mrp with flexible constraints a fuzzy mathematical programming approach in manufacturing environments with complex product structures and multiple production stages material requirements planning mrp systems are the most commonly used for production planning and material supply decision making however in practice diverse difficulties arise such as uncertainty in market demand resources with limited available capacities uncertainty in capacity data or uncertain costs likewise classical procedures of resolution applied in mrp environments do not optimize production decisions this paper provides a new linear programming model for medium term production planning in a capacity constrained mrp multi product multi level and multi period manufacturing environment then this model is transformed into three fuzzy models with flexibility in the objective function in the market demand and in the available capacity of resources the main goal is to determine the master production schedule the mrp for each raw component in each period stock levels delayed demand and capacity usage levels over a given planning horizon in such a way as to hedge against uncertainty finally the model is tested using real data from an automobile seat manufacturer

fuzzy non linear integer program by parametric programming approach in this paper we propose a heuristic algorithm to analyze a differentiable nonlinear integer program when the fuzzy inequality constraint is considered by defining a membership function the fuzzy inequality constraint can be transformed into a parametric inequality of which a hybrid solution procedure incorporating a genetic algorithm is developed theoretical analysis and experimental investigation are presented

bi criteria multi facility location problem in fuzzy environment abstract this paper presents a fuzzy goal programming model for locating multiple new facilities on a plane bounded by a convex polygon under the criteria 1 minimize the sum of all the transportation costs and 2 minimize the maximum distances from the facilities to the demand points it has also been proved that the methodology given will always give a nondominated solution recti linear distances have been taken as the scenario may be thought of in an urban setting a numerical example has been given to illustrate the solution methodology

a simplex based algorithm for the lexicographically extended linear maxmin problem abstract let f x cx c where c is a real m n matrix c a constant m vector and x a variable n vector g x denotes the vector with components f i x in order of ascending magnitude the problem is to maximize g x over x in the lexicographic sense where x is the set of all points such that ax b x 0 a a real r n matrix it is known that this problem can be reduced to a sequence of m maximum problems where the maxmin problems can be solved by the well known simplex method of linear optimization a prerequisite such a procedure is the knowledge of at least one common index per stage it is shown that these common indices can readily be obtained from the respective optimal simplex tableaux some simple tests help to obtain more than just one common index per stage

kummer congruence for the bernoulli numbers of higher order the authors studied the properties of bernoulli numbers of higher order appl math comput in press bull aust math 65 2002 59 for q 1 we can also find their results proc jangjeon math soc 1 2000 97 arch math 76 2001 190 proc jangjeon math soc 1 2000 161 adv stud contemp math 2 2000 9 proc jangjeon math soc 2 2001 23 j math phys a 34 2001 l643 proc jangjeon math soc 2 2001 19 proc jangjeon math soc 2 2001 9 proc jangjeon math soc 3 2001 63 the authors suggested the question to inquire the proof of kummer congruence for bernoulli numbers of higher order appl math comput in press in this paper we give a proof of kummer type congruence for the bernoulli numbers of higher order which is an answer to a part of the question in appl math comput in press

fuzzy optimization for supply chain planning under supply demand and process uncertainties in today s global marketplace individual firms do not compete as independent entities rather as an integral part of a supply chain this paper proposes a fuzzy mathematical programming model for supply chain planning which considers supply demand and process uncertainties the model has been formulated as a fuzzy mixed integer linear programming model where data are ill known and modelled by triangular fuzzy numbers the fuzzy model provides the decision maker with alternative decision plans for different degrees of satisfaction this proposal is tested by using data from a real automobile supply chain

duality results and a dual simplex method for linear programming problems with trapezoidal fuzzy variables linear programming problems with trapezoidal fuzzy variables fvlp have recently attracted some interest some methods have been developed for solving these problems by introducing and solving certain auxiliary problems here we apply a linear ranking function to order trapezoidal fuzzy numbers then we establish the dual problem of the linear programming problem with trapezoidal fuzzy variables and hence deduce some duality results in particular we prove that the auxiliary problem is indeed the dual of the fvlp problem having established the dual problem the results will then follow as natural extensions of duality results for linear programming problems with crisp data finally using the results we develop a new dual algorithm for solving the fvlp problem directly making use of the primal simplex tableau this algorithm will be useful for sensitivity or post optimality analysis when using primal simplex tableaus

