extensions of the analytic hierarchy process in fuzzy environment abstract the aim of this paper is to extend the analytic hierarchy process ahp to the fuzzy environment firstly starting from the viewpoint of actual application considering the method of group decision making in complex systems and the fuzziness of judgment in pairwise comparison of projects we construct the fuzzy judgement matrix by using a set valued statistics method on continuous judgment matrix and prove that every element of the fuzzy judgment scale can be represented by a positive bounded closed fuzzy number after a discussion of the properties of a positive bounded closed fuzzy number the fuzzy weight vector of projects can be introduced by applying the fuzzy extension principle to the altered gradient eigenvector method thus the rank ordering of projects is given finally we also give an example to apply our model

fuzzy based linguistic patterns as a tool for the flexible assessment of a priority vector obtained by pairwise comparisons a novel approach to the assessment of a hierarchy vector generated by an expert pairwise comparison matrix is proposed the quality of a given hierarchy is measured using so called linguistic patterns i e specially constructed logical expressions and an expert assessment matrix the idea of this approach is based on a concept proposed previously as a flexible criterion for the domain of facility layout problems the adaptation of the linguistic pattern approach to the assessment of a hierarchy vector also by analogy to the analytic hierarchy process or ahp facilitates evaluations of expert consistency the proposed idea is shown to be less sensitive to potential expert errors than both the classical and fuzzy ahp techniques the proposed approach is described in detail and illustrated with simple examples some features of the proposed approach were studied using computer simulation experiments and the results are reported together with the conclusions obtained

pairwise comparison matrix with fuzzy elements on alo group this paper deals with pairwise comparison matrices with fuzzy elements fuzzy elements of the pairwise comparison matrix are applied whenever the decision maker is not sure about the value of his her evaluation of the relative importance of elements in question in comparison with pairwise comparison matrices with crisp elements investigated in the literature here we investigate pairwise comparison matrices with elements from abelian linearly ordered group alo group over a real interval we generalize the concept of reciprocity and consistency of crisp pairwise comparison matrices to matrices with triangular fuzzy numbers pcfn matrices we also define the concept of priority vector which is a generalization of the crisp concept such an approach allows for a generalization dealing both with the pcfn matrices on the additive multiplicative and also fuzzy alo groups it unifies several approaches known from the literature moreover we also deal with the problem of measuring the inconsistency of pcfn matrices by defining corresponding indexes the first index called the consistency grade g is the maximal alpha of alpha cut such that the corresponding pcfn matrix is still alpha consistent on the other hand the consistency index i of the pcfn matrix measures the distance of the pcfn matrix to the closest ratio matrix if the pcfn matrix is crisp and consistent then g is equal to 1 and the consistency index i is equal to the identity element e of the alo group otherwise g is less than 1 or i is greater than e four numerical examples are presented to illustrate the concepts and derived properties finally we show that the properties of reciprocity and consistency of pcfn matrices are saved by special transformations based on fuzzy extensions of isomorphisms between two alo groups

a new multiplicative consistency based method for decision making with triangular fuzzy reciprocal preference relations abstract triangular fuzzy reciprocal preference relations tfrprs are good choices to denote the decision makers fuzzy preferences consistency analysis is a very important topic to preference relations that can avoid the illogical ranking order however all previous multiplicative consistency concepts for tfrprs cannot well address this type of preference relations to address this issue this paper introduces a new multiplicative consistency concept for tfrprs which can be seen as a natural extension of tanino s multiplicative consistency concept for reciprocal preference relations based on the new consistency concept 0 1 mixed programming models for judging the multiplicative consistency of tfrprs are constructed and an approach for deriving multiplicative consistent tfrprs is provided considering the incomplete case programming models to determine missing values are constructed subsequently a multiplicative consistency based method for decision making with tfrprs is presented that can address the inconsistent and incomplete cases meanwhile illustrative examples are provided to show the concrete application of the proposed procedure and comparison analysis is also made

a parametric model for determining consensus priority vectors from fuzzy comparison matrices abstract we consider a group decision making problem where a set of alternatives have to be ranked according to fuzzy preference judgments given by multiple experts we assume that expert assessments are expressed in the form of fuzzy multiplicative preference relations or fuzzy comparison matrices in this paper we propose a general model to generate crisp priority weights of the alternatives from possibly inconsistent and conflicting fuzzy preference relations we express our model in terms of matrix approximations to address the consistency problem and we use weighted metrics to simulate the group dynamics matrix approximation techniques for deriving a common crisp consistent matrix are extended to work with fuzzy matrices by using the concept of cuts in the aggregation process the importance of each expert is taken into consideration according to the agreement of the group with the expert this results in a parametric optimization problem for which a computational formulation is given

on fuzzy ratio comparisons in hierarchical decision models abstract this article develops a method for the interactive analysis of fuzzy pairwise comparisons in hierarchical weighting models in the proposed method fuzzy ratios of relative importance are employed as constraints on the membership values of local priorities so that fuzzy weights for the alternatives can be computed through linear programming moreover graph displays of fuzzy relations are used as a means of illustrating the dominance implications of the specified statements

an improved theory of asymptotic unfoldings abstract an asymptotic unfolding of a dynamical system near a rest point is a system with additional parameters such that every one parameter deformation of the original system can be embedded in the unfolding preserving all properties that can be detected by asymptotic methods asymptotic unfoldings are computed using normal and hypernormal form methods we present a simplified and improved method of computing such unfoldings that can be used in any normal form style

on consistency and ranking of alternatives in fuzzy ahp abstract in fuzzy ahp where pairwise comparisons are described by membership values the treatment of consistency has largely been sketchy this paper proposes a fuzzy consistency definition with consideration of a tolerance deviation essentially the fuzzy ratios of relative importance allowing certain tolerance deviation are formulated as constraints on the membership values of the local priorities the fuzzy local and global weights are determined via the extension principle the alternatives are ranked on the basis of the global weights by application of a maximum minimum set ranking method our analysis uses only the data which are considered as consistent the proposed approach is computationally efficient as it consists of solving only linear programs an example is included for illustrative purpose

