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

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

multi item lot sizing with joint set up costs we consider a multi item lot sizing problem with joint set up costs and constant capacities apart from the usual per unit production and storage costs for each item a set up cost is incurred for each batch of production where a batch consists of up to c units of any mix of the items in addition an upper bound on the number of batches may be imposed under widely applicable conditions on the storage costs namely that the production and storage costs are nonspeculative and for any two items the one that has a higher storage cost in one period has a higher storage cost in every period we show that there is a tight linear program with o mt 2 constraints and variables that solves the joint set up multi item lot sizing problem where m is the number of items and t is the number of time periods this establishes that under the above storage cost conditions this problem is polynomially solvable for the problem with backlogging a similar linear programming result is described for the uncapacitated case under very restrictive conditions on the storage and backlogging costs computational results are presented to test the effectiveness of using these tight linear programs in strengthening the basic mixed integer programming formulations of the joint set up problem both when the storage cost conditions are satisfied and also when they are violated

facets and algorithms for capacitated lot sizing the dynamic economic lot sizing model which lies at the core of numerous production planning applications is one of the most highly studied models in all of operations research and yet capacitated multi item versions of this problem remain computationally elusive we study the polyhedral structure of an integer programming formulation of a single item capacitated version of this problem and use these results to develop solution methods for multi item applications in particular we introduce a set of valid inequalities for the problem and show that they define facets of the underlying integer programming polyhedron computational results on several single and multiple product examples show that these inequalities can be used quite effectively to develop an efficient cutting plane branch and bound procedure moreover our results show that in many instances adding certain of these inequalities a priori to the problem formulation and avoiding the generation of cutting planes can be equally effective

computing isogeny volcanoes of composite degree abstract isogeny volcanoes are an interesting structure that have had several recent applications an isogeny volcano is a connected component of a larger graph called a cordillera in this paper we further explore properties of how to compute volcanoes given that we have already computed one of a different degree this allows us to compute volcanoes of composite degree more efficiently than a direct construction using modular polynomials

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

a polynomial time algorithm to solve the single item capacitated lot sizing problem with minimum order quantities and concave costs abstract this paper deals with the single item capacitated lot sizing problem with concave production and storage costs and minimum order quantity clsp moq in this problem a demand must be satisfied at each period t over a planning horizon of t periods this demand can be satisfied from the stock or by a production at the same period when a production is made at period t the produced quantity must be greater to than a minimum order quantity l and lesser than the production capacity u to solve this problem optimally a polynomial time algorithm in o t 5 is proposed and it is computationally tested on various instances

computing the structure of a finite abelian group we present an algorithm that computes the structure of a finite abelian group g from a generating system m the algorithm executes o m g group operations and stores o g group elements

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 study of a numerical solution of the steady two dimensions navier stokes equations in a constricted channel problem by a compact fourth order method we present a numerical solution for the steady 2d navier stokes equations using a fourth order compact type method the geometry of the problem is a constricted symmetric channel where the boundary can be varied via a parameter from a smooth constriction to one possessing a very sharp but smooth corner allowing us to analyse the behaviour of the errors when the solution is smooth or near singular the set of non linear equations is solved by the newton method results have been obtained for reynolds number up to 500 estimates of the errors incurred have shown that the results are accurate and better than those of the corresponding second order method

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

capacitated multi item lot sizing problems with time windows this research concerns a new family of capacitated multi item lot sizing problems namely lot sizing problems with time windows two classes of the problem are analyzed and solved using different lagrangian heuristics capacity constraints and a subset of time window constraints are relaxed resulting in particular single item time window problems that are solved in polynomial time other relaxations leading to the classical wagner whitin problem are also tested several smoothing heuristics are implemented and tested and their results are compared the gaps between lower and upper bounds for most problems are very small less than 1 moreover the proposed algorithms are robust and do not seem to be too affected when different parameters of the problem are varied

on the number of isogeny classes of pairing friendly elliptic curves and statistics of mnt curves we give an upper bound on the number of finite fields over which elliptic curves of cryptographic interest with a given embedding degree and small complex multiplication discriminant may exist and present some heuristic arguments which indicate that this bound is tight we also refine some heuristic arguments on the total number of so called mnt curves with prime cardinalities which have been recently presented by various authors

