Unique Preference Aggregation in Design and Decision Making

Preference aggregation is a core operation in multi-objective design optimisation and group decision-making, as it determines the best-fit-for-common-purpose alternative within complex socio-technical contexts. Therefore, their aggregation requires a rigorous measurement-theoretic foundation to ensure mathematical validity, interpretability, and uniqueness. PFM establishes the principal axioms of unique preference aggregation, providing a rigorous basis on which aggregation can be demonstrated. In this paper, it is shown that commonly used aggregation approaches in MCDM - such as weighted arithmetic and geometric means, as well as weighted distance-based optimisation methods - often fail to produce consistent rankings and are therefore unsuitable for pure MCDM. In contrast, the unique preference aggregation presented here clarifies the mathematical limits of valid aggregation and provides a principled, implementable foundation for robust multi-criteria decision analysis (MCDA) and multi-objective design optimisation (MODO) in multi-faceted problems.

💡 Research Summary

The paper “Unique Preference Aggregation in Design and Decision Making” presents a rigorous measurement‑theoretic foundation for aggregating preferences in multi‑criteria decision making (MCDM) and multi‑objective design optimisation (MODO). The authors adopt Barzilai’s Preference Function Modelling (PFM) framework, which treats preference scores as points in a one‑dimensional affine space where only differences (and ratios of differences, the so‑called k‑ratios) have meaning. Consequently, absolute zero points do not exist, and any admissible transformation of the raw scores must be affine (p′ = a p + b, a > 0), preserving all k‑ratios.

Four PFM‑based axioms are introduced:

1. Preference Preservation (Δ‑meaningfulness) – only affine transformations are allowed, guaranteeing invariance of k‑ratios.
2. Comparable Criteria – all criteria must be expressed on a common interval scale with commensurate units, preventing a criterion from dominating merely because of its numeric range.
3. Meaningful Zero‑Reference – a stable, common zero reference (the mean) must be defined for every criterion; non‑linear operations such as multiplication, exponentiation, or distance‑based optimisation are prohibited.
4. Uniqueness – any two preference systems that generate identical judgments must be related by an affine transformation, ensuring that the aggregated ranking is unique up to such transformations.

To satisfy these axioms, the authors construct a Linear Preference Space (LPS). Raw preference scores p_{i,j} for alternative i on criterion j are standardised by z‑score normalisation:

z_{i,j} = (p_{i,j} – μ_j) / σ_j

where μ_j and σ_j are the arithmetic mean and standard deviation of the raw scores for criterion j. This transformation is affine (a_j = 1/σ_j, b_j = –μ_j/σ_j) and yields, for every criterion, a mean of zero and a standard deviation of one. Hence all criteria share a common origin (μ = 0) and unit (σ = 1), fulfilling the comparability and zero‑reference requirements.

Within the LPS, the only aggregation operator that respects the affine structure is the weighted linear centroid (weighted average):

P_i* = Σ_j w_j · z_{i,j}, with w_j ≥ 0, Σ_j w_j = 1

The authors derive this result by formulating a weighted least‑squared distance (WLSD) problem, showing that the minimiser is precisely the weighted centroid. However, they stress that the intermediate squared‑difference terms have no preference meaning; they serve merely as a mathematical device to reveal the unique linear solution admissible in the affine space.

The paper then critiques commonly used MCDM aggregation methods. Weighted arithmetic means, weighted geometric means, and distance‑based optimisation (e.g., minimising weighted Euclidean distance) violate at least one of the PFM axioms: they either introduce non‑linear transformations, rely on absolute scales, or treat preferences as distances rather than differences. Through a concrete example with four alternatives and three criteria (functionality, footprint, cost) and weights (0.4, 0.1, 0.5), the authors demonstrate that after z‑normalisation the weighted centroid yields a consistent ranking (A2 > A1 > A4 > A3). In contrast, applying a weighted geometric mean or a distance‑based objective produces rank reversals, illustrating the lack of invariance under affine transformations.

For interpretability, the aggregated scores can be linearly rescaled to a 0‑100 interval using a min‑max transformation:

P_i_scaled = (P_i* – min(P*)) / (max(P*) – min(P*)) · 100

Because this transformation is also affine, it preserves the ordering and all k‑ratios.

Key contributions of the work are:

• Theoretical Clarification – It formalises the notion that preferences are differences in an affine space, providing a solid measurement foundation for MCDM/MODO.
• Uniqueness Proof – It proves that the weighted linear centroid is the only aggregation operator invariant under admissible affine transformations, guaranteeing a unique ranking up to affine scaling.
• Critical Evaluation of Existing Methods – It analytically and empirically shows why popular aggregation techniques can produce inconsistent rankings.
• Practical Procedure – It offers a step‑by‑step pipeline (z‑normalisation → weighted centroid → optional linear scaling) ready for implementation in design and decision‑support tools.

Overall, the paper delivers a coherent, mathematically sound framework that bridges the gap between qualitative problem structuring and quantitative decision analysis. By ensuring that every step respects the affine nature of preferences, the proposed approach enables transparent, reproducible, and robust multi‑criteria evaluations, especially in complex socio‑technical systems where human judgment plays a central role.