lp metric sensitivity analysis for single and multi attribute decision analysis abstract analyzing the sensitivity of decisions to probability estimation error in single and multi attribute problems and to errors in estimating additive multi attribute value models in multi attribute problems is an integral part of decision analysis this paper presents an intuitive and tractable approach to this sensitivity analysis here a decision is considered insensitive if 1 the probabilities or multi attribute weights required for any other alternative to become preferred are not close to the original estimated probabilities and weights and 2 the rank order of states implied by the probabilities or the rank order of attributes implied by the additive multi attribute weights must change for any other alternative to become preferred the sensitivity analysis is conducted using straight forward linear programming models an example is used to demonstrate their application

regularity of potential functions of the optimal transportation problem the potential function of the optimal transportation problem satisfies a partial differential equation of monge ampere type in this paper we introduce the notion of a generalized solution and prove the existence and uniqueness of generalized solutions of the problem we also prove the solution is smooth under certain structural conditions on the cost function

the monge ampere equation and its link to optimal transportation we survey old and new regularity theory for the monge ampere equation show its connection to optimal transportation and describe the regularity properties of a general class of monge ampere type equations arising in that context

nearly round spheres look convex we prove that a riemannian manifold m g close enough to the round sphere in the c 4 topology has uniformly convex injectivity domains so m appears uniformly convex in any exponential chart the proof is based on the ma trudinger wang nonlocal curvature tensor which originates from the regularity theory of optimal transport

a perturbation argument for a monge ampere type equation arising in optimal transportation we prove some interior regularity results for potential functions of optimal transportation problems with power costs the main point is that our problem is equivalent to a new optimal transportation problem whose cost function is a sufficiently small perturbation of the quadratic cost but it does not satisfy the well known condition a 3 guaranteeing regularity the proof consists in a perturbation argument from the standard monge ampere equation in order to obtain first interior c1 1 estimates for the potential and second interior holder estimates for second derivatives in particular we take a close look at the geometry of optimal transportation when the cost function is close to quadratic in order to understand how the equation degenerates near the boundary

ranking with partial information a method and an application a method is presented for ranking multiattributed alternatives using a weighted additive evaluation function with partial information about the weighting scaling constants the method is applied to evaluate materials for use in nuclear waste containment the paper derives conditions to determine whether a pair of alternatives can be ranked given the partial information about weighting constants and presents an algorithm that partially rank orders the complete set of alternatives based on the pairwise ranking information

the utility efficient set and its interactive reduction we assume a decision situation under risk with incomplete information on preferences modelled as a vector utility function we consider an additive aggregation of its components and partial information on the scaling constants we develop the concept of utility efficiency to identify efficient strategies in discrete problems when the information about the scaling constants of the decision maker is in the form of a polyhedral cone a characterization of the utility efficient set provides a practical way to compute such efficient strategies we then discuss an interactive method based on the assessment of the scaling constants via an interactive paired comparison with its convergence the method is complemented by a procedure to reduce the utility efficient set to aid in the process of reaching a final strategy

judgemental modelling based on geometric least square abstract the analytic hierarchy process developed by t l saaty has received widespread attention in the literature however this has not been without some criticisms including the problems of meaning of consistency and large data requirements this paper addresses the last two problems and presents a new method of calculating preference vectors to the user the procedure is essentially the same but the data requirements can be made less onerous and at the same time feedback is provided permitting a greater understanding of the data inputs preliminary results indicate the acceptability of the proposed methodology

mean field theory for usp 4 sime so 5 the simple lie algebra usp 4 so 5 is a ten dimensional algebra that contains the angular momentum algebra su 2 in nuclear structure physics the algebra so 5 describes beta rigid collective modes in the bohr mottelson and interacting boson models the so 5 dual space consists of density matrices which are defined by the expectations of so 5 generators a coadjoint orbit is a common level surface in the dual space of the two so 5 casimirs this paper develops mean field theory on any coadjoint orbit of so 5 densities when the densities of a set of quantum states lie on one orbit the system is said to have a weak dynamical symmetry a lax pair determines the dynamics of so 5 densities on each coadjoint orbit analytic solutions are reported for rotating so 5 densities in equilibrium for a particular energy function

