María Teresa Escobar

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.

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.

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

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

A collaborative platform for cognitive decision making in the Knowledge Society

n-1

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

n-1 judgments that are required to derive the priorities. In this local AHP context, using the row geometric mean as the prioritisation procedure, this paper presents an extension of the CCM, the precise consensus consistency matrix (PCCM), which significantly mitigates this problem. By identifying precise values in the common consistency intervals, the PCCM automatically allows the number of entries in the CCM to be increased. The PCCM provides more informed and participative GDM and offers more accurate estimations for the group’s priorities. It can also be used as a starting point for posterior negotiation processes between the actors and it can be employed in global AHP-GDM contexts (hierarchies). The new decisional tool has been applied to a real-life experience concerned with the analysis of the integral viability of public investment projects, more specifically, the economic valuation of social aspects.