Review of ship arrangement design using optimization methods

Abstract

This study presents a comprehensive review of various research efforts aimed at optimizing ship arrangement design. A detailed comparative analysis was conducted on studies that approached ship arrangement design from an optimization perspective. Therefore, the fundamental components of an optimization problem (design variables, objective functions, and constraints) were systematically summarized. Ship arrangement design problems encompass a broad range of applications, including compartments, cabins, and equipment, and the problem definitions for each were compared and analysed. In ship arrangement design, the design variables subject to optimization primarily include the positions of bulkheads, the order of compartment assignments, and, in the case of cabins, elements such as corridors connecting cabins. For equipment arrangement, design variables consider the location of the equipment and, for multilayer arrangement, the floor on which the equipment is to be placed. Regarding objective functions, various studies proposed and applied different functions to optimize operability for improving operational efficiency, stability for enhancing survivability, and cost efficiency, which is particularly critical for merchant vessels. Finally, constraints were considered, including a range of physical constraints and regulatory constraints that vary by ship type. Using these definitions, ship arrangement optimization problems were applied across different ship types, and the study discusses how future research in ship arrangement optimization may develop further.

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1. Introduction

1.1 Research background

From ancient times, the arrangement design process has been one of the critical components in ship design. However, research on this process has not been as extensively conducted as in other design areas, such as hull form and structural designs (Shields et al., 2018). This is primarily because most arrangement designs carried out after determining major specifications and hull form design must comply with various international regulations, design codes, and owner requirements, leading to many constraints. Consequently, designers have traditionally performed arrangement designs manually, meticulously reviewing all aspects. Consequently, designers have traditionally performed arrangement designs manually, meticulously reviewing all aspects. Even today, most of the shipbuilding industry still relies on manual arrangement design (Kim, 2019). Therefore, there is a continuous demand for more efficient methods of conducting arrangement designs in the ship design industry. There is a persistent need to optimize arrangement designs to maximize efficiency within the limited space of the ship. In response, this study aims to comprehensively review existing research on ship arrangement design and introduce methods and directions that have been explored to date.

1.2 Arrangement design process

Generally, unlike vehicles, ships are not purchased as mass-produced items but are built to order based on the customer's specifications. Therefore, negotiations between the shipowner and the shipbuilder are conducted based on a conceptual design. Once a construction contract is signed, the ship is produced according to negotiated requirements and specifications. During this process, the ship must undergo various design stages, including initial design, detail design, and production design, before reaching the production phase (Roh & Lee, 2017). Specific requirements and key deliverables of each design stage are illustrated in Fig. 1.

The ship design process has traditionally followed a series of steps from a shipowner's requirements to an estimation of the ship's cost, guided by the design spiral (Luco Salman et al., 2019), the process of progressively refining the ship design through iterative work. Briefly, initial steps in practice involve distributing preliminary planning documents and work requests to designers. This allows for the creation of a preliminary general arrangement plan that estimates the lightship weight based on the shipowner's requirements. Subsequently, to achieve a more accurate estimate and lightship weight calculation, requirements are compared with those of similar existing ships. Necessary equipment and outfitting items are then selected accordingly. This process enables a more precise estimation of the lightship weight. Following this, intact and damage stability are calculated using the finalized hull form and general arrangement plan. A structural design is then conducted based on these calculations. This process can be diagrammatically presented in Fig. 2, facilitating the preparation of an initial cost estimate.

Ideally, according to the design spiral, the design proposal could be revised after completing each design cycle before moving on to the next spiral. However, in practice, each phase involves iterative improvements to address issues before proceeding to the next stage. If further problems or opportunities for improvement arise, they are addressed in the subsequent spiral. Recently, large shipyards have streamlined this process by leveraging extensive experience from numerous reference projects. They base their cost estimates on a mother ship and make only detailed adjustments, resulting in highly accurate estimates. This estimation process is also expedited, typically taking only 2–3 weeks.

Arrangement design is defined as ‘the arrangement of space for all the required functions and equipment, properly coordinated for location and access’ (Tapscott, 1969). In the process of ship design following the design spiral, most stages can be designed using well-established theoretical methods, with only detailed adjustments made manually. However, the arrangement design process is unique in that it is predominantly conducted manually. Fortunately, recent practices involve reviewing and modifying only changes based on many reference projects. Arrangement design remains one of the most labour-intensive design processes. Detailed research in this area is relatively less compared to others, primarily because arrangement design contributes less to the hydrodynamic and structural performances of the ship. However, as technologies in hydrodynamic and structural performances have advanced and other design processes have become more sophisticated, there is a growing demand for optimization and automation in the arrangement design stage. This study investigates, compares, and analyses research aimed at improving arrangement design from perspectives of optimization and automation in the early stages of ship design. Figure 3 compares the current arrangement design process with the optimal arrangement design process pursued by various studies.

2. Optimization of Compartment Arrangement Design

In ship design, arrangement design refers to the process of designing internal compartments of a hull, also known as compartment design (Roh & Lee, 2017). Compartments fundamentally include spaces necessary for the ship's operation, such as the engine, auxiliary, and electrical rooms. They also encompass tanks essential for the ship's operation, including fuel tanks, ballast tanks, and freshwater tanks. Depending on the type of ship, various compartments are required. For instance, in merchant ships, compartments can include cargo tanks for carrying goods or cargo holds for specialized cargo such as those on LNG (Liquified Natural Gas) carriers. In the case of naval ships, compartments include armories, ammunition rooms, command and control centres, and living quarters for the crew designed for operational purposes.

Research studies related to arrangement design have primarily focused on the early stages of arrangement design, specifically conceptual design, aiming to optimize the arrangement design process. This is because major equipment and outfitting items have not been selected yet at the initial stages, allowing for relatively high spatial arrangement flexibility.

An optimization problem fundamentally consists of three components: design variables, objective functions, and constraints (Mykel & Tim, 2019). Design variables are a set of parameters that represent the system. In the context of arrangement design, design variables include the position and number of decks or bulkheads as well as names of allocated compartments. Objective functions are criteria used to evaluate the quality of various alternatives in an optimization problem. For arrangement design, objective functions might include maximizing operability or minimizing vulnerability. Depending on the number of objective functions, the optimization problem can be classified as a single-objective or a multi-objective. Most engineering problems aim to satisfy multiple objectives. Arrangement design problems are typically set up to meet various objective functions. Constraints refer to conditions that design variables must satisfy. For example, if the position of a deck is a design variable, its position cannot exceed the ship's depth. Constraints can be set as inequality constraints or equality constraints. There are usually multiple constraints in an optimization problem, similar to objective functions. Most research studies on arrangement design have employed optimization problems.

Figure 4 illustrates design variables, objective functions, and constraints typically considered in the optimization of a ship arrangement design. However, research often suggests incorporating additional methodologies to evaluate objective functions or calculate constraints rather than relying solely on optimization problems. Thus, this section discusses the following aspects considered in relevant studies for optimizing ship compartment arrangement design: design variables, objective functions, and constraints, as well as their respective characteristics.

2.1 Design variables for the optimal arrangement of compartments

Design variables in an optimization problem refer to the independent parameters that define the design and whose values are adjusted to achieve an optimal solution (Arora, 2017). Ship arrangement design requires optimization for various objectives involving both qualitative and quantitative factors (Papanikolaou, 2010). To meet these objectives, it is crucial to select variables that influence the evaluation of the goals. For optimizing ship compartment design, design variables can be divided into those for hull compartment arrangements and those for cabin configurations, with each study often introducing specific design variables tailored to its focus.

2.1.1 Design variables for the compartment arrangement

The hull occupies the majority of the space in a ship, particularly in merchant vessels where most compartments are located within it. In warships, key facilities, aside from crew accommodations, are also found inside the hull. Therefore, many studies have established design variables to optimize the hull compartments. The primary structure involves determining the placement of bulkheads and decks (Luco Salman et al., 2019). Most research selects transverse and vertical bulkhead positions, as well as deck locations, as key design variables to divide the hull into functional compartments.

2.1.1.1 Defining the compartment arrangement as a one-dimensional problem

Parsons et al. (2008) and Nick (2008) adopted a two-dimensional (2D) approximation of the hull, defining compartments by pre-determined decks and transverse bulkheads, which were then modelled as ‘zone-decks.’ These zone-decks were represented as design variables in a one-dimensional (1D) vector format. Although this method allows for the reallocation of compartments, it lacks flexibility in optimizing spatial geometry. However, it has the advantage of accommodating both standardized rectangular shapes and arbitrary geometries when spaces have already been pre-defined, considering the hull form.

Lyu et al. (2015) further divided the hull internally using transverse watertight bulkheads. The subdivisions were represented as 1D vectors, simplifying the compartmentalization of the hull structure. This approach facilitates straightforward compartmental subdivision but focuses primarily on position rather than detailed geometric optimization.

A 1D design variable can be expressed as shown in Equation (1) and can be interpreted in various ways depending on the application. This variable serves as a simplified representation of complex systems, encapsulating key parameters into a vector format that can be manipulated within optimization frameworks. By reducing multiple factors into a single dimension, this approach allows for flexibility in design exploration while maintaining computational efficiency

where |${{x}_i}$| represents the position of the ith transverse watertight bulkhead, and n is the total number of bulkheads under consideration in Lyu et al. (2015). On the other hand, it can be used in Parsons et al. (2008) and Nick (2008), where |${{x}_i}$| represents assigned zone-deck for space i, n is the total number of spaces, and each |${{x}_i}$| can take values from 1 to K, where K is the total number of zone-decks.

