Ben owns a cafe in a busy area of town. It is open Tuesday to Sunday from 7:00 am to 6:00 pm. He needs help with scheduling his back-of-house employees. Here is some information about his back-of-house staff and the requirements for the schedule we must create.

Ben is the owner who can work between 40 and 60 hours.
Maria is a salaried manager who can work between 40 and 55 hours.
Emily is an assistant manager who can work up to 30 hours a week.
Sarah is an hourly employee who works up to 40 hours per week.
George is an hourly employee who works up to 30 hours per week.
Please keep in mind these requirements for the back-of-house staff:

A manager must always be on duty (Ben, Maria, or Emily).
There must be two crew members in the kitchen at all times, from 4:00 a.m., when the kitchen prep starts until the kitchen closes at 6:00 p.m.
One manager must be scheduled until 7:00 p.m. each day to help the front of the house with their closing duties, supervise the cashouts, and create the work list for the next day.
Every kitchen member will need a day off. The cafe is closed on Monday, but each staff member, including Ben, requires an additional day off during the week.
Always be sure to overlap shifts so there can be a smooth transition. Shifts should overlap by at least a half hour to ensure a smooth transition and to keep two people working in the kitchen at all times. (For example, you can only have one person leaving at 11 and another entering at 11 a.m. if two other staff are working.)

You want to schedule 5 employees so that there are always 2 working 6 days a week from 4 am to 6 pm, with a manager working from 6–7 pm. Each can work no more than 5 days in a 6-day period, and any coming or going (except the last of the kitchen staff) must be accomplished when two others are working. Hour limits are given, and a manager must be on duty at all times.

Half-days

In a 6-day period with 2 staff on duty, there will be 24 half-day work periods. The hourly employees can work a total of 9 half-days, leaving 15 half-days for the 3 managers.

Emily can work 4 half-days, so the other two must work 11 half-days. Since their work schedule is limited to 5 days a week, one of the managers must work a full day.

The attached schedule shows a proposal. Overlaps are a minimum of 1/2 hour, and there are always two working in the kitchen when one comes in or leaves—except at the end of the day. The overlaps are generally from 11 to 11:30 am. (We might prefer overlaps during the lunch hours.)

Unfortunately, Ben works a full 15-hour day on the day before his day off.

All but Ben and Maria get their days off consecutively.

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Additional comment

The numbers at the bottom are the total work hours for the week. The shaded numbers correspond to the maximum work hours for that person.

Maria has the unfortunate situation of having to work an evening shift followed by a morning shift the next day. We did this so Ben's long day would be followed by a day off. This could be alleviated if Emily were to work the afternoon Thursday, while Maria worked the morning that day.

Working out the math is quite different from working the politics and interpersonal relationships. In a real cafe, those would tend to take priority.