=Paper=
{{Paper
|id=Vol-2845/Paper_7
|storemode=property
|title=Systems for Checking and Testing the Quality of Knowledge Based on Fuzzy Inference
|pdfUrl=https://ceur-ws.org/Vol-2845/Paper_7.pdf
|volume=Vol-2845
|authors=Roman Ponomarenko
|dblpUrl=https://dblp.org/rec/conf/iti2/Ponomarenko20
}}
==Systems for Checking and Testing the Quality of Knowledge Based on Fuzzy Inference==
https://ceur-ws.org/Vol-2845/Paper_7.pdf
Systems for Checking and Testing the Quality of Knowledge
Based on Fuzzy Inference
Roman Ponomarenko
Taras Shevchenko National University of Kyiv, Volodymyrska str.,60, Kyiv, 01033, Ukraine
Abstract
In this article the problems of systems for assessing the quality of knowledge based on test
control. The approaches to the development of intelligent systems for testing the quality of
knowledge are examined, the functioning of which is based on the apparatus of fuzzy
inference. A knowledge assessment model for fuzzy testing systems based on a four-point
assessment system is proposed. Also presented are fuzzy systems of testing. In particular,
adaptive systems, the advantages of using the fuzzy logic apparatus in building intelligent
testing systems designed to improve the accuracy of testing and identifying the quality of
knowledge by students. The method of complex assessment of students knowledge based on
the Type 2 Tagaki-Sugeno fuzzy model are proposed.
Keywords 1
Testing systems, fuzzy inference, intelligent teaching systems, fuzzy rule, grading scales
1. Introduction and discussion
In the modern educational industry, automated methods are increasingly being used to identify and
test the quality of students' knowledge. In particular, knowledge testing systems are becoming more
and more popular, moreover, gradually moving from an auxiliary tool to the main form of knowledge
quality control. Knowledge testing systems have several advantages: the speed of knowledge testing,
a unified approach to examiners, the ability of a student to take direct part in the examination process,
and compare their results with similar results of their colleagues [1, 2].
Considering various grading scales (100-point, 5-point, 7-point, 12-point, etc.), we can note their
common feature - not depending on the degree of graduation, most of them have a linguistic scale:
“Excellent”, “Good”, “Satisfactory”, “Unsatisfactory”. Moreover, it is not always possible to
accurately determine the transition boundary between two neighboring estimates. It can be argued that
the process of assessing the quality of knowledge is intellectual in itself, and systems that automate
these processes are humanistic systems [3], in which human judgments and the operation of quality
indicators play a large role.
Testing systems can have open and closed questions. Open-ended questions suggest an arbitrary
answer from the examiner, checking the degree of conformity to the standard, while closed-ended
questions have a fixed number of possible answers selected from the available list. Closed questions
include: selecting one or more options, drawing up a logical sequence, determining correspondences
in response groups, etc.
As a rule, the calculation of the number of points for the completed task is based on the arithmetic
calculation of the correct and incorrect selected answer options [4].
There are various approaches to improving the quality of identification and testing of knowledge
in the creation of testing systems [5]. We list some of them:
- systems with different levels of task complexity (multisession systems);
- adaptive systems in which the next task (or level) is selected based on previous answers;
- simulation systems for testing knowledge;
IT&I-2020 Information Technology and Interactions, December 02–03, 2020, KNU Taras Shevchenko, Kyiv, Ukraine
EMAIL: ponomarenko_roman@ukr.net
ORCID: 0000-0001-9681-2297
© 2020 Copyright for this paper by its authors.
Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
CEUR Workshop Proceedings (CEUR-WS.org)
65
- systems with a combination of open and closed questions.
Classical testing systems, in comparison with full-time examinations, have a number of
disadvantages: the examiner's inability to apply an individual approach to each student, a fixed list of
questions, difficulties in choosing the next difficulty level and suitable question options, and the main
question is how to use the teacher’s experience directly during testing.
Considering the fact that in each specific field of knowledge, the teacher’s experience and skills
can be quite narrow and specialized, we can talk about the need to model their expert reasoning,
which, in conditions of incomplete and inaccurate answers to open questions by students, entailed the
creation of various intelligent fuzzy systems testing, designed to reproduce the train of thought of the
teacher in assessing the knowledge of students [2, 6, 7, 8].
The aim of this work is to consider the principles of building intelligent testing systems using a
fuzzy logic apparatus, as well as based on fuzzy logic inference algorithms in order to improve the
quality of knowledge identification and assessment.
