Discrete Optimization
An agent-based approach to the two-dimensional guillotine bin packing problem

https://doi.org/10.1016/j.ejor.2007.10.020Get rights and content

Abstract

The two-dimensional guillotine bin packing problem consists of packing, without overlap, small rectangular items into the smallest number of large rectangular bins where items are obtained via guillotine cuts. This problem is solved using a new guillotine bottom left (GBL) constructive heuristic and its agent-based (A–B) implementation. GBL, which is sequential, successively packs items into a bin and creates a new bin every time it can no longer fit any unpacked item into the current one. A–B, which is pseudo-parallel, uses the simplest system of artificial life. This system consists of active agents dynamically interacting in real time to jointly fill the bins while each agent is driven by its own parameters, decision process, and fitness assessment. A–B is particularly fast and yields near-optimal solutions. Its modularity makes it easily adaptable to knapsack related problems.

Introduction

Several real life industrial applications require the allocation of a set of small objects (or items or pieces) into the smallest number of large objects (or stock units or bins). For instance, in glass, plastic or metal industries, rectangular components have to be cut from the fewest large sheets of material. Similarly, when filling the pages of a newspaper, the editor has to arrange articles and advertisements, presented as rectangular areas, on pages of fixed dimensions. In the shipping and transportation industries, packages of identical heights have to be positioned in the minimum number of rectangular bins.

This frequently occurring cutting and packing problem is known in the literature as the two-dimensional single bin-size bin packing (2BP) problem [27]. Formally, it consists of allocating, without overlap, a set N = {1,  , n} of small rectangular items into the minimum number of identical large rectangular bins. An item i  N is characterized by its length li and width wi, and a bin is characterized by its length L and width W.

Variants of the 2BP arise when additional constraints are imposed. Generally, the constraints reflect conditions of the arrangement of items into the bins and/or technological constraints [25] such as limiting the number of cuts, requiring guillotine cuts because of the use of automated cutting machines [15], imposing an orientation on the items, etc.

This paper focuses on two-variants of the guillotine 2BP.

  • The first variant imposes a fixed orientation of the items; i.e., a piece of length l and width w is different from a piece of length w and width l (when l  w). It is denoted 2BP∣O∣G, where O refers to oriented items and G to the requirement that the cuts be guillotine. It is of particular interest in rubber cutting [24], [15], newspaper paging, and in apparel industries in particular when items are cut from decorated or patterned material.

  • The second variant allows the items to be rotated by 90°; i.e., a piece of length l and width w may be positioned either with its length or with its side along the length L of the bin. It is denoted 2BP∣R∣G where R refers to the feasibility of rotation of the items by 90° and G to the requirement that the cuts be guillotine. It is applicable in the steel and shipping industries [26], [15].

The 2BP∣★∣G (where ★ = O or R) is a difficult combinatorial optimization problem. In fact, the 2BP problem is NP-hard in the strong sense since it is an extension of the well-known one-dimensional bin packing (1BP) problem. Therefore, solving 2BP∣★∣G using exact approaches is generally impossible; especially when instances involve a large number of items. Indeed, the search for any exact solution is time consuming without any guarantee of a sufficiently good convergence to optimum [25]. One alternative is to search for an approximate local extremum by sorting the items and packing them in the bins according to some prescribed order [25]. This paper uses this line of approach to solve the 2BP∣★∣G problem. Specifically, it proposes two approximate algorithms: a new guillotine bottom left (GBL) constructive heuristic, and its agent-based (A–B) implementation.

GBL positions items in the bottom left (or south-west) most available position of the bin, and decides whether the first cut of the strip containing the packed item is horizontal or vertical. It subsequently updates the two unused areas of the bin, and continues the filling process until no unpacked item fits in the bin. It creates a new bin every time the current one is saturated. It repeats this process until it packs all items.

A–B, which is based upon distributed artificial intelligence, has a more global vision of the packing process since it allows the items and the bins to communicate. It uses a dynamic pseudo-parallel agent-based system of active interacting agents; each with its own characteristics (parameters), fitness, and decision process (or rules or logic of work) whose basis is GBL. It uses the principle of self-organization of the agents to construct a feasible solution p. It then intensifies the search in the immediate neighborhood of p exploiting the information it learned. It maintains the “good” parts of p and rebuilds the “weaker” ones. Finally, it diversifies the search by using different orderings of the items.

A–B mimics agent-based economic systems [21]. Its agents’ interaction forms a model of the environment and assesses the potential of each activity. Agents use this information to simulate variants of their activity, and subsequently readjust their activity, their decision process, and their characteristics. Results of agents’ activity are disseminated in the system, and fed back to the agents [21]. The process is reiterated until the system reaches steady state. Subsequently, the process launches group breaking actions in a greedy manner in search for a local optimum.

