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 $1,2,5,7$

M $4,5,6$

N $3,4,6$

O $2,4,5$

P $1,5,6,7$

Q $1,5,7$

R $1,3,5$

$($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$.$

Min

Correct: Your answer is correct.

s$.$t$.$

Area $1$

Incorrect: Your answer is incorrect.

Area $2$

Incorrect: Your answer is incorrect.

Area $3$

Incorrect: Your answer is incorrect.

Area $4$

Incorrect: Your answer is incorrect.

Area $5$

Incorrect: Your answer is incorrect.

Area $6$

Incorrect: Your answer is incorrect.

Area $7$

Incorrect: Your answer is incorrect.

$($b$)$

Solve the problem in part $($a$).$ What locations should be given substations? $($Select all that apply.$)$