Greedy Algorithms for Packing Unequal Spheres into a Cuboidal Strip or a Cuboid

Given a finite set of spheres of different sizes, we study the three-dimensional Strip Packing Problem (3D-SPP) as well as the three-dimensional Knapsack Problem (3D-KP). The 3D-SPP asks for a placement of all spheres within a cuboidal strip of fixed width and height so that the variable length of the cuboidal strip is minimized. The 3D-KP requires packing of a subset of the spheres in a given cuboid so that the wasted space is minimized. To solve these problems two greedy algorithms were developed which adapt the algorithms proposed by Huang et al. (2005) to the 3D case with some important enhancements. The resulting methods were tested using the instances provided by Stoyan et al. (2003). Additionally, two series of 12 instances each for the 3D-SPP and for the 3D-KP are introduced and results for these new instances are also reported.

[1]  Mhand Hifi,et al.  Approximate algorithms for constrained circular cutting problems , 2004, Comput. Oper. Res..

[2]  Christian Maaß,et al.  Marktversagen auf dem Softwaremarkt? Zur Förderung der quelloffenen Softwareentwicklung , 2004 .

[3]  Andranik Tangian.  Historical background of the mathematical theory of democracy , 2003 .

[4]  Daniel Mack,et al.  Eine Heuristik für das mehrdimensionaleBin Packing Problem , 2008 .

[5]  Alfred Endres,et al.  Inducing Technical Change by Standard Oriented Environmental Policy: The Role of Information ∗ , 2007 .

[6]  Jie Wang,et al.  Packing of Unequal Spheres and Automated Radiosurgical Treatment Planning , 1999, J. Comb. Optim..

[7]  Andranik S. Tangian.  Equalizing regional unemployment indices in West and East Germany , 2003 .

[8]  Guido Giese,et al.  Endogenous uncertainty and optimal monetary policy , 2008 .

[9]  Volker Arnold.  Competitive versus cooperative Federalism: Is a fiscal equalization scheme necessary from an allocative point of view? , 2005 .

[10]  Christian Maaß,et al.  Web 2.0 als Mythos, Symbol und Erwartung , 2007 .

[11]  Helmut Wagner.  Pension Reforms in the New EU Member States , 2004 .

[12]  Wolfram Berger,et al.  Monetary Policy and Asset Prices: The Impact of Globalization on Monetary Policy Trade-Offs , 2007 .

[13]  Thomas Mazzoni.  Import-Penetration und der Kollaps derPhillips-Kurve , 2006 .

[14]  Giselher Pankratz,et al.  Planen, Lernen, Optimieren: Beiträge zu Logistik und E-Learning. Festschrift zum 60. Geburtstag von Hermann Gehring , 2003 .

[15]  Thomas Mazzoni.  Expected A Posteriori Estimation in Financial Applications , 2008 .

[16]  Yang Dai,et al.  Global Optimization Approach to Unequal Global Optimization Approach to Unequal Sphere Packing Problems in 3D , 2002 .

[17]  Jiabin Chen,et al.  UDUT Continuous-Discrete Unscented Kalman Filtering , 2008, 2008 Second International Symposium on Intelligent Information Technology Application.

[18]  Y. Li,et al.  Greedy algorithms for packing unequal circles into a rectangular container , 2005, J. Oper. Res. Soc..

[19]  J. A. George,et al.  Packing different-sized circles into a rectangular container , 1995 .

[20]  Gotthard Pietsch,et al.  Modernisierung im Bund: Akteursanalyse hat Vorrang , 2007 .

[21]  Johan Eyckmans,et al.  Coalition formation in a global warming game: how the design of protocols affects the success of environmental treaty-making , 2008 .

[22]  Guido Giese,et al.  Graphical Analysis of the New Neoclassical Synthesis , 2006 .

[23]  Yu.G Stoyan,et al.  A mathematical model and a solution method for the problem of placing various-sized circles into a strip , 2004, Eur. J. Oper. Res..

[24]  Michael Finus.  Modesty Pays: Sometimes! , 2004 .

