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*Corresponding author. E-mail: [anonimizat]
This is an Open Access article distributed under the terms of the Creative Commons Attribution License ( http://creativecommons.
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and source are credited.© 2019 The Author(s). Published by VGTU PressTechnological and Economic Development of Economy
ISSN: 2029-4913 / eISSN: 2029-4921
Article in press
https://doi.org/10.3846/tede.2019.10551
DETECTING FINANCIAL SUSTAINABILITY RISK OF THE ASSETS
USING MAMDANI FUZZY CONTROLLER
Marcel-Ioan BOLOȘ1, Ioana-Alexandra BRADEA2,
Claudia Diana SABĂU-POPA3*, Laurențiu-Andrei ILIE4
1, 3Department of Finance-Accounting, Faculty of Economic Sciences,
University of Oradea, Oradea, Romania
2Department of Informatics and Economic Cybernetics,
Bucharest University of Economic Studies, Bucharest, Romania
4Faculty of Geography, University of Bucharest, Bucharest, Romania
Received 13 October 2018; accepted 05 May 2019
Abstract. The paper aims to develop a MAMDANI fuzzy controller for detecting the financial
sustainability risk of the assets owned by the company. This type of risk indicates when an asset no
longer produces economic benefits to the company, or the benefits are small enough to no longer
justify the asset maintaining in working order. The proposed fuzzy controller has as input variables
the asset operating expenses and the variation of this category of expenses from one analysis period
to another. The controller’s objective function is to keep operating costs at their initial state and
thus reducing the financial sustainability risk. The controller’s output variable is represented by the
economic benefits variation, considered to be an essential component in the financial sustainability
risk analysis. The obtained results were interpreted taking into account the objective function of the
controller as well as the evolution of the input variables. Two simulations for fuzzy controllers were
made, with the mention that the variation ranges for the input variables were delimited. In practice,
fuzzy controllers can be generated according to company policies to keep under control the expense
categories that accompany the asset exploitation.
Keywords: MAMDANI fuzzy controller, financial sustainability risk, assets, simulation.
JEL Classification: C63, G32.
Introduction
The company’s decision to invest in tangible assets represents a complex process that involves
long-term capital assets. The cost of capital assets is often quite significant, especially when
this capital comes from financial creditors. That is why the investment decision process must
take into account the profitability of the capital invested in assets regardless of their nature.
Profitability as an indicator is demanded intensely by shareholders, as well as by the financial
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2 M.-I. Boloș et al. Detecting financial sustainability risk of the assets using MAMDANI fuzzy controller
institutions which are interested in recoverin g their capital as quickly as possible and with
minimum risk.
By their content, the economic benefits, for each tangible asset, are a source of “added
value” , from which the financial creditors are remunerated on the basis of the interest paid
by the company and shareholders can also be paid with dividends. Every purchased asset is
a carrier of “added value” for the company and contributes to the company’s financial wealth,
for example for the emergence of the gross surplus before interest, taxes and depreciation
known as EBITDA.
The acquisition of a tangible asset becomes a management decision where not only the
asset acquisition cost, but also its economic performance, matters. The asset economic per –
formance depends on the different technical parameters such as: production capacity, asset
running costs, maintenance costs, and so on. The best combination of the acquisition cost
and their economic performance builds on the process of asset acquisition under economic
efficiency. Economic benefits are the result of the assets economic performance, between
them being a direct proportionality relationship. Thus, the economic performance can de –
termine economic benefits and return on investments duration compatible with shareholders
expectations.
Regardless of the studies and analyzes on which the asset acquisition process is based,
three elements are important in the decision making process, namely:
The assets acquisition cost that determines the amount of financial resources to be dis –
posed of in the investments made by the company and which must then be recovered in a
limited time. If the financing source is a bank loan, then interest should be taken into account
also as a cost item.
The asset economic performance depends on the technical and economic parameters.
They are set up with the initial purchase of assets. Economic performance has a direct impact
on economic benefits generated by assets.
The asset economic benefits are dependent on the initial economic performance of assets.
They have direct impact on the recovery investment rate.
Regardless the combination of acquisition cost, economic performance and benefits, the
financial sustainability represent an indicator of asset behavior analysis, which is important
throughout the assets life cycle. Thus, it is important to establish through financial sustain –
ability the moment when the tangible asset no longer produces the economic benefits that
underpin its maintenance in company.
1. Literature review
The literature has advanced the idea that a tangible asset should no longer be regarded as
a capital asset for a limited period of time and therefore as a cost element that irreversibly
depreciates (Ambrose & Megginson, 1992). A tangible asset becomes a capital asset that is
expected to produce economic benefits to the company and its acquisition cost can be reliably
measured. (Ardeleanu-Popa & Miheș, 2007)
It was introduced a new notion for tangible assets that is very important for the decision-
making process, namely the economic benefits that result from the assets participation in the
company’s operational activity (Galbreath, 2005; Bolos & Sabau-Popa, 2017).
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Technological and Economic Development of Economy. Article in press 3
The use of fuzzy techniques for analyzing the assets financial sustainability in a company
has been less studied, even if the applicability and efficiency of fuzzy methods for solving
multi-criteria group decision making problems were demonstrated in different research area
(Krishankumar et al., 2018).
