numerical scales generated individually for analytic hierarchy process abstract because individual interpretations of the analytic hierarchy process ahp linguistic scale vary for each user this study proposes a novel framework that ahp decision makers can use to generate numerical scales individually based on the 2 tuple linguistic modeling of ahp scale problems by using the concept of transitive calibration individual characteristics in understanding the ahp linguistic scale are first defined an algorithm is then proposed for detecting the individual characteristics from the linguistic pairwise comparison data that is associated with each of the ahp individual decision makers finally a nonlinear programming model is proposed to generate individual numerical scales that optimally match the obtained individual characteristics two well known numerical examples are re examined using the proposed framework to demonstrate its validity

a cosine maximization method for the priority vector derivation in ahp abstract the derivation of a priority vector from a pair wise comparison matrix pcm is an important issue in the analytic hierarchy process ahp the existing methods for the priority vector derivation from pcm include eigenvector method ev weighted least squares method wls additive normalization method an logarithmic least squares method lls etc the derived priority vector should be as similar to each column vector of the pcm as possible if a pair wise comparison matrix pcm is not perfectly consistent therefore a cosine maximization method cm based on similarity measure is proposed which maximizes the sum of the cosine of the angle between the priority vector and each column vector of a pcm an optimization model for the cm is proposed to derive the reliable priority vector using three numerical examples the cm is compared with the other prioritization methods based on two performance evaluation criteria euclidean distance and minimum violation the results show that the cm is flexible and efficient

the lax pairs for the holt system by using the known relation between the holt system and the henon heiles system the lax pairs for all the integrable cases of the holt system are constructed from the known lax representations for the henon heiles system

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

max algebra and pairwise comparison matrices abstract this paper is a continuation of our 2004 paper max algebra and pairwise comparison matrices in which the max eigenvector of a symmetrically reciprocal matrix was used to approximate such a matrix by a transitive matrix this approximation was based on minimizing the maximal relative error in a later paper by dahl a different error measure was used and led to a slightly different approximating transitive matrix here some geometric properties of this approximation problem are discussed these lead among other results to a new characterization of a max eigenvector of an irreducible nonnegative matrix the case of toeplitz matrices is discussed in detail and an application to music theory that uses toeplitz symmetrically reciprocal matrices is given

deriving weights from pairwise comparison ratio matrices an axiomatic approach abstract this paper examines the problem of extracting object or attribute weights from a pairwise comparison ratio matrix this problem is approached from the point of view of a distance measure on the space of all such matrices a set of axioms is presented which such a distance measure should satisfy and the uniqueness of the measure is proven the problem of weight derivation is then shown to be equivalent to that of finding a totally transitive matrix which is a minimum distance from the given matrix this problem reduces to a goal programming model finally it is shown that the problem of weight derivation is related to that of ranking players in a round robin tournament the space of all binary tournament matrices is proven to be isometric to a subset of the space of ratio matrices

eigenvector ranking method as a measuring tool formulas for errors abstract the eigenvector method em is well known to derive information from pairwise comparison matrices in decision making processes however this method is logically incomplete since its actual numerical error is unknown and its reliability is doubted by such phenomena as right left asymmetry rank reversal and reversal of order of intensity of preference in this paper we associate em with some standard measuring procedure analyze this procedure from the viewpoint of measurement theory and find the actual em error we show that the above phenomena have the same cause and are eliminated when the em errors are taken into account the full decision support tool which has all components of a standard measuring tool is composed of pairwise comparisons as an initial measuring procedure em as a data processor and the obtained formulas for em errors as an error indicator we consider two versions of this tool based on the right and the left principal eigenvectors of a pairwise comparison matrix both versions are equally suitable to measure and rank any comparable elements with positive numerical values and have the same mean relative errors equal to the square root of the double saaty s consistency index using the mean relative error we find the simple upper estimate for maximum permissible errors not impeding the reliable ranking this estimate imposes tight restrictions on inconsistency of expert judgements in decision making processes with a large number of alternatives these restrictions are much stronger than previously thought

fiedler ptak scaling in max algebra abstract this is essentially the text of my talk on fiedler ptak scaling in max algebra delivered in the invited minisymposium in honor of miroslav fiedler at the 17th ilas conference in braunschweig germany

