numerical integration of non linear two point boundary value problems using iterated deferred corrections i abstract in several recent papers the idea has been proposed of using implicit runge kutta formulae for the approximate integration of non linear two point boundary value problems these formulae often take a very special form so that computational efficiencies are obtained in certain well defined circumstances in this paper we examine the relative merits of these different classes of formulae and discuss their efficient implementation in a deferred correction framework a theoretical comparison of these formulae is important firstly because they call for different computational tasks to be performed and secondly because they are not always applicable to exactly the same classes of problems the most efficient of these formulae are then compared with methods based on finite differences and with methods based on spline collocation for some simple smooth test functions on the basis of operation counts and numerical experimentation it is concluded that runge kutta methods can be generally competitive with the other methods considered

on a definition of random sequences with respect to conditional probability we study a universal martin lof test with respect to a computable probability on a product space then we define random sequences with respect to a conditional probability by using a section of the set of random points of product space we show that 1 our definition is consistent with fubini s theorem and 2 it is equivalent to the relative notion of randomness under a condition this is an extension of lambalgen s theorem 1987 to a correlated probability

applying ant system for solving unequal area facility layout problems ant colony optimization aco is a young metaheuristic algorithm which has shown promising results in solving many optimization problems to date a formal aco based metaheuristic has not been applied for solving unequal area facility layout problems ua flps this paper proposes an ant system as one of the aco variants to solve them as a discrete optimization algorithm the proposed algorithm uses slicing tree representation to easily represent the problems without too restricting the solution space it uses several types of local search to improve its search performance it is then tested using several case problems with different size and setting overall the proposed algorithm shows encouraging results in solving ua flps

dynamic stabilization of systems with input delays abstract we examine the problem of dynamic stabilization of systems with input delays and show a construction of the compensator which is deduced from a state space representation of the control system itself

embedding formulations and complexity for unions of polyhedra it is well known that selecting a good mixed integer programming mip formulation is crucial for effectively obtaining a solution with state of the art solvers although best practices and guidelines for constructing good formulations abound there is rarely a single systematic construction approach that leads to the best possible formulation here we introduce embedding formulations and complexity as a new mip formulation paradigm for systematically constructing formulations for disjunctive constraints that are optimal with respect to size more specifically they yield the smallest possible ideal formulation i e one whose lp relaxation has integral extreme points among all formulations that only use 0 1 auxiliary variables we use the paradigm to characterize optimal formulations for special ordered sets of type 2 and certain piecewise linear functions of two variables we also show that the resultant formulations can provide a significant computational advantage over all known formulations for pi

an adaptive uniformly convergent numerical method for a semilinear singular perturbation problem a singularly perturbed semilinear two point boundary value problem is considered without any restriction on its turning points a difference scheme is presented for solving this problem on an arbitrary locally quasi uniform mesh it is shown that the solution of the scheme is first order accurate uniformly in the perturbation parameter in a discrete l 1 norm numerical results are presented for an adaptive method based on the scheme

a hybrid genetic algorithm with decomposition phases for the unequal area facility layout problem abstract we address the unequal area facility layout problem ua flp which aims to dimension and locate rectangular facilities in an unlimited floor space without overlap while minimizing the sum of distances among facilities weighted by material handling flows we introduce two algorithmic approaches to address this problem a basic genetic algorithm ga and a ga combined with a decomposition strategy via partial solution deconstructions and reconstructions to efficiently decompose the problem we impose a solution structure where no facility should cross the x or y axis although this restriction can possibly deteriorate the value of the best achievable solution it also greatly enhances the search capabilities of the method on medium and large problem instances for most such instances current exact methods are impracticable as highlighted by our experiments the resulting algorithm produces solutions of high quality for the two classic datasets of the literature improving six out of the eight best known solutions from the first set with up to 125 facilities and all medium and large scale instances from the second set for some of the largest instances of the second set with 90 or 100 facilities the average solution improvement goes as high as 6 percent or 7 percent when compared to previous algorithms in less cpu time we finally introduce additional instances with up to 150 facilities on this benchmark the decomposition method provides an average solution improvement with respect to the basic ga of about 9 percent and 1 3 percent on short and long runs respectively

