Case $5-4($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 $108-$bed hospital has averaged $7,000$ operations annually. Last year, it had a record year and performed nearly $7,500$ 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 take 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 below table shows the number of operations with $108$ Beds $(36$ patients per day$).$ Each row in the table shows the number of patients who checked in on a given day. The columns indicate the number of patients in the hospital on a given day. Patients check in to the hospital the day before their operation is scheduled and stay there for three days.

CHECK$-$IN DAY BEDS REQUIRED

MONDAY TUESDAY WEDNESDAY THURSDAY FRIDAY SATURDAY SUNDAY

Monday $3636360000$

Tuesday $0363636000$

Wednesday $0036363600$

Thursday $0003636360$

Friday $0000000$

Saturday $0000000$

Sunday $3636000036$

Total $72108108108723636$

For example, the first row of the table shows that $36$ people checked in on Monday and were in the hospital for Monday, Tuesday, and Wednesday. By summing the column of the table for Wednesday, we can find that there are $108$ patients staying in the hospital that day.

The medical facilities at Shouldice consist of five operating rooms, a patient recovery room, a laboratory, and six examination rooms. An operation at Shouldice Hospital is performed by one of the $12$ full$-$time surgeons assisted by one of seven part$-$time assistant surgeons. Surgeons generally take about one hour to prepare for and perform each hernia operation.

Now look at the effect of increasing the number of beds by $50$ percent. Although financial data are sketchy, an estimate from a construction company indicates that adding bed capacity would cost about $$100,200$ per bed. In addition, the rate charged for the hernia surgery varies between about $$800$ and $$1,900($U$.$S$.$ dollars$),$ with an average rate of $$1,100$ per operation. The surgeons are paid a flat $$500$ per operation.

How many weeks would it take the hospital to payback its investments?

Note: Round your answer to $1$ decimal place.$\backslash$table$[[$Ch $5$ Cases i$,,,,,,$Saved,$],[\backslash$table$[[$patients who checked in on a given day. The columns indicate the number of patients in the hospital$],[$to the hospital the day before their operation is scheduled and stay there for three days.$]]],[\backslash$table$[[25],[$points$]],,$BEDS REQUIRED$],[$CHECK$-$IN DAY,MONDAY,TUESDAY,WEDNESDAY,THURSDAY,FRIDAY,SATURDAY,SUNDAY$],[$Monday$,36,36,36,,,,],[$Tuesday$,,36,36,36,,,],[$Wednesday$,,,36,36,36,,],[$Thursday$,,,,36,36,36,],[$eBook$,$Friday,,,,,,,$],[$Saturday$,,,,,,,],[$Sunday$,36,36,,,,,36],[$Total$,72,108,108,108,72,36,36]]$

For example, the first row of the table shows that $36$ people checked in on Monday and were in the hospital for Monday, Tuesday, and Wednesday. By summing the column of the table for Wednesday, we can find that there are $108$ patients staying in the hospital that day.

The medical facilities at Shouldice consist of five operating rooms, a patient recovery room, a laboratory, and six examination rooms. An operation at Shouldice Hospital is performed by one of the $12$ full$-$time surgeons assisted by one of seven part$-$time assistant surgeons. Surgeons generally take about one hour to prepare for and perform each hernia operation.

Now look at the effect of increasing the number of beds by $50$ percent. Although financial data are sketchy, an estimate from a construction company indicates that adding bed capacity would cost about $$100,200$ per bed. In addition, the rate charged for the hernia surgery varies between about $$800$ and $$1,900($U$.$S$.$ dollars$),$ with an average rate of $$1,100$ per operation. The surgeons are paid a