post optimality analysis on the membership functions of a fuzzy linear programming problem models of linear programming problems with fuzzy constraints are very well known in the current literature in almost all cases to solve these problems linear membership functions are used because they have very good properties and are very easy to manipulate in some cases however because of the knowledge that the decision maker has such membership functions could be modeled as nonlinear although the complexity of the problem could increase this paper considers the use of nonlinear membership functions in fuzzy linear programming problems to show that the corresponding solution to be obtained can be derived from a parallel linear model moreover it is easier to solve than the nonlinear model making use of a similar procedure to that of post optimal analysis in classical linear programming the case in which these membership functions are defined by means of piecewise linear approximations is also considered and analyzed

a note on q multiple zeta functions the purpose of this letter is to give the value of the q multiple zeta function at negative integers which is an answer to a part of the problem in a previous publication kim t park d w and rim s h 2001 j phys a math gen 34 7633

pareto optimal solutions in fuzzy multi objective linear programming the problem of solving multi objective linear programming problems by assuming that the decision maker has fuzzy goals for each of the objective functions is addressed several methods have been proposed in the literature in order to obtain fuzzy efficient solutions to fuzzy multi objective programming problems in this paper we show that in the case that one of our goals is fully achieved a fuzzy efficient solution may not be pareto optimal and therefore we propose a general procedure to obtain a non dominated solution which is also fuzzy efficient two numerical examples illustrate our procedure

selecting preferred solutions in the minimax approach to dynamic programming problems under flexible constraints abstract dynamic programming is a powerful approach to the optimization of sequential or multistage decision processes e g in planning or in system control in this paper we consider both theoretical and algorithmic issues in sequential decision processes under flexible constraints such processes must attain a given goal within some tolerance tolerances or preferences also apply to the values the decision variables may take or on the action chosen at each step such problems boil down to maximin optimization unfortunately this approach suffers from the so called drowning effect lack of discrimination and the optimality principle of dynamic programming is not always verified in this context we introduce a general framework for refined minimax optimization procedures in order to compare and select preferred alternatives this framework encompasses already introduced methods such as leximin and discrimin but it allows their extension to the comparison of vectors of unequal lengths we show that these refined comparisons restore compatibility with the optimality principle and that classical algorithms can be adapted to compute such preferred solutions by exploiting existing results on idempotent semirings

river assimilative capacity analysis via fuzzy linear programming abstract the assimilative capacity of a river that is the so called allowable pollution loading in a river basin is a problem of water quality management involving optimization subject to constraints of standards of water quality and their models this study is an up to date evaluation of assimilative capacity in a river basin with consideration of the imprecision of the water environment i e water quality and river flowrate the purposes of this study in comparison with crisp linear programming lp is to select and evaluate four well known fuzzy linear programming flp approaches and to apply them to the assimilative vapacity of a river basin tou chen river basin in taiwan the results show the models of assimilative capacity of flp are better than those of a crisp one which reflect the flexible characteristics of the former approaches the emphases in this paper are on how to formulate problems of assimilative vapacity of a river according to flp models and on how to solve them according to algorithms of crisp lp furthermore the results of river analysis serve as criteria for regulatory agencies to implement the control of pollution sources and to accomplish finally the sustainable use of water resources

generalized spectral projections on symmetric spaces of noncompact type paley wiener theorems abstract on a symmetric space x g k of noncompact type we consider the formulas formula formula where is the spherical function on x taken together they represent the synthesis and decomposition formulas for appropriate functions f on x in terms of joint eigenfunctions of the invariant differential operators on x the focus of this paper is the characterization of the range of p on c c x in the case g has real rank one this result extends and generalizes a result due to strichartz on odd dimensional real hyperbolic space and provides a spectral reformulation of the paley wiener theorem for fourier transform on x due to helgason as an application we provide a fairly general support result for the spherical mean operator on x

performance evaluation of nonlinear optimization methods via pairwise comparison and fuzzy numbers in this paper we extend the deterministic performance evaluation of nonlinear optimization methods we carry out a pairwise comparison using fuzzy estimates of the performance ratios to obtain fuzzy final scores of the methods under consideration the key instrument is the concept of fuzzy numbers with triangular membership functions the algebraic operations on them are simple extensions of the operations on real numbers they are exact in the parameters lower modal and upper values not necessarily exact in the shape of the membership function we illustrate the fuzzy performance evaluation by the ranking and rating of five methods geometric programming and four general methods for solving geometric programming problems using the results of recent computational studies some general methods appear to be leading an outcome which is not only due to their performance under subjective criteria like domain of applications and conceptual simplicity of use they also score higher under more objective criteria like robustness and efficiency

