Special Issue Article
A three-layer parallel computing system for shipbuilding project scheduling optimization
Advances in Mechanical Engineering 2017, Vol. 9(10) 1–17 Ó The Author(s) 2017 DOI: 10.1177/1687814017723297 journals.sagepub.com/home/ade
Duanfeng Han, Boxin Yang, Jinghua Li, Miaomiao Sun, Qinghua Zhou and Jiaxuan Wang
Abstract To solve the large-scale scheduling problem more efficiently within the requirements of the contract in shipyard, a threelayer parallel computing system was proposed. An optimized model for shipbuilding project scheduling problem was constructed under the condition of taking time and resource constraints into account. Moreover, the key techniques of proposed system were elaborated and the main steps were designed. In the first computing layer, the problem was decomposed into small parts in heterogeneous systems, reducing the problem scale; then, in the second layer, a coevolution strategy for multi-populations was put forward to improve the algorithm robustness; in the third layer, a massive parallel computing method was performed under the Graphic Processing Unit structure. Finally, through two simulation examples, the robustness and outperforming others of the improved algorithm were verified. Keywords Shipbuilding project, scheduling, parallel computing system, Graphic Processing Unit calculation, genetic algorithm
Date received: 31 January 2017; accepted: 3 July 2017 Academic Editor: Peter Nielsen
Introduction As a complex system, a shipbuilding project is characterized by its long production period, a great quantity of activities, and a concrete completion date signed in the contract. More importantly, production for a shipbuilding project is highly order-oriented, reflecting the nature of single-item or small-batch production. Little amount of pipeline operation further makes the production quite a contrast to those of conventional projects. Under this condition, schedule control is of ever greater significance. Even though project evaluation and review technique (PERT) and critical path method (CPM) techniques contribute a lot to optimize the schedule, both of them deal only with the time without any consideration of resource constraints. Then, on the basis of CPM, Lu and Li1 and Kim and de la Garza2 studied on the scheduling problem with limited resources, which promotes the research on resource-constrained
project scheduling problem (RCPSP). Mo¨hring and Schulz3 mentioned that RCPSP is one of the most intractable problems in operations research. In a resource-constrained project, each activity has duration and renewable resource requirements, and the precedence relationships also exist in different activities, which is shown in Figure 1. The sample project consists of 16 activities with the limited capacity of one renewable resource type. Node 1 and node 16 are dummy activities representing the start and end of the project, respectively. In Figure 1(a), it illustrates the
College of Shipbuilding Engineering, Harbin Engineering University, Harbin, China Corresponding author: Jinghua Li, College of Shipbuilding Engineering, Harbin Engineering University, Harbin 150001, China. Email:
[email protected]
Creative Commons CC-BY: This article is distributed under the terms of the Creative Commons Attribution 4.0 License (http://www.creativecommons.org/licenses/by/4.0/) which permits any use, reproduction and distribution of the work without further permission provided the original work is attributed as specified on the SAGE and Open Access pages (https://us.sagepub.com/en-us/nam/ open-access-at-sage).
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Figure 1. Example of project schedule model: (a) project network, (b) typical RCPSP schedule, and (c) activity list representation.
precedence constraints. Besides, Figure 1(b) and (c) gives a feasible solution in which the permutation sequence is given by Figure 1(c) according to the start time of activities. In shipbuilding project, as a result of long period and enormous amount of activities, the dummy activities are not limited to the start and end of the project but they are important nodes like milestones. Because of the large scale of the problem, the method for conventional RCPSP has limited optimization effect on the shipbuilding, which is mainly reflected in the speed of problem solving and quality of the optimal solution. Bearing the above observations in mind, we apply a three-layer parallel computing system to the resourceconstrained shipbuilding project scheduling problem, and the rest of this article is organized as follows. In section ‘‘Literature review,’’ some related works are outlined based on literature. Section ‘‘Shipbuilding project scheduling problem’’ describes the problem and illustrates and sets up a mathematical model. Next, a
three-layer parallel computing system model is built in section ‘‘Proposed 3-layer parallel computing GA system model.’’ In section ‘‘Proposed algorithm,’’ the steps of the proposed algorithm are designed. Section ‘‘Simulation examples’’ provides two simulation examples to verify the computing system and discuss the differences of genetic algorithm (GA), hybrid GA, and the proposed parallel GA (PGA). Finally, in section ‘‘Conclusions and future work,’’ some advantages and limitations of our parallel computing system are summarized, which points out some remaining future work in the light of this.
