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

contractivity of linear fractional transformations one possible approach to exact real arithmetic is to use linear fractional transformations lfts to represent real numbers and computations on real numbers recursive expressions built from lfts are only convergent i e denote a well defined real number if the involved lfts are sufficiently contractive in this paper we define a notion of contractivity for lfts it is used for convergence theorems and for the analysis and improvement of algorithms for elementary functions

real number computability and domain theory we present the different constructive definitions of real number that can be found in the literature using domain theory we analyse the notion of computability that is substantiated by these definitions and we give a definition of computability for real numbers and for functions acting on them this definition of computability turns out to be equivalent to other definitions given in the literature using different methods domain theory is a useful tool to study higher order computability on real numbers an interesting connection between scott topology and the standard topologies on the real line and on the space of continuous functions on reals is stated an important result in this paper is the proof that every computable functional on real numbers is continuous w r t the compact open topology on the function space

the continuous ordered weighted geometric operator and its application to decision making the aim of this paper is to develop a continuous ordered weighted geometric c owg operator which is based on the continuous ordered weighted averaging c owa operator recently introduced by the author and the geometric mean we study some desirable properties of the c owg operator and present its application to decision making with interval multiplicative preference relation and finally an illustrative example is pointed out

admissible digit sets we examine a special case of admissible representations of the closed interval namely those which arise via sequences of a finite number of mobius transformations we regard certain sets of mobius transformations as a generalized notion of digits and introduce sufficient conditions that such a digit set yields an admissible representation of 0 furthermore we establish the productivity and correctness of the homographic algorithm for such admissible digit sets we present the stern brocot representation and a modification of same as a working example throughout

speculatively redundant continued logarithm representation continued logarithms as originally introduced by gosper represent a means for exact rational arithmetic but their application to exact real arithmetic is limited by the uniqueness of their representation this is quite unfortunate as this representation seems promising for efficient hardware implementation we propose an idea of making the representation redundant using speculative recognition of noncomputable cases this approach solves the problem of real number computability preserves most of the beneficial properties of continued logarithms and only moderately affects complexity of arithmetic algorithms thus keeping the prospect of efficient implementation

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

fuzzy logarithmic least squares ranking method in analytic hierarchy process abstract we xu and zhai 1992 have discussed the extensions of analytic hierarchy process in fuzzy environment and presented a procedure for constructing the fuzzy judgment matrix this paper continues the discussion and goes further into the problem about extracting the fuzzy weights from the fuzzy judgment matrix by logarithmic least squares method which is a main ranking method in analytic hierarchy process first we define the metric of the fuzzy judgment space and develop the expression form of logarithmic least squares method in fuzzy environment next we derive the associated normal equations from the problem of fuzzy logarithmic least squares and prove that the solutions of fuzzy logarithmic least squares model are equivalent to the solutions of its normal equations finally for every t 0 1 we can get its explicit solutions by solving the normal equations the technique proposed in this paper is rather simple to realize and can be used for dealing with actual problems

semiglobal uniform asymptotic stability of an easy to implement pll like sensorless observer for induction motors in this work stability properties of a novel and easy to implement sensorless observer for induction motors are investigated the considered solution is inspired by phase locked loop strategies and it takes advantage from the rotor flux derivatives to define a rotating vector to align with even if such variables are not directly measured input to state stability properties are proven for suitably defined electrical and magneto mechanical observation error subsystems time scale separation is then exploited to invoke singular perturbation arguments and guarantee boundedness of the observation errors trajectories finally the small gain theorem is adopted to prove semiglobal uniform asymptotic stability of the presented scheme some additional interesting properties are discussed beside the main stream of the proposed analysis simulations indicate the observer can be profitably exploited for induction machines speed sensorless control applications

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 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

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

exact real computer arithmetic with continued fractions a representation of the computable real numbers by continued fractions is introduced this representation deals with the subtle points of undecidable comparison and integer division as well as representing the infinite 1 0 and undefined 0 0 numbers two general algorithms for performing arithmetic operations are introduced the algebraic algorithm which computes sums and products of continued fractions as a special case basically operates in a positional manner producing one term of output for each term of input the transcendental algorithm uses a general formula of gauss to compute the continued fractions of exponentials logarithms trigonometric functions and a wide class of special functions a prototype system has been implemented in lelisp and the performance of these algorithms is promising

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

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

power geometric operators and their use in group decision making the power average pa operator and the power ordered weighted average powa operator are the two nonlinear weighted average aggregation tools whose weighting vectors depend on the input arguments in this paper we develop a power geometric pg operator and its weighted form which are on the basis of the pa operator and the geometric mean and develop a power ordered geometric pog operator and a power ordered weighted geometric powg operator which are on the basis of the powa operator and the geometric mean and study some of their properties we also discuss the relationship between the pa and pg operators and the relationship between the powa and powg operators then we extend the pg and powg operators to uncertain environments i e develop an uncertain pg upg operator and its weighted form and an uncertain power ordered weighted geometric upowg operator to aggregate the input arguments taking the form of interval of numerical values furthermore we utilize the weighted pg and powg operators respectively to develop an approach to group decision making based on multiplicative preference relations and utilize the weighted upg and upowg operators respectively to develop an approach to group decision making based on uncertain multiplicative preference relations finally we apply both the developed approaches to broadband internet service selection

fuzzy least squares priority method in the analytic hierarchy process the least squares priority method is an effective priority method in the analytic hierarchy process but it is not appropriate under fuzziness this paper is concerned with the question of estimation of weights of factors by least squares from a fuzzy judgment matrix first the expression form of least squares under fuzziness is given then the associated system of equation is developed after some properties of the system are investigated the explicit solution is derived finally it is shown that the solution of the system is the solution of the fuzzy least squares in this paper an example is also given to illustrate the procedure

