a splitting method for complex symmetric indefinite linear system abstract in this paper not requiring that the hermitian part of the complex symmetric linear system must be hermitian positive definite a class of splitting methods is established by the modified positive negative stable splitting pns of the coefficient matrix and is called the mpns method theoretical analysis shows that the mpns method is absolutely convergent under proper conditions some useful properties of the corresponding mpns preconditioned matrix are obtained numerical experiments are reported to illustrate the efficiency of both the mpns method and the mpns preconditioner

fixed points of asymptotic contractions abstract let r r be a contractive gauge function in the sense that is continuous s 0 and if f m m satisfies d f x f y d x y for all x y in a complete metric space m d then f always has a unique fixed point it is proved that if t m m satisfies d t n x t n y n d x y x y m where each n is continuous and n uniformly on the range of d then t has a unique fixed point and moreover all of the picard iterates of t converge to this fixed point

two level linear programming decentralized planning has long been recognized as an important decision making problem many approaches based on the concepts of large scale system decomposition have generally lacked the ability to model the type of truly independent subsystems which often exist in practice multilevel programming models partition control over decision variables among ordered levels within a hierarchical planning structure a planner at one level of the hierarchy may have his objective function and set of feasible decisions determined in part by other levels however his control instruments may allow him to influence the policies at other levels and thereby improve his own objective function this paper examines the special case of the two level linear programming problem geometric characterizations and algorithms are presented with some examples the goal is to demonstrate the tractability of such problems and motivate a wider interest in their study

best proximity points for asymptotic cyclic contraction mappings abstract in this paper we prove the existence and convergence of best proximity points for asymptotic cyclic contractions in metric spaces with the property uc as well as for asymptotic proximal pointwise contractions in uniformly convex banach spaces moreover we consider a generalized cyclic contraction mapping and prove the existence of best proximity points for such a mapping in banach spaces which do not necessarily satisfy the geometric property uc

multiobjective programming in optimization of interval objective functions a generalized approach abstract we generalize known concepts those introduced by ishihuchi and tanaka 1 and rommerlfanger et al 2 of the solution of the linear programming problem with interval coefficients in the objective function based on preference relations between intervals we unify all the discussed concepts as well as the corresponding solution methods into one general framework

a splitting iterative method for the discrete dynamic linear systems this paper is concerned with a splitting iterative method for a class of complex symmetric linear systems from an n degree of freedom n dof discrete system this splitting iterative method is established by the complex symmetric and skew hermitian splitting of the coefficient matrix and is called the css method which is from the classical state space formulation of frequency analysis of the discrete dynamic linear systems the convergence properties of the css method are obtained the corresponding css preconditioner is proposed and some useful properties of the preconditioned matrix are established the presented numerical examples are to illustrate the efficiency of both the css method and the css preconditioner

on a conjugate gradient type method for solving complex symmetric linear systems abstract we consider large sparse linear systems ax b with complex symmetric coefficient matrices a a t which arise e g from the discretization of partial differential equations with complex coefficients for the solution of such systems we present a new conjugate gradient type iterative method csym which is based on unitary equivalence transformations of a to symmetric tridiagonal form an analysis of csym shows that its convergence depends on the singular values of a and that it has both the minimal residual property and constant costs per iteration step we compare the algorithm with other methods for solving large sparse complex symmetric systems

a labeling algorithm for the fuzzy assignment problem abstract this paper concentrates on the assignment problem where costs are not deterministic numbers but imprecise ones here the elements of the cost matrix of the assignment problem are subnormal fuzzy intervals with increasing linear membership functions whereas the membership function of the total cost is a fuzzy interval with decreasing linear membership function by the max min criterion suggested by bellman and zadeh the fuzzy assignment problem can be treated as a mixed integer nonlinear programming problem we show that this problem can usually be simplified into either a linear fractional programming problem or a bottleneck assignment problem here we propose an efficient algorithm based on the labeling method for solving the linear fractional programming case the algorithm begins with primal feasibility and proceeds to obtain dual feasibility while maintaining complementary slackness until the primal optimal solution is found the computational results show that the proposed labeling algorithm offers an effective and efficient way for handling the fuzzy assignment problem

