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

semilinear copulas a family of copulas called semilinear is constructed starting with some assumptions about the linearity of the copulas along some segments of the unit square this family contains some other known families of copulas e g cuadras auge frechet and has a nice statistical interpretation several construction methods are provided especially concerning aggregation of semilinear copulas and a special form of ordinal sum construction is introduced some results about related families of quasi copulas and semicopulas are hence given

a consistency improving method in the analytic hierarchy process in this paper we propose a method to modify a given comparison matrix by which the consistency ratio cr value of the modified matrix is less than that of the original one and give an algorithm to derive a positive reciprocal matrix with acceptable consistency i e cr 0 1 then the convergence theorem for the given algorithm is established and its practicality is shown by some examples

the hellinger correlation in this paper the defining properties of any valid measure of the dependence between two continuous random variables are revisited and complemented with two original ones shown to imply other usu

a heuristic method to rectify intransitive judgments in pairwise comparison matrices abstract this paper investigates the effects of intransitive judgments on the consistency of pairwise comparison matrices statistical evidence regarding the occurrence of intransitive judgements in pairwise matrices of acceptable consistency is gathered by using a monte carlo simulation which confirms that relatively high percentage of comparison matrices satisfying saaty s cr criterion are ordinally inconsistent it is also shown that ordinal inconsistency does not necessarily decrease in the group aggregation process in contrast with cardinal inconsistency a heuristic algorithm is proposed to improve ordinal consistency by identifying and eliminating intransitivities in pairwise comparison matrices the proposed algorithm generates near optimal solutions and outperforms other tested approaches with respect to computation time

shock models with dependence and asymmetric linkages abstract this paper introduces a new class of copulas and shows its relevance for applications in particular a stochastic interpretation in terms of a system of dependence components affected by a global shock is given as a main feature of the model the global shock has an opposite effect on the different components of the system copulas generated by this mechanism are characterized in the bivariate case and their main properties are illustrated connections with concepts like semilinear copulas and conic aggregation functions are also highlighted moreover a high dimensional extension is presented

a multi step goal programming approach for group decision making with incomplete interval additive reciprocal comparison matrices abstract this article presents a goal programming framework to solve group decision making problems where decision makers judgments are provided as incomplete interval additive reciprocal comparison matrices iarcms new properties of multiplicative consistent iarcms are put forward and used to define consistent incomplete iarcms a two step goal programming method is developed to estimate missing values for an incomplete iarcm the first step minimizes the inconsistency of the completed iarcms and controls uncertainty ratios of the estimated judgments within an acceptable threshold and the second step finds the most appropriate estimated missing values among the optimal solutions obtained from the previous step a weighted geometric mean approach is proposed to aggregate individual iarcms into a group iarcm by employing the lower bounds of the interval additive reciprocal judgments a two step procedure consisting of two goal programming models is established to derive interval weights from the group iarcm the first model is devised to minimize the absolute difference between the logarithm of the group preference and that of the constructed multiplicative consistent judgment the second model is developed to generate an interval valued priority vector by maximizing the uncertainty ratio of the constructed consistent iarcm and incorporating the optimal objective value of the first model as a constraint two numerical examples are furnished to demonstrate validity and applicability of the proposed approach

a characterization of the orthogonal grid constructions of copulas a framework for constructing copulas that can be regarded as a patchwork like assembly of arbitrary copulas with non overlapping rectangles as patches is studied in de baets b de meyer h 2007 orthogonal grid constructions of copulas ieee trans fuzzy syst 15 1053 1062 where the authors provide sufficient conditions to derive a family of construction methods our aim in this note is to provide a characterization of such constructions msc primary 60e05 secondary 62h20

potential symmetries and direct reduction methods of order two for partial differential equations written in conservative form a remarkable link between potential symmetries and direct reduction methods of order two is enlightened

characterization of all copulas associated with non continuous random variables we introduce a constructive method by means of a doubly stochastic measure to describe all the copulas that in view of sklar s theorem are able to connect a bivariate distribution to its marginals we use this to give the lower and upper optimal bounds for all the copulas that extend a given subcopula

ortholinear and paralinear semi copulas abstract a new method to construct semi copulas is introduced these semi copulas are called ortholinear resp paralinear semi copulas and their construction is based on linear interpolation on segments that are perpendicular resp parallel to the diagonal of the unit square the classes of ortholinear and paralinear quasi copulas are characterized as well

new variable separation approach application to nonlinear diffusion equations the concept of the derivative dependent functional separable solution ddfss as a generalization to the functional separable solution is proposed as an application it is used to discuss the generalized nonlinear diffusion equations based on the generalized conditional symmetry approach as a consequence a complete list of canonical forms for such equations which admit the ddfss is obtained and some exact solutions to the resulting equations are described

