The Experts below are selected from a list of 31959 Experts worldwide ranked by ideXlab platform
Bice Cavallo - One of the best experts on this subject based on the ideXlab platform.
-
mathcal g g distance and mathcal g g decomposition for improving mathcal g g consistency of a Pairwise Comparison matrix
Fuzzy Optimization and Decision Making, 2019Co-Authors: Bice CavalloAbstract:Pairwise Comparisons have been a long standing technique for comparing alternatives/criteria and their role has been pivotal in the development of modern decision making methods. Since consistency ensures rational decisions, in literature several approaches are proposed for the revision of the Pairwise Comparison Matrix in order to improve its consistency. In order to obtain general results, suitable for several kinds of Pairwise Comparison Matrices proposed in literature, we focus on matrices defined over a general unifying framework, that is an Abelian linearly ordered group. In this context, firstly, we provide $$\mathcal {G}$$ -distance between Pairwise Comparison Matrices and $$\mathcal {G}$$ -decomposition of a Pairwise Comparison Matrix in its $$\mathcal {G}$$ -consistent and totally $$\mathcal {G}$$ -inconsistent components. Then, we show how a $$\mathcal {G}$$ -inconsistent Pairwise Comparison Matrix can be revised according to the associated $$\mathcal {G}$$ -consistent component; the revision process takes into account $$\mathcal {G}$$ -distance from the former in order to better represent decision maker’s preferences.
-
A general unified framework for interval Pairwise Comparison matrices
International Journal of Approximate Reasoning, 2018Co-Authors: Bice Cavallo, Matteo BrunelliAbstract:Abstract Interval Pairwise Comparison Matrices have been widely used to account for uncertain statements concerning the preferences of decision makers. Several approaches have been proposed in the literature, such as multiplicative and fuzzy interval matrices. In this paper, we propose a general unified approach to Interval Pairwise Comparison Matrices, based on Abelian linearly ordered groups. In this framework, we generalize some consistency conditions provided for multiplicative and/or fuzzy interval Pairwise Comparison matrices and provide inclusion relations between them. Then, we provide a concept of distance between intervals that, together with a notion of mean defined over real continuous Abelian linearly ordered groups, allows us to provide a consistency index and an indeterminacy index. In this way, by means of suitable isomorphisms between Abelian linearly ordered groups, we will be able to compare the inconsistency and the indeterminacy of different kinds of Interval Pairwise Comparison Matrices, e.g. multiplicative, additive, and fuzzy, on a unique Cartesian coordinate system.
-
about a consistency index for Pairwise Comparison matrices over a divisible alo group
International Journal of Intelligent Systems, 2012Co-Authors: Bice Cavallo, Livia Dapuzzo, Massimo SquillanteAbstract:Pairwise Comparison matrices (PCMs) over an Abelian linearly ordered (alo)-group have been introduced to generalize multiplicative, additive and fuzzy ones and remove some consistency drawbacks. Under the assumption of divisibility of , for each PCM A=(aij), a ⊙-mean vector can be associated with A and a consistency measure , expressed in terms of ⊙-mean of -distances, can be provided. In this paper, we focus on the consistency index . By using the notion of rational power and the related properties, we establish a link between and . The relevance of this link is twofold because it gives more validity to and more meaning to ; in fact, it ensures that if is close to the identity element then, from a side A is close to be a consistent PCM and from the other side is close to be a consistent vector; thus, it can be chosen as a priority vector for the alternatives. © 2011 Wiley Periodicals, Inc.
-
deriving weights from a Pairwise Comparison matrix over an alo group
Soft Computing, 2012Co-Authors: Bice Cavallo, Livia DapuzzoAbstract:In this paper, at first, we provide some results on the group of vectors with components in a divisible Abelian linearly ordered group, the related subgroup of $$\odot$$ -normal vectors, the relation of $$\odot$$ -proportionality and the corresponding quotient group. Then, we apply the achieved results to the groups of reciprocal and consistent matrices over divisible Abelian linearly ordered groups; this allows us to deal with the problem of deriving a weighting ranking for the alternatives from a Pairwise Comparison matrix. The proposed weighting vector has several advantages; it satisfies, for instance, the independence of scale-inversion condition.
-
characterizations of consistent Pairwise Comparison matrices over abelian linearly ordered groups
Journal of intelligent systems, 2010Co-Authors: Bice Cavallo, Livia DapuzzoAbstract:We consider the framework of Pairwise Comparison matrices over abelian linearly ordered groups. We introduce the notion of o-proportionality that allows us to provide new characterizations of the consistency, efficient algorithms for checking the consistency and for building a consistent matrix. Moreover, we provide a new consistency index. © 2010 Wiley Periodicals, Inc.
Sándor Bozóki - One of the best experts on this subject based on the ideXlab platform.