towards the smoothness of optimal maps on riemannian submersions and riemannian products of round spheres in particular the variant a3w of ma trudinger and wang s condition for regularity of optimal transportation maps is implied by the non negativity of a pseudo riemannian curvature which we call cross curvature induced by the transportation cost for the riemannian distance squared cost it is shown that 1 cross curvature non negativity is preserved for products of two manifolds 2 both a3w and cross curvature non negativity are inherited by riemannian submersions as is domain convexity for the exponential maps and 3 the n dimensional round sphere satisfies cross curvature non negativity from these results a large new class of riemannian manifolds satisfying cross curvature non negativity thus a3w is obtained including many whose sectional curvature is far from constant all known obstructions to the regularity of optimal maps are absent from these manifolds making them a class for which it is natural to conjecture that regularity holds this conjecture is confirmed for certain riemannian submersions of the sphere such as the complex projective spaces cp n

stability of a 4th order curvature condition arising in optimal transport theory abstract a certain curvature condition introduced by ma trudinger and wang in relation with the regularity of optimal transport is shown to be stable under gromov hausdorff limits even though the condition implicitly involves fourth order derivatives of the riemannian metric two lines of reasoning are presented with slightly different assumptions one purely geometric and another one combining geometry and probability then a converse problem is studied prove some partial regularity for the optimal transport on a perturbation of a riemannian manifold satisfying a strong form of the ma trudinger wang condition

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

ordinal criteria in stochastic multicriteria acceptability analysis smaa abstract we suggest a method for providing descriptive information about the acceptability of decision alternatives in discrete co operative group decision making problems the new smaa o method is a variant of the stochastic multicriteria acceptability analysis smaa smaa o is designed for problems where criteria information for some or all criteria is ordinal that is experts or decision makers have ranked the alternatives according to each ordinal criterion considerable savings can be obtained if rank information for some or all the criteria is sufficient for making decisions without significant loss of quality the approach is particularly useful for group decision making when the group can agree on the use of an additive decision model but only partial preference information or none at all is available

spherical harmonics and basic coupling coefficients for the group so 5 in an so 3 basis an easily programmable algorithm is given for the computation of so 5 spherical harmonics needed to complement the radial beta wave functions to form an orthonormal basis of wave functions for the five dimensional harmonic oscillator it is shown how these functions can be used to compute the clebsch gordan a k a wigner coupling coefficients for combining pairs of irreps in this space to other irreps this is of particular value for the construction of the matrices of hamiltonians and transition operators that arise in applications of nuclear collective models tables of the most useful coupling coefficients are given in the appendix

compatible weighting method with rank order centroid maximum entropy ordered weighted averaging approach in a situation where imprecise attribute weights such as a rank order are captured various approximate weighting methods have been proposed to aid multiattribute decision analysis among others it is well known that the rank order centroid roc weights result in the highest performance in terms of the identification of the best alternative under the ranked attribute weights in this paper we aim to reinterpret the meaning of the roc weights and to develop a compatible weighting method that is based on other well established academic disciplines the ordered weighted averaging owa method is a nonlinear aggregation method in that the weights are associated with the objects reordered according to their magnitudes in the aggregation process some interesting semantics can be attached to the approximate weights in view of the measure developed in the owa method furthermore the weights generated by the maximum entropy method show equally compatible performance with the roc weights under some condition which is demonstrated by theoretical and simulation analysis

reference point approach for multiple decision makers we consider multiple criteria decision making problems where a group of decision makers wants to find the most preferred solution from a discrete set of alternatives we develop a method that uses achievement functions for charting subsets of reference points that would support a certain alternative to be the most preferred one the resulting descriptive information is provided to the decision makers in the form of reference acceptability indices and central reference points for each decision alternative then the decision makers can compare this information with their own preferences we demonstrate the use of the method using a strategic multiple criteria decision model for an electricity retailer

classifying efficient alternatives in smaa using cross confidence factors abstract stochastic multicriteria acceptability analysis smaa is a family of methods for aiding multicriteria group decision making these methods are based on exploring the weight space in order to describe the preferences that make each alternative the most preferred one the main results of the analysis are rank acceptability indices central weight vectors and confidence factors for different alternatives the rank acceptability indices describe the variety of different preferences resulting in a certain rank for an alternative the central weight vectors represent the typical preferences favouring each alternative and the confidence factors measure whether the criteria data are sufficiently accurate for making an informed decision in some cases when the problem involves a large number of efficient alternatives the analysis may fail to discriminate between them this situation is revealed by low confidence factors in this paper we develop cross confidence factors which are based on computing confidence factors for alternatives using each other s central weight vectors the cross confidence factors can be used for classifying efficient alternatives into sets of similar and competing alternatives these sets are related to the concept of reference sets in data envelopment analysis dea but generalized for stochastic models forming these sets is useful when trying to identify one or more most preferred alternatives or suitable compromise alternatives the reference sets can also be used for evaluating whether criteria need to be measured more accurately and at which alternatives the measurements should be focused this may cause considerable savings in measurement costs we demonstrate the use of the cross confidence factors and reference sets using a real life example

