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

velocity induced numerical solutions of reaction diffusion systems on continuously growing domains reaction diffusion systems have been widely studied in developmental biology chemistry and more recently in financial mathematics most of these systems comprise nonlinear reaction terms which makes it difficult to find closed form solutions it therefore becomes convenient to look for numerical solutions finite difference finite element finite volume and spectral methods are typical examples of the numerical methods used most of these methods are locally based schemes we examine the implications of mesh structure on numerically computed solutions of a well studied reaction diffusion model system on two dimensional fixed and growing domains the incorporation of domain growth creates an additional parameter the grid point velocity and this greatly influences the selection of certain symmetric solutions for the adi finite difference scheme when a uniform square mesh structure is used domain growth coupled with grid point velocity on a uniform square mesh stabilises certain patterns which are however very sensitive to any kind of perturbation in mesh structure we compare our results to those obtained by use of finite elements on unstructured triangular elements

an efficient trajectory method for probabilistic production inventory distribution problems we consider a supply chain operating in an uncertain environment the customers demand is characterized by a discrete probability distribution a probabilistic programming approach is adopted for constructing an inventory production distribution plan over a multiperiod planning horizon the plan does not allow the backlogging of the unsatisfied demand and minimizes the costs of the supply chain while enabling it to reach a prescribed nonstockout service level it is a strategic plan that hedges against undesirable outcomes and that can be adjusted to account for possible favorable realizations of uncertain quantities a modular integrated and computationally tractable method is proposed for the solution of the associated stochastic mixed integer optimization problems containing joint probabilistic constraints with dependent right hand side variables the concept of p efficiency is used to construct a finite number of demand trajectories which in turn are employed to solve problems with joint probabilistic constraints we complement this idea by designing a preordered set based preprocessing algorithm that selects a subset of promising p efficient demand trajectories finally to solve the resulting disjunctive mixed integer programming problem we implement a special column generation algorithm that limits the risk of congestion in the resources of the supply chain the methodology is validated on an industrial problem faced by a large chemical supply chain and turns out to be very efficient it finds a solution with a minimal integrality gap and provides substantial cost savings

aggregation and mixed integer rounding to solve mips in this paper we discuss the use of mixed integer rounding mir inequalities to solve mixed integer programs mir inequalities are essentially gomory mixed integer cuts however as we wish to use problem structure we insist that mir inequalities be generated from constraints or simple aggregations of constraints of the original problem this idea is motivated by the observation that several strong valid inequalities based on specific problem structure can be derived as mir inequalities here we present and test a separation routine for such mir inequalities that includes a heuristic row aggregation procedure to generate a single knapsack plus continuous variables constraint complementation of variables and finally the generation of an mir inequality inserted in a branch and cut system the results suggest that such a routine is a useful additional tool for tackling a variety of mixed integer programming problems

capacitated facility location separation algorithms and computational experience we consider the polyhedral approach to solving the capacitated facility location problem the valid inequalities considered are the knapsack cover flow cover effective capacity single depot and combinatorial inequalities the flow cover effective capacity and single depot inequalities form subfamilies of the general family of submodular inequalities the separation problem based on the family of submodular inequalities is np hard in general for the well known subclass of flow cover inequalities however we show that if the client set is fixed and if all capacities are equal then the separation problem can be solved in polynomial time for the flow cover inequalities based on an arbitrary client set and general capacities and for the effective capacity and single depot inequalities we develop separation heuristics an important part of these heuristics is based on the result that two specific conditions are necessary for the effective cover inequalities to be facet defining the way these results are stated indicates precisely how structures that violate the two conditions can be modified to produce stronger inequalities the family of combinatorial inequalities was originally developed for the uncapacitated facility location problem but is also valid for the capacitated problem no computational experience using the combinatorial inequalities has been reported so far here we suggest how partial output from the heuristic identifying violated submodular inequalities can be used as input to a heuristic identifying violated combinatorial inequalities we report on computational results from solving 60 medium size problems

a multi item production planning model with setup times algorithms reformulations and polyhedral characterizations for a special case we study a special case of a structured mixed integer programming model that arises in production planning for the most general case of the model called pi we have earlier identified families of facet defining valid inequalities l s inequalities introduced for the uncapacitated lot sizing problem by barany van roy and wolsey cover inequalities and reverse cover inequalities pi is hard in this paper we focus on a special case called pic we describe a polynomial algorithm for pic and we use this algorithm to derive an extended formulation of polynomial size for pic projecting from this extended formulation onto the original space of variables we show that l s inequalities cover inequalities and reverse cover inequalities suffice to solve the special case pic by linear programming we also describe fast combinatorial separation algorithms for cover and reverse cover inequalities for pic finally we discuss the relationship between our results for pic and a model studied previously by goemans

