real valued flows in a network with fuzzy arc capacities abstract the problem of the maximal minimal flow in a network with fuzzy capacity constraints is considered a theorem being a natural direct generalization of the ford fulkerson theorem relating the flow value with cut capacities in the network is proved a simple algorithm based on the mentioned theorem for the determination of optimal real valued flows is developed

enumerating solutions to p a q b r c s d let p q r s be polynomials with integer coefficients this paper presents a fast method using very little temporary storage to find all small integers a b c d satisfying p a q b r c s d numerical results include all small solutions to a 4 b 4 c 4 d 4 all small solutions to a 4 b 4 c 4 d 4 and the smallest positive integer that can be written in 5 ways as a sum of two coprime cubes

decompositions and range for additive fuzzy measures the aim of the present paper is to prove the existence of hahn and lebesgue decompositions for additive fuzzy measures in the sense of 3 it is shown that the radon nikodym theorem can be carried over from classical measures to additive fuzzy measures with their corresponding integrals these results are applied to obtain a generalization of liapounoff s convexity theorem 12 whose utility in the theory of fuzzy games is argued in the last section

approximation of pareto optima in multiple objective shortest path problems we study methods for approximating the set of pareto optimal paths in multiple objective shortest path problems known generalizations of standard shortest path methods will compute this set but can suffer from rapidly increasing computational and storage demands as problem size increases in an effort to avoid such difficulties we develop approximation methods that can estimate the pareto optima to any required degree of accuracy the approximation methods are fully polynomial that is they operate in time and space bounded by a polynomial in problem size and accuracy of approximation the greater the accuracy the more time required to reach a solution we show how approximation methods may be applied to yield fully polynomial approximation schemes for a variety of np complete single objective problems

fuzzy shortest paths abstract shortest paths are one of the simplest and most widely used concepts in nonfuzzy networks fuzzy shortest paths also have a variety of applications but are hampered by the possibility that the fuzzy shortest length may not correspond to an actual path in the network in this paper we present new models based on fuzzy shortest paths that circumvent this problem and we also give a general algorithm based on dynamic programming to solve the new models in addition we analyze the fuzzy shortest path algorithms in terms of submodular functions the analysis concepts developed for fuzzy shortest paths are also discussed in terms of general fuzzy mathematical programming

fuzzy logic based algorithms for maximum covering location problems this paper concerns a class of maximum covering location problems in networks in uncertain environments it is assumed that a relative weights of demand nodes are either deterministic or imprecise described by linguistic expressions and b potential facility site locations are limited to network nodes the concept of coverage is extended to include a degree of node coverage which means that the borders between the subset of covered demand nodes and the subset of uncovered demand nodes are inexact the acceptable service distance travelling times from a facility site to demand nodes are modelled by fuzzy sets three new algorithms for choosing the best facility locations are developed which assume that 1 demands at all nodes are equally important 2 relative weights of demand at nodes are deterministic and 3 weights of demand at nodes are imprecise and described by linguistic terms respectively the algorithms are based on searching among potential facility nodes by applying comparison operations on discrete fuzzy sets it is shown how to extend the proposed algorithms from one site to multi site covering problems illustrative examples of selecting locations for logistics centres in a distribution company are given

the fuzzy p median problem a global analysis of the solutions abstract we apply fuzzy techniques to incorporate external data into p median problems so we can detect certain solutions that would be discarded by usual crisp and fuzzy algorithms but that contrasted with this additional information can be advantageous this usually reveals a pathology of the model and hence our methods provide some fuzzy validation criteria for p median models

the maximum fuzzy weighted matching models and hybrid genetic algorithm a matching of a graph is an independent subset of the edges set and a maximum matching is a matching with as many edges in it as possible the maximum weighted matching problem is to find a maximum matching in a given graph such that the sum of the weights of the edges in it is maximum in this paper the concepts of expected maximum fuzzy weighted matching the maximum fuzzy weighted matching and the most maximum fuzzy weighted matching are initialized according to various decision criteria the maximum fuzzy weighted matching problem is formulated as expected value model chance constrained programming and dependent chance programming by using the credibility theory and the crisp equivalents are also given furthermore a hybrid genetic algorithm is designed for solving the proposed fuzzy programming models finally a numerical example is given

