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

resource constrained assignment scheduling many resource constrained assignment scheduling problems can be modeled as 0 1 assignment problems with side constraints apsc unlike the well known assignment problem of linear programming apsc is np complete in this paper we define a branch and bound algorithm for solving apsc to optimality the algorithm employs a depth first polychotomous branching strategy in conjunction with a bounding procedure that utilizes subgradient optimization we also define a heuristic procedure for obtaining approximate solutions to apsc the heuristic uses subgradient optimization to guide the search for a good solution as well as to provide a bound on solution quality we present computational experience with both procedures applied to over 400 test problems the algorithm is demonstrated to be effective across three different classes of resource constrained assignment scheduling problems the heuristic generates solutions for these problems that are on average within 0 8 of optimality

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

regular packing of congruent polygons on the rectangular sheet this paper deals with the problem of searching for the fragment of dense lattice dense double lattice packing of congruent polygons on the rectangular sheet which is optimal by the dense factor in doing so we consider only those packings for which the polygons form rows columns parallel to one of the sheet sides the problem arises in many branches of industry in particular in footwear industry the mathematical model of the problem is built simplification of the mathematical model reducing of the problem dimensionality and its decomposition on a number of independent subproblems is done on the basis of analysis of the properties of the mathematical model the approximate method to solve the problem is suggested the example of solving of the real problem is given

an n tet graph approach for non guillotine packings of n dimensional boxes into an n container abstract in this paper we propose a simple recursive uniform algorithm for the problem of packing n dimensional boxes into an n container we are particularly concerned about the special case n 3 where the boxes can be packed in a given subset of their six possible positionings our method studies symmetries in the packings by the use of an ordered set of three directed graphs with the same edges a 3 tet or triad and induced smaller structures of the same kind named minors with the method degeneracy and symmetry issues which curtail the implicit enumeration to practically acceptable running times become transparent in order to illustrate the performance of the algorithm computational results from solving randomly generated 3 d examples are presented and compared with the ones of a layers and knapsack approach the present study has real world applications for the problems of pallet and container loading

on multi index assignment polytopes we investigate an integer programming model for multi dimensional assignment problems this model enables us to establish the dimension for entire families of assignment polytopes thus unifying and generalising previous results in particular we establish the dimension of the linear assignment polytope as well as that of every axial and planar assignment polytope further for the axial polytopes we identify a family of clique facets we also give a necessary condition for the existence of a solution for assignment problems

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

solving irregular strip packing problems by hybridising simulated annealing and linear programming abstract in this paper a hybrid algorithm to solve irregular strip packing problems is presented the metaheuristic simulated annealing is used to guide the search over the solution space while linear programming models are solved to generate neighbourhoods during the search process these linear programming models which are used to locally optimise the layouts derive from the application of compaction and separation algorithms computational tests were run using instances that are commonly used as benchmarks in the literature the best results published so far have been improved by this new hybrid packing algorithm

a 2 exchange heuristic for nesting problems abstract this paper describes a new heuristic for the nesting problem based on a 2 exchange neighbourhood generation strategy this mechanism guides the search through the solution space consisting of the sequences of pieces and relies on a low level placement heuristic to actually convert one sequence in a feasible layout the placement heuristic is based on a bottom left greedy procedure with the ability to fill holes in the middle of the layout at a later stage several variants of the 2 exchange nesting heuristic were implemented and tested with different initial solution ranking criteria different strategies for selecting the next solution and different neighbourhood sizes the computational experiments were based on data sets published in the literature in most of the cases the 2 exchange nesting algorithm generated better solutions than the best known solutions

