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

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

minimax programming under p r invexity abstract minimax programming problems involving p r invex functions with respect to are considered parametric and non parametric necessary and sufficient optimality conditions and duality results for a class of minimax programming problems are obtained under p r invexity assumption on objective and constraint functions some different types of dual problems are introduced and duality theorems is established for each dual problem

staged and constrained two dimensional guillotine cutting problems an and or graph approach abstract an and or graph approach has been proposed by morabito et al to solve non staged and unconstrained two dimensional guillotine cutting problems basically this approach consists of representing cutting patterns as complete paths in an and or graph where nodes and arcs correspond to rectangles and cuts respectively and choosing a search strategy to traverse or enumerate the nodes of the graph in this present paper the and or graph approach is extended to solve staged and constrained problems computational experiments with examples from the literature are performed and indicate that this approach generates good and fast solutions using a microcomputer in addition to the fact that the and or graph approach can handle important constraints it can be easily extended to solve cutting and packing problems with multiple dimensions

an improved bl algorithm for genetic algorithm of the orthogonal packing of rectangles this paper proposes an improved bl algorithm for genetic algorithm of the orthogonal packing of rectangles some improvements on the fitness function of genetic algorithm for the orthogonal packing of rectangles are also suggested solutions of two numerical examples show the effectiveness of these improvements

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

new optimality conditions and duality results of g type in differentiable mathematical programming abstract in this paper a new class of differentiable functions called g invex functions with respect to is introduced by extending the definition of invex functions new necessary optimality conditions of g f john and g karush kuhn tucker type are obtained for differentiable constrained mathematical programming problems the g invexity concept introduced is used to prove the sufficiency of these necessary optimality conditions further a so called g mond weir type dual is formulated and various duality results are also established by assuming the functions involved to be g invex with respect to the same function

equivalence of saddle points and optima and duality for a class of non smooth non convex problems abstract the equivalence between saddle points and optima and duality theorems are established for a much larger class of non smooth non convex problems in which functions are locally lipschitz and are satisfying invex type conditions of hanson and craven

the role of critical exponents in blow up theorems the sequel abstract in 27 fujita showed that for positive solutions the initial value problem in r n for u t u u p with p 1 exhibited the following behavior if p p c 1 2 n then the initial value problem does not have any nontrivial non negative solution existing on r n 0 a global solution whereas if p p c there exist global small data positive solutions as well as solutions which are non global we call such a result a blow up theorem of fujita type in 50 levine discussed the various theorems of this type that had appeared in the literature prior to 1990 in this paper we revisit the literature since 1990

algorithms for packing squares a probabilistic analysis this paper gives a probabilistic performance analysis of simple algorithms for packing lists l n of n squares into a strip or a set of bins it is assumed that the square sizes are drawn independently from the uniform distribution on 0 1 the strip packing problem is to pack the squares into a strip of width 1 so as to minimize the height of the packing let operatorname opt l n denote the height of an optimal strip packing for list l n a simple o n log n approximation algorithm called algorithm a is described and it is proven that the height a l n of a packing of list l n satisfies a l n operatorname opt l n leqq tfrac 1 4 with probability 1 o e c 0 n for a positive constant c 0 it is also shown that e operatorname opt l n 3n 8 theta n 1 3 the bin packing problem is to pack the squares into square bins of size 1 so as to minimize the number of bins let operatorname opt b l n denote the number of bins in an opti

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

interactive procedures in large scale two dimensional cutting stock problems abstract this paper discusses basic problem formulation and solution procedures for solving a particular large scale two dimensional cutting stock problem at a furniture factory in recent years there has been an explosion of interests in this application area the cutting stock problems are met in many branches of industry there is a large economical incentive to find more effective solution procedures and it is easy to compare alternative solution procedures and to identify potential benefits of using a proposed procedure the paper concentrates on the case in which both stock and ordered sizes are rectangular we present a mathematical model of the problem propose an optimization algorithm and describe its basic properties the mathematical model of the optimization problem relies on the gilmore and gomory approach to decrease the total computing time we propose an interactive procedure based on the forecasting of the criterion functions values