-
the logarithmic least squares optimality of the arithmetic geometric mean of weight vectors calculated from all spanning trees for incomplete additive multiplicative Pairwise Comparison matrices
International Journal of General Systems, 2019Co-Authors: Sándor Bozóki, Vitaliy V TsyganokAbstract:ABSTRACTComplete and incomplete additive/multiplicative Pairwise Comparison matrices are applied in preference modelling, multi-attribute decision making and ranking. The equivalence of two well kn...
-
the logarithmic least squares optimality of the arithmetic geometric mean of weight vectors calculated from all spanning trees for incomplete additive multiplicative Pairwise Comparison matrices
International Journal of General Systems, 2019Co-Authors: Sándor Bozóki, Vitaliy V TsyganokAbstract:Complete and incomplete additive/multiplicative Pairwise Comparison matrices are applied in preference modelling, multi-attribute decision making and ranking. The equivalence of two well known meth...
-
Efficient weight vectors from Pairwise Comparison matrices
European Journal of Operational Research, 2018Co-Authors: Sándor Bozóki, János FülöpAbstract: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.
-
Efficiency analysis of double perturbed Pairwise Comparison matrices
2018Co-Authors: Kristóf Ábele-nagy, Sándor Bozóki, Örs RebákAbstract:Efficiency is a core concept of multi-objective optimisation problems and multi-attribute decision-making. In the case of Pairwise Comparison matrices, a weight vector is called efficient if the approximations of the elements of the Pairwise Comparison matrix made by the ratios of the weights cannot be improved in any position without making it worse in some other position. A Pairwise Comparison matrix is called double perturbed if it can be made consistent by altering two elements and their reciprocals. The most frequently used weighting method, the eigenvector method is analysed in the paper, and it is shown that it produces an efficient weight vector for double perturbed Pairwise Comparison matrices.
-
an application of incomplete Pairwise Comparison matrices for ranking top tennis players
European Journal of Operational Research, 2014Co-Authors: Sándor Bozóki, Laszlo Csato, Jozsef TemesiAbstract:Pairwise Comparison is an important tool in multi-attribute decision making. Pairwise Comparison matrices (PCM) have been applied for ranking criteria and for scoring alternatives according to a given criterion. Our paper presents a special application of incomplete PCMs: ranking of professional tennis players based on their results against each other. The selected 25 players have been on the top of the ATP rankings for a shorter or longer period in the last 40 years. Some of them have never met on the court. One of the aims of the paper is to provide ranking of the selected players, however, the analysis of incomplete Pairwise Comparison matrices is also in the focus. The eigenvector method and the logarithmic least squares method were used to calculate weights from incomplete PCMs. In our results the top three players of four decades were Nadal, Federer and Sampras. Some questions have been raised on the properties of incomplete PCMs and remains open for further investigation.
Geert Molenberghs - One of the best experts on this subject based on the ideXlab platform.
-
generalized Pairwise Comparison methods to analyze non prioritized composite endpoints
Statistics in Medicine, 2019Co-Authors: Johan Verbeeck, Ernest Spitzer, T De Vries, G A Van Es, William N Anderson, N M Van Mieghem, Martin B Leon, Geert MolenberghsAbstract:In the analysis of composite endpoints in a clinical trial, time to first event analysis techniques such as the logrank test and Cox proportional hazard test do not take into account the multiplicity, importance, and the severity of events in the composite endpoint. Several generalized Pairwise Comparison analysis methods have been described recently that do allow to take these aspects into account. These methods have the additional benefit that all types of outcomes can be included, such as longitudinal quantitative outcomes, to evaluate the full treatment effect. Four of the generalized Pairwise Comparison methods, ie, the Finkelstein-Schoenfeld, the Buyse, unmatched Pocock, and adapted O'Brien test, are summarized. They are compared to each other and to the logrank test by means of simulations while specifically evaluating the effect of correlation between components of the composite endpoint on the power to detect a treatment difference. These simulations show that prioritized generalized Pairwise Comparison methods perform very similarly, are sensitive to the priority rank of the components in the composite endpoint, and do not measure the true treatment effect from the second priority-ranked component onward. The nonprioritized Pairwise Comparison test does not suffer from these limitations and correlation affects only its variance.
Sylvain Kubler - One of the best experts on this subject based on the ideXlab platform.
