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

least square completion and inconsistency repair methods for additively consistent fuzzy preference relations in this paper we explore the group decision making gdm problems with incomplete additively consistent fuzzy preference relations some properties of additively consistent fuzzy preference relations are also discussed a sufficient and necessary condition is proposed to keep the additive consistency of fuzzy preference relations methods for determining the priority weights of fuzzy preference relations are provided least square completion and inconsistency repair methods are developed to deal with incomplete and inconsistent fuzzy preference relations some numerical examples are also given to illustrate the proposed approaches

julia a fresh approach to numerical computing bridging cultures that have often been distant julia combines expertise from the diverse fields of computer science and computational science to create a new approach to numerical computing julia is designed to be easy and fast and questions notions generally held to be laws of nature by practitioners of numerical computing beginlist item high level dynamic programs have to be slow item one must prototype in one language and then rewrite in another language for speed or deployment item there are parts of a system appropriate for the programmer and other parts that are best left untouched as they have been built by the experts endlist we introduce the julia programming language and its design a dance between specialization and abstraction specialization allows for custom treatment multiple dispatch a technique from computer science picks the right algorithm for the right circumstance abstraction which is what good computation is really about recognizes what remains the same after dif

optimal aggregation of fuzzy preference relations with an application to broadband internet service selection this paper investigates the aggregation of multiple fuzzy preference relations into a collective fuzzy preference relation in fuzzy group decision analysis and proposes an optimization based aggregation approach to assess the relative importance weights of the multiple fuzzy preference relations the proposed approach that is analytical in nature assesses the weights by minimizing the sum of squared distances between any two weighted fuzzy preference relations relevant theorems are offered in support of the proposed approach multiplicative preference relations are also incorporated into the approach using an appropriate transformation technique an eigenvector method is introduced to derive the priorities from the collective fuzzy preference relation the proposed aggregation approach is tested using two numerical examples a third example involving broadband internet service selection is offered to illustrate that the proposed aggregation approach provides a simple effective and practical way of aggregating multiple fuzzy preference relations in real life situations

data driven fuzzy preference analysis from an optimization perspective abstract how to leverage massive online data to understand consumer preferences over products and services has accumulated significant attention in business in this paper we analyze consumer preferences by modeling it as a decision making problem of ranking alternatives with consumers online ratings we propose a data driven fuzzy preference analysis d fpa method to obtain the priorities of alternatives we show that the d fpa is tractable and with high computation efficiency in addition we propose a natural indicator to measure the reliability of the derived ranking results and suggest thresholds of this indicator for better control of the method a real world application about online film rating is provided to illustrate the d fpa demonstrating that the derived ranking results converge rapidly and remain stable with the observed empirical data finally we show how to build up an effective recommendation system with empirical data from movielens

consistency measures for hesitant fuzzy linguistic preference relations hesitant fuzzy linguistic term sets hfltss are used to deal with situations in which the decision makers dms think of several possible linguistic values or richer expressions than a single term for an indicator alternative variable etc compared with fuzzy linguistic approaches they are more convenient and flexible to reflect the dms preferences in decision making for further applications of hfltss to decision making we develop a concept of hesitant fuzzy linguistic preference relations hflprs as a tool to collect and present the dms preferences due to the importance of the consistency measures using preference relations in decision making we develop some consistency measures for hflprs to ensure that the dms are being neither random nor illogical a consistency index is defined to establish the consistency thresholds of hflprs to measure whether an hflpr is of acceptable consistency for hflprs with unacceptable consistency we develop two optimization methods to improve the consistency until they are acceptable several illustrative examples are given to validate the consistency measures and the optimization methods

poincare series of compressed local artinian rings with odd top socle degree abstract we define a notion of compressed local artinian ring that does not require the ring to contain a field let r m be a compressed local artinian ring with odd top socle degree s at least five and socle r m s 1 m s we prove that the poincare series of all finitely generated modules over r are rational sharing a common denominator and that there is a golod homomorphism from a complete intersection onto r

analytic hierarchy process hesitant group decision making in this paper we consider that the judgments provided by the decision makers dms cannot be aggregated and revised then define them as hesitant judgments to describe the hesitancy experienced by the dms in decision making if there exist hesitant judgments in analytic hierarchy process group decision making ahp gdm then we call it ahp hesitant group decision making ahp hgdm as an extension of ahp gdm based on hesitant multiplicative preference relations hmprs to collect the hesitant judgments we develop a hesitant multiplicative programming method hmpm as a new prioritization method to derive ratio scale priorities from hmprs the hmpm is discussed in detail with examples to show its advantages and characteristics the practicality and effectiveness of our methods are illustrated by an example of the water conservancy in china

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

approaches to collective decision making with fuzzy preference relations abstract recent experimental studies show that the predictive accuracy of many of the solution concepts derived from the collective decision making theory leaves much to be desired in a previous paper the author attempted to explain some of the inaccuracies in terms of the fuzzy indifference regions of the individuals participating in the voting game this paper gives straightforward generalizations of the solutions concepts in terms of the fuzzy social or individual preference relations it turns out that some of these new solution concepts cotain their nonfuzzy counterparts as subsets others in turn are subsets of their nonfuzzy counterparts we also discuss a method of aggregating individual nonfuzzy preferences so as to get a fuzzy social preference relation and furthermore a nonfuzzy social choice set

modular centralizer algebras corresponding to p groups abstract we study the loewy structure of the centralizer algebra k p q for p a p group with subgroup q and k a field of characteristic p here k p q is a special type of hecke algebra the main tool we employ is the decomposition k p q k c p q i of k p q as a split extension of a nilpotent ideal i by the group algebra k c p q we compute the loewy structure for several classes of groups investigate the symmetry of the loewy series and give upper and lower bounds on the loewy length of k p q several of these results were discovered through the use of magma especially the general pattern for most of our computations as a final application of the decomposition we determine the representation type of k p q

