Statistical Analysis of the Weight and Center-of-Gravity Position of an Empty Container Ship

Abstract

For the correct execution of the preliminary design of a transport ship, among other things, approximate formulas enabling the calculation of the weight of the unladen ship and the location of the center of gravity are necessary. The aim of the conducted research was to develop approximate formulas for calculating the weight and center of gravity of an empty container ship with a size ranging from 270 TEU to 3100 TEU, depending on the basic design parameters: ship speed V, deadweight DWT, and number of TEU containers. Since the weight of an unladen container ship has a very large impact on the ship’s operating parameters, an additional aim was to obtain regression formulas with greater accuracy than similar formulas published in the literature. Simple and multiple regression methods were used to develop regression formulas. The obtained results were verified on the basis of experimentally measured parameters obtained from built ships. The regression formulas presented in this article are characterized by high accuracy, greater than that of similar formulas published in the literature, and were developed for container ships currently under construction. A novelty of this study is the development of regression formulas for weight classes, which make up the total weight of an unladen ship.

1. Introduction

1.1. Preliminary Ship Design

Along with inexpensiveness, modernly designed transport ships must be characterized by a limited impact on the natural environment. One such parameter is a ship’s energy efficiency design index (EEDI) [1], which determines exhaust emissions (including CO₂) per unit of transport work. The value of this index is influenced, among other things, by the weight of the empty ship. Research is also being carried out on unmanned ships that will be characterized by even lower CO₂ emissions [2].

The ship design process, carried out iteratively (Figure 1, [3]), begins with an analysis of the shipowner’s assumptions. For a transport vessel, the basic assumed parameters are

• Ship speed V;
• Deadweight tonnage DWT or TEU container capacity (for container ships);
• Cruising range R.

Additional assumed factors include the shipping line, sailing restrictions (bridges, canals), type of engine, type of fuel, additional equipment, etc. For these assumptions, the design should

• Comply with all regulations and conventions regarding the construction and strength of a ship’s hull and equipment and its navigational safety and meet environmental protection requirements;
• Have the best economic parameters regarding

  ○

  The lowest cost of building the ship;

  ○

  The lowest cost of ship operation, i.e., the lowest propulsion power and fuel consumption for the predicted speed V and deadweight tonnage (or TEU capacity).

• Be accepted by the shipowner as the ship expected to make profits.

The design process begins with a preliminary design stage (Figure 1), which will include the basic geometric dimensions and technical and operational parameters, as well as the relevant drawings: theoretical lines of the hull and a general plan including a functional–spatial division.

From the assumptions (DWT, TEU) presented earlier, only the size of a ship’s holds can be determined. The geometric dimensions thus determined are insufficient because of the following:

• In order for a ship to have the assumed deadweight tonnage or cargo capacity, its displacement (volume of the submerged part of the hull) must be greater because the hull

  ○

  Has its own weight;

  ○

  Contains the power plant and various auxiliary systems;

  ○

  Supports the superstructure, deck equipment, etc.

• The engine room must accommodate the propulsion engine with all its equipment, with power sufficient for the newly built ship to develop an assumed speed V; the ship’s hull must also contain fuel tanks for the ship to travel in the assumed range R.
• The power output of the engine room plant depends on the resistance and speed V of the ship, and the resistance depends on the dimensions and form of the hull containing the engine room—the dimensions and weight of the engine room at this stage are not known, since the propulsion power is not yet known.
• At this stage, under these assumptions, the design cannot be optimized.

To create a preliminary design, the designer should use approximate relationships between the basic dimensions of the ship and its technical and operational parameters and the design assumptions made. Using such relationships, we can create an iterative design of the ship, (Figure 1), while at the same time optimizing its parameters.

Examples of approximate relationships useful in preliminary ship design, used in the specific steps shown in Figure 1, are as follows:

①

Relationships between main particulars: vessel length L, breadth B, draft T, and depth H as a function of DWT tonnage or TEU container capacity, $L,B,T,H=f(DWT/TEU)$;

②

Relationship between the lightship weight WLS and the center of gravity coordinates CG as a function of DWT or TEU: $WLS,CG=f(DWT/TEU)$;

③

Relationship between propulsion power N_C as a function of ship speed V: $N_C=f\left(V\right).$

These relationships are established based on a statistical analysis of existing ship parameters. Many publications contain equations in the form of ① and ③ for different types and sizes of manned ships built in different periods [4,5,6,7,8,9,10,11]. These equations may differ in the scope of application and accuracy of approximation of basic geometric and technical-operational parameters. In recent publications [12,13], the mentioned equations allow performing design calculations with high accuracy.

However, approximate formulas in the form of ② for the weight and center of gravity of a light ship are needed to perform full calculations as part of the preliminary design and to check various criteria during the design process, e.g., buoyancy, freeboard and stability. The literature on this subject is scarce, and the published relationships are either very inaccurate or useless at the preliminary design stage when the dimensions and hull structure of the ship are not known.

Having functional relationships in the form of ①, ② and ③ developed for manned vessels, we can work out similar relationships also for unmanned vessels.

