an automatic quadrature for cauchy principal value integrals an automatic quadrature is presented for computing cauchy principal value integrals q f c faf t t c dt a c b for smooth functions f t after subtracting out the singularity we approximate the function f t by a sum of chebyshev polynomials whose coefficients are computed using the ftt the evaluations of q f c for a set of values of c in a b are efficiently accomplished with the same number of function evaluations numerical examples are also given

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

constructing the set of efficient objective values in linear multiple objective transportation problems abstract a forward dynamic programming approach is utilized to find an algebraic representation for a polyhedron in objective space associated with a multiple objective transportation problem having k linear objectives this polyhedron has the same efficient structure as the set of all feasible objective values and moreover all of its vertices are efficient the algebraic representation of this polyhedron is of the form y r k hy ua vd where the matrices h u and v are independent of the vector a of availabilities and vector d of demands the procedure is illustrated by the numerical example of isermann h isermann the enumeration of all efficient solutions for a linear multiple objective transportation problem naval research logistics quarterly 26 1979 123 139

a new fuzzy multi objective programming entropy based geometric programming and its application of transportation problems in this paper we introduce a fuzzy mathematical programming with generalized fuzzy number as objective coefficients we also examine a transportation problem with additional restriction there is an additional entropy objective function in the transportation problem besides transportation cost objective function using new fuzzy mathematical programming this multi objective entropy transportation problem with generalized trapezoidal fuzzy number costs has been reduced to a primal geometric programming problem pareto optimal solution of the transportation model is found numerical examples have been provided to illustrate the problem

on the fuzzy multi objective linear programming problem goal programming approach abstract for a fuzzy multi objective linear programming problem we propose a substitute problem the problem is given as the minimization problem the objective function of which is the distance between the l ary restricted optimal values at each optimal solutions kuwano et al 1994 and l ary objective functions with fuzzy parameters

the biobjective undirected two commodity minimum cost flow problem we address the two commodity minimum cost flow problem considering two objectives we show that the biobjective undirected two commodity minimum cost flow problem can be split into two standard biobjective minimum cost flow problems using the change of variables approach this technique allows us to develop a method that finds all the efficient extreme points in the objective space for the two commodity problem solving two biobjective minimum cost flow problems in other words we generalize the hu s theorem for the biobjective undirected two commodity minimum cost flow problem in addition we develop a parametric network simplex method to solve the biobjective problem

solving fuzzy solid transportation problems by an evolutionary algorithm based parametric approach abstract the solid transportation problem stp arises when bounds are given on three item properties the fuzzy solid transportation problem fstp appears when the nature of the data problem is fuzzy this paper deals with the fstp in the case in which the fuzziness affects the constraint set and a fuzzy solution to the problem is required moreover an arbitrary linear or nonlinear objective function is considered in order to find a fuzzy solution to the problem a parametric approach is used to obtain an auxiliary parametric solid transportation problem pstp associated to the original problem as there are no well known solution methods proposed in literature to solve effectively the pstp in this paper an evolutionary algorithm ea based solution method is proposed to solve it which can finally be applied to find a good fuzzy solution to the fstp comparisons with another conventional method are presented and the results show the ea based approach to be better as a whole

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

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

application of fuzzy theories to multiple objective decision making in system design abstract much decision making in the real world takes place in an environment in which the goals constraints and consequences of possible actions are not known precisely to deal with such imprecision in a quantitative manner the tools of fuzzy set theory can be advantageously applied the use and effectiveness of fuzzy theories in the formulation and solution of design problems are developed and described via an application in the engineering design of high speed mechanisms an integrated approach to the design of high speed planar mechanisms is formulated as a multiple objective nonlinear programming problem which combines both the kinematic and dynamic synthesis aspects of mechanism design to model the vague and imprecise information in the problem formulation the tools of fuzzy set theory have been used a novel method of solving the resulting fuzzy multiple objective problem using mathematical programming techniques is presented the outlined procedure should be useful in engineering and management design situations where uncertainty or ambiguity arises about the preciseness of permissible parameters degree of credibility and correctness of statements and judgements