combining wang landau sampling algorithm and heuristics for solving the unequal area dynamic facility layout problem abstract the dynamic facility layout problem dflp is the problem of placing facilities in a certain plant floor for multiple stages so that facilities do not overlap and the sum of the material handling and rearrangement costs are minimized we describe a model where the facilities have unequal areas and the layout for each stage is produced on the continuous plant floor the most difficulty of solving this problem consists in the lack of a powerful optimization method wang landau wl sampling algorithm is an improved monte carlo method and has been successfully applied to solve many optimization problems in this paper we combine the wl sampling algorithm and some heuristic strategies to solve the unequal area dflp in the wl sampling algorithm a vacant point strategy is applied to update layout at one stage to prevent overlapping of facilities and reduce the empty space among facilities a pushing strategy and a pressuring strategy are applied we have tested the proposed algorithm on four groups of cases and the computational results show that the proposed algorithm is effective in solving the unequal area dflp

blockwise estimate of the fundamental matrix of linear singularly perturbed differential systems with small delay and its application to uniform asymptotic solution a singularly perturbed system of linear differential equations with a small delay is considered estimates of blocks of the fundamental matrix solution to this system uniformly valid for all sufficiently small values of the parameter of singular perturbations are obtained in the cases of time independent and time dependent coefficients of the system in the first case the system is considered on an infinite time interval while in the second case it is considered on a finite one these estimates are applied to justify a uniform asymptotic solution of an initial value problem for this system in both cases

h control of linear singularly perturbed systems with small state delay an infinite horizon h state feedback control problem for singularly perturbed linear systems with a small state delay is considered an asymptotic solution of the hybrid system of riccati type algebraic ordinary differential and partial differential equations with deviating arguments associated with this problem is constructed based on this asymptotic solution conditions for the existence of a solution of the original h problem independent of the singular perturbation parameter are derived a simplified controller with parameter independent gain matrices solving the original problem for all sufficiently small values of this parameter is obtained an illustrative example is presented

genetic algorithms function optimization and facility layout design abstract the use of genetic algorithms to solve function optimization problems requires a coding scheme that represents solutions as strings of finite length in this paper a coding scheme designed for facility layout is presented a solution is represented by the postorder sequence of the nodes in a slicing tree new solutions are generated by applying the various genetic operators in each generation the genetic algorithm approach is compared with a local search technique in solving problems with sizes ranging from 12 to 30 facilities the results demonstrate that a genetic algorithm can be a viable tool to solve large scale layout problems

when does randomness come from randomness a result of shen says that if f 2 n 2 n is an almost everywhere computable measure preserving transformation and y 2 n is martin lof random then there is a martin lof random x 2 n such that f x y answering a question of bienvenu and porter we show that this property holds for computable randomness but not schnorr randomness these results combined with other known results imply that the set of martin lof randoms is the largest subset of 2 n satisfying this property and also satisfying randomness conservation if f 2 n 2 n is an almost everywhere computable measure preserving map and if x 2 n is random then f x is random

the descriptive complexity of brownian motion abstract a continuous function x on the unit interval is a generic brownian motion when every probabilistic event which holds almost surely with respect to the wiener measure is reflected in x provided that the event has a suitably effective description we show that a generic one dimensional brownian motion can be computed from an infinite binary string which is complex in the sense of kolmogorov chaitin conversely one can construct a kolmogorov chaitin random string from the values at the rational numbers of a generic brownian motion in this way we construct a recursive isomorphism between encoded versions of generic brownian motions and kolmogorov chaitin random reals

controllability of nonstandard singularly perturbed systems with small state delay a nonstandard singularly perturbed linear time dependent control system with multiple point and distributed small delays in state variables is considered two much simpler parameter free systems the reduced order and the boundary layer ones associated with the original system are introduced due to the nonstandard nature of the original system the reduced order one is descriptor differential algebraic a connection between the properties of controllability of these systems and such a property of the original system itself robust in the small parameter of singular perturbations is established a criterion of the impulse free controllability with respect to the slow variable of the reduced order system is obtained and the invariance of this criterion to an auxiliary gain matrix is shown

improving discrete model representations via symmetry considerations in this paper we focus on a useful modeling concept that is frequently ignored while formu lating discrete optimization problems very often there exists a natural symmetry inherent in the problem itself that if propagated to the model can hopelessly mire a branch and bound solver by burdening it to explore and eliminate such alternative symmetric solutions we discuss three applications where such a symmetry arises a telecommunications network design problem a noise pollution problem and a machine procurement and operation problem for each case we identify the indistinguishable objects in the model that create the problem symmetry and show how imposing certain decision hierarchies within the model significantly enhances its solvability while using a popular modern day commercial branch and cut software cplex 6 5