2.1.1.2 Defining the compartment arrangement as a two-dimensional problem

Chung et al. (2011) defined the design variables for optimizing the inner hull structure of a submarine. To facilitate optimization, the hull was divided into three vertical regions. Following this, the positions of the transverse bulkheads were expressed as proportions of the overall hull length, allowing for a scalable representation of the internal structure. The optimization framework included 13 transverse bulkhead positions alongside the submarine's length and diameter, as well as the allocation of 11 compartments. In total, 28 parameters were used to represent the pressurized hull of the submarine.

Similarly, Kim & Roh (2016) approached submarine hull optimization in three stages, focusing on structural efficiency. In the first two stages, they optimized the positions of transverse bulkheads and decks, as well as the compartment allocations. In the second stage, they refined the allocation of internal subcompartments, treating these assignments as additional design variables. This hierarchical approach ensured a more detailed and systematic optimization of the submarine's internal pressure hull

Equation (2), as expressed in the study by Kim & Roh (2016), represents the design variables where |${{x}_i}$| denotes the position of each transverse bulkhead, |${{z}_i}$| indicates the height of each deck, and |${{c}_i}$| represents the order of compartment assignments within the spaces divided by bulkheads and decks. Here, while |${{x}_i}$| and |${{z}_i}$| are continuous real-valued variables, |${{c}_i}$| is treated as a permutation variable, composed of a set of non-repeating integers, to manage the assignment order of compartments.

2.1.1.3 Defining the compartment arrangement as a three-dimensional problem

The previously defined 1D or 2D arrangement design simplifies the design problem, making it easier to generate alternatives and thus facilitate the derivation of optimal solutions. However, real-world arrangement design problems are more complex and, therefore, cannot be reduced to just one or two dimensions. As such, it is necessary to consider the design in three dimensions, and various studies have analysed the characteristics of hull designs defined by three-dimensional (3D) variables.

In the study by Jung et al. (2018), the transverse bulkhead, longitudinal bulkhead, and deck were designated as design variables, enabling the definition of spaces within the hull based on 3D segmentation. Guan et al. (2018) approached the inner shell design differently by defining variables for the inner shell, which consists of numerous knuckle planes. The knuckle position is determined based on factors such as the cargo hold, water ballast tanks, floating status, and the minimum distance between the inner and outer shells. Since the bow and stern sections of the ship undergo rapid shape transitions, the knuckle plane shifts accordingly, with its form varying based on the type of cargo being transported. Despite these variations, the inner shell can be categorized into components like the inner shell plate, inner bottom plate, hopper sloping plate, and topside wing sloping plate, with parameters set for each to represent the ship's inner shell.

Jafaryeganeh et al. (2019, 2020) focused on cargo tank arrangement within the hull, defining design variables for the overall length of the cargo tank, the positions of transverse and longitudinal bulkheads, and the location of the collision bulkhead. They also included the width of each wind tank and the height of double-bottom tanks in each longitudinal subdivision, optimizing the precise shape of 3D cargo tanks.

Additionally, in the recent study by Milatz et al. (2023), which integrated parametric design with the optimization process for arrangement design, the cargo hold aft, forward bulkhead, fuel oil tank bulkheads, and pump room bulkhead were considered as longitudinal parameters. The double hull width, pipe tunnel position, and water ballast bulkhead were treated as transverse parameters. For the vertical parameters, the positions of the tank top, tween deck, main deck, fuel oil tanks, pump room, and openings were included in the optimization process.

In holistic design concepts, such as the work by Papanikolaou (2014), research has continued to optimize ship design by considering all design phases simultaneously. Studies like those by Papanikolaou et al. (2010), Koutroukis et al. (2013), Nikolopoulos & Boulougouris (2018), Priftis et al. (2018), and Vassalos et al. (2022) are representative examples. These studies define a more diverse set of design variables for initial design and arrangement design optimization than previous research. The studies explore variables such as LOA (Length OverAll), B (Beam), CoB (Centre of Buoyancy), block coefficient, and depth, as well as detailed parameters for internal arrangement optimization like inner bottom height, side shell width, hopper plate angle and width, bulkhead positions, and the number of frames per tank. This approach enables extensive design and optimization across a broad range of parameters, including compartmental arrangement and principal ship dimensions.

Table 1 summarizes the 3D design variables used in studies employing a holistic design approach, categorized by ship type. These studies utilized such diverse design variables to optimize not only the compartment arrangement design but also the main dimensions and propulsion performance, thereby achieving comprehensive ship design optimization across the vessel's entire life cycle.

2.1.2 Design variables for the cabin arrangement

Section 2.1.1 analysed the design variables related to the arrangement of compartments inside the main deck of the hull. In this section, the focus shifts to analysing the design variables set in studies aimed at optimizing the arrangement of cabins – the areas above the deck where the crew resides and that are essential for ship operations. The goal is to review the variables used in the optimization of the cabin arrangement and ship operational spaces.

Lee et al. (2002) defined each cabin as a ‘facility.’ In their study, the design variables included the area of each cabin, the location of vertical and horizontal passages, and the arrangement order of cabins, specifically addressing cabin type allocation. Lee et al. (2005) expanded on this earlier work by applying the same design variables to a multi-layer problem involving three decks, thereby extending the problem definition.

Parsons et al. (2008) and Nick (2008), as explained in Section 2.1.1, initially approached compartment arrangement design as a 1D problem and then created a detailed deck plan in a second step. For cabin arrangement design, they defined ‘zone-decks,’ starting with the damage control deck and detailing the compartments previously arranged in the first stage. The cabin arrangement accounted for not only rectangular shapes but also varied shapes like T, L, C, and Z. A ‘three-box element’ was used for arrangement design, but design variables for each zone-deck were expressed in one dimension, and a 2D arrangement was derived using a 1D vector.

Additionally, AnKim et al. (2014) simplified cabin arrangement design into a 1D allocation problem. The method of transforming a 2D spatial layout problem into a 1D allocation problem has been widely used in solving complex spatial layout problems. Notably, Bozer et al. (1994) proposed a spatial layout algorithm that simplifies spatial layout problems using a space-filling curve (SFC). A typical SFC is the Hilbert curve (Sagan, 1994), which is a 1D curve that fills a 2D grid with sides of length as powers of two. In their study, the cabin arrangement across three decks of a small ship was similarly defined as a 1D allocation problem.

After performing the hull arrangement design in two stages in Section 2.1.1, Jung et al. (2018) proceeded with the cabin arrangement design. In this stage, the design variables were selected under the assumption that compartments had already been arranged. The design variables included the allocation order of each cabin and the location of passages in the x- and y-directions on the deck.

To optimize cabin arrangement for improving passenger evacuation efficiency, In the study by Li et al. (2019), the Level of Service (LOS) approach (Choi et al., 1988) was introduced into arrangement design. Unlike other studies, the arc (x_i, y_i) connecting the LOS nodes in the topological network was designated as the design variable.

To optimize cabin arrangement for improving passenger evacuation efficiency, Hu & Cai (2020) assumed symmetry between the port and starboard sides, excluding the central cabins on the deck. The cabin arrangement problem was defined as the arrangement of 33 modules, such as stair modules, transverse corridors, and passenger cabins. The design variables were represented as a 1D vector, where cabin type, orientation, and size were specified. The door and wall positions were also determined for passenger cabins, while staircases required decisions on location and entry direction.

Wang et al. (2023) formulated cabin arrangement design as an integer programming problem to facilitate interactive design. Each cabin was defined by five design variables: the upper-left coordinates, width, height, and cabin type. Non-rectangular cabins were decomposed into multiple rectangular shapes, allowing the earlier-defined design variables to be applied. Similarly, Wang et al. (2023) adopted a 1D design variable approach for optimizing cabin arrangement. Integer permutations were assigned as design variables, and decoding them resulted in the allocation of pre-defined spaces to the variables. They also introduced a binary vector for corridor placement, allowing the determination of corridor locations within the deck.

Figure 5 illustrates the design variables related to the general cabin arrangement proposed by Jung et al. (2018). Cabin arrangement design typically includes the longitudinal and transverse corridor positions, and in some studies, even the corridor width is specified. Furthermore, the design variables also include the allocation of cabins based on their function and certain fixed design variables required by the operational needs. In Fig. 5, the yellow text and arrows indicate the position of longitudinal corridors, the green text and arrows show the transverse corridors, the blue text represents the cabin allocation numbers, and the black text indicates fixed cabins that cannot be relocated.

2.2 Objective functions for the optimal arrangement of compartments

Objective functions varied among studies, with many focusing on ensuring the fundamental stability of the ship while also enhancing operability to improve the efficiency of the arrangement. Moreover, since many studies dealing with cabin arrangements targeted warships, several have defined objective functions related to survivability. Given the complexity of ship arrangement design problems, which must take into account multiple objectives, it is necessary to consider the major objective functions outlined earlier. For this reason, most studies address ship arrangement design optimization using multi-objective functions (Kim & Roh, 2016). This section analyses the key objective functions discussed in previous studies and highlights the unique objective functions considered in each.

2.2.1 Operability

Operability in arrangement design primarily refers to the spatial relationships between different areas, defined through criteria such as adjacency and antagonism. By improving operability, factors such as the ease of logistical movement and the comfort of both crew and passengers can be enhanced.

A common objective function for operability is to assess the adjacency and antagonism between spaces and then evaluate how well the arrangement satisfies pre-defined adjacency and antagonism metrics. For instance, Lee et al. (2002) focused on the efficiency of logistical movement by optimizing cabin placement based on the positions of spaces that require logistics and also considered adjacency to improve cabin arrangement. Similarly, Parsons et al. (2008) considered factors such as the preferred location of each space, adjacency, and separation between spaces. They also evaluated satisfaction with space size, efficient area utilization, and the relative importance of each space, thereby applying a diversified approach to operability.

