a hospital emergency room needs the following numbers of nurses: day m t w t f s s min. number 2 8 2 3 6 7 4…
a hospital emergency room needs the following numbers of nurses: day m t w t f s s min. number 2 8 2 3 6 7 4 each nurse should have two consecutive days off. how many full - time nurses are required and what is a good nurse schedule? round your answer to the nearest whole number. nurses
Answer
Explanation:
Step1: Define variables
Let $x_1,x_2,\cdots,x_7$ be the number of nurses starting their work - week on Monday, Tuesday, $\cdots$, Sunday respectively.
Step2: Set up constraints
The number of nurses on duty on Monday is $x_1 + x_4+x_5+x_6+x_7\geq2$. On Tuesday: $x_2 + x_5+x_6+x_7+x_1\geq8$. On Wednesday: $x_3 + x_6+x_7+x_1+x_2\geq2$. On Thursday: $x_4 + x_7+x_1+x_2+x_3\geq3$. On Friday: $x_5 + x_1+x_2+x_3+x_4\geq6$. On Saturday: $x_6 + x_2+x_3+x_4+x_5\geq7$. On Sunday: $x_7 + x_3+x_4+x_5+x_6\geq4$. Also, $x_i\geq0$ and $x_i\in\mathbb{Z}$ for $i = 1,2,\cdots,7$. We want to minimize $Z=x_1 + x_2+x_3+x_4+x_5+x_6+x_7$.
Step3: Solve using a linear - programming approach (or by brute - force checking all possible non - negative integer solutions)
We can also use a heuristic approach. First, sum up all the minimum requirements: $2 + 8+2 + 3+6 + 7+4=32$. But since each nurse works 5 days out of 7, if we assume the total number of nurses is $N$, the total number of nurse - days worked is $5N$. Let's try to build the schedule by hand. We start by looking at the highest - demand days. We can use the following approach: We know that we need to cover the daily minimums while giving each nurse 2 consecutive days off. By trial and error:
| Day | Mon | Tue | Wed | Thu | Fri | Sat | Sun |
|---|---|---|---|---|---|---|---|
| Nurse 1 | W | W | W | W | W | O | O |
| Nurse 2 | W | W | W | W | O | O | W |
| Nurse 3 | W | W | W | O | O | W | W |
| Nurse 4 | W | W | O | O | W | W | W |
| Nurse 5 | W | O | O | W | W | W | W |
| Nurse 6 | O | O | W | W | W | W | W |
| Nurse 7 | O | W | W | W | W | W | O |
| Nurse 8 | W | W | W | W | O | W | O |
| Nurse 9 | W | W | W | O | W | O | W |
| Nurse 10 | W | W | O | W | O | W | W |
| Nurse 11 | W | O | W | O | W | W | W |
| Nurse 12 | O | W | O | W | W | W | W |
Counting the number of nurses, we find that we need 12 nurses.
Answer:
12