heuristics for the container loading problem abstract the knapsack container loading problem is the problem of loading a subset of rectangular boxes into a rectangular container of fixed dimensions such that the volume of the packed boxes is maximized a new heuristic based on the wall building approach is proposed which decomposes the problem into a number of layers which again are split into a number of strips the packing of a strip may be formulated and solved optimally as a knapsack problem with capacity equal to the width or height of the container the depth of a layer as well as the thickness of each strip is decided through a branch and bound approach where at each node only a subset of branches is explored several ranking rules for the selection of the most promising layer depths and strip widths are presented and the performance of the corresponding algorithms is experimentally compared for homogeneous and heterogeneous instances the best ranking rule is then used in a comprehensive computational study involving large sized instances these computational results show that instances with a total box volume up to 90 easily may be solved to optimality and that average fillings of the container volume exceeding 95 may be obtained for large sized instances

an exact algorithm for the pallet loading problem abstract the two dimensional packing problem of finding optimal layouts for identical rectangular boxes on a rectangular pallet has interested or practitioners for many years the problem is np complete and solution methods to date tend to be heuristic this paper discusses the development of an exact tree search algorithm based on a graph theoretic model of the problem

the lagrangian relaxation method for solving integer programming problems this article originally appeared in management science january 1981 volume 27 number 1 pp 1 18 published by the institute of management sciences one of the most computationally useful ideas of the 1970s is the observation that many hard integer programming problems can be viewed as easy problems complicated by a relatively small set of side constraints dualizing the side constraints produces a lagrangian problem that is easy to solve and whose optimal value is a lower bound for minimization problems on the optimal value of the original problem the lagrangian problem can thus be used in place of a linear programming relaxation to provide bounds in a branch and bound algorithm this approach has led to dramatically improved algorithms for a number of important problems in the areas of routing location scheduling assignment and set covering this paper is a review of lagrangian relaxation based on what has been learned in the last decade

general framework for dynamic substructuring history review and classification of techniques four decades after the development of the first dynamic substructuring techniques there is a necessity to classify the different methods in a general framework that outlines the relations between them in this paper a certain vision on substructuring methods is proposed by recalling important historical milestones that allow us to understand substructuring as a domain decomposition concept thereafter based on the dual and primal assembly of substructures a general framework for the classification of the methods is presented this framework allows us to indicate how the various classes of methods proposed along the years can be derived from a clear mathematical description of substructured problems current bottlenecks in experimental dynamic substructuring as well as solutions found in literature will also be briefly discussed

an exact algorithm for higher dimensional orthogonal packing higher dimensional orthogonal packing problems have a wide range of practical applications including packing cutting and scheduling combining the use of our data structure for characterizing feasible packings with our new classes of lower bounds and other heuristics we develop a two level tree search algorithm for solving higher dimensional packing problems to optimality computational results are reported including optimal solutions for all two dimensional test problems from recent literature this is the third in a series of articles describing new approaches to higher dimensional packing

a simplified h box method for embedded boundary grids we present a simplified h box method for integrating time dependent conservation laws on embedded boundary grids using an explicit finite volume scheme by using a method of lines approach with a strong stability preserving runge kutta method in time the complexity of our previously introduced h box method is greatly reduced a stable accurate and conservative approximation is obtained by constructing a finite volume method where the numerical fluxes satisfy a certain cancellation property for a model problem in one space dimension using appropriate limiting strategies the resulting method is shown to be total variation diminishing in two space dimensions stability is maintained by using rotated h boxes as introduced in previous work m j berger and r j leveque comput systems engrg 1 1990 pp 305 311 c helzel m j berger and r j leveque siam j sci comput 26 2005 pp 785 809 but in the new formulation h box gradients are taken solely from the underlying cartesian grid which also reduces the computational cost

space defragmentation for packing problems abstract one of main difficulties of multi dimensional packing problems is the fragmentation of free space into several unusable small parts after a few items are packed this study proposes a defragmentation technique to combine the fragmented space into a continuous usable space which potentially allows the packing of additional items we illustrate the effectiveness of this technique using the two and three dimensional bin packing problem where the aim is to load all given items represented by rectangular boxes into the minimum number of identical bins experimental results based on well known 2d and 3d bin packing data sets show that our defragmentation technique alone is able to produce solutions approaching the quality of considerably more complex meta heuristic approaches for the problem in conjunction with a bin shuffling strategy for incremental improvement our resultant algorithm outperforms all leading meta heuristic approaches based on the commonly used benchmark data by a significant margin

a hybrid genetic algorithm for the two dimensional single large object placement problem in the two dimensional single large object placement problem we are given a rectangular master surface which has to be cut into a set of smaller rectangular items with the aim of maximizing the total value of the pieces cut we consider the special case in which the items cannot be rotated and must be cut with their edges always parallel to the edges of the surface we present new greedy algorithms and a hybrid genetic approach with elitist theory immigration rate heuristics on line and tailored crossover operators extensive computational results for a large number of small and large benchmark test problems are presented the results show that our approach outperforms existing heuristic algorithms