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

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

exact bounds for the stochastic upward matching problem we draw at random independently and according to the uniform distribution two sets of n points of the unit square we consider a maximum matching of points of the first set with points of the second set with the restriction that a point can be matched only with a point located at its upper right then with probability close to one the number of unmatched points is of order nl 2 log n 3 4

an exact algorithm for constrained two dimensional two staged cutting problems theconstrained two dimensional cutting c tdc problem consists of determining acutting pattern of a set ofn small rectangular piece types on a rectangular stock plates with lengthl and widthw to maximize the sum of the profits of the pieces to be cut each piece typei i 1 n is characterized by a lengthl i a widthw i a profit or weight c i and an upper demand valueb i the upper demand value is the maximum number of pieces of typei that can be cut ons in this paper we study the two staged c tdc problem noted c 2tdc it is a classical variant of the c tdc where each piece is produced in the final cutting pattern by at most two cuts we solve the c 2tdc problem using an exact algorithm that is mainly based on a bottom up strategy we introduce new lower and upper bounds and propose new strategies that eliminate several duplicate patterns we evaluate the performance of the proposed exact algorithm on problem instances extracted from the literature and compare it to the performance of an existing exact algorithm

a reduction approach for solving the rectangle packing area minimization problem abstract in the rectangle packing area minimization problem rpamp we are given a set of rectangles with known dimensions we have to determine an arrangement of all rectangles without overlapping inside an enveloping rectangle of minimum area the paper presents a generic approach for solving the rpamp that is based on two algorithms one for the 2d knapsack problem kp and the other for the 2d strip packing problem spp in this way solving an instance of the rpamp is reduced to solving multiple spp and kp instances a fast constructive heuristic is used as spp algorithm while the kp algorithm is instantiated by a tree search method and a genetic algorithm alternatively all these spp and kp methods have been published previously finally the best variants of the resulting rpamp heuristics are combined within one procedure the guillotine cutting condition is always observed as an additional constraint the approach was tested on 15 well known rpamp instances above all mcnc and gsrc instances and new best solutions were obtained for 10 instances the computational effort remains acceptable moreover 24 new benchmark instances are introduced and promising results are reported

atm vp based network design abstract we consider the problem of designing an atm vp based leased line backbone network given point to point communication demands having predefined sizes in a network the problem is to find configurations of demand routes and link facilities installed on each edge satisfying all demands at minimum cost under some constraints one of the most important constraints is that a single demand cannot be split over multiple link facilities this is a sort of bin packing constraint we propose an integer programming formulation of the problem and an algorithm to solve it an efficient column generation technique to solve the linear programming relaxation is proposed and a valid inequality is used to strengthen the integer programming formulation the algorithm incorporates the column generation technique and the cutting plane approach into a branch and bound scheme we test the proposed algorithm on some real problems the results show that the algorithm can be used to solve the problems within reasonably small computing times

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

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

fujita type results for parabolic inequalities with gradient terms abstract in this paper we give some fujita type results for strongly p coercive quasilinear parabolic differential inequalities with both a diffusion term and a dissipative term whose prototype is given by u t p u a x u q b x u m u s in r n r u 0 u x 0 u 0 x 0 in r n where p 1 q 0 0 m q 0 s p q m q 1 and a b nonnegative weights which could be singular or degenerate we prove the existence of fujita type exponents q f such that nonexistence of solutions for the inequality occurs when q q f

approximation schemes for covering and packing problems in image processing and vlsi a unified and powerful approach is presented for devising polynomial approximation schemes for many strongly np complete problems such schemes consist of families of approximation algorithms for each desired performance bound on the relative error e o with running time that is polynomial when e is fixed though the polynomiality of these algorithms depends on the degree of approximation e being fixed they cannot be improved owing to a negative result stating that there are no fully polynomial approximation schemes for strongly np complete problems unless np p the unified technique that is introduced here referred to as the shifting strategy is applicable to numerous geometric covering and packing problems the method of using the technique and how it varies with problem parameters are illustrated a similar technique independently devised by b s baker was shown to be applicable for covering and packing problems on planar graphs

a hybrid genetic algorithm for the container loading problem abstract this paper presents a hybrid genetic algorithm ga for the container loading problem with boxes of different sizes and a single container for loading generated stowage plans include several vertical layers each containing several boxes within the procedure stowage plans are represented by complex data structures closely related to the problem to generate offspring specific genetic operators are used that are based on an integrated greedy heuristic the process takes several practical constraints into account extensive test calculations including procedures from other authors vouch for the good performance of the ga above all for problems with strongly heterogeneous boxes