fuzzy decision networks and deconvolution abstract three kinds of networks namely fuzzy minimal spanning tree fuzzy pert and fuzzy shortest path are analyzed by the use of a recently developed fuzzy ranking method to handle the various fuzzy quantities to overcome the problem of double inclusion of the amounts of uncertainties involved in the fuzzy quantities when extended subtraction is used for problems such as fuzzy pert fuzzy deconvolution is used since we are only interested in the ranking and aggregation of the results the existence or nonexistence of the results from deconvolution does not influence the resulting analysis a technique based on the ranking method is developed to handle negative spreads in the resulting fuzzy quantities with the use of this fuzzy ranking method the structure of the network is maintained and conventional algorithms can be applied with appropriate modifications emphasis is placed on the use of the subjective decision maker s opinion in the proposed fuzzy network analysis approach numerical examples are given to illustrate the approach

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

fuzzy shortest path problems incorporating interactivity among paths abstract this paper deals with a shortest path problem on a network in which a fuzzy number instead of a real number is assigned to each arc length such a problem is ill posed because each arc cannot be identified as being either on the shortest path or not therefore based on the possibility theory we introduce the concept of degree of possibility that an arc is on the shortest path every pair of distinct paths from the source node to any other node is implicitly assumed to be noninteractive in the conventional approaches this assumption is unrealistic and involve inconsistencies to overcome this drawback we define a new comparison index between the sum of fuzzy numbers by considering interactivity among fuzzy numbers an algorithm is presented to determine the degree of possibility for each arc on a network finally this algorithm is evaluated by means of large scale numerical examples consequently we can find this approach is efficient even for real world practical networks

the diophantine equation 2 4 we show that within the hypercube x y z w 2 5 10 6 the diophantine equation x 4 2y 4 z 4 4w 4 admits essentially one and only one nontrivial solution namely 1 484 801 1 203 120 1 169 407 1 157 520 the investigation is based on a systematic search by computer

fuzzy network technique for technological forecasting abstract the stochastic network technique is known to be a powerful tool carrying out a technological forecast of complex systems a network dealt with is characterized by a tetrad of essential elements logical nodes with some inputs and outputs probabilistics activity branches feedback loops and multiple sources and sinks a set of network parameters is defined for each element and their values are estimated for practical analysis of the network in the case where the system to be treated is very large and or complex it cannot always be represented by a definite network and therefore forecasted values of parameters are inevitably indefinite themselves a conventional probabilistic approach is sometimes inadequate in such a case in the light of these facts the paper proposes a fuzzy network technique in which among activity branches emanating from a node a branch to be undertaken once the node is realized belongs to a fuzzy set and the time required to complete an activity branch belongs to a fuzzy set operations of maximum and minimum for sum and product of fuzzy sets take the place of manipulations of addition and multiplication for probabilities respectively although the operations are somewhat formal the obtained results seem interesting a numerical example is attached to show a comparison of the proposed technique with the conventional one

the shortest path problem with discrete fuzzy arc lengths in the past the fuzzy shortest path problem in a network has attracted attention from many researchers for its importance to various applications in this paper we propose a new algorithm to deal with the fuzzy shortest path problem it is composed of fuzzy shortest path length procedure and similarity measure the former is presented to determine the fuzzy shortest path length from source node to the destination node in the network and the latter is used to measure the similarity degree between fuzzy length sets this algorithm not only can yield shortest length but also can offer the actual shortest path to decision makers an illustrative example is also included to demonstrate our proposed algorithm

