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

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

fuzzy analytic hierarchy process fallacy of the popular methods abstract the fuzzy analytic hierarchy process fuzzy ahp is a very popular decision making method and literally thousands of papers have been published about it however we find the basic logic of this approach has problems from its methodology the definition and operational rules of fuzzy numbers not only oppose the main logic of fuzzy set theory but also oppose the basic principles of the ahp in dealing with the outcomes fuzzy ahp does not give a generally accepted method to rank fuzzy numbers and a way to check the validity of the results besides we discuss the validity of the analytic hierarchy network process ahp anp in complex and uncertain environments and find that fuzzy anp is a false proposition because there is no fuzzy priority in the super matrix which provides the basis for the anp although fuzzy ahp has been applied in many cases and cited hundreds of times we hoped that those who use fuzzy ahp would understand the problems associated with this method

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

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

a multi step goal programming approach for group decision making with incomplete interval additive reciprocal comparison matrices abstract this article presents a goal programming framework to solve group decision making problems where decision makers judgments are provided as incomplete interval additive reciprocal comparison matrices iarcms new properties of multiplicative consistent iarcms are put forward and used to define consistent incomplete iarcms a two step goal programming method is developed to estimate missing values for an incomplete iarcm the first step minimizes the inconsistency of the completed iarcms and controls uncertainty ratios of the estimated judgments within an acceptable threshold and the second step finds the most appropriate estimated missing values among the optimal solutions obtained from the previous step a weighted geometric mean approach is proposed to aggregate individual iarcms into a group iarcm by employing the lower bounds of the interval additive reciprocal judgments a two step procedure consisting of two goal programming models is established to derive interval weights from the group iarcm the first model is devised to minimize the absolute difference between the logarithm of the group preference and that of the constructed multiplicative consistent judgment the second model is developed to generate an interval valued priority vector by maximizing the uncertainty ratio of the constructed consistent iarcm and incorporating the optimal objective value of the first model as a constraint two numerical examples are furnished to demonstrate validity and applicability of the proposed approach

note on some models for deriving the priority weights from interval fuzzy preference relations abstract deriving accurate interval weights from interval fuzzy preference relations is key to successfully solving decision making problems xu and chen 2008 proposed a number of linear programming models to derive interval weights but the definitions for the additive consistent interval fuzzy preference relation and the linear programming model still need to be improved in this paper a numerical example is given to show how these definitions and models can be improved to increase accuracy a new additive consistency definition for interval fuzzy preference relations is proposed and novel linear programming models are established to demonstrate the generation of interval weights from an interval fuzzy preference relation

closed finitely generated ideals in algebras of holomorphic functions and smooth to the boundary in strictly pseudoconvex domains define in a m a a natural estructure of banach resp fr6chet algebra in this paper we are interested in the closed finitely generated ideals of the algebras am as much as we know there are very few papers on the ideal structure of these algebras in dimension n 1 in 22 nagel proved that if d is piecewise real analytic and g ok are holomorphic in a neighbourhood o f the ideal in a generated by g gk is closed our first main result establishes a sort of reciprocal of nagel s theorem

acceptable consistency of aggregated comparison matrices in analytic hierarchy process abstract the analytic hierarchy process is a method for solving multiple criteria decision problems as well as group decision making the weighted geometric mean method is appropriate when aggregation of individual judgements is used this paper presents a new proof which confirms the property that if the comparison matrices of all decision makers are of acceptable consistency then the weighted geometric mean complex judgement matrix wgmcjm also is of acceptable consistency this property was presented and first proved by xu 2000 but lin et al 2008 rejected the proof we also discuss under what conditions the wgmcjm is of acceptable consistency when not all comparison matrices of decision makers are of acceptable consistency for this case we determine the sufficient condition for the wgmcjm to be of acceptable consistency and provide numerical examples for a special case of two decision makers with 3 3 comparison matrices we find out some additional conditions for the wgmcjm to be of acceptable consistency

the geometric consistency index approximated thresholds abstract crawford and williams journal of mathematical psychology 29 1985 387 suggested for the row geometric mean method rgmm one of the most extended ahp s priorization procedure a measure of the inconsistency based on stochastic properties of a subjacent model in this paper we formalize this inconsistency measure hereafter called the geometric consistency index gci and provide the thresholds associated with it these thresholds allow us an interpretation of the inconsistency tolerance level analogous to that proposed by saaty multicriteria decision making the analytic hierarchy process new york 1980 for the consistency ratio cr used with the right eigenvector method in conventional ahp

