=Paper=
{{Paper
|id=None
|storemode=property
|title=Quantitative Estimation of Competency as a Fuzzy Set
|pdfUrl=https://ceur-ws.org/Vol-1000/ICTERI-2013-p-187-193.pdf
|volume=Vol-1000
|dblpUrl=https://dblp.org/rec/conf/icteri/VasylevychI13
}}
==Quantitative Estimation of Competency as a Fuzzy Set==
https://ceur-ws.org/Vol-1000/ICTERI-2013-p-187-193.pdf
Quantitative Estimation of Competency as a Fuzzy Set
Leonid Vasylevych1 and Ivan Iurtyn1
1
Borys Grinchenko Kyiv University,
Department of Information Technology and Mathematical Sciences
lvasilevich@mail.ru, yurtyn@ukr.net
Abstract. The authors of this paper have used the assessment of competence as
a fuzzy discrete set consisting of essential capacities. There has been proposed a
procedure of competence quantitative estimation on the basis of discrimination
index of discrete fuzzy sets fixed on one totality. A linguistic variable “Compe-
tency coefficient” has been used here for making appropriate decisions on the
grounds of competency quantitative estimation. Assessment of a person’s com-
petency is proposed as a fuzzy discrete set consisting of necessary abilities as its
values. Using such competency assessment allows to estimate persons’ compe-
tency quantitatively and to compare them.
Keywords. Competency-oriented education, competence, capacities, fuzzy dis-
crete set, linguistic variable, membership functions, fuzzification, scalar capac-
ity of any fuzzy discrete set
Key terms. MathematicalModelling, MathematicalModel, FormalMethod
1 Introduction
The analysis of world education development tendencies demonstrates [1,3] compe-
tency-oriented education trend increase. Moreover, competency, which is not only
defined by knowledge, abilities, skills but also by considerably greater quantity of
factors (coefficients), becomes a major category both in education system and in the
job-market. Competency also includes the ability to obtain, to analyze and to revise
information; to learn through one’s lifetime; to change in compliance with the job-
market demands [1].
Thus, the quantitative estimation of competency necessary for making appropriate
decisions is a multicriterial problem, and therefore we need here to derive an integral
estimation of competency. Since there is no methodology of working out this prob-
lem, it makes the article topical for in it an integral index of a person’s competency
coefficient is estimated on the basis of the new competency assessment as a fuzzy
discrete set of which essential capacities are values.
Published works analysis. In the work [1], the key competencies concept has been
considered and three key competencies have been analyzed (specified by the Organi-
zation of Economic Cooperation and Development (OECD) representatives), which
are: autonomous activity; interactive facility use; ability to work in socially hetero-
188 L. Vasylevych and I. Iurtyn
genic groups. Federal Statistics Department of Switzerland and National Center of
Education Statistics of the USA and Canada within the program named Definition and
Selection of Competencies-- Theoretical and Conceptual principles (“DeSeCo”)”
summarized respective scientific results and different countries’ practices. In the work
[3] we give a review of works on the topic. But in all those works the qualitative ap-
proach to the named subject is solely used , but methods of quantitative competency
estimation have never been given.
Thus, the aim of this work is to develop methods of quantitative estimation of
competency on the grounds of its assessment as a fuzzy discrete set [2] consisting of
essential capacities.
Main results. Competency is defined in UNESCO publications as a combination of
knowledge, abilities, values and attitudes used in everyday life. Therefore qualitative
assessment of a person’s competency means his (her) ability to perform professional
duties or some functions efficiently. But this definition does not give a possibility to
estimate expert’s competency on quantitative basis. That is why we proposed to use
the following person’s competency definition.
Definition 1. A person’s competency is a finite discrete fuzzy set consisting of
abilities necessary for a job position or functions necessary for a respective position.
Membership functions of the set elements characterize the level of this competency
innateness to the person.
Definition 2. Abilities are necessary features, characteristics, faculties, qualities,
knowledge, techniques, skills and other traits which a person needs to perform duties
or functions at a respective position efficiently.
Thereby, in the beginning, we need to define at the discrete set of abilities
Y { y j : j 1, m} membership functions D yi 0 ; 1 of the fuzzy set D “Require-
ments necessary to perform duties or functions at a respective position efficiently”.
These membership functions characterize credibility, priority and importance of a
respective ability for a respective position or function.
Further we will use the notation of the discrete fuzzy set D in the form [2]:
Table 3. Designation D discrete fuzzy set
yi y1 y2 y3 … yn
D=
D(yi) D(y1) D(y2) D(y3) … D(yn)
or D y1 / D y1 ; y 2 / D y 2 ; y3 / D y3 ; ... y1 / D y n .
The set of abilities and respective membership functions will be different for each
position. When we specify the Y set we need to apply the Pareto principle, which
points that 20% of factors define 80% of the result. In practice, implementation of this
principle will lead to the effect that abilities with membership functions less than 0.5
will not be included into the D set. The task of specifying the set of abilities and
respective membership functions refers to the task of knowledge estimation by ex-
perts and demands creation of respective questionnaires.
