a group decision making model based on a generalized ordered weighted geometric average operator with interval preference matrices abstract this paper presents a model for a group decision making problem with interval preference matrices first a new definition of interval additive reciprocal matrices with multiplicative consistency is given transformation methods between interval additive and multiplicative reciprocal matrices are proposed and applied to homogenize interval preference matrices second the consistency of interval multiplicative reciprocal matrices is utilized to propose a generalized ordered weighted geometric averaging operator which permits the aggregation of interval multiplicative reciprocal matrices in such a way that more important weight is given to that with more consistency in order to avoid a misleading solution the consistency and acceptable consistency of the collective interval multiplicative reciprocal matrix are studied in detail finally a new algorithm is presented to solve the group decision making problem with interval preference matrices numerical results are carried out to illustrate the given definitions methods and algorithm respectively

an infinity norm bound for the inverse of dashnic zusmanovich type matrices with applications abstract an upper bound for the infinity norm for the inverse of dashnic zusmanovich type matrices is given it is proved that the upper bound is sharper than the well known varah s bound for strictly diagonally dominant matrices by introducing a new subclass of p matrices dashnic zusmanovich type b matrices dz type b matrices and using the proposed infinity norm bound an error bound is given for the linear complementarity problems of dz type b matrices we also give a new pseudospectra localization to measure the distance to instability

group preference aggregation methods employed in ahp an evaluation and an intrinsic process for deriving members weightages abstract the analytic hierarchy process ahp is one of the popular and powerful techniques for decision making a detailed survey of the literature has revealed that there exists no formal evaluation of the group preference aggregation methods currently employed in ahp this paper provides such an evaluation using well established social choice axioms which govern the process of combining individual opinions to obtain a single group opinion the geometric mean method gmm and the weighted arithmetic mean method wamm are the two methods evaluated it is shown using counter examples that the gmm does not always satisfy the pareto optimality axiom which is one of the prominent and widely accepted social choice axioms this finding is significant as the gmm has been the most commonly used method in ahp for combining individual opinions to form a group opinion the other method viz wamm has satisfied all the axioms except the independence of irrelevant alternatives axiom in order to use the wamm one has to find the weightages importance to be assigned to the members of the group this is often a difficult task especially so if the group is large as in the case of public policy decisions and when judgements are elicited through the use of questionnaires these situations need an objective method to derive members weightages but only a few studies are available in the literature to address such a situation we propose a simple and intuitively appealing eigenvector based method to intrinsically determine the weightages for group members using their own subjective opinions the superiority of the proposed method over the previous methods is brought out in the paper

estimation of the eigenvalues and the smallest singular value of matrices abstract the purpose of this paper is to discuss the estimation of eigenvalues and the smallest singular value of matrices firstly we prove that all the eigenvalues of arbitrarily complex matrix are located in one closed disk around trm n of radius n 1 n m f 2 trm 2 n max 1 k n 1 b k n k f c n k k f 2 secondly we present a lower bound for the smallest singular value of matrices finally we give some numerical examples which will show the effectiveness of our results

spectra of digraphs in this primarily expository paper we survey classical and some more recent results on the spectra of digraphs equivalently the spectra of 0 1 matrices with emphasis on the spectral radius

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

the mathematical equivalence of the spanning tree and row geometric mean preference vectors and its implications for preference analysis abstract pairwise comparison is a widely used approach to elicit comparative judgements from a decision maker dm and there are a number of methods that can be used to then subsequently derive a consistent preference vector from the dm s judgements while the most widely used method is the eigenvector method the row geometric mean approach has gained popularity due to its mathematical properties and its ease of implementation in this paper we discuss a spanning tree method and prove the mathematical equivalence of its preference vector to that of the row geometric mean approach this is an important finding due to the fact that it identifies an approach for generating a preference vector which has the mathematical properties of the row geometric mean preference vector and yet in its entirety the spanning tree method has more to offer than the row geometric mean method in that it is inherently applicable to incomplete sets of pairwise comparison judgements and also facilitates the use of statistical and visual techniques to gain insights into inconsistency in the dm s judgements

a heuristic method to rectify intransitive judgments in pairwise comparison matrices abstract this paper investigates the effects of intransitive judgments on the consistency of pairwise comparison matrices statistical evidence regarding the occurrence of intransitive judgements in pairwise matrices of acceptable consistency is gathered by using a monte carlo simulation which confirms that relatively high percentage of comparison matrices satisfying saaty s cr criterion are ordinally inconsistent it is also shown that ordinal inconsistency does not necessarily decrease in the group aggregation process in contrast with cardinal inconsistency a heuristic algorithm is proposed to improve ordinal consistency by identifying and eliminating intransitivities in pairwise comparison matrices the proposed algorithm generates near optimal solutions and outperforms other tested approaches with respect to computation time

a lower bound for the smallest singular value abstract a lower bound for the smallest singular value of a n x n is given in terms of the determinant and the 2 norm of the columns and the rows of a examples are given to show that this bound performs well for a large class of matrices

