extended vikor method in comparison with outranking methods the vikor method was developed to solve mcdm problems with conflicting and noncommensurable different units criteria assuming that compromising is acceptable for conflict resolution the decision maker wants a solution that is the closest to the ideal and the alternatives are evaluated according to all established criteria this method focuses on ranking and selecting from a set of alternatives in the presence of conflicting criteria and on proposing compromise solution one or more the vikor method is extended with a stability analysis determining the weight stability intervals and with trade offs analysis the extended vikor method is compared with three multicriteria decision making methods topsis promethee and electre a numerical example illustrates an application of the vikor method and the results by all four considered methods are compared

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

deriving rankings from incomplete preference information a comparison of different approaches abstract volume based methods for decision making under incomplete information like the smaa family of methods provide rich probabilistic information to support decision making however they usually do not directly generate a unique ranking of alternatives methods to create such a unique ranking from incomplete preference information typically select one parameter vector either by mathematical programming approaches or by averaging and then apply a preference model using this parameter vector in the present paper we develop several models to infer a complete ranking or a complete preorder of alternatives directly from the probabilistic information provided by volume based methods without singling out a specific parameter vector we compare the results obtained by these models to those obtained with a single parameter approach in a computational study results indicate small but significant differences in the performance of methods as well as in the probability that additional preference information might worsen rather than improve the results

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

incorporating the uncertainty of decision judgements in the analytic hierarchy process abstract the uncertainty in the relative weights of a pairwise comparison matrix in the analytic hierarchy process ahp is caused by the uncertainty in our decision judgements and in many cases can not be avoided in this paper it is explicitly shown how such uncertainties can be incorporated within the framework of ahp and how the resulting uncertainties in the relative priorities of the decision alternatives can be computed the required algorithm and the computational procedures are also developed and illustrated with examples uncertainty is introduced as a fundamental concept independent of the concept of consistency with a view to extend the ahp as a decision analysis procedure

preference assessment by imprecise ratio statements the pairs method developed in this paper introduces imprecise preference statements into value trees the assessment of attribute weights in pairs extends the well known smart technique so that in addition to exact statements the decision maker can enter interval judgments which indicate ranges for the relative importance of the attributes the interval judgments and the possibly range valued information about the outcomes of the alternatives are processed with linear programming into value intervals and dominance relations as the decision maker refines the description of his preferences either by entering new statements or by tightening his earlier judgments these results become more detailed and convey more information about which alternatives are preferred throughout the interactive refinement process pairs supports the decision maker by deriving and displaying the consequences of his earlier judgments

establishing dominance between alternatives with incomplete information in a hierarchically structured attribute tree abstract a model for establishing dominance with incomplete information is proposed in a hierarchically structured attribute tree the linear additive value function under certainty is used in the model the model allows the decision maker to provide any type of linear partial information on the attribute weights and the alternative values the incompletely specified information constructs a feasible region of linear constraints and therefore pairwise dominance relationship between alternatives reduces to intractable non linear programming hence to handle this difficulty we propose a solution technique also to handle the tree structure we break down the attribute tree into sub trees due to the recursive structure of the solution technique the optimization results from sub trees can be utilized in computing the value interval on the topmost attribute based on the solution technique an algorithm for obtaining the value interval is proposed and dominance relation is established

modeling subjective evaluation for fuzzy group multicriteria decision making this paper presents a new fuzzy multicriteria decision making mcdm approach for evaluating decision alternatives involving subjective judgements made by a group of decision makers a pairwise comparison process is used to help individual decision makers make comparative judgements and a linguistic rating method is used for making absolute judgements a hierarchical weighting method is developed to assess the weights of a large number of evaluation criteria by pairwise comparisons to reflect the inherent imprecision of subjective judgements individual assessments are aggregated as a group assessment using triangular fuzzy numbers to obtain a cardinal preference value for each decision alternative a new fuzzy mcdm algorithm is developed by extending the concept of the degree of optimality to incorporate criteria weights in the distance measurement an empirical study of aircraft selection is presented to illustrate the effectiveness of the approach

a quadratic time algorithm for the minmax length triangulation it is shown that a triangulation of a set of n points in the plane that minimizes the maximum edge length can be computed in time o n 2 the algorithm is reasonably easy to implement and is based on the theorem that there is a triangulation with minmax edge length that contains the relative neighborhood graph of the points as a subgraph with minor modifications the algorithm works for arbitrary normed metrics

