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

compression of low entropy strings with lempel ziv algorithms we compare the compression ratio of the lempel ziv algorithms with the empirical entropy of the input string this approach makes it possible to analyze the performance of these algorithms without any assumption on the input and to obtain worst case results we show that in this setting the standard definition of optimal compression algorithm is not satisfactory in fact although lempel ziv algorithms are optimal according to the standard definition there exist families of low entropy strings which are not compressed optimally more precisely the compression ratio achieved by lz78 resp lz77 can be much higher than the zeroth order entropy h0 resp the first order entropy h1 for this reason we introduce the concept of lambda optimal algorithm an algorithm is lambda optimal with respect to hk if loosely speaking its compression ratio is asymptotically bounded by lambda times the kth order empirical entropy hk we prove that lz78 cannot be lambda optimal with respect to any hk with k geq 0 then we describe a new algorithm which combines lz78 with run length encoding rle and is 3 optimal with respect to h0 finally we prove that lz77 is 8 optimal with respect to h0 and that it cannot be lambda optimal with respect to hk for any k geq 1

limited memory bfgs systems with diagonal updates abstract in this paper we investigate a formula to solve systems of the form b i x y where b is a limited memory bfgs quasi newton matrix and is a positive constant these types of systems arise naturally in large scale optimization such as trust region methods as well as doubly augmented lagrangian methods we show that provided a simple condition holds on b 0 and the system b i x y can be solved via a recursion formula that requires only vector inner products this formula has complexity m 2 n where m is the number of l bfgs updates and n m is the dimension of x

enhancing data consistency in decision matrix adapting hadamard model to mitigate judgment contradiction cardinal and ordinal inconsistencies are important and popular research topics in the study of decision making with pair wise comparison matrices pcms few of the currently employed tactics are capable of simultaneously dealing with both cardinal and ordinal inconsistency issues in one model and most are heavily dependent on the method chosen for weight priorities derivation or the obtained closest matrix by optimization method that may change many of the original values in this paper we propose a hadamard product induced bias matrix model which only requires the use of the data in the original matrix to identify and adjust the cardinally inconsistent element s in a pcm through graph theory and numerical examples we show that the adapted hadamard model is effective in identifying and eliminating the ordinal inconsistencies also for the most inconsistent element identified in the matrix we develop innovative methods to improve the consistency of a pcm the proposed model is only dependent on the original matrix is independent of the methods chosen to derive the priority vectors and preserves most of the original information in matrix a since only the most inconsistent element s need s to be modified our method is much easier to implement than any of the existing models and the values it recommends for replacement outperform those derived from the literature it significantly enhances matrix consistency and improves the reliability of pcm decision making

global convergence of a cass of quasi newton methods on convex problems we study the global convergence properties of the restricted broyden class of quasi newton methods when applied to a convex objective function we assume that the line search satisfies a standard sufficient decrease condition and that the initial hessian approximation is any positive definite matrix we show global and superlinear convergence for this class of methods except for dfp this generalizes powell s well known result for the bfgs method the analysis gives us insight into the properties of these algorithms in particular it shows that dfp lacks a very desirable self correcting property possessed by bfgs

how to make a decision the analytic hierarchy process abstract this paper serves as an introduction to the analytic hierarchy process a multicriteria decision making approach in which factors are arranged in a hierarchic structure the principles and the philosophy of the theory are summarized giving general background information of the type of measurement utilized its properties and applications

the linear algebra of block quasi newton algorithms abstract the quasi newton family of algorithms for minimizing functions and solving systems of nonlinear equations has achieved a great deal of computational success and forms the core of many software libraries for solving these problems in this work we extend the theory of the quasi newton algorithms to the block case in which we minimize a collection of functions having a common hessian matrix or we solve a collection of nonlinear equations having a common jacobian matrix this paper focuses on the linear algebra update formulas positive definiteness least change secant properties relation to block conjugate gradient algorithms finite termination for quadratic function minimization or solving linear systems and the use of the quasi newton matrices as preconditioners

