extensions of the analytic hierarchy process in fuzzy environment abstract the aim of this paper is to extend the analytic hierarchy process ahp to the fuzzy environment firstly starting from the viewpoint of actual application considering the method of group decision making in complex systems and the fuzziness of judgment in pairwise comparison of projects we construct the fuzzy judgement matrix by using a set valued statistics method on continuous judgment matrix and prove that every element of the fuzzy judgment scale can be represented by a positive bounded closed fuzzy number after a discussion of the properties of a positive bounded closed fuzzy number the fuzzy weight vector of projects can be introduced by applying the fuzzy extension principle to the altered gradient eigenvector method thus the rank ordering of projects is given finally we also give an example to apply our model

a consistency improving method in the analytic hierarchy process in this paper we propose a method to modify a given comparison matrix by which the consistency ratio cr value of the modified matrix is less than that of the original one and give an algorithm to derive a positive reciprocal matrix with acceptable consistency i e cr 0 1 then the convergence theorem for the given algorithm is established and its practicality is shown by some examples

the continuous ordered weighted geometric operator and its application to decision making the aim of this paper is to develop a continuous ordered weighted geometric c owg operator which is based on the continuous ordered weighted averaging c owa operator recently introduced by the author and the geometric mean we study some desirable properties of the c owg operator and present its application to decision making with interval multiplicative preference relation and finally an illustrative example is pointed out

nilpotent orbits and commutative elements abstract let w be a simply laced coxeter group with generating set s and let w c denote the subset consisting of those elements whose reduced expressions have no substrings of the form sts for any non commuting s t s we give a root system characterization of w c and in the case where w corresponds to a finite weyl group show that w c is a union of spaltenstein springer steinberg cells the latter is valid also for affine weyl groups of type a but not for type d or e

incorporating the uncertainty of decision judgements in the analytic hierarchy process abstract the uncertainty in the relative weights of a pairwise comparison matrix in the analytic hierarchy process ahp is caused by the uncertainty in our decision judgements and in many cases can not be avoided in this paper it is explicitly shown how such uncertainties can be incorporated within the framework of ahp and how the resulting uncertainties in the relative priorities of the decision alternatives can be computed the required algorithm and the computational procedures are also developed and illustrated with examples uncertainty is introduced as a fundamental concept independent of the concept of consistency with a view to extend the ahp as a decision analysis procedure

on the issue of obtaining owa operator weights we first investigate the issue of obtaining the weights associated with the owa aggregation in the situation when we have observed data on the arguments and the aggregated value we next introduce a family of owa operators called exponential owa operators finally we look at a simple procedure for generating the weights given a required degree of orness

a new method of obtaining the priority weights from an interval fuzzy preference relation in analyzing a multiple criteria decision making problem the decision maker may express her his opinions as an interval fuzzy or multiplicative preference relation then it is an interesting and important issue to investigate the consistency of the preference relations and obtain the reliable priority weights in this paper a new consistent interval fuzzy preference relation is defined and the corresponding properties are derived the transformation formulae between interval fuzzy and multiplicative preference relations are further given which show that two preference relations consistent interval fuzzy and multiplicative preference relations can be transformed into each other based on the transformation formula the definition of acceptably consistent interval fuzzy preference relation is given furthermore a new algorithm for obtaining the priority weights from consistent or inconsistent interval fuzzy preference relations is presented finally three numerical examples are carried out to compare the results using the proposed method with those using other existing procedures the numerical results show that the given procedure is feasible effective and not requisite to solve any mathematical programing

compatible weighting method with rank order centroid maximum entropy ordered weighted averaging approach in a situation where imprecise attribute weights such as a rank order are captured various approximate weighting methods have been proposed to aid multiattribute decision analysis among others it is well known that the rank order centroid roc weights result in the highest performance in terms of the identification of the best alternative under the ranked attribute weights in this paper we aim to reinterpret the meaning of the roc weights and to develop a compatible weighting method that is based on other well established academic disciplines the ordered weighted averaging owa method is a nonlinear aggregation method in that the weights are associated with the objects reordered according to their magnitudes in the aggregation process some interesting semantics can be attached to the approximate weights in view of the measure developed in the owa method furthermore the weights generated by the maximum entropy method show equally compatible performance with the roc weights under some condition which is demonstrated by theoretical and simulation analysis

a survey of chandler davis chandler davis is a very unusual interesting and complex person this short sketch however is restricted to his contributions to linear algebra and to operator theory although we begin with a brief biography