approximate method for the numerical solution of singular perturbation problems we present an approximate method for the numerical solution of linear singularly perturbed two point boundary value problems in ordinary differential equations with a boundary layer on the left end of the underlying interval it is motivated by the asymptotic behavior of singular perturbation problems the original problem is divided into inner and outer region problems the reduced problem is solved to obtain the terminal boundary condition then a new inner region problem is created and solved as a two point boundary value problem in turn the outer region problem is also modified and the resulting problem is efficiently treated by employing the trapezoidal formula coupled with discrete invariant imbedding algorithm the proposed method is iterative on the terminal point some numerical experiments have been included to demonstrate its applicability

a convex optimisation framework for the unequal areas facility layout problem the unequal areas facility layout problem is concerned with finding the optimal arrangement of a given number of non overlapping indivisible departments with unequal area requirements within a facility we present a convex optimisation based framework for efficiently finding competitive solutions for this problem the framework is based on the combination of two mathematical programming models the first model is a convex relaxation of the layout problem that establishes the relative position of the departments within the facility and the second model uses semidefinite optimisation to determine the final layout aspect ratio constraints frequently used in facility layout methods to restrict the occurrence of overly long and narrow departments in the computed layouts are taken into account by both models we present computational results showing that the proposed framework consistently produces competitive and often improved layouts for well known large instances when compared with other approaches in the literature

higher order uniformly convergent methods for singular perturbation problems abstract until now the best way to construct high order uniformly convergent schemes for singular perturbation problems was to apply exponentially fitted compact difference schemes we propose the following approach firstly solve an auxiliary problem with piecewise or nearly piecewise constant coefficients secondly improve the approximation iteratively using the defect correction idea and piecewise polynomial approximations of higher order an important advantage of our approach lies in the fact that it is possible to start from a classical or weak formulation of a boundary value problem therefore the approach is useful for singular perturbations related to ordinary as well as partial differential equations

computability of probability measures and martin lof randomness over metric spaces in this paper we investigate algorithmic randomness on more general spaces than the cantor space namely computable metric spaces to do this we first develop a unified framework allowing computations with probability measures we show that any computable metric space with a computable probability measure is isomorphic to the cantor space in a computable and measure theoretic sense we show that any computable metric space admits a universal uniform randomness test without further assumption

a computational method for solving singular perturbation problems using exponentially fitted finite difference schemes abstract a computational method is presented for solving linear singularly perturbed two point boundary value problems with a boundary layer on the left end of the interval the original problem is divided into inner and outer region problems the zeroth order asymptotic expansion is used to obtain the terminal boundary condition then a new inner region problem irp is created and solved as a two point boundary value problem tpbvp in turn the outer region problem orp is also solved as a tpbvp both these problems tpbvps are efficiently solved by employing uniform and optimal exponentially fitted finite difference schemes

stabilisation of second order nonlinear equations with variable delay for a wide class of second order nonlinear non autonomous models we illustrate that combining proportional state control with the feedback that is proportional to the derivative of the chaotic signal allows to stabilise unstable motions of the system the delays are variable which leads to more flexible controls permitting delay perturbations only delay bounds are significant for stabilisation by a delayed control the results are applied to the sunflower equation which has an infinite number of equilibrium points

strong reductions in effective randomness we study generalizations of demuth s theorem which states that the image of a martin lof random real under a tt reduction is either computable or turing equivalent to a martin lof random real we show that demuth s theorem holds for schnorr randomness and computable randomness answering a question of franklin but that it cannot be strengthened by replacing the turing equivalence in the statement of the theorem with wtt equivalence we also provide some additional results about the turing and tt degrees of reals that are random with respect to some computable measure

mathematical metaphysics of randomness abstract we consider various mathematical refinements of the notion of randomness of an infinite sequence and relations between them on the base of the game approach a new possible refinement of the notion of randomness is introduced unpredictability the case of finite sequences is also considered

asymptotic expansion for the initial value problem of the sunflower equation abstract in this paper we consider the initial value problem ivp for the sunflower equation ex t ax t b sin x t e 0 with x t t t e 0 x 0 y0 we prove that the solution of the above mentioned ivp has an asymptotic expansion uniformly valid on e for e sufficiently small