1.2. Weight Light Ship

The ship displacement D is equal to [14,15]:

$$D=WLS+DWT$$

(1)

where

WLS—weight light ship, [t],

DWT—deadweight tonnage, [t].

As a rule of thumb, the weight light ship WLS is equal [15,16,17]:

$$WLS=M_1+M_2+M_3+M_4+M_5+M_6+M_7+M_8$$

(2)

where M₁, …, M₈ are the weights in various classes shown in

Table 1. Weight classes and weight indexes for a light ship.

Class No.	Weight Classes	Weight Indices
1	M1 Hull, superstructure, funnel	C1 = M1/LBH
2	M2 Deck equipment with hatch cover and crane	C2 = M2/LB
3	M3 Accommodation and painting	C3 = M3/J
4	M4 Machinery	C4 = M4/NC
5	M5 Piping systems	C5 = M5/LBH
6	M6 Electric equipment	C6 = M6/Ne
7	M7 Special equipment	C7 = M7/LBH
8	M8 Inventories	C8 = M8/LBH

. Each of the listed classes from M₁ to M₈ can still be subdivided into subclasses from 1 to 8. The weight light ship WLS has thus a tabular form with rows from 1 to 8 and columns from 1 to 8. The individual weights in Table 1 are most often shown in the form of corresponding indices defined as follows:

$$\mathrm{e}.\mathrm{g}., { C}_k={\displaystyle \frac{M_k}{P_k}}$$

(3)

where

C_k—index for class k,

M_k—class k weight, [t],

P_k—ship parameter appropriate for class k (the number of containers, volume, propulsion power, number of crew, power of electrical loads, etc.; commonly used in many shipbuilding yards; parameters for individual weight classes are given in Table 1).

• where

M₁–M₈—weights in classes 1 through 8, [t],

C₁–C₈—weight indexes for classes 1 through 8,

L—length, [m],

B—breadth, [m],

H—depth, [m],

LB—product $L\cdot B$, [m²],

LBH—product $L\cdot B\cdot H$, (volumetric module), [m³],

J—number of crew, [-],

N_C—maximum power of the main engine, [kW],

Ne—power of generating sets, [kW].

The chart in Figure 2 shows the average statistical share of each weight class for the container ships under study. The weight of a container ship mostly depends on the hull (class M₁) and to a lesser extent on deck equipment (class M₂) and the engine room (class M₄). The other weight classes have less or minimal impact on the weight light a container ship, but in the case of unmanned container ship design, they too will have a significant impact on the final weight of an unmanned empty container ship.

In addition to the weight indices (Table 1), sometimes the total weight light ship WLS or its index in the form given below is sufficient at the preliminary design stage:

$$C_{PS}={\displaystyle \frac{WLS}{DWT}}$$

(4a)

or

$$C_{PS}={\displaystyle \frac{WLS}{TEU}}$$

(4b)

For each class and subclass, the position of the center of gravity CG is specified (Figure 3). Since the ship has a longitudinal plane of symmetry and cannot list to one side while sailing, the longitudinal coordinate x_CG and vertical coordinate z_CG are generally given (in cases where some machinery is mounted asymmetrically the transverse coordinate y_CG is given).

The coordinates of the center of gravity for class 1 through 8 are first calculated from the coordinates of the individual component weights, then the resultant position of the center of gravity for the entire light ship is determined.

The center of gravity of a light ship is calculated based on the centers of gravity of each class (Table 1):

$$x_{CG}={\displaystyle \frac{\sum _{k=1}^8{x}_{Gk}{\cdot M}_k}{\sum _{k=1}^8{M}_k}}$$

(5a)

$$y_{CG}={\displaystyle \frac{{\sum }_{k=1}^8{y}_{Gk}{\cdot M}_k}{{\sum }_{i=1}^8{M}_k}}$$

(5b)

$$z_{CG}={\displaystyle \frac{{\sum }_{k=1}^8{z}_{Gk}{\cdot M}_k}{{\sum }_{k=1}^8{M}_k}}$$

(5c)

where

x_CG, y_(CG), z_CG—coordinates of the center of gravity of a light ship (it is aimed that y_CG = 0—the ship must not list during operation), Figure 3, [m],

x_Gk, y_(Gk), z_Gk—coordinates of the centers of gravity in each class (Table 1),

M_k—weight in each class (Table 1), [t].

Ship speed V, not present in the above formulas and in Table 1, can also affect the weight of the ship (or its class/es).

During the design of a ship, the weights and coordinates of the center of gravity of the various parts of the ship’s structure and its equipment are calculated at the final stage, after the technical and working design is done. A heel test is carried out after the ship’s construction, which will experimentally verify the calculations of the weights and center of gravity of a light ship. However, at the preliminary design stage of the ship, the individual weight classes of the ship are not known (Table 1), and the positions of the centers of gravity of the classes that make up the weight light ship are not known.

