# Fuzzy Logic in Selection of Maritime Search and Rescue Units
## 3 Materials and Methods  
### Standard MCDA  
The Multi-Criteria Decision Analysis (MCDA) is a tool facilitating the decision-making process. Using mathematical models, the MCDA sorts the available solutions taking into consideration the decision-maker's preferences. The author proposes to use the method in the process of vessel selection for SAR operations. Vessels are selected based on an analysis of their characteristics and parameters. The decision-maker (typically, the SAR coordinator) specifies preferences and defines them as vessel evaluation criteria. Tested by means of the MCDA mathematical models, the available alternatives (here: vessels) are sorted, in a descending order, by the decision-maker's preferences. One of the great merits of the MCDA is the ability to use a set of diversified criteria, expressed in various units of measurement (e.g., m/s or h), each assigned with its own weight and evaluation coefficient. The result of the analysis is typically shown as a ranking of alternatives. The MCDA uses numerous mathematical models, such as AHP, ANP, UTA, Promethee II, Electre III, Oreste, etc. The procedure of analysis for Promethee II (Preference Ranking Organisation Method for Enrichment Evaluations) is discussed below. Pairs of available solutions (here: vessels) are compared for their compliance with the evaluation criteria. The criteria are minimised or maximised and assigned weights and preference functions. The preference function determines the difference for a criterion between the evaluations obtained by two possible decisions into a preference degree ranging from 0 to 1. The method uses six standard preference functions and is implemented in four phases [16]. Phase 1 consists of the calculation of the degree of preference for each pair of alternatives and the value for each criterion. Where \(g_{i}\) (_a_) is the \(i\) criterion value for the \(a\) alternative, the \(d_{i}\) (_a_, _b_) difference between the \(i\) criterion value for two alternatives, \(a\) and \(b\), can be determined. \[d_{i}\) (_a_, _b_) = \(g_{i}\) (_a_) - \(g_{i}\) (_b_)  
The value of the degree of preference of the \(i\) criterion for two decisions, \(a\) and \(b\), is denoted as \(H_{i}\) (_a_, _b_). The preference function, used for the calculation of the degree of preference, is defined as follows:  
\[H_{i}\) (_a_, _b_) = \(F(d_{i}\) (_a_, _b_)) \(with\)\(\forall x\in\)[\(-\infty\), \(\infty\)], \(0\leq F(x)\leq 1\) (2)  
In phase 2, the degrees of preference of all the criteria are summed up and a global preference index is determined for each pair of alternatives. Where \(C\) denotes the criterion and _ij_ its weight, the global preference index for each pair of alternatives, \(a\) and \(b\), is calculated as follows:  
\[\pi(a,b)=\sum_{i\in\mathrm{C}}w_{i}*\ H_{i}(a,b) \tag{3}\]  
Phase 3 consists of the calculation of outranking flows. The positive flow \(\varphi^{+}(a)\) and the negative flow \(\varphi^{-}(a)\) are determined for all the alternatives. Where \(A\) is a set of alternatives and \(n\)--the number of possible alternatives, the following formulas apply:  
\[\varphi^{+}(a)=(n-1)^{-1}=\sum_{x\in A}\pi(a,x) \tag{4}\]  
\[\varphi^{-}(a)=(n-1)^{-1}=\sum_{x\in A}\pi(x,a) \tag{5}\]  
In phase 4, the final evaluation (net flow) is calculated for each alternative with the use of the previously calculated outranking flows. The final evaluation determines the positions of alternatives in the ultimate ranking. The net flow is calculated with the following formula:  
\[\varphi(a)=\varphi^{+}(a)-\varphi^{-}(a) \tag{6}\]  
The higher an alternative's position is in the ranking, the better its compliance with the decision-maker's preferences. The ranking can support the SAR coordinator in the selection of vessels. They may decide to choose one or more top-ranking vessels. Further examination of the pre-selected vessels can be done via, e.g., radio communications.