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

the character covering numbers of the alternating groups we prove that the character covering number of an n 6 is n n where x is the smallest integer not less than x

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

an exposition of the ahp in reply to the paper remarks on the analytic hierarchy process it is a fact that people make decisions and have been making decisions for a very long time contrary to what some of us who are interested in decision making may like to believe most people do not take seriously the existence of theories which purport to set their thinking and feeling right they claim to know their own value system and what they want they may wonder how anyone else can know well enough to tell them how best to organize their thinking in order to make better choices yet research has shown that complex decisions are beyond the capacity of the brain to synthesize intuitively and efficiently since decision making is a natural characteristic of people how do we describe what they do so that an ordinary mortal can understand what we are saying we do not wish to legislate the method with which people should make decisions but only to describe it even when it is prescribed by some method in the process we may learn things that can help people make better decisions how the analytic hierarchy process ahp forman et al harker 1986 harker and vargas 1987 saaty 1986 1988a b saaty and vargas 1987 xu 1988 golden et al 1989 saaty and alexander 1989 is a theory of measurement when applied in decision making it assists one to describe the general decision operation by decomposing a complex problem into a multi level hierarchic structure of objectives criteria subcriteria and alternatives the ahp provides a fundamental scale of relative magnitudes expressed in dominance units to represent judgments in the form of paired comparisons a ratio scale of relative magnitudes expressed in priority units is then derived from each set of comparisons an overall rcatio scale of priorities is then synthesized to obtain a ranking of the alternatives from its axioms to its procedures the ahp has turned out to be historically and theoretically a different and independent theory of decision making from utility theory much as a dialogue evolved in mathematics around the consistency of different geometries and around absolute and relative space and time in physics both to dispel absolute notions those who believe that only utility theory can tell us the absolute truth about man s decision making might take a close look at the ahp it has found varied and serious applications it also has a particular way of generating ratio scales and dealing with inconsistency in judgment that have contributed to its effectiveness in resource allocation and in the setting of priorities by a group of decision makers utility theory is a normative process the ahp as a descriptive theory encompasses procedures leading to outcomes as would be ranked by a normative theory but it must go beyond to deal with outcomes not accounted for by the demanding assumptions of a normative theory we must briefly describe the ahp to enable the reader to see that a practicable theory based on ratio scales need not dilute itself to satisfy expectations of people who derive their understanding from a theory based on interval scales this is particularly true if the rival theory in aspiring for generality also makes unrealistic assumptions for example about the transitivity and consistency of preferences and the difficult use of lotteries

fast noniterative algorithm for designing high accuracy lowpass fir filters with the evaluation of errors we present an analytical approach to the design of zero phase fir digital passband filters we describe an easy to implement finite noniterative algorithm for determining low pass filters whose cost of calculations is o nlog sub 2 n we give the evaluation of errors taking into account the transition band from which there follows the main result of this paper wherein our filters are nearly optimal we compare them with optimal filters optimal kaiser s 1974 filters it results that the errors of our filters are slightly greater than kaiser s filters however a small modification of the presented algorithm may lead to filters that in numerical tests have errors like kaiser s filters or even smaller if the cut off frequency is not very close to 0 or spl pi our algorithms are especially efficient for the design of very accurate i e very long filters in numerical calculations we constructed using 15 digit floating point arithmetic filters with errors up to 10 sup 13 i e 260 db the main limitation of the filters in question is the negligence of weighting functions

