ranking fuzzy numbers with integral value abstract ranking fuzzy numbers is important in decision making since very often the alternatives are evaluated by fuzzy numbers in a vague environment a comparison between these fuzzy numbers is indeed a comparison between alternatives this paper proposes a method of ranking fuzzy numbers with integral value the method which is independent of the type of membership functions used and the normality of the functions can rank more than two fuzzy numbers simultaneously it is relatively simple in computation especially in ranking triangular and trapezoidal fuzzy numbers further an index of optimism is used to reflect the decision maker s optimistic attitude discussion on comparative advantages is included

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

entropic lattice boltzmann method for microflows abstract a new method for the computation of flows at the micrometer scale is presented it is based on the recently introduced minimal entropic kinetic models both the thermal and isothermal families of minimal models are presented and the simplest isothermal entropic lattice bhatnagar gross krook elbgk is studied in detail in order to quantify its relevance for microflow simulations elbgk is equipped with boundary conditions which are derived from molecular models diffusive wall a map of three dimensional kinetic equations onto two dimensional models is established which enables two dimensional simulations of quasi two dimensional flows the elbgk model is studied extensively in the simulation of the two dimensional poiseuille channel flow results are compared to known analytical and numerical studies of this flow in the setting of the bhatnagar gross krook model the elbgk is in quantitative agreement with analytical results in the domain of weak rarefaction characterized by knudsen number kn the ratio of mean free path to the hydrodynamic scale up to kn 0 01 which is the domain of many practical microflows moreover the results qualitatively agree throughout the entire knudsen number range demonstrating knudsen s minimum for the mass flow rate at moderate values of kn as well as the logarithmic scaling at large kn the present results indicate that elbm can complement or even replace computationally expensive microscopic simulation techniques such as kinetic monte carlo and or molecular dynamics for low mach and low knudsen number hydrodynamics pertinent to microflows

wall pressure and shear stress spectra from direct simulations of channel flow wall pressure and shear stress spectra from direct numerical simulations of turbulent plane channel flow are presented in this paper simulations have been carried out at a series of reynolds numbers up to re 1440 which corresponds to re 6 92 x 10 4 based on channel width and centerline velocity single point and two point statistics for velocity pressure and their derivatives have been collected including velocity moments up to fourth order the results have been used to study the reynolds number dependence of wall pressure and shear stress spectra it is found that the point spectrum of wall pressure collapses for re 360 under a mixed scaling for frequencies lower than the peak frequency of the frequency weighted spectrum and under viscous scaling for frequencies higher than the peak point spectra of wall shear stress components are found to collapse for re 360 under viscous scaling the normalized mean square wall pressure increases linearly with the logarithm of reynolds number the rms wall shear stresses also increase with reynolds number over the present range but suggest some leveling off at high reynolds number

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

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

a third order multistep time discretization for a chebyshev tau spectral method a time discretization scheme based on the third order backward difference formula has been embedded into a chebyshev tau spectral method for the navier stokes equations the time discretization is a variant of the second order backward scheme proposed by krasnov et al 2008 3 high resolution direct numerical simulations of turbulent incompressible channel flow have been performed to compare the backward scheme to the runge kutta scheme proposed by spalart et al 1991 2 it is shown that the runge kutta scheme leads to a poor convergence of some third order spatial derivatives in the direct vicinity of the wall derivatives that represent the diffusion of wall tangential vorticity the convergence at the wall is shown to be significantly improved if the backward scheme is applied

a new approach for ranking fuzzy numbers by distance method abstract many ranking methods have been proposed so far however there is yet no method that can always give a satisfactory solution to every situation some are counterintuitive not discriminating some use only the local information of fuzzy values some produce different rankings for the same situation for overcoming the above problems we propose a new method for ranking fuzzy numbers by distance method our method is based on calculating the centroid point where the distance means from original point to the centroid point x 0 y 0 and the x 0 index is the same as murakami et al s x 0 however the y 0 index is integrated from the inverse functions of an lr type fuzzy number thus we use ranking function r a x 2 y 2 distance index as the order quantities in a vague environment our method can rank more than two fuzzy numbers simultaneously and the fuzzy numbers need not be normal furthermore we also propose the coefficient of variation cv index to improve lee and li s method comput math appl 15 1988 887 896 lee and li rank fuzzy numbers based on two different criteria namely the fuzzy mean and the fuzzy spread of the fuzzy numbers and they pointed out that human intuition would favor a fuzzy number with the following characteristics higher mean value and at the same time lower spread however when higher mean value and at the same time higher spread or lower mean value and at the same time lower spread exists it is not easy to compare its orderings clearly our cv index is defined as cv standard error mean which can overcome lee and li s problem efficiently in this way our proposed method can also be easily calculated by the mathematica package to solve problems of ranking fuzzy numbers at last we present three numerical examples to illustrate our proposed method and compare with other ranking methods

