I am dealing with data corresponding to demands for $N$ products. The data consists of demand $d$ of each product that vary by month, and the quantity $m$ of that demand that is met. I also have different $d$ and $m$ data by store. Generalizing, $d$ and $m$ vary by different "independent" variables (the product, the month, the store, and others). I want an intuitive average of $m/d$, and the average will take place over different subsets of the data by freezing certain independent variables. In case it helps, this is the SQL group-by function, with counterpart idioms in Matlab and probably other analysis environments.
I want to weight the averaging by a scarcity weighting, which is basically one over the quantity of the particular product that is ordered each month in central stores. Since there are $N$ products, each $d$ and $m$ value will be weighted by one of $N$ different scarcity weights $w_i$. In the simplest possible form, an average on two values for $d$ and $m$ would be:
$$ \frac{w_1 m_1 + w_2 m_2}{w_1 d_1 + w_2 d_2} $$
I thought that it would be simple to show the equivalence with the usual weighted average:
$$ \frac{ w_1 m_1 / d_1 + w_2 m_2 / d_2 }{ w_1 + w_2 } $$
It turns out that they are not equivalent. It's certainly not a geometric average. It also doesn't seem to correspond to a weighted harmonic average. Yet it seems so intuitive. Can it be related in some intuitive way to a classically defined average?
The supply/demand problem that I used to provide context to the average is just an analogy, as the actual problem is different. So while I understand why respondents may question the nature of the problem, it's the actualy math that I'm hoping to rationalize in terms of classical definitions of averages.
Thanks to Rahul for the answer. The average is known as a weighted mediant.