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.
© 2020 Copyright for this paper by its authors. Use permitted under Creative CPWrEooUrckReshdoinpgs IhStpN:/c1e6u1r3-w-0s.o7r3g CCoEmUmoRns WLiceonrsekAsthtriobuptioPnr4o.0cIneteerdnaitniognasl ((CCC EBYU4R.0)-.WS.org) = ( 1) + ⋯ + ( ) = ∑ ( ) . (3)
1 =1
Linguistic variables are also used in fuzzy systems.
Linguistic variables are the statements of natural language, which are descriptions of fuzzy sets defined on a specific space. Fuzzy systems may be used in many ifelds. 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 diferent 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 , , resTpheectoivveelrya.ll membership function is presented as fol- , ( ≤ ≤ ≤ ) has to be replaced by individual numbers typical to each of the specific membership lows functions. For example to count the membership of argument (relevant to the age) to fuzzy set "middle", if < the parameters , , have to be replaced by 20, 40, 60 respectively. The next step to cost out the risk is building IF-THEN rules. ⎧ ⎪0 ⎪ ⎪ ⎪⎪( − )/( − ) if ∈ [, ] ⎪ /ℎℎ = ⎨1 if ∈ [, ] ⎪ ⎪ ⎪( − )/( − ) if ∈ [, ] ⎪ ⎪⎪⎩0 if > ⎪⎪⎧0 if < ⎪ ⎪ ⎪( − )/( − ) if ∈ [, ] = ⎪⎪⎨( − )/( − ) if ∈ [, ] ⎪⎪⎪0 if > ⎩ . (4) .
(5) 1 and 2
2 and … then where is a variable of the consequence whose value is inferred.
Taking into consideration circumstances mentioned above, we get undermentioned rules: 1. If the driver is young and the power of the enData: age of driver, power of engine, car’s
production date Result: risk
; ; ; Form the membership function for every linguistic variables; [27]; ℎ [27]; Check the membership of each data to appropriate membership functions ; ∶= 0; ∶= 0; ∶= 0; for i < 3 do for j < 3 do for k<3 do [ ] = ( end end ) ⋅ ( ) ⋅ (
); end for i< 27 do
Assign value of result for each rules; end for i<27 do
end
end for i<27 do ∶= 0; + = [ ] ⋅ ℎ ∶= 0; + = [ ];
[ ]; ∶= ⋅ 100/ Display amount of ;
Algorithm 1: Fuzzy system in insurance.
; gine is low and the car is new, then the risk is low. 2. If the driver is young and the power of the engine is low and the car is middle-age, then the risk is medium-low. 3. If the driver is young and the power of the engine is low and the car is old then the risk is medium. 4. If the driver is young and the power of the engine is medium and the car is new, then the risk is medium-low. 5. If the driver is young and the power of the engine is medium and the car is middle-age, then the risk is medium. 6. If the driver is young and the power of the engine is medium and the car is old, then the risk is medium-high. 7. If the driver is young and the power of the engine is high and the car is new, then the risk is medium. 8. If the driver is young and the power of the engine is high and the car is middle-age, then the risk is medium-high. 9. If the driver is young and the power of the engine is high and the car is old, then the risk is high. 10. If the driver is in middle age and power of the engine is low and the car is new, then the risk is low. 11. If the driver is in middle age and power of the engine is low and the car is middle-age, then the risk is medium-low. 12. If the driver is in middle age and power of the engine is low and the car is old, then the risk is medium. 13. If the driver is in middle age and power of the engine is medium and the car is new, then the risk is low. 14. If the driver is in middle age and power of the engine is medium and the car is middle-age, then the risk is medium-low. 15. If the driver is in middle age and power of the engine is medium and the car is old, then the risk is medium-high. 16. If the driver is in middle age and power of the engine is high and the car is new, then the risk is medium-low. 17. If the driver is in middle age and power of the 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 engine is high and the car is old, then the risk is medium-high. 19. If the driver is old and the power of the engine is
low and the car is new, then the risk is medium. 20. If the driver is old and the power of the engine is low and the car is middle-age, then the risk is medium-high. 21. If the driver is old and the power of the engine
is low and the car is old, then the risk is high. 22. If the driver is old and the power of the engine is medium and the car is new, then the risk is medium. young middle
old step - inference. Here for each of rules, we use a for- linguistic variables as well as modifying values. . The result of that formula is a percentage of risk. Fuzzy system algorithm is presented in Alg. 1.
As part of checking the propriety of the proposed solution, for the selected data (see Tab. 1) we described average steps described in Tab. 2 . The table shows only results for 8 rules, but we should note that the implemented application has to check each of the rules respectively and count each of the elements. The sample data shows information that was used for analyzing the proposed system. A presented solution was implemented in C# language.
In Tab. 2, a sample output result was presented. In these experiments, we analyze the results output for a drive aged 25 and a six-year car with a capacity of 100.
It is easy to notice that the more information about the high.
is high. mula The result clusion.
26. If the driver is old and the power of the engine is high and the car is middle-age, then the risk is high and then a car is old, the risk is high.
The results of the implication can be presented as a membership function (see Fig. 2).
By having rules and values, we can go to the next 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 is medium-high.
The reason for that is the defuzzification – we have 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 sysmedium and the car is old, then the risk is high. 25. If the driver is old and the power of the engine is tems work more efectively when we have more linguistic variables. However, more variables contribute high and the car is new, then the risk is medium- to modeling more rules and analyzing relationships 27. If the driver is old and the power of the engine the ability to model fuzzy rules and their multitude (, ,
) = ( ) ⋅ ( ) ⋅ ( ). means the activation level of each con(6)
There are many diferent methods of defuzzification available. In this paper, assigning weight every conclusion holds by using the center of area (COA).
for The general visualization of the proposed system is presented in Fig. 1.
In this paper, we proposed a solution for calculating the risk of causing a car accident and directly the possibility of insurance. Proposed rules and used system based on If-Then rules shows a large dependence on the number of variables. The more variables, the more between data.
Obtained results show the possibilities of using the Takagi-Sugeno system for analyzing risk for insurance companies. In conducted experiments, we analyzed (see Tab. 3). During modeling the technique, it was also noticed that the biggest problem with this approach is rule modeling. During the simulation, it was noticed that it is a flexible system in terms of use, which was reflected in the possibilities of introducing new (a) (b) (c) Figure 3: Charts of membership functions for each of liguistic variables.
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These are undoubted advantages, although rule modeling is a time-consuming activity that needs to be analyzed without using self-adaptive methods.
Based on the obtained results, this approach is very promising. In future works, we plan to analyze and model a solution that can generate rules automatically. tems,
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