A best fit algorithm for the three dimensional **bin** **packing** problem
Authors: C. Paquay 1 , M. Schyns 1 and S. Limbourg 1
Affiliation: 1 University of Liege (ULg), HEC Management School, QuantOM
In this work, we consider the problem of selecting containers in order to pack a set of cuboid boxes in minimizing the unused space inside the selected containers. The set of boxes is highly heterogeneous while there are few types of containers to select. In the literature, this problem is called a three dimensional Multiple **Bin** Size **Bin** **Packing** Problem (MBSBPP). In addition to the geometry constraints, some additional constraints encountered in practical **packing** situations are considered: the container weight limit, the orientation constraints, the load stability, the load-bearing strength or fragility of the boxes and the weight distribution within a container. Moreover, as the original problem is an air cargo application, we extend the definition of the MBSBPP to include situations in which the bins may be truncated parallelepipeds. Indeed, in this context, containers are called Unit Load Devices (ULD) and may have specific shapes to fit inside aircraft.

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IV.2. The Double-Flow Algorithm .
We are now able to describe the double-flow algorithm, which deals with the 2D-**Bin** **Packing** Problem with Length Minimization. This algorithm works as follows: any time we enter the main loop of this algorithm, some subset W of V has been “inserted” in some rectangle Rect(0, 0, L, H), which means that some no circuit double flow vector (F-H, F-L) has been computed, related to the triple (W, L, H). Also, a linear extension of the no circuit arc set E-H E-L together with a linear extension of the no circuit arc set E-L E-H - have been computed, E-H and E-L being respectively the support arc set associated with W, F-H and F-L. So we pick up v 0 in V – W, and we try to “insert” v 0 into W, which means that we try to compute F-H and F-L in such a way that they define a no circuit double flow vector related to the triple (W {v 0 }, L, H).

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Abstract
This paper considers the three-dimensional Multiple **Bin** Size **Bin** **Packing** Problem which consists in **packing** a set of cuboid boxes into containers of various shapes with minimising unused space. The problem is extended to air cargo where bins are Unit Load Devices, especially designed for tting in aircraft. We developed a fast constructive heuristic able to manage the dierent constraints met in transportation. The heuristic is split into two distinct phases. The rst phase deals with the **packing** of boxes into identical bins using an extension of the Extreme Points. During this phase, the fragility, stability and orientations of the boxes are taken into account as well as the special shape of the bins and their weight capacity. The second phase takes into account the multiple types of available bins. If necessary, the best found loading pattern is nally enhanced with respect to weight distribution in a post processing. After parametrisation, computational experiments have been performed on data sets especially designed for this application. The heuristic requires really short computational times to achieve promising results.

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gourgand@isima.fr, grangeon@isima.fr, klement@isima.fr
Abstract. The **bin** **packing** problem aims to pack a set of items in a min- imum number of bins, with respect to the size of the items and capacity of the bins. This is an NP-hard problem. Several approach methods have been developed to solve this problem. In this paper, we propose a new encoding scheme which is used in a hybrid resolution: a metaheuristic is matched with a list algorithm (Next Fit, First Fit, Best Fit) to solve the **bin** **packing** problem. Any metaheuristic can be used but in this paper, our proposition is implemented on a single solution based metaheuris- tic (stochastic descent, simulated annealing, kangaroo algorithm). This hybrid method is tested on literature instances to ensure its good results.

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particulaire est ´etudi´ee plus en d´etail. Plusieurs strat´egies de d´eplacement d’une particule sont propos´ees : la premi`ere approche utilise une d´efinition classique des op´erateurs alors que la seconde se base sur une interpr´etation du principe mˆeme de la PSO. Plusieurs am´eliorations sont ´egalement propos´ees afin d’´eviter une convergence trop rapide vers des optimums locaux. Les notions de dispersion et de mutation sont adapt´ees `a nos m´ethodes. Toutes ces m´ethodes peuvent ˆetre appliqu´ees au probl`eme du **bin** **packing** avec in- compatibilit´es. En fonction de la m´ethode utilis´ee, l’adaptation n’est pas la mˆeme. Si les m´etaheuristiques bas´ees individu sont utilis´ees, le syst`eme de voisinage est red´efini : il faut s’assurer du respect de la contrainte d’incompatibilit´e lors de la g´en´eration du voisin. Si la recherche est effectu´ee dans l’ensemble Ω de toutes les solutions, le calcul du d´epassement de contraintes doit int´egrer la contrainte d’incompatibilit´e.

