Fuzzy Expert System of the Decision Making Support on Foreign Direct
Investment
Eugene E. Fedorov
1 a
, Liubov O. Kibalnyk
2 b
, Lesya O. Petkova
1 c
,
Maryna M. Leshchenko
1 d
and Vladyslav M. Pasenko
1 e
1
Cherkasy State Technological University, 460 Shevchenko Blvd., Cherkasy, 18006, Ukraine
2
The Bohdan Khmelnytsky National University of Cherkasy, 81 Shevchenko Blvd., Cherkasy, 18031, Ukraine
Keywords:
Fuzzy Expert Decision Support System, Foreign Direct Investment, Swarm Metaheuristics, Optimization
Methods, Operator.
Abstract:
The fuzzy expert decision support system for foreign direct investment was developed in the research. A
quality criterion was chosen for the proposed fuzzy expert system, which considers the created fuzzy expert
system’s specifics and allows assessing the probability of future decisions. A metaheuristic method was cre-
ated based on an adaptive gravitational search algorithm to determine the parameters of the proposed fuzzy
expert system. A numerical study was carried out; the parameters of membership functions for linguistic
input variables were determined; the parameters of the membership functions for the values of the linguistic
output variable were determined. The proposed optimization method based on swarm metaheuristics and a
fuzzy expert system make it possible to intellectualize the technology of making decisions on foreign direct
investment.
1 INTRODUCTION
The decision-making systems for foreign direct in-
vestment are very popular nowadays. The regres-
sion (Milovanovi
´
c and Markovi
´
c, 2022) and auto-
regressive (Kurecic and Kokotovic, 2017) methods
are usually used to create decision-making systems
for foreign direct investment based on machine learn-
ing. The construction of only linear models is the
disadvantage of such methods. The knowledge base
(most often in the form of production rules) and
an inference mechanism are used to create decision-
making systems for foreign direct investment based
on expert systems (
ˇ
Samanovi
´
c et al., 2010). The dis-
advantages of such systems include the fact that they
operate only with quantitative estimates, while it is
easier for the operator to work with qualitative esti-
mates.
The fuzzy expert systems are currently used to
simplify the interaction between a human and a com-
a
https://orcid.org/0000-0003-3841-7373
b
https://orcid.org/0000-0001-7659-5627
c
https://orcid.org/0000-0003-4519-3726
d
https://orcid.org/0000-0002-0210-9582
e
https://orcid.org/0000-0002-7411-2625
puter system. These expert systems usually use the
Larsen, Mamdani, Tsukamoto, and Sugeno fuzzy
inference mechanisms (Ruan, 1997; Tsoukalas and
Uhrig, 1997).
The disadvantages of such systems include the
fact that the procedure for determining their param-
eters is not automated (Abe, 1997; Rotshtein et al.,
2001). The optimization methods are currently ac-
tively used to determine the parameters of fuzzy ex-
pert systems.
Modern optimization methods suffer from one or
more of the following disadvantages:
• have high computational complexity;
• fall into a local extremum with a high probability;
• do not guarantee convergence.
In this regard, there is an actual problem of opti-
mization methods’ insufficient efficiency.
Metaheuristics (modern heuristics) are used to
speed up finding a quasi-optimal solution to optimiza-
tion problems and reduce the probability of hitting a
local extremum (Talbi, 2009; Engelbrecht, 2007; Yu
and Gen, 2010; Nakib and Talbi, 2017; Yang, 2018a;
Subbotin et al., 2016). Metaheuristics expand the pos-
sibilities of heuristics by combining heuristic methods
based on a high-level strategy (Blum and Raidl, 2016;
Fedorov, E., Kibalnyk, L., Petkova, L., Leshchenko, M. and Pasenko, V.
Fuzzy Expert System of the Decision Making Support on Foreign Direct Investment.
