fixed points of weakly inward multivalued maps on considere le probleme suivant etant donnes un espace de banach reel x un convexe borne clos k x et f k 2 x o satisfaisant f est semi continue superieurement f x est convexe clos x k f est condensante et f x k x o x k f a t elle un point fixe

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

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

saint venant end effects in multilayered piezoelectric laminates the saint venant end effects and stress decay in multilayered laminates of piezoelectric materials are studied a state space approach is developed in the context of generalized plane strain by means of matrix algebra and transfer matrix the characteristics of the stress decay in 2 d piezoelectric strips and laminates are examined their elastic counterparts are included as special cases evaluation of the characteristic decay length for typical materials shows that the saint venant end effects are significant it also indicates that the analysis based on the plane strain assumption is misleading

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

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

equilibria of set valued maps on nonconvex domains we present new theorems on the existence of equilibria or zeros of convex as well as nonconvex set valued maps defined on compact neighborhood retracts of normed spaces the maps are subject to tangency conditions expressed in terms of new concepts of normal and tangent cones to such sets among other things we show that if k is a compact neighborhood retract with nontrivial euler characteristic in a banach space e and 1 k 2e is an upper hemicontinuous set valued map with nonempty closed convex values satisfying the tangency condition x n tk x 0 for all x c k then there exists xo c k such that 0 c 4 xo here tk x denotes a new concept of retraction tangent cone to k at x suited for compact neighborhood retracts when k is locally convex at x tk x coincides with the usual tangent cone of convex analysis special attention is given to neighborhood retracts having lipschitzian behavior called l retracts below this class of sets is very broad it contains compact homeomorphically convex subsets of banach spaces epi lipschitz subsets of banach spaces as well as proximate retracts our results thus generalize classical theorems for convex domains as well as recent results for nonconvex sets

on new symplectic approach for exact bending solutions of moderately thick rectangular plates with two opposite edges simply supported the symplectic geometry method is introduced for exact bending solutions of moderately thick rectangular plates with two opposite edges simply supported the basic equations for the plates are first transferred into hamilton canonical equations the whole state variables are then separated using the method of eigenfunction expansion in the symplectic geometry typical examples for plates with selected boundary conditions are solved and exact bending solutions obtained since only the basic elasticity equations of the plates are used this method eliminates the need to pre determine the deformation function and is hence more reasonable than conventional methods numerical results were presented to demonstrate the validity and accuracy of this approach as compared to those reported in other literatures

the behaviour of elastic surface waves polarized in a plane of material symmetry i general analysis in all reported instances of surface wave transmission in anisotropic elastic media in which the speed of propagation is supersonic in relation to the slowest branch of homogeneous plane waves the reference plane of the surface wave is a plane of material symmetry of the medium and the displacement is polarized in this plane it is proved herein that whenever the symmetry group of the transmitting material admits a plane of reflexional symmetry a surface wave having as its reference plane excites no displacement orthogonal to and subject to specified conditions may travel with a speed in the supersonic range a general existence uniqueness theorem for such waves is established and the determination of the displacement field and the traction on planes parallel to the boundary is shown to require only the evaluation of four definite integrals the location of the real zero of a combination of these integrals and the solution of a quartic equation the analysis developed in part i is applied in parts ii and iii to media with monoclinic orthorhombic and cubic symmetries

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

on rayleigh waves in incompressible orthotropic elastic solids in this paper the secular equation for the rayleigh wave speed in an incompressible orthotropic elastic solid is obtained in a form that does not admit spurious solutions it is then shown that inequalities on the material constants that ensure positive definiteness of the strain energy function guarantee existence and uniqueness of the rayleigh wave speed finally an explicit formula for the rayleigh wave speed is obtained

