a fusion approach for managing multi granularity linguistic term sets in decision making the aim of this paper is to present a fusion approach of multi granularity linguistic information for managing information assessed in different linguistic term sets multi granularity linguistic term sets together with its application in a decision making problem with multiple information sources assuming that the linguistic performance values given to the alternatives by the different sources are represented in linguistic term sets with different granularity and or semantic in this context a decision process based on two steps is proposed with a view to obtaining the solution set of alternatives first the fusion of the multi granularity linguistic performance values is carried out in order to obtain collective performance evaluations in this step on the one hand the multi granularity linguistic information is made uniform using a linguistic term set as the uniform representation base the basic linguistic term set on the other hand the collective performance evaluations of the alternatives are obtained by means of an aggregation operator being fuzzy sets on the basic linguistic term set second the choice of the best alternative s from the collective performance evaluations is performed to do that a fuzzy preference relation is computed from the collective performance evaluations using a ranking method of pairs of fuzzy sets in the setting of possibility theory applied to fuzzy sets on the basic linguistic term set then a choice degree may be applied on the preference relation in order to rank the alternatives

allowing agents to be imprecise a proposal using multiple linguistic terms in this paper we propose a decision making procedure where the agents judge the alternatives through linguistic terms such as very good good acceptable etc if the agents are not confident about their opinions they can use a linguistic expression formed by several consecutive linguistic terms to obtain a ranking on the set of alternatives the method consists of three different stages the first stage looks for the alternatives in which the overall opinion is closer to the ideal assessment the overall opinion is developed by a distance based process among the individual assessments the next two stages form a tie breaking process firstly by using a dispersion index based on the gini coefficient and secondly by taking into account the number of best assessments the main characteristics of the proposed decision making procedure are analyzed

selecting the individual numerical scale and prioritization method in the analytic hierarchy process a 2 tuple fuzzy linguistic approach the validity of the priority vector used in the analytic hierarchy process ahp relies on two factors the selection of a numerical scale and the selection of a prioritization method the traditional ahp selects only one numerical scale e g the saaty scale and one prioritization method e g the eigenvector method for each particular problem for this traditional selection approach there is disagreement on which numerical scale and prioritization method is better in deriving a priority vector in fact the best numerical scale and the best prioritization method both rely on the content of the pairwise comparison data provided by the ahp decision makers by defining a set of concepts regarding the scale function and the linguistic pairwise comparison matrices lpcms of the priority vector and by using lpcms to unify the format of the input and output of ahp this paper extends the ahp prioritization process under the 2 tuple fuzzy linguistic model based on the extended ahp prioritization process we present two performance measure criteria to evaluate the effect of the numerical scales and prioritization methods we also use the performance measure criteria to develop a 2 tuple fuzzy linguistic multicriteria approach to select the best numerical scales and the best prioritization methods for different lpcms in this paper we call this type of selection the individual selection of the numerical scale and prioritization method we also compare this individual selection with traditional selection by using both random and real data and show better results with individual selection

non autonomous second order hamiltonian systems abstract we study the existence of periodic solutions for a second order non autonomous dynamical system containing variable kinetic energy terms our assumptions balance the interaction between the kinetic energy and the potential energy with neither one dominating the other we study sublinear problems and the existence of non constant solutions

an optimization based approach to adjusting unbalanced linguistic preference relations to obtain a required consistency level in certain real decision making situations decision makers often feel more comfortable providing their knowledge and preferences in linguistic terms unbalanced linguistic term sets may be used in decision problems with preference relations however the lack of consistency in decision making with linguistic preference relations can lead to inconsistent conclusions based on the consistency measure of unbalanced linguistic preference relations this paper proposes an optimization based approach to improving the consistency level of unbalanced linguistic preference relations this consistency improving model preserves the utmost original knowledge and preferences in the process of improving consistency furthermore it guarantees that the elements in the optimal adjusted unbalanced linguistic preference relation are all simple unbalanced linguistic terms finally we propose a mixed 0 1 linear programming aimed to obtain the optimum solution to the proposed consistency improving model and to demonstrate its practicability