an outranking based general approach to solving group multi objective optimization problems abstract this paper presents a general approach to solving multi objective programming problems with multiple decision makers the proposal is based on optimizing a bi objective measure of collective satisfaction group satisfaction is understood as a reasonable balance between the strengths of an agreeing and an opposing coalition considering also the number of decision makers not belonging to any of these coalitions accepting the vagueness of collective satisfaction even the vagueness of person satisfaction fuzzy outranking relations and other fuzzy logic models are used our method transforms a group multi objective optimization problem into a group choice problem on a decision set composed of a relatively small set of alternatives this set contains the possible acceptable consensuses in the parameter space once such a set has been identified other well known techniques can be used to reach the final choice main advantages a each individual decision maker is concerned with his her own multi objective optimization problem only sharing decision variables own constraints and own mapping between decision variables and objective space are allowed b the search for the best agreement is not limited to portions of the pareto frontiers c no voting rule is used by the optimization algorithm no to some extent arbitrary way of handling collective preferences is needed d no assumptions of transitivity and comparability of preference relations are needed and e the concepts of satisfaction non satisfaction do not depend on distance measures or other to some extent arbitrary norms very good performance of the whole proposal is illustrated by a real size example

semiquadratic copulas based on horizontal and vertical interpolation we introduce several families of semiquadratic copulas i e copulas that are quadratic in any point of the unit square in at least one coordinate of which the diagonal and or opposite diagonal sections are given functions these copulas are constructed by quadratic interpolation on segments connecting the diagonal opposite diagonal and sides of the unit square all interpolations are therefore performed horizontally or vertically for each family we provide the necessary and sufficient conditions on the given diagonal and or opposite diagonal functions and two auxiliary real functions to obtain a copula that has these diagonal and or opposite diagonal functions as diagonal and or opposite diagonal sections just as the product copula is a central member of all families of semilinear copulas based on horizontal and vertical interpolation it turns out that the farlie gumbel morgenstern family of copulas is included in all families of semiquadratic copulas introduced and characterized here

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

conic aggregation functions inspired by the notion of conic t norms we introduce conic aggregation functions such aggregation functions are completely characterized by their zero set special classes of binary conic aggregation functions such as conic quasi copulas and conic copulas are considered we provide the necessary and sufficient conditions on the boundary curve of the zero set of a conic aggregation function to obtain a conic quasi copula and conclude that the class of conic copulas is a proper subclass of the class of conic quasi copulas moreover we characterize the class of singular conic copulas we investigate some aggregations of conic quasi copulas some examples are also provided

direct reduction and differential constraints direct reductions of partial differential equations to systems of ordinary differential equations are in one to one correspondence with compatible differential constraints the differential constraint method is applied to prove that a parabolic evolution equation admits infinitely many characteristic second order reductions but admits a non characteristic second order reduction if and only if it is linearizable

a fuzzy extension of saaty s priority theory we present a fuzzy method for choosing among a number of alternatives under conflicting decision criteria a fuzzy version of saaty s pairwise comparison method 1980 extended by de graan 1980 and lootsma 1981 each ratio expressing the relative significance of a pair of factors is displayed in a matrix from which suitable weights can be extracted since these ratios are essentially fuzzy they express the opinion of a decision maker on the importance of a pair of factors we have adapted the above mentioned method in such a way that decision makers are asked to express their opinions in fuzzy numbers with triangular membership functions we apply the method at two distinct levels first to find fuzzy weights for the decision criteria and second to find fuzzy weights for the alternatives under each of the decision criteria by a suitable combination of these results we obtain fuzzy scores for the alternatives as well as their sensitivities using these fuzzy scores the decision makers should be able to make a choice for one of the alternatives

group decision making based on incomplete intuitionistic multiplicative preference relations the incomplete intuitionistic multiplicative preference relation incomplete impr whose elements are intuitionistic multiplicative numbers is suitable for describing the asymmetric preference information and allows the decision makers to skip some dubious comparisons flexibly in this paper the concept of incomplete impr is introduced and the approaches for group decision making gdm with incomplete imprs is investigated to do this the consistency property of an impr is discussed especially the acceptable consistency based on it two approaches to complement all missing elements of incomplete imprs are presented which involve two steps estimating step and adjusting step firstly a geometric mean method is given for estimating initial values of the missing elements then two different methods are developed for adjusting those initial values one adopts the local optimization models which is time saving the other utilizes an iterative method which can operate the whole optimization process conveniently and flexibly after that two procedures for gdm with incomplete imprs are proposed and illustrated by a problem of venue selection for communication drills compared with the previous work the proposed methods have better performance on the computational complexity and the speed for processing a large amount of data

