simultaneous column and row generation for large scale linear programs with column dependent rows in this paper we develop a simultaneous column and row generation algorithm that could be applied to a general class of large scale linear programming problems these problems typically arise in the context of linear programming formulations with exponentially many variables the defining property for these formulations is a set of linking constraints which are either too many to be included in the formulation directly or the full set of linking constraints can only be identified if all variables are generated explicitly due to this dependence between columns and rows we refer to this class of linear programs as problems with column dependent rows to solve these problems we need to be able to generate both columns and rows on the fly within an efficient solution approach we emphasize that the generated rows are structural constraints and distinguish our work from the branch and cut and price framework we first characterize the underlying assumptions for the proposed column and row generation algorithm these assumptions are general enough and cover all problems with column dependent rows studied in the literature up until now to the best of our knowledge we then introduce in detail a set of pricing subproblems which are used within the proposed column and row generation algorithm this is followed by a formal discussion on the optimality of the algorithm to illustrate our approach the paper is concluded by applying the proposed framework to the multi stage cutting stock and the quadratic set covering problems

a reoptimization algorithm for the shortest path problem with time windows abstract the shortest path problem with time windows spptw occurs in the construction of vehicle routes and schedules for which time window constraints must be satisfied the spptw is repeatedly solved to produce routes with disjoint schedules which form the solution of the original problem the cost of repeatedly solving the spptw can be reduced by reusing part of the solution of the preceding problem a primal dual reoptimization method which runs in pseudo polynomial time is proposed it can solve problems with up to 2500 nodes

an exact algorithm for the pickup and delivery problem with time windows the pickup and delivery problem with time windows pdptw is a generalization of the vehicle routing problem with time windows in the pdptw a set of identical vehicles located at a central depot must be optimally routed to service a set of transportation requests subject to capacity time window pairing and precedence constraints in this paper we present a new exact algorithm for the pdptw based on a set partitioning like integer formulation and we describe a bounding procedure that finds a near optimal dual solution of the lp relaxation of the formulation by combining two dual ascent heuristics and a cut and column generation procedure the final dual solution is used to generate a reduced problem containing only the routes whose reduced costs are smaller than the gap between a known upper bound and the lower bound achieved if the resulting problem has moderate size it is solved by an integer programming solver otherwise a branch and cut and price algorithm is used to close the integrality gap extensive computational results over the main instances from the literature show the effectiveness of the proposed exact method

set covering in fuel considered vehicle routing problems the paper studies set covering in fuel considered vehicle routing problems fvrp firstly we study the fvrp with distance constraint and time windows fvrp tw whose objective is to find a set covering with the minimum cardinality which means the number of used vehicles is minimized and hence the fuel consumption is minimized in the real logistics we give a bicriteria approximation algorithm for this problem secondly we study the set covering in the fvrp with distance constraint and constant time windows fvrp ctw which has a constant number of the time windows provided by logistics companies we give a bicriteria approximation algorithm with 2 o log 1 for this problem in which the first term is the approximation ratio on the distance constraint and the second term is the approximation ratio on the cardinality of the covering set thirdly we study the set covering in general fvrp and propose a lower bound for this problem which is based on total unimodularity finally we design an algorithm framework based on the lower bound for solving the set covering in general fvrp simulation results demonstrate the effectiveness of the algorithms for solving the set covering in fvrp tw and general fvrp

lifted polynomial size formulations for the homogeneous and heterogeneous vehicle routing problems abstract we propose compact formulations for the symmetric and asymmetric capacitated vehicle routing problems these formulations are obtained by lifting using the reformulation linearization technique a novel polynomial size ordered path based formulation we show that the strongest proposed formulation is equivalent to the strongest multi commodity flow formulation presented so far in addition we propose polynomial size valid inequalities that aim at further tightening the proposed formulations also the tightest formulation is extended to model the fleet size and mix vehicle routing problem with fixed costs we show that the new derived model is not comparable with a state of the art polynomial length formulation we present the results of computational experiments that demonstrate the tightness of the proposed formulations and the impact of the valid inequalities

new developments in the primal dual column generation technique abstract the optimal solutions of the restricted master problems typically leads to an unstable behavior of the standard column generation technique and consequently originates an unnecessarily large number of iterations of the method to overcome this drawback variations of the standard approach use interior points of the dual feasible set instead of optimal solutions in this paper we focus on a variation known as the primal dual column generation technique which uses a primal dual interior point method to obtain well centered non optimal solutions of the restricted master problems we show that the method converges to an optimal solution of the master problem even though non optimal solutions are used in the course of the procedure also computational experiments are presented using linear relaxed reformulations of three classical integer programming problems the cutting stock problem the vehicle routing problem with time windows and the capacitated lot sizing problem with setup times the numerical results indicate that the appropriate use of a primal dual interior point method within the column generation technique contributes to a reduction of the number of iterations as well as the running times on average furthermore the results show that the larger the instance the better the relative performance of the primal dual column generation technique

vehicle routing with time windows two optimization algorithms we describe two optimization methods for vehicle routing problems with time windows these are a k tree relaxation with time windows added as side constraints and a lagrangian decomposition in which variable splitting is used to divide the problem into two subproblems a semi assignment problem and a series of shortest path problems with time windows and capacity constraints we present optimal solutions to problems with up to 100 customers

an optimization algorithm for the vehicle routing problem with time windows based on lagrangian relaxation our paper presents a new optimization method for the vehicle routing problem with time windows vrptw the vrptw is a generalization of the vehicle routing problem where the service of a customer must start within a given time interval a so called time window our method is based on a lagrangian relaxation of the constraint set requiring that each customer must be serviced the master problem consists of finding the optimal lagrangian multipliers and the subproblem is a shortest path problem with time windows and capacity constraints the optimal multipliers are found using a method exploiting the benefits of subgradient methods as well as a bundle method the method has been implemented and tested on a series of well known benchmark problems of size up to 100 customers our algorithm turns out to be very competitive compared to algorithms considered in the literature and we have succeeded in solving several previously unsolved problems

an efficient algorithm for the schrodinger poisson eigenvalue problem we present a new implementation of the two grid method for computing extremum eigenpairs of self adjoint partial differential operators with periodic boundary conditions a novel two grid centered difference method is proposed for the numerical solutions of the nonlinear schrodinger poisson sp eigenvalue problem we solve the poisson equation to obtain the nonlinear potential for the nonlinear schrodinger eigenvalue problem and use the block lanczos method to compute the first k eigenpairs of the schrodinger eigenvalue problem until they converge on the coarse grid then we perform a few conjugate gradient iterations to solve each symmetric positive definite linear system for the approximate eigenvector on the fine grid the rayleigh quotient iteration is exploited to improve the accuracy of the eigenpairs on the fine grid our numerical results show how the first few eigenpairs of the schrodinger eigenvalue problem are affected by the dopant in the schrodinger poisson sp system moreover the convergence rate of eigenvalue computations on the fine grid is o h 3