existence of infinitely many periodic solutions for ordinary p laplacian systems abstract in this paper we study the existence and multiplicity of non trivial periodic solutions of ordinary p laplacian systems by using the minimax technique in critical point theory we also give an example to illustrate that the obtained results are new even in the case p 2

a novel group ranking model for revealing sequence and quantity knowledge abstract the aggregation of individuals preferences into a consensus ranking is a group ranking problem which has been widely utilized in various applications such as decision support systems recommendation systems and voting systems gathering the comparison of preferences and aggregate them to gain consensuses is a conventional issue for example b c d a indicates that b is favorable to c and c d is somewhat favorable but not fully favorable to d a where and are comparators and a b c and d are items recently a new type of ranking model was proposed to provide temporal orders of items the order b c a means that b and c can occur simultaneously and are also before a although this model can derive the order ranking of items the knowledge about quantity related items is also of importance to approach more real life circumstances for example when enterprises or individuals handle their portfolios in financial management two considerations the sequences and the amount of money for investment objects should be raised initially in this study we propose a model for discovering consensus sequential patterns with quantitative linguistic terms experiments using synthetic and real datasets showed the model s computational efficiency scalability and effectiveness

abstract critical point theorems and applications to some nonlinear problems with strong resonance at infinity critical point theorems and applications to some nonlinear problems with strong resonance at infinity p bartolo and v benci istituto di matematica applicata via re david x0 bari ital and istituto di analisi matematica palazzo ateneo l ia scolai 2 bari italy received in revised form 20 december 1952

consistency of hesitant fuzzy linguistic preference relations an interval consistency index abstract the study of hesitant consistency is very important in decision making with hesitant fuzzy linguistic preference relations hflprs and generally the normalization method is used as a tool to measure the consistency degree of a hflpr in this paper we propose a new hesitant consistency measure called interval consistency index to estimate the consistency range of a hflpr the underlying idea of the interval consistency index consists of measuring the worst consistency index and the best consistency index of a hflpr furthermore by comparative study a connection is shown between the interval consistency index and the normalization method demonstrating that the normalization method should be considered as an approximate average consistency index of a hflpr

a general class of simple majority decision rules based on linguistic opinions in this paper we have introduced a class of decision rules related to simple majority by considering individual intensities of preference these intensities will be shown by means of linguistic labels in order to compare the amount of opinion obtained by each alternative we have considered the total ordered monoid generated by the sums of the original labels according to an addition and an ordering in this general framework different sets of linguistic labels can be employed and these sets can be represented by means of diverse mathematical objects moreover on these mathematical representations of linguistic labels several orderings can be considered thus flexibility is an important feature of this new class of group decision making procedures some examples of putting in practice the simple majority decision rules based on linguistic labels are provided and the main properties of these voting systems are analyzed it is worth emphasizing that these properties are satisfied for any total ordered monoid regardless of the mathematical representation of linguistic labels or the ordering used to compare collective opinions

periodic second order superlinear hamiltonian systems abstract we consider periodic solutions for superlinear second order non autonomous dynamical systems including both kinetic and potential terms we study the existence of nontrivial and ground state solutions

periodic solutions of second order nonlinear difference systems with φ laplacian a variational approach t periodic solutions of systems of difference equations of the form u n 1 u f n u n h n n z where strictly convex is a homeomorphism of rn onto the ball ba rn or of ba onto rn are considered under various conditions upon f and h the approach is variational 2011 elsevier ltd all rights reserved

subharmonic solutions of sublinear second order systems with impacts abstract we mainly consider subharmonic bouncing solutions of sublinear second order systems with an obstacle based on variational method

