Pisinger, David

Trivella, Alessio

The Bin-Packing Problem (BPP) is one of the most investigated and applicable combinatorial optimization problems. The problem consists of packing objects of different sizes into a finite number of similar bins, such that the number of used bins is minimized. Applications of the bin-packing problem apper in a wide range of disciplines, including transportation and logistics, computer science, engineering, economics and manufacturing. The problem is well-known to be N P-hard and difficult to solve in practice, especially when dealing with the multi-dimensional cases. Closely connected to the BPP is the Container Loading Problem (CLP), which addresses the optimization of a spacial arrangement of cargo inside a container or transportation vehicle, with the objective to maximize the value of the cargo loaded or the volume utilization. The CLP focuses on a single container, and has been extended in the literature to handle a variety of different constraints arising from real-world problems. Consider for example the problem of arranging items into an aircraft cargo area such that the barycenter of the loaded plane is as close as possible to an ideal point given by the aircraft’sspecifications. The position of the barycenter has an impact on the flight performance in terms of safety and efficiency, and even a minor displacement from the ideal barycenter can lead to a high increase of fuel consumption [1]. Similar considerations apply when loading trucks and container ships. The aim of this work is to integrate realistic constraints related to e.g. load balancing, cargo stability and weight limits, in the multi-dimensional BPP. The BPP poses additionalchallenges compared to the CLP due to the supplementary objective of minimizing the number of bins. In particular, in section 2 we discuss how to integrate bin-packing and load balancing of items. The problem has only been considered in the literature in simplified versions, e.g. balancing a single bin or introducing a feasible region for the barycenter. In section 3 we generalize the problem to handle cargo stability and weight constraints

