optimality conditions for linear programming problems with fuzzy coefficients the optimality conditions for linear programming problems with fuzzy coefficients are derived in this paper two solution concepts are proposed by considering the orderings on the set of all fuzzy numbers the solution concepts proposed in this paper will follow from the similar solution concept called the nondominated solution in the multiobjective programming problem under these settings the optimality conditions will be naturally elicited

weighted coefficients in two phase approach for solving the multiple objective programming problems abstract two phase approach with equal weighted coefficients has been proposed to yield an efficient solution for multiple objective programming problems in this note we will show that the two phase approach as long as the weighted coefficients are positive not necessarily equal will generate an efficient solution a counterexample is given to the case that some weighted coefficients are zero

bias reduction in exponential family nonlinear models in firth 1993 biometrika it was shown how the leading term in the asymptotic bias of the maximum likelihood estimator is removed by adjusting the score vector and that in canonical link generalized linear models the method is equivalent to maximizing a penalized likelihood that is easily implemented via iterative adjustment of the data here a more general family of bias reducing adjustments is developed for a broad class of univariate and multivariate generalized nonlinear models the resulting formulae for the adjusted score vector are computationally convenient and in univariate models they directly suggest implementation through an iterative scheme of data adjustment for generalized linear models a necessary and sufficient condition is given for the existence of a penalized likelihood interpretation of the method an illustrative application to the goodman row column association model shows how the computational simplicity and statistical benefits of bias reduction extend beyond generalized linear models

singularities of energy minimizing maps from the ball to the sphere we study maps from the unit ball b in r3 to the unit sphere s2 in r3 which minimize dirichlet s energy integral varepsilon varphi int b nabla varphi 2 dv

confidence intervals for multinomial logistic regression in sparse data logistic regression is one of the most widely used regression models in practice but alternatives to conventional maximum likelihood estimation methods may be more appropriate for small or sparse samples modification of the logistic regression score function to remove first order bias is equivalent to penalizing the likelihood by the jeffreys prior and yields penalized maximum likelihood estimates ples that always exist even in samples in which maximum likelihood estimates mles are infinite ples are an attractive alternative in small to moderate sized samples and are preferred to exact conditional mles when there are continuous covariates we present methods to construct confidence intervals ci in the penalized multinomial logistic regression model and compare ci coverage and length for the ple based methods to that of conventional mle based methods in trinomial logistic regressions with both binary and continuous covariates based on simulation studies in sparse data sets we recommend profile cis over asymptotic wald type intervals for the ples in all cases furthermore when finite sample bias and data separation are likely to occur we prefer ple profile cis over mle methods

pictorial representations of fuzzy connectives part ii cases of compensatory operators and self dual operators abstract as the continuation of part i in which pictorial representations of t norms t conorms and averaging operators were made with the aid of a computer this paper discusses compensatory operators and proposes generalized compensatory operators which can be obtained from averaging operators self dual operators are discussed which can be defined by using t norms t conorms and averaging operators finally symmetric sums are reviewed which are also self dual operators the pictorial representations of these operators are also made

multi objective optimization with a max t norm fuzzy relational equation constraint in this paper we consider minimizing multiple linear objective functions under a max t norm fuzzy relational equation constraint since the feasible domain of a max archimedean t norm relational equation constraint is generally nonconvex traditional mathematical programming techniques may have difficulty in yielding efficient solutions for such problems in this paper we apply the two phase approach utilizing the min operator and the average operator to aggregate those objectives to yield an efficient solution a numerical example is provided to illustrate the procedure

interactive multiple objective programming subject to flexible constraints abstract an interactive decision support system is introduced which aids in solving multiple objective programming problems subject to strict and flexible constraints integral part is an extension of a well known fuzzy sets approach evaluating possible solutions by their degrees of membership to objectives and constraints this approach is linked to classical multiple objective programming models if the decision maker cannot determine membership functions a priori the system suggests functions dependent on the given information and interactive modifications are allowed

