An improved heuristic algorithm for 2D rectangle packing area minimization problems with central rectangles

https://doi.org/10.1016/j.engappai.2017.08.012Get rights and content

Highlights

  • A special RPAMP named CR-RPAMP is described.

  • An improved heuristic algorithm called IHACR is proposed for solving CR-RPAMP.

  • IHACR includes three new strategies for packing rectangles.

  • Experiment results prove the advantage of IHACR compared with HACR.

  • IHACR is used to solve the layout problem of semi-submersible production platform.

Abstract

Be different with traditional 2D rectangle packing area minimization problem (RPAMP), a specific RPAMP named CR-RPAMP includes one or more central rectangles, which must be located in the center of the final layout. Besides, for CR-RPAMP, the length and width of the final layout are not fixed, but can be changed within a reasonable length–width ratio scope. In this paper, based on HACR (heuristic algorithm for CR-RPAMP), an improved heuristic algorithm called IHACR is proposed in order to improve the performance of HACR, especially to decrease the computational complexity of HACR. Compared with HACR, IHACR includes three more rectangle placement mechanisms, which are strategy of combining rectangles, strategy of leaving biggest inner space and strategy of eliminating unnecessary comparisons. Then, a set of schematic descriptions is used to describe the difference between HACR and IHACR. Simulation results based on 34 benchmark instances show that computing time obtained by IHACR is much shorter than that obtained by HACR when solving CR-RPAMP, and the filling rate of final layout obtained by IHACR is bigger than that obtained by HACR while the number of rectangles of the instance is appropriate large. It means that IHACR is more effective and efficient than HACR. Finally, HACR and IHACR are used for solving the equipment layout problem of oil–gas–water treating system of semi-submersible production platform, and the results show that the performance of IHACR is better than that of HACR. The wonderful final layout obtained by IHACR satisfies the requirements of semi-submersible production platform and verifies the practicality and high-efficiency of IHACR.

Introduction

As a kind of NP-hard problem, rectangle packing area minimization problem (RPAMP) He et al., (2015), Wei et al. (2011) has been lucubrated and applied in many industry fields, such as textile, apparel, automobile, aerospace and chemical industries He and Wu (2013), Lodi et al. (2002), Alvarez-Valdes et al. (2009), Funke et al. (2016). According to the features of the packing zone, RPAMP can be divided into strip packing problem (SPP) and rectangle packing problem (RPP) Beasley (2004), Yu et al. (2016), Jansen and Prädel (2016), Martello and Vigo (1998). According to whether the orientation of candidate rectangles is fixed and whether the guillotine cutting is required, RPAMP can be categorized into four types: OF, RF, OG, RG, in which “O” means orientation of candidate rectangles is fixed, “R” means candidate rectangles can be rotated by 90°, “F” means guillotine cutting is not required and “G” means guillotine cutting is required.

Due to the high complexity of NP-hard problems, it is not possible to solve RPAMP using exact approaches which are insufficient for large scale problems Martello and Vigo (1998), Horta et al. (2016). Actually, exact approaches attempt to find the rectangle packing positions by solving a set of linear equations which constructs some optimization certain criteria Wang and Chen (2015), Wannakrairot and Phumchusri (2016). In order to find the satisfactory solutions in an efficient manner, many researchers combined heuristic algorithms and meta-heuristic algorithms to solve RPAMP (Wu et al., 2016a). Usually, heuristic algorithms are used to confirm the packing rules and strategies, and meta-heuristic algorithms are used to find the best packing order for rectangles Bennell et al. (2013), Wei and Chen (2009). As the earliest and most famous heuristic algorithm, bottom-left (BL) proposed by Brenda et al. (1980), and its derivative methods are the headstone of existing heuristic algorithms Bernard (1983), Liu and Teng (1999). After that, floor-ceiling (FC) method and touching perimeter (TP) method were reported by Lodi et al. (2002). Wu et al. (2002) introduced the less flexibility first principle to determine the packing rule. Zhang et al. (2006) proposed a new heuristic recursive algorithm which arranged the rectangles by a recursive structure. Huang et al., (2007) presented an effective heuristic algorithm with two important concepts, called the corner-occupying action and caving degree. Cui et al. (2008) combined a new heuristic recursive algorithm with branch-and-bound techniques to solve RPAMP. Martello and Monaci (2015) provided an ILP (Integer Linear Programming) model. Wei and Chen (2009) first presented a least wasted first strategy. By dynamically determining the dimensions of the enveloping rectangles, a dynamic reduction algorithm that transforms an original problem instance to a series of RPP instances was presented by He et al., (2015). Based on the existing solution methods, Bortfeldt (2013) presented a generic procedure for the RPAMP. As for meta-heuristic algorithms, the most used ones are simulated annealing (SA), genetic algorithm (GA), artificial neural network, tabu search (TS), greedy randomized adaptive search procedure (GRASP), iterative maximal area (IMA), and so on Burke et al. (2004), Harwig et al. (2006), Wu et al. (2015). The integration of heuristic algorithms and meta-heuristic algorithms makes the solving process for RPAMP to be more effective and efficient (Saraswat et al., 2015). However, all the above-mentioned algorithms are used to solve the traditional RPAMP.

Recently, Wu et al. (2016b) introduced a new kind of RPAMP called CR-RPAMP, which is a kind of “RF” problem. Be different with traditional RPAMP, CR-RPAMP can be described as follows. (1) There are one or more specific rectangles which are called central rectangles among the packing rectangles π1,π2,,πn. In the final layout, the central rectangles must be located in or near the center of the layout. (2) The length–width ratio of the final layout is not fixed, but should be within a reasonable scope. At the same time, the length and width of the final layout can be changes legitimately.