preference simulation and preference programming robustness issues in priority derivation abstract decision makers often resist having to make what appears to them as precise numerical judgements in fuzzy situations pairwise verbal comparisons used in the ahp are fuzzy in the sense that decision maker s need not relate verbal judgment to precise numbers because of the redundancy inherent in each set of judgments accurate priorities can be derived from such fuzzy verbal judgments another way of making fuzzy judgments is to express each judgment as a numerical interval this paper explores two new approaches for priority derivation when preferences are expressed as interval judgments one based on a simulation approach and the other based on mathematical programming the first approach assumes that the interval judgments are uniformly distributed and proceeds to derive the priority vectors and their underlying rank order by randomly sampling from these distribution this approach provides in addition to the priority vectors a measure of robustness given by the probability of rank reversal the second approach generates a region if one exists that encloses all priority vectors derived from inequalities representing the original interval judgments the two approaches are described and illustrated through a numerical example

zero linear copulas abstract we introduce a new method for constructing copulas c a special type of aggregation functions based on the idea that c u v 0 if u g v for any continuous and decreasing function g 0 1 0 1 such that g 1 0 and c is linear if u g v 1 we also study the relationship between these copulas and other known families of copulas we provide some salient properties and measure the dependence and asymmetry in this class of copulas

copulas diagonals and tail dependence we present some known and novel aspects about bivariate copulas with prescribed diagonal section by highlighting their use in the description of the tail dependence moreover we present the tail concentration function which depends on the diagonal section of a copula as a tool to give a description of tail dependence at finite scale the tail concentration function is hence used to introduce a graphical tool that can help to distinguish different families of copulas in the copula test space moreover it serves as a basis to determine the grouping structure of different financial time series by taking into account their pairwise tail behavior

new similarity reductions of the boussinesq equation some new similarity reductions of the boussinesq equation which arises in several physical applications including shallow water waves and also is of considerable mathematical interest because it is a soliton equation solvable by inverse scattering are presented these new similarity reductions including some new reductions to the first second and fourth painleve equations cannot be obtained using the standard lie group method for finding group invariant solutions of partial differential equations they are determined using a new and direct method that involves no group theoretical techniques

on the weak symmetry groups of partial differential equations abstract we define an algorithm to characterize all weak symmetry groups of a partial differential equation new invariant solutions to the fokker planck equation are found

zero sets of copulas abstract we study conditions on sets in order to be zero sets of semi copulas quasi copulas and with special attention of copulas we find necessary and sufficient conditions for characterizing the zero sets of absolutely continuous copulas and copulas whose support coincides with the closure of the complementary of the zero set moreover we study several topological properties and the lattice theoretic structure and characterize the zero sets of the class of archimedean copulas

nonclassical symmetry solutions and the methods of bluman cole and clarkson kruskal nonclassical symmetry solutions of physically relevant partial differential equations are considered via the reduction methods of bluman and cole and clarkson and kruskal consistency conditions will be provided to show that if satisfied these two methods are equivalent in the sense that they lead to the same symmetry solutions the boussinesq equation and burgers equation are used as illustrative examples exact solutions one of which is new will be presented for burgers equation obtained from the bluman and cole method yet not obtainable by clarkson and kruskal s method

similarity reductions of partial differential equations the author determine the relation between two methods of reduction the similarity reduction method using nonclassical symmetry groups and the direct approach used by clarkson and kruskal 1989 they prove that the solutions which are obtained by similarity in correspondence to nonclassical groups constitute a larger family than the one obtained by the method of clarkson and kruskal the two procedures are equivalent only if the generators zeta x t u tau x t u eta x t u of the nonclassical groups are such that zeta tau is independent of u to explain these results he proves the existence of families of solutions of the burgers equation which are found by means of nonclassical symmetry reduction and which cannot be determined via the general reduction form of clarkson and kruskal

managing non homogeneous information in group decision making within the frame of decision aid literature decision making problems with multiple sources of information have drawn the attention of researchers from a wide spectrum of disciplines in decision situations with multiple individuals each one has his own knowledge on the alternatives of the decision problem the use of information assessed in different domains is not a seldom situation this non homogeneous information can be represented as values belonging to domains with different nature as linguistic numerical and interval valued or can be values assessed in label sets with different granularity multi granular linguistic information decision processes for solving these problems are composed by two phases aggregation and exploitation the main problem to deal with non homogeneous contexts is how to aggregate the information assessed in these contexts in this paper taking as base the 2 tuple fuzzy linguistic representation model we shall develop an aggregation process for dealing with non homogeneous contexts in first place we shall develop an aggregation process for combining numerical interval valued and linguistic information afterwards we shall propose different extensions of this process to deal with contexts in which can appear other type of information as intuitionistic fuzzy sets or multi granular linguistic information

