Application of Graded Fuzzy Preconcept Lattices in Risk Analysis
M
¯
aris Krastin¸
ˇ
s
1
, Ingr
¯
ıda Ul¸jane
1,2
and Alexander
ˇ
Sostak
1,2
1
Department of Mathematics, University of Latvia, Jelgavas iela 3, Riga, Latvia
2
Institute of Mathematics and Computer Science University of Latvia, Rain¸a bulv
¯
aris 29, Riga, Latvia
Keywords:
Risk Assessment, Risk Factors, Fuzzy Context, Graded Fuzzy Concept Lattice, Fuzzy Preconcept, Gradation
of Fuzzy Preconcept Lattices, Pandemic Scenario.
Abstract:
Fuzzy logic has a wide range of applications in the risk assessment based on expert opinion. Several methods
have been used for this purpose with fuzzy implication systems being among the most popular. While these
standard tools have proven their usefulness, we propose the application of an alternative approach. In this
paper we analyse the possibility of using the graded fuzzy preconcept lattices in introducing the risk assess-
ment model. We have also provided the necessary information on the theoretical basis of the graded fuzzy
preconcept lattices introduced by the authors earlier.
1 INTRODUCTION
The risk assessment process is a crucial part of the
risk management. It is carried out in different activi-
ties from a simple daily behaviours, like participation
in the traffic, up to the different industries, science and
managing of a large scale projects. The risk assess-
ment process often relies on statistic data, but in many
instances these data should be further combined with
expert opinions on specific and previously unknown
circumstances and their impact. There are also sce-
narios of unique and previously unknown conditions
when historic statistic data are either not available
or could be hardly applicable. Therefore the experts
should evaluate any possible developments and pro-
pose concrete actions based on their experience and
judgements. These provisions have provided a solid
basis for many studies on application of fuzzy logic in
the risk assessment, see, e.g. (Chan and Wang, 2013),
(Jones, 2009).
For the purposes of constructing the risk assess-
ment model, first we consider the risk hierarchy
with more general risks including further specific
risks which, in turn, are characterised by risk factors
(those can be considered as subclasses of the specific
risks) which are assessed by corresponding risk lev-
els. These risk levels are obtained via evaluations
which are usually based on the fuzzy inference sys-
tems, with Mamdani and Sugeno systems being the
most visible examples, see, e.g. (Lilly, 2010). In our
research we propose the risk assessment model based
on the fuzzy preconcept lattices which, in our opin-
ion, provides a good illustrative basis for modelling
of different scenarios and also allows to study certain
mathematical properties embedded in this model.
Formal concept analysis or just concept analysis
for short was developed mainly in eighties of the pre-
vious century. The principles and fundamental re-
sults of concept analysis were exposed in the paper
(Wille, 1992) and further expanded in (Ganter and
Wille, 1999). The concept analysis starts with the
notion of a (formal) context that is a triple (X,Y, R)
where X and Y are sets and R ⊆ X × Y is a relation
between the elements of these sets. The elements of
X are interpreted as some abstract objects, the ele-
ments of Y are interpreted as some abstract properties
or attributes, and the entry (x, y) ∈ R means that an ob-
ject x has attribute y. The idea of the concept analysis
is to reveal all pairs (A, B) of sets A ⊆ X and B ⊆ Y
(called concepts) such that every object x ∈ A has all
properties y ∈ B and every property y ∈ B holds for
all objects x ∈ A. In the first decade of the 21
st
cen-
tury different fuzzy counterparts of the formal con-
cept were introduced. The most important work in
the first decade of the 21
st
century in fuzzy concept
analysis was carried out by R. B
˘
elohl
´
avek, see, e.g.
(B
˘
elohl
´
avek, 1999), (B
˘
elohl
´
avek, 2004), (B
˘
elohl
´
avek
and Vychodil, 2005).
Since its inception, crisp concept analysis has
found important applications in the study of different
“real-world” problems. Starting with illustrative ex-
amples of application of crisp lattices given in (Wille,
KrastiÅ ˛Eš, M., Ul¸jane, I. and Šostak, A.
Application of Graded Fuzzy Preconcept Lattices in Risk Analysis.
DOI: 10.5220/0010656500003063
In Proceedings of the 13th International Joint Conference on Computational Intelligence (IJCCI 2021), pages 177-184
ISBN: 978-989-758-534-0; ISSN: 2184-3236
Copyright © 2023 by SCITEPRESS – Science and Technology Publications, Lda. Under CC license (CC BY-NC-ND 4.0)
177