an empirical investigation of meta heuristic and heuristic algorithms for a 2d packing problem abstract in this paper we consider the two dimensional 2d rectangular packing problem where a fixed set of items have to be allocated on a single object two heuristics which belong to the class of packing procedures that preserve bottom left bl stability are hybridised with three meta heuristic algorithms genetic algorithms ga simulated annealing sa naive evolution ne and local search heuristic hill climbing this study compares the hybrid algorithms in terms of solution quality and computation time on a number of packing problems of different size in order to show the effectiveness of the design of the different algorithms their performance is compared to random search rs and heuristic packing routines

an effective quasi human based heuristic for solving the rectangle packing problem abstract in this paper we introduce an effective deterministic heuristic less flexibility first for solving the classical np complete rectangle packing problem many effective heuristics implemented for this problem are cpu intensive and non deterministic in nature others including the polynomial approximation methodology j assoc comput mach 32 1 1985 130 are too laborious for practical problem sizes the technique we propose is inspired and developed by enhancing some rule of thumb guidelines resulting from the generation long work experience of human professionals in ancient days although the less flexibility first heuristic is a deterministic algorithm the results are very encouraging this algorithm can consistently produce packing densities of around 99 on most randomly generated large examples as compared with the recent results of a well known simulated annealing based rectangle packing rp algorithm ieee trans computer aided design integrated circuits systems 17 1 1998 60 the results are much better both in less dead space 2 4 vs 6 7 and much less cpu time 9 57 vs 331 78 seconds experimenting our heuristics on a public rectangle packing data set covering instances of 16 97 rectangles the average unpack ratio is quite satisfactory 0 92 for bounding boxes limited to be optimum and 2 68 for the completed packing while most cases spend only a few minutes in cpu time

on blow up results for solutions of inhomogeneous evolution equations and inequalities abstract the main purpose of this work is to obtain blow up results for solutions of the inequality u t u u u q x on the half space r 1 r n where n 1 0 q 1 and r n 0 is a nontrivial locally integrable function and in particular to establish that for n 2 the critical blow up exponent q n 1 n 2 belongs to the blow up case

lp models for bin packing and cutting stock problems abstract we review several linear programming lp formulations for the one dimensional cutting stock and bin packing problems namely the models of kantorovich gilmore gomory onecut models as in the dyckhoff stadtler approach position indexed models and a model derived from the vehicle routing literature we analyse some relations between the corresponding lp relaxations and their relative strengths and refer how to derive branching schemes that can be used in the exact solution of these problems using branch and price

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

on minimax fractional optimality conditions with f ρ convexity abstract we establish the sufficient conditions for generalized fractional programming in the framework of f convex functions when the sufficient conditions are utilized one parametric dual problem and two parametric free dual problems may be formulated and duality results are derived

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

on the critical fujita exponents for solutions of quasilinear parabolic inequalities abstract this work is devoted to the study of critical blow up phenomena for wide classes of quasilinear parabolic equations and inequalities the model example for this treatment is well known and comes from the theory of turbulent diffusion u t div x u m 1 u x 2 u m 1 u x u q 1 u m 1 1 we obtain the existence of critical blow up exponents for nonnegative weak solutions of inequalities of the type that belong only locally to the corresponding sobolev spaces in the half space r 1 r n n 1 we impose no conditions on the behavior of solutions on the hyperplane t 0 so that the results obtained in the paper are of liouville type the approach developed herein is directly applicable to the study of analogous problems for systems of quasilinear elliptic and parabolic equations and inequalities and its elliptic analogue has recently been developed by the authors

optimality and duality for minmax problems involving arcwise connected and generalized arcwise connected functions abstract in this paper we establish necessary optimality conditions for a static minmax programming problem of the form min max y y x y subject tog x 0 in terms of the right derivatives of the functions with respect to the same arc various theorems giving sufficient optimality conditions are proved a mond weir type dual is proposed and duality results are established under arcwise connectedness and generalized arcwise connectedness assumptions

best first search methods for constrained two dimensional cutting stock problems best first search is a widely used problem solving technique in the field of artificial intelligence the method has useful applications in operations research as well here we describe an application to constrained two dimensional cutting stock problems of the following type a stock rectangle s of dimensions l w is supplied there are n types of demanded rectangles r1 r2 rn with the ith type having length li width wi value vi and demand constraint bi it is required to produce from the stock rectangle s ai copies of ri 1 i n to maximize a1v1 a2v2 anvn subject to the constraints ai bi only orthogonal guillotine cuts are permitted all parameters are integers a best first tree search algorithm based on wang s bottom up approach is described that guarantees optimal solutions and is more efficient than existing methods

