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

fusion categories arising from semisimple lie algebras using tilting modules we equip certain semisimple categories with a reduced tensor product structure the fusion rules for this tensor product are determined via known character formulas for the involved modules

reduced standard modules and cohomology first cohomology groups of finite groups with nontrivial irreducible coefficients have been useful in several geometric and arithmetic contexts including wiles s famous paper 1995 internal to group theory 1 cohomology plays a role in the general theory of maximal subgroups of finite groups as developed by aschbacher and scott 1985 one can pass to the case where the group acts faithfully and the underlying module is absolutely irreducible in this case r guralnick 1986 conjectured that there is a universal constant bounding all of the dimensions of these cohomology groups this paper provides the first general positive results on this conjecture proving that the generic 1 cohomology h 1 gen g l lim h 1 g q l see cline parshall scott and van der kallen 1977 of a finite group g q of lie type with absolutely irreducible coefficients l in the defining characteristic of g is bounded by a constant depending only on the root system in all cases we are able to improve this result to a bound on h 1 g q l itself still depending only on the root system the generic h 1 result and related results for ext 1 emerge here as a consequence of a general study of interest in its own right of the homological properties of certain rational modules red red indexed by dominant weights for a reductive group g the modules red and red arise naturally from irreducible representations of the quantum enveloping algebra u of the same type as g at a pth root of unity where p 0 is the characteristic of the defining field for g finally we apply our ext 1 bounds results of bendel nakano and pillen 2006 as well as results of sin 1993 1992 1994 on the ree and suzuki groups to obtain the non generic bounds on h 1 g q l

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

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

p filtrations of dual weyl modules abstract let g be a semisimple algebraic group over a field of characteristic p 0 we prove that the dual weyl modules for g all have p filtrations when p is not too small moreover we give applications of this theorem to p n filtrations for n 1 to modules containing the steinberg module as a tensor factor and to the donkin conjecture on modules having p filtrations

generating alternative mixed integer programming models using variable redefinition dropping the complicating constraints in a mixed integer linear program often yields a special structure subproblem that can be reformulated using a different set of decision variables once the new variables have been identified the entire problem can be reformulated in terms of the new variables we develop a theory of variable redefinition based on relating the two sets of decision variables by a linear transformation and describe methods for reformulating the special structure problem the reformulated models have a more accurate linear relaxation than the problems from which they were derived an important property within the context of linear programming based branch and bound modeling approaches

the splitting of variables and constraints in the formulation of integer programming models the splitting of variables in an integer programming model into the sum of other variables can allow the constraints to be disaggregated leading to a more constrained tighter linear programming relaxation well known examples of such reformulations are quoted from the literature they can be viewed as instances of some general methods of performing such reformulations namely disjunctive formulations partial network reformulations and a method based on the introduction of auxiliary variables

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

facets and reformulations for solving production planning with changeover costs we study a scheduling problem with changeover costs and capacity constraints the problem is np complete and combinatorial algorithms for solving it have not performed well we identify a general class of facets that subsumes as special cases some known facets from the literature we also develop a cutting plane based procedure and reformulation for the problem and we obtain optimal solutions to problem instances with up to 600 integer variables without resorting to branch and bound procedures

approximation formulations for the single product capacitated lot size problem we consider two approximation formulations for the single product capacitated lot size problem one formulation restricts the number of production policies and the other rounds demands up to multiples of a constant after briefly reviewing the literature within a new framework we discuss the relations between these approximation formulations next we provide relative error bounds and algorithms for solving the approximation problems we demonstrate that these approximation formulations require a significantly smaller number of calculations than the original formulation and that the relative error bounds are satisfactory for practical purposes

strong formulations for mixed integer programs valid inequalities and extended formulations we examine progress over the last fifteen years in finding strong valid inequalities and tight extended formulations for simple mixed integer sets lying both on the easy and hard sides of the complexity frontier most progress has been made in studying sets arising from knapsack and single node flow sets and a variety of sets motivated by different lot sizing models we conclude by citing briefly some of the more intriguing new avenues of research