On the other hand the MAMDANI fuzzy controller has captured the interest of the re –
searchers, according to Web of Science database, 651 ISI articles being published over time,
but none of them targeted the economic area, most of them being framed in the follow –
ing research areas: computer science, engineering, automation control systems, robotics and
mathematics. Thus, this type of fuzzy controller was applied in: industry, for designing fuzzy
control systems focused on industrial applications (Precup & Hellendoorn, 2011; Huang et
al., 2018); nonlinear dynamics, for developing new learning laws for Mamdani and other type
fuzzy neural networks based on input-to-state stability approach (Yu & Li, 2004; Mann, Hu,
& Gosine, 1999); designing genetic fuzzy systems that manage the interpretability-accuracy
tradeoff (Cordon, 2011); uncertainty management (Wu, Z. Y ang, Wang, Li, & Y . Y ang, 2018),
or approximating real continuous control functions on a compact set to arbitrary accuracy
(Galichet & Foulloy, 1995).
The most cited studies focused on engineering domain. Wang, Chen, and Dai (2007)
proposed a direct adaptive fuzzy tracking control for a class of perturbed strict-feedback
nonlinear systems, with unknown virtual control coefficients. Their architecture illustrated
that all the signals in the resulting closed-loop system are uniform bounded and the tracking
error converges to an arbitrarily small neighborhood of the origin. Another relevant study in
this field developed a new method that uses the Kalman filter to optimize a Mamdani fuzzy
PID controller (Ahn & Truong, 2009). The hybrid model adjusts the controller parameters
automatically during the operation process of any system applying the controller to minimize
the control error.
The analysis of analytical structure and stability of a fuzzy PID controller was conducted
by Mohan and Sinha (2008) in order to demonstrate the effectiveness of the simplest fuzzy
PID controller in different area. Qin, Sun, Hua, and Liu (2018) managed to improve the PID
controller performance based on fuzzy logic by identifying a linear model based on the least
squares method, optimizing the PID parameters and designing a fuzzy adaptive PID control –
ler based on these parameters.
Regarding company’s investment decision in assets, Mannasoo and Maripuu (2015) stud –
ied its influence on performance. They divided this decision in short-term investments in
current assets and long-term productivity-enhancing investments in tangible and intangible
assets and discovered that there are patterns influenced by macroeconomic fluctuations with
impact on the company’s financial strength and sustainability. On the other hand, Harris
(2017) studied how to increase sustainability through assets management.
The relationship between strategies and the assets conditions was studied by Liu, Chen,
and Xie (2018). Using a sustainability analysis, this study tried to establish the incidence
mechanism of assets on strategies.
Reichardt (2006) analyzed how companies grow by acquisition and conduct due dili –
gence assessments in order to identify the opportunities and risks associated with potential
acquisitions of new assets. He highlights the limitations of conventional assessments based
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4 M.-I. Boloș et al. Detecting financial sustainability risk of the assets using MAMDANI fuzzy controller
solely on financial risk and presents practical recommendations on structuring due diligence
assessments to consider the full risk profile and the lifecycle of an asset.
This paper targets to propose a fuzzy method in which asset regardless of their nature is
analyzed based on the financial sustainability risk defined as the risk category, in which the
company’s investments in that asset are no longer warranted from the economic efficiency
point of view.
2. The concept of assets financial sustainability
The assets financial sustainability has been studied in the literature through various models,
each of which has as a common element, namely the establishment of the optimal moment
of putting the asset out of use. Determining this time point ( T) represents a complex deci –
sion that must take into account the fact that another asset must replace the original asset,
which in turn involves costs. The analysis model for replacing an asset, or for determining
the optimal moment of decommissioning, is the one that conceptually analyzes assets as a
source of cash-flow generator.
The cash flow generated by an asset is based on a few basic elements, namely:
The asset production capacity ( Q) based on its economic performance, on which depends
the value of the output obtained with the help of the asset per unit of time. The output ob –
tained with the assets will have a certain unitary cost of production ( Cu), a profit margin ( rp)
and a selling price ( p). In these conditions the volume of production is determined by the
production capacity and the sale price of the goods obtained by the form: Qp∑× .
The asset operating costs ( Chf) take different forms: from the utilities cost necessary for its
operation up to specific costs for the assets operation. All these costs depend on the dynamics
of the asset’s operation but also on how it is used in the production process.
Maintenance costs ( Chmen) are expenditures required to maintain the asset’s operating
condition, which in turn depends on the frequency of failures, the asset maintenance plan,
and the time required for its technical revisions.
Theorem no. 1: Any asset that is used in the operational activity of a company generates
a total cash-flow over the life cycle of that asset [0, T], of the form:
() () ()
0.T
it iT
ta a aV V t Ch t e VR t e I−− = − ×+ × −∫ (1)
Demonstration: The annual cash flows for the life of the asset [0; T] is formed from the
difference between the income generated by an asset ( Va) and its total operating and main –
tenance expenses ( Cha):
() ()
( )00 0 01:
1aa
at V t Ch t
r−×
+;
() ()
( )11 1 11:
1aa
at V t Ch t
r−×
+;
……………………………………………..
