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

a new multiplicative consistency based method for decision making with triangular fuzzy reciprocal preference relations abstract triangular fuzzy reciprocal preference relations tfrprs are good choices to denote the decision makers fuzzy preferences consistency analysis is a very important topic to preference relations that can avoid the illogical ranking order however all previous multiplicative consistency concepts for tfrprs cannot well address this type of preference relations to address this issue this paper introduces a new multiplicative consistency concept for tfrprs which can be seen as a natural extension of tanino s multiplicative consistency concept for reciprocal preference relations based on the new consistency concept 0 1 mixed programming models for judging the multiplicative consistency of tfrprs are constructed and an approach for deriving multiplicative consistent tfrprs is provided considering the incomplete case programming models to determine missing values are constructed subsequently a multiplicative consistency based method for decision making with tfrprs is presented that can address the inconsistent and incomplete cases meanwhile illustrative examples are provided to show the concrete application of the proposed procedure and comparison analysis is also made

gleason s theorem for rectangular jbw triples a jbw triple b is said to be rectangular if there exists a w algebra a and a pair p q of centrally equivalent elements of the complete orthomodular lattice of projections in a such that b is isomorphic to the jbw triple paq any weak closed injective operator space provides an example of a rectangular jbw triple the principal order ideal of the complete lattice of centrally equivalent pairs of projections in a w algebra a generated by p q forms a complete lattice that is order isomorphic to the complete lattice of weak closed inner ideals in b and to the complete lattice of structural projections on b although not itself in general orthomodular possesses a complementation that allows for definitions of orthogonality centre and central orthogonality to be given a less familiar notion in lattice theory that is well known in the theory of jordan algebras and jordan triple systems is that of rigid collinearity of a pair e2 f2 and e2 f2 of elements of this is defined and characterized in terms of properties of a w algebra a is sometimes thought of as providing a model for a statistical physical system in this case b or equivalently paq may be thought of as providing a model for a fixed sub system of that represented by a therefore may be considered to represent the set consisting of a particular kind of sub system of that represented by paq central orthogonality and rigid collinearity of pairs of elements of may be regarded as representing two different types of disjointness the former classical disjointness and the latter decoherence of the two sub systems it is therefore natural to consider bounded measures m on that are additive on centrally orthogonal and rigidly collinear pairs of elements using results of j d m wright it is shown that provided that neither of the two hereditary sub w algebras pap and qaq of a has a weak closed ideal of type i2 such measures are precisely those that are the restrictions of bounded sesquilinear functionals m on pap qaq with the property that the action of the centroid z b of b commutes with the adjoint operation when b is a complex hilbert space of dimension greater than two this result reduces to gleason s theorem

the central hull and central kernel in jbw triples abstract the complete lattice i a of weak closed inner ideals in a jbw triple a has as its centre the complete boolean algebra z i a of weak closed ideals in a the annihilator l of the subset l of a consists of elements b of a for which l b a is equal to zero and the kernel ker l of l consists of those elements b in a for which l b l is equal to zero for each element j of i a j also lies in i a and a enjoys the generalized peirce decomposition a j m j j 1 where j 1 is the intersection of the kernels of j and j to investigate the properties of the weak closed subspace j 1 of a which is not in general a subtriple the notions of the central hull c l and central kernel k l of a subspace l are introduced these are respectively the smallest element of z i a containing l and the largest element of z i a contained in l for any element j of i a the relationships that exist between the central hull and central kernel of j and j are examined and it is shown that j 1 j is the weak closed ideal k j that j 1 j is the weak closed ideal k j and when j is a peirce inner ideal that j 1 is the weak closed ideal k j m k j

the lattice of weak closed inner ideals in a w algebra the family of centrally equivalent pairs of projections in a w algebra a forms a complete lattice that is order isomorphic to the complete lattice of weak closed inner ideals in a and to the complete lattice of structural projections on a furthermore the set of symmetric elements of is order isomorphic to the complete orthomodular lattice of projections in a although not itself in general orthomodular possesses a complementation that allows for definitions of orthogonality centre and central orthogonality to be given a less familiar notion in lattice theory that is well known in the theory of jordan algebras and jordan triple systems is that of rigid collinearity of a pair e1 f1 and e2 f2 of elements of this is defined and characterized in terms of properties of

