notes on order preservation and consistency in ahp abstract the pairwise comparisons method is a convenient tool used when the relative order among different concepts alternatives needs to be determined one popular implementation of the method is based on solving an eigenvalue problem for the pairwise comparisons matrix in such cases the ranking result for the principal eigenvector of the pairwise comparisons matrix is adopted while the eigenvalue is used to determine the index of inconsistency a lot of research has been devoted to the critical analysis of the eigenvalue based approach one of them is the work of bana e costa and vansnick 2008 in their work the authors define the conditions of order preservation cop and show that even for sufficiently consistent pairwise comparisons matrices this condition cannot be met the presented work defines more precise criteria for determining when the cop is met to formulate the criteria an error factor is used describing how far the input to the ranking procedure is from the ranking result the relationship between the saaty consistency index and cop is also discussed

uncertainty index based interval assignment by interval ahp abstract in a multi attribute decision making problem indigenous values are assigned to attributes based on a decision maker s subjective judgments the given judgments are often uncertain because of the uncertainty of situations and intuitiveness of human judgments in order to reflect the uncertainty in the assigned values they are denoted as intervals whose widths represent the possibilities of attributes since it is difficult for a decision maker to assign values directly to attributes in case of more than two attributes he she gives a pairwise comparison matrix by comparing two attributes at one occasion the given matrix contains two kinds of uncertainty one is inconsistency among comparisons and the other is incompleteness of comparisons this paper proposes the models to obtain intervals of attributes from the given uncertain pairwise comparison matrix at first the uncertainty indexes of a set of intervals are defined from the viewpoints of entropy in probability sum or maximum of widths or ignorance then considering that too uncertain information is not useful the intervals of attributes are obtained by minimizing their uncertainty indexes

arithmetic monodromy groups pinkham 12 showed that the monodromy groups of the 14 exceptional unimodular singularities can be characterized arithmetically using results of m kneser we proved in 4 that this is true for all singularities of arnold s list with certain exceptions see below the object of this note is to show that this is even true for almost all isolated hypersurface singularities more precisely we prove the following theorem

algorithms for improving consistency or consensus of reciprocal 0 1 valued preference relations we investigate the consistency and consensus of reciprocal 0 1 valued preference relations also called fuzzy preference relations by many authors based on the multiplicative consistency property which is an important issue in fuzzy set theory an algorithm is first developed to improve the consistency level of a reciprocal 0 1 valued preference relation and the corresponding algorithm for the incomplete reciprocal 0 1 valued preference relation is also developed we further propose the consensus improving algorithms for individual reciprocal 0 1 valued preference relations or incomplete ones the convergence and robustness of the algorithms are proven and some important conclusions are obtained in addition the proposed algorithms can improve the consistency or consensus of reciprocal 0 1 valued preference relations with less interactions with the decision makers which can save a lot of time and obtain the results quickly

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

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

aggregation of direct and indirect judgments in pairwise comparison matrices with a re examination of the criticisms by bana e costa and vansnick in a recent paper by bana e costa and vansnick c a bana e costa j c vansnick a critical analysis of the eigenvalue method used to derive priorities in ahp european journal of operational research 187 3 2008 1422 1428 analytic hierarchy process ahp particularly its eigenvector method em used for deriving priorities from pairwise comparison matrices was criticized for the violation of a so called condition of order preservation cop due to this violation the em was considered to have a serious fundamental weakness which makes the use of ahp as a decision support tool very problematic the consistency ratio cr index in the ahp was also criticized for its failure to act as an alert of this violation of cop in this paper we look into decision makers overall judgments which can be obtained through the aggregation of their direct and indirect judgments and then re examine bana e costa and vansnick s numerical examples with a detailed analysis to show the invalidity of their criticisms

on the stable exact model matching problem abstract the stable exact model matching problem semmp is investigated we state and prove a number of equivalent necessary and sufficient conditions for the existence of proper solutions to the exact model matching problem that are also stable i e have no poles inside a symmetric forbidden subset of the finite complex plane c these results can be viewed as the counterpart of the results in 3 and 9 for the case of the ring of proper and stable rational functions

jie ke versus alphago a ranking approach using decision making method for large scale data with incomplete information abstract ranking historical players in sports is challenging since some players have never played against each other it is even more complex in go because of alphago a project based on artificial intelligence who became the world s number 1 after it defeated the 528th and the 4th human go players alphago is ranked high in the current go ranking system because it is undefeated the objective of this paper is to propose a new ranking method for large scale go players by means of incomplete fuzzy pair wise comparison matrix whose priority vector is derived using a cosine similarity measure using match results provided by go4go net experiments are designed to rank top go players in the past 45 years and examine the change in ranking after alphago faced off against jie ke furthermore the proposed method was applied to rank all 1544 go players available at go4go net to illustrate its efficiency in handling large scale data