2. Fuzzy inference systems
Fuzzy inference systems (FS) include the following main stages of their work [9]:
1. Fuzzification of input data.
2. Aggregation of fuzzy rule subcontracts and calculation of their consequents.
3. Accumulation of subcontracts of the entire block of fuzzy rules.
4. Fuzzy inference.
5. Defuzzyfication output values.
2.1. Mamdani Type 1 Fuzzy Inference.
In general, a fuzzy system can be expressed as follows [10]:
𝑘 𝑛𝑗
⋃ ((⋂ 𝑥𝑖 = 𝑎𝑖,𝑗 ) → 𝑦 = 𝑏𝑗 ) , ̅̅̅̅̅
𝑗 = 1, 𝑁,
𝑗=1 𝑖=1
where 𝑁 – number of rules in the IF-THEN fuzzy rule block, 𝑏𝑗 – fuzzy rule conclusion, 𝑥𝑖 = 𝑎𝑖,𝑗 –
correspondence of a variable 𝑥𝑖 to a fuzzy 𝑎𝑖,𝑗 , ⋃ ⋂– fuzzy disjunction (conjunction) operations.
Then the conclusion of the fuzzy inference according to the Mamdani algorithm [9, 10] (using the
defuzzification procedure by the gravity center method):
𝑀𝑎𝑥
∫𝑀𝑖𝑛 𝑥∙𝜇(𝑥)𝑑𝑥
𝑦= 𝑀𝑎𝑥 , (1)
∫𝑀𝑖𝑛 𝜇(𝑥)𝑑𝑥
where 𝜇 (𝑥 ) is the membership functions of the fuzzy variables 𝑥 to the corresponding fuzzy terms,
𝜇 (𝑥 ) → [0,1].
2.2. Takagi-Sugeno Type 1 Fuzzy Inference.
The knowledge bases of Takagi-Sugeno fuzzy inference systems contain blocks of fuzzy if-then
rules [9, 11–12] (Figure 1). Fuzzy zero-order rules of Takagi-Sugeno systems are distinguished by the
presence of a zero degree polynomial in the consequent rules:
𝑅𝑚 ∶ 𝐼𝐹 𝑥1 𝑖𝑠 𝐴𝑚 𝑚 𝑚 𝑚 𝑚
1 𝐴𝑁𝐷 𝑥2 𝑖𝑠 𝐴2 𝐴𝑁𝐷 … 𝐴𝑁𝐷 𝑥𝑝 𝑖𝑠 𝐴𝑝 𝑇𝐻𝐸𝑁 𝑦 = 𝑎0 , (2)
where 𝑥1 , 𝑥2 , … , 𝑥𝑝 are the fuzzified values of a set of input variables; 𝐴𝑚 𝑚 𝑚
1 , 𝐴2 , . . . , 𝐴𝑝 are the fuzzy
sets of antecedent of each rule 𝑚; 𝑚 – the number of fuzzy rule; 𝑎0𝑚 are the subconclusion of a fuzzy
rule, represented as a constant value.
66
Conventional Takagi-Sugeno fuzzy systems (first order) operate on the basis of if-then fuzzy rules
of the form:
𝑅𝑚 ∶ 𝐼𝐹 𝑥1 𝑖𝑠 𝐴𝑚 𝑚 𝑚
1 𝐴𝑁𝐷 𝑥2 𝑖𝑠 𝐴2 𝐴𝑁𝐷 … 𝐴𝑁𝐷 𝑥𝑝 𝑖𝑠 𝐴𝑝
𝑇𝐻𝐸𝑁 𝑦 𝑚 = 𝑔𝑚 (𝑥1 , 𝑥2 , … , 𝑥𝑝 ), 𝑚 = 1,2, … . , 𝑁, (3)
where is the function in the consequent fuzzy rule: 𝑔 𝑚 (𝑥1 , 𝑥2 , … , 𝑥𝑝 ) = 𝑎0 + 𝑎1 𝑥1 + 𝑎2 𝑥2 + ⋯ +
𝑎𝑝 𝑥𝑝 consists in the form of a linear functional dependence on a set of non-fuzzy values of the input
variables, 𝑁 – the number of fuzzy rules.
The conclusion of the Takagi-Sugeno fuzzy system (which is a numerical value) is calculate (4):
∑𝑁 𝑚 min 𝜇 𝑚(𝑥 )
𝑚=1 𝑔 𝑚 𝑖 𝑖
𝑖=1…𝑝
𝑦= ∑𝑁 𝑚 .
