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

new lower bounds for bin packing problems with conflicts abstract the bin packing problem with conflicts bpc consists of minimizing the number of bins used to pack a set of items where some items cannot be packed together in the same bin due to compatibility restrictions the concepts of dual feasible functions dff and data dependent dual feasible functions ddff have been used in the literature to improve the resolution of several cutting and packing problems in this paper we propose a general framework for deriving new ddff as well as a new concept of generalized data dependent dual feasible functions gddff a conflict generalization of ddff the gddff take into account the structure of the conflict graph using the techniques of graph triangulation and tree decomposition then we show how these techniques can be used in order to improve the existing lower bounds

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

approximation algorithms for the oriented two dimensional bin packing problem given a set of rectangular items which may not be rotated and an unlimited number of identical rectangular bins we consider the problem of packing each item into a bin so that no two items overlap and the number of required bins is minimized the problem is strongly np hard and finds practical applications in cutting and packing we discuss a simple deterministic approximation algorithm which is used in the initialization of a tabu search approach we then present a tabu search algorithm and analyze its average performance through extensive computational experiments

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

an exact algorithm for the two dimensional strip packing problem this paper considers the two dimensional strip packing problem 2sp in which a set of rectangular items have to be orthogonally packed without overlapping into a strip of a given width and infinite height by minimizing the overall height of the packing the 2sp is np hard in the strong sense and finds many practical applications we propose reduction procedures lower and upper bounds and an exact algorithm for the 2sp the new lower bounds are both combinatorial bounds and bounds derived from different relaxations of mathematical formulations of the 2sp the new upper bounds are obtained by constructive heuristics based on different strategies to place the items into the strip the new exact method is based on a branch and bound approach computational results on different sets of test problems derived from the literature show the effectiveness of the new lower and upper bounds and of the new exact algorithm

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

integer linear programming models for 2 staged two dimensional knapsack problems we are given a unique rectangular piece of stock material s with height h and width w and a list of m rectangular shapes to be cut from s each shape s type i i 1 m is characterized by a height a width a profit and an upper bound ub i indicating the maximum number of items of type i which can be cut we refer to the two dimensional knapsack tdk as the problem of determining a cutting pattern of s maximizing the sum of the profits of the cut items in particular we consider the classical variant of tdk in which the maximum number of cuts allowed to obtain each item is fixed to 2 and we refer to this problem as 2 staged tdk 2tdk for the 2tdk problem we present two new integer linear programming models we discuss their properties and we compare them with other formulations in terms of the lp bound they provide finally both models are computationally tested within a standard branch and bound framework on a large set of instances from the literature by reinforcing them with the addition of linear inequalities to eliminate symmetries

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

symplectic manifolds with contact type boundaries an example of a 4 dimensional symplectic manifold with disconnected boundary of contact type is constructed a collection of other results about symplectic manifolds with contact type boundaries are derived using the theory ofj holomorphic spheres in particular the following theorem of eliashberg floer mcduff is proved if a neighbourhood of the boundary of v is symplectomorphic to a neighbourhood ofs2n 1 in standard euclidean space and if vanishes on all 2 spheres inv thenv is diffeomorphic to the ballb2n

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

floer novikov homology and applications to lagrangian submanifolds we use a non hamiltonian version of lagrangian floer homology to prove that an exact lagrangian submanifold in the cotangent bundle of the 3 torus t 3 must be diffeomorphic to t 3 this improves a previous result of fukaya seidel and smith

a hybrid heuristic algorithm for the 2d variable sized bin packing problem in this paper we consider the two dimensional variable sized bin packing problem 2dvsbpp with guillotine constraint 2dvsbpp is a well known np hard optimization problem which has several real applications a mixed bin packing algorithm mixpacking which combines a heuristic packing algorithm with the best fit algorithm is proposed to solve the single bin problem and then a backtracking algorithm which embeds mixpacking is developed to solve the 2dvsbpp a hybrid heuristic algorithm based on iterative simulated annealing and binary search named hha is then developed to further improve the results of our backtracking algorithm computational experiments on the benchmark instances for 2dvsbpp show that hha has achieved good results and outperforms existing algorithms

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

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

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

polynomial algorithms for guillotine cutting of a rectangle into small rectangles of two kinds abstract in this paper the problem of optimally guillotine cutting a rectangle a b into small rectangles of two kinds is considered rectangles of the first kind c a i i i have the same width and their heights can be various rectangles of the second kind b j d j j have the same height and their widths can be various the number of occurrences of each small rectangle in a cutting pattern is not restricted similar problems often appear in the furniture industry this cutting problem is reduced to the shortest path problem in a special rectangular grid for which a linear time algorithm is suggested this approach generalizes the approach of e girlich a g tarnowski on polynomial solvability of two multiprocessor scheduling problems mathematical methods of operations research 50 1999 27 51 a g tarnowski advanced polynomial time algorithm for guillotine generalized pallet loading problem in the international scientific collection decision making under conditions of uncertainty cutting packing problems ufa state aviation technical university 1997 pp 93 124 and allows us to construct polynomial algorithms for the guillotine cutting problem considered with a fixed number of small rectangles of two kinds