a least deviation method to obtain a priority vector of a fuzzy preference relation based on the transfer relationship between fuzzy preference relation and multiplicative preference relation this paper proposes a least deviation method to obtain a priority vector of a fuzzy preference relation and studies its desirable properties a simple convergent iterative algorithm is given and its practicality is shown by two numerical examples

remarks on the analytic hierarchy process the analytic hierarchy process ahp is flawed as a procedure for ranking alternatives in that the rankings produced by this procedure are arbitrary this paper provides a brief review of several areas of operational difficulty with the ahp and then focuses on the arbitrary rankings that occur when the principle of hierarchic composition is assumed this principle requires that the weights on the higher levels of a hierarchy can be determined independently of the weights on the lower levels virtually all of the published examples of the use of the ahp to evaluate alternatives relative to a set of criteria have assumed this principle the key to correcting this flaw is the synthesis of the ahp with the concepts of multiattribute utility theory

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

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

model matching and decoupling for continuous and discrete time linear time varying systems solutions to the exact model matching and block decoupling problems for both continuous and discrete time linear time varying systems are presented the parametrisation of the whole class of proper solutions is given for the decoupling the minimal delay problem is also considered in a time varying setting the approach is algebraic and based on the smith macmillan form at infinity of a transfer matrix of a time varying system which has been recently introduced in systems theory this avoids the difficulties related to the inversion of the transfer matrices with entries in non commutative fields over which the determinants of dieudonne or ore type are much more complicated the solutions presented here involve only standard matrix computations excluding direct matrix inversions and are thus easy to implement in practice examples are treated in detail to illustrate the theoretical results and the way in which the computations are done and a physical example is also shown

structure and smith macmillan form of a rational matrix at infinity the use of special row and column operations in the reduction of a rational matrix to its smith macmillan form at infinity is investigated the connections between this reduction procedure and the valuation approach are established a graphical method for finding the smith macmillan form of a rational matrix at infinity from its bode magnitude array and some new results on realization theory for polynomial matrices are presented

reconstruction methods for ahp pairwise matrices habitually decision makers are exposed to situations that require a lot of knowledge and expertise therefore they need tools to help them choose the best possible alternatives analytic hierarchical process ahp is one of those tools and it is widely used in many fields while the use of ahp is very simple there is a situation that becomes complex the consistency of the pairwise matrices in order to obtain the consistent pairwise matrix from the inconsistent one reconstruction methods can be used but they cannot guarantee getting the right matrix according to the judgments of the decision maker this situation does not allow proper evaluation of methods reliability i e it is not possible to obtain a reliable ranking of alternatives based on an inconsistent matrix in this work a new way to evaluate the reliability of matrix reconstruction methods is proposed this technique uses a novel measure for alternatives ranking comparison based on element positions and distances which is introduced in order to compare several matrix reconstruction methods finally in order to demonstrate the extensibility of this new reliability measure two reconstruction methods based on bio inspired models a genetic algorithm and the firefly algorithm are presented and evaluated by using the aforementioned reliability measure

proper rational matrix diophantine equations and the exact model matching problem relying on the theory behind the smith macmillan form of a rational matrix at s infty a necessary and sufficient condition is derived from the solvability of matrix diophantine equations of the form a s x s b s y s m s where a s b s and m s are given proper rational matrices and x s and y s are unknown proper rational matrices it is shown that the above result can be used in order to resolve in a new illuminating way the exact model matching problem emmp if a solution to emmp exists then the family of all solutions is parametrized

when is the condition of order preservation met abstract this article explores the relationship between inconsistency in the pairwise comparisons method and conditions of order preservation a pairwise comparisons matrix with elements from an alo group an abelian linearly ordered group is investigated this approach allows for a generalization of previous results sufficient conditions for order preservation based on the properties of elements of pairwise comparisons matrix are derived elucidating numerical examples are presented

a simple method to improve the consistency ratio of the pair wise comparison matrix in anp tests of consistency for the pair wise comparison matrices have been studied extensively since ahp was introduced by saaty in 1970s however existing methods are either too complicated to be applied in the revising process of the inconsistent comparison matrix or are difficult to preserve most of the original comparison information due to the use of a new pair wise comparison matrix those methods might work for ahp but not for anp as the comparison matrix of anp needs to be strictly consistent to improve the consistency ratio this paper proposes a simple method which combines the theorem of matrix multiplication vectors dot product and the definition of consistent pair wise comparison matrix to identify the inconsistent elements the correctness of the proposed method is proved mathematically the experimental studies have also shown that the proposed method is accurate and efficient in decision maker s revising process to satisfy the consistency requirements of ahp anp

