The Normality Conditions of Twofold Interval Priority Weight Vectors and Their Estimation under an Interval Pairwise Comparison Matrix

Abstract

Interval pairwise comparison matrices (IPCMs) are useful to represent the aggregated pairwise comparison matrix of group members and the PCM with vague evaluations. The lower interval priority weight vector (LIPWV) and upper interval priority weight vector (UIPWV) are estimated from a given IPCM. As they are estimated independently, the inclusion relation between them is not always satisfied. In this paper, we consider the estimation problem of a twofold interval priority weight vector (TFIPWV), a nested pair of LIPWV and UIPWV, from an IPCM. Introducing the inclusion constraint between LIPWV and UIPWV to the conventional estimation problem is the simplest way to obtain a TFPWV. In this problem, the normality condition (NC) of TFIPWV is treated by the NCs of LIPWV and UIPWV. However, this NC does not guarantee that any IPWV between LIPWV and UIPWV satisfies the NC. The selection arbitrarity of IPWV for the estimated TFIPWV is not satisfied. We define a NC satisfying the selection arbitrarity and call it a strong NC as it is rather restrictive. An equivalent condition of the strong NC is given. We define also a weak NC by the minimum requirement and show its equivalent condition. We show that the estimation problems of a TFIPWV satisfying those strong and weak NCs are reduced to linear programming problems. By a numerical experiment, we examine the differences among the strong and weak NCs of the TFIWV, and the NCs of the LIPWV and UIPWV in the estimation problems of TFPWVs.