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140
5. Пивоваров Ю. С., Кулешов В. В. Искусственный интеллект в науке и образовании:
перспективы и угрозы // Образование и наука. – 2020. – №3(87). – С. 20–29.
ROBUST APPROACH FOR MULTICRITERIA DECISION-MAKING BASED ON THE
INTUITIONISTIC FUZZY CHOQUET INTEGRAL OPERATOR
Muhamedieva Dilnoz Tulkunovna
“Tashkent Institute of Irrigation and Agricultural Mechanization Engineers” National Research
University, Professor at the department “Digital technologies and artificial intelligence”
Khidirova Charos Murodilloevna
Tashkent University of Information Technologies named after Muhammad al-Khwarizmi,
Doctoral student, PhD., dosent
khidirova@tuit.uz
Annotation:
This paper presents the application of intuitionistic fuzzy sets and aggregation
operators in solving multi-criteria decision-making problems. The study focuses on the use of the
intuitionistic fuzzy Choquet integral operator, which effectively handles the interdependencies
among criteria while aggregating linguistic evaluations expressed as intuitionistic fuzzy values.
Key words:
intuitionistic fuzzy set, agregation operator, multi-criteria design-making,
Choquet integral, Sugeno integral, ordered weighted averaging.
НАДЕЖНЫЙ ПОДХОД К МНОГОКРИТЕРИАЛЬНОМУ ПРИНЯТИЮ РЕШЕНИЙ
НА ОСНОВЕ ИНТУИЦИОНИСТСКОГО НЕЧЕТКОГО ОПЕРАТОРА ИНТЕГРАЛА
ШОКЕ
Аннотация
: В статье рассматривается применение интуиционистских нечётких
множеств и агрегирующих операторов для решения задач многокритериального принятия
решений. Основное внимание уделено использованию интуиционистского нечёткого
интегрального оператора Шоке, который эффективно учитывает взаимосвязи между
критериями при агрегировании лингвистических оценок, выраженных интуиционистскими
нечёткими значениями.
Ключевые слова:
интуиционистские нечёткие множества, агрегирующие
операторы, многокритериальное принятие решений, интеграл Шоке, интеграл Сугено,
упорядоченное взвешенное усреднение.
INTUITSION NORAVSHAN CHOQUET INTEGRAL OPERATORIGA ASOSLANGAN
KO‘P MEZONLI QAROR QABUL QILISH UCHUN ISHONCHLI YONDASHUV
Annotatsiya
: Mazkur maqolada ko‘p mezonli qaror qabul qilish masalalarini yechishda
intuitsionistik noravshan to‘plamlar hamda agregatsiya operatorlaridan foydalanish imkoniyatlari
tadqiq etiladi. Asosiy e’tibor mezonlararo o‘zaro bog‘liqliklarni samarali hisobga oluvchi
intuitsionistik noravshan Choquet integral operatoriga qaratilgan bo‘lib, u lingvistik baholash
natijalarini intuitsionistik noravshan qiymatlar ko‘rinishida ifodalash imkonini beradi.
Kalit so‘zlar:
intuitsion noravshan to‘plam, agregatsiya operatori, ko‘p mezonli qaror
qabul qilish, Choquet integrali, Sugeno integrali, tartiblangan vaznli o‘rtacha.
Introduction.
Decision-making processes often involve multiple criteria, where alternatives
must be evaluated based on various attributes or factors. In many practical situations, such
evaluations are inherently uncertain and imprecise due to the subjective nature of human judgment.
Современные проблемы интеллектуальных систем. Республиканская научно-практическая конференция. Джизак, 18-19 апреля 2025 г.
141
To address this challenge, Zadeh's fuzzy set theory has been widely adopted, but it only considers
membership degrees. Atanassov's intuitionistic fuzzy sets (IFS), which incorporate both
membership and non-membership degrees alongside hesitancy margin, provide more
comprehensive framework for modeling uncertainty [1].
Aggregation operators play a pivotal role in combining these evaluations into a unified result
that reflects the overall performance of each alternative. Traditional aggregation operators like
arithmetic mean, weighted sum, and ordered weighted averaging (OWA) have been extended to
fuzzy environments. However, when criteria exhibit interdependencies, advanced tools such as
fuzzy integrals – particularly the Choquet integral – prove indispensable.