contribution of individual judgments toward inconsistency in pairwise comparisons pairwise comparison pc is a well established method to assist decision makers in estimating their preferences in pcs the acquired judgments are used to construct a pc matrix pcm that is used to check whether the inconsistency in judgments is acceptable or requires revision the use of consistency ratio cr a widely used measure for inconsistency has been widely debated and the literature survey has identified a need for a more appropriate measure considering this need a new measure termed congruence is proposed in this paper the measure is shown to be useful in finding the contribution of individual judgments toward overall inconsistency of a pcm and therefore can be used to detect and correct cardinally inconsistent judgments the proposed measure is applicable to incomplete sets of pc judgments without modification unlike cr which requires a complete set of pc judgments to address ordinal inconsistency another measure termed dissonance is proposed as a supplement to the congruence measure the two measures appear useful in detecting both outliers and the phenomenon of consistency deadlock where all judgments equally contribute toward the overall inconsistency

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

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

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

a critical analysis of the eigenvalue method used to derive priorities in ahp abstract a lot of research has been devoted to the critical analysis of the analytic hierarchy process ahp from various perspectives however as far as we know no one has addressed a fundamental problem discussed in this paper concerning the meaning of the priority vector derived from the principal eigenvalue method used in ahp the role of ahp s consistency ratio is also analysed

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

minimal sensitivity perfect model matching control a simple and direct procedure for the design of a minimal sensitivity perfect model matching control system is proposed the key observation is that in addition to permitting of perfect model matching control the design parameters leave a great deal of flexibility that can be utilized to reduce the effect of external disturbances an example is included for illustration

sparse and distributed analytic hierarchy process abstract the analytic hierarchy process ahp is a de facto standard technique in centralized decision making consider a situation where there is a need to rank a set of elements or alternatives based on their value or utility of which we just know pairwise relative information i e the ratio of their values ahp proved an effective tool to retrieve the value of each element being able to handle also relative information affected by distortions subjective biases and intransitivity a downside of ahp however is that it requires complete information i e knowledge on all pairs in this paper we extend the applicability of the ahp technique to the case of sparse information i e when only a limited amount of information is available and such an information corresponds to an undirected connected graph we complement our sparse framework by developing novel criteria and metrics to evaluate the degree of consistency of the data at hand moreover exploiting the proposed framework we also provide a distributed formulation of ahp in which a set of agents interacting through an undirected graph are able to compute their own values e g for ranking or leader election purposes by only knowing the ratio of their values with respect to their neighbors to this end we develop a novel algorithm to let each agent i compute the dominant eigenvalue and the i th component of the corresponding eigenvector of the sparse ahp matrix we conclude the paper with a simulation campaign that numerically demonstrates the effectiveness of the proposed approach

a novel numerical method for exact model matching problem with stability in system design the exact model matching problem with stability consists of compensating a given system using a realizable control law of a specified structure in order to ensure the stability of the compensated system and achieve a target closed loop transfer function in this paper we develop a novel numerical method to verify the solvability of the problem for left invertible systems and further construct a desired solution our method has a complexity which is cubic in the state dimension of the system and the desired model and can be implemented in a numerically reliable way

inconsistency and non additive capacities the analytic hierarchy process in the framework of choquet integration we examine the ahp in the framework of choquet integration and we propose an extension of the standard ahp aggregation scheme on the basis of the shapley values associated with the criteria in our model a measure of dominance inconsistency between criteria is defined in terms of the totally inconsistent matrix associated with the main pairwise comparison matrix of the ahp the measure of dominance inconsistency is then used to construct a non additive capacity whose associated shapley values reduce to the standard ahp priority weights in the consistency case in the general inconsistency case however the extended aggregation scheme based on the shapley weighted mean tends to attenuate resp emphasize the priority weights of the criteria with higher resp lower average dominance inconsistency with respect to the other criteria

a technical note on two inconsistency indices for preference relations a case of functional relation abstract different representations of valued preferences have been studied and used over the years however one common factor has been the prominence of the concept of consistency inconsistency indices have been introduced to estimate the deviation of preferences from a fully consistent form in this note we shall recall two types of preference relations reciprocal relations and multiplicative preference relations and show that two inconsistency indices introduced in these two different frameworks are functionally related besides this main result some reflections on the consequences of being functionally related are presented

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

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

fixed zeros in the model matching problem for systems over semirings in this paper solution existence conditions for the model matching problem are studied for systems over semirings which are used in many applications such as queueing systems communication networks and manufacturing systems the main contribution is the discovery of fixed pole structure in solutions to the model matching problem this fixed pole structure provides essential information contained in all the solutions to the model matching problem for a discrete event dynamic system example a common petri net component in the solutions of the model matching problem can be discovered from the fixed pole structure

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