statistical ranking and combinatorial hodge theory we propose a technique that we call hodgerank for ranking data that may be incomplete and imbalanced characteristics common in modern datasets coming from e commerce and internet applications we are primarily interested in cardinal data based on scores or ratings though our methods also give specific insights on ordinal data from raw ranking data we construct pairwise rankings represented as edge flows on an appropriate graph our statistical ranking method exploits the graph helmholtzian which is the graph theoretic analogue of the helmholtz operator or vector laplacian in much the same way the graph laplacian is an analogue of the laplace operator or scalar laplacian we shall study the graph helmholtzian using combinatorial hodge theory which provides a way to unravel ranking information from edge flows in particular we show that every edge flow representing pairwise ranking can be resolved into two orthogonal components a gradient flow that represents the l 2 optimal global ranking and a divergence free flow cyclic that measures the validity of the global ranking obtained if this is large then it indicates that the data does not have a good global ranking this divergence free flow can be further decomposed orthogonally into a curl flow locally cyclic and a harmonic flow locally acyclic but globally cyclic these provides information on whether inconsistency in the ranking data arises locally or globally when applied to statistical ranking problems hodge decomposition sheds light on whether a given dataset may be globally ranked in a meaningful way or if the data is inherently inconsistent and thus could not have any reasonable global ranking in the latter case it provides information on the nature of the inconsistencies an obvious advantage over the np hardness of kemeny optimization is that hodgerank may be easily computed via a linear least squares regression we also discuss connections with well known ordinal ranking techniques such as kemeny optimization and borda count from social choice theory

algorithms for improving consistency or consensus of reciprocal 0 1 valued preference relations we investigate the consistency and consensus of reciprocal 0 1 valued preference relations also called fuzzy preference relations by many authors based on the multiplicative consistency property which is an important issue in fuzzy set theory an algorithm is first developed to improve the consistency level of a reciprocal 0 1 valued preference relation and the corresponding algorithm for the incomplete reciprocal 0 1 valued preference relation is also developed we further propose the consensus improving algorithms for individual reciprocal 0 1 valued preference relations or incomplete ones the convergence and robustness of the algorithms are proven and some important conclusions are obtained in addition the proposed algorithms can improve the consistency or consensus of reciprocal 0 1 valued preference relations with less interactions with the decision makers which can save a lot of time and obtain the results quickly

the undirected graph and estimates of matrix singular values abstract with the help of the undirected graph of a matrix we establish new theorems for estimating the singular values the location of singular values of a matrix a is described in terms of its deleted absolute row sums and column sums the formulation has an appearance analogous to brauer s and brualdi s theorems for eigenvalues examples show that in many cases our results can provide more precise estimates than those in the related literature

upper bounds for the infinity norm of the inverse of sdd and s sdd matrices an easily calculable upper bound for the infinity norm of the inverse of s strictly diagonally dominant s sdd matrices introduced by gao and wang in 1992 is obtained in this paper applying that bound to sdd matrices we improve the varah bound we also introduce an iterative procedure for obtaining sharper bounds accompanied with some open questions

scaled pivots and scaled partial pivoting strategies scaled pivots for gaussian elimination of an n n matrix are introduced they are used to obtain bounds for the skeel condition number of the resulting upper triangular matrix and for a growth factor which has been introduced by amodio and mazzia bit 39 1999 pp 385 402 a bound of this growth factor for row scaled partial pivoting strategies is also included it is proved that the skeel condition number of an n n upper triangular matrix which is strictly diagonally dominant by rows is bounded above by a number which is independent of n it is also shown that the calculation of the n scaled pivots associated with a pivoting strategy presenting nice properties when applied to nonsingular n n m matrices adds mathcal o n elementary operations to the complete cost of this pivoting strategy

two particle cluster integral in the expansion of the dielectric constant we study in detail the two particle cluster integral in the cluster expansion for the effective dielectric constant of a suspension of spherically symmetric polarizable inclusions embedded in a uniform medium although our form for the integrand differs from that derived earlier by finkel berg and by jeffrey we show that the integral is equivalent the two body dielectric problem for particles with an arbitrary radial dependence of the dielectric constant is solved by an expansion in spherical harmonics numerical results for some special models illustrate the importance of multipole contributions to the effective dielectric constant

spectral functions for two dimensional random conductors abstract we collect together a number of results for the spectral function for two dimensional random conductors exact solutions are available for the conductivity of sheets containing a few elliptical inclusions we show that exact results can also be obtained for polygonal holes using conformal mapping techniques for polygonal inclusions with arbitrary conductivity the conductivity can be expressed in terms of a spectral function which we have obtained numerically using a boundary element method in this method the singularities associated with the sharp corners are explicitly taken care of spectral functions can also be used to obtain an exact solution for the conductivity when the effect of pairs of circular inclusions is also included throughout this paper we express the conductivity of the composite in terms of the total induced dipole moment and exploit the simplifications due to the reciprocity theorem in two dimensions

max norm estimation for the inverse of block matrices abstract maximum norm bound of the inverse of a given matrix is an important issue in a wide range of applications motivated by this fact we will extend the list of matrix classes for which upper bounds for max norms can be obtained these classes are subclasses of block h matrices and they stand in a general position with corresponding point wise classes efficiency of new results will be illustrated by numerical examples

generalized analytic network process the analytic network process anp is a methodology for multi criteria decision making used to derive priorities of the compared elements in a network hierarchy where the dependences and feedback within and between the elements can be considered however the anp is limited to the input preferences as crisp judgments which is often unfavorable in practical applications as an extension of the anp a generalized analytic network process g anp is developed to allow multiple forms of preferences such as crisp fuzzy judgments interval interval fuzzy judgments hesitant hesitant fuzzy judgments and stochastic stochastic fuzzy judgments in the g anp a concept of complex comparison matrices ccms is developed to collect decision makers preferences in the multiple forms from a stochastic point of view we develop an eigenvector method based stochastic preference method evm spm to derive priorities from ccms the main steps of the g anp are summarized and the implementation of the g anp in matlab and excel environments are given in detail which is also a prototype for a decision support system a real life example of the piracy risk assessment to the energy channels of china is proposed to demonstrate the g anp

a note on a lower bound for the smallest singular value abstract we give a short proof of a lower bound for the smallest singular value mentioned by hong and pan one of our results improves their theorem we obtain another lower bound that is sharp an example shows that these results can be quite accurate