finite dominating sets for network location problems a research theme involving location on networks since its inception has been the identification of a finite dominating set fds or a finite set of points to which an optimal solution must belong we attempt to unify and generalize results of this sort we survey the literature and then prove some theorems that subsume most previous results and that are at the same time more general than previous results the paper is aimed primarily at investigators who wish to know whether an fds exists for a specific problem

a shortest path problem on a network with fuzzy arc lengths abstract we concentrate on a shortest path problem on a network in which a fuzzy number instead of a real number is assigned to each arc length introducing an order relation between fuzzy numbers based on fuzzy min a nondominated path or pareto optimal path from the specified node to every other node is defined an algorithm for solving the problem is developed on the basis of the multiple labeling method for a multicriteria shortest path as a result a number of nondominated paths can be obtained and is offered to a decision maker however a number of nondominated paths derived from large scale network may be too numerous for him to choose a preferable path due to this situation we propose a method to reduce the number of paths according to a possibility level the proposed algorithm is numerically evaluated on large scale random networks

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

the computational complexity of the criticality problems in a network with interval activity times abstract the paper analyzes the criticality in a network with interval activities duration times a natural generalization of the criticality notion for a path an activity and an event for the case of network with interval activity duration times is given the computation complexity of five problems linked to the introduced criticality notion is presented

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

on combinatorial optimization problems on matroids with uncertain weights in this paper the combinatorial optimization problem on weighted matroid is considered it is assumed that the weights in the problem are ill known and they are modeled as fuzzy intervals the optimality of solutions and the optimality of elements are characterized this characterization is performed in the setting of possibility theory a method of choosing a solution under uncertainty is also proposed

generalized path finding algorithms on semirings and the fuzzy shortest path problem a new dioid structure k e e is proposed to solve a path finding problem in a fuzzy graph this algebraic structure is adapted precisely to solve the problem of the k best fuzzy shortest paths we demonstrate that the generalized gauss seidel s algorithm always converges for the solving of the k best fuzzy shortest paths problem on a valued fuzzy graph without cycles of negative weight this work starts a safe extension of the path algebra paradigm to valued fuzzy graphs for the shortest path finding problem

rational points on diagonal quartic surfaces we searched up to height 107 for rational points on diagonal quartic surfaces the computations fill several gaps in earlier lists computed by pinch swinnerton dyer and bright

critical path analysis in the network with fuzzy activity times a natural generalization of the criticality notion in a network with fuzzy activity times is given it consists in direct application of the extension principle of zadeh to the notion of criticality of a path an activity an event treated as a function of the activities duration times in the network there are shown some relations between the notion of fuzzy criticality introduced in the paper and the notion of interval criticality criticality in the network with interval activity times proposed by the authors in another paper two methods of calculation of the path degree of criticality according to the proposed concept of fuzzy criticality are presented

application of the multigrid method to calculations of transonic potential flow about wing fuselage combinations abstract the multigrid mg method has been applied to the calculation of transonic potential flow fields about arbitrary three dimensional wing body combinations numerical results for iterative convergence rate are in good agreement with those predicted by a local mode analysis and show substantial improvement over conventional relaxation algorithms

a dynamic approach to multiple objective resource allocation problem a new approach using dynamic programming is developed for solving the multiple objective resource allocation problem there are two key issues being addressed in this approach the first one is to develop a methodology of fuzzy evaluation and fuzzy optimization for multiple objective systems the procedure of getting the marginal evaluation for each objective and aggregating them synthetically into a global evaluation is presented in this paper the second one is to design a dynamic optimization algorithm by incorporating the method of fuzzy evaluation and fuzzy optimization with the conventional dynamic programming technique a characteristic feature of the approach presented is that various objectives are synthetically considered by the fuzzy systematic technique instead of the frequently employed weighted average method numeric examples are also given to clarify the developed approach and to demonstrate its effectiveness

