A hybrid GRASP/VND algorithm for two- and three-dimensional bin packing

The three-dimensional bin packing problem consists of packing a set of boxes into the minimum number of bins. In this paper we propose a new GRASP algorithm for solving three-dimensional bin packing problems which can also be directly applied to the two-dimensional case. The constructive phase is based on a maximal-space heuristic developed for the container loading problem. In the improvement phase, several new moves are designed and combined in a VND structure. The resulting hybrid GRASP/VND algorithm is simple and quite fast and the extensive computational results on test instances from the literature show that the quality of the solutions is equal to or better than that obtained by the best existing heuristic procedures.     

New lower bounds for the three-dimensional orthogonal bin packing problem

In this paper, we consider the three-dimensional orthogonal bin packing problem, which is a generalization of the well-known bin packing problem. We present new lower bounds for the problem from a combinatorial point of view and demonstrate that they theoretically dominate all previous results from the literature. The comparison is also done concerning asymptotic worst-case performance ratios. The new lower bounds can be more efficiently computed in polynomial time. In addition, we study the non-oriented model, which allows items to be rotated.     

The Three-Dimensional Pallet Chart: An Analysis of the Factors Affecting the Set of Feasible Layouts for a Class of Two-Dimensional Packing Problems

The two-dimensional orthogonal packing problem of packing identical rectangles into a large containing rectangle is important in pallet loading and has recently received much attention in O.R. publications. In this paper, we examine the conditions under which the set of feasible layouts remains unchanged and show that these conditions can be represented by a series of planes in three-dimensional space. We call this representation the three-dimensional pallet chart because it is an extension of the two-dimensional pallet charts presently used in many practical situations. The strength of this result is demonstrated by three examples of its use. Accurate two-dimensional charts are produced from the three-dimensional version with a minimum of calculation, and a complete sensitivity analysis to changes in box and pallet dimensions can be carried out visually by viewing the chart at different angles. Finally, the result is used to generate a new procedure for determining the maximum number of rectangles which can be fitted. This method is shown to be accurate for over 90% of observations in a random sample of 5000—an improvement of 20% over previous methods.     

A global optimization approach for solving three-dimensional open dimension rectangular packing problems

The three-dimensional open dimension rectangular packing problem (3D-ODRPP) aims to pack a set of given rectangular boxes into a large rectangular container of minimal volume. This problem is an important issue in the shipping and moving industries. All the boxes can be any rectangular stackable objects with different sizes and may be freely rotated. The 3D-ODRPP is usually formulated as a mixed-integer non-linear programming problem. Most existing packing optimization methods cannot guarantee to find a globally optimal solution or are computationally inefficient. Therefore, this paper proposes an efficient global optimization method that transforms a 3D-ODRPP as a mixed-integer linear program using fewer extra 0–1 variables and constraints compared to existing deterministic approaches. The reformulated model can be solved to obtain a global optimum. Experimental results demonstrate the computational efficiency of the proposed approach in globally solving 3D-ODRPPs drawn from the literature and the practical applications.     

New lower bounds for the three-dimensional finite bin packing problem

The three-dimensional finite bin packing problem (3BP) consists of determining the minimum number of large identical three-dimensional rectangular boxes, bins, that are required for allocating without overlapping a given set of three-dimensional rectangular items. The items are allocated into a bin with their edges always parallel or orthogonal to the bin edges. The problem is strongly NP-hard and finds many practical applications. We propose new lower bounds for the problem where the items have a fixed orientation and then we extend these bounds to the more general problem where for each item the subset of rotations by 90° allowed is specified. The proposed lower bounds have been evaluated on different test problems derived from the literature. Computational results show the effectiveness of the new lower bounds.     

A new upper bound for unconstrained two-dimensional cutting and packing

A new upper bound for the unconstrained two-dimensional cutting or packing problem is proposed in this paper. The proposed upper bound can be calculated for any size of plate by solving just two knapsack problems at the beginning of the algorithm. In this research, the proposed upper bound was applied to the well known exact cutting algorithm, although it can be used for both cutting and packing applications. The experimental results demonstrate that the new upper bound is very efficient, and reduces the search time required to find an optimal solution.     

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.     