least squares estimators of the mode of a unimodal regression function in this paper we consider nonparametric least squares estimators of the mode of an unknown unimodal regression function we establish almost sure convergence of these estimators with nearly optimal convergence rates under the assumption of the exponential tail for the error distributions

global stochastic optimization with low dispersion point sets this study concerns a generic model free stochastic optimization problem requiring the minimization of a risk function defined on a given bounded domain in a euclidean space smoothness assumptions regarding the risk function are hypothesized and members of the underlying space of probabilities are presumed subject to a large deviation principle however the risk function may well be nonconvex and multimodal a general approach to finding the risk minimizer on the basis of decision observation pairs is proposed it consists of repeatedly observing pairs over a collection of design points principles are derived for choosing the number of these design points on the basis of an observation budget and for allocating the observations between these points in both prescheduled and adaptive settings on the basis of these principles large deviation type bounds of the minimizer in terms of sample size are established

decoherence functionals for von neumann quantum histories boundedness and countable additivity gell mann and hartle have proposed a significant generalisation of quantum theory in which decoherence functionals perform a key role verifying a conjecture of isham linden schreckenberg the author analysed the structure of bounded finitely additive decoherence functionals for a general von neumann algebra a where a has no type i2 direct summand isham et al had already given a penetrating analysis for the situation where a is finite dimensional the assumption of countable additivity for a decoherence functional may seem more plausible physically than that of boundedness the results of this note are obtained much more generally but when specialised to l h the algebra of all bounded linear operators on a separable hilbert space h give

stability analysis of tree structured decision functions in multicriteria decision problems many values must be assigned such as the importance of the different criteria and the values of the alternatives with respect to subjective criteria since these assignments are approximate it is very important to analyze the sensitivity of results when small modifications of the assignments are made when solving a multicriteria decision problem it is desirable to choose a decision function that leads to a solution as stable as possible we propose here a method based on genetic programming that produces better decision functions than the commonly used ones the theoretical expectations are validated by case studies

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

pairwise comparison based interval analysis for group decision aiding with multiple criteria interval ahp analytic hierarchy process was proposed to obtain interval weights from a given pairwise comparison matrix showing relative importance between criteria in this paper interval ahp is applied to group decision problems interval ahp is first revised suitably for comparing alternatives from the viewpoint that the interval weight vector shows the set of agreeable weight vectors for the decision maker under individual interval weight vectors obtained from individual pairwise comparison matrices three approaches to obtaining a consensus interval weight vector are proposed one is the perfect incorporation approach that obtains consensus interval weight vectors including all individual interval weight vectors by this approach we can count out indubitably inferior alternatives the second is the common ground approach that obtains consensus interval weight vectors included in all individual interval weight vectors by this approach we can find agreeable group preference between alternatives when all individual opinions are similar the third is the partial incorporation approach that obtains consensus interval weight vectors intersecting all individual interval weight vectors by this approach we can find agreeable group preference between alternatives when individual opinions are not similar the usefulness of the proposed three approaches is demonstrated by simple numerical examples

the structure of decoherence functionals for von neumann quantum histories gell mann and hartle have proposed a significant generalization of quantum theory in which decoherence functionals perform a key role isham linden schreckenberg have given a penetrating analysis of decoherence functionals for l h where h is finite dimensional dimension greater than 2 in this note their conjecture on the significance of boundedness is verified in particular it is shown that when d is a bounded decoherence functional associated with a von neumann algebra a then provided a has no direct summand of type i2 d can be represented as the difference between semi innerproducts on a

generalized analytic network process the analytic network process anp is a methodology for multi criteria decision making used to derive priorities of the compared elements in a network hierarchy where the dependences and feedback within and between the elements can be considered however the anp is limited to the input preferences as crisp judgments which is often unfavorable in practical applications as an extension of the anp a generalized analytic network process g anp is developed to allow multiple forms of preferences such as crisp fuzzy judgments interval interval fuzzy judgments hesitant hesitant fuzzy judgments and stochastic stochastic fuzzy judgments in the g anp a concept of complex comparison matrices ccms is developed to collect decision makers preferences in the multiple forms from a stochastic point of view we develop an eigenvector method based stochastic preference method evm spm to derive priorities from ccms the main steps of the g anp are summarized and the implementation of the g anp in matlab and excel environments are given in detail which is also a prototype for a decision support system a real life example of the piracy risk assessment to the energy channels of china is proposed to demonstrate the g anp