group decision making with a fuzzy linguistic majority abstract the determination of solutions in group decision making is considered the point of departure is a collection of individual fuzzy preference relations a solution is derived either directly from the individual fuzzy preference relations or by constructing first a social fuzzy preference relation as opposed to conventional approaches in which a crisp threshold type majority rule is used we employ a fuzzy majority rule specified by a fuzzy linguistic quantifier e g most much more than 50 etc a calculus of linguistically quantified propositions is applied using the fuzzy majority various solution concepts are derived mainly of the type of core minimax opposition set and consensus winner

a chi square method for obtaining a priority vector from multiplicative and fuzzy preference relations abstract decision makers dms preferences on decision alternatives are often characterized by multiplicative or fuzzy preference relations this paper proposes a chi square method csm for obtaining a priority vector from multiplicative and fuzzy preference relations the proposed csm can be used to obtain a priority vector from either a multiplicative preference relation i e a pairwise comparison matrix or a fuzzy preference relation or a group of multiplicative preference relations or a group of fuzzy preference relations or their mixtures theorems and algorithm about the csm are developed three numerical examples are examined to illustrate the applications of the csm and its advantages

performance evaluation an integrated method using data envelopment analysis and fuzzy preference relations in a multi attribute decision making madm context the decision maker needs to provide his preferences over a set of decision alternatives and constructs a preference relation and then use the derived priority vector of the preference to rank various alternatives this paper proposes an integrated approach to rate decision alternatives using data envelopment analysis and preference relations this proposed approach includes three stages first pairwise efficiency scores are computed using two dea models the ccr model and the proposed cross evaluation dea model second the pairwise efficiency scores are then utilized to construct the fuzzy preference relation and the consistent fuzzy preference relation third by use of the row wise summation technique we yield a priority vector which is used for ranking decision making units dmus for the case of a single output and a single input the preference relation can be directly obtained from the original sample data the proposed approach is validated by two numerical examples

quantifiers in the formulation of multiple objective decision functions abstract we discuss the interpretation of linguistically quantified statements using fuzzy sets we indicate that there are two methods of interpreting these statements the algebraic approach and the substitution approach we then use quantified statements to express overall decision functions involving multiple objectives these statements are of the type a solution should satisfy many of the objectives using the algebraic and substitution approach we develop a whole family of forms for combining multiple objectives we show that algebraic approach corresponds to situations in which we are unconcerned about the allocation of satisfaction amongst the objectives but just interested in getting much satisfaction the substitution approach on the other hand leads to forms which are dependent upon the allocation of satisfaction between the objectives

the ordinal consistency of a fuzzy preference relation in this paper we first analyze the rationality of weak transitivity of a fuzzy preference relation defined by tanino fuzzy preference orderings in group decision making fuzzy sets and systems 12 1984 117 131 we then propose a revised definition of weak transitivity we call it ordinal consistency we propose the ordinal consistency index oci to measure the degree of ordinal consistency of a fuzzy preference relation which is to count the unreasonable 3 cycles in a directed graph that represents the fuzzy preference relation afterwards a procedure to compute the order consistency index and to locate each cycle as well as to find the inconsistent judgments in the fuzzy preference relation is proposed in order to repair the inconsistency of a fuzzy preference relation an algorithm is developed to find and remove 3 cycles in the graph the algorithm eliminates 3 cycles in a graph more effectively and the proposed method for improving consistency method aims to preserve the initial preference information of the decision maker furthermore the method can be used not only for a strict fuzzy preference relation but also for non strict fuzzy preference relation finally we provide some examples to show the effectiveness and validity of the proposed method

novel basic operational laws for linguistic terms hesitant fuzzy linguistic term sets and probabilistic linguistic term sets in the process of decision making people sometimes may feel more comfortable to express their preferences by linguistic terms instead of the quantitative form however as the basic premise of operations the existing operational laws of linguistic terms and the extended linguistic term sets are very unreasonable in order to overcome this issue in this paper we redefine some more logical operational laws for linguistic terms hesitant fuzzy linguistic elements hfles and probabilistic linguistic term sets pltss based on two equivalent transformation functions these novel operational laws can not only avoid the operation values exceeding the bounds of ltss but also keep the operation results more reasonable in decision making with linguistic information furthermore the operational laws can keep the probability information complete when computing with pltss additionally lots of properties of the operational laws are discussed and some three dimensional figures are drawn to show the regions of different operational laws of linguistic terms more vividly

asymmetric hesitant fuzzy sigmoid preference relations in the analytic hierarchy process this study considers the practical phenomena in the process of preference elicitation and proposes an asymmetric sigmoid numerical scale asns based on a generalized sigmoid function it also offers proof of the scale s asymmetry variability consistency and diminishing utility properties further this study introduces the hesitant fuzzy preference format and defines the hesitant fuzzy continuous preference term based on this approach the asymmetric hesitant fuzzy sigmoid preference relation ahspr is developed and used in the analytic hierarchy process ahp the results show that the ahspr is a general and optimal preference relation additionally this study constructs a discrete fitting technology and an approximate translation method as the applied bases of the new scale and the preference relation following this a model framework of the ahspr in the ahp is provided finally this study re examines a well known numerical example in order to demonstrate the application and advantages of the proposed numerical scale the preference format and the modeling framework