Literature review RCPSP and the extension of RCPSP have gained widespread attention for the last few years. Kone´ et al.4 proposed the generalization of two existing mixed integer linear programming models for RCPSP considering storage resources that can be produced or consumed by
Han et al. activities. Kopanos et al.5 developed two discrete-time formulations based on the definition of binary variables and two continuous-time formulations based on the concept of overlapping of activities for the RCPSP. Sahli et al.6 presented four mixed integer linear programming models inspired by RCPSP when activities requiring renewable resources are replaced by events consuming or producing non-renewable resources. Christodoulou7 included a comprehensive literature review on RCPSP and discussed two directions of schedule generation schemes. As the author states, RCPSP can be solved by use of either a serial-schedule or a parallel-schedule generation scheme. The serial method is activity-based, while the parallel method is time-based. Almeida et al.8 put forward a heuristic framework to solve the multi-skill RCPSP based on parallel scheduling scheme. Maghsoudlou et al.9 extended RCPSP taking into account multi-skill and multi-mode that integrated with discrete time-costquality trade-off. And a new chromosome structure was developed to solve the problem. In their survey, Kreter et al.10 noted that the break-calendars were necessary to be considered for many operative RCPSP applications and generally different from different renewable resources. Then, they gave the models and solution procedures for the problem. Husbands and Mill11 described a framework of coevolution to solve a highly generalized version of the manufacturing scheduling problem. In their research, a PGA search was used. Nevertheless, the devising parallel implementations provide a more robust search rather than speed up as most previous PGAs work. The author stated that the model presented can be generalized. Jiang et al.,12 Korayem et al.,13 and Yu et al.14 used the co-evolutionary algorithm to solve different engineering problem. For the large-scale project scheduling optimization, robust performance is one aspect of the consideration, and the other is the computation speed. Wang15 designed a PGA computing application and conducted in a single system image cluster. GAs are natural parallel evolution algorithms and can greatly improve the computing efficiency in a good parallel environment. Xu et al.,16 Chandio et al.,17 and Xu et al.18 studied on the parallel computing system for task scheduling problem and analyzed the efficiency. With the rapid development of the hardware, an increasing number of people began to use graphics processing unit (GPU) for computing. Iturriaga et al.19 presented a parallel implementation on central processing unit (CPU)/GPU of two variants of a stochastic local search method to efficiently solve the scheduling problem. Zhao et al.20 provided a parallel immune algorithm in the study of traveling salesman problem (TSP). The algorithm was based on GPU and proved to greatly improve the computing efficiency.
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Shipbuilding project scheduling problem Parallel scheduling model based on MAG Scheduling optimization for shipbuilding projects is a special kind of RCPSP, standing in a great contrast to the regular ones. In regular RCPSP, much attention was paid on activity sequencing and resource allocating, aiming to achieve the duration minimization. However, in terms of shipbuilding project with the feature of single-piece production, different projects are normally in different construction stages so as to avoid the conflict of key resources. Meanwhile, the completion time for each important stage has been determined at the time of signing the contract so as to ensure the effective use of key resources. With extensive studies and practices of lean shipbuilding mode, just-in-time (JIT) model has been introduced into shipbuilding RCPSP for increases in effective resource utilization rate and decreases in idle time of semi-finished products. To lessen the scheduling control complexity, a series of milestones are set for shipbuilding projects alongside their long project timelines. By this way, the overall completion scheduling control problem is broken down and distributed on these milestones. Figure 2 is an illustrative network diagram of a shipbuilding project, with several milestones (nodes 5, 8, and 14) considered. The gray nodes represent dummy activities while dotted boxes indicate activity groups. An activity group ending up with a milestone is also named after the milestone. For short, an activity group with a milestone is here marked by MAG (Milestone Activity Group), such as MAG 5 is the activity group whose last activity node is 5. Relationships between two MAG are serial (e.g. 5 and 8 in Figure 2) or parallel (e.g. 5 and 14 in Figure 2) connections. For the sake of simplified computation, there are four hypothesis: 1. 2.