stats a slicing tree and tabu search based heuristic for the unequal area facility layout problem in this paper a slicing tree based tabu search heuristic for the rectangular continual plane facility layout problem flp is presented in addition to the incorporation of facilities with unequal areas we also integrate the possibility to specify various requirements regarding rectangular shape and dimensions of each individual facility by using bounding curves therefore it is possible to solve problems containing facilities of fixed and facilities of flexible shapes at the same time we present a procedure that calculates the layout corresponding to a given slicing tree on the basis of bounding curves these layouts are slicing structures which are able to contain empty spaces to guarantee that stringent shape restrictions of facilities are kept due to these features this approach is better suited for practical use than so far existing ones the effectiveness of our approach in terms of objective function value is shown by comparing our results to those found in the literature even a large problem instance comprised of 62 facilities has been solved

optimal order nonnested multigrid methods for solving finite element equations iii on degenerate meshes in this paper we consider several model problems where finite element triangular meshes with arbitrarily small angles high aspect ratios are utilized to deal with anisotropy interfaces or singular perturbations the constant rate independent of the number of unknowns the smallest angle the interface discontinuity the singular perturbation parameter etc convergence of some special nonnested multigrid methods for solving the finite element systems on such degenerate meshes will be proved numerical data are provided to support the analysis in each case

schnorr randomness for noncomputable measures abstract this paper explores a novel definition of schnorr randomness for noncomputable measures we say x is uniformly schnorr random if t x for all lower semicomputable functions t x such that t x d x is computable we prove a number of theorems demonstrating that this is the correct definition which enjoys many of the same properties as martin lof randomness for noncomputable measures nonetheless a number of our proofs significantly differ from the martin lof case requiring new ideas from computable analysis

lmi based stability for singularly perturbed nonlinear impulsive differential systems with delays of small parameter abstract in this paper a class of singularly perturbed nonlinear impulsive delay differential systems is considered where the time delays include a small parameter based on lyapunov krasovskii functional and free weighting matrix method some novel delay dependent global asymptotic stability results are derived in terms of linear matrix inequalities lmis which can be easily solved using efficient convex optimization algorithms the results show that the system stability is very sensitive to the singular perturbation and the small parameter in time delays a numerical example is given to show the effectiveness of the proposed results

uniform high order difference schemes for a singularly perturbed two point boundary value problem on construit une famille de schemas aux differences finies uniformement precis pour le probleme du type eu a x u b x u f x on donne une analyse complete de l erreur de discretisation basee sur les resultats de stabilite on presente des essais numeriques

singular perturbation analysis of boundary value problems for differential difference equations v small shifts with layer behavior in this paper an investigation is initiated of boundary value problems for singularly perturbed linear second order differential difference equations with small shifts i e where the second order derivative is multiplied by a small parameter and the shift depends on the small parameter similar boundary value problems are associated with expected first exit times of the membrane potential in models for neurons in particular this paper focuses on problems with solutions that exhibit layer behavior at one or both of the boundaries the analyses of the layer equations using laplace transforms lead to novel results it is shown that the layer behavior can change its character and even be destroyed as the shifts increase but remain small in the companion paper siam j appi math 54 1994 pp 273283 similar boundary value problems with solutions that exhibit rapid oscillations are studied

the linear quadratic optimal control problem with delays in state and control variables a state space approach this paper generalizes the results of vinter and kwong to a control operator with a more general delay structure than the one in a ichikawa and to delay systems with both finite and infinite memories the linear quadratic optimal control problem over both finite and infinite time horizons is treated several forms of associated abstract riccati differential and algebraic equations are presented the properties of the kernel of the solution are studied and a coupled set of matrix ordinary and partial differential equations is obtained

a nonlinear optimization approach for solving facility layout problems abstract facility layout research has many subproblems for which optimization models can be developed although modeling these problems is often simple the resulting large size of these models can make solution by decision makers very difficult this paper is concerned with the development of new heuristics to be used as aids in finding good solutions to the facility layout problem in particular nonlinear programming nlp techniques are developed that minimize the material handling cost descriptions of the problems to be solved are presented along with the formulations of the nlp models to be optimized and the methods used to solve them also presented are computational results and a comparison of these results to those of other layout algorithms the heuristics developed are shown to produce solutions comparable to and in many cases better than those of other algorithms the approach is particularly effective for problems with unequal size departments as well the models are shown to be computationally practical for solving real world problems

