a goal driven approach to the 2d bin packing and variable sized bin packing problems abstract in this paper we examine the two dimensional variable sized bin packing problem 2dvsbpp where the task is to pack all given rectangles into bins of various sizes such that the total area of the used bins is minimized we partition the search space of the 2dvsbpp into sets and impose an order on the sets and then use a goal driven approach to take advantage of the special structure of this partitioned solution space since the 2dvsbpp is a generalization of the two dimensional bin packing problem 2dbpp our approach can be adapted to the 2dbpp with minimal changes computational experiments on the standard benchmark data for both the 2dvsbpp and 2dbpp shows that our approach is more effective than existing approaches in literature

generalized selection combining based on the log likelihood ratio we propose a generalized selection combining gsc scheme for binary signaling in which a subset of diversity branches providing the largest magnitude of log likelihood ratio llr are selected and combined it is shown that the bit error probability with maximum ratio combining mrc or square law combining of l branches is identical to that with llr based gsc of l 2 branches we also propose a simple but suboptimal gsc based on a noncoherent envelope detection and discuss its potential advantages over the conventional signal to noise ratio based gsc and mrc

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

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

a two dimensional strip cutting problem with sequencing constraint we study a strip cutting problem that arises in the production of corrugated cardboard in this context rectangular items of different sizes are obtained by machines called corrugators that cut strips of large dimensions according to particular schemes containing at most two types of items because of buffer restrictions these schemes have to be sequenced in such a way that at any moment at most two types of items are in production and not completed yet sequencing constraint we show that the problem of finding a set of schemes of minimum trim loss that satisfies an assigned demand for each item size is strongly np hard even if the sequencing constraint is relaxed then we present two heuristics for the problem with the sequencing constraint both based on a graph characterization of the feasible solutions the first heuristic is a two phase procedure based on a mixed integer linear programming model the second heuristic follows a completely combinatorial approach and consists of solving a suitable sequence of minimum cost matching problems for both procedures an upper bound on the number of schemes setups is found finally a computational study comparing the quality of the heuristic solutions with respect to an lp lower bound is reported

an integrated cutting stock and sequencing problem in this paper an integrated problem formulated as an integer linear programming problem is presented to find an optimal solution to the cutting stock problem under particular pattern sequencing constraints the solution uses a lagrangian approach the dual problem is solved using a modified subgradient method a heuristic for the integrated problem is also presented the computational results obtained from a set of unidimensional instances that use these procedures are reported

a near optimal solution to a two dimensional cutting stock problem we present an asymptotic fully polynomial approximation scheme for strip packing or packing rectangles into a rectangle of fixed width and minimum height a classicalnp hard cutting stock problem the algorithm based on a new linear programming relaxation finds a packing ofn rectangles whose total height is within a factor of 1 e of optimal up to an additive term and has running time polynomial both in n and in 1 e

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

a two stage intelligent search algorithm for the two dimensional strip packing problem this paper presents a two stage intelligent search algorithm for a two dimensional strip packing problem without guillotine constraint in the first stage a heuristic algorithm is proposed which is based on a simple scoring rule that selects one rectangle from all rectangles to be packed for a given space in the second stage a local search and a simulated annealing algorithm are combined to improve solutions of the problem in particular a multi start strategy is designed to enhance the search capability of the simulated annealing algorithm extensive computational experiments on a wide range of benchmark problems from zero waste to non zero waste instances are implemented computational results obtained in less than 60 seconds of computation time show that the proposed algorithm outperforms the supposedly excellent algorithms reported recently on average it performs particularly better for large instances