larger lower bounds on the obdd complexity of integer multiplication integer multiplication as one of the basic arithmetic functions has been in the focus of several complexity theoretical investigations and ordered binary decision diagrams obdds are one of the most common dynamic data structures for boolean functions recently the question whether the obdd complexity of the most significant bit of integer multiplication is exponential has been answered affirmatively in this paper a larger general lower bound is presented using a simpler proof furthermore we prove a larger lower bound for the variable order assumed to be one of the best ones for the most significant bit moreover the best known lower bound on the obdd complexity for the so called graph of integer multiplication is improved

diffusive logistic equations with indefinite weights population models in disrupted environments the dynamics of a population inhabiting a strongly heterogeneous environment are modelledby diffusive logistic equations of the form u t d u m x cu u in 0 where u represents the population density c d 0 are constants describing the limiting effects of crowding and the diffusion rate of the population respectively and m x describes the local growth rate of the population if the environment is bounded and is surrounded by uninhabitable regions then u 0 on 0 the growth rate m x is positive on favourablehabitats and negative on unfavourable ones the object of the analysis is to determine how the spatial arrangement of favourable and unfavourable habitats affects the population being modelled the models are shown to possess a unique stable positive steady state implying persistence for the population provided l d where is the principle positive eigenvalue for the problem m x in 0 on analysis of how depends on m indicates that environments with favourable and unfavourable habitats closely intermingled are worse for the population than those containing large regions of uniformly favourable habitat in the limit as the diffusion rate d 0 the solutions tend toward the positive part of m x c and if m is discontinuous develop interior transition layers the analysis uses bifurcation and continuation methods the variational characterisation of eigenvalues upper and lower solution techniques and singular perturbation theory

diffusive lotka volterra competition models on periodically evolving domains abstract the paper deals with the asymptotic dynamics of a diffusive lotka volterra competition model with periodic coefficients and zero flux boundary conditions acting on a spatially isotropic and temporally periodic evolving domain we determine sufficient conditions for the global stability of the spatially homogeneous positive periodic solution moreover the phenomenon of competitive exclusion is investigated

the teacher assignment problem a special case of the fixed charge transportation problem a basic model of the task of assigning classes to professors in such a way that the average number of distinct subjects assigned to each professor is minimized is formulated as a mixed integer program it turns out that the problem is a special case of the fixed charge transportation problem which in some cases corresponds to finding a basic solution of a transportation problem which is as degenerate as possible we present an equivalent alternative formulation of the problem which makes it easy to prove that it is np hard in the strong sense and an exact branch and bound algorithm for its solution based on this alternative formulation is outlined computational experiments with data from a concrete problem concludes the paper

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

subset coefficient reduction cuts for 0 1 mixed integer programming we describe a method for generating cuts for mixed integer 0 1 programs these cuts are designed to tighten an integer program prior to applying linear programming based branch and bound algorithms the method involves two basic ideas subset selection and coefficient reduction coefficient reduction is a process of reducing the coefficients of the 0 1 variables subset selection is combined with coefficient reduction by applying the coefficient reduction process to the coefficients of a subset of variables from the constraints in the problem formulation the paper exploits these two simple ideas to derive a broad class of cuts for integer programs with both 0 1 and continuous variables it also reports on the use of this methodology in solving a variety of fixed charge problems

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

the merchant subtour problem we consider the problem of a travelling merchant who makes money by buying commodities where they are cheap and selling them in other places where he can make a profit the merchant ships commodities of his own choice in a van of fixed capacity given the prices of all the commodities in all of the places and the cost of driving from one place to another the problem the merchant faces each day is to select a subset of the cities that he can visit in a day and to determine the order in which the cities are visited such that the total profit is maximised we call this problem the merchant subtour problem the msp models the pricing problem of a rather complex pickup and delivery problem that was given to us by the dutch logistics company van gend loos we show that a special case of the msp has a totally unimodular constraint matrix this knowledge enables us to develop a tabu search algorithm for finding good feasible solutions to the msp and a branch and price and cut algorithm for solving the msp to optimality the relaxations solved in each node of the branch and bound tree are strengthened by lifted knapsack inequalities lifted cycle inequalities and mod k cuts we present computational results on data sets derived from our main instance of the van gend loos pickup and delivery problem

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

generating cuts in integer programming with families of special ordered sets abstract this paper describes families of cuts which can be added to integer programming formulations incorporating special ordered sets ways in which the cuts can be used are discussed and some computational experience is presented