a novel risk decision making based on decision theoretic rough sets under hesitant fuzzy information decision theoretic rough sets dtrss play a crucial role in risk decision making problems with respect to the minimum expected risk dtrss deduce the rules of three way decisions considering the new expression of evaluation information with hesitant fuzzy sets hfss we introduce hfss into dtrss and explore their decision mechanisms more specifically we take into account the losses of dtrss with hesitant fuzzy elements and propose a new model of hesitant fuzzy decision theoretic rough sets hfdtrss some properties of the expected losses and their corresponding scores are carefully investigated under the hesitant fuzzy information three way decisions and the associated cost of each object are further derived with the above analysis a novel risk decision making method with the aid of hfdtrss is developed besides the three way decisions with dtrss the method investigates the ranking and resource allocation by utilizing the associated costs of alternatives and multiobjective 0 1 integer programming our study also offers a solution in the aspect of determining losses of dtrs and extends the range of applications

combining numerical and linguistic information in group decision making abstract people give information about their personal preferences in many different ways depending on their background this paper deals with group decision making problems in which the solution depends on information of a different nature i e assuming that the experts express their preferences with numerical or linguistic values the aim of this paper is to present a proposal for this problem we introduce a fusion operator for numerical and linguistic information this operator combines linguistic values assessed in the same label set with numerical ones assessed in the interval 0 1 it is based on two transformation methods between numerical and linguistic values which are defined using the concept of the characteristic values proposed in this paper its application to group decision making problems is illustrated by means of a particular fusion operator guided by fuzzy majority considering that the experts express their opinions by means of fuzzy or linguistic preference relations this operator is used to develop a choice process for the alternatives allowing solutions to be obtained in line with the majority of the experts opinions

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

compressed algebras artin algebras having given socle degrees and maximal length j emsalem and the author showed in 18 that a general polynomialf of degree j in the ring sp k yl yr has jr rl 1 linearly independent partial derivates of order i for i 0 1 t j 2 here we generalize the proof to show that the various partial derivates of s polynomials of specified degrees are as independent as possible given the room available using this result we construct and describe the varieties g e and z e parametrizing the graded and nongraded compressed algebra quotients a r i of the power series ring r k xl sxr having given socle type e these algebras are artin algebras having maximal length dimk a possible given the embedding degree r and given the socle type sequence e els s es where ei is the number of generators of the dual module a of a having degree i the variety z e is locally closed irreducible and is a bundle over g e fibred by affine spaces an whose dimension is known we consider the compressed algebras a new class of interesting algebras and a source of examples many of them are nonsmoothable have no deformation to k k for dimension reasons for some choices of the sequence e d buchsbaum d eisenbud and the author have shown that the graded compressed algebras of socle type e have almost linear minimal resolutions over r with ranks and degrees determined by e other examples have given type e dimk socle a and are defined by an ideal i with certain given numbers of generators in r

the loewy structure of certain fixpoint algebras part i abstract we investigate the loewy structure of the fixpoint algebra of the group algebra of the additive group of a finite field f under the action of a subgroup of the multiplicative group of f

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

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

solving multi period project selection problems with fuzzy goal programming based on topsis and a fuzzy preference relation project portfolio managers are multi objective decision makers dms who are expected to select the best mix of projects by maximizing profits and minimizing risks over a multi period planning horizon however project portfolio decisions are complex multi objective problems with a high number of projects from which a subset has to be chosen subject to various constraints and a multitude of priorities and preferences we propose a goal programming gp approach for project portfolio selection that embraces conflicting fuzzy goals with imprecise priorities a fuzzy goal with an aspiration level and a predefined membership function is defined for each objective the impreciseness in the priorities of the membership values of the fuzzy goals is modeled with fuzzy relations this leads to type ii fuzzy sets since fuzzy relations are organized between the membership values of the fuzzy goals which are themselves fuzzy sets the proposed model is based on the technique for order preference by similarity to ideal solution topsis and fuzzy preference relations topsis is used to reduce the multi objective problem into a bi objective problem the resulting bi objective problem is solved with fuzzy gp fgp the fuzzy preference relations are used to help dms express their preferences with respect to the membership values of the fuzzy goals the proposed approach is used to solve a real life problem characterized as a fuzzy multi objective project selection with multi period planning horizon mops mpph the performance of the proposed approach is compared with a competing method in the literature we show that our approach generates high quality solutions with minimal computational efforts

additional criteria in fuzzy two level pattern recognition abstract pattern recognition with fuzzy sets is considered with regard to medical diagnostic problems an alternative interpretation of membership grade is suggested which expresses the degree of illness of a patient with respect to a certain disease a formal statement of a two level fuzzy classifier is presented the main emphasis is put on additional criteria designed to evaluate a fuzzy diagnostic decision for an unknown object two groups of criteria are formed decision making connected and domain dependent the decision making group of criteria can be viewed as related to the soft consensus measure in group decisions theory interconnections between criteria are discussed the notions proposed are illustrated with real medical data from the fields of aviation medicine and neonatology

hesitant fuzzy linguistic vikor method and its application in qualitative multiple criteria decision making the hesitant fuzzy linguistic term set hflts has turned out to be a powerful and flexible technique in representing decision makers qualitative assessments in the processes of decision making the aim of this paper is to develop a method to solve the multicriteria decision making mcdm problem within the context of hflts in which the criteria conflict with each other to do so the concepts of ideal solutions for a hfl mcdm problem have been introduced in addition in order to represent the closeness of one solution to the ideal one we propose a sort of hesitant fuzzy linguistic measures such as the hesitant fuzzy linguistic group utility measure the hesitant fuzzy linguistic individual regret measure and the hesitant fuzzy linguistic compromise measure based on these measures we develop a hesitant fuzzy linguistic vikor hfl vikor method which is motivated by the traditional vikor method the general procedures for the hfl vikor method are given some numerical examples are provided to demonstrate the advantages and practicality of our method finally we make some discussions on the advantages of the hfl vikor method as well as future work

multiple attribute decision making based on fuzzy preference information on alternatives ranking and weighting this paper investigates the multiple attribute decision making madm problem with fuzzy preference information on alternatives and proposes an eigenvector method to rank them three optimization models are introduced to assess the relative importance weights of attributes in a madm problem which integrate subjective fuzzy preference relations and objective information in different ways since the decisions these models lead to are based on both subjective and objective information they are more balanced and credible than those that consider only one type of information an example is provided to illustrate the application of the proposed models

