helpQ$1$: As a water supply engineer, you are tasked to evaluate a water distribution system for a

rapidly growing urban area. The community of this area was reported to have a problem with the

low$-$pressure water supply system. The system must meet various conflicting requirements to

ensure sustainability, cost$-$efficiency, and reliability. The key parameters and requirements are as

follows:

The water source is located at an elevation of $200$ meters above sea level.

There are three communities at different elevations: Community $x$ at $300$ meters,

Community $Y$ at $400$ meters, and Community $Z$ at $500$ meters above sea level.

The water distribution pipes have varying diameters, roughness coefficients, and

lengths as they traverse the urban landscape.

The flow rate required for each community is as follows: Community $x-0\mathrm{.}15\frac{m^3}{s},$

Community $Y-0\mathrm{.}25\frac{m^3}{s},$ and Community Z $-0\mathrm{.}2\frac{m^3}{s}.$

The local environmental regulations require minimizing energy consumption, reducing

pipe material usage, and avoiding excessive pressure in the system.

The budget constraints necessitate a cost$-$effective design with the least amount of

pipe used and minimal pump operation.

There are existing land use restrictions that limit the path and size of pipes in some

areas.