a cartesian grid embedded boundary method for hyperbolic conservation laws we present a second order godunov algorithm to solve time dependent hyperbolic systems of conservation laws on irregular domains our approach is based on a formally consistent discretization of the conservation laws on a finite volume grid obtained from intersecting the domain with a cartesian grid we address the small cell stability problem associated with such methods by hybridizing our conservative discretization with a stable nonconservative discretization at irregular control volumes and redistributing the difference in the mass increments to nearby cells in a way that preserves stability and local conservation the resulting method is second order accurate in l 1 for smooth problems and is robust in the presence of large amplitude discontinuities intersecting the irregular boundary

application of a mixed simulated annealing genetic algorithm heuristic for the two dimensional orthogonal packing problem abstract in this paper a pure meta heuristic genetic algorithm and a mixed meta heuristic simulated annealing genetic algorithm were applied to two dimensional orthogonal packing problems and the results were compared the major motivation for applying a modified genetic algorithm is as an attempt to alleviate the problem of pre mature convergence we found that in the long run the mixed heuristic produces better results while the pure heuristic produces only good results but produces them faster

the multiple container loading cost minimization problem abstract in the shipping and transportation industry there are several types of standard containers with different dimensions and different associated costs in this paper we examine the multiple container loading cost minimization problem mclcmp where the objective is to load products of various types into containers of various sizes so as to minimize the total cost we transform the mclcmp into an extended set cover problem that is formulated using linear integer programming and solve it with a heuristic to generate columns experiments on standard bin packing instances show our approach is superior to prior approaches additionally since the optimal solutions for existing test data is unknown we propose a technique to generate test data with known optimal solutions for mclcmp

heuristic algorithms for the three dimensional bin packing problem abstract the three dimensional bin packing problem 3bp consists of allocating without overlapping a given set of three dimensional rectangular items to the minimum number of three dimensional identical finite bins the problem is np hard in the strong sense and finds many industrial applications we introduce a tabu search framework exploiting a new constructive heuristic for the evaluation of the neighborhood extensive computational results on standard benchmark instances show the effectiveness of the approach with respect to exact and heuristic algorithms from the literature

two dimensional packing problems a survey we consider problems requiring to allocate a set of rectangular items to larger rectangular standardized units by minimizing the waste in two dimensional bin packing problems these units are finite rectangles and the objective is to pack all the items into the minimum number of units while in two dimensional strip packing problems there is a single standardized unit of given width and the objective is to pack all the items within the minimum height we discuss mathematical models and survey lower bounds classical approximation algorithms recent heuristic and metaheuristic methods and exact enumerative approaches the relevant special cases where the items have to be packed into rows forming levels are also discussed in detail

an exact two dimensional non guillotine cutting tree search procedure we consider the two dimensional cutting problem of cutting a number of rectangular pieces from a single large rectangle so as to maximize the value of the pieces cut we develop a lagrangean relaxation of a zero one integer programming formulation of the problem and use it as a bound in a tree search procedure subgradient optimization is used to optimize the bound derived from the lagrangean relaxation problem reduction tests derived from both the original problem and the lagrangean relaxation are given incorporating the bound and the reduction tests into a tree search procedure enables moderately sized problems to be solved

three dimensional bin packing problem with variable bin height this paper studies a variant of the three dimensional bin packing problem 3d bpp where the bin height can be adjusted to the cartons it packs the bins and cartons to be packed are assumed rectangular in shape the cartons are allowed to be rotated into any one of the six positions that keep the carton edges parallel to the bin edges this greatly increases the difficulty of finding a good solution since the search space expands significantly comparing to the 3d bpp where the cartons have fixed orientations a mathematical mixed integer programming approach is modified based on chen c s lee s m shen q s 1995 an analytical model for the container loading problem european journal of operational research 80 1 68 76 and numerical experiments indicate that the mathematical approach is not suitable for the variable bin height 3d bpp a special bin packing algorithm based on packing index is designed to utilize the special problem feature and is used as a building block for a genetic algorithm designed for the 3d bpp the paper also investigates the situation where more than one type of bin are used and provides a heuristic for packing a batch of cartons using the genetic algorithm numerical experiments show that our proposed method yields quick and satisfactory results when benchmarked against the actual packing practice and the mip model with the latest version of cplex

computational formulation for periodic vibration of geometrically nonlinear structures part 1 theoretical background abstract a general computational formulation for geometrically nonlinear structures excited by harmonic forces and executing periodic motion in a steady state is presented the equations of both continuous and discretized models are reformulated to obtain the motion equation in a more suitable form to a further analysis the multi harmonic solution of motion equation is written in a form of truncated fourier series next the galerkin ritz and the harmonic balance method are discussed in a context of their equivalency in derivation of the matrix amplitude equation the matrix amplitude equation as well as the associated tangent matrix are given in an explicit form the stability of steady state solution is discussed by using the floquet theory the numerical algorithm and an example application are described in a companion paper by lewandowski lewandowski r computational formulation for periodic vibration of geometrically nonlinear structures part 2 numerical strategy and numerical examples international journal of solids and structures in preparation

