# Analysis of the Relationships between Factors Contributing to Marine casualties
## 2 Materials and Methods
### Mcda
The indifference threshold is the largest deviation, which is considered as insignificant by the decision-maker and no outranging between the actions occurs. Both actions are considered to be the same. When the \(r\) is below this threshold, the function value is equal to 0. The V-shape function has the indifference threshold always set to zero (\(q=0\)). The preference threshold (\(p\)) is the smallest deviation which is considered as sufficient in generating a full preference. When the \(r\) is greater than this threshold, the better action will receive a maximum score of 1. The next step is to determine the outranking flows. There are three flows: the leaving flow (positive), the entering flow (negative), and the net flow. The leaving flow \(\varphi^{+}(a)\) represents the preference of a given action over others. It is a global measurement of the strengths of action \(a\) (Formula (2)). The entering flow \(\varphi^{-}(a)\) represents the preference of all other variants over a given action. The negative preference flow measures how much other actions are preferred to action \(a\). It is a global measurement of the weaknesses of action \(a\) (Formula (3)). The net preference flow \(\varphi(a)\) aggregates previous flows. Net flow can be positive or negative. The higher the net flow, the stronger the action (Formula (4)). \[\varphi^{+}(a)=\sum_{i=1}\pi(a,b) \tag{2}\]  
Figure 1: Linear type of preference function. \[\varphi^{-}(a)=\sum\pi(b,a) \tag{3}\] \[\varphi(a)=\varphi^{+}(a)+\varphi^{-}(a) \tag{4}\]  
The ranking of actions (outcome) is built based on the net flow. Where \(\varphi(a)>\varphi(b)\), action \(a\) is better (stronger or more preferred) over action \(b\), the result is presented as a partial ranking (Promethee I-single assessment) or complete ranking (Promethee II-multicriteria single assessment). In addition to the results obtained above, the GAIA (Geometrical Analysis for Interactive Aid) approach can be applied. The GAIA approach is based on the dimension-reduction technique PCA (Principal Components Analysis). The GAIA approach uses the graphical solution and allows the major features of decision problems to be represented. The following information can be found:  
* How much are the actions similar to each other? * Are there subsets of similar actions? * Is the impact of the action the same in each criterion? * Are there any dominant criteria? The GAIA plan is a two-dimensional representation of the multi-criteria problem. It uses two components (\(U\) and \(V\)) that are computed and displayed. This method defines a series of orthogonal dimensions (principal components) that keep as much information as possible on the relative positions of the actions in the \(k\)-dimensional space. The method uses the unicriterion net flow table, which is similar to the multi-criteria table, but contains more data. The rows of the table contain the actions (\(a_{1}-a_{n}\)), the columns of the table contain the unicriterion net flow (\(\phi_{j}\)). The net preference flow (\(\varnothing\)) is calculated according to Formulas (5) and (6):  
\[\varnothing_{j}(a)=\frac{1}{n-1}\sum_{b\
eq a}\left[P_{j}(a,b)-P_{j}(b,a)\right] \tag{5}\]  
\[\varnothing(a)=\sum_{j=1}^{k}w_{j}\times\varnothing_{j}(a) \tag{6}\]  
where,  
\(\varnothing_{j}(a)\)-unicriterion net flow. \(P_{j}(a,b)\)-preference function for the \(j\)th criterion, when action \(a\) is compared to \(b\). \(w_{j}\)-the weight allocated to \(j\)th criterion. \(j\)-set of criteria. \(\varnothing(a)\)-net preference flow. In the Gaia plane, the actions are represented by points, criteria are represented by the axes, and the weight of the criteria are represented by the decision axes. The actions that are similar to each other appear close to each other on the plane. Those actions that are very different from each other appear far away from each other. The subsets of similar action can be identified. The criteria are the axes drawn from the centre of the plane.