on stabilization of nonstandard singularly perturbed systems with small delays in state and control a nonstandard singularly perturbed linear time invariant system with a general type of small delay in state and control variables is considered two much simpler parameter free systems the reduced order and the boundary layer ones associated with the original system are introduced since the original system is nonstandard the reduced order one is descriptor differential algebraic a composite feedback control exponentially stabilizing the original system is constructed based on stabilizing controls of the reduced order and boundary layer systems an illustrative example is presented

convergence of the weiszfeld algorithm for weber problems using a generalized distance function this paper considers a generalization of the single and multisource weber problem for the case when the distance function is some power k of the usual ip distance properties of the generalized problem are established and an appropriate generalization of the weiszfeld iterative approach is given a convergence proof is supplied for an approximation to the original problem under certain restrictions on p and k

on computing the maximal delay intervals for stability of linear delay systems this note is concerned with stability properties of linear time invariant delay systems the authors consider delay systems of both retarded and neutral types expressed in state space forms the author s main goal is to provide a computation oriented method for computing the maximal delay intervals over which the systems under consideration maintain stability the author s results show that this can be accomplished by computing the generalized eigenvalues of certain frequency dependent matrices based on these results the author also states a necessary and sufficient condition concerning stability independent of delay for each of the retarded and neutral systems the author s results can be readily implemented and appear suitable for analyzing systems with high dimensions and many delay units

dynamics of a generic brownian motion recursive aspects we study the local fluctuations of brownian motions which are represented by infinite binary strings which are random in the sense of kolmogorov chaitin we show how the dynamical properties of such a brownian motion at a point depend on its recursive properties

on mathematical optimization for the visualization of frequencies and adjacencies as rectangular maps in this paper we address the problem of visualizing a frequency distribution and an adjacency relation attached to a set of individuals we represent this information using a rectangular map i e a subdivision of a rectangle into rectangular portions so that each portion is associated with one individual their areas reflect the frequencies and the adjacencies between portions represent the adjacencies between the individuals due to the impossibility of satisfying both area and adjacency requirements our aim is to fit as well as possible the areas while representing as many adjacent individuals as adjacent rectangular portions as possible and adding as few false adjacencies i e adjacencies between rectangular portions corresponding to non adjacent individuals as possible we formulate this visualization problem as a mixed integer linear programming milp model we propose a matheuristic that has this milp model at its heart our experimental results demonstrate that our matheuristic provides rectangular maps with a good fit in both the frequency distribution and the adjacency relation

enhanced model formulations for optimal facility layout this paper presents an improved mixed integer programming mip model and effective solution strategies for the facility layout problem and is motivated by the work of meller et al 1999 this class of problems seeks to determine a least cost layout of departments having various size and area requirements within a rectangular building and it is challenging even for small instances the difficulty arises from the disjunctive constraints that prevent departmental overlaps and the nonlinear area constraints for each department which existing models have failed to approximate with adequate accuracy we develop several modeling and algorithmic enhancements that are demonstrated to produce more accurate solutions while also decreasing the solution effort required we begin by deriving a novel polyhedral outer approximation scheme that can provide as accurate a representation of the area requirements as desired we also design alternative methods for reducing problem symmetry evaluate the performance of several classes of valid inequalities explore the construction of partial convex hull representations for the disjunctive constraints and investigate judicious branching variable selection priority schemes the results indicate asubstantial increase in the accuracy of the layout produced while at the same time providing adramatic reduction in computational effort in particular three previously unsolved test problems from the literature for which meller et al s algorithm terminated prematurely after 24 cpu hours of computation on a sun ultra 2 workstation with 390 mb ram with respective optimality gaps of 10 14 26 45 and 40 have been solved to exact optimality with reasonable effort using our proposed approach

a survey of numerical techniques for solving singularly perturbed ordinary differential equations this survey paper contains a surprisingly large amount of material and indeed can serve as an introduction to some of the ideas and methods of singular perturbation theory starting from prandtl s work a large amount of work has been done in the area of singular perturbations this paper limits its coverage to some standard singular perturbation models considered by various workers and the numerical methods developed by numerous researchers after 1984 2000 the work done in this area during the period 1905 1984 has already been surveyed by the first author of this paper see appl math comput 30 1989 223 for details due to the space constraints we have covered only singularly perturbed one dimensional problems

