A best-fit branch-and-bound heuristic for the unconstrained two-dimensional non-guillotine cutting problem

Highlights

• We study the the unconstrained two-dimensional non-guillotine cutting problem.

• We propose a best-fit branch-and-bound algorithm for this problem.

• We suggest a greedy heuristic is to generate a complete solution.

• We give optimality certificates for 50 out of 95 instances.

Abstract

This paper studies the unconstrained two-dimensional non-guillotine cutting problem, in which the objective is to select and pack a set of rectangles into a sheet with fixed sizes and maximize the profit of the selected rectangles. The orientation of the rectangle is fixed and the available number of each rectangle is not limited. We present a staircase based best-fit branch-and-bound method to solve this problem. To speed up the process, a greedy heuristic is used to generate a complete solution from a partial one and an iterative application of the branch-and-bound method is introduced. The results on the well-known instances show that the proposed approach gives optimality certificates for 50 out of 95 instances and improves the results for 29 instances.

Introduction

In the two-dimensional non-guillotine cutting problem (2DNGCP), we are given a set R of n rectangles with dimensions w_i × h_i, profit p_i and available number u_i, $i=1,\dots ,n$. The task is to select and orthogonally cut the selected set of rectangles from a large rectangular sheet with width W and height H to maximize the profit of the selected rectangles. In addition, the size of the sheet and the input rectangles are assumed both integers, and the orientation of each rectangle is fixed. The guillotine cut (requires the cut to be parallel to the sides of the sheet and divide it into two completely separated sheets) constraint is not considered here.

Under the improved typology of cutting and packing problems by Wäscher, Haußner, and Schumann (2007), the 2DNGCP is two-dimensional rectangular single large object packing problem (SLOPP). Following the classification of Fayard, Hifi, and Zissimopoulos (1998), four versions of 2DNGCP can be distinguished based on the properties of the available number of each rectangle and its associated profit:

• Unconstrained unweighted (UU-2DNGCP): the available number u_i for each rectangle i is unlimited and the profit p_i is equal to its area w_i × h_i.

• Unconstrained weighted (UW-2DNGCP): the available number u_i for each rectangle i is unlimited and the profit p_i is independent of its area.

• Constrained unweighted (CU-2DNGCP): the available number u_i for each rectangle i is limited (i.e., at least some u_i are strictly lower than ⌊W/w_i⌋ × ⌊H/h_i⌋, the maximum number of piece i that could be packed into the sheet) and the profit p_i is equal to its area.

• Constrained weighted (CW-2DNGCP): for each rectangle i, the available number u_i is limited and the profit p_i is independent of its area.

In this paper, the unconstrained case U*-2DNGCP including UU-2DNGCP and UW-2DNGCP is studied. For other variants of cutting problems, the reader is referred to Bak et al. (2010), Clautiaux, Jouglet, and Moukrim (2013), Cui (2014), Côté, Gendreau, and Potvin (2014) and Wei and Lim (2015).

Most of the papers in literature consider the constrained case of 2DNGCP. The CU-2DNGCP is often called the two-dimensional rectangular packing (2DRP) problem. Beasley (1985b) proposed a tree search procedure to exactly solve the C*-2DNGCP. Lesh, Marks, McMahon, and Mitzenmacher (2004) introduced exhaustive approaches for the perfected case (no waste space) of the CU-2DNGCP. Besides, the two-level approach was used by Caprara and Monaci (2004), Baldacci and Boschetti (2007) and Fekete, Schepers, and van der Veen (2007). The two-level approach first selects the set of rectangles, then tries to pack the selected rectangles into the sheet. Recently, Belov and Rohling (2013) proposed an interval-graph algorithm for the check of the orthogonal-packing feasibility. Besides the exact approach, several metaheuristics have also been tried, including a population heuristic by Beasley (2004), a tabu search heuristic by Alvarez-Valdés, Parreño, and Tamarit (2007), a genetic algorithm by Bortfeldt and Winter (2009) and a hybrid simulated annealing approach by Leung, Zhang, Zhou, and Wu (2012). In addition, Huang, Chen, and Xu (2007) proposed a new heuristic algorithm based on the concept of caving degree of a corner-occupying action for the 2DRP. Wei, Zhang, and Chen (2009) suggested a least wasted first (LWF) strategy and first solved the 21 instances proposed by Hopper and Turton (2001) to optimality. Egeblad and Pisinger (2009) presented a new heuristic for the two-dimensional knapsack problem based on the sequence pair representation. Recently, Wei, Oon, Zhu, and Lim (2011) introduced a skyline based heuristic and got the best results for the 2DRP on the benchmark test data.

