Mathematical Problems in Engineering

Volume 2015, Article ID 541931, 14 pages

http://dx.doi.org/10.1155/2015/541931

## A Hybrid Demon Algorithm for the Two-Dimensional Orthogonal Strip Packing Problem

^{1}Software School, Xiamen University, Xiamen 361005, China^{2}School of Economics and Business Administration, Chongqing Key Laboratory of Logistics, Chongqing University, Chongqing 400044, China

Received 3 September 2014; Accepted 23 December 2014

Academic Editor: Anders Eriksson

Copyright © 2015 Bili Chen 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

This paper develops a hybrid demon algorithm for a two-dimensional orthogonal strip packing problem. This algorithm combines a placement procedure based on an improved heuristic, local search, and demon algorithm involved in setting one parameter. The hybrid algorithm is tested on a wide set of benchmark instances taken from the literature and compared with other well-known algorithms. The computation results validate the quality of the solutions and the effectiveness of the proposed algorithm.

#### 1. Introduction

Cutting and packing are a very active field of research within operational research, computer science, mathematics, and management science. The two-dimensional cutting and packing problem is widely applied in optimally cutting raw materials such as glass, textile, steel, and paper and transportation and logistics fields. For example, in textile or glass industries, rectangular components have to be cut from large sheets of material. In warehousing, goods have to be placed on shelves. In newspapers paging, articles and advertisements have to be arranged in pages [1]. In order to raise the profitability of the manufacturing or logistics company, the consumption of the raw materials or the cost of transportation should be minimized. Obviously, if an enterprise designs a production scheme using the least waste raw material, it can reduce the manufacturing costs and increase the product’s competitiveness in the market.

The two-dimensional orthogonal strip packing problems (2SP) addressed in this paper consist of packing rectangular pieces into a large rectangular sheet of fixed width and unlimited height in order to minimize the used height, where the rectangular pieces are placed orthogonally without overlap and no rotations are allowed. This problem is of significance both from a theoretical and a practical point of view because it arises in various production processes and has many applications in the glass, steel, paper, and textile industries, and they also have indirect applications in other fields [2] such as layout designing, transportation, and logistics. More extensive survey and classification on cutting and packing problems may refer to Lodi et al. [1] and Wäscher et al. [3].

2SP is known to be NP-hard, some exact algorithms are proposed by Martello et al. [4] and Kenmochi et al. [5], but the size of instances that exact algorithms can handle tends to be small. Therefore, many heuristic algorithms have been suggested in the literature. Baker et al. [6] proposed a bottom-left-fill (BLF) algorithm for 2SP and variants of BLF [7]. Different types of construction heuristics have also been proposed recently, for example, the best-fit heuristic [8], a recursive heuristic [9], a bricklaying heuristic [10], a least waste heuristic [11], and a scoring heuristic [12].

In comparison to the literature on construction heuristic algorithms to packing problems, metaheuristic algorithms are paid more and more attention recently. Bortfeldt [13] proposed a genetic algorithm. The above two algorithms explore placements directly and allow infeasible solutions. However, most metaheuristic algorithms have been developed by incorporating a construction heuristic to improve the quality of solutions. Hopper and Turton [14] implemented a simulated annealing, tabu search, and genetic algorithm by incorporating BLF, respectively. Lesh et al. [15] proposed new heuristic and interactive approaches based on BLF for 2SP. Alvarez-Valdes et al. [16] presented a greedy randomized adaptive search procedure (GRASP) that involves learning some instances to determine the desirable parameter settings for 2SP. Wei et al. [11] presented a least waste algorithm by combining a simulated annealing algorithm for rectangle packing problem. Burke et al. [17] implemented a simulated annealing, tabu search, and genetic algorithm by incorporating BF. Leung et al. [18] proposed a simulated annealing algorithm based on a scoring rule heuristic. Some other algorithms based on different types of strategies have also been proposed, for example, SVC [19] and SWL [20]. Zhang et al. [21] developed a binary search heuristic algorithm based on randomized local search for the rectangular strip packing problem. In particular, an efficient algorithm for strip packing problem can be extended to solve other problems such as bin packing problem [22, 23]. In this paper, we present a hybrid demon algorithm for 2SP which combines a demon algorithm with local search and an improved heuristic. This paper mainly has three contributions: firstly, a new scoring rule is presented; secondly, a least waste strategy is proposed; at last, a demon algorithm with fewer parameters than simulated annealing algorithm is applied to solve 2SP.

The remainder of the paper is organized as follows. Section 2 describes the hybrid algorithm: an improved scoring rule and demon algorithm. Section 3 investigates the effect of the parameter of demon algorithm and the least waste strategy. Section 4 reports the experimental results. Section 5 summarizes the conclusions and proposes future work.

#### 2. Hybrid Algorithm

##### 2.1. Improved Construction Heuristic Algorithm

It has been reported that construction heuristic algorithm is one of the best heuristics while combining with a simulated annealing algorithm [18]. Construction heuristic algorithm is stated as follows: give a rectangle piece sequence, the algorithm finds an unplaced piece with the maximum score for the lowest and the most left space, and then place it. Repeat the above process until all the pieces are placed. Leung et al. [18] proposed the scoring rules as Table 1 which is very important to select one unplaced piece, where is the width of the available space and and are the height of the left and right, respectively, wall of the available space. and denote width and length, respectively, of rectangular piece , denotes the change number of spaces, and is fitness value as Leung et al. [18]. However, they do not explain why case (3) in Table 1 has higher score than case (4). In fact, the two cases have the same ; namely, the number of available spaces does not increase. Due to the fact that the objective of the problem is to minimize the height of the sheet, the result obtained by the scoring rule in Table 1 may be bad, while case (3) has higher score than case (4). For example, given a piece sequence: red, blue, yellow, and green, the packing result is shown in Figure 1, the yellow piece will be first placed according to the original scoring rule after the red piece and blue piece are placed, and then the green piece is placed as Figure 1(a), so the height obtained is 11. However, the height obtained is 9 if the new scoring rule is used that the green piece should be placed earlier than the yellow piece. The above example shows that the new scoring rule may lead to a better result than the original scoring rule. So the new scoring rule for as shown in Table 1 is used in this paper; for , the new scoring rule can be calculated similarly.