Multiplicative consistency of intuitionistic reciprocal preference relations and its application to missing values estimation and consensus building

Abstract

The mathematical modelling and representation of Tanino’s multiplicative transitivity property to the case of intuitionistic reciprocal preference relations (IRPRs) is derived via Zadeh’s extension principle and the representation theorem of fuzzy sets. This result guarantees the correct generalisation of the multiplicative transitivity property of reciprocal preference relations (RPRs), and it allows the multiplicative consistency (MC) property of IRPRs to be defined. The MC property used in decision making problems is threefold: (1) to develop a consistency based procedure to estimate missing values in IRPRs using an indirect chain of alternatives; (2) to quantify the consistency index (CI) of preferences provided by experts; and (3) to build a novel consistency based induced ordered weighted averaging (MC-IOWA) operator that associates a higher contribution in the aggregated value to the more consistent information. These three uses are implemented in developing a consensus model for GDM problems with incomplete IRPRs in which the level of agreement between the experts’ individual IRPRs and the collective IRPR, which is referred here as the proximity index (PI), is combined with the CI to design a feedback mechanism to support experts to change some of their preference values using simple advice rules that aim at increasing the level of agreement while, at the same time, keeping a high degree of consistency. In the presence of missing information, the feedback mechanism implements the consistency based procedure to produce appropriate estimate values of the missing ones based on the given information provided by the experts. Under the assumption of constant CI values, the feedback mechanism is proved to converge to unanimous consensus when all experts are provided with recommendations and these are fully implemented. Additionally, visual representation of experts’ consensus position within the group before and after implementing their feedback advice is also provided, which help an expert to revisit his evaluations and make changes if considered appropriate to achieve a higher consensus level. Finally, an IRPR fuzzy majority based quantifier-guided non-dominance degree based prioritisation method using the associated score reciprocal preference relation is proposed to obtain the final solution of consensus.