The Intelligent Layout of the Ship Piping System Based on the Optimization Algorithm

Abstract

The ship piping layout is one of the essential tasks in the detailed design stage of a ship. Traditional manual expert design has disadvantages such as low efficiency, reliance on experience, and subjective influence. Therefore, this paper systematically proposes an intelligent arrangement method for ships’ single, parallel, and branch pipelines. Firstly, the traditional genetic algorithm is improved and combined with the A* algorithm to solve the intelligent arrangement problem of a single pipeline in ships. Then, the parallel pipeline and branch pipeline are split into multiple single pipelines by combining with the connection point strategy to solve the arrangement problem of parallel pipeline and branch pipeline. Finally, the optimized A*-genetic algorithm proposed in this paper is compared with the A* algorithm, particle swarm algorithm, and the labyrinth-genetic algorithm used in previous research through simulation experiments. The results show that the A*-genetic algorithm of this paper is optimal in six indexes, including length, number of elbows, energy value, fitness value, number of optimal solutions, and average number of convergence generations, in the arrangement of the single pipeline. In solving the parallel pipeline and branch pipeline arrangement problems, the all-around performance of this paper’s algorithm is better than that of A*-genetic algorithm and maze–genetic algorithm, respectively. The A*-genetic algorithm of this paper considers the quality of pipeline arrangement and the solution’s efficiency. It verifies the adaptability and superiority of the algorithm for the intelligent arrangement of various types of pipelines in ship pipelines.

1. Introduction

The piping system in a ship is responsible for supplying and transporting many necessary substances, such as oil, water, gas, and electricity [1], which significantly impacts the ship’s functionality, safety, and economy [2,3]. The workload of ship piping layout accounts for more than half of the workload in the detailed design stage of a ship. Completing the piping layout for a medium-sized ship requires 3000–4000 h of work from a 10-person design team [4,5]. The data published by the American Bureau of Shipping show that labor costs account for more than 60% of the total cost of ship design and production. In addition, traditional manual experts in ship piping arrangement have disadvantages such as low efficiency, high dependence on experience, susceptibility to subjectivity, and increased economic cost.

In recent years, optimization algorithms for solving piping layout problems have become a hot research topic for related researchers. At present, the optimization algorithms that have been widely used are maze algorithm [6,7], escape algorithm [8,9], A* algorithm [10], genetic algorithm [11,12], ant colony algorithm [13,14], and particle swarm algorithm [15]. However, the current research is still in the stage of simulation papers, and there are no examples of practical application to real projects yet.

Transforming relevant ship piping arrangement rules and artificial expert design experience into parametric constrained embedding optimization algorithms to form automatic ship piping arrangement systems has become an inevitable trend in solving the ship piping arrangement problem. The research on ship piping arrangement problems can be divided into three categories according to the adopted methods: deterministic algorithm, stochastic algorithm, and hybrid algorithm, which is a fusion of the two.

In terms of the determination algorithm, Lu et al. [16] proposed a method of partitioning and layered sequential arrangement based on an escape algorithm and maze algorithm to realize the automatic arrangement of ship pipelines. Still, the algorithm needs to be more general and consider the cost of pipeline length without considering other factors such as elbow and equipment type. Asmara [17] studied the problem of ship pipeline arrangement using an optimized Dijkstra algorithm, but the effect of this method is more general for branch pipelines with complex structures. Bian et al. [18] improved the traditional A* algorithm by introducing the “block iteration” method and dynamic weights to make it applicable to the ship piping arrangement problem and enhance the solution efficiency. However, the likelihood of finding the optimal piping arrangement could be higher. In addition, Burdorf et al. [19] and Kantat [20] have also used the corresponding deterministic algorithm to study the piping arrangement problem.

In terms of heuristic algorithms, Dong and Lin [21] optimized the coding method in the particle swarm algorithm to fixed-length particle coding and used the variational operation for the solution process of the particle swarm algorithm, which effectively improved the solution efficiency of the algorithm. However, the simulation showed that the quality of the pipeline arrangement could have been better, and there was still much room for improvement. Jiang and Lin [22] proposed a multi-ant colony optimization algorithm that introduced the concept of pheromone expansion. The disadvantage of this algorithm is that the efficiency and quality of the pipeline arrangement decrease to different degrees when the population size increases. Lin et al. [23] studied the ship piping arrangement problem based on the cuckoo search algorithm and proposed a more compatible and grid-independent coding method, accelerating the piping arrangement efficiency. However, the arrangement of parallel piping still needs to be studied. Wang et al. [24] and Niu et al. [25] also studied the piping arrangement problem using heuristic algorithms such as genetic and cuckoo differential evolution.

In terms of hybrid algorithms, Sui et al. [26] fused the Maze Algorithm (MA) and the Non-Dominated Sorting Genetic Algorithm II (NSGA-II) to complete the ship piping arrangement problem, where the Maze Algorithm was mainly responsible for path expansion and backtracking, and the method achieved good piping arrangement results. However, the overall complexity of the algorithm was high, leading to a long solution time. Paulo et al. [27] proposed a pathfinding solution for circuit piping inside the ship cabin based on Lee’s algorithm and genetic algorithm, although the study considered fewer actual piping constraint rules.

In contrast, the advantage of the deterministic algorithm is that the solution efficiency is relatively faster. Still, the piping arrangement result obtained from each search is fixed and unique, and there is no guarantee that the result will be the optimal global solution [28]. The advantage of the stochastic algorithm is that each run can obtain multiple piping design solutions for the designer to choose from, but its solution time is often too long [29]. The hybrid algorithm can combine the advantages of the deterministic and stochastic algorithms, and it works best in ship piping arrangement.

Based on the previous research, this paper optimizes the traditional genetic algorithm. It combines the genetic algorithm with the A* algorithm to propose an intelligent arrangement method of ship pipeline using the A*-genetic algorithm. It systematically solves the intellectual arrangement problems of the single pipeline, parallel pipeline, and branch pipeline of various types of ships. While ensuring the quality of the piping arrangement, the algorithm-solving efficiency is further improved.

