goal programming for decision making an overview of the current state of the art there have been significant advances in the theory of goal programming gp in recent years particularly in the area of intelligent modelling and solution analysis the intention of this paper is to provide an overview of these developments to detail and assess the current state of the art in the subject and to highlight areas which seem promising for future research modelling techniques such as detection and restoration of pareto efficiency normalisation redundancy checking and non standard utility function modelling are overviewed the connection between gp and other multi objective programming techniques as well as a utility interpretation of gp are examined the rationality of ranking multi criteria decision making techniques and of placing gp in such a ranking is discussed

a parametric model for determining consensus priority vectors from fuzzy comparison matrices abstract we consider a group decision making problem where a set of alternatives have to be ranked according to fuzzy preference judgments given by multiple experts we assume that expert assessments are expressed in the form of fuzzy multiplicative preference relations or fuzzy comparison matrices in this paper we propose a general model to generate crisp priority weights of the alternatives from possibly inconsistent and conflicting fuzzy preference relations we express our model in terms of matrix approximations to address the consistency problem and we use weighted metrics to simulate the group dynamics matrix approximation techniques for deriving a common crisp consistent matrix are extended to work with fuzzy matrices by using the concept of cuts in the aggregation process the importance of each expert is taken into consideration according to the agreement of the group with the expert this results in a parametric optimization problem for which a computational formulation is given

max algebra and pairwise comparison matrices abstract this paper is a continuation of our 2004 paper max algebra and pairwise comparison matrices in which the max eigenvector of a symmetrically reciprocal matrix was used to approximate such a matrix by a transitive matrix this approximation was based on minimizing the maximal relative error in a later paper by dahl a different error measure was used and led to a slightly different approximating transitive matrix here some geometric properties of this approximation problem are discussed these lead among other results to a new characterization of a max eigenvector of an irreducible nonnegative matrix the case of toeplitz matrices is discussed in detail and an application to music theory that uses toeplitz symmetrically reciprocal matrices is given

linear rank tests for interval censored data with application to pcb levels in adipose tissue of transformer repair workers linear rank statistics are described for testing for differences between groups when the data are interval censored the statistics are closely related to those described by prentice for right censored data problems in calculating the statistics are discussed and several approaches to computation including estimation of the efficient rank scores are described results from a small simulation study are presented the methods are applied to data from a study relating tissue levels of pcbs to occupational exposure

deriving weights from pairwise comparison ratio matrices an axiomatic approach abstract this paper examines the problem of extracting object or attribute weights from a pairwise comparison ratio matrix this problem is approached from the point of view of a distance measure on the space of all such matrices a set of axioms is presented which such a distance measure should satisfy and the uniqueness of the measure is proven the problem of weight derivation is then shown to be equivalent to that of finding a totally transitive matrix which is a minimum distance from the given matrix this problem reduces to a goal programming model finally it is shown that the problem of weight derivation is related to that of ranking players in a round robin tournament the space of all binary tournament matrices is proven to be isometric to a subset of the space of ratio matrices

fiedler ptak scaling in max algebra abstract this is essentially the text of my talk on fiedler ptak scaling in max algebra delivered in the invited minisymposium in honor of miroslav fiedler at the 17th ilas conference in braunschweig germany

a multipoint flux mixed finite element method we develop a mixed finite element method for single phase flow in porous media that reduces to cell centered finite differences on quadrilateral and simplicial grids and performs well for discontinuous full tensor coefficients motivated by the multipoint flux approximation method where subedge fluxes are introduced we consider the lowest order brezzi douglas marini bdm mixed finite element method a special quadrature rule is employed that allows for local velocity elimination and leads to a symmetric and positive definite cell centered system for the pressures theoretical and numerical results indicate second order convergence for pressures at the cell centers and first order convergence for subedge fluxes second order convergence for edge fluxes is also observed computationally if the grids are sufficiently regular

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

weighted logrank tests for interval censored data when assessment times depend on treatment we consider weighted logrank tests for interval censored data when assessment times may depend on treatment and for each individual we only use the two assessment times that bracket the event of interest it is known that treating finite right endpoints as observed events can substantially inflate the type i error rate under assessment treatment dependence atd but the validity of several other implementations of weighted logrank tests score tests permutation tests multiple imputation tests has not been studied in this situation with a bounded number of unique assessment times the score test under the grouped continuous model retains the type i error rate asymptotically under atd however although the approximate permutation test based on the permutation central limit theorem is not asymptotically valid under every atd scenario we show through simulation that in many atd scenarios it retains the type i error rate better than the score test we show a case where the approximate permutation test retains the type i error rate when the exact permutation test does not we study and modify the multiple imputation logrank tests of huang lee and yu 2008 statistics in medicine 27 3217 3226 showing that the distribution of the rank like scores asymptotically does not depend on the assessment times we show through simulations that our modifications of the multiple imputation logrank tests retain the type i error rate in all cases studied even with atd and a small number of individuals in each treatment group simulations were performed using the interval r package us government work in the public domain

