modeling interaction phenomena using fuzzy measures on the notions of interaction and independence interaction phenomena among elements of a finite set can be modeled by a discrete fuzzy measure the notion of marginal interaction between two elements is at the root of the definition of interaction we first extend this notion by defining that of marginal interaction among pairwise disjoint subsets and we study some of its properties then we generalize the notion of interaction index to pairwise disjoint subsets finally we formalize the intuitive notion of independence and we propose the concept of measure of marginal amount of interaction in order to fill the gap between the notions of interaction and independence

a methodology to determine optimal transmissivity measurement locations in groundwater quality management models with scarce field information a new criterion to determine transmissivity measurement location for groundwater quality management models is proposed the transmissivity is represented as a random field with known mean and covariance function conditioned to existing measurements values the criterion is the minimization of the coefficient of variation of the optimal remediation cost distribution this distribution is computed numerically by means of monte carlo simulations in which each optimal remediation cost is evaluated by solving a deterministic management problem where the transmissivity is a single realization of the random field the coefficient of variation is equivalent to the relative error in the prediction of the optimal cost and is shown to be invariant with respect to the actual unknown measurement value according to some limiting conditions in the more general case the coefficient of variation of the optimal cost is less sensitive than the expected optimal cost to the actual measurement value the proposed criterion becomes a useful tool to determine the best transmissivity measurement location when only few measurements have been performed

dynamic analysis of continuous plane trusses with equidistant supports the natural and forced vibrations of a continuous warren truss resting on equidistant roller supports are studied in the analyses the longitudinal inertial forces are neglected in the equation of motion and the mass of the truss is replaced by two sets of lumped masses at the nodes in the forced vibration case the truss is subjected to transverse harmonic loads acting at the nodes the truss can be considered as an equivalent system with cyclic bi periodicity periodicities due to the repetitive units of the truss and the supports respectively therefore the governing equation for such a system can be uncoupled by applying u transformation twice to form a set of single degree of freedom equations such equations can be used to solve for the natural frequencies of the trusses or their responses due to external excitations typical warren trusses are taken as examples to demonstrate the advantages of the proposed method and the accuracy that can be achieved the response of a truss with six substructures and four supports with a harmonic load acting at the centre node is investigated

a simplified expression of the shapley function for fuzzy game abstract in this paper a simplified expression of the shapley function for games with fuzzy coalition is proposed which can be regarded as the generalization of shapley functions defined in some particular games with fuzzy coalition the simplified expression of the shapley function is compared with two definitions established by butnariu tsurumi et al a conclusion is drawn that the simplified expression of the shapley function is equivalent to butnariu s definition when characteristic function is a game with proportional values and is equivalent to tsurumi s definition when characteristic function is a game with choquet integral forms furthermore from an angle of interaction between two participation levels the properties of the two games defined by butnariu and tsurumi are respectively studied

on the set of imputations induced by the k additive core an extension to the classical notion of core is the notion of k additive core that is the set of k additive games which dominate a given game where a k additive game has its mobius transform or harsanyi dividends vanishing for subsets of more than k elements therefore the 1 additive core coincides with the classical core the advantages of the k additive core is that it is never empty once k geq 2 and that it preserves the idea of coalitional rationality however it produces k imputations that is imputations on individuals and coalitions of at most k individuals instead of a classical imputation therefore one needs to derive a classical imputation from a k order imputation by a so called sharing rule the paper investigates what set of imputations the k additive core can produce from a given sharing rule

stochastic determination of well capture zones conditioned on regular grids of transmissivity measurements a stochastic approach is adopted for characterizing the uncertainty in time related well capture zones in heterogeneous formations transmissivity is modeled as a random space function conditioned on regular grids of evenly distributed transmissivity measurements monte carlo analysis is used to infer the probability distribution of the resulting stochastic capture zones three measures of conditioning performance are defined and evaluated for several combinations of conditioning density transmissivity variance and transmissivity integral scale furthermore a connection is made with previous work on transport theory simulation results show that conditioning performance increases with an increasing conditioning density an increasing transmissivity integral scale and a decreasing transmissivity variance however there is a threshold combination of conditioning density and integral scale beyond which conditioning performance stagnates the practical implications of the results in general and the threshold in particular are discussed

