José María Moreno-Jiménez

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.

Consensus Building in AHP-Group Decision Making: A Bayesian Approach

This paper examines consensus building in AHP-group decision making from a Bayesian perspective. In accordance with the multicriteria procedural rationality paradigm, the methodology employed in this study permits the automatic identification, in a local context, of “agreement” and “disagreement” zones among the actors involved. This approach is based on the analysis of the pairwise comparison matrices provided by the actors themselves. In addition, the study integrates the attitudes of the actors implicated in the decision-making process and puts forward a number of semiautomatic initiatives for establishing consensus. This information is given to the actors as the first step in the negotiation processes. The knowledge obtained will be incorporated into the system via the learning process developed during the resolution of the problem. The proposed methodology, valid for the analysis of incomplete or imprecise pairwise comparison matrices, is illustrated by an example.

A probabilistic study of preference structures in the analytic hierarchy process with interval judgments

Here we study the problem of determining the ranking of the alternatives that one should infer when decision makers use interval judgments rather than point estimates in the Analytic Hierarchy Process. How many rankings could one infer from the matrix of interval judgments, and which is the most likely to be selected?

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.

Mixed valuation methods: a combined AHP-GP procedure for individual and group multicriteria agricultural valuation

This paper introduces a new assessment method classification, in which a third procedure, mixed valuation, is jointly included with the traditional economic and non-economic methodologies. The paper considers a case of multiple actors (from a previous work by the same authors—Aznar et al. (Estudios de Economía Aplicada, 25(2):389–409, 2007), in which a new technique for multicriteria agriculture valuation (MAVAM) was proposed. The method is specifically designed for situations in which scarce information about the elements being compared (quantified or not) is available. It works in individual and group decision making contexts and attempts to both obtain and incorporate the objective information associated with the tangible aspects of the problem and the subjective knowledge associated with the human factor into the valuation process. It combines two of the most extended multicriteria decision making techniques: the Analytic Hierarchy Process (AHP) and Goal Programming (GP). The first of these enables tangible and intangible information stemming from known elements to be collected by using pairwise comparisons; the second allows the scarce information available and the personal approach to the valuation to be included in the valuation process. The proposed methodology is illustrated by means of its application to a case of individual and group valuation of an agricultural asset in the La Ribera district, Valencia (Spain).

A linkage between the Analytic Hierarchy Process and the Compromise Programming Models

In the search for a unifying theory in the field of multicriteria decision making, we establish a relationship in the discrete case between two of the most extended multicriteria decision making approaches, namely the Analytic Hierarchy Process (AHP) and the Compromise Programming (CP) Models. In order to link them, we consider a hierarchical modelling of the CP where the order of the norm equals one, and we employ the regret associated with the AHP priorities measured in absolute terms. This regret can be considered as the anchor that allows us to link the relativity of the AHP with the goal-seeking behaviour of the CP. In general, the establishment of a unified framework in this field may clarify the controversies that exist between the different schools of thought, which will undoubtedly result in the improvement of both the theoretical and practical applications.

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