improving linguistic pairwise comparison consistency via linguistic discrete regions linguistic pairwise comparison matrices are widely used in decision making procedures however the matrices often give conflicting results when there are multiple criteria under consideration despite intensive research achieving consistency of such matrices remains a daunting task in this paper a novel approach based on linguistic discrete region is proposed to address the challenge unlike existing methods that require a single value for each comparison our approach allows a comparison to be expressed by a discrete region with multiple linguistic terms such front end gives users more freedom to express their opinions in the back end we propose an iterative searching algorithm that is able to achieve approximate optimal consistency for the comparison matrices with discrete region values the final results are single value matrices that not only guarantee approximate optimal consistency but comply with evaluators intentions a well as our approach does not modify any linguistic values like many existing methods we have conducted extensive evaluations and our empirical study confirms that the linguistic discrete region based approach significantly improves the consistency of linguistic pairwise comparison matrices

lp metric sensitivity analysis for single and multi attribute decision analysis abstract analyzing the sensitivity of decisions to probability estimation error in single and multi attribute problems and to errors in estimating additive multi attribute value models in multi attribute problems is an integral part of decision analysis this paper presents an intuitive and tractable approach to this sensitivity analysis here a decision is considered insensitive if 1 the probabilities or multi attribute weights required for any other alternative to become preferred are not close to the original estimated probabilities and weights and 2 the rank order of states implied by the probabilities or the rank order of attributes implied by the additive multi attribute weights must change for any other alternative to become preferred the sensitivity analysis is conducted using straight forward linear programming models an example is used to demonstrate their application

a cosine maximization method for the priority vector derivation in ahp abstract the derivation of a priority vector from a pair wise comparison matrix pcm is an important issue in the analytic hierarchy process ahp the existing methods for the priority vector derivation from pcm include eigenvector method ev weighted least squares method wls additive normalization method an logarithmic least squares method lls etc the derived priority vector should be as similar to each column vector of the pcm as possible if a pair wise comparison matrix pcm is not perfectly consistent therefore a cosine maximization method cm based on similarity measure is proposed which maximizes the sum of the cosine of the angle between the priority vector and each column vector of a pcm an optimization model for the cm is proposed to derive the reliable priority vector using three numerical examples the cm is compared with the other prioritization methods based on two performance evaluation criteria euclidean distance and minimum violation the results show that the cm is flexible and efficient

the mathematical equivalence of the spanning tree and row geometric mean preference vectors and its implications for preference analysis abstract pairwise comparison is a widely used approach to elicit comparative judgements from a decision maker dm and there are a number of methods that can be used to then subsequently derive a consistent preference vector from the dm s judgements while the most widely used method is the eigenvector method the row geometric mean approach has gained popularity due to its mathematical properties and its ease of implementation in this paper we discuss a spanning tree method and prove the mathematical equivalence of its preference vector to that of the row geometric mean approach this is an important finding due to the fact that it identifies an approach for generating a preference vector which has the mathematical properties of the row geometric mean preference vector and yet in its entirety the spanning tree method has more to offer than the row geometric mean method in that it is inherently applicable to incomplete sets of pairwise comparison judgements and also facilitates the use of statistical and visual techniques to gain insights into inconsistency in the dm s judgements

lemma for linear feedback shift registers and dfts applied to affine variety codes in this paper we establish a lemma in algebraic coding theory that frequently appears in the encoding and decoding of e g reed solomon codes algebraic geometry codes and affine variety codes our lemma corresponds to the nonsystematic encoding of affine variety codes and can be stated by giving a canonical linear map as the composition of an extension through linear feedback shift registers from a grobner basis and a generalized inverse discrete fourier transform we clarify that our lemma yields the error value estimation in the fast erasure and error decoding of a class of dual affine variety codes moreover we show that systematic encoding corresponds to a special case of erasure only decoding the lemma enables us to reduce the computational complexity of error evaluation from o n3 using gaussian elimination to o qn2 with some mild conditions on n and q where n is the code length and q is the finite field size

stochastic group preference modelling in the multiplicative ahp a model of group consensus this paper proposes a model for the representation of a consensus seeking group decision making process under the multiplicative variant of the analytic hierarchy process ahp in particular it proposes a group preference model which expresses the group s preference intensity judgements as random variables with associated probability distributions and the structure of the multiplicative ahp is exploited to derive interval judgements for the alternatives final impact scores as perceived by the group these interval judgements can be used to determine the probability that a rank reversal may occur amongst alternatives due to the group preference uncertainty i e to assess the stability of the final impact score vector which in turn offers information about the degree of consensus within the group about the decision