2DCPackGen: A problem generator for two-dimensional rectangular cutting and packing problems

Cutting and packing problems have been extensively studied in the literature in recent decades, mainly due to their numerous real-world applications while at the same time exhibiting intrinsic computational complexity. However, a major limitation has been the lack of problem generators that can be widely and commonly used by all researchers in their computational experiments. In this paper, a problem generator for every type of two-dimensional rectangular cutting and packing problems is proposed. The problems are defined according to the recent typology for cutting and packing problems proposed by Wäscher, Haußner, and Schumann (2007) and the relevant problem parameters are identified. The proposed problem generator can significantly contribute to the quality of the computational experiments run with cutting and packing problems and therefore will help improve the quality of the papers published in this field.     

Improved local search algorithms for the rectangle packing problem with general spatial costs

The rectangle packing problem with general spatial costs is to pack given rectangles without overlap in the plane so that the maximum cost of the rectangles is minimized. This problem is very general, and various types of packing problems and scheduling problems can be formulated in this form. For this problem, we have previously presented local search algorithms using a pair of permutations of rectangles to represent a solution. In this paper, we propose speed-up techniques to evaluate solutions in various neighborhoods. Computational results for the rectangle packing problem and a real-world scheduling problem exhibit good prospects of the proposed techniques.     

A hybrid placement strategy for the three-dimensional strip packing problem

This paper presents a hybrid placement strategy for the three-dimensional strip packing problem which involves packing a set of cuboids (‘boxes’) into a three-dimensional bin (parallelepiped) of fixed width and height but unconstrained length (the ‘container’). The goal is to pack all of the boxes into the container, minimising its resulting length. This problem has potential industry application in stock cutting (wood, polystyrene, etc. – minimising wastage) and also cargo loading, as well as other applications in areas such as multi-dimensional resource scheduling. In addition to the proposed strategy a number of test results on available literature benchmark problems are presented and analysed. The results of empirical testing of the algorithm show that it out-performs other methods from the literature, consistently in terms of speed and solution quality-producing 28 best known results from 35 test cases.     

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.     

Solving the pickup and delivery problem with three-dimensional loading constraints and reloading ban

In this paper, we extend the classical Pickup and Delivery Problem (PDP) to an integrated routing and three-dimensional loading problem, called PDP with three-dimensional loading constraints (3L-PDP). We are given a set of requests and a homogeneous fleet of vehicles. A set of routes of minimum total length has to be determined such that each request is transported from a loading site to the corresponding unloading site. In the 3L-PDP, each request is given as set of rectangular boxes and the vehicle capacity is replaced by a 3D loading space.This paper is the second one in a series of articles on 3L-PDP. As in the first paper we are dealing with constraints which guarantee that no reloading effort will occur. Here the focus is laid on the reloading ban, a packing constraint that ensures identical placements of same boxes in different packing plans. The reloading ban allows for better solutions in terms of travel distance than a routing constraint that was used in the first paper to preclude any reloading effort. To implement this packing constraint a new type of packing procedure is needed that is capable to generate a series of interrelated packing plans per route. This packing procedure, designed as tree search algorithm, and the corresponding concept of packing checks is the main contribution of the paper at hand. The packing procedure and a large neighborhood search procedure for routing form a hybrid algorithm for the 3L-PDP. Computational experiments were performed using 54 3L-PDP benchmark instances.     

装箱问题的算法及最新进展

装箱问题在经济社会发展中扮演着重要的角色,该问题研究的是寻找较好的布局方式,尽可能实现利益的最大化.装箱问题具有NP-难性质,其理论和应用研究存在一定的挑战,但因其有广泛的应用背景而受到研究者高度的关注.本文主要总结近几十年来装箱问题的研究成果,特别针对一维、二维和三维单目标装箱问题和算法,以及多目标装箱问题的算法进行概括和总结,并提出装箱问题算法上有待进一步的研究工作.     

Quasi-human seniority-order algorithm for unequal circles packing

In the existing methods for solving unequal circles packing problems, the initial configuration is given arbitrarily or randomly, but the impact of different initial configurations for existing packing algorithm to the speed of existing packing algorithm solving unequal circles packing problems is very large. The quasi-human seniority-order algorithm proposed in this paper can generate a better initial configuration for existing packing algorithm to accelerate the speed of existing packing algorithm solving unequal circles packing problems. In experiments, the quasi-human seniority-order algorithm is applied to generate better initial configurations for quasi-physical elasticity methods to solve the unequal circles packing problems, and the experimental results show that the proposed quasi-human seniority-order algorithm can greatly improve the speed of solving the problem.     