The rest of this paper is organized as follows. Section 2 introduces the proposed optimal A*-genetic algorithm in detail and its specific application in solving the intelligent arrangement of the single pipeline of ships. Section 3 offers the parallel channel and branch pipeline arrangement method based on the single pipeline arrangement method; Section 4 describes the simulation experiments conducted to verify the effectiveness of the algorithm in this paper and analyzes the results. Section 5 summarizes the whole article. Finally, Section 5 concludes the paper.

2. Optimization A*-Genetic Algorithm for Intelligent Arrangement of Single Pipeline

This section proposes a single-pipe path intelligent arrangement method for ships using an A*-genetic algorithm. The traditional genetic algorithm is optimized and combined with the A* algorithm to solve the brilliant arrangement of single-pipe paths of ships by adding an interpolation test function in the initialization of the population to avoid generating individuals that interpolate with equipment obstacles, introducing a concentration value in the selection process to judge the probability of individuals entering the next generation from the concentration value and the fitness value. It also introduces a “shearing” strategy in the crossover and mutation process to shear the generated duplicate paths. Only the crossover or variation individuals with better quality than the corresponding parent individuals are retained. The A* algorithm in the deterministic algorithm is introduced to search for the subpaths in the crossover and variation links to accelerate the overall solving efficiency of the algorithm. The above approach enriches the population diversity, improves the quality of individuals, and solves the problems wherein the traditional genetic algorithm tends to fall into local optimal solutions and low solution efficiency when completing the ship pipeline arrangement.

2.1. Code

In the internal space of a ship, the optimization algorithm cannot solve the problem directly. Still, it must first encode the problem to be solved into genetic string data that the optimization algorithm can recognize. Encoding is the first step of a genetic algorithm to solve the actual situation, and adopting a suitable way to encode for pipeline intelligence arrangement is essential. The common encoding methods are floating-point encoding, multi-parameter cross-coding, binary encoding, Gray code, and multi-parameter cascade encoding. In this paper, three-dimensional decimal coordinates are used for variable length coding. Firstly, the arrangement space of the ship pipeline is enveloped by the axis-parallel envelope box method. Then, the whole arrangement space is equally divided into a certain number of grids. A grid in the arrangement space is selected as the origin to establish a 3D spatial coordinate system, and each grid has its coordinate value to characterize its spatial position. The specific grid size can be selected according to the relative size between the layout space and the pipe diameter. The code of each grid can be obtained from its coordinates without additional work, improving the solution’s efficiency.

Connecting two grids, all the points between two genes represent a chromosome, an individual in a population. For example, {(5,3,1) (4,3,1) (3,3,1) (2,3,1) (2,2,1) (2,1,1) (2,1,0)} represents a chromosome connecting (5,3,1) and (2,1,0), i.e., a pathway individual. Since different pipelines correspond to different starting and target grids, their generated pipeline individuals connecting the two are of different lengths, so the variable-length encoding method is used.

2.2. Population Initialization

Ship pipelines should not interfere with internal equipment and instruments, so it is necessary to establish a relatively accurate shape model for these equipment obstacles. In this paper, we propose a “segmentation-modelling-combination method”, which splits the equipment obstacles into multiple independent subobstacles according to their shape characteristics, simplifies the modeling of each subobstacle based on the axis-parallel enveloping box method, and finally combines the subobstacles together, which takes into account the efficiency and accuracy of the equipment obstacle modeling. The specific process is shown in Figure 1. In addition, the maintenance space can also be enveloped in a similar way for obstacle processing.

The role of population initialization is to randomly generate several feasible solutions within a given solution space, which together form the initial population. In past studies of using optimization algorithms to solve the intelligent arrangement of ship pipelines, Wang et al. [30], Fan et al. [31], and others artificially reduced the probability that the paths intersecting with equipment obstacles were selected to enter the next generation by adding a subtraction term to the adaptation value calculation formula to penalize the path individuals crossing the equipment obstacles. While this approach reduces the probability that the pipeline paths derived from the search will intersect with equipment obstacles, it does not provide a 100 percent guarantee that interpenetration between equipment obstacles will occur.

This paper sets up an interpolation test function to check the position relationship between the searched grid nodes and equipment obstacles, i.e., whether interpolation occurs. In the process of generating paths in the initialization phase of the population, the interpolation test function is called once for each path grid searched to determine whether interpolation occurs for equipment obstacles. If so, the grid is discarded. In the above way, the quality of pipeline individuals obtained in the population initialization process can be guaranteed.

2.3. Objective Function and Fitness Function

When arranging a ship’s pipeline, it is necessary to meet various pipeline design specifications, relevant conventions, and experts’ experiences. After summarizing, the rules and constraints to be followed when arranging the pipeline are as follows:

(1)

Pipelines should be connected to specific pipeline interfaces to ensure the connectivity and legality of pipelines.

(2)

If there is no special requirement, the pipeline should be arranged orthogonally.

(3)

It is necessary to ensure that there is no physical interpolation between pipelines and equipment obstacles, as well as between pipelines and pipelines.

(4)

The length of the pipeline should be as short as possible to reduce the cost of the pipe.

(5)

The number of elbows in the pipeline should be as few as possible to reduce the cost of elbows.

(6)

In principle, the pipeline should be installed along the bulkhead, bilge, and supportable equipment surface as far as possible to reduce the installation cost. The pipeline should be far away from dangerous equipment to reduce the impact of equipment vibration on the pipeline and the probability of risk. For example, steam, oil, and water pipes should not be installed above and behind electrical equipment, generators, and essential instruments.

Rules (1)~(5) are strict rules, that must be satisfied in the process of pipeline arrangement, and rules (6) are weak rules that should be satisfied in the pipeline arrangement as much as possible.