3.
4.
Each milestone is an end point of a sub-project which is composed of the milestone’s MAG. The level of MAGs is the same, indicating that a MAG only has one milestone in its realm (not include the start activity node). The relation between each pair of MAGs is serial or parallel, constituting the overall workflow. If MAG m is parallel to MAG n, then there is no intersection in their resource utilization sets. In other words, there is no identical resource constraints for these two MAGs.
Classifying activities into multiple groups at this stage lays a foundation for further constructing a parallel structure. Regarding existing means of activity group allocation, Tsai et al.21 conducted a comparison in view of the communication costs between activities. In the
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Figure 2. An example of shipbuilding project: (a) a simplified network diagram of shipbuilding project and (b) MAGs grouping.
light of milestone activities, this article naturally turns the entire large-scale structure into small operable and independent activity groups for computation. In fact, this method is feasible and sensible for the following three reasons. To begin with, to split the complex issue into independent small-scale activity sequences is conducive to reducing an iteration of computation for scheduling. Second, it retains the natural allocation logic of activities, thereby each MAG carrying its own and concrete meaning. Furthermore, the logic helps to avoid unreasonable computation conditions, resulting in a considerable decrease in the computation burden.
Mathematic model On the top of extensive research on RCPSP and JIT, a mathematic model is proposed in consideration of shipbuilding project nature. Take MAG as an example, there are N 2 activities, which are not milestones. The symbol for the ith activity is ai . The dummy a1 and aN represent the start and end of an MAG. To mark a milestone activity, we define the following isMilestonei =
1, ai is milestone activity 0, others
We defined that ai is completed at the time point of ti . Then, the actual completion time of the MAG will be tN = maxfti g(i = 2, 3, . . . , N 1), which means tN is decided by the finish time of the last completed activity. Because the date of each milestone and project completion are specified in contract, we use Tci to represent the specified time if isMilestonei equals 1. Therefore, based on the JIT model, the optimization goal of this article is to obtain a solution and achieve that the completion time of milestones in solution is equal to the completion time of milestones signed in contract. That is to say, we need to ensure that jTcN tN j has a minimum value. In most circumstance, jTcN tN j = 0. Suppose that there are many milestones in the whole project and the number of activities is M. So, the fitness function is min
M X
isMilestonei vi jTci ti j
ð1Þ
i=1
where vi is the weight coefficient and vi varies from stages to stages in a product duration s:t: M X i=1
isMilestonei vi = 1
ð2Þ
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Figure 3. Computing system structure.
ti di tp , ap 2 PSi M X
ð3Þ
rik xit Rk , t 2 1, 2, . . . , Tf , k 2 f1, 2, . . . , rg
i=1
xit 2 f0, 1g,
i = 2, 3, . . . , M 1, t 2 1, 2, . . . , Tf
ð4Þ ð5Þ
Rk 0, k 2 f1, 2, . . . , rg
ð6Þ
Equation (3) means that ap is in the direct pre-active set PSi of ai and di is the duration of ai . Different kinds of manpower, machine, material, or the place can be regarded as resources. Equation (4) indicates that each activity arranged should satisfy r types of resource constrain. Tf is the actual completion time of the project, Rk is the ability of a shipyard to supply the resource k per day, and rik is the demand for resources k per day of ai , k = 1, 2, . . . , r. Equation (5) is a relation indicator between activity and process, where xit = 1 means that ai is in the process at the time point of t. Otherwise, xit = 0.
