# New Orleans’s Mt. Sinai Hospital is a large, private, 600-bed facility complete with laboratories, operating rooms, and X-ray equipment. In seeking to increase revenues, Mt.Sinai’s administration has decided to make a 90-bedadditiononaportionofadjacent

land currently used for staff parking. The administrators feel that the labs, operating rooms, and X-ray department are not being fully utilized at present and do not need to be expanded to handle additional patients. The addition of 90 beds, however, involves
deciding how many beds should be allocated to the medical staﬀ (for medical patients) and how many to the surgical staﬀ (for surgical patients).
The hospital’s accounting and medical records departments have provided the following pertinent information. The average hospital stay for a medical patient is 8 days, and the average medical patient generates \$2,280 in revenues. The average surgical patient
is in the hospital 5 days and generates \$1,515 in revenues. The laboratory is capable of handling 15,000 tests per year more than it was handling. The average medical patient requires 3.1 lab tests, the average surgical patient 2.6 lab tests. Furthermore,the average medical patient uses 1 X-ray, the average surgical patient 2 X-rays. If the hospital were expanded by 90 beds, the X-ray department could handle up to 7,000 X-rays without signiﬁcant additional cost. Finally, the administration estimates that up
to 2,800 additional operations could be performed in existing operating-room facilities. Medical patients, of course, require no surgery, whereas each surgical patient generally
has one surgery performed.Formulate this problem so as to determine how many medical beds and how many
surgical beds should be added to maximize revenues. Assume that the hospital is open 365 days per year.

## To formulate this problem, we need to determine the number of medical beds and surgical beds that should be added to maximize revenues for Mt. Sinai Hospital.

Let's define the decision variables:
X = Number of medical beds to be added
Y = Number of surgical beds to be added

The objective is to maximize revenue, which can be expressed as:
Maximize: 2,280X + 1,515Y

Now let's consider the constraints:

1. Hospital capacity constraint:
The total number of beds after the addition should not exceed the hospital's total capacity of 600 beds.
X + Y ≤ 600

2. Laboratory capacity constraint:
The laboratory can handle 15,000 more tests per year. Considering the average number of lab tests for medical and surgical patients, we have:
3.1X + 2.6Y ≤ 15,000

3. X-ray department capacity constraint:
The X-ray department can handle up to 7,000 X-rays without significant additional cost. Considering the average number of X-rays for medical and surgical patients, we have:
X + 2Y ≤ 7,000

4. Operating room capacity constraint:
The existing operating-room facilities can handle up to 2,800 additional operations. Each surgical patient requires one surgery, so the number of surgical patients and surgeries performed should not exceed the capacity:
Y ≤ 2,800

5. Non-negativity constraints:
The number of medical beds (X) and surgical beds (Y) should be non-negative:
X ≥ 0
Y ≥ 0

Using these constraints, we can setup the linear programming problem as follows:

Maximize: 2,280X + 1,515Y

Subject to:
X + Y ≤ 600
3.1X + 2.6Y ≤ 15,000
X + 2Y ≤ 7,000
Y ≤ 2,800
X ≥ 0
Y ≥ 0

By solving this linear programming problem, the optimal values of X and Y will determine the number of medical beds and surgical beds that should be added to maximize revenues for Mt. Sinai Hospital.