φ 1 concavity and fuzzy multiple objective decision making concave properties play a dominate role in solving both classic and fuzzy optimization problems however since fuzzy problems are generally represented by sets not crisp numbers various aggregation schemes are needed to manipulate and to combine the different elements in a fuzzy optimization problem based on these different aggregations various concavity properties can be formulated and explored in this paper the intersection aggregation and the convex combination aggregation are explored based on the email protected 1 concave fuzzy sets first the concept of f 1 convexity which covers a wider class of sets and functions is extended to fuzzy sets email protected 1 concave and email protected 1 quasiconcave fuzzy sets are then introduced and some useful aggregation and composition rules are developed based on these aggregation and composition rules and the generalized concave properties fuzzy multiple objective decision making problems are formulated and the conditions to ensure local global maximum property are discussed

jackknife bias reduction for polychotomous logistic regression despite theoretical and empirical evidence that the usual mles can be misleading in finite samples and some evidence that bias reduced estimates are less biased and more efficient they have not seen a wide application in practice one can obtain bias reduced estimates by jackknife methods with or without full iteration or by use of higher order terms in a taylor series expansion of the log likelihood to approximate asymptotic bias we provide details of these methods for polychotomous logistic regression with a nominal categorical response we conducted a monte carlo comparison of the jackknife and taylor series estimates in moderate sample sizes in a general logistic regression setting to investigate dichotomous and trichotomous responses and a mixture of correlated and uncorrelated binary and normal covariates we found an approximate two step jackknife and the taylor series methods useful when the ratio of the number of observations to the number of parameters is greater than 15 but we cannot recommend the two step and the fully iterated jackknife estimates when this ratio is less than 20 especially when there are large effects binary covariates or multicollinearity in the covariates

fuzzy multiple objective programming and compromise programming with pareto optimum abstract a fuzzy multiple objective decision making approach is proposed based on the desirable features of compromise programming and the fuzzy set theory the proposed two phase approach guarantees both nondominated and balanced solutions for solving both the crisp and the fuzzy multiple objective decision making problems furthermore it is shown that the original compromise programming does not guarantee nondominated solutions if the distance parameter is assumed to be infinite and the resulting programming problem s solution is not unique

expected value of fuzzy variable and fuzzy expected value models this paper will present a novel concept of expected values of fuzzy variables which is essentially a type of choquet integral and coincides with that of random variables in order to calculate the expected value of general fuzzy variable a fuzzy simulation technique is also designed finally we construct a spectrum of fuzzy expected value models and integrate fuzzy simulation neural network and genetic algorithms to produce a hybrid intelligent algorithm for solving general fuzzy expected value models

an integrated approach for deriving priorities in analytic network process a multiple objective programming approach for the analytic network process anp is proposed to obtain all local priorities for crisp or interval judgments at one time even in an inconsistent situation the weakness of the anp and fuzzy anp fanp is that the complexity of generating priorities is equal to the number of comparison matrices in the proposed approach all sets of crisp priorities for each pairwise comparison matrix can be obtained directly moreover from the outcomes of three examples the power to reach a limiting supermatrix is less than or equal to the power of the fanp thus the proposed approach can be regarded as an efficient alternative of the fuzzy anp

an interactive fuzzy satisficing method for multiobjective nonlinear programming problems with fuzzy parameters abstract this paper presents an interactive fuzzy satisficing method for multiobjective nonlinear programming problems with fuzzy numbers on the basis of the level sets of the fuzzy numbers the concept of multiobjective nonlinear programming and local m pareto optimality is introduced through interaction with the decision maker dm the fuzzy goals of the dm for each of the objective functions in multiobjective nonlinear programming are quantified by eliciting the corresponding membership functions after determining the membership functions in order to generate a candidate for the satisficing solution which is also locally m pareto optimal if the dm specified the degree of the level sets and the reference membership values the augmented minimax problem is solved and the dm is supplied with the corresponding local m pareto optimal solution together with the trade off rates among the values of the membership functions and the degree then by considering the current values of the membership functions and as well as the trade off rates the dm responds by updating his reference membership values and or the degree in this way the local satisfying solution for the dm can be derived efficiently from among an m pareto optimal solution set based on the proposed method an interactive computer program has been written and an illustrative numerical example is given along with the corresponding computer outputs

regularity of harmonic maps to trees we derive regularity theorems and local finiteness results in this paper we first prove the regularity type theorem for harmonic maps to r trees that is when the order of a harmonic map at a point is sufficiently close to 1 then this point is a regular point we then prove a regularity theorem near higher order points we show that if the image of a fixed ball of a properly normalized harmonic map is sufficiently close to a subtree then the image of a smaller ball of this map lies in that subtree as an application we prove that the local image of a harmonic map to an r tree lies in a finite homogeneous subtree