fast approximation algorithms for the one warehouse multi retailer problem under general cost structures and capacity constraints we consider a well studied multi echelon deterministic inventory control problem known in the literature as the one warehouse multi retailer owmr problem we propose a simple and fast 2 approximation algorithm for this np hard problem by recombining the solutions of single echelon relaxations at the warehouse and at the retailers we then show that our approach remains valid under quite general assumptions on the cost structures and under capacity constraints at some retailers in particular we present the first approximation algorithms for the owmr problem with nonlinear holding costs truckload discount on procurement costs or with capacity constraints at some retailers in all cases the procedure is purely combinatorial and can be implemented to run in low polynomial time

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

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

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

an efficient algorithm for the capacitated single item dynamic lot size problem abstract a dynamic programming based algorithm is developed for the single item lot size problem with concave costs and arbitrary capacities by making use of the extreme point properties of the problem first the set of all feasible cumulative production levels that may occur in an optimal solution is generated in the second stage a dynamic programming procedure is carried out over this set the worst case computational effort is equal to that of the standard dynamic programming approach but extensive computational tests with the algorithm indicate that for t period problems the computational effort does not exceed o t4 the performance of the algorithm is compared with the performance of the existing procedures in the literature for the general the constant capacity and the constant unit cost problems the computational results demonstrate that our algorithm is at least three times faster than the other procedures for all problem types considered

multi level lot sizing problem evaluation of a simulated annealing heuristic abstract the multi level lot sizing problem mlp is the problem of determining production quantities in multi stage production settings such that the sum of set up and holding costs is minimized this type of problem is hard to solve to optimality thereby compelling one to use heuristic approaches in this paper we investigate heuristics based on a stochastic search method experimental results concerning the quality and efficiency of these methods for the mlp are presented and compared to the quality and efficiency of heuristic methods which are based on applying single level heuristics on a level by level basis

the complexity of class polynomial computation via floating point approximations we analyse the complexity of computing class polynomials that are an important ingredient for cm constructions of elliptic curves via complex floating point approximations of their roots the heart of the algorithm is the evaluation of modular functions in several arguments the fastest one of the presented approaches uses a technique devised by dupont to evaluate modular functions by newton iterations on an expression involving the arithmetic geometric mean it runs in time o d log 5 d log log d o d 1 epsilon o h 2 epsilon for any epsilon 0 where d is the cm discriminant and h is the degree of the class polynomial another fast algorithm uses multipoint evaluation techniques known from symbolic computation its asymptotic complexity is worse by a factor of log d up to logarithmic factors this running time matches the size of the constructed polynomials the estimate also relies on a new result concerning the complexity of enumerating the class group of an imaginary quadratic order and on a rigorously proven upper bound for the height of class polynomials

modular exponentiation via the explicit chinese remainder theorem fix pairwise coprime positive integers p 1 p 2 p a we propose representing integers u modulo m where m is any positive integer up to roughly p 1 p 2 p s as vectors u mod p 1 u mod p 2 u mod p s we use this representation to obtain a new result on the parallel complexity of modular exponentiation there is an algorithm for the common crcw pram that given positive integers x e and m in binary of total bit length n computes x e mod m in time o n lglgn using n o 1 processors for comparison a parallelization of the standard binary algorithm takes superlinear time adleman and kompella gave an o lg n 3 expected time algorithm using exp o n lg n processors von zur gathen gave an nc algorithm for the highly special case that m is polynomially smooth

economic lot sizing an o n log n algorithm that runs in linear time in the wagner whitin case we consider the n period economic lot sizing problem where the cost coefficients are not restricted in sign in their seminal paper h m wagner and t m whitin proposed an o n2 algorithm for the special case of this problem where the marginal production costs are equal in all periods and the unit holding costs are nonnegative it is well known that their approach can also be used to solve the general problem without affecting the complexity of the algorithm in this paper we present an algorithm to solve the economic lot sizing problem in o n log n time and we show how the wagner whitin case can even be solved in linear time our algorithm can easily be explained by a geometrical interpretation and the time bounds are obtained without the use of any complicated data structure furthermore we show how wagner and whitin s and our algorithm are related to algorithms that solve the dual of the simple plant location formulation of the economic lot sizing problem

