notes on order preservation and consistency in ahp abstract the pairwise comparisons method is a convenient tool used when the relative order among different concepts alternatives needs to be determined one popular implementation of the method is based on solving an eigenvalue problem for the pairwise comparisons matrix in such cases the ranking result for the principal eigenvector of the pairwise comparisons matrix is adopted while the eigenvalue is used to determine the index of inconsistency a lot of research has been devoted to the critical analysis of the eigenvalue based approach one of them is the work of bana e costa and vansnick 2008 in their work the authors define the conditions of order preservation cop and show that even for sufficiently consistent pairwise comparisons matrices this condition cannot be met the presented work defines more precise criteria for determining when the cop is met to formulate the criteria an error factor is used describing how far the input to the ranking procedure is from the ranking result the relationship between the saaty consistency index and cop is also discussed

energy functionals and canonical kahler metrics yau y2 has conjectured that a fano manifold admits a kahler einstein metric if and only if it is stable in the sense of geometric invariant theory there has been much progress on this conjecture by tian t1 t2 donaldson do1 do2 and others the mabuchi energy functional plays a central role in these ideas we study the ek functionals introduced by x x chen and g tian ct1 which generalize the mabuchi energy we show that if a fano manifold admits a kahler einstein metric then the functional e1 is bounded from below and modulo holomorphic vector fields is proper this answers affirmatively a question raised by chen c2 in fact we show that e1 is proper if and only if there exists a kahler einstein metric giving a new analytic criterion for the existence of this canonical metric with possible implications for the study of stability we also show that on a fano kahler einstein manifold all of the functionals ek are bounded below on the space of metrics with nonnegative ricci curvature

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

an exposition of the ahp in reply to the paper remarks on the analytic hierarchy process it is a fact that people make decisions and have been making decisions for a very long time contrary to what some of us who are interested in decision making may like to believe most people do not take seriously the existence of theories which purport to set their thinking and feeling right they claim to know their own value system and what they want they may wonder how anyone else can know well enough to tell them how best to organize their thinking in order to make better choices yet research has shown that complex decisions are beyond the capacity of the brain to synthesize intuitively and efficiently since decision making is a natural characteristic of people how do we describe what they do so that an ordinary mortal can understand what we are saying we do not wish to legislate the method with which people should make decisions but only to describe it even when it is prescribed by some method in the process we may learn things that can help people make better decisions how the analytic hierarchy process ahp forman et al harker 1986 harker and vargas 1987 saaty 1986 1988a b saaty and vargas 1987 xu 1988 golden et al 1989 saaty and alexander 1989 is a theory of measurement when applied in decision making it assists one to describe the general decision operation by decomposing a complex problem into a multi level hierarchic structure of objectives criteria subcriteria and alternatives the ahp provides a fundamental scale of relative magnitudes expressed in dominance units to represent judgments in the form of paired comparisons a ratio scale of relative magnitudes expressed in priority units is then derived from each set of comparisons an overall rcatio scale of priorities is then synthesized to obtain a ranking of the alternatives from its axioms to its procedures the ahp has turned out to be historically and theoretically a different and independent theory of decision making from utility theory much as a dialogue evolved in mathematics around the consistency of different geometries and around absolute and relative space and time in physics both to dispel absolute notions those who believe that only utility theory can tell us the absolute truth about man s decision making might take a close look at the ahp it has found varied and serious applications it also has a particular way of generating ratio scales and dealing with inconsistency in judgment that have contributed to its effectiveness in resource allocation and in the setting of priorities by a group of decision makers utility theory is a normative process the ahp as a descriptive theory encompasses procedures leading to outcomes as would be ranked by a normative theory but it must go beyond to deal with outcomes not accounted for by the demanding assumptions of a normative theory we must briefly describe the ahp to enable the reader to see that a practicable theory based on ratio scales need not dilute itself to satisfy expectations of people who derive their understanding from a theory based on interval scales this is particularly true if the rival theory in aspiring for generality also makes unrealistic assumptions for example about the transitivity and consistency of preferences and the difficult use of lotteries

on consistency and ranking of alternatives in fuzzy ahp abstract in fuzzy ahp where pairwise comparisons are described by membership values the treatment of consistency has largely been sketchy this paper proposes a fuzzy consistency definition with consideration of a tolerance deviation essentially the fuzzy ratios of relative importance allowing certain tolerance deviation are formulated as constraints on the membership values of the local priorities the fuzzy local and global weights are determined via the extension principle the alternatives are ranked on the basis of the global weights by application of a maximum minimum set ranking method our analysis uses only the data which are considered as consistent the proposed approach is computationally efficient as it consists of solving only linear programs an example is included for illustrative purpose

on fuzzy ratio comparisons in hierarchical decision models abstract this article develops a method for the interactive analysis of fuzzy pairwise comparisons in hierarchical weighting models in the proposed method fuzzy ratios of relative importance are employed as constraints on the membership values of local priorities so that fuzzy weights for the alternatives can be computed through linear programming moreover graph displays of fuzzy relations are used as a means of illustrating the dominance implications of the specified statements

uncertainty index based interval assignment by interval ahp abstract in a multi attribute decision making problem indigenous values are assigned to attributes based on a decision maker s subjective judgments the given judgments are often uncertain because of the uncertainty of situations and intuitiveness of human judgments in order to reflect the uncertainty in the assigned values they are denoted as intervals whose widths represent the possibilities of attributes since it is difficult for a decision maker to assign values directly to attributes in case of more than two attributes he she gives a pairwise comparison matrix by comparing two attributes at one occasion the given matrix contains two kinds of uncertainty one is inconsistency among comparisons and the other is incompleteness of comparisons this paper proposes the models to obtain intervals of attributes from the given uncertain pairwise comparison matrix at first the uncertainty indexes of a set of intervals are defined from the viewpoints of entropy in probability sum or maximum of widths or ignorance then considering that too uncertain information is not useful the intervals of attributes are obtained by minimizing their uncertainty indexes