on convex and concave fuzzy mappings abstract in this paper we introduce the concepts of convex subsets of f 0 the set of all fuzzy numbers and monotonic fuzzy mappings on r2 and show that a monotonic fuzzy mapping f on a convex set c r1 or r2 is quasi convex and quasi concave we establish two characterizations for convex fuzzy mappings and two characterizations for concave fuzzy mappings we also prove the least upper bound property

fuzzy shortest path problem with finite fuzzy quantities we discuss the shortest path problem from a specified node to every other nodes on a network in which a positive fuzzy quantity with finite support is assigned to each arc as its arc length we define an order relation between fuzzy quantities with finite supports then by applying hansen s multiple labeling method and dijkstra s shortest path algorithm we propose a new algorithm for finding the set of non dominated paths with respect to the extension principle moreover we show that the only existing approach for this problem klein s algorithm may lead to a dominated path in the sense of extension principle

unified approach fuzzy graph problems we present a taxonomy of fuzzy graphs that treats fuzziness in vertex existence edge existence edge connectivity and edge weight within that framework we formulate some standard graph theoretic problems shortest paths and minimum cut for fuzzy graphs using a unified approach distinguished by its uniform application of guiding principles such as the construction of membership grades via the ranking of fuzzy numbers the preservation of membership grade normalization and the collapsing of fuzzy sets of graphs into fuzzy graphs finally we provide algorithmic solutions to these problems with examples

solution of the euler equations for two dimensional transonic flow by a multigrid method the assumption of potential o w implies that the o w is irrotational this is not strictly correct when shock waves are present an exact description of transonic inviscid o w requires the solution of the euler equations the numerical solution of the euler equations for steady transonic o ws is therefore a problem of great interest to the aeronautical community it also presents a testing challenge to applied mathematicians and numerical analysts

the shortest path problem on networks with fuzzy parameters shortest path problem where the costs have vague values is one of the most studied problems in fuzzy sets and systems area however due to its high computational complexity previously published algorithms present peculiarities and problems that need to be addressed e g they find costs without an existing path they determine a fuzzy solution set but do not give any guidelines to help the decision maker choose the best path they can only be applied in graphs with fuzzy non negative parameters etc in this paper one proposes an iterative algorithm that assumes a generic ranking index for comparing the fuzzy numbers involved in the problem in such a way that each time in which the decision maker wants to solve a concrete problem s he can choose or propose the ranking index that best suits that problem this algorithm that solves the above remarked drawbacks is based on the ford moore bellman algorithm for classical graphs and in concrete it can be applied in graphs with negative parameters and it can detect whether there are negative circuits for the sake of illustrating the performance of the algorithm in the paper it has been here developed using only certain order relations but it is not restricted at all to use these comparison relations exclusively the proposed algorithm is easy of understanding as the theoretical base of a decision support system oriented to solving this kind of problems

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

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

fuzzy quantities in networks one of possible applications of the fuzzy sets theory in network analysis is described and discussed it is shown that the uncertainty connected with the length of activities forming the network can be represented by fuzzy quantities summing these fuzzy quantities it is possible to derive the uncertainty connected with more complex paths and reserves an illustrative numerical example is presented at the end of the paper

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

shortest paths algorithms we conduct an extensive computational study of shortest paths algorithms including some very recent algorithms we also suggest new algorithms motivated by the experimental results and prove inter

a fuzzy shortest path with the highest reliability this paper concentrates on a shortest path problem on a network where arc lengths costs are not deterministic numbers but imprecise ones here costs of the shortest path problem are fuzzy intervals with increasing membership functions whereas the membership function of the total cost of the shortest path is a fuzzy interval with a decreasing linear membership function by the max min criterion suggested in r e bellman l a zade decision making in a fuzzy environment management science 17b 1970 141 164 the fuzzy shortest path problem can be treated as a mixed integer nonlinear programming problem we show that this problem can be simplified into a bi level programming problem that is very solvable here we propose an efficient algorithm based on the parametric shortest path problem for solving the bi level programming problem an illustrative example is given to demonstrate our proposed algorithm