generalized means as model of compensative connectives abstract empirical investigations have shown that instead of strong logical and and or connectives there is often some degree of compensation in between in this paper the generalized means are proposed as connective operators for fuzzy set theory which easily allow for modelling the degree of compensation in a natural manner including the arithmetic and geometric means as well as the maximum and minimum operators as special cases numerical examples with empirical data form an illustration of this concept some aspects of applications to decision making are discussed

fuzzy analytic hierarchy process fallacy of the popular methods abstract the fuzzy analytic hierarchy process fuzzy ahp is a very popular decision making method and literally thousands of papers have been published about it however we find the basic logic of this approach has problems from its methodology the definition and operational rules of fuzzy numbers not only oppose the main logic of fuzzy set theory but also oppose the basic principles of the ahp in dealing with the outcomes fuzzy ahp does not give a generally accepted method to rank fuzzy numbers and a way to check the validity of the results besides we discuss the validity of the analytic hierarchy network process ahp anp in complex and uncertain environments and find that fuzzy anp is a false proposition because there is no fuzzy priority in the super matrix which provides the basis for the anp although fuzzy ahp has been applied in many cases and cited hundreds of times we hoped that those who use fuzzy ahp would understand the problems associated with this method

analytic hierarchy process ahp in group decision making and its optimization with an allocation of information granularity in group decision making one strives to reconcile differences of opinions judgments expressed by individual members of the group fuzzy decision making mechanisms bring a great deal of flexibility by admitting membership degrees we are offered flexibility to exploit different aggregation mechanisms and navigate a process of interaction among decision makers to achieve an increasing level of consistency within the group while the studies reported so far exploit more or less sophisticated ways of adjusting transforming initial judgments preferences of individuals in this paper we bring forward a concept of information granularity here information granularity is viewed as an essential asset which offers a decision maker a tangible level of flexibility using some initial preferences conveyed by each individual that can be adjusted with the intent to reach a higher level of consensus our study is concerned with an extension of the well known analytic hierarchy process to the group decision making scenario more specifically the admitted level of granularity gives rise to a granular matrix of pairwise comparisons the granular entries represented e g by intervals or fuzzy sets supply a required flexibility using the fact that we select the most suitable numeric representative of the reciprocal matrix the proposed concept of granular reciprocal matrices is used to optimize a performance index which comes as an additive combination of two components the first one expresses a level of consistency of the individual pairwise comparison matrices by exploiting the admitted level of granularity we aim at the minimization of the corresponding inconsistency index the second part of the performance index quantifies a level of disagreement in terms of the individual preferences the flexibility offered by the level of granularity is used to increase the level of consensus within the group given an implicit nature of relationships between the realizations of the granular pairwise matrices and the values of the performance index we consider using particle swarm optimization as an optimization vehicle two scenarios of allocation of granularity among decision makers are considered namely a uniform allocation of granularity and nonuniform distribution of granularity where the levels of allocated granularity are also subject to optimization a number of numeric studies are provided to illustrate an essence of the method

applications of the extent analysis method on fuzzy ahp in this paper a new approach for handling fuzzy ahp is introduced with the use of triangular fuzzy numbers for pairwise comprison scale of fuzzy ahp and the use of the extent analysis method for the synthetic extent value si of the pairwise comparison by applying the principle of the comparison of fuzzy numbers that is v m1 m2 1 iff m1 m2 v m2 m1 hgt m1 m2 m1 d the vectors of weight with respect to each element under a certaine criterion are represented by d ai min v si sk k 1 2 n k i this decision process is demonstrated by an example

fuzzy hierarchical analysis the lambda max method abstract we present a new method of finding the fuzzy weights in fuzzy hierarchical analysis which is the direct fuzzification of the max method used by saaty in the analytical hierarchical process an example is presented where there are five criteria and three alternatives

hesitant analytic hierarchy process abstract in traditional analytic hierarch process ahp decision makers dms are required to provide crisp judgments over paired comparisons of objectives to construct comparison matrices to enhance the modeling ability of traditional ahp we propose hesitant ahp h ahp that can consider the hesitancy experienced by the dms in decision h ahp is characterized by hesitant judgments where each hesitant judgment can be represented by several possible values different probability distributions can be used to further describe hesitant judgments according to the dms preferences based on a hesitant comparison matrix hcm that consists of hesitant judgments we define two indices to measure the consistency degree and the consensus degree of the hcm respectively from a stochastic point of view a new prioritization method is developed to derive priorities from hcms where the results are with probability interpretations we provide a step by step procedure for h ahp and demonstrate this new method with a real life decision making problem

a semiring based study of judgment matrices properties and models in decision making and group decision making multiplicative reciprocal judgment matrices and additive reciprocal judgment matrices are used as two kinds of important preference information in this paper semirings are applied to the discussion of judgment matrix properties first two special semirings are constructed second the properties of the consistent judgment matrices are given as a set of equations all in the semiring sense which include idempotency equations and fixed point equations we find that there exists one and only one specially constrained fixed point as the priority vector of a consistent judgment matrix third optimization models for inconsistent judgment matrices are presented finally we offer some simple illustrative examples

a note on the fuzzy extension of saaty s priority theory we consider the problem of the fuzzy extension of saaty s priority theory by van laarhoven and pedrycz in this paper since this problem has not been solved yet we apply the generalised pseudoinverse approach to find a structure of the general solution we recall the arithmetic of triangular fuzzy numbers and show links with the interval analysis by moore sample calculations are included

note on some models for deriving the priority weights from interval fuzzy preference relations abstract deriving accurate interval weights from interval fuzzy preference relations is key to successfully solving decision making problems xu and chen 2008 proposed a number of linear programming models to derive interval weights but the definitions for the additive consistent interval fuzzy preference relation and the linear programming model still need to be improved in this paper a numerical example is given to show how these definitions and models can be improved to increase accuracy a new additive consistency definition for interval fuzzy preference relations is proposed and novel linear programming models are established to demonstrate the generation of interval weights from an interval fuzzy preference relation

