[Paper Review] Aggregation functions for decision making

Jean‐Luc Marichal|ArXiv.org|Jan 27, 2009

Multi-Criteria Decision Making91 references8 citations

TL;DR

This paper presents an axiomatic framework for selecting and characterizing aggregation functions in decision-making, focusing on their properties under different measurement scales. It identifies lattice polynomials and their transformations as the core families of meaningful, nondecreasing, or continuous functions for ordinal scales, with symmetric lattice polynomials equivalent to order statistics like the median.

ABSTRACT

Aggregation functions are generally defined and used to combine several numerical values into a single one, so that the final result of the aggregation takes into account all the individual values in a given manner. Such functions are widely used in many well-known disciplines such as statistics, economics, finance, and computer science. In this paper we confine ourselves to the use of aggregation functions in decision making.

Motivation & Objective

• To provide a systematic axiomatic classification of aggregation functions used in decision-making contexts.
• To identify which aggregation functions are meaningful under specific measurement scales, particularly ordinal, interval, and ratio scales.
• To characterize nondecreasing and continuous aggregation functions that preserve ordinal structure in inputs and outputs.
• To clarify the role of lattice polynomials and their transformations in modeling meaningful aggregation under ordinal scales.
• To establish conditions under which aggregation functions reduce to order statistics or transformed projections, especially under symmetry and idempotency.

Proposed method

• Uses an axiomatic approach to classify aggregation functions based on properties like continuity, symmetry, monotonicity, and scale invariance.
• Applies the concept of meaningfulness to ensure function behavior is invariant under permissible transformations of the scale.
• Characterizes functions using lattice polynomials—expressions built from min and max operations on variables.
• Introduces the use of fuzzy measures (γ ∈ Γ_N) to represent the structure of lattice polynomials in disjunctive normal form.
• Analyzes the impact of transformations g: ℝ → ℝ on lattice polynomials to generalize the class of meaningful functions.
• Applies results from Choquet and Sugeno integrals to unify and extend classical aggregation modes under non-additive measures.

Experimental results

Research questions

• RQ1 Which aggregation functions are meaningful when inputs and outputs are measured on the same ordinal scale?
• RQ2 How can nondecreasing or continuous aggregation functions be fully characterized under ordinal measurement scales?
• RQ3 What is the role of lattice polynomials in representing aggregation functions that are invariant under ordinal transformations?
• RQ4 Under what conditions do aggregation functions reduce to order statistics (e.g., median) or their transformations?
• RQ5 How do symmetry and idempotency constrain the class of meaningful aggregation functions on ordinal scales?

Key findings

• Nondecreasing and meaningful aggregation functions for the same input-output ordinal scales are exactly the lattice polynomials L_γ for some γ ∈ Γ_N.
• Continuous and meaningful aggregation functions for the same input-output ordinal scales are precisely the lattice polynomials L_γ.
• Nondecreasing and meaningful functions that are transformed by a constant or strictly monotone function g are of the form A = g ∘ L_γ.
• Continuous and meaningful functions under ordinal scales are exactly those of the form A = g ∘ L_γ with g continuous and strictly monotone.
• Symmetric, nondecreasing, and meaningful functions on ordinal scales are equivalent to g ∘ OS_k for some k ∈ N and g constant or strictly increasing.
• Symmetric, continuous, and meaningful functions on ordinal scales are exactly those of the form g ∘ OS_k with g continuous and strictly monotone.