a generalization of column generation to accelerate convergence this paper proposes a generalization of column generation reformulating the master problem with fewer variables at the expense of adding more constraints the sub problem structure does not change it shows both analytically and computationally that the reformulation promotes faster convergence to an optimal solution in application to a linear program and to the relaxation of an integer program at each node in the branch and bound tree further it shows that this reformulation subsumes and generalizes prior approaches that have been shown to improve the rate of convergence in special cases

technical note an exact algorithm for the time constrained traveling salesman problem the time constrained traveling salesman problem is a variation of the familiar traveling salesman problem that includes time window constraints on the time a particular city or cities may be visited this note presents a concise formulation of the time constrained traveling salesman problem the model assumes that the distances of the problem are symmetrical and that the triangle inequality holds additionally the model allows the salesman to wait at a city if necessary for a time window to open the dual of the formulation is shown to be a disjunctive graph model which is well known from scheduling theory a longest path algorithm is used to obtain bounding information for subproblems in a branch and bound solution procedure computational results are presented for several small to moderate size problems

branch and price column generation for solving huge integer programs we discuss formulations of integer programs with a huge number of variables and their solution by column generation methods i e implicit pricing of nonbasic variables to generate new columns or to prove lp optimality at a node of the branch and bound tree we present classes of models for which this approach decomposes the problem provides tighter lp relaxations and eliminates symmetry we then discuss computational issues and implementation of column generation branch and bound algorithms including special branching rules and efficient ways to solve the lp relaxation we also discuss the relationship with lagrangian duality

on the distance constrained vehicle routing problem we analyze the vehicle routing problem with constraints on the total distance traveled by each vehicle two objective functions are considered minimize the total distance traveled by vehicles and minimize the number of vehicles used we demonstrate a close relationship between the optimal solutions for the two objective functions and perform a worst case analysis for a class of heuristics we present a heuristic that provides a good worst case result when the number of vehicles used is relatively small

on the flexibility of constraint programming models from single to multiple time windows for the traveling salesman problem one of the major strengths of constraint programming is the flexibility and expressiveness of models in that computational paradigm which make it easy to add problem dependent constraints without having to modify the solution strategy we show here what needs to be done in order to adapt a constraint programming algorithm for the traveling salesman problem with time windows so that it can handle multiple time windows computational results are also presented on a set of instances created for that little studied problem

classification of dantzig wolfe reformulations for binary mixed integer programming problems abstract in this note we provide a classification of dantzig wolfe reformulations for binary mixed integer programming problems we specifically focus on modeling the binary conditions in the convexification approach to the dantzig wolfe decomposition for a general binary mixed integer programming problem an extreme point of the overall problem does not necessarily correspond to an extreme point of the subproblem therefore the binary conditions cannot in general be imposed on the new master problem variables but must be imposed on the original binary variables in some cases however it is possible to impose the binary restrictions directly on the new master problem variables the issue of imposing binary conditions on the original variables versus the master problem variables has not been discussed systematically for mip problems in general in the literature and most of the research has been focused on the pure binary case the classification indicates in which cases you can and cannot impose binary conditions on the new master problem variables

discrete lotsizing and scheduling by batch sequencing the discrete lotsizing and scheduling problem for one machine with sequence dependent setup times and setup costs is solved as a single machine scheduling problem which we term the batch sequencing problem the relationship between the lotsizing problem and the batch sequencing problem is analyzed the batch sequencing problem is solved with a branch bound algorithm which is accelerated by bounding and dominance rules the algorithm is compared with recently published procedures for solving variants of the dlsp and is found to be more efficient if the number of items is not large

hybrid optimization methods for time dependent sequencing problems abstract in this paper we introduce novel optimization methods for sequencing problems in which the setup times between a pair of tasks depend on the relative position of the tasks in the ordering our proposed methods rely on a hybrid approach where a constraint programming model is enhanced with two distinct relaxations one discrete relaxation based on multivalued decision diagrams and one continuous relaxation based on linear programming both relaxations are used to generate bounds and enhance constraint propagation experiments conducted on three variants of the time dependent traveling salesman problem indicate that our techniques substantially outperform general purpose methods such as mixed integer linear programming and constraint programming models

local search with annealing like restarts to solve the vrptw abstract in this paper we propose a metaheuristic based on annealing like restarts to diversify and intensify local searches for solving the vehicle routing problem with time windows vrptw using the solomon s benchmark instances for the problem our method obtained seven new best results and equaled 19 other best results extensive comparisons indicate that our method is comparable to the best in published literature this approach is flexible and can be extended to handle other variants of vehicle routing problems and other combinatorial optimization problems

a cluster based optimization approach for the multi depot heterogeneous fleet vehicle routing problem with time windows this paper presents a novel three phase heuristic algorithmic approach for the multi depot routing problem with time windows and heterogeneous vehicles it has been derived from embedding a heuristic based clustering algorithm within a vrptw optimization framework to this purpose a rigorous milp mathematical model for the vrptw problem is first introduced likewise other optimization approaches the new formulation can efficiently solve case studies involving at most 25 nodes to optimality to overcome this limitation a preprocessing stage clustering nodes together is initially performed to yield a more compact cluster based milp problem formulation in this way a hierarchical hybrid procedure involving one heuristic and two algorithmic phases was developed phase i aims to identifying a set of cost effective feasible clusters while phase ii assigns clusters to vehicles and sequences them on each tour by using the cluster based milp formulation ordering nodes within clusters and scheduling vehicle arrival times at customer locations for each tour through solving a small milp model is finally performed at phase iii numerous benchmark problems featuring different sizes clustered random customer locations and time window distributions have been solved at acceptable cpu times

on the effectiveness of set covering formulations for the vehicle routing problem with time windows the vehicle routing problem with time windows vrptw is one of the most important problems in distribution and transportation a classical and recently popular technique that has proven effective for solving these problems is based on formulating them as a set covering problem the method starts by solving its linear programming relaxation via column generation and then uses a branch and bound strategy to find an integer solution to the set covering problem a solution to the vrptw an empirically observed property is that the optimal solution value of the set covering problem is very close to its linear programming relaxation which makes the branch and bound step extremely efficient in this paper we explain this behavior by demonstrating that for any distribution of service times time windows customer loads and locations the relative gap between fractional and integer solutions of the set covering problem becomes arbitrarily small as the number of customers increases

multi phase dynamic constraint aggregation for set partitioning type problems dynamic constraint aggregation is an iterative method that was recently introduced to speed up the linear relaxation solution process of set partitioning type problems this speed up is mostly due to the use at each iteration of an aggregated problem defined by aggregating disjoint subsets of constraints from the set partitioning model this aggregation is updated when needed to ensure the exactness of the overall approach in this paper we propose a new version of this method called the multi phase dynamic constraint aggregation method which essentially adds to the original method a partial pricing strategy that involves multiple phases this strategy helps keeping the size of the aggregated problem as small as possible yielding a faster average computation time per iteration and fewer iterations we also establish theoretical results that provide some insights explaining the success of the proposed method tests on the linear relaxation of simultaneous bus and driver scheduling problems involving up to 2 000 set partitioning constraints show that the partial pricing strategy speeds up the original method by an average factor of 4 5