computational fluid flow solution over endothelial cells inside the capillary vascular system computational blood flow analysis through glycocalyx located over the endothelial cells in human capillary vascular system is performed stable numerical schemes are developed for obtaining approximate solutions to the mixed problem to the partial differential equation with variable space operator to model the blood flow numerical techniques are developed by applying a procedure of the solution to the linear difference equation with matrix coefficients the flow equations inside the core flow region and porous region are established discretization is done and the flow velocities in both regions are calculated the effect of the flow over the glycocalyx is investigated the wall shear stresses and the drag force over the glycocalyx are calculated copyright 2007 john wiley sons ltd

an interactive possibilistic programming approach for multiple objective supply chain master planning providing an efficient production plan that integrates the procurement and distribution plans into a unified framework is critical to achieving competitive advantage in this paper we consider a supply chain master planning model consisting of multiple suppliers one manufacturer and multiple distribution centers we first propose a new multi objective possibilistic mixed integer linear programming model mopmilp for integrating procurement production and distribution planning considering various conflicting objectives simultaneously as well as the imprecise nature of some critical parameters such as market demands cost time coefficients and capacity levels then after applying appropriate strategies for converting this possibilistic model into an auxiliary crisp multi objective linear model molp we propose a novel interactive fuzzy approach to solve this molp and finding a preferred compromise solution the proposed model and solution method are validated through numerical tests computational results indicate that the proposed fuzzy method outperforms other fuzzy approaches and is very promising not only for solving our problem but also for any molp model especially multi objective mixed integer models

combinatorial algorithms for the minimum interval cost flow problem the aim of minimum the interval cost flow problem micfp is to find the least cost of the shipment of a commodity through a capacitated network in order to satisfy demands at certain nodes from available supplies at other nodes where there exists some vague in vector cost of problem interval cost is a common event in uncertainty environments where statistical data are applied moreover they almost play an essential role in fuzzy programming specially in the case of using their cuts in this paper a complete order on intervals is defined and efficient combinatorial algorithms for micfp are proposed digital simulation results show the performance of the proposed algorithms compared with real scenarios

two new classes of nonlinear transformations for accelerating the convergence of infinite integrals and series two new classes of nonlinear transformations the d transformation to accelerate the convergence of infinite integrals and the d transformation to accelerate the convergence of infinite series are presented in the course of the development of these transformations two interesting asymptotic expansions one for infinite integrals and the other for infinite series are derived the transformations d and d can easily be applied to infinite integrals 0f t dt whose integrands f t satisfy linear differential equations of the form f t s m k 1p k t f k t and to infinite series s r 1f r w terms f r satisfy a linear difference equation of the form f r s m k 1p k r d kf r such that in both cases the p k have asymptotic expansions in inverse powers of their arguments in order to be able to apply these transformations successfully one need not know explicitly the differential equation that the integrand satisfies or the difference equation that the terms of the series satisfy mere knowledge of the existence of such a differential or difference equation and its order m is enough this broadens the areas to which these methods can be applied the connection between the d and d transformations with some known transformations in shown the use and the remarkable efficiency of the d and d transformations are demonstrated through several numerical examples the computational aspects of these transformations are described in detail

fuzzy programming approach to multiobjective solid transportation problem abstract the linear multiobjective solid transportation problem is a special type of vector minimum problem in which constraints are all equally type and the objectives are conflicting in nature a generalization of the linear multiobjective solid transportation problem in which the supply demand and capacity constraints are not only of equality type but also of inequality type is considered this paper presents an application of fuzzy linear programming to the linear multiobjective solid transportation problem it gives efficient solutions as well as an optimal compromise for this method a fortran program has been developed based on the fuzzy linear programming algorithm this method is compared with some existing algorithms using numerical example

an algorithm based on the fft for a generalized chebyshev interpolation an algorithm for a generalized chebyshev interpolation procedure increasing the number of sample points more moderately than doubling is pre sented the fft for a real sequence is incorporated into the algorithm to enhance its efficiency numerical comparison with other existing algorithms is given