3d symplectic expansion for piezoelectric media the symplectic series expansion method is extended to three dimensional problem for transversely isotropic piezoelectric media the governing equations are first derived in hamiltonian form and symplectic eigensolutions are directly obtained through analytical method all solutions of the problem are reduced to finding eigenvalues and eigensolutions the classical st venant solutions are described by zero eigensolutions and the localized solutions are depicted by non zero eigensolutions symplectic relationships of the ortho normalization are used end conditions are rewritten by eigensolutions and a numerical scheme is formed analytically some numerical examples are given copyright 2007 john wiley sons ltd

the mode iii stress electric intensity factors and singularities analysis for edge cracked circular piezoelectric shafts abstract this paper introduces a new analytical method to determine the stress and electric intensity factors for edge cracked circular piezoelectric shafts using hamiltonian formalism the singularities near the crack tip are represented in terms of exponential series that can show the boundary layer effects effectively a symplectic system is established directly by introducing dual vectors in terms of the symplectic eigenfunctions the coefficients of the series are determined from the lateral boundary conditions along the crack faces and the outer boundary conditions along the exterior geometric domain the intensity factors are determined by the first two coefficients of non zero eigenvalue solutions numerical examples for various boundary conditions are given the influencing parameters on the intensity factors are investigated

the poincare miranda theorem and viability condition abstract the aim of this note is to discuss the relation between the assumptions of the poincare miranda theorem and the viability condition first used by nagumo to prove existence of a solution to odes under state constraints viable solutions an interesting consequence of this observation is an extension of the poincare miranda theorem to arbitrary convex compact sets in locally convex hausdorff vector spaces instead of a parallelotope in an euclidean space we also recall a very short proof of this extension based on a ky fan inequality this proof is not new but seems to have passed unnoticed in the literature devoted to the poincare miranda theorem in fact recent variations of this theorem in l 2 follow then in a simple straightforward way the above extension also implies a generalization of the lax intermediate value theorem to infinite dimensional spaces

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

the behaviour of elastic surface waves polarized in a plane of material symmetry i addendum this addition to a recent paper by chadwick proc r soc lond a 430 213 1990 hereafter referred to as part i has been prompted mainly by the discovery of secluded supersonic surface waves propagating in configurations of transversely isotropic elastic media in which the reference plane is not a plane of material symmetry and coexisting with a subsonic surface wave the occurrence of a supersonic surface wave travelling in a direction e 1 with speed v text s implies that there are two homogeneous plane waves with slowness vectors s text i and s text r such that s text i cdot e 1 s text r cdot e 1 v text s 1 which comprise the incident and reflected waves in a case of simple reflection at the traction free boundary supersonic surface waves may therefore be found by searching within a suitably defined space of simple reflection boldsymbol scr r this is the approach which has led to the new results mentioned above and the principal conclusions of part i are re examined here from the same point of view it is found that whereas the secluded supersonic surface waves in transversely isotropic media correspond to isolated points on a curvilinear projection of boldsymbol scr r which does not intersect the curve representing subsonic surface waves the symmetric surface waves studied in part i define a curve which may lie partly inside and partly outside a projection of boldsymbol scr r in the form of a region the interior points representing supersonic and the exterior points subsonic surface waves this discussion is preceded by a simplification of the existence uniqueness theorem proved in part i and followed by a reconsideration of the possibility that an inhomogeneous plane elastic wave can qualify as a surface wave such one component surface waves do exist but a symmetric surface wave necessarily contains two inhomogeneous plane waves

an experimental comparison of different approaches to determining weights in additive utility models several studies this past decade have examined differences between holistic and decomposed approaches to determining weights in additive utility models some have argued that it matters little which procedure is used whereas others strongly favored particular methods in this paper we address this controversy experimentally by comparing five conceptually different approaches in terms of their weights and predictive ability the five methods are 1 multiple linear and non linear regression analyses of ten and fifteen holistic assessments 2 direct decomposed tradeoffs as proposed by keeney and raiffa keeney r l h raiffa 1977 decisions with multiple objectives wiley new york 3 a recent eigen vector technique of saaty saaty t l 1977 a scaling method for priorities in hierarchical structures j math psych 15 3 234 281 involving redundant pairwise comparisons of attributes 4 a straightforward allocation of hundred importance points and 5 unit weighting i e equal weighting after standardizing the attributes the decision task involved college admissions subjects were asked to evaluate hypothetical college applicants on the basis of verbal sat quantitative sat high school grade point average and a measure of extra curricular activity linear as well as nonlinear attribute utility functions were used in constructing the additive models the nonlinear functions were specified graphically by the subjects through selection from five different shapes i e one per attribute to test the predictive ability of the various models each subject made twenty separate pairwise comparisons of alternatives including direction and strength of preference the prediction criteria were percentage correct predictions as well as correlations using these twenty pairs seventy subjects were tested using an order controlled within subject design in comparing the different methods of weight determination monetary incentives were used to enhance motivation in terms of findings the methods generally differed systematically concerning the weights given to the various attributes as well as the variances of the resulting predictions on average however the methods predicted about equally well except for unit weighting which was clearly inferior the findings differ in this regard from the general literature furthermore nonlinear models were found to be inferior to linear ones finally subjects judged the methods to differ significantly in difficulty and trustworthiness which were found to correlate inversely the overall results raise various applied and theoretical issues which are discussed