a markov chain monte carlo em algorithm for analyzing interval censored data under the cox proportional hazards model summary this paper proposes a monte carlo em mcem algorithm for fitting the proportional hazards model for interval censored failure time data the algorithm generates orderings of the failures from their probability distribution under the model we maximize the average of the log likelihoods from these completed data sets to obtain updated parameter estimates as with the standard cox model this algorithm does not require the estimation of the baseline hazard function the performance of the algorithm is evaluated using simulations and the method is applied to data from aids and cancer studies our results indicate that our method produced more precise and unbiased estimates than methods of right and midpoint imputation

a semiring based study of judgment matrices properties and models in decision making and group decision making multiplicative reciprocal judgment matrices and additive reciprocal judgment matrices are used as two kinds of important preference information in this paper semirings are applied to the discussion of judgment matrix properties first two special semirings are constructed second the properties of the consistent judgment matrices are given as a set of equations all in the semiring sense which include idempotency equations and fixed point equations we find that there exists one and only one specially constrained fixed point as the priority vector of a consistent judgment matrix third optimization models for inconsistent judgment matrices are presented finally we offer some simple illustrative examples

a method for approximating symmetrically reciprocal matrices by transitive matrices the problem of approximating symmetrically reciprocal matrices by transitive matrices has received some attention recently this problem has applications in multicriteria decision theory several approximation approaches have been suggested and analyzed we here suggest another approach called the multiplicative approach we show that the optimal approximation in this sense may be found efficiently by transforming the problem into a known combinatorial optimization problem the minimum cycle mean problem for which efficient and simple combinatorial algorithms exist

a nonparametric two sample test of the failure function with interval censoring case 2 for the setting of interval censoring case 2 a nonparametric two sample test based on a smooth functional of the nonparametric maximum pseudolikelihood estimator is developed to compare the failure functions of two populations monte carlo simulation studies on weibull distributions show that this test performs quite well in many situations in comparison with the parametric likelihood ratio test a real dataset from an study is used to illustrate the new test

a nine point scheme for the approximation of diffusion operators on distorted quadrilateral meshes a nine point scheme is presented for discretizing diffusion operators on distorted quadrilateral meshes the advantage of this method is that highly distorted meshes can be used without the numerical results being altered remarkably and it treats material discontinuities rigorously and offers an explicit expression for the face centered flux moreover it has only the cell centered unknowns we prove that the method is stable and has first order convergence on distorted meshes numerical experiments show that the method has second order or nearly second order accuracy on distorted meshes

an improvement of the two point flux approximation scheme on polygonal meshes abstract a new diffusion scheme satisfying the maximum principle on two dimensional 2d polygonal meshes is proposed being different from the existing schemes which are based on the nonlinear combination of the two one side fluxes the scheme in this paper is constructed by a modification of the conventional linear two point flux approximation tpfa scheme in which the two side flux is constructed to improve its accuracy on 2d skewed meshes the tpfa scheme is rewritten by adding a tangential part i e the tangential component of the flux multiplied with a nonlinear factor a new method for the discretization of the tangential part is presented so that both the conservation condition and the maximum principle are satisfied while the tpfa part remains unchanged moreover by virtue of the harmonic averaging points for the calculation of the tangential part the scheme has a minimal stencil similar to the tpfa scheme finally some numerical examples are presented to verify that it is second order accurate on 2d polygonal meshes for anisotropic diffusion problems

tropical implementation of the analytical hierarchy process decision method abstract we apply methods and techniques of tropical optimization to develop a new theoretical and computational framework for the implementation of the analytic hierarchy process in multi criteria problems of rating alternatives from pairwise comparison data in this framework we first consider the minimax chebyshev approximation of pairwise comparison matrices by consistent matrices in the logarithmic scale recasting this approximation problem as a problem of tropical pseudo quadratic programming we then write out a closed form solution to it this solution might be either a unique score vector up to a positive factor or a set of different score vectors to handle the problem when the solution is not unique we develop tropical optimization techniques of maximizing and minimizing the hilbert seminorm to find those vectors from the solution set that are the most and least differentiating between the alternatives with the highest and lowest scores and thus are well representative of the entire solution set

rank invariant tests for interval censored data under the grouped continuous model this paper creates rank invariant score tests for grouped or interval censored data this generalizes finkelstein 1986 biometrics 42 845 854 who derived score tests for interval censored data assuming proportional hazards we frame the problem as a linear rank test of a shift in location with a known error distribution we discuss adjustments to the test that may be required when the number of observation times is large we offer a graphical test of the assumption of the location shift model and discuss an alternative interpretation of the test using the logistic error when the location shift assumption does not hold

modifying the power method in max algebra in the max algebra system the eigenequation for an n n irreducible nonnegative matrix a aij is a x a x here a x i maxjaijxj and a is the maximum circuit geometric mean the complexity of the power method given in l elsner p van den driessche linear algebra appl to appear to compute a and x is considered under some assumptions on the critical matrix it is shown that the algorithm may have time complexity o n4 a modified power method based on karp s formula is presented for this new algorithm with no assumptions on the critical matrix a and x can be computed in o n3 time furthermore this algorithm can be used to compute all linearly independent eigenvectors corresponding to a