applying a direct multi granularity linguistic and strategy oriented aggregation approach on the assessment of supply performance supply performance has the active continuity behaviors which covers the past present and future of time horizons thus supply performance possesses distinct uncertainty on individual behavior which is inadequate to assess with quantification this study utilizes the linguistic variable instead of numerical variable to offset the inaccuracy on quantification and employs the fitting linguistic scale in accordance with the characteristic of supply behavior to enhance the applicability furthermore the uniformity is introduced to transform the linguistic information uniformly from different scales finally the linguistic ordered weighted averaging operator with maximal entropy applies in direct to aggregate the combination of linguistic information and product strategy to ensure the assessment results meeting the enterprise requirements and then to emulate mental decision making in humans by the linguistic manner

connecting the linguistic hierarchy and the numerical scale for the 2 tuple linguistic model and its use to deal with hesitant unbalanced linguistic information the 2 tuple linguistic representation model is widely used as a basis for computing with words cw in linguistic decision making problems two different models based on linguistic 2 tuples i e the model of the use of a linguistic hierarchy and the numerical scale model have been developed to address term sets that are not uniformly and symmetrically distributed i e unbalanced linguistic term sets ultss in this study we provide a connection between these two different models and prove the equivalence of the linguistic computational models to handle ultss further we propose a novel cw methodology where the hesitant fuzzy linguistic term sets hfltss can be constructed based on ultss using a numerical scale in the proposed cw methodology we present several novel possibility degree formulas for comparing hfltss and define novel operators based on the mixed 0 1 linear programming model to aggregate the hesitant unbalanced linguistic information

on weighted unbalanced linguistic aggregation operators in group decision making we generalize linguistic evaluation values and their weights in group decision making gdm problems based on unbalanced linguistic terms the gdm problems include two types of weights belief degrees of linguistic evaluation values and experts weights due to the time pressure and conflict of multi source of information etc experts may have various evaluations values on alternatives the belief degree is used to represent the confidence level of the values the experts weight is used to represent the differences among experts importance which caused by experts experience or knowledge distinction we propose the weighted unbalanced linguistic aggregation operators to synthesize linguistic evaluation value belief degree and experts weights some desired properties of the operator are then studied these properties show that the operator extends the linguistic weighted averaging operator lwa and the linguistic ordered weighted averaging lowa operator finally an illustrative example of human resource performance appraisal based on the linguistic aggregation operator is provided

linguistic multiperson decision making based on the use of multiple preference relations in multiperson decision making situations it is quite natural that different decision makers who may have different background and knowledge will provide their preferences by different kinds of preference relations this paper proposes a linguistic multiperson decision making model lmdmm based on linguistic preference relations integrating fuzzy preference relations different types of multiplicative preference relations and multigranular linguistic preference relations in the lmdmm several transformation functions are first obtained to relate fuzzy preference relations and different types of multiplicative preference relations with multigranular linguistic preference relations then we design the selection process of the lmdmm based on the use of the fuzzy majority and the extended ordered weighted averaging operator or the 2 tuple ordered weighted averaging operator and discuss conditions under which the proposed selection process satisfies the social choice axioms at last we analyze the internal consistency of the proposed transformation functions the results in this paper are helpful to complete chiclana et al s fuzzy decision model integrating three representation models in fuzzy multipurpose decision making based on fuzzy preference relations fuzzy sets and systems 97 1998 33 48

periodic solutions for second order systems with not uniformly coercive potential abstract the existence and multiplicity of periodic solutions are obtained for the nonautonomous second order systems with locally coercive potential that is f t x as x for a e t in some positive measure subset of 0 t by using an analogy of egorov s theorem the properties of subadditive functions the least action principle and a three critical point theorem proposed by brezis and nirenberg