novel types of sensitivity analysis for additive mcdm methods three novel types of sensitivity analysis are presented they determine the following 1 the sensitivity of a ranking to specific changes in the evaluations of all alternatives on certain criteria 2 the influence of specific changes in certain criterion scores of an alternative 3 the minimum modification of the weights required to make an alternative ranked first the use of these instruments is demonstrated by the results of a simulation experiment results show that the first and second type of sensitivity analysis enable to apply multiple criteria decision making mcdm methods in dynamic circumstances the third type of sensitivity analysis is demonstrated to be a tool to analyze the total weight space

the regularity of mappings with a convex potential in this work we apply the techniques developed in cl to the problem of mappings with a convex potential between domains that is given two bounded domains q q2 of rn and two nonnegative real functions fi defined in qi that are bounded away from zero and infinity we want to study the map vf v w for a lipschitz convex v such that v maps ql onto q 2 in the a e sense and in some weak sense 1 f2 vyv det dij v f1 x in recent work y brenier showed existence and uniqueness of such a map provided that jaqil 0 under the obvious compatibility condition

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

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

a subjective and objective integrated approach to determine attribute weights this paper proposes an integrated approach to determine attribute weights in the multiple attribute decision making madm problems the approach makes use of the subjective information provided by a decision maker and the objective information to form a two objective programming model thus the resultant attribute weights and rankings of alternatives reflect both the subjective considerations of a decision maker dm and the objective information an example is used to illustrate the applicability of the proposed approach

robustness analysis in multi objective mathematical programming using monte carlo simulation abstract in most multi objective optimization problems we aim at selecting the most preferred among the generated pareto optimal solutions a subjective selection among objectively determined solutions in this paper we consider the robustness of the selected pareto optimal solution in relation to perturbations within weights of the objective functions for this task we design an integrated approach that can be used in multi objective discrete and continuous problems using a combination of monte carlo simulation and optimization in the proposed method we introduce measures of robustness for pareto optimal solutions in this way we can compare them according to their robustness introducing one more characteristic for the pareto optimal solution quality in addition especially in multi objective discrete problems we can detect the most robust pareto optimal solution among neighboring ones a computational experiment is designed in order to illustrate the method and its advantages it is noteworthy that the augmented weighted tchebycheff proved to be much more reliable than the conventional weighted sum method in discrete problems due to the existence of unsupported pareto optimal solutions

on fuzzy multiattribute decision making models and methods with incomplete preference information in the paper by li fuzzy sets and systems 106 1999 113 a minimum average weighted deviation method was investigated to determine the weights of attributes using incomplete preference information it is shown in this paper that the method is flawed and can only be meaningful if it is revised two general ways are presented to revise it and are justified by using lagrange multiplier method and hessian matrix four specific revised models and approaches are investigated and examined through a numerical example different membership functions are suggested and utilized to normalize decision matrix so that a credible ranking order can be achieved

on the problem of weights in multiple criteria decision making the noncompensatory approach abstract this paper shows how the notion of relative importance of attributes can be defined within the framework of the noncompensatory approach to multiple criteria decision making the problem of weights then appears as a problem of functional representation of relations we state some theoretical results concerning this problem and outline a practical decision aid tactic based on the ideas introduced in the paper

implementation of the centroid method of solymosi and dombi multiple objective decision support either for selection among finite alternatives or for combining multiple functions in mathematical programming often involves estimating a set of weights of relative importance recently solymosi and dombi proposed a means of estimating such weights based upon ordinal preference information this method relies upon the centroid of bounded weights analytic hierarchy process uses ratio estimates of pairwise comparisons using only ordinal information the centroid of weights is developed as an alternative means of weighting hierarchical elements these weights could be used in selection among alternatives directly in concordance analysis or in multiple objective linear programming formulation for the centroid of feasible weight values is presented with a table of values for cases where the preference ranking includes no ties comparison of the technique on a student job selection decision indicates slight but insignificant decline in accuracy relative to the analytic hierarchy based method

