Juan Aguarón

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.

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.

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.

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.α problem), and the ranking of all the alternatives (P.γ 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.α and P.γ problems, respectively). With respect to these three situations (judgement, alternative and matrix) and two problems (P.α and P.γ) 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.

Some extensions of the precise consistency consensus matrix

The Precise Consensus Consistency Matrix (PCCM) is an AHP-Group Decision Making (AHP-GDM) tool, defined by Aguaron et al. 2] and developed in a local context (a single criterion) in which the decision makers are assigned the same weights. Using the Row Geometric Mean as the prioritisation procedure, consensus is sought between the different decision makers when the modifications of their initial positions or judgements are guaranteed to be within the range of values accepted for a given inconsistency level. This paper upgrades the algorithm initially proposed for obtaining the PCCM in two ways: (i) it considers the case of different weights for the decision makers; and (ii) it strengthens the idea of consistency in the design of the algorithm. One of the drawbacks of this decisional tool is that it is sometimes impossible to achieve a complete matrix. To address this, we propose a procedure for attaining a complete common consensus judgement matrix or, at least, a matrix with the minimum number of entries that are required to derive the priorities. Finally, we compare the results obtained when applying the extensions of the PCCM with those obtained using the two traditional procedures (AIJ and AIP) usually employed in AHP-GDM. In order to do this, we use a set of indicators that measure the violations in consistency of the group pairwise matrices and the compatibility between the individuals and group positions in four cases associated with two scenarios (weighted and non-weighted decision makers) and two situations (complete and incomplete PCCMs). New method for consensus building in AHP-group decision makingMinor changes from individual pairwise comparison matricesChanges are consistent with the individual positions.Several measures to compare group consensus building methodsConsistency and compatibility of complete and incomplete group consensus matrices

Notes on Dependent Attributes in TOPSIS

Abstract TOPSIS is a multicriteria decision making technique based on the minimization of geometric distances that allows the ordering of compared alternatives in accordance with their distances from the ideal and anti-ideal solutions. The technique, that usually measures distances in the Euclidean norm, implicitly supposes that the contemplated attributes are independent. However, as this rarely occurs in practice, it is necessary to adapt the technique to the new situation. Using the Mahalanobis distance to incorporate the correlations among the attributes, this paper proposes a TOPSIS extension that captures the dependencies among them, but, in contrast to the Euclidean distance, does not require the normalization of the data. Results obtained by the new proposal have been compared by means of the three Minkowski norms most commonly employed for the calculation of distance: (i) the Manhattan distance (p=1); (ii) the Euclidean distance (p=2); and (iii) the Tchebycheff distance (p=∞). Furthermore, simulation techniques are used to analyse the connection between the TOPSIS results traditionally obtained with the Euclidean distance and those obtained with the Mahalanobis distance.

The precise consistency consensus matrix in a local AHP-group decision making context

A new decisional tool, the consensus consistency matrix (CCM) has recently been proposed for dealing with AHP-group decision making (AHP-GDM) in a local context (a single criterion). Each entry of this matrix, based on the property of consistency, corresponds to the range of values or interval in which all the decision makers are simultaneously consistent in their initial matrices. The main limitation of the CCM is that, on many occasions, it is not possible to obtain a matrix with the minimum

PRIOR-W&K: A Collaborative Tool for Decision Making in the Knowledge Society

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