the finite volume scheme preserving extremum principle for diffusion equations on polygonal meshes we construct a new nonlinear finite volume scheme for diffusion equation on polygonal meshes and prove that the scheme satisfies the discrete extremum principle our scheme is locally conservative and has only cell centered unknowns numerical results are presented to show how our scheme works for preserving discrete extremum principle and positivity on various distorted meshes

the analytic hierarchy process max algebra and multi objective optimisation the analytic hierarchy process ahp is widely used for decision making involving multiple criteria elsner and van den driessche 2004 2010 10 11 introduced a max algebraic approach to the single criterion ahp we extend this to the multi criteria ahp by considering multi objective generalisations of the single objective optimisation problem solved in these earlier papers we relate the existence of globally optimal solutions to the commutativity properties of the associated matrices we relate min max optimal solutions to the generalised spectral radius and we prove that pareto optimal solutions are guaranteed to exist

comparing several score tests for interval censored data i create a general model to perform score tests on interval censored data special cases of this model are the score tests of finkelstein sun and fay although sun s was derived as a test for discrete data and finkelstein s and fay s tests were derived under a grouped continuous model by writing all tests under one general model we see that as long as the regularity conditions hold any of these three classes of tests may be applied to either grouped continuous or discrete data i show the equivalence between the weighted logrank form of the general test and the form with a term for each individual the form often used with permutation tests from the weighted logrank form of the tests we see that sun s and finkelstein s test are similar giving constant or approximately constant weights to differences in survival distributions over time in contrast the proportional odds model fay s model with logistic error gives more weight to early differences

on the optimal consistent approximation to pairwise comparison matrices abstract consistency retrieval from a biased relative preference table is an imperative task in decision theory this paper considers the least squares approximation of a pairwise comparison matrix by consistent matrices it is observed that the highly nonlinear manifold of consistent matrices can be changed into a linear subspace by the componentwise logarithmic transformation a first order optimality condition therefore can be described in terms of coordinates in the linear subspace this approach facilitates the otherwise much more complicated optimality condition if working with the variables in the original manifold fast nonlinear equation solvers can be employed to solve the problem efficiently

the mimetic finite difference discretization of diffusion problem on unstructured polyhedral meshes we study the mimetic finite difference discretization of diffusion type problems on unstructured polyhedral meshes we demonstrate high accuracy of the approximate solutions for general diffusion tensors the second order convergence rate for the scalar unknown and the first order convergence rate for the vector unknown on smooth or slightly distorted meshes on non matching meshes and even on meshes with irregular shaped polyhedra with flat faces we show that in general the meshes with non flat faces require more than one flux unknown per mesh face to get optimal convergence rates

a two sample test for stochastic ordering with interval censored data a new class of asymptotically nonparametric two sample approximate tests for discrete interval censored data is proposed this class extends the family developed by pepe and fleming 1989 biometrics 45 497 507 for right censored data and is based on an integrated weighted difference in survival function estimates the tests are sensitive to the alternative of stochastic ordering and use turnbull s 1976 journal of the royal statistical society series b 38 290 295 estimator of the survival function the choice of appropriate weight functions is investigated simulation results indicate that for interval censored data the new procedure with an appropriate weight function can be more powerful than an omnibus likelihood ratio test and mantel s 1967 biometrics 23 65 78 test under a broad range of stochastically ordered alternatives for small sample sizes n 50 with moderate censoring 47 and 70 the size of the test is close to the nominal size

monotone finite volume schemes for diffusion equations on unstructured triangular and shape regular polygonal meshes we consider a non linear finite volume fv scheme for stationary diffusion equation we prove that the scheme is monotone i e it preserves positivity of analytical solutions on arbitrary triangular meshes for strongly anisotropic and heterogeneous full tensor coefficients the scheme is extended to regular star shaped polygonal meshes and isotropic heterogeneous coefficients

maximum likelihood estimation for interval censored data using a weibull based accelerated failure time model the accelerated failure time regression model is most commonly used with right censored survival data this report studies the use of a weibull based accelerated failure time regression model when left and interval censored data are also observed two alternative methods of analysis are considered first the maximum likelihood estimates mles for the observed censoring pattern are computed these are compared with estimates where midpoints are substituted for left and interval censored data midpoint estimator or mde simulation studies indicate that for relatively large samples there are many instances when the mle is superior to the mde for samples where the hazard rate is flat or nearly so or where the percentage of interval censored data is small the mde is adequate an example using framingham heart study data is discussed

rank based inference in the proportional hazards model for interval censored data summary a marginal likelihood approach to fitting the proportional hazards model to interval censored or grouped data is proposed this approach maximises a likelihood that is the sum over all rankings of the data that are consistent with the observed censoring intervals as in the usual proportional hazards model the method does not require specification of the baseline hazard function the score equations determining the maximum marginal likelihood estimator can be written as the expected value of the score of the usual proportional hazards model with respect to a certain distribution of rankings a gibbs sampling scheme is given to generate rankings from this distribution and stochastic approximation is used to solve the score equations simulation results under various censoring schemes give point estimates that are close to estimates obtained using actual failure times