generalized asymmetric linguistic term set and its application to qualitative decision making involving risk appetites abstract the linguistic term set is an applicable and flexible technique in qualitative decision making qdm to further develop the linguistic term set this paper proposes a generalized asymmetric linguistic term set galts based on the asymmetric sigmoid semantics which belongs to an asymmetric and non uniform linguistic term set and can be used to address the qdm problems involving risk appetites of the decision maker dm then a value at risk fitting varf approach is designed for obtaining the risk appetite parameters of the galts and six desirable properties of the galts are analyzed i e asymmetry non uniformity generality variability range consistency and diminishing utility based on the above approaches and the generalized asymmetric linguistic preference relations galprs a qdm process involving risk appetites of the dm is designed because the galprs consist of subjective information provided by the dm the process is not perfectly consistent and is usually difficult to change or repeat thus a transitivity improvement approach is investigated and the corresponding calculation steps are provided finally an example dealing with the problem of investment decision making is provided and the results fully demonstrate the validity of the proposed methods for qdm involving risk appetites

evaluating the best main battle tank using fuzzy decision theory with linguistic criteria evaluation abstract to face the reality of practical multiple criteria problems usually possessing characters of fuzziness and to consider group decision making with various subjective objective backgrounds usually participating in decision making process in this paper the experts opinions are described by linguistic terms which can be expressed in trapezoidal or triangular fuzzy numbers to make the consensus of the experts consistent we utilize fuzzy delphi method to adjust the fuzzy rating of every expert to achieve the consensus condition for the aggregate of many experts opinions we take the operation of fuzzy numbers to get the mean of fuzzy rating x ij and the mean of weight w j in multi alternatives and multi attributes cases the fuzzy decision matrix x x ij m n is constructed by the mean of the fuzzy rating x ij then we can derive the aggregate fuzzy numbers by multiplying the fuzzy decision matrix with the corresponding fuzzy attribute weights the final results become a problem of ranking fuzzy numbers we also propose an easy procedure of using fuzzy numbers to rank aggregate fuzzy numbers a i in this way we can obtain the best selection for evaluating system for practical application we propose an algorithm for evaluating the best main battle tank by fuzzy decision theory and compare it with other method

periodic solutions for some nonautonomous second order hamiltonian systems the existence and multiplicity of periodic solutions are obtained for nonautonomous second order hamiltonian systems by the minimax methods in critical point theory

late time asymptotics for the wave equation on spherically symmetric stationary spacetimes abstract we derive precise late time asymptotics for solutions to the wave equation on spherically symmetric stationary and asymptotically flat spacetimes including as special cases the schwarzschild and reissner nordstrom families of black holes we also obtain late time asymptotics for the time derivatives of all orders and for the radiation field along null infinity we show that the leading order term in the asymptotic expansion is related to the existence of the conserved newman penrose quantities on null infinity as a corollary we obtain a characterization of all solutions which satisfy price s polynomial law 3 as a lower bound our analysis relies on physical space techniques and uses the vector field approach for almost sharp decay estimates introduced in our companion paper in the black hole case our estimates hold in the domain of outer communications up to and including the event horizon our work is motivated by the stability problem for black hole exteriors and strong cosmic censorship for black hole interiors

periodic solutions of non autonomous second order systems we obtain an existence theorem of periodic solutions of non autonomous second order systems with classical theorems of variational calculus

periodic non autonomous second order dynamical systems we study the existence of periodic solutions for a second order non autonomous dynamical system we give three sets of hypotheses which guarantee the existence of non constant solutions we were able to weaken the hypotheses considerably from those used previously for such systems we employ a saddle point theorem using linking methods

induced 2 tuple linguistic generalized aggregation operators and their application in decision making the induced 2 tuple linguistic generalized ordered weighted averaging 2 tilgowa operator is presented this new aggregation operator extends previous approaches by using generalized means order inducing variables in the reordering of the arguments and linguistic information represented with the 2 tuple linguistic approach its main advantage is that it includes a wide range of linguistic aggregation operators thus its analyses can be seen from different perspectives and we obtain a much more complete picture of the situation considered and are able to select the alternative that best fits with our interests or beliefs we further generalize the 2 tilgowa by using quasi arithmetic means and choquet integrals the result is the quasi 2 tilowa operator and the 2 tuple linguistic induced quasi arithmetic choquet integral aggregation we conclude this paper by analysing the applicability of this new approach in a multi person linguistic decision making problem concerning product management