on the extent analysis method for fuzzy ahp and its applications abstract in a paper by chang d y chang applications of the extent analysis method on fuzzy ahp european journal of operational research 95 1996 649 655 an extent analysis method on fuzzy ahp was proposed to obtain a crisp priority vector from a triangular fuzzy comparison matrix it is found that the extent analysis method cannot estimate the true weights from a fuzzy comparison matrix and has led to quite a number of misapplications in the literature in this paper we show by examples that the priority vectors determined by the extent analysis method do not represent the relative importance of decision criteria or alternatives and that the misapplication of the extent analysis method to fuzzy ahp problems may lead to a wrong decision to be made and some useful decision information such as decision criteria and fuzzy comparison matrices not to be considered we show these problems to avoid any possible misapplications in the future

a note on treating a second order cone program as a special case of a semidefinite program it is well known that a vector is in a second order cone if and only if its arrow matrix is positive semidefinite but much less well known is about the relation between a second order cone program socp and its corresponding semidefinite program sdp the correspondence between the dual problem of socp and sdp is quite direct and the correspondence between the primal problems is much more complicated given a sdp primal optimal solution which is not necessarily arrow shaped we can construct a socp primal optimal solution the mapping from the primal optimal solution of sdp to the primal optimal solution of socp can be shown to be unique conversely given a socp primal optimal solution we can construct a sdp primal optimal solution which is not an arrow matrix indeed in general no primal optimal solutions of the socp related sdp can be an arrow matrix

fuzzy analytic hierarchy process a performance analysis of various algorithms abstract analytical hierarchical process ahp along with fuzzy set theory has been used extensively in the multi criteria decision making mcdm process in which fuzzy numbers are utilized to represent human judgments more realistically over the past couple of decades numerous articles have been published proposing algorithms through which priority vector or weight vector can be calculated from fuzzy comparison matrices the aim of this study is to conduct a comprehensive performance analysis of the most popular algorithms proposed in this domain in terms of accuracy of weights calculated from fuzzy comparison matrices such an analysis is much needed by the researchers and practitioners however none is available an experimental analysis is conducted and the performance of various algorithms are evaluated with varying three parameters i e the size of the comparison matrix the level of fuzziness and the level of inconsistency we found that modified logarithmic least squares method and fuzzy inverse of column sum method ficsm generally outperformed other algorithms while fuzzy extent analysis fea which is the most frequently used algorithm in the literature provides the least accurate results furthermore it was observed that a modified version of fea method significantly improved its performance

pairwise comparison based interval analysis for group decision aiding with multiple criteria interval ahp analytic hierarchy process was proposed to obtain interval weights from a given pairwise comparison matrix showing relative importance between criteria in this paper interval ahp is applied to group decision problems interval ahp is first revised suitably for comparing alternatives from the viewpoint that the interval weight vector shows the set of agreeable weight vectors for the decision maker under individual interval weight vectors obtained from individual pairwise comparison matrices three approaches to obtaining a consensus interval weight vector are proposed one is the perfect incorporation approach that obtains consensus interval weight vectors including all individual interval weight vectors by this approach we can count out indubitably inferior alternatives the second is the common ground approach that obtains consensus interval weight vectors included in all individual interval weight vectors by this approach we can find agreeable group preference between alternatives when all individual opinions are similar the third is the partial incorporation approach that obtains consensus interval weight vectors intersecting all individual interval weight vectors by this approach we can find agreeable group preference between alternatives when individual opinions are not similar the usefulness of the proposed three approaches is demonstrated by simple numerical examples

stochastic and fuzzy pert the key issue in project planning under uncertainty is the choice of a time schedule for further action such a time schedule which is necessary for the subcontractors because they often have to order materials and labour in advance will not automatically be generated by the analysis of the randomness and the imprecision in the project it is the result of a decision the obvious importance of the time schedule will extensively be dicussed in the sections to follow we present a fuzzy version of the project evaluation and review technique pert which is computationally easier to use than the original probabilistic version the fuzzy version also avoids several shortcomings of the probabilistic version

tropical implementation of the analytical hierarchy process decision method abstract we apply methods and techniques of tropical optimization to develop a new theoretical and computational framework for the implementation of the analytic hierarchy process in multi criteria problems of rating alternatives from pairwise comparison data in this framework we first consider the minimax chebyshev approximation of pairwise comparison matrices by consistent matrices in the logarithmic scale recasting this approximation problem as a problem of tropical pseudo quadratic programming we then write out a closed form solution to it this solution might be either a unique score vector up to a positive factor or a set of different score vectors to handle the problem when the solution is not unique we develop tropical optimization techniques of maximizing and minimizing the hilbert seminorm to find those vectors from the solution set that are the most and least differentiating between the alternatives with the highest and lowest scores and thus are well representative of the entire solution set

fuzzy eigenvector method for obtaining normalized fuzzy weights from fuzzy pairwise comparison matrices proper formulas for obtaining the fuzzy maximal eigenvalue and the corresponding fuzzy maximal eigenvector of a fuzzy pairwise comparison matrix are proposed in this paper first the formulas for obtaining the fuzzy maximal eigenvalue of a fuzzy pairwise comparison matrix proposed by csutora buckley 2001 and by ishizaka 2014 are reviewed and the flaws in their formulas regarding the violation of the reciprocity of pairwise comparisons are pointed out new formulas for obtaining the fuzzy maximal eigenvalue preserving the reciprocity of pairwise comparisons are then proposed after a fuzzy extension of saaty s consistency index and consistency ratio is introduced in order to verify an acceptable level of inconsistency of a fuzzy pairwise comparison matrix further the methods for obtaining the fuzzy maximal eigenvector corresponding to the fuzzy maximal eigenvalue of a fuzzy pairwise comparison matrix proposed by wang chin 2006 and by ishizaka 2014 are reviewed the flaws in ishizaka s method are pointed out and wang chin s method is studied and modified in order to preserve the reciprocity of pairwise comparisons the fuzzy maximal eigenvalues and the corresponding fuzzy maximal eigenvectors obtained by the new formulas are confronted with those obtained by methods proposed by csutora buckley 2001 wang chin 2006 and ishizaka 2014 and three numerical examples are given for better illustration a proper fuzzy extension of the maximal eigenvector method is proposed the new fuzzy maximal eigenvalue preserves the reciprocity of pairwise comparisons by employing the reciprocity into the method redundant information is eliminated the elements of the fuzzy maximal eigenvector are normalized fuzzy weights

