Computer Aided Drafting, Design and Manufacturing Volume 26, Number 2, June 2016, Page 34

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1 Computer Aided Drafting, Design and Manufacturing Volume 26, Number 2, June 2016, Page 34 CADDM Pipe-assembly approach for ships using modified NSGA-II algorithm Sui Haiteng, Niu Wentie, Niu Yaxiao, Zhou Chongkai, Gao Weigao Key Laboratory of Mechanism Theory and Equipment Design of Ministry of Education, Tianjin University, Tianjin , China Abstract: Pipe-routing for ship is formulated as searching for the near-optimal pipe paths while meeting certain objectives in an environment scattered with obstacles. Due to the complex construction in layout space, the great number of pipelines, numerous and diverse design constraints and large amount of obstacles, finding the optimum route of ship pipes is a complicated and time-consuming process. A modified NSGA-II algorithm based approach is proposed to find the near-optimal solution to solve the problem. By simplified equipment models, the layout space is firstly divided into three dimensional (3D) grids to build its mathematical model. In the modified NSGA-II algorithm, the concept of auxiliary point is introduced to improve the search range of maze algorithm (MA) as well as to guarantee the diversity of chromosomes in initial population. Then the fix-length coding mechanism is proposed, Fuzzy set theory is also adopted to select the optimal solution in Pareto solutions. Finally, the effectiveness and efficiency of the proposed approach is demonstrated by the contrast test and simulation. The merit of the proposed algorithm lies in that it can provide more appropriate solutions for the designers while subject certain constrains. Key words: pipe routing; fix-length coding; maze algorithm; modified NSGA-II algorithm; ship industry 1 Introduction As an important part in pipe-assembly planning, pipe routing is a technique of developing collision-free and optimal routes for pipes in an environment scattered with obstacles. Especially, it plays a prominent role at the detailed design stage of a ship. However, duo to the complexity and diversity of constraints in piping systems, achieving favorable layout is time-consuming and inefficient. Therefore, investigating automatic pipe routing method becomes extremely significant. Studies in route planning have been carried out for several decades. Dijkstra algorithm [1] is a well-known methodology for path optimization with shortest length. Based on Dijkstra algorithm, A* algorithm is developed by Hart et al. [2] to improve its search efficiency. Lee [3] also proposed maze algorithm to solve connecting problem of two points. However, huge data storage is needed when handling large-size problems, which results in low efficiency. Numerous researches [4-7] have been conducted to overcome the drawbacks. Recently, studies on pipe routing have been promoted by modern optimization algorithms such as genetic algorithm (GA) [8-13], ant colony optimization (ACO) [14] and particle swarm optimization (PSO) [15-16]. Typically, GA is applied to optimize pipe design in two-dimensional space using chromosomes with variable-length. But there exists several problems such as low performance, local optimization, etc. Fan et al. [17] proposed a variable-length encoding technique applied in 3D pipe layout, and the search efficiency is improved. However, similar problems still remain unsolved. In addition, the mentioned routing methods usually transform the multi-objective vector into one single objective, which makes solutions highly sensitive to weight vector given by designers and fails to provide various references needed in finding optimal layout in practice. In this paper, NSGA-II [18] is adopted to find the optimal pipe paths in the layout space. The objectives discussed include minimizing the length of pipes and the amount of pipe turns, placing pipes close to ship hull and far from specific devices, sharing pipe supports as much as possible while avoiding obstacles. Mathematical model of the optimal problem is firstly built by simplifying the equipment models in workspace which is divided using grids. Then an improved MA is integrated to NSGA-II algorithm to Project item: Supported by National Nature Science Foundation of China (Grant No: ). Corresponding author: Niu Wentie, Female, Ph.D., niuwentie@tju.edu.cn.