[25]  Martin Steinrücke.  Zur Abbildungsgenauigkeit des Gini-Koeffizienten bei relativer wirtschaftlicher Konzentration , 2002 .

[26]  Michael Bitz.  Banken als Einrichtungen zur Risikotransformation , 2006 .

[27]  Teodor Gabriel Crainic,et al.  Parallel Strategies for Meta-Heuristics , 2003, Handbook of Metaheuristics.

[28]  G. Scheithauer,et al.  Packing of Various Radii Solid Spheres into a Parallelepiped , 2001 .

[29]  Hendrik Sonnabend,et al.  A note on Nash equilibrium in soccer , 2007 .

[30]  Christian Maaß,et al.  Web 2.0 auf dem Prüfstand: Zur Bewertung von Internet-Unternehmen , 2008 .

[31]  G. Garrod,et al.  Economic Valuation of the Environment , 1999 .

[32]  Thomas Mazzoni,et al.  Computational aspects of continuous–discrete extended Kalman-filtering , 2008, Comput. Stat..

[33]  Hermann Singer,et al.  Bayesian Estimation of Volatility with Moment-Based Nonlinear Stochastic Filters , 2006 .

[34]  Hermann Gehring,et al.  New Large Benchmark Instances for the Two-Dimensional Strip Packing Problem with Rectangular Pieces , 2006, Proceedings of the 39th Annual Hawaii International Conference on System Sciences (HICSS'06).

[35]  M. Steinrücke.  Auftragsterminierung und Materialflussplanung bei Werkstattfertigung , 2006 .

[36]  Fachbereich Wirtschaftswissenschaft.  Auf dem Prüfstand: Die geldpolitische Strategie der EZB , 2003 .

[37]  E. Scherm,et al.  Netzwerkökonomik - Eine vernachlässigte theoretische Perspektive in der Strategie-/ Marketingforschung? , 2005 .

[38]  Giselher Pankratz,et al.  Dynamic Planning of Pickup and Delivery Operations by means of Genetic Algorithms , 2004 .

[39]  Wolfram Berger,et al.  Monetary Policy and Asset Prices: More Bad News for Benign Neglect , 2007 .

[40]  Vangelis Th. Paschos,et al.  A simulated annealing approach for the circular cutting problem , 2004, Eur. J. Oper. Res..

[41]  Harald Wirtz,et al.  Übernahmeangebote und sonstige öffentliche Angebote zum Erwerb von Aktien – Ausgestaltungsmöglichkeiten und deren Beschränkung durch das Wertpapiererwerbs- und Übernahmegesetz , 2004 .

[42]  W. Visscher,et al.  Random Packing of Equal and Unequal Spheres in Two and Three Dimensions , 1972, Nature.

[43]  Thomas Mazzoni.  Zeitdiskrete vs. zeitstetige Modellierung von Preismechanismen zur Regulierung von Angebots- und Nachfragemengen , 2004 .

[44]  Michael Finus,et al.  Do Abatement Quotas Lead to More Successful Climate Coalitions?* , 2007 .

[45]  Ewald Scherm,et al.  Mobile Learning – Modetrend oder wesentlicher Bestandteil lebenslangen Lernens? , 2005 .

[46]  Wolfram Berger,et al.  Why Do Monetary Policymakers Lean with the Wind During Asset Price Booms? , 2009 .

[47]  Thomas Mazzoni,et al.  Zeitstetige Modellbildung am Beispiel einer volkswirtschaftlichen Produktionsstruktur , 2005 .

[48]  Hermann Singer.  Maximum entropy inference for mixed continuous-discrete variables , 2010 .

[49]  Hermann Singer,et al.  An alternative derivation of the Black-Scholes Formula , 2007 .

[50]  Lutz Altenburg,et al.  Effort, Trade, and Unemployment , 2008 .

[51]  Hermann Gehring,et al.  Parallel greedy algorithms for packing unequal circles into a strip or a rectangle , 2009, Central Eur. J. Oper. Res..