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

guided local search and its application to the traveling salesman problem abstract the traveling salesman problem tsp is one of the most famous problems in combinatorial optimization in this paper we are going to examine how the techniques of guided local search gls and fast local search fls can be applied to the problem gls sits on top of local search heuristics and has as a main aim to guide these procedures in exploring efficiently and effectively the vast search spaces of combinatorial optimization problems gls can be combined with the neighborhood reduction scheme of fls which significantly speeds up the operations of the algorithm the combination of gls and fls with tsp local search heuristics of different eiciency and effectiveness is studied in an effort to determine the dependence of gls on the underlying local search heuristic used comparisons are made with some of the best tsp heuristic algorithms and general optimization techniques which demonstrate the advantages of gls over alternative heuristic approaches suggested for the problem

a variable neighborhood search for the capacitated vehicle routing problem with two dimensional loading constraints abstract this paper addresses the capacitated vehicle routing problem with two dimensional loading constraints 2l cvrp which is a generalized capacitated vehicle routing problem in which customer demand is a set of two dimensional rectangular weighted items the objective is to design the route set of minimum cost for a homogenous fleet of vehicles starting and terminating at a central depot to serve all the customers all the items packed in one vehicle must satisfy the two dimensional orthogonal packing constraints a variable neighborhood search is proposed to address the routing aspect and a skyline heuristic is adapted to examine the loading constraints to speed up the search process an efficient data structure trie is utilized to record the loading feasibility information of routes but also to control the computational effort of the skyline spending on the same route the effectiveness of our approach is verified through experiments on widely used benchmark instances involving two distinct versions of loading constraints unrestricted and sequential versions numerical experiments show that the proposed method outperforms all existing methods and improves or matches the majority of best known solutions for both problem versions

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

construction heuristics for two dimensional irregular shape bin packing with guillotine constraints the paper examines a new problem in the irregular packing literature that has many applications in industry two dimensional irregular convex bin packing with guillotine constraints due to the cutting process of certain materials cuts are restricted to extend from one edge of the stock sheet to another called guillotine cutting this constraint is common place in glass cutting and is an important constraint in two dimensional cutting and packing problems in the literature various exact and approximate algorithms exist for finding the two dimensional cutting patterns that satisfy the guillotine cutting constraint however to the best of our knowledge all of the algorithms are designed for solving rectangular cutting where cuts are orthogonal with the edges of the stock sheet in order to satisfy the guillotine cutting constraint using these approaches when the pieces are non rectangular practitioners implement a two stage approach first pieces are enclosed within rectangle shapes and then the rectangles are packed clearly imposing this condition is likely to lead to additional waste this paper aims to generate guillotine cutting layouts of irregular shapes using a number of strategies the investigation compares three two stage approaches one approximates pieces by rectangles the other two approximate pairs of pieces by rectangles using a cluster heuristic or phi functions for optimal clustering all three approaches use a competitive algorithm for rectangle bin packing with guillotine constraints further we design and implement a one stage approach using an adaptive forest search algorithm experimental results show the one stage strategy produces good solutions in less time over the two stage approach

the bottomn left bin packing heuristic an efficient implementation we study implementations of the bottom left heuristic for two dimensional bin packing to pack n rectangles into an infinite vertical strip of fixed width the strategy considered here places each rectangle in turn as low as possible in the strip in a left justified position for reasons of simplicity and good performance the bottom left heuristic has long been a favorite in practical applications however the best implementations found so far require a number of steps o n3 in this paper we present an implementation of the bottom left heuristic which requires linear space and quadratic time the algorithm is fairly practical and we believe that even for relatively small values of n it gives the most efficient implementation of the heuristic to date it proceeds by first determining all the possible locations where the next rectangle can fit then selecting the lowest of them it is optimal among all the algorithms based on this exhaustive strategy and its generality makes it adaptable to different packing heuristics

improved local search algorithms for the rectangle packing problem with general spatial costs the rectangle packing problem with general spatial costs is to pack given rectangles without overlap in the plane so that the maximum cost of the rectangles is minimized this problem is very general and various types of packing problems and scheduling problems can be formulated in this form for this problem we have previously presented local search algorithms using a pair of permutations of rectangles to represent a solution in this paper we propose speed up techniques to evaluate solutions in various neighborhoods computational results for the rectangle packing problem and a real world scheduling problem exhibit good prospects of the proposed techniques

on the symbol error probability of maximum selection diversity reception schemes over a rayleigh fading channel the symbol error probability of two selection schemes namely maximum signal to noise ratio m gamma selection and maximum output mo selection for m ary multidiversity reception over a rayleigh fading channel are discussed the symbol error probability of the mo scheme is lower than that of the m gamma scheme the more diversity receptions that are used the larger is the difference a simple expression of crossover average signal to noise ratio per bit is presented as a guideline for increasing the number of diversity receptions