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

assessing priority weights from subsets of pairwise comparisons in multiple criteria optimization problems using a set covering concept we explore the possibility and flexibility of selecting a subset of all possible pairs of comparisons to assess priority weights of the decision criteria different techniques offer different priority weights with different properties they are however interrelated when special subsets such as basic coverings elementary cycles and complete covering are considered this paper provides a framework for unifying existing assessment methods and offers some fundamental theorems which reveal the intricate relationships among the methods over the subsets

deriving priorities from pairwise comparison matrices with a novel consistency index abstract it is important to measure the inconsistency level of a pairwise comparison matrix pcm and derive the priority vector in the analytic hierarchy process ahp in the present study a new consistency index is proposed by using the cosine similarity measures of two row column vectors in a pcm it is called as the double cosine similarity consistency index dcsci since the row and column vectors are all considered some interesting properties of dcsci are investigated and the thresholds for inconsistency tolerance level are discussed in detail then following the idea of dcsci we provide a new method for obtaining the priority vector from a pcm through maximizing the sum of the cosine similarity measures between the priority vector and the row column vectors the priority vector is derived by solving the constructed optimization problem by analyzing the proposed double cosine similarity maximization dcsm method it is found that the existing cosine maximization cm method can be retrieved by carrying out numerical examples some comparisons with the existing methods show that the proposed index and method are effective and flexible

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

the perron frobenius operators invariant measures and representations of the cuntz krieger algebras for a transformation f on a measure space x we show that the perron frobenius operator of f can be written by a representation l2 x of the cuntz krieger algebra oa associated with f when f satisfies some assumption especially when oa is the cuntz algebra on and l2 x in the above is some irreducible representation of on then there is an f invariant measure on x which is absolutely continuous with respect to

novel types of sensitivity analysis for additive mcdm methods three novel types of sensitivity analysis are presented they determine the following 1 the sensitivity of a ranking to specific changes in the evaluations of all alternatives on certain criteria 2 the influence of specific changes in certain criterion scores of an alternative 3 the minimum modification of the weights required to make an alternative ranked first the use of these instruments is demonstrated by the results of a simulation experiment results show that the first and second type of sensitivity analysis enable to apply multiple criteria decision making mcdm methods in dynamic circumstances the third type of sensitivity analysis is demonstrated to be a tool to analyze the total weight space

a geometric averaging procedure for constructing supertransitive approximation to binary comparison matrices abstract the usefulness of encoding the fuzzy evaluations of alternatives and the importance weights of criteria in a multiple objective decision problem through binary comparison matrices or pairwise judgment matrices is receiving considerable attention the methodology for identifying the best alternative in a given decision problem involves the computation of the principal eigenvectors of the binary comparison matrices the eigenvectors transform the fuzzy evaluations of the importance of the criteria and the ratings of the alternatives into a ratio scale a difficulty that is often experienced in using this approach in practice is the inconsistency of the binary evaluations this paper proposes a simple averaging procedure to construct a supertransitive approximation to a binary comparison matrix where inconsistency is a problem it is further suggested that such an adjustment might be necessary to more closely reflect the inherent fuzziness of the evaluations contained in a binary comparison matrix the procedure is illustrated by means of examples