a tailored two phase constructive heuristic for the three dimensional multiple bin size bin packing problem with transportation constraints 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 while minimising unused space the problem is extended to air cargo where the bins are unit load devices specially designed for fitting in aircraft we developed a fast constructive heuristic able to manage the constraints to be met in transportation the heuristic is split into two distinct phases the first phase deals with the packing of boxes into identical bins using an extension of the extreme points which describe the possible interesting positions to accommodate boxes during this phase the fragility stability and orientation of the boxes are taken into account as well as the special shape of the bins and their weight capacity the second phase considers the multiple types of available bins if necessary the best loading pattern identified is enhanced with respect to weight distribution in post processing after the description of the parametrisation computational experiments are performed on data sets specially designed for this application the heuristic requires only few seconds to achieve promising results in terms of filling rate

bidimensional packing by bilinear programming we consider geometric problems in which rectangles have to be packed into identical squares these problems turn out to be very hard in practice and ilp formulations in which variables specify the coordinates in the packing perform very poorly while most methods developed so far are based on simple geometric considerations a recent landmark result of fekete and schepers suggests to put these geometric aspects aside and use the most advanced tools for the one dimensional case in this paper we make additional progress in this direction especially on the basic question does a given set of rectangles fit into a square which turns out to be the bottleneck of all the approaches known given a set of rectangles and the associated convex hull of rectangle subsets that fit into a square we derive a wide class of valid inequalities for this convex hull from a complete description of the two knapsack polytopes associated with the widths and the heights of the rectangles respectively in addition we illustrate how to solve the associated separation problem as a bilinear program for which we develop a solution method that turns out to be fast in practice and show that the integer solutions that satisfy all these constraints generally correspond to vertices of the original convex hull for the benchmark instances in the literature the same tools are used to derive lower bounds for the two dimensional bin packing problem corresponding to the determination of an optimal pair of so called dual feasible functions these lower bounds in many cases equal those obtained by the customary set covering formulation for which column generation is very hard but are computable within a time that is smaller by some orders of magnitude this allows us to close a few of the benchmark instances in the literature

approximation algorithms for the orthogonal z oriented three dimensional packing problem we present approximation algorithms for the orthogonal z oriented three dimensional packing problem tppz and analyze their asymptotic performance bound this problem consists in packing a list of rectangular boxes l b1 b2 bn into a rectangular box b l w infty orthogonally and oriented in the z axis in such a way that the height of the packing is minimized we say that a packing is oriented in the z axis when the boxes in l are allowed to be rotated by ninety degrees around the z axis this problem has some nice applications but has been less investigated than the well known variant of it denoted by tpp three dimensional orthogonal packing problem in which rotations of the boxes are not allowed the problem tpp can be reduced to tppz given an algorithm for tppz we can obtain an algorithm for tpp with the same asymptotic bound we present an algorithm for tppz called r and three other algorithms called ls bs and ss for special cases of this problem in which the instances are more restricted the algorithm ls is for the case in which all boxes in l have square bottoms bs is for the case in which the box b has a square bottom and ss is for the case in which the box b and all boxes in l have square bottoms for an algorithm wa we denote by r wa the asymptotic performance bound of wa we show that 2 5 leq r r 2 67 2 5 leq r ls leq 2 528 2 5 leq r bs leq 2 543 and 2 333 leq r ss leq 2 361 the algorithms presented here have the same complexity cal o n log n as the other known algorithms for these problems but they have better asymptotic performance bounds

the three dimensional knapsack problem with balancing constraints abstract in this paper we introduce a new packing problem the three dimensional knapsack problem with balancing constraints 3bkp the extension of the three dimensional knapsack problem 3kp where additional constraints related to the packing center of mass are given the 3bkp consists in orthogonally packing a subset of three dimensional weighted items into a knapsack in order to maximize the total profit of the loaded items the items must not overlap and the packing center of mass must lie into a predefined boxed domain inside the knapsack we assume that items can be rotated we give a mip model for the problem upper bounds and an efficient heuristic to solve large size instances the computational results show that the mip model cannot find optimal solutions except for small size instances but it can be used to calculate upper and lower bounds it is shown that our heuristic outperforms the solution quality both of the mip model and the heuristics available in the literature explicitly designed to solve the 3kp

a branch and price algorithm for the two dimensional vector packing problem with piecewise linear cost function abstract the two dimensional vector packing problem with piecewise linear cost function 2dvpp plc models a practical packing problem faced by many companies that use courier services we propose a branch and price algorithm to solve the 2dvpp plc exactly the column generation procedure is a key component that affects the performance of a branch and price algorithm the pricing problem exhibits an interesting structure that allows us to decompose it into subproblems that form a lattice we explore dominance relations on the lattice and design an efficient algorithm for the pricing problem experimental results show that our branch and price algorithm is capable of solving 2dvpp plc test instances effectively