multi attribute decision making a simulation comparison of select methods abstract several methods have been proposed for solving multi attribute decision making problems madm a major criticism of madm is that different techniques may yield different results when applied to the same problem the problem considered in this study consists of a decision matrix input of n criteria weights and ratings of l alternatives on each criterion the comparative performance of some methods has been investigated in a few mostly field studies in this simulation experiment we investigate the performance of eight methods electre topsis multiplicative exponential weighting mew simple additive weighting saw and four versions of ahp original vs geometric scale and right eigenvector vs mean transformation solution simulation parameters are the number of alternatives criteria and their distribution the solutions are analyzed using twelve measures of similarity of performance similarities and differences in the behavior of these methods are investigated dissimilarities in weights produced by these methods become stronger in problems with few alternatives however the corresponding final rankings of the alternatives vary across methods more in problems with many alternatives although less significant the distribution of criterion weights affects the methods differently in general all ahp versions behave similarly and closer to saw than the other methods electre is the least similar to saw except for closer matching the top ranked alternative followed by mew topsis behaves closer to ahp and differently from electre and mew except for problems with few criteria a similar rank reversal experiment produced the following performance order of methods saw and mew best followed by topsis ahps and electre it should be noted that the electre version used was adapted to the common madm problem and therefore it did not take advantage of the method s capabilities in handling problems with ordinal or imprecise information

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

on complete congruence lattices of complete lattices the lattice of all complete congruence relations of a complete lattice is itself a complete lattice in this paper we characterize this lattice as a com plete lattice in other words for a complete lattice l we construct a complete lattice k such that l is isomorphic to the lattice of complete congruence re lations of k regarding k as an infinitary algebra this result strengthens the characterization theorem of congruence lattices of infinitary algebras of g grat zer and w a lampe in addition we show how to construct k so that it will also have a prescribed automorphism group

on the optimal consistent approximation to pairwise comparison matrices abstract consistency retrieval from a biased relative preference table is an imperative task in decision theory this paper considers the least squares approximation of a pairwise comparison matrix by consistent matrices it is observed that the highly nonlinear manifold of consistent matrices can be changed into a linear subspace by the componentwise logarithmic transformation a first order optimality condition therefore can be described in terms of coordinates in the linear subspace this approach facilitates the otherwise much more complicated optimality condition if working with the variables in the original manifold fast nonlinear equation solvers can be employed to solve the problem efficiently

a fuzzy extension of saaty s priority theory we present a fuzzy method for choosing among a number of alternatives under conflicting decision criteria a fuzzy version of saaty s pairwise comparison method 1980 extended by de graan 1980 and lootsma 1981 each ratio expressing the relative significance of a pair of factors is displayed in a matrix from which suitable weights can be extracted since these ratios are essentially fuzzy they express the opinion of a decision maker on the importance of a pair of factors we have adapted the above mentioned method in such a way that decision makers are asked to express their opinions in fuzzy numbers with triangular membership functions we apply the method at two distinct levels first to find fuzzy weights for the decision criteria and second to find fuzzy weights for the alternatives under each of the decision criteria by a suitable combination of these results we obtain fuzzy scores for the alternatives as well as their sensitivities using these fuzzy scores the decision makers should be able to make a choice for one of the alternatives

approximate articulation of preference and priority derivation a comment abstract in a recent paper by a arbel 1989 the standard analytic hierarchy process ahp model is extended to the case where preference strength is expressed in terms of a range of scale values rather than a single value while this extension holds and is natural for the consistent case it is shown that the proposed solution for the more common inconsistent case is erroneous