implementation of diffuse reflection boundary conditions in a thermal lattice boltzmann model with flux limiters we discuss three new implementation versions of diffuse reflection boundary conditions in a thermal lattice boltzmann model their accuracy is investigated in the case of couette flow by considering the slip regime the best results are recovered with versions 2 and 3 which rely on outgoing fluxes to express the particle distribution functions in the ghost nodes outside the flow domain version 2 is found to be more economical since it involves no interpolation procedure this version was thereafter used to investigate the temperature profile in couette flow for various values of prandtl number as well as the capability of the thermal lb model to capture the knudsen minimum in poiseuille flow

lattices for the lattice boltzmann method a recently introduced theory of higher order lattice boltzmann models chikatamarla and karlin phys rev lett 97 190601 2006 is elaborated in detail a general theory of the construction of lattice boltzmann models as an approximation to the boltzmann equation is presented new lattices are found in all three dimensions and are classified according to their accuracy degree of approximation of the boltzmann equation the numerical stability of these lattices is argued based on the entropy principle the efficiency and accuracy of many new lattices are demonstrated via simulations in all three dimensions

higher order galilean invariant lattice boltzmann model for microflows single component gas we introduce a scheme which gives rise to additional degree of freedom for the same number of discrete velocities in the context of the lattice boltzmann model we show that an off lattice d3q27 model exists with correct equilibrium to recover galilean invariant form of navier stokes equation without any cubic error in the first part of this work we show that the present model can capture two important features of the microflow in a single component gas knudsen boundary layer and knudsen paradox finally we present numerical results corresponding to couette flow for two representative knudsen numbers we show that the off lattice d3q27 model exhibits better accuracy as compared to more widely used on lattice d3q19 or d3q27 model finally our construction of discrete velocity model shows that there is no contradiction between entropic construction and quadrature based procedure for the construction of the lattice boltzmann model

accuracy of higher order lattice boltzmann methods for microscale flows with finite knudsen numbers accuracy of the lattice boltzmann lb method for microscale flows with finite knudsen numbers is investigated we employ up to the eleventh order gauss hermite quadrature for the lattice velocities and diffuse scattering boundary condition for fluid wall interactions detailed comparisons with the direct simulation monte carlo dsmc method and the linearized boltzmann equation are made for planar couette and poiseuille flows all higher order lb methods considered here give improved results as compared with the standard lb method the accuracy of the lb hierarchy however does not monotonically increase with the order of the gauss hermite quadrature at moderate and large knudsen numbers the results also show the sensitivity to a quadrature chosen even when the gauss hermite quadratures have the same order of formal accuracy among the schemes investigated here d2q16 is the most efficient method and offers a quantitative prediction in the slip and transition regimes the higher order lb methods predict the knudsen layer up to kn o 0 1 the knudsen layer however rapidly disappears when the knudsen number approaches unity due to a finite number of the lattice velocities while it is still present for kn o 1 in the boltzmann equation it is also found that the higher order lb methods adopted here do not capture the asymptotic behavior of the boltzmann equation at large knudsen numbers

general solution of lattices for cartesian lattice bhatanagar gross krook models we give the general solutions of lattices i e velocity sets and weights for the lattice bhatanagar gross krook lbgk models on two and three dimensional cartesian grids the solutions define the necessary and sufficient conditions so that the resulting lbgk model can accurately capture the dynamics of the moments retained in the distribution function in the parameter space of the weights the general solutions form low dimensional linear spaces from which minimal velocity sets are identified for the degrees of precision that are most relevant to the construction of high order lbgk models all well known lbgk lattices are found to be special cases of the given general solutions

high order lattice boltzmann models for wall bounded flows at finite knudsen numbers we analyze a large number of high order discrete velocity models for solving the boltzmann bhatnagar gross krook equation for finite knudsen number flows using the chapman enskog formalism we prove for isothermal flows a relation identifying the resolved flow regimes for low mach numbers although high order lattice boltzmann models recover flow regimes beyond the navier stokes level we observe for several models significant deviations from reference results we found this to be caused by their inability to recover the maxwell boundary condition exactly by using supplementary conditions for the gas surface interaction it is shown how to systematically generate discrete velocity models of any order with the inherent ability to fulfill the diffuse maxwell boundary condition accurately both high order quadratures and an exact representation of the boundary condition turn out to be crucial for achieving reliable results for poiseuille flow we can reproduce the mass flow and slip velocity up to the knudsen number of 1 moreover for small knudsen numbers the knudsen layer behavior is recovered