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

an iterative algorithm for searching a scaling matrix for diagonal dominance diagonal dominance of matrices and m matrices and h matrices play an important role in stability theory of dynamical systems but it is difficult to determine the scaling matrix g diag g 1 g n g 1 g n 0 with ag being a strictly diagonally dominant matrix in this paper an convergent iterative algorithm for determine the scaling matrix for an irreducible m matrix h matrix a is presented

fractional pseudospectra and their localizations abstract motivated by the growing successful use of fractional differential equations in the modeling of different important phenomena in this paper we derive tools for practical analysis of the robust asymptotic stability of a incommensurate fractional order linear system first the concept of fractional pseudospectra is introduced second driven by the simplicity and usefulness of spectral localizations in the analysis of various matrix properties we introduce adequate localization techniques using the ideas that come from diagonally dominant matrices in order to localize the fractional pseudospectra in such way many theoretical and practical applications of pseudospectra robust stability transient behavior nonnormal dynamics etc in fractional order differential systems can be linked to the specificity of the matrix entries allowing one to understand certain phenomena in practice better third we consider the fractional distance to instability in l l 1 and l 2 norms and determine efficient lower bounds finally this novel approach is implemented on the realistic model of empirical food web to link the stability that incorporates hereditary dynamics of living organisms with the empirical data and their uncertainty limitations

the geometric consistency index approximated thresholds abstract crawford and williams journal of mathematical psychology 29 1985 387 suggested for the row geometric mean method rgmm one of the most extended ahp s priorization procedure a measure of the inconsistency based on stochastic properties of a subjacent model in this paper we formalize this inconsistency measure hereafter called the geometric consistency index gci and provide the thresholds associated with it these thresholds allow us an interpretation of the inconsistency tolerance level analogous to that proposed by saaty multicriteria decision making the analytic hierarchy process new york 1980 for the consistency ratio cr used with the right eigenvector method in conventional ahp

empirical properties of group preference aggregation methods employed in ahp theory and evidence abstract we study various methods of aggregating individual judgments and individual priorities in group decision making with the ahp the focus is on the empirical properties of the various methods mainly on the extent to which the various aggregation methods represent an accurate approximation of the priority vector of interest we identify five main classes of aggregation procedures which provide identical or very similar empirical expressions for the vectors of interest we also propose a method to decompose in the ahp response matrix distortions due to random errors and perturbations caused by cognitive biases predicted by the mathematical psychology literature we test the decomposition with experimental data and find that perturbations in group decision making caused by cognitive distortions are more important than those caused by random errors we propose methods to correct the systematic distortions

boundary properties of the inconsistency of pairwise comparisons in group decisions abstract this paper proposes an analysis of the effects of consensus and preference aggregation on the consistency of pairwise comparisons we define some boundary properties for the inconsistency of group preferences and investigate their relation with different inconsistency indices some results are presented on more general dependencies between properties of inconsistency indices and the satisfaction of boundary properties in the end given three boundary properties and nine indices among the most relevant ones we will be able to present a complete analysis of what indices satisfy what properties and offer a reflection on the interpretation of the inconsistency of group preferences

local stability intervals in the analytic hierarchy process abstract this paper provides a sensitivity analysis of the judgements used in the analytic hierarchy process ahp in relation to the rank reversal produced in two different situations the selection of the best alternative p problem and the ranking of all the alternatives p problem in both cases under the supposition that we employ the row geometric mean method rgmm to determine the local priorities we obtain in the case of a single criterion a local stability interval for each judgement for each alternative and for the paired comparisons matrix which allow us to guarantee the best alternative and the ranking of all of them the p and p problems respectively with respect to these three situations judgement alternative and matrix and two problems p and p we also calculate a local stability index to detect the critical values of the resolution process both the local stability intervals and indexes are used as management tools in the final stage of the decision making process that is to say the exploitation phase especially in the negotiation process and the search for consensus between the actors

accurate singular values and differential qd algorithms in september 1991 j w demmel and w m kahan were awarded the second siam prize in numerical linear algebra for their paper accurate singular values of bidiagonal matrices 1 referred to as dk hereafter among several valuable results was the observation that the standard bidiagonal qr algorithm used in linpack 2 and in many other svd programs can be simplified when the shift is zero and of greater importance no subtractions occur the last feature permits very small singular values to be found with almost all the accuracy permitted by the data and at no extra cost

