hub location as the minimization of a supermodular set function this paper highlights how a general class of hub location problems can be modeled as the minimization of a real valued supermodular set function well known problems such as uncapacitated hub location p hub median and hub arc location among others are shown to be particular cases of this class two integer programming formulations are introduced and compared one uses path based variables frequently employed in hub location whereas the other exploits properties of supermodular functions in addition several worst case bounds for a greedy and a local improvement heuristic are obtained for the general class and for some particular cases in which sharper bounds can be devised computational experiments are performed to compare both formulations when used with a general purpose solver computational results obtained on benchmark instances confirm the superiority of the supermodular formulation

on the optimal control problem for linear schrodinger equation in this paper two estimates have been proven for the solution of the optimal control problem which is set for the linear schrodinger equation appearing in quantum mechanics acquiring necessary and sufficient conditions the existence and uniqueness of the solution have also been shown

solving the uncapacitated multiple allocation hub location problem by means of a dual ascent technique this paper deals with the uncapacitated multiple allocation hub location problem the dual problem of a four indexed formulation is considered and a heuristic method based on a dual ascent technique is designed this heuristic which is reinforced with several specifical subroutines and does not require any external linear problem solver is the core tool embedded in an exact branch and bound framework besides the heuristic provides the branch and bound algorithm with good lower bounds for the nodes of the branching tree the results of the computational experience with the classical cab and ap data sets are included showing the great effectiveness of this approach instances with up to 120 nodes are solved

on an optimal control problem in this paper we applied the finite differences method to the solution of variational problem of an inverse problem for the scrodinger equation with a final functional these types of problems arise in various fields in quantum mechanical nuclear physics and modern physics 2 11 also we prove two estimates for the differences scheme and convergence speed of difference approximations according to the functional the inverse problems for the schrodinger equation having different variational formulation were investigated in 7 12 13

optimal design of a distributed network with a two level hierarchical structure abstract this paper deals with the topological design of a network with a two level hierarchical structure where the embedded backbone network is full meshed and the local networks attached to it are of star type the whole problem unlike the conventional approaches partitioning it into two subproblems is directly handled in such a general setting that both a backbone network and local access networks are to be simultaneously determined we formulate this problem as a quadratic 0 1 programming model and also propose an equivalent linearized version of the model based on the observation that the linearized model is a variant of the well established uncapacitated facility location problem a dual based solution procedure is developed and shown to be efficient from computing experiments which were done on a wide variety of problems with up to 50 backbone nodes and 50 user nodes

the tree of hubs location problem this paper presents the tree of hubs location problem it is a network hub location problem with single assignment where a fixed number of hubs have to be located with the particularity that it is required that the hubs are connected by means of a tree the problem combines several aspects of location network design and routing problems potential applications appear in telecommunications and transportation systems when set up costs for links between hubs are so high that full interconnection between hub nodes is prohibitive we propose an integer programming formulation for the problem furthermore we present some families of valid inequalities that reinforce the formulation and we give an exact separation procedure for them finally we present computational results using the well known ap and cab data sets

efficient solution procedure and reduced size formulations for p hub location problems in this paper we consider the uncapacitated p hub location problems where multiple and single allocation cases are considered in the multiple allocation problem a nonhub node may be allocated to more than one hub we show that the multiple allocation problem can be solved efficiently by the shortest path algorithm when p is fixed also this algorithm can be applied to the cases in which some other hubbing policies are used in the single allocation problem each nonhub node must be allocated to exactly one of the p hubs we provide a reduced size formulation and a mixed integer formulation for the model with fixed hub locations where fixed costs for opening links are considered computational experience is provided using the data given in the literature and the instances having fixed costs for opening links

