$x_{i}be$ the $0-1$ which $is1if$

a camera $is$ installed $at$ opening $i,$ and $0$ otherwise, for $i=1,2,$dots,$13.Mi\frac{n}{b}ar(x_1+x_2+x_3+x_4+x_5+x_6+x_7+x_8+x_9+x_{10}+x_{11}+x_{12}+x_{13})$

$s.t.$

Room $1$

Room $(\frac{x_6+x_8+x_{12}}{2}1)$

Room $3x_1+x_2+x_{3}1$

Room $(\frac{x_3+x_4+x_5+x_7}{4}1)$

Room $(\frac{x_7+x_8+x_9+x_{10}}{5}1)$

Room $6$

Room $7x_2+x_5+x_9+x_{11}1$

Room $8x_{11}+x_{13}1$

$x_i=0,1,$ for $i=1,2,$dots,$13$

$(b)$ Solve the model formulated $in$ part $(a)to$ determine how many two$-$way cameras $to$ purchase and where they should $be$ located.

The qallery should install$|$

~~ cameras with Room $7x_2+x_5+x_9+x_{11}1$

Room $8,x_{11}+x_{13}1$

$x_i=0,1,$ for $i=1,2,$dots,$13$

$(b)$ Solve the model formulated $in$ part $(a)to$ determine how many two$-$way cameras $to$ purchase and where they should $be$ located.

The gallery should install

cameras with

$,$ cameras with $(x_1,x_2,x_3,x_{4^{\text{'}}},x_{5^{\text{'}}}{x}_{6^{\text{'}}}{x}_7,x_{8^{\text{'}}}{x}_{9^{\text{'}}}{x}_{{10}^{\text{'}}}{x}_{{11}^{\text{'}}},x_{12},x_{13})=(1,7,11,12x)$

$(c)$ Suppose that management wants $to$ provide additional security coverage for room $7.$ Specifically, management wants room $7tobe$ covered $by$ two cameras. Which constraint

would have $to$ change?

Room $1$

Room $2$

Room $3$

Room $4$

Room $5$

Room $6$

Room $7$

Room $8$

What should the new constraint $be?$

$x_2+x_5+x_9+x_{11}2$

$(d)$ With the policy restriction specified $in$ part $(c),$ determine how many two$-$way camera systems will need $tobe$ purchased and where they will $be$ located.

The gallery should install

cameras with