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

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

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

a note on archimax copulas and their representation by means of conic copulas abstract the aim of this note is to prove the equivalence between the class of conic copulas and the class of archimax copulas based on the lower frechet hoeffding bound w we demonstrate the transition from arbitrary conic copula to appropriate pickands dependence function and vice versa our results show that archimax copulas are isomorphic transforms of either ev copulas or conic copulas

a note on a goal programming model for incomplete interval multiplicative preference relations and its application in group decision making abstract in a recently published paper by liu et al liu f zhang w g wang z x 2012 a goal programming model for incomplete interval multiplicative preference relations and its application in group decision making european journal of operational research 218 747 754 two equations are introduced to define consistency of incomplete interval multiplicative preference relations imprs and employed to develop a goal programming model for estimating missing values this note illustrates that such consistency definition and estimation model are technically incorrect new transitivity conditions are proposed to define consistent imprs and a two stage goal programming approach is devised to estimate missing values for incomplete imprs

consistency analysis of triangular fuzzy reciprocal preference relations abstract in order to simulate the uncertainty associated with impression or vagueness a decision maker may give her his judgments by means of triangular fuzzy reciprocal preference relations in the process of decision making the study of their consistency becomes a very important aspect to avoid a misleading solution based on the reciprocity property this paper proposes a new definition of consistent triangular fuzzy reciprocal preference relations the new definition is different from that reduced by consistent fuzzy reciprocal preference relations proposed by buckley 1985 the properties of consistent triangular fuzzy reciprocal preference relations in the light of the new definition are studied in detail in addition the shortcomings of the proof procedure of the proposition given by wang and chen 2008 are pointed out and the proposition is reproved by using the new definition of consistent triangular fuzzy reciprocal preference relations finally using the n 1 restricted comparison ratios a method for obtaining consistent triangular fuzzy reciprocal preference relations is proposed and an algorithm is shown to make a consistent decision ranking numerical results are further calculated to illustrate the new definition and the obtained algorithm

uncertainty and rank order in the analytic hierarchy process abstract the analytic hierarchy process uses paired comparisons to derive a scale of relative importance for alternatives we investigate the effect of uncertainty in judgment on the stability of the rank order of alternatives the uncertainty experienced by decision makers in making comparisons is measured by associating with each judgment an interval of numerical values the approach leads to estimating the probability that an alternative or project exchanges rank with other projects these probabilities are then used to calculate the probability that project would change rank at all we combine the priority of importance of each project with the probability that it does not change rank to obtain the final ranking

the distribution of the probability mass of biconic copulas we study the probability mass distribution of biconic copulas provide a counterexample to a result given in jwaid et al 2012 29 concerning a characterization of singular biconic copulas and provide a new one

a method based on pso and granular computing of linguistic information to solve group decision making problems defined in heterogeneous contexts abstract group decision making is a type of decision problem in which multiple experts acting collectively analyze problems evaluate alternatives and select a solution from a collection of alternatives as the natural language is the standard representation of those concepts that humans use for communication it seems natural that they use words linguistic terms instead of numerical values to provide their opinions however while linguistic information is readily available it is not operational and thus it has to be made usable though expressing it in terms of information granules to do so granular computing which has emerged as a unified and coherent framework of designing processing and interpretation of information granules can be used the aim of this paper is to present an information granulation of the linguistic information used in group decision making problems defined in heterogeneous contexts i e where the experts have associated importance degrees reflecting their ability to handle the problem the granulation of the linguistic terms is formulated as an optimization problem solved by using the particle swarm optimization in which a performance index is maximized by a suitable mapping of the linguistic terms on information granules formalized as sets this performance index is expressed as a weighted aggregation of the individual consistency achieved by each expert

incomplete pairwise comparison and consistency optimization this paper proposes a new method for calculating the missing elements of an incomplete matrix of pairwise comparison values for a decision problem the matrix is completed by minimizing a measure of global inconsistency thus obtaining a matrix which is optimal from the point of view of consistency with respect to the available judgements the optimal values are obtained by solving a linear system and unicity of the solution is proved under general assumptions some other methods proposed in the literature are discussed and a numerical example is presented

a remarkable class of non classical symmetries of the steady two dimensional boundary layer equations we study the steady two dimensional boundary layer equations in the flat and axisymmetric case to show that some similarity reductions already found in the literature with ad hoc new methods of reduction are indeed invariant solutions under the action of non classical symmetries as introduced by bluman and cole moreover we show that the celebrated von mises transformation that reduces the two dimensional flat boundary layer equations to a second order evolution equation is indeed a backlund transformation related to a non classical symmetry

a new fuzzy programming method to derive the priority vector from an interval reciprocal comparison matrix in this paper we discuss the interval reciprocal comparison matrices and suggest a new fuzzy programming method nfpm to derive the priority vector from an interval reciprocal comparison matrix based on mikhailov s membership function we first give a membership function which is used to measure the decision maker s satisfaction degree for the priority vector derived from each interval constraint then by using the membership function we propose a new fuzzy programming method to get the most satisfaction degree of each priority vector the nfpm has many prominent characteristics especially it is capable of deriving the priority vector from a consistent or inconsistent interval reciprocal comparison matrix moreover the nfpm can overcome some drawbacks of mikhailov s fuzzy programming method fpm for inconsistent interval reciprocal comparison matrices finally several examples are given to illustrate the applicability and advantage of the proposed method