DOI: 10.5220/0011930700003432
In Proceedings of 10th International Conference on Monitoring, Modeling Management of Emergent Economy (M3E2 2022), pages 15-22
ISBN: 978-989-758-640-8; ISSN: 2975-9234
Copyright
c
2023 by SCITEPRESS – Science and Technology Publications, Lda. Under CC license (CC BY-NC-ND 4.0)
15
Glover and Kochenberger, 2003; Yang, 2018b; Mart
´
ı
et al., 2018; Gendreau and Potvin, 2019).
Modern metaheuristics suffer from one or more of
the following disadvantages:
• insufficient method accuracy (Alba et al., 2013);
• there is only an abstract description of the method
or the description of the method is focused on
solving only a specific problem (Doerner et al.,
2007);
• the procedure for determining parameter values is
not automated (Grygor et al., 2019);
• the influence of the iteration number on the so-
lution search process is not taken into account
(Bozorg-Haddad, 2017);
• there is no possibility to solve problems of condi-
tional optimization (Fedorov et al., 2019);
• there is no possibility to use non-binary potential
solutions (Radosavljevi
´
c, 2018);
• the method convergence is not guaranteed
(Chopard and Tomassini, 2018).
In this regard, the problem of constructing effi-
cient metaheuristic optimization methods arises (Du
and Swamy, 2016; Brownlee, 2011).
One of the popular metaheuristics is the gravita-
tional search algorithm (Rashedi et al., 2009), which
belongs to swarm metaheuristics.
The task of building fuzzy expert systems that use
the method of parametric identification for adaptation
and tuning is actual for our research.
The goal of this research is to improve the effi-
ciency of decisions on foreign direct investment by
creating a fuzzy expert system trained based on meta-
heuristics.
The following tasks were set and solved:
1) to develop a fuzzy expert decision support system
for foreign direct investment;
2) to select a quality criterion for the proposed fuzzy
expert system;
3) to create a metaheuristic method based on an
adaptive gravitational search algorithm to deter-
mine the proposed fuzzy expert system parame-
ters;
4) to conduct numerical research.
2 THE FUZZY EXPERT
DECISION SUPPORT SYSTEM
FOR FOREIGN DIRECT
INVESTMENT
The foreign direct investment analysis is based on
the data of the GDP per capita volume, inflation rates,
goods and services exports volume, and labor force
indicators. To make decisions on foreign direct in-
vestment, a fuzzy expert system is proposed. It in-
volves the following steps:
1) linguistic variables formation;
2) fuzzy knowledge base formation;
3) Mamdani fuzzy inference mechanism formation:
• fuzzification;
• sub-conditions aggregation;
• conclusions activation;
• aggregation of conclusions;
• defuzzification.
4) identification of parameters based on metaheuris-
tics.
2.1 Linguistic Variables Formation
The following input variables were chosen:
• the volume of gross domestic product (GDP) per
capita (per year, US dollars), x
1
;
• the inflation indicator (according to the consumer
price index, which reflects the annual percentage
change in the cost for the average consumer of
purchasing a goods and services basket, per year,
%), x
2
;
• the volume of goods and services export indicator
(total volume, per year, USD), x
3
;
• the labor force indicator (labor force is people
aged 15 and over who provide labor for the pro-
duction of goods and services, per year, number
of people), x
4
.