lot size models with back logging strong reformulations and cutting planes we examine mixed integer programming reformulations of the uncapacitated lot sizing problem with backlogging first we consider the effect of using a standard reformulation technique for fixed charge network flow problems which involves the introduction of new variables leading to a known plant location reformulation and a shortest path reformulation each of these reformulations is strong in the sense that its linear programming relaxation solves the lot sizing problem secondly we attempt to treat the problem in the space of the original variables we give an implicit description of the convex hull of solutions and show how the problem of finding a violated cutting plane can be solved as a linear program we also describe a family of strong valid inequalities which can be generated rapidly by a heuristic and which have proved effective in a cut generation algorithm the efficiency of both the shortest path formulation and the cutting plane algorithm have been tested on a series of multi item capacitated lot sizing problems with backlogging near optimal solutions have been found to problems with 8 periods and up to 100 times

metabelian lie powers of the natural module for a general linear group abstract consider a free metabelian lie algebra m of finite rank r over an infinite field k of prime characteristic p given a free generating set m acquires a grading its group of graded automorphisms is the general linear group g l r k so each homogeneous component m d is a finite dimensional g l r k module the homogeneous component m 1 of degree 1 is the natural module and the other m d are the metabelian lie powers of the title this paper investigates the submodule structure of the m d in the main result a composition series is constructed in each m d and the isomorphism types of the composition factors are identified both in terms of highest weights and in terms of steinberg s twisted tensor product theorem their dimensions are also given it turns out that the composition factors are pairwise non isomorphic from which it follows that the submodule lattice is finite and distributive by the birkhoff representation theorem any such lattice is explicitly recognizable from the poset of its join irreducible elements the poset relevant for m d is then determined by exploiting a 1975 paper of yu a bakhturin on identical relations in metabelian lie algebras

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

good l filtrations for q gl 3 k abstract let k be an algebraically closed field of characteristic p possibly zero and g q gl 3 k the quantum group of three by three matrices as defined by dipper and donkin we may also take g to be gl 3 k we first determine the extensions between simple g modules for both g and g 1 the first frobenius kernel of g we then determine the submodule structure of certain induced modules z for the infinitesimal group g 1 b we induce this structure to g to obtain a good l filtration of certain induced modules for g we also determine the homomorphisms between these induced modules for g

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

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

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 polyhedral study of multiechelon lot sizing with intermediate demands in this paper we study a multiechelon uncapacitated lot sizing problem in series m uls where the output of the intermediate echelons has its own external demand and is also an input to the next echelon we propose a polynomial time dynamic programming algorithm which gives a tight compact extended formulation for the two echelon case 2 uls next we present a family of valid inequalities for m uls show its strength and give a polynomial time separation algorithm we establish a hierarchy between the alternative formulations for 2 uls in particular we show that our valid inequalities can be obtained from the projection of the multicommodity formulation our computational results show that this extended formulation is very effective in solving our uncapacitated multi item two echelon test problems in addition for capacitated multi item multiechelon problems we demonstrate the effectiveness of a branch and cut algorithm using the proposed inequalities

linear programming extended formulations for the single item lot sizing problem with backlogging and constant capacity recently several authors 8 10 have argued for the use of extended formulations to tighten production planning models in this work we present two linear programming extended formulations of the constant capacity lot sizing problem with backlogging the first one applies to the problem with a general cost function and has o n3 variables and constraints this improves on the more straightforward o n4 florian and klein 2 type formulation the second one applies when the costs satisfy the wagner whitin property but it has o n2 variables and o n3 constraints as a by product we positively answer an open question of miller and wolsey 4 about the tightness of an extended formulation for the continuous mixing set

an extended formulation approach to the edge weighted maximal clique problem we consider an extended formulation approach to the edge weighted maximal clique problem the problem is formulated by using additional variables for the set of nodes with the natural variables for the set of edges we show that the proposed formulation is superior to the natural formulation both theoretically and practically by using the projection technique we can also derive new classes of facet defining inequalities for the lower dimensional polytope of the natural variables computational results are reported