nonstationary stochastic analysis in well capture zone design using first order taylor s series approximation 1 nonstationarity of flow fields due to pumping wells and its impact on advective transport is of particular interest in well capture zone design and wellhead protection however techniques based on monte carlo methods to characterize the associated capture zone uncertainty are time consuming and cumbersome this paper introduces an alternative approach the mean and covariance of system state variables i e head pore water velocity and particle trajectory are approximated using a first order taylor s series with sensitivity coefficients estimated from the adjoint operator for a system of discrete equations the approach allows nonstationarity due to several sources e g transmissivity pumping boundary conditions to be treated by employing numerical solution methods it is able to handle irregular geometry varying boundary conditions complicated sink source terms and different covariance functions all of which are important factors for real world applications a comparison of results for the taylor s series approximation with those from monte carlo analysis showed in general good agreement for most of the tested particles particle trajectory variance calculated using taylor s series approximation is then used to predict well capture zone probabilities under the assumption of normality of the mass transport s state variables verification of this assumption showed that not all particle trajectories depending on their starting location are normally or log normally distributed however the risk of using the first order method to delineate the confidence interval of a well capture zone is minimal since it marginally overestimates the 2 5 probability contour furthermore this should be balanced against its greater computation efficiency over the monte carlo approach

semianalytical uncertainty estimation of well catchments conditioning by head and transmissivity data 1 the goal of the investigations is to estimate the uncertainty in the location of two dimensional steady state catchments of pumping wells due to the uncertainty of the spatially variable transmissivity field the uncertainty is evaluated conditional to measured transmissivity and head data the uncertainty bandwidth of the catchment boundary is approximated in first order by formulating the conditional transversal second moment of the particle displacements along the expected mean particle trajectory however empirical relationships have to be provided for the estimation of the uncertainty in the location of stagnation points applications of the approach are presented for a synthetic test case with two arrangements of measurement stations the results are compared with those from conditional numerical monte carlo simulations the comparison allows an assessment of the accuracy and the applicability of the method the approximation corresponds quite well with the monte carlo simulations

weighted banzhaf power and interaction indexes through weighted approximations of games the banzhaf power index was introduced in cooperative game theory to measure the real power of players in a game the banzhaf interaction index was then proposed to measure the interaction degree inside coalitions of players it was shown that the power and interaction indexes can be obtained as solutions of a standard least squares approximation problem for pseudo boolean functions considering certain weighted versions of this approximation problem we define a class of weighted interaction indexes that generalize the banzhaf interaction index we show that these indexes define a subclass of the family of probabilistic interaction indexes and study their most important properties finally we give an interpretation of the banzhaf and shapley interaction indexes as centers of mass of this subclass of interaction indexes

on the structure of some families of fuzzy measures the generation of fuzzy measures is an important question arising in the practical use of these operators in this paper we deal with the problem of developing a random generator of fuzzy measures more concretely we study some of the properties that any random generator should satisfy these properties lead to some theoretical problems concerning the group of isometries that we tackle in this paper for some subfamilies of fuzzy measures

determination of weights of interacting criteria from a reference set abstract in this paper we present a model allowing to determine the weights related to interacting criteria this is done on the basis of the knowledge of a partial ranking over a reference set of alternatives prototypes a partial ranking over the set of criteria and a partial ranking over the set of interactions between pairs of criteria

