a gradient algorithm for chance constrained nonlinear goal programming abstract decision environments involve high degrees of uncertainty as well as multiple conflicting objectives often sampling information is available as a means of describing uncertainty this description can be utilized in the form of chance constraints goal programming offers a means of considering multiple conflicting objectives a nonlinear goal programming algorithm is presented based upon the gradient method utilizing an optimal step length for chance constrained goal programming models the resulting algorithm requires assumptions of convex solution sets differentiable and monotonic nonlinear constraints and normally distributed variance of stochastic parameters the algorithm is evaluated for generality reliability and precision sensitivity to parameters and data preparational and computational effort and convergence model sensitivity to parameters is identified as well as appropriate adjustment the algorithm was found to require minimal preparational effort favorable computation time and rapid convergence to optimal solution with the exception of models containing high degrees of nonlinearity

the numerical analysis of ordinary differential equations runge kutta and general linear methods mathematical and computational introduction the euler method and its generalizations analysis of runge kutta methods general linear methods bibliography

zero stability properties of the three ordinate variable stepsize variable formula methods consider the systemy f x y y a x a b y r n wheref is continuous and lipschitzian with respect to the second argument very often linear multistep variable stepsize variable formula methods lm vsvfm s are used to computey k y xk on the points of the grid a x 0 x1 x2 xn b the general lm vsvfm is based on formulae of the following type y k sum limits i 1 s k alpha i bar h k sk y k i sum limits i 0 s k h k i beta i bar h k s k f x k i y k i whereh k xk xk 1 bar h k h k h k 1 ldots h k s k s k k k 1 1 n the coefficients i and i depend on the lasts k 1 stepsizes and on the formula used at stepk only the zero stability properties of some special classes of lm vsvfm s as for example those based on adams formulae were investigated in the literature a class of three ordinate lm vsvfm s is defined in this paper some results concerning the zero stability properties of these methods are proved it is shown that some well known results are simple corollaries of the results found for the three ordinate lm vsvfm s it is easily seen that similar results hold for the corresponding one leg vsvfm s finally the use of the theoretical results in the practical implementations is briefly discussed

quasilinear multistep methods and variable step predictor corrector methods for neutral functional differential equations the theory of quasilinear multistep methods and predictor corrector methods for neutral functional differential equations is developed the theorem about the asymptotic behaviour of the global discretization error of quasilinear methods which forms a theoretical basis for step control strategy for predictor corrector methods for neutral equations is given some problems of implementation of these methods are also addressed and the results of numerical experiments are presented

a robust riks like path following method for strain actuated snap through phenomena in soft solids abstract in this paper a practical and robust riks like path following method is developed to automatically trace the nonlinear equilibrium path for initial finite strain induced snap through phenomena in soft solids the deformation gradient is decomposed into an elastic part and an initial strain part where the initial strain is characterized through a prescribed pattern tensor and an unknown strain factor describing the magnitude in the non linear finite element fe solution scheme an equivalent body force proportional to the increment of the strain factor is derived which varies in each iteration step due to its dependence on deformed configuration and stress both the displacement and the strain factor increments are introduced as unknowns in the linearized equation which permits to predict the critical equilibrium states before and after the snap through several examples with various configurations and strain histories are constructed including multilayer beams and film substrate systems which demonstrate the effectiveness and robustness of the algorithm to capture the strain induced snap through phenomena

the parameter space investigation method of multiple objective nonlinear programming a computational investigation in this paper we study the parameter space investigation method of i m sobol and r b statnikov the method is a random hitting procedure for solving the kinds of small but difficult multiple objective nonlinear programming problems often encountered in engineering design and optimal control sobol and statnikov have argued the effectiveness of their approach and its application possibilities in this paper we computationally test the parameter space investigation method to examine the extent to which claims about their method can be substantiated

stability restrictions on time stepsize for numerical integration of first order partial differential equations mathematical models described by partial differential equations appear often when different phenomena related to atmospheric transport for example advection and diffusion processes are studied truncated fourier series can be used in the discretization of the space derivatives resulting in a pseudospectral method in this way the partial differential equation is transformed into a system of ordinary differential equations this system is normally solved by so called step by step integration methods the stability properties of these methods are studied some bounds for the time stepsize are derived several formulae and predictor corrector schemes with large stability intervals on the imaginary axis are constructed numerical experiments with eight time integration algorithms are carried out some recommendations concerning the choice of the time integration algorithms in three different situations are given it is emphasized that in a package for the solution of partial differential equations describing transport processes in the atmospheric environment one should have several time integration algorithms each of which can be chosen optionally finally the possibility of using variable stepsize variable formula time integration algorithms is briefly discussed

