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

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

on minima of the absolute value of certain random exponential sums let t n x n j 1 e 2 ij 2 x where stands for a random choice of sign with equal probability it is shown here that with high probability min x e 0 1 t n x n provided n is large and t n x n j 1 r j e 2 ij d x where d 2 3 is fixed and with standard normal r j we show that the minima are typically on the order of n d with high probability and for large n

lower bounds for the absolute value of random polynomials on a neighborhood of the unit circle abstract let t x n 1 j 0 eijx where stands for a random choice of sign with equal probability the first author recently showed that for any 0 and most choices of sign minx 0 2 t x 0 and large n most choices of sign satisfy minx 0 2 t x n 1 2 furthermore we study the case of more general random coefficients and applications of our methods to complex zeros of random polynomials

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

an aggregation disaggregation approach to obtain robust conclusions with electre tri electre tri is a well known method to assign a set of alternatives to a set of predefined categories considering multiple criteria using this method requires setting many parameters which is often a difficult task we consider the case where the decision makers dms in the decision process are unsure of which values should each parameter take which may result from uncertain imprecise or inaccurately determined information as well as from lack of consensus among them this paper discusses the synergy between two approaches developed independently to deal with this difficulty the first approach infers the value of parameters from assignment examples provided by the dms as an elicitation aid each assignment example originates mathematical constraints that the parameter values should satisfy the second approach considers a set of constraints on the parameter values reflecting the imprecise information that the dms are able to provide then it computes the best and worst categories for each alternative compatible with constraints in order to present robust conclusions both approaches avoid asking for precise values for the parameters rather they proceed to solve the problem in a way that requires from the dms much less effort by integrating these two approaches this paper proposes a new interactive approach where the insight obtained during robustness analyses guides the dms during the elicitation phase

a sensitivity analysis algorithm for hierarchical decision models in this paper a comprehensive algorithm is developed to analyze the sensitivity of hierarchical decision models hdm including the analytic hierarchy process and its variants to single and multiple changes in the local contribution matrices at any level of the decision hierarchy the algorithm is applicable to all hdm that use an additive function to derive the overall contribution vector it is independent of pairwise comparison scales judgment quantification techniques and group opinion combining methods the allowable range region of perturbations contribution tolerance operating point sensitivity coefficient total sensitivity coefficient and the most critical decision element at a certain level are identified in the hdm sa algorithm an example is given to demonstrate the application of the algorithm and show that hdm sa can reveal information more significant and useful than simply knowing the rank order of the decision alternatives 2007 elsevier b v all rights reserved

the pair correlation function of fractional parts of polynomials we investigate the pair correlation function of the sequence of fractional parts of n d n 1 2 n where d 2 is an integer and an irrational we conjecture that for badly approximable the normalized spacings between elements of this sequence have poisson statistics as n

link function selection in stochastic multicriteria decision making models a stochastic formulation of the analytic hierarchy process ahp using an approach based on bayesian categorical data models has been developed however in categorical data models it is known that the selection of the link function may have an impact on the model estimates in particular the selection of the probit link implies an assumption that model error terms are normally distributed and this normality assumption is regularly utilized in other related methods such as the multiplicative ahp we examine model performance with respect to the choice of two model link functions with regard to point estimates it is found that the logit formulation is better able to replicate the estimates obtained by the eigenvector decomposition associated with the original formulation of the ahp by contrast the probit link produces priorities which are consistently more moderate than those of the ahp the results suggest that the logit formulation will be preferred by decision makers who wish to replicate the ahp priorities as closely as possible the results also suggest that the unexamined use of the normality assumption in other stochastic ahp methods may have an impact on priority estimates and thus is worthy of further attention

