Two-layer heuristic for the three-dimensional bin design and packing problem

In the three-dimensional (3D) strip packing problem, we are given a set of 3D rectangular items and a 3D box $B$. The goal is to pack all the items in $B$ such that the height of the packing is minimized. We consider the most basic version of the problem, where the items must be packed with their edges parallel to the edges of $B$ and cannot be rotated. Building upon Caprara's work for the two-dimensional (2D) bin packing problem, we obtain an algorithm that, given any $\epsilon>0$, achieves an approximation of $T_{\infty}+\epsilon\approx1.69103+\epsilon$, where $T_{\infty}$ is the well-known number that occurs naturally in the context of bin packing. Our key idea is to establish a connection between bin packing solutions for an arbitrary instance $I$ and the strip packing solutions for the corresponding instance obtained from $I$ by applying the harmonic transformation to certain dimensions. Based on this connection, we also give a simple alternate proof of the $T_{\infty}+\epsilon$ approximation for 2D bin packing due to Caprara. In particular, we show how his result follows from a simple modification of the asymptotic approximation scheme for 2D strip packing due to Kenyon and Rémila.

This paper considers the three-dimensional container packing problem (3DCPP) under home delivery service, which predetermines an ordering of delivering ordered cargo boxes to customers, and aims to load a subset of those boxes according to the delivery ordering into a 3D car container with maximal utilization ratio of the container space while prohibiting the process of delivering cargos to each customer from unloading and subsequent reloading of boxes. In practice, cargos can be taken out from not only the rear door of the delivery container but also the side door. Nonetheless, the previous research only considered the case with only a single door. Therefore, this paper establishes the model for the two-door 3DCPP under home delivery service. Furthermore, the rule of invisible and untouchable items and the subvolume-based approach for the one-door problem are extended to resolve the two-door problem. Last, simulation analysis verifies performance of our approach.

The bin packing problem is one of the most fundamental optimization problems. Owing to its hardness as a combinatorial optimization problem class and its wide range of applications in different domains, different variations of the problem are emerged and many heuristics have been proposed for obtaining approximate solutions. In this paper, we solve a Multi-Level Bin Packing (MLBP) problem in the real make-to-order industry scenario. Existing solutions are not applicable to the problem due to: 1. the final packing may consist multiple levels of sub-packings; 2. the geometry shapes of objects as well as the packing constraints may be unknown. We design an automatic packing framework which extracts the packing knowledge from historical records to support packing without geometry shape and constraint information. Furthermore, we propose a dynamic programming approach to find the optimal solution for normal size problems; and a heuristic multi-level fuzzy-matching algorithm for large size problems. An inverted index is used to accelerate strategy search. The proposed auto packing framework has been deployed in Huawei Process & Engineering System to assist the packing engineers. It achieves a performance of accelerating the execution time of processing 5,000 packing orders to about $8$ minutes with an average successful packing rate as $80.54%$, which releases at least $30%$ workloads of packing workers.

Approximation and online algorithms for multidimensional bin packing: A survey

This paper considers packing problems with balancing conditions and items consisting of clusters of parallelepipeds (mutually orthogonal, i.e. tetris-like items). This issue is quite frequent in space engineering and a real-world application deals with the Automated Transfer Vehicle project (funded by the European Space Agency), at present under development. A Mixed Integer Programming (MIP) approach is proposed. The three-dimensional single bin packing problem is considered. It consists of orthogonally placing, with possibility of rotation, the maximum number of parallelepipeds into a given parallelepiped. A MIP formulation of the problem is reported together with a MIP-based heuristic approach. Balancing conditions are furthermore examined, as well as the orthogonal placement (with rotation) of tetris-like items into a rectangular domain.

We consider two- and three-dimensional bin-packing problems where 90° rotations are allowed. We improve all known asymptotic performance bounds for these problems. In particular, we show how to combine ideas from strip packing and two-dimensional bin packing to give a new algorithm for the three-dimensional strip packing problem where boxes can only be rotated sideways. We propose to call this problem “This side up”. Our algorithm has an asymptotic performance bound of 9/4.