an approximation method for high order fractional derivatives of algebraically singular functions the fractional derivative d qf s email protected email protected 1 of a given function f s with a positive non integer q is defined in terms of an indefinite integral we propose a uniform approximation scheme to d qf s for algebraically singular functions f s s ag s a 1 with smooth functions g s the present method consists of interpolating g s at sample points t j in 0 1 by a finite sum of the chebyshev polynomials we demonstrate that for the non negative integer m such that m q m 1 some numerical examples in the simplest case 1

uncertain solid transportation problems the solid transportation problem arises when bounds are given on three item properties usually these properties are source destination and type of product or mode of transport and often are given in a uncertain way this paper deals with two of the ways in which uncertainty can appear in the problem interval solid transportation problem and fuzzy solid transportation problem the first arises when data problem are expressed as intervals instead of point values and the second when the nature of the information is vague both models are treated in the case in which the uncertainty affects only the constraint set for interval case an auxiliary problem is obtained in order to find a solution this auxiliary problem is a standard solid transportation problem which can be solved with the efficient methods existing for fuzzy case a parametric approach which makes it possible to find a fuzzy solution to the former problem is used

a class of rough multiple objective programming and its application to solid transportation problem this paper establishes a rough multiple objective programming model for a solid transportation problem furthermore a general model for rough multiple objective programming problem is presented properties of the feasible and efficient solutions of rough multiple objective programming problems are investigated in addition compromise solutions are obtained by using the interactive fuzzy satisfying method to solve the rough multiple objective solid transportation problem the rough simulation based genetic algorithm is proposed in which the rough simulation is embedded into the genetic algorithm finally an application of the solid transportation problem at xiluodu hydropower station is provided as an illustration

complete solutions of multiple objective transportation problems with possibilistic coefficients abstract this paper deals with the complete solutions of multiple objective transportation problems with possibilistic coefficients we consider the problem by incorporating possibilistic data into the coefficients of objectice functions a solution concept that is attractive from the standpoint of efficiency is specified a necessary and sufficient condition for such a solution is established a relationship between solutions of possibilistic levels is constructed the parametric analysis is used to decompose the parametric space of the equivalent problem a numerical example is given to illustrate the aspects of the developed results

the stability of difference schemes of second order of accuracy for hyperbolic parabolic equations a nonlocal boundary value problem for hyperbolic parabolic equations in a hilbert space h is considered difference schemes of second order of accuracy difference schemes for approximate solution of this problem are presented stability estimates for the solution of these difference schemes are established

approximating the singular integrals of cauchy type with weight function on the interval it is known that the solutions of characteristic singular integral equations sies are expressed in terms of singular integrals of cauchy type with weight functions w x 1 x n 1 x m where n 12 m 12 new quadrature formulas qfs are presented to approximate the singular integrals sis of cauchy type for all solutions of characteristic sie on the interval 1 1 linear spline interpolation modified discrete vortex method and product quadrature rule are utilized to construct the qfs estimation of errors are obtained in the classes of functions h a 1 1 a and c 1 1 1 it is found that the numerical results are very stable even for the cases of semi bounded and unbounded solutions of singular integral equation of the first kind

bector chandra type duality in fuzzy linear programming with exponential membership functions in this paper we study a pair of fuzzy primal dual linear programming problems and calculate duality results using an aspiration level approach we use an exponential membership function which is in contrast to the earlier works that relied on a linear membership function as the fuzzy environment causes a duality gap we investigate how choosing the exponential membership function impacts this gap this issue is particularly important for fuzzy linear programming where in general the primal and dual objective values may not be bounded

