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

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

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

model selection and akaike s information criterion aic the general theory and its analytical extensions during the last fifteen years akaike s entropy based information criterion aic has had a fundamental impact in statistical model evaluation problems this paper studies the general theory of the aic procedure and provides its analytical extensions in two ways without violating akaike s main principles these extensions make aic asymptotically consistent and penalize overparameterization more stringently to pick only the simplest of the true models these selection criteria are called caic and caicf asymptotic properties of aic and its extensions are investigated and empirical performances of these criteria are studied in choosing the correct degree of a polynomial model in two different monte carlo experiments under different conditions

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

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

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

a 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

finite volume distance field and its application to medial axis transforms accurate and efficient computation of the nearest wall distance d or level set is important for many areas of computational science engineering differential equation based distance level set algorithms such as the hyperbolic natured eikonal equation have demonstrated valuable computational efficiency here in the context as an auxiliary equation to the main flow equations the eikonal equation is solved efficiently with two different finite volume approaches the cell vertex and cell centered the application of the distance solution is studied for various geometries moreover a procedure using the differential field to obtain the medial axis transform mat for different geometries is presented the latter provides a skeleton representation of geometric models that has many useful analysis properties as an alternative to other methods the current d mat procedure bypasses difficulties that are usually encountered by pure geometric methods e g the voronoi approach especially in three dimensions and provides better accuracy than pure thinning methods it is also shown that the d mat approach provides the potential to sculpt control the mat form for specialized solution purposes

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

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

application of model selection criteria to some problems in multivariate analysis a review of model selection criteria is presented with a view toward showing their similarities it is suggested that some problems treated by sequences of hypothesis tests may be more expeditiously treated by the application of model selection criteria consideration is given to application of model selection criteria to some problems of multivariate analysis especially the clustering of variables factor analysis and more generally describing a complex of variables

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

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

decomposing combinatorial auctions and set packing problems combinatorial auctions allow bidders to bid on bundles of items rather than just on single items the winner determination problem in combinatorial auctions is the problem of determining the allocation of items to bidders such that the sum of the accepted bid prices is maximized this problem is equivalent to the well known maximum weight set packing problem even though these problems are np hard in general they can be solved in polynomial time on instances whose associated item graphs have bounded treewidth called structured item graphs however the tractability of determining whether for a given problem instance a structured item graph of fixed treewidth exists and if so computing one efficiently was an open problem in this article we solve this problem by proving that deciding the existence of structured item graphs is computationally intractable even for treewidth 3 motivated by this unfavorable complexity result we investigate other structural restrictions and we show that the notion of hypertree decomposition a well studied measure of hypergraph cyclicity turns out to be most useful here indeed we show that the winner determination problem is solvable in polynomial time on instances whose dual auction hypergraphs have bounded hypertree width our solution method is based on encoding winner determination via a constraint satisfaction optimization problem and on exhibiting an algorithm to solve this latter problem efficiently for such structurally restricted instances the class of tractable instances identified by our approach while being efficiently recognizable properly contains the class of instances having a structured item graph moreover on the larger class our method solves winner determination with the same asymptotic complexity as the best algorithm proposed in the literature for the subclass of structured item graphs hypertree decompositions can equally profitably be applied to the maximum weight independent set problem which is the dual problem of maximum weight set packing

multilayer airfoil ice accretion simulations using a level set method with b spline representation this paper presents a partial differential equation approach to model the icing front evolution on airfoils via the level set equation offering a high fidelity alternative to common algebraic appr

grid generation algorithms for complex glaze ice shapes reynolds averaged navier stokes simulations the paper presents the developments of novel mesh generation algorithms over complex glaze ice shapes containing multicurvature ice accretion geometries such as single double ice horns the twofold approaches tackle surface geometry discretization as well as field mesh generation first an adaptive curvilinear curvature control algorithm is constructed solving a one dimensional elliptic partial differential equation with periodic source terms this method controls the arc length grid spacing so that high convex and concave curvature regions around ice horns are appropriately captured and is shown to effectively treat the grid shock problem second a novel blended method is developed by defining combinations of source terms with two dimensional elliptic equations the source terms include two common control functions sorenson and spekreijse and an additional third source term to improve orthogonality this blended method is shown to be very effective for improving grid quality metrics for complex g

a mixture model for distributions of correlation coefficients an old problem in personnel psychology is to characterize distributions of test validity correlation coefficients the proposed model views histograms of correlation coefficients as observations from a mixture distribution which for a fixed sample sizen is a conditional mixture distributionh r n j j h r j n wherer is the correlation coefficient j are population correlation coefficients and j are the mixing weights the associated marginal distribution ofr is regarded as the parent distribution underlying histograms of empirical correlation coefficients maximum likelihood estimates of the parameters j and j can be obtained with an em algorithm solution and tests for the number of componentst are achieved after the one component density ofr is replaced with a tractable modeling densityh r j n two illustrative examples are provided

grasp for set packing problems abstract the principles of the greedy randomized adaptative search procedure grasp metaheuristic are instantiated for the set packing problem we investigated several construction phases and evaluated improvements based on advanced strategies these improvements include a self tuning procedure using reactive grasp an intensification procedure using path relinking and a procedure involving the diversification of the selection using a learning process two sets of various numerical instances were used to perform the computational experiments the first set contains randomly generated instances while the second includes instances relating to real problems in railway planning no metaheuristic has previously been applied to this combinatorial problem consequently we have discussed grasp s performances both in relation to lower upper bounds and to the results obtained with cplex when such results are available our analysis based on the average performances observed shows the impact of the suggested strategies and indicates the configuration that produces the best results

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

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

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