Pipe route design plays a prominent role in ship design. Due to the complex configuration in layout space with numerous pipelines, diverse design constraints, and obstacles, it is a complicated and time-consuming process to obtain the optimal route of ship pipes. In this article, an optimized design method for branch pipe routing is proposed to improve design efficiency and to reduce human errors. By simplifying equipment and ship hull models and dividing workspace into three-dimensional grid cells, the mathematic model of layout space is constructed. Based on the proposed concept of pipe grading method, the optimization model of pipe routing is established. Then an optimization procedure is presented to deal with pipe route planning problem by combining maze algorithm (MA), nondominated sorting genetic algorithm II (NSGA-II), and cooperative coevolutionary nondominated sorting genetic algorithm II (CCNSGA-II). To improve the performance in genetic algorithm procedure, a fixed-length encoding method is presented based on improved maze algorithm and adaptive region strategy. Fuzzy set theory is employed to extract the best compromise pipeline from Pareto optimal solutions. Simulation test of branch pipe and design optimization of a fuel piping system were carried out to illustrate the design optimization procedure in detail and to verify the feasibility and effectiveness of the proposed methodology.

Since the 1970s, pipe routing design has been studied in various industrial fields, like aeroengine, large-scale integrated circuit, ship, and so forth. Pipe routing design, which is related to other tasks, is one of the most important processes at the detailed design stage of a ship. However, duo to the complexity of piping system and the diversity of constraints in ship piping system, it is time-consuming and difficult to achieve feasible layout. Therefore, it is significant to investigate automatic pipe routing method.

Systematic studies in route path planning have been carried out by researchers for several decades. Dijkstra algorithm [

At present, the optimization algorithm research on pipe route planning mainly concentrates on the case with two terminals, while multibranch pipe design is rarely studied. Park and Storch [

Owing to the diameter differences of branch pipelines, the concept of pipe grading is defined in this paper. In consideration of the number differences of connecting points in each grade, a new algorithm is proposed by combining maze algorithm (MA) [

The rest of this paper is organized as follows. The problem of ship pipe route design is formulated in Section

Due to the complex ship hull structure and diverse equipment with various shapes, it is time-consuming and inefficient to describe all geometric information in detail for pipe routing design in ship piping layout space. Therefore, it is essential to simplify the environment of ship piping layout. To construct a reasonable workspace model representing the essential geometric information of the equipment and ship hull structure, several principles should be obeyed in the environment simplification of piping layout.

The geometrical properties of the model should be simple;

Spatial position of the model should be expressed accurately;

The accurate spatial positions of the pipeline terminals should be guaranteed.

Informationally complete models are proposed in literature [

An example for simplification of the equipment model.

Construct the AABB of the equipment model.

Divide AABB by using the nonuniform grid of cells according to the characteristics of the equipment.

For each cell, find the polygons of the model that lie inside or intersect with it.

Construct the AABB of the polygons by Step

Clip the AABB with the cell itself.

Loop to Step

To represent the geometric information of ship hull structure, components, and pipeline terminals of layout space, the space division is necessary. The irregular workspace is approximately represented by being divided into numbers of cube grids, and the detailed processes are as follows.

The working space is considered as a cuboid space divided into

Mathematical model of workspace. (a) Grid method. (b) An example of approximate representation for the simplified ship hull.

Since the inlet/outlet of equipment is involved in the simplified model, the equipment simplification will result in failed connection of the pipeline terminals in model space. To solve this problem, the actual inlet/outlet involved in one or several grids is extended to the adjacent grids outside the simplified model along its normal. And the adjacent grid cell passed by the axis of inlet/outlet is defined as new connecting point.

Base on the above space-dividing method, a connected pipe path is defined as one continuous path connecting a starting point and a goal point, which contains a series of adjacent grid cells. The encoding of path is represented by a sequence containing a series of coordinates of the grid cells, and an example is shown in Figure

Definition of pipe path.

