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

deriving weights from pairwise comparison ratio matrices an axiomatic approach abstract this paper examines the problem of extracting object or attribute weights from a pairwise comparison ratio matrix this problem is approached from the point of view of a distance measure on the space of all such matrices a set of axioms is presented which such a distance measure should satisfy and the uniqueness of the measure is proven the problem of weight derivation is then shown to be equivalent to that of finding a totally transitive matrix which is a minimum distance from the given matrix this problem reduces to a goal programming model finally it is shown that the problem of weight derivation is related to that of ranking players in a round robin tournament the space of all binary tournament matrices is proven to be isometric to a subset of the space of ratio matrices

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

general affine surface areas abstract two families of general affine surface areas are introduced basic properties and affine isoperimetric inequalities for these new affine surface areas as well as for l affine surface areas are established

on the minimum violations ranking of a tournament this paper examines the problem of rank ordering a set of players or objects on the basis of a set of pairwise comparisons arising from a tournament the criterion for deriving this ranking is to have as few cases as possible where player i is ranked above j while i was actually defeated by j in the tournament such a situation is referred to as a violation the objective therefore is to determine the minimum violations ranking mvr while there are situations where this ranking would be allowed to contain ties among subsets of objects we will concern ourselves herein with linear ordering no ties a series of examples are given where this requirement would seem to be appropriate in order to put the mvr problem into proper perspective we introduce the concept of a distance on the set of tournaments a set of natural axioms is presented which any such distance measure should obey and it is proven that in the presence of these axioms a unique such measure exists it is then shown that the mvr problem is equivalent to the minimum distance problem which can be represented in several forms in particular as a problem of determining the minimum feedback edge set in a graph and as a mixed integer generalized network problem this opens up a wide scope of possible solution procedures for the mvr problem an optimal algorithm is presented along with computational results in addition various heuristics are discussed including an improved heuristic referred to as the iterated kendall method

fuzzy analytic hierarchy process a performance analysis of various algorithms abstract analytical hierarchical process ahp along with fuzzy set theory has been used extensively in the multi criteria decision making mcdm process in which fuzzy numbers are utilized to represent human judgments more realistically over the past couple of decades numerous articles have been published proposing algorithms through which priority vector or weight vector can be calculated from fuzzy comparison matrices the aim of this study is to conduct a comprehensive performance analysis of the most popular algorithms proposed in this domain in terms of accuracy of weights calculated from fuzzy comparison matrices such an analysis is much needed by the researchers and practitioners however none is available an experimental analysis is conducted and the performance of various algorithms are evaluated with varying three parameters i e the size of the comparison matrix the level of fuzziness and the level of inconsistency we found that modified logarithmic least squares method and fuzzy inverse of column sum method ficsm generally outperformed other algorithms while fuzzy extent analysis fea which is the most frequently used algorithm in the literature provides the least accurate results furthermore it was observed that a modified version of fea method significantly improved its performance

a multicriteria evaluation of methods for obtaining weights from ratio scale matrices abstract ratio scale matrices are commonly used as a vehicle to distinguish among competing alternatives or conflicting criteria the issue of extracting weights out of such matrices has attracted increasing attention in recent years and resulted in the development of several scaling methods this paper presents an attempt to evaluate some of the more known techniques in this area the evaluation procedure includes random generation of a large number of representative matrices which are used to evaluate the various scaling methods according to the specified criteria the performance of the different techniques is analyzed with respect to the proposed criteria and conclusions are drawn

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

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

sylvester s question the probability that n points are in convex position for a convex body k in the plane let p n k denote the probability that n random independent and uniform points from k are in convex position that is none of them lies in the convex hull of the others here we determine the asymptotic behavior of p n k by showing that as n goes to infinity n 2 n root p n k tends to a finite and positive limit

a canonical decomposition theory for metrics on a finite set abstract we consider specific additive decompositions d d 1 d n of metrics defined on a finite set x where a metric may give distance zero to pairs of distinct points the simplest building stones are the slit metrics associated to splits i e bipartitions of the given set x while an additive decomposition of a hamming metric into split metrics is in no way unique we achieve uniqueness by restricting ourselves to coherent decompositions that is decompositions d d 1 d n such that for every map f x r with f x f y d x y for all x y x there exist maps f 1 f n x r with f f 1 f n and f i x f i y d i x y for all i 1 n and all x y x these coherent decompositions are closely related to a geometric decomposition of the injective hull of the given metric a metric with a coherent decomposition into a weighted sum of split metrics will be called totally split decomposable tree metrics and more generally the sum of two tree metrics are particular instances of totally split decomposable metrics our main result confirms that every metric admits a coherent decomposition into a totally split decomposable metric and a split prime residue where all the split summands and hence the decomposition can be determined in polynomial time and that a family of splits can occur this way if and only if it does not induce on any four point subset all three splits with block size two

singular value decomposition in ahp abstract the analytic hierarchy process ahp has been accepted as a leading multiattribute decision aiding model both by practitioners and academics the foundation of the ahp is the saaty s eigenvector method em and associated inconsistency index that are based on the largest eigenvalue and associated eigenvector of an n n positive reciprocal matrix the elements of the matrix are the decision maker s dm numerical estimates of the preference of n alternatives with respect to a criterion when they are compared pairwise using the 1 9 ahp fundamental comparison scale the components of the normalized eigenvector provide approximations of the unknown weights of the criteria alternatives and the deviation of the largest eigenvector from n yields a measure of how inconsistent the dm is with respect to the pairwise comparisons singular value decomposition svd is an important tool of matrix algebra that has been applied to a number of areas e g principal component analysis canonical correlation in statistics the determination of the moore penrose generalized inverse and low rank approximation of matrices in this paper using the svd and the theory of low rank approximation of a pairwise comparison matrix we offer a new approach for determining the associated weights we prove that the rank one left and right singular vectors that is the vectors associated with the largest singular value yield theoretically justified weights we suggest that an inconsistency measure for these weights is the frobenius norm of the difference between the original pairwise comparison matrix and one formed by the svd determined weights how this measure can be applied in practice as a means of measuring the confidence the dm should place in the svd weights is still an open question we illustrate the svd approach and compare it to the em for some numerical examples

