Discrete Dynamics in Nature and Society

Volume 2015, Article ID 841637, 6 pages

http://dx.doi.org/10.1155/2015/841637

## A Hybrid Heuristic Algorithm for Ship Block Construction Space Scheduling Problem

^{1}School of Economics and Management, Harbin Institute of Technology, Weihai 264209, China^{2}Department of Mathematics, Harbin Institute of Technology, Weihai 264209, China^{3}School of Software, Sun Yat-sen University, Guangzhou 510275, China

Received 27 December 2014; Accepted 17 January 2015

Academic Editor: Shuenn-Ren Cheng

Copyright © 2015 Shicheng Hu 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.

#### Abstract

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.

#### 1. Introduction

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 [1]. Generally, the goal is to get the shortest project makespan under the available resources and precedence constraints. The methods can be classified into two categories: exact methods and heuristic methods. Hartmann [2] puts forward that RCPSP belongs to NP-hard problem because it is always used to extend machine scheduling problem [3, 4]. So with the augment of problem scale, the computational complexity will increase rapidly. Many researchers have used exact algorithm to solve RCPSP [5, 6]; however, most of them proposed that exact algorithm is not feasible in reality. Priority rules proposed by Kelley [7] indicated that RCPSP can be solved by heuristic algorithms. Liu and Wang [8] tried to reduce the project makespan by heuristic algorithms and achieved good results. Bhaskar et al. [9] utilized parallel methods and priority rules to solve RCPSP with fuzzy activity times. Lee et al. [10] proposed a ship block construction space scheduling problem and described this problem theoretically. Koh et al. [11] solved the scheduling problem in shipbuilding company by heuristic algorithm.

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 [12], Bottom-Left-Fill (BLF) algorithm [13]. In addition, Belov et al. [14] considered adapting one-dimensional problem for solving two-dimensional problems. Chan et al. [15] tried to solve two-dimensional problems by heuristics with stochastic neighborhood structures. For three-dimensional problems, the most popular 3BF [16], proposed by Silvano Martello in 2000, figure out the problem of how to choose the most suitable cubes. Alvarez-Valdes et al. [17] used a GRASP/Path relinking algorithm to solve multiple bin-size bin packing problems. Liao and Hsu [18] found new lower bounds to improve the efficiency of three-dimensional problems.

This paper is organized as follows. After introduction, Section 2 presents a mathematical model. Section 3 gives the hybrid heuristic algorithm for this problem. In Section 4, we conduct simulation experiments to verify the effectiveness of the algorithm. Section 5 proposes general conclusions.

#### 2. Ship Block Construction Space Scheduling Problem

Ship block construction space scheduling problem can be described as a project which includes activities ; place is required to process activities. activity can be defined as , where and represent the length and width of place which activity needs. The duration of activity is . and are the start and finish time of activity . During the project, there are places which can be defined as .

During the project, every activity is under precedence constraints, and we propose that is the predecessors of activity . So can not be started if any one of its predecessors in has not been finished. We assume that the start time of the whole project is 0. For the convenience of modeling, we also introduce two dummy nodes: activity 0 and . They do not need time and space. 0 is the predecessor of all the activities in the project; meanwhile, is the successor of all the activities. So is the makespan of the project. In addition, we regard an activity as a cube which is represented as . All the activities can be rotated in horizontal with 90 degree, and and 0, indicating that activity is rotated and not, respectively. The model of this problem is shown in what follows:

In the model, formula (1) is the objective of the problem, (2) proposes the precedence constraints, and (3) means that two cubes cannot overlap. Formulae (4)–(7) denote that each cube should be completed within available place. The cube can be rotated horizontally and 0 and 1 represent if the cube is rotated, as formula (8) has shown.

#### 3. A Hybrid Heuristic Algorithm

##### 3.1. Initial Solution Method

In this paper, we apply BLF to get the initial solution. BLF, presented by Chazelle [13] in 1983, belongs to heuristic algorithm. In this algorithm, the method of placing cubes is determined by corners [19].

###### 3.1.1. Corner

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 , where indicates the location of the corners and represents the available space of the point in -- dimensions. See Figure 1; and denote the length and width of the place and represents the duration of the project.