fuzzy multi attribute selection among transportation companies using axiomatic design and analytic hierarchy process the basic concept in axiomatic design ad is the existence of design axioms first of these axioms is the independence axiom and the second one is the information axiom information axiom proposes the selection of the best alternative that has minimum information analytic hierarchy process ahp is another multi attribute method which is a decision making method for selecting the best among a set of alternatives given some criteria the method has been extensively applied especially in large scale problems where many criteria must be considered and where the evaluation of alternatives is mostly subjective multi attribute transportation company selection is a very important activity for effective supply chain selection of the best company under determined criteria such as cost time damage loss flexibility and documentation ability using both multi attribute ad and ahp will be realized in this study the fuzzy multiattribute ad approach is also developed and it is compared by one of fuzzy ahp methods in the literature the selection process has been accomplished by aiding a software that includes crisp ad and fuzzy ad

multiobjective transportation problem with interval cost source and destination parameters in this paper we focus on the solution procedure of the multiobjective transportation problem motp where the cost coefficients of the objective functions and the source and destination parameters have been expressed as interval values by the decision maker this problem has been transformed into a classical motp where to minimize the interval objective function the order relations that represent the decision maker s preference between interval profits have been defined by the right limit left limit centre and half width of an interval the constraints with interval source and destination parameters have been converted into deterministic ones finally the equivalent transformed problem has been solved by fuzzy programming technique numerical examples have been provided to illustrate the solution procedure for three possible cases of the original problem

fuzzy programming technique to solve multi objective transportation problems with some non linear membership functions abstract the multi objective transportation problem refers to a special class of vector minimum linear programming problems in which the constraints are of equality type and all the objectives are non commensurable and conflict with each other all the methods either generate a set of non dominated solutions or find a compromise solution in this paper we use a special type of non linear hyperbolic and exponential membership functions to solve the multi objective transportation problem it gives an optimal compromise solution the obtained result has been compared with the solution obtained by using a linear membership function to illustrate the methodology a numerical example is presented

approximating cauchy type singular integral by an automatic quadrature scheme an automatic quadrature scheme is developed for the approximate evaluation of the product type indefinite integral q f x y c x y r t k t c f t dt 1 x y 1 1

an algorithm for a special case of a generalization of the richardson extrapolation process a special case of a generalization of the richardson extrapolation process is considered and its complete solution is given in closed form using this an algorithm for implementing the extrapolation is devised it is shown that this algorithm needs a very small amount of arithmetic operations and very little storage convergence and stability properties for some cases are also considered

the biobjective minimum cost flow problem abstract in this paper we study the single commodity flow problem optimizing two objectives simultaneously we propose a method that finds all the efficient extreme points in the objective space this method is based on the well known method of lee and pulat which also computes some efficient points of the objective space that are not extreme points our alternative method corrects this particularity by using the idea of adjacency among efficient extreme points in the objective space and introduces computational improvements that are shown in an experimental study

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

a note on the second order of accuracy difference schemes for hyperbolic parabolic equations abstract the nonlocal boundary value problem for hyperbolic parabolic equation d 2 u t d t 2 au t f t 0 t 1 d u t d t au t g t 1 t 0 u 1 u 0 1 0 1 in a hilbert space h is considered the second order of accuracy difference schemes approximately solving this boundary value problem are presented the stability estimates for the solution of these difference schemes are established in applications the stability estimates for the solutions of the difference schemes of the mixed type boundary value problems for hyperbolic parabolic equations are obtained the theoretical statements for the solution of these difference schemes for hyperbolic parabolic equation are supported by the results of numerical experiments

bicriteria network flow problems integer case abstract we consider a bicriteria network flow problem with flow variables restricted to integer values the set of efficient solutions to the problem can be obtained by adopting a bicriteria integer programming technique the purpose of this paper is to present a new technique which capitalizes on the network structure and the unimodular property of the incidence matrix the procedure uses efficient solutions to the continuous version of the bicriteria network flow problem and parametric programming in order to generate the efficient solutions to the bicriteria integer network flow problem the performance of the algorithm is also discussed

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 integer transportation problem an algorithm has been proposed which solves the transportation problem with fuzzy supply and demand values and the integrality condition imposed on the solution this algorithm is exact and computationally effective although the problem is formulated in the general way i e its fuzzy supply and demand values can differ from each other and be fuzzy numbers of any type