a clarification of remarks on the analytic hierarchy process the replies by saaty 1990 and by harker and vargas 1990 to my remarks on the analytic hierarchy process dyer 1990 are useful in focusing attention on areas of disagreement regarding the ahp and provide readers with sufficient information to form their own opinions the purpose of this response is not to extend the debate but to clarify the two methods proposed in my paper to resolve the problem of rank reversal in the ahp harker and vargas 1989 misinterpreted my first proposed method method 1 in their reply the key observation they missed is that both the weights on the criteria and the scores of the alternatives on the criteria must be normalized with respect to the same range of alternative values in method 1 the decision maker must specify the ranges over which he assumed the alternatives might vary on each criterion when he determined the weights an example may be useful in clarifying the key ideas suppose each alternative in the harker and vargas 1990 example represents the choice of a decision maker among sums of money deposited in two different bank accounts and each criterion is simply the amount of money deposited in a single bank account obviously the decision maker would use the sum of the accounts as the basis for ranking the alternatives and would select b from the three alternatives a b and c let us see how the appropriate use of method 1 would lead to this result without recourse to the supermatrix technique or to the concept of dependence of the criteria on the alternatives the decision maker would consider the deposits in each bank account to be equally important if he assumed that the ranges of the possible deposits were equal for example suppose the decision maker assumes that the values for each bank account criterion vary over the range 1 80 he responds that changes from 1 to 80 on each criterion are equally important so the normalized eigenvector for the ahp criterion weights is 1 2 1 2 using method 1 since the range of values on criterion c1 is 75 80 and the decision maker assumed a range of 1 80 in determining the equal criterion weights we must introduce a dummy alternative with the value of 1 on criterion c1 this dummy alternative does not need to be explicitly introduced with values on the other criteria but it could be and would not affect the results as long as its assumed values on the other criteria are in the range 1 80 similarly a dummy alternative with a value of 80 must be introduced for criterion c2 the normalized eigenvector for criterion c1 would become 80 235 79 235 75 235 1 235 for example where the last component 1 235 is the score for the dummy alternative we normalize each resulting eigenvector by subtracting its smallest component and then dividing by the largest component the appropriate scores for the three alternatives are

eigenvector ranking method as a measuring tool formulas for errors abstract the eigenvector method em is well known to derive information from pairwise comparison matrices in decision making processes however this method is logically incomplete since its actual numerical error is unknown and its reliability is doubted by such phenomena as right left asymmetry rank reversal and reversal of order of intensity of preference in this paper we associate em with some standard measuring procedure analyze this procedure from the viewpoint of measurement theory and find the actual em error we show that the above phenomena have the same cause and are eliminated when the em errors are taken into account the full decision support tool which has all components of a standard measuring tool is composed of pairwise comparisons as an initial measuring procedure em as a data processor and the obtained formulas for em errors as an error indicator we consider two versions of this tool based on the right and the left principal eigenvectors of a pairwise comparison matrix both versions are equally suitable to measure and rank any comparable elements with positive numerical values and have the same mean relative errors equal to the square root of the double saaty s consistency index using the mean relative error we find the simple upper estimate for maximum permissible errors not impeding the reliable ranking this estimate imposes tight restrictions on inconsistency of expert judgements in decision making processes with a large number of alternatives these restrictions are much stronger than previously thought

openness results for uniform k stability assume that a projective variety together with a polarization is uniformly k stable if the polarization is canonical or anti canonical then the projective variety is uniformly k stable with respects to any polarization sufficiently close to the original polarization

some comments on saaty s ahp the purpose of this short paper is to help clarify some questions which have arisen with respect to the suitability or even the correctness of the way saaty s ahp method handles criteria weights sometimes causing the rank reversal phenomenon the position set forth in this paper is that this undesirable effect does not per se invalidate that method but it does make it necessary to identify the kind of situations in which the method is suitable

a criterion for the properness of the k energy in a general kahler class in this paper we give a criterion for the properness of the k energy in a general kahler class of a compact kahler manifold by using song weinkove s result in commun pure appl math 61 2 210 229 2008 as applications we give some kahler classes on mathbb c mathbb p 2 3 overline mathbb c mathbb p 2 and mathbb c mathbb p 2 8 overline mathbb c mathbb p 2 in which the k energy is proper finally we prove song weinkove s result on the existence of critical points of hat j functional by the continuity method

fuzzy analytic hierarchy process fallacy of the popular methods abstract the fuzzy analytic hierarchy process fuzzy ahp is a very popular decision making method and literally thousands of papers have been published about it however we find the basic logic of this approach has problems from its methodology the definition and operational rules of fuzzy numbers not only oppose the main logic of fuzzy set theory but also oppose the basic principles of the ahp in dealing with the outcomes fuzzy ahp does not give a generally accepted method to rank fuzzy numbers and a way to check the validity of the results besides we discuss the validity of the analytic hierarchy network process ahp anp in complex and uncertain environments and find that fuzzy anp is a false proposition because there is no fuzzy priority in the super matrix which provides the basis for the anp although fuzzy ahp has been applied in many cases and cited hundreds of times we hoped that those who use fuzzy ahp would understand the problems associated with this method

are inconsistent decisions better an experiment with pairwise comparisons this paper tries to contribute to the well known and lengthy discussion on whether we should allow decision makers to be inconsistent or to put it in another way if intransitive decisions better represent their preferences we will focus specifically on the multiple criteria decision making problem since it is one of the most usual stances where this intransitivity appears results from an experiment carried out with graduate students shows that when intransitivities are removed the preferences of decision makers are not better represented and therefore that the effort to do so may not be justified

scalar curvature and projective embeddings i we prove that a metric of constant scalar curvature on a polarised kahler manifold is the limit of metrics induced from a specific sequence of projective embeddings satisfying a condition introduced by h luo this gives as a corollary the uniqueness of constant scalar curvature kahler metrics in a given rational cohomology class the proof uses results in the literature on the asymptotics of the bergman kernel the arguments are presented in a general framework involving moment maps for two different group actions

stochastic group preference modelling in the multiplicative ahp a model of group consensus this paper proposes a model for the representation of a consensus seeking group decision making process under the multiplicative variant of the analytic hierarchy process ahp in particular it proposes a group preference model which expresses the group s preference intensity judgements as random variables with associated probability distributions and the structure of the multiplicative ahp is exploited to derive interval judgements for the alternatives final impact scores as perceived by the group these interval judgements can be used to determine the probability that a rank reversal may occur amongst alternatives due to the group preference uncertainty i e to assess the stability of the final impact score vector which in turn offers information about the degree of consensus within the group about the decision