effective dimension of points visited by brownian motion we consider the individual points on a martin lof random path of brownian motion we show that 1 khintchine s law of the iterated logarithm holds at almost all points and 2 there exist points besides the trivial example of the origin having effective dimension 1 the proof of 1 shows that for almost all times t the path f is martin lof random relative to t and so the effective dimension of t f t is 2

brief a descriptor system approach to nonlinear singularly perturbed optimal control problem this paper considers the infinite horizon nonlinear quadratic optimal control problem for a singularly perturbed system which is nonlinear in both the slow and the fast variables the relationship between this problem and the analogous one for a descriptor system is investigated parameter independent controllers are constructed that solve the problem for the descriptor system and lead the full order system to the near optimal performance estimates on the closeness of the cost under near optimal controllers to the optimal one are obtained

multigrid applied to singular perturbation problems the solution of the singular perturbation problem email protected u b x u f 0 0 u 0 u 0 u 1 u 1 by a multigrid algorithm is considered theoretical and experimental results for a number of different discretizations are presented the theoretical and observed rates agree with the results developed in an earlier work of kamowitz and parter in addition the rate of convergence of the algorithm when the coarse grid operator is the natural finite difference analogue of the fine grid operator is presented this is in contrast to the case in the previous work where the galerkin choice i h hl hi h h was used for the coarse grid operators

efficient models for the facility layout problem abstract in this paper two new models of the facility layout problem are presented linear continuous with absolute values in the objective function and constraints and linear mixed integer the linear mixed integer models have lesser number of integer variables than any other existing formulation for the facility layout problem while most other linear mixed integer models available in the literature have been obtained through a linearization of the quadratic assignment problem the ones presented in this paper are not the continuous models have an even more compact form an advantage of the formulations presented in this paper is that the location of sites need not be known a priori more importantly two of the formulations model the layout problem with facilities of unequal area solving the models presented with an unconstrained optimization algorithm yields good quality suboptimal solutions in a relatively low computation time the continuous models appear to be more useful for solving the facility layout problem than other models published in the literature

monotone difference schemes for singular perturbation problems we consider numerical solutions of the boundary value problem varepsilon y f y b x y 0 0 leqq x leqq 1 y 0 a y 1 b varepsilon 0 b y geqq delta 0 with monotone difference schemes on nonuniform grids we prove general convergence results and show that the engquist osher monotone scheme will reproduce essential properties of the true solution for any grid for the inversion of the nonlinear scheme we suggest implicit time relaxation with variable time steps

some improvements of the nonequidistant engquist osher scheme two nonequidistant finite difference schemes for quasilinear singular perturbation problems are given and their properties are discussed and compared one of them corresponds to the equidistant lorenz modification of the engquist osher scheme the other one switches in dependence on the cell reynolds number both schemes show quadratic l 1 accuracy but the second one gives better pointwise results

euclidean space controllability of singularly perturbed linear systems with state delay abstract a singularly perturbed linear time dependent control system with small point and distributed delays in state variables is considered connections between the properties of controllability of the reduced order and boundary layer systems associated with the original one and such a property of the original system itself are established

global uniformly convergent schemes for a singularly perturbed boundary value problem using patched base spline functions abstract for a self adjoint second order model singular perturbation problem global uniformly convergent schemes are constructed the basic idea consists in replacing the coefficients by piecewise polynomials and solving the resulting problems exactly

a fast adaptive numerical method for stiff two point boundary value problems we describe a robust adaptive algorithm for the solution of singularly perturbed two point boundary value problems many different phenomena can arise in such problems including boundary layers dense oscillations and complicated or ill conditioned internal transition regions working with an integral equation reformulation of the original differential equation we introduce a method for error analysis which can be used for mesh refinement even when the solution computed on the current mesh is underresolved based on this method we have constructed a black box code for stiff problems which automatically generates an adaptive mesh resolving all features of the solution the solver is direct and of arbitrarily high order accuracy and requires an amount of time proportional to the number of grid points

application of multidimensional scaling techniques to facilities layout abstract guidelines in the form of scatter diagrams can be valuable and helpful in facilities layout decisions this paper demonstrates that a direct application of non metric multidimensional scaling mds techniques to layout problems can conveniently provide such guidelines although two layout specific algorithms develop scatter diagrams as well our proposal is significant for two reasons first mds packages for microcomputers and mainframes that generate low cost solutions are readily available second because non metric mds programs accept various forms of input data without restrictions allow different distance measures and can offer solutions in three dimensions they provide users with flexibility in the type of layout problems that can be solved