fractional programming approach to fuzzy weighted average this paper proposes a fractional programming approach to construct the membership function for fuzzy weighted average based on the cut representation of fuzzy sets and the extension principle a pair of fractional programs is formulated to find the cut of fuzzy weighted average owing to the special structure of the fractional programs in most cases the optimal solution can be found analytically consequently the exact form of the membership function can be derived by taking the inverse function of the cut for other cases a discrete but exact solution to fuzzy weighted average is provided via an efficient solution method examples are given for illustration

gromov witten invariants on grassmannians we prove that any three point genus zero gromov witten invariant on a type a grassmannian is equal to a classical intersection number on a two step flag variety we also give symplectic and orthogonal analogues of this result in these cases the two step flag variety is replaced by a sub maximal isotropic grassmannian our theorems are applied in type a to formulate a conjectural quantum littlewood richardson rule and in the other classical lie types to obtain new proofs of the main structure theorems for the quantum cohomology of lagrangian and orthogonal grassmannians

preference simulation and preference programming robustness issues in priority derivation abstract decision makers often resist having to make what appears to them as precise numerical judgements in fuzzy situations pairwise verbal comparisons used in the ahp are fuzzy in the sense that decision maker s need not relate verbal judgment to precise numbers because of the redundancy inherent in each set of judgments accurate priorities can be derived from such fuzzy verbal judgments another way of making fuzzy judgments is to express each judgment as a numerical interval this paper explores two new approaches for priority derivation when preferences are expressed as interval judgments one based on a simulation approach and the other based on mathematical programming the first approach assumes that the interval judgments are uniformly distributed and proceeds to derive the priority vectors and their underlying rank order by randomly sampling from these distribution this approach provides in addition to the priority vectors a measure of robustness given by the probability of rank reversal the second approach generates a region if one exists that encloses all priority vectors derived from inequalities representing the original interval judgments the two approaches are described and illustrated through a numerical example

on weighted unbalanced linguistic aggregation operators in group decision making we generalize linguistic evaluation values and their weights in group decision making gdm problems based on unbalanced linguistic terms the gdm problems include two types of weights belief degrees of linguistic evaluation values and experts weights due to the time pressure and conflict of multi source of information etc experts may have various evaluations values on alternatives the belief degree is used to represent the confidence level of the values the experts weight is used to represent the differences among experts importance which caused by experts experience or knowledge distinction we propose the weighted unbalanced linguistic aggregation operators to synthesize linguistic evaluation value belief degree and experts weights some desired properties of the operator are then studied these properties show that the operator extends the linguistic weighted averaging operator lwa and the linguistic ordered weighted averaging lowa operator finally an illustrative example of human resource performance appraisal based on the linguistic aggregation operator is provided

new decision making techniques and their application in the selection of financial products we develop a new approach that uses the ordered weighted averaging owa operator in the selection of financial products in doing so we introduce the ordered weighted averaging distance owad operator and the ordered weighted averaging adequacy coefficient owaac operator these aggregation operators are very useful for decision making problems because they establish a comparison between an ideal alternative and available options in order to find the optimal choice the objective of this new model is to manipulate the attitudinal character of previous methods based on distance measures so that the decision maker can select financial products according to his or her degree of optimism which is also known as the orness measure the main advantage of using the owa operator is that we can generate a parameterized family of aggregation operators between the maximum and the minimum thus the analysis developed in the decision process by the decision maker is much more complete because he or she is able to select the particular case in accordance with his or her interests in the aggregation process the paper ends with an illustrative example that shows results obtained by using different types of aggregation operators in the selection of financial products

subjective and objective information in linguistic multi criteria group decision making abstract linguistic decision making systems represent situations that cannot be assessed with numerical information but it is possible to use linguistic variables this paper introduces new linguistic aggregation operators in order to develop more efficient decision making systems the linguistic probabilistic weighted average lpwa is presented its main advantage is that it considers subjective and objective information in the same formulation and considering the degree of importance that each concept has in the aggregation a key feature of the lpwa operator is that it considers a wide range of linguistic aggregation operators including the linguistic weighted average the linguistic probabilistic aggregation and the linguistic average further generalizations are presented by using quasi arithmetic means and moving averages an application in linguistic multi criteria group decision making under subjective and objective risk is also presented in the context of the european union law