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

local stability intervals in the analytic hierarchy process abstract this paper provides a sensitivity analysis of the judgements used in the analytic hierarchy process ahp in relation to the rank reversal produced in two different situations the selection of the best alternative p problem and the ranking of all the alternatives p problem in both cases under the supposition that we employ the row geometric mean method rgmm to determine the local priorities we obtain in the case of a single criterion a local stability interval for each judgement for each alternative and for the paired comparisons matrix which allow us to guarantee the best alternative and the ranking of all of them the p and p problems respectively with respect to these three situations judgement alternative and matrix and two problems p and p we also calculate a local stability index to detect the critical values of the resolution process both the local stability intervals and indexes are used as management tools in the final stage of the decision making process that is to say the exploitation phase especially in the negotiation process and the search for consensus between the actors

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

a note on a goal programming model for incomplete interval multiplicative preference relations and its application in group decision making abstract in a recently published paper by liu et al liu f zhang w g wang z x 2012 a goal programming model for incomplete interval multiplicative preference relations and its application in group decision making european journal of operational research 218 747 754 two equations are introduced to define consistency of incomplete interval multiplicative preference relations imprs and employed to develop a goal programming model for estimating missing values this note illustrates that such consistency definition and estimation model are technically incorrect new transitivity conditions are proposed to define consistent imprs and a two stage goal programming approach is devised to estimate missing values for incomplete imprs

consistency analysis of triangular fuzzy reciprocal preference relations abstract in order to simulate the uncertainty associated with impression or vagueness a decision maker may give her his judgments by means of triangular fuzzy reciprocal preference relations in the process of decision making the study of their consistency becomes a very important aspect to avoid a misleading solution based on the reciprocity property this paper proposes a new definition of consistent triangular fuzzy reciprocal preference relations the new definition is different from that reduced by consistent fuzzy reciprocal preference relations proposed by buckley 1985 the properties of consistent triangular fuzzy reciprocal preference relations in the light of the new definition are studied in detail in addition the shortcomings of the proof procedure of the proposition given by wang and chen 2008 are pointed out and the proposition is reproved by using the new definition of consistent triangular fuzzy reciprocal preference relations finally using the n 1 restricted comparison ratios a method for obtaining consistent triangular fuzzy reciprocal preference relations is proposed and an algorithm is shown to make a consistent decision ranking numerical results are further calculated to illustrate the new definition and the obtained algorithm

eigenfunctions and heat kernels of super maass laplacians on the super poincare upper half plane heat kernels of super maass laplacians are explicitly constructed on super poincare upper half plane by a serious treatment of a complete set of eigenfunctions by component decomposition an explicit treatment can be done for arbitrary weight and a knowledge of classical maass laplacians becomes helpful the result coincides with that of aoki commun math phys 117 405 1988 which was obtained by solving differential equations

aggregation of utility based individual preferences for group decision making abstract multi attribute utility theory maut elicits an individual decision maker s preferences for single attributes and develops a utility function by mathematics formulation to add up the preferences of the entire set of attributes when assessing alternatives a common aggregation method of maut for group decisions is the simple additive weighting saw method which does not consider the different preferential levels and preferential ranks for individual decision makers assessments of alternatives in a decision group and thus seems too intuitive in achieving the consensus and commitment for group decision aggregation in this paper the preferential differences denoting the preference degrees among different alternatives and preferential priorities denoting the favorite ranking of the alternatives for each decision maker are both considered and aggregated to construct the utility discriminative values for assessing alternatives in a decision group a comparative analysis is performed to compare the proposed approach to the saw model and a satisfaction index is used to investigate the satisfaction levels of the final two resulting group decisions in addition a feedback interview is conducted to understand the subjective perceptions of decision makers while examining the results obtained from these two approaches for the second practical case both investigation results show that the proposed approach is able to achieve a more satisfying and agreeable group decision than that of the saw method

group decision making using bilateral agreement matrix abstract group decision making based on a bilateral agreement matrix is investigated in this paper to derive a group decision from the opinions of individual decision makers pairs of decision makers are allowed to negotiate with one another and provide their bilateral negotiation values two algorithms are provided to help pairs of decision makers reach bilateral agreements which enables the construction of the bilateral agreement matrix the definition of a consistent bilateral agreement matrix is introduced to measure the credibility of the final result and its properties are investigated based on which a method is provided to construct a consistent bilateral agreement matrix from an inconsistent one models are then given to derive the weight vector from the bilateral agreement matrix based on which a consistency index of the matrix is defined two algorithms are then given to improve the level of consistency of the bilateral agreement matrix the final group decision is obtained by aggregating individual opinions using the weight vector the proposed methods are further extended to deal with the problem of incomplete bilateral agreement matrices several examples are also given to illustrate the proposed approach

