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

a difference scheme for a nonlinear partial integrodifferential equation a difference method for the numerical integration of a nonlinear partial integrodifferential equation is considered the integral term is treated by means of a convolution quadrature suggested by lubich some results from lubich s discretized fractional calculus play a crucial role in proving consistency the verification of stability and convergence is based on the nonnegative character of the real quadratic form associated with the convolution quadrature a stability result is derived that is applicable to equations and numerical methods far more general than those treated in this paper

decay rates for inverses of band matrices spectral theory and classical approximation theory are used to give a new proof of the exponential decay of the entries of the inverse of band matrices the rate of decay of a 1 can be bounded in terms of the essential spectrum of aa for general a and in terms of the essential spectrum of a for positive definite a in the positive definite case the bound can be attained these results are used to establish the exponential decay for a class of generalized eigenvalue problems and to establish exponential decay for certain sparse but nonbanded matrices we also establish decay rates for certain generalized inverses

the boolean pivot operation m matrices and reducible matrices abstract let a be a boolean n n matrix and let g n u be the corresponding diagraph where n 1 n is the set of vertices and u n n is the set of arcs of g for any r n combining the steps r r of warshall s algorithm for determining the reachability matrix of a yields the boolean pivot operation b r the matrix b r a is the so called r reachability matrix of a a ij 1 if and only if aij 1 or there is a connection between i and j via vertices belonging to r we also have b r b s b s b r b r s for any r s n the boolean pivot operation is closely related to the principal pivotal operation for real matrices so we obtain the boolean analogues to the formula for inverting a real block matrix and for the sherman morrison woodbury formulae for updating the inverse of a real matrix using the close connection between nonsingular m matrices and the corresponding boolean matrices we obtain a flexible algorithm for excluding and including vertices one at a time in g while retaining the original connections between the vertices of the current diagraph we derive some criteria for irreducibility of boolean block matrices and of partitioned z matrices and give a condensed form of warshall s algorithm for testing whether a boolean matrix is irreducible moreover we correct two recent results by r l smith on irreducibility of real block matrices

a generalization of n matrices abstract ky fan defines an n matrix to be a matrix of the form a ti b b 0 t b where b is the spectral radius of b and is the maximum of the spectral radii of all principal submatrices of b of order n 1 in this paper we define the closure n 0 matrices of n matrices by letting t it is shown that if a z and a 1 a n certain inequalities of n matrices are shown to hold for n 0 matrices and a method for constructing an n matrix from an m matrix is given

cholesky factorization of semidefinite toeplitz matrices abstract it can be shown directly from consideration of the schur algorithm that any n n semidefinite rank r toeplitz matrix t has a factorization t c r c t r with c r c 11 c 12 0 0 where c 11 is r r and upper triangular this paper explores the reliability of computing such a decomposition with o nr complexity using the schur algorithm and truncating the cholesky factor after computing the first r rows the theoretical conclusion is that as with gaussian elimination with partial pivoting the algorithm is stable in a qualified sense the backward error bounds can be expressed purely in terms of n and r and are independent of the condition number of the leading r r submatrix t 11 of t but they grow exponentially in r thus the algorithm is completely reliable when r is small but assessing its reliability for larger r requires making a judgment as to how realistic the error bounds are in practice experimental results are presented which suggest that the bounds are pessimistic and the algorithm is stable for many types of semidefinite toeplitz matrices the analysis and conclusions are similar to those in a paper by higham dealing with general semidefinite matrices

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

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

second order bounds for linear recurrences with negative coefficients this paper introduces a generalization of fibonacci and pell polynomials in order to obtain optimal second order bounds for general linear recurrences with negative coefficients an important aspect of the derived bounds is that they are applicable and easily computable the results imply bounds on all entries in inverses of triangular matrices as well as on coefficients of reciprocals of power series

monotone convex sequences and cholesky decomposition of symmetric toeplitz matrices this paper studies off diagonal decay in symmetric toeplitz matrices it is shown that if the generating sequence of the matrix is monotone positive and convex then the monotonicity and positivity are maintained through triangular decomposition the work is motivated by recent results on explicit bounds for inverses of triangular matrices

decay rates of inverses of banded m matrices that are near to toeplitz matrices the decaying behavior of inverses of positive definite band matrices is analysed for m matrices that are in some sense close to toeplitz matrices estimates based on the factorization are derived that are better than existing ones in particular for nonsymmetric matrices some examples are given

a dual approach to solve the fuzzy linear programming problem abstract a concept of fuzzy objective based on the fuzzification principle is presented in accordance with this concept the fuzzy linear mathematical programming problem is easily solved a relationship of duality among fuzzy constraints and fuzzy objectives is given the dual problem of a fuzzy linear programming problem is also defined

satisfying solutions for a possibilistic linear program abstract a branch and bound scheme is described for tackling a linear program under discrete possibilistic data we grapple also with the continuous case via recent results on inexact and semiinfinite programming it is shown that solutions yielded by these techniques are satisfying i e possibly and or necessarily feasible and optimal to a great extent finally approaches reported here and elsewhere are appraised in the light of their decision making philosophies and some directions for further developments indicated