a new lattice construction the box product in a recent paper the authors have proved that for lattices a and b with zero the isomorphism conc a otimes b cong conc a otimes conc b holds provided that the tensor product satis es a very natural condition of being capped implying that a otimes b is a lattice in general a otimes b is not a lattice for instance we proved that m 3 otimes f 3 is not a lattice in this paper we introduce a new lattice construction the box product for arbitrary lattices the tensor product construction for complete lattices introduced by g n raney in 1960 and by r wille in 1985 and the tensor product construction of a fraser in 1978 for semilattices bear some formal resemblance to the new construction for lattices a and b while their tensor product a otimes b as semilattices is not always a lattice the box product a bp b is always a lattice furthermore the box product and some of its ideals behave like an improved tensor product for example if a and b are lattices with unit then the isomorphism conc a bp b cong conc a otimes conc b holds there are analogous results for lattices a and b with zero and for a bounded lattice a and an arbitrary lattice b a join semilattice s with zero is called 0 representable if there exists a lattice l with zero such that conc l cong s the above isomorphism results yield the following consequence the tensor product of two 0 representable semilattices is 0 representable

an eigenvector method for generating normalized interval and fuzzy weights crisp comparison matrices produce crisp weight estimates it is logical for an interval or fuzzy comparison matrix to give an interval or fuzzy weight estimate in this paper an eigenvector method em is proposed to generate interval or fuzzy weight estimate from an interval or fuzzy comparison matrix which differs from csutora and buckley s lambda max method in several aspects first the proposed em produces a normalized interval or fuzzy eigenvector weight estimate through the solution of a linear programming model while the lambda max method gives a series of nonnormalized interval eigenvector weight estimate through an iterative solution process next the em directly solves the principal right eigenvector of an interval or fuzzy comparison matrix while the lambda max method needs transforming a fuzzy comparison matrix into a series of interval comparison matrices using a level sets and the extension principle and therefore needs the solution of a series of eigenvalue problems finally the lambda max method needs the help of the principal right eigenvector of a crisp comparison matrix to determine the final interval weights while the em has no such requirement it is also found that not all interval or fuzzy comparison matrices can generate normalized interval or fuzzy eigenvector weights situations where the em is inapplicable are analyzed and the aggregation of local interval or fuzzy weights into global interval or fuzzy weights is also discussed three numerical examples including a hierarchical structure decision making problem are examined using the em to demonstrate its applications

rank from comparisons and from ratings in the analytic hierarchy network processes rank preservation and reversal are important subjects in multi criteria decision making particularly if a theory uses only one of two ways of creating priorities rating alternatives one at a time with respect to an ideal or standard or comparing them in pairs it is known that our minds can do both naturally and without being tutored when rating alternatives they must be assumed to be independent and rank should be preserved when comparing alternatives they must be assumed to be dependent and rank may not always be preserved however even in making comparisons rank can be preserved if one uses idealization instead of normalization with the original set of alternatives and preserves that ideal from then on unless that ideal itself is deleted for some reason so often it is a matter of judgment as to whether it is desirable to force rank preservation or allow rank to adjust as necessary examples are given to illustrate the foregoing ideas

a goal programming method for obtaining interval weights from an interval comparison matrix crisp comparison matrices lead to crisp weight vectors being generated accordingly an interval comparison matrix should give an interval weight estimate in this paper a goal programming gp method is proposed to obtain interval weights from an interval comparison matrix which can be either consistent or inconsistent the interval weights are assumed to be normalized and can be derived from a gp model at a time the proposed gp method is also applicable to crisp comparison matrices comparisons with an interval regression analysis method are also made three numerical examples including a multiple criteria decision making mcdm problem with a hierarchical structure are examined to show the potential applications of the proposed gp method

implementation of the centroid method of solymosi and dombi multiple objective decision support either for selection among finite alternatives or for combining multiple functions in mathematical programming often involves estimating a set of weights of relative importance recently solymosi and dombi proposed a means of estimating such weights based upon ordinal preference information this method relies upon the centroid of bounded weights analytic hierarchy process uses ratio estimates of pairwise comparisons using only ordinal information the centroid of weights is developed as an alternative means of weighting hierarchical elements these weights could be used in selection among alternatives directly in concordance analysis or in multiple objective linear programming formulation for the centroid of feasible weight values is presented with a table of values for cases where the preference ranking includes no ties comparison of the technique on a student job selection decision indicates slight but insignificant decline in accuracy relative to the analytic hierarchy based method