Quantitative Estimation of Competency as a Fuzzy Set 189
As an example, let us specify an IT teacher’s information technology competency
in the form of a fuzzy set D:
Table 4. Example D representation of discrete fuzzy set
Yi y1 y2 y3 y4
D=
µD(yi) 1 0.9 0.7 0.8
in which y1 - ability to work in Word environment; y2 is the technique of work in
Excel environment; y3 is the technique of work in Excess environment; y4 is special
software skills (e.g, working out optimization tasks).
To perform a quantitative estimation of a particular teacher’s competency it is nec-
essary to estimate his abilities yi. To do so, tests, interviews, exams, respective lessons
control and other means can be recommended. Competency grades (their membership
functions estimation) can be shown on the scale from 0 to 1. This process is called
fuzzification. Hereby, for each person (teacher), we can define in the form of a fuzzy
set his personal fuzzy vector of abilities, which defines his competency. To define
μA(yi) a group of experts can be used who, after analyzing the person, answer the
question: “Is ability yi attributable to the person?” If the LD expert of L experts give a
positive answer, then
LD
μ А yi . (2)
L
As a rule this question does not have a single-value answer, so, experts can use
both binary logic (μAγ(yi) is either 0 or 1, where γ is an expert’s number) and fuzzy
logic (multiple-valued verity scale). In so doing they index the value of μAγ(yi)
0 ; 1 (subjective estimate). If quantity of the experts is L, then in the capacity of
μA(yi) we accept weighted arithmetic mean value of these estimates:
k А ( yi )
μ А yi L
, (3)
k
1
where kγ is the γ expert’s competency estimate.
For quantitative comparison of different persons’ competencies we need, firstly, to
compare in pairs finite discrete fuzzy sets D “Demands necessary to perform duties or
functions at a particular position efficiently” and Aj “the j person’s competency”
which are specified at one totality Y.
To compare these finite discrete fuzzy sets in pairs it is possible to use the estimate
P(D,Aj) of difference between D and Aj , which is reduced to the estimate of the trav-
erse of D Aj or D AJ [2]:
D Aj D (4)
РD , AJ ,
D
190 L. Vasylevych and I. Iurtyn
where the ... sign means scalar capacity of any fuzzy discrete set B [2]:
B μ B yi (5)
х Х
operation B of complementing the fuzzy set B is defined by the membership func-
tion [2]
μ _ y 1 μ B y , y Y (6)
B
operation of two fuzzy sets unification (C B K) has the membership function [2]
μ С y maxμ B y ; μ K y , y Y . (7)
In so doing P(D,Aj) as a rule is not equal to P(Aj,D). This attribute is used to com-
pare fuzzy sets specified at one totality: if P(D,Aj) > P(Aj,D), then the fuzzy set D <
Aj and vice-versa.
A person’s abilities, which have some membership functions’ value greater than
the value of respective abilities’ membership functions in the D set, must not compen-
sate small values of the Aj set membership functions. To avoid this it is necessary to
perform the Aj set normalization: membership functions values of the Aj set which
exceed respective values in the D set have to be equated to respective membership
functions’ values of the D set. Thereby it is necessary to insert the normalized fuzzy
set Ajн into the (3) formula.
Let us perform a comparison of two persons’ competencies. Let us define one per-
son’s competency by means of a fuzzy set А1 (y1/0.6); (y2/0.9); (y3/0.7); (y4/0.9)
and the other person’s competency by means of a fuzzy set А2 (y1/0.8); (y2/1);
(y3/0.5); (y4/0.9).
After normalization we have: А1н =А1 (y1/0.6); (y2/0.9); (y3/0.7); (y4/0.9);
А2н (y1/0.8); (y2/0.9); (y3/0.5); (y4/0.8).
The estimate of the difference P(D,A1) is equal to (3):
0,6 0.9 0.7 0.9 0.6
PD, A1 0.735.
1 0.9 0.7 0.9
The estimate of the difference P(A1,D) is equal to:
1 0 .9 0 .7 0 .8 0 .9
P А1 , D 0.806 .
0 .6 0 .9 0 .7 0 .9
We propose to calculate competency coefficient K is as the normalized estimate of
differences:
minP A, D ; PD, A (8)
K
P A, D
Quantitative Estimation of Competency as a Fuzzy Set 191
This coefficient always belongs to [0;1] interval. If P(A,D) > P(D,A) then K<1,
and if P(A,D) < P(D,A) then K=1.
After inserting computed estimates into the (7) formula we have:
min0.806;0.735
K 0.912 .
0.806
Calculation of competence coefficient K for the second person will give values
described below:
3.2 0.6 3.4 0.8
PD, A2 0.765 ; P A2 , D 0.813;
3.4 3.2
minP A, D ; PD, A min0.813;0.765
K 0.941.