on a multicriteria shortest path problem abstract multicriteria shortest path problems have not been treated intensively in the specialized literature despite their potential applications in fact a single objective function may not be sufficient to characterize a practical problem completely for instance in a road network several parameters as time cost distance etc can be assigned to each arc clearly the shortest path may be too expensive to be used nevertheless the decision maker must be able to choose some solution possibly not the best for all the criteria in this paper we present two algorithms for this problem one of them is an immediate generalization of the multiple labelling scheme algorithm of hansen for the bicriteria case based on this algorithm it is proved that any pair of nondominated paths can be connected by nondominated paths this result is the support of an algorithm that can be viewed as a variant of the simplex method used in continuous linear multiobjective programming a small example is presented for both algorithms

paths of least resistance in possibilistic production systems abstract we introduce the idea of possibilistic production systems we provide a general algorithm for finding a path from an initial node to a goal node we show that the use of a heuristic function can help in the efficient implementation of the search process we see that a significant role in determining potential optimal paths is played by concentrating on the most difficult activity to accomplish

fuzzy multicriteria integer programming via fuzzy generalized networks recent advances in network based mathematical programming methods have given rise to practical economical and computationally efficient solution methodologies for truly large scale problems today a wide variety of real world problems have been solved in such a manner even though on the surface they do not have an apparent network structure one type of network the generalized network has been successfully applied to solve large scale zero one or mixed integer mathematical programming models however one drawback to such approaches is their typical focus on but a single objective the purpose of this paper then is to document the results of an ongoing research effort which combines the generalized network concept with the technique of fuzzy programming the resultant hybrid approach has been found to provide a computationally efficient approach to multi objective zero one or integer programming problems

fuzzy dynamic programming main developments and applications abstract we survey major developments and applications of fuzzy dynamic programming which is here advocated as a promising attempt at making dynamic programming models more realistic by a relaxation of often artificial assumptions of precision as to the constraints goals states and their transitions termination time etc issues related to numerical efficiency are considered applications in r d planning health care and medicine socioeconomic regional development energy systems water and environmental systems chemical engineering etc are surveyed

minmax regret approach and optimality evaluation in combinatorial optimization problems with interval and fuzzy weights this paper deals with a general combinatorial optimization problem in which closed intervals and fuzzy intervals model uncertain element weights the notion of a deviation interval is introduced which allows us to characterize the optimality and the robustness of solutions and elements the problem of computing deviation intervals is addressed and some new complexity results in this field are provided possibility theory is then applied to generalize a deviation interval and a solution concept to fuzzy ones

multi grid calculation of three dimensional transonic potential flows a multi grid algorithm has been developed to speed the iterative convergence of calculations for the transonic potential flow past swept wings and wing fuselage combinations the method is based upon a fully conservative finite volume approximation to the steady potential equation which is second order accurate everywhere in the flow field except near shock waves the multi grid scheme is incorporated within the framework of an alternating successive line overrelaxation slor solver of the difference equations computed results confirm the second order accuracy of the scheme and demonstrate the effectiveness of the multi grid procedure

shortest path problem with uncertain arc lengths uncertainty theory provides a new tool to deal with the shortest path problem with nondeterministic arc lengths with help from the operational law of uncertainty theory this paper gives the uncertainty distribution of the shortest path length also it investigates solutions to the a shortest path and the most shortest path in an uncertain network it points out that there exists an equivalence relation between the a shortest path in an uncertain network and the shortest path in a corresponding deterministic network which leads to an effective algorithm to find the a shortest path and the most shortest path roughly speaking this algorithm can be broken down into two parts constructing a deterministic network and then invoking the dijkstra algorithm

an exact algorithm for the fuzzy p median problem in this paper we propose a fuzzy version of the classical p median problem we consider a fuzzy set of constraints so that the decision maker will be able to take into account solutions which provide significantly lower costs by leaving a part of the demand uncovered we propose an algorithm for solving the problem which is based on hakimi s works and we compare the crisp and the fuzzy approach by means of an example