These are generally used in computational intelligence where linguistically expressed pieces
of information are used together. Some well-known examples are arithmetic mean [2], weighted
minimum and maximum [3], weighted sum [4], median [5], and ordered weighted averaging
(OWA) operators [6]. Also, fuzzy integrals such as Choquet and Sugeno integrals are used as
aggregation operator. We will discuss some of the well-known fuzzy aggregating operators.
Methods and results.
Multi-criteria decision-making (MCDM) problem is a method for
finding the best alternative among all the alternatives evaluated using a set of attributes/criteria.
Alternatives are evaluated on the basis of criterion or attribute. Let
𝐵 = {𝑏
1
,
𝑏
2
, 𝑏
3
, … , 𝑏
𝑚
}
be a
set of attributes and
𝐶 = {𝑐
1
, 𝑐
2
, 𝑐
3
, … , 𝑐
𝑛
}
be a set of alternatives. Partial evaluation of the
alternatives,
𝑐
𝑖
(𝑖 = 1,2,3, … , 𝑛)
, is carried out with respect to the attributes or criteria,
𝑏
𝑗
(𝑗 = 1,2,3, … , 𝑚)
. Partial evaluation,
𝑐
𝑖𝑗
, is expressed using intuitionistic fuzzy values,
𝑐
𝑖𝑗
=
(𝜇
𝑖𝑗
, 𝜈
𝑖𝑗
)
, where
𝜇
𝑖𝑗
is the satisfaction degree which means that
𝑐
𝑖
satisfies the criterion
𝑏
𝑗
and
𝜈
𝑖𝑗
is the dissatisfaction degree which means that
𝑐
𝑖
does not satisfy the criterion
𝑏
𝑗
with the condition
0 ≤ 𝜇
𝑖𝑗
≤ 1, 0 ≤ 𝜈
𝑖𝑗
≤ 1
. A multi-attribute decision-making problem is expressed in matrix
form:
𝐷 =
[
𝑏
1
𝑏
2
𝑐
1
(𝜇
11
, 𝜈
11
)
(𝜇
12
, 𝜈
12
)
𝑐
2
(𝜇
21
, 𝜈
21
)
(𝜇
22
, 𝜈
22
)
𝑏
3
…
𝑏
𝑚
(𝜇
13
, 𝜈
13
)
…
(𝜇
1𝑛
, 𝜈
1𝑚
)
(𝜇
23
, 𝜈
23
) … (𝜇
2𝑛
, 𝜈
2𝑚
)
⋮
⋮
⋮
𝑐
𝑛
(𝜇
𝑛1
, 𝜈
𝑛1
) (𝜇
𝑛2
, 𝜈
𝑛2
)
⋮
⋱
⋯
(𝜇
𝑛3
, 𝜈
𝑛3
)
… (𝜇
𝑛𝑚
, 𝜈
𝑛𝑚
)]
Score function,
𝑆(𝑐
𝑖𝑗
)
, of the partial evaluation
𝑐
𝑖𝑗
of the alternative
𝑐
𝑖
is evaluated to rank
𝑐
𝑖𝑗
. If there is no difference between the two score functions, then accuracy function,
𝐻(𝑐
𝑖𝑗
)
, is
used to rank
𝑐
𝑖𝑗
based on the accuracy.
Example of decision-making problem to find an expert supplier based on supplier's
competencies in making a machine is given. Suppose there are three suppliers and their ability is
judged by the attributes of the machine such as (i) innovative level, (ii) longevity, and (iii) cost.
We are to take a decision, which supplier is to be selected for ordering a machine. Attributes are
denoted as
(𝑏
1
, 𝑏
2
, 𝑏
2
)
and the three suppliers, which are the alternatives, are denoted as
(𝑐
1
, 𝑐
2
, 𝑐
3
)
. To evaluate the competencies of the experts, 10 candidates are invited. Suppose there
are six candidates who judge the attribute
𝑏
1
of the expert
𝑐
1
as strong and other three candidates
who judge the attribute
𝑏
1
of the expert
𝑐
1
as not strong and the remaining one candidate does not
judge the candidate as strong or not strong. Then, evaluating value of the attribute
𝑏
1
of
𝑐
1
may be
expressed using an intuitionistic fuzzy value
𝑐
𝑖𝑗
= (0.6, 0.3)
, where
𝑐
𝑖𝑗
is the partial evaluation of
any alternatives,
𝑐
𝑖
with respect to attributes,
𝑏
𝑗
. Likewise, the results of 10 candidates to the 3
experts according to the 3 criteria (attributes) together are performed and an intuitionistic fuzzy
decision matrix of the experts is formed:
Современные проблемы интеллектуальных систем. Республиканская научно-практическая конференция. Джизак, 18-19 апреля 2025 г.