enhancing data consistency in decision matrix adapting hadamard model to mitigate judgment contradiction cardinal and ordinal inconsistencies are important and popular research topics in the study of decision making with pair wise comparison matrices pcms few of the currently employed tactics are capable of simultaneously dealing with both cardinal and ordinal inconsistency issues in one model and most are heavily dependent on the method chosen for weight priorities derivation or the obtained closest matrix by optimization method that may change many of the original values in this paper we propose a hadamard product induced bias matrix model which only requires the use of the data in the original matrix to identify and adjust the cardinally inconsistent element s in a pcm through graph theory and numerical examples we show that the adapted hadamard model is effective in identifying and eliminating the ordinal inconsistencies also for the most inconsistent element identified in the matrix we develop innovative methods to improve the consistency of a pcm the proposed model is only dependent on the original matrix is independent of the methods chosen to derive the priority vectors and preserves most of the original information in matrix a since only the most inconsistent element s need s to be modified our method is much easier to implement than any of the existing models and the values it recommends for replacement outperform those derived from the literature it significantly enhances matrix consistency and improves the reliability of pcm decision making

a parametric gp model dealing with incomplete information for group decision making abstract we consider a group decision making problem where preferences given by the experts are articulated into the form of pairwise comparison matrices in many cases experts are not able to efficiently provide their preferences on some aspects of the problem because of a large number of alternatives limited expertise related to some problem domain unavailable data etc resulting in incomplete pairwise comparison matrices our goal is to develop a computational method to retrieve a group priority vector of the considered alternatives dealing with incomplete information for that purpose we have established an optimization problem in which a similarity function and a parametric compromise function are defined associated to this problem a logarithmic goal programming formulation is considered to provide an effective procedure to compute the solution moreover the parameters involved in the method have a clear meaning in the context of group problems

the mathematical equivalence of the spanning tree and row geometric mean preference vectors and its implications for preference analysis abstract pairwise comparison is a widely used approach to elicit comparative judgements from a decision maker dm and there are a number of methods that can be used to then subsequently derive a consistent preference vector from the dm s judgements while the most widely used method is the eigenvector method the row geometric mean approach has gained popularity due to its mathematical properties and its ease of implementation in this paper we discuss a spanning tree method and prove the mathematical equivalence of its preference vector to that of the row geometric mean approach this is an important finding due to the fact that it identifies an approach for generating a preference vector which has the mathematical properties of the row geometric mean preference vector and yet in its entirety the spanning tree method has more to offer than the row geometric mean method in that it is inherently applicable to incomplete sets of pairwise comparison judgements and also facilitates the use of statistical and visual techniques to gain insights into inconsistency in the dm s judgements

a heuristic method to rectify intransitive judgments in pairwise comparison matrices abstract this paper investigates the effects of intransitive judgments on the consistency of pairwise comparison matrices statistical evidence regarding the occurrence of intransitive judgements in pairwise matrices of acceptable consistency is gathered by using a monte carlo simulation which confirms that relatively high percentage of comparison matrices satisfying saaty s cr criterion are ordinally inconsistent it is also shown that ordinal inconsistency does not necessarily decrease in the group aggregation process in contrast with cardinal inconsistency a heuristic algorithm is proposed to improve ordinal consistency by identifying and eliminating intransitivities in pairwise comparison matrices the proposed algorithm generates near optimal solutions and outperforms other tested approaches with respect to computation time

on the minimum violations ranking of a tournament this paper examines the problem of rank ordering a set of players or objects on the basis of a set of pairwise comparisons arising from a tournament the criterion for deriving this ranking is to have as few cases as possible where player i is ranked above j while i was actually defeated by j in the tournament such a situation is referred to as a violation the objective therefore is to determine the minimum violations ranking mvr while there are situations where this ranking would be allowed to contain ties among subsets of objects we will concern ourselves herein with linear ordering no ties a series of examples are given where this requirement would seem to be appropriate in order to put the mvr problem into proper perspective we introduce the concept of a distance on the set of tournaments a set of natural axioms is presented which any such distance measure should obey and it is proven that in the presence of these axioms a unique such measure exists it is then shown that the mvr problem is equivalent to the minimum distance problem which can be represented in several forms in particular as a problem of determining the minimum feedback edge set in a graph and as a mixed integer generalized network problem this opens up a wide scope of possible solution procedures for the mvr problem an optimal algorithm is presented along with computational results in addition various heuristics are discussed including an improved heuristic referred to as the iterated kendall method

backlund transformations for many body systems related to kdv we present backlund transformations bts with parameter for certain classical integrable n body systems namely the many body generalized henon heiles garnier and neumann systems our construction makes use of the fact that all these systems may be obtained as particular reductions stationary or restricted flows of the kdv hierarchy alternatively they may be considered as examples of the reduced sl 2 gaudin magnet the bts provide exact time discretizations of the original continuous systems preserving the lax matrix and hence all integrals of motion and satisfy the spectrality property with respect to the backlund parameter