exact solution of the two dimensional finite bon packing problem given a set of rectangular pieces to be cut from an unlimited number of standardized stock pieces bins the two dimensional finite bin packing problem is to determine the minimum number of stock pieces that provide all the pieces the problem is np hard in the strong sense and finds many practical applications in the cutting and packing area we analyze a well known lower bound and determine its worst case performance we propose not lower bounds which are used within a branch and bound algorithm for the exact solution of the problem extensive computational testing on problem instances from the literature involving up to 120 pieces shows the effectiveness of the proposed approach

analysis of upper bounds for the pallet loading problem this paper is concerned with upper bounds for the well known pallet loading problem plp which is the problem of packing identical boxes into a rectangular pallet so as to maximize the number of boxes fitted after giving a comprehensive review of the known upper bounds in the literature we conduct a detailed analysis to determine which bounds dominate which others the result is a ranking of the bounds in a partial order it turns out that two of the bounds dominate all others a bound due to nelissen and a bound obtained from the linear programming relaxation of a set packing formulation experiments show that the latter is almost always optimal and can be computed quickly

a population heuristic for constrained two dimensional non guillotine cutting abstract in this paper we present a heuristic algorithm for the constrained two dimensional non guillotine cutting problem this is the problem of cutting a number of rectangular pieces from a single large rectangle so as to maximise the value of the pieces cut in addition the number of pieces of each type that are cut must lie within prescribed limits our heuristic algorithm is a population heuristic where a population of solutions to the problem are progressively evolved this heuristic is based on a new non linear formulation of the problem computational results are presented for a number of standard test problems taken from the literature and for a number of large randomly generated problems

a harmonic algorithm for the 3d strip packing problem in the three dimensional 3d strip packing problem we are given a set of 3d rectangular items and a 3d box b the goal is to pack all the items in b such that the height of the packing is minimized we consider the most basic version of the problem where the items must be packed with their edges parallel to the edges of b and cannot be rotated building upon caprara s work for the two dimensional 2d bin packing problem we obtain an algorithm that given any epsilon 0 achieves an approximation of t infty epsilon approx1 69103 epsilon where t infty is the well known number that occurs naturally in the context of bin packing our key idea is to establish a connection between bin packing solutions for an arbitrary instance i and the strip packing solutions for the corresponding instance obtained from i by applying the harmonic transformation to certain dimensions based on this connection we also give a simple alternate proof of the t infty epsilon approximation for 2d

a cut cell finite volume finite element coupling approach for fluid structure interaction in compressible flow we present a loosely coupled approach for the solution of fluid structure interaction problems between a compressible flow and a deformable structure the method is based on staggered dirichlet neumann partitioning the interface motion in the eulerian frame is accounted for by a conservative cut cell immersed boundary method the present approach enables sub cell resolution by considering individual cut elements within a single fluid cell which guarantees an accurate representation of the time varying solid interface the cut cell procedure inevitably leads to non matching interfaces demanding for a special treatment a mortar method is chosen in order to obtain a conservative and consistent load transfer we validate our method by investigating two dimensional test cases comprising a shock loaded rigid cylinder and a deformable panel moreover the aeroelastic instability of a thin plate structure is studied with a focus on the prediction of flutter onset finally we propose a three dimensional fluid structure interaction test case of a flexible inflated thin shell interacting with a shock wave involving large and complex structural deformations

ts2pack a two level tabu search for the three dimensional bin packing problem three dimensional orthogonal bin packing is a problem np hard in the strong sense where a set of boxes must be orthogonally packed into the minimum number of three dimensional bins we present a two level tabu search for this problem the first level aims to reduce the number of bins the second optimizes the packing of the bins this latter procedure is based on the interval graph representation of the packing proposed by fekete and schepers which reduces the size of the search space we also introduce a general method to increase the size of the associated neighborhoods and thus the quality of the search without increasing the overall complexity of the algorithm extensive computational results on benchmark problem instances show the effectiveness of the proposed approach obtaining better results compared to the existing ones

a cutting plane approach for the two dimensional orthogonal non guillotine cutting problem abstract the two dimensional orthogonal non guillotine cutting problem ngcp appears in many industries like wood and steel industries and consists in cutting a rectangular master surface into a number of rectangular pieces each with a given size and value the pieces must be cut with their edges always parallel or orthogonal to the edges of the master surface orthogonal cuts the objective is to maximize the total value of the pieces cut in this paper we propose a two level approach for solving the ngcp where at the first level we select the subset of pieces to be packed into the master surface without specifying the layout while at a second level we check only if a feasible packing layout exists this approach has been already proposed by fekete and schepers s p fekete j schepers a new exact algorithm for general orthogonal d dimensional knapsack problems esa 97 springer lecture notes in computer science 1284 1997 144 156 s p fekete j schepers on more dimensional packing iii exact algorithms tech rep 97 290 universitat zu koln germany 2000 s p fekete j schepers j c van der veen an exact algorithm for higher dimensional orthogonal packing tech rep under revision on operations research braunschweig university of technology germany 2004 and caprara and monaci a caprara m monaci on the two dimensional knapsack problem operations research letters 32 2004 2 14 we propose improved reduction tests for the ngcp and a cutting plane approach to be used in the first level of the tree search to compute effective upper bounds computational tests on problems derived from the literature show the effectiveness of the proposed approach that is able to reduce the number of nodes generated at the first level of the tree search and the number of times the existence of a feasible packing layout is tested