robust mixed integer linear programming models for the irregular strip packing problem abstract two dimensional irregular strip packing problems are cutting and packing problems where small pieces have to be cut from a larger object involving a non trivial handling of geometry increasingly sophisticated and complex heuristic approaches have been developed to address these problems but despite the apparently good quality of the solutions there is no guarantee of optimality therefore mixed integer linear programming mip models started to be developed however these models are heavily limited by the complexity of the geometry handling algorithms needed for the piece non overlapping constraints this led to pieces simplifications to specialize the developed mathematical models in this paper to overcome these limitations two robust mip models are proposed in the first model dtm the non overlapping constraints are stated based on direct trigonometry while in the second model nfp cm pieces are first decomposed into convex parts and then the non overlapping constraints are written based on nofit polygons of the convex parts both approaches are robust in terms of the type of geometries they can address considering any kind of non convex polygon with or without holes they are also simpler to implement than previous models this simplicity allowed to consider for the first time a variant of the models that deals with piece rotations computational experiments with benchmark instances show that nfp cm outperforms both dtm and the best exact model published in the literature new real world based instances with more complex geometries are proposed and used to verify the robustness of the new models

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 boundary value technique for singular perturbation problems abstract a new way to solve singular perturbation problems is introduced it is designed for the practicing engineer or applied mathematician who needs a practical tool for these problems easy to use modest problem preparation and ready computer implementation as with other methods the original problem is partitioned into inner and outer solution differential equation systems however asymptotic solution techniques are not employed the method is distinguished by the following facts the inner solution problem is solved as a two point boundary problem where the terminal boundary conditions are supplied by the solution of the outer solution problem in turn the outer solution differential equations are solved as an initial value system where the initial conditions are supplied by the inner solution at its terminal point the method is iterative on the terminal point of the inner solution problem three numerical examples are included

complexity of a 3 dimensional assignment problem abstract we show that a certain 3 dimensional assignment problem is np complete to do this we show that the following problem is np complete given bipartite graphs g 1 g 2 with the same sets of vertices do there exist perfect matchings m 1 m 2 of g 1 g 2 respectively such that m 1 m 2 o

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

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

algorithms for two dimensional cutting stock and strip packing problems using dynamic programming and column generation abstract we investigate several two dimensional guillotine cutting stock problems and their variants in which orthogonal rotations are allowed we first present two dynamic programming based algorithms for the rectangular knapsack rk problem and its variants in which the patterns must be staged the first algorithm solves the recurrence formula proposed by beasley the second algorithm for staged patterns also uses a recurrence formula we show that if the items are not so small compared to the dimensions of the bin then these algorithms require polynomial time using these algorithms we solved all instances of the rk problem found at the or library including one for which no optimal solution was known we also consider the two dimensional cutting stock problem we present a column generation based algorithm for this problem that uses the first algorithm above mentioned to generate the columns we propose two strategies to tackle the residual instances we also investigate a variant of this problem where the bins have different sizes at last we study the two dimensional strip packing problem we also present a column generation based algorithm for this problem that uses the second algorithm above mentioned where staged patterns are imposed in this case we solve instances for two three and four staged patterns we report on some computational experiments with the various algorithms we propose in this paper the results indicate that these algorithms seem to be suitable for solving real world instances we give a detailed description a pseudo code of all the algorithms presented here so that the reader may easily implement these algorithms

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

a method of optimal lattice packing of congruent oriented polygons in the plane abstract an approach based on an application of the function apparatus is proposed for constructing a mathematical model of the problem of optimal lattice packing of congruent oriented polygons in the plane the feasible solution region is contained in the set of pairs of points of the function 0 level surface of the polygon the region is partitioned into a finite number of subregions corresponding to pairs of linear segments of the function 0 level surface the frontiers of the subregions are generated by the conditions of non overlapping of polygons only four such conditions imply non overlapping of each pair of polygons in a lattice packing the solution can be attained only in a finite number of points on the frontiers of these subregions hence it is possible to use an organized sorting of such points with discarding of subregions whose objective function upper bound is lower than the record

mathematical model and method of searching for a local extremum for the non convex oriented polygons allocation problem abstract this work deals with the problem of optimal allocation of objects of a so called irregular form the objects are allocated on a strip of given width and with defects this problem is insufficiently studied but it is typical for many industries and is also interesting for developing the theory of solving cutting and packing problems an analytical model of the problem using only continuous variables is written in terms of classical mathematical programming and it is constructed on the basis of the original theory of functions and structures of linear inequalities the presented theory allows one to easily describe the conditions of mutual non overlapping of objects and their allocation in the stock region the exact method for searching a local minimum of the problem from any feasible initial point is based on the application of the active set strategy ideas a number of examples of solving practical problems are considered