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 method for approximating symmetrically reciprocal matrices by transitive matrices the problem of approximating symmetrically reciprocal matrices by transitive matrices has received some attention recently this problem has applications in multicriteria decision theory several approximation approaches have been suggested and analyzed we here suggest another approach called the multiplicative approach we show that the optimal approximation in this sense may be found efficiently by transforming the problem into a known combinatorial optimization problem the minimum cycle mean problem for which efficient and simple combinatorial algorithms exist

the periodic toda chain and a matrix generalization of the bessel function recursion relations the authors obtain the quantization conditions of the periodic toda lattice in the baxter form lambda u q u inq u i i nq u i lambda is the transfer matrix containing the information about the spectrum and q is an integral operator commuting with lambda the logarithms of the matrix elements of q are the generating functions of the canonical backlund transformation the requirement that q is analytic and vanishes when u goes to infinity completely determines the spectrum of lambda

a multicriteria evaluation of methods for obtaining weights from ratio scale matrices abstract ratio scale matrices are commonly used as a vehicle to distinguish among competing alternatives or conflicting criteria the issue of extracting weights out of such matrices has attracted increasing attention in recent years and resulted in the development of several scaling methods this paper presents an attempt to evaluate some of the more known techniques in this area the evaluation procedure includes random generation of a large number of representative matrices which are used to evaluate the various scaling methods according to the specified criteria the performance of the different techniques is analyzed with respect to the proposed criteria and conclusions are drawn

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

modifying the power method in max algebra in the max algebra system the eigenequation for an n n irreducible nonnegative matrix a aij is a x a x here a x i maxjaijxj and a is the maximum circuit geometric mean the complexity of the power method given in l elsner p van den driessche linear algebra appl to appear to compute a and x is considered under some assumptions on the critical matrix it is shown that the algorithm may have time complexity o n4 a modified power method based on karp s formula is presented for this new algorithm with no assumptions on the critical matrix a and x can be computed in o n3 time furthermore this algorithm can be used to compute all linearly independent eigenvectors corresponding to a

canonical transformations of the extended phase space toda lattices and the stackel family of integrable systems we consider some examples of canonical transformations of the extended phase space which map a completely integrable system into another completely integrable system the proposed transformations are closed to the known maupertuis jacobi mappings and to the kepler change of the time using the kepler transformation we construct new integrable systems related to the toda lattices and the stackel family of integrable systems

the analytic hierarchy process max algebra and multi objective optimisation the analytic hierarchy process ahp is widely used for decision making involving multiple criteria elsner and van den driessche 2004 2010 10 11 introduced a max algebraic approach to the single criterion ahp we extend this to the multi criteria ahp by considering multi objective generalisations of the single objective optimisation problem solved in these earlier papers we relate the existence of globally optimal solutions to the commutativity properties of the associated matrices we relate min max optimal solutions to the generalised spectral radius and we prove that pareto optimal solutions are guaranteed to exist

the theory of ratio scale estimation saaty s analytic hierarchy process the analytic hierarchy process developed by saaty saaty t l 1980 the analytic hierarchy process mcgraw hill new york has proven to be an extremely useful method for decision making and planning however some researchers in these areas have raised concerns over the theoretical basis underlying this process this paper addresses currently debated issues concerning the theoretical foundations of the analytic hierarchy process we also illustrate through proof and through examples the validity or fallaciousness of these criticisms

on poincare type integral inequalities abstract the aim of the present paper is to establish some multidimensional poincare type integral inequalities involving two functions and their first order partial derivatives our analysis is more elementary and provide easy proofs of some poincare type integral inequalities

a note on deriving weights from pairwise comparison ratio matrices abstract this paper addresses the cook and kress method regarding the use of mathematical programming to derive weights from pairwise comparison ratio matrices we note that the cook and kress formulation has the shortcoming that alternative optimal solutions may occur which would lead to an infinite set of possible weights this paper proposes to resolve the problem by formulating a phase ii optimization procedure to be solved using quadratic programming this mathematical formulation is based on the determination of a weight vector that is statistically most likely to cause the associated pairwise comparison matrix it is concluded that the cook and kress method accompanied with the proposed phase ii approach can always derive a unique set of weights from a pairwise comparison matrix

duality between integrable stackel systems canonical transformations of the extended phase space are applied to the integrable stackel systems all these transformations may be associated with an ambiguity of the abel map on the corresponding hyperelliptic curve for some stackel systems with two degrees of freedom the 2 2 lax representations and the dynamical r matrix algebras are constructed

