mixed integer programming model formulations for dynamic multi item multi level capacitated lotsizing abstract recently there have been two new proposals to model the dynamic multi item multi level capacitated lotsizing problem by variable redefinitions based on either a shortest route or a simple plant location representation of the underlying decision problem here we will introduce an extended version of the well known inventory and lotsize model followed by a new model formulation based on modeling the changes of end of period inventory levels explicitly secondly we will compare the suitability of different model formulations when solving problem instances with up to 40 items and 16 periods utilizing state of the art standard mip software on a personal computer

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

a hybrid optimization approach for multi level capacitated lot sizing problems abstract solving multi level capacitated lot sizing problems is still a challenging task in spite of increasing computational power and faster algorithms in this paper a new approach combining an ant based algorithm with an exact solver for mixed integer linear programs is presented a max min ant system is developed to determine the principal production decisions a lp mip solver is used to calculate the corresponding production quantities and inventory levels two different local search methods and an improvement strategy based on reduced mixed integer problems are developed and integrated into the ant algorithm this hybrid approach provides superior results for small and medium sized problems in comparison to the existing approaches in the literature for large scale problems the performance of this method is among the best

a novel scheme for the solution of the time domain integral equations of electromagnetics a new method to numerically solve time domain integral equations pertinent to electromagnetic surface scattering phenomena is presented the method uses approximate prolate spheroidal wave functions and standard rao wilton glisson basis functions to effect the temporal and spatial discretization of the integral equations respectively because the temporal basis functions are noncausal an extrapolation scheme is used to construct a system of equations that can be solved by marching on in time numerical results show that the proposed method is stable and that its solutions converge exponentially fast with the time bandwidth product of the approximate prolate spheroidal wave functions to results from a frequency domain method of moments solver that uses spatial basis functions and integration rules identical to those in the time domain solver

elastodynamic direct boundary element methods with enhanced numerical stability properties evidence of numerical instabilities in two dimensional time domain direct boundary element methods is presented the e ects of numerical versus analytical integration of spatial integrals on stability are shown and two new time stepping algorithms are introduced and compared to existing formulations the so called new direct half step scheme and the epsilon scheme are shown to improve the numerical stability of direct boundary element methods copyright 1999 john wiley sons ltd

an accurate scheme for the solution of the time domain integral equations of electromagnetics using higher order vector bases and bandlimited extrapolation despite the numerous advances made in increasing the computational efficiency of time domain integral equation tdie based solvers the stability and accuracy of tdie solvers remain problematic this paper introduces a new numerical method for the accurate solution of tdies for scattering from arbitrary perfectly conducting surfaces the work described in this paper uses the higher order divergence conforming basis functions of graglia et al for spatial discretization and bandlimited interpolation functions for the temporal discretization of the relevant integral equations since the basis functions used for the temporal representation are noncausal an extrapolation scheme is employed to recover the ability to solve the problem by marching on in time numerical results demonstrate that the proposed method is stable and that it exhibits superlinear convergence with regard to the spatial discretization and exponential convergence with respect to the temporal discretization

is gauss quadrature better than clenshaw curtis we compare the convergence behavior of gauss quadrature with that of its younger brother clenshaw curtis seven line matlab codes are presented that implement both methods and experiments show that the supposed factor of 2 advantage of gauss quadrature is rarely realized theorems are given to explain this effect first following o hara and smith in the 1960s the phenomenon is explained as a consequence of aliasing of coefficients in chebyshev expansions then another explanation is offered based on the interpretation of a quadrature formula as a rational approximation of log z 1 z 1 in the complex plane gauss quadrature corresponds to pade approximation at z infty clenshaw curtis quadrature corresponds to an approximation whose order of accuracy at z infty is only half as high but which is nevertheless equally accurate near 1 1

lagrangean relaxation based heuristics for lot sizing with setup times we consider a lot sizing problem with setup times where the objective is to minimize the total inventory carrying cost only the demand is dynamic over time and there is a single resource of limited capacity we show that the approaches implemented in the literature for more general versions of the problem do not perform well in this case we examine the lagrangean relaxation lr of demand constraints in a strong reformulation of the problem we then design a primal heuristic to generate upper bounds and combine it with the lr problem within a subgradient optimization procedure we also develop a simple branch and bound heuristic to solve the problem computational results on test problems taken from the literature show that our relaxation procedure produces consistently better solutions than the previously developed heuristics in the literature

