two level method based on finite element and crank nicolson extrapolation for the time dependent navier stokes equations a fully discrete two level finite element method the two level method is presented for solving the two dimensional time dependent navier stokes problem the method requires a crank nicolson extrapolation solution u h tau 0 p h tau 0 on a spatial time coarse grid j h tau 0 and a backward euler solution u h tau p h tau on a space time fine grid j h tau the error estimates of optimal order of the discrete solution for the two level method are derived compared with the standard crank nicolson extrapolation method the one level method based on a space time fine grid j h tau the two level method is of the error estimates of the same order as the one level method in the h1 norm for velocity and the l2 norm for pressure however the two level method involves much less work than the one level method

a parametric gp model dealing with incomplete information for group decision making abstract we consider a group decision making problem where preferences given by the experts are articulated into the form of pairwise comparison matrices in many cases experts are not able to efficiently provide their preferences on some aspects of the problem because of a large number of alternatives limited expertise related to some problem domain unavailable data etc resulting in incomplete pairwise comparison matrices our goal is to develop a computational method to retrieve a group priority vector of the considered alternatives dealing with incomplete information for that purpose we have established an optimization problem in which a similarity function and a parametric compromise function are defined associated to this problem a logarithmic goal programming formulation is considered to provide an effective procedure to compute the solution moreover the parameters involved in the method have a clear meaning in the context of group problems

on a higher order accurate fully discrete galerkin approximation to the navier stokes equations we consider approximating the solution of the initial and boundary value problem for the navier stokes equations in bounded twoand three dimensional domains using a nonstandard galerkin finite element method for the space discretization and the third order accurate three step backward differentiation method coupled with extrapolation for the nonlinear terms for the time stepping the resulting scheme requires the solution of one linear system per time step plus the solution of five linear systems for the computation of the required initial conditions all these linear systems have the same matrix the resulting approximations of the velocity are shown to have optimal rate of convergence in l2 under suitable restrictions on the discretization parameters of the problem and the size of the solution in an appropriate function space

on a multicommodity stochastic flow network with unreliable nodes subject to budget constraint from the quality management and decision making view point reliability and unreliability are important indices to measure the quality level for a stochastic flow network in a multicommodity stochastic flow network with unreliable nodes the branches and nodes all have several possible capacities and may fail different types of the commodity which are transmitted through the same network simultaneously compete the capacities of branches and nodes in this paper we first define the system capacity as a vector for a multicommodity stochastic flow network with unreliable nodes then we design a performance index which is the probability that the upper bound of the system capacity is a given pattern subject to the budget constraint it can be applied to evaluate the quality level for such a network a simple approach based on minimal cuts is thus presented to evaluate the performance index

assessing priority weights from subsets of pairwise comparisons in multiple criteria optimization problems using a set covering concept we explore the possibility and flexibility of selecting a subset of all possible pairs of comparisons to assess priority weights of the decision criteria different techniques offer different priority weights with different properties they are however interrelated when special subsets such as basic coverings elementary cycles and complete covering are considered this paper provides a framework for unifying existing assessment methods and offers some fundamental theorems which reveal the intricate relationships among the methods over the subsets

an alternating crank nicolson method for decoupling the ginzburg landau equations the time dependent ginzburg landau model of superconductors consists of coupled nonlinear partial differential equations which presents difficulties in the numerical solution we present an alternating crank nicolson method for this model that leads to two decoupled algebraic subsystems one is linear and the other is semilinear both have nice numerical properties and can be solved by efficient matrix solvers we show the stability and convergence and derive error estimates for this scheme numerical results are also reported

multiparameter solutions of yang baxter equation from braid group representations on the basis of weight conservation of the an case yang baxter relations are solved without the assumption that the upper left triangle of each sub block of s matrix vanishes a family of braid group representations having no transpose symmetry are obtained and shown to construct link polynomials finally a sequence of solutions of the yang baxter equation with multiparameters are obtained via the yang baxterization procedure