a linear optimization problem to derive relative weights using an interval judgement matrix abstract in this paper we propose a new decision making model enabling to assess a finite number of alternatives according to a set of bounds on the preference ratios for the pairwise comparisons between alternatives that is an interval judgement matrix in the case in which these bounds cannot be achieved by any assessment vector we analyze the problem of determining of an efficient or pareto optimal solution from a multi objective optimization problem this multi objective formulation seeks for assessment vectors that are near to simultaneously fulfil all the bound requirements imposed by the interval judgement matrix our new model introduces a linear optimization problem in order to define a consistency index for the interval matrix by solving this optimization problem it can be associated a weakly efficient assessment vector to the consistency index in those cases in which the bound requirements are infeasible otherwise this assessment vector fulfils all the bound requirements and has geometrical properties that make it appropriate as a representative assessment vector of the set of feasible weights

a characterization of the logarithmic least squares method abstract we provide an axiomatic characterization of the logarithmic least squares method sometimes called row geometric mean used for deriving a preference vector from a pairwise comparison matrix this procedure is shown to be the only one satisfying two properties correctness in the consistent case which requires the reproduction of the inducing vector for any consistent matrix and invariance to a specific transformation on a triad that is the weight vector is not influenced by an arbitrary multiplication of matrix elements along a 3 cycle by a positive scalar

computing interval weights for incomplete pairwise comparison matrices of large dimension a weak consistency based approach multiple criteria decision making and evaluation problems dealing with a large number of objects are very demanding particularly when the use of pairwise comparison pc techniques is required a major drawback arises when it is not possible to obtain all the pcs due to time or cost limitations or to split the given problem into smaller subproblems in such cases two tools are needed to find acceptable weights of objects an efficient method for partially filling a pairwise comparison matrix pcm and a suitable method for deriving weights from this incomplete pcm this paper presents a novel interactive algorithm for large dimensional problems guided by two main ideas the sequential optimal choice of the pcs to be performed and the concept of weak consistency the proposed solution significantly reduces the number of needed pcs by adding information implied by the weak consistency after the input of each pc providing sets of feasible values for all missing pcs interval weights of objects are computed from the resulting incomplete weakly consistent pcm adapting the methodology for calculating fuzzy weights from fuzzy pcms the computed weight intervals thus cover all possible weakly consistent completions of the incomplete pcm the algorithm works both with saaty s pcms and fuzzy preference relations the performance of the algorithm is illustrated by a numerical example and a real life case study the performed simulation demonstrates that the proposed algorithm is capable of reducing the number of pcs required in pcms of dimension 15 and greater by more than 60 on average

continuity of the generalized spectral radius in max algebra abstract let be an induced matrix norm associated with a monotone norm on r n and be the collection of all nonempty closed and bounded subsets of n n nonnegative matrices under this matrix norm for the hausdorff metric h between and is given by h max sup a inf b a b sup b inf a a b the max algebra system consists of the set of nonnegative numbers with sum a b max a b and the standard product ab for a b 0 for n n nonnegative matrices a b their product is denoted by a b where a b ij max 1 k n a ik b kj for each the max algebra version of the generalized spectral radius of is limsup m sup a m a 1 m where m a 1 a 2 a m a i here a is the maximum circuit geometric mean in this paper we prove that the max algebra version of the generalized spectral radius is continuous on the hausdorff metric space h the notion of the max algebra version of simultaneous nilpotence of matrices is also proposed necessary and sufficient conditions for the max algebra version of simultaneous nilpotence of matrices are presented as well

spectral properties of matrix polynomials in the max algebra we consider the spectral properties of matrix polynomials over the max algebra in particular we show how the perron frobenius theorem for the max algebra extends to such polynomials and illustrate the relevance of this for multistep difference equations in the max algebra we also present a number of inequalities for the largest max eigenvalue of a matrix polynomial

on the continuity of the generalized spectral radius in max algebra abstract given a bounded set of n n non negative matrices let and denote the generalized spectral radius of and its max version respectively we show that sup t 0 n 1 t 1 t where t denotes the hadamard power of we apply this result to give a new short proof of a known fact that is continuous on the hausdorff metric space h of all nonempty compact collections of n n non negative matrices