existence and multiplicity results for nonlinear nonautonomous second order systems in this paper we consider nonlinear periodic systems driven by the ordinary p laplacian and having a nonsmooth potential function under minimal and natural hypotheses on the nonsmooth potential and using variational methods based on the nonsmooth critical point theory we prove four existence theorems and a multiplicity theorem we conclude the paper with an existence theorem for the scalar problem in which the energy functional is indefinite i e it is unbounded both above and below

an axiomatizable logical foundation for lattice ordered qualitative linguistic approach for reasoning with words it is commonsense that people do express think reason and make decision directly using linguistic terms in natural language rather than using numerical quantification how to establish the formalized approach imitating the common way of human being s to manipulate directly linguistic terms without numerical approximation to provide the rational decision is still an open research area investigations on the algebraic structure of linguistic term set for varied decision making applications especially in social science still lack a formalism for development of strict linguistic valued logic system as a theoretical foundation and its approximate reasoning scheme in practice to attain this goal we characterize and construct a typical structure of commonly used linguistic term sets in natural language by a lattice ordered algebra structure lattice implication algebra lia where lukasiewicz implication algebra as a special case of lia plays a substantial role by using lukasiewicz logic s axiomatizability in terms of pavelka type fuzzy logic we propose a new axiomatizable lattice ordered qualitative linguistic truth valued logic system based on lia to place an important foundation for further establishing formal linguistic truth valued logic based approximate reasoning and decision making with applications this proposed logic system has a distinct feature of handling comparable or incomparable linguistic terms directly without numerical quantification will be especially beneficial for perception based decision making processes it attempts to enhance the quantitative theory of decision science with qualitative algebraic and logic oriented approaches to achieve reasoning with words

notes on periodic solutions of subquadratic second order systems some existence theorems are obtained for periodic solutions of the subquadratic second order systems by the minimax methods in critical point theory

direct approach processes in group decision making using linguistic owa operators abstract in a linguistic framework several group decision making processes by direct approach are presented these processes are designed using the linguistic ordered weighted averaging lowa operator to do so first a study is made of the properties and the axiomatic of lowa operator showing the rationality of its aggregation way and secondly we present the use of lowa operator to solve group decision making problems from individuals linguistic preference relations

new existence of periodic solutions for second order non autonomous hamiltonian systems abstract by using mountain pass theorem and local link theorem some existence theorems are obtained for periodic solutions of second order non autonomous hamiltonian systems under local superquadratic condition and other suitable conditions

periodic solutions for some nonautonomous second order systems some existence theorems are obtained for periodic solutions of nonautonomous second order systems by using the least action principle and the minimax methods

existence of periodic solutions for p t laplacian systems abstract in this paper by using minimax methods in critical point theory we obtain some existence theorems for periodic solutions of p t laplacian system u t p t 2 u t f t u t 0 u 0 u t u 0 u t 0 and our results generalize some existed theorems

a fuzzy linguistic methodology to deal with unbalanced linguistic term sets many real problems dealing with qualitative aspects use linguistic approaches to assess such aspects in most of these problems a uniform and symmetrical distribution of the linguistic term sets for linguistic modeling is assumed however there exist problems whose assessments need to be represented by means of unbalanced linguistic term sets i e using term sets that are not uniformly and symmetrically distributed the use of linguistic variables implies processes of computing with words cw different computational approaches can be found in the literature to accomplish those processes the 2 tuple fuzzy linguistic representation introduces a computational model that allows the possibility of dealing with linguistic terms in a precise way whenever the linguistic term set is uniformly and symmetrically distributed in this paper we present a fuzzy linguistic methodology in order to deal with unbalanced linguistic term sets to do so we first develop a representation model for unbalanced linguistic information that uses the concept of linguistic hierarchy as representation basis and afterwards an unbalanced linguistic computational model that uses the 2 tuple fuzzy linguistic computational model to accomplish processes of cw with unbalanced term sets in a precise way and without loss of information

