Research Article Ship Pipe Routing Design Using NSGA-II .

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Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016, Article ID 7912863, 21 h ArticleShip Pipe Routing Design Using NSGA-II andCoevolutionary AlgorithmWentie Niu, Haiteng Sui, Yaxiao Niu, Kunhai Cai, and Weiguo GaoKey Laboratory of Mechanism Theory and Equipment Design of Ministry of Education, Tianjin University, Tianjin 300072, ChinaCorrespondence should be addressed to Wentie Niu; niuwentie@tju.edu.cnReceived 9 July 2016; Revised 6 November 2016; Accepted 30 November 2016Academic Editor: Francisco ChicanoCopyright 2016 Wentie Niu et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.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 humanerrors. By simplifying equipment and ship hull models and dividing workspace into three-dimensional grid cells, the mathematicmodel of layout space is constructed. Based on the proposed concept of pipe grading method, the optimization model of piperouting is established. Then an optimization procedure is presented to deal with pipe route planning problem by combining mazealgorithm (MA), nondominated sorting genetic algorithm II (NSGA-II), and cooperative coevolutionary nondominated sortinggenetic algorithm II (CCNSGA-II). To improve the performance in genetic algorithm procedure, a fixed-length encoding methodis presented based on improved maze algorithm and adaptive region strategy. Fuzzy set theory is employed to extract the bestcompromise pipeline from Pareto optimal solutions. Simulation test of branch pipe and design optimization of a fuel piping systemwere carried out to illustrate the design optimization procedure in detail and to verify the feasibility and effectiveness of the proposedmethodology.1. IntroductionSince the 1970s, pipe routing design has been studied in various industrial fields, like aeroengine, large-scale integratedcircuit, ship, and so forth. Pipe routing design, which isrelated to other tasks, is one of the most important processesat the detailed design stage of a ship. However, duo tothe complexity of piping system and the diversity of constraints in ship piping system, it is time-consuming anddifficult to achieve feasible layout. Therefore, it is significantto investigate automatic pipe routing method.Systematic studies in route path planning have beencarried out by researchers for several decades. Dijkstraalgorithm [1] proposed in 1959 is a well-known algorithmfor path optimization with shortest length. Based on Dijkstraalgorithm, ๐ด algorithm is developed by Hart et al. [2]to improve search efficiency. In 1961, Lee [3] proposedmaze algorithm to solve connecting problem of two points.However, huge data storage is essential to handle large-sizeoptimization problems, which results in low search efficiency.To overcome the mentioned drawbacks, numerous researcheshave been conducted in references [4โ€“8]. Typically, the escapealgorithm is proposed by Hightower [4] to improve searchefficiency, but an optimal solution also cannot be guaranteed.Recently, researches on piping route path planning havebeen promoted by the development of modern optimizationalgorithms such as genetic algorithm [9โ€“17], ant colonyalgorithm [18โ€“23], and particle swarm optimization [24โ€“27].Genetic algorithm is used by Ito [10] to optimize pipe routingin two-dimensional space, in which the chromosome of routepath with variable-length is firstly defined. Afterward, thechromosome with fixed length [12] is redefined to solve theproblems such as low efficiency in GA. A variable-lengthencoding technique is proposed by Fan et al. [13] and appliedto optimize the ship pipe paths in three-dimensional space,which leads to the complexity of algorithm design process. Inaddition, expert systems [28โ€“31] are also applied in ship piperoute design.At present, the optimization algorithm research on piperoute planning mainly concentrates on the case with two

