Case $4-3($Algo$)$

Shouldice Hospital in Canada is widely known for one thing$$hernia repair! In fact, that is the only operation it performs, and it performs a great many of them. Over the past two decades this small $78-$bed hospital has averaged $11,800$ operations annually. Last year, it had a record year and performed nearly $12,300$ operations.

A hernia repair operation at Shouldice Hospital is performed by one of the $12$ full$-$time surgeons assisted by one of seven part$-$time assistant surgeons. The first operations begin at $7$:$30$ AM each day, Monday through Friday. Surgeons generally take about one hour to prepare for and perform each hernia operation, and they operate on an average of at most four patients per day. This four patient per day limit on the average number of operations performed per surgeon has been found to be the best operating level for the hospital as it takes into account time the surgeons need for patient exams and consultations, updating medical charts, writing reports, traveling to professional conferences, vacations, and other times when they are performing other duties or are not available to perform surgeries. A given surgeon may perform more than four surgeries on a given day, but the average cannot exceed four without having adverse effects on overall hospital operations. The surgeons$$ day ends at $4$ p$.$m$.$ Although hernia repair operations are performed only five days a week, the remainder of the hospital is in operation continuously to attend to recovering patients.

The table below shows the room occupancy plan for the existing system. Each row in the table follows the patients who checked in on a given day. The columns indicate the number of patients in the hospital on a given night.