determination of stress intensity factors by the finite element discretized symplectic method abstract when rewriting the governing equations in hamiltonian form analytical solutions in the form of symplectic series can be obtained by the method of separation of variable satisfying the crack face conditions in theory there exists sufficient number of coefficients of the symplectic series to satisfy any outer boundary conditions in practice the matrix relating the coefficients to the outer boundary conditions is ill conditioned unless the boundary is very simple e g circular in this paper a new two level finite element method using the symplectic series as global functions while using the conventional finite element shape functions as local functions is developed with the available classical finite elements and symplectic series the main unknowns are no longer the nodal displacements but are the coefficients of the symplectic series since the first few coefficients are the stress intensity factors post processing is not required a number of numerical examples as well as convergence studies are given

the polarization vector and secular equation for surface waves in an anisotropic elastic half space abstract the displacement at the free surface of an anisotropic elastic half space x2 0 generated by a surface wave propagating in the direction of the x1 axis traces an elliptic path it is represented by the polarization vector a r e 1 i e 2 where e 1 e 2 are the conjugate radii of the ellipse on the polarization plane the displacement traces the ellipse in the direction from e 1 to e 2 we present explicit expressions of e 1 e 2 and the secular equation without computing the stroh eigenvalues p and the associated eigenvectors a and b after presenting the expressions for a general anisotropic elastic material the special cases are studied separately for monoclinic materials with the symmetry plane at x3 0 the secular equation and the conjugate radii e 1 e 2 are identical to that for orthotropic materials when s 16 0 but s 26 need not vanish for monoclinic materials with the symmetry plane at x1 0 e 1 is along the x1 axis while e 2 is on the plane x1 0 if the symmetry plane is at x2 0 e 1 is on the plane x2 0 while e 2 is along the negative x2 axis in both cases e 1 e 2 are the principal radii of the ellipse we also present the derivative of a r with respect to x2 the depth from the free surface that provides information on i whether the conjugate radii of the ellipse increase as x2 increases and ii whether the polarization plane rotates as x2 increases new secular equations are obtained for monoclinic materials with the symmetry plane at x1 0 or x2 0

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

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

a new pcca method idra abstract idra intercriteria decision rule approach a new mcda multiple criteria decision aid method which adopts the pcca pairwise criterion comparison approach methodolgoy is presented it is based on two principal hypotheses about intercriteria information furnished by the decision maker the mixed utility function i e it is assumed that in the decision process both tradeoff and importance intercriteria information are considered and bounded consistency i e no constraint is imposed on the consistency of the intercriteria information

the analytic hierarchy process in an uncertain environment a simulation approach abstract traditionally decision makers were forced to converge ambiguous judgments to a single point estimate in order to describe a pairwise relationship between alternatives relative to some criterion for use in the analytic hierarchy process ahp since many circumstances exist which make such a convergence difficult confidence in the outcome of an ensuing ahp synthesis may be reduced likewise when a group of decision makers cannot arrive at a consensus regarding a judgment some members of the group may simply lose confidence in the overall synthesis if they lack faith in some of the judgments the ahp utilizes point estimates in order to derive the relative weights of criteria sub criteria and alternatives which govern a decision problem however when point estimates are difficult to determine distributions describing feasible judgments may be more appropriate using simulation we will demonstrate that levels of confidence can be developed expected weights can be calculated and expected ranks can be determined it will also be shown that the simulation approach is far more revealing than traditional sensitivity analysis