study on multiobjective transportation problem with fuzzy numbers abstract in this paper we investigate the efficient solutions and stability of multiobjective transportation problem fmtp with fuzzy coefficient c ij r c r and or fuzzy supply quantities a i and or fuzzy demands quantities b j the concept of fuzzy efficient was introduced in which the ordinary efficient solution is extended based on the level of fuzzy numbers a necessary and sufficient condition for such a solution is established the basic notions as the solvability set and the stability set of the first kind are defined and characterized the parametric analysis is used to characterize the parametric optimal solutions to the auxiliary problems an algorithm for the determination of the stability set is proposed finally a numerical example is given

a block alternating splitting iteration method for a class of block two by two complex linear systems optimization problems with partial differential equations as constraints arise widely in many areas of science and engineering in this paper we focus on solving a class of block two by two complex linear systems arising from the distributed optimal control with time periodic parabolic equations a new block alternating splitting bas iteration method is presented for solving the class of complex linear systems the convergence theory and the spectral properties of the bas iteration method are discussed numerical experiments are presented to illustrate the efficiency of the bas iteration as a solver as well as a preconditioner for krylov subspace methods

production planning problem with sequence dependent setups as a bilevel programming problem abstract each of n products is to be processed on two machines in order to satisfy known demands in each of t periods only one product can be processed on each machine at any given time each switch from one item to another requires sequence dependent setup time the objective is to minimize the total setup time and the sum of the costs of production storage and setup we consider the problem as a bilevel mixed 0 1 integer programming problem the objective of the leader is to assign the products to the machines in order to minimize the total sequence dependent setup time while the objective of the follower is to minimize the production storage and setup cost of the machine we develop a heuristics based on tabu search for solving the problem at the end some computational results are presented

the mixed integer linear bilevel programming problem a two person noncooperative game in which the players move in sequence can be modeled as a bilevel optimization problem in this paper we examine the case where each player tries to maximize the individual objective function over a jointly constrained polyhedron the decision variables are variously partitioned into continuous and discrete sets the leader goes first and through his choice may influence but not control the responses available to the follower for two reasons the resultant problem is extremely difficult to solve even by complete enumeration first it is not possible to obtain tight upper bounds from the natural relaxation and second two of the three standard fathoming rules common to branch and bound cannot be applied fully in light of these limitations we develop a basic implicit enumeration scheme that finds good feasible solutions within relatively few iterations a series of heuristics are then proposed in an effort to strike a balance between accuracy and speed the computational results suggest that some compromise is needed when the problem contains more than a modest number of integer variables

on comparing interval numbers abstract this paper first presents a brief survey of the existing works on comparing and ranking any two interval numbers on the real line and then on the basis of this gives two approaches to compare any two interval numbers the first one describes a value judgement index along with a discussion on its strength and weakness over the other approaches this index has also been called an optimistic decision maker s preference index the other approach defines strict and fuzzy preference ordering between any two interval numbers from a pessimistic decision maker s point of view choice of different decision makers with different degrees of pessimism has also been considered

an iterative method for the helmholtz equation an iterative algorithm for the solution of the helmholtz equation is developed the algorithm is based on a preconditioned conjugate gradient iteration for the normal equations the preconditioning is based on an ssor sweep for the discrete laplacian numerical results are presented for a wide variety of problems of physical interest and demonstrate the effectiveness of the algorithm

fuzzy programming in multiobjective linear programming a parametric approach abstract zimmermann s fuzzy approach to the compromise solution concept in the multi objective linear programming problem is considered it is shown that given a class of membership functions of fuzzy goals assigned to objective functions in the problem wider than primarily proposed by zimmermann the use of classical linear programming methods to solve and analyse the problem is also possible as a result of application of the method based on the parametric programming technique one can obtain a fuzzy solution of the problem this solution is a certain fuzzy subset of the set of weakly efficient solutions of the problem the solution which belongs to this set to the highest degree is a maximizing one in bellman and zadeh s terminology it may also be obtained by use of the usual non parametric simplex method

linear programming with fuzzy objectives abstract this paper presents a new method for solving linear programming problems with fuzzy parameters in the objective function to determine a compromise solution the authors do not reduce the set of infinitely many objective functions to a single compromise objective function as done in classical deterministic or stochastic procedures here the problem is only reduced to a few extreme objective functions the information contained in the membership functions can be used to any extent by a method called level related pair formation moreover if different level sets are considered the stability of the optimal solution can be tested

