a review of some methods for ranking fuzzy subsets this paper deals with the problem of ranking n fuzzy subsets of the unit interval a number of methods suggested in the literature is reviewed and tested on a group of selected examples where the fuzzy sets can be nonnormal and or nonconvex the ranking is obtained from i the index of strict preference defined by watson ii three indexes proposed by yager iii the algorithm used by chang iv three versions of the a preference index suggested by adamo v the index defined by baas and kwakernaak vi three modified versions used by baldwin and guild vii the method proposed by kerre viii three forms of the index suggested by jain ix the four grades of dominance studied by dubois and prade in simple cases the results are good for all the methods with some exceptions in questionable cases where the decision must be probably modelled in accordance with the context in which it is imbedded the best indexes seem to be the dominances suggested by dubois and prade these indexes do not force any particular choice but clearly describe the situation hence allowing the decision maker himself to make his best choice

fuzzy optimization models for the design of wdm networks the wavelength division multiplexing network design problem is considered in this paper the literature is replete with prescriptive integer programming models for various versions of wavelength division multiplexing network design problems but in no case they consider the real situation of solving them taking into account their inherent vagueness in this paper this vagueness is assumed and handled by means of fuzzy sets based methodologies in particular two fuzzy integer linear programming models are proposed to be used as auxiliary models for solving the former problem as an original way to effectively solve these models obtaining cost effective network designs we use three metaheuristic procedures that are compared among them

relating different approaches to solve linear programming problems with imprecise costs abstract linear programming problems involving coefficients in the objective function with some lack of precision are usual to solve them several approaches have been proposed in this paper the different methods studied to solve the problem are briefly described next it is shown that the different solutions obtained from them may be obtained by solving a multiobjective linear programming problem

estimates and bounds of dynamic permeability of granular media this paper presents a study of dynamic permeability of porous media combining homogenization of periodic media hpm and the self consistent method scm by taking advantage of the physical principles identified with hpm the application of scm leads to the determination of two physically admissible dynamic permeability assessments both different from that given by the cell model a comparison with numerical modeling demonstrates the fairly good reliability of the three estimates for granular media consisting of a periodic array of spherical grains furthermore the self consistent values enable exact bounds for the dynamic permeability of a wide class of porous media to be derived with a clear identification of their microstructure grain and fluid size distribution

on the sensitivity of membership functions for fuzzy linear programming problems abstract for a fuzzy linear programming problem it is usually assumed that the decision maker exactly knows the shape of the membership functions taking part in it for the sake of simplicity often linear membership functions are supposed but this seems a poor reason here it is shown that in the case of fuzzy linear programming problems whether or not a fuzzy optimal solution has been found by using linear membership functions modeling the constraints possible further changes of those membership functions do not affect the former optimal solution this sensitivity analysis performed for those membership functions and the corresponding solutions shows the convenience of using linear functions instead of other more complicated ones

bandwidth packing a tabu search approach the bandwidth packing bwp problem is a combinatorially difficult problem arising in the area of telecommunications the problem consists of assigning calls to paths in a capacitated graph such that capacities are not violated and the total profit is maximized in this paper we discuss the development of a tabu search ts method for the bwp problem the method makes use of an efficient implementation of the k shortest path algorithm that allows the identification of a controlled set of feasible paths for each call a tabu search is then performed to find the best path assignment for each call the ts method developed here incorporates a number of features that have proved useful for obtaining optimal and near optimal solutions to difficult combinatorial problems we establish the effectiveness of our approach by comparing its performance in speed and solution quality to other specialized heuristics and to a standard optimization package applied to a 0 1 integer programming formulation of the problem

an integer programming approach to the bandwidth packing problem we consider the bandwidth packing problem arising from telecommunication networks the problem is to determine the set of calls and an assignment of them to the paths in an arc capacitated network to maximize profit we propose an algorithm to solve the integer programming formulation of the problem an efficient column generation technique to solve the linear programming relaxation is proposed and a modified cover inequality is used to strengthen the ip formulation the algorithm incorporates the column generation technique and the strong cutting plane approach into a branch and bound scheme we test the proposed algorithm on some random problems the results show that the algorithm can be used to solve the problems within reasonably small time limits