Among them, rule (1) can be realized by searching the route from the starting grid to the target grid through the genetic algorithm logic itself. Rule (2) is realized by setting the search direction, which is specified in this paper that each grid can only be searched in six directions, i.e., a list is set in the program ([1, 0, 0], [0, 1, 0], [0, 0, 1], [−1, 0, 0], [0, −1. 0], [0, 0, −1]), where each element of the list corresponds to a search direction. For example, the search direction represented by [1, 0, 0] is the positive X-axis direction, and the grid (5,5,5) reaches the grid (6,5,5) after searching once in the positive X-axis direction. Rule (3) is implemented by continuously calling the interpolation detection function described in the previous section when searching for the pipeline route. Rules (4) to (6) are implemented by setting the length cost, elbow number cost, energy value cost, etc., and assigning appropriate weight coefficients to these three objectives. This simplifies the complex multi-objective optimization problem into a single-objective optimization problem, thus reducing the complexity of the problem and speeding up the solution rate.

Each mesh in the layout space is assigned an energy property, and the energy value of each mesh is set according to the rules. The energy cost of an individual pipe is obtained by adding up the energy values of all the grids through which it passes. The principle of assigning energy value to each grid is as follows: if the rules require the pipeline to be installed along the bulkhead, bilge, or supportable obstacle surface as much as possible, the network corresponding to the bulkhead, bilge, or supportable obstacle surface is assigned a lower energy value. The grid corresponding to the area where the pipeline is expected not to pass as much as possible, such as the generator, is assigned a higher energy value. Meanwhile, the grid corresponding to other common passable areas with no special requirements is assigned a medium energy value. In a word, the lower the energy value, the higher the priority of the grid in the piping arrangement.

In genetic algorithms, the magnitude of the fitness value characterizes the level of difference between the individuals in the population and the best individuals. The fitness value often determines the genetic algorithm solution’s convergence speed and quality level. For the problem of intelligent arrangement of ship pipelines, it is essential to design a suitable fitness value-solving function. In this paper, the proposed adaptation value F_i is calculated as follows:

$$F_i=T-\alpha ·L_i+\beta ·B_i+\gamma ·E_i$$

(1)

where i is the current individual number. T is a suitably large constant determined by the maximum objective function value and simulation experiments. L_i is the total length of the pipeline represented by i. B_i is the total number of elbows of the pipeline represented by i. E_i is the total energy value of the pipeline represented by i. α, β, and γ are all constants, indicating the weight coefficients corresponding to the pipeline length, number of elbows, and energy value, respectively.

2.4. Select Operation

The selection operation mimics the elimination of winners and losers in Darwinian evolution by selecting some individuals of higher quality from the population and using them in the next generation. The commonly used selection methods are tournament selection, roulette wheel method, expected value method, best individual retention method, ranking selection method, competition method, and linear standardization method. Among them, the roulette method is the most widely used. In the traditional roulette method, the probability of an individual being selected is determined by the ratio of its fitness value to the sum of the fitness values of all individuals in the population. The disadvantage of this method is that most of the individuals in the final population are highly similar, and the diversity of the population is low. The intelligent arrangement of the ship pipeline will lead to the pipeline paths obtained from the search being concentrated in a specific region, and the optimal global solution cannot be obtained.

Concentration value is introduced in this paper to solve the drawbacks mentioned above in the traditional roulette selection method, and the probability of individuals being selected is no longer determined by the fitness value only. Still, it is obtained by the combined judgment of concentration value and fitness value to enhance the population diversity and obtain the optimal global solution of pipeline paths while ensuring the quality of individuals in the population.

The concentration value C_i indicates the proportion of individuals with similar fitness values to the individual i. The concentration value C_i of individual i is calculated by the following formula:

$$C_i=\frac{1}{n}{\displaystyle \sum _{j=1}^n\left|F_j-F_i\right|} (\left|F_j-F_i\right|\le \epsilon )$$

(2)

where ε is an actual number indicating the threshold of similarity between individuals within the population, obtained from multiple simulations; d indicates the population size in the genetic algorithm; F_i indicates the fitness value of individual i; and F_j (j = 1, 2, …, d) denotes the fitness value of individual j.

The selection probability operator P_si for individual i is calculated as

$$P_{si}=\frac{F_i}{{\displaystyle \sum _{j=1}^d{F}_j}}-\lambda ·C_k$$

(3)

where λ denotes the influence coefficient of the concentration value.

2.5. Crossover Operations

Crossover is the process of regrouping selected individuals in a population by exchanging some gene fragments to produce new population individuals, and it determines the global merit-seeking ability of genetic algorithms. The standard crossover methods are single-point crossover, two-point crossover, multi-point crossover, partial-mapped crossover, uniform crossover, and order crossover, position-based crossover, order-based crossover, cycle crossover, subpath exchange crossover, subtour exchange crossover, etc.

Random point crossover divides the selected individuals of the population into groups of two individuals. The crossover operation is carried out within each group. Two genes are selected as crossover points in two individuals to generate a subpathway. The part of the parent individual P₁ before the crossover point, the part of the subpathway, and the part of the parent individual P₂ after the crossover point form the offspring individual S₁. Similarly, the part of the parent individual P₁ after the crossover point, the part of the subpathway, and the part of the parent individual P₂ before the crossover point forms the offspring individual S₂. The specific process is shown in Figure 1 The process is shown in Figure 2, where the gene locus (6,4,5) of the parent individual P₁ and (5,2,4) of the parent individual P₂ are randomly selected as crossover points to generate the subpathway C₁ between them and then recombined to generate the offspring individuals S₁ and S₂.