preconditioned hermitian and skew hermitian splitting methods for non hermitian positive semidefinite linear systems for the positive semidefinite system of linear equations of a block two by two structure by making use of the hermitian skew hermitian splitting iteration technique we establish a class of preconditioned hermitian skew hermitian splitting iteration methods theoretical analysis shows that the new method converges unconditionally to the unique solution of the linear system moreover the optimal choice of the involved iteration parameter and the corresponding asymptotic convergence rate are computed exactly numerical examples further confirm the correctness of the theory and the effectiveness of the method

a note on a modified simplex approach for solving bilevel linear programming problems abstract we analyze the article a modified simplex approach for solving bilevel linear programming problems ejor 67 126 135 we point out some problems in its theoretical analysis moreover the algorithm proposed may not find a global solution as it is claimed we give some examples in order to illustrate these remarks

algorithms for linear programming problems with interval objective function coefficients this paper presents three algorithms for solving linear programming problems in which some or all of the objective function coefficients are specified in terms of intervals which algorithm is applicable depends upon a the number of interval objective function coefficients b the number of nonzero objective function coefficients and c whether or not the feasible region is bounded the algorithms output all extreme points and unbounded edge directions that are multiparametrically optimal with respect to the ranges placed on the objective function coefficients the algorithms are most suitable to linear programs in which the objective function coefficients are deterministic but are likely to vary from time period to time period as for example in blending problems

computational difficulties of bilevel linear programming we show using small examples that two algorithms previously published for the bilevel linear programming problem blp may fail to find the optimal solution and thus must be considered to be heuristics a proof is given that solving blp problems is np hard which makes it unlikely that there is a good exact algorithm

computation of the optimal tolls on the traffic network abstract the present paper is devoted to the computation of optimal tolls on a traffic network that is described as fuzzy bilevel optimization problem as a fuzzy bilevel optimization problem we consider bilinear optimization problem with crisp upper level and fuzzy lower level an effective algorithm for computation optimal tolls for the upper level decision maker is developed under assumption that the lower level decision maker chooses the optimal solution as well the algorithm is based on the membership function approach this algorithm provides us with a global optimal solution of the fuzzy bilevel optimization problem

convex fuzzy mapping with differentiability and its application in fuzzy optimization abstract the concepts of differentiability convexity generalized convexity and minimization of a fuzzy mapping are known in the literature the purpose of this present paper is to extend and generalize these concepts to fuzzy mappings of several variables using buckley feuring approach for fuzzy differentiation and derive karush kuhn tucker condition for the constrained fuzzy minimization problem

a nonlinear interval number programming method for uncertain optimization problems in this paper a method is suggested to solve the nonlinear interval number programming problem with uncertain coefficients both in nonlinear objective function and nonlinear constraints based on an order relation of interval number the uncertain objective function is transformed into two deterministic objective functions in which the robustness of design is considered through a modified possibility degree the uncertain inequality and equality constraints are changed to deterministic inequality constraints the two objective functions are converted into a single objective problem through the linear combination method and the deterministic inequality constraints are treated with the penalty function method the intergeneration projection genetic algorithm is employed to solve the finally obtained deterministic and non constraint optimization problem two numerical examples are investigated to demonstrate the effectiveness of the present method

fuzzy programming approach to multi level programming problems multi level programming is characterized as mathematical programming to solve decentralized planning problems we have considered a multi level linear programming problem and applied fuzzy mathematical programming fmp approach to obtain the solution of the system we have suggested fmp method for the minimization of the objectives using linear membership functions fmp is a supervised search procedure supervised by the higher level decision maker dm the higher level dm provides the preferred values of decision variables under his control to enable the lower level dm to search for his optimum in a wider feasible space and the bounds of his objective function to direct the lower level dm to search for his solutions in the right direction

asymptotic pointwise contractions a simple and elementary without using the ultrapower technique proof for the existence of a fixed point of an asymptotic pointwise contraction is provided the existence of fixed points for asymptotic pointwise nonexpansive mappings in a uniformly convex banach space is also investigated

kkt transformation approach for multi objective multi level linear programming problems abstract the earlier karush kuhn tucker kkt transformation method has been applied to multi level decentralized programming problems ml d pps when the decision variable set was divided into subsets where each decision maker dm of the system controlled only a particular subset but had no control over any decision variables of some other subset in this paper we give the mathematical formulation and corresponding development of ml d pps by kkt transformation when dms have absolute control over certain decision variables but some variables may be shared and hence controlled by two or more dms