nodal aggregation of resource constraints in a shortest path problem the shortest path problem with resource constraints consists of finding the minimum cost path between two specified points while respecting constraints on resource consumption its solving by a dynamic programming algorithm requires a computation time increasing with the number of resources with the aim of producing rapidly a good heuristic solution we propose to reduce the state space by aggregating resources our approach consists of projecting the resources on a vector of smaller dimension and then to dynamically adjust the projection matrix to get a better approximation of the optimal solution we propose an adjustment based on lagrangian and surrogate relaxations in a column generation framework in which the sub problems are shortest path problems with resource constraints we adjust the multipliers only one time at each column generation iteration this permit to obtain good solutions of the scheduling problem in few time

a new optimization algorithm for the vehicle routing problem with time windows the vehicle routing problem with time windows vrptw is a generalization of the vehicle routing problem where the service of a customer can begin within the time window defined by the earliest and the latest times when the customer will permit the start of service in this paper we present the development of a new optimization algorithm for its solution the lp relaxation of the set partitioning formulation of the vrptw is solved by column generation feasible columns are added as needed by solving a shortest path problem with time windows and capacity constraints using dynamic programming the lp solution obtained generally provides an excellent lower bound that is used in a branch and bound algorithm to solve the integer set partitioning formulation our results indicate that this algorithm proved to be successful on a variety of practical sized benchmark vrptw test problems the algorithm was capable of optimally solving 100 customer problems this problem size is six times larger than any reported to date by other published research

quasialgebra structure of the octonions abstract we show that the octonions are a twisting of the group algebra of z 2 z 2 z 2 in the quasitensor category of representations of a quasi hopf algebra associated to a group 3 cocycle in particular we show that they are quasialgebras associative up to a 3 cocycle isomorphism we show that one may make general constructions for quasialgebras exactly along the lines of the associative theory including quasilinear algebra representation theory and an automorphism quasi hopf algebra we study the algebraic properties of quasialgebras of the type which includes the octonions further examples include the higher 2n onion cayley algebras and examples associated to hadamard matrices

exact algorithms for the vehicle routing problem based on spanning tree and shortest path relaxations we consider the problem of routing vehicles stationed at a central facility depot to supply customers with known demands in such a way as to minimize the total distance travelled the problem is referred to as the vehicle routing problem vrp and is a generalization of the multiple travelling salesman problem that has many practical applications we present tree search algorithms for the exact solution of the vrp incorporating lower bounds computed from i shortest spanning k degree centre tree k dct and ii q routes the final algorithms also include problem reduction and dominance tests computational results are presented for a number of problems derived from the literature the results show that the bounds derived from the q routes are superior to those from k dct and that vrps of up to about 25 customers can be solved exactly

selected topics in column generation dantzig wolfe decomposition and column generation devised for linear programs is a success story in large scale integer programming we outline and relate the approaches and survey mainly recent contributions not yet found in textbooks we emphasize the growing understanding of the dual point of view which has brought considerable progress to the column generation theory and practice it stimulated careful initializations sophisticated solution techniques for the restricted master problem and subproblem as well as better overall performance thus the dual perspective is an ever recurring concept in our selected topics

on the nucleolus of the basic vehicle routing game in the vehicle routing cost allocation problem the aim is to find a good cost allocation method i e a method that according to specified criteria allocates the cost of an optimal route configuration among the customers we formulate this problem as a co operative game in characteristic function form and give conditions for when the core of the vehicle routing game is nonempty one specific solution concept to the cost allocation problem is the nucleolus which minimizes maximum discontent among the players in a co operative game the class of games we study is such that the values of the characteristic function are obtained from the solution of a set of mathematical programming problems we do not require an explicit description of the characteristic function for all coalitions instead by applying a constraint generation approach we evaluate information about the function only when it is needed for the computation of the nucleolus

an outer approximate subdifferential method for piecewise affine optimization piecewise affine functions arise from lagrangian duals of integer programming problems and optimizing them provides good bounds for use in a branch and bound method methods such as the subgradient method and bundle methods assume only one subgradient is available at each point but in many situations there is more information available we present a new method for optimizing such functions which is related to steepest descent but uses an outer approximation to the subdifferential to avoid some of the numerical problems with the steepest descent approach we provide convergence results for a class of outer approximations and then develop a practical algorithm using such an approximation for the compact dual to the linear programming relaxation of the uncapacitated facility location problem we make a numerical comparison of our outer approximation method with the projection method of conn and cornuejols and the bundle method of schramm and zowe

solving the asymmetric travelling salesman problem with time windows by branch and cut many optimization problems have several equivalent mathematical models it is often not apparent which of these models is most suitable for practical computation in particular when a certain application with a specific range of instance sizes is in focus our paper addresses the asymmetric travelling salesman problem with time windows atsp tw from such a point of view the real world application we aim at is the control of a stacker crane in a warehouse we have implemented codes based on three alternative integer programming formulations of the atsp tw and more than ten heuristics computational results for real world instances with up to 233 nodes are reported showing that a new model presented in a companion paper outperforms the other two models we considered at least for our special application and that the heuristics provide acceptable solutions

a binary decision diagram based algorithm for solving a class of binary two stage stochastic programs we consider a special class of two stage stochastic integer programming problems with binary variables appearing in both stages the class of problems we consider constrains the second stage variables to belong to the intersection of sets corresponding to first stage binary variables that equal one our approach seeks to uncover strong dual formulations to the second stage problems by transforming them into dynamic programming dp problems parameterized by first stage variables we demonstrate how these dps can be formed by use of binary decision diagrams which then yield traditional benders inequalities that can be strengthened based on observations regarding the structure of the underlying dps we demonstrate the efficacy of our approach on a set of stochastic traveling salesman problems

spectral collocation and a two level continuation scheme for dipolar bose einstein condensates we exploit the high accuracy of spectral collocation methods in the context of a two level continuation scheme for computing ground state solutions of dipolar bose einstein condensates bec where the first kind chebyshev polynomials and fourier sine functions are used as the basis functions for the trial function space the governing gross pitaevskii equation or schrodinger equation can be reformulated as a schrodinger poisson sp type system 13 the two level continuation scheme is developed for tracing the first solution curves of the sp system which in turn provide an appropriate initial guess for the newton method to compute ground state solutions for 3d dipolar bec extensive numerical experiments on 3d dipolar bec and dipolar bec in optical lattices are reported