A layer-building algorithm for the three-dimensional multiple bin packing problem: a case study in an automotive company

With the growing demand for sustainable and optimal packaging solutions, this study proposes a novel two-stage algorithm for the multi-container three-dimensional bin packing problem. The research addresses this problem within the context of a real-world industrial scenario and implements several practical constraints including: full shipment, customer positioning requirements, and product geometric interlocking, for increased stability and with the purpose of minimizing the use of plastic wrapping and/or additional supporting surfaces. The main optimization target is to minimize the total number of containers used in the palletization process of custom orders with varying degrees of complexity. The proposed algorithm includes two stages/phases of processing. In the first phase, the algorithm uses constructive heuristics to generate homogeneous product layers. The layers are then stacked to produce blocks, which are then placed on individual containers or pallets. The second phase packs the leftover items using a genetic algorithm. The performance of the proposed solution is benchmarked using real-world industrial data, as well as a more classic academic benchmark. It is demonstrated, across a very large set of orders, that the algorithm always achieves solutions for full palletization of the orders. The analysis shows that the approach is generic and the quality of the solutions generated is relatively even for both small and large, homogeneous and heterogeneous problem instances.

Packing problems deal with loading of a set of items (objects) into a set of boxes (containers) in order to optimize a performance criterion under various constraints. With the advance of RFID technologies and investments in IT infrastructures companies now have access to the necessary data that can be utilized in cost reduction of packing processes. Therefore bin packing and container loading problems are becoming more popular in recent years. In this research we propose a beam search algorithm to solve a packing problem that we encountered in a real world project. The 3D-MBSBPP (Multiple Bin Sized Bin Packing Problem) that we present and solve has not been analyzed in literature before, to the best of our knowledge. We present the performance of our proposed beam search algorithm in terms of both cost and computational time in comparison to a greedy algorithm and a tree search enumeration algorithm.

In this paper, the three-dimensional sphere packing problem is solved by using a dichotomous search-based heuristic. An instance of the problem is defined by a set of n unequal spheres and an object of fixed width and height and, unlimited length. Each sphere is characterized by its radius and the aim of the problem is to optimize the length of the object containing all spheres without overlapping. The proposed method is based upon beam search, in which three complementary phases are combined: (i) a greedy selection phase which determines a series of eligible search subspace, (ii) a truncated tree search, using a width-beam search, that explores some promising paths, and (iii) a dichotomous search that diversifies the search. The performance of the proposed method is evaluated on benchmark instances taken from the literature where its obtained results are compared to those reached by some recent methods of the literature. The proposed method is competitive and it yields promising results.

A hybrid genetic approach is proposed for container loading in order to meet the up-to-date customer requirements in the logistics industry. Container loading is usually modeled as a three-dimensional packing problem which is known to be NP-complete. By combining an order-based genetic algorithm with a novel heuristic placement method, the complicated and highly constrained packing problem is transformed into a simple permutation problem with a much smaller searching domain. It is guaranteed that every box is stable after placement inside the container. From the simulation results, it is demonstrated that optimal results can be duly obtained in a reasonably short period of time.

On Lazy Bin Covering and Packing problems

In this paper, the NP-complete problem of 1D bin packing is considered. In the 'bin packing problem', a set of numbers or elements that need to be loaded in the bins is considered. The total load of the elements in any bin must not exceed the bin size or bin capacity. These elements must be packed or loaded in the bins in such a way that there is minimum amount of space left in each bin and the number of bins to be used is minimised. A new algorithm, which helps in tackling the bin packing problem, by giving optimal solutions in minimum time, is presented. The algorithms currently in use, requires the elements to be sorted before the loading. The algorithm discussed in this paper avoids the sorting operation before the loading while giving the same best fit results.

In this paper, our objective is to develop a mathematical formulation of solving the Bin Packing Problem (BPP) with different sizes of box. It has become one of the most important mathematical applications throughout the time. In our study, Modified Branch and Bound Algorithm (MBBA) is developed to generate all the feasible packing patterns of given boxes to required containers for One-Dimensional BPP. Further development of algorithms was made to ascertain the locations of each box within the containers by using Cartesian coordinate system. Developed algorithms are coded and programmed in the Python programming environment to generate feasible packing patterns.

The three-dimensional multiple bin-size bin packing problem (3D-MBSBPP) has many practical applications in the logistic problems such as warehouse management, transportation planning and container loading. An efficient solution to the problem can have significant effects on reducing the transportation costs, improving the status and increasing the productivity and profitability of the companies. The mathematical models proposed for this problem are few in number and can, therefore, be expanded and improved much further. In this paper, based on the needs of a spare-part company in Iran, we developed a mathematical model for 3D-MBSBPP. This model takes into account the rotation of the boxes and the maximum weight constraint of the bins and then, we used a new heuristic and a GRASP algorithm for solving the model. The obtained answers in comparison with the exact method, confirm the speed and efficiency of the proposed algorithms, especially in solving the large-scale and real-sized problems.