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

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

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

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

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

a unified formalism for elastostatics or steady state motion of compressible or incompressible anisotropic elastic materials abstract for a two dimensional deformation of an anisotropic elastic material the lekhnitski formalism applies to elastostatics only it does not apply to steady state motion such as surface waves or a moving line dislocation the stroh formalism applies to elastostatics as well as steady state motion however it does not apply to an incompressible material because some of the elastic stiffnesses become unbounded in the incompressible limit modifying a new formalism studied recently a more explicit and remarkably simple expression of the matrices in the new formalism is given and a unified formalism that is applicable to elastostatics as well as steady state motion of a compressible or incompressible anisotropic elastic material is obtained as an application the secular equation is presented for surface waves propagating in a monoclinic material with the symmetry plane at x 3 0 that has a more explicit expression than what is available in the literature for a general anisotropic elastic material explicit expressions are obtained for the sextic equation for the eigenvalues p and the eigenvectors b and a for any steady state wave speed the explicit expressions recover results related to one component surface waves when the material is incompressible it is shown that the eigenvalues p remain complex for elastostatics we also show that when the material is incompressible a monoclinic material with the symmetry plane at x 3 0 behaves like that of an orthotropic material

explicit secular equations for surface waves in monoclinic materials with the symmetry plane x1 0 x2 0 or x3 0 the traditional way of deriving the secular equation for surface waves propagating in the direction of the x1axis in an anisotropic elastic halfspace x20 is to find a general steadystate solution f

an extended rule based inference for general decision making problems complex decision making problems are usually modeled in terms of a number of decisive variables that are related hierarchically pieces of evidence are aggregated in a bottom up way to determine the final decision goal for dealing with such problems ishizuka et al 5 6 developed a rule based inference procedure by using fuzzy sets to represent imprecise data and employed the dempster shafer theory to aggregate evidence for structural damage assessment this paper highlights the necessity of applying the rules with all possible consequent clauses in the procedure with the considerations of information quality and quantity when applying the procedure to more general decision making problems the use of such rules can provide high quality information which contains much less ignorance in the resultant information also it can afford more information due to the decrease in total ignorance a simple example with two cases is used to compare the results based on the two different presentations of rules in the inference procedure for illustrating the importance of considering the rules proposed in this paper

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

fixed point theory and nonlinear problems introduction among the most original and far reaching of the contributions made by henri poincare to mathematics was his introduction of the use of topological or qualitative methods in the study of nonlinear problems in analysis his starting point was the study of the differential equations of celestial mechanics and in particular of their periodic solutions his work on this topic began with his thesis in 1879 and was developed in detail in his great three volume work methodes nouvelles de la mechanique celeste which appeared in the early 1890s and summarized his many memoirs of the intervening period it continued until his memoir shortly before his death in 1912 in which he put forward the unproved fixed point result usually referred to as poincare s last geometric theorem the ideas introduced by poincare include the use of fixed point theorems the continuation method and the general concept of global analysis the writer s acquaintance with poincare s influence came through contact with solomon lefschetz and marston morse both of whom were very explicit as to the role of poincare as an initiator in this direction of mathematical development in 1934 in the foreword to his colloquium volume on the calculus of variations in the large morse had put this forward very forcefully in the first paragraph

a state space formalism for piezothermoelasticity a state space formalism for electrothermoelastic analysis of a linear piezoelectric body is developed a novel feature of the formalism is that by proper grouping of the field variables and using matrix notations the three dimensional equations of piezothermoelasticity are concisely formulated into a state equation and an output equation which bear a striking resemblance to their elastic counterparts the formalism is remarkably simple with which one deals with only three vector quantities with the 13 independent electromechanical variables as their components and six submatrices that represent all the material constants for a piezoelectric material of the most general kind in this work emphasis is placed on the state space formulation and solution to the generalized plane problem exact solutions for a piezoelectric half space under a line of electromechanical loading and an infinite piezoelectric plate with an elliptical notch subjected to inplane loads are determined with relative ease in many cases piezoelectric solutions can be obtained directly from the corresponding elastic solutions by a simple replacement of the corresponding matrices on the basis of the formalism

relative measurement and the power function abstract relative measurement is considered in the context of nomology a theory that defines and relates categories of activity that arise in qualitative decision making a nomological variable is defined to correspond to the amount weight or importance of any naturally forming cluster or category of qualitative information this article shows that in general the mind measures nomological variables relatively and changes in a nomological variable relative to its current value the way the mind assesses the relationship between two related nomological variables generally should be modelled as a power function and it is appropriate to use a geometric mean to average multiple observations empirical evidence from psychophysics is shown to support these results when synthesising direct and indirect relative comparisons of nomological variables it is proven that the indirect comparisons should be given a weight that is inversely proportional to the number of relative measurements made in the indirect comparison this result is shown to be consistent with but not identical with the use of the row geometric means approach for synthesising pairwise relative comparison matrices such as in the analytical hierarchy process in the case of more than three variables other terms should be included in the synthesis the article reports on empirical research that endorses the use of nomological structuring and relative weighting of criteria trees it shows that decision makers generally prefer scoring within intervals when comparing alternatives but can also use relative measurement where there are difficulties with identifying intervals the former leads to modelling by an additive value function the latter to a power function

