New and improved level heuristics for the rectangular strip packing and variable-sized bin packing problems

We consider two types of orthogonal, oriented, rectangular, two-dimensional packing problems. The first is the strip packing problem, for which four new and improved level-packing algorithms are presented. Two of these algorithms guarantee a packing that may be disentangled by guillotine cuts. These are combined with a two-stage heuristic designed to find a solution to the variable-sized bin packing problem, where the aim is to pack all items into bins so as to minimise the packing area. This heuristic packs the levels of a solution to the strip packing problem into large bins and then attempts to repack the items in those bins into smaller bins in order to reduce wasted space. The results of the algorithms are compared to those of seven level-packing heuristics from the literature by means of a large number of strip-packing benchmark instances. It is found that the new algorithms are an improvement over known level-packing heuristics for the strip packing problem. The advancements made by the new and improved algorithms are limited in terms of utilised space when applied to the variable-sized bin packing problem. However, they do provide results faster than many existing algorithms.     

Resource augmentation in two-dimensional packing with orthogonal rotations

We consider the problem of packing two-dimensional rectangles into the minimum number of unit squares, when 90^° rotations are allowed. Our main contribution is a polynomial-time algorithm for packing rectangles into at most OPT bins whose sides have length (1+ε), for any positive ε. Additionally, we show near-optimal packing results for a number of related packing problems.     

Heuristic approaches to large-scale periodic packing of irregular shapes on a rectangular sheet

The nesting problem is a two-dimensional cutting and packing problem where the small pieces to cut have irregular shapes. A particular case of the nesting problem occurs when congruent copies of one single shape have to fill, as much as possible, a limited sheet. Traditional approaches to the nesting problem have difficulty to tackle with high number of pieces to place. Additionally, if the orientation of the given shape is not a constraint, the general nesting approaches are not particularly successful. This problem arises in practice in several industrial contexts such as footwear, metalware and furniture. A possible approach is the periodic placement of the shapes, in a lattice way. In this paper, we propose three heuristic approaches to solve this particular case of nesting problems. Experimental results are compared with published results in literature and additional results obtained from new instances are also provided.     

A family of genetic algorithms for the pallet loading problem

This paper is concerned with a family of genetic algorithms for the pallet loading problem. Our algorithms differ from previous applications of genetic algorithms to two-dimensional packing problems in that our coding contains all the information needed to produce the packing it represents, rather than relying on a packing algorithm to decode each individual solution. We experiment with traditional one-dimensional string representations, and a two-dimensional matrix representation which preserves the notion of closeness between positions on the pallet. Two new crossover operators are introduced for the two-dimensional case. Our definition of solution space includes both feasible and infeasible solutions and we suggest a number of different fitness functions which penalise infeasibility in different ways and a repair operator which allows our populations to maintain feasibility. The results of experiments designed to test the effectiveness of these features are presented.     

A goal-driven approach to the 2D bin packing and variable-sized bin packing problems

In this paper, we examine the two-dimensional variable-sized bin packing problem (2DVSBPP), where the task is to pack all given rectangles into bins of various sizes such that the total area of the used bins is minimized. We partition the search space of the 2DVSBPP into sets and impose an order on the sets, and then use a goal-driven approach to take advantage of the special structure of this partitioned solution space. Since the 2DVSBPP is a generalization of the two-dimensional bin packing problem (2DBPP), our approach can be adapted to the 2DBPP with minimal changes. Computational experiments on the standard benchmark data for both the 2DVSBPP and 2DBPP shows that our approach is more effective than existing approaches in literature.     

An approximation algorithm for square packing

We consider the problem of packing squares into bins which are unit squares, where the goal is to minimize the number of bins used. We present an algorithm for this problem with an absolute worst-case ratio of 2, which is optimal provided P≠NP.     

A heuristic for the circle packing problem with a variety of containers

In this paper we present a heuristic algorithm based on the formulation space search method to solve the circle packing problem. The circle packing problem is the problem of finding the maximum radius of a specified number of identical circles that can be fitted, without overlaps, into a two-dimensional container of fixed size. In this paper we consider a variety of containers: the unit circle, unit square, rectangle, isosceles right-angled triangle and semicircle. The problem is formulated as a nonlinear optimization problem involving both Cartesian and polar coordinate systems.Formulation space search consists of switching between different formulations of the same problem, each formulation potentially having different properties in terms of nonlinear optimization. As a component of our heuristic we solve a nonlinear optimization problem using the solver SNOPT.Our heuristic improves on previous results based on formulation space search presented in the literature. For a number of the containers we improve on the best result previously known. Our heuristic is also a computationally effective approach (when balancing quality of result obtained against computation time required) when compared with other work presented in the literature.     