evolutionary algorithms for the optimization of advective control of contaminated aquifer zones 1 simple genetic algorithms sgas and derandomized evolution strategies dess are employed to adapt well capture zones for the hydraulic optimization of pump and treat systems a hypothetical contaminant site in a heterogeneous aquifer serves as an application template on the basis of the results from numerical flow modeling particle tracking is applied to delineate the pathways of the contaminants the objective is to find the minimum pumping rate of up to eight recharge wells within a downgradient well placement area both the well coordinates and the pumping rates are subject to optimization leading to a mixed discrete continuous problem this article discusses the ideal formulation of the objective function for which the number of particles and the total pumping rate are used as decision criteria boundary updating is introduced which enables the reorganization of the decision space limits by the incorporation of experience from previous optimization runs throughout the study the algorithms capabilities are evaluated in terms of the number of model runs which are needed to identify optimal and suboptimal solutions despite the complexity of the problem both evolutionary algorithm variants prove to be suitable for finding suboptimal solutions the des with weighted recombination reveals to be the ideal algorithm to find optimal solutions though it works with real coded decision parameters it proves to be suitable for adjusting discrete well positions principally the representation of well positions as binary strings in the sga is ideal however even if the sga takes advantage of bookkeeping the vital high discretization of pumping rates results in long binary strings which escalates the model runs that are needed to find an optimal solution since the sga string lengths increase with the number of wells the des gains superiority particularly for an increasing number of wells as the des is a self adaptive algorithm it proves to be the more robust optimization method for the selected advective control problem than the sga variants of this study exhibiting a less stochastic search which is reflected in the minor variability of the found solutions

the combination of evidence in the transferable belief model a description of the transferable belief model which is used to quantify degrees of belief based on belief functions is given the impact of open and closed world assumption on conditioning is discussed the nature of the frame of discernment on which a degree of belief will be established is discussed a set of axioms justifying dempster s rule for the combination of belief functions induced by two distinct evidences is presented

stochastic capture zone delineation within the generalized likelihood uncertainty estimation methodology conditioning on head observations a stochastic methodology to evaluate the predictive uncertainty in well capture zones in heterogeneous aquifers with uncertain parameters is presented the approach is based on the generalized likelihood uncertainty estimation methodology the hydraulic conductivity is modeled as a random space function allowing for the uncertainty that stems from the imperfect knowledge of the parameters of the correlation structure parameters are sampled from prior distributions and are used for the generation of a large number of hydraulic conductivity fields which are subsequently used to solve the groundwater flow equation a likelihood is calculated for every simulation based on some goodness of fit measure between simulated heads and available observations using inverse particle tracking a capture zone is determined which is assigned the likelihood calculated for that particular simulation statistical analysis of the ensemble of all simulations enables the predictive uncertainty of the well capture zones to be defined results are presented for a hypothetical test case and different likelihood definitions used in the conditioning process the results show that the delineated capture zones are most sensitive to the mean hydraulic conductivity and the variance whereas the integral scale of the variogram is the parameter with the smallest influence for all likelihood measures the prior uncertainty is reduced considerably by introducing the observation heads but the reduction is most effective for the very selective likelihood definition the method presented can be used in real applications to quantify the uncertainty in the location and extent of well capture zones when little or no information is available about the hydraulic properties through the conditioning on head observations

k intolerant capacities and choquet integrals we define an aggregation function to be at most k intolerant if it is bounded from above by its kth lowest input value applying this definition to the discrete choquet integral and its underlying capacity we introduce the concept of k intolerant capacities which when varying k from 1 to n cover all the possible capacities on n objects just as the concepts of k additive capacities and p symmetric capacities have been previously introduced essentially to overcome the problem of computational complexity of capacities k intolerant capacities are proposed here for the same purpose but also for dealing with intolerant or tolerant behaviors of aggregation we also introduce axiomatically indices to appraise the extent to which a given capacity is k intolerant and we apply them on a particular recruiting problem

exact solution for vibration analysis of rectangular cable networks with periodically distributed supports abstract rectangular cable networks supported by periodically distributed posts can be regarded as a structure with bi periodicity in two orthogonal directions by considering an equivalent system with cyclic bi periodicity and applying the double u transformation technique twice the harmonic vibration equation can be uncoupled into a set of single variable equations and that leads to the exact solutions as an example a square cable network with a 6 6 mesh and 2 2 internal supports is considered the solutions of natural and forced vibrations are worked out by using the formulas obtained in the present study