𝑚=1∙ min𝑚 𝜇𝑖 (𝑥𝑖 )
𝑖=1…𝑝
(4)
where 𝜇𝑖𝑚 (𝑥𝑖 ) – membership functions in the antecedent of a fuzzy rule, where the operation of
finding the minimum is used as a conjunction.
Figure 1: Schema of Type 1 fuzzy inference system
2.3. Takagi-Sugeno Type 2 Fuzzy Inference
Takagi-Sugeno systems of Type 2 (T2) [13] are characterized by the presence in the antecedent of
fuzzy rules of fuzzy sets, for which the value of primary belonging is a fuzzy set (fuzzy sets of the
second type, invented by L. Zadeh [14]).
The Karnik-Mendel algorithm of fuzzy inference for Takagi-Sugeno T2 systems based on fuzzy
sets with interval secondary functions was developed in [15, 16]. This algorithm has a slightly lower
computational complexity in comparison with the analogous algorithm for Mamdani Type 2 fuzzy
systems [22]. In [12, 17], a parallel algorithm of fuzzy inference for high-order Takagi-Sugeno
systems was proposed.
The continuous T2 fuzzy sets have the form:
𝑓𝑥 (𝑢)
[∫ ]
𝑢
̃ (𝑥)
𝜇𝐴 𝐽𝑢
𝐴̃ = ∫ =∫ 𝑥
, 𝐽𝑥𝑢 = {(𝑥, 𝑢): 𝑢 ∈ [𝜇𝐴̃ (𝑥 ), 𝜇𝐴̃ (𝑥)]} ⊆ [0,1]. (5)
𝑥 𝑥
𝑋
𝑋
The discrete second type fuzzy sets are presented accordingly:
𝑁 𝑁 [∑𝑀𝑗 𝑓𝑥 (𝑢𝑖 )]
𝜇𝐴̃ (𝑥𝑗 ) 𝑖=1 𝑢𝑖
𝐴̃ = ∑ =∑ ,
𝑥𝑗 𝑥𝑗
𝑗=1 𝑗=1
(6)
𝑢𝑖 ∈ 𝐽𝑥𝑢 ⊆ 𝑢 ∈ [𝜇𝐴̃ (𝑥 ), 𝜇𝐴̃ (𝑥 )] ⊆ [0,1], 𝑥𝑗 ∈ 𝑋,
where 𝑓𝑥 – a secondary membership functions, μà (x) is the upper value of the primary membership
functions:
67
𝜇𝐴̃ (𝑥 ) = sup 𝐽𝑥𝑢 , 𝑥 ∈ 𝑋. (7)
𝜇𝐴̃ (𝑥 ) is the lower value of the primary membership functions:
𝜇𝐴̃ (𝑥 ) = inf 𝐽𝑥𝑢 , 𝑥 ∈ 𝑋. (8)
Takagi-Sugeno T2 FS assumes the use of interval T2 fuzzy sets [22] in the antecedents of fuzzy
rules of the following form:
𝑅𝑚 : 𝐼𝑓 𝑥1 𝑖𝑠 𝐴̃
𝑚 ̃𝑚
1 𝑎𝑛𝑑 … 𝑎𝑛𝑑 𝑥𝑝 𝑖𝑠 𝐴𝑝
𝑇ℎ𝑒𝑛 𝑔(𝑥)𝑚 = 𝑤0𝑚 + 𝑤1𝑚 𝑥1 + ⋯ + 𝑤𝑝𝑚 𝑥𝑝 , (9)
where 𝐴̃ 𝑚 ̃
𝑚
1 … 𝐴𝑝 – interval T2 fuzzy sets, 𝑚 is the number of the rule. Interval T2 fuzzy sets
have the form:
1
[∫ ]
𝑢
̃ (𝑥)
𝜇𝐴 𝐽𝑢
𝐴̃ = ∫ =∫ 𝑥
, 𝐽𝑥𝑢 = {(𝑥, 𝑢): 𝑢 ∈ [𝜇𝐴̃ (𝑥 ), 𝜇𝐴̃ (𝑥 )]} ⊆ [0,1]. (10)
𝑥 𝑥
𝑋
𝑋
2.4. Iterative Karnik-Mendel algorithm for Takagi-Sugeno Type 2 fuzzy
inference system
The Karnik-Mendel algorithm [15, 16] assumes the use of constant secondary membership
functions. The initial step of this algorithm is to execute the activation of Type 2 consequents of a
𝑚 𝑚
fuzzy rule base (𝑔(𝑥 )𝑚 , 𝑚 = 1 … 𝑁 ) and finding for each rule the intervals [𝑓 (𝑥), 𝑓 (𝑥)].
