Consistency and optimized priority weight analytical solutions of interval multiplicative preference relations

Abstract

This paper focuses on answering the following question: How to estimate the quality of off-diagonal judgments in an interval multiplicative preference relation (IMPR) and obtain an analytical solution to the optimized interval priority weight vector of an IMPR. By generalizing IMPRs, an interval-multiplication-based transitivity equation is devised to define perfect consistency of generalized IMPRs. A basic interval multiplicative weight (IMW) vector is introduced to characterize different interval weight vectors with equivalency and used as a priority weight benchmark of a generalized IMPR. Two logarithmic least square models are established for determining basic IMWs of a generalized IMPR, and their analytical solutions are found by using the Lagrangian multiplier method. The paper demonstrates that any IMPR with uncertainty has no perfect consistency in terms of interval arithmetic, and constructs a generalized IMPR from an IMPR to introduce consistency of off-diagonal judgments in an IMPR. Two computation formulas are developed to obtain optimized basic IMWs of an IMPR, which can entirely capture off-diagonal judgments if the IMPR is consistent. Numerical examples are offered to illustrate how to apply the proposed models and a comparative study is made to show that off-diagonal judgments in an inconsistent IMPR are properly captured by the analytical solution based IMWs.