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

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

a note on the tensor product of restricted simple modules for algebraic groups let g be a semisimple simply connected algebraic group over an algebraically closed field of positive characteristic p denote by g1 its first frobenius kernel in this note we determine for which group g the restriction to g1 of any indecomposable g summand of the tensor product of any two restricted simple g modules remains indecomposable

tensor products of quantized tilting modules letuk denote the quantized enveloping algebra corresponding to a finite dimensional simple complex lie algebral assume that the quantum parameter is a root of unity ink of order at least the coxeter number forl also assume that this order is odd and not divisible by 3 if typeg2 occurs we demonstrate how one can define a reduced tensor product on the familyf consisting of those finite dimensional simpleuk modules which are deformations of simplel and which have non zero quantum dimension this together with the work of reshetikhin turaev and turaev wenzl prove that uk f is a modular hopf algebra and hence produces invariants of 3 manifolds also by recent work of duurhus jakobsen and nest it leads to a general topological quantum field theory the method of proof explores quantized analogues of tilting modules for algebraic groups

homomorphisms between weyl modules for sl3 k we classify all homomorphisms between weyl modules for sl3 k when k is an algebraically closed field of characteristic at least three and show that the hom spaces are all at most one dimensional as a corollary we obtain all homomorphisms between specht modules for the symmetric group when the labelling partitions have at most three parts and the prime is at least three we conclude by showing how a result of fayers and lyle on hom spaces for specht modules is related to earlier work of donkin for algebraic groups let g be a reductive algebraic group over an algebraically closed field of characteristic p 0 an important class of modules for such a group are the weyl modules labelled by dominant weights these can be constructed relatively explicitly and their heads provide a full set of simple modules for g equivalently one can study the duals of these modules denoted which have the advantage of being induced from one dimensional modules for a borel subgroup in determining the structure of such modules or indeed their cohomology the calculation of hom spaces between them is a useful tool relatively little is known in general about such hom spaces in type a when andare related by a suitable single reflection explicit non zero homomorphisms from to were constructed with some restrictions by carter and lusztig 3 and more generally by carter and payne 4 the corresponding cases in other types were considered by franklin 14 while it is clear that there should be a hierarchy of families of homomorphisms corresponding to different powers of p the only case where the above results provide a complete classification is when g is sl2 where it is relatively easy to determine all hom spaces exactly 6 the only other general results in this area by andersen 1 and koppinen 18 concern homo morphisms between modules labelled by weights which are close together typically such results show that certain hom spaces are non zero or in some cases one dimensional for weights which are far apart and not related by a single reflection almost nothing is known in this paper we will determine all homomorphisms between weyl modules for sl3 when p 3 and provide a recursive procedure for determining the composition factors arising in the image or kernel of such maps in most cases from these results we will also classify all homomorphisms between specht modules for the symmetric groups corresponding to three part partitions when p 3 after a section of preliminaries we review the sl3 data concerning p filtrations that we will need from 21 this describes certain filtrations of induced modules which will allow us to proceed by induction together with the set of p good homomorphisms which will be fundamental in our later constructions we also recall a theorem of carter and payne 4 on the existence of certain homomorphisms these will be the two key sets of data which we need to determine all possible homomorphisms with the notation developed up to that point in place in section 4 we can give the strategy to be followed in the remainder of the paper and in particular the translation functor arguments