two genetic algorithms for solving the uncapacitated single allocation p hub median problem this paper deals with the uncapacitated single allocation p hub median problem usaphmp two genetic algorithm ga approaches are proposed for solving this np hard problem new encoding schemes are implemented with appropriate objective functions both approaches keep the feasibility of individuals by using specific representation and modified genetic operators the numerical experiments were carried out on the standard orlib hub data set both methods proved to be robust and efficient in solving usaphmp with up to 200 nodes and 20 hubs the second ga approach achieves all previously known optimal solutions and achieves the best known solutions on large scale instances

linearization strategies for a class of zero one mixed integer programming problems this paper is concerned with a new linearization strategy for a class of zero one mixed integer programming problems that contains quadratic cross product terms between continuous and binary variables and between the binary variables themselves this linearization scheme provides an equivalent mixed integer linear programming problem which yields a tighter continuous relaxation than that obtainable via the alternative linearization techniques available in the literature moreover the proposed technique provides a unifying framework in the sense that all the alternate methods lead to formulations that are accessible through appropriate surrogates of the constraints of the new linearized formulation extensions to various other types of mixed integer nonlinear programming problems are also discussed

an optimal control problem for two dimensional schrodinger equation abstract in this paper we consider an optimal control problem controlled by three functions which are in the coefficients of a two dimensional schrodinger equation after proving the existence and uniqueness of the optimal solution we get the frechet differentiability of the cost functional using hamilton pontryagin function then we state a necessary condition to an optimal solution in the variational inequality form using the gradient

a difference method for solving parabolic equations of order 2n in this paper a finite difference method is used to approximate for the solution of the parabolic partial differential equation of order 2n and error of the method is determined the resulting system is solved by efficient implicit iterations in some numerical examples maple modules are designed for the purpose of testing and using the method

on tabu search for the location of interacting hub facilities abstract a new heuristic method based on tabu search is developed for the problem of locating p interacting hub facilities among n interacting nodes in a network the method treats equally the problem of locating hub facilities as well as the problem of allocating the nodes to one and only one hub the algorithm obtained improved solutions to problems from the standard test set from literature which has been used in this study

a general variable neighborhood search for solving the uncapacitated single allocation p hub median problem we present a new general variable neighborhood search approach for the uncapacitated single allocation p hub median problem in networks this np hard problem is concerned with locating hub facilities in order to minimize the traffic between all origin destination pairs we use three neighborhoods and efficiently update data structures for calculating new total flow in the network in addition to the usual sequential strategy a new nested strategy is proposed in designing a deterministic variable neighborhood descent local search our experimentation shows that general variable neighborhood search based heuristics outperform the best known heuristics in terms of solution quality and computational effort moreover we improve the best known objective values for some large australia post and planetlab instances results with the new nested variable neighborhood descent show the best performance in solving very large test instances

on a quadratic integer program for the location of interacting hub facilities o kelly 1987 considers the following hub location problem given n interacting nodes flows between pairs of nodes denoted by w u w i 0 by assumption the transportation cost c u of a unit of flows between nodes i and j with cii 0 and p the number of hub facilities to be located p n let x k 1 if node i is serviced by a hub at k and 0 otherwise further let x 1 if node i is a hub and 0 otherwise the problem is to find the locations of the hub facilities and the node assignments that minimize the total transportation cost this problem is formulated as given below

hub location and the p hub median problem hub facilities serve as switching and transshipment points in transportation and communication networks hub networks concentrate flows on the hub to hub links and benefit from economies of scale in interhub transportation most hub location research has focused on problems where each origin destination is allocated to a single hub however multiple allocation to more than one hub is necessary to minimize total transportation costs this paper defines a p hub median analogous to a p median and presents integer programming formulations for the multiple and single allocation p hub median problems two new heuristics for the single allocation p hub median problem are evaluated these heuristics derive a solution to the single allocation p hub median problem from the solution to the multiple allocation p hub median problem computational results are presented for problems with 10 40 origins destinations and up to eight hubs the new heuristics generally perform well in comparison with other heuristics