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 new method of obtaining the priority weights from an interval fuzzy preference relation in analyzing a multiple criteria decision making problem the decision maker may express her his opinions as an interval fuzzy or multiplicative preference relation then it is an interesting and important issue to investigate the consistency of the preference relations and obtain the reliable priority weights in this paper a new consistent interval fuzzy preference relation is defined and the corresponding properties are derived the transformation formulae between interval fuzzy and multiplicative preference relations are further given which show that two preference relations consistent interval fuzzy and multiplicative preference relations can be transformed into each other based on the transformation formula the definition of acceptably consistent interval fuzzy preference relation is given furthermore a new algorithm for obtaining the priority weights from consistent or inconsistent interval fuzzy preference relations is presented finally three numerical examples are carried out to compare the results using the proposed method with those using other existing procedures the numerical results show that the given procedure is feasible effective and not requisite to solve any mathematical programing

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

measures on the lattice of closed inner ideals in a spin triple two elements j and k of the complete lattice i a of weak closed inner ideals in a jbw triple a are said to be centrally orthogonal if there exists a weak closed ideal i in a such that a2 j a2 i and a2 k a0 i and are said to be rigidly collinear when a2 j a1 k and a2 k a1 j where for j equal to 0 1 or 2 aj i aj j and aj k are the components in the generalized peirce decomposition of a relative to the weak closed inner ideals i j and k respectively a measure m on i a is a mapping from i a to c such that if j and k are either centrally orthogonal or rigidly collinear thenmj k mj mk a complex hilbert space a endowed with a conjugation possesses a triple product and norm with respect to which it forms a jbw triple known as a spin triple in this paper the structure of the complete lattice i a of closed inner ideals in a spin triple a and the measures on it are investigated it is shown that if the dimension of a is greater than 5 then there are no non zero measures on i a when the dimension of a is 5 non zero measures exist and up to multiplication by a constant a unique measure exists that is invariant under automorphisms of a when the dimension of a is 4 then a is triple isomorphic to the w algebra of 2 2 complex matrices in this case results of bunce and wright are used to show that there is an uncountable number of measures on i a the situation when the dimension of a is less than 4 is also described

establishing dominance between alternatives with incomplete information in a hierarchically structured attribute tree abstract a model for establishing dominance with incomplete information is proposed in a hierarchically structured attribute tree the linear additive value function under certainty is used in the model the model allows the decision maker to provide any type of linear partial information on the attribute weights and the alternative values the incompletely specified information constructs a feasible region of linear constraints and therefore pairwise dominance relationship between alternatives reduces to intractable non linear programming hence to handle this difficulty we propose a solution technique also to handle the tree structure we break down the attribute tree into sub trees due to the recursive structure of the solution technique the optimization results from sub trees can be utilized in computing the value interval on the topmost attribute based on the solution technique an algorithm for obtaining the value interval is proposed and dominance relation is established

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

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

optimal two stage procedures for estimating location and size of the maximum of a multivariate regression function we propose a two stage procedure for estimating the location and size m of the maximum of a smooth d variate regression function f in the first stage a preliminary estimator of obtained from a standard nonparametric smoothing method is used at the second stage we zoom in near the vicinity of the preliminary estimator and make further observations at some design points in that vicinity we fit an appropriate polynomial regression model to estimate the location and size of the maximum we establish that under suitable smoothness conditions and appropriate choice of the zooming the second stage estimators have better convergence rates than the corresponding first stage estimators of and m more specifically for smooth regression functions the optimal nonparametric rates n 1 2 d and n 2 d at the first stage can be improved to n 1 2 and n 1 2 respectively for 1 1 d 2 these rates are optimal in the class of all possible sequential estimators interestingly the two stage procedure resolves the curse of the dimensionality problem to some extent as the dimension d does not control the second stage convergence rates provided that the function class is sufficiently smooth we consider a multi stage generalization of our procedure that attains the optimal rate for any smoothness level 2 starting with a preliminary estimator with any power law rate at the first stage