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

on fuzzy tournaments and their solution concepts in group decision making abstract assuming as the point of departure a set of individual or collective fuzzy tournaments complete and reciprocal binary relations representing the preferences of the particular individuals or their group as a whole respectively we define some new solution concepts in group decision making as counterparts of the respective solution concepts in the case of nonfuzzy tournaments complete and asymmetric binary relations basically the new solution concepts are mainly counterparts of some sets of undominated options uncovered sets and banks sets we consider the cases of both the conventional nonfuzzy and fuzzy majorities

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

symmetric local algebras and small blocks of finite groups one of the major problems in modular representation theory is the deter mination of the numbers k b of ordinary and z b of modular irreducible characters for the block b of a finite group g with defect group d not much is known about the opposite problem of classifying the defect group by the numbers k b and 1 b of course there is brauer s result that k b 1 is equivalent to d 1 the next two cases have been solved in a recent paper of j brandt 11 he showed as a corollary of more general results that k b 2 or 1 b 1 and k b 3 imply 1 d i k b the case i b 2 and k b 3 seems to be of a quite different nature therefore in this paper we restrict our attention to the case f b 1 and extend brandt s result to the case k b 4 it turns out that this theorem follows from some general properties of symmetric local algebras so we state our result in this more general context 1 from blocks to symmetric local algebras we are going to prove the following theorem a theorem let b be a block of the finite group g with defect group d if z b 1 and k b 4 then idi k b the possible defect groups are therefore cyclic or a klein four group dade has determined the general structure of a block with cyclic defect group and brauer of a block with klein four defect group however we will not use their results we first show how this theorem follows from a more general theorem on symmetric local algebras for us a symmetric algebra is a pair a a where 190

some issues on consistency of fuzzy preference relations in decision making in order to avoid misleading solutions the study of consistency when the decision makers express their opinions by means of preference relations becomes a very important aspect in order to avoid misleading solutions in decision making problems based on fuzzy preference relations the study of consistency is associated with the study of the transitivity property in this paper a new characterization of the consistency property defined by the additive transitivity property of the fuzzy preference relations is presented using this new characterization a method for constructing consistent fuzzy preference relations from a set of n 1 preference data is proposed applying this method it is possible to assure better consistency of the fuzzy preference relations provided by the decision makers and in such a way to avoid the inconsistent solutions in the decision making processes additionally a similar study of consistency is developed for the case of multiplicative preference relations 2002 elsevier b v all rights reserved

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

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

building consensus in group decision making with an allocation of information granularity abstract consensus is defined as a cooperative process in which a group of decision makers develops and agrees to support a decision in the best interest of the whole it is a questioning process more than an affirming process in which the group members usually modify their choices until a high level of agreement within the group is achieved given the importance of forming an accepted decision by the entire group the consensus problem has attained a great attention as it is a major goal in group decision making in this study we propose the concept of the information granularity being regarded as an important and useful asset supporting the goal to reach consensus in group decision making by using fuzzy preference relations to represent the opinions of the decision makers we develop a concept of a granular fuzzy preference relation where each pairwise comparison is formed as a certain information granule say an interval fuzzy set rough set and alike instead of a single numeric value as being more abstract the granular format of the preference model offers the required flexibility to increase the level of agreement within the group using the fact that we select the most suitable numeric representative of the fuzzy preference relation

fuzzy rationality measures fuzzy preference relations formalize intensity of individual preferences over fixed sets of alternatives it is therefore natural to extend to fuzzy preferences the notion of rationality or consistency with this goal in mind we address in this paper the problem of giving an axiomatic basis for defining the concept of fuzzy rationality specifically we establish a collection of conditions that any fuzzy rationality measure must satisfy some examples of fuzzy rationality measures are then given and analyzed

a linear programming method for multiple criteria decision making with probabilistic linguistic information abstract the hesitant fuzzy linguistic term set hflts is applicable in expressing people s hesitant qualitative preference information however it may lose preference information as all the terms in the hflts are assigned the same weight the probabilistic linguistic term set plts can avoid this drawback by allowing the decision makers dms to give the weight of each linguistic term as a probability due to the flexibility of the plts the multiple criteria decision making mcdm with probabilistic linguistic information is now becoming a hot research topic the aim of this paper is to propose a linear programming method to solve the mcdm problems in which the dms preferences over the alternatives with respect to the given criteria are expressed as the pltss we first give the weighted deviation square measure for the normalized probabilistic linguistic positive ideal solution and each normalized probabilistic linguistic evaluation vector of each alternative then we define the inconsistency index and the consistency index according to the objective and subjective evaluations given by the dms based on which a linear programming model is then built to derive the weight of each criterion furthermore a linear programming method is developed to solve the mcdm problems with probabilistic linguistic information and a case study about the evaluation of hospitals is conducted to illustrate the proposed method finally some comparative analyses with the aggregation operator based method and the probabilistic linguistic preference ranking organization method for enrichment evaluation pl promethee method are made to show the advantages of the proposed method