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1 Introduction
Nowadays, **packing** boxes into containers is a daily process in many fields such as truck or air transport. This process has to be conducted as fast and profitable as possible. Indeed, it is important to pack a maximum number of boxes into a mini- mum number of containers such that the costs can be reduced. This minimization problem is called the **bin** **packing** problem (BPP) or container loading problem. **Bin** **packing** doesn’t only concern the transport; it can be applied in different fields. For instance, trying to encode some given electronical data on a minimum number of DVDs is also considered as **bin** **packing**. There exist some variants of the BPP: it can be considered in one, two or three dimensions and with one or several con- tainers. If there are several containers, they can have the same or different shapes (Multi-Container Loading Problem) and so do the boxes. The knapsack problem is a related problem: only one container is given and each box is associated to a weight and a value. The aim is to load the boxes into the container maximizing the value of the loaded set without exceeding a given maximal weight. [5] developed a more complete clas- sification of the cutting and **packing** problem. Some specific constraints inherent to the intended application can be added to the initial problem. For example, the cost of booking the containers could be taken into account ([2, 14]).

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Keywords: **Bin**-**packing**, unit load devices, 3D-BPP
1 Problem statement
Nowadays, **packing** boxes into containers is a daily process in many fields such as truck or air transport. This process has to be conducted as fast and profitable as possible. Indeed, it is important to pack a maximum number of boxes into a minimum number of containers such that the costs can be reduced. This minimization problem is called the **bin** **packing** problem (BPP) or container loading problem. There exist some variants of the BPP: it can be considered in one, two or three dimensions and with one or several containers. If there are several containers, they can have the same or different shapes (Multi-Container Loading Problem) and so do the boxes. There could be different constraints imposed by the operator or by the transportation. A more complete classification of the **packing** problem is developed in [6].

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Lemma 8 Let b ∗ and b ∗ e be the optimal numbers of bins for BPCP on a general graph G and the **Bin** **Packing** problem (E, q), respectively. Then, it holds that b ∗ e ≤ (∆ + 1) · b ∗ , where ∆ is the maximum degree of the input graph G.
Proof. Consider an optimal solution S ∗ of BPCP on G using b ∗ bins. Starting from S ∗ , we construct a feasible solution S 0 , for the **Bin** **Packing** problem (E, q), as follows. We use a ∆ + 1 edge coloring that we know always exists (see any text book in graph theory). This coloring is used only for the sake of the analysis and not by Algorithm 6 . For each **bin** B k of S ∗ , 1 ≤ k ≤ b ∗ , we create ∆ + 1 new bins B k 1 , . . . , B ∆+1 k in S 0 . **Bin** B k c contains all edges with color c whose end vertices are both included in B k . Each pair of colocated vertices appears in at least one **bin** B k and so each edge appears in **bin** B k c if it is of color c. Because all edges of the same color form a matching (i.e they consist of disjoint vertices), the weight of the edges in B c k is at most the sum of the vertex weights in B k and, thus, not greater than q. Therefore, we have constructed a feasi- ble solution of the **Bin** **Packing** problem (E, q), with at most (∆+1)·b ∗ bins and the lemma follows.

