=Paper= {{Paper |id=Vol-2698/paper03 |storemode=property |title=Fuzzy System Model For Insurance Industry |pdfUrl=https://ceur-ws.org/Vol-2698/p03.pdf |volume=Vol-2698 |authors=Karolina Tatarczyk |dblpUrl=https://dblp.org/rec/conf/ivus/Tatarczyk20 }} ==Fuzzy System Model For Insurance Industry== https://ceur-ws.org/Vol-2698/p03.pdf

Fuzzy System Model For Insurance Industry
Karolina Tatarczyka
a Faculty of Applied Mathematics, Silesian University of Technology), Kaszubska 23, 44-100 Gliwice, Poland

Abstract
The insurance industry has to perform an accident risk assessment when insurance is purchased by drivers. Unfortunately,
but this type of assessment can be quite often problematic because it depends on many parameters. The most common criteria
are the age of the driver, the engine power of the car and its year of manufacture. This paper presents the Takagi-Sugeno
system model to assess risk for driver insurance companies. The proposed model presents a fuzzy approach to using these
three parameters. The results were presented and discussed due to the pros and cons of the proposed solution and practical
use.

Keywords
Takagi-Sugeno system, fuzzy logic, insurance problem

1. Introduction that can be useful for a doctor for analyzing some pa-
tient’ data [12, 13, 14, 15].
Fuzzy logic belongs to one of the most popular ar- A deep learning also finds possible use in security,
tificial intelligence techniques next to artificial neu- which is described in a blockchain approach for ana-
ral networks. Recent years have allowed for great de- lyzing weights in neural systems [16].
velopment, not only theoretical but also practical, as In this paper, we propose a system model for risk
evidenced by numerous studies and implementations. analysis in companies insuring drivers using three pa-
This is particularly evident in the area of research on rameters. The described solution is based on a fuzzy
uncertain rule-based fuzzy models and hesitant fuzzy system using If-Then rules and was tested for sample
data [1, 2]. Also, many generalizations of this type of data and discussed compared to the pros and cons of
logic is analyzed for a much broader practical appli- this solution in practice.
cation. A special case is the analysis of generalized
orthopair fuzzy sets in [3]. Again the authors of [4]
compare different types of fuzzy logic systems used for 2. Problem analysis
control problems. All of these theoretical and compar-
ative studies are very important but practical use has Fuzzy systems find practical use in many areas of life.
a big impact on today’s industry. By using fuzzy systems we can define the membership
The main application of the fuzzy approach and the to the selected set. In comparison to the classic system,
artificial intelligence, in general, is making decisions where the only values are 0 and 1, the fuzzy system
about the information which is given to the system [5, value of the argument can be a number between 0 and
6]. In [7], the authors presented a detection and classi- 1. Moreover, one argument can belong to many sets at
fication solution based on neural networks and fuzzy various levels. Fuzzy set 𝐴 is a set of elements, which
controllers that decide about the clustering of images. partly belongs to it.
An interesting solution was described in [8], where the
idea of smartphone-based intelligent indoor position- 𝐴 = {(𝑋 , 𝜇𝐴 (𝑥)) ∶ 𝑥 ∈ 𝑋 }, (1)
ing was described in detail. Fuzzy time series fore- where
casting is a less popular use of these solutions [9] or 𝜇𝐴 ∶ 𝑋 → [0, 1]. (2)
public support for insurgency and terrorism [10]. An-
other possible application is a multiword search over and 𝑋 is a space with complete number of elements,
encrypted data [11] and medicine. In medicine, the ar- 𝑋 = {𝑥1 , 𝑥2 , … , 𝑥𝑛 }
tificial intelligence use it is a very important solution 𝑛
𝜇 (𝑥1 ) 𝜇 (𝑥𝑛 ) 𝜇 (𝑥𝑖 )
𝐴= 𝐴 +⋯+ 𝐴 =∑ 𝐴 . (3)
𝑥1 𝑥𝑛 𝑖=1 𝑥𝑖
IVUS 2020: Information Society and University Studies, 23 April 2020,
KTU Santaka Valley, Kaunas, Lithuania Linguistic variables are also used in fuzzy systems.
" karotat400@student.polsl.pl (K. Tatarczyk)