finite element approximation of the nonstationary navier stokes problem part ii stability of solutions and error estimates uniform in time in this paper assumptions about the stability of a solution are introduced in the numerical analysis of the nonstationary navier stokes problem for the purpose of extending local a priori error estimates and local a posteriori error estimates globally in time

jie ke versus alphago a ranking approach using decision making method for large scale data with incomplete information abstract ranking historical players in sports is challenging since some players have never played against each other it is even more complex in go because of alphago a project based on artificial intelligence who became the world s number 1 after it defeated the 528th and the 4th human go players alphago is ranked high in the current go ranking system because it is undefeated the objective of this paper is to propose a new ranking method for large scale go players by means of incomplete fuzzy pair wise comparison matrix whose priority vector is derived using a cosine similarity measure using match results provided by go4go net experiments are designed to rank top go players in the past 45 years and examine the change in ranking after alphago faced off against jie ke furthermore the proposed method was applied to rank all 1544 go players available at go4go net to illustrate its efficiency in handling large scale data

the approximation of a crank nicolson scheme for the stochastic navier stokes equations in this paper we prove the convergence of stochastic navier stokes equations driven by white noise a linearized version of the implicit crank nicolson scheme is considered for the approximation of the solutions to the n s equations the noise is defined as the distributional derivative of a wiener process and approximated by using the generalized l 2 projection operator optimal strong convergence error estimates in the l 2 norm are obtained

modeling uncertainty in multi criteria decision analysis abstract his paper provides a review of multiple criteria decision analysis mcda for cases where attribute evaluations are uncertain the main aim is to identify different tools which can be used to represent uncertain evaluations and to broadly survey the available decision models that can be used to support uncertain decision making the review includes models using probabilities or probability like quantities explicit risk measures such as quantiles and variances fuzzy numbers and scenarios the practical assessment of uncertain outcomes and preferences associated with these outcomes is also discussed

extended state model and group approach to new polynomials the extended kauffman state model is presented by introducing the annihilation diagrams this new state model enables the authors to establish a connection between the akutsu wadati type polynomials 1987 and the group approach of witten 1988

optimal convergence analysis of crank nicolson extrapolation scheme for the three dimensional incompressible magnetohydrodynamics abstract this paper considers a crank nicolson extrapolation scheme based on mixed finite element method to solve the three dimensional incompressible magnetohydrodynamics mhd equations we prove that the fully discrete scheme is almost unconditionally stable and convergent i e stable and convergent when the time step is less than or equal to a constant by a new negative norm technique the optimal error estimates in l 2 norm are derived meanwhile the numerical investigations provide a sufficient support for the theoretical analysis

a fuzzy extension of saaty s priority theory we present a fuzzy method for choosing among a number of alternatives under conflicting decision criteria a fuzzy version of saaty s pairwise comparison method 1980 extended by de graan 1980 and lootsma 1981 each ratio expressing the relative significance of a pair of factors is displayed in a matrix from which suitable weights can be extracted since these ratios are essentially fuzzy they express the opinion of a decision maker on the importance of a pair of factors we have adapted the above mentioned method in such a way that decision makers are asked to express their opinions in fuzzy numbers with triangular membership functions we apply the method at two distinct levels first to find fuzzy weights for the decision criteria and second to find fuzzy weights for the alternatives under each of the decision criteria by a suitable combination of these results we obtain fuzzy scores for the alternatives as well as their sensitivities using these fuzzy scores the decision makers should be able to make a choice for one of the alternatives

stabilized finite element method based on the crank nicolson extrapolation scheme for the time dependent navier stokes equations this paper provides an error analysis for the crank nicolson extrapolation scheme of time discretization applied to the spatially discrete stabilized finite element approximation of the two dimensional time dependent navier stokes problem where the finite element space pair x h m h for the approximation u n h p n h of the velocity u and the pressure p is constructed by the low order finite element the q 1 p 0 quadrilateral element or the p 1 p 0 triangle element with mesh size h error estimates of the numerical solution u n h p n h to the exact solution u t n p t n with t n 0 t are derived