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

an integral equation method for spherical stefan problems abstract many applications in solidification or melting are described by a two phase stefan problem with spherical symmetry using green s representation theorem the heat equation is reformulated as a system of integral equations for the unknown fluxes on the interface the unknown radius of the solid liquid interface is determined from the stefan condition the integral equation is discretized with the nystrom method for which special singularity corrected quadrature rules are discussed numerical examples are presented that demonstrate the effectiveness of the method

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

fuzzy location problems on networks location problems concern a wide set of fields where it is usually assumed that exact data are known however in real applications the location of the facility considered can be full of linguistic vagueness that can be appropriately modelled using networks with fuzzy values thus fuzzy location problems on networks arise this paper deals with their general formulation and the description of the ways to solve them namely we show the variety of problems that can be considered in this context and for some of them we propose the most operative approaches for their solution

a tabu search algorithm for the optimisation of telecommunication networks abstract a tabu search ts algorithm is presented for topological optimisation of broad band communication networks whose structure is based on a single exchange and on a number of multiplexing centres considering the high cost of the fiber optic cable the network architecture is a minimum spanning tree the optimisation problem is reduced to choose the number and the locations of the multiplexing centres this leads to model the network by a binary string in which ones and zeroes indicate activation or non activation of multiplexing centres in the nodes a ts algorithm has been implemented using a dynamically changing tabu tenure a frequency based long term memory structure and an aspiration criterion extensive testing over networks of various sizes and configurations is made in order to choose the best values of the parameters the performances of the algorithm are compared to those of other more traditional techniques

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

finite difference methods with increased accuracy and correct initialization for one dimensional stefan problems although the numerical solution of one dimensional phase change or stefan problems is well documented a review of the most recent literature indicates that there are still unresolved issues regarding the start up of a computation for a region that initially has zero thickness as well as how to determine the location of the moving boundary thereafter this paper considers the so called boundary immobilization method for four benchmark melting problems in tandem with three finite difference discretization schemes we demonstrate a combined analytical and numerical approach that eliminates completely the ad hoc treatment of the starting solution that is often used and is numerically second order accurate in both time and space a point that has been consistently overlooked for this type of moving boundary problem

atm vp based network design abstract we consider the problem of designing an atm vp based leased line backbone network given point to point communication demands having predefined sizes in a network the problem is to find configurations of demand routes and link facilities installed on each edge satisfying all demands at minimum cost under some constraints one of the most important constraints is that a single demand cannot be split over multiple link facilities this is a sort of bin packing constraint we propose an integer programming formulation of the problem and an algorithm to solve it an efficient column generation technique to solve the linear programming relaxation is proposed and a valid inequality is used to strengthen the integer programming formulation the algorithm incorporates the column generation technique and the cutting plane approach into a branch and bound scheme we test the proposed algorithm on some real problems the results show that the algorithm can be used to solve the problems within reasonably small computing times

using fuzzy numbers in network design optimization problems the fluctuations that have taken place in the worldwide economy over the past few years make it difficult to estimate the costs of the equipment that have to be used in order to design a telecommunications network the literature is replete with prescriptive integer programming models for various network design problems that do not take into account the problems inherent vagueness due to equipment costs this paper considers the real world wavelength division multiplexing wdm network design problem the vagueness that appears in this problem is assumed and handled by means of fuzzy set based methodologies a fuzzy integer linear programming model is then proposed to be used as an auxiliary model to solve the former problem with the purpose to obtain cost effective network designs this model is then solved using cplex which is a commercial high performance optimization engine

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

tabu search for the multilevel generalized assignment problem abstract the multilevel generalized assignment problem mgap differs from the classical gap in that agents can perform tasks at more than one efficiency level important manufacturing problems such as lot sizing can be formulated as mgaps however the large number of variables in the related 0 1 integer program makes the use of commercial optimization packages impractical in this paper we present a heuristic approach to the solution of the mgap which consists of a novel application of tabu search ts our ts method employs neighborhoods defined by ejection chains that produce moves of greater power without significantly increasing the computational effort

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

bandwidth packing with queuing delay costs bounding and heuristic solution procedures in this paper we propose a new formulation for the bandwidth packing problem bwp in telecommunications networks this problem is one of selecting calls from a list of requests to be routed in the telecommunications network we consider both revenue losses and costs associated with communications delay as parts of the objective an efficient lagrangean relaxation based heuristic procedure for finding bounds and problem solutions is demonstrated computational results from a large array of instances are reported we demonstrate that the procedure is efficient in finding good solutions while expending a modest amount of computational effort

