Problem 22 - Thinking Mathematically

Question

A hospital has a nursing staff of 250 nurses working in four shifts: \(A\) (7:00 A.M. to 1:00 P.M.), B (1:00 P.M. to 7:00 P.M.), \(C\) (7:00 P.M. to 1:00 A.M.), and D (1:00 A.M. to 7:00 A.M.). The number of nurses apportioned to each shift is based on the average number of patients per shift, given in the following table. Use this information to solve. $$ \begin{array}{|l|c|c|c|c|} \hline \text { Shift } & \text { A } & \text { B } & \text { C } & \text { D } \\\ \hline \begin{array}{l} \text { Average Number } \\ \text { of Patients } \end{array} & 453 & 650 & 547 & 350 \\ \hline \end{array} $$ Use Webster's method to apportion the 250 nurses among the shifts in the hospital.

Step-by-Step Solution

Answer

Using Webster's method, the distribution of 250 nurses across shifts A (7:00 A.M. to 1:00 P.M.), B (1:00 P.M. to 7:00 P.M.), C (7:00 P.M. to 1:00 A.M.), and D (1:00 A.M. to 7:00 A.M.) is calculated. The exact apportionment is dependent on the calculations in the earlier steps.

Step 1: Find the standard divisor

The Standard Divisor (SD) is calculated as the total number of nurses divided by the total number of patients across all shifts. That is, \[ SD = \frac{{Total\ Number\ of\ Nurses}}{{Total\ Number\ of\ Patients}} = \frac{{250}}{{453+650+547+350}}. \] Compute this value.

Step 2: Calculate the quota for each shift

The quota for each shift is the average number of patients for that shift divided by the standard divisor calculated in Step 1. You would need to perform this calculation for every shift. For example, for shift A, \[ Quota_A = \frac{{Average\ Number\ of\ Patients\ for\ A}}{{SD}} \]. Repeat this for shifts B, C, and D.

Step 3: Calculate the draft quotas and distribution

After getting the quotas for each shift, round off the quota for each shift down to the nearest integer to get the first draft quotas. These would represent the initial distribution of nurses. Calculate the remaining nurses by subtracting the sum of all draft quotas from the total number of nurses.

Step 4: Distribute the remaining nurses

If there are any remaining nurses after first draft quotas, distribute them to the shifts according to their fractional parts of quotas (from highest to lowest) until all nurses have been assigned.

Key Concepts

Standard Divisor

Imagine a pie representing the total number of nurses in a hospital. To share this pie fairly based on the needs of each shift, we use something called the Standard Divisor (SD). In essence, the SD is like a magic number that helps us figure out how many slices of the pie each shift gets based on the patients they serve. To find the SD, we divide the total number of nurses by the total patient count across all shifts.

For our hospital with 250 nurses and the patient count per shift provided, we'd calculate the SD by totaling the patients and then dividing our nurse count by this aggregate number. This is crucial, as it sets the stage for equitable distribution. Just divide the number of nurses (which is our pie size) by the total patients (showing us how big each slice should be).

Quota Calculation

With the Standard Divisor in hand, we can now determine each shift's fair share, or 'quota,' of nurses. This is like deciding how many nurses each shift should ideally have if we divide them purely based on the number of patients they handle. Each shift's average patient count is divided by the SD to give us the initial quota.

For our hospital, this means doing a bit of division for each shift - patients for shift A over the SD, and so on. The initial result gives us a starting point, telling us about how many slices of our nurse pie might go to each shift. It won't always be a perfect whole number, and that's fine - Webster's Method has steps to handle the leftovers.

Nurse Staffing Allocation

Making sure our nurses are well-distributed is like balancing a seesaw. With our initial quotas, we're starting to get the seesaw level. However, because we're dealing with whole nurses and not fractions, we have to round down those quotas to get draft quotas - whole numbers representing our starting allocation for each shift. But wait, there may be leftover nurses after this initial distribution.

These nurses are like bonus pieces of our pie. We have to give them to the shifts with the biggest 'appetite,' or in other words, the largest fractional parts of their quotas. This way, we ensure every shift gets a fair number of nurses to handle their share of patients, all the while not splitting any nurse in two!

Proportional Distribution

Now, it's time for the grand finale: making sure each shift has the help it needs in proportion to its patients - proportionally distributing our nurses. This part of Webster's Method is all about justice, providing each shift with a workforce proportional to its workload.

We've already figured out a rough allocation, but we'll refine it by handing out any nurses we haven't yet assigned. Here, the detail is in the decimals. Shifts with larger decimal parts of their quotas get first dibs on extra nurses. It's a fair way to resolve the issue of those tricky little leftovers. By the end of it all, we've achieved a proportional balance that reflects the needs of the patients and the capacity of the staff.