Citation: Lei, T.; Li, R.Y.M.; Deeprasert, J.; Fu, H. Dynamics and Complexity Analysis of Fractional- Order Inventory Management System Model. Fractal Fract. 2024, 8, 258. https://doi.org/10.3390/ fractalfract8050258 Academic Editor: António Lopes Received: 5 February 2024 Revised: 19 April 2024 Accepted: 22 April 2024 Published: 26 April 2024 Copyright: © 2024 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). fractal and fractional Article Dynamics and Complexity Analysis of Fractional-Order Inventory Management System Model Tengfei Lei 1,2, * , Rita Yi Man Li 3 , Jirawan Deeprasert 2 and Haiyan Fu 1, * 1 Collaborative Innovation Center of Memristive Computing Application (CICMCA), Qilu Institute of Technology, Jinan 250200, China 2 Rattanakosin International College of Creative Entrepreneurship, Rajamangala University of Technology Rattanakosin, Rattanakosin, Bangkok 10400, Thailand; jirawan.dee@rmutr.ac.th 3 Sustainable Real Estate Research Center, Department of Economics and Finance, Hong Kong Shue Yan University, Hong Kong 999077, China; ymli@hksyu.edu * Correspondence: leitengfei2017@qlit.edu.cn (T.L.); fuhaiyan2018@qlit.edu.cn (H.F.) Abstract: To accurately depict inventory management over time, this paper introduces a fractional inventory management model that builds upon the existing classical inventory management frame- work. According to the definition of fractional difference equation, the numerical solution and phase diagram of an inventory management system are obtained by MATLAB simulation. The influence of parameters on the nonlinear behavior of the system is analyzed by a bifurcation diagram and largest Lyapunov exponent (LLE). Combined with the related indexes of time series, the complex characteristics of a quantization system are analyzed using spectral entropy and C0. This study concluded that the changing law of system complexity is consistent with the LLE of the system. By analyzing the influence of order on the system, it is found that the inventory changes will be periodic in some areas when the system is fractional, which is close to the actual conditions of the company’s inventory situation. The research results of this paper provide useful information for inventory managers to implement inventory and facility management strategies. Keywords: fractional-order discrete system; bifurcation diagram; complexity; chaos 1. Introduction In management operations, queuing, inventory, planning and scheduling systems produce chaos under different management decision rules. Inventory shows strong chaotic characteristics with time; that is, inventory is neither periodic nor random [1]. Chaotic behavior makes forecasting more difficult, leading to new tools development, to investigate whether the time series data are chaotic [2–4]. As an important part of the supply chain, inventory is directly related to the interests of enterprises. The ultimate goal of logistics is to minimize costs such as inventory ex- penses. Lei et al. [5] simplified a three-dimensional discrete system into a two-dimensional discrete system for the discrete inventory management model and analyzed the nonlinear characteristics of the inventory management model using fractional complexity. Many professional scholars have studied this model [6–9]. In 2001, Yao et al. [6] adopted the stability theory of differential equations and imple- mented feedback control with a variable parameter structure to manage multi-parameter inventory. They successfully controlled the chaotic model of inventory management. In 2003, Yao et al. [6] and Chen et al. [7] analyzed the chaotic and periodic characteristics of the inventory management system using a phase diagram. They improved the adaptive control method based on the Lyapunov approach and used it to effectively control the chaotic inventory management system. Hua et al. [8] conducted a study on a specific inven- tory management model and proved that the system produces Neimark–Sacker bifurcation and the asymptotic expression of the invariant ring at the fixed point, using discrete-time Fractal Fract. 2024, 8, 258. https://doi.org/10.3390/fractalfract8050258 https://www.mdpi.com/journal/fractalfract