g. see [35] (Chapter 6).
There is a commonly used in FL approach to
represent the fuzzy data by the coordinates (xc, yc)
of the COG of the level’s area between the graph of
the corresponding membership function and the OX
axis [39]. In our case, keeping the same notation as
for the GPA index, it can be shown that the
coordinates of the COG are calculated by the
formulas
xc =
1
2
(y1+3y2+5y3+7y4+9y5) (13)
yc =
1
2
(y12+y22+y32+y42+y52) (14)
[3] (Section 4)
It can be also shown the following result [3]
(Section 4):
Assessment Criterion:
Between two groups, the group with the
greater xc demonstrates the better
performance.
For two groups with the same value of xc, if
xc≥2.5 the group with the greater value of yc
performs better, and if xc<2.5 the group
with the lower value of yc performs better.
Combining equations (12) and (13) one finds that
xc =
1
2
(2GPA + 1) or
xc = GPA +
1
2
(15)
Thus, with the help of the first case of the
previous criterion, one concludes that, if the GPA
value of two student groups is different, then the
RFAM and the GPA index give the same outcomes
concerning the assessment of the qualitative
performance of the two groups. If the GPA index,
however, is the same for the two groups, then one
MUST apply the RFAM to see which group
performs better.
4. The Hybrid Assessment Model
A hybrid method is applied in this Section for the
assessment of a student group’s MM skills with
qualitative grades. Namely, SSs are used as tools for
a parametric assessment of the group’s performance,
the calculation of the GPA index and the RFAM are
applied for evaluating the group’s qualitative
performance, GNs are used as tools for assessing the
group’s mean performance and NSs are used when
the teacher is not sure about the individual grades
assigned to some (or all) students.
4.1 Parametric Assessment Using Soft Sets
Assume that a mathematics teacher wants to assess
the MM skills of a group U = {S1, S2, .…., Sn} of n
students, n≥2. Let E = {A, B, C, D, E} be the set of
the parameters A=excellent, B=very good, C=good,
D=mediocre and F=unsatisfactory. Assume further
that the first four students of the group demonstrated
excellent performance, the next five very good, the
following 7 good, the next eight mediocre and the
rest of them unsatisfactory performance. Let f be the
map assigning to each parameter of E the subset of
students whose performance was assessed by this
parameter. Then, the overall student performance
can be represented mathematically by the SS
(f, E) = {(A, {S1, S2, S3}), (B, {S4, S5,…, S8}), (C,
{S9, S10, …, S15}), (D, {S16, S17,…., S23}), (F, {S24,
S25,…, Sn})} (16)
The use of SSs also enables the representation of
each student’s individual performance at each step
of the MM process. For this, let V = {S1, S2, S3, S4,
S5} be the set of the steps of the MM process
according to Voskoglou’s model presented in
Section 2 (Fig. 3). Consider a particular student of
the group U and define a map f: E
Δ(V) assigning
to each parameter of E the subset of V consisting of
the steps of the MM process assessed by this
parameter with respect to the chosen student. For
example, the SS
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(f, E) = {(A, {S1, S3}), (B, {S5}), (C, {S4}), (D,
{S2}), (F, ∅)} (17)
represents the profile of a student who demonstrated
excellent performance at the steps of analysis of the
problem and solution of the model, very good
performance at the step of implementation of the
solution, good performance at the step of validation
and mediocre performance at the step of
mathematizing (he/she faced difficulties, but he/she
finally came through).
4.2 Use of the COG Technique and the
RFAM for Assessing a Group’s Qualitative
Performance
The following example illustrates this method:
Example 1: The students of two classes obtained
the following grades in a test involving MM
problems: Class I: A=5 students, B=3, C=7, D=0,
F=5, Class II: A=4, B=4, C=7, D=1, F=4. Which
class demonstrated the better qualitative
performance?
Solution: Equation (11) gives that GPAI = GPA2 =
43
20
. The RFAM model must be used, therefore, for
comparing the two classes’ qualitative performance.
Thus, by equation (13) one gets that
I
C
x
=
II
C
x
=
53
20
>
5
2
. But equation (14) gives that
I
C
y
= 54 and
II
C
y
= 49, therefore, by the second case of the
RFAM assessment criterion, one concludes that
Class I demonstrated a better qualitative
performance. Further, since GPAI = GPA2 =
43
20
>2,
both groups demonstrated satisfactory qualitative
performance.
4.3 Use of Grey Numbers for Evaluating a
Group’s Mean Performance.
When the student individual assessment is realized
with qualitative grades, a student group’s mean
performance cannot be assessed with the classical
method of calculating the mean value of the student
scores. To overcome this difficulty, using the
numerical climax 1-100 we assign to each of the
student qualitative grades a closed real interval
(GN), denoted for simplicity with the same letter, as
follows: A = [85, 100], B = [75, 84], C = [60, 74], D
= [50, 59] and F= [0, 49].