on the stability of the solution in an optimal control problem for a schrodinger equation this research paper deals with the frechet differentiability and necessary condition for optimality in a problem governed by the coefficient of a schrodinger equation in order to obtain a stabile solution by means of strong convergence of any minimizer we have used a more regular space and considered second adjoint problem for obtaining the gradient of the cost functional proving the lipschitz continuity we have stated the necessary condition for optimal solution

an optimal control problem governed by the potential of a linear schrodinger equation in this paper for optimal control of the potential in a quantum mechanical system described by the linear schrodinger equation the soundness of the problem statement is analyzed and a computing solution algorithm is specified since this problem is ill posed using the tichonov regularization the problem is solved numerically also the results are discussed on two simple problems

solving large single allocation p hub problems with two or three hubs abstract in this paper we present an efficient approach for solving single allocation p hub problems with two or three hubs two different variants of the problem are considered the uncapacitated single allocation p hub median problem and the p hub allocation problem we solve these problems using new mixed integer linear programming formulations that require fewer variables than those formerly used in the literature the problems that we solve here are the largest single allocation problems ever solved the numerical results presented here will demonstrate the superior performance of our mixed integer linear programs over traditional approaches for large problems finally we present the first mixed integer linear program for solving single allocation hub location problems that requires only o n 2 variables and o n 2 constraints that is valid for any number of hubs

tight linear programming relaxations of uncapacitated p hub median problems abstract the problem of locating hub facilities and allocating non hub nodes to those hubs arises frequently in the design of communication networks airline passenger flow and parcel delivery networks in this paper we consider uncapacitated multiple and single allocation p hub median problems we develop new mixed 0 1 linear formulations with tight linear programming relaxations the approach is tested on a well known and heavily used benchmark data set of real world problems with resulting lp relaxations ranging from 10010 to 391 250 variables and from 2 101 to 31 901 constraints which proved to be difficult linear programs yet this approach proved to be very effective in almost all instances the linear programming solution was integer in cases with fractional solutions the integrality was achieved by adding a small partial set of integrality constraints therefore we extended the range of optimally solvable instances of these np hard hub location problems which have defied researchers for the last ten years as an additional result for the single allocation case we were able to establish optimality of all heuristic solutions obtained via tabu search algorithm from a previous study for the more difficult single allocation p hub median problem we also used the best known heuristic solution as a guidance in adding integrality constraints this novel linkage between optimal and heuristic solutions has a potential impact in a number of other problem settings where efficient heuristic solutions exist and are probably but not provably optimal

an inverse problem of determining the implied volatility in option pricing in the black scholes world there is the important quantity of volatility which cannot be observed directly but has a major impact on the option value in practice traders usually work with what is known as implied volatility which is implied by option prices observed in the market in this paper we use an optimal control framework to discuss an inverse problem of determining the implied volatility when the average option premium namely the average value of option premium corresponding with a fixed strike price and all possible maturities from the current time to a chosen future time is known the issue is converted into a terminal control problem by green function method the existence and uniqueness of the minimum of the control functional are addressed by the optimal control method and the necessary condition which must be satisfied by the minimum is also given the results obtained in the paper may be useful for those who engage in risk management or volatility trading

a quadratic integer program for the location of interacting hub facilities abstract this paper reports a new formulation of a general hub location model as a quadratic integer program non convexity of the objective function makes the problem difficult a variety of alternative solution strategies are discussed computational results from two simple heuristics are presented for the task of siting 2 3 or 4 hubs to serve interactions between sets of 10 15 20 and 25 u s cities the effects of different computational shortcuts are examined

lagrangian relaxation based approaches to capacitated hub and spoke network design problem abstract the main purpose of this paper is to introduce the capacitated hub and spoke network design problem in which hubs have limited capacity for channelling flows between the nodes served by the system the problem is formulated under a networking policy allowing both direct nonstop and hub connected one hub stop and two hub stop services between the nodes a branch and bound algorithm and a heuristic procedure partitioning the set of solutions on the basis of hub locations are presented in both algorithms after identifying a set of hub locations the problem is reduced into a smaller routing problem subgradient optimization is then applied to a lagrangian relaxation of the reduced problem the model and the proposed algorithms were applied to the air transportation system using data on passenger flows between the top forty us cities in 1989 accounting for 73 32 of the surveyed passenger flow computational results from 141 problems suggest that both algorithms are effective at finding good solutions to this problem