transitive matrices and their applications abstract transitive matrices and symmetrically reciprocal sr matrices are defined and spectral properties of certain sr perturbations of transitive matrices are studied the results are applied to a multicriteria decision making method the analytic hierarchy process ahp which uses an sr matrix with positive entries a proof is given that rank reversal is inherent in this method if its input matrix is perturbed

considerations of the existence of interfacial stoneley waves in bonded anisotropic elastic half spaces the questions of uniqueness and existence of subsonic stoneley waves in bonded anisotropic linear elastic half spaces are settled by using the notion of the interface impedance tensor which is a simple linear combination of the hermitian surface impedance tensors of the separate half spaces a definite existence criterion is presented in a form that proves most useful in numerical searches for stoneley waves in the sense that such searches need not be conducted when a stoneley wave mode

preference programming through approximate ratio comparisons abstract in the context of hierarchical weighting this paper operationalizes interval judgments which allow the decision maker to enter ambiguous preference statements by indicating the relative importance of factors as intervals of values on a ratio scale through such judgments the decision maker can capture the subjective uncertainty in his preferences and thus avoid the often cumbersome elicitation of exact ratio estimates after each new statement the interval judgments are synthesized into dominance relations on the alternatives by solving a series of linear programming problems this leads to an interactive process of preference programming which provides more detailed results as the decision maker gradually enters a more specific preference description moreover the overall effort of preference elicitation is smaller than in the analytic hierarchy process because the most preferred alternative can usually be identified before all possible comparisons between pairs of factors have been completed

the behaviour of elastic surface waves polarized in a plane of material symmetry iii orthorhombic and cubic media the results obtained in part ii for symmetric elastic surface waves in monoclinic media are specialized first to orthorhombic then to cubic symmetry a complete account is given of the configurations in which the s tensors and hence the surface wave function and the secular equation can be determined explicitly the theoretical developments are complemented by extensive numerical computations presented in the context of elastodynamic classifications of orthorhombic and cubic materials evolved by musgrave and chadwick smith respectively the transitions between subsonic and supersonic surface wave propagation encountered in part ii recur in considerable abundance but stand in a clear relationship to the classification only in the case of cubic media

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

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

the polarization vectors at the interface and the secular equation for stoneley waves in monoclinic bimaterials stoneley waves propagating in the direction of the x 1 axis in a bimaterial consisting of two half spaces x 2 0 and x 2 0 of dissimilar anisotropic elastic materials are considered several invariants independent of x 2 that relate the displacement and the stress are obtained for a general anisotropic elastic bimaterial we then study the case when both materials have the symmetry plane at x 2 0 or x 1 0 the displacement and the traction at the interface x 2 0 describe two separate elliptic paths they are polarized on planes that contain the x 2 axis or x 1 axis moreover the x 2 axis or x 1 axis is one of the principal axes of the ellipse in the rest of the paper we consider special monoclinic bimaterials with the symmetry plane at x 2 0 x 1 0 or x 3 0 the elastic constants for the two monoclinic materials are identical except that those elastic constants that would vanish if the material were orthotropic have the opposite sign for the two materials it is shown that one of the ellipses degenerates into a line along a coordinate axis while the other ellipse is on a coordinate plane normal to this coordinate axis when the symmetry plane is at x 3 0 both ellipses degenerate into lines along the x 1 and x 2 axes explicit expressions of the polarization vectors and the secular equation are presented for all three monoclinic bimaterials

a methodology for priority setting with application to software development process abstract in this paper we briefly review the basic ideas behind the analytic hierarchy process ahp based on these ideas we introduce the concept of comparison interval and propose a methodology based on stochastic optimization to achieve global consistency and to accommodate the fuzzy nature of the comparison process we also discuss systems with feedback for systems with inter component dependence we review the super matrix technique for systems with intra component dependence we propose a systematic approximation scheme to compute the weight vector for priority setting applications of software development process and a numerical example are given to illustrate our methodology and results