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

logarithmic least squares prioritization and completion methods for interval fuzzy preference relations based on geometric transitivity abstract this article introduces the notion of geometric transitivity which may be used to define consistent interval fuzzy preference relations some useful properties are presented for consistent interval fuzzy preference relations a close investigation reveals that existing methods for determining the analogous property of consistency are not robust to permutations of the pairwise judgments and not able to reflect the hesitancy in the decision maker s preference a parametric transformation formula is put forward to convert a normalized interval weight vector into a consistent interval fuzzy preference relation by minimizing the squared difference between the logarithm of the ratio of the original judgment and the logarithm of the ratio of the converted consistent one a logarithmic least squares model is developed to derive interval weights from any interval fuzzy preference relation based on geometric transitivity a logarithmic least squares model is established to rectify inconsistency for complete and inconsistent interval fuzzy preference relations and a logarithmic least squares completion approach is further developed to estimate missing values for incomplete interval fuzzy preference relations numerical examples are furnished to show the validity of the proposed models and compare with other existing methods

consistency and consensus improving methods for pairwise comparison matrices based on abelian linearly ordered group the aim of a valued pairwise comparison matrix is to derive the priority structure over a set of criteria or alternatives in decision making the consistency and consensus of a pairwise comparison matrix should be measured and improved to avoid a misleading priority structure the basic entries of a pairwise comparison matrix can be described in different forms hence different consistency and consensus methods should be developed for different types of matrices to provide a general framework the pairwise comparison matrix based on abelian linearly ordered group is first introduced a consistency index is defined by constructing the nearest consistent pairwise comparison matrix from an inconsistent one and two consistency improving methods are introduced a group pairwise comparison matrix is derived a consensus index of individual pairwise comparison matrices is defined and two consensus improving methods are developed by introducing a general aggregation operator based on abelian linearly ordered group the proposed consistency and consensus methods are convergent and can provide a general framework for existing methods

δ sequences and evaluation codes defined by plane valuations at infinity we introduce the concept of sequence a sequence generates a well ordered semigroup s in z or r we show how to construct and compute parameters for the dual code of any evaluation code associated with a weight function defined by from the polynomial ring in two indeterminates to a semigroup s as above we prove that this is a simple procedure which can be understood by considering a particular class of valuations of function fields of surfaces called plane valuations at infinity we also give algorithms to construct an unlimited number of sequences of the different existing types and so this paper provides the tools to know and use a new large set of codes

the probability weighting function a probability weighting finction w p is a prominent feature of several non expected utility theories including prospect theoiy and rank dependent models empirical esti mates indicate that w p is regressive first w p p then iv p p s shaped first concave then convex and asymmetrical intersecting the diagonal at about 1 3 the paper states axioms for several w p forms including the compound invariant w p exp ln p 1 0 a 1 which is regressive s shaped and with an invariant fixed point and inflection point at i e 37 the common ratio effect allais 1953 refers to the obselvation that the more risky of two simple prospects becomes relatively more attractive when the probability of winning is reduced by equal proportion in both prospects thus a person who prefers a sure gain of 100 000 over a coin toss for 300 000 or nothing might also prefer a one in a million lottery ticket for 300 000 over a two in a million lottery ticket for 100 000 this contradicts expected utility but not common sense there is a world of difference between certainty and a 50 50 shot the difference between one or two chances in a million is negligible the example has all the force and simplicity of classical arguments for nonlinear utility where the utility interval from say 10 and 20 is taken as self evidently greater than the utility interval from 1 000 010 and 1 000 020 both arguments appeal directly to our intuitions about numbers a major point of difference is that the money argument involves a constant money intelval and the probability argument a constant probability ratio a minor difference is that in the money domain a constant interval has less impact as the numbers get larger while in the probability domain a constant ratio has less impact as the numbers get smaller putting the two demonstrations side by side like this raises an interesting question namely whether an account of nonlinearity in probabilities can be somehow adapted from classical utility theory modulo a log transformation converting statements about probability ratios into statements about utility intervals indeed in their seminal paper on prospect theory kahneman and tversky 1979 explained the common ratio effect by means of a nonlinear transformation of probabilities into decision weights p w p with log w