implementation of a variable stepsize variable formula method in the time integration part of a code for treatment of long range transport of air pollutants abstract a mathematical model consisting of two partial differential equations is used to study the long range transport of sulphur di oxide and sulphate over europe the discretization of the first order space derivatives the advection terms is carried out by a pseudospectral fourier algorithm a special technique is applied in the discretization of the second order space derivatives the diffusion terms two large systems of ordinary differential equations are solved at each time step it is shown that these systems can efficiently be treated by a variable stepsize variable formula method vsvfm based on the use of predictor corrector schemes the stepsize selection strategy and the formula selection strategy are discussed in detail an attempt to carry out both an accuracy control and a stability control is made at each time step the great efficiency of the vsvfm implemented in our software as well as the reliability of the results are illustrated by numerical experiments in which real meteorological data for 1979 at the grid points of a space domain covering the whole of europe were used the main ideas implemented in the time integration part might be applied in many other situations where the systems of ordinary differential equations arising after the space discretization are only moderately stiff so that the stability requirements are dominant over the accuracy requirements on a large part of the time interval but the use of implicit time integration algorithms that require solving systems of algebraic equations at each time step is not justified as an illustration only it should be mentioned that such an application has been carried out in connection with models describing long range transport of nitrogen pollutants over europe

how to select and how to rank projects the promethee method abstract in this paper we present the promethee methods a new class of outranking methods in multicriteria analysis their main features are simplicity clearness and stability the notion of generalized criterion is used to construct a valued outranking relation all the parameters to be defined have an economic signification so that the decision maker can easily fix them two ways of treatment are proposed it is possible to obtain either a partial preorder promethee i or a complete one promethee ii both on a finite set of feasible actions a comparison is made with the electre iii method the stability of the results given by the two methods is analysed numerical applications are given in order to illustrate the properties of the new methods and some further problems are discussed

some computational aspects of the stability analysis of nonlinear structures abstract the perturbation method and the continuation method are the two most popular techniques for the solution of finite element equations that describe instability phenomena this paper presents a summary of the principles involved and describes some trends in the development of these techniques

istmo an interval reference point based method for stochastic multiobjective programming problems in this paper we present an interactive algorithm istmo for stochastic multiobjective problems with continuous random variables this method combines the concept of probability efficiency for stochastic problems with the reference point philosophy for deterministic multiobjective problems the decision maker expresses her his references by dividing the variation range of each objective into intervals and by setting the desired probability for each objective to achieve values belonging to each interval these intervals may also be redefined during the process this interactive procedure helps the decision maker to understand the stochastic nature of the problem to discover the risk level s he is willing to assume for each objective and to learn about the trade offs among the objectives

a sampling based method for generating nondominated solutions in stochastic momp problems chapters 4 and 6 presented methods for incorporating risk in the project selection optimization problem this chapter presents a method for generating nondominated solutions for stochastic multiobjective mathematical programming problems that is applicable to both continuous and zero one variables the method is based on the assumption that the objective function coefficients are random variables with probability distributions that are known or can be approximated the method results in solutions that are nondominated in terms of the expected value of each objective and the probability that each objective meets or exceeds a specified target value a method for generating a set of such solutions is presented and illustrated with examples the method presented here is more general in that it allows for any probability distribution it makes no behavioral assumptions about the decision maker it can accommodate integer zero one or real valued variables and it can accommodate linear or nonlinear constraints and objective functions

solution approaches for the multiobjective stochastic programming abstract we survey in this paper various solution approaches for multiobjective stochastic problems where random variables can be in both objectives and constraints parameters once a problem requires a stochastic formulation a first step consists in transforming the problem into its deterministic formulation we propose to classify and evaluate such transformations with regards to the many proposed concepts of efficiency the paper addresses also some applications of the multiobjective stochastic programming models

stochastic approach versus multiobjective approach for obtaining efficient solutions in stochastic multiobjective programming problems abstract in this work we deal with obtaining efficient solutions for stochastic multiobjective programming problems in general these solutions are obtained in two stages in one of them the stochastic problem is transformed into its equivalent deterministic problem and in the other one some of the existing generating techniques in multiobjective programming are applied to obtain efficient solutions which involves transforming the multiobjective problem into a problem with only one objective function our aim is to determine whether the order in which these two transformations are carried out influences in any way the efficient solution obtained our results show that depending on the type of stochastic criterion followed and the statistical characteristics of the initial problem the order can have an influence on the final set of efficient solutions obtained for a given problem