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

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

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

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

performance bounds for orthogonal oriented two dimensional packing algorithms two orthogonal oriented two dimensional packing algorithms are presented and their behavior is analyzed under several assumptions for one of the algorithms we show that the ratio of the height used by the algorithm to the optimal height is asymptotically bounded by frac 4 3 this bound is an improvement over similar bounds for previously proposed algorithms

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

using tree search bounds to enhance a genetic algorithm approach to two rectangle packing problems a popular approach when using a genetic algorithm in the solution of constrained problems is to exploit problem specific information by representing individuals as ordered lists a construction heuristic is then often used as a decoder to produce a solution from each ordering in such a situation further information is often available in the form of bounds on the partial solutions this paper uses two two dimensional packing problems to illustrate how this information can be incorporated into the genetic operators to improve the overall performance of the search our objective is to use the packing problems as a vehicle for investigating a series of modifications of the genetic algorithm approach based on information from bounds on partial solutions

a dynamic adaptive local search algorithm for the circular packing problem abstract this paper studies the circular packing problem cpp which consists of packing n non identical circles ci of known radius ri i n 1 n into the smallest containing circle c the objective is to determine the coordinates xi yi of the center of ci i n as well as the radius r and center x y of c this problem which is a variant of the two dimensional open dimension problem is solved using a two step dynamic adaptive local search algorithm at each iteration the algorithm identifies the set of potential best local positions of a circle ci i n given the positions of the previously packed circles and determines for each of these positions the coordinates and radius of the smallest containing circle the best local position minimizes the radius of the current containing circle that is every time an additional circle is packed both the center and the radius of the containing circle are dynamically updated and the smallest containing circle is known the experimental results reflect the good performance of the algorithm

parameterized complexity of strip packing and minimum volume packing abstract we study the parameterized complexity of minimum volume packing and strip packing in the two dimensional version the input consists of a set of rectangles s with integer side lengths in the minimum volume packing problem given a set of rectangles s and a number k the goal is to decide if the rectangles can be packed in a bounding box of volume at most k in the strip packing problem we are given a set of rectangles s numbers w and k the objective is to find if all the rectangles can be packed in a box of dimensions w k we prove that the 2 dimensional volume packing is in fpt by giving an algorithm that runs in 2 2 k k o 1 time we also show that strip packing is w 1 hard even in two dimensions and give an fpt algorithm for a special case of strip packing some of our results hold for the problems defined in higher dimensions as well

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

a strip packing algorithm with absolute performance bound 2 this paper proposes a new approximation algorithm m for packing rectangles into a strip with unit width and unbounded height so as to minimize the total height of the packing it is shown that for any list l of rectangles m l leq 2 cdot mbox opt l where m l is the strip height actually used by the algorithm m when applied to l and opt l is the minimum possible height within which the rectangles in l can be packed

applying self adaptive evolutionary algorithms to two dimensional packing problems using a four corners heuristic abstract this paper proposes a four corners heuristic for application in evolutionary algorithms eas applied to two dimensional packing problems the four corners fc heuristic is specifically designed to increase the search efficiency of eas experiments with the fc heuristic are conducted on 31 problems from the literature both with rotations permitted and without rotations permitted using two different ea algorithms a self adaptive parallel recombinative simulated annealing prsa algorithm and a self adaptive genetic algorithm ga results on bin packing problems yield the smallest trim losses we have seen in the published literature with the fc heuristic zero trim loss was achieved on problems of up to 97 rectangles a comparison of the self adaptive ga to fixed parameter gas is presented and the benefits of self adaption are highlighted

lyons pemantle peres monotonicity problem for high biases the speed v of a biased random walk on a galton watson tree without leaves is increasing for 1160 2014 wiley periodicals inc

