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Here's a seemingly simple pondering. If one item is more valuable the higher it is (i.e., $a=5$ is worth more than $a=2$) and another item is more valuable the lower it is (i.e., $b=2$ is worth more than $b=5$), what is an equation that will calculate how "good" the item pair is?

A couple of approaches:

  • The perfect combination will be at $0$.
  • Higher result is better.
  • Lower result is better.

Here's a physical example, bicycles:

  • The lower the weight of a bicycle, the faster it performs. Also, the higher the gear ratio, the faster it performs. So:
    • One bike, bike a, has a weight of $29$ and the highest gear (i.e., left*right, basically the same as gear ratio for our purposes) of 24.
    • Another, bike b has a weight of $26$ and the highest gear of $25$.

Which bike, assuming that weight and gear ratio matter the exact same in determining bike speed, will offer a faster speed?

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What about the cost? the cost of a property being optimised. Other wise this model would only reflect a bike made out of the most expensive components. Is there a ratio of goodness per dollar? for example a 8GB USB stick is cheaper than a 16GB one, but buying 2 x 8GB sticks is cheaper than buying 1 , 16GB stick. –  Arjang Dec 29 '12 at 8:55
    
@Arjang There are obviously a bunch more variables that could be thrown in here; for example, price, weight distribution, lubrication of chain, etc. I'm more curious about if we were to take this exactly at face value—weight and gear. Is that doable? –  tehsockz Dec 29 '12 at 9:00
    
For more general problems, look at goal programming. There are a couple more ideas people have had in links under strengths and weaknesses. –  Daryl Dec 29 '12 at 9:06
    
@Arjang Definitely. But is there some sort of formula one could write to calculate, say, how "good" a particular set was? Assuming that the lightest weight was 1/∞ and the highest gear was ∞. –  tehsockz Dec 29 '12 at 9:08
    
@tehsockz : then the optimum would happen at w=0 and gear=$ \inf $, is there any more constraints that we can use? –  Arjang Dec 29 '12 at 9:13
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1 Answer

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To do problems like these, you need to come up with a mathematical model that describes the situation. Many might argue that math, the whole of math, consists largely of the idea of creating models to describe different situations. I'll create a certain model here, and perhaps you'll see what you do and don't like, and then create your own. As posed, this problem leaves a lot of room as to what the model might be (meaning the situation is imprecise).

We know that having a better gear ratio makes a bike go faster, and a lower great ratio makes it go slower. We know that having a lighter bike is faster, a heavier bike is slower. Let's suppose (sort of plucking numbers out of the air) that there is a 'boring bike' that is our standard: suppose that a 20 pound bike with a gear ratio of 1 allows someone to travel at a whopping 10 miles an hour.

Now we have something to compare different bikes' gears and weight to. But we need to decide how having different weights and gears changes performance. Let's suppose that halving the weight of a bike lets us go twice as fast (this is another time where I'm picking numbers out of the air). Similarly, doubling a weight makes us go half as fast. You said that in this model, we should assume that gear ratios and weights have the same type of model, so we will assume that doubling the gear ratio results in twice the speed, and halving the gear ratio halves the speed. So doubling the weight and doubling the gear ratio will not change the speed.

Then the speed of a bike $B(w,g)$ with weight $w$ and gear ratio $g$ will go $\dfrac{20}{w}\dfrac{g}{1} \cdot 10$ miles per hour. If this isn't clear, let me know and I can clarify how this ended up being the model from the situation I described.

Now you ask something that amounts to: "What is best?" In terms of the model we have, we want the smallest weight and highest gear ratio, and in particular we want to maximize $\frac{g}{w}$.

But there is more that could be looked at and modeled here, if we spend the time and thought. It's not true that the weight and gear ratio are completely distinct. There's probably some relationship. Let's suppose our boring bike standard corresponds to the basic bike out there - increasing the gear ratio will end up increasing the weight of the bike. We might imagine that we started with a fixed gear, and to build another bike we are sticking on a different gear onto the wheel so that the ratio changes. The gear ratio changes linearly with a change in the radius of the new gear, but the weight added by sticking on a new gear changes with the volume added by the new gear. If we suppose we won't need thicker gears for larger gears, then the weight changes quadratically with the radius of the gear.

So we might have two related relationships: the gear ratio $g$ follows some relationship with the radius $r$ of the new gear like $g(r) = c_1r + g_0$ for some constant $c_1$ and 'base ratio' $g_0$, and the weight follows some relationship with the radius like $w(r) = c_2r^2 + c_3r + w_0$ for constants and 'base weight' $w_0$. (I'm trying to italicize every time I make a choice of model for later reference, so here's another time).

Then what we're actually optimizing is the ratio $\dfrac{g(r)}{w(r)} = \dfrac{c_1r + g_0}{c_2r^2 + c_3r + w_0}$ - we want it as large as possible. And if we have a model for the coefficients, then one can solve this using calculus of a single variable.

The idea is that one can create a model and follow it through. Everywhere I used italic text corresponded to a choice of model for this situation. As you can see, I had a lot of choice. I also have no justification for the model I chose, meaning that it's probably not a very good one. If this is a problem that really interests you, you might spend some time with a few different bikes and create your own model. Perhaps having a few different bikes and measuring their gear ratios and weights will allow you to come up with a better model, or might allow you to fit certain bits to curves (and find coefficients for my constants at the end).

I hope this gives you an idea of how to look at such problems, and that this answer helps in some way.

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