multilevel capacitated lotsizing complexity and lp based heuristics abstract this paper presents the first heuristics capable of solving multilevel lotsizing problems with capacity constraints on more than one level moreover the form of the heuristics is quite general so that they can easily be extended to solve a variety of problems if one wants to solve these problems on a routine basis in a real environment one needs to find fast and easy algorithms however we show that for certain problem classes this is a very difficult task far more difficult than has been suggested in the literature for problems with setup times we show that finding a feasible solution is np complete even without setup times testing for feasibility can be very difficult just how time consuming such heuristics must be is demonstrated this leaves little chance to build fast and easy heuristics except for the most simple cases our exploration of the complexity issues points to mathematical programming as a potential source of heuristics for these problems this paper presents a new and general approach based on rounding an lp solution for the problem without setup times the methods use different information and patterns evident in the lp solution are explored the approach is tested on a large set of problems the main contributions of this paper are the way in which we distinguish between the easy and hard lotsizing problems the lp based heuristics and the test set of capacitated multilevel lotsizing problems

a finite difference delay modeling approach to the discretization of the time domain integral equations of electromagnetics a new method for solving the time domain integral equations of electromagnetic scattering from conductors is introduced this method called finite difference delay modeling appears to be completely stable and accurate when applied to arbitrary structures the temporal discretization used is based on finite differences specifically based on a mapping from the laplace domain to the z transform domain first and second order unconditionally stable methods are derived spatial convergence is achieved using the higher order divergence conforming vector bases of graglia et al low frequency instability problems are avoided with the loop tree decomposition approach numerical results will illustrate the accuracy and stability of the technique

non algebraic integrability of the chew low reversible dynamical system of the cremona type and the relation with the 7th hilbert problem non resonant case abstract a smooth reversible dynamical system srds and a system of nonlinear functional equations defined by a certain rational quadratic cremona mapping and arising from the static model of the dispersion approach in the theory of strong interactions the chew low equations for p wave n scattering are considered this srds is splitted into 1 and 2 dimensional ones an explicit cremona transformation that completely determines the exact solution of the two dimensional system is found this solution depends on an odd function satisfying a nonlinear autonomous 3 point functional equation non algebraic integrability of srds under consideration is proved using the method of poincare normal forms and the siegel theorem on biholomorphic linearization of a mapping at a non resonant fixed point the proof is based on the classical feldman baker theorem on linear forms of logarithms of algebraic numbers which in turn relies upon solving the 7th hilbert problem by a i gel fond and t schneider and new powerful methods of a baker in the theory of transcendental numbers the general theorem following from the feldman baker theorem on applicability of the siegel theorem to the set of the eigenvalues c n of a mapping at a non resonant fixed point which belong to the algebraic number field a is formulated and proved the main results are presented in theorems 1 3 5 7 8 and remarks 3 7

a lagrangean relaxation approach for very large scale capacitated lot sizing in this paper we develop a lagrangean relaxation based heuristic procedure to generate near optimal solutions to very large scale capacitated lot sizing problems clsp with setup times and limited overtime our computational results show that large problems involving several thousand products and several thousand 0 1 integer variables can be solved in a reasonable amount of computer time to within one percent of their optimal solution the proposed procedure is general enough to be applied directly or with slight modification to real life production problems

space time discretized retarded potential boundary integral operators quadrature for collocation methods we propose a novel discretization method for time domain boundary integral equations of the three dimensional wave equation the basic idea of the discussed approach is to treat time as if it were

a lagrangian based heuristic for the capacitated lot sizing problem in parallel machines this paper addresses the capacitated lot sizing problem involving the production of multiple items on unrelated parallel machines a production plan should be determined in order to meet the forecast demand for the items without exceeding the capacity of the machines and minimize the sum of production setup and inventory costs a heuristic based on the lagrangian relaxation of the capacity constraints and subgradient optimization is proposed initially the heuristic is tested on instances of the single machine problem and results are compared with heuristics from the literature for parallel machines and small problems the heuristic performance is tested against optimal solutions and for larger problems it is compared with the lower bound provided by the lagrangian relaxation

