Modification of the Proximity Indexed Value Method Based on Distance-Based Normalization to Improve the Stability of Alternative Rankings
The Proximity Indexed Value (PIV) method provides a proximity-based mechanism for ranking alternatives in multi-criteria decision-making; however, its ranking results can be affected by the normalization process and the relative characteristics of criterion values. This study proposes a Distance-Based Proximity Indexed Value (D-PIV) method by incorporating distance-based normalization into the PIV framework to represent the proximity of alternatives to criterion-specific ideal conditions more explicitly. The proposed method was evaluated using an alternative location selection dataset comprising eight alternatives and six criteria, including one cost criterion and five benefit criteria. The evaluation included comparative ranking analysis using 36 systematic criterion-weight perturbation scenarios with changes of ±0.05, ±0.10, and ±0.15. The results show that D-PIV changes the ordering of Loc 3 and Loc 8 compared with conventional PIV, resulting in a ranking that fully corresponds to the reference ranking. The Spearman correlation increased from 0.9762 for PIV to 1.0000 for D-PIV. However, the sensitivity analysis indicates that the distance-based modification does not uniformly improve ranking stability across all alternatives. D-PIV increases the stability of Loc 8 from 69.44% to 72.22%, while lower stability is observed for Loc 3 and Loc 6 compared with PIV. These findings demonstrate that D-PIV contributes a reference-oriented normalization mechanism that improves baseline ranking consistency while redistributing, rather than uniformly eliminating, sensitivity to criterion-weight variations. The study therefore provides both a methodological contribution through the integration of distance-based normalization into PIV and an empirical contribution through systematic evidence of its effects on ranking agreement and stability.
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