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

scalar curvature and stability of toric varieties we define a stability condition for a polarised algebraic variety and state a conjecture relating this to the existence of a kahler metric of constant scalar curvature the main result of the paper goes some way towards verifying this conjecture in the case of toric surfaces we prove that under the stability hypothesis the mabuchi functional is bounded below on invariant metrics and that minimising sequences have a certain convergence property in the reverse direction we give new examples of polarised surfaces which do not admit metrics of constant scalar curvature the proofs use a general framework developed by guillemin and abreu in which invariant kahler metrics correspond to convex functions on the moment polytope of a toric variety this paper is a step towards the solution of the general problem of finding conditions under which a complex projective variety admits a kahler metric of constant scalar curvature the pattern of the answer one expects is that this differential geometric condition should be equivalent to some notion of stability in the sense of geometric invariant theory this expectation is probably now an item of folklore going back to suggestions put forward by yau in the case of kahlereinstein metrics and the many results of tian and others in this case reinforced by a detailed formal picture which makes clear the analogy with the well established relation between the stability of vector bundles and yang mills connections 5 here we begin the investigation of

an evaluation of the eigenvalue approach for determining the membership values in fuzzy sets abstract the membership values of the elements of a fuzzy set of key importance in any theoretical or practical application of fuzzy set theory although there are many methods that evaluate membership values the methd proposed by saaty 8 9 based on matrix of pairwise comparisons and eigenvalue theory is the backbone of many other methods in this paper we evaluate the above method by using a forward error analysis approach with the assumption that the true membership values in a fuzzy set are continuous in the interval 0 1 the results reveal that the eigenvalue method is dramatically inaccurate even for fuzzy sets with few members

a probabilistic interpretation of the final rankings in ahp abstract the analytic hierarchy process ahp has become a popular and practical tool for dealing with complex decision problems it provides a ranking for the decision alternatives this article recommends treating the pairwise comparison input data as random variables this will allow the determination of whether the differences between alternatives are statistically significant

on donaldson s flow of surfaces in a hyperkahler four manifold we prove some basic properties of donaldson s flow of surfaces in a hyperkahler 4 manifold when the initial submanifold is symplectic with respect to one kahler form and lagrangian with respect to another we show that certain kinds of singularities cannot form and we prove a convergence result under a condition related to one considered by m t wang for the mean curvature flow

avoiding rank reversal in ahp decision support models abstract extending the work of a previous paper which showed that rank reversal in ahp is caused by eigenvector normalization this paper shows why methods proposed for avoiding rank reversal work when they are unnecessary and when they are futile the paper also discusses the implicit often unwarranted assumption that the dm s preference structure can be represented by an additive value function and examines the ramifications when a nonadditive value function is required

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

on the convergence and singularities of the j flow with applications to the mabuchi energy the j flow of s k donaldson and x x chen is a parabolic flow on kahler manifolds with two kahler metrics it is the gradient flow of the j functional that appears in chen s formula for the mabuchi energy we find a positivity condition in terms of the two metrics that is both necessary and sufficient for the convergence of the j flow to a critical metric we use this result to show that on manifolds with ample canonical bundle the mabuchi energy is proper on all kahler classes in an open neighborhood of the canonical class defined by a positivity condition this improves previous results of chen and of the second author we discuss the implications of this for the problem of the existence of constant scalar curvature kahler metrics we also study the singularities of the j flow and under certain conditions which always hold for dimension 2 derive some estimates away from a subvariety we discuss the conjectural remark of donaldson that if the j flow does not converge on a kahler surface then it should blow up over some curves of negative self intersection 2007 wiley periodicals inc

a unified approach to ahp with linking pins abstract recent work demonstrates the need for paired comparison estimates of criteria in the analytic hierarchy process ahp to follow prescribed rules and for the subsequent analysis to be contingent on these rules many different approaches are valid each reflecting a different rule for criteria comparison a generalized formula from which valid approaches may be generated is presented and the connection between these methods and saaty s supermatrix approach is explored

understanding local ignorance and non specificity within the ds ahp method of multi criteria decision making abstract ds ahp is a nascent technique for multi criteria decision making inspired by the analytic hierarchy process and whose mathematical foundation is based on the dempster shafer theory of evidence dst one aspect a consequence of the utilisation of dst in ds ahp is the allowance of a decision maker to make preference judgements on groups of decision alternatives das over different criteria in this paper developments are undertaken on ds ahp with respect to the measurement of the level of local ignorance in the judgements made together with the elucidation of an associated measure of non specificity through the evaluation of the limits on these measures subsequent index values are constructed these index values are used to introduce the concept of a vagueness plot within ds ahp the vagueness plot is shown to aid in gauging the results of interactive changes in the preference judgements made the index values and the vagueness plot presented enable the notion of the anchoring and adjustment aspect to decision making in ds ahp to be investigated

a comparison of confidence interval methods for the intraclass correlation coefficient summary different methods of obtaining confidence intervals for the intraclass correlation coefficient p in the unbalanced one way random effects model are investigated focusing on applications to family studies methods based on simple modifications of formulas for the case of equal group sizes are found to provide adequate coverage at small to moderate values of p a method based on the large sample standard error of the sample intraclass correlation as derived by smith 1956 annals of human genetics 21 363 373 is shown to provide consistently good coverage at all values of p a method proposed by thomas and hultquist 1978 annals of statistics 6 582 587 also provides consistently good coverage but generates mean interval widths substantially greater than those generated by smith s method at values of p likely to arise in practice

