Metro Stations are to be located over an area consisting of $8$ zones. $12$

potential sites are available for locating the metro stations along with populations $($in thousands$)$

of each zone. The construction cost of one metro station is $c$.$

Zone #

Metro Stations Sites

$123456789101112$

$13\mathrm{.}506003\mathrm{.}8002\mathrm{.}6302\mathrm{.}5$

$21\mathrm{.}7503\mathrm{.}21\mathrm{.}201\mathrm{.}7503\mathrm{.}10000$

$300003\mathrm{.}503\mathrm{.}20023\mathrm{.}50$

$403\mathrm{.}303002\mathrm{.}7501\mathrm{.}75013\mathrm{.}75$

$5341\mathrm{.}500002\mathrm{.}501\mathrm{.}510$

$600003\mathrm{.}523\mathrm{.}402\mathrm{.}3002$

$7000000000200$

$801\mathrm{.}22\mathrm{.}4002040000$

a$)[4$ marks$]$ Determine the locations of the metro stations based on a maximum of $4$ metro

stations. $($show all steps$)$

b$)[7$ marks$]$ Provide a full formulation $($do not solve$)$ for the above problem considering

that the metro stations should serve all the given zones. In addition, the following

restrictions should be considered: only two metro stations of $10,11,$ and $12$ can be

selected. If metro station $2$ is selected, then metro stations $4,5$ and $6$ should be selected.

Zone $1$ must be covered by at least $2$ metro stations.