1   Introduction

Facility layout planning (FLP) is the process of producing optimum facility layout configurations to achieve efficiency in some operations, such as the production of goods and public services. In naval engineering, FLP can be applied to design shipyard layouts for efficient ship production. In general, the process and flow of ship production are still carried out based on the experience of ship designers. With FLP, the ship production flow can be simulated by a computer through a systematic layout design using various methods and algorithms.

Researchers have adopted FLP to find the optimum facility layout. Liu and Liu (2019) conducted a literature survey of FLP and found that 25 out of 26 papers on FLP aimed to minimize material handling costs (MHCs). Braglia et al. (2003) stated that excellent facility layout design can increase material handling efficiency between facilities and reduce MHC by approximately 10%–30%. Matson et al. (1992) stated that MHC contributes around 30%–40% of total production costs and can increase to 70% in some industry types. Some of the widely used research methods for FLP are heuristic algorithms, such as genetic algorithms (Besbes et al., 2020; 2021; Deep, 2020; Gonçalves and Resende, 2015; Kochhar et al., 1998; Mak et al., 1998; Palomo-Romero et al., 2017; Wang et al., 2005), ant colony algorithm (Hani et al., 2007; Komarudin and Wong, 2010; Kulturel-Konak and Konak, 2011; Liu and Liu, 2019; Wong and Komarudin, 2010; Yu-Hsin Chen, 2013; Zouein and Kattan, 2022), simulated annealing (Allahyari and Azab, 2018; Palubeckis, 2017; Sulaiman et al., 2021; Turgay, 2018), and particle swarm algorithm (Guan et al., 2019; Liu et al., 2018; Muhayat and Utamima, 2024; Samarghandi et al., 2010). To study the application of FLP in shipyard layout, Choi et al. (2017) carried out a two-step optimization consisting of topology design and geometry design. For topology design, a genetic algorithm is used to obtain the optimum layout while minimizing MHCs. The results are then converted into a geometry design using a stochastic growth algorithm. The ultimate goal is to achieve alignment conditions in departments that must be close together. Junior et al. (2023) expounded on this study by adding the ELECTRE method and local search to the stochastic growth algorithm for geometry design. If some departments still do not meet the proximity constraint, then the local search engine is executed. Gunawan et al. (2021) used a genetic algorithm to minimize the MHC of 25 topological departments, several of which have fixed positions. Türk et al. (2022) compared the performance of several types of operators and methods in genetic algorithms to produce the most effective technique in determining topology designs for shipyards.

The fundamental aim of a material handling system is to efficiently transport the appropriate blend of tools and materials (including raw materials, components, and finished products) to the correct location, in the proper state orientation, and precisely on schedule while minimizing the overall cost (Rosenblatt, 2013). According to this explanation, MHC can be defined in various ways depending on priorities: volume of material transported, duration of material delivery, and location between facilities. In the shipyard layout optimization studies conducted by some of researchers (Choi et al., 2017; Gunawan et al., 2021; Junior et al., 2023; Junior and Azzolini, 2019), the MHC calculation only considers the volume of material and distance between departments. The duration of material delivery (weighted distance calculation) is not taken into account. Therefore, the current work addresses this gap by adding material delivery duration as an objective. The two objectives will be "work-based" MHC, which considers the volume of material and distance between departments, and "duration-based" MHC, which considers the duration of materials delivery between departments. Although the actual layout of ship production workshops is a continuous case, research on shipyard layout optimization using a heuristic approach introduces a novel methodology that allows for the discretization of the problem. Taking into account the unique size of each workshop during the optimization, the present work employs a topological approach for shipyards with unequal areas. Previous research using a topological approach assumes that workshops are of equal size throughout the optimization. This work aims to address this gap by employing a topological approach for unequal areas.

2   Problem and methods

2.1   Problem description

The data and conditions of the shipyard in this study are displayed in Tables 1‒4. Table 1 provides information on the size of each department in the shipyard, and Table 2 gives a classification of these departments according to their functionality. Table 3 provides information regarding material flow between departments, and Table 4 provides information on transporters or equipment used to send materials between departments.

In Tables 3 and 4, "t" means tons. In Table 1, the position of each department is defined by position category. If a department has the status "Free", then it can be anywhere in the equal area layout. However, if a department has the status "Fix", then it must be in a particular square in the topological layout. Figure 1 depicts the initial conditions of the topological layout in this study.