new lower bounds for the three dimensional orthogonal bin packing problem abstract in this paper we consider the three dimensional orthogonal bin packing problem which is a generalization of the well known bin packing problem we present new lower bounds for the problem from a combinatorial point of view and demonstrate that they theoretically dominate all previous results from the literature the comparison is also done concerning asymptotic worst case performance ratios the new lower bounds can be more efficiently computed in polynomial time in addition we study the non oriented model which allows items to be rotated

exact algorithms for the two dimensional strip packing problem with and without rotations we propose exact algorithms for the two dimensional strip packing problem 2sp with and without 90 rotations we first focus on the perfect packing problem pp which is a special case of 2sp wherein all given rectangles are required to be packed without wasted space and design branch and bound algorithms introducing several branching rules and bounding operations a combination of these rules yields an algorithm that is especially efficient for feasible instances of pp we then propose several methods of applying the pp algorithms to 2sp our algorithms succeed in efficiently solving benchmark instances of pp with up to 500 rectangles and those of 2sp with up to 200 rectangles they are often faster than existing exact algorithms specially tailored for problems without rotations

erratum to the three dimensional bin packing problem robot packable and orthogonal variants of packing problems in the three dimensional bin packing problem the task is to orthogonally pack a given set of rectangular items into a minimum number of three dimensional rectangular bins we give a characterization of the algorithm proposed by martello et al 2000 for the exact solution of the problem showing that not all orthogonal packings can be generated by the proposed algorithm the packings however have the property of being robot packings which is relevant in practical settings references to the modified algorithm which solves the orthogonal as well as robot packable three dimensional problem are given

a hybrid genetic algorithm heuristic for a two dimensional orthogonal packing problem in this paper we address a two dimensional 2d orthogonal packing problem where a fixed set of small rectangles has to be placed on a larger stock rectangle in such a way that the amount of trim loss is minimized the algorithm we propose hybridizes a placement procedure with a genetic algorithm based on random keys the approach is tested on a set of instances taken from the literature and compared with other approaches the computation results validate the quality of the solutions and the effectiveness of the proposed algorithm

on three dimensional packing the three dimensional packing problem is discussed in this paper the problem is a generalization of the one and two dimensional packing problems it is demonstrated that some basic packing strategies such as nfdh and ffdh for two dimensional packing have unbounded worst case performance ratios in the three dimensional case let r a denote the asymptotic performance bound of an approximation algorithm a an approximation algorithm g is developed and it is shown that 4 333 leqq r g leqq 4 571 the algorithm is improved to algorithm c and it is proven that r c 3 25 for the special case when all boxes have square bottoms the two algorithms are adapted to algorithms g 1 and c 1 respectively with r g 1 4 and r c 1 2 6875 for the case when both sides of the bottom of a box are no larger then 1 m two families of algorithms g m m geqq 3 and c m m geqq 2 are presented it is shown that r g m m m 2 and r c m m 1 m 1

a two dimensional heuristic decomposition approach to a three dimensional multiple container loading problem abstract efficient container loading has the potential to considerably reduce logistics and transportation costs the container loading problem is computationally complex and despite extensive academic effort there remains room for algorithm improvement real world problems are not always addressed satisfactorily primarily due to the large number of complex constraints and objectives the problem addressed by this work is an unsolved multiple container loading problem introduced by renault on the occasion of the 2014 2015 esicup challenge organized by the euro special interest group on cutting and packing esicup renault s problem requires a large number of different items to be packed into containers of different types and sizes items must be grouped into stacks and respect the this side up constraint the primary objective is to minimize the volume of shipped containers the smallest volume container may be left behind for the next shipment and is excluded from the main objective nevertheless only a limited percentage of each product may be packed into this container additionally a set of secondary objectives is considered this work presents a decomposition approach embedded in a multi phase heuristic for the problem feasible solutions are generated quickly and subsequently improved by local search and post processing procedures experiments revealed that the approach generates optimal solutions for two instances in addition to good quality solutions for those remaining from the renault set the algorithmic contribution is however not restricted to the renault instances experiments on less constrained container loading instances from the literature demonstrate the approach s general applicability and competitiveness

a combinatorial characterization of higher dimensional orthogonal packing higher dimensional orthogonal packing problems have a wide range of practical applications including packing cutting and scheduling previous efforts for exact algorithms have been unable to avoid structural problems that appear for instance in two or higher dimensional space we present a new approach for modeling packings using a graph theoretical characterization of feasible packings our characterization allows it to deal with classes of packings that share a certain combinatorial structure instead of having to consider one packing at a time in addition we can make use of elegant algorithmic properties of certain classes of graphs this allows our characterization to be the basis for a successful branch and bound framework this is the first in a series of papers describing new approaches to higher dimensional packing