multi attribute decision making a simulation comparison of select methods abstract several methods have been proposed for solving multi attribute decision making problems madm a major criticism of madm is that different techniques may yield different results when applied to the same problem the problem considered in this study consists of a decision matrix input of n criteria weights and ratings of l alternatives on each criterion the comparative performance of some methods has been investigated in a few mostly field studies in this simulation experiment we investigate the performance of eight methods electre topsis multiplicative exponential weighting mew simple additive weighting saw and four versions of ahp original vs geometric scale and right eigenvector vs mean transformation solution simulation parameters are the number of alternatives criteria and their distribution the solutions are analyzed using twelve measures of similarity of performance similarities and differences in the behavior of these methods are investigated dissimilarities in weights produced by these methods become stronger in problems with few alternatives however the corresponding final rankings of the alternatives vary across methods more in problems with many alternatives although less significant the distribution of criterion weights affects the methods differently in general all ahp versions behave similarly and closer to saw than the other methods electre is the least similar to saw except for closer matching the top ranked alternative followed by mew topsis behaves closer to ahp and differently from electre and mew except for problems with few criteria a similar rank reversal experiment produced the following performance order of methods saw and mew best followed by topsis ahps and electre it should be noted that the electre version used was adapted to the common madm problem and therefore it did not take advantage of the method s capabilities in handling problems with ordinal or imprecise information

the theory of ratio scale estimation saaty s analytic hierarchy process the analytic hierarchy process developed by saaty saaty t l 1980 the analytic hierarchy process mcgraw hill new york has proven to be an extremely useful method for decision making and planning however some researchers in these areas have raised concerns over the theoretical basis underlying this process this paper addresses currently debated issues concerning the theoretical foundations of the analytic hierarchy process we also illustrate through proof and through examples the validity or fallaciousness of these criticisms

confidence intervals for intraclass correlation coefficients in a nonlinear dose response meta analysis this work is motivated by a meta analysis case study on antipsychotic medications the michaelis menten curve is employed to model the nonlinear relationship between the dose and d2 receptor occupancy across multiple studies an intraclass correlation coefficient icc is used to quantify the heterogeneity across studies to interpret the size of heterogeneity an accurate estimate of icc and its confidence interval is required the goal is to apply a recently proposed generic beta approach for construction the confidence intervals on iccs for linear mixed effects models to nonlinear mixed effects models using four estimation methods these estimation methods are the maximum likelihood second order generalized estimating equations and two two step procedures the beta approach is compared with a large sample normal approximation delta method and bootstrapping the confidence intervals based on the delta method and the nonparametric percentile bootstrap with various resampling strategies failed in our settings the beta approach demonstrates good coverages with both two step estimation methods and consequently it is recommended for the computation of confidence interval for iccs in nonlinear mixed effects models for small studies

a weak order solution to a group ranking and consistency driven pairwise comparisons a novel non numerical algorithm based on a partial order for solving a group ranking problem is compared with the more traditional consistency driven approach both approaches are based on the fundamental concept of pairwise comparisons potential applications are discussed

estimating equations for parameters in means and covariances of multivariate discrete and continuous responses generalized estimating equations are introduced in an ad hoc fashion for the covariance matrix of a multivariate response these equations are to be solved jointly with score equations from a generalized linear model for mean parameters a class of quadratic exponential models is used to develop joint estimating equations for mean and covariance parameters in a more systematic fashion and proposals for the use of such equations are developed comments on the relative merits of the ad hoc and model based approaches to estimation are given and a regression illustration with a bivariate response is provided

better bootstrap confidence intervals abstract we consider the problem of setting approximate confidence intervals for a single parameter in a multiparameter family the standard approximate intervals based on maximum likelihood theory can be quite misleading in practice tricks based on transformations bias corrections and so forth are often used to improve their accuracy the bootstrap confidence intervals discussed in this article automatically incorporate such tricks without requiring the statistician to think them through for each new application at the price of a considerable increase in computational effort the new intervals incorporate an improvement over previously suggested methods which results in second order correctness in a wide variety of problems in addition to parametric families bootstrap intervals are also developed for nonparametric situations

implementation of the centroid method of solymosi and dombi multiple objective decision support either for selection among finite alternatives or for combining multiple functions in mathematical programming often involves estimating a set of weights of relative importance recently solymosi and dombi proposed a means of estimating such weights based upon ordinal preference information this method relies upon the centroid of bounded weights analytic hierarchy process uses ratio estimates of pairwise comparisons using only ordinal information the centroid of weights is developed as an alternative means of weighting hierarchical elements these weights could be used in selection among alternatives directly in concordance analysis or in multiple objective linear programming formulation for the centroid of feasible weight values is presented with a table of values for cases where the preference ranking includes no ties comparison of the technique on a student job selection decision indicates slight but insignificant decline in accuracy relative to the analytic hierarchy based method

ranking fuzzy multicriteria alternatives with respect to a decision maker s fuzzy goal abstract a two stage method for ranking fuzzy multicriteria alternatives with respect to a decision maker s fuzzy goal is presented in the first stage the joint membership function of each fuzzy alternative and the membership function of the fuzzy goal are projected orthogonally from the multicriteria space to the one dimensional preference subspace of the decision maker in the second stage a ranking index is used to rank the fuzzy projections in the decision maker s preference subspace a numerical example demonstrates the method

an empirical analysis of the use of the analytic hierarchy process for estimating membership values in a fuzzy set abstract the analytic hierarchy process ahp is now a popular multicriteria decision making method and has been applied in many diverse areas despite the debate on its theoretical foundations many researchers have been using the pairwise comparison matrix and the eigenvector method suggested by saaty for obtaining priority ratings of the decision alternatives in an ahp hierarchy these priority ratings have been used as estimates of fuzzy membership values of the elements called alternatives in ahp of a fuzzy set triantaphyllou and mann have argued that saaty s scale and the eigenvector method are inadequate for estimating fuzzy membership values this paper points out the limitations in the arguments of triantaphyllou and mann about the eigenvector method it primarily concentrates on two aspects the first pertains to saaty s condition on deleting or combining two alternatives which are near copies alternatives whose priority values are close to each other within a 10 range over all criteria the second aspect concerns the use of the eigenvector method for estimating the membership values of elements of a fuzzy set the use of the eigenvector method for estimating fuzzy membership values in a fuzzy set has been justified using the empirical analysis together with theoretical support it has been amply demonstrated in this paper that the problem lies in saaty s scale and not in the eigenvector method

