###### Abstract
It is also imperative to highlight that machine learning and modern simulation approaches are revolutionizing the optimization of water distribution networks, turning traditional systems into intelligent, adaptive infrastructures. Such technologies are seen as the brains behind the operation, allowing the network to not just function but to think and respond in real time. In this context, [PERSON] et al. [16] implemented deep reinforcement learning (DRL) to control pump operations. Their DRL agent learns optimal pump speeds based on instantaneous nodal pressure data, outperforming conventional methods with over 98% efficiency while doubling the speed. It is akin to having a seasoned operator who anticipates and adjusts to system demands instantly, enhancing both efficiency and responsiveness. On another front, [PERSON] et al. [17] tackled the persistent issue of leakage in urban water networks using multiscale neural networks combined with density-based clustering. By reducing complex pipe networks into manageable zones, they significantly improved leakage detection accuracy--by 78% compared with methods like SVM and KNN. This can be thought of as the transformation of a tangled web into organized neighborhoods, making it easier to pinpoint and address leaks promptly, thereby conserving water and reducing losses. Furthermore, [PERSON] et al. [18] explored the incorporation of big data and IoT sensors within underwater management systems. Their method utilizes real-time data collection through IoT to proactively monitor water usage and quality, employing a SCADA system for sustainable management. This is like giving the water network a sensory system,allowing it to detect and respond to issues before they escalate, promoting operational safety and sustainability in smart cities. Despite the advancements in optimizing water distribution networks, there remains a notable gap in existing models that study the performance of WDNs with fixed nodes and inflows--specifically those with constant consumption rates for each node. This study aims to fill this gap by proposing a novel mathematical model tailored to analyze the performance of WDNs characterized by fixed nodes and consumption rates. The model incorporates various parameters, including cost, demand, head losses, node elevations, design pressure, and water velocity, to assess their effects on optimizing these proposed WDNs. Additionally, this research introduces an optimization model that delves into the complex interplay between pressure and velocity within the network. By scrutinizing these dynamics, the study elucidates pathways to achieving optimal configurations that enhance operational efficacy and economic viability. The insights garnered from this analysis significantly advance the understanding of WDN behavior, offering substantive and practical implications for engineers and planners in the field. Ultimately, this work aims to inform more nuanced design and management strategies for urban water distribution systems, thereby contributing to the development of resilient and efficient infrastructures. The remaining parts of the paper are as follows. The methodology is described in Section 2. Section 3 discusses the results and the conclusions are pointed out in Section 4.