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

three models of fuzzy integer linear programming abstract in this paper we study some models for dealing with fuzzy integer linear programming problems which have a certain lack of precision of a vague nature in their formulation we present methods to solve them with either fuzzy constraints or fuzzy numbers in the objective function or fuzzy numbers defining the set of constraints these methods are based on the representation theorem and on fuzzy number ranking methods

a procedure for ordering fuzzy subsets of the unit interval abstract we introduce a function to help in the ordering of fuzzy subsets of the unit interval this function is the integral of the mean of the level sets associated with the fuzzy subsets various properties of this function are studied

on the use of the ranking function approach to solve fuzzy matrix games in a direct way abstract a general method of solving fuzzy matrix games is presented this method is suitable when players choose their fuzzy number ranking procedures in a wide class i e that which may be represented by means of a linear ranking functions in the first case described both players use the same criterion to rank fuzzy numbers and in the second case each player uses different criteria several situations of players knowledge are analyzed in the second case and game solutions are provided in all of them

possibilistic linear programming a brief review of fuzzy mathematical programming and a comparison with stochastic programming in portfolio selection problem in this paper we review some fuzzy linear programming methods and techniques from a practical point of view in the rst part the general history and the approach of fuzzy mathematical programming are introduced using a numerical example some models of fuzzy linear programming are described in the second part of the paper fuzzy mathematical programming approaches are compared to stochastic programming ones the advantages and disadvantages of fuzzy mathematical programming approaches are exemplied in the setting of an optimal portfolio selection problem finally some newly developed ideas and techniques in fuzzy mathematical programming are briey reviewed c 2000 elsevier science b v all rights reserved

intelligent local search strategies for solving facility layout problems with the quadratic assignment problem formulation abstract in this paper we implement a tabu search heuristic a probabilistic tabu search heuristic a simulated annealing heuristic and a hybrid tabu search heuristic to solve facility layout problems with the quadratic assignment problem formulation classical problems are used as a benchmark for performance comparison extensive computational experiments were conducted and the results were statistically analyzed to observe the behavior of each heuristic and to discriminate the performance of each heuristic the computational results strongly support the competitiveness of intelligent heuristics for the design of facilities the hybrid tabu search heuristic combining the advantages from both tabu search and simulated annealing can generally outperform their parent algorithms

tabu search for fuzzy optimization and applications in this paper we present a tabu search technique to approximately solve fuzzy optimization problems we demonstrate the performance of the proposed method by applying it to an elementary fuzzy optimization problem other applications of the method to fuzzy linear programming fuzzy regression and the training of fuzzy neural networks are also presented

the numerical solution of one phase classical stefan problem in this paper variable space grid and boundary immobilisation techniques based on the explicit finite difference are applied to the one phase classical stefan problem it is shown that all the results obtained by the two methods are in good agreement with the exact solution and exhibit the expected convergence as the mesh size is refined

periodic homogenization and consistent estimates of transport parameters through sphere and polyhedron packings in the whole porosity range this paper presents a study of transport parameters diffusion dynamic permeability thermal permeability trapping constant of porous media by combining the homogenization of periodic media hpm and the self consistent scheme scm based on a bicomposite spherical pattern the link between the hpm and scm approaches is first established by using a systematic argument independent of the problem under consideration it is shown that the periodicity condition can be replaced by zero flux and energy through the whole surface of the representative elementary volume consequently the scm solution can be considered as a geometrical approximation of the local problem derived through hpm for materials such that the morphology of the period is close to the scm pattern these results are then applied to derive the estimates of the effective diffusion the dynamic permeability the thermal permeability and the trapping constant of porous media these scm estimates are compared with numerical hpm results obtained on periodic arrays of spheres and polyhedrons it is shown that scm estimates provide good analytical approximations of the effective parameters for periodic packings of spheres at porosities larger than 0 6 while the agreement is excellent for periodic packings of polyhedrons in the whole range of porosity