l p l p estimates for overdetermined radon transforms we prove several variations on the results of f ricci and g travaglini 2001 concerning l p l p bounds for convolution with all rotations of arc length measure on a fixed convex curve in r 2 estimates are obtained for averages over higher dimensional convex nonsmooth hypersurfaces smooth k dimensional surfaces and nontranslation invariant families of surfaces we compare ricci and travaglini s approach based on average decay of the fourier transform with an approach based on l 2 boundedness of fourier integral operators and show that essentially the same geometric condition arises in proofs using the two techniques

short rational generating functions for solving some families of fuzzy integer programming problems in this paper we present new complexity results for solving single and multiobjective fuzzy integer programs with soft constraints and fuzzy objective function coefficients our method is based on the use of barvinok s short rational generating functions of convenient transformations of the fuzzy problems to crisp ones this analysis allows us to provide new insights on the structure of these families of fuzzy programs

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

an exponential membership function for fuzzy multiple objective linear programming abstract the operator min is one of the most frequently used aggregation operators in fuzzy decision however this operator is the softest operator and no allowance is made for any compensation the product and other operators some of them may be compensatory are seldom used because of the nonlinearity of the resulting problem in this paper an exponential instead of linear membership function is proposed the advantages of using exponential membership are two fold first the resulting problems can be transformed to linear ones when the product and several other nonlinear aggregate operators are used secondly exponential representation is more realistic than the linear ones usually used for some practical applications

a fuzzy approach to the transportation problem abstract the transportation problem with fuzzy supply values of the deliverers and with fuzzy demand values of the receivers is analysed for the solution of the problem the technique of parametric programming is used this makes it possible to obtain not only the maximizing solution according to the bellman zadeh criterion but also other alternatives close to the optimal solution

process planning in a fuzzy environment abstract mixed integer optimization models for chemical process planning typically assume that model parameters can be accurately predicted as precise forecasts are difficult to obtain process planning usually involves uncertainty and ambiguity in the data this paper presents an application of fuzzy programming to process planning the forecast parameters are assumed to be fuzzy with a linear or triangular membership function the process planning problem is then formulated in terms of decision making in a fuzzy environment with fuzzy constraints and fuzzy net present value goals the model is transformed to a deterministic mixed integer linear program or mixed integer nonlinear program depending on the type of uncertainty involved in the problem for the nonlinear case a global optimization algorithm is developed for its solution this algorithm is applicable to general possibilistic programs and can be used as an alternative to the commonly used bisection method illustrative examples and computational results for a petrochemical complex with 38 processes and 24 products illustrate the applicability of the developed models and algorithms

fuzzy linear programming based on statistical confidence interval and interval valued fuzzy set abstract in this paper we use statistical data and statistical confidence interval to derive level 1 fuzzy numbers 0 1 1 interval valued fuzzy numbers 0

linear programming with fuzzy random variable coefficients abstract linear programming with fuzzy random variable coefficients is introduced by discussing a practical engineering problem in order to study the solution of the linear programming with fuzzy random variable coefficients we discuss the simplex algorithm for linear programming with random variable coefficients furthermore the solution and distribution problem of this new fuzzy random programming are studied

a two phase approach for multi objective programming problems with fuzzy coefficients in this study a two phase procedure is introduced to solve multi objective fuzzy linear programming problems the procedure provides a practical solution approach which is an integration of fuzzy parametric programming fpp and fuzzy linear programming flp for solving real life multiple objective programming problems with all fuzzy coefficients the interactive concept of the procedure is performed to reach simultaneous optimal solutions for all objective functions for different grades of precision according to the preferences of the decision maker dm the procedure can be also performed to obtain lexicographic optimal and or additive solutions if it is needed in the first phase of the procedure a family of vector optimization models is constructed by using fpp then in the second phase each model is solved by flp the solutions are optimal and each one is an alternative decision plan for the dm

the use of parametric programming in fuzzy linear programming in this paper the possibility of the identification of a complete fuzzy decision not only the maximizing alternative in fuzzy linear programming by use of the parametric programming technique is presented also it is shown that this fact can be useful in the zimmermann approach to multiple objective linear programming the presented remarks are illustrated by some numerical examples

analysis of structure in fuzzy linear programs abstract fuzzy linear programming problems are analyzed within the fuzzy set context to uncover redundancies infeasibilities variables whose values are fixed and implied bounds on rows and columns applications include pre analysis of perturbed constraint matrices and post optimization analysis fuzzy linear programming problems

