Uncertainty Modeling in the Process of SMEs Financial Mechanism
Using Intuitionistic Fuzzy Estimations
George L. Shahpazov, Lyubka A. Doukovska and Vassia K. Atanassova
Institute of Information and Communication Technologies, Bulgarian Academy of Sciences,
Acad. G. Bonchev str., bl. 2, 1113 Sofia, Bulgaria
Keywords: SMEs Financial Mechanism, Credit Risk, Creditworthiness, Intuitionistic Fuzzy Sets, i-Fuzzification.
Abstract: In the present paper, we discuss the mechanism of bank support of small and medium-sized enterprises
(SMEs). Analysis is made of the effectiveness of the bank’s internal financial structural unit and hierarchy,
and it is shown how the concept of intuitionistic fuzzy sets can be applied to the process of evaluating
creditworthiness of the SMEs applications for bank loans, from the bank’s perspective. The presented
approach aims to yield estimations of the effectiveness of the process, taking consideration of the aspects of
uncertainty, which is an inherent part of the processes of evaluation of applications for bank support and
evaluation of the process itself.
1 INTRODUCTION
Supporting emerging and present legal entities as
making a form of investment, such as financing
SME sector involves substantial risk in general and
particularly in emerging markets like Bulgaria. A
significant portion of this risks results from the lack
of business ethics in the market and a legislation,
which doesn’t support in particular this kind of
investments. Results published in paper (Shahpazov,
Doukovska, 2012), shows that the timing for
financial support in Bulgarian SMEs from the
manufacturing sector is perfect. The actual result
lays on deep analysis of the sector, which forecasted
a faster growth in the sector than local GDP growth
during a 5-8 year period spread.
Over the same period, the share of service sector
output in GDP is expected to raise from 61.5% -
63.4%.
Local agriculture sector is experiencing a boost
in the last few years, and falls under the program of
rehabilitation and modernization of value creating
industries, as the main focus is to overturn present
trade situation where the country imports more
goods than it exports. The overall aim is to utilize
the EU accession and its supportive instruments,
local Government programs assistance, and financial
institution involvement into accelerating growth
processes and SMEs further development.
The above mentioned facts allow us to look for
new techniques for intelligent analysis of the process
of SMEs financial mechanism.
In paper (Shahpazov, Doukovska, 2013), an
application of the apparatus of generalized nets is
proposed for modeling of the mechanism of
financial support of the SMEs.
The present work traces the most important steps
of the process of evaluation of a business project
proposal, applying for bank financing. It is a conti-
nuation of our previous research (Shahpazov,
Doukovska, 2013). The research model is offered
how the concept of intuitionistic fuzziness can be
applied to the process of evaluating creditworthiness
of the SMEs.
The evaluation follows a predefined hierarchy of
the levels of the bank’s decision makers, and
sophisticated policies and procedures.
For the needs of our discussion, we make a
relatively simple model, which takes into account
which levels of the bank hierarchy receive and
process the business applications for bank loans,
which levels make funding decisions, and in case of
uncertainty, which upper levels of the hierarchy are
these applications directed to, for taking a decision
at the higher level. This model is schematically
illustrated on Figure 1.
In this highly regulated process, for each level of
the bank’s decision making hierarchy, we are
interested to estimate and interpret in terms of
271
Shahpazov G., Doukovska L. and Atanassova V.
Uncertainty Modeling in the Process of SMEs Financial Mechanism Using Intuitionistic Fuzzy Estimations.
DOI: 10.5220/0005427002710275
In Proceedings of the Fourth International Symposium on Business Modeling and Software Design (BMSD 2014), pages 271-275
ISBN: 978-989-758-032-1
Copyright
c
 2014 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved

intuitionistic fuzzy sets the share of successfully
approved applications, the share of rejected app-
lications and the share of those applications, which
for various reasons, may exhibit certain uncertainty
(e.g. high risk / high return of investment) and thus
get forwarded from lower to upper level of bank
hierarchy, being a higher authority in the decision
making process.
Manage
ment
Board
Credit
Council
Headquarters
Bank Branch
Supervisory
Board
Accepted
applications
Rejected
a
pp
lications
Newapplication
Figure 1: Diagram of the process of bank loan applictions
review along the bank’s decision making hierarchy.
2 SHORT REMARKS ON
INTUITIONISTIC FUZZY SETS
Intuitionistic fuzzy sets (IFSs) were initially
proposed by Atanassov in 1983 (Atanassov, 1983;
Atanassov, 1986) as an extension of the concept of
fuzzy sets, introduced by Zadeh in 1965 (Zadeh,
1965). The theory of IFSs has been extensively
developed by the author in (Atanassov, 1991;
Atanassov, 2012) and further developed by many
other researchers worldwide.
