approximation algorithms for the capacitated multi item lot sizing problem via flow cover inequalities we study the classical capacitated multi item lot sizing problem with hard capacities there are n items each of which has a specified sequence of demands over a finite planning horizon of t discrete periods the demands are known in advance but can vary from period to period all demands must be satisfied on time each order incurs a time dependent fixed ordering cost regardless of the combination of items or the number of units ordered but the total number of units ordered cannot exceed a given capacity c on the other hand carrying inventory from period to period incurs holding costs the goal is to find a feasible solution with minimum overall ordering and holding costs we show that the problem is strongly np hard and then propose a novel facility location type lp relaxation that is based on an exponentially large subset of the well known flow cover inequalities the proposed lp can be solved to optimality in polynomial time via an efficient separation procedure for this subset of inequalities moreover the optimal solution of the lp can be rounded to a feasible integer solution with cost that is at most twice the optimal cost this provides a 2 approximation algorithm which is the first constant approximation algorithm for the problem we also describe an interesting on the fly variant of the algorithm that does not require solving the lp a priori with all the flow cover inequalities as a by product we obtain the first theoretical proof regarding the strength of flow cover inequalities in capacitated inventory models

how porous is the graph of brownian motion we prove that the graph of brownian motion is almost surely porous and determine the hausdorff dimension of sets with a given porosity index in particular we show that the porosity index of the graph is yo 0 6948

primal dual algorithms for deterministic inventory problems we consider several classical models in deterministic inventory theory the single item lot sizing problem the joint replenishment problem and the multistage assembly problem these inventory models have been studied extensively and play a fundamental role in broader planning issues such as the management of supply chains for each of these problems we wish to balance the cost of maintaining surplus inventory for future demand against the cost of replenishing inventory more frequently for example in the joint replenishment problem demand for several commodities is specified over a discrete finite planning horizon the cost of maintaining inventory is linear in the number of units held but the cost incurred for ordering a commodity is independent of the size of the order furthermore there is an additional fixed cost incurred each time a nonempty subset of commodities is ordered the goal is to find a policy that satisfies all demands on time and minimizes the overall holding and ordering cost we shall give a novel primal dual framework for designing algorithms for these models that significantly improve known results in several ways the performance guarantees for the quality of the solutions improve on or match previously known results the performance guarantees hold under much more general assumptions about the structure of the costs and the algorithms and their analysis are significantly simpler than previous known results finally our primal dual framework departs from the structure of previously studied primal dual approximation algorithms in significant ways and we believe that our approach may find applications in other settings more specifically we provide 2 approximation algorithms for the joint replenishment problem and for the assembly problem and solve the single item lot sizing problem to optimality the results for the joint replenishment and the lot sizing problems also hold for their generalizations with back orders allowed as a byproduct of our work we prove known and new upper bounds on the integrality gap of some linear programming lp relaxations of the abovementioned 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

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

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

fully polynomial approximation schemes for single item capacitated economic lot sizing problems np hard cases of the single item capacitated lot sizing problem have been the topic of extensive research and continue to receive considerable attention however surprisingly few theoretical results have been published on approximation methods for these problems to the best of our knowledge until now no polynomial approximation method is known that produces solutions with a relative deviation from optimality that is bounded by a constant in this paper we show that such methods do exist by presenting an even stronger result the existence of fully polynomial approximation schemes the approximation scheme is first developed for a quite general model which has concave backlogging and production cost functions and arbitrary monotone holding cost functions subsequently we discuss important special cases of the model and extensions of the approximation scheme to even more general models

on splittable and unsplittable flow capacitated network design arc set polyhedra we study the polyhedra of splittable and unsplittable single arc set relaxations of multicommodity flow capacitated network design problems we investigate the optimization problems over these sets and the separation and lifting problems of valid inequalities for them in particular we give a linear time separation algorithm for the residual capacity inequalities 19 and show that the separation problem of c strong in equalities 7 is np hard but can be solved over the subspace of fractional variables only we introduce two classes of inequalities for the unsplittable flow problems we present a summary of computational experiments with a branch and cut algorithm for multicommodity flow capacitated network design problems to test the effectiveness of the results presented here empirically

valid inequalities and facets of the capacitated plant location problem recently several successful applications of strong cutting plane methods to combinatorial optimization problems have renewed interest in cutting plane methods and polyhedral characterizations of integer programming problems in this paper we investigate the polyhedral structure of the capacitated plant location problem our purpose is to identify facets and valid inequalities for a wide range of capacitated fixed charge problems that contain this prototype problem as a substructure

lot sizing with constant batches formulation and valid inequalities we consider the classical lot sizing problem with constant production capacities lcc and a variant in which the capacity in each period is an integer multiple of some basic batch size lcb we first show that the classical dynamic program for lcc simplifies for lcb leading to an on2 min n c algorithm where n is the number of periods and c the batch size using this new algorithm we show how to formulate both problems as linear programs with on3 variables and constraints a class of so called k l s i inequalities are described for lcb which capture both the dynamic nature of the problem as well as the capacity aspects for lcb we prove that these inequalities are the only facet defining inequalities of a certain form for lcc we show that these inequalities include all the known classes of valid inequalities finally we discuss several open questions and possible extensions

