Cloud seeding optimization under uncertainty: A Markov chain approach in a two-stage fuzzy-stochastic framework

The occurrence of sequential droughts and various forms of water shortages globally underscores the urgent need for sustainable water management solutions. In this context, cloud seeding has gained attention for its potential to enhance precipitation, yet its effectiveness is often uncertain due to complex cloud microphysics and atmospheric conditions. Acknowledging the inherent uncertainty in this endeavor, in this study, we employ a two-stage stochastic framework, integrating strategic decisions (facility location and network design) and operational realizations (seeding planning according to storm trajectories). Additionally, our model also considers fuzzy nature of seeding parameters. Above all, we develop a Markov chain procedure to mathematically model the prediction of expected increase in precipitation across cloud seeding decision-making processes. The integration of these stochastic methods into existing deterministic models from the literature results in a multi-objective Mixed-Integer Linear Programming (MILP) model designed to maximize rain probability and coverage while minimizing system-wide costs. To enhance the scalability and efficiency of the model, valid inequalities are developed to reduce the domain of binary variables. Additionally, a Lagrangian relaxation technique is proposed, yielding exact optimal solutions within reasonable timeframes and facilitating the handling of continuous space instances. Finally, a real-world case study in Iran demonstrates significant enhancements in precipitation predictions, with the Markov chain procedure showing an average 55 % increase in expected rain probability based on optimized seeding decisions. Scenario-based stochastic programming yields an 11.7 % value of stochastic solution and 16.5 % expected value of perfect information for cloud seeding initiatives.