on the structural stability of the euler voigt and navier stokes voigt models abstract we consider the euler voigt equations and the navier stokes voigt equations which are obtained by an inviscid regularization from the corresponding equations the main result we show is the structural stability of the system in terms of the variations of both viscosity and regularization parameters

least square completion and inconsistency repair methods for additively consistent fuzzy preference relations in this paper we explore the group decision making gdm problems with incomplete additively consistent fuzzy preference relations some properties of additively consistent fuzzy preference relations are also discussed a sufficient and necessary condition is proposed to keep the additive consistency of fuzzy preference relations methods for determining the priority weights of fuzzy preference relations are provided least square completion and inconsistency repair methods are developed to deal with incomplete and inconsistent fuzzy preference relations some numerical examples are also given to illustrate the proposed approaches

closed embeddings of c in c 2 part i we consider closed curves c c in the affine plane s c2 that admit a good or very good asymptote by this we mean a curve l c in s that in suitable coordinates for s is linear and tangent to c at infinity and meets c once or not at all at finite distance we classify these curves up to automorphism of s relying on the theory of open algebraic surfaces we first determine the possibilities for the singularities of c at infinity and then proceed to give explicit equations

a cosine maximization method for the priority vector derivation in ahp abstract the derivation of a priority vector from a pair wise comparison matrix pcm is an important issue in the analytic hierarchy process ahp the existing methods for the priority vector derivation from pcm include eigenvector method ev weighted least squares method wls additive normalization method an logarithmic least squares method lls etc the derived priority vector should be as similar to each column vector of the pcm as possible if a pair wise comparison matrix pcm is not perfectly consistent therefore a cosine maximization method cm based on similarity measure is proposed which maximizes the sum of the cosine of the angle between the priority vector and each column vector of a pcm an optimization model for the cm is proposed to derive the reliable priority vector using three numerical examples the cm is compared with the other prioritization methods based on two performance evaluation criteria euclidean distance and minimum violation the results show that the cm is flexible and efficient

modeling country risk ratings using partial orders abstract in order to evaluate the creditworthiness of various countries a learning model is induced from the 1998 standard and poor s country risk ratings using the 1998 values of nine economic and three political indicators this learning model allows the construction of a partially ordered set describing the relative superiority of countries on the basis of their creditworthiness and it is shown that the condorcet linear extensions of this poset match closely the s p ratings moreover the ratings derived from the model correlate highly with those of other rating agencies the model is shown to provide excellent ratings even when applied to the following years data or to the ratings of previously unrated countries rating changes implemented by s p in subsequent years resolved most of the few discrepancies between the constructed poset and s p s initial ratings

a group decision making model based on a generalized ordered weighted geometric average operator with interval preference matrices abstract this paper presents a model for a group decision making problem with interval preference matrices first a new definition of interval additive reciprocal matrices with multiplicative consistency is given transformation methods between interval additive and multiplicative reciprocal matrices are proposed and applied to homogenize interval preference matrices second the consistency of interval multiplicative reciprocal matrices is utilized to propose a generalized ordered weighted geometric averaging operator which permits the aggregation of interval multiplicative reciprocal matrices in such a way that more important weight is given to that with more consistency in order to avoid a misleading solution the consistency and acceptable consistency of the collective interval multiplicative reciprocal matrix are studied in detail finally a new algorithm is presented to solve the group decision making problem with interval preference matrices numerical results are carried out to illustrate the given definitions methods and algorithm respectively

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

optimal aggregation of fuzzy preference relations with an application to broadband internet service selection this paper investigates the aggregation of multiple fuzzy preference relations into a collective fuzzy preference relation in fuzzy group decision analysis and proposes an optimization based aggregation approach to assess the relative importance weights of the multiple fuzzy preference relations the proposed approach that is analytical in nature assesses the weights by minimizing the sum of squared distances between any two weighted fuzzy preference relations relevant theorems are offered in support of the proposed approach multiplicative preference relations are also incorporated into the approach using an appropriate transformation technique an eigenvector method is introduced to derive the priorities from the collective fuzzy preference relation the proposed aggregation approach is tested using two numerical examples a third example involving broadband internet service selection is offered to illustrate that the proposed aggregation approach provides a simple effective and practical way of aggregating multiple fuzzy preference relations in real life situations

nonlinear evolution equations and the euler flow abstract a new theorem on abstract nonlinear equations of evolution is proved as an application the existence uniqueness regularity and continuous dependence on the data are proved for the solution of the euler equation for incompressible fluids in a bounded domain in r m