the smallest grammar problem this paper addresses the smallest grammar problem what is the smallest context free grammar that generates exactly one given string spl sigma this is a natural question about a fundamental object connected to many fields such as data compression kolmogorov complexity pattern identification and addition chains due to the problem s inherent complexity our objective is to find an approximation algorithm which finds a small grammar for the input string we focus attention on the approximation ratio of the algorithm and implicitly the worst case behavior to establish provable performance guarantees and to address shortcomings in the classical measure of redundancy in the literature our first results are concern the hardness of approximating the smallest grammar problem most notably we show that every efficient algorithm for the smallest grammar problem has approximation ratio at least 8569 8568 unless p np we then bound approximation ratios for several of the best known grammar based compression algorithms including lz78 b isection sequential longest match greedy and re pair among these the best upper bound we show is o n sup 1 2 we finish by presenting two novel algorithms with exponentially better ratios of o log sup 3 n and o log n m sup where m sup is the size of the smallest grammar for that input the latter algorithm highlights a connection between grammar based compression and lz77

on the behavior of broyden s class of quasi newton methods this paper analyzes algorithms from the broyden class of quasi newton methods for nonlinear unconstrained optimization this class depends on a parameter phi k for which the choices phi k 0 and phi k 1 give the well known bfgs and dfp methods this paper examines algorithms that allow for negative values of the parameter phi k it shows that severe restrictions have to be imposed on the selection of phi k to guarantee q superlinear convergence it is argued that negative values of phi k are desirable and conditions on phi k that guarantee superlinear convergence are given however practical algorithms that preserve the excellent properties of the bfgs method are not easy to design

a broyden class of quasi newton methods for riemannian optimization this paper develops and analyzes a generalization of the broyden class of quasi newton methods to the problem of minimizing a smooth objective function f on a riemannian manifold a condition on vector transport and retraction that guarantees convergence and facilitates efficient computation is derived experimental evidence is presented demonstrating the value of the extension to the riemannian broyden class through superior performance for some problems compared to existing riemannian bfgs methods in particular those that depend on differentiated retraction

a simulation approach for handling uncertainty in the analytic hierarchy process abstract uncertainty considerations are introduced into the analytic hierarchy process ahp the rank order of decision alternatives depends on two types of related uncertainties 1 uncertainty regarding the future characteristics of the decision making environment described by a set of scenarios and 2 uncertainty associated with the decision making judgment regarding each pairwise comparison a simulation approach for handling both types of related uncertainties in the ahp is described the example introduced by saaty and kearns 1985 is extended here to include uncertainty considerations

a multicriteria evaluation of methods for obtaining weights from ratio scale matrices abstract ratio scale matrices are commonly used as a vehicle to distinguish among competing alternatives or conflicting criteria the issue of extracting weights out of such matrices has attracted increasing attention in recent years and resulted in the development of several scaling methods this paper presents an attempt to evaluate some of the more known techniques in this area the evaluation procedure includes random generation of a large number of representative matrices which are used to evaluate the various scaling methods according to the specified criteria the performance of the different techniques is analyzed with respect to the proposed criteria and conclusions are drawn

boosting textual compression in optimal linear time we provide a general boosting technique for textual data compression qualitatively it takes a good compression algorithm and turns it into an algorithm with a better compression performance guarantee it displays the following remarkable properties a it can turn any memoryless compressor into a compression algorithm that uses the best possible contexts b it is very simple and optimal in terms of time and c it admits a decompression algorithm again optimal in time to the best of our knowledge this is the first boosting technique displaying these properties technically our boosting technique builds upon three main ingredients the burrows wheeler transform the suffix tree data structure and a greedy algorithm to process them specifically we show that there exists a proper partition of the burrows wheeler transform of a string s that shows a deep combinatorial relation with the kth order entropy of s that partition can be identified via a greedy processing of the suffix tree of s with the aim of minimizing a proper objective function over its nodes the final compressed string is then obtained by compressing individually each substring of the partition by means of the base compressor we wish to boost our boosting technique is inherently combinatorial because it does not need to assume any prior probabilistic model about the source emitting s and it does not deploy any training parameter estimation and learning various corollaries are derived from this main achievement among the others we show analytically that using our booster we get better compression algorithms than some of the best existing ones that is lz77 lz78 ppmc and the ones derived from the burrows wheeler transform further we settle analytically some long standing open problems about the algorithmic structure and the performance of bwt based compressors namely we provide the first family of bwt algorithms that do not use move to front or symbol ranking as a part of the compression process