The following indicators were chosen as linguistic
input variables. They are qualitative indicators:
• the GDP volume ˜x
1
with values
e
α
11
= little,
e
α
12
= medium,
e
α
13
= much, where the
ranges are fuzzy sets
˜
A
11
= {(x
1
,µ
˜
A
11
(x
1
))},
˜
A
12
= {(x
1
,µ
˜
A
12
(x
1
))},
˜
A
13
= {(x
1
,µ
˜
A
13
(x
1
))};
• the inflation indicator ˜x
2
with values
e
α
21
=
little,
e
α
22
= medium,
e
α
23
= much, where the
ranges are fuzzy sets
˜
A
21
= {(x
2
,µ
˜
A
21
(x
2
))},
˜
A
22
= {(x
2
,µ
˜
A
22
(x
2
))},
˜
A
23
= {(x
2
,µ
˜
A
23
(x
2
))};
M3E2 2022 - International Conference on Monitoring, Modeling Management of Emergent Economy
16
• the volume of goods and services export indica-
tor ˜x
3
with values
e
α
31
= little,
e
α
32
= medium,
e
α
33
= much, where the ranges are fuzzy sets
˜
A
31
= {(x
3
,µ
˜
A
31
(x
3
))},
˜
A
32
= {(x
3
,µ
˜
A
32
(x
3
))},
˜
A
33
= {(x
3
,µ
˜
A
33
(x
3
))};
• the labor force indicator ˜x
4
with values
e
α
41
= little,
e
α
42
= medium,
e
α
43
= much, where
the ranges are fuzzy sets
˜
A
41
= {(x
4
,µ
˜
A
41
(x
4
))},
˜
A
42
= {(x
4
,µ
˜
A
42
(x
4
))},
˜
A
43
= {(x
4
,µ
˜
A
43
(x
4
))}.
The volume of foreign direct investment (net flows
for the year, USD) was chosen as a clear output vari-
able ˜y. It is a qualitative indicator.
The volume of foreign direct investment was
chosen ˜y with its values
e
β
1
= little,
e
β
2
=
medium,
e
β
3
= much, where the ranges are fuzzy
sets
˜
B
1
= {(y,µ
˜
B
1
(y))},
˜
B
2
= {(y,µ
˜
B
42
(y))},
˜
B
3
= {(y,µ
˜
B
3
(y))};
2.2 Fuzzy Knowledge Base Formation
Fuzzy knowledge is represented as the following
fuzzy rules that contain a linguistic output variable
R
n
: IF ˜x
1
is ˜a
1i
AND ˜x
2
is ˜a
2 j
AND ˜x
3
is ˜a
3k
AND ˜x
4
is ˜a
4p
then ˜y is
˜
B
m
In the case of linguistic variables specific values,
fuzzy knowledge is presented in relational form in ta-
ble 1.
Table 1: Relational form of fuzzy knowledge representa-
tion.
The rule ˜x
1
˜x
2
˜x
3
˜x
4
˜y
R
1
e
α
11
e
α
21
e
α
31
e
α
41
e
α
1
R
2
e
α
12
e
α
21
e
α
31
e
α
41
e
α
1
R
3
e
α
13
e
α
21
e
α
31
e
α
41
e
α
2
R
4
e
α
11
e
α
22
e
α
31
e
α
41
e
α
2
... ... ... ... ... ...
R
81
e
α
13
e
α
23
e
α
33
e
α
43
e
α
3
2.3 Mamdani Fuzzy Inference
Mechanism Formation
2.3.1 Fuzzification
We will determine the truth degree of each sub-
condition of each rule, using the membership function
µ
˜
A
i j
(x
i
).
As membership functions of sub-conditions, we
chose:
• piecewise linear Z-shaped function, i.e.
µ
˜
A
i1
(x
i
) =
1, x
i
≤ a
i
b
i
−x
i
b
i
−a
i
, a
i
< x
i
< b
i
0, x
i
≥ b
i
,i ∈ 1, 4
• piecewise linear Π-shaped function, i.e.
µ
˜
A
i2
(x
i
) =
0, x
i
≤ a
i
x
i
−a
i
b
i
−a
i
, a
i
≤ x
i
≤ b
i
1, b
i
≤ x
i
≤ c
i
d
i
−x
i
d
i
−c
i
, c
i
≤ x
i
≤ d
i
0, x
i
≥ d
i
,i ∈ 1, 4
• piecewise linear S-shaped function, i.e.
µ
˜
A
i3
(x
i
) =
0, x
i
≤ c
i
x
i
−c
i
d
i
−c
i
, c
i
< x
i
< d
i
1, x
i
≥ d
i
,i ∈ 1, 4,
where a
i
, b
i
, c
i
, d
i
- membership function param-
eters.