algebraic geometry codes the theory of error correcting codes derived from curves in an algebraic geometry was initiated by the work of goppa as generalizations of bose chaudhuri hocquenghem bch reed solomon rs and goppa codes the development of the theory has received intense consideration since that time and the purpose of the paper is to review this work elements of the theory of algebraic curves at a level sufficient to understand the code constructions and decoding algorithms are introduced code constructions from particular classes of curves including the klein quartic elliptic and hyperelliptic curves and hermitian curves are presented decoding algorithms for these classes of codes and others are considered the construction of classes of asymptotically good codes using modular curves is also discussed

unitary isomorphism of fock spaces of bosons and fermions arising from a representation of the cuntz algebra o2 bosons and fermions are described by using canonical generators of cuntz algebras on any permutative representation according to branching laws associated with these descriptions a certain representation of the cuntz algebra o2 induces fock representations hb and hf of bosons and fermions simultaneously from this a unitary operator u from hb to hf is obtained we show the explicit formula of the action of u on the standard basis of hb it is shown that u preserves the particle number of hb and hf

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

sensitivity analysis of additive multiattribute value models we present two simple computational procedures for sensitivity analysis of additive multiattribute value models that yield variations in attribute weights scaling constants for the first or entropy based procedure the point of departure is the concept of equal weights for all attributes for the second or least squares procedure the point of departure is a set of arbitrary weights for attributes in each procedure we either a calculate the closest set of weights that equates the multiattribute value for a pair of alternatives that alternative whose overall value is optimal for the original attribute weights and any other specific nondominated alternative or b calculate the closest set of weights required for the specific alternative to exceed the optimal alternative by a specified amount

on the transitivity and consistency approximated thresholds of some consistency indices for pairwise comparison matrices abstract several measures have been proposed in the literature to verify the rationality and consistency of judgements expressed by means of pairwise comparisons however some of these indices lack a meaningful interpretation because of the absence of thresholds associated with them in this paper we first introduce the consistency approximated thresholds for the salo hamalainen index by deriving an inconsistency tolerance level which is analogous to that proposed for the consistency ratio by saaty for the geometric consistency index by aguaron moreno jimenez and for the consistency measure by koczkodaj then via a simulation we propose a transitivity threshold for n n matrices which may give meaningful information about the degree of misclassification and the reliability of preferences while avoiding the need to revise the judgements on varying n we derive the transitivity thresholds for salo hamalainen and for the other consistency indices proposed in literature

computing the feng rao distances for codes from order domains abstract we compute the feng rao distance of a code coming from an order domain with a simplicial value semigroup the main tool is the apery set of a semigroup that can be computed using a grobner basis

inconsistency in the ordinal pairwise comparisons method with and without ties abstract comparing alternatives in pairs is a well known method used to create ranking experts are asked to perform a series of binary comparisons and then using mathematical methods the final ranking is constructed experts conduct a series of single assessments however they may not always be consistent the level of inconsistency among individual assessments is widely accepted as a measure of the ranking quality the higher the ranking quality the higher its credibility one of the earliest and most widespread inconsistency indices is the consistency coefficient defined by kendall and babington smith in their work the authors consider binary pairwise comparisons i e those where the result of an individual comparison can only be better or worse in the presented work the maximal number of inconsistent triads in the set of ordinal pairwise comparisons with ties of arbitrary size is determined formula 14 this in turn opens the possibility of effectively extending the kendall and babington smith index to pairwise comparisons where the result of an individual comparison can be better worse or equal hence this effectively extends the use of this index to the analytic hierarchy process and other quantitative methods based on comparing alternatives in pairs the work also introduces the notions of a generalized tournament and a double tournament as graphs that model ordinal pairwise comparisons with ties and the maximally inconsistent set of pairwise comparisons with ties respectively the relationship between the most inconsistent set of pairwise comparisons with ties and the set cover problem is also shown

universal fermionization of bosons on permutative representations of the cuntz algebra o2 bosons and fermions are described by using canonical generators of cuntz algebras on any permutative representation we show a fermionization of bosons which universally holds on any permutative representation of the cuntz algebra o2 as examples we show fermionizations on the fock space and the infinite wedge