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

priority structure in fuzzy goal programming abstract this paper introduces priority structure in fuzzy goal programming fgp this approach utilizes the lexicographic order of goal programming gp and yields an efficient computational algorithm for solving fgp fuzzy priorities are also discussed a numerical example is given to highlight the procedure concluding remarks are also made

quadrature formulas based on rational interpolation we consider quadrature formulas based on interpolation using the ba sis functions 1 1 tkx k 1 2 3 on 1 1 where tk are parameters on the interval 1 1 we investigate two types of quadratures quadrature formulas of maximum accuracy which correctly integrate as many basis functions as possible gaussian quadrature and quadrature formulas whose nodes are the zeros of the orthogonal functions obtained by orthogonalizing the system of basis functions or thogonal quadrature we show that both approaches involve orthogonal polynomials with modified or varying weights which depend on the number of quadrature nodes the asymptotic distribution of the nodes is obtained as well as various interlacing properties and monotonicity results for the nodes

a multi objective transportation problem under fuzziness this paper presents a fuzzy programming approach to determine the optimal compromise solution of a multi objective transportation problem motp its fuzzy membership function is defined the approach performance is tested by measuring the degree of closeness of the compromise solution to the ideal solution using a family of distance functions in addition the paper shows that the fuzzy approach outperforms the interactive procedure as the number of objectives and constraints increases two illustrative numerical examples are provided to illustrate the approach

an interactive fuzzy satisficing method for multiobjective linear fractional programming problems abstract we present a new interactive fuzzy decision making method for solving multiobjective linear fractional programming problems by assuming that the decision maker dm has fuzzy goals for each of the objective functions through the interaction with the dm the fuzzy goals of the dm are quantified by eliciting the corresponding membership functions including nonlinear functions after determining the membership functions if the dm specifies reference membership values the minimax problem is solved by combined use of the bisection method and the linear programming method and the dm is supplied with the corresponding pareto optimal solution together with the trade off rates between the membership functions then by considering the current values of the membership functions as well as the trade off rates the dm responds by updating his her reference membership values in this way the compromise or satisficing solution for the dm can be derived efficiently from among a pareto optimal solution set by updating his her reference membership values on the basis of the proposed method a time sharing computer program is written and an illustrative numerical example is demonstrated along with the computer outputs

computing improved optimal solutions to max min flexible constraint satisfaction problems abstract the formal framework for decision making in a fuzzy environment is based on a general max min bottleneck like optimization problem proposed by zadeh it is also the basis for extending the constraint satisfaction paradigm of artificial intelligence to accommodating flexible or prioritized constraints this paper surveys refinements of the ordering of solutions supplied by the max min formulation namely the discrimin partial ordering and the leximin complete preordering a general algorithm is given which computes all maximal solutions in the sense of these relations it also sheds light on the structure of the set of best solutions moreover classes of problems for which there is a unique best discrimin and leximin solution are exhibited namely continuous problems with convex domains and so called isotonic problems noticeable examples of such problems are fuzzy linear programming problems and fuzzy pert like scheduling problems

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

an algorithm for a generalization of the richardson extrapolation process in this paper we present a recursive method designated the w m algorithm for implementing a generalization of the richardson extrapolation process that has been ntroduced in 8 compared to the direct solution of the linear systems of equations defining the extrapolation procedure this method requires a small number of arithmetic operations and very little storage the technique is also applied to solve recursively the coefficient problem associated with the rational approximations obtained by applying the d transformation of 6 13 to power series in the course of development a new recursive algorithm for implementing a very general extrapolation procedure is introduced which is similar to that given in 2 4 for solving the same problem a fortran program for the w m algorithm is also appended