Proposed three-layer parallel computing GA system model The overall structure of computing system Currently, most of the solutions to solve the project scheduling are accomplished in a single system, or adopting parallel tools like MATLAB toolbox to improve efficiency, which is convenient to realize, but it cannot achieve the best effect of computational performance on some specific issues. Therefore, this article innovatively proposes a parallel computing system with three built-in layers, namely, sub-problem-solving control layer, population behavior control layer, and individual behavior control layer. This structure is illustrated in Figure 3 with much details where the three-layer structure is displayed clearly and each layer are embedded in a specific algorithm to achieve their function. The sub-problem-solving control layer is built on the basis of MAG referred before. The next two layers are designed to further achieve optimized solutions for each sub-problem (MAG) and use a method of two-layer PGA. This system is implemented in the MATLAB operating environment, as illustrated in Figure 4. A variety of
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Figure 4. Structure diagram of system realization.
Figure 5. Schematic diagram of distributed computing.
mathematical methods are provided in class library of MATLAB, which allows the user to pay more attention to the design of computing systems and algorithms. Moreover, C++ and CUDA are also applied to improve the system structure and implement key functions of GPU computing. The first layer and the second layer are achieved through MATLAB, in which the first layer adopts the C++ to obtain the various site computing status and then updates their own status. The data interaction is achieved using the network to access the shared database. The third layer uses the GPU to calculate; the data interaction of which with
the second layer is realized through the copy of memory between host and device.
Design of the first computing layer and the third computing layer Some scholars tried to design a parallel computing system to solve the scheduling problem, such as heterogeneous computing in distributed computing sites,18,21,22 which assigns the tasks required to be calculated to the nodes that need to be calculated, as shown in Figure 5. The way that solves the subtask scheduling problem of
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Figure 6. Schematic diagram of CPU-GPU calculation.
task in different calculating nodes is more suitable to deal with the needs of dynamic planning and handle offsite scheduling problem. As the MAG grouping proposed in this article is based on the milestone nodes in the contract, department is taken as units to form different MAGs. The design of second layer structure is based on this idea. It consists of a novel grouping strategy of parallel computing, which is shown in section ‘‘Parallel scheduling model based on MAG’’ and particularly designed for rapidly solving large-scale problems. To put it brief, the group strategy decomposes the target problem into sub-problems and then addresses them in the heterogeneous computing system (HCS). In this regard, the problem scale is reduced and the problemsolving efficiency is improved. Such a strategy is of great significance for shipbuilding industry wherein the number of tasks amounts to ten thousands for a single project and even hundreds of thousands for a cruise ship project. The third layer is an individual behavior control layer, which targets at GA computation on the unitary population. In fact, GA has parallel attributes in its nature. However, in most cases, computational systems adopt the serial design model or realize the parallel computing
to a limited extent. With an increase in the problem’s scale, operations on individual crossover, mutation, decoding, and fitness calculation are difficult to carry out, let alone the huge amount of computing resources to consume. As a result, computation time would dramatically rise in case of the serial design model. Yang et al.23 proposed a CPU-GPU-based GA to solve the scheduling problem, in which a large number of similar repetitive work is very consistent with the calculation characteristics of GPU. Wang and Shen24 designed the structure of the GPU calculation, as displayed in Figure 6 where one block handles one task and one thread handles one activity plan of itself. According to the settings of different GPU dimensions, the required computing structure can be designed. In this sense, a large amount of work, such as population initialization, coding, decoding, and genetic operation, is all placed on the third layer, and a GPU-based individual control layer is designed and illustrated in Figure 7.