rapid solution of the wave equation in unbounded domains in this paper we propose and analyze a new fast method for the numerical solution of time domain boundary integral formulations of the wave equation we employ lubich s convolution quadrature method for the time discretization and a galerkin boundary element method for the spatial discretization the coefficient matrix of the arising system of linear equations is a triangular block toeplitz matrix possible choices for solving the linear system arising from the above discretization include the use of fast fourier transform fft techniques and the use of data sparse approximations by using fft techniques the computational complexity can be reduced substantially while the storage cost remains unchanged and is typically high using data sparse approximations the gain is reversed i e the computational cost is approximately unchanged while the storage cost is substantially reduced the method proposed in this paper combines the advantages of these two approaches first the discrete convolution related to the block toeplitz system is transformed into the discrete fourier image thereby arriving at a decoupled system of discretized helmholtz equations with complex wave numbers a fast data sparse e g fast multipole or panel clustering method can then be applied to the transformed system additionally significant savings can be achieved if the boundary data are smooth and time limited in this case the right hand sides of many of the helmholtz problems are almost zero and hence can be disregarded finally the proposed method is inherently parallel we analyze the stability and convergence of these methods thereby deriving the choice of parameters that preserves the convergence rates of the unperturbed convolution quadrature we also present numerical results which illustrate the predicted convergence behavior

heuristics for multilevel lot sizing with a bottleneck in this paper we present a heuristic method based on lagrangian relaxation for multilevel lot sizing when there is a single bottleneck facility a series of lagrangian relaxations one for each item in the product structure is imbedded in a branch and bound procedure the objective is to find a production schedule that fits within available capacity at minimum cost the method has two solution phases dual and primal in the dual phase of the procedure implied costs of setups and production are determined based on a tentative schedule the primal phase is repeated with these new prices and we iterate to reach a good solution the solution procedure is first tested on two special cases uncapacitated multilevel lot sizing and the capacitated single level multi item lot sizing problem the results show that the solution procedure can provide better solutions than some heuristics designed especially for those problems test results on the bottleneck problem indicate that good feasible solutions are found for problems too difficult to solve with exact methods

a hybrid adaptive large neighborhood search heuristic for lot sizing with setup times abstract this paper presents a hybrid of a general heuristic framework and a general purpose mixed integer programming mip solver the framework is based on local search and an adaptive procedure which chooses between a set of large neighborhoods to be searched a mixed integer programming solver and its built in feasibility heuristics is used to search a neighborhood for improving solutions the general reoptimization approach used for repairing solutions is specifically suited for combinatorial problems where it may be hard to otherwise design suitable repair neighborhoods the hybrid heuristic framework is applied to the multi item capacitated lot sizing problem with setup times where experiments have been conducted on a series of instances from the literature and a newly generated extension of these on average the presented heuristic outperforms the best heuristics from the literature and the upper bounds found by the commercial mip solver ilog cplex using state of the art mip formulations furthermore we improve the best known solutions on 60 out of 100 and improve the lower bound on all 100 instances from the literature

high order discretization of a stable time domain integral equation for 3d acoustic scattering abstract we develop a high order explicit method for acoustic scattering in three space dimensions based on a combined field time domain integral equation the spatial discretization of nystrom type uses gaussian quadrature on panels combined with a special treatment of the weakly singular kernels arising in near neighbor interactions in time a new class of convolution splines is used in a predictor corrector algorithm experiments on a torus and a perturbed torus are used to explore the stability and accuracy of the proposed scheme this involved around one thousand solver runs at up to 8th order and up to around 20 000 spatial unknowns demonstrating 5 9 digits of accuracy in addition we show that parameters in the combined field formulation chosen on the basis of analysis for the sphere and other convex scatterers work well in these cases

