Problem 10 - Thinking Mathematically

Question

An HMO has 70 doctors to be apportioned among six clinics. The HMO decides to apportion the doctors based on the average weekly patient load for each clinic, given in the following table. Use Jefferson's method to apportion the 70 doctors. $$ \begin{array}{|l|c|c|c|c|c|c|} \hline \text { Clinic } & \text { A } & \text { B } & \text { C } & \text { D } & \text { E } & \text { F } \\ \hline \begin{array}{l} \text { Average Weekly } \\ \text { Patient Load } \end{array} & 316 & 598 & 396 & 692 & 426 & 486 \\ \hline \end{array} $$

Step-by-Step Solution

Answer

Jefferson's method will give the final apportionment of the number of doctors per clinic. The step with adjusting the modified divisor is particularly important as it ensures the total numbers of doctors apportioned equals 70.

Step 1: Calculate the Standard Divisor

Calculate the Standard Divisor (SD) which is the Total Population (sum of all weekly patient loads) divided by Number of Seats (number of doctors), i.e., SD = (316 + 598 + 396 + 692 + 426 + 486) / 70.

Step 2: Calculate the Standard Quotas

Calculate the Standard Quotas (SQ) which is the population of each state divided by the standard divisor. So, the SQ's for clinic A, B, C, D, E, and F are 316/SD, 598/SD, 396/SD, 692/SD, 426/SD, 486/SD respectively.

Step 3: Determine Lower Quotas

The Lower Quotas (LQ) are the whole number parts of the Standard Quotas. If the LQs total to less than the number of seats, we decrease the divisor. If it totals more, we increase the divisor. In either case, adjust until the sum of the LQs equals the total number of seats (70 doctors). That will be our modified divisor.

Step 4: Final Apportionment

Using the Modified Divisor, recalculate the quotas and keep only the whole number part. This will give the final apportionment of doctors among the six clinics.

Key Concepts

Jefferson's method

Jefferson's method, named after Thomas Jefferson, is a way to fairly distribute resources or seats among various groups based on their population. This method is particularly useful in scenarios where the total number of things to be distributed (like seats or doctors) cannot be exactly divided according to each group's size. Jefferson's method uses a process of modifying divisors to adjust quotas, ensuring that the total remains constant and resources are distributed as closely as possible to the needs or sizes of each group.
Here's how it works in simple steps:

• Calculate initial quotas using a standard divisor, which reflects a fair share but could be fractional.
• Round these quotas down to whole numbers (lower quotas).
• Adjust the divisor if the total is not equal to the quantity of resources.
• Continue adjusting until you achieve the correct allocation.

This method always makes sure smaller portions are adjusted in a way that when added up, they reflect a logical and fair distribution. It's especially helpful in political contexts where equal and fair representation is critical.

Standard Divisor

The Standard Divisor is a crucial element in apportionment problems, acting as a baseline for calculating quotas. It represents the average amount of resources each group should logically receive, assuming a perfectly divisible distribution. Calculating the standard divisor is your first step toward understanding each group's fair share in a given context.
To compute the Standard Divisor:

• First, find the total population or amount involved. In this exercise, it's the sum of all weekly patient loads across the clinics.
• Next, divide this total by the number of resources you are apportioning. Here, that number is 70 doctors.

Once you have the Standard Divisor, you use it to determine each group's initial quota. It's like a measuring stick that sets the standard for how resources are calculated and initially allocated before any adjustments are made.

Standard Quota

Standard Quota represents the initial allocation each group would receive if fractional values were allowable. It is calculated by dividing each group's population by the standard divisor. In this exercise, the standard quotas measure the number of doctors each clinic deserves based on its patient load.
To calculate the Standard Quota for each clinic:

• Take each clinic's average weekly patient load.
• Divide it by the Standard Divisor.

The result is a number, often a fraction, that shows what portion of the total resources each group would receive in an ideal scenario. While these quotas provide an initial estimate, they usually need adjustment since resources (like doctors here) can only be assigned in whole numbers.

Lower Quota

When using methods like Jefferson's for apportionment, the Lower Quota is the whole number part of each standard quota. Once you have calculated the standard quotas for each group, rounding them down to the nearest whole number provides the lower quotas. These play a critical role in the method because they ensure that the total allocation does not exceed the total number of resources available.
The lower quotas act like a safety net, guaranteeing that everyone receives at least a basic minimum, derived from their initial fair share. However, they usually do not altogether equal the total resources required, especially when you're starting with them. That's where Jefferson's method of adjusting the divisor comes into play, modifying allocation until the total aligns with the available quantity.