a new approach for ranking fuzzy numbers by distance method abstract many ranking methods have been proposed so far however there is yet no method that can always give a satisfactory solution to every situation some are counterintuitive not discriminating some use only the local information of fuzzy values some produce different rankings for the same situation for overcoming the above problems we propose a new method for ranking fuzzy numbers by distance method our method is based on calculating the centroid point where the distance means from original point to the centroid point x 0 y 0 and the x 0 index is the same as murakami et al s x 0 however the y 0 index is integrated from the inverse functions of an lr type fuzzy number thus we use ranking function r a x 2 y 2 distance index as the order quantities in a vague environment our method can rank more than two fuzzy numbers simultaneously and the fuzzy numbers need not be normal furthermore we also propose the coefficient of variation cv index to improve lee and li s method comput math appl 15 1988 887 896 lee and li rank fuzzy numbers based on two different criteria namely the fuzzy mean and the fuzzy spread of the fuzzy numbers and they pointed out that human intuition would favor a fuzzy number with the following characteristics higher mean value and at the same time lower spread however when higher mean value and at the same time higher spread or lower mean value and at the same time lower spread exists it is not easy to compare its orderings clearly our cv index is defined as cv standard error mean which can overcome lee and li s problem efficiently in this way our proposed method can also be easily calculated by the mathematica package to solve problems of ranking fuzzy numbers at last we present three numerical examples to illustrate our proposed method and compare with other ranking methods

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

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

on the convergence of the central path in semidefinite optimization the central path in linear optimization always converges to the analytic center of the optimal set this result was extended to semidefinite optimization in d goldfarb and k scheinberg siam j optim 8 1998 pp 871 886 in this paper we show that this latter result is not correct in the absence of strict complementarity we provide a counterexample where the central path converges to a different optimal solution this unexpected result raises many questions we also give a short proof that the central path always converges in semidefinite optimization by using ideas from algebraic geometry

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

on the structure of nilpotent normal form modules abstract the set of vector fields that are in normal form with respect to a given linear part has the structure of a module and is best described by giving the stanley decomposition of that module an algorithm is presented that produces a stanley decomposition for the module of equivariants of the flow of a nilpotent linear vector field given a stanley decomposition for the ring of invariants this reduces the study of nilpotent normal forms to classical invariant theory plus this one additional algorithm both the inner product normal form elphick and the sl 2 normal form cushman sanders are covered within a single theory and simplified non equivariant versions of each normal form are presented

a modified smoothing and regularized newton method for monotone second order cone complementarity problems in this paper we propose a globally and quadratically convergent newton type algorithm for solving monotone second order cone complementarity problems denoted by soccps this algorithm is based on smoothing and regularization techniques by incorporating smoothing newton s method many newton type methods with smoothing and regularization techniques have been studied for solving nonlinear complementarity problems ncps and box constrained variational inequalities bvis our algorithm is regarded as an extension of those methods to soccp however it is different from the existing methods because we solve soccp by treating both the smoothing parameter m and the regularization parameter e as independent variables in addition numerical experiments indicate that the proposed method is quite effective

an eigenvector method for generating normalized interval and fuzzy weights crisp comparison matrices produce crisp weight estimates it is logical for an interval or fuzzy comparison matrix to give an interval or fuzzy weight estimate in this paper an eigenvector method em is proposed to generate interval or fuzzy weight estimate from an interval or fuzzy comparison matrix which differs from csutora and buckley s lambda max method in several aspects first the proposed em produces a normalized interval or fuzzy eigenvector weight estimate through the solution of a linear programming model while the lambda max method gives a series of nonnormalized interval eigenvector weight estimate through an iterative solution process next the em directly solves the principal right eigenvector of an interval or fuzzy comparison matrix while the lambda max method needs transforming a fuzzy comparison matrix into a series of interval comparison matrices using a level sets and the extension principle and therefore needs the solution of a series of eigenvalue problems finally the lambda max method needs the help of the principal right eigenvector of a crisp comparison matrix to determine the final interval weights while the em has no such requirement it is also found that not all interval or fuzzy comparison matrices can generate normalized interval or fuzzy eigenvector weights situations where the em is inapplicable are analyzed and the aggregation of local interval or fuzzy weights into global interval or fuzzy weights is also discussed three numerical examples including a hierarchical structure decision making problem are examined using the em to demonstrate its applications

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

performance evaluation of nonlinear optimization methods via pairwise comparison and fuzzy numbers in this paper we extend the deterministic performance evaluation of nonlinear optimization methods we carry out a pairwise comparison using fuzzy estimates of the performance ratios to obtain fuzzy final scores of the methods under consideration the key instrument is the concept of fuzzy numbers with triangular membership functions the algebraic operations on them are simple extensions of the operations on real numbers they are exact in the parameters lower modal and upper values not necessarily exact in the shape of the membership function we illustrate the fuzzy performance evaluation by the ranking and rating of five methods geometric programming and four general methods for solving geometric programming problems using the results of recent computational studies some general methods appear to be leading an outcome which is not only due to their performance under subjective criteria like domain of applications and conceptual simplicity of use they also score higher under more objective criteria like robustness and efficiency

fuzzy multi attribute selection among transportation companies using axiomatic design and analytic hierarchy process the basic concept in axiomatic design ad is the existence of design axioms first of these axioms is the independence axiom and the second one is the information axiom information axiom proposes the selection of the best alternative that has minimum information analytic hierarchy process ahp is another multi attribute method which is a decision making method for selecting the best among a set of alternatives given some criteria the method has been extensively applied especially in large scale problems where many criteria must be considered and where the evaluation of alternatives is mostly subjective multi attribute transportation company selection is a very important activity for effective supply chain selection of the best company under determined criteria such as cost time damage loss flexibility and documentation ability using both multi attribute ad and ahp will be realized in this study the fuzzy multiattribute ad approach is also developed and it is compared by one of fuzzy ahp methods in the literature the selection process has been accomplished by aiding a software that includes crisp ad and fuzzy ad

