Hybrid Approach for Solving Real-World Bin Packing Problem Instances Using Quantum Annealers

Efficient packing of items into bins is a common daily task. Known as Bin Packing Problem, it has been intensively studied in the field of artificial intelligence, thanks to the wide interest from industry and logistics. Since decades, many variants have been proposed, with the three-dimensional Bin Packing Problem as the closest one to real-world use cases. We introduce a hybrid quantum-classical framework for solving real-world three-dimensional Bin Packing Problems (Q4RealBPP), considering different realistic characteristics, such as: i) package and bin dimensions, ii) overweight restrictions, iii) affinities among item categories and iv) preferences for item ordering. Q4RealBPP permits the solving of real-world oriented instances of 3dBPP, contemplating restrictions well appreciated by industrial and logistics sectors.Comment: 9 pages, 24 figure

Online dynamic bin packing

In this thesis we study online algorithms for dynamic bin packing. An online algorithm is presented with input throughout time and must make irrevocable decisions without knowledge of future input. The classical bin packing problem is a combinatorial optimization problem in which a set of items must be packed into a minimum number of uniform-sized bins without exceeding their capacities. The problem has been studied since the early 1970s and many variants continue to attract researchers’ attention today. The dynamic version of the bin packing problem was introduced by Coffman, Garey and Johnson in 1983. The problem is a generalization of the bin packing problem in which items may arrive and depart dynamically. In this setting, an online algorithm for bin packing is presented with one item at a time, without knowledge of its departure time, nor arrival and departure times of future items, and must decide in which bin the item should be packed. Migration of items between bins is not allowed, however rearrangement of items within a bin is permitted. The objective of problem is to minimize the maximum number of bins used over all time. In multi-dimensional generalizations of the problem, multi-dimensional items must be packed without overlap in multi-dimensional bins of uniform size in each dimension. In this work, we study the setting where items are oriented and cannot be rotated. We first consider online one-dimensional dynamic bin packing and present a lower bound of 8/3 ∼ 2.666 on the achievable competitive ratio of any deterministic online algorithm, improving the best known 2.5-lower bound. Since the introduction of the problem by Coffman, Garey and Johnson, the progress on the problem has focused on improving the original lower bound of 2.388 to 2.428, and to the best known 2.5-lower bound. Our improvement from 2.5 to 8/3 ∼ 2.666 makes a big step forward in closing the gap between the lower bound and the upper bound, which currently stands at 2.788. Secondly we study the online two- and three-dimensional dynamic bin packing problem by designing and analyzing algorithms for special types of input. Bar-Noy et al. initiated the study of the one-dimensional unit fraction bin packing problem, a restricted version where all sizes of items are of the form 1/k, for some integer k > 0. Another related problem is for power fraction items, where sizes are of the form 1/2k, for some integer k ≥ 0. We initiate the study of online multi-dimensional dynamic bin packing of unit fraction items and power fraction items, where items have lengths unit fraction and power fraction in each dimension, respectively. While algorithms for general input are suitable for unit fraction and power fraction items, their worst-case performance guarantees are the same for special types of input. For unit fraction and power fraction items, we design and analyze online algorithms that achieve better worst-case performance guarantees compared to their classical counterparts. Our algorithms give careful consideration to unit and power fraction items, which allows us to reduce the competitive ratios for these types of inputs. Lastly we focus on obtaining lower bounds on the performance of the family of Any- Fit algorithms (Any-Fit, Best-Fit, First-Fit, Worst-Fit) for online multi-dimensional dynamic bin packing. Any-Fit algorithms are classical online algorithms initially studied for the one-dimensional version of the bin packing problem. The common rule that the algorithms use is to never pack a new item to a new bin if the item can be packed in any of the existing bins. While the family of Any-Fit algorithms is always O(1)-competitive for one-dimensional dynamic bin packing, we show that this is no longer the case for multi-dimensional dynamic bin packing when using Best-Fit and Worst-Fit, even if the input consists of power fraction items or unit fraction items. For these restricted inputs, we prove that Best-Fit and Worst-Fit have unbounded competitive ratios, while for First-Fit we provide lower bounds that are higher than the lower bounds for any online algorithm. Furthermore, for general input we show that all classical Any-Fit algorithms are not competitive for online multi-dimensional dynamic bin packing

Bin Packing and Related Problems: General Arc-flow Formulation with Graph Compression

We present an exact method, based on an arc-flow formulation with side constraints, for solving bin packing and cutting stock problems --- including multi-constraint variants --- by simply representing all the patterns in a very compact graph. Our method includes a graph compression algorithm that usually reduces the size of the underlying graph substantially without weakening the model. As opposed to our method, which provides strong models, conventional models are usually highly symmetric and provide very weak lower bounds. Our formulation is equivalent to Gilmore and Gomory's, thus providing a very strong linear relaxation. However, instead of using column-generation in an iterative process, the method constructs a graph, where paths from the source to the target node represent every valid packing pattern. The same method, without any problem-specific parameterization, was used to solve a large variety of instances from several different cutting and packing problems. In this paper, we deal with vector packing, graph coloring, bin packing, cutting stock, cardinality constrained bin packing, cutting stock with cutting knife limitation, cutting stock with binary patterns, bin packing with conflicts, and cutting stock with binary patterns and forbidden pairs. We report computational results obtained with many benchmark test data sets, all of them showing a large advantage of this formulation with respect to the traditional ones

Online Bin Covering with Limited Migration

Semi-online models where decisions may be revoked in a limited way have been studied extensively in the last years. This is motivated by the fact that the pure online model is often too restrictive to model real-world applications, where some changes might be allowed. A well-studied measure of the amount of decisions that can be revoked is the migration factor beta: When an object o of size s(o) arrives, the decisions for objects of total size at most beta * s(o) may be revoked. Usually beta should be a constant. This means that a small object only leads to small changes. This measure has been successfully investigated for different, classical problems such as bin packing or makespan minimization. The dual of makespan minimization - the Santa Claus or machine covering problem - has also been studied, whereas the dual of bin packing - the bin covering problem - has not been looked at from such a perspective. In this work, we extensively study the bin covering problem with migration in different scenarios. We develop algorithms both for the static case - where only insertions are allowed - and for the dynamic case, where items may also depart. We also develop lower bounds for these scenarios both for amortized migration and for worst-case migration showing that our algorithms have nearly optimal migration factor and asymptotic competitive ratio (up to an arbitrary small epsilon). We therefore resolve the competitiveness of the bin covering problem with migration