compatible subtriples of jordan triples abstract the purpose of this paper is to extend the notion of compatibility of tripotents in an anisotropic jordan triple a to that of subtriples of a for a subtriple b of a define the kernel ker b and the annihilator b of b to be the subsets a a b a b 0 and a a b a a 0 respectively the peirce spaces b i i 0 1 2 of b are defined in terms of the kernel and the annihilator by b 0 b b 1 ker b ker b and b 2 b a pair b c of subtriples of a is said to be compatible if a coincides with i j 0 1 2 b i c j the subtriple b is said to be self compatible if it is compatible with itself a self compatible subtriple b is complemented in that b ker b coincides with a which implies that b is an inner ideal in a it follows that a pair u v of tripotents in a is compatible if and only if the ranges im p 2 u im p 2 v of the peirce projections p 2 u and p 2 v form a compatible pair of subtriples moreover for any tripotent u im p 2 u is a self compatible subtriple of a for the special case of a jbw triple a it is shown that a subtriple i is compatible with every self compatible subtriple j if and only if i is a complemented ideal in a these results are then applied to abelian jbw triples w algebras and hilbert spaces

an eigenvector method for generating normalized interval and fuzzy weights crisp comparison matrices produce crisp weight estimates it is logical for an interval or fuzzy comparison matrix to give an interval or fuzzy weight estimate in this paper an eigenvector method em is proposed to generate interval or fuzzy weight estimate from an interval or fuzzy comparison matrix which differs from csutora and buckley s lambda max method in several aspects first the proposed em produces a normalized interval or fuzzy eigenvector weight estimate through the solution of a linear programming model while the lambda max method gives a series of nonnormalized interval eigenvector weight estimate through an iterative solution process next the em directly solves the principal right eigenvector of an interval or fuzzy comparison matrix while the lambda max method needs transforming a fuzzy comparison matrix into a series of interval comparison matrices using a level sets and the extension principle and therefore needs the solution of a series of eigenvalue problems finally the lambda max method needs the help of the principal right eigenvector of a crisp comparison matrix to determine the final interval weights while the em has no such requirement it is also found that not all interval or fuzzy comparison matrices can generate normalized interval or fuzzy eigenvector weights situations where the em is inapplicable are analyzed and the aggregation of local interval or fuzzy weights into global interval or fuzzy weights is also discussed three numerical examples including a hierarchical structure decision making problem are examined using the em to demonstrate its applications

preference programming and inconsistent interval judgments the problem of derivation of the weights of altematives from pairwise comparison matrices is long standing in this paper lexicographic goal programming lgp has been used to find out weights from pairwise inconsistent interval judgment matrices a number of properties and advantages of lgp as a weight determination technique have been explored an algorithm for identification and modification of inconsistent bounds is also provided the proposed technique has been illustrated by means of numerical examples

a goal programming method for obtaining interval weights from an interval comparison matrix crisp comparison matrices lead to crisp weight vectors being generated accordingly an interval comparison matrix should give an interval weight estimate in this paper a goal programming gp method is proposed to obtain interval weights from an interval comparison matrix which can be either consistent or inconsistent the interval weights are assumed to be normalized and can be derived from a gp model at a time the proposed gp method is also applicable to crisp comparison matrices comparisons with an interval regression analysis method are also made three numerical examples including a multiple criteria decision making mcdm problem with a hierarchical structure are examined to show the potential applications of the proposed gp method

interval articulation of superiority and precise elicitation of priorities this paper develops an approach to obtaining precise point estimates of priorities of objects in particular coefficients of importance of criteria it is assumed that the judgements regarding the preference superiority of one object over another are given in the forms of intervals that is by indicating the lower and upper bounds the suggested approach is suitable both in the case of consistent and inconsistent preferences in the former case the available intervals are processed by contracting them to the maximum degree in the latter by extending them to the minimum degree the degrees of contraction or extension are regarded as the indexes of indeterminacy or inconsistency of preferences and treated as multiple objective functions a method of transformation of such a multicriterial problem to a sequence of several linear programming problems is suggested