the multi item capacitated lot sizing problem with setup times and shortage costs we address a multi item capacitated lot sizing problem with setup times and shortage costs that arises in real world production planning problems demand cannot be backlogged but can be totally or partially lost the problem is np hard a mixed integer mathematical formulation is presented our approach in this paper is to propose some classes of valid inequalities based on a generalization of miller et al a j miller g l nemhauser m w p savelsbergh on the polyhedral structure of a multi item production planning model with setup times mathematical programming 94 2003 375 405 and marchand and wolsey h marchand l a wolsey the 0 1 knapsack problem with a single continuous variable mathematical programming 85 1999 15 33 results we also describe fast combinatorial separation algorithms for these new inequalities we use them in a branch and cut framework to solve the problem some experimental results showing the effectiveness of the approach are reported

a fourth order accurate method for the incompressible navier stokes equations on overlapping grids a method is described to solve the time dependent incompressible navier stokes equations with finite differences on curvilinear overlapping grids in two or three space dimensions the scheme is fourth order accurate in space and uses the momentum equations for the velocity coupled to a poisson equation for the pressure the boundary condition for the pressure is taken as u 0 extra numerical boundary conditions are chosen to make the scheme accurate and stable the velocity is advanced explicitly in time any standard time stepping scheme such as runge kutta can be used the poisson equation is solved using direct or iterative sparse matrix solvers or by the multigrid algorithm computational results in two and three space dimensions are given

high accuracy solutions of incompressible navier stokes equations abstract in recent years we have developed high accuracy finite difference approximations for partial differential equations of elliptic type with particular emphasis on the convection diffusion equation these approximations are of compact type have a local truncation error of fourth order and allow the use of standard iterative schemes to solve the resulting systems of algebraic equations in this paper we extend these high accuracy approximations to the solution of navier stokes equations solutions are obtained for the model problem of driven cavity and are compared with solutions obtained using other approximations and those obtained by other authors it is discovered that the high order approximations do indeed produce high accuracy solutions and have a potential for use in solving important problems of viscous fluid flows

a p adic algorithm to compute the hilbert class polynomial classically the hilbert class polynomial p z x of an imaginary quadratic discriminant a is computed using complex analytic techniques in 2002 couveignes and henocq suggested a p adic algorithm to compute p unlike the complex analytic method it does not suffer from problems caused by rounding errors in this paper we give a detailed description of the algorithm in the paper by couveignes and henocq and our careful study of the complexity shows that if the generalized riemann hypothesis holds true the expected runtime of the p adic algorithm is o log 8 e instead of o 1 e we illustrate the algorithm by computing the polynomial p 639 using a 643 adic algorithm

a lagrangian heuristic for an integrated lot sizing and fixed scheduling problem this paper presents a novel approach for solving an integrated production planning and scheduling problem in theory as well as in practice because of their complexity these two decision levels are most of the time treated sequentially scheduling largely depends on the production quantities lot sizes computed at the production planning level and ignoring scheduling constraints in planning leads to inconsistent decisions integrating production planning and scheduling is therefore important for efficiently managing operations an integrated model and an iterative solution procedure were proposed in earlier research papers the approach has limitations in particular when solving the planning problem in this paper a new formulation is proposed to determine a feasible optimal production plan i e lot sizes for a fixed sequence of operations on the machines when setup costs and times are taken into account capacity constraints correspond to paths of the conjunctive graph associated to the sequence an original lagrangian relaxation approach is proposed to solve this np hard problem a lower bound is derived and an upper bound is calculated using a novel constructive heuristic the quality of the approach is tested on numerous problem instances

the discrete lot sizing and scheduling problem abstract the discrete lot sizing and scheduling problem consists in scheduling several products on a single machine so as to meet the known dynamic demand and to minimize the sum of inventory and setup cost the planning interval is phased into many short periods e g shifts or days and setups may occur only at the beginning of a period a branch and bound procedure is presented using lagrangean relaxation for determining both lower bounds and feasible solutions the relaxed problems are solved by dynamic programming computational results on a personal computer are reported for various examples from the literature with up to 12 products and 122 periods or 3 products and 250 periods the procedure yields optimal solutions or at least feasible solutions with tight lower bounds in a few minutes the results are compared with those obtained by solving the usual capacitated lot sizing problem