1992), there appeared many serious works in which
concept lattices were used. Concept analysis has
found a very wide range of applications in medical
research and teaching of medical students (see, e.g.
(Hu et al., 2019), (Keller et al., 2012), (Liu et al.,
2010), (Yazdani and Hoseini, 2018)), in particlar in
the process of basic studies of nurses (see e.g. (Rod-
hers et al., 2018)). Methods of concept analysis are
used also in the research of problems related to bi-
ology (Raza, 2017), (Hashikami et al., 2013), geol-
ogy (B
˘
elohl
´
avek, 2004, p. 214), social type prob-
lems (Missaoui et al., 2017), software engineering
(Tonella, 2004) and in other applied sciences. On the
other hand, we found only a few works, where fuzzy
concept analysis is used in the research of any practi-
cal problems. In our opinion, the problem to use the
fuzzy concept lattices in case when the scale value L
is continuous (like L = [0, 1]) or lattice having many,
probably incomparable elements, is that the request
in the concept analysis of the precise correspondence
between a fuzzy set A of objects and a fuzzy set B
of attributes in ”real-world” situations is (almost) im-
practicable. In this case one sooner has to deal with
the weaker request asking that the correspondence be-
tween A and B must hold up to a certain degree. In
order to provide a theoretical basis for the research in
this situation, in (
ˇ
Sostak et al., 2021), we introduced
the notion of a graded preconcept lattice, more flex-
ible than the notion of a concept lattice and initiated
the study of the properties of graded preconcept lat-
tices. In this paper we provide a further insight in the
practical application of the theoretical models in the
assessment of risks related to spread of pandemic cri-
sis.
This paper is organized as follows. In the second
section we briefly remind the notions related to
lattices, residuated lattices or quantales, fuzzy sets
and fuzzy relations. Besides, we remind the concept
of a fuzzy inclusion of fuzzy sets that lies in the base
of gradation of fuzzy preconcepts. In the third section
we define fuzzy preconcepts, introduce partial order
relation  on the family of all fuzzy preconcepts of
a given fuzzy context (X ,Y, L, R) and show that the
resulting structure (P(X ,Y, L, R), ) is a lattice. Fur-
ther in this section we propose a method allowing to
determine the grade of the distinction of a fuzzy pre-
concept from “being a real fuzzy concept” and study
the corresponding graded preconcept lattices. In the
fourth section we explain the possible application of
fuzzy preconcept lattices in the analysis of pandemic
scenario and proceed with practical calculations in
modelling the spread of Covid-19 pandemics. In the
last, conclusion section, we briefly summarize main
results obtained and survey some directions for
the future work.
2 BACKGROUND
2.1 Lattices, Quantales and Residuated
Lattices
We use the standard terminology accepted in the-
ory of lattices, see, e.g. (Birkhoff, 1995), (Gierz et
al., 2003), (Morgan and Dilworth, 1939) for details.
For the reader convenience, we make clarification of
some, possibly less known terms.
A complete lattice L = (L, ≤, ∧, ∨) is called in-
finitely distributive if a ∧ (
W
i∈I
b
i
) =
W
i∈I
(a ∧ b
i
) for
every a ∈ L and every {b
i
| i ∈ I} ⊆ L. A com-
plete lattice L is called infinitely co-distributive if
a ∨ (
V
i∈I
b
i
) =
V
i∈I
(a ∨ b
i
) for every a ∈ L and every
{b
i
| i ∈ I} ⊆ L. A complete lattice is called infinitely
bi-distributive if it is infinitely and co-infinitely dis-
tributive. It is known, that every completely distribu-
tive lattice is infinitely bi-distributive.
Let L be a complete lattice and ∗ : L × L → L be a
binary associative commutative monotone operation.
Then the tuple (L, ≤, ∧, ∨, ∗) is called a (commuta-
tive) quantale (Rosenthal, 1990) if ∗ distributes over
arbitrary joins: a ∗ (
W
i∈I
b
i
) =
W
i∈I
(a ∗ b
i
).
A quantale is called integral if the top element 1
acts as the unit, that is 1∗a = a∗1 = a for every a ∈ L.
In what follows saying quantale we mean a com-
mutative integral quantale. A typical example of a
quantale is the unit interval endowed with a lower
semi-continuous t-norm, see, e.g. (Klement et al.,
2000).
In a quantale a further binary operation 7→: L ×
L → L, the residuum, can be introduced as associ-
ated with operation ∗ of the quantale (L, ≤, ∧, ∨, ∗)
via the Galois connection, that is a ∗ b ≤ c ⇐⇒ a ≤
b 7→ c for all a, b, c ∈ L. A quantale (L, ≤, ∧, ∨, ∗)
provided with the derived operation 7→, that is the tu-
ple (L, ≤, ∧, ∨, ∗, 7→), is known also as a (complete)
residuated lattice (Morgan and Dilworth, 1939).
2.2 Fuzzy Sets and Fuzzy Relations
Given a set X, its L-fuzzy subset is a mapping A : X →
L. The lattice and the quantale structure of L is ex-
tended point-wise to the L-exponent of X, that is to
the set L
X
of all L-fuzzy subsets of X. Specifically,
the union and intersection of a family of L-fuzzy sets
{A
i
|i ∈ I} ⊆ L are defined by their join
W
i∈I
A
i
and
FCTA 2021 - 13th International Conference on Fuzzy Computation Theory and Applications
178