an approach to multiple attribute decision making based on fuzzy preference information on alternatives this paper investigates the multiple attribute decision making madm problem with preference information on alternatives a new approach is proposed to solve the madm problem where the decision maker gives his her preference on alternatives in a fuzzy relation to reflect the decision maker s preference information an optimization model is constructed to assess the attribute weights and then to select the most desirable alternatives finally an example is used to illustrate the proposed approach

a method for determining the weights of criteria the centralized weights abstract in multicriteria analysis many methods use weights to represent the relative importance of criteria the noncompensatory aggregation procedure used in outranking methods enables comparisons of subsets of criteria from this point of view an interactive method is presented which requests only ordinal comparisons from the decision maker the relation more important than is assumed to be a semiorder therefore the indifference part of this relation is not necessarily transitive the judgements can be formulated as linear inequalities which constrain the set of feasible weights and threshold values all vertices of this polyhedron are determined the threshold and the unknown real weights are estimated by the centroid some characteristics a possible refinement in the estimation and an illustrative example are also given

regularity of optimal transport on compact locally nearly spherical manifolds given a couple of smooth positive measures of same total mass on a compact connected riemannian manifold m we look for a smooth optimal transportation map g pushing one measure to the other at a least total squared distance cost directly by using the continuity method to produce a classical solution of the elliptic equation of monge ampere type satisfied by the potential function u such that g exp grad u this approach boils down to proving an textit a priori upper bound on the hessian of u which was done on the flat torus by the first author the recent local c 2 estimate of ma trudinger wang enabled loeper to treat the standard sphere case by overcoming two difficulties namely in collaboration with the first author he kept the image g m of a generic point m in m uniformly away from the cut locus of m he checked a fourth order inequality satisfied by the squared distance cost function proving the uniform positivity of the so called c curvature of m in the present paper we treat along the same lines the case of manifolds with curvature sufficiently close to 1 in c 2 norm specifying and proving a conjecture stated by trudinger

vector coherent state theory of the generic representations of so 5 in an so 3 basis for applications of group theory in quantum mechanics one generally needs explicit matrix representations of the spectrum generating algebras that arise in bases that reduce the symmetry group of some hamiltonian of interest here we use vector coherent state techniques to develop an algorithm for constructing the matrices for arbitrary finite dimensional irreps of the so 5 lie algebra in an so 3 basis the so 3 subgroup of so 5 is defined by regarding so 5 as linear transformations of the five dimensional space of an so 3 irrep of angular momentum two a need for such irreps arises in the nuclear collective model of quadrupole vibrations and rotations the algorithm has been implemented in maple and some tables of results are presented

a multiple attribute utility theory approach to ranking and selection managers of large industrial projects often measure performance by multiple attributes for example our paper is motivated by the simulation of a large industrial project called a land seismic survey in which project performance is based on duration cost and resource utilization to address these types of problems we develop a ranking and selection procedure for making comparisons of systems e g project configurations that have multiple performance measures the procedure combines multiple attribute utility theory with statistical ranking and selection to select the best configuration from a set of possible configurations using the indifference zone approach we apply our procedure to results generated by the simulator for a land seismic survey that has six performance measures and describe a particular type of sensitivity analysis that can be used as a robustness check

continuity of optimal transport maps and convexity of injectivity domains on small deformations of 2 given a compact riemannian manifold we study the regularity of the optimal transport map between two probability measures with cost given by the squared riemannian distance our strategy is to define a new form of the so called ma trudinger wang condition and to show that this condition together with the strict convexity on the nonfocal domains implies the continuity of the optimal transport map moreover our new condition again combined with the strict convexity of the nonfocal domains allows us to prove that all injectivity domains are strictly convex too these results apply for instance on any small c4 deformation of the 2 sphere 2009 wiley periodicals inc

multicriteria estimation of probabilities on basis of expert non numeric non exact and non complete knowledge abstract a new method of alternatives probabilities estimation under deficiency of expert numeric information obtained from different sources is proposed the method is based on the bayesian model of uncertainty randomization additional non numeric non exact and non complete expert knowledge nnn knowledge nnn information is used for final estimation of the alternatives probabilities an illustrative example demonstrates the proposed method application to forecasting of oil shares price with the use of nnn information obtained from different experts investment firms