single item lot sizing with non decreasing capacities we consider the single item lot sizing problem with capacities that are non decreasing over time when the cost function is i non speculative or wagner whitin for instance constant unit production costs and non negative unit holding costs and ii the production set up costs are non increasing over time it is known that the minimum cost lot sizing problem is polynomially solvable using dynamic programming when the capacities are non decreasing we derive a compact mixed integer programming reformulation whose linear programming relaxation solves the lot sizing problem to optimality when the objective function satisfies i and ii the formulation is based on mixing set relaxations and reduces to the known convex hull of solutions when the capacities are constant over time we illustrate the use and potential effectiveness of this improved lp formulation on a few test instances including instances with and without wagner whitin costs and with both non decreasing and arbitrary capacities over time

determining the 2 sylow subgroup of an elliptic curve over a finite field in this paper we describe an algorithm that outputs the order and the structure including generators of the 2 sylow subgroup of an elliptic curve over a finite field to do this we do not assume any knowledge of the group order the results that lead to the design of this algorithm are of inductive type then a right choice of points allows us to reach the end within a linear number of successive halvings the algorithm works with abscissas so that halving of rational points in the elliptic curve becomes computing of square roots in the finite field efficient methods for this computation determine the efficiency of our algorithm

solving multi item lot sizing problems using strong cutting planes we consider two classes of multi item lot sizing problems the first is a class of single stage problems involving joint machine capacity constraints and or start up costs and the second is a class of multistage problems with general product structure the problems are solved as mixed integer programs based on i an appropriate choice of the initial problem formulation and ii the addition of cuts which are generated automatically by a mathematical programming system mpsarx our results extend and complement those of karmarkar and schrage 1985 afentakis and gavish 1986 eppen and martin 1987 and van roy and wolsey 1987 a major advantage of this approach is its robustness or flexibility by using just a matrix generator and a mathematical programming system with automatic cut generation routines we can formulate and solve model variants without incurring the costs of adapting an algorithm

relaxations for two level multi item lot sizing problems we consider several variants of the two level lot sizing problem with one item at the upper level facing dependent demand and multiple items or clients at the lower level facing independent demands we first show that under a natural cost assumption it is sufficient to optimize over a stock dominant relaxation we further study the polyhedral structure of a strong relaxation of this problem involving only initial inventory variables and setup variables we consider several variants uncapacitated at both levels with or without start up costs uncapacitated at the upper level and constant capacity at the lower level constant capacity at both levels we finally demonstrate how the strong formulations described improve our ability to solve instances with up to several dozens of periods and a few hundred products

computing modular polynomials in quasi linear time we analyse and compare the complexity of several algorithms for computing modular polynomials we show that an algorithm relying on floating point evaluation of modular functions and on interpolation which has received little attention in the literature has a complexity that is essentially up to logarithmic factors linear in the size of the computed polynomials in particular it obtains the classical modular polynomials phi ell of prime level ell in time o ell 3 log 4 ell log log ell besides treating modular polynomials for gamma 0 ell which are an important ingredient in many algorithms dealing with isogenies of elliptic curves the algorithm is easily adapted to more general situations composite levels are handled just as easily as prime levels as well as polynomials between a modular function and its transform of prime level such as the schlafli polynomials and their generalisations our distributed implementation of the algorithm confirms the theoretical analysis by computing modular equations of record level around 10000 in less than two weeks on ten processors

a single item economic lot sizing problem with a non uniform resource approximation we study a generalization of the classical single item capacitated economic lot sizing problem to the case of a non uniform resource usage for production the general problem and several special cases are shown to be non approximable with any polynomially computable relative error in polynomial time an optimal dynamic programming algorithm and its approximate modification are presented for the general problem fully polynomial time approximation schemes are developed for two np hard special cases 1 cost functions of total production are separable and holding and backlogging cost functions are linear with polynomially related slopes and 2 all holding costs are equal to zero