1.3. The Aim and Scope of the Study

Few scientific publications deal with the estimation of ship weight or ship weight classes. The known methods can be divided into two groups:

• methods for calculating ship weight from basic geometric or operational-technical parameters—these methods can be used at the stage of preliminary ship design,
• methods for estimating ship weight from the design of hull or its elements—these methods are not used at the preliminary design stage, because at this point the ship’s structure (thickness of plating, type of stiffeners, etc.) is not yet known.

The simplest formulas for estimating the weight of a ship’s hull have been developed in the form of coefficients depending on the ship’s displacement D or deadweight tonnage DWT: [18,19,20,21]. These formulas, in the form of a single coefficient, linearly dependent on D or DWT are not very accurate. Besides, they were developed for old types of ships. Most often, formulas are given for calculating only certain weight groups, e.g., hull, machinery and engine room weights. The remaining weights, shown in Table 1, are taken into account as weight reserve (the sum of these three groups and the reserve weight makes up the weight light ship). Approximate formulas for calculating hull weight and engine room weight are contained in [14,15,18,19,22,23]. Publications [14,15,24,25,26,27,28,29,30] provide relationships to determine the hull weight for various types of ships. The study [6] contains formulas for hull weight of container ships.

Some authors present methods for calculating the weight of a ship based on its structure, which is not yet known at the preliminary design stage. Also shipbuilding computer programs (AVEVA [31], NAPA [32]) for the design of ship structures contain modules for direct calculation of the weight and position of the center of gravity of the designed hull section. However, these methods and computer tools are not useful in the preliminary design of a container ship.

There are very few scientific papers on methods of determining the position of the ship’s center of gravity based on the basic geometric parameters of the ship. The only approximate formulas that have been found are in scripts or unpublished auxiliary materials prepared for shipbuilding students [23]. The results of calculating the coordinates of the center of gravity for individual classes and for an entire light container ship (2700 TEU) are also contained in [16], and approximate formulas for calculating the position of the center of gravity for the hull (class M₁) are contained in [17].

Based on the analysis of available articles and research works, it was found that there are no approximation formulas for the weight light ship (WLS) and the center of gravity for modern container ships in the range of 270–3100 TEU. There is also a complete lack of approximation formulas for all classes (Table 1), which are also useful for designing container ships.

The main objective of our research was therefore to develop a set of regression formulas for calculating the weight and center of gravity coordinates of container ships in the range of 270 to 3100 TEU.

This range was established for two reasons:

• 270 to 3000 TEU capacity range covers container ships classified as Small Feeder, Regional Feeder and Feedermax, which together make up the largest group of container ships under construction [33], (graph in Figure 4 is based on the database of ships—Sea-web Ships built in 2000–2024, a total of 50.9% of all container ships),
• design and research work is continued on unmanned, remotely or autonomously controlled container ships; the first unmanned container ships will belong to the Small or Regional Feeder group.

The knowledge of accurate regression formulas for determining the weight and position of the center of gravity of a light container ship will be very useful for designing both manned and unmanned vessels, especially at the preliminary design stage. The results of the ongoing research presented in the article are novel in the absence of such regression formulas in the available literature.

During preliminary design, the basic characteristic parameters of the ship are determined based on the most important functions of the ship. These functions include: deadweight tonnage, displacement, speed, propulsion power, cruising range. To optimize the ship design that meets the listed functional characteristics, we need to know the weight and position of the center of gravity of the designed ship. Therefore, the main objective of this research was to develop regression formulas in the form of:

• weight indices for each class (3):

$$C_k={\displaystyle \frac{M_k}{P_k}}=f(P_k)$$

(6)

• hull weight light ship (4):

$$WLS=f\left(P_{PS}\right)$$

(7)

• ship’s center of gravity coordinates (5):

$$x_{CG}={\displaystyle \frac{{\sum }_{k=1}^8{x}_{Gk}{\cdot M}_k}{{\sum }_{k=1}^8{M}_k}}=\mathrm{f}(DWT/TEU/LBH)$$

(8a)

$$y_{CG}={\displaystyle \frac{{\sum }_{k=1}^8{y}_{Gk}\cdot M_k}{{\sum }_{k=1}^8{M}_k}}=\mathrm{f}(DWT/TEU/LBH)$$

(8b)

$$z_{CG}={\displaystyle \frac{{\sum }_{k=1}^8{z}_{Gk}{\cdot M}_k}{{\sum }_{k=1}^8{M}_k}}=\mathrm{f}(DWT/TEU/LBH)$$

(8c)

where

P_PS—parameter of the vessel, e.g., DWT, TEU or LBH.

2. Materials and Method

2.1. Materials

In accordance with the assumed research goal to develop regression formulas for the weight and center of gravity of light container ships ranging from 270 TEU to 3100 TEU, data were collected for modern container ships. This data was divided into two sets due to the number of available parameters for individual container ships:

• one set contains technical and operational parameters and weight light ship for 96 container ships, (the range of technical parameters is provided in

  Table 2. Range of basic technical parameters of container ships for the first set (96 container ships).