meet
V
i∈I
A
i
respectively. An L-fuzzy relation be-
tween two sets X and Y is an L-fuzzy subset of the
product X ×Y , that is a mapping R : X ×Y → L, see,
e.g. (Valverde, 1985), (Zadeh, 1971).
2.3 Measure of Inclusion of L-fuzzy Sets
The gradation of a preconcept lattice presented below
is based on the fuzzy inclusion between fuzzy sets.
We present here a brief introduction into this field.
In order to fuzzify the inclusion relation A ⊆ B “a
fuzzy set A is a subset of a fuzzy set B”, we have to
interpret it as a certain fuzzy relation → based on ”if
- then” rule, that is on some implication ⇒ defined
on the lattice L. In the result we come to the formula
A → B = inf
x∈X
(A(x) ⇒ B(x)).
In this paper we use implication ⇒ defined by
means of the residuum: a ⇒ b =
de f
a 7→ b.
Definition 2.1. By setting A → B =
V
x∈X
(A(x) 7→
B(x)) for all A, B ∈ L
X
, we obtain a mapping →:
L
X
× L
X
→ L. We call A → B by the (L-valued) mea-
sure of inclusion of the L-fuzzy set A into the L-fuzzy
set B. We denote A
∼
=
B =
de f
(A → B)∧ (B → A) and
view it as the degree of equality of L-fuzzy sets A and
B.
Proposition 2.2. (see, e.g. (Han and
ˇ
Sostak, 2016),
(Han and
ˇ
Sostak, 2018)) Mapping →: L
X
× L
X
→ L
satisfies the following properties:
(1) (
W
i
A
i
) → B =
V
i
(A
i
→ B) for all {A
i
| i ∈ I} ⊆
L
X
and for all B ∈ L
X
;
(2) A → (
V
i
B
i
) =
V
i
(A → B
i
) for all A ∈ L
X
and for
all {B
i
| i ∈ I} ⊆ L
X
;
(3) A → B = 1
L
whenever A ≤ B;
(4) 1
X
→ A =
V
x
A(x) for all A ∈ L
X
where 1
X
: X → L
is a constant function with the value 1
L
∈ L;
(5) (A → B) ≤ (A ∗C → B ∗C) for all A, B,C ∈ L
X
;
(6) (A → B) ∗ (B → C) ≤ (A → C) for all A, B,C ∈
L
X
;
(7) (
V
i
A
i
) → (
V
i
B
i
) ≥
V
i
(A
i
→ B
i
) for all {A
i
: i ∈
I}, {B
i
: i ∈ I} ⊆ L
X
;
(8) (
W
i
A
i
) → (
W
i
B
i
) ≥
V
i
(A
i
→ B
i
) for all {A
i
: i ∈
I}, {B
i
: i ∈ I} ⊆ L
X
.
3 GRADED PRECONCEPT
LATTICES
3.1 Preconcepts and Preconcept
Lattices
Let L be a complete lattice. Further, let X,Y be sets
and R : X × Y → L be a fuzzy relation. Following
terminology accepted in the theory of (fuzzy) con-
cept lattices, see, e.g. (Wille, 1992), (B
˘
elohl
´
avek,
1999), (B
˘
elohl
´
avek, 2004), (B
˘
elohl
´
avek and Vy-
chodil, 2005), we refer to the tuple (X,Y, L, R) as a
fuzzy context.
Definition 3.1. Given a fuzzy context (X,Y, L, R), a
pair P = (A, B) ∈ L
X
×L
Y
is called a fuzzy preconcept.
On the set L
X
× L
Y
of all fuzzy preconcepts we
introduce a partial order  as follows. Given P
1
=
(A
1
, B
1
) and P
2
= (A
2
, B
2
), we set P
1
 P
2
if and
only if A
1
≤ A
2
and B
1
≥ B
2
. Let (P, ) be the
set L
X
× L
Y
endowed with this partial order. Fur-
ther, given a family of fuzzy concepts {P
i
= (A
i
, B
i
) :
i ∈ I} ⊆ L
X
× L
Y
, we define its join (supremum) by
Y
i∈I
P
i
= (
W
i∈I
A
i
,
V
i∈I
B
i
) and its meet (infimum) as
Z
i∈I
P
i
= (
V
i∈I
A
i
,
W
i∈I
B
i
).
Theorem 3.2. P is a complete lattice. Besides, if L is
a infinitely bi-distributive lattice, then (P, , Z, Y) is
also a infinitely bi-distributive lattice.
In the sequel we write just P or (P, ) instead
of (P, , Z, Y) if no misunderstanding is possible, or
(P, X,Y, L, R) in case when we need to specify the
fuzzy context we are working in.
3.2 Operators R
↑
and R
↓
on L-powersets
and Fuzzy Concept Lattices
Let X and Y be sets and let R : X ×Y → L be a fuzzy
relation, where L is a fixed quantale. Given a fuzzy
context (X,Y, L, R), we define operators R
↑
: L
X
→ L
Y
and R
↓
: L
Y
→ L
X
as follows:
Definition 3.3. (see, e.g. (B
˘
elohl
´
avek and Vychodil,
2005)) Given A ∈ L
X
and B ∈ L
Y
, we define A
↑
∈ L
Y
and B
↓
∈ L
X
by setting
A
↑
(y) =
V
x∈X
(A(x) 7→ R(x, y)) ∀y ∈ Y ,
B
↓
(x) =
V
y∈Y
(B(y) 7→ R(x, y)) ∀x ∈ X.
By changing A over L
X
and B over L
Y
, we get opera-
tors R
↑
: L
X
→ L
Y
and R
↓
: L
Y
→ L
X
.
In the crisp case, that is when A ⊆ X, B ⊆ Y and
R : X ×Y → {0, 1}, this definition is obviously equiv-
alent to the original definition of operators A −→ A
0
and B −→ B
0
in (Wille, 1992). Informally these sets
can be described as follows. Let a set X of objects
with a subset A ⊆ X and a set Y of properties with a
subset B ⊆ Y be given. Further, let R ⊆ X ×Y be a re-
lation where (x, y) ∈ R means “object x has property
y”. Now A
↑
is the set of such properties y ∈ Y which
have all objects x ∈ A while B
↓
is the set of such ob-
jects x ∈ X which have all properties y ∈ B. From the
properties of the residuum one can easily justify the
following:
Application of Graded Fuzzy Preconcept Lattices in Risk Analysis
179