2 Sui Haiteng et al., Pipe-assembly approach for ships using modified NSGA-II algorithm 35 construct the pipe routing methodology. The concept of auxiliary point is introduced to improve the search ability of MA; Based on the improved MA, a fix-length coding mechanism and corresponding genetic operators are proposed; Fuzzy set theory is also adopted to select the optimal solution among Pareto solutions. Finally, the proposed approach is demonstrated by simulation tests. 2 Problem formulation 2.1 Representation of layout space The representation of the layout space is defined as the mathematical description of the equipment and pipe models in the workspace, and space division is the essential work. Axis aligned bounding box (AABB) is used to enclose the workspace in a cuboid that is further divided into MNL uniform 3D grids. Each grid is given a specific coordinate depending on its row, column and layer, and one vertex grid of the cuboid is selected as origin whose coordinate is set as ( 1,1,1 ). In addition, each grid is also assigned a tag value that represents it status. The default tag value of each grid is set as 0, which represents the feasible search space. If the tag values of grids occupied by obstacles, such as ship equipment and generated pipes, are set as #, which represents the infeasible search space. Status of each grid belongs to the following three situations: empty, partially occupied, and fully occupied. To make the problem manageable, grids partially occupied are regarded as fully occupied. Representation of informationally-complete models [19] for layout space usually demands huge data storage, which may influence the search efficiency. Therefore, simplification technique is employed when solving routing problems using optimization algorithms. Traditional technique used in simplifying the equipment models is AABB, which reduces the complexity of model structures largely [20]. However, the over-simplified models waste much extra space, which may result in failing to find optimal route. A separation-simplification-recombination technique is adopted in this paper to simplify the equipment model. The model is firstly separated into several solo parts, each part is then simplified using AABB, and these simplified parts are finally recombined together as the simplified model. Fig.1 illustrates an example for simplifying a ship equipment using the abovementioned technique. Apparently, it can not only reduce the data storage, but improve the accuracy in representing equipment models. Fig. 1 An example for simplification of equipment model. Based on the space division technique, a pipe route is defined as one rectilinear path connecting starting point and goal point and containing a serious of adjacent grids. Coding of a path is represented by a sequence consisting of the coordinates of involved adjacent grids. Actually, the starting/goal point is the inlet/outlet of equipment, but it cannot be reached as the inlet/outlet which is involved in the simplified model. To solve this problem, the actual inlet/outlet is extended to the adjacent grids outside the simplified model along its axes. To eliminate the direction constrain of inlet/outlet, the connection point is then extended by a minimum distance for bending. 2.2 Spatial potential energy Using the concept of spatial potential energy [8], the degree of access to the wall or other pipelines and keep away from the special equipment is quantitatively calculated. Each grid is given a potential energy value according to the distance from wall, equipment and pipe models. It is assumed that route paths passing through cells with lower potential value cost less. Rules to define the potential energy value are as follows: (1) A lower value is given to the cells closer to the wall or generated pipe, to make pipes go along the wall and share pipe supports with other pipelines;

3 36 Computer Aided Drafting, Design and Manufacturing (CADDM), Vol.26, No.2, Jun (2) A higher value is given to the cells closer to special equipment, such as the one that releases of much heat, to make pipes keep away from that equipment. Value E is given to the cells occupied by the obstacles, while the cells in feasible space are given a default value E. And each cell in the layout space can be valued and distributed to a potential value E i based on the rules mentioned above. Fig. 2 shows three examples for distribution of space potential energy. Fig. 2 Distribution of space potential energy. 2.3 Mathematical modeling Based on workspace modeling and distribution of space potential energy, objective functions of the optimization problem are determined by combing the evaluation criteria. Obj f Obj f Obj f ( 1) Lp ( 2) Bp ( 3) Ep (1) (2) (3) where L p represents the length of a pipe route, namely the number of grids involved in pipeline, B p donates the number of bends, E p represents the sum of potential value in the grids of a pipe route. 3 Proposed pipe routing method Using the optimization algorithm only cannot guarantee the optimal solution directly applied in practice. However, it enables a designer to find the optimal path in a constrained environment by providing several near-optimal references. In this study, MA and NSGA-II are combined together to find the near-optimal route paths. The improved MA is used in optimization procedure to generate favorable chromosomes so as to improve the performance of optimization. Then fuzzy set theory is adopted to select the optimal solution among Pareto solutions, which builds a basis for verifying the performance of the proposed approach. Flow chart of the proposed approach is shown in Fig. 3. Fig. 3 Flow chat of the proposed pipe routing methodology. 3.1 Improved maze algorithm Overview of maze algorithm Maze algorithm is a classical algorithm for route searching, mainly including two processes: extended search and retracing. Extended search is the expansion process starting from one grid to the adjacent grid, and the searched grid is specified by its tag value marked in the process. The process will be continued until the goal grid is reached, in which the rules of expansion and marking tag value are as follows. (1) Initial grid is marked as 1 ;