142
𝑏
1
𝑏
2
𝑏
3
𝑐
1
(0.6, 0.3) (0.5, 0.4) (0.7, 0.2)
𝑐
2
(0.4, 0.3) (0.6, 0.2) (0.5, 0.2)
𝑐
3
(0.4, 0.3) (0.3, 0.5) (0.6, 0.3)
Partial evaluation
𝑐
𝑖𝑗
of candidate supplier
𝑐
𝑖
with respect to the attributes or criteria,
𝑏
𝑗
, is
reordered such that
𝑐
𝑖(𝑗)
≤ 𝑐
𝑖(𝑗+1)
.
Reordering is done on the basis of score function [6,7]. As has been said, the partial
evaluation of the alternatives
𝑐
𝑖
with respect to the attributes
𝑏
𝑗
is made by intuitionistic fuzzy
values
𝑐
𝑖𝑗
= (𝜇
𝑖𝑗
, 𝜈
𝑖𝑗
)
and the decision matrix
(𝐷)
is formed:
𝐷 = [
𝑐
11
𝑐
12
𝑐
13
𝑐
21
𝑐
22
𝑐
23
𝑐
31
𝑐
32
𝑐
33
]
Then, score on
𝑐
𝑖𝑗
is given as:
𝑆(𝑐
𝑖𝑗
) = 𝜇
𝑐(𝑖𝑗)
− 𝜈
𝑐(𝑖𝑗)
and
𝑆(𝑐) ∈ [−1,1].
It represents the difference of membership and non-membership values. If the score values
are similar, the accuracy degree is evaluated.
𝐻(𝑐
𝑖𝑗
) = 𝜇
𝑐
𝑖𝑗
+ 𝜈
𝑐
𝑖𝑗
, where
𝐻(𝑐) ∈ [0,1].
After reordering, which is done on the basis of score function
𝑐
𝑖(𝑗)
≤ 𝑐
𝑖(𝑗+1)
, we get:
𝑐
1(1)
= (0.5,0.4), 𝑐
1(2)
= (0.6,0.3), 𝑐
1(3)
= (0.7,0.2)
𝑐
2(1)
= (0.4,0.3), 𝑐
2(2)
= (0.5,0.2), 𝑐
2(3)
= (0.6,0.2)
𝑐
3(1)
= (0.3,0.5), 𝑐
3(2)
= (0.4,0.3), 𝑐
3(3)
= (0.6,0.3)
Let the fuzzy measure of the criterion
𝑏
1
, 𝑏
2
, 𝑏
3
or group of criteria i.e, the importance of
each criterion, be given as:
𝜇(𝑏
1
) = 0.4, 𝜇(𝑏
2
) = 0.2, 𝜇(𝑏
3
) = 0.3.
Using equation
𝜆 + 1 = ∏
(1 + 𝜆𝑔
𝑖
)
𝑛
𝑖=1
, we obtain the value of
𝜆
and we get,
𝜇(𝑏
1
, 𝑏
2
) = 0.63, 𝜇(𝑏
2
, 𝑏
3
) = 0.52, 𝜇(𝑏
1
, 𝑏
3
) = 0.7445, 𝜇(𝑏
1
, 𝑏
2
, 𝑏
3
) = 1,
where
𝜇
is a fuzzy measure.
Using intuitionistic fuzzy Choquet integral operator:
𝑐
𝑖
= 𝐼𝐹𝐶
𝜇
(𝑐
𝑖1
, 𝑐
𝑖2
, 𝑐
𝑖3
, … , 𝑐
𝑖𝑛
) = (1 − ∏
𝑛
𝑗=1
(1 − 𝜇
𝑐
𝑖(𝑗)
)
𝜇(𝐴
𝑗
)−𝜇(𝐴
𝑗+1
)
, ∏
𝑛
𝑗=1
𝜈
𝑐
𝑖(𝑗)
𝜇(𝐴
𝑗
)−𝜇(𝐴
𝑗+1
)
)
where
𝐴
(𝑗)
= {𝑏
(𝑗)
, … , 𝑏
(3)
}, 𝐴
(3+1)
= ∅
. We are to aggregate
𝑐
𝑖𝑗
corresponding to the supplier
𝑐
𝑖
(𝑖 = 1,2,3)
.