The unconstrained case of 2DNGCP is less studied in the literature. As mentioned in Birgin, Lobato, and Morabito (2012), the unconstrained 2DNGCP is often encountered in cutting stock applications with large-scale production and weakly heterogeneous items. More specifically,there are relatively few piece types, but many copies per type. Birgin et al. (2012) proposed a recursive partitioning approach (RPA) for the U*2DNGCP. The RPA combines both the recursive five-block Heuristic (Morabito & Morales, 1998) and the L-approach (Birgin, Morabito, Nishihara, 2005, Lins, Lins, Morabito, 2003). The RPA found an optimal solution of a large number of instances with moderate sizes, and a counterexample that the RPA fails to find known optimal solutions were not found. So, Birgin et al. (2012) raised a question that whether the L-approach is an optimal method to solve the U*2DNGCP. Recently, Alves de Queiroz, Keidi Miyazawa, and Wakabayashi (2014) answered this question by finding an instance for which the L-approach fail to find an optimal solution.

Two common structures skyline and staircase are used to represent the packing pattern while solving the 2D cutting problem. The skyline representation of packing pattern was first proposed by Burke, Kendall, and Whitwell (2004) and used by many researchers later (Leung, Zhang, Zhou, Wu, 2012, Wei, Oon, Zhu, Lim, 2011, Wei, Qin, Cheang, Xu, 2016).

The staircase representation was first proposed by Martello, Pisinger, and Vigo (2000) to solve the three-dimensional bin packing problem. A good property of the staircase representation is that there always exists an optimal solution achievable only by extending staircase representations (Martello, Monaci, Vigo, 2003, Martello, Pisinger, Vigo, 2000). So many exact algorithms based on the staircase were proposed to solve different packing problems. Martello et al. (2003) and Kenmochi, Imamichi, Nonobe, Yagiura, and Nagamochi (2009) introduced exact algorithms based on the staircase for the two-dimensional strip packing (2DSP) problem. Ahn, Park, and Yoon (2015) suggested a best-fit branch-and-bound method based on the staircase structure for the pallet-loading problem. In addition, this structure was also used in the heuristic as the representation of the solution. Wei et al. (2009) and Zhang, Wei, Leung, and Chen (2013) introduced this structure in their heuristic for the 2DSP. Recently, Tao and Wang (2015) used the staircase based algorithm to solve the sub-problem of the three-dimensional loading capacitated vehicle routing problem.

In this paper, an exact approach is proposed to solve the U*-2DNGCP. The staircase is used to represent the partial packing state. A best-fit branch-and-bound method is proposed to solve this problem. In order to improve the lower bound quickly, a greedy heuristic is introduced to get a complete state from a partial state. To limit the number of states generated, we use a parameter to control the number of points in the staircase and enlarger this parameter iteratively. Experiments on the well-known instances show that our approach gives optimality certificates for half of the instances (48 out of 95 instances) and improves the results for most of the remaining instances (29 out of 47 instances).

There are three main contributions of this paper. Firstly, we propose a best-fit branch-and-bound method to solve the unconstrained two-dimensional non-guillotine cutting problem. Second, a greedy heuristic is introduced to generate a complete solution from a partial state and an iterative application of the branch-and-bound method is suggested to speed up the search process. Third, our approach gives optimality certificates for half of the instances and improves the results for many other instances.

The remainder of this paper is organized as follows. Section 2 gives a detailed description of the addressed branch-and-bound method. Computational results of the proposed approach are described in Section 3. The paper is concluded in Section 4.