an improved benders decomposition algorithm for the tree of hubs location problem abstract the tree of hubs location problem is a particularly hard variant of the so called hub location problems when solving this problem by a benders decomposition approach it is necessary to deal with both optimality and feasibility cuts while modern implementations of the benders decomposition method rely on pareto optimal optimality cuts or on rendering feasibility cuts based on minimal infeasible subsystems a new cut selection scheme is devised here under the guiding principle of extracting useful information even when facing infeasible subproblems the proposed algorithm outperforms two other modern variants of the method and it is capable of optimally solving instances five times larger than the ones previously reported on the literature

allocation strategies in hub networks in this paper we study allocation strategies and their effects on total routing costs in hub networks given a set of nodes with pairwise traffic demands the p hub median problem is the problem of choosing p nodes as hub locations and routing traffic through these hubs at minimum cost this problem has two versions in single allocation problems each node can send and receive traffic through a single hub whereas in multiple allocation problems there is no such restriction and a node may send and receive its traffic through all p hubs this results in high fixed costs and complicated networks in this study we introduce the r allocation p hub median problem where each node can be connected to at most r hubs this new problem generalizes the two versions of the p hub median problem we derive mixed integer programming formulations for this problem and perform a computational study using well known datasets for these datasets we conclude that single allocation solutions are considerably more expensive than multiple allocation solutions but significant savings can be achieved by allowing nodes to be allocated to two or three hubs rather than one we also present models for variations of this problem with service quality considerations flow thresholds and non stop service

path relinking approach for multiple allocation hub maximal covering problem we consider the multiple allocation hub maximal covering problem mahmcp considering a serviced o d flow was required to reach the destination optionally passing through one or two hubs in a limited time cost or distance what is the optimal way to locate p hubs to maximize the serviced flows by designing a new model for the mahmcp we provide an evolutionary approach based on path relinking the computational experience of an ap data set was presented and a special application on hub airports location of chinese aerial freight flows between 82 cities in 2002 was introduced

integer programming formulations of discrete hub location problems abstract hubs are facilities that serve as transshipment and switching points for transportation and telecommunication systems with many origins and destinations this paper present integer programming formulations for four types of discrete hub location problems the p hub median problem the uncapacitated hub location problem p hub center problems and hub covering problems hub median and uncapacitated hub location problems have received limited attention from researchers this paper introduces discrete hub center and hub covering problems basic formulations and formulations with flow thresholds for spokes are presented

hub location under competition abstract hubs are consolidation and dissemination points in many to many flow networks hub location problem is to locate hubs among available nodes and allocate non hub nodes to these hubs the mainstream hub location studies focus on optimal decisions of one decision maker with respect to some objective s even though the markets that benefit hubbing are oligopolies therefore in this paper we propose a competitive hub location problem where the market is assumed to be a duopoly two decision makers or firms sequentially decide locations of their hubs and then customers choose one firm with respect to provided service levels each decision maker aims to maximize his her own market share we propose two problems for the leader former decision maker and follower latter decision maker r x p h u b m e d i a n o i d and r p h u b c e n t r o i d problems respectively both problems are proven to be np complete linear programming models are presented for these problems as well as exact solution algorithms for the r p h u b c e n t r o i d problem the performance of models and algorithms are tested by computational analysis conducted on cab and tr data sets

