Multiplicative consistency analysis of interval-valued fuzzy preference relations

Interval-valued fuzzy preference relations (IVFPRs) have been applied to many real-life decision-making problems. However, most definitions of consistency of IVFPRs do not satisfy invariability to compared objects' labels. To overcome this drawback, this paper mainly focuses on the multiplicative consistency analysis of interval-valued fuzzy preference relations (IVFPRs). Firstly, this paper proposes a new multiplicative consistency of complete IVFPRs. It is proved that this new multiplicative consistency is robust and invariable to compared objects' labels. Then, the definition of acceptable incomplete IVFPRs (In-IVFPRs) is presented. To make full use of all direct and indirect evaluations of decision-makers, an algorithm is devised to evaluate the missing elements of an acceptable incomplete In-IVFPR. To comprehensively describe the closeness between any two complete IVFPRs, the total deviation of two complete IVFPRs is defined based on the p-norm of a vector. By minimizing the total deviation of two complete IVFPRs, a programming model is built to determine an interval weight vector from a complete IVFPR. Subsequently, a novel decision-making method with an In-IVFPR is proposed. Lastly, three practical and numerical examples and simulation-based comparative analyses are provided to further validate the practicability and advantages of the proposed method.