analysis of three dimensional transient acoustic wave propagation using the boundary integral equation method this paper describes a boundary integral equation boundary element method for the solution of a variety of transient acoustic problems the spatial and temporal discretization employs quadratic isoparametric elements with high order gauss quadrature and the ensuing equations are implicit the implicit formulation both eliminates the instabilities reported in explicit treatments and permits a freedom of choice of timestep which can reduce costs dramatically the accuracy of the approach is demonstrated by comparison with the analytical solution for a sphere results for more demanding sphere cone sphere geometries extending to seven wavelengths long are presented and compared to those obtained from a related frequency domain approach

integrable mappings and soliton equations ii abstract some simple solutions of discrete soliton equations are shown to satisfy 2d mappings we show that these belong to a recently introduced 18 parameter family of integrable reversible mappings of the plane thus lending weight to a previous conjecture we also give an example of an integrable mapping occuring in an exactly solvable model in statistical mechanics finally we discuss the notion of generalized reversibility

a galerkin scheme for the time domain integral equation of acoustic scattering from a hard surface the transient acoustic scattered wave from a hard surface is computed using a space time integral equation a variational formulation of this integral equation is obtained from an energy identity leading to a galerkin type numerical procedure by choosing an appropriate basis for the discretized functional space this scheme is of the marching in time type with matrix coefficients this technique is applied to a sphere a box and dihedron targets

grasp heuristic with path relinking for the multi plant capacitated lot sizing problem this paper addresses the independent multi plant multi period and multi item capacitated lot sizing problem where transfers between the plants are allowed this is an np hard combinatorial optimization problem and few solution methods have been proposed to solve it we develop a grasp greedy randomized adaptive search procedure heuristic as well as a path relinking intensification procedure to find cost effective solutions for this problem in addition the proposed heuristics is used to solve some instances of the capacitated lot sizing problem with parallel machines the results of the computational tests show that the proposed heuristics outperform other heuristics previously described in the literature the results are confirmed by statistical tests

lot sizing in general assembly systems with setup costs setup times and multiple constrained resources we introduce a heuristic method for finding good feasible solutions for multiproduct lot sizing problems with general assembly structures multiple constrained resources and nonzero setup costs and setup times we evaluate the performance of this heuristic by comparing its solutions to optimal solutions of small randomly generated problems and to time truncated optimization subroutine library osl solutions of medium sized randomly generated problems in the first case the heuristic locates solutions averaging 4 percent worse than optimal in less than 1 percent of time required by osl the heuristic solutions to medium sized problems are approximately 26 percent better than solutions osl finds after 10 000 cpu seconds and the heuristic finds these solutions in approximately 10 percent of osl time

using lagrangean techniques to solve hierarchical production planning problems this paper proposes and tests a procedure for decomposing a large scale production planning problem modeled as a mixed integer linear program we interpret this decomposition in the context of hax and meal s hierarchical framework for production planning the procedure decomposes the production planning problem into two subproblems which correspond to the aggregate planning subproblem and a disaggregation subproblem in the hax meal framework the linking mechanism for these two subproblems is an inventory consistency relationship which is priced out by a set of lagrange multipliers the best values for the multipliers are found by an iterative procedure which may be interpreted as a feedback mechanism in the hax meal framework at each iteration the procedure finds both a lower bound on the optimal value to the production planning problem and a feasible solution from which an upper bound is obtained our computational tests show that the best feasible solution found from this procedure is very close to o

runge kutta convolution quadrature for operators arising in wave propagation an error analysis of runge kutta convolution quadrature is presented for a class of non sectorial operators whose laplace transform satisfies besides the standard assumptions of analyticity in a half plane re s 0 and a polynomial bound operatorname o s mu 1 there the stronger polynomial bound operatorname o s mu 2 in convex sectors of the form operatorname arg s leq pi 2 theta for 0 the order of convergence of the runge kutta convolution quadrature is determined by 2 and the underlying runge kutta method but is independent of 1 time domain boundary integral operators for wave propagation problems have laplace transforms that satisfy bounds of the above type numerical examples from acoustic scattering show that the theory describes accurately the convergence behaviour of runge kutta convolution quadrature for this class of applications our results show in particular that the full classical order of the runge kutta method is attained away from the scattering boundary