In fact, CR-RPAMP is very common in modern industry. The layout problem of drilling equipment in semi-submersible drilling platforms is a typical CR-RPAMP (Wu et al., 2016b). As the most important equipment, the drilling floor must be located in the center of the main deck of the semi-submersible drilling platform, and other facilities and modules should be placed around the drilling floor. Besides, the layout problem of factory production facilities can be abstracted as CR-RPAMP. For the sake of economy, one or more facilities which are the key production modules should be placed at the center of the workshop, and other facilities which can be seen as the supply modules should be placed around the center. Moreover, there are many other RPAMPs that are similar to the layout problem of drilling equipment and layout problem of specific factory production facilities. Apparently, it is significant to research the heuristic algorithm for solving the CR-RPAMP.

Based on the first heuristic algorithm for CR-RPAMP (HACR) proposed by Wu et al. (2016b), this paper introduces an improved heuristic algorithm, named IHACR. As mentioned in literature (Wu et al., 2016b), the major disadvantage of HACR is time-consuming. In IHACR, several rectangle placement mechanisms are imported to overcome the shortcomings of HACR. The mechanisms include strategy of combining rectangles, strategy of leaving biggest inner space and strategy of eliminating unnecessary comparisons. Compared with HACR, IHACR has faster solving speed. The computation time obtained by IHACR is less than those obtained by HACR when solving the 34 benchmark instances, and the filling rate of final layout obtained by IHACR is bigger than that obtained by HACR when the number of rectangles of the instance is appropriate large.

The subsequent sections are organized as follows: Section 2 gives the descriptions of CR-RPAMP and HACR. Section 3 indicates the analysis of computational complexity for HACR. In Section 4, an improved heuristic algorithm IHACR is stated, and a set of schematic descriptions is used to distinguish HACR and IHACR. Section 5 presents simulation experiment and results. In Section 6, IHACR is used to solve equipment layout problem of oil–gas–water treating system of semi-submersible production platform. Conclusions are summarized in Section 7.

Section snippets

Descriptions of CR-RPAMP and HACR

To describe CR-RPAMP, Wu et al. (2016b) introduced some definitions and packing rules which are shown in Table 1. Detailed descriptions of these definitions and rules can be found in literature (Wu et al., 2016b).

Based on these definitions, the CR-RPAMP can be formulated as follows:

Given a set of n rectangulars with each item πi(1in) having width wi and height hi. Evidently, the area of each rectangle can be expressed as areaπi=wihi. A two-dimensional Cartesian Reference Frame is established

Analysis of computational complexity for HACR

In this section, analysis of computational complexity for HACR is given in order to explain why HACR is time-consuming. Actually, the computing time of heuristic algorithm is decided by the complexity of solving procedure, especially by the quantity of the repetitive computations (Zhao and Shen, 2016). In HACR, the repetitive computations take the dominant time of the whole solving procedure. Analyzing the computational process of HACR, the repetitive computations are as follows:

IHACR: an improved heuristic algorithm for CR-RPAMP

In this section, several strategies are imported into HACR in order to decrease the computational complexity when solving CR-RPAMP.

Experiments and results

For the purpose of finding the most suitable value of ENCR and verifying the advantage of IHACR, experiments are implemented in this section. Two groups of benchmark instances are introduced. The first group is made up of 13 instances in which the numbers of candidate rectangles are from 10 to 3125, and the detailed information can be found in literature (Burke et al., 2004). The second group includes 21 instances whose numbers of candidate rectangles are from 16 to 297, and the detailed

Application in solving equipment layout problem of semi-submersible production platform

Semi-submersible production platform is one of the key equipment for the deep-water oil–gas exploiting. The major functions of semi-submersible production platform are oil–gas treatment, storage and transmission. Therefore, the most characteristic system of semi-submersible production platform is oil–gas–water treating system which is used for separating oil from mixture of the well flow, by way of the degassing, dehydration, desalting, impurities removal and crude oil stabilization, achieving

Conclusion

In this paper, an improved heuristic algorithm for 2D rectangle packing area minimization problems with central rectangles (CR-RPAMP) was proposed, and named IHACR. HACR is the first appropriative heuristic algorithm for CR-RPAMP. However, analysis of computational complexity of HACR shows that there are many repetitive computations during the computational process. This is the reason why the computing time is so long when solving large-scale CR-RPAMP by HACR. In IHACR, three rectangle

Acknowledgments

This work was supported in part by the major project foundation of high technology scientific research for ship of China Ministry of Industry and Information Technology “Research of gordian technique of deep-water semi-submersible platforms”, and the national scientific research plan project for high technological vessel of china “Brand project for self-elevating drilling platform”.

References (32)

Cited by (15)

  • Improved dynamic adaptive ant colony optimization algorithm to solve pipe routing design

    2022, Knowledge-Based Systems
    Citation Excerpt :

    Thus, some novel mechanisms are proposed to advance the effectiveness and efficiency of ACO in this study. In the oil and gas industry, the semi-submersible production platform is one of the vital equipment for deep-water oil and gas exploitation [39]. The primary functions of semi-submersible production platform are oil and gas treatment, storage and transmission [40].

  • NHACR: A novel heuristic approach for 2D rectangle packing area minimization problem with central rectangle

    2021, Engineering Applications of Artificial Intelligence
    Citation Excerpt :

    However, the major shortcoming of HACR is time-consuming. Therefore, an improved HACR (IHACR) and a modified HACR (MHACR) were proposed later (Wu et al., 2017, 2018b). The main contributions of IHACR and MHACR are the developments of new strategies for monitoring the aspect ratio, filling inner space, combining rectangles, and decreasing computational complexity.

View all citing articles on Scopus
View full text