Linguistic variables are the statements of natural lan-
© 2020 Copyright for this paper by its authors. Use permitted under Creative guage, which are descriptions of fuzzy sets defined on
Commons License Attribution 4.0 International (CC BY 4.0).
CEUR
Workshop
Proceedings
http://ceur-ws.org
ISSN 1613-0073 CEUR Workshop Proceedings (CEUR-WS.org) a specific space. Fuzzy systems may be used in many
Figure 1: Flow chart of the fuzzy logic system.

fields. In this paper, we focus on using fuzzy systems
in the car insurance industry, especially in the case of
calculating the risk.

2.1. Fuzzy system
As an example in car insurance, the linguistic variables
that are used are describing the age of a driver, the
power of the engine and the car’s production date. For
each of them, there are three different labels describing
the state. For the age: "young", "middle", "old". For
the engine’s power: "low", "medium", "high". For the Figure 2: Chart of a membership function.
car’s production date: "old", "middle", "new". Charts
3 are presenting membership functions for the age of
the driver, power of engine and car’s production date To count the membership each set’s parameters 𝑎, 𝑏,
respectively. 𝑐, 𝑑 (𝑎 ≤ 𝑏 ≤ 𝑐 ≤ 𝑑) has to be replaced by individual
The overall membership function is presented as fol- numbers typical to each of the specific membership
lows functions. For example to count the membership of
argument 𝑥 (relevant to the age) to fuzzy set "middle",
⎧
⎪ 0 if 𝑥 < 𝑎 the parameters 𝑎, 𝑏, 𝑐 have to be replaced by 20, 40, 60
⎪
⎪
⎪ (𝑥 − 𝑎)/(𝑏 − 𝑎) if 𝑥 ∈ [𝑎, 𝑏] respectively. The next step to cost out the risk is build-
⎪
⎪ ing IF-THEN rules.
𝜇𝑙𝑜𝑤/ℎ𝑖𝑔ℎ = ⎨1 if 𝑥 ∈ [𝑏, 𝑐] . (4)
⎪
⎪
⎪
⎪ (𝑑 − 𝑥)/(𝑑 − 𝑐) if 𝑥 ∈ [𝑐, 𝑑] If 𝑥1 𝑖𝑛 𝐴1 and 𝑥2 𝑖𝑛 𝐴2 and … then 𝑦
⎪
⎩0 if 𝑥 > 𝑑
⎪
where 𝑦 is a variable of the consequence whose value
⎧ is inferred.
⎪ 0 if 𝑥 < 𝑎
⎪
⎪ Taking into consideration circumstances mentioned
⎪(𝑥 − 𝑎)/(𝑏 − 𝑎) if 𝑥 ∈ [𝑎, 𝑏]
⎪
above, we get undermentioned rules:
𝜇𝑚𝑒𝑑𝑖𝑢𝑚 = ⎨ . (5)
⎪
⎪ (𝑐 − 𝑥)/(𝑐 − 𝑏) if 𝑥 ∈ [𝑏, 𝑐]
⎪ 1. If the driver is young and the power of the en-
⎪
⎪ 0 if 𝑥 > 𝑐
⎩