dielectric function for a material containing hyperspherical inclusions to o c 2 i multipole expansions by averaging over pairs of hyperspheres we have obtained the dielectric function for a binary mixture containing hyperspherical inclusions up to order c 2 where c is the volume fraction of inclusions the method used is based on multipole expansions for the potential of two spheres in a uniform eld and is a generalization of the method of jerey to d dimensional space numerical results are presented for the second order coecient in the low c expansion of the dielectric constant for arbitrary d these verify earlier known results as well as showing the dependence of on dimensionality which is particularly simple as d 1 and as d 1 ever since the time of maxwell 1873 the study of the properties of random mix tures or suspensions containing a low volume fraction of inclusions has been of much interest the problems concerned are applicable to the study of electrical conduc tion thermal conduction electric permittivity magnetic permeability and others by virtue of the universality of laplace s equation in this paper we will concentrate on the case of dielectric inclusions other cases being easily obtainable by renaming the symbols maxwell 1873 provided the exact rst order coecient to o c for a system of spherical inclusions other similar problems too c not within the univer sal class have also attracted much interest for example the o c coecient for the viscosity of a suspension containing a system of hard spheres was found by einstein 1906 more recent exact results for the o c coecient of electrical conductivity for inclusions of other shapes in two dimensions were obtained by thorpe 1992 using conformal mapping a method favoured by maxwell it took nearly 100 years before any serious quantitative studies were made of the second order o c 2 coecient the original work being done by batchelor 1972 and batchelor green 1972 who unfortunately undertook initially to study the more complex fluid suspensions problem to this day few exact results are known batchelor 1974 1977 we remark here that the second order coecient we are interested in is related to the so called huggins coecient h for fluid suspensions i e the expansion of the viscosity is given by 0 1 c h c 2

monotonicity and discretization error estimates precise bounds of supremum norms are given for the inverse of monotone matrices this result sharpens an earlier result by varga matrix iterative analysis prentice hall englewood cliffs nj 1962 for h matrices that are monotone various algebraic approaches to prove monotonicity based on an extended version of the weak regular splitting theorem are presented and applied to some important examples of finite difference and finite element matrices this together with the above mentioned bounds of the norm of the inverse operator shows that the discretization error in supremum and least square norms for self adjoint problems can be bounded by a constant times the norm of the truncation error where the best constant is available

inclusion intervals of singular values and applications abstract in this paper we improve the brauer type inclusion interval and present a modified brauer type theorem for singular values numerical examples show that our estimation for singular values is more precise than those corresponding results in recent literature moreover we also consider the marginal singular value on the brauer type inclusion interval some applications of our results are presented

the conductivity of a sheet containing inclusions with sharp corners we have written down an integral equation for the surface charge on the interface between an n sided regular polygonal inclusion with conductivity o 1 and a host with conductivity 0 the conductivity of a sheet with few inclusions can be determined from the induced dipole associated with this charge distribution the integral equation is solved numerically and detailed results are given for n 3 4 and 6 the circle limit is approached for large n we show that our results are consistent with the reciprocity theorem for two dimensional conducting media and we reproduce the known analytic results in the limits when the inclusion is either a hole or a superconducting region we also find the conductivity of a sheet containing a general n pointed star analytic fits to our numerical results are given which are used as input for effective medium theories the results given in this paper for the electrical conductivity apply also to the thermal conductivity and to dielectric media

bounds for the extremal eigenvalues of a class of symmetric tridiagonal matrices with applications abstract we consider a class of symmetric tridiagonal matrices which may be viewed as perturbations of toeplitz matrices the toeplitz structure is destroyed since two elements on each off diagonal are perturbed based on a careful analysis we derive sharp bounds for the extremal eigenvalues of this class of matrices in terms of the original data of the given matrix in this way we also obtain a lower bound for the smallest singular value of certain matrices some numerical results indicate that our bounds are extremely good

estimation for matrix singular values abstract we present in this note two types of estimates for matrix singular values which are analogous to the well known ky fan s theorem and brauer s theorem for eigenvalues respectively

a note on ahp group consistency for the row geometric mean priorization procedure abstract this work analyses the consistency in group decision making for the analytic hierarchy process ahp when using the weighted geometric mean method wgmm as the aggregation procedure the row geometric mean method rgmm as the priorization procedure and the geometric consistency index as the inconsistency measure the paper proves that the inconsistency of the group is smaller than the largest individual inconsistency this result complements that obtained by xu eur j oper res 126 2000 683 for the eigenvector priorization method em and its associated consistency index saaty multicriteria decision making the analytic hierarchy process mcgraw hill 1980 moreover our result guarantees that by using the rgmm priorization procedure the group priorities obtained through the aggregation of the individual priorities verify the requirement of consistency proposed in ahp methodology if the individual priorities also verify this requirement

bounds for norms of the matrix inverse and the smallest singular value abstract in the first part we obtain two easily calculable lower bounds for a 1 where is an arbitrary matrix norm in the case when a is an m matrix using first row sums and then column sums using those results we obtain the characterization of m matrices whose inverses are stochastic matrices with different approach we give another easily calculable lower bounds for a 1 and a 1 1 in the case when a is an m matrix in the second part using the results from the first part we obtain our main result an easily calculable upper bound for a 1 1 in the case when a is an sdd matrix thus improving the known bound all mentioned norm bounds can be used for bounding the smallest singular value of a matrix

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

equality conditions for lower bounds on the smallest singular value of a bidiagonal matrix abstract several lower bounds have been proposed for the smallest singular value of a square matrix such as johnson s bound brauer type bound li s bound and ostrowski type bound in this paper we focus on a bidiagonal matrix and investigate the equality conditions for these bounds we show that the former three bounds give strict lower bounds if all the bidiagonal elements are non zero for the ostrowski type bound we present an easily verifiable necessary and sufficient condition for the equality to hold

cavities vis a vis rigid inclusions elastic moduli of materials with polygonal inclusions in this paper we explore the correspondence between rigid inclusions and cavities dundurs 1989 j appl mech 56 786 790 as applied to the effective elastic moduli of materials with polygonal rigid inclusions and cavities in the analysis we use a complex variable method of elasticity and a conformal transformation to solve for the stress field due to a single rigid inclusion then we use a far field approach to obtain the effective elastic constants of composites with a dilute concentration of rigid polygonal inclusions by employing the dundurs correspondence we can automatically obtain the result for the effective elastic moduli of materials with cavities finally we use effective medium theories to predict the elastic moduli of materials containing a finite concentration of inclusions