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

theoretical estimation of the probability of weight rank reversal in pairwise comparisons abstract pairwise comparison is a key component in multi criteria decision making the probability of rank reversal is a useful measure for evaluating the impact of uncertainty on the final outcome in the context of this paper the type of uncertainty considered is related to the fact that experts have different opinions or that they may perform inconsistent pairwise comparisons we provide a theoretical model for estimating the probability of the consequent rank reversal using the multivariate normal cumulative distribution function the model is applied to two alternative weight extraction methods frequently used in the literature the geometric mean and the eigenvalue method we introduce a reasonable framework for incorporating uncertainty in the decision making process and calculate the mean value and cross correlation of the average weights which are required in the application of the model the theoretical results are compared against numerical simulations and a very good agreement is observed we further show how our model can be extended in applications of a full multi criteria decision making analysis such as the analytic hierarchy process we also discuss how the theoretical model can be used in practice where the statistical properties of the uncertainty induced perturbations are unknown and the only information provided by the pairwise comparison matrices of a small group of experts the methodology presented here can be used to extend the pairwise comparison framework in order to provide some information on the credibility of its outcome

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

robust estimation of priorities in the ahp abstract a pairwise comparison matrix of the analytic hierarchy process ahp is considered as a contingency table that helps to identify unusual or false data elicited from a judge special techniques are suggested for robust estimation of priority vectors they include transformation of a saaty matrix to matrix of shares of preferences and solving an eigenproblem designed for the transformed matrices we also introduce an optimizing objective that produces robust priority estimation numerical results are compared using the ahp with these differing approaches the comparison demonstrates that robust estimations yield priority vectors not prone to influence of possible errors among the elements of a pairwise comparison matrix

almost sure optimality of sliding window lempel ziv algorithm and variants revisited the sliding window lempel ziv swlz algorithm has been studied extensively in the information theory literature and has been used in several commercial compression packages the asymptotic behavior of this algorithm has been studied by wyner and ziv and by shields using different approaches shields results proving asymptotic optimality for all individual sequences with respect to the class of all finite state encoders is known to imply the almost sure asymptotic optimality for the class of stationary and ergodic sources we establish the almost sure asymptotic optimality without any mixing conditions by extending the ideas of wyner and ziv using results on recurrence times from ergodic theory a nonuniversal fixed shift version of the algorithm is introduced to gain more insight into its behavior pointwise optimality results are also obtained for several variants based on match length functions introduced by gavish and lempel we also propose variants of the algorithm for sources with countable alphabet and demonstrate their pointwise optimality

modelling pairwise comparisons on ratio scales abstract many real world decision problems have multiple criteria on qualitative domains given the expectation that more and more of such problems will be modelled as mcdm problems involving scoring on subjective qualitative domains then there is a class of significant problems for which the evaluator will require an evaluation framework which handles occurrences of seeming intransitivity and inconsistency these phenomena are likely to appear in qualitative subjective domains where the decision making environment is characterized by ambiguity and vagueness in this paper we present a model for exploring the implications of pairwise comparisons on ratio scales this model provides a mathematical explanation for the phenomena of intransitivity and inconsistency that sometimes appear in situations involving scoring on a subjective qualitative domain using this framework we develop methods for generating consistent intervals on ratio scales we also present a two stage action learning process for generating consistent points on ratio scales that provides both behavioral and mathematical convergence

a new methodology for sensitivity and stability analysis of analytic network models abstract in this paper we develop a methodology to study the sensitivity and the stability of models built using the analytic network process we study two types of stability core and solution stability the former deals with finding the region of the perturbation space in which the initial solution i e the alternative that is ranked first obtained from the anp model remains most preferred the latter deals with finding the regions of the perturbation space in which the solutions that were not initially most preferred i e alternatives that were not ranked first become most preferred i e they are ranked first the methodology consists of three stages generation of the perturbation space finding the boundaries of the regions in the perturbation space in which the different alternatives are ranked first and finding the stability regions

a multi criteria agency model with incomplete preference information abstract we analyze an agency model in which the principal has only partial information on the agent s preferences consequently the principal can predict the agent s reaction to an incentive system only stochastically we first present the analytical solution for a simple decision problem in which the agent can choose only between two alternatives from this simple model we derive several hypotheses on how the preference information available to the principal influences her net results these hypotheses are then tested via a simulation model for more complex decision problems and we show that not all results of the simple model hold for the more general case most notably it is not generally the case that more precise information on the agent s preferences improves the principal s situation