an additional result of monsuur s paper about intrinsic consistency threshold for reciprocal matrices abstract the paper addresses ranking of factors with a use of the pairwise comparison method in sense of the analytic hierarchy process ahp the aim of this study is to show that the logarithmic least squares method used to approximate judgement matrix in order to calculate the ranking of factors does not cause the reduction of weight of a new alternative as the eigenvector method does

a least deviation method to obtain a priority vector of a fuzzy preference relation based on the transfer relationship between fuzzy preference relation and multiplicative preference relation this paper proposes a least deviation method to obtain a priority vector of a fuzzy preference relation and studies its desirable properties a simple convergent iterative algorithm is given and its practicality is shown by two numerical examples

a characterization of the logarithmic least squares method abstract we provide an axiomatic characterization of the logarithmic least squares method sometimes called row geometric mean used for deriving a preference vector from a pairwise comparison matrix this procedure is shown to be the only one satisfying two properties correctness in the consistent case which requires the reproduction of the inducing vector for any consistent matrix and invariance to a specific transformation on a triad that is the weight vector is not influenced by an arbitrary multiplication of matrix elements along a 3 cycle by a positive scalar

computing interval weights for incomplete pairwise comparison matrices of large dimension a weak consistency based approach multiple criteria decision making and evaluation problems dealing with a large number of objects are very demanding particularly when the use of pairwise comparison pc techniques is required a major drawback arises when it is not possible to obtain all the pcs due to time or cost limitations or to split the given problem into smaller subproblems in such cases two tools are needed to find acceptable weights of objects an efficient method for partially filling a pairwise comparison matrix pcm and a suitable method for deriving weights from this incomplete pcm this paper presents a novel interactive algorithm for large dimensional problems guided by two main ideas the sequential optimal choice of the pcs to be performed and the concept of weak consistency the proposed solution significantly reduces the number of needed pcs by adding information implied by the weak consistency after the input of each pc providing sets of feasible values for all missing pcs interval weights of objects are computed from the resulting incomplete weakly consistent pcm adapting the methodology for calculating fuzzy weights from fuzzy pcms the computed weight intervals thus cover all possible weakly consistent completions of the incomplete pcm the algorithm works both with saaty s pcms and fuzzy preference relations the performance of the algorithm is illustrated by a numerical example and a real life case study the performed simulation demonstrates that the proposed algorithm is capable of reducing the number of pcs required in pcms of dimension 15 and greater by more than 60 on average

incomplete pairwise comparison and consistency optimization this paper proposes a new method for calculating the missing elements of an incomplete matrix of pairwise comparison values for a decision problem the matrix is completed by minimizing a measure of global inconsistency thus obtaining a matrix which is optimal from the point of view of consistency with respect to the available judgements the optimal values are obtained by solving a linear system and unicity of the solution is proved under general assumptions some other methods proposed in the literature are discussed and a numerical example is presented

finite element approximation of the nonstationary navier stokes problem i regularity of solutions and second order error estimates for spatial discretization this is the first part of a work dealing with the rigorous error analysis of finite element solutions of the nonstationary navier stokes equations second order error estimates are proven for spatial discretization using conforming or nonconforming elements the results indicate a fluid like behavior of the approximations even in the case of large data so long as the solution remains regular the analysis is based on sharp a priori estimates for the solution particularly reflecting its behavior as t to 0 and as t to infty it is shown that the regularity customarily assumed in the error analysis for corresponding parabolic problems cannot be realistically assumed in the case of the navier stokes equations as it depends on nonlocal compatibility conditions for the data the results which are presented here are independent of such compatibility conditions which cannot be verified in practice

multicriteria decision analysis with fuzzy pairwise comparisons abstract we present a modification of the fuzzy multi criteria method proposed by van laarhoven and pedrycz 1983 first the weights of the decision criteria are calculated by the minimization of a logarithmic regression function next the weights of the decision alternatives are calculated for each criterion separately lastly the fuzzy final scores of the alternatives are determined by an appropriate aggregationm of the calculated weights we will show that our modified method in contrast with the original one yields weights which are optimal with respect to the logarithmic regression function we also present a version of our method with geometric ratio scales rather than saaty s equi distant scale to quantify the gradations of human comparative judgement this approach also enables us to develop an efficient procedure for investigating the scale sensitivity of the calculated weights and the final scores of the alternatives