4 Sui Haiteng et al., Pipe-assembly approach for ships using modified NSGA-II algorithm 37 (2) Only six neighbors of the current grid could be marked, whose tag value is that of current grid plus 1; (3) If the adjacent grid has been marked, the smaller tag value is selected. Retracing process is an anti-search from goal point towards the girds with smaller tag values, which is complete when starting point is finally found. Coordinates of grids in the pipe route construct a chromosome with no repeated nodes Auxiliary point Route paths generated by using MA can cover the space S constructed by taking starting point and goal point as diagonal coordinates, but the space outer S cannot be searched. Although the optimal path usually exists in the space S when considering only the factors of length and bends, a pipe found outer S could be a better solution while taking other criteria into consideration, such as placing closer to the wall, share of pipe supports, etc. Therefore, the concept of auxiliary point is introduced to improve maze search ability, and space S is also enlarged into S' by expanding along axes. Fig.4 shows an example for introduction of auxiliary point. As the anti-search can only reach grids with smaller tag values, the feasible route paths mainly lie in the top right area of search space, as the light red path shown in Fig.4, while the lower left area cannot be covered. To enlarge the covered area, an auxiliary point chosen randomly inside S' is added. Pipe route path 1 is generated by taking original starting point and auxiliary point as connection points, while path 2 is generated by taking auxiliary point and original goal point as connection points. The route path lying in the lower left area can be obtained by connecting these two pipes together. Obviously, the generated path is longer than the original path, but it has the priority to run together and share pipe supports with a pipe path: Generated path, which may reduce the total cost on the contrary. Fig. 4 Introduction of auxiliary point. The introduction of auxiliary point can not only improve the search range of MA, but also guarantee a variety of chromosomes in initial population. It is proved that using priority vector alone would generate chromosomes with same tendencies in terms of direction [8], which is similar in maze searching, so that the route paths converged by optimization operations are likely to be local optimization. The introduction of auxiliary point deals with the problem effectively as its random selection strategy, and the overall workspace can be fully covered. 3.2 Definition of fixed-length chromosome A fixed-length coding method is presented in this section based on the improved MA. In a limit workspace, there exists a longest pipe among all the pipes. Theoretically, auxiliary point can cover the overall workspace as its randomness, and the maximum value max 1 can be determined easily. Actually, as only few initial individuals are needed, the generated pipes can hardly cover the overall workspace. Therefore, the pipes generated by taking eight vertexes of workspace as auxiliary point respectively are measured to find another maximum value max 2. Compare max 1 with max 2, the larger one is selected as the fixed length of chromosome. If the length of chromosome generated in genetic operation is smaller than or equal to the fixed length, the chromosome will be reserved and its empty nodes will be supplemented by (0,0,0). Otherwise, if a certain chromosome is overlong, which means that too many repeated nodes exists in the chromosome, it will be removed directly.

