What is the algebraic formula to reduce the final result if it exceeds a preset value due to higher value of a particular parameter? I run a call center and I want to craft an incentive plan for taking as many calls as my agents can take, but I want to encourage them to keep the total time spent on call (aka AHT) between 4 to 10 minutes and I want to reduce the incentive if they increase the total number of calls by reducing the AHT lower than 4 minutes.
My current formula is:
$$USD = \frac{TC \cdot TT}{100} = \frac{TC \cdot (AHT \cdot TC)}{100}$$
where,
$TC$ (total calls) = total calls taken in a day,
$TT$ (total talktime) = total number of minutes spent with customer on call in a day,
$AHT$ (average handling time) $= TT/TC$,
$TT = AHT \cdot TC$
I want to reward those who keep their AHT between 4 minutes to 10 minutes and I want to punish those who keep their AHT below 4 minutes.
So if we take the ideal situation where $TC = 96, AHT = 5, TT = 480$, then total daily incentive is $USD = 460.80$
Now if their AHT is more than 4, I want to reward them as per the formula, however if their AHT becomes lower than 4 minutes, I want to reduce their incentive by any percentage corresponding to extra talktime they have achieved, so if they keep their AHT 3 minutes, then as per formula their incentive is USD = 768, in this case I want to deduct some 20% of extra (768 - 460) from the incentive they would have achieved in an ideal case.
I want an algebraic formula that can reduce the result (USD) if AHT becomes lower than 4 minutes, and the result should be reduced by 20% of extra USD (total USD - ideal case USD).
Can you help me with such a formula in algebraic form?, because my associates don't understand integration, differentiation. They can understand algebra.
 A: Rather than having a single master formula, perhaps you can find a formula that computes the incentive of a given call, and then the total incentive is the sum of all incentives for all calls taken. So if there are calls $c_1, c_2, c_3, c_4 \cdots$, each with value $v(c_1), v(c_2), v(c_3), v(c_4), \cdots$, then the total incentive is $v(c_1) + v(c_2) + v(c_3) + v(c_4) + \cdots$. Ok, so what should the incentive of a call be?
I made the following figure to show a potential function. The x-axis is the number of minutes of the call, and the y-axis is the incentive value. For the first four minutes, the incentive grows slowly, between four and ten it grows more quickly, and then after ten it grows slowly again. Why might such a shape be a good idea?

*

*If the line always has the same slope, there is no imperative to ending a call and starting a new one, as each gives the same total incentive.

*By having the slope increase between 4 and 10, you are given an incentive to have a call in this range. Afterwards, the the line returns to the same slope as in the beginning, so it makes sense to switch to a new call, where after another 4 minutes the slope picks up again. Still, if the curve flatlined at 10 there would be a very strong incentive to hang up just then.

*Given this curve, is there an incentive to try and get closer to 10, and then hang up shortly after? Not quite. After you hit 6, there is no long-term advantage in waiting to 10 (and then starting a new call), or starting a new call immediately. Thus, going for anywhere between 6 and 10 gives the same amount of total incentive, so within that range the agent can focus on the customer and not on their incentive. Indeed, with this strategy, the agent is encouraged (but not too aggressively) to keep the calls within 6 and 10 minutes.

Now, you can tweak this with a little work to move the ideal call time from [6,10] to [4,10], by having two different slopes in the [4,10] range. I'll leave that as a fun exercise.