conditional estimation of solute travel time in heterogeneous formations impact of transmissivity measurements solute transport through heterogeneous formations is modeled by the travel time approach i e the time it takes a solute particle to travel from the source to a control plane due to uncertainty is a random variable characterized by its cumulative probability density function g this function is used by regulatory agencies in pollution site assessment the main aim of the present study is to examine the impact of data namely measured transmissivities upon g and reduction of uncertainty this is achieved for two dimensional steady flow of average uniform head gradient by using a lagrangian approach developed in the past dagan 1982 1984 1989 rubin 1990 1991 the impact of data is seen in conditioning on measured values and in estimation of parameters characterizing flow and formation properties the approach is illustrated by a simulation based on synthetic data the results show how uncertainty is reduced by increasing the density of measurements for a control plane sufficiently far from the source

exact analysis of localized modes in two dimensional bi periodic mass spring systems with a single disorder abstract the frequency equation and localized modes in two dimensional bi periodic mass spring systems with one disordered subsystem are exactly analyzed by means of double u transformation at first a plane distributed mass spring system with 2 n 1 2 n 2 subsystems and cyclic periodicity in x and y directions is considered then by adopting a limiting process with n 1 n 2 approaching to infinity the limiting solution is applicable for the plane distributed bi periodic mass spring systems with boundary at infinity the explicit frequency equation and localized modes are derived some specific systems are taken as examples to demonstrate how to apply the formulas and equations obtained in the present paper in order to find the localized modes and corresponding frequencies

random generation of k interactive capacities abstract the theory of capacities provides powerful formal methodology to account for criteria dependencies in multiple criteria decision problems the discrete choquet and sugeno integrals aggregate criteria valuations accounting for criteria synergies and redundancies we address an important problem of randomly generating capacities of special classes for simulation studies and for capacity learning through evolutionary algorithms we discuss two efficient methods suitable for k interactive capacities the results are supported by the extensive numerical evidence and provide a useful tool for large scale simulations

the application of fuzzy integrals in multicriteria decision making abstract this paper presents a synthesis on the application of fuzzy integrals as an innovative tool for criteria aggregation in decision problems the main point is that fuzzy integrals are able to model interaction between criteria in a flexible way the methodology has been elaborated mainly in japan and has been applied there successfully in various fields such as design reliability evaluation of goods etc it seems however that this technique is still very little known in europe it is one of the aims of this review to disseminate this emerging technology in many industrial fields

bayesian methodology to stochastic capture zone determination conditioning on transmissivity measurements 1 a methodology to determine the uncertainty associated with the delineation of well capture zones in heterogeneous aquifers is presented the log transmissivity field is modeled as a random space function and the bayesian paradigm accounts for the uncertainty that stems from the imperfect knowledge about the parameters of the stochastic model unknown parameters are treated as random quantities and characterized by a prior probability distribution log transmissivity measurements are incorporated into bayes theorem updating the prior distribution and yielding posterior estimates of the mean value and the covariance parameters of the log transmissivity conditional simulations of the log transmissivity field are generated using samples from the posterior distribution of the parameters yielding samples from the predictive distribution of the log transmissivity field the uncertainty in the model parameters is propagated to the predictive uncertainty of the capture zone by solving numerically the groundwater flow equation followed by a semianalytical particle tracking algorithm the method is applied to a set of hypothetical flow fields for various sampling densities and assuming different levels of parameter uncertainty simulation results for all the sampling densities show no univocal relation between the predictive uncertainty of the capture zones and the level of parameter uncertainty however in general the predictive uncertainty increases when parameter uncertainty is taken into account