  Parameter	Loa [m]	L [m]	B [m]	H [m]	T [m]	DWT [t]	TEU [-]	V [kn]
  Max.	207.4	195.4	30.60	16.75	11.95	41,583	2506	23.7
  Min.	93.5	85.0	15.85	6.40	4.90	3950	270	13.5

  ), built in the years 2000–2024; technical parameters and weights light of individual container ships were collected from the Sea-web Ships database [33];
• the other set contains, in addition to technical and operational parameters, accurate weight data for all classes (Table 1), weight light ship and coordinates of the center of gravity position for 13 container ships—the data come from shipyard documentation, calculated during the design and construction of these ships, (the range of technical parameters is provided in

  Table 3. Range of basic technical parameters of container ships for the second set (13 container ships).

  Parameter	Loa [m]	L [m]	B [m]	H [m]	T [m]	DWT [t]	TEU [-]	V [kn]
  Max.	220.0	210.2	32.25	18.7	12.1	45,000	3100	22.3
  Min.	123.0	113.7	15.50	9.0	6.4	6300	420	13.0

  )—Sea-web Ships database does not contain this type of data.

The number of the first group (Table 2) is not large, but these are all container ships of the same type (functional and structural and with the same equipment for a given range of transported containers) built in the years 2000–2024, the parameters of which are included in the database [33]. The research did not include older container ships, because the models were supposed to be for the newest ones. The number of the second group (Table 3) is a small population, but these are reliable and representative data, because they contain detailed weight parameters developed during the design and construction of these container ships in the shipyard.

For two container ships, the results of measurements (after construction and sea trials) of the weight and position of the center of gravity are also available—these data were used to verify the obtained test results.

The container ships selected for the study were of the same functional and structural type, and each container ship had:

• a single propeller, one rudder located behind the propeller and one bow tunnel thruster, the hull with a bulbous bow and transom stern,
• an engine room and superstructure at the stern,
• hatch covers on which the containers are stacked,
• unloading cranes (2 or 3 depending on the ship size),
• however, the container ships differed mainly in size: main dimensions, displacement and container capacity.

A general plan of an example container ship is shown in Figure 3.

2.2. Research Method

Statistical analysis methods were used to obtain mathematical relationships between weight indexes, weight light ship and center of gravity position and geometric parameters, propulsion power, speed, TEU capacity or displacement for manned container ships.

The process proceeded as follows:

• selection of substantively relevant parameters—affecting or related to the variable,
• selection of the form of the estimation function,
• selection of statistically significant parameters,
• checking the accuracy of the fit.

The general form of simple and multiple regression functions and statistics for evaluating their accuracy are known and reported in many publications, e.g., [34,35]. It is also often used for finding mathematical relationships between ship parameters, [36,37].

The required approximation function had to meet the criteria of high accuracy and simplicity. Finding the approximation function in the form suitable for the quantity under study began with simple regression, where the predicted quantity depends on a single independent variable. The approximation relationships were first in the simplest form, that is, a class of linear functions (linear regression). When fit errors were too large, other solutions were sought in the group of nonlinear functions (polynomial, power, exponential functions). Simple regression was used to estimate the various approximated parameters and evaluate their variations depending on the parameters taken as independent variables. For the obtained functional relationships, the coefficient of determination R², i.e., the ratio of explained variation to total variation, was considered to assess their accuracy. Also examined was the standard error of estimation, which informs of the average magnitude of deviations of the dependent variable values from the values calculated from the approximating function.

Based on the analysis of the obtained simple regressions, to increase the accuracy of the predicted values, multiple regressions were developed (for two or more independent variables) starting with a linear and then non-linear multiple approximation function.

Statistical verification of functional relationships in multiple estimation was based on the coefficient of determination R² (goodness of fit), as well as the mean squared error and a scatter plot of predicted versus observed values (a scatter plot of values obtained from approximation versus benchmarks). Also tested were the significance of the parameters (independent variables) in the estimation function model and the Pearson correlation coefficient, which determines the level of linear dependence between the parameters.

To search for formula equations and their coefficients, the computer program NdCurveMaster (version 8.2, SigmaLab, Mumbai, India, 2021) [38] was used, which uses heuristic curve-fitting and data analysis techniques and incorporates scientific machine learning-based algorithms such as random search. This program provides tools for evaluating the validity and usefulness of the resulting regression relationships.

3. Research Results

3.1. Results of Analyses from Simple Nonlinear Regression

To estimate the weight light of a container ship, simple nonlinear regression methods were used in the first step, because the developed approximation formulas are easy to use and are a function of only one ship parameter: V, TEU, DWT, J, Nc, Ne.

Statistical analysis using simple regression for light container ships was conducted for:

• weight indices for each class (Table 1)—Formula (6), where the independent variables used include the parameter resulting from the definition for a given weight index (Table 1), LBH, DWT and TEU, J, N_C and Ne.
• weight of the whole hull of weight light ship—Formula (7), where LBH, DWT and TEU are independent variables,
• coordinates of the center of gravity of a light empty container ship—Formula (8), with LBH, DWT and TEU taken as independent variables.