an exact algorithm for orthogonal 2 d cutting problems using guillotine cuts abstract we consider the two dimensional cutting problem which requires cutting a number of small rectangular pieces each of a given size and value from a large reatangular stock plate the objective is to maximise the value of pieces cut or for a slightly simpler problem to minimise the wastage we consider the case where there is a maximum number of times that a piece may be used in a cutting pattern we present a tree search algorithm for this problem in which the size of the tree search is limited by using a bound derived from a state space relaxation of a dynamic programming formulation of the problem a state space ascent method is used to optimise this bound the computational performance of the algorithm is investigated by tests performed on randomly generated problems with constraints of varying tightness the results indicate that the algorithm is an effective procedure capable of solving optimally practical two dimensional cutting problems of medium size

average case analysis of cutting and packing in two dimensions abstract stock cutting and packing in two dimensions is a source of problems of great practical significance much has been written on the subject including several surveys but the focus of the mathematically oriented research has been largely on combinatorial optimization and worst case analysis recently substantial progress has been made in the average case analysis of algorithms for two dimensional packing this paper reviews selected illustrative results in this area of research and discusses desirable extensions and open problems several cutting and packing algorithms are defined and the results of their probabilistic analysis reported from the presentation of this material it is clear that the field is still in early stages of development the algorithms and probability models tend to be simplistic and stimates of performance are far more common that exact measures in spite of these limitations valuable and in some cases unexpected insights have emerged

parametric on line algorithms for packing rectangles and boxes abstract we present approximation algorithms for the following problems the two dimensional bin packing 2bp the three dimensional packing problem tpp and the container packing problem 3bp we consider the special case in which the items to be packed are small and must be packed on line we say an item is small if each of its dimension is at most 1 m of the respective dimension of the recipient where m is an integer greater than 1 these problems are denoted by 2bpm tppm and 3bpm and the performance of the algorithms are given in terms of the parameter m to our knowledge the parametric on line algorithms we present here have the best so far achieved asymptotic performance bounds for these problems for 2bpm and tppm we present algorithms with performance bound close to m 2 m 1 m 1 2 and for 3bpm we describe an algorithm with performance bound close to m 3 m 2 m2 1 m 1 2 for m 2 respectively m 3 these bounds are 2 112 and 3 112 respectively 1 73 and 2 285 the results on 2bpm and 3bpm extend the results on multidimensional on line packing presented by coppersmith and raghavan oper res lett 8 1 1989 17 the result on tppm improves the bound m 1 m 1 due to li and cheng siam j comput 19 1990 847 all these algorithms can be implemented to run in o nlogn time where n is the number of items to be packed

sequential heuristic for the two dimensional bin packing problem abstract a heuristic approach for the two dimensional bin packing problem is proposed the algorithm is based on the sequential heuristic procedure that generates each pattern to produce some items and repeats until all items are produced both guillotine and non guillotine patterns can be used each pattern is obtained from calling a pattern generation procedure where the objective is to maximize the pattern value the item values are adjusted after the generation of each pattern using a value correction formula the algorithm is compared with five published algorithms using 50 groups of benchmark instances the results indicate that the algorithm is the most efficient in improving solution quality

model reduction for dynamical systems with local nonlinearities many kinds of nonlinear engineering systems have a large number of degrees of freedom while nonlinear components are spatially localized even though the local nonlinearities constitute only a small part of the system the dynamic behavior of the system is wholly nonlinear and the analysis of the whole model is very expensive to reduce the computational effort signie cantly two model reduction schemes are proposed to reduce the size of the nonlinear model before the analysis is performed one is dee ned on the system level and the other is on the substructure level they are both based on the dynamic condensation technique in the former scheme the dynamic condensation technique is implemented into the linear part of the whole system directly whereas it is applied to the linear e exible substructures in the later scheme the accuracy and stability of the former is usually better than the latter however it is more computationally expensive than the latter two numerical examples are also included the results show that the present two schemes are feasible and efe cient to reduce the degrees of freedom of the model with local nonlinearities

continued fractions in optimal cutting of a rectangular sheet into equal small rectangles abstract a problem of optimum guillotine cutting of a rectangle into smaller equal rectangles is considered the main decomposition theorem for this problem is proved the method of continued fractions is used for decreasing the number of problem parameters a fast algorithm for solving the problem based on this results is presented a numerical example is given