a fuzzy multicriteria group decision making approach to select configuration items for software development selection of configuration items in software configuration management is important to determine the software quality and reduce the development time and cost the objective of this research is to develop a multicriteria group decision making model based on fuzzy set theory to improve the configuration items selection process since most information available in this stage is not numerical fuzzy set theory is used to represent the evaluation ratings of candidate items the developed model ranks candidate items into partial or complete orders that can assist decision makers in selecting more proper set of configuration items the consensus measures are also developed to determine the group acceptability of the obtained ranking orders in addition sensitivity analysis can be performed to examine the solution robustness an example of flight simulator development project is used to illustrate the concept developed

the group consensus based evidential reasoning approach for multiple attributive group decision analysis abstract many multiple attribute decision analysis problems include both quantitative and qualitative attributes with various kinds of uncertainties such as ignorance fuzziness interval data and interval belief degrees an evidential reasoning er approach developed in the 1990s and in recent years can be used to model these problems in this paper the er approach is extended to group consensus gc situations for multiple attributive group decision analysis problems in order to construct and check the gc a compatibility measure between two belief structures is developed first considering two experts utilities the compatibility between their assessments is naturally constructed using the compatibility measure based on the compatibility between two experts assessments the gc at a specific level that may be the attribute level the alternative level or the global level can be constructed and reached after the group analysis and discussion within specified times under the condition of gc we conduct a study on the forming of group assessments for alternatives the achievement of the aggregated utilities of assessment grades and the properties and procedure of the extended er approach an engineering project management software selection problem is solved by the extended er approach to demonstrate its detailed implementation process and its validity and applicability

group decision analysis based on fuzzy preference relations logarithmic and geometric least squares methods logarithmic and geometric least squares methods llsm and glsm are respectively applied to deal with the group decision analysis problems with fuzzy preference relations where multiplicative preference relations if any are transformed into fuzzy preference relations through proper transformation technique distance between any two fuzzy preference relations and the average distance from one fuzzy preference relation to all the others are defined and used to measure the relative importance of each fuzzy preference relation a numerical example involving multiple fuzzy and multiplicative preference relations is examined using the proposed methods it is shown that llsm and glsm provide two analytical and effective ways of modelling multiple fuzzy preference relations

softer optimization and control models via fuzzy linguistic quantifiers a softening of a basic formulation of multicriterion optimization and control multistage decision making is presented for optimization instead of seeking an optimal solution that best satisfies all the fuzzy objectives as has been done so far we seek an optimal solution that best satisfies most much more than 50 etc a linguistic quantifier in general of the fuzzy objectives for control we seek in turn an optimal sequence of controls that best satisfies the fuzzy constraints and fuzzy goals at most much more than 50 etc of the control stages a calculus of linguistically quantified statements based upon fuzzy sets and possibility theory is used some applications to softer evidence aggregation in expert systems are also indicated

probabilistic linguistic term sets in multi attribute group decision making when expressing preferences in qualitative setting several possible linguistic terms with different weights represented by probabilities may be considered at the same time the probabilistic distribution is usually hard to be provided completely and ignorance may exist in this paper we first propose a novel concept called probabilistic linguistic term set plts to serve as an extension of the existing tools then we put forward some basic operational laws and aggregation operators for pltss after that we develop an extended topsis method and an aggregation based method respectively for multi attribute group decision making magdm with probabilistic linguistic information and apply them to a practical case concerning strategy initiatives finally the strengths and weaknesses of our methods are clarified by comparing them with some similar techniques

an intrinsic consistency threshold for reciprocal matrices abstract in this paper we present a new intrinsic and scale independent consistency threshold for reciprocal matrices in the analytic hierarchy process ahp its derivation is based upon the relation between the consistency measure of the ahp and the reduction of weight of a new alternative we compare this standard with an existing one

a method for repairing the inconsistency of fuzzy preference relations this paper investigates the inconsistency problems of preference information about alternatives expressed as a fuzzy preference relation by a decision maker an analysis method is presented to identify the inconsistency and weak transitivity of a fuzzy preference relation and to repair its inconsistency to reach weak transitivity first several definitions are given on additive consistency inconsistency and weak transitivity of a fuzzy preference relation next two methods derived from graph theory are presented to judge whether a fuzzy preference relation has weak transitivity or not then an algorithm is developed to repair an inconsistent fuzzy preference relation and to make it become one with weak transitivity via a synthesis matrix which reflects the relationship between the fuzzy preference relation with additive consistency and the initial one given by a decision maker a convergence theorem is also given for the algorithm lastly two numerical examples are shown to illustrate the proposed method

on reciprocity indexes in the aggregation of fuzzy preference relations using the owa operator chiclana herrera and herrera viedma studied conditions under which the reciprocity property is maintained in the aggregation of reciprocal fuzzy preference relations using the owa operator guided by a relative linguistic quantifier in this note we focus on the reciprocity in the aggregation of fuzzy preference relations that is the additive reciprocity is not assumed using the owa operator by defining a reciprocity index for measuring the non reciprocity degree of fuzzy preference relations we show that a sufficient but not necessary condition under which a collective fuzzy preference relation obtained by aggregating a set of individual fuzzy preference relations using an owa operator guided by a linguistic quantifier with parameter a and b has a reciprocity index no greater than the largest of the corresponding individual reciprocity indexes our result is helpful to complete chiclana et al s decision model f chiclana f herrera e herrera viedma integrating three representation models in fuzzy multipurpose decision making based on fuzzy preference relations fuzzy sets and systems 97 1998 33 48