-
Measuring inconsistency and deriving priorities from fuzzy Pairwise Comparison matrices using the knowledge-based consistency index
Knowledge-Based Systems, 2018Co-Authors: Sylvain Kubler, William Derigent, Alexandre Voisin, Jérérmy Robert, Yves Le Traon, Enrique Herrera ViedmaAbstract:The fuzzy analytic hierarchy process (AHP) is a widely applied multiple-criteria decision-making (MCDM) technique , making it possible to tackle vagueness and uncertainty arising from decision makers, especially in a Pairwise Comparison process. Indeed, as the human brain reasons with uncertain rather than precise information, Pairwise Comparisons may involve some degree of inconsistency, which must be correctly managed to guarantee a coherent result/ranking. Several consistency indexes for fuzzy Pairwise Comparison matrices (FPCMs) have been proposed in the literature. However, some scholars argue that most of these fail to be axiomatically grounded, which may lead to misleading results. To overcome this lack of an axiomatically grounded index, a new index is proposed in this paper, referred to as the knowledge-based consistency index (KCI). A comparative study of the proposed index with an existing one is carried out, and the results show that KCI contributes to substantially reducing the computation time. In addition, the different fuzzy weights derived from the initial FPCM (for KCI computation purposes) can also be employed to find a crisp set of weights that corresponds to an optimal solution to the MCDM problem according to the decision maker's viewpoint and expertise.
-
knowledge based consistency index for fuzzy Pairwise Comparison matrices
IEEE International Conference on Fuzzy Systems, 2017Co-Authors: Sylvain Kubler, William Derigent, Alexandre Voisin, Jeremy Robert, Yves Le TraonAbstract:Fuzzy AHP is today one of the most used Multiple Criteria Decision-Making (MCDM) techniques. The main argument to introduce fuzzy set theory within AHP lies in its ability to handle uncertainty and vagueness arising from decision makers (when performing Pairwise Comparisons between a set of criteria/alternatives). As humans usually reason with granular information rather than precise one, such Pairwise Comparisons may contain some degree of inconsistency that needs to be properly tackled to guarantee the relevance of the result/ranking. Over the last decades, several consistency indexes designed for fuzzy Pairwise Comparison matrices (FPCMs) were proposed, as will be discussed in this article. However, for some decision theory specialists, it appears that most of these indexes fail to be properly “axiomatically” founded, thus leading to misleading results. To overcome this, a new index, referred to as KCI (Knowledge-based Consistency Index) is introduced in this paper, and later compared with an existing index that is axiomatically well founded. The Comparison results show that (i) both indexes perform similarly from a consistency measurement perspective, but (ii) KCI contributes to significantly reduce the computation time, which can save expert's time in some MCDM problems.
Jaroslav Ramík - One of the best experts on this subject based on the ideXlab platform.
-
Isomorphisms between fuzzy Pairwise Comparison matrices
Fuzzy Optimization and Decision Making, 2015Co-Authors: Jaroslav RamíkAbstract:This paper deals with Pairwise Comparison matrices. In Comparison with Pairwise Comparison matrices with crisp elements investigated in the literature, here we investigate also Pairwise Comparison matrices with fuzzy elements of Abelian linearly ordered group (Alo-group) over a real interval. We generalize the concept of reciprocity and consistency of crisp Pairwise Comparison matrices to matrices with triangular fuzzy numbers (PCFN matrices). We also define the concept of priority vector which is a generalization of the crisp concept. Doing this we unify several approaches known from the literature. Finally, we show that the properties of reciprocity and consistency of PCFN matrices are saved by special transformations based on fuzzy extensions of isomorphisms between two Alo-groups.
-
Pairwise Comparison matrix with fuzzy elements on alo group
Information Sciences, 2015Co-Authors: Jaroslav RamíkAbstract:This paper deals with Pairwise Comparison matrices with fuzzy elements. Fuzzy elements of the Pairwise Comparison matrix are applied whenever the decision maker is not sure about the value of his/her evaluation of the relative importance of elements in question. In Comparison with Pairwise Comparison matrices with crisp elements investigated in the literature, here we investigate Pairwise Comparison matrices with elements from abelian linearly ordered group (alo-group) over a real interval. We generalize the concept of reciprocity and consistency of crisp Pairwise Comparison matrices to matrices with triangular fuzzy numbers (PCFN matrices). We also define the concept of priority vector which is a generalization of the crisp concept. Such an approach allows for a generalization dealing both with the PCFN matrices on the additive, multiplicative and also fuzzy alo-groups. It unifies several approaches known from the literature. Moreover, we also deal with the problem of measuring the inconsistency of PCFN matrices by defining corresponding indexes. The first index called the consistency grade G is the maximal alpha of alpha-cut, such that the corresponding PCFN matrix is still alpha-consistent. On the other hand, the consistency index I of the PCFN matrix measures the distance of the PCFN matrix to the closest ratio matrix. If the PCFN matrix is crisp and consistent, then G is equal to 1 and the consistency index I is equal to the identity element e of the alo-group, otherwise, G is less than 1, or I is greater than e. Four numerical examples are presented to illustrate the concepts and derived properties. Finally, we show that the properties of reciprocity and consistency of PCFN matrices are saved by special transformations based on fuzzy extensions of isomorphisms between two alo-groups.
