Question

Nurse Scheduling. Hospital administrators must schedule nurses so that the hospital’s patients are provided adequate care. At the same time, careful attention must be paid to keeping costs down. From historical records, administrators can project the minimum number of nurses required to be on hand for various times of day and days of the week. The objective is to find the minimum total number of nurses required to provide adequate care.

Nurses start work at the beginning of one of the four-hour shifts given below except for shift 6, and work for 8 consecutive hours. Hence, possible start times are the start of shifts 1 through 5. Also, assume that the projected required number of nurses factors in time for each nurse to have a meal break.
Formulate and solve the nurse scheduling problem as an integer program for one day for the data given below.
Hint: Note that exceeding the minimum number of needed nurses in each shift is acceptable so long as the total number of nurses overall shifts is minimized.
Shift Time Minimum Number of Nurses Needed
1 12:00 A.M. – 4:00 A.M. 10
2 4:00 A.M. – 8:00 A.M. 24
3 8:00 A.M. – 12:00 P.M. 18
4 12:00 P.M. – 4:00 P.M. 10
5 4:00 P.M. – 8:00 P.M. 23
6 8:00 P.M. – 12:00 A.M. 17
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Answer 1

To solve the nurse scheduling problem, we formulate an integer program using variables to represent the number of nurses at each shift. The objective is to minimize the total number of nurses, subject to constraints based on the minimum number of nurses needed for each shift. By solving this integer program, we can find the optimal schedule.

Step-by-step explanation:

To formulate the nurse scheduling problem as an integer program, we need to assign variables to represent the number of nurses starting at each shift. Let's say x1, x2, x3, x4, and x5 represent the number of nurses starting at shifts 1, 2, 3, 4, and 5 respectively. Since we want to minimize the total number of nurses overall shifts, our objective function is:

Minimize Z = x1 + x2 + x3 + x4 + x5

We also have the following constraints:

x1 + x6 ≥ 10

x2 + x1 ≥ 24

x3 + x2 ≥ 18

x4 + x3 ≥ 10

x5 + x4 ≥ 23

x1 + x5 ≥ 17

All variables are non-negative integers.

By solving this integer program using a solver such as Excel's Solver, we can find the minimum total number of nurses required to provide adequate care.