() ()
( )1:
1n an an n
at V t Ch t
r−×
+. (2)
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Technological and Economic Development of Economy. Article in press 5
The present value of the future cash flows determined by the asset operation over its
entire life cycle will be as follows:
() () ( )
( )( ) ( )0111 1.
11 1T an an n
aa aV V t Ch t
rr r
 = − × + +…+++ + (3)
If ra is the same, than
( ) 1.1lim
1n
it
nni ae
r−
→∞=→
+∑
The amount of operational cash flows generated by the company’s assets over their life
cycle will be written as:
() ()
0.T
it
tGross a aV V t Ch t e− = −×∫ (4)
The operational cash flows are important because they contribute to generate “added
value” across the company. If from the cash flow value is deducted the acquisition value of
the asset (I) and is added the residual value ( VRa) is obtained the value of the total cash flow
generated by an active during its lifetime.
() () ()
0.T
it iT
tNet a a a V V t Ch t e VR t e I−− = − ×+ × −∫ (5)
The company is interested in determining the economic life of the assets, the period on
which they will invest in tangible assets in terms of economic efficiency. During the economic
life of assets, the economic benefits have a logarithmic evolution, being initially higher, after
which, with the passage of time, they begin to decrease as a result of the increase in main –
tenance or operating expenses.
The economic benefits over the lifetime of the asset: () ()
0 T
it
ea aB V t Ch t e− = −×∫ rep-
resent the most important indicator providing information about the economic life of an
asset. When the cumulative economic benefits are equal to the cost of the tangible asset, the
asset’s economic lifetime has been reached. Beyond that time, the operation cost of the asset
will generate low-value of economic benefits or even losses, that no longer justify keeping it
in working order.
Theorem no. 2: The economic life of an asset is determined by the operating cash flow of
the asset and its discount rates according to the formula:
() 00 *years .aa a
cha vaV Ch iVR tTrr−−= − (6)
Demonstration: The economic lifetime is obtained with the first-order derivative in rela –
tion to T for the total cash-flow equation, namely:
0.tV
T∂=∂ (7)
By performing the calculations, we obtain:
() () () () 0iT iT iT
aa a aV t Ch t e iVR t e VR t e− ′ −−  − ×− ×+ ×=
or
() () () ()0.aa a aV t Ch t iVR t VR t′  − − += (8)
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6 M.-I. Boloș et al. Detecting financial sustainability risk of the assets using MAMDANI fuzzy controller
In determining the optimum asset lifetime or the duration in which the value of the
economic benefits equals the cost of the investment, the following assumptions are made:
Hypothesis no. 1: Income generated by tangible assets has an annual actualization rate ( rva)
and total expenses have an annual actualization rate ( rcha) that is recorded over the life of
assets:
–for the operational income of the asset: () ( ) 01t
a va Vt V r = + ;
–for the operational expenses of the asset: () ( ) 01t
a cha Ch t Ch r = + .
Hypothesis no. 2: The remaining value of the asset is dependent on its decreasing depreciation
during [0, T], of the form:
()4iT
aVR t Ie−
= cu ()4
4iT
aiVR t Ie−
′= − . (9)
Considering the residual value of the asset as a market-dependent constant at time T, it
is obtained after solving the equation: () () () ()0aa a aV t Ch t iVR t VR t′  − − += the following
expression for determining T:
() 00 *years .aa a
cha vaV Ch iVR tTrr−−= − (10)
In practice, a number of models have been developed to determine the optimal replace –
ment time for an asset, such as: the annual average cost of the machine (Kaufmann, 1963),
the equivalent annual cost minimization method, the minimal negative method (Terborgh,
1949), the replacement algorithm, the cost minimization model, models based on finite-ho –
rizon technical progress, or models based on Pontryagin principle (Pontryagin, Boltyanskii,
Gamkrelidze, & Mishchenko, 1962). All these models aim to determine the optimal moment
for replacing the asset motivated by the fact that the actual investment is no longer efficient.
Almost all of these models are based on the concepts of: financial asset sustainability based
on asset-generated cash flow, the residual value and the initial investment cost.
3. The concept of financial sustainability risk
Is there a risk of financial sustainability? The answer to this question is affirmative as long as
the economic benefits of the asset are the result of the influence of external factors such as
the tear of the asset, the evolution of technical progress, the technical condition of the asset
and the examples could continue. The financial sustainability risk of an asset indicates the
situation where any assets investment, whatever its nature, can generate losses for the com –
pany’s economic activity. These categories of assets cause deterioration in economic benefits
and may generate long-term losses for the company. For any assets investment is risen the
question of the return on investment. The basic condition for identifying the financial sus –
tainability risk of an asset, when the amounts are discrete, is that the net present value of the
benefit plus the residual value of the asset must be equal to the cost of its initial investment
after a relationship of the form:
() ()
( )()
( )1.