a geometrical approach to finding multivariate approximate lcms and gcds abstract in this article we present a new approach to compute an approximate least common multiple lcm and an approximate greatest common divisor gcd of two multivariate polynomials this approach uses the geometrical notion of principal angles whereas the main computational tools are the implicitly restarted arnoldi method and sparse qr decomposition upper and lower bounds are derived for the largest and smallest singular values of the highly structured macaulay matrix this leads to an upper bound on its condition number and an upper bound on the 2 norm of the product of two multivariate polynomials numerical examples are provided

inclusion sets for singular values abstract in this paper the following inclusion sets are under certain conditions presented for singular values of a matrix a a ij c n n d a c a 0 i z a i i b b ii and k p a i 1 n j p i a z 0 z a i z a j b b ii b b jj where a i a ii and p i a j a ij 0 or a ji 0 j i for any i 1 2 n b and c a denote the perron root of the nonnegative matrix b b ij satisfying a ij a ji b ij for all i j 1 2 n with i j and the set of all nontrivial circuits in a the associated directed digraph of a respectively subsequently relations between these two results are investigated theoretic analysis and numerical examples show that these estimates are more precise than recent corresponding results in some cases

euclidean norm estimates of the inverse of some special block matrices in this paper we start from the known lower bounds for minimal singular value of the matrices possessing certain kind of the diagonal dominance property and derive euclidean norm estimates of the inverses of several new subclasses of the block h matrices the motivation comes from applications where the matrix in question has distinguished block structure which can be exploited to obtain useful information an example arising from ecological modeling illustrates the benefits of the presented approach

diagnosis with dependent symptoms bayes theorem and the analytic hierarchy process judgments are needed in medical diagnosis to determine what tests to perform given certain symptoms for many diseases what information to gather on symptoms and what combination of symptoms lead to a given disease are not well known even when the number of symptoms is small the required number of experiments to generate adequate statistical data can be unmanageably large there is need in diagnosis for an integrative model that incorporates both statistical data and expert judgment when statistical data are present but no expert judgment is available one property of this model should be to reproduce results obtained through time honored procedures such as bayes theorem when expert judgment is also present it should be possible to combine judgment with statistical data to identify the disease that best describes the observed symptoms here we are interested in the analytic hierarchy process ahp framework that deals with dependence among the elements or clusters of a decision structure to combine statistical and judgmental information it is shown that the posterior probabilities derived from bayes theorem are part of this framework and hence that bayes theorem is a sufficient condition of a solution in the sense of the ahp an illustration is given as to how a purely judgment based model in the ahp can be used in medical diagnosis the application of the model to a case study demonstrates that both statistics and judgment can be combined to provide diagnostic support to medical practitioner colleagues with whom we have interacted in doing this work

consistency stability intervals for a judgement in ahp decision support systems abstract this paper focuses on the need to evaluate the consistency of human judgements in decision support systems dss in this sense and in the context of one of the most applied multicriteria methodologies the analytic hierarchy process ahp we obtain the consistency stability interval csi associated with each judgement that is to say the interval in which this can oscillate without exceeding a value of the consistency measure fixed in advance to calculate these csis we consider the row geometric mean method as the priorisation procedure the geometric consistency index as the consistency measure and a local situation with one criterion the proposed procedure has been implemented in a module that is easily adaptable to any decision support system based on ahp

the schur complement of ph matrices abstract in this paper we consider the class of p h matrices a subclass of h matrices and using scaling characterization we show that this class is closed under taking the schur complement we show that under certain conditions the perron complement of a p h matrix is a p h matrix we also present a way of constructing a scaling matrix for the given p h matrix and we give eigenvalue localization for the schur complement of a p h matrix using only the entries of the original matrix we illustrate this by numerical examples

bayesian decision process for cost efficient dynamic ranking via crowdsourcing rank aggregation based on pairwise comparisons over a set of items has a wide range of applications although considerable research has been devoted to the development of rank aggregation algorithms one basic question is how to efficiently collect a large amount of high quality pairwise comparisons for the ranking purpose because of the advent of many crowdsourcing services a crowd of workers are often hired to conduct pairwise comparisons with a small monetary reward for each pair they compare since different workers have different levels of reliability and different pairs have different levels of ambiguity it is desirable to wisely allocate the limited budget for comparisons among the pairs of items and workers so that the global ranking can be accurately inferred from the comparison results to this end we model the active sampling problem in crowdsourced ranking as a bayesian markov decision process which dynamically selects item pairs and workers to improve the ranking accuracy under a budget constraint we further develop a computationally efficient sampling policy based on knowledge gradient as well as a moment matching technique for posterior approximation experimental evaluations on both synthetic and real data show that the proposed policy achieves high ranking accuracy with a lower labeling cost

dependence and independence from linear hierarchies to nonlinear networks abstract most decisions need to be free from assumptions of independence to be faithful to the complex problems in which they arise this paper illustrates how to generate priorities for decisions involving general types of dependence of criteria on alternatives criteria on criteria and alternatives on alternatives it is based on the feedback system framework of the analytic hierarchy process of which a hierarchy is a special case

