Fuzzy TOPSIS Ranking Stability under Membership-Function Variations across Alternative-Closeness Scenarios

Abstract

This study examines how alternative closeness and membership-function geometry jointly affect ranking stability in Fuzzy TOPSIS. The Energy Efficiency Dataset was normalized into three criteria, and sliding windows of 20 alternatives were selected to represent strong dominance, moderate competition, and tight competition. Triangular, trapezoidal, and Gaussian membership functions were compared under identical linguistic centers and equal criterion weights. Stability was evaluated using rank displacement, Spearman's correlation, and Kendall's tau. Under strong dominance, the mean rank displacement was 1.45 positions and all pairwise correlations were high. Moderate competition produced the largest mean displacement (4.20 positions), whereas tight competition produced a maximum shift of 10 positions. In the tightest scenario, triangular and Gaussian rankings remained nearly identical (Spearman's $\rho=0.9985$; Kendall's $\tau=0.9894$), while pairs involving the trapezoidal function declined to $\tau\approx0.56$. The plateau of the trapezoidal function reduced discrimination among adjacent values and amplified rank changes. These results show that ranking stability depends on the interaction between data structure and membership-function shape. Density diagnostics and cross-function stability checks should therefore accompany Fuzzy TOPSIS applications involving closely matched alternatives.