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

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

on the weak computability of a four dimensional orthogonal packing and time scheduling problem this paper proposes a four dimensional orthogonal packing and time scheduling problem the problem differs from the classical packing problems in that the position and orientation of each item in the container can be changed over time in this way the four dimensional space time problem better uses the container time also we consider a general case that all parameters are real numbers which makes the problems more difficult to solve this paper proposes an algorithm and proves that the algorithm could solve the problem optimally by a finite number of operations we say this problem is weak computational meaning that if there exists a universal machine that could represent real numbers and could do unit arithmetic or logical operation on real numbers in finite time then the algorithm could find optimal solutions in finite time this paper also presents a proof of the weak computability over a general case of the three dimensional orthogonal packing problem where all parameters are positive real numbers

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

computing closely matching upper and lower bounds on textile nesting problems we consider an industrial cutting problem in textile manufacturing and report on algorithms for computing cutting images and lower bounds on waste for this problem for the upper bounds we use greedy strategies based on hodographs and global optimization based on simulated annealing for the lower bounds we use branch and bound methods for computing optimal solutions of placement subproblems that determine the performance of the overall subproblem the upper bounds are computed in less than an hour on a common day workstation and are competitive in quality with results obtained by human nesters the lower bounds take a few hours to compute and are within 0 4 of the upper bound for certain types of clothing e g for pants

assignment problems a golden anniversary survey abstract having reached the 50th golden anniversary of the publication of kuhn s seminal article on the solution of the classic assignment problem it seems useful to take a look at the variety of models to which it has given birth this paper is a limited survey of what appear to be the most useful of the variations of the assignment problem that have appeared in the literature over the past 50 years the intention here is not to identify every such paper of which there have been hundreds nor to identify the best solution procedure for each variation rather the intention is to identify what these variations are and what they are called so as to make it easier for a researcher trying to develop some variation of the assignment problem for a particular application to find the relevant literature

a mathematical model and a solution method for the problem of placing various sized circles into a strip abstract a mathematical model of the problem of placing different radii circles into a strip is constructed several peculiarities of the mathematical model are considered on the ground of these peculiarities an original method of transition from one local minimum to another one to provide a decrease of the objective function value is suggested the method is based on the idea of increasing the problem dimension and a reduced gradient method as well as on the concept of active inequalities and the newton method an algorithm to solve the problem is described in detail numerical examples which demonstrate the possibilities of the method are also given

a tree search algorithm for the p median problem abstract in this paper we present two lower bounds for the p median problem the problem of locating p facilities medians on a network these bounds are based on two separate lagrangean relaxations of a zero one formulation of the problem with subgradient optimisation being used to maximise these bounds penalty tests based on these lower bounds and a heuristically determined upper bound to the problem are developed and shown to result in a large reduction in problem size the incorporation of the lower bounds and the penalty tests into a tree search procedure is described and computational results are given for problems with an arbitrary number of medians and having up to 200 vertices a comparison is also made between these algorithms and the dual based algorithm of erlenkotter

extensions to a lagrangean relaxation approach for the capacitated warehouse location problem abstract in this paper we present a lower bound for the capacitated warehouse location problem based upon the lagrangean relaxation of a mixed integer formulation of the problem where we use subgradient optimisation in an attempt to maximise this lower bound problem reduction tests based upon this lower bound and the original problem are given incorporating this bound and the reduction tests into a tree search procedure enables us to solve problems involving up to 50 warehouses and 150 customers