on kahler einstein metrics on certain kahler manifolds with c1 m 0 on demontre qu il existe une metrique de kahler einstein sur une hypersurface de fermat a m dimensions de degre superieur a m 1

comparison of the rembrandt system with analytic hierarchy process abstract this study compared the use of the rembrandt system to analytic hierarchy process ahp in a group multiple criteria selection problem the decision was to select an operations management text and the decision was the responsibility of a faculty group the primary purpose of this study is to apply rembrandt to the same problem that was supported by ahp there are no noticeable differences between the techniques from the perspective of the users as the same input is used important technical differences between ahp and rembrandt are demonstrated including 1 different ratio input scales 2 alternative calculation of impact scores and 3 a different aggregation procedure rembrandt was found to recommend the same decision as ahp when the geometric mean was used for aggregation but a different decision was given by conventional ahp using arithmetic mean aggregation

remarks on the analytic hierarchy process the analytic hierarchy process ahp is flawed as a procedure for ranking alternatives in that the rankings produced by this procedure are arbitrary this paper provides a brief review of several areas of operational difficulty with the ahp and then focuses on the arbitrary rankings that occur when the principle of hierarchic composition is assumed this principle requires that the weights on the higher levels of a hierarchy can be determined independently of the weights on the lower levels virtually all of the published examples of the use of the ahp to evaluate alternatives relative to a set of criteria have assumed this principle the key to correcting this flaw is the synthesis of the ahp with the concepts of multiattribute utility theory

decision making with the ahp why is the principal eigenvector necessary in this paper it is shown that the principal eigenvector is a necessary representation of the priorities derived from a positive reciprocal pairwise comparison judgment matrix a aij when a is a small perturbation of a consistent matrix when providing numerical judgments an individual attempts to estimate sequentially an underlying ratio scale and its equivalent consistent matrix of ratios near consistent matrices are essential because when dealing with intangibles human judgment is of necessity inconsistent and if with new information one is able to improve inconsistency to near consistency then that could improve the validity of the priorities of a decision in addition judgment is much more sensitive and responsive to large rather than to small perturbations and hence once near consistency is attained it becomes uncertain which coefficients should be perturbed by small amounts to transform a near consistent matrix to a consistent one if such perturbations were forced they could be arbitrary and thus distort the validity of the derived priority vector in representing the underlying decision

on some useful properties of the perron eigenvalue of a positive reciprocal matrix in the context of the analytic hierarchy process abstract a positive n x n matrix r rij is said to be reciprocal if its entries verify fji 1 rij 0 for all 1 2 decision alternatives on an arbitrary ratio scale saaty in journal of mathematical psychology 1977 proposed to use z ama x n n 1 0 a linear transform of the perron eigenvalue max of r as a measure of the cardinal consistency in an agent s responses and posed the problem of determining how it might vary as a function of the rij s he also suggested that an upper bound could be found for that consistency index when the entries of r are restricted to take their values in a bounded set both of these questions are answered here using classical results from linear algebra keywords analytic hierarchy process decision theory pairwise comparisons perron frobenius theo rem reciprocal matrices i introduction saaty 20 has introduced a procedure for pri oritising decision alternatives based on an n x n positive matrix r

non parametric bootstrap confidence intervals for the intraclass correlation coefficient the intraclass correlation coefficient rho plays a key role in the design of cluster randomized trials estimates of rho obtained from previous cluster trials and used to inform sample size calculation in planned trials may be imprecise due to the typically small numbers of clusters in such studies it may be useful to quantify this imprecision this study used simulation to compare different methods for assigning bootstrap confidence intervals to rho for continuous outcomes from a balanced design data were simulated for combinations of numbers of clusters 10 30 50 intraclass correlation coefficients 0 001 0 01 0 05 0 3 and outcome distributions normal non normal continuous the basic bootstrap t percentile bias corrected and bias corrected accelerated bootstrap intervals were compared with new methods using the basic and bootstrap t intervals applied to a variance stabilizing transformation of rho the standard bootstrap methods provided coverage levels for 95 per cent intervals that were markedly lower than the nominal level for data sets with only 10 clusters and only provided close to 95 per cent coverage when there were 50 clusters application of the bootstrap t method to the variance stabilizing transformation of rho improved upon the performance of the standard bootstrap methods providing close to nominal coverage

the dirichlet problem for a complex monge ampere type equation on hermitian manifolds abstract we are concerned with fully nonlinear elliptic equations on hermitian manifolds and search for technical tools to overcome difficulties in deriving a priori estimates which arise due to the nontrivial torsion and general non pseudoconvex boundary data we present our methods which work for more general equations by considering a specific equation which resembles the complex monge ampere equation in many ways but with crucial differences our work is motivated by recent increasing interests in fully nonlinear equations on complex manifolds from geometric problems

transitive calibration of the ahp verbal scale abstract one of the strengths of the analytic hierarchy process ahp is that it allows decision makers to specify their preferences using a verbal scale yet as is well known a strict reliance on the corresponding saaty 1 9 numeric scale can induce some inconsistency hence we argue that in certain situations it may be appropriate to calibrate the verbal scale the result of this calibration is a geometric scale based on a single parameter we present some limited evidence that this geometric scale marginally outperforms the saaty 1 9 scale moreover we suggest that this calibration can be used to do a simple sensitivity analysis in cases where judgements are uncertain

existence of twisted constant scalar curvature kahler metrics with a large twist suppose that there exist two kahler metrics omega and alpha such that the metric contraction of alpha with respect to omega is constant i e varlambda omega alpha text const we prove that for all large enough r 0 there exists a twisted constant scalar curvature kahler metric omega in the cohomology class omega satisfying s omega r varlambda omega alpha text const we discuss its implication to k stability and the continuity method recently proposed by chen