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 multiple criteria sorting method where each category is characterized by several reference actions the electre tri nc method this paper presents electre tri nc a new sorting method which takes into account several reference actions for characterizing each category this new method gives a particular freedom to the decision maker in the co construction decision aiding process with the analyst to characterize the set of categories while there is no constraint for introducing only one reference action as typical of each category like in electre tri c almeida dias et al 2010 as in such a sorting method this new sorting method is composed of two joint rules electre tri nc also fulfills a certain number of natural requirements additional results on the behavior of the new method are also provided in this paper namely the ones with respect to the addition or removal of the reference actions used for characterizing a certain category a numerical example illustrates the manner in which electre tri nc can be used by a decision maker a comparison with some related sorting procedures is presented and it allows to conclude that the new method is appropriate to deal with sorting problems

assessing a set of additive utility functions for multicriteria decision making the uta method abstract the purpose of the method presented in this paper is to assess additive utility functions which aggregate multiple criteria in a composite criterion using the information given by a subjective ranking on a set of stimuli or actions weak order comparison judgments and the multicriteria evaluations of these actions it is an ordinal regression method using linear programming to estimate the parameters of the utility function stability and sensitivity analysis leads to the assessment of a set of utility functions by means of post optimality analysis techniques in linear programming finally a simple illustrative example is presented and some extensions of the method are proposed

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

the analytic hierarchy process in an uncertain environment a simulation approach abstract traditionally decision makers were forced to converge ambiguous judgments to a single point estimate in order to describe a pairwise relationship between alternatives relative to some criterion for use in the analytic hierarchy process ahp since many circumstances exist which make such a convergence difficult confidence in the outcome of an ensuing ahp synthesis may be reduced likewise when a group of decision makers cannot arrive at a consensus regarding a judgment some members of the group may simply lose confidence in the overall synthesis if they lack faith in some of the judgments the ahp utilizes point estimates in order to derive the relative weights of criteria sub criteria and alternatives which govern a decision problem however when point estimates are difficult to determine distributions describing feasible judgments may be more appropriate using simulation we will demonstrate that levels of confidence can be developed expected weights can be calculated and expected ranks can be determined it will also be shown that the simulation approach is far more revealing than traditional sensitivity analysis

learning intransitive reciprocal relations with kernel methods abstract in different fields like decision making psychology game theory and biology it has been observed that paired comparison data like preference relations defined by humans and animals can be intransitive intransitive relations cannot be modeled with existing machine learning methods like ranking models because these models exhibit strong transitivity properties more specifically in a stochastic context where often the reciprocity property characterizes probabilistic relations such as choice probabilities it has been formally shown that ranking models always satisfy the well known strong stochastic transitivity property given this limitation of ranking models we present a new kernel function that together with the regularized least squares algorithm is capable of inferring intransitive reciprocal relations in problems where transitivity violations cannot be considered as noise in this approach it is the kernel function that defines the transition from learning transitive to learning intransitive relations and the kronecker product is introduced for representing the latter type of relations in addition we empirically demonstrate on two benchmark problems one in game theory and one in theoretical biology that our algorithm outperforms methods not capable of learning intransitive reciprocal relations

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

pairwise comparison methods in multiple objective programming with applications in a long term energy planning model abstract we propose to use pairwise comparisons within the framework of ideal point or reference point methods for multi objective programming the decision makers are requested to estimate the ratios which are acceptable for deviations from the ideal vector thereafter we seek the nearest feasible solution using the tchebycheff norm in this paper we sketch the pairwise comparison methods some aspects of magnitude scaling and the ideal point methods we show the results of our numerical experiments in long term energy planning with nine objective functions finally we present a preliminary evaluation of the combined method on the basis of its possible contribution to interactive decision analysis

a promethee based classification method using concordance and discordance relations and its application to bankruptcy prediction outranking relation theory has been widely used to study pattern classification here we propose a classification method with concepts from the flows used in promethee methods which are extensively applied in multi criteria decision aids promethee uses a flow generated on the basis of a preference index and measured by various preference functions for each criterion to represent the preference intensity for one pattern over another pattern however only criteria that are concordant with the preference contribute to a preference index in the present study the opinions from discordant criteria are also taken into account the proposed method newly defines an overall preference index using both concordance and discordance relations for ordinal sorting problems the final classification decision for a new pattern depends on its net flow the criteria weights are determined using a genetic algorithm based approach empirical results obtained for a real world problem regarding bankruptcy prediction demonstrate that the proposed method performs well compared to other well known classification methods

a new approach to multi criteria sorting based on fuzzy outranking relations the theseus method abstract in this paper we propose the theseus method a new approach based on fuzzy outranking relations to multi criteria sorting problems compared with other outranking based methods theseus is inspired by another view of multi criteria classification problems it utilizes a new way of evaluating the assignment of an object to an element of the set of ordered categories that were previously defined this way is based on comparing every possible assignment with the information from various preference relations that are derived from a fuzzy outranking relation defined on the universe of objects the appropriate assignment is determined by solving a simple selection problem the capacity of a reference set for making appropriate assignments is related to a good characterization of the categories a single reference action characterizing a category may be insufficient to achieve well determined assignments in this paper the reference set capacity to perform appropriate assignments is characterized by some new concepts this capacity may be increased when more objects are added to the reference set theseus is a method for handling the preference information contained in such larger reference sets

