The work is based on estimation of competence as a fuzzy event, the reliability of which is determined by the membership function; the method of competence evaluation based on fuzzy statements is proposed, which creates a system of compositional rules of fuzzy products. The final verbal assessment of competence is proposed to determine basing on the linguistic variable "Competence", defined on the set of values of membership function of competence.
Training specialists that meet modern requirements, promotion of individuals to positions should be based on competence approach, which must consider not only the availability of necessary knowledge, habits and skills but also such psychological peculiarities of subjects as their creativity, ability to work in a team and so on. The quantitative estimation of competence necessary for making appropriate decisions is a multicriterial problem, and therefore we need here to derive an integral estimation of competence. Since there is no generally accepted methodology of working out this problem, it makes the article topical.
Published works analysis. In article [
The aim of this paper is to develop a methodology for the quantitative assessment
of competence based on fuzzy statements [
Main results
Hereinafter the paper presents and will further use the following definitions and their characteristics.
Definition 1. Competence is a fuzzy event, characterized by the membership function, the completion of which designates the ability of a subject to effectively perform his duties on a certain position which is a result of his professional training and lifelong self-learning.
Definition 2. Quantitative evaluation of competence is an integral characteristic of knowledge, skills, motivation of a subject from the totality of the abilities, defined by the membership function k0;1, and it characterizes the degree of ability to effectively perform the responsibilities of a certain office position.
Definition 3. Verbal evaluation of competence is an integral characteristics of knowledge, skills, motivation of a subject, which is defined by the terms of a linguistic variable "Competence", that is set on a multitude of values of the membership function k0;1, and it characterizes a degree of his ability to effectively perform the duties of a certain position.
Terms of this linguistic variable can be: «Incompetent person»; «Little competent person»; «Average competent person»; «Well competent person»; «Highly competent person».
Definition 4. Ability is one of many components of competence, that is
characterized by a dynamic combination of a special job knowledge, habits and skills of a
subject. Obtaining the integral characteristics of competence (its membership function
k0;1 ) is offered to base on the system of compositional rules of fuzzy
production Fi, i=1, n in the form of implications [
F : IF А , THEN i i
B ;i 1, n, i (1) where Ai;i=1, n - is a fuzzy statement that is a condition (antecedent); Bi, i=1, n - is a fuzzy statement, that is a conclusion (the consequent) of fuzzy production.
For example, the system of compositional rules can be the following: - F1: «IF the subject possesses required knowledge(A11), habits and skills (A12), and is motivated to perform his responsibilities(A13), that are related to corresponding office positions, THEN he is competent (В1)»;
- F2: «IF the subject graduated from the university with an appropriate specialization (A21) and has good grades (A22), THEN he possesses required knowledge (В2=A11)»; - F3: «IF the subject knows how to use a personal computer for tackling essential job tasks (A31), and he has an experience for relevant position not less than a year (A32), THEN he has required skills (В3=A12)»;
- F4: «IF the subject likes his profession (A41) and earns a good salary (A42), THEN he is motivated (В4=A13)»;
- F5: «IF the subject is communicative (A5), THEN he has one of the required skills (В5=A12)»; and so on.
In some rules the conditions and conclusions can be complex (consist of simple
fuzzy statements connected by logical operations of negation, conjunction and
disjunction). For instance, in rules F1,F2,F3,F4 the conditions are complex: they are
composed of simple conditions connected by logical conjunction (^): A1=
A11^A12^A13; A2=A21^A22; A3=A31^A32; A4=A41^A42. The conclusion can also be
complex: consist of simple conclusions often connected by logical disjunction
operations. In some compositional rules both the condition and the conclusion may be in a
form of fuzzy linguistic statements [
The system of composition rules is created by relevant experts. Such rules can range from tens to more than a hundred.
To find the membership functions of complex statements the following formulas
can be applied [
_ 1. The membership function of negation A of the fuzzy statement А is found by formula
A
1 A 2. The membership function of disjunction А1+А2 of fuzzy statements А1 and А2 can be found by algebraic formula: or the formula of maximum: μ A1 A2 A1 A2 A1 A2 μ A1 A2 max( A1 ; A2 ) (2) (3) (4) 3. The membership function of conjuction А1^А2 of fuzzy statements А1 and А2 can be found by algebraic formula: μ
A1A2 A1 A2 (5) or the formula of minimum:
The condition of some of the compositional rules is defined by core abilities: knowledge, capabilities and skills, which characterize the competence. The condition of other rules is defined by position or professional abilities, and their consequences can be core abilities.
Competences have hierarchical structure (fig.1). One competence might be general for one position, and special or professional for another, at the same time. Understanding the hierarchical character of competences allows you to specify the compositional rules that are necessary for defining the membership function of a corresponding conclusion. This process is called activation of the compositional rules.
In the example that is considered, for quantitative assessment of the competence it is necessary to activate compositional rules F1, F2, F3, F4, F5. For each fuzzy compositional rule Fi the experts define a degree of importance Fi, i=1, n . As in the system of compositional rules it is necessary to apply important rules, so their membership functions are usually close to one. This corresponds to the Pareto principle.
For each subject of evaluation the membership function Аj of fuzzy statements of conditions Aij is determined by analyzing resumes, interviewing, testing and setting professional problem tasks. To find the membership functions of complex statements of conditions Ai formulas (2-6) can be applied.
The central place in competence assessment is taken by a method of conclusion: finding the membership functions of conclusions of compositional rules Bj.
