A Structurally Consistent Risk-Adjusted Z-Number AHP-TOPSIS Framework for Multi-Criteria Decision Making

Abstract

Decision-making under Z-number environments faces persistent challenges, including inconsistent uncertainty aggregation and rank instability. This paper proposes a risk-adjusted Z- number multi-criteria decision-making framework for Z-AHP–TOPSIS. The method replaces the traditional linear convex combination with a multiplicative, reliability-qualified magnitude: M(Z) ={C(A)C(B)}γ exp[︁−λ(︁σ2 A +ησ2 B )︁]︁ , where reliability qualifies rather than compensates restriction, and uncertainty is penalized through an exponential variance term inspired by Pratt’s constant absolute risk aversion frame work. The formulation is non-additive and Choquet-inspired; it is not claimed to be formally equivalent to a Choquet integral. The magnitude is embedded in Z-AHP for criteria weighting and Z-TOPSIS for alternative ranking. Mathematical properties are analyzed, including monotonicity in central tendency, anti-monotonicity in variance, and non-homogeneity under scaling. Three classical rank-reversal benchmarks are examined: Belton and Gear’s alternative addition problem and Triantaphyllou’s distributive- and ideal-mode problems. In these tests, the proposed method yields internally consistent simultaneous and pairwise rankings. However, because it is a new risk-adjusted aggregation, it can change the original distributive or ideal mode rankings. Thus, the results demonstrate suppression of simultaneous–pairwise reversal on the tested benchmarks, not universal rank-reversal immunity. Sensitivity analysis over λ, γ, and η shows that criteria-weight ordering remains invariant in the tested ranges. The method preserves Z-number restriction and reliability information without fuzzy conversion. Limitations include parameter specification, triangular fuzzy assumptions, a small benchmark set, and the absence of a substantive real-world case. The framework provides a mathematically consistent basis for reliability-aware MCDM, but broader validation is needed.