a lift and project cutting plane algorithm for mixed 0 1 programs we propose a cutting plane algorithm for mixed 0 1 programs based on a family of polyhedra which strengthen the usual lp relaxation we show how to generate a facet of a polyhedron in this family which is most violated by the current fractional point this cut is found through the solution of a linear program that has about twice the size of the usual lp relaxation a lifting step is used to reduce the size of the lp s needed to generate the cuts an additional strengthening step suggested by balas and jeroslow is then applied we report our computational experience with a preliminary version of the algorithm this approach is related to the work of balas on disjunctive programming the matrix cone relaxations of lovasz and schrijver and the hierarchy of relaxations of sherali and adams

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

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

hybridising tabu search with optimisation techniques for irregular stock cutting sequential meta heuristic implementations for the irregular stock cutting problem have highlighted a number of common problems the literature suggests a consensus that it is more efficient to allow configurations with overlapping pieces in the solution space and to penalise these in the evaluation function however depending on the severity of the penalty this relaxation results in a tendency to converge toward infeasible solutions or to seek out feasible solutions at the expense of overall quality a further problem is encountered in defining a neighbourhood search strategy that can deal with the infinite solution space inherent in the irregular stock cutting problem the implementation in this paper adopts a hybrid tabu search approach that incorporates two very different optimisation routines that utilise alternative neighbourhoods to address the described problems

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

a method for the cutting stock problem with different qualities abstract an operations research project at a swedish company is reported the project deals with the problem of allocating wood products with different quality requirements to be cut from wooden boards which in turn consists of a number of quality zones a major difficulty is the existence of distinct time limits since the cross cutting of the boards is done in a real time process in this paper we derive a mathematical model which includes a set packing problem that accurately describes the real world problem this problem is then solved using a combination of lagrangean relaxation and subgradient optimization numerical results for practical problems are reported the results indicate a valid model and satisfactory efficiency of the proposed solution procedure

a reduction approach to the repeated assignment problem abstract we consider the repeated assignment problem rap which is a k fold repetition of the n n linear assignment problem lap with the additional requirement that no assignment can be repeated more than once in actual applications k is typically much smaller than n first we derive upper and lower bounds respectively by a heuristic together with local search and an efficient method solving the continuous relaxation the latter also solves a lagrangian relaxation such that the related pegging test to fix variables at zero or one decomposes into k independent pegging tests to laps these can be solved exactly by transforming them into all pairs shortest path problems together with these procedures we also employ a virtual pegging test and reduce rap in size numerical experiments show that the reduced instances with k n can be solved exactly using standard mip solvers

a bidirectional building approach for the 2d constrained guillotine knapsack packing problem abstract this paper investigates the 2d guillotine knapsack packing problem in which the objective is to select and cut a set of rectangles from a sheet with fixed size and maximize the total profit of the selected rectangles the orientation of the rectangles is fixed and the guillotine cut in which the cut must be parallel to the sides of the sheet to divide it into two completely separated sheets is required two well known methods namely the top down and bottom up approaches are combined into a coherent algorithm to address this problem computational experiments on benchmark test sets show that the approach finds the optimal solution for almost all moderately sized instances and outperforms all existing approaches for larger instances

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

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

compaction and separation algorithms for non convex polygons and their applications abstract given a two dimensional non overlapping layout of convex and non convex polygons compaction can be thought of as simulating the motion of the polygons resulting from applied forces by moving many polygons simultaneously compaction can improve the material utilization of even tightly packed layouts compaction is hard finding the tightest layout that a valid motion can reach is pspace hard and even compacting to a local optimum can require an exponential number of moves our first compaction algorithm uses a new velocity based optimization model based on existing physical simulation approaches the performance of this algorithm illustrates that these approaches cannot quickly compact tightly packed layouts we then present a new position based optimization model this model represents the forces as a linear objective function and it permits direct calculation via linear programming of new non overlapping polygon positions at a local minimum of the objective the new model yields a translational compaction algorithm that runs two orders of magnitude faster than physical simulation methods we also consider the problem of separating overlapping polygons using the least amount of motion prove optimal separation to be np complete and show that our position based model also yields an efficient algorithm for finding a locally optimal separation the compaction algorithm has improved cloth utilization of human generated layouts of trousers and other garments given a database of human generated markers and with an efficient technique for matching the separation algorithm can automatically generate layouts that approach human performance