Proposition 3.4. Operators R
↑
: L
X
→ L
Y
and R
↓
:
L
Y
→ L
X
are decreasing: A
1
≤ A
2
⇒ A
↑
1
≥ A
↑
2
, B
1
≤
B
2
⇒ B
↓
1
≥ B
↓
2
.
In the sequel we write A
↑↓
instead of (A
↑
)
↓
and B
↓↑
instead of (B
↓
)
↑
. We write also R
↑↓
for the composi-
tion R
↓
◦ R
↑
: L
X
→ L
X
and R
↓↑
for the composition
R
↑
◦ R
↓
: L
Y
→ L
Y
.
Proposition 3.5. (cf, e.g. (Wille, 1992) in crisp case,
(B
˘
elohl
´
avek, 2004)) A
↑↓
≥ A for every A ∈ L
X
and
B
↓↑
≥ B for every B ∈ L
Y
.
Proposition 3.6. (cf, e.g. (Wille, 1992) in crisp case,
(B
˘
elohl
´
avek, 2004)) A
↑
= A
↑↓↑
for every A ∈ L
X
and
B
↓
= B
↓↑↓
for every B ∈ L
Y
.
Example 3.7. Let a fuzzy context (X,Y, L, R) be
given and let A ⊆ X.
1
Then for every y ∈ Y A
↑
(y) =
V
x∈X
A(x) 7→ R(x, y) =
V
x∈A
R(x, y). In the same way
we prove that if B ⊆ Y , then B
↓
(x) =
V
y∈B
R(x, y).
Hence, even in case when A ⊆ X, B ⊆ Y a pair (A, B)
can be a concept (either crisp or fuzzy) only in case
when R is also crisp, that is R : X ×Y → {0, 1}. This
shows the limitation for the use of concept lattices in
the case of a fuzzy context and gives an additional
evidence in favour of the graded approach to fuzzy
preconcept lattices.
Continuing the previous example we calculate A
↑↓
and B
↓↑
in case of crisp sets A and B:
A
↑↓
(x) =
V
y∈Y
(
V
x
0
∈A
(R(x
0
, y) 7→ R(x, y))) ,
B
↓↑
(y) =
V
x∈X

V
y
0
∈B
(R(x, y
0
) 7→ R(x, y))

