Water of sufficient quantity and quality is indispensable for multiple purposes like domestic use, irrigated agriculture, hydropower generation and ecosystem functioning. However, in many regions of the world water availability is limited and even declining. Moreover, water availability is variable in space and time so that it does not match with the spatio-temporal use pattern of the water consumers. To overcome the temporal discrepancy between availability and consumption, reservoirs are constructed. Monitoring and predicting the water available in the reservoirs, the needs of the consumers and the losses throughout the water distribution system are necessary requirements to fairly allocate the available water for the different consumers. In this article, the water allocation problem is considered as a Network Flow Optimization Problem (NFOP) to be solved by a spatio-temporal optimization approach using Mixed Integer Linear Programming (MILP) techniques

The allocation of water flowing through a river-with-reservoirs system to optimally meet spatially distributed and temporally variable demands can be conceived as a network flow optimization (NFO) problem and addressed by linear programming (LP). In this paper, we present an extension of the strategic NFO-LP model of our previous model to a mixed integer linear programming (MILP) model to simultaneously optimize the allocation of water and the location of one or more new reservoirs; the objective function to minimize only includes two components (floods and water demand), whereas the extended LP-model described in this paper, establishes boundaries for each node (reservoir and river segments) and can be considered closer to the reality. In the MILP model, each node is called a “candidate reservoir” and corresponds to a binary variable (zero or one) within the model with a predefined capacity. The applicability of the MILP model is illustrated for the Machángara river basin in the Ecuadorian Andes. The MILP shows that for this basin the water-energy-food nexus can be mitigated by adding one or more reservoirs.

Water in sufficient quantity and quality is indispensable for multiple purposes like domestic and industrial use, irrigated agriculture, hydropower generation and ecosystem functioning. In many regions of the world, water availability is limited and even declining. Moreover, water availability is variable in space and time and often does not match with the spatio-temporal demand pattern. To overcome the temporal discrepancy between availability and consumption, reservoirs are constructed. Monitoring and predicting the water available in the reservoirs, the needs of the consumers and the losses throughout the river and water distribution system are necessary requirements to fairly allocate the available water to the different users, prevent floods and ensure sufficient water flow in the river. In this paper, this surface water allocation problem is considered a Network Flow Optimisation Problem (NFOP) solved by spatio-temporal optimisation using linear programming techniques.

Efficient surface water allocation in reservoir-equipped basins is essential for balancing competing demands within the Water–Energy–Food (WEF) nexus. This study investigated the applicability of Particle Swarm Optimization (PSO) for optimizing water distribution in the Machángara River Basin, Ecuador; a complex, constraint-rich hydrological system. Implemented via the Pymoo package in Python, the PSO model was evaluated across calibration, validation, and execution phases, and benchmarked against exact methods, including Linear Programming (LP) and Mixed Integer Linear Programming (MILP). The results revealed that standard PSO struggled to satisfy equality constraints and yielded suboptimal solutions, with elevated penalty costs. Despite incorporating MILP-inspired encoding and repair functions, the algorithm failed to identify feasible solutions that met operational requirements. The execution phase, which includes reservoir construction decisions, resulted in a total penalty exceeding EUR 164.95 billion, with no improvement observed from adding reservoirs. Comparative analysis confirmed that LP and MILP outperformed PSO in constraint compliance and penalty minimization. Nonetheless, the study contributes a reproducible implementation framework and a comprehensive benchmarking strategy, including synthetic test functions, performance metrics, and diagnostic visualizations. These tools can facilitate systematic evaluation of PSO’s behavior in high-dimensional, nonlinear environments and provide a foundation for future hybrid or adaptive heuristic models. The findings underscore the limitations of standard PSO in hydrological optimization but also highlight its potential when enhanced through hybridization. Future research should explore PSO variants that integrate exact solvers, adaptive control mechanisms, or cooperative search strategies to improve feasibility and convergence. This work advances the methodological understanding of metaheuristics in environmental resource management and supports the development of robust optimization tools under the WEF-nexus paradigm.