cell model calculations of dynamic drag parameters in packings of spheres an external flow approach is used to predict the viscous drag due to oscillating flow in an air filled stack of fixed identical rigid spheres analytical expressions for dynamic and direct current dc permeability high frequency limit of tortuosity and the characteristic viscous dimension are derived using a cell model with an adjustable cell radius which allows for hydrodynamic interactions between the spherical particles the resulting theory requires knowledge of two fixed parameters the volume porosity and the particle radius the theory also requires a value for the cell radius use of the cell radius corresponding to that of the sphere circumscribing a unit cell of a cubic lattice arrangement is proposed this is found to enable good agreement between predictions of the new theory and both published data and numerical results for simple cubic and random spherical packings

a study of the ranking function approach through mean values abstract the average value was introduced to help in the ordering of fuzzy numbers and it was defined by means of an integrating process of a parametric function representing the position of every cut in the real line by considering the relation between a particular case of the average value and the notion of expectation for fuzzy numbers we interpret the general ranking function approach as a ranking process using a mean value and an interval relation some well known indices are included in this schema using the general average value a more general mean value for a fuzzy number is defined finally some properties of the general mean value are studied as well as its behaviour on several particular cases

application of standard and refined heat balance integral methods to one dimensional stefan problems the work in this paper concerns the study of conventional and refined heat balance integral methods for a number of phase change problems these include standard test problems both with one and two phase changes which have exact solutions that enable us to test the accuracy of the approximate solutions we also consider situations where no analytical solution is available and compare these to numerical solutions it is popular to use a quadratic profile as an approximation of the temperature but we show that a cubic profile seldom considered in the literature is far more accurate in most circumstances in addition the refined integral method can give greater improvement still and we develop a variation on this method which turns out to be optimal in some cases we assess which integral method is better for various problems showing that it is largely dependent on the specified boundary conditions

algorithms of discrete optimization and their application to problems with fuzzy coefficients an approach to solving optimization problems with fuzzy coefficients in objective functions and constraints is described it consists in formulating and solving one and the same problem within the framework of mutually related models with constructing equivalent analogs with fuzzy coefficients in objective functions alone it enables one to maximally cut off dominated alternatives from below as well as from above since the approach is applied within the context of fuzzy discrete optimization problems several modified algorithms of discrete optimization are discussed these algorithms are associated with the method of normalized functions are based on a combination of formal and heuristic procedures and allow one to obtain quasi optimal solutions after a small number of steps thus overcoming the computational complexity posed the np completeness of discrete optimization problems the subsequent contraction of the decision uncertainty regions is associated with reduction of the problem to multiobjective decision making in a fuzzy environment with using techniques based on fuzzy preference relations the techniques are also directly applicable to situations in which the decision maker is required to choose alternatives from a set of explicitly available alternatives the results of the paper are of a universal character and can be applied to the design and control of systems and processes of different purposes as well as the enhancement of corresponding cad cam systems and intelligent decision making systems the results of the paper are already being used to solve problems of power engineering

bandwidth packing with priority classes the bandwidth packing problem is defined as the selection and routing of messages from a given list of messages with prespecified requirements on demand for bandwidth the messages have to be routed over a network with given topology so that the generated revenue is maximized messages to be routed are classified into two priority classes an integer programming based formulation of this problem is proposed and a lagrangean relaxation based methodology is described for solving this problem a general purpose heuristic is then developed for generating feasible solutions of good quality several numerical experiments are conducted using a number of problem parameters such as number of messages ratio of messages for lower and higher priority classes capacity of links and demand distribution of messages belonging to different classes and high quality solutions to the priority bandwidth packing problem are generated under the different situations

boolean programming problems with fuzzy constraints abstract boolean linear programming problems for which there exist some lack of precision of vague nature in the constraints are considered an cut based approach is considered to solve them then an algorithm providing a fuzzy solution is proposed and analyzed the linking with other known solution methods is also studied

choosing robust solutions in discrete optimization problems with fuzzy costs in this paper a wide class of discrete optimization problems which can be formulated as a 0 1 linear programming problem is discussed it is assumed that the objective function costs are not precisely known this uncertainty is modeled by specifying a finite set of fuzzy scenarios under every fuzzy scenario the costs are given as fuzzy intervals possibility theory is then applied to chose a solution in such a problem and mixed integer linear programming models are proposed to compute this solution

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

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