Jung et al. (2018) and Kim & Roh (2016) employed an adjacency index to improve the efficiency of connections between compartments or cabins. In Kim et al. (2014), a more detailed analysis of operability was conducted using the Systematic Layout Planning (SLP) method (Abotaleb et al., 2016) to optimize the living efficiency of crew members.

Research has also considered crew and passenger comfort as part of operability. Vassalos et al. (2022), for example, used a holistic approach to optimize ship arrangement design. Although they focused deeply on damage stability protection, they also considered open spaces for amenities and entertainment alongside essential safety zones to maximize operational efficiency.

In Wang et al. (2023), passenger comfort and mobility were evaluated by assessing the impact of the arrangement on daily life. They employed functions to ensure adequate cabin area, distribution, and correlation between spaces with similar functions. Wang et al. (2024), similar to previous studies, considered circulation intensity and adjacency as objective functions but added the unique consideration of crew comfort, specifically examining ergonomics, noise and vibration sensitivity, natural lighting, and ventilation performance based on cabin arrangement.

Equations (3) and (4) below represent the adjacency index, which is commonly used to account for operability in arrangement design (Kim et al., 2017)

where |${{q}_{ij}}$| represents the adjacency index between the ith and jth cabins or compartments, and |${{d}_{ij}}$| denotes the distance between the ith and jth cabins or compartments. Thus, this objective function guides the arrangement by placing compartments with high adjacency values closer together while those with low adjacency values are placed farther apart.

2.2.2 Cost efficiency

In many studies focusing on the optimization of ship arrangement design, cost efficiency has been considered as an objective function similar to operability. While operability indirectly contributes to cost efficiency by optimizing the movement of goods within the ship, this section comprehensively analyses studies that directly consider cost efficiency as an objective function.

Maximizing the cargo hold volume can increase the operational expenditures (OPEX) of the ship by improving its operational profitability. For example, Guan et al. (2018) focused on maximizing the cargo hold volume to enhance cost efficiency. Similarly, Jafaryeganeh et al. (2019) also set the objective function to maximize cargo volume. Additionally, Koutroukis et al. (2013) aimed to increase the ship's competitiveness by minimizing the required freight rate (RFR). Nikolopoulos & Boulougouris (2018) pursued a similar approach but aimed at minimizing both the RFR and OPEX directly. Priftis et al. (2018), addressing updated international regulations, employed a holistic design approach that included minimizing the RFR while maximizing cargo capacity and stowage ratio, thus maximizing the owner's profits.

On the other hand, some studies focused on minimizing the capital expenditures (CAPEX) associated with building the ship rather than maximizing the profit from its operation. For instance, Chung et al. (2011) set the objective function to minimize the volume of the pressure hull in a submarine, thereby reducing production costs. Papanikolaou et al. (2010) aimed to reduce CAPEX by generating a structural model during the arrangement design phase and directly calculating the steel weight in the cargo space area, with the goal of minimizing it. Nikolopoulos & Boulougouris (2018) also performed parametric optimization of the cargo hold to minimize CAPEX, similar to the earlier studies.

Equations (5) and (6) represent common methods for calculating OPEX, specifically through RFR and the Energy Efficiency Design Index (EEDI; IMO, 2022). The RFR is a direct measure of OPEX, representing the unit cost of transporting cargo (Watson, 2002). On the other hand, EEDI indirectly calculates OPEX by measuring the carbon emissions produced during cargo transportation, making it a standard for assessing the environmental impact and energy efficiency of the ship's operation

where PW is the present worth of the respective cost. In calculating EEDI, the term emissions reduction in the numerator refers to the amount of CO₂ removed by recently installed pollution abatement technologies. These technologies are designed to reduce the overall environmental impact of the vessel by capturing or minimizing carbon emissions during operations.

2.2.3 Stability and survivability

Among the various studies on arrangement design optimization, many have been applied to military purposes, specifically warships. For warships, it is essential to carry out compartment arrangement design focused on ensuring stability and survivability, and thus, many objective functions have been set to address these requirements. Even for commercial vessels, enhancing stability and meeting regulatory standards is critical. Therefore, studies that considered stability and survivability as objective functions have been comparatively analysed.

A study that specifically evaluated survivability for warships is Kim et al. (2014). This study used the SLP method (Abotaleb et al., 2016) to optimize the ship's arrangement, with the objective of maximizing survivability through vulnerability assessments. To quantify the ship's vulnerability, failure mode and effect analysis (FMEA; Tay & Lim, 2008) and fault tree analysis (FTA; Andrews & Tolo, 2023) were performed on the equipment placed within compartments. The vulnerabilities of the equipment were aggregated to derive the overall vulnerability of each compartment.

In Lyu et al. (2015), the objective function for arrangement design optimization considered anti-wind capacity, optimizing the placement of watertight bulkheads for subdivision. Anti-wind capacity refers to the maximum wind speed that ensures damaged ship stability in the presence of waves. To compute this, the maximum wind heeling arm and maximum roll angle in a damaged ship under wave conditions were determined. Using non-linear square damping, the roll motion equation was calculated, while water ingress was simplified using Bernoulli's equation. The probabilities of different damage scenarios were used to compute the rated wind velocity and the weighting factor for each compartment, ultimately calculating the average anti-wind capacity as the objective function.

Jung et al. (2018) performed arrangement optimization for warships, considering both stability and survivability. Stability was assessed based on three intact stability criteria and five damage stability criteria specific to naval vessels. Survivability was assessed by evaluating the vulnerability of bulkheads and compartments to damage. The bulkhead damage vulnerability was calculated by assessing the probability and extent of damage from explosive forces along the ship's longitudinal axis. Similarly, room damage vulnerability considered probabilities of damage in various directions and the importance of equipment in each compartment.

Stability is also a crucial factor for commercial vessels, especially as it is mandated by IMO (International Maritime Organization) regulations. Hence, many studies have considered stability and survivability as objective functions for optimizing commercial ship arrangement design. In the case of oil tankers, the amount of oil spilled during accidents can vary depending on internal arrangement changes. Papanikolaou et al. (2010) calculated the mean oil outflow, a non-dimensional coefficient under MARPOL (International Convention for the Prevention of Pollution from Ships) regulations, and minimized it as an objective function. Similarly, Jafaryeganeh et al. (2019, 2020) aimed to minimize the mean oil outflow and the still water bending moment (SWBM) to enhance survivability after damage. Since weight distribution can change due to arrangement modifications, which affect the SWBM, minimizing the SWBM was also included as an objective to maintain structural integrity.

Some studies have focused on safety measures directly related to the crew and passengers. Li et al. (2019) used the arcs of the LOS to configure the cabin arrangement, with the objective function set to minimize the total evacuation time of passengers within the ship. In addition, Hu & Cai (2020) optimized the internal arrangement of the ship to minimize passenger evacuation time and maximize evacuation efficiency, using a cellular automaton model (Toffoli & Margolus, 1987) to simulate passenger escape according to IMO regulations. During the operational phase, the internal arrangement must accommodate passengers and cargo while adhering to safety regulations.

Vassalos et al. (2022) emphasized the importance of balancing operational arrangements for profit optimization, such as open spaces for amenities and entertainment, with essential safety measures like watertight compartments. Their design process included measures to enhance damage stability and survivability in the event of hull breaches. The arrangement was optimized to facilitate swift and efficient evacuation and rescue operations, complying with international standards such as SOLAS (safety of life at sea), which stipulates specific requirements for compartmentation and stability. Thus, their approach helps select an arrangement design that enhances safety without compromising operational capabilities.

Milatz et al. (2023), in order to satisfy the probabilistic damage stability criteria typically applied to merchant vessels, referenced the requirements set by IMO (2020) and included an objective function to maximize the attained subdivision index (A-index).

2.2.4 Designers’ preferences

As previously described, ship arrangement design is primarily conducted manually by design experts. These experts typically possess extensive knowledge of reference projects, allowing them to review and modify only the additional requirements and changes needed for the current ship based on information from the mother ship. They also ensure that these modifications comply with ship design regulations. Thus, the expertise and know-how of the designer play a crucial role in the arrangement design process. From this perspective, several studies have considered the expertise of professionals, such as their knowledge and know-how, as objective functions in optimizing ship arrangement design.

Chung et al. (2011) used a knowledge-based system for the optimization of the internal arrangement of a submarine's pressure hull. To construct the knowledge-based system for internal compartment arrangement, they employed the Java Expert System Shell (Friedman-Hill, 1997), a rule-based expert system. Knowledge for the system was gathered by conducting interviews with design experts and referencing relevant regulations and projects. This knowledge was expressed in IF–THEN statements within the rule-based system. The knowledge gathered from experts was then applied as a penalty function during the arrangement optimization process, ensuring that the arrangement closely aligned with expert knowledge.

Similarly, Kim & Roh (2016) aimed to incorporate expert knowledge into the optimization of arrangement design by utilizing their custom inference engine (Kim et al., 2015). This expert system comprised a space list, representing the knowledge required for individual spaces or equipment, and a relation list, representing the relationships between spaces or equipment. These lists were used to formulate the feasibility index in the optimization problem. The objective was to maximize the feasibility index during the arrangement design process, ensuring that the expert's intent was fully reflected in the final arrangement.

Wang et al. (2023) explored an interactive approach to optimizing the cabin arrangement of a ship. This study introduced an interactive expert system into the ship arrangement optimization process, which could propose alternative solutions to the user at various stages of the optimization. This allowed the user to modify the design iteratively, refining the arrangement based on expert guidance throughout the process.