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

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

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

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

online packing of arbitrary sized items into designated and multipurpose bins abstract we consider an online multitype bin packing problem with two item types this setting can be motivated by transport applications in which some items may be shipped either in dry shipping containers or more costly refrigerated ones while other items can only be transported in refrigerated containers the problem was introduced by goldberg and karhi omega 71 85 92 2017 who focused on the case of up to two item types three bin types and only two item sizes for this special case a tight also known as optimal absolute competitive ratio was shown of approximately 1 618 here we consider the general problem of arbitrary item sizes and show a lower bound on the absolute competitive ratio of any online algorithm that is a function of the bin costs this bound in the worst case is approximately 1 781 we then extend the first fit method to our problem setting and prove an absolute competitive ratio bound that is a function of the bin costs in the worst case this upper bound is approximately 1 930 in addition an upper bound of 1 750 is established on the worst case asymptotic as the number of items grows large competitive ratio

an investigation into two bin packing problems with ordering and orientation implications this paper considers variants of the one dimensional bin packing and stock cutting problem in which both the ordering and orientation of items in a container influences the validity and quality of a solution two new real world problems of this type are introduced the first that involves the creation of wooden trapezoidal shaped trusses for use in the roofing industry the second that requires the cutting and scoring of rectangular pieces of cardboard in the construction of boxes to tackle these problems two variants of a local search based approximation algorithm are proposed the first that attempts to determine item ordering and orientation via simple heuristics the second that employs more accurate but costly branch and bound procedures we investigate the inevitable trade off between speed and accuracy that occurs with these variants and highlight the circumstances under which each scheme is advantageous

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

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

on sufficiency of the kuhn tucker conditions if all the functions fo x f x f x are convex on c then these conditions are also sufficient various classes of functions have been defined for the purpose of weakening this limitation of convexity in mathematical programming mangasarian 11 has speculated that pseudo convexity of x and quasiconvexity off x are the weakest conditions that can be imposed so that the above conditions are sufficient for optimality it will be shown that there are other wide classes of functions for which the conditions are sufficient 545

the optimality conditions for generalized minimax programming in this paper we first weaken some assumptions in jiao et al 2005 9 and discuss the continuity and the directional differentiability of the objective function furthermore we investigate the optimality conditions for unconstrained and constrained generalized minimax programming these optimality conditions are mixed ones that include different conditions at different directions which is suitable for the non differentiable objective function for constrained problem we exploit two kinds of analysis methods to obtain some different optimality conditions moreover we show these optimality conditions by some examples

an efficient approximation scheme for variable sized bin packing in the classical bin packing problem one is required to pack a given list of items into the smallest possible number of unit sized bins because this problem is np complete researchers have tried to find efficient approximation algorithms that solve this problem in a reasonable amount of time if we let a i be the number of bins used by algorithm a to pack a list of items i and textit opt i be the minimum number of bins necessary to pack i one defines the asymptotic performance ratio of a to be a i textit opt i as textit opt i tends to infinity de la vega and lueker presented an approximation scheme which for any in 0 yields an approximation algorithm with performance ratio 1 varepsilon this scheme has time complexity polynomial in n the number of items but exponential in 1 varepsilon karmarkar and karp extended this scheme to one that has time complexity polynomial in n and 1 varepsilon in the variable sized bin packing problem one

wolfe type duality involving b η invex functions for a minmax programming problem abstract a sufficient optimality theorem is proved for a certain minmax programming problem under the assumptions of proper b invexity conditions on the functions involved in the objective and in the constraints next a dual is presented for such a problem and duality theorems relating the primal and the dual are proved the dual for a minimax generalized fractional programming problem is presented as special case of the main problem considered in the paper

