A Decision Making Model Based on Intuitionistic Multiplicative Preference relations with Acceptable Consistency

Two categories of hesitancy often exist in an intuitionistic multiplicative preference relation (IMPR). One is inconsistency among three or more preferences in an IMPR, and the other is hesitancy of a non-diagonal intuitionistic multiplicative judgment itself. Capturing such hesitancy plays an important role for deriving priority weights of IMPRs and measuring inconsistency degrees of IMPRs. This article first analyzes a recent consistency definition of IMPRs and reveals its deficiency. A normalized intuitionistic multiplicative weight vector is introduced and utilized as a benchmark of different intuitionistic multiplicative weight vectors with equivalency. A new notion of consistent IMPRs is then proposed by building the link between an IMPR and its normalized intuitionistic multiplicative weights. In order to find the closest IMPR with consistency to an original IMPR, the paper establishes a minimization model, in which the hesitancy difference between the two IMPRs is regarded as a constraint and the optimal goal function value is used to determine an inconsistency index of the original IMPR. The minimization model is further transformed into a least square model and its analytical solution is discovered by the Lagrangian multiplier method. Based on the analytical solution, an intuitionistic multiplicative judgment based index is devised to measure inconsistency of IMPRs, and an approach with checking acceptable consistency is developed for multi-criteria decision making with IMPRs. The presented models are illustrated and validated by three numerical examples including a hierarchical multi-criteria decision making problem.