Logic for rate approximation - algorithm

Question

I am looking for some logic to solve the below problem.
There are n transaction amounts : T1,T2,T3.. Tn.
Commission for these transactions are calculated using a rate table provided as below.
if amount between 0 and A1 -> rate is r1
if amount between A1 and A2 -> rate is r2
if amount between A2 and A1 -> rate is r3
...
...
if amount greater than An -> rate is r4
So if T1 < A1 then rate table returns r1 else if r1 < T1 < r2;it returns r2.
So,lets says the rate table results for T1,T2 and T3 are r1,r2 and r3 respectively.
Commission C = T1 * r1 + T2 * r2 + T3 * r3
e.g; if rate table is defined(rates are in %)
0 - 2500 -> 1
2501 - 5000 -> 2
5001 - 10000 -> 4
10000 or more-> 6
If T1 = 6000,T2 = 3000, T3 = 2000, then
C= 6000 * 0.04 + 3000* 0.02 + 2000 * 0.01 = 320
Now my problem is whether we can approximate the commission amount if instead of individual values of T1,T2 and T3 we are provided with T1+T2+T3 (T)
In the above example if T (11000) is applied to the rate tablewe would get 6% and which would result in a commision of 600.
Is there a way to approximate the commission value given T instead of individual values of T1,T2,T3?

Answer 1

Come up with some probabilities for the percentage of transactions that fall into each bracket.
P(r1) = x // The probability of a transaction falling into a rate 1 bracket
P(r2) = y // The probability of a transaction falling into a rate 2 bracket
P(r3) = z // The probability of a transaction falling into a rate 3 bracket
You can therefore estimate the amounts from the transaction that fell into each bracket
Estimated amount of commission at
rate 1: x * (T1 + T2 + T3) * r1
rate 2: y * (T1 + T2 + T3) * r2
rate 3: z * (T1 + T2 + T3) * r3
You can get estimates for x y and z by looking at historic transactions. The amount of money from a transaction you are actually charging at any rate could be less than the requirement for the bracket, but it doesn't matter. On average transactions will accrue the average rate of commission across all bands.
Though I'd like to disclaim, this will give you an estimate but the accuracy of the estimate for a single transaction depends on the standard deviation of your data set, but will on average give you a fair estimate across a large number of transactions. You need to decide if this is sensible in your use case.