Towards achieving consistent opinion fusion in group decision making with complete distributed preference relations

On Indeterminacy of Interval Multiplicative Pairwise Comparison Matrix

The interval multiplicative pairwise comparison matrix (IMPCM) is widely used to model human judgments affected by uncertainty and/or ambiguity. To improve the quality of an IMPCM, consistency is not sufficient. The indeterminacy should also be within an acceptable threshold because a consistent IMPCM may be deemed unacceptable due to high indeterminacy. Regarding indeterminacy, two metrics have been proposed in the literature: the indeterminacy ratio and the indeterminacy index. The former is from a local view, and the latter is from a global view. We have proposed an acceptable IMPCM model, which guarantees that an inconsistent IMPCM can be transformed into a consistent IMPCM, and the maximal indeterminacy ratio can be reduced. However, there is still a research gap. That is, a concomitant question naturally arises: can the indeterminacy index be reduced as well? In this paper, we further prove that the indeterminacy index of an originally inconsistent IMPCM can be reduced under the proposed model. Three numerical examples are presented to illustrate the feasibility and superiority of the proposed model. We also flowcharted the proposed model from a pragmatic view such that we can judiciously reduce the indeterminacy index of the IMPCM to a certain satisfactory level. That is, by applying the proposed model once, the original inconsistent IMPCM can be transformed into a consistent IMPCM that will possess less indeterminacy than the original one has. Consequently, by successively applying the proposed model, we can reduce or even eventually eliminate the indeterminacy of the IMPCM. In other words, we can/may obtain an MPCM rather than an IMPCM. In addition to mathematical proofs, we present experimental results of computer simulations to corroborate our argument. In summary, this model is not only effective but also efficient because it only requires arithmetic operations without solving complex optimization problems.

Identification of Homogeneous Groups of Actors in a Local AHP-Multiactor Context with a High Number of Decision- Makers: A Bayesian Stochastic Search

The identification of homogeneous groups of actors in a local AHP-multiactor context based on their preferences is an open problem, particularly when the number of decision-makers is high. To solve this problem in the case of using stochastic AHP, this paper proposes a new Bayesian stochastic search methodology for large-scale problems (number of decision-makers greater than 20). The new methodology, based on Bayesian tools for model comparison and selection, takes advantage of the individual preference structures distributions obtained from stochastic AHP to allow the identification of homogeneous groups of actors with a maximum common incompatibility threshold. The methodology offers a heuristic approach with several near-optimal partitions, calculated by the Occam’s window, that capture the uncertainty that is inherent when considering intangible aspects (AHP). This uncertainty is also reflected in the graphs that show the similarities of the decision-maker’s opinions and that can be used to achieve representative collective positions by constructing agreement paths in negotiation processes. If a small number of actors is considered, the proposed algorithm (AHP Bayesian clustering) significantly reduces the computational time of group identification with respect to an exhaustive search method. The methodology is illustrated by a real case of citizen participation based on e-Cognocracy.