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

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

updating the inverse of a matrix the sherman morrison woodbury formulas relate the inverse of a matrix after a small rank perturbation to the inverse of the original matrix the history of these fomulas is presented and various applications to statistics networks structural analysis asymptotic analysis optimization and partial differential equations are discussed the sherman morrison woodbury formulas express the inverse of a matrix after a small rank perturbation in terms of the inverse of the original matrix this paper surveys the history of these formulas and we examine some applications where these formulas are helpful

non additive multi attribute fuzzy target oriented decision analysis as an emerging area considering behavioral aspects of decision making target oriented decision model lies in the philosophical root of bounded rationality as well as represents the s shaped value function this paper deals with multi attribute decision analysis from target oriented viewpoint first the basic random target oriented decision model is extended to involve three types of target preferences benefit target cost target and equal target next since applying fuzzy set theory in decision analysis allows the decision maker to specify imprecise aspiration levels fuzzy target oriented decision analysis is formulated to model three typical types of fuzzy targets fuzzy min fuzzy max and fuzzy equal also different attitudes are used to derive target achievement functions which can be viewed as a support for probability as psychological distance furthermore we have proved that multi attribute target oriented decision analysis has a similar structure with discrete fuzzy measure and choquet integral hence we propose using discrete fuzzy measure and choquet integral to model non additive multi attribute target oriented decision analysis in particular the l measure is applied to reduce the difficulty of collecting information via a designed bisection search algorithm finally a new product development example is used to illustrate the effectiveness and advantages of our model the main advantages of our target oriented decision model are its abilities to model the fuzzy uncertainty of targets as well as capture the non additive behaviors among targets by means of discrete fuzzy measure and choquet integral

a solution algorithm for fuzzy linear programming with piecewise linear membership functions abstract this paper deals with the fuzzy linear program with continuous piecewise linear membership functions a fuzzy linear program with piecewise linear membership functions has been considered by hannan and nakamura hannan s method can solve the problems only when all membership functions are concave in the range 0 1 and nakamura s method needs to use the linear programming technique repetitionally we propose a technique to solve the problem using a standard linear programming when membership functions are strictly quasiconcave and the minimum operator is adopted for aggregating fuzzy goals the repetitional use of linear programming technique is not required in our method

some notes on z matrices abstract an inequality is obtained relating the eigenvalue of minimum modulus of an n0 matrix a to the eigenvalue of minimum modulus of a a11 the schur complement of a11 in a where a11 is an m matrix similar results are obtained for irreducible m matrices and irreducible nonsingular f0 matrices circulant n0 matrices of order three are characterized in terms of their spectrum matrices whose inverses are f0 matrices are shown to have nonpositive almost principal minors

on finding compromise solutions in multicriteria problems using the fuzzy min operator abstract this paper presents an application of fuzzy approaches to the linear vectormaximum problem it shows that using the fuzzy min operator together with linear as well as special nonlinear membership functions the obtained solutions are always compromise solutions of the original multicriteria problem

fuzzy approaches for multiple objective linear fractional optimization abstract the problem of finding a solution to a multiple objective linear fractional program arises in several real world situations in this paper we advocate that fuzzy sets theory provides a basis for solving this problem with sufficient consistency and rigorousness after representing imprecise aspirations of the decision maker by structured linguistic variables or converting the original problem via approximations or change of variables into a multiple objective linear program techniques of fuzzy linear programming may be used to reach a satisfactory solution it is shown that under reasonable restrictions this solution is efficient pareto optimal for the original problem numerical examples are also included for illustration

fuzzy goal programming with nested priorities abstract this paper proposes a new approach to formulating fuzzy priorities in a goal programming problem the proposed methodology remedies certain shortcomings of the composite membership function approach discussed in previous works 7 10 the principal advantage of the proposed method is that it leads to a formulation in which tradeoffs between goals more closely reflect the decision maker s intentions than in other noninteractive approaches 8 9 10 14 in some of which a fixed hierarchy of goals is assumed

a new approach to fuzzy programming abstract a fuzzy program is defined in the usual way as a sequence of statements instruction which are considered as functions possibly fuzzy functions and fuzzy predicates defined on the given input domain the essential difference in the approach presented in this paper is the new interpretation of the execution of fuzzy programs and a new method of evaluating fuzzy predicates the result of the fuzzy program execution is an appropriate fuzzy subset in the output domain