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

the constrained bottleneck problem in networks we consider problems on networks that are captured by two performance measures one performance measure is any general cost function of a solution the other is a bottleneck measure that describes the worst maximum cost component of the solution the paper contains algorithms to solve three problems in one problem we minimize the bottleneck subject to a constraint on the generalized cost in the second problem we minimize the generalized cost subject to a constraint on the bottleneck in the third problem we consider the two criteria simultaneously and find all the pareto optimum solutions the major result is that the introduction of the bottleneck measure changes the complexity of the original general cost problem by a factor which is at most linear in the number of links

the line segmentation problem this paper describes a line segmentation problem in a multistage multimachine production system the production facility can concurrently produce several types of circuit boards because each production stage consists of multiple machines the items produced are categorized into families and items belonging to the same family share the common major setup while switching over from one family to another requires a major setup the line segmentation problem determines an allocation of machines at each production stage to families so as to minimize the time to complete all jobs as a result of segmenting the line several minilines are formed which are dedicated to the production of items in each family forming dedicated minilines and producing the items in a family on the same line captures the benefits of group technology and focused factory we first formalize the line segmentation problem as a quadratic integer programming problem and establish its np completeness since the problem is np complete we

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 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

an algorithm for the planar three index assignment problem abstract a branch and bound algorithm for the planar three index assignment problem is presented the branching operation is performed on sets of variables belonging to a common constraint rather than on a single variable the algorithm comprises an upper and lower bound procedure the upper bound procedure consists of two parts one for finding an initial solution and one for improving it the local improvement rule takes advantage of the latin square structure of the solution the lower bound procedure combines a new dual heuristic and a lagrangean relaxation scheme solved by subgradient optimization the use of reduced costs in the evaluation of bounds is another feature of the algorithm computational experience both on the exact algorithm and on the primal heuristics as standalone procedures is reported

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

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

solution approaches to irregular nesting problems this paper reviews some of the approaches which have been adopted in the solution of problems involving the nesting of irregularly shaped pieces such problems arise in a wide variety of application areas and the purpose of this paper is to bring together problems and solution approaches from a number of diverse domains

implementation and testing of a primal dual algorithm for the assignment problem state of the art computational results have shown that primal simplex algorithms are more efficient than primal dual algorithms for general minimum cost network flow problems there is however some controversy concerning their relative merits for solving assignment problems this paper presents a detailed development for a computationally efficient primal dual algorithm and extensive computational comparisons to primal simplex algorithms

general local search methods abstract this paper is a tutorial introduction to three recent yet widely used general heuristics simulated annealing tabu search and genetic algorithms a relatively precise description and an example of application are provided for each of the methods as well as a tentative evaluation and comparison from a pragmatic point of view

an algorithm for the two dimensional assortment problem abstract in this paper we consider the two dimensional assortment problem this is the problem of choosing from a set of stock rectangles a subset which can be used for cutting into a number of smaller rectangular pieces constraints are imposed upon the number of such pieces which result from the cutting a heuristic algorithm for the guillotine cutting version of the problem is developed based on a greedy procedure for generating two dimensional cutting patterns a linear program for choosing the cutting patterns to use and an interchange procedure to decide the best subset of stock rectangles to cut computational results are presented for a number of test problems which indicate that the algorithm developed produces good quality results both for assortment problems and for two dimensional cutting problems

