Assume one owns different products, , usually . Each product has a cost , , associated with this ownership, . Let , and be the powerset of , i.e., the set of all subsets. The powerset has elements.
There are two real-valued functions and operating on , i.e., . Function denotes the sum of the cost of the product set given by , is therefore easy to evaluate. I.e.,
Function denotes the buying cost if one can dispose the set of products given by . This means, it costs money to get rid of the products given by set . As the products have interrelationships, is more costly to evaluate than and is given by some table lookup. Function does not depend on .
Some notes on : Assume a non-negative matrix , where denotes the cost to decouple the -th product from the -th. Then
Usually , if , i.e., is upper triangular, because decoupling product from product does not need to be done again for decoupling the other direction from to . More zeros in above sum are necessary if decoupling product is sufficient if done once and once only. In practical application cases depends on a real-valued parameter giving different solution scenarios.
Example: For three products let , thus cost of ownership is , , , , , and so on for the rest of sets. Function gives positive values in a similar fashion, although, as mentioned before, these values are not related to .
2. Problem statement
Find the set of products which gives the least cost, i.e., find for
Rationale: One invests to get rid of the products in set but saves ownership costs given by .
does not need to be unique. If depends on then will likely also depend on .
As can be “large”, evaluating all possible combinations becomes prohibitive on todays computers. Maybe one of our gentle readers knows a method to find the optimum, or maybe just a nearby solution to above problem, or a probabilistic approach. Any comment is welcome.