relative measurement and the power function abstract relative measurement is considered in the context of nomology a theory that defines and relates categories of activity that arise in qualitative decision making a nomological variable is defined to correspond to the amount weight or importance of any naturally forming cluster or category of qualitative information this article shows that in general the mind measures nomological variables relatively and changes in a nomological variable relative to its current value the way the mind assesses the relationship between two related nomological variables generally should be modelled as a power function and it is appropriate to use a geometric mean to average multiple observations empirical evidence from psychophysics is shown to support these results when synthesising direct and indirect relative comparisons of nomological variables it is proven that the indirect comparisons should be given a weight that is inversely proportional to the number of relative measurements made in the indirect comparison this result is shown to be consistent with but not identical with the use of the row geometric means approach for synthesising pairwise relative comparison matrices such as in the analytical hierarchy process in the case of more than three variables other terms should be included in the synthesis the article reports on empirical research that endorses the use of nomological structuring and relative weighting of criteria trees it shows that decision makers generally prefer scoring within intervals when comparing alternatives but can also use relative measurement where there are difficulties with identifying intervals the former leads to modelling by an additive value function the latter to a power function

acceptability measurement and priority weight elicitation of triangular fuzzy multiplicative preference relations based on geometric consistency and uncertainty indices this paper investigates consistency and uncertainty measurements as well as acceptability checking for triangular fuzzy multiplicative preference relations tfmprs a geometric consistency index is proposed to measure the consistency degree of a tfmpr and thresholds of acceptable consistency are discussed for tfmprs uncertainty indices are introduced to measure indeterminacy degrees of triangular fuzzy judgments and of tfmprs and a notion of acceptable tfmprs are put forward by considering both acceptable consistency and acceptable uncertainty in tfmprs we also explore the elicitation of triangular fuzzy weights from acceptable tfmprs and the aggregation of local triangular fuzzy weights for solving hierarchical multi criteria decision making problems a geometric mean based equation and a logarithmic least square model are developed to elicit normalized acceptable triangular fuzzy multiplicative weights from an acceptable tfmpr an equation and a linear program are established to transform triangular fuzzy multiplicative weights into triangular fuzzy additive weights a geometric weighting based equation and a pair of linear programs are devised to generate global priority fuzzy weights from local triangular fuzzy multiplicative weights three numerical examples including a hierarchical multi criteria decision making problem are used to illustrate how to apply the developed framework and comparative studies are provided to show its availability and advantages

jump transformations and an embedding of o into o2 a measurable map f on a measure space induces a representation f of a cuntz algebra on when f satisfies a certain condition for such two maps f and g and representations f and g associated with them we show that g is the restriction of f when g is a jump transformation of f especially the gauss map 1 and the farey map 1 induce representations 1 of o and that 1 of o2 respectively and 1 1 o with respect to a certain embedding of o into o2

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

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

an intrinsic consistency threshold for reciprocal matrices abstract in this paper we present a new intrinsic and scale independent consistency threshold for reciprocal matrices in the analytic hierarchy process ahp its derivation is based upon the relation between the consistency measure of the ahp and the reduction of weight of a new alternative we compare this standard with an existing one

multicriteria procedural rationality on sisdema abstract this paper considers the application under the multicriteria procedural rationality paradigm mprp of two environmental selection tools included in the priorization module prior of the sisdema decisional system preference structures and stability intervals these tools apart from being used in analysing the robustness of the model allow us to obtain some critical points of the environmental decision making process dmp which will be employed in the negotiation and consensus building process between the actors involved in the resolution of the problem

reciprocal distributions in the analytic hierarchy process abstract the paper analyses a special kind of probability distributions used to capture the existing uncertainty for the judgements in the analytic hierarchy process ahp these distributions called reciprocal satisfy the reciprocity axiom of the ahp methodology first we define and characterise the reciprocal distributions and present some results and properties associated with them we then study the relationship between the reciprocal and symmetrical random variables and the distribution of the priorities vector considering the eigenvector method egv in the consistent case and the row geometric mean method rgm as priorization methods and assuming that the judgements follow reciprocal distributions we prove that the elements of the priorities vector are distributed according to reciprocal random variables the knowledge of these distributions will allow us to obtain the probability of two interesting events in practical selection problems the probability that a given alternative would be ranked in the first position and the probability of a given ranking preference structure for the whole set of alternatives in particular we find the analytic expressions for these probabilities when a reciprocal random variable such as the lognormal is used