. 2
Proposition 3.8. (cf, e.g. (Wille, 1992) for the crisp
case, (B
˘
elohl
´
avek, 2004)) Given a family {A
i
| i ∈
I} ⊆ L
X
, we have (
W
i∈I
A
i
)
↑
=
V
i∈I
A
↑
i
. Given a family
{B
i
| i ∈ I} ⊆ L
Y
, we have (
W
i∈I
B
i
)
↓
=
V
i∈I
B
↓
i
.
3.3 Concepts and Concept Lattices
Referring to the definition of a (fuzzy) concept given
in (Wille, 1992), (B
˘
elohl
´
avek, 2004) we reformulate
it as follows:
Definition 3.9. A fuzzy preconcept (A, B) is called a
(formal) fuzzy concept if A
↑
= B and B
↓
= A.
Let C = C(X,Y, L, R) be the subset of P =
P(X,Y, L, R) consisting of fuzzy concepts (A, B) and
let  be the partial order on C induced by the par-
tial order  from the lattice (P, ). Then (C, ) is
a partially ordered subset of the lattice (P, ). How-
ever generally (C, ) is not a sublattice of the lattice
(P, , Z, Y) and we need to define joins and meets in
1
Here and in the sequel we do not distinguish between
a crisp set A ⊆ X and its characteristic function χ
A
: X →
{0, 1}.
(C, ) differently from Z and Y. To do this first we
show the following simple lemma:
Lemma 3.10. Let (A
1
, B
1
), (A
2
, B
2
) be fuzzy con-
cepts. If A
1
≤ A
2
, then B
1
≥ B
2
and if B
1
≥ B
2
then
A
1
≤ A
2
.
Corollary 3.11. Let (A
1
, B
1
), (A
2
, B
2
) ∈ C. Then
(A
1
, B
1
)  (A
2
, B
2
) if and only if A
1
≤ A
2
if and only
if B
1
≥ B
2
.
Proposition 3.12. If A ∈ L
X
, then (A
↑↓
, A
↑
) is the
smallest concept containing A as the fuzzy set of ob-
jects. If B ∈ L
Y
, then (B
↓
, B
↓↑
) is the smallest concept
containing B as the fuzzy set of attributes.
Theorem 3.13. Let (X,Y, L, R) be a fuzzy context and
let  be the partial order induced from the lattice
P(X,Y, L, R, ). Then C(X,Y, L, R ) is a complete
lattice.
Taking into account that in a fuzzy concept (A
i
, B
i
)
it holds A
↑
i
= B
i
and B
↓
i
= A
i
, we get the following
corollary:
Corollary 3.14. Let C = {C
i
= (A
i
, B
i
) | i ∈ I} ⊆ C
be a family of fuzzy concepts. Then
f
i∈I
C
i
=

V
i∈I
A
i
, (
V
i∈I
A
i
)
↑

is its infimum in the
lattice (C, ),
g
i∈I
C
i
= ((
V
i∈I
B
i
)
↓
,
V
i∈I
B
i
) is its supremum in the
lattice (C, ).
3.4 Conceptuality Degree of a Fuzzy
Preconcept
Let (X,Y, L, R) be a fuzzy context and (A, B) ∈
P(X,Y, L, R).
Definition 3.15. The degree of contentment of the
fuzzy set A of objects by the fuzzy set B of attributes or
the degree object based contentment of the preconcept
(A, B) for short is defined by D
↑
(A, B) =
de f
A
↑
∼
=
B.
Definition 3.16. The degree of contentment of the
fuzzy set B of of attributes by the fuzzy set A of
objects or the attribute based contentment of the
preconcept (A, B) is defined by D
↓
(A, B) =
de f
A
∼
=
B
↓
.
Definition 3.17. The degree of conceptuality of the
preconcept (A, B) in the preconcept lattice P is de-
fined by D(A, B) = D
↑
(A, B) ∧ D
↓
(A, B).
Changing pairs (A, B) ∈ P, we obtain mappings
D
↑
: P → L, D
↓
: P → L and D : P → L.
Referring to our previous paper “Graded Fuzzy
Preconcept Lattices: Theoretical Basis”, we illus-
trate the evaluation of conceptuality degree in the
fuzzy context (X,Y, L, R) in case when A and B are
crisp sets. Besides, to make exposition as simple as
FCTA 2021 - 13th International Conference on Fuzzy Computation Theory and Applications
180