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

a riemannian symmetric rank one trust region method the well known symmetric rank one trust region method where the hessian approximation is generated by the symmetric rank one update is generalized to the problem of minimizing a real valued function over a d d dimensional riemannian manifold the generalization relies on basic differential geometric concepts such as tangent spaces riemannian metrics and the riemannian gradient as well as on the more recent notions of first order retraction and vector transport the new method called rtr sr1 is shown to converge globally and d 1 d 1 step q superlinearly to stationary points of the objective function a limited memory version referred to as lrtr sr1 is also introduced in this context novel efficient strategies are presented to construct a vector transport on a submanifold of a euclidean space numerical experiments rayleigh quotient minimization on the sphere and a joint diagonalization problem on the stiefel manifold illustrate the value of the new methods

eigenvector and logarithmic least squares abstract the eigenvector method deals with two questions simultaneously closeness and order together they belong to the field topology of order the metric idea of closeness is inadequate to judge what is a good approximation to data involving order relations there is usually a condition which relates order in the derived scale to order in the data this is precisely the kind of condition captured by the eigenvector but not by the logarithmic least squares estimate

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

on solving large scale limited memory quasi newton equations abstract we consider the problem of solving linear systems of equations arising with limited memory members of the restricted broyden class of updates and the symmetric rank one sr1 update in this paper we propose a new approach based on a practical implementation of the compact representation for the inverse of these limited memory matrices numerical results suggest that the proposed method compares favorably in speed and accuracy to other algorithms and is competitive with several update specific methods available to only a few members of the broyden class of updates using the proposed approach has an additional benefit the condition number of the system matrix can be computed efficiently

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

geodesic finite elements for cosserat rods we introduce geodesic finite elements as a new way to discretize the non linear configuration space of a geometrically exact cosserat rod these geodesic finite elements naturally generalize standard one dimensional finite elements to spaces of functions with values in a riemannian manifold for the special orthogonal group our approach reproduces the interpolation formulas of crisfield and jelenic geodesic finite elements are conforming and lead to objective and path independent problem formulations we introduce geodesic finite elements for general riemannian manifolds discuss the relationship between geodesic finite elements and coefficient vectors and estimate the interpolation error then we use them to find static equilibria of hyperelastic cosserat rods using the riemannian trust region algorithm of absil et al we show numerically that the discretization error depends optimally on the mesh size copyright 2009 john wiley sons ltd

statistical quasi newton a new look at least change a new method for quasi newton minimization outperforms bfgs by combining least change updates of the hessian with step sizes estimated from a wishart model of uncertainty the hessian update is in the broyden family but uses a negative parameter outside the convex range that is usually regarded as the safe zone for broyden updates although full newton steps based on this update tend to be too long excellent performance is obtained with shorter steps estimated from the wishart model in numerical comparisons to bfgs the new statistical quasi newton sqn algorithm typically converges with about 25 fewer iterations functions and gradient evaluations on the top 1 3 hardest unconstrained problems in the cute library typical improvement on the 1 3 easiest problems is about 5 the framework used to derive sqn provides a simple way to understand differences among various broyden updates such as bfgs and dfp and shows that these methods do not preserve accuracy of the hessian in a certain sense while the new method does in fact bfgs dfp and all other updates with nonnegative broyden parameters tend to inflate hessian estimates and this accounts for their observed propensity to correct eigenvalues that are too small more readily than eigenvalues that are too large numerical results on three new test functions validate these conclusions

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

reciprocal distributions in the analytic hierarchy process abstract the paper analyses a special kind of probability distributions used to capture the existing uncertainty for the judgements in the analytic hierarchy process ahp these distributions called reciprocal satisfy the reciprocity axiom of the ahp methodology first we define and characterise the reciprocal distributions and present some results and properties associated with them we then study the relationship between the reciprocal and symmetrical random variables and the distribution of the priorities vector considering the eigenvector method egv in the consistent case and the row geometric mean method rgm as priorization methods and assuming that the judgements follow reciprocal distributions we prove that the elements of the priorities vector are distributed according to reciprocal random variables the knowledge of these distributions will allow us to obtain the probability of two interesting events in practical selection problems the probability that a given alternative would be ranked in the first position and the probability of a given ranking preference structure for the whole set of alternatives in particular we find the analytic expressions for these probabilities when a reciprocal random variable such as the lognormal is used

interval estimation of priorities in the ahp this paper extends and modifies the analytic hierarchy process ahp and the synthetic hierarchy method shm of priority estimation to accommodate random data in the pairwise comparison matrices it employs a cauchy distribution to describe the pairwise comparison of alternatives in saaty matrices and shows how to modify these matrices in order to handle random data the use of random data yields saaty matrices that are not reciprocally symmetrical several variants of the ahp are then modified i to accommodate reciprocally asymmetric matrices and ii to allow each priority estimate to be expressed on an interval of possible values rather than as a single discrete point the merits of interval estimation are illustrated by an example