Design of the second computing layer Unlike the differences described in section ‘‘Design of the first computing layer and the third computing
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Figure 7. Schematic diagram of GPU computing structure.
layer,’’ the design of second-level parallel computing does not directly improve the computational efficiency, but rather improve the convergence rate by increasing the solution quality of the algorithm. Even though efforts (four hypotheses in section ‘‘Shipbuilding project scheduling problem’’) have been done in the milestone tasks’ set to decrease the grouping complexity and reduce their mutual interference, the number of tasks inside the MAG still reaches as much as around one thousand. This is apprehensible since coupling relationships become too complicated to tackle if the granularity of MAG is very small, on which the research does not meant to attach much emphasis. In accordance with the partitioned MAGs, there is a large solution space, easy to incur precocious phenomena. Satisfied with the present optimum, the algorithm often sees a premature convergence away from the global optimized solution. There are many solutions for avoidance of precocious phenomena. Mutation merely does help to escape from the local optimum but at a cost of tremendous computing time. Besides, several well-developed methods have been applied to counteract with precocious phenomena, among which hybrid algorithms are believed to be the best choice.25–28 When combined with strong local searching algorithms, GA as an excellent global optimization algorithm can rapidly converge to the global best. Figure 8 depicts such a combined approach. However, as the large-scale problem at present has burdened much on computing, it is inevitable to spend even longer time for conducting the sequentially hybrid algorithm. To overcome these barriers, this article puts forward a parallel computing layer defined as population
behavior control layer. The population behavior control layer is shown in Figure 9. This layer is based on the co-evolutionary algorithm which is proven to have a more robust performance and whose computation time can be significantly reduced in the parallel computing environment.29 The substance of co-evolutionary lies in exchanging individuals across populations’ border to maintain the population diversity. By design of competing and cooperating policies, it is available for the algorithm to converge to the global optimum as fast as possible without influences on the original population evolution.
Proposed algorithm Algorithm strategy While exact methods are available for obtaining optimal solutions for small-scale problems, metaheuristic algorithms are required as the scale and complexity of the problem increase.30 For the large-scale problems, it is prerequisite to be much stricter with algorithm design, seeking optimization in either quality or efficiency. In alignment with the articulated three-layer parallel computing system, algorithm proposed in this article can be divided into two parts. On one hand, it explains that problems decompose into sub-problem based on MAG, which is shown in Figure 10. As a large-scale problem, project is broken down into a number of subproblems. According to the hypothesis in section ‘‘Parallel scheduling model based on MAG,’’ each
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Figure 8. GA combines with local searching.
Figure 9. Parallel GA with co-evolution strategy.
MAG is capable to get solutions independently and complete the entire project planning according to the networked synchronization information. On the other hand, a parallel computing for solving a single sub-
problem is presented, which is displayed in Figure 11. In Figure 11(a), appropriate number of populations are generated, and various populations that get evolution dependently are working on obtaining a same goal.
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Proposed GA steps Chromosome encoding. The chromosome encoding scheme is based on the sequence of activities composing genes in a single MAG. Generate a code chain randomly with the length of N which satisfies the constraints of the sequence of activities. The order of activities in the code chain represents the scheduling priority and each coding chain corresponds to a certain schedule. Therefore, the scheme has the features of fast generating and easy operating.
Figure 10. The first-layer parallel computing system.
Figure 11(b) is a sub-flowchart of ‘‘Population Evolution’’ in Figure 11(a), which illustrates an evolutionary process for a single population. And ‘‘External
Figure 11. The second- and third-layer parallel computing system.
Population initialization. According to individual number C of the MAG population and the gene length N , make the individual activities meet the requirement of constraint through the population initialization algorithm.
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Figure 12. Crossover operation.
Figure 13. Mutation operation: (a) common mutation method and (b) rebuild mutation method.
1. 2. 1) 2)
3) 3. 4.
Set the cycle counter initial value i = 1. Generate an individual of which the length is N. Set the cycle counter initial value r = 1. And put activity 1 into the selectable set SS. Choose an activity from SS as the rth gene of the individual i in accordance with certain rules from SS. And remove the activity from the SS. If all predecessors of the successor are selected, add them into SS. If r \ N, then r = r + 1 and go to step 2). Put the individual into the initial population. If i\C, then i = i + 1, and turn to step 2.