a rational approach to handling fuzzy perceptions in route choice abstract the purpose of this paper is to develop a heuristic way for handling fuzzy perceptions in explaining route choice behavior from behavioral point of view a hybrid model where route choice decision making is described in a hierarchy uses concepts from fuzzy logic and the analytical hierarchy process ahp is proposed for making possible a more proper description of route choice behavior in transportation systems teodorovic and kikuchi s transportation route choice model using fuzzy inference technique paper presented at the first international symposium on uncertainty modeling and analysis fuzzy reasoning probabilistic models and risk management university of college park maryland 1990 p 140 fuzzy if then rules are adopted to represent a typical driver s psychology for capturing essential preferences pairwise among alternatives that a driver may consider the ahp is then incorporated in this model to capture the imaginary psychological process that represent underlying observable behavior to estimate drivers preference allotment among the alternatives this new procedure is applied in a real world sample based on stated values of subjects findings show that this method provides intuitively and statistically promising results

inverse inconsistency in analytic hierarchies abstract this paper discusses a fundamental problem in the analytic hierarchy process of t l saaty vis inverse inconsistency previously addressed only indirectly in the literature a mechanism for handling this is suggested more importantly it is demonstrated how the problem is less significant in the modified analytic hierarchy process devised by donegan et al the related problem of dependence upon the irrelevant alternative is also examined and a list of statistically significant random indexes for the modified process is presented

fully nonlinear elliptic equations for conformal deformations of chern ricci forms abstract we study a general class of fully nonlinear elliptic equations of second order on hermitian manifolds we derive a priori estimates and prove some existence results for these equations in particular we prove interior estimates for the complex hessian which in general do not hold for other types of equations we apply our results to draw geometric conclusions on conformal deformations of the mixed chern ricci forms a notion we introduced in the article our work is motivated by the close connections of these equations to problems in non kahler geometry and the fact that there have been increasing interests in fully nonlinear pde s beyond the complex monge ampere equation from complex geometry

quantifying heterogeneity in a meta analysis the extent of heterogeneity in a meta analysis partly determines the difficulty in drawing overall conclusions this extent may be measured by estimating a between study variance but interpretation is then specific to a particular treatment effect metric a test for the existence of heterogeneity exists but depends on the number of studies in the meta analysis we develop measures of the impact of heterogeneity on a meta analysis from mathematical criteria that are independent of the number of studies and the treatment effect metric we derive and propose three suitable statistics h is the square root of the chi2 heterogeneity statistic divided by its degrees of freedom r is the ratio of the standard error of the underlying mean from a random effects meta analysis to the standard error of a fixed effect meta analytic estimate and i2 is a transformation of h that describes the proportion of total variation in study estimates that is due to heterogeneity we discuss interpretation interval estimates and other properties of these measures and examine them in five example data sets showing different amounts of heterogeneity we conclude that h and i2 which can usually be calculated for published meta analyses are particularly useful summaries of the impact of heterogeneity one or both should be presented in published meta analyses in preference to the test for heterogeneity

a comparative study of the numerical scales and the prioritization methods in ahp abstract in the analytic hierarchy process ahp a decision maker first gives linguistic pairwise comparisons then obtains numerical pairwise comparisons by selecting certain numerical scale to quantify them and finally derives a priority vector from the numerical pairwise comparisons in particular the validity of this decision making tool relies on the choice of numerical scale and the design of prioritization method by introducing a set of concepts regarding the linguistic variables and linguistic pairwise comparison matrices lpcms and by defining the deviation measures of lpcms we present two performance measure algorithms to evaluate the numerical scales and the prioritization methods using these performance measure algorithms we compare the most common numerical scales the saaty scale the geometrical scale the ma zheng scale and the salo hamalainen scale and the prioritization methods the eigenvalue method and the logarithmic least squares method in addition we also discuss the parameter of the geometrical scale develop a new prioritization method and construct an optimization model to select the appropriate numerical scales for the ahp decision makers the findings in this paper can help the ahp decision makers select suitable numerical scales and prioritization methods

on the efficacy of modeling multi attribute decision problems using ahp and sinarchy abstract ahp is a multi attribute decision making methodology widely used by both practitioners and researchers in the 1980s critics had raised questions regarding its proper use there were quite a few suggested modifications to overcome the supposed limitations of ahp these modifications are themselves limited as they typically impede the applicability of ahp in this paper we revisit some of the earlier criticisms we have two objectives 1 to articulate the proper use of ahp by highlighting the assumptions and implications underlying ahp and 2 to show that sinarchy can be used to address the earlier criticisms while maintaining the applicability of the ahp framework we identify that in ahp tradeoffs between criteria vary amongst individual alternatives and are dependent on the alternative s proportion of contribution towards each criterion for problems where tradeoffs between criteria are in terms of their relative measurements sinarchy should be used it is also shown that sinarchy can prevent rank reversal illustrative examples are included throughout

ds ahp method a mathematical analysis including an understanding of uncertainty abstract the analytic hierarchy process ahp was developed to aid decision makers to rank or sort information based on a number of criteria a recent advance is the ds ahp method which incorporates the dempster shafer theory of evidence with ahp this method allows judgements on groups of decision alternatives da to be made it also offers a measure of uncertainty in the final results in this paper a mathematical analysis of ds ahp is included constructing the functional form of the preference weightings given to groups of da these functions allow an understanding of the appropriateness of the rating scale values used in the ds ahp method through evaluating the range of uncertainty able to be expressed by the decision maker

choice functions and mechanisms for linguistic preference relations abstract the problem of finding a solution set of alternatives when a linguistic preference relation represents a collective preference is analyzed following two research lines choice functions and mechanisms four classical choice sets of alternatives for a linguistic preference relation are presented and some relations between them are pointed out the concept of linguistic choice function as a tool to derive linguistic choice sets of alternatives is introduced and some particular linguistic choice functions are presented together with a study of their rationality properties on the other hand two types of linguistic choice mechanisms to derive solution sets of alternatives from linguistic choice functions are introduced simple and composite ones the concept of linguistic covering relation is introduced with a view to allow us the design of consistent linguistic choice mechanisms which may achieve more precise and coherent solution sets of alternatives