root theoretic young diagrams and schubert calculus planarity and the adjoint varieties abstract we study root theoretic young diagrams to investigate the existence of a lie type uniform and nonnegative combinatorial rule for schubert calculus we provide formulas for co adjoint varieties of classical lie type this is a simplest case after the co minuscule family where a rule has been proved by h thomas and the second author using work of r proctor our results build on earlier pieri type rules of p pragacz j ratajski and of a buch a kresch h tamvakis specifically our formula for og 2 2 n is the first complete rule for a case where diagrams are non planar yet the formulas possess both uniform and non uniform features using these classical type rules as well as results of p e chaput n perrin in the exceptional types we suggest a connection between polytopality of the set of nonzero schubert structure constants and planarity of the diagrams this is an addition to work of a klyachko and a knutson t tao on the grassmannian and of k purbhoo f sottile on cominuscule varieties where the diagrams are always planar

uncertainty and rank order in the analytic hierarchy process abstract the analytic hierarchy process uses paired comparisons to derive a scale of relative importance for alternatives we investigate the effect of uncertainty in judgment on the stability of the rank order of alternatives the uncertainty experienced by decision makers in making comparisons is measured by associating with each judgment an interval of numerical values the approach leads to estimating the probability that an alternative or project exchanges rank with other projects these probabilities are then used to calculate the probability that project would change rank at all we combine the priority of importance of each project with the probability that it does not change rank to obtain the final ranking

a combinatorial rule for co minuscule schubert calculus we prove a root system uniform concise combinatorial rule for schubert calculus of minuscule and cominuscule flag manifolds g p the latter are also known as compact hermitian symmetric spaces we connect this geometry to the poset combinatorics of proctor thereby giving a generalization of schutzenberger s jeu de taquin formulation of the littlewood richardson rule that computes the intersection numbers of grassmannian schubert varieties our proof introduces cominuscule recursions a general technique to relate the numbers for different lie types

generalized ordered weighted utility averaging hyperbolic absolute risk aversion operators and their applications to group decision making abstract this paper develops a new class of aggregation operator based on utility function which introduces the risk attitude of decision makers dms in the aggregation process first under the general framework of utility function we provide a new operator called the generalized ordered weighted utility averaging gowua operator and study its properties which are suitable for any utility function then under the hyperbolic absolute risk aversion hara utility function we propose another new operator named as the generalized ordered weighted utility averaging hyperbolic absolute risk aversion gowua hara operator and further investigate its families including a wide range of aggregation operators to determine the gowua hara operator weights we put forward an orness measure of the gowua hara operator and analyze its properties considering that different dms may have different opinions toward decision making and their opinions can be characterized by different orness measures we construct a new optimization model to determine the optimal weights which can aggregate all the individual sets of weights into an overall set of weights furthermore based on the gowua hara operator a method for the multiple attribute group decision making magdm is developed finally an example is given to illustrate the application of the gowua hara operator to the magdm

lower bounds for matrices abstract in a recent paper 3 lyons has discovered an interesting lower bound for the cesaro matrix c his result says that cx 6 x for every x l 2 satisfying x 1 x 2 0 the purpose of this note is to establish analogous lower bounds for arbitrary matrices with nonnegative entries acting on arbitrary l p spaces 1 p

aggregation operators for selection problems this paper describes and compares aggregation techniques for multi expert multi criteria decision making the process of selecting applicants for a ph d program using different aggregation operators is presented as a case study the focus is on applying these operators using only ordinal sorted information a heuristic for determining weights in these situations is discussed the results obtained with different aggregation methods are compared and analyzed

group decision making with expertons and uncertain generalized probabilistic weighted aggregation operators abstract expertons and uncertain aggregation operators are tools for dealing with imprecise information that can be assessed with interval numbers this paper introduces the uncertain generalized probabilistic weighted averaging ugpwa operator it is an aggregation operator that unifies the probability and the weighted average in the same formulation considering the degree of importance that each concept has in the aggregation moreover it is able to assess uncertain environments that cannot be assessed with exact numbers but it is possible to use interval numbers thus we can analyze imprecise information considering the minimum and the maximum result that may occur further extensions to this approach are presented including the quasi arithmetic uncertain probabilistic weighted averaging operator and the uncertain generalized probabilistic weighted moving average we analyze the applicability of this new approach in a group decision making problem by using the theory of expertons in strategic management

