Heuristic algorithm for RPAMP with central rectangle and its application to solve oil–gas treatment facility layout problem

https://doi.org/10.1016/j.engappai.2018.04.008Get rights and content

Highlights

  • A special 2D rectangle packing problem named CR-RPAMP is described.

  • A modified heuristic algorithm called MHACR is proposed for solving CR-RPAMP.

  • MHACR includes three new strategies.

  • Experimental results based on 34 benchmark instances show the advantages of MHACR.

  • MHACR is used to solve the facility layout problem of oil–gas treatment factory.

Abstract

In this study, we address a specific 2D rectangle packing area minimization problem with central rectangle (CR-RPAMP). The most distinguishing feature of CR-RPAMP is that there exists at least one central rectangle among the candidate rectangles. The proposed MHACR (modified heuristic algorithm for CR-RPAMP) includes three new strategies which are strategy of monitoring the aspect ratio, strategy of decreasing computational complexity and strategy of filling the marginal inner space. The computational complexity analyses show that MHACR has a lower computational complexity than the existing algorithms IHACR (improved heuristic algorithm for CR-RPAMP) and HACR (heuristic algorithm for CR-RPAMP). Then, a series of experiments based on 34 benchmark instances are carried out, and the experimental results show that MHACR can produce better solutions as compared with IHACR and HACR, especially under a narrow limited scope of aspect ratio. Finally, MHACR is used to solve the facility layout problem of oil–gas treatment factory, and the final layout generated by MHACR is more suitable than that obtained by IHACR. Therefore, we can conclude that MHACR is a remarkable heuristic algorithm to solve CR-RPAMP.

Introduction

The two-dimensional rectangle packing area minimization problem (RPAMP) is one of the most difficult problems in combinational optimization Fleszar, 2016, Alvarez-Valdes et al., 2009, and has been applied in many industries such as metal, wood, glass, paper, automobile, aerospace, and textiles Wei et al., 2011, He and Wu, 2013. As shown in the current literatures, there are many classification methods for RPAMP: (1) it can be divided into the strip packing problem (SPP) and the rectangle packing problem (RPP) based on the feature of the packing zone; (2) according to whether the candidate rectangles are in the oriented case, it can be classed as “O” (the items are oriented rectangles) and “R” (the items can be rotated by 90°); (3) in terms of guillotine cutting, RPAMP can be categorized into “G” and “F”, “G” means that guillotine cutting is required, “F” means that guillotine cutting is not required Zhao et al., 2016, Alvarez-Valdes et al., 2008.

Since RPAMP is a kind of NP-hard problem, it is very difficult to solve it by using exact approaches (Horta et al., 2016). Therefore, lots of heuristic and meta-heuristic methods were presented in order to obtain a good solution in an efficient way Beasley, 2004, Yu et al., 2016, Jansen and Prädel, 2016. The bottom-left (BL) fill method proposed by Brenda et al. (1980) and Bernard (1983) is one of the earliest and most famous heuristic algorithm. In BL, rectangles are placed into the container one by one, and every item is placed at the bottom-most and left-most location in the current layout (Liu and Teng, 1999). Besides, in order to find a satisfactory solution for RPAMP, Lodi et al. (2002) presented floor-ceiling (FC) method and touching perimeter (TP) method. Valenzuela and Wang employed genetic algorithm when solving the rectangle packing problem (Valenzuela and Wang, 2001). Moreover, the simulated annealing approach was used to iteratively improve the solutions, and to achieve good results on small size problem Lai and Chan, 1997, Faina, 1999. For the purpose of finding a minimum height for two-dimensional SPP, a heuristic algorithm with a recursive structure was proposed by Zhang et al. (2006). To improve the performance of heuristic algorithm when solving big scale RPAMP, a heuristic algorithm was recommended based on the corner-occupying action and caving degree (Huang and Chen, 2007). In order to find a satisfactory solution for RPAMP, Wei et al. (2009) introduced the least wasted first (LWF) strategy. Edmund et al. (2001) proposed a packing method based on squeaky wheel optimization.

In last several years, many approaches have been presented to solve RPAMP. Ji et al. (2017) proposed an iterative merging packing algorithm. However, this algorithm just can address the soft rectangle packing problem, which is a variant of RPAMP. Wei et al. introduced an improved best-fit heuristic (ISH) to solve the 2D strip packing problem (Wei et al., 2017). Funke et al. (2016) introduced a new algorithm designed to solve floorplanning problem optimally. To address the strip packing problem, a hybrid metaheuristic that combines an improved heuristic algorithm with a variable neighborhood search was proposed by Zhang et al. (2016). Besides, the integer linear programming (ILP) model was provided by Martello and Monaci (2015). Dynamic reduction algorithm is another method presented by He and Ji (2015), this algorithm can transform the original problem to many RPP instances by dynamically determining the dimensions of the enveloping rectangles. Wang and Chen (2015) introduced a heuristic algorithm with a policy: maximize the residual space during packing. Bortfeldt (2013) presented a generic approach based on two algorithms, one for the 2D Knapsack Problem (KP), and the other for the 2D SPP. Based on the action space, He et al. (2013) proposed a deterministic heuristic algorithm to solve rotatable 2D strip packing problem. Bennell et al. (2013) utilized genetic algorithm for searching the solution space. Verstichel et al. (2013) proposed an improved BF heuristic named three-way best-fit heuristic. Leung et al. (2011) introduced a novel scoring rule to enhance the BF (Wei et al., 2017).