17
gine is medium and the car is middle-age, then
Data: age of driver, power of engine, car’s the risk is medium.
production date 6. If the driver is young and the power of the en-
Result: risk gine is medium and the car is old, then the risk
𝑎𝑔𝑒; is medium-high.
𝑝𝑜𝑤𝑒𝑟; 7. If the driver is young and the power of the en-
𝑦𝑒𝑎𝑟; gine is high and the car is new, then the risk is
Form the membership function for every medium.
linguistic variables; 8. If the driver is young and the power of the en-
𝑤[27]; gine is high and the car is middle-age, then the
𝑤𝑒𝑖𝑔ℎ𝑡[27]; risk is medium-high.
Check the membership 𝜇 of each data to 9. If the driver is young and the power of the en-
appropriate membership functions ; gine is high and the car is old, then the risk is
𝑖 ∶= 0; high.
𝑗 ∶= 0; 10. If the driver is in middle age and power of the
𝑘 ∶= 0; engine is low and the car is new, then the risk is
for i < 3 do low.
for j < 3 do 11. If the driver is in middle age and power of the
for k<3 do engine is low and the car is middle-age, then the
𝑤[𝑙] =
risk is medium-low.
𝜇𝐴𝑖 (𝑎𝑔𝑒) ⋅ 𝜇𝐵𝑗 (𝑝𝑜𝑤𝑒𝑟) ⋅ 𝜇𝐶𝑘 (𝑦𝑒𝑎𝑟);
12. If the driver is in middle age and power of the
end
engine is low and the car is old, then the risk is
end medium.
end 13. If the driver is in middle age and power of the
for i< 27 do engine is medium and the car is new, then the
Assign value of result for each rules; risk is low.
end
14. If the driver is in middle age and power of the
𝑛𝑢𝑚𝑒𝑟𝑎𝑡𝑜𝑟 ∶= 0;
engine is medium and the car is middle-age, then
for i<27 do
the risk is medium-low.
𝑛𝑢𝑚𝑒𝑟𝑎𝑡𝑜𝑟+ = 𝑤[𝑖] ⋅ 𝑤𝑒𝑖𝑔ℎ𝑡[𝑖];
end 15. If the driver is in middle age and power of the
𝑑𝑒𝑛𝑜𝑚𝑖𝑛𝑎𝑡𝑜𝑟 ∶= 0; engine is medium and the car is old, then the
for i<27 do risk is medium-high.
𝑑𝑒𝑛𝑜𝑚𝑖𝑛𝑎𝑡𝑜𝑟+ = 𝑤[𝑖]; 16. If the driver is in middle age and power of the
end engine is high and the car is new, then the risk
𝑟𝑖𝑠𝑘 ∶= 𝑛𝑢𝑚𝑒𝑟𝑎𝑡𝑜𝑟 ⋅ 100/𝑑𝑒𝑛𝑜𝑚𝑖𝑛𝑎𝑡𝑜𝑟; is medium-low.
Display amount of 𝑟𝑖𝑠𝑘; 17. If the driver is in middle age and power of the
Algorithm 1: Fuzzy system in insurance. engine is high and the car is middle-age, then
the risk is medium.
18. If the driver is in middle age and power of the
gine is low and the car is new, then the risk is engine is high and the car is old, then the risk is
low. medium-high.
2. If the driver is young and the power of the en- 19. If the driver is old and the power of the engine is
gine is low and the car is middle-age, then the low and the car is new, then the risk is medium.
risk is medium-low. 20. If the driver is old and the power of the engine
3. If the driver is young and the power of the en- is low and the car is middle-age, then the risk is
gine is low and the car is old then the risk is medium-high.
medium. 21. If the driver is old and the power of the engine
4. If the driver is young and the power of the en- is low and the car is old, then the risk is high.
gine is medium and the car is new, then the risk 22. If the driver is old and the power of the engine
is medium-low. is medium and the car is new, then the risk is
5. If the driver is young and the power of the en- medium.