The disadvantages of the above crossover method are mainly three-fold: firstly, there may be genetic duplication in a section of the child individuals generated by random point crossover, and the corresponding consequence in the actual pipeline arrangement is that the pipeline will have invalid loops in a section, which increases the pipeline cost. Secondly, the generation of child paths by the random search for path nodes cannot guarantee the quality of child paths, which leads to the subsequent need to iterate more. The third disadvantage is that the quality of the generated subpaths is poor or excellent but inferior to their corresponding parents, which causes population degradation to a certain extent.

In order to solve the above problems, the following optimization methods are proposed for the crossover operation: First, the crossover process in this paper is based on random point crossover, and a “cutting” link is introduced to detect the repeatability of the new individuals generated by random point crossover. If a gene is found to be repeated in the same individual, the gene fragment between these two same gene points is cut and discarded. Suppose a gene is found to be duplicated in the same individual. In that case, all the gene fragments between the two duplicated gene sites are cut and discarded, thus improving the quality of individuals while ensuring the diversity of the population, as shown in Figure 3. The shearing operation removes the gene fragments between the two gene loci (5,4,4) to regenerate the offspring individual S₃. Second, in this paper, the subpaths are generated by the optimized A* algorithm in the literature [32] to ensure that the newly generated subpaths meet the actual requirements of the pipeline arrangement, thus improving the quality of the newly generated offspring individuals and speeding up the convergence rate. Third, after each crossover operation, the relationship between the fitness values of the newly generated children and their corresponding parents is compared once. Only when the fitness value of the offspring is greater than that of the parent can the offspring replace the corresponding parent in the new population. Otherwise, the newly generated offspring is discarded, and the original parent is retained.

2.6. Mutation Operation

The mutation operation mimics the mutation process in biological evolution to increase the diversity of the population, which determines the algorithm’s feasible solution local search capability. The standard mutation methods include Simple Mutation, Uniform Mutation, Boundary Mutation, Non-Uniform Mutation, and Gaussian Approximation Mutation.

Similar to the crossover method described above, this paper uses random point mutation combined with a “cut” strategy, i.e., two genes are selected as mutation points in an individual, a subpath is generated to replace the corresponding original path, and if a gene is found to be repeated in an individual, all the gene fragments between these two repeated gene points are cut and discarded. The process is shown in Figure 4, where the gene loci (6,1,2) and (7,4,2) of the parent individual P₃ are randomly selected as variant points, and the subpath C1 between them is generated to replace the original pathway to generate the offspring individual S₄. Again, the subpathway generation process is based on the optimized A* algorithm in the literature [32]. After each mutation operation, the relationship between the size of the offspring individual and the corresponding parent individual fitness value should be compared. If the fitness value of the offspring is greater than that of the parent, the offspring can replace the parent in the new population. Otherwise, the new offspring is discarded, and the original parent is retained.

2.7. The Overall Process of Intelligent Arrangement of a Single Pipeline

The specific process of the optimized A*-genetic algorithm proposed in this paper for solving the intelligent arrangement of the single pipeline of a ship is shown in Figure 5. The initial parameters to be set are threshold value of similarity between individuals within the population ε, influence coefficient of concentration value λ, population size d, crossover probability P_c, variation probability P_m, selection probability P_s, maximum number of iterations g_max, pipeline length weighting coefficient α, number of elbows weighting coefficient β, energy value weighting coefficient γ, and constant T.

3. Intelligent Arrangement Methods for Parallel Lines and Branch Pipelines

In addition to a single pipeline, the ship piping system arrangement problem contains many parallel and branch pipelines. Based on the proposed optimal A*-genetic algorithm, the parallel and branch pipelines are split into multiple single pipelines by combining the connection point strategy. Then, the parallel and branch pipeline arrangement problem is solved.

3.1. Parallel Piping

The parallel pipeline comprises multiple single pipelines arranged at appropriate distances, characterized by the high degree of proximity of every single pipeline’s starting and ending positions in a parallel pipeline and the high degree of similarity of pipeline functions. Based on the optimized A*-genetic algorithm for the intelligent arrangement of single pipelines above, an “auxiliary connection point” is introduced to solve the ships’ parallel pipeline arrangement problem.

Firstly, it is necessary to determine the arrangement order of each single pipeline in the parallel pipeline. Combined with past research and interviews with the ship piping designers, the influencing factors of piping arrangement order are as follows: (1) the piping with a larger diameter has higher arrangement priority; (2) among multiple parallel pipings, the piping nearer to the middle position has higher arrangement priority; (3) the piping in the same compartment has higher priority than the piping across compartments; (4) the piping with a higher risk of transporting work material, for example, the arrangement priority of an oil pipe is higher than that of water pipe. Each pipeline is scored according to the ranking of these four indexes, and its total score is obtained by weighting together to determine the arrangement order of each pipeline. If the scores of all or some of the pipelines are the same, the arrangement order will be decided by random selection.

The pipeline with the highest layout priority is selected and designed according to the single pipeline layout method proposed in Section 2. After that, according to the position relationship between other single pipeline and this arranged pipeline, a certain number of auxiliary connection points are randomly generated at a certain distance from the arranged pipeline in batches, and then the optimized A*-genetic algorithm is used to connect the auxiliary connection points inside each pipeline to be arranged in turn. It should be noted that the interpolation test function should be invoked to determine the auxiliary connection points, and each pipeline will be added to the obstacle list as a virtual obstacle to prevent physical interference between pipelines.

The optimal A*-genetic algorithm based on the introduction of auxiliary connection points is used to solve the intelligent arrangement of parallel pipelines in the following order of operation:

Step 1: Set the initial parameters of the algorithm.

Step 2: Input the starting node information [S₁, S₂, S₃,…, S_n] and the target node information [T₁, T₂, T₃, …, T_n] of each single pipeline [C₁, C₂, C₃, …, C_n] in a set of parallel pipelines.

Step 3: Determine the arrangement order of each single pipeline in the parallel pipeline [Q_c1, Q_c2, Q_c3, …, Q_cn]. The more advanced the single pipeline, the higher the arrangement priority. For example, in [Q_c1, Q_c2, Q_c3, …, Q_cn], the single pipe with the highest priority is C1.

