Consider the scheduling problem solved in class:

A$3$: Scheduling Problem

$\backslash$table$[[$Period$,$Time,$\backslash$table$[[$Minimum number of$],[$officers on duty$]]],[1,12a-04a,5],[2,04a-08a,7],[3,08a-12p,15],[4,12p-04p,8],[5,04p-08p,12],[6,08p-12a,9]]$

Union contracts stipulate that all employees work $8$ consecutive hours.

Objective: Hire the minimum number of officers need to cover all shifts.

The LP formulation of this problem is given below.

Decision Variables:

X$1$: Number of employees who start at $12$ a$.$m$.$

X$2$: Number of employees who start at $4$ a$.$m$.$

X$3$: Number of employees who start at $8$ a$.$m$.$

X$4$: Number of employees who start at $12$ p$.$m$.$

X$5$: Number of employees who start at $4$ p$.$m$.$

X$6$: Number of employees who start at $8$ p$.$m$.$

Objective Function:

Min. $Z=x1+x2+x3+x4+x5+x6$

Constraints:

$1$

$x1+x65$

$x1+x27$

$x2+x315$

$x3+x48$

$x4+x512$

$x5+x69$

$x1,x2,$dots,$x60$ and integer

Solve this problem using AMPL. Include the content of AMPL $.$mod file $($and AMPL $.$dat file if you use it$)$ and the AMPL output that indicates the optimal solution and the optimal value into the PDF file which you will submit through Teams.

Note:

To define an integer variable in AMPL, use the command "integer".

E$.$g$.$: var x integer $0$;