11a
aa tT
aaVR tV t Ch t I
rr −× + =++ (11)
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Technological and Economic Development of Economy. Article in press 7
Starting from the basic condition of financial sustainability, the financial sustainability
risk ( Rsf) can be quantified, for discrete values, as:
() ()
( )()
( )1
11100 % .a
aa tT
aa
sfVR tV t Ch t
rrRI −× +++= ×  (12)
For continuous values, the financial sustainability risk of an asset can be quantified as:
() () ()
0100 % .Tit iT
aa a
sfV t Ch t e VR t e
RI−−  − ×+ ×= × ∫ (13)
From the quantification of the financial sustainability risk, it is clear that an essential
component of the risk is the economic benefit generated by the asset. Thus, high economic
benefits will result in: optimal economic life for assets that are compatible with the expec –
tations of the company’s shareholders and controllable values for sustainability risk. Small
economic benefits will generate long economic lifetimes that can cause the replacement of the
asset due to the fact that it is no longer a source of added value for the company, and hence
generates high values for the financial sustainability risk. What is relevant for the financial
sustainability risk is the influence of operational and maintenance costs on the economic
benefits. These spending categories directly affect the level of economic benefits and the
financial sustainability risk.
Highlighting the impact of these spending categories on the economic benefits generated
by companies’ assets can be achieved by the economic elasticity coefficient ( ebe) established
according to the variance in economic benefits ( DBe), as well as the variation in operating
expenses ( DCha) after a relationship of the form:
.be
a
aBe
BeeCh
ChD
=D (14)
There are several situations, namely:
1) If ebe → 1, most of the economic benefit is influenced by the variation in the assets
operating and maintenance costs and hence the exposure to financial sustainability
risk is increased.
2) If ebe → 0, there is a slight influence of the assets operation and maintenance costs on
the economic benefits and therefore the exposure to financial sustainability risk is low.
Regardless of the form of the financial sustainability risk and its quantification, it is im –
portant to determine the impact of this category of risk on the economic life of an asset. Once
the financial sustainability risk has been manifested, especially when it has values close to 1,
it is important to determine whether the investment in the asset continues or it is necessary
its replacement.
4. The Mamdani Fuzzy Controller to detect
the assets financial sustainability risk
The MANDAMI fuzzy controller has been built around the elasticity coefficient that best
reflects the financial sustainabili ty risk. The first step in designing the fuzzy controller is to
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8 M.-I. Boloș et al. Detecting financial sustainability risk of the assets using MAMDANI fuzzy controller
identify the input fuzzy variables that will be set as the objective function, as well as the out –
put variable for each is tracked the impact on it when the input variables varies. The input
variables of the MAMDANI fuzzy controller are:
a) The asset operating and maintenance costs ( Cha) are expressed as a percentage of the
asset’s value using the following linguistic terms shown in the Table 1:
Table 1. The linguistic terms and the variation range of the fuzzy variable − operating costs
Linguistic terms Symbol Variation range
Operating costs
NEGATIVE ChaN 0% ÷ (–pCha %*)
Operating costs
ZERO ChaZ 0%
Operating costs
POSITIVE ChaP 0% ÷ (+pCha %*)
Note : * – *) % 100
aChapChaV= ×  – the percentage of expense in the asset value.
b) The variance of the asset operating costs ( DCha) being expressed as percentage changes
from one analysis period to the another, using the following linguistic terms (Table 2):
Table 2. The linguistic terms and the variation range of the fuzzy variable – operating cost variation
Linguistic terms Symbol Variation range
Operating cost variation
NEGATIVEDChaN 0% ÷ (–pDCha%*)
Operating cost variation
ZERODChaZ 0%
Operating cost variation
POSITIVEDChaP 0% ÷ (+pDCha%*)
Note : * – 1*) % 100 –t
tChap ChaCha+D= × the variation cost from one period to another.
c) The output variable is the economic benefits variation ( DBe) when there is a change in
operating expenses from one period to another. It is determined their impact on the
change in net benefits expressed in linguistic terms (Table 3):
Table 3. The linguistic terms and the variation range of the fuzzy variable – the economic benefits
variation
Linguistic terms Symbol Variation range
The economic benefits variation
NEGATIVEDBeN 0% ÷ (–pDBe%*)
The economic benefits variation
ZERODBeZ 0%
The economic benefits variation
POSITIVEDBeP 0% ÷ (+pDCha%*)
Note : * – 1*) % 100 –t
tBep BeBe+D= × the economic benefits variation from one period to another.
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Technological and Economic Development of Economy. Article in press 9
The second step is to establish the rules base of the fuzzy controller. For this, it starts from
the objective function of the fuzzy controller, which is the deviations minimization from the
initial asset operating costs for its life cycle, of the form:
() ()*0a aa e Ch Ch Ch k = −→ . (15)
This means that any deviation from the initial operating costs *()aCh will cause a devia –
tion of the asset’s economic benefits that are mostly influenced by the income generated by
the asset (Table 4).
Table 4. The fuzzy rule base of the controller
e (Cha)
De(Cha) ChaN ChaZ ChaP
DChaN N N Z
DChaZ N Z P
DChaP Z P P
Rule no. 1:
() ()*
a aa e Ch Ch Ch k = − is N → () ()**0 ;aa a aCh Ch k Ch k Ch−→
() () () () ()** ( ) ( 1 ) 0 1 0a aa aa a a e Ch Ch Ch k Ch Ch k is N Ch k Ch kD = − − − − < → −− <
and () ()1aaCh k Ch k >− . (16)
Conclusion : () aCh k records an increase value at time ( k) versus time ( k − 1) and is higher
than the initial reference expense* aCh. This means that the impact of this increase will cause
a reduction in economic benefits, which means () ()1 and henceeeBk Bk <− () eBk ND< .