Step 4: Select line C₁ with the highest arrangement priority from the arrangement order list [Q_c1, Q_c2, Q_c3, …, Q_cn] and arrange it based on the optimized A*-genetic algorithm.

Step 5: Add the path nodes of pipeline C₁ to the obstacle list.

Step 6: Select the pipeline C_k with the highest priority among the remaining unarranged pipelines from the pipeline arrangement order list. Randomly generate a certain number of auxiliary connection points at equidistant positions of pipeline C₁ according to the position relationship between this pipeline C_k to be arranged and the arranged pipeline C₁.

Step 7: Use the optimized A*-genetic algorithm to connect the connection points inside pipeline C_k in order.

Step 8: Add the path nodes of pipeline C_k to the obstacle list.

Step 9: Check whether there is a single pipeline in the group of parallel pipelines that have yet to be arranged. If yes, go to step 6. If not, go to step 10.

Step 10: Output the intelligent arrangement result of the group of parallel pipelines.

The flow of the A*-genetic algorithm based on the introduction of auxiliary connection points to solve the intelligent arrangement of parallel pipelines is shown in Figure 6.

3.2. Branch Piping

Branch piping can be considered a one-to-many problem, formed by combining several single pipelines with a common starting point and different ending points. More than 70% of the piping types in ship piping systems are branch piping [2]. In previous studies, such as Jiang et al. [33], Wu et al. [34], and Dong et al. [35], every single pipeline forming the branch pipeline is regarded as an independent population, and the various populations are treated as a cooperative evolution to solve the branch pipeline arrangement problem. However, this solution method is generally less efficient and prone to combinatorial explosion, which cannot guarantee the quality of the piping arrangement.

The branch connection points of a Branch Pipeline are often referred to as divergence points or branch points. These branch points are located in the piping network and are used to connect the main line to branch piping in order to direct fluids in different directions. These branch points may be located at pipe crossings, supports, or other locations where diversion is required to meet the functional and performance needs of the system. The branch pipeline can be divided into two categories according to whether the diameter is the same, and its structure is shown in Figure 7.

In this paper, when solving the intelligent arrangement of branch pipelines, the optimal A*-genetic algorithm with the “branch connection point” strategy is introduced. Firstly, the branch pipeline is decomposed into several single pipelines, which are divided into different levels. The first level is the single core pipeline. The second level is the second single pipeline, which is often directly connected with the single core pipeline through a certain “branch connection point”. Then, the other levels of single pipelines, in turn, are not directly connected with the single core pipeline but are often connected with their corresponding superior single pipeline. The A*-genetic algorithm is used to arrange the core single pipe. Then, the branch connection points between the core single pipe and the secondary single pipe are searched for, and the pipeline paths between each branch connection point and its corresponding secondary single-pipe target node are arranged in turn based on the optimal A*-genetic algorithm. Similarly, other single-pipe levels are arranged in turn.

The specific steps of the optimal A*-genetic algorithm based on the introduction of the “branch connection point” strategy to solve the intelligent arrangement of Branch Pipelines are as follows:

Step 1: Set the initial parameters of the algorithm.

Step 2: Input the coordinate information of a starting point and multiple endpoints of a group of branch pipelines.

Step 3: Decompose a set of branch pipelines into multiple single pipelines with one branch connection point (new starting point) corresponding to one endpoint according to the internal connection relationship of branch pipelines.

Step 4: Split the branch pipeline into different levels of a single pipeline. The first level is the single core pipeline. The second level is the second single secondary pipeline, which is often directly connected with the single core pipeline through a certain “branch connection point”. Then, the other levels of a single pipeline, in turn, are not directly connected with the single core pipeline but are often connected with its corresponding.

Step 5: Determine the order of the single pipes within each level.

Step 6: Complete the arrangement of the core single pipe using the optimized A*-genetic algorithm.

Step 7: Find the branch connection points between the core single pipe and the secondary single pipe based on the specified connection strategies, such as the shortest path strategy, the least number of bends strategy, or the lowest energy value strategy.

Step 8: Search the pipeline path between each branch connection point and its corresponding secondary single-pipe target node in turn based on the optimized A*-genetic algorithm.

Step 9: Complete the arrangement of other single pipelines at all levels using the above method.

Step 10: Output the intelligent arrangement result of the branch pipeline.

It should be noted that when solving the intelligent arrangement of branch pipelines, the principles for determining the arrangement order of every single pipeline obtained from the decomposition of branch pipelines are as follows: (1) the higher the level of a single pipeline, the higher the arrangement priority. For example, the single core pipeline has a higher arrangement priority than the single secondary pipeline. (2) Within the same grade pipeline, the pipeline with a larger diameter has a higher arrangement priority, and the branch pipeline in the same compartment has a higher priority than the branch pipeline arranged across compartments. The pipeline with more branch connection points has a higher arrangement priority. (3) If all pipes or some pipes have the same priority, the arrangement order is decided by random selection. In addition, the branch connection points between different pipelines are searched by calling the A* algorithm several times, and the strategies include and are not limited to the shortest path strategy, the strategy with the least number of bends, and the strategy with the lowest energy value. The A* algorithm must search several times to find suitable branch connection points.

4. Simulation Experiments and Analysis

To test the effectiveness of the intelligent arrangement method of the pipe system proposed in this paper, the proposed optimized A*-genetic algorithm was compiled with Pycharm2021.3 compiler and Python 3.7 64-bit in the 64-bit operating system of Windows 10. The simulation was verified for a single pipeline, parallel pipeline, and branch pipeline cases, and the results were discussed and analyzed.

4.1. Spatial Model

A simulation space model of ship piping arrangement containing obstacles is established. Obstacles refer to equipment inside the ship that cannot be interposed with the piping, and each equipment obstacle is simplified to one or more rectangular combinations using the axis-enveloping box method. The spatial model is set as a cube with a side length of 100, and it is uniformly divided into 1 million grids based on the raster method. The specific obstacle coordinate information is shown in

Table 1. Obstacle information coordinate table.