a revised framework for deriving preference values from pairwise comparison matrices in a recent study choo and wedley proposed a framework to categorize and conduct simulations to compare methods for deriving preference values from pairwise comparison matrices this study demonstrates that some pairs in the examined methods are identical and some methods can be solved in better ways additionally the simulations of choo and wedley are somewhat idealized thus this study provides additional simulations the simulation results lead to different method ranking and conclusions from those of choo and wedley

on sobolev type integral inequalities on etablit de nouvelles inegalites integrales du type de sobolev comportant des fonctions de plusieurs variables independantes on obtient de nouvelles estimations pour ce type d inegalite

the he non heiles system revisited abstract the known integrable cases of the henon heiles system are shown to be closely related to the stationary flows of the known and only integrable fifth order single component and polynomial nonlinear evolution equations this is further evidence that these are the only integrable cases of the henon heiles system lax pairs are deduced for each of the integrable cases and used to construct the constants of motion a curious lax operator recently found by the painleve method is explained

relative measurement and the power function abstract relative measurement is considered in the context of nomology a theory that defines and relates categories of activity that arise in qualitative decision making a nomological variable is defined to correspond to the amount weight or importance of any naturally forming cluster or category of qualitative information this article shows that in general the mind measures nomological variables relatively and changes in a nomological variable relative to its current value the way the mind assesses the relationship between two related nomological variables generally should be modelled as a power function and it is appropriate to use a geometric mean to average multiple observations empirical evidence from psychophysics is shown to support these results when synthesising direct and indirect relative comparisons of nomological variables it is proven that the indirect comparisons should be given a weight that is inversely proportional to the number of relative measurements made in the indirect comparison this result is shown to be consistent with but not identical with the use of the row geometric means approach for synthesising pairwise relative comparison matrices such as in the analytical hierarchy process in the case of more than three variables other terms should be included in the synthesis the article reports on empirical research that endorses the use of nomological structuring and relative weighting of criteria trees it shows that decision makers generally prefer scoring within intervals when comparing alternatives but can also use relative measurement where there are difficulties with identifying intervals the former leads to modelling by an additive value function the latter to a power function

transitive matrices and their applications abstract transitive matrices and symmetrically reciprocal sr matrices are defined and spectral properties of certain sr perturbations of transitive matrices are studied the results are applied to a multicriteria decision making method the analytic hierarchy process ahp which uses an sr matrix with positive entries a proof is given that rank reversal is inherent in this method if its input matrix is perturbed

a multiple criteria decision model with ordinal preference data abstract this paper develops a model for aggregating ordinal preference ranking across multiple criteria criteria weights are developed as part of the optimization process the model utilizes three levels of ordinal priorities 1 preferences among alternatives on the basis of each criterion 2 preferences among the criteria to reflect the importance of those criteria and 3 preferences among the criteria which express the degree to which the respondent can clearly distinguish among the alternatives the model is based upon a data envelopment analysis structure which permits a flexible weighting mechanism for each alternative being ranked

a matrix based ranking method with application to tennis abstract we introduce and investigate a ranking methodology which may be of interest in sports like tennis the approach may also be of interest in decision making situations based on pairwise comparisons the method is based on linear algebra and one computes a score for each player by solving a certain linear system of equations from these scores one finds the ranking the input is a set of matches and weights representing the importance of the matches this is represented by a weighted directed graph we prove a number of properties of the method including uniqueness of scores connection to m matrices and combinatorial interpretations a case study from ranking in professional tennis is discussed in detail

extremal properties of tropical eigenvalues and solutions to tropical optimization problems abstract an unconstrained optimization problem is formulated in terms of tropical mathematics to minimize a functional that is defined on a vector set by a matrix and calculated through multiplicative conjugate transposition for some particular cases the minimum in the problem is known to be equal to the tropical spectral radius of the matrix we examine the problem in the common setting of a general idempotent semifield a complete direct solution in a compact vector form is obtained to this problem under fairly general conditions the result is extended to solve new tropical optimization problems with more general objective functions and inequality constraints applications to real world problems that arise in project scheduling are presented to illustrate the results obtained numerical examples are also provided

properties of the canonical transformations of the time for the toda lattice and the henon heiles system for the toda lattice and the henon heiles system we consider properties of the canonical transformations of the extended phase space which preserve integrability at the special values of integrals of motion the integral trajectories separated variables and the action variables are invariant under change of the time on the other hand mapping of the time induces a shift of the generating function of the backlund transformation

a note on sobolev type inequalities in two independent variables abstract the aim of the present note is to establish some new integral inequalities of the sobolev type involving functions of two independent variables and their first order partial derivatives the analysis used in the proofs is quite elementary and based on some simple observations and applications of the fundamental inequalities