Rule no. 2:
() ()*
a aa e Ch Ch Ch k = − is N → () ()**0 ;aa a aCh Ch k Ch k Ch−→
() () () () ()** ( ) ( 1 ) 0 1 0a aa aa a a e Ch Ch Ch k Ch Ch k is Z Ch k Ch kD = − − − − = → −− =
and () ()1aaCh k Ch k = − . (17)
Conclusion : () aCh k has the same value at time ( k) as in moment ( k − 1) and is greater
than the initial operating cost* aCh. This means that the impact of this increase will cause a
reduction in economic benefits, which means that () ( 1) and henceeeB k Bk <− () eBk ND< .
Rule no. 3:
() ()*
a aa e Ch Ch Ch k = − is N → () ()**0 ;aa a aCh Ch k Ch k Ch−→
() () () () ()** ( ) ( 1 ) 0 1 0a aa aa a a e Ch Ch Ch k Ch Ch k is P Ch k Ch kD = − − − − > → −− >
and () ()1aaCh k Ch k <− . (18)
Conclusion : () aCh k records a decrease value at time ( k) from time ( k − 1) and is higher
than the initial reference expense* aCh. This means that the impact of this increase will result
in a maintenance of the economic benefits, which means that () ()1eeBk Bk = − and hence
() .eBk ZD=
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10 M.-I. Boloș et al. Detecting financial sustainability risk of the assets using MAMDANI fuzzy controller
Rule no. 4:
() ()*
a aa e Ch Ch Ch k = − is Z → () ()**0 ;aa a aCh Ch k Ch k Ch−= →=
() () () () ()** ( ) ( 1 ) 0 1 0a aa aa a a e Ch Ch Ch k Ch Ch k is N Ch k Ch kD = − − − − < → −− <
and () ()1.aaCh k Ch k >− (19)
Conclusion: () aCh k records an increase value at time ( k) from time ( k − 1) and is equal
to initial reference expense* aCh. This means that the impact of this increase will cause a re –
duction in economic benefits, which means that () ()1eeBk Bk <− and hence () eBk ND< .
Rule no. 5:
() ()*
a aa e Ch Ch Ch k = − is Z → () ()**0 ;aa a aCh Ch k Ch k Ch−= →=
() () () () ()** ( ) ( 1 ) 0 1 0a aa aa a a e Ch Ch Ch k Ch Ch k is Z Ch k Ch kD = − − − − = → −− =
and () ()1.aaCh k Ch k = − (20)
Conclusion: () aCh k records the same value at time ( k) versus time ( k − 1) and is the
same as the initial reference expense* aCh. This means that the impact of this increase will
cause a stagnation of the economic benefits which means that () ()1eeBk Bk = − and hence
() eBk ZD= .
Rule no. 6:
() ()*
a aa e Ch Ch Ch k = − is Z → () ()**0 ;aa a aCh Ch k Ch k Ch−= →=
() () () () ()** ( ) ( 1 ) 0 1 0a aa aa a a e Ch Ch Ch k Ch Ch k is N Ch k Ch kD = − − − − > → −− >
and () ()1aaCh k Ch k <− . (21)
Conclusion: () aCh k records a decrease in value at time ( k) from moment ( k − 1) and is
the same as the initial reference expense* aCh. This means that the impact of this increase will
result in an increase in economic benefits, which means that () ()1eeBk Bk >− and hence
() eBk ZD> .
Rule no. 7:
() ()*
a aa e Ch Ch Ch k = − is P → () ()**0 ;aa a aCh Ch k Ch k Ch− >→ <
() () () () ()** ( ) ( 1 ) 0 1 0a aa aa a a e Ch Ch Ch k Ch Ch k is N Ch k Ch kD = − − − − < → −− <
and () ()1aaCh k Ch k >− . (22)
Conclusion: () aCh k records an increase value at time ( k) from time ( k − 1) and is less
than the initial reference expense* aCh. This means that the impact of this increase will cause a
stagnation of economic benefits, which means that () ()1eeBk Bk = − and hence () .eBk ZD=
Rule no. 8:
() ()*
a aa e Ch Ch Ch k = − is P → () ()**0 ;aa a aCh Ch k Ch k Ch− >→ <
() () () () ()** ( ) ( 1 ) 0 1 0a aa aa a a e Ch Ch Ch k Ch Ch k is Z Ch k Ch kD = − − − − = → −− =
and () ()1aaCh k Ch k = − . (23)
Conclusion: () aCh k records the same value at time ( k) versus time ( k − 1) and is less than
the initial reference expense* aCh. This means that the impact of this increase will result in an
increase in economic benefits, which means that () ()1eeBk Bk >− and hence () .eBk PD>
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Technological and Economic Development of Economy. Article in press 11
Rule no. 9:
() ()*
a aa e Ch Ch Ch k = − is P → () ()**0 ;aa a aCh Ch k Ch k Ch− >→ <
() () () () ()** ( ) ( 1 ) 0 1 0a aa aa a a e Ch Ch Ch k Ch Ch k is P Ch k Ch kD = − − − − > → −− >
and () ()1aaCh k Ch k <− . (24)
Conclusion: () aCh k records a decrease in the value at time ( k) from moment ( k − 1) and
is lower than the initial reference expense* aCh. This means that the impact of this increase
will result in an increase in economic benefits, which means that () () 1eeBk Bk >− and hence
() .eBk ZD>
The inference operation of the fuzzy rule-based of the controller which is elaborated to
detect the asset’s financial sustainability risk is of the type min-max. The formula of the impli –
cation relation used for the mathematical description of each rule from the fuzzy rule-based
is of type min − MAMDANI. The aggregating method of the partial conclusions generated
by the rules in the fuzzy rule-based is of type max.