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

an evolutionary approach to construction of outranking models for multicriteria classification the case of the electre tri method outranking methods constitute an important class of multicriteria classification models often however their implementation is cumbersome due to the large number of parameters that the decision maker must specify past studies tried to address this issue using linear and nonlinear programming to elicit the necessary preferential information from assignment examples in this study an evolutionary approach based on the differential evolution algorithm is proposed in the context of the electre tri method computational results are given to test the effectiveness of the methodology and the quality of the obtained models

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

convergence properties and practical estimation of the probability of rank reversal in pairwise comparisons for multi criteria decision making problems abstract in this paper we address the impact of uncertainty introduced when the experts complete pairwise comparison matrices in the context of multi criteria decision making we first discuss how uncertainty can be quantified and modeled and then show how the probability of rank reversal scales with the number of experts we consider the impact of various aspects which may affect the estimation of probability of rank reversal in the context of pairwise comparisons such as the uncertainty level alternative preference scales and different weight estimation methods we also consider the case where the comparisons are carried out in a fuzzy manner it is shown that in most circumstances augmenting the size of the expert group beyond 15 produces a small change in the probability of rank reversal we next address the issue of how this probability can be estimated in practice from information gathered simply from the comparison matrices of a single expert group we propose and validate a scheme which yields an estimate for the probability of rank reversal and test the applicability of this scheme under various conditions the framework discussed in the paper can allow decision makers to correctly choose the number of experts participating in a pairwise comparison and obtain an estimate of the credibility of the outcome

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

decision making with the ahp why is the principal eigenvector necessary in this paper it is shown that the principal eigenvector is a necessary representation of the priorities derived from a positive reciprocal pairwise comparison judgment matrix a aij when a is a small perturbation of a consistent matrix when providing numerical judgments an individual attempts to estimate sequentially an underlying ratio scale and its equivalent consistent matrix of ratios near consistent matrices are essential because when dealing with intangibles human judgment is of necessity inconsistent and if with new information one is able to improve inconsistency to near consistency then that could improve the validity of the priorities of a decision in addition judgment is much more sensitive and responsive to large rather than to small perturbations and hence once near consistency is attained it becomes uncertain which coefficients should be perturbed by small amounts to transform a near consistent matrix to a consistent one if such perturbations were forced they could be arbitrary and thus distort the validity of the derived priority vector in representing the underlying decision

uncertainty and rank order in the analytic hierarchy process abstract the analytic hierarchy process uses paired comparisons to derive a scale of relative importance for alternatives we investigate the effect of uncertainty in judgment on the stability of the rank order of alternatives the uncertainty experienced by decision makers in making comparisons is measured by associating with each judgment an interval of numerical values the approach leads to estimating the probability that an alternative or project exchanges rank with other projects these probabilities are then used to calculate the probability that project would change rank at all we combine the priority of importance of each project with the probability that it does not change rank to obtain the final ranking

multigroup discriminant analysis using linear programming in this paper we introduce a nonparametric linear programming formulation for the general multigroup classification problem previous research using linear programming formulations has either been limited to the two group case or required complicated constraints and many zero one variables we develop general properties of our multigroup formulation and illustrate its use with several small example problems and previously published real data sets a comparative analysis on the real data sets shows that our formulation may offer an interesting robust alternative to parametric statistical formulations for the multigroup discriminant problem