fuzzy sets as a basis for a theory of possibility the theory of possibility described in this paper is related to the theory of fuzzy sets by defining the concept of a possibility distribution as a fuzzy restriction which acts as an elastic constraint on the values that may be assigned to a variable more specifically if f is a fuzzy subset of a universe of discourse u u which is characterized by its membership function f then a proposition of the form x is f where x is a variable taking values in u induces a possibility distribution tx which equates the possibility of x taking the value u to f u the compatibility of u with f in this way x becomes a fuzzy variable which is associated with the possibility distribution tx in much the same way as a random variable is associated with a probability distribution in general a variable may be associated both with a possibility distribution and a probability distribution with the weak connection between the two expressed as the possibility probability consistency principle a thesis advanced in this paper is that the imprecision that is intrinsic in natural languages is in the main possibilistic rather than probabilistic in nature thus by employing the concept of a possibility distribution a proposition p in a natural language may be translated into a procedure which computes the probability distribution of a set of attributes which are implied by p several types of conditional translation rules are discussed and in particular a translation rule for propositions of the form x is f is possible where is a number in the interval 0 1 is formulated and illustrated by examples

k additivity and c decomposability of bi capacities and its integral k additivity is a convenient way to have less complex bi capacities this paper gives a new characterization of k additivity introduced by grabisch and labreuche of bi capacities and contrasts between the existing characterization and the new one that differs from the one of saminger and mesiar besides in the same way for capacities a concept of c decomposability distinct from the proposal of saminger and mesiar but closely linked to k additivity is introduced for bi capacities moreover the concept of c decomposability applies to the choquet integral with respect to bi capacities

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

simulation of dispersion in heterogeneous porous formations statistics first order theories convergence of computations this paper discusses the results of numerical analysis of dispersion of passive solutes in two dimensional heterogeneous porous formations statistics of flow and transport variables the accuracy and the role of approximations implicit in existing first order theories and the convergence of computational results are investigated the results suggest that quite different rates of convergence with monte carlo runs hold for different spatial moments and that over 1000 realizations are required to stabilize second moments even for relatively mild heterogeneity sigma y 2 1 6 this has implications for the extent of the spatial domain for single realization numerical studies of the same type a comparison of the variance of plumes with the results of linear theories 0 05 sigma y 2 1 6 shows an unexpectedly broad validity field for the theoretical solution obtained from a suitable linearization of flow and transport reformulation of the same problem linearizing in tum the flow or the transport equations shows opposite deviations from the linear theory the interesting consequence is that the errors induced by linearizations in the flow or the transport equations have different signs and their effects on the moments of dispersing plumes are compensating thereby yielding consistent formulations unexpected features of the statistics of probability distributions of longitudinal and transverse velocities and travel times are also computed and discussed

k order additive discrete fuzzy measures and their representation abstract in order to face with the complexity of discrete fuzzy measures we propose the concept of k orderadditive fuzzy measure including usual additive measures and fuzzy measures every discrete fuzzy measure is a k order additive fuzzy measure for a unique k a related topic of the paper is to introduce an alternative representation of fuzzy measures called the interaction representation which sets and extends in a common framework the shapley value and the interaction index proposed by murofushi and soneda

an axiomatic approach of the discrete choquet integral as a tool to aggregate interacting criteria the most often used operator to aggregate criteria in decision making problems is the classical weighted arithmetic mean however in many problems the criteria considered interact and a substitute to the weighted arithmetic mean has to be adopted we show that under rather natural conditions the discrete choquet integral is an adequate aggregation operator that extends the weighted arithmetic mean by taking into consideration of the interaction among criteria the axiomatic that supports the choquet integral is presented and an intuitive approach is proposed as well

robust optimization of the choquet integral we study the problem of the choquet integral maximization for the case when preferences of the decision maker do not define a unique capacity but rather some convex set we introduce a robust version of the problem based on the minimax regret criterion and construct a global optimization algorithm it is shown that the decision rule based on regret minimization over a set of capacities is equivalent to maximization of the choquet integral with respect to a certain capacity belonging to this set we compare our method to various capacity identification approaches developed in the literature and provide a semantic interpretation in terms of information value

