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Multi-Weight Ranking for Multi-Criteria Decision Making

Andreas H Hamel, Daniel Kostner

TL;DR

This work addresses the dilemma of ranking among non-comparable multivariate alternatives in MCDM by introducing a cone-based ranking framework derived from cone distribution functions. The authors define point and set rankings, $r_{X,w}$ and $r_{X,C}$ (and their set analogues), that aggregate over a family of linear scalarizations rather than fixing a single weight vector, yielding a worst-case, monotone ordering with respect to a convex cone preorder $≤_C$. A key contribution is the explicit treatment of rank reversals under the cone-based ranking, along with conditions and interpretations, and an extension to ranking sets of alternatives, linking set optimization concepts to set-valued MCDM approaches. The framework also sketches practical pathways for machine learning, including level-set based classification, preference learning via cone alignment (e.g., SVM) and potential semi-supervised methods, thereby broadening the interaction between MCDM, statistics, and data-driven decision making.

Abstract

Cone distribution functions from statistics are turned into Multi-Criteria Decision Making tools. It is demonstrated that this procedure can be considered as an upgrade of the weighted sum scalarization insofar as it absorbs a whole collection of weighted sum scalarizations at once instead of fixing a particular one in advance. As examples show, this type of scalarization--in contrast to a pure weighted sum scalarization-is also able to detect ``non-convex" parts of the Pareto frontier. Situations are characterized in which different types of rank reversal occur, and it is explained why this might even be useful for analyzing the ranking procedure. The ranking functions are then extended to sets providing unary indicators for set preferences which establishes, for the first time, the link between set optimization methods and set-based multi-objective optimization. A potential application in machine learning is outlined.

Multi-Weight Ranking for Multi-Criteria Decision Making

TL;DR

This work addresses the dilemma of ranking among non-comparable multivariate alternatives in MCDM by introducing a cone-based ranking framework derived from cone distribution functions. The authors define point and set rankings, and (and their set analogues), that aggregate over a family of linear scalarizations rather than fixing a single weight vector, yielding a worst-case, monotone ordering with respect to a convex cone preorder . A key contribution is the explicit treatment of rank reversals under the cone-based ranking, along with conditions and interpretations, and an extension to ranking sets of alternatives, linking set optimization concepts to set-valued MCDM approaches. The framework also sketches practical pathways for machine learning, including level-set based classification, preference learning via cone alignment (e.g., SVM) and potential semi-supervised methods, thereby broadening the interaction between MCDM, statistics, and data-driven decision making.

Abstract

Cone distribution functions from statistics are turned into Multi-Criteria Decision Making tools. It is demonstrated that this procedure can be considered as an upgrade of the weighted sum scalarization insofar as it absorbs a whole collection of weighted sum scalarizations at once instead of fixing a particular one in advance. As examples show, this type of scalarization--in contrast to a pure weighted sum scalarization-is also able to detect ``non-convex" parts of the Pareto frontier. Situations are characterized in which different types of rank reversal occur, and it is explained why this might even be useful for analyzing the ranking procedure. The ranking functions are then extended to sets providing unary indicators for set preferences which establishes, for the first time, the link between set optimization methods and set-based multi-objective optimization. A potential application in machine learning is outlined.
Paper Structure (7 sections, 4 theorems, 13 equations, 12 figures)

This paper contains 7 sections, 4 theorems, 13 equations, 12 figures.

Table of Contents

  1. The ranking problem
  2. Cone ranking functions
  3. Rank reversal for cone ranking functions
  4. Discussion of the cone
  5. Extension to rankings of sets
  6. Classification models
  7. Conclusion and perspective

Key Result

Proposition 2.2

The functions $r_w$ and $r_C$ are strict ranking functions. Moreover, one has for an invertible matrix $A \in \mathrm{I R}^{d \times d}$ and a vector $b \in \mathrm{I R}^d$ where $AX+b = \{Ax^1+b, \ldots, Ax^N+b\}$. Finally, if $D \subseteq \mathrm{I R}^d$ is another closed convex cone with $C \subseteq D$, then $r_{X, C}(z) \leq r_{X, D}(z)$ for all $z \in \mathrm{I R}^d$; i

Figures (12)

  • Figure 2.1: Non-convex Pareto frontier
  • Figure 3.1: Rank reversal 1
  • Figure 3.2: Rank reversal 2
  • Figure 3.3: Student ranking with TOPSIS
  • Figure 3.4: Student ranking with $r_{X,\mathrm{I R}^2_+}$
  • ...and 7 more figures

Theorems & Definitions (6)

  • Definition 2.1
  • Proposition 2.2
  • Theorem 2.3
  • Definition 5.1
  • Proposition 5.2
  • Proposition 5.3