The results of these analyses are presented in the form of functional relationships, including R² coefficients, and in graphs:

• for indices C₁–C₈ (classes 1–8), Formulas (9)–(24), Figure 5, Figure 6, Figure 7, Figure 8, Figure 9, Figure 10, Figure 11, Figure 12, Figure 13, Figure 14, Figure 15, Figure 16, Figure 17, Figure 18, Figure 19 and Figure 20, (second set of container ships, Table 3),
• for the weight light ship WLS, Formulas (25)–(27), Figure 21, Figure 22 and Figure 23, (first set of container ships, Table 2),
• for the coordinates of the ship’s center of gravity x_CG, z_CG, Formulas (28)–(33), Figure 24, Figure 25, Figure 26, Figure 27, Figure 28 and Figure 29, (second set of container ships, Table 3).

Index C₁:

$$C_1=0.1652\cdot {LBH}^{-0.054} R^2=0.1001$$

(9)

$$C_1=0.155\cdot {DWT}^{-0.053} R^2=0.1132$$

(10)

$$C_1=0.177\cdot {TEU}^{-0.092} R^2=0.2634$$

(11)

Index C₂:

$$C_2=-2\cdot {10}^{-5}\cdot LB+0.4037 R^2=0.0849$$

(12)

Index C₃:

$$C_3=-0.0954\cdot J+13.007 R^2=0.0154$$

(13)

$$C_3=0.40927\cdot {TEU}^{0.45} R^2=0.9569$$

(14)

Index C₄:

$$C_4=0.0419{\cdot N}_C^{0.0345} R^2=0.0180$$

(15)

Index C₅:

$$C_5=0.0118\cdot e^{-1\cdot {10}^{-5}\cdot LBH} R^2=0.6883$$

(16)

$$C_5=0.0099\cdot e^{-4\cdot {10}^{-4}\cdot TEU} R^2=0.4742$$

(17)

Index C₆:

$$C_6=4.2761\cdot {Ne}^{-0.576} R^2=0.3640$$

(18)

Index C₇:

$$C_7=-4\cdot {10}^{-9}\cdot LBH+0.0007 R^2=0.3285$$

(19)

$$C_7=-1\cdot {10}^{-8}\cdot DWT+0.0007 R^2=0.3795$$

(20)

$$C_7=-2\cdot {10}^{-7}\cdot TEU+0.0007 R^2=0.3250$$

(21)

Index C₈:

$$C_8=-2\cdot {10}^{-9}\cdot LBH+0.0005 R^2=0.0487$$

(22)

$$C_8=-6\cdot {10}^{-9}\cdot DWT+0.0005 R^2=0.0901$$

(23)

$$C_8=-1\cdot {10}^{-7}\cdot TEU+0.0005 R^2=0.0910$$

(24)

The weight light ship WLS:

$$WLS=3.321\cdot {DWT}^{0.7711} R^2=0.9110$$

(25)

$$WLS=4.0726\cdot TEU+1150.9 R^2=0.8874$$

(26)

$$WLS=0.1022\cdot LBH+1216.9 R^2=0.9197$$

(27)

Coordinates of the center of gravity of the whole ship x_CG, z_CG (Figure 2):

$$x_{CG}=2.6063\cdot {DWT}^{0.3267} R^2=0.8935$$

(28)

$$x_{CG}=6.9392\cdot {TEU}^{0.3171} R^2=0.7139$$

(29)

$$x_{CG}=1.3668\cdot {LBH}^{0.3567} R^2=0.9259$$

(30)

$$z_{CG}=1.9769\cdot {DWT}^{0.1661} R^2=0.6825$$

(31)

$$z_{CG}=2.8088\cdot {TEU}^{0.182} R^2=0.6820$$

(32)

$$z_{CG}=8.3298\cdot e^{3\cdot {10}^{-6}\cdot LBH} R^2=0.7019$$

(33)

The estimation formulas obtained from simple regression (dependencies, among others, on parameters used in shipyards) for the C_k indices, Equations (9)–(24), Figure 5, Figure 6, Figure 7, Figure 8, Figure 9, Figure 10, Figure 11, Figure 12, Figure 13, Figure 14, Figure 15, Figure 16, Figure 17, Figure 18, Figure 19 and Figure 20, do not meet the expected accuracy of fit (low R² value). This is especially true for the C₇ and C₈ indices, where the degree of fit was below 0.4. These dependencies concern the weights with the smallest percentage share in the total weight of the ship (M₇~0.33%; M₈~0.24%), so the error made in their determination will not have such a significant impact on the total weight. The results of the calculations of the weight light ship WLS, Equations (25)–(27), and the coordinates of the center of gravity x_CG, z_CG, Equations (28)–(33), are much better—the value of the R² index is definitely higher.