computational complexity of the capacitated lot size problem in this paper we study the computational complexity of the capacitated lot size problem with a particular cost structure that is likely to be used in practical settings for the single item case new properties are introduced classes of problems solvable by polynomial time algorithms are identified and efficient solution procedures are given we show that special classes are np hard and that the problem with two items and independent setups is np hard under conditions similar to those where the single item problem is easy topics for further research are discussed in the last section

capacitated lot sizing and scheduling by lagrangean relaxation abstract development of new models and solution procedures for production planning has been of research interest for several decades implementation of these models has resulted in lower production costs by reducing inventories number of setups and labor costs in this paper we develop several optimal near optimal procedures for the capacitated lot sizing and scheduling problem clsp with setup times limited regular time and limited overtime we formulate a mixed integer linear programming model of the problem and solve it by lagrangean relaxation we experiment with alternative lagrangean relaxations and develop new procedures to solve these relaxations overall the capacity constraints relaxation seems to be superior to the demand constraints relaxation our results show that large problems can be solved in reasonable computer times and within one percent accuracy of the optimal solutions we solved 99 8 i e 99 items and 8 periods 50 12 and 50 8 problems in 30 61 36 25 and 12 65 seconds of cdc cyber 730 computer time respectively our procedures are general enough to be applied directly or with slight modifications in real life production settings

a galerkin method for retarded boundary integral equations with smooth and compactly supported temporal basis functions we consider retarded boundary integral formulations of the three dimensional wave equation in unbounded domains our goal is to apply a galerkin method in space and time in order to solve these problems numerically in this approach the computation of the system matrix entries is the major bottleneck we will propose new types of finite dimensional spaces for the time discretization they allow variable time stepping variable order of approximation and simplify the quadrature problem arising in the generation of the system matrix substantially the reason is that the basis functions of these spaces are globally smooth and compactly supported in order to perform numerical tests concerning our new basis functions we consider the special case that the boundary of the scattering problem is the unit sphere in this case explicit solutions of the problem are available which will serve as reference solutions for the numerical experiments

a tabu search heuristic for the capacitated lot sizing problem with set up carryover this paper presents a tabu search heuristic for the capacitated lot sizing problem clsp with set up carryover this production planning problems allows multiple items to be produced within a time period and setups for items to be carried over from one period to the next two interrelated decisions sequencing and lot sizing are present in this problem our tabu search heuristic consists of five basic move types three for the sequencing decisions and two for the lot sizing decisions we allow infeasible solutions to be generated at a penalty during the course of the search we use several search strategies such as dynamic tabu list adaptive memory and self adjusting penalties to strengthen our heuristic we also propose a lower bounding procedure to estimate the quality of our heuristic solution we have also modified our heuristic to produce good solutions for the clsp without set up carryover the computational study conducted on a set of 540 test problems indicates that on average our heuristic solutions are within 12 of a bound on optimality in addition for the set of test problems our results indicate an 8 reduction in total cost through set up carryover

solving uncapacitated multilevel lot sizing problems using a particle swarm optimization with flexible inertial weight the multilevel lot sizing mlls problem is a key production planning problem in materials requirements planning mrp system the mlls problem deals with determining the production lot sizes of various items appearing in the product structure over a given finite planning horizon to minimize the production cost the inventory carrying cost the back ordering cost and etc this paper proposed a particle swarm optimization pso algorithm for solving the uncapacitated mlls problem with assembly structure all the mathematical operators in our algorithm are redefined and the inertial weight parameter can be either a negative real number or a positive one the feasibility and effectiveness of our algorithm are investigated by comparing the experimental results with those of a genetic algorithm ga

meta heuristics for dynamic lot sizing a review and comparison of solution approaches proofs from complexity theory as well as computational experiments indicate that most lot sizing problems are hard to solve because these problems are so difficult various solution techniques have been proposed to solve them in the past decade meta heuristics such as tabu search genetic algorithms and simulated annealing have become popular and efficient tools for solving hard combinatorial optimization problems we review the various meta heuristics that have been specifically developed to solve lot sizing problems discussing their main components such as representation evaluation neighborhood definition and genetic operators further we briefly review other solution approaches such as dynamic programming cutting planes dantzig wolfe decomposition lagrange relaxation and dedicated heuristics this allows us to compare these techniques understanding their respective advantages and disadvantages gives insight into how we can integrate elements from several solution approaches into more powerful hybrid algorithms finally we discuss general guidelines for computational experiments and illustrate these with several examples