fuzzy programming with fuzzy decisions and fuzzy simulation based genetic algorithm this paper provides a spectrum of chance constrained programming as well as chance constrained multiobjective programming and chance constrained goal programming with fuzzy rather than crisp decisions which will seek a fuzzy set from the given reference collection as an optimal solution the technique of fuzzy simulation is also presented to check fuzzy chance constraints and to handle fuzzy objective and goal constraints finally a fuzzy simulation based genetic algorithm for solving these models will be designed and illustrated by some numerical examples

improving small sample inference in group randomized trials with binary outcomes group randomized trials grts randomize groups of people to treatment or control arms instead of individually randomizing subjects when each subject has a binary outcome over dispersed binomial data may result quantified as an intra cluster correlation icc typically grts have a small number bin of independent clusters each of which can be quite large treating the icc as a nuisance parameter inference for a treatment effect can be done using quasi likelihood with a logistic link a wald statistic which under standard regularity conditions has an asymptotic standard normal distribution can be used to test for a marginal treatment effect however we have found in our setting that the wald statistic may have a variance less than 1 resulting in a test size smaller than its nominal value this problem is most apparent when marginal probabilities are close to 0 or 1 particularly when n is small and the icc is not negligible when the icc is known we develop a method for adjusting the estimated standard error appropriately such that the wald statistic will approximately have a standard normal distribution we also propose ways to handle non nominal test sizes when the icc is estimated we demonstrate the utility of our methods through simulation results covering a variety of realistic settings for grts

an interactive fuzzy satisficing method for generalized multiobjective linear programming problems with fuzzy parameters abstract a new interactive fuzzy satisficing method for multiobjective linear programming problems with fuzzy parameters is proposed in general two types of fuzziness of human judgements should be incorporated in multiobjective programming problems one is the experts ambiguous understanding of the nature of the parameters in the problem formulation process and the other is the fuzzy goals of the decision maker for each of the objective functions in order to cope with both types of fuzziness multiobjective linear programming problems with fuzzy parameters which reflect the experts ambiguous understanding in the problem formulation are formulated and the concept of generalized multiobjective linear programming and m pareto optimality is introduced in our interactive fuzzy satisficing method the satisficing solution of the decision maker is derived efficiently from among m pareto optimal solutions on the basis of the proposed method an interactive computer program is written to implement man machine interactive procedures finally an illustrative numerical example for multiobjective linear programming problems with fuzzy parameters is demonstrated along with the corresponding computer outputs

a two phase possibilistic linear programming methodology for multi objective supplier evaluation and order allocation problems in this study an integration of the analytic hierarchy process ahp and a multi objective possibilistic linear programming moplp technique is developed to account for all tangible intangible quantitative and qualitative factors which are used to evaluate and select suppliers and to define the optimum order quantities assigned to each a multi objective linear programming technique is first employed to solve the problem to model the uncertainties encountered in the integrated supplier evaluation and order allocation methodology fuzzy theory is adopted hence possibilistic linear programming plp is proposed for solving the problem as it is believed to be the best approach for absorbing the imprecise nature of the real world in the supplier evaluation phase environmental criteria are also considered

evaluate fuzzy optimization problems based on biobjective programming problems abstract the embedding theorem shows that the set of fuzzy numbers can be embedded into a banach space inspired by this embedding theorem we propose a solution concept of the fuzzy optimization problem by solving a biobjective optimization problem which is transformed from the original fuzzy optimization problem using the embedding theorem we then also consider the fuzzy optimization problem with fuzzy coefficients i e the coefficients are assumed to be fuzzy numbers under a setting of core values of fuzzy numbers we show that the optimal solution of its corresponding crisp optimization problem the usual optimization problem is also an optimal solution of the original fuzzy optimization problem

fuzzy multiple level programming the use of fuzzy set concept to overcome the computational difficulties of multi level optimization for large systems appears to be a powerful approach and can solve practical large problems in a very simplified manner furthermore unlike the duo ploy or multi ploy approaches the proposed interactive concept provides flexibility which is essential due to the different styles of management in different organizations the approach is formulated based on exploring the fuzziness or indefiniteness which is almost always present in a decentralized large hierarchy organization the advantage of this approach is that not only the problem is much more simplified for solution purposes the original problem is also much more realistically represented