Dynamic multi-dimensional bin packing

A natural generalization of the classical online bin packing problem is the dynamic bin packing problem introduced by Coffman et al. (1983) [7]. In this formulation, items arrive and depart and the objective is to minimize the maximal number of bins ever used over all times. We study the oriented multi-dimensional dynamic bin packing problem for two dimensions, three dimensions and multiple dimensions. Specifically, we consider dynamic packing of squares and rectangles into unit squares and dynamic packing of three-dimensional cubes and boxes into unit cubes. We also study dynamic d-dimensional hypercube and hyperbox packing. For dynamic d-dimensional box packing we define and analyze the algorithm NFDH for the offline problem and present a dynamic version. This algorithm was studied before for rectangle packing and for square packing and was generalized only for multi-dimensional cubes. We present upper and lower bounds for each of these cases.     

Exact algorithms for the two-dimensional strip packing problem with and without rotations

We propose exact algorithms for the two-dimensional strip packing problem (2SP) with and without 90° rotations. We first focus on the perfect packing problem (PP), which is a special case of 2SP, wherein all given rectangles are required to be packed without wasted space, and design branch-and-bound algorithms introducing several branching rules and bounding operations. A combination of these rules yields an algorithm that is especially efficient for feasible instances of PP. We then propose several methods of applying the PP algorithms to 2SP. Our algorithms succeed in efficiently solving benchmark instances of PP with up to 500 rectangles and those of 2SP with up to 200 rectangles. They are often faster than existing exact algorithms specially tailored for problems without rotations.     

Two-Dimensional Finite Bin-Packing Algorithms

Given a set of rectangular pieces, the two-dimensional bin-packing problem is to place the pieces into an open-ended bin of infinite height such that the height of the resulting packing is minimized. In this paper we analyse the performance of two-dimensional bin-packing heuristics when applied to the special case of packing into finite bins. We develop new bin-packing heuristics by adapting the bottom-left packing method and the next-fit, first-fit and best-fit level-oriented packing heuristics to the finite-bin case. We present our implementation of these algorithms, and compare them to other finite-bin heuristics. Our computational results indicate that the heuristics presented in this paper are suitable for practical use, and behave in a manner which reflects the proven worst-case bounds for the two-dimensional open-ended bin-packing problem.     

On solving multiobjective bin packing problems using evolutionary particle swarm optimization

The bin packing problem is widely found in applications such as loading of tractor trailer trucks, cargo airplanes and ships, where a balanced load provides better fuel efficiency and safer ride. In these applications, there are often conflicting criteria to be satisfied, i.e., to minimize the bins used and to balance the load of each bin, subject to a number of practical constraints. Unlike existing studies that only consider the issue of minimum bins, a multiobjective two-dimensional mathematical model for bin packing problems with multiple constraints (MOBPP-2D) is formulated in this paper. To solve MOBPP-2D problems, a multiobjective evolutionary particle swarm optimization algorithm (MOEPSO) is proposed. Without the need of combining both objectives into a composite scalar weighting function, MOEPSO incorporates the concept of Pareto’s optimality to evolve a family of solutions along the trade-off surface. Extensive numerical investigations are performed on various test instances, and their performances are compared both quantitatively and statistically with other optimization methods to illustrate the effectiveness and efficiency of MOEPSO in solving multiobjective bin packing problems.     

Construction heuristics for two-dimensional irregular shape bin packing with guillotine constraints

The paper examines a new problem in the irregular packing literature that has many applications in industry: two-dimensional irregular (convex) bin packing with guillotine constraints. Due to the cutting process of certain materials, cuts are restricted to extend from one edge of the stock-sheet to another, called guillotine cutting. This constraint is common place in glass cutting and is an important constraint in two-dimensional cutting and packing problems. In the literature, various exact and approximate algorithms exist for finding the two dimensional cutting patterns that satisfy the guillotine cutting constraint. However, to the best of our knowledge, all of the algorithms are designed for solving rectangular cutting where cuts are orthogonal with the edges of the stock-sheet. In order to satisfy the guillotine cutting constraint using these approaches, when the pieces are non-rectangular, practitioners implement a two stage approach. First, pieces are enclosed within rectangle shapes and then the rectangles are packed. Clearly, imposing this condition is likely to lead to additional waste. This paper aims to generate guillotine-cutting layouts of irregular shapes using a number of strategies. The investigation compares three two-stage approaches: one approximates pieces by rectangles, the other two approximate pairs of pieces by rectangles using a cluster heuristic or phi-functions for optimal clustering. All three approaches use a competitive algorithm for rectangle bin packing with guillotine constraints. Further, we design and implement a one-stage approach using an adaptive forest search algorithm. Experimental results show the one-stage strategy produces good solutions in less time over the two-stage approach.     