2Mathematical Problems in Engineeringterminals, while multibranch pipe design is rarely studied.Park and Storch [29] developed a cell-generation method forpipe routing in a ship engine room, in which the branchpipeline is regarded as a compound of two simple forms: endforked and middle-forked. Using maze algorithm and Steinertree theory, Fan et al. [7] conduct the research on pipe routingproblem of aeroengine. Coevolutionary algorithm is appliedby Wu et al. [32] for multibranch pipe routing in ship. Steinerminimal tree and particle swarm optimization are combinedby Liu and Wang [33] to solve the optimization problemof multibranch pipe routing in aeroengine. However, fewerresearchers have taken the differences of pipeline diametersinto consideration. When pulsating pressure is existing inpipeline, the excitation force will be generated at the branchpoint because of the big difference in diameter value of theconnected pipes, which further influences working periodand usage security of pipeline [34]. Therefore, the diameterdifferences should be taken into consideration in optimization algorithm.Owing to the diameter differences of branch pipelines,the concept of pipe grading is defined in this paper. Inconsideration of the number differences of connecting pointsin each grade, a new algorithm is proposed by combiningmaze algorithm (MA) [3], nondominated sorting geneticalgorithm II (NSGA-II) [35], and cooperative coevolutionarynondominated sorting genetic algorithm II (CCNSGA-II)[36]. To improve the performance in genetic algorithmprocedure, a fixed-length encoding method is presentedbased on the improved maze algorithm and adaptive regionstrategy. In addition, fuzzy set theory is employed to extractthe best compromise pipeline from Pareto optimal solutions,by which the imprecise nature of decision makerโ€™s judgmentis avoided.The rest of this paper is organized as follows. The problemof ship pipe route design is formulated in Section 2. Theproposed pipe routing algorithm is presented in Section 3. Acase study is conducted to verify the proposed algorithm inSection 4. Finally, conclusions are drawn in Section 5.2. Problem Formulation2.1. Representation of Piping Layout Space. Due to the complex ship hull structure and diverse equipment with variousshapes, it is time-consuming and inefficient to describe allgeometric information in detail for pipe routing design inship piping layout space. Therefore, it is essential to simplifythe environment of ship piping layout. To construct a reasonable workspace model representing the essential geometricinformation of the equipment and ship hull structure, severalprinciples should be obeyed in the environment simplification of piping layout.(1) The geometrical properties of the model should besimple;(2) Spatial position of the model should be expressedaccurately;(3) The accurate spatial positions of the pipeline terminals should be guaranteed.2.1.1. Simplification of Equipment Model. Informationallycomplete models are proposed in literature [37] for designconstraints based on an analysis of geometric and nongeometric properties of the related space volumes, which issuitable for component layout problem. For the problem ofpiping system layout, since the equipment locations havealready been determined, the simplification of componentsis just required, and the simplest approach is to establish axisaligned bounding box (AABB). However, if the equipmentmodel is complex, it is difficult for AABB to meet therequirements for accurate expression of spatial location. Asshown in Figure 1, the equipment model simplified by AABBis illustrated in Figure 1(b), in which the model information oforiginal equipment in Figure 1(a) is not described accurately.In this paper, the optimized subdivision boundary boxmethod described in literature [38] is introduced to simplifythe models of ship equipment, an example of simplificationis shown in Figure 1(c), and other equipment involved inlayout space are simplified by the same way. The procedures of optimized subdivision boundary box method are asfollows.Step 1. Construct the AABB of the equipment model.Step 2. Divide AABB by using the nonuniform grid of cellsaccording to the characteristics of the equipment.Step 3. For each cell, find the polygons of the model that lieinside or intersect with it.Step 4. Construct the AABB of the polygons by Step 3.Step 5. Clip the AABB with the cell itself.Step 6. Loop to Step 3 until all cells in the grid are processed.2.1.2. Mathematical Model of Workspace. To represent thegeometric information of ship hull structure, components,and pipeline terminals of layout space, the space division isnecessary. The irregular workspace is approximately represented by being divided into numbers of cube grids, and thedetailed processes are as follows.The working space is considered as a cuboid space dividedinto ๐‘€ ๐‘ ๐ฟ uniform 3D cubic grid cells, and an exampleis shown in Figure 2(a). To build the mathematical model oflayout space, one vertex cell of the cuboid should be selectedas the origin whose coordinate is (1, 1, 1) to establish a coordinate system, and each cell will be given a specific coordinate(๐‘ฅ, ๐‘ฆ, ๐‘ง) based on the row, column, and floor in the space.The default tag value of each cell is set as โ€œ0โ€ representingthe feasible search space. The cell occupied by obstacles isset as โ€œ#โ€ representing the infeasible search space. Since theobstacles mainly include ship hull, spaces outside the hull,equipment, and the generated pipelines can be described bythe diagonal coordinates of one or more cuboids. Grid cellsoccupied partially by cuboids are approximately regarded asbeing occupied completely. Figure 2(b) shows an exampleof approximate representation for part of the simplified shiphull.