exploiting special structures in constructing a hierarchy of relaxations for 0 1 mixed integer problems a new hierarchy of relaxations is presented that provides a unifying framework for constructing a spectrum of continuous relaxations spanning from the linear programming relaxation to the convex hull representation for linear mixed integer 0 1 problems this hierarchy is an extension of the reformulation linearization technique rlt of sherali and adams 1990 1994a and is particularly designed to exploit special structures specifically inherent special structures are exploited by identifying particular classes of multiplicative factors that are applied to the original problem to reformulate it as an equivalent polynomial programming problem and subsequently this resulting problem is linearized to produce a tighter relaxation in a higher dimensional space this general framework permits us to generate an explicit hierarchical sequence of tighter relaxations leading up to the convex hull representation a similar hierarchy can be constructed for polynomial mixed integer 0 1 problems additional ideas for further strengthening rlt based constraints by using conditional logical implications as well as relationships with sequential lifting are also explored several examples are presented to demonstrate how underlying special structures including generalized and variable upper bounding covering partitioning and packing constraints as well as sparsity can be exploited within this framework for some types of structures low level relaxations are exhibited to recover the convex hull of integer feasible solutions

the impact of distribution system characteristics on computational tractability mathematical programming models that seek optimal design and operational plans for distribution systems can be computationally intractable this paper examines the extent to which distribution configuration and demand characteristics affect the ease of obtaining an optimal solution problem characteristics which are reflected in a model by the parameter values can render the model for one distribution scenario to be computationally intractable and that for another to yield an optimal solution easily we introduce echelon flow based valid inequalities and use them to explicate the extent to which problem characteristics impact computational tractability

numerical solution of singular perturbation problems via deviating arguments we present a new approach to numerically solving linear singularly perturbed two point boundary value problems in ordinary differential equations with a boundary layer on the left end of the interval the original problem is divided into outer and inner region problems a terminal boundary condition in the implicit form is derived then the outer region problem is solved as a two point boundary value problem tpbvp and an explicit terminal boundary condition is obtained in turn a new inner region problem is obtained and solved as a tpbvp using the explicit terminal boundary condition the proposed method is iterative on the terminal point of the inner region some numerical examples have been solved to demonstrate the applicability of the method

extensions to stats for practical applications of the facility layout problem abstract we consider a very general case of the facility layout problem which allows incorporating various aspects appearing in real life applications these aspects include loose requirements on facilities footprints each of which only needs to be of rectangular shape and can optionally be restricted concerning the surface area or the aspect ratio compared to former approaches other generalizations of practical relevance are multiple not necessarily rectangular workshops exclusion zones in workshops predefined positions of facilities the consideration of aisles and the adherence of further restrictions such as the enforced placement of certain facilities next to an exterior wall or a minimum distance between certain pairs of facilities although different objectives could be applied we especially focus on the most relevant one in practice the minimization of transportation costs we show that this problem can heuristically be solved using an extension of the slicing tree and tabu search stats based approach the application of this algorithm on practical data shows its effectiveness the paper concludes with a step by step guide for the application of stats in practice

a β accurate linearization method of euclidean distance for the facility layout problem with heterogeneous distance metrics abstract most existing research on facility layout problems flps considers a single distance metric mainly rectilinear distance in the calculation of the material handling cost between departments however there are many industrial cases in which heterogeneous distance metrics may need to be used simultaneously to cater for different styles of material handling such as the euclidean distance metric for conveyor belts and the tchebychev distance metric for overhead cranes in this paper we study the unequal area facility layout problem with heterogeneous distance metrics ua flp hdm considering a hybrid use of three metrics i e rectilinear euclidean and tchebychev as distance measures of different styles of material handling in the production system we propose a accurate linearization method that uses a set of tangent planes to convert the non linear euclidean distance constraint into a set of linear constraints that guarantee the approximation error within a given percentage e g as small as 0 01 in our experiments and develop linear constraints for the tchebychev distance metric as well based on these contributions we present a mixed integer linear programming milp model for the ua flp hdm computational experiments are carried out to test the performance of the milp model with five benchmark problems in the literature and compare the layout designs using different distance metrics numerical results indicate that different distance metrics may lead to significantly different solutions and a hybrid use of heterogeneous distance metrics fits better for real industrial applications