# Fuzzy Logic in Selection of Maritime Search and Rescue Units
## 3 Materials and Methods
### Fuzzy Logic for Criteria Evaluation  
Imprecise information is very common. We naturally use imprecise (non-numerical, unreliable) notions to explain the occurring phenomena--hence the term 'fuzzy informa \begin{table}
\begin{tabular}{c c} \hline \hline
**Criterion C\({}_{0}\)** & **Grade** \\ \hline full compliance with the expectation/ & Level 1 \\ or the highest range value & \\ \hline average compliance with the expectation/ & Level 2 \\ or the medium range value & \\ \hline low compliance with the expectation/ & \\ or the lowest range value & Level 3 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Methodology of criterion evaluation. tion'. In the maritime industry, examples of fuzzy information are expressions such as 'high speed','medium range' or 'good saworthiness'. Although they do not refer to any particular values, they can effectively convey an important message. We also tend to describe the nature of mechanisms or processes by means of rules, such as, e.g., 'If the search speed is high, the search effectiveness is high'. Interestingly, imprecise terms can be described using the language of mathematics. Understood by computers, the language of mathematics can be applied in decision-making support systems. [PERSON] [17] proposes fuzzy sets as a way of presenting a problem. [PERSON] [18] has developed a way of employing such sets in decision-making and process control. In fuzzy logic, there is an array of intermediate values between false (0) and true (1), determining the elements' degree of membership in a set. Finding how close an element is to 0 or 1 can convey a sufficiently significant piece of information. Determining the degree to which a vessel complies with a certain criterion can be applied at the stage of planning an SAR operation (selecting the most suitable vessels for a certain type of distress). Selection of vessels based on their features and parameters can improve the effectiveness of SAR operations and accelerate the decision-making process. The procedure discussed below \((8,9)\) is used to produce the mathematical description. A fuzzy set (\(A\)) in a certain (not empty) space \(X\), denoted as \(A\) in \(X\), is a set of pairs [19]:  
\[A=\{(x,\;\mu_{A}(x));x\in X\} \tag{7}\]  
where  
\[\mu_{A}:x\;\rightarrow[0,1] \tag{8}\]  
is the membership function of fuzzy set \(A\). Each element is assigned a degree of membership in set \(A\). A well-defined fuzzy set requires a properly constructed membership function. The function serves as a tool encrypting the expert's knowledge (their perception of the phenomenon). Some of the typical membership functions are (Figure 2):  
* \(s\)-type, modelling (many, a lot of, fast, e.g., high speed);
* \(z\)-type, modelling (few, little, slowly, e.g., small range);
* trapezoidal, modelling (about, on average, e.g., average height). Fuzzy sets are typically defined as trapezoidal or triangular curves. Each value has a slope where it rises, falls or peaks (1). The system is controlled using fuzzy reasoning rules and fuzzy controllers. Fuzzy reasoning rules are built on knowledge and experience of a relevant process or phenomenon. In an SAR operation, experts are individuals with work experience at sea, knowledge of ships' construction and operation, and advanced skills of organising and carrying out operations at sea. The reasoning rules are typically expressed by experts in imprecise utterances using vague terms (linguistic variables), such as, e.g., 'When the vessel slightly drifts off course, make a small correction of the rudder angle'. Fuzzy reasoning values are represented by If-Then sentences:  
* IF x is A THEN y is B
* IF x1 is A1 AND x2 is A2 AND,..., THEN y is B
* IF x1 is A1 OR x2 is A2 OR,..., THEN y is B  
Figure 2: Typical membership functions. Where A and B represent linguistic values defined as fuzzy sets, X and Y represent universes, x represents the input variable and y--the output variable. Another example of a fuzzy reasoning rule is presented below:  
* IF speed is excellent OR altitude is medium THEN suitability is high.