2.3.2 Sub-Condition Aggregation
The condition membership functions for each rule R
n
are determined based on the minimum value method:
µ
S
4
i=1
˜
A
i, f (n,i)
(x
1
,x
2
,x
3
,x
4
) = min
i∈1,4
n
µ
˜
A
i, f (n,i)
(x
i
)
o
,
where f – a function that returns the value number of
the i-th linguistic input variable of the n-th rule and is
determined on the basis of table 1. For example, if the
linguistic input variable ˜x
1
rules R
81
matters
e
α
13
, then
f (81,1) = 3.
2.3.3 Activation of Conclusions
The membership functions of the conclusion for each
rule R
n
are determined based on the minimum value
method (based on the Mamdani rule):
µ
˜
B
g(n)
(y) = min
µ
˜
U
4
i=1
A
i, f (n,i)
(x
1
,x
2
,x
3
,x
4
),µ
˜
B
g(n)
(y)
,
where g – a function that returns the value number of
the linguistic output variable of n-th rule and deter-
mined on the basis of table 1.
For example, if the linguistic output variable ˜y of
the rule R
81
is
e
β
3
, then g(81) = 3.
A piecewise linear triangular function was chosen
as the membership functions of the conclusions, i.e.
µ
˜
B
m
(y) =
0, y ≤ e
m
y−e
m
u
m
−e
m
, e
m
≤ y ≤ u
m
v
m
−y
v
m
−u
m
, u
m
≤ y ≤ v
m
0, y ≥ v
m
,m ∈ 1, 3,
where e
m
,u
m
,v
m
– membership function parameters.
In the case of such a membership function, the
kernel of each fuzzy set
˜
B
m
is:
ker
˜
B
m
= {y ∈ Y |µ
˜
B
m
(y) = 1} = {u
m
}.
Fuzzy Expert System of the Decision Making Support on Foreign Direct Investment
17
2.3.4 Aggregation of Conclusions
The membership functions of the final conclusion are
defined, which contains a linguistic output variable
based on the maximum value method:
µ
˜
B
m
(Y ) = max
n∈1,81
{µ
˜
B
g
(n)
(y)}
2.3.5 Defuzzification
The volumes of foreign direct investment are deter-
mined basedon the centroid method:
y
∗
=
∑
y∈Y
µ
˜
B
(y)y
∑
y∈Y
µ
˜
B
(y)
,Y =
1,2,3
3 QUALITY CRITERION FOR
THE PROPOSED FUZZY
EXPERT SYSTEM
The objective function is chosen as a quality crite-
rion, representing the accuracy as probability of cor-
rect foreign direct investment
F =
1
P
P
∑
p=1
,[y
p
= d
p
] → max
θ
,
[p = q] =
1, p = q
0, p ̸= q
,
(1)
where d
p
– test foreign direct investment,
y
p
– foreign direct investment received as a result
of fuzzy inference,
P – number of test implementations,
θ = (a
1
,b
1
,c
1
,d
1
,...,a
4
,b
4
,c
4
,d
4
,e
1
,u
1
,v
1
,...,e
3
,
u
3
,v
3
) – parameter vector of membership functions.
4 METAHEURISTIC METHOD
BASED ON AN ADAPTIVE
GRAVITATIONAL SEARCH
ALGORITHM FOR
DETERMINING THE
PARAMETERS OF THE
PROPOSED FUZZY EXPERT
SYSTEM
The particle velocity (not the gravitational constant)
depends on the iteration number in this method, which
provides control over the convergence rate of the
method, as well as providing a global search at the
initial iterations, and a local search at the final itera-
tions. The parameter vector of membership functions
corresponds to the position vector of one particle x.
The quality criterion is used as the goal function (1).
