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In order to stop and reverse land degradation and curb the loss of biodiversity, the United Nations 2030 Agenda for Sustainable Development proposes to combat desertification. In this paper, a fractional vegetation–water model in an arid flat environment is studied. The pattern behavior of the fractional model is much more complex than that of the integer order. We study the stability and Turing instability of the system, as well as the Hopf bifurcation of fractional order α , and obtain the Turing region in the parameter space. According to the amplitude equation, different types of stationary mode discoveries can be obtained, including point patterns and strip patterns. Finally, the results of the numerical simulation and theoretical analysis are consistent. We find some novel fractal patterns of the fractional vegetation–water model in an arid flat environment. When the diffusion coefficient, d, changes and other parameters remain unchanged, the pattern structure changes from stripes to spots. When the fractional order parameter, β , changes, and other parameters remain unchanged, the pattern structure becomes more stable and is not easy to destroy. The research results can provide new ideas for the prevention and control of desertification vegetation patterns. [ABSTRACT FROM AUTHOR]

As consumers become ever more conscious of environmental issues, socially responsible corporate practices, and government regulations, companies are increasingly motivated to incorporate reverse logistics (RLs) into their operations, thus raising the question of provider selection. In previous studies, the food industry generally lacked a systematic reference method for RLs provider selection, especially during the post-COVID-19 pandemic. This study aims to develop a comprehensive approach that combines a technique for order preference by similarity to ideal solution (TOPSIS) and multi-segment goal programming (MSGP) models to select optimal RLs providers. Furthermore, this method will enable decision makers (DMs) to evaluate and select the best RLs provider considering the limited resources of the business. This approach allows DMs to consider both qualitative and quantitative criteria, set multiple target segmentation expectations, and achieve optimal RLs provider selection. This study also provides case studies of applications by food manufacturers. The main finding is that considering multiple criteria in making a decision produces better results than using a single criterion. [ABSTRACT FROM AUTHOR]

We developed a maximum profit inventory model with a generalized deterioration rate where the selling rate is dependent on the inventory level that is an extension of two published papers. A complete solution structure is provided to decide the optimal solution with reasonable conditions supported by numerical examples, and then we prove that the optimal solution is independent of the demand pattern. Numerical examples are provided to illustrate our findings. In a previously published paper, three examples had symmetric conditions to decide the local maximum solution. Our approach provides a reasonable explanation for this symmetric phenomenon. Our findings will help researchers develop new inventory models in the future. [ABSTRACT FROM AUTHOR]