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

the coolidge nagata conjecture let e p2 be a complex rational cuspidal curve contained in the projective plane the coolidge nagata conjecture asserts that e is cremona equivalent to a line that is it is mapped onto a line by some birational transformation of p2 the second author recently analyzed the log minimal model program run for the pair x 12d where x d p2 e is a minimal resolution of singularities and as a corollary he proved the conjecture in the case when more than one irreducible curve in p2 e is contracted by the process of minimalization we prove the conjecture in the remaining cases

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

note on some models for deriving the priority weights from interval fuzzy preference relations abstract deriving accurate interval weights from interval fuzzy preference relations is key to successfully solving decision making problems xu and chen 2008 proposed a number of linear programming models to derive interval weights but the definitions for the additive consistent interval fuzzy preference relation and the linear programming model still need to be improved in this paper a numerical example is given to show how these definitions and models can be improved to increase accuracy a new additive consistency definition for interval fuzzy preference relations is proposed and novel linear programming models are established to demonstrate the generation of interval weights from an interval fuzzy preference relation

singular limits in bounded domains for quasilinear symmetric hyperbolic systems having a vorticity equation abstract a short time existence theorem is proven for the initial boundary value problem for a class of quasilinear symmetric hyperbolic systems containing a constant coefficient spatial operator multiplied by a large parameter solutions u are shown to remain bounded independently of for a time independent of provided that u 0 and u l 0 are bounded independently of and certain structural conditions are satisfied this result is applied to the quasigeostrophic approximation of the shallow water equations and the constant pressure approximation in combustion and solutions for these cases are shown to converge to solutions of limiting equations as

the coolidge nagata conjecture part i let e p2 be a complex rational cuspidal curve contained in the projective plane and let x d p2 e be the minimal log resolution of singularities applying the log minimal model program to x 12d we prove that if e has more than two singular points or if d which is a tree of rational curves has more than six maximal twigs or if p2 e is not of log general type then e is cremona equivalent to a line i e the coolidge nagata conjecture for e holds we show also that if e is not cremona equivalent to a line then the morphism onto the minimal model contracts at most one irreducible curve not contained in d

a chi square method for obtaining a priority vector from multiplicative and fuzzy preference relations abstract decision makers dms preferences on decision alternatives are often characterized by multiplicative or fuzzy preference relations this paper proposes a chi square method csm for obtaining a priority vector from multiplicative and fuzzy preference relations the proposed csm can be used to obtain a priority vector from either a multiplicative preference relation i e a pairwise comparison matrix or a fuzzy preference relation or a group of multiplicative preference relations or a group of fuzzy preference relations or their mixtures theorems and algorithm about the csm are developed three numerical examples are examined to illustrate the applications of the csm and its advantages

aggregation of direct and indirect judgments in pairwise comparison matrices with a re examination of the criticisms by bana e costa and vansnick in a recent paper by bana e costa and vansnick c a bana e costa j c vansnick a critical analysis of the eigenvalue method used to derive priorities in ahp european journal of operational research 187 3 2008 1422 1428 analytic hierarchy process ahp particularly its eigenvector method em used for deriving priorities from pairwise comparison matrices was criticized for the violation of a so called condition of order preservation cop due to this violation the em was considered to have a serious fundamental weakness which makes the use of ahp as a decision support tool very problematic the consistency ratio cr index in the ahp was also criticized for its failure to act as an alert of this violation of cop in this paper we look into decision makers overall judgments which can be obtained through the aggregation of their direct and indirect judgments and then re examine bana e costa and vansnick s numerical examples with a detailed analysis to show the invalidity of their criticisms

singular value decomposition in additive multiplicative and logistic forms singular value decomposition svd is widely used in data processing reduction and visualization applied to a positive matrix the regular additive svd by the first several dual vectors can yield irrelevant negative elements of the approximated matrix we consider a multiplicative svd modification that corresponds to minimizing the relative errors and produces always positive matrices at any approximation step another logistic svd modification can be used for decomposition of the matrices of proportions when a regular svd can yield the elements beyond the zero one range while the modified svd decomposition produces all the elements within the correct range at any step of approximation several additional modifications of matrix approximation are also considered

