I'm applying an assignment algorithm to pair objects A with objects B. The result is a matching with a sum of minimum cost. I want to find out whether it makes sense to restrict the assignments in specific ways. The difficulty here is that I will exclude pairings from the matching which do not fulfill these constraints but how can I take into account the number of pairings with respect to their cost?
For example (maximum number 1000 pairings):
Restriction 1 => 800 pairings, cost: 125
Restriction 2 => 650 pairings, cost: 74
Restriction 3 => 902 pairings, cost: 170
A mean or a median does not consider the gained number of pairings but deviding the costs with the number of pairings is not meaningful:
125 / 800 = 0.15625
74 / 650 = 0.11
170 / 902 = 0.188
Maybe (cost * 1000) / number of pairings? Or better a penalty term for each missing pairing?
cost + missing pairs?
On gap penalty are some useful tools and probably the penalty terms from the information criteria are worth a look too.