group decision analysis based on fuzzy preference relations logarithmic and geometric least squares methods logarithmic and geometric least squares methods llsm and glsm are respectively applied to deal with the group decision analysis problems with fuzzy preference relations where multiplicative preference relations if any are transformed into fuzzy preference relations through proper transformation technique distance between any two fuzzy preference relations and the average distance from one fuzzy preference relation to all the others are defined and used to measure the relative importance of each fuzzy preference relation a numerical example involving multiple fuzzy and multiplicative preference relations is examined using the proposed methods it is shown that llsm and glsm provide two analytical and effective ways of modelling multiple fuzzy preference relations

projective market model approach to ahp decision making abstract in this paper we describe market in the projective geometry language and give the definition of a matrix of market rate which is related to the matrix rate of return and the matrix of judgements in the analytic hierarchy process ahp we use these observations to extend the ahp model to the projective geometry formalism and generalise it to an intransitive case we give financial interpretations of such a generalised model and propose its simplification the unification of the ahp model and projective aspect of portfolio theory suggests a wide spectrum of new applications for such an extended model

a matrix based ranking method with application to tennis abstract we introduce and investigate a ranking methodology which may be of interest in sports like tennis the approach may also be of interest in decision making situations based on pairwise comparisons the method is based on linear algebra and one computes a score for each player by solving a certain linear system of equations from these scores one finds the ranking the input is a set of matches and weights representing the importance of the matches this is represented by a weighted directed graph we prove a number of properties of the method including uniqueness of scores connection to m matrices and combinatorial interpretations a case study from ranking in professional tennis is discussed in detail

consistent clustering of entries in large pairwise comparison matrices abstract in multi attribute decision making the number of decision elements under consideration may be huge especially for complex real world problems typically these elements are clustered and then the clusters organized hierarchically to reduce the number of elements to be simultaneously handled these decomposition methodologies are intended to bring the problem within the cognitive ability of decision makers however such methodologies have disadvantages and it may happen that such a priori clustering is not clear and or the problem has previously been addressed without any grouping action this is the situation for the case study we address in which a panel of experts gives opinions about the operation of 15 previously established district metered areas in a real water distribution system large pairwise comparison matrices may also be found when building comparisons of elements using large bodies of information in this paper we address a consistent compression of an ahp comparison matrix that collapses the judgments corresponding to a given number of compared elements as a result an a posteriori clustering of various elements becomes possible in our case study such a clustering offers several added benefits including the identification of hidden or unknown criteria to cluster the considered elements of the problem

finite element approximations of the nonstationary navier stokes problem part iv error estimates for second order time discretization this paper provides an error analysis for the crank nicolson method of time discretization applied to spatially discrete galerkin approximations of the nonstationary navier stokes equations second order error estimates are proven locally in time under realistic assumptions about the regularity of the solution for approximations of an exponentially stable solution these local error estimates are extended uniformly in time as text t to infty

the logarithmic least squares and the generalized pseudoinverse in estimating ratios abstract this paper concerns the pairwise comparison method used in analytic hierarchy process ahp the logarithmic least square method is one of the methods used to rank a finite number of stimuli based on their pairwise comparison in the case of one decision maker the problem can be solved using the geometric mean method it is then assumed that the solution is geometrically normalized in the case of multiple decision makers a set of linear equations is obtained and if we have a different number of judgments for each pair of the compared objects the geometric normalization assumption can not be used directly the aim of this paper is to show that applying the generalized pseudoinverse we obtain the solution that is geometrically normalized and consistent with the case of one decision maker to define the pseudoinverse the spectral decomposition is used the structure of the general solution is presented and the existence of the general solution is discussed