5 38 Computer Aided Drafting, Design and Manufacturing (CADDM), Vol.26, No.2, Jun Genetic operator Fuzzy set theory used in selection The elitism non-dominated sorting and crowed tournament selection that were proposed in NSGA-II [18] are used to obtain the evolution population. And fuzzy set theory [21] is employed to get the best compromise individual in each iteration to verify effectiveness of the optimization algorithm. For an optimal chromosome sets with N obj objectives and M individuals, a membership function i denotes the ith objective function of an individual in Pareto optimal solutions, which is defined as: where 1, max Fi Fi i max min Fi Fi max i 0,, min i Fi min max i i i max i Fi F F F F F (4) F and F min denote the maximum and i minimum values of the ith fitness function, respectively. For each non-dominated solution k, the normalized membership function k is expressed as: k N obj i1 M N obj j1 i1 k i j i (5) In Eq. (5), larger k indicates better compromise individual. Therefore, a priority list of non-dominated solutions is obtained by descending sort of k, and the optimal individual is easily determined Crossover A new crossover operator with fixed-length chromosome is presented in this section, whose principle is shown in Fig.5. Given two randomly selected parent chromosomes Parent 1 and Parent 2, and a crossover node is randomly selected in each chromosome. The coordinates of nodes in this case are assumed as (1,7,4) and (5,7,4). Using improved MA, an assistant path Mid-path 1 is generated by taking these two nodes as connection points. Then the recombination between parent chromosomes and assistant path is performed as shown in Fig.5, and two child chromosomes Child 1 and Child 2 are obtained. Apparently, the length of Child 1 is no larger than the fixed length, and the empty nodes are supplemented by (0,0,0). While Child 2 should be removed as it is overlong. It is noted that the MA adopted is not the same as the one used in generating initial populations. The main difference lies in retracing strategy: a priority direction vector is firstly determined according to the coordinates of two connection points. A certain priority direction is then randomly selected as the original anti-search direction that will not change before cells on obstacles are reached. But if the feasible route cannot be found since the search direction has changed several times, the crossover nodes will be reselected and the procedures described above will also be repeated. Fig. 5 An example for crossover operation with fixed-length chromosomes Mutation A mutation strategy with fixed-length chromosome is given in this section. Fig.6 shows an example for mutation operation, in which the parent chromosome Parent 3 is already given. Two mutation nodes, nodes (1,7,4) and (5,8,6), are randomly selected from Parent 3 in this case. Using the same anti-searching technique adopted in crossover operation, an assistant path Mid-path 2 is generated by taking the selected nodes as connection points. Then the genes between two crossover nodes are replaced by Mid-path 2, and a child chromosome Child 3 is finally generated. Apparently, Child 3 is reserved as its length is within the fixed length, and its empty nodes are supplemented by (0,0,0).

6 Sui Haiteng et al., Pipe-assembly approach for ships using modified NSGA-II algorithm 39 Fig. 6 An example for mutation operation with fixed-length chromosomes. 4 Simulation and result The proposed pipe routing methodology is verified in this section by conducting simulation and comparative tests, and parameters used in the optimization algorithm are obtained by experience and a large number of experiments. 4.1 Layout space modeling A workspace model whose diagonal coordinates are (1,1,1) and (50,50,50) is constructed as shown in Fig.7, where seven obstacles that represent simplified equipment and two connection points with specific directions are given. Fig.7 represent the inlet/outlet direction. Considering the minimum distance for bending, the two coordinates are extended as (3,2,2) and (48,49,49). Using the modified NSGA-II whose parameters are shown in Table 2, and Pareto optimal solutions of pipe routes are obtained as shown in Fig.8(a), which verifies the validity and global search ability of the proposed approach. In each iteration, individuals in Pareto optimal solutions are all non-dominated, and there exists no absolute optimal individual. But a best compromise individual of each iteration is needed to show convergence of the algorithm. Combing with elitism strategy, fuzzy set theory is employed to achieve the goal, and the convergence curve is shown in Fig.8(b). The figure shows that individuals converge to the appropriate route after 58th generation, which verifies the convergence of the algorithm. Compared with the results of the similar simulation test conducted by Fan et al. [17], the proposed approach could acquire more feasible solutions that will provide the designer with more reliable references, while the convergence generations are all about 60. Table 1 Diagonal coordinates of obstacles. Fig. 7 3D solid model of simulation based layout space. The workspace is firstly divided into cuboid grids, and the diagonal coordinates of the cuboid involved in the obstacles are determined as shown in Table 1. A default potential value E is distributed to grids of feasible workspace, while grids near the wall are given specific value E i determined by distance. In this case, value E is set as 10 and value range of E i is (2,8). 4.2 Simulation and discussion The coordinates of connection points are assumed as (1,2,2) and (50,49,49), and the red arrows shown in ID number of obstacles Diagonal coordinates 1 (1,4,1;20,16,11) 2 (1,1,34;11,20,50) (4,30,1;16,44,16) 3 (4,35,15;13,44,25) (30,1,1;45,10,6) 4 (30,11,1;45,25,19) 5 (26,38,1;38,46,25) (35,1,35;20,15,50) 6 (30,3,40;35,11,45) 7 (40,20,30;50,49,40) Table 2 Parameters of NSGA-II. Parameter NSGA-II Population size 20 Number of generation 100 Crossover probability 0.8 Mutation probability 0.05