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

four short stories about toeplitz matrix calculations the stories told in this paper are dealing with the solution of finite infinite and biinfinite toeplitz type systems a crucial role plays the off diagonal decay behavior of toeplitz matrices and their inverses classical results of gelfand et al on commutative banach algebras yield a general characterization of this decay behavior we then derive estimates for the approximate solution of bi infinite toeplitz systems by the finite section method showing that the approximation rate depends only on the decay of the entries of the toeplitz matrix and its condition number furthermore we give error estimates for the solution of doubly infinite convolution systems by finite circulant systems finally some quantitative results on the construction of preconditioners via circulant embedding are derived which allow us to provide a theoretical explanation for numerical observations made by some researchers in connection with deconvolution problems

a polynomial algorithm for testing the nonnegativity of principal minors of z matrices abstract a z matrix is a square matrix with nonpositive off diagonal elements we give a polynomial algorithm for testing the nonnegativity of principal minors of z matrices

linear programming under randomness and fuzziness this paper is concerned with the use of fuzzy sets theory well suited for nuanced human behavior to extend the applicability of linear programming in a way to let both randomness and fuzziness uncertainties be taken into account a fuzzy stochastic linear programming problem is considered and ways to handle it suggested the basic idea behind our approaches is to translate the original problem in deterministic terms via tools provided by fuzzy sets theory to get a classical one some conditions for the resulting problems to be solved without extra computational effort are stressed and illustrative examples given

two sided bounds on the inverses of diagonally dominant tridiagonal matrices abstract we establish upper and lower bounds for the entries of the inverses of diagonally dominant tridiagonal matrices these bounds improve the bounds recently given by shivakumar and ji moreover we show how to improve our bounds iteratively for an n n m matrix this iterative refinement yields the exact inverse after n 1 steps

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

a polynomial time test for m matrices abstract a real square matrix with nonpositive off diagonal elements is a z matrix and a z matrix with nonnegative principal minors is an m matrix using principal pivoting we derive a polynomial time algorithm for testing whether a z matrix is an m matrix our algorithm is much simpler than a recent one proposed by ramamurthy we also study the zero nonzero structure of the inverse of a nonsingular m matrix and compare a 1 and b 1 in the case that a and b are nonsingular m matrices and b a

chance constrained programming with fuzzy parameters this paper extends chance constrained programming from stochastic to fuzzy environments analogous to stochastic programming some crisp equivalents of chance constraints in fuzzy environments are presented we also propose a technique of fuzzy simulation for the chance constraints which are usually hard to be converted to their crisp equivalents finally a fuzzy simulation based genetic algorithm is designed for solving this kind of problems and some numerical examples are discussed

spectral factorization of bi infinite multi index block toeplitz matrices in this paper we formulate a theory of lu and cholesky factorization of bi infinite block toeplitz matrices a ai j i j zd indexed by i j z d and develop two numerical methods to compute such factorizations 2002 elsevier science inc all rights reserved

some improvements for two sided bounds on the inverse of diagonally dominant tridiagonal matrices abstract in this paper we obtain lower and upper bounds for the entries of the inverses of diagonally dominant tridiagonal matrices first of all we derive the bounds for off diagonal elements of the inverse as a function of the diagonal ones then we improve the two sided bounds for the diagonal entries obtaining sharper lower and upper bounds for all the elements of the inverse

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

fuzzy reasoning for solving fuzzy mathematical programming problems abstract we interpret fuzzy linear programming flp problems with fuzzy coefficients and fuzzy inequality relations as multiple fuzzy reasoning schemes mfr where the antecedents of the scheme correspond to the constraints of the flp problem and the fact of the scheme is the objective of the flp problem then the solution process consists of two steps first for every decision variable x r n we compute the fuzzy value of the objective function max x via sup min convolution of the antecedents constraints and the fact objective then an optimal solution to flp problem is any point which produces a maximal element of the set max x x r n in the sense of the given inequality relation we show that our solution process for a classical crisp lp problem results in a solution in the classical sense and under well chosen inequality relations and objective function coincides with those suggested by buckley fuzzy sets and systems 31 1989 329 341 delgado et al control and cybernetics 16 1987 114 121 fuzzy sets and systems 26 1988 49 62 negoita fuzzy sets and systems 6 1981 261 269 ramik and rimanek fuzzy sets and systems 16 1985 123 138 verdegay fuzzy information and decision processes north holland amsterdam 1982 control and cybernetics 13 1984 230 239 and zimmermann int j general systems 2 1976 209 215 furthermore we show how to extend the proposed solution principle to non linear programming problems with fuzzy coefficients we illustrate our approach by some simple examples

a unified approach to one parametric general quadratic programming a method is proposed for finding local minima to the parametric general quadratic programming problem where all the coefficients are linear or polynomial functions of a scalar parameter the local minimum vector and the local minimum value are determined explicitly as rational functions of the parameter a numerical example is given

fuzzy linear programming problems with fuzzy numbers abstract linear programming problems with fuzzy parameters are formulated by fuzzy functions the ambiguity considered here is not randomness but fuzziness which is associated with the lack of a sharp transition from membership to nonmembership parameters on constraint and objective functions are given by fuzzy numbers in this paper our object is the formulation of a fuzzy linear programming problem to obtain a reasonable solution under consideration of the ambiguity of parameters this fuzzy linear programming problem with fuzzy numbers can be regarded as a model of decision problems where human estimation is influential

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