## 1 Introduction
In particular, longitudinal and transverse cargo sliding are the movements studied in the present paper to be counteracted. As it is known, in real practice, many times the cargo securing is carried out according to the crew staff's own experience, so it can result in inefficient securing practices, which can negatively affect the sea passage and cargo safety because of the sliding or tilting cargo. Moreover, in many cases, the shipping lines and shipmasters try to reduce the time in port and the economic costs, reducing the number of lashing lines employed to secure the cargo items. However, this decision, which is sometimes made considering the best weather forecasts, can jeopardize the safety of maritime navigation if the weather conditions worsen or the destination port is changed, having to sail through other different sea conditions. What is more, in our opinion, the key is to obtain new mathematical models proposing a novel methodology that allows using the minimum number of lashing lines but at the same time maintaining overall safety at any moment, complying with the international regulation of the International Maritime Organization (IMO), which is reflected in the CSS Code [21]. In the same way that other scholars have performed previously [22], the numerical simulations were carried out and were compared with the CSS Code, proposed by the IMO. This empirical-based criterion is an international standard and is one of the most referenced codes in the world [22; 23]. Furthermore, as other authors carried out for optimizing the container securing system on container vessels [6], in the present work, the mathematical models were developed for a particular vessel and cargo unit, although afterward, the results were compared and discussed for vessels with different dimensions. Consequently, the goal of this research is to provide the missing tools (a new methodology to obtain useful mathematical models and 3D maps) for optimization of the stowage and securing procedure of non-standardized cargo items by calculating the balance forces in the lashing lines meeting the requirements of the CSS Code, from a safety and economical point of view. The remainder of the paper is structured as follows: Section 2 presents the followed data provided by the IMO and includes our simulations' condition for the selected case studied. Section 3 shows the different novel mathematical models obtained. In the first sub-section is represented the model for a correlation between fy and fx values, being the correlation values for the securing capacity in transverse (fy) and longitudinal (fx) directions, respectively, which depend on the friction coefficient (\(\upmu\)) and the vertical (\(\upalpha\)) and horizontal angle (\(\upbeta\)) of the lashing lines. The second one studies the tendency of these values along the ship's length, analyzing the influence of the vertical storage position and the ship's speed. In the last sub-section are developed the most interesting mathematical models, accompanied by new and useful 3D maps. Section 4 discusses the obtained results of all models shown in the previous sub-sections, commenting on the results and differences when other variables are employed in the simulations. Section 5 concludes the paper. In this study, the transverse tipping was not analyzed because, in the balance of forces, the \(\upalpha\) angle of the lashing lines has to be known in advance to calculate the corresponding lever-arm of securing force.