insdecm an interactive procedure for stochastic multicriteria decision problems abstract a new interactive technique for a discrete stochastic multiattribute decision making problem is proposed in this paper it is assumed that performance probability distribution for each action on each attribute is known two concepts are combined in the procedure stochastic dominance and interactive approach the first one is employed for generating efficient actions and constructing rankings of actions with respect to attributes the second concept is used when the communication between the dm and the model is conducted it is assumed that decision maker s restrictions are defined by specifying minimal or maximal values of scalar criteria measuring either expected outcome or variability of outcomes as such restrictions are in general not consistent with stochastic dominance rules we suggest verifying this consistency and asking the decision maker to redefine inconsistent restrictions

convergence properties and practical estimation of the probability of rank reversal in pairwise comparisons for multi criteria decision making problems abstract in this paper we address the impact of uncertainty introduced when the experts complete pairwise comparison matrices in the context of multi criteria decision making we first discuss how uncertainty can be quantified and modeled and then show how the probability of rank reversal scales with the number of experts we consider the impact of various aspects which may affect the estimation of probability of rank reversal in the context of pairwise comparisons such as the uncertainty level alternative preference scales and different weight estimation methods we also consider the case where the comparisons are carried out in a fuzzy manner it is shown that in most circumstances augmenting the size of the expert group beyond 15 produces a small change in the probability of rank reversal we next address the issue of how this probability can be estimated in practice from information gathered simply from the comparison matrices of a single expert group we propose and validate a scheme which yields an estimate for the probability of rank reversal and test the applicability of this scheme under various conditions the framework discussed in the paper can allow decision makers to correctly choose the number of experts participating in a pairwise comparison and obtain an estimate of the credibility of the outcome

boundedness of completely additive measures with application to 2 local triple derivations we prove a jordan version of dorofeev s boundedness theorem for completely additive measures and use it to show that every not necessarily linear nor continuous 2 local triple derivation on a continuous jbw triple is a triple derivation 2 local triple derivations are well understood on von neumann algebras jbw triples which are properly defined in section i are intimately related to infinite dimensional holomorphy and include von neumann algebras as special cases in particular continuous jbw triples can be realized as subspaces of continuous von neumann algebras which are stable for the triple product xy z zy x and closed in the weak operator topology

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

the range of a structural projection abstract let a be a jbw triple a linear subspace j of a is called an inner ideal in a provided that the subspace j a j is contained in j a subtriple b in a is said to be complemented if a b ker b where ker b a a b a b 0 a complemented subtriple in a is a weak closed inner ideal a linear projection on a is said to be structural if for all elements a b and c in a formula the range of a structural projection is a complemented subtriple and conversely a complemented subtriple is the range of a unique structural projection we analyze the structure of the weak closed inner ideal generated by two arbitrary tripotents in a jbw triple in terms of the simultaneous peirce spaces of three suitably chosen pairwise compatible tripotents this result is then used to show that every weak closed inner ideal j in a jbw triple a is a complemented subtriple in a and therefore the range of a unique structural projection on a as an application structural projections on w algebras are considered

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

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

the evidential reasoning approach for multi attribute decision analysis under interval uncertainty abstract the evidential reasoning er approach is a method for multiple attribute decision analysis mada under uncertainties it improves the insightfulness and rationality of a decision making process by using a belief decision matrix bdm for problem modelling and the dempster shafer d s theory of evidence for attribute aggregation the d s theory provides scope and flexibility to deal with interval uncertainties or local ignorance in decision analysis which is not explored in the original er approach and will be investigated in this paper firstly interval uncertainty will be defined and modelled in the er framework then an extended er algorithm ier is derived which enables the er approach to deal with interval uncertainty in assessing alternatives on an attribute it is proved that the original er algorithm is a special case of the ier algorithm the latter is demonstrated using numerical examples