The next steps of the algorithm are the operation of lowering the type and finding the interval
output of the fuzzy system according to formulas (11–13). In [18], a study of interval output values in
fuzzy systems of the second type is given.
𝐺 (x) = [𝑔𝑙 (x), 𝑔𝑟 (x)], (11)
∑𝑁 𝑚 𝑚
𝑚=1 𝑓𝑗 (x)𝑔 (x)
𝑔𝑙 (x) = min 𝑚 ∑𝑁 𝑚 , (12)
𝑓𝑗𝑚 (𝑥)∈[𝑓 𝑚(𝑥),𝑓 (𝑥)] 𝑚=1 𝑓𝑗 (x)
𝑚=1,…,𝑁
∑𝑁 𝑚 𝑚
𝑚=1 𝑓𝑗 (x)𝑔 (x)
𝑔𝑟 (x) = max 𝑚 ∑𝑁 𝑚 . (13)
𝑓𝑗𝑚 (𝑥)∈[𝑓𝑚(𝑥),𝑓 (𝑥)] 𝑚=1 𝑓𝑗 (x)
𝑚=1,…,𝑁
𝑚
Thus, multiple calculation of fuzzy rule consequents on the interval [𝑓 𝑚 (𝑥 ), 𝑓 (𝑥 )] is
performed, since the obtained values may differ. The final output value of the Takagi-Sugeno FS T2
is calculated according to (14):
𝑔(x) = 1⁄2 (𝑔𝑙 (x) + 𝑔𝑟 (x)). (14)
Figure 2 shows the structure of a Type 2 fuzzy system model, where x = (𝑥1 , 𝑥2 , … , 𝑥𝑝 ) – vector
of crisp input values, x̃ = (𝑥̃1 , 𝑥̃2 , … , 𝑥̃𝑝 ) – a set of fuzzy input variables obtained as a fuzzyfication
result.
68
3. Adaptive fuzzy inference systems for testing students based on fuzzy
selection of the next level of complexity
One of the problems associated with the development of testing systems is the intellectualization
of the algorithm for choosing the next level of difficulty when passing a multi-level test by a student.
Knowledge
x = (𝑥1 , 𝑥2 , … , 𝑥𝑝 ) base
Fuzzy 𝑔(x)
Defazzification
Fuzzing reduction
block
block block
Fuzzy
inference
x̃ = (𝑥̃1 , 𝑥̃2 , … , 𝑥̃𝑝 )
Fuzzy System Type 2
Figure 2: Fuzzy inference system Type 2
The approach proposed in [4] involves the use of the methodology of a specific teacher or expert
in the automation of multilevel testing systems. Thus, the fuzzy system is not directly involved in
passing the test, however, as an intermediate link between the levels.
The input data vector is information about the last level of difficulty of the test task passed by the
student, the success of his passage, and the average assessment of passing all the tests at the previous
stages can also be taken into account.
The output is the value of the selected difficulty level at the next stage of testing. Figure 3, 4 show
the linguistic variables “complexity” and “correctness” (relative to the last test performed) by type 2
triangular membership functions with interval secondary membership functions (the numerical scale
of difficulty levels depends on the specific system and is not given here):
The knowledge base can be represented by the following block of fuzzy rules (fragment):
𝑅1 : 𝐼𝐹𝑥1 = 𝐻𝑖𝑔ℎ 𝐴𝑁𝐷𝑥2 = 𝐺𝑜𝑜𝑑
𝑇𝐻𝐸𝑁 𝑦1 = 𝐻𝑖𝑔ℎ;
𝑅2 : 𝐼𝐹𝑥1 = 𝐿𝑜𝑤 𝐴𝑁𝐷𝑥2 = 𝐸𝑥𝑐𝑒𝑙𝑙𝑒𝑛𝑡
𝑇𝐻𝐸𝑁 𝑦2 = 𝐴𝑣𝑒𝑟𝑎𝑔𝑒;
…
𝑅𝑛 : 𝐼𝐹𝑥1 = 𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝐴𝑁𝐷𝑥2 = 𝑆𝑎𝑡𝑖𝑠𝑓𝑎𝑐𝑡𝑜𝑟𝑦
𝑇𝐻𝐸𝑁 𝑦𝑛 = 𝐿𝑜𝑤;
Thus, this approach can be applied to multilevel testing systems for which a set of difficulty levels
of tasks passed by a student is a non-trivial component of identifying the level of knowledge.