The final step of the fuzzy controller is to determine the value of the output variable ob –
tained after the fuzzy inference operation. This operation is known in the literature as the
centroid deafuzzification or the Center of Aria (COA) method. The results obtained are a
control curve without sudden variations. The formula for determining the value of the output
variable based on the output fuzzy set is as follows (Figure 1):
–For continuous variables:
()
()*
*%
%*
%
%.p
Bep
p
BepBe Be d Be
Be
Be d Be+
D−
+
D−Dµ D D
D=
µ DD∫
∫ (25)
–For discrete variables:
()
()*
*%
%*
%
%.p
Bep
p
BepBe Be
Be
Be+
D−
+
D−Dµ D
D=
µD∑
∑ (26)
The output variable value of the fuzzy controller for ( DBe*) gives us information about the
variable trend when there is a change in the value of the input variables: ( Cha) and ( DCha).
Figure 1. The output fuzzy set Z* for establishing the output variablemDBe
Z*
z*mDBe1
mDBe2
DBe
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12 M.-I. Boloș et al. Detecting financial sustainability risk of the assets using MAMDANI fuzzy controller
5. Validation of the resu lts obtained with the fuzzy controller
The output variable of the fuzzy controller sets the economic benefit variation ranges ( DBe)
when a change in the value of the input variables mentioned above occurs. These output
values obtained for the fuzzy variable must be converted to values that can be used to analyze
the asset financial sustainability risk. From the fuzzy rule base of the fuzzy controller can be
distinguished three situations namely:
Situation 1: When 0Be> or 1 0ttBe Be+−> results that 1ttBe Be+> . This assumes that
the economic benefits increase from time to time and the financial sustainability risk is sig –
nificantly reduced. This risk reduction is possible either as a result of the use of assets under
appropriate technical conditions or as a result of compliance with the technical maintenance
program. The financial sustainability risk with converted and discrete values will be deter –
mined as follows:
( )
( )1%
.1%a
aa
aBe p Be
BeersfCh p Ch
ChD +D
=D +D (27)
Situation 2: When 0Be= or 1 0ttBe Be+−= results that 1 . ttBe Be+= This assumes that
the economic benefits are constant from one period to another. This situation, though rarely,
catch the influence of the asset’s operating expenses variation on the financial sustainability
risk. In this situation the financial sustainability risk with values converted from the control –
ler will be determined with the relation:
( ) .1%a
aa
aBe
BeersfCh p Ch
ChD
=D ±D (28)
Situation 3 : When 0BeD< and 1 0ttBe Be+−< results that1 ttBe Be+< . This implies a
diminution of the economic benefits from one period to another. The situation may indi –
cate exposure to financial sustainability risk, either as a result of the decrease in production
capacity due to technical reasons, or as a result of the uncontrolled increase in the operat –
ing expenses. Regardless of the nature of these causes, assets must be carefully monitored
to avoid situations where asset investments become economically inefficient. The financial
sustainability risk with values converted from the controller will be as follows:
( )
( )1%
.1%a
aa
aBe p Be
BeersfCh p Ch
ChD −D
=D ±D (29)
6. Simulation of the Mamdani Fuzzy Controller in the financial
sustainability risk analysis for ski slopes
The simulation was conducted for a company that manages a ski area in the Southern Car –
pathians of Romania. The focus was pointed on tangible assets such as artificial snow instal –
lations and Ratrak equipment.
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Technological and Economic Development of Economy. Article in press 13
The MAMDANI fuzzy controller developed works having as input variables the operat –
ing expenses of the asset ( Cha) and their variation ( DCha). For the operating costs ( Cha)
where established variation ranges from (−50%, +50%). This means that two asset life cycles
have been tested, namely: a period during which the operating expenses decrease (a good
asset behavior) and another period when the asset operating expenses grow with impact on
economic benefits. Such a period shall be recorded, in particular, when the asset no longer
complies with the technical conditions of operation. During these periods, technical defects
are frequent.