weight conservation condition and structure of the braid group representation a weight conservation condition for the s matrix of the braid group representation from which non vanishing elements of the s matrix can be determined is introduced a method of weight lattice analysis of representations of lie algebra is suggested and used to give structures of braid group representations in various cases this makes it feasible for braid group representations to be obtained by solving the spectral parameter independent yang baxter equation directly which leads to several new braid group representations

the stabilized extrapolated trapezoidal finite element method for the navier stokes equations abstract we consider a fully discrete stabilized finite element method for the navier stokes equations which is unconditionally stable and has second order temporal accuracy of o k 2 hk spatial error the method involves a simple artificial viscosity stabilization of the linear system for the approximation of the new time level connected to anti diffusion of its effects at the old time level lowering the cell reynolds number of the oseen problem to o 1 the method requires only the solution of one linear system arising from an oseen problem per time step the effectiveness of the method is illustrated in several numerical experiments

a family of second order time stepping methods for the darcy brinkman equations abstract this study presents an efficient accurate effective and unconditionally stable time stepping scheme for the darcy brinkman equations in double diffusive convection the stabilization within the proposed method uses the idea of stabilizing the curvature for velocity temperature and concentration equations accuracy in time is proven and the convergence results for the fully discrete solutions of problem variables are given several numerical examples including a convergence study are provided that support the derived theoretical results and demonstrate the efficiency and the accuracy of the method

a second order in time uniquely solvable unconditionally stable numerical scheme for cahn hilliard navier stokes equation we propose a novel second order in time numerical scheme for cahn hilliard navier stokes phase field model with matched density the scheme is based on second order convex splitting for the cahn hilliard equation and pressure projection for the navier stokes equation we show that the scheme is mass conservative satisfies a modified energy law and is therefore unconditionally stable moreover we prove that the scheme is unconditionally uniquely solvable at each time step by exploring the monotonicity associated with the scheme thanks to the simple coupling of the scheme we design an efficient picard iteration procedure to further decouple the computation of cahn hilliard equation and navier stokes equation we implement the scheme by the mixed finite element method ample numerical experiments are performed to validate the accuracy and efficiency of the numerical scheme

a two level method with backtracking for the navier stokes equations we consider a two level method for resolving the nonlinearity in finite element approximations of the equilibrium navier stokes equations the method yields l2 and h1 optimal velocity approximations and an l2 optimal pressure approximation the two level method involves solving one small nonlinear coarse mesh system one oseen problem hence linear with positive definite symmetric part on the fine mesh and one linear correction problem on the coarse mesh the algorithm we study produces an approximate solution with the optimal asymptotic in h accuracy for any fixed reynolds number we do not consider the behavior of the error for fixed h as re rightarrow infty i e for flows in transition to turbulence

the braid group representations associated with some non fundamental representations of lie algebras the braid group representations associated with the eight dimensional representation of b3 and six dimensional representation of a2 including the quantum lie algebra connected with the latter are explicitly calculated

efficient and accurate numerical schemes for a hydro dynamically coupled phase field diblock copolymer model in this paper we consider numerical approximations of a hydro dynamically coupled phase field diblock copolymer model in which the free energy contains a kinetic potential a gradient entropy a ginzburglandau double well potential and a long range nonlocal type potential we develop a set of second order time marching schemes for this system using the invariant energy quadratization approach for the double well potential the projection method for the navierstokes equation and a subtle implicit explicit treatment for the stress and convective term the resulting schemes are linear and lead to symmetric positive definite systems at each time step thus they can be efficiently solved we further prove that these schemes are unconditionally energy stable various numerical experiments are performed to validate the accuracy and energy stability of the proposed schemes

on conforming finite element methods for the inhomogeneous stationary navier stokes equations we consider the stationary navier stokes equations written in terms of the primitive variables in the case where both the partial differential equations and boundary conditions are inhomogeneous under certain conditions on the data the existence and uniqueness of the solution of a weak formulation of the equations can be guaranteed a conforming finite element method is presented and optimal estimates for the error of the approximate solution are proved in addition the convergence properties of iterative methods for the solution of the discrete nonlinear algebraic systems resulting from the finite element algorithm are given numerical examples using an efficient choice of finite element spaces are also provided