minuscule elements of weyl groups the numbers game andd complete posets abstract certain posets associated to a restricted version of the numbers game of mozes are shown to be distributive lattices the posets of join irreducibles of these distributive lattices are characterized by a collection of local structural properties which form the definition of d complete poset in representation theoretic language the top minuscule portions of weight diagrams for integrable representations of simply laced kac moody algebras are shown to be distributive lattices these lattices form a certain family of intervals of weak bruhat orders these bruhat lattices are useful in studying reduced decompositions of minuscule elements of weyl groups and their associated schubert varieties the d complete posets have recently been proven to possess both the hook length and the jeu de taquin properties

parabolic kazhdan lusztig r polynomials for quasi minuscule quotients abstract we give explicit combinatorial formulas for the parabolic kazhdan lusztig r polynomials of the quasi minuscule quotients of the classical weyl groups as an application of our results we obtain explicit combinatorial formulas for certain sums and alternating sums of ordinary kazhdan lusztig r polynomials

on the spectrum of the bergman hilbert matrix abstract the bergman hilbert matrix is the matrix of the hilbert hankel operator on the bergman space l2a of the disk we obtain estimates on its norm and prove the existence of eigenvalues

on equivariant quantum schubert calculus for g p we show a z2 filtered algebraic structure and a quantum to classical principle on the torus equivariant quantum cohomology of a complete flag variety of general lie type generalizing earlier works of leung and the second author we also provide various applications on equivariant quantum schubert calculus including an equivariant quantum pieri rule for partial flag variety fln1 nk n 1 of lie type a

quantum cohomology of g p and homology of affine grassmannian let g be a simple and simply connected complex algebraic group p g a parabolic subgroup we prove an unpublished result of d peterson which states that the quantum cohomology qh g p of a flag variety is up to localization a quotient of the homology h gr g of the affine grassmannian gr g of g as a consequence all three point genus zero gromov witten invariants of g p are identified with homology schubert structure constants of h gr g establishing the equivalence of the quantum and homology affine schubert calculi

minuscule elements of weyl groups abstract we provide several characterizations of the minuscule elements of weyl groups studied by d peterson and r a proctor and extend proctor s classification from the simply laced case to the general case

integrating three representation models in fuzzy multipurpose decision making based on fuzzy preference relations abstract the purpose of this paper is to study a fuzzy multipurpose decision making problem where the information about the alternatives provided by the experts can be of a diverse nature the information can be represented by means of preference orderings utility functions and fuzzy preference relations and our objective is to establish a general model which cover all possible representations firstly we must make the information uniform using fuzzy preference relations as uniform preference context secondly we present some selection processes for multiple preference relations based on the concept of fuzzy majority fuzzy majority is represented by a fuzzy quantifier and applied in the aggregation by means of an owa operator whose weights are calculated by the fuzzy quantifier we use two quantifier guided choice degrees of alternatives a dominance degree used to quantify the dominance that one alternative has over all the others in a fuzzy majority sense and a non dominance degree that generalises orlovski s non dominated alternative concept the application of the two above choice degrees can be carried out according to two different selection processes a sequential selection process and a conjunction selection process

quantum pieri rules for tautological subbundles abstract we give quantum pieri rules for quantum cohomology of grassmannians of classical types expressing the quantum product of chern classes of the tautological subbundles with general cohomology classes we derive them by showing the relevant genus zero three pointed gromov witten invariants coincide with certain classical intersection numbers

a parametric approach to fuzzy linear programming in this paper we discuss how to deal with decision problems that are described with lp models and formulated with elements of imprecision and uncertainty more precisely we will study lp models in which the parameters are not fully known but only with some degree of precision even with incomplete information the model builder or model user is normally able to give a realistic interval for the parameters of an lp model for the constraint vector this is combined with some wishes or some leeway on the constraints even with ambiquity in the objective function there is normally some preference ordering to be found among alternative ways of action we will demonstrate that these modelling complications can be handled with the help of some results developed in the theory of fuzzy sets after an overview of some central contributions to fuzzy linear programming we will develop an lp model in which the parameters are not fully known only with some degree of precision and show that the model can be parametrised in such a way that the optimal solution becomes a function of the degree of precision the fuzzy lp model derived in this way appears to be fairly easy to handle computationally which is demonstrated with a numerical example

on the quantum cohomology of adjoint varieties we study the quantum cohomology of quasi minuscule and quasi cominuscule homogeneous spaces the product of any two schubert cells does not involve powers of the quantum parameter higher than 2 with the help of the quantum to classical principle we give presentations of the quantum cohomology algebras these algebras are semi simple for adjoint non coadjoint varieties and some properties of the induced strange duality are shown