3.2. Results of Analyses from Multiple Nonlinear Regression

In further studies, multiple estimation was used and other ship parameters were used than those indicated in Table 1—such as the basic dimensions of the ship L, B, H, the block coefficient C_B and the speed V, which is a parameter assumed by the shipowner, determined at the preliminary design stage. The best results of the analyses performed are presented in the form of functional dependencies, Formulas (34)–(44) with the determination coefficient R² and in Figure 30, Figure 31, Figure 32, Figure 33, Figure 34, Figure 35, Figure 36, Figure 37, Figure 38, Figure 39 and Figure 40 presenting the dispersion of observed values (measured data) in relation to the predicted values (calculated from the developed regression dependencies).

$$C_1=0.0913757\cdot {DWT}^{0.1}-0.00304364\cdot {\left(ln(TEU)\right)}^2 R^2=0.9978$$

(34)

$$C_2=-1.17147\cdot {10}^7\cdot {DWT}^{-2}+259.94\cdot {TEU}^{-0.9}-3.60709\cdot {10}^9{\cdot (LB)}^{-3} R^2=0.9828$$

(35)

$$C_3=-0.241143{\cdot (\ln \left(DWT\right))}^2+0.891088\cdot {TEU}^{-1/10}\cdot {LBH}^{0.4} R^2=0.9762$$

(36)

$$C_4=761505\cdot V^{-6}-1.22616\cdot {10}^6\cdot N_C^{-2}+0.000588802\cdot V^5{\cdot N}_C^{-1.1} R^2=0.9913$$

(37)

$$C_5=0.00671665\cdot {TEU}^{1/8}-0.0000422513\cdot {DWT}^{0.55} R^2=0.9779$$

(38)

$$C_6=1.30598\cdot {10}^9\cdot J^{-8}+0.000355803\cdot J^{2.6}\cdot {Ne}^{-0.5} R^2=0.9606$$

(39)

$$C_7=-33711.5\cdot {LBH}^{-1.8}+0.229357\cdot {DWT}^{-0.6} R^2=0.9493$$

(40)

$$C_8=0.0614659\cdot {LBH}^{-1/2}-1.89401\cdot {10}^{15}\cdot {TEU}^{-7}+0.0000307182\cdot {LBH}^4\cdot {TEU}^{-6} R^2=0.8517$$

(41)

$$WLS=0.106819\cdot DWT+0.0585666\cdot LBH+78.816\cdot V R^2=0.9886$$

(42)

$$x_{GC}=0.197216\cdot L^{1.15}-0.0000244495\cdot N_C^{1.3}+{0.0000866675\cdot DWT}^{1.1} R^2=0.9989$$

(43)

$$z_{GC}=4.18089+0.0000242958\cdot LBH+0.253695\cdot V\hspace{1em}{R}^2=0.9443$$

(44)

The obtained approximation formulas using multiple nonlinear regression are characterized by a much higher fit coefficient R² than from simple regression. This was influenced by the use of other ship parameters than those previously used, presented in Table 1.

4. Analysis of the Test Results

The search for functional relationships between the weight indices (C₁–C₈), the light ship weight (WSL) and the coordinates of the light ship centre of gravity (x_CG, z_CG) and the design parameters, presented in Section 3.1, was performed using simple non-linear regression. The ship parameters P_k, presented in Table 1, were used to develop the C_k indices. The parameters P_k for each class were in the past commonly used in shipyards to determine C_k indices.

The number of ship crew J or the number of TEU containers were taken as independent variables in the simple regression of the weight indices C₁–C₈, the weight light ship WLS and the coordinates of the center of gravity (x_GC, z_GC), which seems logical and correct. However, an analysis of these two parameters showed that the actual number of crewmembers J has little relation to the size of the ship (the number of crew is derived from the relevant regulations and not from the size of the ship). Also, the number of TEU containers (capacity) is not closely related to the size of the container ship (container ships can also carry empty containers, which has little effect on the DWT tonnage of the container ship). However, due to increasingly better ship design methods using optimization models, as well as increasing automation in ship operation, the parameters P_k in Table 1 for currently designed ships may not always be appropriate when developing regression formulas. This causes the resulting regression formulas for the C_k weight indices for each class (Table 1) to have very low accuracy (R² value). The low value of the coefficient of determination R² is also due to:

• a small number of analyzed container ships (the analysis was conducted for built container ships, for which the weight values for each class were known, calculated on the basis of the technical and working documentation used for building the ship),
• analyzed container ships were built over 25 year period, the latest ones may have an improved design (less steel used in the construction),
• although the container ships were of the same type (Figure 2), they differed in operating speed and sailing range, which affects the weight of the engine room, the volume of fuel tanks, and thus the weight of the hull.

Therefore, further research analyses were performed to determine P_K parameters other than those indicated in Table 1, to check whether a better fit of the R² index was possible.

The results of calculating the weight light ship WLS (25–27) or for the coordinates of of the resultant center of gravity of a light container ship x_CG, z_CG (28–33) have become already much better—the value of the coefficient R² is already acceptable.

In further studies, using multiple nonlinear estimation, other ship parameters were used than those indicated in Table 1.

The results obtained from multiple regression are more accurate than in the case of simple regression. This is particularly visible for the weight indices C₁–C₈.

