I thought of this problem recently.
Suppose there is a town with 5 regions. Each of these regions has multiple bakeries - however, people can only visit bakeries within their region. Thus, we know the ratio of customers to bakery in any of these regions. Furthermore, we assume that each person visits a bakery, each person can only visit one bakery and each bakery is equally likely to be visited by any person in that region. For argument sake, we also assume that all bakeries in all regions are identical - they have the same number of staff, the same amount of equipment and supplies - some of these bakeries are simply being under-utilized.
Now, suppose that many new people arrive in these 5 regions. The ratio of customers to bakeries now increases in each of these 5 regions. The same assumptions still apply : people can only visit bakeries within their regions, people must visit a bakery within their region, people can only visit one bakery within the region and each bakery is equally likely to be visited by any person in that region.
Suppose this is how the data looks like (ratios are customer-to-bakery ratio, e.g. Region 1 had 5.1 customers per each bakery, and now has 17.3 customers per each bakery):
Region Old_Ratio New_Ratio
1 Region 1 5.1 17.3
2 Region 2 31.3 36.2
3 Region 3 28.7 39.1
4 Region 4 55.8 60.2
5 Region 5 41.5 52.2
If we were to look at the percent change in ratio (i.e. (New_Ratio/Old_Ratio) / Old_Ratio) ) , this would look something like this:
Region Old_Ratio New_Ratio Percent_Change
1 Region 1 5.1 17.3 239.215686
2 Region 2 31.3 36.2 15.654952
3 Region 3 28.7 39.1 36.236934
4 Region 4 55.8 60.2 7.885305
5 Region 5 41.5 52.2 25.783133
This analysis would make it seem like there is a real alarming problem in Region 1 - the customer to bakery ratio has increased by 239% ! However, when you look at the numbers more closely, even AFTER this massive increase in customer to bakery ratio, bakeries in Region 1 are still dealing with significantly less customers on average when compared to bakeries BEFORE the change in population! Even though bakeries in Region 1 will now be working a lot harder than before, bakeries in Region 1 are still not fully pulling their weight compared to the bakeries in other regions!
To address this problem, I had the following idea. Prior to the change in population, the average number of customers per bakery was 32.48, we can see this here: (5.1 + 31.3 + 28.7 + 55.8 + 41.5)/5 = 32.48 . What if we used this average number as a reference point. That is, we define a "scaled ratio" as (New_Ratio - 32.48)/(32.48) and compare the results:
Region Old_Ratio New_Ratio Percent_Change Scaled_Change
1 Region 1 5.1 17.3 239.215686 -46.73645
2 Region 2 31.3 36.2 15.654952 11.45320
3 Region 3 28.7 39.1 36.236934 20.38177
4 Region 4 55.8 60.2 7.885305 85.34483
5 Region 5 41.5 52.2 25.783133 60.71429
If we compare these numbers, this now looks like this:
The goal of my problem is to try and find a method to compare the Old Ratio and the New Ratio in a way such that these "alarming" changes are "mitigated". For example, given the assumptions about the bakeries in my problem (as unlikely as they might be in real life) - I would like to find some comparison method which shows "where the biggest changes are happening but also draws attention away from changes like those in Region 1".
I know that ideally it would be great to have some "limiting capacity", i.e. the number of customers any bakery can comfortably handle - but with this information being absent, what kind of comparison methods can I use for this kind of problem?
Thanks!