Figure 6 represents the most commonly used expert system for reflecting the designer's know-how, which uses IF–THEN rules to evaluate specific design knowledge and apply it as an objective function. In this case, the expert system in Fig. 6 incorporates the designer's knowledge with the rule that ‘the accommodation and machinery should ideally be around 5 m apart.’ This knowledge is then processed within the expert system to guide the optimization of the arrangement, ensuring that such design constraints are followed during the arrangement design process.

Le Poole et al. (2022, 2023) recently proposed a collaborative design rationale method to be applied in the early stages of ship arrangement design. The method provides a structured format of guidelines to ensure that expert considerations are effectively integrated and consistently reused throughout the design process. Designers are encouraged to organize their rationales according to these guidelines, thereby systematizing and unifying design knowledge. This approach facilitates the training of designers and establishes a feedback system during the arrangement process, allowing for iterative revisions and the continuous refinement of arrangement design knowledge.

This method is fundamentally based on the research of DeNucci (2012). His contributions to ship arrangement design emphasize capturing and utilizing design rationale to enhance the conceptual design process. DeNucci (2012) introduces a methodology focused on expressing, structuring, storing, and reusing design rationale. This approach employs a parametric description grounded in packing problems, allowing for simultaneous adjustments to various ship components, including the hull, superstructure, decks, bulkheads, and system arrangements, without requiring manual intervention.

2.3 Constraints for the optimal arrangement of compartments

The most critical aspect of arrangement design optimization is determining the objective function, which defines the purpose of the arrangement to be achieved. Thus, Section 2.2 dealt with the objective functions. However, equally important in arrangement design optimization is the establishment of constraints.

Constraints are the restrictions imposed on the design or decision variables in an optimization problem, which must be satisfied for the solution to be feasible. They can take the form of equalities, inequalities, or bounds (Bazaraa et al., 2006). Constraints ensure that the solution to an optimization problem is not only mathematically optimal but also practically feasible. They reflect real-world limitations, such as physical availability, safety standards, regulatory restrictions, or performance criteria. Without constraints, the optimization could result in unrealistic or impractical solutions. Therefore, this section provides a detailed comparative analysis of the constraints used in ship arrangement design optimization.

2.3.1 Physical constraints

Most of the studies previously discussed are inherently constrained by physical limitations. The various design variables used for arrangement design, as explained in Section 2.1, such as the positions of bulkheads and decks, or the order of compartment allocations, represent physical parameters. These design variables naturally have boundaries or applicable ranges within the ship's hull, meaning they come with inherent physical constraints.

Lee et al. (2002, 2005) assumed each deck in the ship's cabins to be rectangular and carried out the arrangement design while considering the edges of these shapes. Constraints were introduced to ensure that the boundaries and assigned numbers for each facility did not exceed the total number of available facilities, ensuring reasonable solutions. Jung et al. (2018) proposed a arrangement design optimization for surface vessels, optimizing the positions of bulkheads, decks, and tanks, followed by cabin arrangement optimization. The positions of bulkheads were constrained so as not to exceed the ship's width and length, and similarly, the deck positions were constrained to ensure they did not surpass the ship's depth. These constraints were similarly applied in studies by Chung et al. (2011) and Kim & Roh (2016).

Wang et al. (2024), while optimizing cabin arrangements for passenger comfort, constrained the total area of the cabins so as not to exceed the total deck area. In addition, structural stability and the connectivity of cabins to hallways or stairs were also restricted to ensure a realistic arrangement.

Guan et al. (2018), in their study to maximize the tank volume within a commercial ship, introduced unique constraints to ensure compliance with GM (metacentric height), average draft, trim, and propeller immersion criteria. Similarly, Hu & Cai (2020) addressed essential area or volume constraints when designing cabin or compartment arrangement.

In studies utilizing a holistic approach, such as Koutroukis et al. (2013), physical boundaries for cargo hold modelling were constrained to ensure the outer hull, deck, double bottom ceiling, and stringer were not exceeded. Nikolopoulos & Boulougouris (2018) used CAESES (CAE System Empowering Simulation) (Harries & Abt, 2019) for parametric design, considering the positions of bulkheads, frame spacing, topside tank dimensions, lower stool height, and double bottom height, all of which were constrained by physical limits.

Parsons et al. (2008), similarly to other studies, used physical constraints in their arrangement optimization but incorporated fuzzy membership functions (Goguen, 1985). This approach allowed for finding better alternatives even when constraints were not perfectly satisfied, making it particularly useful for solving arrangement optimization problems, which often involve numerous conflicting conditions.

2.3.2 Constraints for regulations

As mentioned earlier, in practical arrangement design, physical constraints are imposed because compartments or the parameters defining them cannot exceed the hull, and cabins must remain within their designated compartments. Aside from these physical constraints, one of the most important constraints in arrangement design optimization comes from regulatory requirements. When designing a ship, key regulations such as IMO's SOLAS, ICLL (International Convention on Load Lines), and MARPOL must be followed. Naturally, these regulations play a significant role in ship arrangement design as well.

A prominent constraint arising from regulatory requirements is related to stability. For example, in the study by Jung et al. (2018), the internal compartment arrangement of a warship was optimized in two stages, and the design was constrained to meet three intact stability criteria and five damage stability criteria according to U.S. Navy regulations. Similarly, for military submarines, Kim & Roh (2016) imposed a constraint to ensure the submarine remained in a neutrally buoyant condition, adding a constraint based on the evaluation of the equilibrium polygon, ensuring that all three loading conditions remained within the polygon.

In merchant vessel arrangement design, Jafaryeganeh et al. (2019) applied constraints to meet safety regulations and environmental standards, adding further constraints to comply with IMO regulations for damage stability. Similarly, Hu & Cai (2020) performed arrangement optimization to maximize passenger evacuation efficiency while ensuring that the arrangement met the IMO standards for passenger evacuation times.

Several holistic approaches to arrangement design optimization have also incorporated constraints from various regulations. In the initial study by Papanikolaou et al. (2010), constraints were added to ensure compliance with MARPOL regulations. Building on this, Koutroukis et al. (2013) added constraints to ensure compliance with IMO visibility and stability regulations for merchant ships (IMO, 1998). Priftis et al. (2018) incorporated constraints to meet the stricter regulations introduced by the IMO in 2012. More recently, Vassalos et al. (2022) included the latest SOLAS regulations for passenger safety as part of their constraints in arrangement design optimization. Milatz et al. (2023) set a constraint to ensure that the attained index (A-index), used to satisfy probabilistic damage stability for merchant vessels, must always be greater than or equal to the required index (R-index).

Figure 7 and Table 2 present the intact stability regulations commonly applied to merchant vessels. These stability criteria serve as constraints for compartment arrangement design, ensuring the ship's stability under non-damaged conditions. Additionally, damage stability regulations must also be considered, providing further constraints to ensure the ship remains safe and stable in case of damage or flooding. These regulations, which are enforced by IMO (2008), are critical for optimizing ship arrangement while ensuring compliance with international safety standards.

In Fig. 7, Area A is the area under the righting arm curve between a heel angle of 0° and 30°, representing the initial stability of the vessel. Area B is the area under the righting arm curve between a heel angle of 30° and the minimum of 40° or the angle at which openings in the hull submerge, denoted as ϕ_f. Additionally, ϕ_f refers to the heel angle at which hull openings, such as hatches, begin to submerge, while ϕ_m is the heel angle where the maximum righting arm occurs. These parameters are crucial for assessing the vessel's overall stability and resistance to capsizing.

2.4 Optimization technique for compartment arrangement design

As described in Section 1, ship arrangement design is still predominantly carried out manually by experts. This is due to the complex nature of the problem, where a large number of design variables and factors must be considered simultaneously, resulting in a multitude of possible alternatives that make selection challenging. Given the inherent complexity of ship arrangement design, most studies treat it as a multi-objective optimization problem, as described in Section 2.3, while considering numerous constraints. This complexity makes it difficult to select the optimal solution using traditional optimization methods, leading to the use of heuristic approaches, one of the most prominent being genetic algorithms (GA). In this study, the different optimization methods used in various research efforts, specifically considering the unique challenges of arrangement design optimization, are compared and analysed.

Figure 8 shows the flow of the optimization process followed in most studies. Based on the design variables generated by the optimization algorithm, the values are converted into practical variables used for actual design execution in each study. The hull or cabin modelling is then conducted using these design variables. Following this, the satisfaction of various objective functions and constraints is calculated. The results from this evaluation of design alternatives are then fed back into the optimization algorithm, enabling it to generate better alternatives and further refine the optimization process.

As mentioned earlier, GA-based optimization methods are widely employed for arrangement design problems. As discussed in Section 2.1, various types of design variables have been specified for optimizing the arrangement of compartments and cabins, ranging from continuous variables to integers and permutations. The most common GA algorithms start with the research of Lee et al. (2002), followed by Lyu et al. (2015) and Hu & Cai (2020). These studies used basic forms of GA while incorporating custom gene encoding methods to suit their design variables. A similar approach, but with the addition of hybrid agents to GA, was also utilized by Parsons et al. (2008).

Another widely used heuristic method for multi-objective optimization is the NSGA-II (Non-Dominated Sorting Genetic Algorithm-II) (Deb et al., 2002). NSGA-II, developed in 2002, is still extensively used due to its effectiveness in optimizing multi-objective problems. By incorporating elitism and non-dominated sorting, NSGA-II excels at finding optimal solutions across a broad spectrum, offering users a pareto optimal set for high-dimensional decision-making. Jung et al. (2018), Chung et al. (2011), and Kim & Roh (2016) applied NSGA-II to ship arrangement design using a combination of real-valued, integer, and permutation-based variables. To optimize these variables simultaneously, they enhanced standard GA operators. Similarly, Koutroukis et al. (2013) applied NSGA-II in their holistic approach to optimizing the ship's arrangement design, given the large number of diverse design variables and multiple objectives.