Figure  1  Illustration of equal area layout and sequence of algorithm

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2.2   Comparison method

Table 5 compares methods and problems in this work with those of Choi et al. (2017) and Junior et al. (2023). The fixed position department already has a geometrically fixed position. In other words, the total area and shape used for the shipyard have been regulated, along with several departments that must be in certain positions. The remaining empty land will become the area of the optimized departments.

In this study, the total area used for the shipyard is not determined in advance, and the departments that must be in a fixed position are also only within the topological design review. For example, Figure 1 shows that department 17 must be in the top proper position (square 25). Its coordinates in the geometric design depend on the placement results from the proposed method. Hence, its natural position has yet to be determined at the start.

The following table shows the coordinates of the center points of several fixed departments based on Figure 3. Figure 3 further explains that the geometric position of the fixed department can vary depending on the solution produced, in this case, the solution by the algorithm.

Figure  2  Initial layout of the shipyard (example from Choi et al. (2017), Junior et al. (2013))

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Figure  3  Several possible solutions for fixed departments

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2.3   Methodology

The configuration of the shipyard layout is based on the sequence of genes (departments) within the NSGA-11 individual. After the individual algorithms have been fully formed, the sequence is converted into a sequence in the shipyard layout. Given that the shipyard layout is a continuous problem, the solution is limited to the factorial value of the number of existing departments and can go to infinity. A suitable approach is warranted to simplify the process of finding the optimum solution to this continuous problem. This study uses NSGA-Ⅱ as an algorithm to produce department sequences. These sequences are then converted into a shipyard layout using the horizontal layer method, also called a semiflexible bay structure. After the layout has been formed, the fitness values, namely "work-based" MHC and "duration-based" MHC, are calculated. In this study, the shipyard's layout is considered a 2D building, namely, from the top view.

2.3.1   NSGA-Ⅱ

First introduced in 2002 by Kalyanmoy Deb, NSGA-Ⅱ is an optimization algorithm for multiobjective cases. Similar to the genetic algorithm, NSGA-Ⅱ uses the principles of natural selection among living creatures for the survival of the fittest. In essence, NSGA-Ⅱ works differently in each case. The following is the pseudocode of NSGA-Ⅱ, which is also used in this study.

2.3.2   Layout design

The departments in the shipyard layout are grouped into several bays or layers. In the flexible bay structure (FBS), each bay does not need to have the same width and number of departments. The width of each bay is automatically adjusted by the number of departments it contains. In addition, the bays in FBS can be depicted vertically or horizontally. In this study, the semiflexible bay structure method is used, namely, a modification of FBS with each bay having the same number of departments.

In this study, a separation model is used in horizontal bays or horizontal layers. Each bay contains five departments, and the distance between adjacent departments is 1 m. Each department has four sides, and each bay has four sides. Figures 5 and 6 provide an overview of the representation of departments and bays.

Figure  4  FBS model representation

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Figure  5  Representation of the department's form

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Figure  6  Representation of the bay's form

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Parameters and variables:

n = Total number of bays = 25

n_i = Total number of departments in bay i = 5

S_NSGA−Ⅱ=x₁, x₂, x₃, x₄, …, x_n

Kj = The sets of departments in bay j

rx = Closest side distance of 2 departments horizontally in the same bay = 1

ry = Distance between layer = 1

D_{j, z}= Department in bay j sequence z

Wb_i = Width of bay i

Lb_i = Length of bay i

Wd_i = Width of department i

Ld_i = Length of department i

Cx_{j, z}= Centroid x of department in bay j in order to z

Cy_{j, z}= Centroid y of department in bay j in order to z

Models:

If j = 1:

If j > 1:

According to the equations above, for bay 1, the S1 part of the department intersects with the S1 part of the bay. For the other bays, the S3 section of the department overlaps with the S3 section of the bay. This placement method also provides evidence that the width and length of each bay constantly change with each modification in the solution of the algorithm. Ultimately, the departments with fixed positions will have geometric places that are also continually evolving.