on the spectrum of pairwise comparison matrices abstract this paper deals with the spectrum of symmetrically reciprocal sr perturbations of transitive matrices of nonzero complex entries by introducing the so called pseudo circulant matrices the spectral decomposition of specific sr perturbations of n n transitive matrices and that of arbitrary 3 3 sr matrices are provided closed forms are given for all eigenvalues and their associated eigenvectors of these sr matrices also known as pairwise comparison matrices pcms to improve the evaluations at a given pcm as well as the priority ranking of the alternatives these results are applied to the analytic hierarchy process ahp using positive sr matrices explicit intervals for the ranges of possible rank reversals between the alternatives are established the spectrum of certain sr perturbations of transitive matrices with nonzero complex entries is also derived which can be applied in vehicle system dynamics three application examples are included as well

bbtopsis a bag based technique for order preference by similarity to ideal solution abstract this contribution explores the way of accomplishing a topsis ranking of various submitted variants taking into account a our newly developed notions of variant superiority inferiority and noninferiority multiattribute canonical fuzzy bags defined on attributes and b making use of bag metrics to compute the relative closeness of the different variants to an ideal solution in particular we prove the existence and the uniqueness of a bag metric fulfilling a number of interesting properties

a max version of the generalized spectral radius theorem abstract let be a bounded set of n n nonnegative matrices in max algebra in this paper we propose the notions of the max algebra version of the generalized spectral radius of and the max algebra version of the joint spectral radius of the max algebra version of the generalized spectral radius theorem is established we propose the relationship between the generalized spectral radius of in the sense of daubechies and lagarias and its max algebra version moreover a generalization of elsner and van den driessche s lemma is presented as well

pairwise ranking choice of method can produce arbitrarily different rank order abstract we examine three methods for ranking by pairwise comparison perronrank principal eigenvector hodgerank and tropicalrank we show that the choice of method can produce arbitrarily different rank order to be precise for any two of the three methods and for any pair of rankings of at least four items there exists a comparison matrix for the items such that the rankings found by the two methods are the prescribed ones we discuss the implications of this result in practice study the geometry of the methods and state some open problems

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

representing the strengths and directions of pairwise comparisons we consider the problem of finding weights that well represent a set of pairwise multiplicative comparisons of a set of objects as in the ahp and other methods our main contribution is a method for deriving such weights that takes into consideration not only the strengths of the pairwise comparisons but also their directions for example if the comparison directions satisfy transitivity then the weights produced by our method also satisfy transitivity this is not always true for other methods we also present a set of reasonable axioms for which our method is the essentially unique solution our method and axioms are closely related to those of cook and kress eur j oper res 37 1988 335 our method like theirs reduces to solving a linear program hence it is different from the approach used in the ahp for the special case that the comparison directions satisfy transitivity our method is quite simple and reduces to performing a forward pass as in the critical path method

axiomatic foundation of the analytic hierarchy process this paper contains an axiomatic treatment of the analytic hierarchy process ahp the set of axioms corresponding to hierarchic structures are a special case of axioms for priority setting in systems with feedback which allow for a wide class of dependencies the axioms highlight 1 the reciprocal property that is basic in making paired comparisons 2 homogeneity that is characteristic of people s ability for making comparisons among things that are not too dissimilar with respect to a common property and hence the need for arranging them within an order preserving hierarchy 3 dependence of a lower level on the adjacent higher level 4 the idea that an outcome can only reflect expectations when the latter are well represented in the hierarchy the ahp neither assumes transitivity or the stronger condition of consistency nor does it include strong assumptions of the usual notions of rationality a number of facts are derived from these axioms providing an operational basis for the ahp

on backlund transformations for many body systems using the n particle periodic toda lattice and the relativistic generalization due to ruijsenaars of the elliptic calogero moser system as examples we revise the basic properties of the backlund transformations bts from the hamiltonian point of view the analogy between the bt and baxter s quantum q operator pointed out by pasquier and gaudin is exploited to produce a conjugated variable for the parameter of the bt such that belongs to the spectrum of the lax operator as a consequence the generating function of the composition of n bts gives rise to another canonical transformation separating variables for the model for the toda lattice the dual bt parametrized by is introduced