the bilinear assignment problem complexity and polynomially solvable special cases in this paper we study the bilinear assignment problem bap with size parameters m and n m le n bap is a generalization of the well known quadratic assignment problem and the three dimensional assignment problem and hence np hard we show that bap cannot be approximated within a constant factor unless p np even if the associated quadratic cost matrix q is diagonal further we show that bap remains np hard if m o root r of n for some fixed r but is solvable in polynomial time if m o left frac log n log log n right when the rank of q is fixed bap is observed to admit fptas and when this rank is one it is solvable in polynomial time under some additional restrictions we then provide a necessary and sufficient condition for bap to be equivalent to two linear assignment problems a closed form expression to compute the average of the objective function values of all solutions is presented whereas the median of the solution values cannot be identified in polynomial time unless p np we then provide polynomial time heuristic algorithms that find a solution with objective function value no worse than that of m 1 n 1 solutions however computing a solution whose objective function value is no worse than that of m n lceil frac m beta rceil lceil frac n beta rceil solutions is np hard for any fixed rational number beta 1

efficient algorithms with performance guarantees for some problems of finding several discrete disjoint subgraphs in complete weighted graph some hard to solve combinatorial problems of finding several disjoint discrete structures in complete weighted graph are considered efficient algorithms with performance guarantees are constructed for the euclidean m peripatetic salesman problem m weighted clique problem and m layer planar three index assignment problem

fast neighborhood search for two and three dimensional nesting problems abstract in this paper we present a new heuristic solution method for two dimensional nesting problems it is based on a simple local search scheme in which the neighborhood is any horizontal or vertical translation of a given polygon from its current position to escape local minima we apply the meta heuristic method guided local search the strength of our solution method comes from a new algorithm which is capable of searching the neighborhood in polynomial time more precisely given a single polygon with m edges and a set of polygons with n edges the algorithm can find a translation with minimum overlap in time o mn log mn solutions for standard test instances are generated by an implementation and a comparison is done with recent results from the literature the solution method is very robust and most of the best solutions found are also the currently best results published our approach to the problem is very flexible regarding problem variations and special constraints and as an example we describe how it can handle materials with quality regions finally we generalize the algorithm for the fast neighborhood search and present a solution method for three dimensional nesting problems

a variant of time minimizing assignment problem a time minimizing assignment problem tmap dealing with the allocation of n jobs to m n persons is considered in this paper one job is to be allocated to exactly one person and each person does at least one job all the persons start working on the jobs simultaneously if a person is to do more than one job he does them one after the other in any order the aim of the present study is to find that feasible assignment which minimizes the total time for completing all the jobs a lexi search approach is proposed to find an optimal feasible assignment

packing different sized circles into a rectangular container abstract this paper is motivated by the problem of fitting pipes of different diameters into a shipping container here we study the subproblem of fitting circles of different sizes into a rectangle since that problem is a central part of the larger problem we formulate this situation as a nonlinear mixed integer programming problem and develop a number of heuristic procedures for approximately solving this problem the heuristics are based on a variety of solution building rules that emulate the process of packing a container some of these methods including a genetic algorithm were based on a more structured design intended to provide solutions which are stable from a stowage viewpoint these heuristics are described in detail and their relative performances are compared for a sample set of 66 randomly generated problems based on this sample the best performing heuristic methods were a quasi random technique and a genetic algorithm of the stable solution structure

an application of simulated annealing to the cutting stock problem we solve a two dimensional cutting stock problem by applying a general global optimization algorithm the simulated annealing our algorithms applied to the cutting problems involving both the guillotine and non guillotine constraints underlying that the latter is to be preferred for a big number of items several tests prove the validity of the algorithms

a recursive exact algorithm for weighted two dimensional cutting gilmore and gomory s algorithm is one of the better actually known exact algorithms for solving unconstrained guillotine two dimensional cutting problems herz s algorithm is more effective but only for the unweighted case we propose a new exact algorithm adequate for both weighted and unweighted cases which is more powerful than both algorithms the algorithm uses dynamic programming procedures and one dimensional knapsack problem to obtain efficient lower and upper bounds and important optimality criteria which permit a significant branching cut in a recursive tree search procedure recursivity computational power adequateness to parallel implementations and generalization for solving constrained two dimensional cutting problems are some important features of the new algorithm