continuous generalized owa operator and its application to decision making in this paper we present a new class of operators called the continuous generalized ordered weighted averaging c gowa operators which extends the continuous ordered weighted averaging operator to adapt to uncertain and complex situations in decision making we use differentiated aggregation method and produce special forms of c gowa operator in addition we apply the c gowa operator to the aggregation of multiple interval arguments and obtain a wide range of aggregation operators we especially discuss the controlling parameter and the selection of basic unit interval monotonic function introduced in the information aggregation to make it more feasible we further generalize the previous approaches by using choquet integral and quasi arithmetic means obtaining the combined continuous generalized choquet integral aggregation operator and the combined continuous quasi arithmetic choquet integral aggregation operator we also present some further extension by using hierarchical aggregation rules obtaining the ordered hierarchies of the combined c gowa operator meanwhile some desirable properties of these operators are investigated finally we give a numerical example to illustrate the application of these operators to group decision making with interval arguments

a grobner basis for kazhdan lusztig ideals it kazhdan lusztig ideals a family of generalized determinantal ideals investigated in woo yong 08 provide an explicit choice of coordinates and equations encoding a neighborhood of a torus fixed point of a schubert variety on a type a flag variety our main result is a gr o bner basis for these ideals this provides a single geometric setting to transparently explain the naturality of pipe dreams on the it rothe diagram of a permutation and their appearance in begin itemize item combinatorial formulas fomin kirillov 94 for schubert and grothendieck polynomialsof lascoux sch u tzenberger 82 item the equivariant k theory specialization formula of buch rim a nyi 04 and item a positive combinatorial formula for multiplicities of schubert varieties in good cases including those for which the associated kazhdan lusztig ideal is homogeneous under the standard grading end itemize our results generalize with alternate proofs knutson miller 05 s gr o bner basis theorem for schubert determinantal ideals and their geometric interpretation of the monomial positivity of schubert polynomials we also complement recent work of knutson 08 09 on degenerations of kazhdan lusztig varieties in general lie type as well as work of goldin 01 on equivariant localization and of lakshmibai weyman 90 rosenthal zelevinsky 01 and krattenthaler 01 on grassmannian multiplicity formulas

soft computing based aggregation methods for human resource management abstract we are interested in the personnel selection problem we have developed a flexible decision support system to help managers in their decision making functions this dss simulates experts evaluations using ordered weighted average owa aggregation operators which assign different weights to different selection criteria moreover we show an aggregation model based on efficiency analysis to put the candidates into an order

induced aggregation operators we introduce the induced ordered weighted averaging iowa operator in these operators the argument ordering process is guided by a variable called the order inducing value a procedure for learning the weights from data is described we suggest a number of applications of these induced owa aggregation operators first we show its possibilities in modeling nearest neighbor rules next it is applied to the aggregation of complex objects such as matrices it is also used to establish a new class of information fusion models called best yesterday models finally we extend the idea of order induced aggregation to the choquet aggregation resulting in what we call the induced choquet ordered averaging i coa operator

aggregation with generalized mixture operators using weighting functions this paper regards weighted aggregation operators in multiple attribute decision making and its main goal is to investigate ways in which weights can depend on the satisfaction degrees of the various attributes criteria we propose and discuss two types of weighting functions that penalize poorly satisfied attributes and reward well satisfied attributes we discuss in detail the characteristics and properties of both functions moreover we present an illustrative example to clarify the use and behaviour of such weighting functions comparing the results with those of standard weighted averaging operators

abelian ideals in a borel subalgebra of a complex simple lie algebra let mathfrak g be a complex simple lie algebra and mathfrak b a fixed borel subalgebra of mathfrak g we describe the abelian ideals in mathfrak b in a uniform way that is independent of the classification of complex simple lie algebras as an application we derive a formula for the maximal dimension of a commutative lie subalgebra of mathfrak g

quantum pieri rules for isotropic grassmannians we study the three point genus zero gromov witten invariants on the grassmannians which parametrize non maximal isotropic subspaces in a vector space equipped with a nondegenerate symmetric or skew symmetric form we establish pieri rules for the classical cohomology and the small quantum cohomology ring of these varieties which give a combinatorial formula for the product of any schubert class with certain special schubert classes we also give presentations of these rings with integer coefficients in terms of special schubert class generators and relations

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