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

a hermite hadamard type inequality with applications to the estimation of moments of continuous random variables we present a new hadamard type inequality for quasi convex functions our results have applications to the estimation of moments of random variables our work generalizes some recent results published in 2014 in amc in this paper we present a hadamard type inequality for quasi convex functions with applications to the estimation of moments of continuous random variables with compact support the main result obtained here generalizes the inequality obtained by alomari et al 2010

an axiomatic approach to noncompensatory sorting methods in mcdm i the case of two categories in the literature on mcdm many methods have been proposed in order to sort alternatives evaluated on several attributes into ordered categories most of them were proposed on an ad hoc basis the purpose of this paper is to contribute to a recent trend of research aiming at giving these methods sound theoretical foundations using tools from conjoint measurement we provide an axiomatic analysis of the partitions of alternatives into two categories that can be obtained using what we call noncompensatory sorting models these models have strong links with the pessimistic version of electre tri our analysis allows to pinpoint what appears to be the main distinctive features of electre tri when compared to other sorting methods it also gives hints on the various methods that have been proposed to assess the parameters of electre tri on the basis of assignment examples

improving generalization of fuzzy if then rules by maximizing fuzzy entropy when fuzzy if then rules initially extracted from data have not a satisfying performance we consider that the rules require refinement distinct from most existing rule refinement approaches that are based on the further reduction of training error this paper proposes a new rule refinement scheme that is based on the maximization of fuzzy entropy on the training set the new scheme which is realized by solving a quadratic programming problem is expected to have the advantages of improving the generalization capability of initial fuzzy if then rules and simultaneously overcoming the overfitting of refinement experimental results on a number of selected databases demonstrate the expected improvement of generalization capability and the prevention of overfitting by a comparison of both training and testing accuracy before and after the refinement

eigenvector and logarithmic least squares abstract the eigenvector method deals with two questions simultaneously closeness and order together they belong to the field topology of order the metric idea of closeness is inadequate to judge what is a good approximation to data involving order relations there is usually a condition which relates order in the derived scale to order in the data this is precisely the kind of condition captured by the eigenvector but not by the logarithmic least squares estimate

multicriteria decision analysis with fuzzy pairwise comparisons abstract we present a modification of the fuzzy multi criteria method proposed by van laarhoven and pedrycz 1983 first the weights of the decision criteria are calculated by the minimization of a logarithmic regression function next the weights of the decision alternatives are calculated for each criterion separately lastly the fuzzy final scores of the alternatives are determined by an appropriate aggregationm of the calculated weights we will show that our modified method in contrast with the original one yields weights which are optimal with respect to the logarithmic regression function we also present a version of our method with geometric ratio scales rather than saaty s equi distant scale to quantify the gradations of human comparative judgement this approach also enables us to develop an efficient procedure for investigating the scale sensitivity of the calculated weights and the final scores of the alternatives

some inequalities for differentiable convex and concave mappings abstract several inequalities are obtained for some differentiable convex mappings that are connected with the celebrated hermite hadamard integral inequality also a parallel development is made for concave functions

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

some inequalities for differentiable convex mapping with application to weighted trapezoidal formula and higher moments of random variables abstract several inequalities for differentiable convex wright convex and quasi convex mapping are obtained respectively that are connected with the celebrated hermite hadamard integral inequality also some error estimates for weighted trapezoid formula and higher moments of random variables are given

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

induction of fuzzy decision trees abstract most decision tree induction methods used for extracting knowledge in classification problems do not deal with cognitive uncertainties such as vagueness and ambiguity associated with human thinking and perception in this paper cognitive uncertainties involved in classification problems are explicitly represented measured and incorporated into the knowledge induction process a fuzzy decision tree induction method which is based on the reduction of classification ambiguity with fuzzy evidence is developed fuzzy decision trees represent classification knowledge more naturally to the way of human thinking and are more robust in tolerating imprecise conflict and missing information

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

electre tri c a multiple criteria sorting method based on characteristic reference actions in this paper a new sorting method following a decision aiding constructive approach is proposed this method is called electre tri c as a sorting method a set of categories must be defined to represent the way in which the actions that are going to be assigned to each of them should further be processed this method is appropriate to deal with decision aiding contexts where the categories are completely ordered and each of them is defined through a single characteristic reference action the set of characteristic actions should be co constructed through an interactive process between the analyst and the decision maker electre tri c has been conceived to verify a set of natural structural requirements conformity homogeneity monotonicity and stability which can be viewed as its fundamental properties this method is composed of two joint rules called descending rule and ascending rule which must be used conjointly and not separately each one of these rules selects only one category or a range of possible categories for a possible assignment of an action this assignment depends on the comparison of such an action to the characteristic actions according to a chosen credibility level numerical examples are also presented in order to illustrate the main theoretical results provided by the method