a comparison of time discretization linearization approaches for the incompressible navier stokes equations abstract this paper presents a numerical study of two ways for discretizing and linearizing the time dependent incompressible navier stokes equations one approach consists in first applying a semi discretization in time by a fully implicit scheme then in each discrete time the equations are linearized by a fixed point iteration the number of iterations to reach a given stopping criterion is a priori unknown in this approach in the second approach rosenbrock schemes with s stages are used as temporal discretization the non linearity of the navier stokes equations is treated internally in the rosenbrock methods in each discrete time exactly s linear systems of equations have to be solved the numerical study considers five two dimensional problems with distinct features four implicit time stepping schemes and five rosenbrock methods are involved

a modified fuzzy logarithmic least squares method for fuzzy analytic hierarchy process this paper revisits the fuzzy logarithmic least squares method llsm in the analytic hierarchy process and points out its incorrectness in the normalization of local fuzzy weights infeasibility in deriving the local fuzzy weights of a fuzzy comparison matrix when the lower bound value of a non normalized fuzzy weight turns out to be greater than its upper bound value uncertainty of local fuzzy weights for incomplete fuzzy comparison matrices and unreality of global fuzzy weights a modified fuzzy llsm which is formulated as a constrained nonlinear optimization model is therefore suggested to tackle all these problems a numerical example is examined to show the applicability of the modified fuzzy llsm and its advantages

a monte carlo investigation of incomplete pairwise comparison matrices in ahp the analytic hierarchy process ahp is a decision analysis technique used to evaluate complex multiattribute alternatives among one or more decision makers it imposes a hierarchical structure on any complex multicriterion problem however a major drawback of the ahp is that a large number of pairwise comparisons is needed to calibrate the hierarchy when there are a few levels and sublevels the ahp can be applied in a straightforward manner to derive the weights relative preference for each alternative as the size of the hierarchy increases the number of pairwise comparisons increases rapidly it is well established in the marketing and consumer behavior literature that in a very long interview even under the best circumstances the respondent is likely to suffer from information overload recognition of this problem was the motivation which led to the investigation of a modification of ahp which required less data collection i e a reduction in the threat of information overload the first question to be answered is the effect on ahp weights due to different patterns of missing data likely to result from reallife data collection in this study a monte carlo simulation was conducted which uses the incomplete pairwise comparisons ipc algorithm 14 to investigate the effect of reduced sets of pairwise comparisons in the ahp data for the study were generated with known structure and comparisons made between complete and incomplete matrices the results of the simulation suggest that incomplete sets of pairwise comparison matrices can capture the attribute level weights without significant loss of accuracy and independent of decision model form and amount of error considered

computational scales of sobolev norms with application to preconditioning this paper provides a framework for developing computationally efficient multilevel preconditioners and representations for sobolev norms specifically given a hilbert space v and a nested sequence of subspaces v 1 c v 2 c c v we construct operators which are spectrally equivalent to those of the form a k k q k q k 1 here k k 1 2 are positive numbers and q k is the orthogonal projector onto v k with q 0 0 we first present abstract results which show when a is spectrally equivalent to a similarly constructed operator a defined in terms of an approximation q k of q k for k 1 2 we show that these results lead to efficient preconditioners for discretizations of differential and pscudo differential operators of positive and negative order these results extend to sums of operators for example singularly perturbed problems such as i e can be preconditioned uniformly independently of the parameter e we also show how to precondition an operator which results from tikhonov regularization of a problem with noisy data finally we describe how the technique provides computationally efficient bounded discrete extensions which have applications to domain decomposition

multiperson decision making based on multiplicative preference relations abstract a multiperson decision making problem where the information about the alternatives provided by the experts can be presented by means of different preference representation structures preference orderings utility functions and multiplicative preference relations is studied assuming the multiplicative preference relation as the uniform element of the preference representation a multiplicative decision model based on fuzzy majority is presented to choose the best alternatives in this decision model several transformation functions are obtained to relate preference orderings and utility functions with multiplicative preference relations the decision model uses the ordered weighted geometric operator to aggregate information and two choice degrees to rank the alternatives quantifier guided dominance degree and quantifier guided non dominance degree the consistency of the model is analysed to prove that it acts coherently