a cocr method for solving complex symmetric linear systems the conjugate orthogonal conjugate gradient cocg method has been recognized as an attractive lanczos type krylov subspace method for solving complex symmetric linear systems however it sometimes shows irregular convergence behavior in practical applications in the present paper we propose a conjugate a orthogonal conjugate residual cocr method which can be regarded as an extension of the conjugate residual cr method numerical examples show that cocr often gives smoother convergence behavior than cocg

a general model for fuzzy linear programming abstract in the current literature there are several models of fuzzy linear programming problems the aim of this paper is first to describe the more important problems in fuzzy linear programming and second to give a general model of fuzzy linear programming problems involving all of the above ones finally a resolution method for that general model is proposed

a concept of the optimal solution of the transportation problem with fuzzy cost coefficients abstract in the paper a definition of the optimal solution of the transportation problem with fuzzy cost coefficients as well as an algorithm determining this solution are proposed

optimization of linear problems subjected to the intersection of two fuzzy relational inequalities defined by dubois prade family of t norms abstract in this paper optimization of a linear objective function with fuzzy relational inequality constraints is investigated in doing so the feasible region is formed by the intersection of two inequality fuzzy systems and dubois prade family of t norms which are considered as fuzzy composition the most well known continuous t norms are archimedean such as frank yager hamacher sugeno weber and schweizer sklar family an interesting family of t norms that is not archimedean has been introduced by dubois and prade in this paper the resolution of the feasible region of the problem is initialy investigated when it is defined with max duboise prade composition a necessary and sufficient condition along with three other necessary conditions are derived for determining the feasibility of the problem moreover two procedures have also been presented with the aim of simplifying the current linear problems a method is proposed to generate random feasible max dubois prade fuzzy relational inequalities and an algorithm is accordingly presented to solve the problem finally an example is described to illustrate this algorithm

application of particle swarm optimization algorithm for solving bi level linear programming problem bi level linear programming is a technique for modeling decentralized decision it consists of the upper level and lower level objectives this paper attempts to develop an efficient method based on particle swarm optimization pso algorithm with swarm intelligence the performance of the proposed method is ascertained by comparing the results with genetic algorithm ga using four problems in the literature and an example of supply chain model the results illustrate that the pso algorithm outperforms ga in accuracy

a note on a paper of i d arandelovic on asymptotic contractions abstract w a kirk w a kirk fixed points of asymptotic contractions j math anal appl 277 2003 645 650 defined the notion of an asymptotic contraction on a metric space and using ultrapower techniques he gave a nonconstructive proof of an asymptotic version of the boyd wong fixed point theorem subsequently i d arandelovic i d arandelovic on a fixed point theorem of kirk j math anal appl 301 2005 384 385 established somewhat more general version of kirk s result and he gave an elementary proof of it however our purpose is to show that there is an error in this proof and moreover arandelovic s theorem is false we also explain how to correct this result

the generalized hss method for solving singular linear systems for the singular non hermitian and positive semidefinite linear systems we propose an alternating direction iterative method with two parameters based on the hermitian and skew hermitian splitting the semi convergence analysis and the quasi optimal parameters of the proposed method are discussed moreover the corresponding preconditioner based on the splitting is given to improve the semi convergence rate of the gmres method numerical examples are given to illustrate the theoretical results and the efficiency of the generalized hss method either as a solver or a preconditioner for gmres

extended procedure for computing the values of the membership function of a fuzzy solution to a class of fuzzy linear optimization problems in 1994 chanas and kuchta suggested one approach to determine the membership function of a fuzzy solution to the fuzzy linear optimization problem with fuzzy coefficients in the objective function based on computing the sum of lengths of certain intervals in 2012 dempe and ruziyeva introduced a methodology for realizing chanas and kuchta s idea and derived explicit formulas for computing the endpoints of the suggested intervals in the particular case of triangular fuzzy numbers the purpose of this paper is to extend dempe and ruziyeva s approach by handling the possible degeneracy of basic feasible solutions and derive new formulas for computing the values of the membership function the special example considered by dempe and ruziyeva is again used to illustrate the relevance of the extended solving procedure and new formulas