dependence and independence from linear hierarchies to nonlinear networks abstract most decisions need to be free from assumptions of independence to be faithful to the complex problems in which they arise this paper illustrates how to generate priorities for decisions involving general types of dependence of criteria on alternatives criteria on criteria and alternatives on alternatives it is based on the feedback system framework of the analytic hierarchy process of which a hierarchy is a special case

peirce inner ideals in jordan triples abstract a subspace j of an anisotropic jordan triple a is said to be an inner ideal if the subspace j a j is contained in j an inner ideal j in a is said to be complemented if a is equal to the sum of j and the kernel ker j of j defined to be the subspace of a consisting of elements a in a for which j a j is equal to 0 the annihilator j of an inner ideal j in a is the inner ideal consisting of elements a in a such that j a a is equal to 0 when both j and j are complemented a can be decomposed into the direct sum of j ker j ker j and j modulo six of the generalized peirce relations this decomposition is a grading of a of peirce type since an inner ideal in a jbw triple is complemented if and only if it is weak closed the result described above applies to all weak closed inner ideals j in a jbw triple a furthermore it can be shown that in this case all except five of the generalized peirce relations hold and an example is given of a weak closed inner ideal in a jbw triple for which all five fail to hold thereby showing that the result is the best possible it is also shown that the condition that a weak closed inner ideal in a jbw triple a leads to a grading of a which is of peirce type is equivalent to several other conditions all of a topological rather than algebraic nature these results are applied to w algebras spin triples and the bi cayley triple

nonparametric estimation of homothetic and homothetically separable functions for vectors x and w let r x w be a function that can be nonparametrically estimated consistently and asymptotically normally we provide consistent asymptotically normal estimators for the functions g and h where r x w h g x w g is linearly homogeneous and h is monotonic in g this framework encompasses homothetic and homothetically separable functions such models reduce the curse of dimensionality provide a natural generalization of linear index models and are widely used in utility production and cost function applications extensions to related functional forms include a generalized partly linear model with unknown link function we provide simulation evidence on the small sample performance of our estimator and we apply our method to a chinese production dataset

criteria importance theory for multicriterial decision making problems with a hierarchical structure abstract we develop a new model of multicriterial decision analysis with a hierarchical criteria structure the suggested methodology is based on a rigorously defined specification of qualitative and quantitative importance of criteria of all levels of the hierarchy and enables the comparison of decision alternatives for different types of criteria scale

a goal programming approach to deriving interval weights in analytic form from interval fuzzy preference relations based on multiplicative consistency abstract this paper focuses on how to find an analytic solution of optimal interval weights from consistent interval fuzzy preference relations ifprs and obtain approximate solution based interval weights in analytic form from inconsistent ifprs the paper first analyzes the popularly used interval weight additive normalization model and illustrates its drawbacks on the existence and uniqueness for characterizing 0 1 valued interval weights obtained from ifprs by examining equivalency of 0 1 valued interval weight vectors a novel framework of multiplicatively normalized interval fuzzy weights mnifws is then proposed and used to define multiplicatively consistent ifprs the paper presents significant properties for multiplicatively consistent ifprs and their associated mnifws these properties are subsequently used to establish two goal programming gp models for obtaining optimal mnifws from consistent ifprs by the lagrangian multiplier method analytic solutions of the two gp models are found for consistent ifprs the paper further devises a two step procedure for deriving approximate solution based mnifws in analytic form from inconsistent ifprs two visualized computation formulas are developed to determine the left and right bounds of approximate solution based mnifws of any ifpr the paper shows that this approximate solution is an optimal solution if an ifpr is multiplicatively consistent three numerical examples including three ifprs and comparative analyses are offered to demonstrate rationality and validity of the developed model