Reproduction. GA accumulates best genes by ‘‘survival of the fitness’’ and obtains the optimum during evolution. A new generation is generated by two ways: crossover and mutation. Crossover. There are several methods to make a crossover. In this article, a single-point crossover is applied, as shown in Figure 12. Mutation. Mutation results in new genes. There are also a number of methods to mutate, such as insertion, swap, and reversion (see Figure 13(a)). However, some
of them might undermine the precedence constraints between activities. For this regard, this research adopts the rebuild method, namely, regenerating the gene section before the mutation point. The process is shown in Figure 13(b), where the last two sites 2 and 7 are unchanged, but regenerate other parts of the chromosome. Decoding rule. Prior to fitness calculation, it is necessary to decode the chromosome. By decoding, basic information of each activity is acquired, including begin to end time, resource consumption alongside the time line, and so forth. The decoding algorithm consists of i = 1, . . . , J steps, in each step an activity j is selected and scheduled at the earliest feasible start time with respect to the precedence relationship and resource availability. All scheduled activities are in the set PSi . Let P ;Rk (t) = Rk j2PSi xjt rjk be the remaining capacity of resource type k at time instant t and let Fi = ffj jj 2 PSi g be the set of all finish times. The start time of activity j is calculated by determining the earliest precedence feasible start time ESj . The activity list is represented by l = fj1 , . . . , ji , . . . , jJ g.
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Table 1. Configurations of simulation environment.
Initialization i = 1, PSi = [, and Fi = [. If jPSi j\J , then turn to step 3, else end. Calculated by following formula Fi ; Rk ðtÞ; j : = ji ; ESj = MAXi2PSj fFi g; Sj = MIN tjESj t, rj k ;Rk ðtÞ,
Parameter name
Parameter content
1 2 3 4 5 6 7 8
Simulation platform Operation system CPU Memory GPU GPU memory Development environment Simulation engine
PC Windows 7 i7-4790 3.6 GHz 8 GB GTX 1060 1.6 GHz 6 GB VC++ 2015, CUDA 8.0 MATLAB 2016b
Simulation examples
t = t, . . . , t + dj , k = 1, . . . , rg
Fi + 1
Index
fj = Sj + dj : = Fi [ fj ; PSi : = PSi [fj g
i : = i + 1 turn to step 2.
Co-evolutionary strategy among populations To maintain diversity of the population and avoid mature convergence, two strategies inserted into the coevolutionary model are articulated as follows.
In this section, two simulation examples are given to validate the proposed algorithm based on the threelayer parallel computing system. The first simulation example intends to illustrate the feasibility and efficiency of the proposed algorithm. Then, two MAGs in the first simulation group are selected to conduct the second experiment, by which to what degree the proposed two-layer parallel GA (2LPGA) outperforms others can be seen. Configurations of the simulation environment are shown in Table 1.
Simulation example 1 Competitive strategy. Let a constant Ncom denotes the generation count and suppose that all populations experience a cross-population completion at the interval of Ncom generation. Every population contributes its current best individual to make up the global optimum set. Then, the populations are ranked according to the average fitness and by order select optimal solution from global optimum set other than their contributed one. This process is intended to speed up the evolutionary of better populations. Corporative strategy. When a population traps into the local best in consecutive Ncor generations, the average fitness of each population at present is calculated. By comparing these results, the population B with the closest number to A’s average fitness is found. Furthermore, one or several individuals in B would be selected to exchange as long as the individual fitness nearest to the best fitness of A.