continuity of the generalized spectral radius in max algebra abstract let be an induced matrix norm associated with a monotone norm on r n and be the collection of all nonempty closed and bounded subsets of n n nonnegative matrices under this matrix norm for the hausdorff metric h between and is given by h max sup a inf b a b sup b inf a a b the max algebra system consists of the set of nonnegative numbers with sum a b max a b and the standard product ab for a b 0 for n n nonnegative matrices a b their product is denoted by a b where a b ij max 1 k n a ik b kj for each the max algebra version of the generalized spectral radius of is limsup m sup a m a 1 m where m a 1 a 2 a m a i here a is the maximum circuit geometric mean in this paper we prove that the max algebra version of the generalized spectral radius is continuous on the hausdorff metric space h the notion of the max algebra version of simultaneous nilpotence of matrices is also proposed necessary and sufficient conditions for the max algebra version of simultaneous nilpotence of matrices are presented as well

spectral properties of matrix polynomials in the max algebra we consider the spectral properties of matrix polynomials over the max algebra in particular we show how the perron frobenius theorem for the max algebra extends to such polynomials and illustrate the relevance of this for multistep difference equations in the max algebra we also present a number of inequalities for the largest max eigenvalue of a matrix polynomial

on the continuity of the generalized spectral radius in max algebra abstract given a bounded set of n n non negative matrices let and denote the generalized spectral radius of and its max version respectively we show that sup t 0 n 1 t 1 t where t denotes the hadamard power of we apply this result to give a new short proof of a known fact that is continuous on the hausdorff metric space h of all nonempty compact collections of n n non negative matrices

enforcing positivity with conservation for nine point scheme of nonlinear diffusion equations abstract the nine point scheme nps is a finite volume scheme for solving nonlinear diffusion equations on quadrilateral grids it s well known that the solution of the scheme on distorted grids cannot preserve positivity when all given data are nonnegative here for the nps of nonlinear diffusion problems we propose a conservative enforcing positivity algorithm cepa which is posterior correction of the discrete solution and allows the new solution preserves positivity as well as local conservation at each step of nonlinear iteration numerical results are presented to demonstrate that the accuracy of our algorithm of enforcing positivity with conservation is higher than some common procedures of positivity correction

a test for comparing current status survival data with crossing hazard functions and its application to immunogenicity of biotherapeutics several tests have been recently implemented in the nonparametric comparison of current status survival data however they are not suited for the situation of crossing hazards in this setting we propose a new test specifically designed for crossing hazards alternatives the proposed test is compared to classical implemented tests through simulations mimicking crossing hazards situations with various schemes of censoring the results show that the proposed test has a correct type i error and generally outperforms the existing methods the application of the proposed test on a real dataset on immunogenicity of interferon among multiple sclerosis patients highlights the interest of the proposed test

monotone finite volume schemes for diffusion equations on polygonal meshes we construct a nonlinear finite volume fv scheme for diffusion equation on star shaped polygonal meshes and prove that the scheme is monotone i e it preserves positivity of analytical solutions for strongly anisotropic and heterogeneous full tensor coefficients our scheme has only cell centered unknowns and it treats material discontinuities rigorously and offers an explicit expression for the normal flux numerical results are presented to show how our scheme works for preserving positivity on various distorted meshes for both smooth and non smooth highly anisotropic solutions and numerical results show that our scheme appears to be approximate second order accuracy for the solution and first order accuracy for the flux

an improved monotone finite volume scheme for diffusion equation on polygonal meshes we construct a new nonlinear monotone finite volume scheme for diffusion equation on polygonal meshes the new scheme uses the cell edge unknowns instead of cell vertex unknowns as the auxiliary unknowns in order to improve the accuracy of monotone scheme our scheme is locally conservative and has only cell centered unknowns numerical results are presented to show how our scheme works for preserving positivity on various distorted meshes specially numerical results show that the new scheme is robust and more accurate than the existing monotone scheme on some kinds of meshes

elimination of adaptive grid interface errors in the discrete cell centered pressure equation classical cell centered discretization of the reservoir simulation pressure equation on anh adaptive grid results in ano 1 h leading truncation error at the grid interface a new flux continuous finite volume correction is presented together with an improved version of a previously proposed correction while both corrections eliminate the leading error the new correction exhibits the best convergence rates and has the following properties the resulting matrix is in general symmetric positive definite diagonally dominant for locally isotropic spatially varying coefficients convergence too h is demonstrated and support of the standard approximation is retained ensuring a fully implicit implementation results computed by the uncorrected classical scheme and both correction schemes are compared for two phase flow simulations with multilevel dynamic local grid refinement

transitive matrices and their applications abstract transitive matrices and symmetrically reciprocal sr matrices are defined and spectral properties of certain sr perturbations of transitive matrices are studied the results are applied to a multicriteria decision making method the analytic hierarchy process ahp which uses an sr matrix with positive entries a proof is given that rank reversal is inherent in this method if its input matrix is perturbed