an intelligent decomposition of pairwise comparison matrices for large scale decisions abstract a pairwise comparison matrix pcm has been used to compute for relative priorities of elements and are integral components in widely applied decision making tools the analytic hierarchy process ahp and its generalized form the analytic network process anp however pcms suffer from several issues limiting their applications to large scale decision problems these limitations can be attributed to the curse of dimensionality that is a large number of pairwise comparisons need to be elicited from a decision maker this issue results to inconsistent preferences due to the limited cognitive powers of decision makers to address these limitations this research proposes a pcm decomposition methodology that reduces the elicited pairwise comparisons a binary integer program is proposed to intelligently decompose a pcm into several smaller subsets using interdependence scores among elements since the subsets are disjoint the most independent pivot element is identified to connect all subsets to derive the global weights of the elements from the original pcm as a result the number of pairwise comparison is reduced and consistency is of the comparisons is improved the proposed decomposition methodology is applied to both ahp and anp to demonstrate its advantages

a two stage logarithmic goal programming method for generating weights from interval comparison matrices a two stage logarithmic goal programming tlgp method is proposed to generate weights from interval comparison matrices which can be either consistent or inconsistent the first stage is devised to minimize the inconsistency of interval comparison matrices and the second stage is developed to generate priorities under the condition of minimal inconsistency the weights are assumed to be multiplicative rather than additive in the case of hierarchical structures a nonlinear programming method is used to aggregate local interval weights into global interval weights a simple yet practical preference ranking method is investigated to compare the interval weights of criteria or rank alternatives in a multiplicative aggregation process the proposed tlgp is also applicable to fuzzy comparison matrices when they are transformed into interval comparison matrices using a level sets and the extension principle six numerical examples including a group decision analysis problem with a group of comparison matrices a hierarchical decision problem and a fuzzy decision problem using fuzzy comparison matrix are examined to show the applications of the proposed methods comparisons with other existing procedures are made whenever possible

models of mathematical programming for intuitionistic multiplicative preference relations in order to capture model uncertainty associated with imprecision or vagueness a decision maker may express her his judgments in terms of intuitionistic multiplicative preference relation impr two important research topics with this regard are studied in the paper 1 checking consistency of impr and 2 generating weights on the basis of this relation a new definition of consistent impr is proposed in light of this new definition the properties of consistent impr are studied in detail a transformation formula is proposed to construct a consistent impr from a given normalized intuitionistic fuzzy weight vector by minimizing the differences between the constructed consistent impr and the given impr some fractional programming models are developed to derive the intuitionistic fuzzy weight vector moreover the models are also extended to group decision making finally two numerical examples are provided to illustrate the effectiveness and practical relevance of the proposed models

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

injectivity of operator spaces we study the structure of injective operator spaces and the existence and uniqueness of the injective envelopes of operator spaces we give an easy example of an injective operator space which is not completely isometric to any c algebra this answers a question of wittstock 23 furthermore we show that an operator space e is injective if and only if there exists an injective c algebra a and two projections p and q in a such that e is completely isometric to paq

nonparametric matching and efficient estimators of homothetically separable functions for vectors z and w and scalar v let r v z w be a function that can be non parametrically estimated consistently and asymptotically normally such as a distribution density or conditional mean regression function we provide consistent asymptotically normal nonparametric estimators for the functions g and h where r v z w h vg z w and some related models this framework encompasses homothetic and homothetically separable functions and transformed partly additive models r v z w h v g z w for unknown functions g and h such models reduce the curse of dimensionality provide a natural generalization of linear index models and are widely used in utility production and cost function applications we also provide an estimator of g that is oracle efficient achieving the same performance as an estimator based on local least squares when h is known

post optimality analysis of priority vectors derived from interval comparison matrices by lexicographic goal programming abstract in a recent paper by wang y m wang on lexicographic goal programming method for generating weights from inconsistent interval pair wise comparison matrices applied mathematics and computation 173 2006 985 991 it is shown that lexicographic goal programming method results in different rankings for upper and lower triangular interval judgments arranged in a positive reciprocal pair wise comparison matrix this was unexpected from a prioritization point of view because upper and lower triangular entries in reciprocal interval pair wise comparison matrices essentially provide the same judgment information a research question was posed at the same article which ranking is true or more reliable motivated by this question in this paper we provide a post optimality analysis to detect the most reliable priority vector among m vectors derived by lexicographical goal programming for an inconsistent interval pair wise comparison matrix of order n the method is based on violations in the order of preferences among the resultant priorities and runs in o mn 2 time when there exists a tie between some vectors we also illustrate a tie breaking algorithm running in o mn 4 time based on violations in the order of the preference intensities among the resultant priorities we re visited the examples discussed in the afore mentioned article and validate the practical usability of this method

the mackey gleason problem let a be a von neumann algebra with no direct summand of type i 2 and let p a be its lattice of projections let x be a banach space let m p a x be a bounded function such that m p q m p m q whenever p and q are orthogonal projections the main theorem states that m has a unique extension to a bounded linear operator from a to x in particular each bounded complex valued finitely additive quantum measure on p a has a unique extension to a bounded linear functional on a