7 40 Computer Aided Drafting, Design and Manufacturing (CADDM), Vol.26, No.2, Jun (a) (b) Fig. 8 Routing result of simulation test. (a) Routing result; (b) Convergence curve. To further verify the performance of the proposed algorithm, two more simulation tests are designed with other conditions unchanged: TEST-1: Based on the given workspace, obstacle number 6 is assumed as an equipment releasing too much heat; TEST-2: Based on TEST-1, an straight pipe: Generated pipe is added to the workspace, where the pipe code is [(24,1,1);(24,1,2);(24,1,3)...(24,1,49)]. Simulation results are shown in Fig. 9 and Fig. 10, respectively. Compared with pipes shown in Fig. 8(a), the results shown in Fig. 9(a) and Fig. 10(a) demonstrate that the proposed algorithm enables pipes to keep away from the specific equipment and run together with other pipelines, which proves the validity of the proposed algorithm powerfully. (a) (b) Fig. 9 Routing result of TEST-1. (a) Routing result; (b) Convergence curve. (a) (b) Fig. 10 Routing result of TEST-2. (a) Routing result; (b) Convergence curve.

8 Sui Haiteng et al., Pipe-assembly approach for ships using modified NSGA-II algorithm 41 The 3D parametric CAD model of pipe can be automatically created in SolidWorks using the Application Programming Interface (API). SolidWorks abstracts the the optimization result data from MATLAB in which the optimization procedure is performed, and creates solid models of pipe and equiment for 3D visualition as shown in Fig.11. Fig. 11 3D visualization model of routing result obtained in TEST-2. 5 Conclusion This paper presents a modified NSGA-II approach for effective design of ship pipe system while meeting certain objectives in an environment scattered with obstacles, which mainly includes model simplification, objective function, improvement of MA, coding, genetic operator and simulation test. The equipment models are firstly simplified using separation-simplification-recombination technique, and workspace is divided into grids, then the mathematical model of the optimization problem is built. The pipe routing methodology is proposed by combing MA with NSGA-II. In the approach, the concept of auxiliary point is introduced to improve the search ability of MA; based on the improved MA, a fix-length coding mechanism and corresponding genetic operator are proposed; fuzzy set theory is also adopted to select the best compromise solution from Pareto solutions. The simulation results demonstrate the effectiveness and efficiency of the proposed method. Future work will be carried out to research the optimization approach for the problem of multi-pipe system and to develop an intelligent piping design system of ships. References [1] Dijkstra E W. A note on two problems in connexion with graphs numerische mathematic [J]. Numerische Mathematik, 1959, 1(1): [2] Hart P E, Nilsson N J, Raphael B. A formal basis for the heuristic determination of minimum cost paths [J]. Systems Science & Cybernetics IEEE Transactions on, 1968, 4(2): [3] Lee C Y. An algorithm for path connections and its application [J]. IRE Transactions on Electronic Computer, 1961, EC-10(3): [4] Hightower D W. A solution to the line routing problem on a continuous plane [C]// Design Automation Conference. New York: ACM press, 1969: [5] Rourke P W. Development of a three-dimensional pipe routing algorithm [D]. Bethlehem: Thesis Lehigh University, [6] Fan J, Mei M A, Yang X G. Research on automatic laying out for external pipeline of aeroengine [J]. Machine Design, 2003, 20(7): [7] Wang C, Liu Q. Projection and geodesic-based pipe routing algorithm [J]. IEEE Transactions on Automation Science & Engineering, 2011, 8(3): [8] Ito T. Genetic algorithm approach to piping route path planning [J]. Journal of Intelligent Manufacturing, 1999, 10(1): [9] Sandurkar S, Chen W. GAPRUS genetic algorithms based pipe routing using tessellated objects [J]. Computers in Industry, 2000, 38(3): [10] Ito T. Route planning wizard: basic concept and its implementation [C]// International Conference on Industrial and Engineering Applications of Artificial Intelligence and Expert Systems: Developments in Applied Artificial Intelligence. Berlin: Springer-Verlag, 2002: [11] Wang H, Zhao C, Yan W, et al. Three-dimensional multi-pipe route optimization based on genetic algorithms [C]// Knowledge Enterprise: Intelligent Strategies in Product Design, Manufacturing, and Management, Proceedings of PROLAMAT 2006, IFIP TC5 International Conference. Boston: Springer, 2006:

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