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I have a set of items: {s1, s2, s3}. There are two functions that rate these items: f1() and f2(). For example f1(s1) = 4.

f1() returs values in [0,15], but f2() returns values in [0,100]. Moreover, the raiting given by f1() is more important than the one given by f2().

How do I combine the raitings returned by f1() and f2() into one raiting?

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Any way you want to. To make a real question out of this, you have to decide what range you want the new rating to have, and how much more important the first rating is, compared to the second. Twice as important? Three times as important? – Gerry Myerson May 11 '12 at 13:19
up vote 3 down vote accepted

This is a constant problem with no simple solution; it's one of the reasons engineering is interesting. Suppose you need to buy a suit to wear to your brother's wedding. The Armani suit looks great, but costs \$2000. The Ralph Lauren suit also looks good, and costs only \$400. Or you could go to the place around the corner and get a polyester suit that does not look good, but that is serviceable, for \$89. How do you combine the two ratings (suit quality and cost) into one? Well, it depends on a lot of things, and there is no simple solution that will always work. The answers will be different for different people. Perhaps if Fred goes into debt to buy Armani, his rich uncle will be so impressed at the wedding that he will give Fred a cushy job at his logistics firm. The mathematics of Fred's uncle has not yet been worked out.

For your particular problem the first thing to try is probably to change the scale of rating $f_1$ from $[0,15]$ to $[0,100]$ by multiplying by $100\over15$. Now both ratings are scaled the same.

A great many combined ratings $r$ are of the form $r = a\cdot f_1 + b\cdot f_2$, where $a$ and $b$ represent the weights that we assign to the two ratings. When $b=0$, we ignore $f_2$ completely; when $a=0$, we ignore $f_1$ completely. If $a>b$, then $f_1$ is more important than $f_2$. If $a$ is twice as big as $b$, then one point of $f_1$ rating is worth as much a two points of $f_2$ rating. The larger we make $\frac ab$, the more weight we give to $f_1$ over $f_2$. So pick an $a$ and a $b$ with $a>b$ and see if it gives the results you want. If the combined rating seems to be favoring $f_2$ more than you like, adjust $a$ upwards a bit, or $b$ downwards, until you get what you want.

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