Exponential Price discount on quantity

SamCKayak

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Nov 29, 2018
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I've tried to create an exponential price discount based on quantity of purchase. I've run into an expected surprise that the total units times the discounted price can peak, then dip down as the quantity increases.

Maybe my formula is wrong. Maybe it is an expected result of a linear function times an exponential decay.

Here's what I'm working with:

Single unit price is $11, discounts to a floor of $1.00 when you hit about 50,000 units

Can I get a formula with variables for the single unit price, floor price and the 50,000 quantity to which the floor is pinned?

Unit price = Decay function(single unit price), Total Price = Unit Price * Unit Quantity

We want Total Price to be a smoothly increasing value with no dips.

Hopefully, I've attached an image showing my formula issues.

Thanks,

Sam
 
I've tried to create an exponential price discount based on quantity of purchase. I've run into an expected surprise that the total units times the discounted price can peak, then dip down as the quantity increases.

Maybe my formula is wrong. Maybe it is an expected result of a linear function times an exponential decay.

Here's what I'm working with:

Single unit price is $11, discounts to a floor of $1.00 when you hit about 50,000 units

Can I get a formula with variables for the single unit price, floor price and the 50,000 quantity to which the floor is pinned?

Unit price = Decay function(single unit price), Total Price = Unit Price * Unit Quantity

We want Total Price to be a smoothly increasing value with no dips.

Hopefully, I've attached an image showing my formula issues.

Thanks,

Sam
It is impossible to say what is wrong with your formula if you do not tell us what it is. It is also difficult to know how to answer if we do not know what grade you are in and what you are studying.

If this is not a classroom exercise, what is the level of mathematical knowledge of those who are expected to use the formula? Most people will not be able to handle exponential functions reliably. Or is this formula going to be calculated by a computer?

Is what is below the problem?

\(\displaystyle q = \text { units in order, where } q \text { is a positive integer.}\)

\(\displaystyle p = \text { price per unit for order size of } q \text { units.}\)

\(\displaystyle t(q) = \text { total amount due for order of q units before tax and shipping.}\)

\(\displaystyle \therefore t(q) = pq.\)

\(\displaystyle q = 1 \implies p = a > 0.\)

In other words, the price for an order of 1 unit is a.

\(\displaystyle q \ge z > 1 \implies p = b \text { such that } 0 < b < a.\)

In other words, additional discounting stops when the quantity ordered equals or exceeds z, and the unit price where discounting stops is b.

You want a formula or an exponential formula, not clear which is required, such that

\(\displaystyle p = f(q) \text { for integer } q > 0.\)

\(\displaystyle f(1) = a;\)

\(\displaystyle q \ge z \implies f(q) = b;\text { and}\)

\(\displaystyle 1 \le x < y \le z \implies f(x) > f(y) \text { but } t(x) < t(y).\)

If this is not a classroom problem, how are you going to handle fractional pennies? How you propose to do so may cause "dips."
 
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