evolution equations invariant surface conditions and functional separation of variables abstract this paper is devoted to a discussion of the reduction methods for evolution equations based on invariant surface conditions related to functional separation of variables the relationship of these methods with nonclassical and weak point symmetries is stressed applications to diffusion equations with an inhomogeneous reaction term or with saturating dissipation are provided

the distribution of the probability mass of conic copulas in this note we study the probability mass distribution of conic copulas provide counterexamples to a result given in jwaid et al 21 concerning a characterization of singular conic copulas and provide a new one

consistency analysis and group decision making based on triangular fuzzy additive reciprocal preference relations a positive triangular fuzzy number is expressed by a cross ratio expressed triplet operational laws are established for cross ratio expressed triangular fuzzy numbers cretfns multiplicative consistency is analyzed for triangular fuzzy additive reciprocal preference relations a triangular fuzzy weighted geometric operator is developed to aggregate cretfnsa novel method is put forward for comparing two cretfns triangular fuzzy numbers are effective in modeling imprecise and uncertain information and have been widely applied in decision making this paper uses a cross ratio expressed triplet to characterize a positive triangular fuzzy number and introduces notions of cross ratio expressed triangular fuzzy numbers cretfns and triangular fuzzy additive reciprocal preference relations tfarprs we present transformation methods between tfarprs and triangular fuzzy multiplicative reciprocal preference relations and develop operational laws of cretfns such as complement addition multiplication and power a cross ratio expressed triangular fuzzy multiplication based transitivity equation is established to define multiplicative consistency of tfarprs the new consistency captures tanino s multiplicative consistency among the cross ratio expressed modal values and geometric consistency of the interval fuzzy preference relation constructed from lower and upper support values of cross ratio expressed triangular fuzzy judgments some desirable properties are furnished for multiplicatively consistent tfarprs we propose a cross ratio expressed triangular fuzzy weighted geometric operator to aggregate cretfns and extend it to fuse tfarprs score and uncertainty index functions are defined and employed to devise a novel comparison method for cretfns a detailed procedure is put forward to solve group decision making problems with tfarprs six numerical examples are provided to illustrate the validity and applicability of the proposed models

multivariate archimedean copulas d monotone functions and l1 norm symmetric distributions it is shown that a necessary and sufficient condition for an archimedean copula generator to generate a d dimensional copula is that the generator is a d monotone function the class of d dimensional archimedean copulas is shown to coincide with the class of survival copulas of d dimensional l 1 norm symmetric distributions that place no point mass at the origin the d monotone archimedean copula generators may be characterized using a little known integral transform of williamson duke math j 23 1956 189 207 in an analogous manner to the well known bernstein widder characterization of completely monotone generators in terms of the laplace transform these insights allow the construction of new archimedean copula families and provide a general solution to the problem of sampling multivariate archimedean copulas they also yield useful expressions for the d dimensional kendall function and kendall s rank correlation coefficients and facilitate the derivation of results on the existence of densities and the description of singular components for archimedean copulas the existence of a sharp lower bound for archimedean copulas with respect to the positive lower orthant dependence ordering is shown

the optimal group continuous logarithm compatibility measure for interval multiplicative preference relations based on the cowga operator the calculation of compatibility measures is an important technique employed in group decision making with interval multiplicative preference relations in this paper a new compatibility measure called the continuous logarithm compatibility which considers risk attitudes in decision making based on the continuous ordered weighted geometric averaging cowga operator is introduced we also develop a group continuous compatibility model gcc model by minimizing the group continuous logarithm compatibility measure between the synthetic interval multiplicative preference relation and the continuous characteristic preference relation furthermore theoretical foundations are established for the proposed model such as the sufficient and necessary conditions for the existence of an optimal solution the conditions for the existence of a superior optimal solution and the conditions for the existence of redundant preference relations in addition we investigate certain conditions for which the optimal objective function of the gcc model guarantees its efficiency as the number of decision makers increases finally practical illustrative examples are examined to demonstrate the model and compare it with previous methods

a note on an idempotent transformation of absolutely continuous archimedean copulas abstract in de baets et al 6 the authors define for a copula c an idempotent transformation d c which in some cases generates new copulas in this note we give a negative answer to a question posed in 6 showing that d c is not necessarily a copula when c is an absolutely continuous archimedean copula

weak transversality and partially invariant solutions new exact solutions are obtained for several nonlinear physical equations namely the navier stokes and euler systems an isentropic compressible fluid system and a vector nonlinear schrodinger equation the solution method makes use of the symmetry group of the system in situations when the standard lie method of symmetry reduction is not applicable