second order interactions in a medium containing perfect conducting hyperspherical inclusions we consider the interaction between two hyperspherical inclusions surrounded by a medium experiencing an e field in the perfect conducting limit by the use of d dimensional dipole moments we show that the dielectric function of the medium can be calculated to any order n in particular d n the coefficient of o c 2 in the series expansion for the dielectric function is determined in terms of a dimensional dependence which even though it is mathematically complex proves to be superior in convergence to other methods we calculate the potential difference between the two hyperspheres for various limits including the all important closely packed limit using the theory of continued fractions we investigate the convergence of the interaction terms between the two inclusions and obtain results that reduce the enormous number of calculations that need to be computed as n the latter may be useful in the pursuit of a theory that resums the complicated interaction terms present in the two body medium problem with a view towards an improved effective medium theory

further lower bounds for the smallest singular value abstract in an earlier paper of the first author gersgorin s theorem was used in a novel way to give a simple lower bound for the smallest singular value of a general complex matrix that lower bound was stronger than previous published bounds here we use three variants of gersgorin s theorem in a similar way to give further lower bounds each of the new bounds is more complicated but generally stronger than the pure gersgorin based bound the three new bounds are mutually noncomparable

linear system based approach for solving some related problems of m matrices abstract this paper develops a novel linear system based approach for computing a 1 the skeel condition number of an m matrix a and the positive diagonal matrices d guaranteeing that ad be a strictly diagonally dominant matrix theoretic analysis and simulation results justify the validity of the proposed approach moreover the proposed linear system model can be implemented by application specific integrated circuits and then it has asynchronous parallel processing ability and can achieve high computing performance

a natural norm successive constraint method for inf sup lower bounds we present a new approach for the construction of lower bounds for the inf sup stability constants required in a posteriori error analysis of reduced basis approximations to anely parametrized partial dierential equations we combine the linearized inf sup statement of the natural norm approach with the approximation procedure of the successive constraint method scm the former natural norm provides an economical parameter expansion and local concavity in parameter a small er optimization problem which enjoys intrinsic lower bound properties the latter scm provides a systematic optimization framework a linear program lp relaxation which readily incorporates continuity and stability constraints the natural norm scm requires a parameter domain decomposition we propose a greedy algorithm for selection of the scm control points as well as adaptive construction of the optimal subdomains the ecacy of the natural norm scm is illustrated through numerical results for two types of non coercive problems the helmholtz equation for acoustics elasticity and electromagnetics and the convection diusi on equation

lower bounds for the smallest singular value abstract let a aij be an n n complex matrix suppose that g a the undirected graph of a has no isolated vertex let e be the set of edges of g a we prove that the smallest singular value of a n satisfies n min ij i j e where gij ai aj ai aj 2 ri ci rj cj 1 2 2 with ai aii and ri ci are the ith deleted absolute row sum and column sum of a respectively the result simplifies and improves that of johnson and szulc n mini j ij see 1

some simple estimates for singular values of a matrix abstract some simple estimation theorems for singular values of a rectangular matrix a are given they only use the elements of a itself and in some cases they yield better results than does the gerschgorin theorem applied to a a a bound for the condition number of a may be obtained from them when a is square a bound is derived which explains why scaling improves the performance of gauss elimination when row or column norms differ widely in magnitude their application to perturbation theory of singular values is also discussed

a soft computing method for multi criteria decision making with dependence and feedback in this paper the decision making problems with the dependence and the feedback effects are considered although the analytic network hierarchy process anp ahp has been proposed to deal with the problems above several problems make the method impractical in this paper we proposed the fuzzy decision maps fdm which incorporates the eigenvalue method the fuzzy cognitive maps fcm and the weighting equation to overcome the problem of preferential independent and the shortcomings of the anp in addition two numerical examples are used to demonstrate the proposed method on the basis of the numerical results we can conclude that the proposed method can soundly deal with the decision making problems with the dependence and the feedback effects

dielectric function for a material containing hyperspherical inclusions to o c2 ii method of images we have developed the method of images for the solution of two hyperspheres in d dimensions in a uniform field in order to study the o c 2 coefficient in the dielectric constant of the medium the method employs extensions of image theory for classical charges and dipoles for dielectric spheres which produce novel line charge and line dipole densities numerical results for with arbitrary integer d are consistent with our previous paper i this issue and demonstrate that the image method shows superior convergence properties although computationally less convenient the procedure is in principle calculable term by term and leads to useful analytic results in the weak scattering limit this work also produces a simple algebraic approximation that is useful everywhere

optimal gersgorin style estimation of extremal singular values our purpose is to identify matrices that extremize the maximum and minimum singular values among all matrices that share one of two types of gersgorin data all matrices sharing one type of information are treated equally and no other information is taken into account knowledge of extremizers provides optimal estimates examples are given as the largest singular value is also an operator norm induced by an absolute vector norm we mention analogous facts for other operator norms

a universal voting system based on the potential method abstract pairwise comparison of various objects alternatives is common in many procedures related to decision making potential method pm uses a weighted preference graph as the basic structure generated by such comparisons to obtain a value function potential on the set of alternatives the potential is calculated as a solution of a system of equations involving the laplacian matrix of the preference graph in multiple criteria decision analysis or group decision making each criterion or decision maker is represented by its preference graph a multigraph obtained by joining these graphs aggregates the individual preferences and generates the group potential moreover the influence weight of each criterion or decision maker on the group potential may be adjusted in this article the aggregation of preference graphs is applied to voting systems many different forms of voting ballots can generate preference graphs which gives us a universal ballot independent voting system as an illustration of the pm in the voting context we have chosen the eurovision song contest and the analysis of the cookie type ballots where the voters allocate a certain number of points cookies to the candidates

