Find single parameter value in multi-degree equation to match certain value when all other parameters are locked

Question

Aka there might be a simple name for that question, but here goes:

So I'm trying to score objects on certain parameters, one of which is price. The way I calculate the score is I multiply each parameter together. Pretty simple stuff.

Ex: score = Param1 * Param2 * Param3 * Param4 * price

Each parameter could individually be assigned an exponent (ranging from 0.5 to 3.5 to give it more or less weight in the equation). I do not know if that influences the complexity of what I am asking for.

Now, I have an ordered list where items are filtered from descending score. All parameters are set in stone, but price might vary; each item's price could potentially be discussed.

What I'm trying to do is to suggest a bargained price for each item so it matches the item with the highest score.

For the moment, those 3 suggested prices have been manually found (and aren't very accurate of course).

The idea is that basically, an item might be worth giving priority over a higher ranked item if we can manage to bargain a good price. We're on the buyer side, so better score means cheaper price.

So the final question is: what equation(s) do I need to come up with to find a price whose score would match the best score ?

Edit: I should've explained things a bit more clearly, sorry about that. I thought the following wouldn't have any impact on what I was asking, but it does seeing the answers.

Turns out I misled the question by withholding some information regarding the parameters; they are all composites, ranging from 0 to positive infinite. The way I calculate them is based on certain criteria so that they don't go too far off the 0.5 - 2.5 range. This holds true for the price parameter as well. If it goes above a certain threshold, I return the following value:

$\left(\frac{threshold}{price}\right)^5$

Given a threshold of $5000$ and a price of $5700$, the returned value will be $0.88$. The 5th exponent is there to make each dollar above the threshold have a more dramatic effect on the total score.

On the other hand, if the price is below the threshold, the composite is calculated in the following manner:

$\frac{threshold}{price}$

So either a parameter increases the score, or it decreases it slightly (since they're all multiplied between each other).

As an example, here's the formula with the item with the highest score's parameters:

$1.5 * 1.99 * 450.00 * 1.44 * 1.18 = 2272$

And one with a lower score (due to the item's higher price):

$2.00 * 2.92 * 350.00 * 1.00 * 0.88 = 1790$

As you can see, it could've score higher if it weren't for its steeper price. That's why I'm trying to find a formula that suggest a price to make the item worth it. So as the buyer, you could try and discuss to get to that price.

Answer 1

what equation(s) do I need to come up with to find a price whose score would match the best score ?

If I understood your right then for object $i$ its score $score_i$ equals $s_i\cdot f\left(\frac{threshold}{price_i}\right)$, where $s_i$ is set in stone, that is constant for given $i$ and the function $f$ is defined as follows

$$f(x)=\cases{ x , \mbox{if }x\le 1,\\ x^5, \mbox{if }x\ge 1 }.$$

Thus if we wish to have $suggestedscore_i=bestscore$ then we should change $price_i$ to $suggestedprice_i$ as follows:

$$bestscore=suggestedscore_i = s_i\cdot f\left(\frac{threshold}{suggestedprice_i }\right).$$

Let $g$ be the inverse function for $f$, that is

$$g(y)=\cases{ y , \mbox{if }y\le 1,\\ y^{1/5}, \mbox{if }y\ge 1 }.$$

Then $g\circ f(x)=f\circ g(x)=x$ for each $x\ge 0$ and the above equality implies

$$suggestedprice_i=\frac{threshold}{g\left(\frac{bestscore}{s_i}\right)}.$$

In other words,

$$ suggestedprice_i =\cases{ threshold\cdot \frac{s_i}{bestscore} , \mbox{if }s_i\ge bestscore,\\ threshold\cdot\left(\frac{s_i}{bestscore}\right)^{1/5}, \mbox{if } s_i \le bestscore }.$$

Answer 2

If you are looking at the mathematical solution to the problem you describe, the answer by Alex Ravsky is accurate. However, there seems to be a contradiction in your preference model. The equation provided for the score is :

Ex: score = Param1 * Param2 * Param3 * Param4 * price

Each parameter could individually be assigned an exponent (ranging from 0.5 to 3.5 to give it more or less weight in the equation).

Or :

$score = (param1)^a\times (param2)^b \times (param3)^c \times (param4)^d \times (cost)^e$

With $a, b, c, d, e >0$

This function is increasing with regard to the price (the more costly the item is, the higher score it gets). This is always true if $e$ is positive.

Then, the question states :

We're on the buyer side, so better score means cheaper price.

This is in contradiction with the scoring formula.

What can you do

The easiest solution would be to make $e$ negative. This way, the score function decreases with prices, which is the desired behaviour. If you do that, the answer by Alex Ravsky is still good.

However

You might want to revise that formula because the score can be overly affected by one parameter. Say you have an object that scores near 0 on param1, but is near perfect with regard to all other parameters. Even if you assign a low exponent to param1, the overall score will be very low, perhaps lower than you expect.

A perhaps better way to score items would be with a weighted sum, i.e.

$score = a\times param2 + b\times param2 + c\times param3 + d \times param4 + e \times cost$

With a, b, c, d, and e playing the same role as before (the higher the value, the more important the parameter). Note that $e$ should be negative so the score decreases when the cost increases. If you want to see what should be the price of an item in order to obtain score $X$, all else being fixed, you can use the formula :

$desired\_cost = \frac{X - a\times param2 - b\times param2 - c\times param3 - d \times param4}{e}$