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

evidential reasoning approach for multiattribute decision analysis under both fuzzy and interval uncertainty many multiple attribute decision analysis mada problems are characterized by both quantitative and qualitative attributes with various types of uncertainties incompleteness or ignorance and vagueness or fuzziness are among the most common uncertainties in decision analysis the evidential reasoning er and the interval grade er ier approaches have been developed in recent years to support the solution of mada problems with interval uncertainties and local ignorance in decision analysis in this paper the er approach is enhanced to deal with both interval uncertainty and fuzzy beliefs in assessing alternatives on an attribute in this newly developed fuzzy ier fier approach local ignorance and grade fuzziness are modeled under the integrated framework of a distributed fuzzy belief structure leading to a fuzzy belief decision matrix a numerical example is provided to illustrate the detailed implementation process of the fier approach and its validity and applicability

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

spectral sequences and parametric normal forms abstract this paper extends recent developments on spectral sequence approach on the simplest normal form theory to parametric cases in this paper the method is first applied to two simple examples parametric single zero and parametric resonant saddle singularities and then to parametric generalized hopf singularity we provide the simplest parametric normal form for these cases

deriving priority vector from pairwise comparisons matrix with fuzzy elements abstract this paper deals with the pairwise comparisons matrix with fuzzy elements fpcm fuzzy elements of the fpcm are applied whenever the decision maker is not sure about the value of his her evaluation of the relative importance of elements in question we investigate the pairwise comparisons matrices with elements from abelian linearly ordered group alo group over a real interval introduce the concept of reciprocity and weak consistency of fpcms and derive their properties such an approach unifies several approaches known from the literature moreover we propose a new general procedure for deriving the priority vector from the fpcm enabling ranking or rating a given set of alternatives

fuzzy hierarchical analysis revisited abstract we present a new method of finding the fuzzy weights in fuzzy hierarchical analysis which is the direct fuzzification of the original method used by saaty in the analytic hierarchy process we test our new procedure in two cases where there are formulas for the crisp weights an example is presented where there are five criteria and three alternatives

interval weight generation approaches based on consistency test and interval comparison matrices a simple yet pragmatic method of consistency test is developed to check whether an interval comparison matrix is consistent or not linear programming method is used to derive consistent interval weights from consistent interval comparison matrices and to aggregate local interval weights to generate global interval weights in cases where an interval comparison matrix is inconsistent an eigenvector method based nonlinear programming nlp approach is developed to generate interval weights that can meet pre determined consistency requirements a simple and effective preference ranking method is utilized to compare the interval weights of criteria or rank alternatives three numerical examples including a hierarchical ahp decision problem are provided to illustrate the validity and practicality of the proposed methods

polynomial convergence of primal dual algorithms for the second order cone program based on the mz family of directions in this paper we study primal dual path following algorithms for the second order cone programming socp based on a family of directions that is a natural extension of the monteiro zhang mz family for semidefinite programming we show that the polynomial iteration complexity bounds of two well known algorithms for linear programming namely the short step path following algorithm of kojima et al and monteiro and adler and the predictor corrector algorithm of mizuno et al carry over to the context of socp that is they have an o loge 1 iteration complexity to reduce the duality gap by a factor of e where n is the number of second order cones since the mz type family studied in this paper includes an analogue of the alizadeh haeberly and overton pure newton direction we establish for the first time the polynomial convergence of primal dual algorithms for socp based on this search direction

a new fuzzy multiple attributive group decision making methodology and its application to propulsion manoeuvring system selection problem in this paper a new fuzzy multiple attribute decision making fmadm method which is suitable for multiple attributive group decision making gdm problems in fuzzy environment is proposed to deal with the problem of ranking and selection of alternatives since the subjectivity imprecision and vagueness in the estimates of a performance rating enter into multiple attribute decision making madm problems fuzzy set theory provides a mathematical framework for modelling vagueness and imprecision in the proposed approach an attribute based aggregation technique for heterogeneous group of experts is employed and used for dealing with fuzzy opinion aggregation for the subjective attributes of the decision problem the propulsion manoeuvring system selection as a real case study is used to demonstrate the versatility and potential of the proposed method for solving fuzzy multiple attributive group decision making problems the proposed method is a generalised model which can be applied to great variety of practical problems encountered in the naval architecture from propulsion manoeuvring system selection to warship requirements definition

acceptability measurement and priority weight elicitation of triangular fuzzy multiplicative preference relations based on geometric consistency and uncertainty indices this paper investigates consistency and uncertainty measurements as well as acceptability checking for triangular fuzzy multiplicative preference relations tfmprs a geometric consistency index is proposed to measure the consistency degree of a tfmpr and thresholds of acceptable consistency are discussed for tfmprs uncertainty indices are introduced to measure indeterminacy degrees of triangular fuzzy judgments and of tfmprs and a notion of acceptable tfmprs are put forward by considering both acceptable consistency and acceptable uncertainty in tfmprs we also explore the elicitation of triangular fuzzy weights from acceptable tfmprs and the aggregation of local triangular fuzzy weights for solving hierarchical multi criteria decision making problems a geometric mean based equation and a logarithmic least square model are developed to elicit normalized acceptable triangular fuzzy multiplicative weights from an acceptable tfmpr an equation and a linear program are established to transform triangular fuzzy multiplicative weights into triangular fuzzy additive weights a geometric weighting based equation and a pair of linear programs are devised to generate global priority fuzzy weights from local triangular fuzzy multiplicative weights three numerical examples including a hierarchical multi criteria decision making problem are used to illustrate how to apply the developed framework and comparative studies are provided to show its availability and advantages

applications of second order cone programming in a second order cone program socp a linear function is minimized over the intersection of an affine set and the product of second order quadratic cones socps are nonlinear convex problems that include linear and convex quadratic programs as special cases but are less general than semidefinite programs sdps several efficient primaldual interior point methods for socp have been developed in the last few years after reviewing the basic theory of socps we describe general families of problems that tan be recast as socps these include robust linear programming and robust leastsquares problems problems involving sums or maxima of norms or with convex hyperbolic constraints we discuss a variety of engineering applications such as filter design antenna array weight design truss design and grasping forte optimization in robotics we describe an efficient primaldual interior point method for solving socps which shares many of the features of primaldual interior point methods for linear program