using a bayesian approach to quantify scale compatibility bias this paper proposes a new analytical framework to quantify and correct for scale compatibility bias in the assessment of trade off weights in multiattribute value analysis the procedure is demonstrated with an application to a fisheries management problem trade off judgments are elicited from a group of fisheries experts with management responsibility in the lake erie basin then we use a bayesian method to compute posterior probability distributions of attribute weights in computing the bayesian weights our measurement model assumes that the weight ratios produced by each respondent s judgments are subject to random error and an unknown scale compatibility bias ratios are log transformed and analyzed by a bayesian linear model with a noninformative prior distribution posterior distributions are then developed for the weights and the bias we estimate the compatibility bias for each person and in most cases it is found to be large and in the predicted direction suggesting the importance of its consideration in deriving trade off weights in addition the bayesian framework is shown to be useful for quantifying the value of additional information about multiattribute weights finally a simple heuristic procedure for assessing the weights appears to be effective in eliminating the bias

racah s method for general subalgebra chains coupling coefficients of so 5 in canonical and physical bases it is shown that the method of infinitesimal generators racah s method can be broadly and systematically formulated as a method applicable to the calculation of reduced coupling coefficients for a generic subalgebra chain g h provided the reduced matrix elements of the generators of g and the recoupling coefficients of h are known the calculation of so 5 so 4 reduced coupling coefficients is considered as an example and a procedure for transformation of reduced coupling coefficients between canonical and physical subalegebra chains is presented the problem of calculating coupling coefficients for generic irreps of so 5 reduced with respect to any of its subalgebra chains is completely resolved by this approach

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

robust and stochastically weighted multiobjective optimization models and reformulations we introduce and study a family of models for multiexpert multiobjective criteria decision making these models use a concept of weight robustness to generate a risk averse decision in particular the multiexpert multicriteria robust weighted sum approach mcrow introduced in this paper identifies a robust pareto decision that minimizes the worst case weighted sum of objectives over a given weight region the corresponding objective value called the robust value of a decision is shown to be increasing and concave in the weight set we study compact reformulations of the mcrow model with polyhedral and conic descriptions of the weight regions the mcrow model is developed further for stochastic multiexpert multicriteria decision making by allowing ambiguity or randomness in the weight region as well as the objective functions the properties of the proposed approach are illustrated with a few textbook examples the usefulness of the stochastic mcrow model is demonstrated using a disaster planning example and an agriculture revenue management example

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

a note on the subjective and objective integrated approach to determine attribute weights abstract this note points out that the objective weight introduced by ma et al eur j oper res 112 1999 397 can be very different from the objective weight obtained by the entropy method we also query the motive and the rationale of the objective weight of ma et al

a riemannian interpolation inequality a la borell brascamp and lieb a concavity estimate is derived for interpolations between l 1 m mass densities on a riemannian manifold the inequality sheds new light on the theorems of prekopa leindler borell brascamp and lieb that it generalizes from euclidean space due to the curvature of the manifold the new riemannian versions of these theorems incorporate a volume distortion factor which can however be controlled via lower bounds on ricci curvature the method uses optimal mappings from mass transportation theory along the way several new properties are established for optimal mass transport and interpolating maps on a riemannian manifold

smaa 2 stochastic multicriteria acceptability analysis for group decision making stochastic multicriteria acceptability analysis smaa is a multicriteria decision support method for multiple decision makers in discrete problems in smaa the decision makers need not express their preferences explicitly or implicitly instead the method is based on exploring the weight space in order to describe the valuations that would make each alternative the preferred one inaccurate or uncertain criteria values are represented by probability distributions from which the method computes confidence factors describing the reliability of the analysis in this paper we introduce the smaa 2 method which extends the original smaa by considering all ranks in the analysis in situations where the elitistic smaa may assess large acceptability only for extreme alternatives without sufficient majority support the more holistic smaa 2 analysis can be used to identify good compromise candidates the results are presented graphically we consider also situations where partial preference information is available we demonstrate the new method using a real life decision problem