multi criteria group decision making based on bilateral agreements abstract in multi criteria group decision making a key issue is aggregating individual preference into group one in this study we employ a bilateral agreement to conduct group evaluation of alternatives the bilateral agreement between a pair of individuals could be on the weight of a criterion a criterion evaluation function or willingness to pay any one of three types of bilateral agreements can derive the group utility to express the relationship between the bilateral agreement and individual evaluations the quasi arithmetic mean is used which can ensure the consistency property of the pairwise comparison matrices the minimum requirements are explored to obtain the group preference which shows that n 1 pairs of bilateral agreements are necessary finally several examples are provided to illustrate the proposed methods

integrating three representation models in fuzzy multipurpose decision making based on fuzzy preference relations abstract the purpose of this paper is to study a fuzzy multipurpose decision making problem where the information about the alternatives provided by the experts can be of a diverse nature the information can be represented by means of preference orderings utility functions and fuzzy preference relations and our objective is to establish a general model which cover all possible representations firstly we must make the information uniform using fuzzy preference relations as uniform preference context secondly we present some selection processes for multiple preference relations based on the concept of fuzzy majority fuzzy majority is represented by a fuzzy quantifier and applied in the aggregation by means of an owa operator whose weights are calculated by the fuzzy quantifier we use two quantifier guided choice degrees of alternatives a dominance degree used to quantify the dominance that one alternative has over all the others in a fuzzy majority sense and a non dominance degree that generalises orlovski s non dominated alternative concept the application of the two above choice degrees can be carried out according to two different selection processes a sequential selection process and a conjunction selection process

biconic aggregation functions we introduce a new method to construct aggregation functions these aggregation functions are called biconic aggregation functions with a given diagonal resp opposite diagonal section and their construction is based on linear interpolation on segments connecting the diagonal resp opposite diagonal of the unit square to the points 0 1 and 1 0 resp 0 0 and 1 1 subclasses of biconic aggregation functions such as biconic semi copulas biconic quasi copulas and biconic copulas are studied in detail

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

short communication a note on the notion of singular copula we clarify the link between the notion of singular copula and the concept of support of the measure induced by a copula

characterization of copulas with given diagonal and opposite diagonal sections in recent years special attention has been devoted to the problem of finding a copula the diagonal section and opposite diagonal section of which are known for given diagonal function and opposite diagonal functions we provide necessary and sufficient conditions for the existence of a copula to have these functions as diagonal and opposite diagonal sections we make use of techniques related to interpolation between the diagonals the construction of checkerboard copulas and linear programming this result allows us to solve two open problems to characterize the class of copulas where the knowledge of diagonal and opposite diagonal sections determines the copula in a unique way and to formulate necessary and sufficient conditions for each pair of such functions to be the diagonal and opposite diagonal sections of a unique copula

group theoretical framework for potential symmetries of evolution equations we develop algebraic approach to the problem of classification of potential symmetries of nonlinear evolution equations it is essentially based on the recently discovered fact r zhdanov j math phys 50 053522 2009 that any such symmetry is mapped into a contact symmetry the approach enables using the classical results on classification of contact symmetries of nonlinear evolution equations by sokolov and magadeev to classify evolution equations admitting potential symmetries we construct several examples of new nonlinear fourth order evolution equations admitting potential symmetries since the symmetries obtained depend on nonlocal variables they cannot be derived by the infinitesimal lie approach

new classes of symmetries for partial differential equations new classes of symmetries for partial differential equations are introduced by writing a given partial differential equation s in a conserved form a related system t with potentials as additional dependent variables is obtained the lie group of point transformations admitted by t induces a symmetry group of s new symmetries may be obtained for s that are neither point nor lie backlund symmetries they are determined by a completely algorithmic procedure significant new symmetries are found for the wave equation with a variable wave speed and the nonlinear diffusion equation

note on group consistency in analytic hierarchy process abstract we study the paper of xu z xu on consistency of the weighted geometric mean complex judgement matrix in ahp european journal of operational research 126 2000 683 687 for the group consistency in analytic hierarchy process of multicriteria decision making the purpose of this note is threefold first we point out the questionable results in this paper second for three by three comparison matrices we provide a patchwork for his method third we constructed a counter example to show that in general his method is wrong numerical examples are provided to illustrate our findings if there are four or more alternatives then we may advise researchers to ignore his results to avoid questionable estimation of group consistency

on consistency of the weighted geometric mean complex judgement matrix in ahp abstract the weighted geometric mean method wgmm is the most common group preference aggregation method in the analytic hierarchy process this paper reports on research concerning the consistency of wgmm and proves that the weighted geometric mean complex judgement matrix wgmcjm is of acceptable consistency in saaty s t l saaty the analytic hierarchy process mcgraw hill new york 1980 opinion a consistency ratio cr of 0 1 or less is acceptable under the condition that all judgement matrices given by experts for the same problem of decision making are of acceptable consistency accordingly a theoretic basis has been developed for the application of the wgmm in group decision making

