To illustrate the difficulties of coming up with a good general allocation formulation consider the case in which there is a single market demand that a number of suppliers are trying to satisfy. This is a many to one allocation problem.
A common approach to this problem is to assign to each supplier a fraction 'f' of the demand, according to the simple formula:
f[i] = (a[i]/ sum(a[i!]))
where
f[i] = fraction of demand won by supplier i (in other words, supplier i's market share)
a[i] = attractiveness of supplier i's offering (if price were the only difference among the products, a[i] might be 1/price[i])
sum(a[i!]) is the sum over all the attractiveness values.
This formula has many desirable properties, but it has a major shortcoming: it is possible for a tiny supplier to capture the entire market. For example, suppose the demand is the total global demand for cars. Suppose, further that there are only two competitors: GM, and JoeBob's. GM is a world-wide giant; JoeBob's is two cousins hand building cars in their garage. If Joe and Bob manage to build a better car than GM, then the above formula will immediately award a majority of the market to JoeBob's.
We can overcome this problem in two ways. One would be to include measures of visibility in the attractiveness number. In that case, JoeBob's product might be best, but its attractiveness would be tiny, because of lack of word-of-mouth and advertising. This approach is tricky as we are really overloading the meaning of attractiveness as a surrogate measure for capacity.
As this example suggests, the allocation problem can be quite tricky.
