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New Approaches to Generalized Logistic Equation with Bifurcation Graph Generation Tool

Ćmil Michał, Strzałka Dominik, Grabowski Franciszek, Kuraś Paweł

Advances in Science and Technology. Research Journal

2024

Vol. 18, no 6

1--12

This paper propose two new generalizations of the logistic function, each drawing on non-extensive thermodynamics, the q-logistic equation and the logistic equation of arbitrary order respectively. It demonstrate the impact of chaos theory by integrating it with logistics equations and reveal how minor parameter variations will change system behavior from deterministic to non-deterministic behavior. As well, this work presents BifDraw – a Python program for making bifurcation diagrams using classical logistic function and its generalizations illustrating the diversity of the system's response to the changes in the conditions. The research gives a pivotal role to the logistic equation's place in chaos theory by looking at its complicated dynamics and offering new generalizations that may be new in terms of thermodynamic basic states and entropy. Also, the paper investigates dynamics nature of the equations and bifurcation diagrams in it which present complexity and the surprising dynamic systems features. The development of the BifDraw tool exemplifies the practical application of theoretical concepts, facilitating further exploration and understanding of logistic equations within chaos theory. This study not only deepens our comprehension of logistic equations and chaos theory but also introduces practical tools for visualizing and analyzing their behaviors.

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.