absolutely continuous copulas with given sub diagonal section abstract recently durante and jaworski 2008 6 have proved that the class of absolutely continuous copulas with a given diagonal section is non empty in case that the diagonal function is such that the set of points where this coincides with the identity function has null measure in this paper we show that if we consider sub diagonals or super diagonals then the framework changes concretely for each sub diagonal function there exists an absolutely continuous copula having this function as sub diagonal section

localized excitations in 2 1 dimensional systems by means of a special variable separation approach a common formula with some arbitrary functions has been obtained for some suitable physical quantities of various 2 1 dimensional models such as the davey stewartson ds model the nizhnik novikov veselov nnv system asymmetric nnv equation asymmetric ds equation dispersive long wave equation broer kaup kupershmidt system long wave short wave interaction model maccari system and a general n m component ablowitz kaup newell segur akns system selecting the arbitrary functions appropriately one may obtain abundant stable localized interesting excitations such as the multidromions lumps ring soliton solutions breathers instantons etc it is shown that some types of lower dimensional chaotic patterns such as the chaotic chaotic patterns periodic chaotic patterns chaotic line soliton patterns chaotic dromion patterns fractal lump patterns and fractal dromion patterns may be found in higher dimensional soliton systems the interactions between the traveling ring type soliton solutions are completely elastic the traveling ring solitons pass through each other and preserve their shapes velocities and phases some types of localized weak solutions peakons are also discussed especially the interactions between two peakons are not completely elastic after the interactions the traveling peakons also pass through each other and preserve their velocities and phases however they completely exchange their shapes

piecewise linear aggregation functions based on triangulation we introduce a method to construct piecewise linear binary aggregation functions on the unit interval based on a triangulation of the unit square with one additional vertex we derive conditions under which such piecewise linear aggregation functions possess additional interesting properties such as idempotence symmetry lipschitz continuity and 2 monotonicity this construction method can also be used to approximate binary aggregation functions in this way copulas and quasi copulas are approximated by singular copulas

on the class of bivariate archimax copulas under constraints abstract the class of bivariate archimax copulas spans a broad range of dependence however its size is much reduced when the degree of association is prespecified the extent of this reduction is studied for a large set of measures of association the cases of exchangeable and non exchangeable archimax copulas are both considered

derivative dependent functional separable solutions for the kdv type equations abstract a complete list of canonical forms for a special kdv type of equations which admit derivative dependent functional separable solutions is obtained by using the generalized conditional symmetry approach some exact solutions to the resulting equations are explicitly obtained

nonclassical potential symmetries for the burgers equation abstract in this work classes of symmetries for partial differential equations which can be written in a conserved form are found these nonclassical potential symmetries are realized as nonclassical symmetries of the associated potential system and are neither classical potential symmetries realized as lie symmetries of a related auxiliary system nor nonclassical symmetries of the considered equation some of these symmetries are carried out for the burgers equation these symmetries are realized as the nonclassical symmetries of the associated potential system and yield many new nonlocal generators and new explicit solutions of the burgers equation

conservation laws and hierarchies of potential symmetries for certain diffusion equations abstract we show that the so called hidden potential symmetries considered in a recent paper m l gandarias new potential symmetries for some evolution equations physica a 387 2008 2234 2242 are ordinary potential symmetries that can be obtained using the method introduced by bluman and collaborators g w bluman s kumei symmetries and differential equations springer new york 1989 g w bluman g j reid s kumei new classes of symmetries for partial differential equations j math phys 29 1988 806 811 in fact these are simplest potential symmetries associated with potential systems which are constructed with single conservation laws having no constant characteristics furthermore we classify the conservation laws for classes of porous medium equations and then using the corresponding conserved potential systems we search for potential symmetries this is the approach one needs to adopt in order to determine the complete list of potential symmetries the provenance of potential symmetries is explained for the porous medium equations by using potential equivalence transformations point and potential equivalence transformations are also applied to deriving new results on potential symmetries and corresponding invariant solutions from known ones in particular in this way the potential systems potential conservation laws and potential symmetries of linearizable equations from the classes of differential equations under consideration are exhaustively described infinite series of infinite dimensional algebras of potential symmetries are constructed for such equations

on relation between potential and contact symmetries of evolution equations we prove that any evolution equation that can be written in the form of a conservation law reduces to an evolution equation admitting the group of contact symmetries leaving the temporal variable invariant and vice versa one of the consequences of this fact is that any evolution equation admitting the potential symmetries can be transformed into another evolution equation so that the potential symmetries map into contact symmetries of the latter based on this fact is out group approach to classification of evolution equations possessing nonlocal symmetries we present several examples of classifications of second order evolution equations admitting potential symmetries