The collected, best results of regression analyses for both methods are presented in

Table 4. Functional relationships of the obtained regression analyses results and the coefficient of determination R².

Simple Regression		Multiple Regression
Formula	R2	Formula	R2
$C_1=0.177{\cdot TEU}^{-0.092}$	0.2634	$C_1=0.0913757\cdot {DWT}^{0.1}-0.00304364\cdot {\left(ln(TEU)\right)}^2$	0.9978
$C_2=-2\cdot {10}^{-5}LB+0.4037$	0.0849	$C_2=-1.17147\cdot {10}^7\cdot {DWT}^{-2}+259.94\cdot {TEU}^{-0.9}-3.60709\cdot {10}^9{(LB)}^{-3}$	0.9828
$C_3=0.40927\cdot {TEU}^{0.45}$	0.9569	$C_3=-0.241143{\cdot (\ln \left(DWT\right))}^2+0.891088{\cdot TEU}^{-1/10}\cdot {LBH}^{0.4}$	0.9762
$C_4=0.0419\cdot N_C^{0.0345}$	0.0180	$C_4=761505\cdot V^{-6}-1.22616\cdot {10}^6\cdot N_C^{-2}+0.000588802\cdot V^5\cdot N_C^{-1.1}$	0.9913
$C_5=0.0118\cdot e^{-1\cdot {10}^{-5}\cdot LBH}$	0.6883	$C_5=0.00671665\cdot {TEU}^{1/8}-0.0000422513\cdot {DWT}^{0.55}$	0.9779
$C_6=4.2761{\cdot Ne}^{-0.576}$	0.3640	$C_6=1.30598\cdot {10}^9\cdot J^{-8}+0.000355803\cdot J^{2.6}\cdot {Ne}^{-0.5}$	0.9606
$C_7=-1\cdot {10}^{-8}\cdot DWT+0.0007$	0.3795	$C_7=-33711.5\cdot {LBH}^{-1.8}+0.229357\cdot {DWT}^{-0.6}$	0.9493
$C_8=-1\cdot {10}^{-7}\cdot TEU+0.0005$	0.0910	$C_8=0.0614659\cdot {LBH}^{-1/2}-1.89401\cdot {10}^{15}\cdot {TEU}^{-7}+0.0000307182\cdot {LBH}^4\cdot {TEU}^{-6}$	0.8517
$WLS=0.1022\cdot LBH+1216.9$	0.9197	$WLS=0.106819\cdot DWT+0.0585666\cdot LBH+78.816\cdot V$	0.9886
$x_{CG}=1.3668\cdot {LBH}^{0.3567}$	0.9259	$x_{GC}=0.197216\cdot L^{1.15}-0.0000244495\cdot N_C^{1.3}+{0.0000866675\cdot DWT}^{1.1}$	0.9989
$z_{CG}=8.3298\cdot e^{3\cdot {10}^{-6}\cdot LBH}$	0.7019	$z_{GC}=4.18089+0.0000242958\cdot LBH+0.253695\cdot V$	0.9443

. Based on the R² index, it can be stated that the obtained formulas for multiple regression achieve high accuracy.

For the functional dependencies obtained from multiple nonlinear regression, their qualitative and quantitative verification was carried out.

The quantitative verification was carried out by comparing the results of ship weight calculations with measurements made for two container ships after their construction and performance of appropriate tests. The basic technical and operational parameters of both container ships are given in

Table 5. Parameters of the container ships used for the verification of the regression formulas.

Parameter	Container Ship A [39].	Container Ship B [16].
L [m].	154.0	196.0
B [m].	25.30	32.26
H [m].	13.5	19.0
T [m].	10.5	10.0
DWT [t].	20,275	39,128
TEU [-].	1334	2700
V [kn].	19.1	21.0

.

The results of the verifications performed using data of the existing container ships (Table 5) are shown in

Table 6. Comparison of values measured for each class of container ship weights with calculations according to regression formulas.

	Container Ship A [39].			Container Ship B [16].
	Measured Values	Calculations as per Reg. Formulas (34)–(41)	Error of Calculation [%]	Measured Values	Calculations as per Reg. Formulas (34)–(41)	Error of Calculation [%]
M1	4153.8	4462.7	−7.4%	9331.0 t	8778.71 t	5.9%
M2	1137.2	1092.1	4.0%	1367.0 t	1202.8 t	12.0%
M3	236.6	275.6	−16.5%	416.0 t	430.3 t	−3.4%
M4	738.7	695.9	5.8%	1073.0 t	1022.4 t	4.7%
M5	385.6	348.9	9.5%	505.0 t	462.8 t	8.4%
M6	112.0	105.3	6.0%	170.0 t	-	-
M7	21.9	25.8	−17.8%	41.0 t	45 t	−10.9%
M8	15.4	16.28	−5.7%	31.5 t	23.3 t	26.1%

(weights for each class) and

Table 7. Comparison of measured values of weight light ship and position of center of gravity of container ships with calculations according to regression formulas.

