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REDUCE – A Python Module for Reducing Inconsistency in Pairwise Comparison Matrices

Kuraś Paweł, Strzałka Dominik, Kowal Bartosz, Mazurek Jiří

Advances in Science and Technology. Research Journal

2023

Vol. 17, no 4

227--234

This paper introduces REDUCE, a Python module designed to minimize inconsistency in multiplicative pairwise comparisons (PC), a fundamental technique in Multi-Criteria Decision Making (MCDM). Pairwise comparisons are extensively used in various fields, including engineering science and numerical simulation methods, to compare different options based on a set of criteria. However, human errors in perception and judgment often lead to inconsistencies in pairwise comparison matrices (PCM). REDUCE addresses this issue by implementing several algorithms that identify and correct inaccurate data in PCMs, thereby reducing the inconsistency ratio. These algorithms do not require expert intervention, making REDUCE a valuable tool for both scientific research and small to medium enterprises that may not have access to costly commercial software or dedicated decision-making experts. The main functionality of the module is incorporating iterative algorithms for inconsistency reduction. The REDUCE library, written in Python and utilizing auxiliary libraries such as NumPy, SciPy, and SymPy, offers 21 functions categorized into data input helpers, consistency ratio (CR) reduction algorithms, PCM indexes, and support functions. Performance testing indicates that the library can efficiently handle matrices of varying sizes, particularly those ranging from 3x3 to 10x10, and its use significantly accelerates the process compared to spreadsheets, especially when dealing with large quantities of matrices. The library has already been used in several research papers and application tools, and its availability as a free resource opens up opportunities for small and medium-sized enterprises to leverage multi-criteria decision-making methods. Currently, there are no publicly available libraries for this solution. The authors believe that the proposed module may contribute to do better decision-making process in pairwise comparisons, not only for the circle of scientists but also for small and medium enterprises that usually cannot afford expensive commercial software and do not employ full-time experts in decision-making as they rely on the experience of their employees and free online resources. It should also contributes to the transition to Industry 4.0 and advances research in fields such as fuzzy logic, preference programming, and constructive consistent approximations.

The limit of inconsistency reduction in pairwise comparisons

Koczkodaj W. W., Szybowski J.

International Journal of Applied Mathematics and Computer Science

2016

Vol. 26, no. 3

721--729

This study provides a proof that the limit of a distance-based inconsistency reduction process is a matrix induced by the vector of geometric means of rows when a distance-based inconsistent pairwise comparisons matrix is transformed into a consistent PC matrix by stepwise inconsistency reduction in triads. The distance-based inconsistency indicator was defined by Koczkodaj (1993) for pairwise comparisons. Its convergence was analyzed in 1996 (regretfully, with an incomplete proof) and finally completed in 2010. However, there was no interpretation provided for the limit of convergence despite its considerable importance. This study also demonstrates that the vector of geometric means and the right principal eigenvector are linearly independent for the pairwise comparisons matrix size greater than three, although both vectors are identical (when normalized) for a consistent PC matrix of any size.