interval articulation of superiority and precise elicitation of priorities this paper develops an approach to obtaining precise point estimates of priorities of objects in particular coefficients of importance of criteria it is assumed that the judgements regarding the preference superiority of one object over another are given in the forms of intervals that is by indicating the lower and upper bounds the suggested approach is suitable both in the case of consistent and inconsistent preferences in the former case the available intervals are processed by contracting them to the maximum degree in the latter by extending them to the minimum degree the degrees of contraction or extension are regarded as the indexes of indeterminacy or inconsistency of preferences and treated as multiple objective functions a method of transformation of such a multicriterial problem to a sequence of several linear programming problems is suggested

a method for determining the weights of criteria the centralized weights abstract in multicriteria analysis many methods use weights to represent the relative importance of criteria the noncompensatory aggregation procedure used in outranking methods enables comparisons of subsets of criteria from this point of view an interactive method is presented which requests only ordinal comparisons from the decision maker the relation more important than is assumed to be a semiorder therefore the indifference part of this relation is not necessarily transitive the judgements can be formulated as linear inequalities which constrain the set of feasible weights and threshold values all vertices of this polyhedron are determined the threshold and the unknown real weights are estimated by the centroid some characteristics a possible refinement in the estimation and an illustrative example are also given

a multi criteria agency model with incomplete preference information abstract we analyze an agency model in which the principal has only partial information on the agent s preferences consequently the principal can predict the agent s reaction to an incentive system only stochastically we first present the analytical solution for a simple decision problem in which the agent can choose only between two alternatives from this simple model we derive several hypotheses on how the preference information available to the principal influences her net results these hypotheses are then tested via a simulation model for more complex decision problems and we show that not all results of the simple model hold for the more general case most notably it is not generally the case that more precise information on the agent s preferences improves the principal s situation

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

evidential reasoning approach for multiattribute decision analysis under both fuzzy and interval uncertainty many multiple attribute decision analysis mada problems are characterized by both quantitative and qualitative attributes with various types of uncertainties incompleteness or ignorance and vagueness or fuzziness are among the most common uncertainties in decision analysis the evidential reasoning er and the interval grade er ier approaches have been developed in recent years to support the solution of mada problems with interval uncertainties and local ignorance in decision analysis in this paper the er approach is enhanced to deal with both interval uncertainty and fuzzy beliefs in assessing alternatives on an attribute in this newly developed fuzzy ier fier approach local ignorance and grade fuzziness are modeled under the integrated framework of a distributed fuzzy belief structure leading to a fuzzy belief decision matrix a numerical example is provided to illustrate the detailed implementation process of the fier approach and its validity and applicability

a new fuzzy multiple attributive group decision making methodology and its application to propulsion manoeuvring system selection problem in this paper a new fuzzy multiple attribute decision making fmadm method which is suitable for multiple attributive group decision making gdm problems in fuzzy environment is proposed to deal with the problem of ranking and selection of alternatives since the subjectivity imprecision and vagueness in the estimates of a performance rating enter into multiple attribute decision making madm problems fuzzy set theory provides a mathematical framework for modelling vagueness and imprecision in the proposed approach an attribute based aggregation technique for heterogeneous group of experts is employed and used for dealing with fuzzy opinion aggregation for the subjective attributes of the decision problem the propulsion manoeuvring system selection as a real case study is used to demonstrate the versatility and potential of the proposed method for solving fuzzy multiple attributive group decision making problems the proposed method is a generalised model which can be applied to great variety of practical problems encountered in the naval architecture from propulsion manoeuvring system selection to warship requirements definition

preference programming through approximate ratio comparisons abstract in the context of hierarchical weighting this paper operationalizes interval judgments which allow the decision maker to enter ambiguous preference statements by indicating the relative importance of factors as intervals of values on a ratio scale through such judgments the decision maker can capture the subjective uncertainty in his preferences and thus avoid the often cumbersome elicitation of exact ratio estimates after each new statement the interval judgments are synthesized into dominance relations on the alternatives by solving a series of linear programming problems this leads to an interactive process of preference programming which provides more detailed results as the decision maker gradually enters a more specific preference description moreover the overall effort of preference elicitation is smaller than in the analytic hierarchy process because the most preferred alternative can usually be identified before all possible comparisons between pairs of factors have been completed