Parameter	Container Ship A [39]			Container Ship B [16]
	Measurement	Formulas (42)–(44)		Measurement	Formulas (42)–(44)
		Value	Error		Value	Error
WLS	6824 t	6752 t	0.9%	12,935 t	12,871 t	0.5%
xCG	62.4 m	63.8 m	−2.1%	80.6 m	84.4 m	−4.7%
zCG	10.74 m	10.3 m	4.1%	12.7 m	12.4 m	2%

(weight light ship and position of the center of gravity).

The values of the weights M_k (Table 1) for each class were determined from the transformation of relation (6).

From the comparative analyses results, contained in Table 6, we can see that the accuracy of calculations of individual weight classes ranges from 4% to 26% compared to the measured values. The weight of the whole ship is mostly affected by the classes M₁, M₂ and M₄. For these major classes, the calculation error from regression formulas oscillates between 6% and 12%.

Table 7 shows the results of the accuracy of the calculation of the weight light ship WLS and the coordinates of the center of gravity. For the weight WLS, the accuracy of the calculations is less than 1%, while for the position of the center of gravity it is less than 5%. These are exceptionally good results confirming very good approximations using multiple regression formulas.

The developed method and the formulas obtained from multiple regression allow at the preliminary design stage to predict with high accuracy the weight WLS of a light container ship and the position of the center of gravity.

The qualitative verification was carried out by comparing the accuracy of the weight calculations for class M₁ (weight of the ship’s hull with superstructure without equipment) and for class M₄ (weight of the engine room) of a light ship from the developed formulas with the corresponding formulas published in the scientific literature [14,15,29]. The results of these analyses are shown in

Table 8. Weight of empty ship without equipment (weight group 1—M₁)—Comparison of calculations with measurements for different methods.

	Measurement M1	Regression (34)		Weight M1 Calculated from the Formulas in the Publication:
				Chapman [14].		Benford [29].		Murry [15].		Sato [15].
		v	e	v	e	v	e	v	e	v	e
A	4153.8	4462.7	−7.4%	3888.3	6.4%	4671.6	−12.5%	4364.7	−5.1%	2349.7	43.4%
B	9331.0	8778.7	5.9%	8029.7	13.9%	9514.4	−2.0%	8219.6	11.9%	5383.8	42.3%

v—value, e—error.

, (measured data for container ship A [39] and B [16] were used for these analyses). The predicted weight of class M₁ from the formulas in the publications varies from 2% to 43% compared to the measurement (ships A and B). The results of these analyses are also shown graphically (Figure 41) where the M₁ class weight was estimated for the second set of container ships (Table 3).

Table 9. Engineroom weight (weight group 4—M₄)—Comparison of calculations with measurements for different methods.

	Measurement M4	Regression (37)		Weight M4 Calculated from the Formulas in the Publication:
				Watson [25].		Barras [15].		Molland [22].		Papanikolaou [14].		Benford [29].
		v	e	v	e	v	e	v	e	v	e	v	e
A	738.7	695.9	5.8%	1187	−60.7%	1299	−75.8%	959	−29.8%	1874	153.7%	1387	−87.8%
B	1073.0	1022.4	4.7%	1739	−62.1%	1930	−79.9%	1404	−30.8%	2898	−170.1%	1973	−83.9%

shows the results of calculating the weight of a container ship’s engine room (class M₄) obtained from formulas in the publications [14,15,22,25,29] for ship A and B and from the regression Formula (35). The values for estimating the weight of class M₄ from the formulas given in scientific publications are much less accurate than from the developed Formula (35).

The comparison of the calculations shows that the developed regression formulas in Table 4 are, for the most part, more accurate than those presented in the literature. One reason for this is the fact that those methods were developed in the 1960s, i.e., for ships much older than those used in the herein presented analyses.

5. Summary and Conclusions

In the preliminary design of cargo ships, one of the most important steps is the prediction of the weight light ship and the coordinates of the center of gravity position. Formulas useful for calculating weights based on design assumptions and basic ship dimensions are scarce in the publications, and those available are very inaccurate.

During the research, the results of which are presented in the article, appropriate regression formulas were developed for relatively new container ships, for which relevant weight data were collected. The publication uses accurate weight group calculations for container ships built in a shipyard, of which post-construction measurement data were also used.

The authors’ regression formulas are characterized by very high accuracy in predicting not only the weight light ship and the coordinates of the center of gravity position, but also the individual weight groups into which the ship’s structure and equipment are divided.

The functional relationships presented in the article are a complete novelty in terms of application and accuracy in predicting the weight of a ship at the preliminary design stage. They are directly applicable to the design of manned container ships in the 270 to 3100 TEU range.

By developing functional relationships applicable to the whole ship and to specific weight groups, it will be possible to use them, after appropriate modification and analysis of the structure and equipment, for the preliminary design of unmanned container ships. On this basis, a method for designing unmanned container ships will be prepared, as well as a method for evaluating the benefits of their operation, particularly for reducing fuel consumption and emissions of CO₂ and other exhaust gases.