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

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

polyhedral combinatorics of multi index axial transportation problems abstract this paper studies integer points ip and integer vertices iv of the p index axial transportation polytope p atp of order n 1 n 2 n p n 1 n 2 n p 2 p 2 defined by integer vectors as well as noninteger vertices of the 3 atp in particular for the p atp we establish criteria for the minimum and maximum number of ips and describe the class of polytopes for which the number of ips coincides with the number of ivs for the 3 atp of order n n n we prove the theorem on the exponential growth of denominators of fractional components of the polytope vertices three conjectures are stated regarding the maximum number of vertices of the p atp the maximum number of ivs and the structure of the nondegenerate polytopes with the maximum number of ips

lower bounds for the axial three index assignment problem this paper describes new bounding methods for the axial three index assignment problem 3ap for calculating 3ap lower bounds we use a projection method followed by a hungarian algorithm based on a new lagrangian relaxation we also use a cost transformation scheme which iteratively transforms 3ap costs in a series of equivalent 3aps which provides the possibility of improving the 3ap lower bound these methods produce efficiently computed relatively tight lower bound

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

heuristic approaches to large scale periodic packing of irregular shapes on a rectangular sheet the nesting problem is a two dimensional cutting and packing problem where the small pieces to cut have irregular shapes a particular case of the nesting problem occurs when congruent copies of one single shape have to fill as much as possible a limited sheet traditional approaches to the nesting problem have difficulty to tackle with high number of pieces to place additionally if the orientation of the given shape is not a constraint the general nesting approaches are not particularly successful this problem arises in practice in several industrial contexts such as footwear metalware and furniture a possible approach is the periodic placement of the shapes in a lattice way in this paper we propose three heuristic approaches to solve this particular case of nesting problems experimental results are compared with published results in literature and additional results obtained from new instances are also provided

on linear programs with random costs we consider linear programs in which the objective function cost coefficients are independent non negative random variables and give upper bounds for the random minimum cost one application shows that for quadratic assignment problems with such costs certain branch and bound algorithms usually take more than exponential time

the dh kd algorithm a hybrid approach for unconstrained two dimensional cutting problems abstract we study both weighted and unweighted unconstrained two dimensional guillotine cutting problems we develop a hybrid approach which combines two heuristics from the literature the first one dh uses a tree search procedure introducing two strategies depth first search and hill climbing the second one kd is based on a series of one dimensional knapsack problems using dynamic programming techniques the dh kd algorithm starts with a good initial lower bound obtained by using the kd algorithm at each level of the tree search the proposed algorithm uses also the kd algorithm for constructing new lower bounds and uses another one dimensional knapsack for constructing refinement upper bounds the resulting algorithm can be seen as a generalization of the two heuristics and solves large problem instances very well within small computational time our algorithm is compared to morabito et al s algorithm the unweighted case and to beasley s 2 approach the weighted case on some examples taken from the literature as well as randomly generated instances

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

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

the multiperiod assignment problem a multicommodity network flow model and specialized branch and bound algorithm abstract the multiperiod assignment problem a specialization of the three dimensional assignment problem is an optimization model that describes the situation of assigning people to activities or jobs over time we consider the most general case which has both a cost of assigning a person to an activity in each time period and a cost of transferring the person from one activity in each period to another activity in the next period in general the number of time periods need not equal the number of persons and activities we present a new formulation of the multiperiod assignment problem that of an integer multicommodity network flow model the special structure of the model allows us to find a good feasible solution relatively quickly by a shortest path heuristic algorithm we discuss a new branch and bound algorithm for solving this problem to optimality the subproblems of the branch and bound tree are evaluated by solving a set of special structured shortest path problems all of which can be solved in order n 2 t time where n is the number of persons and activities and t is the number of time periods we present computational tests of the algorithm on moderately sized problems