a robust and efficient finite volume scheme for the discretization of diffusive flux on extremely skewed meshes in complex geometries in this paper an improved finite volume scheme to discretize diffusive flux on a non orthogonal mesh is proposed this approach based on an iterative technique initially suggested by khosla p k khosla s g rubin a diagonally dominant second order accurate implicit scheme computers and fluids 2 1974 207 209 and known as deferred correction has been intensively utilized by muzaferija s muzaferija adaptative finite volume method for flow prediction using unstructured meshes and multigrid approach ph d thesis imperial college 1994 and later fergizer and peric j h fergizer m peric computational methods for fluid dynamics springer 2002 to deal with the non orthogonality of the control volumes using a more suitable decomposition of the normal gradient our scheme gives accurate solutions in geometries where the basic idea of muzaferija fails first the performances of both schemes are compared for a poisson problem solved in quadrangular domains where control volumes are increasingly skewed in order to test their robustness and efficiency it is shown that convergence properties and the accuracy order of the solution are not degraded even on extremely skewed mesh next the very stable behavior of the method is successfully demonstrated on a randomly distorted grid as well as on an anisotropically distorted one finally we compare the solution obtained for quadrilateral control volumes to the ones obtained with a finite element code and with an unstructured version of our finite volume code for triangular control volumes no differences can be observed between the different solutions which demonstrates the effectiveness of our approach

ignorability and coarse data we present a general statistical model for data coarsening which includes as special cases rounded heaped censored partially categorized and missing data formally with coarse data observations are made not in the sample space of the random variable of interest but rather in its power set grouping is a special case in which the degree of coarsening is known and nonstochastic we establish simple conditions under which the possible stochastic nature of the coarsening mechanism can be ignored when drawing bayesian and likelihood inferences and thus the data can be validly treated as grouped data the conditions are that the data be coarsened at random a generalization of the condition missing at random and that the parameters of the data and the coarsening process be distinct applications of the general model and the ignorability condition are illustrated in a numerical example and described briefly in a variety of special cases

a reconstruction algorithm with flexible stencils for anisotropic diffusion equations on 2d skewed meshes a new reconstruction algorithm is developed to obtain diffusion schemes with cell centered unknowns only the main characteristic of the new algorithm is the flexibility of stencils when the auxiliary unknowns are reconstructed with cell centered unknowns the stencils are selected depending on the mesh geometry and discontinuities of diffusion coefficients moreover an explicit expression is derived for interpolating the auxiliary unknowns in terms of cell centered unknowns and the auxiliary unknowns can be defined at any point on the edge the algorithm is applied to construct several new diffusion schemes whose effectiveness is illustrated by numerical experiments for anisotropic problems with or without discontinuities nearly second order accuracy is achieved on skewed meshes

interval censoring identifiability and the constant sum property the constant sum property given in oller et al 2004 for censoring models justifies the use of a simplified likelihood to obtain the nonparametric maximum likelihood estimator of the lifetime distribution in this paper we study the relevance of the constant sum property in the identifiability of the lifetime distribution we show that the lifetime distribution is not identifiable outside the class of constant sum models we also show that the lifetime probabilities assigned to the observable intervals are identifiable inside the class of constant sum models we illustrate all these notions with several examples

finite volume method for 2d linear and nonlinear elliptic problems with discontinuities in this paper we study the approximation of solutions to linear and nonlinear elliptic problems with discontinuous coefficients in the discrete duality finite volume ddfv framework this family of schemes allows very general meshes and inherits the main properties of the continuous problem in order to take into account the discontinuities and to prevent consistency defect in the scheme we propose to modify the definition of the numerical fluxes on the edges of the mesh where the discontinuity occurs we first illustrate our approach by the study of the 1d situation then we show how to design our new scheme called m ddfv and we propose its analysis we also describe an iterative solver whose convergence is proved which can be used to solve the nonlinear discrete equations defining the finite volume scheme finally we provide numerical results which confirm that the m ddfv scheme significantly improves the convergence rate of the usual ddfv method for both linear and nonlinear problems

a monotone finite volume scheme for diffusion equations on general non conforming meshes a nonlinear monotone finite volume scheme on general non conforming meshes for diffusion equations is introduced which deals with discontinuous tensor coefficients rigorously since the expression of normal flux depends on auxiliary unknowns defined at cell vertex including hanging nodes we propose a new method to eliminate vertex unknown by using primary unknowns at the centers of the cells sharing the vertex especially the unknowns defined on hanging nodes are eliminated by flux continuous conditions the resulting scheme is monotone and preserves positivity of analytical solutions for strongly anisotropic and heterogeneous full tensor coefficient problems numerical results show that the convergent order of the monotone scheme by different methods of eliminating vertex unknowns will vary remarkably and our new method can assure that it has almost second order accuracy and more accurate than some existing methods

local flux mimetic finite difference methods we develop a local flux mimetic finite difference method for second order elliptic equations with full tensor coefficients on polyhedral meshes to approximate the velocity vector variable the method uses two degrees of freedom per element edge in two dimensions and n degrees of freedom per n gonal mesh face in three dimensions to approximate the pressure scalar variable the method uses one degree of freedom per element a specially chosen quadrature rule for the l 2 product of vector functions allows for a local flux elimination and reduction of the method to a cell centered finite difference scheme for the pressure unknowns under certain assumptions first order convergence is proved for both variables and second order convergence is proved for the pressure the assumptions are verified on simplicial meshes for a particular quadrature rule that leads to a symmetric method for general polyhedral meshes non symmetric methods are constructed based on quadrature rules that are shown to satisfy some of the assumptions numerical results confirm the theory