axiomatic foundation of the analytic hierarchy process this paper contains an axiomatic treatment of the analytic hierarchy process ahp the set of axioms corresponding to hierarchic structures are a special case of axioms for priority setting in systems with feedback which allow for a wide class of dependencies the axioms highlight 1 the reciprocal property that is basic in making paired comparisons 2 homogeneity that is characteristic of people s ability for making comparisons among things that are not too dissimilar with respect to a common property and hence the need for arranging them within an order preserving hierarchy 3 dependence of a lower level on the adjacent higher level 4 the idea that an outcome can only reflect expectations when the latter are well represented in the hierarchy the ahp neither assumes transitivity or the stronger condition of consistency nor does it include strong assumptions of the usual notions of rationality a number of facts are derived from these axioms providing an operational basis for the ahp

towards identification of wiener systems with the least amount of a priori information iir cases in this paper we investigate what constitutes the least amount of a priori information on the nonlinearity so that the linear part is identifiable in the non gaussian input case under the white noise input three types of a priori information are considered quadrant information point information and monotonic information in all three cases identifiability has been established and the corresponding nonparametric identification algorithms are developed along with their convergence proofs

on consistency of the weighted geometric mean complex judgement matrix in ahp abstract the weighted geometric mean method wgmm is the most common group preference aggregation method in the analytic hierarchy process this paper reports on research concerning the consistency of wgmm and proves that the weighted geometric mean complex judgement matrix wgmcjm is of acceptable consistency in saaty s t l saaty the analytic hierarchy process mcgraw hill new york 1980 opinion a consistency ratio cr of 0 1 or less is acceptable under the condition that all judgement matrices given by experts for the same problem of decision making are of acceptable consistency accordingly a theoretic basis has been developed for the application of the wgmm in group decision making

distributed estimation of gaussian correlations we study a distributed estimation problem in which two remotely located parties alice and bob observe an unlimited number of i i d samples corresponding to two different parts of a random vector alice can send k bits on average to bob who in turn wants to estimate the cross correlation matrix between the two parts of the vector in the case where the parties observe jointly gaussian scalar random variables with an unknown correlation rho we obtain two constructive and simple unbiased estimators attaining a variance of 1 rho 2 2k ln 2 which coincides with a known but non constructive random coding result of zhang and berger we extend our approach to the vector gaussian case which has not been treated before and construct an estimator that is uniformly better than the scalar estimator applied separately to each of the correlations we then show that the gaussian performance can essentially be attained even when the distribution is completely unknown this in particular implies that in the general problem of distributed correlation estimation the variance can decay at least as o 1 k with the number of transmitted bits this behavior however is not tight we give an example of a rich family of distributions for which local samples reveal essentially nothing about the correlations and where a slightly modified estimator attains a variance of 2 omega k

estimation of an upper bound for a email protected and a lower bound for the smallest singular value for the weakly diagonally dominant matrices we introduce here which are an extended matrix class of the strictly diagonally dominant matrices are presented examples show that the new bounds improve the bounds in varah j m varah a lower bound for the smallest singular value linear algebra appl 11 1975 3 5 johnson c r johnson a gersgorin type lower bound for the smallest singular value linear algebra appl 112 1989 1 7 and johnson szulc c r johnson t szulc further lower bounds for the smallest singular value linear algebra appl 272 1998 169 179

effective conductivity of a two dimensional medium containing elliptical inhomogeneities two effective medium theories the maxwell method and the differential method are used to estimate the effective conductivity of a two dimensional medium consisting of a random distribution of elliptical inclusions in an otherwise homogeneous matrix results are found for arbitrary values of the matrix inclusion conductivity ratio and arbitrary values of the inclusion aspect ratio both predictions are shown to satisfy keller s reciprocity relation the differential prediction is found always to lie within the hashin shtrikman bounds whereas the maxwell prediction typically violates the bounds for sufficiently large inclusion concentrations comparison with three sets of experimental numerical data shows that both theories offer an improvement over the first order dilute concentration results however it does not seem that either method is superior to the other in general furthermore neither method is reliable near the percolation limit

improving bounds for eigenvalues of complex matrices using traces abstract let a be a complex matrix of order n with eigenvalues j j 1 2 n and m be any integer satisfying rank a m n the bound for j 2 by kress de vries and wegmann is strengthened furthermore new bounds are presented to estimate the spectral radius of a using m and traces of a a 2 a a and a a aa we also improve some wolkowicz styan bounds and previous localization of eigenvalues in rectangular or elliptic regions using traces several simple lower bounds for the spectral radius are given involving tr a tr a 2 tr a 3 and m

ranking methods based on valued preference relations a characterization of the net flow method abstract this paper deals with the problem of ranking several alternatives on the basis of a valued preference relation a system of three independent axioms is shown to characterize a ranking method based on net flows which contains as particular cases the rules of copeland and borda and is used in one of the promethee methods

new learning methods for supervised and unsupervised preference aggregation in this paper we present a general treatment of the preference aggregation problem in which multiple preferences over objects must be combined into a single consensus ranking we consider two instances of this problem unsupervised aggregation where no information about a target ranking is available and supervised aggregation where ground truth preferences are provided for each problem class we develop novel learning methods that are applicable to a wide range of preference types specifically for unsupervised aggregation we introduce the multinomial preference model mpm which uses a multinomial generative process to model the observed preferences for the supervised problem we develop a supervised extension for mpm and then propose two fully supervised models the first model employs svd factorization to derive effective item features transforming the aggregation problems into a learning to rank one the second model aims to eliminate the costly svd factorization and instantiates a probabilistic crf framework deriving unary and pairwise potentials directly from the observed preferences using a probabilistic framework allows us to directly optimize the expectation of any target metric such as ndcg or err all the proposed models operate on pairwise preferences and can thus be applied to a wide range of preference types we empirically validate the models on rank aggregation and collaborative filtering data sets and demonstrate superior empirical accuracy