interval valued hesitant fuzzy linguistic sets and their applications in multi criteria decision making problems an interval valued hesitant fuzzy linguistic set ivhfls can serve as an extension of both a linguistic term set and an interval valued hesitant fuzzy set this new set combines quantitative evaluation with qualitative evaluation these can describe the real preferences of decision makers and reflect their uncertainty hesitancy and inconsistency this work focuses on multi criteria decision making mcdm problems in which the criteria are in different priority levels and the criteria values take the form of interval valued hesitant fuzzy linguistic numbers ivhflns the new approach to solving these problems is based on the prioritized aggregation operators of ivhflns having reviewed the relevant literature we provide interval valued hesitant fuzzy linguistic operations and apply some linguistic scale functions which have been improved on the basis of psychological theory and prospect theory ultimately two kinds of prioritized aggregation operators of ivhflns are developed which extend to a grouping prioritized situation and are applied to mcdm problems finally an example is provided to illustrate and verify the proposed approach in two separate situations which are then compared to other representative methods

a statistical comparative study of different similarity measures of consensus in group decision making an essential aim in group decision making gdm problems is to achieve a high level of consensus among experts consensus is defined as general or widespread agreement and it is usually modelled mathematically via a similarity function measuring how close experts opinions or preferences are similarity functions are defined based on the use of a metric describing the distance between experts opinions or preferences in the literature different metrics or distance functions have been proposed to implement in consensus models but no study has been carried out to analyse the influence the use of different distance functions can have in the gdm process this paper presents a comparative study of the effect of the application of some different distance functions for measuring consensus in gdm by using the nonparametric wilcoxon matched pairs signed ranks test it is concluded that different distance functions can produce significantly different results moreover it is also shown that their application also has a significant effect on the speed of achieving consensus finally these results are analysed and used to derive decision support rules based on a convergent criterion that can be used to control the convergence speed of the consensus process using the compared distance functions

a generalization of fuzzy multistage decision making and control via linguistic quantifiers abstract multistage incision making control under fuzziness is considered at each control stage a fuzzy constraint is imposed on the control applied and a fuzzy goal is imposed on the state attained an optimal sequence of controls is sought which maximizes the membership function of the fuzzy decision which is assumed to be the intersection of fuzzy constraints and fuzzy goals thin is basically equivalent to the determination of an optimal sequence of controls which best satisfies the fuzzy constraints and fuzzy goals at nil the control stages the generalization proposed in the paper replaces the strict all the something milder for example most thus we seek an optimal sequence of controls which best satisfies the fuzzy constraints and fuzzy goals at for example most of the control stages a calculus of linguistically quantified statements based upon fuzzy sets and possibility theory is used first the algebraic method is employed the resulting control problem is solved by dy

fuzzy quantification a state of the art abstract quantified sentences are a very powerful notion for modelling statements in natural language nl but in practice they have been used to solve several problems this paper is intended to offer a global view of the development on this branch until now focusing in the different approaches dealing with quantification specially those involving imprecision called fuzzy quantification we put attention to the different mechanisms for defining them the evaluation methods for measuring their fulfilment as well as the properties they should satisfy

a goal programming approach to group decision making based on multiplicative preference relations and fuzzy preference relations abstract this paper proposes a goal programming approach to solve group decision making gdm problems where the preference information on alternatives provided by decision makers is represented in two different formats i e multiplicative preference relations and fuzzy preference relations in order to narrow the gap between the collective opinion and each decision maker s opinion a linear goal programming model is constructed to integrate the two different formats of preference relations and to compute the collective ranking values of the alternatives thus the ranking of alternatives or selection of the most desirable alternative s is obtained directly from the computed collective ranking values a numerical example is also used to illustrate the applicability of the proposed approach

seclusion factor method to solve fuzzy multiple criteria decision making problems this paper incorporates the basic concepts of subjectivity with the objective of decision theory to develop a method which intellectualizes ambiguity into multiple criteria decision making mcdm problems an outranking intensity is introduced to determine the degree of overall outranking between competing alternatives which are represented by fuzzy numbers the comparison of these degrees is made through the concept of overall existence ranking index comparative examples as well as a case study are given to illustrate the approach

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

poincare series of modules over local rings of small embedding codepth or small linking number abstract let s n k be a commutative noetherian local ring and m be a finitely generated s module then the poincare series of m is rational if any of the following holds edim s depth s 3 edim s depth s 4 and s is gorenstein s is one link from a complete intersection s is two links from a complete intersection and s is gorenstein to prove this result we assume without loss of generality that s r i with r m regular local and the key to the argument is to produce a factorization of the canonical map r s as a composition of a complete intersection r c with a golod map c s this is accomplished by invoking a theorem of avramov and backelin once one has existence of a dg algebra structure on a minimal r free resolution of s together with detailed knowledge of the structure of the induced homology algebra tor r s k h k s linkage theory provides the main technical tool for this analysis

choice functions and mechanisms for linguistic preference relations abstract the problem of finding a solution set of alternatives when a linguistic preference relation represents a collective preference is analyzed following two research lines choice functions and mechanisms four classical choice sets of alternatives for a linguistic preference relation are presented and some relations between them are pointed out the concept of linguistic choice function as a tool to derive linguistic choice sets of alternatives is introduced and some particular linguistic choice functions are presented together with a study of their rationality properties on the other hand two types of linguistic choice mechanisms to derive solution sets of alternatives from linguistic choice functions are introduced simple and composite ones the concept of linguistic covering relation is introduced with a view to allow us the design of consistent linguistic choice mechanisms which may achieve more precise and coherent solution sets of alternatives