theory and methodology abstract the hierarchial network design problem with transshipment facilites was first formulated in current j 1988 the design of a hierarchical transportation network with transshipment facilities transportation science 22 270 277 the objective of the fc hndp is to identify a least cost two level network the network must include a primary path from a prespecified origin node to a prespecified terminus node all nodes not on the primary path must be connected to that path via secondary arcs in addition transshipment facilities are required at the intersections of the two network levels in this paper we present two efficient heuristics for solving the fc hndp these heuristics were based upon a lagrangian relaxation of the problem to facilitate this relaxation we have reformulated the original problem results for 240 randomly generated problems are summarized the heuristics yielded the optimal solution in 72 of these problems the mean gap between the lagrangian lower bound and the feasible solution value was 0 38 and the maximum gap was 4 33 of the lower bound

weight stability intervals in multicriteria decision aid abstract in this paper we study the stability of the results given by a wide class of mcda methods to changes of the weights of the criteria three aspects of stability are discussed in each case we propose to construct stability intervals for the weights of either single criteria or groups of criteria a necessary and sufficient condition for stability is also considered the use of these instruments is demonstrated with a numerical example

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

contribution of individual judgments toward inconsistency in pairwise comparisons pairwise comparison pc is a well established method to assist decision makers in estimating their preferences in pcs the acquired judgments are used to construct a pc matrix pcm that is used to check whether the inconsistency in judgments is acceptable or requires revision the use of consistency ratio cr a widely used measure for inconsistency has been widely debated and the literature survey has identified a need for a more appropriate measure considering this need a new measure termed congruence is proposed in this paper the measure is shown to be useful in finding the contribution of individual judgments toward overall inconsistency of a pcm and therefore can be used to detect and correct cardinally inconsistent judgments the proposed measure is applicable to incomplete sets of pc judgments without modification unlike cr which requires a complete set of pc judgments to address ordinal inconsistency another measure termed dissonance is proposed as a supplement to the congruence measure the two measures appear useful in detecting both outliers and the phenomenon of consistency deadlock where all judgments equally contribute toward the overall inconsistency

syzygies of affine toric varieties we give a method for computing the syzygies of the coordinate ring r of an affine toric variety we show how the method works for dimension one and two cases cohen macaulay semigroups and for computing minimal generators of the defining ideal we show how to compute the depth of r and generalize a criterion for cohen macaulayness

axiomatic property based consistency analysis and decision making with interval multiplicative reciprocal preference relations abstract this paper focuses on obtaining an interval extension of saaty s consistency and eliciting normalized interval weights in analytic form from interval multiplicative reciprocal preference relations imrprs as well as checking acceptability of imrprs four axiomatic properties are presented to characterize multiplicative consistency of imrprs the paper shows that six existing consistency definitions fail to satisfy all the four axiomatic properties a constrained interval multiplication based transitivity equation is devised and an interval matrix is constructed to capture original preferences and row uncertainty proportionalities of an imrpr based on the constructed interval matrix an ordinary interval multiplication based transitivity equation is developed to define consistency of imrprs a logarithmic least square model is established to find normalized interval weights from imrprs and its analytic solution is obtained by the lagrangian multiplier method the paper introduces an index to measure uniformity of row uncertainties in an imrpr and proposes a geometric consistency index to measure inconsistency of imrprs a novel method is put forward to determine acceptability of imrprs by checking acceptable consistency and acceptable uncertainty the proposed models are illustrated by eight numerical examples and a hierarchical multi criteria decision making problem dealing with recommending undergraduate students to graduate admission