2.3.3   Calculation of objectives

Choi et al. (2017), Junior et al. (2023), Gunawan et al. (2021), Junior and Azzolini (2019) performed MHC calculations when the departments were still in equal areas; that is, all departments have the same size. For the present study, the calculation of the two objectives, namely "work-based" MHC and "duration-based" MHC, is carried out when the departments are already in an unequal area and considered to have different sizes (Table 1). The Manhattan distance is used to measure the distance between two departments. For the first objective, namely "work-based" MHC or "weighted distance, " only the volume of transported material and the delivery distance are considered. For the second objective, namely "duration-based" MHC, the following diagram explains the scheme (suppose the transporter will deliver materials from workshop x to r).

Figure  7  Conversion of the algorithm to the unequal area layout

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• Each transporter can only send material according to its capacity on each trip (Table 4).

• Each transporter can run at the same time.

• The trips considered in duration-based MHC are only those on the red line, specifically when carrying material loads.

• The blue line and dashed blue line represent idle capacity, meaning the transporter is not carrying any load. These trips are not included in duration-based MHC.

The following section explains the calculations of the objectives used in the mathematical model.

Figure  8  Schematic of transporter trip

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Parameters and variables:

dm_ab = Manhattan distance between Department a and Department b

f_{a, b} = Material flow from department a to department b (given in Table 2)

N_x = number of transporters = 2

v_i = speed of transporter i (given in Table 3)

C_i= Capacity of transporter i (t)

W_{a, b} = Amount of materials that needs to be transported from dept a to dept b (t)

T_{a, b} = The time required to trip from dept a to dept b

Y_{a, b} = Total time required to trip from dept a to dept b when

P_{a, b} = Number of trips required from dept a to dept b

Models:

Objectives:

k represents the index of the row in the Table 2, ${i_k}~\mathop {and}\limits_{.~.~.} $ j_k denote the workshops 'From' and 'To' in the kth row, respectively.

This model aims to optimize material transportation between departments by minimizing the total distance and travel time. According to Equations (8) and (9), the Manhattan distance is calculated by summing the absolute differences in the x and y coordinates. Equation (10) combines these results to obtain dm_ab. Equation (11) calculates the travel time (T_{a, b}) based on the distance and the transporter's speed. Equation (12) determines the number of trips (P_{a, b}) required based on the transporter's capacity and the amount of material. Equation (13) then calculates the total travel time (Y_{a, b}) by multiplying the travel time by the number of trips. The first objective is to minimize the total distance by multiplying the material flow with the Manhattan distance and summing it for all department pairs. The second objective is to minimize the total travel time by summing the total time required for all trips. With this model, we can achieve operational efficiency in material transportation between departments.

3   Optimization results

Optimization is carried out using the Python 3 programming language. NSGA-Ⅱ has the following characteristics:

1) Performs 500 generations/iterations

2) Has 30 individuals in each population (Pn)

3) The crossover process uses a cyclic operator.

4) In the mutation process, random methods are taken at each iteration, namely, the swap, flip, and slide methods.

For multiobjective optimization, each iteration can have multiple optimal solutions, which are collectively part of the Pareto optimal solutions. However, the solution points from each iteration depicted in Figure 9 are selected based on the highest crowding distances.

Figure  9  Illustration of the distance between two departments

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4   Analysis and discussion

This study fundamentally differs from similar papers by considering the duration of material delivery between departments. The optimal layout of the shipyard should not only involve weighted distance but should also consider the time of content delivery. Figure 9 shows that the algorithm model converges relatively quickly at around 500 iterations. This finding indicates that the developed algorithm model is capable of reaching an optimal solution in a short amount of time. The figure also shows that the work-based MHC and duration-based MHC are approaching their respective minimum values throughout the iteration.

Figure  10  3D scatter of fitness performance

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Figure  11  Optimum configuration

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5   Conclusions

This research presents an optimization of shipyard layout configuration by considering the weighted distance and total duration of material delivery between departments for each production cycle using NSGA-Ⅱ. This study aims to simultaneously minimize weighted distance and the duration of material delivery between departments to maximize production efficiency in the shipyard. Results showed that NSGA-Ⅱ solves this multiobjective problem while providing optimum results.

The following conclusions can be drawn from this research: 1) NSGA-Ⅱ can solve multiobjective problems for production efficiency in shipyard layout configuration; 2) The optimal results obtained after 500 iterations show that the work-based MHC reaches 163 670.0 WBM-units, and the duration-based MHC reaches 34.750 DBM-units; 3) Starting from a random solution, the efficiency of NSGA-Ⅱ is 50.19% for work-based MHC and 48.58% for duration-based MHC.