During the ship pipe route planning, different diameters of pipelines may be required to connect different equipment; for example, diameter values of the pipes connecting fuel oil storage tanks are larger than those of pipes connecting marine main engines. In consideration of the differences between pipeline diameters, the concept of pipe grading is introduced. If all pipes in a route path are sorted by the diameter values, the top one, namely, the pipe with largest diameter value, is defined as pipe grade 1, whose terminals are defined as points-grade 1, and the remaining pipes can be graded successively according to the ranked list.

Since the workspace is approximately described by cube grid method, the accuracy depends on the size of cube grid. In this paper, the minimum round pipe diameter of all pipelines in the design space is chosen as the cube side length. For a pipeline, including different diameter of pipes, to simplify the path routing algorithm, the border cells of obstacles including the generated pipe lines are extended outward by an appropriate cell number. By this method, the encoding of subpipe grade with larger diameter remains the same with that of the subpipe grade with smaller diameter. The extended cell number is determined by largest round pipe diameter of current pipeline, as shown in

Ship pipe route planning is a typical multiobjective optimization problem; that is, it aims to achieve optimization for the best compromise pipeline under several given constraints based on the discrete mathematical model. The considered constraints and evaluation criteria are as follows.

Avoiding obstacles;

Shortest length of route path;

Minimum bends;

Maximum overlapping length of subpipe route paths.

Then the objective functions

According to the connection relationships of equipment as depicted in schematic diagram of piping system, the terminal set of each pipeline is obtained. In consideration of the effect of pipeline diameter on layout result and usage safety, the pipe grading is conducted, and then diameter value of subpipe grade 1 is taken as the typical diameter value of each pipeline with certain subpipes. And then the rules for piping layout sequence are decided as follows.

The pipeline with larger diameter is arranged in priority.

For subpipes of a pipeline, the one with higher grade is arranged in priority.

Branch point is generated on the subpipe with adjacent higher grade.

An example of a pipeline with three subpipes is shown in Figure

Selection of branch point location.

According to the mentioned rules for pipe routing design, a ship pipe routing methodology is proposed by combining MA, NSGA-II, and CCNSGA-II. The flow chart of the proposed method is shown in Figure

The flow chart of ship pipe routing algorithm based on pipe grading.

For case with two terminals, MA-NSGA-II is used for pipe routing design as shown in Figure

The flow chart of MA-NSGA-II.

For cases with three or more terminals, MA-CCNSGA-II is used for pipe routing design as shown in Figure

The schematic diagram of MA-CCNSGA-II.

The key connecting point is defined as follows. For points-grade 1, the sum of Euclidean distances between a certain point of points-grade 1 and others is, respectively, calculated by (

An example for the key connecting point selection.

Due to the huge piping layout space, global search for optimal pipeline will result in operation with large storage and low efficiency. So the adaptive region strategy [

Adaptive region in 3D space.

Extended search is the expansion process starting from one grid to the adjacent grid, and a grid is specified by its tag value marked in the process. The rules of marking tag value are as follows.

Initial grid is marked as “1.”

Only six neighbors of the current grid could be marked, whose tag value is that of current grid plus 1.

If the adjacent grid has been marked, the smaller tag value is selected.

An expansion process diagram is illustrated in Figure

An example for the extended search process of maze algorithm.

Retracing process is antisearch from goal point to starting point. Figure

An example for retracing process of maze algorithm.

When the workspace is marked by using depth-first search strategy, the adaptive region cannot be reached entirely by the retracing process, which limits the diversity of population and affects the optimization efficiency, and then the auxiliary point is introduced to solve this problem. An example for two-dimensional maze search is shown in Figure

Introduction of auxiliary point.

As to the introduction of auxiliary point, the distribution scope of pipe paths is expanded and the diversity of initial population is increased, which provides the basis for better optimization.

