delphic hierarchy process dhp a methodology for priority setting derived from the delphi method and analytical hierarchy process identifying of criteria and objectives is of paramount importance to a decision making process and is the basis for a sound decision a systematic approach therefore is needed so that the objectives of the organization not only can be identified but also prioritized so that the resources will be allocated to the relative importance of the objectives and how well the alternatives satisfy them the methodology proposed here uses the delphi method and integrates it with the analytic hierarchy process it assists the decision maker s to systematically identify the organizational objectives and then to set priorities among them the application of the proposed model is illustrated by a case study

group preference aggregation methods employed in ahp an evaluation and an intrinsic process for deriving members weightages abstract the analytic hierarchy process ahp is one of the popular and powerful techniques for decision making a detailed survey of the literature has revealed that there exists no formal evaluation of the group preference aggregation methods currently employed in ahp this paper provides such an evaluation using well established social choice axioms which govern the process of combining individual opinions to obtain a single group opinion the geometric mean method gmm and the weighted arithmetic mean method wamm are the two methods evaluated it is shown using counter examples that the gmm does not always satisfy the pareto optimality axiom which is one of the prominent and widely accepted social choice axioms this finding is significant as the gmm has been the most commonly used method in ahp for combining individual opinions to form a group opinion the other method viz wamm has satisfied all the axioms except the independence of irrelevant alternatives axiom in order to use the wamm one has to find the weightages importance to be assigned to the members of the group this is often a difficult task especially so if the group is large as in the case of public policy decisions and when judgements are elicited through the use of questionnaires these situations need an objective method to derive members weightages but only a few studies are available in the literature to address such a situation we propose a simple and intuitively appealing eigenvector based method to intrinsically determine the weightages for group members using their own subjective opinions the superiority of the proposed method over the previous methods is brought out in the paper

harnack estimates and extinction profile for weak solutions of certain singular parabolic equations we establish an intrinsic harnack estimate for nonnegative weak solutions of the singular equation 1 1 below for m in the optimal range n 2 n 1 intrinsic means that due to the singularity the space time dimensions in the parabolic geometry must be rescaled by a factor determined by the solution itself consequences are sharp supestimates on the solutions and decay rates as t approaches the extinction time analogous results are shown for p laplacian type equations

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

interactive group decision making procedure under incomplete information this paper deals with multiple criteria decision making problem with incomplete information when multiple decision makers multiple criteria group decision making mcgdm are involved usually decision makers dms are willing or able to provide only incomplete information because of time pressure lack of knowledge or data and their limited expertise related with problem domain there have been just a few studies considering incomplete information in group settings this incompletely specified information constructs region of linear constraints and therefore pairwise dominance relationship between alternatives reduces to intractable nonlinear programmings hence to handle this difficulty we suggest a method utilizing individual decision results to form group consensus final group consensus ranking toward more agreement of participants can be built through solving a series of linear programmings using individual decision results under group members possibly different weight constraints

renormalization and transition to chaos in area preserving nontwist maps abstract the problem of transition to chaos i e the destruction of invariant circles or kam kolmogorov arnold moser curves in area preserving nontwist maps is studied within the renormalization group framework nontwist maps are maps for which the twist condition is violated along a curve known as the shearless curve in renormalization language this problem is that of finding and studying the fixed points of the renormalization group operator r that acts on the space of maps a simple period two fixed point of r whose basin of attraction contains the nontwist maps for which the shearless curve exists is found also a critical period 12 fixed point of r with two unstable eigenvalues is found the basin of attraction of this critical fixed point contains the nontwist maps for which the shearless curve is at the threshold of destruction this basin defines a new universality class for the transition to chaos in area preserving maps

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

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

canonical fuzzy bags and bag fuzzy measures as a basis for madm with mixed non cardinal data abstract this contribution introduces what will be called sine analysis a new approach to multiple attribute decision making madm with mixed noncardinal data this approach is based upon the notion of canonical fuzzy bags and their scalar evaluations by bag fuzzy measures

relationships between l 1 metrics on linear ranking spaces in this paper we examine various l 1 distance measures on linear ranking spaces beginning with the absolute position measure of blin we construct two offspring measures one based on relative rank positioning the other on relative object positioning the first is shown to be equivalent to the permutation vector method of cook and seiford the other is equivalent to the preference matrix method of kemeny and snell we also prove that the kemeny and snell method can be formulated as a quadratic assignment problem

a novel group ranking model for revealing sequence and quantity knowledge abstract the aggregation of individuals preferences into a consensus ranking is a group ranking problem which has been widely utilized in various applications such as decision support systems recommendation systems and voting systems gathering the comparison of preferences and aggregate them to gain consensuses is a conventional issue for example b c d a indicates that b is favorable to c and c d is somewhat favorable but not fully favorable to d a where and are comparators and a b c and d are items recently a new type of ranking model was proposed to provide temporal orders of items the order b c a means that b and c can occur simultaneously and are also before a although this model can derive the order ranking of items the knowledge about quantity related items is also of importance to approach more real life circumstances for example when enterprises or individuals handle their portfolios in financial management two considerations the sequences and the amount of money for investment objects should be raised initially in this study we propose a model for discovering consensus sequential patterns with quantitative linguistic terms experiments using synthetic and real datasets showed the model s computational efficiency scalability and effectiveness

