A two-stage linear goal programming approach to eliciting interval weights from additive interval fuzzy preference relations

This article presents a linear goal programming framework to obtain normalized interval weights from interval fuzzy preference relations (IFPRs). A parameterized transformation equation is put forward to convert a normalized interval weight vector into IFPRs with additive consistency. Based on a linearization approximate relation of the transformation equation, a two-stage linear goal programming approach is developed to elicit interval weights and determine an appropriate parameter value from an additive IFPR. The first stage devises a linear goal programming model to generate optimal interval weight vectors by minimizing the absolute deviation between sides of the parameterized linearization approximate relation. The second stage aims to find a benchmark among the optimal solutions derived from the previous stage by minimizing the absolute deviation between the parameter and 1. The obtained benchmark is the closest to the original IFPR and can sufficiently reflect uncertainty of original judgments. A procedure is further proposed for solving group decision making problems with IFPRs. Two numerical examples including a comparative study with existing approaches are provided to illustrate validity and practicality of the proposed model.