ordinal judgments in multiattribute decision analysis abstract the article discusses the contradiction between the ambiguity of human judgment in a multicriterion environment and the exactness of the assessments required in the majority of the decision making methods preferential information from the decision makers in the ordinal form e g more preferable less preferable etc is argued to be more stable and more reliable than cardinal input ways of obtaining and using ordinal judgments for rank ordering of multiattribute alternatives are discussed the effectiveness of the step wise procedure of using ordinal tradeoffs for comparison of alternatives is evaluated we introduce the notion of ordinal tradeoffs presentation of ordinal tradeoffs as a flexible three stage process a paired joint ordinal scale pjos and evaluation of the effectiveness of the three stage process simulation results examine the sensitivity of the number of pairwise comparisons required for given numbers of criteria and categories within criteria as well as the number of alternatives analyzed this simulation shows that ordinal pairwise comparisons provide sufficient power to discriminate between 75 and 80 of the alternatives compared while the proportional number of pairwise comparisons relative to the maximum possible decreases with the number of criteria and categories the method is relatively insensitive to the number of alternatives considered

comparison of weighting judgments in multiattribute utility measurement this paper compares four weighting methods in multiattribute utility measurement the ratio method the swing weighting method the tradeoff method and the pricing out method 200 subjects used these methods to weight attributes for evaluating nuclear waste repository sites in the united states the weighting methods were compared with respect to their internal consistency convergent validity and external validity internal consistency was measured by the degree to which ordinal and cardinal or ratio responses agreed within the same weighting method convergent validity was measured by the degree of agreement between the weights elicited with different methods external validity was determined by the degree to which weights elicited in this experiment agreed with weights that were elicited with managers of the department of energy in terms of internal consistency the tradeoff method fared worst in terms of convergent validity the pricing out method turned out to be an outlier in terms of external validity the pricing out method showed the best results while the ratio and swing methods are quite consistent and show a fair amount of convergent validity their external validity problems cast doubt on their usefulness the main recommendation for applications is to improve the internal consistency of the tradeoff method by careful interactive elicitation and to use it in conjunction with the pricing out method to enhance its external validity

decision aid for discrete multiple criteria decision making problems with imprecise data abstract we describe ways of aiding decision making with a discrete set of alternatives in many decision situations it is not possible to obtain explicit preference information from the decision makers instead useful decision aid can be provided to the decision makers by describing what kind of weighting of the criteria result in certain choices of the alternatives the suggested treatment is based on the basic ideas of the electre iii method the modelling of the preferences by pseudo criteria is especially helpful in case the data that is the criterion values are imprecise unlike electre iii no ranking of the alternatives is produced based on a minimum procedure in the exploitation of the outranking relations we provide information about the weights of the criteria that make a certain alternative the best we also present an interactive searching procedure in the weighting space the auxiliary optimization problems to be solved are nondifferentiable cases with both single and multiple decision makers are considered

regularity of optimal transport in curved geometry the nonfocal case we explore some geometric and analytic consequences of a curvature condition introduced by ma trudinger and wang in relation to the smoothness of optimal transport in curved geometry we discuss a conjecture according to which a strict version of the ma trudinger wang condition is sufficient to prove regularity of optimal transport on a riemannian manifold we prove this conjecture under a somewhat restrictive additional assumption of nonfocality at the same time we establish the striking geometric property that the tangent cut locus is the boundary of a convex set partial extensions are presented to the case when there is no pure focalization on the tangent cut locus

on policy capturing with fuzzy measures policy capturing methods generally apply linear regression analysis to model human judgment in this paper we examine the application of fuzzy set and fuzzy measure theories to obtain subjective descriptions of cue importance for policy capturing at the heart of the approach is a method of learning fuzzy measures the shapley values associated with the fuzzy measures provide a basis for comparison with the results of linear regression however the fuzzy measure theoretical approach provides additional insight into interaction effects corresponding to the nonlinear noncompensatory nature of the underlying decision model to illustrate the methodology we estimated the importance of factors and the interactions among them that influence decisions related to strategic investments in telecommunications infrastructure and compared the results from the fuzzy approach to those obtained from traditional statistical methods