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

the evidential reasoning approach for multi attribute decision analysis under interval uncertainty abstract the evidential reasoning er approach is a method for multiple attribute decision analysis mada under uncertainties it improves the insightfulness and rationality of a decision making process by using a belief decision matrix bdm for problem modelling and the dempster shafer d s theory of evidence for attribute aggregation the d s theory provides scope and flexibility to deal with interval uncertainties or local ignorance in decision analysis which is not explored in the original er approach and will be investigated in this paper firstly interval uncertainty will be defined and modelled in the er framework then an extended er algorithm ier is derived which enables the er approach to deal with interval uncertainty in assessing alternatives on an attribute it is proved that the original er algorithm is a special case of the ier algorithm the latter is demonstrated using numerical examples

complete congruence lattices of complete distributive lattices abstract in 1983 wille raised the question is every complete lattice l isomorphic to the lattice of complete congruence relations of a suitable complete lattice k in 1988 this was answered in the affirmative by the first author a number of papers have been published on this problem by freese johnson lakser teo and the authors in the present paper we prove that k can always be chosen as a complete distributive lattice in fact we prove the following more general result theorem let m be a regular cardinal 0 every m algebraic lattice l can be represented as the lattice of m complete congruence relations of an m complete distributive lattice k

approximating uniform triangular meshes in polygons we consider the problem of triangulating a convex polygon using n steiner points under the following optimality criteria 1 minimizing the overall edge length ratio 2 minimizing the maximum edge length and 3 minimizing the maximum triangle perimeter we establish a relation of these problems to a certain extreme packing problem based on this relationship we develop a heuristic producing constant approximations for all the optimality criteria above provided n is chosen sufficiently large that is the produced triangular mesh is uniform in these respects the method is easy to implement and runs in o n2log n time and o n space the observed runtime is much less moreover for criterion 1 the method works within the same complexity and approximation bounds for arbitrary polygons with possible holes and for criteria 2 and 3 it does so for a large subclass

why triangular membership functions abstract in this note we look at a certain theoretically sound motivation behind the common use of triangular and trapezoidal membership functions the studies are completed within a conceptual framework of fuzzy modelling whose structure comprises of input and output interfaces linked with a single transformation module aimed at processing linguistic information it is shown that under some additional mild assumptions these triangular fuzzy sets comply with a request for a uniformly excited codebook in the case of the input interfaces and a satisfaction of a zero error reconstruction criterion being formulated for the output interface

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

a two stage logarithmic goal programming method for generating weights from interval comparison matrices a two stage logarithmic goal programming tlgp method is proposed to generate weights from interval comparison matrices which can be either consistent or inconsistent the first stage is devised to minimize the inconsistency of interval comparison matrices and the second stage is developed to generate priorities under the condition of minimal inconsistency the weights are assumed to be multiplicative rather than additive in the case of hierarchical structures a nonlinear programming method is used to aggregate local interval weights into global interval weights a simple yet practical preference ranking method is investigated to compare the interval weights of criteria or rank alternatives in a multiplicative aggregation process the proposed tlgp is also applicable to fuzzy comparison matrices when they are transformed into interval comparison matrices using a level sets and the extension principle six numerical examples including a group decision analysis problem with a group of comparison matrices a hierarchical decision problem and a fuzzy decision problem using fuzzy comparison matrix are examined to show the applications of the proposed methods comparisons with other existing procedures are made whenever possible

a quasi polynomial time approximation scheme for minimum weight triangulation the minimum weight triangulation problem is to find a triangulation t of minimum length for a given set of points p in the euclidean plane it was one of the few longstanding open problems from the famous list of twelve problems with unknown complexity status published by garey and johnson 1979 very recently the problem was shown to be np hard by mulzer and rote 2006 in this article we present a quasi polynomial time approximation scheme for minimum weight triangulation

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

a method for dealing with inconsistencies in pairwise comparisons abstract the pairwise comparison method is a powerful inference tool for assessing the relative importance of a set of items formally its objective is to make compatible decision maker s assignments paired comparison with properties needed for obtaining an overall rank in this paper we propose a distance based framework for analysing this kind of compatibility in this context goal programming is proposed as an attractive and flexible tool

compromise solution by mcdm methods a comparative analysis of vikor and topsis the multiple criteria decision making mcdm methods vikor and topsis are based on an aggregating function representing closeness to the ideal which originated in the compromise programming method in vikor linear normalization and in topsis vector normalization is used to eliminate the units of criterion functions the vikor method of compromise ranking determines a compromise solution providing a maximum group utility for the majority and a minimum of an individual regret for the opponent the topsis method determines a solution with the shortest distance to the ideal solution and the greatest distance from the negative ideal solution but it does not consider the relative importance of these distances a comparative analysis of these two methods is illustrated with a numerical example showing their similarity and some differences 2003 elsevier b v all rights reserved