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

a new consensus ranking approach for correlated ordinal information based on mahalanobis distance the cohesiveness among experts opinions is addressed from a global point of view a new procedure to codify ordinal information is provided and characterized a measurement of the degree of dissensus designed for possibly correlated alternatives is defined a procedure to obtain a social consensus solution based on the mahalanobis distance is investigated we investigate from a global point of view the existence of cohesiveness among experts opinions we address this general issue from three basic essentials the management of experts opinions when they are expressed by ordinal information the measurement of the degree of dissensus among such opinions and the achievement of a group solution that conveys the minimum dissensus to the experts group accordingly we propose and characterize a new procedure to codify ordinal information we also define a new measurement of the degree of dissensus among individual preferences based on the mahalanobis distance it is especially designed for the case of possibly correlated alternatives finally we investigate a procedure to obtain a social consensus solution that also includes the possibility of alternatives that are correlated in addition we examine the main traits of the dissensus measurement as well as the social solution proposed the operational character and intuitive interpretation of our approaches are illustrated by an explanatory example

heuristic aggregation of individual judgments in ahp group decision making using simulated annealing algorithm sa algorithm is used as aggregation procedure in ahp group decision making the group priority vector is obtained by minimization of the euclidean distance objective search for maximum consensus between individuals within the group proposed procedure is invariant to any prioritization method proposed procedure performs better or at least equally to several other procedures there are various aggregation procedures for obtaining a group priority vector within analytic hierarchy process ahp supported decision making processes this paper will introduce a heuristic aggregation procedure based on simulated annealing sa algorithm to be used for the purposes of obtaining a group priority vector at any node of an ahp hierarchy the proposed procedure performs its aggregation process by minimizing the group euclidean distance ged consensus measure across group weights and judgments and the group vector obtained in this way is invariant to any prioritization method in other words there is no need to have individual priority vectors as is required by some other aggregation procedures along with sa minimization of the ged the group rank reversal minimum violation criterion is implemented as a control mechanism as well as the consensus measure based on the ranks of alternatives the presented procedure is compared with several reported combinations of different prioritization methods and group aggregation procedures five examples from literature are used to show that the proposed procedure performs better or at least equally to several other well known combinations of prioritization and aggregation in ahp group decision making frameworks

a fuzzy bag approach to choosing the best multiattributed potential actions in a multiple judgement and non cardinal data context abstract the present work is concerned with showing how a fuzzy bag construct along with the concept of a positive outstandingness predicate and that of a strict t norm operator can be used to resolve conflicts and aggregation problems in decisions involving multiple judgements and non cardinal data it is also shown how the differential expertness degrees of the judges may be taken into account in the aggregation process the choice device proposed is decisive i e it excludes incomparability

information and preference in partial orders a bimatrix representation this paper presents a bimatrix structure for examining ordinal partial rankings a set of axioms is given similar to those of kemeny and snell 1962 and bogart 1973 which uniquely determines the distance between any pair of such rankings thel 1 norm is shown to satisfy this set of axioms and to be equivalent to the kemeny and snell distance on their subspace of weak orderings consensus formation is discussed

renormalization group operators for maps and universal scaling of universal scaling exponents renormalization group rg methods provide a unifying framework for understanding critical behaviour such as transition to chaos in area preserving maps and other dynamical systems which have associated with them universal scaling exponents recently de la llave et al 2007 10 have formulated the principle of approximate combination of scaling exponents pacse for short which relates exponents for different criticalities via their combinatorial properties the main objective of this paper is to show that certain integrable fixed points of rg operators for area preserving maps do indeed follow the pacse

empirical properties of group preference aggregation methods employed in ahp theory and evidence abstract we study various methods of aggregating individual judgments and individual priorities in group decision making with the ahp the focus is on the empirical properties of the various methods mainly on the extent to which the various aggregation methods represent an accurate approximation of the priority vector of interest we identify five main classes of aggregation procedures which provide identical or very similar empirical expressions for the vectors of interest we also propose a method to decompose in the ahp response matrix distortions due to random errors and perturbations caused by cognitive biases predicted by the mathematical psychology literature we test the decomposition with experimental data and find that perturbations in group decision making caused by cognitive distortions are more important than those caused by random errors we propose methods to correct the systematic distortions

the golden number and fibonacci sequences in the design of voting structures abstract some distinguished types of voters as vetoes passers or nulls as well as some others play a significant role in voting systems because they are either the most powerful or the least powerful voters in the game independently of the measure used to evaluate power in this paper we are concerned with the design of voting systems with at least one type of these extreme voters and with few types of equivalent voters with this purpose in mind we enumerate these special classes of games and find out that its number always follows a fibonacci sequence with smooth polynomial variations as a consequence we find several families of games with the same asymptotic exponential behavior except for a multiplicative factor which is the golden number or its square from a more general point of view our studies are related with the design of voting structures with a predetermined importance ranking