average contaminant concentration and mass flow in aquifers from time dependent pumping well data analytical framework 1 conventional water samples are representative for small subsurface volumes relative to the scale of many natural geological heterogeneities we present an analytical framework for estimation of representative field scale average concentrations and mass flows on the basis of much larger sampling volumes that are obtained through so called integral pumping tests the contaminant concentration is then measured as a function of time in a pumping well and used for estimation of the conditions in the aquifer prior to and after pumping increasing the observation scale to the size of the well capture zone this method complements and provides an alternative to conventional monitoring grids where mass flow and concentration may be misinterpreted or plumes even missed because of problems related to grid spacing the not measured initial spatial concentration distribution and the time dependent concentration measured at the well are related through a volterra integral equation of the first kind for limiting cases of short and long dimensionless pumping duration two closed form analytical solutions are given from which the mass flow and average concentration can be evaluated furthermore a new solution for evaluating integral pumping tests of any duration is provided and used for investigating the applicability of the simple analytical closed form solutions for interpreting test results from seven large scale contaminated sites in europe

applying young diagrams to 2 symmetric fuzzy measures with an application to general fuzzy measures abstract in this paper we apply young diagrams a well known object appearing in combinatorics and group representation theory to study some properties of the polytope of 2 symmetric fuzzy measures with respect to a given partition a subpolytope of the set of fuzzy measures the main result in the paper allows to build a simple and fast algorithm for generating points on this polytope in a random fashion besides we also study some other properties of this polytope as for example its volume in the last section we give an application of this result to the problem of identification of general fuzzy measures

extreme points of some families of non additive measures non additive measures are a valuable tool to model many different problems arising in real situations however two important difficulties appear in their practical use the complexity of the measures and their identification from sample data for the first problem additional conditions are imposed leading to different subfamilies of non additive measures related to the second point in this paper we study the set of vertices of some families of non additive measures namely k additive measures and p symmetric measures these extreme points are necessary in order to properly apply a new method of identification of non additive measures based on genetic algorithms whose cross over operator is the convex combination we solve the problem through techniques of linear programming

a bayesian approach to stochastic capture zone delineation incorporating tracer arrival times conductivity measurements and hydraulic head observations this paper presents a methodology to invert tracer arrival times and to incorporate travel time data in the delineation of well capture zones within a bayesian framework the observed arrival times are used to obtain probability based weights for each realization of the hydraulic conductivity field realizations that closely reproduce the observed arrival times are more likely to represent the true or real conductivity field than realizations yielding worse predictions and are consequently characterized by a higher conditional probability in the prediction of the capture zones the realizations are weighted by their respective probability we combine the arrival time data with conductivity measurements and head observations to delineate stochastic capture zones the conductivity measurements update the prior distributions specified for the unknown structural parameters of the conductivity field and are used as conditioning data in the generation of conditional conductivity fields the parameter values used to generate the conductivity realizations are sampled by monte carlo from the updated parameter distributions the head and travel time observations are subsequently used to assign probability based weights to the conductivity realizations by applying bayes theorem the hyperparameters of the error model are assumed unknown and their effect is accounted for by marginalization a synthetic flow setup consisting of a single extraction well in a fully confined aquifer under a regional gradient is used to illustrate the method we evaluate the relative worth of the different data types by introducing them sequentially in the inverse methodology results indicate that the different data types are complementary and that the travel time data are essential to improve the predictions of the capture zones this is confirmed by the results for the case where uncertainty in the homogeneous porosity is accounted for

subjective probability and expected utility without additivity comonotic acts are defined and an axiom of comonotomic independence 19 introduced that weakens the von newmann morgenstern axiom of independence if a nondegenerate continuous and monotonic state independent weak order over acts satisfies comonotonic independence then it induces a unique non necessarily additive probability and a von neumann morgenstern utility an extension of theexpected utility theory covers situations as the ellsberg paradox which are inconsistent with additive expected utility the concept of uncertainty aversion and interpretation of comonotonic independence in the context of social welfare functions are included

k maxitive aggregation functions abstract inspired by the idea of k maxitive measures we introduce and study k maxitive aggregation functions in particular 1 maxitive aggregation functions are shown to be just maxitive aggregation functions i e maxima of distorted inputs we introduce among others a representation of symmetric k maxitive aggregation functions moreover we show that for any k maxitive capacity and the smallest universal integral based on an arbitrary fixed semicopula including the sugeno and shilkret integrals the resulting aggregation function is k maxitive