an accelerated branch and price algorithm for multiple runway aircraft sequencing problems abstract this paper presents an effective branch and price b p algorithm for multiple runway aircraft sequencing problems this approach improves the tractability of the problem by several orders of magnitude when compared with solving a classical 0 1 mixed integer formulation over a set of computationally challenging instances central to the computational efficacy of the b p algorithm is solving the column generation subproblem as an elementary shortest path problem with aircraft time windows and non triangular separation times using an enhanced dynamic programming procedure we underscore in our computational study the algorithmic features that contribute in our experience to accelerating the proposed dynamic programming procedure and hence the overall b p algorithm

an interpolation approximation for the mean workload in a gi g 1 queue this paper develops a closed form approximation for the mean steady state workload or virtual waiting time in a gi g 1 queue using the first two moments of the service time distribution and the first three moments plus the density at the origin of the interarrival time distribution with default values provided in case information is unavailable the approximation is based on light and heavy traffic limiting behavior the essential ideas are exposed by using s l brumelle s formula to relate the mean workload to the mean waiting time and k t marshall s formula to relate the mean waiting time to the first two moments of the idle period both formulas extend to general single server models without independence conditions so this approach provides a basis for extensions but a convenient exact expression for the second order heavy traffic term is evidently not possible even for gi g 1 for the gi g 1 second order heavy traffic term an approximation is proposed based on the relatively nice expression est

rendezvous search on the line with more than two players suppose n blind speed one players are placed by a random permutation onto the integers 1 to n and each is pointed randomly to the right or left what is the least expected time required form m n of them to meet together at a single point if they must all use the same strategy we call this time the symmetric rendezvous value rn ms otherwise the asymmetric value rn ma we show that r3 2a 47 48 and that rn ns is asymptotic to n 2 these results respectively extend those for two players given by alpern and gal alpern s s gal 1995 rendezvous search on the line with distinguishable players siam j control optim 33 1270 1276 and anderson and essegaier anderson e j s essegaier 1995 rendezvous search on the line with indistinguishable players siam j control optim 33 1637 1642

weak hopf algebras i integral theory and c structure we give an introduction to the theory of weak hopf algebras proposed as a coassociative alternative of weak quasi hopf algebras we follow an axiomatic approach keeping as close as possible to the classical theory of hopf algebras the emphasis is put on the new structure related to the presence of canonical subalgebras al and ar in any weak hopf algebra a that play the role of non commutative numbers in many respects a theory of integrals is developed in which we show how the algebraic properties of a such as the frobenius property or semisimplicity or innerness of the square of the antipode are related to the existence of non degenerate normalized or haar integrals in case of c weak hopf algebras we prove the existence of a unique haar measure h a and of a canonical grouplike element g a implementing the square of the antipode and factorizing into left and right elements g glg 1r gl al gr ar further discussion of the c case will be presented in part ii

asymmetry matters dynamic half way points in bidirectional labeling for solving shortest path problems with resource constraints faster abstract with their paper symmetry helps bounded bi directional dynamic programming for the elementary shortest path problem with resource constraints discrete optimization 3 2006 pp 255 273 righini and salani introduced bounded bidirectional dynamic programming dp as an acceleration technique for solving variants of the shortest path problem with resource constraints spprc spprcs must be solved iteratively when vehicle routing and scheduling problems are tackled via lagrangian relaxation or column generation techniques righini and salani and several subsequent works have shown that bounded bidirectional dp algorithms are often superior to their monodirectional counterparts since the former can mitigate the fact that the number of labels increases strongly with the path length bidirectional dp has become a quasi standard for solving spprcs with general resource extension functions in computational experiments however one can still observe that the number of forward and backward label extensions is very unbalanced despite a symmetric bounding of a critical resource in the middle of its feasible domain we exploit this asymmetry in forward and backward label extensions to reduce the overall workload by introducing a so called dynamic half way point which is a dynamic bounding criterion based on the current state of the simultaneously solved forward and backward dps experiments with the standard and the electric vehicle routing problem with time windows as well as the vehicle routing and truck driver scheduling problem confirm that dynamic half way points better balance forward and backward labeling and reduce the overall runtime

lp models for bin packing and cutting stock problems abstract we review several linear programming lp formulations for the one dimensional cutting stock and bin packing problems namely the models of kantorovich gilmore gomory onecut models as in the dyckhoff stadtler approach position indexed models and a model derived from the vehicle routing literature we analyse some relations between the corresponding lp relaxations and their relative strengths and refer how to derive branching schemes that can be used in the exact solution of these problems using branch and price

reachability cuts for the vehicle routing problem with time windows this paper introduces a class of cuts called reachability cuts for the vehicle routing problem with time windows vrptw reachability cuts are closely related to cuts derived from precedence constraints in the asymmetric traveling salesman problem with time windows and to k path cuts for the vrptw in particular any reachability cut dominates one or more k path cuts the paper presents separation procedures for reachability cuts and reports computational experiments on well known vrptw instances the computational results suggest that reachability cuts can be highly useful as cutting planes for certain vrptw instances

branch and price algorithms for the solution of the multi trip vehicle routing problem with time windows we investigate the exact solution of the vehicle routing problem with time windows where multiple trips are allowed for the vehicles in contrast to previous works in the literature we specifically consider the case in which it is mandatory to visit all customers and there is no limitation on duration we develop two branch and price frameworks based on two set covering formulations a traditional one where columns variables represent routes that is a sequence of consecutive trips and a second one in which columns are single trips one important difficulty related to the latter is the way mutual temporal exclusion of trips can be handled it raises the issue of time discretization when solving the pricing problem our dynamic programming algorithm is based on concept of groups of labels and representative labels we provide computational results on modified small sized instances 25 customers from solomon s benchmarks in order to evaluate and compare the two methods results show that some difficult instances are out of reach for the first branch and price implementation while they are consistently solved with the second

a clifford algebra is a weak hopf algebra in a suitable symmetric monoidal category abstract it is well known that clifford algebras are group algebras deformed by a 2 cocycle furthermore these algebras which are not commutative in the usual sense can be viewed as commutative algebras in certain symmetric monoidal categories of graded vector spaces in this note we invent a clifford process for coalgebras that will allow us to show that clifford algebras have also cocommutative coalgebra structures and consequently commutative and cocommutative weak braided hopf algebras structures within the same symmetric monoidal categories where they lie as commutative algebras also we will show that they are selfdual weak braided hopf algebras monoidal frobenius algebras and monoidal cofrobenius coalgebras

record breaking optimization results using the ruin and recreate principle a new optimization principle is presented solutions of problems are partly but significantly ruined and rebuilt or recreated afterwards performing this type of change frequently one can obtain astounding results for classical optimization problems the new method is particularly suited for more complex optimization problems discontinuous ones problems with hard to find admissible solutions problems with complex objectives or many constraints the method is an all purpose heuristic numerical results are given for the traveling salesman problem for the vehicle routing problem with time windows and for network optimization numerical evidence for the quality of the proposed principle is given for most of the instances of a research library of problems the ruin and recreate r r implementation achieved the best published results for many instances better or much better solutions could be found