possible, we assume that the relation R : X × Y → L
satisfies the following two conditions:
(†
R
BA
)
W
y∈B
c
(
V
x∈A
(R(x, y)) = 0;
(†
R
AB
)
W
y∈A
c
(
V
x∈B
(R(x, y)) = 0.
Note that if B = Y then (†
R
BA
) is satisfied and if
A = X then (†
R
AB
) is satisfied.
Example 3.18. Let A ⊆ X , B ⊆ Y, let (L, ≤, ∧, ∨, ∗)
be an arbitrary quantale, 7→: L × L → L its residuum,
and R : X ×Y → L a fuzzy relation. Then
A
↑
→ B = 1;
B → A
↑
=
V
y∈B
V
x∈A
R(x, y);
D
↑
(A, B) =
V
x∈A,y∈B
R(x, y).
A → B
↓
=
V
x∈A,y∈B
R(x, y);
B
↓
→ A = 1;
D
↓
(A, B) =
V
x∈A,y∈B
R(x, y)
and hence D(A, B) =
V
x∈A,y∈B
R(x, y).
Example 3.19. Let now B ⊆ Y , a ∈ (0, 1) and fuzzy
set A : X → L = [0, 1] be defined by A(x) = a ∀x ∈ X.
Then B → A
↑
=
V
y∈B,x∈X
(a 7→ R(x, y)); A
↑
→ B = 1;
hence D
↑
(A, B) =
V
y∈B,x∈X
(a 7→ R(x, y)).
A → B
↓
=
V
x∈X

V
y∈B
R(x, y) 7→ a

; B
↓
→ A = 1
and hence D
↓
(A, B) =
V
x∈X

V
y∈B
R(x, y) 7→ a

and
D(A, B) =

V
y∈B,x∈X
(a 7→ R(x, y))

∧

V
x∈X

V
y∈B
R(x, y 7→ a)

.
We demonstrate the formulas obtained in the
previous example for calculating D(A, B) for the
three basic t-norms ∗ on [0, 1]: ∗
∧
= ∧ - the minimum
t-norm, ∗
L
- the Łukasiewicz t-norm and ∗
P
- the
product t-norm, see, e.g. (Klement et al., 2000).
Besides, we restrict to the case when X
a
= X and
B = Y .
(1) In the case of Łukasiewicz t-norm
D
↑
(A, B) =

V
x∈X
(1 − a +
V
y∈Y
R(x, y))

∧ 1;
D
↓
(A, B) =
V
x∈X

1 −
V
y∈Y
R(x, y) + a

∧ 1 and
hence
D(A, B) =
^
x∈X,y∈Y
(1 − |a − R(x, y)|).
(2) In the case of product t-norm for describing
D(A, B) we denote
X
1
= {x ∈ X | a ≤
V
y∈B
R(x, y)},
X
2
= {x ∈ X | a ≥
V
y∈B
R(x, y)}. Then D(A, B) =

V
x∈X
1
a
V
y∈B
R(x,y)

∧

V
x∈X
2
V
y∈B
R(x,y)
a

if X
1
∪ X
2
6=
/
0 and D(A, B) = 1 otherwise.
(3) In the case of minimum t-norm applying
the notations from the previous example we have
D(A, B) =

V
x∈X
2
,y∈Y
R(x, y) if X
2
6=
/
0;
a otherwise .
2
3.5 D-graded Preconcept Lattices
Given a fuzzy context (X,Y, L, R), let (P(X,Y, L, R), 
) be the corresponding fuzzy preconcept lattice, and
let D
↑
, D
↓
be the operators defined in the previous
subsection. Then the tuple (P, , D
↑
, D
↓
) will be re-
ferred to as a D -graded fuzzy preconcept lattice of the
fuzzy preconcept (X ,Y, L, R). In the next theorems we
provide the most important properties of the opera-
tors of D-gradation and D-graded fuzzy preconcept
lattices.
Theorem 3.20. Let P = (P,  Y, Z) be a fuzzy pre-
concept lattice. Given a family of fuzzy preconcepts
P = {P
i
= (A
i
, B
i
) | i ∈ I} ⊆ P it holds
D
↑
(Y
i∈I
P
i
) ≥
^
i∈I
D
↑
(P
i
)
Theorem 3.21. Given a family of fuzzy preconcepts
P = {P
i
= (A
i
, B
i
) | i ∈ I} ⊆ P it holds
D
↓
(Z
i∈I
P
i
) ≥
^
i∈I
D
↓
(P
i
).
4 RISK ASSESSMENT AND
FUZZY PRECONCEPT
LATTICES
4.1 Risk Assessment Model
The risk assessment process often allows a combi-
nation of statistical data and qualitative evaluations
based on expert opinions. When dealing with any
risks of rare occurrence the whole society usually
faces unique circumstances with no historic patterns.
Such circumstances are often called the tail events
in the terminology of the risk management. These
events are characterised by low possibility, but huge
impact. Pandemic scenario is among the most obvi-
ous examples. Therefore the current spread of Covid-
19 pandemic provides a unique opportunity in study-
ing possible pandemic developments from different
angles. While a lot of statistical data have been avail-
able for almost a year, we should still combine the
statistics with expert based opinions on possible fur-
ther developments related to sudden virus outbreaks
and mutations in order to propose the actions for, e.g.
national governments or other decision making in-
stitutions. In its essence, the whole Covid-19 crisis
management is dependent on fuzzy logic based de-
cisions and provides a favourable preconditions for
enhancing the fuzzy logic methods. Several medical
studies based on fuzzy logic applications have already
been carried out, see, e.g. (Orozco-del-Castillo et al.,
Application of Graded Fuzzy Preconcept Lattices in Risk Analysis
181