contribution of individual judgments toward inconsistency in pairwise comparisons pairwise comparison pc is a well established method to assist decision makers in estimating their preferences in pcs the acquired judgments are used to construct a pc matrix pcm that is used to check whether the inconsistency in judgments is acceptable or requires revision the use of consistency ratio cr a widely used measure for inconsistency has been widely debated and the literature survey has identified a need for a more appropriate measure considering this need a new measure termed congruence is proposed in this paper the measure is shown to be useful in finding the contribution of individual judgments toward overall inconsistency of a pcm and therefore can be used to detect and correct cardinally inconsistent judgments the proposed measure is applicable to incomplete sets of pc judgments without modification unlike cr which requires a complete set of pc judgments to address ordinal inconsistency another measure termed dissonance is proposed as a supplement to the congruence measure the two measures appear useful in detecting both outliers and the phenomenon of consistency deadlock where all judgments equally contribute toward the overall inconsistency

modelling multicriteria value interactions with reasoning maps idiographic causal maps are extensively employed in operational research to support problem structuring and complex decision making processes they model means end or causal discourses as a network of concepts connected by links denoting influence thus enabling the representation of chains of arguments made by decision makers there have been proposals to employ such structures to support the structuring of multicriteria evaluation models within an additive value measurement framework however a drawback of this multi methodological modelling is the loss of richness of interactions along the means end chains when evaluating options this has led to the development of methods that make use of the structure of the map itself to evaluate options such as the reasoning maps method which employs ordinal scales and ordinal operators for such evaluation however despite their potential reasoning maps cannot model explicitly value interactions nor perform a quantitative ranking of options limiting their applicability and usefulness in this article we propose extending the reasoning maps approach through a multilinear evaluation model structure built with the macbeth multicriteria method the model explicitly captures the value interactions between concepts along the map and employs the macbeth protocol of questioning to assess the strength of influence for each means end link the feasibility of the proposed approach to evaluate options and to deal with multicriteria interactions is tested in a real world application to support the construction of a population health index

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

a simulation based evaluation of the approximate and the exact eigenvector methods employed in ahp abstract the analytic hierarchy process ahp is a popular multi criteria decision making method it provides ratio scale measurements of the priorities of elements in various levels of a hierarchy these priorities are obtained through pairwise comparisons of elements in one level with reference to each element in the immediate higher level saaty has suggested two methods viz the approximate eigenvector method aev and the exact eigenvector method eev for calculating the priorities research on ahp has led to conflicting views on the use of these two methods this paper presents a simulation analysis to evaluate them the simulation analysis uses the concept of approximating a continuous pairwise comparison cpc matrix by its closest discretized pairwise comparison dpc matrix this exercise supports saaty s theoretical statements and empirically concludes that the eev method is to be preferred over the aev method for the calculation of priority vectors

bootstrapping clustered data summary various bootstraps have been proposed for bootstrapping clustered data from one way arrays the simulation results in the literature suggest that some of these methods work quite well in practice the theoretical results are limited and more mixed in their conclusions for example mccullagh reached negative conclusions about the use of non parametric bootstraps for one way arrays the purpose of this paper is to extend our understanding of the issues by discussing the effect of different ways of modelling clustered data the criteria for successful bootstraps used in the literature and extending the theory from functions of the sample mean to include functions of the between and within sums of squares and non parametric bootstraps to include model based bootstraps we determine that the consistency of variance estimates for a bootstrap method depends on the choice of model with the residual bootstrap giving consistency under the transformation model whereas the cluster bootstrap gives consistent estimates under both the transformation and the random effect model in addition we note that the criteria based on the distribution of the bootstrap observations are not really useful in assessing consistency

the analytic hierarchy process an exposition this exposition on the analytic hierarchy process ahp has the following objectives 1 to discuss why ahp is a general methodology for a wide variety of decision and other applications 2 to present brief descriptions of successful applications of the ahp and 3 to elaborate on academic discourses relevant to the efficacy and applicability of the ahp vis a vis competing methodologies we discuss the three primary functions of the ahp structuring complexity measurement on a ratio scale and synthesis as well as the principles and axioms underlying these functions two detailed applications are presented in a linked document athttp mdm gwu edu formangass pdf

a fuzzy approach to deriving priorities from interval pairwise comparison judgements abstract in this paper we study the problem of priority elicitation in the analytic hierarchy process and propose a new approach to deriving crisp priorities from interval pairwise comparison judgements by introducing linear or non linear membership functions representing the decision maker s degree of satisfaction with various crisp priority vectors the interval judgements are transformed into fuzzy inequality constraints the interval prioritisation problem is then formulated as a fuzzy mathematical programming problem for obtaining an optimal crisp priority vector that maximises the overall degree of satisfaction the proposed approach yields linear or non linear mathematical programs capable of deriving priorities from consistent and inconsistent interval judgements the presence of a consistency index that measures the level of inconsistency of interval judgements is an attractive feature of our approach another feature which does not exist in the known prioritisation methods is the opportunity for additional prioritisation of the initial judgements numerical examples are given and comparisons with other interval prioritisation methods are carried out

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

highlights and critical points in the theory and application of the analytic hierarchy process abstract this paper provides a detailed discussion with references on the fundamentals of the analytic hierarchy process and in particular of relative measurement the points discussed are grouped under the following categories structure in the ahp hierarchies and networks scales of measurement judgments consistency and the eigenvector synthesis and normative vs descriptive the paper also includes a discussion of rank and a number of citations of rank reversals attributed to a variety of factors ranging from intransitivity to procedure invariance that are thought to be unexplained by utility theory with its underlying principle to always preserve rank it is shown that when there is synergy due to the number of elements the ahp can be used to both preserve rank when it is desired to preserve it and allow it to reverse when it should reverse