on the blow up of heat flow for conformal 3 harmonic maps using a comparison theorem chang ding and ye 1992 proved a finite time derivative blow up for the heat flow of harmonic maps from d 2 a unit ball in r 2 to s 2 a unit sphere in r 3 under certain initial and boundary conditions we generalize this result to the case of 3 harmonic map heat flow from d 3 to s 3 in contrast to the previous case our governing parabolic equation is quasilinear and degenerate technical issues such as the development of a new comparison theorem have to be resolved

a fuzzy control model fcm for dynamic portfolio management abstract in the present study we continue previous work by recasting the portfolio management problem into a control theoretic form still retaining a recursive programming structure through time the uncertainty of future prices and risk levels is recognized by fuzzy numbers our system has interesting directions for future research improving performance through expert systems support for forecasting using the experience from the model in efforts to theory design solving the fuzzy control model fcm when the state objectives return predictions and the level of economic friction are fuzzy

an interactive fuzzy programming system abstract an interactive system is introduced which supports a decision maker in solving programming models with crisp or fuzzy constraints and crisp or fuzzy goals one part of the system is the determination of membership functions representing goals to this purpose fuzzy extreme solutions are computed and are presented to the decision maker these and each of the proposed compromise solutions are fuzzy efficient

fuzzy approaches to multicriteria de novo programs abstract multicriteria de novo programming is a promising tool for optimal system design however there exist no algorithms for solving a general multicriteria de novo program only special cases have been discussed this paper proposes a two step fuzzy approach based on the ideal and negative ideal solutions it is shown that this approach is very efficient and is applicable to the general multicriteria de novo programming the fuzzy version of this problem is also formulated and analyzed numerical examples are given to illustrate the approaches

fuzzy goal programming with different importance and priorities abstract this paper formulates fuzzy goal programming fgp incorporating different importance and preemptive priorities by using an additive model to maximize the sum of achievement degrees of all fuzzy goals in contrast to previous works the proposed approach allows the decision maker to determine a desirable achievement degree for each fuzzy goal to reflect explicitly the relative importance of these goals this approach can generate a set of achievement degrees consistent with the decision maker s expectations even though the relative importance of the goals may change furthermore in this paper we incorporate the decision maker s preemptive priority structure into a single formulation the resulting solutions satisfy both the preemptive priority structure and have the maximum achievement degrees in sum the proposed approaches effectiveness and computational superiority over the existing approaches are demonstrated and compared with examples from the literature

fuzzy convexity and multiobjective convex optimization problems in this paper based on the more restrictive definition of fuzzy convexity due to ammar and metz 1 several useful composition rules are developed the advantages in using the more restrictive definition of fuzzy convexity are that local optimality implies global optimality and that any convex combination of such convex fuzzy sets is also a convex fuzzy set as shown in this paper these properties are laking in the usual convex fuzzy sets in addition to illustrate the applications in fuzzy convex optimization two examples in multiple objective programming are considered

an interactive algorithm for decomposing the parametric space in fuzzy multiobjective dynamic programming problem the aim of this chapter is to study the stability of multi objective dynamic programming modp problems with fuzzy parameters in the objective functions and in the constraints these fuzzy parameters are characterized by fuzzy numbers for such problems the concept and notion of the stability set of the first kind in parametric nonlinear programming problems are redefined and analyzed qualitatively under the concept of pareto optimality an interactive fuzzy decision making algorithm for the determination of any subset of the parametric space that has the same corresponding pareto optimal solution is proposed a numerical example is given to illustrate the method developed in the chapter

simulation of a supply chain game with multiple fuzzy goals supply chain management scm issues have been popularly discussed in recent years however few papers in past literature have been dedicated to the use of fuzzy multi objective game framework for analyzing how the strategic partnership influences individual and global performance in a supply chain this study formulates a game framework for the investigation of strategic behavior of supply chain partners based on fuzzy multi objective programming the alliance matrix and achievement level aspiration degree this work assumes the partners as having multiple objectives and the supply chain is multi product and multi material to validate our model we simulate the micro and macro performance of a supply chain by the achievement level under various partner alliances in addition a sensitivity analysis is conducted finally some useful lessons are summarized

