Multidimensional Bin Packing Revisited
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We consider the NP-complete problem of bin packing. Given a set of numbers, and a set of bins of fixed capacity, find the minimum number of bins needed to contain all the numbers, such that the sum of the numbers assigned to each bin does not exceed the bin capacity. We present a new algorithm for optimal bin packing. Rather than considering the different bins that each number can be placed into, we consider the different ways in which each bin can be packed. Our algorithm appears to be asymptotically faster than the best existing optimal algorithm, and runs more that a thousand times faster on problems with 60 numbers.
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This paper studies a new type of 3D bin packing problem (BPP), in which a number of cuboid-shaped items must be put into a bin one by one orthogonally. The objective is to find a way to place these items that can minimize the surface area of the bin. This problem is based on the fact that there is no fixed-sized bin in many real business scenarios and the cost of a bin is proportional to its surface area. Based on previous research on 3D BPP, the surface area is determined by the sequence, spatial locations and orientations of items. It is a new NP-hard combinatorial optimization problem on unfixed-sized bin packing, for which we propose a multi-task framework based on Selected Learning, generating the sequence and orientations of items packed into the bin simultaneously. During training steps, Selected Learning chooses one of loss functions derived from Deep Reinforcement Learning and Supervised Learning corresponding to the training procedure. Numerical results show that the method proposed significantly outperforms Lego baselines by a substantial gain of 7.52%. Moreover, we produce large scale 3D Bin Packing order data set for studying bin packing problems and will release it to the research community.
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Analysis of Algorithms In the extensible bin packing problem we are asked to pack a set of items into a given number of bins, each with an original size. However, the original bin sizes can be extended if necessary. The goal is to minimize the total size of the bins. We consider the problem with unequal (original) bin sizes and give the complete analysis on a list scheduling algorithm (LS). Namely we present tight bounds of LS for every collection of original bin sizes and every number of bins. We further show better on-line algorithms for the two-bin case and the three-bin case. Interestingly, it is proved that the on-line algorithms have better competitive ratios for unequal bins than for equal bins. Some variants of the problem are also discussed.
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We continue the study of two recently introduced bin packing type problems, called bin packing with clustering, and online bin packing with delays. A bin packing input consists of items of sizes not larger than 1, and the goal is to partition or pack them into bins, where the total size of items of every valid bin cannot exceed 1. In bin packing with clustering, items also have colors associated with them. A globally optimal solution can combine items of different colors in bins, while a clustered solution can only pack monochromatic bins. The goal is to compare a globally optimal solution to an optimal clustered solution, under certain constraints on the coloring provided with the input. We show close bounds on the worst-case ratio between these two costs, called "the price of clustering", improving and simplifying previous results. Specifically, we show that the price of clustering does not exceed 1.93667, improving over the previous upper bound of 1.951, and that it is at least 1.93558, improving over the previous lower bound of 1.93344. In online bin packing with delays, items are presented over time. Items may wait to be packed, and an algorithm can create a new bin at any time, packing a subset of already existing unpacked items into it, under the condition that the bin is valid. A created bin cannot be used again in the future, and all items have to be packed into bins eventually. The objective is to minimize the number of used bins plus the sum of waiting costs of all items, called delays. We build on previous work and modify a simple phase-based algorithm. We combine the modification with a careful analysis to improve the previously known competitive ratio from 3.951 to below 3.1551.
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We consider the NP Hard problems of online Bin Covering and Packing while requiring that larger (or longer, in the one dimensional case) items be placed at the bottom of the bins, below smaller (or shorter) items — we call such a version, the LIB version of problems. Bin sizes can be uniform or variable. We look at computational studies for both the Best Fit and Harmonic Fit algorithms for uniform sized bin covering. The Best Fit heuristic for this version of the problem is introduced here. The approximation ratios obtained were well within the theoretical upper bounds. For variable sized bin covering, a more thorough analysis revealed definite trends in the maximum and average approximation ratios. Finally, we prove that for online LIB bin packing with uniform size bins, no heuristic can guarantee an approximation ratio better than 1.76 under the online model considered.
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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.
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