a monte carlo investigation of incomplete pairwise comparison matrices in ahp the analytic hierarchy process ahp is a decision analysis technique used to evaluate complex multiattribute alternatives among one or more decision makers it imposes a hierarchical structure on any complex multicriterion problem however a major drawback of the ahp is that a large number of pairwise comparisons is needed to calibrate the hierarchy when there are a few levels and sublevels the ahp can be applied in a straightforward manner to derive the weights relative preference for each alternative as the size of the hierarchy increases the number of pairwise comparisons increases rapidly it is well established in the marketing and consumer behavior literature that in a very long interview even under the best circumstances the respondent is likely to suffer from information overload recognition of this problem was the motivation which led to the investigation of a modification of ahp which required less data collection i e a reduction in the threat of information overload the first question to be answered is the effect on ahp weights due to different patterns of missing data likely to result from reallife data collection in this study a monte carlo simulation was conducted which uses the incomplete pairwise comparisons ipc algorithm 14 to investigate the effect of reduced sets of pairwise comparisons in the ahp data for the study were generated with known structure and comparisons made between complete and incomplete matrices the results of the simulation suggest that incomplete sets of pairwise comparison matrices can capture the attribute level weights without significant loss of accuracy and independent of decision model form and amount of error considered

axiomatic property based consistency analysis and decision making with interval multiplicative reciprocal preference relations abstract this paper focuses on obtaining an interval extension of saaty s consistency and eliciting normalized interval weights in analytic form from interval multiplicative reciprocal preference relations imrprs as well as checking acceptability of imrprs four axiomatic properties are presented to characterize multiplicative consistency of imrprs the paper shows that six existing consistency definitions fail to satisfy all the four axiomatic properties a constrained interval multiplication based transitivity equation is devised and an interval matrix is constructed to capture original preferences and row uncertainty proportionalities of an imrpr based on the constructed interval matrix an ordinary interval multiplication based transitivity equation is developed to define consistency of imrprs a logarithmic least square model is established to find normalized interval weights from imrprs and its analytic solution is obtained by the lagrangian multiplier method the paper introduces an index to measure uniformity of row uncertainties in an imrpr and proposes a geometric consistency index to measure inconsistency of imrprs a novel method is put forward to determine acceptability of imrprs by checking acceptable consistency and acceptable uncertainty the proposed models are illustrated by eight numerical examples and a hierarchical multi criteria decision making problem dealing with recommending undergraduate students to graduate admission

consistent completion of incomplete judgments in decision making using ahp decision making dm processes are becoming increasingly complex the reasons are manifold dm usually involves many aspects some are purely technical while others are subjective and derived from social political and environmental factors among others as a result they involve items that are not easily comparable under the same units of measurement problems are made even more complex by the fact that current governance processes tend to involve all the stakeholders in the dm process in this paper we consider the ahp methodology analytic hierarchy process which is used to build consistent aggregate results from data provided by decision makers as some of the actors involved may not be completely familiar with all the criteria under consideration it is common that the body of opinion expressed in terms of pairwise comparison is incomplete to overcome this weakness we propose a framework that enables users to provide data on their preferences in a partial and or incomplete way and at different times this article is an advance towards a dynamic model of ahp the authors have addressed the problem of adding a new criterion or deleting obsolete criteria here we address the consistent completion of a reciprocal matrix as a mechanism to obtain a consistent body of opinion issued in an incomplete manner by a specific actor this feature is incorporated into a process of linearization previously introduced by the authors which is concisely presented finally we provide an application for leakage control in a water supply company the adoption of suitable control leakage policies in water supply is a problem of enormous interest in the water industry particularly in urban hydraulics

preference elicitation from inconsistent judgments using multi objective optimization abstract several decision making techniques involve pairwise comparisons to elicit the preferences of a decision maker dm this paper proposes a new approach for prioritization from pairwise comparisons using the concept of indirect judgments no method exists that simultaneously minimizes deviations from both direct and indirect judgments in order to estimate preferences it is sensible to consider both the acquired judgments and the other judgments latent in the dm s mind hence a technique is developed here to minimize the deviations from both types of judgments estimated preferences are generally evaluated based on two criteria their deviation from the provided judgments and the number of judgments that have been ordinally violated here it is proposed to optimize three objectives simultaneously the deviations from both direct and indirect judgments and the number of judgments violated a prototype application has been developed to generate all non dominated solutions using a multiple objective evolutionary algorithm the new approach is shown to offer users greater flexibility than all other tested methods

on two inequalities of the serrin type abstract we establish two new multidimensional integral inequalities of the serrin type involving a function and its first order partial derivatives the analysis used in the proofs is quite elementary and provides new estimates on this type of inequality

scaling binary comparison matrices a comment on narasimhan s proposal and other methods abstract recently narasimhan has proposed an adjustment to saaty s solution for scaling square judgment matrices consisting of pairwise ratio comparisons he recommends that prior to the application of saaty s solution the judgment matrix be adjusted to satisfy the supertransitivity requirement the present note shows that the proposed adjustment is equivalent to an alternative solution to this problem based upon calculation of the geometric means of the matrix rows thus the proposed adjustment should be regarded as an alternative to saaty s solution rather than a preliminary stage finally four solutions to this scaling problem are compared and the need for further research in this area is reiterated