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1. Introduction
We study a variant of the classical **bin** **packing** problem, called the hierar- chically structured **bin** **packing** (HSBP) problem. In this problem, the items to be packed into bins are the leaves of a tree. The objective of the **packing** is to minimize the total number of bins into which the descendants of an inter- nal node are packed, summed over all internal nodes. Such a **packing** problem has applications in document organization and retrieval [1], and sparse matrix computations domain [2]. Both of these papers investigate an approximation algorithm, which is claimed to be a 3/2 approximation on a variant of the prob- lem. Our main contributions are two folds: (i) to show that this approximation does not hold unless a particular condition is met by the given tree; (ii) to

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5 Conclusion
In this paper,we have considered three versions of the on-line **bin**-**packing** prob- lem. For each one, we used an approximation criterion called differential ratio to analyse the competitivity behavior of algorithms developed to solve it. In both n − steps and 2 − steps versions, we proved that no algorithm can do better than the ones here developed. So, n−steps and 2−steps on-line **bin**-**packing** problems are well solved under the differential approximation criterion. Our hardness re- sults also imply that the n-steps and 2-steps **bin**-**packing** problem do not admit a differential approximation schema, contrary to the off-line framework. We also studied the on-line **bin**-**packing** problem with the additional constraint that the items have to be placed in bins in the order of their length. The longest item is required to be placed at the bottom (the LIB constraint). For this problem, we prove that, the n − steps version (items are revealed one by one) is well studied with the differential approximation, since δ F F ≥ 1/2 and no algorithm can do

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Communication Title: Several constructive heuristics for the three dimensional Multiple **Bin** Size **Bin** **Packing** Problem with air transportation constraints
Authors: C. Paquay 1 , S. Limbourg 1 , J.F. Oliveira 2 and M. Schyns 1
Affiliation: 1 University of Liege (ULg), HEC Management School, QuantOM

0 in the model. The additional cardinality/flow based filtering we introduce is well suited when those lower bounds are also constrained initially min(L j ) > 0.
2 Cardinality reasoning for **bin**-**packing**
Existing filtering algorithms for the BinP acking constraint do not make use of the cardinality information inside each **bin** (i.e.. the number of items packed inside each **bin**). However this information can be very valuable in some situa- tions. Consider the extreme case where every item has an equal weight (assume a weight of 1) such that the BinP acking constraint reduces to a GCC. It is clear that the filtering algorithms for the BinP acking are very weak compared to the global arc consistent filtering for the GCC in such a situation. Of course this situation rarely happens in practice but in many applications the weights of the items to place are not so different and it is preferable not to lose completely the reasoning offered by a cardinality constraint (the flow reasoning). Our idea is to introduce one redundant GCC in the modelling of the BinP acking:

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Abstract: The present work deals with the Hospital Group of Territory problem. The objective of the cooperation between these health institutions is to provide a better treatment offer. To do so, these entities pool their means together. Our goal is to propose efficient methods to assign the different operations to the periods and resources, considering resources compatibilities and due dates. We consider this problem as an extension of the classical **Bin** **Packing** Problem. We propose a Particle Swarm Optimization to solve this problem using a hybridization proposed by Klement et al. (2017). The results show the interest of the proposed PSO for this kind of problem. Copyright c
2019 IFAC

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Department of Informatics & Telecommunications, National and Kapodistrian University of Athens
Abstract. Motivated by an assignment problem arising in MapReduce computations, we investigate a generalization of the **Bin** **Packing** problem which we call **Bin** **Packing** with Colocations Problem. We are given a weigthed graph G = (V, E), where V represents the set of items with positive integer weights and E the set of related (to be colocated) items, and an integer q. The goal is to pack the items into a minimum number of bins so that (i) for each **bin**, the total weight of the items packed in this **bin** is at most q, and (ii) for each edge (i, j) ∈ E there is at least one **bin** containing both items i and j.

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Keywords: **Bin** **Packing**, Metaheuristic, Particle Swarm Optimization, Hospital Community. 1. INTRODUCTION
In France, in 2014, Hospital Group of Territory (HGT) has been introduced. It is an evolution of Hospital Community of Territory (HCT), previously defined in 2008. It is a group of distinct places which aim at improving their efficiency by putting together means from different places (Gourgand et al., 2014a). A pool of human resources is shared on several distant hospitals belonging to the same group. The involved problem is to find a hospital assignment for the patients and their operations and, for each operation, to assign the needed resources. The hospitals are distant, so the patients and human resources have to take into account transportation times. The goal is to improve the productivity by pooling human resources and patients within the community. Other applications could be imagined (production sites with shared machines, multi-site time table, ...), rising yet the interest of taking into account the resource transport in a project scheduling context (Laurent et al., 2017).