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

the diffusive logistic model with a free boundary in heterogeneous environment abstract in this paper we study the population dynamics of an invasive species in heterogeneous environment which is modeled by a diffusive logistic equation with free boundary condition to understand the effect of the dispersal rate d and the parameter the ratio of the expansion speed of the free boundary and the population gradient at the expanding front on the dynamics of this model we divide the heterogeneous environment into two cases strong heterogeneous environment and weak heterogeneous environment by choosing d and as variable parameters we derive sufficient conditions for species spreading resp vanishing in the strong heterogeneous environment while in the weak heterogeneous environment we obtain sharp criteria for the spreading and vanishing moreover when spreading happens we give an estimate for the asymptotic spreading speed of the free boundary these theoretical results may have important implications for prediction and prevention of biological invasions

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

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 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

o n log n procedures for tightening cover inequalities abstract we present two procedures for tightening cover induced inequalities in 0 1 programs by using knapsack constraints plus some other information from the program the tightening is obtained by solving successive knapsack problems with all 0 1 objective function coefficients and using dominance criteria to avoid the explicit solving of some knapsack problems the new constraints are 0 1 equivalent to and lp tighter than the original ones both procedures have o n log n complexity where n is the number of variables with nonzero coefficients in the knapsack constraint however one of them can strongly reduce the computational effort some computational experience is reported

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

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

hierarchical design of an integrated production and 2 echelon distribution system abstract this study uses the method of lagrangean relaxation in the hierarchical design of an integrated model of production distribution functions in a 2 echelon system a mixed integer mathematical model is developed with a centralized planning perspective to address production and distribution decisions simultaneously in order to solve the resulting large scale problem the lagrangean relaxation is used to decouple the imbedded distribution and production subproblems and subgradient optimization is implemented to coordinate the information flow between these in a hierarchical manner this corresponds to a decentralized organizational design where a central agent coordinates the information exchange between the distribution and production organizational units a forward heuristic designed to solve the distribution subproblem is shown to provide good solutions hierarchical interdependency is incorporated into the lagrangean heuristic such that distribution decisions are placed in the top level to restrict the solution of the production subproblem in the lower level

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

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

time stepping schemes for moving grid finite elements applied to reaction diffusion systems on fixed and growing domains in this paper we illustrate the application of time stepping schemes to reaction diffusion systems on fixed and continuously growing domains by use of finite element and moving grid finite element methods we present two schemes for our studies namely a first order backward euler finite differentiation formula coupled with a special form of linearisation of the nonlinear reaction terms 1 sbem and a second order semi implicit backward finite differentiation formula 2 sbdf with no linearisation of the reaction terms our results conclude that for the type of reaction diffusion systems considered in this paper the 1 sbem is more stable than the 2 sbdf scheme and that the 1 sbem scheme has a larger region of stability at least by a factor of 10 than that of the 2 sbdf scheme as a result the 1 sbem scheme becomes a natural choice when solving reaction diffusion problems on continuously deforming domains

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

the diffusive logistic equation on periodically evolving domains abstract a diffusive logistic equation on n dimensional periodically and isotropically evolving domains is investigated we first derive the model and present the eigenvalue problem on evolving domains then we prove that the species persists if the diffusion rate d is below the critical value d 0 while the species become extinct if it is above the critical value d 0 finally we analyze the effect of domain evolution rate on the persistence of a species precisely it depends on the average value 2 where t is the domain evolution rate and 2 1 t 0 t 1 2 t d t if 2 1 the periodical domain evolution has a negative effect on the persistence of a species if 2 1 the periodical domain evolution has a positive effect on the persistence of a species if 2 1 the periodical domain evolution has no effect on the persistence of a species numerical simulations are also presented to illustrate the analytical results

a benders decomposition based heuristic for the hierarchical production planning problem abstract in this paper a heuristic procedure for determining good feasible solutions to a multi item dynamic production planning problem is described and evaluated the problem which is modelled as a mixed integer linear program has a hierarchical structure where production items are aggregated into families and families into product types a decomposition of the original problem creates a trivial subproblem providing feasible solutions on the type level and a master problem which separates into one uncapacitated lot sizing problem for each family the latter problems can be solved by dynamic programming with very little computational effort besides the nice structure of both the sub and master problem the dual information from the subproblem can be given interesting economic interpretations as distinguished from hax and meal s hierarchical framework for production planning and also they hybrid approach suggested by graves in 1982 there exists a direct information flow from the family level problem to the type level problem in form of a production plan the heuristic is essentially an application of benders decomposition the only difference being that the benders cuts are relaxed using lagrangean multipliers a subgradient procedure is used to update the lagrangean multipliers one iteration of this heuristic requires less computational effort compared to grave s algorithm this is due to the fact that the benders subproblem can be solved by inspection while graves algorithm involves solving an lp problem on the type level for thirtysix medium size test problems the average deviation from optimum is 2 34 and the deviation ranges between 0 00 and 5 95