fuzzy φ convexity and fuzzy decision making the notion of convexity and its various generalization is important for quantitative and qualitative studies in operations research or applied mathematics it has also been considered by many authors in fuzzy set theory in this paper we study the concept of f convex and f quasiconvex fuzzy sets which was proposed bychen et al 1 and develop some useful extrema properties of these fuzzy sets we prove that any local maximizer of a f convex fuzzy set is also a global maximizer and that any strict local maximizer of a f quasiconvex fuzzy set is also a global maximizer we also study the class of strictly f convex respectively strictly f quasiconvex fuzzy sets that is more restricted than the class of f convex respectively f quasiconvex fuzzy sets we prove for both families of strictly f convex and strictly f quasiconvex fuzzy sets that every local maximizer is also the unique global maximizer in addition some applications to fuzzy decision making are discussed

interactive compensatory fuzzy programming for decentralized multi level linear programming dmllp problems abstract this paper presents interactive compensatory fuzzy programming for decentralized multi level linear programming dmllp problems by adjusting the cooperative decision making process between the different levels and also between the decision makers of the same level our aim is to obtain a preferred compensatory compromise pareto optimal solution for dmllp for this the weights of objectives at each level are assigned by the next upper level decision maker dm by using analytic hierarchy process ahp or any other weighting methods the weight of any objective for whole system is equal to the product of the weights on the path tying it to the top decision maker dm 0 using these weights equivalence is established such that the satisfactory levels of all objectives are proportional to their own weights werners compensatory fuzzy and operator is offered to solve dmllp problem the most important idea to be emphasized is that equivalence is established such that the satisfactory levels of all objectives are proportional to their own weights thanks to this equivalence dmllp problem has been transformed to the multi objective linear programming molp problem at level 0 the equivalence is reflected to the compensatory model within the constraints and the equivalence also enables all dms to obtain proportional satisfactions with their weights as much as possible so in our compensatory model a reduction on equivalent satisfactory level of one dm can be compensated for by an increase in the equivalent satisfactory level of another dm furthermore being developed a finite interactive iterative procedure with maximum interaction step a set of compensatory solutions which are also pareto optimal is obtained depending on compensation parameter giving a theorem we will show that the solutions generated by werners compensatory fuzzy and operator do guarantee pareto optimality for our dmllp problem and comparing it with some other computational efficient compensatory fuzzy aggregation operators we will conclude that this operator is more appropriate for dmllp illustrative numerical example is provided to demonstrate the feasibility and efficiency of the proposed interactive fuzzy compensatory method for dmllp

a parametric approach to fuzzy linear programming in this paper we discuss how to deal with decision problems that are described with lp models and formulated with elements of imprecision and uncertainty more precisely we will study lp models in which the parameters are not fully known but only with some degree of precision even with incomplete information the model builder or model user is normally able to give a realistic interval for the parameters of an lp model for the constraint vector this is combined with some wishes or some leeway on the constraints even with ambiquity in the objective function there is normally some preference ordering to be found among alternative ways of action we will demonstrate that these modelling complications can be handled with the help of some results developed in the theory of fuzzy sets after an overview of some central contributions to fuzzy linear programming we will develop an lp model in which the parameters are not fully known only with some degree of precision and show that the model can be parametrised in such a way that the optimal solution becomes a function of the degree of precision the fuzzy lp model derived in this way appears to be fairly easy to handle computationally which is demonstrated with a numerical example

interdependence in fuzzy multiple objective programming abstract in multiple objective programs mop application functions are establised to measure the degree of fulfillment of the decision maker s requirements achievement of goals nearness to an ideal point satisfaction etc about the objective functions see e g delgado et al 1990 zimmermann 1978 and are extensively used in the process of finding good compromise solutions we have earlier demonstrated that the use of interdependences among objectives of a mop in the definition of the application functions provides for more correct solutions and faster convergence in this paper generalizing the principle of application functions to fuzzy multiple objective programs fmop with interdependent objectives we define a large family of application functions for fmop and illustrate our ideas by a simple three objective program

fuzzy goal programming with multiple priorities via generalized varying domain optimization method this paper proposes a generalized varying domain optimization method for fuzzy goal programming incorporating multiple priorities according to the three possible styles of the objective function the varying domain optimization method and its generalization are corresponding proposed in contrast to the previous method the proposed method can make that the higher priority achieving the higher satisfaction degree in this way the decision maker can get the optimal solution as well as guarantee the priorities of the multiple objective optimization problem we demonstrate the power of this proposed method by three illustrative examples and a practice application

