## 4 Discussion
### About the Novel Mathematical Modelling  
The novel mathematical models obtained for the presented case study using the response surface technique can be considered useful and easily applied by the lashing designers, the crew members on board, or by other people working in the shipping industry of stowage and lashing (shipping lines or stevedoring company) of non-standardized cargo on board. For that mission, it is only necessary to program the corresponding mathematical model in spreadsheet programming. In the present paper, for practical purposes and to avoid cluttering the paper with multiple Figures, only some of them were presented as a representative of the model obtained. However, these graphs are an indication that the results of the models are reasonable and very easy to understand, an aspect very important in the shipping industry due to the need for quick solutions. Furthermore, the biggest advantage of the presented novel models is that, during their programming, we only needed to know the longitudinal (expressed as a percentage of the ship's length) and vertical stowage positions (different decks above the keel) and the CS of the lashing lines available on board.  
Once the different mathematical models were obtained by applying the new proposed methodology, it was possible to calculate the optimization of the fx or fy values in any case. As a representative case, and according to mathematical models of Equations (14) and (18) for fy and fx, respectively (Code 2; high deck), a calculation was carried out for 100 < CS < 400 kN, and for any LSP, to obtain the minimum and maximum values. The results are shown in Table 6.  
\begin{table}
\begin{tabular}{c c c} \hline \hline  & **Minimum** & **Maximum** \\ \hline fy & 0.286625 & 1.204201 \\ \hline fx & 0.099525 & 0.404149 \\ \hline \hline \end{tabular}
\end{table}
Table 6: Minimum and maximum values of fx and fy for models of Equations (14) and (18).  
In order to find these optimization points (minimum and maximum of both values), these results are found between the runs carried out in the numerical simulations to note the variables LSP and CS, which determine this optimization. These data are included in Table 7, where it is observed, on one hand, that the minimum requirements from both components (fx and fy) are registered in the mid-length ship and using the highest CS. On the other hand, the required maximum values of both variables are demanded when the cargo item is stowed near the forward perpendicular and with the lowest CS. This logical conclusion shows that the obtained mathematical models work very well.  
The reason is that, from a longitudinal point of view, these positions are the furthest away from the ship's center of gravity (rotation). In a vertical sense, considering that the ship's center of gravity will be, approximately, middle height of the cargo hold, the high deck is the position furthest away from this center of rotation. Moreover, the stowage positions above the main deck are affected by the wind pressure and sea sloshing, which increase the demands of fx and fy values.  
What is more, single values of minimum and maximum values of fx and fy are also obtained, but not in combination with each other. For instance, when LSP = 0.5 and CS = 350 kN, an fx minimum is produced, although the corresponding fy is 0.310925.  
With regards to the optimal hashing arrangement, in the industry, it is assumed that a vertical angle (\(\alpha\)) greater than 60\({}^{\circ}\) reduces the effectiveness of the lashing line with respect to the sliding of the cargo unit. Moreover, the horizontal angle (\(\beta\)) should not exceed 30\({}^{\circ}\) to not exclude the lashing line from the transverse sliding. Finally, from a combination point of view, the alternative method considers that an \(\alpha\) angle less than 45\({}^{\circ}\) in combination with a \(\beta\) angle greater than 45\({}^{\circ}\) should not be used in the balance of transverse tipping, although this movement is not studied in the present paper.