Optimization-driven group decision making with XOR comparison matrices

The “exclusive-or”(XOR) logic describes a situation where only one of multiple options can be chosen, which is used to give the XOR number for quantifying some uncertainty in pairwise comparisons. It is interesting to develop group decision making (GDM) model with XOR comparison matrices by investigating consistency of judgements and consensus of experts. First, the consistency/acceptable consistency of XOR comparison matrices is defined by considering the basic principle of XOR logic. A mathematical programming model is constructed to check the consistency/acceptable consistency of XOR comparison matrices. Second, for improving consistency of XOR comparison matrices, an optimization model is built to obtain a multiplicative preference relation with acceptable consistency from XOR comparison matrices. The model is further developed according to the idea of minimizing the number of adjusted variables. Third, a consensus index is established for reaching consensus in GDM, which is utilized to propose an iterative algorithm for improving the consensus level. An algorithm for GDM with XOR comparison matrices is developed, which yields an optimal solution under acceptable consistency and acceptable consensus. Finally, a case study with discussion comparison is reported to illustrate the applicability of the proposed model. The results help to construct an optimization-driven mechanism of reaching consensus in GDM under XOR environments.