This paper is organized as follows. Section 2 surveys methods used to solve the 2BP problem. Section 3 details the constructive approach GBL and the approximate algorithm A–B. Section 4 evokes issues related to A–B’s implementation, and assesses GBL’s and A–B’s performance in terms of run time and solution quality. Finally, Section 5 summarizes the paper highlighting A–B’s advantages, and discussing its application to other combinatorial optimization problems.

Section snippets

Literature review

The 2BP problem has been widely studied [20], [18]. Lodi et al. [16], [17] provide a survey of recent advances on the subject; summarizing available bounds, and common exact and approximate approaches. One of the novel exact methods is proposed by Dell Amico et al. [7] who introduce a lower bound to the 2BP problem with rotation, and embed it into an exact branch-and-bound.

Most approximate algorithms [2], [4], [13], [14], [15] use a two-phase approach [2]. The first phase packs all items into a

An agent-based approach

A–B, an approach based upon distributed artificial intelligence, solves the 2BP∣★∣G problem using a new sequential packing heuristic GBL as part of its decision process. Section 3.1 details GBL, whereas Section 3.2 describes A–B.

Computational results

This section discusses some implementation issues of A–B, and studies GBL’s and A–B’s performance. Both GBL and A–B are coded in Borland Delphi 6.0 with Object Pascal and run on an Athlon XP 2800.

Conclusion

This paper solves the guillotine two-dimensional bin packing problem, which is NP-hard, using a constructive sequential guillotine bottom left heuristic and a pseudo-parallel agent-based system. GBL successively packs items into a bin and creates a new bin every time it can no longer fit any unpacked item into the current one. It competes well with existing constructive approaches: It is very fast and yields good solutions. A–B, which is based on the concept of distributed artificial

Acknowledgements

The authors thank Aida F. Valeeva, Denis V. Popov, Elita A. Mukhacheva, and three anonymous referees for their invaluable comments.

References (27)

  • F.G. Vasko et al.

    A practical solution to a fuzzy two-dimensional cutting stock problem

    Fuzzy Sets and Systems

    (1989)
  • G. Wäscher et al.

    An improved typology of cutting and packing problems

    European Journal of Operational Research

    (2007)
  • J.O. Berkey et al.

    Two-dimensional finite bin packing algorithms

    Journal of the Operational Research Society

    (1987)
  • Cited by (41)

    • Just-in-time two-dimensional bin packing

      2021, Omega (United Kingdom)
    • A quasi-human strategy-based improved basin filling algorithm for the orthogonal rectangular packing problem with mass balance constraint

      2017, Computers and Industrial Engineering
      Citation Excerpt :

      The solution to this problem is multi-optional, for example, the problem can be two-dimensional or three-dimensional, the container can be circular, rectangular, or polygonal, the way of placing objects can be orthogonal or arbitrary, and the constraints can include mass balance, inertia, and stability. In recent years, scholars have done research on the orthogonal rectangular packing problem (ORPP) and put forward algorithms to solve it, including exact algorithms such as the graph theory method (Macedo, Alves, & Carvalho, 2010), the dynamic planning algorithm (Birgin, Lobato, & Morabito, 2012), the branch-and-bound method (Clautiaux, Jouglet, Carlier, & Moukrim, 2008; Cui, Yang, Cheng, & Song, 2008; Hifi, 1998; Lesh, Marks, McMahon, & Mitzenmacher, 2004), heuristic methods such as the method based on the action space (He & Huang, 2014; He, Huang, & Jin, 2012; He, Jin, & Huang, 2013; Wang & Yin, 2015) and other heuristics (Chan, Alvelos, Silva, & Valério, 2013; Charalambous & Fleszar, 2011; Liu, Zhang, Yao, Xue, & Guan, 2016; Martello & Monaci, 2015; Polyakovsky & M’Hallah, 2009), meta-heuristic algorithms such as evolutionary approaches (Khebbache, Prins, & Yalaoui, 2008; Li & Dong, 2011), the energy landscape paving method (Liu, Huang et al., 2016), the Wang-Landau sampling method (Liu, Hao et al., 2016), hybrid approaches (Bortfeldt, 2013). This article studies the ORPP with mass balance constraint (Feng, Wang, Wang, & Teng, 1999), which differs from the general rectangular packing problem.

    • A priority heuristic for the guillotine rectangular packing problem

      2016, Information Processing Letters
      Citation Excerpt :

      To verify the performance of PH for the single bin packing problem, 21 instances from Hopper and Turton [8] were used. Polyakovsky and M'Hallah [16] reported computational results of GBL and A-B for OG and RG problems. Table 3 reports the filling rates of three algorithms.

    View all citing articles on Scopus
    View full text