repair and all irreducible grammars are upper bounded by high order empirical entropy irreducible grammars are a class of context free grammars with well known representatives such as repair with a few tweaks longest match greedy and sequential we show that a grammar based compression method described by kieffer and yang 2000 is upper bounded by the high order empirical entropy of the string when the underlying grammar is irreducible specifically given a string s over an alphabet of size sigma we prove that if the underlying grammar is irreducible then the length of the binary code output by this grammar based compression method is bounded by s h k s o s log sigma for any k in o log sigma s where h k s is the k order empirical entropy of s this is the first bound encompassing the whole class of irreducible grammars in terms of the high order empirical entropy with coefficient 1

descent algorithms on oblique manifold for source adaptive ica contrast a riemannian manifold optimization strategy is proposed to facilitate the relaxation of the orthonormality constraint in a more natural way in the course of performing independent component analysis ica that employs a mutual information based source adaptive contrast function despite the extensive development of manifold techniques catering to the orthonormality constraint only a limited number of works have been dedicated to oblique manifold ob algorithms to intrinsically handle the normality constraint which has been empirically shown to be superior to other riemannian and euclidean approaches imposing the normality constraint implicitly in line with the ica definition essentially guarantees a substantial improvement in the solution accuracy by way of increased degrees of freedom while searching for an optimal unmixing ica matrix in contrast with the orthonormality constraint designs of the steepest descent conjugate gradient with hager zhang or a hybrid update parameter quasi newton and cost effective quasi newton methods intended for ob are presented in this paper their performance is validated using natural images and systematically compared with the popular state of the art approaches in order to assess the performance effects of the choice of algorithm and the use of a riemannian rather than euclidean framework we surmount the computational challenge associated with the direct estimation of the source densities using the improved fast gauss transform in the evaluation of the contrast function and its gradient the proposed ob schemes may find applications in the offline image signal analysis wherein on one hand the computational overhead can be tolerated and on the other the solution quality holds paramount interest

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 graphical analysis of ratio scaled paired comparison data one of the building blocks of the analytic hierarchy process saaty saaty t l 1977 a scaling method for priorities in hierarchical structures math psych 15 234 281 is the ratio scaled assessment of an agent s preferences between pairs of alternatives this article shows how it is possible to visualize such data and to detect cardinal and ordinal inconsistencies in the respondent s judgments the graphical method used to display preferences is not new in itself it dates back at least to the work of gower gower j c 1977 the analysis of asymmetry and orthogonality j r barra f brodeau g romier b van cutsem eds recent developments in statistics north holland amsterdam 109 123 and many variants of his technique can be found in the literature but gower plots provide an especially valuable diagnostic tool in the context of ahp as discussed and abundantly illustrated herein

a thurstonian view of the analytic hierarchy process abstract the application of deterministic decision models in situations characterized by noise and uncertainty is likely to produce results of questionable value in this paper some very simple probabilistic models are developed and substituted for the deterministic scales used in the analytic hierarchy process ahp it is shown that the use of these probabilistic models can extend the domain of ahp to situations such as consensual or group decision making that possess significant amounts of uncertainty in addition explicit measures of the variation present in the evaluation of decision alternatives and attributes are obtained

visualizing and rectifying different inconsistencies for fuzzy reciprocal preference relations abstract consistency is related with reasonableness this paper proposes a graphical method to visualize and rectify different inconsistencies for fuzzy reciprocal preference relations fprs and the consistency includes the ordinal and cardinal consistencies ordinal inconsistency implies cardinal inconsistency but not vice versa and cardinal consistency of fprs includes additive consistency and multiplicative consistency therefore there are four types of inconsistencies includes 1 the additive inconsistency 2 the multiplicative inconsistency 3 the ordinal and additive inconsistencies and 4 the ordinal and multiplicative inconsistencies first gower plots are proposed to visualize the ordinal and cardinal inconsistencies on a two dimensional plane then some optimization models are formulated to rectify the four different types of inconsistences respectively some examples are illustrated to show how the models work and comparisons with the existing methods are also offered to show the advantages of the proposed methods

automatic preconditioning by limited memory quasi newton updating this paper proposes a preconditioner for the conjugate gradient method cg that is designed for solving systems of equations ax bi with different right hand side vectors or for solving a sequence of slowly varying systems ak x bk the preconditioner has the form of a limited memory quasi newton matrix and is generated using information from the cg iteration the automatic preconditioner does not require explicit knowledge of the coefficient matrix a and is therefore suitable for problems where only products of a times a vector can be computed numerical experiments indicate that the preconditioner has most to offer when these matrix vector products are expensive to compute and when low accuracy in the solution is required the effectiveness of the preconditioner is tested within a hessian free newton method for optimization and by solving certain linear systems arising in finite element models