In this part, a feasibility graph to represent shipbuilding project scheduling was set up using the authors’ selfdeveloped program in MATLAB. There are 9700 activities grouped in 11 MAGs. Details about each MAG are shown in Table 2. Parameters’ value exerts a great impact on the solution quality and searching speed. In fact, the coefficient weight attached on each MAG varies for different project stages and scheduling needs. In this simulation test, an equal weight was assigned to every MAG, namely, 1/11. Considering the simulation environment, the number of populations for co-evolution was initialized as 6. Table 3 compares computation conditions of the proposed GA driven by GPU and driven by CPU in terms of the problem scale and the population size. The GA driven by GPU is named third-layer PGA which applies the third computing layer. The GA driven by CPU can run directly through .m files in MATLAB. For each condition, these two kinds of algorithms were applied for five times, during which results of both the
Table 2. Information of MAG. Total MAG 1 MAG 2 MAG 3 Activities no. 9700 900 Contract duration 896 339 Key resource 1–11 1–4
700 322 6–8
MAG 4 MAG 5 MAG 6 MAG 7 MAG 8 MAG 9 MAG 10 MAG 11
800 1000 900 1200 301 341 341 402 10 and 11 1 and 2 3 and 4 6
1000 389 7
1300 403 8
200 105 5
600 171 9
1100 389 10 and 11
1300
600
GA
200
The third-layer PGA
GA
The third-layer PGA
GA
The third-layer PGA
Method
Activities nos
86 105 84 90 93 86 115 148 87 107 100 99 159 111 136 121 124 103
54 49 60 50 69 53 99 116 142 80 93 122 200 389 408 179 346 353
56 76 58 37 38 34 419 522 312 266 239 243 1934 1384 1708 1054 1083 903
87 92 123 93 82 110 125 97 94 115 109 113 154 114 93 107 126 129
24 52 14 19 27 10 2 21 79 11 19 33 1 6 60 2 3 11
Fitness
Iteration
T (s)
Iteration
Fitness
Population size = 100
Population size = 50
Table 3. Computation conditions of the usual GA and the third-layer PGA.
139 151 224 94 80 107 1044 748 652 633 600 624 4486 3270 2127 2183 2579 2630
T (s) 131 104 120 122 121 115 154 155 143 139 147 152 139 109 114 121 106 107
Iteration 0 8 1 0 0 0 0 0 4 0 1 0 0 4 0 0 2 2
Fitness
Population size = 200
474 361 431 270 266 253 2650 3018 2426 1553 1633 1694 7393 5864 6302 4377 3871 3907
T (s) 96 148 89 90 113 97 106 151 164 112 133 122 140 151 129 139 127 133
Iteration 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0
Fitness
Population size = 500
865 1513 781 487 621 530 5201 6779 7887 3510 4201 3827 22016 19825 25461 14901 13580 14267
T (s) 105 117 137 103 111 122 112 113 104 100 103 116 136 90 183 122 104 111
Iteration 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 1 0
Fitness
1863 2057 2534 1048 1148 1277 10452 10276 7956 5899 6053 6819 33767 20571 50153 20293 17322 18456
T (s)
Population size = 1000
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Figure 14. Convergence curve.
best and the worst value were trimmed. It is apparently seen that the proposed third-layer PGA performed well in a short computation time. Drawn from previous experience, if a population size is too small, the search space gets much smaller and the solution quality cannot be guaranteed. In contrary, if the size is too big, computation consumes more time, resulting in defects in calculation efficiency. Through this experiment, it was observed that the optimal solution could be achieved with proper population size, and increasing population size to above 200 would undermine the calculation time more than the solution quality. In this sense, the population size was determined to be 200. With such a condition (200 being the population size), the large-scale problem of 600 activities and 1300 activities were calculated. Taking a group of typical data as the source, a convergence diagram was
Figure 15. Simulation result.
Advances in Mechanical Engineering generated (shown in Figure 14). In general, the convergence curves see several transient local convergences. For the 600 activities experimental group, the convergence speed was slow down in two periods. One is from generation 17 to generation 26, the other is generations beyond generation 36. In comparison, the 1300 experimental group experienced a rather slow convergence speed in three periods. They were generation 21 to generation 30, generation 32 to generation 50, and generation 51 to generation 78. However, it encountered a local optimum after generation 118. In lights of these clues, communication operations across populations were found to be necessary when there was a trend toward premature convergence. By means of statistics for the above group data, the authors ultimately selected to apply the Ncom = 10 and Ncor = 3. The simulation result of the optimal MAG scheduling is illustrated as Figure 15 and the fitness value is 0.