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

symplectic fixed points and holomorphic spheres letp be a symplectic manifold whose symplectic form integrated over the spheres inp is proportional to its first chern class on the loop space ofp we consider the variational theory of the symplectic action function perturbed by a hamiltonian term in particular we associate to each isolated invariant set of its gradient flow an abelian group with a cyclic grading it is shown to have properties similar to the homology of the conley index in locally compact spaces as an application we show that if the fixed point set of an exact diffeomorphism onp is nondegenerate then it satisfies the morse inequalities onp we also discuss fixed point estimates for general exact diffeomorphisms

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

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

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

1 formes fermees singulieres et groupe fondamental we study the influence of the 1 of a closed manifoldm n n 3 on the foliations ofm defined by closed differential 1 forms with morse singularities of index 0 n every nonexact form is cohomologous to a weakly complete one that is one whose leaf space is of the same type as that of a nonsingular form generically a form has compact leaves or is weakly complete if 1 m has no quotient isomorphic to then every nonexact form onm is weakly complete we also say a form is complete if every path inm is homotopic to either a path transverse to or a path contained in a leaf of completeness of depends only on its de r ham cohomology class the set of complete cohomology classes depends only on 1 m and is related to finitely generated normal subgroups of 1 m with quotient if 1 m is nilpotent or even polycyclic every nonexact form onm is complete on irreducible 3 manifolds a form is complete iff it is cohomologous to a nonsingular one

models and algorithms for three stage two dimensional bin packing we consider the three stage two dimensional bin packing problem 2bp which occurs in real world applications such as glass paper or steel cutting we present new integer linear programming formulations models for a restricted version and the original version of the problem are developed both only involve polynomial numbers of variables and constraints and effectively avoid symmetries those models are solved using cplex furthermore a branch and price b p algorithm is presented for a set covering formulation of the unrestricted problem which corresponds to a dantzig wolfe decomposition of the polynomially sized model we consider column generation stabilization in the b p algorithm using dual optimal inequalities fast column generation is performed by applying a hierarchy of four methods a a fast greedy heuristic b an evolutionary algorithm c solving a restricted form of the pricing problem using cplex and finally d solving the complete pricing problem using cplex computational experiments on standard benchmark instances document the benefits of the new approaches the restricted version of the integer linear programming model can be used to quickly obtain near optimal solutions the unrestricted version is computationally more expensive column generation provides a strong lower bound for 3 stage 2bp the combination of all four pricing algorithms and column generation stabilization in the proposed b p framework yields the best results in terms of the average objective value the average run time and the number of instances solved to proven optimality

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

an approach for constructing nonisospectral hierarchies of evolution equations for a given isospectral lambda t 0 hierarchy of evolution equations the author proposes a simple method of constructing its corresponding nonisospectral lambda t lambda n n or 0 hierarchy of evolution equations closely related to tau symmetries it is crucial to find an initial lax operator w0 and an initial vector field g0 satisfying the key equation w0 l l g0 i in which l i are spectral and identity operators respectively as examples the author presents the corresponding nonisospectral hierarchies of equations and displays the fundamental relations generating symmetry algebras for the kdv hierarchy the akns hierarchy and a new integrable hierarchy

k symmetries and tau symmetries of evolution equations and their lie algebras on the basis of tu s work 1951 the author establishes a more general skeleton on k symmetries and tau symmetries of evolution equations and their lie algebraic structures and discuss carefully the corresponding symmetries and lie algebras for kdv and jaulent miodek hierarchies

combinatorial benders cuts for the strip packing problem we study the strip packing problem in which a set of two dimensional rectangular items has to be packed in a rectangular strip of fixed width and infinite height with the aim of minimizing the height used the problem is important because it models a large number of real world applications including cutting operations where stocks of materials such as paper or wood come in large rolls and have to be cut with minimum waste scheduling problems in which tasks require a contiguous subset of identical resources and container loading problems arising in the transportation of items that cannot be stacked one over the other the strip packing problem has been attacked in the literature with several heuristic and exact algorithms nevertheless benchmark instances of small size remain unsolved to proven optimality in this paper we propose a new exact method that solves a large number of the open benchmark instances within a limited computational effort our method is based on a benders decomposition in whi

a block based layer building approach for the 2d guillotine strip packing problem we examine the 2d strip packing problems with guillotine cut constraint where the objective is to pack all rectangles into a strip with fixed width and minimize the total height of the strip we combine three most successful ideas for the orthogonal rectangular packing problems into a single coherent algorithm 1 packing a block of rectangles instead of a single rectangle in each step 2 dividing the strip into layers and pack layer by layer and 3 unrolling and repacking the top portion of the solutions where usually wasted space occurs computational experiments on benchmark test sets suggest that our approach rivals existing approaches

a simulated annealing algorithm for the capacitated vehicle routing problem with two dimensional loading constraints abstract this paper studies the well known capacitated vehicle routing problem with two dimensional loading constraints 2l cvrp it requires designing a set of min cost routes starting and terminating at the central depot to satisfy customer demands which involve a set of two dimensional rectangular weighted items a simulated annealing algorithm with a mechanism of repeatedly cooling and rising the temperature is proposed to solve the four versions of this problem with or without the lifo constraint and allowing rotation of goods or not an open space based heuristic is employed to identify the feasible loading patterns in addition the data structure trie is used to accelerate the procedure by keeping track of the packing feasibility information of routes examined and also by controlling the effort spent on different routes the proposed algorithm is tested on the widely used instances of 2l cvrp the results show that our approach outperforms all existing algorithms on the four problem versions and reaches or improves the best known solutions for most instances furthermore we compared the impact of different loading constraints and observed some interesting results