With different application requirements from industrial processes, the constraints and objectives are different when solving RPAMP Wannakrairot and Phumchusri, 2016, Zhao and Shen, 2016. Recently, a specific two-dimensional rectangle packing area minimization problem with central rectangle named CR-RPAMP, was introduced. In CR-RPAMP, there exists at least one specific item (central rectangles) which must be placed near the center of the final layout, and the length–width ratio of the final layout is variable rather than fixed, but should be within a reasonable scope (Wu et al., 2017). Actually, CR-RPAMP widely occurs in industry, it is very significant to do deeply research on how to solve CR-RPAMP. Due to the particularity of CR-RPAMP, a specific heuristic algorithm named HACR was proposed (Wu et al., 2016). In HACR, a series of definitions and rules were introduced to form a basic framework on solving CR-RPAMP. Firstly, in order to make sure that the central rectangle is near the center of the final layout, it is advisable to constrain the betweenness centrality of central rectangle. Secondly, it is feasible to constrain the aspect ratio of the final enveloping rectangle to ensure the length–width ratio of the final layout within the limited scope. Besides, based on the priorities of candidate rectangles and strategy of padding inner space, the filling rate of the final layout can be in a high level. However, the main disadvantage of HACR is time-consuming. Thus, an improved HACR (IHACR) was presented to overcome the shortcomings of HACR (Wu et al., 2017). The experimental results show that the filling rate obtained by IHACR is higher than HACR, and the computing time of IHACR is much shorter. Although the new strategies included in IHACR can decrease the computational complexity, the computing time of IHACR is too long to solve large scale instance. Therefore, further improvements should be designed to solve CR-RPAMP.

The main focus of this study is to propose a preferable heuristic algorithm for CR-RPAMP. Based on the analyses of IHACR and HACR, some issues are illuminated based on specific instances. Actually, IHACR and HACR are powerless if the limited scope of aspect ratio of final layout is quite narrow. Therefore, novel strategy of monitoring the aspect ratio is presented so that it is noticed once the length–width ratio is not within the limited scope and then necessary operations will be carried out. In addition, another mechanism named strategy of decreasing computational complexity is introduced to effectively reduce the computational complexity of algorithm. Besides, for the purpose of increasing the filling rate of final layout, strategy of filling the marginal inner space is introduced. After integrating all the three strategies, a modified heuristic algorithm for CR-RPAMP called MHACR is presented. Then, the comparisons between MHACR and IHACR are performed based on a specific instance, and computational complexity analyses of MHACR are given to take an insight into the proposed algorithm. Then, we evaluate the proposed algorithm MHACR on a set of 34 benchmark instances, the experimental results show that MHACR has better performance compared with IHACR and HACR, especially under a narrow limited scope of aspect ratio. Finally, MHACR is employed to solve a real-word layout problem, the facility layout problem of oil–gas treatment factory.

The following sections are organized as shown below. Section 2 introduces the descriptions of CR-RPAMP. In Section 3, three strategies are described in detail and MHACR is stated, and then schematic descriptions showing the differences between MHACR and IHACR are introduced. The computational complexity analyses of MHACR are also shown in Section 3. The experimental results and comparisons are shown in Section 4. In Section 5, MHACR is employed to solve facility layout problem of oil–gas treatment factory. At last, the final section ends with a conclusion.

Section snippets

Descriptions of CR-RPAMP

Based on features of CR-RPAMP, it can be described as follows:

Given n rectangles with each item πi(1in) having width wi and height hi, the rectangle’s area is calculated as areaπi=wihi. Then, based on the layout plane, a two-dimensional Cartesian Reference Frame is established. For every rectangle πi, (xi1,yi1) and (xi2,yi2) are the bottom-left vertex coordinate and upper-right vertex coordinate (Wu et al., 2016). minArean=HnWns.t.(xi2xi1,yi2yi1)wi,hi,(hi,wi)max(xi1xj2,xj1xi2,yi1yj2,yj

MHACR: A modified heuristic algorithm for CR-RPAMP

In this section, a modified heuristic algorithm named MHACR including three new strategies (strategy of monitoring the aspect ratio, strategy of decreasing computational complexity and strategy of filling the marginal inner space) is illuminated. Actually, one of the main advantages of MHACR is that it can monitor the aspect ratio of transitional layout, the strategy of monitoring the aspect ratio can effectively improve the length–width ratio of the final layout especially there exists a

Experimental results and comparisons

In order to test the performance of the proposed algorithm MHACR, a series of experiments based on 34 benchmark instances are carried out in this section. The first 13 instances with the number of candidate rectangles from 10 to 3125, can be found in literature (Burke et al., 2004). The other 21 instances with the number of candidate rectangles from 16 to 297, can be found in literature (Hopper and Turton, 2001). To ensure these instances to be CR-RPAMP, requisite changes for some instances

Application to solve the facility layout problem of oil–gas treatment factory

The facility layout problem is widely applied in industry. A good layout not only can save space, but also can promote the production efficiency. In this section, MHACR is used to solve a real-world facility layout problem. In order to verify the advantage of MHACR, IHACR is used for comparison.

The real-world instance comes from the oil & gas industry. As we know, after collecting the oil–gas–water mixture from the oil well, the next key work is to separate oil from the mixture. Actually, the

Conclusions

In this study, a modified heuristic algorithm named MHACR is presented in order to solve a specific 2D rectangle packing area minimization problems with central rectangles (CR-RPAMP). In detail, three strategies, which are strategy of monitoring the aspect ratio, strategy of decreasing computational complexity, strategy of filling the marginal inner space, are introduced and integrated into MHACR. Since IHACR and HACR are powerless if the limited scope of aspect ratio is narrow, a new mechanism

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 (40)

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    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.

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