polynomial time algorithms for stochastic uncapacitated lot sizing problems in 1958 wagner and whitin published a seminal paper on the deterministic uncapacitated lot sizing problem a fundamental model that is embedded in many practical production planning problems in this paper we consider a basic version of this model in which problem parameters are stochastic the stochastic uncapacitated lot sizing problem we define the production path property of an optimal solution for our model and use this property to develop a backward dynamic programming recursion this approach allows us to show that the value function is piecewise linear and right continuous we then use these results to show that a full characterization of the optimal value function can be obtained by a dynamic programming algorithm in polynomial time for the case that each nonleaf node contains at least two children moreover we show that our approach leads to a polynomial time algorithm to obtain an optimal solution to any instance of the stochastic uncapacitated lot sizing problem regardless of the structur

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

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

computing the height of volcanoes of l isogenies of elliptic curves over finite fields the structure of the volcano of l isogenies l prime of elliptic curves over finite fields has been extensively studied over recent years previous works present some results and algorithms concerning the height of such volcanoes in the case of isogenies whose kernels are generated by a rational point the main goal of this paper is to extend such works to the case of l isogenies whose kernels are defined by a rational subgroup in particular the height of such volcanoes is completely characterized and can be computationally obtained

factoring integers with elliptic curves this paper is devoted to the description and analysis of a new algorithm to factor positive integers it depends on the use of elliptic curves the new method is obtained from pollard s p 1 method proc cambridge philos soc 76 1974 521 528 by replacing the multiplicative group by the group of points on a random elliptic curve it is conjectured that the algorithm determines a non trivial divisor of a composite number n in expected time at most k p log n 2 where p is the least prime dividing n and k is a function for which log k x 2 o 1 log x log log x for x x o in the worst case when n is the product of two primes of the same order of magnitude this is

lagrangian relaxation based solution procedures for a multiproduct capacitated facility location problem with choice of facility type this paper presents exact and heuristic solution procedures for a multiproduct capacitated facility location mpcfl problem in which the demand for a number of different product families must be supplied from a set of facility sites and each site offers a choice of facility types exhibiting different capacities mpcfl generalizes both the uncapacitated or simple facility location ufl problem and the pure integer capacitated facility location problem we define a branch and bound algorithm for mpcfl that utilizes bounds formed by a lagrangian relaxation of mpcfl which decomposes the problem into ufl subproblems and easily solvable 0 1 knapsack subproblems the ufl subproblems are solved by the dual based procedure of erlenkotter we also present a subgradient optimization lagrangian relaxation based heuristic for mpcfl computational experience with the algorithm and heuristic are reported the mpcfl heuristic is seen to be extremely effective generating solutions to the test problems that are on average within 2 of optimality and the branch and bound algorithm is successful in solving all of the test problems to optimality

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

finding suitable curves for the elliptic curve method of factorization using the parametrizations of kubert we show how to produce in nite families of elliptic curves which have prescribed nontrivial torsion over q and rank at least one these curves can be used to speed up the ecm factorization algorithm of lenstra we also brie y discuss curves with complex multiplication in this context

modelling practical lot sizing problems as mixed integer programs in spite of the remarkable improvements in the quality of general purpose mixed integer programming software the effective solution of a variety of lot sizing problems depends crucially on the development of tight formulations for the special problem features occurring in practice after reviewing some of the basic preprocessing techniques for handling safety stocks and multilevel problems we discuss a variety of aspects arising particularly in small and large bucket time period models such as start ups changeovers minimum batch sizes choice of one or two set ups per period etc a set of applications is described that contains one or more of these special features and some indicative computational results are presented finally to show another technique that is useful a slightly different supply chain application is presented for which the a priori addition of some simple mixed integer inequalities based on aggregation leads to important improvements in the results

single item lot sizing problem for a warm cold process with immediate lost sales abstract we consider the dynamic lot sizing problem with finite capacity and possible lost sales for a process that could be kept warm at a unit variable cost for the next period t 1 only if more than a threshold value q t has been produced and would be cold otherwise production with a cold process incurs a fixed positive setup cost k t and setup time s t which may be positive setup costs and times for a warm process are negligible we develop a dynamic programming formulation of the problem establish theoretical results on the structure of the optimal production plan in the presence of zero and positive setup times with wagner whitin type cost structures we also show that the solution to the dynamic lot sizing problem with lost sales are generated from the full commitment production series improved via lost sales decisions in the presence of a warm cold process