the ordinal consistency of an incomplete reciprocal preference relation the ordinal consistency is the usual weak transitivity condition that a logical and consistent person should use if he she does not want to express inconsistent opinions and therefore becomes the minimum requirement condition that a consistent reciprocal preference relation should verify in this paper we define and study the ordinal consistency of an incomplete reciprocal preference relation we then develop an algorithm to judge whether an incomplete reciprocal preference relation is ordinally consistent this proposed algorithm can also find all cycles of length 3 to n in the incomplete digraph of the incomplete reciprocal preference relation based on this proposed algorithm and two rules we develop another algorithm to repair an inconsistent incomplete reciprocal preference relation and to convert it to one with ordinal consistency our algorithm eliminates the cycles of length 3 to n in the digraph of an incomplete reciprocal preference relation most effectively our proposed method can preserve the initial preference information as much as possible furthermore the proposed method can be used for an incomplete reciprocal preference relation with strict comparison and non strict comparison information finally the effectiveness and validity of the proposed method are illustrated with examples

group choice under fuzzy information the group choice problem under fuzzy information about the criteria the choice has to satisfy or and about the elements among which the choice has to be made is analyzed two ways of making choice on a set of elements a are considered in detail 1 non deterministic choice the group or the individual is able to select only some subset of elements s mostly preferred with respect to given criteria without indicating any preference intensity on s the elements of s are called significant 2 deterministic choice the group or the individual is able to order all elements in a according to the strength of the expressed preferences socially satisfactory group choice function ssgcf under fuzzy information is defined as a choice function compatible with four elementary requirements formulated similarly to the classical conditions u p i and d firstly presented by the founder of the modern group choice theory k arrow two theorems of existence of such a function are proved and used for formulating a principle called incoherence principle in group choice a deterministic group choice is incompatible with deterministic individual choices either a non deterministic group choice can be obtained on the basis of deterministic individual choices or a deterministic group choice on the basis of non deterministic individual choices conformably to a phenomenon known in the group choice theory as manipulating the individual choices it is proved that there is a ssgcf resistant against manipulating and mapping the individual preference orderings into a non deterministic choice of the group an individual has a right to be decisive if his her significant elements are recognised to be significant for all the group it is proved that there is a group choice function mapping the individual choices into a non deterministic choice of the group and compatible with conditions u p i and the requirement each individual to be decisive

group consensus algorithms based on preference relations in many group decision making situations decision makers preferences for alternatives are expressed in preference relations including fuzzy preference relations and multiplicative preference relations an important step in the process of aggregating preference relations is to determine the importance weight of each preference relation in this paper we develop a number of goal programming models and quadratic programming models based on the idea of maximizing group consensus our models can be used to derive the importance weights of fuzzy preference relations and multiplicative preference relations we further develop iterative algorithms for reaching acceptable levels of consensus in group decision making based on fuzzy preference relations or multiplicative preference relations finally we include an illustrative example

a consensus process for group decision making with probabilistic linguistic preference relations abstract in group decision making gdm process consensus is a fundamental problem the information provided by the experts in a gdm problem is usually expressed as preferences and the linguistic preference relation lpr is one of the most frequently used structures to model the experts preferences as a new type of lpr the probabilistic linguistic preference relation plpr not only allows the experts to provide more than one linguistic term about linguistic variables but also reflects different importance degrees of the possible preference values this paper focuses on the consensus reaching process for gdm with plprs an index for measuring the consensus degree is defined first then as for the expert group with unacceptable consensus degree a consensus improving method is introduced based on the consistency and consensus criteria moreover the whole gdm process is introduced based on the aggregation operators for the probabilistic linguistic term sets pltss finally an application case about medical information presented on search engine is discussed

multiperson decision making based on multiplicative preference relations abstract a multiperson decision making problem where the information about the alternatives provided by the experts can be presented by means of different preference representation structures preference orderings utility functions and multiplicative preference relations is studied assuming the multiplicative preference relation as the uniform element of the preference representation a multiplicative decision model based on fuzzy majority is presented to choose the best alternatives in this decision model several transformation functions are obtained to relate preference orderings and utility functions with multiplicative preference relations the decision model uses the ordered weighted geometric operator to aggregate information and two choice degrees to rank the alternatives quantifier guided dominance degree and quantifier guided non dominance degree the consistency of the model is analysed to prove that it acts coherently

minimizing adjusted simple terms in the consensus reaching process with hesitant linguistic assessments in group decision making in some real world decision processes decision makers may prefer to provide their opinions using linguistic expressions instead of a single linguistic term particularly they may hesitate between several linguistic terms in this paper we deal with the consensus issue in the hesitant linguistic group decision making gdm problem firstly a novel distance based consensus measure is proposed then using this consensus measure we develop an optimization based consensus model in the hesitant linguistic gdm which minimizes the number of adjusted simple terms in the consensus building furthermore a two stage model is displayed to further optimize the solutions to the proposed consensus model through which we obtain the unique optimal adjustment suggestion to support the consensus reaching process in the hesitant linguistic gdm finally several desirable properties are proposed to justify the proposal and two examples are used to demonstrate the validity of the models

a sequential selection process in group decision making with a linguistic assessment approach in this paper a sequential selection process in group decision making under linguistic assessments is presented where a set of linguistic preference relations represents individuals preferences a collective linguistic preference is obtained by means of a defined linguistic ordered weighted averaging operator whose weights are chosen according to the concept of fuzzy majority specified by a fuzzy linguistic quantifier then we define the concepts of linguistic non dominance linguistic dominance and strict dominance degrees as parts of the sequential selection process the solution alternative s is obtained by applying these concepts