bi criteria evolution strategy in estimating weights from the ahp ratio scale matrices abstract the problem of deriving weights from ratio scale matrices in an analytic hierarchy process ahp is addressed by researchers worldwide there are various ways to solve the problem that are generally grouped into simple matrix and optimization methods all methods have received criticism regarding the accuracy of derived weights and different criteria are in use to compare the weights obtained from different methods because the set of pareto non dominated solutions weights is unknown and for inconsistent matrices is indefinite a bi criterion optimization approach is proposed for manipulating such matrices the problem specific evolution strategy algorithm esa is implemented for a robust stochastic search over a feasible indefinite solution space the fitness function is defined as a scalar vector function composed of the common error measure i e the euclidean distance and a minimum violation error that accounts for no violation of the rank ordering the encoding scheme and other components of the search engine are adjusted to preserve the imposed constraints related to the required normalized values of the weights the solutions generated by the proposed approach are compared with solutions obtained by five well known prioritization techniques for three judgment matrices taken from the literature in these and other test applications the prioritization method that uses the entitled weights estimation by evolution strategy algorithm weesa appears to be superior to other methods if only two the most commonly used methods are applied the euclidean distance and minimum violation exclusion criteria

representing the strengths and directions of pairwise comparisons we consider the problem of finding weights that well represent a set of pairwise multiplicative comparisons of a set of objects as in the ahp and other methods our main contribution is a method for deriving such weights that takes into consideration not only the strengths of the pairwise comparisons but also their directions for example if the comparison directions satisfy transitivity then the weights produced by our method also satisfy transitivity this is not always true for other methods we also present a set of reasonable axioms for which our method is the essentially unique solution our method and axioms are closely related to those of cook and kress eur j oper res 37 1988 335 our method like theirs reduces to solving a linear program hence it is different from the approach used in the ahp for the special case that the comparison directions satisfy transitivity our method is quite simple and reduces to performing a forward pass as in the critical path method

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

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

preference elicitation from inconsistent judgments using multi objective optimization abstract several decision making techniques involve pairwise comparisons to elicit the preferences of a decision maker dm this paper proposes a new approach for prioritization from pairwise comparisons using the concept of indirect judgments no method exists that simultaneously minimizes deviations from both direct and indirect judgments in order to estimate preferences it is sensible to consider both the acquired judgments and the other judgments latent in the dm s mind hence a technique is developed here to minimize the deviations from both types of judgments estimated preferences are generally evaluated based on two criteria their deviation from the provided judgments and the number of judgments that have been ordinally violated here it is proposed to optimize three objectives simultaneously the deviations from both direct and indirect judgments and the number of judgments violated a prototype application has been developed to generate all non dominated solutions using a multiple objective evolutionary algorithm the new approach is shown to offer users greater flexibility than all other tested methods

a thurstonian view of the analytic hierarchy process abstract the application of deterministic decision models in situations characterized by noise and uncertainty is likely to produce results of questionable value in this paper some very simple probabilistic models are developed and substituted for the deterministic scales used in the analytic hierarchy process ahp it is shown that the use of these probabilistic models can extend the domain of ahp to situations such as consensual or group decision making that possess significant amounts of uncertainty in addition explicit measures of the variation present in the evaluation of decision alternatives and attributes are obtained

affine perimeter and limit shape it is proved here that as n infinity almost all convex 1 n z 2 lattice polygons lying in a given convex compact set k subset of r 2 are very close to a fixed convex set k 0 the distinguishing property of k 0 is that its affine perimeter is the largest among all convex sets contained in k

estimating criterion weights using eigenvectors a comparative study abstract in addition to saaty s eigenvector there are infinitely many eigen weight vectors which can be constructed for any given data of estimated weight ratios as the judgment of the ratios are dependent on personal experience learning situations and state of mind inconsistencies and degree of easiness or judgment on these individual ratios can be different we study the properties of the different eigen weight vectors including that of saaty and that recently proposed by cogger and yu a general framework for the construction of eigen weight vectors incorporating that confidence in obtaining the individual ratios can be different will be proposed and discussed

general l_p affine isoperimetric inequalities sharp l p affine isoperimetric inequalities are established for the entire class of l p projection bodies and the entire class of l p centroid bodies these new inequalities strengthen the l p petty projection and the l p busemann petty centroid inequality

efficient weight vectors from pairwise comparison matrices abstract pairwise comparison matrices are frequently applied in multi criteria decision making a weight vector is called efficient if no other weight vector is at least as good in approximating the elements of the pairwise comparison matrix and strictly better in at least one position a weight vector is weakly efficient if the pairwise ratios cannot be improved in all non diagonal positions we show that the principal eigenvector is always weakly efficient but numerical examples show that it can be inefficient the linear programs proposed test whether a given weight vector is weakly efficient and in case of strong inefficiency an efficient strongly dominating weight vector is calculated the proposed algorithms are implemented in pairwise comparison matrix calculator available at pcmc online