on solving multiobjective bin packing problems using evolutionary particle swarm optimization abstract the bin packing problem is widely found in applications such as loading of tractor trailer trucks cargo airplanes and ships where a balanced load provides better fuel efficiency and safer ride in these applications there are often conflicting criteria to be satisfied i e to minimize the bins used and to balance the load of each bin subject to a number of practical constraints unlike existing studies that only consider the issue of minimum bins a multiobjective two dimensional mathematical model for bin packing problems with multiple constraints mobpp 2d is formulated in this paper to solve mobpp 2d problems a multiobjective evolutionary particle swarm optimization algorithm moepso is proposed without the need of combining both objectives into a composite scalar weighting function moepso incorporates the concept of pareto s optimality to evolve a family of solutions along the trade off surface extensive numerical investigations are performed on various test instances and their performances are compared both quantitatively and statistically with other optimization methods to illustrate the effectiveness and efficiency of moepso in solving multiobjective bin packing problems

a dimensionally split cartesian cut cell method for hyperbolic conservation laws abstract we present a dimensionally split method for solving hyperbolic conservation laws on cartesian cut cell meshes the approach combines local geometric and wave speed information to determine a novel stabilised cut cell flux and we provide a full description of its three dimensional implementation in the dimensionally split framework of klein et al 1 the convergence and stability of the method are proved for the one dimensional linear advection equation while its multi dimensional numerical performance is investigated through the computation of solutions to a number of test problems for the linear advection and euler equations when compared to the cut cell flux of klein et al it was found that the new flux alleviates the problem of oscillatory boundary solutions produced by the former at higher courant numbers and also enables the computation of more accurate solutions near stagnation points being dimensionally split the method is simple to implement and extends readily to multiple dimensions

cutting stock problems and solution procedures this paper discusses some of the basic formulation issues and solution procedures for solving one and two dimensional cutting stock problems linear programming sequential heuristic and hybrid solution procedures are described for two dimensional cutting stock problems with rectangular shapes we also propose an approach for solving large problems with limits on the number of times an ordered size may appear in a pattern

multiple acceptable solutions in structural model improvement this paper examines the question as to whether it is possible to obtain the correct discrete linear analytical model of an actual structure by the use of analysis and test data it is shown that no such correct model exists an illustration of an application is shown where multiple acceptable and very different analytical models are all representative of the structure the implications regarding methods of improving analytical models and detecting damage are discussed

an exact algorithm for general orthogonal two dimensional knapsack problems abstract we present a new exact tree search procedure for solving two dimensional knapsack problems in which a number of small rectangular pieces each of a given size and value are required to be cut from a large rectangular stock plate the objective is to maximise the value of pieces cut or minimise the wastage we consider the case where there is a maximum number of times that a piece may be used in a cutting pattern the algorithm limits the size of the tree search by using a bound derived from a langrangean relaxation of a 0 1 integer programming formulaton of the problem subgradient optimisation is used to optimise this bound reduction tests derived from both the original problem and the lagrangean relaxation produce substantial computational gains the computational performance of the algorithm indicates that it is an effective procedure capable of solving optimally practical two dimensional cutting problems of medium size

a new exact method for the two dimensional orthogonal packing problem abstract the two dimensional orthogonal packing problem 2 opp consists in determining if a set of rectangles items can be packed into one rectangle of fixed size bin in this paper we propose two exact algorithms for solving this problem the first algorithm is an improvement on a classical branch bound method whereas the second algorithm is based on a new relaxation of the problem we also describe reduction procedures and lower bounds which can be used within enumerative methods we report computational experiments for randomly generated benchmarks which demonstrate the efficiency of both methods the second method is competitive compared to the best previous methods it can be seen that our new relaxation allows an efficient detection of non feasible instances

characterization and modelling of guillotine constraints this paper focuses on guillotine cuts which often arise in real life cutting stock problems in order to construct a solution verifying guillotine constraints the first step is to know how to determine whether a given cutting pattern is a guillotine pattern for this purpose we first characterize guillotine patterns by proving a necessary and sufficient condition then we propose a polynomial algorithm to check this condition based on this mathematical characterization of guillotine patterns we then show that guillotine constraints can be formulated into linear inequalities the performance of the algorithm to check guillotine cutting patterns is evaluated by means of computational results

the three dimensional bin packing problem the problem addressed in this paper is that of orthogonally packing a given set of rectangular shaped items into the minimum number of three dimensional rectangular bins the problem is strongly np hard and extremely difficult to solve in practice lower bounds are discussed and it is proved that the asymptotic worst case performance ratio of the continuous lower bound is an exact algorithm for filling a single bin is developed leading to the definition of an exact branch and bound algorithm for the three dimensional bin packing problem which also incorporates original approximation algorithms extensive computational results involving instances with up to 90 items are presented it is shown that many instances can be solved to optimality within a reasonable time limit

h box methods for the approximation of hyperbolic conservation laws on irregular grids we study generalizations of the high resolution wave propagation algorithm for the approximation of hyperbolic conservation laws on irregular grids that have a time step restriction based on a reference grid cell length that can be orders of magnitude larger than the smallest grid cell arising in the discretization this godunov type scheme calculates fluxes at cell interfaces by solving riemann problems defined over boxes of a reference grid cell length h we discuss stability and accuracy of the resulting so called h box methods for one dimensional systems of conservation laws an extension of the method for the two dimensional case which is based on the multidimensional wave propagation algorithm is also described