deriving priorities from pairwise comparison matrices with a novel consistency index abstract it is important to measure the inconsistency level of a pairwise comparison matrix pcm and derive the priority vector in the analytic hierarchy process ahp in the present study a new consistency index is proposed by using the cosine similarity measures of two row column vectors in a pcm it is called as the double cosine similarity consistency index dcsci since the row and column vectors are all considered some interesting properties of dcsci are investigated and the thresholds for inconsistency tolerance level are discussed in detail then following the idea of dcsci we provide a new method for obtaining the priority vector from a pcm through maximizing the sum of the cosine similarity measures between the priority vector and the row column vectors the priority vector is derived by solving the constructed optimization problem by analyzing the proposed double cosine similarity maximization dcsm method it is found that the existing cosine maximization cm method can be retrieved by carrying out numerical examples some comparisons with the existing methods show that the proposed index and method are effective and flexible

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 well posedness theorem for non homogeneous inviscid fluids via a perturbation theorem on applique un theoreme de perturbation pour montrer le caractere bien pose d equations hyperboliques non lineaires application a une equation d ecoulement

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

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

multiplicative data envelopment analysis cross efficiency and stochastic weight space acceptability analysis for group decision making with interval multiplicative preference relations abstract to deal with group decision making gdm with interval multiplicative preference relations imprs this paper proposes a novel method based on multiplicative data envelopment analysis dea cross efficiency and stochastic weight space acceptability analysis we first develop a multiplicative dea model to evaluate the relative efficiency of all alternatives of a given multiplicative preference relation mpr then we present a method free from consistency adjustment to derive a priority vector using the multiplicative dea cross efficiency with respect to the given mpr for gdm with imprs we consider the decision makers weights as a uniform distribution for acceptability analysis a modified unacceptability index is further defined to measure the unlikeliness for a particular alternative in a particular rank finally we develop an assignment problem model to achieve an optimal ranking by minimizing the total rank unacceptability and to compute the expected priority vector of all alternatives numerical examples are provided to show the applicability and justifications of the proposed gdm method

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

a review on some contributions to perturbation theory singular limits and well posedness abstract in the beginning of the 1990s we devoted a sequence of papers to perturbation theory singular limits and well posedness problems in particular the strong well posedness of the initial boundary value problem for the compressible euler equations was demonstrate for the first time our method also allowed singular limit results in the strong norm even under assumptions weaker than the current ones in the literature where the strong norm is not reached it is worth noting that until now the above method and results have not been substantially improved hence an introduction to it still looks timely actually in a forthcoming paper by returning to this method we improve in a very substantial way some important results recently appeared in the literature

multiple attribute decision making based on fuzzy preference information on alternatives ranking and weighting this paper investigates the multiple attribute decision making madm problem with fuzzy preference information on alternatives and proposes an eigenvector method to rank them three optimization models are introduced to assess the relative importance weights of attributes in a madm problem which integrate subjective fuzzy preference relations and objective information in different ways since the decisions these models lead to are based on both subjective and objective information they are more balanced and credible than those that consider only one type of information an example is provided to illustrate the application of the proposed models

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

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

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

how pooling failure data may reverse increasing failure rates abstract although mixtures of decreasing failure rate dfr distributions are always dfr some mixtures of increasing failure rate ifr distributions can also be ultimately dfr in this article various types of discrete and continuous mixtures of ifr distributions are considered and conditions are developed for such mixtures to be ultimately dfr these conditions lead to an interesting result that certain mixtures of ifr distributions even those with very rapidly increasing failure rates e g weibull truncated extreme ultimately become dfr distributions it is common practice to pool data from several different ifr distributions to enlarge sample size for instance the results of this article sound a warning that such pooling may actually reverse the ifr property of the individual samples to a dfr property

a general multiple attribute decision making approach for integrating subjective preferences and objective information a general multiple attribute decision making madm approach is proposed to integrate subjective preferences and objective information and to assess the relative weights of attributes where subjective preferences are expressed either by a fuzzy preference relation on decision alternatives or by a pairwise comparison matrix on the relative weights of attributes or by both of them and objective information is expressed by a decision matrix three special cases of the approach are investigated which are the weighted least deviation norm wldn the weighted least square deviation norm wlsdn and the weighted minimax deviation norm wmdn approaches the extension of the general approach to group decision making and the choice of decision parameters are also discussed a numerical example is examined to show the applications of the proposed madm approaches