stability of multistep methods on variable grids this paper is concerned with the stability of multistep methods for ordinary initial value problems on grids with variable mesh sizes a necessary and sufficient condition for stability is given from which generalizations of recent results by gear et al and by zlatev can be obtained as special cases as an application the stability of the variable bdf formulas is treated

mathematical model for studying the sulphur pollution over europe the long range transport of sulphur pollutants in the atmosphere is an important environmental problem a mathematical model consisting of a system of partial differential equations has been developed at the danish agency of environmental protection for studying this phenomenon the model is handled numerically a pseudospectral fourier algorithm is used in the space discretization phase a variable stepsize variable formula method vsvfm based on predictor corrector schemes may be specified during the time discretization part the time integration formulae are especially constructed so that a the predictor corrector schemes have good absolute stability properties on the imaginary axis and b the fundamental properties consistency zero stability and convergence of the global vsvfm are not affected by the changes of the stepsize and or the predictor corrector scheme both the accuracy requirements and the stability requirements are taken into account in the stepsize selection strategy as well as in the formula selection strategy this double check leads to a great efficiency in the numerical treatment of problems arising in atmospheric environments this is demonstrated by many numerical results the question of the reliability of the results obtained in runs of real physical problems is discussed some results obtained in the comparison of calculated results with measurements are reported an illustration where the model is applied in studying the air pollution in the area around denmark due to danish sources is presented

numerical solution of ordinary differential equations is there anything left to do two decades ago many people thought that there was very little left to occupy the researcher in the numerical initial value problem for ordinary differential equations but there has since been an enormous increase in the effort devoted to this problem and related software is everything finally done so we can turn to more difficult areas this paper examines the development of numerical solution techniques from the identification of a problem to the never final preparation of automatic codes for the solution of classes of similar problems it discusses a number of ode problems that are only a part of the way along this path of development these problems include ones with highly oscillatory solutions ones with frequent discontinuities and ones in which very little accuracy is needed and which should be integrated at low cost there is a spectrum of problems from ones that can now be solved using very robust software to ones that we still do not know how to solve effectively the paper then surveys s

comparison of methodologies for fuzzy and stochastic multi objective programming abstract methods for solving a multicriteria linear program with coefficients of the objective functions and the constraints being flat fuzzy numbers and those dealing with a stochastic multicriteria linear program where imprecision of some data is modelled by probability distributions are surveyed the methodologies are compared and evaluated some ideas for future research are emphasized

satisfactory solution concepts and their relations for stochastic multiobjective programming problems abstract this work deals with the concept of satisfactory solution for stochastic multiobjective programming smp problems based on previous literature we will introduce different concepts of satisfactory solutions for smp problems define a new concept of solution where the decision maker dm sets his her preferences in terms of two aspiration levels for the stochastic objective and two probabilities to reach those levels and establish some relationship between these concepts the results will aim at featuring these concepts and determine the differences between them moreover the paper proposes a new step by step procedure to exchange information between the analyst and dm prior to solving the problem thus the dm will be able to choose the transformation criterion for each stochastic objective and the aspiration level

revising the owa operator for multi criteria decision making problems under uncertainty abstract multi criteria decision making mcdm problems are usually under uncertainty one of these uncertain parameters is the decision maker dm s degree of optimism which has an important effect on the results fuzzy linguistic quantifiers are used to obtain the assessments of this parameter from dm and then because of its uncertainty it is assumed to have stochastic nature a new approach entitled fsrowa is introduced to combine the fuzzy and stochastic features into a revised owa operator if the dm is not aware of the risk in decision then the decision objective is to maximize the expected combined goodness measure if the dm cares only about the risk then minimizing the variance of the combined goodness measure is his her objective the results of the fsrowa provide the expected value and the variance of the combined goodness measure for each alternative in order to combine these two attributes a composite goodness measure is introduced this measure gives more robust decision outcomes the theoretical results are illustrated in a watershed management problem

the convergence of variable stepsize variable formula multistep methods in this paper variable stepsize variable formula multistep methods for the numerical solution of ordinary differential equations are considered some more general results on stability than are given in the papers of gear tu siam j numer anal 11 1974 pp 1025 1043 and gear watanabe siam j numer anal 11 1974 pp 1044 1058 are obtained further the asymptotic behavior of the error in the variable stepsize fixed formula case is studied finally the above study is extended to predictor corrector methods

distributional efficiency in multiobjective stochastic linear programming several concepts of distributional efficiency are proposed for the multiobjective stochastic linear programming mslp problem in contexts where the probability distribution of random parameters is known and the decision maker dm has an unknown multi attribute utility function belonging to a given glass u we present a general efficient set the u admissible solutions and two subsets the u unanimous and u advocated solutions the latter being particularly relevant to the case of a single dm we show how advocated solutions can be generated and or tested when u is the class of non decreasing additive concave functions