In classical set theory, the membership of
elements in a set is evaluated binary terms as either
‘true’ or ‘false’: an element either belongs or does
not belong to the set. As an extension, fuzzy set
theory permits the gradual assessment of the mem-
bership of elements in a set; this is described with
the aid of a membership function valued in the real
unit interval [0, 1].
The theory of intuitionistic fuzzy sets further
extends both concepts by allowing the assessment of
the elements by two functions, µ for the degree of
membership and ν for the degree of non-mem-
bership, with which belong the element belongs to a
set, where both these degrees and their sum are
numbers in the [0, 1] - interval.
Speaking formally, if we have a fixed universe E
and A is a subset of E, we can construct the
intuitionistic fuzzy set A*, so that:
A* = {〈x, µ
A
(x), ν
A
(x)〉 | x ∈ E},
where 0 ≤ µ
A
(x), ν
A
(x), µ
A
(x) + ν
A
(x) ≤ 1. In the case
of strict inequality to the right, i.e.:
0 ≤ µ
A
(x) + ν
A
(x) < 1,
there is a non-negative complement of the sum of
membership and non-membership to 1, and this
complement is denoted by π
A
(x) = 1 – µ
A
(x) – ν
A
(x)
and usually called degree of uncertainty or hesitancy
margin.
IFSs represent a true generalization of fuzzy sets,
since in the partial case when the non-membership
function fully complements the membership func-
tion to 1, not leaving room for any degree of
uncertainty, is practically the case of fuzzy sets.
IFSs have different graphic representations, for
instance linear, which bears resemblance with the
graphic representation of fuzzy sets, radar-chart, or
triangular, which reflects the specifics of the IFS.
The standard linear graphic representation has the
form of Figure 2, where both functions µ and ν are
visualized as is.
Figure 2: Standard graphical interpretation of IFSs.
However, together with the standard linear
representation, a small modification of this graphics,
as shown in Figure 3, was introduced (Atanassov,
1991) representing not the exact function ν, but the
function ν* = 1 – ν. It plots the non-membership
function not in ‘bottom-up’ manner like the mem-
bership function µ, but in ‘top-down’ manner using
its mirror image. Thus, we can very already well
distinguish the formed in-between ‘belt of
uncertainty’, which for every x ∈ E complements the
μ
ν
1
0
x
∈
E
Fourth International Symposium on Business Modeling and Software Design
272

sum of µ
A
(x) and ν
A
(x) to 1. This modified linear
representation of IFSs is probably the one most often
used in practice.
Figure 3: Modified graphical interpretation of IFSs.
3 MAIN RESULTS
As we mentioned above, the process of evaluation of
every bank loan application passes through one or
more (rarely more than three) levels of the bank’s
decision making hierarchy. Usually the decision
about the approval or rejection of the applications is
taken on the Branch level or the Headquarters level,
however in certain cases when lower levels cannot
take a categorical decision, the application is sent to
the upper level.
Hence, it is of particular interest to trace the
degrees of acceptance, rejection and uncertainty in
taking the decisions on every bank hierarchy level,
and for this purpose we can use a simple i-fuzzifi-
cation procedure, analogous to the one given in
(Atanassova, 2013), where from crisp data sets we
can construct intuitionistic fuzzy data sets.
We can introduce intuitionistic fuzziness in these
estimations, using two possible schemes, which are
mathematically identical and can be used inter-
changeably, although visually they produce rather
different results. In both cases, we will denote the
levels of the bank’s decision making hierarchy with
the following denotations:
• Level 0 represents bank loan applicants,
• Level 1 is ‘Branch’ level,
• Level 2 is ‘Headquarters’ level,
• Level 3 is ‘Credit Council’ level,
• Level 4 is ‘Management Board’ level,
• Level 5 is ‘Supervisory Board’ level.
We will also agree to denote with µ
i
, ν
i
and π
i
respectively, the number of applications, which on
the i-th level are accepted, rejected or forwarded for
decision to the level (i + 1), and with t – the total
number of applications submitted for evaluation.
Obviously, in the top level of the Supervisory
Board, π
5
= 0, as all applications that have reached
this level must there get final resolution.
The whole process, interpreted in terms of IF
estimations can be graphically illustrated in the
following Figure 4.
µ
1
ν
1
π
1
µ
2
ν
2
π
2
µ
3
ν
3
π
3
µ
4
ν
4
π
4
µ
5
ν
5
All submitted
applications: t
Figure 4: IF estimations of the performance of the
different levels of decision making hierarchy during the
bank loan applications review process.