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 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

a method for fuzzy aggregation based on group expert evaluations this paper introduces an approach to group decision making based on the coordination index and similarity of the finite collections of fuzzy sets the mathematical apparatus of this approach based on the concept of maximal coordination is constructed in the metric lattice of fuzzy sets a specific kind of fuzzy aggregation operator is proposed a method for fuzzy aggregation is introduced thoroughly discussed and its algorithm determined the practical example of application of the introduced method is offered as well

global observability analysis of sensorless induction motors the current problems to successfully apply sensorless controllers for induction motors are the existence of operation regimes for which the performance is remarkably deteriorated due to the difficulties in estimating correctly motor speed and flux and the lack of a theoretical explanation for this kind of behavior in this paper a global observability analysis for these machines is carried out it is first shown that all indistinguishable trajectories of the system i e pairs of state trajectories with the same input output behavior can be described by a differential equation on a manifold named here the indistinguishable dynamics studying the stability properties of this latter system it can be shown that the induction motor is not completely observable nor detectable in a local or in a global sense and for every set of parameters this implies that it is impossible to construct a state observer for the motor that converges for every trajectory of the system moreover the indistinguishable dynamics provides a systematic method to study understand and explain particular operation regimes and this is illustrated by some case studies of practical relevant operating conditions

real functions incrementally computable by finite automata this is an investigation into exact real number computation using the incremental approach of potts ph d thesis department of computing imperial college 1998 edalat and potts electronic notes in computer science vol 6 elsevier science publishers amsterdam 2000 nielsen and kornerup j universal comput sci 1 7 1995 527 and vuillemin ieee trans on comput 39 8 1990 1087 where numbers are represented as infinite streams of digits each of which is a mobius transformation the objective is to determine for each particular system of digits which functions r r can be computed by a finite transducer and ultimately to search for the most finitely expressible mobius representations of real numbers the main result is that locally such functions are either not continuously differentiable or equal to some mobius transformation this is proved using elementary properties of finite transition graphs and mobius transformations applying the results to the standard signed digit representations we can classify functions that are finitely computable in such a representation and are continuously differentiable everywhere except for finitely many points they are exactly those functions whose graph is a fractured line connecting finitely many points with rational coordinates

a speed sensorless indirect field oriented control for induction motors based on high gain speed estimation the authors design a new speed sensorless output feedback control for the full order model of induction motors with unknown constant load torque which guarantees local asymptotic tracking of smooth speed and rotor flux modulus reference signals and local asymptotic field orientation on the basis of stator current measurements only the proposed nonlinear controller exploits the concept of indirect field orientation no flux estimation is required in combination with a new high gain speed estimator based on the torque current tracking error the estimates of unknown load torque and time varying rotor speed converge to the corresponding true values under a persistency of excitation condition with a physically meaningful interpretation basically equivalent to non null synchronous frequency stability analysis of the overall dynamics has been performed exploiting the singular perturbation method the proposed control algorithm is a true industrial sensorless solution since no simplifying assumptions flux and load torque measurements are required simulation and experimental tests show that the proposed controller is suitable for medium and high performance applications

pcf extended with real numbers we extend the programming language pcf with a type for total and partial real numbers by a partial real number we mean an element of a cpo of intervals whose subspace of maximal elements single point intervals is homeomorphic to the euclidean real line we show that partial real numbers can be considered as continuous words concatenation of continuous words corresponds to refinement of partial information the usual basic operations cons head and tail used to explicitly or recursively define functions on words generalize to partial real numbers we use this fact to give an operational semantics to the above referred extension of pcf we prove that the operational semantics is sound and complete with respect to the denotational semantics a program of real number type evaluates to a head normal form iff its value is different from b if its value is different from b then it successively evaluates to head normal forms giving better and better partial results converging to its value author s abstract

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

fuzzy multiple attributes and multiple hierarchical decision making abstract a procedure is proposed to solve the multiple attributes and multiple hierarchical system under fuzzy environment the approach is based on 1 1 fuzzy representation 2 2 hierarchical performance evaluation structure 3 3 gradient eigenvector method for rating the fuzzy criteria weighting and 4 4 using the max min paired elimination method for aggregation to illustrate the approach an example on the evaluation of teaching performance in higher education is solved

fuzzy mcdm based on ideal and anti ideal concepts this paper presents a novel fuzzy multiple criteria decision making mcdm based on the concepts of ideal and anti ideal points the concepts of fuzzy set theory and hierarchical structure analysis are used to develop a weighted suitability decision matrix to evaluate the weighted suitability of different alternatives versus various criteria the distance of different alternatives versus positive ideal solution and negative ideal solution are then obtained by using the proposed ranking method finally the relative approximation values of various alternatives versus positive ideal solution are ranked to determine the best alternative

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

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

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