Although GA is the most representative heuristic algorithm, other similar heuristic methods have also been employed for arrangement optimization. Kim et al. (2014), for example, used differential evolution (DE, Rocca et al., 2011), a more intuitive method than GA, for optimizing cabin allocation order, which can be expressed as a 1D problem. Additionally, Guan et al. (2018) performed detailed optimization for cargo hold arrangement using the improved particle swarm optimization (IPSO) method (Li & Coster, 2022), an enhanced version of particle swarm optimization (PSO), known for its faster convergence speed and higher solution accuracy. While GA uses crossover and mutation operators, IPSO improves solutions by adjusting the velocity and position of particles. Generally, IPSO has a simpler improvement strategy than GA, resulting in faster convergence; however, GA may be better suited for finding Pareto optimal solutions in problems requiring deep multi-objective exploration.

As discussed, while many studies use various types of design variables, integer programming is often used for space allocation problems. Integer programming evaluates fewer alternatives than GA, leading to faster convergence. For example, Wang et al. (2023) considered the location, size, and type of cabins, as well as the shape of corridors, as design variables. Therefore, a mix of continuous and integer variables was involved. Mixed Integer Linear Programming was used to rapidly derive optimal solutions, with the Gurobi solver (Gleeson & Ryan, 1990) used to run the optimization algorithm.

In summary, ship arrangement design optimization presents a complex challenge that requires the consideration of numerous variables and constraints, leading researchers to employ a variety of heuristic methods, such as GA, NSGA-II, DE, and IPSO, depending on the specific characteristics and objectives of the design problem.

3. Optimization of Equipment Arrangement Design

As previously mentioned, arrangement design in ships generally refers to compartment arrangement design (Roh & Lee, 2017). Typically, arrangement design must satisfy various regulatory standards. It is often conducted by modifying information from previous reference projects or mother ships without specific performance indicators. These regulatory standards are revised over time, making it challenging to optimize arrangement design as modifications must align with updated regulations. Unlike terrestrial structures, ships are designed to maximize cargo hold size to function effectively as transport vessels, leaving minimal space for other compartments.

The basis for equipment allocation starts with determining which equipment or systems will be installed within each compartment. Therefore, the process accompanying compartment arrangement is the arrangement of equipment within these compartments. By optimizing the equipment arrangement within a compartment, it is possible to allocate the compartment more efficiently, potentially increasing cargo hold space. Thus, equipment arrangement is also a crucial step in compartment arrangement design.

Figure 9 shows design variables, objective functions, and constraints commonly considered when optimizing equipment arrangement in ships. This section compares and analyses studies that have optimized equipment arrangement within compartments and those that have integrated both compartment and equipment arrangement optimizations.

In particular, as with the previously discussed optimization methods for compartment and cabin arrangement design, a comparative analysis was conducted on the design variables, objective functions, and constraints related to optimizing the placement of equipment within compartments or cabins. This involved examining how these variables are defined, how objectives like minimizing space usage or improving accessibility are prioritized, and how various constraints, such as spatial limitations or regulatory compliance, are applied in the optimization process.

3.1 Design variables for the optimal arrangement of equipment

In Section 2.1, the design variables for compartment arrangement optimization were mainly concerned with the external positions of compartments, such as the placement of bulkheads and decks, or how compartments were allocated for specific purposes. However, for equipment within these compartments, the size of the equipment is usually determined by the vendor's specifications, so design variables related to the equipment's position and orientation are more critical than those related to size.

Since equipment is typically placed on decks, it is common to consider equipment positions in two dimensions. For example, Shin et al. (2002) assumed the engine room to be a 2D plane and selected the x and y positions of equipment as design variables, though the study did not account for rotation or multi-level placements. In the Kim & Roh (2016) study on optimizing compartment and equipment arrangement for submarines, a multi-stage optimization was performed, with the final stage focusing on equipment arrangement within compartments. Here, design variables included the x and y positions of equipment within the compartments, as well as the rotation of the equipment. The study assumed that equipment was generally aligned orthogonally, using Boolean variables where 0 indicated 0° orientation and 1 indicated 90° orientation.

In Ha et al. (2023), a multi-deck arrangement design of the ship's engine room was conducted. Similar to Kim & Roh (2016), they defined the deck positions and orientations of equipment as design variables. However, since the study aimed to optimize pipe routing, the z-direction position of the equipment was also considered as a design variable. Similarly, Li et al. (2019) focused on the position of equipment in two dimensions but introduced variables representing the type of equipment and the relative positions between successive pieces of equipment, making it easier to incorporate relationships between equipment into the objective function or constraints.

The arrangement model of the cabin and equipment is shown in Fig. 10. In the figure, the lower left corner of the cabin is designated as the origin. The variable |${{l}_j}$| represents the length of equipment j, |${{h}_{gk}}$| denotes the minimum horizontal spacing between equipment g and k, and |${{h}_{j0}}$| specifies the minimum horizontal distance between equipment j and the cabin boundary. Additionally, |${{\Delta }_j}$| represents the net distance between equipment j and either equipment j−1 or the boundary. The variable s indicates the row spacing between devices, while |${{s}_0}$| represents the distance from the first row of equipment to the boundary of the workshop. Finally, |${{x}_j}$| and |${{y}_j}$| denote the x and y coordinates of the centre of equipment j, respectively.

Typically, equipment that performs similar functions or requires sequential processes is grouped into systems or modules. This concept of modularization was introduced in the study of Cort & Hills (1987), and has been adopted in recent studies, including Gunawan et al. (2020, 2021) and Arta et al. (2020). These studies applied the modularization concept, where a module refers to a standardized set of equipment across different ships. To modularize engine room equipment, the commonalities and functions of the equipment were analysed, and the design structure matrix (Browning, 2016) was used to visualize dependencies within the module. They represented the engine room using a fine grid, with the module positions defined by the deck and the grid number where the modules were located.

As previously mentioned, most equipment is arranged within compartments, and their positions are expressed on a 2D plane (deck). However, equipment located in the superstructure above the main deck requires consideration of its 3D position and shape. This is especially important for warships, where armaments are often housed in the superstructure. In Hwang et al. (2022), a 3D arrangement optimization for equipment was conducted, considering the shapes of the equipment. The hull was represented using a 3D axis-aligned bounding box (AABB) model (Majercik et al., 2018), and the main equipment positions were allowed to change within predefined ranges of x, y, and z coordinates from a reference position.

3.2 Objective functions for the optimal arrangement of equipment

In equipment arrangement design, objective functions, much like in compartment arrangement design, typically focus on the relationships between equipment, especially their adjacency. However, for equipment, it is often possible to account for actual physical connections, such as pipes or wires, between different pieces of equipment. This allows for the inclusion of connection costs in the optimization process, which is a common practice when setting up objective functions for equipment arrangement.

This section provides a comparative analysis of the general objective functions used for optimizing equipment arrangement within compartments, as well as specific objective functions considered in various studies. These objective functions might also consider factors such as minimizing installation costs, improving operational efficiency, or reducing the length of connecting pipes and cables between equipment to minimize material usage and installation time.

In essence, optimizing equipment arrangement design extends beyond simple adjacency by incorporating considerations of physical interconnections and their associated costs, providing a more comprehensive approach to the design process.

3.2.1 Adjacency and interconnectivity

Among the studies that set connection costs between equipment as an objective function, Shin et al. (2002) and Kim & Roh (2016) calculated the connection distance using the simple rectilinear distance between equipment. However, in Shin et al. (2002), an additional objective function was considered to ensure that the maintenance space requirements for engine room equipment arrangement were met. In contrast, Kim & Roh (2016) included not only the direct connection relationships between equipment but also introduced an adjacency index to maximize the proximity and minimize antagonism between adjacent equipment systems that were not directly connected.

The study by Ha (2023) is very similar to Kim & Roh (2016) in terms of design variables and the consideration of connection distance as an objective function. However, Ha's study performed direct pipe routing to reflect more accurate connection distances between equipment. Thus, the objective function was further refined to minimize the production hours and material quantities required for connections by minimizing the length of pipe connections. Specifically, the objective functions were set to minimize the length of pipes and ducts and the number of bends while maximizing space availability for operation and maintenance. In the second stage, a 3D pathfinding algorithm, the jumping point search algorithm (Harabor & Grastien, 2012), was used to directly calculate the 3D connection relationships between equipment and confirm arrangement feasibility.

In studies focusing on the arrangement of equipment modules, such as those by Gunawan et al. (2020, 2021) and Arta et al. (2020), the direct connection relationships between individual equipment were not considered, but objective functions were set to minimize the pipe connection costs between modules and to maximize module similarity. The pipe connection costs were calculated based on the module location, height difference, and offset on the 2D plan. For similarity, normalized values of module locations from a series of ships were used to maintain consistent module placement across the ship series.

Since the adjacency of equipment is easier to analyse compared to the qualitative adjacency of compartments and can be represented by direct connection relationships, some studies focus on this aspect. For instance, Li et al. (2019) applied the systematic facility layout planning (SLP) method (Van Donk & Gaalman, 2004; Liu et al., 2016; Benitez et al., 2018), commonly used in other fields of arrangement optimization to analyse the adjacency of the equipment. Unlike traditional SLP methods, which primarily focus on the transportation factor, this study emphasized the adjacency relationships between equipment. Thus, the objective functions were set to maximize the connection strength between equipment and circulation intensity.

3.2.2 Stability and survivability

Stability and survivability are typically considered in compartment arrangement optimization because, in most ship arrangement designs, compartment planning precedes equipment placement. Since the equipment that will be installed in each compartment is pre-determined, the relocation of equipment within a compartment generally does not significantly impact the overall stability and survivability of the ship. However, in the case of the engine room, where the heaviest and most numerous equipment is located, some studies have considered stability in equipment arrangement design.