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

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

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

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

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

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

a multicriteria classification approach based on pairwise comparisons abstract classification refers to the assignment of a set of alternatives into predefined homogenous classes most of the existing classification methodologies are based on absolute comparisons among the alternatives and some reference profiles cut off points that discriminate the classes this paper proposes a new approach that involves pairwise comparisons based on the multicriteria decision aid mcda paradigm the basis of the methodology is a preference relation that is used to perform pairwise comparisons among the alternatives the criteria weights used to construct the preference relation are specified using a reference set of alternatives training sample on the basis of linear programming techniques

improvement and further generalization of inequalities for differentiable mappings and applications based on the celebrated hermite hadamard integral inequality for convex functions some inequalities for differentiable convex and concave mappings are generalized furthermore the results obtained are examined in the context of special means for real numbers

a linear programming approach for learning non monotonic additive value functions in multiple criteria decision aiding abstract a new framework for preference disaggregation in multiple criteria decision aiding is introduced the proposed approach aims to infer non monotonic additive preference models from a set of indirect pairwise comparisons the preference model is presented as a set of marginal value functions and the discriminatory power of the inferred preference model is maximized against its complexity to infer a value function that is compatible with the supplied preference information the proposed methodology leads to a linear programming optimization problem that is easy to solve the applicability and effectiveness of the new methodology is demonstrated in a thorough experimental analysis covering a broad range of decision problems

refinements of hadamard type inequalities for quasi convex functions with applications to trapezoidal formula and to special means in this paper some inequalities of hadamard s type for quasi convex functions are given some error estimates for the trapezoidal formula are obtained applications to some special means are considered

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

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

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

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

consequences of uncertainty in the analytic hierarchy process a simulation approach a methodology for the propagation of uncertainty in the analytic hierarchy process ahp is described the sole source of the uncertainty is assumed to lie in the elements of the preference matrices the uncertainty in decision alternative rankings and the probability of a rank reversal as functions of the number of alternatives and hierarchy depth are examined ranking uncertainty is found to decrease as the number of alternatives or the depth of the hierarchy is increased the observed behaviour of ranking uncertainty is partially explained using the central limit theorem

note a preference ranking organisation method the promethee method for multiple criteria decision making principles for a new family of outranking methods are given the main aim of the proposed promethee approach is to be as easily understood as possible by the decision maker it is based on extensions of the notion of criterion six possible extensions are considered these extensions can easily be identified by the decision maker because the parameters to be defined at most 2 have an economic significance a valued outranking graph is constructed by using a preference index two possibilities are considered to solve the ranking problem by using this valued graph promethee i provides a partial preorder and promethee ii a total preorder on the set of the possible actions

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

integrated framework for preference modeling and robustness analysis for outranking based multiple criteria sorting with electre and promethee a novel framework for preference modeling and robustness analysis is proposed we apply a sorting rule deriving from the semantic meaning of an outranking relation we ensure input output correspondence with respect to three sorting perspectives the generality of our framework is illustrated with its two specific implementations preference modeling can be simplified when using model instances providing precise assignments we introduce an integrated framework for preference modeling and robustness analysis in outranking based multiple criteria sorting the preference information supplied by the decision maker dm is composed of parts of desired recommendation and imprecise requirements that the delivered outcomes should satisfy its allowed forms are imprecise assignment examples desired class cardinalities and assignment based pairwise comparisons the exploitation of all instances of the outranking model compatible with these preferences results in three types of outcomes these raise robustness concerns in terms of the stability of a suggested assignment for each alternative an observed size of each class and a comparison between recommendation delivered for pairs of alternatives the correspondence between different types of inputs and outputs facilitates the dialogue with the dm and enhances her his confidence in the suggested recommendation while referring only to a semantic meaning of an outranking relation we present the procedures for translating the preference information into parameters of compatible outranking model instances and for deriving various kinds of sorting results then we implement this general framework in the context of outranking models specific for electre and promethee finally we show how preference modeling and robustness analysis can be performed and greatly simplified with a set of preference model instances providing precise assignments for the alternatives the framework proposed for this case is based on mixed integer linear programming milp being independent of the underlying model and method application of the approaches is demonstrated on the case of classifying polish research units into four quality classes