periodic solutions of a class of non autonomous second order systems some existence theorems are obtained by the least action principle for periodic solutions of nonautonomous second order systems with a potential which is the sum of a subconvex function and a subquadratic function

periodic solutions for a class of nonautonomous subquadratic second order hamiltonian systems some existence theorems are obtained for periodic solutions of a class of unbounded nonautonomous nonconvex subquadratic second order hamiltonian systems by using the minimax methods in critical point theory

hypothesis testing for means in connection with fuzzy rating scale based data algorithms and applications abstract the fuzzy rating scale was introduced as a tool to measure intrinsically ill defined imprecisely valued attributes in a free way thus users do not have to choose a value from a class of prefixed ones like it happens when a fuzzy semantic representation of a linguistic term set is considered but just to draw the fuzzy number that better represents their valuation or measurement the freedom inherent to the fuzzy rating scale process allows users to collect data with a high level of richness accuracy expressiveness diversity and subjectivity what is especially valuable for statistical purposes this paper presents an inferential approach to analyze data obtained by using the fuzzy rating scale more concretely the paper is to be focused on testing different hypothesis about means on the basis of a sound methodology which has been stated during the last years all the procedures that have been developed to this aim will be presented in an algorithmic way adapted to the usual generic fuzzy rating scale based data and they will be illustrated by means of a real life example

some existence results on periodic solutions of ordinary p laplacian systems some existence theorems are obtained for periodic solutions of ordinary p laplacian systems by the minimax methods in critical point theory

some critical point theorems and their applications to periodic solution for second order hamiltonian systems abstract some critical point theorems without the compactness assumptions are obtained by the reduction method the perturbation argument and the least action principle as applications some existence results of periodic solutions are obtained for nonautonomous second order hamiltonian systems which unify and generalize some recent corresponding results

a tool for aggregation with words the need of computing with words has become an important topic in many areas dealing with vague information the aim of this paper is to present different tools which support computing with words especially we are concerned with the weighted aggregation of linguistic term sets whose results are just the words themselves without using the fuzzy numbers that represent the semantics of their linguistic terms we propose a new aggregation operator referred to as the symbolic weighted median that computes the most representative element from an ordered collection of weighted linguistic terms this operator aggregates the linguistic labels such that its result is expressed in terms of the initial linguistic term set though is modified by using dedicated tools called the generalized symbolic modifiers one advantage of this proposal is that the expression domain does not change we increase or decrease the granularity only where it becomes necessary additionally this new operator exhibits several interesting mathematical properties

periodic solutions of second order differential inclusions systems with p laplacian some existence results are obtained for periodic solutions of nonautonomous second order differential inclusions systems with p laplacian

periodic and subharmonic solutions of a class of subquadratic second order hamiltonian systems some solvability conditions of periodic and subharmonic solutions are obtained for a class of subquadratic nonautonomous second order hamiltonian systems by the minimax methods in critical point theory

multiple periodic solutions for hamiltonian systems with not coercive potential under an appropriate oscillating behavior of the nonlinear term the existence of infinitely many periodic solutions for a class of second order hamiltonian systems is established moreover the existence of two non trivial periodic solutions for hamiltonian systems with not coercive potential is obtained and the existence of three periodic solutions for hamiltonian systems with coercive potential is pointed out the approach is based on critical point theorems

periodic solutions for a class of nonautonomous second order hamiltonian systems abstract in this paper some existence theorems of periodic solutions of a class of the nonautonomous second order hamiltonian systems x t b t x t h t x t 0 t 0 t are obtained by a mountain pass theorem and a local link theorem

periodic solutions of non autonomous second order systems with γ quasisubadditive potential abstract the existence of periodic solutions to non autonomous second order systems with quasisubadditive potential is proved with classical theorems of variational calculus