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Downloaded from orbit.dtu.dk on: Apr 10, 2018Bin-packing problems with load balancing and stability constraintsTrivella, Alessio; Pisinger, DavidPublication date:2017Document VersionPeer reviewed versionLink back to DTU OrbitCitation (APA):Trivella, A., & Pisinger, D. (2017). Bin-packing problems with load balancing and stability constraints. Paperpresented at INFORMS Transportation and Logistics Society 2017, Chicago, United States.Bin-packing problems with load balancing andstability constraintsAlessio Trivella*, David PisingerDepartment of Management EngineeringTechnical University of Denmark, Kgs. Lyngby, Denmark∗Email: atri@dtu.dk1 IntroductionThe Bin-Packing Problem (BPP) is one of the most investigated and applicable combi-natorial optimization problems. The problem consists of packing objects of different sizesinto a finite number of similar bins, such that the number of used bins is minimized.Applications of the bin-packing problem apper in a wide range of disciplines, includingtransportation and logistics, computer science, engineering, economics and manufacturing.The problem is well-known to be NP-hard and difficult to solve in practice, especiallywhen dealing with the multi-dimensional cases.Closely connected to the BPP is the Container Loading Problem (CLP), which addressesthe optimization of a spacial arrangement of cargo inside a container or transportationvehicle, with the objective to maximize the value of the cargo loaded or the volume uti-lization. The CLP focuses on a single container, and has been extended in the literatureto handle a variety of different constraints arising from real-world problems. Consider forexample the problem of arranging items into an aircraft cargo area such that the barycen-ter of the loaded plane is as close as possible to an ideal point given by the aircraft’sspecifications. The position of the barycenter has an impact on the flight performance interms of safety and efficiency, and even a minor displacement from the ideal barycentercan lead to a high increase of fuel consumption [1]. Similar considerations apply whenloading trucks and container ships.The aim of this work is to integrate realistic constraints related to e.g. load balancing,cargo stability and weight limits, in the multi-dimensional BPP. The BPP poses additionalchallenges compared to the CLP due to the supplementary objective of minimizing thenumber of bins. In particular, in section 2 we discuss how to integrate bin-packing andload balancing of items. The problem has only been considered in the literature in simpli-fied versions, e.g. balancing a single bin or introducing a feasible region for the barycenter.In section 3 we generalize the problem to handle cargo stability and weight constraints.2 The integrated packing and balancing problemPacking and balancing the load of a set of items represent two conflicting objectives and itcan be convenient to incorporate them in a single problem. In [2] we develop an integratedapproach for solving the multi-dimensional bin-packing problem with load balancing. Thegoal is to arrange the items in the smallest number of bins, while ensuring the best overallload balancing of the used bins, i.e. ensuring that the average barycenter of the loadedbins falls as close as possible to an ideal point, for instance the center of the bin. We use:• a Mixed-Integer Program (MIP) model. Despite having a quadratic number of variablesand constraints, the model is considerably more complex than the BPP as several newsets of support binary variables and conditional constraints are necessary to optimizeload balancing. The model is able to solve only instances involving up to 20 items.• a multi-level local search heuristic able to deal with large instances. The algorithmtakes advantage of the Fekete-Schepers representation of feasible packings in terms ofparticular classes of interval graphs [3], and iteratively improves the load balancing ofa bin-packing solution using three different search levels:1. The first level explores the space of transitive orientations (TROs) of the com-plement graphs associated with the packing. This is made possible by exploitingnice theoretical properties of interval graphs and a relation TRO-packing.2. The second level modifies the structure of the interval graphs.3. The third level exchanges items between bins by repacking proper n-tuples ofweakly balanced bins, coded inside a variable neighborhood search framework.Extended computational experiments can be found in [2]. The results reveal that aneffective load balancing can be obtained, with a running time rarely exceeding 3-5 minuteseven for difficult instances with up to 200 items (C program on an Intel Core i5 with 8GBRAM). When the optimal barycenter is the geometric center of the bin, for instance, morethan 95% of the initial imbalance can be removed by using the three search phases.In Table 1 is a summary of the main features of the two approaches and the differencesbetween them. It will be useful when considering additional constraints in the next section.Table 1: MIP vs. local search.MIP Local searchalgorithm type: exact heuristicobjective function: only L1 any, e.g. Lpideal barycenter: any anyinstances handled: small medium/large3 Including stability and weight constraintsIn practical contexts, additional properties are relevant when packing containers to ensurethe stability of the cargo (see e.g. [4, 5]). Mechanical stability can be enforced at a staticor dynamic level, related to obtaining a stable motionless or moving cargo, respectively.Moreover, a number of weight constraints may apply depending on the specific transportapplication. In the following, we limit the discussion to static stability, which is by itselfchallenging, and provide an overview of the different stability and weight requirementsconsidered.Table 2: Static stability and weight constraints.Property DescriptionLower gapless Each box touches another box (or the bin) below itFull base support The entire base of each box is in contact to other boxes (or the bin)Partial base support A fix percentage of the box base is in contact to other boxes (or bin)Barycenter support The box barycenter is located above the surface of a supporting boxMechanical equilibrium The sum of external forces and torque acting on each box is zeroLoad bearing Maximum pressure that can be applied over the top face of a boxWeight limit Items packed in each bin cannot exceed a given maximum weightWeight distribution The weight is distributed within the bin according to certain criteriaIn the previous section, we provided two tools to solve different instances of the load-balanced bin-packing problem: an MIP model and a heuristic based on local search.The goal now is to understand whether it is possible (and straightforward) to embed theadditional constraints of Table 2 into the two algorithms. In the remainder of the section,we briefly discuss two of the properties; the others will be examined more in detail in thepaper following this abstract.3.1 Gapless propertyThe local-search algorithm can be easily adapted to lower gapless requirements. Indeed,each TRO of the z-interval graph complement can be associated to a lower gapless packing.In other words, we can simply collapse the bunch of packings associated to a TRO to asingle lower gapless solution, narrowing in this way the search space.The gapless property is, in contrast, very hard to model in the MIP framework. However,if the ideal barycenter location is the center of the bin’s base, than enforcing the lowergapless property is actually not needed, as an optimal solution necessarily satisfies it. Ifnot, it would be possible to move downwards at least one box in one bin, keeping theothers fixed, decreasing the total cost.3.2 Weight limitThe MIP model contains binary variables cij = 1 if and only if item i is in bin j. Usingthese variables, it is easy to identify the total load of a bin and impose a limit by addingfor each bin j the constraints∑i cijmi ≤ Wmax, where mi is the mass of item i. Theweight limit can also be formulated here as bin-dependent Wmax =Wmaxj .Integrating weight limits in the heuristic algorithm affects both the construction phaseand external search (recombination of bins). The two phases are in a sense similar, as theexternal search consists of applying a constructive heuristic to a certain number of bins.Thus, both phases can be modified by closing earlier bins where no more boxes can beadded without violating the weight limit. If n bins cannot be recombined in the samenumber n of bins, each fulfilling the weight limit, we discard the solution and go to thenext local search move.3.3 Further workFuture work includes quantifying the impact of additional stability constraints on thevolume utilization of containers and on the objective function of the integrated load-balanced bin-packing problem.References[1] Mongeau M. and Bes C., Optimization of aircraft container loading, IEEE Transac-tions on Aerospace and Electronic Systems, 39, 140-150, 2003.[2] Trivella A. and Pisinger D., The load-balanced multi-dimensional bin-packing prob-lem, Computers & Operations Research, 74, 152-164, 2016.[3] Fekete S.P. and Schepers J., A Combinatorial Characterization of Higher-DimensionalOrthogonal Packing, Mathematics of Operations Research 29(2), 353-368, 2004.[4] Ramos A.G, Oliveira J. F., Gonc¸alves J.F. and Lopes M.P., A container loading al-gorithm with static mechanical equilibrium stability constraints. Transportation Re-search Part B, 91, 565-581, 2016.[5] Junqueira L., Morabito R. and Yamashita D.S., Three-dimensional container loadingmodels with cargo stability and load bearing constraints, Computers & OperationsResearch, 39, 74-85, 2012.

Bin-packing problems with load balancing and stability constraints

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Bin-packing problems with load balancing and stability constraints

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