the general solution of three dimensional problems in piezoelectric media abstract the governing equations of three dimensional problems in transversely isotropic piezo electric material are composed of four second order partial differential equations in which the displacement components and electric potential functions are the essential unknowns in this paper the so called general solutions of these equations are obtained with the aid of a set of new potential functions which are used to express the components of displacements and the electric potential functions as an application of the general solution the problem of a half space piezoelectrics acted by a concentrated lateral shear force is solved

a comparison of the analytic hierarchy process and a simple multi attribute value function abstract the paper reviews the applicability of approaches to multiple criteria decision making to aiding in the selection of a preferred option from a short list of alternatives in the light of a wealth of information about those alternatives it is concluded that saaty s analytic hierarchy process and a simple multi attributed value function are the approaches best suited to this problem and the most widely used in practice the paper goes on to compare these approaches in detail from both a theoretical and a practical standpoint the underlying assumptions of each are examined and considered in the light of practical decision aiding the practical implications of the use of each approach are discussed reference is made to experimental work carried out by the author which has examined both practical and theoretical aspects of the use of each approach in isolation and in comparative studies the paper will be of relevance to anyone facing a decision involving a finite number of alternatives and multiple criteria and considering the use of a formal approach

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

interval priorities in ahp by interval regression analysis abstract this paper proposes an interval approach for obtaining interval weights of priorities in the analytic hierarchy process ahp reflecting inconsistency of pairwise comparison ratios given by a decision maker even if pairwise comparison values with respect to preference are given as crisp values the priority weights should be estimated as intervals because of a decision maker s uncertainty of judgments we deal with two types of comparison matrices represented as crisp and interval elements this allows some uncertainty of estimations made by a decision maker the proposed approach can obtain the estimated interval priorities since interval regression analysis is based on the concept of possibility the interval weights obtained by the proposed method can be interpreted as the possible ranges estimated from the given data the sum of widths of the obtained intervals can be regarded as the degree of inconsistency included in the given data it is emphasized that interval data of preference can obtain the lower and upper models for the estimated weight intervals

constrained topological degree and positive solutions of fully nonlinear boundary value problems abstract in the first part of the paper we provide a construction of an abstract homotopy invariant detecting zeros of maps of the form a f where a d a e is a densely defined m accretive operator in a banach space e and f u e is a tangent field defined on an open subset u of a neighborhood retract m being invariant with respect to the resolvents of a the construction is performed under the assumption that resolvents of a are completely continuous in the second part we derive index formulae for isolated zeros and apply them to show the existence of nontrivial positive steady state solutions for a class of nonlinear reaction diffusion equations and equations with one dimensional p laplacian with possibly non positive perturbations as well as some controlled neumann like problems

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

weight stability intervals in multicriteria decision aid abstract in this paper we study the stability of the results given by a wide class of mcda methods to changes of the weights of the criteria three aspects of stability are discussed in each case we propose to construct stability intervals for the weights of either single criteria or groups of criteria a necessary and sufficient condition for stability is also considered the use of these instruments is demonstrated with a numerical example

the boundary layer phenomena in two dimensional transversely isotropic piezoelectric media by exact symplectic expansion a separable variable method is introduced to find the exact homogeneous solutions of a two dimensional transversely isotropic piezoelectric media to handle general boundary conditions the usual method of separable variables for partial differentiation equations cannot be readily applicable due to the tangling of the unknowns and their derivatives introducing dual variables of stresses we obtain a set of first order hamiltonian equations whose eigensolutions are symplectic spanning over the solution space to cover all possible boundary conditions the solutions consist of two parts the first part is the derogative zero eigenvalue solutions of the saint venant type together with all their jordan chains the second part is the decaying non zero eigenvalue solutions describing the boundary layer effects the classical solutions are actually the zero eigenvalue solutions representing the simple extension bending equipotential field and the uniform electric displacement on the other hand the non zero eigenvalue solutions represent the localized solutions which are sensitive to the boundary conditions and are decaying rapidly with respect to the distance from the boundaries some rate of decay curves of the newly found non zero eigenvalue solutions are shown by numerical examples finally the complete boundary layer effects are quantified for the first time

completeness and nonuniqueness ofgeneral solutions of transversely isotropic elasticity abstract in this paper we give general solutions of transversely isotropic elasticity their completenessand nonuniqueness are proved we point out that famous lekhnitskii hu nowacki solutions and elliott lodge solutions are complete if the elastic region is z convex