quadrature rule for indefinite integral of algebraic logarithmic singular integrands an automatic quadrature method is presented for approximating the indefinite integral of functions having algebraic logarithmic singularities q x y c f email protected x yf t t c alog t c dt 1 1 within a finite range 1 1 for some smooth function f t that is approximated by a finite sum of chebyshev polynomials we expand the given indefinite integral in terms of chebyshev polynomials by using auxiliary algebraic logarithmic functions present scheme approximates the indefinite integral q x y c f uniformly namely bounds the approximation error independently of the value c as well x and y this fact enables us to evaluate the integral transform q x y c f with varied values of x y and c efficiently some numerical examples illustrate the performance of the present quadrature scheme

a new approach to some possibilistic linear programming problems abstract in practice the unit costs profits of new products or new projects the lending and borrowing interest rates and cash flows are always imprecise we provide an auxiliary multiple objective linear programming model to solve a linear programming problem with imprecise objective and or constraint coefficients our strategy is to maximize the most possible value of the imprecise profit at the same time we would like to minimize the risk of obtaining lower profit and maximize the possibility of obtaining higher profit this strategy is equivalent to the practical considerations of financial problems in this paper a numeric investment problem is solved for illustrating the new approach

a bi objective column generation algorithm for the multi commodity minimum cost flow problem we present a column generation algorithm for solving the bi objective multi commodity minimum cost flow problem this method is based on the bi objective simplex method and dantzig wolfe decomposition the method is initialised by optimising the problem with respect to the first objective a single objective multi commodity flow problem which is solved using dantzig wolfe decomposition then similar to the bi objective simplex method our algorithm iteratively moves from one non dominated extreme point to the next one by finding entering variables with the maximum ratio of improvement of the second objective over deterioration of the first objective our method reformulates the problem into a bi objective master problem over a set of capacity constraints and several single objective linear fractional sub problems each over a set of network flow conservation constraints the master problem iteratively updates cost coefficients for the fractional sub problems based on these cost coefficients an optimal solution of each sub problem is obtained the solution with the best ratio objective value out of all sub problems represents the entering variable for the master basis the algorithm terminates when there is no entering variable which can improve the second objective by deteriorating the first objective this implies that all non dominated extreme points of the original problem are obtained we report on the performance of the algorithm on several directed bi objective network instances with different characteristics and different numbers of commodities

refinements of the maximin approach to decision making in a fuzzy environment abstract the most popular approach to decision making in the setting of fuzzy sets is the maximin ranking of solutions this method is natural when interpreting the fuzzy sets as flexible constraints that cannot compensate with one another however the obtained ranking of solutions is very coarse two kinds of refinements to this ordering are introduced a partial ordering according to the least satisfied discriminating constraint and a lexicographical ranking the latter refines the former and combines utilitarist and egalitarist points of view on the aggregation of feasibility degrees these orderings are characterized in several ways and their representation by means of two place numerical functions is studied dual refinements of the maximax ranking are provided

an operator difference scheme for abstract cauchy problems an abstract cauchy problem for second order hyperbolic differential equations containing the unbounded self adjoint positive linear operator a t with domain in an arbitrary hilbert space is considered a new second order difference scheme generated by integer powers of a t is developed the stability estimates for the solution of this difference scheme and for the first and second order difference derivatives are established in hilbert norms with respect to space variable to support the theoretical statements for the solution of this difference scheme the numerical results for the solution of one dimensional wave equation with variable coefficients are presented

fuzzy total transportation cost measures for fuzzy solid transportation problem abstract the solid transportation problem considers the supply the demand and the conveyance to satisfy the transportation requirement in a cost effective manner this paper develops a method that is able to derive the fuzzy objective value of the fuzzy solid transportation problem when the cost coefficients the supply and demand quantities and conveyance capacities are fuzzy numbers based on the extension principle the fuzzy solid transportation problem is transformed into a pair of mathematical programs that is employed 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 approximated since the objective value is fuzzy the values of the decision variables derived in this paper are fuzzy as well an example is illustrated for the whole idea