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

backpropagation multi layer perceptron for incomplete pairwise comparison matrices in analytic hierarchy process abstract analytic hierarchy process ahp is a widely used decision making method in many areas such as management sciences the performance ratings of multiple criteria and alternatives can be elicited from pairwise comparisons obtained by expressing the decision maker s perceive however it may be difficult for the decision maker to prudently assign values to comparisons for large number of criteria or alternatives since there exist distinct relationships between any two elements in the real world the relationship between the missing comparison and the assigned comparisons should be taken into account the aim of this paper is to propose a novel method using a well known regression tool namely the backpropagation multi layer perceptron i e mlp to realize the above implicit relationship so as to estimate a missing pairwise judgment from the other assigned entries a computer simulation is employed to demonstrate that the proposed method can effectively find a missing entry of an incomplete pairwise matrix such that its consistency index is minimized

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

linear rank tests for doubly censored data this paper establishes the asymptotic properties of the non parametric maximum likelihood estimator of the survival function based on data censored both from the left and the right we show that the sample mean based on the survival estimator converges to a normal distribution and we derive non parametric tests for the differences in survival functions the methods are applied to a study of dementia associated with parkinson s disease

on the spectrum of pairwise comparison matrices abstract this paper deals with the spectrum of symmetrically reciprocal sr perturbations of transitive matrices of nonzero complex entries by introducing the so called pseudo circulant matrices the spectral decomposition of specific sr perturbations of n n transitive matrices and that of arbitrary 3 3 sr matrices are provided closed forms are given for all eigenvalues and their associated eigenvectors of these sr matrices also known as pairwise comparison matrices pcms to improve the evaluations at a given pcm as well as the priority ranking of the alternatives these results are applied to the analytic hierarchy process ahp using positive sr matrices explicit intervals for the ranges of possible rank reversals between the alternatives are established the spectrum of certain sr perturbations of transitive matrices with nonzero complex entries is also derived which can be applied in vehicle system dynamics three application examples are included as well

extremal properties of tropical eigenvalues and solutions to tropical optimization problems abstract an unconstrained optimization problem is formulated in terms of tropical mathematics to minimize a functional that is defined on a vector set by a matrix and calculated through multiplicative conjugate transposition for some particular cases the minimum in the problem is known to be equal to the tropical spectral radius of the matrix we examine the problem in the common setting of a general idempotent semifield a complete direct solution in a compact vector form is obtained to this problem under fairly general conditions the result is extended to solve new tropical optimization problems with more general objective functions and inequality constraints applications to real world problems that arise in project scheduling are presented to illustrate the results obtained numerical examples are also provided

simulation study and implementation of the tests based on weighted turnbull s estimators for interval censored data several tests for comparing k groups of interval censored data based on turnbull s estimator of a survival function are compared by a monte carlo simulation study the tests under consideration include the iwd test proposed by petroni and wolfe the score test derived by finkelstein and a logrank type test suggested by sun with a large sample size and in a two sample comparison simulation results suggest that the iwd test is applicable for early and crossing hazard difference alternatives whereas finkelstein s score test sun s test and the logrank test are slightly more powerful than the iwd tests under late hazard difference alternatives the iwd test is designed for the two sample testing problem with a large sample size and in a k sample comparison finkelstein s score test is slightly better than the other tests under a proportional hazards model while sun s test and the logrank test are slightly better than the score test under early and crossing hazard difference alternatives however none of the test statistics is uniformly better than the others for the alternatives considered in this paper with both non overlapping and overlapping interval censored data

a max version of the generalized spectral radius theorem abstract let be a bounded set of n n nonnegative matrices in max algebra in this paper we propose the notions of the max algebra version of the generalized spectral radius of and the max algebra version of the joint spectral radius of the max algebra version of the generalized spectral radius theorem is established we propose the relationship between the generalized spectral radius of in the sense of daubechies and lagarias and its max algebra version moreover a generalization of elsner and van den driessche s lemma is presented as well

pairwise comparisons simplified this study examines the notion of generators of a pairwise comparisons matrix such approach decreases the number of pairwise comparisons from n n 1 to n 1 an algorithm of reconstructing of the pc matrix from its set of generators is presented

pairwise ranking choice of method can produce arbitrarily different rank order abstract we examine three methods for ranking by pairwise comparison perronrank principal eigenvector hodgerank and tropicalrank we show that the choice of method can produce arbitrarily different rank order to be precise for any two of the three methods and for any pair of rankings of at least four items there exists a comparison matrix for the items such that the rankings found by the two methods are the prescribed ones we discuss the implications of this result in practice study the geometry of the methods and state some open problems