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 modified lekhnitskii formalism a la stroh for anisotropic elasticity and classifications of the 6 6 matrix n the stroh formalism for two dimensional deformations of anisotropic elastic materials computes the eigenvalue p and the eigenvector a from an eigenrelation the vector b is then determined from a depending on the number of repeated eigenvalues p and the number of independent eigenvectors a b the system has it can be classified into six groups the lekhnitskii formalism has no eigenrelation to speak of we present a modified lekhnitskii formalism that computes the eigenvalue p and the eigenvector b from an eigenrelation the vector a is then determined from b thus the modified lekhnitskii formalism is a dual to the stroh formalism not only does the modified formalism enable us to do the classifications it is much simpler than using the stroh formalism the six groups are the sp ss d1 d2 ed and es groups the es group does not exist for a real material the ss group that has p 1 p 2 p 3 and the d2 group that has p 1 p 2 p 3 can be identified without computing the eigenvalues p and the eigenvectors a b we show that the repeated eigenvalue p 1 p 2 in the ss and d2 groups is simply a root of the quadratic equation l 2 0 we present an explicit expression of p 3 for the ss group the ed group that has three identical p can also be identified without computing p and a b however we do present an explicit expression of p we show that monoclinic materials with the symmetry plane at x 3 0 cannot belong to the ed group the identification of the sp and d1 groups is the only one that requires computation of p but not a b for special classes of materials however they can be identified without computing p in all cases the eigenvectors and the generalized eigenvectors are obtained explicitly

a spectrally formulated plate element for wave propagation analysis in anisotropic material a new spectrally formulated plate element is developed to study wave propagation in composite structures the element is based on the classical lamination plate theory recently developed method based on singular value decomposition svd is used in the element formulation along with this a new strategy based on the method of solving polynomial eigenvalue problem pep is proposed in this paper which significantly reduces human intervention and thus human error in the element formulation the developed element has an exact dynamic stiffness matrix as it uses the exact solution of the governing elastodynamic equation of plate in frequency wavenumber domain as the interpolating functions due to this the mass distribution is modeled exactly and as a result a single element captures the exact frequency response of a regular structure and it suffices to model a plate of any dimension thus the cost of computation is dramatically reduced compared to the cost of conventional finite element analysis the fast fourier transform fft and fourier series are used for inversion to time space domain this element is used to model plate with ply drops and to capture the propagation of lamb waves

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

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

on the convergence of multiattribute weighting methods abstract the convergent validity of five multiattribute weighting methods is studied in an internet experiment this is the first experiment where the subjects created the alternatives and attributes themselves each subject used five methods to assess attribute weights one version of the analytic hierarchy process ahp direct point allocation simple multiattribute rating technique smart swing weighting and tradeoff weighting they can all be used following the principles of multiattribute value theory furthermore smart swing and ahp ask the decision makers to give directly the numerical estimates of weight ratios although the elicitation questions are different in earlier studies these methods have yielded different weights our results suggest that the resulting weights are different because the methods explicitly or implicitly lead the decision makers to choose their responses from a limited set of numbers the other consequences from this are that the spread of weights and the inconsistency between the preference statements depend on the number of attributes that a decision maker considers simultaneously

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

general solutions for coupled equations for piezoelectric media abstract three general solutions are obtained for the coupled dynamic equations for a transversely isotropic piezoelectric medium these solutions are expressed in terms of the two functions and f where satisfies a second degree partial differential equation and f a sixth degree partial differential equation respectively if the terms concerning the derivatives of time are removed the results become three general solutions for the corresponding equilibrium equations in which the function f can be represented by functions fi i 1 2 3 each of which satisfies a second degree partial differential equation by utilizing a generalized almansi theorem and the solution wang and zheng int j solids structures32 105 115 1995 obtained is proved to be consistent with one case of one of the three general solutions when the constants e15 e31 e33 0 the piezo electric coupling is absent then two of the solutions reduce to the elasticity general solutions for a transversely isotropie medium one of which is the result hu acta scientia sin 2 2 145 151 1953 obtained the other one has not been published last the solution in the limiting explicit form for the problem for a half space with concentrated loads at the boundary is obtained by utilizing the general solutions