In the study by Li et al. (2019), which thoroughly evaluated the connections between equipment using the SLP method, an additional objective function was introduced to minimize the distance between the equipment and the ship's centreline, with the goal of improving the ship's stability. However, unlike the compartment arrangement design, this study did not strictly apply IMO regulations for stability.

Equations (7) and (8) represent the objective functions applied in Li et al. (2019) to enhance stability and ensure the uniform placement of equipment. By defining the objective function as shown in Equation (7), the equipment can be positioned closer to the centreline of the hull, thereby improving the vessel's stability. Additionally, Equation (8) is designed to promote the even distribution of equipment within the engine room to prevent the free surface effect, which can cause an imbalance in the moment of inertia during the normal operation of auxiliary machines, thus contributing to improved ship stability. The symbols used in both equations are employed with the same meanings as those in Fig. 10

In the study by Hwang et al. (2022), which optimized the arrangement of equipment in the superstructure of warships, particularly equipment related to armaments, the ship's external 3D shape was directly addressed as part of the objective function. The survivability of the equipment arrangement was assessed using radar cross-section analysis (Balanis, 2012). During this process, a line distribution space division (LDSD) method was developed and applied. The objective function aimed to maximize the survivability of the ship, with evaluations based on the susceptibility and vulnerability of the equipment.

3.2.3 Designers’ preferences

Similar to compartment arrangement design, equipment arrangement design is also carried out within limited space, meaning the available room for equipment movement is very restricted, with little potential for variation. As a result, the designer's expertise plays a significant role, with much of it acquired through experience. Some studies have incorporated objective functions that account for this expertise in optimizing equipment arrangement.

In the studies by Kim & Roh (2016) and Ha (2023), an expert system was used to integrate the designer's knowledge and rules, which are difficult to express as objective functions, into the optimization process. The concept of object information was employed to apply expert knowledge relevant to each piece of equipment. In contrast, relation information was used to incorporate the expert's knowledge regarding the relationships between two pieces of equipment. Although object information and relation information are quite similar to the approaches used in Section 2.2 for compartment arrangement design, the difference lies in their application to equipment.

However, in Ha (2023), the application method was further diversified, with the positions of equipment within the compartments represented as node positions, allowing a broader range of expert knowledge to be applied during the optimization process.

3.3 Constraints for the optimal arrangement of equipment

Similar to the constraints analysed in Section 2.3 for compartment arrangement optimization, constraints are also essential for the realistic optimization of equipment arrangement within compartments. In equipment arrangement optimization, the primary constraints are physical constraints. Since equipment must be located within the designated compartments, this type of constraint is a fundamental consideration in all studies. This section provides a comparative analysis of the various constraints, including physical and additional constraints considered in equipment arrangement optimization studies.

As previously mentioned, the most common constraints in equipment arrangement optimization are physical constraints. A typical set of requirements for physical constraints can be found in the study by Kim & Roh (2016). First, to avoid interference between equipment, the location, length, and width of each piece of equipment are used for calculations. Second, a standard method is employed to ensure that the equipment remains within the boundaries of the compartment. Li et al. (2019) also applied constraints to prevent overlap between equipment, similar to previous studies. In this study, since the sequence of equipment is used as a design variable, an additional constraint was introduced to ensure that each piece of equipment is assigned only once. Additionally, as explained in Section 2.2, an objective function was introduced to bring the centre of gravity closer to the centreline of the ship to enhance stability. An additional constraint was also set to ensure that the moment difference caused by the weight of the equipment on the port and starboard sides does not exceed a certain threshold.

While previous studies assumed that equipment or modules were placed on a 2D plane, Hwang et al. (2022) performed a 3D arrangement optimization for armaments located in the superstructure of warships. Although physical constraints were applied, the method differed. Since armaments are placed in 3D space and have specific shapes, it is challenging to place them arbitrarily within the space. Therefore, Hwang et al. (2022) constrained the range of movement in the x-, y-, and z-directions for each piece of equipment based on its pre-defined position, ensuring that the arrangement remained realistic.

In the more recent study by Ha (2023), constraints were set similarly to previous studies to prevent interference between the equipment and ensure the equipment remained in installable positions. However, in addition to the objective function that considered expert system evaluation scores, the study also incorporated expert system evaluations into the constraints. For example, certain spaces critical for equipment operation and maintenance, as well as spaces for crew movement in the engine room, were considered mandatory based on expert knowledge. Since these conditions had to be strictly met, they were treated as constraints rather than objective functions, and any alternatives failing to meet these conditions were excluded from the solution set.

3.4 Optimization technique for equipment arrangement design

Equipment arrangement design, like compartment arrangement design, employs a wide variety of design variables and typically applies multiple objective functions in most studies. Given the wide range of constraints involved, solving the problem using conventional optimization methods is challenging. Therefore, most equipment arrangement optimization studies employ heuristic algorithms, with GA being the most commonly used approach. However, the application methods vary slightly across different studies.

As previously mentioned, GA is commonly used for generating basic optimization alternatives. In Shin et al. (2002), however, the objective function was not treated as a Boolean function but as a continuous function, leading the researchers to incorporate fuzzy logic into GA to evaluate the satisfaction of the objective function. In Li et al. (2019), instead of directly treating the position of each piece of equipment, the position difference between successive equipment was used as a design variable, resulting in a chromosome design in GA that differs from other studies. Similarly, in the studies by Gunawan et al. (2020, 2021) and Arta et al. (2020), which focused on module arrangement design, design variables were defined based on deck order, module order, and position, leading to a distinct gene structure compared to other research.

In Ha (2023), the arrangement optimization problem was set up similarly to the study by Kim & Roh (2016), but while Kim & Roh (2016) used only NSGA-II for optimization, Ha (2023) also applied other heuristic algorithms known for their superior performance, comparing and analysing the results. The SPEA2 (Strength Pareto Evolutionary Algorithm 2) (Zitzler et al., 2001) and SMPSO (Speed-constrained Multi-objective Particle Swarm Optimization algorithm) (Nebro et al., 2009) were evaluated alongside NSGA-II, and after comparing the pros and cons of each algorithm, SMPSO was found to be the most suitable for the study. As a result, Ha (2023) ultimately used the SMPSO algorithm to derive the final results of the study.

This study reviews and organizes previous research that directly optimizes the arrangement of ship compartments and equipment using optimization methods, as discussed earlier. However, taking a different approach, Shields et al. (2017) emphasized a systematic and structured approach to developing ship arrangements. Their method focused on optimizing the spatial arrangement of ship components to balance functionality, cost, and safety. By integrating computational techniques with design theory, they employed optimization algorithms and simulations to create effective and adaptable designs. This approach enabled the evaluation of multiple design alternatives, facilitating the selection of arrangements that meet specific operational and regulatory requirements.

The origins of this method can be traced to traditional naval architecture principles and engineering design theory, as well as advancements in computational design. Earlier methods often relied on experience-based heuristics and linear processes, which limited adaptability. Shields et al. (2017) built upon these foundations by incorporating optimization algorithms and modular design principles, allowing for a more flexible, multi-objective approach that aligns with modern demands for increased efficiency and sustainability (Dowling et al., 2024).

Although not explicitly applied directly to ship arrangement design, one example of a parametric approach for early ship design is the ‘packing approach,’ which enables the creation and exploration of numerous 3D ship arrangements at an early stage, helping designers identify optimal configurations (van Oers, 2011). This approach employs a parametric description to adjust multiple ship components simultaneously, supported by a search algorithm that generates diverse configurations. Its transparent selection process combines objective data with designer insight, resulting in practical and well-informed decisions. Successfully applied to complex vessels like drillships and frigates, the packing approach enhances efficiency by allowing designers to select configurations that meet specific performance needs, ultimately improving early-stage design for competitive, functional ships.

4. Applications of Optimal Arrangement Design

Ships exhibit various characteristics depending on their type. For example, in general merchant ships, the majority of the vessel's arrangement is dedicated to the cargo hold, and the design aims to maximize this area (Papanikolaou, 2014). In the case of VLCCs (very large crude carriers), the primary focus is on maximizing the size of the oil tanks. Similarly, for container ships, the arrangement is designed to accommodate the maximum number of containers within the grid-like cargo hold, organized by rows, tiers, and bays. In particular, for container ships, a large portion of the cargo is often stowed on the deck as well. Therefore, factors such as the visibility from the living quarters and the position of the engine casing must be carefully considered to meet regulatory requirements. On the other hand, warships prioritize operational performance and survivability over the cargo capacity considerations typical of merchant ships. Thus, it is challenging to apply a uniform arrangement design across all ship types, and the specific characteristics of each ship type must be taken into account. Therefore, this chapter analyses how various studies have applied the arrangement optimization techniques described in Sections 2 and 3 based on the characteristics of different ship types.

4.1 Merchant ships

4.1.1 Tankers

Tankers are one of the most common types of merchant ships, and as previously mentioned, maximizing the size of the cargo hold is prioritized. Given that the cargo is in liquid form, there is considerable flexibility in adjusting the cargo hold design. Consequently, numerous studies have been conducted on optimizing the arrangement of tankers.

Guan et al. (2018) conducted a study to maximize the cargo hold volume by optimizing the arrangement of the inner shell in tankers. The inner shell of tankers consists of several knuckle planes, and the positioning of these knuckles must account for factors such as the cargo hold, water ballast tanks, floating status, and the minimum distance between the inner and outer shells. Since the shape of the tanker changes sharply at the bow and stern, the knuckle planes also shift dramatically in these areas, and their form varies depending on the type of cargo being transported. However, the general structure remains consistent and can be classified into components such as the inner shell, inner bottom, hopper sloping, and topside wind sloping plates, which can be represented by several parameters. To optimize the tanker's shape, design variables were defined as the cross-section parameters at each knuckle position, and the objective function was set to maximize the volume of the cargo hold. Constraints were imposed to satisfy the ship's GM (metacentric height), average draft, trim, and propeller immersion. The optimization process used the IPSO method (Li & Coster, 2022).

