In this note, we point out an error to the proof of Theorem 3.4 in Zhang and Xu (Int J Intell Syst 2014;29(12):1061–1078) by a counterexample. We find that the inequality (i.e., ) with respect to the degrees of indeterminacy of any three Pythagorean fuzzy numbers in the proof of Theorem 3.4 in Zhang and Xu's paper is not valid. A new proof is provided in this note.

The scientific and reasonable evaluation of the carrying capacity of water resources is of guiding significance for solving the issues of water resource shortages and pollution control. It is also an important method for realizing the sustainable development of water resources. Aiming at an evaluation of the carrying capacity of water resources, an evaluation model based on the cloud model theory and evidential reasoning approach is studied. First, based on the existing indicators, a water resources evaluation index system based on the pressure-state-response (PSR) model is constructed, and a classification method of carrying capacity grade is designed. The cloud model theory is used to realize the transformation between the measured value of indicators and the degree of correlation. Second, to obtain the weight of the evaluation index, the weight method of the index weights model based on the entropy weight method and evidential reasoning approach is proposed. Then, the reliability distribution function of the evaluation index and the graded probability distribution of the carrying capacity of water resources are obtained by an evidential reasoning approach. Finally, the evaluation method of the carrying capacity of water resources is constructed, and specific steps are provided. The proposed method is applied to the evaluation of water resources carrying capacity for Hunan Province, which verifies the feasibility and effectiveness of the method proposed in the present study. This paper applies this method of the evaluation of the water resources carrying capacity of Hunan Province from 2010 to 2019. It is concluded that the water resources carrying capacity of Hunan Province belongs to III~V, which is between the critical state and the strong carrying capacity state. The carrying capacity of the province’s water resources is basically on the rise. This shows that the carrying capacity of water resources in Hunan Province is in good condition, and corresponding protective measures should be taken to continue the current state.