a linear program for the two hub location problem abstract this paper considers the discrete two hub location problem we need to choose two hubs from a set of nodes the remaining nodes are to be connected to one of the two hubs which act as switching points for internodal flows a configuration which minimizes the total flow cost needs to be found we show that the problem can be solved in polynomial time when the hub locations are fixed since there are at most 1 2 n n 1 ways to choose the hub locations the two hub location problem can be solved in polynomial time we transform the quadratic 0 1 integer program of the single allocation problem in the fixed two hub system into a linear program and show that all extreme points of the polytope defined by the lp are integral also the problem can be transformed into a minimum cut problem which can be solved efficiently by any polynomial time algorithm

route planning for a mixed delivery system in long distance transportation and comparison with pure delivery systems the long distance routing problems are divided into three kinds of pure delivery systems where every order is allocated the same distribution strategy pure delivery systems have been generally independently and widely studied this research provides a solution to help pallet and package delivery companies in decision making considering a mixed delivery system to improve the use of resources it returns the route planning after allocating to each order the distribution strategy that best fits to the global scene and proves if a mixed delivery system achieves best results than a pure one and under which circumstances

lower bounds for the hub location problem we present a new lower bound for the hub location problem hlp where distances satisfy the triangle inequality our lower bound is based on a linearization of the problem and its modification obtained by incorporating the knowledge of a known heuristic solution a lower bound was computed for some standard data sets from the literature ranging between 10 and 25 nodes with 2 3 and 4 hubs and for different values for the parameter representing the discount for the flow between hubs the novel approach of using a known heuristic solution to derive a lower bound in all cases reduced the difference between the upper and lower bound this difference measures the quality of the best known heuristic solution in percentages above the best lower bound as a result of this research for smaller problems all instances with 10 and 15 nodes the average difference is reduced to 3 3 for larger sets 20 and 25 nodes the average difference is reduced to 5 9

an exact solution procedure for a cluster hub location problem abstract this paper proposes a new mip model formulation and a new solution procedure for the hub network design problem under a non restrictive policy introduced by sung and jin sung c s jin h w 2001 dual based approach for a hub network design problem under non restrictive policy european journal of operational research 132 1 88 105 the model formulation contains significantly fewer variables so that optimal solutions for the lp relaxation of the model can be determined for large instances using standard procedures for lp models furthermore the lp relaxation provides very tight lower bounds computational results are given which demonstrate that the new model formulation allows for solving much larger instances it turned out that the new exact solution procedure which utilises the new model formulation is faster than the heuristic proposed by sung and jin 2001 it is also shown that the problem is np hard

on the optimal control problem for schrodinger equation with complex potential the existence and uniqueness for the solution of the problem of determining the v x t potential in the schrodinger equation i j t xa 0 x j x a x j iv x t j f x t from the measured final data j x t y x is investigated for the objective functional j a v j x t v y x l 2 0 2 a v w w 2 0 1 w 2 it is proven that the problem has at least one solution for a 0 and has a unique solution for a 0 the necessary condition for solvability the problem is stated as the variational principle

exact and heuristic algorithms for the uncapacitated multiple allocation p hub median problem in this paper new milp formulations for the multiple allocation p hub median problem are presented these require fewer variables and constraints than those traditionally used in the literature an efficient heuristic algorithm based on shortest paths is described lp based solution methods as well as an explicit enumeration algorithm are developed to obtain exact solutions computational results are presented for well known problems from the literature which show that exact solutions can be found in a reasonable amount of computational time our algorithms are also benchmarked on a different data set this data set which includes problems that are larger than those used in the literature is based on a postal delivery network and has been treated by the authors in an earlier paper

computer experiments on quadratic programming algorithms abstract this paper compares the computational performance of five quadratic programming algorithms these include wolfe s simplex method lemke s complementary pivot method convex simplex method and quadratic differential algorithm execution time and iteration count are used as the major criteria for comparison since lemke s algorithm out performed all other methods in the study a detailed statistical analysis was performed to determine the relative importance of problem parameters on the efficiency of lemke s algorithm an analysis of variance showed that the number of variables the percent of positive linear terms in the objective the number of constraints and their interactions were the significant factors for both iteration count and execution time finally regression equations for iteration count and execution time are derived as a function of fifteen problem parameters