reference direction interactive method for solving multiobjective fuzzy programming problems abstract a reference direction interactive method called rdim is proposed in this paper for solving multiobjective fuzzy programming problems the method combines the reference direction approach for multiobjective linear programming of korhonen and laakso 1986 a visual interactive method for solving multiple criteria problems european journal of operational research 24 277 287 with the interactive satisficing method of sakawa and yano 1989 an interactive fuzzy satisficing method for multiobjective nonlinear programming problem with fuzzy parameters fuzzy sets and systems 30 221 238 the proposed method is applicable to nonsymmetric linear as well as nonlinear mathematical models wherein the objectives are modeled as fuzzy goals and the satisfaction of fuzzy constraints is considered at different level sets of the fuzzy parameters involved at each interactive phase the decision maker dm specifies a reference direction in the membership function space a discrete parametric programming problem is then solved in this direction to provide the dm with a number of alternatives from amongst which he she can select the most preferred solution on the basis of his her preference structure illustrative examples taken from literature have been solved using the proposed method

using analytic hierarchy process and fuzzy set theory to rate and rank the disability abstract any discrimination against qualified disabled people in all employment practices is violating the americans with disabilities act ada and it must be prohibited assigning qualified disabled people to a proper job or making reasonable accommodations for disabled employees is not only ethical but also necessary for reducing the social economic burden therefore in order to place qualified disabled people in appropriate job positions a multidimensional assessment method should be developed for measuring and interpreting the disabled people s residual capabilities with respect to their strengths weaknesses and their compatibility to job requirements and work environments thus the primary purpose of this study is to discuss how to structure a hierarchy related to the problem of deriving an overall disability index odi for measuring the individual s disability the analytic hierarchy process along with the entropy theory and fuzzy set theory is applied for eliciting the weights among the attributes and aggregating the multiple attributes into a single measurement we believe that the resultant odi is meaningful and useful for both the evaluator who supervises a rehabilitation program and the human resource director who engages in a hiring process

two phase approach for solving the fuzzy linear programming problems we propose a two phase approach to solve the fuzzy linear programming problem although several methods in the literature have been proposed to treat this problem the two phase approach has the merit stated as if the decision maker is seeking an efficient solution which can improve the max min operators solution so that each membership degree should be improved then the two phase method automatically attains this desire if there is room to improve an interpretation of this result is that the two phase method not just pursue the highest membership degree in the objective but also pursue a better utilization of each constrained resource

neural net solutions to fuzzy linear programming we show how a neural net with sign restrictions on its weights can be trained to produce approximate solutions to fuzzy linear programming problems the neural net approximates the joint solution buckley 1995 to the fuzzy linear program

an interactive algorithm for large scale multiple objective programming problems with fuzzy parameters through topsis approach in this paper we extend technique for order preference by similarity ideal solution topsis for solving large scale multiple objective programming problems involving fuzzy parameters these fuzzy parameters are characterized as fuzzy numbers for such problems the pareto optimality is introduced by extending the ordinary pareto optimality on the basis of the level sets of fuzzy numbers an interactive fuzzy decision making algorithm for generating pareto optimal solution through topsis approach is provided where the decision maker dm is asked to specify the degree and the relative importance of objectives finally a numerical example is given to clarify the main results developed in the paper

compensatory operators in fuzzy linear programming with multiple objectives previously h j zimmermann 11 presented a fuzzy approach to solve the linear vectormaximum problem in the mentioned approach the author uses the minimum or the product operators to translate the semantic meaning of the and the aim of this paper is to reconsider this approach by using operators which allow some degree of compensation between aggregated membership functions firstly the c operator already defined in 12 is used to combine our fuzzy objectives it is shown that solutions obtained by this way are always efficient unfortunately the substitute problem is not generally computationally feasible we suggest then the min bounded sum operator which is a compensatory one has some desired properties and is supported by empirical arguments it allows a considerable decrease of the computational burden in the substitute problem and leads to a solution which is attractive from the stand point of efficiency

an interactive satisfying method based on alternative tolerance for multiple objective optimization with fuzzy parameters an interactive satisfying method based on alternative tolerance is presented for the multiple objective optimization problem with fuzzy parameters using the alpha level sets of the fuzzy numbers all the objectives are modeled as the fuzzy goals and the tolerances of the objectives are iteratively changed according to a decision maker for a satisfying solution via a specific attainable point programming model the membership functions can be modified and then a lexicographic two phase programming procedure is constructed correspondingly to find the final solution in a special case the objective constraint is added instead of changing the membership functions therefore the dissatisfying objectives for the decision maker can be improved step by step the presented method not only acquires the alpha pareto optimal or weak alpha pareto optimal solution of the fuzzy multiple objective optimization but also satisfies the progressive preference of the decision maker a numerical example shows its power

