Optimizing consistency and consensus in group decision making based on relative projection between multiplicative reciprocal matrices

Individual consistency and group consensus are the two protagonists in group decision-making (GDM). The former is considered as the logical basis of decision information. The latter is regarded as the guarantee of obtaining a widely accepted solution. In this study, two novel indexes for measuring consistency and consensus levels are defined using relative projection between multiplicative reciprocal matrices (MRMs) originating in Analytic Hierarchy Process (AHP). They are called the relative projection consistency index (RPCI) and the geometric relative projection consensus index (GRPCI), respectively. In order to test acceptable consistency of MRMs, the thresholds of RPCI are determined using the simulation method. Then, an optimization method for improving consistency of MRMs is proposed. By considering acceptable consistency of individual MRMs, a model of optimizing group consensus is further established by minimizing the GRPCI. Finally, a novel GDM algorithm is elaborated on, where the degrees of individual consistency and group consensus are optimized. A case study is carried out to illustrate the proposed models, where the Gaussian quantum behavior particle swarm optimization (GQPSO) algorithm is used to solve the constructed optimization models. Comparison with these previous studies reveals that a novel perspective is offered to simultaneously measure consistency degree of MRMs and group consensus level in GDM. The constructed optimization models are effective for improving consistency and reaching consensus in GDM with MRMs.