# A New Understanding and Modelling of TSP and BP Indices Compared to Safety IMO Ship Requirements
## 3 Results  
### Modelling  
As it was mentioned in previous sections, many people with interest involved in the towing industry, even in the escort sector, associate the performance and capabilities of tugs directly with BP. After a previous analysis of normality and homoscedasticity of the more common variables employed to define the BP, some of them were identified by an ANOVA study to model BP. These variables were Type of Tug; Length (m); Beam (m); Depth (m); Draft (m); Speed (knots); Horsepower (HP); and Year Built. Therefore, in the first stage of this research work, we developed a response surface to obtain a mathematical model able to calculate the estimated BP with the correlation between each of the previous variables. The model (1) obtained is shown in Equation (1). \[ \begin{array}{l}\text{BP}=-1937+12.73\cdot\text{TT}+2.036\cdot\text{L}-0.155 \cdot\text{B}-0.17\cdot\text{De}-\\ -7.03\cdot\text{Dr}+0.03\cdot\text{TS}+0.00681\cdot\text{BHP}+0.958\cdot\text{ YB}.\end{array}  \tag{1}\]  
Units of BP obtained are in tons and, as it can be concluded from Figure 1, which represents the real BP of samples with BP obtained with this Model 1, a high precision is obtained: (R-square, 93.40%). \begin{table}
\begin{tabular}{c c} \hline \hline
**Type of Tug** & **Code** \\ \hline Tractor Voith & 1 \\ Azimuth Stern Drive (ASD) & 2 \\ Tractor-Z & 3 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Codification of Type of Tug variable. Considering that in the literature, we can find many different types of BP (as sustained BP, maximum static BP, marketing BP, or Brazilian BP) [11], the proposed model will be useful in order to calculate, with a very simple equation, the real BP. These data can be used in the next stage of the present research in the strange case that this information is not included in the tug's particulars. Therefore, although BP is usually published by tug owners, as it was commented previously, the same does not happen with TSP data. Therefore, here, we propose another response surface modelling in order to calculate the TSP of tugs at 10 knots, and considering the same variables of the previous model, which are usually known, i.e. : Type of Tug: Length (m); Beam (m); Draft (m); Depth (m); Tug Speed (Knots); Horsepower (BHP); and Year Built. The regression equation of mathematical Model 2 obtained is included in Equation (2):  
\[ \begin{array}{l}\text{TSP}(10\text{knots})=-7388570-5828.47\cdot\text{TT}-253.6 89\cdot\text{L}+64.9967\cdot\text{B}-366.268\cdot\text{De}-\\ -1143.01\cdot\text{Dr}-29.2185\cdot\text{TS}+0.161301\cdot\text{BHP}+7375.24\cdot \text{YB}+1002.89\cdot\text{TT}^{2}+3.21910\cdot\text{L}^{2}-\\ -5.98118\cdot\text{B}^{2}+24.8924\cdot\text{De}^{2}+73.5723\cdot\text{Dr}^{2}+ 1.20324\cdot\text{TS}^{2}-0.000010\cdot\text{BHP}^{2}-1.83738\cdot\text{YB}^{ 2}+\\ +12.2277\cdot\text{TT}\cdot\text{L}+57.7064\cdot\text{TT}\cdot\text{B}+75.4063 \cdot\text{TT}\cdot\text{De}+181.199\cdot\text{TT}\cdot\text{Dr}\end{array}  \tag{2}\]  
where the units TSP are tons, and a precision of 100% was obtained in R square. Figure 2 shows the good relation between real and estimated data of tugs' TSP. Figure 1: Representation of tugs’ BP obtained with Model 1. Figure 2: Representation of tugs’ TSP real and obtained with Model 2. Once the equations of BP and TSP are modelled with the tug's usual data, a new regression equation was obtained including BP as a variable. This regression equation of Model 3 allows calculating the TSP developed by a tug at 10 knots as a function of only two independent variables: Tug Speed (knots) and BP (tons). The result of this model is Equation (3). \[\text{TSP}(10\text{knots})=563+4.93\text{-BP}-113.3\text{-TS}-0.0192\text{-BP}^ {2}+4.93\text{-TS}^{2}-0.081\text{-BP}\text{-TS}. \tag{3}\]  
A determination factor of 80.01% for R-square indicates good precision. It can be also concluded from Figure 3, which plots the real TSP and the estimated TSP of tugs as per Model 3.