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

error bounds for gaussian quadrature of analytic functions for gaussian quadrature rules over a finite interval applied to analytic or meromorphic functions we develop error bounds from contour integral representations of the remainder term as in previous work on the subject we consider both circular and elliptic contours in contrast with earlier work however we attempt to determine exactly where on the contour the kernel of the error functional attains its maximum modulus we succeed in answering this question for a large class of weight distributions including all jacobi weights when the contour is a circle in the more difficult case of elliptic contours we can settle the question for certain special jacobi weight distributions with parameters pm frac 1 2 and we provide empirical results for more general jacobi weights we further point out that the kernel of the error functional at any complex point outside the interval of integration can be evaluated accurately and efficiently by a recursive procedure the same procedure is useful also t

quadrature formulas for functions with poles near the interval of integration on examine la construction des regles de quadrature de type interpolatoire utilisant uniquement des valeurs de fonctions reelles pour des fonctions ayant des paires complesex conjuguees de poles pres de l intervalle d integration

on the numerical solution of the diffusion equation with variable space operator in present paper numerical schemes are developed for obtaining approximate solutions to the mixed problem for one dimensional diffusion equation with variable space operator the method of lines semidiscretization approach is used to transform the model partial differential equation into a system of first order linear ordinary differential equations the partial derivative with respect to the space variable is approximated by the first and second order finite difference approximation for the solution of the resulting system of first order ordinary differential equations we apply the first and second order of accuracy difference schemes stability estimates for the solution of these difference schemes are established numerical techniques are developed by applying a procedure of the solution of first order linear difference equation with matrix coefficients the algorithms are tested on a model problem in biofluid mechanics two regions are considered close to the endothelial cells ec which can be modeled taking the mixed problem for one dimensional diffusion equation with variable space operator

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

fuzzy programming approach to multicriteria decision making transportation problem abstract the linear multiobjective transportation problem is a special type of vector minimum problem in which constraints are all equality type and the objectives are conflicting in nature all existing methods generate a set of nondominated solutions or construct a single compromise solution this paper presents an application of fuzzy linear programming to the linear multiobjective transportation problem it gives efficient solutions as well as an optimal compromise solution for this method an efficient fortran program has been developed based on the fuzzy linear programming algorithm which is an extended version of the simplex algorithm this method is compared with two existing interactive algorithms using two numerical examples

hilbert and hadamard transforms by generalized chebyshev expansion abstract an automatic quadrature is presented for approximating hadamard finite part fp integrals of a smooth function with a double pole singularity within the range of integration the quadrature rule is derived from the differentiation of an approximation to a cauchy principal value integral or the hilbert transform the approximation to the fp integral is represented as a function of the value of pole by using chebyshev polynomials of the second kind since the error can be estimated independently of the value of pole a set of integrals for a set of values of pole can be efficiently approximated to a required tolerance with the same number of function evaluations numerical examples are also included to illustrate the performance of the methods

application of a modified fft to product type integration abstract an automatic integration scheme is proposed for evaluating the so called product type indefinite integral q k y x k t t dt 1 x y where t is assumed to be a smooth function and k t are some singular or badly behaved functions typical examples for k t are 1n t c t c 1 cauchy principal value 1 t c and ei t 1 the function t is approximated by a truncated chebyshev series pn t of degree n whose coefficients are efficiently computed using the fft the approximation qn k to the integral q k is given by q k pn the sequence pn t is recursively generated until the required tolerance for the integral is satisfied to enhance the efficiency of the automatic quadrature the degree n is increased more slowly than doubling which is usually the case the evaluations of qn k q k pn for a set of x y c can be efficiently made by using recurrence relations for the singular kernels k t above numerical examples for the algebraic singular kernel k t t c 1 are included

stability of difference schemes for hyperbolic parabolic equations the stable difference schemes approximately solving the nonlocal boundary value problem for hyperbolic parabolic equationd 2u t dt 2 au t f t 0 t 1 u 1 au m bu l fdu t dt au t g t 1 t 0 a b 1 0 m l 1in a hilbert space h with self adjoint positive definite operator a are presented the stability estimates for the solutions of the difference schemes of the mixed type boundary value problems for hyperbolic parabolic equations are obtained the theoretical statements for the solution of these difference schemes for hyperbolic parabolic equation are supported by the results of numerical experiments

generalized chebyshev interpolation and its application to automatic quadrature a generalized chebyshev interpolation procedure increasing a fixed number of sample points at a time is developed and analvzed it is incorporated into an efficient automatic quadrature scheme of clenshaw curtis tvpe numerical examples indicate that the present method is efficient not onlv for well behaved functions but for those w ith discontinuous low order derivatives by virtue of adequate error estimation as wvell as saving of sample points