Obstacle	X-Range	Y-Range	Z-Range
1	(50, 60)	(60, 75)	(0, 60)
2	(60, 80)	(85, 100)	(0, 50)
3	(20, 40)	(65, 73)	(0, 90)
4	(80, 100)	(30, 45)	(0, 100)
5	(80, 100)	(55, 70)	(0, 30)
6	(5, 40)	(0, 30)	(0, 50)
7	(20, 40)	(45, 53)	(0, 90)
8	(20, 40)	(65, 73)	(0, 90)
9	(20, 40)	(90, 100)	(80, 100)

. Two single pipelines, a set of parallel pipelines, and a set of branch pipelines are set up, and the specific starting and ending coordinates of the pipelines are shown in

Table 2. Pipeline starting and ending information in the simulation case.

Case Number	Piping Type	Numbering of Pipework	Starting Point Coordinates	Endpoint Coordinates	Transported Medium	Pipe Diameter/cm
Case 1	Single	P1	(100, 100, 0)	(0, 0, 100)	Water	Φ4.8 × 0.4
Case 2	Single	P2	(0, 100, 100)	(100, 20, 0)	Gas	Φ4.8 × 0.4
Case 3	Parallel	P3	(100, 100, 0)	(0, 0, 100)	Water	Φ2.2 × 0.2
		P4	(100, 98, 0)	(2, 0, 100)	Water	Φ2.2 × 0.2
		P5	(100, 96, 0)	(4, 0, 100)	Water	Φ2.2 × 0.2
Case 4	Branch	P6	(100, 20, 0)	(0, 100, 100)	Oil	Φ3.4 × 0.25
		P7		(70, 15, 0)	Oil	Φ2.2 × 0.2
		P8		(35, 45, 0)	Oil	Φ2.2 × 0.2
		P9		(30, 30, 0)	Oil	Φ2.2 × 0.2

.

The energy value of the piping layout space is set as follows: the energy value of the grid adjacent to attachable objects such as bulkheads, ground, and equipment obstacles is 0. The energy value of the grid increases by 5 for each increase in the grid length from the attachable objects.

According to the set space size and equipment obstacle location information, the final space diagram of the pipeline arrangement is shown in Figure 8.

4.2. Single Pipeline Simulation

4.2.1. Single Pipeline Parameter Setting

The selection of control parameters in the algorithm is very critical, and the different selection of parameters will have a very significant impact on the performance of the algorithm. The ideal range of algorithm parameters should be able to ensure that the algorithm can converge to a better solution in a reasonable time while maintaining the diversity of the population to avoid premature convergence and falling into local optimal solutions. Determining the ideal range of algorithm parameters usually requires experimentation and tuning. Based on previous studies and hundreds of simulation experiments simulation experiments, the optimal range of the algorithm parameters in this paper is determined. Among them, the ideal size of the population size d is 20–150. The value of the maximum iteration number g_max should be taken to ensure that it is not less than the average value of the iteration number at convergence. The ideal size of the crossover probability P_c is 0.6–1.0. The ideal size of the variation probability P_m is 0.005–0.15. The ideal value of the selection probability P_s is 1.0 to ensure that the population size does not change during the iterative evolutionary process. The ideal range of the similarity threshold ε is 2–4. The ideal range of the concentration value influence coefficient λ is 0.001–0.2.

To verify the optimized A*-genetic algorithm proposed in this paper in the actual pipeline intelligent arrangement, the optimized A*-genetic algorithm, the A* algorithm of Bian [18], the particle swarm algorithm of Dong [21], and the maze–genetic algorithm of Sui [26] are used to search for the optimal solution of the pipeline in the set pipeline arrangement space. The specific basic parameter settings of each algorithm are shown in

Table 3. Solving several algorithm parameter values of a single pipeline.

Parameters	PSO	MA-GA	A*-Bian	A*-GA
d	50	50	/	50
gmax	500	500	/	500
Pc	0.85	0.85	/	0.85
Pm	0.05	0.05	/	0.05
Ps	1.0	1.0	/	1.0
α	0.2	0.2	0.2	0.2
β	0.4	0.4	0.4	0.4
γ	0.4	0.4	0.4	0.4
T	400	400	400	400
ε	/	/	/	2.5
λ	/	/	/	0.012

.

4.2.2. Single-Pipe Simulation Results and Analysis

The optimal pipeline path information in the two cases obtained based on the optimized A*-genetic algorithm proposed in this paper is shown in

Table 4. The optimal path information of a single pipeline based on the optimal A* genetic algorithm.

Parameters	Pipe 1	Pipe 2
Optimal path length	300.0	280.0
The optimum number of path bends	3.0	4.0
Optimum path energy values	0	0
Optimal path adaptation value	338.80	342.40
Optimal path-solving time/s	44.7	30.9
Optimal path convergence algebra	42	36

.

The optimal path layout effects of Case 1 and Case 2 based on the optimized A*-genetic algorithm are shown in Figure 9 and Figure 10, respectively. It can be found the algorithm of this paper can find the shortest length, the least number of elbows, and the lowest-energy-value (attached to the low-energy-value areas, such as walls) pipeline arrangement route connecting the starting and ending points, which proves the feasibility of the algorithm of this paper.

The optimized A*-genetic algorithm proposed in this paper, the A* algorithm of Bian [18], the particle swarm algorithm of Dong [21], and the maze–genetic algorithm of Haiteng Sui [26] are each repeatedly run 15 times in the set pipeline arrangement space for the intelligent arrangement of pipelines. The specific results are shown in

Table 5. The single pipeline layout information obtained based on each algorithm.