a column generation heuristic for the two dimensional two staged guillotine cutting stock problem with multiple stock size abstract we consider a two dimensional cutting stock problem where stock of different sizes is available and a set of rectangular items has to be obtained through two staged guillotine cuts we propose a heuristic algorithm based on column generation which requires as its subproblem the solution of a two dimensional knapsack problem with two staged guillotines cuts a further contribution of the paper consists in the definition of a mixed integer linear programming model for the solution of this knapsack problem as well as a heuristic procedure based on dynamic programming computational experiments show the effectiveness of the proposed approach which obtains very small optimality gaps and outperforms the heuristic algorithm proposed by cintra et al 3

linear models for the approximate solution of the problem of packing equal circles into a given domain abstract the linear models for the approximate solution of the problem of packing the maximum number of equal circles of the given radius into a given closed bounded domain g are proposed we construct a grid in g the nodes of this grid form a finite set of points t and it is assumed that the centers of circles to be packed can be placed only at the points of t the packing problems of equal circles with the centers at the points of t are reduced to 0 1 linear programming problems a heuristic algorithm for solving the packing problems based on linear models is proposed this algorithm makes it possible to solve packing problems for arbitrary connected closed bounded domains independently of their shape in a unified manner numerical results demonstrating the effectiveness of this approach are presented

an integer programming model for two and three stage two dimensional cutting stock problems abstract in this paper an integer programming model for two dimensional cutting stock problems is proposed in the problems addressed it is intended to cut a set of small rectangular items of given sizes from a set of larger rectangular plates in such a way that the total number of used plates is minimized the two stage and three stage exact and non exact problems are considered other issues are also addressed as the rotation of items the length of the cuts and the value of the remaining plates the new integer programming model can be seen as an extension of the one cut model proposed by dyckhoff for the one dimensional cutting stock problem in the proposed model each decision variable is associated with cutting one item from a plate or from a part of a plate resulting from previous cuts residual plates comparative computational results of the proposed model and of models from the literature are presented and discussed

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

some experiments with simulated annealing techniques for packing problems abstract experiments with a variety of combinatorial optimisation problems have shown that the simulated annealing algorithm is an effective local search heuristic method the basic requirements of the algorithm are a neighbourhood structure on the set of feasible solutions and a number of parameters which govern the acceptance or rejection of new solutions generated during the search the quality of the solution is very sensitive to both these factors this paper is concerned with the application of the simulated annealing approach to packing problems and describes a series of experiments carried out to ascertain the effectiveness of the method for such problems and the most appropriate neighbourhood structure to use and the best parameters to apply

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

optimized selection diversity for rayleigh fading channels this paper considers the choice of the number of diversity branches for minimizing the bit error rate of a selection diversity system using noncoherent binary frequency shift keying modulation for transmission over a rayleigh fading gaussian channel an exact expression for the optimum number of branches l sel ast is given the resulting probability of bit error decreases exponentially with the square root of the energy per bit

new and improved level heuristics for the rectangular strip packing and variable sized bin packing problems we consider two types of orthogonal oriented rectangular two dimensional packing problems the first is the strip packing problem for which four new and improved level packing algorithms are presented two of these algorithms guarantee a packing that may be disentangled by guillotine cuts these are combined with a two stage heuristic designed to find a solution to the variable sized bin packing problem where the aim is to pack all items into bins so as to minimise the packing area this heuristic packs the levels of a solution to the strip packing problem into large bins and then attempts to repack the items in those bins into smaller bins in order to reduce wasted space the results of the algorithms are compared to those of seven level packing heuristics from the literature by means of a large number of strip packing benchmark instances it is found that the new algorithms are an improvement over known level packing heuristics for the strip packing problem the advancements made by the new and improved algorithms are limited in terms of utilised space when applied to the variable sized bin packing problem however they do provide results faster than many existing algorithms

an optimization algorithm for the ordered open end bin packing problem the ordered open end bin packing problem is a variant of the bin packing problem in which the items to be packed are sorted in a given order and the capacity of each bin can be exceeded by the last item packed into the bin we present a branch and price algorithm for its exact optimization the pricing subproblem is a special variant of the binary knapsack problem in which the items are ordered and the last one does not consume capacity we present a specialized optimization algorithm for this subproblem the speed of the column generation algorithm is improved by subgradient optimization steps allowing for multiple pricing and variable fixing computational results are presented on instances of different sizes and items with different correlations between their size and their position in the given order

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