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

a clarification of remarks on the analytic hierarchy process the replies by saaty 1990 and by harker and vargas 1990 to my remarks on the analytic hierarchy process dyer 1990 are useful in focusing attention on areas of disagreement regarding the ahp and provide readers with sufficient information to form their own opinions the purpose of this response is not to extend the debate but to clarify the two methods proposed in my paper to resolve the problem of rank reversal in the ahp harker and vargas 1989 misinterpreted my first proposed method method 1 in their reply the key observation they missed is that both the weights on the criteria and the scores of the alternatives on the criteria must be normalized with respect to the same range of alternative values in method 1 the decision maker must specify the ranges over which he assumed the alternatives might vary on each criterion when he determined the weights an example may be useful in clarifying the key ideas suppose each alternative in the harker and vargas 1990 example represents the choice of a decision maker among sums of money deposited in two different bank accounts and each criterion is simply the amount of money deposited in a single bank account obviously the decision maker would use the sum of the accounts as the basis for ranking the alternatives and would select b from the three alternatives a b and c let us see how the appropriate use of method 1 would lead to this result without recourse to the supermatrix technique or to the concept of dependence of the criteria on the alternatives the decision maker would consider the deposits in each bank account to be equally important if he assumed that the ranges of the possible deposits were equal for example suppose the decision maker assumes that the values for each bank account criterion vary over the range 1 80 he responds that changes from 1 to 80 on each criterion are equally important so the normalized eigenvector for the ahp criterion weights is 1 2 1 2 using method 1 since the range of values on criterion c1 is 75 80 and the decision maker assumed a range of 1 80 in determining the equal criterion weights we must introduce a dummy alternative with the value of 1 on criterion c1 this dummy alternative does not need to be explicitly introduced with values on the other criteria but it could be and would not affect the results as long as its assumed values on the other criteria are in the range 1 80 similarly a dummy alternative with a value of 80 must be introduced for criterion c2 the normalized eigenvector for criterion c1 would become 80 235 79 235 75 235 1 235 for example where the last component 1 235 is the score for the dummy alternative we normalize each resulting eigenvector by subtracting its smallest component and then dividing by the largest component the appropriate scores for the three alternatives are

are inconsistent decisions better an experiment with pairwise comparisons this paper tries to contribute to the well known and lengthy discussion on whether we should allow decision makers to be inconsistent or to put it in another way if intransitive decisions better represent their preferences we will focus specifically on the multiple criteria decision making problem since it is one of the most usual stances where this intransitivity appears results from an experiment carried out with graduate students shows that when intransitivities are removed the preferences of decision makers are not better represented and therefore that the effort to do so may not be justified

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

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

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

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

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

avoiding rank reversal in ahp decision support models abstract extending the work of a previous paper which showed that rank reversal in ahp is caused by eigenvector normalization this paper shows why methods proposed for avoiding rank reversal work when they are unnecessary and when they are futile the paper also discusses the implicit often unwarranted assumption that the dm s preference structure can be represented by an additive value function and examines the ramifications when a nonadditive value function is required

a unified approach to ahp with linking pins abstract recent work demonstrates the need for paired comparison estimates of criteria in the analytic hierarchy process ahp to follow prescribed rules and for the subsequent analysis to be contingent on these rules many different approaches are valid each reflecting a different rule for criteria comparison a generalized formula from which valid approaches may be generated is presented and the connection between these methods and saaty s supermatrix approach is explored

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

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

a weak order solution to a group ranking and consistency driven pairwise comparisons a novel non numerical algorithm based on a partial order for solving a group ranking problem is compared with the more traditional consistency driven approach both approaches are based on the fundamental concept of pairwise comparisons potential applications are discussed

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

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

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

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

on some useful properties of the perron eigenvalue of a positive reciprocal matrix in the context of the analytic hierarchy process abstract a positive n x n matrix r rij is said to be reciprocal if its entries verify fji 1 rij 0 for all 1 2 decision alternatives on an arbitrary ratio scale saaty in journal of mathematical psychology 1977 proposed to use z ama x n n 1 0 a linear transform of the perron eigenvalue max of r as a measure of the cardinal consistency in an agent s responses and posed the problem of determining how it might vary as a function of the rij s he also suggested that an upper bound could be found for that consistency index when the entries of r are restricted to take their values in a bounded set both of these questions are answered here using classical results from linear algebra keywords analytic hierarchy process decision theory pairwise comparisons perron frobenius theo rem reciprocal matrices i introduction saaty 20 has introduced a procedure for pri oritising decision alternatives based on an n x n positive matrix r

