Axiomatic property based consistency analysis and decision making with interval multiplicative reciprocal preference relations

This paper focuses on obtaining an interval extension of Saaty's consistency and eliciting normalized interval weights in analytic form from interval multiplicative reciprocal preference relations (IMRPRs) as well as checking acceptability of IMRPRs. Four axiomatic properties are presented to characterize multiplicative consistency of IMRPRs. The paper shows that six existing consistency definitions fail to satisfy all the four axiomatic properties. A constrained interval multiplication based transitivity equation is devised and an interval matrix is constructed to capture original preferences and row uncertainty proportionalities of an IMRPR. Based on the constructed interval matrix, an ordinary interval multiplication based transitivity equation is developed to define consistency of IMRPRs. A logarithmic least square model is established to find normalized interval weights from IMRPRs and its analytic solution is obtained by the Lagrangian multiplier method. The paper introduces an index to measure uniformity of row uncertainties in an IMRPR and proposes a geometric consistency index to measure inconsistency of IMRPRs. A novel method is put forward to determine acceptability of IMRPRs by checking acceptable consistency and acceptable uncertainty. The proposed models are illustrated by eight numerical examples and a hierarchical multi-criteria decision making problem dealing with recommending undergraduate students to graduate admission. (C) 2019 Elsevier Inc. All rights reserved.