Case Number	Algorithm Name	Optimal Number of Solutions	Average Solution Time/s	Average Convergence Algebra	Average Length	Average Number of Bends	Average Energy Value	Average Fitness Value
Case 1	PSO	3	5.7	298.5	321.5	7.8	20	324.58
	MA-GA	9	110.9	71.6	304.8	3.7	0	337.56
	A*	7	2.1	/	302.7	4.3	0	337.74
	A*-GA	12	51.4	49.2	301.9	3.5	0	338.22
Case 2	PSO	2	4.6	291.7	296.1	9.1	25	327.14
	MA-GA	8	82.5	64.6	283.5	4.9	0	341.34
	A*	7	2.4	/	282.6	5.1	0	341.44
	A*-GA	11	34.6	41.5	280.7	4.6	0	342.02

Note: Bolded data in the table are optimal values.

. The following conclusions can be obtained from the comparative analysis of the pipeline arrangement information obtained in Table 5:

(1)

The quality of the intelligent arrangement of the ship pipeline: firstly, the average length of the pipeline path obtained by the A*-genetic algorithm is the shortest, the average number of elbows is the smallest, the average energy value is the lowest, and the average adaptability value of path is the highest, i.e., it best meets the actual requirements of ship pipeline arrangement. Secondly, the number of times that A*-genetic algorithm searches for the optimal solution ranks first among the four algorithms, i.e., it has the highest probability of searching for the optimal solution of pipeline arrangement. The probability of obtaining the optimal solution of the piping arrangement is the highest. Thus, the optimized A* -genetic algorithm performs better than the other three algorithms in the quality of intelligent arrangement of ship pipelines.

(2)

Ship pipeline intelligent arrangement efficiency: the average solution time of the four algorithms is MA-GA, A*-genetic, PSO, and A* in the order of longest to shortest. MA-GA takes longer to grid-generate individuals and then complete the population initialization through the more complex maze algorithm, so its average solution time is the longest. Compared with it, the optimized A*-GA proposed in this paper completes the crossover and mutation process through the A* algorithm with higher search efficiency, which further improves the search efficiency of the traditional genetic algorithm.

(3)

Convergence of ship pipeline intelligence arrangement: the average number of generations of convergence of MA-GA, A*-Genetic algorithm, and PSO increases sequentially, and the A*-Genetic algorithm performs better than PSO and MA-GA in the average number of generations of convergence.

In summary, compared with the other three algorithms, the proposed algorithm obtains the optimal value in six indexes, such as length of the pipeline, number of elbows, energy value, fitness value, number of optimal solutions, and average number of convergence generations, and the performance in the average solution time is also better than that of the labyrinth-genetic algorithm, which is also a hybrid algorithm. Although the optimized A*-genetic algorithm performs better than PSO and A* in terms of the average solution time, the PSO and A* algorithms still differ from this paper’s algorithm regarding the quality of the pipeline arrangement by a large margin. In addition, A* is a deterministic algorithm, which can only provide a single arrangement result and cannot specifically provide multiple solutions for preference. Compared with MA-GA, PSO, and A*, the optimized A*-genetic algorithm proposed in this paper can effectively improve the piping arrangement efficiency while ensuring the piping arrangement quality, which verifies the adaptability and superiority of the algorithm for the intelligent arrangement of single piping of ships.

4.3. Parallel Pipeline Simulation Test

4.3.1. Parallel Pipeline Parameter Setting

Simulation experiments are performed for Case 3 using the optimized A*-genetic algorithm with the introduction of auxiliary connection points and the A* algorithm of Bian [18], and the specific parameter settings are shown in

Table 6. The parameter values of several algorithms for solving parallel pipelines.

Parameters	A*	A*-GA
d	/	60
gmax	/	300
Pc	/	0.90
Pm	/	0.10
Ps	/	1.0
α	0.1	0.1
β	0.5	0.5
γ	0.4	0.4
T	400	400
ε	/	2.4
λ	/	0.03
Am	/	4

. It should be noted that the algorithm parameters that need to be set in the parallel pipeline have more auxiliary connection points than the single pipeline. The order of arrangement is P4 ⟶ P3 ⟶ P5 based on the piping information.

4.3.2. Parallel Pipeline Simulation Results and Analysis

Simulation experiments were conducted for Case 3 using the optimized A*-genetic algorithm with the introduction of auxiliary connection points and the A* algorithm of Bian [18]. The parallel piping arrangement information obtained is shown in

Table 7. The parallel pipeline layout information of Case 3 is obtained based on each algorithm.

Algorithm Name	Optimal Number of Solutions	Average Solution Time/s	Average Length	Average Number of Bends	Average Energy Value	Average Fitness Value
A*	8	41.52	899.5	12.5	5	301.80
A*-GA	11	1298.14	892.6	10.1	0	305.69

Note: Bolded data in the table are optimal values.

, and the best arrangement of parallel piping obtained for Case 3 is shown in Figure 11.

A comparative analysis of the parallel piping arrangement information obtained in Table 7 leads to the following conclusions.

(1)

Compared with the A* algorithm proposed by Bian, the optimized A*-genetic algorithm proposed in this paper obtains a shorter average path length, fewer average elbows, lower average energy value, and higher average path adaptation value in solving the parallel piping arrangement problem, which shows that the algorithm in this paper can obtain a higher quality of piping arrangement in solving the parallel piping arrangement problem.

(2)

Compared with the A* algorithm proposed by Bian, the optimal A*-genetic algorithm proposed in this paper has more time to obtain the optimal solution when solving the parallel piping arrangement problem, which shows that the algorithm in this paper has a higher probability of obtaining the optimal solution.

(3)

In terms of parallel piping arrangement efficiency, the A* algorithm proposed by Bian performs better, and the main reason for this is that the A* algorithm is a deterministic first search. However, the A*-GA algorithm proposed in this paper can be iteratively optimized continuously according to the evaluation function to obtain a variety of arrangement schemes, which is more selective.

