Measuring Weak Consistency and Weak Transitivity of Pairwise Comparison Matrices

Area of Science:

• Decision Analysis
• Operations Research
• Behavioral Economics

Background:

• Rational decision-making assumes transitive preferences, yet measuring these properties in real-world scenarios is challenging.
• Pairwise Comparison Matrices (PCMs) are central to methods like the Analytic Hierarchy Process (AHP), but assessing their consistency and transitivity is crucial.
• Existing methods for quantifying preference transitivity in PCMs have limitations.

Purpose of the Study:

• To develop and present novel methods for measuring weak consistency (w-consistency) and weak transitivity (w-transitivity) in PCMs.
• To investigate the properties of PCMs exhibiting w-consistency and w-transitivity.
• To propose an optimization model for modifying non-transitive PCMs and establish a decision-making framework based on w-transitivity.

Main Methods:

• Studied inherent properties of PCMs with w-consistency and w-transitivity.
• Developed novel quantification indices for w-consistency and w-transitivity.
• Utilized Particle Swarm Optimization (PSO) to solve an optimization model for modifying PCMs.
• Established a decision-making model prioritizing w-transitivity.

Main Results:

• Proposed efficient methods for computing w-consistency and w-transitivity indices.
• Demonstrated that the new indices effectively reflect inherent relations within PCMs.
• The optimization model successfully modified non-transitive PCMs to achieve desired transitivity properties.
• Numerical examples validated the effectiveness of the developed methods and models.

Conclusions:

• The developed quantification indices for w-consistency and w-transitivity are computationally efficient and accurately represent PCM properties.
• The proposed optimization model and decision-making framework offer practical tools for improving decision-making processes.
• The study contributes novel metrics and methodologies for assessing preference transitivity in AHP and related decision-making contexts.