box products in nilpotent normal form theory the factoring method abstract let n be a nilpotent matrix and consider vector fields x n x v x in normal form then v is equivariant under the flow e n t for the inner product normal form or e m t for the sl 2 normal form these vector equivariants can be found by finding the scalar invariants for the jordan blocks in n or m taking the box product of these to obtain the invariants for n or m itself and then boosting the invariants to equivariants by another box product these methods developed by murdock and sanders in 2007 are here given a self contained exposition with new foundations and new algorithms yielding improved simpler stanley decompositions for the invariants and equivariants ideas used include transvectants from classical invariant theory stanley decompositions from commutative algebra and integer cones from integer programming this approach can be extended to covariants of sl 2 k for k 1 known as slocc in quantum computing

perturbation analysis of second order cone programming problems we discuss first and second order optimality conditions for nonlinear second order cone programming problems and their relation with semidefinite programming problems for doing this we extend in an abstract setting the notion of optimal partition then we state a characterization of strong regularity in terms of second order optimality conditions this is the first time such a characterization is given for a nonpolyhedral conic problem

a fuzzy seasonal arima model for forecasting this paper proposes a fuzzy seasonal arima fsarima forecasting model which combines the advantages of the seasonal time series arima sarima model and the fuzzy regression model it is used to forecast two seasonal time series data of the total production value of the taiwan machinery industry and the soft drink time series the intention of this paper is to provide business which are affected by diversified management with a new method to conduct short term forecasting this model includes both interval models with interval parameters and the possible distribution of future value based on the results of practical application it can be shown that this model makes good forecasts and is realistic furthermore this model makes it possible for decision makers to forecast the best and worst estimates based on fewer observations than the sarima model

applying fuzzy linguistic preference relations to the improvement of consistency of fuzzy ahp the lack of consistency in decision making can lead to inconsistent conclusions in fuzzy analytic hierarchy process fuzzy ahp method it is difficult to ensure a consistent pairwise comparison furthermore establishing a pairwise comparison matrix requires nx n 1 2 judgments for a level with n criteria alternatives the number of comparisons increases as the number of criteria increases therefore the decision makers judgments will most likely be inconsistent to alleviate inconsistencies this study applies fuzzy linguistic preference relations fuzzy linprera to construct a pairwise comparison matrix with additive reciprocal property and consistency in this study the fuzzy ahp method is reviewed and then the fuzzy linprera method is proposed finally the presented method is applied to the example addressed by kahraman et al c kahraman d ruan i dogan fuzzy group decision making for facility location selection information sciences 157 2003 135 153 this study reveals that the proposed method yields consistent decision rankings from only n 1 pairwise comparisons which is the same result as in kahraman et al research the presented fuzzy linguistic preference relations method is an easy and practical way to provide a mechanism for improving consistency in fuzzy ahp method

vector potential normal form classification for completely integrable solenoidal nilpotent singularities abstract we introduce an sl 2 invariant family of polynomial vector fields with an irreducible nilpotent singularity in this paper we are concerned with characterization and normal form classification of these vector fields we show that the family is a lie subalgebra and each vector field from this family is volume preserving completely integrable and rotational all such vector fields share a common quadratic invariant we provide a poisson structure for the lie subalgebra from which the second invariant for each vector field can be readily derived we show that each vector field from this family can be uniquely characterized by two alternative representations which can be found in applications one uses a vector potential while the other uses two functionally independent clebsch potentials our normal form results are designed to preserve these structures and representations

on the normalization of interval and fuzzy weights the normalization of interval and fuzzy weights is often necessary in multiple criteria decision analysis mcda under uncertainty especially in analytic hierarchy process ahp with interval or fuzzy judgements the existing normalization methods based on interval arithmetic and fuzzy arithmetic are found flawed and need to be revised this paper presents the correct normalization methods for interval and fuzzy weights and offers relevant theorems in support of them numerical examples are examined to show the correctness of the proposed normalization methods and their differences from those existing normalization methods

the logarithmic least squares and the generalized pseudoinverse in estimating ratios abstract this paper concerns the pairwise comparison method used in analytic hierarchy process ahp the logarithmic least square method is one of the methods used to rank a finite number of stimuli based on their pairwise comparison in the case of one decision maker the problem can be solved using the geometric mean method it is then assumed that the solution is geometrically normalized in the case of multiple decision makers a set of linear equations is obtained and if we have a different number of judgments for each pair of the compared objects the geometric normalization assumption can not be used directly the aim of this paper is to show that applying the generalized pseudoinverse we obtain the solution that is geometrically normalized and consistent with the case of one decision maker to define the pseudoinverse the spectral decomposition is used the structure of the general solution is presented and the existence of the general solution is discussed

a fuzzy ranking method with range reduction techniques for ranking alternatives based on pairwise comparisons current analytic hierarchy process ahp methods are difficult to use to generate useful information to assist decision makers in specifying their preferences this study proposes a novel method incorporating fuzzy preferences and range reduction techniques modified from the concept of data envelopment analysis dea the proposed approach is not only capable of treating incomplete preference matrices but also provides reasonable ranges to help decision makers to rank decision alternatives confidently

constrained fuzzy hierarchical analysis for portfolio selection under higher moments marginal impacts of assets on portfolio higher moments are characterized by triangular fuzzy numbers and then evaluated by fuzzy ranking procedures in order to assemble fuzzy reciprocal matrices that are needed for the constrained fuzzy analytic hierarchy process ahp methods the proposed methodology increases the scope for emphasizing objective quantitative measures thus alleviating the influence of subjective qualitative factors employing constrained fuzzy arithmetic during fuzzy ahp application produces greater precision and reliability compared with applications of the standard fuzzy arithmetic by reference to higher moments the investor is able to strategize portfolios so that there is a reduction not only in exposure to normal risk i e volatility but also to risk of asymmetry skewness and the risk of fat tails kurtosis the efficiency of the proposed approach is highlighted in the ability to handle investor preferences regarding higher moment risks more optimal values for a particular risk can be obtained if that risk is of more concern moreover in terms of obtaining portfolio diversification the method is shown to be more useful than conventional optimization models