mean value of higher order generalized bernoulli numbers gauss sums and general kloosterman sums the main purpose of this paper is using the properties of higher order gen eralized bernoulli numbers and the mean value theorems of dirichlet l function to study the mean value of gauss sums general kloosterman sums and higher order generalized bernoulli numbers and give two interesting asymptotic formulae for it

solving mathematical programs with fuzzy equilibrium constraints this work deals with mathematical programs with fuzzy equilibrium constraints it shows that solving the fuzzy mpec is equivalent to solving a fuzzy complementarity constrained optimization problem by using the tolerance approach we show that the fuzzy complementarity constrained optimization problem can be converted to a regular nonlinear programming problem a new smoothing approach based on entropic regularization is developed for solving the resulting optimization problem numerical examples are also included to illustrate the solution procedure

support theorems on r n and non compact symmetric spaces we consider convolution equations of the type f t g where f g is an element of l p r n and t is a compactly supported distribution under natural assumptions on the zero set of the fourier transform of t we show that f is compactly supported provided g is similar results are proved for non compact symmetric spaces as well c 2010 elsevier inc all rights reserved

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

applications of fuzzy set theory to mathematical programming mathematical programming is one of the areas to which fuzzy set theory has been applied extensively primarily based on bellman and zadeh s model of decision in fuzzy environments models have been suggested which allow flexibility in constraints and fuzziness in the objective function in traditional linear and nonlinear programming in integer and fractional programming and in dynamic programming these models in turn have been used to offer computationally efficient approaches for solving vector maximum problems this paper surveys major models and theories in this area and offers some indication on future developments which can be expected

the complete decision set of the generalized symmetrical fuzzy linear programming problem abstract the generalized symmetrical fuzzy linear programming sflp problem is defined in this paper as involving both fuzzy and nonfuzzy constraints of which fuzzy constraints are represented by linear and nonlinear membership functions fuzzy intervals are proposed to represent fuzzy equality constraints the general form of fuzzy constraints is proposed by defining the membership function d as an explicit function of min z rather than an explicit function of the solution set x the complete solution set and its existence criterion are studied the improvement of objective values based on the obtained complete solution set is also discussed a truck fleet problem presented in the literature is cited to explain the proposed method and to prove the correctness of the solution

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

interactive fuzzy linear programming abstract in this study an interactive fuzzy linear programming approach is investigated to improve the flexibility and robustness of linear programming lp technique this paper provides a practical modelling method which is a symmetric integration of zimmermann s werners s vergegay s and chanas s fuzzy lp approaches and additionally provides a decision support system for solving a specific domain of practical lp problems the interactive concept provides a learning process about the system whereby the decision maker can learn to recognize good solutions relative importance of factors in the system and then design a high productivity system instead of optimizing a given system this interactive fuzzy lp system provides integration oriented adaptation and learning features by considering all the possibilities of a specific domain of lp problems which are integrated in logical order using an if then rule

on higher order generalized bernoulli numbers in this paper we define higher order generalized bernoulli numbers in order to give the values of a series attached to dirichlet character at non positive integers and investigate the arithmetic properties of them in particular we obtain euler summation formula involving higher order generalized bernoulli numbers also in the p adic cyclotomic field we construct witt s type formula involving higher order generalized bernoulli numbers

decisions and evaluations by hierarchical aggregation of information usually in preparing a decision a hierarchy of criteria for evaluating alternatives can he agreed upon by the decision makers the problem addressed in this paper is the appropriate modelling of such evaluations and their aggregation in order to arrive at a proper overall evaluation function by which the decision alternatives can be ordered easily representing the criteria subjective categories by fuzzy sets it is possible to use aggregation operators as models for the amalgamation of those criteria in the framework of an empirical research project a hierarchy of criteria was developed for the evaluation of credit worthiness of customers 50 credit applications were evaluated by a number of credit managers of banks and their judgement could be appropriately modelled by a four level hierarchy of criteria

linear programming with multiple fuzzy goals abstract this paper describes the use of fuzzy set theory in goal programming gp problems in particular it is demonstrated how fuzzy or imprecise aspirations of the decision maker dm can be quantified through the use of piecewise linear and continuous functions models are then presented for the use of fuzzy goal programming with preemptive priorities with archimedean weights and with the maximization of the membership function corresponding to the minimum goal examples are also provided