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

boundary conditions for the upwind finite difference lattice boltzmann model evidence of slip velocity in micro channel flow we conduct a systematic study of the effect of various boundary conditions bounce back and three versions of diffuse reflection for the two dimensional first order upwind finite difference lattice boltzmann model simulation of couette flow in a micro channel using the diffuse reflection boundary condition reveals the existence of a slip velocity that depends on the knudsen number e l l where l is the mean free path and l is the channel width for walls moving in opposite directions with speeds u w the slip velocity satisfies u s l i p 2 eu w a l l 1 2 e in the case of poiseuille flow in a micro channel the slip velocity is found to depend on the lattice spacing ds and knudsen number e to both first and second order the best results are obtained for diffuse reflection boundary conditions that allow thermal mixing at a wall located at half lattice spacing outside the boundary nodes

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

consistent lattice boltzmann modeling of low speed isothermal flows at finite knudsen numbers in slip flow regime ii application to curved boundaries gaseous flows inside microfluidic devices often fall in the slip flow regime according to this theoretical description the navier stokes model remains applicable in bulk while at solid walls a slip velocity boundary model shall be considered physically it is well established that to properly account for the wall curvature the wall slip velocity must be determined by the shear stress rather than the normal component of the velocity derivative alone as commonly applied to planar surfaces it follows that the numerical transcription of this type of boundary condition is generally a challenging task for standard computational fluid dynamics cfd techniques this paper aims to show that the realization of the slip velocity condition on arbitrarily shaped boundaries can be accomplished in a natural way with the lattice boltzmann method lbm to substantiate this conclusion this work undertakes the following three studies first we examine the conditions under which the generic reflection type boundary rules used by lbm become consistent models for the slip velocity boundary condition this effort makes use of the second order chapman enskog expansion method where we address both planar and curved boundaries the analysis also clarifies the capabilities and limitations behind the considered reflection type slip schemes second we revisit the family of parabolic accurate lbm slip boundary schemes originally formulated in phys rev e 96 013311 2017 2470 004510 1103 physreve 96 013311 on the basis of the multireflection framework and discuss their characteristics when operating on curved boundaries as well as the limitations of other less accurate lbm slip boundary formulations such as the linearly accurate slip schemes and the widely popular kinetic based boundary schemes in addition we also discuss the numerical stability of the parabolic slip schemes previously developed providing an heuristic strategy to improve their stable range of operation third we evaluate the performance of the several slip boundary schemes debated in this paper the numerical tests correspond to two classical 2d benchmark flow problems of slip over non planar solid surfaces namely i the velocity profile of the cylindrical couette flow and ii the permeability of a slow rarefied gas over a periodic array of circular cylindrical obstacles the obtained numerical results confirm the competitiveness of the lbm when equipped with slip boundary schemes of parabolic accuracy as cfd tool to simulate slippage phenomena over arbitrarily non planar surfaces indeed although operating on a simple uniform mesh discretization the lbm yields a similar or even superior level of accuracy compared to state of the art fem simulations conducted on hardworking body fitted meshes this conclusion establishes the lbm as a very appealing cfd technique for simulating microfluidic flows in the slip flow regime a result that deserves further exploration in future studies

spectral methods for the navier stokes equations with one infinite and two periodic directions abstract two numerical methods were designed to solve the time dependent three dimensional incompressible navier stokes equations in boundary layers method a semi infinite domain and mixing layers or wakes method b fully infinite domain their originality lies in the use of rapidly decaying spectral basis functions to approximate the vertical dependence of the solutions combined with one method a or two method b slowly decaying extra functions for each wave vector that exactly represent the irrotational component of the solution at large distances both methods eliminate the pressure term as part of the formulation thus avoiding fractional step time integration they yield rapid convergence and are free of spurious modes in the orr sommerfeld spectra they are also efficient although the operation count is of order n 2 n is the number of modes in the infinite direction these methods have been used for extensive direct numerical simulations of transition and turbulence a new time integration scheme with low storage requirements and good stability properties is also described