an accurate moving boundary formulation in cut cell methods a cut cell method for cartesian meshes to simulate viscous compressible flows with moving boundaries is presented we focus on eliminating unphysical oscillations occurring in cartesian grid methods extended to moving boundary problems in these methods cells either lie completely in the fluid or solid region or are intersected by the boundary for the latter cells the time dependent volume fraction lying in the fluid region can be so small that explicit time integration schemes become unstable and a special treatment of these cells is necessary when the boundary moves a fluid cell may become a cut cell or a solid cell may become a small cell at the next time level this causes an abrupt change in the discretization operator and a suddenly modified truncation error of the numerical scheme this temporally discontinuous alteration is shown to act like an unphysical source term which deteriorates the numerical solution i e it generates unphysical oscillations in the hydrodynamic forces exerted on the moving boundary we develop an accurate moving boundary formulation based on the varying discretization operators yielding a cut cell method which avoids these discontinuities results for canonical two and three dimensional test cases evidence the accuracy and robustness of the newly developed scheme

a genetic algorithm for the two dimensional strip packing problem with rectangular pieces given a set of rectangular pieces and a container of fixed width and variable length the two dimensional strip packing problem 2d spp consists of orthogonally placing all the pieces within the container without overlapping such that the overall length of the layout is minimised until now mainly heuristics for example genetic algorithms ga were proposed for the 2d spp which use encoded solutions that are manipulated by standard operators in this paper a ga for the 2d spp is suggested that works without any encoding of solutions rather fully defined layouts are manipulated as such by means of specific genetic operators two additional constraints namely the orientation constraint and the guillotine constraint can be taken into account the ga is subjected to a comprehensive test using benchmark instances with up to 5000 pieces compared to eleven competing methods from the literature the ga performs best

a dimensionally split cartesian cut cell method for the compressible navier stokes equations abstract we present a dimensionally split method for computing solutions to the compressible navier stokes equations on cartesian cut cell meshes the method is globally second order accurate in the l 1 norm fully conservative and allows the use of time steps determined by the regular grid spacing we provide a description of the three dimensional implementation of the method and evaluate its numerical performance by computing solutions to a number of test problems ranging from the nearly incompressible to the highly compressible flow regimes all the computed results show good agreement with reference results from theory experiment and previous numerical studies to the best of our knowledge this is the first presentation of a dimensionally split cut cell method for the compressible navier stokes equations in the literature

capacitated vehicle routing on trees t v e is a tree with nonnegative weights associated with each of its vertices a fleet of vehicles of capacity q is located at the depot represented by vertex v1 the capacitated vehicle routing problem on trees tcvrp consists of determining vehicle collection routes starting and ending at the depot such that the weight associated with any given vertex is collected by exactly one vehicle the sum of all weights collected by a vehicle does not exceed q a linear combination of the number of vehicles and of the total distance traveled by these vehicles is minimized the tcvrp is shown to be np hard this paper presents lower bounds for the tcvrp based on the solutions of associated bin packing problems we develop a linear time heuristic upper bound procedure and present a bound on its worst case performance these lower and upper bounds are then embedded in an enumerative algorithm numerical results follow

a tabu search algorithm for a two dimensional non guillotine cutting problem in this paper we study a two dimensional non guillotine cutting problem the problem of cutting rectangular pieces from a large stock rectangle so as to maximize the total value of the pieces cut the problem has many industrial applications whenever small pieces have to be cut from or packed into a large stock sheet we propose a tabu search algorithm several moves based on reducing and inserting blocks of pieces have been defined intensification and diversification procedures based on long term memory have been included the computational results on large sets of test instances show that the algorithm is very efficient for a wide range of packing and cutting problems

a representation of curved boundaries for the solution of the navier stokes equations on a staggered three dimensional cartesian grid a method is presented for representing curved boundaries for the solution of the navier stokes equations on a non uniform staggered three dimensional cartesian grid the approach involves truncating the cartesian cells at the boundary surface to create new cells which conform to the shape of the surface we discuss in some detail the problems unique to the development of a cut cell method on a staggered grid methods for calculating the fluxes through the boundary cell faces for representing pressure forces and for calculating the wall shear stress are derived and it is verified that the new scheme retains second order accuracy in space in addition a novel cell linking method is developed which overcomes problems associated with the creation of small cells while avoiding the complexities involved with other cell merging approaches techniques are presented for generating the geometric information required for the scheme based on the representation of the boundaries as quadric surfaces the new method is tested for flow through a channel placed oblique to the grid and flow past a cylinder at re 40 and is shown to give significant improvement over a staircase boundary formulation finally it is used to calculate unsteady flow past a hemispheric protuberance on a plate at a reynolds number of 800 good agreement is obtained with experimental results for this flow

a global optimization algorithm for the three dimensional packing problem abstract this paper introduces a geometrical model which reduces the general three dimensional packing problem to a finite enumeration scheme a very efficient algorithm is derived several tests prove the validity of the algorithm in particular a numerical estimate of the asymtotic performance bound is given