mixed 0 1 programming by lift and project in a branch and cut framework we investigate the computational issues that need to be addressed when incorporating general cutting planes for mixed 0 1 programs into a branch and cut framework the cuts we use are of the lift and project variety some of the issues addressed have a theoretical answer but others are of an experimental nature and are settled by comparing alternatives on a set of test problems the resulting code is a robust solver for mixed 0 1 programs we compare it with several existing codes on a wide range of test problems it performs as well as or better than some of the best currently available mixed integer programming codes

the asymptotic analysis of an insect dispersal model on a growing domain abstract this paper is concerned about a reaction diffusion equation on n dimensional isotropically growing domain which describes the insect dispersal the model for growing domains is first derived and the comparison principle is then presented the asymptotic behavior of the solution to the reaction diffusion problem is given by constructing upper and lower solutions our results show that the growth of domain takes a positive effect on the asymptotic stability of positive steady state solution while it takes a negative effect on the asymptotic stability of the trivial solution numerical simulations are also performed to illustrate the analytical results

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

valid linear inequalities for fixed charge problems many problems in the operations research management science literature can be formulated with both zero one and continuous variables however the exact optimization of such mixed zero one models remains a computational challenge in this paper we propose to study mixed problems from a mathematical point of view that is similar in spirit to recent research on purely combinatorial problems that has investigated systems of defining linear inequalities or facets of the underlying polytope at least two numerical studies have validated this line of research computationally and the advances in the problem solving capabilities are considerable we expect that similar gains are possible for the mixed zero one problem more precisely we consider the mixed integer programs whose feasible region x is composed of i a simple additive constraint in the continuous variables xj for j 1 2 n and ii constraints 0 xj mjyj defined by binary variables yj for j 1 2 n this type of feasible region arises in a variety of mixed integer problems and particularly in network problems with fixed charges on the arcs we derive two classes of facet defining linear inequalities of the convex hull of x and show that the second of these classes gives a complete description of the convex hull when mj m for all j we also develop methods to detect violated inequalities from these classes so that these facets can be used as cutting planes to strengthen the formulations of certain mixed integer problems

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

dynamic knapsack sets and capacitated lot sizing a dynamic knapsack set is a natural generalization of the 0 1 knapsack set with a continuous variable studied recently for dynamic knapsack sets a large family of facet defining inequalities called dynamic knapsack inequalities are derived by fixing variables to one and then lifting surprisingly such inequalities have the simultaneous lifting property and for small instances provide a significant proportion of all the facet defining inequalities we then consider single item capacitated lot sizing problems and propose the joint study of three related sets the first models the discrete lotsizing problem the second the continuous lot sizing problem with wagnerwhitin costs and the third the continuous lot sizing problem with arbitrary costs the first set that arises is precisely a dynamic knapsack set the second an intersection of dynamic knapsack sets and the unrestricted problem can be viewed as both a relaxation and a restriction of the second it follows that the dynamic knapsack inequalities and their generalizations provide strong valid inequalities for all three sets some limited computation results are reported as an initial test of the effectiveness of these inequalities on capacitated lot sizing problems

optimization based production planning with hybrid dynamics and constraints optimizing production planning has tremendous economic impact for many industrial production systems in this paper the planning problem of a class of production systems with hybrid dynamics and constraints is considered with practical background of power generation planning and other applications the problem is solved within the lagrangian relaxation framework with the system wide demand and resource limit constraints relaxed by lagrange multipliers a new method is developed in this paper to obtain the exact optimal solutions to the subproblems with hybrid dynamics and constraints efficiently without discretizing the continuous production levels or introducing intermediate levels of relaxation a novel definition of the discrete state associated with a consecutive time span is introduced so that solving each subproblem is converted into solving a number of continuous optimization problems and a discrete optimization problem separately an efficient double dynamic programming dp method is developed to solve these subproblems and the principle of optimality is guaranteed for both the continuous and discrete problem the production levels in a consecutive running span with non convex piecewise linear cost functions are determined in a dp forward sweep without discretization the dp method is then applied to determine the optimal discrete operating states across time efficiently numerical testing results demonstrate that the new method is efficient and effective for optimization based production planning with the complex hybrid dynamics and constraints