4. Adaptive fuzzy testing systems based on fuzzy processing of student's
answers
Another approach to creating adaptive testing systems is the intellectualization of the process of
evaluating results [19–21].
69
Figure 5 shows a diagram of an adaptive knowledge testing system based on the use of fuzzy
inference algorithms when processing student test answers and deriving the final knowledge score.
Methods for constructing hierarchical fuzzy systems are given in [22–24].
Figure 3: Linguistic variable «complexity of the last completed task
Figure 4: Linguistic variable «correctness of the last completed task» on a 100-point scale
A model of fuzzy assessment on a 4-point scale is proposed due to its versatility and implicit
presence in most modern scales. In this diagram (Figure 5) there are n inputs and N = 4 the number of
options for sub-connections. Where the number n corresponds to the number of test tasks, N – to the
set of possible options for evaluating the completed task:
N = {“Unsatisfactory”, “Satisfactory”, “Good”, “Excellent”}.
The input vector X x1 , x2 ,..., xn is the set of results of answers to many test questions. The
membership functions of the input data to a particular fuzzy term for a 100-point scale can be similar
to the membership functions shown in Figure 4.
The conclusion of the fuzzy system is a qualitative indicator of a student’s knowledge of the set of
N linguistic terms (which, however, can, if necessary, be reduced to a quantitative form). The fuzzy
inference algorithm can be selected depending on the way the fuzzy rules consequents are presented.
The knowledge base of the intellectual testing system is presented in the form of fuzzy predicate
rules IF-THEN, in which such assessment rules can be displayed that are inherent to a particular
teacher, taking into account the field of knowledge.
The fragment of a block of fuzzy rules is presented below:
𝑅1 : 𝐼𝐹𝑥1 = 𝐺𝑜𝑜𝑑 𝐴𝑁𝐷𝑥2 = 𝐺𝑜𝑜𝑑
𝑇𝐻𝐸𝑁 𝑦1 = 𝐺𝑜𝑜𝑑;
𝑅2 : 𝐼𝐹𝑥3 = 𝑆𝑎𝑡𝑖𝑠𝑓𝑎𝑐𝑡𝑜𝑟𝑦 𝐴𝑁𝐷𝑥3 = 𝐺𝑜𝑜𝑑
𝑇𝐻𝐸𝑁 𝑦2 = 𝑆𝑎𝑡𝑖𝑠𝑓𝑎𝑐𝑡𝑜𝑟𝑦;
…
𝑅𝑛 : 𝐼𝐹𝑥1 = 𝐸𝑥𝑐𝑒𝑙𝑙𝑒𝑛𝑡 𝐴𝑁𝐷𝑥2 = 𝑆𝑎𝑡𝑖𝑠𝑓𝑎𝑐𝑡𝑜𝑟𝑦
𝑇𝐻𝐸𝑁 𝑦𝑛 = 𝐸𝑥𝑐𝑒𝑙𝑙𝑒𝑛𝑡.
70
Figure 5: The scheme of a fuzzy system for testing the quality of knowledge
It is worth noting that in this model the consequents of fuzzy rules are also fuzzy values.
The output value of the fuzzy Mamdani system for estimates on a four-point scale is written as
follows:
𝑌 = {𝑦1 , 𝑦2 , 𝑦3 , 𝑦4 };
𝑦1 = ⋃ 𝜇 (𝑥 );
𝑈𝑛𝑠𝑎𝑡𝑖𝑠𝑓𝑎𝑐𝑡𝑜𝑟𝑦
𝑦2 = ⋃ 𝜇 (𝑥 );
𝑆𝑎𝑡𝑖𝑠𝑓𝑎𝑐𝑡𝑜𝑟𝑦
𝑦3 = ⋃ 𝜇 (𝑥 );
𝐺𝑜𝑜𝑑
𝑦4 = ⋃ 𝜇 (𝑥 ).
𝐸𝑥𝑐𝑒𝑙𝑙𝑙𝑒𝑛𝑡
There ⋃ are determines the fuzzy conjunction operation when performing the accumulation
operation of the subclauses of fuzzy Mamdani rules. Сonclusion of the fuzzy system will be produced
by defuzzing the output variable Y according to (1).
In the case of using the fuzzy Takagi-Sugeno knowledge base, the ⋃ operation is accordingly
replaced by the summation operation
𝑦𝑖 = ∑ 𝑥𝜇(𝑥 ),
𝑁𝑖
and the output value is calculated according to (4).