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

holder continuity and injectivity of optimal maps consider transportation of one distribution of mass onto another chosen to optimize the total expected cost where cost per unit mass transported from x to y is given by a smooth function c x y if the source density f x is bounded away from zero and infinity in an open region u subset mathbf r n and the target density f y is bounded away from zero and infinity on its support overline v subset mathbf r n which is strongly c convex with respect to u and the transportation cost c satisfies the mathbf a3 rm w condition of trudinger and wang ann sc norm super pisa cl sci 5 8 1 143 174 2009 we deduce the local holder continuity and injectivity of the optimal map inside u so that the associated potential u belongs to c 1 alpha loc u here the exponent 0 depends only on the dimension and the bounds on the densities but not on c our result provides a crucial step in the low interior regularity setting in a sequel figalli et al j eur math soc jems 1131 1166 2013 we use it to establish regularity of optimal maps with respect to the riemannian distance squared on arbitrary products of spheres three key tools are introduced in the present paper namely we first find a transformation that under mathbf a3 rm w makes c convex functions level set convex as was also obtained independently from us by liu calc var partial diff eq 34 435 451 2009 we then derive new alexandrov type estimates for the level set convex c convex functions and a topological lemma showing that optimal maps do not mix the interior with the boundary this topological lemma which does not require mathbf a3 rm w is needed by figalli and loeper calc var partial diff eq 35 537 550 2009 to conclude the continuity of optimal maps in two dimensions in higher dimensions if the densities f are holder continuous our result permits continuous differentiability of the map inside u in fact c 2 alpha loc regularity of the associated potential to be deduced from the work of liu et al comm partial diff eq 35 1 165 184 2010

estimating aspiration levels from discrete choices computational techniques and experiences abstract this paper discusses the problem of estimating aspiration or reference levels in criteria space from choices between alternatives characterized by multiple criteria several methods for such estimations are developed and compared taking into account both solution quality and computational efficiency methods based on mixed integer linear programming which provide optimal solutions require unacceptable computing times for application in interactive systems an alternative method based on a direct search algorithm is shown to be an effective way of generating high quality estimates with small computational effort

a generalized decomposable multiattribute utility function in this paper a generalized decomposable multiattribute utility function mauf is developed it is demonstrated that this new mauf structure is more general than other well known mauf structures such as additive multiplicative and multilinear therefore it is more flexible and does not require that the decision maker be consistent with restrictive assumptions such as preferential independence conditions about his her preferences we demonstrate that this structure does not require any underlying assumption and hence solves the interdependence among attributes hence there is no need for verification of its structure several useful extensions and properties for this generalized decomposable mauf are developed which simplify its structure or assessment the concept of utility efficiency is developed to identify efficient alternatives when there exists partial information on the scaling constants of an assumed mauf it is assumed that the structure decomposition of the mauf is known and the partial information about the scaling constants of the decision maker is in the form of bounds or constraints for the generalized decomposable structure linear programming is sufficient to solve all ensuing problems some examples are provided

compromise solution by mcdm methods a comparative analysis of vikor and topsis the multiple criteria decision making mcdm methods vikor and topsis are based on an aggregating function representing closeness to the ideal which originated in the compromise programming method in vikor linear normalization and in topsis vector normalization is used to eliminate the units of criterion functions the vikor method of compromise ranking determines a compromise solution providing a maximum group utility for the majority and a minimum of an individual regret for the opponent the topsis method determines a solution with the shortest distance to the ideal solution and the greatest distance from the negative ideal solution but it does not consider the relative importance of these distances a comparative analysis of these two methods is illustrated with a numerical example showing their similarity and some differences 2003 elsevier b v all rights reserved

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

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

highlights and critical points in the theory and application of the analytic hierarchy process abstract this paper provides a detailed discussion with references on the fundamentals of the analytic hierarchy process and in particular of relative measurement the points discussed are grouped under the following categories structure in the ahp hierarchies and networks scales of measurement judgments consistency and the eigenvector synthesis and normative vs descriptive the paper also includes a discussion of rank and a number of citations of rank reversals attributed to a variety of factors ranging from intransitivity to procedure invariance that are thought to be unexplained by utility theory with its underlying principle to always preserve rank it is shown that when there is synergy due to the number of elements the ahp can be used to both preserve rank when it is desired to preserve it and allow it to reverse when it should reverse

an algebraic approach to problems with polynomial hamiltonians on euclidean spaces explicit expressions are given for the actions and radial matrix elements of basic radial observables on multi dimensional spaces in a continuous sequence of orthonormal bases for unitary su 1 1 irreps explicit expressions are also given for so n reduced matrix elements of basic orbital observables these developments make it possible to determine the matrix elements of polynomial and other hamiltonians analytically to within so n clebsch gordan coefficients and to select an optimal basis for a particular problem such that the expansion of eigenfunctions is most rapidly convergent

