## 1 Introduction  
Today, with thousands of vessels used to transport goods by sea, a substantial part of international trade is handled by container ships. The loading of this type of vessel is very fast, versatile, and efficient. The largest container ships carry cargo on intercontinental routes. This ensures the fast transport of containers around the world and reduces the costs but increases the manoeuvring problems of these big vessels [1]. Containers to and from small ports are transported using feeders--smaller ships, often equipped with their handling equipment. With a capacity of up to about 1000 TEU, feeders handle ports that larger vessels cannot reach due to their manoeuvrability. Container ships are specialized vessels, adapted to carry only containers. However, the diverse construction of containers makes it possible to use different ways of transporting goods without having to adapt the ship's construction to them [2; 3; 4; 5]. The biggest challenge in container carriage is the loading plan. This is related to the stability of the ship, which directly translates into the safety of the carriage itself [6; 7; 8]. Incorrect positioning of containers on the ship and misdeclared container weight are factors that can have a negative effect on the ship's system, leading to stack collapse or incorrect separation of dangerous cargo, which can lead to fire [9]. A combination of these factors can lead to damage, loss of cargo, crew fatalities, and even total loss of stability and sinking [10; 11; 12]. It is therefore essential to plan the loading of container vessels using a loading calculator. Planning and correct loading of vessels is an important tool for determining the safety of ship operation [13; 14]. The mandatory stability requirements specified by the IMO [15; 16] are contained in the International Intact Stability Code (Table 1). They are based on experience from the design and accident analysis of seagooing ships. The criteria presented in the Code apply to typical conditions [17]. These criteria are mandatory for all ships covered by the SOLAS Convention or the Load Line Convention. For some ships, classification societies may recommend increasing certain requirements [2; 18; 19]. For example, the initial metacentric height in the loading condition with containers on deck shall not be less than 0.15 m, but the Polish Register of Shipping recommends that this height should not be less than 0.20 m [20]. The crew of each ship, when planning loading, take into account the criteria listed in Table 1, which are understood as follows:  
* The criterion for areas under the righting arm curve. The area found under the righting arm curve shall not be less than 0.055 m\(\cdot\)rad, calculated from 0\({}^{\circ}\) to 30\({}^{\circ}\) of the heel, not less than 0.090 m\(\cdot\)rad, calculated from 0\({}^{\circ}\) to 40\({}^{\circ}\) of the heel or up to the angle of flooding (the angle of flooding, in this case, being the angle of heel at which the inner spaces of the ship are flooded with seawater through openings) if the angle of flooding is less than 40\({}^{\circ}\), and not less than 0.030 m\(\cdot\)rad, calculated for 30\({}^{\circ}\) to 40\({}^{\circ}\) of the heel, or from 30\({}^{\circ}\) to the angle of flooding if the angle of flooding is less than 40\({}^{\circ}\)[17]. * Righting arm curve, where the righting arm is at least 0.2 m at an angle of heel equal to or greater than 30\({}^{\circ}\). The maximum righting arm must occur at an angle of the heel of not less than 25\({}^{\circ}\). The righting arm of the vessel in the upright position must