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

theoretical estimation of the probability of weight rank reversal in pairwise comparisons abstract pairwise comparison is a key component in multi criteria decision making the probability of rank reversal is a useful measure for evaluating the impact of uncertainty on the final outcome in the context of this paper the type of uncertainty considered is related to the fact that experts have different opinions or that they may perform inconsistent pairwise comparisons we provide a theoretical model for estimating the probability of the consequent rank reversal using the multivariate normal cumulative distribution function the model is applied to two alternative weight extraction methods frequently used in the literature the geometric mean and the eigenvalue method we introduce a reasonable framework for incorporating uncertainty in the decision making process and calculate the mean value and cross correlation of the average weights which are required in the application of the model the theoretical results are compared against numerical simulations and a very good agreement is observed we further show how our model can be extended in applications of a full multi criteria decision making analysis such as the analytic hierarchy process we also discuss how the theoretical model can be used in practice where the statistical properties of the uncertainty induced perturbations are unknown and the only information provided by the pairwise comparison matrices of a small group of experts the methodology presented here can be used to extend the pairwise comparison framework in order to provide some information on the credibility of its outcome

properties of ladder tournaments abstract ladder tournaments use work trials to screen workers for positions in organizations despite being common in many industries their properties remain poorly understood especially in organizations characterized by complex complementarities and externalities in their production process in this study we formalize the ranking rule generated by a ladder tournament and derive measures of relative performance that can be used to compare workers when their true inputs cannot be observed we show that this rule is reflexive but not generally transitive unless the organization has fewer than three layers of hierarchy or is complete it follows that complete organizations have an embedded merit based hierarchical structure we also study the relationship between a worker s tournament rank and his utility or usefulness as measured by the frequency with which he plays a pivotal role we show that a worker s likelihood of being pivotal is a weakly increasing function of his rank this finding implies that in complete organizations ladder tournaments discriminate more effectively between workers than do mechanisms based on pivotability or marginal productivity

accurate algorithms for identifying the median ranking when dealing with weak and partial rankings under the kemeny axiomatic approach abstract preference rankings virtually appear in all fields of science political sciences behavioral sciences machine learning decision making and so on the well known social choice problem consists in trying to find a reasonable procedure to use the aggregate preferences or rankings expressed by subjects to reach a collective decision this turns out to be equivalent to estimate the consensus central ranking from data and it is known to be a np hard problem a useful solution has been proposed by emond and mason in 2002 through the branch and bound algorithm bb within the kemeny and snell axiomatic framework as a matter of fact bb is a time demanding procedure when the complexity of the problem becomes untractable i e a large number of objects with weak and partial rankings in presence of a low degree of consensus as an alternative we propose an accurate heuristic algorithm called fast that finds at least one of the consensus ranking solutions found by bb saving a lot of computational time in addition we show that the building block of fast is an algorithm called quick that finds already one of the bb solutions so that it can be fruitfully considered to speed up even more the overall searching procedure if the number of objects is low simulation studies and applications on real data allows to show the accuracy and the computational efficiency of our proposal

ranking in tournaments and group decisionmaking this paper fully discusses methods for ranking a set of alternatives in the fairest possible way according to a minimum violations criterion new methods based on finding paths and circuits in graphs are presented for ranking participants in round robin and generalized tournaments and for consensus and group decisionmaking problems the objective of the paper is to review existing methods for tackling these problems and compare them with the new methods according to a fairness criterion and the amount of computing required to reach a solution it is shown that the new methods often exceed the existing methods in both fairness and reduced computing requirements in particular the new methods are generally more versatile than existing methods this allows organizations to obtain the fairest ranking of a number of alternatives according to their managers or employees wishes particular attention is given to incomplete rankings where insufficient exposure of an individual to some alternatives restricts that individual to ranking only the remaining alternatives

universal properties of the transition from quasi periodicity to chaos in dissipative systems abstract an exact renormalization group transformation is developed for dissipative systems which describes how the transition to chaos may occur in a continuous and universal manner if the frequency ratio in the quasi periodic regime is held at a fixed irrational value our approach is a natural extension of k a m theory to strong coupling most of our analysis is for analytic circle maps we have found a strong coupling fixed point where invertibility is lost which describes the universal features of the transition to chaos we find numerically that any two such critical maps with the same winding number are c 1 conjugate it follows that the low frequency peaks in an experimental spectrum are universal and we determine how their envelope scales with frequency when the winding number has a periodic continued fraction our renormalization transform has a fixed point and spectra are self similar in addition for a set of non periodic winding numbers with full measure our renormalization transformation yields an ergodic trajectory in a sub space of all critical maps physically one finds singular and universal spectra that do not scale