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

on the vanishing viscosity limit of 3d navier stokes equations under slip boundary conditions in general domains we consider the vanishing viscosity limit for the navier stokes equations with certain slip without friction boundary conditions in a bounded domain with non flat boundary in particular we are able to show convergence in strong norms for a solution starting with initial data belonging to the special subclass of data with vanishing vorticity on the boundary the proof is obtained by smoothing the initial data and by a perturbation argument with quite precise estimates for the equations of the vorticity and for that of the curl of the vorticity

generalized golden sections repeated bisections and aesthetic pleasure abstract we generalize the golden section in the sense that we partition a given line segment into a finite sequence of diminishing subsegments in such a way that the ratio of any pair of adjacent subsegments equals the ratio of the whole and the first subsegment when the number of subsegments increases the partitioning approximates the subdivision of the line segment via repeated bisections this is the basis of the geometric scales in multi criteria decision analysis introduced in order to model the gradations of comparative preferential judgement since the golden section plays an important role in the analysis of beauty and harmony we suppose that it is an aesthetic pleasure to observe the resulting pattern of preference modelling

classification of planar rational cuspidal curves ii log del pezzo models let e p be a complex curve homeomorphic to the projective line the negativity conjecture asserts that the kodaira iitaka dimension of k x 1 2 d where x d p e is a minimal log resolution is negative we prove structure theorems for curves satisfying this conjecture and we finish their classification up to a projective equivalence by describing the ones whose complements admit no c fibration as a consequence we show that they satisfy the strong rigidity conjecture of flenner zaidenberg the proofs are based on the almost minimal model program the obtained list contains one new series of bicuspidal curves

the compressible euler equations in a bounded domain existence of solutions and the incompressible limit a short time existence theorem is proven for the euler equations for nonisentropic compressible fluid flow in a bounded domain and solutions with low mach number and almost incompressible initial data are shown to be close to corresponding solutions of the equations for incompressible flow

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

singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids many interesting problems in classical physics involve the limiting behavior of quasilinear hyperbolic systems as certain coefficients become infinite using classical methods the authors develop a general theory of such problems this theory is broad enough to study a wide variety of interesting singular limits in compressible fluid flow and magneto fluid dynamics including new constructive local existence theorems for the time singular limit equations in particular the authors give an entirely self contained classical proof of the convergence of solutions of the compressible fluid equations to their incompressible limits as the mach number becomes small the theory depends upon a balance between certain inherently nonlinear structural conditions on the matrix coefficients of the system together with appropriate initialization procedures similar results are developed also for the compressible and incompressible navier stokes equations with periodic initial data independent of the viscosity coefficients as they tend to zero

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

the geometry of sporadic c embeddings into c2 abstract a closed algebraic embedding of c c 1 0 into c 2 is sporadic if for every curve a c 2 isomorphic to an affine line the intersection with c is at least 2 non sporadic embeddings have been classified there are very few known sporadic embeddings we establish geometric and algebraic tools to classify them based on the analysis of the minimal log resolution x d p 2 u where u is the closure of c on p 2 we show in particular that one can choose coordinates on c 2 in which the type at infinity of the c and the self intersection of its proper transform on x are sharply limited

the synthetic hierarchy method an optimizing approach to obtaining priorities in the ahp abstract a new method of synthesizing local and criteria priorities into global priorities is suggested this approach is a development of the analytic hierarchy process enabling the united consideration of all horizontal and vertical connections of a hierarchical system in a single optimizing objective function based on statistical models of the synthesis process the solution can be reduced to a linear system or to an eigenproblem of a special matrix constructed as a combination of kronecker s sums and products of pairwise judgement matrices a numerical example shows that the optimizing approach produces a ranking of global priorities that may be different from the ranking produced by the classical ahp