Mathematical Problems in Engineering3(b)(a)(c)Figure 1: An example for simplification of the equipment model.Ship hullzyxCells on obstacle(a)(b)Figure 2: Mathematical model of workspace. (a) Grid method. (b) An example of approximate representation for the simplified ship hull.2.2. Description of Pipe Information2.2.1. Redefinition of Connecting Point. Since the inlet/outletof equipment is involved in the simplified model, the equipment simplification will result in failed connection of thepipeline terminals in model space. To solve this problem, theactual inlet/outlet involved in one or several grids is extendedto the adjacent grids outside the simplified model along itsnormal. And the adjacent grid cell passed by the axis ofinlet/outlet is defined as new connecting point.2.2.2. Definition of Pipe Path. Base on the above spacedividing method, a connected pipe path is defined as onecontinuous path connecting a starting point and a goalpoint, which contains a series of adjacent grid cells. Theencoding of path is represented by a sequence containinga series of coordinates of the grid cells, and an example isshown in Figure 3. The coordinate system is establishedafter the layout space is meshed, in which a green pipe pathconnecting two object points is generated by taking bluegrid cell 1 as a starting point and blue grid cell 2 as thegoal point. Apparently, coordinates of grid cells 1 and 2 are,respectively, (1, 2, 5) and (9, 7, 1), and the illustrated path codeis {(1 2 5), (1 3 5), (1 4 5), (1 5 5), (1 6 5), (1 7 5),(1 7 4), (1 7 3), (1 7 2), (1 7 1), (2 7 1), (3 7 1),(4 7 1), (5 7 1), (6 7 1), (8 7 1), (9 7 1)}.

4Mathematical Problems in Engineeringaccording to the parameters of next pipeline that need to berouted.1zyx2.3. Problem Formulation of Pipe Routing. Ship pipe routeplanning is a typical multiobjective optimization problem;that is, it aims to achieve optimization for the best compromise pipeline under several given constraints based on thediscrete mathematical model. The considered constraints andevaluation criteria are as follows.(i) Avoiding obstacles;(ii) Shortest length of route path;(iii) Minimum bends;2(iv) Maximum overlapping length of subpipe route paths.Object pointThen the objective functions Obj(๐‘“) for pipeline designare determined as follows:Route pathmin Obj (๐‘“1 ) ๐ฟ ๐‘ƒ ,(2)min Obj (๐‘“2 ) ๐ต๐‘ƒ ,(3)Obj (๐‘“3 ) ๐‘‚๐‘ƒ ,(4)Figure 3: Definition of pipe path.max2.2.3. Representation of Pipeline. During the ship pipe routeplanning, different diameters of pipelines may be requiredto connect different equipment; for example, diameter valuesof the pipes connecting fuel oil storage tanks are largerthan those of pipes connecting marine main engines. Inconsideration of the differences between pipeline diameters,the concept of pipe grading is introduced. If all pipes in aroute path are sorted by the diameter values, the top one,namely, the pipe with largest diameter value, is defined as pipegrade 1, whose terminals are defined as points-grade 1, and theremaining pipes can be graded successively according to theranked list.Since the workspace is approximately described by cubegrid method, the accuracy depends on the size of cube grid. Inthis paper, the minimum round pipe diameter of all pipelinesin the design space is chosen as the cube side length. Fora pipeline, including different diameter of pipes, to simplifythe path routing algorithm, the border cells of obstaclesincluding the generated pipe lines are extended outward byan appropriate cell number. By this method, the encoding ofsubpipe grade with larger diameter remains the same withthat of the subpipe grade with smaller diameter. The extendedcell number is determined by largest round pipe diameter ofcurrent pipeline, as shown in๐ทmax{{int ( 2๐ฟ ) 1,๐‘› {๐ท{int ( max ) ,{2๐ฟ๐ทmax๐ท int ( max ) 0.5,2๐ฟ2๐ฟ๐ทmax๐ทmax int () 0.5,2๐ฟ2๐ฟ(1)where ๐‘› represents the cell number that needs to be extendedby obstacles, ๐ทmax represents the largest diameter of currentpipeline, and ๐ฟ denotes side length of cells. Once a pipelineis routed, the generated pipeline will be set as obstacles.Then, the mathematical model of workspace will be updatedwhere ๐ฟ ๐‘ƒ represents the length of the pipeline, namely, thenumber of grid cells passed through by the axis of pipelinefrom starting point to end point, ๐ต๐‘ƒ represents the number ofbends, and ๐‘‚๐‘ƒ denotes the overlapped length between currentpipe grade and other pipe grades.3. Proposed Pipe Routing Method3.1. Overview of the Proposed Method. According to the connection relationships of equipment as depicted in schematicdiagram of piping system, the terminal set of each pipeline isobtained. In consideration of the effect of pipeline diameteron layout result and usage safety, the pipe grading is conducted, and then diameter value of subpipe grade 1 is takenas the typical diameter value of each pipeline with certainsubpipes. And then the rules for piping layout sequence aredecided as follows.(1) The pipeline with larger diameter is arranged inpriority.(2) For subpipes of a pipeline, the one with higher gradeis arranged in priority.(3) Branch point is generated on the subpipe with adjacent higher grade.An example of a pipeline with three subpipes is shown inFigure 4. Regardless of the difference of subpipe diameters,the routing result is shown in Figure 4(a). When the pulsatingpressure exists in pipeline, the excitation force will be generated at the branch point due to the large difference betweendiameter values of subpipe grade 1 and subpipe grade 3,which will result in vibration influencing working period andusage security of the pipeline. By considering the diameterdifference of subpipes, the routing result is improved basedon the order of arrangement, which is shown in Figure 4(b).