For computing
𝑐
1
, we require to compute
𝜇(𝐴
(𝑗)
)
. We have
𝑐
1(1)
= (0.5, 0.4), 𝑐
1(2)
=
(0.6, 0.3), 𝑐
1(3)
= (0.7, 0.2)
. So,
(1) = 2, (2) = 1, (3) = 3
.
Hence, we get.
𝜇(𝐴
(1)
) = 1, 𝜇(𝐴
(2)
) = 0.7445, 𝜇(𝐴
(3)
) = 0.3.
𝑐
1
= 𝐼𝐹𝐶
𝜇
(𝑐
11
, 𝑐
12
, 𝑐
13
) = (1 − ∏
3
𝑗=1
(1 − 𝜇
𝑐
1(𝑗)
)
𝜇(𝐴
(𝑗)
)−𝜇(𝐴
(𝑗+1)
)
, ∏
3
𝑗=1
𝜈
𝑐
1(𝑗)
𝜇(𝐴
(𝑗)
)−𝜇(𝐴
(𝑗+1)
)
),
⇒ 𝑐
1
= [(1 − (1 − 0.5)
1−0.7445
× (1 − 0.6)
0.7445−0.3
× (1
− 0.7)
0.3
, 0.4
1−0.74
× 0.3
0.74−0.33
× 0.2
0.3
)]
= (1 − 0.5
0.255
× 0.4
0.4445
× 0.3
0.3
, 0.4
0.255
× 0.3
0.4445
× 0.2
0.3
) = (0.6115, 0.2964)
.
Likewise, for computing
𝑐
2
, we have
𝑐
2(1)
= (0.4, 0.3),
𝑐
2(2)
= (0.5, 0.2),
𝑐
2(3)
= (0.6, 0.2)
Proceeding as above, we obtain
𝜇(𝐴
1
) = 1,
𝜇(𝐴
2
) = 0.52,
𝜇(𝐴
3
) = 0.2.
𝑐
2
= (0.4781, 0.2430)
Likewise, for computing
𝑐
3
, we have
Современные проблемы интеллектуальных систем. Республиканская научно-практическая конференция. Джизак, 18-19 апреля 2025 г.
143
𝑐
3(1)
= (0.3, 0.5),
𝑐
3(2)
= (0.4, 0.3),
𝑐
3(3)
= (0.6, 0.3)
Thus, we get
𝜇(𝐴
1
) = 1
,
𝜇(𝐴
2
) = 0.7445
,
𝜇(𝐴
3
) = 0.3
;
𝑐
3
= (0.4474, 0.3418)
.
So,
𝑐
1
= (0.6115, 0.2964), 𝑐
2
= (0.4781, 0.2430), 𝑐
3
= (0.4474, 0.3418)
Using the score function (difference between membership and non-membership values),
ranking is done as:
𝑐
1
> 𝑐
2
> 𝑐
3
. Thus,
𝑐
1
supplier is the best.
Conclusion.
The application of intuitionistic fuzzy sets and aggregation operators
provides a powerful framework for addressing multi-criteria decision-making problems under
uncertainty. The results underscore the importance of considering both membership and non-
membership degrees in decision-making, as well as the need to account for interactions among
criteria. The proposed methodology not only enhances the precision of decisions but also
accommodates the inherent subjectivity and ambiguity in human judgments.
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РОЛЬ ИНФОРМАЦИОННЫХ ТЕХНОЛОГИЙ В СОВРЕМЕННОЙ МЕДИЦИНЕ
Эргашов Садиёрбек Акмалжон угли
Студент 1 курса направления Фармация, филиал КФУ в г. Джизаке
ergashovsadiyorbek@gmail.com
Хожаниёзов Муроджон Ражаббой угли
Студент 2 курса направления Лечебное дело, филиал КФУ в г. Джизаке
xojaniyozovm9@gmail.com
Научный руководитель:
Юсуфова Сабина Георгиевна
Ассистент преподаватель кафедры Медицинских наук филиала КФУ в г. Джизаке
Аннотация:
Современный этап развития социума характеризуется влиянием
компьютерных технологий на все сферы человеческой деятельности. За последние два
десятилетия уровень применения компьютеров в медицине значительно возрос.
Медицинская информатика - прикладная дисциплина, решающая проблемы планирования