serial correlation estimation through the imprecise goal programming model abstract the goal programming gp model was used as a time series analysis tool that incorporates a serial correlation where the dependent variable is considered as precise this formulation does not take into consideration the decision maker s preferences however the dependent variable can be imprecise and its value can be expressed through an interval the aim of this paper is to develop a new formulation of the gp model for regression with serial correlation where the dependent variable is imprecise the proposed model will also integrate explicitly the decision maker s preferences a numerical example was used to illustrate our model

optimization with stochastic preferences based on a general class of scalarization functions it is of crucial importance to develop risk averse models for multicriteria decision making under uncertainty a major stream of the related literature studies optimization problems that feature multivariate stochastic benchmarking constraints these problems typically involve a univariate stochastic preference relation often based on stochastic dominance or a coherent risk measure such as conditional value at risk which is then extended to allow the comparison of random vectors by the use of a family of scalarization functions all scalarized versions of the vector of the uncertain outcomes of a decision are required to be preferable to the corresponding scalarizations of the benchmark outcomes while this line of research has been dedicated almost entirely to linear scalarizations the corresponding deterministic literature uses a wide variety of scalarization functions that among other advantages offer a high degree of modeling flexibility in this paper we aim to incorporate these scalarizations i

a generalized stochastic goal programming model in this paper we show how one can get stochastic solutions of stochastic multi objective problem smop using goal programming models in literature it is well known that one can reduce a smop to deterministic equivalent problems and reduce the analysis of a stochastic problem to a collection of deterministic problems the first sections of this paper will be devoted to the introduction of deterministic equivalent problems when the feasible set is a random set and we show how to solve them using goal programming technique in the second part we try to go more in depth on notion of smop solution and we suppose that it has to be a random variable we will present stochastic goal programming model for finding stochastic solutions of smop our approach requires more computational time than the one based on deterministic equivalent problems due to the fact that several optimization programs which depend on the number of experiments to be run needed to be solved on the other hand since in our approach we suppose that a smop solution is a random variable according to the central limit theorem the larger will be the sample size and the more precise will be the estimation of the statistical moments of a smop solution the developed model will be illustrated through numerical examples

the stability of variable stepsize nordsieck methods conditions are given for the stability of multistep methods that vary the stepsize using the interpolation technique of nordsieck if the stepsize selection function is variation bounded then these methods are stable alternatively there exist constants a b c depending on the method and satisfying a leqq b 1 c such that if the stepsize ratio h n 1 h n satisfies either 0 h n 1 h n leqq a or b h n 1 h n leqq c then the method is stable as the frequency of stepsize changing decreases the methods become more stable in the sense that the intervals defined by a b and c are larger tables and graphs are given for the adams and backward differentiation methods indicating how stability changes with frequency of changing stepsize in particular it is shown that k 1 value adams formulas k leqq 6 are stable for 0 h n 1 h n leqq c that is it is shown that stability is maintained whenever the stepsize is decreased by an arb

dominance conditions for multivariate utility functions stochastic dominance conditions are given for n variate utility functions when k variate risk aversion is assumed for k 1 2 n these conditions are expressed through a comparison of distribution functions as in the well known univariate case and through a comparison of random variables defined on the same probability space

high order predictor corrector algorithms new predictor corrector algorithms are presented for the computation of solution paths of non linear partial differential equations the predictors and the correctors are based on perturbation techniques and pade approximants this extends the asymptotic numerical method anm which is an efficient high order continuation technique without corrector the efficiency and the reliability of the new technique are assessed by several examples within thin shell theory and navier stokes equations many variants have been tested to establish an optimal algorithm

a nonlinear finite element model for the stress analysis of soft solids with a growing mass abstract the paper presents a new finite element fe model for the stress analysis of soft solids with a growing mass based on the work of lubarda and hoger 2002 contrary to the traditional numerical methods emphasizing on the influence of growth on constitutive equations an equivalent body force is firstly detected which is resulted from the linearization of the nonlinear equation and acts as the driver for material growth in the numerical aspect in the algorithm only minor correction on the traditional tangent modulus is needed to take the growth effects into consideration and its objectivity could be guaranteed comparing with the traditional method to solve the resulted equation in time domain both explicit and implicit integration algorithms are developed where the growth tensor is updated as an internal variable of gauss point the explicit updating scheme shows higher efficiency while the implicit one seems to be more robust and accurate the algorithm validation and its good performance are demonstrated by several two dimensional examples including free growth constrained growth and stress dependent growth