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

lattice boltzmann modeling of microchannel flow in slip flow regime we present the lattice boltzmann equation lbe with multiple relaxation times mrt to simulate pressure driven gaseous flow in a long microchannel we obtain analytic solutions of the mrt lbe with various boundary conditions for the incompressible poiseuille flow with its walls aligned with a lattice axis the analytical solutions are used to realize the dirichlet boundary conditions in the lbe we use the first order slip boundary conditions at the walls and consistent pressure boundary conditions at both ends of the long microchannel we validate the lbe results using the compressible navier stokes ns equations with a first order slip velocity the information preservation direct simulation monte carlo ip dsmc and dsmc methods as expected the lbe results agree very well with ip dsmc and dsmc results in the slip velocity regime but deviate significantly from ip dsmc and dsmc results in the transition flow regime in part due to the inadequacy of the slip velocity model while still agreeing very well with the slip ns results possible extensions of the lbe for transition flows are discussed

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

diffuse reflection boundary condition for high order lattice boltzmann models with streaming collision mechanism for lattice boltzmann lb models their particle feature is ensured by the streaming collision mechanism however this mechanism is often discarded for high order lb models due to multi speed lattices here we propose a new way of implementing the diffuse reflection boundary condition that can maintain the characteristic streaming collision mechanism for high order models it is then tested on four lattice models namely d2q16 d2q17 d2q37 and d3q121 for isothermal and thermal couette flows and lid driven cavity flows our implementation is shown to be able to achieve second order accuracy globally

kinetic boundary conditions in the lattice boltzmann method derivation of the lattice boltzmann method from the continuous kinetic theory x he and l s luo phys rev e 55 r6333 1997 x shan and x he phys rev lett 80 65 1998 is extended in order to obtain boundary conditions for the method for the model of a diffusively reflecting moving solid wall the boundary condition for the discrete set of velocities is derived and the error of the discretization is estimated numerical results are presented which demonstrate convergence to the hydrodynamic limit in particular the knudsen layer in the kramers problem is reproduced correctly for small knudsen numbers

analytic solution for a higher order lattice boltzmann method slip velocity and knudsen layer we present the analysis of a higher order lattice boltzmann lb method based on the fourth order gauss hermite quadrature with emphasis on the slip velocity and the knudsen layer the exact solution of the slip velocity for the higher order lb equation is obtained for poiseuille flows with finite knudsen numbers due to increased accuracy in velocity space discretization the higher order scheme gives much improved slip coefficients as compared with the standard lb method based on the third order gauss hermite quadrature a multiple relaxation time model is investigated to show the effects of the relaxation times for higher order moments on the slip phenomena

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 procedure for ranking fuzzy numbers using fuzzy relations abstract this paper presents a method to give fuzzy order relations between fuzzy numbers these are founded on the concept of comparison function defined in the paper and the use of fuzzy measures related with the same numbers the link of such relations with fuzzy operations is investigated too

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

accuracy analysis of high order lattice boltzmann models for rarefied gas flows in this work we have theoretically analyzed and numerically evaluated the accuracy of high order lattice boltzmann lb models for capturing non equilibrium effects in rarefied gas flows in the incompressible limit the lb equation is shown to be able to reduce to the linearized bhatnagar gross krook bgk equation therefore when the same gauss hermite quadrature is used lb method closely resembles the discrete velocity method dvm in addition the order of hermite expansion for the equilibrium distribution function is found not to be directly correlated with the approximation order in terms of the knudsen number to the bgk equation for incompressible flows meanwhile we have numerically evaluated the lb models for a standing shear wave problem which is designed specifically for assessing model accuracy by excluding the influence of gas molecule surface interactions at wall boundaries the numerical simulation results confirm that the high order terms in the discrete equilibrium distribution function play a negligible role in capturing non equilibrium effect for low speed flows by contrast appropriate gauss hermite quadrature has the most significant effect on whether lb models can describe the essential flow physics of rarefied gas accurately our simulation results where the effect of wall gas interactions is excluded can lead to conclusion on the lb modeling capability that the models with higher order quadratures provide more accurate results for the same order gauss hermite quadrature the exact abscissae will also modestly influence numerical accuracy using the same gauss hermite quadrature the numerical results of both lb and dvm methods are in excellent agreement for flows across a broad range of the knudsen numbers which confirms that the lb simulation is similar to the dvm process therefore lb method can offer flexible models suitable for simulating continuum flows at the navier stokes level and rarefied gas flows at the linearized boltzmann model equation level

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