Mathematical Problems in Engineering5Grade 1Grade 1Grade 2Grade 2Grade 3Grade 3(a)(b)Figure 4: Selection of branch point location.According to the mentioned rules for pipe routingdesign, a ship pipe routing methodology is proposed bycombining MA, NSGA-II, and CCNSGA-II. The flow chartof the proposed method is shown in Figure 5, where MANSGA-II denotes the combination of MA and NSGA-II,and MA-CCNSGA-II represents the combination of MA andCCNSGA-II.For case with two terminals, MA-NSGA-II is used forpipe routing design as shown in Figure 6. The search strategyof tendencies in direction is mainly adopted by researchersto generate initial population, which might lead to thegeneration of repeated nodes in a chromosome. In this paper,maze algorithm is used to generate the initial individualsto avoid generating repeated nodes. Since fewer individualsare required to compose the initial population, the searchefficiency of maze algorithm becomes acceptable. The Paretooptimal set is obtained after optimizing the initial populationby using NSGA-II algorithm, and fuzzy set theory is used toselect the best compromise solution of current subpipe, whichwill be referred during the optimization procedure of othersubpipes.For cases with three or more terminals, MA-CCNSGAII is used for pipe routing design as shown in Figure 7.Decomposition method, namely, the optimization problem,is divided into several subproblems to be optimized, respectively, and is adopted to deal with multibranch pipelinerouting problem. The key connecting point is taken as thestarting point, and other points are taken as end points,respectively. The initial subpopulations are generated in turnby using the same method in the cases with two terminals.And cooperative coevolutionary strategy is applied in optimization algorithm, in which the subproblems are optimizedcooperatively. Therefore, the optimal individual of eachsubpopulation is selected to be shared during the solutionprocedures, and an individual of a certain subpopulation isevaluated by calculating the fitness function of the solutionconstructed by the individual and optimal individuals ofother subpopulations. The evolution and cooperation of eachpopulation are performed continually until the evolution iscompleted, and the final optimal individuals are combinedtogether as an optimal solution of the multibranch pipes.The key connecting point is defined as follows. For pointsgrade 1, the sum of Euclidean distances between a certainpoint of points-grade 1 and others is, respectively, calculatedby (5), and the point with minimum sum is selected as keyconnecting point of pipe grade 1; for points-grade ๐‘– (๐‘– 1),the key connecting point is generated in points-grade ๐‘– 1.The sum of Euclidean distances between a certain point ofpoints-grade ๐‘– 1 and the connecting points of points-grade๐‘– is calculated by (5), respectively. Then, the point of pointsgrade ๐‘– 1 with minimum sum is taken as key connectingpoint of pipe grade ๐‘–.๐‘›๐ฟ ๐‘ƒ๐‘– ๐ฟ ๐‘ƒ๐‘– ๐‘ƒ๐‘— , ๐‘– 1, 2, . . . , ๐‘›; ๐‘— 1, 2, . . . , ๐‘›,๐‘— 1(5)where ๐ฟ ๐‘ƒ๐‘– represents the sum of Euclidean distances whenthe connecting point ๐‘ƒ๐‘– is taken as the starting point; ๐ฟ ๐‘ƒ๐‘– ๐‘ƒ๐‘—denotes the Euclidean distance between connecting points ๐‘ƒ๐‘– ;and ๐‘ƒ๐‘— ; ๐‘› represents the number of connecting points. Anexample for the key connecting point selection of pipelinewith three subpipe grades is shown in Figure 8. As shownin Figure 8, according to the concept of pipe grading, theconnecting points ๐‘ƒ1 and ๐‘ƒ2 are defined as points-grade 1,the connecting points ๐‘ƒ3 and ๐‘ƒ4 are defined as points-grade2, and ๐‘ƒ5 is defined as points-grade 3. Then, according tothe definition of key connecting point, ๐‘ƒ1 is selected as keyconnecting point of subpipe grade 1 and subpipe grade 2, and๐‘ƒ4 is selected as key connecting point of subpipe grade 3.3.2. Adaptive Region. 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

6Mathematical Problems in Engin

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 xed-length encoding method is presented based on improved maze algorithm and adaptive region strategy.

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