efficient cm constructions of elliptic curves over finite fields we present an algorithm that on input of an integer n 1 together with its prime factorization constructs a finite field f and an elliptic curve e over f for which e f has order n although it is unproved that this can be done for all n a heuristic analysis shows that the algorithm has an expected run time that is polynomial in 2 n log n where n is the number of distinct prime factors of n in the cryptographically relevant case where n is prime an expected run time o logn 4 e can be achieved we illustrate the efficiency of the algorithm by constructing elliptic curves with point groups of order n 10 2004 and n nextprime 102004 10 2004 4863

explicit bounds for primality testing and related problems many number theoretic algorithms rely on a result of ankeny which states that if the extended riemann hypothesis erh is true any nontrivial multiplicative subgroup of the integers modulo m omits a number that is o log2 m this has been generalized by lagarias montgomery and odlyzko to give a similar bound for the least prime ideal that does not split completely in an abelian extension of number fields this paper gives a different proof of this theorem in which explicit constants are supplied the bounds imply that if the erh holds a composite number m has a witness for its compositeness in the sense of miller or solovay strassen that is at most 2 log2m

elliptic curves and primality proving this report describes the theory and implementation of the elliptic curve primality proving algorithm ecpp algorithm this includes the relationships between representing primes by quadratic forms and the explicit construction of class fields of imaginary quadratic fields the theory of elliptic curves with complex multiplication over the field of complex numbers as well as over finite fields we then use this theory to design a very powerful primality proving algorithm half of the paper is devoted to the description of its implementation in particular we give the currently best algorithms to speed up each part of the program the resulting program is very fast we can prove the primality of 100 digit numbers in less than five minutes on a sun 3 60 workstation and we can treat all numbers with less than 1000 digits in a reasonable amount of time using a distributed implementation

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 simple forward algorithm to solve general dynamic lot sizing models with n periods in 0n log n or 0n time this paper is concerned with the general dynamic lot size model or generalized wagner whitin model let n denote the number of periods into which the planning horizon is divided we describe a simple forward algorithm which solves the general model in 0n log n time and 0n space as opposed to the well known shortest path algorithm advocated over the last 30 years with 0n2 time a linear i e 0n time and space algorithm is obtained for two important special cases a models without speculative motives for carrying stock i e where in each interval of time the per unit order cost increases by less than the cost of carrying a unit in stock b models with nondecreasing setup costs we also derive conditions for the existence of monotone optimal policies and relate these to known planning horizon and other results from the literature

modular polynomials via isogeny volcanoes we present a new algorithm to compute the classical modular polynomial l in the rings z x y and z mz x y for a prime l and any positive integer m our approach uses the graph of l isogenies to eciently compute l mod p for many primes p of a suitable form and then applies the chinese remainder theorem crt under the generalized riemann hypoth esis grh we achieve an expected running time of o l 3 logl 3 log logl and compute l mod m using o l 2 logl 2 l 2 logm space we have used the new algorithm to compute l with l over 5000 and l mod m with l over 20000 we also consider several modular functions g for which g is smaller than l allowing us to handle l over 60000

some experiments with high order compact methods using a computer algebra software part i abstract we consider a procedure for obtaining a compact fourth order method to the steady 2d navier stokes equations in the streamfunction formulation using the computer algebra system maple the resulting code is short and from it we obtain the fortran program for the method to test the procedure we have solved many cavity type problems which include one with an analytical solution and the results are compared with results obtained by second order central differences to moderate reynolds numbers

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

an optimization framework for solving capacitated multi level lot sizing problems with backlogging this paper proposes two new mixed integer programming models for capacitated multi level lot sizing problems with backlogging whose linear programming relaxations provide good lower bounds on the optimal solution value we show that both of these strong formulations yield the same lower bounds in addition to these theoretical results we propose a new effective optimization framework that achieves high quality solutions in reasonable computational time computational results show that the proposed optimization framework is superior to other well known approaches on several important performance dimensions