A Method for Solving Container Packing for a Single Size of Box

There is an extensive literature on heuristic algorithms for two-dimensional cutting problems and three-dimensional packing, but there seems to be very little on the three-dimensional single-box type packing problem. This paper gives a structure for dealing with that problem as a heuristic. It also presents a set of upper bounds on the optimal fit. Finally, the paper compares a particular application of the algorithmic structure with the George and Robinson multiple-box type heuristic.     

Consecutive Ones Matrices for Multi-dimensional Orthogonal Packing Problems

The multi-dimensional orthogonal packing problem (OPP) is a well studied decisional problem. Given a set of items with rectangular shapes, the problem is to decide whether there is a non-overlapping packing of these items in a rectangular bin. The rotation of items is not allowed. A powerful caracterization of packing configurations by means of interval graphs was recently introduced. In this paper, we propose a new algorithm using consecutive ones matrices as data structure. This new algorithm is then used to solve the two-dimensional orthogonal knapsack problem. Computational results are reported, which show its effectiveness.     

Finished-vehicle transporter routing problem solved by loading pattern discovery

This work addresses a new transportation problem in outbound logistics in the automobile industry: the finished-vehicle transporter routing problem (FVTRP). The FVTRP is a practical routing problem with loading constraints, and it assumes that dealers have deterministic demands for finished vehicles that have three-dimensional irregular shapes. The problem solution will identify optimal routes while satisfying demands. In terms of complex packing, finished vehicles are not directly loaded into the spaces of transporters; instead, loading patterns matching finished vehicles with transporters are identified first by mining successful loading records through virtual and manual loading test procedures, such that the packing problem is practically solved with the help of a procedure to discover loading patterns. This work proposes a mixed-integer linear programming (MILP) model for the FVTRP considering loading patterns. As a special class of routing models, the FVTRP is typically difficult to solve within a manageable computing time. Thus, an evolutionary algorithm is designed to solve the FVTRP. Comparisons of the proposed algorithm and a commercial MILP solver demonstrate that the proposed algorithm is more effective in solving medium- and large-scale problems. The proposed scheme for addressing the FVTRP is illustrated with an example and tested with benchmark instances that are derived from well-studied vehicle routing datasets.     

An Algorithm to Optimize the Layout of Boxes in Pallets

The problem of Optimum Pallet Layout at Rowntree Mackintosh Ltd is discussed. Production constraints reduce this three-dimensional box packing problem into the two-dimensional problem of packing large numbers of identical rectangles orthogonally into fixed-size containing rectangles. The exact solution procedure was found to require prohibitive amounts of computer time. A non-exact procedure is therefore described, its validity demonstrated, and the results obtained exhibited graphically in the form of a Pallet Layout Chart. This work has been implemented at Rowntree Mackintosh Ltd where improved pallet layouts and better design of new box shapes have resulted in cash savings.     

A heuristic algorithm for the strip packing problem

The two-dimensional strip packing problem is to pack a given set of rectangles into a strip with a given width and infinite height so as to minimize the required height of the packing. From the computational point of view, the strip packing problem is an NP-hard problem. With the B*-tree representation, this paper first presents a heuristic packing strategy which evaluates the positions used by the rectangles. Then an effective local search method is introduced to improve the results and a heuristic algorithm (HA) is further developed to find a desirable solution. Computational results on randomly generated instances and popular test instances show that the proposed method is efficient for the strip packing problem.     

Packing strong subgraph in digraphs

In this paper, we study two types of strong subgraph packing problems in digraphs, including internally disjoint strong subgraph packing problem and arc-disjoint strong subgraph packing problem. These problems can be viewed as generalizations of the famous Steiner tree packing problem and are closely related to the strong arc decomposition problem. We first prove the NP-completeness for the internally disjoint strong subgraph packing problem restricted to symmetric digraphs and Eulerian digraphs. Then we get inapproximability results for the arc-disjoint strong subgraph packing problem and the internally disjoint strong subgraph packing problem. Finally we study the arc-disjoint strong subgraph packing problem restricted to digraph compositions and obtain some algorithmic results by utilizing the structural properties.