branch and price and cut for the multiple traveling repairman problem with distance constraints abstract in this paper we extend the multiple traveling repairman problem by considering a limitation on the total distance that a vehicle can travel the resulting problem is called the multiple traveling repairmen problem with distance constraints mtrpd in the mtrpd a fleet of identical vehicles is dispatched to serve a set of customers each vehicle that starts from and ends at the depot is not allowed to travel a distance longer than a predetermined limit and each customer must be visited exactly once the objective is to minimize the total waiting time of all customers after the vehicles leave the depot to optimally solve the mtrpd we propose a new exact branch and price and cut algorithm where the column generation pricing subproblem is a resource constrained elementary shortest path problem with cumulative costs an ad hoc label setting algorithm armed with bidirectional search strategy is developed to solve the pricing subproblem computational results show the effectiveness of the proposed method the optimal solutions to 179 out of 180 test instances are reported in this paper our computational results serve as benchmarks for future researchers on the problem

note on the complexity of the shortest path models for column generation in vrptw in this note we prove that the relaxation approach in designing the subproblem of pricing out only the feasible routes for the set partition formulation of the vrptw is justified on complexity grounds that is the first dynamic programming model presented in m desrochers j desrosiers and m solomon 1992 that is able to price out all feasible routes is np hard in the strong sense

dynamic constraint and variable aggregation in column generation abstract starting from the improved primal simplex ips decomposition introduced by elhallaoui et al 2011 to tackle degeneracy in general linear programs we introduce and discuss the mathematical foundations of improved column generation decompositions icg that are better than the standard column generation decomposition when degeneracy is an issue we also present an improved dynamic constraint aggregation idca which is a specialization of icg to efficiently solve set partitioning problems we show that idca improves the dynamic constraint aggregation dca and the multiphase dynamic constraint aggregation mpdca algorithms elhallaoui et al 2005 2010 that not only reduce degeneracy but also profit from it to efficiently solve set partitioning problems idca solves at each iteration a complementary problem cp to obtain a group of variables that can be aggregated or pivoted into the basis to bypass degenerate pivots and decrease the objective value and to obtain more central dual solutions to generate new columns our numerical results show reduction factors in the solution times exceeding 10 for idca compared to a state of the art standard column generation solver

real time vehicle rerouting problems with time windows this paper introduces and studies real time vehicle rerouting problems with time windows applicable to delivery and or pickup services that undergo service disruptions due to vehicle breakdowns in such problems one or more vehicles need to be rerouted in real time to perform uninitiated services with the objective to minimize a weighted sum of operating service cancellation and route disruption costs a lagrangian relaxation based heuristic is developed which includes an insertion based algorithm to obtain a feasible solution for the primal problem a dynamic programming based algorithm solves heuristically the shortest path problems with resource constraints that result from the lagrangian relaxation computational experiments show that the developed lagrangian heuristic performs very well

new route relaxation and pricing strategies for the vehicle routing problem in this paper we describe an effective exact method for solving both the capacitated vehicle routing problem cvrp and the vehicle routing problem with time windows vrptw that improves the method proposed by baldacci et al baldacci r n christofides a mingozzi 2008 an exact algorithm for the vehicle routing problem based on the set partitioning formulation with additional cuts math programming115 2 351 385 for the cvrp the proposed algorithm is based on the set partitioning sp formulation of the problem we introduce a new route relaxation called ng route used by different dual ascent heuristics to find near optimal dual solutions of the lp relaxation of the sp model we describe a column and cut generation algorithm strengthened by valid inequalities that uses a new strategy for solving the pricing problem the new ng route relaxation and the different dual solutions achieved allow us to generate a reduced sp problem containing all routes of any optimal solution that is finally solved by an integer programming solver the proposed method solves four of the five open solomon s vrptw instances and significantly improves the running times of state of the art algorithms for both vrptw and cvrp

a branch and price algorithm for scheduling parallel machines with sequence dependent setup times abstract we consider the problem of scheduling n independent jobs on m unrelated parallel machines with sequence dependent setup times and availability dates for the machines and release dates for the jobs to minimize a regular additive cost function in this work we develop a new branch and price optimization algorithm for the solution of this general class of parallel machines scheduling problems a new column generation accelerating method termed primal box and a specific branching variable selection rule that significantly reduces the number of explored nodes are proposed the computational results show that the approach solves problems of large size to optimality within reasonable computational time

probabilistic analysis of a combined aggregation and math programming heuristic for a general class of vehicle routing and scheduling problems we propose and analyze a heuristic that uses region partitioning and an aggregation scheme for customer attributes load size time windows etc to create a finite number of customer types a math program is solved based on these aggregated customer types to generate a feasible solution to the original problem the problem class we address is quite general and defined by a number of general consistency properties problems in this class include vrps with general distance norms capacitated problems time window vrps pick up and delivery problems combined inventory control and routing problems and arc routing we provide a probabilistic analysis of this heuristic under very general probabilistic assumptions in particular we do not require independence between customer locations and their various attributes the heuristic is a s optimal as the number of customers n tends to infinity further it runs in o n log n time for a fixed relative error and can be designed to be asymptotically optimal w

vehicle routing problem with fuzzy time windows in this paper a vehicle routing problem with fuzzy time windows vrpftw is proposed and solved in the transportation business time windows are not always strictly obeyed and the deviation of service time from the customer specific time window determines the customer s satisfaction level which can also be regarded as the supplier s service level this paper applies fuzzy membership functions to characterize the service level issues associated with time window violation in a vehicle routing problem and proposes vrpftw vrpftw is formulated as a multi objective model with two goals 1 to minimize the travel distance and 2 to maximize the service level of the supplier to customers to solve this multi objective model a two stage algorithm is developed to obtain a pareto solution for vrpftw using the two stage algorithm vrpftw is decomposed into two subproblems namely a traditional vehicle routing problem with time windows email protected and a service improvement problem and each of the objectives is sequentially solved the service improvement problem is solved under two different scenarios when the fuzzy membership function is linear it is shown that the service improvement problem can be solved by the cutting plane algorithm within finite iterations and when the fuzzy membership function is concave the service improvement problem can be solved by a subgradient based algorithm moreover an alternative formulation of vrpftw is also proposed and analyzed experiments are conducted to compare different models of vehicle routing problems with time windows in situations where violation of time windows is allowed and the results show that the vrpftw model can achieve considerable cost savings while at the same time maintaining an acceptable service level

parallel machine scheduling linear programming and parameter list scheduling heuristics in this paper we consider a class of parallel machine scheduling problems and their associated set partitioning formulations we show that the tightness of the linear programming relaxation of these formulations is directly related to the performance of a class of heuristics called parameter list scheduling heuristics this makes it possible to characterize the worst possible gap between optimal solutions for the scheduling problems and the corresponding linear programming relaxations in the case of the classical parallel machine weighted completion time model we also show that the solution to the linear programming relaxation of the set partitioning formulation is asymptotically optimal under mild assumptions on the distribution of job weights and processing times finally we extend most of the results to the time discretized formulation of machine scheduling problems