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

a heuristic approach for big bucket multi level production planning problems multi level production planning problems in which multiple items compete for the same resources frequently occur in practice yet remain daunting in their difficulty to solve in this paper we propose a heuristic framework that can generate high quality feasible solutions quickly for various kinds of lot sizing problems in addition unlike many other heuristics it generates high quality lower bounds using strong formulations and its simple scheme allows it to be easily implemented in the xpress mosel modeling language extensive computational results from widely used test sets that include a variety of problems demonstrate the efficiency of the heuristic particularly for challenging problems

computing l l isogenies in polynomial time on jacobians of genus 2 curves in this paper we compute l isogenies between abelian varieties over a field of characteristic different from 2 in polynomial time in l when l is an odd prime which is coprime to the characteristic we use level n symmetric theta structure where n 2 or n 4 in a second part of this paper we explain how to convert between mumford coordinates of jacobians of genus 2 hyperelliptic curves to theta coordinates of level 2 or 4 combined with the preceding algorithm this gives a method to compute l l isogenies in polynomial time on jacobians of genus 2 curves

a rigorous subexponential algorithm for computation of class groups let c d denote the gauss class group of quadratic forms of a negative discriminant d or equivalently the class group of the imaginary quadratic field q a we give a rigorous proof that there exists a las ve gas algorithm that will compute the structure of c d with an expected running time of l d vf2 1 bit operations where l d exp vlog d log log d thus of course also includes the computation of the class number h d the cardinality of c d ibm research division almaden research center k53 802 650 harry road san jose california 95 120 6099 current address k mccurley organization 1423 sandia national laboratories albuquerque new mexico 87185 this content downloaded from 207 46 13 180 on thu 08 sep 2016 05 47 29 utc all use subject to http about jstor org terms

choosing the correct elliptic curve in the cm method we give an elementary way to distinguish between the twists of an ordinary elliptic curve e over f p in order to identify the one with p 1 2u points when p u 2 dv 2 with 2u 2v and e is constructed using the cm method for finding elliptic curves with a prescribed number of points our algorithms consist in most cases of reading off simple congruence conditions on u and v modulo 4

high order compact scheme for the steady stream function vorticity equations a higher order compact scheme that is o h4 on the nine point 2 d stencil is formulated for the steady stream function vorticity form of the navier stokes equations the resulting stencil expressions are presented and hence this new scheme can be easily incorporated into existing industrial software we also show that special treatment of the wall boundary conditions is required the method is tested on representative model problems and compares very favourably with other schemes in the literature

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

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

lot sizing with start up times many practical applications of lot sizing and scheduling problems involve start up times operations research literature contains but few studies of lot sizing models that take start up times explicitly into account here we review some of these studies discuss the models and their complexity and we propose further models we consider in particular a single stage single mode multi item lot sizing model with continuous set ups and sequence independent start up times which we solve using an integer programming column generation algorithm and we develop a dynamic programming procedure for the single item subproblem that treats the initial stock as a decision variable we also use cutting planes developed by constantino for the multi item polyhedra by combining column and cut generation the lower bounds that we obtain before branching are on average less than 2 from an optimal solution our algorithm solves instances with 3 to 5 items and 24 periods in an average of 50 seconds on a modern workstation and problems with 36 periods in an average of 750 seconds solutions guaranteed to be within 2 of optimality are obtained in less than 75 of these times

a space efficient algorithm for group structure computation we present a new algorithm for computing the structure of a finite abelian group which has to store only a fixed small number of group elements independent of the group order we estimate the computational complexity by counting the group operations such as multiplications and equality checks under some plausible assumptions we prove that the expected run time is o n with n denoting the group order and we explicitly determine the o constants we implemented our algorithm for ideal class groups of imaginary quadratic orders and present experimental results

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

computing hilbert class polynomials with the chinese remainder theorem we present a space efficient algorithm to compute the hilbert class polynomial h d x modulo a positive integer p based on an explicit form of the chinese remainder theorem under the generalized riemann hypothesis the algorithm uses o d 1 2 e log p space and has an expected running time of o d 1 e we describe practical optimizations that allow us to handle larger discriminants than other methods with d as large as 10 13 and h d up to 10 6 we apply these results to construct pairing friendly elliptic curves of prime order using the cm method