multistep and multistage convolution quadrature for the wave equation algorithms and experiments we describe how a time discretized wave equation in a homogeneous medium can be solved by boundary integral methods the time discretization can be a multistep runge kutta or a more general multistep multistage method the resulting convolutional system of boundary integral equations belongs to the family of convolution quadratures of lubich the aim of this work is twofold it describes an efficient robust and easily parallelizable method for solving the semidiscretized system the resulting algorithm has the main advantages of time stepping methods and of fourier synthesis at each time step a system of linear equations with the same system matrix needs to be solved yet computations can easily be done in parallel the computational cost is almost linear in the number of time steps and only the laplace transform of the time domain fundamental solution is needed the new aspect of the algorithm is that all this is possible without ever explicitly constructing the weights of the convolution quadrature this approach also readily allows the use of modern data sparse techniques to efficiently perform computation in space we investigate theoretically and numerically to which extent hierarchical matrix mathcal h matrix techniques can be used to speed up the space computation the second aim of this article is to perform a series of large scale 3d experiments with a range of multistep and multistage time discretization methods the backward difference formula of order 2 bdf2 the trapezoid rule and the 3 stage radau iia methods are investigated in detail one of the conclusions of the experiments is that the radau iia method often performs overwhelmingly better than the linear multistep methods especially for problems with many reflections yet in connection with hyperbolic problems bdfs have so far been predominant in the literature on convolution quadrature

a galerkin bem for transient acoustic scattering by an absorbing obstacle this paper deals with the numerical resolution of the so called time domain boundary integral equations the scattering problem by an absorbing obstacle serves as the model for this discussion a new system of retarded potential boundary integral equations rpbie is set up to solve the problem using an energy identity we are able to prove the unconditional stability for standard conforming galerkin approximation of the rpbie the actual space time boundary elements are described in some detail and their implementation has confirmed that stability in a wide range of situations and for large run times never reached before

the multi item capacitated lot size problem error bounds of manne s formulations we discuss an approximation scheme for the multi item lot size problem it is based on an optimal basic solution of a linear programming problem derived from the original problem the approximate solution is obtained by taking a linear convex combination of the optimal solution of the linear programming problem we express error bounds of the approximation as a function of some parameters that can be easily estimated in practice when set up times are positive the approximation may result in an infeasible solution we take the same approach to show that the infeasibility of the approximation is small the analysis is extended to a variable capacity problem with overtime as an auxiliary result we provide a bound on the duality gap of the lagrangian dual problem

fast convolution quadrature for the wave equation in three dimensions this work addresses the numerical solution of time domain boundary integral equations arising from acoustic and electromagnetic scattering in three dimensions the semidiscretization of the time domain boundary integral equations by runge kutta convolution quadrature leads to a lower triangular toeplitz system of size n this system can be solved recursively in an almost linear time o n log 2 n but requires the construction of o n dense spatial discretizations of the single layer boundary operator for the helmholtz equation this work introduces an improvement of this algorithm that allows to solve the scattering problem in an almost linear time the new approach is based on two main ingredients the near field reuse and the application of data sparse techniques exponential decay of runge kutta convolution weights w n h d outside of a neighborhood of d n h where h is a time step allows to avoid constructing the near field i e singular and near singular integrals for most of the discretizations of the single layer boundary operators near field reuse the far field of these matrices is compressed with the help of data sparse techniques namely h matrices and the high frequency fast multipole method numerical experiments indicate the efficiency of the proposed approach compared to the conventional runge kutta convolution quadrature algorithm