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

on the asymptotic stability of nonnegative matrices in max algebra abstract in the max algebra system for an n n nonnegative matrix a a ij the eigenequation for max eigenvalue and corresponding max eigenvector x is a x x where a x i max 1 j n a ij x j and a is the maximum circuit geometric mean it is shown that the following conditions are mutually equivalent i a r n ii a 1 iii a lim k a k 0 where a max x 1 x 0 a x and a limsup k a k 1 k

on the max version of the generalized spectral radius theorem abstract let be a bounded set of n n non negative matrices recently the max algebra version of the generalized spectral radius of was introduced we show that lim t i t 1 t where denotes the generalized spectral radius and t the hadamard power of this provides a description of that uses no max terminology as an application we give a short proof of the max version of the generalized spectral radius theorem

approximation of positive matrices by transitive matrices a newton kantorovich algorithm used in the approximation of symmetrically reciprocalpositive matrices by transitive matrices is found to be more generally useful in the approximation of merely positive matrices the occurrence of multiple stationary values when the approximated matrix has certain symmetries is investigated

a max version of the perron frobenius theorem abstract if a an n n nonnegative irreducible matrix then there exists a 0 and a positive vector x such that maxjaijxj a xi i 1 2 n furthermore a is the maximum geometric mean of a circuit in the weighted directed graph corresponding to a this theorem which we refer to as the max version of the perron frobenius theorem is well known in the context of matrices over the max algebra and also in the context of matrix scalings in the present work which is partly expository we bring out the intimate connection between this result and the perron frobenius theory we present several proofs of the result some of which use the perron frobenius theorem structure of max eigenvalues and max eigenvectors is described possible ways to unify the perron frobenius theorem and its max version are indicated some inequalities for a are proved

estimating criterion weights using eigenvectors a comparative study abstract in addition to saaty s eigenvector there are infinitely many eigen weight vectors which can be constructed for any given data of estimated weight ratios as the judgment of the ratios are dependent on personal experience learning situations and state of mind inconsistencies and degree of easiness or judgment on these individual ratios can be different we study the properties of the different eigen weight vectors including that of saaty and that recently proposed by cogger and yu a general framework for the construction of eigen weight vectors incorporating that confidence in obtaining the individual ratios can be different will be proposed and discussed

hodgerank is the limit of perron rank in the context of pairwise comparison ranking we show that hodgerank row geometric mean is the limit of perron rank ranking with principal eigenvector as a certain parameter k goes to 0 this result provides a novel mathematical link between two important pairwise ranking methods it complements the known result that as k approaches infinity perron rank converges to tropical rank thus these three pairwise ranking methods belong to the same parametrized family our proof technique is useful for mathematical comparison of these methods as a sample application we show that for ranking models with i i d noise hodgerank is a linear approximation of perron rank in this particular setup for large numbers of items with sufficiently large score differences the two methods yield identical ordinal rankings

on the power method in max algebra abstract the eigenvalue problem for an irreducible nonnegative matrix a a ij in the max algebra system is a otimes x lambda x where a otimes x i mathop max j a ij x j and lambda turns out to be the maximum circuit geometric mean mu a a power method algorithm is given to compute mu a and eigenvector x the algorithm is developed by using results on the convergence of max powers of a which are proved using nonnegative matrix theory in contrast to an algorithm developed in 4 this new method works for any irreducible nonnegative a and calculates eigenvectors in a simpler and more efficient way some asymptotic formulas relating mu a the spectral radius and norms are also given

efficient weight vectors from pairwise comparison matrices abstract pairwise comparison matrices are frequently applied in multi criteria decision making a weight vector is called efficient if no other weight vector is at least as good in approximating the elements of the pairwise comparison matrix and strictly better in at least one position a weight vector is weakly efficient if the pairwise ratios cannot be improved in all non diagonal positions we show that the principal eigenvector is always weakly efficient but numerical examples show that it can be inefficient the linear programs proposed test whether a given weight vector is weakly efficient and in case of strong inefficiency an efficient strongly dominating weight vector is calculated the proposed algorithms are implemented in pairwise comparison matrix calculator available at pcmc online