Similarly, Jafaryeganeh et al. (2019) proposed a multi-objective optimization method for shuttle tankers, aiming to maximize cargo capacity while minimizing SWBM and enhancing safety to ensure survivability after damage. They used a parametric model to designate the internal arrangement with a fixed hull shape and regulatory framework. The design variables were set as the positions of watertight bulkheads, the height of the double bottom, and the width of the wing tanks. The objective functions were to minimize oil spill parameters for environmental protection, maximize cargo capacity for economic benefits, and minimize longitudinal bending moments for operational safety. Safety regulations and environmental standards were incorporated to ensure compliance with IMO's damage stability requirements. The optimization was performed using DAKOTA (Adams et al., 2021), a software that supports multi-objective optimization. In a follow-up study (Jafaryeganeh et al., 2020), this research was slightly expanded. The methodology remained similar, but additional constraints were introduced to comply with SOLAS and MARPOL (The International Convention for the Prevention of Pollution from Ships). The focus was on selecting the final design from the Pareto optimal solutions (Mathur, 1991) derived from the multi-objective optimization process. A weighted sum method was used to track the optimal solutions by adjusting the influence of various objectives, and technique for order preferences by similarity to an ideal solution (Hwang & Yoon, 2012) was applied to select the best design from the Pareto optimal solutions.

There are also cases where a holistic design approach was used for tanker arrangement optimization. Papanikolaou et al. (2010) began their design optimization using a reference design as the base, closely resembling real ship design processes. The necessary regulations for the arrangement design were referenced from MARPOL 73/78 (Julian, 2000). For hull form and arrangement design, they used NAPA (Agiwal et al., 2021), which supports parametric design tools. The NAPA software's features were utilized to perform calculations on ship performance, oil outflow, and intact and damage stability. Additionally, POSEIDON (Tanny et al., 2017), provided by DNV-GL (Det Norske Veritas-Germanischer Lloyd), was used to generate a structural model of the design, which was then used to calculate scantling requirements and steel weight in the cargo space. They defined 41 design parameters, including the number of bulkheads within compartments and the positions of transverse bulkheads, and used these to generate the hull form and internal compartments in NAPA. The objectives were to maximize cargo capacity, minimize oil outflow in the event of an accident according to MARPOL, and to minimize the structural steel weight. The problem was set to satisfy a variety of constraints as per MARPOL regulations. Given the numerous conflicting objectives and constraints, the solution space was vast and complex. To solve this, GAs were used for optimization.

4.1.2 Bulk carriers

Bulk carriers, like tankers, do not have a fixed cargo shape, allowing for flexible cargo hold design, which leads to significant demand for arrangement design optimization. Nikolopoulos & Boulougouris (2018) proposed a holistic design method for bulk carrier arrangement optimization that takes into account uncertainty. The study considered three types of uncertainties: environmental, economic, and methodological uncertainties. To control geometric properties, the Lackenby variation (Han et al., 2012) was used for hull design, and the CAESES program (Harries & Abt, 2019) was employed for the parametric design of the cargo hold. Design parameters included the positions of bulkheads, frame spacing, topside tank dimensions, lower stool height, and double bottom height, which were used to model the internal arrangement. Calculations were then performed for calm water resistance, added resistance, and fouling-related resistance for lifetime operations. The selection of the main engine, as well as calculations of lightweight, deadweight, and stability, were also performed. To account for uncertainty, environmental parametric modelling assumed that speed deviation and environmental loads change during ship operation, and fouling margins were also assumed to vary. Economic uncertainty involved factors such as OPEX, CAPEX, RFR, IRR, and EEOI, with assumptions that RFR and EEOI would change over the ship's lifetime. The performance margins and uncertainties were calculated using Clarkson's data (Clarkson et al., 2015) to determine performance deviations over time. Optimization was then conducted with design variables such as the ship's main specifications, bilge, and propeller characteristics. The objectives were to reduce RFR, OPEX, and CAPEX while also minimizing EEOI and ballast water for environmental considerations.

4.1.3 Container ships

In contrast, container ships, with their standardized cargo shape (containers), face relatively more constraints during arrangement optimization. To optimize the arrangement of container ships, Koutroukis et al. (2013) developed the friendship framework (FFW) for ship design. The Lackenby transformation (Han et al., 2012) was embedded into FFW to design the hull form, followed by cargo hold modelling. The study considered two main factors: in-hold storage and deck stowage. Based on the ship's hull form and main dimensions, the system calculated the maximum container capacity. The in-hold arrangement was constrained by the ship's outer structure, and factors such as deck, double bottom ceiling, and stringer were considered. For on-deck cargo, the vertical and transverse stowage arrangement was calculated, taking into account the visibility line as specified by IMO (1998). The engine room location was also optimized to improve cargo hold arrangement, ensuring compliance with IMO stability criteria and maximizing the EEDI (Trivyza et al., 2020) while minimizing the RFR. The optimization process used multi-stage chain optimization to effectively explore the design space and methodology sensitivity, with NSGA-II as the optimization algorithm.

In a more recent study, Priftis et al. (2018) proposed a parametric design method for container ships to meet evolving international regulations. Under the IMO regulations revised in 2012, ships built after 2017 must minimize ballast water carriage and ensure proper treatment for discharge to protect the global ecosystem. This has significantly impacted the arrangement design of container ships that carry large amounts of ballast water. Additionally, with the development of new intact stability criteria by IMO, designs must also meet these new stability standards. As a result, ship design must comprehensively consider all stages, including hull design, compartment design, and outfitting. A holistic approach was used for arrangement optimization, with parametric models employed for both the hull and internal arrangement. The hull was optimized using Lackenby transformation and fine-tuned using optimized B-splines (Lepine et al., 2001). The arrangement of the cargo hold and superstructure was designed using internal modules in CAESES. Parameters such as the number of decks, double bottom height, and double side distance were used to configure the deckhouse and cargo hold arrangement. On-deck cargo stowage was also optimized using a similar approach, ensuring compliance with IMO visibility regulations. The objective functions for the final ship design included minimizing RFR, maximizing capacity ratio, minimizing EEDI, maximizing stowage ratio, and minimizing ship resistance.

4.2 Naval ships

Ship arrangement design fundamentally concerns the efficient placement of various compartments and equipment within a limited space. In merchant ships, the cargo hold is largely empty, making subcompartments and equipment relatively small compared to the overall size of a ship. However, naval ships have entirely different purposes compared to merchant ships, resulting in distinctly different internal arrangement designs. Naval ships carry numerous weapon systems for operational purposes, necessitating additional considerations for the arrangement of these weapons. Furthermore, the number of personnel required to operate and maintain these systems is significantly higher than that on merchant ships, making the arrangement design of crew compartments and facilities a critical aspect.

4.2.1 Surface vessel

As previously explained, enhancing survivability is crucial for warships. Kim et al. (2014) simplified the 2D compartment arrangement problem into a 1D allocation problem using the SLP method (Abotaleb et al., 2016) to improve the survivability of warships. The objective functions for optimizing the compartment arrangement were set to evaluate vulnerability, operational performance, and habitability based on the relative relationships between compartments. To quantify the ship's vulnerability, FMEA (Tay & Lim, 2008) and FTA (Andrews & Tolo, 2023) were performed on the equipment placed in the compartments. The vulnerabilities of the equipment were aggregated to determine the vulnerability of each compartment. The DE method (Rocca et al., 2011) was used for optimization.

Warships, compared to commercial vessels, must operate in rougher seas for missions. Lyu et al. (2015) conducted research to optimize the subdivision positions within the warship to enhance survivability under such conditions. The optimization objective function considered anti-wind capacity, with design variables set as the positions of watertight bulkheads for subdivision arrangement. Anti-wind capacity is defined as the maximum wind speed that ensures damage stability through dynamic analysis in waves under damaged conditions. To calculate this, the maximum wind heeling arm and the maximum roll angle of the damaged ship in waves were determined. The roll motion equation was calculated, considering non-linear square damping, and seawater ingress due to damage was simplified using Bernoulli's equation. To minimize anti-wind capacity across various damage scenarios, the rated wind velocity and weighting factors for each compartment were estimated, based on the probability distribution of the damage, to calculate the average anti-wind capacity. GA was used to optimize the positions of bulkheads for subdivision.

Key requirements for warships include operability, stability, and survivability. Jung et al. (2018) sought to maximize these through warship arrangement optimization. Stability was evaluated based on three intact stability criteria and five damage stability criteria for warships. Operability was calculated using an adjacency index, which considered the distances and proximity between compartments. Finally, survivability was assessed based on bulkhead damage vulnerability and room damage vulnerability. Bulkhead damage vulnerability was calculated by assessing the probability of damage at different longitudinal positions based on various damage scenarios, and determining the damage extent on bulkheads due to explosion pressures. Similarly, room damage vulnerability was calculated by considering the probability of damage in longitudinal, transverse, and vertical directions, as well as the importance of the equipment within each compartment. The ship arrangement optimization was conducted in two stages: the first stage optimized the x and y positions of bulkheads and decks, and the second stage optimized the positions of main compartments and corridors. The NSGA-II was used to find the global optimal solution.