scaling binary comparison matrices a comment on narasimhan s proposal and other methods abstract recently narasimhan has proposed an adjustment to saaty s solution for scaling square judgment matrices consisting of pairwise ratio comparisons he recommends that prior to the application of saaty s solution the judgment matrix be adjusted to satisfy the supertransitivity requirement the present note shows that the proposed adjustment is equivalent to an alternative solution to this problem based upon calculation of the geometric means of the matrix rows thus the proposed adjustment should be regarded as an alternative to saaty s solution rather than a preliminary stage finally four solutions to this scaling problem are compared and the need for further research in this area is reiterated

acceptability analysis and priority weight elicitation for interval multiplicative comparison matrices existing research on acceptability of pairwise interval comparison matrices focuses on acceptable consistency by controlling their inconsistency levels to within a certain threshold however a perfectly consistent but highly indeterminate interval comparison matrix can be unacceptable as it contains little sometimes no useful decision information this paper first analyzes the current definition of acceptable consistency for interval multiplicative comparison matrices imcms and shows its technical deficiencies we then introduce a new notion of acceptable imcms considering both inconsistency and indeterminacy levels in imcms a geometric mean based index is proposed to measure the indeterminacy ratio of an imcm and useful properties are derived for consistent imcms and acceptable imcms an indeterminacy ratio and geometric mean based transformation equation is subsequently put forward to convert normalized acceptable interval multiplicative weights into an acceptable imcm with consistency by introducing an auxiliary constraint a logarithmic least square model is established to generate interval multiplicative weights from acceptable imcms a geometric mean based possibility degree formula is designed to compare and rank normalized interval multiplicative weights two numerical examples are presented to illustrate how to utilize the proposed framework

group decision making based on multiplicative consistent reciprocal preference relations the main aim of this paper is to investigate the consistency and consensus of multiplicative consistent reciprocal preference relations based on the multiplicative consistency property a straightforward method to obtain the priority vector for multiplicative consistent reciprocal preference relation is introduced together with some of its desired properties in addition some interesting properties on consistency and consensus for reciprocal preference relations are further investigated a consensus optimization model for group decision making with reciprocal preference relations is proposed which enables the decision makers to reach consensus with the highest overall consensus level numerical examples are provided to make a discussion and comparison with some existing similar methods

stochastic decision making using multiplicative ahp abstract the analytic hierarchy process ahp has found a number of applications in decision making problems its multiplicative version called the multiplicative ahp mahp has been proposed to overcome some of the criticisms of the conventional version both these methodologies operate by obtaining expert judgements on the ratios of perceived importance of objects under consideration the literature on mahp in dealing with these judgements when they are specified without uncertainty is well developed however stochastic aspects of these judgements have not received much consideration in the literature so far stochastic judgements are considered in this paper for use in mahp the fact that weight derivation in mahp can be handled using mathematical programming is exploited and the literature on stochastic programming is adapted to the mahp context

a technical note on two inconsistency indices for preference relations a case of functional relation abstract different representations of valued preferences have been studied and used over the years however one common factor has been the prominence of the concept of consistency inconsistency indices have been introduced to estimate the deviation of preferences from a fully consistent form in this note we shall recall two types of preference relations reciprocal relations and multiplicative preference relations and show that two inconsistency indices introduced in these two different frameworks are functionally related besides this main result some reflections on the consequences of being functionally related are presented

sensitivity analysis for product design selection with an implicit value function abstract in product design selection the decision maker dm often does not have enough information about the end users needs to state the preferences with precision as is required by many of the existing selection methods we present for the case where the dm gives estimates of the preferences a concept for calculating a robustness index the concept can be used with any iterative selection method that chooses a trial design for each iteration and uses the dm s preference parameters at that trial design to eliminate some design options which have lower value than the trial design such methods like our previously published method are applicable to cases where the dm s value function is implicit our robustness index is a metric of the allowed variation between the actual and estimated preferences for which the set of non eliminated trial designs which could be singleton will not change the dm through experience can use the robustness index and other information generated in calculating the index to determine what action to take make a final selection from the present set of non eliminated designs improve the precision of the preference estimates or otherwise cope with the imprecision we present an algorithm for finding the robustness index and demonstrate and verify the algorithm with an engineering example and a numerical example