modelling pairwise comparisons on ratio scales abstract many real world decision problems have multiple criteria on qualitative domains given the expectation that more and more of such problems will be modelled as mcdm problems involving scoring on subjective qualitative domains then there is a class of significant problems for which the evaluator will require an evaluation framework which handles occurrences of seeming intransitivity and inconsistency these phenomena are likely to appear in qualitative subjective domains where the decision making environment is characterized by ambiguity and vagueness in this paper we present a model for exploring the implications of pairwise comparisons on ratio scales this model provides a mathematical explanation for the phenomena of intransitivity and inconsistency that sometimes appear in situations involving scoring on a subjective qualitative domain using this framework we develop methods for generating consistent intervals on ratio scales we also present a two stage action learning process for generating consistent points on ratio scales that provides both behavioral and mathematical convergence

a note on ahp group consistency for the row geometric mean priorization procedure abstract this work analyses the consistency in group decision making for the analytic hierarchy process ahp when using the weighted geometric mean method wgmm as the aggregation procedure the row geometric mean method rgmm as the priorization procedure and the geometric consistency index as the inconsistency measure the paper proves that the inconsistency of the group is smaller than the largest individual inconsistency this result complements that obtained by xu eur j oper res 126 2000 683 for the eigenvector priorization method em and its associated consistency index saaty multicriteria decision making the analytic hierarchy process mcgraw hill 1980 moreover our result guarantees that by using the rgmm priorization procedure the group priorities obtained through the aggregation of the individual priorities verify the requirement of consistency proposed in ahp methodology if the individual priorities also verify this requirement

a generalized network formulation of the pairwise comparison consensus ranking model one of the best known and most widely referenced models for representing ordinal preferences is that due to kemeny and snell kemeny j g l j snell 1962 preference ranking an axiomatic approach mathematical models in the social sciences glnn new york 9 23 this model is designed to accommodate pairwise comparison data with an l1 norm used to measure voter disagreement while this model possesses many of the necessary properties for a social choice function solution procedures developed to date have been capable of handling only small problems due to the difficulty of modelling the transitivity requirements of an optimal consensus ranking this paper shows how the consensus formation problem for strict linear orderings can be modelled as a generalized network since efficient computer codes already exist for handling this special structure this approach will permit the solution of much larger problems than has been the case previously

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

linguistic multiperson decision making based on the use of multiple preference relations in multiperson decision making situations it is quite natural that different decision makers who may have different background and knowledge will provide their preferences by different kinds of preference relations this paper proposes a linguistic multiperson decision making model lmdmm based on linguistic preference relations integrating fuzzy preference relations different types of multiplicative preference relations and multigranular linguistic preference relations in the lmdmm several transformation functions are first obtained to relate fuzzy preference relations and different types of multiplicative preference relations with multigranular linguistic preference relations then we design the selection process of the lmdmm based on the use of the fuzzy majority and the extended ordered weighted averaging operator or the 2 tuple ordered weighted averaging operator and discuss conditions under which the proposed selection process satisfies the social choice axioms at last we analyze the internal consistency of the proposed transformation functions the results in this paper are helpful to complete chiclana et al s fuzzy decision model integrating three representation models in fuzzy multipurpose decision making based on fuzzy preference relations fuzzy sets and systems 97 1998 33 48

an empirical analysis of the use of the analytic hierarchy process for estimating membership values in a fuzzy set abstract the analytic hierarchy process ahp is now a popular multicriteria decision making method and has been applied in many diverse areas despite the debate on its theoretical foundations many researchers have been using the pairwise comparison matrix and the eigenvector method suggested by saaty for obtaining priority ratings of the decision alternatives in an ahp hierarchy these priority ratings have been used as estimates of fuzzy membership values of the elements called alternatives in ahp of a fuzzy set triantaphyllou and mann have argued that saaty s scale and the eigenvector method are inadequate for estimating fuzzy membership values this paper points out the limitations in the arguments of triantaphyllou and mann about the eigenvector method it primarily concentrates on two aspects the first pertains to saaty s condition on deleting or combining two alternatives which are near copies alternatives whose priority values are close to each other within a 10 range over all criteria the second aspect concerns the use of the eigenvector method for estimating the membership values of elements of a fuzzy set the use of the eigenvector method for estimating fuzzy membership values in a fuzzy set has been justified using the empirical analysis together with theoretical support it has been amply demonstrated in this paper that the problem lies in saaty s scale and not in the eigenvector method