multiple criteria robust interactive decision analysis mcrid for optimizing public policies abstract important public policy decisions are often very complex involving many factors large uncertainties and multiple conflicting objectives decision analysis which is a formal framework for dealing with such issues has been applied to a variety of public sector problems over the last two decades a limitation of using decision analysis is the often onerous assessment burden it places on the decision maker dm we have developed a novel interactive procedure and decision support system dss for performing decision analysis on the personal computer called robust interactive decision analysis rid which avoids the difficult problems of precisely assessing a dm s utility and state probability functions associated with traditional decision treee analysis the rid method permits a dm to voluntarily and interactively express strong viz sure binary preferences for actions partial decision functions and full decision functions and only imprecise probability and utility function assessments the inputs are used with various dominance operators via a mathematical programming framework to prune the state probability utility and decision space until there is convergence on a relatively small subset of efficient strategies or an optimal choice strategy conceptually the operation of the rid method can be regarded as a state pruning system and its computer implementation as a dss in this paper we extended the rid concept and dss to deal with multiple criteria decisions obviating the need to precisely and completely assess attribute weights tradeoffs consequences and or value utility functions the approach which we call multiple criteria robust interactive decision analysis mcrid is developed illustrated and shown to be effective in eliminating many inefficient alternatives however since the aggregate utility of each alternative is interval valued it then becomes difficult to intuitively choose an optimal alternative from among the remaining efficient ones since the intervals are not defined by any specific probability distributions we have tentatively shown that aggregate utility intervals ultimately converge toward a normal density function and can be so approximated with relatively few attributes hence stochastic dominance is employed to further filter out alternatives an interactive goal programming gp technique is then developed to assist the dm in choosing an optimal alternative

multiple criteria decision making multiattribute utility theory the next ten years management science and decision science have grown exponentially since midcentury two closely related fields central to this growth are multiple criteria decision making mcdm and multiattribute utility theory maut this paper comments on the history of mcdm and maut and discusses topics we believe are important in their continued development and usefulness to management science over the next decade our aim is to identify exciting directions and promising areas for future research

fuzzy mcdm based on ideal and anti ideal concepts this paper presents a novel fuzzy multiple criteria decision making mcdm based on the concepts of ideal and anti ideal points the concepts of fuzzy set theory and hierarchical structure analysis are used to develop a weighted suitability decision matrix to evaluate the weighted suitability of different alternatives versus various criteria the distance of different alternatives versus positive ideal solution and negative ideal solution are then obtained by using the proposed ranking method finally the relative approximation values of various alternatives versus positive ideal solution are ranked to determine the best alternative

computing a consistent approximation to a generalized pairwise comparisons matrix abstract this paper presents an algorithm for computing a consistent approximation to a generalized pairwise comparisons matrix that is without the reciprocity property or even 1s on the main diagonal the algorithm is based on a logarithmic transformation of the generalized pairwise comparisons matrix into a linear space with the euclidean metric it uses both the row and reciprocals of column geometric means and is thus a generalization of the ordinary geometric means method the resulting approximation is not only consistent but also closest to the original matrix i e deviates least from an expert s original judgments the computational complexity of the algorithm is o n 2

linking validation a search for coherency within the supermatrix abstract the consistency check within each pairwise comparison matrix is an important step in an analytic network process anp decision in an anp network there is both the ability and the need to test for additional levels of consistency or coherency among the priority vectors examples are used to highlight cases where a supermatrix with priority vectors that were obtained from either perfect or nearly perfect consistent pairwise comparison matrices generates suboptimal decisions simulations are used to further demonstrate the frequency of these occurrences in general anp networks a form of cross validation within the supermatrix called linking validation is developed and demonstrated the linking validation method allows decision makers to use the priority vectors within the supermatrix to validate other priority vectors within the supermatrix the linking validation method involves generating linking estimates the linking estimates are compared against each other to identify the most incoherent priority vector by calculating the linking coherency index lci scores the decision maker can then update the specified priority vector and repeat this process until the lci score for every linking estimate is below the given threshold the use of linking validation to test for coherency further improves the validity of anp models