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

group preference aggregation in the multiplicative ahp the model of the group decision process and pareto optimality abstract a recent paper has focused awareness on group aggregation procedures in the ahp showing that geometric mean aggregation violates the desirable social choice axiom of pareto optimality we show that this violation can be attributed to the representation used to model the group decision process thereby questioning the legitimacy of the pareto optimality axiom we furthermore propose a geometric mean group aggregation procedure which satisfies all the social choice axioms suggested

on the relativity of relative measures accommodating both rank preservation and rank reversals in the ahp abstract the analytic hierarchy process ahp has been criticized for allowing the introduction of new alternatives to cause rank reversals among existing alternatives while in many cases this is a perfectly valid phenomenon there are also many cases where rank should be preserved to address both contingencies the ahp now supports two types of synthesis procedures one that preserves rank and one that does not this paper proposes guidelines for the selection of one synthesis mode over the other it also proposes several modifications to software implementations of the ahp to better support both synthesis modes and to reduce user stress over the choice

stochastic decision making using multiplicative ahp abstract the analytic hierarchy process ahp has found a number of applications in decision making problems its multiplicative version called the multiplicative ahp mahp has been proposed to overcome some of the criticisms of the conventional version both these methodologies operate by obtaining expert judgements on the ratios of perceived importance of objects under consideration the literature on mahp in dealing with these judgements when they are specified without uncertainty is well developed however stochastic aspects of these judgements have not received much consideration in the literature so far stochastic judgements are considered in this paper for use in mahp the fact that weight derivation in mahp can be handled using mathematical programming is exploited and the literature on stochastic programming is adapted to the mahp context

a sequential selection process in group decision making with a linguistic assessment approach in this paper a sequential selection process in group decision making under linguistic assessments is presented where a set of linguistic preference relations represents individuals preferences a collective linguistic preference is obtained by means of a defined linguistic ordered weighted averaging operator whose weights are chosen according to the concept of fuzzy majority specified by a fuzzy linguistic quantifier then we define the concepts of linguistic non dominance linguistic dominance and strict dominance degrees as parts of the sequential selection process the solution alternative s is obtained by applying these concepts

kahler einstein metrics with positive scalar curvature in this paper we prove that the existence of kahler einstein metrics implies the stability of the underlying kahler manifold in a suitable sense in particular this disproves a long standing conjecture that a compact kahler manifold admits kahler einstein metrics if it has positive first chern class and no nontrivial holomorphic vector fields we will also establish an analytic criterion for the existence of kahler einstein metrics our arguments also yield that the analytic criterion is satisfied on stable kahler manifolds provided that the partial c 0 estimate posed in t6 is true

multiple criteria robust interactive decision analysis mcrid for optimizing public policies abstract important public policy decisions are often very complex involving many factors large uncertainties and multiple conflicting objectives decision analysis which is a formal framework for dealing with such issues has been applied to a variety of public sector problems over the last two decades a limitation of using decision analysis is the often onerous assessment burden it places on the decision maker dm we have developed a novel interactive procedure and decision support system dss for performing decision analysis on the personal computer called robust interactive decision analysis rid which avoids the difficult problems of precisely assessing a dm s utility and state probability functions associated with traditional decision treee analysis the rid method permits a dm to voluntarily and interactively express strong viz sure binary preferences for actions partial decision functions and full decision functions and only imprecise probability and utility function assessments the inputs are used with various dominance operators via a mathematical programming framework to prune the state probability utility and decision space until there is convergence on a relatively small subset of efficient strategies or an optimal choice strategy conceptually the operation of the rid method can be regarded as a state pruning system and its computer implementation as a dss in this paper we extended the rid concept and dss to deal with multiple criteria decisions obviating the need to precisely and completely assess attribute weights tradeoffs consequences and or value utility functions the approach which we call multiple criteria robust interactive decision analysis mcrid is developed illustrated and shown to be effective in eliminating many inefficient alternatives however since the aggregate utility of each alternative is interval valued it then becomes difficult to intuitively choose an optimal alternative from among the remaining efficient ones since the intervals are not defined by any specific probability distributions we have tentatively shown that aggregate utility intervals ultimately converge toward a normal density function and can be so approximated with relatively few attributes hence stochastic dominance is employed to further filter out alternatives an interactive goal programming gp technique is then developed to assist the dm in choosing an optimal alternative

deriving priorities from fuzzy pairwise comparison judgements a new approach for deriving priorities from fuzzy pairwise comparison judgements is proposed based on cuts decomposition of the fuzzy judgements into a series of interval comparisons the assessment of the priorities from the pairwise comparison intervals is formulated as an optimisation problem maximising the decision maker s satisfaction with a specific crisp priority vector a fuzzy preference programming method which transforms the interval prioritisation task into a fuzzy linear programming problem is applied to derive optimal crisp priorities aggregating the optimal priorities which correspond to different cut levels enables overall crisp scores of the prioritisation elements to be obtained a modification of the linear fuzzy preference programming method is also proposed to derive priorities directly from fuzzy judgements without applying cut transformations the formulation of the prioritisation problem as an optimisation task is similar to the previous approach but it requires the solution of a non linear optimisation program the second approach also derives crisp priorities and has the advantage that it does not need additional aggregation and ranking procedures both proposed methods are illustrated by numerical examples and compared to some of the existing fuzzy prioritisation methods

geometry of decision making and the vector space formulation of the analytic hierarchy process we extend the conventional analytic hierarchy process ahp to an euclidean vector space and develop formulations for aggregation of the alternative preferences with the criteria preferences relative priorities obtained from such a formulation are almost identical with the ones obtained using conventional ahp each decision is represented by a preference vector indicating the orientation of the decision maker s mind in the decision space spanned by the decision alternatives this adds a geometric meaning to the decision making processes we utilise the measure of similarity between any two decision makers and apply it for analysing decisions in a homogeneous group we propose an aggregation scheme for calculating the group preference from individual preferences using a simple vector addition procedure that satisfies pareto optimality condition the results agree very well with the ones of conventional ahp

aggregation of analytic hierarchy process models based on similarities in decision makers preferences abstract the analytic hierarchy process is widely used in both individual and group decision making environments in this paper we investigate its applicability to model a specific class of decentralized decision problems where many decision makers take individual subjective decisions using locally available information in such subjective decision making environments it is neither possible nor appropriate to use group preference aggregation techniques to model the problem as a single group decision problem an approach to identify homogeneous subgroups of decision makers based on similarities in preferences and to aggregate preferences within each subgroup is proposed this approach is validated using employment preferences of 70 subjects modeled using the analytic hierarchy process

the j flow and stability we study the j flow from the point of view of an algebro geometric stability condition in terms of this we give a lower bound for the natural associated energy functional and we show that the blowup behavior found by fang and lai 11 is reflected by the optimal destabilizer finally we prove a general existence result on complex tori