on the ordinal equivalence of the johnston banzhaf and shapley power indices abstract in this paper we characterize the games in which johnston shapley shubik and penrose banzhaf coleman indices are ordinally equivalent meaning that they rank players in the same way we prove that these three indices are ordinally equivalent in semicomplete simple games which is a newly defined class that contains complete games and includes most of the real world examples of binary voting systems this result constitutes a twofold extension of diffo lambo and moulen s result diffo lambo and moulen 2002 in the sense that ordinal equivalence emerges for three power indices not just for the shapley shubik and penrose banzhaf coleman indices and it holds for a class of games strictly larger than the class of complete games

a multiple criteria ranking procedure based on distance between partial preorders abstract in this paper a multiple criteria ranking procedure based on distance between partial preorders is proposed this method consists of two phases in the first phase the decision maker is asked to rank alternatives with a preorder complete or partial for each criterion and provide complete or partial linear information about the relative importance weights of the criteria in the second phase we introduce a distance procedure to aggregate the above individual rankings into a global ranking a partial preorder an algorithm for the aggregation procedure is proposed followed by a numerical illustration

achievable hierarchies in voting games with abstention abstract it is well known that he influence relation orders the voters the same way as the classical banzhaf and shapley shubik indices do when they are extended to the voting games with abstention vga in the class of complete games moreover all hierarchies for the influence relation are achievable in the class of complete vga the aim of this paper is twofold firstly we show that all hierarchies are achievable in a subclass of weighted vga the class of weighted games for which a single weight is assigned to voters secondly we conduct a partial study of achievable hierarchies within the subclass of h complete games that is complete games under stronger versions of influence relation

the use of domains in multicriteria decision making abstract this paper describes a domain criterion for a multicriteria problem given decision maker it then outlines possibilities for aggregating individual preference structures in the case of multiple decision makers

a new measure for analyzing and fusing sequences of objects this work is related to the combinatorial data analysis problem of seriation used for data visualization and exploratory analysis seriation re sequences the data so that more similar samples or objects appear closer together whereas dissimilar ones are further apart despite the large number of current algorithms to realize such re sequencing there has not been a systematic way for analyzing the resulting sequences comparing them or fusing them to obtain a single unifying one we propose a new positional proximity measure that evaluates the similarity of two arbitrary sequences based on their agreement on pairwise positional information of the sequenced objects furthermore we present various statistical properties of this measure as well as its normalized version modeled as an instance of the generalized correlation coefficient based on this measure we define a new procedure for consensus seriation that fuses multiple arbitrary sequences based on a quadratic assignment problem formulation and an efficient way of approximating its solution we also derive theoretical links with other permutation distance functions and present their associated combinatorial optimization forms for consensus tasks the utility of the proposed contributions is demonstrated through the comparison and fusion of multiple seriation algorithms we have implemented using many real world datasets from different application domains

a cardinal dissensus measure based on the mahalanobis distance abstract in this paper we address the problem of measuring the degree of consensus dissensus in a context where experts or agents express their opinions on alternatives or issues by means of cardinal evaluations to this end we propose a new class of distance based consensus model the family of the mahalanobis dissensus measures for profiles of cardinal values we set forth some meaningful properties of the mahalanobis dissensus measures finally an application over a real empirical example is presented and discussed

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

dominance based rough set approach for group decisions the objective of this paper is to propose an approach to support group multicriteria classification the approach is composed of three phases the first phase exploits the knowledge provided by each decision maker to individually approximate the decision classes using rough approximation the second phase seeks to combine the outputs of individual approximation phase into a collective decision table by using an appropriate aggregation procedure the third phase uses the collective decision table in order to infer a set of collective decision rules which synthesize the judgements and perspectives of the different decision makers and to permit the classification of all decision objects the proposed approach relies on the dominance based rough set approach drsa which is used at two different levels first the drsa is used during the first phase to approximate the input data relative to each decision maker second the drsa is used during the third phase to approximate the collective decision table and generate the collective decision rules this paper presents the theoretical foundation of the proposed approach three case studies using real world data and a comparative study of recent similar proposals

universal scalings of universal scaling exponents in the last decades renormalization group rg ideas have been applied to describe universal properties of different routes to chaos quasi periodic period doubling or tripling siegel disc boundaries etc each of the rg theories leads to universal scaling exponents which are related to the action of certain rg operators the goal of this announcement is to show that there is a principle that organizes many of these scaling exponents we give numerical evidence that the exponents of different routes to chaos satisfy approximately some arithmetic relations these relations are determined by combinatorial properties of the route and become exact in an appropriate limit

