Optimization of Water Management for Reducing Health Risks in the MENA Region: A Lagrange Multiplier Approach

The effective management of essential resources like water and public health presents a significant challenge, particularly in the Middle East and North Africa (MENA) region. Faced with growing environmental and health pressures, there is a critical need to optimize intervention strategies to minimize costs while maximizing societal and environmental benefits. This study develops a rigorous theoretical framework, using the Lagrange multiplier optimization method, to guide public decisions on pollution control and health risk management expenditures. The model minimizes total costs—combining control investments and residual damage costs—within technical and economic constraints, yielding explicit optimal solutions. Results indicate that optimal spending is proportional to the initial contamination level and hinges on both technological efficiency and the marginal costs of damages. A minimal level of residual contamination always remains, reflecting a necessary cost–benefit trade-off. The optimal total cost depends quadratically on the initial pollution, highlighting the economic imperative of early action. This approach provides an operational framework for policymakers, emphasizing investment in advanced technologies and balanced prevention–remediation strategies tailored to the specific constraints of MENA nations, thereby supporting sustainable improvements in quality of life and the environment.