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Ship block construction space is an important bottleneck resource in the process of shipbuilding, so the production scheduling optimization is a key technology to improve the efficiency of shipbuilding. With respect to ship block construction space scheduling problem, a hybrid heuristic algorithm is proposed in this paper. Firstly, Bottom-Left-Fill (BLF) process is introduced. Next, an initial solution is obtained by guiding the sorting process with corners. Then on the basis of the initial solution, the simulated annealing arithmetic (SA) is used to improve the solution by offering a possibility to accept worse neighbor solutions in order to escape from local optimum. Finally, the simulation experiments are conducted to verify the effectiveness of the algorithm.

Space is the key resource in ship block construction process. How to minimize the makespan of the project under space resource and precedence constraints is a complicated scheduling problem. As for this problem, two related problems are involved: resource constrained project scheduling problem (RCPSP) and bin packing problem.

RCPSP can be described as a problem which should be scheduled under the limits of technology and other constraints to meet the objective of a project [

The bin packing problem is putting more boxes into a limited bin in order to minimum the height. This problem can be classified into two categories, two-dimensional and three-dimensional problems. For the former one, researchers tend to solve bin packing problems by heuristic methods. They are Bottom-Left (BL) algorithm [

This paper is organized as follows. After introduction, Section

Ship block construction space scheduling problem can be described as a project which includes

During the project, every activity is under precedence constraints, and we propose that

In the model, formula (

In this paper, we apply BLF to get the initial solution. BLF, presented by Chazelle [

In BLF, it is important to find corners to place the activity. Firstly, we will try the lowest and leftmost point (lowest point first); if this placement can match the activity, then place it in the position and update corners; otherwise, try next point until a corner is found. The corners can be represented as

Corners.

In this paper, we represent the activities as cubes whose length, width, and height are limited. For these cubes, the length and width denote the length and width of the place they need, and the height represents the time they need. Meanwhile, we regard the unrestricted height as timeline

We embed the place (the platform in ship block construction space) in a three-dimensional coordinate system, put the bottom-left-rear point on the origin (0, 0, 0), and put length, width, and height on

Set 5 sets:

Move the activities which can be scheduled from

If an activity in

Select the higher

Proposed by Steinbrunn et al. [

Chan et al. [

Initialization: the initial and final temperature is

Select a new solution

Calculate the incremental

If

We use a set of activities to verify our algorithm. The parameters of these activities are shown in Table

The parameters of activities.

Activity | Duration | Length, width | Place |
---|---|---|---|

1 | 0 | 0, 0 | 1 |

2 | 8 | 2, 1 | 1 |

3 | 4 | 2, 2 | 1 |

4 | 6 | 2, 1 | 2 |

5 | 3 | 2, 2 | 2 |

6 | 8 | 2, 1 | 2 |

7 | 5 | 2, 1 | 2 |

8 | 9 | 3, 1 | 1 |

9 | 2 | 3, 2 | 1 |

10 | 3 | 1, 1 | 1 |

11 | 7 | 2, 2 | 1 |

12 | 2 | 2, 2 | 2 |

13 | 7 | 3, 1 | 1 |

14 | 9 | 2, 1 | 2 |

15 | 4 | 3, 2 | 1 |

16 | 6 | 2, 1 | 2 |

17 | 3 | 2, 1 | 1 |

18 | 0 | 0, 0 | 2 |

The network diagram of activities.

We get the initial solution as follows.

At first,

Next, activity 5 will be scheduled on place 2. We put the bottom-left-rear point of activity 5 at the origin of place 2. Then new corners

Detect the corners of

At time 0, activity 4 will be put in order. Detect

At time 0, space is still available, but it is not enough for any remaining activity. So next time point is

According to this train of thought, we get an initial solution

The status of place 1 while activity 3 is put.

The status of place 2 while activity 5 is put.

The status of place 1 while activity 2 is put.

The status of place 2 while activity 4 is put.

The status of place 1 while activity 15 is put.

The status while all activities are put.

Place 1

Place 2

In SA, the order of activities has great influence on the duration of the project. So by searching the neighborhood and increasing the size of neighborhood dynamically, we manage to use SA and improve the initial solution.

We show a step of improvement as follows.

If

If

Select

Compare

Generate a number

After this improvement, the optimal solution reduces from 25 to 24. So it is proved that the method is effective.

In this paper, we combine BLF and SA to solve ship block construction space scheduling problem. During the procedure of scheduling activities, we guide the sorting process with corners. Then, the sorting of initial solution can be changed by SA. However, how to improve the searching efficiency will be the future research, especially when the number of blocks is very large.

The authors declare that they have no financial and personal relationships with other people or organizations that can inappropriately influence their work, and they also declare that there is no conflict of interests regarding the publication of this paper.

This work was supported in part by NSFC under Grant no. 61202345, NSFS under Grant no. ZR2012FM006, and SDPW under Grant no. IMZQWH010016. The authors thank the 2 reviewers and handling editor for their meticulous reading of the paper and constructive comments which greatly improved the paper.