concordance and consensus this paper deals with the measurement of concordance and the construction of consensus in preference data either in the form of preference rankings or in the form of response distributions with likert items we propose a set of axioms of concordance in preference orderings and a new class of concordance measures the measures outperform classic measures like kendall s t and w and spearman s r in sensitivity and apply to large sets of orderings instead of just to pairs of orderings for sets of n orderings of n items we present very efficient and flexible algorithms that have a time complexity of only o nn 2 remarkably the algorithms also allow for fast calculation of all longest common subsequences of the full set of orderings we experimentally demonstrate the performance of the algorithms a new and simple measure for assessing concordance on likert items is proposed

robust and stochastically weighted multiobjective optimization models and reformulations we introduce and study a family of models for multiexpert multiobjective criteria decision making these models use a concept of weight robustness to generate a risk averse decision in particular the multiexpert multicriteria robust weighted sum approach mcrow introduced in this paper identifies a robust pareto decision that minimizes the worst case weighted sum of objectives over a given weight region the corresponding objective value called the robust value of a decision is shown to be increasing and concave in the weight set we study compact reformulations of the mcrow model with polyhedral and conic descriptions of the weight regions the mcrow model is developed further for stochastic multiexpert multicriteria decision making by allowing ambiguity or randomness in the weight region as well as the objective functions the properties of the proposed approach are illustrated with a few textbook examples the usefulness of the stochastic mcrow model is demonstrated using a disaster planning example and an agriculture revenue management example

regularity and existence of solutions of elliptic equations with p q growth conditions for some positive constants m a4 and for exponents q 2 p 2 2 under 1 2 1 3 and some other assumptions by assuming also that the quotient q p is sufficiently close to one in dependence on n precisely if q p c n n 2 then we prove that every weak solution to 1 1 of class 4 4 0 is locally lipschitz continuous in q moreover there are positive constants 8 c and 0 2 1 such that

a general framework for distance based consensus in ordinal ranking models abstract the problem of aggregating a set of ordinal rankings of n alternatives has given rise to a number of consensus models among the most common of these models are those due to borda and kendall which amount to using average ranks and the l1 and l2 distance models a common criticism of these approaches is their use of ordinal rank position numbers directly as the values of being ranked at those levels this paper presents a general framework for associating value or worth with ordinal ranks and develops models for deriving a consensus based on this framework it is shown that the lp distance models using this framework are equivalent to the conventional ordinal models for any p 1 this observation can be seen as a form of validation of the practice of using ordinal data in a manner for which it was presumably not designed in particular it establishes the robustness of the simple borda kendall and median ranking models

c1 α local regularity of weak solutions of degenerate elliptic equations abstract it is demonstrated the local c 1 alpha nature of weak solutions of elliptic equations of the type 1 1 in the introduction under the degeneracy or singularity assumptions a sub 1 a sub 3

a flexible approach to ranking with an application to mba programs we develop a model for flexibly ranking multi dimensional alternatives units into preference classes via mixed integer programming we consider a linear aggregation model but allow the criterion weights to vary within pre specified ranges this allows the individual alternatives units to play to their strengths we illustrate the use of the model by considering the financial times global mba program rankings and discuss the implications we argue that in many applications neither the data nor the weights or the aggregation model itself is precise enough to warrant a complete ranking providing an argument for sorting or what we call flexible ranking

group aggregation of pairwise comparisons using multi objective optimization in group decision making multiple decision makers dms aim to reach a consensus ranking of alternatives in a decision problem the differing expertise experience and potentially conflicting interests of the dms will result in the need for some form of conciliation to achieve consensus pairwise comparisons are commonly used to elicit values of preference of a dm the aggregation of the preferences of multiple dms must additionally consider potential conflict between dms and how these conflicts may result in a need for compromise to reach group consensus we present an approach to aggregating the preferences of multiple dms utilizing multi objective optimization to derive and highlight underlying conflict between the dms when seeking to achieve consensus extracting knowledge of conflict facilitates both traceability and transparency of the trade offs involved when reaching a group consensus further the approach incorporates inconsistency reduction during the aggregation process to seek to diminish adverse effects upon decision outcomes the approach can determine a single final solution based on either global compromise information or through utilizing weights of importance of the dms within multi criteria decision making we present a case study within the analytical hierarchy process from which we derive a richer final ranking of the decision alternatives

distance based and ad hoc consensus models in ordinal preference ranking abstract this paper examines the problem of aggregating ordinal preferences on a set of alternatives into a consensus this problem has been the subject of study for more than two centuries and many procedures have been developed to create a compromise or consensus we examine a variety of structures for preference specification and in each case review the related models for deriving a consensus two classes of consensus models are discussed namely ad hoc methods evolving primarily from parliamentary settings over the past 200 years and distance or axiomatic based methods we demonstrate the levels of complexity of the various distance based models by presenting the related mathematical programming formulations for them we also present conditions for equivalence that is for yielding the same consensus ranking for some of the methods finally we discuss various extensions of the basic ordinal ranking structures paying specific attention to partial ranking voting member weighted consensus ranking with intensity of preference and rank correlation methods as alternative approaches to deriving a consensus suggestions for future research directions are given