It is of worth noting that, although the GNs
assigned to the qualitative grades satisfy commonly
accepted standards, the previous assignment is not
unique, depending on the teacher’s personal goals.
For a more strict assessment, for example, the
teacher could choose A = [90, 100], B = [80, 89], C
= [70, 79], D = [60, 69], F= [0, 59], etc.
The estimation of a group’s mean performance
with the help of the previously defined GNs is
illustrated with the following example:
Example 2: Reconsider Example 1. Which class
demonstrated the better mean performance?
Solution: Under the light of equation (10), it is
logical to accept that the GNs
MI=
1
20
(5A+3B+7C+0D+5F) and
MII=
1
20
(4A+4B+7C+1D+4F) respectively can be
used for estimating the two classes’ mean
performance. Straightforward calculations with the
help of equations (8) and (9) give that
MI=
1
20
[1070, 1515] = [53.5, 75.75] and
MII=
1
20
[1110, 1509] = [55.5, 75.45].
Equation (7) gives, therefore, that V(MI) = 64.625
and V(MII) = 64.75. Thus, both classes
demonstrated good (C) mean performance, with the
mean performance of Class II being slightly better.
4.4 Using Neutrosophic Sets for Student
Assessment
In many cases the teacher has doubts about the
grades assigned to some (or all) students of the
group under assessment. In such cases the use of
NSs is more appropriate for estimating the student
group overall performance. This process is
illustrated in the following example:
Example 3: Let {s1, s2, …. , s20} be a class of 20
students. The teacher of the class is not sure about
the grades obtained by them in a test involving MM
problems, because some of the students did not give
proper explanations about their solutions. The
teacher decides, therefore, to characterize the
students who demonstrated excellent performance in
the test by using neutrosophic triplets 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 for all the other students (0, 0, 1). This
means that the teacher is absolutely sure that s1
demonstrated excellent performance, 90% sure that
s2 demonstrated excellent performance too, but at
the same time has a 10% doubt about it and also a
10% belief that s2 did not demonstrate excellent
performance, etc. For the last 10 students the teacher
International Journal of Computational and Applied Mathematics & Computer Science
DOI: 10.37394/232028.2022.2.15
Michael Gr. Voskoglou
E-ISSN: 2769-2477
112
Volume 2, 2022
is absolutely sure that they did not demonstrate
excellent performance. What should be the teacher’s
conclusion about the class’s mean performance in
this case?
Solution: It is logical to accept that the class’s
mean performance can be estimated by the
neutrosophic triplet
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 (8) and (9) is equal to
1
20
(4.5, 5.3, 16.3) = (0.225, 0.265, 0.815). This
means that the performance of a random student of
the class has a 22.5% probability to be characterized
as excellent, however, there exist also a 26.5%
doubt about it and an 81.5% probability to be
characterized as not excellent. Obviously this
conclusion is characterized by inconsistency, which
is an expected outcome due to the teacher’s
uncertainty for the grades assigned to students.
The teacher could work in the same way by
considering the NSs of students who demonstrated
very good, good, mediocre and unsatisfactory
performance in the test, thus obtaining analogous
conclusions.
5. Discussion and Conclusions
A hybrid assessment method was applied in this
work for assessing student MM skills under fuzzy
conditions (with qualitative grades). The whole
process followed leads to the following conclusions:
SSs can be used for realizing a parametric
assessment of the student group’s overall
performance.
The qualitative performance of a student
group (where greater coefficients are
assigned to the higher grades) can be
measured either by the classical method of
calculating the GPA index, or by applying
the RFAM, which is based on the COG
defuzzification technique. When two groups
have the same GPA index, however, then
the RFAM model must be applied to find
which group demonstrates the better
performance.
In case of using qualitative grades for
assessing the student performance, the
assessment of a student group’s mean
performance cannot be realized by the
classical way of calculating the mean value
of the student individual scores. The student
mean performance in this case can be
estimated by using GNs (closed real
intervals).
When the teacher has doubts for the grades
assigned to some (or all) students, NSs is
more appropriate to be used for assessing
the overall performance of a student group.
Our experience from the present and earlier
works implies that hybrid methods, like the previous
one, usually give better and more complete results,
not only in the assessment processes, but also in
decision-making, in tackling the existing in real
world uncertainty and possibly in various other
human or machine activities. This is, therefore, an
interesting area for further research.
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113
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International Journal of Computational and Applied Mathematics & Computer Science
DOI: 10.37394/232028.2022.2.15
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E-ISSN: 2769-2477
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