# Risk Assessment in the Design of Railroad Control Command and Signaling Devices Using Fuzzy Sets
## 3 Application of Fuzzy Sets  
Risk analysis is very difficult to implement in investment projects. It is not easy to see and adequately quantify risk factors in an increasingly volatile environment. Therefore, this article proposes the use of fuzzy sets. The great advantage of the fuzzy set theory is that it allows imprecise concepts to be described in mathematical language and manipulated and decisions to be made [38; 40]. An analysis of data collected on the basis of the results of surveys and workshop meetings, which are input data for the use of fuzzy sets, was carried out. The analysis was carried out in the Matlab R2022a environment as part of the toolbox shown in Figure 1.  
The fuzzy set X is defined by a membership function \(\mu(x_{i})\), which associates each value within X with a real number from 0 to 1, denoting the degree of membership in X. The membership function must satisfy one condition: the membership degree for all fuzzy elements of the set should range from 0 to 1. Within this constraint, membership functions can take any form (triangular, sigmoid, stepwise, Gaussian, etc.) depending on the dataset and context. Fuzzification, or blurring, is the determination of the degree of belonging of a value to a set. The specificity of the fuzzification process lies in the fact that it allows a measure of imprecision to be added for three factors, represented by the membership function. For each category, each factor can be assigned an inaccuracy defined in words such as \"more than,\" \"between,\" or \"less than.\" This way, inaccuracy is incorporated into the process and allows for a better analysis of possible choices. The scoring and inference phases for a specific investment begin with a subjective assessment of each indicator. Instead of defining the risk categories and calculating the \(R_{CR}\) by multiplying the \(P_{CR}\) by the \(T_{CR}\) and \(W_{CR}\), the proposed method transforms the language terms related to these aspects into fuzzy numbers using previously invoked Gaussian membership functions, trapezoidal \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Risk Category** & **Risk Value (\(R_{CR}\))** & **Description** & **Alerts** \\ \hline Tolerated & \(\leq\)11 & Acceptable risk level—actions taken on a voluntary basis, require monitoring & None or in the form of recommendations \\ Unacceptable & (11 \(\div\) 25) & Unacceptable level of risk—the introduction of actions is necessary, but it may be postponed in time, requires constant monitoring & Mandatory to perform \\ Critical & \(\geq\)25 & Intolerable level of risk—requires immediate action & & Original \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Categories and numerical values of investment risk. For each of the belonging functions (urgency, difficulty, and greatness), a set of real number values covering a range from 1 to 4 was proposed, similar to the traditional FMEA method. The difference is that the specific values could also take the form of fractions (adapted to respondents' answers). Various membership functions (Gaussian, trapezoidal, and triangular) were then applied, and the resulting values were compared with each other. Then, an expert system was built, which consisted of using the tool designed and presented in Ref. [1]. Meetings were held again with the same experts, who are long-time designers and producers of railway traffic control systems. The experts were asked the same questions using a fuzzy rule base in which conditional tasks are a combination of  
Figure 4: Triangular urgency membership function graph (source: own elaboration). Figure 3: Trapezoidal urgency membership function graph (source: own elaboration). linguistic terms and logical operators triggering subsequent, more detailed risk categories. A set of rules was created for combinations of \(P_{CR}\), \(T_{CR}\), and \(W_{CR}\) values, whose product results were classified according to the breakdown in the classical method in Table 4. The authors thus transferred the risk valuation used in the classical method to the fuzzy method.