a generalized insertion heuristic for the traveling salesman problem with time windows this article describes a generalized insertion heuristic for the traveling salesman problem with time windows in which the objective is the minimization of travel times the algorithm gradually builds a route by inserting at each step a vertex in its neighbourhood on the current route and performing a local reoptimization this is done while checking the feasibility of the remaining part of the route backtracking is sometimes necessary once a feasible route has been determined an attempt is made to improve it by applying a post optimization phase based on the successive removal and reinsertion of all vertices tests performed on 375 instances indicate that the proposed heuristic compares very well with alternative methods and very often produces optimal or near optimal solutions

a new dantzig wolfe reformulation and branch and price algorithm for the capacitated lot sizing problem with setup times although the textbook dantzig wolfe decomposition reformulation for the capacitated lot sizing problem as already proposed by manne manne a s 1958 programming of economic lot sizes management sci 4 2 115 135 provides a strong lower bound it also has an important structural deficiency imposing integrality constraints on the columns in the master program will not necessarily give the optimal integer programming solution manne s model contains only production plans that satisfy the wagner whitin property and it is well known that the optimal solution to a capacitated lot sizing problem will not necessarily satisfy this property the first contribution of this paper answers the following question unsolved for almost 50 years if manne s formulation is not equivalent to the original problem what is then a correct reformulation we develop an equivalent mixed integer programming mip formulation to the original problem and show how this results from applying the dantzig wolfe decomposition to the original mip formulation the set of extreme points of the lot size polytope that are needed for this mip dantzig wolfe reformulation is much larger than the set of dominant plans used by manne we further show how the integrality restrictions on the original setup variables translate into integrality restrictions on the new master variables by separating the setup and production decisions our new formulation gives the same lower bound as manne s reformulation second we develop a branch and price algorithm for the problem computational experiments are presented on data sets available from the literature column generation is accelerated by a combination of simplex and subgradient optimization for finding the dual prices the results show that branch and price is computationally tractable and competitive with other state of the art approaches found in the literature

dynamic aggregation of set partitioning constraints in column generation column generation is often used to solve problems involving set partitioning constraints such as vehicle routing and crew scheduling problems when these constraints are in large numbers and the columns have on average more than 8 12 nonzero elements column generation often becomes inefficient because solving the master problem requires very long solution times at each iteration due to high degeneracy to overcome this difficulty we introduce a dynamic constraint aggregation method that reduces the number of set partitioning constraints in the master problem by aggregating some of them according to an equivalence relation to guarantee optimality this equivalence relation is updated dynamically throughout the solution process tests on the linear relaxation of the simultaneous vehicle and crew scheduling problem in urban mass transit show that this method significantly reduces the size of the master problem degeneracy and solution times especially for larger problems in fact for an instance involving 1 600 set partitioning constraints the master problem solution time is reduced by a factor of 8

worst case analyses linear programming and the bin packing problem in this paper we consider the familiar bin packing problem and its associated set partitioning formulation we show that the optimal solution to the bin packing problem can be no larger than 4 3 zlp wherezlp is the optimal solution value of the linear programming relaxation of the set partitioning formulation an example is provided to show that the bound is tight a by product of our analysis is a new worst case bound on the performance of the well studied first fit decreasing and best fit decreasing heuristics

minimization of travel time and weighted number of stops in a traffic light network abstract the time constrained shortest path problem is an important generalization of the shortest path problem recently a model called traffic light control model was introduced by chen and yang transport res b 34 2000 241 to simulate the operations of traffic light control in a city the constraints of the model consist of a repeated sequence of time windows and each window allows only certain routes to pass through a node in this paper we introduce a new kind of network called on off time switch network in which an arc is associated with a sequence of windows with status on or off analogous to go or wait we show that both networks have the same mathematical structure in the sense that a path in one network corresponds to a path in the other one since chen and yang have developed algorithms to find the minimum total time path in the previous paper we include one more criterion in this paper weighted number of stops to solve this bi criteria path problem we transform the traffic light network into the on off time switch network which allows us to take advantages of the special structure to design more efficient algorithms by this transformation finding the bi criteria shortest path in the traffic light network can be done in time o wn3 where n is the number of nodes and w is a given maximum number of weighted stops

an optimal algorithm for the traveling salesman problem with time windows this paper presents the development of new elimination tests which greatly enhance the performance of a relatively well established dynamic programming approach and its application to the minimization of the total traveling cost for the traveling salesman problem with time windows the tests take advantage of the time window constraints to significantly reduce the state space and the number of state transitions these reductions are performed both a priori and during the execution of the algorithm the approach does not experience problems stemming from increasing problem size wider or overlapping time windows or an increasing number of states nearly as rapidly as other methods our computational results indicate that the algorithm was successful in solving problems with up to 200 nodes and fairly wide time windows when the density of the nodes in the geographical region was kept constant as the problem size was increased the algorithm was capable of solving problems with up to 800 nodes for these problems the cpu time increased linearly with problem size these problem sizes are much larger than those of problems previously reported in the literature

a branch and price algorithm for the generalized assignment problem the generalized assignment problem examines the maximum profit assignment of jobs to agents such that each job is assigned to precisely one agent subject to capacity restrictions on the agents a new algorithm for the generalized assignment problem is presented that employs both column generation and branch and bound to obtain optimal integer solutions to a set partitioning formulation of the problem

a two grid discretization scheme for semilinear elliptic eigenvalue problems we present a two grid finite element discretization scheme with a two loop continuation algorithm for tracing solution branches of semilinear elliptic eigenvalue problems first we use the predictor corrector continuation method to compute an approximating point for the solution curve on the coarse grid then we use this approximating point as a predicted point for the solution curve on the fine grid in the corrector step we solve the first and the second order approximations of the nonlinear pde to obtain corrections for the state variable on the fine grid and the coarse grid respectively the continuation parameter is updated by computing the rayleigh quotient on the fine space to guarantee the approximating point we just obtained lies on the solution curve we perform newton s method we repeat the process described above until the solution curve on the fine space is obtained we show how the singular points such as folds and bifurcation points can be well approximated comprehensive numerical experiments show that the two grid finite element discretization scheme with a two loop continuation algorithm is efficient and robust for solving second order semilinear elliptic eigenvalue problems

a column generation based decomposition algorithm for a parallel machine just in time scheduling problem we propose a column generation based exact decomposition algorithm for the problem of scheduling n jobs with an unrestrictively large common due date on m identical parallel machines to minimize total weighted earliness and tardiness we first formulate the problem as an integer program then reformulate it using dantzig wolfe decomposition as a set partitioning problem with side constraints based on this set partitioning formulation a branch and bound exact solution algorithm is developed for the problem in the branch and bound tree each node is the linear relaxation problem of a set partitioning problem with side constraints this linear relaxation problem is solved by column generation approach where columns represent partial schedules on single machines and are generated by solving two single machine subproblems our computational results show that this decomposition algorithm is capable of solving problems with up to 60 jobs in reasonable cpu time