For the operating costs variation ( DCha) where also established variation ranges from
(−30% to +30%). The change in operating expenses between two periods ( k) and ( k + 1) was
considered positive, respectively: ( DCha > 0) and then negative ( DCha < 0). Two situations
are recorded: operating expenses that increase between two consecutive periods with a direct
impact on reducing economic benefits and operating expenses that diminish from one period
to another with an impact on increasing the economic benefits and reducing the financial
sustainability risk. Two scenarios were simulated:
Table 5. The first fuzzy controller simulation with
progressive increase of the variation intervals
Cha% DCha% DBN% (esrf)
−50 −30 −20.9 1.13
−45 −27 −20.2 1.09
−40 −24 −19.5 1.06
−35 −21 −18.9 1.03
−30 −18 −18.3 1.00
−25 −15 −17.8 0.97
−20 −12 −17.3 0.94
−15 −9 −16.9 0.91
−10 −6 −16.6 0.89
−5 −3 −16.4 0.86
0 0 −1.6 0.98
5 3 −1.61 0.96
15 6 −1.67 0.93
20 9 −1.72 0.90
25 12 −1.8 0.88
30 15 −1.9 0.85
35 18 −2.04 0.83
40 21 −5.71 0.87
45 24 −2.5 0.79
50 27 −1.82 0.77Table 6. Simulation of the fuzzy controller no. 2
with alternating variation intervals
Cha% DCha% DBN% (esrf)
−45 −25 −23.5 1.02
−30 −25 −22.7 1.03
−15 −10 −19.2 0.90
−5 −10 −17.8 0.91
−25 15 19.2 1.04
−15 30 0 0.77
−5 0 −1.6 0.98
−5 10 17.8 1.07
−5 20 20.9 1.01
5 −25 −1.42 1.31
10 −20 −4.93 1.19
20 −5 18.3 1.25
40 −10 22.3 1.36
50 −25 0 1.33
0 20 −1.82 0.82
10 5 −1.67 0.94
20 5 −1.9 0.93
40 5 −4.93 0.91
50 20 0 0.83
5 5 −1.64 0.94
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14 M.-I. Boloș et al. Detecting financial sustainability risk of the assets using MAMDANI fuzzy controller
The first scenario , with the first fuzzy controller, consists in input variables that progres –
sively change. The variation ranges of the input variables in the fuzzy controller were split
using local gradients in 5-point subintervals, being formed 20 pairs of values presented in
the table below. The tangible asset taken into account generates operating expenses of 4.000
per month and an operational income of 5.000 per month with a net benefit of 1.000 per
month (Table 5).
Variation ranges were divided into two categories, namely:
Negative variation ranges with decreasing operating costs ( )0aCh< and decreasing oper –
ating costs variations ( )0aChD< . This means that operating costs are higher than those origi –
nally established ()* aCh , but also from one period to the next, respectively () () 1aaCh k Ch k >−
. The direct consequence is the diminution of the economic benefits obtained with the asset
over its lifecycle. On these variation ranges where the asset’s operating expenses can regis –
ter values within the range 0%, 50%aCh∈− and the operating expenses variation from
one period to the next may register values in the range 0%, 30% ,aChD∈ −  the variance
of the economic benefits obtained with the fuzzy controller can take values in the range
16.4%, 20.4%BeD ∈− . Greater variations in economic benefits than those generated using
fuzzy controller determine the emergence of the financial sustainability risk, with immediate
consequences on the efficiency of investments regardless of their nature.
Positive variation ranges of operating expenses ( )0aCh> and positive operating costs
variations ( )0aChD< . This assumes that operating costs are lower than those originally set
()*
aCh and decrease from one period to the next () () 1aaCh k Ch k <− . The immediate con –
sequence is that there are insignificant variations in economic benefits from one operating
period to another. The ranges of operating cost variation are between 0%, 50% ,aCh∈+
while the operating cost variation range 0%, 30% .aChD∈ +  The variation in economic
benefits is in the range 1.61%, 5.71%BeD ∈− − . It can be argued that on these types of
intervals the financial sustainability risk interferes in a small extent.
The conclusion is that it is possible to measure the asset financial sustainability risk at
every moment of its operation. The deviation of the economic benefits from the initial value
is a measure of the intensity of this type of risk. The reliability of the fuzzy controller can be
measured with the help of the indicator ( esrf), whose values are closer to 1, fact that confirms
the usefulness of the fuzzy rule base and of the controller (Figure 2).
The second simulation scenario is the one in which are alternating ranges with positive
values and ranges with negative values. The variation ranges of the input variables were
subtracted by 5%, in order to draw conclusions about the economic benefits evolution and
0%, 50%aCh∈±, 0%, 30%aChD∈ ± . The results are presented in the Table 6.
