covariate information matrix for sufficient dimension reduction building upon recent research on the applications of the density information matrix we develop a tool for sufficient dimension reduction sdr in regression problems called covariate information m

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

variable window width kernel estimates of probability densities kernel density estimators which allow different amounts of smoothing at different locations are studied modifications of estimators proposed by breiman meisel and purcell 1977 and abramson 1982a which have variable window widths are seen to have very fast rates of convergence these rates have traditionally been obtained using a less natural higher order kernel which has the disadvantage of allowing an estimator which takes on negative values

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

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

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

a class of invariant procedures for assessing multivariate normality summary distribution theory pertaining to a class of invariant procedures for assessing multivariate normality is described a cramer von mises type statistic belonging to this class is investigated in greater detail its asymptotic distribution is tabulated and compared with a monte carlo simulation of its finite sample distribution and a set of quadrivariate data is examined with it

fourier analysis on a hyperbolic supermanifold with constant curvature the fourier inversion theorem is proved for a rank one noncompact homogeneous space the hyperbolic superplane the proof makes use of some novel features of perfectly graded superspaces which are not encountered in classical geometric analysis an application to quasi one dimensional disordered one electron systems is given

adaptive smoothing and density based tests of multivariate normality abstract methods of adaptive smoothing of density estimates where the amount of smoothing applied varies according to local features of the underlying density are investigated the difficulties of applying taylor series arguments in this context are explored simple properties of the estimates are investigated by numerical integration and compared with the fixed kernel approach optimal smoothing strategies based on the multivariate normal distribution are derived as an application of these techniques two tests of multivariate normality one based on integrated squared error and one on entropy are developed and some power calculations are carried out

testing multivariate distributions in garch models in this paper we consider testing distributional assumptions in multivariate garch models based on empirical processes using the fact that joint distribution carries the same amount of information as the marginal together with conditional distributions we first transform the multivariate data into univariate independent data based on the marginal and conditional cumulative distribution functions we then apply the khmaladze s martingale transformation k transformation to the empirical process in the presence of estimated parameters the k transformation eliminates the effect of parameter estimation allowing a distribution free test statistic to be constructed we show that the k transformation takes a very simple form for testing multivariate normal and multivariate t distributions the procedure is applied to a multivariate financial time series data set

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

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

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

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

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

testing the parametric null model hypothesis in semiparametric partial and generalized spline models on etend les resultats de cox et koh 1986 aux modeles generalises de wahba 1985 et a divers autres modeles splines partiels on montre que la statistique de test a une relation etroite avec le comportement de la fonction de validation croisee generalisee

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

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

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

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 euclidean fourier inversion on super hyperbolic space for the super hyperbolic space in any dimension we introduce the non euclidean helgason fourier transform we prove an inversion formula exhibiting residue contributions at the poles of the harish chandra c function signalling discrete parts in the spectrum the proof is based on a detailed study of the spherical superfunctions using recursion relations and localization techniques to normalize them precisely careful estimates of their derivatives and a rigorous analysis of the boundary terms appearing in the polar coordinate expression of the invariant integral

on the use of nonparametric regression for model checking summary the use of nonparametric regression is explored to check the fit of a parametric regression model the principal aim is to check the validity of the regression curve rather than necessarily to detect outliers a pseudo likelihood ratio test is developed to provide a global assessment of fit and simulation bands are used to indicate the nature of departures from the model the types of data considered include discrete response variables where standard diagnostic techniques are often not appropriate and first order autoregressive series several numerical examples are given nonparametric regression can be used in an informal graphical way to assess the relationship between a response and an explanatory variable in this paper we aim to develop more formal methods of assessing the assumptions of a parametric model in particular when regression diagnostics of the type developed for normal linear models are not readily available the principal aim is to check the validity of the systematic part of the model by comparing a nonparametric estimate of the regression curve with a parametric one such a comparison may also identify outliers although the distinction between outliers and model inadequacy is not always easy two techniques are used to assess the fit of a parametric model in 2 confidence bands are constructed around the fitted regression curve by simulation a comparison of these with the nonparametric curve gives an indication of the nature of any departures from the model in 3 a pseudo likelihood ratio test is developed this provides a quantitative global assessment of fit in applying these ideas special emphasis is given to discrete data and notably logistic regression because of the difficulty in applying standard residual based model checking techniques to this type of response variable a poisson regression example is discussed in 4 however the underlying ideas have wider applications autoregressive time series of order 1 are discussed in 6 sections 5 and 7 discuss general issues we first discuss the context of binary regression with a single covariate and the difficulties caused by the discreteness of the response variable the observed data are assumed to be of the form xi yi ni where xi is a covariate value and yi has a binomial