single and multi objective defensive location problems on a network abstract this paper considers a new optimal location problem called defensive location problem dlp in the dlps a decision maker locates defensive facilities in order to prevent her his enemies from reaching an important site called a core for example a government of a country locates self defense bases in order to prevent her his aggressors from reaching the capital of the country it is assumed that the region where the decision maker locates her his defensive facilities is represented as a network and the core is a vertex in the network and that the facility locater and her his enemy are an upper and a lower level of decision maker respectively then the dlps are formulated as bilevel 0 1 programming problems to find stackelberg solutions in order to solve the dlps efficiently a solving algorithm for the dlps based upon tabu search methods is proposed the efficiency of the proposed solving methods is shown by applying to examples of the dlps moreover the dlps are extended to multi objective dlps that the decision maker needs to defend several cores simultaneously such dlps are formulated as multi objective programming problems in order to find a satisfying solution of the decision maker for the multi objective dlp an interactive fuzzy satisfying method is proposed and the results of applying the method to examples of the multi objective dlps are shown

application of fuzzy minimum cost flow problems to network design under uncertainty this paper deals with fuzzy quantities and relations in multi objective minimum cost flow problem when t norms and t conorms are available the goal programming is applied to minimize the deviation among the multiple costs of fuzzy flows and the given targets when the fuzzy supplies and demands are satisfied to obtain the most optimistic and the most pessimistic satisficing solutions of this problem two polynomial time algorithms are introduced applying some network transformations to demonstrate the performance of this approach in actual substances network design under fuzziness is considered and an efficient scheme is proposed including genetic algorithm together with fuzzy minimum cost flow problem this scheme is applied on a pilot network for more description

on fuzzy tree definition abstract the connectedness and acyclicity levels are presented for a fuzzy graph several fuzzy tree definitions are given too and their relations with previous studies on the matter are analyzed

fuzzy optimal flow on imprecise structures abstract the main aim of this paper is to study the maximum flow problem when the underlying associated structure is not well defined and must be modeled as a fuzzy graph however to deal with the problem it is necessary to develop several well known concepts from classical theory in the fuzzy graph context such as flow network and admissible flow the above maximum flow problem is formulated and solved by using algorithms previously developed by the authors

fuzzy boolean programming problems with fuzzy costs a general study abstract this paper deals with boolean linear programming problems involving coefficients in the objective function as fuzzy numbers in the study of these problems different approaches can be proposed to use ranking fuzzy numbers methods and fuzzy preference relations obtaining auxiliary classical boolean programming problems and to use the representation theorem obtaining a convex set with extreme points defined by the lower or upper bound of the a level sets of the fuzzy coefficients and consequently an auxiliary interval boolean programming problem in this paper we develop and link the different approaches

logistics distribution centers location problem and algorithm under fuzzy environment distribution centers location problem is concerned with how to select distribution centers from the potential set so that the total relevant cost is minimized this paper mainly investigates this problem under fuzzy environment consequentially chance constrained programming model for the problem is designed and some properties of the model are investigated tabu search algorithm genetic algorithm and fuzzy simulation algorithm are integrated to seek the approximate best solution of the model a numerical example is also given to show the application of the algorithm

project network analysis with fuzzy activity times abstract triangular fuzzy numbers are used to represent project activities whose duration times are uncertain and cannot be represented stochastically a comprehensive path analysis method is devised using fuzzy arithmetic and a fuzzy number comparison method to determine fuzzy project completion time and the degrees of criticality of each network path possibility theory is then used to determine the possibilities of the project completion given the fuzzy project completion time the existing composite method and the new comprehensive comparison method are compared and contrasted