solving complex valued linear systems via equivalent real formulations most preconditioned iterative methods apply to both real and complex valued linear systems at the same time most iterative linear solver packages available today focus exclusively on real valued systems or deal with complex valued systems as an afterthought by recasting the complex problem in a real formulation a real valued solver can be applied to the equivalent real system on one hand real formulations have been dismissed due to their unfavorable spectral properties on the other hand using an equivalent preconditioned real formulation can be very effective we give theoretical and experimental evidence that an equivalent real formulation is useful in a number of practical situations furthermore we show how to use the advanced features of modern solver packages to formulate equivalent real preconditioners that are computationally efficient and mathematically identical to their complex counterparts the effectiveness of equivalent real formulations is demonstrated by solving ill conditioned complex valued linear systems for a variety of large scale applications moreover the circumstances under which certain equivalent real formulations are competitive is more clearly delineated

on a fixed point theorem of kirk abstract w a kirk j math anal appl 277 2003 645 650 first introduced the notion of asymptotic contractions and proved the fixed point theorem for this class of mappings in this note we present a new short and simple proof of kirk s theorem

a new heuristic method for the flow shop sequencing problem abstract a new heuristic method is presented for solving the m machine n job flow shop scheduling problem this method named spirit is composed of two phases the first finds an initial sequence using an analogy with the travelling salesman problem and the second tries to improve this solution using taboo search techniques the results of the heuristic are compared with those from other well known methods

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

designing robust coverage networks to hedge against worst case facility losses in order to design a coverage type service network that is robust to the worst instances of long term facility loss we develop a facility location interdiction model that maximizes a combination of initial coverage by p facilities and the minimum coverage level following the loss of the most critical r facilities the problem is formulated both as a mixed integer program and as a bilevel mixed integer program to solve the bilevel program optimally a decomposition algorithm is presented whereby the original bilevel program is decoupled into an upper level master problem and a lower level subproblem after sequentially solving these problems supervalid inequalities can be generated and appended to the upper level master in an attempt to force it away from clearly dominated solutions computational results show that when solved to optimality the bilevel decomposition algorithm is up to several orders of magnitude faster than performing branch and bound on the mixed integer program

a new approach for solving linear bilevel problems using genetic algorithms bilevel programming involves two optimization problems where the constraint region of the first level problem is implicitly determined by another optimization problem this paper develops a genetic algorithm for the linear bilevel problem in which both objective functions are linear and the common constraint region is a polyhedron taking into account the existence of an extreme point of the polyhedron which solves the problem the algorithm aims to combine classical extreme point enumeration techniques with genetic search methods by associating chromosomes with extreme points of the polyhedron the numerical results show the efficiency of the proposed algorithm in addition this genetic algorithm can also be used for solving quasiconcave bilevel problems provided that the second level objective function is linear

iterative system solvers for the frequency analysis of linear mechanical systems abstract the paper deals with the numerical treatment of the direct frequency domain dfd analysis of linear mechanical systems attention is mainly focused on the solution of the complex system of equations needed for each analyzed frequency strategies for transforming the system into the shifted form t i z d are proposed and discussed where is related to the frequency for two formulations of the shifted system the performance of some krylov sub space iterative solvers is tested and compared to that of a multi frontal direct method advantage is taken of the shifted form in solving simultaneously a large number of systems resulting from different values of the shift frequency numerical experiments on some prototype structural dynamics problems are reported the results shown demonstrate how the devised strategies for the iterative solution can outperform in many cases the direct solver

a simple tabu search method to solve the mixed integer linear bilevel programming problem abstract multilevel programming is characterized as mathematical programming to solve decentralized planning problems the models partition control over decision variables among ordered levels within a hierarchical planning structure of which the linear bilevel form is a special case of a multilevel programming problem in a system with such a hierarchical structure the high level decision making situations generally require inclusion of zero one variables representing yes no decisions we provide a mixed integer linear bilevel programming formulation in which zero one decision variables are controlled by a high level decision maker and real value decision variables are controlled by a low level decision maker an algorithm based on the short term memory component of tabu search called simple tabu search is developed to solve the problem and two supplementary procedures are proposed that provide variations of the algorithm computational results disclose that our approach is effective in terms of both solution quality and efficiency

new method for solving fuzzy transportation problems with lr flat fuzzy numbers transportation problems have various applications in logistics and supply chains for reducing costs in real life situations the parameters of transportation problems may not be known precisely because of uncontrollable factors herein we propose a new method for solving fuzzy transportation problems ftps in which the transportation costs and supply and demand are represented by non negative lr flat fuzzy numbers the ftp is converted into four transportation problems which are solved using standard transportation simplex algorithms the advantages of the proposed method over existing methods are discussed in the context of two application examples the results show that the proposed method is simpler and computationally more efficient than existing methods in the literature