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

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

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

interactive solutions for the linear multiobjective transportation problem abstract mathematical programming problems that exhibit the mathematical structure of a transportation problem often arise in settings with multiple conflicting objectives existing procedures for analyzing these problems fall into two general categories these methods either generate all nondominated solutions or they construct a single compromise solution this paper presents two interactive algorithms which take advantage of the special form of the multiple objective transportation problem two examples are included to illustrate these algorithms and to demonstrate their viability

an additive fuzzy programming model for multiobjective transportation problem abstract all existing methods generate a set of nondominated solutions or construct a single compromise solution for the linear multiobjective transportation problem in which objectives are equally important in this paper an additive fuzzy programming model for the multiobjective transportation problem is presented the method aggregates the membership functions of the objectives to construct the relevant decision function weights and priorities for nonequivalent objectives are also incorporated in the method this model gives a nondominated solution which is nearer to the best compromise solution a fortran program has been developed based on simple and weighted additive algorithms the method is illustrated with a numerical example

the use of rational functions in numerical quadrature quadrature problems involving functions that have poles outside the interval of integration can profitably be solved by methods that are exact not only for polynomials of appropriate degree but also for rational functions having the same or the most important poles as the function to be integrated constructive and computational tools for accomplishing this are described and illustrated in a number of quadrature contexts the superiority of such rational polynomial methods is shown by an analysis of the remainder term and documented by numerical examples

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

non l r type fuzzy parameters in mathematical programming problems the triangular norm based operations in fuzzy logic usually lead to non l r type fuzzy sets this study considers mathematical programming problems with non l r type fuzzy parameters it shows that the fuzzy solutions to such problems can be obtained by solving an optimization problem on a mixed domain the necessary and sufficient conditions for solving the resulting optimization problems are investigated by employing the theory of convex optimization on mixed domains this is the first attempt to solve the fuzzy optimization problem with non l r type membership functions in view of optimization problems on a mixed domain

strongly p positive operators and explicit representations of the solutions of initial value problems for second order differential equations in banach space abstract the initial value problems for two second order differential equations with an unbounded operator coefficient a in a banach space are considered using a linear fractional transform the cayley transform of the operator a we give explicit formulas for the solution of these problems if the spectrum of a is situated inside of a parabola these formulas also provide the algorithmic representations of the operator cosine function and of the operator bessel function with the generator a being a basis for approximate solutions for which error estimates are given one of the important properties of our approach is the following the accuracy of the approximate solutions depends automatically on the regularity of the initial data

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

fuzzy programming technique to solve multi objective geometric programming problems abstract in this paper fuzzy programming technique is used to solve a multi objective geometric programming problem as a vector minimum problem a fuzzy membership function is defined for the multi objective geometric programming problem two numerical examples are presented to illustrate the method

numerical integration of functions with poles near the interval of integration abstract an automatic quadrature scheme is presented for approximating integrals of functions that are analytic in the interval of integration but contain pole or poles of order 2 i e a double pole on the real axis or a complex conjugate pair of double poles near the interval of integration the present scheme is based on product integration rules of interpolatory type using function values of the abscissae only in the interval of integration the integral is approximated and evaluated by using recurrence relations and some extrapolation method after the smooth part of the integrand is expanded in terms of the chebyshev polynomials the fast fourier transform fft technique is used to generate efficiently the sequence of the finite chebyshev series expansions until an approximation of the integral satisfying the required tolerance is obtained with an adequate estimate of the error numerical examples are included to illustrate the performance of the method

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