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Inst3 Mˆemes param`etres que pour la deuxi`eme in- stance, mais le pourcentage d’objets d´ej` a plac´es est de 90% ou 95%.
Ceci r´ev`ele que certains types d’instances sont mieux adapt´ees `a R0 et RMin, tandis que d’autres sont mieux adapt´ees `a RMax. Les r´eductions R25, R50 et R75 ne sont jamais meilleures, en moyenne, que RMin et RMax. C’est pourquoi ces r´eductions interm´ediaires ne sont plus utilis´ees dans les exp´eriences suivantes. Comparaison sur des donn´ ees de la lit´ erature. Pour que l’analyse soit plus pr´ecise, nous comparons le comportement des trois r´eductions propos´ees sur des instances r´eelles. Des algorithmes CP ont ´et´e ex´ecut´es sur les instances SALBP-1 de Scholl [7] et sur les instances de **bin** **packing** de Scholl [8] (premier jeu de donn´ees avec n=50 et n=100), et `a chaque changement du domaine des variables, la solution partielle courante a ´et´e extraite. 30 000 instances ont ´et´e s´electionn´ees al´eatoirement parmi celles-ci, pour chaque jeu de donn´ees. Dans le second cas, seules des instances pour lesquelles au moins un des filtres d´etectait une inconsistance ont ´et´e s´electionn´ees. Les trois r´eductions ont ´et´e appliqu´ees aux instances s´electionn´ees, avec L 3 . La figure 4 donne un sch´ema

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drawn uniformly from the set f 1 = 11 2 = 11 : : : 8 = 11 g , and also when the items
are drawn uniformly from the set f 1 = 12 2 = 12 : : : 9 = 12 g .
1.2 The result
Best-t emergesas the winner among the various on-line algorithms: it is sim- ple, behaves well in practice, and no algorithm is known which beats it both in the worst-case and in the average uniform case. But the worst-case per- formance ratio and the uniform-distribution performance ratio are not quite satisfactory measures for evaluating on-line **bin**-**packing** algorithms. More- over, it appears that studying given distributions accurately is an extremely challenging problem.

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[1] Bortfeldt, A. and G. Wäscher (2013). Constraints in container loading - A State-of-the-Art Review. European Journal of Operational Research 229, 1 20.
[2] Paquay, C., Schyns, M., and Limbourg, S. A Mixed Integer Programming formulation for the three dimensional **bin** **packing** problem deriving from an air cargo application. Submitted to International Transactions in Operational Research.

Han et al (1994) studied 2-VSVBP optimization problem, with several types of available bins and the aim is to minimize the sum of **bin** costs. They proposed exact and heuristic approaches along with a process to improve lower bounds.
In the classical First Fit Decreasing (FFD) heuristic, one has to select the largest item and then pack it into a **bin**. Hence, if one generalizes this heuristic to the multidimensional case, it has to be determined how to measure and compare items. Panigrahy et al (2011) presented a generalization of the classical First Fit Decreasing (FFD) heuristic to VBP and experimented several measures. A promising measure is the DotP roduct which defines the largest item as the item that maximizes some weighted dot product between the vector of remaining capacities and the vector of requirements for the item.

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This thesis aims to solve the problem of **packing** a set of boxes into containers of various shapes without wasting loading space. The goal is to select the best set of ULDs to pack all the boxes achieving a minimum unused volume. As for all the **packing** problems, geometric constraints have to be satisfied: items cannot overlap and have to lie entirely within the bins. The richness of this application is to manage additional and common constraints: the **bin** weight limit, rotations, stability and fragility of the boxes, and weight distribution within a ULD. In practice, this problem is manually solved with no strict guarantee that the constraints are met.

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