two step interactive satisfactory method for fuzzy multiple objective optimization with preemptive priorities this paper presents a two step interactive satisfactory optimization method for fuzzy multiple objective optimization with preemptive priorities in contrast to previous works this proposed approach guarantees the order of satisfactory degrees consistent with priorities the decision maker not only acquires satisfactory solution of all the objectives but also realizes the preemptive priority requirement among them the originally complex optimization problem is simplified and divided into two subproblems that are solved in sequence numerical examples and actual application show the proposed method s effectiveness flexibility and efficiency in comparison with results from the literature

fuzzy goal programming an additive model abstract an additive model to solve fuzzy goal programming fgp is formulated the method uses arithmetic addition to aggregate the fuzzy goals to construct the relevant decision function cardinal and ordinal weights for nonequivalent fuzzy goals are also incorporated in the method the solution procedure is illustrated with a numerical example

solving the multiobjective possibilistic linear programming problem in conventional multiobjective decision making problems the estimation of the parameters of the model is often a problematic task normally they are either given by the decision maker dm who has imprecise information and or expresses his considerations subjectively or by statistical inference from past data and their stability is doubtful therefore it is reasonable to construct a model reflecting imprecise data or ambiguity in terms of fuzzy sets for which a lot of fuzzy approaches to multiobjective programming have been developed in this paper we propose a method to solve a multiobjective linear programming problem involving fuzzy parameters fp molp whose possibility distributions are given by fuzzy numbers estimated from the information provided by the dm as the parameters intervening in the model are fuzzy the solutions will be also fuzzy we propose a new pareto optimal solution concept for fuzzy multiobjective programming problems it is based on the extension principle and the joint possibility distribution of the fuzzy parameters of the problem the method relies on cuts of the fuzzy solution to generate its possibility distributions these ideas are illustrated with a numerical example

an l p regularity theory for harmonic maps motivated by the harmonic map heat flow we consider maps between riemannian manifolds such that the tension field belongs to an lp space under an appropriate smallness condition a certain degree of regularity follows for suitable solutions of the harmonic map heat flow we have a partial regularity result as a consequence

fuzzy programming with nonlinear membership functions piecewise linear approximation one promising approach that has been proposed for dealing with multicriteria i e multiple objective programming models has been fuzzy linear programming flp in essence the flp approach involves the replacement of the multiple objectives with goals by means of assigning an aspiration level to each objective fuzzy membership functions are then introduced to represent the measure of the achievement of the actual solution for each goal relative to its aspired level it has been shown that if the flp problem is linear and if linear membership functions are employed to reflect goal achievement then the flp problem may be transformed into a conventional i e single objective linear programming model as such one may use any conventional linear programming algorithm to solve the transformed model however in the more general case of nonlinear membership functions the transformation process becomes considerably more involved and has in the past typically led to less desirable formulations in this paper we present what we believe to be a straightforward and computationally efficient procedure for dealing with the flp problem with any general class of nonlinear membership functions conversion of such a flp will result in either a regular linear programming model or a linear integer programming model depending upon the specific characteristics of the membership function in either case existing commercial software is readily available to solve such models

a comparative investigation of methods for logistic regression with separated or nearly separated data in logistic regression analysis of small or sparse data sets results obtained by classical maximum likelihood methods cannot be generally trusted in such analyses it may even happen that the likelihood meets the convergence criteria while at least one parameter estimate diverges to infinity this situation has been termed separation and it typically occurs whenever no events are observed in one of the two groups defined by a dichotomous covariate more generally separation is caused by a linear combination of continuous or dichotomous covariates that perfectly separates events from non events separation implies infinite or zero maximum likelihood estimates of odds ratios which are usually considered unrealistic i provide some examples of separation and near separation in clinical data sets and discuss some options to analyse such data including exact logistic regression analysis and a penalized likelihood approach both methods supply finite point estimates in case of separation profile penalized likelihood confidence intervals for parameters show excellent behaviour in terms of coverage probability and provide higher power than exact confidence intervals general advantages of the penalized likelihood approach are discussed