P A, D 0.813
Thereby, we can conclude that the second person’s competency is greater than the
first one’s.
To define a person’s competency level basing on the competency coefficient value
it is necessary to specify a linguistic variable (LV) [2] “A person’s competency coef-
ficient”, which we will determine by means of a tuple
E , E j , j 1,5; E j ( x) 0;1; x K 0;1; .
Terms of “Competency” LV can be: E1 – very low competency; E2 – low compe-
tency; E3 – medium competency; E4 – high competency; E5 – very high competency.
Trapezoidal membership functions of terms can be defined by experts by means of
four numbers a : b : c : d , which define each term.
Using trapezoidal membership functions of terms and considering Harrington’s
scale it is possible to specify “Competency” LV as follows: E1 0 : 0 : 0.1 : 0.2 ;
E 2 0 .1 : 0 .2 : 0 .3 : 0 .4 ; E 3 0 . 3 : 0 . 4 : 0, 6 : 0, 7 ; E4 0.6 : 0.7 : 0,8 : 0,9 ;
E5 0.8 : 0.9 : 1 : 1 .
Let estimate Ej of a term by an γ expert amounts E j aj ; bj ; cj ; dj , then in
the capacity of membership function Ej of the term we accept a fuzzy quantity
1 L 1 L 1 L 1 L
Ej aj ; bj ; cj ; dj . (9)
L 1 L 1 L 1 L 1
To specify terms more appropriately Delphi technique can be applied.
Specifying membership functions lateral branches by straight line segments does
not reduce persons’ competency estimate’s generality but simplifies mathematical
operations over fuzzy quantities considerably [4]. In so doing the left l (x) and the
right r (x) lateral branches of the membership linear function have analytical form
respectively:
xa
l ( x) ; x a; b , (10)
ba
192 L. Vasylevych and I. Iurtyn
dx
r ( x) ; x c; d . (11)
d c
For the just made example, we have ascertained that the competence coefficient
K1=x=0.912 belongs to E5 term (very high competency) with membership function
(verity) one, and the competence coefficient K2=x=0.88 belongs to E4 term (high
competence with membership function 0.2 and to E5 term with membership function
0.8.
Algorithm of a person’s competency estimation consists of six stages: the prepara-
tory (1 to 4) and operational (5, 6) ones.
Y { y j : j 1, m}
1. Specifying the set of abilities for a position or functions.
2. Abilities’ membership functions assessment (Specifying D fuzzy set) ( (1) and (2)
formulae are applied).
3. Ai fuzzy set assessment – “A person’s competency”.
j E, E , j 1,5; j ( x) 0;1; x 0;1;
E
4. “A person’s competency” LV assessment: .
5. A person’s competency coefficient computing ((3); (4); (5); (6) and (7) formulae
are applied).
Computing a person’s competency coefficient’s membership functions to respec-
tive terms LV “A person’s competency” (formulae (9) and (10)).
At the preparatory stage experts are used, who define the notion “Competency” as
a discrete fuzzy set, the values of which are abilities necessary for a particular posi-
tion or functions.
Point 3 demands creating respective techniques, tests, problems and tasks that al-
low estimating various abilities of a person (to find membership functions of each
ability).
At the stage of receiving a person’s competency quantitative estimate points 5 and
6 are performed.
To specify A fuzzy set of an expert’s antecedent characteristics the expert’s ques-
tionnaire data, his (her) tests, interviews can be used.
The examined competency estimation methodology based upon using fuzzy sets
and a linguistic variable allows resolving several problems: conversion from current
qualitative competency assessments to quantitative estimation; multicriteriality of
competency estimation problem; impossibility of quantitative measuring certain par-
ticular indexes of competency; impossibility of real experiments to estimate different
persons’ competency.
2 Conclusions
1. Assessment of a person’s competency is proposed as a fuzzy discrete set consisting
of necessary abilities as its values. Using such competency assessment allows to
estimate persons’ competency quantitatively and to compare them.
2. A methodology of a person’s competency quantitative estimation is proposed.
Quantitative Estimation of Competency as a Fuzzy Set 193
3. It is proposed to estimate quantitatively persons’ competency on the basis of dif-
ference coefficient of finite discrete fuzzy sets D “Demands necessary to perform
duties or functions at a particular position efficiently”.
4. It is proposed to specify an expert’s competency coefficient in the form of linguis-
tic variable “Competency”.
References
1. Key Competencies: A Developing Concept in General Compulsory Education.
EURYDICE. The Information Network on Education in Europe, p. 224 (2002)
2. Pospelov B.A. (ed.): Fuzzy Sets in Management and Artificial Intelligence Models. Sci-
ence, Moscow (1986)
3. Sysoyeva S.O.: Education and Personality in Post-Industrial World. Monograph. KSPA,
Khmelnytsky (2008)
4. Vasylevych, L.F. Malovik, K.N., Smirnov, S.B.: Quantitative Methods of Making Deci-
sions in Terms of Risk. SNUNEP, Sevastopol (2007)