While previous studies focused on the interior of warships, Hwang et al. (2022) proposed an optimization method aimed at improving the ship's detectability, which is influenced by the shape of the hull's exterior and the placement of equipment such as weapons and radar. The study used an AABB model (Majercik et al., 2018) and damaged ellipsoid volume (Djukic et al., 2020) to perform equipment arrangement design. The survivability of the arrangement was evaluated using radar cross-section analysis (Balanis, 2012). GA was used to optimize the equipment arrangement, maximizing survivability, and the LDSD method was developed and applied during the process. The design variables included the susceptibility of each compartment and the coordinates of weapon placements. The objective function was to maximize compartment survivability, evaluated based on susceptibility and vulnerability. Constraints were set to limit the arrangement area and prevent overlapping between compartments and their heights.

4.2.2 Submarine

Additionally, among warships, submarines require particularly complex design due to their underwater operations, with a high density of internal compartments, making arrangement optimization essential. To address this, Kim & Roh (2016) defined a topology to represent the compartment information of submarines and established an arrangement template model for the arrangement design of the pressure hull. They also developed an expert system using a custom inference engine (Kim et al., 2015). The expert system was composed of a space list to represent the knowledge required for individual spaces or equipment, and a relation list to represent the knowledge of the relationships between spaces or equipment. This system was used to formulate a feasibility index for the optimization problem.

The arrangement design problem was structured into three stages. In the first stage, vertical and horizontal partitions were defined, dividing the compartments based on these partitions. In the second stage, internal partitions were created within the divided spaces, further subdividing them into subcompartments, which included various tanks. In the final stage, the main equipment within the subcompartments was arranged in 3D space. Design variables included the position and rotation of each piece of equipment, and the objective functions were based on equipment adjacency, connection cost, and the evaluation score from the expert system. Constraints were imposed to prevent equipment from interfering with each other or exceeding the compartment boundaries.

This method was applied to areas with high equipment density, such as the command and control room, main machinery, and auxiliary machinery, demonstrating its effectiveness. Expert knowledge required for arrangement design was gathered through interviews with designers, related regulations, and reference projects, applying dozens of expert insights at each stage. For optimization, the study used NSGA-II, similar to previous studies.

4.3 Passenger ships

Passenger ships are designed for the purpose of transporting passengers and their accompanying cargo. However, since the number of passengers is much higher than that of merchant ships, the design must comply with passenger evacuation regulations as per SOLAS (Klüpfel et al., 2010). Numerous considerations need to be addressed for passenger comfort and safety. Therefore, studies that have proposed the arrangement design of passenger ships are compared and analysed.

Passenger ships must comply with SOLAS regulations (IMO, 2003) concerning passenger evacuation, making adherence to these rules mandatory. Therefore, Hu & Cai (2020) proposed an approach using a cellular automaton model (Toffoli & Margolus, 1987) combined with GAs to optimize the arrangement for efficient evacuation on passenger ships. In their study, the cellular automaton model was used to simulate pedestrian evacuation for each arrangement, with the goal of minimizing evacuation time through GAs. The design variables included the placement and location of cabins on the deck, the width and location of passageways, and the position of staircases. The objective functions were to minimize evacuation time and maximize evacuation efficiency. During this process, IMO regulations related to evacuation procedures were strictly adhered to, and arrangement designs that violated the structural or functional requirements of the ship were excluded. Additionally, constraints were imposed to ensure that areas such as cabins and staircases met the minimum size requirements for safety and convenience.

Staircases, which connect the decks, are one of the most crucial facilities for passenger evacuation. Wang et al. (2022) conducted a study to optimize the arrangement of staircases on passenger ships to ensure the most efficient evacuation. To simulate passenger evacuation in various designs, they used FDX + Evac (Korhonen et al., 2010). The study used the ‘Yong Xing Dao’ ship, capable of carrying approximately 1000 passengers, and a case study with nine scenarios was performed. Each case varied the number of bow and aft staircases, staircase width, and passengers’ understanding of the ship's structure. The optimization results showed that placing staircases toward the bow reduced evacuation time, and adding aft staircases further decreased the evacuation time.

Given that passenger ships carry many passengers for extended periods, comfort is also a key factor. Wang et al. (2023) proposed an interactive approach to optimize cabin arrangement, focusing on passenger comfort and mobility. This approach is based on integer programming (Karp, 1972) and evaluates how cabin arrangement affects passengers’ daily lives. The design variables included cabin location, size, and type, along with the unique consideration of passageway design and shape. The study used a cabin area function to ensure that cabin sizes met industry standards, a coverage function to ensure an even distribution of cabins across the available space, and a correlation function to place compartments with similar functions in close proximity. An interactive expert system was introduced to propose alternative design solutions to the user at various stages of the optimization process, allowing iterative arrangement modifications. The Gurobi solver (Gleeson & Ryan, 1990) was used for optimization.

An important consideration for passenger ships, similar to other ship types, is stability. Vassalos et al. (2022) proposed a method to optimize the arrangement design with a focus on satisfying damage stability requirements. During the operational phase, the internal arrangement must accommodate passengers and cargo while complying with stringent safety regulations. The study considered a balance between operational arrangement designed for profit optimization (such as open spaces for amenities and entertainment) and critical safety zones like watertight compartments. The design included measures to enhance the ship's damage stability and survivability in the event of hull breaches, helping to control damage and maintain stability. The arrangement was optimized for efficient evacuation and rescue operations, incorporating strategically placed escape routes, lifeboats, and other lifesaving equipment. The design process ensured compliance with international standards such as SOLAS, which stipulate specific requirements for compartmentation and stability. This approach aids in selecting a arrangement design that enhances safety features without compromising the ship's operational capabilities.

5. Conclusions

This study provides a comparative analysis of research on ship arrangement design. Section 1.2 explains the actual arrangement design processes currently being applied. In Sections 2 and 3, the methods proposed for ship arrangement optimization are compared and analysed. Since optimization problems generally consist of design variables, objective functions, and constraints, the study focused on analysing which elements were considered in ship arrangement optimization. Finally, Section 4 discusses how the previously mentioned arrangement optimization problems have been applied across different ship types.

In Section 2.1, the design variables for optimizing ship compartment arrangement were discussed. Most studies set the positions of bulkheads and the order of compartment assignments as design variables and considered these variables in 1D, 2D, and 3D optimization approaches. In particular, studies utilizing a holistic approach aimed to optimize not only the arrangement but also the ship's main dimensions and basic design. Ships also contain various cabins within compartments, so additional design variables for cabin arrangement optimization were mentioned. Similar to compartment design, the positions of partition walls and the order of assignments are considered design variables, but for cabins, additional variables such as the positions and widths of corridors and staircases are added, as these are spaces where people move.

The most crucial aspect of optimization is the objective function, as the ultimate goal of an optimization problem is to find a solution that minimizes or maximizes the objective function. In Section 2.2, the objective functions considered in arrangement design optimization were analysed. First, to improve operability, adjacency relationships between compartments were considered. Second, cost efficiency was emphasized, especially for merchant ships, with various objective functions proposed to minimize OPEX and CAPEX. Third, since the ship's centre of gravity can shift depending on the arrangement, impacting its stability, objective functions to improve stability and survivability were considered, particularly in warships. Lastly, to reflect the reliance of arrangement design on expert knowledge, objective functions incorporating designer preferences were considered, often quantified through expert systems.

Constraints are as important as objective functions in optimization problems. Section 2.3 provides a comparative analysis of the various constraints applied in arrangement design. Beginning with the physical constraints considered in most studies, additional regulatory constraints required for ship design, such as those set by IMO under SOLAS, ICLL, and MARPOL, were also included in many studies.

Section 2.4 discusses the methods used to solve optimization problems and find optimal alternatives. Due to the complexity of arrangement optimization, with many design variables of different types, it is difficult to solve these problems using traditional gradient-based methods. Therefore, most studies use heuristic algorithms, with GA, particularly NSGA-II, being the most commonly employed.

Section 3 expands on the arrangement optimization of various equipment within compartments, similar to the compartment arrangement optimization discussed in Section 2. The design variables, objective functions, and constraints for optimizing equipment arrangement were compared and analysed. Most of the research focused on optimizing equipment arrangement in the engine room, as it is common to all ship types and critical to the ship's functionality.

Section 4 compares how the design variables, objective functions, and constraints discussed in the previous sections were applied across different ship types. Studies were categorized into merchant ships, warships, and passenger ships, and the arrangement and equipment optimization for each type were discussed, highlighting the specific characteristics of each study.

However, most of the studies presented in Section 4 have limitations, as they are difficult to apply to detailed design or complex compartment arrangement, making them suitable mainly for conceptual or initial design phases. More recent studies are focusing more on practical applications. For example, the optimization of stair placement to minimize passenger evacuation time, introduced earlier, represents a specialized study with a practical focus.

In this study, a multi-faceted analysis of arrangement optimization methods for various ship types was conducted. Recent topics in ship design include digital twins (Assani et al., 2022), eco-friendly ships (Hessevik, 2022), and autonomous ships (Kim et al., 2020). Few studies have incorporated arrangement optimization for eco-friendly ship systems. Nubli et al. (2022) optimized the fuel gas supply system arrangement for LNG vessels, focusing on cost and safety requirements, while Souflis-Rigas et al. (2023) explored uncertainties in integrating methanol-fuelled power systems, analysing their impact on engine room length and connection costs. It is expected that more research focusing on these three key areas will emerge, particularly with a focus on integrating eco-friendly equipment into ship arrangement, as seen in the studies mentioned.

Conflicts of interest statement

The authors declare no conflict of interest.

Author Contributions

Ki-Su Kim: Conceptualization, Data curation, Investigation, Visualization, Writing-original draft. Myung-Il Roh: Supervision, Writing-Review & editing, Project administration.

Funding

This work was supported by the 2022 Research Fund of the University of Ulsan, Republic of Korea.

References