$*$Please answer fully and clearly$**($Multi$-$objective optimization, Pareto Frontier, $40\%)$

The State Highway Patrol $($SHP$)$ is responsible

for managing highway infrastructure across rural and suburban areas of the state. With a limited

number of maintenance crews available, the SHP needs to strategically allocate them to different

highway segments to ensure the safety and functionality of the infrastructure. In a small town in New

York, SHP has only $60$ officers to patrol $j=1,$dots,$10$ major highway segments. 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 require 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 $c_{i,j},$ the traffic density on segment $j$ during shift $i.$ A second is $a_{i,j},$ 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, and total coverage of

high accident shifts and coverage.

$($a$)$ Formulate the problem as a bi$-$objective integer linear programming with nonnegative integer

decision variables $x_{ij},$ the number of officers assigned to shifts i and segment $j.$

$($b$)$ Find the Pareto efficiency frontier by using the weighted objectives method.