To overcome the drawbacks of variable-length encoding technique, such as the complexity in dealing with chromosomes and generating chromosomes with repeated nodes in genetic operation, a fixed-length encoding method is proposed based on the improved maze algorithm and extended adaptive region strategy, which improves the performance in genetic algorithm procedure.

Two-dimensional diagram shown in Figure

Fixed-length encoding method.

Due to the large layout space in a ship and separate distribution of obstacles, we assume that the extended space is a workspace without obstacles. Therefore, the length of chromosome in 3D space is determined as the maximum tag value plus

Fitness value is the evaluation criterion for superiority of a chromosome and the basis of nondominated sorting and selection operation in nondominated sorting genetic algorithm. The objective function formulated in Section

An example of pipeline with three subpipe grades is shown in Figure

An example for fitness evaluation.

Only the pipe path chromosome, which is composed of a series of nodes with continuous coordinates, can be regarded as a valid one. By using traditional genetic operator directly in crossover and mutation operation, the connectivity of nodes in a chromosome cannot be guaranteed after genetic operation, which may lead to the generation of invalid individuals. Based on the fixed-length encoding method mentioned in Section

The crossover point is selected randomly in the traditional single-point crossover operation, and then two offspring individuals are generated by exchanging the right parts of two parent chromosomes. A crossover strategy with fixed-length chromosomes is proposed by improving single-point crossover operator. For two chromosomes, two crossover nodes,

The generation method for assistant paths is different from that for initial population. To improve the operation efficiency of the algorithm and avoid generating repeated nodes, the following method is applied: the extended search process of maze algorithm is still used to encode the workspace composed by start and goal points. Then the prior search direction is determined by the position relationship between two points, and an initial search direction is chosen randomly to search towards the grid cells with decreasing values. The search direction will not be changed unless the grid cell on obstacle is searched, and process will be continued until the terminal point is reached.

As shown in Figure

Crossover based on fixed-length encoding method.

Similar to the crossover operation, the connectivity of pipe path cannot be guaranteed by using traditional mutation operation method directly. A mutation strategy of fixed-length chromosome is given in this part. For the parent chromosome, two mutation points are randomly selected as the starting and end points, respectively, and an assistant path is generated by the same way as that in crossover operation. Then the original nodes between two mutation points are replaced by the assistant path, which brings the new offspring chromosome. Similarly, if the length of offspring chromosome exceeds the limited length, it will be removed, or else the empty nodes will be supplemented by “0.”

An example for mutation operation is shown in Figure

Mutation based on fixed-length encoding method.

For Pareto optimal chromosome sets with

In (

To verify the feasibility and efficiency of the proposed algorithm, the proposed approach is utilized to execute a branch pipe design example of our previous work [

The pipe routing space is set as 500 mm × 500 mm × 500 mm and is divided by using the cube whose side length is 10 mm. the working space is then divided into 50 × 50 × 50 uniform 3D cubic grid cells. Obstacles in the workspace are represented by the cuboids whose diagonal coordinates are determined as follows:

Information of piping route path and connecting points.

Route path name | Coordinates of connecting points | Pipe specification (mm) |
---|---|---|

Pipe grade 1 | (2, 2, 2), (10, 44, 46) | 20 |

Pipe grade 2 | (39, 12, 48) | 15 |

Pipe grade 3 | (45, 46, 15) | 10 |

In this example, each pipe grade includes two connecting points, and MA-NSGA-II is adopted as the pipe routing algorithm. Based on the concept of key connecting point given in Section

Parameters of MA-NSGA-II.

Parameter | Value |
---|---|

Population size | 50 |

Number of generation | 100 |

Crossover probability | 0.8 |

Mutation probability | 0.05 |

The evolution graphs of average and minimum values of the two objects versus generation are, respectively, illustrated in Figures

Evolutionary graphs of pipe grade 1. (a) Evolution process of length; (b) evolution process of bend numbers; (c) Pareto optimal set.

Evolutionary graphs of pipe grade 2. (a) Evolution process of length; (b) evolution process of bend numbers; (c) Pareto optimal set.