linguistic hesitant fuzzy power aggregation operators and their applications in multiple attribute decision making as a generalization of hesitant fuzzy sets hfss and hesitant fuzzy linguistic term sets hfltss linguistic hesitant fuzzy sets lhfss establish a proper interpretation of the hesitancy inconsistency and uncertainty of experts in evaluating decision alternatives in this paper based on the cloud model we discuss multiple attribute decision making problems with linguistic hesitant fuzzy information to extend the applicability of lhfss we introduce the concept of a comprehensive cloud of lhfss that is based on which form of lhfss with different length can be unified utilizing the comprehensive cloud we define the distance measure between two lhfss which makes the support measure of two lhfss possible by extending the power operators to the linguistic hesitant fuzzy environment a series of linguistic hesitant fuzzy power aggregation operators are established we also study the properties and applications of these aggregation operators and develop two approaches for linguistic hesitant fuzzy multiple attribute decision making finally two practical examples are given to demonstrate the applicability and effectiveness of the proposed approach

a priori identification of preferred alternatives of owa operators by relational analysis of arguments we present a reverse decision aiding method that is distinct from previously reported ordered weighted averaging owa aggregation methods the proposed method is implemented in two phases in the first phase potentially best alternative defined as one having any weighting vector that enables it to be at least as good as the others is identified in the second phase the maximum and the minimum attitudinal characters for such an alternative are computed as the highest and the lowest values which it can attain under the weights set identified in the first phase these two phases are governed only by the relational analysis of input arguments without soliciting the decision maker to supply a specific attitudinal character the proposed method can be applied to cases when it is difficult to obtain a precise attitudinal character and when even if a precise one is obtained the owa operator weights are different depending on the weights generating methods adopted further if uncertain attitudinal character in the form of the interval number is available its projection into the results of the proposed method yields less alternatives of consideration in some cases a single best alternative the proposed method also allows for a priori identification of alternatives prone to change at a particular range of attitudinal character

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

consistency stability intervals for a judgement in ahp decision support systems abstract this paper focuses on the need to evaluate the consistency of human judgements in decision support systems dss in this sense and in the context of one of the most applied multicriteria methodologies the analytic hierarchy process ahp we obtain the consistency stability interval csi associated with each judgement that is to say the interval in which this can oscillate without exceeding a value of the consistency measure fixed in advance to calculate these csis we consider the row geometric mean method as the priorisation procedure the geometric consistency index as the consistency measure and a local situation with one criterion the proposed procedure has been implemented in a module that is easily adaptable to any decision support system based on ahp

multi criteria group decision making based on bilateral agreements abstract in multi criteria group decision making a key issue is aggregating individual preference into group one in this study we employ a bilateral agreement to conduct group evaluation of alternatives the bilateral agreement between a pair of individuals could be on the weight of a criterion a criterion evaluation function or willingness to pay any one of three types of bilateral agreements can derive the group utility to express the relationship between the bilateral agreement and individual evaluations the quasi arithmetic mean is used which can ensure the consistency property of the pairwise comparison matrices the minimum requirements are explored to obtain the group preference which shows that n 1 pairs of bilateral agreements are necessary finally several examples are provided to illustrate the proposed methods

completeness relations for maass laplacians and heat kernels on the super poincare upper half plane simple completeness relations are proposed for maass laplacians with the help of these completeness relations correct heat kernels of super maass laplacians are derived on the super poincare upper half plane