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

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

an l p theory for the n dimensional stationary compressible navier stokes equations and the incompressible limit for compressible fluids the equilibrium solutions in this paper we study the system 1 1 1 3 which describes the stationary motion of a given amount of a compressible heat conducting viscous fluid in a bounded domain ofr n n 2 hereu x is the velocity field x is the density of the fluid x is the absolute temperature f x andh x are the assigned external force field and heat sources per unit mass andp is the pressure in the physically significant case one hasg 0 we prove that for small data f g h there exists a unique solution u of problem 1 1 1 3 1 in a neighborhood of 0 m 0 for arbitrarily large data the stationary solution does not exist in general see sect 5 moreover we prove that for barotropic flows the stationary solution of the navier stokes equations 1 8 is the incompressible limit of the stationary solutions of the compressible navier stokes equations 1 7 as the mach number becomes small finally in sect 5 we will study the equilibrium solutions for system 4 1 for a more detailed explanation see the introduction

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

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

group decision making based on multiplicative consistent reciprocal preference relations the main aim of this paper is to investigate the consistency and consensus of multiplicative consistent reciprocal preference relations based on the multiplicative consistency property a straightforward method to obtain the priority vector for multiplicative consistent reciprocal preference relation is introduced together with some of its desired properties in addition some interesting properties on consistency and consensus for reciprocal preference relations are further investigated a consensus optimization model for group decision making with reciprocal preference relations is proposed which enables the decision makers to reach consensus with the highest overall consensus level numerical examples are provided to make a discussion and comparison with some existing similar methods

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

deviation square priority method for distinct preference structures based on generalized multiplicative consistency preference structures such as fuzzy preference relations incomplete fuzzy preference relations utility values and preference orderings are very useful in depicting the decision makers preferences over the considered objects in group decision making they have been attracting great interest from both researchers and practitioners in recent decades an important issue on these preference structures is to investigate the techniques of deriving priority weights of the objects from them in this paper based on generalized multiplicative consistency we establish a general nonlinear optimization model with fuzzy preference relations incomplete fuzzy preference relations utility values and preference orderings by solving the model we get a system of nonlinear equations and prove the uniqueness of its solution then a convergent iterative algorithm is devised to derive the priority weights from this system of equations the model and the algorithm include a variety of special cases suitable for group decision making with one or several of the aforementioned preference structures a practical case illustration concerning a manufacturing company that is searching for the best compression ignition engine design is provided and the numerical results of the developed models and the algorithm and the existing ones are also compared and analyzed in order to demonstrate the advantages and practicality of our models and algorithm

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

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

perturbation theorems for linear hyperbolic mixed problems and applications to the compressible euler equations the main result of this paper which is completely new apart from our previous and less general result proved in reference 9 states that the nonlinear system of equations 1 11 or equivalently 1 10 that describes the motion of an inviscid compressible barotropic fluid in a bounded domain gives rise to a strongly well posed problem in the hadamard classical sense in spaces hk k 3 see theorem 1 4 below roughly speaking if an n a in hk hk and if fn f in 2 0 t hk then vn gn v g in 0 t hk hk the method followed here see also 8 also applies to the non barotropic case p p s see 10 and to other nonlinear problems these results are based upon an improvement of the structural stability theorem for linear hyperbolic equations see theorem 1 2 below added in proof the reader is referred to 29 part i for a concise explanation of some fundamental points in the method followed here 1993 john wiley sons inc

on rational cuspidal curves let c subset bold p 2 be a rational curve of degree d which has only one analytic branch at each point denote by m the maximal multiplicity of singularities of c it is proven in ms that d alpha m and this estimate is sharp the main tool used here is the logarithmic version of the bogomolov miyaoka yau inequality for curves as above we give an interpretation of this inequality in terms of the number of parameters describing curves of a given degree and the number of conditions imposed by singularity types

data aggregation and simpson s paradox gauged by index numbers abstract simpson s paradox is a phenomenon occurring in data aggregation in complex systems it consists in the increase decrease of the rate in the data aggregate at the higher level with the simultaneous decrease increase of the rate in each subgroup of the lower levels of the hierarchy although the nature of this paradox is known it is difficult to interpret it without gauging the reasons of its occurrence to capture the causes of the paradox we elaborated some measures of index analysis we suggest to apply these measures for estimation of changes due to the partial rates and the structural effects using numerical examples from the marketing research field we show how the elaborated gauges evaluate absolute and relative changes helping to interpret the incidence cases