fuzzy multi level minimum cost flow problems both multiple objectives and multiple hierarchies minimum cost flow problems with fuzzy costs and fuzzy capacities in the arcs are investigated to reduce the complexity a possibility programming is used to handle the vagueness in the parameters fuzzy approach can considerably simplify the problem and thus a fairly general multi level problem can be solved reasonably easily in spite of the fact that the multi level problem is np hard and very difficult to solve several numerical examples are considered to illustrate the approach

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

on inexact hermitian and skew hermitian splitting methods for non hermitian positive definite linear systems we study theoretical properties of two inexact hermitian skew hermitian splitting ihss iteration methods for the large sparse non hermitian positive definite system of linear equations in the inner iteration processes we employ the conjugate gradient cg method to solve the linear systems associated with the hermitian part and the lanczos or conjugate gradient for normal equations cgne method to solve the linear systems associated with the skew hermitian part respectively resulting in ihss cg lanczos and ihss cg cgne iteration methods correspondingly theoretical analyses show that both ihss cg lanczos and ihss cg cgne converge unconditionally to the exact solution of the non hermitian positive definite linear system moreover their contraction factors and asymptotic convergence rates are dominantly dependent on the spectrum of the hermitian part but are less dependent on the spectrum of the skew hermitian part and are independent of the eigenvectors of the matrices involved optimal choices of the inner iteration steps in the ihss cg lanczos and ihss cg cgne iterations are discussed in detail by considering both global convergence speed and overall computation workload and computational efficiencies of both inexact iterations are analyzed and compared deliberately

solving fuzzy transportation problems based on extension principle abstract transportation models play an important role in logistics and supply chain management for reducing cost and improving service this paper develops a procedure to derive the fuzzy objective value of the fuzzy transportation problem in that the cost coefficients and the supply and demand quantities are fuzzy numbers the idea is based on the extension principle a pair of mathematical programs is formulated to calculate the lower and upper bounds of the fuzzy total transportation cost at possibility level from different values of the membership function of the objective value is constructed two different types of the fuzzy transportation problem are discussed one with inequality constraints and the other with equality constraints it is found that the membership function of the objective value of the equality problem is contained in that of the inequality problem since the objective value is expressed by a membership function rather than by a crisp value more information is provided for making decisions

solving bi level linear programming problem through hybrid of immune genetic algorithm and particle swarm optimization algorithm bi level linear programming consisting of upper level and lower level objectives is a technique for modeling decentralized decision this study presents a hybrid of immune genetic algorithm and vector controlled particle swarm optimization igvpso to solve the bi level linear programming problem blpp it is applied to a supply chain model that is a blpp using four problems from the literature and the supply chain distribution models the computational results indicate that the proposed method is superior to some algorithms

on the calculation of a membership function for the solution of a fuzzy linear optimization problem in the present paper the fuzzy linear optimization problem with fuzzy coefficients in the objective function is considered recent concepts of fuzzy solution to the fuzzy optimization problem based on the level cut and the set of pareto optimal solutions of a multiobjective optimization problem are applied chanas and kuchta suggested one approach to determine the membership function values of fuzzy optimal solutions of the fuzzy optimization problem which is based on calculating the sum of lengths of certain intervals the purpose of this paper is to determine a method for realizing this idea we derive explicit formulas for the bounds of these intervals in the case of triangular fuzzy numbers and show that only one interval needs to be considered

two block triangular preconditioners for asymmetric saddle point problems in this paper two block triangular preconditioners for the asymmetric saddle point problems with singular 1 1 block are presented the spectral characteristics of the preconditioned matrices are discussed in detail theoretical analysis shows that all the eigenvalues of the preconditioned matrices are strongly clustered numerical experiments are reported to the efficiency of the proposed preconditioners

optimal parameter in hermitian and skew hermitian splitting method for certain two by two block matrices the optimal parameter of the hermitian skew hermitian splitting hss iteration method for a real two by two linear system is obtained the result is used to determine the optimal parameters for linear systems associated with certain two by two block matrices and to estimate the optimal parameters of the hss iteration method for linear systems with n by n real coefficient matrices numerical examples are given to illustrate the results

