Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Lexicographic optimization-based approaches to learning a representative model for multi-criteria sorting with non-monotonic criteria

Zhen Zhang, Zhuolin Li, Wenyu Yu

TL;DR

The paper tackles multi-criteria sorting with non-monotonic criteria by learning a representative model through two lexicographic optimization-based approaches within a preference-disaggregation framework. It introduces transformation functions to map marginal values and category thresholds into a UTAdIS-like space, preserves sorting results, and builds constraint sets to model non-monotonicity. A consistency-checking mechanism and a minimum-adjustment procedure ensure reliable reference judgments, followed by two-stage lexicographic models that balance model complexity and discriminative power. Through a real-data illustrative example and extensive simulations, the approaches outperform several non-monotonic modeling baselines, demonstrating improved accuracy and robustness, with insights into parameter effects and practical applicability.

Abstract

Deriving a representative model using value function-based methods from the perspective of preference disaggregation has emerged as a prominent and growing topic in multi-criteria sorting (MCS) problems. A noteworthy observation is that many existing approaches to learning a representative model for MCS problems traditionally assume the monotonicity of criteria, which may not always align with the complexities found in real-world MCS scenarios. Consequently, this paper proposes some approaches to learning a representative model for MCS problems with non-monotonic criteria through the integration of the threshold-based value-driven sorting procedure. To do so, we first define some transformation functions to map the marginal values and category thresholds into a UTA-like functional space. Subsequently, we construct constraint sets to model non-monotonic criteria in MCS problems and develop optimization models to check and rectify the inconsistency of the decision maker's assignment example preference information. By simultaneously considering the complexity and discriminative power of the models, two distinct lexicographic optimization-based approaches are developed to derive a representative model for MCS problems with non-monotonic criteria. Eventually, we offer an illustrative example and conduct comprehensive simulation experiments to elaborate the feasibility and validity of the proposed approaches.

Lexicographic optimization-based approaches to learning a representative model for multi-criteria sorting with non-monotonic criteria

TL;DR

The paper tackles multi-criteria sorting with non-monotonic criteria by learning a representative model through two lexicographic optimization-based approaches within a preference-disaggregation framework. It introduces transformation functions to map marginal values and category thresholds into a UTAdIS-like space, preserves sorting results, and builds constraint sets to model non-monotonicity. A consistency-checking mechanism and a minimum-adjustment procedure ensure reliable reference judgments, followed by two-stage lexicographic models that balance model complexity and discriminative power. Through a real-data illustrative example and extensive simulations, the approaches outperform several non-monotonic modeling baselines, demonstrating improved accuracy and robustness, with insights into parameter effects and practical applicability.

Abstract

Deriving a representative model using value function-based methods from the perspective of preference disaggregation has emerged as a prominent and growing topic in multi-criteria sorting (MCS) problems. A noteworthy observation is that many existing approaches to learning a representative model for MCS problems traditionally assume the monotonicity of criteria, which may not always align with the complexities found in real-world MCS scenarios. Consequently, this paper proposes some approaches to learning a representative model for MCS problems with non-monotonic criteria through the integration of the threshold-based value-driven sorting procedure. To do so, we first define some transformation functions to map the marginal values and category thresholds into a UTA-like functional space. Subsequently, we construct constraint sets to model non-monotonic criteria in MCS problems and develop optimization models to check and rectify the inconsistency of the decision maker's assignment example preference information. By simultaneously considering the complexity and discriminative power of the models, two distinct lexicographic optimization-based approaches are developed to derive a representative model for MCS problems with non-monotonic criteria. Eventually, we offer an illustrative example and conduct comprehensive simulation experiments to elaborate the feasibility and validity of the proposed approaches.
Paper Structure (25 sections, 4 theorems, 35 equations, 11 figures, 24 tables, 4 algorithms)

This paper contains 25 sections, 4 theorems, 35 equations, 11 figures, 24 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Threshold-based value-driven sorting procedure
  3. Problem formulation
  4. The proposed approaches
  5. Transformation functions
  6. Modeling the non-monotonic criteria in MCS problems
  7. Consistency check and preference adjustment
  8. Consistency check
  9. Preference adjustment
  10. The lexicographic optimization-based approaches
  11. The proposed algorithm
  12. Robustness analysis
  13. Illustrative example
  14. Implementation of the proposed approaches
  15. Further discussions
  16. ...and 10 more sections

Key Result

Proposition 1

By employing the transformation functions $f_v(\cdot)$ and $f_b(\cdot)$, the sorting result for alternatives remain unchanged after transformation.

Figures (11)

  • Figure 1: The flow chart of the proposed algorithm
  • Figure 2: The transformed marginal value functions obtained by Approach 1
  • Figure 3: The transformed marginal value functions obtained by Approach 2
  • Figure 4: The transformed marginal value functions obtained by Approach 1 ($s_j=5$)
  • Figure 5: The transformed marginal value functions obtained by Approach 1 ($s_j=6$)
  • ...and 6 more figures

Theorems & Definitions (13)

  • Definition 1
  • Definition 2
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Remark 1
  • Proposition 3
  • proof
  • Proposition 4
  • ...and 3 more