Simulation example 2 In order to differentiate the proposed algorithm from hybrid GA (with simulated annealing algorithm), the hybrid GA is abbreviated as HGA and the proposed algorithm is represented by 2LPGA. In this part, GA, HGA, and 2LPGA were applied to solve the sub-problem of MAG 8 and MAG 10 from simulation example 1. The same simulation environment was arranged for these three algorithms: the number of chromosomes in the population was 200, the crossover probability was 0.7, the mutation probability was 0.3, and the max iterative time was 200. In addition, if the fitness value is 0 or the best fitness does not change in 20 loops, the algorithm would be terminated. In order to avoid the random error, each algorithm was run 20 times and calculated the average values.
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Table 4. Comparison of GA, HGA, and 2LPGA. Algorithms GA HGA 2LPGA
MAG 8 MAG 10 MAG 8 MAG 10 MAG 8 MAG 10
Terminate generation
Percent of best solution (%)
Computing time (s)
123 148 90 88 98 72
78.3 81.7 89.2 90.3 95.1 96.8
6377 2479 5099 1816 3807 1413
GA: genetic algorithm; HGA: hybrid genetic algorithm; 2LPGA: two-layer parallel genetic algorithm.
Figure 16. Convergence trend of three algorithms: (a) fitness value comparison and (b) average fitness value comparison.
The terminate generation, percent of best solution p, and computing time of three P algorithms were shown in Table 4. Precisely, p = n= N , and n is the number of individuals whose fitness values are equal to 0, while P N is the number of total individuals in all populations.
At last, the convergence trend of three algorithms was shown in Figure 16. It was observed that the performance of 2LPGA was better than GA and HGA. Seen from Figure 16, the convergence trend of 2LPGA and HGA is nearly equivalent and
16 considerably faster than GA. However, 2LPGA outweighs HGA in the computation time and the quality of solution (reflects in the average fitness). To summarize, the 2LPGA performs better than GA and HGA in terms of searching speed, higher solution quality, and robustness.
Conclusion and future work Since resource-constrained shipbuilding project scheduling problem is an non-deterministic polynomialtime hard (NP-hard) combinatorial optimization problem, an efficient heuristic approach is required. In this article, a three-layer parallel computing system is proposed to solve this problem. In the first layer, a JIT scheduling model is designed with respect to requirements of shipbuilding projects. It helps to decompose the project into small parts, reducing the problem scale. However, these small sub-problems are still in large scale for the common RCPSP. On this account, the remaining two layers are arranged for computing with 2LPGA, targeting at conversions of the original serial computation into parallel counterparts without influences on the computing quality. In the second layer, a competitive strategy and a cooperative strategy for multi-populations are put forward to improve the algorithm robustness. In the last layer, each individual is regarded as an independent thread to operate generic and decoding processes. Together with the adopted GPU framework in this layer, computational efficiency increases remarkably. The quality and convergence rate are shown to be more ideal. Moreover, the proposed algorithm is proven to be much competent to generate a solution in line with expectations written in contract already for the large-scale problem of shipbuilding project scheduling in a virtual shipyard with reasonable amount of time. The proposed algorithm stands out remarkably when the project involves a large number of activities. However, further improvements are still required. First, MAG scale remains quite large, waiting to be decomposed into smaller groups. The exponential relation between coupling influence and the group finegrain would be the first barrier to overcome. Therefore, a corresponding algorithm is needed to undertake this task. Second, since the solution satisfying constraints is not unique, resource balance models are considered to be necessarily designed and added to the research. Besides, application of the parallel system to other NPhard combinatorial optimization problems is also an important research orientation for the authors. Declaration of conflicting interests The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.
Advances in Mechanical Engineering Funding The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This study was supported by the National Natural Science Foundation of China (No. 51679059) and the High Technology for Ship Scientific Research Program of Ministry of Industry and Information Technology of the People’s Republic of China ((2016) No. 543).
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