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

regression towards the mode we propose a semi parametric mode regression estimator for the case in which the dependent variable has a continuous conditional density with a well defined global mode the estimator is semi parametric in that the conditional mode is specified as a parametric function but only mild assumptions are made about the nature of the conditional density of interest we show that the proposed estimator is consistent and has a tractable asymptotic distribution

a statistical approach to the analytic hierarchy process with interval judgements i distributions on feasible regions abstract this paper addresses the problem of extracting preferences for alternatives from interval judgement matrices in the analytic hierarchy process ahp the method of arbel for extracting such preferences which is based on the assumption that the interval judgements specify a feasible region in the weight space of the alternatives is critically appraised from a statistical perspective and some new ideas emanating from this approach are developed and discussed in particular it is proposed that a distribution for the weights on the feasible region which is both tractable and meaningful be adopted the mean of the distribution can then be used as an assessment of the overall ranking of the alternatives and quantities of interest such as probabilities and marginal distributions can immediately be quantified two specific distributions on the feasible region the uniform distribution and the distribution of random convex combinations with coefficients which are uniform spacings are examined in some detail and the ideas which emerge are illustrated by means of selected examples

rough approximation of a preference relation by a multi attribute stochastic dominance for determinist and stochastic evaluation problems abstract let a be a set of actions evaluated by a set of attributes two kinds of evaluations will be considered in this paper determinist or stochastic in relation to each attribute the multi attribute stochastic dominance msd r for a reduced number of attributes will be suggested to model the preferences in this kind of problem the case of mixed data where we have the attributes of different natures is not well known in the literature although it is essential from a practical point of view to apply the msd r the subset r of attributes from which approximation of the global preference is valid should be known the theory of rough sets gives us an answer on this issue allowing us to determine a minimal subset of attributes that enables the same classification of objects as the whole set of attributes in our approach these objects are pairs of actions in order to represent preferential information we shall use a pairwise comparison table this table is built for subset b a described by stochastic dominance sd relations for particular attributes and a total order for the decision attribute given by the decision maker dm using a rough set approach to the analysis of the subset of preference relations a set of decision rules is obtained and these are applied to a set a b of potential actions the rough set approach of looking for the reduction of the set of attributes gives us the possibility of operating with msd r

the classification of decoherence functionals an analog of gleason s theorem gell mann and hartle have proposed a significant generalization of quantum theory with a scheme whose basic ingredients are histories and decoherence functionals within this scheme it is natural to identify the space up of propositions about histories with an orthoalgebra or lattice this raises the important problem of classifying the decoherence functionals in the case where up is the lattice of projectors p v in some hilbert space v in effect one seeks the history analog of gleason s famous theorem in standard quantum theory in the present article the solution to this problem for the case where v is finite dimensional is presented in particular it is shown that every decoherence functional d p v can be written in the form d trv v x for some operator x on the tensor product space v v

the analytic hierarchy process with stochastic judgements the analytic hierarchy process ahp is a widely used method for multicriteria decision support based on the hierarchical decomposition of objectives evaluation of preferences through pairwise comparisons and a subsequent aggregation into global evaluations the current paper integrates the ahp with stochastic multicriteria acceptability analysis smaa an inverse preference method to allow the pairwise comparisons to be uncertain a simulation experiment is used to assess how the consistency of judgements and the ability of the smaa ahp model to discern the best alternative deteriorates as uncertainty increases across a range of simulated problems results indicate that according to conventional benchmarks judgements are likely to remain consistent unless uncertainty is severe but that the presence of uncertainty in almost any degree is sufficient to make the choice of best alternative unclear