asymmetric fuzzy preference relations based on the generalized sigmoid scale and their application in decision making involving risk appetites numerical preference relations constitute a useful decision making technique and can be classified into two types on the basis of the 0 1 0 9 and 1 9 scales however these numerical scales cannot fully describe some desired properties such as asymmetry consistency variability and diminishing utility in accordance with the preference relations to address this issue we first develop a generalized sigmoid function and define a continuous preference set on the basis of which we propose the generalized sigmoid scale then we introduce the optimal discrete fitting and risk preference selection approaches to solve the risk appetite parameters in the generalized sigmoid scale using these new methods we further propose the asymmetric fuzzy preference relation afpr examine its additive transitivity property and some weak transitivity properties and design an approximate consistency test the corresponding five step modeling process under an asymmetric fuzzy preference environment is constructed which can be applied in decision making involving different risk appetites finally two examples are used to demonstrate the properties and advantages of these new methods the first example is simple and shows the differences between the traditional fuzzy preference relations and the afprs while the second is a practical case and is used to illustrate the feasibility and reasonability of the afprs

poincare series of modules over compressed gorenstein local rings abstract given two positive integers e and s we consider gorenstein artinian local rings r whose maximal ideal m satisfies m s 0 m s 1 and rank r m m m 2 e we say that r is a compressed gorenstein local ring when it has maximal length among such rings it is known that generic gorenstein artinian algebras are compressed if s 3 we prove that the poincare series of all finitely generated modules over a compressed gorenstein local ring are rational sharing a common denominator a formula for the denominator is given when s is even this formula depends only on the integers e and s note that for s 3 examples of compressed gorenstein local rings with transcendental poincare series exist due to bogvad

deviation square priority method for distinct preference structures based on generalized multiplicative consistency preference structures such as fuzzy preference relations incomplete fuzzy preference relations utility values and preference orderings are very useful in depicting the decision makers preferences over the considered objects in group decision making they have been attracting great interest from both researchers and practitioners in recent decades an important issue on these preference structures is to investigate the techniques of deriving priority weights of the objects from them in this paper based on generalized multiplicative consistency we establish a general nonlinear optimization model with fuzzy preference relations incomplete fuzzy preference relations utility values and preference orderings by solving the model we get a system of nonlinear equations and prove the uniqueness of its solution then a convergent iterative algorithm is devised to derive the priority weights from this system of equations the model and the algorithm include a variety of special cases suitable for group decision making with one or several of the aforementioned preference structures a practical case illustration concerning a manufacturing company that is searching for the best compression ignition engine design is provided and the numerical results of the developed models and the algorithm and the existing ones are also compared and analyzed in order to demonstrate the advantages and practicality of our models and algorithm

endomorphism rings of permutation modules over maximal young subgroups abstract let k be a field of characteristic two and let be a two part partition of some natural number r denote the permutation module corresponding to the maximal young subgroup in r by m we construct a full set of orthogonal primitive idempotents of the centraliser subalgebra s k 1 s k 2 r 1 end k r m of the schur algebra s k 2 r these idempotents are naturally in one to one correspondence with the 2 kostka numbers

monotonicity based ranking on the basis of multiple partially specified reciprocal relations abstract the aggregation of rankings is a recurrent task in several fields of application in a recent work by rademaker and de baets a ranking rule based on a natural monotonicity property was proposed in the context of social choice theory this rule is built on the premise that for a ranking a b c to represent a group s opinion it would be natural that the strength with which a c is supported should not be less than both the strength with which a b and the strength with which b c are supported a first approach to this ranking rule considering totally specified monotone reciprocal relations on a bipolar qualitative scale has already been taken in this paper a more general setting is considered each voter is allowed to provide a partially specified reciprocal relation that may not be monotone on the unit interval additionally new ways of measuring the cost of imposing monotonicity are introduced

a fuzzy approach for modelling visual texture properties abstract in this paper we address the problem of simulating the human perception of texture properties in images in particular we have focused our study on the properties of coarseness contrast and directionality that play a fundamental role in the human perception of texture the objective is not to precisely identify classify or discriminate between different textures as a whole but to be able to assess the presence of each texture property in the image for this purpose fuzzy sets defined on the domain of different groups of measures are employed in order to model each property by using parametric membership functions the corresponding parameters are obtained by learning a functional relationship between the computational values given by the measures and the human perception the performance of each fuzzy set is analyzed and tested with the human assessments and a ranking of subsets of measures is obtained according to their ability to represent the perception of the property allowing us to identify the most suitable combination of measures

the w reciprocity rule abstract given a small number of choices a matrix is a convenient way to represent a fuzzy preference relation usually this matrix is reciprocal the w reciprocity rule is presented here as a way to relax the rigidity of the reciprocal assumption and to embody linguistic meanings for indifference and for the representation of the converse preference

group decision making based on acceptable multiplicative consistency and consensus of hesitant fuzzy linguistic preference relations abstract in this paper we propose a new group decision making gdm method based on the acceptable multiplicative consistency and consensus of hesitant fuzzy linguistic preference relations hflprs first an approach for improving the consistency of hflprs is proposed to generate an acceptable multiplicative consistent hflpr then a consensus index of hflprs is defined and an optimization model is presented to meet an acceptable consensus requirement under the premise of the acceptable multiplicative consistency and the smallest information loss where it yields adjusted hflprs with an acceptable consistency and consensus then the weights of decision makers dms are calculated based on the obtained adjusted hflprs moreover we propose a new gdm method based on hflprs finally the proposed gdm method is illustrated by an application example and comparative analyses are conducted to show the performance and the superiority of the proposed gdm method

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