Q$3$a$.$ The State Highway Patrol $($SHP$)$ has only $60$ officers to patrol $j=1,$dots,$10$ major highway segments in rural and suburban areas of the state. Officers work $i=1,$dots,$7$

different shift patterns across a $24$ hour, $7$ day work week. Exceptions can be made on late$-$night shifts $i=1$ and $2,$ but all other shifts reguire at least $1$ officer on each

highway segment, and no segment ever receives more than $3.$ An SHP goal is to concentrate the patrol resources in times and places of greatest need. One measure of

this is $c_{\text{i}\text{j}\text{j}}$ the traffic density on segment $j$ during shift $i.$ A second is $a_{ij},$ the relative accident rate on segment $j$ during shift $i.$ The following table shows values derived from

history for both across shifts and segments.

SHP would like to decide how to feasibly allocate its officers to shifts and segments to maximize total coverage of congested shifts and segments, as well as total coverage

of high accident shifts and coverage.

$($a$)$ Formulate the problem as a multi$-$ objective ILP over nonnegative integer decision variables

$x_{i,j}=$ the number of officers assigned to each shift i and segment $j.($Closed form is enough, you can define indexed parameters $)$

$($b$)$ Show how to modify your model of $($a$)$ as a goal ILP with targets for total congestion $=120$ and for total accident level $=150.$ Weight under$-$satisfaction of the two

goals equallv. define all new decision variables. and annotate new obiectives and constraints to show their meanings.