1. Initialization.
1.1. Setting the gravitational constant G, the max-
imum number of iterations N, the population
size K, the length of the particle position vec-
tor M (it corresponds to the length of the pa-
rameter vector of membership functions and is
equal to 25), the minimum and maximum val-
ues for the position vector x
min
j
, x
max
j
, j ∈ 1,M,
the minimum and maximum values for the ve-
locity vector v
min
j
, v
max
j
, j ∈ 1,M .
1.2. The best position vector randomly generating
x
∗
= (x
∗
1
,...,x
∗
M
),
x
∗
j
= x
min
j
+ (x
max
j
− x
min
j
)U(0,1),
where U(0,1) – a function that returns a uni-
formly distributed random number in a range
[0,1].
1.3. The initial population creation
1.3.1. Particle number k = 1, P =
/
0.
1.3.2. A position vector at random x
k
generating
x
k
= (x
k1
,...,x
kM
),
x
k j
= x
min
j
+ (x
max
j
− x
min
j
)U(0,1).
1.3.3. Random velocity vector v
k
generating
v
k
= (v
k1
,...,v
kM
),
v
i j
= v
min
j
+ (v
max
j
− v
min
j
)U(0,1).
1.3.4. If (x
k
,v
k
) /∈ P, then P = P ∪ {(x
k
,v
k
)},
k = k + 1.
1.3.5. If k ≤ K, then go to step 1.3.2.
2. Iteration number n = 1.
3. The computation of the best and worst particle of
a population from a target function
l = arg min
k
F(x
k
),x
best
= x
l
,
l = arg max
k
F(x
k
),x
worst
= x
l
.
4. The computation of all particles masses.
5. The computation of the gravitational force acting
between all pairs of particles
5.1. m
k
= G
F(x
k
)−F(x
worst
)
F(x
best
)−F(x
worst
)
,k ∈ 1,K.
5.2. M
k
=
m
k
∑
K
s=1
m
s
,k ∈ 1,K.
6. The computation of the gravitational force acting
between all pairs of particles
f
kl
= G
M
k
M
l
d(x
k
,x
l
) + ε
(x
l
− x
k
),k,l ∈ 1, K,
where d(x
k
,x
l
) – distance between particles k and
l (e.g. Euclid distance).
M3E2 2022 - International Conference on Monitoring, Modeling Management of Emergent Economy
18
7. The computation of the resulting force acting on
all particles
r
kl
= U(0,1), k, l ∈ 1, K
f
k
=
K
∑
l = 1
l ̸= k
r
kl
f
kl
,k ∈ 1,K
8. Modification of the acceleration of all particles
a
k
=
f
k
M
k
,k ∈ 1,K
9. Speed modification of all particles
r
k
= U(0,1), k ∈ 1,K
v
k
= r
k
v
k
+ a
k
,k ∈ 1,K
10. Modification all of the particles’ position, taking
into account the iteration number
10.1. x
k
= x
k
+ v
k
1 −
n
N
,k ∈ 1,K
10.2. x
k j
= max{x
min
j
,x
k j
},x
k j
=
min{x
max
j
,x
k j
}, j ∈ 1,M,k ∈ 1,K
11. If n < N, then n = n +1, go to step 3
The result is x
∗
.
5 NUMERICAL RESEARCH
Numerical research was carried out using the Keras
submodule of the TensorFlow module. The Pandas
module was used to fill in missing values through lin-
ear interpolation, as well as for tabular data I/O oper-
ations. The Scikit-fuzzy module was used to create a
fuzzy expert system.
The fuzzy expert system was researched using
the World Bank economic indicators database (https:
//databank.worldbank.org/home.aspx). The economic
indicators of 145 countries for 10 years were used.
The size of the original sample was 1450.
For the proposed adaptive gravity search algo-
rithm, the gravity constant G was 100, the maximum
number of iterations was 1000, and the population
size was 50.
The comparison results of the proposed fuzzy ex-
pert system with the operator are presented in table 2.
Table 2: Comparison results of the proposed fuzzy expert
system with an operator.