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

an intelligent decomposition of pairwise comparison matrices for large scale decisions abstract a pairwise comparison matrix pcm has been used to compute for relative priorities of elements and are integral components in widely applied decision making tools the analytic hierarchy process ahp and its generalized form the analytic network process anp however pcms suffer from several issues limiting their applications to large scale decision problems these limitations can be attributed to the curse of dimensionality that is a large number of pairwise comparisons need to be elicited from a decision maker this issue results to inconsistent preferences due to the limited cognitive powers of decision makers to address these limitations this research proposes a pcm decomposition methodology that reduces the elicited pairwise comparisons a binary integer program is proposed to intelligently decompose a pcm into several smaller subsets using interdependence scores among elements since the subsets are disjoint the most independent pivot element is identified to connect all subsets to derive the global weights of the elements from the original pcm as a result the number of pairwise comparison is reduced and consistency is of the comparisons is improved the proposed decomposition methodology is applied to both ahp and anp to demonstrate its advantages

heat kernels and super determinants of laplace operators on super riemann surfaces heat kernels of laplacians on superfields of arbitrary tensor weight on super riemann surfaces are constructed and are used to compute the super determinants of these operators in terms of the selberg super zeta function

a universal voting system based on the potential method abstract pairwise comparison of various objects alternatives is common in many procedures related to decision making potential method pm uses a weighted preference graph as the basic structure generated by such comparisons to obtain a value function potential on the set of alternatives the potential is calculated as a solution of a system of equations involving the laplacian matrix of the preference graph in multiple criteria decision analysis or group decision making each criterion or decision maker is represented by its preference graph a multigraph obtained by joining these graphs aggregates the individual preferences and generates the group potential moreover the influence weight of each criterion or decision maker on the group potential may be adjusted in this article the aggregation of preference graphs is applied to voting systems many different forms of voting ballots can generate preference graphs which gives us a universal ballot independent voting system as an illustration of the pm in the voting context we have chosen the eurovision song contest and the analysis of the cookie type ballots where the voters allocate a certain number of points cookies to the candidates

on consistency of the weighted geometric mean complex judgement matrix in ahp abstract the weighted geometric mean method wgmm is the most common group preference aggregation method in the analytic hierarchy process this paper reports on research concerning the consistency of wgmm and proves that the weighted geometric mean complex judgement matrix wgmcjm is of acceptable consistency in saaty s t l saaty the analytic hierarchy process mcgraw hill new york 1980 opinion a consistency ratio cr of 0 1 or less is acceptable under the condition that all judgement matrices given by experts for the same problem of decision making are of acceptable consistency accordingly a theoretic basis has been developed for the application of the wgmm in group decision making

group decisions with multiple criteria we consider a decision problem where a group of individuals evaluates multiattribute alternatives we explore the minimal required agreements that are sufficient to specify the group utility function a surprising result is that under some conditions a bilateral agreement among pairs of individuals on a single attribute is sufficient to derive the multiattribute group utility the bilateral agreement between a pair of individuals could be on the weight of an attribute on an attribute evaluation function or on willingness to pay we focus on the case in which each individual s utility function is additive we show that the group utility can be represented as the weighted sum of group attribute weights and more remarkably of attribute evaluation functions these group attribute evaluation functions are in turn weighted sums of individual attribute evaluation functions

the analytic hierarchy process with stochastic judgements the analytic hierarchy process ahp is a widely used method for multicriteria decision support based on the hierarchical decomposition of objectives evaluation of preferences through pairwise comparisons and a subsequent aggregation into global evaluations the current paper integrates the ahp with stochastic multicriteria acceptability analysis smaa an inverse preference method to allow the pairwise comparisons to be uncertain a simulation experiment is used to assess how the consistency of judgements and the ability of the smaa ahp model to discern the best alternative deteriorates as uncertainty increases across a range of simulated problems results indicate that according to conventional benchmarks judgements are likely to remain consistent unless uncertainty is severe but that the presence of uncertainty in almost any degree is sufficient to make the choice of best alternative unclear

note on group consistency in analytic hierarchy process abstract we study the paper of xu z xu on consistency of the weighted geometric mean complex judgement matrix in ahp european journal of operational research 126 2000 683 687 for the group consistency in analytic hierarchy process of multicriteria decision making the purpose of this note is threefold first we point out the questionable results in this paper second for three by three comparison matrices we provide a patchwork for his method third we constructed a counter example to show that in general his method is wrong numerical examples are provided to illustrate our findings if there are four or more alternatives then we may advise researchers to ignore his results to avoid questionable estimation of group consistency

boundary properties of the inconsistency of pairwise comparisons in group decisions abstract this paper proposes an analysis of the effects of consensus and preference aggregation on the consistency of pairwise comparisons we define some boundary properties for the inconsistency of group preferences and investigate their relation with different inconsistency indices some results are presented on more general dependencies between properties of inconsistency indices and the satisfaction of boundary properties in the end given three boundary properties and nine indices among the most relevant ones we will be able to present a complete analysis of what indices satisfy what properties and offer a reflection on the interpretation of the inconsistency of group preferences