computing a fuzzy shortest path in a network with mixed fuzzy arc lengths using α cuts we are concerned with the design of a model and an algorithm for computing a shortest path in a network having various types of fuzzy arc lengths first we develop a new technique for the addition of various fuzzy numbers in a path using a cuts by proposing a linear least squares model to obtain membership functions for the considered additions then using a recently proposed distance function for comparison of fuzzy numbers we present a dynamic programming method for finding a shortest path in the network examples are worked out to illustrate the applicability of the proposed model

the use of fuzzy variables in pert abstract in this paper we present a method called fpert for estimating a project completion time in the situation when activity duration times in the project network model are given in the form of fuzzy variables fuzzy sets on time space theoretical foundations of the method as well as results of calculations derived from a simple example are included

maximum cut in fuzzy nature models and algorithms the maximum cut max cut problem has extensive applications in various real world fields such as network design and statistical physics in this paper a more practical version the max cut problem with fuzzy coefficients is discussed specifically based on credibility theory the max cut problem with fuzzy coefficients is formulated as an expected value model a chance constrained programming model and a dependent chance programming model respectively according to different decision criteria when these fuzzy coefficients are represented by special fuzzy variables like triangular fuzzy numbers and trapezoidal fuzzy numbers the crisp equivalents of the fuzzy max cut problem can be obtained finally a genetic algorithm combined with fuzzy simulation techniques is designed for the general fuzzy max cut problem under these models and numerical experiment confirms the effectiveness of the designed genetic algorithm

maximum flow in a network with fuzzy arc capacities abstract our main concern is the maximum flow in a network in which an excess over the beforehand fixed quota of arc capacity is admissible the problem is represented as a partially fuzzy linear programming task a theorem equivalent to the ford and fulkerson one concerning the classic task of maximum flow is proved in the paper an algorithm for searching maximum flow assuming integer values of flows on network arcs is presented

rational points of bounded height on fano varieties a prime pe7z let v be an algebraic variety defined over f and li a metrized line bundle on v i e a system l consisting of a line bundle l and a family of banach v adic metrics on l f for all v s satisfying well known conditions cf 1 chap vi then we can define a height function hl x f i r by the formula h x is x 1 where s is a local section of l non vanishing

the fuzzy shortest path problem and its most vital arcs abstract the shortest path problem is to find the shortest distance between two specified nodes in a network an arc is called a single most vital arc in the network if its removal from the network results in the greatest increase in the shortest distance the availability of arcs plays a key role in network problems the most vital arcs problems provide a means by which the importance of arc s availability can be measured in the traditional shortest path problem the arc lengths are assumed to be crisp numbers in this paper we consider the case that the arc lengths are fuzzy numbers in particular we derive the membership function of the shortest distance by using a fuzzy linear programming approach based on this result we then propose an algorithm for finding the single most vital arc in a network

ranking fuzzy numbers in the setting of possibility theory the arithmetic manipulation of fuzzy numbers or fuzzy intervals is now well understood equally important for application purposes is the problem of ranking fuzzy numbers or fuzzy intervals which is addressed in this paper a complete set of comparison indices is proposed in the framework of zadeh s possibility theory it is shown that generally four indices enable one to completely describe the respective locations of two fuzzy numbers moreover this approach is related to previous ones and its possible extension to the ranking of n fuzzy numbers is discussed at length finally it is shown that all the information necessary and sufficient to characterize the respective locations of two fuzzy numbers can be recovered from the knowledge of their mutual compatibilities

a new algorithm for the discrete fuzzy shortest path problem in a network in a network the arc lengths may represent time or cost in practical situations it is reasonable to assume that each arc length is a discrete fuzzy set we called it the discrete fuzzy shortest path problem there were several methods reported to solve this kind of problem in the open literature in these methods they can obtain either the fuzzy shortest length or the shortest path in this paper we propose a new algorithm that can obtain both of them the discrete fuzzy shortest length method is proposed to find the fuzzy shortest length and the fuzzy similarity measure is utilized to get the shortest path an illustrative example is given to demonstrate our proposed algorithm