2020), (Shaban et al., 2021). From a different per-
spective, it is always important to introduce the mod-
els for taking different decisions under these circum-
stances, such as modelling of availability of medical
services, limiting or restricting public activities and
services etc. We propose the analysis of data contain-
ing estimated total maximum number of infected in-
habitants vs estimated total maximum number of hos-
pitalised inhabitants at any fixed time moment as ex-
plained in the Table 1. It should be admitted that the
estimated total number of infected inhabitants can be
replaced with any other parameter characterizing the
spread of pandemics, e.g. estimated cumulative cases,
while the estimated total maximum number of hos-
pitalised inhabitants at any fixed time moment is the
most important parameter in the model characteris-
ing the crisis severity. In many instances such model
could be expanded further to the analysis of mortal-
ity rate, but this stage illustrates the efficiency of the
healthcare system and its ability to save the lives of in-
habitants. In the example we also consider the values
of total number of inhabitants of particular country
and the maximum admissible number of hospitalised
inhabitants, but these values are used just for the il-
lustrative reference purposes as the benchmark values
for assessment of severity of the pandemics. We intro-
duce the model containing the following objects and
attributes: set X contains estimated quantities of in-
fected inhabitants and set Y contains estimated quan-
tities of hospitalised inhabitants. In this context we
treat X as a set of objects expressed in a vector for-
mat x = (x
1
, x
2
, . . . , x
m
) and Y as a set of attributes
expressed in a vector format y = (y
1
, y
2
, . . . , y
n
). Fur-
thermore, we introduce a fuzzy relation R(x
i
, y
j
) = α
i j
which is a matrix containing corresponding expert
opinion:
R(x
i
, y
j
) =




α
11
α
12
··· α
1n
α
21
α
22
··· α
2n
.
.
.
.
.
.
.
.
.
.
.
.
α
m1
α
m2
··· α
mn




This model can be applied for various purposes,
including illustration of expert assessments, compar-
ing of these assessments against real statistics, ad-
justing these assessments depending on new circum-
stances and taking decisions on implementation of
public restrictions. For the sake of clarity, it should
be noted that the proposed setup can be used for de-
veloping the model for any risk assessment, but we
have chosen the tail events and Covid-19 crisis in par-
ticular for highlighting the importance of expert opin-
ion. In the following section we show how the fuzzy
preconcept lattice can be applied for estimation of the
possible impact of Covid-19 crisis on the healthcare
system in Latvia.
Table 1: The data for modelling of pandemic scenario. The
following data are provided in the respective columns: 1 -
Total Number of Country Inhabitants w, 2 - Estimated Total
Maximum Number of Infected Inhabitants x
i
, 3 - Estimated
Total Maximum Number of Hospitalised Inhabitants y
j
, 4 -
Maximal Admissible Number of Hospitalised Inhabitants z.
1 2 3 4
x
1
y
1
x
2
y
2
w
.
.
.
.
.
. z
x
m
y
n
Table 2: The data for modelling of pandemic scenario in
Latvia. The following data are provided in the respective
columns: 1 - Total Number of Inhabitants, 2 - Estimated
Total Maximum Number of Infected Inhabitants, 3 - Esti-
mated Total Maximum Number of Hospitalised Inhabitants,
4 - Maximal Admissible Number of Hospitalised Inhabi-
tants.
1 2 3 4
50,000
80,000 500
1,920,000 110,000 1,000 3,000
140,000 1,500
170,000
4.2 Assessment of Possible Covid-19
Impact on the Healthcare System in
Latvia
The management of Covid-19 crisis has created al-
most similar challenges in all countries around the
world. Different public restrictions have been imple-
mented and lifted based on actual rates of infected in-
habitants, risks of new Covid-19 mutations and vac-
cination rates. In our opinion, the management of the
public restrictions can be based on the proposed risk
assessment model. Therefore the corresponding rates
for Latvia provided in the Table 2 can be replaced
with corresponding numbers for any country.
In the example we link the estimated total maxi-
mum number of infected inhabitants x with estimated
total maximum number of hospitalised inhabitants y
by applying the following fuzzy relation reflecting hy-
pothetical expert opinion
2
:
R(x, y) =