an interpolation approximation for queueing systems with poisson input for many queueing systems of practical interest it is difficult to obtain exact analytical results it is however often possible to obtain asymptotic results for light and heavy traffic heavy traffic limit theorems yield expressions for normalized quantities of interest in light traffic we can obtain in addition to limits more sensitive information by calculating what are effectively derivatives of the quantity of interest with respect to the arrival rate we can then combine the light and heavy traffic results to yield a polynomial in the arrival rate as an approximation to the normalized quantity of interest for instance by utilizing the heavy traffic limit the light traffic limit and the first derivative we can obtain a quadratic approximation then by reversing the normalization process we can obtain an approximation for the original quantity of interest in this paper we present the details of the above approximation focusing via several examples on applications of the method we then compare the results of the approximation for the examples considered with exact and simulation results in addition we compare our results to some related approximations for the examples considered the approximation works extremely well

subset row inequalities applied to the vehicle routing problem with time windows this paper presents a branch and cut and price algorithm for the vehicle routing problem with time windows the standard dantzig wolfe decomposition of the arc flow formulation leads to a set partitioning problem as the master problem and an elementary shortest path problem with resource constraints as the pricing problem we introduce the subset row inequalities which are chvatal gomory rank 1 cuts based on a subset of the constraints in the master problem applying a subset row inequality in the master problem increases the complexity of the label setting algorithm used to solve the pricing problem because an additional resource is added for each inequality we propose a modified dominance criterion that makes it possible to dominate more labels by exploiting the step like structure of the objective function of the pricing problem computational experiments have been performed on the solomon benchmarks where we were able to close several instances the results show that applying subset row inequalities in the master problem significantly improves the lower bound and in many cases makes it possible to prove optimality in the root node

branch price and cut for the mixed capacitated general routing problem with time windows abstract the mixed capacitated general routing problem with time windows mcgrptw is defined over a mixed graph for which some nodes arcs and edges have strictly positive demand and must be serviced the problem consists of determining a set of least cost vehicle routes that satisfy this requirement while respecting pre specified time windows and without exceeding the vehicle capacity in this work we transform the mcgrptw into an equivalent node routing problem over a directed graph thus we solve the equivalent problem by using a branch price and cut algorithm which relies on some effective techniques introduced in the field computational experiments over instances derived from the capacitated arc routing problem with time windows and from the mixed capacitated general routing problem are presented the article also describes experiments over benchmark instances

the integration of an interior point cutting plane method within a branch and price algorithm this paper presents a novel integration of interior point cutting plane methods within branch and price algorithms unlike the classical method columns are generated at a central dual solution by applying the analytic centre cutting plane method accpm on the dual of the full master problem first we introduce some modifications to accpm we propose a new procedure to recover primal feasibility after adding cuts and use for the first time a dual newton s method to calculate the new analytic centre after branching second we discuss the integration of accpm within the branch and price algorithm we detail the use of accpm as the search goes deep in the branch and bound tree making full utilization of past information as a warm start we exploit dual information from accpm to generate incumbent feasible solutions and to guide branching finally the overall approach is implemented and tested for the bin packing problem and the capacitated facility location problem with single sourcing we compare against cplex mip 7 5 as well as a classical branch and price algorithm

the weak braided hopf algebra structure of some cayley dickson algebras abstract it has been shown by albuquerque and majid that a class of unital k algebras not necessarily associative obtained through the cayley dickson process can be viewed as commutative associative algebras in some suitable symmetric monoidal categories in this note we will prove that they are moreover commutative and cocommutative weak braided hopf algebras within these categories to this end we first define a cayley dickson process for coalgebras we then see that the k vector space of complex numbers of quaternions of octonions of sedenions etc fit to our theory hence they are all monoidal coalgebras as well and therefore weak braided hopf algebras

a neural network based approach to the double traveling salesman problem the double traveling salesman problem is a variation of the basic traveling salesman problem where targets can be reached by two salespersons operating in parallel the real problem addressed by this work concerns the optimization of the harvest sequence for the two independent arms of a fruit harvesting robot this application poses further constraints like a collision avoidance function the proposed solution is based on a self organizing map structure initialized with as many artificial neurons as the number of targets to be reached one of the key components of the process is the combination of competitive relaxation with a mechanism for deleting and creating artificial neurons moreover in the competitive relaxation process information about the trajectory connecting the neurons is combined with the distance of neurons from the target this strategy prevents tangles in the trajectory and collisions between the two tours results of tests indicate that the proposed approach is efficient and reliable for harvest sequence planning moreover the enhancements added to the pure self organizing map concept are of wider importance as proved by a traveling salesman problem version of the program simplified from the double version for comparison

lagrangian relaxation methods for solving the minimum fleet size multiple traveling salesman problem with time windows we consider the problem of finding the minimum number of vehicles required to visit once a set of nodes subject to time window constraints for a homogeneous fleet of vehicles located at a common depot this problem can be formulated as a network flow problem with additional time constraints the paper presents an optimal solution approach using the augmented lagrangian method two lagrangian relaxations are studied in the first one the time constraints are relaxed producing network subproblems which are easy to solve but the bound obtained is weak in the second relaxation constraints requiring that each node be visited are relaxed producing shortest path subproblems with time window constraints and integrality conditions the bound produced is always excellent numerical results for several actual school busing problems with up to 223 nodes are discussed comparisons with a set partitioning formulation solved by column generation are given

algorithms for the vehicle routing and scheduling problems with time window constraints this paper considers the design and analysis of algorithms for vehicle routing and scheduling problems with time window constraints given the intrinsic difficulty of this problem class approximation methods seem to offer the most promise for practical size problems after describing a variety of heuristics we conduct an extensive computational study of their performance the problem set includes routing and scheduling environments that differ in terms of the type of data used to generate the problems the percentage of customers with time windows their tightness and positioning and the scheduling horizon we found that several heuristics performed well in different problem environments in particular an insertion type heuristic consistently gave very good results