Accuracy
fuzzy expert system operator
0.98 0.8
The comparison results of the proposed fuzzy ex-
pert system with the proposed meta-heuristic adaptive
gravitational search algorithm (AGSA) and the tradi-
tional meta-heuristic adaptive gravitational search al-
gorithm (AGSA) operator are presented in table 3.
Table 3: Comparison results of the proposed fuzzy expert
system of the proposed meta-heuristic and the traditional
meta-heuristic.
Accuracy
GSA AGSA
0.93 0.98
Figure 1 shows the accuracy for the proposed
fuzzy expert system trained based on the proposed
meta-heuristic adaptive gravitational search algorithm
(AGSA) and on the proposed meta-heuristic gravita-
tional search algorithm (GSA).
Figure 1: Accuracy of the proposed fuzzy expert system
with GSA and AGSA.
The comparison results of the proposed fuzzy ex-
pert system trained on the basis of back-propagation
(BP) and the proposed meta-heuristic adaptive gravi-
tational search algorithm (AGSA) are presented in ta-
ble 4.
Figure 2: Accuracy of the proposed fuzzy expert system
with BP and AGSA.
Figure 2 shows the accuracy for the proposed
Fuzzy Expert System of the Decision Making Support on Foreign Direct Investment
19
Table 4: Comparison results of the proposed fuzzy expert
system based on the back-propagation method and proposed
meta-heuristic.
Accuracy
BP AGSA
0.90 0.98
fuzzy expert system trained on the basis of back-
propagation (BP) and the proposed meta-heuristic
adaptive gravitational search algorithm (AGSA).
Figures 3-7 shows the membership functions for
the values of linguistic variables ˜x
1
, ˜x
2
, ˜x
3
, ˜x
4
and y.
Figure 3: Membership functions for linguistic variable val-
ues ˜x
1
.
Figure 4: Membership functions for linguistic variable val-
ues ˜x
2
.
6 DISCUSSION
The traditional non-automatic approach to assessing
the foreign direct investment effectiveness reduces the
accuracy of a correct assessment (table 2). The pro-
posed method eliminates this disadvantage.
The traditional method of the gravitational search
algorithm ignores the iteration number during the par-
ticle position calculating; this reduces the accuracy of
Figure 5: Membership functions for linguistic variable val-
ues ˜x
3
.
Figure 6: Membership functions for linguistic variable val-
ues ˜x
4
.
finding a solution (table 3); requires a large number of
parameters associated with the gravitational constant
calculating. The proposed method eliminates these
shortcomings.
The traditional approach to training a fuzzy expert
system based on back propagation reduces the prob-
ability of correct estimation (table 4). The proposed
method eliminates this disadvantage.
7 CONCLUSIONS
1. Relevant optimization methods and expert sys-
tems were investigated as part of the decision-
making technology for foreign direct investment.
The research results showed that the most effec-
tive is the use of fuzzy expert systems, the param-
eters of which are identified by means of meta-
heuristic methods today.
2. A fuzzy expert decision support system for for-
eign direct investment has been developed. The
proposed system simplifies the interaction be-
tween the operator and the computer system
M3E2 2022 - International Conference on Monitoring, Modeling Management of Emergent Economy
20
Figure 7: Membership functions for linguistic variable val-
ues ˜y.
through the use of qualitative indicators, and also
allows to identify its parameters using the pro-
posed swarm metaheuristics.
3. A quality criterion is proposed; it considers the
specifics of the created fuzzy expert system and
allows assessing of the decisions accuracy.
4. A swarm metaheuristic algorithm based on an
adaptive gravitational search algorithm has been
created; it provides control over the rate of method
convergence, as well as providing global search at
the initial iterations, and local search at the final
iterations due to adaptive control of the particle
velocity.
5. The proposed optimization method based on
swarm metaheuristics and a fuzzy expert system
make it possible to intellectualize the technol-
ogy of making decisions on foreign direct invest-
ment. Prospects for further research involve test-
ing the proposed method and system on a wider
test database set.
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