This is a "Restaurant work scheduling problem" where we need to determine the minimum number of waiters and busboys required for each shift to meet the staffing requirements while minimizing total staff.

The problem provides six time periods and the minimum number of workers needed for each period. The workers start their shifts at specific times (3 A.M., 7 A.M., 11 A.M., 3 P.M., 7 P.M., 11 P.M.) and work for 8-hour shifts.

Let's define $X_i$ as the number of workers starting their shifts in period $i$, where $i = 1, 2, 3, 4, 5, 6$. Based on this, we can set up equations to minimize the total number of workers subject to meeting the staffing requirements for each period.

Here is how each time period depends on the shifts:

• Period 1 (3 A.M. - 7 A.M.) requires 3 workers. Only workers starting at 3 A.M. contribute to this period.
• Period 2 (7 A.M. - 11 A.M.) requires 12 workers. Workers starting at 3 A.M. and 7 A.M. cover this period.
• Period 3 (11 A.M. - 3 P.M.) requires 16 workers. Workers starting at 7 A.M. and 11 A.M. cover this period.
• Period 4 (3 P.M. - 7 P.M.) requires 9 workers. Workers starting at 11 A.M. and 3 P.M. cover this period.
• Period 5 (7 P.M. - 11 P.M.) requires 9 workers. Workers starting at 3 P.M. and 7 P.M. cover this period.
• Period 6 (11 P.M. - 3 A.M.) requires 3 workers. Workers starting at 7 P.M. and 11 P.M. cover this period.

We can set up the following system of linear inequalities based on the table:

1. $X_1 \geq 3$
2. $X_1 + X_2 \geq 12$
3. $X_2 + X_3 \geq 16$
4. $X_3 + X_4 \geq 9$
5. $X_4 + X_5 \geq 9$
6. $X_5 + X_6 \geq 3$

The objective is to minimize the total number of workers:

$\text{Minimize } X_1 + X_2 + X_3 + X_4 + X_5 + X_6$

I can now solve this system to find the optimal staffing solution. Would you like me to proceed with solving these inequalities or would you like to explore another part of the problem?