In the direct method of drawing a conclusion, based on using a fuzzy
generalization of the rule of fuzzy modus ponens [
As several conclusions of compositional rules determine the conclusion of the compositional rule that is located on the upper hierarchical level, then competence will be determined by the product of fuzzy binary relations, which is a fuzzy binary relation on the Decart product of conditions and conclusions of corresponding compositional rules.
In the example that is being considered, the competence (conclusion В1) is determined by condition А1, which in turn, is determined by conditions (abilities) B2; B3 and B4. Condition А5 determines conclusion B4=А12. The hierarchical of fuzzy statements, that determine competence, are shown in fig.2.
According to this hierarchy, first the membership functions of conditions A2, A3, A4 and A5 are found. On the basis of them the membership functions of the conclusions В2, В3, B4 and В5 are found. The membership functions of these conclusions determine the membership functions of the condition A1, on the basis of which the membership function of conclusion В1 is found.
The membership function) Bi (y) of the fuzzy conclusion Вi is frequently found by
max- min composition [
Вi ( y) mxaxmin( A (x); R (x; y)) where Ai(x) - the membership function of conditions; R(x;y) - the membership function of fuzzy relation R of the Decart product of all conditions Ai=xiX, and conclusions Вi=yiY.
Sometimes, the max-prod composition is used [
Вi ( y) mxax A (x) * R (x; y)
To account for a degree of a risk, that is taken by a person, who uses competence assessment, it is proposed to apply the derived max – min composition of the type: μ B y max (1- ) min μ A x; μ R x , y max( A (x); R (x, y)) хХ x x where [0;1] - the coefficient that characterizes the degree of a risk that is taken when applying the competence assessment.
With zero degree of the risk (λ= 0) statement (9) transforms into statement (7). The maximum degree of the risk is λ= 1.
In the problem considered the membership functions of importance of
compositional rules µFi act as the membership function R(x;y) of a fuzzy relation R. As
psychologists established quantitative information is verbally encoded and stored in a
human’s brain in the form of terms of linguistic variables (LV) [
Using trapezoidal membership functions of terms and considering Harrington’s
scale it is possible to specify LV “Competence” as follows [
Trapezoidal membership functions of terms can be defined by experts by means of four numbers <a; b; c; d>, which define each term.
Thus, the main stages for evaluating the competence is shown in fig.3. In the evaluation there should be two main stages: the stage of obtaining knowledge and the stage of evaluation of competence.
The preparatory stage of obtaining knowledge involves the implementation of items 1; 2; 3; 4. At this stage the experts in the field, in which competence of the subject will be evaluated, are used.
At the next stage (performing item 5) in order to evaluate each subject, the membership function of fuzzy statements of conditions, that are arguments in multicriterial evaluation of competence, are determined by analyzing resumes, interviewing, testing and giving problem professional tasks.
Design stage of assessment of competence includes items 6;7;8. The given method of assessment of experts’ competence, which is based on the use of fuzzy logic, allows to overcome such problems as multicriteriality of the task to evaluate the competence of the subject; incomplete character of necessary information for this task; the impossibility to measure quantitatively certain partial indicators of competence. This method is more adequate to represent the knowledge base of corresponding experts.
Let us imagine we received the following assessments for a particular subject: А21=1; А22=0,8; А31=0,9; А32=1; А41=0,6; А42=0,65; А5=0,5; А6=1.
The reliability of the compositional rules, which are considered in the example, was estimated by the experts as the following values of the membership functions: F1=0,5; F2=0,8; F3=0,9; F4=0,9; F5=1; F6=0,85.
It is needed to find the membership function Bi of conclusion Bi regarding the competence of the subjection the basis of which a verbal evaluation of his competence can be made.
When using the formula (6) to find the membership functions of complex fuzzy statements of conditions, we have A2=min (1;0,8)=0,8; A3=0,9; A4=0,6. where symbol means some pair – wise operation between the corresponding elements vector of the raw and vector of the column. Accordingly, A12=0,74; A13=0,78.
The membership function of complex fuzzy statement of condition А1 is A2=min (0,8;0,74;0,78)=0,74 Then the conclusion В1 is determined by the membership function
В1=min (А1; F1)=min(0,74;0,95)=0,74, that corresponds to term E4 that is high competence with the degree of confidence 1. 3 Conclusions 1. Assessment of a person’s competence is proposed as a fuzzy event, realization of which means the ability of the subject of assessment to effectively perform responsibilities of a certain office position that is a result of his professional training and lifelong self-learning. Quantitative evaluation of competence is an integral characteristic of knowledge, skills, motivation of the subject from the totality of the abilities, defined by the membership function k[0;1], and it characterizes the degree of person’s ability to effectively perform responsibilities of a certain office position. Verbal evaluation of competence is an integral feature of knowledge, skills, motivation of the subject, which is defined by the terms of a linguistic variable "Competence», which is set on the multitude of values of the membership function, and characterizes a degree of his ability to effectively perform the duties of the certain office position. 2. A methodology of quantitative estimation of a person’s competency is proposed.
The method of quantitative evaluation of competence is proposed on the basis of fuzzy statements that create a system of compositional rules. The paper shows the use of production algorithms on the basis of max-min and multiplicative convolution of membership functions and the algorithm that account for the risk taken by a person who uses this competence assessments. The method of assessment of experts’ competence that is based on the application of fuzzy logic allows to overcome such problems as multicriteriality of the task of evaluating competence of the subject, incomplete character of necessary information for this task, impossibility to measure quantitatively certain partial indicators of competence; and is more adequate to represent the experts’ knowledge base.