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

agent preferences and the topology of networks in this paper a different framework to study weighed networks is introduced the idea behind this methodology is to consider that each node of the network is an agent that desires to satisfy his her preferences in an economic sense moreover the formation of a link between two agents depends on the benefits and costs associated with this link therefore an edge between two given nodes will arise only if the tradeoff between satisfaction and cost for building it is jointly positive using a computational framework i intend to show that depending on the agents combination of benefits and costs some very well known networks can naturally arise

a polyhedral study of the cardinality constrained knapsack problem a cardinality constrained knapsack problem is a continuous knapsack problem in which no more than a specified number of nonnegative variables are allowed to be positive this structure occurs for example in areas such as finance location and scheduling traditionally cardinality constraints are modeled by introducing auxiliary 0 1 variables and additional constraints that relate the continuous and the 0 1 variables we use an alternative approach in which we keep in the model only the continuous variables and we enforce the cardinality constraint through a specialized branching scheme and the use of strong inequalities valid for the convex hull of the feasible set in the space of the continuous variables to derive the valid inequalities we extend the concepts of cover and cover inequality commonly used in 0 1 programming to this class of problems and we show how cover inequalities can be lifted to derive facet defining inequalities we present three families of non trivial facet defining inequalities that are lifted cover inequalities finally we report computational results that demonstrate the effectiveness of lifted cover inequalities and the superiority of the approach of not introducing auxiliary 0 1 variables over the traditional mip approach for this class of problems

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

on some free boundary problems of the prey predator model abstract in this paper we investigate some free boundary problems for the lotka volterra type prey predator model in one space dimension the main objective is to understand the asymptotic behavior of the two species prey and predator spreading via a free boundary we prove a spreading vanishing dichotomy namely the two species either successfully spread to the entire space as time t goes to infinity and survive in the new environment or they fail to establish and die out in the long run the long time behavior of solution and criteria for spreading and vanishing are also obtained finally when spreading successfully we provide an estimate to show that the spreading speed if exists cannot be faster than the minimal speed of traveling wavefront solutions for the prey predator model on the whole real line without a free boundary

on the polyhedral structure of a multi item production planning model with setup times we present and study a mixed integer programming model that arises as a substructure in many industrial applications this model generalizes a number of structured mip models previously studied and it provides a relaxation of various capacitated production planning problems and other fixed charge network flow problems we analyze the polyhedral structure of the convex hull of this model as well as of a strengthened lp relaxation among other results we present valid inequalities that induce facets of the convex hull under certain conditions we also discuss how to strengthen these inequalities by using known results for lifting valid inequalities for 0 1 continuous knapsack problems

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 mathematical programming model and solution for scheduling production orders in shanghai baoshan iron and steel complex in this paper we investigate the production order scheduling problem derived from the production of steel sheets in shanghai baoshan iron and steel complex baosteel a deterministic mixed integer programming mip model for scheduling production orders on some critical and bottleneck operations in baosteel is presented in which practical technological constraints have been considered the objective is to determine the starting and ending times of production orders on corresponding operations under capacity constraints for minimizing the sum of weighted completion times of all orders due to large numbers of variables and constraints in the model a decomposition solution methodology based on a synergistic combination of lagrangian relaxation linear programming and heuristics is developed unlike the commonly used method of relaxing capacity constraints this methodology alternatively relaxes constraints coupling integer variables with continuous variables which are introduced to the objective function by lagrangian multipliers the lagrangian relaxed problem can be decomposed into two sub problems by separating continuous variables from integer ones the sub problem that relates to continuous variables is a linear programming problem which can be solved using standard software package osl while the other sub problem is an integer programming problem which can be solved optimally by further decomposition the subgradient optimization method is used to update lagrangian multipliers a production order scheduling simulation system for baosteel is developed by embedding the above lagrangian heuristics computational results for problems with up to 100 orders show that the proposed lagrangian relaxation method is stable and can find good solutions within a reasonable time

solving the fixed charge problem with lagrangian relaxation and cost allocation heuristics abstract a new heuristic algorithm is presented for solving large general fixed charge problems based on langrangian relaxation and cost allocation heuristics limited computational results indicate the new algorithm provides good solutions to large problems with reasonable computational requirements it demonstrates significant improvement over current methods for solving large general fixed charge problems

