Let us define AM$3$ to represent number of workers starting work at $3$ AM$.$ Note that we have six possible starting times. So our variables would be AM$3,$ AM$7,$ AM$11,$ PM$3,$ PM$7,$ and PM$11.$ Each worker works eight hours or covers two four work periods. The easiest way to model scheduling problems like this one is to first build a work coverage table. The columns represent the decision variables or the shift identification. The rows represent various work periods. An entry of $1$ in the table shows a worker in a particular shift covers that work period. A blank means that a worker does not work in that period.

Work Periods$/$Shifts $$ AM$3$ AM$7$ AM$11$ PM$3$ PM$7$ PM$11$ Requirements

$3$ AM to $7$ AM $113$

$7$ AM to $11$ AM $1112$

$11$ AM to $3$ PM $1116$

$3$ PM to $7$ PM $119$

$7$ AM to $11$ PM $1111$

$11$ PM to $3$ AM $114$