monte carlo simulation techniques for group decision making with incomplete information abstract in this paper we deal with group decision making problems where several decision makers elicit their own preferences separately the decision makers preferences are quantified using a decision support system which admits incomplete information concerning the decision makers responses to the questions they are asked consequently each decision maker proposes classes of utility functions and attribute weight intervals for the different attributes we introduce an approach based on monte carlo simulation techniques for aggregating decision maker preferences that could be the starting point for a negotiation process if necessary the negotiation process would basically involve the decision maker tightening the imprecise component utilities and weights to output more meaningful results and achieve a consensus alternative we focus on how attribute weights and the component utilities associated with a consequence are randomly generated in the aggregation process taking into account the decision makers preferences i e their respective attribute weight intervals and classes of utility functions finally an application to the evaluation of intervention strategies for restoring a radionuclide contaminated lake illustrates the usefulness and flexibility of this iterative process

an experiment on the consistency of aggregated comparison matrices in ahp the analytic hierarchy process can be used for group decision making by aggregating individual judgments or individual priorities the most commonly used aggregation methods are the geometric mean method and the weighted arithmetic mean method while it is known that the weighted geometric mean comparison matrix is of acceptable consistency if all individual comparison matrices are of acceptable consistency this paper addresses the following question under what conditions would an aggregated geometric mean comparison matrix be of acceptable consistency if some or all of the individual comparison matrices are not of acceptable consistency using monte carlo simulation results indicate that given a sufficiently large group size consistency of the aggregate comparison matrix is guaranteed regardless of the consistency measures of the individual comparison matrices if the geometric mean is used to aggregate this result implies that consistency at the aggregate level is a non issue in group decision making when group size exceeds a threshold value and the geometric mean is used to aggregate individual judgments this paper determines threshold values for various dimensions of the aggregated comparison matrix

a new group ranking approach for ordinal preferences based on group maximum consensus sequences group ranking problems involve aggregating individual rankings to generate group ranking which represents consolidated group preference group ranking problems are commonly applied in real world decision making problems however supporting a group decision making process is difficult due to the existence of multiple decision makers each with his her own opinions hence determining how to best aid the group ranking process is an important consideration this study aims to determine a total ranking list which meets group consensus preferences for group ranking problems a new group consensus mining approach based on the concept of tournament matrices and directed graphs is first developed an optimization model involving maximum consensus sequences is then constructed to achieve a total ranking list compared to previous methods the proposed approach can generate a total ranking list involving group consensus preferences it can also determine maximum consensus sequences without the need for tedious candidate generation processes while also providing flexibility in solving ranking problems using different input preferences that vary in format and completeness in addition consensus levels are adjustable

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

decision making with incomplete information abstract decision situations with incomplete information are characterized by a decision maker without a precisely defined stable preference structure by probability distributions not known completely or by an inexact evaluation of consequences within the paper a general framework for decision making with incomplete information is presented which shows how to solve problems from descriptive as well as prescriptive decision theory based on this framework an overview of existing methods which are particularly suitable for handling incomplete information is given

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

priority ranking and consensus formation the case of ties this paper investigates the problem of obtaining a compromise consensus from a set of ordinal rankings of n objects supplied by m committee members earlier work by cook and seiford cook wade d lawrence m seiford 1978 priority ranking and consensus formation management sci 24 16 1721 1732 dealt with the problem of consensus when attention was restricted to complete rankings only that is no ties were permitted this paper examines the general problem which allows for tied preferences a convex polyhedral representation is given of the feasible solution space and a branch and bound procedure is developed for determining an optimal ranking computational results for various problem sizes are presented generalizations and directions for further research are discussed

the critical renormalization fixed point for commuting pairs of area preserving maps we prove the existence of the critical fixed point f g for mackay s renormalization operator for pairs of maps of the plane the maps f and g commute are area preserving reversible real analytic and they satisfy a twist condition

renormalization for breakup of invariant tori we present renormalization group operators for the breakup of invariant tori with winding numbers that are quadratic irrationals we find the simple fixed points of these operators and interpret the map pairs with critical invariant tori as critical fixed points coordinate transformations on the space of maps relate these fixed points and also induce conjugacies between the corresponding operators

group preference aggregation in the multiplicative ahp the model of the group decision process and pareto optimality abstract a recent paper has focused awareness on group aggregation procedures in the ahp showing that geometric mean aggregation violates the desirable social choice axiom of pareto optimality we show that this violation can be attributed to the representation used to model the group decision process thereby questioning the legitimacy of the pareto optimality axiom we furthermore propose a geometric mean group aggregation procedure which satisfies all the social choice axioms suggested