Evolutionary graphs of pipe grade 2. (a) Evolution process of length; (b) evolution process of bend numbers; (c) Pareto optimal set.

As shown in Figure

Some best solutions are listed in Table

Some solutions of test example.

Solution | | | | | | | | | | | | |
---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 197 | 8 | 94 | 3 | 118 | 5 | 74 | 4 | 50 | — | 39 | 0.093095 |

2 | 197 | 8 | 94 | 3 | 120 | 5 | 74 | 4 | 51 | — | 40 | 0.091905 |

3 | 205 | 8 | 94 | 3 | 128 | 4 | 74 | 4 | 43 | — | 40 | 0.090952 |

4 | 197 | 8 | 94 | 4 | 120 | 4 | 74 | 4 | 51 | — | 40 | 0.085119 |

5 | 197 | 8 | 94 | 3 | 140 | 5 | 74 | 4 | 61 | — | 50 | 0.08 |

6 | 197 | 8 | 94 | 4 | 132 | 4 | 74 | 4 | 58 | — | 45 | 0.077976 |

7 | 205 | 8 | 94 | 3 | 140 | 5 | 74 | 4 | 53 | — | 50 | 0.075119 |

8 | 197 | 8 | 94 | 4 | 140 | 4 | 74 | 4 | 61 | — | 50 | 0.073214 |

9 | 197 | 8 | 94 | 3 | 152 | 5 | 74 | 4 | 67 | — | 56 | 0.072857 |

10 | 197 | 8 | 94 | 3 | 156 | 5 | 74 | 4 | 69 | — | 58 | 0.070476 |

The routing result of solution 1.

As shown in Figures

Figure

A schematic diagram of the fuel piping system in a ship engine room.

By using 3D CAD software SolidWorks, the solid model of related equipment is established by the method presented in Section

Diagonal coordinates of main parts of the simplified models.

Equipment name | Diagonal coordinate 1 | Diagonal coordinate 2 |
---|---|---|

Fuel oil storage tank 1 | (1, 67, 23) | (40, 107, 103) |

| ||

Fuel oil storage tank 2 | (1, 67, 183) | (40, 107, 263) |

| ||

Steam boiler | (90, 1, 290) | (115, 17, 311) |

(115, 1, 283) | (152, 83, 320) | |

| ||

Hot water boiler | (135, 15, 1) | (152, 55, 17) |

(152, 19, 5) | (163, 29, 13) | |

| ||

Diesel generator 1 | (122, 15, 97) | (198, 17, 127) |

(123, 17, 104) | (139, 41, 116) | |

(128, 17, 104) | (46, 33, 118) | |

(160, 17, 103) | (188, 42, 117) | |

| ||

Diesel generator 2 | (122, 15, 163) | (198, 23, 188) |

(123, 17, 170) | (139, 41, 181) | |

(138, 17, 170) | (157, 33, 183) | |

(160, 17, 169) | (188, 42, 183) | |

| ||

Marine main engine 1 | (110, 11, 42) | (122, 35, 65) |

(146, 9, 47) | (175, 15, 60) | |

(136, 15, 47) | (187, 46, 66) | |

(122, 42, 40) | (196, 54, 55) | |

| ||

Marine main engine 2 | (110, 11, 222) | (122, 35, 246) |

(146, 9, 228) | (175, 15, 241) | |

(136, 15, 228) | (187, 46, 247) | |

(122, 42, 220) | (196, 54, 245) | |

| ||

Fuel transfer pump 1 | (17, 47, 105) | (33, 56, 111) |

| ||

Fuel transfer pump 2 | (41, 45, 105) | (57, 53, 111) |

According to the schematic diagram, the fuel piping system is classified into six pipelines. The coordinates, representing the locations of connecting points and the diameter parameters of pipes which the connecting point belongs to, are summarized in Table

Information of piping route path and connecting points.

