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 “Competency coefficient” has been used here for making appropriate decisions on the grounds of competency quantitative estimation. 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.
The analysis of world education development tendencies demonstrates [
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 problem, 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 [
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 [
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 “Requirements 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 [
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 experts and demands creation of respective questionnaires.
As an example, let us specify an IT teacher’s information technology competency in the form of a fuzzy set D:
D= 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 necessary 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
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: μ А yi LD .
L μ А yi k А ( yi )
L 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 traverse of
D A j or D AJ
[
D A j D
operation B of complementing the fuzzy set B is defined by the membership
function [
B
μС y maxμ B y; μ K y, y Y .
where the ... sign means scalar capacity of any fuzzy discrete set B [
In so doing P(D,Aj) as a rule is not equal to P(Aj,D). This attribute is used to compare 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 compensate 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 person’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):
PD, A1 0,6 0.9 0.7 0.9 0.6 1 0.9 0.7 0.9 0.735.
The estimate of the difference P(A1,D) is equal to: PА1 , D 1 0.9 0.7 0.8 0.9 0.806 .
0.6 0.9 0.7 0.9
We propose to calculate competency coefficient K is as the normalized estimate of differences:
K minPA, D; PD, A
PA, D
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: K min0.806;0.735 0.912 .
0.806
Calculation of competence coefficient K for the second person will give values described below:
PD, A2 3.2 0.6 0.765 ; PA2, D 3.4 0.8 0.813;
3.4 3.2 K minPA, D; PD, A min0.813;0.765 0.941.
PA, 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) [
Terms of “Competency” LV can be: E1 – very low competency; E2 – low competency; 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 ; E2 0.1 : 0.2 : 0.3 : 0.4 ; the capacity of membership function Ej of the term we accept a fuzzy quantity E3 0.3 : 0.4 : 0,6 : 0,7 ;
E4 0.6 : 0.7 : 0,8 : 0,9 ; E j 1 L 1 L 1 L 1 L
L 1 aj ; L 1 bj ; L 1 cj ; L 1 dj .
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 [
(9) r (x) d x d c ; x c; d .
(11)
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 preparatory (1 to 4) and operational (5, 6) ones. 1. Specifying the set of abilities Y {y j : j 1, m} 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”. E, E j , j 1,5; Ej (x) 0;1; x 0;1; 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 respective 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 position or functions.
Point 3 demands creating respective techniques, tests, problems and tasks that allow 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 questionnaire 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 particular 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.