0.9 0.9 0.4
0.9 0.8 0.4
0.6 0.5 0.3
0.5 0.4 0.3
0.4 0.3 0.2





2
The values closer to 0 indicate lower possibility while
the values closer to 1 indicate higher possibility.
FCTA 2021 - 13th International Conference on Fuzzy Computation Theory and Applications
182

This matrix is further analysed in such way that
we consider the values of R(x, y) for each particular
row. First of all we analyse scenario when the val-
ues of x are considered as precise and calculate the
values of D(A, B) as provided in the Example 2.21.
In such case a = 1 and we obtain the following val-
ues for corresponding rows for the Łukasiewicz t-
norm, the product t-norm and the minimum t-norm
(all results are equal for the corresponding t-norms in
this case, indices of D(A, B) denote the correspond-
ing rows for which the corresponding values have
been calculated): D
1
(A, B) = 0.4, D
2
(A, B) = 0.4,
D
3
(A, B) = 0.3, D
4
(A, B) = 0.3, D
5
(A, B) = 0.2.
Furthermore we can assume that the values of x
are not precise. In such case we fuzzify these val-
ues by applying different values of a. The follow-
ing example shows the values of D (A, B) calculated
for a = 0.8. For the Łukasiewicz t-norm we obtain
the following values: D
1
(A, B) = 0.6, D
2
(A, B) = 0.6,
D
3
(A, B) = 0.5, D
4
(A, B) = 0.5, D
5
(A, B) = 0.4. For
the product t-norm we obtain the following values:
D
1
(A, B) = 0.5, D
2
(A, B) = 0.5, D
3
(A, B) = 0.375,
D
4
(A, B) = 0.375, D
5
(A, B) = 0.25. For the minimum
t-norm we obtain the following values: D
1
(A, B) =
0.4, D
2
(A, B) = 0.4, D
3
(A, B) = 0.3, D
4
(A, B) =
0.3, D
5
(A, B) = 0.2. Another example shows the
values of D(A, B) calculated for a = 0.4. Conse-
quently, for the Łukasiewicz t-norm we obtain the
following values: D
1
(A, B) = 0.5, D
2
(A, B) = 0.5,
D
3
(A, B) = 0.8, D
4
(A, B) = 0.9, D
5
(A, B) = 0.8. For
the product t-norm we obtain the following values:
D
1
(A, B) = 0.44, D
2
(A, B) = 0.44, D
3
(A, B) = 0.67,
D
4
(A, B) = 0.75, D
5
(A, B) = 0.5. For the minimum
t-norm we obtain the following values: D
1
(A, B) =
0.4, D
2
(A, B) = 0.4, D
3
(A, B) = 0.3, D
4
(A, B) = 0.3,
D
5
(A, B) = 0.2.
These results reflect differences in the impact
of the t-norms used in the above calculations. On
the practical side we can foresee that application of
Łukasiewicz t-norm returns higher values indicating
possibility of more severe pandemic impact. At the
same time we should admit that these results should
not be interpreted as correct or incorrect, but just as il-
lustration of different expert opinions and application
of different t-norms in their analysis.
5 CONCLUSIONS
The research has been focused on the application of
an alternative method in risk assessment that is based
on application of graded fuzzy preconcept lattices.
Graded fuzzy preconcept lattices were introduced and
studied in the paper (
ˇ
Sostak et al., 2021), their basic
properties are expounded also in our earlier papers.
The use of graded preconcept lattices could be more
appropriate for practical problems, in particular, those
ones that are related to risk assessment, than fuzzy
concept lattices due to the possibility to establish a
more flexible relationship between the fuzzy set of ob-
jects and corresponding fuzzy set of attributes. In this
paper, graded fuzzy preconcept lattices are applied for
the analysis of pandemic spread by introducing the
model of estimated total number of infected inhab-
itants vs estimated levels of hospitalised inhabitants
which have been assessed based on expert opinion.
The application of different t-norms in the calculation
of the degree of conceptuality revealed differences in
the impact of the selected t-norms on the assessment
of the crisis severity. The obtained results can be fur-
ther applied in the implementation of public restric-
tions aimed at containment of Covid-19 pandemics.
The results of our research can be extended to com-
paring opinions from different experts. The aggrega-
tion of multiple opinions analysed by means of fuzzy
preconcept lattices is subject to our further research.
ACKNOWLEDGEMENTS
The second and the third named authors are thankful
for the partial financial support from the project No.
Lzp-2020/2-0311 by the Latvian Council of Science.
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