Question $2$

Table $2\mathrm{.}1$ shows the viable connection and distance $($in $km)$ between water storage

facilities $($nodes $a,b,$dots,$g)$ that serves the community within its vicinity. Note that the

distance values are symmetrical about the diagonal, e$.$g$.,$ distance between nodes a and

$b$ is the same as between $b$ and $a,$ so the latter is omitted. Underground pipe has yet to

be laid and it is the city planner's objective to ensure that all water storage facilities

should be connected so that water supply is available to everyone in the community,

but it should cost as little as possible.

Table $2\mathrm{.}1$

Question $2$a

Implement an appropriate network model using the information in Table $2\mathrm{.}1$ by

drawing the network model. Recommend how the nodes should be connected to achieve

the city planner's objective by drawing the final network model.

$(17$ marks$)$

Question $2$b

Consider the following scenarios:

i$.$ The engineering team has informed that the soil condition in the original connection

between nodes $f$ and $g$ is unstable. A new direct connection between the nodes has to

be built around it and the new distance is $x.$ Verify the range of values for $x$ such that

the original solution in $Q2(a)$ remains unchanged.

$(4$ marks$)$

ii$.$ A new water storage facility, node $Z,$ with viable connections to nodes a and $d$ has

been proposed. The distance between node $Z$ and node a is $2km,$ while the connection

between node $Z$ and node $d$ is $km.$ Verify the range of values for $y$ such that the

original solution in $Q2(a)$ is still part of the optimal solution for the new system.

$(4$ marks$)$