distribution planning decisions using interactive fuzzy multi objective linear programming this work develops an interactive fuzzy multi objective linear programming i fmolp method for solving the fuzzy multi objective transportation problems with piecewise linear membership function the proposed i fmolp method aims to simultaneously minimize the total distribution costs and the total delivery time with reference to fuzzy available supply and total budget at each source and fuzzy forecast demand and maximum warehouse space at each destination additionally the proposed method provides a systematic framework that facilitates the fuzzy decision making process enabling a decision maker dm to interactively modify the fuzzy data and related parameters until a set of satisfactory solutions is obtained an industrial case is presented to demonstrate the feasibility of applying the proposed method to real transportation problems consequently the proposed method yields an efficient solution and overall degree of dm satisfaction with the determined objective values especially several significant management implications regarding the practical application of the proposed i fmolp method are presented

a modified simplex approach for solving bilevel linear programming problems abstract a method for solving bilevel linear programming problems is proposed the inner problem is represented by its kuhn tucker conditions and the complementary slackness equations are moved to the outer objective as a penalty function the global optimum of the resulting quadratic programme is reached by iterative application of a modified simplex algorithm

a new nonlinear interval programming method for uncertain problems with dependent interval variables this paper proposes a new nonlinear interval programming method that can be used to handle uncertain optimization problems when there are dependencies among the interval variables the uncertain domain is modeled using a multidimensional parallelepiped interval model the model depicts single variable uncertainty using a marginal interval and depicts the degree of dependencies among the interval variables using correlation angles and correlation coefficients based on the order relation of interval and the possibility degree of interval the uncertain optimization problem is converted to a deterministic two layer nesting optimization problem the affine coordinate is then introduced to convert the uncertain domain of a multidimensional parallelepiped interval model to a standard interval uncertain domain a highly efficient iterative algorithm is formulated to generate an efficient solution for the multi layer nesting optimization problem after the conversion three computational examples are given to verify the effectiveness of the proposed method

the existence of best proximity points in metric spaces with the property uc abstract eldred and veeramani in a a eldred p veeramani existence and convergence of best proximity points j math anal appl 323 2006 1001 1006 mr2260159 proved a theorem which ensures the existence of a best proximity point of cyclic contractions in the framework of uniformly convex banach spaces in this paper we introduce a notion of the property uc and extend the eldred and veeramani theorem to metric spaces with the property uc

a bilevel bottleneck programming problem this paper studies a bilevel programming problem with linear constraints and in which the objective functions at both levels are concave bottleneck functions which are to be minimized the problem is a non convex programming problem it is shown that an optimal solution to the problem is attainable at an extreme point of the underlying region the outer level objective function values are ranked in increasing order until a value is reached one of the solutions corresponding to which is feasible for the problem this solution is then the required global optimal solution

on conjugate gradient type methods and polynomial preconditioners for a class of complex non hermitian matrices we consider conjugate gradient type methods for the solution of large linear systemsa x b with complex coefficient matrices of the typea t i i wheret is hermitian and a real scalar three different conjugate gradient type approaches with iterates defined by a minimal residual property a galerkin type condition and an euclidean error minimization respectively are investigated in particular we propose numerically stable implementations based on the ideas behind paige and saunder s symmlq and minres for real symmetric matrices and derive error bounds for all three methods it is shown how the special shift structure ofa can be preserved by using polynomial preconditioning and results on the optimal choice of the polynomial preconditioner are given also we report on some numerical experiments for matrices arising from finite difference approximations to the complex helmholtz equation

asymmetric hermitian and skew hermitian splitting methods for positive definite linear systems in this paper efficient iterative methods for the large sparse non hermitian positive definite systems of linear equations based on the hermitian and skew hermitian splitting of the coefficient matrix are studied these methods include an asymmetric hermitian skew hermitian ahss iteration and its inexact version the inexact asymmetric hermitian skew hermitian iahss iteration which employs some krylov subspace methods as its inner process we theoretically study the convergence properties of the ahss method and the iahss method moreover the contraction factor of the ahss iteration is derived numerical examples illustrating the effectiveness of both ahss and iahss iterations are presented