acceptability analysis and priority weight elicitation for interval multiplicative comparison matrices existing research on acceptability of pairwise interval comparison matrices focuses on acceptable consistency by controlling their inconsistency levels to within a certain threshold however a perfectly consistent but highly indeterminate interval comparison matrix can be unacceptable as it contains little sometimes no useful decision information this paper first analyzes the current definition of acceptable consistency for interval multiplicative comparison matrices imcms and shows its technical deficiencies we then introduce a new notion of acceptable imcms considering both inconsistency and indeterminacy levels in imcms a geometric mean based index is proposed to measure the indeterminacy ratio of an imcm and useful properties are derived for consistent imcms and acceptable imcms an indeterminacy ratio and geometric mean based transformation equation is subsequently put forward to convert normalized acceptable interval multiplicative weights into an acceptable imcm with consistency by introducing an auxiliary constraint a logarithmic least square model is established to generate interval multiplicative weights from acceptable imcms a geometric mean based possibility degree formula is designed to compare and rank normalized interval multiplicative weights two numerical examples are presented to illustrate how to utilize the proposed framework

contact symmetries and solutions by reduction of partial differential equations the correspondence between a contact symmetry of a second order pde e and a point symmetry of the equivalent first order system s is used to determine a class of solutions for e which may not always be invariant under contact transformation we name these solutions pseudo invariant because they are determined from a point symmetry of s via a reduction method invariant solutions under the related contact transformation are included in the family of pseudo invariant solutions

copulas with continuous strictly increasing singular conditional distribution functions abstract using iterated function systems induced by so called modifiable transformation matrices t and tools from symbolic dynamical systems we first construct mutually singular copulas a t with identical possibly fractal or full support that are at the same time singular with respect to the lebesgue measure 2 on 0 1 2 afterwards the established results are utilized for a simple proof of the existence of singular copulas a t with full support for which all conditional distribution functions y f x a t y are continuous strictly increasing and have derivative zero almost everywhere this result underlines the fact that conditional distribution functions of copulas may exhibit surprisingly irregular analytic behavior finally we extend the notion of empirical copula to the case of non i i d data and prove uniform convergence of the empirical copula e n corresponding to almost all orbits of a markov process usually referred to as chaos game to the singular copula a t several examples and graphics illustrate both the chosen approach and the main results

acceptable consistency analysis of interval reciprocal comparison matrices when a decision maker expresses his her opinions by means of an interval reciprocal comparison matrix the study of consistency becomes a very important aspect in decision making in order to avoid a misleading solution in the present paper an acceptably consistent interval reciprocal comparison matrix is defined which can be reduced to an acceptably consistent crisp reciprocal comparison matrix when the intervals become exact numbers an interval reciprocal comparison matrix with unacceptable consistency can be easily adjusted such that the revised matrix possesses acceptable consistency utilizing a convex combination method a family of crisp reciprocal comparison matrices with acceptable consistency can be obtained whose weights are further found to exhibit a style of convex combination and aggregated to obtain interval weights from an acceptably consistent interval reciprocal comparison matrix a novel simple yet effective formula of possibility degree is presented to rank interval weights numerical results are calculated to show the quality and quantity of the proposed approaches and compare with other existing procedures

the role of generalized convexity in conic copula constructions abstract inspired by the notion of a conic semi copula we introduce upper conic lower conic and biconic semi copulas with a given section such semi copulas are constructed by linear interpolation on segments connecting the graph of a strict negation operator to the points 0 0 and or 1 1 the important subclasses of upper conic lower conic and biconic quasi copulas with a given section are characterized the notion of generalized convexity turns out to play a key role in this characterization some examples are also provided

boundary properties of the inconsistency of pairwise comparisons in group decisions abstract this paper proposes an analysis of the effects of consensus and preference aggregation on the consistency of pairwise comparisons we define some boundary properties for the inconsistency of group preferences and investigate their relation with different inconsistency indices some results are presented on more general dependencies between properties of inconsistency indices and the satisfaction of boundary properties in the end given three boundary properties and nine indices among the most relevant ones we will be able to present a complete analysis of what indices satisfy what properties and offer a reflection on the interpretation of the inconsistency of group preferences

integrating multiplicative preference relations in a multipurpose decision making model based on fuzzy preference relations the aim of this paper is to study the integration of multiplicative preference relation as a preference representation structure in fuzzy multipurpose decision makingproblems assumingfuzzy multipurpose decision makingproblems under di4erent preference representation structures ordering utilities and fuzzy preference relations and using the fuzzy preference relations as uniform representation elements the multiplicative preference relations are incorporated in the decision problem by means of a transformation function between multiplicative and fuzzy preference relations a consistency study of this transformation function which demonstrates that it does not change the informative content of multiplicative preference relation is shown as a consequence a selection process based on fuzzy majority for multipurpose decision makingproblems under multiplicative preference relations is presented to design it an aggregation operator of information called ordered weighted geometric operator is introduced and two choice degrees the quanti7er guided dominance degree and the quanti7er guided non dominance degree are de7ned for multiplicative preference relations c 2001 elsevier science b v all rights reserved