existence and nonexistence results for higher order semilinear evolution inequalities with critical potential we prove nonexistence results for higher order semilinear evolution equations and inequalities of the form ku tk u x 2u u qin rn 0 where n 22 2 this problem can be seen as a higher order evolution version of the nonlinear wheeler de witt equation which appears in the theory of quantum cosmology in order to show that our result is sharp in the parabolic case we establish the existence of positive solutions to the semilinear equation u t u x 2u uqin rn 0 for 0 the nonexistence results are based on the test function method developed by mitidieri pohozaev tesei and veron the existence result is established by the construction of an explicit global solution of the semilinear parabolic inequality

shelf algorithms for two dimensional packing problems this paper studies two approximation algorithms for packing rectangles using the two dimensional packing model of baker coffman and rivest siam j comput 9 1980 pp 846 855 the algorithms studied are called next fit and first fit shelf algorithms respectively they differ from previous algorithms by packing the rectangles in the order given the previous algorithms required sorting the rectangles by decreasing height or width before packing them which is not possible in some applications the shelf algorithms are a modification of the next fit and first fit decreasing height level algorithms of coffman garey johnson and tarjan siam j comput 9 1980 pp 808 826 each shelf algorithm takes a parameter r it is shown that by choosing r appropriately the asymptotic worst case performance of the shelf algorithms can be made arbitrarily close to that of the next fit and first fit level algorithms without the restriction that items must be packed in order of decreasing height nonasymptoti

nonsmooth minimax programming under locally lipschitz φ ρ invexity abstract minimax programming problems involving locally lipschitz invex functions are considered the parametric and non parametric necessary and sufficient optimality conditions for a class of nonsmooth minimax programming problems are obtained under nondifferentiable invexity assumption imposed on objective and constraint functions when the sufficient conditions are utilized parametric and non parametric dual problems in the sense of mond weir and wolfe may be formulated and duality results are derived for the considered nonsmooth minimax programming problem with the reference to the said functions we extend some results of optimality and duality for a larger class of nonsmooth minimax programming problems

dynamic reduction heuristics for the rectangle packing area minimization problem abstract the rectangle packing area minimization problem is a key sub problem of floorplanning in vlsi design this problem places a set of axis aligned two dimensional rectangular items of given sizes onto a rectangular plane such that no two items overlap and the area of the enveloping rectangle is minimized this paper presents a dynamic reduction algorithm that transforms an instance of the original problem to a series of instances of the rectangle packing problem by dynamically determining the dimensions of the enveloping rectangle we define an injury degree to evaluate the possible negative impact for candidate placements and we propose a least injury first approach for solving the rectangle packing problem next we incorporate a compacting approach to compact the resulting layout by alternatively moving the items left and down toward a bottom left corner such that we may obtain a smaller enveloping rectangle we also show the feasibility compactness non inferiority and halting properties of the compacting approach comprehensive experiments were conducted on 11 mcnc and gsrc benchmarks and 28 instances reported in the literature the experimental results show the high efficiency and effectiveness of the proposed dynamic reduction algorithm especially on large scale instances with hundreds of items

fuzzy bin packing problem this paper deals with the fuzzy bin packing problem that is a packing problem of non rigid rectangles into an open rectangular bin this problem is different from the conventional bin packing problem which considers only rigid rectangles the goal of the fuzzy bin packing problem is to minimize both the height of a packing and the extra cost due to the reduction of each piece the total cost of the problem is represented as the sum of the height cost and the extra cost due to reductions of the pieces which is called reduction cost because the conventional bin packing problem itself is an np hard problem the presented optimization method assumes that an initial packing for non reduced pieces has already been found a closed form solution is presented for fuzzy bin packing problems in which fuzzy numbers are triangular and the reduction cost is given by a quadratic function

variable sized bin packing in the classical bin packing problem one seeks to pack a list of pieces in the minimum space using unit capacity bins this paper addresses the more general problem in which a fixed collection of bin sizes is allowed three efficient approximation algorithms are described and analyzed they guarantee asymptotic worst case performance bounds of 2 3 2 and 4 3