18
Table 1 In conclusion to get the final result, all data have to
Tables with examples of linguistic variables values and their be apllied to the formula:
affiliation to each set.
𝑛
Driver’s age young middle old
20 1 0 0
∑ 𝛿𝑖 ⋅ 𝜇𝑅 (𝑥𝑖 , 𝑦𝑖 , 𝑧𝑖 )
21 0.95 0.05 0 𝑦 = 𝑖=1𝑛 . (7)
25 0.75 0.25 0
30 0.5 0.5 0
∑ 𝜇𝑅 (𝑥𝑖 , 𝑦𝑖 , 𝑧𝑖 )
35 0.25 0.75 0
𝑖=1
40 0 1 0 The result of that formula is a percentage of risk. Fuzzy
44 0 0.8 0.2
50 0 0.5 0.5 system algorithm is presented in Alg. 1.
80 0 0 1
Engine’s power low medium high
50
75
1
0.9
0
0.1
0
0
3. Experiments
80 0.8 0.2 0
100 0.4 0.6 0 As part of checking the propriety of the proposed so-
110
145
0.2
0
0.8
0.5
0
0.5
lution, for the selected data (see Tab. 1) we described
160 0 1 0 average steps described in Tab. 2 . The table shows
180 0 0.2 0.8 only results for 8 rules, but we should note that the
190 0 0 1
Year of production old medium new implemented application has to check each of the rules
1998 1 0 0 respectively and count each of the elements. The sam-
2001
2004
0.875
0.5
0.125
0.5
0
0
ple data shows information that was used for analyz-
2006 0.25 0.75 0 ing the proposed system. A presented solution was
2008 0 1 0 implemented in C# language.
2009 0 0.857 0.143
2010 0 0.714 0.286 In Tab. 2, a sample output result was presented. In
2012 0 0.429 0.571 these experiments, we analyze the results output for a
2017 0 0 1
drive aged 25 and a six-year car with a capacity of 100.
It is easy to notice that the more information about the
23. If the driver is old and power of the engine is driver, the better and more accurate are the results.
medium and the car is middle-age, then the risk The reason for that is the defuzzification – we have
is medium-high. more specific elements of summation in the last step
24. If the driver is old and the power of the engine is (called defuzzification). It allows saying that fuzzy sys-
medium and the car is old, then the risk is high. tems work more effectively when we have more lin-
25. If the driver is old and the power of the engine is guistic variables. However, more variables contribute
high and the car is new, then the risk is medium- to modeling more rules and analyzing relationships
high. between data.
26. If the driver is old and the power of the engine Obtained results show the possibilities of using the
is high and the car is middle-age, then the risk Takagi-Sugeno system for analyzing risk for insurance
is high. companies. In conducted experiments, we analyzed
27. If the driver is old and the power of the engine the ability to model fuzzy rules and their multitude
is high and then a car is old, the risk is high. (see Tab. 3). During modeling the technique, it was
also noticed that the biggest problem with this approach
The results of the implication can be presented as a is rule modeling. During the simulation, it was no-
membership function (see Fig. 2). ticed that it is a flexible system in terms of use, which
By having rules and values, we can go to the next was reflected in the possibilities of introducing new
step - inference. Here for each of rules, we use a for- linguistic variables as well as modifying values.
mula
𝜇𝑅 (𝑥, 𝑦, 𝑧) = 𝜇𝐴 (𝑥) ⋅ 𝜇𝐵 (𝑦) ⋅ 𝜇𝐶 (𝑧). (6)
The result 𝜇𝑅 means the activation level of each con- 4. Conclusion
clusion.
In this paper, we proposed a solution for calculating
There are many different methods of defuzzifica-
the risk of causing a car accident and directly the pos-
tion available. In this paper, assigning weight 𝛿𝑖 for
sibility of insurance. Proposed rules and used system
every conclusion holds by using the center of area (COA).
based on If-Then rules shows a large dependence on
The general visualization of the proposed system is
the number of variables. The more variables, the more
presented in Fig. 1.

19
(a)

(b)

20
Table 2
Table of selected rules and the conclusion, when drivers are 21, the engine’s power is 100 and the year of production is 2014
(other rules will give us zero results).
Number of rule 𝑖 𝜇𝑎𝑔𝑒 (25) 𝜇𝑝𝑜𝑤𝑒𝑟 (100) 𝜇𝑦𝑒𝑎𝑟 (2014) 𝜇𝑅 result of rule 𝜇𝑅 ⋅ 𝛿𝑖
1. 0.75 0.4 0.875 0.2625 low 0.0525
2. 0.75 0.4 0.125 0.0375 medium-low 0.015
4. 0.75 0.6 0.875 0.39375 medium-low 0.1575
5. 0.75 0.6 0.125 0.05625 medium 0.028125
10. 0.25 0.4 0.875 0.0875 low 0.0175
11. 0.25 0.4 0.125 0.0125 medium-low 0.005
13. 0.25 0.6 0.875 0.13125 low 0.02625
14. 0.25 0.6 0.125 0.01875 medium-low 0.0075
SUM 1 0.309375

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2
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𝑛2
𝑛 3 775.
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⋮ ⋮ Lessening stress and anxiety-related behaviors
𝑘 𝑘𝑛
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