kronecker products characters partitions and the tensor square conjectures abstract we study the remarkable saxl conjecture which states that tensor squares of certain irreducible representations of the symmetric groups s n contain all irreducibles as their constituents our main result is that for sufficiently large n they contain representations corresponding to young diagrams of hooks two row and diagrams obtained from hooks and two rows by adding a finite number of squares for that we develop a new sufficient condition for the positivity of kronecker coefficients in terms of characters and apply this tool using combinatorics of rim hook tableaux in combination with known results on unimodality of certain partition functions we also present connections and speculations on random characters of s n

the asymptotics of optimal equiripple filters for equiripple filters the relation among the filter length n 1 the transition bandwidth spl delta spl omega and the optimal passband and stopband errors spl delta sub p and spl delta sub s has been a secret for more than 20 years this paper is aimed at solving this mystery we derive the exact asymptotic results in the weight free case spl delta sub p spl delta sub s spl delta which enables us to interpret and improve the existing empirical formulas our main results are finally combined into a formula in the transition band the filter response is discovered to be asymptotically close to a scaled error function the main tools are potential theory in the complex plane and asymptotic analysis

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

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

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

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

meteor a constraint based fir filter design program it is proposed to specify a filter only in terms of upper and lower limits on the response find the shortest filter length which allows these constraints to be met and then find a filter of that order which is farthest from the upper and lower constraint boundaries in a minimax sense the simplex algorithm for linear programming is used to find a best linear phase fir filter of minimum length as well as to find the minimum feasible length itself the simplex algorithm while much slower than exchange algorithms also allows the incorporation of more general kinds of constraints such as concavity constraints which can be used to achieve very flat magnitude characteristics examples are given to illustrate how the proposed and common approaches differ and how the proposed approach can be used to design filters with flat passbands filters which meet point constraints minimum phase filters and bandpass filters with controlled transition band behavior

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

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 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 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

multiplicity free kronecker products of characters of the symmetric groups we provide a classification of multiplicity free inner tensor products of irreducible characters of symmetric groups thus confirming a conjecture of bessenrodt concurrently we classify all multiplicity free inner tensor products of skew characters of the symmetric groups we also provide formulae for calculating the decomposition of these tensor products

on the relativity of relative measures accommodating both rank preservation and rank reversals in the ahp abstract the analytic hierarchy process ahp has been criticized for allowing the introduction of new alternatives to cause rank reversals among existing alternatives while in many cases this is a perfectly valid phenomenon there are also many cases where rank should be preserved to address both contingencies the ahp now supports two types of synthesis procedures one that preserves rank and one that does not this paper proposes guidelines for the selection of one synthesis mode over the other it also proposes several modifications to software implementations of the ahp to better support both synthesis modes and to reduce user stress over the choice

on almost extreme components in kronecker products of characters of the symmetric groups abstract using a recursion formula due to dvir we obtain information on maximal and almost maximal components in kronecker products of characters of the symmetric groups this is applied to confirm a conjecture made by bessenrodt and kleshchev in 1999 which classifies all such kronecker products with only three or four components

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

powers of characters of finite groups letg be a finite group and a complex character ofg define irr to be the set of all irreducible constituents of andirr g to be the set of all irreducible characters ofg thecharacter covering number of a finite groupg ccn g is defined as the smallest positive integer m such thatirr m irr g for all irr g 1g if no such positive integer exists we say that the character covering number ofg is infinite in this article we show that a finite nontrivial groupg has a finite character covering number if and only ifg is simple and non abelian and ifg is a nonabelian simple group thenccn g k2 3k 4 wherek is the number of conjugacy classes ofg then we show using the classification of the finite simple groups that the only finite group with a character covering number equal to two is the smallest janko s group j1 these results are analogous to results obtained previously concerning the covering of groups by powers of conjugacy classes other related results are shown

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

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