brownian motion reflected on brownian motion we study brownian motion reflected on an independent brownian path we prove results on the joint distribution of both processes and the support of the parabolic measure in the space time domain bounded by a brownian path we show that there exist two different natural local times for a brownian path reflected on a brownian path

on capacitated network design cut set polyhedra this paper provides an analysis of capacitated network design cut set polyhedra we give a complete linear description of the cut set polyhedron of the single commodity single facility capacitated network design problem then we extend the analysis to single commodity multifacility and multicommodity multifacility capacitated network design problems valid inequalities described here are applicable to directed network design problems with any number of facility types and any level of capacities we report results from a computational study done for testing the effectiveness of the new inequalities

q difference operators for orthogonal polynomials in this work we apply a q ladder operator approach to orthogonal polynomials arising from a class of indeterminate moment problems we derive general representation of first and second order q difference operators and we study the solution basis of the corresponding second order q difference equations and its properties the results are applied to the stieltjes wigert and the q laguerre polynomials

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

a constant approximation algorithm for the one warehouse multiretailer problem deterministic inventory theory provides streamlined optimization models that attempt to capture trade offs in managing the flow of goods through a supply chain we will consider two well studied deterministic inventory models called the one warehouse multiretailer owmr problem and its special case the joint replenishment problem jrp and give approximation algorithms with worst case performance guarantees that is for each instance of the problem our algorithm produces a solution with cost that is guaranteed to be at most 1 8 times the optimal cost this is called a 1 8 approximation algorithm our results are based on an lp rounding approach we provide the first constant approximation algorithm for the owmr problem and improve the previous results for the jrp

stable processes have thorns let x t be the symmetric stable process in r 0 2 d 2 for f dvtx 0 1 0 let d f be the thorn x r dvtxx1 0 1 allowbreak x2 xd f x1 we give an integral criterion in terms of f for the existence of a random time s such that x t remains in x s d f for all t s s 1

on brownian slow points it is shown that for a wiener process x t both the quantities mathop inf limits t mathop overline lim limits h text to text 0 x t text text h x t sqrt h text and mathop text sup limits t mathop underline lim limits h text to text 0 text x t text text h x t sqrt h are almost surely equal to 1

lp based approximation algorithms for capacitated facility location in the capacitated facility location problem with hard capacities we are given a set of facilities mathcal f and a set of clients mathcal d in a common metric space each facility i has a facility opening cost f i and capacity u i that specifies the maximum number of clients that may be assigned to this facility we want to open some facilities from the set mathcal f and assign each client to an open facility so that at most u i clients are assigned to any open facility i the cost of assigning client j to facility i is given by the distance c ij and our goal is to minimize the sum of the facility opening costs and the client assignment costs the only known approximation algorithms that deliver solutions within a constant factor of optimal for this np hard problem are based on local search techniques it is an open problem to devise an approximation algorithm for this problem based on a linear programming lower bound or indeed to prove a constant integrality gap for any lp relaxation we make progress on this question by giving a 5 approximation algorithm for the special case in which all of the facility costs are equal by rounding the optimal solution to the standard lp relaxation one notable aspect of our algorithm is that it relies on partitioning the input into a collection of single demand capacitated facility location problems approximately solving them and then combining these solutions in a natural way

fixed dimensional stochastic dynamic programs an approximation scheme and an inventory application we study fixed dimensional stochastic dynamic programs in a discrete setting over a finite horizon under the primary assumption that the cost to go functions are discrete l convex we propose a pseudo polynomial time approximation scheme that solves this problem to within an arbitrary prespecified additive error of e 0 the proposed approximation algorithm is a generalization of the explicit enumeration algorithm and offers us full control in the trade off between accuracy and running time the main technique we develop for obtaining our scheme is approximation of a fixed dimensional l convex function on a bounded rectangular set using only a selected number of points in its domain furthermore we prove that the approximation function preserves l convexity finally to apply the approximate functions in a dynamic program we bound the error propagation of the approximation our approximation scheme is illustrated on a well known problem in inventory theory the single product problem with lost sales and lead times we demonstrate the practical value of our scheme by implementing our approximation scheme and the explicit enumeration algorithm on instances of this inventory problem

the convex hull of two core capacitated network design problems the network loading problem nlp is a specialized capacitated network design problem in which prescribed point to point demand between various pairs of nodes of a network must be met by installing loading a capacitated facility we can load any number of units of the facility on each of the arcs at a specified arc dependent cost the problem is to determine the number of facilities to be loaded on the arcs that will satisfy the given demand at minimum cost

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

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