inconsistency of pair wise comparison matrix with fuzzy elements based on geometric mean in this paper inconsistency of pair wise comparison matrix with fuzzy elements is investigated inconsistency is based on the distance of the matrix to a special matrix measured by a particular metric new inconsistency index of reciprocal matrix with fuzzy elements is introduced and newly designed method of logarithmic least squares for eliciting associated weights is applied some basic properties of the index are derived simple illustrating examples and results of numerical simulation for comparing the new index with saaty s consistency ratio are supplied

comments on a note on applying fuzzy linguistic preference relations to the improvement of consistency of fuzzy ahpź a very recent note by pandey and kumar 3 pointed out some mathematical incorrect assumptions with respect to the statement and proof of the results given by wang and chen 8 furthermore the note had modified the statement and proof of the results this paper shows that the modified results are technically and logically incorrect new properties of fuzzy linguistic preference relations are developed to correct flaws in the existing results and a triangular fuzzy addition based transitivity equation is introduced to define additively consistent fuzzy linguistic preference relations

fuzzy group decision making for facility location selection the selection of a facility location among alternative locations is a multicriteria decision making problem including both quantitative and qualitative criteria the conventional approaches to facility location problem tend to be less effective in dealing with the imprecise or vagueness nature of the linguistic assessment under many situations the values of the qualitative criteria are often imprecisely defined for the decision makers the aim of the paper is to solve facility location problems using different solution approaches of fuzzy multi attribute group decision making the paper includes four different fuzzy multi attribute group decision making approaches the first one is a fuzzy model of group decision proposed by blin the second is the fuzzy synthetic evaluation the third is yager s weighted goals method and the last one is fuzzy analytic hierarchy process although four approaches have the same objective of selecting the best facility location alternative they come from different theoretic backgrounds and relate differently to the discipline of multi attribute group decision making these approaches are extended to select the best facility location alternative by taking into account quantitative and qualitative criteria a short comparative analysis among the approaches and a numeric example to each approach are given

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

fuzzy cognitive network process comparisons with fuzzy analytic hierarchy process in new product development strategy fuzzy analytic hierarchy process f ahp has increasingly been applied in many areas however as the perception and cognition toward the semantic representation for the linguistic rating scale used by the fuzzy pairwise comparison in f ahp are still open to discuss f ahp very likely produces misapplications this research proposes the fuzzy cognitive network process f cnp as an ideal alternative to f ahp a new product development application using f ahp is revised using f cnp this study shows that the proposed f cnp yields better results due to the appropriate mathematical definition of the fuzzy paired interval scale for the human perception of paired difference

a modified fuzzy logarithmic least squares method for fuzzy analytic hierarchy process this paper revisits the fuzzy logarithmic least squares method llsm in the analytic hierarchy process and points out its incorrectness in the normalization of local fuzzy weights infeasibility in deriving the local fuzzy weights of a fuzzy comparison matrix when the lower bound value of a non normalized fuzzy weight turns out to be greater than its upper bound value uncertainty of local fuzzy weights for incomplete fuzzy comparison matrices and unreality of global fuzzy weights a modified fuzzy llsm which is formulated as a constrained nonlinear optimization model is therefore suggested to tackle all these problems a numerical example is examined to show the applicability of the modified fuzzy llsm and its advantages

strong transitivity rationality and weak monotonicity in fuzzy pairwise comparisons strong transitivity and weak monotonicity are two important axioms in the theory of value and utility measurement in this paper we define strong transitivity and weak monotonicity for fuzzy pairwise comparisons matrices using the requirements of strong transitivity we develop a metric for measuring the degree of strong transitivity in fuzzy pairwise comparisons we further demonstrate that a matrix which satisfies the metric of strong transitivity will satisfy the weak monotonicity axiom as well

the estimation of normalized fuzzy weights abstract the estimation of a normalized set of positive fuzzy weights constitutes the most important aspects in the fuzzy multiple attribute decision making process a systematic treatment of this problem is carried out in this paper the concept of fuzzy normalization is first defined and the meaning of consistency in a fuzzy environment is discussed based on these definitions and discussions the various approaches in the literature are examined and several improvements or new approaches are proposed numerical examples are used to evaluate and to compare the various existing and the newly proposed approaches

a fuzzy approach to deriving priorities from interval pairwise comparison judgements abstract in this paper we study the problem of priority elicitation in the analytic hierarchy process and propose a new approach to deriving crisp priorities from interval pairwise comparison judgements by introducing linear or non linear membership functions representing the decision maker s degree of satisfaction with various crisp priority vectors the interval judgements are transformed into fuzzy inequality constraints the interval prioritisation problem is then formulated as a fuzzy mathematical programming problem for obtaining an optimal crisp priority vector that maximises the overall degree of satisfaction the proposed approach yields linear or non linear mathematical programs capable of deriving priorities from consistent and inconsistent interval judgements the presence of a consistency index that measures the level of inconsistency of interval judgements is an attractive feature of our approach another feature which does not exist in the known prioritisation methods is the opportunity for additional prioritisation of the initial judgements numerical examples are given and comparisons with other interval prioritisation methods are carried out

decision analysis with incomplete utility and probability information an approach to the solution of decision analysis problems under uncertainty with imprecise and incomplete information is presented the methodology is designed for cases in which payoffs conditional on the state of nature are known precisely but only limited or imprecise probability and utility information is available regarding a decision maker s beliefs and tastes a decision maker provides conditional payoffs optionally bounds on state probabilities bounds on the certainty equivalent for a simple lottery any known relationships between probabilities of states of nature and a series of strict preferences between pairs of vectors of conditional payoffs we assume an exponential utility function with unknown parameter the method proceeds by sequentially eliciting preferences new bounds on probabilities and or the certainty equivalent and new relationships among probabilities until the problem is solved the methodology is demonstrated through a two state and a three state example which illust