multiobjective optimization through a series of single objective formulations this work deals with bound constrained multiobjective optimization mop of nonsmooth functions for problems where the structure of the objective functions either cannot be exploited or are absent typical situations arise when the functions are computed as the result of a computer simulation we first present definitions and optimality conditions as well as two families of single objective formulations of mop next we propose a new algorithm called for the biobjective optimization bop problem i e mop with two objective functions the property that pareto points may be ordered in bop and not in mop is exploited by our algorithm generates an approximation of the pareto front by solving a series of single objective formulations of bop these single objective problems are solved using the recent mesh adaptive direct search algorithm for nonsmooth optimization the pareto front approximation is shown to satisfy some first order necessary optimality conditions based on the clarke calculus finally is tested on problems from the literature designed to illustrate specific difficulties encountered in biobjective optimization such as a nonconvex or disjoint pareto front local pareto fronts or a nonuniform pareto front

convergence and existence results for best proximity points we provide a positive answer to a question raised by eldred and veeramani a a eldred p veeramani existence and convergence of best proximity points j math anal appl 323 2006 1001 1006 about the existence of a best proximity point for a cyclic contraction map in a reflexive banach space moreover we introduce a new class of maps called cyclic contractions which contains the cyclic contraction maps as a subclass convergence and existence results of best proximity points for cyclic contraction maps are also obtained 2008 elsevier ltd all rights reserved

a modified pso algorithm for linear optimization problem subject to the generalized fuzzy relational inequalities with fuzzy constraints fri fc in this paper optimization of a linear objective function subject to a generalized fuzzy relational inequalities is investigated in which an arbitrary continuous t norm is considered as fuzzy composition and fuzzy inequality replaces ordinary inequality in the constraints unlike most optimization algorithms in fuzzy relational inequalities with fuzzy constraints fri fc we find a near feasible solution having a better objective value than those resulting from the resolution of the similar problems with crisp ordinary inequality constraints such solutions are called super optima in this paper subsequently an algorithm is proposed to find a super optimum with pre specified desirable infeasibility for this purpose we firstly study some structural properties of the fri fc problem and present a new formulization that is independent of t norms used in the constraints of the problem this new formulation converts the primary problem into an equivalent problem with simple constraints without considering any penalty parameters however it is proved that the transformed equivalent problem enables our algorithm to distinguish unfavorable points and generate a sequence of solutions converging to a super optimum under a certain sufficient condition finally a modified pso is presented in which the ability of pso to solve unconstrained problems with continuous domain the structure of the transformed problem and some fuzzy structural modifications are combined and form an efficient algorithm to solve the generalized fri fc problems the modified pso algorithm has been applied to the generalized fri fc problem defined with ten well known continuous t norms additionally an idea of the fri fc problems as outer approximators for fri problems with ordinary inequalities is also investigated

existence and convergence of best proximity points abstract consider a self map t defined on the union of two subsets a and b of a metric space and satisfying t a b and t b a we give some contraction type existence results for a best proximity point that is a point x such that d x t x dist a b we also give an algorithm to find a best proximity point for the map t in the setting of a uniformly convex banach space

an algorithm for solving the general bilevel programming problem the conflict that naturally arises in a hierarchical system can often be modeled as a multistage optimization problem this paper considers the sequential uncooperative problem in which two decision makers wish to maximize their own objective functions over a feasible region defined by interactive strategy sets the resultant problem is known as the bilevel program such programs are inherently nonconvex and resistant to standard nonlinear programming solution techniques such as piecewise linearization and convex underestimating envelopes alternatively a grid search algorithm is offered which exhibits the desirable property of monotonicity the algorithm is based on two sets of necessary conditions previously developed and combined here to provide an operational check for stationarity and local optimality potential solutions are obtained from a parameterized master program whose feasible region approximates that of the original problem as the one dimensional parameter is varied over the unit interval

joint solution to fuzzy programming problems we propose a new solution concept for fuzzy programming problems it is based on our new method of solving fuzzy equations 101 for simplicity we discuss in detail only fuzzy linear programming in this paper we define and obtain the basic properties of the joint solution a fuzzy vector in rn and the optimal value of the objective function a fuzzy number three examples are presented illustrating these concepts

multiobjective programming in optimization of the interval objective function abstract in this paper the mathematical programming problem whose objective function has interval coefficients is investigated to maximize the interval objective function the order relations which represent the decision maker s preference between interval profits are defined by the right limit the left limit the center and the width of an interval the maximization problem with the interval objective function is converted into a multiobjective problem using the order relations the minimization problem is also converted into a multiobjective problem using the order relations which represent the decision maker s preference between interval costs