Grave City is considering the relocation of several police substations to obtain better enforcement in high$-$crime areas. The locations under consideration together with the areas that can be covered from these locations are given in the following table.

Potential Locations

for Substations Areas Covered

L $1,3,5$

M $4,5,6$

N $1,5,7$

O $3,4,6$

P $1,5,6,7$

Q $1,2,5,7$

R $2,4,5$

$($a$)$

Formulate an integer programming model that could be used to find the minimum number of locations necessary to provide coverage to all areas.

Let xi $=$

$1$ if a substation is located at area i

$0$ otherwise

for

i $=$ L$,$ M$,$ N$,$ O$,$ P$,$ Q$,$ R$.$