an efficient and accurate reconstruction algorithm for the formulation of cell centered diffusion schemes a new reconstruction algorithm is proposed for constructing cell centered diffusion schemes on distorted meshes its main feature is that edge unknowns are defined at certain balance points the locations of which depend on the diffusion coefficient and the skewness of grid cells so as to obtain a two point reconstruction stencil implementing the new algorithm for the approximation of gradients we extend the idc improved deferred correction scheme which was proposed by traore et al p traore y ahipo c louste a robust and efficient finite volume scheme for the discretization of diffusive flux on extremely skewed meshes in complex geometries j comput phys 228 2009 5148 5159 to handle diffusion problems with discontinuous coefficients numerical results demonstrate the accuracy and efficiency of the extended scheme

ignorability and coarse data some biomedical examples heitjan and rubin 1991 annals of statistics 19 2244 2253 define data to be coarse when one observes not the exact value of the data but only some set a subset of the sample space that contains the exact value this definition covers a number of incomplete data problems arising in biomedicine including rounded heaped censored and missing data in analyzing coarse data it is common to proceed as though the degree of coarseness is fixed in advance in a word to ignore the randomness in the coarsening mechanism when coarsening is actually stochastic however inferences that ignore this randomness may be seriously misleading heitjan and rubin 1991 have proposed a general model of data coarsening and established conditions under which it is appropriate to ignore the stochastic nature of the coarsening the conditions are that the data be coarsened at random a generalization of missing at random rubin 1976 biometrika 63 581 592 and that the parameters of the data and the coarsening process be distinct this article presents detailed applications of the general model and the ignorability conditions to a variety of coarse data problems arising in biomedical statistics a reanalysis of the stanford heart transplant data crowley and hu 1977 journal of the american statistical association 72 27 36 reveals significant evidence that censoring of pretransplant survival times by transplantation was nonignorable suggesting a greater benefit from cardiac transplantation than previous analyses had found

a method for dealing with inconsistencies in pairwise comparisons abstract the pairwise comparison method is a powerful inference tool for assessing the relative importance of a set of items formally its objective is to make compatible decision maker s assignments paired comparison with properties needed for obtaining an overall rank in this paper we propose a distance based framework for analysing this kind of compatibility in this context goal programming is proposed as an attractive and flexible tool

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

preference elicitation from inconsistent judgments using multi objective optimization abstract several decision making techniques involve pairwise comparisons to elicit the preferences of a decision maker dm this paper proposes a new approach for prioritization from pairwise comparisons using the concept of indirect judgments no method exists that simultaneously minimizes deviations from both direct and indirect judgments in order to estimate preferences it is sensible to consider both the acquired judgments and the other judgments latent in the dm s mind hence a technique is developed here to minimize the deviations from both types of judgments estimated preferences are generally evaluated based on two criteria their deviation from the provided judgments and the number of judgments that have been ordinally violated here it is proposed to optimize three objectives simultaneously the deviations from both direct and indirect judgments and the number of judgments violated a prototype application has been developed to generate all non dominated solutions using a multiple objective evolutionary algorithm the new approach is shown to offer users greater flexibility than all other tested methods

on the asymptotic stability of nonnegative matrices in max algebra abstract in the max algebra system for an n n nonnegative matrix a a ij the eigenequation for max eigenvalue and corresponding max eigenvector x is a x x where a x i max 1 j n a ij x j and a is the maximum circuit geometric mean it is shown that the following conditions are mutually equivalent i a r n ii a 1 iii a lim k a k 0 where a max x 1 x 0 a x and a limsup k a k 1 k

mimetic finite difference methods for diffusion equations on non orthogonal non conformal meshes mimetic discretizations based on the support operators methodology are derived for non orthogonal locally refined quadrilateral meshes the second order convergence rate on non smooth meshes is verified with numerical examples

on the max version of the generalized spectral radius theorem abstract let be a bounded set of n n non negative matrices recently the max algebra version of the generalized spectral radius of was introduced we show that lim t i t 1 t where denotes the generalized spectral radius and t the hadamard power of this provides a description of that uses no max terminology as an application we give a short proof of the max version of the generalized spectral radius theorem

approximation of positive matrices by transitive matrices a newton kantorovich algorithm used in the approximation of symmetrically reciprocalpositive matrices by transitive matrices is found to be more generally useful in the approximation of merely positive matrices the occurrence of multiple stationary values when the approximated matrix has certain symmetries is investigated

a new nonlinear finite volume scheme preserving positivity for diffusion equations in this paper we present a new nonlinear finite volume scheme preserving positivity for diffusion equations the main feature of the scheme is the assumption that the values of auxiliary unknowns are nonnegative is avoided two nonnegative parameters are introduced to define a new nonlinear two point flux in which one point is the cell center and the other is the midpoint of cell edge the final flux on the edge is obtained by the continuity of normal flux numerical results show that the accuracy of both solution and flux for our new scheme is superior to that of some existing monotone schemes

a max version of the perron frobenius theorem abstract if a an n n nonnegative irreducible matrix then there exists a 0 and a positive vector x such that maxjaijxj a xi i 1 2 n furthermore a is the maximum geometric mean of a circuit in the weighted directed graph corresponding to a this theorem which we refer to as the max version of the perron frobenius theorem is well known in the context of matrices over the max algebra and also in the context of matrix scalings in the present work which is partly expository we bring out the intimate connection between this result and the perron frobenius theory we present several proofs of the result some of which use the perron frobenius theorem structure of max eigenvalues and max eigenvectors is described possible ways to unify the perron frobenius theorem and its max version are indicated some inequalities for a are proved

