Grey Relational Entropy Calculation and Fractional Prediction of Water and Economy in the Beijing-Tianjin-Hebei Region

The implementation of the Beijing-Tianjin-Hebei coordinated development strategy promotes the rapid development of the regional economy, but the consequent water shortage has become a major concern. How to optimize the allocation of water resources, promote the cooperation of water resources among various water-using departments, and maximize the water efficiency of the limited water resources in the region has become the main issue of research. Thus, this paper mainly studies the entropy value and the entropy difference of the grey relational entropy between water resources and economic systems. First, use the grey correlation entropy method to calculate the existing data to explore the relationship between the two systems, then use the FGM(1, 1) model to predict the grey correlation entropy value of Beijing-Tianjin-Hebei in 2020–2024, and finally, calculate the entropy difference of the grey relation entropy for the region from 2015 to 2024. The results show the following: (i) The connection between the water resources system and the economic system in the Beijing-Tianjin-Hebei region is poor, the entropy value between the two will continue to decrease from 2015 to 2024, and the degree of coordination has shown a decreasing trend. (ii) The entropy change value between the water resources system and the economic system in the Beijing-Tianjin-Hebei region reflects a gradual and orderly change trend. The research results can provide reasonable suggestions for improving the correlation between water resources and economic systems for government departments, local residents, and industrial enterprises in the Beijing-Tianjin-Hebei region, ultimately realizing the sustainable development of water resources and economic systems.