Route path name | Included cabins and equipment with | Pipe specification (mm) |
---|---|---|

Path 1 | Fuel oil tank: (58, 21, 6), (58, 21, 304) | |

Fuel transfer pump 1: (39, 49, 106) | | |

Fuel transfer pump 2: (15, 52, 106) | | |

| ||

Path 2 | Fuel transfer pump 1: (39, 49, 113) | |

Fuel transfer pump 2: (15, 52, 113) | | |

Fuel oil storage tank 1: (33, 70, 104) | | |

Fuel oil storage tank 2: (33, 70, 182) | | |

| ||

Path 3 | Fuel oil storage tank 1: (20, 61, 69) | |

Fuel oil storage tank 2: (20, 61, 216) | | |

Diesel generator 1: (181, 33, 122), (185, 33, 122) | | |

Diesel generator 2: (181, 33, 188), (185, 33, 188) | | |

| ||

Path 4 | Fuel oil storage tank 1: (20, 61, 63) | |

Fuel oil storage tank 2: (20, 61, 223) | | |

Marine main engine 1: (139, 40, 67), (144, 40, 67) | | |

Marine main engine 2: (139, 40, 248), (144, 40, 248) | | |

| ||

Path 5 | Fuel oil storage tank 1: (20, 61, 56) | |

Fuel oil storage tank 2: (20, 61, 229) | | |

Hot water boiler: (159, 23, 14) | | |

Steam boiler: (101, 13, 289) | | |

| ||

Path 6 | Fuel oil storage tank 1: (20, 61, 31) | |

Fuel oil storage tank 2: (20, 61, 254) | | |

Hot water boiler: (161, 21, 14) | | |

Steam boiler: (96, 5, 289) | |

Based on the established space model, the fuel piping system could be designed by using the algorithm proposed in Section

Pipe path 3 is taken as an example to illustrate the algorithm procedure of pipe route planning. According to the concept of pipe grading defined in Section

Genetic algorithm parameters.

MA-NSGA-II | MA-CCNSGA-II | |
---|---|---|

Population size | 30 | 40 |

Number of generation | 100 | 200 |

Crossover probability | 0.8 | 0.8 |

Mutation probability | 0.05 | 0.05 |

According to the results obtained by using the proposed algorithm, the parametric CAD Model of the fuel pipe system is established by utilizing the SolidWorks API and VB.NET. The routing results of the pipelines are summarized in Table

Route paths of fuel path system.

Route path name | 3D solid model of pipe | Parameters | |
---|---|---|---|

Grade 1 | Grade 2 | ||

Path 1 | | | — |

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Path 2 | | | — |

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Path 3 | | | |

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Path 4 | | | |

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Path 5 | | | |

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Path 6 | | | — |

Final routing result of the fuel pipe system in a ship engine room.

In order to improve design efficiency and to reduce human errors, an optimized design method for branch pipe routing is proposed in this paper. To cope with the diameter differences of branch pipelines, the concept of pipe grading is proposed in this paper. In consideration of the number differences of connecting points in each grade, a pipe routing optimization procedure is proposed by combining MA, NSGA-II, and CCNSGA-II. Based on the concept of pipe grading, the requirement of usage safety can be considered at initial design stage, which lay a solid foundation for detailed design. To improve the performance of genetic algorithm procedure, fixed-length encoding method is employed in our proposed design optimization procedure. By utilizing the proposed pipe routing optimization procedure, optimal solution set of branch pipe is obtained; then the best compromise solution can be selected by taking full account of the important indicators of pipeline, such as bending number, subpipe length, and overlapped length as well as other complicated engineering constraints.

A design example of branch pipeline is conducted and design optimization of a fuel piping system in an actual ship is implemented to verify the feasibility and effectiveness of the proposed methodology. Further work will be carried out to develop a comprehensive and user-friendly computer-aided pipe routing system, which employs the proposed pipe routing optimization method.

The authors declare that they have no competing interests.

This work is funded by the National Natural Science Foundation of China (Grant no. 51275340).