And-like-uninorm based consistency analysis and optimized fuzzy weight closed-form solution of triangular fuzzy additive preference relations

Abstract

This paper focuses on multiplicative consistency of triangular fuzzy additive preference relations (TFAPRs) and deriving a closed-form solution of optimized triangular fuzzy weights (TFWs) from TFAPRs. And-like-uninorm based indices are introduced to measure increasing part vagueness, decreasing part vagueness and overall vagueness for ]0,1[-valued triangular fuzzy preferences and TFAPRs. An index is then defined to measure row vagueness proportionality of TFAPRs. An and-like-uninorm based method is further proposed to generate multiplicatively consistent TFAPRs from ]0,1[-valued TFWs. By discussing equivalency of ]0,1[-valued TFW vectors, the paper presents two frameworks of normalized TFWs called multiplicatively modal-value-normalized TFWs and multiplicatively support-interval-normalized TFWs. Based on crucial properties of consistent TFAPRs, a logarithmic least square (LLS) model is established to seek multiplicatively support-interval-normalized TFWs from TFAPRs. By decomposing its goals and constraints, the LLS model is transformed into two least square models whose closed-form solutions are found by the Lagrange multiplier method. On basis of the obtained closed-form solution of TFWs, an algorithm including acceptable consistency checking is developed for decision making with TFAPRs. The rationality and advantages of the presented models are illustrated by a numerical example with five TFAPRs and a comparative analysis.