three dimensional eigenfunction analysis of a crack in a piezoelectric material abstract a three dimensional analysis of a semi infinite crack embedded in a transversely isotropic piezoelectric material was performed by means of the eigenfunction expansions method the results show the characteristic 1 r singular behavior of the stress tensor in the vicinity of the crack a similar behavior is also revealed for the electric field in addition the study demonstrates that coupling or decoupling of the mechanical and electrical variables is a function of the distance to the crack tip as well as of the crack orientation

a model for the relative importance of the criteria in the multiplicative ahp and smart abstract we define the relative importance of any pair of criteria as the substitution rate between the relative gains and losses of the alternatives when we move along an indifference curve under the geometric mean aggregation rule in the multiplicative ahp and under the arithmetic mean aggregation rule in smart the relative not the marginal substitution rate depends neither on the performance of the alternatives under the remaining criteria nor on the units of performance measurement hence it provides a sound argument for distributed decision making processes where those who judge the criteria are not the same actors as those who assess the performance of the alternatives the definition has a plausible basis in the psycho physical research on the relationship between physical stimuli and sensory responses which shows that human beings are sensitive not to marginal but to relative changes of the stimulus intensities

solutions for transversely isotropic piezoelectric infinite body semi infinite body and bimaterial infinite body subjected to uniform ring loading and charge abstract based on the fundamental solutions for transversely isotropic piezoelectric materials the fundamental solutions of axisymmetric problems are derived by integration and explicit expressions for three possible cases of different characteristic roots and multiple roots are all presented in the case of s 1 s 2 s 3 s 1 based on the greens functions for semi infinite piezoelectric body and bimaterial infinite piezoelectric body the greens functions for axisymmetric problems of semi infinite body and bimaterial infinite body are obtained taking pzt 4 as an example numerical computations are conducted by use of the fundamental solutions to axisymmetric problems comparison of the calculated results with those of fem shows good agreement between them

the bolzano mean value theorem and partial differential equations abstract we study the existence of solutions to abstract equations of the form 0 a u f u u k e where a is an abstract differential operator acting in a banach space e k is a closed convex set of constraints being invariant with respect to resolvents of a and a perturbation f satisfies a certain tangency condition such problems are closely related to the so called poincare miranda theorem being the multi dimensional counterpart of the celebrated bolzano intermediate value theorem in fact our main results should be regarded as infinite dimensional variants of bolzano and miranda poincare theorems along with single valued problems we deal with set valued ones yielding the existence of the so called constrained equilibria of set valued maps the abstract results are applied to show existence of strong steady state solutions to some weakly coupled systems of drift reaction diffusion equations or differential inclusions of this type in particular we get the existence of strong solutions to the dirichlet neumann and periodic boundary problems for elliptic partial differential inclusions under the presence of state constraints of different type certain aspects of the bernstein theory for bvp for second order ode are studied too no assumptions concerning structural coupling monotonicity cooperativity are undertaken

fixed points of weakly inward multis soit x un espace de banach reel k x convexe ferme borne f k 2 x semicontinue superieurement et condensant f x convexe ferme x k et f faiblement interieur sur k alors f a un point fixe

on the solution sets of differential inclusions and the periodic problem in banach spaces abstract in this paper the topological structure of the solution set of a constrained semilinear differential inclusion in a banach space e is studied it is shown that the set of all mild solutions with values in a closed and in general thin subset d e is an r set provided natural boundary conditions and appropriate geometrical assumptions on d which hold e g when d is convex are satisfied applications to the periodic problem and to the existence of equilibria are given

inconsistency and non additive capacities the analytic hierarchy process in the framework of choquet integration we examine the ahp in the framework of choquet integration and we propose an extension of the standard ahp aggregation scheme on the basis of the shapley values associated with the criteria in our model a measure of dominance inconsistency between criteria is defined in terms of the totally inconsistent matrix associated with the main pairwise comparison matrix of the ahp the measure of dominance inconsistency is then used to construct a non additive capacity whose associated shapley values reduce to the standard ahp priority weights in the consistency case in the general inconsistency case however the extended aggregation scheme based on the shapley weighted mean tends to attenuate resp emphasize the priority weights of the criteria with higher resp lower average dominance inconsistency with respect to the other criteria

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