## 3 Materials and Methods
An important economic aspect directly related to vessel delays is the financial loss due to the daily maintenance of vessels. Ships waiting to enter the port continuously generate costs associated with their maintenance. Also, ships that have completed loading and unloading operations occupy the quay, causing the port to be unable to service another ship. The maintenance cost depends on a number of factors, i.e., the size and age of the vessel, the type of shipping, or the cargo specification. Assessing reliable daily maintenance costs for a ship is not an easy task since shipowners consider this information as highly confidential. Based on the available literature on the subject [3,4], the daily cost of maintaining each of the ships participating in the study was calculated. In order to get an overview of the scale of the problem of financial losses caused by ship delays in the studied waterway, calculations of expected costs in the annual prediction were carried out. The results obtained in the process of analysing traffic intensity and causes of delays were subjected to statistical modelling in order to examine the correlation between the studied factors and to assess financial losses. To analyse the relationship between traffic intensity, ship delays, and the resulting financial losses, univariate and multivariate regression analyses were employed. The univariate regression analysis focused on identifying the individual impact of each factor on ship delays, while the multivariate regression analysis aimed to determine the combined effect of multiple factors on ship delays and financial losses. The multivariate regression model was chosen for analysis due to its ability to evaluate the combined effect of multiple independent variables on one or more dependent variables, such as ship delays and financial losses. To ensure the robustness and reliability of the statistical models, various validation techniques were employed:  
1. Model diagnostics: Goodness-of-fit and validity of assumptions were assessed through residual plots, Q-Q plots, and goodness-of-fit statistics such as AIC and BIC. 2. Cross-validation: K-fold cross-validation was applied to evaluate model performance on unseen data. 3. Out-of-sample testing: Independent datasets were used to validate predictive accuracy and model generalizability. The constructed multiple regression model was developed by an adopted formula in the following form [39]:  
\[y=a_{0}+a_{1}x_{1}+a_{2}x_{2}+\dots+a_{n}x_{n}+\varepsilon \tag{1}\]  
where:  
\(y\)--explained variable (hours)  
\(a_{0}\)--intercept  
\(x_{1}\dots\ x_{n}\)--explanatory variables  
\(a_{1}\dots\ a_{n}\)--coefficients  
\(\varepsilon\)--error of estimation A sensitivity analysis was also conducted to evaluate the robustness of findings and assess the impact of changes in input parameters on the results. This involved identifying key parameters, selecting parameter ranges, varying parameter values, and analysing the results. This process provided additional insights into the robustness and reliability of the study's findings, helping decision-makers better understand potential uncertainties and limitations associated with the multivariate regression models.