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

paley wiener theorems for hyperbolic spaces we prove a topological paley wiener theorem for the fourier transform defined on the real hyperbolic spaces soo p q soo p 1 q for p q 2n without restriction to k types we also obtain paley wiener type theorems for l schwartz functions 0 2 for fixed k types

test of significance based on wavelet thresholding and neyman s truncation abstract traditional nonparametric tests such as the kolmogorov smirnov test and the cramer von mises test are based on the empirical distribution functions although these tests possess root n consistency they effectively use only information contained in the low frequencies this leads to low power in detecting fine features such as sharp and short aberrants as well as global features such as high frequency alternations the drawback can be repaired via smoothing based test statistics in this article we propose two such kind of test statistics based on the wavelet thresholding and the neyman truncation we provide extensive evidence to demonstrate that the proposed tests have higher power in detecting sharp peaks and high frequency alternations while maintaining the same capability in detecting smooth alternative densities as the traditional tests similar conclusions can be made for two sample nonparametric tests of distribution functions in that case the traditional linear rank tests such as th

testing independence and goodness of fit in linear models we consider a linear regression model and propose an omnibus test to simultaneously check the assumption of independence between the error and predictor variables and the goodness of fit of the parametric model our approach is based on testing for independence between the predictor and the residual obtained from the parametric fit by using the hilbert schmidt independence criterion gretton et al 2008 the proposed method requires no user defined regularization is simple to compute based on only pairwise distances between points in the sample and is consistent against all alternatives we develop distribution theory for the proposed test statistic under both the null and the alternative hypotheses and devise a bootstrap scheme to approximate its null distribution we prove the consistency of the bootstrap scheme a simulation study shows that our method has better power than its main competitors two real datasets are analysed to demonstrate the scope and usefulness of our method

supersymmetry and localization we study conditions under which an odd symmetry of the integrand leads to localization of the corresponding integral over a super manifold we also show that in many cases these conditions guarantee exactness of the stationary phase approximation of such integrals

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

kernels degrees of freedom and power properties of quadratic distance goodness of fit tests in this article we study the power properties of quadratic distance based goodness of fit tests first we introduce the concept of a root kernel and discuss the considerations that enter the selection of this kernel we derive an easy to use normal approximation to the power of quadratic distance goodness of fit tests and base the construction of a noncentrality index an analogue of the traditional noncentrality parameter on it this leads to a method akin to the neyman pearson lemma for constructing optimal kernels for specific alternatives we then introduce a midpower analysis as a device for choosing optimal degrees of freedom for a family of alternatives of interest finally we introduce a new diffusion kernel called the pearson normal kernel and study the extent to which the normal approximation to the power of tests based on this kernel is valid supplementary materials for this article are available online

on assessing multivariate normality summary we discuss procedures for assessing multivariate normality based on properties of radii and angles of the multivariate normal distribution and suggest that different procedures may be efficiently combined in order to achieve omnibus tests the assumption of population multivariate normality underlies many multivariate data analytic procedures unfortunately however there are relatively few formal methods available for assess ing the validity of this underlying assumption gnanadesikan 1977 section 5 4 2 and cox and small 1978 have extensively reviewed various formal and informal procedures proposed for testing the multinormality hypothesis and it is not our purpose to survey the procedures anew rather we suggest in this note that different characteristics of multivariate normality can be examined with different procedures which may thereafter be combined to obtain omnibus tests for the multinormality hypothesis

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

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

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

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

a 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 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

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

on bandwidth variation in kernel estimates a square root law we consider kernel estimation of a smooth density f at a point but depart from the usual approach in admitting an adaptive dependence of the sharpness of the kernels on the underlying density proportionally varying the bandwidths like f 1 2 at the contributing readings lowers the bias to a vanishing fraction of the usual value and makes for performance seen in well known estimators that force moment conditions on the kernel and so sacrifice positivity of the curve estimate issues of equivariance and variance stabilitization are treated

normality based validation for crisp clustering we introduce a new validity index for crisp clustering that is based on the average normality of the clusters unlike methods based on inter cluster and intra cluster distances this index emphasizes the cluster shape by using a high order characterization of its probability distribution the normality of a cluster is characterized by its negentropy a standard measure of the distance to normality which evaluates the difference between the cluster s entropy and the entropy of a normal distribution with the same covariance matrix the definition of the negentropy involves the distribution s differential entropy however we show that it is possible to avoid its explicit computation by considering only negentropy increments with respect to the initial data distribution where all the points are assumed to belong to the same cluster the resulting negentropy increment validity index only requires the computation of covariance matrices we have applied the new index to an extensive set of artificial and real problems where it provides in general better results than other indices both with respect to the prediction of the correct number of clusters and to the similarity among the real clusters and those inferred

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

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

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

a quality control approach to consistency paradoxes in ahp abstract the analytic hierarchy process ahp requires a specific consistency check of the pairwise comparisons in order to ensure that the decision maker is being neither inconsistent nor random in his or her pairwise comparisons however there are many situations where the decision maker has been reasonable logical and non random in making the pairwise comparison and yet will fail the consistency check this paper argues against the use of the standard consistency check if a consistency test is to be done a quality control approach is recommended

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