The output linguistic variable is shown in Figure 6. We can observe a rather high degree of
vagueness of the “Good” and “Excellent” ratings in order to increase the objectivity of knowledge
control.
Figure 6 shows the result of the accumulation of the three fuzzy rules presented above and finding
the final student grade by the center of gravity method.
This approach to building knowledge quality control systems helps to a large extent to bring the
process of evaluating test results closer to that in which pedagogical experience and the methodology
for testing the quality of knowledge by an expert teacher are involved.
71
5. The method of complex assessment of students' knowledge based on the
Type 2 Tagaki-Sugeno fuzzy model
A method of fuzzy assessment of the quality of knowledge has been developed to obtain a
comprehensive characteristic of a student for a training course (module). The Takagi-Sugeno fuzzy
inference model with interval fuzzy membership functions of type 2 was taken as a basis. This model
allows one to take into account the vague nature of the boundaries of linguistic estimates. Thus,
giving at the output a more objective characteristic of knowledge (using the Karnik-Mendel fuzzy
inference algorithm according to formulas (11-14).
The Takagi-Sugeno fuzzy model, the fuzzy rule consequents of which are presented in the form of
functional dependencies, was not chosen by chance. Since this model allows you to form an expert
opinion based on the numerical rating points given to the student during the course.
Figure 6: Accumulation of fuzzy rule subclauses and application of the center of gravity
defuzzification procedure
Figure 7 schematically shows the organization of the fuzzy rule calculations when using the
proposed method. Every fuzzy rule 𝑅𝑘 of rule block 𝑅 = {𝑅1 , 𝑅2 , … , 𝑅𝑛 } as input parameters 𝑋 =
{𝑥1 , 𝑥2 , … , 𝑥𝑚 } in the antecedent are accepts the evaluation 𝑥𝑖 for each lesson (topic), where m is
number of lessons.
The block of rules for fuzzy inference is drawn up by an expert teacher and can take into account
the nonlinear dependencies of a student's knowledge for individual lessons (topics). This may take
into account the incompleteness of the student's knowledge, as well as the subjective methodology of
teaching and assessing certain academic disciplines.
6. Final reasoning
The linguistic nature of fuzzy mathematics, which makes it possible to operate with qualitative
quantities, makes it possible to introduce fuzzy characteristics into the system. This helps the teacher
to more accurately formulate and evaluate the requirements regarding the complexity of the tasks, as
well as improve the interaction of the system with students during the tests.
The introduction of a 100-point rating scale did not solve the problem of the accuracy and
objectivity of the knowledge assessment system, but rather made it more fragile and vulnerable.
Indeed, the verge of transition of the Satisfactory score (74 points) to the Good score (75 points) is 1
point.
Fuzzy systems can eliminate this drawback by establishing a varying degree of fuzzy transitions
from one estimate to another. And also introducing additional quality indicators, such as “pretty
good”, “almost satisfactory”, “not very good”, “brilliant” and others.
72
Sometimes a useful feature of fuzzy testing systems is the adjustment of the severity of evaluating
the results, with the ability for students to give fuzzy-logical answers when passing through the testing
procedure used in [2].
𝜇(𝑥1 )
Lesson 1 TS Type 2 Fuzzy Rulek
1
Grade for the lesson 1
𝑥1
0
𝑔 𝑘 = 𝑎0 + 𝑎1 𝑥 1 + ⋯
𝑚𝑖𝑛 𝜇𝐴̃ (𝑥 )
+ 𝑎𝑚 𝑥 𝑚
… 𝑚𝑖𝑛 𝜇𝐴̃ (𝑥 )
𝜇(𝑥𝑚 )
Lesson m
1 Grade for the lesson m
0 𝑥𝑚
Figure 7: Organization scheme of the Takagi-Sugeno Type 2 fuzzy rule in the complex assessment of
student knowledge
7. Conclusions
The apparatus of fuzzy logic in the design of testing systems allows you to more accurately
identify gaps in student knowledge. And at the same time, given the incompleteness of answers
during the tests, to identify quantitative and qualitative indicators of existing knowledge, without
requiring the student to give knowingly false answers in the absence of them.
The paper considers testing systems for identifying and checking the quality of students'
knowledge, operating on the basis of the fuzzy inference methods. The method of complex
assessment of students' knowledge based on the Type 2 Takagi-Sugeno fuzzy model are proposed. A
knowledge assessment model for fuzzy testing systems based on a four-point assessment system are
proposed.
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