solving mixed integer programming problems using automatic reformulation in this paper we describe computational experience in solving mixed 0 1 programming problems using strong valid inequalities as cutting planes in particular we report on the solution to optimality of 18 medium to large size problems including production planning problems with setup costs and capacity constraints multilevel distribution planning problems drainage and heating system design problems and electricity generator scheduling problems the solution approach uses the theory of strong valid inequalities that we developed in a series of earlier papers here we report specifically on the implementation of an experimental system mpsarx which consists of the sciconic mathematical programming system and an automatic reformulation executor arx that use this theory and on the results obtained with this system

on the complexity of vlsi implementations and graph representations of boolean functions with application to integer multiplication lower bound results on boolean function complexity under two different models are discussed the first is an abstraction of tradeoffs between chip area and speed in very large scale integrated vlsi circuits the second is the ordered binary decision diagram obdd representation used as a data structure for symbolically representing and manipulating boolean functions the lower bounds demonstrate the fundamental limitations of vlsi as an implementation medium and that of the obdd as a data structure it is shown that the same technique used to prove that any vlsi implementation of a single output boolean function has area time complexity at sup 2 omega n sup 2 also proves that any obdd representation of the function has omega c sup n vertices for some c 1 but that the converse is not true an integer multiplier for word size n with outputs numbered 0 least significant through 2n 1 most significant is described for the boolean function representing either output i 1 or output 2n i 1 where 1

modeling and solving the two facility capacitated network loading problem this paper studies a topical and economically significant capacitated network design problem that arises in the telecommunications industry in this problem given point to point communication demand in a network must be met by installing loading capacitated facilities on the arcs loading a facility incurs an arc specific and facility dependent cost this paper develops modeling and solution approaches for loading facilities to satisfy the given demand at minimum cost we consider two approaches for solving the underlying mixed integer program a lagrangian relaxation strategy and a cutting plane approach that uses three classes of valid inequalities that we identify for the problem we show that a linear programming formulation that includes these inequalities always approximates the value of the mixed integer program at least as well as the lagrangian relaxation bound our computational results on a set of prototypical telecommunication data show that including these inequalities considerably improves the gap between the integer programming formulation and its linear programming relaxation from an average of 25 to an average of 8 these results show that strong cutting planes can be an effective modeling and algorithmic tool for solving problems of the size that arise in the telecommunications industry

different transformations for solving non convex trim loss problems by minlp in the present paper trim loss problems often named the cutting stock problem connected to the paper industry are considered the problem is to cut out a set of product paper rolls from raw paper rolls such that the cost function including the trim loss as well as the costs for the over production is minimized the problem is non convex due to certain bilinear constraints the problem can however be transformed into linear or convex form the resulting transformed problems can thereafter be solved as mixed integer linear programming problems or convex mixed integer non linear programming problems the linear and convex formulations are attractive from a formal point of view since global optimal solutions to the originally non convex problem can be obtained however as the examples considered will show the numerical efficiency of the solutions from the different transformed formulations varies considerably an example based on a trim optimization problem encountered daily at a finnish paper converting mill is finally presented in order to demonstrate differences in the numerical solutions

using cutting planes in an interactive reference point approach for multiobjective integer linear programming problems we propose an interactive approach for multiple objective integer linear programming moilp problems that combines the use of the tchebycheff metric with cutting plane techniques at each interaction the method computes the nondominated solution for the moilp problem that is closest to a reference point according to the tchebycheff metric the information provided by the decision maker in each dialogue phase is used to adjust the next reference point through a sensitivity analysis stage cutting plane techniques enable the method to take advantage of computations performed at previous iterations to solve the next scalarizing integer program we address both theoretical issues and the computational implementation

cutting planes for integer programs with general integer variables we investigate the use of cutting planes for integer programs with general integer variables we show how cutting planes arising from knapsack inequalities can be generated and lifted as in the case of 0 1 variables we also explore the use of gomory s mixed integer cuts we address both theoretical and computational issues and show how to embed these cutting planes in a branch and bound framework we compare results obtained by using our cut generation routines in two existing systems with a commercially available branch and bound code on a range of test problems arising from practical applications 1998 the mathematical programming society inc published by elsevier science b v

effective heuristics for multiproduct partial shipment models motivated by real applications we consider the problem of shipping products to multiple customers from limited inventory after formulating the optimization problems under different restrictions on partial shipments we find that commercially available packages applied directly are unsatisfactory as are simple greedy approaches we develop a scheme of heuristics that enables the user to select a good balance between computation time and effectiveness a detailed computational study of one and two period industrial sized problems indicates that these heuristics are computationally practical and generate solutions that are on average within 34 of the optimum