First Scheme of i-Fuzzification. In the first scheme
of i-fuzzification, on every level of the bank’s
decision making hierarchy, at a given moment of
time, we estimate what percentage of the total
number of submitted applications for evaluation
have been approved, and, respectively, hitherto
rejected. Let us denote these by
11
,,
ii
M
N i = 1, …, 5,
hence:
1
i
k
k
i
M
t
μ
=
=
∑
,
1
i
k
k
i
N
t
ν
=
=
∑
.
Second Scheme of i-Fuzzification. In the second
scheme of i-fuzzification, on every level of the
bank’s decision making hierarchy, at a given
moment of time, we estimate what percentage of the
applications for evaluation, received from the lower
level are approved, and, respectively, rejected, on
that level. Let us denote these by
22
,,
ii
M
N i = 1, …,
5, hence:
2
1
i
i
i
M
μ
π
−
=
,
2
1
i
i
i
N
ν
π
−
=
.
Numerical Example. Graphical Interpretation of
the Two Proposed i-Fuzzification Schemes. Let us
μ
ν
* = 1 ‒
ν
1
0
x ∈ E
zero
uncertainty
high
uncertainty
Uncertainty Modeling in the Process of SMEs Financial Mechanism Using Intuitionistic Fuzzy Estimations
273

give the following numerical example, which will
make the differences between both proposed
schemes easy to follow.
In given moment of time, let the following
exemplary distribution of project applications along
the levels in the bank’s decision making hierarchy
be observed, as shown on Figure 5.
25
48
27
6
18
3
1
1
1
0
0
1
1
0
All submitted
applications: 100
Figure 5: IF estimations for the numerical example.
Applying the first scheme of i-fuzzification over
these data, will give the results in the following
Table 1, as illustrated in Figure 6.
Table 1: Application of the first i-fuzzification scheme
over the data from Figure 5.
µ
i
ν
i
π
i
Level 1 25/100 = 0.25 48/100 = 0.48 27/100 = 0.27
Level 2 (25 + 6)/100 = 0.31
(48 + 18)/100 =
0.66
3/100 = 0.3
Level 3
(31 + 1)/100
= 0.32
(66 + 1)/100 =
0.67
1/100 = 0.01
Level 4
(32 + 0)/100
= 0.32
(67 + 0)/100 =
0.67
1/100 = 0.01
Level 5
(32 + 1)/100
= 0.33
(67 + 0)/100 =
0.67
0/100 = 0.00
0
0.2
0.4
0.6
0.8
1
Level1 Level2 Level3 Level4 Level5
µ ν*
Figure 6: Interpretation of the first i-fuzzification scheme.
Applying the second scheme of i-fuzzification
over these data, will give the results in the following
Table 2, as illustrated in Figure 7.
Table 2: Application of the second i-fuzzification scheme
over the data from Figure 5.
µ
i
ν
i
π
i
Level 1 25/100 = 0.25 48/100 = 0.48 27/100 = 0.27
Level 2 6/27 = 0.22 18/27 = 0.67 3/27 = 0.11
Level 3 1/3 = 0.33 1/3 = 0.33 1/3 = 0.33
Level 4 0/1 = 0.00 0/1 = 0.00 0/1 = 0.00
Level 5 1/1 = 1.00 0/1 = 0.00 0/1 = 0.00
0
0.2
0.4
0.6
0.8
1
Level1 Level2 Level3 Level4 Level5
µ ν*
Figure 7: Interpretation of the second i-fuzzification scheme.
4 CONCLUSION
The comparison between both i-fuzzification sche-
mes shows well that in the first scheme, at every
level i, the [0, 1] - interval corresponds to the initial
number of t submitted bank loan applications, and
11
,,
ii
M
N i = 1, …, 5, are cumulative. In comparison,
in the second scheme, on every upper level i we only
operate with the IF evaluations for that level, and
every time the degree of uncertainty from the lower
(i – 1)
-th
level is again re-normed to match the [0, 1]
- interval (see the grey dotted lines).
Both approaches can be used interchangeably,
and may prove useful in different situations, when it
is necessary to evaluate the effectiveness of the
different bank’s internal financial structural unit as
levels of the bank’s decision making hierarchy.
Fourth International Symposium on Business Modeling and Software Design
274

ACKNOWLEDGEMENTS
The research work reported in the paper is partly
supported by the project AComIn “Advanced
Computing for Innovation”, grant 316087, funded
by the FP7 Capacity Programme (Research Potential
of Convergence Regions) and partially supported by
the European Social Fund and Republic of Bulgaria,
Operational Programme “Development of Human
Resources” 2007-2013, Grant № BG051PO001-
3.3.06-0048.
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Uncertainty Modeling in the Process of SMEs Financial Mechanism Using Intuitionistic Fuzzy Estimations
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