representation of the five dimensional harmonic oscillator with scalar valued u 5 so 5 so 3 coupled vcs wave functions vector coherent state methods which reduce the u 5 so 5 so 3 subgroup chain are used to construct basis states for the five dimensional harmonic oscillator algorithms are given to calculate matrix elements in this basis the essential step is the construction of so 5 so 3 irreps of type v 0 the methodology is similar to that used in two recent papers except that one dimensional as opposed to multidimensional vector valued wave functions are used to give conceptually simpler results another significant advance is a canonical resolution of the so 5 so 3 multiplicity problem

clebsch gordan coefficients for u 8 o 8 su 3 the group chain u 8 o 8 su 3 plays an important role in many particle systems whenever the fundamental particles have eight degrees of freedom as a particular example the systems of many gluons and pairs of quark antiquark are discussed which can be coupled to a flavor octet in order to determine the explicit structure of states and decay probabilities the calculation of the clebsch gordan coefficients cgc of this group chain is indispensable in this contribution the polynomial states of the u 8 chain are constructed and also the isoscalar factors of the cgc s tables of isoscalar factors are presented the method shown serves as an example for higher rank groups

the effects of splitting attributes on weights in multiattribute utility measurement this study examined how weights in multiattribute utility measurement change when objectives are split into more detailed levels subjects were asked to weight attributes in value trees containing three objectives which were specified by either three four five or six attributes the robust finding was that the more detailed parts of the value tree were weighted significantly higher than the less detailed ones this overweighting bias was found for several weighting techniques but the techniques that used holistic judgments to derive weights were affected somewhat less than techniques that used decomposed attribute weights this bias is interpreted in terms of the increased salience and availability of attributes that are spelled out in more detail

clebsch gordan coefficients of su 3 in su 2 and so 3 bases new algorithms are developed for the purpose of optimizing the efficient calculation of su 3 clebsch gordan coefficients in both su 2 and so 3 coupled bases the new algorithms make use of the fact that highest weight states in a tensor product space are easily identified by vector coherent state methods the methods are developed for su 3 but apply to other compact semi simple lie groups

decision quality using ranked attribute weights three published approximation formulae for selecting the best multiattribute alternative based on rank ordered weights are evaluated all formulae are surprisingly efficacious in determining the best multiattribute alternative rank order centroid roc weights are more accurate than the other rank based formulae furthermore the roc formula generalizes to incorporate both other forms of partial information about attribute weights and partial rank order information as well because a roc based analysis is so straightforward and efficacious it provides an appropriate implementation tool

fuzzy outranking for environmental assessment case study iron and steel making industry recent years have seen a world wide change in the environmental policy towards integrated pollution prevention taking into account all environmental media consequently the environmental assessment of production techniques has to consider multiple criteria which cannot be aggregated to one single index the concepts of multi criteria decision making mcdm seem a suitable means in order to implement life cycle assessment lca into integrated decision processes the outranking methods as a special subgroup of mcdm methods are particularly suitable for integral decision making through the notion of weak preference and incomparability which better represent the real decision situation as demonstrated in this paper especially the outranking method promethee brings together flexibility and simplicity for the user and is therefore chosen for the enhancement towards the evaluation of fuzzy data on preferences scores and weights

polar factorization and monotone rearrangement of vector valued functions given a probability space x p and a bounded domain r in rd equipped with the lebesgue measure 1 i normalized so that 10 i i it is shown under additional technical assumptions on x and q that for every vector valued function u e lp x p rd there is a unique polar factorization u v s where is a convex function defined on r and s is a measure preserving mapping from x p into q i i provided that u is nondegenerate in the sense that p u e 0 for each lebesgue negligible subset e of rd through this result the concepts of polar factorization of real matrices helmholtz decomposition of vector fields and nondecreasing rearrangements of real valued functions are unified the monge amgre equation is involved in the polar factorization and the proof relies on the study of an appropriate monge kantorovich problem

the geometry of optimal transportation introduction 113 1 summary of main results 120 2 background on optimal measures 126 part i strictly convex costs 133 3 existence and uniqueness of optimal maps 133 4 characterization of the optimal map 137 part ii costs which are strictly concave as a function of d i s t a n c e 141 5 the role of optimal maps 141 6 uniqueness of optimal solutions 144 part iii appendices 148 a legendre transforms and conjugate costs 148 b examples of c concave potentials 152 c regularity of c concave potentials 154 references 159