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

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

a note on splitting of variables in integer programming models abstract the splitting of variables in an integer programming model is a tool to obtain a more constrained tighter linear programming relaxation a known general method for knapsack constraints the partial network reformulation method produces an equivalent set of constraints that has only integer solutions we show how to modify this method to reduce substantially the number of split variables and in most cases the number of disaggregated constraints further the modified method can implicitly handle upper bounds on as well as fixed charges for at least two variables of the considered constraint

a strong cutting plane algorithm for production scheduling with changeover costs changeover costs and times are central to numerous manufacturing operations these costs arise whenever work centers capable of processing only one product at a time switch from the manufacture of one product to another although many researchers have contributed to the solution of scheduling problems that include changeover costs due to the problem s combinatorial explosiveness optimization based methods have met with limited success in this paper we develop and apply polyhedral methods from integer programming for a dynamic version of the problem computational tests with problems containing one to five products and up to 225 integer variables show that polyhedral methods based upon a set of facet inequalities developed in this paper can effectively reduce the gap between the value of an integer programming formulation of the problem and its linear programming relaxation by a factor of 94 to 100 these results suggest the use of a combined cutting plane branch and bound procedure as a solution approach in a test with a five product problem this procedure when compared with a standard linear programming based branch and bound approach reduced computation time by a factor of seven

a fully polynomial approximation scheme for single product scheduling in a finite capacity facility this paper considers a version of the economic lot sizing problem for a single product produced in a facility of finite capacity over a finite time horizon with specifiable start and end conditions a set of algorithms is presented that will approximate the optimal production schedule to a given allowable error e algorithms with computation time bounds of o1 e2 are presented which allow for setups of finite length setups with or without direct cash flow quite general cost and demand functions and a wide variety of production policy constraints the procedures make no a priori assumptions about the form of the optimal solution numerical results are included

stabilization of stochastic singular nonlinear hybrid systems abstract this paper deals with the control of the class of singular nonlinear stochastic hybrid systems under some appropriate assumptions results on stochastic stability and stochastic stabilization are developed two state feedback controllers linear and nonlinear that stochastically stabilize the class of systems we are considering are designed lmi sufficient conditions are developed to compute the gains of these controllers

approximate extended formulations mixed integer programming mip formulations are typically tightened through the use of a separation algorithm and the addition of violated cuts using extended formulations involving new variables is a possible alternative but this often results in prohibitively large mips where even the linear programming relaxations are hard or impossible to solve in this paper we demonstrate how in certain cases it is possible and interesting to define approximate extended formulations in all the examples considered our description involves a single control parameter k large values of k result in strong but large formulations in particular when k takes its maximum value the approximate formulation is identical to the complete extended formulation through this approximation parameter the user has control over the tradeoff between the strength and the size of the formulation

valid inequalities for the single item capacitated lot sizing problem with step wise costs this paper presents a new class of valid inequalities for the single item capacitated lot sizing problem with step wise production costs ls sw we first provide a survey of different optimization methods proposed to solve ls sw then flow cover and flow cover inequalities derived from the single node flow set are described in order to generate the new class of valid inequalities the single node flow set can be seen as a generalization of some valid relaxations of ls sw a new class of valid inequalities we call mixed flow cover is derived from the integer flow cover inequalities by a lifting procedure the lifting coefficients are sequence independent when the batch sizes v and the production capacities p are constant and if v divides p when the restriction of the divisibility is removed the lifting coefficients are shown to be sequence independent we identify some cases where the mixed flow cover inequalities are facet defining a cutting plane algorithm is proposed for these three classes of valid inequalities the exact separation algorithm proposed for the constant capacitated case runs in polynomial time finally some computational results are given to compare the performance of the different optimization methods including the new class of valid inequalities

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

on good p r filtrations for rational g modules abstract in this paper we investigate donkin s p r filtration conjecture and present two proofs of the if direction of the statement when p 2 h 2 one proof involves the investigation of when the tensor product between the steinberg module and a simple module has a good filtration one of our main results shows that this holds under suitable requirements on the highest weight of the simple module the second proof involves recasting donkin s conjecture in terms of the identifications of projective indecomposable g r modules with certain tilting g modules and establishing necessary cohomological criteria for the p r filtration conjecture to hold