The following master production schedule $($MPS$)$ for a group of three products planned to be produced in a month has been obtained.

$\backslash$table$[[$Product$,$Week,$\backslash$table$[[$Total$],[($units$)]]],[32,33,34,35],[1,180,180,,,360],[2,,,180,36,216],[3,,,,144,144],[$Total $($units$),180,180,180,180,720]]$

It has been observed that the above MPS is not feasible in terms of assembly capacity since:

The weekly capacity of the assembly department is $58$ hours, and

The unit assembly times for products $1,2,$ and $3$ are $0\mathrm{.}342,0\mathrm{.}294,$ and $0\mathrm{.}258$ hours, respectively.

Without changing the total planned production quantity of each product in the month $($i$.$e$.,$ without changing the quantities $360,216,$ and $144),$

$($a$)$ Formulate a mixed$-$integer linear programming model for obtaining a feasible solution, which allows the weekly production quantities to be changed, with an objective of minimizing the maximum unused capacity among four weeks.

$($b$)$ Using GAMS, solve your model in part $($a$),$ and tabulate the results and comment.