linking validation a search for coherency within the supermatrix abstract the consistency check within each pairwise comparison matrix is an important step in an analytic network process anp decision in an anp network there is both the ability and the need to test for additional levels of consistency or coherency among the priority vectors examples are used to highlight cases where a supermatrix with priority vectors that were obtained from either perfect or nearly perfect consistent pairwise comparison matrices generates suboptimal decisions simulations are used to further demonstrate the frequency of these occurrences in general anp networks a form of cross validation within the supermatrix called linking validation is developed and demonstrated the linking validation method allows decision makers to use the priority vectors within the supermatrix to validate other priority vectors within the supermatrix the linking validation method involves generating linking estimates the linking estimates are compared against each other to identify the most incoherent priority vector by calculating the linking coherency index lci scores the decision maker can then update the specified priority vector and repeat this process until the lci score for every linking estimate is below the given threshold the use of linking validation to test for coherency further improves the validity of anp models

quasi newton methods on grassmannians and multilinear approximations of tensors in this paper we proposed quasi newton and limited memory quasi newton methods for objective functions defined on grassmannians or a product of grassmannians specifically we defined bfgs and limited memory bfgs updates in local and global coordinates on grassmannians or a product of these we proved that when local coordinates are used our bfgs updates on grassmannians share the same optimality property as the usual bfgs updates on euclidean spaces when applied to the best multilinear rank approximation problem for general and symmetric tensors our approach yields fast robust and accurate algorithms that exploit the special grassmannian structure of the respective problems and which work on tensors of large dimensions and arbitrarily high order extensive numerical experiments are included to substantiate our claims

capturing group preferences in a multicriteria decision abstract this paper examines the extent to which methods used for analyzing judgments from a group of decision makers result in preference rankings which are consistent with the group member s subjective beliefs about their preferences in a laboratory group decision setting the analytic hierarchy process a newly developed probabilistic extension to the ahp and two pairwise voting methods are used to develop separate preference rankings based on input from the group members the probabilistic version of the ahp in its first independent test is found to provide more insight into the group decision while requiring fewer a priori assumptions a commonly used and easily implemented pairwise voting procedure is found to be significantly inferior to the other methods in its ability to capture the group s preference rankings this underscores the importance of using appropriate decision models for developing a full understanding of group preferences

consequences of uncertainty in the analytic hierarchy process a simulation approach a methodology for the propagation of uncertainty in the analytic hierarchy process ahp is described the sole source of the uncertainty is assumed to lie in the elements of the preference matrices the uncertainty in decision alternative rankings and the probability of a rank reversal as functions of the number of alternatives and hierarchy depth are examined ranking uncertainty is found to decrease as the number of alternatives or the depth of the hierarchy is increased the observed behaviour of ranking uncertainty is partially explained using the central limit theorem

pick s inequality and tournaments abstract let t be a tournament on n nodes and let a be its adjacency matrix a brauer and i gentry observed that an inequality due to g pick implies that im 1 2 cot 2n for all eigenvalues of a we say that t is a pick or p tournament if equality holds for some we determine when equality holds in pick s inequality for arbitrary real matrices and use this to show that the p tournaments can all be constructed from the transitive tournament m by reversing the arcs between the sets of certain node partitions or cuts u the cuts are specified by 1 n vectors u such that utw 0 where w 1 n 1 t e i n this links the cuts to cyclotomic polynomials there is at least one p tournament on n nodes if and only if n 2k k 1 up to isomorphism there is precisely one p tournament on n nodes if and only if n p or n 2p for some odd prime p for odd n up toisomorphism the only regular p tournament on n nodes has matrix zn circ 0 1 1 0 0 a composition rule is used on the matrices zp to form all p tournaments on n 2kpl nodes l 1

an approach to using the analytic hierarchy process for solving multiple criteria decision making problems abstract in this paper we present a procedure for solving multiple criteria decision making problems our procedure which is an extension of the analytic hierarchy process addresses the issue of fuzziness in the weighing information provided by the decision maker the consequent stability of the ahp ranking of the alternatives and the sensitivity of such rankings to small perturbations these issues are especially important when the concept of optimality incorporates parameters or weights that reflect subjective preferences the procedure provides a consistent and systematic method for carrying out goal seeking sensitivity analysis that captures the decision maker s preference structure using his her indifference region