A hybrid genetic algorithm for the two-dimensional single large object placement problem

In the two-dimensional single large object placement problem, we are given a rectangular master surface which has to be cut into a set of smaller rectangular items, with the aim of maximizing the total value of the pieces cut. We consider the special case in which the items cannot be rotated and must be cut with their edges always parallel to the edges of the surface. We present new greedy algorithms and a hybrid genetic approach with elitist theory, immigration rate, heuristics on-line and tailored crossover operators. Extensive computational results for a large number of small and large benchmark test problems are presented. The results show that our approach outperforms existing heuristic algorithms.     

Railway delay management with passenger rerouting considering train capacity constraints

Delay management for railways is concerned with the question of whether a train should wait for a delayed feeder train or depart on time. The answer should not only depend on the length of the delay but also consider other factors, such as capacity restrictions. We present an optimization model for delay management in railway networks that accounts for capacity constraints on the number of passengers that a train can effectively carry. While limited capacities of tracks and stations have been considered in delay management models, passenger train capacity has been neglected in the literature so far, implicitly assuming an infinite train capacity. However, even in open systems where no seat reservation is required and passengers may stand during the journey if all seats are occupied, physical space is naturally limited, and the number of standing seats is constrained for passenger safety reasons. We present a mixed-integer nonlinear programming formulation for the delay management problem with passenger rerouting and capacities of trains. Our model allows the rerouting of passengers missing their connection due to delays or capacity constraints. We linearize the model in exact and approximate ways and experimentally compare the different approaches with the solution of a reference model from the literature that neglects capacity constraints. The results demonstrate that there is a significant impact of considering train capacity restrictions in decisions to manage delays.     

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

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.     

TSPACK: A two-level tabu search for the three-dimensional bin packing problem

Three-dimensional orthogonal bin packing is a problem NP-hard in the strong sense where a set of boxes must be orthogonally packed into the minimum number of three-dimensional bins. We present a two-level tabu search for this problem. The first-level aims to reduce the number of bins. The second optimizes the packing of the bins. This latter procedure is based on the Interval Graph representation of the packing, proposed by Fekete and Schepers, which reduces the size of the search space. We also introduce a general method to increase the size of the associated neighborhoods, and thus the quality of the search, without increasing the overall complexity of the algorithm. Extensive computational results on benchmark problem instances show the effectiveness of the proposed approach, obtaining better results compared to the existing ones.     

Three-dimensional bin packing problem with variable bin height

This paper studies a variant of the three-dimensional bin packing problem (3D-BPP), where the bin height can be adjusted to the cartons it packs. The bins and cartons to be packed are assumed rectangular in shape. The cartons are allowed to be rotated into any one of the six positions that keep the carton edges parallel to the bin edges. This greatly increases the difficulty of finding a good solution since the search space expands significantly comparing to the 3D-BPP where the cartons have fixed orientations. A mathematical (mixed integer programming) approach is modified based on [Chen, C. S., Lee, S. M., Shen, Q. S., 1995. An analytical model for the container loading problem. European Journal of Operational Research 80 (1), 68–76] and numerical experiments indicate that the mathematical approach is not suitable for the variable bin height 3D-BPP. A special bin packing algorithm based on packing index is designed to utilize the special problem feature and is used as a building block for a genetic algorithm designed for the 3D-BPP. The paper also investigates the situation where more than one type of bin are used and provides a heuristic for packing a batch of cartons using the genetic algorithm. Numerical experiments show that our proposed method yields quick and satisfactory results when benchmarked against the actual packing practice and the MIP model with the latest version of CPLEX.     

Upper bounds for the homogeneous case of a two-dimensional packing problem

A method for determining an upper bound for the homogeneous case of a two-dimensional packing problem is presented in this paper. It is based on an analysis of the problem's structure and can be evaluated as the optimal solution of a non-convex minimization problem which can be transformed to a piecewise linear problem by using its special properties. Finally a comparative analysis of solution quality and time complexity is presented.

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Zusammenfassung In dieser Arbeit wird ein Verfahren zur Bestimmung oberer Schranken für ein homogenes zweidimensionales Packproblem vorgestellt. Auf der Grundlage von Analysen der Problemstruktur kann man eine obere Schranke als optimale Lösung eines nichtkonvexen Minimierungsproblems ermitteln, das unter Ausnutzung spezieller Eigenschaften in ein stückweise lineares Problem transformiert werden kann. Den Abschluß dieser Arbeit bildet eine vergleichende Analyse von Lösungsqualität und Rechenzeitbedarf.