a new transvectant algorithm for nilpotent normal forms we offer an algorithm to determine the form of the normal form for a vector field with a nilpotent linear part when the form of the normal form is known for each jordan block of the linear part taken separately the algorithm is based on the notion of transvectant from classical invariant theory

asymptotic unfoldings of dynamical systems by normalizing beyond the normal form abstract systematic methods are given for simplification of the normal form of a perturbed dynamical system near a rest point it then becomes easy to write down an unfolding of the dynamical system which can be rigorously justified as a universal unfolding with respect to asymptotic equivalence these unfoldings always have finite codimension they are both weaker and stronger than topological unfoldings they may fail to be universal unfoldings in the topological sense because of behavior which escapes detection by asymptotic methods and they also may distinguish asymptotic behaviors that are topologically equivalent the method is illustrated by computing an unfolding of codimension 14 as well as some standard examples of codimension 2 and 3

a consistency test for rational weights in multi criterion decision analysis with fuzzy pairwise comparisons abstract we present some results and extensions of the use of fuzzy pairwise comparisons in multi criterion decision analysis we address the problem encountered in earlier articles by van laarhoven and pedrycz 1983 and boender et al 1989 in which normalized weights derived from estimates based on ratio scales can exhibit an irrational ordering we derive conditions for a rational ordering and illustrate that the irrational ordering is the result of relative preferences provided by the decision makers we provide a test for consistency that will insure a rational ordering of the normalized weights when using the methods of van laarhoven and pedrycz and boender et al for normalization

consistency analysis and priority derivation of triangular fuzzy preference relations based on modal value and geometric mean existing consistency notions of triangular fuzzy preference relations tfprs are illustrated to be non robust new consistency and acceptable consistency definitions are put forward for tfprs we propose a notion of normalized triangular fuzzy multiplicative weights ntfmws transformation formulae are provided to convert ntfmws into consistent tfprs a logarithmic least square model is developed to derive a ntfmw vector from a tfpr triangular fuzzy preference relation tfpr is an effective framework to model pairwise estimations with imprecision and vagueness in order to obtain a reliable and rational decision result it is important to investigate consistency and priority derivation of tfprs the paper analyzes existing definitions and properties of consistent tfprs and illustrates that they have no invariance with respect to permutations of decision alternatives a new triangular fuzzy arithmetic based transitivity equation is introduced to define consistent tfprs the new transitivity equation reflects multiplicative consistency of modal values and multiplicative consistency of geometric means of triangular fuzzy estimations some properties are presented for consistent tfprs and a notion of acceptable consistency is put forward for tfprs geometric mean and uncertainty ratio based transformation formulae are devised to convert normalized triangular fuzzy multiplicative weights into consistent tfprs a logarithmic least square model is further established for deriving a normalized triangular fuzzy multiplicative weight vector from a tfpr with acceptable consistency a geometric mean based method is developed to compare and rank triangular fuzzy multiplicative weights three numerical examples including a group decision making problem are examined to demonstrate validity and advantages of the proposed models

the analytic hierarchy process with stochastic judgements the analytic hierarchy process ahp is a widely used method for multicriteria decision support based on the hierarchical decomposition of objectives evaluation of preferences through pairwise comparisons and a subsequent aggregation into global evaluations the current paper integrates the ahp with stochastic multicriteria acceptability analysis smaa an inverse preference method to allow the pairwise comparisons to be uncertain a simulation experiment is used to assess how the consistency of judgements and the ability of the smaa ahp model to discern the best alternative deteriorates as uncertainty increases across a range of simulated problems results indicate that according to conventional benchmarks judgements are likely to remain consistent unless uncertainty is severe but that the presence of uncertainty in almost any degree is sufficient to make the choice of best alternative unclear

deriving priorities from fuzzy pairwise comparison judgements a new approach for deriving priorities from fuzzy pairwise comparison judgements is proposed based on cuts decomposition of the fuzzy judgements into a series of interval comparisons the assessment of the priorities from the pairwise comparison intervals is formulated as an optimisation problem maximising the decision maker s satisfaction with a specific crisp priority vector a fuzzy preference programming method which transforms the interval prioritisation task into a fuzzy linear programming problem is applied to derive optimal crisp priorities aggregating the optimal priorities which correspond to different cut levels enables overall crisp scores of the prioritisation elements to be obtained a modification of the linear fuzzy preference programming method is also proposed to derive priorities directly from fuzzy judgements without applying cut transformations the formulation of the prioritisation problem as an optimisation task is similar to the previous approach but it requires the solution of a non linear optimisation program the second approach also derives crisp priorities and has the advantage that it does not need additional aggregation and ranking procedures both proposed methods are illustrated by numerical examples and compared to some of the existing fuzzy prioritisation methods

the possibilistic moments of fuzzy numbers and their applications this paper introduces the nearest weighted interval and point approximations of a fuzzy number and then suggests weighted possibilistic moments about these points of fuzzy numbers the possibilistic moments play an important role in fuzzy sets and systems specifically in physics mathematics and statistics we provide the definition of the moments of fuzzy numbers as well as the definition of moments in probability theory some of their applications are mentioned

an extension to fuzzy mcdm an extension to the fuzzy multiple criteria decision making mcdm model is suggested in this work where the ratings of alternatives versus criteria and the importance weights of all criteria are assessed in linguistic values represented by fuzzy numbers moreover values of alternatives under objective criteria are normalized by a suggested approach meanwhile the membership function of the final fuzzy evaluation value of each alternative can be developed in addition a riemann integral based mean of removals is suggested to rank all the final fuzzy evaluation values for final decision making so that the ranking procedure can be clearly formulated finally a numerical example demonstrates the feasibility of the proposed model

a simple global characterization for normal forms of singular vector fields we derive a new global characterization of the normal forms of amplitude equations describing the dynamics of competing order parameters in degenerate bifurcation problems using an appropriate scalar product in the space of homogeneous vector polynomials we show that the resonant terms commute with the group generated by the adjoint of the original critical linear operator this leads to a very efficient constructive method to compute both the nonlinear coefficients and the unfolding of the normal form explicit examples and results obtained when there are additional symmetries are also presented

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

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