robust convergence of multi point flux approximation on rough grids this paper establishes the convergence of a multi point flux approximation control volume method on rough quadrilateral grids by rough grids we refer to a family of refined quadrilateral grids where the cells are not required to approach parallelograms in the asymptotic limit in contrast to previous convergence results for these methods we consider here a version where the flux approximation is derived directly in the physical space and not on a reference cell as a consequence less regular grids are allowed however the extra cost is that the symmetry of the method is lost

nonparametric comparison of survival functions based on interval censored data with unequal censoring nonparametric comparison of survival functions is one of the most commonly required task in failure time studies such as clinical trials and for this many procedures have been developed under various situations this paper considers a situation that often occurs in practice but has not been discussed much the comparison based on interval censored data in the presence of unequal censoring that is one observes only interval censored data and the distributions of or the mechanisms behind censoring variables may depend on treatments and thus be different for the subjects in different treatment groups for the problem a test procedure is developed that takes into account the difference between the distributions of the censoring variables and the asymptotic normality of the test statistics is given for the assessment of the performance of the procedure a simulation study is conducted and suggests that it works well for practical situations an illustrative example is provided copyright 2017 john wiley sons ltd

hodgerank is the limit of perron rank in the context of pairwise comparison ranking we show that hodgerank row geometric mean is the limit of perron rank ranking with principal eigenvector as a certain parameter k goes to 0 this result provides a novel mathematical link between two important pairwise ranking methods it complements the known result that as k approaches infinity perron rank converges to tropical rank thus these three pairwise ranking methods belong to the same parametrized family our proof technique is useful for mathematical comparison of these methods as a sample application we show that for ranking models with i i d noise hodgerank is a linear approximation of perron rank in this particular setup for large numbers of items with sufficiently large score differences the two methods yield identical ordinal rankings

combining minsum and minmax a goal programming approach a number of methods for multiple objective optimization problems mop give as solution to mop the set of optimal solutions for some single objective optimization problems associated with it well known examples of these single objective optimization problems are the minsum and the minmax in this note we propose a new parametric single objective optimization problem associated with mop by means of goal programming ideas we show that the minsum and minmax are particular instances so we are somehow combining minsum and minmax by means of a parameter moreover such parameter has a clear meaning in the value space applications of this parametric problem to classical models in locational analysis are discussed

finding the observed information matrix when using the em algorithm a procedure is derived for extracting the observed information matrix when the em algorithm is used to find maximum likelihood estimates in incomplete data problems the technique requires computation of a complete data gradient vector or second derivative matrix but not those associated with the incomplete data likelihood in addition a method useful in speeding up the convergence of the em algorithm is developed two examples are presented

cell centered finite volume methods with flexible stencils for diffusion equations on general nonconforming meshes a cell centered finite volume method is presented for discretizing diffusion operator on general nonconforming meshes the node values are accurately approximated using a new weighted interpolation formula in which the calculation of the weight is adaptive to both geometric parameters and diffusion coefficients it follows that an explicit expression composed of cell centered unknowns only is obtained for the discretization of normal flux numerical results demonstrate that linear solutions are reproduced exactly on the nonconforming random grids and that the convergence rate is close to second order for non linear or discontinuous problems

on the power method in max algebra abstract the eigenvalue problem for an irreducible nonnegative matrix a a ij in the max algebra system is a otimes x lambda x where a otimes x i mathop max j a ij x j and lambda turns out to be the maximum circuit geometric mean mu a a power method algorithm is given to compute mu a and eigenvector x the algorithm is developed by using results on the convergence of max powers of a which are proved using nonnegative matrix theory in contrast to an algorithm developed in 4 this new method works for any irreducible nonnegative a and calculates eigenvectors in a simpler and more efficient way some asymptotic formulas relating mu a the spectral radius and norms are also given

efficient weight vectors from pairwise comparison matrices abstract pairwise comparison matrices are frequently applied in multi criteria decision making a weight vector is called efficient if no other weight vector is at least as good in approximating the elements of the pairwise comparison matrix and strictly better in at least one position a weight vector is weakly efficient if the pairwise ratios cannot be improved in all non diagonal positions we show that the principal eigenvector is always weakly efficient but numerical examples show that it can be inefficient the linear programs proposed test whether a given weight vector is weakly efficient and in case of strong inefficiency an efficient strongly dominating weight vector is calculated the proposed algorithms are implemented in pairwise comparison matrix calculator available at pcmc online

computationally simple accelerated failure time regression for interval censored data summary an approach is presented for fitting the accelerated failure time model to interval censored data that does not involve computing the nonparametric maximum likelihood estimate of the distribution function at the residuals the approach involves estimating equations computed with the examination times from the same individual treated as if they had actually been obtained from different individuals the dependence between different measurements obtained from the same individual is then accounted for in the calculation of the standard error of the regression coefficients the approach is applicable to interval censored data in settings in which examinations continue to occur regardless of whether the failure time has occurred simulations are presented to assess the behaviour of the approach and the methodology is illustrated through an application to data from an aids clinical trial