Thus four value ranges are distinguished:
a) Negative variation interval, in which both ( )0aCh< and ( )0aChD< record nega –
tive values assuming that 17.8%, 23.5%BeD ∈− − . Beyond these economic benefits
variations, it is possible that the company’s investments in assets to be made under
conditions of economic inefficiency, so that the asset needs to be replaced;
b) Negative variation interval for operating costs ( )0aCh< and positive values for
(D ( )0aChD> . The economic significance of these variations shows that operating
costs are higher than those initially recorded, however they are decreasing from one
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Technological and Economic Development of Economy. Article in press 15Figure 2. Evolution of economic benefits and financial sustainability risk
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16 M.-I. Boloș et al. Detecting financial sustainability risk of the assets using MAMDANI fuzzy controller
period to the next. The immediate consequence is that the change in economic benefits
is positive, which implies a reduction in the financial sustainability risk.
c) Positive variation interval for operating costs ( )0aCh> and negative values for
(D ( )0aChD< . The economic significance of these variations is that operating costs
are lower than those initially recorded, while the same categories of expenditure are
increasing from one period to the next. The variation in economic benefits can be posi –
tive or negative. For periods in which the change in economic benefits is positive, the
financial sustainability risk decreases, while negative variations in economic benefits
increase the financial sustainability risk.
d) Positive variation interval, for both ( )0aCh> and ( )0aChD> . In this situation, the
variation in economic benefits remains almost unchanged, which confirms the reli –
ability of the elaborated fuzzy controller. The recovery periods of the invested capital
are in line with the expectations of the company’s shareholders.
The MAMDANI fuzzy controller was tested on the assets of the company managing the
skiing area, especially those assets that are essential to the business (Figure 3). Based on the
technical capacity of assets, were considered in the analysis the operational expenses and
revenues generated for 12 periods of time.
It could be noticed that the real net benefits of the company are above the values gener –
ated by the MAMDANI fuzzy controller for the first two periods, illustrating that the com –
pany is not exposed to the financial sustainability risk. For the others 10 periods of time the
net benefits values are situated below the optimal values calculated by the fuzzy controller,
indicating a susceptible risk area and thus requiring management intervention in real-time.
Figure 3. Testing the MAMDANI fuzzy controller on the artificial
snow installations and Ratrak equipment–200–150–100–50050100150200
No risk exp osur e
Risk zo ne
BN
BN co ntroller125175
137.74 139.1 144.48150.03 151.51 157.55165.8174.38
177.12
–144–106–5037107143
94
37
–5031132.51127.48
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Technological and Economic Development of Economy. Article in press 17
Conclusions
Asset regardless of their nature need to be analyzed based on the financial sustainability
risk defined as the risk category in which the company’s investments in that asset are no
longer warranted from the economic efficiency point of view. To measure and identify this
risk category, a fuzzy controller has been developed that is based on two input variables that
are determinant for any asset, namely: the operating costs and the operating cost variations
between two time points. The exit variable was established to be the variation in the asset
economic benefits considered to be important in defining the asset financial sustainability.
The controller’s operation identifies four categories of important situations, namely:
Situation no. 1: Negative operating costs ( )0aCh< and negative operating cost varia –
tions ( )0aChD< . The economic significance of this situation indicates that the operating
expenses of the asset are higher than the initial operating expenses ( )*
aaCh Ch> , but also
that these spending categories are higher from one period to another () ()1aaCh k Ch k >− .
The impact on economic benefits is negative ( DBe <0), which implies an increase in the
financial sustainability risk.
Situation no. 2: Positive operating costs ( )0aCh> and negative operating cost variations
( )0aChD< . The economic significance of this situation indicates that the operating costs are
lower than the initial operating expenses ( )*
aaCh Ch< but still increasing from one period
to another () () 1.aaCh k Ch k >− The impact on economic benefits is zero ( DBe = 0), while
keeping the financial sustainability risk under control.
Situation no. 3: Negative operating costs ( )0aCh< and positive operating cost varia –
tions ( )0.aChD> This situation reflects that the operating costs are higher than the initial
expenses ( )*
aaCh Ch> but still decreasing from time to time () ()1aaCh k Ch k <− . The im –
pact on economic benefits is zero ( DBe = 0), while keeping the financial sustainability risk
under control.
Situation no. 4: Positive operating costs ( )0aCh> and positive operating cost variations
( )0.aChD> This situation indicated that operating expenses are lower than the initial op –
erating expenses ( )*
aaCh Ch< while the change in operating expenses from one period to
another is decreasing () () ( )1.aaCh k Ch k <− The impact on economic benefits is positive
(DBe > 0), with the reduction of the financial sustainability risk.
The major advantage of the fuzzy controller is that it measures the financial sustainability
of the asset at every desired time. Any value recorded by the economic benefit, outside the
range of variation indicated by the MAMDANI fuzzy controller, indicates that the asset is
exposed to financial sustainability risk and generates investments that no longer add value
to the company. For assets chains, the use of the fuzzy controller involves measuring the in –
dividual financial sustainability risk of each asset, in order to determine which of the assets
should be replaced as a result of their inefficiency.
Funding
No grants were used for the research and publication of this article.
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18 M.-I. Boloș et al. Detecting financial sustainability risk of the assets using MAMDANI fuzzy controller
Author contributions
The paper is a result of a collaborative work. Bradea Ioana and Diana Sabău-Popa did the
literature review, Bolos Marcel and Bradea Ioana worked at the methodology, Boloș Marcel
and Ilie Laurențiu did the valorification the results and the simulation. Boloș Marcel, Bradea
Ioana and Ilie Laurențiu prepare the first draft and Diana Sabău-Popa worked at the abstract,
language and final revisions.
Disclosure statement
The authors declare no conflict of interest.
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