partial information dominance and potential optimality in multiattribute utility theory when a multiattribute utility function is only partially specified by prior preference statements what can be said about the relative desirability of actual alternatives this question is addressed for the cases of additively separable cardinal utility with unknown scaling constants multiplicatively separable cardinal utility with unknown scaling constants and additively separable ordinal utility we review previous approaches and treat issues of consistency is the prior information consistent dominance does the prior preference information imply that one outcome is preferred to another and potential optimality are there utility functions of the given form consistent with prior preference information under which a particular outcome is preference optimal in the additive cases a key relationship between dominance and potential optimality may be derived the paper concludes by presenting an example application to a well known nuclear siting study

deriving priorities from fuzzy pairwise comparison judgements a new approach for deriving priorities from fuzzy pairwise comparison judgements is proposed based on cuts decomposition of the fuzzy judgements into a series of interval comparisons the assessment of the priorities from the pairwise comparison intervals is formulated as an optimisation problem maximising the decision maker s satisfaction with a specific crisp priority vector a fuzzy preference programming method which transforms the interval prioritisation task into a fuzzy linear programming problem is applied to derive optimal crisp priorities aggregating the optimal priorities which correspond to different cut levels enables overall crisp scores of the prioritisation elements to be obtained a modification of the linear fuzzy preference programming method is also proposed to derive priorities directly from fuzzy judgements without applying cut transformations the formulation of the prioritisation problem as an optimisation task is similar to the previous approach but it requires the solution of a non linear optimisation program the second approach also derives crisp priorities and has the advantage that it does not need additional aggregation and ranking procedures both proposed methods are illustrated by numerical examples and compared to some of the existing fuzzy prioritisation methods

ranking fuzzy numbers with maximizing set and minimizing set abstract up to now these are five methods of ranking n fuzzy numbers in order but these methods contain some confusions and occasionally conflict with intuition this paper introduces the concept of maximizing set and minimizing set to decide the ordering value of each fuzzy number and uses these values to determine the order of the n fuzzy numbers in addition we give a method for calculating the ordering value of each fuzzy number with triangular trapezoidal and two sided drum like shaped membership 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

effects of imprecise weighting in hierarchical preference programming abstract preference programming is a general term for multi criteria decision analytical approaches allowing incomplete preference information in the pairs method interval judgments are assigned to weight ratios between attributes to model imprecision in multi attribute value trees this paper studies the effects of a hierarchical model structure on the overall imprecision as the form of the hierarchy also affects the form of imprecision that can be assigned to the model the aim is to find out good procedural practices for reducing overall imprecision descending inherently from the model structure the study provides simulation results about the ability of various weighting schemes to identify dominated alternatives which are discussed with respect to other issues related to the weighting process according to the results a hierarchical model is structurally somewhat more unable to identify dominances than a corresponding nonhierarchical model but its cognitive advantages often cancel out this the results also suggest paying reasonable attention to the precision of the lower level judgments and to identifying possible correlations between the criteria

the possible values of critical points between varieties of lattices abstract we denote by con c l the 0 semilattice of all finitely generated congruences of a lattice l for varieties i e equational classes v and w of lattices such that v is contained neither in w nor in its dual and such that every simple member of w contains a prime interval we prove that there exists a bounded lattice a v with at most 2 elements such that con c a is not isomorphic to con c b for any b w the bound 2 is optimal as a corollary of our results there are continuum many congruence classes of locally finite varieties of bounded modular lattices

representing congruence lattices of lattices with partial unary operations as congruence lattices of lattices i interval equivalence abstract in part i of this paper we introduced a method of making two isomorphic intervals of a bounded lattice congruence equivalent in this paper we make one interval dominate another one let l be a bounded lattice let a b and c d be intervals of l and let be a homomorphism of a b onto c d we construct a bounded convex extension k of l such that a congruence of l has an extension to k iff x y implies that x y for a x y b in which case has a unique extension to k this result presents a lattice k whose congruence lattice is derived from the congruence lattice of l in a new way different from the one presented in part i the main technical innovation is the 2 3 boolean triple construction which owes its origin to the boolean triple construction of g gratzer and f wehrung