strange an interactive method for multi objective linear programming under uncertainty abstract in the field of investment planning within a time horizon problems typically involve multiple objectives and basic data are uncertain in a large number of cases these decision problems can be written as linear programming problems in which time dependent uncertainties affect the coefficients and the right hand side of constraints given the possibility of defining plausible scenarios on basic data discrete sets of such coefficients are given each with its subjective probability of occurrence the corresponding structure is then characteristic for multi objective stochastic linear programming moslp in the paper an interactive procedure is described to obtain a best compromise for such a moslp problem this algorithm called strange extends the stem method to take the random aspects into account it involves in particular the concepts of stochastic programming with recourse in its interactive steps the efficiency projection techniques are used to provide the decision maker with detailed graphical information on efficient solution families as an illustration of the successive steps a didactic example is solved in some detail and the results of a case study in energy planning are given

advance in the theory of variable stepsize variable formula methods for ordinary differential equations the theory of the constant stepsize constant formula methods cscfm s for solving initial value problems for ordinary differential equations ode s is well developed however the integration methods are often used as variable stepsize variable formula methods vsvfm s in the codes for automatic solution of ode s this fact explains why it is necessary to develop a rigorous theory for the vsvfm s also results obtained until now in the efforts to develop a theory for the vsvfm s are presented in a systematic way and compared with the well known results for cscfm s the comparison is used to outline the directions in which further efforts in the attempt to improve the theoretical results and to obtain new results are most needed the integration methods can be separated into two large groups implicit methods and explicit methods including here the predictor corrector schemes it is discussed what kind of results are needed for each of these two groups assume that a set f of cscfm s is given from a practical point of view it is highly desirable that the cscfm s belonging to set f have the following property if every cscfm from set f is consistent zero stable and convergent then any vsvfm induced by the cscfm s of set f is also consistent zero stable and convergent for a sufficiently large class of grids a theorem concerning a class of cscfm s with this property is presented the application of this theorem in the selection of methods for solving large systems of ode s arising after the space discretization of air pollution models is discussed it is explained why the vsvfm s based on this class provide a very efficient computational process in the treatment of air pollution models

an interactive fuzzy satisficing method for multiobjective stochastic linear programming problems through an expectation model abstract for decision making problems involving uncertainty both stochastic programming as an optimization method based on the theory of probability and fuzzy programming representing the ambiguity by fuzzy concept have been developing in various ways in this paper we focus on multiobjective linear programming problems with random variable coefficients in objective functions and or constraints for such problems as a fusion of these two approaches after incorporating fuzzy goals of the decision maker for the objective functions we propose an interactive fuzzy satisficing method for the expectation model to derive a satisficing solution for the decision maker an illustrative numerical example is provided to demonstrate the feasibility of the proposed method

an arc length method including line searches and accelerations this paper describes a method for introducing line searches into the arc length solution procedure such line searches may be used at each iteration to calculate an optimum scalar step length which scales the normal iterative vector in practice a loose tolerance is provided so that on many iterations the line searches are avoided however on difficult iterations the line searches are shown to lead to a substantial improvement in the convergence characteristics a simple single parameter acceleration is also developed using line search concepts the new arc length method is applied to both the geometrically nonlinear analysis of shallow shells and the materially nonlinear analysis of reinforced concrete beams and slabs significant improvements are demonstrated in relation to the standard arc length method

on the mechanics of solids with a growing mass a general constitutive theory of the stress modulated growth of biomaterials is presented with a particular accent given to pseudo elastic soft living tissues the governing equations of the mechanics of solids with a growing mass are revisited within the framework of finite deformation continuum thermodynamics the multiplicative decomposition of the deformation gradient into its elastic and growth parts is employed to study the growth of isotropic transversely isotropic and orthotropic biomaterials an explicit representation of the growth part of the deformation gradient is given in each case which leads to an effective incremental formulation in the analysis of the stress modulated growth process the rectangular components of the instantaneous elastic moduli tensor are derived corresponding to selected forms of the elastic strain energy function physically appealing structures of the stress dependent evolution equations for the growth induced stretch ratios are proposed

a compromise solution for the multiobjective stochastic linear programming under partial uncertainty abstract this paper solves the multiobjective stochastic linear program with partially known probability we address the case where the probability distribution is defined by crisp inequalities we propose a chance constrained approach and a compromise programming approach to transform the multiobjective stochastic linear program with linear partial information on probability distribution into its equivalent uniobjective problem the resulting program is then solved using the modified l shaped method we illustrate our results by an example