multilevel lot sizing with setup times and multiple constrained resources internally rolling schedules with lot sizing windows in this paper a new time oriented decomposition heuristic is proposed to solve the dynamic multi item multilevel lot sizing problem in general product structures with single and multiple constrained resources as well as setup times while lot sizing decisions are made sequentially within an internally rolling planning interval or lot sizing window capacities are always considered over the entire planning horizon for each submodel a model formulation based on the simple plant location representation is developed these mixed integer linear submodels are solved by standard mathematical programming software even for relatively large test instances extensive computational tests show that the heuristic proposed provides a better solution quality than a well known special purpose heuristic

nonalgebraic integrability of one reversible dynamical system of the cremona type a reversible dynamical system rds and a system of nonlinear functional equations defined by a certain rational quadratic cremona mapping and arising from the static model of the dispersion approach in the theory of strong interactions the chew low type equations with crossing symmetry matrix a l 1 are considered this rds is split into one and two dimensional ones an explicit cremona transformation that completely determines the exact solution of the two dimensional system is found this solution depends on an odd function satisfying a nonlinear autonomous three point functional equation nonalgebraic integrability of rds under consideration is proved using the method of poincare normal forms and the siegel theorem on biholomorphic linearization of a mapping at a nonresonant fixed point

a lagrangean based heuristic for dynamic multilevel multiitem constrained lotsizing with setup times in this paper a heuristic approach for the dynamic multilevel multiitem lotsizing problem in general product structures with multiple constrained resources and setup times is proposed with the help of lagrangean relaxation the capacitated multilevel multiitem lotsizing problem is decomposed into several uncapacitated single item lotsizing problems from the solutions of these single item problems lower bounds on the minimum objective function value are derived upper bounds are generated by means of a heuristic finite scheduling procedure the quality of the approach is tested with reference to various problem groups of differing sizes

solving multi item capacitated lot sizing problems using variable redefinition mixed integer programming models are typically not used to solve realistic sized production scheduling problems because they require exorbitant solution times we impose a useful taxonomy on production scheduling problems and develop alternative formulations for a wide variety of problems within the taxonomy the linear programming relaxation of the new models is very effective in generating bounds we show that these bounds are equal to those that could be generated using lagrangian relaxation or column generation the linear programming bounds increase in effectiveness as the problems become larger perhaps of greatest significance is that practitioners can obtain our results using only standard off the shelf codes such as lindo or mpsx 370 we report computational experience in several computing environments hardware and software on problems with up to 200 products and 10 time periods 2000 0 1 variables

cremona transformation and general solution of one dynamical system of the static model abstract the general solution to equations for s matrix elements of the three channel static model with the crossing symmetry matrix a 1 1 in their famous formulation as a system of nonlinear difference equations is obtained for the first time this three dimensional dynamical system is defined essentially by the two dimensional subsystem given by the mapping that is the quadratic cremona transformation for the latter all invariant algebraic curves are found on their basis the cremona transformation is constructed which transforms the subsystem to a very simple form of the mapping with a zero fixed hyperbolic point and a jacobian not equal to one which enables one to find an invariant nonalgebraic curve and to investigate its properties we have found the invariant measure and cremona transformation reducing the subsystem to the form of area preserving mapping using the results of birkhoff and moser for the mapping we transform the initial difference equations into an easily integrable form and obtain two types of the general solution the phase curves of the subsystem for different values of the invariant of the cremona transformation are computed

adaptive time discretization for retarded potentials in this paper we will present advanced discretization methods for solving retarded potential integral equations we employ a c infty c partition of unity method in time and a conventional boundary element method for the spatial discretization one essential point for the algorithmic realization is the development of an efficient method for approximation the elements of the arising system matrix we present here an approach which is based on quadrature for non analytic c infty c functions in combination with certain chebyshev expansions furthermore we introduce an a posteriori error estimator for the time discretization which is employed also as an error indicator for adaptive refinement numerical experiments show the fast convergence of the proposed quadrature method and the efficiency of the adaptive solution process

primal dual approach to the single level capacitated lot sizing problem abstract the lagrangean relaxation of the single level capacitated dynamic lot sizing problem can be solved using the primal dual method the algorithm has monotone and finite convergence properties it works as a steepest ascent method a variant of this approach is also studied a heuristic routine used to obtain a feasible solution in each iteration is presented computational experiences show that this method usually yields better solutions than the subgradient method although it requires greater cpu times