## 3 Results
### Novel Mathematical Modelling
Regarding the models obtained for the fx, the response surface also showed a high precision of 0.9587 in R-square, obtaining the following models for each vertical stowage position:  
\[\text{Code1}:\] \[\text{fx}=0.7079-0.5537\cdot\text{LSP}-0.001837\cdot\text{CS}+0.6879 \cdot\text{LSP}^{2}+0.000002\cdot\text{CS}^{2}-0.000288\cdot\text{LSP}\cdot\text {CS} \tag{17}\] \[\text{Code2}:\] \[\text{fx}=0.4876-0.5537\cdot\text{LSP}-0.001364\cdot\text{CS}+0.6879 \cdot\text{LSP}^{2}+0.000002\cdot\text{CS}^{2}-0.000288\cdot\text{LSP}\cdot\text {CS}\] (18) \[\text{Code3}:\] \[\text{fx}=0.0556-0.5537\cdot\text{LSP}-0.000438\cdot\text{CS}+0.6879 \cdot\text{LSP}^{2}+0.000002\cdot\text{CS}^{2}-0.000288\cdot\text{LSP}\cdot \text{CS}\] (19) \[\text{Code4}:\] \[\text{fx}=-0.0668-0.5537\cdot\text{LSP}-0.000175\cdot\text{CS}+0.6879 \cdot\text{LSP}^{2}+0.000002\cdot\text{CS}^{2}-0.000288\cdot\text{LSP}\cdot \text{CS} \tag{20}\]  
In Figure 16 are also inserted the 3D maps for the mathematical models of fx when the cargo items are stowed on high deck (Code 1) and on lower hold (Code 4), where relevant differences in tendency between them and with the corresponding models of Figure 16 are shown. For the cases when the vertical stowage position is not known in advance by lashing planners, it is very interesting to develop novel mathematical models that allow calculating the fx and fy values considering the three most important variables in play. For that mission, a new response surface methodology for fx and fy was developed, where no categorical  
Figure 16: 3D maps for the obtained fx models. (**a**) fx vs. CS; LSP for VSP = 1 (high deck); (**b**) fx vs. CS; LSP for VSP = 4 (lower hold). \[\text{Code1}:\]0 3D maps for the obtained fy models. (**a**) fx vs. CS; LSP for VSP = 1 (high deck); (**b**) fx vs. CS; LSP for VSP = 4 (lower hold). factor was defined. That is, all variables were defined as continuous factors and in the same range as the results obtained in the previous simulations. The obtained mathematical model of fy, shown in the following Equation, has an important high precision of 0.9201 for R-square:  
\[\text{Code1}:\]1  
\[\text{Code1}:\]2  
where VSP corresponds to the vertical stowage position. In Figure 17 are plotted the combinations by pairs of the three continuous variables, and for practical purposes, a fixed value of the third variable was selected in each graph. Concerning the model corresponds to fx, the obtained R-square of 0.9285 also indicates a high precision of the following Equation:  
\[\text{Code1}:\]3  
\[\text{Code1}:\]4  
In the same way as above, Figure 18 represents the combinations between pairs of variables for the fx model, considered a certain value of the third variable. The combination between both Figures 17 and 18 once the hashing planner determines one of the variables (CS, VSP, or LSP) provides valuable information about the fx and fy components. \[\text{Code1}:\]5 3D maps for the obtained fy model. (**a**) fy vs. CS; VSP for LSP = 0.3 from aft perpendicular; (**b**) fy vs. LSP; VSP for a CS = 300 kN; (**c**) fy vs. CS; LSP for a VSP = 1 (high deck). In real practice on board ships, many times it is possible to fix the \(\alpha\) or \(\beta\) angle of the hashing line in advance, so it is important to know what is the best corresponding \(\beta\) or \(\alpha\) angle to fulfill the balance of forces with the minimum hashing lines. Therefore, the mathematical models where fx or fy is calculated as a dependent variable of the fy or fx, respectively, can be very useful. In this sense, on one hand, the following Equation of fy model was developed, which depends on the fx variable, among others:  
\[