simultaneous multigrid techniques for nonlinear eigenvalue problems solutions of the nonlinear schrodinger poisson eigenvalue problem in two and three dimensions algorithms for nonlinear eigenvalue problems ep s often require solving self consistently a large number of ep s convergence difficulties may occur if the solution is not sought in an appropriate region if global constraints have to be satisfied or if close or equal eigenvalues are present multigrid mg algorithms for nonlinear problems and for ep s obtained from discretizations of partial differential ep have often been shown to be more efficient than single level algorithms this paper presents mg techniques and a mg algorithm for nonlinear schr odinger poisson ep s the algorithm overcomes the above mentioned difficulties combining the following techniques a mg simultaneous treatment of the eigenvectors and nonlinearity and with the global constrains mg stable subspace continuation techniques for the treatment of nonlinearity and a mg projection coupled with backrotations for separation of solutions these techniques keep the solutions in an appropriate region where the algorithm converges fast and reduce the large number of self consistent iterations to only a few or one mg simultaneous iteration the mg projection makes it possible to efficiently overcome difficulties related to clusters of close and equal eigenvalues computational examples for the nonlinear schr odinger poisson ep in two and three dimensions presenting special computational difficulties that are due to the nonlinearity and to the equal and closely clustered eigenvalues are demonstrated for these cases the algorithm requires o qn operations for the calculation of q eigenvectors of size n and for the corresponding eigenvalues one mg simultaneous cycle per fine level was performed the total computational cost is equivalent to only a few gauss seidel relaxations per eigenvector an asymptotic convergence rate of 0 15 per mg cycle is attained

a branch first cut second approach for locomotive assignment the problem of assigning locomotives to trains consists of selecting the types and number of engines that minimize the fixed and operational locomotive costs resulting from providing sufficient power to pull trains on fixed schedules the force required to pull a train is often expressed in terms of horsepower and tonnage requirements rather than in terms of number of engines this complicates the solution process of the integer programming formulation and usually creates a large integrality gap furthermore the solution of the linearly relaxed problem is strongly fractional to obtain integer solutions we propose a novel branch and cut approach the core of the method consists of branching decisions that define on one branch the projection of the problem on a low dimensional subspace there the facets of the polyhedron describing a restricted constraint set can be easily derived we call this approach branch first cut second we first derive facets when at most two types of engines are used we then extend the branching rule to cases involving additional locomotive types we have conducted computational experiments using actual data from the canadian national railway company simulated test problems involving two or more locomotive types were solved over 1 2 and 3 day rolling horizons the cuts were successful in reducing the average integrality gap by 52 for the two type case and by 34 when more than 25 types were used furthermore the branch first cut second approach was instrumental in improving the best known solution for an almost 2 000 leg weekly problem involving 26 locomotive types it reduced the number of locomotives by 11 or 1 1 at an equivalent savings of 3 000 000 per unit additional tests on different weekly data produced almost identical results

solving a class of stochastic mixed integer programs with branch and price we begin this paper by identifying a class of stochastic mixed integer programs that have column oriented formulations suitable for solution by a branch and price algorithm b p we then survey a number of examples and use a stochastic facility location problem sflp for a detailed demonstration of the relevant modeling and solution techniques computational results with a scenario representation of uncertain costs demands and capacities show that b p can be orders of magnitude faster than solving the standard formulation by branch and bound we also demonstrate how b p can solve sflp exactly as exactly as a deterministic mixed integer program when demands and other parameters can be represented as certain types of independent random variables e g independent normal random variables with integer means and variances

robust branch and cut and price for the capacitated vehicle routing problem the best exact algorithms for the capacitated vehicle routing problem cvrp have been based on either branch and cut or lagrangean relaxation column generation this paper presents an algorithm that combines both approaches it works over the intersection of two polytopes one associated with a traditional lagrangean relaxation over q routes the other defined by bound degree and capacity constraints this is equivalent to a linear program with exponentially many variables and constraints that can lead to lower bounds that are superior to those given by previous methods the resulting branch and cut and price algorithm can solve to optimality all instances from the literature with up to 135 vertices this more than doubles the size of the instances that can be consistently solved

an adaptive multigrid scheme for bose einstein condensates in a periodic potential we present a novel multigrid continuation method for treating parameter dependent problems the proposed algorithm which can be flexibly implemented is a generalization of the two grid discretization schemes c s chien b w jeng a two grid discretization scheme for semilinear elliptic eigenvalue problems siam j sci comput 27 2006 1287 1304 that is approximating points on a solution curve do not necessarily lie on the same fine grid we apply the algorithm to compute energy levels and superfluid densities of bose einstein condensates bec in a periodic potential both positive and negative scattering lengths are considered in our numerical experiments for positive scattering length if the chemical potential is large enough and the domain is properly chosen the results show that the number of peaks of the first few energy states of the 2d bec in a periodic potential depends on the wave number of the periodic potential moreover for bright solitons the number of peaks of the ground state solutions is 1d 1 2 and 1d 2 where the periodic potential is expressed in terms of the sine or the cosine functions respectively however these formulae do not hold if the scattering length is negative the numerical study is extended to the two component 1d and 2d bec in a periodic potential

dynamic programming strategies for the traveling salesman problem with time window and precedence constraints the traveling salesman problem with time window and precedence constraints tsp twpc is to find an hamiltonian tour of minimum cost in a graph g x a of n vertices starting at vertex 1 visiting each vertex i x during its time window and after having visited every vertex that must precede i and returning to vertex 1 the tsp twpc is known to be np hard and has applications in many sequencing and distribution problems in this paper we describe an exact algorithm to solve the problem that is based on dynamic programming and makes use of bounding functions to reduce the state space graph these functions are obtained by means of a new technique that is a generalization of the state space relaxation for dynamic programming introduced by christofides et al christofides n a mingozzi p toth 1981b state space relaxation for the computation of bounds to routing problems networks 11 145 164 computational results are given for randomly generated test problems

vehicle routing problem with time windows and a limited number of vehicles abstract this paper introduces a variant of the vehicle routing problem with time windows where a limited number of vehicles is given m vrptw under this scenario a feasible solution is one that may contain either unserved customers and or relaxed time windows we provide a computable upper bound to the problem to solve the problem we propose a tabu search approach characterized by a holding list and a mechanism to force dense packing within a route we also allow time windows to be relaxed by introducing the notion of penalty for lateness in our approach customer jobs are inserted based on a hierarchical objective function that captures multiple objectives computational results on benchmark problems show that our approach yields solutions that are competitive to best published results on vrptw on m vrptw instances experiments show that our approach produces solutions that are very close to computed upper bounds moreover as the number of vehicles decreases the routes become more densely packed monotically this shows that our approach is good from both the optimality as well as stability point of view

three stage approaches for optimizing some variations of the resource constrained shortest path sub problem in a column generation context this paper presents a unified three stage approach tsa comprising preprocessing setup and iterative solution stages for solving several variations of the resource constrained shortest path problem rcsp tsa is designed specially for column generation applications in which sub problems must be solved repetitively the first two stages are implemented one time and only the third stage need be applied repetitively in a companion paper the authors proposed a tsa for solving rcsp on an acyclic graph with upper bound resource limitation constraints this paper shows that a tsa can be designed to solve each of several related problems shortest path with equality resource limitation constraints shortest path with resource windows resource constrained k shortest path and multiple resource multiple choice knapsack a numerical example demonstrates application of a tsa to design an international assembly system and its supply chain using branch and price with multiple choice knapsack sub problems computational results show that our tsa can solve this sub problem effectively in such a column generation environment