[Paper Review] New solution principles of multi-criteria problems based on comparison standards

V. O. Groppen|ArXiv.org|Jan 21, 2005

Advanced Data Processing Techniques3 references3 citations

TL;DR

This paper introduces a novel method for solving multi-criteria optimization problems by transforming them into single-criterion models using comparison standards and interval-based rules, without requiring additional preference information. The approach ensures Pareto optimality of solutions through theoretical proofs and demonstrates universality and efficiency via illustrative examples.

ABSTRACT

Proposed is a new formal approach for solution of extreme multi-criteria problems transforming them into single-criterion mathematical models, without any additional information. Transforming rules are based on comparison standards and intervals between these standards and goal function values, corresponding to problem solution. Pareto-optimality of these solutions is proved by a number of theorems. Examples illustrating efficiency and universality of this approach are presented.

Motivation & Objective

• To address the challenge of solving extreme multi-criteria optimization problems where traditional methods require extensive preference information.
• To develop a formal transformation technique that converts multi-criteria problems into single-criterion models using comparison standards.
• To ensure the resulting solutions are Pareto-optimal without relying on additional decision-maker input.
• To demonstrate the universality and efficiency of the proposed method through concrete examples.
• To provide a theoretically grounded framework for multi-criteria decision-making based on interval comparisons.

Proposed method

• The method introduces comparison standards as reference points for evaluating objective function values in multi-criteria problems.
• It defines intervals between these standards and actual solution values to guide the transformation process.
• The transformation rules map multi-criteria problems into equivalent single-criterion models using interval-based comparisons.
• The approach relies on a formal mathematical structure that preserves optimality properties under the transformation.
• Pareto optimality of the solutions is established through a series of theorems based on the defined comparison framework.
• The method operates without requiring additional preference information, such as weights or utility functions.

Experimental results

Research questions

• RQ1 How can multi-criteria optimization problems be transformed into single-criterion problems without additional preference data?
• RQ2 What formal rules based on comparison standards ensure the preservation of Pareto optimality in the transformed model?
• RQ3 How can interval-based comparisons between standards and solution values guide effective problem transformation?
• RQ4 In what ways does the proposed method demonstrate universality and efficiency across diverse problem instances?
• RQ5 What theoretical foundations support the Pareto optimality of solutions derived via this comparison-based approach?

Key findings

• The proposed method successfully transforms extreme multi-criteria problems into single-criterion models using only comparison standards and interval relationships.
• Pareto optimality of the solutions is rigorously proven through multiple theorems, establishing theoretical validity.
• The approach does not require any additional preference information, such as weights or utility functions, enhancing its practical applicability.
• Examples in the paper illustrate the method’s efficiency and broad applicability across different types of multi-criteria problems.
• The transformation process is consistent and systematic, enabling reliable solution generation under the given framework.
• The method provides a universal solution principle applicable to a wide range of multi-criteria optimization scenarios.