railroad blocking a network design application in this study we formulate the railroad blocking problems as a network design problem with maximum degree and flow constraints on the nodes and propose a heuristic lagrangian relaxation approach to solve the problem the newapproach decomposes the complicated mixed integer programming problem into two simple subproblems so that the storage requirement and computational effort are greatly reduced a set of inequalities are added to one subproblem to tighten the lower bounds and facilitate generating feasible solutions subgradient optimization is used to solve the lagrangian dual an advanced dual feasible solution is generated to speed up the convergence of the subgradient method the model is tested on blocking problems from a major railroad and the results show that the blocking plans generated have the potential to reduce the railroad s operating costs by millions of dollars annually

letter to the editor an efficient linearization technique for mixed 0 1 polynomial problem this paper addresses a new and efficient linearization technique to solve mixed 0 1 polynomial problems to achieve a global optimal solution given a mixed 0 1 polynomial term z c tx 1x 2 x ny where x 1 x 2 x n are binary 0 1 variables and y is a continuous variable also c t can be either a positive or a negative parameter we transform z into a set of auxiliary constraints which are linear and can be solved by exact methods such as branch and bound algorithms for this purpose we will introduce a method in which the number of additional constraints is decreased significantly rather than the previous methods proposed in the literature as is known in any operations research problem decreasing the number of constraints leads to decreasing the mathematical computations extensively thus research on the reducing number of constraints in mathematical problems in complicated situations have high priority for decision makers in this method each n auxiliary constraints proposed in the last method in the literature for the linearization problem will be replaced by only 3 novel constraints in other words previous methods were dependent on the number of 0 1 variables and therefore one auxiliary constraint was considered per 0 1 variable but this method is completely independent of the number of 0 1 variables and this illustrates the high performance of this method in computation considerations the analysis of this method illustrates the efficiency of the proposed algorithm

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

fixed charge transportation problem facets of the projection polyhedron in this paper we consider the well known fixed charge transportation problem to send any flow from source si to destination tj we incur a unit variable shipping cost of cij and a fixed cost fij here we study the structure of the projection polyhedron of this problem in the space of 0 1 variables associated with fixed charges and we develop several classes of valid inequalities and derive conditions under which they are facet defining in some cases if the conditions are not satisfied we show how they can be lifted to define facets several heuristics for generating and adding these facets are presented using these results we develop a computationally effective algorithm for solving the problem the computational results clearly indicate the usefulness of this approach

valid inequalities and projecting the multicommodity extended formulation for uncapacitated fixed charge network flow problems multicommodity extended formulations of single source uncapacitated fixed charge network flow problems have significantly sharper linear programming relaxations than the standard flow formulations however the tradeoff is the introduction of many new constraints and variables to accomodate a sink oriented flow disaggregation in this paper we introduce a new family of dicut collection inequalities and show that they completely describe the projection of the multicommodity formulation onto the original variables a simple subclass is seen to include a variety of known inequalities for particular models and combinatorial separation is examined for some special cases

a lagrangean relax and cut approach for the two stage capacitated facility location problem abstract the two stage capacitated facility location problem is to find the optimal locations of depots to serve customers with a given demand the optimal assignment of customers to depots and the optimal product flow from plants to depots in order to solve this problem a lagrangean heuristic is proposed which is based on the relaxation of the capacity constraints the resulting lagrangean subproblem is an uncapacitated facility location problem with an aggregate capacity constraint and can be solved efficiently by branch and bound methods the best bound available from this relaxation is computed by means of a weighted dantzig wolfe decomposition approach feasible solutions are constructed from those of the lagrangean subproblems by applying simple reassignment procedures furthermore the relaxation is strengthened by adding and dualizing valid inequalities which are violated by the fractional primal solution of the dual master program computational results and comparisons with some other bounds and heuristics for this problem are presented

toward verifying nonlinear integer arithmetic we eliminate a key roadblock to efficient verification of nonlinear integer arithmetic using cdcl sat solvers by showing how to construct short resolution proofs for many properties of the most widely used multiplier circuits such short proofs were conjectured not to exist more precisely we give no 1 size regular resolution proofs for arbitrary degree 2 identities on array diagonal and booth multipliers and no log n size proofs for these identities on wallace tree multipliers

lifted flow cover inequalities for mixed 0 1 integer programs we investigate strong inequalities for mixed 0 1 integer programs derived from flow cover inequalities flow cover inequalities are usually not facet defining and need to be lifted to obtain stronger inequalities however because of the sequential nature of the standard lifting techniques and the complexity of the optimization problems that have to be solved to obtain lifting coefficients lifting of flow cover inequalities is computationally very demanding we present a computationally efficient way to lift flow cover inequalities based on sequence independent lifting techniques and give computational results that show the effectiveness of our lifting procedures

strong formulations for mixed integer programming a survey we attempt to motivate and survey recent research on the use of strong valid inequalities and reformulation to solve mixed integer programming problems