a critical analysis of the eigenvalue method used to derive priorities in ahp abstract a lot of research has been devoted to the critical analysis of the analytic hierarchy process ahp from various perspectives however as far as we know no one has addressed a fundamental problem discussed in this paper concerning the meaning of the priority vector derived from the principal eigenvalue method used in ahp the role of ahp s consistency ratio is also analysed

stability and convergence of the crank nicolson adams bashforth scheme for the time dependent navier stokes equations in this paper we study the stability and convergence of the crank nicolson adams bashforth scheme for the two dimensional nonstationary navier stokes equations a finite element method is applied for the spatial approximation of the velocity and pressure the time discretization is based on the crank nicolson scheme for the linear term and the explicit adams bashforth scheme for the nonlinear term moreover we present optimal error estimates and prove that the scheme is almost unconditionally stable and convergent i e stable and convergent when the time step is less than or equal to a constant

representations of a braid group with transpose symmetry and the related link invariants under a weight conservation condition when q is not a root of unity yang baxter relations are solved without the assumption that the upper left triangle of each sub block of the s matrix of the braid group representation bgr vanishes a general form of bgr associated with bn cn and dn is obtained and shown to construct link invariants the many to one correspondence between bgrs and link invariants is given

an application of incomplete pairwise comparison matrices for ranking top tennis players pairwise comparison is an important tool in multi attribute decision making pairwise comparison matrices pcm have been applied for ranking criteria and for scoring alternatives according to a given criterion our paper presents a special application of incomplete pcms ranking of professional tennis players based on their results against each other the selected 25 players have been on the top of the atp rankings for a shorter or longer period in the last 40 years some of them have never met on the court one of the aims of the paper is to provide ranking of the selected players however the analysis of incomplete pairwise comparison matrices is also in the focus the eigenvector method and the logarithmic least squares method were used to calculate weights from incomplete pcms in our results the top three players of four decades were nadal federer and sampras some questions have been raised on the properties of incomplete pcms and remains open for further investigation

acceptable consistency analysis of interval reciprocal comparison matrices when a decision maker expresses his her opinions by means of an interval reciprocal comparison matrix the study of consistency becomes a very important aspect in decision making in order to avoid a misleading solution in the present paper an acceptably consistent interval reciprocal comparison matrix is defined which can be reduced to an acceptably consistent crisp reciprocal comparison matrix when the intervals become exact numbers an interval reciprocal comparison matrix with unacceptable consistency can be easily adjusted such that the revised matrix possesses acceptable consistency utilizing a convex combination method a family of crisp reciprocal comparison matrices with acceptable consistency can be obtained whose weights are further found to exhibit a style of convex combination and aggregated to obtain interval weights from an acceptably consistent interval reciprocal comparison matrix a novel simple yet effective formula of possibility degree is presented to rank interval weights numerical results are calculated to show the quality and quantity of the proposed approaches and compare with other existing procedures

integrating multiplicative preference relations in a multipurpose decision making model based on fuzzy preference relations the aim of this paper is to study the integration of multiplicative preference relation as a preference representation structure in fuzzy multipurpose decision makingproblems assumingfuzzy multipurpose decision makingproblems under di4erent preference representation structures ordering utilities and fuzzy preference relations and using the fuzzy preference relations as uniform representation elements the multiplicative preference relations are incorporated in the decision problem by means of a transformation function between multiplicative and fuzzy preference relations a consistency study of this transformation function which demonstrates that it does not change the informative content of multiplicative preference relation is shown as a consequence a selection process based on fuzzy majority for multipurpose decision makingproblems under multiplicative preference relations is presented to design it an aggregation operator of information called ordered weighted geometric operator is introduced and two choice degrees the quanti7er guided dominance degree and the quanti7er guided non dominance degree are de7ned for multiplicative preference relations c 2001 elsevier science b v all rights reserved