## 1 Introduction  
As a critical element for sustaining ecological balance and supporting socio-economic development, the rational allocation of water resources constitutes the central challenge in achieving regional sustainable development [1, 2]. The interplay of global climate change, urbanization, and rapid economic growth has exacerbated the imbalance between water supply and demand, while issues such as water pollution and the degradation of aquatic ecosystems have become increasingly prominent [3; 4; 5; 6]. This situation is particularly severe in northern China. For example, in Jinzhong City, Shanxi Province, 2023 statistical data reveal that its per capita water availability falls below one-fifth of the national average, with intense competition between industrial and agricultural water use, and ecological baseflows are difficult to maintain [7; 8]. Therefore, under the constraint of limited total water resources, devising scientifically sound allocation strategies to meet the diverse water demands of society and the economy has become a pressing issue in water resource management. Water resource optimization, as a critical technical approach for sustainable water management, is characterized by multi-dimensional coupling, dynamic interactions, and spatial heterogeneity. Its core lies in constructing multi-objective optimization models to seek Pareto-optimal solutions within complex solution spaces defined by resource constraints, competitive user demands, and ecological redlines. Since the 1970s, academia has gradually established a theoretical framework for water resource system analysis, marked by [PERSON] multi-objective decision-making framework [9; 10] and [PERSON] stochastic programming models [11], forming a technical pathway of \"system modeling-scenario simulation-solution prioritization.\" In the early stages of research, water resource allocation typically focused on single-objective optimization targeting economic benefits, using methods such as linear programming [12] and dynamic programming [13]. These single-objective models did not consider social and ecological factors, lacking the capacity for comprehensive water resource development and utilization. As research has progressed, water resource optimization has shifted from pursuing a single objective to maximizing the comprehensive benefits of multiple objectives. Consequently, water resource system modeling has evolved from linear programming to mixed-integer nonlinear programming, with solution algorithms undergoing multiple transformations: initially, classical algorithms like the simplex method and dynamic programming laid the computational foundation; in the intermediate phase, intelligent algorithms, such as genetic algorithm (GA) [14; 15], particle swarm optimization algorithm (PSO) [16; 17], and simulated annealing algorithm (SA) [18; 19], were introduced to address high-dimensional non-convex problems; currently, we are in a phase of multi-algorithm integration and innovation, with multi-objective optimization algorithms, like the non-dominated sorting genetic algorithm-II (NSGA-II) [20], multiple objective particle swarm optimization (MOPSO) [21], and multi-objective evolutionary algorithm based on decomposition (MOEA/D) [22], gaining attention for their advantages in solving complex multi-objective optimization problems and being widely used in multi-objective water resource optimization studies. It is worth noting that, in recent years, such algorithms have been practically validated in typical regions of Shanxi Province. [PERSON] et al. [23] proposed a hybrid algorithm that integrates NSGA-II with an Adaptive Real-coded Simulated Binary Crossover (ARSBX) strategy for the optimal allocation of water resources in Taiyuan City. The resulting scheme achieved a coordination degree exceeding 0.8, significantly alleviating the city's water scarcity. [PERSON] et al.