4.4. Branch Pipeline Simulation Test

4.4.1. Branch Pipeline Parameter Setting

The optimal A*-genetic algorithm with the introduction of the “branch connection point” strategy and the maze–genetic algorithm of Sui [26] is used for the simulation of Case 4, and the specific parameters are shown in

Table 8. The parameter values of several algorithms for solving branch pipelines.

Parameters	MA-GA	A*-GA
d	60	60
gmax	300	300
Pc	0.92	0.92
Pm	0.08	0.08
Ps	1.0	1.0
α	0.1	0.1
β	0.5	0.5
γ	0.4	0.4
T	400	400
ε	/	2.1
λ	/	0.05
Ab	/	3

. It should be noted that the algorithm parameters to be set in the parallel pipeline have more auxiliary connection points A_b than those in the single pipeline. The order of arrangement is derived from the piping information as P6 ⟶ P9 ⟶ P7 ⟶ P8.

4.4.2. Branch Pipeline Simulation Results and Analysis

The optimized A*-genetic algorithm with the introduction of the “branch connection point” strategy and the maze–genetic algorithm of Sui [26] is used to simulate the experiment of Case 4. The information on branch piping arrangement obtained by the two algorithms is shown in

Table 9. Case 4 branch pipeline layout information obtained based on each algorithm.

Algorithm Name	The Optimal Number of Solutions	Average Solution Time/s	Average Convergence Algebra	Average Length	Average Number of Bends	Average Energy Value	Average Fitness Value
MA-GA	9	632.51	125.9	306.0	5.6	8	363.40
A*-GA	12	289.36	51.5	303.8	4.5	0	367.37

Note: Bolded data in the table are optimal values.

. The best arrangement of the branch pipeline for Case 4 obtained by the algorithm of this paper is shown in Figure 12.

A comparative analysis of the branch piping arrangement information obtained in Table 9 leads to the following conclusions.

(1)

The optimized A*-genetic algorithm proposed in this paper, when solving the branch piping arrangement problem, searches for the obtained paths with shorter average length, lower average number of elbows, lower average energy value, and higher average path adaptation value. Thus, the algorithm in this paper can obtain a higher-quality piping arrangement when solving the parallel piping arrangement problem.

(2)

Compared with the maze–genetic algorithm proposed by Sui, the optimized A*-genetic algorithm proposed in this paper further improves the pipeline’s optimization efficiency while ensuring the quality of pipeline arrangement when solving the parallel pipeline arrangement problem.

(3)

Compared with the maze–genetic algorithm proposed by Sui, the optimal A*-genetic algorithm proposed in this paper has more time to obtain the optimal solution and a higher probability of obtaining the optimal solution when solving the branch piping arrangement problem.

5. Conclusions

In this paper, we propose an optimal A*-genetic algorithm to systematically solve the problems of a single pipeline, parallel pipeline, and branch pipeline. This paper proposes an optimized A*-genetic algorithm for systematically solving the intelligent arrangement of various types of ship piping, such as single piping, parallel piping, and branch piping. This has some reference and guidance for actual ship pipeline arrangement. Combined with the simulation experiment results, the following conclusions are obtained:

(1)

As shown by Ito et al. [11], the traditional genetic algorithm easily falls into the optimal local solution and inefficiently solves the piping arrangement. To solve the above problems, the traditional genetic algorithm is improved and combined with the A* algorithm to solve the intelligent arrangement of a single pipeline of ships by adding the interpolation test function in the initialization of the population to avoid the generation of individuals that interpolate with the equipment obstacles. The concentration value is introduced in the selection process to judge the probability of individuals entering the next generation by the concentration value and the fitness value. Further, “shear” is introduced in the crossover and variation. Moreover, in the crossover and variation section, a “cut” strategy is introduced to cut the generated duplicate paths, and the crossover or variation individuals need to be compared with the corresponding original parents, and only the crossover or variation individuals with better quality than the corresponding parents are retained. The A* algorithm in the deterministic algorithm is introduced to search for subpaths in the crossover and variation section to speed up the overall solving efficiency of the algorithm.

(2)

Combined with the connection point strategy, the parallel and branch pipelines are split into multiple single pipelines. The optimal A*-genetic algorithm proposed in this paper is used to complete the arrangement of every single pipeline in turn, which reduces the complexity of the ship pipeline arrangement problem and completes the intelligent arrangement of the parallel pipeline and branch pipeline.

(3)

The optimized A*-genetic algorithm proposed in this paper is compared and verified with other related algorithms through simulation experiments. The results show that in the single-pipe arrangement, compared with the A* algorithm, particle swarm algorithm, and maze–genetic algorithm, the A*-genetic algorithm of this paper obtains the optimal value in six indexes, such as the length of the pipe, number of elbows, energy value, fitness value and number of optimal solutions and average convergence generations, etc., and also outperforms the maze–genetic algorithm, which is a hybrid algorithm in terms of its average solution time. In solving the parallel pipe and branch pipe arrangement problems, the all-around performance of this paper’s algorithm is also better than that of the A* and maze–genetic algorithms, respectively. The A*-genetic algorithm considers the quality of pipeline arrangement and the efficiency of the solution, which verifies the adaptability and superiority of the algorithm for the intelligent arrangement of various types of ship pipelines.

(4)

Compared with the traditional optimization algorithm, the A*-genetic algorithm proposed in this paper has a more substantial global search capability and shorter search time. The appropriate optimization method proposed in this paper can provide some reference for other optimization algorithms in solving the problem of intelligent arrangement of ship pipelines.

(5)

In subsequent work, we will continue to study how to quantify more ship piping layout rules into parameters that can be embedded in the pathfinding algorithm by computer recognition and explore the development of a highly automated piping intelligent layout system that integrates equipment and layout space modeling, piping layout rules parameterization, intelligent piping pathfinding, and 3D CAD display of layout effect. This system will provide a reference for relevant ship piping designers.