a gersgorin type lower bound for the smallest singular value abstract for an n by m matrix a aij n m we show that the smallest singular value of a is bounded below by min 1 i n a ii 1 2 j 1 j i n a ij j 1 j i n a ji this improves upon two known lower bounds without increasing information requirements or complexity of calculation

remarks on equality in johnson s lower bounds for the smallest singular value of a matrix abstract a necessary condition for johnson s lower bound for the smallest singular value to hold with equality is derived in addition the notions of i inverse radial matrix and also ii unitary similarity transformation to a maximally diagonally dominant matrix are discussed

on bootstrapping panel factor series this paper studies the asymptotic validity of sieve bootstrap for nonstationary panel factor series two main results are shown firstly a bootstrap invariance principle is derived pointwise in i obtaining an upper bound for the order of truncation of the ar polynomial that depends on n and t consistent estimation of the long run variances is also studied for n t secondly joint bootstrap asymptotics is also studied investigating the conditions under which the bootstrap is valid in particular the extent of cross sectional dependence which can be allowed for is investigated whilst we show that for general forms of cross dependence consistent estimation of the long run variance and therefore validity of the bootstrap is fraught with difficulties however we show that one cross sectional unit at a time resampling schemes yield valid bootstrap based inference under weak forms of cross sectional dependence

scaling technique for partition nekrasov matrices it is well known that for a given h matrix a there exists a diagonal nonsingular matrix that scales a by multiplying it from the right to a strictly diagonally dominant sdd matrix there are subclasses of h matrices that can be fully characterised by the form of the corresponding diagonal scaling matrices however for some applications it is not necessary to have such full characterisation it is sufficient to find at least one scaling matrix that will do the job the aim of this paper is to present a way of constructing a diagonal scaling matrix for one special subclass of h matrices called partition nekrasov matrices as an application of this scaling approach we obtain eigenvalue localisation for the corresponding schur complement matrix using only the entries of the original matrix

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

the conductivity of a sheet containing a few polygonal holes and or superconducting inclusions i use conformal mapping techniques to determine the change in the conductivity of a sheet containing a few well separated holes the hole shapes studied are the equilateral triangle square pentagon and regular n gons i show that the conductivity can be written as sigma sigma 0 1 alpha n f o f 2 where f is the area fraction of the inclusions and the coefficient alpha n frac tan pi n 2 pi n gamma 4 left frac 1 n right gamma 2 left frac 2 n right which is 2 5811 2 1884 2 0878 for triangles squares and pentagons and tends to the circle limit of 2 as n rightarrow infty the coefficient alpha n is proportional to the induced dipole moment around the polygonal hole which can be found using an appropriate conformal mapping i have also examined and compared the results for long thin needle like holes in the shape of diamonds rectangles and ellipses in all cases the conductivity parallel to the needles has the limiting form sigma sigma 0 1 f while for the perpendicular conductivity i find that sigma sigma 0 1 n pi a 2 where 2a is the length of the needle and n is the number of needles per unit area for thicker needles the shape becomes important and i compare the results with recent analog experiments and computer simulations because of the reciprocity theorem all the results found here apply equally well to superconducting inclusions

bounds for the infinity norm of the inverse for certain m and h matrices abstract the paper presents new two sided bounds for the infinity norm of the inverse for the so called pm matrices which form a subclass of the class of nonsingular m matrices and contain the class of strictly diagonally dominant matrices these bounds are shown to be monotone with respect to the underlying partitioning of the index set and the equality cases are analyzed also an upper bound for the infinity norm of the inverse of a ph matrix whose comparison matrix is a pm matrix is derived the known ostrowski ahlberg nilson varah and moraca bounds are shown to be special cases of the upper bound obtained

note on group consistency in analytic hierarchy process abstract we study the paper of xu z xu on consistency of the weighted geometric mean complex judgement matrix in ahp european journal of operational research 126 2000 683 687 for the group consistency in analytic hierarchy process of multicriteria decision making the purpose of this note is threefold first we point out the questionable results in this paper second for three by three comparison matrices we provide a patchwork for his method third we constructed a counter example to show that in general his method is wrong numerical examples are provided to illustrate our findings if there are four or more alternatives then we may advise researchers to ignore his results to avoid questionable estimation of group consistency

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

an upper bound for a 1 of strictly diagonally dominant m matrices abstract let a be a real strictly diagonally dominant m matrix we give a sharp upper bound for a 1 furthermore the lower bound of the smallest eigenvalue q a is established

singularity wielandt s lemma and singular values in this study some upper and lower bounds for singular values of a general complex matrix are investigated according to singularity and wielandt s lemma of matrices especially some relationships between the singular values of the matrix a and its block norm matrix are established based on these relationships one may obtain the effective estimates for the singular values of large matrices by using the lower dimension norm matrices in addition a small error in piazza 2002 g piazza t politi an upper bound for the condition number of a matrix in spectral norm j comput appl math 143 1 2002 141 144 is also corrected some numerical experiments on saddle point problems show that these results are simple and sharp under suitable conditions