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

progressive interval heuristics for multi item capacitated lot sizing problems we consider a family of n items that are produced in or obtained from the same production facility demands are deterministic for each item and each period within a given horizon of t periods if in a given period an order is placed setup costs are incurred the aggregate order size is constrained by a capacity limit the objective is to find a lot sizing strategy that satisfies the demands for all items over the entire horizon without backlogging and that minimizes the sum of inventory carrying costs fixed order costs and variable order costs all demands cost parameters and capacity limits may be time dependent in the basic joint setup cost js model the setup cost of an order does not depend on the composition of the order the joint and item dependent setup cost jis model allows for item dependent setup costs in addition to the joint setup costs we develop and analyze a class of so called progressive interval heuristics a progessive interval heuristic solves a js or jis problem over a progressively larger time interval always starting with period 1 but fixing the setup variables of a progressively larger number of periods at their optimal values in earlier iterations different variants in this class of heuristics allow for different degrees of flexibility in adjusting continuous variables determined in earlier iterations of the algorithm for the js model and the two basic implementations of the progressive interval heuristics we show under some mild parameter conditions that the heuristics can be designed to be e optimal for any desired value of e 0 with a running time that is polynomially bounded in the size of the problem they can also be designed to be simultaneously asymptotically optimal and polynomially bounded a numerical study covering both the js and jis models shows that a progressive interval heuristic generates close to optimal solutions with modest computational effort and that it can be effectively used to solve large scale problems

fixed charge transportation on a path optimization lp formulations and separation in the fixed charge transportation problem the goal is to optimally transport goods from depots to clients when there is a fixed cost associated to transportation or equivalently to opening an arc in the underlying bipartite graph we further motivate its study by showing that it is both a special case and a strong relaxation of the big bucket multi item lot sizing problem and a generalization of a simple variant of the single node flow set this paper is essentially a polyhedral analysis of the polynomially solvable special case in which the associated bipartite graph is a path we give a mathcal o n 3 time optimization algorithm and a mathcal o n 2 size linear programming extended formulation we describe a new class of inequalities that we call path modular inequalities we give two distinct proofs of their validity the first one is direct and crucially relies on sub and super modularity of an associated set function the second proof is by showing that the projection of the extended linear programming formulations onto the original variable space yields exactly the polyhedron described by the path modular inequalities thus we also show that these inequalities suffice to describe the convex hull of the set of feasible solutions

a polyhedral approach to single machine scheduling problems we report new results for a time indexed formulation of nonpreemptive single machine scheduling problems we give complete characterizations of all facet inducing inequalities with integral coefficients and right hand side 1 or 2 for the convex hull of the set of feasible partial schedules i e schedules in which not all jobs have to be started furthermore we identify conditions under which these facet inducing inequalities are also facet inducing for the original polytope which is the convex hull of the set of feasible complete schedules i e schedules in which all jobs have to be started to obtain insight in the effectiveness of these classes of facet inducing inequalities we develop a branch and cut algorithm based on them we evaluate its performance on the strongly np hard single machine scheduling problem of minimizing the weighted sum of the job completion times subject to release dates

on the 0 1 knapsack polytope this paper deals with the 0 1 knapsack polytope in particular we introduce the class ofweight inequalities this class of inequalities is needed to describe the knapsack polyhedron when the weights of the items lie in certain intervals a generalization of weight inequalities yields the so called weight reduction principle and the class of extended weight inequalities the latter class of inequalities includes minimal cover and l k configuration inequalities the properties of lifted minimal cover inequalities are extended to this general class of inequalities

reformulation of the multiperiod milp model for capacity expansion of chemical processes the problem of selecting processes and capacity expansion policies for a chemical complex consisting of continuous chemical processes can be formulated as a multiperiod mixed integer linear programming milp problem based on a variable disaggregation technique which exploits lot sizing substructures we propose two reformulations of the conventional milp model the first one is an nlp reformulation which very quickly yields good suboptimal solutions the second is an milp reformulation for exact solutions which leads to up to an order of magnitude faster computational results for large problems due to its tighter linear programming relaxation

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

capacitated facility location valid inequalities and facets we examine the polyhedral structure of the convex hull of feasible solutions of the capacitated facility location problem in particular we derive necessary and sufficient conditions for a family of effective capacity inequalities to be facet defining and further results on a more general family called submodular inequalities