**1. Introduction**

Due to population growth, economic development and consumption upgrade, global water consumption has increased by six times, and it has been continuing to grow steadily at an annual rate of about 1% during the past 100 years [1]. All of these would lead to the water shortage problem that is already pessimistically even severer, and seriously hinders the sustainable development of social economy. Managing water resources is an effective way to deal with the above challenges. However, in the process of management, experts and governors have encountered a lot of problems [2–6], such as dynamic variability and uncertainty, which are thorny and inevitable. Besides, in areas with water shortage, when the available water cannot meet the needs, unreasonable water allocation will lead to conflicts among users [7]. Therefore, it is definitely necessary to put forward a comprehensive model to deal with dynamic variabilities and uncertainties in water resources system as well as the contradiction between different users, so as to improve the management efficiency and the users' satisfaction.

The water resources system is of great complexities involving many uncertain factors, such as water use efficiency, water demand, pollutant discharge, water supply capacity and so on, and these uncertain factors could affect the structure for the optimal allocation model of water resources and resulting solutions [8–11]. Previously, scholars in related fields have got fruitful achievements in dealing with uncertainties in water resources management. For the optimization under uncertainties, mathematical methods that are commonly used

Fuzzy-Interval Dynamic Optimization Model for Regional Water Resources Allocation under Uncertainty. *Sustainability* **2022**, *14*, 1096. https://doi.org/10.3390/ su14031096

**Citation:** Suo, M.; Xia, F.; Fan, Y. A

Academic Editors: Julian Scott Yeomans and Mariia Kozlova

Received: 12 November 2021 Accepted: 12 January 2022 Published: 18 January 2022

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**Copyright:** © 2022 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/).

include stochastic programming [12], fuzzy programming [13], interval programming [14], and various coupling programming methods [15]. Among them, fuzzy programming can deal with the conflicts among multi-objective functions well, making it a new flexible planning problem [16], while the interval programming can reflect the uncertain coefficients in the form of intervals, and convert the uncertain planning problems into deterministic planning problems [17]. Both of them are helpful to solve uncertain planning problems. Based on these models, a large number of integrated models have been developed to optimize the allocation of water resources, such as uncertain two-stage stochastic water resources optimal allocation model [18], improved interval linear optimal allocation model [19], chance constrained water resources optimal allocation model [20], multi-objective interval linear water resources optimal allocation model [21], and so on [22–24]. Li et al. [25] proposed a multi-objective water resources optimal allocation model under uncertainties by integrating constrained programming, semi-infinite programming, integer programming and interval linear programming. Suo et al. [26] presented an approach for interval multiobjective planning by coupling fuzzy programming and improved two-step method, and then proved the objectivity and stability of this method by comparing it with the weighted sum method. Li et al. [27] formulated a new two-stage random interval parameter fuzzy planning strategy model by considering various uncertainties in planning and management of water resources and water environment systems, which was then applied to reveal the relationship between local economic goals and environmental goals. The above-mentioned models can well deal with the uncertainty in data acquisition in the system. However, they are insufficient to handle the dynamic features in the allocation process of water resources.

In the process of water resources optimization, it is essential to give a full consideration to dynamic characteristics, and thus provide the best scheme for water distribution at different stages of the planning period [28]. Dynamic programming cannot only solve the optimization problem of multi-stage decision process in water resources allocation [29], but also obtain the optimal strategy of the whole process and the optimal sub-strategy of each stage [30–34]. Peng [35] established a multi-objective dynamic water resources allocation model to achieve a dynamic balance for the optimal water resources allocation by using a modified simplex method with the addition of a time variable. Feng [36] set up a multiobjective dynamic water resources optimization configuration model and introduced the satisfaction function to realize the dynamic balance of optimal allocation of water resources on the time scale. Ramírez et al. [37] used stochastic dynamic programming to provide release decisions for each stage, and combined genetic algorithm and reservoir operation simulation program to obtain the annual release curve. These models proposed above are able to solve the multi-stage decision-making problem and get satisfactory allocation results. However, they took less consideration for uncertainties in water resources system.

Therefore, in order to comprehensively consider uncertainties and dynamic variability in the water resources system, a fuzzy-interval dynamic (FIDP) optimal allocation model is proposed in this study by integrating fuzzy-interval linear programming (FILP) and dynamic programming (DP) into a general framework. In addition, in order to realize the fairness of water resources allocation, the satisfaction function is added as one objective function to reduce the contradictions among users. The main innovative points of this study can be summarized as: (i) By introducing FILP into the FIDP model, the uncertainty coefficients and constraints, such as water use efficiency and water demand in water resources system, can be reflected in the form of interval numbers, which would make the results more accurate and reasonable. (ii) By introducing DP into FIDP model, not only the annual optimization scheme, but also the detailed water distribution scheme of each stage in planning year can be obtained. (iii) For FIDP model, the principle of fairness for water users is added to the objective function, which can reduce the contradictions between government and water users, as well as among different water users. This model is then applied to Handan City, Hebei Province, China, where water volume is small and uneven, to pursue the maximization for social benefits, overall satisfaction of water users, and environmental benefits. Finally, the FIDP model is compared with the traditional FILP

model, to prove the dynamic superiority of the proposed model with stage changes. It is expected that this model would be helpful to optimize the allocation of regional water resources under uncertainties and dynamics to reduce water shortage and conflict, and promote sustainable development of local society and economy.

This paper is composed of the following parts. Section 2 expounds the generation process of the general FIDP method. Section 3 puts forward a specific FIDP model suitable for Handan city. Section 4 presents the result analysis, which briefly expounds the water consumption characteristics of users at different stages of the planning year, and then compares the proposed FIDP model with FILP model. Section 5 is the summary of this study.