consensus building in a local context for the ahp gdm with the individual numerical scale and prioritization method the methodology for selecting the individual numerical scale and prioritization method has recently been presented and justified in the analytic hierarchy process ahp in this study we further propose a novel ahp group decision making gdm model in a local context a unique criterion based on the individual selection of the numerical scale and prioritization method the resolution framework of the ahp gdm with the individual numerical scale and prioritization method is first proposed then based on linguistic euclidean distance led and linguistic minimum violations lmv the novel consensus measure is defined so that the consensus degree among decision makers who use different numerical scales and prioritization methods can be analyzed next a consensus reaching model is proposed to help decision makers improve the consensus degree in this consensus reaching model the led based and lmv based consensus rules are proposed and used finally a new individual consistency index and its properties are proposed for the use of the individual numerical scale and prioritization method in the ahp gdm simulation experiments and numerical examples are presented to demonstrate the validity of the proposed model

a new correlation coefficient for comparing and aggregating non strict and incomplete rankings abstract we introduce a correlation coefficient that is designed to deal with a variety of ranking formats including those containing non strict i e with ties and incomplete i e unknown preferences the correlation coefficient is designed to enforce a neutral treatment of incompleteness whereby no assumptions are made about individual preferences involving unranked objects the new measure which can be regarded as a generalization of the seminal kendall tau correlation coefficient is proven to satisfy a set of metric like axioms and to be equivalent to a recently developed ranking distance function associated with kemeny aggregation in an effort to further unify and enhance both robust ranking methodologies this work proves the equivalence of an additional distance and correlation coefficient pairing in the space of non strict incomplete rankings these connections induce new exact optimization methodologies a specialized branch and bound algorithm and an exact integer programming formulation moreover the bridging of these complementary theories reinforces the singular suitability of the featured correlation coefficient to solve the general consensus ranking problem the latter premise is bolstered by an accompanying set of experiments on random instances which are generated via a herein developed sampling technique connected with the classic mallows distribution of ranking data associated experiments with the branch and bound algorithm demonstrate that as data becomes noisier the featured correlation coefficient yields relatively fewer alternative optimal solutions and that the aggregate rankings tend to be closer to an underlying ground truth shared by a majority

preference relations on actions and criteria in multicriteria decision making abstract let a be a finite set of feasible actions which are judged following several criteria an outranking relation is defined on a by considering preference of the decision maker as a weak order on each criterion and the relation among criteria as a semi order on the given set of criteria several ways of constructing outranking relations have been proposed one of the most popular introduced by b roy for instance electre s is based on the use of weights related to criteria in our approach the knowledge of weights is replaced by the existence of a semi order a case study is developed it deals with a computer selection problem

a new proposal for fusing individual preference orderings by rank ordered agents a generalization of the yager s algorithm abstract the problem of the aggregation of multi agents preference orderings has received considerable attention in the scientific literature because of its importance for different fields of research yager 2001 proposed an algorithm for addressing this problem when the agents importance is expressed through a rank ordering instead of a set of weights the algorithm by yager is simple and automatable but is subject to some constraints which may limit its range of application i preference orderings should not include incomparable and or omitted alternatives and ii the fused ordering may sometimes not reflect the majority of the multi agent preference orderings the aim of this article is to present a generalized version of the algorithm by yager which overcomes the above limitations and in general is adaptable to less stringent input data a detailed description of the new algorithm is supported by practical examples

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

geometry of decision making and the vector space formulation of the analytic hierarchy process we extend the conventional analytic hierarchy process ahp to an euclidean vector space and develop formulations for aggregation of the alternative preferences with the criteria preferences relative priorities obtained from such a formulation are almost identical with the ones obtained using conventional ahp each decision is represented by a preference vector indicating the orientation of the decision maker s mind in the decision space spanned by the decision alternatives this adds a geometric meaning to the decision making processes we utilise the measure of similarity between any two decision makers and apply it for analysing decisions in a homogeneous group we propose an aggregation scheme for calculating the group preference from individual preferences using a simple vector addition procedure that satisfies pareto optimality condition the results agree very well with the ones of conventional ahp

aggregation of analytic hierarchy process models based on similarities in decision makers preferences abstract the analytic hierarchy process is widely used in both individual and group decision making environments in this paper we investigate its applicability to model a specific class of decentralized decision problems where many decision makers take individual subjective decisions using locally available information in such subjective decision making environments it is neither possible nor appropriate to use group preference aggregation techniques to model the problem as a single group decision problem an approach to identify homogeneous subgroups of decision makers based on similarities in preferences and to aggregate preferences within each subgroup is proposed this approach is validated using employment preferences of 70 subjects modeled using the analytic hierarchy process

universal small scale structure near the boundary of siegel disks of arbitrary rotation number abstract we present numerical evidence that for analytic maps of the complex plane with a siegel disk of any irrational rotation number and a quadratic critical point on its boundary the small scale structure near the critical point can be described asymptotically by a universal two parameter family we propose an explanation in terms of an ergodic attractor for a renormalisation operator