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

asymptotic methods in the probabilistic analysis of sequencing and packing heuristics recently there has been considerable interest in the average case performance of heuristics this paper pursues that interest where it concerns sequencing and packing problems in particular we survey the methods that have been used to obtain formal probabilistic analyses of heuristics for makespan scheduling and one dimensional bin packing in so doing we present many of the key results in these research areas

an aptas for generalized cost variable sized bin packing bin packing is a well known problem which has a large number of applications classical bin packing is a simple model in which all bins are identical in the bin packing problem with variable sized bins we are given a supply of a variety of sizes this latter model assumes however that the cost of a bin is always defined to be its exact size in this paper we study the more general problem where an available bin size is associated with a fixed cost which may be smaller or larger than its size the costs of different bin sizes are unrelated this generalized problem has various applications in storage and scheduling in order to generalize previous work we design new rounding and allocation methods our main result is an asymptotic polynomial time approximation scheme for the generalized problem

algorithms for the variable sized bin packing problem in this paper we consider the variable sized bin packing problem where the objective is to minimize the total cost of used bins when the cost of unit size of each bin does not increase as the bin size increases two greedy algorithms are described and analyzed in three special cases a the sizes of items and bins are divisible respectively b only the sizes of bins are divisible and c the sizes of bins are not divisible here we say that a list of numbers a1 a2 am are divisible when aj exactly divides aj 1 for each 1 j m in the case of a the algorithms give optimal solutions and in the case of b each algorithm gives a solution whose value is less than 119c b 4119 where c b is the optimal value in the case of c each algorithm gives a solution whose value is less than 32c b 1

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

strong and weak convexity of sets and functions in this paper we study two classes of sets strongly and weakly convex sets for each class we derive a series of properties which involve either the concept of supporting ball an obvious extension of the concept of supporting hyperplane or the normal cone to the set we also study a class of functions denoted convex which satisfy for arbitrary points x1 and x2 and any value 0 1 the classical inequality of convex functions up to a term 1 x1 x2 2 depending on the sign of the constant the function is said to be strongly or weakly convex we provide characteristic properties of this class of sets and we relate it to strongly and weakly convex sets via the epigraph and the level sets finally we give three applications a separation theorem a sufficient condition for global optimum of a nonconvex programming problem and a sufficient geometrical condition for a set to be a manifold

an and or graph approach to the solution of two dimensional non guillotine cutting problems this work is concerned with unconstrained two dimensional non guillotine cutting problems where a rectangular plate is cut into a number of rectangular pieces in such a way as to optimize an objective function we reduce the state space of the problem by considering non guillotine cutting patterns which are combinations of guillotine and simple non guillotine cuts these cutting patterns are represented as complete paths in an and or graph and a branch and bound method is described we provide rules and heuristics that reduce the graph search and present computational results from randomly generated examples as well as from the literature

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

data set generation for rectangular placement problems abstract this paper describes a recursive process for generating data sets of rigid rectangles that can be placed into rectangular regions with zero waste the generation procedure can be modified to guarantee that the aspect and area ratios of the rectangles in the generated data sets satisfy user specified parameters this recursive process can thus be employed to create a variety of data sets that can be used to evaluate the efficiency and scalability of rectangular cutting and packing algorithms

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

asymptotic bounds of solutions for a periodic doubly degenerate parabolic equation abstract this paper is concerned with a doubly degenerate parabolic equation with logistic periodic sources we are interested in the discussion of the asymptotic behavior of solutions of the initial boundary value problem in this paper we first establish the existence of non trivial nonnegative periodic solutions by a monotonicity method then by using the moser iterative method we obtain an a priori upper bound of the nonnegative periodic solutions by means of which we show the existence of the maximum periodic solution and asymptotic bounds of the nonnegative solutions of the initial boundary value problem we also prove that the support of the non trivial nonnegative periodic solution is independent of time

proper efficiency conditions and duality results for nonsmooth vector optimization in banach spaces under φ ρ invexity abstract in this paper a new class of nonsmooth nonconvex multiobjective programming problems in banach spaces is considered the sufficient optimality conditions for efficiency and proper efficiency are established under nonsmooth invexity and generalized invexity notions after extending these concepts to the functions between banach spaces further mond weir duality theorems are also established under invexity for the considered vector optimization problems in banach spaces

an asymptotic approximation scheme for the concave cost bin packing problem abstract we consider a generalized one dimensional bin packing model in which the cost of a bin is a nondecreasing concave function of the utilization of the bin we show that for any given positive constant there exists a polynomial time approximation algorithm with an asymptotic worst case performance ratio of no more than 1